阅读说明：本技术 用于偏转和聚焦光线的设备 (Device for deflecting and focusing light ) 是由 阿提姆·鲍里斯金 奥克萨那·什拉姆科娃 劳伦特·布隆德 于 2018-08-27 设计创作，主要内容包括：本公开涉及一种用于从入射电磁波产生和引导纳米喷射波束的设备,所述入射电磁波在所述设备上。所述设备的特征在于它包括：-第一介电材料层,其具有第一折射率值(n<Sub>1</Sub>)；-第二介电材料层,其具有第二折射率值(n<Sub>2</Sub>),其中所述第二折射率值大于所述第一折射率值；以及其中所述第一层和第二层之间的第一边界包括第一单台阶形状结构,所述第一单台阶形状结构是根据与接近第一倾斜角的倾斜角相关联的第一直线明显限定的层级变化,其中所述设备还包括被包含在嵌入层中的介电元件,所述嵌入层是所述第一层或所述第二层,所述介电元件具有高于所述嵌入层的折射率的折射率(n<Sub>3</Sub>),以及其中所述介电元件和所述嵌入层之间的第二边界包括第二单台阶形状结构,所述第二单台阶形状结构是明显地由与接近第二倾斜角的倾斜角相关联第二直线限定的层级变化,并且其中所述第二单台阶形状结构位于所述第一单台阶形状结构附近,用于将由所述第一和第二单台阶形状结构中的每一个产生的纳米喷射波束聚焦在第一聚焦点周围。(The present disclosure relates to a device for generating and directing a nanojet beam from incident electromagnetic waves, which are incident on the device. Said device being characterized in that it comprises: -a first layer of dielectric material having a first refractive index value (n) 1 ) (ii) a -a second layer of dielectric material having a second refractive index value (n) 2 ) Wherein the second refractive index value is greater than the first refractive index value; and wherein a first boundary between the first and second layers comprises a first single-step shape structure that is a distinctly defined level variation according to a first line associated with a tilt angle close to a first tilt angle, wherein the device further comprises a dielectric element comprised in an embedded layer, the embedded layer being the first layer or the second layer, the dielectric element havingHaving a refractive index (n) higher than that of the embedding layer 3 ) And wherein a second boundary between the dielectric element and the embedding layer includes a second single-step shaped structure that is a level variation defined significantly by a second straight line associated with a tilt angle near a second tilt angle, and wherein the second single-step shaped structure is located in proximity to the first single-step shaped structure for focusing a nanojet beam produced by each of the first and second single-step shaped structures around a first focal point.)

1. An apparatus for generating and directing a nanojet beam from an incident electromagnetic wave, said incident electromagnetic wave being on said apparatus, wherein said apparatus is characterized in that it comprises:

-a first layer of dielectric material having a first refractive index value (n)₁)；

-a second layer of dielectric material having a second refractive index value (n)₂) Wherein the second refractive index value is greater than the first refractive index value; and

wherein a first boundary between the first and second layers comprises a first single-step shape structure that is a distinctly defined level variation according to a first line associated with a tilt angle close to a first tilt angle, wherein the device further comprises a dielectric element comprised in an embedded layer, the embedded layer being the first layer or the second layer, the dielectric element having a refractive index (n) higher than the refractive index of the embedded layer₃) And an

Wherein a second boundary between the dielectric element and the embedding layer includes a second single-step shape structure that is a level variation defined significantly by a second straight line associated with a tilt angle near a second tilt angle, and wherein the second single-step shape structure is located proximate to the first single-step shape structure for focusing a nanojet beam produced by each of the first and second single-step shape structures around a first focal point.

2. The apparatus of claim 1, wherein the first boundary further comprises a third single-step shape structure that is a level variation defined significantly by a third straight line associated with an inclination angle having an absolute value close to the first inclination angle, wherein the first and third single-step shape structures face each other and are a first distance from each other, and wherein the second boundary further comprises a fourth single-step shape structure that is a level variation defined significantly by a fourth straight line associated with an inclination angle having an absolute value close to the second inclination angle, and wherein the second and fourth single-step shape structures face each other and are a second distance from each other that is less than the first distance, and wherein the fourth single-step shape structure is located in proximity to the third single-step shape structure, for focusing the nanojet beams produced by each of the third and fourth single-step shaped structures around a second focal point.

3. The apparatus of claim 2, wherein the incident electromagnetic wave is a plane wave perpendicular to the apparatus, and wherein the first and second angles are equal to 90 degrees, and wherein the first and second focal points are close to each other.

4. The apparatus of claims 2 and 3, wherein the first and third single-step shaped structures have the same first height H₁And wherein the second and fourth single-step shaped structures have the same second height H₂And wherein said second height H₂Has a value of about

5. The apparatus of claim 2, wherein the incident electromagnetic wave is a plane wave perpendicular to the first layer and the second layer, and wherein the first angle and the second angle are equal to an angle a, and wherein the first focal point and the second focal point are close to each other.

6. The apparatus of claims 2 and 5, wherein the first and third single-step shaped structures have the same first height H₁And wherein the second and fourth single-step shaped structures have the same second height H₂And wherein said second height H₂Has a value of aboutWherein d is₁Is the first distance, and γ₁Is aboutWherein Θ is_B1Is the angle of radiation of the nano-jet beam produced by the first and third single-step shaped structures, wherein

7. The apparatus of claim 2, wherein the incident electromagnetic wave is at an angle of incidence Θ_iA plane wave that impinges the apparatus, and wherein the first and second angles are equal to 90 degrees, and wherein the first and second focal points are proximate to each other.

8. The apparatus of claims 2 and 7, wherein the index of refraction of the second layer is equal to the index of refraction of the dielectric element, and wherein the first and third single step-shaped structures have the same first height H₁And wherein said second distance d₂Is defined according to the following equation:

9. The apparatus of any of claims 1-8, wherein the index of refraction of the dielectric element is less than an index of refraction of a layer of the first and second layers that is not the embedded layer.

10. The device according to any one of claims 1 to 9, wherein said incident electromagnetic waves have a wavelength equal to a value comprised between 390 and 700 nm.

11. The apparatus of any one of claims 1 to 10, wherein the dielectric element has a shape belonging to a group of shapes, the group of shapes comprising:

-a cubic shape;

-a cylindrical shape;

-a prism shape.

Technical Field

The present disclosure relates to techniques for guiding electromagnetic waves, particularly visible light waves.

Background

This section is intended to introduce the reader to various aspects of art, which may be related to various aspects of the present invention that are described and/or claimed below. This discussion is believed to be helpful in providing the reader with background information to facilitate a better understanding of the various aspects of the present invention. Accordingly, it should be understood that these statements are to be read in this light, and not as admissions of prior art.

The development of techniques for guiding visible light as electromagnetic radiation is a busy research topic. In fact, the use of light is widespread in our society and can be applied in a wide range of applications. For example, in our daily lives, people are using more and more devices that are able to convey information to our eyes. From see-through glasses to head-mounted devices, and through smart phones including display structures (LCD or LED display devices or OLED display devices) to television sets, devices capable of producing information by light are a very important piece of many electronic equipment. Furthermore, in other devices, such as optical fibers, or in image sensors, the guiding of light is mandatory in order to obtain effective results that may meet certain expectations. As explained in PCT patent applications PCT/EP2017/057129 and PCT/EP17/057131, which have not yet been published at the time of filing the present patent application, the guiding of visible light (in terms of deflection or in terms of focusing effect) can be achieved by using a single step-like structure within a dielectric layer (dielectric layer), wherein the choice of the value of the height of said step and the value of the base angle (i.e. related to the slope of said step) has an effect on the deflection of the incident light (i.e. these parameters enable the control of complex electromagnetic phenomena associated with light diffraction on the edges of the step). Single step nano-jet (NJ) lenses may be used to mention the optical architectures or systems described in PCT patent applications PCT/EP2017/057129 and PCT/EP 17/057131.

However, it remains a pending problem to obtain techniques that can improve the focusing efficiency (i.e., produce field strength enhancement in the focused spot) of single-step nanojet lenses while retaining all other advantages of single-step nanojet lenses, such as wavelength order size, sub-wavelength resolution, and low dispersion behavior.

The present disclosure proposes a solution that can achieve this desire.

Disclosure of Invention

References in the specification to "one embodiment," "an example embodiment," indicate that the embodiment described may include a particular feature, structure, or characteristic, but every embodiment may not necessarily include the particular feature, structure, or characteristic. Moreover, such phrases are not necessarily referring to the same embodiment. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is submitted that it is within the knowledge of one skilled in the art to affect such feature, structure, or characteristic in connection with other embodiments whether or not explicitly described.

Thus, in one embodiment of the present disclosure, it is proposed to combine two single-step structures described in PCT patent applications PCT/EP2017/057129 and PCT/EP17/057131 into a multi-step structure to enhance the intensity of the resulting focused beam.

Accordingly, in one embodiment of the present disclosure, a nano-projection lens is presented that includes a first step and a second step within a material layer, wherein the second step is stacked on top of the first step with an offset, and wherein a height of the second step and a width of the second step are dimensioned according to an incident angle of incident light on the nano-projection lens and/or a base angle associated with each of the first and second steps and/or a width and a height of the first step so as to focus a nano-projection beam generated from the first step and the second step on a point or area. Thus, the proposed technology enables to obtain a stronger focused nanojet beam compared to the nanojet lenses described in PCT patent applications PCT/EP2017/057129 and PCT/EP 17/057131.

More specifically, the present disclosure relates to a device for generating and directing a nanojet beam from an incident electromagnetic wave on the device. The device is characterized in that it comprises:

-a first layer of dielectric material having a first refractive index value;

-a second layer of dielectric material having a second refractive index value, wherein the second refractive index value is greater than the first refractive index value; and wherein a first boundary between the first and second layers comprises a first single-step structure that is a change of level that is substantially defined according to a first line associated with a tilt angle near a first tilt angle, wherein the device further comprises a dielectric element contained in an embedded layer, the embedded layer being either the first layer or the second layer, the dielectric element having a higher refractive index (n) than a refractive index from the embedded layer₃) And is and

wherein a second boundary between the dielectric element and the embedding layer includes a second single-step shaped structure that is a level variation defined significantly by a second straight line associated with a tilt angle near a second tilt angle, and wherein the second single-step shaped structure is located proximate to the first single-step shaped structure for focusing a nanojet beam produced by each of the first and second single-step shaped structures around a first focal point.

In a preferred embodiment, the first boundary further comprises a third single-step shape structure being a height variation defined sensibly by a third straight line associated with an inclination angle close to the first inclination angle in absolute value, wherein the first and third single-step shape structures face each other and are distant from each other by a first distance, and wherein the second boundary further comprises a fourth single-step shape structure being a level variation defined sensibly by a fourth straight line associated with an inclination angle close to the second inclination angle in absolute value, and wherein the second and fourth single-step shape structures face each other and are distant from each other by a second distance smaller than the first distance, and wherein the fourth single-step shape structure is located in the vicinity of the third single-step shape structure, for focusing the nanojets produced by each of the third and fourth single-step shaped structures around a second focal point.

In a preferred embodiment, the incident electromagnetic wave is a plane wave perpendicular to the device, and wherein the first and second angles are equal to 90 degrees, and wherein the first and second focal points are close to each other.

In a preferred embodiment, the first and third single-step shaped structures have the same first height H₁And wherein the second and fourth single-step shaped structures have the same second height H₂And wherein said second height H₂Has a value of about

Wherein d is₁Is the first distance, and γ₁Is about

Wherein Θ is_B1Is the angle of radiation of the nano-jet beam produced by the first and third single-step shaped structures, wherein

WhereinIs a first critical angle of refraction, n₁Is the first refractive index value, n₂Is the refractive index of the second layer, and wherein

Has a value of about H₂/γ₂Wherein d is₂Is said second distance, γ₂Is about

Wherein Θ is_B2For the angle of radiation of the nano-jet beam generated by the second and fourth single-step shaped structures, wherein

WhereinIs the first critical angle of refraction, n₃Is the refractive index of the dielectric element.

In a preferred embodiment, the incident electromagnetic wave is a plane wave perpendicular to the first and second layers, and wherein the first and second angles are equal to an angle a, and wherein the first and second focal points are close to each other.

In a preferred embodiment, the first and third single-step shaped structures have the same first height H₁And wherein the second and fourth single-step shaped structures have the same second height H₂And wherein said second height H₂Has a value of about

Wherein d is₁Is the first distance, and γ₁Is aboutWherein Θ is_B1Is the angle of radiation of the nano-jet beam produced by the first and third single-step shaped structures, whereinWherein

Is a first critical angle of refraction, n₁Is the first refractive index value, n₂Is the refractive index of the second layer, and wherein

Has a value of about H₂/γ₂Wherein d is₂Is said second distance, γ₂Is aboutWherein Θ is_B2For the angle of radiation of the nano-jet beam generated by the second and fourth single-step shaped structures, wherein

Wherein

Is the first critical angle of refraction, n₃Is the refractive index of the dielectric element.

In a preferred embodiment, the incident electromagnetic wave is at an angle of incidence Θ_iA plane wave that impinges the apparatus, and wherein the first and second angles are equal to 90 degrees, and wherein the first and second focal points are proximate to each other.

In a preferred embodiment, the refractive index of the second layer is equal to the refractive index of the dielectric element, and wherein the first and third single step-shaped structures have the same first height H₁And wherein said second distance d₂Is defined according to the following equation:whereinAnd is

Wherein d is₁Is the first distance, and is,is a first critical angle of refraction, n₁Is the first refractive index value, n₂Is the refractive index of the second layer.

In a preferred embodiment, the refractive index of the dielectric element is less than the refractive index of the layer of the first and second layers that is not the embedding layer.

In a preferred embodiment, said incident electromagnetic wave has a wavelength equal to a value comprised between 390 and 700 nm.

In a preferred embodiment, the dielectric element has a shape belonging to a group of shapes comprising:

-a cubic shape;

-a cylindrical shape;

-a prism shape.

Drawings

The above and other aspects of the present invention will become more apparent by describing in detail exemplary embodiments thereof with reference to the attached drawings, in which:

fig. 1(a), 1(b) and 1(c) show schematic diagrams of three possible variations of a two-step nanojet lens according to different embodiments of the present disclosure, where the following relationship exists between the refractive indices of the dielectric layers/elements: n is₂＝n₃(FIG. 1(a)), or n₃＞n₂(FIG. 1(b)) or n₃＜n₂(FIG. 1 (c));

FIG. 2 illustrates a cross-section of a two-step nanojet lens according to one embodiment of the present disclosure;

FIG. 3 illustrates the evolution of the optical path according to the variation of the angle of radiation of the nanojet beam for single layer nanojet lenses having different heights;

FIGS. 4(a) and 4(b) show the peak power density and Z coordinate of the nanojet hot spot of a stepped system illuminated by a unit amplitude plane wave at λ 550nm with the following parameters, respectively, relative to W₂The relationship of (1): n is₁＝1，n₂＝n₃＝1.49，W₁＝275nm，H₁＝350nm，H₂＝250nm；

Fig. 5(a) and (b) show the power density distribution along the x-axis at λ 550nm for a system with the following parameters: n is₁＝1，n₂＝n₃＝1.49，W₁＝275nm，W₂120nm, wherein, for (a), H₁+H₂620nm, for (b), H₁＝350nm；

FIG. 6 shows for different ratiosPower density distribution of

Fig. 7 shows the power density distribution along the z-axis at λ 550nm for a system with the following optimization parameters: w₁＝275nm，n₁＝1，H₁At 350nm, the system also has the following set of parameters (indicated by the dashed curve): n is₂＝n₃＝1.3，W₂＝150nm，H₂410 nm; the system also has the following set of parameters (indicated by the dash-dot line): n is₂＝1.3，n₃＝1.49，W₂＝140nm，H₂410 nm; the system also has the following set of parameters (represented by the solid curves): n is₂＝n₃＝1.49，W₂＝120nm，H₂＝268nm；

FIGS. 8(a) and (b) show schematic cross-sectional views of a possible embodiment of a two-step nanojet lens placed on a dielectric substrate for which the following relation n of the refractive indices of the layers/elements is satisfied₁＜n₃＜n₂；

FIGS. 9(a) - (d) show schematic diagrams of possible embodiments of a two-step nanojet lens;

fig. 10(a) and (b) show the peak power density of the two-step nano-jet lens and the Z-coordinate of the nano-jet heat spot with respect to W with the following parameters irradiated with a unit-amplitude plane wave at λ 450nm (solid curve), λ 550nm (dotted curve), and λ 650nm (dashed-dotted curve), respectively₂The relationship of (1): n is₁＝1，n₂＝n₃＝1.8，W₁＝550nm，H₁＝400nm，H₂＝250nm；

11(a) - (c) show cross-sectional views of a two-step nanojet lens with non-vertical edges;

FIG. 12 shows the peak power density versus W for a system having the following parameters₂The relationship of (1): n is₁＝1，n₂＝n₃＝1.5，W₁＝275nm，H₁＝200nm；(a)-H₂＝200nm,α＝87.2°；(b)-H₂100nm, α -80.1 °. system consisting of λ -550 nm and Y₀Plane wave illumination of 0;

13(a) - (c) show some cross-sectional views of a two-step nanojet lens according to various embodiments of the present disclosure when an oblique electromagnetic wave strikes the two-step nanojet lens;

FIG. 14 shows a method for a program having a parameter n₁＝1，n₂＝n₃＝1.5，W₁＝275nm，H₁＝200nm，H₂100nm system consisting of λ 550nm, Y ₀0 and Θ_i(a) -Peak Power Density and W of stepped NanoEjection lens irradiated with plane wave of 15 °₂The relationship of (1); (b, c) -Hot Spot position of NanoEjection vs. Width W₂The relationship of (1);

FIG. 15 depicts a method for a program having a parameter n₁＝1，n₂＝n₃＝1.5，W₁＝275nm，H₁＝200nm，H₁100nm system consisting of λ 550nm, Y ₀0 and Θ_i(a) -Peak Power Density and W of 15 ° plane wave irradiated double-step NanoEjection lens₂The relationship of (1); (b, c) -Hot Spot location of NanoEjection relative to X₁The relationship (2) of (c). Reference scheme (with same general)A single step nano-jet lens of height) corresponds to 400.2 nm.

Detailed Description

The present disclosure relates to techniques for focusing electromagnetic waves (and visible light therein), and more particularly, to techniques for near-field focusing and beamforming. The present disclosure provides a specific technique for generating a focused optical nanojet beam by means of a purely dielectric microstructure. In fact, this technique relies on complex electromagnetic phenomena related to the diffraction of light on the edges of stepped dielectric microstructures embedded in a carrier medium having a refractive index lower than that of said microstructures, as already identified and explained in the documents in PCT patent applications PCT/EP2017/057129 and PCT/EP 17/057131.

As depicted in fig. l (a) - (c), the present disclosure relates to modifying the topology of the nanojet lenses detailed in PCT patent applications PCT/EP2017/057129 and PCT/EP17/057131 to more complex shapes with at least two steps in vertical cross-section.

In the following, nano-jet lenses with such a topology are referred to as layered stepped nano-jet lenses, in contrast to the single-layer or single-step nano-jet lenses proposed in PCT patent applications PCT/EP2017/057129 and PCT/EP 17/057131. This switching results in an increase in field strength in the focused spot compared to a single-step nanojet lens. As explained below, the increase in field strength may be about 10% to 25%, depending on the lens size and the material used for each layer. It should be noted that for a lens of about one wavelength in size, this increase in field strength should be observed. For larger sized lenses, greater enhancement can be achieved by multi-parameter optimization of stepped lens topology and materials.

More precisely, in one embodiment of the present disclosure, it is proposed to transform the shape of the nanojet lens in such a way that, when the angle of incidence is equal to 90 °, all nanojet beams originating from the different edges (associated with the different layers) of said stepped microstructure will recombine and contribute to the formation of a single high-intensity nanojet beam, which can be located on the axis of symmetry of the system (as shown in fig. 1(a) - (c)). As schematically shown in fig. l (a) - (c), the desired effect can be achieved using a multi-layered step structure of the focusing element and combining two or more materials having different refractive indices. It should be noted that in another embodiment of the present disclosure, only half of the structure depicted in fig. 1(a) - (c) may be used (i.e., in this case, there is no symmetry in the structure).

It should also be noted that an additional advantage of the proposed two-step nanojet lens is the increased nanojet beam length due to the partial contribution of the multiple nanojet beams associated with the different layers. The characteristics of these nano-jet spray beams are controlled by the parameters of the respective layers, i.e. the refractive index ratio between the lens and the carrier medium, the base angle and the size/shape of the steps.

The proposed disclosure has been numerically validated by full-wave electromagnetic analysis of a two-step nano-jet lens in the form of an infinite rib (rib) (fig. 4-7, 10) or a prism (fig. 12) or cylinder (fig. 14 and 15). For simplicity, we assume that all materials are lossless and non-dispersive. The analysis has revealed that diffraction of plane waves on a two-step nanojet lens with a step (layer) material having a refractive index higher than that of the carrier medium can lead to the formation of a stronger focused light beam (so-called nanojet). The intensity, size and shape of each individual beam can be controlled by varying the step size, shape (e.g. base angle), material and position of the second step (i.e. we should use oblique incidence for an asymmetric system).

The general topology of the double stepped nano-jet lens is shown in fig. 2. The cross-sectional view may correspond to an embedding at the refractive index n₁＜n₂Double-layer ribs, cubes or cylinders of arbitrary cross-section in a homogeneous dielectric carrier medium. Hereafter, we assume that we have a refractive index n₃The material and dimensions of the second layer can be arbitrarily selected and optimized according to the parameters of the first layer in order to achieve the maximum field strength enhancement of the beam generated in the near zone due to the recombination of the nano-jet beam associated with the edge of the stepped nano-jet lens. The second layerThe effect of the size and refractive index of (a) on the intensity and length of the nano-jets produced will be described below.

In the following, we consider a structure with a vertical edge parallel to the z-axis and a top/bottom surface parallel to the xy-plane, where the xy-plane corresponds to a bottom angle α of 90 degrees. However, some prism structures (with arbitrary base angles) may also be used. The variation of the base angle value provides an additional degree of freedom in controlling the radiation direction of the nano-jet beam. Such variations are described later herein.

In a first approximation, the focal length of the stepped nanojet lens can be characterized as a function of the size (width or radius) and the refractive index ratio of the medium inside and outside the microstructure. In the following, we propose a set of equations to estimate the optimal dimensions of the layer for maximum enhancement of the field strength of the nano-jets produced.

It should be noted that the beam shape and hot spot location are sensitive to the size/form of the complex stepped lens. This effect is explained by the interference of the nanojet beam associated with the bottom edge of the first step (first layer) and the nanojet beam associated with the bottom edge of the second step (second layer) of the system. In this case, the two beams are input into the total generated beam.

The total response of an element with dimensions several wavelengths larger than the lens in the medium may represent the interaction between nanojet and fresnel diffraction phenomena.

As demonstrated in PCT patent applications PCT/EP2017/057129 and PCT/EP17/057131, the beam forming phenomenon is only related to the edges of the system, and the nano-jet beam radiation angle is defined by snell's law.

It can therefore be determined as a function of the ratio between the refractive indices of the carrier medium and the step material and the base angle of the element. Fig. 2 shows a structure with vertical edges (i.e., base angle equal to 90 °).

For n₂＝n₃Can be determined using the following approximate formula_B1And Θ_B2：

WhereinIs the critical angle of refraction.

Thus, the focal length of the step can be estimated by the following equation:

F_j＝W_jγ

where j is l,2,and W_jIs the half width (radius) of a single element or step.

In order to increase the intensity in the nano-jet hot spot, it is proposed to adjust the focal length of the constituent elements as follows:

F₁＝H₁+F₂

here H_jIs the height of the corresponding step (see fig. 2). Assuming that the maximum intensity of the nano-jet hot spot corresponds to an element having a total height equal to the focal length, we can obtain the formula for the size of the top layer:

H₂≈W₁γ-H₁

it should be noted that if the materials of the steps are different, we should use this approximation formula only for a preliminary estimate that considers the size of the second step, taking into account the fact that for n₃＞n₂＞n₁，Θ_B1＜Θ_B2And for n₃＜n₂(n₂＞n₁And n is₃＞n₁)，Θ_B1＞Θ_B2. This means that for proper adjustment, the size of the top step should be corrected: for n₃＞n₂The total width (radius) of the top step should be increased for n₃＜n₂SaidThe total width (radius) of the top step should be less than optimal.

Note the intensity of the generated nanojet beam versus the height of the underlayer (H)₁) Is very sensitive. The maximum of the intensity occurs at the optical path difference between the nanojet beam and the wave propagating in the first step, which is formed by

Given, where k is an integer. For the proposed system, the optical path difference in point a (see fig. 2) is equal to

The values for four different values H are shown in FIG. 3₁The relationship between the optical path difference and the radiation angle of the nano-jet beam (reference curve 301 represents the relationship for the height H₁1000nm, the relation between the radiation angle of the nano jet beam and the distance of the focusing point; reference curve 302 shows the curve for the height H₁The relation between the radiation angle of the nanometer jet beam and the distance of the focusing point is 500 nm; reference curve 303 shows the curve for the height H₁300nm, the relation between the radiation angle of the nano jet beam and the distance of the focusing point; reference curve 304 shows the curve for the height H₁100nm, nano-jet beam radiation angle versus focal spot distance). The vertical dotted line indicates the ratio to the refractive index

The angle of radiation of the nano-jet beam of some specific values. In FIG. 3, we have n ₁1, and β₁＝1.15,β₂＝1.3,β₃1.49 and β₄1.8. It can be seen that the equation cannot be fully satisfied for a nano-jet lens having a size of about one wavelength

Analysis of Δ in FIG. 3 shows that for small values of β, the optical path difference is minimal (Δ → 0). this indicates that with small differences between the refractive indices of the lens and the carrier medium material, a two-step nanojet can be expectedThe maximum relative field strength in the focal spot of the lens increases.

Let us present the data obtained using the electromagnetic field simulation software package CST MICROWAVE study. The lens is assumed to be infinite (ribbed) along the y-axis and is illuminated by a linearly polarized plane wave E ═ 0,1, 0. The material of the steps of the nano-spray lens may be the same (n)₂＝n₃). To illustrate the effect of the nanojet lens topology on the parameters of the nanojet beam, we consider a system with the following parameters: n is₁＝1，n₂＝n₃1.49 (and γ 2.25), W₁＝275nm，H₁350nm (the size of the underlayer is arbitrary). The proposed simulation is performed for 2D problems.

By using the previously established formula H₂＝W₁γ-H₁And

the optimum size of the second step can be obtained: w₂107.5nm and H₂＝268.8nm。

The maximum power density is shown in fig. 4(a) for the half width (radius) W of the second layer₂The dependence of (c). The black horizontal dashed line shows the reference scheme for a single step microlens. It can be seen that the maximum value of the power distribution can be obtained if the parameters of the second layer are close to optimal. For having W₂Maximum power density was observed for 120nm double step nano-spray lenses. By varying the size of the second layer, we can vary the location of the nanojet hot spot. FIG. 4(b) shows Z coordinate pairs W for NJ hot spots₂The dependence of (c). The black horizontal dashed line shows the position of the top of the system.

It can be seen that in all cases the beam width at half power (BWHP) is about 200nm, which is below the diffraction limit, which predicts that the smallest possible focal spot size in the carrier medium is about half a wavelength. As expected, for the best parameters (W)₂120nm) with a focal spot BWHP of about 170nm, which is partly due to the improved focusing of the double-step nano-jet lensThe focal power and the shift of the focal spot inside the lens with a higher refractive index value than the refractive index value of the carrier medium.

As we can see in fig. 5(a) - (b), the power density distribution is very sensitive to the height of the steps of the nano-jet lens. For an optimum height H of the approach step₁350nm and H₂Peak power density was observed at 270 nm. As a result, we can conclude that for H_1,2< lambda/4 case, equation H for two nanometer jet hot spot adjustment₂＝W₁γ-H₁And

and does not work. In this case, we observed two nano-jet hot spots.

To evaluate the effect of the material of the layer on the nanojet properties, modified ratios have been used

To simulate a power density distribution (see fig. 6) it can be seen that the peak of the power density increases with β, but in this case the beam length decreases in fig. 6, the reference curve 601 corresponds to β ═ 1.8 (W)₂＝100nm，H₂164nm), reference curve 602 corresponds to β ═ 1.49 (W)₂＝120nm，H₂268nm), reference curve 603 corresponds to β -1.3 (W)₂＝150nm，H₂410nm), reference curve 604 corresponds to β ═ 1.15 (W)₂＝l82nm，H₂＝691nm)。

Comparison of the curves for different combinations of refractive indices for the layers in fig. 7 shows that the combination of different materials qualitatively changes the power distribution along the z-axis. That is, we can observe that when n is₃＞n₂We can obtain a higher peak (up to 25%) of the power density distribution at a smaller width of the second layer. But in this case the length of the nanojet beam will be almost equal to that for a beam having n₃＝n₂1.49 and optimized smaller size stepped nano-jet lens.

One advantage of the proposed technique is that it enables an additional degree of freedom for steering the nanojet beam in the near zone, and in particular, it enhances the field strength in the focal spot of the double-stepped nanojet lens. These "degrees of freedom" are provided by adding additional steps that create independent nano-jet beams that together contribute to the formation of a high intensity secondary jet beam.

In this case, the characteristics of the secondary nanojet beam (i.e., field strength, beam width, and orientation) are determined by the partial contributions of the multiple independent beams associated with the different steps.

In the case of a symmetrical structure (such as the one shown in fig. l (b) - (c)), the secondary beam is on the axis of the lens, however, its orientation can be modified by breaking the symmetry of the structure (e.g. changing the bottom angle of the selected step).

Some possible implementations of stepped nano-jet lenses are shown in fig. 8(a) - (b) and fig. 9(a) - (d). Such nano-jet lenses may be embedded in a carrier medium or placed on a dielectric substrate acting as a support layer. The material of the substrate may be arbitrarily selected. It may be the same or different material as the wider step. Furthermore, such microstructures can be realized by standard photolithographic techniques.

It should be noted that such a structure may be illuminated from the top or the bottom, however, in order to provide the desired focusing function, the material properties and dimensions of the step have to be adapted accordingly (see fig. 7).

To illustrate the effect of the stepped topology on the parameters of the nanojet beam in the case of a lens with a larger size, we consider a system with the following parameters: n is₁＝1，n₂＝n₃＝1.8，W₁＝550nm，H ₁400 nm. The proposed simulation is performed for 2D problems. The nanojet lens is assumed to be infinite (rib-type) along the y-axis and illuminated by a linearly polarized plane wave E ═ 0,1, 0.

Using equation H₂＝W₁γ-H₁And

we obtain the optimum dimension of the second step as W₂336.2nm and H₂689.8 nm. FIGS. 10(a) and (b) show the maximum power density and Z coordinate of the nanojet hot spot versus the half width (radius) W of the second layer at 3 different wavelengths₂The dependence of (c).

Numerical simulations show that, in this case, equation H₂＝W₁γ-H₁Andand does not work. First, to locate the focal point outside the nanolject lens, we should use the much lower height of the second step. In the case shown, it is only H ₂250 nm. Furthermore, for smaller size W₂Maximum power density was observed. It must also be noted that this system is very sensitive to the wavelength of the electromagnetic waves. But it can be seen that the same height single step system (W)₁＝W₂500nm) gives a stronger overall response of the system (for W in the case of λ 450nm, for a two-step topology)₂200nm, up to 200%).

Let us assume α to be the base angle of a single-step/layer system (see FIG. 11 (a)). 11(a) - (c) show the general topology of single-step and double-step nanojet lenses₁＜n₂A prismatic system in a homogeneous dielectric carrier medium.

It has been found that for systems with non-vertical edges, the nanojet beam radiation angle can be determined using the following approximate formula:

wherein Θ'_TIRIs the critical angle of refraction from the non-vertical edge. To give Θ'_TIRWe must take into account the variation in the position of the edge. As a result, the nano-jet beam radiation angle can be estimated as:

to increase the intensity in the nano-jet hot spot we can adjust the nano-jet beam associated with the steps of the system. As a result, the formula for the radius/half width of the bottom edge of the top layer can be written as:

W₂≈W₁-H₁tanΘ_B1

we have assumed that the bottom corners of the top layer coincide with the bottom corners of the bottom layer. We should mention that in order to estimate the focal length of the step, we should consider the electromagnetic wave refraction phenomenon at the top surface of the layer. For this geometry, equation F used previously_j＝W_jIn addition, all given formulas are only for α > 90 ° - Θ_B1Is effective.

For a stepped system with stepped materials of different refractive index and different base angle components (see fig. 11(c)), we should use the same principles as discussed previously.

The proposed concept was numerically verified by full-wave electromagnetic analysis of a stepped nano-jet lens in the form of a prism, the cross-section of which is shown in fig. 11 (b). The proposed simulation is performed for 3D problems. The lens is illuminated by a linearly polarized plane wave E ═ {0,1,0 }. The steps of the nano-jet lens are made of the same material (n)₂＝n₃) The base angles of the components are the same. To illustrate the effect of the modified nanojet lens topology on the parameters of nanojets, we consider a system with the following parameters: n is₁＝1，n₂＝n₃＝1.5，W₁275nm and H₁＝200nm。

By using the formula W₂≈W₁-H₁tanΘ_B1We can determine the optimum for the second step with a base angle α of 87.2Radius is W₂179.6 nm. FIG. 12(a) shows the peak of the power density versus radius W₂The dependence of (c). Selected height (H) for second step₂200nm) we observed nano-jet hot spots inside the structure/system. We can see that the maximum power density corresponds to W₂The black dashed curve in fig. 12(a) corresponds to the reference scheme, i.e. non-vertical edge (α ═ 87.2 °) and W with the same total height (400nm) for a stepped nano-jet lens 205nm₁275nm single step nano-jet lens the peak power density versus W for the system with the lower base angle (α ═ 80.1 °) is shown in fig. 12(b)₂The dependence of (c). To obtain a focal spot outside the system, we use H ₂100 nm. It is obtained that the optimum radius of the second step for such a system corresponds to W₂164 nm. For having W₂Maximum power density can be observed for the 180nm double step nano-spray lens the dashed curve in fig. 12(b) corresponds to the single step reference scheme (α 80.1 °, W₁＝275nm，H ₁300 nm). We can therefore conclude that by choosing the optimal parameters for the layers of a two-step nano-jet lens with non-vertical edges, we can get a stronger overall response of the system.

Let us now consider the effect of the angle of incidence of the plane wave on the properties of the produced nanojet beam.

We assume that Θ_iIs the incident angle of the electromagnetic wave.

For oblique incidence of plane waves at refractive index n₂We should consider that the radiation angles of the opposite edges of the system are not equal (see fig. 13 (a)). As a result, for a single step element, we can have

It has to be noted that for a nanojet lens with a focused spot (hot spot of the total produced nanojet) within the system, the angle of the total nanojet deflection can be determined as:

therefore, to determine the focal length (F) of a nanojet lens with a hot spot outside the system₁) We should consider additional electromagnetic wave refraction at the top edge of the system. As a result, the angle at which we obtain the total nano-jet deflection will be approximately equal to the incident angle Θ_i。

To increase the intensity in the nano-jet hot spot, we should adjust the focal lengths of the constituent elements of the stepped system as it was done before considering the difference between the nano-jet beam radiation angles at the opposite edges of the system. As a result, the two-step system (fig. 13(b)) will not be symmetrical. For a single material two-step nano-jet lens we have:

L₁＝H₁tanΘ′_B1

L₂＝H₁tanΘ″_B1

in this case, the additional nano-jets associated with the second step will be superimposed with the nano-jets for the first step.

Thus, the parameters of the second step may be determined using the following approximate formula:

X₁＝W₁-L₂-W₂

wherein X₁Is the offset between the axes of symmetry of the steps. Proposed formula for normal incidence angle and L₂＜L₁Is effective. At a negative angle theta_iIn case of (2), we get L₂＞L₁。

For nano-jet beam radiation in the case of plane waves obliquely incident on elements with non-vertical edges, we should use only the approximation equation for the modification of the radiation angle:

it should be noted that if the material of the step is different (fig. 13(c)), it is considered that for n₃＞n₂＞n₁，Θ′_B1＜Θ′_B2And Θ_B1＜Θ″_B2And for n₃＜n₂(n_2,3＞n₁)，Θ′_B1＞Θ′_B2And Θ_B1＞Θ″_B2We should use these approximation formulas only for a preliminary estimate of the second step size. This means that, for proper adjustment, the size of the top step should be corrected: for n₃＞n₂In terms of n, the total width (radius) of the top step should be increased₃＜n₂In other words, the total width (radius) of the top step should be less than the optimum width. Furthermore, we should consider the refraction phenomena associated with the top edges of the first and second steps (first and second layers), which would result in a significant reduction of the focal length of the proposed system.

Therefore, in order to determine the optimal geometry of a dual-step nanojet lens with different step materials, we should consider that the angle of nanojet deflection caused by the two steps is approximately equal to the angle of incidence Θ_i. Therefore, in this case, in order to obtain a proper adjustment of the focus point, it is important to determine the offset between the symmetry axes of the steps, since

X₁＝H₁tanΘ_i，

And also the size of the second layer.

Through full-wave electromagnetic analysis of cylindrical stepped nano-jet lensThe proposed concept was numerically verified and the cross-section of the lens is shown in fig. 13 (b). The proposed simulation is performed for 3D problems. The lens is illuminated by a linearly polarized plane wave E ═ {0,1,0 }. The material of the step of the nano-spray lens is the same (n)₃＝n₂). To illustrate the effect of the nanojet lens topology on the parameters of the nanojet beam in case of oblique plane wave incidence, we consider a system with the following parameters: n is₁＝1，n₂＝n₃＝1.5，W₁275nm and H₁＝200nm。

Using equation L₁＝H₁tanΘ′_B1，L₂＝H₁tanΘ″_B1，

And X₁＝W₁-L₂-W₂We can determine the optimum radius of the second step, which is W₂183.7nm and an offset between the symmetry axes of the layers of X₁＝31.7nm。

FIG. 14(a) shows the offset X for a fixed offset₁W at 30nm₂Six different values, peak power density distribution along the z-axis. To get a focal spot outside the system, we use H ₂100 nm. For W₂Maximum power density was observed for a stepped nano-spray lens of 200 nm. We should mention that a change in the size of the additional layer results in a shift in the position of the nanojet hot spot (see fig. 14(b) and 14 (c)). The dashed curves in fig. 14(a) - (c) correspond to the reference solution, i.e. a single step nano-jet lens with the same total height. The maximum power density is given in FIG. 15(a) for the power having W₂Parameter X of the second layer of 200nm₁The dependence of (c). The horizontal dashed line shows the reference scheme for a single step nano-jet lens. It can be seen that if the size and position of the second layer is close to the optimum for oblique plane wave incidence, we can get the maximum value of the power distribution.