Abstract

We discuss the Strehl ratio of systems with a Gaussian pupil and determine the range of validity of its approximate expression based on the aberration variance. The results given are equally applicable to propagation of Gaussian beams. The uniform and weakly truncated pupils are considered as limiting cases of a Gaussian pupil. We show that the approximate expression for Strehl ratio in terms of the aberration variance yields a good estimate of the true value for a strongly truncated pupil but a much smaller value for a weakly truncated pupil.

© 2005 Optical Society of America

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  1. K. Strehl, “Über Luftschlieren und Zonenfehler,” Z. Instrumentenkd. 22, 213–217 (1902).
  2. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE Press, 2004).
  3. M. Born, E. Wolf, Principles of Optics, 7th ed. (Oxford, New York, 1999).
    [CrossRef]
  4. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. 72, 1258–1266 (1982).
    [CrossRef]
  5. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils: errata,” J. Opt. Soc. Am. A 10, 2092 (1993).
    [CrossRef]
  6. D. D. Lowenthal, “Maréchal intensity criteria modified for Gaussian beams,” Appl. Opt. 13, 2126–2133 (1974).
    [CrossRef] [PubMed]
  7. D. D. Lowenthal, “Maréchal intensity criteria modified for Gaussian beams: errata,” Appl. Opt. 13, 2774 (1974).
    [CrossRef]
  8. V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73, 860–861 (1983).
    [CrossRef]
  9. R. Herloski, “Strehl ratio for untruncated Gaussian beams,” J. Opt. Soc. Am. A 2, 1027–1030 (1985).
    [CrossRef]
  10. V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A 3, 470–485 (1986).
    [CrossRef]
  11. A. E. Siegman, An Introduction to Lasers and Masers (McGraw Hill, 1971).
  12. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (McGraw Hill, 1978).
  13. V. N. Mahajan, “Axial irradiance of a focused beam,” J. Opt. Soc. Am. A 22, XXX – XXXX (2005).
    [CrossRef]
  14. S. Szapiel, “Aberration balancing techniques for radially symmetric amplitude distributions; a generalization of the Maréchal approach,” J. Opt. Soc. Am. 72, 947–956 (1982).
    [CrossRef]
  15. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
    [CrossRef]
  16. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” Errata, J. Opt. Soc. Am. 71, 1408 (1981).
    [CrossRef]
  17. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” JOSA Communication, J. Opt. Soc. Am. A 1, 685 (1984).
    [CrossRef]
  18. V. N. Mahajan, “Zernike-Gauss polynomials for optical systems with Gaussian pupils,” Appl. Opt. 34, 8057–8059 (1995).
    [CrossRef] [PubMed]
  19. V. N. Mahajan, “Zernike polynomials and aberration balancing,” Proc. SPIE 5173, 1–7 (2003).
    [CrossRef]
  20. G. Martial, “Strehl ratio and aberration balancing,” J. Opt. Soc. Am. A 8, 164–170 (1991).
    [CrossRef]
  21. V. N. Mahajan, “Symmetry properties of aberrated point-spread functions,” J. Opt. Soc. Am. A 11, 1993–2003 (1994).
    [CrossRef]
  22. V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A 2, 833–846 (1985).
    [CrossRef]

2005 (1)

V. N. Mahajan, “Axial irradiance of a focused beam,” J. Opt. Soc. Am. A 22, XXX – XXXX (2005).
[CrossRef]

2003 (1)

V. N. Mahajan, “Zernike polynomials and aberration balancing,” Proc. SPIE 5173, 1–7 (2003).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1991 (1)

1986 (1)

1985 (2)

1984 (1)

1983 (1)

1982 (2)

1981 (2)

1974 (2)

1902 (1)

K. Strehl, “Über Luftschlieren und Zonenfehler,” Z. Instrumentenkd. 22, 213–217 (1902).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Oxford, New York, 1999).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (McGraw Hill, 1978).

Herloski, R.

Lowenthal, D. D.

Mahajan, V. N.

V. N. Mahajan, “Axial irradiance of a focused beam,” J. Opt. Soc. Am. A 22, XXX – XXXX (2005).
[CrossRef]

V. N. Mahajan, “Zernike polynomials and aberration balancing,” Proc. SPIE 5173, 1–7 (2003).
[CrossRef]

V. N. Mahajan, “Zernike-Gauss polynomials for optical systems with Gaussian pupils,” Appl. Opt. 34, 8057–8059 (1995).
[CrossRef] [PubMed]

V. N. Mahajan, “Symmetry properties of aberrated point-spread functions,” J. Opt. Soc. Am. A 11, 1993–2003 (1994).
[CrossRef]

V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils: errata,” J. Opt. Soc. Am. A 10, 2092 (1993).
[CrossRef]

V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A 3, 470–485 (1986).
[CrossRef]

V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A 2, 833–846 (1985).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” JOSA Communication, J. Opt. Soc. Am. A 1, 685 (1984).
[CrossRef]

V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73, 860–861 (1983).
[CrossRef]

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

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” Errata, J. Opt. Soc. Am. 71, 1408 (1981).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE Press, 2004).

Martial, G.

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw Hill, 1971).

Strehl, K.

K. Strehl, “Über Luftschlieren und Zonenfehler,” Z. Instrumentenkd. 22, 213–217 (1902).

Szapiel, S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Oxford, New York, 1999).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (8)

Proc. SPIE (1)

V. N. Mahajan, “Zernike polynomials and aberration balancing,” Proc. SPIE 5173, 1–7 (2003).
[CrossRef]

Z. Instrumentenkd. (1)

K. Strehl, “Über Luftschlieren und Zonenfehler,” Z. Instrumentenkd. 22, 213–217 (1902).

Other (4)

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE Press, 2004).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Oxford, New York, 1999).
[CrossRef]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw Hill, 1971).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (McGraw Hill, 1978).

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Figures (8)

Fig. 1
Fig. 1

Fractional power P trans transmitted by the exit pupil as a function of γ = a ω . The curve labeled “Beam” is for a Gaussian beam incident on a pupil that has uniform transmission, and that labeled “Apodized pupil” is for uniform illumination of a pupil with a Gaussian transmission.

Fig. 2
Fig. 2

Strehl ratio of a Gaussian beam as a function of the defocus wave aberration coefficient B d in units of wavelength λ, showing how it increases as γ increases. For a certain value of the Strehl ratio, the value of B d is obtained from this figure which, in turn, is used to obtain the corresponding value of the depth of focus from Eq. (21).

Fig. 3
Fig. 3

Strehl ratio of a beam aberrated by spherical aberration A s in units of wavelength λ and balanced by defocus aberration B d . (a) Uniform beam ( γ = 0 ) . (b) Gaussian beam with γ = 1 . (c) Weakly truncated Gaussian beam with γ = 3 .

Fig. 4
Fig. 4

Strehl ratio for balanced spherical aberration as a function of defocus aberration δ, which represents the deviation from optimum balancing of spherical aberration for minimum variance. Both A s and δ are in units of wavelength λ. (a) Uniform beam ( γ = 0 ) and δ = B d + A s . (b) Gaussian beam with γ = 1 and δ = B d + 0.933 A s . (c) Gaussian beam with γ = 3 and δ = B d + ( 4 γ ) A s .

Fig. 5
Fig. 5

Strehl ratio of beam aberrated by coma A c in units of wavelength λ. (a) Uniform beam ( γ = 0 ) . (b) Gaussian beam with γ = 1 . (c) Weakly truncated Gaussian beam with γ = 3 .

Fig. 6
Fig. 6

Strehl ratio of beam aberrated by astigmatism A a in units of wavelength λ. (a) Uniform beam ( γ = 0 ) . (b) Gaussian beam with γ = 1 . (c) Weakly truncated Gaussian beam with γ = 3 .

Fig. 7
Fig. 7

Strehl ratio of a beam aberrated by a primary aberration with and without balancing as a function of its standard deviation σ w in units of wavelength λ. (a) Uniform beam ( γ = 0 ) with a primary aberration. (b) Uniform beam with a balanced primary aberration. (c) Gaussian beam with γ = 1 and a primary aberration. (d) Gaussian beam with γ = 1 and a balanced primary aberration.

Fig. 8
Fig. 8

Strehl ratio of a Gaussian beam aberrated by a primary aberration compared with its value for a corresponding balanced primary aberration as a function of standard deviation σ w in units of wavelength λ. S, spherical; BS, balanced spherical; C, coma; BC, balanced coma; A, astigmatism; and BA, balanced astigmatism. (a) γ = 2 . (b) Weakly truncated beam with γ = 3 .

Tables (6)

Tables Icon

Table 1 Primary Aberrations and Their Standard Deviations for Optical Systems with Gaussian Pupils a

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Table 2 Balanced Primary Aberrations

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Table 3 Standard Deviation of Balanced Primary Aberrations

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Table 4 Factor by Which the Standard Deviation of a Seidel Aberration Is Reduced When Optimally Balanced With Other Aberrations

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Table 5 Diffraction Focus

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Table 6 Strehl Ratio Expressions for Uniform and Weakly Truncated Gaussian Beams in Terms of the Standard Deviation σ i of an Aberration a

Equations (25)

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A ( ρ ) = A 0 exp ( γ ρ 2 ) , 0 ρ 1 ,
γ = ( a ω ) 2
P inc = 2 S p A 0 2 0 exp ( 2 γ ρ 2 ) ρ d ρ = S p A 0 2 2 γ .
P p = 2 S p A 0 2 0 1 exp ( 2 γ ρ 2 ) ρ d ρ = ( S p 2 γ ) A 0 2 [ 1 exp ( 2 γ ) ] .
P trans = P p P inc = 1 exp ( 2 γ ) .
P inc = A 0 2 S p .
P trans = P p P inc = 1 exp ( 2 γ ) 2 γ .
A 0 2 = 2 γ P trans S p [ 1 exp ( 2 γ ) ] .
S = { γ π [ 1 exp ( γ ) ] } 2 0 1 0 2 π exp ( γ ρ 2 ) exp [ i Φ ( ρ , θ ) ] ρ d ρ d θ 2 ,
S 1 > ( 1 σ Φ 2 2 ) 2 ,
S 2 1 σ Φ 2 ,
S 3 exp ( σ Φ 2 ) ,
σ Φ 2 = Φ 2 Φ 2
Φ n = 0 1 0 2 π A ( ρ ) [ Φ ( ρ , θ ) ] n ρ d ρ d θ 0 1 0 2 π A ( ρ ) ρ d ρ d θ = γ π [ 1 exp ( γ ) ] 0 1 0 2 π exp ( γ ρ 2 ) [ Φ ( ρ , θ ) ] n ρ d ρ d θ
A ( ρ ) = ( 2 γ P trans S p ) 1 2 exp ( γ ρ 2 ) .
S = ( γ π ) 2 0 0 2 π exp ( γ ρ 2 ) exp [ i Φ ( ρ , θ ) ] ρ d ρ d θ 2 ,
Φ n = γ π 0 0 2 π exp ( γ ρ 2 ) [ Φ ( ρ , θ ) ] n ρ d ρ d θ ,
S = [ γ 1 exp ( γ ) ] 2 0 1 exp [ ( γ i B d ) x ] d x 2 = [ γ 1 exp ( γ ) ] 2 1 γ 2 + B d 2 [ 1 exp ( 2 γ ) 2 exp ( γ ) cos B d ] .
S = ( sin B d 2 B d 2 ) 2 ,
S = 1 1 + ( B d γ ) 2 ,
B d ( z ) = π N ( R z 1 ) ,
B d ( z ) = π N ( 1 z R ) .
Δ = 8 B d F 2 ,
S = [ γ 1 exp ( γ ) ] 2 0 1 exp ( γ x ) f ( x ) d x 2 ,
f ( x ) = { exp [ i ( A s x 2 + B d x ) ] Spherical + defocus J 0 ( A c x 3 2 + B t x 1 2 ) Coma + tilt exp [ i ( 0.5 A a + B d ) x ] J 0 ( 0.5 A a x ) Astigmatism + defocus } .

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