Abstract

Expressions for minimal astigmatism in image and pupil planes in off-axis afocal reflective telescopes formed by pairs of spherical mirrors are presented. These formulae which are derived from the marginal ray fan equation can be used for designing laser cavities, spectrographs and adaptive optics retinal imaging systems. The use, range and validity of these formulae are limited by spherical aberration and coma for small and large angles respectively. This is discussed using examples from adaptive optics retinal imaging systems. The performance of the resulting optical designs are evaluated and compared against the configurations with minimal wavefront RMS, using the defocus-corrected wavefront RMS as a metric.

© 2009 OSA

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  1. J. Liang, D. R. Williams, and D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14(11), 2884–2892 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-14-11-2884 .
    [CrossRef]
  2. J. Rha, R. S. Jonnal, K. E. Thorn, J. Qu, Y. Zhang, and D. T. Miller, “Adaptive optics flood-illumination camera for high speed retinal imaging,” Opt. Express 14(10), 4552–4569 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-10-4552 .
    [CrossRef] [PubMed]
  3. B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29(18), 2142–2144 (2004), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-29-18-2142 .
    [CrossRef] [PubMed]
  4. R. J. Zawadzki, S. M. Jones, S. S. Olivier, M. T. Zhao, B. A. Bower, J. A. Izatt, S. Choi, S. Laut, and J. S. Werner, “Adaptive-optics optical coherence tomography for high-resolution and high-speed 3D retinal in vivo imaging,” Opt. Express 13(21), 8532–8546 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8532 .
    [CrossRef] [PubMed]
  5. E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
    [CrossRef] [PubMed]
  6. Y. Zhang, J. T. Rha, R. S. Jonnal, and D. T. Miller, “Adaptive optics parallel spectral domain optical coherence tomography for imaging the living retina,” Opt. Express 13(12), 4792–4811 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4792 .
    [CrossRef] [PubMed]
  7. D. Merino, C. Dainty, A. Bradu, and A. G. Podoleanu, “Adaptive optics enhanced simultaneous en-face optical coherence tomography and scanning laser ophthalmoscopy,” Opt. Express 14(8), 3345–3353 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-8-3345 .
    [CrossRef] [PubMed]
  8. C. E. Bigelow, N. V. Iftimia, R. D. Ferguson, T. E. Ustun, B. Bloom, and D. X. Hammer, “Compact multimodal adaptive-optics spectral-domain optical coherence tomography instrument for retinal imaging,” J. Opt. Soc. Am. A 24(5), 1327–1336 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-5-1327 .
    [CrossRef]
  9. A. Roorda, F. Romero-Borja, W. Donnelly Iii, H. Queener, T. J. Hebert, and M. C. W. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10(9), 405–412 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-9-405 .
    [PubMed]
  10. Y. Zhang, S. Poonja, and A. Roorda, “MEMS-based adaptive optics scanning laser ophthalmoscopy,” Opt. Lett. 31(9), 1268–1270 (2006), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-9-1268 .
    [CrossRef] [PubMed]
  11. D. X. Hammer, R. D. Ferguson, C. E. Bigelow, N. V. Iftimia, T. E. Ustun, and S. A. Burns, “Adaptive optics scanning laser ophthalmoscope for stabilized retinal imaging,” Opt. Express 14(8), 3354–3367 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3354 .
    [CrossRef] [PubMed]
  12. D. C. Gray, W. Merigan, J. I. Wolfing, B. P. Gee, J. Porter, A. Dubra, T. H. Twietmeyer, K. Ahamd, R. Tumbar, F. Reinholz, and D. R. Williams, “In vivo fluorescence imaging of primate retinal ganglion cells and retinal pigment epithelial cells,” Opt. Express 14(16), 7144–7158 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7144 .
    [CrossRef] [PubMed]
  13. 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(5), 1313–1326 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-5-1313 .
    [CrossRef]
  14. R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28(7), 488–492 (1981).
    [CrossRef]
  15. R. H. Webb, G. W. Hughes, and F. C. Delori, “Confocal scanning laser ophthalmoscope,” Appl. Opt. 26(8), 1492–1499 (1987).
    [CrossRef] [PubMed]
  16. H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
    [CrossRef]
  17. W. T. Foreman, “Lens Correction of Astigmatism in a Czerny-Turner Spectrograph,” Appl. Opt. 7(6), 1053–1059 (1968), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-7-6-1053 .
    [CrossRef] [PubMed]
  18. G. R. Rosendahl, “Contributions to the Optics of Mirror Systems and Gratings with Oblique Incidence. III. Some Applications,” J. Opt. Soc. Am. 52(4), 412–415 (1962), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-52-4-412 .
    [CrossRef]
  19. A. E. Conrady, Applied Optics and Optical Design, (Dover Publications Inc., New York, 1960), Chap XII, part 2.
  20. D. Malacara and Z. Malacara, Handbook of Lens Design, (Marcel Dekker Inc., New York. 2004), Chap. 5.
  21. R. Kingslake, Lens Design Fundamentals, (Academic Press, San Diego, 1978), Chap 10.
  22. D. A. Atchison, A. Bradley, L. N. Thibos, and G. Smith, “Useful variations of the Badal optometer,” Optom. Vis. Sci. 72(4), 279–284 (1995).
    [CrossRef] [PubMed]
  23. G. Smith and D. A. Atchison, The eye and visual optical instruments, (Cambridge University Press, Cambridge, U.K., 1997), Chap. 30.

2007 (2)

2006 (5)

2005 (3)

2004 (1)

2002 (1)

1997 (1)

1995 (1)

D. A. Atchison, A. Bradley, L. N. Thibos, and G. Smith, “Useful variations of the Badal optometer,” Optom. Vis. Sci. 72(4), 279–284 (1995).
[CrossRef] [PubMed]

1987 (1)

1981 (1)

R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28(7), 488–492 (1981).
[CrossRef]

1972 (1)

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

1968 (1)

1962 (1)

Ahamd, K.

Ahnelt, P.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

Artal, P.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29(18), 2142–2144 (2004), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-29-18-2142 .
[CrossRef] [PubMed]

Atchison, D. A.

D. A. Atchison, A. Bradley, L. N. Thibos, and G. Smith, “Useful variations of the Badal optometer,” Optom. Vis. Sci. 72(4), 279–284 (1995).
[CrossRef] [PubMed]

Bigelow, C. E.

Bloom, B.

Bower, B. A.

Bradley, A.

D. A. Atchison, A. Bradley, L. N. Thibos, and G. Smith, “Useful variations of the Badal optometer,” Optom. Vis. Sci. 72(4), 279–284 (1995).
[CrossRef] [PubMed]

Bradu, A.

Burns, S. A.

Campbell, M. C. W.

Choi, S.

Dainty, C.

Delori, F. C.

Dienes, A.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Donnelly Iii, W.

Drexler, W.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29(18), 2142–2144 (2004), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-29-18-2142 .
[CrossRef] [PubMed]

Dubra, A.

Elsner, A. E.

Fercher, A. F.

Ferguson, D.

Ferguson, R. D.

Fernández, E. J.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29(18), 2142–2144 (2004), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-29-18-2142 .
[CrossRef] [PubMed]

Foreman, W. T.

Gee, B. P.

Gray, D. C.

Hammer, D. X.

Hebert, T. J.

Hermann, B.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29(18), 2142–2144 (2004), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-29-18-2142 .
[CrossRef] [PubMed]

Hughes, G. W.

R. H. Webb, G. W. Hughes, and F. C. Delori, “Confocal scanning laser ophthalmoscope,” Appl. Opt. 26(8), 1492–1499 (1987).
[CrossRef] [PubMed]

R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28(7), 488–492 (1981).
[CrossRef]

Iftimia, N. V.

Ippen, E. P.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Izatt, J. A.

Jones, S. M.

Jonnal, R. S.

Kogelnik, H. W.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Laut, S.

Leitgeb, R.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

Liang, J.

Merigan, W.

Merino, D.

Miller, D. T.

Olivier, S. S.

Podoleanu, A. G.

Poonja, S.

Porter, J.

Povazay, B.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

Prieto, P. M.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29(18), 2142–2144 (2004), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-29-18-2142 .
[CrossRef] [PubMed]

Qu, J.

Queener, H.

Reinholz, F.

Rha, J.

Rha, J. T.

Romero-Borja, F.

Roorda, A.

Rosendahl, G. R.

Sattmann, H.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29(18), 2142–2144 (2004), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-29-18-2142 .
[CrossRef] [PubMed]

Shank, C. V.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Smith, G.

D. A. Atchison, A. Bradley, L. N. Thibos, and G. Smith, “Useful variations of the Badal optometer,” Optom. Vis. Sci. 72(4), 279–284 (1995).
[CrossRef] [PubMed]

Thibos, L. N.

D. A. Atchison, A. Bradley, L. N. Thibos, and G. Smith, “Useful variations of the Badal optometer,” Optom. Vis. Sci. 72(4), 279–284 (1995).
[CrossRef] [PubMed]

Thorn, K. E.

Tumbar, R.

Twietmeyer, T. H.

Unterhuber, A.

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29(18), 2142–2144 (2004), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-29-18-2142 .
[CrossRef] [PubMed]

Ustun, T. E.

Webb, R. H.

R. H. Webb, G. W. Hughes, and F. C. Delori, “Confocal scanning laser ophthalmoscope,” Appl. Opt. 26(8), 1492–1499 (1987).
[CrossRef] [PubMed]

R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28(7), 488–492 (1981).
[CrossRef]

Werner, J. S.

Williams, D. R.

Wolfing, J. I.

Zawadzki, R. J.

Zhang, Y.

Zhao, M. T.

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

R. H. Webb and G. W. Hughes, “Scanning laser ophthalmoscope,” IEEE Trans. Biomed. Eng. BME-28(7), 488–492 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Express (7)

A. Roorda, F. Romero-Borja, W. Donnelly Iii, H. Queener, T. J. Hebert, and M. C. W. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10(9), 405–412 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-9-405 .
[PubMed]

Y. Zhang, J. T. Rha, R. S. Jonnal, and D. T. Miller, “Adaptive optics parallel spectral domain optical coherence tomography for imaging the living retina,” Opt. Express 13(12), 4792–4811 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4792 .
[CrossRef] [PubMed]

D. Merino, C. Dainty, A. Bradu, and A. G. Podoleanu, “Adaptive optics enhanced simultaneous en-face optical coherence tomography and scanning laser ophthalmoscopy,” Opt. Express 14(8), 3345–3353 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-8-3345 .
[CrossRef] [PubMed]

J. Rha, R. S. Jonnal, K. E. Thorn, J. Qu, Y. Zhang, and D. T. Miller, “Adaptive optics flood-illumination camera for high speed retinal imaging,” Opt. Express 14(10), 4552–4569 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-10-4552 .
[CrossRef] [PubMed]

R. J. Zawadzki, S. M. Jones, S. S. Olivier, M. T. Zhao, B. A. Bower, J. A. Izatt, S. Choi, S. Laut, and J. S. Werner, “Adaptive-optics optical coherence tomography for high-resolution and high-speed 3D retinal in vivo imaging,” Opt. Express 13(21), 8532–8546 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8532 .
[CrossRef] [PubMed]

D. X. Hammer, R. D. Ferguson, C. E. Bigelow, N. V. Iftimia, T. E. Ustun, and S. A. Burns, “Adaptive optics scanning laser ophthalmoscope for stabilized retinal imaging,” Opt. Express 14(8), 3354–3367 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3354 .
[CrossRef] [PubMed]

D. C. Gray, W. Merigan, J. I. Wolfing, B. P. Gee, J. Porter, A. Dubra, T. H. Twietmeyer, K. Ahamd, R. Tumbar, F. Reinholz, and D. R. Williams, “In vivo fluorescence imaging of primate retinal ganglion cells and retinal pigment epithelial cells,” Opt. Express 14(16), 7144–7158 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7144 .
[CrossRef] [PubMed]

Opt. Lett. (2)

Optom. Vis. Sci. (1)

D. A. Atchison, A. Bradley, L. N. Thibos, and G. Smith, “Useful variations of the Badal optometer,” Optom. Vis. Sci. 72(4), 279–284 (1995).
[CrossRef] [PubMed]

Vision Res. (1)

E. J. Fernández, B. Povazay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45(28), 3432–3444 (2005).
[CrossRef] [PubMed]

Other (4)

A. E. Conrady, Applied Optics and Optical Design, (Dover Publications Inc., New York, 1960), Chap XII, part 2.

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

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

G. Smith and D. A. Atchison, The eye and visual optical instruments, (Cambridge University Press, Cambridge, U.K., 1997), Chap. 30.

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Figures (6)

Fig. 1
Fig. 1

Geometry of an off-axis reflective afocal telescope: I 1 and I 2 are the angles of incidence of the principal ray (in red) onto the mirrors m 1 and m 2 respectively, and the angle θ between the incidence planes. The object is a distance s from the first mirror, and the image formed by the second mirror is a distance s2,θ' from it.

Fig. 2
Fig. 2

Notation used to describe an afocal telescope formed by a pair of off-axis spherical mirrors in the finite conjugate case, with s = r1/2. This particular configuration corresponds to θ = 0.

Fig. 3
Fig. 3

Notation used to describe an afocal telescope formed by a pair of off-axis spherical mirrors in the infinite conjugates case. This particular arrangement corresponds to the case θ=0.

Fig. 4
Fig. 4

Each column describes a different off-axis reflective afocal (4f) telescope, with the parameters indicated at the top. The object point is at the front focal point of the first mirror (Fig. 2). The top row shows the angle of incidence of the principal ray onto the second mirror as a function of the angle of incidence onto the first mirror. The second row shows the distance between the second mirror and the image point as a function of the first angle of incidence for the same configurations. The bottom row shows the wavefront RMS values for the (defocus-corrected) corresponding configurations. The RMS of the planar (θ = 0°) configuration is also plotted for comparison. The red dashed lines indicate Marechal’s diffraction limit.

Fig. 5
Fig. 5

Each column describes a different off-axis reflective afocal (4f) telescope, with the parameters indicated at the top. The object point is at the front focal point of the first mirror (see Fig. 3). The top row shows the angle of incidence of the principal ray onto the second mirror as a function of the angle of incidence onto the first mirror. The second row shows the power (vergence) between the second mirror and the image point in diopters (D) as a function of the first angle of incidence for the same configurations. The bottom row shows the wavefront RMS values for the (defocus-corrected) corresponding configurations. The RMS of the planar (θ = 0°) configuration is also plotted for comparison. The red dashed lines indicate Marechal’s diffraction limit.

Fig. 6
Fig. 6

Data from a pair of off-axis reflective afocal (4f) telescopes, with astigmatism simultaneously corrected in pupil and image planes. The left plot shows the angle of incidence of the principal ray onto the second mirror as a function of the angle of incidence onto the first mirror, for the linear approximation, and the four different optimal configurations sketched on the left side of Table 1. The middle plot shows the distance between the second mirror and the image point as a function of the angle of incidence. The plot on the right shows the wavefront RMS values for the (defocus-corrected) corresponding configurations. The RMS of the planar (θ = 0°) configuration is also plotted for comparison. The red dashed line in the last plot indicates Marechal’s diffraction limit.

Tables (1)

Tables Icon

Table 1 Wavefront RMS of the most compact configurations of two different pairs of off-axis telescopes with minimal aberrations, for I 1 = 5°. The numbers on the top row are the mirror radii of curvature in mm. The entrance beam diameter considered was 8mm and λ = 500nm.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

1s+1s'=1f ,
1s+1sθ'=2cosIr(1cos2θsin2I) ,
1ss+1ss'=2cosIr ,
1st+1st'=2rcosI.
ϕ+ϕθ'=2cosIr(1cos2θsin2I).
f(I)=r[3+cos(2I)]8cosI.
f(I)r2.
Δs=r2(sin2θsin2I21)   [sr1r12scosI1+d]r1cosI2(d+sr1r12scosI1)+r2(sin2θsin2I21)                                r2(cos2θsin2I21)   [sr1cosI1r1cosI12s+d]2cosI2(d+sr1cosI1r1cosI12s)+r2(cos2θsin2I21).
ΔsM r1cos(2θ)2(1+M2Msr1)2I222s2M2r1I12.
Δϕ=r1ϕ2  cos  I1d(r1ϕ2  cos  I1)+r1+2ϕr1  cos  I1(1+ϕd)  cos  I1r12d                                      +2  cos  I2r2(cos2θsin2I21)+2  cos  I2r2(1r2sin2θsin2I2),
Δϕ2cos(2θ)r2I22+8r1[r1ϕ(1+M)2M]2I12.
I2Mcos(2θ)I1.
Δsr1M2I1ε1.
I21Mcos(2θ)I1.
Δϕ4I1M(ε2r2+ε1r1).
I2M12cos(2θ12)I1 ,
I4M34cos(2θ34)I3.
I3=1M12r3(1M122)cos(2θ34)r1{M342sin[2(θ12+θ23)]sin(2θ34)+(1M342)   cos[2(θ12+θ23)]cos(2θ34)}I1.
I31M12±r3r1(1M1221M342)I1.
Δsr1Mcos(2θ)2I22r1M22I12+[2M2cos(2θ)I222M2I12]εs
Δϕ2cos(2θ)r1MI22+2r1M2I12+2(1M3+1M2)I12εϕ.
Δs2M2(1+M)I12εs ,
Δϕ21M2(1M+1)I12εϕ.
Δϕ={r32[r2M12+r3(M121)]cos[2(θ12+θ23)]cos(2θ34)          r32r4(1M12)cos[2(θ12+θ23+θ34)]r2r42M12cos[2(θ12+θ23+θ34)]}           ×2(1+M12)r2M123M342{r32cos[2(θ12+θ23)]cos(2θ34)r42cos[2(θ12+θ23+θ34)]} I12 εϕ,
Δd1M3r1I14+(1+M)(211+59M)180r1I16+3+2Mr12I16εd.

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