Abstract

Propagation invariance is extended in the paraxial regime, leading to a generalized self-imaging effect. These wave fields are characterized by a finite number of transverse self-images that appear, in general, at different orientations and scales. They possess finite energy and thus can be accurately generated. Necessary and sufficient conditions are derived, and they are appropriately represented in the Gauss–Laguerre modal plane. Relations with the following phenomena are investigated: classical self-imaging, rotating beams, eigen-Fourier functions, and the recently introduced generalized propagation-invariant wave fields. In the paraxial regime they are all included within the generalized self-imaging effect that is presented. In this context we show an important relation between paraxial Bessel beams and Gauss–Laguerre beams.

© 2000 Optical Society of America

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  1. Part of this study appeared in R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” presented at the DO’97, EOS Topical Meeting on Diffractive Optics, Savonlinna, Finland, July 7–9, 1997.
  2. K. Patorski, “Self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Pergamon, Oxford, UK, 1989), Vol. XXVII, pp. 3–108.
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  4. G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
    [CrossRef]
  5. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [CrossRef]
  6. P. Szwaykowski, J. Ojeda-Castaneda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
    [CrossRef]
  7. J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
    [CrossRef]
  8. G. Indebetouw, “Quasi-self-imaging using aperiodic sequences,” J. Opt. Soc. Am. A 9, 549–558 (1992).
    [CrossRef]
  9. V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
    [CrossRef]
  10. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  11. E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
    [CrossRef]
  12. Y. Y. Schechner, R. Piestun, J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
    [CrossRef]
  13. Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
    [CrossRef]
  14. R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997).
    [CrossRef] [PubMed]
  15. V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
    [CrossRef]
  16. R. Piestun, J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
    [CrossRef]
  17. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).
  18. R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
    [CrossRef]
  19. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  20. W. D. Montgomery, “Algebraic formulation of diffraction applied to self-imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [CrossRef]
  21. J. M. Cowley, A. F. Moodie, “Fourier images. III. Finite sources,” Proc. Phys. Soc. London Sect. B 70, 505–513 (1957).
    [CrossRef]
  22. A. Kalestynski, B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
    [CrossRef]
  23. A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. (USSR) 44, 208–212 (1978).
  24. R. Moignard, J. L. de Bougrenet de laTocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
    [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  26. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  27. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  28. Y. Y. Schechner, “Rotation phenomena in waves,” MSc. Thesis (Technion–Israel Institute of Technology, Haifa, Israel, 1996).
  29. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  30. Y. Y. Schechner, J. Shamir, “Parameterization and orbital angular momentum of anisotropic dislocations,” J. Opt. Soc. Am. A 13, 967–973 (1996).
    [CrossRef]
  31. Note that there is a change of sign in Eq. (33) compared with that equation in Ref. 16. This is simply due to a different convention for the direction of positive rotations (γ).
  32. Note that in this case the condition n→∞ that appears in expression (29) is redundant, since it is implicit in Eq. (24) together with the rest of the conditions of expression (29). In effect, in the limit (Δz/z0)→0, we obtain a uniform periodicity in z and, according to Eq. (24), for γ=0 we have nj-n1≈zˆ→02πNj(z0/Δz)→zˆ→0∞ for all j. Thus nj→∞.
  33. G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic-oscillators,” Phys. Rev. A 48, 656–665 (1993).
    [CrossRef] [PubMed]
  34. A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
    [CrossRef]
  35. M. J. Caola, “Self-Fourier functions,” J. Phys. A 24, L1143–L1144 (1991).
    [CrossRef]
  36. S. G. Lipson, “‘Self-Fourier objects and other self-transform objects’: comment,” J. Opt. Soc. Am. A 10, 2088–2089 (1993).
    [CrossRef]
  37. M. W. Coffey, “Self-reciprocal Fourier functions,” J. Opt. Soc. Am. A 9, 2453–2455 (1994).
    [CrossRef]
  38. G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
    [CrossRef]
  39. M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  40. G. Slekys, K. Staliunas, C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
    [CrossRef]
  41. M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1965).

1999 (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

1998 (2)

1997 (2)

R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997).
[CrossRef] [PubMed]

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

1996 (4)

Y. Y. Schechner, R. Piestun, J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[CrossRef]

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

Y. Y. Schechner, J. Shamir, “Parameterization and orbital angular momentum of anisotropic dislocations,” J. Opt. Soc. Am. A 13, 967–973 (1996).
[CrossRef]

R. Moignard, J. L. de Bougrenet de laTocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
[CrossRef]

1995 (1)

G. Slekys, K. Staliunas, C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

1994 (1)

M. W. Coffey, “Self-reciprocal Fourier functions,” J. Opt. Soc. Am. A 9, 2453–2455 (1994).
[CrossRef]

1993 (4)

S. G. Lipson, “‘Self-Fourier objects and other self-transform objects’: comment,” J. Opt. Soc. Am. A 10, 2088–2089 (1993).
[CrossRef]

G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic-oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

1992 (4)

G. Indebetouw, “Quasi-self-imaging using aperiodic sequences,” J. Opt. Soc. Am. A 9, 549–558 (1992).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

1991 (3)

M. J. Caola, “Self-Fourier functions,” J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

P. Szwaykowski, J. Ojeda-Castaneda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

1989 (1)

1988 (1)

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

1987 (1)

1982 (1)

1978 (2)

A. Kalestynski, B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. (USSR) 44, 208–212 (1978).

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

1968 (1)

1967 (1)

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1957 (1)

J. M. Cowley, A. F. Moodie, “Fourier images. III. Finite sources,” Proc. Phys. Soc. London Sect. B 70, 505–513 (1957).
[CrossRef]

Abramochkin, E.

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

Allen, L.

G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic-oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Bouchal, Z.

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

Caola, M. J.

M. J. Caola, “Self-Fourier functions,” J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

Cincotti, G.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

Coffey, M. W.

M. W. Coffey, “Self-reciprocal Fourier functions,” J. Opt. Soc. Am. A 9, 2453–2455 (1994).
[CrossRef]

Cowley, J. M.

J. M. Cowley, A. F. Moodie, “Fourier images. III. Finite sources,” Proc. Phys. Soc. London Sect. B 70, 505–513 (1957).
[CrossRef]

de Bougrenet de laTocnaye, J. L.

R. Moignard, J. L. de Bougrenet de laTocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
[CrossRef]

Durnin, J.

Friberg, A. T.

Goodman, J. W.

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

Gori, F.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

Honkanen, M.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

Horak, R.

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

Indebetouw, G.

Kalestynski, A.

A. Kalestynski, B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

Lautanen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lipson, S. G.

Lohmann, A. W.

Mendlovic, D.

Moignard, R.

R. Moignard, J. L. de Bougrenet de laTocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
[CrossRef]

Montgomery, W. D.

Moodie, A. F.

J. M. Cowley, A. F. Moodie, “Fourier images. III. Finite sources,” Proc. Phys. Soc. London Sect. B 70, 505–513 (1957).
[CrossRef]

Mukunda, N.

Nazarathy, M.

Nienhuis, G.

G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic-oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Ojeda-Castaneda, J.

P. Szwaykowski, J. Ojeda-Castaneda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

Patorski, K.

K. Patorski, “Self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Pergamon, Oxford, UK, 1989), Vol. XXVII, pp. 3–108.

Piestun, R.

R. Piestun, J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[CrossRef]

R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997).
[CrossRef] [PubMed]

Y. Y. Schechner, R. Piestun, J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[CrossRef]

Part of this study appeared in R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” presented at the DO’97, EOS Topical Meeting on Diffractive Optics, Savonlinna, Finland, July 7–9, 1997.

Santarsiero, M.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

Schechner, Y. Y.

R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997).
[CrossRef] [PubMed]

Y. Y. Schechner, R. Piestun, J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[CrossRef]

Y. Y. Schechner, J. Shamir, “Parameterization and orbital angular momentum of anisotropic dislocations,” J. Opt. Soc. Am. A 13, 967–973 (1996).
[CrossRef]

Y. Y. Schechner, “Rotation phenomena in waves,” MSc. Thesis (Technion–Israel Institute of Technology, Haifa, Israel, 1996).

Part of this study appeared in R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” presented at the DO’97, EOS Topical Meeting on Diffractive Optics, Savonlinna, Finland, July 7–9, 1997.

Shamir, J.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Simon, R.

Slekys, G.

G. Slekys, K. Staliunas, C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

Smirnov, A. P.

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. (USSR) 44, 208–212 (1978).

Smolinska, B.

A. Kalestynski, B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

Soskin, M. S.

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Staliunas, K.

G. Slekys, K. Staliunas, C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

Szwaykowski, P.

P. Szwaykowski, J. Ojeda-Castaneda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

Turunen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

Vasara, A.

Vasnetsov, M. V.

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Volostnikov, V.

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

Wagner, J.

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

Weiss, C. O.

G. Slekys, K. Staliunas, C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

J. Mod. Opt. (6)

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

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996).
[CrossRef]

V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (10)

J. Phys. A (2)

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

M. J. Caola, “Self-Fourier functions,” J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

Opt. Acta (1)

A. Kalestynski, B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
[CrossRef]

Opt. Commun. (4)

R. Moignard, J. L. de Bougrenet de laTocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
[CrossRef]

P. Szwaykowski, J. Ojeda-Castaneda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

G. Slekys, K. Staliunas, C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. (USSR) 44, 208–212 (1978).

Phys. Rev. A (1)

G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic-oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (1)

Y. Y. Schechner, R. Piestun, J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Proc. Phys. Soc. London Sect. B (1)

J. M. Cowley, A. F. Moodie, “Fourier images. III. Finite sources,” Proc. Phys. Soc. London Sect. B 70, 505–513 (1957).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Other (8)

Note that there is a change of sign in Eq. (33) compared with that equation in Ref. 16. This is simply due to a different convention for the direction of positive rotations (γ).

Note that in this case the condition n→∞ that appears in expression (29) is redundant, since it is implicit in Eq. (24) together with the rest of the conditions of expression (29). In effect, in the limit (Δz/z0)→0, we obtain a uniform periodicity in z and, according to Eq. (24), for γ=0 we have nj-n1≈zˆ→02πNj(z0/Δz)→zˆ→0∞ for all j. Thus nj→∞.

Y. Y. Schechner, “Rotation phenomena in waves,” MSc. Thesis (Technion–Israel Institute of Technology, Haifa, Israel, 1996).

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

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Part of this study appeared in R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” presented at the DO’97, EOS Topical Meeting on Diffractive Optics, Savonlinna, Finland, July 7–9, 1997.

K. Patorski, “Self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Pergamon, Oxford, UK, 1989), Vol. XXVII, pp. 3–108.

M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (6)

Fig. 1
Fig. 1

Examples of GL modes: (a) intensity, (b) phase.

Fig. 2
Fig. 2

GL modal domain and generalized self-imaging: Each mode that composes the wave is represented by a point within a grid. The support of generalized self-imaging waves lies upon uniformly spaced parallel lines (see text).

Fig. 3
Fig. 3

Particular cases of the GSI waves: (a) Scaled self-imaging waves have GL modes lying on horizontal parallel lines. (b) Eigen-Fourier functions generate waves that are self-reproduced in the far field, with GL modes lying on parallel lines at a distance 4, leading to four possible families depicted by different dashed lines. (c) A mode possesses an invariant transverse intensity distribution during propagation; all the constituent GL modes have the same n number. (d) Rotating beams are composed of GL modes lying on a single line.

Fig. 4
Fig. 4

Spiral trajectory of a point in the transverse intensity distributions of a rotating beam. (a) Detail of the waist, showing maximal rotation rate. (b) Coarser view of the spiral, showing that the rotation rate tends to zero toward the far field. (c) Projection of the spiral onto a transverse plane. The axial distance between consecutive marks (+) is z0.

Fig. 5
Fig. 5

Example of scale-rotated self-imaging. (a) The transverse intensity distribution is self-reproduced at different positions with different orientations and scales (see text). (b) Representation of the changes of scale along the propagation. (c) The intermediate cross sections are different but are also self-reproduced with scaling and rotation.

Fig. 6
Fig. 6

Example of a rotating beam with an odd azimuthal symmetry. (a) The scaled transverse intensity distribution is invariant on propagation, except for a change in orientation. Note the slow rotation rate that leads to a total rotation of 30° from waist to far field. (b) The phase is invariant on propagation, except for a change in orientation and scale and the addition of a quadratic phase term. Note the rotating (first-order) phase dislocations. From left to right in (a) and (b), zˆ0, 0.25, 5.04.

Equations (59)

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U(r, t)=U(r)exp(-iωt),
Of=μf,
O=PS[s]R[γ]D[Δz],
|Of|2=|μf|2.
D[Δz]=F-1 expik1-(λξ)22ΔzF,
U(r, t)=u(r)exp[i(kz-ωt)].
unm(r)=G(ρˆ, zˆ)Rnm(ρˆ)Φm(ϕ)Zn(zˆ),
G(ρˆ, zˆ)=w0w(zˆ)exp(-ρˆ2)exp(iρˆ2zˆ)exp[-iψ(zˆ)],
Rnm(ρˆ)=(2ρˆ)|m|L(n-|m|)/2|m|(2ρˆ2),
Φm(ϕ)=exp(imϕ),
Zn(zˆ)=exp[-inψ(zˆ)],
n=|m|,|m|+2,|m|+4,|m|+6,.
(Δx)2=-x2uu*dxdy-uu*dxdy.
tan(θbeam)=n+1(λ/πw0).
u(r)=j=1QAjunj,mj(r),
I(r)=|G(ρˆ)|2j=1Q|Aj|2Rnj, mj2(ρˆ)+j=1Qp=j+1Q2|Aj||Ap|Rnj,mj(ρˆ)Rnp,mp(ρˆ)×cos[Δmjpϕ-Δnjpψ(zˆ)-θjp],
s=w(zˆ2)w(zˆ1)=1+zˆ221+zˆ121/2.
|u(ρ, ϕ, z1)|2=Sw(zˆ2)w(zˆ1)R[γ]D[Δz]u(ρ, ϕ, z1)2,
I(ρˆ, ϕ, z1)=I(ρˆ, ϕ+γ, z2)
cos[Δmjpϕ-Δnjpψ(zˆ1)-θjp]
=cos[Δmjp(ϕ+γ+2πN)-Δnjpψ(zˆ2)-θjp].
Δmjpϕ-Δnjpψ(zˆ1)-θjp
=Δmjp(ϕ+γ+2πN)-Δnjpψ(zˆ2)-θjp+2πN
ΔnjpΔψ=Δmjpγ+2πNjp
Δψ=ψ(zˆ2)-ψ(zˆ1).
nj=n1+γ(mj-m1)+2πNjΔψ
u(ρˆ, ϕ, zˆ1)=w0w(zˆ1)exp(-ρˆ2)exp[-i(1+n1)ψ(zˆ1)]×exp(iρˆ2zˆ1)exp(im1ϕ)j=1QAjRj(ρˆ)×exp{i[(mj-m1)ϕ-(nj-n1)ψ(zˆ1)]},
u(ρˆ, ϕ+γ, z2)=w0w(zˆ2)exp(-ρˆ2)×exp[-i(1+n1)ψ(zˆ2)]×exp(iρˆ2zˆ2)exp[im1(ϕ+γ)]×j=1QAjRj(ρˆ)exp{i[(mj-m1)×(ϕ+γ)-(nj-n1)ψ(zˆ2)]}.
arg[u(ρˆ, ϕ+γ, zˆ2)]=arg[u(ρˆ, ϕ, zˆ1)]+m1γ-(1+n1)Δψ+(zˆ2-zˆ1)ρˆ2.
z02+z12-2Δzs2-1z1-Δz2s2-1=0,
z/z00,ρ/w00,n,
n-|m|n,λn/w0C2=const.,
unmlimit(r)=c(m)J|m|2nρw0exp(imϕ)exp-izz0n,
unmlimit(r)exp[i(kz-ωt)]=c(m)J|m|(αρ)exp(imϕ)×expik-α22kz×exp(-iωt),
βk2-α2k1-α22k2.
β=βm,N=β0-mγ+2πNΔz,0βk,
β=k-n1z0+γm1Δz-mγ+2πNΔz.
u00 ρ/w001,
nj=n1+2πNjΔψforallj,
u(ρ, ϕ, z1)=u(ρ, ϕ, z2)*,
Ff=μf.
μ4=1μ=exp(iLπ/2),L=0, 1, 2,,
f(x)=exp(ia)g(x)+G(x)+exp(i3a)g(-x)+exp(i2a)G(-x),
u=exp[i(1+n1)π/2]=i(1+n1),
nj=n1+4Njforallj.
R[γ]F f=μf;
nj=n1+2γπ(mj-m1)+4Njforallj.
|u(ρ, ϕ, z1)|2
=Sw(z)w(z1)R[γ(z)]D[z-z1]u(ρ, ϕ, z1)2
dϕdzˆjp=ΔnjpΔmjpdψ(zˆ)dzˆ.
nj=V1mj+V2,j=1, 2,,
dγdzˆ=dϕ(zˆ)dzˆ=ΔnΔmdψ(zˆ)dzˆ=V11+zˆ2.
ρ=ρ01+zˆ2,
ϕ=ϕ0+V1ψ(zˆ),
Δϕtotal=V1(π/2),
(π/λ)tan(θbeam)=n+1w0nnw0.
λn/w0C2=const.
η-bLηb(χ/η)ηχ-b/2Jb(2χ),
nj-n1zˆ02πNj(z0/Δz)zˆ0

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