Abstract

A single analytical formulation for evaluating the focal shift in any apodized nontelecentric focusing setup is reported. The formulation is also useful in the case of imaged paraxial beams. We show explicitly that the magnitude of the focal shift is determined by only one parameter that depends on the effective width of the pupil filter and its axial position. To illustrate our approach we examine different focusing setups.

© 2001 Optical Society of America

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References

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  1. Y. Li and E. Wolf, Opt. Commun. 39, 211 (1981).
    [CrossRef]
  2. Y. Li and E. Wolf, J. Opt. Soc. Am. A 11, 801 (1994).
  3. W. Wang and E. Wolf, Opt. Commun. 119, 453 (1995).
    [CrossRef]
  4. B. Lü, B. Zhang, and B. Cai, J. Mod. Opt. 42, 289 (1995).
    [CrossRef]
  5. Y. Li, J. Opt. Soc. Am. A 14, 1297 (1997).
    [CrossRef]
  6. G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, Appl. Opt. 36, 8091 (1997).
    [CrossRef]
  7. C. J. R. Sheppard and P. Török, Opt. Lett. 23, 1803 (1998).
    [CrossRef]
  8. S. de Nicola, D. Anderson, and H. Lisak, Pure Appl. Opt. 7, 1249 (1998).
    [CrossRef]
  9. M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and M. Kowalczyk, J. Mod. Opt. 45, 1671 (1998).
    [CrossRef]
  10. P. L. Greene and D. G. Hall, Opt. Express 4, 411 (1999); http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  11. I. G. Palchikova and S. G. Rautian, Opt. Commun. 174, 1 (2000).
    [CrossRef]
  12. C. J. R. Sheppard and K. Larkin, J. Opt. Soc. Am. A 17, 772 (2000).
    [CrossRef]
  13. B. Lü and X. Ji, Opt. Commun. 189, 47 (2001).
    [CrossRef]
  14. C. J. R. Sheppard and P. Török, J. Opt. Soc. Am. A 15, 3016 (1998).
    [CrossRef]
  15. A detailed definition of telecentricity can be found in M. Born and E. Wolf, eds., Principles of Optics, 7th ed. (Cambridge University, Cambridge, 1999), p. 200.
  16. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  17. M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and E. Silvestre, J. Opt. Soc. Am. A 15, 449 (1998).
    [CrossRef]
  18. J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, Opt. Commun. 114, 189 (1995).
    [CrossRef]
  19. P. W. Milonni and J. M. Eberly, Lasers (Wiley, New York, 1998).

2001 (1)

B. Lü and X. Ji, Opt. Commun. 189, 47 (2001).
[CrossRef]

2000 (2)

I. G. Palchikova and S. G. Rautian, Opt. Commun. 174, 1 (2000).
[CrossRef]

C. J. R. Sheppard and K. Larkin, J. Opt. Soc. Am. A 17, 772 (2000).
[CrossRef]

1999 (1)

1998 (5)

1997 (2)

1995 (3)

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, Opt. Commun. 114, 189 (1995).
[CrossRef]

W. Wang and E. Wolf, Opt. Commun. 119, 453 (1995).
[CrossRef]

B. Lü, B. Zhang, and B. Cai, J. Mod. Opt. 42, 289 (1995).
[CrossRef]

1994 (1)

Y. Li and E. Wolf, J. Opt. Soc. Am. A 11, 801 (1994).

1981 (1)

Y. Li and E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Anderson, D.

S. de Nicola, D. Anderson, and H. Lisak, Pure Appl. Opt. 7, 1249 (1998).
[CrossRef]

Andrés, P.

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and E. Silvestre, J. Opt. Soc. Am. A 15, 449 (1998).
[CrossRef]

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and M. Kowalczyk, J. Mod. Opt. 45, 1671 (1998).
[CrossRef]

Beijersbergen, M. W.

Cai, B.

B. Lü, B. Zhang, and B. Cai, J. Mod. Opt. 42, 289 (1995).
[CrossRef]

de Nicola, S.

S. de Nicola, D. Anderson, and H. Lisak, Pure Appl. Opt. 7, 1249 (1998).
[CrossRef]

Eberly, J. M.

P. W. Milonni and J. M. Eberly, Lasers (Wiley, New York, 1998).

Escalera, J. C.

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, Opt. Commun. 114, 189 (1995).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Greene, P. L.

Hall, D. G.

Ji, X.

B. Lü and X. Ji, Opt. Commun. 189, 47 (2001).
[CrossRef]

Karman, G. P.

Kowalczyk, M.

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and M. Kowalczyk, J. Mod. Opt. 45, 1671 (1998).
[CrossRef]

Larkin, K.

Li, Y.

Y. Li, J. Opt. Soc. Am. A 14, 1297 (1997).
[CrossRef]

Y. Li and E. Wolf, J. Opt. Soc. Am. A 11, 801 (1994).

Y. Li and E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Lisak, H.

S. de Nicola, D. Anderson, and H. Lisak, Pure Appl. Opt. 7, 1249 (1998).
[CrossRef]

Lü, B.

B. Lü and X. Ji, Opt. Commun. 189, 47 (2001).
[CrossRef]

B. Lü, B. Zhang, and B. Cai, J. Mod. Opt. 42, 289 (1995).
[CrossRef]

Martínez-Corral, M.

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and E. Silvestre, J. Opt. Soc. Am. A 15, 449 (1998).
[CrossRef]

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and M. Kowalczyk, J. Mod. Opt. 45, 1671 (1998).
[CrossRef]

Milonni, P. W.

P. W. Milonni and J. M. Eberly, Lasers (Wiley, New York, 1998).

Ojeda-Castañeda, J.

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, Opt. Commun. 114, 189 (1995).
[CrossRef]

Palchikova, I. G.

I. G. Palchikova and S. G. Rautian, Opt. Commun. 174, 1 (2000).
[CrossRef]

Rautian, S. G.

I. G. Palchikova and S. G. Rautian, Opt. Commun. 174, 1 (2000).
[CrossRef]

Sheppard, C. J. R.

Silvestre, E.

Török, P.

van Duijl, A.

Wang, W.

W. Wang and E. Wolf, Opt. Commun. 119, 453 (1995).
[CrossRef]

Woerdman, J. P.

Wolf, E.

W. Wang and E. Wolf, Opt. Commun. 119, 453 (1995).
[CrossRef]

Y. Li and E. Wolf, J. Opt. Soc. Am. A 11, 801 (1994).

Y. Li and E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Yzuel, M. J.

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, Opt. Commun. 114, 189 (1995).
[CrossRef]

Zapata-Rodríguez, C. J.

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and E. Silvestre, J. Opt. Soc. Am. A 15, 449 (1998).
[CrossRef]

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and M. Kowalczyk, J. Mod. Opt. 45, 1671 (1998).
[CrossRef]

Zhang, B.

B. Lü, B. Zhang, and B. Cai, J. Mod. Opt. 42, 289 (1995).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (2)

B. Lü, B. Zhang, and B. Cai, J. Mod. Opt. 42, 289 (1995).
[CrossRef]

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, and M. Kowalczyk, J. Mod. Opt. 45, 1671 (1998).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (5)

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, Opt. Commun. 114, 189 (1995).
[CrossRef]

W. Wang and E. Wolf, Opt. Commun. 119, 453 (1995).
[CrossRef]

I. G. Palchikova and S. G. Rautian, Opt. Commun. 174, 1 (2000).
[CrossRef]

B. Lü and X. Ji, Opt. Commun. 189, 47 (2001).
[CrossRef]

Y. Li and E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Pure Appl. Opt. (1)

S. de Nicola, D. Anderson, and H. Lisak, Pure Appl. Opt. 7, 1249 (1998).
[CrossRef]

Other (3)

A detailed definition of telecentricity can be found in M. Born and E. Wolf, eds., Principles of Optics, 7th ed. (Cambridge University, Cambridge, 1999), p. 200.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

P. W. Milonni and J. M. Eberly, Lasers (Wiley, New York, 1998).

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Figures (3)

Fig. 1
Fig. 1

Geometry of the focusing setup.

Fig. 2
Fig. 2

Relative focal shift versus ζN for the circular aperture, assuming N=2.

Fig. 3
Fig. 3

Relative focal shift versus ζN for the Gaussian ring. The parameters for the calculations are N=4 and Ω=0.205. In the inset the exact values of the focal shift that correspond to the circular aperture (dashed curve) and to the selected Gaussian ring (dotted curve) are represented.

Equations (12)

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

U0r0=exp-ik2ζr021Mtr0M,
hz=2πiλz+ζ01Mtr0M×exp-i2πz2λζz+ζr02r0dr0,
INu=1-2λζu2-+qμ/M2exp-i2πμudμ2-+qμ/M2dμ2.
u=z2λζz+ζ.
INu1-4λζu+4λ2ζ2-π2M4σ2u2,
umax=signζN2λfζN2π2N2-ζN,
INumax=π2N2π2N2-ζN.
N=σλfζN,
zmaxf=signζNπ2N2.
zmaxf=λ2f2ζNπ2w04,
qμ=exp-μ2/Ω2rectμ.
σ=rmax2ΩπΩ-exp-1/4 Ω2erf1/2Ω/2π1/21/2,

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