Theory of laser beam apodization with a graded random phase window.
Analysis indicates that graded random phase modulation can be used to apodize a laser beam. In the case of an obscuration or hard edge, it can prevent the formation of Fresnel diffraction ripples. For example, suppose laser radiation of wavelength lambda interacts with a central obscuration of halfwidth a followed immediately by a window. If the exit surface of the window is randomly modulated with a Gaussian amplitude transverse correlation length l and a mean square amplitude t 2(r) that decreases smoothly from a peak height at the obstacle of ~(2) with a transverse scale length L > a >> l, then the Fresnel diffraction ripples normally produced by the obscuration are eliminated. Following the apodizer the laser beam consists of two components. One is widely scattered with mean intensity I (s). The other is essentially tunscattered and apodized with mean intensity I (a). The scattered light has an angular divergence proportional to lambda/l. The apodizer intensity transmission is T(a) = exp[-alpha(2)t (2)(r)], where alpha = 2pir(n - 1)/lambda and n is the window refractive index. Tayloring t (2)(r) can produce desired apodization profiles for hard edges and obscurations. Interference between the scattered and unscattered radiation produces intensity fluctuations on the laser beam that have the characteristics of laser speckle. The magnitude of these fluctuations is greatest near the apodizer and at the edge of the apodized beam. The local intensity probability density of the fluctuations is approximately the modified Rician density. Using this distribution the probability of intensity fluctuations exceeding a given value I/I (a) is calculated for several values of r = I (a)/I (s). It is found that for high power applications minimization of optical damage requires r greater, similar 100. This may be achieved by minimizing the correlation length l of the apodizer surface modulation, spatial filtering the beam, and/or placing optical components a sufficient distance from the apodizer.