Abstract

Imaging systems with circular and annular pupils aberrated by primary aberrations are considered. Both classical and balanced (Zernike) aberrations are discussed. Closed-form solutions are derived for the Strehl ratio, except in the case of coma, for which the integral form is used. Numerical results are obtained and compared with Maréchal’s formula for small aberrations. It is shown that, as long as the Strehl ratio is greater than 0.6, the Maréchal formula gives its value with an error of less than 10%. A discussion of the Rayleigh quarter-wave rule is given, and it is shown that it provides only a qualitative measure of aberration tolerance. Nonoptimally balanced aberrations are also considered, and it is shown that, unless the Strehl ratio is quite high, an optimally balanced aberration does not necessarily give a maximum Strehl ratio.

© 1982 Optical Society of America

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References

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  1. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981); J. Opt. Soc. Am. 71, 1408 (1981).
    [Crossref]
  2. Some of this work was presented at the 1981 Annual Meeting of the Optical Society of America, Kissimee, Florida, October 1981. See V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results,” J. Opt. Soc. Am. 71, 1561(A) (1981).
  3. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.
  4. A. Maréchal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. d’Opt. 26, 257–277 (1947). See also Ref. 3, p. 469.
  5. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. Thesis (University of Groningen, Groningen, The Netherlands, 1942), p. 17.
  6. See Ref. 3, p. 464.
  7. G. C. Steward, The Symmetrical Optical System (Cambridge U. Press, 1928), Chaps. VI and VII.
  8. The corresponding result for circular pupils has been given by B. R. A. Nijboer in “The diffraction theory of optical aberrations,” Physica 14, 590–608 (1949).
    [Crossref]
  9. V. N. Mahajan, “Luneburg apodization problem I,” Opt. Lett. 5, 267–269 (1980).
    [Crossref] [PubMed]
  10. D. A. Holmes, J. E. Korka, and P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
    [Crossref] [PubMed]
  11. A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
    [Crossref]
  12. M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
    [Crossref]
  13. Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [Crossref]
  14. Rayleigh, Philos. Mag. 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 432–435.
  15. Ref. 3, p. 468.
  16. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 207.
  17. R. Barakat, “Rayleigh wavefront criterion,” J. Opt. Soc. Am. 55, 572–573 (1965).
    [Crossref]
  18. R. Barakat, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965).
    [Crossref]
  19. See Ref. 5, p. 48.
  20. R. Barakat and A. Houston, “Transfer function of an annular aperture in the presence of spherical aberration,” J. Opt. Soc. Am. 55, 538–541 (1965). Strehl-ratio numbers in Table 3 of this paper are also in error. They need to be divided by (1 − ∊2)2.
    [Crossref]
  21. R. Barakat, “Diffraction effects of coma,” J. Opt. Soc. Am. 54, 1084–1088 (1964), Fig. 12.
    [Crossref]
  22. W. B. King, “Dependence of the Strehl ratio on the magnitude of the variance of the wave aberration,” J. Opt. Soc. Am. 58, 655–661 (1968).
    [Crossref]
  23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 483, Eq. 11.3.9.

1981 (2)

1980 (1)

1977 (1)

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

1976 (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[Crossref]

1972 (1)

1968 (1)

1965 (3)

1964 (1)

1949 (1)

The corresponding result for circular pupils has been given by B. R. A. Nijboer in “The diffraction theory of optical aberrations,” Physica 14, 590–608 (1949).
[Crossref]

1947 (1)

A. Maréchal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. d’Opt. 26, 257–277 (1947). See also Ref. 3, p. 469.

1879 (1)

Rayleigh, Philos. Mag. 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 432–435.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 483, Eq. 11.3.9.

Arimoto, A.

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[Crossref]

Avizonis, P. V.

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.

Gusinow, M. A.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

Holmes, D. A.

Houston, A.

King, W. B.

Korka, J. E.

Li, Y.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[Crossref]

Mahajan, V. N.

Maréchal, A.

A. Maréchal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. d’Opt. 26, 257–277 (1947). See also Ref. 3, p. 469.

Nijboer, B. R. A.

The corresponding result for circular pupils has been given by B. R. A. Nijboer in “The diffraction theory of optical aberrations,” Physica 14, 590–608 (1949).
[Crossref]

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. Thesis (University of Groningen, Groningen, The Netherlands, 1942), p. 17.

Palmer, M. A.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

Rayleigh,

Rayleigh, Philos. Mag. 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 432–435.

Riley, M. E.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 483, Eq. 11.3.9.

Steward, G. C.

G. C. Steward, The Symmetrical Optical System (Cambridge U. Press, 1928), Chaps. VI and VII.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 207.

Wolf, E.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.

Appl. Opt. (1)

J. Opt. Soc. Am. (6)

Opt. Acta (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[Crossref]

Opt. Commun. (1)

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[Crossref]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

Philos. Mag. (1)

Rayleigh, Philos. Mag. 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 432–435.

Physica (1)

The corresponding result for circular pupils has been given by B. R. A. Nijboer in “The diffraction theory of optical aberrations,” Physica 14, 590–608 (1949).
[Crossref]

Rev. d’Opt. (1)

A. Maréchal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. d’Opt. 26, 257–277 (1947). See also Ref. 3, p. 469.

Other (9)

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. Thesis (University of Groningen, Groningen, The Netherlands, 1942), p. 17.

See Ref. 3, p. 464.

G. C. Steward, The Symmetrical Optical System (Cambridge U. Press, 1928), Chaps. VI and VII.

Some of this work was presented at the 1981 Annual Meeting of the Optical Society of America, Kissimee, Florida, October 1981. See V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results,” J. Opt. Soc. Am. 71, 1561(A) (1981).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.

Ref. 3, p. 468.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 207.

See Ref. 5, p. 48.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 483, Eq. 11.3.9.

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Figures (10)

Fig. 1
Fig. 1

Strehl ratio for spherical aberration.

Fig. 2
Fig. 2

Strehl ratio for balanced spherical aberration.

Fig. 3
Fig. 3

Strehl ratio for coma.

Fig. 4
Fig. 4

Strehl ratio for balanced coma.

Fig. 5
Fig. 5

Strehl ratio for astigmatism.

Fig. 6
Fig. 6

Strehl ratio for balanced astigmatism.

Fig. 7
Fig. 7

Percent error 100(SS1)/S as a function of S. (a) = 0, (b) = 0.50, (c) = 0.75. Spherical, —; coma, -·-; balanced coma, – - – astigmatism, ····; balanced astigmatism, - - - - -. In (a), the curves for spherical and balanced spherical aberrations are identical. In (b), the curve for balanced spherical aberration is shown by -·-·-. In (c), this curve exists for S ≳ 0.81 and overlaps the others in this region.

Fig. 8
Fig. 8

Variation of a primary aberration coefficient for 10% error with obscuration ratio.

Fig. 9
Fig. 9

Strehl ratio for a quarter-wave aberration as a function of obscuration ratio. Ai = λ/4. S, spherical; BS, balanced spherical; C, coma; BC, balanced coma; A, astigmatism; BA, balanced astigmatism. The right-hand-side vertical scale is only for coma.

Fig. 10
Fig. 10

Strehl ratio for circular pupils ( = 0) aberrated by spherical aberration as a function of the deviation of focus from its optimum balancing value (Δ = AdAs) for several values of the aberration coefficient As. The curves are symmetric about the origin. The right-hand-side vertical scale is for As = 3λ, 4λ.

Tables (6)

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Table 1 Standard Deviation and Strehl Ratio for Primary Aberrationsa

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Table 2 Aberration Coefficient, Absolute Peak Value, and Peak-to-Peak Value for Primary Aberrations ( = 0)

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Table 3 Strehl Ratio for a Quarter-Wave Absolute Peak Value of a Primary Aberration (|Wp| = λ/4)a

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Table 4 Strehl Ratio for a Quarter-Wave Peak-to-Peak Value of a Primary Aberration (Wpp = λ/4)a

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Table 5 Aberration Coefficient Ai, Absolute Peak Value |Wp|, and Peak-to-Peak Value Wpp, All in Units of Wavelength, for a Strehl Ratio of 0.80 ( = 0)

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Table 6 Strehl Ratio for Annular Pupils Aberrated with One Wave of Spherical Aberration Optimally Balanced with Defocus for Annular and Circular Pupils

Equations (29)

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I ( r ) = 1 λ 2 R 2 A ( ρ ) exp [ i Φ ( ρ ) ] exp ( - 2 π i r · ρ / λ R ) d ρ 2 ,
S = I ( 0 ) Φ / I ( 0 ) Φ = 0 = exp ( i Φ ) 2 ,
f = A ( ρ ) f ( ρ ) d ρ / A ( ρ ) d ρ .
S = exp [ i ( Φ - Φ ) ] 2 .
S = cos ( Φ - Φ ) 2 - sin ( Φ - Φ ) 2
S cos ( Φ - Φ ) 2 ,
S ( 1 - ½ σ Φ 2 ) 2 ,
S 1 ( 1 - ½ σ Φ 2 ) 2
S 2 1 - σ Φ 2 .
A ( ρ ) = 1 , ρ 1 = 0 , otherwise ,
S = 1 π 2 ( 1 - 2 ) 2 | 1 0 2 π exp [ i Φ ( ρ , θ ) ] ρ d ρ d θ 2 .
σ Φ 2 = 1 π ( 1 - 2 ) 1 0 2 π Φ 2 ( ρ , θ ) ρ d ρ d θ - [ 1 π ( 1 - 2 ) 1 0 2 π Φ ( ρ , θ ) ρ d ρ d θ ] 2 .
Φ ( ρ ) = A ˙ s ρ 4 - A d ρ 2 ,
S = [ π / 2 A s ( 1 - 2 ) 2 ] × { [ C ( a + ) + C ( a - ) ] 2 + [ S ( a + ) + S ( a - ) ] 2 } ,
a ± = [ ( 1 - 2 ) A s ± Δ ] / ( 2 π A s ) 1 / 2
Δ = A d - A s ( 1 + 2 ) .
Φ A ( ρ ; ) = 2 π [ ρ 4 - ( 1 + 2 ) ρ 2 ]
Φ B ( ρ ; ) = 2 π ( ρ 4 - ρ 2 ) ,
S A ( ) = [ C 2 ( 1 - 2 ) + S 2 ( 1 - 2 ) ] / ( 1 - 2 ) 2
S B ( ) = { [ C ( 1 ) + C ( 1 - 2 2 ) ] 2 + [ S ( 1 ) + S ( 1 - 2 2 ) ] 2 } / 4 ( 1 - 2 ) 2 .
Φ ( ρ , θ ) = A ρ 2 cos 2 θ
Φ ( ρ , θ ) = ½ A ρ 2 ( 1 + cos 2 θ ) .
S = 1 π 2 ( 1 - 2 ) 2 | 1 exp ( i A ρ 2 / 2 ) ρ d ρ × 0 2 π exp ( ½ i A ρ 2 cos 2 θ ) d θ | 2 .
S = 4 ( 1 - 2 ) 2 | 1 exp ( i A ρ 2 / 2 ) J 0 ( A ρ 2 / 2 ) ρ d ρ | 2 ,
0 z exp ( i t ) J 0 ( t ) d t = z exp ( i z ) [ J 0 ( z ) - i J 1 ( z ) ] ,
S = ( 1 - 2 ) - 2 { H 2 ( A / 2 ) + 2 H 2 ( 2 A / 2 ) - 2 2 H ( A / 2 ) H ( 2 A / 2 ) cos [ ( 1 / 2 ) ( 1 - 2 ) A - α ( A / 2 ) + α ( 2 A / 2 ) ] } ,
H ( a ) = [ J 0 2 ( a ) + J 1 2 ( a ) ] 1 / 2
α ( a ) = tan - 1 [ J 1 ( a ) / J 0 ( a ) ] .
S = J 0 2 ( A / 2 ) + J 1 2 ( A / 2 ) .