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 [ VII-23 ] and [ VII-23 ] from a previous lecture set entitled Nonlinear Optics I -- i.e.

[ IX-1 ]

[ IX-2 ]

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

[ IX-3 ]

where and, thus, Equation [ IX-1 ] simplifies to

[ IX-4 ]

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 time-localized 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

[ IX-5 ]

where is a wave or propagation number to be associated with and, thus, Equation [ IX-4 ] becomes

[ IX-6]

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

[ IX-7a ]

so that

[ IX-7b ]

To use perturbation theory, we first reduce Equation [ IX-7a ] to a solvable linear problem by writing

[ IX-8a ]

[ IX-8b ]

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

[ IX-9 ]

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 dielectric-slab guidewave propagation

Following an earlier discussion, we now presume that -- i.e., we take the slowly vary amplitude or paraxial approximation -- so that Equation [ IX-7b ] reduces to

[ IX-10a ]

where and, thus,

[ IX-10b ]

For equation [ IX-7a ] to be completely satisfied in first order, we must have

[ IX-11 ]

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

[ IX-12 ]

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

[ IX-13 ]

In this approximation, Equation [ IX-10 ] becomes

[ IX-14a ]

If we take

[ IX-15 ]

Equation [ IX-10 ] in the frequency domain reduces to

[ IX-14b ]

which implies the following time domain equation for the pulse envelope:

[ IX-16 ]

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 [ IX-16 ] becomes [3]

[ IX-17 ]

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 less-than-ideal 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 [ IX-16 ] into a normalized, standard form by introducing

[ IX-18 ]

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

[ IX-18

where .

Pulse Solutions of Linear Schrödinger Equation

If in Equation [ IX-10 ] we set and neglect the loss and nonlinear terms, we see that satisfies the following differential equation:

[ IX-19 ]

For minimal dispersion -- viz. if -- Equation [ IX-19 ] becomes

[ IX-20 ]

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

[ IX-21 ]

where is the group velocity of the pulse.

When limitation to first order dispersion is an adequate approximation, Equation [ IX-19 ] expressed in surfer's coordinates becomes

[ IX-22 ]

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 [ IX-22 } the first-order pulse dispersion equation -- viz.

Let us write a Fourier transform for this modulation in terms of "surfer time" -- i.e.

[ IX-23a ]

where

[ IX-23b ]

Thus, the pulse dispersion equation reduces to an ordinary differential equation -- viz.

[ IX-24 ]

for the Fourier transform and thus we have the simple solution

[ IX-25 ]

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

[ IX-26a ]

where

[ IX-26b ]

Let us suppose that we have a "chirped" Gaussian at -- i.e.

[ IX-27a ]

so that

[ IX-27b ]

where characterizes an "up-chirp" and a "down-chirp" pulse.

A Gaussian pulse with a frequency "down-chirp"

Inverting the transform, we see that

[ IX-28a ]

or rationalizing

[ IX-28b ]

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

[ IX-29]

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 [ IX-17 ] reduces to

[ IX-30 ]

and if we take we obtain

[ IX-31 ]

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

[ IX-32a ]

where

[ IV-32b ]

and

[ IX-32c ]

This interesting result shows that so called "self-phase 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

[ IV-33 ]

Note that the pulse spectrum is "red-shifted" on the leading edge of a pulse and "blue-shifted" on the trailing edge of the pulse. If we suppose the initial pulse to be a super-Gaussian of mth-order -- i.e.,

[ IV-34 ]

-- then the instantaneous optical frequency shift would be given by

[ IV-35 ]

Super-Gaussian Envelope Shapes SPM Induced Frequency Chirp

A Very Brief Discussion of Solitons

Hokusai's "Soliton"

The Soliton Home Page from Heriot-Watt 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.

KdV 1-soliton propagation movie