Slowly varying envelope approximation

The slowly varying envelope approximation (SVEA) for calculating and modeling light propagation can be used to simplify the scalar form of Maxwell’s equations that describe the propagation of electromagnetic waves through a material or vacuum. SVEA is valid only in a narrow range of propagation angles.

In the scalar form of Maxwell’s equations, the electric field can be written as

E(x,y,z,t) = φ(x,y,z,t) exp[ -iωt ] + c.c.

However, the phase of φ(x,y,z,t) varies rapidly along the propagation direction (usually specified as the z-direction). This rapid phase variation is not important for many types of problems and can be removed by introducing a slowly varying field A(x,y,z,t) and constant k, such that

φ(x,y,z,t) = A(x,y,z,t) exp[ ikz ]

resulting in the following SVEA

E(x,y,z,t) = A(x,y,z,t) exp[ ikz -iωt ] + c.c.

It is assumed that A(x,y,z,t) varies slowly in the propagation direction z as well as the other two directions and time t,

| ∂A/∂x | << k | A |
| ∂A/∂y | << k | A |
| ∂A/∂z | << k | A |
| ∂A/∂t | << ω | A |

which implies, for example, that

2A/∂z2 ≈ 0

This reduces a second order differential equation to a first order equation that is easier to solve. The resulting form of the first-order scalar Maxwell’s equations is called the slowly varying envelope approximation (or the paraxial approximation, if the back-propagating term A*(x,y,z,t) is left out, since it is valid only in a narrow range of angles).



See also: Paraxial approximation, Nonparaxial approximation.

References
[1] P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics, Cambridge U. Press (1990).


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