This paper proposes a method to design an incident-angle-insensitive polarization-independent polarization rotator. This polarization rotator is composed of layers of impedance-matched anisotropic metamaterial (IMAM) with each layer’s optical axes gradually rotating an angle. Numerical simulation based on the generalized 4 × 4 transfer matrix method is applied, and the results reveal that the IMAM rotator is not only polarization-independent but also insensitive to the angle of incidence. A 90° polarization rotation with tiny ellipticity variation is still available at a wide range of incident angles from 0 to 40°, which is further confirmed with a microwave bi-split-ring resonator (bi-SRR) rotator. This may be valuable for the design of optoelectronic and microwave devices.
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As an important property of transverse EM waves, polarization has been widely applied in engineering and scientific researches. Many approaches have been employed to manipulate the polarization of light. Traditional anisotropic crystals and chiral liquid crystals  are commonly used as wave retarders and polarization rotators. However, the polarization rotator made of anisotropic crystal is generally polarization-dependent and incident-angle-sensitive, and the polarization rotator composed of liquid crystals is also incident-angle-sensitive. These shortcomings limit their applications in microwave and optoelectronic devices. To our knowledge, how to design and fabricate an incident-angle-insensitive and polarization-independent polarization rotator still remains a challenge.
Thanks to the advent of metamaterials, this gives a chance to overcome the challenge. Metamaterials are artificial structures, which usually have periodic arrangements and exhibit exotic electromagnetic properties [2–8]. These manmade structures provide completely new mechanisms and novel methods to control light. Early and ongoing researches on metamaterials have shown that it is possible to obtain strong anisotropy or chirality via deliberate design and fabrication. This can be used in the design of polarization devices. One of them is polarization beam splitter achieved by anomalous reflection and transmission [9–12]. Another application is the polarization rotator based on modes coupling, extraordinary optical transmission (EOT), chirality of the structure, or phase mutation at resonance frequency [5,13–17]. Up till now, wide-angle polarizer , splitter [10,11] and absorber  have been realized with these artificial structures. In this paper, we proposed a method to design an incident-angle-insensitive and polarization-independent polarization rotator. The polarization rotator is composed of layered impendence-matched anisotropic metamaterials (IMAM) with the crystal axes rotated, as shown in Fig. 1 . The cross polarization conversion becomes polarization-independent and insensitive to the incident direction of the light beam. It can be proved that only two layers of IMAMs are enough to construct a cross-polarization rotator, which greatly alleviates the difficulty in fabrication. It might be valuable for the design of optoelectronic and microwave devices.
2. General formalism of IMAM polarization rotator
To study the transmission and reflection of plane waves in an anisotropic slab, a generalized 4 × 4 transfer-matrix method is applied . By transforming the permittivity and permeability of IMAM with a rotation matrix: , where21]. To simplify the discussion, we restrict the incidence in x-z plane, i.e. and one of the crystal axes is chosen as z axis.
When the polarization of the transmitted wave is perpendicular to a linearly-polarized incident wave one can obtain
For a polarization-independent rotator, the following condition must be satisfied: Generally, it cannot be fulfilled with a single anisotropic medium but with a layered structure. To give an intuitionistic discussion, we choose an IMAM slab with the following permittivity and permeability tensors:
Since the impendence-matched condition is satisfied, reflections at the interface either between slab and surroundings or between slabs disappear. Then we can give an explicit expression on the transmission coefficient of the stratified slabs. When the plane wave is normally incident onto x-y plane, the projections of the transmitted wave polarization on the crystal axes (denoted by suffix 1, 2) of the (n + 1)th slab can be written as
As shown in Fig. 2 , δ is the angle between the x axis and the crystal axis 1 of the first layer; ϕ is the angle of the crystal axes between adjacent layers. We once again project onto the coordinate axes and the Jones matrix can be obtained:
By carefully tuning the thickness of IMAM slab that makes it as a half-wave retarder, one can obtain When n is odd, the Jones matrix can be simplified as follows:
It is clear that, if T is not considered, the Jones matrix is a coordinate transformation matrix by rotating an angle of anticlockwise. Hence the polarization rotation angle can be tuned dynamically and is independent of φ and When
This is the condition for the cross polarization conversion. It is clear that only two (n = 1) IMAM slabs are enough to construct a polarization-independent rotator.
Equation (14) indicates that the eigen polarization states of the rotator are left and right circular polarizations. Although it is the same for planar chiral structures with four-fold symmetry, it is very difficult to achieve linear polarization rotation in planar metallic chiral structures due to dichroism; while in dielectric chiral structures, one can achieve linear polarization rotation but this property strongly depends on incident direction, as has been pointed out .
3. Incident-angle-insensitivity and polarization independency
It is more interesting to find that the IMAM rotator is not only polarization-independent, but also incident-angle-insensitive. This is distinct from conventional anisotropic or chiral rotators. Without loss of generality, we choose air as surroundings, and a lossless IMAM bilayered rotator with parameters as The thickness h of each layer is chosen as so that is satisfied at normal incidence and each layer of slab acts as a half-wave retarder. To construct a polarization-independent rotator, two half-wave plates are aligned together in the above mentioned way, i.e. To study the effects of incident direction on the performance of the polarization rotator, incident angle θ (denoted by), angle δ and incident polarization angle φ should be considered. Figures 3(a) to 3(d) show the polarization changes of the transmitted light. The ellipticity Δ and the polarization rotation angleof the transmitted light are defined as
Since we have restricted the incident beam in x-z plane, the variation of δ is actually equivalent to the change of incidence in polar direction. Now we look about how the performance of the polarization rotator depends on δ when φ = 45°. Figure 3(a) is the projection of the three-dimensional (3D) curvature of ellipticity Δ as a function of δ and onto the plane; Fig. 3(b) is the projection of the 3D curvature of polarization rotation angle as a function of δ and onto the plane. It is clear that, at normal incidence (), and the bilayered structure acts as a perfect cross-polarization rotator. As the incident angle increases, the variation ranges of Δ and become larger but still acceptable within a broad range of incident angles. It can be seen in Fig. 3(a) and Fig. 3(b) that when δ ranges from 0° to 360°, (indicating a value of 0.006 as an intensity contrast of two polarization components) and even at a 40° incident angle (). As the incident angle decreases to 30°, variations can drop to and
Figures 3(c) and 3(d) depict the effect of incident polarization angle φ on the polarization rotation at the oblique incidence of 40°. The insets plot the variations of Δ and with respect to δ and φ at a 40° incident angle. The projections of these curvatures onto and planes are shown in these figures. When φ changes from −90° to 90°, the maximum variation range of Δ is while the maximum variation of is After studying the effects of incident direction under various φ and we confirm that the IMAM rotator is more insensitive to incident direction when the incident angle is smaller.
Moreover, we studied the effect of material loss. As is expected, the loss has a small impact on the performance of the IMAM rotator that satisfies Eq. (10). We set while other parameters maintain the same. The results show that for a lossy rotator, the corresponding curvatures of Figs. 3(a), 3(b) and 3(c) are similar to the lossless ones; while for the one correspond to Fig. 3(d), the variation range of is enlarged to (−91.66°,-88.37°). Even so, a less than 2° deviation in polarization rotation at 40° incidence is still quite acceptable.
4. Comparison and discussion
To make a comparison, we choose three lossless anisotropic slabs (surrounded by air) with parameters as , and the thickness h of each layer is chosen as , where respectively. Only the case b = 1 is impedance-matched among the three cases. Since for TM waves and = for TE waves, is satisfied for all three half-wave plates at normal incidence; when two identical plates are stacked together in the aforementioned way, i.e. they all act as polarization-independent rotators at normal incidence. However, the impedance-matched conditions are not satisfied at oblique incidence, and this will bring in negative influences on the performance of the rotators. Nevertheless, as is shown above, the structure composed of IMAM slabs (b = 1) is more insensitive to incident angle.
Figures 4(a) to 4(d) demonstrate the cross-polarization transmission coefficients and polarization rotations with respect to when φ = 45°. It is clear that for the impendence-matched polarization rotator, the incident angle region with (and tend to zero, which are not shown here) and is much wider than the impendence-mismatched ones. Thus, the polarization rotation angle and the polarization ellipticity can be realized within a much wider incident angle as well. For the IMAM polarization rotator, when the incident angle varies from 0° to 40° (), and . While for the impendence-mismatched rotators, the polarization rotation and ellipticity become very instable as the incident angle increases. To take the nonmagnetic rotator with b = 2 as an example, the polarization rotation becomes −105° and the ellipticity is around-0.6 at a 40° incidence.
The basic mechanism of the incident-angle-insensitivity of the IMAM polarization rotator can be understood by a simple argument. A lossless incident-angle-insensitive polarization rotator actually requires that can be satisfied for each single slab even at oblique incidence. When the transmission coefficient of TM wave through a homogeneous slab surrounded by air (the transmission coefficient of TE wave can be obtained by duality) can be written as
The key issue is that for the impendence-mismatched slab, and thusCalculations show that (not shown here), the reflection increases quickly and become fluctuating whenmeanwhile, and change periodically with respect to kx, indicating the existence of high-order Fabry-Perot interference. While for the IMAM slab, (i.e. ) can be maintained at a much larger as depicted in Fig. 4(f), and thus and vary smoothly as a function of Moreover, it can be seen from the dispersion curves in Fig. 4(e) that is maintained better in the IMAM polarization rotator as increases, which indicates a more stable phase difference. Hence, can be obtained, and Eq. (14) can be fulfilled even at a large-angle incidence. This is the reason that a bilayered polarization rotator composed of such impedance-matched half-wave retarders will thus exhibit incident-angle-insensitivity.
It should be noted that we mainly consider about the effects of transversal parameters (,,and) in the above discussion. Our calculations disclosed that the ratio between and will also affect the property of angle-insensitivity. Numerical simulations show that the optimized ratio generally locates around one when the surroundings is air, i.e. .Sinceand as the longitudinal parameters and increase, the dispersion curve of the metamaterial will become flatter. Thus, the IMAM polarization rotator will become more insensitive to incident angle and even exhibits a self-collimating property when and are sufficiently small . Then, the ratio deviation betweenand can be larger within the same tolerable variation range of Δ and . To take the lossless IMAM polarization rotator discussed in Fig. 3 as an example, within the same tolerable variation range of Δ and (i.e. |Δ| < 0.077 and at 40° incidence), when the maximum ratio deviation can be up to 20% (i.e. ).
5. Construct a microwave IMAM rotator with bi-SRR structure
To demonstrate an incident-angle-insensitive and polarization-independent polarization rotator, we need to construct an IMAM half-wave retarder first, and then stack two retarders together with their optical axes rotated 45° to form such a polarization rotator. Various metamaterial structures can be employed to fabricate an IMAM retarder; however, metamaterials with both electric and magnetic resonances are preferred, since the effective permeability of nonmagnetic structures generally lies around one, which indicates that high refractive index and impedance-matching condition can hardly be achieved simultaneously. Moreover, spatial dispersion and anisotropy are inevitable because metamaterials are artificial mesostructures . In order to diminish such effects, electrically small non-bianisotropic microwave bi-split-ring resonator (bi-SRR) is selected to construct an IMAM half-wave retarder .
Figures 5(a) and 5(b) are the schematic layouts of the IMAM retarder. The metallic bi-SRR patterns are fabricated on one side of the FR-4 (lossless) substrate characterized as , ,the substrate thickness t = 0.5 mm. The other dimensions of a unit cell are as follows: lattice constant az = ay = 5 mm; the length of metal slices (perfect electric conductor) in Z and Y directions z = y = 4 mm, the separation distance of the metal slices p = 0.12 mm, the gap g = 0.2 mm, the separation distance between adjacent unit cells s = 1.0 mm, and the width w and thickness of metal are 0.2 mm and 0.08mm respectively. The design principle of the IMAM half-wave retarder is that only TM component can excite magnetic resonances while TE wave propagates “quietly” through the structure. Thus nearly full transmission and low effective refractive index can be obtained for TE wave, while high effective index is available for TM component at the impedance-matched frequency near the resonance. By tuning the dimensions or the substrate material, one can change the impedance-matched frequency and the phase difference between the two orthogonal components of the transmitted waves.
The simulation is performed using the software package CST Microwave Studio, in which periodic boundary conditions are applied. Figure 6(a) depicts the amplitudes of co-polarization terms of transmission coefficientand reflection coefficientfor the first retarder, where the cross-polarization terms are negligible. There are two dips in the TM wave reflection coefficient curve corresponding to two possible impedance-matched frequencies at 3.12 GHz and 3.25 GHz. Here the working frequency is chosen to be 3.25GHz. Because the ratio of unit cell size to wavelength is less than 1/9, the effective medium description is valid, and the retrieval effective parameters at normal incidence are: and. Though the structure is not perfectly impedance-matched for TE wave at the working frequency, the transmission is still very high (>99%) and it will have less influence on the transmitted polarization state.
Note that in Fig. 6(b), the phase difference between two polarization states at 3.25GHz is 180°, which indicates that the structure acts as a half-wave retarder. Figure 6(c) reveals the amplitudes of transmission coefficients for TE and TM waves with respect to at 3.25GHz, and Fig. 6(d) shows the transmitted phase difference between the two orthogonal components, in which a 174° phase difference is still available at a 40° incidence (). It is clear that high transmission and a near 180° phase difference can be maintained within a large incident angle.
Then, we construct a polarization rotator by aligning two such half-wave retarders in the aforementioned way, i.e. , as depicted in Fig. 5(c). In the simulation, the polar angle of incident plane is rotated by 45° to perform an equivalent rotation of the IMAM retarder. Thus, we can get the transmission and reflection Jones matrices of the two IMAM retarders,and,directly from the simulation rather than from the effective parameters. The cross-polarization terms of the transmission coefficient for the second retarder satisfy at 3.25GHz, as expected, and are not shown here. The total transmission Jones matrix of the rotator can then be written aswhere which is the small quantity due to multiple reflections between the slabs. The first term of the expression of O(T) represents the first order approximation. The ellipticity and polarization rotation angle with respect to incident wave polarization under different incident angles can then be worked out. For clarity but without loss of generality, we only give out the results for incident polar angle in Figs. 6(e) and 6(f). The results under zeroth order and first order approximations are compared, it is clear that the variation ranges of ellipticity and polarization rotation can be well described under zeroth order approximation. The variations of ellipticity and polarization rotation are still quite acceptable even at the angle of incidence, showing that the structure can act as an incident-angle-insensitive and polarization-independent polarization rotator.
When a lossy substrate with a permittivity is considered in the polarization rotator, it is found that the imaginary term may lead to a stronger absorption for TM wave near resonant frequencies. However, we can still find the two dips in reflection curve of the half-wave retarder. Although they are not as sharp as that in the lossless case, but they are small enough to suppress the influence of multiple reflections between the two retarders in a polarization rotator. For a commercial available low loss substrate withour simulations show that the phase difference has a tiny change (less than 2°), and there is a small decrease in the transmission amplitude of TM wave due to absorption. The influences of the substrate loss are illustrated in Figs. 6(g) and 6(h) when and the polarization rotator still works at 3.25GHz. Compared with the lossless rotator, the variations of ellipticity and polarization rotation angle are slightly increased. But we have to emphasize that the depicted results for the lossy rotator above are just for comparison, without any structural adjustment according to the loss, so the performance can be further improved after a proper structural optimization.
In conclusion, we proposed a design method to realize broad–angle and polarization-independent polarization rotator based on impedance-matched anisotropic metamaterial (IMAM). The mechanism and the influencing factors of the polarization rotator are discussed analytically, in conjunction with the generalized 4 × 4 transfer matrix method. Also, we illustrate how to construct a microwave IMAM rotator by using a bi-SRR structure. Compared with traditional polarization rotators, the IMAM polarization rotator is more insensitive to incident direction. This may offer the possibility to unyoke the limitation of the narrow working angle and enhance the compactness and stability of microwave and optoelectronic systems.
The authors acknowledge the financial support from the National Natural Science Foundation of China (NSFC) under the contracts 10774195, U0834001 and 10974263. The work is also partially supported by Program for New Century Excellent Talents in University and the Chinese National Key Basic Research Special Fund (2010CB923200).
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