Abstract

The effect of fourth-, sixth-, and eighth-order balanced spherical aberrations on the incoherent point-spread function of an optical imaging system with a circular pupil is considered. It is shown that the location of the first minimum remains practically unchanged and its value remains close to zero as aberrations are introduced into the system. Thus, the central Airy disk maintains its size and distinction. Moreover, the aberrations reduce the irradiance distribution inside the Airy disk quite uniformly. The central irradiance, i.e., the Strehl ratio, can be determined quite accurately from the phase aberration variance according to Sexp(σΦ2). Thus, the aberrated spread functions and encircled energy for a given aberration can be determined very quickly from the aberration-free results by multiplying them with the Strehl ratio. For further simplicity, the spread functions are approximated by a Gaussian function appropriately scaled by the Strehl ratio. The approximation is quite good for points lying within a circle of radius roughly half that of the Airy disk. Defocused but otherwise aberration-free spread functions are also considered. It is shown that results similar to those for spherical aberrations are obtained but over a narrower range of Strehl ratio as well as distance from the center of the spread functions.

© 1983 Optical Society of America

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References

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  1. V. N. Mahajan, J. Opt. Soc. Am. 72, 1258 (1982).
    [CrossRef]
  2. V. N. Mahajan, J. Opt. Soc. Am. 73, 860 (1983).
    [CrossRef]
  3. Some of this work was presented at the 1982 Annual Meeting of the Optical Society of America, Tucson, Arizona, October 1982.SeeV. N. Mahajan, J. Opt. Soc. Am. 72, 1812A (1982).
    [CrossRef]
  4. R. Barakat, J. Opt. Soc. Am. 51, 152 (1961);a factor of 4 is missing on the right-hand side of Eqs. (3.1)–(3.8) in this paper.
    [CrossRef]
  5. R. Barakat, M. V. Morello, J. Opt. Soc. Am. 54, 235 (1964).
    [CrossRef]
  6. J. H. Rosen, “Diffraction Patterns Produced By Focused Laser Beams,” Rand Corporation Report No. R-925-ARPA (April1972).
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 482.
  8. R. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965), p. 247.
  9. Ref.8, p. 250.
  10. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., New York, 1944), p. 134;Ref. 17, p. 484.
  11. R. Barakat, A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).
    [CrossRef]
  12. R. Barakat, J. Opt. Soc. Am. 54, 38 (1964).
    [CrossRef]
  13. R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, Methods and Applications, B. R. Frieden, Ed. (Springer, Heidelberg, 1980), pp. 35–80.
    [CrossRef]
  14. B. R. A. Nijboer, Physica 23, 605 (1947).
    [CrossRef]
  15. Ref. 7, Sec. 9.2.
  16. V. N. Mahajan, J. Opt. Soc. Am. 71, 75 (1981);J. Opt. Soc. Am.71, 1408 (1981).
    [CrossRef]
  17. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 887.Only 1-D integral is considered here. The formula for the 2-D integral can be obtained by repeating the formula for a 1-D integral.
  18. Ref. 7, p. 396.
  19. Ref. 7, p. 398.
  20. Ref. 7, Sec. 8.8.
  21. R. E. Stephens, L. E. Sutton, J. Opt. Soc. Am. 58, 1001 (1968).
    [CrossRef]
  22. J. C. Dainty, Opt. Commun. 1, 176 (1969).
    [CrossRef]
  23. V. N. Mahajan, Appl. Opt. 17, 3329 (1978).
    [CrossRef] [PubMed]

1983 (1)

1982 (1)

1981 (1)

1978 (1)

1969 (1)

J. C. Dainty, Opt. Commun. 1, 176 (1969).
[CrossRef]

1968 (1)

1964 (2)

1963 (1)

1961 (1)

1947 (1)

B. R. A. Nijboer, Physica 23, 605 (1947).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 887.Only 1-D integral is considered here. The formula for the 2-D integral can be obtained by repeating the formula for a 1-D integral.

Barakat, R.

Born, M.

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

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965), p. 247.

Dainty, J. C.

J. C. Dainty, Opt. Commun. 1, 176 (1969).
[CrossRef]

Houston, A.

Mahajan, V. N.

Morello, M. V.

Nijboer, B. R. A.

B. R. A. Nijboer, Physica 23, 605 (1947).
[CrossRef]

Rosen, J. H.

J. H. Rosen, “Diffraction Patterns Produced By Focused Laser Beams,” Rand Corporation Report No. R-925-ARPA (April1972).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 887.Only 1-D integral is considered here. The formula for the 2-D integral can be obtained by repeating the formula for a 1-D integral.

Stephens, R. E.

Sutton, L. E.

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., New York, 1944), p. 134;Ref. 17, p. 484.

Wolf, E.

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

Appl. Opt. (1)

J. Opt. Soc. Am. (8)

Opt. Commun. (1)

J. C. Dainty, Opt. Commun. 1, 176 (1969).
[CrossRef]

Physica (1)

B. R. A. Nijboer, Physica 23, 605 (1947).
[CrossRef]

Other (12)

Ref. 7, Sec. 9.2.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 887.Only 1-D integral is considered here. The formula for the 2-D integral can be obtained by repeating the formula for a 1-D integral.

Ref. 7, p. 396.

Ref. 7, p. 398.

Ref. 7, Sec. 8.8.

Some of this work was presented at the 1982 Annual Meeting of the Optical Society of America, Tucson, Arizona, October 1982.SeeV. N. Mahajan, J. Opt. Soc. Am. 72, 1812A (1982).
[CrossRef]

J. H. Rosen, “Diffraction Patterns Produced By Focused Laser Beams,” Rand Corporation Report No. R-925-ARPA (April1972).

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

R. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965), p. 247.

Ref.8, p. 250.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., New York, 1944), p. 134;Ref. 17, p. 484.

R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, Methods and Applications, B. R. Frieden, Ed. (Springer, Heidelberg, 1980), pp. 35–80.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Variation of an aberration Φn(ρ) with ρ. σΦ = 1 for each aberration in this figure.

Fig. 2
Fig. 2

Strehl ratio as a function of wave aberration standard deviation σw. S2, S4, S6, and S8 represent Strehl ratios corresponding to aberrations Φ2, Φ4, Φ6, and Φ8, respectively. S2, —; S4,…; S6,- - -; S8,–·–·–. S g = exp ( σ Φ 2 ) and S m = ( 1 σ Φ 2 / 2 ) 2 represent the Gaussian and Maréchal approximations to the Strehl ratio, respectively.

Fig. 3
Fig. 3

Aberrated PSFs and encircled energy for Φ4 corresponding to Strehl ratios of 0.8, 0.6, 0.4, 0.2 and 0.1. Aberration-free (S = 1) curves are included for comparison. The PSF approximated by a Gaussian function is also included. The parameters r and r0 are in units of λF, where F = R/D is the focal ratio of the system. Airy radius is 1.22 in these units.

Fig. 4
Fig. 4

Aberrated PSFs for Φ4 normalized to unity at the center and corresponding normalized encircled energy. S = 1, —; S = 0.8, – – –; S = 0.6, -·-; S = 0.4, - - -; S = 0.2,…; S = 0.1; -·-·-·-. The Gaussian PSF and encircled energy are also included.

Fig. 5
Fig. 5

Same as Fig. 4 except that the aberration is Φ6.

Fig. 6
Fig. 6

Same as Fig. 4 except that the aberration is Φ8.

Fig. 7
Fig. 7

Aberration-free PSF, encircled energy, and their Gaussian approximations.

Fig. 8
Fig. 8

Same as Fig. 4 except that the aberration is Φ2 and r and r0 are in units of λz/D, where z is the distance of the observation (target) plane from the pupil plane. The radius of curvature of the reference sphere is R or the beam is focused at a distance R. Note that SS2.

Equations (43)

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I ( r ) = ( I 0 / λ 2 R 2 ) | | ρ | a exp [ i Φ ( ρ ) ] exp ( 2 π i r · ρ / λ R ) d ρ | 2 ,
I ( r ) = ( 2 π / λ R ) 2 I 0 | 0 a exp [ i Φ ( ρ ) ] J 0 ( 2 π r ρ / λ R ) ρ d ρ | 2 ,
r n = r / λ F ,
ρ n = ρ / a ,
I n ( r n ) = I ( r ) / [ P A / λ 2 R 2 ] ,
I n ( r n ) = 4 | 0 1 exp [ i Φ n ( ρ n ) ] J 0 ( π r n ρ n ) ρ n d ρ n | 2 .
I ( r ) = 4 | 0 1 exp [ i Φ ( ρ ) ] J 0 ( π r ρ ) ρ d ρ | 2 .
I ( r ) = 4 0 1 0 1 ρ s cos [ Φ ( ρ ) Φ ( s ) ] J 0 ( π r ρ ) J 0 ( π r s ) d ρ d s .
S = 4 0 1 0 1 ρ s cos [ Φ ( ρ ) Φ ( s ) ] d ρ d s .
E ( r 0 ) = ( π 2 / 2 ) 0 r 0 I ( r ) r d r .
E ( r 0 ) = 2 π 2 0 1 0 1 ρ s cos [ Φ ( ρ ) Φ ( s ) ] Q ( ρ , s ; r 0 ) d ρ d s ,
Q ( ρ , s ; r 0 ) = 0 r 0 J 0 ( π r s ) J 0 ( π r s ) r d r = ( r 0 2 / 2 ) [ J 0 2 ( π ρ r 0 ) + J 1 2 ( π ρ r 0 ) ] if ρ = s ,
= [ r 0 / π ( ρ 2 s 2 ) ] [ ρ J 1 ( π ρ r 0 ) J 0 ( π s r 0 ) s J 1 ( π s r 0 ) J 0 ( π ρ r 0 ) ] if ρ s ,
E ( r 0 ) ( π r 0 / 2 ) 2 S ,
Φ 4 ( ρ ) = 5 ( 6 ρ 4 6 ρ 2 + 1 ) σ Φ ,
Φ 6 ( ρ ) = 7 ( 20 ρ 4 30 ρ 4 + 12 ρ 2 1 ) σ Φ ,
Φ 8 ( ρ ) = 3 ( 70 ρ 8 140 ρ 6 + 90 ρ 4 20 ρ 2 + 1 ) σ Φ .
S 4 = [ C 2 ( b ) + S 2 ( b ) ] / b ,
b = 3 5 σ Φ / π ,
C ( b ) = 0 b cos ( π x 2 / 2 ) d x ,
S ( b ) = 0 b sin ( π x 2 / 2 ) d x .
0 1 0 1 f ( ρ , s ) d ρ d s = ( 1 / 4 ) i = 1 N j = 1 N w i w j f ( 1 + x i 2 , 1 + x j 2 ) ,
w i = 2 1 x i 2 [ d P N ( x i ) d x ] 2 .
0 1 0 1 f ( ρ , s ) d ρ d s = ( 1 / 4 ) i = 1 N w i 2 f ( 1 + x i 2 , 1 + x i 2 ) + ( 1 / 2 ) i = 2 N j = 1 i 1 w i w j f ( 1 + x i 2 , 1 + x j 2 ) .
S g exp ( σ Φ 2 ) .
I g ( r ) = ( 1 / 2 π σ 2 ) exp ( r 2 / 2 σ 2 ) ,
I ( r ) = ( π / 4 ) [ 2 J 1 ( π r ) / π r ] 2 ,
σ = 2 / π .
I g ( r ) = ( π / 4 ) exp [ ( π r / 2 ) 2 ] .
E g ( r 0 ) = 2 π 0 r 0 I g ( r ) rdr
= 1 exp [ ( π r 0 / 2 ) 2 ] .
I g ( r ; S ) = ( π / 4 ) S exp [ ( π r / 2 ) 2 ] ,
E g ( r 0 ; S ) = S { 1 exp [ ( π r 0 / 2 ) 2 ] } .
I g ( r ; σ Φ ) = ( π / 4 ) exp ( σ Φ 2 ) exp [ ( π r / 2 ) 2 ] ,
E g ( r 0 ; σ Φ ) = exp ( σ Φ ) 2 { 1 exp [ ( π r 0 / 2 ) 2 ] } ,
S = I g ( 0 ; σ Φ ) / I g ( 0 ; 0 ) .
I ( r ) = [ 2 J 1 ( π r ) / π r ] 2 .
E ( r 0 ) = 1 J 0 2 ( π r 0 ) J 1 2 ( π r 0 ) .
Φ 2 ( ρ ) = π λ ( 1 z 1 R ) ρ 2 , 0 ρ a .
I ( r ; z ) = ( 2 π / λ z ) 2 I 0 | 0 a exp [ i Φ 2 ( ρ ) ] J 0 ( 2 π r ρ / λ z ) ρ d ρ | 2 .
Φ 2 ( ρ ) = 3 ( 2 ρ 2 1 ) σ Φ , 0 ρ 1 ,
σ Φ = π 2 3 λ a 2 | 1 z 1 R |
S 2 = [ sin ( 3 σ Φ ) / 3 σ Φ ] 2 .

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