Slowly varying envelope approximation

"SVEA" redirects here. For other uses, see Svea.

In physics, slowly varying envelope approximation ^[1] (SVEA, sometimes also called slowly varying asymmetric 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 is 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.

Example

For example, consider the electromagnetic wave equation:

$${\displaystyle \nabla ^{2}E-{\frac {1}{c^{2}}}{\frac {\partial ^{2}E}{\partial t^{2}}}=0\,,}$$

where ${\displaystyle c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}~.}$

If k₀ and ω₀ are the wave number and angular frequency of the (characteristic) carrier wave for the signal E(r,t), the following representation is useful:

$${\displaystyle E(\mathbf {r} ,t)=\operatorname {\operatorname {Re} } \left[E_{0}(\mathbf {r} ,t)\,e^{i(\mathbf {k} _{0}\cdot \mathbf {r} -\omega _{0}t)}\right],}$$

where ${\displaystyle \operatorname {Re} [\,\cdot \,]}$ denotes the real part of the quantity between brackets, and ${\displaystyle i^{2}\equiv -1.}$

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 E(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]

${\displaystyle \left|\nabla ^{2}E_{0}\right|\ll \left|\mathbf {k} _{0}\cdot \nabla E_{0}\right|}$  and  ${\displaystyle \left|{\frac {\partial ^{2}E_{0}}{\partial t^{2}}}\right|\ll \left|\omega _{0}\,{\frac {\partial E_{0}}{\partial t}}\right|,}$  with  ${\displaystyle k_{0}\equiv \left|\mathbf {k} _{0}\right|.}$

Full approximation

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

$${\displaystyle 2i\mathbf {k} _{0}\cdot \nabla E_{0}+{\frac {2i\omega _{0}}{c^{2}}}{\frac {\partial E_{0}}{\partial t}}-\left(k_{0}^{2}-{\frac {\omega _{0}^{2}}{c^{2}}}\right)E_{0}=0~.}$$

It is convenient to choose k₀ and ω₀ such that they satisfy the dispersion relation:

$${\displaystyle k_{0}^{2}-{\frac {\omega _{0}^{2}}{c^{2}}}=0~.}$$

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

$${\displaystyle \mathbf {k} _{0}\cdot \nabla E_{0}+{\frac {\omega _{0}}{c^{2}}}\,{\frac {\partial E_{0}}{\partial t}}=0~.}$$

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 ${\displaystyle \Delta _{\perp }\equiv \partial ^{2}/\partial x^{2}+\partial ^{2}/\partial y^{2}}$ is the Laplace operator in the x×y plane, the result is: ^[3]

$${\displaystyle k_{0}{\frac {\partial E_{0}}{\partial z}}+{\frac {\omega _{0}}{c^{2}}}{\frac {\partial E_{0}}{\partial t}}-{\frac {1}{2}}\,i\,\Delta _{\perp }E_{0}=0~.}$$

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.

Alternative limit of validity

In the one-dimensional case, another sufficient condition for the SVEA validity is

${\displaystyle \ell _{\mathsf {g}}\gg \lambda }$  and  ${\displaystyle \ell _{\mathsf {p}}\gg \lambda \left(1-{\frac {v}{c}}\right)\,,}$  with  ${\displaystyle \lambda ={\frac {2\pi }{k_{0}}}\,,}$

where ${\displaystyle \ell _{\mathsf {g}}}$ is the length over which the radiation pulse is amplified, ${\displaystyle \ell _{\mathsf {p}}}$ is the pulse width and ${\displaystyle v}$ is the group velocity of the radiating system. ^[4]

These conditions are much less restrictive in the relativistic limit where ${\displaystyle {\frac {v}{c}}}$ is close to 1, as in a free-electron laser, compared to the usual conditions required for the SVEA validity.

See also

• Ultrashort pulse
• WKB approximation

References

1. Arecchi, F.; Bonifacio, R. (1965). "Theory of optical maser amplifiers". IEEE Journal of Quantum Electronics . 1 (4): 169–178. Bibcode:1965IJQE....1..169A. doi:10.1109/JQE.1965.1072212.
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 . Vol. 12. North Holland. pp. 23–25. ISBN   0-444-10571-9.
4. Bonifacio, R.; Caloi, R.M.; Maroli, C. (1993). "The slowly varying envelope approximation revisited". Optics Communications . 101 (3–4): 185–187. Bibcode:1993OptCo.101..185B. doi:10.1016/0030-4018(93)90363-A.