Soliton Propagation Through Photorefractive Media
123
transformations
q =
γ 0 T
2
0
1
2
|k ∗ |
E
ξ =
z|k
|
T
2
0
(6)
τ =
T
T 0
Redefining ξ as z and τ as t, we get the NLSE,
iq x − sgn(k
)q tt + 2|q
2
|q = 0
( 7 )
where T 0 represents the width of the incident pulse, z and t are the normalized
distance and time along the direction of propagation and q, the normalized envelope.
Interchanging t and x, the standard form of NLSE is obtained as
iq t + q xx ± 2|q|
2 q = 0 (q ∈ c)
(8)
If we consider normal dispersion regime [negative sign in (11)], the solution take
“tanh” form and the intensity profile associated with such solutions shows dip in a
uniform background and thus the solutions are called “dark solitons” [9].
5 Coupled Nonlinear Schrödinger Equation (CNLSE)
The formation of optical soliton is due to interplay between spreading of pulse and
nonlinear response of medium [Kerr effect] which leads to an intensity dependent
phase change described by self-phase modulation [SPM] of incident pulse. In the
case of birefringent fibers in addition to SPM one has to consider cross phase modulation [XPM] which leads to a phase dependence of each mode on the intensity of
co-propagating modes. The resulting propagation equation is a set of coupled nonlinear Schrödinger equations [CNLSE] [10]. This equation was developed by S. V.
Manakov in 1973 known as the Manakov Model. This equation describes two-mode
propagation in optical fiber.
iq 1x + c 1 q 1tt + 2(α|q 1 |
2
+ β|q 1 |
2
)q 2 = 0
( 9 )
iq 1x + c 1 q 1tt + 2(α|q 1 |
2
+ β|q 2 |
2
)q 1 = 0
c 1 , c 2 , α, β real parameters.
For the two specific parametric choices, Manakov system is defined by
123
transformations
q =
γ 0 T
2
0
1
2
|k ∗ |
E
ξ =
z|k
|
T
2
0
(6)
τ =
T
T 0
Redefining ξ as z and τ as t, we get the NLSE,
iq x − sgn(k
)q tt + 2|q
2
|q = 0
( 7 )
where T 0 represents the width of the incident pulse, z and t are the normalized
distance and time along the direction of propagation and q, the normalized envelope.
Interchanging t and x, the standard form of NLSE is obtained as
iq t + q xx ± 2|q|
2 q = 0 (q ∈ c)
(8)
If we consider normal dispersion regime [negative sign in (11)], the solution take
“tanh” form and the intensity profile associated with such solutions shows dip in a
uniform background and thus the solutions are called “dark solitons” [9].
5 Coupled Nonlinear Schrödinger Equation (CNLSE)
The formation of optical soliton is due to interplay between spreading of pulse and
nonlinear response of medium [Kerr effect] which leads to an intensity dependent
phase change described by self-phase modulation [SPM] of incident pulse. In the
case of birefringent fibers in addition to SPM one has to consider cross phase modulation [XPM] which leads to a phase dependence of each mode on the intensity of
co-propagating modes. The resulting propagation equation is a set of coupled nonlinear Schrödinger equations [CNLSE] [10]. This equation was developed by S. V.
Manakov in 1973 known as the Manakov Model. This equation describes two-mode
propagation in optical fiber.
iq 1x + c 1 q 1tt + 2(α|q 1 |
2
+ β|q 1 |
2
)q 2 = 0
( 9 )
iq 1x + c 1 q 1tt + 2(α|q 1 |
2
+ β|q 2 |
2
)q 1 = 0
c 1 , c 2 , α, β real parameters.
For the two specific parametric choices, Manakov system is defined by