Ultrathin wave plates based on bi-resonant silicon Huygensโ metasurfaces
Zhiyuan Fan1 and Gennady Shvets1,*
1School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853
*Corresponding author: [email protected]
Abstract
An all-dielectric Huygens metasurface supporting electric and magnetic resonances is a promising platform for a variety of optical applications that require a combination of extremely high transmission and broad-range control of the phase of the transmitted light. A highly efficient wave plate is one example of such an application. In combination with high spectral selectivity and strong optical energy concentration, such phase plates are desirable for precision sensing and high efficiency nonlinear optics. We propose a novel approach to realizing such ultra-thin optical plates: an anisotropic Fano-resonant optical metasurface (AFROM) employing the combination of a spectrally sharp electric and a relatively broadband magnetic resonance. The phase shift coverage approaching ๐๐ is achieved through judicious choice of the geometric parameters of a complex unit cell. A new methodology based on eigenvalue simulations of leaky magnetic/electric resonances enables rapid computational design of such metasurfaces.
1. INTRODUCTION Polarization is a fundamental degree of freedom of a light wave, along with its intensity and phase. The ability to manipulate the state of polarization (SOP) is crucial for a variety of optical application such as glare reduction, surface characterization, detection of chiral molecular enantiomers, and even improved laser machining [ 1], to name just a few. Changing lightโs polarization requires changing the phase relationship between the two orthogonal components of lightโs electric field components, i.e. ๐ธ๐ฅ and ๐ธ๐ฆ for the light waves propagating in the
๐ง โdirection. This is usually accomplished by wave plates (also known as phase retarders) that employ natural birefringent materials with different refractive indices ๐๐ฅ and ๐๐ฆfor the two linear polarizations.
The need for novel approaches to polarization manipulation of light is particularly urgent in the mid-wave infrared (mid-IR) spectral range, where fewer standard phase-retarding optical components are available than in the visible rage despite the rapidly growing interest in mid-IR applications such as free-space optical communications [ 2, 3], remote gas sensing [ 4], and Stokes polarimetry [ 5, 6, 7].
Some of the most common polarization conversion applications involve transforming linearly polarized (LP) light into either left- or right-hand circularly polarized (LCP or RCP) states (quarter-wave plates or three-quarter-wave plates), or rotating the LP light by 90โ(half-wave plates). For example, LP-to-CP conversion is important in high-power applications that involve laser processing of materials (e.g., hole drilling), as well as for suppressing specular reflections. Similarly, half-wave plates, in combination with polarizing beam splitters, can be used for variable light attenuation.
One obvious limitation of standard wave plates is their thickness. The
typically small refractive index contrast ฮ๐ = |๐๐ฅ โ ๐๐ฆ| cases the
wave plates thickness โ to be larger than the targeted wavelength of light ๐ . Metamaterials and metasurfaces [ 8, 9, 10] are natural
candidates for enhancing the effective birefringence and, therefore, shrinking the size of the wave plates. A number of distinct types of metasurfaces (reflective, transmissive, metallic, or dielectric) have been recently deployed for polarization control and conversion [ 8, 9, 10, 11, 12, 13]. While plasmonic metasurfaces can be made extremely thin (tens of nanometers), they have limited conversion efficiency [ 10] and low laser damage threshold. Many of the reported birefringent dielectric metasurfaces [ 14, 15, 16, 17, 12] have the thickness โ โผ ๐. This presents a particular challenge to mid-IR applications that would require fairly thick metasurfaces.
In addition to their sub-wavelength dimensions, several key requirements must be met by polarization-controlling transmitting metasurfaces in order to make them appealing to the broadest range of applications. First, many high-power applications require extremely high transmittance ๐ because even the smallest amount of loss or reflection can, correspondingly, damage the metasurface or destabilize the high-power laser system. For any metasurface deployed for such applications, the starting design point must be ๐ = 1 at the operating wavelength because the inevitable imperfections of the photonic structure will reduce the transmission. Stringent transmittance requirements rule out promising plasmonic metasurface concepts [ 11, 10, 18, 19] because of their Ohmic loss at optical frequencies.
Second, polarization-converting metasurfaces have to be extremely anisotropic in order to provide the largest possible range of phase differences ฮ๐ = ๐๐ฆ โ ๐๐ฅ between the two polarizations. For
example, it requires the phase difference coverage of at least ๐/2 <ฮ๐ < 3๐/2 range to convert LP light into either one of the two CP polarizations, or to rotate its polarization by 90โ . Ideally, an even broader range of ฮ๐ (close to 2๐) would be desirable to enable the transformation of LP light into arbitrarily elliptically polarized one. While many successful metasurface designs were shown to produce large phase retardations for one or both polarization [ 13, 20, 21, 22, 23], the challenge remains to produce a large range of ฮ๐ , all the while
maintaining subwavelength thickness of the metasurface combined with ultra-high transmittances ๐๐ฅ,๐ฆ for both polarizations.
One approach to achieving such high transmittance is to utilize a recently introduced [ 23, 24] concept of a dielectric Huygens (i.e. perfectly transmitting) metasurface that relies on combining a pair of crossed electric dipole active (E-) and magnetic dipole active (M-) modes that are simultaneously excited by an incident light wave. Such structures can be based on silicon resonant meta-atoms that combine the benefits of CMOS compatibility with the ease of single-layer lithographic fabrication. Si-based metasurfaces have negligible Ohmic losses at mid-IR, while the reflection is suppressed due to the simultaneous excitation of the electric and magnetic dipole active modes [ 25, 26, 27, 28]. It has been demonstrated [ 20, 23] that matching of the frequencies and lifetimes of the E- and M-modes of Si disk arrays is sufficient for a unity transmission with a nearly-2๐ phase change. However, the proposed designs are isotropic and do not lend themselves to making Huygens phase plates.
2. SUMMARY OF THE RESULTS In this article, we show a new approach to designing Huygens wave plates using anisotropic metasurfaces comprised of silicon meta-molecules with broken mirror symmetry. This asymmetry enables resonant excitation of high quality factor (high-Q) E- and M-modes by only one linear polarization of light (e.g., the y-polarization). The orthogonal (non-resonant) polarization of light does not significantly interact with the metasurface, so its transmission phase ๐๐ฅ is nearly frequency-independent while its transmittance ๐๐ฅ โ 1.
Using a coupled mode theory (CMT) [ 29, 30, 31, 32, 33], we show that for any set of complex-valued resonant frequencies ๏ฟฝฬ๏ฟฝ๐ธand ๏ฟฝฬ๏ฟฝ๐ of the E- and M-modes, respectively, there exists a real-valued Huygens frequency ๐ โก ๐๐ป for which the transmittance of the resonant polarization is unity: ๐๐ฆ(๐๐ป) = 1 . The phase ๐๐ฆ of the transmitted
resonant polarization at the Huygens frequency can assume any value in the 0 < ๐๐ฆ < 2๐ range depending on the relative spectral positions
of the E- and M-resonances. Therefore, by varying the resonant frequencies of the electric and magnetic dipole resonances of a meta-molecule with respect to each other, we can achieve near-100% transmittance with an arbitrary phase difference between the two polarizations, thus enabling a wide variety of functional phase plates. The reductive description of the Huygens metasurface in terms of its two resonances and their respective frequencies ๐๐ธ(๐) and radiative
lifetimes ๐พ๐ธ(๐)โ1 enables rapid search for the geometry of the
metasurfaces that provide the desired SOP of the transmitted light. While the described wave plate is inherently narrowband due to its
resonant nature, it has an important advantage of deeply subwavelength (โ < ๐/4) thickness which is important for a variety of mid-IR applications that do not require broadband operation (e.g., remote gas sensing). Moreover, as we show in the Supplemental Materials (SM) section, the bandwidth can be significantly broadened by straightforward design modifications. Transmissive metamaterial pixels with different Huygens frequencies can also be spatially multiplexed because their sharp resonant response does not rely on long-range interaction between metamolecules [ 34].
3. COUPLED MODE THEORY (CMT) OF A RESONANT WAVE PLATE In the framework of the CMT, the general description of the excitation of the two dipole-active modes by the ๐ฆ โ polarized light (which is assumed to be the resonant linear polarization) is given by the following equation:
(๐๐ธ
๐๐) = โ๐ (
๐ โ ๏ฟฝฬ๏ฟฝ๐ธ 00 ๐ โ ๏ฟฝฬ๏ฟฝ๐
)โ1
(๐ผ๐ธ๐ธ๐ฆ
๐ผ๐๐ป๐ฅ), (1)
where the two modes are, respectively, an electric mode resonating at ๐๐ธ and a magnetic mode resonating at ๐๐ . Here ๏ฟฝฬ๏ฟฝ๐ธ(๐) = ๐๐ธ(๐) โ
๐๐พ๐ธ(๐) are the complex-valued frequencies of the E- and M-modes,
respectively. Neglecting the non-radiative (Ohmic) losses, the inverse lifetimes of the modes are directly related to the modesโ far field coupling coefficients ๐ผ๐ธ(๐) according to ๐พ๐ธ(๐) โก ๐ผ๐ธ(๐)
2 . Without loss
of generality, we assume that the metasurface is surrounded by air on both sides and that the incident wave is planar (๐ธ๐ฆ = ๐ป๐ฅ).
To simplify the analysis, we further assume that that the electric dipole mode is much more narrow-band than the magnetic dipole mode, i.e. ๐พ๐ธ โช ๐พ๐ . The reported results do not rely on the above relationship between the lifetimes of the modes, which is used here to simplify the expression for the Huygens frequency. One specific design that is the focus of this paper is shown in Fig.2. Due to the mirror symmetry of the metamolecules, ๐ฅ โ and ๐ฆ โaxes form the principal axes. Thus, no cross-polarization conversion between ๐ฅ โ and ๐ฆ โpolarized light takes place, and the E- and M-modes shown in Figs. 2(b,c) are not excited by the ๐ฅ โpolarized light. The spectrally sharp E-mode emerges as a result of Fano interference [ 34, 35, 36] of the โdarkโ electric quadrupole mode with a โbrightโ electric dipole mode [ 34].
Under these assumptions, an explicit general form of the complex-valued transmission coefficient ๐ก๐ฆ โก ๐ก(๐) (which includes the phase
and transmittance information: ๐ก = โ๐๐๐๐ , where ๐(๐) โก ๐๐ฆ and
๐(๐) โก ๐๐ฆ) can be obtained [23, 37, 38] :
๐ก(๐) = 1 โ๐๐พ๐
๐ โ ๏ฟฝฬ๏ฟฝ๐
โ๐๐พ๐ธ
๐ โ ๏ฟฝฬ๏ฟฝ๐ธ
. (2)
Huygens transmission can be understood from Eq. (2) by observing that there always exists a frequency ๐๐ป such that ๐ก(๐๐ป) = ๐๐๐๐ป . This frequency, further referred to as the Huygens Frequency (HF), is given by the following simple expression:
๐๐ป(๏ฟฝฬ๏ฟฝ๐, ๏ฟฝฬ๏ฟฝ๐ธ) = ๐พ๐ธ๐๐ โ ๐พ๐๐๐ธ
๐พ๐ธ โ ๐พ๐โ ๐๐ธ , (3)
where the final simplification is obtained in the ๐พ๐ธ โช ๐พ๐ limit.
Fig. 1. Light transmission through a phenomenological bi-resonant metasurface: (a) amplitude, (b) phase, (c) complex-valued transmission coefficient. The metasurface is defined by Eqs.(1,2); two examples are considered: ๏ฟฝฬ๏ฟฝ๐/๐๐ป = 1.2 โ 0.2๐ (blue curves) and ๏ฟฝฬ๏ฟฝ๐/๐๐ป = 0.8 โ0.2๐ (red curves). The electric resonance is assumed to be narrow-band ( ๐พ๐ธ = 0.01๐๐ธ ); frequencies are normalized to the Huygens frequency ๐๐ป kept the same for both examples. Hollow (solid) circles correspond to ๐ = 0.9๐๐ป ( ๐ = 1.1๐๐ป ). (d) Transmission phases
๐(๐๐ป , ๏ฟฝฬ๏ฟฝ๐(1,2)
) at perfect transmission from Eq.(4). Dashed line: the
Huygens frequency. Black dashed line indicates the necessary condition for HSA ๐๐ป โ ๐๐ธ . The arrows indicate the fractional frequency detuning ๐ฅ = (๐๐ โ ๐๐ธ)/๐๐ธ between electric and magnetic modes.
The transmittance spectra ๐(๐) are plotted in Fig. 1(a) for two
examples (blue and red solid lines) corresponding to ๏ฟฝฬ๏ฟฝ๐(1)
/๐๐ธ(1)
=
1.2 โ 0.2๐ (blue curves) and ๏ฟฝฬ๏ฟฝ๐(2)
/๐๐ธ(2)
= 0.8 โ 0.2๐ (red curves).
The corresponding frequencies of the electric resonances, ๐๐ธ(1)
and ๐๐ธ(2)
, were selected in such a way that unitary transmission is
achieved at the same HF: ๐ = ๐๐ป . It follows from Eq. (3) that, in the
limit of a high-Q electric resonance, ๐๐ป โ ๏ฟฝฬ๏ฟฝ๐ธ(1)
โ ๏ฟฝฬ๏ฟฝ๐ธ(2)
is very close to
that of the electric resonance, but far from the magnetic resonance
frequencies satisfying |๐๐(1)
โ ๐๐(2)
| โผ ๐พ๐(1,2)
โซ ๐พ๐ธ .
Under the perfect transmission assumption, the Huygens phase ๐๐ป โก ๐(๐ = ๐๐ป , ๏ฟฝฬ๏ฟฝ๐) is given by
๐๐ป(๐, ๏ฟฝฬ๏ฟฝ๐) = โ2 tanโ1 (๐พ๐
ฯ โ ๐๐), (4)
where the interpretation of Eq.(4) is as follows: for any values of ๐ and ๏ฟฝฬ๏ฟฝ๐ , there exists a complex-valued electric resonance frequency ๏ฟฝฬ๏ฟฝ๐ธ , calculated from Eq. (3), such that ๐ = ๐๐ป . For those parameters, the transmission is perfect, and the transmitted phase is given by Eq. (4) at ๐ = ๐๐ป . Note that the zeros of the transmission amplitudes
๐(1,2)(๐) shown in Fig. 1(a) occur at ๐๐ธ(1)
and ๐๐ธ(2)
, respectively.
The plots of the transmitted phases ๐(1,2)(๐) in Fig.1(b)
demonstrate that the transmission phases ๐(1)(๐) โ ๐(2)(๐) for all frequencies (including ๐ = ๐๐ป) in the two examples corresponding to the M-modesโ resonance on the blue (blue curve) and the red (red curve) sides of the E-modeโs resonance. Specifically, it can be observed
from Fig.1(c) that ๐๐ป โก ๐๐ป(1)
= ฯ/2 for the first example, while ๐๐ป โก
๐๐ป(2)
= 3ฯ/2 for the second example. Therefore, the phase shift of
180โ can be accomplished by sweeping the frequency of the magnetic resonance across a sharp electric resonance that determines the Huygens frequency.
Because perfect transmission is achieved only for ๐ = ๐๐ป , the
intersection of the ๐๐ป(๐, ๏ฟฝฬ๏ฟฝ๐(1)
) and ๐๐ป(๐, ๏ฟฝฬ๏ฟฝ๐(2)
) plots (solid line) and
the ๐ = ๐๐ป (dashed line) in Fig.1(d) recovers the Huygens phases,
๐๐ป(1)
and ๐๐ป(2)
, of the perfectly-transmitted light. Clearly, by choosing a
family of structures with ๐๐(1)
< ๐๐(๐)
< ๐๐(2)
, we can provide the
phase shift coverage of ๐/2 < ๐๐ป(๐)
< 3๐/2. Note that the difference
between ๐๐ธ(1)
and ๐๐ธ(2)
is of order ๐พ๐ธ(1,2)
โช ๐พ๐(1,2)
, so the frequencies of
the two E-modes are indistinguishable in Fig.1(d).
Fig. 2. A specific implementation of dielectric anisotropic Fano-resonant optical metasurface supporting magnetic and electric resonances. (a) A schematic and geometric parametersโ definitions of a unit cell of a Si Huygens metasurface. Color-coded ๐ธ๐ฆ for an electric (b)
and magnetic (c) resonances. Red: ๐ธ๐ฆ > 0, blue: ๐ธ๐ฆ < 0.
The phenomenological examples analyzed in Fig.1 show that, by sweeping the frequency of the magnetic resonance across that of the electric one by approximately one radiative linewidth of the former, we
can achieve more than 180 degrees of phase shift. While the idea of bringing the magnetic and electric resonances close to each other has been discussed in the past [ 20, 23] , the concept of swapping the spectral order the E-mode and M-mode for obtaining arbitrary phase shifts is discussed here for the first time.
The example presented above points to a viable and so far unexplored strategy for designing a lossless Huygens metasurface capable of imparting an essentially arbitrary phase shift to transmitted light. Our strategy involves starting with a sufficiently complex (see, for example, Fig.2) unit cell that supports a sharp electric and a broad magnetic resonances, and then perturbing the dimensions of the unit cell so as to keep the frequency of the electric resonances fixed at ๐๐ธwhile significantly varying the frequency of the magnetic resonance ๐๐ . If the topology of the unit cell possesses a sufficient number of degrees of freedom, then this strategy can be easily implemented using the following straightforward procedure (see SI for more details). For each ๐ โth set of unit cellโs geometric parameters, we numerically
calculate the corresponding frequency triplets ๐(๐) โก (๐๐ธ(๐)
, ๐๐(๐)
, ๐พ๐(๐)
),
and then down-select only those members of the set that correspond to
the same electric dipole frequency ๐๐ธ(๐)
โก ๐๐ป .
4. EXAMPLE OF A SPECIFIC DESIGN: FANO-RESONANT ANISOTROPIC METASURFACE
One specific design of a dielectric-based AFROM satisfying the ๐พ๐ธ โช๐พ๐ condition is shown in Fig.2, where the unit cell is comprised of two unequal (straight and C-shaped) Si antennas. The asymmetry between the antennas gives rise to the high-Q E-mode shown in Fig.2(b). The finite thickness โ of the metasurface is responsible for the low-Q M-mode shown in Fig.2(c). The high-Q electric dipole mode emerges as a result of a mirror-symmetry breaking that hybridizes the โdarkโ electric quadrupole mode with a โbrightโ electric dipole mode [ 34]. The new hybridized electric mode is modeled here using a very small radiative decay constant ๐พ๐ธ. This modesโ hybridization can be interpreted as their Fano interference [ 34, 35].
For the unit cell shown in Fig.2, the geometric parameters set is given
by ๐ โก {๐ฟ๐ฅ , ๐ฟ๐ฆ, ๐ค1, ๐ค2, ๐} , where the dimensions are labelled in
Fig.2(a). The corresponding frequency vectors ๐(๐)are obtained from the eigenvalue simulations using the COMSOL Multiphysicsยฎ [ 39] software. The simulations also reveal the field distributions of ๐ธ๐ฆ
corresponding to the electric (shown in Fig. 2(b)) and magnetic (shown in Fig.2(c)) resonances. The magnetic mode has a magnetic dipole moment in the ๐ฅ-direction, and the electric mode has an electric dipole moment in the y-direction. The AFROMโs optical response to ๐ฅ -polarized light can be neglected because of the high aspect ratio of each other two antennas.
We numerically confirmed that the assumption of the sharpness of the electric resonance (๐พ๐ธ โช ๐พ๐) is indeed satisfied for a very broad range of geometric parameters. In the rest of the paper we keep the metasurface thickness โ fixed because of the practical convenience of fabricating planar structures vs the three-dimensional ones. Therefore, in what follows all frequencies are normalized to ๐0โ = 2๐๐/โ . In addition, to ensure that a metasurface does not become too large or too small, the unit cells are arranged in a square lattice with a fixed period ๐. Through the rest of the paper, we assume ๐ = 3๐๐ and โ = 1.2๐๐, and further confine the parameters space to the mid-infrared (mid-IR) spectral range, so that the normalized frequencies ๐/๐0โ โผ 0.25 , which implies that the thickness of the AFROM is of order ๐๐ป/4 of an operating wavelength ๐๐ป (i.e. it is strongly sub-wavelength).
To illustrate that the five-component geometric parameters vector ๐ provides sufficient freedom for keeping the electric resonance frequency constant while sweeping the magnetic resonance, we have reduced it even further by fixing the values of ๐ฟ๐ฆ , ๐ค1, and ๐ค2 (listed in
the caption to Fig.3), and varied the remaining two parameters: ๐ฟ๐ฅ and ๐. The results of finding the electric/magnetic modesโ eigenvalues as a function of these two design parameters are presented in Fig.3, where
the constant frequency contours ๐๐ธ(๐ฟ๐ฅ, ๐)(solid lines) and ๐๐(๐ฟ๐ฅ, ๐) (dashed lines) are plotted.
In addition, the calculated HP from Eq.(4) is presented in the same figure as a color-coded ๐๐ป(๐ฟ๐ฅ , ๐) map. For example, the ๐๐ธ =0.265๐๐โ contour is intercepted by all four ๐๐ = ๐๐๐๐ ๐ก contours. Based on the color-coded HP, we conclude that a phase difference ฮด๐๐ป โ 2 ๐๐๐ can be obtained by varying these two structure parameters while keeping the electric resonance frequency (and, to the lowest approximation according to Eq.(3), the Huygens frequency ๐๐ป โ ๐๐ธ) constant. Of course, all five dimensions can be independently varied. Therefore, much larger tunability of the Huygens phase can be achieved at a given operating frequency.
Fig. 3. Resonance frequencies (contours) and color-coded Huygens phase shift of air-suspended all-Si AFROM. White solid lines: ๐๐ธ(๐, ๐ฟ๐ฅ) = const. contours, white dotted lines: ๐๐ธ(๐, ๐ฟ๐ฅ) = const. contours. The complex-valued resonant frequencies were obtained from eigenvalue simulations using COMSOL Multiphysics software. The frequencies are normalized to ๐0โ and marked on the constant frequency contours. Fixed dimensions: ๐ = 3๐๐ , โ = 1.2๐๐ , ๐ค1 =๐ค2 = 0.4๐๐ , ๐ฟ๐ฆ = 1.8๐๐ . Huygens phase is obtained from Eq.(4).
Dielectric permittivity of Si: ๐Si = 12 . Geometric parameters are
defined in Fig.2.
Additional qualitative conclusions can be drawn from Fig.3. First, if relative variation of ๐ and ๐ฟ๐ฅ are to be kept reasonably small (as exemplified by the ranges shown in Fig.3), then there is an approximate relationship between the thickness of the metasurface and the operating wavelength that provides for the largest variance of ฮด๐๐ป with respect to the variations of ๐ and ๐ฟ๐ฅ. Specifically, by inspecting Fig. 3, we can deduce that the optimal operating frequency range is around ๐/๐0โ โผ 0.25, which corresponds to metasurface thickness of order ๐/4. Thinner metasurfaces would require much more extreme variations of all geometric parameters, and will not be considered further in our designs. We note that most of the earlier proposed phase plates designs that did not utilize electric and magnetic resonances either had to relax the transmission requirements [ 13], or to employ thicker metasurfaces [ 12].
More detailed simulations that include the variation of all five geometry parameters are presented in the SI section, and illustrated in Fig.S1. The 0.24 < ๐/๐๐โ < 0.28 frequency range is found to be optimal for constructing a family of metasurfaces that spans the ฮด๐๐ป โ5 ๐๐๐ range of Huygens phases. The calculation is based on the following two-step procedure: (i) extracting the complex-valued eigen frequencies ๏ฟฝฬ๏ฟฝ๐ธ and ๏ฟฝฬ๏ฟฝ๐ of the electric and magnetic resonances for the metasurface shown in Fig.2, and (ii) substituting them into Eq.(4) to determine the Huygens phase ๐๐ป(๐ = ๐๐ป , ๏ฟฝฬ๏ฟฝ๐). The effectiveness of this approach was verified using driven (i.e. with a normally incident EM wave) COMSOL simulations. Such driven simulations extract the phase of transmitted wave as function of the frequency of incident light. Phase
coverage of ๐ฟ๐๐ป โ 2ฯ phase is obtained for transmission efficiencies ๐ > 95% as shown in Fig.S3 of the SI.
So far our discussion was limited to calculating the phase shift ๐๐ฆ โก
๐๐ป of the ๐ฆ โpolarized light. However, the design of the AFROM shown in Fig. 2 is deliberately anisotropic in order to produce a polarization-transforming phase plate. We now proceed to demonstrate their performance by including their non-resonant response to ๐ฅ โpolarized light. Therefore, the search in the space of the five-component geometric parameters recipe needs to take into account the phase ๐๐ฅof the ๐ฅ-polarized light. We carried out the eigenmode-based search for those structures that support high-Q ๐ฆ โ polarized E-modes in the desired frequency range, and conduct full wave simulation that accepts only those geometries that satisfy the following conditions: (a) ๐๐ฅ, ๐๐ฆ >
0.95, and (b) ฮ๐ = ๐๐ฆ โ ๐๐ฅcorresponds to the desired polarization of
the transmitted light.
Fig. 4. Performance of four Si AFROM designs characterized by the Stokes parameters of the transmitted light: air-bridged metasurfaces operating as quarter-wave plates (a), half-wave plates (b), and three quarter-wave plates (c). (d) Same as (c), but with a quartz substrate and air superstrate. Horizontal axes: frequency shifts from ๐๐ป =0.272๐๐โ . Insets: the polarization state of transmitted beams. The incident beam is linearly polarized at 45โwith respect to the principal axes. Thickness of the metasurface: โ = 1.2๐๐. Huygens wavelength: ๐๐ป = 2๐๐/๐๐ป = 4.41๐๐. Other geometric sizes: see Table S3 of the SI.
The performance characteristics of three such designs that transform
linearly polarized incident (LP) light (๐ฌinc = ๐๐ + ๐๐) into left handed
circularly polarized light (LCP: quarter-wave plate), orthogonally polarized LP light (half-wave plate), and right handed circularly polarized light (RCP: three quarter-wave plate), are shown in Figs. 4(a-c), respectively. The geometric dimensions of the three wave plates, which operate at ๐๐ป = 0.272๐๐โ , are listed in Table S3 of the SI. Specifically, the three relevant Stokes parameters 2๐0(๐) โก (๐+45โ +๐โ45โ) โก (๐RCP + ๐LCP) (black dashed line), ๐2/๐0 โก (๐+45โ โ๐โ45โ)/(๐+45โ + ๐โ45โ) (blue dashed line), and ๐3/๐0 โก (๐RCP โ๐LCP)/(๐RCP + ๐LCP) (red solid line) are plotted in Figs. 4(a-c). By definition of the Huygens frequency, ๐0(๐๐ป) = 1, as shown in Figs.4(a-c) for all three wave plate designs. The effect of a low-index substrate (neglected in Figs. 3(a-c) , where air-bridged metasurfaces were assumed) is included in a representative design of a three quarter-wave design shown in Fig.4(d). The Si AFROM is assumed to be placed on a quartz substrate (๐SiO2
= 2.25). Note that the total transmission is still
very close to perfect, and the normalized Stokes parameter ๐3 โ โ1 signifying that the transmitted light is LCP.
5. CONCLUSIONS
We have demonstrated a new approach to designing perfectly transmitting (Huygens) phase plates based on utilizing two electromagnetic resonances of a high-index anisotropic metasurface. The examples presented in this article demonstrated that essentially any polarization state of light can be obtained from the incident LP wave. Although many high-intensity laser applications do not require broadband operation, we note that the specific design that was used as an example in this article has a very narrow spectral width (๐ โผ 500), However, the same approach of using unequal asymmetric dielectric antennas can be used for designing AFROMs with much lower spectral selectivity (๐ โผ 60) as shown in Fig. S6 of the SI.
Funding sources and acknowledgments. This work was supported by the Office of Naval Research Office Award N00014-17-1-2161.
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