Abstract

The main constraint of classical off-axis reflecting systems is the primary astigmatism that has long been a research topic of interest. This astigmatism in off-axis spherical reflective imaging systems can be eliminated by using the proper configuration. These configurations could be derived from the marginal ray fans equation, and they are valid for small angles of incidence. The conditions for the astigmatism compensation in configurations with two and three off-axis mirrors have been derived and analyzed, which have not been reported previously. The expression that defines the conditions for primary astigmatism compensation in a four-mirror system is presented. This shows that the marginal ray fan equation can be used to obtain the condition for astigmatism compensation of a reflective system with any number of mirrors. The developed methodology is verified by ray-tracing analysis of some examples.

© 2011 Optical Society of America

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References

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  1. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).
  2. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).
  3. J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 554, 76–81 (1985).
  4. J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061(1994).
    [CrossRef]
  5. G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence. III. some applications,” J. Opt. Soc. Am. 52, 412–415 (1962).
    [CrossRef]
  6. S. Chang, J. H. Lee, S. P. Kim, H. Kim, W. J. Kim, I. Song, and Y. Park, “Linear astigmatism of confocal off-axis reflective imaging systems and its elimination,” Appl. Opt. 45, 484–488 (2006).
    [CrossRef] [PubMed]
  7. R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28, 488–492 (1981).
    [CrossRef]
  8. R. H. Webb, G. W. Hughes, and F. C. Delori, “Confocal scanning laser ophthalmoscope,” Appl. Opt. 26, 1492–1499 (1987).
    [CrossRef] [PubMed]
  9. S. A. Burns, R. Tumbar, A. E. Elsner, D. Ferguson, and D. X. Hammer, “Large-field-of-view, modular, stabilized, adaptive-optics-based scanning laser ophthalmoscope,” J. Opt. Soc. Am. A 24, 1313–1326 (2007).
    [CrossRef]
  10. A. Gómez-Vieyra, A. Dubra, D. Malacara-Hernández, and D. R. Williams, “First-order design of off-axis reflective ophthalmic adaptive optics systems using afocal telescopes,” Opt. Express 17, 18906–18919 (2009).
    [CrossRef]
  11. A. E. Conrady, Applied Optics and Optical Design, Part 2(Dover, 1960), Chap. 12.
  12. D. Malacara and Z. Malacara, Handbook of Lens Design(Marcel Dekker, 2004), Chap. 5.
  13. R. Kingslake, Lens Design Fundamentals (Academic, 1978), Chap. 10.

2009

2007

2006

1994

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

1987

1985

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 554, 76–81 (1985).

1981

R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28, 488–492 (1981).
[CrossRef]

1962

Buchroeder, R. A.

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).

Burns, S. A.

Chang, S.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part 2(Dover, 1960), Chap. 12.

Delori, F. C.

Dubra, A.

Elsner, A. E.

Ferguson, D.

Gómez-Vieyra, A.

Hammer, D. X.

Hughes, G. W.

R. H. Webb, G. W. Hughes, and F. C. Delori, “Confocal scanning laser ophthalmoscope,” Appl. Opt. 26, 1492–1499 (1987).
[CrossRef] [PubMed]

R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28, 488–492 (1981).
[CrossRef]

Kim, H.

Kim, S. P.

Kim, W. J.

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, 1978), Chap. 10.

Lee, J. H.

Malacara, D.

D. Malacara and Z. Malacara, Handbook of Lens Design(Marcel Dekker, 2004), Chap. 5.

Malacara, Z.

D. Malacara and Z. Malacara, Handbook of Lens Design(Marcel Dekker, 2004), Chap. 5.

Malacara-Hernández, D.

Park, Y.

Rogers, J. R.

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 554, 76–81 (1985).

Rosendahl, G. R.

Sasian, J. M.

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

Song, I.

Thompson, K. P.

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).

Tumbar, R.

Webb, R. H.

R. H. Webb, G. W. Hughes, and F. C. Delori, “Confocal scanning laser ophthalmoscope,” Appl. Opt. 26, 1492–1499 (1987).
[CrossRef] [PubMed]

R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28, 488–492 (1981).
[CrossRef]

Williams, D. R.

Appl. Opt.

IEEE Trans. Biomed. Eng.

R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28, 488–492 (1981).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

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

Opt. Express

Proc. SPIE

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 554, 76–81 (1985).

Other

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).

A. E. Conrady, Applied Optics and Optical Design, Part 2(Dover, 1960), Chap. 12.

D. Malacara and Z. Malacara, Handbook of Lens Design(Marcel Dekker, 2004), Chap. 5.

R. Kingslake, Lens Design Fundamentals (Academic, 1978), Chap. 10.

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Figures (8)

Fig. 1
Fig. 1

Gaussian and marginal image formation by an off-axis reflective spherical mirror.

Fig. 2
Fig. 2

Schematic of general off-axis two-spherical-mirror system.

Fig. 3
Fig. 3

Possible configurations for a two-spherical-mirror system.

Fig. 4
Fig. 4

Schematic of general off-axis three- spherical-mirror system with one folding plane.

Fig. 5
Fig. 5

Possible configurations for a three-spherical-mirror system.

Fig. 6
Fig. 6

Schematic for general off-axis four- spherical-mirror system with one folding plane.

Fig. 7
Fig. 7

Three-mirror system ( r 1 = 1000 mm , r 2 = 2000 mm , r 3 = 1500 mm , s = 800 mm , d 1 = 1500 mm , d 2 = 1750 mm , I 1 = 3 ° , I 2 = 5 ° , and I 3 = 3.93947 ° ). (a) Spot diagram in image plane without astigmatism correction ( θ = 0 ° ). (b) Spot diagram in image plane without astigmatism correction at θ = 90 ° .

Fig. 8
Fig. 8

Four-mirror system ( r 1 = 1000 mm , r 2 = 2000 mm , r 3 = 1000 mm , r 4 = 200 , s = 800 mm , d 1 = 1500 mm , d 2 = 1500 mm , d 3 = 1500 mm , I 2 = 3 ° , I 2 = 5 ° , I 3 = 4 , and I 4 = 14.2842 ° ). (a) Spot diagram in image plane without astigmatism correction ( θ = 0 ° ). (b) Spot diagram in image plane without astigmatism correction at θ = 90 ° .

Tables (2)

Tables Icon

Table 1 Configuration Conditions for a Two-Spherical-Mirror System (Fig. 3)

Tables Icon

Table 2 Configuration Conditions for a Three-Spherical-Mirror System (Fig. 5)

Equations (23)

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1 s θ + 1 s θ = 2 cos I r ( 1 cos 2 θ sin 2 I ) ,
1 s s + 1 s s = 2 cos I r ,
1 s t + 1 s t = 2 r cos I .
ϕ θ + ϕ θ = 2 cos I r ( 1 cos 2 θ sin 2 I ) .
Δ = s s s t ,
s i ( θ ) = r [ cos 2 ( θ ) sin 2 ( I ) 1 ] s i ( θ ) r 2 s i ( θ ) cos ( I ) r cos 2 ( θ ) sin 2 ( I ) ,
s 1 ( 0 ) = r 1 s 1 ( 0 ) [ sin 2 ( I 1 ) 1 ] r 1 2 s 1 ( 0 ) cos ( I 1 ) r 1 sin 2 ( I 1 ) ,
s 1 ( π / 2 ) = r 1 s 1 ( π / 2 ) r 1 2 s i ( π / 2 ) cos ( I 1 ) ,
s 2 ( θ 2 ) = r 2 s 2 ( θ 2 ) [ cos 2 ( θ 2 ) sin 2 ( I 2 ) 1 ] r 2 2 s 2 ( θ 2 ) cos ( I 2 ) r 2 cos 2 ( θ 2 ) sin 2 ( I 2 ) ,
s 2 ( θ 2 + π / 2 ) = r 2 s 2 ( θ 2 + π / 2 ) [ cos 2 ( θ 2 + π / 2 ) sin 2 ( I 2 ) 1 ] r 2 2 s 2 ( θ 2 + π / 2 ) cos ( I 2 ) r 2 cos 2 ( θ 2 + π / 2 ) sin 2 ( I 2 ) .
s 2 ( θ ) = { 2 s + r 1 cos ( I 1 ) 2 d s ( d + s ) r 1 cos ( I 1 ) + 2 cos ( I 2 ) r 2 [ 1 cos 2 ( θ ) sin 2 ( I 2 ) ] } 1 ,
s 2 ( θ + π / 2 ) = { 2 s cos ( I 1 ) + r 1 2 d s cos ( I 1 ) ( d + s ) r 1 + 2 cos ( I 2 ) r 2 [ 1 sin 2 ( θ ) sin 2 ( I 2 ) ] } 1 ;
Δ = r 2 [ 1 + sin 2 ( θ ) sin 2 ( I 2 ) ] [ 2 d s cos ( I 1 ) ( d + s ) r 1 ] 2 [ 2 d s cos ( I 1 ) ( d + s ) r 1 ] cos ( I 2 ) + r 2 [ 1 + sin 2 ( θ ) sin 2 ( I 2 ) ] [ 2 s cos ( I 1 ) r 1 ] + r 2 [ 1 + cos 2 ( θ ) sin 2 ( I 2 ) ] [ 2 d s sec ( I 1 ) ( d + s ) r 1 ] 2 [ 2 d s sec ( I 1 ) ( d + s ) r 1 ] cos ( I 2 ) + r 2 [ 1 + cos 2 ( θ ) sin 2 ( I 2 ) ] [ 2 s sec ( I 1 ) r 1 ] .
Δ = 2 s 2 r 1 r 2 2 I 1 2 { 4 d s 2 s r 2 + r 1 [ 2 ( d + s ) + r 2 ] } 2 + 2 { cos ( 2 θ ) [ 2 d s + ( d + s ) r 1 ] 2 r 2 } I 2 2 { 4 d s 2 s r 2 + r 1 [ 2 d s + ( d + s ) r 1 ] } 2 .
I 2 = s r 1 r 2 I 1 ( 2 d s + ( d + s ) r 1 ) cos [ 2 θ ] .
r 1 = 2 d s d + s ,
| θ | = 45 ° and 135 ° .
I 2 = r 1 r 2 I 1 ( 2 d + ( d + s ) r 1 ϕ ) cos [ 2 θ ] ,
I 3 = r 2 r 3 s 2 r 1 r 2 I 1 2 + [ 2 s d 1 + ( s + d 1 ) r 1 ] 2 I 2 2 { d 1 ( 2 s r 1 ) ( 2 d 2 r 2 ) + s r 1 r 2 + d 2 [ 2 s r 2 + r 1 ( 2 s + r 2 ) ] } cos ( 2 θ ) .
s 2 r 1 r 2 I 1 2 + [ 2 s d 1 + ( s + d 1 ) r 1 ] 2 I 2 2 ,
r d 1 ( 2 s r 1 ) ( 2 d 2 r 2 ) + s r 1 r 2 + d 2 [ 2 s r 2 + r 1 ( 2 s + r 2 ) ] = 0 ,
| θ | = 45 ° and 135 ° .
I 4 = { [ sec ( 2 θ ) r 3 r 4 ( 4 s 2 { d 1 2 r 2 r 3 I 2 2 + [ 2 d 1 d 2 + ( d 1 + d 2 ) r 2 ] 2 I 3 2 } + s r 1 { r 2 r 3 [ s r 2 I 1 2 4 d 1 ( s + d 1 ) I 2 2 ] 4 [ 2 d 1 d 2 ( d 1 + d 2 ) r 2 ] [ 2 ( s + d 1 ) d 2 ( s + d 1 + d 2 ) r 2 ] I 3 2 } + r 1 2 { ( s + d 1 ) 2 r 2 r 3 I 2 2 + [ 2 ( s + d 1 ) d 2 + ( s + d 1 + d 2 ) r 2 ] 2 I 3 2 } ) ] } / ( d 2 [ r 1 ( 2 s r 2 ) + 2 s r 2 ] ( 2 d 3 r 3 ) + s r 1 r 2 r 3 d 1 ( 2 s r 1 ) * [ d 2 ( 4 d 3 2 r 3 ) + r 2 r 3 2 d 3 ( r 2 + r 3 ) ] + d 3 { 2 s r 2 r 3 + r 1 [ 2 s r 3 + r 2 ( 2 s + r 3 ) ] } ) .

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