Abstract

We show that the Fresnel field at a fraction of the Talbot distance behind a complex transmittance grating is conveniently described by a matrix operator. We devote special attention to a discrete-type grating, whose basic cell (of length d) is formed with a finite number (Q) of intervals of length d/Q, each with a constant complex transmittance. Ignoring the physical units of the optical field, we note that the transmittance of the discrete grating and its Fresnel field belong to a common Q-dimensional complex linear space (VQ). In this context the Fresnel transform is recognized as a linear operator that is represented by a Q × Q matrix. Several properties of this matrix operator are derived here and employed in a discussion of different issues related to the fractional Talbot effect. First, we review in a simple manner the field symmetries in the Talbot cell. Second, we discuss novel Talbot array illuminators. Third, we recognize the eigenvectors of the matrix operator as discrete gratings that exhibit self-images at fractions of the Talbot distance. And fourth, we present a novel representation of the Fresnel field in terms of the eigenvectors of the matrix operator.

© 1996 Optical Society of America

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References

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  1. L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 80.
  2. F. Gori, “Why is the Fresnel transform so little known?” in Trends in Optics, J. C. Dainty, ed. (Academic, New York, 1994), pp. 139–148.
  3. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  4. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  5. P. Pellat-Finet, “Fresnel diffraction and the fractional Fourier transform,” Opt. Lett. 19, 1388–1390 (1995).
    [CrossRef]
  6. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–382 (1965).
    [CrossRef]
  7. J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
    [CrossRef]
  8. R. E. Ioseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. (USSR) 55, 544–547 (1983).
  9. V. Arrizón, J. Ojeda-Castaneda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
    [CrossRef] [PubMed]
  10. W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [CrossRef]
  11. V. Arrizón, J. Ojeda-Castañeda, “Fresnel diffraction of substructed gratings: matrix description,” Opt. Lett. 20, 118–120 (1995).
    [CrossRef]
  12. J. Westerholm, J. Turunen, J. Huttunen, “Fresnel diffraction in fractional Talbot planes: a new formulation,” J. Opt. Soc. Am. A 11, 1283–1290 (1994).
    [CrossRef]
  13. V. Arrizón, J. Ojeda-Castañeda, “Irradiance at fractional Talbot planes,” J. Opt. Soc. Am. A 10, 1801–1806 (1992).
    [CrossRef]
  14. G. E. Shilov, Linear Algebra (Dover, New York, 1977), p. 263.
  15. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics plates at fractional-Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [CrossRef] [PubMed]

1995

1994

1993

1992

V. Arrizón, J. Ojeda-Castañeda, “Irradiance at fractional Talbot planes,” J. Opt. Soc. Am. A 10, 1801–1806 (1992).
[CrossRef]

1990

1983

R. E. Ioseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. (USSR) 55, 544–547 (1983).

1971

J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

1968

1965

Arrizón, V.

Gori, F.

F. Gori, “Why is the Fresnel transform so little known?” in Trends in Optics, J. C. Dainty, ed. (Academic, New York, 1994), pp. 139–148.

Guigay, J. P.

J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

Huttunen, J.

Ioseliani, R. E.

R. E. Ioseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. (USSR) 55, 544–547 (1983).

Leger, J. R.

Lohmann, A. W.

Mendlovic, D.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 80.

Montgomery, W. D.

Ojeda-Castaneda, J.

Ojeda-Castañeda, J.

V. Arrizón, J. Ojeda-Castañeda, “Fresnel diffraction of substructed gratings: matrix description,” Opt. Lett. 20, 118–120 (1995).
[CrossRef]

V. Arrizón, J. Ojeda-Castañeda, “Irradiance at fractional Talbot planes,” J. Opt. Soc. Am. A 10, 1801–1806 (1992).
[CrossRef]

Ozaktas, H. M.

Pellat-Finet, P.

Shilov, G. E.

G. E. Shilov, Linear Algebra (Dover, New York, 1977), p. 263.

Swanson, G. J.

Turunen, J.

Westerholm, J.

Winthrop, J. T.

Worthington, C. R.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

Opt. Lett.

Opt. Spectrosc. (USSR)

R. E. Ioseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. (USSR) 55, 544–547 (1983).

Other

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 80.

F. Gori, “Why is the Fresnel transform so little known?” in Trends in Optics, J. C. Dainty, ed. (Academic, New York, 1994), pp. 139–148.

G. E. Shilov, Linear Algebra (Dover, New York, 1977), p. 263.

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Figures (2)

Fig. 1
Fig. 1

One period of (a) a discrete phase grating G(x) with Q equally spaced subintervals with constant complex amplitudes and (b) the binary amplitude grating with opening ratio 1/Q, used in the construction of a basis for the vector space VQ of discrete gratings.

Fig. 2
Fig. 2

Basic cell of a continuous grating, sampled at points xn = x0 + nd/Q (n = 0, …, Q − 1), where 0 x 0 < d / Q.

Tables (1)

Tables Icon

Table 1 Eigenvalues and Orthogonal Eigenvectors of Matrix D for the Distance z = Zi/8

Equations (64)

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u ( x , z ) = exp [ i ( 2 π λ z π 4 ) ] ( λ z ) 1 / 2 × G ( x ) exp [ i π λ z ( x x ) 2 ] d x .
u ( x , Z t / N ) = exp ( i π 4 ) ( N 2 d 2 ) 1 / 2 ɛ 2 d ɛ D ξ G ( x ξ ) × exp ( i π N ξ 2 / 2 d 2 ) L = exp ( i 2 π N L ξ / d ) .
L = exp ( i 2 π N L ξ / d ) = d N L = δ ( ξ L d / N ) .
u ( x , Z t / N ) = exp ( i π / 4 ) 2 N L = 0 2 N 1 G ( x L d / N ) × exp ( i π L 2 / 2 N ) .
u ( x , Z t / N ) = exp ( i π / 4 ) 2 N L = 0 N 1 G ( x L d / N ) × exp ( i π L 2 / 2 N ) [ 1 + i N ( 1 ) L ] .
u ( x , Z t / N ) = n = 0 Q 1 c n G ( x n d / Q ) ,
c n = 1 Q exp [ i π ( n 2 Q 1 4 ) ]
c n = 1 N exp [ i ( n 2 2 N π Δ ϕ ) ] ,
Δ ϕ = { 0 , N 1 2 π / 2 , otherwise and n both even or both odd .
u ( x , M Z t / N ) = n = 0 Q 1 c n G ( x n d / Q ) ,
c n = 1 N q = 0 N 1 exp ( i 2 π q Q n ) exp ( i 2 π M N q 2 )
| c n | = 1 / Q ,
B ( x ) = q = rect ( x q d d / Q ) ,
G ( x ) = m = 0 Q 1 a m B ( x m d / Q ) .
B = { B ( x m d / Q ) for m = 0 , 1 , , Q 1 } .
u B ( x , Z t / N ) = n = 0 Q 1 c n B [ x ( m + n ) d / Q ] = n = 0 Q 1 d n m B ( x n d / Q ) ,
d n m = { c n m + Q , 0 n m 1 c n m , m n Q 1 .
u ( x , Z t / N ) = n = 0 Q 1 c n m = 0 Q 1 a m B [ x ( m + n ) d / Q ] = m = 0 Q 1 a m { n = 0 Q 1 c n B [ x ( m + n ) d / Q ] } = m = 0 Q 1 a m n = 0 Q 1 d n m B ( x n d / Q ) = n = 0 Q 1 b n B ( x n d / Q ) ,
b n = m = 0 Q 1 d n m a m ,
β = D α ,
α = ( a 0 a 1 a Q 1 ) , β = ( b 0 b 1 b Q 1 ) , D = [ c 0 c Q 1 c 1 c 1 c 0 c 2 c Q 1 c Q 2 c 0 ] .
I u ( x , Z t / N ) = n = 0 Q 1 | b n | 2 B ( x n d / Q ) .
I G ( x , z = 0 ) = n = 0 Q 1 | a n | 2 B ( x n d / Q ) .
n = 0 Q 1 | a n | 2 = n = 0 Q 1 | b n | 2 .
u ( x , M Z t / N ) = n = 0 Q 1 e n B ( x n d / Q ) ,
η = D · D D α M times = D M α .
u ( x m d / Q , M Z t / N ) = n = 0 Q 1 e n B ( x n d / Q ) ,
e n = { e n m + Q , 0 n m 1 e n m , m n Q 1 .
D M = [ c 0 c Q 1 c 1 c 1 c 0 c 2 c Q 1 c Q 2 c 0 ] ,
x n = x 0 + n d / Q , n = 0 , 1 , . . . , Q 1 ,
u [ x = x 0 + n ( d / Q ) , Z t / N ) ] = m = 0 Q 1 c m G [ x 0 + ( n m ) d / Q ] ,
u [ x = x 0 + n ( d / Q ) , Z t / N ) ] = m = 0 Q 1 d n m G ( x 0 + m d / Q ) ,
d n m = { c n m , 0 m n c n m + Q , n + 1 m Q 1 .
( u 0 u Q 1 ) = D ( G 0 G Q 1 ) .
c Q L = c L .
D t = D ,
n = 0 Q 1 c n = 1 .
n = 0 Q 1 | c n | 2 = 1 .
D N = T ,
D 1 = D N 1 .
D 1 = ( D t ) * = D * .
α = D 1 β = D * β .
η = ( d 0 , 0 d 1 , 0 d Q 1 , 0 ) , η = ( d Q m , 0 d Q 1 , 0 d 0 , 0 d Q m 1 , 0 ) .
u B ( x , M Z t / N ) = n = 0 Q 1 d n , 0 B ( x n d / Q ) .
u B ( x , M Z t / N ) = u ( x , M Z t / N ) = n = 0 Q 1 c n B ( x n d / Q ) = n = 0 Q 1 c n B ( x Fnd / Q ) ,
d n , 0 = { c k if n = k F , k = 0 , . . . , Q 1 0 otherwise .
β * = D N 1 α *
α * = D β * .
η * = D N M α * ,
α * = D M η * ,
d 00 = d 40 = c 0 = 1 / 8 exp ( i π / 4 ) , d 10 = d 70 = c 1 = 1 / 8 exp ( i π / 8 ) , d 20 = d 60 = c 2 = 1 / 8 exp ( i π / 4 ) , d 30 = d 50 = c 3 = 1 / 8 exp ( i 7 π / 8 ) .
[ c 0 c 1 c 2 c 3 c 0 c 3 c 2 c 1 c 1 c 0 c 1 c 2 c 3 c 0 c 3 c 2 c 2 c 1 c 0 c 1 c 2 c 3 c 0 c 3 c 3 c 2 c 1 c 0 c 1 c 2 c 3 c 0 c 0 c 3 c 2 c 1 c 0 c 1 c 2 c 3 c 3 c 0 c 3 c 2 c 1 c 0 c 1 c 2 c 2 c 3 c 0 c 3 c 2 c 1 c 0 c 1 c 1 c 2 c 3 c 0 c 3 c 2 c 1 c 0 ] ( 1 0 0 0 0 0 0 0 ) = ( c 0 c 1 c 2 c 3 c 0 c 3 c 2 c 1 ) .
D 2 = [ c 0 0 c 1 0 c 2 0 c 1 0 0 c 0 0 c 1 0 c 2 0 c 1 c 1 0 c 0 0 c 1 0 c 2 0 0 c 1 0 c 0 0 c 1 0 c 2 c 2 0 c 1 0 c 0 0 c 1 0 0 c 2 0 c 1 0 c 0 0 c 1 c 1 0 c 2 0 c 1 0 c 0 0 0 c 1 0 c 2 0 c 1 0 c 0 ] ,
c 0 = 1 2 exp ( i π / 4 ) , c 1 = 1 2 , c 2 = 1 2 exp ( i 3 π / 4 ) .
α = ( 1 0 0 1 0 0 0 0 ) , η = ( c 0 c 1 c 1 c 0 c 2 c 1 c 1 c 2 ) .
D α = K α .
α p · α = ( const ) n = 0 Q 1 a n = 0 .
1 2 [ exp ( i π / 4 ) 1 exp ( i 3 π / 4 ) 1 1 exp ( i π / 4 ) 1 exp ( i 3 / 4 ) exp ( i 3 π / 4 ) 1 exp ( i π / 4 ) 1 1 exp ( i 3 π / 4 ) 1 exp ( i π / 4 ) ] × ( a 0 a 1 a 2 a 3 ) = K ( a 0 a 1 a 2 a 3 ) .
2 ( 1 / 2 0 1 / 2 0 ) + i 2 ( 0 1 / 2 0 1 / 2 ) = ( 1 i 1 i ) ,
G ( x ) = n = 0 Q 1 a n E n ,
u ( x , Z t / N ) = n = 0 Q 1 a n K n E n ,
u ( x , M Z t / N ) = n = 0 Q 1 a n ( K n ) M E n .
( 1 / 2 , 0 , 1 / 2 , 0 )
( 0 , 1 / 2 , 0 , 1 , 2 )

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