3,

4.

B. S. A b r a m o v i c h and V. V. Tamoikin, Fiz. P l a s m y , 6, No. 3, 531 (1980). V. I. Klyatskin, Statistical D e s c r i p t i o n of Dynamic S y s t e m s with Fluctuating P a r a m e t e r s [in R u s s i a n ] , Nauka, Moscow (1975).

WAVES

IN NONSTATIONARY

NONDISPERSING

NONHOMOGENEOUS

MEDIA

S. I . A v e r k o v

and

V.

P.

Boldin

UDC 538.574:538.56:519.25

Some p r o b l e m s a r e c o n s i d e r e d in the linear theory of propagation of e l e c t r o m a g n e t i c w a v e s through nondispersing media with t r a v e l i n g p a r a m e t e r s , s p e c i a l c a s e s of s u c h media being nonhomogeneous ones and nonstationary homogeneous ones. It is d e m o n s t r a t e d that fields in media with t r a v e l i n g p a r a m e t e r s can be uniquely r e p r e s e n t e d as s p e c t r a of independently propagating a n h a r m o n l c w a v e s with v a r i a b l e amplitudes and phase velocities. The s t a t e m e n t by Shelkunoff [11] about the indistinguishability between standing and t r a v e l i n g w a v e s in nonhomogeneous m e dia is a l s o d i s c u s s e d . The d i s c o v e r y of the l a s e r has stimulated the publication of many studies pertaining to i n t e r a c t i o n of m a t t e r with s t r o n g optical fields [1]. In r e c e n t y e a r s , following the development of new r a d i o e n g i n e e r i n g m a t e r i a l s , much has been done t c ~ a r d constructing a theory of e l e c t r o m a g n e t i c shock w a v e s in nonlinear media [2]. Meanwhile, not enough attention has been paid in the technical l i t e r a t u r e on this subject to the principal p r o b l e m s in the linear theory of wave propagation through nonstationary nonhomogeneous nondispersing media, concepts about which a r e v e r y i m p o r t a n t for applications in the range of r a d i o frequencies with periods much longer than the r e l a x a t i o n time of m a t t e r (on the o r d e r of 10 -14 sec). The r e l a t i o n between field intensities E, H and inductions D, B can be defined in t e r m s of quantities and ~, which a r e explicit functions of s p a c e coordinates and t i m e , making it in many c a s e s possible to obtain exact solutions to r a t h e r g e n e r a l p r o b l e m s of both t h e o r e t i c a l and p r a c t i c a l i n t e r e s t . On the b a s i s of such s o l u tions there has been, in e a r l i e r studies, d e m o n s t r a t e d the feasibility of converting m o n o c h r o m a t i c signals in said kinds of media to pulse signals [3], of propagating T E , TM, and TEM waves [4, 5], and of other i n t e r esting a c h i e v e m e n t s . S e v e r a l p r o b l e m s have not y e t been explored sufficiently, however, and among t h e m the methodologically i m p o r t a n t p r o b l e m of unique r e p r e s e n t a t i o n of fields in such media as a s u p e r p o s i t i o n of independently p r o p a gating a n h a r m o n i c w a v e s . In 1947 Kofink [12] proved the validity of this r e p r e s e n t a t i o n in the c a s e of s t a t i o n a r y nonhomogeneous media (with the d i e l e c t r i c p e r m i t t i v i t y v a r y i n g a c c o r d i n g to the law ~ = p~ + ~ w h e r e p = p(z) and k = eonst)

inwhieho,2p-l/~(z)exp (ik~ p(z)clz~i~t]l

1

[L p" P"

waves (x = eonst) can p r o p -

agate. No attention was in subsequent y e a r s paid to the n e c e s s i t y of examining the validity in the g e n e r a l c a s e , evidently on the s t r e n g t h of Shelkunoff's s t a t e m e n t [11] about the indistinguishability, in principle, between t r a v e l i n g and standing w a v e s in a nonhomogeneous medium. In this study h e r e we will d i s c u s s that s t a t e m e n t and d e m o n s t r a t e the possibility of said s p e c t r a l r e p r e s e n t a t i o n of fields in media with t r a v e l i n g p a r a m e t e r s . 1. S h e l k u n o f f ' s

Identity

In o r d e r to c o n f i r m the validity of his s t a t e m e n t about the indistinguishability between a standing wave and a t r a v e l i n g wave in a nonhomogeneous m e d i u m , S. A. Shelkunoff pointed to the identity* cos ~ z + ~ e i~z = V (1 + ~)~ cos ~-~ z + ~-~sin ~~'z exp i arctg ~

tg ~ z ,

*The notation h e r e is that in [6]. Gorki State University. T r a n s l a t e d f r o m I z v e s t i y a Vysshikh Uehebnykh Zavedenii, Radiofizika, Vol. 23, No. 9, pp. 1060-1066, S e p t e m b e r , 1980. Original a r t i c l e submitted July 18, 1979.

0033-8443/80/2309- 0705507.50 9 1981 Plenum Publishing C o r p o r a t i o n

705

w h e r e e = const, /~ : const, and z is the local coordinate. The left-hand side of this identity describes a wave* which is principally a standing one (when e << 1), inasmuch as the f i r s t t e r m is predominant here. The e x p r e s sion of the right-hand side is that for a traveling wave. The last s t a t e m e n t is based essentially on the concept of an impossible unique r e p r e s e n t a t i o n of the r e a l part of expressions in the f o r m y(z) = V(z)ei~'(z) (or V(z) exp [i#(z) - i~t]) when V(z) and t,(z) a r e a r b i t r a r y functions. In the case of physically o c c u r r i n g vibrations such as those d e s c r i b e d by the equation y" + W(z)y = 0, e.g., functions V(z) and ~,(z) a r e not a r b i t r a r y [10], however, and in dealing with the p r o b l e m of a notation for such vibrations which will make them distinguishable one must consider their c h a r a c t e r i s t i c s determined by the fundamental equations. Seeking the equation which cos fiz + eeifl z will satisfy, a c c o r d i n g l y , we find y,, + flZy = 0, W(z) = f12. We conclude, t h e r e f o r e , that the e x p r e s s i o n s on the right-hand side and on the lefthand side of the identity describe waves in a s t a t i o n a r y homogeneous medium. Consequently, V(z) = 4(1 + e)2 cos2 flz + e2 sin2 flz = const, which e = - 1 / 2 . The identity becomes a trivial one, m o r e o v e r , and does not c o n f i r m Shelkunoff's statement. Without dwelling on and discussing analogous statements in other studies, we now proceed to prove the validity of the said r e p r e s e n t a t i o n of fields in media under consideration here. 2. Wave

Equation

We will limit the analysis to E and H fields in a nonstationary nonhomogeneous nondispersing and l o s s less medium which can be described by one-dimensional equations t OE Ox ....

1 -c

O~H Ot "

OH Ox

1 c

O~E Ot '

(1)

If the magnetic permeability ~ and the dielectric permittivity e of the medium a r e a s s u m e d to be functions of the traveling p a r a m e t e r 7/= t - x / a (a denoting the velocity of the traveling p a r a m e t e r ) , then the variables in Eqs. (1) a r e generally not separable. Using the t r a n s f o r m a t i o n in [8, 9] f o r the purpose of making the s e p a r a tion of variable possible, on the other hand, does not yield simple enough new equations. In o r d e r to o v e r c o m e (as far as possible) the attendant difficulties, we will r e p l a c e x and t in Eqs. (1) with new variables. F i r s t , as in the e a r l i e r study [3], we will change to coordinates which move at the velocity of the p a r a m e t e r wave z'~x,

-q=t----

x

,

(2)

and then we will replace E and H with the quantities E,_~ ~.~_E_ l__m, C

H " = t~ H - - 1 E ,

a

C

(3)

a

related to them linearly and identical, except for the constant coefficient, to the field intensities described in the moving s y s t e m of coordinates (2). Relations (1)-(3)yield

1 OE'

-OE' -+

0"~

1

1

OX'

a

OX'

=0, (4)

v" OH" O~

1

{t~ OE' 1 c Ox" +

1 ,732

1 OH' ----0.

a

c)x'

a2

Eliminating H' f r o m Eqs. (4) and changing in the thus obtained equation for E ' to coordinates a s s o c i a t e d with the c h a r a c t e r i s t i c s ~=x'--

~ tl

1--

a~ ]

da,

~ - ~ - i d~,

(5)

*Upon multiplication by the time f a c t o r e -i~~ ~Waves in t r a n s m i s s i o n lines with variable linear inductance and capacitance as well as waves in strings with variable density and tension can, as is well known, be described by analogous equations. T h e r e f o r e , the expressions which will be derived here apply also to such s y s t e m s . 706

where v = c

/

~

and p = qu(7?)/e0?), we obtain for E ' the equation

O~~

p-2 (~)

= O.

(6)

w h e r e p(r) = p[~)(r)]. A wave equation of the f o r m (6) is often encountered in physics. However, as f a r as these authors know, it has n e v e r before been applied to media with t r a v e l i n g p a r a m e t e r s . In the s p e c i a l c a s e o f p(T) = const the solution to Eq. (6) is

and r e l a t i o n s (2)-(5) E(x, t ) - -

1 V

P

F,

1

x--

a

1

1

v

a

+

__1 + 1 ~ T

(s

F2 x +

__1 + 1__

~

~

)

9

(7)

a

This e x p r e s s i o n is the s a m e as the one obtained e a r l i e r [3]. 3.

General

Solution

to the

Problem

F o r p(T) v a r y i n g a c c o r d i n g to any a r b i t r a r y law, a solution of Eq. (6) by the method of s e p a r a t i o n of v a r i a b l e s yields t~ (~,.) = 9 (~) [r i~ + c~ e-ik~ 1 ,

(8)

w h e r e cl, c2 a r e integration c o n s t a n t s , k is the s e p a r a t i o n constant, and function 6(r) s a t i s f i e s the equation of vibrations d' r d~l + kao*(~)q) = 0.

(9)

The solution to Eq. (9) is conveniently sought in the f o r m [10] r (~) = dL~V(~) exp ( • ikpo ~ V - : ( ~ ) d , ) ,

(10)

w h e r e d,, 2 a r e a r b i t r a r y c o n s t a n t s , P0 is the value of o(r) for a m e d i u m with constant p a r a m e t e r s , and the a m plitude f a c t o r V(r) is d e t e r m i n e d f r o m the equation d2V -- - k ' p ~ V -a + k~ p2(~)V = 0,

(11)

d ~a

E x p r e s s i o n s of the (10) Mnd, in the f o r m of the W e n t z e l - K r a m e r s - B r i l l o u i n solution, have been e x amined in other studies [6, 12] of northomogeneous media for d e t e r m i n i n g how the m e d i u m p a r a m e t e r s v a r y when the field is given.* It has been d e m o n s t r a t e d in the other e a r l i e r study [10] that, upon elimination of extraneous solutions to Eq. (11) [ w h e n o ( r ) = const, e.g., then V(r) # eonst is physically meaningless], the amplitude factor V(T)is d e t e r m i n e d uniquely. I n s e r t i n g e x p r e s s i o n (10) into e x p r e s s i o n (8) and taking into account r e l a t i o n s (2)-(5), we obtain E~,(x, t)-----

vo

[

} V+

v~ [

v,( v,)]

pV

--

ika

1--

-~

C,keik~--+

[

V--

v,( pV

ika

1--

j

~

Co.ke~k~+

(12)

'

a ~

w h e r e V' = dV /d ~, ~ • = x •

S "~

-T

1 - - _-7,

d ~q and C l k , C2k a r e a r b i t r a r y constants.

Thus, the uniqueness of both the amplitude factor V(T) and the phase f a c t o r 5 V-Z(z)d~ in solution (10) m a k e s it possible in a m e d i u m with t r a v e l i n g p a r a m e t e r s to r e s o l v e the total field into independently p r o p a g a ting waves with v a r i a b l e amplitudes and phases. *Specifically, e x p r e s s i o n s (10) w e r e t r e a t e d [6, 12] as a d e s c r i p t i o n of independently propagating waves.

707

We note that in r e f e r e n c e s to media such as those under consideration h e r e , the concept of a standing wave is generally less useful than the concept of traveling waves. Indeed, the e x p r e s s i o n for an anharmonic standing wave will be simple only when the traveling waves forming the standing wave a r e identical. In the general case (12) this is not so, however, owing to the different conditions under which traveling waves propagate in the same direction as the traveling p a r a m e t e r and in the opposite direction, r e s p e c t i v e l y . On the other hand, the said s t a t e m e n t does not apply to stationary nonhomogeneous media and nonstationary homogeneous ones. * It has been demonstrated e a r l i e r [5], specifically, that the laws describing the space distribution of fields in nonstationary homogeneous media do not differ f r o m those describing it in media with constant p a r a m e t e r s , which indicates the possibility of using in this case the standing-waves representation. 4.

Variation

to the p(v)

of the

Traveling

Parameter

According

= p0(1 + ~ r ) -2 L a w

Equation (6) admits an exact solution only in a few special c a s e s . As an example, we will consider the case of p(r) varying according to the i n v e r s e - s q u a r e law p(r) = p0(1 + c~r)-2,~ where P0 and c~ a r e constant quantities. Then 1+== {Fx(5__Spd.:)+Fz(r Po

E'(r

S pd=)} "

F r o m h e r e , with the aid of relations (2)-(5), one can find E(x, t) and H(x, t). For E ~ , t), specifically, we have

1

E(x, t) = 1 + ~ (~)

F1(~,-) +

1

1

v

a

1

F2(~+)I_

1 + 1 v

~

j

(1

a

1 )

~oa v=

{Fl + F..,} d ~,

(13)

a'

where F 1 and F 2 are a r b i t r a r y functions defined by the boundary conditions and the initial conditions, and ~+ = x + S (a-~ -+- v-')-t d r,. With initial and boundary conditions such that Ft(~,-)

=

cos ~_,

&(,%) = 0,

e.g., we have 6(x, t) =

1 1+~(~])

cos ~_ 1 1

~ sin ?_ ( 1 1 )

+ Po ~

' 7) z

(14)

aa

The exact solutions (13), which are valid at any rate of change of parameters a and ;i, indicate the possibility of independent propagation of anharmonic waves through the medium. Defining vp = 07• / O,~+_ find that the phase velocities of these waves are -- 0-7 ~ , we ~,'p = + c / V ~ ( ~ ) ~(~) .

According to expressions (13) and (14) given h e r e , the undulatory amplitude modulation in these waves ~aries at e v e r y point in space in s y n c h r o n i s m with the traveling p a r a m e t e r . One can f u r t h e r conclude, on the basis of e x p r e s s i o n (12), that this phenomenon occurs also in media with other laws of p(r) variation. 5.

Two Extreme

Cases

of Variation

of the

Medium

Parameters

Equalities (12) a r e general in form. Inasmuch as the amplitude factor V(r) here cannot be simply d e t e r mined h ' o m the nonlinear equation (11), we will consider the e x p r e s s i o n for fields E and H with V(T) d e t e r mined for two e x t r e m e c a s e s of variation of the medium p a r a m e t e r s . F o r the medium with slowly varying p a r a m e t e r s , we have in the W e n t z e l - K r a m e r s - B r i l l o u i n approximation *Also the media with a standing parameter wave, not considered in this study, v2 ~According to relation (5), moreover, P0?) is determined from the algebraic equation =~~ p-ai20,~st~+ ~/~ ~-i/2= //0

=--~+c' Pc

708

with v 0 = {v}c~=o, c' = const, a n d # = const.

(15) with d~,2 = coast. F r o m e x p r e s s i o n s (15) and (8), with the aid of relations (2)-(5), we obtain 1

(16)

whereupon on the basis of relations (3), (4), and {16) we obtain

Ea(x, t) = ] / 7 '

1 C~a 1 v a

1 + i -k a -V p

1--

+ _1_ + __ 1

v

where p'----d,o'd.,~ and ~+ ----x + S (a-t + v-')-tdq"

a

ika

(17)

V'~-

In this case the resolution of the total field into independently

propagating anharmonic waves is, obviously, unique. Let us now consider the interesting case of e or ~ varying periodically. Let # = P0 and ---- % (1 + ~ sin fl~)-l,

(18)

where e0, ~, a2 a r e constant quantities, and ~ << 1. With the aid of relations (5) and (17), considering only the condition b2 > c~2 where b ~ ~O ~

1 , we define

,o(,1 = Pc

1 + 2~

-b-

1+

-~

V

+

2"

1+-~

tg ~- ~,

where o0 = ~ v0 = c/P0C~0, and fl = P 0 v 0 ~ 2 ~ a . Solution (12) for the amplitude factor V(r) will be sought by the method of a small p a r a m e t e r . Accordingly, in the f i r s t approximation [10], we have V (~) ~ 1 + ~ p sin ~o ~,

fi2

where ~ , = s

O:a, p----[

L ~2 V2ok

v_~--4

as]

(19)

]-'

. Inserting e x p r e s s i o n s (10)and (8)yields

E'k (~:, ~) --- V(~){ Clk exp [ik (~ - - ~ Pc V -~d~)] + C2kexp [ik (~ + S pc V -2 d~)]}

(20)

f r o m which, on the basis of relations {2)-(5), we find

Ek(x, t) =

Cl~po

9

[I+~A1(~)] e'~- +

C~kP____A_o[1+:A2(~)] e~V+,

1

1

1

1

vo

a

vo

a

(21)

where

~• _

+ rot + a 1 _+

cos~2~ 1

l+p

,

as

Ix,

A1.2(~) =

sln ~ -- 2-----~-P(1 - T " --'~-)cos ~

1

v~

ika

Cg~

and the a r b i t r a r y constants Clk, C2k a r e d e t e r m i n e d uniquely f r o m the initial conditions and the boundary conditions.

709

Expressions (21) given here are valid also at fast changes of the traveling parameter. Just as in the previous cases, these expressions describe independently propagating waves with undulatory variation of their amplitudes. In conclusion, we will note that in the analysis of fields in media with a traveling wave of parame te rs, it is expedient to transform the variables in Eqs. (1) as described here so as to a r r i v e at a simple form of the wave equation (6) convenient for the study of complex wave propagation processes in such media by the a m p l i t u d e , p h a s e method [10]. LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

CITED

V S. Butylkin, A. E. Kaplan, Yu. G. Khronopulo, and E. I. Yakubovich, Resonant Interaction of Light and Matter [in Russian], Nauka, Moscow (1977). A . V . Gaponov, L. A. Ostrovskii, and G. I. Freidman, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 10, Nos. 9-10, 1376 (1967). S . I . Averkov and N. S. Stepanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 2, No. 2, 203 (1959). L . I . Ostrovskii, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 4, No. 2, 293 (1961). S . I . Averkov and Yu. G. Khronopulo, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 3, No. 5, 818 (1960). L . I . Brekhovskikh, Waves in Layered Media, Academic P r e s s (1960). N . S . Stepanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 3, No. 4, 672 (1960}. N . S . Stepanov, Izv. Vyssh. Uchebru Zaved., Radiofiz., 5, No. 5, 908 (1962). Yu. M. Sorokin and N. S. Stepanov, Zh. Prikl. Mekh. Tekh. Fiz., No. 1, 31 (1972L S . I . Averkov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 21, No. 6, 835 (1978). S . A . Shelkunoff, Commum Pure Appl. Math., 4, No. 1, 117 (1951}. W. Kofink, Ann. Phys., 1, No. 1, 119 (1947).

CHARACTERISTICS SIGNALS

OF T H E S C A T T E R I N G

BY P E R I O D I C

T. V. G a v r i l o v a

OF M O D U L A T E D

OBSTACLES UDC 538.566.535.421

The diffraction of amplitude- and frequency-modulated signals by periodic obstacles is investigated. Analytical expressions for the fundamental parameters of the scattered signal are derived in the case of quasimonochromatic pulses. The effect of pulse compression for higher spatial harmonics is discerned. The characteristics of the scattered signal are analyzed numerically for a strip grating and a double strip grating. The feasibility of controlling signal characteristics over a wide range by means of resonant periodic structures is demonstrated. The use of amplitude-modulated (AM) and frequency-modulated (FM) electromagnetic pulses in the millimeter and submillimeter ranges for communications, radar, radio-navigation, and other applications, as well as the development of a broad base of hardware components in these ranges require comprehensive investigation of the scattering of such signals by periodic obstacles [1, 2]. The propagation of electromagnetic pulses in dispersive media has been thoroughly studied [3-8]. The corresponding problems have been solved on the basis of the formal machinery of Fourier integrals. In this article we formulate and solve the problem of diffraction of AM and FM pulses by periodic stru c tures, which in particular are strip and double strip gratings. A pivotal feature of the solution of the a sso ciated boundary-value problems is the fact that integral representations for the primary and scattered signals reduce the investigated problems to the key problems of scattering of a monochromatic plane wave by periodic obstacles; these problems can be solved in a rigorous mathematically substantiated setting by the well-developed machinery of diffraction theory for ordinary plane waves [9].

Automobile and Highway Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 23, No. 9, pp. 1067-1074, September, 1980. Original article submitted July 16, 1979.

710

0033-8443/80/2309-071050%50 9 1981 Plenum Pttblishing Corporation