Abstract

Conditions for exactly self-imaging nonparaxial fields that are periodic also in the transverse direction are introduced. The theory is first derived by assuming full coherence and then extended into the domain of partial coherence. Different types of solutions are discussed, and some illustrations of the existence of solutions and intensity distributions of the fields are presented.

© 2004 Optical Society of America

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References

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  1. H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).
  2. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).
    [CrossRef]
  3. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989).
  4. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  5. W. D. Montgomery, “Algebraic formulation of diffraction applied to self-imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [CrossRef]
  6. J. Ojeda-Castañeda, P. Andrés, E. Tepı́chin, “Spatial filters for replicating images,” Opt. Lett. 11, 551–553 (1986).
    [CrossRef] [PubMed]
  7. G. Indebetouw, “Quasi-self-imaging using aperiodic sequences,” J. Opt. Soc. Am. A 9, 549–558 (1992).
    [CrossRef]
  8. J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
    [CrossRef]
  9. A. W. Lohmann, J. Ojeda-Castañeda, N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
    [CrossRef]
  10. V. Kettunen, J. Turunen, “Propagation-invariant spot arrays,” Opt. Lett. 23, 1247–1249 (1998).
    [CrossRef]
  11. N. Guérineau, J. Primot, “Nondiffracting array generator using an N-wave interferometer,” J. Opt. Soc. Am. A 16, 293–298 (1999).
    [CrossRef]
  12. N. Guérineau, B. Harchaoui, J. Primot, K. Heggarty, “Generation of achromatic and propagation-invariant spot arrays by use of continuous self-imaging gratings,” Opt. Lett. 26, 411–413 (2001).
    [CrossRef]
  13. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  14. D. M. Burton, Elementary Number Theory, 2nd ed. (Wm. C. Brown, Dubuque, Ia., 1989).
  15. G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
  16. A. W. Lohmann, J. Ojeda-Castañeda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
  17. G. Indebetouw, “Spatially periodic wave fields: an experimental demonstration of the relationship between the lateral and the longitudinal spatial frequencies,” Opt. Commun. 49, 86–90 (1984).
  18. G. Indebetouw, “Propagation of spatially periodic wave fields,” Opt. Acta 31, 531–539 (1984).
  19. J. Ojeda-Castañeda, J. Ibarra, J. C. Barreiro, “Noncoherent Talbot effect: coherence theory and applications,” Opt. Commun. 71, 151–155 (1989).
  20. 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).
  21. 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).
  22. J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
  23. J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).

2003 (1)

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

2002 (1)

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).

2001 (1)

1999 (1)

1998 (1)

1993 (1)

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).

1992 (1)

1991 (1)

1989 (1)

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

1988 (1)

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

1986 (1)

1984 (2)

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

G. Indebetouw, “Propagation of spatially periodic wave fields,” Opt. Acta 31, 531–539 (1984).

1983 (2)

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

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

1982 (1)

1968 (1)

1967 (1)

1881 (1)

Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).
[CrossRef]

1836 (1)

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Andrés, P.

Barreiro, J. C.

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

Burton, D. M.

D. M. Burton, Elementary Number Theory, 2nd ed. (Wm. C. Brown, Dubuque, Ia., 1989).

Friberg, A. T.

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).

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).

Guérineau, N.

Harchaoui, B.

Heggarty, K.

Ibarra, J.

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

Indebetouw, G.

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

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

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

G. Indebetouw, “Propagation of spatially periodic wave fields,” Opt. Acta 31, 531–539 (1984).

Jahns, J.

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

Kettunen, V.

Knuppertz, H.

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

Lohmann, A. W.

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

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

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

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Montgomery, W. D.

Ojeda-Castañeda, J.

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

J. Ojeda-Castañeda, P. Andrés, E. Tepı́chin, “Spatial filters for replicating images,” Opt. Lett. 11, 551–553 (1986).
[CrossRef] [PubMed]

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

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

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989).

Primot, J.

Rayleigh,

Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).
[CrossRef]

Streibl, N.

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

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Tepi´chin, E.

Tervo, J.

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).

Turunen, J.

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).

V. Kettunen, J. Turunen, “Propagation-invariant spot arrays,” Opt. Lett. 23, 1247–1249 (1998).
[CrossRef]

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).

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).

Vasara, A.

Wolf, E.

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Acta (3)

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

G. Indebetouw, “Propagation of spatially periodic wave fields,” Opt. Acta 31, 531–539 (1984).

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

Opt. Commun. (4)

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

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

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

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).

Opt. Lett. (3)

Philos. Mag. (2)

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).
[CrossRef]

Pure Appl. Opt. (1)

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).

Other (3)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

D. M. Burton, Elementary Number Theory, 2nd ed. (Wm. C. Brown, Dubuque, Ia., 1989).

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Figures (6)

Fig. 1
Fig. 1

Angular spectrum of a transversely periodic field, with allowed spatial frequencies marked by dots. The propagating plane-wave components are located inside the circle with radius equal to k.

Fig. 2
Fig. 2

Example of a set of Montgomery’s rings.

Fig. 3
Fig. 3

Allowed plane-wave components of a self-imaging field. The intersections of the half-sphere and the planes form the set of Montgomery’s rings described in Fig. 2.

Fig. 4
Fig. 4

Allowed plane-wave components of periodic self-imaging field in the case Q=77. For clarity only the first quadrant is shown.

Fig. 5
Fig. 5

Allowed plane-wave components of a periodic self-imaging field in the case Dx=1/89k, Dy=2Dx, and Dz=3Dx.

Fig. 6
Fig. 6

Allowed plane waves of a field with three-dimensionally periodic intensity distribution. Here kx0=5k/21, ky0=4k/21, kz0=20k/21, Dx=Dy=Dz=k/7.

Equations (41)

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U(r, ω)=-A(kx, ky, ω)×exp[i(kxx+kyy+kzz)]dkxdky,
A(kx, ky, ω)=1(2π)2-U(x, y, 0, ω)×exp[-i(kxx+kyy)]dxdy.
kx2+ky2+kz2=k2,
kz=(k2-kx2-ky2)1/2ifk2>kx2+ky2i(kx2+ky2-k2)1/2otherwise.
U(x+d, y+d, z, ω)=U(x, y, z, ω),
U(r, ω)=m=-n=-Amn(ω)×exp[i2π(mx+ny)/d+ikzmn)],
Amn(ω)=1d20d0dU(x, y, 0, ω)×exp[-i2π(mx+ny)/d]dxdy
kzmn=2π(1/λ2-m2/d2-n2/d2)1/2,
U(r, ω)=exp(ikz)m=-n=-Amn(ω)×exp[i2π(mx+ny)/d]×exp[-iπλ(m2+n2)z/d2].
U(x, y, z+zT, ω)=exp(i2πzT/λ)U(x, y, z, ω),
kz,q=kz,0-q2π/zM,
kz,02+m2Dx2+n2Dy2+q2Dz2-2qDzkz,0=k2,
Dx=2π/dx,Dy=2π/dy,Dz=2π/zM.
k2=m2Dx2+n2Dy2+p2Dz2,
p=P-q.
m2+n2+p2=(k/D)2,
I(x+Mdx, y+Ndy, z+PzM, ω)=I(x, y, z, ω)
U(x+dx, y, z, ω)=exp[iξ(r)]U(x, y, z, ω),
-A(kx, ky, ω)exp[i(kxx+kyy+kzz)]×{exp[iξ(r)]-exp(ikxdx)}dkxdky=0.
kx,m=kx,0+mDx,
ky,n=ky,0+nDy.
m2Dx2+n2Dy2+p2Dz2+2(mDxkx,0+nDyky,0-pDzkz,0)=0,
W(r1, r2, ω)=U*(r1, ω)U(r2, ω),
W(r1, r2, ω)=-A(k1, k2, ω)×exp[i(k2xx2-k1xx1)]×exp[i(k2yy2-k1yy1)]×exp[i(k2zz2-k1z*z1)]×dk1xdk1ydk2xdk2y,
A(k1, k2, ω)
=1(2π)4-W(r1, r2, ω)|z1=z2=0×exp[i(k1xx1-k2xx2)]×exp[i(k1yy1-k2yy2)]dx1dx2dy1dy2.
W(r1+d1, r2+d2, ω)=W(r1, r2, ω),
S(r+d, ω)=S(r, ω),
W(r1, r2, ω)
=m=-n=-p=-u=-v=-w=-Am,n,pu,v,w(ω)×exp[i(ux2-mx1)Dx]exp[i(vy2-ny1)Dy]×exp[i(wz2-pz1)Dz],
Am,n,pu,v,w(ω)=0Dx0Dy0DzW(r1, r2, ω)Dx2Dy2Dz2×exp[i(mx1-ux2)Dx]×exp[i(ny1-vy2)Dy]×exp[i(pz1-wz2)Dz]×dx1dx2dy1dy2dz1dz2.
12W(r1, r2, ω)+k2W(r1, r2, ω)=0,
22W(r1, r2, ω)+k2W(r1, r2, ω)=0,
m2Dx2+n2Dy2+p2Dz2=k2,
u2Dx2+v2Dy2+w2Dz2=k2.
-A(k1, k2, ω)exp[i(k2x-k1x)x]
×exp[i(k2y-k1y)y]exp[i(k2z-k1z*)z]×{exp[i(k2x-k1x)dx]-1}dk1xdk1ydk2xdk2y=0,
k2x=k1x+mDx
k2y=k1y+nDy,
k2z=k1z-pDx,
m2Dx2+n2Dy2+p2Dz2+2(mDxkxj+nDykyj-pDzkzj)=0,

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