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Chapter 4:
Systems with Circular Pupils
Author(s): Virendra N. Mahajan
Published: 2013
DOI: 10.1117/3.927341.ch4
Optical systems generally have a circular pupil. The imaging elements of such systems also have a circular boundary. Therefore, they are also represented by circular pupils in fabrication and testing. As a result, the Zernike circle polynomials have been in widespread use since Zernike introduced them in his phase contrast method for testing circular mirrors. They are used in optical design and testing to understand the aberration content of a wavefront. They have also been used for analyzing the wavefront aberrations introduced by atmospheric turbulence on a wave propagating through it. We start this chapter with a brief discussion of the point-spread function (PSF) and the optical transfer function (OTF) of an aberration-free system with a circular pupil. We then consider the effect of primary aberrations on the Strehl ratio of an image. Since the Strehl ratio for small aberrations depends on the variance of an aberration, we balance a classical aberration of a certain order with those of lower orders to reduce its variance. The utility of the Zernike circle polynomial stems from the fact that they are not only orthogonal over a circular pupil, but they also uniquely represent the balanced classical aberrations yielding minimum variance over the pupil [3–6]. Because of their orthogonality, when a circular wavefront is expanded in terms of them, the value of a Zernike expansion coefficient is independent of the number of polynomials used in the expansion. Hence, one or more polynomial terms can be added or subtracted without affecting the other coefficients. The piston coefficient represents the mean value of the aberration function, and the variance of the function is given simply by the sum of the squares of the other expansion coefficients. Given the m-fold symmetry of a Zernike polynomial aberration, we discuss the symmetry of its interferogram, the corresponding aberrated PSF, the real and imaginary parts of the OTF, and the modulation transfer function (MTF). It is shown that the interferogram, the real part of the OTF, and the corresponding MTF are 2m-fold whether m is an even or an odd integer, but the PSF and the imaginary part of the OTF are m-fold when m is odd. Numerical examples are given to illustrate the Zernike aberrations isometrically, interferometrically, and by the corresponding PSFs, OTFs, and MTFs. Relationships between the coefficients of a power series expansion of an aberration function and the corresponding Zernike expansion coefficients are considered. In particular, we discuss how to obtain the Seidel coefficients from the Zernike coefficients of an aberration function. We illustrate by an example how wrong Seidel coefficients are obtained when using only the corresponding Zernike polynomials. Finally, we show how the Zernike coefficients of an aberration function over a circular pupil change as its diameter is reduced.
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Optical transfer functions

Modulation transfer functions

Point spread functions

Imaging systems


Zernike polynomials


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