Manipulating Electromagnetic Wave Polarizations by Anisotropic Metamaterials

Jiaming Hao,1 Yu Yuan,2 Lixin Ran,2 Tao Jiang,2 Jin Au Kong,2,3 C. T. Chan,4 and Lei Zhou1,*1Surface Physics Laboratory and Physics Department, Fudan University, Shanghai 200433, China

2The Electromagnetic Academy at Zhejiang University, Zhejiang University, Hangzhou 310027, China3Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

4Physics Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China(Received 26 February 2007; published 10 August 2007)

We show that the polarization states of electromagnetic waves can be manipulated through reflectionsby an anisotropic metamaterial plate, and all possible polarizations (circular, elliptic, and linear) arerealizable via adjusting material parameters. In particular, a linearly polarized light converts its polar-ization completely to the cross direction after reflection under certain conditions. Microwave experimentswere performed to successfully realize these ideas and results are in excellent agreement with numericalsimulations.

DOI: 10.1103/PhysRevLett.99.063908 PACS numbers: 42.25.Ja, 42.25.Bs, 78.20.Bh, 78.20.Fm

Polarization is an important characteristics of electro-magnetic (EM) waves. It is always desirable to have fullcontrol of the polarization states of EM waves. Con-ventional methods to manipulate polarization include us-ing optical gratings, dichroic crystals, or employing theBrewster and birefringence effects, etc. [1,2]. Here wepropose an alternative approach based on metamaterials[3–6]. Metamaterials have drawn much attention recentlydue to many fascinating properties discovered, such as thenegative refraction [4], the in-phase reflection [5], and theaxially frozen modes [6], etc. Here, we show that a specificmetamaterial reflector can be employed to manipulate thepolarization state of an incident EM wave. In particular, acomplete conversion between two independent linear po-larizations is realizable under certain conditions. We showthe physics to be governed by the unique reflection prop-erties of the metamaterial, and we perform experimentsand finite-difference-time-domain (FDTD) simulations todemonstrate these ideas in the microwave regime.

We start from studying a model system as shown inFig. 1(a), which consists of an anisotropic homogeneousmetamaterial layer (of a thickness d) with a dispersiverelative permeability tensor �$2 (with diagonal elements�xx,�yy,�zz) and a relative permittivity "2, put on top of aperfect metal substrate (with "3 ! �1, �3 � 1). Weconsider the reflection and refraction properties of thestructure, when a monochromatic EM wave with a wavevector ~kin��!=c��sin�cos�x�sin�sin�y�cos�z� and agiven polarization strikes on the surface. According to theMaxwell equations, EM waves should satisfy ~E ���c2=!2"2� ~k� ���

$�12 � �

~k� ~E�� inside the metamateriallayer with ~k the wave vector. Given kx and ky, the dispersionrelation between ! and kz is determined by �!=c�4"2

2��!=c�2"2

Pi��

�1ii ��k

2 � k2i � � �k

2=2�Pi�j�l�

�1ii �

�1jj k

2l �

0, where i; j; l � x; y; z. The above equation has four rootscorresponding to two refracted waves propagating for-wardly and backwardly. The solution inside the second

layer must be a linear combination of these four waves,manifesting the birefringence effect [1,7]. To match theboundary conditions, we must also expand the waves inother regions to linear combinations of four solutions,namely, the forward (backward) waves with s and p polar-izations. The reflected beam thus generally consists of boths and p modes, even if the incident wave possesses onepolarization. To solve these problems, we have extendedthe conventional 2� 2 transfer-matrix-method (TMM) [8]to a 4� 4 matrix version [6,7,9].

We performed numerical calculations based on theTMM assuming that �xx � 1� 70=�12:712 � f2�, �yy �

1� 22=�6:802 � f2�, �zz � 1, "2 � 1, d � 1:3 mm,where f denotes the linear frequency measured in GHz.Suppose the incident wave is s polarized, ~Ein �

e�in�s ei�� ~kin ~r�!t�, the reflected wave can be written as ~Er �

�rsse�r�s � rspe

�r�p �ei��

~kr~r�!t�, where e�in�s , e�r�s , and e�r�p arethe unit E vectors in different cases. We define a polariza-tion conversion ratio (PCR) as PCR � r2

sp=�r2ss � r

2sp�.

Without absorption and diffractions, we get r2ss � r2

sp

1, and thus PCR � r2sp. The calculated results of PCR are

shown as solid lines in Fig. 2(a) for a normal incidence casewith � � 45� [10], and in Fig. 2(b) for � � � � 45�. Wefind the PCR to be strongly enhanced around two frequen-cies,�12:7 and�6:8 GHz, corresponding precisely to thetwo resonances at which �xx or �yy tends to infinity. Inparticular, for the normal incidence case studied inFig. 2(a), our theory predicts that PCR � 100% at thetwo resonance frequencies, indicating that a linearly po-larized light converts its polarization completely after thereflection.

In general, both s and p components exist inside thereflected beam. Define �’sp by rsp=rss � jrsp=rssjei�’sp ;we calculated �’sp versus frequency for the two casesstudied in Fig. 2. From the results (solid lines) recorded inFig. 3, we find that �’sp can take arbitrary values within[� 180�, 180�] depending on the frequency, indicating

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that all possible polarization states (circular, linear, ellip-tic) are realizable for the reflected beam. To illustrate, wechose four cases to calculate the polarization patterns ofthe reflected beams, and found that the resultant polar-izations are linear (6.87 GHz), circular (7.27 GHz), andelliptic (10.9 and 12.46 GHz), correspondingly.

The physics can be understood by a simple argument.Consider the normal incidence case for simplicity. Sup-pose the incident wave is given by ~Ein � �Exx�Eyy�ei��!z=c�!t�, then, after reflection by an ordinary ma-terial, the reflected wave is ~Er � r�Exx� Eyy�e

i�!z=c�!t�

with r the reflection coefficient. However, with anisotropy(�xx � �yy), the reflection coefficients, rx, ry are dif-ferent for incident waves polarized along two directions.Thus, the reflected wave should be ~Er � �rxExx�ryEyy�e

i�!z=c�!t� and the polarization state can be manipu-lated through varying rx, ry. For the configuration studied

in Fig. 2(a) with Ex � Ey, if we tune the material parame-ters to yield rx=ry � �1, the polarization direction of thereflected wave would be�x� y, which is perpendicular tothat of the original wave, x� y. A complete polarizationconversion (CPC) is thus realized [11].

The key issue is then how to control rx and ry. With ametal plate on the back, our structure is always totallyreflecting, i.e., jrxj � jryj 1. However, the phase � of the reflection coefficient, defined by rx�y� � ei� x�y� ,strongly depends on the metamaterial parameters. Thecalculated reflection phases (� x, � y) for the modelsystem have been depicted in Fig. 4 versus frequency. Inmost cases where �xx��yy� is not large, we have � y�x� � 180�, since the metamaterial layer is transparent andlight can directly ‘‘see’’ the metal plate which is reflectingout of phase. However, at the resonances where�xx��yy� ! 1, we get � y�x� � 0 since light is reflecteddirectly by the opaque metamaterial which possesses infi-nite impedance [5]. In general, we can obtain any value of� y�x� through adjusting the values of�xx��yy� and d. Thephysics for the CPC effect is now clear. Near each reso-nance, one of �xx, �yy becomes very large while anotherclose to 1, and there must be a frequency where � x �� y � 180� and thus rx=ry � �1 (see Fig. 4).

We study how the effect depends on the incident angles.Depicted in Fig. 5(a) are the PCR as functions of � and �,calculated at f � 6:85 GHz. Numerical computations re-

0.0

0.5

1.0

PC

R TMM FDTD Expt.

θθθθ = 0o

φφφφ = 45o

6 9 12 150.0

0.5

1.0

Frequency (GHz)

θθθθ = 45o

φφφφ = 45o

FIG. 2 (color online). PCR as the functions of frequency,obtained by TMM calculations on the model system (solid lines),FDTD simulations on realistic structures (solid stars), and ex-perimental measurements (open circles). The incident directionis (a) � � 0�, � � 45�, and (b) � � � � 45�.

FIG. 1 (color online). (a) Geometry of the model systemstudied in this Letter. (b) Image of part of the experimentalsample. (c) A schematic picture of the experiment setup. Startingfrom the laboratory coordinate system fx0; y0; z0g, we first rotatethe sample for an angle of � with respect to the y0�� y1� axis,then for an angle of � with respect to the z1�� z� axis, andfinally arrive at the local coordinate system fx; y; zg attached tothe sample.

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veal that the PCR cannot reach 1 in the arbitrary incidencecase, but the maximum PCR value can still approach 1. As� increases, we find that the value of � to realize amaximum PCR increases toward 90�, indicating that atglancing incidence we can rotate ~E toward x to maximizethe polarization conversion effect [12]. An opposite behav-

ior of � was found for the resonance at f � 12:7 GHz. Tofacilitate easy comparisons with experiments, we show inFig. 5(b) the PCR as a function of � with � � 45�, and inFig. 5(c) the PCR as a function of � with � � 45�.

We employed experiments and FDTD simulations todemonstrate these predictions. To realize the proposedmodel system, we designed a frequency-selective structureas shown in Fig. 1(b), consisting of a periodic array ofH-shaped metallic pattern (lattice constant a � 7 mm,thickness � 0:1 mm), printed on a 1.2 mm thick printedcircuit board slab (with "r � 3:6) with a metal sheet on theback. Other parameters are fixed as b � 5 mm, w �1:0 mm. A single H-shaped metallic structure possesseselectric resonances for both polarizations [13]. When ametal plate is added, currents are induced on the metalplate surface, flowing along a direction opposite to thecurrents induced on the H structures. The entire systemthus exhibits magnetic responses with well defined reso-nance frequencies, but its electric polarization is stronglydiminished [14,15]. The composite material is then per-fectly described by the double-layer model shown inFig. 1(a), with d � 1:3 mm being the thickness of theH-pattern (0.1 mm) plus the inner dielectric layer

6 9 12 15-180

-90

0

90

180

Ref

l. p

has

e ( ∆∆ ∆∆

ΨΨ ΨΨ )

Frequency (GHz)

TMM FDTDE ||

||

X

E y

FIG. 4 (color online). Frequency dependences of the reflectionphase changes � y�x� on the metamaterial reflector surface fornormally incident waves with polarizations ~Ejjx and ~Ejjy, cal-culated by TMM on model system (solids lines) and FDTDsimulations on realistic structures (solid stars).

-180

0

180

Rel

ativ

e p

has

e ( ∆∆ ∆∆

ϕϕ ϕϕ SP

)

TMM FDTD

θθθθ = 0o, φφφφ =45o

(a)

7.27 10.9

6 9 12 15-180

0

180

Frequency (GHz)

θθθθ = 45o, φφφφ = 45o

(b)

6.87

12.46

FIG. 3 (color online). Frequency dependence of the relativephase �’sp between the s- and p-polarized modes inside thereflected beam, calculated by TMM on model systems (solidslines) and FDTD simulations on realistic structures (solidstars). The incident direction is (a) � � 0�, � � 45�, and(b) � � � � 45�. The insets show the polarization patterns ofthe reflected beams calculated at different frequencies.

FIG. 5 (color online). (a) PCR as a function of � and �,calculated by TMM. (b) PCR as a function of � with � fixedas 45�, obtained by TMM (solid lines), FDTD simulation (solidstars), and measurements (open circles). (c) PCR as a function of� with � fixed as 45�, obtained by TMM (solid lines), FDTDsimulations (solid stars) and measurements (open circles). Here,we set f � 6:85 GHz.

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(1.2 mm) [15]. As a demonstration, we employed FDTDsimulations [16,17] to calculate the reflection phase spectra(� x�f�, � y�f�) of the designed composite and showedthe results in Fig. 4. The good agreement with the modelTMM results supports our conclusions [18].

We fabricated a 497� 497 mm2 sample based on thedesigns, and performed microwave experiments. As sche-matically shown in Fig. 1(c), the measurements were car-ried out in an anechoic chamber using a network analyzer(Agilent 8722 ES) and two linearly polarized horn anten-nas. The distance between the source/receiver antenna andthe sample is 6.9 m. The sample was rotated appropriatelyto achieve a desired incidence angle (�, �). Illuminatingthe sample by a linearly polarized signal with ~Ejjy0 (i.e., swave [19]), we measured the reflected signal using ay0-polarized receiver horn (data collected as jrssj2) andan x0-polarized one (data collected as jrspj2 [19]). Withboth jrssj2 and jrspj2 obtained, we calculated the PCR anddrew the results in Figs. 2(a) and 2(b) as open symbols. Wealso performed FDTD simulations [17] to calculate thePCR spectra based on the designed system, and showedthe results as solid stars in the same figures. Quantitativeagreements are found among the results obtained by theTMM, the FDTD simulations and experiments. In particu-lar, both experiments and simulations verified the predictedCPC effects. We next employed FDTD simulations tocompute the relative phase spectra [�’sp�f�], which areagain in good agreement with the model TMM results asshown in Fig. 3, indicating that we can obtain any polar-ization as desired. We finally employed experiments andsimulations to study the angle dependences of the polar-ization conversion effects, and depicted the measured andsimulated results in Figs. 5(b) and 5(c). Satisfactory agree-ments are noted compared with the model TMM results.

Other types of anisotropic metamaterials, say a slab withan anisotropic "$ tensor or an anisotropic �$ tensor, yieldsimilar polarization manipulation effects. However, suchsinge-layer systems are usually not perfectly reflective forEM waves in arbitrary cases, and thus the manipulationefficiency is low. In contrast, our double-layer systemensures 100% reflection for arbitrary EM waves, so thatthe manipulation efficiency is maximized.

In short, we proposed to use metamaterials to manipu-late EM wave polarizations, and showed that a completeconversion between two perpendicular linear polarizationsis realizable under certain conditions. The ideas weresuccessfully demonstrated in the microwave frequencyregime by experiments and simulations, and are believedrealizable also in higher frequency regimes.

This work was supported by the China-973 program(No. 2004CB719800), NSFC (No. 10504003,No. 60531020, and No. 60671003), ZJNSF(No. R105253), Shanghai-STC (No. 05PJ14021,No. 5JC14061), FYTEF, and PCSIRT. C. T. C. is supported

by HKUST3/06C.

*Corresponding author.phzhou@fudan.edu.cn

[1] Max Born and Emil Wolf, Principles of Optics(Cambridge University Press, Cambridge, England, 1999).

[2] Eugene Hecht, Optics (Addison Wesley, New York, 2002).[3] J. B. Pendry et al., IEEE Trans. Microwave Theory Tech.

47, 2075 (1999); D. R. Smith et al., Phys. Rev. Lett. 84,4184 (2000); D. R. Smith, J. B. Pendry, and M. C. K.Wiltshire, Science 305, 788 (2004).

[4] R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77(2001).

[5] D. Sievenpiper et al., IEEE Trans. Microwave TheoryTech. 47, 2059 (1999); Fan Yang and Y. Rahmat-Samii,IEEE Trans. Antennas Propag. 51, 2691 (2003); H.Mosallaei and K. Sarabandi, IEEE Trans. AntennasPropag. 52, 2403 (2004).

[6] A. Figotin and I. Vitebskiy, Phys. Rev. E 68, 036609(2003).

[7] Pochi Yeh, Optical Wave in Layered Media (Wiley, NewYork, 1988).

[8] See, e.g., K. Busch, C. T. Chan, and C. M. Soukoulis, inPhotonic Band Gap Materials, edited by C. M. Soukoulis(Kluwer, Dordrecht, 1996).

[9] Jiaming Hao, Hao Xu, and L. Zhou (unpublished).[10] For normal incidence, � is meaningless to define the

direction of ~kin, but is meaningful to differentiate twopolarizations. We define that the s-polarized wave has~Ejj�� sin�x� cos�y� and the p-polarized one has~Ejj�� cos�x� sin�y�.

[11] The reflected EM wave can also possess a linear polar-ization, which is not perpendicular to, but rather rotated byan angle with respect to the incident one. For example, inthe normal incidence case with � � 45�, the reflectedwave still takes a linear polarization when the conditionrx=ry � �1 is met, but the resultant polarization direc-tion,�Exx� Eyy, is not perpendicular to the original one,Exx� Eyy, since here Ex � Ey.

[12] The width (��) of the PCR peak becomes narrower when�! 90� and reaches zero when � � 90�.

[13] W. J. Wen et al., Phys. Rev. Lett. 89, 223901 (2002).[14] L. Zhou, W. J. Wen, C. T. Chan, and P. Sheng, Appl. Phys.

Lett. 83, 3257 (2003).[15] J. M. Hao, L. Zhou, and C. T. Chan, Appl. Phys. A 87, 281

(2007).[16] K. S. Yee, IEEE Trans. Antennas Propag. 14, 302 (1966).[17] CONCERTO 4.0, Vector Fields Limited, England (2004).

Limited by the computational power, we studied a smallersystem sized 98� 98 mm2 in our simulations.

[18] The double-layer model not only accounts for the reflec-tive properties, but also the surface wave properties ofsuch composite materials (see Ref. [15]).

[19] According to Fig. 1(c), we get z0 � sin� cos�x�sin� sin�y� cos�z. Therefore, a wave with~kinjjz0; ~Ejjy0�y1� is s polarized with respect to the sample,and that with ~kinjjz0; ~Ejjx0 is p polarized.

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