QUASIOPTICS IN ANISOTROPIC AND DISPERSIVE MEDIA

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Submitted on 1 Jan 1979

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QUASIOPTICS IN ANISOTROPIC AND DISPERSIVE MEDIA

V. Milantiev, P. Saikia

To cite this version:

V. Milantiev, P. Saikia. QUASIOPTICS IN ANISOTROPIC AND DISPERSIVE MEDIA. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-613-C7-614. �10.1051/jphyscol:19797297�. �jpa-00219286�

JOURNAL DE PHYSIQUE CoZZoque C7, supple'ment au n07, Tome 40, JuiZZet 1979, page C7- 613

QUASIOPTICS IN ANISOTROPIC AND DISPERSIVE MEDIA

V.P. Milantiev, P. Saikia.

P a t r i c e Lwnwnba U n i v e r s i t y , Moscow W-302 U . S.S. R.

The paper deals with the general the- ory of wave packets in inhomogeneous,non-

stati~nary~anisotropic and dispersive me- dia.Considering the second order geometr- ic optics avproximation,we generalize the Leontovich parabolic equation [I] and obta- in the energy conservation equation.It has been shown that the final expressions for these equations differ for different orderings of the field variables and the varameters of the media and depend on how we consider the inhomogeneity and nonsta- tionarity in the material equation.

1. The electromagnetic field equation for an arbitrary nonmagnetic medium in the absence of external charges and curr-

slowly varying functions oft,* and,in 3

genera1,depend on the s~atial and time --+

derivatives of the eikonal .Further, is supposed to depend only on the first spa- tial and time derivatives of the eikonal

-, + that rep~esent the local rave vector

k,&,t)

and the local f requenc;tdS(@)of the quasi- monochromatic wave (6) respectivel~..Wow we use,the geometric optics approximation

44 3-

(7)

-

_WsT

i -

⁽⁸⁾

the characteristic leng-

2.

th of field in homo gene it^, the correspondfng time scale. T=

/;I/(

^{c b t}

1 ^-

Let

LE*Lw9 L K

be the characteristic va- riation lengC'hs of the dielectic permeab- ility tensor,local frequency and the loc- ents is:

a20"

a1 wave number

respective1y.T' , ,T,

TK are

cuee C U * ~

E + - _{a t 2} ^{= o}

^(I)

the corresponding time scales.

The equation of energy conservation is

Following orderings are considered:

derived from t$e relation:

+

a 5

^{-* $0}

⁺

⁺

^hd ^ L o @ ~ x ~ - ^L ;T'

-Tt,-Tr ^Y) ( 9 )

E z +

0

W = - ~ v . [ ~ x ~ ]

^{(2) IC"}

~eriiwe consider three material equ@

-43

A + + L

.T I.T,T,-.Z-?/~

(10)

3(5t14

= j d t d ~ d ; , E (e

-

^ki

,+

-tA

.i,

t )

i. ^(i,

^,ti

14)

^{( 3}

1 L&-L; L W - L X - ~ ,

^c

-

^&

+ - P A + *

3Rtld) ^.)-

5dt1J'4t,

E(e-.4 :-ti

b r U I L W - L K w L ; T'vrT,*~~T

^)'I(

-a)

t ^2. The amplitude

g @ , u 4

^{d the pol-}

arisation vector -* e=f;(& 4) are decompo-

-00 -- .sed in the power series of the small para-

4 + 4

and express the field V @ C ~ O ~ S

E ,B

and meter . ~ ~ f i n i ~ g

in the form m i(w,z-z:)

ads

ids

~ - ~ ( ~ t , ~ s ~ s ) = ~ d ~ ~ c l ~ E " ( t , ~ = ~ )

e

(12)

&tld)

= 5 &(Z,*, - ^{a~ * vds;.)e} + ^c.c.(~)

^O

+ +

2 =

t - t i ,

y

=

4 - &

where c.c.denotes the complex conjugation,

3

ds -rapid phases(eikonals 1, amplitudes

-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797297

we find g"(g,t,w,,&)f rom (31, (5) and the rel- ation

+ ,

D@la>

( ? 3 ) In the zeroth ordcr acproxim~tion of the perturbation theory, orderings (9)-(71) and eqs. (3)-(5) yield the well-known eikonal equation [3]

.

We observe that for (91,while the eq-

4

uation for is the same,in the first order approximation ,for all material eq- uations,i.t differs for dtfferent material eqs. in the second order avproximation.

In the first case we get

+

IC

vg . ve, ^{= Y E .}

at. _,

⁽⁷⁴⁾

where V* -group velocity,

Y -

^damping

coefficient [2] -c4]

.

NOW onwards index

"s" is dropped.

In the second case,for (3),we obtain the generalized Leontovich parabolic equ- ation Cl] :

.

^9%

d a *

^_t

% + ~ - ~ ~ , = r e ; t ) ~ : v v & - ( x ~

B t

-

ae d a2

t

a2

,

- + a - . - - v K :

^.-

la* aZaw

where,

*,

4 =

a J

a=

e ,

eedQ,

* H " A

--

E ,

^& being the hermitian and antihem-

A

.-+

itian parts of

k'

respecti~el~. %i, %are the group velocity and damping coefficient

the right hand side of (?5).0rderings (lo), (11) lead to more complicated equat- ions for $,and

& .

3.

The general expression for the equat- ion of energy conservation,in the media with rnateri31 equation

(3)

and orderings

(9),(70),is written as

0 6 )

N,

--t

S ,

Q being the usual energy density, Pointing's-vector and the energy dissipa-

4-

-

^-t

tion term respectively,

e,+ .

In the case (I?) ,the energy conserv- ation equation can be obtained only if

-+ -b

E = E o . ~ i l e 'or (51,it differs from ⁽¹⁶⁾ only by the minus sign before the last two terms on the right han3 side,for (4) it does not contain the terms with second

A

derivatives of

& .

Thufl,we conclude that the eouation of energy conservation is more sensitive to the form of the material equation than to the orderings.

R E F E R E N C E S tl] M.A.Leontovich,Iev.Aca.Sci.U.S.S.R., ser.phy~. ,8,16.1944.

c23 Yu.L.Kravtsov, XEJ!P,z, 1470,1968.

[31 B.B.Kadomtsev,"The collective pheno- mena in plasma",~oscow,l976.

t4] V.P.Silin,"An introduction to the kinetic theory of gases" ,Moscow,l971.

corresponding to the change of wave pola- risation.

For

(4),(5)

some new terms appear on