Abstract

Hopkins’s treatment of tolerance theory shows that, in designing quality optical systems, we should aim at minimizing the variance K of a wave-aberration difference function. This suggests that the value of K may serve as a diffraction-based criterion of image quality in the fine-correction stage of automatic optical design. For this purpose, it would be desirable to know the range of states of poorer correction over which K may still be regarded as a useful criterion. On the basis of the present investigation, it appears that a necessary condition to be satisfied is that the relative modulation of the system should exceed the range 0.60–0.69, depending on the type of wave aberration and the azimuth of the grating test object considered.

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  1. H. H. Hopkins, Opt. Acta 13, 343 (1966).
  2. K. Strehl, Z. Instrumentenk. 22, 213 (1902).
  3. A. Maréchal, Rev. Opt. 26, 257 (1947).
  4. W. B. King, J. Opt. Soc. Am. 58, 655 (1968).
  5. W. B. King, Appl. Opt. 7, 489 (1968).
  6. H. H. Hopkins, Proc. Phys. Soc. (London) 70B, 449 (1957).
  7. When M (s,ψ) > 0.8, there is no appreciable relative phase shifting between frequencies up to s = 0.2; the image is merely shifted as a whole by an amount V¯/λ; so for frequencies s< 0.2, it suffices to impose the criterion solely to modulation reduction.
  8. H. H. Hopkins, Proc. Phys. Soc. (London) 70B, 1002 (1957).
  9. The shape of the pupil periphery for an extra-axial pencil was first approximated by a best-fitting ellipse and then scaled to correspond to a unit circle. W. B. King, Appl. Opt. 7, 197 (1968).
  10. Am, n coefficients have been tabulated for a variety of values of s byA. M. Goodbody in Summer School, Appl. Opt., Imperial College, London (1959).
  11. The Am, n coefficients required for our present purpose constitute the lower-order coefficients of a more general case defined in Eq. (17) of a paper byW. B. King and J. Kitchen, Appl. Opt. 7, 1193 (1968).
  12. Here we intend to investigate the relative modulation under the condition that for given values of the secondary-spherical-aberration coefficient, the relative amounts of primary spherical aberration and defocusing are variable.
  13. The plots for the frequencies s = 0.04 and 0.1 were not presented here owing to space limitation.
  14. Cm, n coefficients have been tabulated for a variety of values of s, and for ψ = 0 and 90°, byA. M. Goodbody. See Ref. 10.
  15. The Cm, n coefficients required for our present purpose constitute the lower-order coefficients of a more general case defined in Eq. (18) of a paper byW. B. King and J. Kitchen. See Ref. 11.
  16. The plots for the frequencies s = 0.015, 0.04, and 0.1 are not presented here, owing to space limitation.
  17. Note that the actual wave aberrations corresponding to ψ = 0 and ψ = 90° cases are quite different.
  18. W. B. King and J. Kitchen, Appl. Opt. 7, 1193 (1968).

Goodbody, A. M.

Am, n coefficients have been tabulated for a variety of values of s byA. M. Goodbody in Summer School, Appl. Opt., Imperial College, London (1959).

Cm, n coefficients have been tabulated for a variety of values of s, and for ψ = 0 and 90°, byA. M. Goodbody. See Ref. 10.

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) 70B, 449 (1957).

H. H. Hopkins, Proc. Phys. Soc. (London) 70B, 1002 (1957).

H. H. Hopkins, Opt. Acta 13, 343 (1966).

King, W. B.

W. B. King, J. Opt. Soc. Am. 58, 655 (1968).

W. B. King, Appl. Opt. 7, 489 (1968).

The shape of the pupil periphery for an extra-axial pencil was first approximated by a best-fitting ellipse and then scaled to correspond to a unit circle. W. B. King, Appl. Opt. 7, 197 (1968).

The Am, n coefficients required for our present purpose constitute the lower-order coefficients of a more general case defined in Eq. (17) of a paper byW. B. King and J. Kitchen, Appl. Opt. 7, 1193 (1968).

The Cm, n coefficients required for our present purpose constitute the lower-order coefficients of a more general case defined in Eq. (18) of a paper byW. B. King and J. Kitchen. See Ref. 11.

W. B. King and J. Kitchen, Appl. Opt. 7, 1193 (1968).

Kitchen, J.

W. B. King and J. Kitchen, Appl. Opt. 7, 1193 (1968).

The Cm, n coefficients required for our present purpose constitute the lower-order coefficients of a more general case defined in Eq. (18) of a paper byW. B. King and J. Kitchen. See Ref. 11.

The Am, n coefficients required for our present purpose constitute the lower-order coefficients of a more general case defined in Eq. (17) of a paper byW. B. King and J. Kitchen, Appl. Opt. 7, 1193 (1968).

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

Strehl, K.

K. Strehl, Z. Instrumentenk. 22, 213 (1902).

Other (18)

H. H. Hopkins, Opt. Acta 13, 343 (1966).

K. Strehl, Z. Instrumentenk. 22, 213 (1902).

A. Maréchal, Rev. Opt. 26, 257 (1947).

W. B. King, J. Opt. Soc. Am. 58, 655 (1968).

W. B. King, Appl. Opt. 7, 489 (1968).

H. H. Hopkins, Proc. Phys. Soc. (London) 70B, 449 (1957).

When M (s,ψ) > 0.8, there is no appreciable relative phase shifting between frequencies up to s = 0.2; the image is merely shifted as a whole by an amount V¯/λ; so for frequencies s< 0.2, it suffices to impose the criterion solely to modulation reduction.

H. H. Hopkins, Proc. Phys. Soc. (London) 70B, 1002 (1957).

The shape of the pupil periphery for an extra-axial pencil was first approximated by a best-fitting ellipse and then scaled to correspond to a unit circle. W. B. King, Appl. Opt. 7, 197 (1968).

Am, n coefficients have been tabulated for a variety of values of s byA. M. Goodbody in Summer School, Appl. Opt., Imperial College, London (1959).

The Am, n coefficients required for our present purpose constitute the lower-order coefficients of a more general case defined in Eq. (17) of a paper byW. B. King and J. Kitchen, Appl. Opt. 7, 1193 (1968).

Here we intend to investigate the relative modulation under the condition that for given values of the secondary-spherical-aberration coefficient, the relative amounts of primary spherical aberration and defocusing are variable.

The plots for the frequencies s = 0.04 and 0.1 were not presented here owing to space limitation.

Cm, n coefficients have been tabulated for a variety of values of s, and for ψ = 0 and 90°, byA. M. Goodbody. See Ref. 10.

The Cm, n coefficients required for our present purpose constitute the lower-order coefficients of a more general case defined in Eq. (18) of a paper byW. B. King and J. Kitchen. See Ref. 11.

The plots for the frequencies s = 0.015, 0.04, and 0.1 are not presented here, owing to space limitation.

Note that the actual wave aberrations corresponding to ψ = 0 and ψ = 90° cases are quite different.

W. B. King and J. Kitchen, Appl. Opt. 7, 1193 (1968).

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