Dynamics of Nonlinear Systems: Integrable and Chaotic Solutions
177
(thick) layers of refractive index n 1 , the number of modes the waveguide can support
depends on the waveguide parameter
V = 2π
2L
λ
n
2
0 − n
2
1
(40)
where λ is the wavelength of the light. For total internal reflection, we require a
high index material surrounded on the top and bottom by lower index materials.
The requirement for a wave to propagate through the waveguide is that the angle of
incidence must be greater than the critical angle, sinθ c =
n 2
n 1
where n 1 and n 2 are the
refractive index of the core and cladding, respectively.
For an asymmetric waveguide the cut-off frequency above which the multiple
modes of a particular polarization can propagate is given by
ω c =
π c
d
1
n
2
1 − n
2
2
(41)
The number of modes in a wave guide should always be minimal for the ease of
designing and fabrication. We consider a waveguide single mode which supports a
single-guided mode for a particular polarization (TE or TM).
4.2 Optical Solitons and NLSE
The pulse propagation within a nonlinear waveguide needs a spatial restriction to
reduce the effective area and enhance nonlinear interactions. This helps to reduces
the dispersion thereby controlling the spreading of the pulse. This introduces an
interplay between the nonlinearity and the group velocity dispersion which leads to
the formation of optical solitons [5, 6]. The dispersion length and nonlinear length
where such effects may become dominant are given by Ł 0 =
L
2
0
|β 2 |
and Ł N L =
1
γ P 0
,
respectively. When L D ≤ L and L N L ≥ L the pulse undergoes significant dispersive broadening, but the spectrum remains constant. If L D >> L and L N L ≤ L, the
spectrum will change via self-phase modulation due to nonlinear effects. When both
L D ≤ L and L N L ≤ L the interplay between the dispersion and the nonlinearity
produces different results depending on relative signs of the two effects. If the dispersion is anomalous with β < 0 and n 2 > 0, stable solutions, known as solitons,
forms. Soliton solutions are extremely stable because they can shed excess energy in
the form of a dispersive wave until a stable solution is formed. Inorder to propogate
solitons, the waveguide parameters are to be fixed [6]. These parameters are fixed
using numerical methods. By solving the wave equation and computing the modal
eigenvalue of a waveguide structure, the modal index or the effective refractive index
may be calculated as
177
(thick) layers of refractive index n 1 , the number of modes the waveguide can support
depends on the waveguide parameter
V = 2π
2L
λ
n
2
0 − n
2
1
(40)
where λ is the wavelength of the light. For total internal reflection, we require a
high index material surrounded on the top and bottom by lower index materials.
The requirement for a wave to propagate through the waveguide is that the angle of
incidence must be greater than the critical angle, sinθ c =
n 2
n 1
where n 1 and n 2 are the
refractive index of the core and cladding, respectively.
For an asymmetric waveguide the cut-off frequency above which the multiple
modes of a particular polarization can propagate is given by
ω c =
π c
d
1
n
2
1 − n
2
2
(41)
The number of modes in a wave guide should always be minimal for the ease of
designing and fabrication. We consider a waveguide single mode which supports a
single-guided mode for a particular polarization (TE or TM).
4.2 Optical Solitons and NLSE
The pulse propagation within a nonlinear waveguide needs a spatial restriction to
reduce the effective area and enhance nonlinear interactions. This helps to reduces
the dispersion thereby controlling the spreading of the pulse. This introduces an
interplay between the nonlinearity and the group velocity dispersion which leads to
the formation of optical solitons [5, 6]. The dispersion length and nonlinear length
where such effects may become dominant are given by Ł 0 =
L
2
0
|β 2 |
and Ł N L =
1
γ P 0
,
respectively. When L D ≤ L and L N L ≥ L the pulse undergoes significant dispersive broadening, but the spectrum remains constant. If L D >> L and L N L ≤ L, the
spectrum will change via self-phase modulation due to nonlinear effects. When both
L D ≤ L and L N L ≤ L the interplay between the dispersion and the nonlinearity
produces different results depending on relative signs of the two effects. If the dispersion is anomalous with β < 0 and n 2 > 0, stable solutions, known as solitons,
forms. Soliton solutions are extremely stable because they can shed excess energy in
the form of a dispersive wave until a stable solution is formed. Inorder to propogate
solitons, the waveguide parameters are to be fixed [6]. These parameters are fixed
using numerical methods. By solving the wave equation and computing the modal
eigenvalue of a waveguide structure, the modal index or the effective refractive index
may be calculated as