Wave propagation and group velocity in absorbing media

Volume29$, number 2

WAVE

PHYSICS LETTERS

7 April 1969

P R O P A G A T I O N AND G R O U P V E L O C I T Y IN A B S O R B I N G M E D I A * J. NEUFELD Oak R i d g e National L a b o r a t o r y , Oak R i d g e , T e n n e s s e e USA 37830, USA

Received 15 February 1969

Brillouin's work dealing with the formulation of the group velocity and the velocity of energy transport has been extended to absorbing media.

The c u r r e n t f o r m u l a t i o n a c c o r d i n g to which the d i e l e c t r i c constant ~ depends not only on f r e quency co but a l s o on a f r i c t i o n a l p a r a m e t e r l e a d s to a definition of e n e r g y which i s not c o n s i s t e n t with the m e a n i n g of t h e . c o n c e p t of e n e r g y in g e n e r a l i z e d d y n a m i c s [1]. F r o m the s t a n d point of d y n a m i c s e n e r g y is defined by the p a r a m e t e r s which r e p r e s e n t the s t a t e of a s y s t e m and is not e x p l i c i t l y dependent on any p a r a m e t e r s which a r e d i r e c t l y a s s o c i a t e d with the d i s sipation p r o c e s s . F u r t h e r m o r e , the r e l a t i o n s h i p be tw e en e n e r g y and the p a r a m e t e r s which define the p h y s i c a l s t at e is a s s u m e d to be the s a m e w h e t h e r t h e r e is d i s s i p a t i o n o r not. L e t us take as an i l l u s t r a t i o n a single h a r m o n i c o s c i l l a t o r . The equation of motion without d i s s i p a t i o n is m ~ o + k x o = 0 with m the m a s s , k the s t i f f n e s s , and x ° the d i s p l a c e m e n t . If, h o w e v e r , t h e r e is d i s s i p a t i o n the equation of motioi~ i s m~ + R~ + + k x = 0 w h e r e R is the d i s s i p a t i o n c o e f f i c i e n t and x is the d i s p l a c e m e n t . We a s s u m e that R r e p r e s e n t s a s m a l l p e r t u r b a t i o n and c o n s i d e r x ° t o g e t h e r with ~o and x t o g e t h e r with ~ as r e p r e senting, r e s p e c t i v e l y , the s t a t e of the u n p e r t u r b e d and the p e r t u r b e d s y s t e m . If R = 0, the e n e r g y of an u n p e r t u r b e d and, t h e r e f o r e , c o n s e r v a t i v e s y s t e m is VOCvO,~°) = ½m~O)2 + ½k(xO)2, w h e r e a s when R ¢ 0 the e n e r g y of a p e r t u r b e d and, t h e r e f o r e , d i s s i p a t i v e s y s t e m i s known to be U(x, h) = ½ m~ 2 + ½kx 2 w h e r e , f o r s u f f i c i e n t l y s m a l l R , one has x = x ° e x p ( - R / 2 m). The f o r m u l a t i o n of the e n e r g y of a d i s s i p a t i v e s y s t e m can be extended to the e l e c t r o m a g n e t i c f i e l d t h e o r y by m e a n s of the L a g r a n g i a n f o r m u lation, s i n c e the L a g r a n g e equations of motion apply equally well to d y n a m i c s of m e c h a n i c a l and * Research sponsored by the US Atomic Energy Commission under contract with Union Carbide Corporation. 68

electrical s y s t e m s . It can, therefore, be shown that by analogy the e n e r g y d e n s i t y of an e l e c t r i cal f i el d is defined by the f i e l d i n t en si t y E and is independent of any p a r a m e t e r which r e p r e s e n t s dissipation. U si n g the r e f o r m u l a t e d v e r s i o n of the d i e l e c t r i c constant [1] and a s s u m i n g that conductivity of the m e d i u m r e p r e s e n t s a s m a l l p e r t u r b a t i o n , we g e n e r a l i z e d and extended the B r i l l o u i n t h e o r y of group v e l o c i t y to a b s o r b i n g media. Our p r o c e d u r e is b a s e d on two a s s u m p t i o n s as follows. 1) T h e r e e x i s t s an o p e r a t o r K defined by E = = K E o w h e r e E o i s the e l e c t r i c a l i n t e n s i t y in a n o n - a b s o r b i n g (~ = 0) and t h e r e f o r e u n p e r t u r b e d m e d i u m , w h e r e a s E i s the e l e c t r i c a l i n t e n s i t y in an a b s o r b i n g (~ ¢ 0) and t h e r e f o r e p e r t u r b e d m e dium. 2) If E ° = E ° exp i (kx - wt) is a homogeneous plane wave, then the operation ~= K~ ° is equivalent to a multiplication ~ = K ~ ° where K=exp(otx-fit). Then the wave number and f r e quency for an unperturbed homogeneous plane wave which were expressed by real values of k and co are now expressed by complex values k - ia and co - ifl. Substituting the expression for an unperturbed wave into Maxwell's equations in which a = 0 and the corresponding expression for a perturbed wave into Maxwell's equations in which ~ ¢ 0 and assuming that ]c~]<>/3, we obtain two equations: F = F ( k , o)) = = (c~k2/w) - ~

= 0 and a ( ~ F / ~ k ) + [3(~F/a~)) +

+ 4 ~ = 0 which provide relationships between k, w, a, andfl. If the medium is non-absorbing, an electromagnetic disturbance is usually a specified mixture of homogeneous plane waves ('unperturbed' wave packet). It can then be e x p r e s s e d as E o = = ~ o ( @ ) e x p i ( k x - ~ot) w h e r e exp i ( k x - w t ) i s a c a r r i e r wave, ~o(@) i s a modulating function, ~P = x - V g t , and Vg = d ~ o / d t i s the group v e l o c i t y . Th e c o r r ~ s p o n d i n g f o r m u l a t i o n f o r a ' p e r t u r b e d '

Volume 29A, n u m b e r 2

PHYSICS

LETTERS

7 April 1969

= (exp St) E ° w h e r e E 0 = ~ 0 (~) exp i(kx - wt) and x, t, a n d @ a r e t h e v a l u e s a t t h e p o i n t M. T h e e x p r e s s i o n E = (exp ~x) E ° s h o w s h o w t h e d i s t u r b a n c e l o c a l i z e d a t M d e c a y s w i t h x, w h e r e a s t h e e x p r e s s i o n E = (exp 5 t ) E ° s h o w s h o w i t d e cays with time. The attenuation parameters and 5 are

w a v e p a c k e t (in a n a b s o r b i n g m e d i u m ) i s t h e n obtained by applying the operator/~ to each homogeneous plane wave in the unperturbed wave p a c k e t . T h i s r e s u l t s i n t h e s u b s t i t u t i o n of k - lot f o r k a n d co - i~ f o r ¢o. T h e t o t a l e f f e c t of a b s o r p tion can then be obtained by summing the nonhomogeneous plane waves in the same manner as homogeneous plane waves are summed in the unperturbed wave packet. The perturbed wave p a c k e t t h e n h a s t h e f o r m E = ~ ( ~ ) e x p i (kx - cot), w h e r e ~ (~p) = e x p ( ~ x - f i t ) ~ o ( ~ ) i s t h e m o d u l a t i n g f u n c t i o n a n d e x p ( a x - fit) i s t h e a t t e n u a t i o n factor. In o r d e r t o e x t e n d t h e m e a n i n g of t h e p r o p a gation velocity to absorbing media, we assume that the disturbance can be localized at a point M(XM, tM) a t W h i c h i t s i n t e n s i t y i s m a x i m u m a n d c o n s i d e r t h e v e l o c i t y of t h i s p o i n t a s t h e v e l o c i t y of t h e d i s t u r b a n c e . T h e n t h e v e l o c i t y of t h e p e r turbed wave packet can be shown to be the same a s t h e v e l o c i t y of t h e u n p e r t u r b e d w a v e p a c k e t a n d to b e e q u a l t o v~ = vO~ = d w / d k U s i n g t h e e q u a l i t y ~ = x M - v g t M = 0 a n d e x p r e s s i n g t M in t e r m s of XM o r , c o n v e r s e l y , x M i n t e r m s of t M , w e o b t a i n two a l t e r n a t e r e p r e s e n t a t i o n s f o r a w a v e p a c k e t w h i c h a r e E = (exp k x ) E o and E =

4~a

k = - ~

4~a

~F/~'

and 5-

I F = ( c 2 k 2 / w ) - we]. It h a s b e e n s h o w n i n t h i s a n a l y s i s t h a t b y t a k i n g into account absorption, the energy density becomes ~= U° exp2kx or ~=-ffSexp25t, the energy flow becomes S = S°exp 2kx or S = = S ° exp 25t, a n d t h a t t h e v e l o c i t y of t h e e n e r g y transport vu = ~/U is the same as the kinematic v e l o c i t y Vg = d w / d k , e v e n if t h e r e i s a b s o r p t i o n .

References 1. J. Neufeld, Phys. Rev. 152 (1966) 708; V. L. Ginzburg, The propagation of e l e c t r o m a g n e t i c waves in p l a s m a s (Pergamon P r e s s , New York, 1964) pp. 240-250 and 477-495.

* * * * *

A NEW

APPROACH

TO

THE

STARK

EFFECT

IN

SOLIDS

T. LUI
A new method is used to investigate the Stark effect in solids by m e a n s of a G r e e n ' s function f o r m a l i s m . An original and exact condition is obtained for the energy levels in a solid in the p r e s e n c e of an e l e c t r i c field of a r b i t r a r y s t r e n g t h , which connects t h e s e levels with the lattice p a r a m e t e r s , e l e c t r i c field, and periodic potential. Unlike the usual approximate theory of the Stark effect in solids the r e s u l t p r e d i c t s that (depending on the values of the p a r a m e t e r s ) Stark l a d d e r s m a y only e x i s t for limited r a n g e s of e n ergy, or not e x i s t at all.

W a n n i e r [1] w a s t h e f i r s t to p r e d i c t t h a t a n e l e c t r i c f i e l d f a c t i n g o n a B l o c h e l e c t r o n of c h a r g e e ( m o v i n g in o n e d i m e n s i o n ) w i l l g i v e r i s e to a s e t of d i s c r e t e l e v e l s p a e f w h e r e a i s t h e l a t t i c e s p a c i n g a n d p a n i n t e g e r . W e s h a l l r e f e r to s u c h l e v e l s a s S t a r k l a d d e r s . R e c e n t l y Z a k [2] h a s d r a w n a t t e n t i o n to v a r i o u s d e f e c t s i n p r o o f s t h a t h a v e b e e n g i v e n of t h e e x i s t e n c e of S t a r k l a d d e r s a n d h a s s u g g e s t e d that there are no adequate theoretical grounds for belief in their existence. In t h i s p a p e r w e w i s h to g i v e a p r e l i m i n a r y a c c o u n t of a n e w m e t h o d f o r s t u d y i n g t h e e f f e c t of e l e c tric fields on crystal electrons. * On leave of absence f r o m the Vidyodya U n i v e r s i t y , Ceylon. 69