Infrared broadband quarter-wave and half-wave plates synthesized from anisotropic Bézier metasurfaces

P. E. Sieber; D. H. Werner

doi:10.1364/OE.22.032371

1. Introduction

Electromagnetic metamaterials have gained significant attention in the literature since their inception [1–3]. Yet, widespread adaptation of metamaterials has been limited in many cases due to their inherently lossy and narrowband behavior attributed to their Lorentzian resonances. Thus, especially in the infrared (IR) and visible ranges, techniques to synthesize extended bandwidth planarized matallodielectric structures need to be developed. Once this is achieved, metamaterials become much more attractive, avoiding the limitations of conventional dielectric slab-based optical designs such as unwanted reflections, limited angular response, and a reliance on available materials. In terms of polarization control at IR and visible wavelengths, several different approaches have been utilized ranging from three dimensional (3D) chiral metamaterial structures [4–7], to the more recent transition to two dimensional (2D) metasurfaces such as chiral [8–15] and anisotropic (achiral) structures [16–39]. Much of this work relies on known metasurface geometries; namely, the dipole, the split ring, the L-resonator, and the V-resonator. Additional work has also utilized some of these geometries in reconfigurable applications [40–48]. Yet, the question remains whether these designs represent the most optimal performance that can be achieved.

In this work a new technique for optimizing 2D periodic anisotropic metasurfaces is presented which is shown to exhibit performance significantly exceeding that of the aforementioned commonly used geometries. Specifically, the Bézier surface is introduced and utilized as an effective design tool to obtain very broadband metasurface-enabled IR quarter-wave and half-wave plates. To demonstrate the effectiveness of this Bézier surface synthesis technique, a real valued optimizer, CMA-ES [49, 50], is employed to synthesize the metasurfaces. The computational efficiency and performance of the optimized Bézier surfaces is compared to an existing and well known optimization technique for pixelized structures synthesized via a binary GA [51–58]. When compared to the GA optimized pixelized structures, the synthesis of Bézier surfaces using CMA-ES is shown to not only provide better results but also generates them with an order of magnitude reduction in computation time. In order to further demonstrate the advantages of the proposed Bézier metasurface synthesis technique, a broadband reconfigurable quarter-wave plate is designed for the 8 µm to 12 µm band and subsequently shown to exhibit improved performance when compared to conventional geometries from previous work [29]. Finally, broadband quarter-wave and half-wave plate Bézier metasurface designs are presented with bandwidths surpassing the 3 µm to 5 µm band, once again demonstrating superior performance to that of conventional metasurface geometries.

2. Analytical expressions for reflection from anisotropic metasurface quarter-wave and half-wave plates

For a plane wave normally incident to the surface of a planar metamaterial, the interaction, or more specifically the linear, lossless scattering of polarized light, can be described in a manner similar to Jones matrix calculus [59–62], where the amplitudes of the reflected wave in terms of co-polarization and cross-polarization for a two port system are contained in a 2 by 2 matrix. A reflection matrix is used to define the co-polarized and cross-polarized reflection for the two port system where:

(\begin{matrix} Γ_{x x} & Γ_{x y} \\ Γ_{y x} & Γ_{y y} \end{matrix}) = (\begin{matrix} a & b \\ c & d \end{matrix}) .

Using this formalism, the reflection from an anisotropic metasurface is expressed analytically for a quarter-wave plate and a half-wave plate in the following sections.

2.1 Scattering from an anisotropic metasurface quarter-wave plate

Given the reflection matrix from Eq. (1), the reflection eigenvalues are determined as follows:

\det (\begin{matrix} a - λ^{s} & b \\ c & d - λ^{s} \end{matrix}) = (a - λ^{s}) (d - λ^{s}) - b c = 0,

Since an anisotropic metamaterial can be created by utilizing a geometry with two-fold mirror symmetry, we have

a = d

and

| b | = | c |

[63]. As a result, the eigenvalues reduce to:

λ_{1, 2}^{s} = a \pm | b | e^{i θ},

where the desired phase difference for circular polarization is

θ = \frac{π}{2} .

The power fraction of the scattered wave, S, can be expressed as [63]:

S = - a = \cos^{2} (\frac{θ}{2}),

which, in this case, gives:

a = - 1 / 2.

Further constraints can be imposed on the scattering matrix coefficients. Specifically, energy conservation dictates that [63]:

{| a + \frac{1}{2} |}^{2} + {| b |}^{2} = {(\frac{1}{2})}^{2} .

Therefore, applying

a = - 1 / 2

to Eq. (7) results in

| b | = 1 / 2.

Returning to Eq. (3), we have:

λ_{1, 2}^{s} = \frac{1}{2} (- 1 \pm i) .

In turn, a representative reflection matrix for the metasurface can now be expressed in the form,

Γ_{φ = 0} = \frac{\sqrt{2}}{2} (\begin{matrix} e^{i \frac{3 π}{4}} & 0 \\ 0 & e^{- i \frac{3 π}{4}} \end{matrix}) .

Since the metamaterial is assumed to be anisotropic, the eigenvalues resulting in circular polarization are valid for azimuthal rotations which satisfy

φ = π / 4 + n π / 2

(measured between the incident field and the metamaterial line of symmetry). Equation (9) can be expressed in terms of

φ

as,

Γ (φ) = R (φ) \cdot Γ_{φ = 0} \cdot R (- φ),

where the rotation matrix is given as,

R (φ) = (\begin{matrix} \cos φ & - \sin φ \\ \sin φ & \cos φ \end{matrix}) .

To demonstrate the polarization conversion of the anisotropic metamaterial, a normally incident plane wave,

{\bar{E}}_{i n c} = E_{0} (\hat{x} \cos φ + \hat{y} \sin φ) e^{i (k z - ω t)},

is assumed to be horizontally polarized (i.e.,

φ = 0

) and in phase at the interface of the metasurface such that in matrix form,

{\vec{E}}_{i n c} = (\begin{matrix} 1 \\ 0 \end{matrix}),

where, in order to simplify successive derivations, the amplitude has been normalized while the time and frequency dependence has been suppressed. Moreover, the direction of propagation is retained for the purpose of referencing polarization orientation and is subsequently represented by the vector notation introduced in Eq. (13). With the field incident upon a rotated metasurface, the resultant interaction can be described by the reflection and transmission matrices:

{\overset{\leftarrow}{E}}_{r} = \frac{1}{2} (\begin{matrix} - 1 & i \\ i & - 1 \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix}) = \frac{1}{2} (\begin{matrix} - 1 \\ i \end{matrix}) = \frac{\sqrt{2}}{2} {\overset{\leftarrow}{E}}_{R H C P}

and

{\vec{E}}_{t} = \frac{1}{2} (\begin{matrix} 1 & i \\ i & 1 \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix}) = \frac{1}{2} (\begin{matrix} 1 \\ i \end{matrix}) = \frac{\sqrt{2}}{2} {\vec{E}}_{L H C P} .

Thus, the metasurface scattering consists of a purely circularly polarized reflected wave and a purely circularly polarized transmitted wave of opposite handedness and equal amplitude (see Fig. 1).

Fig. 1 Scattered field handedness, amplitude, and propagation direction for an anisotropic metasurface illuminated by a linearly polarized incident field.

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If the transmitted wave is reflected backwards via the introduction of a conductive sheet with an electrical standoff length of $l = λ / 4$ , then the reflection coefficient becomes $Γ = + 1$ . Once the change in coordinate systems for reverse propagation is accounted for, the reflected electric field is:

{\overset{\leftarrow}{E}}_{r, g r o u n d} = \frac{1}{2} (\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix} 1 \\ i \end{matrix}) = \frac{1}{2} (\begin{matrix} - 1 \\ i \end{matrix}) .

As a result the cumulative reflection from the metasurface will now exhibit the desired quarter-wave plate response. This behavior can be expressed in the following manner:

{\overset{\leftarrow}{E}}_{r, t o t a l} = \sqrt{{(\frac{1}{2} (\begin{matrix} - 1 \\ i \end{matrix}))}^{2} + {(\frac{1}{2} (\begin{matrix} - 1 \\ i \end{matrix}))}^{2}} = \frac{\sqrt{2}}{2} (\begin{matrix} 1 \\ i \end{matrix}) = {\overset{\leftarrow}{E}}_{R H C P},

where the intensities of the two waves are summed, resulting in a right hand circularly polarized (RHCP) reflected wave. While seemingly restrictive, the behavior in Eq. (17) is readily achieved over a considerable bandwidth when the anisotropic metasurface exhibits an orthogonal capacitive and inductive impedance tensor exhibiting low dispersion over the bandwidth of interest. This is achieved by creating a sufficiently high Q-factor resonance above the band of interest and an orthogonal Q-factor resonance below the band of interest for the capacitive and inductive impedance tensors respectively. A quarter-wave plate design covering the 8 µm to 12 µm band which also includes a reconfigurable response is introduced and discussed in Section 4.1. A second design covering the 3 µm to 5 µm band is considered in Section 4.2.

2.2 Reflection from an anisotropic metasurface half-wave plate

In order to determine the desired half-wave plate reflection response, we begin with a general expression for reflection from an anisotropic metasurface at a rotation angle $φ = π / 4$ ,

Γ_{φ = \frac{π}{4}} = (\begin{matrix} \frac{1}{2} Γ_{x} + \frac{1}{2} Γ_{y} & \frac{1}{2} Γ_{x} - \frac{1}{2} Γ_{y} \\ \frac{1}{2} Γ_{x} - \frac{1}{2} Γ_{y} & \frac{1}{2} Γ_{x} + \frac{1}{2} Γ_{y} \end{matrix}) .

As such, the general expression for the desired reflected field is given as:

{\overset{\leftarrow}{E}}_{r} = (\begin{matrix} \frac{1}{2} Γ_{x} + \frac{1}{2} Γ_{y} & \frac{1}{2} Γ_{x} - \frac{1}{2} Γ_{y} \\ \frac{1}{2} Γ_{x} - \frac{1}{2} Γ_{y} & \frac{1}{2} Γ_{x} + \frac{1}{2} Γ_{y} \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ 1 \end{matrix}) .

Solving for the reflection coefficients yields,

Γ_{x} = 1 and Γ_{y} = - 1.

Under these conditions Eq. (19) simplifies to,

{\overset{\leftarrow}{E}}_{r} = \frac{1}{2} (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ 1 \end{matrix}) = {\overset{\leftarrow}{E}}_{y} .

Once again, the behavior in Eq. (21) is readily achieved over a considerable bandwidth when, in this case, an anisotropic metasurface exhibits an orthogonal short circuit impedance (i.e. non resonant) and a very high Q-factor resonance (i.e. open circuit) at the center frequency. A half-wave plate design covering the 3 µm to 5 µm band is presented in Section 4.3.

3. Synthesis of anisotropic Bézier metasurfaces

In previous work, several known anisotropic geometries (the end-loaded dipole, the meander line, and the split ring) were studied and subsequently shown to exhibit broadband quarter-wave plate behavior [29]. Recent work has also investigated the L-resonator [35] and the Y-resonator [39]. Yet, additional methods for synthesizing and subsequently optimizing novel metasurface geometries applicable to not only manipulating polarization but also other applications are currently limited to pixelized structures synthesized via the genetic algorithm (GA) [51–58]. In this work we introduce a synthesis technique where a real valued optimizer CMA-ES [49, 50] is used to optimize the control points of a Bézier surface.

3.1 Mathematical representation of a Bézier surface

Historically, the origin of the Bézier curve and Bézier surface can be traced to Bernstein’s proof of the Weierstrass approximation theorem. The Weierstrass approximation theorem states that polynomials can uniformly approximate any function which is continuous over a given interval. Bernstein’s proof of the Weierstrass theorem employs only basic algebraic functions; thus, its form is conducive to rapid computation. The Bernstein polynomial with any continuous function f(t) is given as [64],

P_{n} (x) = \sum_{k = 0}^{n} f (k / n) B_{k}^{n} (x)

where the Bernstein basis is defined as,

B_{k}^{n} (x) = (\begin{matrix} n \\ k \end{matrix}) x^{k} {(1 - x)}^{(n - k)}, k = 0, ..., n .

In the early 1960s, De Casteljau and Bézier began investigating mathematical tools applicable to automotive body design which would provide intuitive and accurate methods to create and then subsequently convey complex shapes which were not easily defined by simple geometric parameters. Their work lead to an adaptation of the Bernstein polynomial to what is now referred to as the Bézier curve [64]:

P_{n} (x) = \sum_{k = 0}^{n} C_{k} B_{k}^{n} (x)

where C_k represents the amplitude and position for the control points to be manipulated. An example of a Bézier curve is shown in Fig. 2; where only five (i.e., n = 5) control points are used to define the complex curve.

Fig. 2 A Bézier curve (blue) plotted as a function of x with n = 5 control points (black).

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This approach can now be extended to three dimensions via the introduction of a Bézier surface which is simply represented as:

P_{n, m} (x, y) = \sum_{i = 0}^{n} \sum_{j = 0}^{m} C_{i, j} B_{i}^{n} (x) B_{j}^{n} (y) .

The control points are now defined as amplitudes contained within a matrix of arbitrary size (m x n):

C_{i, j} = (\begin{matrix} C_{11} & C_{12} & \dots & C_{1 m} \\ C_{21} & C_{22} & \dots & C_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ C_{n 1} & C_{n 2} & \dots & C_{n m} \end{matrix})

An example of a Bézier surface is shown in Fig. 3(a), where a five by five matrix (i.e., m = 5, n = 5) of control points is used to define the complex surface. Figure 5(b) depicts the positive values of the Bézier surface projected onto the z = 0 plane. This projection illustrates the approach used to synthesize metasurface geometries in following sections.

Fig. 3 Plot of a m = 5, n = 5 Bézier surface. (a) Three dimensional surface plot (color) with control points shown (black). (b) Projection of positive values of the Bézier surface onto the z = 0 axis.

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3.2 Comparison of optimized Bézier and pixelated quarter-wave plate metasurfaces

It can be reasonably inferred that fabricating structures synthesized via Bézier surfaces will have an inherent advantage over the conventionally used pixelated structures when considering relative geometrical complexity, especially for optical metasurface-enabled devices. This is attributed to the greater ease in the manufacturing of the Bézier surface’s continuous contours versus the potential difficulty in realizing the finer features of a pixelated structure, especially in the case of numerous diagonally adjacent pixels. In addition to fabrication complexity of generated designs, another concern when optimizing structures is the overall time it takes to complete the required simulations. Global optimizers generally take thousands of evaluations to arrive at good results; thus, considering that full-wave electromagnetic solvers are notoriously slow, any demonstrable increase in the speed of an optimization is highly desirable. In order to assess the computational efficiency of the Bézier surface optimization, a GA is used as a reference optimizer. The binary GA has been widely used to generate pixelized metasurfaces at radio frequencies (RF) and optical wavelengths for several decades [51–58]. While the GA is a binary algorithm, the Bézier surface is optimized by a real valued algorithm CMA-ES which has been shown to be very efficient and reliable at solving real valued problems [49, 50]. CMA-ES is linked to the Bézier surface through the optimization of a 10 by 5 matrix of control points (see Eq. (26)) with the values of individual control points ranging from −1 to 1. For the algorithms’ comparison, the initial populations for both algorithms are generated by the same 20 seeds.

The use of the same seeds allows for a direct comparison of the two algorithms by ensuring that they have the same initial random populations; whereas the use of multiple seeds helps in assessing the overall algorithm performance. The progress and overall performance of the algorithms are assessed by their ability to minimize the error from ideal performance as a function of the number of completed simulations. The deviation from optimal performance is defined by a fitness function which for this study is defined as:

f i t n e s s = \sum_{f r e q u e n c y} ({(1 - I)}^{2} + {(Q)}^{2} + {(U)}^{2} + {(1 - | V |)}^{2}),

where the Stokes parameters [65] are utilized to assess the degree of circular polarization across the 8 µm to 12 µm band. The Stokes parameter

I

corresponds to the intensity of the reflected or transmitted wave. The Stokes parameter

\pm Q

corresponds to horizontally and vertically oriented linear polarizations respectively. Similarly, the

\pm U

Stokes parameter corresponds to horizontal or vertical linear polarizations rotated 45° degrees counter clockwise. Lastly, the

\pm V

Stokes parameter corresponds to left hand and right hand circular polarization respectively. In Cartesian coordinates, the Stokes parameters are calculated as follows:

I = {| E_{x} |}^{2} + {| E_{y} |}^{2},

Q = {| E_{x} |}^{2} - {| E_{y} |}^{2},

U = 2 Re (E_{x} E_{y}^{*}),

V = - 2 Im (E_{x} E_{y}^{*}) .

Figure 4(a) depicts the performance of each optimization over 20 seeds. For both algorithms the average fitness of all seeds, the mean fitness of all seeds, and the seed with the best fitness are each plotted in terms of fitness value versus the number of function evaluations. Figure 4(b) and Fig. 4(c) depict the best seed’s performance for the GA (pixelized surface) and CMA-ES (Bézier surface) respectively. It is clear that the Bézier surface optimized via CMA-ES not only provides better final results than the GA optimized pixelated surfaces; but, it also achieves superior solutions with more than an order of magnitude reduction in the number of required function evaluations.

Fig. 4 Performance comparison of pixelized structure GA optimizations versus Bézier surface CMA-ES optimizations. (a) Average mean and best fitness over 20 seeds for GA and CMA-ES optimizations. (b) Stokes parameters for the best GA optimized seed. (c) Stokes parameters for the best CMA-ES optimized seed.

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4. Further applications of optimized Bézier metasurfaces for polarization control at IR wavelengths

4.1 Reconfigurable quarter-wave plate Bézier metasurface covering 8 µm to 12 µm band

In this section, the performance of a reconfigurable Bézier metasurface is compared to a reconfigurable split ring (SR) metamaterial structure reported in previous work [29]. The pertinent Stokes parameters are shown plotted in Fig. 5 for both the SR and Bézier metasurfaces. It can be seen that the Bézier metasurface provides a higher intensity as well as a broader band quarter wave response. The unit cells of the respective structures are depicted in the insets. The SR unit cell is 975 nm by 975 nm and the SR itself is comprised of 150 nm thick Au, supported by a 465 nm Ge₂Sb₂Te₅ (GST) substrate, which is backed by a 150 nm Au ground plane. The Bézier unit cell is 1300 nm by 100 nm (a square 1 by 13 unit cell array is shown in Fig. 5(a)) and is comprised of 150 nm thick Au, supported by a 500 nm GST substrate, which is backed by a 150 nm Au ground plane. Figure 5(b) shows the angular response of the Bézier metasurface, demonstrating a very wide field of view (FOV).

Fig. 5 Full wave simulations at normal incidence comparing the SR’s CP response versus that of the optimized Bézier metasurface (a). Full wave simulations of the Bézier metasurface’s CP response as a function of oblique incidence angle (b).

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GST is a chalcogenide glass (ChG) phase change material (PCM) which can be switched between two bistable material phases (amorphous or crystalline) via direct or indirect heating (such as electrically or optically) [66–77]. In the crystalline state the substrate becomes conductive, resulting in a mirror like reflection with no polarization change in the reflected wave. The metasurface functions as a circular polarizer when the GST is in the amorphous state (i.e., high n dielectric). A comparison between the two states as a function of azimuthal rotation is represented by normalized Stokes parameters and shown in Fig. 6(a) and Fig. 6(b) for the amorphous and crystalline states respectively. The measured values utilized for GST at λ = 10µm are n = 4.2, k = 0.01 in the amorphous state and n = 8, k = 4.8 in the crystalline state. The full wave simulations consider all losses using material parameters measured via spectroscopic ellipsometry.

Fig. 6 Normalized Stokes parameters as a function of azimuthal rotation when GST is in (a) the amorphous state and (b) the crystalline state.

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4.2 Quarter-wave plate Bézier metasurface for the 3 µm to 5 µm band

Next, a quarter-wave plate Bézier metasurface is optimized to cover the 3 µm to 5 µm band. The optimized Bézier unit cell is 2050 nm by 200 nm (a square 1 by 10 unit cell array is shown in the Fig. 7(a) inset). The surface is comprised of a 150 nm thick layer of patterned Au, supported by a 370 nm SiO₂ (n = 1.4) substrate, and then backed by a 150 nm Au ground plane. The results are displayed in terms of pertinent Stokes parameters in Fig. 7(a) and the normalized azimuthal response is shown in Fig. 7(b). The results demonstrate an extremely wide band quarter-wave response, far surpassing the targeted 3 µm to 5 µm range.

Fig. 7 Full wave simulation results of an optimized quarter-wave plate Bézier metasurface (inset) expressed in terms of the Stokes parameters (a) as a function of frequency and (b) as a function of azimuthal rotation.

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4.3 Half-wave plate Bézier metasurface for the 3 µm to 5 µm band

Finally, a half-wave plate Bézier metasurface is optimized to also cover the 3 µm to 5 µm band. The optimized Bézier unit cell is 1700 nm by 1550 nm where the surface is comprised of a 150 nm thick patterned Au layer, supported by a 650 nm SiO₂ (n = 1.4) substrate, and then backed by a 150 nm Au ground plane. The results are illustrated in terms of pertinent Stokes parameters in Fig. 8(a) while the normalized azimuthal response is shown in Fig. 8(b). These results demonstrate an extremely wide band half-wave response, once again surpassing the desired 3 µm to 5 µm range and comparing favorably to the findings of previous work [34].

Fig. 8 Full wave simulation results of an optimized half-wave plate Bézier metasurface (inset) expressed in terms of the Stokes parameters (a) as a function of frequency and (b) as a function of azimuthal rotation.

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4. Conclusion

In this work a new technique for the synthesis of 2D periodic anisotropic metasurfaces was presented. Specifically, a global optimization scheme based on the CMA-ES was used to synthesize Bézier metasurfaces and the computational efficiency was shown to be an order of magnitude faster than the Genetic Algorithm. Furthermore, the Bézier metasurface was shown to exhibit superior performance to the pixelized structures and other known metasurface geometries found in the literature. To demonstrate this, a broadband reconfigurable quarter-wave plate Bézier metasurface was synthesized for the 8 µm to 12 µm band that exhibited better performance when compared to other known geometries. Additionally, broadband quarter-wave plate and half-wave plate Bézier metasurface designs were presented for the 3 µm to 5 µm band which also demonstrate significantly improved performance when compared to other known metasurface geometries.

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Infrared broadband quarter-wave and half-wave plates synthesized from anisotropic Bézier metasurfaces

Abstract

1. Introduction

2. Analytical expressions for reflection from anisotropic metasurface quarter-wave and half-wave plates

2.1 Scattering from an anisotropic metasurface quarter-wave plate

2.2 Reflection from an anisotropic metasurface half-wave plate

3. Synthesis of anisotropic Bézier metasurfaces

3.1 Mathematical representation of a Bézier surface

3.2 Comparison of optimized Bézier and pixelated quarter-wave plate metasurfaces

4. Further applications of optimized Bézier metasurfaces for polarization control at IR wavelengths

4.1 Reconfigurable quarter-wave plate Bézier metasurface covering 8 µm to 12 µm band

4.2 Quarter-wave plate Bézier metasurface for the 3 µm to 5 µm band

4.3 Half-wave plate Bézier metasurface for the 3 µm to 5 µm band

4. Conclusion

References and links

Cited By

Figures (8)

Equations (31)

Optics Express