Slowly Varying Envelope Approximation
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Slowly Varying Envelope Approximation

In physics, the slowly varying envelope approximation[1] (SVEA, sometimes also called slowly varying amplitude approximation or SVAA) is the assumption that the envelope of a forward-travelling wave pulse varies slowly in time and space compared to a period or wavelength. This requires the spectrum of the signal to be narrow-banded—hence it also referred to as the narrow-band approximation.

The slowly varying envelope approximation is often used because the resulting equations are in many cases easier to solve than the original equations, reducing the order of--all or some of--the highest-order partial derivatives. But the validity of the assumptions which are made need to be justified.


For example, consider the electromagnetic wave equation:

If k0 and ?0 are the wave number and angular frequency of the (characteristic) carrier wave for the signal E(r,t), the following representation is useful:

where denotes the real part of the quantity between brackets.

In the slowly varying envelope approximation (SVEA) it is assumed that the complex amplitude E0(r, t) only varies slowly with r and t. This inherently implies that E0(r, t) represents waves propagating forward, predominantly in the k0 direction. As a result of the slow variation of E0(r, t), when taking derivatives, the highest-order derivatives may be neglected:[2]

  and     with  

Full approximation

Consequently, the wave equation is approximated in the SVEA as:

It is convenient to choose k0 and ?0 such that they satisfy the dispersion relation:

This gives the following approximation to the wave equation, as a result of the slowly varying envelope approximation:

This is a hyperbolic partial differential equation, like the original wave equation, but now of first-order instead of second-order. It is valid for coherent forward-propagating waves in directions near the k0-direction. The space and time scales over which E0 varies are generally much longer than the spatial wavelength and temporal period of the carrier wave. A numerical solution of the envelope equation thus can use much larger space and time steps, resulting in significantly less computational effort.

Parabolic approximation

Assume wave propagation is dominantly in the z-direction, and k0 is taken in this direction. The SVEA is only applied to the second-order spatial derivatives in the z-direction and time. If is the Laplace operator in the x-y plane, the result is:[3]

This is a parabolic partial differential equation. This equation has enhanced validity as compared to the full SVEA: it represents waves propagating in directions significantly different from the z-direction.

See also


  1. ^ Arecchi, F. T. & Bonifacio, R. IEEE J. Quantum Electron. 1, 169-178 (1965).
  2. ^ Butcher, Paul N.; Cotter, David (1991). The elements of nonlinear optics (Reprint ed.). Cambridge University Press. p. 216. ISBN 0-521-42424-0.
  3. ^ Svelto, Orazio (1974). "Self-focussing, self-trapping, and self-phase modulation of laser beams". In Wolf, Emil (ed.). Progress in Optics. 12. North Holland. pp. 23-25. ISBN 0-444-10571-9.

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