Abstract

The concept of propagation invariance in partially coherent optics is introduced. Explicit expressions are given for the cross-spectral density and the angular correlation function (cross-angular spectrum) characterizing a class of fields that are propagation invariant in the sense that their correlation properties in the space-frequency domain are exactly the same in every transverse plane. The so-called diffraction-free beams are shown to be members of this new, wider class of wave fields, which itself is a subset of a generalized class of partially coherent self-imaging fields. The existence of partially coherent propagation-invariant fields with a sharp correlation peak is verified experimentally by considering radiation from a planar J0 Bessel-correlated source.

© 1991 Optical Society of America

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  3. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [Crossref] [PubMed]
  4. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [Crossref]
  5. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [Crossref] [PubMed]
  6. A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng. 311, 85–89 (1989).
  7. R. G. Cañas, R. W. Smith, “Measurements with surface relief diffractive optical elements,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng.311, 160–163 (1989).
  8. J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
    [Crossref] [PubMed]
  9. S. Y. Cai, A. Bhattacharjee, T. C. Marshall, “Diffraction-free optical beams in inverse free electron laser accelerators,” Nucl. Instrum. Methods Phys. Res. A 272, 481–484 (1988).
    [Crossref]
  10. M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
    [Crossref]
  11. Y. Ohtsuka, Y. Nozoe, Y. Imai, “Acoustically modified spatial coherence in optical Fresnel diffraction region,” Opt. Commun. 35, 157–160 (1980).
    [Crossref]
  12. Y. Ohtsuka, “Non-modified propagation of optical mutual intensity in the Fresnel diffraction region,” Opt. Commun. 39, 283–286 (1981).
    [Crossref]
  13. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross-spectra of steady-state sources,”J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]
  14. F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [Crossref]
  15. G. C. Sherman, “Introduction to the angular-spectrum representation of optical fields,” in Applications of Mathematics in Modern Optics, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.358, 31–38 (1982).
    [Crossref]
  16. E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,”J. Opt. Soc. Am. 62, 379–385 (1972).
    [Crossref]
  17. L. Mandel, E. Wolf, “Complete coherence in space-frequency domain,” Opt. Commun. 36, 247–249 (1981).
    [Crossref]
  18. W. D. Montgomery, “Self-imaging objects of infinite aperture,”J. Opt. Soc. Am. 57, 772–778 (1967).
    [Crossref]
  19. W. D. Montgomery, “Algebraic formulation of diffraction applied to self-imaging,”J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [Crossref]
  20. A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
    [Crossref]
  21. A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
    [Crossref]
  22. G. Indebetouw, “Self-imaging through a Fabry–Perot resonator,” Opt. Acta 30, 1463–1471 (1983).
    [Crossref]
  23. G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
    [Crossref]
  24. G. Indebetouw, “Spatially periodic wavefields: an experimental demonstration of the relationship between the lateral and the longitudinal spatial frequencies,” Opt. Commun. 49, 86–90 (1984).
    [Crossref]
  25. P. Swaykowski, “Self-imaging in polar coordinates,” J. Opt. Soc. Am. A 5, 185–191 (1988).
    [Crossref]
  26. G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
    [Crossref]
  27. J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
    [Crossref]
  28. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [Crossref]
  29. Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
    [Crossref]
  30. P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).
  31. E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
    [Crossref]

1989 (5)

M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
[Crossref]

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng. 311, 85–89 (1989).

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[Crossref]

G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
[Crossref]

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[Crossref] [PubMed]

1988 (6)

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

P. Swaykowski, “Self-imaging in polar coordinates,” J. Opt. Soc. Am. A 5, 185–191 (1988).
[Crossref]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[Crossref] [PubMed]

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[Crossref]

S. Y. Cai, A. Bhattacharjee, T. C. Marshall, “Diffraction-free optical beams in inverse free electron laser accelerators,” Nucl. Instrum. Methods Phys. Res. A 272, 481–484 (1988).
[Crossref]

1987 (4)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

1984 (3)

E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
[Crossref]

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
[Crossref]

G. Indebetouw, “Spatially periodic wavefields: an experimental demonstration of the relationship between the lateral and the longitudinal spatial frequencies,” Opt. Commun. 49, 86–90 (1984).
[Crossref]

1983 (3)

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[Crossref]

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[Crossref]

G. Indebetouw, “Self-imaging through a Fabry–Perot resonator,” Opt. Acta 30, 1463–1471 (1983).
[Crossref]

1982 (1)

1981 (2)

L. Mandel, E. Wolf, “Complete coherence in space-frequency domain,” Opt. Commun. 36, 247–249 (1981).
[Crossref]

Y. Ohtsuka, “Non-modified propagation of optical mutual intensity in the Fresnel diffraction region,” Opt. Commun. 39, 283–286 (1981).
[Crossref]

1980 (1)

Y. Ohtsuka, Y. Nozoe, Y. Imai, “Acoustically modified spatial coherence in optical Fresnel diffraction region,” Opt. Commun. 35, 157–160 (1980).
[Crossref]

1972 (1)

1968 (1)

1967 (1)

Agarwal, G. S.

Barreiro, J. C.

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[Crossref]

Bhattacharjee, A.

S. Y. Cai, A. Bhattacharjee, T. C. Marshall, “Diffraction-free optical beams in inverse free electron laser accelerators,” Nucl. Instrum. Methods Phys. Res. A 272, 481–484 (1988).
[Crossref]

Cai, S. Y.

S. Y. Cai, A. Bhattacharjee, T. C. Marshall, “Diffraction-free optical beams in inverse free electron laser accelerators,” Nucl. Instrum. Methods Phys. Res. A 272, 481–484 (1988).
[Crossref]

Cañas, R. G.

R. G. Cañas, R. W. Smith, “Measurements with surface relief diffractive optical elements,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng.311, 160–163 (1989).

Durnin, J.

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).

Florjanczyk, M.

M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
[Crossref]

Friberg, A. T.

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng. 311, 85–89 (1989).

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[Crossref] [PubMed]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[Crossref] [PubMed]

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

He, Q.

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Ibarra, J.

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[Crossref]

Imai, Y.

Y. Ohtsuka, Y. Nozoe, Y. Imai, “Acoustically modified spatial coherence in optical Fresnel diffraction region,” Opt. Commun. 35, 157–160 (1980).
[Crossref]

Indebetouw, G.

G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
[Crossref]

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[Crossref]

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
[Crossref]

G. Indebetouw, “Spatially periodic wavefields: an experimental demonstration of the relationship between the lateral and the longitudinal spatial frequencies,” Opt. Commun. 49, 86–90 (1984).
[Crossref]

G. Indebetouw, “Self-imaging through a Fabry–Perot resonator,” Opt. Acta 30, 1463–1471 (1983).
[Crossref]

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[Crossref]

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[Crossref]

Mandel, L.

L. Mandel, E. Wolf, “Complete coherence in space-frequency domain,” Opt. Commun. 36, 247–249 (1981).
[Crossref]

Marchand, E. W.

Marshall, T. C.

S. Y. Cai, A. Bhattacharjee, T. C. Marshall, “Diffraction-free optical beams in inverse free electron laser accelerators,” Nucl. Instrum. Methods Phys. Res. A 272, 481–484 (1988).
[Crossref]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).

Montgomery, W. D.

Nozoe, Y.

Y. Ohtsuka, Y. Nozoe, Y. Imai, “Acoustically modified spatial coherence in optical Fresnel diffraction region,” Opt. Commun. 35, 157–160 (1980).
[Crossref]

Ohtsuka, Y.

Y. Ohtsuka, “Non-modified propagation of optical mutual intensity in the Fresnel diffraction region,” Opt. Commun. 39, 283–286 (1981).
[Crossref]

Y. Ohtsuka, Y. Nozoe, Y. Imai, “Acoustically modified spatial coherence in optical Fresnel diffraction region,” Opt. Commun. 35, 157–160 (1980).
[Crossref]

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[Crossref]

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[Crossref]

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[Crossref]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Sherman, G. C.

G. C. Sherman, “Introduction to the angular-spectrum representation of optical fields,” in Applications of Mathematics in Modern Optics, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.358, 31–38 (1982).
[Crossref]

Smith, R. W.

R. G. Cañas, R. W. Smith, “Measurements with surface relief diffractive optical elements,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng.311, 160–163 (1989).

Streibl, N.

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[Crossref]

Swaykowski, P.

Tremblay, R.

M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
[Crossref]

Turunen, J.

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng. 311, 85–89 (1989).

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[Crossref] [PubMed]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[Crossref] [PubMed]

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Vasara, A.

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[Crossref] [PubMed]

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng. 311, 85–89 (1989).

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[Crossref] [PubMed]

Wolf, E.

Appl. Opt. (1)

Holographic Systems, Components, and Applications II (1)

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free beams produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng. 311, 85–89 (1989).

J. Mod. Opt. (1)

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[Crossref]

J. Opt. Soc. Am. (4)

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

Nucl. Instrum. Methods Phys. Res. A (1)

S. Y. Cai, A. Bhattacharjee, T. C. Marshall, “Diffraction-free optical beams in inverse free electron laser accelerators,” Nucl. Instrum. Methods Phys. Res. A 272, 481–484 (1988).
[Crossref]

Opt. Acta (4)

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[Crossref]

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[Crossref]

G. Indebetouw, “Self-imaging through a Fabry–Perot resonator,” Opt. Acta 30, 1463–1471 (1983).
[Crossref]

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
[Crossref]

Opt. Commun. (9)

G. Indebetouw, “Spatially periodic wavefields: an experimental demonstration of the relationship between the lateral and the longitudinal spatial frequencies,” Opt. Commun. 49, 86–90 (1984).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
[Crossref]

Y. Ohtsuka, Y. Nozoe, Y. Imai, “Acoustically modified spatial coherence in optical Fresnel diffraction region,” Opt. Commun. 35, 157–160 (1980).
[Crossref]

Y. Ohtsuka, “Non-modified propagation of optical mutual intensity in the Fresnel diffraction region,” Opt. Commun. 39, 283–286 (1981).
[Crossref]

L. Mandel, E. Wolf, “Complete coherence in space-frequency domain,” Opt. Commun. 36, 247–249 (1981).
[Crossref]

J. Ojeda-Castaneda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Other (3)

G. C. Sherman, “Introduction to the angular-spectrum representation of optical fields,” in Applications of Mathematics in Modern Optics, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.358, 31–38 (1982).
[Crossref]

R. G. Cañas, R. W. Smith, “Measurements with surface relief diffractive optical elements,” in Holographic Systems, Components, and Applications II, Proc. Inst. Electr. Eng.311, 160–163 (1989).

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).

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Figures (6)

Fig. 1
Fig. 1

Nonmodified propagation region (hatched area) of a truncated propagation-invariant field.

Fig. 2
Fig. 2

Experimental setup for generating propagation-invariant fields: H, hologram; L1, L2, lenses; P, Fourier plane.

Fig. 3
Fig. 3

Diffraction-free Bessel-beam profiles measured with a charge-coupled-device line element (a) immediately behind lens L2 in Fig. 2 and (b) at a distance z = 15 m.

Fig. 4
Fig. 4

Diffraction-free speckle patterns (a) behind lens L2 and (b) at a distance z = 8 m.

Fig. 5
Fig. 5

Modified Young interferometer: V, V-shaped aperture; L3, L4, lenses; S, slit; CCD, charge-coupled-device line element.

Fig. 6
Fig. 6

Visibility measured as a function of pinhole separation across a Bessel-correlated light field: +, behind lens L2; ■, at a distance z = 15 m.

Equations (40)

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( j 2 + k 2 ) W ( r 1 , r 2 ) = 0 ,             j = 1 , 2 ,
W ( x 1 , y 1 , z ; x 2 , y 2 , z ) = W ( x 1 , y 1 , 0 ; x 2 , y 2 , 0 ) ,             z 0 ,
W ( r 1 , r 2 ) = 0 0 2 π f 1 f 2 A ( f 1 , θ 1 , f 2 , θ 2 ) × exp [ i 2 π ( x 2 f 2 cos θ 2 - x 1 f 1 cos θ 1 + y 2 f 2 sin θ 2 - y 1 f 1 sin θ 1 ) + i ( z 2 w 2 - z 1 w 1 * ) ] d f 1 d f 2 d θ 1 d θ 2 ,
A ( f 1 , θ 1 , f 2 , θ 2 ) = - W ( x 1 , y 1 , 0 , x 2 , y 2 , 0 ) × exp [ - i 2 π ( x 2 f 2 cos θ 2 - x 1 f 1 cos θ 1 + y 2 f 2 sin θ 2 - y 1 f 1 sin θ 1 ) ] × d x 1 d x 2 d y 1 d y 2
w j = [ k 2 - ( 2 π f j ) 2 ] 1 / 2             if 2 π f j k ,
w j = i [ ( 2 π f j ) 2 - k 2 ] 1 / 2             if 2 π f j > k .
H 0 2 π f 1 f 2 A ( f 1 , θ 1 , f 2 , θ 2 ) exp ( i 2 π ( x 2 f 2 cos θ 2 - x 1 f 1 cos θ 1 + y 2 f 2 sin θ 2 - y 1 f 1 sin θ 1 ) + i z { [ k 2 - ( 2 π f 2 ) 2 ] 1 / 2 - [ k 2 - ( 2 π f 1 ) 2 ] 1 / 2 } ) d f 1 d f 2 d θ 1 d θ 2 = H 0 2 π f 1 f 2 A ( f 1 , θ 1 , f 2 , θ 2 ) × exp [ i 2 π ( x 2 f 2 cos θ 2 - x 1 f 1 cos θ 1 + y 2 f 2 sin θ 2 - y 1 f 1 sin θ 1 ) ] d f 1 d f 2 d θ 1 d θ 2 ,
A ( f 1 , θ 1 , f 2 , θ 2 ) = S ( f 1 , θ 1 , θ 2 ) δ ( f 1 - f 2 ) .
W ( r 1 , r 2 ) = 0 k / 2 π 0 2 π f 2 S ( f , θ 1 , θ 2 ) exp { i ( z 2 - z 1 ) [ k 2 - ( 2 π f ) 2 ] 1 / 2 } × exp [ i 2 π f ( x 2 cos θ 2 - x 1 cos θ 1 + y 2 sin θ 2 - y 1 sin θ 1 ) ] × d f d θ 1 d θ 2 .
W ( r 1 , r 2 ) = m n exp [ i ( m ϕ 1 + n ϕ 2 ) ] 0 k / 2 π a m n ( f ) J m ( 2 π f ρ 1 ) × J n ( 2 π f ρ 2 ) exp { i ( z 2 - z 1 ) [ k 2 - ( 2 π f ) 2 ] 1 / 2 } d f .
W ( r 1 , r 2 ) = U * ( r 1 ) U ( r 2 ) .
A ( f 1 , θ 1 , f 2 , θ 2 ) = A * ( f 1 , θ 1 ) A ( f 2 , θ 2 ) .
A ( f 1 , θ 1 , f 2 , θ 2 ) = ( 4 π 2 / α 2 ) F * ( θ 1 ) F ( θ 2 ) × δ ( f 1 - α / 2 π ) δ ( f 2 - α / 2 π )
U ( r ) = exp ( i β z ) 0 2 π F ( θ ) exp [ i α ( x cos θ + y sin θ ) ] d θ ,
α 2 + β 2 = k 2 .
A ( f 1 , θ 1 , f 2 , θ 2 ) = G ( f 1 , θ 1 ) f 1 - 1 δ ( f 1 - f 2 ) δ ( θ 1 - θ 2 ) ,
W ( r 1 , r 2 ) = 0 k / 2 π 0 2 π f G ( f , θ ) exp { i ( z 2 - z 1 ) [ k 2 - ( 2 π f ) 2 ] 1 / 2 } × exp { i 2 π f [ ( x 2 - x 1 ) cos θ + ( y 2 - y 1 ) sin θ ] } d f d θ .
A ( f 1 , θ 1 , f 2 , θ 2 ) = ( 2 π / α ) G ( θ 1 ) f 1 - 1 δ ( f 1 - α / 2 π ) × δ ( f 2 - α / 2 π ) δ ( θ 1 - θ 2 ) .
W ( r 1 , r 2 ) = exp [ i β ( z 2 - z 1 ) ] 0 2 π G ( θ ) exp { i α [ ( x 2 - x 1 ) cos θ + ( y 2 - y 1 ) sin θ ] } d θ ,
W ( r 1 , r 2 ) = exp [ i β ( z 2 - z 1 ) ] J 0 { α [ ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 ] 1 / 2 } .
U ( r ) = exp ( i β z ) J 0 [ α ( x 2 + y 2 ) 1 / 2 ]
W ( x 1 , y 1 , z 1 ; x 2 , y 2 , z 2 ) = W ( x 1 , y 1 , 0 ; x 2 , y 2 , 0 ) Z ( z 1 - z 2 ) ,
A ( f 1 , θ 1 , f 2 , θ 2 ) = ( 4 π 2 / α 2 ) L ( θ 1 , θ 2 ) δ ( f 1 - α / 2 π ) δ ( f 2 - α / 2 π )
W ( r 1 , r 2 ) = exp [ i β ( z 2 - z 2 ) ] 0 2 π L ( θ 1 , θ 2 ) × exp [ i α ( x 2 cos θ 2 - x 1 cos θ 1 + y 2 sin θ 2 - y 1 sin θ 1 ) ] d θ 1 d θ 2 ,
W ( r 1 , r 2 ) = exp [ i β ( z 2 - z 1 ) ] m n c m n exp [ i ( m ϕ 1 + n ϕ 2 ) ] × J m ( α ρ 1 ) J n ( α ρ 2 ) ,
W ( x 1 , y 1 , z + d ; x 2 , y 2 , z + d ) = W ( x 1 , y 1 , z ; x 2 , y 2 , z ) ,
W ( x 1 , y 1 , z 1 + d ; x 2 , y 2 , z 2 + d ) = W ( x 1 , y 1 , z 1 ; x 2 , y 2 , z 2 ) ,
[ k 2 - ( 2 π f 2 ) 2 ] 1 / 2 - [ k 2 - ( 2 π f 1 ) 2 ] 1 / 2 = q ( 2 π / d ) ;
A ( f 1 , θ 1 , f 2 , θ 2 ) = q S q ( f 1 , θ 1 , θ 2 ) × δ ( f 2 - { λ - 2 - [ ( λ - 2 - f 1 2 ) 1 / 2 + q / d ] 2 } 1 / 2 ) ,
W ( r 1 , r 2 ) = q 0 k / 2 π 0 2 π S q ( f , θ 1 , θ 2 ) × f { λ - 2 - [ ( λ - 2 - f 2 ) 1 / 2 + q / d ] 2 } 1 / 2 × exp [ - i 2 π f ( x 1 cos θ 1 + y 1 sin θ 1 ) ] × exp [ i ( k 2 - { [ k 2 - ( 2 π f ) 2 ] 1 / 2 + 2 π q / d } 2 ) 1 / 2 × ( x 2 cos θ 2 + y 2 sin θ 2 ) ] × exp { i ( z 2 - z 1 ) [ k 2 - ( 2 π f ) 2 ] 1 / 2 + i 2 π q z 2 / d } d f d θ 1 d θ 2 .
W ( r 1 , r 2 ) = l p K l p ( x 1 , y 1 , x 2 , y 2 ) exp [ i 2 π ( p z 2 - l z 1 ) / d ] ,
R l = [ λ - 2 - ( l / d ) 2 ] 1 / 2 .
A ( f 1 , θ 1 , f 2 , θ 2 ) = l p ( R l R p ) - 1 L l p ( θ 1 , θ 2 ) × δ ( f 1 - R l ) δ ( f 2 - R p ) ,
K l p ( x 1 , y 1 , x 2 , y 2 ) = 0 2 π L l p ( θ 1 , θ 2 ) × exp [ i 2 π ( R p x 2 cos θ 2 - R l x 1 cos θ 1 + R p y 2 sin θ 2 - R l y 1 sin θ 1 ) ] d θ 1 d θ 2 .
z max = F D L / D S ;
W ( r 1 , r 2 ) = W T ( x 1 , y 1 , x 2 , y 2 ) W L ( z 1 , z 2 )
W L ( z 1 , z 2 ) = 1.
W out ( x 1 , y 1 , x 2 , y 2 ) W in ( - x 1 , y 1 , - x 2 , y 2 ) + W in ( x 1 , - y 1 , x 2 , - y 2 ) + W in ( - x 1 , y 1 , x 2 , - y 2 ) exp ( i β Δ z ) + W in ( x 1 , - y 1 , - x 2 , y 2 ) exp ( - i β Δ z ) ,
W out ( x 1 , y 1 , x 2 , y 2 ) J 0 { α [ ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 ] 1 / 2 } + cos ( β Δ z ) J 0 { α [ ( x 2 + x 1 ) 2 + ( y 2 + y 1 ) 2 ] 1 / 2 } .
I out ( x , y ) 1 + cos ( β Δ z ) J 0 [ 2 α ( x 2 + y 2 ) 1 / 2 ] .

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