Abstract

We report on an analytical formulation, based on the concept of effective Fresnel number, to evaluate in a simple way the relative focal shift of rotationally nonsymmetric scalar fields that have geometrical focus and moderate Fresnel number. To illustrate our approach, certain previously known results and also some new focusing setups are analytically examined.

© 1998 Optical Society of America

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References

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  1. E. Collet, E. Wolf, “Symmetry properties of focused fields,” Opt. Lett. 5, 264–266 (1980).
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.1.
  3. E. Wolf, Y. Li, “Conditions for the validity of Debye integral representation of focused fields,” Opt. Commun. 39, 205–209 (1981).
    [CrossRef]
  4. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
    [CrossRef] [PubMed]
  5. A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to a circular aperture in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
    [CrossRef]
  6. M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
    [CrossRef]
  7. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  8. J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  9. M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
    [CrossRef]
  10. Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
    [CrossRef]
  11. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]
  12. J. Ojeda-Castañeda, M. Martı́nez-Corral, P. Andrés, A. Pons, “Strehl ratio versus defocus for noncentrally obscured pupils,” Appl. Opt. 33, 7611–7616 (1994).
    [CrossRef] [PubMed]
  13. M. Martı́nez-Corral, V. Climent, “Focal switch: a new effect in low-Fresnel-number systems,” Appl. Opt. 35, 24–27 (1996).
    [CrossRef] [PubMed]
  14. S. Szapiel, “Maréchal intensity formula for small-Fresnel-number systems,” Opt. Lett. 8, 327–329 (1983).
    [CrossRef] [PubMed]
  15. M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, “On-axis diffractional behavior of two-dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
    [CrossRef] [PubMed]
  16. P. Andrés, M. Martı́nez-Corral, J. Ojeda-Castañeda, “Off-axis focal shift for rotationally nonsymmetric screens,” Opt. Lett. 18, 1290–1292 (1993).
    [CrossRef] [PubMed]
  17. G. Goubau, “Optical relations for coherent wave beams,” in Electromagnetic Theory and Antennas, E. C. Jordan, ed. (Macmillan, New York, 1963), Part 2, pp. 907–918.
  18. A. G. van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).
  19. H. Kogelnik, “Imaging of optical modes–resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–594 (1965).
    [CrossRef]
  20. W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
    [CrossRef] [PubMed]
  21. Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [CrossRef]
  22. Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
    [CrossRef]
  23. C. W. McCutchen, “Generalized aperture and three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  24. J. Ojeda-Castañeda, P. Andrés, M. Martı́nez-Corral, “Zero axial irradiance by annular screens with angular variation,” Appl. Opt. 31, 4600–4602 (1992).
    [CrossRef] [PubMed]
  25. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 212–214.
  26. J. F. Nye, “Diffraction by a small unstopped lens,” J. Mod. Opt. 38, 743–754 (1991).
    [CrossRef]
  27. Specifically, it is shown in Refs. 17-22 that the point of maximum axial irradiance is located at a normalized distance zmax/f=-1/(1+π2ω4/λ2f2) from the geometrical focus. We would like to remark that this equation is equivalent to Eq. (28) in the text, except for an additive constant in the denominator. It is easy to show that the difference between the two results is very low for moderate values of Neff.
  28. C. J. R. Sheppard, Z. S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
    [CrossRef]
  29. M. J. Yzuel, J. C. Escalera, J. Campos, “Polychromatic axial behavior of axial apodizing and hyperresolving filters,” Appl. Opt. 29, 1631–1641 (1990).
    [CrossRef] [PubMed]

1996 (1)

1994 (2)

1993 (1)

1992 (2)

1991 (1)

J. F. Nye, “Diffraction by a small unstopped lens,” J. Mod. Opt. 38, 743–754 (1991).
[CrossRef]

1990 (1)

1988 (1)

1983 (3)

1982 (3)

M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
[CrossRef] [PubMed]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

1981 (3)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of Debye integral representation of focused fields,” Opt. Commun. 39, 205–209 (1981).
[CrossRef]

1980 (1)

1977 (1)

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[CrossRef]

1976 (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to a circular aperture in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

1972 (1)

1965 (1)

H. Kogelnik, “Imaging of optical modes–resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–594 (1965).
[CrossRef]

1964 (2)

A. G. van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).

C. W. McCutchen, “Generalized aperture and three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

Andrés, P.

Arimoto, A.

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to a circular aperture in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Avizonis, P. V.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.1.

Campos, J.

Carter, W. H.

Climent, V.

Collet, E.

Escalera, J. C.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 212–214.

Givens, M. P.

M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

Goubau, G.

G. Goubau, “Optical relations for coherent wave beams,” in Electromagnetic Theory and Antennas, E. C. Jordan, ed. (Macmillan, New York, 1963), Part 2, pp. 907–918.

Gusinow, M. A.

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[CrossRef]

Hegedus, Z. S.

Holmes, D. A.

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes–resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–594 (1965).
[CrossRef]

Korka, J. E.

Li, Y.

Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
[CrossRef]

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of Debye integral representation of focused fields,” Opt. Commun. 39, 205–209 (1981).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Mahajan, V. N.

Marti´nez-Corral, M.

McCutchen, C. W.

Nye, J. F.

J. F. Nye, “Diffraction by a small unstopped lens,” J. Mod. Opt. 38, 743–754 (1991).
[CrossRef]

Ojeda-Castañeda, J.

Palmer, M. A.

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[CrossRef]

Platzer, H.

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Pons, A.

Riley, M. E.

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[CrossRef]

Sheppard, C. J. R.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Szapiel, S.

van Nie, A. G.

A. G. van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).

Wolf, E.

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of Debye integral representation of focused fields,” Opt. Commun. 39, 205–209 (1981).
[CrossRef]

E. Collet, E. Wolf, “Symmetry properties of focused fields,” Opt. Lett. 5, 264–266 (1980).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.1.

Yzuel, M. J.

Appl. Opt. (8)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes–resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–594 (1965).
[CrossRef]

J. Mod. Opt. (2)

J. F. Nye, “Diffraction by a small unstopped lens,” J. Mod. Opt. 38, 743–754 (1991).
[CrossRef]

Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (2)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to a circular aperture in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Opt. Commun. (5)

E. Wolf, Y. Li, “Conditions for the validity of Debye integral representation of focused fields,” Opt. Commun. 39, 205–209 (1981).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[CrossRef]

Philips Res. Rep. (1)

A. G. van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).

Other (4)

G. Goubau, “Optical relations for coherent wave beams,” in Electromagnetic Theory and Antennas, E. C. Jordan, ed. (Macmillan, New York, 1963), Part 2, pp. 907–918.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.1.

Specifically, it is shown in Refs. 17-22 that the point of maximum axial irradiance is located at a normalized distance zmax/f=-1/(1+π2ω4/λ2f2) from the geometrical focus. We would like to remark that this equation is equivalent to Eq. (28) in the text, except for an additive constant in the denominator. It is easy to show that the difference between the two results is very low for moderate values of Neff.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 212–214.

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Figures (4)

Fig. 1
Fig. 1

Two typical examples of scalar fields that have geometrical focus: (a) radially nonsymmetric diffracting screen illuminated by a monochromatic spherical wave with focal length f, (b) truncated spherical Gaussian beam.

Fig. 2
Fig. 2

Relative focal shift as a function of the effective Fresnel number of the focused beam.

Fig. 3
Fig. 3

Relative error associated with the focal shift determined by Eq. (18) versus the effective Fresnel number of the focused beam.

Fig. 4
Fig. 4

Amplitude transmittance of the axially superresolving diffracting screen (solid curves) and the axially apodizing parabolic filter (dashed curves): (a) ζ-space representation, (b) radial coordinate representation.

Equations (41)

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U(r, θ)=exp(-ikf)fexp-i k2 fr2t(r, θ),
h(z)=exp(ikz)iλf(f+z)02π0t(r, θ)×exp-i2πz2λf(f+z)r2r drdθ,
h(z)=2π exp(ikz)iλf(f+z)0t0(r)×exp-i2π z2λf(f+z)r2r dr,
t0(r)=12π02πt(r, θ)dθ
ζ=r2,t0(r)=q0(ζ),
h(z)=πλf(f+z)-q0(ζ)exp-i2π z2λf(f+z)ζdζ,
u=z2λf(z+f),
h(z)=Q(u)=πλf2(1-2λfu)-q0(ζ)exp(-i2πuζ)dζ.
IN(u)=(1-2λfu)2 -q0(ζ)exp(-i2πuζ)dζ2-q0(ζ)dζ2.
dIN/du=0
IN(u)=IN(0)+IN(0)u+IN(0)2u2.
IN(u)=1-4λfu+4(λ2f2-π2σ2)u2,
σ=m2m0-m1m021/2
mn=-q0(ζ)ζn dζ,
umax=12λf11-π2Neff2,
IN(umax)=π2Neff2π2Neff2-1,
Neff=σ/λf.
zmaxf=-1π2Neff2.
Δ(zmax)|zmax|=2π2Neff2-1,
ΔImax=2π2Neff2(π2Neff2-1)3.
Neff=(1-2)12a2λf.
N=(1-2)a2λf,
zmaxf=-12π2N2,
IN(zmax)=π2N2π2N2-12,
α=(a/ω)2,
Neff=ω2λf1-α2 exp(α)[1-exp(α)]21/2.
zmaxf=-λfπ2ω2{1-α2 exp(α)/[1-exp(α)]2}.
Neff=ω2/λf.
zmaxf=-λ2f2π2ω4,
q0s(ζ)=4[(ζ/a2)-0.5]2if0ζa20otherwise,
q0a(ζ)=1-q0s(ζ)if0ζa20otherwise,
Neffs=320 a2λf,
Neffa=120 a2λf.
zmaxsf=-203π2λfa22,
zmaxaf=-20π2λfa22.
R(IN; 0, u)=IN(u0)6u3,
IN(u0)IN(0)=96π2σ2λf.
ΔImax=Δ[IN(umax)]=2π2Neff2(π2Neff2-1)3.
zmaxf=1IN(umax)-1.
Δzmaxf=2π2Neff21π2Neff2-1.
Δ(zmax)|zmax|=2π2Neff2-1.

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