Abstract

We propose a configuration of an off-axis three-mirror system for maximum compactness and brightness. The chief ray is arranged to cross three times inside the system, and the system has a round configuration for compactness. We introduced into the design a ray triangle formed by the reflection points of the chief rays at the mirrors. The ray triangle indicates the size and the brightness of the system. Based on the proposed configuration, a design example of a 4° × 4° field of view is shown. The F-number of the system is 2.2, in close agreement with the estimation from the ray triangle.

© 2005 Optical Society of America

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  1. R. B. Johnson, A. Mann, “Evolution of compact, wide field-of-view, unobscured, all-reflective zoom optical system,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 370–375 (1997).
    [CrossRef]
  2. I. R. Abel, M. R. Hatch, “The pursuit of symmetry in wide-angle reflective optical designs,” in 1980 International Lens Design Conference, R. E. Fischer, eds., Proc. SPIE237, 271–280 (1980).
    [CrossRef]
  3. A. G. Dall’Era, “Wide angle, reflective optical systems for infrared applications,” in Lens and Optical System Design, H. Zuegge, ed., Proc. SPIE1780, 817–824 (1992).
  4. J. Ishigaki, T. Okamura, K. Tanikawa, H. Shirahata, M. Ushida, H. Mirofushi, “Designing and testing of off-axis tree-mirror optical system for multi-spectral sensor,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 356–369 (1997).
    [CrossRef]
  5. K. L. Hallam, B. J. Howell, M. E. Wilson, “An all-reflective wide-angle flat-field telescope for space,” in Instrumentation in Astronomy V, A. Boksenberg, D. L. Crawford, eds., Proc. SPIE445, 295–300 (1984).
    [CrossRef]
  6. K. L. Hallam, “Wide-angle flat field telescope,” U.S. patent4,598,981 (8July1986).
  7. D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991).
  8. R. B. Johnson, “Wide field of view three-mirror telescopes having a common optical axis,” Opt. Eng. 27, 1046–1050 (1988).
    [CrossRef]
  9. B. Tatian, “A first look at the computer design of optical systems without any symmetry,” in Metrology of Optoelectronic Systems, E. M. Granger, ed., Proc. SPIE766, 38–47 (1987).
  10. P. J. Sands, “Aberration coefficients of plane symmetric systems,” J. Opt. Soc. Am. 62, 1211–1220 (1972).
    [CrossRef]
  11. J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
    [CrossRef]
  12. B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3297–3307 (1994).
    [CrossRef]
  13. B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
    [CrossRef]
  14. J. M. Howard, B. D. Stone, “Imaging with three spherical mirrors,” Appl. Opt. 39, 3216–3231 (2000).
    [CrossRef]
  15. S. H. Chao, N. C. Evans, D. L. Shealy, R. B. Johnson, “Design of three-mirror telescopes via a differential equation method,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 276–286 (1996).
    [CrossRef]

2000 (1)

1994 (3)

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3297–3307 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
[CrossRef]

1988 (1)

R. B. Johnson, “Wide field of view three-mirror telescopes having a common optical axis,” Opt. Eng. 27, 1046–1050 (1988).
[CrossRef]

1972 (1)

Abel, I. R.

I. R. Abel, M. R. Hatch, “The pursuit of symmetry in wide-angle reflective optical designs,” in 1980 International Lens Design Conference, R. E. Fischer, eds., Proc. SPIE237, 271–280 (1980).
[CrossRef]

Chao, S. H.

S. H. Chao, N. C. Evans, D. L. Shealy, R. B. Johnson, “Design of three-mirror telescopes via a differential equation method,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 276–286 (1996).
[CrossRef]

Dall’Era, A. G.

A. G. Dall’Era, “Wide angle, reflective optical systems for infrared applications,” in Lens and Optical System Design, H. Zuegge, ed., Proc. SPIE1780, 817–824 (1992).

Evans, N. C.

S. H. Chao, N. C. Evans, D. L. Shealy, R. B. Johnson, “Design of three-mirror telescopes via a differential equation method,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 276–286 (1996).
[CrossRef]

Forbes, G. W.

B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3297–3307 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
[CrossRef]

Hallam, K. L.

K. L. Hallam, B. J. Howell, M. E. Wilson, “An all-reflective wide-angle flat-field telescope for space,” in Instrumentation in Astronomy V, A. Boksenberg, D. L. Crawford, eds., Proc. SPIE445, 295–300 (1984).
[CrossRef]

K. L. Hallam, “Wide-angle flat field telescope,” U.S. patent4,598,981 (8July1986).

Hatch, M. R.

I. R. Abel, M. R. Hatch, “The pursuit of symmetry in wide-angle reflective optical designs,” in 1980 International Lens Design Conference, R. E. Fischer, eds., Proc. SPIE237, 271–280 (1980).
[CrossRef]

Howard, J. M.

Howell, B. J.

K. L. Hallam, B. J. Howell, M. E. Wilson, “An all-reflective wide-angle flat-field telescope for space,” in Instrumentation in Astronomy V, A. Boksenberg, D. L. Crawford, eds., Proc. SPIE445, 295–300 (1984).
[CrossRef]

Ishigaki, J.

J. Ishigaki, T. Okamura, K. Tanikawa, H. Shirahata, M. Ushida, H. Mirofushi, “Designing and testing of off-axis tree-mirror optical system for multi-spectral sensor,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 356–369 (1997).
[CrossRef]

Johnson, R. B.

R. B. Johnson, “Wide field of view three-mirror telescopes having a common optical axis,” Opt. Eng. 27, 1046–1050 (1988).
[CrossRef]

R. B. Johnson, A. Mann, “Evolution of compact, wide field-of-view, unobscured, all-reflective zoom optical system,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 370–375 (1997).
[CrossRef]

S. H. Chao, N. C. Evans, D. L. Shealy, R. B. Johnson, “Design of three-mirror telescopes via a differential equation method,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 276–286 (1996).
[CrossRef]

Korsch, D.

D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991).

Mann, A.

R. B. Johnson, A. Mann, “Evolution of compact, wide field-of-view, unobscured, all-reflective zoom optical system,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 370–375 (1997).
[CrossRef]

Mirofushi, H.

J. Ishigaki, T. Okamura, K. Tanikawa, H. Shirahata, M. Ushida, H. Mirofushi, “Designing and testing of off-axis tree-mirror optical system for multi-spectral sensor,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 356–369 (1997).
[CrossRef]

Okamura, T.

J. Ishigaki, T. Okamura, K. Tanikawa, H. Shirahata, M. Ushida, H. Mirofushi, “Designing and testing of off-axis tree-mirror optical system for multi-spectral sensor,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 356–369 (1997).
[CrossRef]

Sands, P. J.

Sasian, J. M.

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Shealy, D. L.

S. H. Chao, N. C. Evans, D. L. Shealy, R. B. Johnson, “Design of three-mirror telescopes via a differential equation method,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 276–286 (1996).
[CrossRef]

Shirahata, H.

J. Ishigaki, T. Okamura, K. Tanikawa, H. Shirahata, M. Ushida, H. Mirofushi, “Designing and testing of off-axis tree-mirror optical system for multi-spectral sensor,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 356–369 (1997).
[CrossRef]

Stone, B. D.

Tanikawa, K.

J. Ishigaki, T. Okamura, K. Tanikawa, H. Shirahata, M. Ushida, H. Mirofushi, “Designing and testing of off-axis tree-mirror optical system for multi-spectral sensor,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 356–369 (1997).
[CrossRef]

Tatian, B.

B. Tatian, “A first look at the computer design of optical systems without any symmetry,” in Metrology of Optoelectronic Systems, E. M. Granger, ed., Proc. SPIE766, 38–47 (1987).

Ushida, M.

J. Ishigaki, T. Okamura, K. Tanikawa, H. Shirahata, M. Ushida, H. Mirofushi, “Designing and testing of off-axis tree-mirror optical system for multi-spectral sensor,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 356–369 (1997).
[CrossRef]

Wilson, M. E.

K. L. Hallam, B. J. Howell, M. E. Wilson, “An all-reflective wide-angle flat-field telescope for space,” in Instrumentation in Astronomy V, A. Boksenberg, D. L. Crawford, eds., Proc. SPIE445, 295–300 (1984).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3297–3307 (1994).
[CrossRef]

Opt. Eng. (2)

R. B. Johnson, “Wide field of view three-mirror telescopes having a common optical axis,” Opt. Eng. 27, 1046–1050 (1988).
[CrossRef]

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Other (9)

B. Tatian, “A first look at the computer design of optical systems without any symmetry,” in Metrology of Optoelectronic Systems, E. M. Granger, ed., Proc. SPIE766, 38–47 (1987).

S. H. Chao, N. C. Evans, D. L. Shealy, R. B. Johnson, “Design of three-mirror telescopes via a differential equation method,” in Current Developments in Optical Design and Engineering VI, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2863, 276–286 (1996).
[CrossRef]

R. B. Johnson, A. Mann, “Evolution of compact, wide field-of-view, unobscured, all-reflective zoom optical system,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 370–375 (1997).
[CrossRef]

I. R. Abel, M. R. Hatch, “The pursuit of symmetry in wide-angle reflective optical designs,” in 1980 International Lens Design Conference, R. E. Fischer, eds., Proc. SPIE237, 271–280 (1980).
[CrossRef]

A. G. Dall’Era, “Wide angle, reflective optical systems for infrared applications,” in Lens and Optical System Design, H. Zuegge, ed., Proc. SPIE1780, 817–824 (1992).

J. Ishigaki, T. Okamura, K. Tanikawa, H. Shirahata, M. Ushida, H. Mirofushi, “Designing and testing of off-axis tree-mirror optical system for multi-spectral sensor,” in Infrared Technology and Applications XXIII, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3061, 356–369 (1997).
[CrossRef]

K. L. Hallam, B. J. Howell, M. E. Wilson, “An all-reflective wide-angle flat-field telescope for space,” in Instrumentation in Astronomy V, A. Boksenberg, D. L. Crawford, eds., Proc. SPIE445, 295–300 (1984).
[CrossRef]

K. L. Hallam, “Wide-angle flat field telescope,” U.S. patent4,598,981 (8July1986).

D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991).

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Figures (14)

Fig. 1
Fig. 1

Geometrical illustration of construction parameters in the plane of symmetry.

Fig. 2
Fig. 2

Illustrations of distances between two mirrors. The tilted mirrors (b) can be placed closer to one another than the parallel mirrors (a).

Fig. 3
Fig. 3

Definition of parameters Din and Dout. Din provides the size of the incident flux, and Dout provides that of the exit flux.

Fig. 4
Fig. 4

Contour map of the F-numbers (a) for the incident flux and (b) for the exit flux. The darkest regions correspond to the lowest F-numbers.

Fig. 5
Fig. 5

Contour map of F-number F at dbf/f = 0.8. The conditions for minimum F are that α = 0.8 and θ2 = 132°.

Fig. 6
Fig. 6

Optimum condition of angle θ2 for the minimum F-number.

Fig. 7
Fig. 7

Optimum condition of the ratio α = d1/2R for the minimum F-number.

Fig. 8
Fig. 8

Minimum F-number versus dbf/f at f/2R = 1. The F-number of the other system is obtainable by multiplication by f/2R.

Fig. 9
Fig. 9

Illustration of focal position with no spherical aberration. The center field rays are arranged to travel from one focal point to another focal point of Cartesian surfaces in this system.

Fig. 10
Fig. 10

Focal displacement between meridional and sagittal focal points. The horizontal axis is the distance of the focal point from the reflection point for mirror 1, l1.

Fig. 11
Fig. 11

The rms spot diameters of the point image versus the order of the highest term added for mirror deformation. The evaluated spots are at the center and the corners of the 4° × 4° field of view.

Fig. 12
Fig. 12

Design example of a three-mirror system. The F-number is 2.2, and the field of view is 4° × 4°.

Fig. 13
Fig. 13

Spot diagrams of the example system at the center and the four corners. The field of view is 4° × 4°.

Fig. 14
Fig. 14

Distortion of the example system. The grid is the image plane map of a 1-deg orthogonal grid of the object plane at infinity, and the points are corresponding positions of the point image.

Tables (2)

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Table 1 Zernike Polynomials and Symbols of Coefficients

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Table 2 Parameters of the Example System

Equations (12)

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d 2 = 2 R sin ( θ 2 - θ 3 ) ,
d 3 = d bf - d 2 [ cos θ 3 - sin θ 3 / tan ( θ 2 + θ 3 ) ] ,
cos θ 1 = - d 1 + ( d 2 / 2 ) cos θ 2 [ d 1 2 + ( d 2 / 2 ) 2 + d 1 d 2 cos θ 2 ] 1 / 2 ,
tan θ 3 = - ( d 1 / 2 ) sin θ 2 d 2 + ( d 1 / 2 ) cos θ 2 ,
D in = - d 2 sin ( θ 1 + θ 2 ) / 2 = d 1 sin θ 1 .
F in = f / D in = f / d 1 sin θ 1 ,
D out = - d 1 sin ( θ 2 + θ 3 ) / 2.
F ex = d bf / D out = - 2 d bf / d 1 sin ( θ 2 + θ 3 ) .
F in = f 2 R { α 2 - α 2 [ 2 α + cos θ 2 sin ( θ 2 - θ 3 ) ] 2 4 α 2 + 4 α cos θ 2 sin ( θ 2 - θ 3 ) + sin 2 ( θ 2 - θ 3 ) } - 1 / 2 ,
F ex = 2 f 2 R d bf f { α sin × [ θ 2 + tan - 1 α sin θ 2 α cos θ 2 + 2 sin ( θ 2 - θ 3 ) ] } - 1 ,
F = max ( F in , F ex ) .
f = l 1 l 2 l 3 ( d 1 - l 1 ) ( d 2 - l 2 ) .

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