Device for deviating and focusing light

Abstract

A device includes a host medium having a first refractive index value; a first layer comprising a first dielectric material having a second refractive index value, wherein the second refractive index value is greater than the first refractive index value, and wherein the first layer comprises a first step structure at a boundary between the first layer and the host medium; and a second layer comprising a second dielectric material and comprising a second step structure, the second layer having a third refractive index value higher than the first refractive index value of the host medium, wherein the second step structure is stacked on the first step structure, and, in response to an incident electromagnetic wave reaching the device, a first nanojet beam generated by the first step structure and a second nanojet beam generated by the second step structure are combined and focused around a first focusing point.

Claims

1. A device for generating and guiding nanojet beams from an incident electromagnetic wave that is incident on the device, the device comprising: a host medium having a first refractive index value; a first layer comprising a first dielectric material having a second refractive index value, wherein the second refractive index value is greater than the first refractive index value, and wherein the first layer comprises a first step structure and a third step structure at a boundary between the first layer and the host medium, wherein the first step structure and the third step structure face each other and are separated from each other by a first distance; and a second layer comprising a second dielectric material and comprising a second step structure and a fourth step structure at a boundary between the second layer and the host medium, the second layer having a third refractive index value higher than the first refractive index value of the host medium, wherein the second step structure and the fourth step structure face each other and are separated from each other by a second distance that is smaller than the first distance, wherein the second step structure is stacked on the first step structure and wherein the fourth step structure is stacked on the third step structure, wherein each of the first step structure, the second step structure, the third step structure, and the fourth step structure is configured such that, in response to the incident electromagnetic wave reaching the device in a direction from the first layer toward the second layer, nanojet beams originating from edges of the first step structure, the second step structure, the third step structure, and the fourth step structure are combined at a focal spot to form a single high-intensity nanojet beam.

2. The device of claim 1, wherein: the incident electromagnetic wave is a plane wave, the device is configured such that the plane wave incident on the device is normal to the first layer, the first step structure includes a first edge, a first bottom surface, and a first base angle between the first edge and the first bottom surface, the second step structure includes a second edge, a second bottom surface, and a second base angle between the second edge and the second bottom surface, and the first base angle and second base angle are equal to 90 degrees.

3. The device of claim 2, wherein the first step structure further comprises a first top surface and the second step structure further comprises a second top surface; wherein the first bottom surface of the first step structure and the first top surface of the first step structure are parallel to a first plane, and the first edge of the first step structure is parallel to an axis perpendicular to the first plane; and wherein the second bottom surface of the second step structure and the second top surface of the second step structure are parallel to the first plane; and wherein the second edge of the second step structure is parallel to the axis.

4. The device of claim 1, wherein the first step structure and the third step structures have a same first height H.sub.1, wherein the second step structure and the fourth step structure have a same second height H.sub.2, wherein the second height H.sub.2 has a value around $(\frac{d_{1}}{2}) .Math._{1} - H_{1},$ wherein d.sub.1 is the first distance, and .sub.1 is around $\frac{1}{\tan .Math._{B 1} .Math.},$ where .sub.B1 is a nanojet beam radiation angle generated by the first step structure and the third step structure, wherein $_{B 1} \frac{9 0^{} -_{{TIR}_{1}}}{2}$ where $_{TIR 1} = \sin^{- 1} (\frac{n_{1}}{n_{2}})$ is a first critical angle of refraction, n.sub.1 is the first refractive index value of the host medium, n.sub.2 is the second refractive index value of the first layer, wherein $(\frac{d_{2}}{2})$ has a value around H.sub.2/.sub.2, wherein d.sub.2 is the second distance, .sub.2 is around $\frac{1}{\tan .Math._{B 2} .Math.},$ where .sub.B2 is a nanojet beam radiation angle generated by the second step structure and the fourth step structure, and wherein $_{B 2} \frac{9 0^{} -_{{TIR}_{2}}}{2}$ where $_{TIR 2} = \sin^{- 1} (\frac{n_{1}}{n_{3}})$ is a second critical angle of refraction, and n.sub.3 is the third refractive index value of the second layer.

5. The device of claim 1, wherein: the device is configured such that the incident electromagnetic wave is a plane wave that is normal to the first layer and the second layer, the first step structure includes a first edge, a first bottom surface, and a first base angle between the first edge and the first bottom surface, the second step structure includes a second edge, a second bottom surface, and a second base angle between the second edge and the second bottom surface, and the first base angle and the second base angle are equal to an angle .

6. The device of claim 5, wherein the first step structure and the third step structure have a same first height H.sub.1, wherein the second step structure and the fourth step structure have a same second height H.sub.2, wherein the second height H.sub.2 has a value around $(\frac{d_{1}}{2}) .Math._{1} - H_{1},$ wherein d.sub.1 is the first distance, and .sub.1 is around $\frac{1}{\tan .Math._{B 1} .Math.},$ where .sub.B1 is a nanojet beam radiation angle generated by the first step structure and the third step structure, wherein $_{B 1} \frac{9 0^{} -_{{TIR}_{1}} +}{2}$ where $_{TIR 1} = \sin^{- 1} (\frac{n_{1}}{n_{2}})$ is a first critical angle of refraction, n.sub.1 is the first refractive index value of the host medium, n.sub.2 is the second refractive index value of the first layer, wherein $(\frac{d_{2}}{2})$ has a value around H.sub.2/.sub.2, wherein d.sub.2 is the second distance, .sub.2 is around $\frac{1}{\tan .Math._{B 2} .Math.},$ where .sub.B2 is a nanojet beam radiation angle generated by the second step structure and the fourth step structure, and wherein $_{B 2} \frac{9 0^{} -_{TIR} 2 +}{2}$ where $_{TIR 2} = \sin^{- 1} (\frac{n_{1}}{n_{3}})$ is a second critical angle of refraction, and n.sub.3 is the third refractive index value of the second layer.

7. The device of claim 5, wherein the first step structure further comprises a first top surface and the second step structure further comprises a second top surface; wherein the first bottom surface of the first step structure and the first top surface of the first step structure are parallel to a first plane, and the first edge of the first step structure is non-perpendicular to the first top surface and the first bottom surface; and wherein the second bottom surface of the second step structure and the second top surface of the second step structure are parallel to the first plane, and the second edge of the second step structure is non-perpendicular to the second top surface and the second bottom surface.

8. The device of claim 1, wherein: the device is configured such that the incident electromagnetic wave is a plane wave that hits the device at an angle of incidence .sub.i, and the first step structure includes a first edge, a first bottom surface, and a first base angle between the first edge and the first bottom surface, the second step structure includes a second edge, a second bottom surface, and a second base angle between the second edge and the second bottom surface, and the first base angle and the second base angle are equal to 90 degrees.

9. The device of claim 8, wherein the second refractive index value of the first layer is equal to the third refractive index value of the second layer, wherein the first step structure and the third step structure have a same first height H.sub.1, and wherein the second distance d.sub.2 is described according to $(\frac{d_{2}}{2}) = - (\frac{d_{1}}{2}) - H_{1} .Math. \frac{\tan_{B 1}^{} + \tan_{B 1}^{}}{2},$ wherein $_{B 1}^{} \frac{9 0^{} -_{TIR}}{2} + \frac{_{i}}{2}, and$ $_{B 1}^{} \frac{9 0^{} -_{TIR}}{2} - \frac{_{i}}{2},$ where d.sub.1 is the first distance, and $_{TIR} = \sin^{- 1} (\frac{n_{1}}{n_{2}})$ is a critical angle of refraction, n.sub.1 is the first refractive index value of the host medium, and n.sub.2 is the second refractive index value of the first layer.

10. The device of claim 1, wherein the third refractive index value of the second layer is smaller than the second refractive index value of the first layer.

11. The device of claim 1, wherein the incident electromagnetic wave has a wavelength equal to a value between 390 and 700 nm.

12. The device of claim 1, wherein the second layer has a shape that is selected from the group consisting of a cuboid shape, a cylinder shape, and a prism shape.

13. The device of claim 1, wherein the host medium comprises a homogenous dielectric host medium.

14. The device of claim 1, wherein the host medium functions as an embedding layer in which at least the second layer comprising the second step structure is embedded.

15. The device of claim 1, wherein the first step structure has a first height, wherein the second step structure has a second height, and wherein the second height is based on the first height, the first refractive index value, and the second refractive index value.

16. The device of claim 1, wherein the first step structure and the third step structure have a first height, wherein the second step structure and the fourth step structure have a second height, and wherein the second height is based on the first height, the first refractive index value, and the second refractive index value.

17. The device of claim 1, wherein the device receives the incident electromagnetic wave as a plane wave normal to the first layer.

18. The device of claim 1, wherein the focal spot is beyond the second layer.

19. The device of claim 1, wherein the first step structure and the third step structure have a same first height, wherein the second step structure and the fourth step structure have a same second height; wherein the second height is based on the first height, the first distance, and a nanojet beam radiation angle generated by the first step structure and the third step structure; and wherein the second distance is based on the second height and a nanojet beam radiation angle generated by the second step structure and the fourth step structure.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The above and other aspects of the invention will become more apparent by the following detailed description of exemplary embodiments thereof with reference to the attached drawings in which:

(2) FIGS. 1(a), 1(b) and 1(c) present schematic views of three possible variant of double-steps nanojets lens according to different embodiment of the disclosure for which the following relationships between refractive indexes of dielectric layers/elements occur: n.sub.2=n.sub.3 (FIG. 1(a)), or n.sub.3>n.sub.2 (FIG. 1(b)) or n.sub.3<n.sub.2 (FIG. 1(c));

(3) FIG. 2 presents a cross-section of a double-steps nanojets lens according to one embodiment of the disclosure;

(4) FIG. 3 presents the evolution of optical path according to the variation of nanojets beam radiation angles for a single layer nanojets lens with different heights;

(5) FIGS. 4(a) and 4(b) present respectively a peak power density and Z-coordinate of nanojets hot spot versus W.sub.2 for a step-like systems with such parameters: n.sub.1=1, n.sub.2=n.sub.3=1.49, W.sub.1=275 nm, H.sub.1=350 nm, H.sub.2=250 nm, that is illuminated by a unit-amplitude plane wave at =550 nm;

(6) FIGS. 5(a) and (b) present the power density distribution along x-axis at =550 nm for the systems with such parameters: n.sub.1=1, n.sub.2=n.sub.3=1.49, W.sub.1=275 nm, W.sub.2=120 nm, with H.sub.1+H.sub.2=620 nm (a) and with H.sub.1=350 nm (b);

(7) FIG. 6 presents the power density distributions for different values of the ratio

(8) 0 $= \frac{n_{2}}{n_{1}};$

(9) FIG. 7 presents the power density distribution along z-axis at =550 nm for the systems with optimized parameters: W.sub.1=275 nm, n.sub.1=1, H.sub.1=350 nm, with the following set of parameters (represented by the dashed curve): n.sub.2=n.sub.3=1.3, W.sub.2=150 nm, H.sub.2=410 nm; with the following set of parameters (represented by the dash-dot curve): n.sub.2=1.3, n.sub.3=1.49, W.sub.2=140 nm, H.sub.2=410 nm; with the following set of parameters (represented by the solid curve): n.sub.2=n.sub.3=1.49, W.sub.2=120 nm, H.sub.2=268 nm;

(10) FIGS. 8(a) and (b) present a schematic cross-section view of possible embodiments of double steps nanojets lenses, placed on a dielectric substrate for which the following relationship for refractive indexes of layers/elements is fulfilled n.sub.1<n.sub.3<n.sub.2;

(11) FIGS. 9(a)-(d) present schematic views of possible embodiments of double-steps nanojets lenses;

(12) FIGS. 10 (a) and (b) present respectively the peak power density and Z-coordinate of nanojets hot spot versus W.sub.2 for a double-steps nanojets lens with such parameters: n.sub.1=1,n.sub.2=n.sub.3=1.8, W.sub.1=550 nm, H.sub.1=400 nm, H.sub.2=250 nm, illuminated by a unit-amplitude plane wave at =450 nm (solid curve), =550 nm (dashed curve) and =650 nm (dash-dot curve);

(13) FIG. 11(a)-(c) presents cross-section views of a double steps nanojets lens with nonvertical edges;

(14) FIG. 12 presents the peak power density versus W.sub.2 for a system with parameters: n.sub.1=1, n.sub.2=n.sub.3=1.5, W.sub.1=275 nm, H.sub.1=200 nm; (a)H.sub.2=200 nm, =87.2; (b) H.sub.2=100 nm, =80.1. System is illuminated by a plane wave with =550 nm and Y.sub.0=0;

(15) FIGS. 13(a)-(c) present some cross-section views of a double-steps nanojets lens according to different embodiments of the disclosure, when an oblique electromagnetic wave hits the double-steps nanojets lens;

(16) FIG. 14 presents (a)Peak power density versus W.sub.2; (b,c)Position of nanojets' hot spots versus the width W.sub.2 for a step-like nanojets lens illuminated by a plane wave at =550 nm, Y.sub.0=0 and .sub.i=15 for the systems with such parameters: n.sub.1=1, n.sub.2=n.sub.3=1.5, W.sub.1=275 nm, H.sub.1=200 nm, H.sub.2=100 nm;

(17) FIG. 15 describes (a) peak power density; (b,c) Position of nanojets' hot spots versus X.sub.1 for a double steps nanojets lens illuminated by a plane wave at =550 nm, Y.sub.0=0 and .sub.i=15 for the systems with such parameters: n.sub.1=1, n.sub.2=n.sub.3=1.5, W.sub.1=275 nm, H.sub.1=200 nm, H.sub.1=100 nm. Z-coordinate of the reference solution (single step nanojets lens of the same total height) corresponds to 400.2 nm.

DETAILED DESCRIPTION

(18) The present disclosure relates to techniques for focusing electromagnetic waves (and among them visible light) and, more particularly, techniques for near-field focusing and beam forming. The present disclosure provides a specific technique for generating condensed optical nanojets beams with the aid of purely dielectric microstructures. Indeed, such technique relies on the complex electromagnetic phenomenon associated with the light diffraction on edges of a step like dielectric microstructures embedded in a host medium with refractive index lower than that of the microstructure, as already identified and explained in document in PCT patent applications PCT/EP2017/057129 and PCT/EP17/057131.

(19) As depicted in FIGS. 1(a)-(c), the disclosure is related to a modification of the topology of the nanojets lenses detailed in PCT patent applications PCT/EP2017/057129 and PCT/EP17/057131 to a more complex shape having at least two steps in a vertical cross-section.

(20) Hereafter, nanojets lenses having such a topology are referred as a layered step-like nanojets lenses, in contrast to the single-layer or single-step nanojets lenses presented in PCT patent applications PCT/EP2017/057129 and PCT/EP17/057131. This transformation results in the increase of the field intensity in the focal spot compared to the single step nanojets lenses. As explained in the following, depending on the lens dimensions and materials used for each layer, the increase of the field intensity can be about 10% to 25%. It should be noted that such increase of the field intensity was observed for the lenses with dimensions of about one wavelength. A larger enhancement can be obtained through a multi-parameter optimization of the step-like lens topology and materials for the lenses with bigger dimensions.

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

(22) It should also be noted that an additional advantage of the proposed double-steps nanojets lens is the increase of the nanojets beam length as the result of the partial contributions of multiple nanojets beams, associated with different layers. The characteristics of these nanojets beams are controlled by the parameters of the corresponding layers (i.e. refractive index ratios between the lens and host medium, base angle and size/shape of the step).

(23) The proposed disclosure has been validated numerically via full-wave electromagnetic analysis of a double-steps nanojets lens in a form of an infinite rib (FIGS. 4-7,10), or a prism (FIG. 12) or a cylinder (FIGS. 14 and 15). For simplicity, we assume that all the materials are lossless and non-dispersive. The analysis has revealed that diffraction of a plane wave on a double-steps nanojets lens, with the materials of the steps (layers) having a higher refractive index than that of the host medium, can result in the formation of more intensive condensed optical beams (so-called nanojets). The intensity, dimensions and shape of each individual beam can be controlled by the variation of the step size, shape (e.g. base angle), material and position of the second step (i.e. the oblique incidence for which we should take nonsymmetrical system).

(24) The general topology of the double-steps-like nanojets lens is illustrated in FIG. 2. This cross-section view may correspond to a double-layer rib, cuboid, or cylinder with an arbitrary cross-section embedded in a homogeneous dielectric host media with a refractive index n.sub.1<n.sub.2. Hereafter, we assume that the material and size of the second layer with refractive index n.sub.3 can be selected arbitrary and optimized depending on the parameters of first layer in order to reach a maximum field intensity enhancement for the beam generated in the near zone as a result of recombination of the nanojets beams associated with the edges of the step-like nanojets lens. The effect of the size and refractive index of the second layer on the intensity and length of the generated nanojets is described in the following.

(25) Hereafter, we consider the structures with vertical edges parallel to z-axis and top/bottom surface parallel to xy-plane, which corresponds to a base angle =90 degrees. However, some prismatic structures (with arbitrary base angles) can also be used. Variation of the base angle value provides additional degree of freedom in the control of the nanojets beam radiation direction. Such variant is described later in this document.

(26) In a first approximation, the focal length of the step-like nanojets lens can be characterized as the function of the size (width or radius) and index ratio of the media inside and outside the microstructure. In the following, we present a set of equations to estimate the optimal dimensions of the layers for maximal enhancement of the field intensity of generated nanojets.

(27) It should be noted that the beam shape and hot spot position are sensitive to the sizes/form of the complex step-like lens. This effect is explained by the interference of the nanojets beams, associated with the bottom edge of the first step (first layer) of the system, and the nanojets beam, associated with the bottom edge of the second step (second layer). In this case, the two beams make an input into the total generated beam.

(28) The total response of the elements with dimensions larger than a few wavelengths in the medium the lens represents the interplay between the nanojets and Fresnel diffraction phenomenon.

(29) As it was demonstrated in PCT patent applications PCT/EP2017/057129 and PCT/EP17/057131, the beam-forming phenomenon is associated solely with the edge of the system and the nanojets beam radiation angle is defined by the Snell's low.

(30) So, it can be determined as a function of the ratio between the refraction indexes of the host media and material of the steps, and the base angle of the element. The FIG. 2 presents a structure with vertical edges (i.e. with base angles being equal to 90).

(31) For a structure where n.sub.2=n.sub.3, the nanojets beam radiation angles .sub.B1 and .sub.B2 can be determined using the approximate formula:

(32) $_{B 1} =_{B 2} =_{B} \frac{9 0^{} -_{TIR}}{2},$
where

(33) $_{TIR} = \sin^{- 1} (\frac{n_{1}}{n_{2}})$
is the critical angle of refraction. Therefore, the focal length of the steps can be estimated via the following equation:
F.sub.j=W.sub.j
where j=1,2,

(34) $= \frac{1}{\tan_{B}},$
and W.sub.j is the half-width (radius) of a single element or step.

(35) In order to increase the intensity in the nanojets hot spot, it is proposed to adjust the focal lengths of the constitutive elements as follows:
F.sub.1=H.sub.1+F.sub.2

(36) Here H.sub.j is the height of the corresponding step (see in FIG. 2). Assuming that the maximal intensity of the nanojets hot spot corresponds to the elements with the total height equal to the focal length, we can get the formulas for the sizes of the top layer:

(37) $H_{2} W_{1} - H_{1} w_{2} \frac{H_{2}}{}$

(38) It is necessary to note, that if materials of the steps are different, we should use this approximate formulas just for preliminary estimation of the second step size taking into account that for n.sub.3>n.sub.2>n.sub.1, .sub.B1<.sub.B2, and for n.sub.3<n.sub.2 (n.sub.2>n.sub.1 and n.sub.3>n.sub.1) .sub.B1>.sub.B2. It means that for a proper adjustment, the dimensions of the top step should be corrected: for n.sub.3>n.sub.2 the total width (radius) of the top step should be increased, for n.sub.3<n.sub.2 the total width (radius) of the top step should be less than the optimal one.

(39) Note that the intensity of the generated nanojets beam is sensitive to the height of the bottom layer (H.sub.1). The maxima of intensity occurs at optical path difference between the nanojets beam and wave propagating in the first step given by

(40) $= 2 k \frac{}{2},$
where k is an integer number. For the proposed system, the optical path difference in the point A (see FIG. 2) is equal to

(41) $= \frac{1 - \cos_{B 1}}{\cos_{B 1}} H_{1} .$
The dependence of the optical path difference on the nanojets beam radiation angle for four different values H.sub.1 is presented in FIG. 3 (the curve referenced 301 represents the relationship between the nanojets beam radiation angle and the distance of the focusing point, for a height H.sub.1 equal to 1000 nm; the curve referenced 302 represents the relationship between the nanojets beam radiation angle and the distance of the focusing point, for a height H.sub.1 equal to 500 nm; the curve referenced 303 represents the relationship between the nanojets beam radiation angle and the distance of the focusing point, for a height H.sub.1 equal to 300 nm; the curve referenced 304 represents the relationship between the nanojets beam radiation angle and the distance of the focusing point, for a height H.sub.1 equal to 100 nm). The vertical dash-dot lines indicate the nanojets beam radiation angles for some particular values of the ratios of refractive indexes

(42) $(= \frac{n_{2}}{n_{1}}) .$
In FIG. 3, we have n.sub.1=1, and .sub.1=1.15, .sub.2=1.3, .sub.3=1.49 and .sub.4=1.8. It can be seen, that for the nanojets lenses with dimensions of about one wavelength, the equation

(43) $= 2 k \frac{}{2}$
can not be fully satisfied. The analysis of in FIG. 3 shows that for small values of the optical path difference is minimal (.fwdarw.0). This suggests that the maximal relative field intensity enhancement in the focal spot of a double-steps nanojets lens can be expected in a case of small difference between the refractive indexes of the lens and the host medium materials.

(44) Let us present the data obtained using the electromagnetic field simulation software package CST MICROWAVE STUDIO. The lens is assumed to be infinite along y-axis (rib-type) and is illuminated by a linearly-polarized plane wave E={0,1,0}. The materials of the steps of the nanojets lens can be the same (n.sub.2=n.sub.3). To illustrate the influence of the nanojets lens topology on the parameters of the nanojets beams, we consider the system with the following parameters: n.sub.1=1, n.sub.2=n.sub.3=1.49 (and =2.25), W.sub.1=275 nm, H.sub.1=350 nm (the dimensions of the bottom layer are arbitrary). The presented simulations were done for 2D problem.

(45) By using the previously established equations H.sub.2W.sub.1H.sub.1 and

(46) $W_{2} \frac{H_{2}}{},$
the optimal dimensions of the second step can be obtained: W.sub.2=107.5 nm and H.sub.2=268.8 nm.

(47) The dependence of the maximal power density on the half-width (radius) of the second layer W.sub.2 is presented in FIG. 4(a). The dashed black horizontal line shows the reference solution for the single step microlens. It can be seen that, if parameters of the second layer are close to the optimal, we can get the maximal value of power distribution. The maximal power density is observed for the double step nanojets lens with W.sub.2=120 nm. Changing the dimensions of the second layer we can change the position of the nanojets hot spot. FIG. 4(b) shows the dependence of Z coordinate of NJ hot spot on W.sub.2. The dashed black horizontal line shows the position of the top of the system.

(48) It can be seen that in all cases the beam width at half power (BWHP) is about 200 nm, which is below the diffraction limit which predicts the smallest possible focal spot size of about of the wavelength in the host medium. As expected, the maximum power density is observed for the nanojets lens with optimal parameters (W.sub.2=120 nm), whose focal spot BWHP is about 170 nm, which is in part explained by the improved focusing ability of the double-step nanojets lens and the shift of the focal spot inside the lens with a refractive index value higher than that of the host medium.

(49) As we can see in FIGS. 5(a)-(b), the power density distribution is sensitive to the heights of the steps of the nanojets lens. The peak power density is observed for H.sub.1=350 nm and H.sub.2=270 nm which are close to the optimal heights of the steps. As the result, we can conclude that the equations H.sub.2W.sub.1H.sub.1 and

(50) 0 $W_{2} \frac{H_{2}}{}$
for two nanojets hot spot adjustment does not work for the case when H.sub.1,2</4. In this case, we observe two nanojets hot spots.

(51) In order to evaluate the effect of the materials of the layers on the nanojets properties, the power density distributions have been simulated at the modified ratios

(52) $= \frac{n_{2}}{n_{1}}$
(see FIG. 6). It is possible to see that peak value of power density rises with , but the length of the beam decreases in this case. In FIG. 6, the curve referenced 601 corresponds to =1.8 (W.sub.2=100 nm, H.sub.2=164 nm), the curve referenced 602 corresponds to =1.49 (W.sub.2=120 nm, H.sub.2=268 nm), the curve referenced 603 corresponds to =1.3 (W.sub.2=150 nm, H.sub.2=410 nm) and the curve referenced 604 corresponds to =1.15 (W.sub.2=182 nm, H.sub.2=691 nm).

(53) Comparison of the curves in FIG. 7 for different combinations of refractive indexes of the layers demonstrates that combinations of different materials qualitatively alter the power distribution along the z-axis. Namely, we can observe that when n.sub.3>n.sub.2 we can get higher peak (up to 25%) of power density distribution at smaller width of the second layer. But in this case the length of the nanojets beam will be nearly equal to the length of the beam obtained for the step-like nanojets lens with n.sub.3=n.sub.2=1.49 and optimized smaller dimensions.

(54) One advantage of the proposed technique is that it enables additional degrees of freedom for the control over nanojets beam in the near zone and, in particular, it enhances the field intensity in the focal spot of the double steps nanojets lens. These degrees of freedom are provided by adding additional steps giving rise to independent nanojets beams that all together contribute to the formation of a high intensity secondary beam.

(55) In such a case, characteristics of the secondary nanojets beam (i.e. field intensity, beam width, and orientation) are determined by the partial contribution of multiple independent beams associated with different steps.

(56) In the case of a symmetrical structure (like the one presented in the FIGS. 1(b)-(c)) the secondary beam is located on axis of the lens, however its orientation can be modified by breaking the symmetry of the structure, e.g. changing the base angles of selected steps.

(57) A few possible realizations of the step-like nanojets lenses are presented in FIGS. 8(a)-(b), and FIGS. 9(a)-(d). Such nanojets lenses can be embedded in a host medium or placed on a dielectric substrate acting as a support layer. Material of the substrate can be selected arbitrary. It can be the same or different from the material of the wider step. In addition, the implementation of such microstructures is possible via standard lithography techniques.

(58) It should be noted that such structure can be illuminated both from the top or bottom, however, to provide the desired focusing function, material properties and dimensions of the steps must be adjusted accordingly (see FIG. 7).

(59) To illustrate the influence of the step-like topology on the parameters of the nanojets beams in the case of lenses with bigger dimensions, we considered the system with the following parameters: n.sub.1=1, n.sub.2=n.sub.3=1.8, W.sub.1=550 nm, H.sub.1=400 nm. The presented simulations were done for 2D problem. The nanojets lens is assumed to be infinite along y-axis (rib-type) and is illuminated by a linearly-polarized plane wave E={0,1,0}.

(60) Using the equations H.sub.2W.sub.1H.sub.1 and

(61) $W_{2} \frac{H_{2}}{}$
we obtain that the optimal dimensions of the second step are W.sub.2=336.2 nm and H.sub.2=689.8 nm. FIGS. 10(a) and (b) show the dependence of the maximal power density and Z coordinate of nanojets hot spot on the half-width (radius) of the second layer W.sub.2 at 3 different wavelengths.

(62) The numerical simulations demonstrate that the equations H.sub.2W.sub.1H.sub.1 and

(63) $W_{2} \frac{H_{2}}{}$
do not work in this case. First of all, to get the focal point outside the nanojets lens we should take much lower height of the second step. In the presented case, it was just H.sub.2=250 nm. Moreover, the maximal power density was observed for the smaller sizes W.sub.2. It is also necessary to note that the system is very sensitive to the wavelength of electromagnetic wave. But it is possible to see that the double steps-like topology gives more intensive total response of the system (up to 200% in the case of =450 nm for W.sub.2=200 nm) compare to the single step system of the same height (W.sub.1=W.sub.2=500 nm).

(64) In the following, we consider the structures with non-vertical edges and top/bottom surface parallel to xy-plane. Let us assume that is the base angle for a single step/layer system (see FIG. 11(a)). The general topologies of the single step and double-steps nanojets lenses are illustrated in FIGS. 11(a)-(c). These cross-section views may correspond to the prismatic systems embedded in a homogeneous dielectric host media with a refractive index n.sub.1<n.sub.2.

(65) It was obtained that for the systems with non-vertical edges the nanojets beam radiation angle can be determined using the approximate formula:

(66) $_{B 1} \frac{9 0^{} -_{TIR}^{}}{2}$
where .sub.TIR is the critical angle of refraction from the nonvertical edge. To get the approximate formula for .sub.TIR, we must take into account the changing of the position of the edge. As the result, the nanojets beam radiation angle can be estimated as:

(67) $_{B 1} 90^{} - \frac{_{TIR} +}{2}$

(68) In order to increase the intensity in the nanojets hot spot, we can adjust nanojets beams related to the steps of the system. As the result, the formula for the radius/half width of the bottom edge of the top layer can be written as:
W.sub.2W.sub.1H.sub.1 tan .sub.B1

(69) We have assumed that the base angle of the top layer coincides with the base angle of a bottom one. We should mention that to estimate the focal length of the steps, we should take into account electromagnetic wave refraction phenomena at the top surface of the layers. For such geometry, the equation F.sub.j=W.sub.j, previously used, gives relatively big calculating error. Moreover, all presented formulas are valid only for >90.sub.B1.

(70) For a step-like system with the different refractive indexes of the materials of the steps and different base angles of the constitutive parts (see FIG. 11(c)), we should use the same principle as it was discussed previously.

(71) The proposed concept is validated numerically via full-wave electromagnetic analysis of a step-like nanojets lens in a form of a prism, whose cross-section is represented in FIG. 11(b). The presented simulations were done for 3D problem. The lens is illuminated by a linearly-polarized plane wave E={0,1,0}. The materials of the steps of the nanojets lens are the same (n.sub.2=n.sub.3), the base angles for the constitutive parts are the same. To illustrate the influence of modified nanojets lens topology on the parameters of the nanojets, we consider the system with the following parameters: n.sub.1=1, n.sub.2=n.sub.3=1.5, W.sub.1=275 nm and H.sub.1=200 nm.

(72) By using the equation W.sub.2W.sub.1H.sub.1 tan .sub.B1, we can determine that the optimal radius of the second step for a base angle =87.2 is W.sub.2=179.6 nm. The FIG. 12(a) shows the dependence of the peak of the power density on the radius W.sub.2. For a chosen height of the second step (H.sub.2=200 nm), we observe the nanojets hot spot inside the structure/system. We can see that the maximal power density corresponds to the step-like nanojets lens with W.sub.2=205 nm. Dashed black curve in FIG. 12(a) corresponds to the reference solution, namely a single step nanojets lens with non-vertical edges (=87.2) of the same total height (400 nm) and with W.sub.1=275 nm. The dependence of peak power density on W.sub.2 for a system with lower base angle (=80.1) is presented in FIG. 12(b). To get the focal spot outside the system we take H.sub.2=100 nm. It was obtained that the optimal radius of the second step for such system corresponds to W.sub.2=164 nm. The maximal power density was observed for the double-steps nanojets lens with W.sub.2=180 nm. The dashed curve in the FIG. 12(b) corresponds to the single step reference solution (=80.1, W.sub.1=275 nm, H.sub.1=300 nm). Hence, we can conclude that by choosing the optimal parameters of the layers for the double-steps nanojets lenses with nonvertical edges, we can get more intensive total response of the system.

(73) Let us now consider the effect of the angle of plane wave incidence on the properties of generated nanojets beam.

(74) We assume that .sub.i is the angle of electromagnetic wave incidence.

(75) To get the approximate formula for nanojets beam radiation angles in the case of plane wave oblique incidence on the element with refractive index n.sub.2, we should take into account that the radiation angles for opposite edges of the system are not equal (see FIG. 13(a)). As the result, for a single-step element we can have

(76) $_{B 1}^{} \frac{9 0^{} -_{TIR}}{2} + \frac{_{i}}{2},_{B 1}^{} \frac{9 0^{} -_{TIR}}{2} - \frac{_{i}}{2} .$

(77) It is necessary to note, that for nanojets lens with the focal spot (hot spot of the total generated nanojets) inside the system, angle of total nanojets deviation can be determined as

(78) $\tan = \frac{\tan_{B 1}^{} + \tan_{B 1}^{}}{2} .$

(79) Therefore, in order to determine the focal length of the nanojets lens with hot spot outside the system (F.sub.1), we should take into account the additional electromagnetic wave refraction at the top edge of the system. As the result, we obtain that angle of total nanojets deviation will be approximately equal to the angle of incidence .sub.i.

(80) To increase the intensity in the nanojets hot spot, we should adjust the focal lengths of the constitutive elements of the step-like system as it was done before considering the difference between the nanojets beam radiation angles for the opposite edges of the system. As the result, the double-step system (FIG. 13(b)) will not be symmetrical. For a single-material double-step nanojets lens we have:
L.sub.1=H.sub.1 tan .sub.B1
L.sub.2=H.sub.1 tan .sub.B1
In this case additional nanojets related to the second step will be superimposed with the nanojets for a first step.
So, the parameters of the second step can be determined using the approximate formulas:

(81) $W_{2} = W_{1} - H_{1} \frac{\tan_{B 1}^{} + \tan_{B 1}^{}}{2}$ $X_{1} = W_{1} - L_{2} - W_{2}$
where X.sub.1 is the shift between the axis of the symmetry for the steps. The presented formulas are valid for positive angles of incidence and L.sub.2<L.sub.1. In the case of negative angles .sub.i, we obtain that L.sub.2>L.sub.1.

(82) For nanojets beam radiation in the case of plane wave oblique incidence on the element with nonvertical edges, we should just use modified approximate equations for the radiation angles

(83) $_{B 1}^{} 9 0^{} - \frac{_{TIR} +}{2} + \frac{_{i}}{2},_{B 1}^{} 9 0^{} - \frac{_{TIR} +}{2} - \frac{_{i}}{2} .$

(84) It is necessary to note, that if materials of the steps are different (FIG. 13(c)), we should use these approximate formulas just for preliminary estimation of the second step size taking into account that for n.sub.3>n.sub.2>n.sub.1, .sub.B1<.sub.B2 and .sub.B1<.sub.B2, and for n.sub.3<n.sub.2 (n.sub.2,3>n.sub.1).sub.B1>.sub.B2 and .sub.B1>.sub.B2. It means that for a proper adjustment, the dimensions of the top step should be corrected: for n.sub.3>n.sub.2, the total width (radius) of the top step should be increased, for n.sub.3<n.sub.2, the total width (radius) of the top step should be less than the optimal one. Moreover, we should take into account the refraction phenomena associated with the top edges of the first and second steps (first and second layers) leading to the considerable decrease of the focal distance for proposed system.

(85) Therefore, in order to determine the optimal geometry of double-steps nanojets lens with different materials of the steps, we should take into account that the angles of nanojets deviation by both steps are approximately equal to the angle of incidence .sub.i. So, to get the proper adjustment of the focal points in this case, it is important to determine the shift between the axis of the symmetry for the steps as
X.sub.1=H.sub.1 tan .sub.i,
and to take into account the dimensions of the second layer.

(86) The proposed concept is validated numerically via full-wave electromagnetic analysis of a step-like nanojets lens in a form of a cylinder, whose cross-section is represented in FIG. 13(b). The presented simulations were done for 3D problem. The lens is illuminated by a linearly-polarized plane wave E={0,1,0}. The materials of the steps of the nanojets lens are the same (n.sub.3=n.sub.2). To illustrate the influence of the nanojets lens topology on the parameters of the nanojets beams in the case of oblique plane wave incidence, we consider the system with the following parameters: n.sub.1=1, n.sub.2=n.sub.3=1.5, W.sub.1=275 nm, H.sub.1=200 nm.

(87) Using the equations L.sub.1=H.sub.1 tan .sub.B1, L.sub.2=H.sub.1 tan .sub.B1,

(88) 0 $W_{2} = W_{1} - H_{1} \frac{\tan_{B 1} + \tan_{B 1}}{2}$
and X.sub.1=W.sub.1L.sub.2W.sub.2 we can determine the optimal radius of the second step which is W.sub.2=183.7 nm and shift between the axes of the symmetry for the layers is X.sub.1=31.7 nm.

(89) The FIG. 14(a) shows the peak power density distribution along z-axis for six different values of W.sub.2 at fixed shift X.sub.1=30 nm. To get the focal spot outside the system we take H.sub.2=100 nm. The maximal power density is observed for the step-like nanojets lens with W.sub.2=200 nm. We should mention that changing of the sizes of an additional layer leads to the shift of the nanojets hot spot position (see FIGS. 14(b) and 14(c)). The dashed curves in the FIGS. 14(a)-(c) correspond to the reference solution, namely a single step nanojets lens of the same total height. The dependence of the maximal power density on the parameter X.sub.1 for second layer with W.sub.2=200 nm is presented in FIG. 15(a). The dashed horizontal line shows the reference solution for the single step nanojets lens. It can be seen, that if for oblique plane wave incidence size and position of the second layer are close to the optimal, we can get the maximal value of power distribution.

Device for deviating and focusing light

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G02B2207/101

PHYSICS

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G02B27/095

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G02B27/56

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International classification

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G02B27/09

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Abstract

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