Generalized reciprocal relations for transmission and reflection of light through a 1D stratified anisotropic metamaterial

Transcription

1 Generalized reciprocal relations for transmission and reflection of light through a 1D stratified anisotropic metamaterial Railing Chang 1 and P. T. Leung 1,* 1 Institute of Optoelectronic Sciences, National Taiwan Ocean University, Keelung, Taiwan, R. O. C. Department of Physics, Portland State University, P. O. Box 751, Portland, OR , U. S. A. Abstract Motivated by the recent demonstration of asymmetric transmission of light through stratified systems of various anisotropic metamaterials, we have established generalized reciprocal relations for both transmission and reflection for these systems referring to the two cases of incidence from opposite directions. These relations reduce to those derived previously for simple isotropic systems which guarantee reciprocal symmetry in both transmission and reflection for non-dissipative medium, and provide constraint for reflection from both sides in the presence of absorption. Illustrations of our generalized relations are provided for different material systems. * Phone: FAX: hopl@pdx.edu Keywords: reciprocity, anisotropic materials, transmission and reflection asymmetry 1

2 Introduction The study of optical reciprocity in the context of classical electrodynamics has a long history [1] including some recent re-formulation of this principle from a more microscopic point of view []. In particular, the reciprocal symmetry in the 1D propagation of light through stratified media is of both fundamental interest and application significance such as in the design of devices like optical isolators [1]. For ordinary isotropic materials, it is well-known that while transmission must be identical for both left-incident and right-incident light, reflection for the two-side incidence will in general be different in the presence of absorption by the medium [3]. Moreover, Agarwal and coworkers have derived in the literature very general constraints for both the amplitude and phase for the reflected light corresponding to incidence from the two opposite sides of the stratified medium [4, 5]. These generalized constraints turn out to depend on the dissipative property of the medium in a unique way, and will lead back to symmetric reflection amplitudes with a simple definite relation for the phases for the two-side incidence in the case of non-dissipative medium. The recent explosive developments in the optics with various metamaterials have led to the observation of many intriguing novel optical phenomena such as negative refraction, perfect lens action, cloaking, reversed Doppler effect, etc [6]. Among these, asymmetric transmission from different sides of a stratified system of metamaterials has drawn great excitement from many researchers in the field, as it has been reported for various systems including D chiral [7], 3D chiral[8], gyrotropic [9], magneto-optical [10] materials as well as materials with excitations of spoof plasmons [11]. While most of the time the breakdown of transmission symmetry in these systems arises from the anisotropic or magnetic response of the material, the effects of

3 which may turn out to depend critically on the polarization of the incident light [1]. This new possibility of asymmetric transmission will have significant modifications to those generalized reciprocal relations established previously in the literature [4, 5]. It is the purpose of the present Communication to provide modified generalized reciprocal relations for both the reflected and transmitted light corresponding to normal incidence from opposite sides on a stratified system of anisotropic materials. This will be in complement to a recent study on the reciprocity relations in the 3D scattering from anisotropic medium using the correlation and Green s function method [13]. The results obtained will be significant in the understanding of the constraints for the 1D propagation of light through these media which will be useful in the design of various non-reciprocal optical devices using these materials. We shall first present the derivation of these modified relations and then illustrate them via consideration of various material systems. To begin, we would like to emphasize that as pointed out previously in the literature, reciprocity and transmission symmetry may have different meaning for anisotropic materials since one can have asymmetric transmission for these systems while Lorentz reciprocity remains valid for the left- and right- incidence of light [8]. Generalized constraints for transmission and reflection Let us consider a 1D stratified system of linear optical response characterized by a dispersive dielectric tensor εω ( ) which may be complex. The lateral extent of the system is assumed to be finite within 1 z, with vacuum on both sides z < 1 and < z. Consider now light is being incident from both sides: E 1 from the left and E from the right side; and subscripts irt,, will be used to indicate the incident, reflected, and transmitted light from the two 3

4 sides of the system. We shall first give a brief review of the derivation of the result in Agarwal and Gupta [4] for an isotropic medium and then generalize the results to the anisotropic case. (i) Isotropic medium For an isotropic stratified system with scalar dielectric function εω (, z) ε( z) (where for simplicity we have suppressed the explicit frequency dependence), the fields on both sides can be described by the following simple scalar wave equation [4]: d dz + k ε ( z) E 0 α =, (1) with α = 1, and k = ω / cbeing the wave number in vacuum. For the fields outside the system, we have simple plane waves as follows: Ee + Ee for z< E1 = Ee ikz ikz 1i 1r 1 ikz 1t for z>, () and E E e + E e for z > = E e ikz ikz i r ikz t for z < 1. (3) By considering the following integral: b d d E E 1 E1 E dz = 0 a dz dz (4) as implied from (1), and for a< 1 and < b, the results in () and (3) easily lead to the following reciprocal relation for the transmitted fields [4]: EE EE =, (5) 1t i 1i t 0 which implies symmetric transmission ( Eαt / Eα i), or identical transmitted fields if one 4

5 assumes unit incident fields on both sides. For the reflected fields from both sides, we consider the wave equation (1) for E 1 and the conjugate equation for E, respectively. Instead of (4), we now consider the following integral [4]: b b * * * * E E 1 E1 E dz = k E1E ( ε ε ) dz. (6) a d dz d dz Again, (6) with the results in () and (3) finally leads to the following reciprocity constraint for the reflected fields [4]: ( ε ) * * b * Im 0 1t r + 1r t + a 1 = E E E E k E E dz, (7) where one sees explicitly the breakdown of the reflection reciprocity symmetry is originated from the dissipative nature of the medium via the imaginary part of ε. It is clear from (5) and (7) that while transmission is always symmetric for isotropic medium, reflection will be asymmetric if absorption takes place in the medium [3, 4]. a (ii) Anisotropic medium To generalize the above results to a stratified anisotropic medium, we start from the two curl equations of Maxwell with no presence of free current source: 1 B E + = 0, (8) c t 1 D H = 0. (9) c t Assuming linear anisotropic response with harmonic time dependence: D= ε E e ω B= µ H e ω i t, (10) i t, (11) one obtains from (8) and (9) the following equations: 5

6 E ikµ H = 0, (1) H + ikε E = 0, (13) which can be recast into the following form: E ik H = ( µ ) 0 H + ik E = ( ε ) 0, (14). (15) Note that in this case one cannot derive similar Helmholtz wave equations for the fields since the single curl terms in (14) and (15) cannot be simplified further in general. In the following, we shall limit ourselves to nonmagnetic anisotropic medium for which (13) and (14) can be combined to lead to the following equation (with µ 1): E ik ik E = 0 E k εe = 0, (16) ( ε ) which can be re-expressed in the form: E+ E k ε E = 0. (17) ( ) Note that (17) cannot be simplified further since one cannot conclude from Gauss s law E = 0 with D= ε E. Now if we further assume the electric field takes the following special form: E= E( zx ) ˆ+ E( zy ) ˆ+ 0zˆ, (18) x y that is, the electric field has only x and y components which vary only in the z direction, then d E = E = E and E = 0, (19) z dz and Eq. (17) reduces to a form similar to Eq. (1) as follows: 6

7 d E + k ε E = 0 dz. (0) Note that in this case the full vector-nature of the electric field is included in the wave equation, since the propagation of an incident plane wave through an anisotropic medium has critical dependence on the polarization of the wave. Note also that in order to ensure the field is of the form as in (18) throughout, we restrict the dielectric tensor in (0) to be of the following block-diagonal form: εxx εxy 0 ε( z) = εyx εyy 0. (1) 0 0 ε zz Other than this specific form, we shall consider no further restriction on (1) and allow each of the non-vanishing elements in (1) to be arbitrary complex functions in general. Thus ε does not have to be hermitian or symmetric at this point and we can hence decompose it into the following two alternate forms: or T T ε + ε ε ε ε = + εs + ε ε ε ε AS ε + ε ε ε = + H + AH, (), (3) with the subscripts S, AS, H, and AH stand for symmetric, anti-symmetric, hermitian, and anti-hermitian, respectively. Within this framework, it is then straightforward to generalize the results in (5) and (7) obtained for the isotropic case, to the case of anisotropic medium of dielectric tensor as given in (1). Thus, for transmission reciprocity between the left- and right- incidence case, we consider (0) for E 1 and the 7

8 transpose equation for E, respectively; while for reflection, we consider the hermitian conjugate equation for E instead. Then through manipulation of similar integrals as those in (4) and (6), we finally obtain the following two generalized reciprocity constraints for each of the transmitted and reflected wave from oppositely incident light in the form: and E E E E ik E E dz, (4) b T T T i 1t t 1i ε AS 1 = 0 a E E E E ik E E dz, (5) b r 1t + t 1r ε AH 1 = 0 a respectively. It is obvious that for scalar ε as in the isotropic case, ε AS = 0 and εah Im( ε) with (4) and (5) reduce back to (5) and (7) as expected. Eqs. (4) and (5) constitute our main results in this Communication and we shall provide illustrations for their application through the following examples. Note that these reciprocal constraints are expressed directly in terms of the permittivity tensor of the medium rather than the more phenomenological quantities such as the Jones matrix of the system [8]. Application Let us explore the implications from (4) and (5) via consideration of the dielectric tensor in the following three special cases where we shall restrict ourselves to a homogeneous system or to an effective dielectric response (assumed well-defined [14]) in case of heterogeneous systems: Case (I): All ε ij = real in (1) 8

9 T In this case we have ε = ε, and conservation of energy requires the dielectric tensor to be both symmetric and hermitian [15]. The results in (4) and (5) hence reduce to the following: E E = E E, (6) T T i 1t t 1i E E = E E. (7) r 1t t 1r To study the implications from (6) and (7), we introduce the following forward and backward transfer (Jones) matrices as used in the literature [7, 8]: E = T E, (8) f 1t 1i E = T E. (9) b t i Substituting (8) and (9) into (6) leads to the following result (de Hoop Reciprocity [1]): T f = b ( T ) T. (30) Note that in general, for an anisotropic medium, (30) is not the same as transmission symmetry f b [7, 8]. However, in this case one can actually show that T = T from the hermiticity of the dielectric tensor and hence the result in (30) implies that the tensor T must be symmetric [16]. Thus symmetric transmission from opposite sides of incident light is guaranteed in this case since asymmetry only arises from the difference in the off-diagonal elements in T [7, 8]. Similarly, we can introduce the following reflection matrices: E = R E, (31) (1) 1r 1i E = R E, (3) () r i Substituting (8), (9), (31), (3) into (7) leads to the following relation: R T R T b f ( ) ( ) 1 1 () + (1) = 0, (33) 9

10 which can be regarded as the corresponding generalization of the result in (7) for the special case of non-dissipative medium, i.e. Im( ε ) = 0. Situations where the results in (30) and (33) are valid will be the transmission and reflection of light from a transparent crystal, for example. In the special case when the transfer matrix is diagonal with equal diagonal elements, (33) yields a () (1) simple relation for the reflection matrices with R = R. Case (II): Not all ε ij = real in (1). In particular, εi j = complex In this case, the two integrals in (4) and (5) do not vanish in general and precise implications from them can be obtained only via calculation of the fields in the medium. To briefly explore some general conclusions, we divide into the following situations: (a) If εxy = ε, then ε = 0. However, yx AS * * εxx ε 0 xy * * ε = εyx εyy 0 ε (8) * 0 0 ε zz in general, even if the diagonal elements are real. Hence ε 0in this case and AH (7) is no longer valid. Then the constraint for reflection must refer back to (5) although one still has (6) to remain valid. An example for this case will be an Ohmic anisotropic conductor [15]. In this case, (6) and (30) will still hold but (7) and (33) have to be modified by the integral in (5) which signifies the breakdown of reflection reciprocal symmetry in a dissipative anisotropic stratified system. (b) If ε xy * = εyx, and the diagonal elements ii ε are real, then we have ε = 0 but ε 0. AH AS * An example for this will be the case of a gyrotropic medium with ε = ε = iε and xy yx g ε xx = ε [17]. Here one will have (7) valid but not (6), and the transmission yy 10

11 constraints has to refer back to (4). Since transmission symmetry is lost, (7) will no longer lead to symmetry in the reflected light. (c) The most general case will be εxy ε and ε yx xy ε, then bothε 0 and ε 0 * yx AS AH in general. This will be the case of a dissipative gyrotropic medium, for example. We then have to implement the full constraints both in (4) and (5). Case (III): Other specific examples Here we would like to extend our discussion on the most general case when the dielectric tensor is not of the block diagonal form as in (1). While the generalized reciprocity relations for this case can be rather complicated, we see that under certain restrictions our results in (4) and (5) can still be applicable to some extent. Two explicit examples are considered in the following: (a) Magneto-optic effects The effects on the propagation of a light wave in a medium with the presence of a quasistatic magnetic field have been actively studied over a century since the discovery of the Faraday rotation effect on the polarization of the wave [18]. In general, these magnetooptic (MO) effects break time-reversal symmetry as well as Lorentz reciprocity, and are caused mainly by the anisotropic response of the specific material medium such as Bismuth iron garnet [19]. Thus the dielectric tensor for these effects can be described by the following general form [0]: εxx εxy + igz εxz ig y εω ( ) = εxy igz εyy εyz + igx. (9) εxz + ig y εyz igx ε zz 11

12 where g = ( gx, gy, gz) is known as the gyration vector. Hence (9) is not of the block diagonal form in (1) in general, and our generalized relations in (4) and (5) cannot be obtained for such a MO system. However, in the special case when g aligns with one of the principal axis (say, the z axis) of the real dielectric tensor ε ( which is defined from (9) with g = g = g = 0 and all x y z other elements εij being real); the dielectric tensor reduces to the following gyrotropic form: ε1 igz 0 εω ( ) = igz ε1 0, (30) 0 0 ε where we have assumed two of the eigenvalues of ε to be degenerate. This will then identify with case (II b) above and the discussion there applies completely to this special case of MO system. In this case, light propagates parallel (perpendicular) to the direction of g will lead to the well-known Faraday (Cotton-Mouton) effect. (b) Liquid crystals Another example which has all the elements in ε non-vanishing is liquid crystals which in general can be described by the following tensor: εxx εxy ε xz ε = εxy εyy εyz, (31) εxz εyz ε zz where each of the elements in (31) is defined in terms of the extraordinary and ordinary refractive index of refraction as well as the director orientation as follows [1]: 1

13 ε = n + ( n n )sin θ cos φ xx 0 e ε = ( n n )sin θ sinφ cosφ xy e εxz = ( ne n0)sinθ0cosθ0cosφ0. (3) ε = n + ( n n )sin θ sin φ yy 0 e ε = ( n n )sinθ cosθ sinφ yz zz e ε = n + ( n n )cos θ 0 e 0 0 Conclusion In (3), θ 0 is the angle between the director and the z-axis, and φ0 is the angle between the projection of the director on the x-y plane and the x-axis. Hence, again, we see that in this case our generalized results in (4) and (5) will not apply except for the special case when θ0 equals to π /. The symmetry property for 1D transmission and reflection of light with respect to incident light through a stratified medium is intriguing, especially for an anisotropic medium. Previously it has been clarified by Agarwal and Gupta [4] for isotropic medium showing explicitly how absorption sets constraint on the reflection amplitudes, while transmission is always symmetric even if time-reversal breaks down in the propagation through a dissipative system. Our present work seeks to generalize the results of [4] to the case of anisotropic medium and we see that reciprocity symmetry can be broken in both transmission and reflection even for a non-dissipative material. This is consistent with many of the recent observations of transmission asymmetry from anisotropic metamaterials, although one has to emphasize again that the breaking of such symmetry is in general not the same as the invalidity of reciprocity symmetry as revealed from our results in (4) and (5). In addition, the implication for the modification of the (non-reciprocal) phase relation derived in [5] will be more involved since the polarization of the incident light will change with an anisotropic medium even for normal 13

14 incidence. In any case, we believe that these results will be useful as guidance for future design of non-reciprocal optical devices using these anisotropic metamaterials. References [1] R. J. Potton, Rep. Prog. Phys. 67 (004) 717. [] M. Mansuripur and D. P. Tsai, Opt. Comm. 84 (011) 707. [3] See, e.g., Pochi Yeh, Optical Waves in Layered Media (John Wiley & Sons, Hoboken, 005). [4] G. S. Agarwal and S. D. Gupta, Opt. Lett. 7 (00) 105. [5] V. S. C. M. Rao, S. D. Gupta and G. S. Agarwal, J. Opt. B 6 (004) 555. [6] V. M. Shalaev, Nature photonics 1 (007) 41. [7] A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, Nano Lett. 8 (008) 940. [8] C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tunnermann, T. Pertsch, and F. Lederer, Phys. Rev. Lett. 104 (010) 5390; C Menzel, C Rockstuhl, F Lederer, Physical Review A 8 (010) See also the following recent review article: Z. Li, M. Mutlu, and E. Ozbay, J. Opt. 15 (013) [9] D. L.Sounas and C. Caloz, Appl. Phys. Lett. 98 (011) [10] Z. Yu, Z. Wang, and S. Fan, Appl. Phys. Lett. 90 (007) [11] A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, Phys. Rev. Lett. 105 (010) [1] M. Kang, J. Chen, H. X. Cui, Y. Li, and H. T. Wang, Opt. Exp. 19 (011)

15 [13] X. Du and D. Zhao, Opt. Comm. 84 (011) [14] See, e.g., Menzel et al in Ref. [8] for a comment on this issue. [15] P.-H. Tsao, Am. J. Phys., 61 (1993) 83. [16] By working in a base where f ε is diagonal, it is easy to show that T b = T by applying the result in Ref. [4]. A unitary transformation of this result will lead to the conclusion that it must remain valid in any base for arbitrary type of polarization of the incident light. [17] J. A. Kong, Electromagnetic wave theory (Cambridge, MA, 008). [18] See, e.g., the review article by M. J. Fraiser, IEEE Transactions on Magnetics 4 (1968) 15; and references therein. [19] Z. Yu, Z. Wang, and S. Fan, Appl. Phys. Lett. 90 (007) [0] [1] I.L. Ho, Y.C. Chang, C.H. Huang and W.Y. Li, Liquid Crystals 38 (011)