On the problem of nonlinear coupled electromagnetic transverse-electric–transverse magnetic wave propagation

On the problem of nonlinear coupled electromagnetic transverse-electric–transverse magnetic wave propagationDmitry V. Valovik Citation: J. Math. Phys. 54, 042902 (2013); doi: 10.1063/1.4799275 View online: http://dx.doi.org/10.1063/1.4799275 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v54/i4 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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JOURNAL OF MATHEMATICAL PHYSICS 54, 042902 (2013)

On the problem of nonlinear coupled electromagnetictransverse-electric–transverse magnetic wave propagation

Dmitry V. Valovika)

Department of Mathematics and Supercomputing, Penza State University,Krasnaya Street 40, Penza 440026, Russia

(Received 17 November 2012; accepted 18 March 2013; published online 12 April 2013)

Coupled electromagnetic TE and TM wave propagation in a nonlinear plane layer isconsidered. Nonlinearity inside the layer is described by Kerr law. Physical problemis reduced to a nonlinear two-parameter eigenvalue problem for a system of (nonlin-ear) ordinary differential equations. It is proved that TE and TM waves that form(nonlinear) coupled TE-TM wave can propagate at different frequencies ωE

and ωM, respectively. These frequencies can be chosen independently. Exis-tence of coupled surface TE and TM waves is proved. Intervals of localiza-tion of coupled eigenvalues are found. C© 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4799275]

I. INTRODUCTION

In this paper nonlinear coupled surface electromagnetic TE-TM wave propagation in a layerwith Kerr nonlinearity is considered. It is known1 that in this case new propagation regime exists. Inthis regime we have sum of two independent waves (TE and TM) that create new (nonlinear) polar-ization (so called coupled TE-TM wave). TE and TM waves that form this nonlinear TE-TM wavewe call pseudopolarizations (pseudo-TE and pseudo-TM, respectively). As it is proved in Ref. 1in this case each pseudopolarization has its own propagation constant (γ E and γ M for pseudo-TE andpseudo-TM waves, respectively). In this paper we prove that even more fascinating case is possible.To be more precise we prove that pseudo-TE and pseudo-TM waves can propagate at their own fre-quencies (ωE and ωM, respectively). The physical problem is reduced to a two-parameter eigenvalueproblem for Maxwell’s equations. The case when ωE = ωM was investigated in detail in Ref. 1.

The problem when ωE �= ωM, as it is known to the author, has not been considered in literaturebefore. This regime is seemed to be interesting for, at least, two reasons: we obtain exact resultsabout existence of coupled TE-TM waves; this model can be applied to study nonlinear interactionof two types of waves (pseudo-TE and pseudo-TM) at different frequencies. We should note thatfrequencies ωE and ωM do not depend on each other.

Short background of nonlinear guided waves in media with Kerr and Kerr-like nonlinearities isgiven in Ref. 1.

During revisions making it has come to the author’s knowledge that problems of nonlinearcoupled TE-TM wave propagation for plane and cylindrical geometries (with ωE = ωM in bothcases) were considered in Refs. 2 and 3. It seems to be true that sources2, 3 are the first messagesabout coupled TE-TM wave propagation. However, in Refs. 2 and 3 researchers give neither strictstatements nor any results of solvability of the problems.

II. STATEMENT OF THE PROBLEM

Let us consider electromagnetic waves propagating through a nonlinear homogeneous isotropicnonmagnetic dielectric layer. The permittivity inside the layer is described by Kerr law. The layer

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042902-2 Dmitry V. Valovik J. Math. Phys. 54, 042902 (2013)

is located between two half-spaces: x < − h and x > h in Cartesian coordinate system Oxyz. Thehalf-spaces are filled with isotropic nonmagnetic media without any sources and characterized bypermittivities ε1 ≥ ε0 and ε3 ≥ ε0, respectively, where ε0 is the permittivity of free space. Everywherebelow μ = μ0 is the permeability of free space.

Maxwell’s equations have the form:4

rot H = ∂tD,

rot E = −∂tB,(1)

where D = εE, B = μH and ∂ t = ∂/∂t; (E, H) represents the total field.From system (1) we obtain

rot E = −∂t(μH),

rot H = ∂t(εE).(2)

We assume that the fields E, H exist at two frequencies ωE and ωM in space

E(x, y, z, t) = E+E (x, y, z) cos ωE t + E−

E (x, y, z) sin ωE t++ E+

M (x, y, z) cos ωM t + E−M (x, y, z) sin ωM t,

H(x, y, z, t) = H+E (x, y, z) cos ωE t + H−

E (x, y, z) sin ωE t++ H+

M (x, y, z) cos ωM t + H−M (x, y, z) sin ωM t,

where ωE, ωM are the circular frequencies; E+E , E−

E , E+M , E−

M , H+E , H−

E , H+M , H−

M are the real requiredfunctions; vectors with indices E or M correspond, respectively, to TE or TM waves. We meanthat TE waves have the form a = (0, ay, 0)T and b = (bx , 0, bz)T for electric and magnetic fields,respectively; TM waves have the form a = (ax , 0, az)T and b = (0, by, 0)T for electric and magneticfields, respectively, where ( · )T is the transposition operation.

Let us form complex amplitudes E, H:

E = EE + EM , H = HE + HM , (3)

where

EE = E+E + iE−

E , EM = E+M + iE−

M ,

HE = H+E + iH−

E , HM = H+M + iH−

M .

It is clear that

E = Re{EE e−iωE t + EM e−iωM t

}, H = Re

{HE e−iωE t + HM e−iωM t

},

where

EE = (0, Ey, 0)T , EM = (Ex , 0, Ez)T ,

HE = (Hx , 0, Hz)T , HM = (0, Hy, 0)T

and

Ex ≡ Ex (x, y, z) , Ey ≡ Ey (x, y, z) , Ez ≡ Ez (x, y, z) ,

Hx ≡ Hx (x, y, z) , Hy ≡ Hy (x, y, z) , Hz ≡ Hz (x, y, z) .

Denote by

EωE ωM := EE e−iωE t + EM e−iωM t ,

HωE ωM := HE e−iωE t + HM e−iωM t .(4)

As it is known (see, for example, Refs. 5–7), Kerr law has the form ε = ε2 + α|E|2, where ε2 is aconstant part of the permittivity, α is the coefficient of nonlinearity, and E is a complex amplitude of



a monochromatic wave Ee−iωt . The medium is supposed to be isotropic. Taking into account that inthe case under consideration the vector EωE ωM represents a sum of TE and TM waves, we obtain thatthe following formula |EωE ωM | = |EE e−iωE t + EM e−iωM t | = |EE + EM | = |E| takes place, where Eis defined by formula (3). Thus, the Kerr law takes the form ε = ε2 + α|E|2, where ε2 and α aredefined above.

In the case under consideration we obtain that Maxwell’s equations depend on t in the sameway as if ε was constant. This allows us to write down Eqs. (2) for fields (4)

rot (EEe−iωEt + EMe−iωMt) = iμωEHEe−iωEt + iμωMHMe−iωMt,

rot (HEe−iωEt + HMe−iωMt) = −iεωEEEe−iωEt − iεωMEMe−iωMt.(5)

Thus, we gave proof of possibility to investigate Eqs. (5) for the complex amplitudes E and Hinstead of Eqs. (1) for the fields E and H.

The complex amplitudes E, H (see (3)) must satisfy Eqs. (5); the continuity condition fortangential field components on the boundaries x = − h, x = h; the radiation condition at infinity,where electromagnetic field exponentially decays as |x| → ∞ in the domains x < − h and x > h

The permittivity in the entire space has the form:

ε =

⎧⎪⎪⎨⎪⎪⎩

ε1, x < −h,

ε2 + α|E|2, −h < x < h,

ε3, x > h.

The solutions to Eqs. (5) are sought for in the entire space.From equations (5) we obtain the following equations:

(e−iωM t∂y Ez − e−iωE t∂z Ey, e−iωM t (∂z Ex − ∂x Ez), e−iωE t∂x Ey − e−iωM t∂y Ex

) == iμ

(ωE Hx e−iωE t , ωM Hye−iωM t , ωE Hze

−iωE t),(

e−iωE t∂y Hz − e−iωM t∂z Hy, e−iωE t (∂z Hx − ∂x Hz), e−iωM t∂x Hy − e−iωE t∂y Hx) =

= −iε(ωM Ex e−iωM t , ωE Eye−iωE t , ωM Eze

−iωM t).

(6)

We can see from (6) that the right-hand sides are sums of TE and TM vectors at the frequenciesωE and ωM, respectively. In order that the same conclusion holds for the left-hand sides it is necessaryand sufficient that the following conditions:

e−iωM t∂y Ez ≡ 0, e−iωM t∂y Ex ≡ 0, e−iωE t∂y Hz ≡ 0, e−iωE t∂y Hx ≡ 0

to be fulfilled. From these conditions we obtain that the components Ez, Ex, Hz, Hx do not dependon y.

It is well known (see, for example, Refs. 4 and 8) that in the case of constant permittivity surfacewave can be represented as a superposition of TE and TM waves. This means that in this case generalsolution to Maxwell’s equations (2) is a linear combination of TE and TM waves. At this rate if wesubstitute TE or TM fields in Eqs. (2) we shall be convinced that the components Hz, Hx (for TEwave) and components Ez, Ex (for TM wave) do not depend on y.

Further, it turns out that (see, for example, Ref. 9) for Kerr nonlinearity and for each polarizationseparately (TE and TM) exists surface waves that do not depend on y. However, it is necessary to keepin mind that for the nonlinear case general solution to Maxwell’s equations cannot be representedas a superposition of TE and TM waves. In this nonlinear case pure TE and TM surface waves arepartial solutions to Maxwell’s equations.



We shall assume that in the nonlinear case under consideration the components Ez, Ex, Hz, Hx

do not depend on y.2, 3, 6 Then from (6) we obtain

(−e−iωE t∂z Ey, e−iωM t (∂z Ex − ∂x Ez), e−iωE t∂x Ey) =

= iμ(ωE Hx e−iωE t , ωM Hye−iωM t , ωE Hze

−iωE t),(−e−iωM t∂z Hy, e−iωE t (∂z Hx − ∂x Hz), e−iωM t∂x Hy

) == −iε

(ωM Ex e−iωM t , ωE Eye−iωE t , ωM Eze

−iωM t).

(7)

We can see that system (7) can be rewritten in the following form:

⎧⎪⎪⎨⎪⎪⎩

−e−iωE t∂z Ey = iμωE e−iωE t Hx ,

e−iωE t∂x Ey = iμωE e−iωE t Hz,

e−iωE t (∂z Hx − ∂x Hz) = −iεωE e−iωE t Ey,

⎧⎪⎪⎨⎪⎪⎩

−e−iωM t∂z Hy = −iεωM e−iωM t Ex ,

e−iωM t∂x Hy = −iεωM e−iωM t Ez,

e−iωM t (∂z Ex − ∂x Ez) = iμωM e−iωM t Hy .

(8)

Now from (8) we obtain

⎧⎪⎪⎨⎪⎪⎩

∂z Ey = −iμωE Hx ,

∂x Ey = iμωE Hz,

∂z Hx − ∂x Hz = −iεωE Ey,

⎧⎪⎪⎨⎪⎪⎩

∂z Hy = iεωM Ex ,

∂x Hy = −iεωM Ez,

∂z Ex − ∂x Ez = iμωM Hy .

(9)

As it is seen from (9) all equations are split into two subsystem. The first subsystem correspondsto pseudo-TE waves, and the second one to pseudo-TM waves.

Waves propagating along the boundary z depend harmonically on z. In other words, dependenceon z for the components of the fields has the form eiγ z, where γ is unknown spectral parameter(propagation constant). It is clear that |Eeiγ z| = |E| does not depend on z if Im γ = 0. That is to say,system (9) depends linearly on eiγ z. On the other hand, continuing this line of reasoning, we see thatfor each pseudopolarizations (TE and TM) different spectral parameters can be chosen. To be moreprecise, we can chose γ E for TE waves and γ M for TM waves1 (see Fig. 1). Thus, the componentsEx, Ey, Ez, Hx, Hy, Hz have the form:

Ex ≡ Ex (x)eiγM z, Ey ≡ Ey(x)eiγE z, Ez ≡ Ez(x)eiγM z,

Hx ≡ Hx (x)eiγE z, Hy ≡ Hy(x)eiγM z, Hz ≡ Hz(x)eiγE z

0

h

−h

ε3

ε1

ε

x

z

TE TM

ωE , γE ωM , γM

FIG. 1. Geometry of the problem.



and the fields E, H have the form:

E = Re{(

Ex (x)ei(γM z−ωM t), Ey(x)ei(γE z−ωE t), Ez(x)ei(γM z−ωM t))T}

,

H = Re{(

Hx (x)ei(γE z−ωE t), Hy(x)ei(γM z−ωM t), Hz(x)ei(γE z−ωE t))T}

.

Now from (9) we obtain⎧⎪⎪⎪⎨⎪⎪⎪⎩

iγE Ey = −iμωE Hx ,

E′y = iμωE Hz,

iγE Hx − H′z = −iεωE Ey,

⎧⎪⎪⎪⎨⎪⎪⎪⎩

iγM Hy = iεωM Ex ,

H′y = −iεωM Ez,

iγM Ex − E′z = iμωM Hy,

(10)

where ( · )′ ≡ d/dx.After simple transformations from (10) we get⎧⎪⎪⎨

⎪⎪⎩iγ 2

M Ex − γM E′z = iεμω2

M Ex ,

γ 2E Ey − E′′

y = εμω2E Ey,

iγM E′x − E′′

z = εμω2M Ez,

(11)

where Hx = − γ E(μωE)− 1Ey, Hz = (iμωE )−1E′y , Hy = γ −1

M εωM Ex , H′y = −iεωM Ez .

Normalize equations (11) accordingly with the formulae x = kx , ddx = k d

dx , γE = γE

k , γM = γM

k ,ε j = ε j

ε0, ( j = 1, 2, 3), α = α

ε0, where k2 = ω2

Mε0μ0. We obtain⎧⎪⎪⎨⎪⎪⎩

γ 2M (iEx ) − γM E′

z = ε(iEx ),

γ 2E Ey − E′′

y = ω2Eω−2

M εEy,

γM (iEx )′ − E′′z = εEz .

Introduce the notation X: = iEx, Y: = Ey, Z; = Ez, τ := ω2Eω−2

M . Omitting the tilde symbol weobtain from the latter system: ⎧⎪⎪⎨

⎪⎪⎩γ 2

M X − γM Z ′ = εX,

γ 2E Y − Y ′′ = τεY,

γM X ′ − Z ′′ = εZ ,

(12)

where ε is described by

ε =

⎧⎪⎪⎨⎪⎪⎩

ε1, x < −h,

ε2 + α(X2 + Y 2 + Z2), −h < x < h,

ε3, x > h.

III. DIFFERENTIAL EQUATIONS

Introduce the notation:

k2E1 = γ 2

E − τε1, k2E3 = γ 2

E − τε3, k2M1 = γ 2

M − ε1, k2M3 = γ 2

M − ε3.

System (12) is linear in the half-spaces x < − h and x > h. Its solutions (in accordance with theradiation condition) have the forms: for x < − h:⎧⎪⎪⎨

⎪⎪⎩X (x) = C (−h)

M e(x+h)kM1 ,

Y (x) = C (−h)E e(x+h)kE1 ,

Z (x) = γ −1M kM1C (−h)

M e(x+h)kM1 ,

(13)



for x > h: ⎧⎪⎪⎨⎪⎪⎩

X (x) = C (h)M e−(x−h)kM3 ,

Y (x) = C (h)E e−(x−h)kE3 ,

Z (x) = −γ −1M kM3C (h)

M e−(x−h)kM3 .

(14)

It should be noticed that the constants C (h)M and C (h)

E are supposed to be known (initial conditions).Let

fX := (X2 + Y 2 + Z2)X, fY := (X2 + Y 2 + Z2)Y, fZ := (X2 + Y 2 + Z2)Z .

Inside the layer − h < x < h system (12) takes the form:⎧⎪⎪⎨⎪⎪⎩

γM (γM X − Z ′) = ε2 X + α fX ,

γ 2E Y − Y ′′ = τε2Y + ατ fY ,

γM X ′ − Z ′′ = ε2 Z + α fZ .

(15)

IV. TRANSMISSION CONDITIONS AND MATHEMATICAL FORMULATIONOF THE PROBLEM

Tangential components of electromagnetic field are known to be continuous at the interfaces.In this case tangential components are Ey, Ez, Hy, Hz. It follows from the continuity of Hy thatZ′ − γ MX is continuous at the interfaces. It follows from above that the transmission conditions forthe functions X, Y, Y′, Z have the form:

[Z ′ − γM X ]∣∣x=−h = [Z ′ − γM X ]

∣∣x=h = 0,

[Y ]|x=−h = [Y ]|x=h = 0,

[Y ′]∣∣x=−h = [Y ′]

∣∣x=h = 0,

[Z ]|x=−h = [Z ]|x=h = 0,

(16)

where [ f ]|x=x0= lim

x→x0−0f (x) − lim

x→x0+0f (x).

Using (13) and (14) we obtain for boundary values of the fields:

X (−h − 0) = C (−h)M , X (h + 0) = C (h)

M ,

Y (−h − 0) = C (−h)E , Y (h + 0) = C (h)

E ,

Y ′(−h − 0) = kE1C (−h)E , Y ′(h + 0) = −kE3C (h)

E ,

Z (−h − 0) = γ −1M kM1C (−h)

M , Z (h + 0) = −γ −1M kM3C (h)

M ,

Z ′(−h − 0) = γ −1M k2

M1C (−h)M , Z ′(h + 0) = γ −1

M k2M3C (h)

M .

(17)

From transmission conditions (16) and formulae (17) we obtain

Z ′(−h + 0) − γM X (−h + 0) = −ε1γ−1M C (−h)

M , Z ′(h − 0) − γM X (h − 0) = −ε3γ−1M C (h)

M ,

Y (−h + 0) = C (−h)E , Y (h − 0) = C (h)

E ,

Y ′(−h + 0) = kE1C (−h)E , Y ′(h − 0) = −kE3C (h)

E ,

Z (−h + 0) = γ −1M kM1C (−h)

M , Z (h − 0) = −γ −1M kM3C (h)

M .

(18)



Definition 1. The pair (γ E, γ M) is called coupled eigenvalues of the problem if for prescribedvalues C (h)

E and C (h)M there are nontrivial functions X, Y, Z such that for − h < x < h they satisfy

system, for h < − h and x > h they defined by (13) and (14), respectively, and they satisfy transmissionconditions (16). The functions X, Y, Z are called eigenfunctions of the problem.

The main problem (problem P) is to prove existence of coupled eigenvalues.

V. INTEGRAL EQUATIONS AND DISPERSION EQUATIONS

From system (15) we obtain

⎧⎪⎪⎪⎨⎪⎪⎪⎩

X = −γM k−2M Z ′ − αk−2

M fX ,

Y ′′ + k2E Y = −ατ fY ,

Z ′′ + k2M Z = −αε−1

2 k2M fZ − αε−1

2 γM f ′X ,

(19)

where k2E := τε2 − γ 2

E , k2M := ε2 − γ 2

M .Using results of Ref. 1 we can represent system (19) as the system of integral equations:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

X (s) = −αγM

ε2

h∫−h

∂s G2(x, s) fZ dx + αγ 2

M

ε2k2M

h∫−h

∂2xs G2(x, s) fX dx − α

γ 2M + ε2

ε2k2M

fX (s)+

+ C (−h)M

kM1

kM

cos kM (s − h)

sin 2kM h+ C (h)

M

kM3

kM

cos kM (s + h)

sin 2kM h,

Y (s) = ατ

h∫−h

G1(x, s) fY dx+

+ C (−h)E

kE1

kE

cos kE (s − h)

sin 2kE h+ C (h)

E

kE3

kE

cos kE (s + h)

sin 2kE h,

Z (s) = αk2

M

ε2

h∫−h

G2(x, s) fZ dx − αγM

ε2

h∫−h

∂x G2(x, s) fX dx−

− C (−h)M

kM1

γM

sin kM (s − h)

sin 2kM h− C (h)

M

kM3

γM

sin kM (s + h)

sin 2kM h,

(20)

where

G1(x, s) =⎧⎨⎩

− cos kE (x+h) cos kE (s−h)kE sin 2kE h , x < s � h,

− cos kE (x−h) cos kE (s+h)kE sin 2kE h , s < x � h,

(21)



G2(x, s) =⎧⎨⎩

− sin kM (x+h) sin kM (s−h)kM sin 2kM h , x < s � h,

− sin kM (x−h) sin kM (s+h)kM sin 2kM h , s < xh,

(22)

∂s G2(x, s) =⎧⎨⎩

− sin kM (x+h) cos kM (s−h)sin 2kM h , x < s � h,

− sin kM (x−h) cos kM (s+h)sin 2kM h , s < x � h,

∂x G2(x, s) =⎧⎨⎩

− cos kM (x+h) sin kM (s−h)sin 2kM h , x < s � h,

− cos kM (x−h) sin kM (s+h)sin 2kM h , s < x � h,

∂2xs G2(x, s) =

⎧⎨⎩

−kMcos kM (x+h) cos kM (s−h)

sin 2kM h , x < s � h,

−kMcos kM (x−h) cos kM (s+h)

sin 2kM h , s < x � h.

Assuming that α = 0 in (20) we obtain the linear case (see Ref. 1).It can be proved1 that the dispersion equations have the form:1

C (h)E gE (h, γE ) = α

QE (h, γE , γM )

sin 2kE h, (23)

C (h)M kM gM (h, γM ) = α

QM (h, γM , γE )

sin 2kM h, (24)

where

gE (h, γE ) = (k2

E − kE1kE3)

sin 2kE h − kE (kE1 + kE3) cos 2kE h,

gM (h, γM ) = (ε1ε3k2

M − ε22kM1kM3

)sin 2kM h − ε2kM (ε1kM3 + ε3kM1) cos 2kM h

and

QE (h, γE , γM ) =

= τ (kE1 cos 2kE h − kE sin 2kE h)

h∫−h

cos kE (x + h) fY dx − τkE1

h∫−h

cos kE (x − h) fY dx,

QM (h, γM , γE ) =

= γM kM (ε2kM1 cos 2kM h − ε1kM sin 2kM h) ×

×h∫

−h

(kM sin kM (x + h) fZ − γM cos kM (x + h) fX ) dx

+ γM kM kM1ε2

h∫−h

(kM sin kM (x − h) fZ − γM cos kM (x − h) fX ) dx+

+ 2γ 2M [(ε2kM1 cos 2kM h − ε1kM sin 2kM h) fX (h − 0) − kM1ε2 fX (−h + 0)] sin 2kM h.

Dispersion equations (23) and (24) are equations with respect to coupled eigenvalues (propaga-tion constants). To derive these dispersion equations formulae (16)–(18) and (20) are used.

Using Eqs. (20) and transmission conditions (16) it is possible to express unknown constantsC (−h)

E and C (−h)M through known constants C (h)

E and C (h)M . In this case system (20) can be rewritten in



the following form:1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

X (s) = α

h∫−h

[γ 2

M

ε2k2M

∂2xs G2(x, s) fX − γM

ε2∂s G2(x, s) fZ

]dx − α

γ 2M + ε2

ε2k2M

fX (s)+

+ αγMkM1

k2M

cos kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM h×

× lims→−h+0

h∫−h

[γM∂2

xs G2(x, s) fX − k2M∂s G2(x, s) fZ

]dx−

− 2αγ 2M

kM1

k2M

cos kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM hfX (−h + 0)+

+ C (h)M

kM3

kM

[ε2kM1 cos kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM h+ cos kM (s + h)

]1

sin 2kM h,

Y (s) = ατ

h∫−h

G1(x, s) fY dx + ατkE1 cos kE (s − h)

kE sin 2kE h − kE1 cos 2kE h

h∫−h

G1(x,−h) fY dx+

+ C (h)E

kE3

kE

[kE1 cos kE (s − h)

kE sin 2kE h − kE1 cos 2kE h+ cos kE (s + h)

]1

sin 2kE h,

Z (s) = α

h∫−h

[−γM

ε2∂x G2(x, s) fX + k2

M

ε2G2(x, s) fZ

]dx+

+ αkM1 sin kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM h×

× lims→−h+0

h∫−h

[−γM

kM∂2

xs G2(x, s) fX + kM∂s G2(x, s) fZ

]dx+

+ 2αγMkM1

kM

sin kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM hfX (−h + 0)−

− C (h)M

kM3

γM

[ε2kM1 sin kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM h+ sin kM (s + h)

]1

sin 2kM h.

(25)

It is convenient to rewrite system (25) in an operator form.Let K(x, s) be the kernel matrix:

K(x, s) = {Knm(x, s)}3n,m=1 = ε−1

2

⎛⎜⎜⎝

γ 2M k−2

M ∂2xs G2 0 −γM∂s G2

0 τε2G1 0

−γM∂x G2 0 k2M G2

⎞⎟⎟⎠ .



Introduce the matrix integral operator Kg =h∫

−hK(x, s)g(x)dx , where g = (g1, g2, g3)T . Further

let J = − γ 2M +ε2

ε2k2M

⎛⎜⎜⎝

1 0 0

0 0 0

0 0 0

⎞⎟⎟⎠,

K(x, s) = {Knm(x, s)

}3n,m=1 =

=

⎛⎜⎜⎝

γM p1(s) ∂2xs G2

∣∣s→−h+0 0 −k2

M p1(s) ∂s G2|s→−h+0

0 q(s) G1|s→−h+0 0

−γM k−1M p2(s) ∂x G2|s→−h+0 0 kM p2(s) G2|s→−h+0

⎞⎟⎟⎠ ,

where

p1(s) = γMkM1

k2M

cos kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM h,

p2(s) = kM1 sin kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM h,

q(s) = τkE1 cos kE (s − h)

kE sin 2kE h − kE1 cos 2kE h.

Define one more matrix integral operator Kg =h∫

−hK(x, s)g(x)dx , where g = (g1, g2, g3)T . Let

also J =

⎛⎜⎜⎝

−2γM p1(s) 0 0

0 0 0

2γM k−1M p2(s) 0 0

⎞⎟⎟⎠ and h = (h1, h2, h3)T , where

h1 = C (h)M

kM3

kM

[ε2kM1 cos kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM h+ cos kM (s + h)

]1

sin 2kM h,

h2 = C (h)E

kE3

kE

[kE1 cos kE (s − h)

kE sin 2kE h − kE1 cos 2kE h+ cos kE (s + h)

]1

sin 2kE h,

h3 = −C (h)M

kM3

γM

[ε2kM1 sin kM (s − h)

ε1kM sin 2kM h − ε2kM1 cos 2kM h+ sin kM (s + h)

]1

sin 2kM h.

It should be noticed that K, K, J, J are linear operators.Introduce two linear operators N := α

(K + J + K + J

)and N1 := K + J + K + J.

Let u = (X, Y, Z )T and |u|2 = X2 + Y 2 + Z2. Then system (25) can be rewritten in the form:

u = αN1(|u|2u

)+ h. (26)

We study Eq. (26) in C[−h, h] = C[−h, h] × C[−h, h] × C[−h, h], with the norm ‖u‖2C =

‖X‖2C + ‖Y‖2

C + ‖Z‖2C , where ‖u‖C = max

x∈[−h,h]|u(x)|.

VI. SOLVABILITY OF PROBLEM P

In order to formulate theorem of existence of coupled eigenvalues it is necessary to give auxiliarystatements about Eq. (26) (for all proofs, see Ref. 1). As Eq. (26) has the same form as analogousequation in Ref. 1, all the statements and theorems given in Ref. 1 are valid for Eq. (26).

The following theorem asserts existence and uniqueness of a solution of Eq. (26).



Theorem 1. Let Br0 ≡ {u : ‖u‖ � r0} be the ball of radius r0 with its centre at zero. Also lettwo conditions q := 3αr2

0 ‖N1‖, αr30 ‖N1‖ + ‖h‖ � r0 hold.

Then a unique solution u ∈ Br0 of Eq. (26) (or system (25)) exists. The sequence of approximatesolutions u(n) ∈ Br0 of Eq. (26) (or system (25)) defined by the iteration process:

u(n+1) = αN1

(∣∣u(n)∣∣2 u(n)

)+ h,

converges in C[−h, h] to a unique exact solution u ∈ Br0 of Eq. (26) (or system (25)) for any initialapproximation u(0) ∈ Br0 with the geometric progression rate q.

The following theorem gives a constraint on the parameter α.

Theorem 2. If α ≤ A2, where A = 23

1‖h‖√3‖N1‖ , then Eq. (26) has a unique solution u in the ball

Br∗ ≡ {u : ‖u‖ � r∗} and u ∈ C[−h, h], ‖u‖ � r∗, where

r∗ = − 2√3 ‖N‖ cos

(1

3arccos

(3√

3

2‖h‖

√‖N‖

)− 2π

3

)

is a root of equation r30 − ‖N‖−1 r0 + ‖h‖ · ‖N‖−1 = 0.

It should be noticed that A > 0 does not depend on α.The following theorem guarantees that the solution of Eq. (26) depends continuously on the

parameters γ E, γ M.

Theorem 3. Let the matrix operator kernels N and right-hand side h of Eq. (26) dependcontinuously on the parameter γ ∈ 0, N(γ ) ⊂ C(0), h(γ ) ⊂ C(0), on certain real segment 0.Let also ‖h‖ � 2

31√

3‖N(γ )‖ .Then the solutions u(γ ) of Eq. (26) for γ ∈ 0 exist, to be unique and continuously depend on

the parameter γ, u(γ ) ⊂ C(0).

Now let us prove solvability of the problem P.Consider system of dispersion equations (23) and (24).If α = 0 we obtain dispersion equations for a constant permittivity in the layer:

gE (γE ) ≡ (k2

E − kE1kE3)

sin 2kE h − kE (kE1 + kE3) cos 2kE h = 0, (27)

gM (γM ) ≡ (ε1ε3k2

M − ε22kM1kM3

)sin 2kM h − ε2kM (ε1kM3 + ε3kM1) cos 2kM h = 0. (28)

Denote by λEs,m := τε2 − j2

s,m

4h2 , λEc,m := τε2 − j2

c,m

4h2 , λMs,m := ε2 − j2

s,m

4h2 , λMc,m := ε2 − j2

c,m

4h2 , where

js, m = πm is mth nonnegative root of equation sin x = 0 and jc,m = π(2m+1)2 is mth positive root of

equation cos x = 0; m = 0, 1, 2, . . . .It is possible to prove (see Ref. 1) that the following statements are valid.

Statement 1. Let Eq. (27) have lE roots γ(1)E , γ

(2)E , . . . , γ

(lE )E . The number lE can be written as

lE = mE + nE, where γ(i+1)E ∈

(√λE

s,i ,

√λE

c,i

)for i = 0, m E − 1 and γ

(i+1)E ∈

(√λE

c,i ,

√λE

s,i+1

)for

i = m E , lE − 1.

Statement 2. Let Eq. (28) have lM roots γ(1)M , γ

(2)M , . . . , γ

(lM )M . The number lM can be written as

lM = mM + nM, where γ(i+1)M ∈

(√λM

s,i ,

√λM

c,i

)for i = 0, mM − 1 and γ

(i+1)M ∈

(√λM

c,i ,

√λM

s,i+1

)for i = mM , lM − 1.

The numbers mE and mM are defined in Ref. 1. It should be noticed that for given parameters ofthe problem it is possible to chose 2h so that Eqs. (27) and (28) have roots. The numbers lE and lMare not independent.



Let the thickness of the layer be 2h and such that equation gE(γ E) = 0 has lE solutions insidethe interval

(√max(τε1, τε3),

√τε2

), and equation gM(γ M) = 0 has lM solutions inside the interval(√

max(ε1, ε3),√

ε2).

It should be noticed that the points√

λEs,i ,

√λM

s, j are poles of Green’s functions (21) and (22).

Let δEi > 0 and δM

j > 0, i = 0, lE , j = 0, lM be sufficiently small values such that

γ(i+1)E ∈

(√λE

s,i + δEi ,

√λE

c,i

), i = 0, m E − 1,

γ(i+1)E ∈

(√λE

c,i ,

√λE

s,i+1 − δEi+1

), i = m E , lE − 1

and

γ(i+1)M ∈

(√λM

s,i + δMi ,

√λM

c,i

), i = 0, mM − 1,

γ(i+1)M ∈

(√λM

c,i ,

√λM

s,i+1 − δMi+1

), i = mM , lM − 1.

Let us construct the segments

Ei+1 :=

[√λE

s,i + δEi ,

√λE

c,i

], i = 0, m E − 1,

Ei+1 :=

[√λE

c,i ,

√λE

s,i+1 − δEi+1

], i = m E , lE − 1

and

Mi+1 :=

[√λM

s,i + δMi ,

√λM

c,i

], i = 0, mM − 1,

Mi+1 :=

[√λM

c,i ,

√λM

s,i+1 − δMi+1

], i = mM , lM − 1.

Under these assumptions the function gE(γ E) has different signs at the different extremities ofthe segments E

i and vanishes at the points γ(i)E , i = 1, lE . The same conclusion takes place for

the function gM(γ M), that is function gM(γ M) has different signs at the different extremities of the

segments Mj and vanishes at the points γ

( j)M , j = 1, lM . Denote by E :=

lE⋃i=1

Ei , M :=

lM⋃i=1

Mi ,

and := E × M.Functions

∣∣∣ QE

sin 2kE h

∣∣∣ and∣∣∣ QM

sin 2kM h

∣∣∣ can be estimated as∣∣∣ QE

sin 2kE h

∣∣∣ � CEr300,

∣∣∣ QM

sin 2kM h

∣∣∣ � CMr300,

where r00 = max(γE ,γM )∈

r∗. Let C = max {CE, CM}.

Denote by

M1E = min

0�p≤lE

∣∣∣gE

(√λE

s,p + δEp

)∣∣∣ , M2E = min

0�p�lE

∣∣∣gE

(√λE

c,p

)∣∣∣ ,M1

M = min0�q�lM

∣∣∣gM

(√λM

s,q + δMq

)∣∣∣ , M2M = min

0�q�lM

∣∣∣gM

(√λM

c,q

)∣∣∣ .Let M = min

{M1

E , M2E , M1

M , M2M

}.

Let also A1 = min(γE ,γM )∈

A(γE , γM ), where A(γE , γM ) = 23‖h‖√3‖N1(γ )‖ .

The following theorem is the main result of this paper

Theorem 4. Let ε2 > max (ε1, ε3) > 0, 0 < α ≤ α0, where α0 = min{

A21,

MCr3

00

}and the

conditions:

λEc,i , λ

Es,i ∈ (max(τε1, τε3), τε2) , i = 0, lE ; λM

c, j , λMs, j ∈ (max(ε1, ε3), ε2) , j = 0, lM .



be fulfilled. Then there are at least lElM coupled eigenvalues(γ

[i]E , γ

[ j]M

), i = 1, lE , j = 1, lM of the

problem P.

From Theorem 4, it follows that, under the above assumptions, there exist coupled propagatingTE-TM waves in a dielectric layer filled with nonmagnetic isotropic medium with Kerr nonlinearity.As in a linear isotropic dielectric layer TE and TM waves cannot be coupled this result shows thatthere is a new regime for waves propagating in a nonlinear layer.

Proof of this theorem follows from the proof of the analogous theorem from Ref. 1. In spite ofthe fact that formulae here look similarly to analogous formulae in Ref. 1, it should be noted that theirsense is different. It is because the multiplier τ = ω2

Eω−2M is contained in formulae here. In Ref. 1

τ = 1. So Theorem 4 proves essentially new result. In other words, here we consider more generalcase. That is Theorem 4 proves that coupled surface TE-TM waves exist and these waves propagateat different frequencies with different propagation constants along the boundaries of an isotropiclayer with Kerr nonlinearity.15

Method of Theorem 4 proving is based on the method of small parameter. Here coefficient ofnonlinearity α is supposed to be small. This approach is possible as it is known10 that the Kerr lawis fulfilled for small α.

VII. CONCLUSION

In spite of the fact that in the half-spaces x < − h and x > h the considered electromagnetic wavecan be represented as a sum of TE and TM waves, the same conclusion does not hold for a (nonlinear)coupled surface TE-TM wave. To be more precise, in a medium with constant permittivity a sum ofTE and TM waves is a solution to Maxwell’s equations. The same conclusion holds for purely TEand TM waves, they are also solutions to Maxwell’s equations. What is more in this case a solutionfor the sum of TE and TM waves can be obtained as a sum of solutions to Maxwell’s equationsfor purely TE and TM waves (linearity of Maxwell’s equations plays key role in this case). Fornonlinear permittivity the situation changes dramatically. Purely nonlinear TE and TM waves and(nonlinear) coupled TE-TM wave separately are solutions to the Maxwell’s equations. However,any sum of purely nonlinear TE and TM waves does not give a (nonlinear) coupled TE-TM wave.So the considered coupled TE-TM wave is a new type of waves (new polarization) that exists onlyin a nonlinear waveguide structure. On the other hand, it is obvious that in some sense this TE-TMwave consists of TE and TM waves. This is the reason why we call them pseudo-TE and pseudo-TMpolarizations.

Two important peculiarities of coupled surface TE-TM waves should be pointed out. First,pseudo-TE and pseudo-TM waves propagate at two different frequencies. Second, these frequenciesdo not depend on each other, that is they can be chosen independently. Availability of two differentfrequencies (ωE and ωM) and two different initial conditions (C (h)

E and C (h)M ) allows us to hope that it

is possible to control some characteristics (for example, propagation constant, amplitude, etc.) of onepseudopolarization by changing characteristics of another one (for example, amplitude, frequency).

Natural question that arises is to check experimentally existence of coupled surface TE-TMwaves.

A conjecture about some generalization should be discussed. As it is known (see, for example,Refs. 9 and 11) and clear from this research in a layer with constant permittivity ε2 the followingtwo inequalities:

max(τε1, τε3) < γ 2E < τε2,

max(ε1, ε3) < γ 2M < ε2

are fulfilled for TE and TM waves, respectively. Under reasonable conditions it is proved9, 12–14 thatpurely nonlinear TE and TM waves can propagate in a layer with Kerr and Kerr-like nonlinearity evenif γ 2

E > τε2 (for TE waves) and γ 2M > ε2 (for TM waves). Here in the proof of existence of coupled

TE-TM wave propagation both of the latter inequalities are required. The conjecture is that couplednonlinear TE-TM waves for Kerr and Kerr-like nonlinearities exist even if γ 2

E > τε2 and/or γ 2M > ε2



at least when all constant permittivities of the nonlinear problem are positive. In other words, if(γ

[k]E , γ

[l]M

)is coupled eigenvalues then probably exist indices k, l such that

(γ

[k]E

)2> τε2 and/or(

γ[l]M

)2> ε2. It is probable that infinite numbers of coupled eigenvalues exist for finite thickness of

the layer in this nonlinear problem (to compare see Refs. 9 and 12–14).Two more facts should be also noted:

• Obtained results can be generalized to the case of tensor (diagonal) permittivity in the layer.This generalization leads just to complication of formulae here.

• Considered method can be applied to study more complicated nonlinearities. For example,permittivity inside the layer can have the form: ε = ε2 + f(|Ex|2, |Ey|2, |Ez|2), where f is,so say, more or less arbitrary function with small norm. Smallness of the norm is necessaryto apply the contracting mapping method. From above it is clear that a diagonal permittivitytensor can be considered instead of scalar ε.

ACKNOWLEDGMENTS

I would like to thank Professor Yu. G. Smirnov for his criticism and attention to my research.This study is supported by Russian Foundation for Basic Research (Grant Nos. 11-07-00330-A and12-07-97010-R A), The Ministry of Education and Science of the Russian Federation (Grant No.14.B37.21.1950), and the Visby program of the Swedish Institute.

1 Y. G. Smirnov and D. V. Valovik, “Coupled electromagnetic TE-TM wave propagation in a layer with kerr nonlinearity,”J. Math. Phys. 53, 123530 (2012).

2 P. N. Eleonskii, L. G. Oganes’yants, and V. P. Silin, “Vector structure of electromagnetic fielding self-focused waveguides,”Sov. Phys. Usp. 15, 524–525 (1973).

3 P. N. Eleonskii, L. G. Oganes’yants, and V. P. Silin, “Structure of three-component vector fields in self-focusing waveg-uides,” Sov. Phys. JETP 36, 282–285 (1973), http://www.jetp.ac.ru/cgi-bin/e/index/e/36/2/p282?a=list.

4 J. A. Stretton, Electromagnetic Theory (McGraw Hill, New York, 1941).5 L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Course of Theoretical Physics

Vol. 8 (Butterworth-Heinemann, Oxford, 1993).6 A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, Third-order Nonlinear Electromagnetic TE and TM

Guided Waves (Elsevier Sci. Publ., 1991) (reprinted from Nonlinear Surface Electromagnetic Phenomena).7 P. N. Eleonskii, L. G. Oganes’yants, and V. P. Silin, “Cylindrical nonlinear waveguides,” Sov. Phys. JETP 35, 44–47 (1972),

http://www.jetp.ac.ru/cgi-bin/e/index/e/35/1/p44?a=list.8 L. A. Vainstein, Electromagnetic Waves (Radio i svyaz, Moscow, 1988).9 Y. G. Smirnov and D. V. Valovik, Electromagnetic Wave Propagation in Nonlinear Layered Waveguide Structures (Penza

State University Press, Penza, Russia, 2011), p. 248.10 N. N. Akhmediev and A. Ankevich, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).11 M. J. Adams, An Introduction to Optical Waveguide (John Wiley and Sons, 1981).12 D. V. Valovik, “Propagation of electromagnetic waves in a nonlinear metamaterial layer,” J. Commun. Technol. Electron.

56, 544–556 (2011).13 D. V. Valovik and Y. G. Smirnov, “Nonlinear effects in the problem of propagation of tm electromagnetic waves in a kerr

nonlinear layer,” J. Commun. Technol. Electron. 56, 283–288 (2011).14 Y. G. Smirnov and D. V. Valovik, “Nonlinear effects of electromagnetic tm wave propagation in anisotropic layer with kerr

nonlinearity,” Adv. Math. Phys. 2012, 1–21 (2012).15 Y. G. Smirnov and D. V. Valovik, “Coupled electromagnetic TE-TM wave propagation in a cylindrical waveguide with

Kerr nonlinearity,” J. Math. Phys. 54, 043506 (2013).


http://dx.doi.org/10.1063/1.4769885

http://dx.doi.org/10.1070/PU1973v015n04ABEH005018

http://www.jetp.ac.ru/cgi-bin/e/index/e/36/2/p282?a=list

http://www.jetp.ac.ru/cgi-bin/e/index/e/35/1/p44?a=list

http://dx.doi.org/10.1134/S1064226911050111

http://dx.doi.org/10.1134/S1064226911030120

http://dx.doi.org/10.1155/2012/609765

http://dx.doi.org/10.1063/1.4799276

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