Chapter 2
Angular Sp ectrum
Representation of Optical Fields
The angular sp ectrum representation is a mathematical technique to describ e optical elds
in homogeneous media. Optical elds are describ ed as a sup erp osition of plane waves and
evanescentwaves which are physically intuitive solutions of Maxwell's equations. The angu-
lar sp ectrum representation is found to b e a very p owerful metho d for the description of laser
b eam propagation and light fo cusing. Furthermore, in the paraxial limit, the angular sp ec-
trum representation b ecomes identical with the framework of Fourier optics which extents
its imp ortance even further. In the subsequent sections we will apply the angular sp ectrum
representation to gain an understanding of the limits of far eld optics.
2.1 Basic Formalism
Under angular sp ectrum representation we understand the series expansion of an arbitrary
eld in terms of plane waves with variable amplitudes and propagation directions. Assume we
know the electric eld Eratany p oint r =x; y ; z in space. For example, Er can b e the
solution of an optical scattering problem as shown in Fig. 2.1 for which E = E + E .In
inc scatt
the angular sp ectrum picture, we draw an arbitrary axis z and consider the eld E in a plane
z = const transverse to the chosen axis. In this plane we can evaluate the two dimensional
Fourier transform of the eld E as
1
Z Z
1