The Strehl ratio for imaging systems with circular and annular pupils aberrated by primary aberrations is considered in terms of the variance σΦ2 of the phase aberration across the pupil. Classical as well as balanced (Zernike) aberrations are considered. By comparing the actual numerical results with the approximate ones, we find that exp(−σΦ2) gives a better approximation for the Strehl ratio than does the Maréchal formula. Whereas the Maréchal formula underestimates the Strehl ratio, exp(−σΦ2) generally overestimates it, especially for annular pupils with a large obscuration.

© 1983 Optical Society of America

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  1. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. 72, 1258–1266 (1982).

1982 (1)

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Figures (3)

Fig. 1
Fig. 1

Strehl ratio for circular pupils ( = 0) aberrated with primary aberrations as a function of their standard deviation σw in units of optical wavelength λ. The approximate results S1, S2, and S3 are shown by the solid curves. The other curves show the results obtained by using the exact expressions for the various aberrations. Spherical, ….. coma, – – – – –; astigmatism, – · – · –. For large values of σw, the classical coma and classical astigmatism give a higher Strehl ratio than that given by the corresponding balanced aberration. For a given value of σw, the Strehl ratios for classical and balanced spherical aberrations are identical. σΦ = (2π/λ)σw.

Fig. 2
Fig. 2

Annular pupils with = 0.5. The Strehl ratio in the case of coma is practically independent of whether it is classical or balanced. For large values of σw, the Strehl ratio for classical astigmatism is larger than that for balanced astigmatism.

Fig. 3
Fig. 3

Annular pupils with = 0.75. For large values of σw, the balanced coma gives a higher Strehl ratio than that given by the classical coma. In the case of astigmatism, it is the classical aberration that gives a slightly higher Strehl ratio. Note that the curves for classical coma and classical astigmatism are practically identical.

Equations (3)

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S 1 ( 1 - σ Φ 2 / 2 ) 2 ,
S 2 1 - σ Φ 2 ,
S 3 exp ( - σ Φ 2 ) .