Devices

Abstract

Consider an optical wave propagating along the +z -direction, so that it may be written as

$$u(x,y,t) = \operatorname{Re} [\tilde u(x,y){e^{i(\omega t - \beta z)}}]$$

(9.1)

The basic problem of optical-beam propagation is: Given the complex wave amplitude ũ _o(x_o, y_o) across an input plane z _o, find the complex amplitude and phase ũ(x, y) of the wave across any later output plane, z. The most common wave used in analysis is one having a Gaussian variation in amplitude across the wavefront:

$$\begin{gathered} \tilde u(x,y) = {\left( {\frac{{2e}}{\pi }} \right)^{1/2}}\frac{1}{\omega }\exp \left( { - \frac{{i\pi {x^2}}}{\lambda }\frac{{{x^2} + {y^2}}}{{\tilde q}}} \right) \hfill \\ \frac{1}{{\tilde q}} = \frac{1}{R} - i\frac{\lambda }{{\pi {\omega ^2}}} \hfill \\ \end{gathered} $$

(9.2)

where R is the radius of the spherical wave and w is the spot size.