Slowly Varying Envelope Approximation

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## Example

### Full approximation

### Parabolic approximation

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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 **k**_{0} 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 *E*_{0}(**r**, *t*) only varies slowly with **r** and *t*. This inherently implies that *E*_{0}(**r**, *t*) represents waves propagating forward, predominantly in the **k**_{0} direction. As a result of the slow variation of *E*_{0}(**r**, *t*), when taking derivatives, the highest-order derivatives may be neglected:^{[2]}

- and with

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

It is convenient to choose **k**_{0} 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 **k**_{0}-direction. The space and time scales over which *E*_{0} 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.

Assume wave propagation is dominantly in the *z*-direction, and **k**_{0} 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.

**^**Arecchi, F. T. & Bonifacio, R. IEEE J. Quantum Electron. 1, 169-178 (1965).**^**Butcher, Paul N.; Cotter, David (1991).*The elements of nonlinear optics*(Reprint ed.). Cambridge University Press. p. 216. ISBN 0-521-42424-0.**^**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.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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