Fresnel-Kirchhoff equation

Kirchhoff derived the Helmholtz-Kirchhoff theorem (Eq. (1.24)) from the Helmholtz equation (Eq. (1.22)) and Green s theorem (Eq. (1.23)). Consequently, the Fresnel-Kirchhoff equation containing the obliquity factor is given by Eq. (1.25),... 

Kirchhoff-Fresnel integro-differentiai equation analog to (5.26). For illustration, the near-held and far-held patterns of an unstable resonator of the type shown in Fig. 5.15a is compared with the diffraction pattern of a circular aperture. 

For the two unstable resonators of Fig. 5.17 the near-field pattern of the outcoupled wave is an annular ring (Fig. 5.18). The spatial far-field intensity distribution can be obtained as a numerical solution of the corresponding Kirchhoff-Fresnel integro-differential equation analog to (5.26). For illustration, the near-field and far-field patterns of an unstable resonator of the type shown in Fig. 5.17a is compared with the diffraction pattern of a circular aperture. 

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