阅读说明：本技术 用于成像系统的光学设计方法以及用此方法设计的光学系统 (Optical design method for imaging system and optical system designed by the method ) 是由 F·杜尔 H·蒂恩庞特 于 2018-12-28 设计创作，主要内容包括：本发明提供了用于通过求解从费马原理推导出的微分方程来设计共轴成像系统的方法。这些成像系统由(可移动或固定)球面或非球面的序列或其组合组成。幂级数求解办法允许计算预定义数目的表面参数,该预定义数目等于要标称地消失的所选像差项的数目。另外,这些方法提供了允许描述任意光线与每个单独表面相交的确切位置的映射函数系数。(The present invention provides a method for designing a coaxial imaging system by solving a differential equation derived from the fermat principle. These imaging systems consist of a sequence of (movable or fixed) spherical or aspherical surfaces or a combination thereof. The power series solution allows for the calculation of a predefined number of surface parameters equal to the number of selected aberration terms to be nominally vanished. In addition, these methods provide mapping function coefficients that allow describing the exact location where any ray intersects each individual surface.)

1. A computer-based method for designing a rotationally symmetric imaging system comprising at least two optical surfaces and an optical axis, and which nominally eliminates:

at least one light aberration coefficient when calculating at least one surface coefficient,

providing all mapping function coefficients of the intersection of the ray with the at least two optical surfaces, and all non-eliminated aberration coefficients up to a given order, the method comprising:

-inputting to the computer a first set of system specifications comprising:

o said at least two (N)_s≧ 2) the sequence of the optical surfaces and their lateral position along the optical axis in the z-direction

Omicron whether each optical surface is reflective or refractive and has a spherical or aspherical shape, and wherein each surface f_iIs expressed as a single radial vector A function of (2), the function having a total number N_cUnknown system surface coefficients

O design wavelength λ₀And corresponding refractive indices before and after each optical surface

Omicron will be located at finite or infinite distance, and in case of finite distance have a flat or curved shape

-a prescribed image as a function of an object variable t, said function describing an object-image relationship

A real image which is the sum of the prescribed image and the expansion of the number of ray aberration levels

A certain number (m) at the system stop_pNot less than 1) pupil plane, each pupil plane being defined by an angleDefinitions, wherein p 1_pThus, represents a compound from said object and is in (x)_p，y_p，0)＝(q_pcos(θ_p)，q_psin(θ_p) 0) the position at which a ray passing through the pupil plane intersects the image plane

Express each surface as a function f of the ray mapping function in (x, y)_i((u_i，p(q_p，t)，v_i，p(q_pT))) describing the position where a ray from the object passing through a fixed pupil plane p intersects each optical surface, and being the pupil and object variables (q)_pNumber of stages of t)

-using the computer, converting the first set of system specifications to differential equations by: applying the Fermat principle to the surface function f_iMapping function u_i，pAnd v_i，pAnd two successive optical path length segments of each pair expressed in the form of said real image, to derive 2N_sA differential equation D_i′(i′＝1...2N_s)(N_sA differential equation D_i′At u_i，pIn the direction of, and N_sA differential equation D_i′At v_i，pIn the direction)

-defining pupil characteristics, wherein

Omicron if said diaphragm and one of the surfacesCoincidence is then formed by (x)_p，y_p) SubstitutionIn (1)Andas diaphragm position

Omicron if an aperture stop is placed at the entrance, between two optical surfaces, or at the exit (before the image), the cross product representing the vector describing any ray path towards the aperture stop and the vector describing any ray path from the aperture stop is zero, so that three additional equations are added to the previously derived differential equation, resulting in D_i′Equation (i' ═ 1.. 2N)_s+3)

-using said computer, converting said 2N by_sOr (2N)_s+3) equations are converted into algebraic equations: by setting k, l equal to 0,1,2, 3.. and the combination order k + l equal to 1,3, 5.. o_mEvaluation ofTo use a power series solution method resulting in for all defined series coefficientsOr 2 (N)_s+3) algebraic equations of the order of the algebraic equations,

-directing at least one light raySelecting the aberration coefficient to be zero; for each aberration coefficient set to zero, one surface coefficient is selected as the (sub-) group M_cAnd each of said pair(s) of light ray aberrations and surface coefficients occur simultaneously in at least one algebraic equation

-inputting to the computer the value of any surface coefficient that has not been used in the previous step to eliminate aberrations, said value being other than the group M_cAll of the surface coefficients of a portion of (a),

-solving an algebraic system of equations using the computer to obtain as output at least one surface coefficient of the optical surface, all mapping function coefficients, and the non-eliminated light aberration coefficients up to a given combined order.

2. The computer-based method of claim 1, wherein the number of optical surfaces is a maximum of 30 or the combination order is a maximum of 19.

3. A computer-based method according to claim 1 or 2, wherein each optical surface is refractive or reflective and has a spherical or aspherical shape of up to 20 orders.

4. A computer-based method according to any preceding claim, wherein the ray aberration series expansion is based on Buchdahl or the like.

5. The computer-based method of any preceding claim, wherein the number of (m) at system stop is_p≧ 1) each of the pupil planes defined by an angleDefinitions, wherein p 1_pThe tangential plane, the sagittal plane and the oblique ray plane.

6. A computer-based method according to any preceding claim, wherein the step of defining the pupil characteristic comprises an aperture stop.

7. A computer-based method according to any preceding claim, wherein an aperture stop is behind the object if the aperture stop is placed at the entrance.

8. A computer-based method according to any preceding claim, wherein a value for any surface coefficient that has not been used in a previous step to remove aberrations is input to the computer, said value being other than group M_cThe input can be done manually or using a computer.

9. A computer-based method according to any preceding claim, wherein the step of using the computer to solve the system of algebraic equations to obtain at least one surface coefficient as an output comprises at least as M_cAll coefficients of a part of.

10. The computer-based method of any preceding claim, further comprising the step of inputting at least a second set of system specifications to the computer system, wherein each set corresponds to a different configuration of zoom system, and each set (c 1, 2.) of zoom system specifications comprises:

lateral position of all surfaces describing zoom movement

-mapping function for each zoom configuration

Aberration function for each zoom configuration

-configuring a defined image as a function of the object for each zoom

-the method comprises:

-using the computer, converting the second set of system specifications to differential equations by: applying the Fermat principle to two consecutive optical path lengths of each pair to derive 2N for each zoom system configuration_sDifferential equation

-defining pupil characteristics, wherein

Omicron if said diaphragm and one of the surfacesCoincidence is then formed by (x)_p，y_p) SubstitutionIn (1)Andas diaphragm position

Omicron if an aperture stop is placed, the cross product of the vector describing any ray path towards said stop and the vector describing any ray path from said stop is zero, adding three additional equations to the previously derived differential equation for each zoom system configuration

-using said computer, converting said 2cN by_sOr c (2N)_s+3) equations are converted into algebraic equations: by setting k, l equal to 0,1,2, 3.. and the combination order k + l equal to 1,3, 5.. o_mEvaluation ofTo use a power series solution method resulting for all defined series coefficients and all zoom system configurations Or 2c (N)_s+3) algebraic equations

-selecting at least one light ray aberration coefficient per zoom system specification as zero; for each aberration coefficient set to zero, one surface coefficient is selected as the (sub-) group M_cAnd each of said pairs of light aberrations and surface coefficients occur simultaneously in at least one algebraic equation,

-inputting to a computer the value of any surface coefficient that has not been used in the previous step to eliminate aberrations, whereas said inputting can be done manually or using a computer

-solving the system of algebraic equations using the computer to obtain M as the optical surface_cAs output, all surface coefficients of a part of, all mapping functions, and the non-eliminated light aberration coefficients up to the given combination order and for all zoom system configurations.

11. The computer-based method of claim 10, wherein the prescribed image as a function of the object describes a change in focal length for each zoom configuration.

12. The computer-based method of claim 10 or 11, wherein defining the pupil characteristic comprises the aperture stop.

13. A computer-based method according to any one of claims 10 to 12, wherein the combination order k + 1-1, 3,5_mIs 19.

14. A computer-based method according to any one of claims 10 to 13, wherein not group M_cAll surfaces of a part ofAnd (4) the coefficient.

15. A computer-based method according to any of claims 10 to 14, wherein inputting to the computer the value of any surface coefficient that has or has not been used to remove aberrations in a previous step comprises not being a group M_cAll surface coefficients of a portion of (a).

16. A computer-based method wherein the rotational symmetry of the imaging system is broken, and wherein the system has (1) no symmetry, (2) one plane of symmetry, or (3) two planes of symmetry, the method comprising:

-inputting to the computer system specifications comprising:

οN_ssequence of more than or equal to 2 (maximum 30) optical surfaces

Omicron whether each optical surface is refractive or reflective and has a spherical, aspherical or free shape, and wherein each free-shape surface is expressed as a function of two variables with a power series coefficient f that is not yet known_i，j，kAnd the total number N_cA system surface coefficient, wherein j, k is 1, 2.. up to 12 and i is 1, 2.. N_s

O design wavelength λ₀And corresponding refractive indices before and after each optical surface

-whether the object is to be located at a finite distance or an infinite distance and in case of a finite distance has a flat or curved shape

Omicron as two variables t_xAnd t_yThe function describing the object-image relationship

-a real image as the sum of said prescribed image plus the ray aberration progression expansion according to the system symmetry considered

O mapping the ray to a function u_i(x_p，y_p，t_x，t_y) And v_i(x_p，y_p，t_x，t_y) Expressed as pupil (x)_p，y_p) Andmatter variable (t)_x，t_y) Number of stages of

Express all surface positions, and any tilt of the object, optical surface, final aperture stop and the image, by a rotation matrix (at least one rotation matrix for symmetry case (1) or (2)); and the rotation matrix is defined by inputting the rotation axis and angle that match the symmetry of the system, which in turn defines the chief ray path of the on-axis field and all the vertices of the mapping function

-using the computer, converting the system specification into a differential equation by: applying the Fermat principle to the surface function f_iMapping function u_iAnd v_iAnd two successive optical path length segments of each pair expressed in the form of said real image, to derive 2N_sA differential equation D_i′(i′＝1...2N_s)(N_sDifferential equation in u_iIn the direction of, and N_sA differential equation at v_iIn the direction)

-defining pupil characteristics, wherein

Omicron if said diaphragm and one of the surfacesCoincidence is then formed by (x)_p，y_p) SubstitutionIn (1)Andas diaphragm position

Omicron if an aperture stop is placed in a given position with a defined orientation (rotation matrix), the cross product of the vector describing any ray path towards said aperture stop and the vector describing any ray path from said aperture stop is zero, so as to attach three to the aperture stopAddition of a power to previously derived D_i′A differential equation (i' ═ 1.. 2N)_s+3)

-using said computer, converting said 2N by_s2N or in the case of an aperture stop_s+3 equations are converted to algebraic equations: by referring to the indices j, k, l, m, 0,1,2, 3.. and the combination order j + k + l + m, 1,2, 3.. o_m(maximum 11) evaluation of all possible combinationsTo use a power series solution method to obtain a system of algebraic equations for all defined series coefficients

-selecting at least one light aberration coefficient to be zero; for each aberration coefficient set to zero, one surface coefficient is selected as the (sub-) group M_cAnd each of said pair(s) of light ray aberrations and surface coefficients occur simultaneously in at least one algebraic equation

-inputting to the computer the value of any surface coefficient that has not been used in the previous step to eliminate aberrations, said value being other than the group M_cAnd the input can be done manually or using a computer

-solving the system of algebraic equations using the computer to obtain at least one surface coefficient of the optical surface (as M)_cAll coefficients of a portion of), all mapping function coefficients, and, up to the given combination order, the non-eliminated light aberration coefficients as output.

17. The computer-based method of claim 16, wherein the number of optical surfaces is a maximum of 30.

18. A computer-based method according to claim 16 or 17, wherein each optical surface is refractive or reflective and has a spherical or aspherical shape of up to 20 orders.

19. A computer-based method according to any of claims 16 to 18, wherein the ray aberration series expansion is based on Buchdahl or the like.

20. The computer-based method of any of the preceding claims, further comprising: simultaneously applying the method to at least a second design wavelength with a defined ray mapping and ray aberration function for each design wavelength, the method further comprising:

-using the computer, converting the system specification into an algebraic equation by: applying the Fermat principle to two consecutive optical path length segments of each pair for each design wavelength

-selecting at least one aberration coefficient to be zero per design wavelength, thereby providing a surface coefficient group M_c

-solving the system of algebraic equations using the computer to obtain at least two surface coefficients (as M)_cAll coefficients of a portion of), all mapping functions, and the non-eliminated light aberration coefficients up to the given combination order and for all design wavelengths as output.

21. A computer-based method according to any of claims 1 to 20, wherein the steps further comprise:

-including chromatic aberration coefficients in the expansion of aberration order in the form of a wavelength variable λ

By adding a series term (λ - λ)₀)^ΛIncluding wavelength dependence in all mapping functions

By means of a refractive index profile n_i-1，i(λ) (e.g., Sellmeier equation) to express each optical path length segment

-using the computer, converting the system specification into a differential equation by: applying the Fermat principle to two successive optical path length sections of each pair (with surface or aperture stops)

-using the computer to transfer the information to the computerThe equations are converted into algebraic equations, wherein the step of solving the method using a power series comprises: will add additional itemsIncluded, where Λ ═ 0,1,2, Λ may optionally have a maximum value of 5, and index Λ is added to the sum of the combining orders

-selecting aberration coefficients to be nominally zero, wherein the step comprises selecting at least one monochromatic (i.e. for Λ ≠ 0) aberration coefficient and one chromatic (Λ ≠ 0) aberration coefficient to be zero, thereby providing a group M of surface coefficients_cAnd each of said pairs of ray aberrations and surface coefficients occurring simultaneously in at least one algebraic equation

-input to a computer other than group M_cOf a part of (a)

-solving the system of algebraic equations using the computer to obtain at least two surface coefficients (as M)_cAll coefficients of a portion of), all mapping function coefficients, and up to a given combination order, the unabated monochromatic and chromatic aberration coefficients as output.

22. A computer-based method according to any preceding claim, wherein the step of selecting aberration coefficient(s) comprises: setting the aberration coefficient(s) to non-zero values, for each aberration coefficient set to non-zero values, selecting one surface coefficient as the (sub-) group M_cAnd each of said pair(s) of light ray aberrations and surface coefficients occur simultaneously in at least one algebraic equation.

23. A computer-based method according to any preceding claim, wherein system specifications and/or not group M are input to the computer_cThe step of surface coefficients of a part of is automated using an optimization method.

24. A computer-based method according to any one of claims 1 to 23, whichCharacterized in that the step of selecting aberration coefficient(s) comprises: coefficient not selected (null group M)_c) And solving the system of algebraic equations using the computer to obtain as output all mapping function coefficients and all light aberration coefficients up to the given combined order.

25. A computer-based system adapted to implement the method of any one of claims 1 to 24.

26. A computer-based method for designing a rotationally symmetric imaging system having an optical surface, the method for nominally eliminating at least one ray aberration, the method comprising:

-inputting to the computer system specifications comprising:

-an imaging system parameter of the imaging system,

o-a wavelength dependence of the wavelength,

the number of surface coefficients of the optical surface,

o a pupil characteristic of the pupil,

-an object space and an image space,

-using the computer to convert the system specification into a system of differential equations by applying Fermat's principle,

-using a set of selection, consistency and distribution rules to input to the computer a subgroup of light ray aberrations to be nominally set to vanish,

-solving said differential equation to obtain surface coefficients of the optical surface and mapping function coefficients defining the ray trajectories and their intersection with the optical surface and thereby defining each clear aperture.

27. The computer-based method of claim 26, wherein the step of inputting the system specifications for the imaging system parameters to the computer comprises: inputting at least one design wavelength lambda₀。

28. The computing-based method of claim 26 or 27A method of inputting system specifications for imaging system parameters to said computer, comprising: input N_sA sequence of optical surfaces, wherein each optical surface has an optical surface profile f_iAnd wherein said N is_sThe sequence of optical surfaces defining the optical axis of the imaging system, and/or

In the step of inputting to the computer a system specification of surface coefficients of an optical surface, comprising: defining each optical surface as spherical or aspherical and wherein each optical surface is expressed as a function of a single radial variable r, and/or

Further expressing the single radial vectorAnd developing each surface to have Taylor coefficient f_i，2jIn which j is 1, 2.. up to 30 and i is 1, 2.. 50.

29. A computer-based method according to any of claims 26 to 28, wherein the step of inputting system specifications of pupil characteristics to the computer comprises:

the position of an input pupil plane along the optical axis, m of the pupil plane being selected_pA fixed pupil plane cross-section p, where m_p≧ 1, where each pupil plane cross-section is defined by an angleDefinitions, wherein p 1_pAnd/or

Wherein each pupil plane cross-section is given a single pupil radial variable q_pDefinition of said pupil radial variable q_pRepresents a signal from said object space and is in (x)_p，y_p，0)＝(q_pcos(θ_p)，q_psin(θ_p) 0) where a ray passing through the pupil plane intersects the first image plane.

30. A computer-based method according to any of claims 26 to 29, wherein the step of inputting system specifications for the object space and the image space to the computer comprises:

selecting an object point located at an infinite distance or at a finite distance, and wherein the step of selecting an object point located at a finite distance further comprises the steps of: selecting an object point in an object plane along the optical axis, and/or

Selecting a first image point in an image plane along said optical axis, and/or

Wherein the step of inputting the system specifications of the object space and the image space to the computer comprises the steps of: expressing the second image point of each fixed pupil plane cross-section as the sum of the first image point plus an optical aberration expressed as a ray aberration order expansion of ray aberration coefficients for each pupil plane cross-section.

31. A computer-based method according to claim 29 or 30, wherein the step of using the computer to convert the system specification into a system of differential equations further comprises the steps of: expressing the optical surface profile as a function of a ray mapping function describing the locations where rays from an object point or at an angle of view and passing through a fixed pupil plane cross-section p intersect each optical surface of the pupil plane cross-section, and/or

Further comprising the steps of: expressing the ray mapping function as (u)_i，p(q_p，t)，v_i，p(q_pT)), where the variable t defines the object and the fixed pupil plane cross-section θ_pHaving pupil coordinates (q)_p，θ_p) So that the optical surface f of the pupil plane cross section_iFunction f expressed as the ray mapping function_i((u_i，p(q_p，t)，v_i，p(q_p，t)))。

32. A computer-based method according to any one of claims 26 to 31,wherein the step of inputting system specifications for the imaging system parameters to the computer comprises: selection of N_sA sequence of +1 materials, each material having a wavelength λ for at least one design wavelength₀Refractive index distribution ofThe object plane, the N_sThe sequence of optical surfaces and said image plane being defined with an optical path length d_iN of (A)_sA sequence of +1 segments, and/or

Wherein the step of converting the system specification to a system of differential equations using the computer further comprises the steps of: applying the Fermat principle to two consecutive segments of each pair as a function of said ray mapping function by mathematically representing the extreme value of the path length between two fixed points along the ray to derive two sets of N_sA differential equation.

33. A computer-based method according to any of claims 26 to 32, wherein the step of inputting to the computer a system specification of surface coefficients of optical surfaces comprises: expanding each optical surface profile to have a surface coefficient f for the ith surface and of order 2j_i，2jIs expanded in series, and/or

Wherein the system specification is converted into two sets of N using the computer_sThe step of differential equations includes: the ray mapping function is expanded in a series expansion of ray mapping function coefficients.

34. A computer-based method according to claim 32 or 33, wherein the step of solving the differential equation further comprises the steps of: finding the two sets of N using a power series approach_sA solution of a differential equation to compute the mapping function and the series coefficients of the optical surface as a function of the subgroup of light aberrations.

35. As claimed in claim 33 or 34The computer-based method, characterized by further comprising the steps of: applying partial derivatives of order k, l to the two sets of N_sA plurality of pupil plane cross-sections, each pupil plane cross-section having a respective optical surface coefficient and a respective ray mapping function coefficient, wherein each pupil plane cross-section has a respective optical surface coefficient and a respective ray mapping function coefficient, and wherein each pupil plane cross-section has a respective aberration matrix.

36. The computer-based method of claim 35, wherein the step of solving the differential equation further comprises the steps of: deriving the surface profile of each optical surface from the calculated coefficients of the optical surface, and/or

Wherein the step of solving the differential equation further comprises the steps of: the clear aperture of each optical surface is thus derived from the series coefficients of the calculated mapping function.

37. A computer-based method according to any of claims 30 to 36, wherein the step of selecting a first image point further comprises the steps of: expressing the first image point with a function specifying an object-image relationship, and wherein the step of expressing the second image further comprises the steps of: expressing the second image in the selected pupil plane section as the sum of the first image plus the light ray aberrations in the x and y directions, wherein the light ray aberrations are related to the known light ray aberration spreads and wave aberration spreads.

38. A computer-based method according to any of claims 29 to 37, wherein the step of inputting system specifications for the pupil characteristics to the computer comprises: inputting a position of an aperture stop along the optical axis, wherein the aperture stop is an entrance pupil, an aperture stop between two optical surfaces, or an exit pupil,

wherein the content of the first and second substances,

-replacing the ray mapping function coordinates of one of the optical surfaces by the pupil coordinates if the aperture stop coincides with said optical surface

-if the aperture stop is between two optical surfaces, then the cross product representing the direction vector from the previous optical surface towards the aperture stop and the direction vector from the aperture stop towards the subsequent optical surface is zero, thereby adding three additional equations to each of the two sets of differential equations;

-if the aperture stop coincides with the exit pupil, it means that the cross product of the direction vector from the last surface to the aperture stop and the direction vector from the stop to the image plane coincide, whereby the cross product of the two vectors is zero and three additional equations are added to the two sets of differential equations,

-if the aperture stop coincides with the entrance pupil, the direction vector from the object to the stop and the direction vector from the stop to the first surface coincide, whereby the cross product of the two vectors is zero and three additional equations are added to the two sets of differential equations.

39. A computer-based method according to any of claims 27 to 38, further comprising the steps of: repeating the method for at least a second design wavelength, and/or

Wherein the step of inputting the system specification of the imaging system parameters to the computer further comprises the steps of: for the at least one design wavelength λ by means of each material₀Refractive index distribution ofTo make each optical path length d_iExpressed as a function of wavelength, and/or

Further comprising the steps of: inputting at least a second set of system specifications to the computer system, wherein each set corresponds to a different configuration of the zoom system, the at least second set of system specifications comprising at least imaging system parameters, pupil characteristics, and object space and image space, the computer-based method of claims 1 to 13 being evaluated for each second set of system specifications.

40. A computer-based method according to any of claims 30 to 39, wherein in the step of inputting system specifications to the computer, the specifications, such as surface coefficients and/or mapping function coefficients, are calculated using an optimization method, such as Monte Carlo optimization.

41. A computer-based method according to any of claims 26 to 40, wherein the method is used to calculate the aberration of light at a first design wavelength up to a maximum order in an optical system comprising a plurality of optical surfaces, each optical surface being defined by a set of surface coefficients f_i，jBy definition, the method comprises the steps of:

calculating all aberration coefficients ∈ for arbitrary pupil plane cross-sections and in ascending order of aberration order using the power series method_{x，p，k，l}And ∈_{y，p，k，l}，

-solving a system of linear equations for each aberration order by

Apply all derivatives for each order and for each pupil plane cross section to the two sets of differential equations,

solving a system of linear equations,

repeat each step until the highest order is reached.

42. A computer-based method according to claim 41, wherein the method is applied to at least two configurations of a zoom system, and/or

Wherein the method is repeated for at least a second design wavelength to calculate monochromatic aberrations for the first and second design wavelengths, and/or

Further comprising the steps of: a refractive index fraction for the at least one design wavelength by means of each materialClothTo make each optical path length d_iExpressed as a function of wavelength to further calculate chromatic aberration of the optical system.

43. A computer-based method according to any one of claims 26 to 42, wherein the method is used in conjunction with an optical design program.

44. A computer-based method for generating an electronic file for a digitally controlled machine used to manufacture a rotationally symmetric imaging system having an optical surface, the method comprising:

-inputting to the computer system specifications comprising:

-an imaging system parameter of the imaging system,

o-a wavelength dependence of the wavelength,

the number of surface coefficients of the optical surface,

o a pupil characteristic of the pupil,

-an object space and an image space,

using a computer to convert the system specification into a system of differential equations by applying the Fermat principle,

-using a set of selection, consistency and distribution rules to input to the computer a subgroup of light ray aberrations to be nominally set to vanish,

-solving said differential equation to obtain surface coefficients of the optical surface and mapping function coefficients defining the ray trajectories and their intersections with the optical surface and thereby defining each clear aperture, and outputting said electronic file comprising the surface coefficients of the optical surface and mapping function coefficients defining the ray trajectories and their intersections with the optical surface and thereby defining each clear aperture.

45. A computer-based method according to any preceding claim, wherein the method is used in conjunction with an optical design program.

46. A computer-based method according to any of claims 1 to 24 and 26 to 45, for generating an electronic file for a numerically controlled machine used to manufacture an imaging system having an optical surface.

47. A computer program product comprising software which, when executed on one or more processing engines, performs any of the methods recited in claims 1-24 and 26-46.

48. A non-transitory signal storage medium storing the computer program product of claim 47.

49. A computer-based system comprising the non-transitory signal storage medium of claim 48.

50. A computer-based system implementing any of the methods of claims 26-46 on a computer.

Technical Field

The present invention relates generally to the field of optical design methods for imaging systems and enables simultaneous correction and/or calculation of optical aberration coefficients, furthermore comprising calculation of ray mapping coefficients providing ray trajectory related information and software for performing the methods. These methods are suitable for determining spherical, aspherical or free-form surfaces.

Background

Today's optical design relies heavily on software implementations of efficient ray tracing and optimization algorithms. Different parameters of the optical system (e.g. the radius and position of the lens) are varied to optimize a defined merit function that measures the image quality for a given field of view (d.p. feder, "Automatic optical design" appl.opt.2, 1209-1226 (1963)). These merit functions typically have many local minima "madly" and there is no guarantee that a local or global optimization algorithm will find a good solution. Thus, one successful and commonly used optimization-based optical design strategy is to select a well-known optical system (e.g., from a patent or published literature) as a starting point and achieve incremental improvements. This optical design approach requires considerable experience, guesswork, and intuition, and is therefore sometimes referred to as "art and science" (Shannon, Robert r. "The art and science of optical design", Cambridge University Press (1997)).

In order to simplify or overcome such laborious and often difficult to reproduce protocols based on optimized optical designs, various optical design approaches have been developed and proposed, the goal of which is to deterministically calculate surfaces or surface coefficients to find an excellent starting point for rapid and final optimization. One known method (and related analytical method) (the simultaneous multi-surface (SMS) design method) is based on constant optical path length conditions for a discrete set of on-axis and off-axis fields that are then imaged perfectly onto the image plane by several surface contours. (J.C.P.Benítez、W.Lin、J.Infante、F.And a. santamaria, "An application of the SMS method for imagingdesigns", opt. express 17,24036-24044 (2009)). The number of imaging fields is usually equal to the number of surfaces, but may in special cases also be greater than the number of surfaces (p.M. nikolic, j.liu, j.infante, f.duerr, "Conditions for perfect focusing multiple point sources with the SMSdesign method", proc. spie 9191,919102 (2014)). These approaches are limited in the number of surfaces that can be counted (up to six today) and do not provide simple scalability to a larger number or surface. Benitez, J.C.M. nikolic, j.liu, j.infante, f.duerr, "Conditions for perfect focus multiple point sources with the SMS design method", proc.spie 9191,919102 (2014)). Related approaches have been used for laser beam shaping systems, but not for imaging System design (f.duerr, h.thienpont, "Optical Zoom System", EP 3147697 a 1). Furthermore, these direct design methods do not use common imaging optics concepts and languages, such as focal length or aberrations.

More generally and well known from the literature, aberration theory can be used to describe and quantify the deviation of light from ideal focus in imaging systems in modern representations, light ray aberration ∈⁽ⁱ⁾Let the deviation intersection point h with the image plane be h₀+∈⁽¹⁾+∈⁽³⁾+∈⁽⁵⁾+ … describes the series expansion of the system's object and pupil variables. There are indeed several concepts and methods that allow to eliminate a certain number of aberration coefficients. The basic idea is always the same, namely to calculate an initial design without specific aberrations and then rely on standard optimization techniques to equalize all aberrations and produce the best overall imaging performance. This design strategy is widely used and very successful, but so far is limited to only a few different low order aberration terms. Gaussian or ABCD matrix opticsOnly first order aberrations are allowed to be eliminated (Shannon, R.R., "The art and science of optical design", Cambridge University Press (1997)). It is also known how spherical aberration can be eliminated for all orders (J.C. Valencia-Estrada, R.B.Flores-Hern a ndez, D.Malaraa-Hern a ndez, "Singles lenses free of all of the spherical aberration", Proc.R.Soc.A. 471,20140608 (2015)). The Abbe Sine Condition (Abbe's Sine Condition), published in 1879, can be used to eliminate coma for all orders, i.e. aberrations linearly related to object variables (m&Photonics News 9(2),56-60 (1998)). By eliminating both spherical aberration and coma, a so-called aplanatic system can be designed (Wassermann, g.d., and e.wolf., "On the same of aplanatic optics systems", Proceedings of the physical society. section B62.1 (1949)). Seidel's formula can be used to calculate The third-order aberrations (Shannon, R.R., "The art and science of optical design", Cambridge University Press (1997)) from The optical path differences of The oblique cones and chief rays of a surface. The Seidel equation has been used or re-expressed to derive several Closed solutions for global binary, ternary or quaternary systems that are Corrected for (part of) third-order aberrations (d.korsch, "Closed Form Solution for Three-Mirror telescales, Corrected for spatial errors, Coma, acquisition, and Field future", applied. operation.11, 2986-2987(1972), d.korsch, "Closed-Form solutions for imaging systems, Corrected for third-order errors, j.operation.soc.am.63, 667-1973), Rakich," Four-Mirror mixtures, set 1: a for space-scale optics ", 10346.46). Schulz has shown how to design three refractive surfaces (two aspheric surfaces and one spherical surface) without any of all five monochromatic Seidel aberrations (g.schulz, "Primary aberration-free imaging by way of three refractive surfaces", j.opt.soc.am.70,1149-1152 (1980)). The same authors have also developed a point-by-point continuous construction method which allows the design of a second-order aplanatic system consisting of four lenses (G Schulz, "high order aplan)"Optics communications 41, 315-. Importantly, the obtained points must still be fitted afterwards to retrieve the closed-form aspheric description. Correction Criteria for secondary field-dependent aberrations have been derived based on tracing the beamlets (Zhao c. and burger j.h., "criterion for correction of temporal field-dependent aberrations", JOSA a 19, 2313-. The "Theory of abortions of central Optical Systems" by Hristov is based on conformal and matrixed transformations of a set of aperture rays in sagittal and tangential planes and paraxial transformations in the Optical axis region to eliminate Seidel Aberrations (Hristov, B.A. Optical Review 20, 395-419 (2013)).

In the case of zoom systems, there are indeed several Gaussian/matrix-optical-based design methods, for example YehM, Shiue S, Lu M, "Two-optical-component method for designing the zoom system", Opti.Eng.34, 1826-1834(1995), Park, SC. & Lee, "partial design method based on and with analytical calculation and analysis application to a needle-group inner-focus system", Journal of the Korea Physical Society 64, 1671-. Gaussian optics and Seidel formulas have been used to design Zoom lenses by applying these formulas to each individual lens module (ParkS, Shannon R.R, "Zoom lens design using lens modules", opt. eng.35,1668-1676 (1996)). If more than one wavelength is considered, there does exist a formula to correct for The dominant (linear dependence on wavelength, field and aperture) longitudinal and transverse chromatic aberrations (Shannon, R.R., "The art and science of optical design", Cambridge University Press (1997)). However, most work on chromatic aberration relates to methods of how to properly select lens material combinations, such as P.Harihanan, "Apochromatographic lens combinations: anovel," Optics & Laser Technology 29,217-219(1997), B.F.Carneir Albuquerque, J.Sasian, F.Luis de Sousa, and A.Silva monomers, "Method of choice for color correlation in optical system design," Opti.express 20,13592-13611(2012), GE Wiese, F.Dumont, "reflective porous object analysis and Method of choice of optical materials thermal", US Patent6,950, 2005 (54), Experimental & optical Technology, 31 (1999).

Despite the numerous design methods proposed, there is no design method that allows the elimination of aberration terms equal in number to the number of calculated surface coefficients for any system. To date, the number of surfaces is a limiting factor, and existing methods can only fully exploit the potential of systems consisting of up to six full spherical surfaces and systems with up to three aspherical surfaces. Some methods (e.g., ABCD or Seidel formulas) are applicable to any number of surfaces, but only allow control of a fixed amount of low order aberrations that does not scale with increasing number of surfaces. Furthermore, all existing design methods have in common that they do not provide position information where a ray intersects each individual surface without performing additional ray tracing for the entire optical system. This information is very important for three main reasons. First, it directly provides the clear aperture of all optical surfaces. Second, it, together with the surface expression, allows immediate verification that total internal reflection does not occur in the entire lens system. Third, and in relation to the second argument (argument), it allows further evaluation of the angle of incidence, which is an important factor when referring to the optical system tolerances. The present invention not only provides these ray intersections for all surfaces, but also scales the number of aberrations removed with the number of optical surfaces used, making it the first overall, deterministic and fully scalable approach to imaging design.

Summary of The Invention

According to an aspect of embodiments of the present invention there is provided (see claim 1) a computer-based method for designing an imaging system having an optical surface for nominally eliminating at least one ray aberration, the method comprising:

-inputting to the computer a first system specification comprising:

a parameter of the imaging system is determined,

o a wavelength dependence of the material,

the type and number of surface coefficients of the optical surface,

a pupil characteristic (diaphragm position),

an object space and an image space characteristic,

-using a computer to convert the first set of system specifications into a system of differential equations by applying Fermat's principle,

-using a computer, converting the differential equation into an algebraic equation by: using a power series method up to a given combination order to solve a power series solution of the derived differential equation,

-inputting at least one light aberration coefficient to zero; for each light aberration coefficient set to zero, a surface coefficient is chosen to be unknown, and the pair of light aberration and surface coefficient must occur simultaneously in at least one algebraic equation,

-inputting to the computer the values of all surface coefficients that have not been applied in the previous step to eliminate aberrations,

solving algebraic equations to calculate at least one surface coefficient of the optical surface, all mapping functions (which define the ray trajectories and their intersection with the optical surface and thus each clear aperture) and the aberration coefficients up to the given combined order.

The dependent claims each define a further embodiment of the invention.

The present invention provides methods for designing imaging systems (including optical zoom systems, both for monochromatic and polychromatic light) and/or for designing freeform or rotationally symmetric imaging systems. These imaging systems may consist of several spherical and/or aspherical and/or free-form lenses or mirrors or combinations thereof, whereby the optical elements are centered along a common axis (coaxially aligned) or rotated relative to each other without a common optical axis. These design methods allow for nominally eliminating at least one, or up to a maximum number of, ray aberration terms of the power series expansion for a given number of unknown optical surface coefficients. The number of calculated surface coefficients is always equal to the number of nominally eliminated aberration terms. Furthermore, any non-eliminated aberration coefficients up to a given order may also be calculated.

The method of embodiments of the present invention provides the user directly with an optical design that is nominally free of aberration groups and can meet the requirements to be obtained. Thus, the method directly provides the best solution or one of the best solutions that may be obtained for the system specification that has been entered into the computer program. This creates an important time gain for the optical designer while ensuring to the user that an optimal solution has been found. Furthermore, it allows structured studies of a given system layout by varying the aberrations to be eliminated and evaluating the obtained imaging performance for a given specification. Advantageously, this allows the optical designer to streamline the common design process: an initial design without specific aberrations is calculated and evaluated, and then all aberrations are equalized by relying on standard optimization techniques to yield the best overall imaging performance.

In another aspect of the present invention, the step of inputting system specifications as imaging system parameters to a computer comprises: input N_sA sequence of optical surfaces, wherein each optical surface has an optical surface profile f_iAnd wherein the N_sThe sequence of optical surfaces defines the optical axis of the imaging system.

In another aspect of the present invention, the step of inputting to the computer system specifications as surface coefficients of the optical surface comprises: defining each optical surface as spherical or aspherical; and wherein each optical surface is expressed as a function of a single radial variable r.

In another aspect of the present invention, the step of inputting the system specification as the pupil characteristic to the computer includes: the position of the input pupil plane along the optical axis.

In another aspect of the present invention, the step of inputting the system specification as the object space and the image space to the computer comprises: a first image point is selected in the image plane along the optical axis.

In another aspect of the present invention, the step of inputting the system specification as the pupil characteristic to the computer includes: the position of an aperture stop along the optical axis, wherein the aperture stop is an entrance pupil, an aperture stop between two optical surfaces, or an exit pupil.

If the aperture stop coincides with one of the optical surfaces, the ray mapping function coordinates of said optical surface are replaced by pupil coordinates. If the aperture stop is between two optical surfaces, the cross product representing the direction vector from the previous optical surface towards the aperture stop and the direction vector from the aperture stop towards the next optical surface is zero, thereby adding three additional equations to the original differential equation. If the aperture diaphragm and the exit pupil coincide; it means that the cross products of the direction vector from the last surface to the aperture stop and the direction vector from the stop to the image plane coincide, whereby the cross product of these two vectors is zero and three additional equations are added to the original differential equation. If the aperture stop coincides with the entrance pupil, the direction vector from the object to the stop and the direction vector from the stop to the first surface coincide, whereby the cross product of these two vectors is zero and three additional equations are added to the original differential equation.

In the case of a static system (fixed focal length), the user can thus further define at least one design wavelength (multi-color case discussed later), the number and type of surfaces (and the number of coefficients per surface). Object, object space and image space at finite or infinite distance, which includes the object-image relationship (effective focal length or magnification or any other functional relationship) and the position of the pupil (surface or individual aperture stop).

In another aspect of the present invention, the step of inputting the system specification as the pupil characteristic to the computer further comprises: selecting m of pupil plane_pA fixed pupil plane cross-section p, where m_pNot less than 1. Each pupil plane cross-section is defined by an angleDefinitions, wherein p 1 … m_p(the order corresponds to tangential, sagittal andadditional oblique ray planes). Alternative sequences are possible.

Each pupil plane cross-section is further defined by a single pupil radial variable q_pDefinition of q_pRepresenting a signal from object space and passing through (x)_p,y_p,0)＝(q_pcos(θ_p),q_psin(θ_p) 0), where the ray of pupil plane coordinates intersects the image plane.

In another aspect of the present invention, the step of inputting the system specification as the object space and the image space to the computer comprises: an object point located at an infinite distance or a finite distance is selected.

The step of selecting object points located at a finite distance further comprises the steps of: an object point is selected in an object plane along an optical axis.

In another aspect of the present invention, the step of converting the first set of system specifications to a system of differential equations using a computer further comprises the steps of: the optical surface is expressed as a function of a light ray mapping function. These functions describe the position where a ray from an object point or at a field angle and passing through a fixed pupil plane cross section p intersects each optical surface of said pupil plane cross section.

The ray mapping function may be further expressed as (u)_i,p(q_p,t),v_i,p(q_pT)), where the variable t defines the object and the pupil plane cross section θ is fixed_pHaving pupil coordinates (q)_p,θ_p) So that the optical surface f of the pupil plane cross section_iFunction f expressed as a ray mapping function_i((u_i,p(q_p,t),v_i,p(q_p,t)))。

In another aspect of the present invention, the step of inputting system specifications as imaging system parameters to a computer comprises: selection of N_sA sequence of +1 materials, each material having a wavelength λ for at least one design wavelength₀Refractive index distribution ofObject plane, N_sSequence definition N of optical surfaces and image planes_s+1 optical path length d_iThe sequence of the segment(s) of (c).

In a further aspect of the present invention, the step of converting the first set of system specifications to differential equations using a computer further comprises the steps of: the fermat principle is applied to the two consecutive optical path length segments of each pair by mathematically representing that the optical path length between two fixed points is an extreme along the ray. Optionally, the fermat principle can be applied to two consecutive segments of each pair as a function of the ray mapping function to derive two sets of differential equations.

Advantageously, by introducing a surface and a mapping function for an arbitrary but fixed pupil plane cross-section, the path of an arbitrary ray passing through the pupil plane cross-section can be described. By applying the fermat principle, two sets of differential equations are derived. The two new sets of differential equations may be identified as the x and y components of the associated ray aberration series. The fermat principle is applied in the same way throughout all embodiments. An arbitrary ray path from an object to an image is expressed as a path length section, and the fermat principle is applied in pairs, thereby obtaining N +1 sections of N optical surfaces. It is very important that the conversion from the system description defined above to differential equations follows the basic rules (A. Friedman and B. McLeod, "optical design of an optical algorithms", architectural for Rational Mechanics and Analysis 99,147-164(1987), B. Van-Brunt, "chemical compatibility of systematic in geometrical optics", JOSA A11, 2905-2914 (1994)). Thus, this means that there is no inherent limit to the number of optical surfaces that can be treated with these methods. For all embodiments of the invention they can be easily scaled to systems with far more than just six optical surfaces. It is assumed that the number of optical surfaces can be limited to 30 or optionally up to 50, since optical systems with more than 30 surfaces are unlikely to occur, or only in very special applications, such as optionally up to 50.

In another aspect of the present invention, the step of inputting the system specification as the object space and the image space to the computer comprises the steps of: the second image point of each fixed pupil plane cross-section is expressed as the sum of the first image point plus the optical aberration. The optical aberration can be expressed as the number of ray aberration steps of the ray aberration coefficient per pupil plane cross section.

An advantage of the present invention is that common optical aberration approaches provide very important indications. Whether wave aberration or light aberration, these functions are usually expressed in terms of a power series expansion of the variables under consideration. The hint leads to the idea of using a power series approach to the derived differential equation to solve for a power series solution of the variable. In general, the solution substitutes the power series representation of all the involved functions into a differential equation to find the recurrence relation of the unknown coefficients. This means that the optical surface, the mapping function and the real image function (ideal object image function plus the optical ray aberration term) are expressed as a power series expansion. Closed recursion relationships of infinite order cannot be deduced generally; however, for a finite order of all the functions involved, the resulting algebraic equations can be solved for the unknown coefficients using this solution. The key element in doing so is by operating in a defined sequence of pupil plane cross-sections (see matrix representations or visualizations in fig. 3 to 9) and by following a given order of pupil plane cross-sections and ray aberrations for the identified x and y components in said pupil plane cross-sections.

In the case where all surface coefficients of a system consisting of N surfaces are well defined and known, the proposed finite power series method is able to calculate all unknown mapping functions and light aberration coefficients for a defined sequence of pupil plane cross sections and in ascending order. Since there is a clear correlation between the light ray aberrations used here for the pupil plane cross-section and the light ray aberrations or wave aberrations well known from the literature, it is easy to switch between these different representations.

In another embodiment of the method, a number of available, but not yet known, surface coefficients are used to select an equal number of aberration coefficients to be nominally eliminated in one or more pupil plane cross-sections. Optionally, this may be done by following the select-consistency-distribution (SCD) rule in some embodiments of the invention. The selected and nominally eliminated aberration coefficients define which orders the finite power series method is applied to and which pupil plane cross sections to calculate all mapping functions, unknown surface coefficients and unabated aberration coefficients.

In the case of a (fully) spherical system, this approach yields a system of non-linear equations for all unknown coefficients, which can be solved using standard methods such as the Newton-Raphson algorithm or other algorithms. For the remaining higher order calculations, due to the additional aspheric coefficients, and in the case of an all-aspheric system, the finite power series method results in a linear system of equations for the unknown coefficients in the cross-section of the pupil plane under consideration in ascending order of the aberration orders, which can be solved using a standard solution of solution, such as gaussian elimination. In the case of an all-aspheric system, only the first aberration order calculation yields a set of non-linear equations that can be solved as explained for the global surface system. These relationships between the sphere/asphere and the derived set of (non-) linear equations hold for all embodiments of the invention, including free-form optical surfaces.

In another aspect of the present invention, the step of inputting to the computer a system specification of surface coefficients of the rotationally symmetric optical surface comprises: expanding each optical surface profile to have a surface coefficient f for the ith surface and of order 2_i,2jIs expanded.

In another aspect of the present invention, the step of inputting to the computer system specifications as surface coefficients of the free-form optical surface comprises: expanding each optical surface to have a surface coefficient f for the ith surface with x being j order and y being k order_i,j,kIs expanded. The monomial expansion may be further converted to alternative free-form surface representations such as Zernike, Chebyshev, Forbes, and the like.

Further, the step of converting the first set of system specifications to differential equations using a computer comprises: the ray mapping function is expanded by a series expansion of the ray mapping function coefficients. Optionally, in some implementationsIn an example, the system specification is converted into two sets of N_SA differential equation.

In a preferred embodiment of the invention, the step of solving the differential equation further comprises the steps of: a power series method is used to find the solution to the differential equation to calculate the mapping function and the series coefficients of the optical surface as a function of the eliminated light aberrations.

Optionally, in some embodiments, partial derivatives of the order k, l are applied to the two sets of N_SA step of differential equations, said order k, l being provided by using for each pupil plane cross section a predefined aberration matrix, wherein each predefined aberration matrix relates each light ray aberration of a subgroup to a partial derivative of the order k, l of the series expansion of light ray aberrations in said pupil plane cross section, to derive a system of equations for optical surface coefficients and light ray mapping function coefficients.

Preferably, the step of solving the differential equation further comprises the steps of: the surface profile or shape of each optical surface is derived from the calculated coefficients of the optical surface.

Advantageously, the step of solving the differential equation further comprises the steps of: the clear aperture of each optical element is derived from the series coefficients of the calculated mapping function.

Further, the step of inputting to the computer system specifications including surface coefficients of the optical surface further comprises: each surface is selected to be spherical or aspherical. Furthermore, the method comprises the steps of: expressing each surface as a single radial variableAnd expanding each surface as a series, such as having a taylor coefficient f_i,2jWherein j is 1,2 … up to 10 or optionally up to 30, and i is 1, 2.. 30 or optionally up to 50.

Each method is based on a (non-) linear system of equations and solutions that define how the optical surface coefficients used to eliminate aberration terms can be calculated as power series coefficients, such as taylor power series coefficients, of up to any order (typically no more than 20 orders, but optionally in some embodiments up to 30 orders). It is assumed that the number of surfaces does not exceed 30, or may optionally be up to 50 in some embodiments, as discussed above.

In another aspect, the step of selecting the first image point further comprises the steps of: the first image point is expressed as a function defining an object-image relationship. The step of expressing the second image may further comprise the steps of: the second image in the selected pupil plane cross section is expressed as the sum of the first image plus the light ray aberrations in the x and y directions, which are related to the known light ray aberration spreads and wave aberration spreads.

In another aspect of the invention, the method further comprises the steps of: at least two different materials are selected for at least two design wavelengths.

In another aspect, the method further comprises the steps of: at least two different materials are selected for a design wavelength to nominally cancel at least one chromatic aberration coefficient.

In another aspect of the invention, the method further comprises the steps of: the method is repeated for at least a second design wavelength, for example to eliminate monochromatic aberrations at more than one wavelength.

In a further aspect of the invention, the method comprising the step of inputting system specifications for imaging system parameters to a computer further comprises the steps of: by means of each material for the at least one design wavelength lambda₀Refractive index distribution ofTo make each optical path length d_iExpressed as a function of wavelength.

In the case of a multi-color design, there are three options similarly available for both static and zoom systems, using at least two different materials (e.g., flint and crown glasses) simultaneously. First, the above-mentioned monochromatic method is used directly without any modification, while the polychromatic performance is controlled only by a suitable choice of lens material. Second, at least two design wavelengths are defined that yield different refractive indices in the optical path length segment, and for each design wavelength two sets of differential equations are derived from the fermat principle. Much like a monochromatic zoom system, the aberration terms to be nominally eliminated are now distributed among the design wavelengths considered. The solutions for the mapping function, aberration coefficients and surface coefficients are again obtained by the proposed power series method, this time for multiple wavelengths. In other words, this allows nominally eliminating the selected monochromatic aberrations at each defined design wavelength, and thereby controlling the multicolor imaging performance.

Third, the aberration power series is extended by multiplying the monochromatic aberration series by the ascending order of λ to also include the dependence on wavelength. The fermat principle is applied as in the case of a monochromatic color for one design wavelength. The only difference is that the refractive index in the optical path length segment is now a function of λ, expressed for example by the Sellmeier equation. In case all surface coefficients (and zoom configurations) are well defined, the proposed power series method allows to calculate all mapping functions and aberration coefficients (including chromatic aberration terms) for a defined sequence of pupil plane cross sections and in ascending order. In another embodiment, an equal number of aberration coefficients are selected using a number of available but not yet known surface coefficients, with at least one chromatic aberration coefficient selected to be nominally eliminated. Optionally, this can be done by following the extended SCD rule for chromatic aberration of the present invention. As before, the unknown map and surface coefficients can be calculated using a power series method.

In a further aspect of the invention, the system specification input to the computer program is calculated or changed using an optimization method.

The step of inputting system specifications into the computer comprises: calculating at least one optical surface (from its vertex f) by means of an optimization method_i,0Defined) an initial position along the optical axis.

In a further aspect, the optimization method is Monte Carlo (Monte Carlo) optimization.

Calculating the starting system configuration based on low order aberration theory (e.g., gaussian optics or Seidel formula) is a well known strategy. This design approach ensures that the calculated system is nominally free of these low order aberrations. In the next step, the final system layout, in which the equalization of all aberrations takes place, is reached using common optimizations.

Thus, embodiments of the present invention allow for fast computation of complex starting systems using a standard optimization process as a final step. This allows for significant time and effort (and thus cost) savings that can be used to address additional product-related issues, such as manufacturability, tolerances, and assembly. Importantly, the calculated ray mapping function and clear aperture allow immediate evaluation of important parameters, such as all optical surfaces and angles of incidence of different fields.

In another embodiment according to the invention, the method further comprises the steps of: at least a second set of system specifications is input to the computer system, wherein each set corresponds to a different configuration of the zoom system. The at least second set of system specifications includes at least imaging system parameters including surface coefficients and surface vertices, pupil characteristics, and object and image spaces, and the computer-based method of the invention is performed for each second set of system specifications.

In the case of a monochrome zoom system, it is necessary to define at least two zoom configurations in addition to the system description defined above; each zoom configuration is characterized by a set of surface positions (which allow movement between different zoom stages) and an object-image relationship (e.g., a change in focal length between the two zoom configurations). The fermat principle is applied as before, resulting in sets (e.g., two sets) of differential equations for each zoom configuration. In case all surface coefficients and zoom configurations are well defined, the proposed power series method is able to calculate all mapping functions and aberration coefficients for a defined sequence of pupil plane cross sections, in ascending order and simultaneously for the zoom configuration. In another embodiment of the method, an equal number of aberration coefficients to be nominally eliminated and distributed among zoom configurations are selected using a number of available, but not yet known, surface coefficients. Optionally, this may be done by following a well-defined selection-consistency-distribution (SCD) rule for the zoom system. Much like a static system, unknown mapping, surface and/or aberration coefficients are calculated for each zoom configuration simultaneously by using a power series approach for a zoom system consisting of spherical and/or aspherical surfaces.

In a further aspect of the invention, the method is used to calculate the aberration of light at a first design wavelength up to a maximum order in an optical system comprising a plurality of optical surfaces, wherein each optical surface is defined by a set of surface coefficients f_i,jBy definition, the method comprises the steps of:

calculating all aberration coefficients ∈ for arbitrary pupil plane cross-sections and in ascending order of aberration order using the power series method_x,p,k,lAnd ∈_y,p,k,l，

-solving, for each aberration order, a system of linear equations by:

all derivatives for each order and for each pupil plane cross section are applied to a differential equation to derive an algebraic equation, for example to two sets of differential equations,

o solving a system of linear equations,

o repeat each step until the highest order is reached.

In an embodiment, the method is applied to at least two configurations of the zoom system.

In another embodiment, the method is repeated for at least a second design wavelength to calculate monochromatic aberrations for the first and second design wavelengths.

In another embodiment, the method further comprises the steps of: by means of a refractive index profile of each material for the at least one design wavelengthTo make each optical path length d_iExpressed as a function of wavelength to further calculate chromatic aberration of the optical system.

Advantageously, the method of the invention can also be used to calculate aberrations in existing optical systems, wherein the optical system can be a static or zoom system. Optionally, for a zoom system, equations related to the zoom system may be used. The calculated aberrations can also be evaluated at different wavelengths and the method of the invention also allows chromatic aberrations to be calculated.

In a further aspect, the method is for use with an optical design program.

In a further aspect, a method for manufacturing an optical element having at least one optical surface defined by at least one surface coefficient is provided. The optical element may be used in an imaging system in which at least one light ray aberration is nominally eliminated, wherein the at least one surface coefficient is obtained by the method according to the invention. Optionally, the method is used in a rotationally symmetric system, but the invention is not limited thereto.

Thus, in a further aspect of the invention, there is provided a computer-based method for generating an electronic file for a digitally controlled machine for use in the manufacture of a rotationally symmetric imaging system having an optical surface, the method comprising:

-inputting to the computer a first set of system specifications comprising:

a parameter of the imaging system is determined,

o a wavelength dependency is determined from the wavelength dependence,

the type and number of surface coefficients of the optical surface,

pupil characteristics (e.g., diaphragm position),

an object space and an image space,

using a computer to convert the system specification into a system of differential equations by applying the Fermat principle,

-using a computer, converting the differential equation into an algebraic equation by: using a power series method up to a given order to solve a power series solution of the derived differential equation,

-inputting at least one light aberration coefficient to zero; for each light aberration coefficient set to zero, a surface coefficient is chosen to be unknown, and the light aberration and surface coefficient of the pair(s) must occur simultaneously in at least one algebraic equation,

-inputting to the computer values of the remaining surface coefficients that have not been applied in the previous step to eliminate aberrations,

solving the differential equations to calculate at least one surface coefficient of the optical surface, and all mapping functions (which define the ray trajectories and their intersection with the optical surface and thus each clear aperture) and the aberration coefficients up to a given order.

Advantageously, the method of the invention is used as a method for manufacturing an optical element, such as a lens or a mirror. In practice, the method provides surface coefficients of the optical elements of the imaging system as output in an electronic file, which surface coefficients can be provided directly as input for manufacturing the optical elements. When implemented in an imaging system using mapping function coefficients, the fabricated optical element will allow the system to be nominally free of the at least one light ray aberration. Thus, the surface coefficients calculated by the method of the present invention can be provided as an electronic file directly to the manufacturing process of lenses, mirrors, etc. The mapping function coefficients further define the ray trajectory and its intersection with the optical surface.

The present invention also provides a computer program product which, when executed on a processing engine, performs any of the methods of the present invention.

The computer program may be stored on a non-transitory signal storage medium such as a compact disc (CD-ROM or DVD-ROM), a magnetic disk, and solid state memory such as flash memory.

The technical effects and advantages of the embodiments according to the invention correspond, mutatis mutandis, to those of the corresponding embodiments of the method according to the invention.

Brief Description of Drawings

These and other technical aspects and advantages of embodiments of the present invention will now be described in more detail, with reference to the accompanying drawings, in which:

fig. 1 illustrates different coordinate systems in the object plane, pupil plane and image plane. The optical system is aligned along the z-axis and the object point as a function of t is along the x-axis, such that the x-z plane is a tangential plane and the y-z plane is a sagittal plane.

FIG. 2 shows a general system layout of a coaxial rotationally symmetric optical system that should be formed by a wave front n₀The depicted object is imaged at the image surface D as an image depicted by the relation t (t).

FIG. 3 illustrates an x-component in a tangential plane in accordance with various embodiments of the invention.

FIG. 4 illustrates the x-component in the sagittal plane according to various embodiments of the invention.

FIG. 5 illustrates the y-component in the sagittal plane according to various embodiments of the invention.

FIG. 6 shows the x-component in the (π/3) oblique ray plane according to various embodiments of the present invention.

FIG. 7 shows the y-component in the (π/3) oblique ray plane according to various embodiments of the present invention.

Fig. 8 shows a joint matrix of combined x and y components in the sagittal plane.

Fig. 9 illustrates the visualization of a desired plane and corresponding black font matrix elements by shaded rectangles, which can be easily extended to any higher aberration order, according to an embodiment of the invention.

Fig. 10 shows the optical layout of a Cooke triplet obtained with the method of the present invention.

FIG. 11 shows a dot diagram associated with the Cooke triplet lens layout of FIG. 10 obtained with the method of the present invention.

Fig. 12 shows the optical layout of a wide-angle lens obtained with the method according to the invention.

Fig. 13 shows a stippling diagram associated with the wide-angle lens of fig. 12 obtained with a method according to the invention.

Fig. 14 shows the optical layout of a zoom lens with a fixed final image position and total system length obtained with the method according to the invention.

Fig. 15 shows a second optical layout of a zoom lens with a fixed final image position and total system length obtained with the method according to the invention.

Fig. 16 shows a diagram of the dots associated with the zoom lens of fig. 14 obtained with the method according to the invention.

Fig. 17 shows a diagram of the dots associated with the second zoom lens of fig. 15 obtained with a method according to the invention.

Fig. 18 shows the optical layout of an all-aspherical Hastings triplet obtained with the method according to the invention.

Fig. 19 shows a dot diagram associated with a Hastings triplet lens obtained with a method according to the invention.

FIG. 20 illustrates the chromatic aberration focus shift obtained with a Hastings triplet lens obtained with the method according to the invention.

FIG. 21 shows an optical layout of a full free-form three mirror imaging design obtained with a method according to an embodiment of the invention.

FIG. 22 shows a dot alignment diagram associated with the full free-form three mirror imaging design obtained with the method of FIG. 21 according to the present invention.

FIG. 23 shows an optical layout of a full free form four mirror imaging design obtained with a method according to an embodiment of the invention.

FIG. 24 shows a dot alignment diagram associated with a full free-form three mirror imaging design obtained with a method according to an embodiment of the invention.

Detailed description of the embodiments

The present invention will be described with respect to particular embodiments and with reference to certain drawings but the invention is not limited thereto but only by the claims. The drawings described are only schematic and are non-limiting. In the drawings, the size of some of the elements may be exaggerated and not drawn on scale for illustrative purposes. Where the term "comprising" is used in the present description and claims, it does not exclude other elements or steps. Furthermore, the terms first, second, third and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequential or chronological order. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments of the invention described herein are capable of operation in other sequences than described or illustrated herein.

The present invention provides a method for designing (centering) an imaging system by solving differential equations derived from the fermat principle. These imaging systems consist of a sequence of (movable or fixed) spherical, aspherical or free-form surfaces, or a combination thereof. The power series solution follows a clear set of rules that allow the calculation of a predefined number of surface parameters equal to the number of selected aberration terms to be nominally vanished. In addition, methods provide mapping function coefficients that allow for describing the exact location where any ray intersects each individual surface (e.g., without requiring ray tracing).

Definition of

The imaging system parameters are parameters that define the properties of the imaging system. These parameters include at least one wavelength or bandwidth for which the imaging system is designed, the number and type of optical surfaces (i.e., spherical, aspherical, and/or free-form). The materials of the optical surfaces, the medium in which light propagates in the imaging system, the location of each or at least one of the optical surfaces (e.g., surface vertices), object and image space, field of view, effective f-number, and the like. The imaging system is designed for at least one design wavelength, and the performance of the imaging system may be, and typically is, wavelength dependent.

An optical surface may be expressed as a function z ═ f_i(x, y) (with or without symmetry), the function can be further decomposed into a finite series, each term of the series weighted by a surface coefficient. For example, the finite series may be a polynomial series or a truncated power series expansion. An example of a polynomial sequence is an even-order rotationally symmetric power series of the general polynomial form:

when the term "pupil" is used throughout, it is equivalent to an exit pupil, as known to the skilled person in the literature. The definition of the exit pupil and the entrance pupil depends on the aperture or aperture stop.

The entrance pupil is an optical image of the aperture stop as seen from a point on the optical axis in the object plane. The exit pupil is an optical image of the aperture stop as seen from a point on the optical axis in the image plane.

The pupil plane cross-section is the pupil plane and a second plane (defined by the angle θ) containing the optical axis z_pCharacterized) and the pupil plane cross-section can be used to evaluate optical aberrations. FIG. 1 illustrates an arbitrary but fixed angle_pCorresponding to a cross-section including at the fixed angle theta_pA plane of rays intersecting the pupil plane.

The pupil plane cross-section may be selected according to the polar angle relationship:

wherein p is 1 … m_p

Where the integer value m_pThe total number of pupil plane cross-sections considered is described, whereas the index p will be the angle θ_pDistributed in the angular domain [0, pi/2 ]]And (4) the following steps. Case p-1 corresponds to the angle θ_pA first plane, the so-called tangential plane, is given by 0 (the direction changes sign from pi to 0, but it describes the same plane). For p 2, the second plane is defined by θ₂Given by pi/2, is commonly referred to as the sagittal plane. For each increment p 3,4, …, the angle θ is increased_pAn additional oblique ray plane is defined.

An aperture or aperture stop is an opening that limits the amount of light that passes through the imaging system. The aperture or aperture stop has a diameter and a position along the local optical axis. The diameter of the aperture stop defines the clear aperture of each surface. The clear aperture defines the unobstructed portion of the optic through which light will pass. What should meet the specification is the diameter or size of the optical component. In some cases, the diaphragm may have an elliptical or rectangular shape.

The optical axis is an imaginary line from the object space to the image space in the imaging system, which passes through the center of curvature of each optical surface and coincides with the symmetry axis of the rotationally symmetric optical system. In the case of tilted optical elements, the local optical axis is usually defined by the chief ray of the on-axis field.

The characteristics of the entrance and exit pupils (or pupils) are shape and size, their position (e.g., optionally along the optical axis), and by definition further depend on the position and diameter of the aperture stop.

The object space is a plane, a shaped surface or a volume comprising a real or virtual object to be imaged by the imaging system. Which is generally perpendicular to the optical axis of, for example, the first optical element. The object may be on or off axis, at an infinite or finite distance, real or virtual, and point-like, extended or three-dimensional.

The object point may be any point to which it belongs if the object is extended; or if the object is a dot, the object point is the object.

The object plane is located at a finite or infinite distance and is described by the variable t (e.g. along the x-axis) as (t,0, z), respectively_O) (wherein z is_OIs a fixed position along the optical axis or is a function of t (in this case not the object plane but the object space) or is described by a direction vector (sin (t),0, cos (t)) for an infinite distance. In the case of non-rotationally symmetric systems, a single variable t is extended to t_xAnd t_yIn a limited case, to (t)_x,t_y,z_O) And at the same time extended to a direction vector for the infinite case

The image plane is the plane in which the image produced by the imaging system is formed. If the object plane is perpendicular to the common optical axis, the image plane will also be perpendicular to this axis. The images may be on-axis or off-axis, oblique, at infinite or finite distances, real or virtual, and point-like, shaped, or three-dimensional. If three-dimensional, the concept of image space is used instead of image plane. The image point may be any point to which it belongs if the image is extended or if the image is punctual.

The deviation of the wavefront from spherical is called wave aberration. The distance of the ray intersection from the center of curvature of the spherical wavefront in the image plane is called the ray aberration. In the case where there is aberration in the optical system, an imperfect image is formed. If all aberrations are zero, the wavefront is spherical, all rays converge to their center of curvature, and a perfect geometric point image (point image) is obtained. There do exist various equivalent representations of ray and Wave Aberration orders and conversion rules between them, which are known to the skilled person from the literature (see, for example, j.l. rays, "Exact relationship between Wave Aberration and ray Aberration", optical Acta,11:2,85-88 (1964)).

Ray aberration expansion represents the displacement of a real (tracked) ray from an ideal image point at the image plane (or image surface). From having coordinates along the x-axisThe intersection of the ray of the object point with the image plane can be written in vector formWhereinMarking the intersection point of paraxial rays.For an object at infinity, t becomes tan (α), where α is the angle of view of the parallel rays entering the optical system, here epsilon ∈⁽ⁱ⁾Are Buchdahl aberration polynomials of order 3,5, and 7, …, respectively. For systems with single or biplane symmetry or without any given symmetry, there do exist similar expansions that include additional aberrations not present in rotationally symmetric systems.

The wave aberration spread of a particular field angle or point object (point object) represents the deviation of its wavefront from spherical at the exit pupil. Without loss of generalityIn the case of a rotationally symmetrical system, the object can be oriented along the x-axis. The wave aberration function can then be represented by the wave aberration coefficient a_(2l+m)nmWritten as (V.Mahajan, "Optical Imaging and Absolutions: Ray geometrical optics" SPIE press (1998)):the order of the wave aberration term is equal to i 2l + m + n and is always even in the case of rotational symmetry. The total number of i-order independent aberration coefficients is given by the formula N_iGiven as (i +2) (i + 4)/8-1.

If all the wave aberrations are zero, the wavefront is spherical, all the rays converge to their center of curvature and a perfect geometric point image is obtained. In the case where there is aberration in the optical system, an imperfect image is formed. Very similarly, the ray aberration represents the displacement of a ray from an ideal image point at the image plane (image surface), as illustrated in fig. 1.

For Systems with one or more planar symmetries or without any given symmetries (all of which are intended to be included within The scope of The present invention), there do exist similar developments that include additional Aberrations not present in rotationally symmetric Systems (see, for example, Richard Barakat, Agnes Houston, "The Aberrations of Non-rotationally symmetric Systems and therapy Effects", Optica Acta,13:1,1-30 (1966)).

Methods according to embodiments of the present invention relate to rotationally symmetric, planar symmetric, and asymmetric optical systems. Thus, the corresponding aberration function is described.

The output of any computer-based method according to any embodiment of the invention is an input to a further optical design program. The output of any embodiment of the invention may be an electronic file defining the implemented design; i.e. the shape or surface of the optical element constituting the design. The output of any embodiment of the invention may be an electronic file for a numerically controlled machine that defines components of a design to be used to manufacture an imaging system having an optical surface. The output may be a computer program product comprising software which, when executed on one or more processing engines, performs any of the methods of embodiments of the present invention. The output may be a non-transitory signal storage medium storing the computer program product.

1.1 aberration function of rotationally symmetric System

From having coordinates along the x-axisThe intersection of the ray of the object point with the image plane can be written in vector formWhereinMarking the intersection point of paraxial rays.For an object at infinity, t becomes tan (α), where α denotes the field angle of parallel rays entering the optical system ∈ is the Buchdahl aberration polynomial of 3 rd order, 5 th order, 7 th order …, respectively, or similar polynomials.

Up to seven orders are given and have indexed aberration coefficients σ, μ, and τ. For the sake of completeness, two coefficients v are added₁(defocus) and v₂(inclined) and these two coefficients are at h₀The case of marking paraxial ray intersections is zero, but these lowest order terms will also be considered below. From σ₁To sigma₅Are the coefficients of third-order spherical aberration, coma, astigmatism, field curvature and distortion, respectively. From mu₁To mu₁₂Only nine of the fifth order aberration coefficients of (d) are independent and there are three additional relations μ₂＝3/2*μ₃，μ₇＝μ₈+μ₉And mu₄＝μ₅+μ₆. Similarly, from τ₁To tau₂₀Only 14 of the 20 seventh order coefficients are independent and there are indeed six additional relations. Any higher order of aberration can be deduced in a similar manner. Optionally in some embodiments, the light aberration coefficients may also be expressed in the form of wavefront aberration coefficients. Generally, the total number of i-order independent light aberration coefficients is given by the formula N_iGiven as (i +1) (i +7)/8, where the aberration order i is always odd in the case of rotational symmetry.

The intersection of rays from an object point (finite distance) or field angle (infinite object) described by t along the x-axis can be written in the vector form h ═ h (h ═ h)_x,h_y)＝h₀+∈⁽¹⁾+∈⁽³⁾+∈⁽⁵⁾+∈⁽⁷⁾+ …, wherein h₀The ideal image point is marked. In Cartesian coordinates, the vector component is given as h₀T (t),0, D), where t (t) describes the ideal image point position as a function of t at the image plane D. In the most general case, the image plane may also be a function d (t) describing a curved image surface. In evaluating optical aberrations, this is usually done in different pupil plane cross-sections that can be selected according to the following polar angle relationship:

where the integer value m_pThe total number of pupil plane cross-sections considered is described, whereas the index p will be the angle θ_pDistributed in the angular domain [0, pi/2 ]]And (4) the following steps. Optionally, the case when p ═ 1 corresponds to the first plane given by the angle ═ pi ═ 0 (the direction changes sign from pi to 0, but it describes the same plane), the so-called tangential plane. For p 2, the second plane is given by pi/2 and is commonly referred to as the sagittal plane. For each increment p +1, one additional plane defined by the angle is added.

For a channel formed by theta_pGiven each pupil plane cross-section, the cartesian pupil coordinates (x, y) of the elements of that plane are (rCos θ)],rSin[θ]) Can be defined by a single pupil variable q_pI.e. in the tangential plane is (q)₁0), in the sagittal plane is (0, q)₂) At the first oblique ray plane (theta)₃Pi/3) areOr generally as:

(q_p,x,q_p,y)＝(q_pCos[θ_p],q_pSin[θ_p])。 (1.2)

optionally, in some embodiments, this applies to all planes for which the given value p 1.

Consider m in more detail_pIn the case of 3, the ray aberration spread can be evaluated in different planesAndwherein the second lower index p denotes the x and y coordinate components for the angle theta_pOf the plane under consideration. For the tangential plane (p ═ 1), the aberration (with x ═ q here)₁) Obtaining:

note that due to symmetry, all y-components of the tangential planesAre all zero. For a sagittal plane (p ═ 2), the aberration (with y ═ q here)₂) For the x-coordinate component, we obtain:

＝tν₂,

and for the y coordinate component:

＝yν₁,

＝y⁵μ₁+t²y³μ₅+t⁴yμ₁₁,for the first oblique ray plane (p ═ 3), (q)₃The aberration of t) is found for the x-coordinate component:

and for the y coordinate component:

advantageously, if more than these three planes (m) are considered_p≧ or) all additional oblique ray planes (with θ) can be used in the same manner_p) To expand the aberration order of the lightAndis expressed as a corresponding variable_pAs a function of (c). An advantage of the present invention is that by using such different planes, the original light or wave aberration coefficients can be determined and separated. This may preferably be done by looking at a single variable q_p(q₁＝x,q₂＝y,q₃Etc.) and/or t is/are performed by applying to the corresponding aberration component partial derivative(s) having a given correlation, these partial derivatives being at q_pAnd t is evaluated at 0. For example,and coefficient sigma₂In proportion toAnd coefficient (3 σ)₃+σ₄) And (4) in proportion. Following this concept, all aberration coefficients (or combinations thereof) for different planes can be expressed by the associated partial derivatives, which are at q_pEvaluated at t 0 and visualized in different matrices. For example, where each column represents the partial derivative with respect to x, starting from 0 in column a to 7 in column h, and where each row represents the partial derivative with respect to t, from 0 in row 1 to 7 in row 8. Using this formalism, the following figures show the derivative corresponding to the ray aberration in the pupil plane, with the variable q_pAnd t. Thus, fig. 3 shows the partial derivatives associated with the x-component in the tangential plane, fig. 4 shows the partial derivatives associated with the x-component in the sagittal plane, and fig. 5 shows the partial derivatives associated with the y-component in the sagittal plane.

FIG. 6 shows the partial derivatives associated with the x-component in the (π/3) oblique ray plane, and FIG. 7 shows the partial derivatives associated with the y-component in the (π/3) oblique ray plane.

The drawn-off derivative matrix elements represent the respective aberration x or y components in the considered plane due to symmetryAnd not on these derivatives. The grey font matrix elements indicate that the corresponding aberration x or y components in the considered plane do depend on these derivatives, but do not provide independent aberration expressions when following the plane order. For example,providing a coma coefficient σ₂And in grey fontThe coma coefficients are also provided, both evaluated at x-y-t-0, since the existing matrix elements of the x and y components in the sagittal plane do not overlap, the two matrices ∈_x,2And ∈_y,2May be combined into a single joint matrix ∈_x,y,2As shown in fig. 8.

As has been described previously in the context of the present invention,combined astigmatism and field curvature coefficient (3 sigma)₃+σ₄) Proportionally, it is therefore necessary to consider one additional plane to determine the two aberrations independently. In thatAnd aberration (3 sigma)₃+σ₄) In the proportional case, the expression is knownAndequivalent to knowing the aberration σ separately₃And σ₄. The first order aberrations require only the tangential plane, the third order (Seidel) aberrations require the tangential plane and the y component in the sagittal plane, and the fifth order aberrations require both the x and y components in the tangential and sagittal planes to determine all aberrations independently. Every fourth additional aberration order requires an additional x or y component of the new skew-ray plane, and therefore every eighth new order requires a new skew-ray plane. The relationship between the aberration order of the ray i and the number of matrices required can be given asWhereinIs the largest integer less than or equal to x. The planar sequence ensures the number N of independent aberration order coefficients for a given aberration order i_i(i +1) (i +7)/8 is equal to the number of diagonal black font matrix elements having the same combination order i in the matrix of the desired plane and the corresponding x and y components.

The visualization of the desired plane and the corresponding black font matrix elements by the shaded rectangles is shown in the matrix of fig. 9, which can be easily extended to any higher aberration order.

On the contrary, since the method according to embodiments of the invention allows to determine all aberrations independently, this means if accordingly (at q) this means_pAt 0) and set to 0, all relevant ray aberrations and wave aberrations will also disappear. The planar order is chosen here primarily for conventional reasons, and any other order can be deduced in a very similar way. The tangential plane is unique in the sense that there is no correlation to the y-component, compared to all other planes. This is also why the third order aberration does require a second plane to determine all aberrations (in particular σ -₃And σ₄) The reason for (1). In this order, any additional oblique ray planes (p ≧ 3) areHas its first and only black font matrix element and the aberration order i is 8 p-17.

1.2 aberration function of non-rotationally symmetric System

A brief discussion of aberration functions for systems with (1) no symmetry, (2) one plane of symmetry, (3) two planes of symmetry is discussed (for details see Richard Barakat, Agnes Houston, "The Aberrations of Non-aberration Symmettic Systems and the same Diffraction Effects ", optical Acta,13:1,1-30 (1966)). Without any symmetry, four independent variables (x) are required in the wavefront series expansion_p,y_p,t_x,t_y) All combinations of (for pupil and object coordinates).

In the case of one plane of symmetry, not all combinations of these four arguments occur in the wavefront aberration development. Assuming symmetry about the x-z plane, the wave aberration function W must be for y_p、t_yAnd y_pt_yHas invariance to the sign change of x_pAnd t_xWithout any further limitation.

Further constraints are imposed on the wave aberration development of these four independent variables if there are two orthogonal symmetry planes. Assuming symmetry about the x-z plane and the y-z plane, the wave aberration function W must be for x_p、t_xAnd x_pt_xAnd y is_p、t_yAnd y_pt_yHas invariance to the sign change of (a). Due to the invariance, it is not difficult to indicate that the wave aberration function W contains only even-order terms in the case of two planes of symmetry. This statement does not hold for the other two cases.

The skilled person knows from the literature how to deduce the corresponding ray aberration expansion from the known wave aberration expansion for each of these three cases. Similar to the rotationally symmetric case, all independent aberration coefficients (or combinations thereof) can be represented by the values at x_p＝y_p＝t_x＝t_yRelative partial derivative evaluated at 0(for ∈_ySimilarly) to be expressed. Since there are four arguments instead of two, visualization is not feasible, but in principle possible. For similar reasons it is in principle possible to use a sequence of pupil planes, but will not be used for any non-rotationally symmetric situation throughout the present invention.

2.1 description of rotationally symmetric coaxial systems

Such an optical system is described since the present invention relates to a computer-based method for designing a rotationally symmetric imaging system with an optical surface. This section encompasses lens-based systems, but mirror-based systems can be similarly processed and included in some embodiments.

A rotationally symmetric, coaxial, lens-based optical system is characterized by a sequence of N refractive optical elements aligned along a common optical axis z. All optical surfaces f_iIs rotationally symmetric and can thus be expressed as a single radial variableWherein f is satisfied_i+1(0)＞f_i(0). Surface function z ═ f_i(x, y) can be used in the most general polynomial form up to 2N_iCoefficient of order f_i,2jExpressed as an even power series:

this representation in the form of an even-order power r also allows the use of other rotationally symmetric surface functions, such as the common aspheric description (with additional spherical terms in many optical design programs), but other types of even-order functions may be used. In the case of a sphere (where there is no conic constant, which can be treated similarly), these coefficients are represented by the radius of curvature R appearing in each order_iThe description, namely:

if desired, by the reaction of (r-r)₀)^2jSubstitution of r in equation (2.1,2.2)^2jTo shift the progression of the series around the origin to different radial offsets r₀To (3).

Furthermore, it is important to introduce a ray mapping function (u) in the x and y directions_i(x,y,t),v_i(x, y, t)), the ray mapping functions are optimized in this wayThe general form describes that a ray from an object defined by the variable t and passing through the pupil coordinates (x, y) will be at point (u)_i,v_i) And f_iThe location of the intersection. As previously described, if different pupil plane cross-sections are used, the ray position at the pupil can be determined by a single pupil coordinate q_pDescribed (see equations (1.1) and (1.2)). Therefore, by the factor u_i,p,k,lAnd v_i,p,l,kMapping the ray to a function (u)_i,v_i) Is expressed as (q)_pT) is sufficient.

All spaces between adjacent surfaces (e.g., filled with air, glass, plastic …) are described by refractive indices, which may be a function of wavelength, temperature, and the like. If only a single wavelength λ is to be considered₀(for a fixed temperature, pressure …), then these indices of refraction become scalar values, which is always the case here unless otherwise stated. Fig. 2 illustrates the most general layout of such a lens-based optical system, wherein the surface f_i(x, y) may be defined as polynomial (aspherical) and/or spherical.

The optical system may or may not include an aperture stop a, which may be placed anywhere and defines a pupil plane within the optical system. For example, in fig. 2, the diaphragm is placed at the entrance of the optical system, in front of the first optical surface.

The ray path from object space to image space may be defined by (n +1) distances d₁..d_n+1And the on-axis distance between the two corresponding surfaces is described by d_i+1＝f_i+1(0)-f_i(0) It is given. In general, the distance between two respective intersecting surfaces along an arbitrary ray path is given by Pythagorean Theorem and can be written as:

wherein n is_i-1,iDenotes the scalar index of refraction between two corresponding surfaces, and (u)_i,p,v_i,p,f_i) And (u)_i-1,p,v_i-1,p,f_i-1Two corresponding intersection points are indicated. First and last distance d₁(spacer) and d_n+1(to images) can be similarly expressed, listed in table 1.

Table 1: distance from object plane to image plane in case of Infinite (IOD) or Finite (FID) object distance

In the first case (1a) of Infinite Object Distance (IOD), distance d₁Comprising a refractive index n_0,1(usually air) is supplied to the reaction chamber,is the normal vector of the wavefront, which is a function of the field angle t (defined herein asVector quantityIndicating the intersection of a ray and a first surfaceAnd isMay be an arbitrary point on the incident wavefront, such as the origin.

In the second case (1b) of Finite Object Distance (FOD), distance d₁Dependent on the refractive index n_0,1Is given as (t,0, z)_O) The position of the object point and the intersection point of the ray and the first surface are given by the pythagorean theorem. This is achieved byHere, t describes the position of the object point in the x-direction, not the angle with respect to the optical axis. The emergent wavefront w from the object is a spherical wavefront. The finite object point may be a planar object (where z is_OIs constant) or is formed by z_O＝z_O(t) a portion of the surface of the shape described.

In both cases 1a and 1b, the final distance d_n+1From the point of intersection (u) on the rearmost surface_n,p,v_n,p,f_n(u_n,p,v_n,p) With the real image point (determined from the coordinates (h) of the cross-section of the pupil plane under consideration)_x,p,h_y,pD) description), Pythagorean theorem between, multiplied by the refractive index_n,n+1To give. In both cases, the image may be in a plane with a constant D, or on a formed image of D ═ D (t). The real image is given by the ideal image (t),0, D) plus aberrations in the x and y directions:

the function T (T) provides an ideal relationship between the field angle T (case 1a) or object coordinates (case 1b) and the corresponding image point at the image plane or image surface. For example, in case 1a, the image function t (t) ═ f_LTan (t) corresponds to the undistorted imaging of the different fields with field angle t at their image points (t),0, D). In case 1b, the function T-M · T generally includes the magnification M (M is negative for a real and inverted image).

Similar to the ray mapping function described above, the ray aberration function ∈_xAnd_ycan be expressed as a variable (q) of any pupil plane cross-section_pT) series expansion:

∈ therein_x,p,k,lAnd ∈_y,p,k,lIs compatible with other knownRepresenting (such as wave aberration, Buchdahl or other) fully correlated aberration coefficients. For example, evaluate for x-t-0∈ is provided_x,1,1,2∈ is made_x,1,1,2With Buchdahl coefficient (3. sigma.)₃+σ₄) This means that several coefficients ∈ are present_x,p,k,lAnd ∈_y,p,k,lAnd is zero due to symmetry.

In some embodiments, aberration coefficients ∈_x,p,k,lAnd ∈_y,p,k,lMay be fully correlated with the "matrix aberration coefficient" (MAC). The matrix aberration coefficients correspond to coefficients obtained from the matrices shown in fig. 3 to 9. These may be expressed in terms of wave aberration, Buchdahl or light aberration coefficients, as explained above. For example, evaluate for x-t-0∈ is provided_x,1,1,2∈ is made_x,1,1,2Is related to the Buchdahl coefficient (3. sigma.)₃+σ₄) Proportional MAC. This means that some MACs are by definition zero.

The expression of table 1 together with equation (2.5) provides the distance d along an arbitrary ray path from (finite or infinite) object space to image space_iThe total sequence of (c). Another situation that can be described in a very similar way is an afocal system, where both the incoming and outgoing wavefronts at the object and image sides are described by plane wavefronts, and the relationship between the incoming and outgoing field angles can be for the distance d_n+1D from case 1a₁Redefined in a similar manner.

Thus, a computer-based method for designing a rotationally symmetric imaging system according to embodiments of the present invention includes the step of inputting to a computer system specifications including:

a parameter of the imaging system is determined,

o a wavelength dependence of the material,

the number of surface coefficients of the optical surface,

a pupil characteristic (diaphragm position),

an object space and an image space,

as described above.

The imaging system parameters may, for example, optionally be one or more or all of the following: the imaging system is designed for at least one wavelength or bandwidth, the number of optical surfaces, the type of optical surfaces, the material of the optical surfaces, the medium in which light propagates in the imaging system, the location of each or at least one optical surface, object space and image space, field of view, effective f-number, etc.

The fermat principle and the differential equation for arbitrary pupil plane cross sections are thus described:

as is known to the skilled person, the fermat principle states that the optical path length between two fixed points is an extreme along the ray. Consider a fixed but arbitrary point (finite or infinite object distance) on the wavefront w (t), and a fixed but arbitrary point (u) on the second surface_2,p,v_2,p,f₂(u_2,p,v_2,p)). Exits from the wavefront w and passes through (u)_2,p,v_2,p,f₂(u_2,p,v_2,p) ) must be such that the added optical path length d₁+d₂Is an extreme value. In the case where the point at the boundary remains fixed, d is to be achieved₁+d₂Is a point (u) on the first (intermediate) surface_1,p,v_1,p,f₁(u_1,p,v_1,p) U at (b) of_1,pOr v_1,p. The fermat principle thus implies:

wherein the partial derivative indicates (u)_2,p,v_2,p,f₂(u_2,p，v_2,p) Is stationary. Following similar arguments, for all defined distances d from object space to image space_iCan deduce twoAnd (4) forming a differential equation.

An optical system comprising n surfaces is formed by n differential equations D_ixAnd n differential equations D_iyDescribing (where i ═ 1.. n), these differential equations are derived directly from the fermat principle for a given but arbitrary pupil plane.

Thus, the computer-based method according to embodiments of the invention further comprises the steps of: using a computer, the system specifications provided in the previous step are converted into, for example, 2N per pupil plane by applying the Fermat principle_sA differential equation.

The following steps of the computer-based method according to some embodiments of the invention may optionally include the steps of: a well-defined selection-consistency-distribution (SCD) rule is used to input to the computer a subgroup of the light aberration coefficients in one or more pupil plane sections that are to be nominally set to vanish. These optional selection rules are further described and explicitly defined in the specification. The selected and nominally eliminated aberration coefficients define which orders the finite power series method is applied to and which pupil plane cross-sections to calculate all mapping functions and unknown surface coefficients.

The next step of the computer-based method according to some optional embodiments of the invention is to solve the differential equations to obtain the surface coefficients of the optical surface and the mapping function coefficients that define the ray trajectories and their intersection with the optical surface and thus each clear aperture.

Aperture stop/pupil coordinates:

there are four options, for example, where the diaphragm in fig. 1 can be placed. (A1) If an optical surface is assumed to act as a diaphragm of the system, its ray mapping function is derived from the equationPupil coordinates of (1.2) are substituted. For example, if the ith surface is a pupil stop, (u)_i,p,v_i,p,f_i(u_i,p,v_i,p) Are (q) in all equations (3.3) and (3.4)_p,x,q_p,y,f_i(q_p,x,q_p,y) ) replacement.

If a separate aperture stop A is added, its z-coordinate z_AThe position of the pupil plane is defined. There are three options for the positioning of such diaphragms: (A2) an entrance diaphragm, (A3) a diaphragm between the two optical surfaces, and (a4) an exit diaphragm in front of the image plane.

The treatment (A2-A4) was always the same. Coordinate (q)_p,x,q_p,y,z_A) The position of an arbitrary ray in the pupil plane at the stop is described. To ensure that each ray passes through the correct position (q)_p,x,q_p,y,z_A) An additional condition must be satisfied. The direction vector from the preceding element towards the diaphragm and the direction vector from the diaphragm to the following element must coincide. One way to ensure this is that the cross product of the two vectors is zero. For example, assume that the system layout is in z between two consecutive surfaces with indices j and k_AWith an aperture stop. Thus, the following equation must be satisfied:

[q_p,x-u_j,p,q_p,y-v_j,p,z_A-f_j(u_j,p,v_j,p)]×[u_k,p-q_p,xv_k,p-q_p,y,f_k(u_k,p,v_k,p)-z_A]＝0(3.5)

equation (3.5) is thus required to describe the system layout in which the aperture stop a is placed between two surfaces. The cross product of equation 3.5 may be, for example, zero for all three of its components simultaneously. By using the same arguments for the two direction vectors to and from the aperture stop, the remaining system layouts (a2) and (a4) also require three additional equations that can be derived using cross products similar to equation (3.5). Thus, all system layouts (IOD or FOD) with a single aperture stop (A2-A4) result in three additional equations, each added to the original equations (3.3) and (3.4).

Algebraic equation:

let us assume a function h_x,p、h_y,p、u_i,p、v_i,pAnd f_iIs a differential equation D in a given pupil plane cross section_ixAnd D_iy(where i 1.. n, or i 1.. n +3 in the case of an additional diaphragm) and a smooth solution, in a preferred embodiment according to the invention, taylor's theorem implies that these functions must be infinitely differentiable and have a power series representation as defined in equations (2.1), (2.2), (2.3), (2.4) and (2.6). In order to solve these differential equations, a known power series method may be employed. The method substitutes a power series representation into a differential equation to determine series coefficients. Function h_x,p、h_y,p、u_i,p、v_i,pAnd f_iCan be calculated by evaluating the following equation or a combination thereof for a given but arbitrary pupil plane cross section:

in some embodiments, these equations may optionally be evaluated in a particular order following a set of selection rules that will be described below. Equations (3.6) and (3.7) are algebraic equations and may be directly related to the ray aberration (or optionally to the aberration matrix in some embodiments):

if the black or gray font elements of the equivalent matrix are applied according to equation (3.6), the chosen derivative with indices k and l defines the leading (highest) order coefficient of all functions the leading aberration is ∈_x,p,k,l，u_i,pIs u_i,p,k,lAnd f is_i(u_i,p,v_i,p) Is f_i,(k+l+1). If the surface is defined as a diaphragm, f_iFirst item system ofNumber f_i,(k+1)。

Equation (3.7) corresponds to the y-component of the aberration for a given pupil plane cross-section if the black or gray font elements of the equivalent matrix are applied according to equation (3.7), the leader MAC is ∈_y,p,k,l，v_i,pIs v_i,p,k,lAnd f is_i(u_i,p,v_i,p) Is f_i,(k+l+1). If defined as a diaphragm, f_iIs f_i,(k+1)。

Coefficient of progression u_i,p,k,lAnd v_i,p,k,lSharing ∈_x,p,k,lAnd ∈_y,p,k,lMeaning the function ∈_x,pAnd ∈_y,pIs directly related to the ray mapping function u by the same index of (k, l) thereof_i,pAnd v_i,pAnd is related to the matrix elements. Thus, any index pair (k, l) that does not correspond to aberration dependence also means u due to symmetry_i,p,k,l＝v_i,p,k,l＝0。

In another optional alternative embodiment, the method further comprises the steps of: applying partial derivatives of order k, l, provided by using a predefined aberration matrix for each pupil plane cross-section, to the two sets of differential equations, wherein each predefined aberration matrix relates each light ray aberration of a subgroup to a partial derivative of order k, l of a light ray aberration series expansion in the pupil plane cross-section, thereby deriving a set of equations for optical surface coefficients and light ray mapping function coefficients.

2.2 solution to rotationally symmetric optical systems

Here, A solution is derived for all lens systems that may be described by A sequence of aspheric and/or spherical surfaces (e.g., S-A-S-. in order of indication). For all rotationally symmetric optical systems, the design process includes the following steps:

designating the imaging system as case 1a (IOD) or 1b (FOD)

Defining the number, type, order and materials of all optical surfaces

Define asphericCoefficient of surface f_i,2j(j ═ 1,2 …) number

T (t) according to equation (2.6), where T (t) must be classC^hWhere h is the highest order derivative to be applied. Defining the number of surfaces

Define all surface vertices f_i,0Initial position of (2), object plane z_o(or object surface z)_o(t)), the position of the image plane D (or the image surface D (t)), and the design wavelength λ₀

Select one of options (A1) - (A4) for the system stop

The number p ≧ 1 of selection matrix elements defining the pupil plane under consideration and the highest combined order o of the aberrations under consideration_mNot (k + l) (see section 1.1)

The number and type of surfaces, pupil planes, stop options (A1) - (A4), and order o_mThe total number of equations calculated according to equations (3.3) - (3.7) is then determined

In the case of an all-aspheric system, at least one ray aberration coefficient is chosen to be zero; for each selected light aberration coefficient set to zero, a surface coefficient is selected as unknown (and later calculated), and the pair of light aberration and surface coefficient must occur simultaneously in at least one derived algebraic equation

Any remaining surface coefficient(s) that have not been used to eliminate the aberration in the previous step must be input as values to an algebraic equation (manually or optimized).

One or more matrix elements of the system of algebraic equations are selected using a computer to obtain as output at least one surface coefficient of the optical surface, all mapping function coefficients, and the light aberration coefficients up to a given order. The entire system of (non-) linear equations can be solved. The entire set of combined equations for the subgroup can be solved simultaneously for all unknown coefficients by using standard methods, such as the Newton-Raphson algorithm or other algorithms.

In the case where at least one surface is aspheric, the system of algebraic equations may be divided into nonlinear and linear portions of unknown coefficients, which portions may then be solved sequentially, while the linear portions may be solved using standard methods such as gaussian elimination or other algorithms.

In the case of an all-aspheric system, only the first combined order results in the nonlinear system solved first. Any combined order considered to be higher than first order results in a linear equation that can be solved successively in ascending order using standard methods.

3.1 general description of rotationally symmetric optical zoom systems

As in section 2, such optical systems are characterized by a sequence of N optical elements aligned along a common optical axis. In the zoom system, the surface shape remains unchanged; however, the coefficient f_i,0Is not constant but can be varied and requires the designation of a coefficient f for at least two different designated zoom positions_i,0. Thus, these surfaces are described by the following equations:

wherein c indicates different zoom configurations with roman numerals I, II …, and wherein each surface f_i ^cThe position of (x, y) can be shifted to different lateral positionsThe definition of the light mapping function in equations (2.3) and (2.4) remains the same, except that an index c is added to indicate the zoom configuration, which means that in the x and y directionsUsing series coefficients in arbitrary pupil plane cross-sectionsAndis defined as:

similarly, the light aberration function ∈_x,pAnd ∈_y,pExpressed as a variable (q) for each zoom configuration and pupil plane cross-section_pT) is expanded in series as follows:

some embodiments of the present invention (described below) use "configuration matrix aberration coefficients" (CMAC).

All distancesThe calculation is done as previously in section 2 for the IOD or FOD case, but this time the calculation is repeated for each zoom configuration indicated by index c. In both cases, the image may be at D^cOn a constant (but eventually displaced for each zoom configuration) plane, or at D^c＝D^c(t) on the surface of the formed image. Again for each defined zoom configuration, the real image point is then represented by the ideal image point (T)^c(t),0,D^c) Plus the aberration power series terms in the x and y directions gives:

function T^c＝T^c(t) an ideal relationship between the field angle t (case 1a) or object coordinate t (case 1b) and the corresponding image point at the image plane or image surface is provided for each zoom configuration. For example, in case 1a, the image functionCorresponding to different zoom configurations and different focal lengthsCorresponding situation. In case 1b, the image function may describe the magnification M of different zoom configurations^c。

Following section 2, the fermat principle can be applied to each defined zoom configuration in sections to calculate a differential equation from object space to image space for each zoom configuration indicated by the added index c:

different pupil plane options (A1-A4) are handled as before. This means that if the ith surface is a pupil stop for a given zoom configuration, thenIn all differential equations (3.17) and (3.18) by (q)_p,x,q_p,y,_i ^c(q_p,x,q_p,y) ) replacement. If a separate aperture stop A is added, its z-coordinateThe pupil plane position for each zoom configuration is defined, which may also shift. The three separate pupil plane options (A2-A4) are processed as before by using cross products, resulting in each differential equation component x and y, and an additional equation for each zoom configuration.

Algebraic equation:

series expansion and function of aberrationAnd f_i ^cThe coefficients of (d) can be calculated by evaluating the following equation:

equations (3.19) and (3.20) may be directly related to the aberration derivative matrix as before, but now for different zoom configurations. Thus, for each zoom configuration, equation (3.19) corresponds to the x-component in a given pupil plane cross-sectionWhile equation (3.20) corresponds to the y-component in a given pupil plane cross-sectionThe respective relation between the applied derivative and the first coefficient of the different functions remains the same, with the difference being the added index c for the different zoom configurations. As before, any index pair (k, l) that does not correspond to aberration dependence means that due to symmetryAndwill also disappear.

3.2 solution for rotationally symmetric optical zoom systems

Here, the solution using embodiments of the method according to the invention is derived for all zoom lens systems that can be described by a sequence of spherical and/or aspherical optical surfaces in a given order. The design process comprises the following steps:

designating the zoom system as case 1a (IOD) or 1b (FOD)

Defining the number, type, order and materials of all optical surfaces

Coefficient f defining the aspherical surface_i,2j(j ═ 1,2 …) number

Define two or more zoom configurations described by the index c ═ I, II …

Defining the ideal pixel relationship T according to equation (3.16)^c(t) of (d). Function T^c(t) must be classC^hDifferentiable, where h is the highest derivative to be applied

All surfaces defining all zoom configurations (i.e. all surface vertices)) Initial position, object plane(or object surface)) Position(s), image plane D^c(or like surface D)^c(t)) position(s), and design wavelength λ₀

Select one of options (A1) - (A4) for the system stop

Defining the number p of pupil planes considered^c≧ 1, and the highest combined order of the considered aberrations for each zoom configuration

The number and type of surfaces, pupil plane, zoom configuration, stop options (A1) - (A4), and orderThe total number of equations calculated from equations (3.19) - (3.20) is then determined

Selecting at least one light aberration coefficient to be zero; for each selected light aberration coefficient set to zero, a surface coefficient is selected as unknown (and later calculated), and the pair of light aberration and surface coefficient must occur simultaneously in at least one derived algebraic equation

Any remaining surface coefficient(s) that have not been used to eliminate aberrations in the previous step must be input as values into an algebraic equation (manually or optimized)

The final step is to solve the system of algebraic equations using a computer to obtain as output at least one surface coefficient of the optical surface, all mapping function coefficients, and the light aberration coefficients up to a given order. The entire system of (non-) linear equations can be solved simultaneously for all unknown coefficients by using standard methods. In the case where at least one surface is aspheric, the system of algebraic equations may be divided into nonlinear and linear portions of unknown coefficients, which portions may then be solved continuously. In the case of an all-aspheric zoom system, only the first combined order results in an initially solved nonlinear system. Any combination order considered higher results in a linear equation that can be solved continuously in ascending order.

For the zoom configuration under consideration, the calculated optical zoom system is nominally free of any light ray aberrations (or combination thereof). If there is more than one solution (e.g., for different initial values), each solution corresponds to a different optical zoom system that nominally satisfies the same application conditions.

4.1 general description of non-rotationally symmetric systems

Embodiments of the present invention relate to a computer-based method for designing a non-rotationally symmetric imaging system having an optical surface, and thus such an optical system is described. This section encompasses lens-based or mirror-based systems; catadioptric systems can be treated similarly. Since the present invention is directed to the common nature of the methods of non-rotationally symmetric systems and rotationally symmetric systems, the main differences will be discussed herein only for three different symmetric scenarios: systems with (1) no symmetry, (2) one plane of symmetry, and (3) two planes of symmetry.

Non-rotationally symmetric (free form) optical surfaces

Free form optical surface f_iIs non-rotationally symmetric and is thus expressed as a function of two independent variables (e.g., x and y). Surface z ═ f_i(x, y) may be up to N_jOrder x and N_kCoefficient f of order y_i,j,kExpressed as a monomial power series, as follows:

unless otherwise stated, this monomial expansion is used for all free form surfaces throughout, but can also be converted to various other free form representations, such as Zernike, Chebyshev, Forbes, and so forth. These three symmetric cases also define which coefficients are present:

(1) and (3) asymmetry: in principle all series coefficients can be used

(2) One plane of symmetry: even order of k only (x-z plane)

(3) Two planes of symmetry: even orders of j, k only (x-z plane and y-z plane)

Ray mapping function (u)_i,v_i) Any ray path is described as before, but now depends on four variables, namely

The ray path from the object to the image is as before defined by the distance d₁..d_n+1The only difference is that the optical surfaces in the symmetrical cases (1) and (2) can be tilted, expressed. Introducing the inclination to each surface f_iOf (3) a rotation matrix R_iSo that the optical path length section is formed by introducing R_i·(u_i,v_i,f_i) To express. The rotation matrix for x-z plane symmetry only allows rotation around the y-axis, whereas in the case of no symmetry arbitrary rotations around all three axes are possible. In the case of two planes of symmetry, no rotation is possible and the system still has one common optical z-axis.

In the case of a finite object, the object consists of (t)_x,t_y,z_O) Given, and for the infinite case, the direction vector of the incident parallel ray is given as

Once the object-image relationship is specified, all optical path length segments can be expressed using the pythagorean theorem or the like as before. The real image is again the sum of the prescribed image plus the aberrations in the x and y directions.

Aberration of light function ∈_xAnd ∈_yThe x and y components of (c) now also depend on four variables (see section 1.2), namely:

∈ therein_x,j,k,l,mAnd ∈_y,j,k,l,mAre defined according to section 1.2, and may be fully correlated with alternative and known representations, such as wave aberration or other aberrations.

Using the same arguments as before, the fermat principle is applied to the two consecutive optical path length segments of each pair by mathematically representing that the optical path length between two fixed points is along the extreme of the ray, resulting in:

thus, the computer-based method according to embodiments of the invention further comprises the steps of: using a computer, the system specification provided in the previous step is converted to 2N by applying the Fermat principle_sA differential equation.

Aperture stop/pupil coordinates

The four options (a1) - (a4) are treated in the same manner as before.

(A1) If the ith surface is a pupil stop, (u)_i,v_i,f_i(u_i,v_i) Is expressed by (x) in all equations (4.6) and (4.7)_p,y_p,f_i(x_p,y_p) Instead, (a2) - (a4) -by using the same arguments as before for both direction vectors to and from the aperture stop; and depending on the symmetry considered, the diaphragm can also pass through the rotation matrix R_AAnd (4) inclining. All system layouts (IOD or FOD) with individual aperture stops (a2) - (a4) result in three additional equations, each added to the original equations (4.6) and (4.7).

Algebraic equation:

as before, the derived differential equation is solved using a known power series method function ∈_x、∈_y、u_i、v_iAnd f_iCan be calculated by evaluating the following equation for a given but arbitrary combination order:

equations (4.8) and (4.9) are algebraic equations that can be directly related to the ray aberrations x and y components as before.

Coefficient of progression u_i,j,k,l,mAnd v_i,j,k,l,mSharing ∈_x,j,k,l,mAnd ∈_y,j,k,l,mThe same symmetry of (a), but any index tuple (j, k, l, m) that does not correspond to the aberration correlation also means u due to symmetry_i,j,k,l,m＝v_i,j,k,l,m＝0。

4.2 solution for non-rotationally symmetric optical systems

Here, the solution for all non-rotationally symmetric optical systems is outlined. For the symmetrical case (1) no symmetry and (2) one plane of symmetry, at least one optical element is tilted with respect to the z-axis according to the symmetry considered. For the symmetrical case (3) there are two planes of symmetry, at least one optical surface having a free shape. The design process comprises the following steps:

designating the imaging system as case 1a (IOD) or 1b (FOD)

Defining the symmetry case (1), (2) or (3)

Defining the number, type, order and materials of all optical surfaces

Number of coefficients defining all optical surfaces

Defining a defined pixel relationship T_x(t_x,t_y) And T_y(t_x,t_y) Wherein T is_xAnd T_yMust be class C^hWhere h is the highest order derivative to be applied

Define the design wavelength λ₀Initial vertex of all surfaces and final rotation R_iObject plane z in the case of (FOD)_o(or object surface z)_o(t_x,t_y) Position and final rotation R of_objImage plane D (or image surface D (t))_x,t_y) Position and final rotation R of_IMAAnd all the introduced rotation matrices are matched to the defined symmetry case

Selecting one of options (A1) - (A4) for the system stop; in the case of (A2) - (A4), the rotation matrix R is defined according to the defined symmetry case_A

Define the highest combined order o of the aberrations considered_m(j + k + l + m) (see section 1.2)

The number and type of surfaces, pupil planes, stop options (A1) - (A4), and order o_mThe total number of equations calculated according to equations (4.6) - (4.9) is then determined

Selecting at least one independent ray aberration coefficient to be zero; for each selected light aberration coefficient set to zero, a surface coefficient is selected as unknown (and later calculated), and the pair of light aberration and surface coefficient must occur simultaneously in at least one derived algebraic equation

Any remaining surface coefficient(s) that have not been used to eliminate aberrations in the previous step must be input as values into an algebraic equation (manually or optimized)

The final step is to solve the system of algebraic equations using a computer to obtain as output at least one surface coefficient of the optical surface, all mapping function coefficients, and the light aberration coefficients up to a given combined order. The entire set of (non-) linear equations can be solved simultaneously for all unknown coefficients by using standard methods such as the Newton-Raphson algorithm or other algorithms.

In the case where at least one surface is aspherical or free-form, the system of algebraic equations may be divided into nonlinear and linear parts of unknown coefficients, which parts may then be solved continuously, while the linear parts may be solved using standard methods such as gaussian elimination or other algorithms.

In the case of an all-aspheric and/or all-free-form system, only the first combined order results in the nonlinear system being solved first. Any combination order considered to be higher than first order results in a linear equation that can be solved continuously in ascending order of combination order using standard methods such as gaussian elimination or other algorithms.

5 extended multicolor solution

So far, in the case of systems with at least one refractive optical surface, all derived methods according to the invention consider only a single wavelength λ₀While the refractive index is a scalar value depending on the choice of material. If the extended wavelength domain λ is considered ═ λ₁,λ₂]The refractive index of the lens material (e.g. glass or plastic) then becomes a function of the wavelength, i.e. n_i-1,i＝n_i-1,i(λ), or a refractive index profile of the material. There are three possible ways how the polychromatic behavior of the previously defined monochromatic optical design method according to any embodiment of the present invention can be considered and controlled during the design process for a given wavelength domain.

5.1 applying existing monochromatic solution schemes using multiple lens materials

A first possible solution is to use at least two different lens materials (such as, for example, flint glass and crown glass). The monochromatic design methods in the previous sections can be used directly and a single reference wavelength λ within the wavelength domain under consideration₀And remains unchanged. If chosen properly, the solution takes advantage of the well-known equalization effect of different dispersion relationships for different materials. The system design is then obtained via one of the previously defined single color design methods by using polychromatic evaluation (e.g., ray tracing), where the initial values of the single color design methods are selected to ensure good polychromatic performance for the wavelength domain under consideration. The initial values of the selection of materials and/or the design method may be automated, for example, by using multi-color based optimization for these initial degrees of freedom.

5.2 Combined Monochromatic solution for multiple wavelengths

A second possible solution is to use one of the monochromatic design methods of the invention and simultaneously address at least two (or more) different wavelengths λ within the wavelength domain under consideration₀₁,λ₀₂… solve the system of equations. For all static systems (no zoom), the solution is very similar to the zoom system described in section 3, and comprises:

here, the configuration with the roman numeral index c corresponds to a plurality of wavelength configurations instead of a plurality of zoom configurations. For each different wavelength λ₀₁,λ₀₂…, distanceOnly the scalar values that vary for each wavelength and different material (at least two materials) are included. In contrast to zoom systems, the original position of all surfacesRemains constant because no lens element is moving. Accordingly, equations (3.12) - (3.20) are defined in the same manner, but now for different wavelength configurations. The only other difference is that the first and second,function T providing an ideal relationship between field angle T (case 1a) or object coordinate T (case 1b) and the corresponding image point at the image plane or image surface^c＝T^cThe (t) ═ t (t) is typically the same for all wavelength configurations, since the optical system should operate in the same way for all wavelengths considered. The general solution for two (or more) configurations and for a system that can be described by a sequence of spherical and/or aspherical optical surfaces then works exactly as described in section 3.2, but for the wavelength configuration.

The calculated optical system is nominally free of any ray aberrations (or combination thereof) for each wavelength considered. If there is indeed more than one solution (for different initial values), each solution corresponds to a different optical system but nominally satisfies the same conditions of application.

In the case of achromatic zoom systems for multiple different wavelengths, the solution is to extend the monochromatic zoom system design approach in section 3, for example, by using two Roma indices, but following the same arguments and solution as previously described. Similar arguments apply equally to any rotationally symmetric and non-rotationally symmetric system.

5.3 solution scheme including chromatic aberration

A third possible solution is to manipulate the wavelength dependence of a system having at least one refractive surface in the form of a well-known chromatic aberration order expansion in any method according to embodiments of the invention. As before, the wave aberration of a particular field angle or point represents the deviation of its wavefront from becoming spherical at the exit pupil, this time also depending on the wavelength. Without loss of generality, the object is oriented along the x-axis. The wave aberration function can then be represented by the aberration coefficient a_(2l+m)nmoIs written as:

and case o-0 designates the monochrome case discussed previously. If all the wave aberrations are zero, the wavefront is spherical and all the rays converge to their curvatureCentered and a perfect geometric point image is obtained for all wavelengths. Similarly, all ray mapping functions and aberration orders in the x and y directions are now expressed as (q) for an arbitrary pupil plane cross-section_pT,) i.e.:

the mathematical description of the surface remains as previously defined. All distances d_iNow using the refractive index n_i-1,i＝n_i-1,i(λ) is expressed as a function of wavelength accordingly (e.g., using the Sellmeier equation or other equation). The ideal image point (T),0, D) is defined as in section 2, since it should generally not depend on the wavelength, but can also be a function T (T, λ) if a certain dispersion on the image side is desired. The fermat principle can be applied as in section 2, resulting in two sets of differential equations, such as equations (2.3) and (2.4), but here expressions (8.1) - (8.4) are used. Again, individual aperture stops/pupil planes may be processed exactly as described in section 3 by using cross products.

Algebraic equation:

following similar arguments as in section 2, function u_i,p、v_i,pAnd f_i,jIs calculated in any pupil plane cross section by evaluating the following equation:

usually in the wavelength domain λ ═ λ₁,λ₂]Is chosen somewhere within the initial reference wavelength lambda₀。

Achromatic aberration order:

original aberration sequence ∈_x,1,∈_x,y,2,∈_x,3,∈_y,3…∈_x,p,∈_y,p(ordered as specified in section 1) the reference wavelength λ is described as before₀Monochromatic aberration of the lower. Additional aberration by each considered chromatic aberration order mTo introduce wavelength dependence. First order λ is formed by multiplying all elements by a termThe same aberration sequence. Similarly, a second order λ corresponds to all elements multiplied by a termThe aberration sequence of (a). In general, for an order of m λ, all elements of the original aberration sequence are multiplied by a term

For almost all imaging systems, it is usually sufficient to correct a few aberration coefficients that have a linear dependence on wavelength (m 1). For this case, a general solution is given in the following two subsections. For all higher orders (m ≧ 2), solutions can be derived in a very similar manner and for all design approaches disclosed in the present invention, including zoom and non-rotationally symmetric systems.

5.3.1 general solution for Achromatic rotationally symmetric optical systems

The design process includes all the steps defined in section 2, with only a small number of adjustments: for whatChoosing a given number of degrees of freedom for aspheric and/or spherical surfaces, the aberration coefficients that would be zero are now according to section 5.3(0 and 1 for m). The calculations from sections 5.1-5.3 are extended to include chromatic aberration (for m-1) and are adapted by using equations (8.1) - (8.6).

The calculated optical system is nominally free of any monochromatic and chromatic light aberrations (or combinations thereof) that have been selected. If more than one solution exists, each solution corresponds to a different optical system that nominally satisfies the same application conditions.

5.3.2 general solution for achromatic rotationally symmetric optical zoom systems

The design process includes the description in section 3.1 and all the defined steps in section 3.2, with only minor adjustments: for a given number of degrees of freedom of the selected aspheric and/or spherical surface, the matrix elements are now according to sections 3.1 and 5.3Defined and selected (for m-0 and 1) for at least two different zoom configurations. Equations (8.1) - (8.4) are extended by the upper index c for the (at least two) different zoom configurations, and equations (8.5) - (8.6) can then be used to calculate all power series coefficients for each zoom configuration. The calculations in sections 7.1-7.3 are extended accordingly to include one or more color difference matrices (for m ═ 1) and adapted by using equations (8.1) - (8.6).

The calculated optical zoom system is nominally free of any monochromatic and chromatic light aberrations (or combinations thereof) that have been selected for each of the zoom configurations under consideration. If more than one solution exists, each solution corresponds to a different optical system that nominally satisfies the same application conditions.

6 additional System features

The following features and possible extensions apply to all previous embodiments of the invention.

6.1 afocal System

Afocal system(without focus) is a system that does not produce any net convergence or divergence of the field for an object at infinity, and thus forms an image at infinity. Its properties are angular magnification and transverse magnification M_αAnd M_LAngle magnification tan (α)_in)＝M_αtan(α_out) Relating the incident and emergent angles of view (object-image relationship), and the lateral magnification M_L＝h_in/h_outThe object and image ray heights are related relative to the optical axis. The afocal system imposes (1) subtle variations and (2) further conditions on the foregoing process:

(1) in deriving the differential equation, the final distance d is for infinite object distance_n+1And d with₁Expressed in the same way. With d_n+1The normal vector of the represented exit wavefront passes through an angular magnification and a sum of the angle t α_inThe defined incident wavefront is correlated.

(2) Achieving a defined lateral magnification requires an additional condition that maps the mapping function u on the first surface_1,pAnd v_1,pWith the mapping function u on the final surface_n,pAnd v_n,pPerforming a correlation, i.e. (a) u_1,p＝M_Lu_n,pAnd (b) v_1,p＝M_Lv_n,p. These two equations are added to the original differential equation and then solved as described in the previous section.

6.2 telecentric System

The imaging system is telecentric in object space and/or image space if the chief rays of all fields are parallel to the optical axis in front of and/or behind the optical system. Thus, the object and/or image must be located at a limited distance. A telecentric system imposes an additional condition per telecentric space that is added to the original system of differential equations for the x component:

(1) in the object space: the mapping function on the first surface must satisfy u_1,1＝t

(2) In image space: the mapping function on the final surface must satisfy u_n,1＝T(t)

This is because only in a single cross-section (e.g. tangential plane) of the pupil plane cross-section is requiredBoth conditions are satisfied simultaneously, since all chief rays pass through the center of the pupil plane. The equation(s) for the telecentric condition are added to the original system of differential equations. They depend only on the object variable t and not on the pupil variable p_qAnd thus is closely related to and processed together with the distortion. For each nominally eliminated distortion aberration in ascending order, one or both conditions may be satisfied by selecting one or two corresponding surface coefficients for each order considered. The additional equation(s) are added to the original system of differential equations and then solved as described in the previous sections.

6.3 mirror-based or catadioptric imaging systems

So far, the condition f has been required for all refractive optical systems_i+1(0)＞f_i(0). This condition is no longer necessary if at least one reflection occurs. In the case of one (or more) reflective surfaces, f_i(0) The values define the sequence of any lens-mirror based optical system. In addition to the changes in the sequence of surface vertices, the system of differential equations is solved as described in the previous sections. The same process can be used for surfaces that undergo Total Internal Reflection (TIR).

6.4 surface coefficients of odd order

Typically, a rotationally symmetric surface is described by only the even-order surface coefficient r. Introducing surface coefficients of odd order (which in the most general case are) In the case of the above-described embodiments), the methods described in the previous sections may be adapted accordingly.

6.5 surface vertices as design parameters

In some cases, at least one surface vertex f is utilized_i,0It may prove beneficial to nominally eliminate at least one aberration term. This applies correspondingly to all methods described in the previous sections.

6.6 calculating the aberrations of the known System

In general, all derived solutions allow to calculate the ray aberration analytically, if all surface coefficients are given. Conversely, any method that allows for the calculation of aberrations may be used to set certain aberrations to zero and calculate the unknown surface coefficients accordingly, as shown in the present invention.

Assuming all surface coefficients f of a system consisting of N refractive surfaces_i，jWell defined and given, the proposed power series method is able to calculate all (monochromatic) chromatic aberration coefficients in ascending order of the combined aberration order. This applies to all possible systems including, but not limited to, static or zoom (non-) rotationally symmetric systems.

Assuming all coefficients of a zoom system consisting of N refractive surfacesBeing well defined and given, the proposed power series method is able to calculate all aberration coefficients for all zoom configurations, for arbitrary pupil plane cross sections and in ascending order of aberration orderAndthe same arguments apply to both static and zoom systems where the chromatic aberration defect is evaluated according to section 5.2 (monochromatic aberration at different wavelengths) and section 5.3 (chromatic aberration).

The solution for calculating the light aberration coefficients of known optical systems always follows that of an all-aspheric system. Since all surface coefficients are known (spherical and/or aspherical), all necessary matrix elements are applied (see fig. 9) step by step and for all required planes. The resulting system of equations for each combined aberration order is always linear in the mapping and aberration coefficients, and can be solved using, for example, gaussian elimination. For each combined aberration order, final zoom configuration and wavelength(s), the corresponding derivative for each combined order is applied to D accordingly_i,xAnd D_i,yAnd solving the resulting system of linear equations. Repeating the calculation with increasing combined aberration orders until a maximum is reachedOrder o_m(typically no greater than 20). As a result, all ray mapping functions and ray aberration coefficients up to that order are calculated for any given system according to any embodiment of the invention. In contrast, prior art direct computation methods are only capable of computing a limited number of low order aberration coefficients (typically up to 3 orders). For higher orders, ray tracing is required to calculate the value of the aberration coefficient. The fully known aberrations up to a high order can be used to estimate the imaging quality of the optical system or as a figure of merit for optimization. Furthermore, the derivative of the aberration coefficients with respect to the surface coefficients can be evaluated, which provides a direct measure of the influence of said coefficients on the respective aberration.

7. Alternative embodiments

In an alternative embodiment, the method of the present invention further comprises the steps of: applying partial derivatives of order k, l to two sets of N_sA differential equation, said order k, l being provided by using for each pupil plane cross section a predefined aberration matrix, wherein each predefined aberration matrix relates each light ray aberration of a subgroup to a partial derivative of the order k, l of the progression of light ray aberrations in the pupil plane cross section, thereby deriving an equation set of optical surface coefficients and light ray mapping function coefficients.

7.1 matrix selection rule for arbitrary pupil plane cross sections

According to equations (2.1) and (2.2), there are, for example, two general types of surfaces. A polynomial (aspherical) surface described by one series coefficient per even-order r, and a spherical surface described by a single variable (radius of curvature) occurring in each even-order r. Any rotationally symmetric system can be described by_AS_SWherein the indices a and S represent the number of each surface type, respectively.

In one embodiment, the following matrix element selection rules apply to all matrices ∈_x,pAnd ∈_y,p：

Has index i and highest order 2N_iEach aspheric surface of (a) allows each available coefficient f_i,2jAt most one black font matrix element is selected,where even number 2j must equal the matrix elementPlus one, i.e. 2j ═ a + b + 1.

In one aspheric surface (with index i)₀) In the case of use as a diaphragm (A1), this allows each available coefficientAt most one black font matrix element is selected, where 2j ═ a +1 is satisfied.

From the radius of curvature R_iEach sphere described allows at most one individual black font matrix element to be selected

In a sphere (with index i)₀) In the case of use as a diaphragm (a1), this allows at most one black font matrix element to be selected, where a ≠ 0 is satisfied.

In this embodiment, the following matrix element consistency rule applies to all pupil plane cross-sections (and thus to all clustered matrices [ ∈ ]_x,1],[∈_x,y,2],[∈_x,3,∈_y,3]…[∈_x,p,∈_y,p])：

For each selected black font matrix element in the pupil plane cross sectionEach adjacent black font matrix element with indices (a-2, b) and (a, b-2) must also be selected (if indeed present).

In this further embodiment of the invention, the following matrix element distribution rule applies for a given system A_pS_qAnd applies to matrix ∈_x,1,∈_x,y,2,∈_x,3,∈_y,3…∈_x,p,∈_y,p(ordered as specified in the specification):

according to selection and consistency rules, topA large number of matrix elements may be sufficient to derive only the tangential plane matrix ∈ from_x,1To select an element.

The maximum number of matrix elements may be distributed among the sorted matrices according to selection and consistency rules, then, the matrix ∈_x/y,pThe highest combination order (a + b) of the selected matrix elements of each row and each column of (where p ≧ 2) cannot exceed the highest combination order of each row and each column, respectively, of the preceding matrix.

In a preferred embodiment according to the invention, these selection rules, consistency rules and distribution rules are implemented in a computer-based method.

7.2 solution for rotationally symmetric optical systems

Here, for sequences that may consist of aspheric and/or spherical optical surfaces (by A)_AS_SAnd for example S-A-S-. to describe all lens systems described in order of indication) to derive A further solution. For all rotationally symmetric optical systems, the design process includes the following steps:

designate the imaging system as having Infinite (IOD) or Finite (FOD) object distance case 1a or 1 b.

The ideal pixel relationship t (t) is defined according to equation (2.6). The function T (t) must be class C^hWhere h is the highest order derivative to be applied.

Define surface A_AS_SThe number of them, their order and the material from object space to image space.

Define surface vertices f_i,0Initial position of (2), object plane z_O(or object surface z)_o(t)), the position of the image plane D (or the image surface D (t)), and the design wavelength λ₀。

One of options (a1) - (a4) of selecting an aperture stop or surface as the system stop.

Specifying the available surface coefficients f for each aspheric surface_i,j(j ═ 2,4, …).

The matrix elements are selected according to selection, consistency and distribution rules.

The number and type of surfaces, the selected diaphragm options (A1) - (A4), and the selection of matrix elements then determine the total number of corresponding equations that can be calculated from equations (3.3) - (3.7).

7.3 for all-aspheric systems (A)_AS₀) General solution scheme of

Further methods according to embodiments of the present invention provide a general solution for all-aspheric systems.

In the case of an all-aspheric system, according to section 4, the calculation scheme follows the order of the diagonal of the elements selected from the derivative matrix or matrices_dI.e. 1 st order, 3 rd order … o_mOrder up to the highest combined order o_m(a + b). By repeating the following calculation routine for o₁、o₃… until o is reached_mThe general solution is calculated in ascending order (d ═ 1,3 …):

(1)o_dthe selected matrix element or elements of the order diagonal are applied to the CEQ by equations (3.6) and (3.7) and the surface coefficient f is obtained for_i,d+1、MAC∈_x,p,k,lAnd ∈_y,p,k,lAnd a mapping coefficient u_i,p,k,lAnd v_i,p,k,lOr a non-linear (d ═ 1) or linear (d > 1) system of equations.

(2) All occurrences of MAC ∈_x,pAnd ∈_y,pAre set to zero. Usable surface coefficient f_i,d+1Minus the number of the selected one or more matrix elements yields the coefficient f that needs to be predefined as an initial value_i,d+1The number of (2). Residual unknown coefficient f_i,d+1、u_i,p,k,lAnd v_i,p,k,lIs then equal to the number of equations and may be for an unknown coefficient f_i,d+1、u_i,p,k,lAnd v_i,p,k,lTo solve a system of (non-) linear algebraic equations.

(3) If there is a selected matrix element(where d ═ a + b) corresponds to gray font matrix elements having the same index in any other matrices and planes under consideration, then these gray font matrix elementsThe element is applied to the CEQ by equations (3.6) and/or (3.7) and yields a mapping coefficient u for the solution that can be solved_i,p,k,lAnd/or v_i,p,k,lA (non-) linear system of equations (iv).

Once the highest order o is solved_mThe calculation routine is stopped. Since the selected matrix element or elements directly correspond to MACs of the same order that have all been set to zero, the resulting optical system is nominally free of any ray aberrations (or combinations thereof) associated with these MACs. If there is more than one solution (e.g., for different initial values), each solution corresponds to a different optical system that nominally satisfies the same application conditions.

7.4 for all spherical systems (A)₀S_S) General solution scheme of

In an embodiment according to the invention, a general solution for an all-spherical system is thus described.

In the case of the global plane system, the calculation scheme is as follows:

(1) all selected matrix element or elements are applied to CEQ by equations (3.6) and (3.7) and the resulting surface coefficient R is obtained_i、∈_x,pAnd ∈_y,pMAC and mapping coefficient u_i,p,k,lAnd v_i,p,k,lAll occurrences of MAC ∈_x,p,k,lAnd ∈_y,p,k,lAre set to zero.

(2) Usable radius R_iMinus the number of the selected one or more matrix elements yields the coefficient R that needs to be predefined as an initial value_iThe number of (2). Residual unknown coefficient R_i、u_i,p,k,lAnd v_i,p,k,lThe number of (c) is then equal to the number of equations (in (1)).

(3) If there is a selected matrix element(where d ═ a + b) corresponds to gray font matrix elements having the same index in any other matrices and planes considered, then these gray font matrix elements should be treated by equations (3.6) and/or (3.7)For CEQ and obtaining the corresponding R_iAnd a mapping coefficient u_i,p,k,lAnd/or v_i,p,k,lThe second set of non-linear equations added to the first set of equations.

(4) All unknown coefficients R can be addressed, typically by using a numerical solver_i、u_i,p,k,lAnd v_i,p,k,lTo solve the entire combined system of nonlinear equations simultaneously.

(5) The calculated optical system is nominally free of any ray aberrations (or combination thereof) associated with a vanishing MAC by the selected and applied matrix element or elements. If more than one solution exists, each solution corresponds to a different optical system that nominally satisfies the same conditions for application.

7.5 for systems with combined aspherical and spherical surfaces (A)_AS_S) General solution scheme of

In another embodiment according to the present invention, the present invention provides a general solution for a system having a combined aspheric and spherical surface.

In the case of a "hybrid" surface system, the calculation scheme is a modified combination of the two calculation schemes in sections 7.4 and 7.5 and works as follows:

(1) identifying a highest combination order o in which at least one matrix element has been selected for a sphere_S. Up to the combination order o_SAll of the selected one or more matrix elements then form a subgroup of the selected one or more matrix elements.

(2) All elements of this subgroup are applied to CEQ by equations (3.6) and (3.7), and the resulting surface coefficient R is obtained_i、f_i,j、∈_x,pAnd ∈_y,pMAC and mapping coefficient u_i,p,k,lAnd v_i,p,k,lAll MAC ∈_x,p,k,lAnd ∈_y,p,k,lAre set to zero.

(3) Available variable R in this subgroup_iPlus f_i,jIs given by the number of one or more matrix elements selected in the subgroup needs to be predefined asCoefficient R of initial value_iAnd/or f_i,jThe number of (2). Residual unknown coefficient R_i、f_i,j、u_i,p,k,lAnd v_i,p,k,lThe number of (2) is then equal to the number of equations (for the subgroup in (2)).

(4) If there is a selected matrix element in a subgroup(where d ═ a + b) corresponds to the gray font matrix elements having the same index in any other matrices and planes considered, then these gray font matrix elements are applied to CEQ by equations (3.6) and/or (3.7) and the result is obtained for R_i、f_i,j、u_i,p,k,lAnd v_i,p,k,lThe second set of non-linear equations added to the first set of equations.

(5) All unknown coefficients R can be addressed, typically by using a numerical solver_i、f_i,j、u_i,p,k,lAnd v_i,p,k,lTo solve the entire combined set of equations for the subgroups simultaneously.

(6) Once the subgroup containing all the one or more matrix elements related to the sphere is solved, the exact solution defined in section 7.4 is followed, from d ═ o_s+2 start until the maximum combination order o is reached_mThe final remaining unknown aspheric coefficients are calculated (from the aspheric matrix elements) in ascending order.

As before, the calculated optical system is nominally free of any ray aberrations (or combinations thereof) associated with the MAC by the selected and applied matrix element or elements. If more than one solution exists, each solution corresponds to a different optical system that nominally satisfies the same application conditions.

In another embodiment according to the present invention, the method may further be applied to the calculation of a rotationally symmetric optical zoom system, as described herein.

7.6 general description of rotationally symmetric optical zoom systems

As in section 2, such optical systems are characterized by alignment along a common optical axis zOf N refractive optical elements. In the zoom system, the surface shapes in equations (2.1) and (2.2) remain unchanged; however, the coefficient f_i,0Is not constant but may vary, and the coefficient f needs to be specified for at least two different specified zoom positions_i,0. In the case of a zoom system, the surface is described by the following equation:

wherein c indicates different zoom configurations with roman numerals I, II …, and wherein the position f of each surface_i ^c(x, y) can be shifted to different lateral positionsThe definition of the light mapping function in equations (2.3) and (2.4) remains the same, except that an index c is added to indicate different zoom configurations, which means that in the x and y directionsUsing series coefficients in arbitrary pupil plane cross-sectionsAndis defined as:

similarly, the light aberration function ∈_x,pAnd ∈_y,pUsing what is now called "configuration matrix aberration coefficients" (CMAC)Andexpressed as a variable (q) for each zoom configuration and pupil plane cross-section_pT) is expanded in series as follows:

all distancesThe calculations are performed as previously in section 2 for the IOD or FOD case, but this time the calculations are repeated for each zoom configuration indicated by index c. In both cases, the image may be at D^cOn a constant (but eventually displaced for each zoom configuration) plane, or at D^c＝D^c(t) on the surface of the formed image. Again for each defined zoom configuration, the real image point is then represented by the ideal image point (T)^c(t),0,D^c) Plus the aberration power series terms in the x and y directions gives:

function T^c＝T^c(t) an ideal relationship between the field angle t (case 1a) or object coordinate t (case 1b) and the corresponding image point at the image plane or image surface is provided for each zoom configuration. For example, in case 1a, the image functionCorresponding to different fields with field angles T at their image points (T)^c(t),0,D^c) In which different zoom configurations correspond to different effective focal lengthsIn case 1b, the image function T^c(t)＝M^cT for example describes different magnifications M for different zoom configurations^c。

Following section 2, the fermat principle can be applied to each defined zoom configuration in sections to compute two sets of differential equations from object space to image space for each zoom configuration indicated by the added index c:

the different pupil plane options (A1-A4) are handled as before. This means that if the ith surface is a pupil stop for a given zoom configuration, thenAre given in all differential equations (6.7) and (6.8)And (4) replacing. If a separate aperture stop A is added, its z-coordinateThe position of the pupil plane for each zoom configuration is defined, which may also be shifted. The three separate pupil plane options (A2-A4) are processed as before by using cross products, resulting in each differential equation component x and y and an additional equation for each zoom configuration.

The conditional equation is as follows:

function(s)And f_i ^cThe taylor series coefficients of (a) may be calculated by evaluating the following equations in a particular order for all defined zoom configurations following a selection rule adaptation set:

equations (6.9) and (6.10) are now referred to as "configuration condition equations" (CCEQ) and can be directly related to the aberration derivative matrix as before, but now for different zoom configurations. This means that for each zoom configuration, equation (6.9) corresponds to the x-component (matrix) in a given pupil plane cross-section) While equation (6.10) corresponds to the y-component (matrix) in a given pupil plane cross-section). Each zoom configuration has a matrix sequence as defined in section 1. The respective relationships between the applied matrix elements and the leading coefficients of the different functions remain the same, except for the index c added for the different zoom configurations. As before, all series coefficients not related by these matricesAndand is zero due to symmetry.

7.7 matrix selection rule:

all the selection, consistency and distribution (SCD) rules previously defined in section 2.2 remain valid for each individual zoom configuration. Now, given the zoom system Z: A_AS_SCan be distributed among different zoom configurations and among different pupil plane cross-section matrices. For each specified zoom configuration (at least two), at least one may be selected according to the SCD ruleThe matrix elements, while the maximum total number of matrix elements selected across all zoom configurations is limited by the total degrees of freedom of the aspheric and spherical surfaces used.

8. Solution for rotationally symmetric optical zoom systems

Here, for sequences that may consist of spherical and/or aspherical optical surfaces (from Z: A)_AS_SAnd for example S-A-S- …) to derive A solution using embodiments of the method according to the invention. For all rotationally symmetric optical systems, the design process includes the following steps:

designate the imaging system as having Infinite (IOD) or Finite (FOD) object distance case 1a or 1 b.

Define two or more zoom configurations described by the index c ═ I, II ….

Defining the ideal pixel relationship T according to equation (6.6)^c(t) of (d). Function T^c(t) must be classC^hWhere h is the highest order derivative to be applied. For IOD, this may be, for example, different focal lengths for all defined zoom configurationsFor FOD, this may be, for example, a different magnification M for all defined zoom configurations^c。

Defining the surface Z A_AS_sThe number of them, their order and the material from object space to image space.

All surfaces defining all zoom configurations (i.e. all optical surface vertices)) Initial position, object plane(or object surface)) Position(s), image plane D^c(orImage surface D^c(t)) position(s), and design wavelength λ₀。

One of options (a1) - (a4) of aperture stop or surface is selected as the system pupil.

Specifying the available surface coefficients f for each aspheric surface_i,j(j ═ 2,4, …).

Select matrix elements according to the SCD rule defined in sections 2.2 and 3.1 for each defined zoom configuration, with at least one selected matrix element per zoom configuration.

The number and type of surfaces, the selected diaphragm options (a1) - (a4), the number of zoom configurations, and the selection of matrix elements then determine the total number of corresponding equations that can be calculated from equations (6.7) - (6.10).

8.1 for all-aspheric zoom systems (Z: A)_AS₀) General solution scheme of

In the case of an all-aspheric zoom system, the calculation scheme using embodiments of the method according to the invention is closely related to section 7.4 and follows the diagonal order o of the elements selected from the derivative matrix for all different zoom configurations_d. Then by repeating the following calculation routine forUntil it reachesThe general solutions are calculated (for at least one zoom configuration) for all zoom configurations for which matrix elements have been selected in ascending order (d ═ 1,3 …):

(1)the selected one or more matrix elements of the order diagonal are applied to CCEQ by equations (6.9) and (6.10) and result for the surface coefficient f_i,d+1、AndCMAC of (1), and mapping coefficientsAndor a non-linear (d ═ 1) or linear (d > 1) system of equations.

(2) All occurring CMACsAndare set to zero. Usable surface coefficient f_i,d+1The number of the selected one or more matrix elements minus the number of the selected one or more matrix elements gives the coefficient f that needs to be predefined as an initial value_i,d+1The number of (2). Residual unknown coefficient f_i,d+1、Andis equal to the number of equations and may be for an unknown coefficient f_i,d+1、Andto solve a system of (non-) linear equations.

(3) If there is a selected matrix element(where d ═ a + b) corresponds to the gray font matrix elements with the same index in any other considered matrices for the same zoom configuration, then these gray font matrix elements are applied to CCEQ by equations (6.9) and/or (6.10) and the result is availableWith mapping coefficients solved for each zoom configurationAnd/orA (non-) linear system of equations (iv).

Once the highest order number is reachedThe calculation routine is stopped. Since the selected matrix elements directly correspond to CMACs of the same order that have been set to zero, the calculated optical zoom system is nominally free of any ray aberrations (or combinations thereof) associated with these CMACs of the zoom configuration under consideration. If there is more than one solution (e.g., for different initial values), each solution corresponds to a different optical zoom system that nominally satisfies the same application conditions.

8.2 for a full sphere zoom system (Z: A)₀S_S) General solution scheme of

In the case of the global plane system, the calculation scheme using embodiments of the method according to the invention is closely related to section 7.5 and is as follows:

(1) all selected matrix element or elements are applied to CCEQ by equations (6.9) and (6.10) and the resulting surface coefficient R is obtained_i、AndCMAC of (1), and mapping coefficientsAndthe first set of equations. All CMACsAndare set to zero.

(2) Usable radius R_iIs subtracted by the number of the selected matrix element or elements gives the coefficient R which needs to be predefined as an initial value_iThe number of (2). Residual unknown coefficient R_i、Andthe number of (c) is then equal to the number of equations (in (1)).

(3) For each zoom configuration, if there is a selected matrix element present(where d ═ a + b) corresponds to the gray font matrix elements in any other considered matrix (different pupil plane cross-sections) with the same index, then these gray font matrix elements are applied to CCEQ by equations (6.9) and/or (6.10) and the result is for R_iAnd mapping coefficientsAnd/orThe second set of non-linear equations added to the first set of equations.

(4) Can be applied to all unknown coefficients R_i、Andto solve the entire combined system of equations simultaneously.

The calculated optical zoom system is nominally free of any light ray aberrations (or combinations thereof) associated with the missing CMAC of each of the considered zoom configurations by the selected and applied matrix elements. If more than one solution exists, each solution corresponds to a different optical system that nominally satisfies the same conditions for application.

8.3 for zoom systems with combined aspherical and spherical surfaces (Z: A)_AS_S) General solution scheme of

In the case of a "hybrid" surface zoom system, the calculation scheme using embodiments of the method according to the invention is closely related to section 7.6, which works as a modified combination of the calculation schemes in sections 8.1 and 8.2 as follows:

(1) identifying a highest combination order in which a single matrix element has been selected for a sphere of at least one zoom configurationFor all zoom configurations up to the combined orderThen form a subgroup of the selected elements.

(2) All elements of this subgroup are applied to CCEQ by equations (6.9) and (6.10) and result for the surface coefficient R_i、f_i,j、AndCMAC of (1), and mapping coefficientsAndthe first set of non-linear equations. CMACAndis set to zero.

(3) Variable R available in this subgroup_iPlus f_i,jThe number of minus the number of selected one or more matrix elements in the subgroup gives the coefficient R which needs to be predefined as an initial value_iAnd/or f_i,jThe number of (2). Residual unknown coefficient R_i、f_i,j、Andthe number of (2) is then equal to the number of equations (for the subgroup in (2)).

(4) For each zoom configuration, if there is a selected subgroup matrix element(where d ═ a + b) corresponds to the gray font matrix elements in any other considered matrix (different pupil plane cross-sections) with the same index, then these gray font matrix elements are applied to CCEQ by equations (6.9) and/or (6.10) and the result is for R_i、f_i,j、And/orThe second set of non-linear equations added to the first set of equations.

(5) Usually using numerical solvers, one can target all unknown coefficients R_i、f_i,j、Andthe entire combined set of equations for the subgroup is solved simultaneously.

(6) Once this subgroup, containing all the matrix elements related to the sphere, is solved, the exact solution defined in section 8.1 is followed, fromStarting until the maximum combination order is reached(from one or more aspheric matrix elements for at least one zoom configuration)

The final remaining aspheric coefficients are calculated in ascending order.

As before, the calculated optical zoom system is nominally free of any light ray aberrations (or combinations thereof) associated with the CMAC of each zoom configuration under consideration by the selected and applied matrix elements. If more than one solution exists, each solution corresponds to a different optical zoom system that nominally satisfies the same application conditions.

Examples of the invention

The following examples illustrate various designs that have been calculated using methods according to embodiments of the present invention.

Example 1: a. the₀S₆Cooke triplet lens

The optical design of the Cooke triplet obtained with the method according to the invention is illustrated in FIG. 10.

The system specification includes: IOD, f_L50mm, F/#5, 40 ° FFOV, designed for λ₀＝587.56nm。

The aberrations to be nominally eliminated by the method according to the invention are two 1 st order and four 3 rd order aberrations in the tangential plane. Six surface curvatures (radii) and associated mapping function coefficients have been calculated by solving the derived nonlinear system of equations using, for example, the Newton-Raphson method (NRM).

Lens data:

#	type (B)	Curvature	Thickness of	Glass	Radius of
						Article (A)	Standard of merit	0	Infinite size		0
1	Standard of merit	0	2.5		9.75380
						2	Standard of merit	0.05064	3.20634	SK16	9.50000
3	Standard of merit	-0.00046	5.48499		9.50000
						4	Standard of merit	-0.03783	0.74697	F2	5.00000
Diaphragm	Standard of merit	0.00000	1.22965	F2	3.85566
						6	Standard of merit	0.05565	4.67249		5.00000
7	Standard of merit	0.01836	3.29917	SK16	7.50000
						8	Standard of merit	-0.04555	40.46451		7.50000
Image	Standard of merit	0	0		18.20285

Fig. 11 shows a dot diagram associated with the optical layout of a Cooke triplet. As known to the skilled person, the dot diagram gives an indication of the image of the dot object. Without aberrations, the point object will converge to a perfect image point. The dot diagrams are shown for three different wavelengths 486.1nm, 587.6nm, and 656.3 nm. The results were obtained without placing any a priori knowledge of the Cooke triplet in the calculations. The results obtained perform very similarly to the known examples in the literature and perform equally well or better after standard optimization in the optical design program.

Example 2: a. the₀S₈Wide-angle lens

Fig. 12 illustrates the optical design of a wide-angle lens obtained with the method according to the invention.

The system specifications are as follows: IOD, f_L3.5mm, F/# ═ 8, 90 ° FFOV, designed for λ₀＝587.56nm。

The aberrations to be nominally eliminated by the method of the invention are as follows:andwhere l is 1,3,5, 7 (tangential plane). By solving the derived system of nonlinear equations using NRM, eight surface curvatures (radii) and associated mapping function coefficients have been calculated.

Lens data:

#	type (B)	Curvature	Thickness of	Glass	Radius of
						Article (A)	Standard of merit	0	Infinite size		0
1	Standard of merit	0	1.00000		2.97903
						2	Standard of merit	0.13444	0.74130	Polycarbonate resin	1.76628
3	Standard of merit	0.66282	0.69130		1.11301
						4	Standard of merit	0.88629	0.75560	COC	0.85664
5	Standard of merit	0.13969	0.28130		0.60689
						Diaphragm	Standard of merit	0	0.24480		0.29047
7	Standard of merit	-0.17572	0.98040	COC	0.33339
						8	Standard of merit	-0.89501	0.63120		0.74739
9	Standard of merit	-0.82912	0.58760	Polycarbonate resin	0.95122
						10	Standard of merit	-0.15886	1.65710		1.54923
Image	Standard of merit	0	0		3.60739

FIG. 13 shows a dot diagram associated with the optical layout of the design. The dot diagrams are shown for three different wavelengths 486.1nm, 587.6nm, and 656.3 nm.

Selected aberrationsAndemphasis is placed on the angle-related terms to be eliminated. It is important to emphasize that such differential aberration cancellation is not possible with any prior art direct design approach. The choice of the invention ensures very good equalization performance in the entire 90 ° field of view (as can be seen in fig. 13). In the final step, the speed of the lens can be increased (decreasing the f-number) and standard optimization in the optical design program provides the final system layout. With only two design steps, time, effort and cost can be reduced substantially.

Example 3: a with fixed final image position and total system length₈S₀Zoom lens

Fig. 14 and 15 illustrate two optical configurations of a zoom lens with a fixed final image position obtained with a method according to an embodiment of the invention.

The system specifications are as follows: IOD, F/# ═ 5.6, designed for λ₀587.56nm, total track length 70mm

Zoom configuration 1: f. of_L1＝9.24mm

Zoom configuration 2: f. of_L2＝27.72mm(f_L13 times of

The aberrations to be nominally eliminated by the method of the present invention for each zoom configuration are the following two 1 st order aberrations (non-linear system solved using NRM) plus (where l ═ 3, 5) (system of linear equations), resulting in a total of 20 aberrations that are nominally eliminated.

Lens data [ configuration 1 ]:

zoom configuration thickness variation:

#	configuration 1	Configuration 2
			3	3.00216	20.58425
5	21.99784	4.41575
			8	4.41878	2.56780
10	15.58122	17.43220

Fig. 16 and 17 illustrate dot diagrams of the first and second zoom configurations, respectively, for a wavelength of 587.6 nm. At this stage, the stippling has shown fairly balanced performance as a direct result of the calculations.

The selected 20 aberrations to be eliminated (10 per zoom configuration) are significantly more than the aberrations that the prior art direct design solution for the zoom system can eliminate. The number of aberrations to be eliminated can be increased even further for l 7,9 …, if desired. The calculated solution provides a very good starting point for the two defined zoom stages in a single calculation scheme, which is not possible using any prior art method. In the final step, standard multi-configuration optimization in the optical design program can be used to increase the lens speed (reduce f-number) and balance the overall performance to achieve the final zoom system layout. Streamlined design processes allow for reduced design time, effort, and thus reduced cost.

Example 4: a. the₄S₀Hastings triplet lens

Fig. 18 shows the optical layout of a hasting triplet obtained with an embodiment of the method according to the invention.

The system parameters are as follows: IOD, f_L20mm, 2, 6 ° FFOV, designed for λ₀＝554.2nm。

The aberrations to be nominally eliminated by the method of the present invention are:andwhere k is 1,3,5, 7,9, 11 (tangential plane). Using NRM in the case of a non-linear system (order 1) and gaussian elimination in the case of all higher order linear systems, all 24 surface coefficients (order 2 to order 12) and the associated mapping function coefficients have been calculated.

Lens data:

#	type (B)	Thickness of	Glass	2 order	4 th order	6 th order	8 th order	10 th order	12 th order
										Article (A)	Standard of merit	Infinite size		0	0	0	0	0	0
1	Standard of merit	1.5		0	0	0	0	0	0
										Diaphragm	EVENASPH	3.5	F2	0.0375	-2.380E-05	-3.795E-07	-2.473E-09	-1.167E-11	-2.050E-14
3	EVENASPH	9.0	K7	0.0817	1.005E-04	-1.992E-06	-4.431E-08	-6.034E-10	-5.896E-12
										4	EVENASPH	1.5	F2	-0.0746	2.096E-04	-5.732E-06	2.746E-08	-1.024E-09	3.405E-12
5	EVENASPH	14.0		-0.0358	4.637E-05	-1.332E-06	1.222E-08	-1.341E-10	7.216E-13
										Image	Standard of merit	0.0		0	0	0	0	0	0

Fig. 19 shows a dot diagram associated with the optical design obtained with the present invention for wavelengths 486.1nm, 587.6nm, and 656.3 nm.

Fig. 20 further illustrates the focus shift (in μm) as a function of wavelength (in μm).

Example 5: three mirror imager with free shaped surface

System parameters: IOD, f_L＝600mm，F/#＝3，4°FFOV

Aberration ∈ nominally eliminated for all index pairs (j, k, l, m) ≧ 0 (where l + m ≦ 1 and j + k + l + m ≦ 5)_x,j,k,l,mAnd ∈_y,j,k,l,m(aspherical Condition, absence of spherical aberration, absence of linear field dependence in x, y)

Of the 40 surface coefficients (no vertices), 38 coefficients and the associated mapping function coefficients have been calculated. The inputs include the focal length in x and y (here 600mm), the surface and image plane vertex positions, and the curvature in x and y on the mirror 2.

Tilt/centrifuge value:

the first mirror rotates-16.665 degrees around the y-axis (rotates the same angle back to the surface)

Second mirror was centrifuged 218.55mm in the x-direction

The second mirror rotates-11.237 degrees about the y-axis (back to surface by the same angle)

The third mirror is decentered 82.382mm in the x-direction

The third mirror is rotated-1.2052 degrees about the y-axis (back to the surface by the same angle)

The image plane is decentered 122.48mm in the x-direction

The image plane rotates 0.012043 degrees around the y-axis (back to surface by the same angle)

Mirror data:

example 6: four mirror imager with free form surface

System parameters: IOD, f_L＝600mm，F/#＝3，6°FFOV

The aberrations that are nominally eliminated for all index pairs (j, k, l, m) ≧ 0, with l + m ≦ 1 for j + k + l + m ≦ 5, l ≦ 2 and m ≦ 0 for j + k + l + m ≦ 5, and m ≦ 2 and 0 for j + k + l + m ≦ 5.

Of the 56 surface coefficients (no vertices), 52 coefficients and the associated mapping function coefficients have been calculated. The inputs include the focal length in x and y (here 600mm), the surface and image plane vertex positions, and the curvature in x and y on mirrors 1 and 3.

Tilt/centrifuge value:

the first mirror rotates-30.462 degrees around the y-axis (rotates the same angle back to the surface)

Second mirror was centrifuged 342.83mm in the x-direction

The second mirror rotates-39.011 degrees about the y-axis (back to surface by the same angle)

The third mirror is decentered-125.76 mm in the x-direction

The third mirror is rotated-68.703 degrees about the y-axis (back to the surface by the same angle)

The fourth mirror is decentered 322.41mm in the x-direction

The fourth mirror rotates-95.066 degrees about the y-axis (rotates the same angle back to the surface)

The image plane is decentered-285.23 mm in the x-direction

The image plane is rotated-75.153 degrees around the y-axis (back to surface by the same angle)

Mirror data:

all the results shown clearly confirm the expected performance according to the selected aberrations to be eliminated. The prior art design method is of course able to eliminate linear chromatic aberration, i.e.(which is called axial chromatic aberration) and(which is known to the skilled person as lateral chromatic aberration). The examples given for the design with the method of the invention enable the elimination of higher orders, i.e.And(here, k is 1,3..11), emphasis is placed on higher-order axial chromatic aberration. This additional control over the chromatic aberration behavior is important, for example, in high speed lenses such as the f/2 lens shown here.

FIG. 21 shows an optical layout of a full freeform three mirror imaging design obtained with a method according to an embodiment of the invention.

FIG. 22 shows a dot alignment diagram associated with the full freeform three mirror imaging design obtained with the method of FIG. 21 according to the present invention.

FIG. 23 shows an optical layout of a four-mirror imaging design of a full freeform shape obtained with a method according to an embodiment of the invention.

FIG. 24 shows a dot alignment diagram associated with a full freeform three mirror imaging design obtained with a method according to an embodiment of the invention.

It is important to note that the present invention is not limited to homogeneous materials. Indeed, embodiments of the invention may include the use of materials having locally varying refractive indices, such as gradient index optical material (GRIN). This can be done by taking into account the varying refractive index in the wavelength dependence.

Furthermore, the present invention is not limited to smooth, uniform surfaces, although the description is directed to such surfaces. In fact, the discontinuous surface may beTo be considered as a collection of parts of a smooth surface. The method should then be applied to each smooth portion and the results obtained should be combined. Thus, function h_x,p、h_y,p、u_i,p、v_i,pAnd f_iMust be infinitely differentiable and have, in part, a power series representation as defined in equations (2.1), (2.2), (2.3), (2.4) and (2.6), and taylor's theorem can be applied on each smooth part, as discussed throughout the specification (e.g., in section 2).

The invention can also be used for active optical surfaces, i.e. optical surfaces that can have different shapes (with e.g. an array of actuators attached to the back side of the mirror). Such actuator arrays are commonly used in deformable mirrors in the field of adaptive optics to compensate for atmospheric distortions, but also in active optics to prevent mirror deformation due to temperature changes, mechanical stress, wind turbulence, etc. For such applications, the surface coefficients of the optical surface will have different values for each configuration of the optical system, and thus, the method according to any embodiment of the invention will be applied to each configuration.

In systems comprising aspherical and/or spherical surfaces, not every surface coefficient of the optical surface can be calculated. The number of calculated surface coefficients and the final residual coefficients that need to be given by the designer as input parameters depend on the optical system under consideration and can be derived by the skilled person directly from the relevant SCD rules. In case surface coefficients are required to be provided as initial values, a Monte Carlo optimization, local or global optimization algorithm or any equivalent method may be used with embodiments of the invention to provide initial values of the surface coefficients before applying the method according to the invention.

In a further aspect according to any embodiment of the invention, the optical design program may be used after the method of the invention has been applied. Such design workflows are quite common to those skilled in the art of optical design. The initial system layout is calculated based on the specific aberrations to be eliminated and in the next step, the final system layout is reached using optimization in the optical design program. The result of the optimization is an equalization process in which the aberrations that were initially eliminated will again reach a non-zero value, while the higher order aberrations will be reduced to obtain an overall equalization effect. This approach has proven its worth in many prior art optical designs, but the direct design approach according to any prior art is only used for a severely limited number of low order aberrations. There is no such limitation on several lower order aberrations in the present invention. Together with optimization in the optical design program, the present invention provides an efficient and streamlined workflow to first compute an excellent initial system layout and second optimize the initial system to reach a final system layout.

In this case, the obtained surface coefficients may be used as input to an optical design program to further adapt the obtained results to the requirements of the application.

A computer-based method according to embodiments of the invention for designing a rotationally symmetric imaging system to nominally eliminate at least one ray aberration, the method comprising the steps of:

-selecting a design wavelength λ₀，

-a selection number N_sAnd a sequence of optical surfaces, each optical surface having a profile f_iN of the group_sThe sequence of optical surfaces defines the optical axis of the imaging system,

-selecting each surface as spherical or aspherical, wherein each surface is expressed as a function of a single radial variable r,

-selecting the angle of freedom(wherein p is 1 … m_p) A defined arbitrary but fixed pupil plane cross-section, where m_pCorresponding to the number of pupil plane cross sections to be considered,

-selecting an object point in an object plane along an optical axis,

-selecting candidate image points in an image plane along an optical axis,

-selecting pupil plane cross-sections θ_pReal image point table inTo the sum of the candidate image points plus the optical aberration, where the aberration is expressed as a ray aberration spread,

-profiling the surface f_iExpressed as functions of ray mapping functions describing a cross-section θ from an object point through a pupil plane_pWith each surface f_iThe position of the intersection is such that,

-selecting N_sA sequence of species of material, each material at λ₀Lower has a refractive indexObject plane, N_sSequence definition N of optical surfaces and image planes_s+1 optical path length d_iThe sequence of the segments of (a) or (b),

-applying the Fermat principle to two consecutive segments of each pair as a function of a ray mapping function by mathematically representing that the path length between two fixed points is an extreme along the ray, to derive two sets of N_sA differential equation of the form (A) and (B),

-selecting a position of the pupil plane along the optical axis,

-expanding each surface contour as a series expansion, each term of the expansion being a surface coefficient expressed as a function of a ray mapping function,

-expanding the ray mapping function by a series expansion of the ray mapping function coefficients,

selecting each optical surface (from its vertex f)_i,0Define) an initial position along the optical axis

-identifying a highest aberration order o to be nominally eliminated, wherein said highest aberration order defines a subgroup of aberration orders up to order o,

-using a power series approach to find the two groups of N_sA solution of a differential equation to compute the series coefficients of the mapping function and the series coefficients of the optical surface by selecting a subgroup of the light aberration coefficients to be nominally eliminated in the imaging system according to a set of selection rules,

applying partial derivatives of order k, l for the subgroup, said order being provided by using a predefined aberration matrix for each pupil plane cross-section, wherein each predefined aberration matrix relates each light aberration of the subgroup to the partial derivatives of order k, l of the light aberration series expansion in said pupil plane cross-section, to derive a system of equations for optical surface coefficients and light mapping function coefficients,

solving a system of equations for the optical surface coefficients and the light ray mapping function coefficients,

-deriving the surface profile of each optical surface and hence each clear aperture from the calculated order coefficients of the optical surfaces.

In embodiments of the invention, the system of equations (which may be linear or non-linear depending on the situation) may be solved using known techniques, such as the Newton Raphson technique, the gaussian elimination technique (for linear systems), and the like.

In an embodiment, the number of surface coefficients corresponding to the subgroup minus the number of selected matrix elements of the predefined matrix of the subgroup corresponds to the number of surface coefficients to be predefined as initial values, and the number of unknown surface and ray mapping coefficients corresponds to the number of equations to be used.

According to another embodiment of the invention, the software may be implemented as a computer program product that has been compiled for execution by a processing engine of any of the methods of the invention, or compiled for execution in an interpreted virtual machine (such as Java, for example)^TMVirtual machines). The device may include logic encoded in media for performing any of the steps of the method according to embodiments of the invention. Logic may include software encoded in a disk or other computer readable medium and/or instructions encoded in an Application Specific Integrated Circuit (ASIC), Field Programmable Gate Array (FPGA), or other processor or hardware. The device will also include a CPU and/or GPU having a processing engine capable of executing the software of the present invention and memory.

The computer program product may be stored on a non-transitory signal storage medium such as a compact disc (CD-ROM or DVD-ROM), digital tape, magnetic disk, solid state memory (such as USB flash memory), ROM, etc.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

a rotationally symmetric imaging system having an optical surface is designed for nominally eliminating at least one ray aberration.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-inputting to the computer system specifications comprising:

a parameter of the imaging system is determined,

o a wavelength dependency is determined from the wavelength dependence,

the number of surface coefficients of the optical surface,

a pupil characteristic is then calculated for each of the plurality of pupil characteristics,

object space and image space.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

using a computer, converting the system specification into a system of differential equations by applying the fermat principle.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-using a set of selection, consistency and distribution rules to input to the computer a subgroup of ray aberrations to be nominally set to vanish.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-solving the differential equations to obtain surface coefficients of the optical surface and mapping function coefficients defining the ray trajectories and their intersections with the optical surface and thereby defining each clear aperture.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-in the step of inputting system specifications of imaging system parameters to the computer comprising: inputting at least one design wavelength lambda₀And/or

-in the step of inputting system specifications of imaging system parameters to the computer comprising: input N_sA sequence of optical surfaces, wherein each optical surface has an optical surface profile f_iAnd wherein the N_sThe sequence of optical surfaces defining the optical axis of the imaging system, and/or

-in the step of inputting to the computer a system specification of surface coefficients of the optical surface comprising: each optical surface is defined as spherical or aspherical and wherein each optical surface is expressed as a function of a single radial variable r.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-in the step of inputting into the computer system specifications of pupil characteristics comprising: position of the input pupil plane along the optical axis, and/or

-in the step of inputting the system specification of pupil characteristics to the computer further comprising: selecting m of pupil plane_pA fixed pupil plane cross-section p, where m_pNot less than 1, and/or

-wherein each pupil plane cross-section is defined by an angle(wherein p is 1 … m_p) Define, and/or

-wherein each pupil plane cross section uses a single pupil radial variable q_pDefinition of q_pRepresenting a signal from object space and passing through (x)_p,y_p,0)＝(q_pcos(θ_p),q_psin(θ_p) 0), where the ray of pupil plane coordinates intersects the first image plane.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-wherein in the step of inputting the system specifications of object space and image space to the computer comprises: selecting object points located at infinite or finite distances, and/or

-wherein the step of selecting object points located at a finite distance further comprises the steps of: an object point is selected in an object plane along an optical axis.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-further comprising the steps of, in the step of converting the system specification to a system of differential equations using a computer: expressing the optical surface profile as a function of ray mapping functions describing the locations where rays from an object point or at an angle of view and passing through a fixed pupil plane cross-section p intersect each optical surface of said pupil plane cross-section, and/or

-comprising the steps of: expressing the ray mapping function as (u)_i,p(q_p,t),v_i,p(q_pT)), wherein the variablest defines the object and fixes the pupil plane cross section theta_pHaving pupil coordinates (q)_p,θ_p) So that the optical surface f of the pupil plane cross section_iFunction f expressed as a ray mapping function_i((u_i,p(q_p,t),v_i,p(q_p,t)))。

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-wherein in the step of inputting the system specifications of object space and image space to the computer comprises: a first image point is selected in the image plane along the optical axis.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-wherein the step of inputting system specifications of imaging system parameters to the computer comprises: selection of N_sA sequence of +1 materials, each material having a wavelength λ for at least one design wavelength₀Refractive index distribution ofObject plane, N_sThe sequence of optical surfaces and image planes being defined to have an optical path length d_iN of (A)_sA sequence of +1 segments.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-the step of converting the system specification into a system of differential equations using a computer further comprises the steps of: the Fermat principle is worked out by mathematically representing the extreme along the ray of the optical path length between two fixed pointsTwo sets of N are derived by applying a function of the ray mapping function to two consecutive segments of each pair_sA differential equation.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-wherein the step of inputting the system specifications of object space and image space to the computer comprises the steps of: expressing the second image point of each fixed pupil plane cross-section as the sum of the first image point plus the optical aberration expressed as a ray aberration order expansion of the ray aberration coefficients of each pupil plane cross-section.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

wherein the step of inputting to the computer a system specification of surface coefficients of the optical surface comprises: expanding each optical surface profile to have a surface coefficient f for the ith surface and of order 2j_i,2jIs expanded.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-wherein the system specification is converted into two sets of N using a computer_sThe step of differential equations includes: the ray mapping function is expanded by a series expansion of the ray mapping function coefficients.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-wherein the step of solving the differential equation further comprises the steps of: finding two sets of N using a power series approach_sA solution of a differential equation to calculate the mapping function and the series coefficients of the optical surface as a function of the subgroup of aberrations of the light rays, and/or

-further comprising the steps of: applying partial derivatives of order k, l to the two groups N_sA differential equation, said order k, l being provided by using for each pupil plane cross-section a predefined aberration matrix, wherein each predefined aberration matrix relates each light aberration of a subgroup to a partial derivative of the order k, l of the series expansion of light aberrations in the pupil plane cross-section to derive a system of equations for optical surface coefficients and light mapping function coefficients, and/or

-wherein the step of solving the differential equation further comprises the steps of: deriving the surface profile of each optical surface from the calculated optical surface coefficients, and/or

-wherein the step of solving the differential equation further comprises the steps of: the clear aperture of each optical surface is thus derived from the series coefficients of the calculated mapping function.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-wherein the step of inputting into the computer system specifications comprising surface coefficients of the optical surface further comprises: selecting each surface to be spherical or aspherical, further comprising the steps of: expressing each surface as a single radial variableAnd each surface is expanded to have a taylor coefficient f_i,2j(where j is 1,2 … up to 30, and i is 1,2, … 50).

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-wherein the step of selecting the first image point further comprises the steps of: expressing the first image point as a function defining an object-image relationship, and wherein the step of expressing the second image further comprises the steps of: the second image in the selected pupil plane cross section is expressed as the sum of the first image plus the light ray aberrations in the x and y directions, which are related to the known light ray aberration spreads and wave aberration spreads.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-wherein the step of inputting into the computer system specifications of pupil characteristics comprises: inputting the position of an aperture stop along the optical axis, wherein the aperture stop is an entrance pupil, an aperture stop between two optical surfaces, or an exit pupil, and/or

-wherein,

when the aperture stop coincides with one of the optical surfaces, the coordinates of the optical surface in the ray mapping function are replaced by replacing the coordinates of the surface with pupil coordinates

If the aperture stop is between two optical surfaces, the cross product representing the direction vector from the previous optical surface towards the aperture stop and the direction vector from the aperture stop towards the next optical surface is zero, adding three additional equations to each of the two sets of differential equations;

if the aperture stop coincides with the exit pupil, then the cross products representing the direction vector from the final surface to the aperture stop and the direction vector from the stop to the image plane coincide, whereby the cross products of the two vectors are zero and three additional equations are added to the two sets of differential equations,

if the aperture stop coincides with the entrance pupil, the direction vector from the object to the stop and the direction vector from the stop to the first surface coincide, whereby the cross product of the two vectors is zero, and three additional equations are added to the two sets of differential equations.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

duplicating the inventive method for at least a second design wavelength, and/or

In the step of inputting system specifications of the imaging system parameters into the computer, the at least one design wavelength λ is further specified for each material₀Refractive index distribution ofTo make each optical path length d_iExpressed as a function of wavelength.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

at least two different materials are selected for at least two design wavelengths.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-selecting at least two different materials for a design wavelength to nominally cancel a chromatic aberration coefficient.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

the method is used for calculating the aberration of light rays at a first design wavelength and up to a maximum order in an optical system comprising a plurality of optical surfaces, each optical surface being defined by a set of surface coefficients f_i,jBy definition, the method comprises the steps of:

calculating all aberration coefficients ∈ for arbitrary pupil plane cross-sections and in ascending order of aberration order using the power series method_x,p,k,lAnd ∈_y,p,k,l，

-solving, for each aberration order, a system of linear equations by:

all derivatives for each order and for each pupil plane cross section are applied to both sets of differential equations,

o solving a system of linear equations,

o repeat each step until the highest order is reached

And/or

-applying the method to at least two configurations of a zoom system

And/or

-repeating the method for at least a second design wavelength to calculate monochromatic aberrations for the first and second design wavelengths

And/or

By means of a refractive index profile of each material for the at least one design wavelengthTo make each optical path length d_iExpressed as a function of wavelength to further calculate chromatic aberration of the optical system.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

calculating system specifications by optimization methods, and/or

Selecting at least one optical surface (from its vertex f)_i,0Define) an initial position along the optical axis, and/or

The optimization method is Monte Carlo optimization.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

-inputting to the computer system at least a second set of system specifications, wherein each set corresponds to a different configuration of the zoom system, the at least second set of system specifications comprising at least imaging system parameters, pupil characteristics, and object space and image space, the computer-based method being evaluated for each second set of system specifications.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

a computer-based method for generating an electronic file for a digitally controlled machine used to manufacture a rotationally symmetric imaging system having an optical surface, the method comprising:

-inputting to the computer system specifications comprising:

a parameter of the imaging system is determined,

o a wavelength dependency is determined from the wavelength dependence,

the number of surface coefficients of the optical surface,

a pupil characteristic is then calculated for each of the plurality of pupil characteristics,

an object space and an image space,

-applying Fermat's principle to convert system specifications into a system of differential equations using a computer,

-using a set of selection, consistency and distribution rules to input to the computer a subgroup of ray aberrations to be nominally set to vanish,

-solving the differential equation to obtain surface coefficients of the optical surface and mapping function coefficients defining the ray trajectories and their intersections with the optical surface and thereby defining each clear aperture, and outputting an electronic file comprising the surface coefficients of the optical surface and the mapping function coefficients defining the ray trajectories and their intersections with the optical surface and thereby defining each clear aperture.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

a computer-based method for designing a rotationally symmetric imaging system comprising at least two optical surfaces and an optical axis, and which nominally eliminates the following:

at least one light aberration coefficient when calculating at least one surface coefficient,

providing all mapping function coefficients of the intersection of the light ray with the at least two optical surfaces, and all non-eliminated aberration coefficients up to a given order, the method comprising:

-inputting to the computer a first set of system specifications comprising:

o at least two (N)_s≧ 2) the sequence of the optical surfaces and their lateral position along the optical axis in the z-direction

Each optical surface is refractive or reflective and has a spherical or aspherical shape, and wherein each surface f_iExpressed as a single radial variableA function of (2), the function having a total number N_cUnknown system surface coefficients

O a design wavelength λ₀And corresponding refractive indices before and after each optical surface

O will be located at a finite or infinite distance and in the case of a finite distance have a flat or curved shape

A defined image as a function of an object variable t, the function describing object-image relationships

O a real image as the sum of the prescribed image plus the series expansion of the light aberration

A certain number (m) of system apertures_pNot less than 1) pupil plane, each pupil plane being defined by an angle(wherein p is 1 … m_p) Is defined so as to mean that the compound is derived from an object and is in (x)_p,y_p,0)＝(q_pcos(θ_p),q_psin(θ_p) 0) position where a ray passing through the pupil plane intersects the image plane

O express each surface as a function f of the ray mapping function in (x, y)_i((u_i,p(q_p,t),v_i,p(q_pT))) that describe the position at which a ray from an object passing through a fixed pupil plane p intersects each optical surface, and a pupil and object variable (q)_pNumber of stages of t)

-using a computer, converting the first set of system specifications into differential equations by: applying Fermat principle to object-by-object, surface function f_iMapping function u_i,pAnd v_i,pAnd two consecutive optical path length segments of each pair expressed in the form of a real image to derive 2N_sA differential equation D_i′(i′＝1…2N_s)(N_sDifferential equation in u_i,pIn the direction of, and N_sA differential equation at v_i,pIn the direction)

-defining pupil characteristics, wherein

If the diaphragm is associated with one of the surfacesCoincidence is then formed by (x)_p,y_p) SubstitutionIn (1)Andas diaphragm position

If an aperture stop is placed at the entrance, between two optical surfaces, or at the exit (like before), the cross product of the vector describing any ray path towards the aperture stop and the vector describing any ray path from the aperture stop is zero, whereby three additional equations are added to the previously derived differential equation, resulting in D_i′Equation (i' ═ 1 … 2N_s+3)

-using a computer, converting said 2N by_sOr (2N)_s+3) equations are converted into algebraic equations: by making k, l equal to 0,1,2,3 … and the combination order k + l equal to 1,3,5 … o_mEvaluation ofTo use a power series solution method to obtain the series coefficients for all definitionsOr 2 (N)_s+3) algebraic equations of the order of the algebraic equations,

-selecting at least one light aberration coefficient to be zero; for each aberration coefficient set to zero, one surface coefficient is selected as the (sub-) group M_cAnd each of said pair(s) of light ray aberrations and surface coefficients occur simultaneously in at least one algebraic equation,

-inputting to the computer the values of any surface coefficients that have not been used in the previous step to eliminate aberrations, these values being other than the group M_cAll of the surface coefficients of a portion of (a),

-solving, using a computer, the system of algebraic equations to obtain as output at least one surface coefficient of the optical surface, all mapping function coefficients, and the non-eliminated light aberration coefficients up to a given combined order.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

inputting at least a second set of system specifications into the computer system, wherein each set corresponds to a different configuration of the zoom system, and each set (c ═ 1,2 …) of zoom system specifications comprises:

lateral position of all surfaces describing zoom movement

-mapping function for each zoom configuration

Aberration function for each zoom configuration

-defining an image as a function of an object for each zoom configuration

-the method comprises:

-using a computer, converting the system specification into a differential equation by:

the Fermat principle is applied to the two consecutive optical path lengths of each pair to derive a 2N for each zoom system configuration_sDifferential equation

-defining pupil characteristics, wherein

If the diaphragm is associated with one of the surfacesCoincidence is then formed by (x)_p,y_p) SubstitutionIn (1)Andas diaphragm position

When the aperture stop is placed, the cross product of the vector describing the arbitrary ray path towards the aperture stop and the vector describing the arbitrary ray path from the aperture stop is zero, thereby adding three additional equations to the previously derived differential equation for each zoom system configuration

-using a computer, converting said 2cN by_sOr c (2N)_s+3) equations are converted into algebraic equations: by making k, l equal to 0,1,2,3 … and the combination order k + l equal to 1,3,5 … o_mEvaluation ofTo use a power series solution method to obtain a power series for all defined series coefficients and all zoom system configurationsOr 2c (N)_s+3) algebraic equations

-selecting at least one light ray aberration coefficient per zoom system specification as zero; for each aberration coefficient set to zero, one surface coefficient is selected as the (sub-) group M_cAnd each of said pairs of light aberrations and surface coefficients occur simultaneously in at least one algebraic equation,

inputting to the computer the value of any surface coefficient that has or has not been used in the previous step to eliminate the aberration (whereas the input can be done manually or using a computer)

-solving algebraic equations using a computer to obtain M as the optical surface_cAll surface coefficients of a part, all mapping functions, and the non-eliminated light aberration coefficients up to a given combination order and for all zoom system configurations as outputs.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

a computer-based method in which the rotational symmetry of an imaging system is broken, and in which the system has (1) no symmetry, (2) one plane of symmetry, or (3) two planes of symmetry, the method comprising:

-inputting to the computer system specifications comprising:

оN_ssequence of more than or equal to 2 (maximum 30) optical surfaces

Each optical surface is refractive or reflective and has a spherical, aspherical or free shape, and wherein each free-shaped surface is expressed as a function of two variables with a power series coefficient f that is not yet known_i,j,k(where j, k is 1,2 … up to 12, and i is 1,2, … N_s) And the total number N_cSurface coefficient of individual system

O a design wavelength λ₀And corresponding refractive indices before and after each optical surface

O will be located at a finite or infinite distance and in the case of a finite distance have a flat or curved shape

O as two object variables t_xAnd t_yThe function describing the object-image relationship

O a real image as the sum of the prescribed image plus the ray aberration series expansion according to the symmetry of the system under consideration

O mapping a ray to a function u_i(x_p,y_p,t_x,t_y) And v_i(x_p,y_p,t_x,t_y) Expressed as pupil (x)_p,y_p) Sum variable (t)_x,t_y) Number of stages of

All surface positions, and any tilt of the object, optical surface, final aperture stop and image are expressed by a rotation matrix (at least one rotation matrix for the symmetric case (1) or (2)); and the rotation matrix is defined by inputting the rotation axes and angles that match the symmetry of the system, which in turn defines the chief ray path of the on-axis field and all the vertices of the mapping function

-using a computer, converting the system specification into a differential equation by: applying Fermat principle to object-by-object, surface function f_iMapping function u_iAnd v_iAnd two consecutive optical path length segments of each pair expressed in the form of a real image to derive 2N_sA differential equation D_i′(i′＝1…2N_s)(N_sDifferential equation in u_iIn the direction of, and N_sA differential equation at v_iIn the direction)

-defining pupil characteristics, wherein

If the diaphragm is associated with one of the surfacesCoincidence is then formed by (x)_p,y_p) SubstitutionIn (1)Andas diaphragm position

If the aperture stop is placed in a given position with a defined orientation (rotation matrix), the cross product of the vector representing any ray path towards the aperture stop and the vector describing any ray path from the aperture stop is zero. Thereby adding three additional equations to the previously derived D_iDifferential equation (i' ═ 1 … 2N_s+3)

-using a computer, converting said 2N by_sAn equation or 2N in the case of an aperture stop_s+3 equations are converted to algebraic equations: by aiming at the indices j, k, l, m, ═ 0,1,2,3 … and the combination order j + k + l + m ═ 1,2,3 … o_m(maximum 11) evaluation of all possible combinationsTo use a power seriesSolving the method to obtain an algebraic equation set for all defined series coefficients

-selecting at least one light aberration coefficient to be zero; for each aberration coefficient set to zero, one surface coefficient is selected as the (sub-) group M_cAnd each of said pair(s) of light ray aberrations and surface coefficients occur simultaneously in at least one algebraic equation

-inputting to the computer the value of any surface coefficient that has not been used in the previous step to eliminate aberrations, i.e. not the group M_cAnd the input may be done manually or using a computer

Solving algebraic equations using a computer to obtain at least one surface coefficient of the optical surface (as M)_cAll coefficients of a portion of), all mapping function coefficients, and, up to a given combination order, the non-eliminated light aberration coefficients as output.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

the computer-based method further comprises simultaneously applying the method to at least a second design wavelength, wherein for each design wavelength there is a defined ray mapping and ray aberration function, the method further comprising:

-using a computer, converting the system specification into an algebraic equation by: applying the Fermat principle to two consecutive optical path length segments of each pair for each design wavelength

-selecting at least one aberration coefficient to be zero per design wavelength, thereby providing a surface coefficient group M_c

Solving algebraic equations using a computer to obtain at least two surface coefficients (as M)_cAll coefficients of a portion of), all mapping functions, and the non-eliminated light aberration coefficients up to a given combination order and for all design wavelengths.

-including chromatic aberration coefficients in the expansion of aberration order in the form of a wavelength variable λ

By additional series terms (λ - λ)₀)^ΛIncluding wavelength dependence in all mapping functions

By means of a refractive index profile n_i-1,i(λ) (e.g., Sellmeier equation) to express each optical path length segment

-using a computer, converting the system specification into a differential equation by: applying the Fermat principle to two successive optical path length sections of each pair (with surface or aperture stops)

-converting the equations into algebraic equations using a computer, wherein the step of solving the method using a power series comprises: will add additional itemsIncluded, where Λ -0, 1,2 …, optionally has a maximum value of 5, and index Λ is added to the sum of the combining orders

-selecting aberration coefficients to be nominally zero, wherein the step comprises selecting at least one monochromatic (i.e. 0 for Λ) and one chromatic (Λ ≠ 0) aberration coefficient to be zero, thereby providing a group M of surface coefficients_cAnd each of said pairs of ray aberrations and surface coefficients occurring simultaneously in at least one algebraic equation

-input to a computer other than group M_cOf a part of (a)

Solving algebraic equations using a computer to obtain at least two surface coefficients (as M)_cAll coefficients of a portion of), all mapping function coefficients, and up to a given combination order, the unabated monochromatic and chromatic aberration coefficients as output.

The software may be embodied in a computer program product adapted to perform the following functions when the software is loaded onto the respective one or more devices and executed on one or more processing engines (such as microprocessors, ASICs, FPGAs, etc.):

in a method for a computer-based system,wherein the step of selecting aberration coefficient(s) comprises: the aberration coefficient(s) is set to a non-zero value. For each aberration coefficient set to a non-zero value, one surface coefficient is selected as the (sub-) group M_cAnd each of said pair(s) of light ray aberrations and surface coefficients occur simultaneously in at least one algebraic equation.

The above-mentioned software may be stored on a non-transitory signal storage medium such as a compact disc (CD-ROM or DVD-ROM), magnetic tape, magnetic disk, ROM, or solid state memory such as USB flash memory, etc.

While the present invention has been described above with reference to several embodiments, this is for the purpose of illustration and not of limitation, the scope of the invention being determined by the appended claims. The skilled person will appreciate that features disclosed herein in connection with various embodiments can be combined with features of other embodiments to achieve the same technical effects and advantages, without departing from the scope of the invention.