The electromagnetic fields within a waveguide or a resonator cannot have arbitrary distributions. The requirements of satisfying Maxwell's equations as well as the boundary conditions specific to the waveguide (or the resonator) confine the distribution to certain shapes and forms. The electromagnetic field distributions that can be sustained within a device are known as its stable modes of oscillation.

When the device and its geometry are simple, the stable modes can be determined analytically. For complex systems and complicated geometries, however, numerical methods must be used to solve Maxwell's equations in the presence of the relevant boundary conditions. The method of Fox and Li is an elegant numerical technique that can be applied to certain waveguides and resonators in order to obtain the operating mode of the device. Instead of solving Maxwell's equations explicitly, the method of Fox and Li uses the Fresnel–Kirchhoff diffraction integral to mimic the physical process of wavefront propagation within the device, thus arriving at its stable mode of operation after several iterations.

To illustrate the method of Fox and Li we focus our attention on the confocal resonator shown in Figure 31.1(a). Let us assume that the two mirrors are aberration-free parabolas with an effective numerical aperture NA = 0.01 and focal length f = 62 500λ0 (λ0 is the vacuum wavelength of the light confined within the cavity). The clear aperture of each mirror will therefore have a diameter of 1250λ0.