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Chapter 2:
Optical Systems with Circular Pupils
In this chapter, we consider the imaging properties of a system with a circular exit pupil by applying the general formulas derived in Chapter 1. We derive expressions for its point-spread function (PSF), optical transfer function (OTF), and encircled, ensquared, and excluded powers. The effect of aberrations on a PSF is first studied in terms of its central value, i.e., its Strehl ratio. An exact expression is obtained for the Strehl ratio and the results for primary aberrations are compared with those obtained from the approximate expressions based on the phase variance of the aberration across the pupil derived in Chapter 1. The tolerance for primary aberrations are given for a certain value of the Strehl ratio. Balanced primary aberrations are also discussed and they are identified with the Zernike circle polynomials. A defocused PSF is considered next and it is shown that systems with large Fresnel numbers, such as photographic systems, have a very small depth of focus. However, systems with small Fresnel numbers, e.g., a laser transmitter, have a large depth of focus. Focused and collimated beams are discussed, and the concept of near- and far-field distances is introduced. It is shown that if the Fresnel number of the beam focused on a target is small, the axial irradiance is maximum at a point that is closer to the focusing optics. However, maximum central irradiance on a target at a given distance is obtained when a beam is focused on it, even though a larger irradiance is obtained at a closer point. Aberrated PSFs for rotationally symmetric aberrations are considered, and numerical results are obtained that show that the size of their central spot is practically independent of the aberration. Symmetry properties of the aberrated PSFs are considered next. Full PSFs for primary aberrations are shown graphically and pictorially. A brief comparison of the diffraction and geometrical PSFs is also given. The line of sight of an optical system is identified with the centroid of its PSF and discussed for aberrated systems. It is shown that only coma-type aberrations change the line of sight. The OTFs for primary aberrations are discussed, and the concept of contrast reversal is described. The OTF aberrated by coma is complex, and we give its real and imaginary parts as well as its modulus and phase. Aberration tolerances are given for a certain value of the Hopkins ratio. The geometrical OTF is also discussed and approximate expressions for it are obtained for the primary aberrations. It gives reasonably accurate results in the region of very low spatial frequencies, a region of practical interest when the MTF of an aberrated system at high frequencies is practically negligible. It is shown that the MTF averaged over all angular orientations of a spatial frequency vector is maximum when the variance of the ray aberration, or the root mean square (rms) radius of the PSF with respect to its centroid, is minimum. The diffraction OTF is shown to approach the geometrical OTF in the limit of zero wavelength. A summary of the differences between the diffraction and geometrical OTFs is also given.
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