Abstract

A reciprocal vector theory for analysis of the Talbot effect of periodic objects is proposed. Using this method we deduce a general condition for determining the Talbot distance. Talbot distances of some typical arrays (a rectangular array, a centered-square array, and a hexagonal array) are derived from this condition. Further, the fractional Talbot effect of a one-dimensional grating, a square array, a centered-square array, and a hexagonal array is analyzed and some simple analytical expressions for calculation of the complex amplitude distribution at any fractional Talbot plane are deduced. Based on these formulas, we design some Talbot array illuminators with a high compression ratio. Finally, some computer-simulated results consistent with the theoretical analysis are given.

© 2008 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).
    [CrossRef]
  2. J. T. Winthrop and C. R. Worthington, "Theory of Fresnel images. I. Plane periodic objects in monochromatic light," J. Opt. Soc. Am. 55, 373-381 (1965).
    [CrossRef]
  3. A. W. Lohmann and J. A. Thomas, "Making an array illuminator based on the Talbot effect," Appl. Opt. 29, 4337-4340 (1990).
    [CrossRef] [PubMed]
  4. C. Zhou and L. Liu, "Simple equations for the calculation of a multilevel phase grating for Talbot array illumination," Opt. Commun. 115, 40-44 (1995).
    [CrossRef]
  5. C. Zhou, L. Wang, and T. Tschudi, "Solutions and analyses of fractional-Talbot array illuminations," Opt. Commun. 147, 224-228 (1998).
    [CrossRef]
  6. C. Zhou, H. Wang, S. Zhao, P. Xi, and L. Liu, "Number of phase levels of a Talbot array illuminator," Appl. Opt. 40, 607-613 (2001).
    [CrossRef]
  7. Y. Lu, C. Zhou, S. Wang, and B. Wang, "Polarization-dependent Talbot effect," J. Opt. Soc. Am. A 23, 2154-2160 (2006).
    [CrossRef]
  8. J. R. Leger and G. J. Swanson, "Efficient array illuminator using binary-optics phase at fractional-Talbot planes," Opt. Lett. 15, 288-290 (1990).
    [CrossRef] [PubMed]
  9. P. Szwaykowski and V. Arrizon, "Talbot array illuminator with multilevel phase grating," Appl. Opt. 32, 1109-1114 (1993).
    [CrossRef] [PubMed]
  10. C. Zhou, S. Stankovic, and T. Tschudi, "Analytic phase-factor equations for Talbot array illuminations," Appl. Opt. 38, 284-290 (1999).
    [CrossRef]
  11. C.-S. Guo, X. Yin, L.-W. Zhu, and Z.-P. Hong, "Analytical expression for phase distribution of a hexagonal array at fractional Talbot planes," Opt. Lett. 32, 2079-2081 (2007).
    [CrossRef] [PubMed]
  12. C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).
  13. P. Xi, C. Zhou, E. Dai, and L. Liu, "Generation of near-field hexagonal array illumination with a phase grating," Opt. Lett. 27, 228-230 (2002).
    [CrossRef]

2007 (1)

2006 (1)

2002 (1)

2001 (1)

1999 (1)

1998 (1)

C. Zhou, L. Wang, and T. Tschudi, "Solutions and analyses of fractional-Talbot array illuminations," Opt. Commun. 147, 224-228 (1998).
[CrossRef]

1996 (1)

C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).

1995 (1)

C. Zhou and L. Liu, "Simple equations for the calculation of a multilevel phase grating for Talbot array illumination," Opt. Commun. 115, 40-44 (1995).
[CrossRef]

1993 (1)

1990 (2)

1965 (1)

1836 (1)

H. F. Talbot, "Facts relating to optical science. No. IV," Philos. Mag. 9, 401-407 (1836).
[CrossRef]

Arrizon, V.

Dai, E.

Guo, C.-S.

Hong, Z.-P.

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).

Leger, J. R.

Liu, L.

Lohmann, A. W.

Lu, Y.

Stankovic, S.

Swanson, G. J.

Szwaykowski, P.

Talbot, H. F.

H. F. Talbot, "Facts relating to optical science. No. IV," Philos. Mag. 9, 401-407 (1836).
[CrossRef]

Thomas, J. A.

Tschudi, T.

C. Zhou, S. Stankovic, and T. Tschudi, "Analytic phase-factor equations for Talbot array illuminations," Appl. Opt. 38, 284-290 (1999).
[CrossRef]

C. Zhou, L. Wang, and T. Tschudi, "Solutions and analyses of fractional-Talbot array illuminations," Opt. Commun. 147, 224-228 (1998).
[CrossRef]

Wang, B.

Wang, H.

Wang, L.

C. Zhou, L. Wang, and T. Tschudi, "Solutions and analyses of fractional-Talbot array illuminations," Opt. Commun. 147, 224-228 (1998).
[CrossRef]

Wang, S.

Winthrop, J. T.

Worthington, C. R.

Xi, P.

Yin, X.

Zhao, S.

Zhou, C.

Zhu, L.-W.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

C. Zhou and L. Liu, "Simple equations for the calculation of a multilevel phase grating for Talbot array illumination," Opt. Commun. 115, 40-44 (1995).
[CrossRef]

C. Zhou, L. Wang, and T. Tschudi, "Solutions and analyses of fractional-Talbot array illuminations," Opt. Commun. 147, 224-228 (1998).
[CrossRef]

Opt. Lett. (3)

Philos. Mag. (1)

H. F. Talbot, "Facts relating to optical science. No. IV," Philos. Mag. 9, 401-407 (1836).
[CrossRef]

Other (1)

C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).

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Figures (4)

Fig. 1
Fig. 1

Sketches of some 2D periodic lattices and the corresponding reciprocal lattices: (a) a rectangular array, (b) a centered-square array, and (c) a hexagonal array.

Fig. 2
Fig. 2

Output lattice structure of a hexagonal array at the fractional Talbot plane with β = 3 . The white spots correspond to the points with an amplitude of zero, and the black spots represent the points with a relative amplitude of 3 3 .

Fig. 3
Fig. 3

Square array (a1) and its fractional Talbot image at β = 2 (a2), β = 3 (a3), and β = 4 (a4). A centered-square array (b1) and its fractional Talbot image at β = 2 (b2), β = 3 (b3), and β = 4 (b4). A hexagonal array (c1) and its fractional Talbot image at β = 2 (c2), β = 3 (c3), and β = 4 (c4).

Fig. 4
Fig. 4

Examples of the TAIs designed according to Eqs. (23a, 23b, 27, 30) using a high fraction parameter of β = 17 , where (a1), (b1), and (c1) are the phase distributions of the TAIs corresponding to a square array, a centered-square array, and a hexagonal array, respectively. When these TAIs are illuminated by a plane wave, the reconstructed intensity distributions at the corresponding fractional Talbot distance are shown in (a2), (b2), and (c2), respectively. The fractional Talbot distance is equal to 65.8 mm (for the square TAI), 67.2 mm (for the centered-square TAI), and 64.5 mm (for the hexagonal TAI).

Tables (1)

Tables Icon

Table 1 Talbot Effect of Some Typical 2D Periodic Arrays a

Equations (36)

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U ( r ) = u 0 ( r ) lattice ( r , R n ) ,
R n = n 1 a 1 + n 2 a 2 ,
F { lattice ( r , R n ) } = 1 S n lattice ( ρ , K h ) ,
K h = h 1 b 1 + h 2 b 2 ,
F 1 { lattice ( ρ , K h ) } = 1 S h lattice ( r , R n ) ,
b 1 = a 2 × a 0 a 0 ( a 1 × a 2 ) , a 1 = b 2 × b 0 b 0 ( b 1 × b 2 ) ,
b 2 = a 0 × a 1 a 0 ( a 1 × a 2 ) , a 2 = b 0 × b 1 b 0 ( b 1 × b 2 ) ,
G ( ρ ) = 1 S n G 0 ( ρ ) lattice ( ρ , K h ) .
U z = 1 S n F 1 { G 0 ( ρ ) lattice ( ρ , K h ) × exp [ i 2 π z λ 1 λ 2 ρ 2 ] } .
U z = 1 S n exp ( i 2 π z λ ) F 1 { G 0 ( ρ ) lattice ( ρ , K h ) × exp ( i π λ z ρ 2 ) } ,
λ z K h 2 = 2 m ( m is an integer ) ,
U z = 1 S n exp ( i 2 π z λ ) F 1 { G 0 ( ρ ) lattice ( ρ , K h ) × exp ( i 2 π m ) } = exp ( i 2 π z λ ) U ( r ) .
U z β = 1 S n exp ( i 2 π z β λ ) F 1 { G 0 ( ρ ) lattice ( ρ , K h ) × F ( h 1 , h 2 , β ) } ,
F ( h 1 , h 2 , β ) = exp ( i π λ z T K h 2 β ) .
U z β = 1 S n exp ( i 2 π z β λ ) F 1 { G 0 ( ρ ) [ C ( ρ ) lattice ( ρ , β K h ) ] } ,
C ( ρ ) = h 2 = 0 β 1 h 1 = 0 β 1 δ ( ρ K h ) F ( h 1 , h 2 , β ) .
U z β = exp ( i 2 π z β λ ) u 0 ( r ) [ c ( r ) lattice ( r , R n β ) ] ,
c ( r ) = 1 β 2 h 2 β 1 h 1 β 1 F ( h 1 , h 2 , β ) × exp ( i 2 π β K h R n ) .
c ( n 1 , n 2 , β ) = 1 β 2 h 2 = 0 β 1 h 1 = 0 β 1 F ( h 1 , h 2 , β ) × exp [ i 2 π β ( n 1 h 1 + n 2 h 2 ) ] .
c ( n 1 , n 2 , β ) = A ( n 1 , n 2 , β ) exp [ i ϕ ( n 1 , n 2 , β ) ] ,
F ( h , β ) = exp ( i 2 π h 2 β ) ,
c ( n , β ) = 1 β h = 0 β 1 exp [ i 2 π β ( h 2 n h ) ] .
A ( n , β ) = { 1 β for odd β 2 β for even β , even β 2 and n 2 β for even β , odd β 2 and n 0 for even β , others } ,
ϕ ( n , β ) = π 2 ( α 1 β ) n 2 ,
α = { 1 2 , β = 4 L 2 , L = 1 , 2 , 3 , 1 , β = 4 L 1 , L = 1 , 2 , 3 , 0 , β = 4 L , L = 1 , 2 , 3 , 1 , β = 4 L + 1 , L = 1 , 2 , 3 , } .
F ( h 1 , h 2 , β ) = exp [ i 2 π ( h 1 2 + h 2 2 ) β ] .
c ( n 1 , n 2 , β ) = 1 β 2 h 2 = 0 β 1 h 1 = 0 β 1 exp [ i 2 π β ( h 1 2 + h 2 2 n 1 h 1 n 2 h 2 ) ] .
A ( n 1 , n 2 , β ) = { 1 β for odd β 2 β for even β ; even β 2 , n 1 , and n 2 2 β for even β ; odd β 2 , n 1 , and n 2 0 for even β , others } ,
ϕ ( n 1 , n 2 , β ) = π 2 ( α 1 β ) ( n 1 2 + n 2 2 ) ,
c ( n 1 , n 2 , β ) = 1 β 2 h 2 = 0 β 1 h 1 = 0 β 1 exp [ i 2 π β ( h 1 2 + 2 h 2 2 2 h 1 h 2 n 1 h 1 n 2 h 2 ) ] .
A ( n 1 , n 2 , β ) = { 1 β for odd β 2 β for even β , even β 2 , odd n 1 , and even n 2 2 β for even β , odd β 2 , even n 1 and n 2 0 for even β , otherwise } ,
ϕ ( n 1 , n 2 , β ) = π 2 ( α 1 β ) ( 2 n 1 2 + n 2 2 + 2 n 1 n 2 ) ,
c ( n 1 , n 2 , β ) = 1 β 2 h 2 = 0 β 1 h 1 = 0 β 1 exp [ i 2 π β ( h 1 2 + h 2 2 h 1 h 2 n 1 h 1 n 2 h 2 ) ] .
A ( n 1 , n 2 , β ) = { 3 β , β = 3 L , n 1 n 2 = 3 m 0 , β = 3 L , n 1 n 2 3 m 1 β , β 3 L } .
ϕ ( n 1 , n 2 , β ) = 2 π 3 ( γ + 1 β ) ( n 1 2 + n 2 2 + n 1 n 2 ) ,
γ = { 0 , β = 3 L , L = 1 , 2 , 3 1 , β = 3 L 1 , L = 1 , 2 , 3 2 , β = 3 L + 1 , L = 1 , 2 , 3 } .

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