On Classical Electromagnetic Fields (cont.)
IX. Optical
Pulse Propagation (pdf copy)
The Electromagnetic Nonlinear Schrödinger
Equation
We begin our discussion of optical pulse propagation[1] with a derivation of the nonlinear Schrödinger
(NLS) equation. To that end, we recall Equations [ VII23 ] and [ VII23
] from a previous lecture set entitled Nonlinear Optics I  i.e.

[ IX1 ] 

[ IX2 ] 
In this treatment we will confine our attention to wave propagation
in uniform, isotropic optical materials  viz., glass
fibers. For such materials, we can write

[ IX3 ] 
where
and, thus, Equation [ IX1 ] simplifies to

[ IX4 ] 
where
. [2]
To proceed, postulate that this nonlinear Helmholtz equation can be treated
by separation of variables methods. In particular, we are looking
for a timelocalized solution (a pulse) with a relatively narrow frequency
spectrum (or "group" of frequencies centered on a frequency
.
Example: A Gaussian Pulse
Thus, we assume a separation of variables solution

[ IX5 ] 
where
is a wave or propagation number to be associated with
and, thus, Equation [ IX4 ] becomes

[ IX6] 
where
. In the linear problem
would be the "separation constant," but in this case we will need a bit more
elaboration. Nevertheless, we shall assume that we can find a set of functions
and values
that satisfy the equation

[ IX7a ] 
so that

[ IX7b ] 
To use perturbation theory, we first reduce Equation [ IX7a
] to a solvable linear problem by writing

[ IX8a ] 

[ IX8b ] 
where
. Thus, to first order we need to solve the linear equation

[ IX9 ] 
which taken together with appropriate boundary conditions defines
the linear eigenvalue problem for propagation in the medium where the functions
F are the eigenfunctions and the values
are the eigenfunctions. For example, in the previous lecture set  i.e.,
VIII. Guided Waves in Planar Structures  we found a set of eigenfunctions
and eigenvalues appropriate to the dielectricslab guidewave propagation
Following an earlier discussion, we now presume that
 i.e., we take the slowly vary amplitude or
paraxial approximation  so that Equation [ IX7b ] reduces
to

[ IX10a ] 
where
and, thus,

[ IX10b ] 
For equation [ IX7a ] to be completely satisfied in first order,
we must have

[ IX11 ] 
Since we are assuming that propagating pulse has a relatively
narrow frequency spectrum, it is reasonable to use a Taylor expansion around
for
 viz.

[ IX12 ] 
where
Near the dispersion minimum in glass fibers (i.e.
) we may, to very good approximation, stop with the quadratic term
and write

[ IX13 ] 
In this approximation, Equation [ IX10 ] becomes

[ IX14a ] 
If we take

[ IX15 ] 
Equation [ IX10 ] in the frequency domain reduces to

[ IX14b ] 
which implies the following time domain equation for
the pulse envelope:

[ IX16 ] 
The last term on the left hand side has been added to incorporate
the effects of various possible loss mechanisms.
Next we transform into a coordinate system which moves with
the "group"  what might be called the surfer's coordinates of
the pulse  i.e.,
where
and
. In terms of these surfer's coordinates, Equation [ IX16 ] becomes [3]

[ IX17 ] 
Obviously, if we omit all of the terms on the right hand side
of this equation so that
, we would have the ideal situation wherein a pulse of any shape
propagates forever without changing shape at a velocity
 the group velocity. In treating the lessthanideal
situation, we will initially neglect the effects of the loss (first) term
and confine our attention to the competing effects of the dispersion (second)
and nonlinear (third)term . We cast the lossless version of Equation
[ IX16 ] into a normalized, standard form by introducing

[ IX18 ] 
where
is the width of the pulse and
is its peak power.
Thus, we, at last, obtain the standard form of the nonlinear
Schrödinger (NLS) equation

[ IX18 
where
.
Pulse Solutions of Linear Schrödinger Equation
If in Equation [ IX10 ] we set
and neglect the loss and nonlinear terms, we see that
satisfies the following differential equation:

[ IX19 ] 
For minimal dispersion  viz. if
 Equation [ IX19 ] becomes

[ IX20 ] 
which is the basic wave equation for minimally dispersive
media with the general solution

[ IX21 ] 
where
is the group velocity of the pulse.
When limitation to first order dispersion is an adequate
approximation, Equation [ IX19 ] expressed in surfer's coordinates becomes

[ IX22 ] 
Amazingly, this pulse dispersion equation
is the, so called, parabolic equation that we saw earlier in connection with
beam propagation  viz. the paraxial wave propagation equation.
Solution of Pulse Dispersion Equation
For convenience, we restate here Equation [ IX22 } the firstorder
pulse dispersion equation  viz.
.
Let us write a Fourier transform for this modulation in terms
of "surfer time"  i.e.

[ IX23a ] 
where

[ IX23b ] 
Thus, the pulse dispersion equation reduces to an ordinary
differential equation  viz.

[ IX24 ] 
for the Fourier transform and thus we have the simple solution

[ IX25 ] 
Thus, we see that the dispersion changes the phase of each
spectral component of the pulse by an amount that depends on the frequency
and the propagated distance. The general solution may be written as

[ IX26a ] 
where

[ IX26b ] 
Let us suppose that we have a "chirped" Gaussian at
 i.e.

[ IX27a ] 
so that

[ IX27b ] 
where
characterizes an "upchirp" and
a "downchirp" pulse.
A Gaussian pulse with a frequency "downchirp"
Inverting the transform, we see that

[ IX28a ] 
or rationalizing

[ IX28b ] 
Hence, the pulse width broadening factor at a given
position z is given by

[ IX29] 
where
.
The spatial evolution of the pulse width of chirped
Gaussian pulse
(For "normal dispersion"  i.e.
)
Pulse Solutions of "Negligible Dispersion" Nonlinear Schrödinger
Equation
If group velocity dispersion can be neglected, Equation [ IX17
] reduces to

[ IX30 ]

and if we take
we obtain

[ IX31 ]

where the characteristic nonlinear length is given by
. A solution to this equation is readily obtain in the form

[ IX32a ]

where

[ IV32b ]

and

[ IX32c ]

This interesting result shows that so called "selfphase modulation"
or SPM gives rise to an intensity dependent phase shifted or chirped pulse
which remains constant in shape as it propagates. The instantaneous optical
frequency shift is given by

[ IV33 ]

Note that the pulse spectrum is "redshifted" on the leading
edge of a pulse and "blueshifted" on the trailing edge of the pulse.
If we suppose the initial pulse to be a superGaussian of mthorder  i.e.,

[ IV34 ]

 then the instantaneous optical frequency shift would be given
by

[ IV35 ]

SuperGaussian Envelope Shapes
SPM Induced Frequency Chirp
A Very Brief Discussion of Solitons
Hokusai's "Soliton"

The Soliton Home Page
from HeriotWatt University, Edinburgh is an interesting with a lot of history
and valuable movies.
A very interesting and novel approach to the subject can be
found at Unification
of Linear and Nonlinear Wave Optics by Allan W. Snyder, D. John Mitchell
and Yuri S. Kivshar at the Institute of Advanced Studies, Australian National
University Canberra ACT 0200, Australia Received 6 October 1995.
We return briefly to Equation [ IX18 ]  the standard
form of the nonlinear Schrödinger (NLS) equation  i.e.,

[ IX18 ]

where
,
,
and
.
In soliton analysis the most common form of NLS equation
is the following:

[ IX36 ]

where
.
If
the following fundamental soliton will propagate undistorted
for an arbitrary distance:

[ IX37 ]

If
the following secondorder soliton will propagate
undistorted for an arbitrary distance:

[ IX38a ]


[ IX38b ]

Two
Soliton Collision Illustration
The following are timelapsed illustration of the propagation
and collision of two solitons  plot here is
For reference in lecture discussions:
Notes on absorption in fiber optic
glass
Notes on dispersion in fiber optic glass
[1] An excellent reference on this subject is Govind
P. Agrawal's Nonlinear Fiber Optics, Academic Press (1989) ISBN 0120451409.
[2] In this simplification, we have taken
[3] Since
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Last updated February 29, 2000