Nonlinear TE Electromagnetic Surface Waves in a Ferrite ... · 220 ـــــــــــــــــــــ “Nonlinear TE Electromagnetic Surface Waves in …” An-Najah Univ

An-Najah Univ. J. Res. (N. Sc.), Vol. 18(2), 2004

Nonlinear TE Electromagnetic Surface Waves in a Ferrite Layered Structure

الموجات السطحية الكهرومغناطسية غير الخطية في تركيب طبقى يحتوي على الفرايت

Khetam El-Wasife, Mohammad Shabat, & Sameer Yassin Department of Physics, Faculty of Science, Islamic University, Gaza, Palestine

E-mail: [email protected]

Received: (15/2/2004), Accepted: (31/10/2004) Abstract

Characteristics of TE electromagnetic surface waves propagating in a nonlinear dielectric film bounded by a ferrite cover are examined theoretically. A dispersion relation based on Jacobian Elliptic Functions is derived, which describes the behaviour of the nonreciprocal nonlinear waves.

Keywords: Nonlinear surface waves, ferrite, dispersion relation.

ملخصفي تركيب يحتوى على ) TE(يتناول البحث دراسة نظرية الخصائص انتشار الموجات الكهرومغناطسية

ىشتقت معادلة التشتت التي وجدت أنها تعتمد علاولقد .شفافية من مادة عازلة غير خطية محاطة بمادة الفرايت .وغير متماثلة التي بدورها وصفت موجات غير خطية) Jacobian Function(دوال جاكوبي

1. Introduction

In several years, the investigation of nonlinear electromagnetic surface and guided waves in nonlinear wave guide structures has been advanced rapidly by many authors (1-10). Recently, the propagation characteristics of nonlinear electromagnetic and magnetostatic surface waves in gyromagnetic (Ferrite materials) wave guide structure have been studied (10-17) as the microwave devices using ferrite materials have unique characteristics; non-reciprocity, and magnetic tenability (17-19). Potential applications of nonlinear electromagnetic waves in designing microwave solid state devices could be utilized by stimulating the study of dispersion relation and the power in a proposed ferrite layered

”… Nonlinear TE Electromagnetic Surface Waves in“ ـــــــــــــــــــــ 216

An-Najah Univ. J. Res. (N. Sc.), Vol. 18(2), 2004 ـــــــــــــــــــــــــــ

structure. In this paper, we investigate theoretically the bahaviour of TE surface waves in a nonlinear dielectric film bounded by a gyromagnetic ferrite cover and a linear dielectric substrate. We derive an exact analytical dispersion equation. The layout of the paper is as follows; section two presents the basic equations of electric and magnetic fields components in each layer of wave guide structure. In section 3, we derive the dispersion equations. In section 4, we compute, illustrate, discuss some numerical results and present our final conclusions. 2. Theory

The analytical model of wave-guide structure is shown in Fig.1. A static biasing magnetic field H0 is applied in the +y direction. We assume that the finite nonlinear film occupies the region dz 0 bounded by the ferrite (YIG) cover of the space dz and a semi-infinite linear dielectric substrate in the region z0.

Figure (1): Analytical model of a layered structure and coordinate system

The magnetic permeability tensor of the gyromagnetic ferrite cover is described as (11-14).

xxxz

B

xzxx

0

00

0

(1a)

1 medium 1

Z

z=d

o

Nonlinear medium 2 2

EyLNL

Ferrite.

3 medium 3

x

Khetam El-Wasife, Mohammad Shabat, & Sameer Yassin ـــــــــــــــــــ 217

ـــــــــــــــــــــــــــ An-Najah Univ. J. Res. (N. Sc.), Vol. 18(2), 2004

Where

220

220

200 ,

)(

m

Bxzm

Bxx i , ω is the angular

frequency of the supported wave, 0000000 , , HH Mm is

the applied magnetic field, 11111076.1 Ts is the gyromagnetic

ratio, 0 M0 is the dc saturation magnetization of the magnetic insulator and B has been introduced as the background, optical magnon permeability. It will be assumed that ferrite cover has frequency-independent dielectric constant 1 and has value 1. The dielectric function of the nonlinear dielectric medium is assumed to be Kerr-like, isotropic and depends on the electric field, then the dielectric function of a nonlinear film is characterized by (1-11):

2

yyL

NL E , (1b)

where L is a frequency- dependent linear part,and is a nonlinear coefficient.

Only TE modes are going to be considered with propagation constant kx along the x-direction, and operating frequency . Electromagnetic fields are assumed to be

)( ,0

)(0 , )( ,0

z,

y

txkie

txkiez

x

HHH

EE

Using Maxwell’s equations and following the usual derivation, then the electric and magnetic field components in each layer can be written as:

(2) (3)



)1(1)1(

1)1(

)1(

1)1( exp

yo

z

o

yx

oy

Ek

H

kiEH

zkEE

2.1. In a linear dielectric medium

The electric and magnetic fields in the dielectric substrate are:

Where, 1

2

2

221

ckk x , 2

1

coo , c is the velocity of light, and

kx= /c,E0 is the electric field in region 1 at z = 0

2.2 In nonlinear medium

The components of electric and magnetic fields are closely follow the treatment given by Boardman and Egan (10) as:

Where2

22

222 c

kk x

, cn, sn and dn are specific Jacobian elliptic

functions, and

2

22

24

1

2242

2 , 4

cCkq

, and p2 = (q2+k2

2)/22

C2 is a constant that can be obtained from the boundary conditions.

mk

H

dniq

H

mE

o

o

y

|zzqcn

zzq zzqsn

|zzqcn

o2(2)

z

oo(2)x

o)2(

(7) (8) (9)

(4) (5) (6)



2.3 In the ferrite cover

The electric and magnetic fields components in the ferrite cover can be written as (20):

where Ed denotes the electric field in region (3) at z= d, k3 = (kx2-

2v2/c2) andv= (xx

2+xz2)/xx, which is the effective permeability.

3. Dispersion Equations

Matching the field components Hx , H z and Ey at the boundaries, z =

0 and z = d, the dispersion equations are:

Where n1=ck1/, E0 and Ed are the values of the electric field at the lower and upper boundaries of the film, respectively. This equation (13) is represented in the form

y

y

Ei

ikkzH

Ei

ikkz

vxxo

xxxxz

vxxo

xzxxx

3(3)z

3(3)x

)(

)(H (11) (12)

2222222

222

][

22

222

22

223

2

2222

1

222

31

22

22

2

dodo

vxx

xzxxxdodo

vxx

xzxxdo

EEQ

EEikkcEn

EEE

ikkcnQ

EE

qdcn

(13)

)(3 3zdk

dy eEE

(10)



(nx, E02/2, , D) = 0,

where nx = ckx/ , D = d/c, and E02/2 is called the optical power

density.

The conditions for the existence of nonlinear surface waves, for the present study are nx2, 1, v 0 It is very clear that the dispersion

equation displays non-reciprocal behavior which is very important in designing microwave devices since

(nx, E02/2, , D) (-nx, E0

2/2, , D)

We examine the dispersion equation in the special case xz = 0 and xx = 1, we get,

which is identical to the results obtained in (10).

After some algebraic manipulations, the relation between E02 and Ed

2 can be derived as 4. Numerical Results, Discussions, and conclusions

The derived nonlinear characteristic dispersion equation Eq. (13) has been computed and simulated numerically. The results are presented in graphical form by using software mathematical program known as MAPLE V. In our computation, the data is used as reported by shabat (17),

.rads 101.76 and 175. , 25.1 , 5. 1-111oB

TTMTH ooo For 0v , the range

of surface waves is from moo fff to mo ff , where f0=0/2, and

2

2

2

2222222

2

222

222

22

2

22

23

221

312

22

22

2

][d

dd

d

EEEEQEnEn

EdE

nQEoE

ooo

o

n

qdcn

(15) 1

22

2

2

212

23113

22

2

2

232

23113

22

dvv

vd

vv

EE

(14)



fm=m/2, This gives result of values corresponds to frequencies 16.234 GHz to 18.9 GHz.

Figure 2 shows dispersion curves as E02/2 versus the effective wave

index nx of Eq. (13) in both wave propagation directions, since (nx = + 1, nx = 1 for propagation in the forward direction, nx = -1 for the backward direction). Both curves represent the non-reciprocal behavior for different values of operating frequency. These values are very important in computing the power flow in the wave guide structures which are now under consideration. The figure illustrates the dependence of the wave index of TEo waves on the optical power density at the lower boundary of the nonlinear layer. The curves are labeled with values of f: curves 1, 2, 3 represent different frequency as 16.40GHz, 16.43GHz and 16.46GHz respectively. In addition the results of the present work exhibit that the power density depends strongly on the frequencies, it might be also used in designing and implementing integrated microwave devices based on non-reciprocal behavior as isolators and switches. The curve of power density versus wave index decreased suddenly from high to lower point.

Figure 3 describes the relation between the power density (E02/2)

and the thickness of the film D for various values of frequency. We notice that the nonlinearly increases in strength as the film thickness increases in the forward and backward wave direction. Both figures also display non-reciprocal behavior.

We proposed here a new approach describing a non- reciprocal behavior of nonlinear surface waves in a three–layer ferrite wave guide structure, which is very promising for designing future microwave devices.



Figure (2): Computed power density versus the effective mode index for various values of frequency. (a) Forward, (b) backward (=1.67x1011 rad.S-1T-1, 0M0=0.175T, 0H0=0.5T,=1,

Ed2/2=. 5,d=30m). Curve (1),=16.40GHz;curve (2),

=16.43GHz;curve (3),=16.46GHz

Figure(3): Computed power density versus the film thickness for various values of the frequency, in two direction (a) forward (b) backward. curve1,=16.40GHz; curve2,=16.43GHz; curve3,=16.46GHz

Power density

Mode Index

(b) backward



References

1) DB. Ostrowsky, and R. Reinish, “Guided Waves Nonlinear Optics”, NATO Series E:214, Kluwer Academic Publishers, Dordecht, (1992).

2) NN. Akhmediev, “Novel Class of Nonlinear Surface Waves: Asymmetric Modes in a Symmetric Layered Structure”, Sov. Phys. Jetp, 56, (1982), 299-303.

3) F. Ledere, U. Langbein, and H.E. Ponath, “Nonlinear Waves Guided by a Dielectric Slab”, Appl. Phys.-B, 31, (1985), 69-73.

4) D. Mihalache, and H. Totia, “On TE- Polarized Nonliear Waves Guided by Dielectric Layered Structures”, Solid State Commun, 45, (1985), 175-177.

5) U. Langbein, F. Lederer, and H.E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab Guided Waves”, Opt. Commun, 53, (1985), 417-420.

6) U. Langbein, F. Lederer, T. Peschel, and HE. Ponath, “Nonlinear Guided Waves in Saturable Nonlinear Media”, Opt. Lett., 10, (1985), 571.

7) D. Mihalache, D. Mazilu, and F. Ledere, “Nonlinear TE-Polarized Surface Plasmon Polaritons Guided by Metal Films”, Op. Commun, 59, (1986), 391-394.

8) GI. Stegeman, E.M. Wright, C.T. Seaton, J. Moloney, T. Shen, A.A. Maradudin, and R.F. Wallis, “Nonlinear Slab GuidedWaves in Non-Kerr-Like Media”, IEEE J. Quantum Electron. QE-32, (1986), 977-983.

9) G.I. Stegman, C.T. Seaton, J. Ariyasu, T.P. Shen, and J.V. Moloney, “Saturation and Power Law Dependence of Nonlinear Waves Guided by a Single Interface”, Opt. Commun, 56, (1986), 365-368.

10) A.D. Boardman and P. Egan, “Optically Nonlinear Waves in Thin Films, IEEE of Quant. Electronics. QE-22(2), February, (1986), 319.

11) A.D. Boardman, M.M. Shabat, and R.F. Wallis, “TE Waves at an Interface between Linear Gyromagnetic and Nonlinear Dielectric Media”, J. Phys. D., 24, (1991), 1702-1707.

12) M.M. Shabat, and D. Jäger, Nonlinear Electromagnetic Surface Waves Guided by a Single Hexagonal Planar Ferrite,”Proceedings of XIVth International Conference on Microwave Ferrite”, Eger, Hugary (1985), 127-130.

13) M.M. Shabat, D. Jäger, “Microwave Nonlinear Characteristics of TE Waves at Ferrite-Ferroelectric Interface,”Proceeding of 10th Microcoll”, Budapest, Hungary, (1999), 343-346.

14) M.M. Shabat, “New Nonlinear Magnetostatic Surface Waves in a Metallised Ferromagnetic Film”, Opt. Applicata, 43(4), (1994), 293-295.



15) Q. Wang, Z. Wu, S. Li., and L. Wang, “Nonlinear Magnetic Surface Waves On the Interface between Ferromagnet and antiferromagnet”, J. App. Phy., 87(4), (2000), 1908-1913.

16) G.N. Burlak, V.V. Grimal’skii, and S.V. Koshevaya, “Nonlinear Waves In a Three Layer Ferroelectric-Ferrite- Ferroelectric System”, Phy. Solid, 35(8), (1993), 1049-1050.

17) M.M. Shabat, “Strongly Nonlinear Magnetostatic Surface Waves In a Ground Ferrite Film”, phy. stat. sol.(a), 149, (1995), 691-696.

18) I. Kasa, “Microwave Integrated Circuits”, Elsevier Publ. Co., Amsterdam, (1991).

19) M.S. Sodha and N. C. Srivastava, “Microwave Propagation in Ferrimagnetics”, Plenum Press, New York, (1981).

Appendix A

We consider a thin, optically nonlinear, dielectric film sandwiched between semi-infinit linear dielectrics. Only TE modes are going to be considered, and these propagate along the x axis with wave number kx and angular frequency . The electric and magnetic field component have the form E= (0,Ey(z),0) exp(ikx-it) and H = (Hx(z),0,Hz(z)) exp(ikx-it). If the optical nonlinearity is of the kerr type, then the dielectric constant of the film is 2+ 2 E2 2 so that if Ei are taken to be real, then they must satisfy the equation:

the first integration of equation (A.1) is derived as follows:

242

22

22

2

2

242

22

22

2

2

42

222

2

2

2

232

22

22

22

c

c

. 2

2

1

2

1

2

1

04

1.2

2

1

2

1

02

EEkzd

dE

EEkzd

dE

zd

Ed

zd

dEk

zd

dE

zd

d

zd

EdE

zd

dEEk

zd

dE

zd

d

zd

dE

oEEkE 2222

222 2 (A.1)

(A.2)



Then Eq. (A.2) gives:

the value of constant C2 is evaluated as:

220

2022

22

2202

12

220

220

202

222

12

220

220

202

221

20

22

12

1222

21

o1o

01121

oo

0121

e , ,

e ,

oz

CEEc

kEc

kE

CEEkc

kE

CEEkkE

CEEkE

EkEkEEE

EEEEE

xx

x

at

Finally we obtain:

Where E02, and Eb

2, are the value of electric fields at the lower and higher boundary, respectively.

222

222

22

22 CEEkE (A.3)

(A.4) (A.5) (A.6)

2

202

1220

2

22 CE

Ec

2

22

322

2

2

2C

EE

cb

b

At z=0 (A.8) At z=d (A.9)

220

202

202

22

20

2202

12

220 CEEE

cEkE

ckE xx

(A.7)



Appendix B

The relationship between E02 and Eb

2 is derived mathematically with details as follows:

Dividing both Eq’s. (A.8) And (A.9), we get;

022

22

2212

22232

2

2232

220212

20

oobb

bb

EEEE

EEEE

Multiply Eq. (B.1) by 2α2 :

02

2

02222

022

02222

13231202

13231

22

222

222

212

2322

2

22

222

2

220212

22

2232

22

20212

22

2232

22

E

bEoEb

oEbob

EoEbEb

oEb

Eb

E

EEE

E

EE

Then, add 23113 2 to both sides of Eq. (B.3), we

have:

231132311313231

20213331

22

222

222

222

2

EEEE bob

Adding the term, 22

22

22

22 obob EEEE in left-hand side of Eq.

(B.3) we got;

(B.1)

(B.2) (B.3)

(B.4)



2311323113

132

22312

2132

2

2312

2

222

22

22

22

22

222

22

2

2

oob

boobobb

EEE

EEEEEEE

23113

2

2

21222

2

2

23222

231132

2122

232

2

23113

21

2223

2221

223

22

2

2

2

ob

ob

obob

EE

E

EEEE

E

At last, we get the relation between E02 and Eb

2 as:

(B.5)

231132312

22

2132

22

2 22 oboEb EEE (B.6) (B.7) (B.8) (B.9)

1

222

2

212

23113

22

2

2

232

23113

22

o

b

E

E

(B.10)



Appendix C

The derivation of the constant C2 in both boundaries:

(1) At z = o

From Eq. (3.46) we have;

222

222

22

22 CEEkE

But )exp( 1)1( zkEE oy

And 21

21

21111 , kEEkEE

Where yyy EEEE )2(1

)1( and

And 2121 and EEEE

Then 2

22

22

222

12

22

222

222

22

21

21

CEEkc

kE

CEEkkE

o

)(1 ,

)(11

)(

)(

22

2

22

2

22

2

qzsnP

E

qzcnP

E

qzcnP

E

qzcnP

E

oo

oo

oo

oo

(C.1)

(C.2)



Where 2

12

221 c

kk

And 2

22

222 c

kk

Equation (4.42) becomes:

2

22

122

22

2C

E

c

E oo

Where Eo is the value of electric field at the lower boundary of the film

(2) At z = d

23

23

2

3

3(3)y

333

3(3)y

exp

EkE

EE

EkE

zdkEE d

Substitute in:

vck

ck

EEEE

CEEkE

32

222

322

222

2

3232

22

22

222

222

k , k

, dzat

(C.3)

(C.4)

We have Then Where

And

But And



Then Eq. (4.44) becomes:

222

22222

222

323 CEE

ckEk

222

222

22

2

222 CEE

ck

ckE ddvfd

Then

,2 2

22

32

2

2

2

CEEc

dvd

Relationship between E02 and Ed

2 :

We drive mathematically the relationship between 20E and 2

dE as

follow:

Dividing Eq. (C.3) by Eq. (C.4) we get:

2

0212

2

0

22

32

2

2

1

2EEEE

dvd

Multiply Eq. (C.7) by 2, it becomes:

02

12

2

12 2

0212

2

0

2

32

2

EEE

vE

dd

0222

2

0212

2

0

22

32

2

EEEE

dvd

0222221

2

0232

2

2

22

02

22

2

EEEE

vdd

(C.5)

(C.6)

(C.7)

(C.8)



022223311

2

021231

2

2

22

02

22

2

vvvddEEEE

Adding 23113 2 vv to the both sides of Eq. (C.7)

then the left-hand side of Eq. (C.7) becomes:

2311313231

2213231

22

222

222

22

2

vvvv

ovvdod EEEE

Also adding the term 22

22

22

22 odod EEEE to the left-hand side,

we obtain:

2311313

22231

2213

22231

22

222

22

22

22

22

222

22

2

vvvovov

dvdoododd

EE

EEEEEEEE

The above equation becomes:

1312

22

2132

22

2 2 vodvod EEEE

Finally the Eq. (C.9) lead to:

23113

212

2332

2212

2332

2

23113

222313

222231

22

22

231

132312

22

2132

22

2

2

2

2

2

vv

ovdovd

vv

ovdvod

v

vvodvod

EEEE

EEEE

EEEE

(C.10)

(C.9)



, Equation (C.10) becomes:

Dividing Eq. (C.11) by 23113 2 vv , we obtain:

1

222

2

212

23113

22

2

2

232

23113

22

o

vv

vb

vv

E

E

By using Eq. (C.12) we plot the relations between 22 oE against

22 dE .

This (E02,Ed

2) relationship can be used to great effect in the nonlinear generalisation of the dispersion relationships of asymmetric and symmetric waveguides that are familiar from linear solid state optics.

To find Eq(13) mahematically

Applying the boundary conditions on E and H at both z = 0 and z=d we obtain:

(C.12)

(C.11)

23113

2

2

21222

2

2

23222

23113

2

212

2

2

332

2

2

2

vvov

d

vvovd

EE

EE



q

kEqzdn

qzdnqikiE

HH

qzcnE

mzqzkE

EE

oo

ooo

y

xx

o

oo

yy

1o

o

1)1(

)2()1(

o

1

)2()1(

qzsn

qzsn

,)(

)(cn )exp(

ddE

EE

d

yy

3o

)3()2(

k exp zdqcn

PEcn

qdqPcnE od

11

and

2

22

24

1

2242 2

, 4c

Ckq

222

22 2 kqP

From Eq’s. (4.19) And (4.21) we have:

)(1 ,

)(

22

2

qzsnP

E

qzcnP

E

oo

oo

where: cn2+sn2=1

(1)

(2)

(3)

(4)

(5)

then

then then

and



Similarly; for the upper boundary, we have:

PEcnq

dqsnP

E

PEcnq

dqcnP

E

od

od

122

2

1

11 ,

1

The boundary condition for the magnetic field at z = d is

oxzxx

vxxd

dvxxo

xzxxo

o

xx

zdqdnPqikk

E

Ei

ikkzdqdnPq

i

HH

o3

3o

)3()2(

zdqsn

zdqsn

and

)2()3()3(

)2()3(

zzxxxxz

zz

HHH

BB

where B = ()H ( the magnetic flux density) .

Substitute about (3)x

(3)z

)2( and , HHH z we get:

then

][qzcn [qd]cn

o

d

o

o

qzqd

zdqPcnE

(6)

(7)

let

oo

dvxxo

xxxzxxd

vxxo

xxxxz zdqPcn

kE

i

ikkE

i

ikk

323 (8)



Using Jacobian Functions for Eq. (4.28) we get (12):

)]z(d [qsn ][qzsn m1

)]qz[(ddn )]z[q(dsn ][qzdn ][qzsn )z[q(dcn ]cn[qz]qd[cn

o2

o2

oooooo

Substitute from Eq’s. (4.20), (4.22), (4.23), (4.24) and (4.25) in Eq. (4.29) we obtain:

2

2

2

2223

12

2

2

2

23

21

2

2

2

2

23

221

2

2

2

2

231

11122][

1111][

111][

111][

P

E

P

EmPq

ikkkqEEqdcn

P

E

P

Em

ikk

q

k

P

EEqdcn

P

E

P

Em

ikk

qP

kEE

P

EEqdcn

P

E

P

Em

Pq

Eikk

Pq

kE

P

E

P

Eqdcn

do

vxx

xzxxdo

do

vxx

xzxxdo

do

vxx

xzxxdodo

dod

vxx

xzxxodo

then Eq. (4.30) becomes:

2222

2222

2

320

21

31

22][

bobod

vxx

xzxx

vxx

xzxxdo

EEEEqE

ikkEk

ikkkqEEqdcn

Multiply the right hand side of Eq. (4.33) by

(9)

(10)

(11)



dodo EEc

EEc

2

22

22

22

2

22

, We get:

2222222

222

][

22

222

22

223

2

2222

1

222

31

22

22

2

dodo

vxx

xzxxdodo

vxx

xzxxdo

EEQ

EEikkcEn

EEE

ikkcnQ

EE

qdcn

(12)

Documents

Nonlinear TE Electromagnetic Surface Waves in a Ferrite ... · 220 ـــــــــــــــــــــ “Nonlinear TE Electromagnetic Surface Waves in …” An-Najah Univ