LITERATURE 1. 2. 3. 4.

CITED

V . P . Gribkovskii, The T h e o r y of Absorption and T ransm i ssi on of Light in Semiconductors [in Russian], pa87)'m i Tektmika, Minsk (1975). V. ~.. Komolov, I. D. Yar0shetskii, and I. N. Yassievich, Fiz. Tekh. Poluprovodn., 11, 85 (1977). F . K . Rutkovskii andV. P. Gribkovskii, Zh. Prikl. Spektrosk., 3, 32 (!965). L . G . Zimin andV. P. Gribkevskii, Fiz. Tekh. Poluprovodn., 7, 1252 (1973).

OPTICAL P.

NONRECIPROCITY A.

Apanasevich

OF GYROTROPIC

a n d N.

V.

MEDIA

Martinovich

UDC 535.51

It has been shown on the basis of the microtheoretical analysis in [1] (also see [2]) that in the linear approximation the constitutive equations f or homogeneous media can be represent ed in the form D~= ea~E~+y.o~VaEf~, Ba ~- ~t~vltl~,

(1)

when spatial dispersion is taken into account, where E, H, D, and B are the positive-frequency (proportional to exp (-ioJt) with oJ > 0) parts of the e l e c t r i c and magnetic field strengths and inductions of the quasimonochromatic components of radiation of frequency co; ~, /3, and ~ are indices which run through the values of x, y, and z; V~ = 5/axe; e ~ and ~ are the permittivity and magnetic permeability t ensors; and ~ 5 = ~ - - - ~ is a tensor which takes account of the nonlocal nature of the polarization of the medium under the action of radiation, i.e., the contributions introduced into the polarization of the medium by the quadrupole e l e c t r i c and dipole magnetic moments of molecules. In the general case the tensors eel, # ~ , and ~/~# a re complex Hermitian quantities. In transparency windows, i,e., under conditions in which the frequency detunIng is appreciably g r e a t e r than the widths of the absorption lines, I ~ i j - coI >> Fij, e~B and #~B are Hermitian tensors, and T ~ p o s s e s s e s the pr oper t y of being anti-Hermitian for the extreme indices, namely, ~ = -~a, as follows from the m i c r o s t r u c t u r a l definition. These sym m et ry properties also follow from the energy conservation law of the electromagnetic field in a nonabsorbing medium. The real t ens or - / ~ = Re ~ o ~ , which is antisymmetric for the extreme indices, descri bes, as is well known [3, 4], the gyrotropy of transparent media - t h e i r ability to rotate the polarization of light beams. At p res en t the t h e o r y of this phenomenon has been well developed [4]. The t e n s o r ~ h ~ is related by the e x p r e s sion %~

= (~g)~,~eoo~ - - (~g)~oe~o ~'

(2)

to t h e gyration tensor g, which is introduced and investigated in [4, 51 and allows the constitutive equations of gyrotropic media to be written in a s y m m e t r i c form with r e s p e c t to the field vectors. The effect on light propagation of polarization, which is taken into account in (1) by the imaginary part of ~0~, has been intensively studied recently both theoretically and experimentally. Most of the attention has been devoted to the antisymmetrical pa r t , which describes the phenomenon of ci rcul ar dichroism. Concerning the symmetrical p a r t , it has been noted in [1] that it can lead to optical nonreciprocity, i.e., to a difference in the refractive indices, and consequently, the phase velocities for opposed waves. The structure of this tensor in different media is discussed and the conditions for which it can result in optical nonreciprocity are exhibited in .this article. In p ar t i cul ar , the magnetoelectric effect (ME) in magnetically ordered crystals [6] can lead to optical nonreciprocity. In the constitutive Eqs. (1) this effect is described by the tensor ~ ME ' ~ = (u~- i ) ~ l # ~ # (#-i~)p5l~p, where u~5 is the tensor usually used to describe ME. In the general case the par t of the tensor I m y c ~ = ~ which is s y m m e t r i c a l for the extreme indices has 18 independent components.* The s y m m e t r y of the structure of media can impose appreciable r e s t r i c tions on the possible values of these components. These rest ri ct i ons can be found by the d i r e c t - c h e c k method [7] ff one notes that y ~ (also ~ is a polar tensor. In a coordinate system whose axes lx, ly, lz are *The ten s o r of the magnetoelectric effect ~ is, as is evident from its definition, specified only by nine independent p a r a m e t e r s . Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 31, No. 5, pp. 807-812, November, 1979. Original article submitted November 23, 1978. 1362

0021-9037/79/3105-1362 $ 07.50 9 1980 Plenum Publishing Corporation

T A B L E 1. T a b l e of t h e T e n s o r s Y~a{3 and 3/~rf3 f o r M e d i a H a v i n g D i f f e r e n t T y p e s of S y m m e t r y Triclinic system Class i

) YZXXYZxg

Vz.,vz?zgxYzyy'~zgzYzz.); '~)zzyYzzzJ

(18)

MonoelJ~e system

Class 2' axis 2 {[ OY .

.

.

.

.

.

.

.Q

9

.

.

.

~ yrr

Class 4

.

. "rr

rr

Class 422

. . . Class 3

C/as.~ 6

.l.r

.f,r

xrigona~ wlstcm Class ~2 .r .f~r

y~r

Hexagonal system Class 622 rr y~r

rr T

, .

.

C/a~ 4 r n ~

Cubic s y s m m

Class 23

~.

9

4).

.

'Q'.

Tet~agonal system Class 4"

.fir .

.

Orthochornbie s y ~ m 2,)~ ~.~ ~aa.~ ~ .(rr

g/a~ m1"tlg

?p

.or

.

Class m, m ~ OY

Class

}.,C/ass "42mr#

Class 8m

yr

rn

Class 6mm ,fr y",

~3m

~/ Class 432 and s media "( Y . withou~ an inversion center .............. t -0 . . . . . . : ~ " ~.r . . . . . /(~/b~. ) . . . . "w~ Cla~es 8,6m 2 and centrally symmetric meaia r ~ r~e,e " =0 -

"

~,

.

=

o r i e n t e d in t h e u s u a l w a y w i t h r e s p e c t to the c r y s t a l l o g r a p h i c a x e s ~7], one c a n r e p r e s e n t the r e s u l t s of t h i s c h e c k in the f o r m of T a b l e 1. T h e a r r a n g e m e n t s c h e m e o f the t e n s o r e l e m e n t s Y~rfl i s c l e a r f r o m the n o t a t i o n f o r a t r i c l i n i c s y s t e m . T h e d o t s in T a b l e 1 d e n o t e t e n s o r e l e m e n t s w h i c h a r e e q u a l to z e r o , and t h e c i r c l e s d e n o t e e l e m e n t s w h i c h m a y be d i f f e r e n t f r o m z e r o , E l e m e n t s c o n n e c t e d by l i n e s a r e e q u a l in m a g n i t u d e a n d s i g n ( c i r c l e s of s a m e c o l o r ) o r o n l y in m a g n i t u d e ( c i r c l e s of o p p o s i t e c o l o r ) . T h e n u m b e r of i n d e p e n d e n t e l e m e n t s i s i n d i c a t e d in p a r e n t h e s e s n e x t to the m a t r i c e s . F o r c o m p a r i s o n , t h e s t r u c t u r e of the t e n s o r Y ~ 3 i s a l s o g i v e n in T a b l e 1. I t i s e a s y to c o n s t r u c t t h i s t e n s o r w i t h Eq. (2) t a k e n into a c c o u n t by m a k i n g u s e of the e x p r e s s i o n s f o r the t e n s o r g g i v e n in [4]. U s i n g t h e e x p l i c i t f o r m f o r the t e n s o r s T' and ~/", l e t u s e l u c i d a t e w h a t e f f e c t s s p a t i a l d i s p e r s i o n d e s c r i b e d by the t e n s o r T" can l e a d t o in the s i m p l e s t c a s e s of p r o p a g a t i o n of p l a n e m o n o c h r o m a t i c w a v e s in u n i a x i a l and c u b i c c r y s t a l s . T h e c h a r a c t e r i s t i c s o f s u c h w a v e s a r e d e t e r m i n e d by the s y s t e m of e q u a t i o n s

In [,1, E])~ + (e~, -- r~'~n~) Z~ + ir'~n~Z~ = O,

(3)

1363

which follows f r o m Maxwell's equations with (1) taken into account and f r o m the e x p r e s s i o n E ~ E exp [ - i ( c t nr)~0/c]. H e r e In, E] is the v e c t o r product of n and E; n = (nx, ny, nz) is the r e f r a c t i o n v e c t o r , whose d i r e c tion gives the d i r e c t i o n of the wave v e c t o r k = ~0n/c, and the length n = [nl is the r e f r a c t i v e index f o r this d i r e c tion; ro~rfl = r We note that besides c r y s t a l s of t r i c l i n i c and monoclinic s y s t e m s the t e n s o r ea[3 is diagonal in the coordinate s y s t e m in which Table 1 is written for the t e n s o r s 1" and 3'" and that always [F~a/3na[ , << s o s in the optical f r e q u e n c y region.

Ir3~n~l

F o r a specified d i r e c t i o n of n Eqs. (3) d e t e r m i n e , as is well known, the r e f r a c t i v e indices and the p o l a r ization states of the two waves (modes) which can exist in the medium. The amplitudes of these w a v e s , and consequently the total e l e c t r o m a g n e t i c field excited in the medium by waves incident f r o m the outside, a r e specified by the boundary conditions. F o r waves p r o p a g a t i n g along the z, axis in cubic c r y s t a l s of class 23 the s y s t e m (3) takes the f o r m (8 - n9 E~ + ( i t R - r x ) nE~ = 0,

(iFa q- r I ) nE~ -F (e - - nz) E,j = 0, E~ = 0,

(4)

where FR,I = F ~ y . The r e f r a c t i v e indices of waves satisfying (4) a r e equal with an a c c u r a c y out to the f i r s t s m a l l t e r m s in F to

n -+ = V ~ , " - V ~ + r2~/2

(5)

and consequently the coordinate components of the field are related as (Ey/Ex) + = u (iF R + FI)/Ir elliptically p o l a r i z e d waves with ellipticity r defined by the relationship = (V

+

-

+

+

+ F~. Thus (6)

a r e solutions of (4). The m a j o r axes of the polarization ellipses of these waves a r e mutually orthogonal and f o r m angles 0+ = +45 ~ with the x axis (with the axis of the second order). A plate made out of the c r y s t a l under d i s c u s s i o n should cause both a rotation of the polarization (for F R g 0) and a variation of its ellipticity (for F l ~ 0). F o r example, when a wave E linearly p o l a r i z e d along the x axis is incident on the c r y s t a l , an elliptically p o l a r i z e d wave should be produced at the exit. The_m a j o r axis of the p o l a r i z a t i o n ellipse of this wave m a k e s an angle 0 defined by the e x p r e s s i o n tan20 = (FR4F~ + r ~ ) . tan 2A with the x axis, and the p o l a r i z a t i o n ellipticity is equal to r 2 = (1 - l a t a n 2 0 1 ) / ( 1 +[atan20l). Here zX= ~0VcF 2 . , + F~(cl~2c) is th e d i f f e r e n c e i n t h e phase advance inthe c r y s t a l t h i c k n e s s d and a = cos 2A. M e a s u r ing the rotation angle 0 a n d the p o l a r i z a t i o n ellipticity of the e m e r g e n t wave r , it is not difficult in this case to find both p a r a m e t e r s F R and F I of the spatial dispersion. When F I = 0, Eqs. (4)-(6) a r e also applicable in c r y s t a l s of c l a s s e s 422, 32, and 622 for the case of waves p r o p a g a t i n g along the optical axis and class 432 c r y s t a l s f o r an a r b i t r a r y propagation direction. This is the well-known case of c i r c u l a r d o u b l e - b e a m r e f r a c t i o n . In c r y s t a l s of c l a s s e s 4, 42m, and 4-3m we have FI. = I~'xzy 0, as is evident f r o m Table 1, and Eqs. (4)(6) d e s c r i b e the case of linear d o u b l e - b e a m r e f r a c t i o n upon propagation of light along the 4 axis. The s y s t e m of equations (3) has the f o r m (e - - nz -- FI n) E~: + iFRnEy = 0,

(7)

iFRnE x - - (e - - n z - - FI n) E u = 0, E~ = 0

f o r waves t r a v e l l i n g along the optical axis in c r y s t a l s of s y m m e t r y 4, 3, and 6. Here FR = F~yz, F R = F"XZX* In this case waves having c i r c u l a r p o l a r i z a t i o n , whose r e f r a c t i v e indices have the values n:~ = V-~--- ~ -1 (r~ _ rR),

(Sa)

upon propagation in the positive direction of the z axis and the values n + = l / [ - k ~-- (r I _ FR)

(8b)

upon r e v e r s a l of the propagation direction, a r e the modes. It follows f r o m a c o m p a r i s o n of (8a) and (Sb) that optical n o n r e c i p r o c i t y , along with rotation of the polarization plane, is possible in such c r y s t a l s . The size of the effect is d e t e r m i n e d by F I.

1364

When r R = 0, the s y s t e m (7) a l s o d e s c r i b e s the p r o p a g a t i o n of e l e c t r o m a g n e t i c w a v e s along the optical axis in c r y s t a l s of 4 m m , 3 m , and 6 m m s y m m e t r y . In this c a s e no rotation of the p o l a r i z a t i o n plane of the light o c c u r s , and optical r e c i p r o c i t y is p o s s i b l e only with a d i f f e r e n c e in the r e f r a c t i v e indices of opposed w a v e s equal to F I. In the g e n e r a l c a s e optical r e c i p r o c i t y caused by the g y r o t r o p y of the m e d i u m and r e l a t e d to a s m a l l c o r r e c t i o n to e which is l i n e a r in k is given by e l e m e n t s of the t e n s o r y" of the f o r m Y~zx, 7~zv, Y~yz, and so on if the signs of the c o r r e s p o n d i n g e l e m e n t s a g r e e (for e x a m p l e , Y~zx and y~zy ). Conse~tubntly,when light p r o p a g a t e s along the z a x i s , the r e c i p r o c i t y effect is p o s s i b l e in c r y s t a l s of s y m m e t r y 4, 3, 6, 4ram, 3 m , and 6ram but only in c r y s t a l s of c l a s s 3 when the light p r o p a g a t e s p e r p e n d i c u l a r l y to the z axis. Since the s y m m e t r i c a l i m a g i n a r y p a r t of yar in (1) is not i n v a r i a n t with r e s p e c t to t i m e inversion, i . e . , with r e s p e c t to the t r a n s f o r m a t i o n t - * - t , then e i t h e r e n e r g y d i s s i p a t i o n o r p r o c e s s e s which change sign upon a time i n v e r s i o n m u s t be p r e s e n t in the i n v e s t i g a t e d s a m p l e in o r d e r f o r it to a r i s e . We will consider s o m e p h y s i c a l m e c h a n i s m s which can lead in Eq. (1) to a p o l a r i z a t i o n d e s c r i b e d by the s y m m e t r i c a l p a r t of I m Ya~. One of the d i s s i p a t i v e p r o c e s s e s which leads to n o n r e c i p r o c i t y is the p a s s a g e of an e l e c t r i c c u r r e n t through a t r a n s p a r e n t g y r o t r o p i c e l e c t r i c a l l y conducting m e d i u m . Such a p h y s i c a l m e c h a n i s m w a s d i s c u s s e d in [8] in connection with the e l e c t r i c a l a n a l o g of the F a r a d a y effect. T h i s effect c o n s i s t s of the f a c t that p o l a r i z a t i o n of the f o r m i~:~o6~V~Ev, a r i s e s in an e x t e r n a l e l e c t r i c field ~ and it is d e s c r i b e d in Eq. (1) by the t e n s o r 7~g~ = 7 e a s ~ , ' w h e r e the r e a l t e n s o r 7eg6~3 is s y m m e t r i c a l With r e s p e c t to the e x t r e m e indices. F r o m the e s t i m a t e of [8] ye ~ 10 -17 CGSE. Such a value of ye c o r r e s p o n d s to a difference in the r e f r a c t i v e indices of opposed w a v e s of a n ~ 10-12J~[. The s i z e of the effect should i n c r e a s e as the f r e q u e n c y of the e l e c t r o m a g n e t i c field a p p r o a c h e s the a b s o r p t i o n lines of g y r o t r o p i c ions of a solution. .H If a f~yrotropic m e d i u m is in an e x t e r n a l m a g n e t i c field ~ , then t h e r e is a t e r m of the f o r m ~To~r6~Ns~7oE3 [9] in the expansion of the p o l a r i z a t i o n of the m e d i u m in p o w e r s of ~ . The optical n o n r e e i p r o c i t y given by this p a r t of the p o l a r i z a t i o n was m e a s u r e d in a c r y s t a l of lithium iodate (LiIO 3) of c l a s s 6 [10]. When ~ is p e r p e n d i c u l a r to the optical axis, the size of the effect lay within the l i m i t s An ~ (1-2) 9 10 -12 N. When ~ is p a r a l l e l to the optical axis, the effect is not o b s e r v e d . This r e s u l t [10] a g r e e s with the conclusions given above. Since in this case 7~g~ = 7Hg6flN6, one should t r e a t the s y s t e m of c r y s t a l + m a g n e t i c field as a c r y s t a l with s y m m e t r y p r o p e r t i e s d e t e r m i n e d a c c o r d i n g to the Curie principle. In the e x p e r i m e n t a l s c h e m e in which g~ , k, a n d t h e optical axis of the lithium iodate plate a r e mutually p e r p e n d i c u l a r , the c r y s t a l has the s y m m e t r y of a t r i c l i n i c s y s t e m (since* 6fTz6~O/m = 1), and optical n o n r e c i p r o c i t y is not forbidden by s y m m e t r y (see T a b l e 1). When ~ i s p a r a l l e l to the optical a x i s , the s y m m e t r y of the s y s t e m c r y s t a l + field a g r e e s with the s y m m e t r y of the LiIO 3 c r y s t a l (6Nl16~o/m = 6), and when the light p r o p a g a t i o n is p e r p e n d i c u l a r to the optical a x i s , n o n r e c i p r o c i t y should not be o b s e r v e d , since the c o r r e s p o n d i n g e l e m e n t s of 7~r~ a r e equal to z e r o . When a s t r o n g wave is p r e s e n t in the m e d i u m , an addition to the p o l a r i z a t i o n a p p e a r s which is produced by cubic nonlinearity. F o r o u r d i s c u s s i o n the t e r m s of this addition of the f o r m ~,

~

0 ~e~m~ 2

~',~

(9)

2

a r e of i n t e r e s t , which d e s c r i b e the nonlocal r e s p o n s e at the frequency ~ of a t e s t signal in the field of a s t r o n g r e s o n a n t wave r 0. 9 The p a r t of (9) which is s y m m e t r i c a l in ~ and ~ should lead to optical n o n r e c i p r o c i t y . the o r d e r of magnitude of • by u s i n g the r e l a t i o n s h i p

One can e s t i m a t e

xNL ~ r c ~ g [E0[~ h (Ac0~§ 2) ' w h e r e the l i n e a r p o l a r i z a b i l i t y of a p a r t i c l e is a L ~ 10 -24 c m 3, and the constant of n a t u r a l optical activity is g ~ 10 -9. Setting the f r e q u e n c y detuning of the s t r o n g wave (w/j - coo) ~ 3F and the width of the r e s o n a n c e line F ~ 0.1 n m , we obtain the following e s t i m a t e of the d i f f e r e n c e in the r e f r a c t i v e indices of opposed w a v e s in the visible region in the CGSE s y s t e m : An ~ 10 -12 [E~2. The value of An will i n c r e a s e as the frequency of the signal acting a p p r o a c h e s the frequency of a t r a n s i t i o n forbidden in the e l e c t r i c - d i p o l e approximation. *This a r b i t r a r y notation indicates that the 1 axis is a c o m m o n e l e m e n t of groups of s y m m e t r y 6 and ~ / m upon the condition that the ~o axis is p e r p e n d i c u l a r to the 6 .axis.

1365

LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

CITED

P . A . Apanasevich and N. V. Martinovich, Zh. Prild. Spektrosk., 25, 493 (1976). P . A . Apanasevich, Principles of the Theory of Interaction of Light with Matter [in Russian], Nauka i Tekhnika, Minsk (1977). L . D . Landau and E. M. Lifshits, The Electrodynamics of Continuous Media, Pergamon. F . I . Fedorov, The Theory of Gyrotropy [in Russian], Nauka i Tekhnika, Minsk (1976). B . V . Bokut', A. N. Serdyukov, and F. I. Fedorov, Kristallografiya, 15, 1002 (1970). V . N . Lyubimov, Dold. Akad. Nauk SSSR, 181, 858 (1968); Kristallografiya, 14, 213 (1969). J . F . Nye, Physical P r o p e r t i e s of Crystals, Oxford Univ. (1957). N . B . Baranova, Yu. V. Bogdanov, and B. Ya. Zel'dovich, Usp. Fiz. Nauk, 123, 349 (1977). V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion Taken into Account and the Theory of Excitons [in Russian], Nauka, Moscow (1965). V . A . Markelov, M. A. Novikov, and A. A. Turkin, Pisma Zh. Eksp. T eor. Fiz., 2___55,404 (1977).

SPECTRAL

LUMINESCENT

PROPERTIES

OF L I G H T

S T A B I L I Z E R S. DERIVATIVES OF 2-PHENYLBENZOTRIAZOLE DERIVATIVES V.

S.

Sivokhin

and A.

A.

Efimov

UDC 535~

Hydroxy derivatives of 2-phenylbenzotriazole are widely used as light=stabilizing additives to polymer materials. The light-protective properties of these compounds were discovered a considerable time ago - at the start of the 16th century, but until now there is no unified view on the nature of their light-protective action. z

It was reported in several papers [1-3] that these compounds, which have a large extinction coefficient (e -> 20,000) in the region of 300-400 nm, emit hardly any light in solutions at room t e m p e r a t u r e , and exhibit weak fluorescence in the frozen state. It has been suggested [4] that one of the possible reasons for such an effective p r o c e s s of deactivation of light energy in the structure could be the t r a n s f e r of a proton in the excited state of the molecule from the hydroxy group to the nitrogen of the triazole ring. This hypothesis is based on the increase in acidity of the hydroxy groups on excitation of the molecule, and also on the anomalously large Stokes shifts of the fluorescence spectra of many aromatic compounds with an intramolecular hydrogen bond

[4-81. Heller et al. [9] attempted to r e c o r d the fluorescence of the form with the t r a n s f e r r e d proton for 2-(2'hydroxyphenol)-benzotriazole, but this f or m did not exhibit luminescence in the region up to 800 nm. In view of this Heller et al. [9] postulated that the f orm with the t r a n s f e r r e d proton is not involved in energy deactivation by these compounds. In the pr e s ent work we investigated the ability of hydroxy derivatives of 2-phenylbenzotriazole to absorb and emit light energy. F o r experimental detection of the form of the molecule with a t r a n s f e r r e d proton (fluorescence with an anomalously large Stokes shift) we used a highly sensitive MPF-4 luminescence s p e c t r o m e t e r (Hitachi, Japan). We investigated solutions of 2-phenylbenzotriazole (tm = 109~ I), 2-(2'-hydroxy-5'-methylphenyl)benzotriazole (tm = 132~ II), and 2-(2'-hydroxy-3' ,5'-ditert-butylphenyl)-benzotriazole (tm = 153~ III) in polar (ethyl alcohol) and nonpolar (n-hexane) solvents. We found that only 2-phenyl-benzotriazole molecules emitted light energy in the f or m of fluorescence at room temperature (Fig. 1). The long-wave absorption band of 2-phenylbenzotriazole in hexane had a coarse vibrational structure with frequency ~980 cm -1 (vibrations of the c a r b o n - c a r b o n bonds in the benzene nucleus). The emission band of solutions of this compound in hexane and alcohol also exhibited the benzene frequencies 540 and 960 cm -1. In view of the high intensity of the longwave absorption band, the vibrational structure in absorption and emission, and the short lifetime of the excited state, the observed luminescence can be attributed to fluorescence of the vv* type. Translated from Zhurnal Prildadnoi Spektroskopii, Vol. 31, No. 5, pp. 813-816, November, 1979. Original article submitted December 15, 1978.

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0021-9037/79/3105- 1366507.50 9 1980 Plenum Publishing Corporation