Abstract

Low-Fresnel number optical systems exhibit significant diffraction effects that cause a shift in the peaks of on-axis irradiance away from the geometric focal point. This is currently interpreted as a change of the focal length of an optical system, leading optical system designers to compensate for the effect by assuming the image plane is coincident with the peak of on-axis irradiance. While this may be an appropriate interpretation for certain applications, I show that despite the shift in peak irradiance away from the geometrical focal point, a change in a system’s optical power will not increase the on-axis irradiance at that distance. This is important for low-light level applications where it is necessary to mitigate diffraction induced transmission losses. I also show that low-Fresnel number systems have increased tolerance on system power at the geometrical focal point and as a result are inherently achromatic.

© 2009 Optical Society of America

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References

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2008 (1)

C. Baranec, “High-order wavefront sensing system for PALM-3000,” Proc. SPIE 7015, 70155M (2008).
[CrossRef]

2007 (1)

2006 (4)

P. Ruffieux, T. Scharf, H. P. Herzig, R. Völkel, and K. J. Weible, “On the chromatic aberration of microlenses,” Opt. Express 14, 4687-4694 (2006).
[CrossRef] [PubMed]

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: application to the SPHERE instrument,” Opt. Express 14, 7515-7534(2006).
[CrossRef] [PubMed]

R. Dekany, A. Bouchez, M. Britton, V. Velur, M. Troy, J. C. Shelton, and J. Roberts, “PALM-3000: visible-light AO on the 5.1-meter Telescope,” Proc. SPIE 6272, 62720G (2006).
[CrossRef]

2005 (1)

2004 (1)

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

2003 (1)

1998 (3)

1987 (1)

1982 (1)

1961 (1)

Andrés, P.

Baranec, C.

C. Baranec, “High-order wavefront sensing system for PALM-3000,” Proc. SPIE 7015, 70155M (2008).
[CrossRef]

Baudoz, P.

Beuzit, J.-L.

Borghi, R.

R. Borghi, M. Santarsiero, and S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243-248(1998).
[CrossRef]

Bouchez, A.

R. Dekany, A. Bouchez, M. Britton, V. Velur, M. Troy, J. C. Shelton, and J. Roberts, “PALM-3000: visible-light AO on the 5.1-meter Telescope,” Proc. SPIE 6272, 62720G (2006).
[CrossRef]

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Brack, G.

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

Britton, M.

R. Dekany, A. Bouchez, M. Britton, V. Velur, M. Troy, J. C. Shelton, and J. Roberts, “PALM-3000: visible-light AO on the 5.1-meter Telescope,” Proc. SPIE 6272, 62720G (2006).
[CrossRef]

Campbell, R.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Carter, W.

Charton, J.

Cheselka, M.

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

Chin, J.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Dekany, R.

R. Dekany, A. Bouchez, M. Britton, V. Velur, M. Troy, J. C. Shelton, and J. Roberts, “PALM-3000: visible-light AO on the 5.1-meter Telescope,” Proc. SPIE 6272, 62720G (2006).
[CrossRef]

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

Dohlen, K.

Fusco, T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Hartman, S.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Herzig, H. P.

Johansson, E.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Kasper, M.

Kibblewhite, E.

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

Lafon, R.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Li, Y.

Martínez-Corral, M.

Mignant, D.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Mouillet, D.

Nicolle, M.

Osterberg, H.

Petit, C.

Petrie, H.

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

Puget, P.

Roberts, J.

R. Dekany, A. Bouchez, M. Britton, V. Velur, M. Troy, J. C. Shelton, and J. Roberts, “PALM-3000: visible-light AO on the 5.1-meter Telescope,” Proc. SPIE 6272, 62720G (2006).
[CrossRef]

Rousset, G.

Ruffieux, P.

Santarsiero, M.

R. Borghi, M. Santarsiero, and S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243-248(1998).
[CrossRef]

Sauvage, J.-F.

Scharf, T.

Shelton, J. C.

R. Dekany, A. Bouchez, M. Britton, V. Velur, M. Troy, J. C. Shelton, and J. Roberts, “PALM-3000: visible-light AO on the 5.1-meter Telescope,” Proc. SPIE 6272, 62720G (2006).
[CrossRef]

Sheppard, C. J. R.

Silvestre, E.

Smith, L. W.

Stomski, P.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Summers, D.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Thicksten, R.

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

Törörk, P.

Trin, T.

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

Troy, M.

R. Dekany, A. Bouchez, M. Britton, V. Velur, M. Troy, J. C. Shelton, and J. Roberts, “PALM-3000: visible-light AO on the 5.1-meter Telescope,” Proc. SPIE 6272, 62720G (2006).
[CrossRef]

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

van Dam, M.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Velur, V.

R. Dekany, A. Bouchez, M. Britton, V. Velur, M. Troy, J. C. Shelton, and J. Roberts, “PALM-3000: visible-light AO on the 5.1-meter Telescope,” Proc. SPIE 6272, 62720G (2006).
[CrossRef]

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

Vicalvi, S.

R. Borghi, M. Santarsiero, and S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243-248(1998).
[CrossRef]

Völkel, R.

Weible, K. J.

Wizinowich, P.

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Zapata-Rodríguez, C. J.

Zhong, Y.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

R. Borghi, M. Santarsiero, and S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243-248(1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (3)

V. Velur, E. Kibblewhite, R. Dekany, M. Troy, H. Petrie, R. Thicksten, G. Brack, T. Trin, and M. Cheselka, “Implementation of the Chicago sum frequency laser at Palomar laser guide star test bed,” Proc. SPIE 5490, 1033-1040 (2004).
[CrossRef]

R. Dekany, A. Bouchez, M. Britton, V. Velur, M. Troy, J. C. Shelton, and J. Roberts, “PALM-3000: visible-light AO on the 5.1-meter Telescope,” Proc. SPIE 6272, 62720G (2006).
[CrossRef]

C. Baranec, “High-order wavefront sensing system for PALM-3000,” Proc. SPIE 7015, 70155M (2008).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

M. van Dam, A. Bouchez, D. Mignant, E. Johansson, P. Wizinowich, R. Campbell, J. Chin, S. Hartman, R. Lafon, P. Stomski, and D. Summers, “The W. M. Keck observatory laser guide star adaptive optics system: performance characterization,” Publ. Astron. Soc. Pac. 118, 310-318 (2006).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (6)

Fig. 1
Fig. 1

Examples of the shift in Z p away from the geometrical focal distance of 0.15 m for optical systems of different Fresnel number, normalized to peak irradiance. All systems assume a uniformly illuminated circular aperture of diameter 600 μm with a thin lens, ϕ = 6.67 m 1 . Different Fresnel numbers are achieved by varying λ from 6 nm to 6 μm .

Fig. 2
Fig. 2

Geometry of a coincident aperture Σ of radius a and a thin lens of power ϕ at P 1 .

Fig. 3
Fig. 3

Relative on-axis irradiance as a function of distance from the exit pupil of a uniformly illuminated ( λ = 635 nm ) system with a 600 μm circular aperture and various optical power lenses. A magnification of the area around z = 0.15 m is shown (upper right).

Fig. 4
Fig. 4

Experimental setup to measure the on-axis irradiance at z = 0.15 m as a function of lens optical power. The λ = 635 nm laser is coupled to a single mode fiber, and the diverging output is collimated by an achromatic lens. The stability of irradiance of the collimated beam is measured with a beam splitter and overfilled photodiode detector. The collimated light is incident on the planar side of a plano-convex lens with the convex side in contact with the 600 μm diameter circular aperture. A CMOS detector is then placed 0.15 m away from the aperture.

Fig. 5
Fig. 5

Relative on-axis irradiance at z = 0.15 m for an optical system comprised of a uniformly illuminated ( λ = 635 nm ) 600 μm diameter circular aperture with various optical power lenses ( 1 / ϕ shown). The measured data are precise to ± 0.002 in relative irradiance as scaled here and to within ± 1 % of 1 / ϕ .

Fig. 6
Fig. 6

Relative on-axis irradiance at z = 0.15 m as a function of optical power for the different Fresnel number systems presented in Fig. 1.

Equations (5)

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FN = ϕ a 2 / λ ,
U ( P 0 ) = 1 j λ Σ U ( P 1 ) exp [ 2 π j λ r ] r cos [ r ] d s ,
U ( P 1 ) = A exp [ 2 π j λ ϕ ( z 2 r 2 ) 2 ]
U ( z ) = A j λ z z 2 + a 2 exp [ 2 π j λ ( r + ϕ 2 ( z 2 r 2 ) ) ] cos [ r ] r d r .
Δ ( 1 / ϕ ) = R o C Δ n ( n - 1 ) 2 ,

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