Abstract

Maréchal’s treatment of tolerance theory shows that, in designing high-quality optical systems, we should aim at minimizing the variance E of the wave aberration. This suggests that the value of E may serve as a diffraction-based criterion of image quality in the fine-correction stage of automatic optical design. For this purpose, it would be desirable to know the range of states of poorer correction over which E may still be regarded as a useful criterion. In the present investigation, it is found that a necessary condition to be satisfied is that the Strehl ratio of the system should exceed 0.5.

© 1968 Optical Society of America

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References

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  1. K. Strehl, Z. Instrumentenk. 22, 213 (1902).
  2. A. Maréchal, Rev. Opt. 26, 257 (1947).
  3. A. Maréchal, thesis, University of Paris (1948). See also: M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1965), 3rd ed., p. 468; E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Publ. Co., Reading, Mass., 1963), p. 62.
  4. The shape of the pupil periphery for an extra-axial pencil was first approximated by a best-fitting ellipse, and then scaled to correspond to a unit-circle. W. B. King, Appl. Opt. 7, 197 (1968).
    [CrossRef] [PubMed]
  5. H. H. Hopkins, Proc. Phys. Soc. (London) B,  70, 1002 (1957).
    [CrossRef]
  6. Here we intend to investigate the Strehl ratio under the condition that the secondary spherical aberration coefficient is fixed, while relative amounts of primary spherical and defocusing are variable. For the even-aberration case, the scaling parameter f is numerically equal to W60.
  7. R. Richter, Z. Instrumentenk. 45, 1 (1925); E. H. Linfoot, Opt. Acta 10, 84 (1963).
  8. This special property also served to check the accuracy of the numerical-integration technique over the range of aberration values employed. Using M= 1 and N= 10, we have found that the numerical results for the Strehl ratio corresponding to these two cases agree to the seventh decimal place; the respective phases are, of course, different.
  9. W. B. King, Appl. Opt. 7, 489 (1968); R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1142 (1965).
    [CrossRef] [PubMed]
  10. H. H. Hopkins, Optica Acta,  13, 343 (1966).
    [CrossRef]

1968 (2)

1966 (1)

H. H. Hopkins, Optica Acta,  13, 343 (1966).
[CrossRef]

1957 (1)

H. H. Hopkins, Proc. Phys. Soc. (London) B,  70, 1002 (1957).
[CrossRef]

1947 (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

1925 (1)

R. Richter, Z. Instrumentenk. 45, 1 (1925); E. H. Linfoot, Opt. Acta 10, 84 (1963).

1902 (1)

K. Strehl, Z. Instrumentenk. 22, 213 (1902).

Hopkins, H. H.

H. H. Hopkins, Optica Acta,  13, 343 (1966).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. (London) B,  70, 1002 (1957).
[CrossRef]

King, W. B.

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

A. Maréchal, thesis, University of Paris (1948). See also: M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1965), 3rd ed., p. 468; E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Publ. Co., Reading, Mass., 1963), p. 62.

Richter, R.

R. Richter, Z. Instrumentenk. 45, 1 (1925); E. H. Linfoot, Opt. Acta 10, 84 (1963).

Strehl, K.

K. Strehl, Z. Instrumentenk. 22, 213 (1902).

Appl. Opt. (2)

Optica Acta (1)

H. H. Hopkins, Optica Acta,  13, 343 (1966).
[CrossRef]

Proc. Phys. Soc. (London) B (1)

H. H. Hopkins, Proc. Phys. Soc. (London) B,  70, 1002 (1957).
[CrossRef]

Rev. Opt. (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

Z. Instrumentenk. (2)

R. Richter, Z. Instrumentenk. 45, 1 (1925); E. H. Linfoot, Opt. Acta 10, 84 (1963).

K. Strehl, Z. Instrumentenk. 22, 213 (1902).

Other (3)

This special property also served to check the accuracy of the numerical-integration technique over the range of aberration values employed. Using M= 1 and N= 10, we have found that the numerical results for the Strehl ratio corresponding to these two cases agree to the seventh decimal place; the respective phases are, of course, different.

A. Maréchal, thesis, University of Paris (1948). See also: M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1965), 3rd ed., p. 468; E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Publ. Co., Reading, Mass., 1963), p. 62.

Here we intend to investigate the Strehl ratio under the condition that the secondary spherical aberration coefficient is fixed, while relative amounts of primary spherical and defocusing are variable. For the even-aberration case, the scaling parameter f is numerically equal to W60.

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Figures (13)

Fig. 1
Fig. 1

Plot of the Strehl ratio I vs B26 for different aberration-scaling factors f, with B46 = −1.5 (curves corresponding to f = 3, 5, 7, 9 are omitted).

Fig. 2
Fig. 2

Plot of the Strehl ratio I vs B26 for different aberration-scaling factors f with B46 = −1.25 (curves corresponding to f = 4, 6, 8, 9 are omitted).

Fig. 3
Fig. 3

Values of Imax (dashed curve) and Jmax (solid curve) are plotted for f = 1, 2, ⋯, 10, with B46 = −1.5.

Fig. 4
Fig. 4

Values of Imax (dashed curve) and Jmax (solid curve) are plotted for a range of f values, with B46 = −1.25.

Fig. 5
Fig. 5

Values of B26 corresponding to Imax are plotted for different f values, with B46 = −1.5. The upper curve is derived from Fig. 1. The reference line B26 = 0.6 satisfies the condition ∂E/∂B26 = 0.

Fig. 6
Fig. 6

Values of B26 corresponding to Imax are plotted for different f values, with B46 = −1.25. The upper curve is derived from Fig. 2. The reference line B20 = 0.35 satisfies the condition ∂E/∂B26 = 0.

Fig. 7
Fig. 7

Plot of the Strehl ratio I vs f.W11 for different aberration-scaling factors f, with W31 = −0.63λ, (curves corresponding to f = 2, 4, 6, 8, 9 are omitted).

Fig. 8
Fig. 8

Plot of the Strehl ratio I vs f.W11 for different aberration-scaling factors f, with W31 = −0.78λ, (curves corresponding to f = 2, 4, 6, 8, 9 are omitted).

Fig. 9
Fig. 9

Plot of the Strehl ratio I vs f.W11 for different aberration-scaling factors f, with W31 = −0.93λ (curves corresponding to f = 2, 4, 6, 8, 9 are omitted).

Fig. 10
Fig. 10

Values of Imax (dashed curve) and Jmax (solid curve) are plotted for a range of f values, with W31 = −0.78λ.

Fig. 11
Fig. 11

Values Imax and Jmax are plotted for different f values. The uppermost curve represents Imax values corresponding to W31 = −0.63λ. The middle curve shows Imax values with W31 = −0.93λ. The lowest curve indicates the Jmax values corresponding to W31 = −0.63λ and −0.93λ.

Fig. 12
Fig. 12

[W11]Imax/W51 vs f curves. Curve A corresponds to W31 = −0.93λ: the upper curve is plotted from Fig. 9 and the reference line has W11/W51 = 0.4538. Curve B corresponds to W31 = −0.78λ: The upper curve is derived from Fig. 8 and the reference line has W11/W51 = 0.30. Curve C corresponds to W31 = −0.63λ: the upper curve is plotted from Fig. 7 and the reference line has W11/W51 = 0.1462. The reference line in each case indicates the condition for ∂E/∂W11 = 0.

Fig. 13
Fig. 13

Plot of Strehl ratio I vs W20 for three different odd-aberrations, with W51 = 0.65λ. The uppermost curve has f = 4, W31 = −0.78λ, and W11 = 0.195λ. The middle curve has f = 4, W31 = 0.93λ, W11 = 0.299λ. The lowest curve has f = 6, W31 = −0.78λ, and W11 = 0.1965λ.

Equations (23)

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I = | Pupil exp [ i k W ( r , ϕ ) ] d A | 2 ,
d A = r · d r · d ϕ / Pupil r · d r · d ϕ .
I = | exp [ i k W ¯ ] Pupil exp { i k [ W ( r , ϕ ) W ¯ ] } d A | 2 .
I = | 1 ( 2 π 2 / λ 2 ) E | 2 ,
E = Pupil W 2 ( r , ϕ ) d A [ Pupil W ( r , ϕ ) d A ] 2
E W 20 = 0 or E W 11 = 0.
I = | 1 2 π t = 0 1 ϕ = 0 2 π exp [ i k f W ( t , ϕ ) ] d t d ϕ | 2 ,
t = t p ± α t , ϕ = ϕ q ± α ϕ .
I = | 2 α t α ϕ π p q exp [ i k f W ( t p , ϕ q ) ] · sin T T · sin Φ Φ | 2 ,
T = k α t f W t ( t p , ϕ q ) Φ = k α ϕ f W ϕ ( t p , ϕ q ) .
ϕ q = ( 2 q 1 2 M ) · π 2 , q = 1 , 2 , , M α ϕ = ( 1 2 M ) · π 2
t p = ( 2 p 1 ) / 2 N ; α t = 1 / 2 N .
W = W 20 r 2 + W 40 r 4 + W 60 r 6 ,
E = ( E ) even = W 20 2 12 + 4 W 40 2 45 + 9 W 60 2 112 + W 20 W 40 6 + W 40 W 60 6 + 3 W 20 W 60 20 .
W 60 = ± 4 λ B 46 = 1.5 B 26 = 0.6 ,
f W = f ( B 26 r 2 + B 46 r 4 + 1.0 r 6 ) .
f 2 E = f 2 [ B 26 2 12 + 4 B 46 2 45 + 9 112 + B 26 B 46 6 + B 46 6 + 3 B 26 20 ] .
J = [ 1 2 π 2 ( f 2 E ) ] 2
W = W ( r , ϕ ) = W 11 r cos ϕ + W 31 r 3 cos ϕ + W 51 r 5 cos ϕ ,
E = ( E ) odd = W 11 2 4 + W 31 2 8 + W 51 2 12 + W 11 W 31 3 + W 11 W 51 4 + W 31 W 51 5 .
W 51 = ± 2.6 λ , W 31 = 1.2 W 51 , W 11 = 0.3 W 51 ,
f W = f ( W 11 r + W 31 r 3 + 0.65 r 5 ) cos ϕ .
f W = f ( W 20 r 2 + W 11 r cos ϕ + W 31 r 3 cos ϕ + W 51 r 5 cos ϕ )