2
0 d
1
Λ =d
1
+d
2
n
1
n
2
z
A
1 1
C
1 2
B
1 1
D
1 2
A
2 1
B
2 1
E
0 d
1
Λ =d
1
+d
2
n
1
n
2
z
A
1 1
C
1 2
B
1 1
D
1 2
A
2 1
B
2 1
E
Figure 1. Structure of 1D photonic crystal with proparation direction z.
The refractive index of the structure can be expressed as :
Λ
= z
a ;
n a
z ;
n z
n
2 1
2
3.1. Matrix transfer method
We assume that the electromagnetic field E propagates to the right and to the left within the layer with refractive index n
1
has amplitudes of A
1
and B
1
, respectively. Whereas the light within the layer of n
2
propagates with amplitudes of C
1
and D
1
, respectively. Therefore, the propagation of light in the photonic crystal structure becomes [6]:
1 d
z 2
ik 1
1 d
z 2
ik 1
z 1
ik 1
z 1
ik 1
e D
e C
z E
e B
e A
z E
− −
− −
+ =
+ =
3 Parameters k
1
and k
2
are called propagation constants k
1
= ω
n
1
and k
2
= ω
n
2
. By applying boundary conditions at z = d
1
and z =L, we obtain [6]:
−
+
−
+
=
−
+
−
+ =
− −
− −
2 2
1 d
1 ik
1 2
1 d
1 ik
1 2
1 d
1 ik
1 2
1 d
1 ik
1 2
1 1
1 1
1 d
1 ik
1 2
1 d
1 ik
1 2
1 d
1 ik
1 2
1 d
1 ik
1 2
1 1
B A
e k
k 1
e k
k 1
e k
k 1
e k
k 1
2 1
D C
D C
e k
k 1
e k
k 1
e k
k 1
e k
k 1
2 1
B A
4
By eliminating of C
1
, D
1
matrix, we obtain :
=
1 1
22 21
12 11
2 2
B A
M M
M M
B A
5 where the components of M are :
+ −
=
−
− =
− =
+ +
=
− −
d k
sin k
k k
k i
2 1
d k
cos e
2 ,
2 M
d k
sin k
k k
k i
2 1
e 1
, 2
M d
k sin
k k
k k
i 2
1 e
2 ,
1 M
d k
sin k
k k
k i
2 1
d k
cos e
1 ,
1 M
2 2
2 1
1 2
2 2
1 d
1 ik
2 2
2 1
1 2
1 d
1 ik
2 2
2 1
1 2
1 d
1 ik
2 2
2 1
1 2
2 2
1 d
1 ik
6
The matrix of M is called matrix transfer of a unit cell. If the structure of photonic crystal consist of N unit cell and the light comes from the left side of the structure and interacts within the structure
leads to waves which propagate to the right and to the left with amplitudes of t and r, respectively, then :
3
+
−
−
+
+
−
−
+
+
−
−
+
=
r
1 k
k 1
2 1
k k
1 2
1 k
k 1
2 1
k k
1 2
1 k
k 1
2 1
k k
1 2
1 k
k 1
2 1
k k
1 2
1 M
k k
1 2
1 k
k 1
2 1
k k
1 2
1 k
k 1
2 1
t
2 2
2 2
1 2
1 2
1 2
1 2
N 1
1 1
1
7
The transmitted light is then expressed by T = |t|
2
. 3.2. Nonlinear coupled mode equation
Propagation of light process in 1D photonic crystal is governed by Maxwell equations. We assume that no charge and electric current sources in the dielectric materials and no magnetic materials.
Therefore, the electromagnetic wave equation can be expressed by [4]:
E z
n c
dz E
d
2 2
2 2
2
= ω
+
8 where
1 ε
µ =
c
is the light velocity in vacuum and nz is the refractive index of the structure. We studied two different structures, i.e. nonlinear Distributed Bragg Reflector DBR and
nonlinear photonic crystal composed of nonlinear optical materials with identical linear refractive index but opposite sign of their nonlinear refractive ind ices.
3.2.1. Nonlinear Distributed Bragg Reflector
Basically, the structure of nonlinear Distributed Bragg Reflector DBR is equal to the structure depicted in Fig. 1, but the layer-1 is made from nonlinear optical material, therefore the refractive index
of the structure becomes :
2 nl
z E
n Gz
cos n
n z
n +
+ =
9 where
n
is the depth variation of the refractive index. S ubtitution of this equation into Eqn. 8 leads to the wave equation :
z E
z E
n n
2 c
z GzE
cos n
n 2
z E
n c
dz E
d
2 nl
2 2
2 2
2 2
2
= ω
+ +
ω +
10 If the electric field and light intensityare defined, respectively, as :
+ +
= +
=
β β
− β
− β
β −
β z
i z
i z
i z
i 2
z i
z i
e B
e A
Be Ae
z E
Be Ae
z E
11 and by using slowly varying amplitude SVA approximation, equation 10 becomes :
[ ]
[ ]
κ +
+ =
κ +
+ −
=
δ δ
− z
i 2
2 2
2 z
i 2
2 2
2
Ae B
B A
2 k
n dz
dA i
Be A
B 2
A k
n dz
dA i
12
where n
2
k =
α and
c 2
n ω
= κ
. Equation 12 is called nonlinear coupled mode equation. By using definition :
z z
z e
B B
e A
A
B A
z B
i z
A i
φ −
φ =
ψ =
=
φ φ
13 and subsituting it into Eqn. 12, then by separating their real and imaginary parts, we obtain :
B A
3 cos
B A
2 B
cos B
A dz
d ;
sin A
dz B
d A
B 2
A cos
A B
dz d
; sin
B dz
A d
2 2
B 2
2 A
κ α
− =
ψ +
α +
ψ κ
= φ
ψ κ
= +
α +
ψ κ
= φ
ψ κ
=
14
4
The transmitted light is defined by :
2 2
2
B A
T −
= 15
Substitution Eqn. 15 into Eqn. 14 leads to :
−
κ α
− −
κ =
2 2
2 2
2 2
2 2
2 2
T A
A 3
1 T
A A
4 dz
A d
16 By substituting the following definitions of x = 2zL, |C|
2
= 23 α
L, y = |A|
2
|B|
2
and I = |T|
2
|C|
2
, where L is the length of the photonic crystal structure and I
is the normalized output light intensity to |T|
2
into Eqn. 16, we obtain :
{ }
2 2
2
y I
y 4
y L
I y
dx dy
− −
κ −
=
17
The solution of Eqn. 17 is a Jacobian elliptic equation :
−
+ =
m 2
x 1
Q 2
nd 1
2 I
x y
18 where
2 2
2 2
2
L I
Q L
I L
m κ
+ =
κ +
κ =
19
If the normalized input light intensity is defined by I
i
= y x = 0, then the relation between output and input intensity is :
m Q
2 nd
1 I
2 I
i
+ =
20
3.2.2. Photonic crystal with equal linear refractive index but opposite sign of nonlinear refractive index
The structure of this 1D nonlinear photonic crystal consist of periodic dielectric materials with
refractive indices :
I n
n n
I n
n n
2 nl
02 2
1 nl
01 1
− =
+ =
21 where n
01
, n
nl1
and n
02
, n
nl2
are linear and nonlinear refractive indices of layer-1 and layer-2, respectively. In order to obtain an analytical expression for the evolution of forward and backward propagating light
inside the structure, we use the nonlinear coupled mode equation by defining A
1
z and A
2
z are amplitudes of the forward and backward propagating light and also by assuming that absorption of the
materials are neglected [7]:
[ ]
[ ]
[ ]
[ ]
z xA
z I
n c
z 2
c n
2 i
exp z
xA d
sin d
i exp
z I
n n
n n
c dz
z dA
i z
xA z
I n
c z
2 c
n 2
i exp
z A
x d
sin d
i exp
z I
n n
n n
c dz
z dA
i
2 nl
1 2
2 2
nl 1
nl 2
1 2
1 nl
2 2
2 2
nl 1
nl 2
1 1
ω +
Λ π
− ω
π Λ
π
Λ
π −
− +
− ω
− =
ω −
Λ π
− ω
π Λ
π
Λ
π −
− +
− ω
=
22
5
where Λ
+ =
2 2
1 1
d n
d n
n and
Λ +
=
2 2
nl 1
1 nl
nl
d n
d n
n are the average linear and nonlinear refractive
indices of the structure, respectively. In this work, we assumed that n
01
= n
02
and n
nl1
= n
nl2
, therefore, the Eqn. 22 becomes :
[ ]
[ ]
Λ π
− ω
− +
π ω
− =
Λ π
− ω
+ π
ω −
= z
2 c
n 2
i exp
z A
z A
z A
n 2
c dz
z dA
z 2
c n
2 i
exp z
A z
A z
A n
2 c
dz z
dA
1 2
2 2
1 nl
2 2
2 2
2 1
nl 1
23
The solution of Eqn. 23 is taken at resonance condition 2 ω
n c =
2πΛ , by applying the boundary
conditions at the position z = L, where L is the length of the structure and A
2
L = 0, i.e. no radiation is incident on the structure from the right and A
1
L = A
1out
. By taking the squared modulus of A
1out
yield the intensity of the forward propagating within the structure :
out nl
out nl
out 2
1
I n
z L
n I
4 cos
2 n
z L
n I
4 cos
1 z
A z
I
Λ
−
Λ
− +
= =
24
where I
out
= |A
1out
|
2
. The input light intensity is obtained at z = 0:
out out
z in
I 1
a I
4 cos
1 2
1 z
I I
+
=
=
=
25
with a = 2n Nn
nl
and N = 2L Λ
is number of layers. Equation 25 is a characteristic equation of optical limiter.
IV. RESULTS AND DISCUSSION 4.1. Nonlinear Distributed Bragg Reflector
