Paraxial approximation

Paraxial appriximation is the scientific concept of treating of the wave equation

\(\!\!\!\!\!\!\!\!\!\!\ (1) ~ ~ ~ \nabla^2 \Psi +k^2 \Psi =0\)

assuming that

\(\!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~ \Psi=\Psi( \vec x, z)=\exp(\mathrm i k z)\psi(\vec x,z)\)

and neglecting the second derivative of \(\psi\) with respect to its last argument.

Equation of Ginzburg-Landau

Substitution of (2) into (1) and neglecting term with second derivative with respect to \(z\) gives

\(\!\!\!\!\!\!\!\!\!\!\ (3) ~ ~ ~ 2 \mathrm i k \dot \psi + \psi''=0\)

where \(\psi=\psi(\vec x,z)\); dot differentiates with respect to the last argument (that has sense of coordinate in which the paraxial wave propagates) and prime differentiates with respect to transversal coordinates \(\vec x\); in the typical case, it is 2-vector. As it is declared in the preamble, the term \(\ddot\psi\) is neglected.

Method of Fourier

The efficient method of solution of equation (3) is related to the Fourier transform. Let

\(\!\!\!\!\!\!\!\!\!\!\ (4) \displaystyle ~ ~ ~ \psi(\vec x, z)=\frac{1}{2 \pi} \iint \exp(\mathrm i \vec p \vec x)~ f_{\vec p}(z) ~\mathrm d ^2 p\)

The substitution to (3) gives the equation

\(\!\!\!\!\!\!\!\!\!\!\ (5) \displaystyle ~ ~ ~ \dot f_{\vec p}(z)=\frac{-\mathrm i p ^2}{2k} f_{\vec p}(z)\)

which gives the solution

\(\!\!\!\!\!\!\!\!\!\!\ (6) \displaystyle ~ ~ ~ f_{\vec p}(z+T) = \exp\left(\frac{-\mathrm i p ^2}{2k} T \right) f_{\vec p}(z)\)

In the usual case, \(T\) is considered as real parameter, that has sense of the length of propagation.

The initial function \(\psi\) in any plane \(z=\mathrm{const}\) can be reconstructed with the inverse Fourier transform.

WIth the efficient numerical implementation of the Fourier transform, the solution \(\psi\) of the paraxial equation can be approximated with good precision in the real time.

Example of the application of the Fourier transform

Consider propagation of waves with wavenumber

\( k=2 \pi /(0.532~\mu) \approx 11.81 ~\mu^{-1}\)

through the set of slits of half-width

\(w=0.5 \mu\)

one may choose the grid step \(d_{\mathrm x}\) for the representation of field \(\psi\) along the \(x\) axis; then the step of representation of its Fourier-transform is

\(\displaystyle d_{\mathrm p}= \frac{2\pi}{N ~d_{\mathrm x}}\)

where \(N\) is number of nodes. This number is the same for both the momentum and the coordinate representation of the wave function. In particular, the choice

\(N=1024\),

\(d_{\mathrm x}=10\mu\)

leads to the estimate for \(d_{\mathrm p}\):

\(d_{\mathrm p}=0.0613592 ~ \mu^{-1}\)

The distribution of field behind the first slit is shown in figure at right. (oops.. Sorry, I still need to plot the figure..)

Application of paraxial approximation

In centuries 19, 20, the most of optic for visible range of electromagnetic waves referred to the paraxial approximation, although the equation (3) is attributed to Ginsgurg and Landau and corresponds to the middle of century 20.

For the beginning of century 21, the most of optics for the electromagnetic waves works far from the paraxial approximation. However, this approximation still works well for the optics of atomic waves.

Keywords

Optics, Diffraction