The author thanks P. A. Apanasevich and A. P. Voitovich for discussing the results obtained in the present work. LITERATURE CITED 1.

2. 3. 4. 5. 6.

.

8.

9.

V. M. Arutyunyan and G. G. Adonts, "Induced dichroism and gyrotropy in a resonant medium," Opt. Spektrosc., 46, 809-813 (1979). A. I. Alekseev, "Polarization phenomena of nonlinear spectroscopy," Zh. Eksp. Teor. Fiz., 58, 2064-2074 (1970). A. M. Shalagin, "Determination of the relaxation characteristics with the polarization technique in nonlinear spectroscopy," Zh. Eksp. Teor. Fiz., 73, 99-111 (1977). B. A. Glushko and V. O. Chaltykyan, "Nonlinear Faraday effect in resonance media," Kvantovaya Elektron., ~, No. 5, 1107-1112 (1978). A. A. Kurbatov and T. Ya. Popova, "Nonlinear polarization phenomena in the spectrum of a gas in a magnetic field," Zh. Prikl. Spektrosc., 31, 922-925 (1979). P. A. Apanasevich and V. G. Dubovets, "Theory of the interaction of powerful polarized radiation with an isotropic resonance medium," Zh. Prikl. Spektrosc., 17, 796-803 (1972). P. A. Apanasevich, Principles of the Theory of the Interaction of Light with Matter [in Russian], Nauka i Tekhnika, Minsk (1977), p. 328-342. P. A. Apanasevich and V. G. Dubovets, "Interaction of waves of various polarizations in Resonance media," Zh. Prikl. Spektrosc., 19, 528-537 (1973). V. M. Arutyunyan, G. A. Popazyan, G. G. Adonts, et al., "Resonance rotation of the plane of polarization in potassium vapors," Zh. Eksp. Teor. Fiz., 68, 44-50 (1975).

PECULIAR WAVES IN NATURALLY GYROTROPIC MEDIA B. V. Bokut', V. V. Gvozdev, and A. N. Serdyukov

UDC 535.56

The peculiar features of the propagation of longitudinal (plasma) and spiral (helicoidal) electromagnetic waves in magnetically active plasmas and metals in a magnetic field were considered in [i, 2]. We study in the present work the possible existence and the excitation of similar waves in an isotropic medium having natural optical activity. The electromagnetic properties of a naturally gyrotropic medium with dispersion will be described by the material equations [3]:

D~-e(co) E+io~(~)H,

B=H--i~x(co) E

(1)

with scalar parameters of the dielectric constant ~(~) and the gyrotropy ~(~) at the frequency m. For E(m) > 0 and ~(m) ~ 0 this medium is characterized by circular birefringence with the following refractive index of isonormal waves [3]:

n+ ((o) = ]/-~--~ ___ o: (co).

(2)

Equation (2) was obtained under the assumption that the dielectric constant e(~) is greater than the absolute value of the gyrotropy parameter ~(~). But in the neighborhood of absorption bands, the dielectric constant may assume values close to zero so that the inequality

0 ~ ~ (~o) ~ (z2 (~o).

(3)

can be satisfied in a certain frequency interval. It is by no means obvious that Eq. (2) can be employed when condition (3) is observed and special considerations are required, because in this case one of the quantities n+(m) or n-(m) becomes negative or vanishes. It is easy to verify that inequality (3) can indeed exist; to this end the example of a spiral model of a naturally gyrotropic medium is considered since for this medium [4]

1981. 460

Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 34, No. 4, pp. 701-706, April, Original article submitted June I0, 1980. 0021-9037/81/3404-0460507.50

9 1981 Plenum Publishing Corporation

2

o0 (co) ---- 1 -t- ~z __ 02,

o~ (co) =

2

amoco c (~2 __ o ~ '

(4)

where a denotes a parameter having the dimension of a length and defining the linear dimensions of the spiral. An elementary calculation shows that for the particular model of the medium, inequality (3) is satisfied to the right of the resonance frequency ~ in the spectral band

(5)

U ~ + ~o~<~ o~<~ v a~ + ,o~ ~ + -ffr For a ~ 10 -7 em and ~o~ ~ 1032 sec -2 [5], the width of this band is ~5 ~ in the optical range. First let us consider the case ~(~) = 0. plane monochromatic waves [6]: m•

It follows from the Maxwell equations for re•

naB=0,

(6)

mD=0

and the material equations (i) that, in a naturally gyrotropic medium, only one transverse wave with right or left circular polarization (depending upon the sign of the gyrotropy parameter a(e) at this frequency) can propagate at the frequency to for which e(m) = 0. The field vectors can be represented in the form

Br ~) (r, l) =

--

io~ (co) E(x) (r, t),

H(x) (r, t) = D
for this wave, where

e(~)

a + ihb

(8)

--

denotes the normalized vector of the circular (right or left, depending upon the value of the sign function % = sgn a(w)) polarization; the vectors a and b, along with the vector of the wave normal n, form a right-hand triple of unit vectors. Since the magnetic field strength is zero in the case under consideration, also the Umov-Poynting vector of such a wave must vanish. Accordingly, an electromagnetic wave of this kind does not transfer energy and the group velocity associated with that wave is zero. The resulting solution (7) can be considered as the analog of the spiral electromagnetic wave in a magnetically active plasma [I]. However, by contrast to the latter wave, which can propagate only in the direction of the external magnetic field, a spiral wave of the type of Eq. (7) can propagate in any direction in an isotropic, naturally gyrotropic medium. Now let us consider the possible existence of transverse waves in a naturally gyrotropic medium under the condition e(~)=~2(~). For the spiral model of the medium of Eq.

(9)

(4), this case corresponds to the frequency

=V~z+~z

(

1+

2c2

2"

When condition (9) is satisfied, the refractive index of Eq. waves is

s~ (~) = 21 = (~)1, n_~ (~) = 0.

(2) of the transverse

(10) (11)

The combined solution of the Maxwell equations (6) and the material equations (i) leads to the following formulas for the field vectors of the transverse wave with circular polarization and the refractive index of Eq. (i0):

461

E (x) (r, t)

E(X)e (x) exp

D (x) (r, t) = 2o~~(r

i

I(

E (x)(r, t),

2 I~z (to)l to n r - -

cot

c

,

)I

12)

B ~x)(r, t) = --2i~z((0)E (~) (r, t),

tt(z) (r, t) = - - ice (co) E(x) (r, t), where e(~) denotes the unit vector of the corresponding circular polarization quency under consideration, the vector being defined by Eq. (8).

at the fre-

Of particular interest is the second case defined by Eq. (ii), which points to the possible existence of a transverse wave with refractive index zero or an infinite phase velocity. When Eq. (ii) is satisfied, the vector of refraction m = n-X(~) of the corresponding wave vanishes so that the Maxwell equations (6) are satisfied for B (-~) (t) ----D (-~) (t) = 0,

(13)

and the material equations (i) form a system of two linear homogeneous electric and magnetic field strengths:

equations

for the

e (r E (-z) (t) -t- ice (r H(-x) (t) = 0, - - ir (to) E (-~) (t) @ n (-~) (t) = 0.

(14)

Since according to Eq. (9), the determinant of Eq. system (14) vanishes, nonzero solutions for E (-x)(t) and H(-%)(t) are possible. These solutions can be represented in the form E(-z) (t) = E(-~)e(-~) exp (-- Rot),

H~ -~) (t) = i~ (co) E(-~) (t), where e (-z) = ( a - -

(15)

i%b)/]/2.

The vector of the energy flux density of the field of Eq.

(15) is

S(_~> =

c Is (o~)l. If(-X)l ~ n. (16) 8st It follows from Eq. (16) that the wave of Eq. (15) is a transverse wave relative to the direction of propagation n of the energy. The wave normal is in this case not defined because the refractive index of Eq. (ii) is equal to zero. It should be noted that solutions of the type of Eq. (15), which represent waves having an energy transfer velocity directed to opposite sides, can make up a wave with refractive index zero and some ellipticity: a -]- ipb . e - " %

E = E

14 = i s (co) E.

(17)

F1 +pZ The vector of the energy flux density, is given by the formula S -

averaged over the period 2~/~ of the wave of Eq. cox (r

4~

z

P

(17),

n.

1 + p~

In an isotropic, naturally gyrotropic medium there may also propagate longitudinal waves of the type of plasma waves at frequencies which are defined by Eq. (9). The solution for these waves is of the form

E(II) (r, l) = En exp [i (knr + (o/)], H(II) (r, t) = i s (~o) E(I/) (r, t),

(18)

D(ll) (r, t) = B(II) (r, t) = 0

with an arbitrary wave number k which is independent of the optical characteristics of the medium. The vector of the energy flux density of the longitudinal wave of Eq. (18) is zero. As in the case of longitudinal waves in nongyrotropic media [i], the wave number or the refractive index of the longitudinal wave of Eq. (18) can be determined in a medium with natural optical activity by taking into account the spatial dispersion of second order. The spatial dispersion is phenomenologically brought into account by replacing electric constant in the material equations (i) b y the tensor

462

the di-

(co, k) = s (o)) -i" 7o (0) k = + (~h (o)) - - ~0 (0)) k. k, where Yo(~) and y~(~) denote phenomenological parameters describing the spatial dispersion of second order. The substitution results in an equation for the electric field strength of the following type:

{ I I ~ - - ~ (~y , This equation

--,0)]m.m-- (1 - - -~~ % ) m Z @ 2 i c ~ m X @ e - - a J E = 0 .

has a longitudinal

solution

for E under the

(19)

condition

02

(20)

(0) -- 0~2 (e)) @ --~ ?i (el)n 2 = O.

Indeed, in this case we obtain from Eq.

(19) with Eq.

l+~(yi--%)

(20) the following equation:

m x+2ioc mxg=0,

which is satisfied by an m which is parallel to the vector E, of refraction,

i.e., for

mXE = 0. The c o n d i t i o n

for

the existence

of the solutions

o f Eq.

(20) m u s t b e c o n s i d e r e d

as a

dispersion equation for the determination of the refractive index of the longitudinal waves :

According to Eq. (21), for y~ > 0, real solutions for n (if)exist only in the frequency range in which E(~) ~ 2 ( m ) . For YI < 0, longitudinal solutions in the form of nonattenuated waves can exist only for ~(~) ~ a 2 ( m ) . Let us consider the problem of the possible excitation of the above peculiar waves in a naturally gyrotropic medium. We return to the solution of the boundary value problem for a semiinfinite medium with normal passage of the waves [7]. The field vectors of the spiral wave of Eq. (7), w h i c h is excited by the incident electromagnetic radiation with the corresponding frequency m (for which ~(~) = 0), has the following form (see [7]):

E(•

t)=(E +-i[nEl)exp [i (+ ~c ~176176

(22)

where E denotes the electric field vector of the incident wave. It follows from Eq. (22) that the incident circular-polarized wave excites a spiral wave with a phase velocity, the wave running either away from the boundary or towards it, depending upon the sign of the parameter ~(~). The direction of the phase velocity of the spiral wave also changes oppositely to the change in the sign of the circular polarization of the incident wave. As indicated above, the wave of Eq. (22) does not transfer energy. Therefore the entire incident energy is carried away with the reflected wave from the boundary. This result agrees with the solution of the boundary value problem in a semiinfinite gyrotropic medium [7]. It follows directly from that solution that R = 1 and T = 0 at s(m) = 0 for the coefficients of energy reflection and passage, respectively. Similarly, in the case of reflection from a gyrotropic medium, transverse waves of Eq. (12) and (13), (15) can be excited, too. The solution of the boundary value problem for a semiinfinite medium is as follows in the case of normal incidence: -

-

n@~

E('~) (0 -- - -

nr--

oI

,

C

(E - - i~ [nE D exp ( - -

ioO, 463

where n denotes the refractive index of the medium having a boundary with the gyrotropic medium. Since such waves transfer energy, the coefficient of energy transmission is nonvanishing in this problem and equal to 4~n (n + a)a" I n t h e c a s e o f i n c i d e n c e f r o m v a c u u m o r f r o m a medium i n w h i c h n ~ l , t h i s small and has a magnitude of the order of the gyrotropy parameter T ~6a<<

coefficient 1.

•

The V a v i l o v - - C e r e n k o v e f f e c t i s o n e o f t h e m e c h a n i s m s w i t h w h i c h t h e l o n g i t u d i n a l waves o f Eq. (18) and t h e r e f r a c t i v e i n d e x o f Eq. (21) c a n b e e x c i t e d . When a s o u r c e e x i s t s i n a naturally g y r o t r o p i c m e d i u m , t h e e q u a t i o n f o r t h e f i e l d E i s as f o l l o w s when t h e s p a t i a l dispersion of second order is taken into account:

__ II

0z - - oc~ +

The parameters c' ~' Y~

,

ca

02

2in

Ota

c

0 Ot V•

and Y~ must be c~

}

E-

%

0 2 ) V2 +

c2

Ota

4~

OJ

ca

Ot

differential ~ 1 7 6

(23)

)' =( ~ I ' i

x

The density of the current J, arising from the point charge e moving with the velocity v is of the form J ---- ev5 ( r - - vt).

(24)

The solution of Eq. (23) with a source of the form of Eq. (24) is as follows for the Fourier components of the longitudinal field when the initial condition is E(r, t) = 0 at t = 0: E(r, t ) - -

4aiek

k2

1--exp[--i(kv--@t] exp [i (kr - - cot)], e (kv) - - ~2 (kv) + "h (kv) k z

(25)

where k and to are related by the dispersion equation (20). The solution of Eq. (25) is a longitudinal wave with an amplitude oscillating in time. When the condition e (kv) -- aa (kv) + ?l (kv) k z ---- 0

(26)

is satisfied, the field of Eq. (25) increases linearly in the course of time: E (r, t) --

-

-

4~ekt

exp [i (kr - - cot)].

(27)

ka. Oe (@/Oco Equation (27) was obtained with neglected derivatives of ~2(~) and ~x(~) with respect to the frequency; these terms were disregarded in comparison with ~c(m)/3m. Thus, Eq. (26) must be considered the condition of Cerenkov radiation and this condition determines the direction of propagation of the longitudinal waves. It follows from Eqs. (21) and (26) that the Cerenkov radiation of such waves is possible under the condition v ~co~I/(ai--e). LITERATURE CITED i. 2. 3. 4. 5. 6.

464

V . L . Ginzburg, Propagation of Electromagnetic Waves in Plasma [in Russian], Fizmatgiz, Moscow (1960), p. 108-223. O . V . Konstantinov and V. I. Perel' , "The possible passage of electromagnetic waves through a metal in a strong magnetic field," Zh. Eksp. Teor. Fiz., 38, 161 (1960). B . V . Bokut', A. N. Serdyukov, and V. V. Shepelevich, "On the phenomenological theory of absorbing optically active media," Opt. Spektrosc., 37, 120 (1974). U. Kozman, Introduction to quantum chemistry [Russian translation], IL, Moscow (1960), p. 468-483. V . M . Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Theory of Excitons [in Russian], Nauka, Moscow (1979), p. 432. F . I . Fedorov, Theory of Gyrotropy [in Russian], Nauka i Tekhnika, Minsk (1976), p. 456.

7.

B. V. Bokut ~ , A. N. Serdyukov, F. I. Fedorov, and N. A. Khilo, "On the boundary conditions in the electrodynamics of optically active media," Kristallografiya, 18, 227233 (1973).

ENERGY TRANSFER IN STRONG INCOHERENT INTERACTIONS A. S. Agabekyan

UDC 535.35

Interest in lasing media with high concentrations of active ions has recently increased as a consequence of the need for miniaturization of laser sources [i~ 2]~ An increase in the concentration of active ions by reducing the distance between them enhances the ion--ion interactions which account for both the migration of the energy and the concentration-dependent attenuation of the radiation so that ion--ion interactions substantially influence the characteristics of optical quantum generators. At the present time more than 50 lasers working with sensitized crystals are known [3]. Owing to the radiationless transfer of the excitation energy by donors of the laser ions in these matrices, the threshold for lasing decreases and the intensity of the laser emission increases. However, a significant use of the energy transfer effect can be reached only in matrices with high concentrations of impurity ions. The classical theory of energy transfer [4] cannot be employed in this concentration range [5] and therefore an in-depth understanding of the transfer processes at high concentrations is certainly required. When the characteristics of the process (kinetics, quantum yield, average lifetime of the states) are considered, the initial conditions of the energy transfer are of particular interest. In the usual formulation of the two-level problem of energy transfer from a donor to an acceptor [5] it is assumed that the donor is excited at t = 0, whereupon the interaction of the ions and the transfer of the excitation energy are considered. The initial conditions depend either upon the direct selective excitation of the donor on the level of operation from which the energy transfer takes place or upon the excitation on high excited levels and subsequent transfer of the excitation to the level of operation [2]. The first case is hard to put into practice because the donor and the acceptor are in resonance and therefore usually an excitation to high donor levels is employed, Naturally, any one of the two interacting ions can take the role of the donor and only the initial conditions specify the donor by indicating where the excitation is at t = 0. The goal of the present work is to take into account the influence of the initial conditions and of the time required for the excitation to arrive on the level of donor operation in resonance energy transfer. We consider a system of two impurity ions D and A which are bound by Coulomb interaction V and are located in a solid-state matrix. We assume that the donor is rapidly excited onto high levels and therefore, as a result of downward transitions in the initial moment t = 0, the donor is on level 3 (see Fig. i), whereas the acceptor is on level 0, i.e., it is not excited. The exchange of the excitation between the ions in the case of resonance occurs between 1 § 0 transitions of the donor and 2 § 0 transitions of the acceptor. In the process which we consider, the excitation from level 3 switches to level 1 during the time T3 (with or without emission of radiation); then, owing to the resonance interaction between D and A, the excitation is transferred from D to A and may be emitted as radiation. When a large number of high levels exist and downward cascade transitions are possible, the time 9 ~ has the meaning of the resulting time required for the excitation to arrive on level i. It is easy to show that T3 depends upon the longest relaxation time of the intermediate levels. The condition imposed below on the applicability of balance equations precludes in-phase oscillations of the populations n~ and n2 but does not rule out the transfer of the excitation to both sides. This means that in the general case, the excitation can return from the acceptor to the donor and that the net result must be assessed with proper regard for this process. Naturally, though the energy transfer is in this case reversible, one can nevertheless speak of transfer because after each transfer event, the donor and the

1981.

Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 34, No. 4, pp. 707-711, April, Original article submitted May 15, 1980.

0021-9037/81/3404-0465507.50

9 1981 Plenum Publishing Corporation

465