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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, slowly varying envelope approximation (SVEA, sometimes also called slowly varying asymmetric approximation or SVAA) is the assumption that the
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a shor ...
of a forward-travelling
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
pulse varies slowly in time and space compared to a
period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
or
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
. This requires the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of the signal to be
narrow-band Narrowband signals are signals that occupy a narrow range of frequencies or that have a small fractional bandwidth. In the audio spectrum, narrowband sounds are sounds that occupy a narrow range of frequencies. In telephony, narrowband is usual ...
ed—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 derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s. But the validity of the assumptions which are made need to be justified.


Example

For example, consider the
electromagnetic wave equation The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form ...
: :\nabla^2 E - \mu_0\, \varepsilon_0\, \frac = 0. If k0 and ''ω''0 are the
wave number In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
and
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the (characteristic)
carrier wave In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has a ...
for the signal ''E''(r,''t''), the following representation is useful: :E(\mathbf,t) = \Re\left\, where \scriptstyle \Re\ denotes the
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
of the quantity between brackets. In the ''slowly varying envelope approximation'' (SVEA) it is assumed that the
complex amplitude Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
''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 k0 direction. As a result of the slow variation of ''E''0(r, ''t''), when taking derivatives, the highest-order derivatives may be neglected: :\displaystyle \left, \nabla^2 E_0 \ \ll \left, \vec k_0\cdot \nabla E_0 \ and \displaystyle \left, \frac \ \ll \left, \omega_0\, \frac \, with k_0 = , \mathbf_0, .


Full approximation

Consequently, the wave equation is approximated in the SVEA as: :2\, i\ \mathbf_0\ \cdot \nabla E_0 + 2\ i\ \omega_0\ \mu_0\ \varepsilon_0\ \frac - \left( k_0^2 - \omega_0^2\ \mu_0\ \varepsilon_0 \right)\ E_0 = 0. It is convenient to choose k0 and ''ω''0 such that they satisfy the
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
: : k_0^2 - \omega_0^2\ \mu_0\ \varepsilon_0 = 0.\ This gives the following approximation to the wave equation, as a result of the slowly varying envelope approximation: :\mathbf_0 \cdot \nabla E_0 + \omega_0\ \mu_0\ \varepsilon_0\, \frac = 0. This is a
hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...
, 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 ''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.


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 \scriptstyle \Delta_\perp=\partial^2/\partial x^2 + \partial^2/\partial y^2 is the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
in the ''x''–''y'' plane, the result is: :k_0 \frac + \omega_0\, \mu_0\, \varepsilon_0\, \frac - \tfrac12\, i\, \Delta_\perp E_0 = 0. This is a
parabolic partial differential equation A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivati ...
. 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 : l_g \gg \lambda and l_p \gg \lambda (1-v/c) ,with \lambda = \frac and c = \frac where ''l''g is the length over which the radiation pulse is amplified, ''l''p is the pulse width and ''v'' is the group velocity of the radiating system. These conditions are much less restrictive in the relativistic limit where ''v''/''c'' is close to 1, as in a
Free-electron laser A free-electron laser (FEL) is a (fourth generation) light source producing extremely brilliant and short pulses of radiation. An FEL functions and behaves in many ways like a laser, but instead of using stimulated emission from atomic or molecula ...
, compared to the usual conditions required for the SVEA validity.


See also

*
WKB approximation In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...
*
Ultrashort pulse In optics, an ultrashort pulse, also known as an ultrafast event, is an electromagnetic pulse whose time duration is of the order of a picosecond (10−12 second) or less. Such pulses have a broadband optical spectrum, and can be created by ...


References

{{reflist Theoretical physics Asymptotic analysis