Abstract

A new bifocal optical system used for distant object tracking is proposed. This system combines a birefringent element with a conventional glass lens so that the spot image size and its variation with the axial distance can be controlled according to the requirement of a distant object tracker. The lens design for the tracking application is discussed and an example is given. The new lens system provides a more uniform spot image size and an extended focal depth compared to a conventional lens with one focus.

© 2005 Optical Society of America

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References

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  1. Carl  Christian Liebe, “Accuracy performance of star tracking-a tutorial,” IEEE Transactions on Aerospace and Electronic Systems 38, 587–599 (2002).
    [CrossRef]
  2. G.  Borghi, D.  Procopio, M.  Magnani, S.  Pieri, S.  Becucci, “Stellar reference unit for CASSINI mission,” Proc. SPIE 2210, 150–161 (1994).
    [CrossRef]
  3. Giancarlo  Rufino, Domenico  Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica 53, 135–147 (2003).
    [CrossRef]
  4. Carl  Christian Liebe, “Star trackers for attitude determination,” IEEE AES Systems Magazines, 10–16 (1995).
    [CrossRef]
  5. K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
    [CrossRef]
  6. J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
    [CrossRef]
  7. E. R.  Dowski, W. T.  Catchy, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. Arthur  Cox, A system of optical design (The Focal Press, London and New York 1964).
  13. S.  Sanyal, A.  Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
    [CrossRef]

2003

Giancarlo  Rufino, Domenico  Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica 53, 135–147 (2003).
[CrossRef]

S.  Mezouari, A. R.  Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28, 771–773 (2003).
[CrossRef] [PubMed]

2002

Carl  Christian Liebe, “Accuracy performance of star tracking-a tutorial,” IEEE Transactions on Aerospace and Electronic Systems 38, 587–599 (2002).
[CrossRef]

2001

2000

S.  Sanyal, A.  Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
[CrossRef]

1996

K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
[CrossRef]

1995

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

E. R.  Dowski, W. T.  Catchy, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
[CrossRef]

1994

G.  Borghi, D.  Procopio, M.  Magnani, S.  Pieri, S.  Becucci, “Stellar reference unit for CASSINI mission,” Proc. SPIE 2210, 150–161 (1994).
[CrossRef]

1992

1990

Jorge  Ojeda-Castaneda, L. R.  Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” App. Opt. 29, 994–997 (1990).
[CrossRef]

Accardo, Domenico

Giancarlo  Rufino, Domenico  Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica 53, 135–147 (2003).
[CrossRef]

Bará, S.

Becucci, S.

G.  Borghi, D.  Procopio, M.  Magnani, S.  Pieri, S.  Becucci, “Stellar reference unit for CASSINI mission,” Proc. SPIE 2210, 150–161 (1994).
[CrossRef]

Berriel-Valdos, L. R.

Jorge  Ojeda-Castaneda, L. R.  Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” App. Opt. 29, 994–997 (1990).
[CrossRef]

Borghi, G.

G.  Borghi, D.  Procopio, M.  Magnani, S.  Pieri, S.  Becucci, “Stellar reference unit for CASSINI mission,” Proc. SPIE 2210, 150–161 (1994).
[CrossRef]

Catchy, W. T.

E. R.  Dowski, W. T.  Catchy, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
[CrossRef]

Chi, W.

Christian Liebe, Carl

Carl  Christian Liebe, “Accuracy performance of star tracking-a tutorial,” IEEE Transactions on Aerospace and Electronic Systems 38, 587–599 (2002).
[CrossRef]

Carl  Christian Liebe, “Star trackers for attitude determination,” IEEE AES Systems Magazines, 10–16 (1995).
[CrossRef]

Cox, Arthur

Arthur  Cox, A system of optical design (The Focal Press, London and New York 1964).

Dowski, E. R.

E. R.  Dowski, W. T.  Catchy, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
[CrossRef]

George, N.

Ghosh, A.

S.  Sanyal, A.  Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
[CrossRef]

Harvey, A. R.

Hasimoto, T.

K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
[CrossRef]

Hills, R. F.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Hirokawa, E.

K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
[CrossRef]

Jaroszewicz, Z.

Kasikawa, R.

K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
[CrossRef]

Kolodziejczyk, A.

Kordas, J. F.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Ledebuhr, A. G.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Lewis, I. T.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Magnani, M.

G.  Borghi, D.  Procopio, M.  Magnani, S.  Pieri, S.  Becucci, “Stellar reference unit for CASSINI mission,” Proc. SPIE 2210, 150–161 (1994).
[CrossRef]

Mezouari, S.

Nielsen, D. P.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Ninomiya, K.

K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
[CrossRef]

Noguchi, K.

K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
[CrossRef]

Ogura, N.

K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
[CrossRef]

Ojeda-Castaneda, Jorge

Jorge  Ojeda-Castaneda, L. R.  Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” App. Opt. 29, 994–997 (1990).
[CrossRef]

Park, H.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Pieri, S.

G.  Borghi, D.  Procopio, M.  Magnani, S.  Pieri, S.  Becucci, “Stellar reference unit for CASSINI mission,” Proc. SPIE 2210, 150–161 (1994).
[CrossRef]

Pleasance, L. D.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Priest, R. E.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Procopio, D.

G.  Borghi, D.  Procopio, M.  Magnani, S.  Pieri, S.  Becucci, “Stellar reference unit for CASSINI mission,” Proc. SPIE 2210, 150–161 (1994).
[CrossRef]

Rufino, Giancarlo

Giancarlo  Rufino, Domenico  Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica 53, 135–147 (2003).
[CrossRef]

Sanyal, S.

S.  Sanyal, A.  Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
[CrossRef]

Sato, K.

K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
[CrossRef]

Shannon, M. J.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Sochacki, J.

Wilson, B. A.

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Acta Astronautica

Giancarlo  Rufino, Domenico  Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica 53, 135–147 (2003).
[CrossRef]

App. Opt.

E. R.  Dowski, W. T.  Catchy, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
[CrossRef]

Jorge  Ojeda-Castaneda, L. R.  Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” App. Opt. 29, 994–997 (1990).
[CrossRef]

S.  Sanyal, A.  Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
[CrossRef]

IEEE Transactions on Aerospace and Electronic Systems

Carl  Christian Liebe, “Accuracy performance of star tracking-a tutorial,” IEEE Transactions on Aerospace and Electronic Systems 38, 587–599 (2002).
[CrossRef]

Opt. Lett.

Proc. SPIE

G.  Borghi, D.  Procopio, M.  Magnani, S.  Pieri, S.  Becucci, “Stellar reference unit for CASSINI mission,” Proc. SPIE 2210, 150–161 (1994).
[CrossRef]

K.  Noguchi, K.  Sato, R.  Kasikawa, N.  Ogura, K.  Ninomiya, T.  Hasimoto, E.  Hirokawa, “CCD star tracker for attitude determination and control of satellite for space VLBI mission,” Proc. SPIE 2810, 190–200 (1996).
[CrossRef]

SPIE

J. F.  Kordas, I. T.  Lewis, B. A.  Wilson, D. P.  Nielsen, H.  Park, R. E.  Priest, R. F.  Hills, M. J.  Shannon, A. G.  Ledebuhr, L. D.  Pleasance, “Star tracker stellar compass for the Clementine mission,” SPIE 2466, 70–83 (1995).
[CrossRef]

Other

Arthur  Cox, A system of optical design (The Focal Press, London and New York 1964).

Carl  Christian Liebe, “Star trackers for attitude determination,” IEEE AES Systems Magazines, 10–16 (1995).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of a lens having two foci

Fig. 2.
Fig. 2.

Spot size versus the value of Δs

Fig. 3.
Fig. 3.

Spot size versus the axial distance

Fig. 4.
Fig. 4.

Schematic of the combination of a bifocal lens

Fig. 5.
Fig. 5.

Configuration of an optical system comprising a birefringent lens

Fig. 6.
Fig. 6.

Spot size for e rays oe-13-1-136-i001.jpg

Fig. 7.
Fig. 7.

Spot size of a bifocal lens system oe-13-1-136-i002.jpg

Equations (19)

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

D H ( Ω ) = D 0 ( Ω ) D g ( Ω ) .
I ( r ) = F 1 [ D 0 ( Ω ) D g ( Ω ) ] = I 0 ( r ) * I g ( r ) ,
I 0 ( r ) = [ 2 · J 1 ( λ · F n · r ) λ · F n · r ] 2 ,
I ( r , Δ s , δ z ) = I 0 ( r ) * I g 1 ( r ) + I 0 ( r ) * I g 2 ( r ) .
I g 1 ( r ) = { 1 π [ 2 F n ( Δ s 2 ) + δ z ] 2 r ( Δ s 2 ) + δ z 2 F n 0 others ,
and I g 2 ( r ) = { 1 π [ 2 F n ( Δ s 2 ) δ z ] 2 r ( Δ s 2 ) + δ z 2 F n 0 others .
I ( r , Δ s , δ z ) = 0 a 1 ( 2 J 1 [ λ . F n . ( r τ ) ] λ . F n . ( r τ ) ) 2 d τ + 0 a 2 ( 2 J 1 [ λ . F n . ( r τ ) ] λ . F n . ( r τ ) ) 2 d τ ,
where a 1 = [ ( Δ s 2 ) + δ z ] 2 F n ,
a 2 = [ ( Δ s 2 ) δ z ] 2 F n .
RED ( c , Δ s , δ z ) = 0 c 2 π · I ( r , Δ s , δ z ) · r · dr 0 2 π · I ( r , Δ s , 0 ) · r · dr .
f 1 o = r 1 ( n o 1 ) ,
f 1 e = r 1 ( n e 1 ) .
Δ d 1 = f 1 o f 1 e .
Δ d 1 = ( n e n o ) · r 1 ( n o 1 ) · ( n e 1 ) .
Δ d = ( x 2 x 2 ) · Δ d 1 ,
f 2 2 = x 2 · x 2 ,
1 f = 1 f 1 e ' + 1 f 2 ( a + f 2 ) f 1 e f 2 .
a = f 1 e x 2 .
Δ d = f 2 ( n e 1 ) ( n e n o ) ( n o 1 ) · r 1 .

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