Abstract

A set of equations is developed which yields the constructional parameters of three-mirror all-reflecting optical systems. An equation whose factors allow the shape of the image surface to be controlled is also derived. A method of optimizing the performance of three-mirror systems by varying the inputs to the design equations is described, and the results are compared with those obtained through a conventional numerical design optimization. The technique described is shown to be markedly superior to the usual optimization method of varying the constructional parameters of the system.

© 1978 Optical Society of America

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References

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  1. M. Paul, Rev. Opt. 14, 169 (1935).
  2. N. J. Rumsey, Optical Instruments and Techniques (Oriel, Newcastle upon Tyne, 1970), p. 516.
  3. J. G. Baker, IEEE Trans. Aerosp. Electron. Syst. AES-5, 261 (1969).
    [CrossRef]
  4. R. V. Shack, A. B. Meinel, J. Opt. Soc. Am. 56, 545 (1966).
  5. N. J. Rumsey, “Telescopic System Utilizing Three Axially Aligned Substantially Hyperbolic Mirrors,” U.S. Patent3,460,886 (1969).
  6. W. B. Wetherell, M. P. Rimmer, Appl. Opt. 11, 2817 (1972).
    [CrossRef] [PubMed]
  7. Both the coefficients and factors of the Petzval condition equation are formed through sums and differences of quantities having large absolute values. Double precision arithmetic may be required to obtain solutions which are not numerical noise.
  8. D. Korsch, J. Opt. Soc. Am. 63, 667 (1973).
    [CrossRef]
  9. G. A. and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 25.
  10. P. N. Robb, J. Opt. Soc. Am. 66, 1037 (1976).
    [CrossRef]
  11. R. Fletcher, C. M. Reeves, Comput J. 7, No. 2, 149 (1964).
    [CrossRef]
  12. L. W. Cornwall, A. K. Rigler, Appl. Opt. 7, 1659 (1972).
    [CrossRef]
  13. ACCOS-V, a proprietary product of Scientific Calculations, Inc., Rochester, N.Y. (1977). Both the damped least squares and Gram-Schmidt orthogonolization methods were used.
  14. J. Meiron, Appl. Opt. 7, 667 (1968).
    [CrossRef] [PubMed]
  15. Note that the image surface introduces vignetting at half-field angles greater than approximately 0.7°. The design illustrated in Fig. 5 was intended to operate with a line field, i.e., as a scanning system. The vignetting produced by the focal plane in this case was insignificant.
  16. The cost to design these systems is very modest; most of the calculations required are to evaluate the figure of merit. Using the merit function of Ref. 10 and computing the optical aberration coefficients out through the seventh order, the time per configuration averaged 15 msec on a Univac 1110 computer. This works out to a cost of 0.54 cents per design configuration.

1976

1973

1972

1969

J. G. Baker, IEEE Trans. Aerosp. Electron. Syst. AES-5, 261 (1969).
[CrossRef]

1968

1966

R. V. Shack, A. B. Meinel, J. Opt. Soc. Am. 56, 545 (1966).

1964

R. Fletcher, C. M. Reeves, Comput J. 7, No. 2, 149 (1964).
[CrossRef]

1935

M. Paul, Rev. Opt. 14, 169 (1935).

Baker, J. G.

J. G. Baker, IEEE Trans. Aerosp. Electron. Syst. AES-5, 261 (1969).
[CrossRef]

Cornwall, L. W.

Fletcher, R.

R. Fletcher, C. M. Reeves, Comput J. 7, No. 2, 149 (1964).
[CrossRef]

Korn, G. A. and T. M.

G. A. and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 25.

Korsch, D.

Meinel, A. B.

R. V. Shack, A. B. Meinel, J. Opt. Soc. Am. 56, 545 (1966).

Meiron, J.

Paul, M.

M. Paul, Rev. Opt. 14, 169 (1935).

Reeves, C. M.

R. Fletcher, C. M. Reeves, Comput J. 7, No. 2, 149 (1964).
[CrossRef]

Rigler, A. K.

Rimmer, M. P.

Robb, P. N.

Rumsey, N. J.

N. J. Rumsey, “Telescopic System Utilizing Three Axially Aligned Substantially Hyperbolic Mirrors,” U.S. Patent3,460,886 (1969).

N. J. Rumsey, Optical Instruments and Techniques (Oriel, Newcastle upon Tyne, 1970), p. 516.

Shack, R. V.

R. V. Shack, A. B. Meinel, J. Opt. Soc. Am. 56, 545 (1966).

Wetherell, W. B.

Appl. Opt.

Comput J.

R. Fletcher, C. M. Reeves, Comput J. 7, No. 2, 149 (1964).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst.

J. G. Baker, IEEE Trans. Aerosp. Electron. Syst. AES-5, 261 (1969).
[CrossRef]

J. Opt. Soc. Am.

Rev. Opt.

M. Paul, Rev. Opt. 14, 169 (1935).

Other

N. J. Rumsey, Optical Instruments and Techniques (Oriel, Newcastle upon Tyne, 1970), p. 516.

N. J. Rumsey, “Telescopic System Utilizing Three Axially Aligned Substantially Hyperbolic Mirrors,” U.S. Patent3,460,886 (1969).

G. A. and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 25.

Both the coefficients and factors of the Petzval condition equation are formed through sums and differences of quantities having large absolute values. Double precision arithmetic may be required to obtain solutions which are not numerical noise.

ACCOS-V, a proprietary product of Scientific Calculations, Inc., Rochester, N.Y. (1977). Both the damped least squares and Gram-Schmidt orthogonolization methods were used.

Note that the image surface introduces vignetting at half-field angles greater than approximately 0.7°. The design illustrated in Fig. 5 was intended to operate with a line field, i.e., as a scanning system. The vignetting produced by the focal plane in this case was insignificant.

The cost to design these systems is very modest; most of the calculations required are to evaluate the figure of merit. Using the merit function of Ref. 10 and computing the optical aberration coefficients out through the seventh order, the time per configuration averaged 15 msec on a Univac 1110 computer. This works out to a cost of 0.54 cents per design configuration.

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Figures (6)

Fig. 1
Fig. 1

Nomenclature of the three-mirror telescope.

Fig. 2
Fig. 2

One-sigma spot radius vs fractional object height. The curves plotted are for (a)– – – – the starting design; (b)— - - —a flat Petzval surface through selection of Fpri; (c)— —selection of B; and (d)— - —selection of D3.

Fig. 3
Fig. 3

Plots of the merit function σ as a function of Fpri and F2 with (A) the logarithm of σ plotted, (B) the reciprocal of σ plotted.

Fig. 4
Fig. 4

One-sigma spot radius vs fractional object height. The curves plotted are for (a)— - - —the starting design, (b)–––after optimizing Fpri, and (c)– – – – –after the final OPD based optimization.

Fig. 5
Fig. 5

Optical layout of the optimized design.

Fig. 6
Fig. 6

Spot diagrams across the field of the optimized design. The Airy disk diameter is shown for a wavelength of 1.0 μm.

Tables (8)

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Table I Coefficients of the Petzval Condition Equation

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Table II Coefficients of the Third-Order Aberration Equations

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Table III Variable Change Table

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Table IV Design Alternatives Obtained from Solution of the Petzval Condition Equation

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Table V Design Prescriptions

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Table VI Parameters for the Design Optimization Experiment

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Table VII Performance of the Design Optimization Techniques

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Table VIII Design Prescription after OPD Based Optimization

Equations (32)

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Z ( ρ ) = c ρ 2 / { 1 + [ 1 ( K + 1 ) c 2 ρ 2 ] 1 / 2 } + d ρ 4 + e ρ 6 + f ρ 8 + g ρ 10 ,
y j = y j 1 + t j 1 u j 1 , i j = y j c j + u j 1 , u j = u j 1 2 i j .
S 0 = A B S ( D 3 ) / D 3 S 1 = A B S ( F 2 ) / F 2 ,
c 1 = 1 / ( 2 F p ) .
d 1 = ( F 2 B ) ( 1 + A 2 ) .
c 2 = ( 1 A 2 ) 2 F 2 ( 2 d 1 c 1 1 ) .
d 2 = d 1 + B + D 3 .
c 3 = 0.5 D 3 [ 1 S 0 S 1 F 2 F 3 ] .
d 3 = A B S ( D 3 ) S 1 F 3 / F 2 .
c 4 = 2 ( c 1 c 2 + c 3 ) .
P = ( N j 1 N j ) c j N j N j 1 .
P z = j = 1 M 2 c j ( 1 ) j .
P z = 2 ( c 1 c 2 + c 3 ) .
2 c 1 2 c 2 + 2 c 3 P z = 0.
F p 2 [ F 2 2 S 0 S 1 + F 2 F 3 ( P z + 1 ) + F 3 D 3 ] + F p [ F 2 2 B S 0 S 1 + F 2 F 3 ( B ( P z + 1 ) + D 3 ) ] + F 2 F 3 D 3 ( B F 2 ) = 0.
α 2 F 2 2 + α 1 F 2 + α 0 = 0
β 2 F 2 2 + β 1 F p + β 0 = 0
γ 1 D 3 + γ 0 = 0 ; δ 1 B + δ 0 = 0 ; η 1 F 3 + η 0 = 0. }
D 3 = γ 0 / γ 1 , B = δ 0 / δ 1 , F 3 = η 0 / η 1 . }
S j = 2 y j ( u j + i j ) ( 1 ) j ,
T j = 2 K j c j 3 y j 4 ( 1 ) j ,
Q j = ( y ¯ / y ) j .
B j = S j i j 2 + T j ,
F j = S j i j i ¯ j + Q j T j ,
C j = S j i ¯ j 2 + Q j 2 T j .
S A 3 = γ j = 1 3 B j ,
C M A 3 = γ j = 1 3 F j ,
A S T 3 = γ j = 1 3 C j ,
γ = 1 / ( 2 u k ) .
A 11 K 1 + A 12 K 2 + A 13 K 3 + A 14 = 0 ,
A 21 K 1 + A 22 K 2 + A 23 K 3 + A 24 = 0 ,
A 31 K 1 + A 32 K 2 + A 33 K 3 + A 34 = 0.

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