We study the behavior of rotation of polarization plane when passing from positive to negative dielectric and magnetic susceptibilities of a gyrotropi...

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The Problem of Equivalence of Two Formulations of Material Equations for Natural Gyrotropic Media H. S. Eritsyan, J. B. Khachatryan, A. A. Papoyan, and H. M. Arakelyan Yerevan State University, Yerevan, Armenia Received June 4, 2012

AbstractWe study the behavior of rotation of polarization plane when passing from positive to negative dielectric and magnetic susceptibilities of a gyrotropic medium for two formulations of material equations in such media. DOI: 10.3103/S1068337212050076 Keywords: gyrotropic media, material equations, polarization plane, negative susceptibility

1. INTRODUCTION At present, in scientific literature, two formulations of the system of material equations of naturally gyrotropic media are used [1, 2]. Both formulations satisfy the energy conservation law, although expressions for Pointing vector, as well as boundary conditions are different. It was shown in [3] that these systems of material equations lead to different wave equations for inhomogeneous media. We show in the present work that rotation of the polarization plane behaves in different ways when changing from positive to negative values of dielectric and magnetic susceptibilities; this allows concluding that the above-mentioned material equations are not equivalent. 2. TWO SYSTEMS OF MATERIAL EQUATIONS FOR HOMOGENEOUS ISOTROPIC GYROTROPIC MEDIUM The system of equations in the form used by authors of monograph [1] has the appearance D E rotE, B H.

(1)

In work [1] the magnetic susceptibility is assumed to equal unity in optical range of frequencies, corresponding to [4]. Therefore, the formulation of problem in this paper assumes passing to the range of lower frequencies where magnetic susceptibility can differ from unity. This corresponds to the frequency range for which artificial media with negative and are designed [5]. Another system of material equations is proposed in [2] and has the form D E rotE, (2) B H rotH. 3. DISPERSION EQUATIONS AND ROTATION OF POLARIZATION PLANE If we use system (1), the dispersion equation has the form 2 4 4 k 4 2 2 4 2 2 k 2 4 2 2 0. c c c Restricting ourselves to the quantities containing g to the power not higher than unity we obtain k2

2 c 1 . c 2 228

(3)

(4)

THE PROBLEM OF EQUIVALENCE

229

Rotation of the polarization plane 1 at the ray path length l equals 1 2 l. 2 c2 If we use the system of material equations (2), the dispersion equation has the form (1)

(5)

4 4 2 2 4 k 4 1 4 2 2 4 2 2 2 2 4 2 2 4 2 k 2 4 2 2 0. (6) c c c c c Restricting ourselves, as above, to the terms not higher than first-order in the gyrotropy parameter, we obtain 2 2 2 . 2 2 c c c Rotation of the polarization plane (2) at the ray path length l equals k2

(2)

(7)

2 l. c2

(8)

4. DISCUSSION According to formula (5) rotation of polarization plane changes its sign if we replace a medium with 0, 0 by a medium with 0, 0, whereas the rotation in formula (8) does not change its sign. At simultaneous change in sign of , , and the gyrotropy parameter, rotation (5) does not change its sign, while rotation (8) does. In this sense systems of material equations (1) and (2) are not equivalent, although they not only obey both the energy conservation law, but also may be made identical by a relation 2. Note that the difference between these systems is present also in media with open surface of wave vectors [6–8]. Results of this work may be used for experimental verification of correspondence to reality of one or another system of material equations written above, as well as for elaboration of artificial gyrotropic media with needed parameters. REFERENCES 1. Agranovich, V.M. and Ginzburg, V.L., Kristallooptika s uchetom prostranstvennoi dispersii i teoriya eksitonov (Crystal Optics with Allowance for Spatial Dispersion and Theory of Excitons), Moscow: Nauka, 1979. 2. Fyodorov, F.I., Teoriya girotropii (Theory of Girotropy), Minsk: Nauka i tekhnika, 1976. 3. Galumyan, A.G. and Arakelyan, O.M., Ucheniye zapiski EGU, 2002, vol. 3, p. 35. 4. Landau, L.D. and Lifshitz, E.M., Elektrodinamika sploshnykh sred (Electrodynamics of Continuous Media), Moscow: Nauka, 1972. 5. Gollub, J.N., Chin, J.Y., Cui, T.J., and Smith, D.R., Opt. Express, 2009, vol. 17, p. 2122. 6. Eritsyan, H.S., Kristallografiya, 1978, vol. 33, p. 461. 7. Eritsyan, H.S., Optika girotropnykh sred i kholestericheskikh zhidkikh kristallov (Optics of Gyrotropic Media and Cholesteric Liquid Crystals), Yerevan: Hayastan, 1988. 8. Eritsyan, H.S., Papoyan, A.A., and Arakelyan, H.M., J. Contemp. Phys. (Armenian Ac. Sci.), 2008, vol. 43, p. 161.

JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

Vol. 47

No. 5

2012