TECHNICAL PHYSICS

VOLUME 44, NUMBER 8

AUGUST 1999

Some features of the polarization characteristics of strongly absorbing helical periodic media A. A. Gevorgian Erevan State University, 375049 Erevan, Armenia

共Submitted July 8, 1997; resubmitted April 6, 1998兲 Zh. Tekh. Fiz. 69, 72–78 共August 1999兲 Results are presented on the polarization characteristics 共rotation of the plane of polarization and polarization ellipticity兲 as a function of the layer thickness and the absorption anisotropy in strongly absorbing media having a helical structure. A strong resonancelike change in the polarization ellipticity is found as a function of the anisotropy of the absorption at frequencies of diffractional interaction of the light with the medium. A change in the sign of rotation of the plane of polarization of the light is observed as the layer thickness is varied. It is established that sign of the rotation also changes as the absorption anisotropy varies. These effects are studied under conditions of interaction of light with a half space and with a layer of medium of finite thickness. Some new features are identified in the previously observed effect wherein the absorption of radiation in media having a periodic structure decreases as the layer thickness increases. © 1999 American Institute of Physics. 关S1063-7842共99兲01408-7兴

characteristics on the layer thickness. We merely note Ref. 12, in which the rotation of the plane of polarization was studied experimentally as a function of the layer thickness. As far as we are aware, no studies have been made of the polarization characteristics of media having a helical structure as a function of the absorption anisotropy. In the present paper we attempt to fill this gap. In order to facilitate our analysis of the mechanisms responsible for the observed behavior for a layer of finite thickness, we first deem it necessary to analyze the interaction of light with a half space.

INTRODUCTION

Studies of the optical properties of strongly absorbing helical periodic media were reported in Refs. 1–5. These media include cholesteric liquid crystals, chiral smectics, helical magnetic media, and artificial ferromagnetic helical structures. These media come within the definition of gyrotropy given by Fedorov,6 although in the natural state they cannot exhibit spatial dispersion and cannot be magnetoactive. Some characteristics of the Faraday effect in helical periodic media under conditions of oblique incidence were studied in Ref. 7. The optical activity spectra were investigated in Ref. 8 the propagation of light in media possessing dielectric and magnetic helicity was examined in Ref. 9. Some features of the amplitude characteristics of strongly absorbing helical media were studied in Refs. 10 and 11, and an effect wherein the absorption decreases with increasing layer thickness and increasing absorption anisotropy. There it was shown that when light is incident normally on a planar layer of helical periodic medium, three diffraction mechanisms take place: diffraction of light by the periodic helicity caused by refraction anisotropy, diffraction of light by the periodic helicity caused by absorption anisotropy, and diffraction of light in a bounded volume caused by the finite nature of the layer thickness. The present paper reports the results of a further study of the properties of strongly absorbing helical periodic media. Some unique effects are identified, i.e., a change in the sign of rotation with varying layer thickness and with varying absorption anisotropy, and also a resonancelike variation of the ellipticity as a function of the absorption anisotropy. Whereas the wavelength dependences of the polarization characteristics of media having a helical structure and also the influence of isotropic and anisotropic absorption on this dependence have been studied in fairly great detail,1–5 the same cannot be said of the dependence of the polarization 1063-7842/99/44(8)/6/$15.00

BOUNDARY-VALUE PROBLEM FOR A HALF SPACE

We shall consider the case of light normally incident on a half space filled with a medium having a helical structure whose axis is perpendicular to the boundary surface. The field in the medium at the distance z from the boundary has the form1,2 ⫹ E共 z,t 兲 ⫽ 兵 关 E ⫹ 1 exp共 ik 1 z 兲 ⫹E 2 exp共 ik 2 z 兲兴 exp共 iaz 兲 n⫹ ⫹ ⫹ 关 ␰ 1E ⫹ 1 exp共 ik 1 z 兲 ⫹ ␰ 2 E 2 exp共 ik 2 z 兲兴

⫻exp共 ⫺iaz 兲 n⫺ 其 exp共 ⫺i ␻ t 兲 ,

共1兲

where

␰ 1,2⫽⫺ ␦ / 关 1⫺ 共 ␹ ⫾b 1,2兲 2 兴 ; k 1,2⫽2 ␲ b 1,2冑␧ m /␭; b 1,2⫽ 冑1⫹ ␹ 2 ⫾ ␥ ;

␦ ⫽␧ a /␧ m ;

a⫽2 ␲ / ␴ ;

␥ ⫽ 冑4 ␹ 2 ⫹ ␦ 2 ;

␹ ⫽␭/ ␴ 冑␧ m ;

␧ m ⫽ 共 ␧ 1 ⫹␧ 2 兲 /2;

␧ a ⫽ 共 ␧ 1 ⫺␧ 2 兲 /2;

共2兲

n⫾ ⫽(x⫾iy)/ 冑2 are the unit vectors of the circular polarizations, ␧ 1 and ␧ 2 are the principal values of the permittivity 935

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tensor in the plane perpendicular to the axis of the medium, ␭ is the wavelength in vacuum, ␴ is the pitch of the helix, ⫹ are determined from the boundary and the amplitudes E 1,2 conditions. The influence of the periodicity of the structure and the helix parameters on the change in the field and also the wavelength dependence of the fields excited in the medium have been analyzed in detail in many studies 共see, in particular, Refs. 1–3兲. In accordance with formula 共1兲, the rotation of the plane of polarization may be expressed in the form

␸ ⫽ ␸ 0 ⫺az,

共3兲

where ␸ 0 is the rotation of the plane of polarization if the field in the medium is expressed in the form ⫹ E共 z,t 兲 ⫽ 兵 关 E ⫹ 1 exp共 ik 1 z 兲 ⫹E 2 exp共 ik 2 z 兲兴 n⫹ ⫹ ⫹ 关 ␰ 1E ⫹ 1 exp共 ik 1 z 兲 ⫹ ␰ 2 E 2 exp共 ik 2 z 兲兴 n⫺ 其

⫻exp共 ⫺i ␻ t 兲 .

共4兲

Since the analytical formulas are cumbersome, the dependences of the polarization characteristics on the distance z from the boundary and also on the imaginary part of the dielectric anisotropy ␧ a⬙ are best analyzed by means of numerical calculations using the formulas put forward above. A single prime will denote the real part of a particular quantity and a double prime will denote the imaginary part. Figure 1a gives the rotation ␸ 0 of the plane of polarization and Fig. 1b gives the polarization ellipticity e as a function of the distance z from the boundary for anisotropic absorption at the following characteristic wavelengths of the incident light: 1 — ␭⫽ ␴ 冑兩 ␧ m 兩 (1⫺ 兩 ␦ 兩 )⬇0.615 ␮ m and 2 — ␭⫽ ␴ 冑兩 ␧ m 兩 (1⫹ 兩 ␦ 兩 )⬇0.635 ␮ m 共near the boundary of the selective reflection region兲, 3 — ␭⫽0.2 ␮ m⬍ ␴ 冑兩 ␧ m 兩 , and 4 — ␭⫽1.5 ␮ m⬎ ␴ 冑兩 ␧ m 兩 共far from the selective reflection region兲. Note that here and subsequently, to be specific, we consider the case ␧ a⬙ ⬎0 when studying the dependences of ␸ 0 and e both on ␧ a⬙ and on the distance z from the boundary 共under conditions of anisotropic absorption兲. It can be seen from Fig. 1 that under conditions of anisotropic absorption ‘‘saturation’’ of the rotation is observed at the frequencies of diffractional interaction of the light with the medium: after passing through a peak the rotation undergoes damped oscillations about a certain value. Saturation of the rotation has also been observed experimentally.12 An interesting pattern is observed near the longwavelength boundary. Here, before reaching saturation the rotation decreases, goes to zero, changes sign, and only then reaches saturation. The calculations show that this pattern only occurs for specific values of the absorption anisotropy. For ␧ a⬙ ⬍0 the opposite pattern is observed. In particular, the change in the sign of the rotation is observed near the shortwavelength boundary of the selective reflection region. It can be seen from the figures that the polarization ellipticity also saturates at the frequencies of diffractional interaction of the light with the medium. In this case, whereas near the short-wavelength boundary of the selective reflection region the ellipticity saturates by decreasing, near the long-wavelength boundary it saturates by increasing.

FIG. 1. Rotation of the plane of polarization 共a兲 and the ellipticity 共b兲 as functions of the parameter z/ ␴ for anisotropic absorption: ␧ 1 ⫽2.29, ␧ 2⬘ ⫽2.143, ␧ ⬙1 ⫽0.1, ␧ ⬙2 ⫽0, and ␴⫽0.42 ␮m.

Figure 2a gives the rotation ␸ 0 and Fig. 2b gives the polarization ellipticity e as a function of the absorption anisotropy 关the parameter ln(2␧⬙a)兴 for the same wavelengths of the incident light as in Fig. 1. These figures demonstrate resonancelike behavior of the ellipticity as a function of the absorption anisotropy at wavelengths near the selective reflection region. It can also be seen that at specific wavelengths the sign of rotation also changes as the absorption anisotropy varies. DISCUSSION

In order to identify the mechanisms responsible for the observed behavior, we shall use an expression for the wave field in the medium. In cases of weak anisotropy the amplitude of one of the natural waves having nondiffracting circular polarization is much smaller than the amplitudes of the other waves, so that Eq. 共4兲 can be expressed in the form ⫹ ⫹ E共 z,t 兲 ⫽ 兵 关 E ⫹ 1 exp共 ik 1 z 兲 ⫹E 2 exp共 ik 2 z 兲兴 n⫹ ⫹ ␰ 2 E 2

⫻exp共 ik 2 z 兲 n⫺ 其 exp共 ⫺i ␻ t 兲 .

共5兲

This is a fairly good approximation and can explain many of the characteristic features of the properties of helical periodic media. In particular, the saturation of the rotation and the ellipticity under conditions of anisotropic absorption

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near the short-wavelength boundary can be attributed to the fact that the different natural waves undergo different absorption. This can be seen from the expressions for k 1 and k 2 . It can also be confirmed analytically using for the imaginary parts of the wave numbers the following approximate expressions obtained under the condition 1⫺ 兩 ␹ 兩 , 兩 ␹ 兩 Ⰷ␧ ⬙a , ⬙: ␧ a⬘ , ␧ m

⬙ / 共 ␭ 冑␧ m⬘ 兲 , k ⬙1 ⬇ ␲ ␧ m k ⬙2 ⬇

␲

⬘ ␭ 冑␧ m

冋

⬙⫺ ␧m

␧ a⬙ ␧ a⬘

⬘ 兩 ␹ 兩 共 1⫺ 兩 ␹ 兩 兲 2␧ m

册

,

⬘ 兲. 兩 ␹ 兩 ⫽␭/ 共 ␴ 冑␧ m 共6兲

The results of an exact calculation of k 1⬙ and k 2⬙ as func⬙ 共Fig. tions of the wavelength for various values of ␧ a⬙ and ␧ m 3兲 confirm what we have said. Near the short-wavelength ⬘ 兩 (1⫺ 兩 ␦ 兩 ) we find k 2⬙ ⬃0 and thus k ⬙1 boundary ␭ 1 ⫽ ␴ 冑兩 ␧ m ⬙ Ⰷk 2 共absorption suppression effect兲. An increase in thickness leads to a rapid decrease in the amplitudes of the natural ⬙ z). Since k 1⬙ Ⰷk 2⬙ , those amwaves, proportional to exp(⫺k1,2 plitudes which are proportional to exp(⫺k1⬙z) decrease rapidly and become negligible as z increases. Consequently, at some distance from the boundary lying near the shortwavelength boundary, the field may be expressed in the form E共 z,t 兲 ⫽E ⫹ 2 exp共 ik 2 z 兲共 n⫹ ⫹ ␰ 2 n⫺ 兲 exp共 ⫺i ␻ t 兲 .

FIG. 2. Rotation of the plane of polarization 共a兲 and the ellipticity 共b兲 as functions of the parameter ln(2␧⬙a) for anisotropic absorption: z⫽50␴ ; the other parameters are the same as in Fig. 1.

共7兲

It therefore follows that the ellipticity is e⫽( 兩 ␰ 2 兩 ⫺1)/ ( 兩 ␰ 2 兩 ⫹1) and does not depend on z. Moreover, since 兩 ␰ 2 兩 ⬇1 near the selective reflection region, we find e⬇0. The rotation of the plane of polarization at some distance from the boundary is determined only by the real and imaginary parts of ␰ 2 and also does not depend on z. After comparatively rapid variations 共in regions of rapidly varying

FIG. 3. Imaginary parts of the nonresonant k 1⬙ and resonant k 2⬙ wave numbers as functions of the wavelength ␭ for various values of the absorption anisotropy ␧ ⬙a and the average absorption ␧ m⬙ : 1 — ␧ a⬙ ⫽␧ m⬙ ⫽0; 2 — ␧ ⬙a ⫽␧ m⬙ ⫽0.05; 3 — ␧ ⬙a ⫽0, ␧ m⬙ ⫽0.05; the other parameters are the same as in Fig. 1; 1–3 — k ⬙1 , 1⬘ –3⬘ — k ⬙2 .

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FIG. 4. Imaginary parts of the nonresonant and resonant wave numbers as functions of the absorption anisotropy: 1 — ␭ 1 ⫽0.615 ␮ m, 2 — ␭ 2 ⫽0.635 ␮ m; the other parameters are the same as in Fig. 1; 1, 2 — k ⬙1 ; 1⬘, 2⬘ — k 2⬙ .

wave amplitudes兲, the rotation goes to saturation. The change in the sign of rotation observed near the longwavelength boundary of the selective reflection region occurs because, under certain conditions of anisotropic absorption, the natural wave, whose amplitude is usually neglected, begins to play an important role. Near the long-wavelength boundary we find k 1⬙ ⬍k ⬙2 共anomalously strong absorption兲. This implies that whereas for small z the field can be represented as the sum 共5兲, as z increases further the amplitudes of the natural waves which are proportional to exp(⫺k⬙2z) decrease more rapidly, and beyond a certain value of z they become smaller than the amplitude of the natural wave whose field was neglected 关i.e., the natural wave with amplitude proportional to ␰ 1 exp(⫺k⬙1z)兴. The amplitude of this wave decreases far more slowly with increasing z. The direction of rotation begins to change near these values of z. As z increases further, the field can be expressed as the sum E共 z,t 兲 ⫽E ⫹ 1 exp共 ik 1 z 兲共 n⫹ ⫹ ␰ 1 n⫺ 兲 exp共 ⫺i ␻ t 兲 .

共8兲

Thus whereas for small z the field is formed by the sum of circular waves 共5兲 and the rotation takes place in one direction, for large z the field has the form 共8兲 and the rotation changes direction, since the fast circular component is replaced by a slow one. Similarly, i.e., by studying the characteristics of the natural waves in the medium, we can explain other features in the dependences of the polarization characteristics on the distance z from the boundary. Characteristic features in the dependences of ␸ 0 and e on the parameter ln(2␧⬙a) can also be explained. Near the short-wavelength boundary, the wave amplitudes proportional to exp(⫺k2⬙z) vary negligibly as ln(2␧⬙a) increases, whereas the amplitude of the other wave, proportional to exp(⫺k⬙1z), decreases rapidly 共in the absence of ab-

sorption this amplitude is greater than the other two兲. Thus resonancelike behavior is observed in the dependence of e on ln(2␧a⬙). Note that the sign of rotation changes with varying absorption anisotropy at ␭⫽0.2 ␮m 共far from the selective reflection region; short-wavelength region兲. This effect is also caused by the different damping of the natural waves in the medium as the absorption anisotropy varies and by the increased effect of this difference as this anisotropy varies. Near the long-wavelength boundary, as ln(2␧⬙a) increases, the wave amplitudes proportional to exp(⫺k2⬙z) begin to decrease more rapidly than those proportional to exp(⫺k1⬙z). Moreover, as near the short-wavelength boundary, this amplitude is greater than the other two in the absence of absorption. Here, however, a peculiarity appears. As the absorption anisotropy increases further, the wave amplitudes proportional to exp(⫺k⬙2z) begin to decrease more slowly than the other wave amplitude, proportional to exp (⫺k⬙1z) 共the diffraction mechanism caused by the absorption anisotropy begins to have an influence, with the result that the mechanism of absorption suppression begins to come into play兲. Figure 4 gives k 1⬙ and k ⬙2 as functions of the absorption anisotropy at wavelengths near the shortwavelength and long-wavelength boundaries of the selective reflection region, which confirm what we have said. Thus above a certain value of ln(2␧⬙a), the amplitude proportional to exp(⫺k1⬙z) becomes smaller than the other two. As a result of this behavior of the natural wave amplitudes as functions of the parameter ln(2␧a⬙), we observe a resonancelike change in the ellipticity and a change in the sign of rotation as functions of this parameter near the long-wavelength boundary of the selective reflection region. This reasoning suggests that the reduced absorption of radiation in periodic media with increasing layer thickness,

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FIG. 5. Results of the absorption of radiation Q in a layer as a function of the parameter d/ ␴ for ␧ ⬙a : 1 — 0.00005, 2 — 0.0005, 3 — 0.005, 4 — 0.05, 5 — 0.25, 6 — 0.5; ␧ m⬙ ⫽␧ a⬙ , ␧ ⬘1 ⫽2.29, ␧ 2⬘ ⫽2.25, ␴⫽0.42 ␮m, and ␭⫽0.635 ␮m.

discovered in Refs. 10 and 11, may also be observed near the long-wavelength boundary of the selective reflection region 共when ␧ a⬙ ⬎0). This reduced absorption is observed at an absorption anisotropy for which the diffraction mechanism attributed to the absorption anisotropy is already appreciable. Calculations made for a layer of finite thickness confirm this. Figure 5 gives the value of Q⫽1⫺(R⫹T) 共characterizing the optical energy absorbed in the medium兲 plotted as a function of the layer thickness 共where R is the reflection coefficient and T is the transmission coefficient兲 for various values of the absorption anisotropy near the long-wavelength boundary of the selective reflection region. It can be seen that above a certain level of absorption anisotropy the value of Q decreases with increasing layer thickness. We also note that, as can be seen from Fig. 5, for the given parameters of the medium the decreasing radiation absorption with increasing layer thickness is observed at ‘‘enormous’’ levels of absorption anisotropy 共near the long-wavelength boundary a similar effect is observed at a much lower level of absorption anisotropy兲. Naturally, the lower the refraction anisotropy 共first diffraction mechanism兲, the sooner 共i.e., at a lower level of absorption anisotropy兲 the decrease in absorption with increasing layer thickness set in near the long-wavelength boundary 共since this effect is a manifestation of the diffraction of light on the periodic structure created by the absorption anisotropy兲. Numerical calculations confirm this statement. In fact, the calculations show that for the parameters ␧ 1⬘ ⫽2.29, ␧ 2⬘ ⫽2.285 共␦⬇0.001兲, and ␴⫽0.42 ␮m, this effect begins to appear for ␧ a⬙ ⫽0.02. To conclude this section, we note that as the calculations have shown, changes in the sign of rotation with varying layer thickness and varying absorption anisotropy for a cholesteric liquid crystal having the parameters ␧ 1⬘ ⫽2.29, ␧ 2⬘ ⫽2.143 共␦⫽0.033兲, and ␴⫽0.42 ␮m are observed for transmission coefficients T of the order of 10⫺11 – 10⫺12, i.e., there is no transmitted wave. However, this does not imply that the identified effects are purely ‘‘theoretical.’’ The calculations show that these effects depend strongly on the refraction anisotropy ␧ a⬘ . As the refraction anisotropy decreases, the transmission coefficient increases rapidly in those regions where the sign of rotation changes. For ex-

FIG. 6. Rotation of the plane of polarization 共a兲 and the ellipticity 共b兲 as functions of the parameter d/ ␴ for various wavelengths of the incident light for anisotropic absorption. The other parameters are the same as in Fig. 1.

ample, for the parameters ␧ 1⬘ ⫽2.29, ␧ 2⬘ ⫽2.285 共␦⫽0.001兲, ␴⫽0.42 ␮m, a change in the sign of rotation is observed for values of the transmission coefficient T of the order of 10⫺1 – 10⫺2 . Thus, these effects are fully ‘‘experimental,’’ i.e., they are amenable to measurement. From this it also follows that the change in rotation identified in this study is a manifestation of the diffraction of light by the periodic helicity caused by the absorption anisotropy.

BOUNDARY-VALUE PROBLEM FOR A LAYER

Let us analyze the normal transmission of light through a layer of helical periodic medium whose axis is perpendicular to the boundary surfaces. Figure 6a gives ␸ 0 and Fig. 6b gives e plotted as functions of the layer thickness d for the same wavelengths of the incident light as in Fig. 1 but for ⬙ in the case when the light is different values of ␧ a⬙ and ␧ m transmitted through the layer. Figure 7a gives ␸ 0 and Fig. 7b gives e plotted as function of the parameter ln(2␧a⬙) for the same wavelengths of the incident light as in Fig. 1. The characteristics and relationships observed in this case can also be attributed to the behavior of the amplitudes

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where l 1,2⫽ ␥ ⫾2, a 1,2⫽cos(k1,2d)⫿iul 1,2 sin(k1,2d)/(k 1,2d), u⫽ ␲ d/ 冑␧ m /␭. A comparison between the curves in Figs. 1a and 1b and those in Figs. 4a and 4b shows that in most cases they are similar. However, there are some important differences. In particular, the dependence of the ellipticity e on z near the short-wavelength boundary under conditions of anisotropic absorption differs substantially from the dependence of e on the layer thickness d under similar conditions. This difference demonstrates once again the difference between the half-space problem and the layer problem 共although the layer is arbitrarily thick兲. We reiterate that the effects identified in this study are observed in periodic helical media near the selective reflection region, and they occur because the interaction between light and these media is accompanied by the excitation of different natural waves whose amplitudes, phase velocities, and attenuation differ. Thus, depending on the variation in the parameters of the medium responsible for these wave characteristics, quite unusual patterns of wave interaction with the medium can be obtained. In conclusion, we note that these results can be applied, in particular, to ellipsometry in the design of various ellipsometric systems using layers of media of finite thickness having a helical structure. The authors are grateful to G. A. Vardanyan and O. S. Eritsyan for valuable discussions. This work was carried out under Topic No. 96-895, financed by the State Centralized Sources of the Republic of Armenia.

FIG. 7. Rotation of the plane of polarization 共a兲 and the ellipticity 共b兲 as functions of the absorption anisotropy ln(2␧a⬙) for anisotropic absorption: d⫽50␴ . The other parameters are the same as in Fig. 1.

1

⫻ 关共 2b 2 ⫺l 2 兲 exp共 ik 2 d 兲 ⫹ 共 2b 2 ⫺l 2 兲

V. A. Belyakov and A. S. Sonin, Optics of Cholesteric Liquid Crystals 关in Russian兴, Nauka, Moscow 共1982兲. 2 V. A. Belyakov, Diffraction Optics of Complex Periodic Structures 关in Russian兴, Nauka, Moscow 共1988兲. 3 O. S. Eritsyan, Optics of Gyrotropic Media and Cholesteric Liquid Crystals 关in Russian兴, AŽastan, Erevan 共1988兲. 4 V. S. Rachkevich, Ber. Bunsenges. Phys. Chem. 93, 1137 共1989兲. 5 V. A. Belyakov, A. A. Gevorgyan, O. S. Eritsyan, and N. V. Shipov, Kristallografiya 33, 574 共1988兲 关Sov. J. Crystallogr. 33, 337 共1988兲兴; Zh. Tekh. Fiz. 57, 1418 共1987兲 关Sov. Phys. Tech. Phys. 32, 843 共1987兲兴. 6 F. I. Feodorov, Theory of Gyrotropy 关in Russian兴, Nauka i Tekhnika, Minsk 共1976兲. 7 V. A. Kienya and I. V. Semchenko, Kristallografiya 39, 514 共1994兲 关Crystallogr. Rep. 39, 457 共1994兲兴. 8 E. K. Galanov and V. G. Medvedev, Opt. Spektrosk. 76, 79 共1994兲 关Opt. Spectrosc. 76, 72 共1994兲兴. 9 V. N. KapshaŽ, V. A. Kienya, and I. V. Semchenko, Kristallografiya 36, 822 共1991兲 关Sov. J. Crystallogr. 36, 459 共1991兲兴. 10 G. A. Vardanyan and A. A. Gevorgyan, Kristallografiya 42, 316 共1997兲 关Crystallogr. Rep. 42, 276 共1997兲兴. 11 G. A. Vardanyan and A. A. Gevorgyan, Kristallografiya 42, 723 共1997兲 关Crystallogr. Rep. 42, 663 共1997兲兴. 12 Yu. V. Denisov, V. A. Kizel’, E. P. Sukhenko et al., Kristallografiya 21, 991 共1976兲 关Sov. J. Crystallogr. 21, 568 共1976兲兴.

⫻exp共 ⫺ik 2 d 兲兴 / 共 2b 2 兲兴 n⫺ 其 exp共 ⫺i ␻ t 兲 / 共 4 ␥ a 1 a 2 兲 ,

Translated by R. M. Durham

and phases of the waves excited in the medium. In this case, however, the field in the medium has the form 共at the second boundary兲 E共 d,t 兲 ⫽ 兵 关共 ␥ ⫺ ␦ ⫺2 ␹ 兲 ⫹ 关共 2b 1 ⫺l 1 兲 exp共 ik 1 d 兲 ⫹ 共 2b 1 ⫹l 1 兲 exp共 ⫺ik 1 d 兲兴 / 共 2b 1 兲 ⫹ 共 ␥ ⫹ ␦ ⫹2 ␹ 兲关共 2b 2 ⫹l 2 兲 exp共 ik 2 d 兲 ⫹ 共 2b 2 ⫺l 2 兲 exp共 ⫺ik 2 d 兲兴 / 共 2b 2 兲兴 n⫹ ⫹ 关共 ␥ ⫺ ␦ ⫹2 ␹ 兲关共 2b 1 ⫺l 1 兲 exp共 ik 1 d 兲 ⫹ 共 2b 1 ⫹l 1 兲 exp共 ⫺ik 1 d 兲兴 / 共 2b 1 兲兴 ⫹ 关共 ␥ ⫹ ␦ ⫺2 ␹ 兲