### Analytical and Numerical Analyses of

### TE

### -Polarized Waves in the Planar

### Optical Waveguides with the Nonlinear Guiding Film

**Yaw-Dong Wu***_{ and Dong-Heng Cai }

Department of Electronic Engineering

National Kaohsiung University of Applied Sciences, Kaohsiung, 807, Taiwan, R. O. C. FAX: 886-7-3811182, E-mail: ydwu@cc.kuas.edu.tw

**Abstract **

The modal theory can be used to analyze the wave propagation in the linear and nonlinear planar optical waveguide. By solving the Maxwell's equation and matching the boundary conditions, we can find the eigenmode of transverse electric field in the planar optical waveguide with nonlinear guiding film. A detail modal analysis of the nonlinear waveguide structure is proposed. The analytical results are accompanied by numerical examples. The numerical simulation results show that the analyses are correct.

**Keywords : modal theory, Jocabian elliptic function, nonlinear guiding film **

**1. Introduction **

A number of papers have been proposed to discuss the waves propagating along the nonlinear slab optical waveguide structures [1-5]. Considerable progress has been made by considering only Kerr nonlinearities with some additional assumptions. Some of those papers [1-4] that analyze the guiding properties of a nonlinear film have already in the literature. The treatment in Ref. [3] results in the wavenumber of propagation being treated as an implicit function of the modulus of a Jocabian elliptic function.

The modal theory [6-7] can be used to analyze the wave propagation in the linear and nonlinear planar waveguide. By solving the Maxwell's equation and matching the boundary conditions, we can find the eigenmode of transverse electric field in the planar waveguide. We propose an analytical method for analyzing the optical waveguide with the nonlinear guiding film. The numerical simulation results show that our analyses are correct.

**2. Analysis **

*In this paper, we will analytically calculate the TE-polarized waves in the three-layer planar *
waveguides with the nonlinear guiding film, as shown in Figure 1, and get the exact dispersion
relation. The structure is composed of the cladding, the film, and the substrate. The thickness of
guiding film is d. The cladding and substrate layers are assumed to extend to infinity in the +x and

-x direction, respectively. The major characteristic of this assumption is that there are no reflections
*in the x-direction to be concerned with, except for those occurring at interface. For the case of TE *
plane waves traveling in the z direction, the Maxwell's wave equation can be reduced to

2
2
2
2
2
*t*
*c*
*n _{j}*

_{y}*y*

_{∂}∂ = ∇ ε ε (1)

with solutions of the form

)]
(
exp[
)
,
(
)
,
,
(*x* *z* *t* *E* *x* *z* *j*

### β

*k*

_{0}

*z*

### ω

*t*

### ε

= −_{ (2) }

The subscripts f, c, and s in Eq. (1) are used to denote the guiding film, cladding, and substrate,
*respectively. For TE waves, the electric field components E*x* and E*z are zero. Note also in Eq. (2)
*that the transverse electric field E(x,z) has no y and z dependence because the planar layers are *
assumed to be infinite in these directions precluding the possibility of reflections and resultant
*standing waves. In the following expressions of the field components, the time factor exp(iwt) and *
*the z-dependence exp(-ik0*β*z) are omitted, where *ω* is the angular frequency, k*0 is the wave number
in the free space, and β is the effective refractive index. For Kerr-type nonlinear medium, the square
of refractive index of guiding film can be expressed as following:

2
*f*
*n* = 2 2
)
*(x*
*E*
*nfo* +α (3)

where *n is the linear refractive index of the film and α is the nonlinear coefficient. By _{f}*

_{0}substituting Eq. (2) and Eq. (3) into the wave equation, we can get

0
)
(
]
[ 2
2
2
=
−*q* *E* *x*
*dx*
*d*
*c* (4)
0
)
(
]
)
(
[ 2 2
0
2
2
2
=
+
+*q* *k* *E* *x* *E* *x*
*dx*
*d*
*f* α (5)
0
)
(
]
)
(
[ 2 2
0
2
2
2
=
+
−*Q* *k* *E* *x* *E* *x*
*dx*
*d*
*f* α (6)
0
)
(
]
[ 2
2
2
=
−*q* *E* *x*
*dx*
*d*
*s* (7)
where
)
( 2 2
2
0
2 _{=} _{−}_{β}
*f*
*f* *k* *n*
*q* , for β <*n _{f}*

)
( 2 2
2
0
2
*f*
*f* *k* *n*
*Q* = β − , for β >*n _{f}*
)
( 2 2
2
0
2

*j*

*j*

*k*

*n*

*q*= β − , j = c, s

By solving the differential equations, the transverse electric fields in cladding and substrate can
be expressed as follows:
)]
(
exp[
)
(*x* *E _{c}*

*p*

_{c}*x*

*d*

*c*= − − ε (8) ] exp[ ) (

*x*

*E*

_{s}*p*

_{s}x*s*= ε (9)

where *Ec* an *Es* are the values of the electric field at the lower and upper boundaries of the film,
respectively. On the other hand, the first integration of Eq. (5) and Eq. (6) gives

)
2
(
)
2
(
2
2
2
2
2
0
2
2
2
2
2
0 *s* *f* *s* *s* *c* *f* *c* *c*
*E*
*n*
*n*
*E*
*k*
*E*
*n*
*n*
*E*
*k*
*K* = − +α = − +α , (10)

and the second integration gives the electric field in the form

]
)
(
[
)
(*x* *bcn* *A* *x* *x*_{0} *l*
*f* = +
ε (11)
]
'
)
(
[
)
( '
0
'
'_{cn}_{A}_{x}_{x}_{l}*b*
*x*
*f* = +
ε (12)
where
2
1
2
0
2
2 _{)]}
2
)(
[(*a* *b* *k*
*A*= + α , _{2} 2 _{2}
*b*
*a*
*b*
*l*
+
= , _{2}
0
2
2
0
4
2 2
*k*
*q*
*K*
*k*
*q*
*a*
α
α +
+
= , _{2}
0
2
2
0
4
2 2
*k*
*q*
*K*
*k*
*q*
*b*
α
α −
+
= ,
2
1
2
0
2
'
2
'
' _{)]}
2
)(
[(*a* *b* *k*
*A* = + α , _{'}_{2} _{'}_{2}
2
'
'
*b*
*a*
*b*
*l*
+
= , _{2}
0
2
2
0
4
2
' 2
*k*
*Q*
*K*
*k*
*Q*
*a*
α
α −
+
= ,
2
0
2
2
0
4
2
' 2
*k*
*Q*
*K*
*k*
*Q*
*b*
α
α +
+
= ,
=
= −
*b*
*E*
*cn*
*x*
*x* ' 1 0
0
0 ,
0
*x and * '
0

*x are the second constants of integration, and cn is a specific Jacobian elliptic function. *
For simplicity, the modulus *l is omitted in the following discussions. Considering the case * β <*n _{f}*
and matching the boundary conditions, we could obtain the following eiqenvalue equation:

−
−
−
−
−
= ( ) ( ){[1 (1 )] ( )(1 _{2} )}
2
2
2
2
*b*
*E*
*Ad*
*lcn*
*b*
*E*
*l*
*Ad*
*dn*
*Ad*
*sn*
*b*
*E*
*Ab*
*q*
*E _{c}*

_{c}

_{s}

_{s}

_{s}2
2
2
2
2_{(} _{)} _{(} _{)]} ( ) ( ) ( )
)[
(
_{+}
−
*b*
*E* *Ab*
*q*
*E*
*Ad*
*dn*
*Ad*
*sn*
*b*
*E*
*Ad*
*cn*
*Ad*
*lsn*
*b*
*E*
*Ad*
*dn*
*Ad*
*cn*
*Ab*
*q*
*E*
*c*
*s*
*s*
*s*
*s*
*s*
*s* _{ (12) }

where sn and dn are Jacobi elliptic functions. The Eq. (12) can be solved numerically. When
constants β* and E*s are determined, all the other constants *q , _{f}*

*q ,*

_{s}*q , K, A, a, b, l,*

_{c}*x , and Ec*

_{0}are also determined.

For the case that β >*n _{f}*

*, on the other hand, we just have to replace the constants A, a, b, l,*and

*x in Eq. (12) with*

_{0}

*',*

_{A}*'*

_{a , }*'*

_{b , }*' '*

_{l and }0

*x , and get these equations with similar expressions. *
When constants β* and E*s are determined, all the other constants *q , _{f}*

*q ,*

_{s}*q , K,*

_{c}*A*',

*a*',

*b*',

*l*',

*x*

_{0}'

*and E*c are also determined.

The power per unit width along the y-axis carried in the z direction by the nonlinear guided wave is obtained by integrating the Poynting vector and time averaging:

## ∫

−∞∞ =*c*

*E*

*x*

*dx*

*P*( ) 2 1

_{2}0β ε (13)

For the linear planar waveguide, the effective refractive index β is independent of the input
power. Whereas for the nonlinear planar waveguide, β is dependent on the input power. Since the
effective refractive index of the nonlinear waveguide is changed gradually with the intensity of the
input light beam, there is an intimate relation between the effective refractive index and power in
the nonlinear waveguide. This relation is so-called the dispersion relation. For a nonlinear film, on
the other hand, bounded by semi-infinite dielectrics, the electric field outside the film must always
decay exponentially at each boundary. Therefore, the electric field inside the film must always
increase from the boundary. In this sense, the field always resembles linear guided waves, and this
is reinforced by the periodic nature of the Jocabin elliptic functions. Because of this periodicity, an
*infinite numbers are possible, we only present the TE*0*, TE*1* and TE*2 waves here as being sufficient
numerical illustrations of the theory and dispersion relations.

**3. Numerical Results **

Consider the case that the refractive index of guiding film is higher than the refractive index of
cladding and substrate. It means that confined modes are possible when the effective refractive
index of the three-layer planar waveguide is higher than the refractive index of cladding and
substrate. Several numerical examples are presented as follows. All numerical results presented here
were calculated with the values: *n _{c}* = =1.55,

*n*

_{s}*n*

_{f}_{0}=1.57,

_{α}

_{=}

_{6}

_{.}

_{3786}

_{µ}

*2*

_{m}_{/}

*2*

_{V}_{, and the free }space wavelength λ=1.3µm. Figure 2(a) shows the dispersion curve of TE0 wave in the three-layer

*nonlinear waveguide with the width d = 0.8µm of the guiding film. Figure 2(b) shows the electric*field distributions for the various input optical power density shown in Figure 2(a). Figure 3(a)

*shows the dispersion curves of TE*0

*and TE*1 waves in the three-layer nonlinear waveguide with the

*width d = 1.2 µm of the guiding film. Figure 3(b) shows the electric field distributions for the *
various input optical power density shown in Figure 3(a). Figure 4(a) shows the dispersion curves
*of TE*0*, TE*1* and TE*2* waves in the three-layer nonlinear waveguide with the width d = 1.6 µm of the *
guiding film. Figure 4(b) shows the electric field distributions for the various input optical power
*density shown in Figure 4(a). Figure 5(a) shows the dispersion curves of TE*0*, TE*1*, TE*2*, TE*3 and
*TE4 waves in the three-layer nonlinear waveguide with the width d = 3 µm of the guiding film. *
Figure 5(b) shows the electric field distributions for the various input optical power density shown
in Figure 5(a). As shown in Figure 2 to Figure 5, as the guided power increases and consequently β
increases, the electric field distributions gradually become narrow, and the optical wave would be
confined in the nonlinear film.

**4. Conclusions **

In this paper, we propose an analytical method for analyzing the optical waveguide with the nonlinear guiding film. We give a detailed modal analysis of the proposed nonlinear waveguide structure. The analytical results are accompanied by numerical examples. The numerical simulation results show that our analyses are correct.

**References **

[1] Fedyanin, V. K. and D. Mikhalake, “P-polarized Nonlinear Surface Polarized in Layered
*Structures,” Z. Phys. B, Vol. 47, 1984, pp. 167-181. *

[2] Langbern, U., F. Legerer, and H. E. Ponath, “A New Type of Nonlinear Slab-Guided Waves,”
*Opt. Commun.*, Vol. 46, 1982, pp. 167-169.

[3] Akhmediev, N. N., K. O. Botar, and V. M. Eleonskii, “Dielectric Optical Waveguide with
*Nonlinear Susceptibility Symmetric Refractive-Index Profile,” Opt. Spectrum (USSR), Vol. *
53(b), 1982, pp. 654-658.

*[4] Boardman, A. D. and P. Egan, “Optically Nonlinear Waves in Thin Films,” IEEE J. Quantum *
*Electron*, Vol. 22, 1986, pp. 319-324.

[5] Danielsen, P. and D. Yevick, “Improved Analysis of the Propagating Beam Method in
*Longitudinally Perturbed Optical Waveguides, ”Appl. Opt., Vol. 21, 1982, pp. 4188-4189. *
[6] Burns, W. K. and A. F. Milton, “Mode Conversion in Planar Dielectric Separating

*Waveguides,” IEEE J. Quantum Electron, Vol. 11, 1975, pp. 32-39. *

[7] Zutsu, M. I., Y. Nakai, and T. Sutena, “Operation Mechanism of the Single-Mode
*Optical-Waveguide Y Junction,” Opt. Lett., Vol. 7, 1982, pp. 136-138. *

*x=d *

*x=0 *

**Cladding**

**Film**

**Substrate**

*(Linear) *

*(Nonlinear) *

*(Linear) *

*x *

*z *

Transverse distance (µm)
Electric field
(µ
v/m)
A
B
C
**D**
A
B
C
**D **

Effective refractive index (a)

*Figure 2. (a) The dispersion relation of TE*0* wave in a three-layer nonlinear waveguide with d*

= 0.8µm. (b) Field distribution at four points A-D in (a).

1.56 1.6 1.64 1.68 1.72
0
0.04
0.08
0.12
0.16
0.2
-8 -4 0 4 8
0
0.1
0.2
0.3
0.4
Power density (
×10
-3 ) (
*W*
/mm)
(b)

A

B C

D

Effective refractive index
(a)
A
B
C
*D *
Transverse distance (µm)
Electric field
(µ
V/
m)

*Figure 3. (a) The dispersion relation of TE*0* and TE*1 wave in a three-layer nonlinear

*waveguide with d = 1.2µm. (b) Field distribution at four points A-D in (a). *

1.56 1.6 1.64 1.68 1.72
0
0.05
0.1
0.15
0.2
0.25
-4 -2 0 2 4 6
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
(b)
Power density (
×10
-3 ) (
*W*
/mm)

Transverse distance (µm) (b) Electric field (µ V/ m)

*Figure 4. (a) The dispersion relation of TE*0*, TE*1* and TE*2 wave in a three-layer nonlinear

waveguide with d=1.6µm. (b) Field distribution at five points A-E in (a).

E
A
*D *
B
C
Power density (
×10
-3 ) (
*W*
/mm)

Effective refractive index
(a)
A
B C
D *E *
1.56 1.6 1.64 1.68 1.72
0
0.2
0.4
0.6
0.8
-4 -2 0 2 4 6
-0.4
0
0.4

A B

C D

E

Effective refractive index (a) Transverse distance (µm) (b) Electric field (µ V/ m)

*Figure 5. (a) The dispersion relation of TE*0*, TE*1*, TE*2*, TE*3* and TE*4 wave in a three-layer

*nonlinear waveguide with d = 3µm. (b) Field distribution at five points A-E in (a).*

Power density (
×10
-3 ) (
*W*
/mm)
A
B
C
D
*E *
1.56 1.6 1.64 1.68
0
0.2
0.4
0.6
0.8
1
-4 0 4 8
-0.4
0
0.4