Abstract

Based on the theory of scalar diffraction, the diffraction of gratings in the deep Fresnel diffraction region is developed, and the general formula of the diffraction intensity of the one-dimensional grating is presented by using the Hankel function. Through numerical calculations, some interesting diffraction phenomena are found. In the deep Fresnel diffraction region, the dominant effects, with increasing propagation distance from the grating, are, in order, the geometrical effect, the quasi-geometrical effect, and the interference and diffraction effects. Furthermore, the diffraction intensities vary periodically in the diffraction effect region with increasing propagation distance. Quasi-Talbot imaging of the grating exists in the interference and diffraction regions, and the intensity distributions most similar to the structure of the grating are not at the exact Talbot distances. These phenomena in the deep Fresnel diffraction region are distinct from those in the Fresnel diffraction region. The formation origin of quasi-Talbot imaging of the grating is also discussed, and the numerical calculations powerfully verify the theoretical results.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, "Phase codes of Talbot array illumination for encoding holographic multiplexing storage," Opt. Commun. 161, 209-211 (1999).
    [CrossRef]
  2. J. Sumaya-Martinez, O. Mata-Mendez, and F. Chavez-Rivas, "Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization," J. Opt. Soc. Am. A 20, 827-835 (2003).
    [CrossRef]
  3. Donald C. O'Shea and Willie S. Rochward, "Light modulation from crossed phase gratings," Opt. Lett. 23, 491-493 (1998).
    [CrossRef]
  4. W. H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).
  5. L. Liu, "Talbot and Lau effects on incident beams of arbitrary wavefront and their use," Appl. Opt. 28, 4668-4678 (1989).
    [CrossRef] [PubMed]
  6. A. W. Lohmann and J. Thomas, "Making an array illuminator based on the Talbot effect," Appl. Opt. 29, 4337-4340 (1990).
    [CrossRef] [PubMed]
  7. L. Liu, "Lau cavity and phase locking of laser arrays," Opt. Lett. 14, 1312-1314 (1989).
    [CrossRef] [PubMed]
  8. S. Teng, L. Liu, J. Zu, Z. Luan, and D. Liu, "Uniform theory of the Talbot effect with partially coherent light illumination," J. Opt. Soc. Am. A 20, 1747-1754 (2003).
    [CrossRef]
  9. J. D. Mills, C. W. J. Hillman, B. H. Blott, and W. S. Brocklesby, "Imaging of free-space interference patterns used to manufacture fiber Bragg gratings," Appl. Opt. 39, 6128-6135 (2000).
    [CrossRef]
  10. P. Latimer and R. Crouse, "Talbot effect reinterpreted," Appl. Opt. 31, 80-89 (1992).
    [CrossRef] [PubMed]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  12. X. P. Wu and F. P. Chiang, "Coherent chromatic speckle," in Proceedings of the International Conference on Advanced Experimental Mechanics (Peiyang S. & T. Development Co., 1988), p. C19-C24.
  13. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 2001).
  14. J. A. Sanchez-Gil and M. Nieto-Vesperinas, "Light scattering from random rough dielectric surfaces," J. Opt. Soc. Am. A 8, 1270-1286 (1991).
    [CrossRef]
  15. M. Abramowitz and I. A. Stegum, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 4th ed. (National Bureau of Standards, 1965).

2003

2000

1999

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, "Phase codes of Talbot array illumination for encoding holographic multiplexing storage," Opt. Commun. 161, 209-211 (1999).
[CrossRef]

1998

1992

1991

1990

1989

1836

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

Abramowitz, M.

M. Abramowitz and I. A. Stegum, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 4th ed. (National Bureau of Standards, 1965).

Blott, B. H.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 2001).

Brocklesby, W. S.

Chavez-Rivas, F.

Chiang, F. P.

X. P. Wu and F. P. Chiang, "Coherent chromatic speckle," in Proceedings of the International Conference on Advanced Experimental Mechanics (Peiyang S. & T. Development Co., 1988), p. C19-C24.

Crouse, R.

Denz, C.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, "Phase codes of Talbot array illumination for encoding holographic multiplexing storage," Opt. Commun. 161, 209-211 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Hillman, C. W. J.

Latimer, P.

Liu, D.

Liu, L.

Lohmann, A. W.

Luan, Z.

Mata-Mendez, O.

Mills, J. D.

Nieto-Vesperinas, M.

O'Shea, Donald C.

Rochward, Willie S.

Sanchez-Gil, J. A.

Stankovic, S.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, "Phase codes of Talbot array illumination for encoding holographic multiplexing storage," Opt. Commun. 161, 209-211 (1999).
[CrossRef]

Stegum, I. A.

M. Abramowitz and I. A. Stegum, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 4th ed. (National Bureau of Standards, 1965).

Sumaya-Martinez, J.

Talbot, W. H. F.

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

Teng, S.

Thomas, J.

Tschuli, T.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, "Phase codes of Talbot array illumination for encoding holographic multiplexing storage," Opt. Commun. 161, 209-211 (1999).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 2001).

Wu, X. P.

X. P. Wu and F. P. Chiang, "Coherent chromatic speckle," in Proceedings of the International Conference on Advanced Experimental Mechanics (Peiyang S. & T. Development Co., 1988), p. C19-C24.

Zhou, C.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, "Phase codes of Talbot array illumination for encoding holographic multiplexing storage," Opt. Commun. 161, 209-211 (1999).
[CrossRef]

Zu, J.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

C. Zhou, S. Stankovic, C. Denz, and T. Tschuli, "Phase codes of Talbot array illumination for encoding holographic multiplexing storage," Opt. Commun. 161, 209-211 (1999).
[CrossRef]

Opt. Lett.

Philos. Mag.

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

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

X. P. Wu and F. P. Chiang, "Coherent chromatic speckle," in Proceedings of the International Conference on Advanced Experimental Mechanics (Peiyang S. & T. Development Co., 1988), p. C19-C24.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 2001).

M. Abramowitz and I. A. Stegum, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 4th ed. (National Bureau of Standards, 1965).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the diffraction of the grating.

Fig. 2
Fig. 2

Gray-scale diagram of the diffraction intensity of the rectangular grating with d = 10 μ m .

Fig. 3
Fig. 3

Gray-scale diagrams of the diffraction intensity of the rectangular gratings with (a) d = 4 μ m and (b) d = 20 μ m .

Fig. 4
Fig. 4

Diffraction intensity distributions of the grating at (a)–(e) the Talbot distances and (e)–(h) the quasi-Talbot distances.

Fig. 5
Fig. 5

Gray-scale diagram of the diffraction intensities near (a) z = z T , (b) z = 2 z T , (c) z = 5 z T , (d) z = 10 z T .

Fig. 6
Fig. 6

Phase angle and the real and imaginary parts of (a) P m and (b) F m near z = 5 z T .

Fig. 7
Fig. 7

Influence of each diffraction order on Talbot imaging.

Fig. 8
Fig. 8

Distributions of complex amplitudes and angles of F 0 , F 1 , and F 3 near the quasi-Talbot distances.

Fig. 9
Fig. 9

Influence of the series expansion of r on the shift of the quasi-Talbot distance.

Fig. 10
Fig. 10

Influence of additional phase on the quasi-Talbot distance.

Fig. 11
Fig. 11

Influence of grating size on deep Fresnel diffraction.

Tables (1)

Tables Icon

Table 1 Comparison of the Talbot Distance and the Quasi-Talbot Distance

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

t ( r 0 ) = a ( r 0 ) exp [ i n k h ( r 0 ) ] ,
u ( r ) = 1 4 π S { u 0 ( r 0 ) G ( r , r 0 ) n G ( r , r 0 ) u 0 ( r 0 ) n } d S ,
u ( r ) = A t ( r 0 ) K ( r , r 0 ) d r 0 ,
K ( r , r 0 ) = 1 4 π G ( r , r 0 ) n
G ( r , r 0 ) = i π H 0 ( 1 ) ( k r ) ,
K ( x , x 0 ) = i 4 k ( z h ) H 1 ( 1 ) ( k r ) r ,
I ( x , z ) = k 2 A 2 16 [ t ( x 0 ) ( z h ) H 1 ( 1 ) ( k r ) r ] d x 0 2 .
t ( x 0 ) = n = 0 N rect ( x 0 n d d M ) ,
t ( x 0 ) = m = C m exp ( i 2 π m x 0 d ) ,
K ( r ; r 0 ) = 1 i z λ exp ( i k z ) exp ( i k 2 z r r 0 2 ) .
u ( x , z ) = A i z λ exp ( i k z ) L 2 L 2 m C m exp ( i 2 π m x 0 d ) exp [ i π ( x + x 0 ) 2 λ z ] d x 0 .
u ( x , z ) = A i z λ exp ( i k z ) m C m exp ( i 2 π m x d ) exp ( i π λ m 2 z d 2 ) .
I ( x , z ) = A 2 z λ exp ( i k z ) m C m exp ( i 2 π m x d ) exp ( i π λ m 2 z d 2 ) 2 .
K ( r , r 0 ) = i 4 2 k π z r 3 2 exp ( i k r i 3 π 4 ) .
I ( x , z ) = k A 2 8 π L 2 L 2 m C m exp ( i 2 π m x 0 d ) z exp ( i k r ) r 3 2 d x 0 2 .
I ( x , z ) = k A 2 8 π m C m exp ( i 2 π m x d ) x L 2 x + L 2 exp ( i 2 π m ξ d ) z exp [ i k ( ξ 2 + z 2 ) 1 2 ] ( ξ 2 + z 2 ) 3 4 d ξ 2 .
F m = x L 2 x + L 2 exp ( i 2 π m ξ d ) z exp [ i k ( ξ 2 + z 2 ) 1 2 ] ( ξ 2 + z 2 ) 3 4 d ξ
z T = λ 1 ( m 2 λ 2 ) d 2 1 ( n 2 λ 2 ) d 2 ,
P m = x L 2 x + L 2 exp ( i 2 π m ξ d ) 1 2 exp ( i k z ) exp [ i k ( ξ 2 + z 2 ) 1 2 ] d ξ ,
1 z exp ( i k z ) exp ( i π λ m 2 z d 2 )
z + ξ 2 2 z ξ 4 8 z 3 + ξ 6 16 z 5 ξ 8 128 z 7 ,
z + ξ 2 2 z ξ 4 8 z 3 + ξ 6 16 z 5 ,
z + ξ 2 2 z ξ 4 8 z 3 , and z + ξ 2 2 z .

Metrics