Abstract

We describe a reciprocal-lattice vector method for analysis of the diffractive self-imaging (or Talbot effect) of a two-dimensional periodic object. Using this method we analyze the fractional Talbot effect of a hexagonal array and deduce a simple analytical expression for calculation of the complex amplitude distribution at any fractional Talbot plane. Based on this new formula, we design a hexagonal array illuminator (HTAI) with a high fractional parameter. A computer simulation for demonstration of the HTAI is also given.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. B. Mortimore and J. W. Arkwright, Appl. Opt. 30, 650 (1991).
    [Crossref] [PubMed]
  2. S. Nemoto and J. Kida, Appl. Opt. 30, 815 (1991).
    [Crossref] [PubMed]
  3. M. Thorhauge, L. H. Frandsen, and P. I. Borel, Opt. Lett. 28, 1525 (2003).
    [Crossref] [PubMed]
  4. L. Liu, X. Liu, and B. Cui, Appl. Opt. 30, 943 (1991).
    [Crossref] [PubMed]
  5. P. Xi, C. Zhou, E. Dai, and L. Liu, Opt. Lett. 27, 228 (2002).
    [Crossref]
  6. A. W. Lohmann and J. A. Thomas, Appl. Opt. 29, 4337 (1990).
    [Crossref] [PubMed]
  7. J. T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
    [Crossref]
  8. M. Testorf, J. Opt. Soc. Am. A 23, 187 (2006).
    [Crossref]

2006 (1)

2003 (1)

2002 (1)

1991 (3)

1990 (1)

1965 (1)

Arkwright, J. W.

Borel, P. I.

Cui, B.

Dai, E.

Frandsen, L. H.

Kida, J.

Liu, L.

Liu, X.

Lohmann, A. W.

Mortimore, D. B.

Nemoto, S.

Testorf, M.

Thomas, J. A.

Thorhauge, M.

Winthrop, J. T.

Worthington, C. R.

Xi, P.

Zhou, C.

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 (2)

Fig. 1
Fig. 1

Sketches of (a) a hexagonal lattice and (b) the corresponding reciprocal lattice.

Fig. 2
Fig. 2

Example of (a) the HTAI designed according to Eq. (17) with a fractional parameter of β = 16 and (b) the simulated output intensity distribution of the HTAI shown in (a) at the fractional Talbot plane of z β = z T 16 . In this example, the wavelength of the input beam is taken as λ = 0.6328 μ m , and the output hexagonal array is designed with a basis vector length of Δ = 256 μ m and a spot size of 16 μ m .

Equations (18)

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

U ( r ) = u 0 ( r ) lattice ( r , R n ) ,
F { lattice ( r , R n ) } = 1 S n lattice ( ρ , K h ) , 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 ) , b 2 = a 0 × a 1 a 0 ( a 1 × a 2 ) , a 1 = b 2 × b 0 b 0 ( b 1 × b 2 ) , a 2 = b 0 × b 1 b 0 ( b 1 × b 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 k z ) U ( r ) .
a 1 = Δ i , a 2 = Δ 2 i + 3 Δ 2 j .
b 1 = 1 Δ i 1 ( 3 Δ ) j , b 2 = 2 ( 3 Δ ) j ,
4 ( h 1 2 + h 2 2 h 1 h 2 ) λ z 3 Δ 2 = m .
z T = 3 Δ 2 2 λ .
U z = 1 S n exp ( i 2 π z λ ) F 1 { G 0 ( ρ ) lattice ( ρ , K h ) exp [ i π 2 β ( h 1 2 + h 2 2 h 1 h 2 ) ] } ,
U z = 1 S n exp ( i 2 π z 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 ) exp [ i 2 π β ( h 1 2 + h 2 2 h 1 h 2 ) ] .
U z = exp ( i 2 π z λ ) u 0 ( r ) [ c ( n 1 , n 2 , β ) lattice ( r , R n β ) ] .
c ( n 1 , n 2 , β ) = A ( n 1 , n 2 , β ) exp [ ϕ ( n 1 , n 2 , β ) ] .
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 ( 1 β + α ) ( n 1 2 + n 2 2 + n 1 n 2 ) 2 α π 2 ,
α = { 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 } .

Metrics