Abstract

Analytic formulas for the variance of an aberration of arbitrary order over a specified exit pupil with either uniform or untruncated Gaussian weighting are derived, and closed-form solutions are presented for the actual Strehl ratio of an untruncated Gaussian-beam system suffering from a primary aberration, except in the case of coma, for which an integral solution is given. These formulas are valid for an arbitrary magitude of the given primary aberration. It is shown that the aberration variance and Strehl ratio solutions for untruncated Gaussian-beam illumination depend on a reference-radius to beam-radius ratio, and judicious choice of this ratio allows one to apply the results of Strehl ratio calculations for uniformly illuminated systems to untruncated Gaussian-beam systems.

© 1985 Optical Society of America

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References

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  1. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73, 860–861 (1983).
    [CrossRef]
  4. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytic results for circular and annular pupils,” J. Opt. Soc. Am. 72, 1258–1266 (1982).
    [CrossRef]
  5. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
    [CrossRef]
  6. S. Szapiel, “Maréchal intensity formula for small-Fresnel-number systems,” Opt. Lett. 8, 327–329 (1983).
    [CrossRef] [PubMed]
  7. S. Szapiel, “Aberration balancing technique for radially symmetric amplitude distributions: a generalization of the Maréchal approach,” J. Opt. Soc. Am. 72, 947–956 (1982).
    [CrossRef]
  8. R. Barakat, “Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: generalization of Zernike polynomials,” J. Opt. Soc. Am. 70, 739–742 (1980).
    [CrossRef]
  9. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
    [CrossRef] [PubMed]
  10. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  11. Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [CrossRef]
  12. A. Yoshida, “Spherical aberration in beam optical systems,” Appl. Opt. 21, 1812–1816 (1982).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. S. M. Selby, CRC Standard Mathematical Tables, 23rd ed. (CRC, Cleveland, Ohio, 1975), p. 436, Eqs. (306) and (307).
  15. Ref. 14, p. 465, Eqs. (662) and (667).
  16. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), p. 303, Eqs. (7.4.22) and (7.4.23); p. 300, Eqs. (7.3.5) and (7.3.6).
  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 707, Eq. (6.611.1).
  18. Ref. 16, p. 486, Eq. (11.4.29).

1984 (1)

1983 (4)

1982 (4)

1981 (1)

1980 (1)

1974 (1)

1972 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), p. 303, Eqs. (7.4.22) and (7.4.23); p. 300, Eqs. (7.3.5) and (7.3.6).

Avizonis, P. V.

Barakat, R.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 707, Eq. (6.611.1).

Holmes, D. A.

Korka, J. E.

Li, Y.

Lowenthal, D. D.

Mahajan, V. N.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 707, Eq. (6.611.1).

Selby, S. M.

S. M. Selby, CRC Standard Mathematical Tables, 23rd ed. (CRC, Cleveland, Ohio, 1975), p. 436, Eqs. (306) and (307).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), p. 303, Eqs. (7.4.22) and (7.4.23); p. 300, Eqs. (7.3.5) and (7.3.6).

Szapiel, S.

Wolf, E.

Yoshida, A.

Appl. Opt. (5)

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Opt. Lett. (1)

Other (5)

S. M. Selby, CRC Standard Mathematical Tables, 23rd ed. (CRC, Cleveland, Ohio, 1975), p. 436, Eqs. (306) and (307).

Ref. 14, p. 465, Eqs. (662) and (667).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), p. 303, Eqs. (7.4.22) and (7.4.23); p. 300, Eqs. (7.3.5) and (7.3.6).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 707, Eq. (6.611.1).

Ref. 16, p. 486, Eq. (11.4.29).

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Tables (2)

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Table 1 Standard Deviation and Strehl Ratio for Primary Aberrations with Uniform Illuminationa

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Table 2 Standard Deviation and Strehl Ratio for Primary Aberrations with Untruncated Gaussian Illuminationa

Equations (26)

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S = | 0 2 π 0 δ g ( ρ ) exp [ j k Φ ( ρ , θ ) ] ρ d ρ d θ | 2 / | 2 π 0 δ g ( ρ ) ρ d ρ | 2 ,
S u = ( π α 2 ) 2 | 0 2 π 0 a exp [ j k Φ ( ρ , θ ) ] ρ d ρ d θ | 2 .
S g = ( π ω 2 ) 2 | 0 2 π 0 exp ( ρ 2 / ω 2 ) × exp [ j k Φ ( ρ , θ ) ] ρ d ρ d θ | 2 .
Φ ( ρ , θ ) = A n m ( ρ / ρ 0 ) n cos m θ ,
S 1 ( 1 ½ k 2 σ Φ 2 ) 2 ,
S 2 1 k 2 σ Φ 2 ,
S 3 exp ( k 2 σ Φ 2 ) ,
σ Φ 2 = Φ 2 Φ 2 ,
Φ n = 0 2 π 0 δ g ( ρ ) Φ n ( ρ , θ ) ρ d ρ d θ / 0 2 π 0 δ g ( ρ ) ρ d ρ d θ ,
Φ 2 = A n m 2 ( 2 m ) ! / [ ( n + 1 ) 2 2 m ( m ! ) 2 ]
Φ = 2 A n m ζ m / ( n + 2 ) ,
ζ m = { 0 m odd m ! / { 2 m [ ( m / 2 ) ! ] 2 } m even .
σ u , nm = A n m { ( 2 m ) ! / [ ( n + 1 ) 2 2 m ( m ! ) 2 ] 4 ζ m 2 / ( n + 2 ) 2 } 1 / 2 .
σ u , n 0 = A n 0 [ 1 / ( n + 1 ) 4 / ( n + 2 ) 2 ] 1 / 2 ,
Φ 2 = A nm 2 ( 1 / κ ) 2 n n ! ( 2 m ) ! / [ 2 2 m ( m ! ) 2 ]
Φ = A nm ( 1 / κ ) n Γ [ ( n + 2 ) / 2 ] ζ m ,
σ g , n m = A n m ( 1 / k ) n { n ! ( 2 m ) ! / [ 2 2 m ( m ! ) 2 ] Γ 2 [ ( n + 2 ) / 2 ] ζ m 2 } 1 / 2 .
( 2 / 3 5 ) A s
( 1 / 2 2 ) A c
| 0 1 J 0 ( k A c x 3 / 2 ) d x | 2
( 1 / 2 3 ) A d
2 5 ( 1 / κ ) 4 A s
3 ( 1 / κ ) 3 A c
| 0 exp ( x ) J 0 [ k A c ( 1 / κ ) 3 x 3 / 2 ] d x | 2
( 1 / 2 ) ( 1 / κ ) 2 A a
( 1 / 2 ) ( 1 / κ ) A t

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