## Abstract

This paper discusses a theoretical approach towards synthesizing broadband circularly polarizing reflectors. The broadband polarizing reflectors consist of birefringent metallo-dielectric metasurfaces which are described by the Jones matrices and verified via full-wave simulations. Specifically, full-wave simulations for candidate designs are presented that target operation in the near infrared band. In addition, reconfigurability is introduced and demonstrated for candidate designs at the long wave infrared band through the inclusion of a bistable phase change chalcogenide glass substrate.

©2013 Optical Society of America

## 1. Introduction

Electromagnetic metamaterials are engineered structures that utilize periodic sub-wavelength patterning to create an effective macroscopic response. The engineered resonances and symmetries of the metamaterial can mimic or facilitate behavior not found in naturally occurring materials [1,2]. Each unit cell is most commonly comprised of a planarized metallic inductor - capacitor (LC) resonant circuit along with a supporting low loss dielectric substrate. This approach, with the appropriate geometry scaling, allows metamaterials to be applied to radio waves, microwaves, terahertz waves, and even into the infrared and optical ranges where fabrication constraints and ohmic losses can become a limiting factor. The metamaterial unit cell geometry is generally treated as a flat (*d = λ*), two-dimensional periodic surface with a sub-wavelength unit cell width (*w ˂ λ*). As a result, propagation distance through the dielectric can generally be ignored, and diffraction is eliminated at normal incidence in addition to a considerable range of oblique angles. Metamaterials can be considered to be weakly spatially dispersive, due to the small periodicity of the unit cells compared to the characteristic wavelength. Thus, due to a reduced field interaction, the sub-wavelength design requires that an efficient, and in turn high Q-factor, LC-type resonator be employed, the most familiar of which is the split ring resonator (SRR) [3].

Recently, interest is transitioning from the inherent narrowband behavior of LC resonators to metamaterials that exhibit broadband behavior. Broadband metamaterial design methodologies represent an emerging technology with a wide range of potential applications [4–8]. For example, they can be utilized in conjunction with existing communication bands or designed to coincide with atmospheric transmission windows. In this paper, a theoretical analysis is presented on the synthesis of broadband circularly polarizing mirrors. Full wave simulations are performed on candidate designs with traditional dielectric substrates in addition to designs which incorporate bistable phase changing substrates, facilitating broadband polarization reconfigurability.

## 2. Scattering from a two dimensional infinitely periodic metamaterial

Let us assume that a planar metamaterial is infinitely periodic along the X - Y plane. A forward traveling plane wave propagates along a vector at an angle, θ, off surface normal and subsequently interacts with the metamaterial. This interaction, or more specifically the linear, lossless scattering of polarized light, can be described in a manner similar to Jones matrix calculus [9–13] as described elsewhere [14]. The forward propagating transmission matrix is given as:

and the corresponding reflection matrix:For convenience, these two lossless scattering relations are collected into the forward propagating scattering matrix [14]:With this information in hand, the scattering for additional incident directions on both sides of the metamaterial can be understood through the application of two electromagnetic principles. First, planar metamaterials can only couple to tangential electric fields and normal magnetic fields; thus, two electromagnetic waves that are described by the same field components must interact with the metamaterial in the same way [14]. Therefore, if the incident vector is rotated 180° around the Y axis, the resultant scattering matrix, as a function of the original scattering parameters, is given by the backwards scattering matrix:In the context of a circularly polarized (CP) wave, Eq. (4) results in [14],Next, in the absence of any static magnetic fields, we can apply the Lorentz reciprocity theorem [15]. When applied to oppositely propagating vectors the Lorentz reciprocity theorem results in the following scattering relationship:

where the subscripts*i*and

*j*designate the scattering matrix indices. In the context of a polarized wave, Eq. (7) results in [14],

Solving for the scattering matrix coefficients, the forward and backward scattering eigenvalues,${\lambda}_{1,2}^{s}$, are determined as follows:

*S*, can be expressed as [17]:resulting in$a=-1/2$for the desired CP condition - a transmitted CP wave and a reflected CP wave of opposite handedness and equal amplitude. Energy conservation dictates that [14]: Therefore, applying $a=-1/2$ to Eq. (22) implies that $\left|b\right|=1/2.$ Returning to Eq. (17), we have:The eigenvalues for forward and reverse propagation must be orthogonal; thus, $\theta =\pi /2.$ This gives:The scattering matrix for Eq. (1) can now be expressed as,

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

*i.e.*${\phi}_{x}=0)$, and in phase at the metamaterial’s interface, simplifying to,The resultant interactions with the reflection and transmission matrices are:

With an additional $\phi =\pm \pi /4$ azimuthal rotation, the reflection can be considered as maximized or minimized respectively. In other words, the metamaterial does not couple with the electric field for $\phi =0;$ but, it becomes resonantly excited for $\phi =\pi /4.$ Thus, a planar electrically resonant anisotropic metamaterial can be represented as:

*ω*gives:

_{0}Alternatively, this can be explained by first recognizing that the incident linearly polarized plane wave can be decomposed into two co-propagating CP waves of opposite handedness. Next, the result of Eq. (30) reveals that the transmission eigenstate for the metamaterial is ${\overrightarrow{E}}_{\pm}$. Applying Eq. (4), the transmission eigenstate for reverse propagation is ${\overleftarrow{E}}_{\mp}$ which also happens to represent the transmitted field after reflection from the ground plane. In such a configuration, where the field transmits through the metamaterial and reflects off a ground plane, the field propagates through the dielectric substrate with negligible net phase advancement and retransmits through the now transparent metamaterial, resulting in the following transmission matrix:

The propagation is phase-delayed due to the inductance attributed to the finite $(d<\lambda /4)$ substrate transmission line length, wherein a properly tuned structure will have a total phase advancement of zero degrees for the round-trip wave propagation through the dielectric [19,20].

The reflection of the composite metasurface can alternatively be described for all azimuthal angles through the use of Jones calculus applied to rotational polarization elements [9–13]. Where, at normal incidence, the metasurface reflection matrix is:

*R*, is given as,

*N*matrix corresponds to the axial reflection coefficients of the birefringent metamaterial at normal incidence:A solution for CP reflection is given by:The y-axis reflection coefficient corresponds to that given by a perfect electrical conductor (PEC); whereas, the x-axis reflection coefficient is purely imaginary corresponding to the surface impedance given by Eq. (33). As demonstrated with the matrices given by Eqs. (1-3), the metasurface reflection matrix is extended to oblique angles of incidence where reasonable agreement is assumed for a considerable field of view (FOV),

At this juncture, several assumptions have been utilized in order to simplify the analysis. For example, a 2D infinite periodic structure has been assumed for simulation purposes. Such an assumption is commonly expected to agree with physical measurements if the metamaterial structure has an overall dimension of several wavelengths in both directions. Additionally, the assumption of a negligible substrate thickness has been refined to be that of insignificant electrical distance. However, the assumption of low loss needs further consideration. For example, Eq. (31) is only partially valid since loss is not taken into account. The well-known Kramers-Kronig relations,

## 3. Broadband circularly polarizing metasurfaces

As previously discussed, the desire for broadband metamaterials is primarily driven by existing communication bands and atmospheric transmission windows. Revisiting Eq. (40), it will now be shown that this condition can exist in between two resonances if the resonances are sufficiently separated in frequency and occur along orthogonal axes. In turn, the resulting dispersive *N* matrix supporting broadband CP reflection is given by:

#### 3.1 Broadband infrared circularly polarizing metasurface simulation and performance metrics

To validate Eq. (48) and emphasize the advantages of metamaterials, two examples are presented for the short wave infrared band, using full wave simulations with dispersive material parameters measured via ellipsometry. To simultaneously convey the reflected wave intensity, bandwidth and polarization, the Stokes parameters are utilized [22]. The Stokes vector is represented by the *I* (intensity), *Q* (linear polarization), *U* (45 degree rotated linear polarization) and *V* (circular) parameters. In Cartesian coordinates, the Stokes parameters are calculated as follows:

The first metallo-dielectric structure is a 2D infinitely periodic metasurface that consists of capacitively end-loaded dipole elements. The unit cell is 480 nm x 480 nm and the dipole is comprised of 75 nm thick Au, supported by a 75 nm polyimide substrate, and backed by a 200 nm Au ground plane. The lower resonance, ${\omega}_{1}$, caused by the decomposition of an incident linear wave parallel to the dipole axis falls into the realm of metamaterial (*i.e.* sub-wavelength) behavior due to the size reduction caused by the capacitive end-loading of adjacent unit cells. However, the orthogonal resonance, ${\omega}_{2}$, does not occur at a sufficiently large wavelength because it is attributed to the formation of a dependent grating which is in turn highly angularly dependent [23]. Consequently, a very broadband CP design is achieved at normal incidence but performance degrades rapidly for increasing oblique angles (Fig. 4(a)
), limiting the utility of the design.

The second metallo-dielectric structure is a 2D infinitely periodic metasurface that consists of split ring (SR) elements. The unit cell is 380 nm x 380 nm and the SR is comprised of 75 nm thick Au, supported by a 50 nm Polyimide substrate, and backed by a 200 nm Au ground plane. The lower resonance, ${\omega}_{1}$, is caused by the decomposition of an incident linear wave perpendicular to the SR gap. The upper orthogonal resonance is a caused by the decomposition of an incident linear wave parallel to the SR gap. Both resonances fall within the realm of metamaterial behavior due to the retention of a sufficiently sub wavelength geometry at both frequencies. This directly results in a broadband CP reflector with a wide field of view (FOV). Referencing Fig. 4(b), a low loss, wide FOV, wide CP band centered at λ = 1.55 µm is evident.

For added clarity the SR metasurface polarization response for a horizontally polarized wave normally incident upon a $\phi =\pi /4$ rotated SR metasurface is represented by additional metrics in Fig. 5 The metrics utilized are axial ratio and polarization conversion ratio (PCR) respectively. These metrics are useful in comparing the published performance for metamaterial based polarizers presented elsewhere across various spectrum [24–34] in addition to conventional dielectric quarter wave plates. Figure 6(a) depicts the complete Stokes parameters at normal incidence and Fig. 6(b) depicts the complete Stokes parameters for $\lambda =1.55\mu m$ as a function of a normally incident field’s azimuthal rotation.

#### 3.2 Broadband reconfigurable circularly polarizing metasurface using phase change materials

The utility of the circularly polarizing metasurface can be increased if reconfigurability is introduced since polarization sensing can offer a lower signal to noise ratio than amplitude modulation. Some examples of reconfigurable metamaterials include the incorporation of a semiconducting substrate [35] or inclusion of a semiconducting device [36,37], the incorporation of cantilevers [38,39] or MEMs [21], and the incorporation of phase change materials (PCM) [39–42]. Chalcogenide glass (ChG) PCMs have gained considerable interest due to their reversible, thermally-induced phase transition between amorphous and crystalline states with a large associated change in electrical and optical properties [43].

To demonstrate the effect of upper and lower resonance Q factor symmetry and unit cell size, three examples are presented for the long wave infrared band. Each design incorporates a Ge_{2}Sb_{2}Te_{5} (GST) ChG PCM substrate and functions as a circular polarizer when the GST is in the amorphous state. In the crystalline state the substrate becomes lossy, resulting in a mirror like reflection with no polarization conversion. 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. As with the near IR designs, full wave simulations consider all losses using material parameters measured via ellipsometry.

The first metallo-dielectric structure is a 2D infinitely periodic metasurface that consists of end-loaded dipole elements. The unit cell is 700 nm x 700 nm and the dipole is comprised of 150 nm thick Au, supported by a 430 nm GST substrate, and backed by a 150 nm Au ground plane. The lower resonance, ${\omega}_{1}$, is caused by the decomposition of an incident linear wave parallel to the dipole axis. The orthogonal resonance, ${\omega}_{2}$, is attributed to the higher Q end load resonance. Since the structure is sufficiently sub-wavelength, good angular performance is exhibited (Fig. 7(a) ); but, the asymmetrical Q factors result in a reduced region of high CP.

The second metallo-dielectric structure is a 2D infinitely periodic metasurface that consists of meander line elements. The unit cell is 900 nm x 1800 nm and the meander line is comprised of 150 nm thick Au, supported by a 400 nm GST substrate, and backed by a 150 nm Au ground plane. While exhibiting a broad bandwidth and reasonably comparable Q factors for the lower and upper resonances (Fig. 7(b)), the large unit cell length results in a deterioration of angular performance. This is exhibited by the emergence of an in-band resonance at larger oblique angles.

The third metallo-dielectric structure is the familiar SR design with appropriate geometry scaling. The unit cell is 975 nm x 975 nm and the SR is comprised of 150 nm thick Au, supported by a 465 nm GST substrate, and backed by a 150 nm Au ground plane. Since the structure is sufficiently sub-wavelength and exhibits highly symmetric Q factors, broad bandwidth and good angular performance is exhibited (Fig. 7(c)). Furthermore, when the GST substrate phase is changed to the crystalline state the structure no longer behaves as a broadband birefringent surface but rather as a metallic mirror (Fig. 7(d)).

## 4. Conclusion

A theoretical analysis utilizing Jones matrices is presented, enabling the synthesis of broadband circularly polarizing mirrors. The theoretical analysis and the validity of the associated assumptions are confirmed through full wave simulations performed on candidate designs with traditional dielectric substrates in addition to designs which incorporate the bistable phase changing substrate GST, facilitating broadband polarization reconfigurability.

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