This is the famous result obtained by Rayleigh, known as quarter-wave Rayleigh criterion [123] stating that an optical system can be considered as perfect if the wavefront is included between two concentric surfaces separated by A/4.

Similarly, denoting ftp4 the primary spherical aberration, the AWpiv tolerance limit is ft = ±0.24 A, (1.92b)

and if this aberration is balanced but choosing the best focus, i.e. considering a W(p, 9) term of the form a1p2 + ftp4, the AWptv tolerance limit is

In the same way, we can find the tolerance limit for other aberrations and balance them with appropriate terms. Table 1.6 displays some AWptv wavefront tolerance limits satisfying either the Strehl criterion S = 0.8 or equivalently the Marechal criterion AWrms = X/14.

Table 1.6 Best sphere fit wavefront tolerances AWptv for obtaining diffraction limited images, i.e. for S = 0.8 or equivalently AWrms = A/14. (after Welford [167])


Aberration polynomial

Coefficients at tolerance limit

Cv 1, defocus



= ±0.25 A

Sphe 3, primary spherical aber.



= ±0.24 A

Sphe 3 with choice of best focus Cv 1

«lP2 + ßlP4

ßl al

= ±0.95 A = T0.95a

Sphe 5 with choice of best Sphe 3 and Cv 1

«lP2 + ßlP4 + 7iP6

Yl ßl al

= ±3.47 A = t5.62a = ±2.24 A

Coma 3, primary coma

ß2p3 cos 6


= ±0.20 A

Astm 3, primary astigmatism



= ±0.l7 A

• Note: Many useful optical systems do not require a diffraction-limited quality in the purpose they have been designed for. With those systems, e.g. photographic objectives, optics of television cameras, etc, the specification is to achieve a given number resolving elements, typically 400-1000 elements in each of x,y directions. Their optical tolerances are based on the concept of the optical transfer function.

1.12 Some Image Processor Options

A review on the theoretical and practical limitation that an astronomical system may reach in observational astrophysics is given by Lena, Rouan et al. [96] for the whole electromagnetic spectrum. Restraining hereafter to image processors in the range from extreme-ultraviolet to infrared, there are many options for which each one induces a single- or multi-telescope system, a preferred telescope design and an optimal telescope focal length. Some associated imaging systems may have two or more processors in series such as, for instance, an atmospheric dispersion compensator plus an adaptive optics module followed by an imaging spectrograph. We hereafter follow some descriptions by Richardson [131] on image processor options.

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