PROPAGATION OF TRANSVERSE-MAGNETIC WAVES IN A WAVEGUIDE WITH ANISOTROPIC AND MAGNETOACTIVE MODULATED INSERT

EUROCON2009

Eduard A. Gevorkyan

Professor of the Moscow State University of Economics, Statistics and Informatics

Contact address: 7, Nezhinskaya str., Moscow, 119501, Russia E-mail: EGevorkyan@teach.mesi.ru, gevor_mesi@mail.ru

Abstract – Propagation of transverse-magnetic (TM) waves in the waveguide with arbitrary cross section, filled with periodically modulated anisotropic and magnetoactive medium is considered. The differential equation for the potential of TM field is found from the Maxwell equations. This equation is solved with help of method, developed in our earlier articles [1-3], [6-7]. The analytical expressions for the magnetic and electric vectors of the TM field are found in the region of weak interaction between the signal wave and the modulation wave in the first approximation for small modulation indexes of the insert. The frequency of strong (resonance) interaction between the signal wave and the modulation wave is expressed in an analytic form. Index Terms: Propagation, electromagnetic waves, anisotropic, magnetoactive, waveguide. In the study [7] the propagation of transverse-electric (TE) waves in a waveguide of arbitrary cross section with periodically modulated anisotropic and magnetoactive insert has been considered. Below a similar problem is solved for the transverse-magnetic (TM) waves. Note, that this investigation is of interest both from the theoretical standpoint and from the standpoint of the possibility of using the waveguides with a periodically modulated anisotropic and magnetoactive insert in various fields of electronics, integral optics, acoustooptics, etc. [4-5]. Let we have a regular waveguide of arbitrary cross section, whose axis coincides with the OZ axis of some rectangular system of coordinates. Suppose, that the tensor permittivity and permeability of the waveguide insert have a form

( )( )

( )���

�

�

���

�

�=

tztz

tz

,000,000,

2

1

1

εε

εε� ,

( )( )

( )���

�

�

���

�

�=

tztz

tz

,000,000,

2

1

1

μμ

μμ� , (1)

where ( ) ( )[ ],cos1, 00

011 1

utkzkmtz −+= εεε (2)

( ) ( )[ ],cos1, 00022 2

utkzkmtz −+= εεε (3)

( ) ( )[ ],cos1, 00011 1

utkzkmtz −+= μμμ (4)

( ) ( )[ ],cos1, 00022 2

utkzkmtz −+= μμμ (5) 02

01

02

01 ,,, μμεε are the constant permittivity and

permeability of the insert in the absence of a modulation wave; 1

1<<εm , 1

2<<εm , 1

1<<μm ,

12

<<μm are the small modulation indexes; 0k is the

modulation-wave number; uk0 is the modulation-wave frequency. Suppose the transverse-magnetic (TM) signal wave unit amplitude and frequency 0ω propagates in the waveguide in a positive direction of an axis OZ . The wave equation for potential zE of the TM field can be obtained from the Maxwell equations in the form

( ) ( )

( ) ( ) 0~

,,

~

,1,~

1200

12

=�

��

∂

∂∂∂−

−�

��

∂

∂∂∂+Δ ⊥

tEtz

ttz

zE

tzztzE

z

zz

μεμε

εε

, (6)

where 2222 // yx ∂∂+∂∂=Δ⊥ ; 0ε is the electric

constant; 0μ is the magnetic constant and

( ) zz EtzE ,~2ε= . (7)

Note, that the equation (4) coincides with the equation (7) of the article [3], when

( )tz,1ε =const and ( )tz,1μ =const. From Maxwell equations the transverse components of TM field can be represented in terms of potential

( ) ( )yxtzEEn

nz ,,0

Ψ=�∞

=

(8)

as follows: ( ) ( )[ ] ( )[ ]yxz

ztzEtzH n

n

nn ,,,

02

0

20 Ψ∇

∂∂−= �

∞

=

− �� ελετ , (9)

( )( ) ( )[ ] ( )yx

ttzEtz

tzE n

n

nn ,,,

,1 2

0

2

1

Ψ∇∂

∂= �∞

=

− ελετ

�, (10)

978-1-4244-3861-7/09/$25.00 ©2009 IEEE76

where );/()/( yjxi ∂∂+∂∂=∇

�� subscript τ refers

to the transverse components; 0z� is the unit of the

OZ axis; ( )yxn ,Ψ and nλ are the orthonormal eigenfunctions and eigenvalues of the first boundary-value problem for the cross section of the waveguide and satisfy the following equation

( ) ( ) ( ) 0,,0,, 2 =Ψ=Ψ+ΨΔ⊥ yxyxyx nnnn λ , (11) where Σ is the contour of the waveguide’s cross section and n� is the normal to Σ.

Let us introduce the new variables

( ) ( )�−

−=−=ξ

μεξμξεβ

ξηξ0

01

01

1121

1, duu

zutz , (12)

( )0022 μεβ u=

into (6). After some algebraic transformations we shall receive

( ) ( )( ) ( )

( ) ( )( ) ( ) .0

~

1

~11~

2

2

01

01

112

2100

01

01

112

12

=∂∂

−−

−��

���

∂∂���

����

�−

∂∂+Δ⊥

ημε

ξμξεβ

ξεξμμε

ξμεξμξεβ

ξεξξε

z

zz

E

EE (13)

If to search for the solution of the equation (13) in the form

( ) ( )�∞

=

Ψ=0

,~~n

nni

z yxEeE ξγη , (14)

taking into account (11), we shall receive the following second order ordinary differential equation for ( )ξnE~

( ) ( )( ) ( ) ( )

( ) ( ) ( ) 0~

1

~11

01

01

112

2

01

01

112

12

=−

+

+��

���

���

����

�−

ξ

μεξμξεβ

χ

ξξ

μεξμξεβ

ξεξξε

nn

n

E

dEd

dd

, (15)

where

( ) ( ) ( ) ( )���

����

�−−= 0

101

112221100

2 1με

ξμξεβλγξμξεμεχ nn . (16)

In terms of new variable

( )( ) ( )�

−=

ξ

μεξμξεβ

ξξεε 0

01

01

112

101

0

12dbks ( 21 β−=b ) (17)

the equation (15) takes the form of the Mathieu-Hill equation

( ) ( )[ ( )

( ) ( ) ] ( ) 0~2cos2cos

4~

21

20

20

01

012

22

0220

02

01

2

2

=+−⋅

⋅���

����

�−+⋅+

sEsmms

ubkdssEd

nn

nnn

εεχ

χμεγχε

ε

�

�

(18)

where

( ) bnn 220

10100

20 λγμεμεχ −= 211 βμε

bmm

l+

= . (19)

The equation (18) can be presented in the form of

( ) ( )�−=

=+1

1

22

2

0~~

kn

Skink

n sEsd

sEd e ��

θ , (20)

where the coefficients n

kθ of the Fourier decompositions of the expression in square brackets of the equation (18) are expressed by formulas

( )2022

002

01

04 nn

bkχ

εεθ = , (21)

( )

( ) ( )21

2022

002

01

20

01

012

2

220

02

01

1

2

2

εεχε

ε

χμεγε

εθ

mmbk

ubk

n

nn

+−

−���

����

�−=± �

. (22)

The solution of the equation (20) we shall search in the form

( ) �−=

⋅=1

1

2~k

skink

sin ee CsE nμ

, (23)

where the quantities nμ may be real, complex, or purely imaginary. Substituting (23) into (20) and being confined to members of second order on small parameters lmmmm ,,,

2211 μεε μ, after

simple transformations we shall receive the dispersion equation of this problem and the expressions for definition of coefficients nC 1± in the form

( )( )

( )( ) n

n

n

nn

nn

n0

2

21

02

21

02

22 θμθ

θμθθμ

−++

−−+= , (24)

( ) .2 0

210

1 nn

nnn �C

θμθ

−±=±

(25)

As is known, at performance of the condition [6]

nn δθ >>− 01 ~ lm ,

2ε ,���

����

�≅

221n

nθδ (26)

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we get into the region of weak interaction between the signal wave and the modulation wave. The quantities nμ and nC 1± in this region have a form

,02 n

n θμ = ( ),14 0

011 n

nnn CC

θθ±

=± (27)

where nC 0 are determined from the normalization condition. If now to substitute (27) in (23) taking into account (2-5) and pass to variables z and t , from (7) we obtain the following analytic expression for the potential ( )tzyxEz ,,,

( ) ( ) ( )utzikk

n k

nk

ntzpinz eVCeyxE

n −∞

= −=

−� �Ψ= 000

0

1

100

2

,1 ω

ε (28)

where the quantities in terms of 0ω expressed as k

n

nk

nn

k

mCCkV ��

�

����

�−+Δ=

222

0

0 ε� , (29)

( ) lubk

bubk

mlnn

n

00

0000

122 1 �

�

�

�

��

�

�−−+=Δ ω

θθε

, (30)

( ) ( )

( )( ) ] ,~1

2

~1~~1

4

22

002

0220

101

2220

2

2012

02

22220

0

βεωβεμ

λβεεωβ

βθ

−−

−���

−−

−=

uk

bu

ukn

n

(31)

( ) ( )

( )( ) ��

���

−−+

+−−−

=

220

02

01

2010

2

2220

2

2012

020

~121

~1~~1

βεμεωβ

λβεεωβ

β

ukb

u

bu

ubp n

n

, (32)

( )

( ) ( )2122

002

20

01

202

00001

0122

002

01

1

2

422

εεεχε

χωθ

μεε

εθ

mmbk

lu

bbukbk

n

nn

n

+−

−��

���

−

−=±

, (33)

( ) bbbuk

n

nn 2

2

0000

10100

20 2

λωθ

μεμεχ −��

�

�

��

�

�−= ,

( ) 202

01

2012~ βε

μεβ = . (34)

As is seen from (28), in the region of weak interaction the fundamental harmonic of the TM field (k=0) is independent of small modulation indexes, while the side harmonics (k= ± 1) are proportional to the small modulation indexes. As has been shown in [1] (see also [6]), when the condition

nn δθ ≤− 01 (35)

is fulfilled, strong interaction between the signal wave and the modulation wave develops. Then for the frequency and frequency width of the strong interaction region the following expressions can be obtained from (35):

( )nsuk ηβ

βω += ~

~20

,0 , 20

02

20141bk

nn ε

λεη += ,

( )n

n

ns

uk δηβ

ηβω ~4

~10,0

+=Δ . (36)

Note, that in the region of strong interaction the dispersion equation (24) has complex solution. In this case we obtain

11 ≅−nV , .,

211 �εε mmV n ≈ (37)

According to (37) in the region of strong interaction the amplitude of the minus-first harmonic does not depend on modulation indexes, while the amplitude of the plus-first harmonic linearly depends on these. In other words, in this case in addition to the fundamental harmonic of the signal wave plays a substantial role reflected minus-first harmonic. REFERENCES .1. K.A. Barsukov, E.A. Gevorkyan. On the theory of

electromagnetic waves in a waveguide filled with a nonstationary and nonhomogeneous dielectric material. Radiotekhnika I Elektronika 28 (2) (1983). pp.237-241.

2. E.A. Gevorkyan. Propagation of electromagnetic waves in a waveguide with a modulated anisotropic insert. Zhurnal Tekhnicheskoy Fiziki 76 (5) (2006). pp. 134-137.

3. E.A. Gevorkyan. The theory of propagation of electromagnetic waves in a waveguide with a magnetoactive anisotropic modulated filling. Radiotekhnika I Elektronika 53 (5) (2008). pp. 565-569.

4. A. Yariv, P. Yeh. Optical waves in crystals: propagation and control of laser radiation, Wiley, New York, 1984; Mir, Moscow, 1987. p. 616.

5. Ch. Elachi, C. Yeh. Periodic strucyures in integrated optics, Journal of Applied Physics 44 (1973). pp. 3146-3152.

6. E.A. Gevorkyan. On the electrodynamics of space-time periodic mediums in the waveguides of arbitrary cross section. Uspekhi Sovremennoy Radioelektroniki, (1) (2006). pp. 3-29.

7. E.A. Gevorkyan. Electromagnetic waves in a waveguide with a magnetoactive anisotropic periodically modulated filling. Proceedings of the 13-th International Seminar/Workshop “On Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory”, Tbilisi, Georgia 22-25 September 2008, pp. 101-105.

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Eduard A. Gevorkyan: Doctor of physico-mathematical science, professor of higher mathematics of the Moscow State University of Economics, Statistics and Informatics (MESI). Area of scientific interests: Propagation of electromagnetic waves in the waveguides with

periodically modulated filling, boundary problems of electrodynamics of the periodic limited mediums.

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