Combined effects of amplitude and phase variations on the irradiance in the image of a sinusoidal complex object are investigated for a partially coherent optical system. For this purpose, the general expression for the irradiance in the image of a sinusoidal complex object that has amplitude transmittance (C + A cos2π x0u) and phase distribution of the form exp(B cos2π x0u) is formulated for a partially coherent optical system. The image irradiance obtained in such an optical system consists of the fundamental frequency and a number of harmonics produced by the nonlinear modulation of the object. The characteristics of the image are assessed in terms of the image contrast, which is defined as the ratio of the coefficient of a harmonic component to the background. By use of the general formula, the image contrast of the fundamental and harmonic components up to sixth order can be calculated for the partially coherent optical system illuminated with bounded and annular illuminations. The typical nonlinear effect occurs in the low-spatial-frequency region. Especially, maxima of the contrast curves occur in the low-frequency region; they may be caused by nonlinearity of the system and effects of phase variations. Analysis and computation showed that, in image formation in a partially coherent optical system, phase variation in the object is extremely influential for the appearance of the image and its contrast.
Y. Ichioka, K. Yamamoto, and T. Suzuki, "Image of a sinusoidal complex object in a partially coherent optical system," J. Opt. Soc. Am. 65, 892-902 (1975)