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1.11.4 Point Spread Function (PSF) and Diffracted Aberrations

The point spread function - or PSF - is the relative function I(x, y)/IQ representing the intensity distribution of the light in the image region of the Gaussian focal plane.

• Spherical wavefront: In the basic case of a perfect spherical wavefront, the normalized diffraction function representing the intensity distribution over the focal plane is the PSF.

For a full circular aperture, the PSF is the Airy function I(r)/I0 in (1.87). If the circular aperture is a ring, then the PSF is ratio I(r)/I0 in (1.89). For a rectangular aperture of sides 2a and 2b, one shows that the PSF is

Important cases of diffracted images are, for instance, with a pupil whose transmission is a function (apodization), or with a polarized input beam. By introducing convenient parameters, the diffraction theory has been mostly developed in a scalar form, thus avoiding the difficulties of the general vectorial form.

• Aberrated wavefront: In the presence of a single primary aberration, many diffraction patterns from an object point have been recorded by K. Nienhuis [116], Francon et al. [61] and others for various aberration amplitudes. The diffraction theory applied to a single primary aberration was treated by F. Zernike and B. Nijboer [180], Nienhuis, Kingslake, who obtained isophotes in meridian sections of the beam and at the focal plane. The theoretical results are in accordance with the diffraction patterns that are observed for single primary aberrations (Fig. 1.32). Details on these analyses and the resulting spatial distributions of intensity are reviewed by Born and Wolf [17].

Hereafter theoretical results on the determination of an optimal diffraction imaging in the presence of primary aberrations allow the definition of optical tolerances.

1.11.5 Diffraction-Limited Criteria and Wavefront Tolerances

An important aspect for highly accurate optical systems is the achievement of a diffraction-limited imaging. A certain freedom in the polished shape and set up of the elements always exists, but these parameters must conform to strict criteria. The raytrace and manufacturing processes have to be matched with respect to criteria.

In the presence of aberrations the point spread function is very complicated in form, but when the aberrations are small the effect on the PSF is that the central intensity decreases, the half-width of the central maximum does not change whereas more light appears in the rings. Thus, we could set a tolerance level on that amount of aberration which produces such a just perceptible change. The first investigations in this field were developed by Rayleigh [123] and more systematically by Strehl [147].

Denoting IAber/I0 the point spread function delivered by an optical system with residual aberrations and ITh/I0 the theoretical PSF if that system would be perfect, Strehl suggested that an appropriate tolerance level for a drop in intensity of the central peak would be such as

Fig. 1.32 Diffraction patterns in the presence of primary aberrations. (Up) Primary spherical aberration at the least confusion focal plane w = A40P4 — § p2) with A40 = 1.4X, 3.7X, 8.4X, 17.5X. (Center) Primary coma in the Gaussian focal plane w = A31P3cos6 with A31 = 0.3X, X, 2.4X, 5X, 10X. (Down) Primary astigmatism at the least confusion plane w = A42P2cos26 with A42 = 1.4X, 2.7X, 3.5X, 6.5X. Image in the plane containing one of the two separated focal lines for w = 2.7Xp2cos20 (after K. Nienhuis [116])

Fig. 1.32 Diffraction patterns in the presence of primary aberrations. (Up) Primary spherical aberration at the least confusion focal plane w = A40P4 — § p2) with A40 = 1.4X, 3.7X, 8.4X, 17.5X. (Center) Primary coma in the Gaussian focal plane w = A31P3cos6 with A31 = 0.3X, X, 2.4X, 5X, 10X. (Down) Primary astigmatism at the least confusion plane w = A42P2cos26 with A42 = 1.4X, 2.7X, 3.5X, 6.5X. Image in the plane containing one of the two separated focal lines for w = 2.7Xp2cos20 (after K. Nienhuis [116])

S = lAber > 0.8 where S is the Strehl intensity ratio, (1.90a)

lTh a tolerance which is generally accepted.

For instance, if a defocus is considered as an aberration, a Strehl ratio S = 0.8 provides a tolerance in agreement with the early result of a X/4 defocus by Rayleigh. For Sphe 3, this Strehl value also corresponds to the quarter-wave Rayleigh criterion. However, the quarter-wave criterion do not satisfy S = 0.8 for Coma3, Astm3 or higher-order aberrations. Therefore, the Strehl definition has been preferred as an extension definition to the general case including all aberration types. ^ If the Strehl intensity ratio of an optical system is at least 0.8 or greater, then the system is well corrected and so-called "diffraction limited."

Considering the general form of the wavefront function, such as defined by (1.47) and (1.48), where (p,9) are the normalized coordinates of a point in a circular pupil, it has been shown by Marechal [103] (cf. also Born and Wolf [17] and Wetherell [168]) that the value of the Strehl ratio is

where the integrations are taken over the normalized area of the pupil (p e [0,1], 9e [0,2n]). When the aberrations are sufficiently small, this equation may be approximated by the two first terms of an expansion as

4n2 2 2

where AW^s and oj are the variances of the wavefront and phase aberration, respectively. When S = 0.8, the so-called Maréchal criterion [103] is oj = 0.2 [in rad2], (1.91a)

and from (1.90c), we obtain a mean square wavefront aberration, or wavefront variance,

thus a root mean square wavefront aberration value

which is valid whatever the aberration type. Hence, Maréchal's criterion provides an equivalence to the Strehl value S = 0.8, and may be enounced as follows.

^ If the root mean square departure of the wavefront to the best fit reference sphere does not exceed X /14, then the system is diffraction limited.

Now considering the wavefront tolerancing of various aberrations, we give hereafter the results published by Welford [167] in his Section "Optical Tolerances." For example, denoting a, ft and y some of the first coefficient al,n,mf}l of the wavefront function W as defined by (1.47) and (1.48), we may consider defocus as an aberration and write it a1p 2 where p = 1 at the pupil full aperture. For a defocus, the Strehl tolerance S = 0.8 leads to a AWptv wavefront deviation ax = ±0.25 X. (1.92a)

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