In this chapter, we describe briefly how the primary aberrations of an optical system can be observed. The emphasis of our discussion is on how to recognize a primary aberration and not on how to measure it precisely. Since the optical frequencies are very high (10¹⁴ - 10¹⁵ Hz), optical wavefronts, aberrated or not, cannot be observed directly; optical detectors simply do not respond at these frequencies. We have seen in Chapter 8 that the image of a monochromatic point object formed by an aberrated system is characteristically different for a different aberration. Another and more powerful way to recognize an aberration is to form an interferogram by combining two parts of a light beam, one of which has been transmitted through the system.

12.2 PRIMARY ABERRATIONS

Consider an optical imaging system with a circular exit pupil of radius a. Letting (r, θ) be the polar coordinates of a point in the plane of its exit pupil, the functional form of the primary phase aberrations may be written

(12-1)

(12-2)

(12-3)

(12-4)

(12-5)

where A_i or B_i is a peak aberration coefficient representing the maximum value of the corresponding aberration across the pupil, and ρ = r/a is a normalized radial variable. When Φ(ρ, θ) = 0 for a certain point object, the wavefront passing through the center of the exit pupil is spherical centered at its Gaussian image point. Let its radius of curvature be R. For an aberrated system, Φ(ρ, θ) represents the optical deviation of the wavefront at a point (ρ, θ) from being spherical.

In Eq. (12-1), when B_d = 0, the aberration is spherical. Nonzero B_d implies that the aberration is combined with defocus; i.e., the aberration is not with respect to a reference sphere centered at the Gaussian image point but with respect to another sphere centered at a distance z from the plane of the exit pupil according to Eq. (8-6). As discussed in Chapter 7, the reference sphere is centered at the marginal image point, center of the circle of least confusion, and the point midway between the marginal and Gaussian image points when B_d / A_s = -2, -1.5, and -1, respectively. The midway point corresponds to minimum variance of the aberration and, therefore, to maximum Strehl ratio (for small aberrations), as may be seen by comparing the aberration thus obtained with the Zernike circle polynomial Z⁰₄(ρ).