ISSN 1068–3372, Journal of Contemporary Physics (Armenian Academy of Sciences), 2008, Vol. 43, No. 4, pp. 161–167. © Allerton Press, Inc., 2008. Original Russian Text © H.S. Eritsyan, A.A. Papoyan, H.M. Arakelyan, 2008, published in Izvestiya NAN Armenii, Fizika, 2008, Vol. 43, No. 4, pp. 252–260.
Media with Nonclassical Form of the Surfaces of Wave Vectors of Classical Optics H. S. Eritsyan, A. A. Papoyan, and H. M. Arakelyan Yerevan State University, Yerevan, Armenia Received August 13, 2007
Abstract—The forms of surfaces of wave vectors (SWVs) of the known media are considered and the systematization of them by the form of SWVs is made. Characteristic features of optical properties of the media caused by the new forms of SWVs are indicated. PACS numbers: 42.25.Bs DOI: 10.3103/S1068337208040026 Key words: surface of wave vector, nonclassical forms, optical properties
1. INTRODUCTION Surfaces of wave vectors, as known, describe the dependence of the wave vector of an electromagnetic wave propagating in a medium on the propagation direction. SWVs are closed centrosymmetrical surfaces. These two properties are considered so natural that they even are not mentioned in the literature [1–5]. At the same time, beginning in 1950s, the works have appeared on the optical properties of various media which are of fundamental interest because there are unusual from the point of view of classical optics. These media, as will be shown below, are described by SWVs with a form which also is unusual in classical optics. A novelty of the form of SWVs is both in non-observance of common properties of SWV (closure and centrosymmetricality) and in other features considered below. It is evident that these peculiarities are caused by the features of optical properties of corresponding media. We list briefly the SWVs unusual for classical optics. a) SWvs which have a close and centrosymmetrical form but are not split into two subsurfaces (corresponding to two polarizations) in spite of the presence of anisotropy. b) Centosymmetrical closed SWVs with a shifted physical center (from which the wave vectors are ploted). Note that in optics commonly the physical center is simultaneously the geometrical center. c) Noncentrosymmetrical closed SWVs. d) Inverse closed centrosymmetricla SWVs (where the outer and inner domains are interchanged) e) Open centrosymmetrical SWVs. f) Open noncentrosymmetrical SWVs. Before proceeding to consideration of media with SWVs listed above, we should note, that below they are disposed according to the degree of difference of the SWV form from a common cloased centrosymmetrical form. The authors of works related to media mentioned in point b) (magnetoelectric media) have also considered the problem of SWV form [6–10]. 2. CLOSED CENTROSYMMETRICAL SWVS WITH THE SAME RADIUS-VECTOR FOR ANY POLARIZATION, IN SPITE OF THE PRESENCE OF POLARIZATION Such SWVs related to media named single-reflecting [3] (see also [6]). However, SWVs were not considered in [3]. The single-refraction implies that the wave in not split into two waves with different polarization, i.e., the wave vector modulus, being dependent on the propagation direction, does not depend on the polarization. Such independence takes place under the following condition between the tensors of the dielectric permittivity ( εˆ ) and magnetic permeability ( μˆ ) : μˆ −1 = pεˆ −1 , (1)
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where tilde stands for the transposition and p is a scalar. In the simplest case, when the tensor εˆ and μˆ can be diagonalized in the same system of coordinates, condition (1) takes the form ε xx ε yy ε zz = = . (2) μ xx μ yy μ zz The dispersion equation has the form μ ⎞ 1⎛ ε μ ⎞ 1⎛ε 1 ω2 k z2 + ⎜ 1 + 1 ⎟ k x2 + ⎜ 2 + 2 ⎟ k y2 − (3) ( ε1μ 2 + ε 2μ1 ) = 0. 2 ⎝ ε3 μ3 ⎠ 2 ⎝ ε3 μ3 ⎠ 2 c2 Equation (3) is not of the forth degree with respect to the components of the vector k but of the second degree and describes and ellipsoid. The independence of k on the polarization has a simple interpretation [3]. Let a wave propagate along z-axis. Then we have ω2 ω2 ε μ , k22 = 2 ε 2μ1. (4) 2 1 2 c c In the general case ε1μ2 ≠ ε2μ1 and therefore k1 ≠ k2 two waves with different polarizations propagate in the same direction and the SVWs is split into two surfaces. In the case of uniaxial medium we have a sphere and ellipsoidal displaying the dependence of the modulus of k on the propagation direction. However, if relation (1) takes palce, then ε1μ2 = ε2μ1 and, hence, k1 = k2: only one wave with the wave vector independent of the polarization propagates in one direction. At the same time k depends on the propagation direction. This phenomenon shows that the birefringence (i.e., the dependence of k on polarization) is caused by not anisotropy of εij (i.e., by (εαα − εββ) / (εαα + εββ)) or of μij (i.e., by (μαα − μββ) / (μαα + μββ)) but by different anisotropy of the tensor εij and μij [3]. Indeed, it is easy to show that under condition (1), or more simply, condition (2) an equality of the anisotropic of εij and μij takes place: ε αα − εββ μαα − μββ . (5) = ε αα + εββ μ αα + μββ k12 =
Figure 1 shows the SWVs for a uniaxial crystal with anisotropic εij and isotropic μij (a), as well as the SWV for simultaneously anisotropic εij and μij satisfying the condition of single-refraction (1).
Fig. 1. Forms of SWVs (a) for a uniaxial crystal with anisotropic εij and isotropic μij and (b) for uniaxial crystal with simultaneously anisotropic εij and μij satisfying the condition of single-reflection.
3. CLOSED CENTROSYMMETRICAL SWVS WITH A SHIFTED PHYSICAL CENTER As is known, magnetoelectric media have such SWVs [7–11]. Material equations for these media have the form. D = εˆ E + νˆ H, B = μˆ H + βˆ E. (6) For a number of uniaxial crystals, due to properties of the tensor νˆ and βˆ , equations (6) are reduced to the form D = εˆ E + [ pˆ H ] , B = μˆ H − [ pˆ E ]. (7) Maxwell’s equations
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ω ω D, [kE ] = − B. c c for such a medium coincide with equations for the medium with D = εˆ E, B = μˆ H.
[kH ] = −
(8) (9)
if we replace k with k + p. Figure 2a presents the form of SWVs for uniaxial crystals without the megnetoelectric effect. The presence of this effect remains unchanged the form of SWVs, but the wave vector k is plotted not from the geometrical center Ogeom but from the geometrical center Ophys shifted by p from Ogeom (Fig. 2 corresponds to such medium).
Fig. 2. Forms of SWVs for uniaxial crystals (a) in the absence and (b) presence of magnetoelectric effect.
Such a shift leads to the irreversibility of waves: wave vectors corresponding to the two opposite directions have different moduli. This takes place for both waves, to which the sphere and ellipsoid correspond (ordinary and extraordinary waves). The irreversibility takes place not only with respect to the wave vector but also with respect to the group velocity vector. 4. CLOSED NONCENTROSYMETRICAL SWVS Such media describe naturally gyrotropic media in the presence of magnetooptical activity. The irreversibility of waves and asymmetry of a number of optical properties are determined by the noncentrosymmetrical SWVs [12–15]. As naturally gyroscopic media we consider the media with a right-left asymmetry of the molecular structure (optically active nonmagnetic and magnetic media), the media with supermolecular helical structure (cholesteric liquid crystals), and helical magnetic structures. We restrict ourselves to the case of a naturally gyroscopic medium in the presence of an external magnetic field (more general cases were considered in [13–15]). This case, being simple, reflects the physical picture of the irreversibility of waves. Material equations for such a medium have the form D = εˆ E + i ⎡⎣ g ( k ) E ⎤⎦ + i ⎡⎣g ( H ext ) E ⎤⎦ , B = μH.
(10)
The second term of the first equation is responsible for the natural optical activity, while the third term for the magnetooptical activity. The components of the natural activity vector g(k) depend linearly on the components of the wave vector, while the components of the magnetooptical activity vector g(Hext) on the components of the external magnetic field (also linearly). Equations (10) can be rewritten in the form D = εˆ E + i ⎡⎣( g ( k ) + g ( H ext ) ) E ⎤⎦ , B = μH.
(11)
D = εE + i γ [kE] + i [ gE ] , B = μH.
(12)
Note that after the replacement k → −k the sum G1 = g ( k ) + g ( H ext ) goes to the difference G2 = −g ( k ) + g ( H ext ) . Therefore this replacement changes the dispersion equation, and, hence, if k is its solution the −k is not a solution. This means that the irreversibility of waves takes place. Below we consider the simplest case of an isotropic medium in the presence of a magnetic field (the general case was studied in [13–15]). The material equations for such a medium can be rewritten as For k we obtain JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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k ±2 =
ω2 ⎡ ⎛ γ 0 g ω ⎞⎤ εμ ⎢1 ± ⎜ + cos α ± ⎟ ⎥ , γ 0 = γ εμ , 2 c c ⎠⎦ ⎣ ⎝ ε ε
(13)
Where α± are the angles between the magnetic field and propagation directions of waves with the wave vectors k+ and k−. After the replacement of α± → α± + π and (α± → α± − π) the sums γ 0 ε −1 + g ε −1 cos α ± go to the differences γ 0 ε −1 − g ε −1 cos α ± . Therefore the modulus of k± change with the replacement of their propagation directions to the opposite ones, i.e., the irreversibility of waves is present. The irreversibility of waves can be interpreted in the following way. Let the waves propagate in the direction of the magnetic field, and the natural and magnetooptical rotations of the polarization plane caused by the natural gyrotropy and magnetooptical activity have the same direction (opposite directions), i.e., the natural and magnetooptical rotations are added (subtracted). Let us now change the propagation direction to the opposite one. Then the natural rotation will change its direction with respect to the magnetic field direction, whereas the magnetooptical rotation will remain unchanged. This known property of two types of rotation follows also from equations (12): the natural rotation is determined by the wave vector direction, while the magnetooptical rotation by the direction of the magnetooptical activity vector g. Therefore tha natural and magnetooptical rotations now will be subtracted (added), i.e., the direct and opposite directions of propagation are nonequivalent.
Fig. 3. Forms of SWVs of an isotropic medium having simultaneously the natural and magnetoelectric medium.
Fig. 4. Forms of SWVs for an isotropic medium in different cases.
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This fact is reflected on the form of SWVs (Fig. 3), which in the simplest form of an isotropic medium, having simultaneously the natural and magnetooptical activities, represents two spheres O1 and O2 are shifted oppositely from the physical center O, in the direction of the external magnetic field. Such a form of the SWVs is explained in the following way [13]. In the absence of the magnetic field and natural gyrotropy the SWV represents a sphere (surface A in Fig. 4). The natural gyrotropy leads to the fact that instead of one sphere we have two concentric spheres with different radii corresponding to the waves with the right and left circular polarizations (surface B in Fig. 4). If one imposes a magnetic field on the isotropic nongyroscopic medium, then, due to the magnetooptical activity, the sphere A breaks up into two spheres with the same radii as that of the sphere Abut shifted by the same magnitude into the opposite directions along the external magnetic field [1] (surface C, asterisks denotes the center of spheres). However, if there are simultaneously both the natural gyrotropy and magnetooptical activity, then we obtain two oppositely shifted spheres with different radii (surface D in Fig. 4). Then, in contrast to the surfaces A, B, and C, the surface C does not possess the center of symmetry. 5. CLOSED CENTROSYMMETRICAL INVERSE SWVS Isotropic media with simultaneously negative dielectric permittivity ε and magnetic permeability μ were considered in [16]. According to Maxwell’s equations ω ω (14) [kH ] = − εE, [kE] = − μH, c c at ε < 0 and μ < 0 the vectors k, E, and H make up a left ternary, therefore the Poynting vector S = c [ EH ] 4π , making a right ternary with E and H is directed oppositely to the wave vector k. One can directly check also that the group velocity vector u defined as ∂ω u= , (15) ∂k is directed oppositely to the vector k. If we take into account the classical result, according to which the group velocity is directed along the outer normal to the SWV then at ε < 0 and μ < 0 the direction of the outer normal is opposite to that for the case of ε > 0 and μ > 0 (Fig. 5). A domain, which is inner at ε > 0 and μ > 0, becomes outer at ε < 0 and μ < 0. In this sense, the SWV can be named inverse at ε < 0 and μ < 0.
Fig. 5. Direction of the outer normal ( n = ∂ω ∂k ) in the case (a) when ε > 0 and μ > 0 and (b) when ε < 0 and μ < 0.
According to [16], in a medium with ε < 0 and μ < 0, instead of the light pressure, an attraction of the reflecting medium to the light source takes place. A focusing of the diverging beam at its reflection from the plane boundary of the medium occurs as well. Note that in [16] SWVs were not considered.
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6. OPEN CENTROSYMMETRICAL SWVS In [17] the optical properties of nonmagnetic crystals with different signs of components of the dielectric permittivity tensor were considered. It was shown that the presence of the negative component (along with the positive one) leads to the fact that the SWV becomes open. Below we consider a simple case of a uniaxial medium. The dispersion equation has the form 2 2 ⎛ 2 ω2 ⎞ ⎛ kex + key kez2 ω2 ⎞ (16) + − ⎟ = 0, ⎜ k0 − 2 ε ⊥ ⎟ ⎜⎜ c ε ⊥ c 2 ⎠⎟ ⎝ ⎠ ⎝ ε& where k0 is the modulus of the wave vector of the ordinary wave, kex, key, and kez are the components of the wave vector of the extraordinary waves, and the z-axis is directed along the optical axis of a crystal. Figure 6 shows the SWVs at different choices of sign of ε ⊥ and ε& .
Fig. 6. Forms of SWVs for a uniaxial crystal in the case when (a) ε ⊥ > and ε& > 0 , (b) ε ⊥ > 0 and ε& < 0 and (c) ε ⊥ < 0 and ε& > 0
Figure 6a corresponds to a crystal with ε ⊥ > 0 and ε& > 0 : SWVs are sphere and ellipsoid of revolution. The case of ε ⊥ > 0 and ε& < 0 is shown in Fig. 6b: SWVs are the sphere and hyperboloid of revolution; both the ordinary wave (the sphere on the SWV corresponds to it) and extraordinary wave (the hyperboloid with the axis along the optical axis of the crystal corresponds to it) can propagate in the crystal. The case of ε ⊥ < 0 and ε& > 0 is presented in Fig. 6c: here the ordinary wave is absent (therefore the sphere is indicated by a dashed line). In both cases ( ε ⊥ > 0, ε& < 0 and ε ⊥ < 0, ε& > 0 ) the extraordinary wave can propagate. However, in contrast to the case of ε ⊥ > 0 and ε& > 0, the propagation directions are restricted by the asymptotes of the hyperboloid of revolution. As an outer domain of the open SWV corresponding to the extraordinary wave, we will consider the domain, into which the Poynting vector is directed. This domain may be determined by a direct calculation [15]. In Figs 6b,c the inner domains are shaded. Note also that the Fresnel ellipsoid and characteristic surfaces of the tensor εij and εij−1 also transforms into hyperboloids [18]. In media with open SWVs also the focusing takes place at the refraction of a diverging beam on a plane boundary [19]. It should be noted that the total reflection from such a medium can occur at the normal incidence, while the refracted beam appears with the increase in the incident angle. Media with different signs of the components of εij were studied also in [20] where, however, the form of SWVs was not considered. 7. OPEN NONCENTROSYMMETRIC SWVS In the case of different signs of the components of εij , the SWV becomes open. As was shown above (section 4), the noncentrosymmetrical form of SWVs arises in the presence of the natural gyrotropy and JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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magnetooptical activity. Therefore media with different signs of the components εij , which possess a natural gyrotropy, in the presence of an external magnetic field should be described by open noncentrosymmetrical SWVs. In this paper we have restricted ourselves to indication of one or other features of media casued by the form of SWVs without consideration of optical properties of these media which is an independent research. REFERENCES 1. Landau, L.D. and Lifshitz, E.M., Electrodinamika sploshnikh sred (Electrodinamics of Continuous Media), Moscow: Nauka, 1970. 2. Born, M. and Wolf, E., Principles of Optics, Oxford: Pergamon Press, 1968. 3. Fedorv, F.I., Teoriya girotropii (Theory of Gyrotropy), Minsk: Nauka I tekhnika, 1976. 4. Arnol’d, V.I., Dopolnitel’nie glavi teorii obiknovenikh differencial’nikh uravnenii (Additional Chapters to the Theory of Ordinary Differential Equations), Moscow: Nauka, 1978. 5. Yariv, A. and Yeh, P., Optic Waves in Crystals, New York: Wiley, 1984. 6. Fedorov, F.I., Optika i spektroscopiya, 1957, vol. 2, p. 514. 7. Lyubimov, V.N., DAN USSR, 1968, vol. 181, p. 858. 8. Lyubimov, V.N., Kristallografiya, 1969, vol. 13, p. 1008. 9. Lyubimov, V.N., FTT, 1968, vol. 13, p. 3502. 10. Lyubimov, V.N., Kristallografiya, 1969, vol. 14, p. 213. 11. Lyubimov, V.N., Kristallografiya, 1967, vol. 12, p. 708. 12. Eritsyan, H.S., Izv. AN Arm. SSR, Fizika, 1968, vol. 3, p. 217. 13. Eritsyan, H.S., UFN, 1982, vol. 138, p. 645. 14. Eritsyan, H.S., FTT, vol. 22, p. 3684. 15. Eritsyan, H.S., Optika girotropnikh sred i kholestericheskikh zhidkikh kristallov (Optics of Gyrotropic Media and Cholesteric Liquid Crystals), Yerevan: Hayastan, 1988. 16. Veselago, V.G., UFN, 1967, vol. 92, p. 517. 17. Eritsyan, H.S., Kristallografiya, 1978, vol. 33, p. 461. 18. Eritsyna, H.S. and Arakelyan, H.M, J. Contemp. Phys. (Armenian Ac. Sci.), 2003, vol. 38, no. 5, p. 28. 19. Eritsyan, H.S., Papoyan, A.A., Arakaelyan, H.M., et al., J. Contemp. Phys. (Armenian Ac. Sci.), 2006, vol. 41, no. 3, p. 14. 20. Ryazanov, M.I., ZhETF, 1993, vol. 103, p. 1840.
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