Abstract

The relation between the Strehl ratio and the wave-front variance is analyzed from the point of view of aberration balancing. For some given set of aberrations the Strehl ratio is calculated along two different parametric paths in the aberration space: one corresponding to the minimum wave-front variance and the other to the maximum Strehl ratio. The case of the combination of different orders of coma, spherical aberration, and astigmatism are shown. A simple graphic is presented to show, at constant wave-front variance, the departure of the Strehl ratio from a constant value. The wave-front-balancing functions corresponding to the maximum Strehl ratio are shown to be discontinuous or stepwise functions.

© 1991 Optical Society of America

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References

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  1. A. Maréchal, M. Françon, Diffraction, Structure des Images (Masson, Paris, 1970), p. 107; Maréchal and Françon wrote that, for D > 0.75, the error on the exact D is not more than 0.01.
  2. W. B. Wetherell, “The calculation of image quality,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Orlando, Fla., 1980), Vol. VIII, pp. 172–315.
  3. 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]
  4. V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,”J. Opt. Soc. Am. 73, 860–861 (1983).
    [CrossRef]
  5. W. B. King, “Dependance of the Strehl ratio of the magnitude of the variance of the wave aberration,”J. Opt. Soc. Am. 58, 655–661 (1968).
    [CrossRef]
  6. D. Kessler, “Image quality criteria in the presence of moderately large aberrations,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).
  7. R. V. Shack, “Fifth-order aberration,” lecture presented at the Optical Sciences Center, University of Arizona, Tucson, Ariz., 1988–1989.
  8. See, for example, S. C. Johnston, “An investigation into the consequence of classifying orthogonal aberrations by degree,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1988); or 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]
  9. First named by K. Strehl, “Definitionshelligkeit,” Z. Instrumentenk. 22, 213 (1902).
  10. See Ref. 2, p. 206.
  11. R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 35–80.
    [CrossRef]
  12. See for more recent references M. Mansuripur, “Certain computational aspects of the vector diffraction,” J. Opt. Soc. Am. A 6, 784–805 (1989).
    [CrossRef]
  13. S. Szapiel, “Point-spread function computation: quasi-digital method,” J. Opt. Soc. Am. A 2, 3–5 (1985).
    [CrossRef]
  14. R. Barakat, A. Houston, “Diffraction effects of coma,”J. Opt. Soc. Am. 54, 1084–1088 (1964), principally Fig. 12.
    [CrossRef]

1989 (1)

See for more recent references M. Mansuripur, “Certain computational aspects of the vector diffraction,” J. Opt. Soc. Am. A 6, 784–805 (1989).
[CrossRef]

1985 (1)

1983 (1)

1982 (1)

1968 (1)

1964 (1)

1902 (1)

First named by K. Strehl, “Definitionshelligkeit,” Z. Instrumentenk. 22, 213 (1902).

Barakat, R.

R. Barakat, A. Houston, “Diffraction effects of coma,”J. Opt. Soc. Am. 54, 1084–1088 (1964), principally Fig. 12.
[CrossRef]

R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 35–80.
[CrossRef]

Françon, M.

A. Maréchal, M. Françon, Diffraction, Structure des Images (Masson, Paris, 1970), p. 107; Maréchal and Françon wrote that, for D > 0.75, the error on the exact D is not more than 0.01.

Houston, A.

Johnston, S. C.

See, for example, S. C. Johnston, “An investigation into the consequence of classifying orthogonal aberrations by degree,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1988); or 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]

Kessler, D.

D. Kessler, “Image quality criteria in the presence of moderately large aberrations,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

King, W. B.

Mahajan, V. N.

Mansuripur, M.

See for more recent references M. Mansuripur, “Certain computational aspects of the vector diffraction,” J. Opt. Soc. Am. A 6, 784–805 (1989).
[CrossRef]

Maréchal, A.

A. Maréchal, M. Françon, Diffraction, Structure des Images (Masson, Paris, 1970), p. 107; Maréchal and Françon wrote that, for D > 0.75, the error on the exact D is not more than 0.01.

Shack, R. V.

R. V. Shack, “Fifth-order aberration,” lecture presented at the Optical Sciences Center, University of Arizona, Tucson, Ariz., 1988–1989.

Strehl, K.

First named by K. Strehl, “Definitionshelligkeit,” Z. Instrumentenk. 22, 213 (1902).

Szapiel, S.

Wetherell, W. B.

W. B. Wetherell, “The calculation of image quality,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Orlando, Fla., 1980), Vol. VIII, pp. 172–315.

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

See for more recent references M. Mansuripur, “Certain computational aspects of the vector diffraction,” J. Opt. Soc. Am. A 6, 784–805 (1989).
[CrossRef]

S. Szapiel, “Point-spread function computation: quasi-digital method,” J. Opt. Soc. Am. A 2, 3–5 (1985).
[CrossRef]

Z. Instrumentenk. (1)

First named by K. Strehl, “Definitionshelligkeit,” Z. Instrumentenk. 22, 213 (1902).

Other (7)

See Ref. 2, p. 206.

R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 35–80.
[CrossRef]

A. Maréchal, M. Françon, Diffraction, Structure des Images (Masson, Paris, 1970), p. 107; Maréchal and Françon wrote that, for D > 0.75, the error on the exact D is not more than 0.01.

W. B. Wetherell, “The calculation of image quality,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Orlando, Fla., 1980), Vol. VIII, pp. 172–315.

D. Kessler, “Image quality criteria in the presence of moderately large aberrations,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

R. V. Shack, “Fifth-order aberration,” lecture presented at the Optical Sciences Center, University of Arizona, Tucson, Ariz., 1988–1989.

See, for example, S. C. Johnston, “An investigation into the consequence of classifying orthogonal aberrations by degree,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1988); or 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]

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Figures (10)

Fig. 1
Fig. 1

Strehl ratio for primary coma W31 and tilt W11 along a path of constant variance ωconstant (black curve) with the corresponding Strehl ratio D eff (lighter curve) and the Strehl approximation D a (lightest curve): (a) D a= 0.40, ωv = 0.152348; (b) D a= 0.25, ωv = 0.187391.

Fig. 2
Fig. 2

Strehl ratio for secondary coma W51 and primary coma W31 (with a tilt set for the minimum wave-front variance) along a path of constant variance ωconstant (black curve) with the corresponding Strehl ratio D eff (lighter curve) and the Strehl approximation D a (lightest curve): (a) D a= 0.40, ωv = 0.152348; (b) D a= 0.25, ωv, = 0.187391.

Fig. 3
Fig. 3

Strehl ratio for primary spherical aberration W40 and defocus W20 along a path of constant variance ωconstant (black curve) with the corresponding Strehl ratio D eff (lighter curve) and the Strehl approximation D a (lightest curve): (a) D a= 0.40, ωv = 0.152348; (b) D a= 0.25, ωv = 0.187391.

Fig. 4
Fig. 4

Strehl ratio for primary astigmatism W22 and field curvature W220 along a path of constant variance ωconstant (black curve) with the corresponding Strehl ratio D eff (lighter curve) and the Strehl approximation D a (lightest curve): (a) D a= 0.40, ωv = 0.152348; (b) D a= 0.25, ωv = 0.187391.

Fig. 5
Fig. 5

Mapping of the Strehl ratio D for the primary coma and tilt. The thick curve in the contour plot corresponds to the path of maximum Strehl ratio.

Fig. 6
Fig. 6

Mapping of the Strehl ratio D for the secondary and primary coma. The tilt is set so that the wave-front variance is a minimum. The thick curve in the contour plot corresponds to the path of maximum Strehl ratio max D.

Fig. 7
Fig. 7

Mapping of the Strehl ratio D for the secondary and primary coma. The tilt is set so that, over this variable, the Strehl lies on a global maximum. The thick curve in the contour plot corresponds to the path of maximum Strehl ratio max D.

Fig. 8
Fig. 8

Mapping of the Strehl ratio D for the primary spherical aberration and the defocus. The thick curve in the contour plot corresponds to the path of maximum Strehl ratio max D.

Fig. 9
Fig. 9

Mapping of the Strehl ratio D for the primary astigmatism and the defocus (here set equal to the field curvature W220). The thick curve in the contour plot corresponds to the path of maximum Strehl ratio max D.

Fig. 10
Fig. 10

All the Strehl ratio values D obtained for Fig. 9 are plotted here as a function of the Petzval curvature ζP.

Tables (3)

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Table 1 Coefficients of the Ellipse of Constant ωV,

Tables Icon

Table 2 Equations of the Paths of Minimum ωv and Maximum D

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Table 3 Relation between the Form of W and the Line of Local Strehl Ratio Maximum

Equations (15)

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W = j , n , m W k , l , m H k ρ l cos ϑ m ,
W l , m = k W k , l , m H k .
ω v 2 = 0 1 0 2 π W ( ρ , ϑ ) 2 d ρ d ϑ - [ 0 1 0 2 π W ( ρ , ϑ ) d ρ d ϑ ] 2 ,
ω v 2 = l , m l , m [ ν ( m + m ) l + l + 2 - ν ( m ) + ν ( m ) ( l + 2 ) ( l + 2 ) ] W l , m W l , m ,
ν ( m ) = { 0 if m is odd m ! 2 m - 1 { m 2 ! } 2 if m is even .
D = 1 π 2 { 0 1 0 2 π exp [ i 2 π W ( ρ , ϑ ) ] ρ d ρ d ϑ } 2 .
D 1 - 4 π 2 ω v 2 .
D exp ( - 4 π 2 ω v 2 ) .
ω v 2 = 1 4 ( W 11 + 2 3 W 31 + 1 2 W 51 ) 2 + 1 72 ( W 31 + 12 10 W 51 ) 2 + 1 1200 W 51 2
ω v 2 = 1 12 ( W 20 + W 40 ) 2 + 1 180 W 40 2
ω v 2 = 1 12 ( W 20 + W 22 ) 2 + 1 16 W 22 2
ω v 2 = W k l 2 a 2 + W k l 2 b 2 ,
W k l { 0 1 0 2 π exp [ i k W ( ρ , ϑ ) ] ρ d ρ d ϑ } 2 = 0.
PSF coma ( r , φ ) = C { 0 1 1 2 π exp [ - 2 π i r ρ cos ( ψ - φ ) ] ρ d ρ d ψ } 2 ,
ρ P = ( W 222 - 2 W 220 ) λ = ζ P λ .

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