Abstract

Monochromatic imaging systems with spatial-frequency filters in the form of Fabry–Perot interferometers, concentric ring masks, and diffractive multifocal lenses are shown to realize the same effect of multiple equidistant imaging. However, the forms of manifestation of this effect are not identical due to the difference in spectral content of generated wave fields. Self-imaging fields with a discrete angular spectrum inherent in the systems with masks and interferometers are found to comprise a subclass of periodically focused fields with a continuous angular spectrum peculiar to the systems with diffractive multifocal lenses. The advantages of the latter systems are the extremely high total light efficiency and the sharply defined longitudinal localization of generated wave fields, which enhance the brightness of the reproduced images and decreases their parasitic diffraction dispersion, background noise, and blurring.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. R. Piestun and J. Shamir, "Generalized propagation-invariant wave fields," J. Opt. Soc. Am. A 15, 3039-3044 (1998).
    [CrossRef]
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    [CrossRef]
  13. J. Tervo and J. Turunen, "Self-imaging of electromagnetic fields," Opt. Express 9, 622-630 (2001).
    [CrossRef]
  14. J. Jahns, H. Knuppertz, and A. W. Lohmann, "Montgomery self-imaging effect using computer-generated diffractive optical elements," Opt. Commun. 225, 13-17 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  19. G. Indebetouw, "Propagation of spatially periodic wave fields," Opt. Acta 31, 531-539 (1984).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. J. Wagner and Z. Bouchal, "Experimental realization of self-reconstruction of the 2D aperiodic objects," Opt. Commun. 176, 309-311 (2000).
    [CrossRef]
  27. A. G. Sedukhin, "Periodically focused propagation-invariant beams with sharp central peak," Opt. Commun. 228, 231-247 (2003).
    [CrossRef]
  28. A. G. Sedukhin, "Generalized periodically focused beams and multiple on-axis lens imaging," Opt. Commun. 229, 39-57 (2004).
    [CrossRef]
  29. V. V. Cherkashin, A. G. Poleshchuk, and A. G. Sedukhin, "Experimental examination of self-imaging effect with the use of a diffractive multifocal lens," in EOS Topical Meeting Diffractive Optics 2005 (European Optical Society, 2005), pp. 70-71.
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    [CrossRef]
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    [CrossRef] [PubMed]
  32. A. G. Sedukhin, "Marginal phase correction of truncated Bessel beams," J. Opt. Soc. Am. A 17, 1059-1066 (2000).
    [CrossRef]
  33. W. Wang, A. T. Friberg, and E. Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A 12, 1947-1953 (1995).
    [CrossRef]
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    [CrossRef]
  35. H. Dammann and K. Gortler, "High-efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
    [CrossRef]
  36. H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
    [CrossRef]

2004

T. Saastamoinen, J. Tervo, P. Vahimaa, and J. Turunen, "Exact self-imaging of transversely periodic fields," J. Opt. Soc. Am. A 21, 1424-1429 (2004).
[CrossRef]

A. G. Sedukhin, "Generalized periodically focused beams and multiple on-axis lens imaging," Opt. Commun. 229, 39-57 (2004).
[CrossRef]

2003

A. G. Sedukhin, "Periodically focused propagation-invariant beams with sharp central peak," Opt. Commun. 228, 231-247 (2003).
[CrossRef]

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

2001

2000

1999

M. Erdélyi, Zs. Bor, W. L. Wilson, M. C. Smayling, and F. K. Tittel, "Simulation of coherent multiple imaging by means of pupil plane filtering in optical microlithography," J. Opt. Soc. Am. A 16, 1909-1914 (1999).
[CrossRef]

S. Khonina, V. Kotlyar, and V. Soifer, "Self-reproduction of multimode Hermite-Gaussian beams," Tech. Phys. Lett. 25, 489-491 (1999).
[CrossRef]

M. V. Berry and E. Bodenschatz, "Caustics, multiply reconstructed by Talbot interference," J. Mod. Opt. 46, 349-365 (1999).
[CrossRef]

1998

1997

M. Erdélyi, Z. L. Horváth, G. Szabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, "Generation of diffraction-free beams for applications in optical microlithography," J. Vac. Sci. Technol. B 15, 287-292 (1997).
[CrossRef]

Z. L. Horváth, M. Erdélyi, G. Szabó, Zs. Bor, F. K. Tittel, and J. R. Cavallaro, "Generation of nearly nondiffracting Bessel beams with a Fabry-Perot interferometer," J. Opt. Soc. Am. A 14, 3009-3013 (1997).
[CrossRef]

1995

1993

J. Turunen and A. T. Friberg, "Self-imaging and propagation-invariance in electromagnetic fields," Pure Appl. Opt. 2, 51-60 (1993).
[CrossRef]

1991

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

1989

1988

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

1987

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

1986

1984

G. Indebetouw, "Propagation of spatially periodic wave fields," Opt. Acta 31, 531-539 (1984).
[CrossRef]

1983

G. Indebetouw, "Self-imaging through a Fabry-Perot interferometer," Opt. Acta 30, 1463-1471 (1983).
[CrossRef]

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

1981

G. Indebetouw, "Unfocused optical correlator. Part 2: Fabry-Perot prefiltering," Optik (Stuttgart) 59, 287-302 (1981).

1980

1977

H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
[CrossRef]

1975

1971

H. Dammann and K. Gortler, "High-efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

1967

1836

H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).

Andrés, P.

Berry, M. V.

M. V. Berry and E. Bodenschatz, "Caustics, multiply reconstructed by Talbot interference," J. Mod. Opt. 46, 349-365 (1999).
[CrossRef]

Bodenschatz, E.

M. V. Berry and E. Bodenschatz, "Caustics, multiply reconstructed by Talbot interference," J. Mod. Opt. 46, 349-365 (1999).
[CrossRef]

Bor, Zs.

Bouchal, Z.

Z. Bouchal and J. Wagner, "Self-reconstruction effect in free propagation of wave field," Opt. Commun. 176, 299-307 (2000).
[CrossRef]

J. Wagner and Z. Bouchal, "Experimental realization of self-reconstruction of the 2D aperiodic objects," Opt. Commun. 176, 309-311 (2000).
[CrossRef]

Carter, W. H.

Cavallaro, J. R.

M. Erdélyi, Z. L. Horváth, G. Szabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, "Generation of diffraction-free beams for applications in optical microlithography," J. Vac. Sci. Technol. B 15, 287-292 (1997).
[CrossRef]

Z. L. Horváth, M. Erdélyi, G. Szabó, Zs. Bor, F. K. Tittel, and J. R. Cavallaro, "Generation of nearly nondiffracting Bessel beams with a Fabry-Perot interferometer," J. Opt. Soc. Am. A 14, 3009-3013 (1997).
[CrossRef]

Cherkashin, V. V.

V. V. Cherkashin, A. G. Poleshchuk, and A. G. Sedukhin, "Experimental examination of self-imaging effect with the use of a diffractive multifocal lens," in EOS Topical Meeting Diffractive Optics 2005 (European Optical Society, 2005), pp. 70-71.

Dammann, H.

H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
[CrossRef]

H. Dammann and K. Gortler, "High-efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Erdélyi, M.

Friberg, A. T.

W. Wang, A. T. Friberg, and E. Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A 12, 1947-1953 (1995).
[CrossRef]

J. Turunen and A. T. Friberg, "Self-imaging and propagation-invariance in electromagnetic fields," Pure Appl. Opt. 2, 51-60 (1993).
[CrossRef]

Gortler, K.

H. Dammann and K. Gortler, "High-efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Horváth, Z. L.

M. Erdélyi, Z. L. Horváth, G. Szabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, "Generation of diffraction-free beams for applications in optical microlithography," J. Vac. Sci. Technol. B 15, 287-292 (1997).
[CrossRef]

Z. L. Horváth, M. Erdélyi, G. Szabó, Zs. Bor, F. K. Tittel, and J. R. Cavallaro, "Generation of nearly nondiffracting Bessel beams with a Fabry-Perot interferometer," J. Opt. Soc. Am. A 14, 3009-3013 (1997).
[CrossRef]

Indebetouw, G.

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

G. Indebetouw, "Propagation of spatially periodic wave fields," Opt. Acta 31, 531-539 (1984).
[CrossRef]

G. Indebetouw, "Self-imaging through a Fabry-Perot interferometer," Opt. Acta 30, 1463-1471 (1983).
[CrossRef]

G. Indebetouw, "Unfocused optical correlator. Part 2: Fabry-Perot prefiltering," Optik (Stuttgart) 59, 287-302 (1981).

G. Indebetouw, "Tunable spatial filtering with a Fabry-Perot interferometer," Appl. Opt. 19, 761-764 (1980).
[CrossRef] [PubMed]

Jahns, J.

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

Khonina, S.

S. Khonina, V. Kotlyar, and V. Soifer, "Self-reproduction of multimode Hermite-Gaussian beams," Tech. Phys. Lett. 25, 489-491 (1999).
[CrossRef]

Klotz, E.

H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
[CrossRef]

Knuppertz, H.

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

Kotlyar, V.

S. Khonina, V. Kotlyar, and V. Soifer, "Self-reproduction of multimode Hermite-Gaussian beams," Tech. Phys. Lett. 25, 489-491 (1999).
[CrossRef]

Lohmann, A. W.

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

A. W. Lohmann and A. S. Marathay, "About periodicities in 3-D wavefields," Appl. Opt. 28, 4419-4423 (1989).
[CrossRef] [PubMed]

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

Marathay, A. S.

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Michette, A. G.

A. G. Michette, Optical Systems for Soft X Rays (Plenum, 1986).
[CrossRef]

Montgomery, W. D.

Ojeda-Castañeda, J.

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

J. Ojeda-Castañeda, P. Andrés, and E. Tepíchin, "Spatial filters for replicating images," Opt. Lett. 11, 551-553 (1986).
[CrossRef] [PubMed]

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

Patorski, K.

K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, E.Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 1-108.
[CrossRef]

Piestun, R.

Poleshchuk, A. G.

V. V. Cherkashin, A. G. Poleshchuk, and A. G. Sedukhin, "Experimental examination of self-imaging effect with the use of a diffractive multifocal lens," in EOS Topical Meeting Diffractive Optics 2005 (European Optical Society, 2005), pp. 70-71.

Saastamoinen, T.

Schechner, Y. Y.

Sedukhin, A. G.

A. G. Sedukhin, "Generalized periodically focused beams and multiple on-axis lens imaging," Opt. Commun. 229, 39-57 (2004).
[CrossRef]

A. G. Sedukhin, "Periodically focused propagation-invariant beams with sharp central peak," Opt. Commun. 228, 231-247 (2003).
[CrossRef]

A. G. Sedukhin, "Marginal phase correction of truncated Bessel beams," J. Opt. Soc. Am. A 17, 1059-1066 (2000).
[CrossRef]

V. V. Cherkashin, A. G. Poleshchuk, and A. G. Sedukhin, "Experimental examination of self-imaging effect with the use of a diffractive multifocal lens," in EOS Topical Meeting Diffractive Optics 2005 (European Optical Society, 2005), pp. 70-71.

Shamir, J.

Smayling, M. C.

Soifer, V.

S. Khonina, V. Kotlyar, and V. Soifer, "Self-reproduction of multimode Hermite-Gaussian beams," Tech. Phys. Lett. 25, 489-491 (1999).
[CrossRef]

Streibl, N.

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

Szabó, G.

Z. L. Horváth, M. Erdélyi, G. Szabó, Zs. Bor, F. K. Tittel, and J. R. Cavallaro, "Generation of nearly nondiffracting Bessel beams with a Fabry-Perot interferometer," J. Opt. Soc. Am. A 14, 3009-3013 (1997).
[CrossRef]

M. Erdélyi, Z. L. Horváth, G. Szabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, "Generation of diffraction-free beams for applications in optical microlithography," J. Vac. Sci. Technol. B 15, 287-292 (1997).
[CrossRef]

Szwaykowski, P.

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

Talbot, H. F.

H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).

Tepíchin, E.

Tervo, J.

Tittel, F. K.

Turunen, J.

Vahimaa, P.

Wagner, J.

Z. Bouchal and J. Wagner, "Self-reconstruction effect in free propagation of wave field," Opt. Commun. 176, 299-307 (2000).
[CrossRef]

J. Wagner and Z. Bouchal, "Experimental realization of self-reconstruction of the 2D aperiodic objects," Opt. Commun. 176, 309-311 (2000).
[CrossRef]

Wang, W.

Wilson, W. L.

Wolf, E.

Appl. Opt.

J. Mod. Opt.

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

M. V. Berry and E. Bodenschatz, "Caustics, multiply reconstructed by Talbot interference," J. Mod. Opt. 46, 349-365 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

M. Erdélyi, Z. L. Horváth, G. Szabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, "Generation of diffraction-free beams for applications in optical microlithography," J. Vac. Sci. Technol. B 15, 287-292 (1997).
[CrossRef]

Opt. Acta

H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensional periodic structures," Opt. Acta 24, 505-515 (1977).
[CrossRef]

G. Indebetouw, "Self-imaging through a Fabry-Perot interferometer," Opt. Acta 30, 1463-1471 (1983).
[CrossRef]

G. Indebetouw, "Propagation of spatially periodic wave fields," Opt. Acta 31, 531-539 (1984).
[CrossRef]

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

Opt. Commun.

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

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

H. Dammann and K. Gortler, "High-efficiency in-line multiple imaging by means of multiple phase holograms," Opt. Commun. 3, 312-315 (1971).
[CrossRef]

Z. Bouchal and J. Wagner, "Self-reconstruction effect in free propagation of wave field," Opt. Commun. 176, 299-307 (2000).
[CrossRef]

J. Wagner and Z. Bouchal, "Experimental realization of self-reconstruction of the 2D aperiodic objects," Opt. Commun. 176, 309-311 (2000).
[CrossRef]

A. G. Sedukhin, "Periodically focused propagation-invariant beams with sharp central peak," Opt. Commun. 228, 231-247 (2003).
[CrossRef]

A. G. Sedukhin, "Generalized periodically focused beams and multiple on-axis lens imaging," Opt. Commun. 229, 39-57 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttgart)

G. Indebetouw, "Unfocused optical correlator. Part 2: Fabry-Perot prefiltering," Optik (Stuttgart) 59, 287-302 (1981).

Philos. Mag.

H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Pure Appl. Opt.

J. Turunen and A. T. Friberg, "Self-imaging and propagation-invariance in electromagnetic fields," Pure Appl. Opt. 2, 51-60 (1993).
[CrossRef]

Tech. Phys. Lett.

S. Khonina, V. Kotlyar, and V. Soifer, "Self-reproduction of multimode Hermite-Gaussian beams," Tech. Phys. Lett. 25, 489-491 (1999).
[CrossRef]

Other

K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, E.Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 1-108.
[CrossRef]

V. V. Cherkashin, A. G. Poleshchuk, and A. G. Sedukhin, "Experimental examination of self-imaging effect with the use of a diffractive multifocal lens," in EOS Topical Meeting Diffractive Optics 2005 (European Optical Society, 2005), pp. 70-71.

A. G. Michette, Optical Systems for Soft X Rays (Plenum, 1986).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(a) Exact angular sampling for two collimated rays I and II of the lth diffraction order of a 1D diffraction grating and (b) the respective approximate sampling under the Talbot effect.

Fig. 2
Fig. 2

4 - f imaging systems for multiple replication of a planar object with amplitude transmittance t ( ρ , ξ ρ ) at the regularly spaced transverse planes z = z n . In the first system (a), a filter in the form of an FP interferometer with transmittance T FP ( θ ) is placed between the object and the first refractive lens L 1 . Here ξ ρ is the azimuth coordinate and θ is the incidence angle of the rays behind the object. In the second system (b), a filter with transmittance T ( r ) is placed in the Fourier plane and is implemented in the form of a CR mask or a DMF lens.

Fig. 3
Fig. 3

Exact angular samplings of rays (a) for Montgomery’s self-imaging condition, (b) for an interference maximum of an FP interferometer, (c) for a true multiple imaging, and (d) for the boundary of a Fresnel zone.

Fig. 4
Fig. 4

(a) Discrete nonlinear samples φ q (solid curves) of a continuous-relief phase function φ ( r ) (dashed curve) of a DMF lens with a fan-out of nine at Q = 16 and (b)–(d) decomposition of the phase function into sequences of the window functions of CR masks with different biased index constants Δ p and phase delays φ q . Three starting periods of the phase function (a) and three sequences of the window functions [(b)–(d)] are shown only at p = 0 , 1, 2 and Δ p = 0 , 1 Q , 2 Q .

Fig. 5
Fig. 5

Schematic representation of the angular spectra of plane waves of propagation-invariant, self-imaging, and periodically focused wave fields. The angular samples and relative phase offsets of respective wave vectors, indicated by bold arrows, are shown at Z λ = 10 . For the periodically focused fields, only four sampled vectors are shown in each angular zone, while the envelopes of spatial positions of the vectors correspond to a selection of nine images. For clarity, Talbot’s samples are shown approximately beyond the paraxial domain.

Fig. 6
Fig. 6

(a) Transmission function of an unbiased CR mask for a multiple imaging system operating at λ = 632.8 nm , N = 3 , P = 6 , Δ p = 0 , Z = 70 mm , Λ r = 1.276 mm , f = 300 mm , R b = 6.03 mm , and σ = 0.443 N = 0.148 . (b) Axial intensity distribution of the field generated by the system with the mask shown in (a). (c) Same as (a) but with σ = 0.0148 . (d) Axial intensity distribution of the field generated by a system with the mask shown in (c). (e) Phase function of a (0, non-π)-binary-phase DMF lens having a fan-out of seven and replacing the masks shown in (a) and (c). (f) Axial intensity distribution of the field generated by a system with the DMF lens shown in (e). Dashed curves in (b), (d), and (f) depict the squared single-zone diffraction envelopes of on-axis intensity maxima, while insets display the total light efficiency, η, and the nonuniformity of useful maxima in intensity, ε.

Equations (33)

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U ( ρ ) = p A p exp [ i ( n k p ρ + φ p ) ] ,
q d l = Z T tan [ arcsin ( l λ d ) ] l λ d π 10 l λ Z T d ,
( q d l ) 2 + Z T 2 = ( Z T + l 2 λ ) 2 l 2 λ Z T Z T 2 + 2 l 2 λ Z T ,
cos θ l 1 l 2 λ Z T ,
cos θ m = m λ Z ,
cos θ m = m λ ( 2 Δ z ) ,
cos θ p = 1 p λ Z ,
φ p = 0 ,
U ( ρ , z ) = 1 P + 1 2 [ 1 2 exp ( i k z ) + p = 1 P J 0 ( α p ρ ) exp ( i z k 2 α p 2 ) ] ,
α p = k sin θ p = 2 π 2 p λ Z p 2 Z 2
U ( 0 , z ) = { 1 2 P + 1 + 2 P 2 P + 1 sincs P ( z Z ) exp [ i π ( P + 1 ) z Z ] } exp ( i k z ) ,
U ( ρ , z ) P 1 = n = PSF n ( ρ , z )
C n = sgn ( z z n ) A [ α ( ρ , z ) α P ] exp [ sgn ( z z n ) i k ρ 2 + ( z z n ) 2 + i ψ ¯ n ] ρ 2 + ( z z n ) 2 .
U ( ρ , z ) = lim Q 1 P Q p = 0 P 1 q = 0 Q 1 A ( α p q α P ) T ( α p q ) J 0 ( α p q ρ ) exp ( i z k 2 α p q 2 ) ,
cos θ p q = 1 ( p + q Q ) λ Z .
U ( ρ , z ) = i C 0 α P A ( α α P ) T ( α ) J 0 ( α ρ ) exp ( i z k 2 α 2 ) α d α k 2 α 2 .
U ( 0 , z ) = E ( z ) sincs P ( z Z ) exp [ i k z i π ( P 1 ) z Z ] ,
E ( z ) = lim Q 1 Q q = 0 Q 1 T ( α 0 q ) exp ( i 2 π z Z q Q ) = 2 α 0 2 0 α 0 2 T ( α ) exp [ i 2 π z Z ( α α 0 ) 2 ] α d α
E ( n Z ) 2 1 2 N + 1 rect [ n 2 N + 1 ] ,
T FP ( θ ) = [ 1 + 4 R 2 ( 1 R ) 2 sin 2 ( 2 π Δ z λ cos θ ) ] 1 ,
r p f [ p + Fr ( 2 Δ z λ ) ] λ Δ z .
η n R 2 n 1 2 ( 1 + R ) ( 1 R ) 2 π ,
p c Δ z R b 2 [ λ ( f + 2 n Δ z ) 2 ] .
T CRM ( r , Δ p ) = p = 0 P + 1 2 W Δ p W p + Δ p w p + Δ p ,
w p + Δ p = rect [ r r p + Δ p δ r p + Δ p 2 δ r p + Δ p ] ,
r p + Δ p = f [ 1 ( p + Δ p ) λ Z ] 2 1 Λ r p + Δ p ,
δ r p + Δ p = Λ r ( p + Δ p + σ p + Δ p )
E p + Δ p ( z ) = 2 Λ r 2 r p + Δ p r p + Δ p + δ r p + Δ p exp ( i 2 π z r 2 Z Λ r 2 ) r dr = σ 2 sinc ( σ z Z ) exp { i π z Z [ 2 ( p + Δ p ) + σ ] } ,
η n = σ 2 sinc 2 ( n σ ) 4 .
T DMFL ( r ) = exp [ i φ ( r ) ] = n = a n exp [ i k r 2 2 ( F n ) ] ,
U ( ρ , z 0 ) = k exp ( i 2 k f ) i f F { T DMFL ( r ) } ω = k ρ f = f exp ( i 2 k f ) Z n = a n n exp ( i k ρ 2 2 n Z ) ,
E ( z ) = 2 Λ r 2 0 Λ r 2 T DMFL ( r ) exp [ i 2 π n z Z ( r Λ r ) 2 ] r d r = n = ( 1 ) n a n sinc ( n z Z ) exp ( i π z Z ) ,
T DMFL ( d ) ( r ) = q = 0 Q 1 T CRM ( r , Δ p = q Q ) exp ( i φ q ) .

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