STOKES RELATIONS IN OPTICALLY ACTIVE MEDIA V. V. Shepelevich

UDC 535.56

It is well known (see, e.g., [I-3]) that an optically active medium may be described using the physical equations OB Og Oi ' H = ~ - ~ B - - ~ - ~ - ,

D=~E--~

where g, B, a, and 8 are tensors of the second rank.* sions take the more compact form D = ~E + iyB,

H =

(i)

For monochromatic waves these expres-

~-lB+iSE,

(2)

in which the magnitudes y and 6 differ from ~ and 8 by scalar factors. The relationship between the tensors e and 8 is not obvious, and in Casimir [1] an experimental method is proposed to discover its possible breakdown. As a demonstration of this connection, either the conservation of energy [2] or the Onsager theorem [4] is used. However, the law of conservation of energy is fulfilled only in transparent media, and the Onsager theorem is based on the principle of microscopic reversibility and is demonstrated within the scope of phenomenological theory. In [5] Gyarmati proposed the possibility of the existence of a purely phenomenological means of proof. The work of Maksimenko and Serdyukov [6] is dedicated to an investigation of this route by requiring invariance of the scattering amplitude of electromagnetic waves by macroscopic particles relative to inversion of time. The principle of macroscopic reversibility is also used to determine the connection between the amplitudes of incident, reflected, and refracted waves (the Stokes relation) at the interface between a vacuum and an isotropie medium [7]. In the present work the Stokes relation is generalized to the case of the interface between an absorptive optically active isotropic medium and is used as evidence of the connection between the magnitudes of 7 and 6 in the absence of an internal magnetic field. Let right or left circularly polarized electromagnetic radiation impinge normally from a vacuum onto a semiinfinite optically active medium or cubic crystal

Eoe•

Eo• =

(3)

where e+ and e_ are right and left circular vectors [3]. Maxwell equations for monochromatic plane waves D=

--m•

Equation (i) combined with the

(4)

B=m•

(where m = nn = (c/e)k are refraction vectors; n, a unit vector of a normal wave; and n, index of refraction), leads to the vector equation

[~-1(m.m -- m z) @ i (y @

6) m x @ e] E = O.

(5)

Here m.m denotes the dyadic product; m x, antisymmetric tensor dual to the refraction vector m [3]; and ~, ~, 7, and 6, scalar magnitudes. From (5) follows the equation of the normals, de%[~-1(mm--m 2 ) + i ( ? @ 8 ) m • @8] =0,

(6)

the solution of which gives two values of the index of refraction of the isonormal waves: n•

F/~

(~

+

6)~

~t (v +

6)

(7)

*The proposed treatment is applicable also in the ease of other physical equations equivalent

to (1) [31.

1979.

240

Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 30, No. 2, pp. 345-348, February, Original article submitted November 18, 1977.

0021-9037/79/3002-0240507.50

9 1979 Plenum Publishing Corporation

Substituting (7) in (5), it is easily ascertained that the isonormal waves corresponding to the solutions n+ and n- of the equation of the normals are polarized right and left circularly, respectively, for any y and 6, excluding the case

=--~,

(8)

which will be examined separately. Employing the usual boundary conditions allows the determination of the maximum coefficients of reflection r• and transmission t• of right and left waves for normal incidence from vacuum on an interface with a semiinflnite optically active medium:

l•177 r~--

l~no •

(9)

,

(lO)

t~ = l q : n o , where n~

--

8--v~_

+__

2

(n)

Examining the normal incidence of right and left circularly polarized waves from an optically active medium on an interface dividing this medium from a vacuum, we arrive at the following expressions for the maximum coefficients of reflection r+T and transmission ti:

r~--

1 • no• l_Fno~

l~ = no_-- no+ l•

(12) (13)

Now we obtain a relation analogous to the Stokes formula [7] for the case of optically active media. For simplicity, let a right (left) circularly polarized plane monochromatic wave of unit amplitude Eo = 1 impinge on the interface separating an optically active medium and a vacuum normally from the side of the latter. Then the amplitudes of the reflected, r• and refracted, t• waves are determined from expressions (9) and (i0). We apply the principle of reversibility; i.e., we make the substitution l---~--t.

(14)

It is easy to show that this substitution is equivalent to the transformations

n - + - - n , e + ~ e _ , E--+E*

(15)

e--+e*, ~--->~*, 7-+--?*, 6---~--6",

(16)

and where E is the complex amplitude of the electromagnetic field of the wave, and the sign * denotes the complex conjugate. According to the principle of reversibility, as a result of the reversal of the direction of propagation of the reflected and refracted waves a reversed incident wave should arise. Using the superposition principle of the electric fields of electromagnetic waves, we obtain

r~F~+t~ t~= I,

(17)

= O.

(18)

t• r •

Here the coefficients of reflection and transmission of the reversed waves are indicated by the symbol ~. In the case of the absence of absorption and optical activity (e = e*j ~ = ~*, y = ~ = O) Eqs. (17) and (18) are transformed into the well-known Stokes relations [7] r z § lt' : l,

(19)

r' @ r = O .

(20)

Here there is no point in using the notation ~, since the parameters e and ~ of a transparent medium do not undergo a change under time reversal.

241

Either of Eqs. (17) and (18) allow the sought relationship between y and 6 to be obtained. Thus, Eq. (18) based on (9), (i0), and (12) takes the form

2 ( l::tXn~-4- , After simplification,

1-4-no~:.) , 1-+-n~• l• ' 1-T-n~

2 l_+~Zo----~== 0 '

(21)

we obtain

rto• = no• Inserting into (22) the values of no• from (ii) and considering ? = ~.

(22) (16), we have (23)

Now we examine the special case (8) of the relationship between y and 6 when the indices of refraction of the isonormal waves coincide. Inserting (7) into (5) and using (8), we find that waves with arbitrary polarizations may propagate through the medium. Thus, the preceding treatment appears formally valid based on the equations n + = n_ = V ~ ,

no• =

-T , , f Z V ~

(24) (25)

Substituting (25) in (22) we arrive at the conclusion that 7 ffi ~ ffi 0, and crossed terms in the physical equations (2) are nonexistent; i.e., relatlon (8) is forbidden in an optically active medium. Thus, the relationship between the physical parameters characterizing the optical activity of an absorbing isotropic medium is established by the direct application of the principle of macroscopic reversibility without departing from the scope of the phenomenological theory. LITERATURE CITED i. 2.

3. 4. 5. 6. 7.

H . B . Casimir, Phillps Res. Rep., 21, 417 (1966). B . V . Bokut', A. N. Serdyukov, and F. I. Fedorov, The Electrodynamlcs of Optically Active Media [in Russian], Preprlnt, Physics Institute, Academy of Sciences of the BSSR, Minsk (1970). F . I . Fedorov, Theory of Gyrotropes [in Russian], Nauka i Tekhnika, Minsk (1976). B . V . Bokut', A. N. Serdyukov, and V. V. Shepelevich, Opt. Spektrosk., 37, 120 (1974). I. Gyarmati, Non-Equilibrium Thermodynamics: Field Theory and Variational Principles, Springer-Verlag (1970). N . V . Maksimenko and A. N. Serdyukov, Zh. Prikl. Spektrosk., 24, 936 (1976). A. Shuster, Introduction to Theoretical Optics [in Russian], ONTI, Leningrad--Moscow

(1935).

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