Abstract

Most optical systems have rotational symmetry. For such systems, we establish a method of finding (a) the maximum attainable modulation transfer function (MTF) at arbitrary frequency ω0; and (b) the required pupil function U(ω0; ρ). Physically, the latter are absorbing films in the pupil of diffraction-limited optics. The method of solution is numerical and iterative, based upon the Newton–Raphson algorithm. Solutions (a) and (b) are established at frequencies ω0 = 0.1, 0.2, ⋯, 0.9 (× optical cutoff). The computed (a) are correct to ±0.0001 over all ω0 indicated. Quantities (b) have an average error over each pupil of ±0.002 for frequencies ω0 ≤ 0.5. With 0.6 ≤ ω0 ≤ 0.9, the error is ±0.01. The curve of maximum MTF(ω0) seems sufficiently smooth to allow for accurate interpolation. Solutions (a) and (b) were also found over the finer subdivision ω0 = 0.05, 0.1, 0.15, ⋯, 0.8 with slightly less accuracy than above, in order to allow for interpolation of pupils U(ω0; ρ) over values ω0. This seems possible for 0.05 ≤ ω0 ≤ 0.40. The maximum MTF(ω0) shows appreciable gain (e.g., 8% at ω0 = 0.2) over the MTF for uncoated, diffraction-limited optics at all ω0 except in the intermediate region 0.4 ≤ ω0 ≤ 0.6. However, in the high-frequency band 0.5 ≤ ω0 ≤ 1.0, the maximum MTF(ω0) shows little gain over the MTF due to an uncoated, diffraction-limited pupil with the proper central obstruction. The light loss due to each U(ω0; ρ) may be measured by the total energy transmittance and the Strehl flux ratio. These are plotted against ω0, and indicate moderate light loss.

© 1969 Optical Society of America

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References

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  1. Limitations of this method are discussed by P. G. Roetling, E. A. Trabka, and R. E. Kinzly, J. Opt. Soc. Am. 58, 342 (1968).
    [CrossRef]
  2. B. R. Frieden, J. Opt. Soc. Am. 58, 1107 (1968).
  3. J. A. MacDonald, Proc. Phys. Soc. (London) 72, 749 (1958).
    [CrossRef]
  4. W. Lukosz, J. Opt. Soc. Am. 52, 827 (1962).
    [CrossRef]
  5. B. P. Hildebrand, J. Opt. Soc. Am. 56, 12 (1966).
    [CrossRef]
  6. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass., 1963).
  7. R. Barakat, J. Opt. Soc. Am. 54, 920 (1964).
    [CrossRef]
  8. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw–Hill Book Co., New York, 1956), p. 343.
  9. P. Jacquinot and B. Roizen–Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964).

1968 (2)

1966 (1)

1964 (1)

1962 (1)

1958 (1)

J. A. MacDonald, Proc. Phys. Soc. (London) 72, 749 (1958).
[CrossRef]

Barakat, R.

Frieden, B. R.

B. R. Frieden, J. Opt. Soc. Am. 58, 1107 (1968).

Hildebrand, B. P.

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw–Hill Book Co., New York, 1956), p. 343.

Jacquinot, P.

P. Jacquinot and B. Roizen–Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964).

Kinzly, R. E.

Lukosz, W.

MacDonald, J. A.

J. A. MacDonald, Proc. Phys. Soc. (London) 72, 749 (1958).
[CrossRef]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass., 1963).

Roetling, P. G.

Roizen–Dossier, B.

P. Jacquinot and B. Roizen–Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964).

Trabka, E. A.

J. Opt. Soc. Am. (5)

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

J. A. MacDonald, Proc. Phys. Soc. (London) 72, 749 (1958).
[CrossRef]

Other (3)

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass., 1963).

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw–Hill Book Co., New York, 1956), p. 343.

P. Jacquinot and B. Roizen–Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964).

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Figures (3)

Fig. 1
Fig. 1

Optimal pupil U(ω0; ρ) (solid) and resulting MTF (ω0; ω) (dashed), for subdivision of frequencies ω0 = 0.1, 0.2, ⋯, 0.9 × optical cutoff. Pupils U are apodizers at ω0 ≤ 0.5, but with nonzero amplitude at the pupil rim. At ω0 > 0.5 the pupils are superresolvers, most of the enhancement of MTF (ω0) being due to the central obstruction shown. The dotted curves are the MTF due to uncoated, circular, diffraction-limited optics. Comparison with the MTF(ω0; ω) shows that the latter are enhanced over an appreciable region about each frequency ω0.

Fig. 2
Fig. 2

Comparison of maximum MTF(ω0, curve (1), with other relevant MTF curves; (2) Maximum possible MTF without constraint of rotational symmetry, due to Lukosz; (3) Asymptotic solution to our problem, for small ω0, due to MacDonald; (4) MTF due to uncoated, circular, diffraction-limited optics; (5) (dashed) for frequencies ω0 > 0.5, MTF due to preceding case with added central obstruction of radius ω0ρ0.

Fig. 3
Fig. 3

Light loss for each pupil U(ω0; ρ) with M = 5, as measured by the total pupil power transmittance T, and the Strehl flux ratio S. Curves T and S for the case M = 50 differ from the above by a maximum of 0.03 at any ω0.

Tables (2)

Tables Icon

Table I Numerical comparison of MTF’s for values of ω0. Maximum MTF(ω0) is shown for M = 50; estimated error (departure from case M → ∞) is ±0.0001 maximum at any ω0 shown. For comparison, values are shown for uncoated, diffraction-limited optics and for a simple obstruction in the uncoated pupil in the high-frequency band 0.5 ≤ ω0 ≤ 0.9. Figures in the first and third MTF columns are nearly identical, indicating that coatings U(ω0; ρ) in the annulus ω0ρ0ρ ≤ 2ρ0 have little effect at these high frequencies.

Tables Icon

Table II Solution B and maximum MTF, for case M = 5, at fine subdivision of ω0. Values of maximum MTF differ from those for M = 50 by, at most, 0.0003 (cf. Table I). Optimal pupils U(ω0; ρ) are generated from the B through Eqs. (2).

Equations (13)

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U ( ω 0 ; ρ ) = 0             for             ρ < ω 0 - ρ 0 ,
ω 0 > ρ 0 .
ρ 1 = { ω 0 - ρ 0 if ω 0 > ρ 0 0 if ω 0 ρ 0 .
U ( ω 0 ; ρ ) = { m = 0 M - 1 b m J 0 ( μ m ρ ) , for ρ 1 ρ ρ 0 , 0 for ρ < ρ 1 and ρ > ρ 0 .
J 1 ( μ m ρ 0 ) = 0 ,             m = 1 , 2 , , M - 1
b 0 , b 1 , , b M - 1 by b ,
MTF ( ω 0 ) / b m b = B = 0 ,             m = 0 , 1 , , M - 1.
W = W ( b ) m = 0 M - 1 [ MTF ( ω 0 ) / b m ] 2 .
Δ b m - W · W / b m / n = 0 M ( W / b n ) 2 , m = 0 , 1 , , M - 1.
0 θ d ϕ U ( ω 0 ; ρ ) = π MTF ( ω 0 ) U ( ω 0 ; ρ ) ,
ρ = ( ρ 2 + ω 0 2 - 2 ρ ω 0 cos ϕ ) 1 2
θ = { cos - 1 [ ( 2 ρ ω 0 ) - 1 ( ρ 2 + ω 0 2 - ρ 0 2 ) ] for ρ ρ 0 - ω 0 π for ρ ρ 0 - ω 0 .
2 π ρ 1 ρ 0 d ρ ρ .