Abstract

Self-imaging means image formation without the help of a lens or any other device between object and image. There are three versions of self-imaging: the classical Talbot effect (1836), the fractional Talbot effect, and the Montgomery effect (1967). Talbot required the object to be periodic; Montgomery realized that quasiperiodic suffices. Classical means that the distance from object to image is an integer multiple of the Talbot distance zT=2p2λ, where p is the grating period. Fractional implies a distance that is a simple fraction of zT: say, zT2,zT4,3zT2 . We explore the most general case of the fractional Montgomery effect.

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  1. H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).
  2. Lord Rayleigh, “On copying diffraction gratings and on some phenomenon connected therewith,” Philos. Mag. 11, 196–205 (1881).
    [CrossRef]
  3. K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
    [CrossRef]
  4. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  5. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  6. J. Cowley, A. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. London 76, 378–384 (1960).
    [CrossRef]
  7. H. Dammann, G. Groh, M. Kock, “Restoration of faulty images of periodic objects by means of self-imaging,” Appl. Opt. 10, 1454–1455 (1971).
    [CrossRef] [PubMed]
  8. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–418 (1973).
    [CrossRef]
  9. R. Ulrich, T. Kamiya, “Resolution of self-image in planar optical waveguides,” J. Opt. Soc. Am. 68, 583–592 (1978).
    [CrossRef]
  10. A. W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik (Stuttgart) 79, 41–45 (1988).
  11. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  12. A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
    [CrossRef]
  13. S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575–1580 (1971).
    [CrossRef] [PubMed]
  14. A. W. Lohmann, “A new Fourier spectrometer consisting of a two-grating-interferometer (resonance effects between two diffraction gratings in series),” in Proceedings of the International Commission for Optics Conference on Optical Instruments (International Commission for Optics, 1962), pp. 58–61.
  15. J. Jahns, E. ElJoudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
    [CrossRef]
  16. G. L. Rogers, “Interesting paradox in Fourier images,” J. Opt. Soc. Am. 62, 917–918 (1972).
    [CrossRef]
  17. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  18. G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
    [CrossRef]
  19. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
    [CrossRef]
  20. J. Tervo, J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. 190, 51–57 (2001).
    [CrossRef]
  21. Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Spatial Fourier-transform polarimetry using space-variant subwavelength metal-stripe polarizers,” Opt. Lett. 26, 1711–1713 (2001).
    [CrossRef]
  22. E. Lau, “Beugungserscheinung an Doppelrastern,” Ann. Phys. 6, 417–427 (1948).
    [CrossRef]
  23. J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  24. J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
    [CrossRef]
  25. A. W. Lohmann, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
    [CrossRef]
  26. P. Szwaykowski, “Self-imaging in polar coordinates,” J. Opt. Soc. Am. A 5, 185–191 (1988).
    [CrossRef]
  27. P. Szwaykowski, K. Patorski, “Moire fringes by evolute gratings,” Appl. Opt. 28, 4679–4681 (1989).
    [CrossRef] [PubMed]

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]

2001 (3)

J. Tervo, J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. 190, 51–57 (2001).
[CrossRef]

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Spatial Fourier-transform polarimetry using space-variant subwavelength metal-stripe polarizers,” Opt. Lett. 26, 1711–1713 (2001).
[CrossRef]

J. Jahns, E. ElJoudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

1999 (1)

1990 (1)

1989 (2)

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
[CrossRef]

P. Szwaykowski, K. Patorski, “Moire fringes by evolute gratings,” Appl. Opt. 28, 4679–4681 (1989).
[CrossRef] [PubMed]

1988 (3)

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

A. W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik (Stuttgart) 79, 41–45 (1988).

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

1979 (1)

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

1978 (1)

1973 (1)

1972 (2)

G. L. Rogers, “Interesting paradox in Fourier images,” J. Opt. Soc. Am. 62, 917–918 (1972).
[CrossRef]

A. W. Lohmann, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

1971 (3)

1967 (1)

1965 (1)

1964 (1)

1960 (1)

J. Cowley, A. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. London 76, 378–384 (1960).
[CrossRef]

1948 (1)

E. Lau, “Beugungserscheinung an Doppelrastern,” Ann. Phys. 6, 417–427 (1948).
[CrossRef]

1881 (1)

Lord Rayleigh, “On copying diffraction gratings and on some phenomenon 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).

Biener, G.

Bomzon, Z.

Bryngdahl, O.

Cowley, J.

J. Cowley, A. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. London 76, 378–384 (1960).
[CrossRef]

Dammann, H.

ElJoudi, E.

J. Jahns, E. ElJoudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

Gori, F.

Groh, G.

Hagedorn, D.

J. Jahns, E. ElJoudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

Hasman, E.

Indebetouw, G.

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

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]

J. Jahns, E. ElJoudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Kamiya, T.

Kinne, S.

J. Jahns, E. ElJoudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

Kleiner, 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]

Kock, M.

Lau, E.

E. Lau, “Beugungserscheinung an Doppelrastern,” Ann. Phys. 6, 417–427 (1948).
[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. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[CrossRef] [PubMed]

A. W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik (Stuttgart) 79, 41–45 (1988).

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

A. W. Lohmann, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

A. W. Lohmann, “A new Fourier spectrometer consisting of a two-grating-interferometer (resonance effects between two diffraction gratings in series),” in Proceedings of the International Commission for Optics Conference on Optical Instruments (International Commission for Optics, 1962), pp. 58–61.

McCutchen, C. W.

Montgomery, W. D.

Moodie, A.

J. Cowley, A. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. London 76, 378–384 (1960).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
[CrossRef]

P. Szwaykowski, K. Patorski, “Moire fringes by evolute gratings,” Appl. Opt. 28, 4679–4681 (1989).
[CrossRef] [PubMed]

Rayleigh, Lord

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

Rogers, G. L.

Silva, D. E.

A. W. Lohmann, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Suzuki, T.

Szwaykowski, P.

Talbot, H. F.

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

Tervo, J.

J. Tervo, J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. 190, 51–57 (2001).
[CrossRef]

Thomas, J. A.

Turunen, J.

J. Tervo, J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. 190, 51–57 (2001).
[CrossRef]

Ulrich, R.

Winthrop, J. T.

Worthington, C. R.

Yokozeki, S.

Ann. Phys. (1)

E. Lau, “Beugungserscheinung an Doppelrastern,” Ann. Phys. 6, 417–427 (1948).
[CrossRef]

Appl. Opt. (4)

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (6)

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

Opt. Commun. (5)

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

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, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

J. Tervo, J. Turunen, “Transverse and longitudinal periodicities in fields produced by polarization gratings,” Opt. Commun. 190, 51–57 (2001).
[CrossRef]

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (2)

J. Jahns, E. ElJoudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

A. W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik (Stuttgart) 79, 41–45 (1988).

Philos. Mag. (2)

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

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

Proc. Phys. Soc. London (1)

J. Cowley, A. Moodie, “Fourier images IV: the phase grating,” Proc. Phys. Soc. London 76, 378–384 (1960).
[CrossRef]

Prog. Opt. (1)

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
[CrossRef]

Other (1)

A. W. Lohmann, “A new Fourier spectrometer consisting of a two-grating-interferometer (resonance effects between two diffraction gratings in series),” in Proceedings of the International Commission for Optics Conference on Optical Instruments (International Commission for Optics, 1962), pp. 58–61.

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Figures (10)

Fig. 1
Fig. 1

Talbot setup. A grating with the period p is the object; the observed image is at distance z.

Fig. 2
Fig. 2

Ronchi grating.

Fig. 3
Fig. 3

Fractional Talbot image of a Ronchi grating at a distance z = z T 2 .

Fig. 4
Fig. 4

Close relative of the Ronchi grating with the duty factor ( 1 3 ) .

Fig. 5
Fig. 5

Lateral field in x direction generated for (a) Talbot and (b) Montgomery objects at the distances z z T = 0 , 0.25, 0.5 (top to bottom); in both cases only odd coefficients are relevant. The coefficients used are sinclike with a duty cycle of 1 2 ; seven harmonics are used. The lateral field in the x direction is generated for (c) Talbot and (d) Montgomery objects in the distances z z T = 0 , 0.25, 0.5 (top to bottom); in both cases odd and even coefficients are relevant. Used coefficients are sinclike with a duty cycle of 1 4 ; seven harmonics are used.

Fig. 6
Fig. 6

(a) Ewald sphere, (b) shifted Ewald sphere, (c) paraboloid, paraxial approximation.

Fig. 7
Fig. 7

Talbot effect in the spatial-frequency domain.

Fig. 8
Fig. 8

Montgomery effect in the spatial-frequency domain.

Fig. 9
Fig. 9

Peaks of the Talbot effect (open circles) and of the Montgomery effect (solid circles); ν 0 is the lateral periodicity; ρ is the longitudinal periodicity.

Fig. 10
Fig. 10

Setup underlying the McCutchen theorem.

Tables (1)

Tables Icon

Table 1 Classification of Self-Imaging Effects

Equations (82)

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z T = 2 p 2 λ , p : grating period .
U T ( x ) = U T ( x + p ) = m A m exp ( 2 π i m x p ) .
U T ( x ) exp ( 2 π i ν x ) d x = U ̃ T ( ν ) = m A m δ ( ν m ν 0 ) .
ν 0 = 1 p .
U m ( x ) = B 0 + m 0 B m exp ( 2 π i m m ν 0 x ) ,
U ̃ M ( ν ) = B 0 δ ( ν ) + m 0 B m δ ( ν m m ν 0 ) .
Δ z = N M z T .
U ( x , 0 ) = U ¯ + m 0 A m exp ( 2 π i m ν 0 x ) = A 0 + u ( x , 0 ) .
U ̃ ( ν , 0 ) = A 0 δ ( ν ) + m 0 A m δ ( ν m ν 0 ) = A 0 δ ( ν ) + u ̃ ( ν , z = 0 ) .
exp ( i π λ z ν 2 ) = exp ( i π λ z m 2 ν 0 2 ) ,
U ̃ ( ν , z ) = A 0 δ ( ν ) + m 0 A m exp ( i π λ z m 2 ν 0 2 ) δ ( ν m ν 0 ) = A 0 δ ( ν ) + u ̃ ( ν , z ) .
i π λ z m 2 ν 0 2 = 2 π i m 2 z z T , z T = 2 λ ν 0 2 = 2 p 2 λ .
U ̃ ( ν , z ) = A 0 δ ( ν ) + m 0 A m exp ( 2 π i m 2 z z T ) δ ( ν m ν 0 ) .
U ( x , z ) = A 0 + m 0 A m exp [ 2 π i ( m ν 0 x m 2 z z T ) ] = U ( x , z + N z T ) .
U ( x , 0 ) = 1 + m 0 sinc ( m 2 ) exp ( 2 π m ν 0 x ) = 1 + Δ U ( x , 0 ) .
A 0 = 1 ; A m = sinc ( m 2 ) ; A 2 m = 0 .
U ( x , z ) = 1 + m 0 sinc ( m 2 ) exp [ 2 π i ( m ν 0 x m 2 z z T ) ] = 1 + Δ U ( x , z ) .
U ( x , z T 2 ) = 1 m 0 sinc ( m 2 ) exp ( 2 π i m ν 0 x ) = 1 Δ U ( x , 0 ) .
exp ( 2 π i m 2 2 ) = 1 ( m odd ) .
U ( x , 0 ) = 1 + Δ U ( x , 0 ) = 1 + { + 1 x m p p 4 p 2 1 otherwise } = { 2 0 } ,
U ( x , z T 2 ) = 1 Δ U ( x , 0 ) = 1 { + 1 1 } = { 0 2 } .
U ( x , z T 2 ) = U ( x + p 2 , 0 ) .
v ( x ) = m B m exp ( 2 π i m ν 0 x ) ,
Δ U ( x , 0 ) = v ( x + p 4 ) v ( x p 4 ) = 2 i m B m sin ( m π 2 ) exp ( 2 π i m ν 0 x ) .
A m A m exp [ 2 π i m 2 ( M N ) ] ,
A m A m exp [ 2 π i m ( M N ) ] .
U ( x , 0 ) U ( x , 0 ) exp ( 2 π i ν 0 x ) ,
A m A m 1 .
U M ( x , z ) = B 0 + m 0 B m exp [ 2 π i ( m m ν 0 x m z z T ) ] ,
U ̃ M ( ν , z ) = B 0 δ ( ν ) + m 0 B m exp ( 2 π i m z z T ) δ ( ν m m ν 0 ) .
U ( x , z T ) = U ( x , 0 ) .
2 U M ( x , z ) x 2 + 2 i k U M ( x , z ) z = 0 .
U M ( x , z T 2 ) = B 0 m 0 B m exp ( 2 π i m m ν 0 x ) .
U M ( x , z T 2 ) = B 0 Δ U M ( x , 0 ) ,
U M ( x , z T 2 ) = 1 m 0 sinc ( m 2 ) cos ( 2 π m ν 0 x ) .
A m = B m = sinc ( m 2 ) .
U T ( x , 0 ) = 1 + m 0 sinc ( m 2 ) exp ( 2 π i m ν 0 x ) ,
U M ( x , 0 ) = 1 + m 0 sinc ( m 2 ) exp ( 2 π i m m ν 0 x ) .
U T ( x , z T 2 ) = 1 m 0 sinc ( m 2 ) exp ( 2 π i m ν 0 x ) ,
U M ( x , z T 2 ) = 1 m 0 sinc ( m 2 ) exp ( 2 π i m m ν 0 x ) .
B 2 m = 0 , except B 0 = U ¯ M .
Δ U M ( x , z T 2 ) = Δ U M ( x , 0 ) .
B 2 m + B 2 m + 1 = U M , S ( x , z ) + U M , A ( x , z ) .
Δ U M , S ( x , z T 2 ) = + Δ U M , S ( x , z T 2 ) ,
Δ U M , A ( x , z T 2 ) = Δ U M , A ( x , z T 2 ) .
Δ U M = B 3 m + B 3 m + 1 + B 3 m 1 ,
exp [ ( 2 π i 3 ) 3 m ] = 1 ,
exp [ ( 2 π i 3 ) ( 3 m + 1 ) ] = exp ( 2 π i 3 ) ,
exp [ ( 2 π i 3 ) ( 3 m 1 ) ] = exp ( + 2 π i 3 ) .
Δ U M ( x , 0 ) = U ¯ M + Δ U M ( x , 0 ) ,
Δ U M ( x , z T 2 ) = U ¯ M Δ U M ( x , 0 ) .
F ̃ ( ν ) = sin [ π ( ν ν 0 ) 2 ] = exp [ + i π ( ν ν 0 ) 2 ] exp [ i π ( ν ν 0 ) 2 ] 2 i .
F ̃ ( ν ) = 1 cos [ π 2 ( ν ν 0 ) 2 ] .
Δ V ( x , y , z ) + k 2 V ( x , y , z ) = 0 , k = 2 π λ .
V ( x , y , z ) = V ̃ ( ν x , ν y , ν z ) exp [ 2 π i ( x ν x + y ν y + z ν z ) ] d ν x d ν y d ν z .
[ ( 2 π ) 2 ( ν x 2 + ν y 2 + ν z 2 ) + k 2 ] V ̃ ( ν ) exp ( 2 π i ν x ) d ν x d ν y d ν z = 0 .
ν x 2 + ν y 2 + ν z 2 = ( 1 λ ) 2 .
V ( x , y , z ) = u ( x , y , z ) exp ( 2 π i z λ ) .
ν x 2 + ν y 2 + ( ν z + 1 λ ) 2 = ( 1 λ ) 2 .
Δ x y z u ( x , y , z ) + 2 i k u ( x , y , z ) z = 0 .
2 u ( x , y , z ) z 2 0 .
Δ x y u ( x , y , z ) + 2 i k u ( x , y , z ) z = 0 .
ν x 2 + ν y 2 + 2 ν z λ = 0 .
ν x 2 + 2 ν z λ = 0 .
U ( x , y ) = n U n ( z ) exp ( 2 π i n ν 0 x ) .
U ̃ ( ν x , ν y ) = n U ̃ n ( ν z ) δ ( ν x n ν 0 ) .
ν z = n 2 λ ν 0 2 2 = n 2 ρ ,
1 ρ = z T = 2 λ ν 0 2 = 2 p 2 λ .
ν z = m ρ .
ν x 2 = 2 m ρ λ .
ν m = m ν 1 = m ρ
U T ( x , z ) = A 0 + m 0 A m exp [ 2 π i ( m ν 0 x m 2 z z T ) ] ,
U M ( x , z ) = B 0 + m 0 B m exp [ 2 π i ( m m ν 0 x m z z T ) ] .
( m , m 2 ) for Talbot and ( m m , m ) for Montgomery .
U T ( x , z = N z T ) = U T ( x , z = 0 ) = A 0 + m 0 A m exp [ 2 π i ( m ν 0 x ) ] ,
U M ( x = 0 , z ) = B 0 + m 0 B m exp ( 2 π i m z z T ) .
A m = B m .
z T p = 2 p λ .
U ( 0 , 0 , z ) = P ̃ ( r 2 ) exp ( i π z λ f 2 r 2 ) d ( r 2 ) .
P ̃ ( r 2 ) = m > 0 A m δ ( r 2 m r 1 2 ) = m > 0 A m δ ( r m r 1 ) .
r m = r 1 m , m = 0 , 1 , 2 .
B m B m exp ( 2 π i m N M ) .

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