1. Introduction

The common method for characterizing free space laser beams is to use a laser beam profiler, which is a cumbersome and expensive device which blocks the beam propagation. In many complex optical setups, especially those requiring continuous monitoring, this approach is impractical, and a more suitable solution can be had; a meta-hologram beam monitor. Metasurfaces are thin artificial surfaces tailored to possess diverse electromagnetic properties that are not exhibited by natural uniform surfaces [1]. Metasurfaces are constructed from large arrays of nano-structures consisting of subwavelength dielectric [2–22] or metallic (plasmonic) [23–43] elements. Dielectric and metallic configurations often utilize the inherent resonances of the nano-structures comprising the metasurface (plasmonic and Mie resonances) to modify the properties of an impinging electromagnetic wave. The ability of metasurfaces to abruptly and locally, alter the phase of the incident wave, over complete 2π range [2], underlies their huge potential for ultrathin planar optical devices. Compared to their plasmonic counterparts, dielectric metasurfaces exhibit low absorption, rendering them attractive for both transmitarrays and reflectarrays [22,44]. Dielectric metasurfaces have been indeed applied in a variety of applications including beam shaping elements [7,21,44], gratings [19,20], mode converters and sorters [2,7,21,32,44], broadband meta-lenses [5,16,18], meta-holograms [4,6,13,16,17], and high efficiency wide angle beam deflection [11].

While shown to be of great use in beam synthesis, the potential of dielectric metasurfaces for beam monitoring and analysis has been only sparsely studied. It was only recently shown, that meta-holograms can be used for in situ mode sorting by monitoring a beam’s orbital and spin angular momentum components [15], i.e. a breakdown of the beam to RHCP/LHCP and topological charge components. It was also shown that 0.05% of the optical power can be converted into a useful image, preserving the original wave front with efficiency exceeding 96%. Other studies demonstrated miniaturization and integration of vortex beam mode-sorters and spatial filters in an attempt to reduce the device footprint [45,46]. Despite the little attention paid to dielectric metasurfaces in beam monitoring applications, the same properties that make these devices attractive for beam shaping (i.e. thin, flat profiles, a wide spectral range of operation, low absorption and potentially high transmission), render them prime candidates for the realization of many other ultrathin in situ optical components.

Here we present a novel approach based on dielectric meta-holograms for performing in situ beam monitoring and characterization. For this purpose, a family of meta-holograms is presented, and is employed to monitor some of the most important properties of optical beams: spot size, astigmatism, lateral position, and waist position along the optical axis.

Figure 1 illustrates the utilization of a dielectric meta-hologram for in situ monitoring of a beam propagating through an optical setup. Conventionally, a beam is often analyzed by deflecting some of the light towards a beam profiler, or by inserting a beam profiler into its path. If the beam is deflected, by a mirror or a beam-splitter, then its characteristics, such as its spatial profile (phase and amplitude), may differ considerably from those at the position of interest. Alternatively, if a beam profiler is inserted at the position of interest within the setup (Fig. 1(b)), it often entails the removal of some optical elements and essentially inhibits the setup operation during the beam characterization. Thus, a simple, accurate, in situ monitoring and analysis capability of optical beams is necessary for many applications and optical systems.

 

Fig. 1 (a) Schematic of an optical setup including a beam monitoring meta-hologram. A portion of the illuminating power is deflected to a screen, forming an image which is used for monitoring the beam. (b) The conventional approach: a beam profiler is inserted at the position of interest, instead of the meta-hologram. Some parts of the setup are subsequently removed to make room for the beam profiler and the beam is blocked.

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2. A transverse-profile beam monitoring meta-hologram

In order to characterize the spatial intensity distribution of an incident beam we designed a transverse-profile beam monitoring meta-hologram (BMMH). The transverse-profile BMMH, depicted in Fig. 2, consists of several spatially separated phase maps, where each individual phase map is assigned an independent image component. The illuminated phase maps in the BMMH project their combined images to the far field to produce a pattern indicating the beam shape and size (Fig. 2(b)). The partitioning scheme of the hologram is designed in the following manner: First, the hologram area is partitioned into a pattern of concentric, equal area, ring patches (Fig. 2(a)). A symmetric Gaussian beam illuminating the center of the pattern causes only the images of the illuminated rings to be projected to the far field, thus providing a clear indication of the spot size. The spacing rings, between the active phase maps, (showing in white in Fig. 2(a)) are introduced to help discriminating between different ranges of the beam radii. Furthermore, each ring is divided into four sectors (each one projecting a different image) in order to facilitate alignment, as well as to impart information regarding the beam astigmatism. We note that the areas of all patches are identical in order to project similar power levels to each image component. The images, illustrated in Fig. 2(b), are chosen to indicate accurately the shape and lateral position (with respect to the hologram center) of the beam. The phase maps are calculated for each patch in the hologram using the Gerchberg-Saxton (GS) algorithm [47], assuming a homogeneous illumination of the patch area. The patches are stitched together to produce the hologram presented in Fig. 2(c). Details of the stitching process are found in Appendix A and the considerations leading to the choice of ring radii are described in Appendix B. The hologram’s far field image under homogeneous illumination (corresponding to its spatial Fourier transform) is shown in Fig. 2(d).

 

Fig. 2 (a) A transverse-profile beam monitoring meta-hologram (BMMH) partitioning scheme, and (b) its corresponding compound image design. All colored patches in (a) have equal areas. The patches are marked by P, with subscript index indicating the ring number (1-3), and superscript label for the direction (up/down/left/right). Their respective image components I, marked in (b), indicate their radii and directions. The remaining area of the hologram, in white, constitutes another patch. Its image can be chosen arbitrarily (a star in this case), as long as it does not overlap any of the other images. (c) A BMMH design, based on the partitioning scheme presented in (a). Each patch is designed using the GS algorithm to form its counterpart image component. (d) The calculated far field image of (c), given by its Fourier transform.

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2.1 Experiments

Following the assembly of the hologram patches into a single rectangular pattern (Fig. 2(c)), a physical hologram was realized as a dielectric metasurface (Fig. 3). Each cell of the metasurface, comprising a 300 nm thick silicon pillar, represents a single pixel of the hologram. The pillar’s radius determines the phase of the pixel. A genetic algorithm that iteratively invokes rigorous coupled-wave analysis (RCWA) simulations was used to optimize the pillar radii [11] (see also Appendix D). The meta-holograms were fabricated using conventional ion-beam lithography followed by deep reactive ion etching of amorphous silicon on a fused-silica substrate. A 300 nm layer of amorphous silicon was deposited on a fused-silica substrate using low pressure chemical vapor deposition at 590 °C (Oerlicon 790). A negative electron beam resist (40 nm, ma-N 2401 diluted with a thinner) was then spin-coated and pre-baked. The pattern was transferred to the resist using focused ion-beam lithography (ionLine) at 35 KeV, and was developed in a resist developer (ma-D 332). The patterned 40 nm ma-N pillars were used as a hard mask for deep reactive ion etching (Plasma-Therm Versaline DSE) of the amorphous silicon using SF₆, and C₄F₈ was used as the passivation layer. Finally, the residues of the ma-N mask were removed by oxygen plasma.

 

Fig. 3 (a) A schematic of the metasurface realization. Each pillar is a cell in the metasurface, representing a single pixel in the hologram. The radii of the 300 nm thick silicon pillars are translated to different phase retardations. (b) An SEM micrograph showing the realization of the phase map. (Inset) A magnification of the edge of the structure. (c) A microscope image of a transverse-profile BMMH based on the phase map design presented in Fig. 2(c). Scale bar length is 100 µm.

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To characterize the meta-hologram, a Gaussian beam laser at telecom wavelengths (λ = 1550 nm) was focused on the sample plane through a telescope providing control over the spot size. The beam diameter was controlled by changing and repositioning the lenses comprising the telescope. The far field image was projected on a screen, positioned approximately 20 cm away from the sample plane and was recorded by an InGaAs IR camera. Roughly 30% of the beam power was diverted towards the image. This figure can be greatly reduced by applying a beam tapping technique employing non-resonant Pancharatnam-Berry phase elements in the manner shown in [15]. A complete description of the experimental setup is given in Appendix C. The beam intensity at the sample plane was also recorded by a knife-edge beam profiler (Newport KEP-3-IR3) for comparison. The beam profiles, images and matching simulations are shown in Fig. 4. Excellent agreement is apparent between the theoretical and the experimental results. More importantly, a clear differentiation between the various beam profiles, as well as the characterization of an astigmatic beam (a_V), is demonstrated. In Table 1 the 1/e² diameter of the beam, recorded by the knife-edge beam profiler, is compared with the inner and outer diameters of the outermost illuminated ring. The agreement between the two sets demonstrates the usability of this device for in situ beam characterization and monitoring.

 

Fig. 4 (a_I)-(a_V) The source beam intensity profiles, obtained from the waist values measured by a beam profiler. The contours of the transverse-profile BMMH patches were drawn over the heat maps in black. (b_I)-(b_V) Simulated far field images. (c_I)-(c_V) The on-screen images recorded by an IR camera. The bright spot observed in (c_I)-(c_V) corresponds to the main transmission lobe. In (c_IV) the screen was cut to remove the main lobe from the image for clarity purposes. A video demonstrating the impact of the source beam shape and size on the obtained image is provided as Visualization 1 in the supplementary material.

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Table 1. Beam diameters (1/e² power criterion), as measured by a knife-edge beam profiler, and the respective range indicated by the transverse-profile BMMH.

View Table

In order to obtain an accurate evaluation of the beam diameter one must align the beam and the hologram center. To facilitate the alignment procedure, the hologram is dissected in four sectors, as shown in Fig. 2(a), corresponding to the four directions up/down/right/left. The image components are designed and assigned to patches such that when the beam is off-axis a guiding arrow emerges from isosceles triangles pointing inwards (see Fig. 5). This shape intuitively informs the user that the hologram is off center, and indicates the direction and approximate distance towards the beam axis. The user can then adjust the hologram position to reach home-position, where the beam and hologram centers are aligned. When home-position is reached, arrows in opposite directions are illuminated with equal intensity. Figure 5(a) illustrates this feedback mechanism. The top panel, (b_I)-(b_V), exemplifies different source beam positions. Below, corresponding simulations (c_I)-(c_V) and experimental images (d_I)-(d_V) are shown.

 

Fig. 5 (a) The transverse-profile BMMH alignment utilizing the feedback mechanism: The beam illuminates the hologram, lighting a guiding image. The operator observes the guiding image and controls the hologram mount to drive it to home position. The images below show different beam positions (b_I)-(b_V), their respective simulations (c_I)-(c_V) and corresponding recorded images (d_I)-(d_V). The bright spot observed in (d_I)-(d_V) corresponds to the main transmission lobe. A video demonstrating the impact of the source beam position on the obtained image is provided as Visualization 2 in the supplementary material.

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A direct application of the transverse-profile BMMH concept is the common task of illumination of a small region of interest (ROI) on a large sample. Generally, it requires imaging capabilities built into the optical setup, and alignment with the optical axis. In the absence of such apparatus, the user often embarks on a blind search for some anticipated output signal. A transverse-profile BMMH can be a useful tool in finding any predefined position on the sample plane, and save experimentalists a great deal of time spent in search. Specifically, the hologram design can be readily modified to accommodate an ROI at its center. This way, it could also provide an estimate of the fraction of the beam illuminating the ROI, given that its area is known, and that the total beam power can be measured (e.g. by a power meter).

3. A z-position beam monitoring meta-hologram

The presented BMMH is clearly useful for in situ monitoring and analysis of beam profiles, as well as for lateral positioning and alignment. For some applications, however, there is a need to accurately locate the position of the beam waist along the optical axis, denoted here as the z axis. For example, when illuminating an array of antennas or diffractive elements to obtain their collective far field emission, elements out of phase may have a destructive contribution. Only when the array is positioned exactly at the waist of the beam will all the contributions of the illuminated elements interfere constructively. For this purpose we design the z-position BMMH, depicted in Fig. 6.

 

Fig. 6 (a) A z-position BMMH partitioning scheme, and (b) its corresponding compound image design. The field exciting the four side-patches has an average phase gradient changing with z-position and directed towards the center. The corresponding circles in (b) slightly move inwards/outwards as the hologram moves away from the beam waist. The middle patch P^c always maintains zero average phase gradient. Its image, the two rings in (b), remains in the same lateral position regardless of the z-position of the BMMH. (c) A z-position BMMH design, based on the partitioning scheme presented in (a). (d) The calculated far field image of (c).

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The z-position BMMH consists of five patches: a center square patch, and four surrounding sectors. The partitioning is shown in Fig. 6(a). The transverse phase profile of a Gaussian beam (at any point along its propagation axis) is parabolic where the radius of curvature at a certain z-position decreases with the distance of that point from the position of the waist. At small longitudinal distances from the waist, the phase profile can be approximated by a piecewise linear profile: At the center of the beam the phase profile is relatively flat while at larger radii the phase gradient can be locally approximated by a linear phase gradient directed towards the center of the beam.

Thus, when the z-position BMMH is located away from the waist, the impact on the central patch is minor, but the four surrounding sectors are illuminated by tilted phase-fronts. This, in turn, causes the image components of these sectors to be steered inwards/outwards with respect to the center depending on the longitudinal position of the hologram relative to the waist. As seen in Fig. 6(b), the image component of the center patch is chosen to be of two concentric rings, setting the reference frame for the remaining image components. The other image components are chosen to be of disks that fit exactly between the two reference rings. As the hologram position is moved away from waist, the images of the disks move inwards or outwards, crossing the rings. Thus, setting the z-position BMMH, such that the images of the disks are located precisely between the boundaries of the rings, guarantees that the hologram is located at the waist. Additionally, if the spot-size is small, only the center patch is illuminated around the waist position. This prevents the disks from appearing in its vicinity. The changing image, as a function of the longitudinal beam position, is shown in Fig. 7. Under the conditions of our experiments these changes allow to evaluate the waist position to within z_R/4.Note that contrary to other techniques used to find the beam waist position, the identification of the waist position does not rely on the “sharpness” of the image. For beams narrower than the central patch of the hologram the resolution is actually lower at the waist than at other positions in its vicinity, and the pattern is used instead.

 

Fig. 7 (a_I) The exact transverse phase profile of a Gaussian beam is presented as a function of its z-position, in units of z_R, the Rayleigh length. (a_II) Its linear piecewise approximate profile, based on the partitioning scheme presented in Fig. 6(a). (a_III) The expected image from the z-position BMMH. (b_I)-(b_III) Experimental results, and (c_I)-(c_III) corresponding simulations. The red dashed circles and green rings in (b_I) and (b_III) highlight the disk positions. The blue arrows in (c_I) and (c_III) point outwards and inwards, respectively, in the direction of the displacement of the disks. A video simulation showing the disk displacement in the image due to varying z-position is shown in Visualization 3 in supplementary material.

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4. Conclusions

We have demonstrated here a unique type of holograms that can be used for evaluating free space optical beam characteristics in situ. These passive devices allow for real-time beam monitoring and analysis with a minimal footprint, no larger than a typical lens or sample mount. Among the characteristics they can extract are the lateral beam position, longitudinal waist position, size, and shape. We note that it is possible to modify the transverse-profile BMMH to differentiate beam radii with better resolution by partitioning to finer rings. In our experiments the transmitted optical beam partially overlapped with the image of the transverse-profile BMMH. This can be easily avoided either by making the image smaller or by increasing the projection angle. Based on the principles outlined here, many more devices can be designed for simple evaluation of additional characteristics, such as the beam’s polarization state profile, asymmetry, level of spatial coherence, etc.

Appendix A BMMH assembly

We define patch holograms as an assembly of independent holograms of arbitrary shapes that form together one structure. Compared to regular holograms, where different illuminating beams produce essentially the same image, in our patch holograms the spatial intensity distribution determines which image components will appear. Figure 8 illustrates this property. Below, we show the design and operation of patch holograms.

 

Fig. 8 Patch versus conventional hologram. (a) An illustration of a wide Gaussian source beam illuminating a regular hologram, and (b) a narrow beam illuminating the same hologram, and the corresponding far field images visible in the imaging plane. The image resulting from the narrow beam is blurred and noisy. (c) A narrow Gaussian beam illuminating a patch hologram creates a different image. Only the highly illuminated patches create image components. Other components of the compound image remain dark.

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The Gerchberg-Saxton (GS) algorithm [47], for example, can be used to produce a computer-generated hologram (CGH) by solving the phase of a wave function with given amplitude distributions in the diffraction and imaging planes. This relation can be expressed by:

(1)$$\left|F\right\{S(x,y)\cdot \exp (i\Phi (x,y))\}{|}^2\propto I(u,v),$$

where $S(x,y)$ is the amplitude distribution of the source beam, $\Phi (x,y)$ is the calculated phase map, $I(u,v)$ is the intensity distribution in the imaging plane, and $F$ denotes the 2D Fourier transform operator.

When the source beam is infinitely wide, Eq. (1) can be further simplified by replacing the source amplitude distribution $S(x,y)$ with constant amplitude.

(2)$$\left|F\right\{\exp (i\Phi (x,y))\}{|}^2\propto I(u,v)$$

Given a partition of the device aperture on the diffraction plane $\Pi \in {ℝ}^2$ into N non-overlapping parts $P_1,P_2,\mathrm{...},P_N$, i.e. ${\displaystyle {\cup }_j{P}_j=\Pi }$ and $\forall j,k,$ $P_j\cap P_k=\varnothing ,$ we can define a set of binary masks $M_1,M_2,\mathrm{...},M_N$ such that

(3)$$M_j(x,y)=\{\begin{array}{cc}1& (x,y)\in P_j\\ 0& (x,y)\notin P_j.\end{array}$$

We note that for any point $(x_0,y_0)$ belonging to $\Pi$ we get ${\displaystyle \sum _{j=1}^N{M}_j(x_0,y_0)=1},$ and we can write

(4)$$|F\left\{{\displaystyle {\sum }_{j=1}^N{M}_j(x,y)\exp (i\Phi (x,y))}\right\}{|}^2\propto I(u,v).$$

Due to the linearity of the transform, we get

(5)$$\begin{array}{l}F\left\{{\displaystyle {\sum }_{j=1}^N{M}_j(x,y)\exp (i\Phi (x,y))}\right\}={\displaystyle {\sum }_{j=1}^{N}F\{M_j(x,y)\exp (i\Phi (x,y))\}}\\ \triangleq {\displaystyle {\sum }_{j=1}^N{A}_j(u,v)\exp (i{\psi }_j(u,v))},\end{array}$$

where we defined $A_j(u,v)\exp (i{\psi }_j(u,v))$ as the field resulting only from illumination of patch j (i.e. via mask $M_j(x,y)$). The image plane intensity distribution is therefore given by

(6)$$\begin{array}{l}|{\displaystyle {\sum }_{j=1}^N{A}_j(u,v)\cdot \exp (i{\psi }_j(u,v))}{|}^2\\ ={\displaystyle {\sum }_{j=1}^N{A}_j^2(u,v)}+{\displaystyle {\sum }_{j=1}^N{\displaystyle {\sum }_{\begin{array}{l}k=1\\ k\ne j\end{array}}^{N}2A_j(u,v)A_k(u,v)\cos ({\psi }_j(u,v)-{\psi }_k(u,v)).}}\end{array}$$

We observe that the resulting image can be broken down into a sum of square terms, and a double sum of cross terms. We denote these $I_j$ and $X_{jk},$ respectively. The term $I_j$ is the intensity image component resulting only from part $P_j$ of the source beam and phase map. The term $X_{jk}$ accounts for the interference between parts $P_j$ and $P_k$. If we choose non-overlapping images, then for any point $(u_0,v_0)$ we get $A_j(u_0,v_0)A_{k\ne j}(u_0,v_0)=0,$ and the cross terms vanish, leaving us with

(7)$$\left|F\right\{\exp (i\Phi (x,y))\}{|}^2\propto {\displaystyle {\sum }_{j=1}^N{I}_j(u,v),}$$

with the condition that $\Phi (x,y)$ must satisfy

(8)$$\left|F\right\{M_j(x,y)\exp (i\Phi (x,y)\}{|}^2\propto I_j(u,v)$$

for every $j.$ To achieve this, we first notice that due to Eq. (3),

(9)$$M_j(x,y)\cdot \exp (i\Phi (x,y))=M_j(x,y)\cdot \exp (i{M}_j(x,y)\Phi (x,y)),$$

and denote $M_j(x,y)\Phi (x,y)\triangleq {\varphi }_j(x,y).$ It is easy to see that

(10)$$\Phi (x,y)={\displaystyle {\sum }_j{\varphi }_j(x,y)}.$$

Each phase map component ${\varphi }_j,$ relating a part, or patch, $P_j$ of the hologram to an image $I_j$ in the imaging plane, can be easily calculated using the GS algorithm. Figure 9 illustrates the elements of the design and their roles in the patch hologram.

 

Fig. 9 An illustration of the design setup. (a) A patch mask M_j is used as the diffraction plane spatial amplitude distribution applied to a homogeneous source beam. An image component I_j, the star shape in this case, is designed on the image plane. The GS algorithm can then be used to find an approximate phase map that relates the two planes by a Fourier transform. (b) The phase map Φ is constructed from the calculated phase map patches. When Illuminated by the source beam S, the phase map patch ɸ_j contributes its counterpart image component I_j to the total image I.

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Explicitly, one needs to design the required mask $M_j,$ and the desired image $I_j$ (which must not overlap with $I_k$ for any $k\ne j$) and use the GS algorithm to calculate the output phase map ${\Phi }^j.$ The phase map component for the hologram is simply ${\varphi }_j(x,y)=M_j(x,y){\Phi }^j(x,y).$ After adding up the different contributions (see Eq. (10)), the phase map can be implemented as a patch hologram, made of independent holographic patches.

Appendix B Gaussian beam illumination of a transverse-profile BMMH

The optical beam used in the experiments described in Section 2.1, and illuminating the BMMHs, was characterized by a knife-edge beam profiler. In Fig. 10 the beam intensity distribution, as measured by the beam profiler, is plotted. Its shape was found to be Gaussian with waist radius w₀ = 44µm.

 

Fig. 10 The beam spatial intensity (in arbitrary units) at the waist position, as measured by a laser beam profiler.

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The beam waist radius is defined as the radius at which the beam intensity is reduced to $1/e^2\cong 13.5\%$ its peak intensity. Within this radius 86.5% of the total beam power is confined. The power confined within a radius $r$ is given by

(11)$$P(r)=P_0\left[1-\exp \left(-\frac{2r^2}{w^2}\right)\right].$$

In designing the transverse-profile BMMH, we found experimentally that to get a clear image component from one ring of outer radius $R,$ and yet to avoid projecting the image component related to the following ring, of radius greater than $R,$ the fraction of optical power $f$ confined within the boundary radius $R$ should be about 98.5%. The corresponding beam waist radius is given by

(12)$$w_0=\frac{R}{\sqrt{-\frac{1}{2}\log (1-f)}}.$$

Using this criterion for our design, the ratio between the outermost illuminated ring and the maximum waist radius of the illuminating beam is simply given by $w_0=R/\mathrm{1.449.}$

Appendix C experimental setup

The experimental setup is presented in Fig. 11. The beam used in our experiments was a laser beam at telecom wavelengths (λ=1550 nm). The source used was an IR fiber optic laser (HP 8164A). The beam polarization was adjusted by a fiber polarization controller and then collimated by a fiber optic collimator. The beam was then directed at an assembly of lenses and focused at the meta-hologram monitor. The light deflected by the meta-hologram was projected on a screen and observed using an InGaAs IR camera. For the measurements presented in Table 1, a laser beam-profiler (Newport KEP-3-IR3) was used instead of the meta-hologram.

 

Fig. 11 The optical setup. The beam is generated by the laser source, polarized by the polarization controller (PC), collimated, and then focused by a lens on a meta-hologram (MH). A pattern is projected from the meta-hologram to a screen and observed using an IR camera.

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Appendix D meta-atom design

Each meta-atom, comprising a single silicon pillar, corresponds to a pixel of the designed phase-map. The phase imparted by the pillar was designed using a genetic algorithm that iteratively invokes rigorous coupled-wave analysis (RCWA) simulations. Figure 12 shows the numerically calculated phase and transmission characteristics as a function of pillar radius. It is evident that these dielectric elements span a 2π range, achieving full phase coverage. More information regarding the meta-atom design method can be found in [11].

 

Fig. 12 The phase and transmission responses vs. the radius of the silicon pillars.

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Funding

XIN Center and by BSF grant 2016388.

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