Achieving arbitrary polarization control using complex birefringent meta-materials

TL;DR

This paper introduces complex birefringent meta-materials capable of arbitrarily controlling pairs of light polarization states, enabling advanced polarization manipulation for applications in sensing and communications.

Contribution

It develops a novel class of non-Hermitian meta-materials that can transform pairs of polarization states, a capability not previously demonstrated.

Findings

Successfully demonstrates polarization control using complex birefringent meta-materials.

Enables detection of small polarization changes in scattering experiments.

Facilitates new polarization multiplexing schemes in communications.

Abstract

We demonstrate that the key to realizing arbitrary control over pairs of polarization states of light, i.e. transforming an arbitrarily polarized pair of input states to an arbitrarily polarized pair of output states, is the ability to generate pairs of states with orthogonal polarizations from non-orthogonal pairs of initial states. We then develop a new class of non-Hermitian meta-materials, termed complex birefringent meta-materials, which are able to do exactly this. Such materials could facilitate the detection of small polarization changes in scattering experiments, as well as enable new polarization multiplexing schemes in communications networks.

Equations

$$|1,1⟩=S_{\theta }|\theta ⟩,$$

$$|1,-1⟩=S_{\theta }|-\theta ⟩.$$

$$|{\theta }_{\alpha }⟩=U_{\alpha }|{\alpha }_1⟩,$$

$$|-{\theta }_{\alpha }⟩=U_{\alpha }|{\alpha }_2⟩,$$

$$S=U_{\beta }^{†}{S}_{{\theta }_{\beta }}^{-1}{S}_{{\theta }_{\alpha }}{U}_{\alpha }.$$

$$\bar{\varepsilon}=\left(\begin{array}[]{ccc}\varepsilon_{xx}&\varepsilon_{xy}&0\\ \varepsilon_{yx}&\varepsilon_{yy}&0\\ 0&0&\varepsilon_{zz}\end{array}\right)\equiv\left(\begin{array}[]{cc}\bar{\varepsilon}_{\perp}&0\\ 0&\varepsilon_{zz}\end{array}\right),$$

$${\epsilon }_{xx}$$

$${\epsilon }_{xy}$$

$${\omega }^2\mu {\bar{\epsilon }}_{\perp }|E_{\pm }^R⟩=k^2|E_{\pm }^R⟩,$$

$$\frac{k_{\pm }^2}{{\omega }^2\mu }$$

$$|E_{\pm }^R⟩$$

$$N_{\pm }^2=2(1-{\tau }^2)\pm 2i\tau \sqrt{1-{\tau }^2},$$

$$|E(z)⟩=e^{i{k}_{-}z}\left(e^{i\Delta kz}{A}_{+}|E_{+}^R⟩+A_{-}|E_{-}^R⟩\right),$$

$$e^{i\Delta kl}=\frac{⟨E_{-}^L|\pm \theta ⟩⟨E_{+}^L|1,\pm 1⟩}{⟨E_{+}^L|\pm \theta ⟩⟨E_{-}^L|1,\pm 1⟩}.$$

$$|E(z)⟩=e^{i{k}_{0}z}\left[\left(A_E+i{A}_J{k}_{0}z\right)|E^R⟩+A_J|J^R⟩\right],$$

$$I(z)=$$

$$+|A_{-}{|}^2⟨E_{-}^R|E_{-}^R⟩+A_{-}^{\ast }{A}_{+}{e}^{i\Delta kz}⟨E_{-}^R|E_{+}^R⟩.$$

$$(\mathcal{M}\mathcal{T})S(\mathcal{M}\mathcal{T})=S^{-1}.$$

Full text

Polarization is one of the fundamental properties of light, and control over the polarization is paramount in many optical communications and imaging applications. In general, the effect of propagation through any media on the polarization of an incident electromagnetic signal can be described as $|\beta\rangle=S(z)|\alpha\rangle$, where $|\alpha\rangle$ and $|\beta\rangle$ are the input and output polarization states, respectively, and $S(z)$ is a $2\times 2$ matrix that depends on the properties of the medium, as well as the propagation distance $z$. Conventionally, the polarization of a signal is manipulated through the use of birefringent materials Imai et al. (1985); Okoshi (1985); Walker and Walker (1990); Zhuang et al. (1999); Hao et al. (2007, 2009); Pors et al. (2011). For lossless birefringent media, with proper choice of material parameters and propagation distance, it is always possible to convert an input polarization $|\alpha_{1}\rangle$ to an arbitrary output polarization $|\beta_{1}\rangle$. However, once the response to $|\alpha_{1}\rangle$ is determined, the output polarization $|\beta_{2}\rangle=S|\alpha_{2}\rangle$ is no longer arbitrary for any other input polarization $|\alpha_{2}\rangle$. This is because $S$ is unitary in lossless media, and thus $\langle\beta_{2}|\beta_{1}\rangle=\langle\alpha_{2}|\alpha_{1}\rangle$.

In this Letter, we seek to overcome the limitation of conventional birefringent media by developing a class of meta-materials which enable arbitrary control over pairs of polarization states. By arbitrary control, we demand that for a pair of arbitrary input polarizations $|\alpha_{1}\rangle$ and $|\alpha_{2}\rangle$, one can generate an arbitrary pair of output polarizations $|\beta_{1}\rangle$ and $|\beta_{2}\rangle$. Achieving such polarization control has significant implications for a wide range of technologies. For example, with this capability one can map two polarizations that are close to each other into two orthogonal polarizations, which may facilitate the detection of small polarization changes, such as those arising from the imaging of biological tissues De Boer and Milner (2002); Adato et al. (2009) and thin films Losurdo et al. (2009). Likewise, the ability to completely separate non-orthogonal polarization states could enable new multiplexing schemes in optical communications networks beyond what is currently possible Hayee et al. (2001); Morant et al. (2014).

We first show that the key step for achieving arbitrary control over pairs of polarization states is to develop a class of meta-materials which are capable of performing the following polarization transformation as denoted by $S_{\theta}$,

[TABLE]

Here we assume propagation along the $z$-axis, and label the polarization states in terms of the electric field components in the $xy$-plane as $|E_{x},E_{y}\rangle$. $|\pm\theta\rangle$ denote the two polarization states that lie on the great circle of the Poincaré sphere passing through $|1,i\rangle$ and $|1,1\rangle$, and are symmetrically placed away from $|1,i\rangle$, subtending an angle of $\pm\theta$ with respect to $|1,i\rangle$. Suppose we can construct a class of materials which can provide $S_{\theta}$ for an arbitrary $\theta$. For an arbitrary pair of input states $|\alpha_{1}\rangle$ and $|\alpha_{2}\rangle$, using conventional lossless birefringent materials, one can achieve the transformation Zhuang et al. (1999)

[TABLE]

where $U_{\alpha}$ is unitary and $\langle\theta_{\alpha}|-\theta_{\alpha}\rangle=\langle\alpha_{1}|\alpha_{2}\rangle$. For the pair of arbitrary output states $|\beta_{1}\rangle$ and $|\beta_{2}\rangle$, one can obtain a similar unitary transformation $U_{\beta}$ that transforms them to $|\pm\theta_{\beta}\rangle$. Therefore, the transformation $S$ from the input states $|\alpha_{1}\rangle$ and $|\alpha_{2}\rangle$ to the output states $|\beta_{1}\rangle$ and $|\beta_{2}\rangle$ is then

[TABLE]

To achieve the transformation as described by Eqs. (1) and (2) in general requires a non-Hermitian meta-material. We now proceed to show that such a transformation can be realized in a class of complex symmetric meta-materials with its dielectric tensor having the form

[TABLE]

in which

[TABLE]

where $\varepsilon_{r},\varepsilon_{i},\varepsilon_{xy}\in\mathbb{R}$. Here, an equal amount of gain and loss has been added to the $x$ and $y$ axes of a conventional birefringent material, a choice inspired by recent developments in optical media with spatially distributed regions containing equal amounts of gain and loss Bender et al. (1999, 2002); Musslimani et al. (2008); Makris et al. (2008); Guo et al. (2009); Klaiman et al. (2008); Longhi (2009); Rüter et al. (2010); Chong et al. (2011); Szameit et al. (2011); Feng et al. (2013); Peng et al. (2014); Chang et al. (2014); Lawrence et al. (2014); Kang and Chong (2015); Cerjan et al. (2016). Henceforth, we refer to materials which obey Eqs. (7) and (8) as complex birefringent meta-materials.

For light propagating along the $z$-axis of such a medium, the allowed wavevectors, $k_{\pm}$, of a monochromatic signal with frequency $\omega$, can be found by solving the right eigenvalue equation Yeh (2005); Haus (1983),

[TABLE]

in which $\mu$ is the scalar magnetic permeability. As such, the allowed wavevectors and right eigen-polarizations of complex birefringent meta-materials are

[TABLE]

in which $\tau=\varepsilon_{i}/\varepsilon_{xy}$ represents a normalized measure of the strength of the gain and loss in the system, and

[TABLE]

is the normalization of the eigenstates. The matrix $\omega^{2}\mu\bar{\varepsilon}_{\perp}$ also has left eigen-polarizations, which are solutions to $\langle E_{\pm}^{L}|\omega^{2}\mu\bar{\varepsilon}_{\perp}=k_{\pm}^{2}\langle E_{\pm}^{L}|$. Together, the left and right eigen-polarizations form a bi-orthogonal basis, and can be normalized such that $\langle E_{m}^{L}|E_{n}^{R}\rangle=\delta_{mn}$, with the choice of $N_{\pm}$ in Eq. (12). In conventional lossless birefringent media $\bar{\varepsilon}_{\perp}$ is Hermitian, and so $\langle E_{\pm}^{L}|=|E_{\pm}^{R}\rangle^{\dagger}$. However, for complex birefringent materials, $\omega^{2}\mu\bar{\varepsilon}_{\perp}$ is complex-symmetric, and the left and right eigen-polarizations are related by $\langle E_{\pm}^{L}|=|E_{\pm}^{R}\rangle^{T}$. Moreover, although the two right eigen-polarizations are linearly independent for complex birefringent materials with $|\tau|\neq 1$, it can be readily seen that they are not orthogonal, $\langle E_{\mp}^{R}|E_{\pm}^{R}\rangle\neq 0$ except for when $\tau=0$ and the system reverts to a conventional birefringent material, or when $|\tau|\rightarrow\infty$ and the system becomes a conventional dichroic material.

The evolution of the polarization of light propagating within a complex birefringent material with $|\tau|\neq 1$ can be expressed in terms of the right eigen-polarizations (11), as

[TABLE]

in which $\Delta k=k_{+}-k_{-}$ is the additional phase accumulated by $|E_{+}^{R}\rangle$ relative to $|E_{-}^{R}\rangle$ per unit length, and the initial amplitudes, $A_{\pm}$, are defined in terms of the left eigen-polarizations as $A_{\pm}=\langle E_{\pm}^{L}|E(0)\rangle$. The resulting polarization dynamics of a complex birefringent material can be visualized by plotting the output polarization as a function of $z$ on the Poincaré sphere. The example shown in Fig. 1(a) illustrates the polarization dynamics when $\tau<1$, for which both allowed wavevectors are real, $k_{\pm}\in\mathbb{R}$. This is analogous to the exact phase in parity-time symmetric systems Musslimani et al. (2008); Makris et al. (2008); Guo et al. (2009). For the sake of comparison, we also plot the polarization dynamics for a conventional Hermitian birefringent material in Fig. 1(b), as described by setting $\varepsilon_{i}=0$ in Eq. (6). For both types of materials as shown in Fig. 1, the eigen-polarizations define the fixed point of the dynamics. So long as the incident polarization is not parallel to one of these eigen-polarizations, the polarization of the initial signal forms a closed trajectory around the eigen-polarizations as $z$ is varied.

When $\tau=0$, which describes a conventional lossless birefrigent material, the two eigen-polarizations are located at $|1,1\rangle$ and $|1,-1\rangle$, corresponding to two linearly polarized states. The two eigenstates are orthogonal to each other, and the polarization trajectories form circles around the axis formed by the two eigenstates, shown in Fig. 1(b). (On the Poincare sphere, orthogonal states are represented by antipodal points.) As $\tau$ is increased, so that $0<\tau<1$, the two eigen-polarizations remain on the great circle connecting $|1,1\rangle$ and $|1,i\rangle$, but are tilted away from the states $|1,\pm 1\rangle$ towards $|1,i\rangle$, which is a manifestation of the non-orthogonality of these two eigenstates. For $-1<\tau<0$, the eigenstates instead tilt away from $|1,\pm 1\rangle$ towards $|1,-i\rangle$. As $k_{\pm}\in\mathbb{R}$, the polarization trajectories are still closed, but are no longer centered on the axis formed by the eigenstates, as shown in Fig. 1(a).

Examining the polarization dynamics of Fig. 1(a), we note that there are two states lying on the great circle connecting $|1,1\rangle$ and $|1,i\rangle$, indicated by the purple crosses, which can be transformed to the two states $|1,1\rangle$ and $|1,-1\rangle$, indicated by purple circles, with a proper choice of the propagation distance $l$. Therefore, complex birefrigent meta-materials with $0<|\tau|<1$ indeed provide the key non-trivial step required for achieving arbitrary control over pairs of polarization states, which is to realize $S_{\theta}$ as defined in Eqs. (1) and (2). Mathematically, for a given pair of input states $|\pm\theta\rangle$, the complex birefringent meta-material which transforms these two states to $|1,\pm 1\rangle$ satisfies

[TABLE]

Here, as $\tau$ appears in both $\Delta k$ and the left eigenstates $\langle E_{\pm}^{L}|$, Eq. (14) represents a complex transcendental equation which can be solved for $\tau$ and $l$. One can show that for any given value of $0<|\tau|<1$, one can achieve orthogonal output states for any choice of $\theta$ with the proper choice of propagation distance $l$. In general, as the choice of initial states become parallel ($\theta\rightarrow 0$), stronger gain and loss ($|\tau|\rightarrow 1$) along with longer propagation distances are required to achieve orthogonal output states.

Up to this point we have shown that, in the regime where $|\tau|<1$, complex birefringent materials, as described in Eqs. (6)-(8), exhibit polarization dynamics that can be used to achieve arbitrary control over pairs of polarization states. This class of metamaterials also exhibits interesting polarization dynamics with $|\tau|\geq 1$. When $|\tau|>1$, both eigenvalues $k_{\pm}$ become complex, with $k_{+}=k_{-}^{*}$. This is analogous to the broken phase in parity-time symmetric systems. In this regime the eigen-polarizations reside along the great circle on the Poincaré sphere connecting $|1,0\rangle$, $|0,1\rangle$, and $|1,\pm i\rangle$. As $z$ varies, the eigenstates correspond to a stable or an unstable fixed point on the Poincaré sphere depending on the sign of $\textrm{Im}[k_{\pm}]$, as shown in Fig. 2(a), and the material provides polarization-dependent attenuation and amplification. As $|\tau|\rightarrow\infty$, the system becomes a conventional dichroic material, with orthogonal eigen-polarizations $|1,0\rangle$ and $|0,1\rangle$.

When $|\tau|=1$, complex birefringent materials possess an exceptional point Kato (1995); Heiss (2004) where both the eigenvalues and eigenvectors coalesce, and the eigenvectors become self-orthogonal, $\langle E_{\pm}^{L}|E_{\pm}^{R}\rangle=0$. This yields two unique properties. First, the expression for the evolution of the electric field (13) is no longer valid as $\omega^{2}\mu\bar{\varepsilon}_{\perp}$ has a non-trivial Jordan normal form. Instead, the evolution of the field must be expressed in terms of the single remaining eigenvector, $|E^{R}\rangle$ and its associated Jordan vector, $|J^{R}\rangle$ Mailybaev and Seyranian (2004); Pick et al. (2016), as

[TABLE]

in which $k_{0}=\omega\sqrt{\mu\varepsilon_{r}}$, and $A_{E},A_{J}$ are the modal amplitudes at $z=0$. The derivation of this equation is provided in the Supplementary Information SI . As can be seen in Eq. (15), the polarization of light flowing through a complex birefringent material at $|\tau|=1$ has only a single fixed point that corresponds to circularly polarized light, to which the polarization of every initial state with $A_{J}\neq 0$ converges to through linear amplification as a function of $z$ and without any attenuation. Second, any initial polarization state converges to the fixed point from a single direction, along the $|1,0\rangle$ to $|1,i\rangle$ contour for $\tau=1$ as shown in Fig. 2(b). This is distinct from what is observed for the stable fixed point when $|\tau|>1$, in which $|E_{-}^{R}\rangle$ can be approached from any direction.

In all three of their phases with $\tau\neq 0$, complex birefringent materials are necessarily active optical structures, and in general do not conserve the intensity of the incident radiation. This fact is immediately evident from the linear and exponential amplification present when $|\tau|\geq 1$. When $|\tau|<1$, $k_{\pm}$ are real, and the intensity is instead periodic as a function of $z$,

[TABLE]

Thus, in this case the total change in the intensity is bounded. Even though the intensity is not conserved in complex birefringent materials, two generalized unitarity relations can be derived which relate the transmission and reflection coefficients of the scattering matrix, $S$ Chong et al. (2011); Ge et al. (2012), as shown in the Supplementary Information SI . These conserved quantities stem from the fact that complex birefringent materials are invariant upon the operation that switches the $x$ and $y$ axes of the system, $\mathcal{M}$, and the time-reversal operation, $\mathcal{T}$, thus

[TABLE]

There are many possible experimental realizations of complex birefringent materials. A dielectric response as described by Eqs. (6)-(8) has been previously realized experimentally in a meta-surface structure Lawrence et al. (2014). However, in order to observe the polarization dynamics effects, and to achieve the capability for arbitrary control over pairs of polarization states, neither of which are considered in Lawrence et al. (2014), it would be interesting to create three-dimensional media where the propagation distance can be varied. Therefore, here instead we discuss the construction of three-dimensional systems with the appropriate dielectric tensor. As an example, one could construct a meta-material consisting of an ordinary birefringent material interspersed with layers containing regions of both gain and loss, as depicted in Fig. 3(a). Transfer matrix calculations of this exact structure without using the effective medium approximation in the propagation direction confirm its ability to separate a pair of initial states with similar polarizations to be nearly orthogonal, shown in Fig. 3(b). Alternatively, there are many methods for adding birefringence to optical fibers geometrically, allowing for $\varepsilon_{xy}\neq 0$. By doping such a birefringent fiber, gain could be added to both $\varepsilon_{xx}$ and $\varepsilon_{yy}$. Then, all that is required to realize complex birefringence is the ability to add loss specifically to $\varepsilon_{yy}$. Regardless of the specific realization chosen, the experimental design of complex birefringent materials benefits from the critical feature that the amount of gain and loss, $\varepsilon_{i}$, necessary to observe significant non-trivial polarization dynamics is set by the anisotropy of the system, $\varepsilon_{xy}$, which can be designed to be quite small. Thus, very little gain or loss is necessary to realize arbitrary control over pairs of polarization states in these materials.

Here, we have focused on meta-materials with $\varepsilon_{xx}=\varepsilon_{yy}^{*}$ as this choice yields a regime of parameter space, $|\tau|<1$, where the eigenvalues of the system are real, and as such the change in intensity of an incident signal is bounded. However, many other choices of $\varepsilon_{xx},\varepsilon_{yy}\in\mathbb{C}$, such as birefringent materials with loss in a single polarization channel, i.e. $\varepsilon_{xx}=\varepsilon_{r}$ and $\varepsilon_{yy}=\varepsilon_{r}+2i\varepsilon_{i}$ with $\varepsilon_{xy}\neq 0$, will still yield non-orthogonal eigenvectors, which can allow for nearly arbitrary control over pairs of polarization states as shown in Fig. S2 in the Supplementary Information SI . Similarly, off-axis propagation in directions which do not conserve $\mathcal{M}$ also results in an entirely complex spectrum even for $|\tau|<1$. Fortunately, the rate at which $k_{\pm}$ acquires an imaginary component for off-axis propagation is slow relative to the change in propagation angle, as shown in Fig. S3, and thus the off-axis components of a wave-packet traveling through a complex birefringent meta-material will experience similar polarization dynamics to the on-axis component.

In conclusion, we have developed a theory of complex symmetric anisotropic dielectric materials and demonstrated that such systems enable arbitrary control over pairs of polarization states. In particular, such complex birefringent materials may have applications in both splitting signals with adjacent polarizations and nearly combining signals with orthogonal polarizations. Experimentally, these materials benefit from the fact that the scale of the gain and loss required to observe these effects is set by the off-diagonal anisotropy in the system.