Abstract

We consider field distributions in fractional Talbot planes behind a periodic two-dimensional complex-amplitude transparency that is illuminated by a unit-amplitude plane wave. In the paraxial approximation the field in various fractional Talbot planes is expressed as a sum of contributions from a finite number of points in the plane of the transparency, yielding compact algebraic formulas for the diffracted field. Given the desired intensity distribution in the fractional Talbot plane, we synthesize the transmission function from nonlinear equations. An experimental illustration that uses a binary phase grating is given.

© 1994 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 449–453.
  2. H. F. Talbot, “Facts relating to optical science. No. IV,” Phil. Mag. Ser. 3 9, 401–407 (1836).
  3. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989) Vol. 27, pp. 1–108.
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Fransisco, Calif., 1968), pp. 48–54.
  5. B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self-imaging,” Opt. Commun. 56, 394–398 (1986).
    [CrossRef]
  6. V. Arrizón, J. Ojeda-Castańẽda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
    [CrossRef]
  7. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [CrossRef] [PubMed]
  8. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992), pp. 379–383.
  9. N. Streibl, “Beam shaping with optical array illuminators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  10. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).
  11. P. Szwaykowski, V. Arrizón, “Talbot array illuminator with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
    [CrossRef] [PubMed]
  12. J. Turunen, “Fractional Talbot imaging setup for high-efficiency real-time diffractive optics,” Pure Appl. Opt. 2, 243–250 (1993).
    [CrossRef]
  13. H. Ichikawa, J. Turunen, “Improvement of the diffraction efficiency of liquid crystal spatial light modulators by the Talbot effect,” Kogaku (Jpn. J. Opt.) 22, 151–155 (1993).
  14. D. Schattschneider, M. C. Escher, Visions of Symmetry (Freeman, New York, 1990), pp. 202–203.
  15. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1965), Eqs. (1.344.2), (1.344.3) and (1.342.1), (1.342.2).

1993 (3)

J. Turunen, “Fractional Talbot imaging setup for high-efficiency real-time diffractive optics,” Pure Appl. Opt. 2, 243–250 (1993).
[CrossRef]

H. Ichikawa, J. Turunen, “Improvement of the diffraction efficiency of liquid crystal spatial light modulators by the Talbot effect,” Kogaku (Jpn. J. Opt.) 22, 151–155 (1993).

P. Szwaykowski, V. Arrizón, “Talbot array illuminator with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
[CrossRef] [PubMed]

1992 (1)

1990 (1)

1989 (1)

N. Streibl, “Beam shaping with optical array illuminators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

1988 (1)

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

1986 (1)

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self-imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

1836 (1)

H. F. Talbot, “Facts relating to optical science. No. IV,” Phil. Mag. Ser. 3 9, 401–407 (1836).

Arrizón, V.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 449–453.

Bryngdahl, O.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self-imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Eschbach, R.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self-imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Escher, M. C.

D. Schattschneider, M. C. Escher, Visions of Symmetry (Freeman, New York, 1990), pp. 202–203.

Flannery, B. R.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992), pp. 379–383.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Fransisco, Calif., 1968), pp. 48–54.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1965), Eqs. (1.344.2), (1.344.3) and (1.342.1), (1.342.2).

Ichikawa, H.

H. Ichikawa, J. Turunen, “Improvement of the diffraction efficiency of liquid crystal spatial light modulators by the Talbot effect,” Kogaku (Jpn. J. Opt.) 22, 151–155 (1993).

Leger, J. R.

Lohmann, A. W.

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

Ojeda-Castan?da, J.

Packross, B.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self-imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989) Vol. 27, pp. 1–108.
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992), pp. 379–383.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1965), Eqs. (1.344.2), (1.344.3) and (1.342.1), (1.342.2).

Schattschneider, D.

D. Schattschneider, M. C. Escher, Visions of Symmetry (Freeman, New York, 1990), pp. 202–203.

Streibl, N.

N. Streibl, “Beam shaping with optical array illuminators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

Swanson, G. J.

Szwaykowski, P.

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science. No. IV,” Phil. Mag. Ser. 3 9, 401–407 (1836).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992), pp. 379–383.

Turunen, J.

J. Turunen, “Fractional Talbot imaging setup for high-efficiency real-time diffractive optics,” Pure Appl. Opt. 2, 243–250 (1993).
[CrossRef]

H. Ichikawa, J. Turunen, “Improvement of the diffraction efficiency of liquid crystal spatial light modulators by the Talbot effect,” Kogaku (Jpn. J. Opt.) 22, 151–155 (1993).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992), pp. 379–383.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 449–453.

Appl. Opt. (1)

J. Mod. Opt. (1)

N. Streibl, “Beam shaping with optical array illuminators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

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

Kogaku (Jpn. J. Opt.) (1)

H. Ichikawa, J. Turunen, “Improvement of the diffraction efficiency of liquid crystal spatial light modulators by the Talbot effect,” Kogaku (Jpn. J. Opt.) 22, 151–155 (1993).

Opt. Commun. (1)

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self-imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Opt. Lett. (1)

Optik (1)

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

Phil. Mag. Ser. 3 (1)

H. F. Talbot, “Facts relating to optical science. No. IV,” Phil. Mag. Ser. 3 9, 401–407 (1836).

Pure Appl. Opt. (1)

J. Turunen, “Fractional Talbot imaging setup for high-efficiency real-time diffractive optics,” Pure Appl. Opt. 2, 243–250 (1993).
[CrossRef]

Other (6)

D. Schattschneider, M. C. Escher, Visions of Symmetry (Freeman, New York, 1990), pp. 202–203.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1965), Eqs. (1.344.2), (1.344.3) and (1.342.1), (1.342.2).

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989) Vol. 27, pp. 1–108.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Fransisco, Calif., 1968), pp. 48–54.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992), pp. 379–383.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 449–453.

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Figures (2)

Fig. 1
Fig. 1

Comparison of focal-plane intensity patterns of diffractive Talbot-type array illuminators (dashed lines), arrays of refractive lenses (solid curves), and single refractive lenses (dotted curves) in fractional Talbot planes zT/4q with q = 1, …, 4. In (c) and (d) the profiles have been truncated [the peaks are I(0, zT/12)= 6 and I(0, zT/16) = 8, respectively] so that they show the structure of the sidelobes more clearly.

Fig. 2
Fig. 2

(a) One period of the desired intensity pattern, (b) one period of the binary-phase Talbot element (with a phase delay of π rad between the black and the white regions) that generates the (a) pattern in the quarter-Talbot plane, (c) experimental reconstruction of the intensity profile.

Equations (51)

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U ( x , y , z = 0 ) = t ( x , y ) U inc ( x , y , z < 0 ) ,
U ( x , y , z = 0 ) = m , n T m n exp [ 2 π i ( m x + n y ) / d ] ,
T m n = 1 d 2 0 d 0 d d x d y U ( x , y , z = 0 ) × exp [ 2 π i ( m x + n y ) / d ]
U ( x , y , z > 0 ) = exp ( ikz ) m , n T m n exp [ 2 π i ( m x + n y ) / d ] × exp [ 2 π i ( m 2 + n 2 ) z / z T ] ,
T m n ( x + d / 2 , y ) = T m n ( x , y ) exp ( i π m ) , T m n ( x , y + d / 2 ) = T m n ( x , y ) exp( i π n ) .
U ( x , y , z T / 2 ) = exp ( i k z T / 2 ) m , n T m n exp [ 2 π i ( m x + n y ) / d ] × exp ( i π m ) exp ( i π n ) = exp ( i k z T / 2 ) U ( x + d / 2 , y + d / 2 , 0 ) ,
U ( x , y , z T / 4 ) exp ( i k z T / 4 ) = m , n T m n ( x , y ) × exp [ 2 π i ( m 2 + n 2 ) / 4 ] .
U ( x , y , z T / 4 ) exp ( i k z T / 4 ) = m even n even T m n ( x , y ) i m odd n even T m n ( x , y ) i m even n odd T m n ( x , y ) m odd n odd T m n ( x , y ) .
U ( x , y , 0 ) + i U ( x + d / 2 , y , 0 ) + i U ( x , y + d / 2 , 0 ) + i 2 U ( x + d / 2 , y + d / 2 , 0 ) = 2 i U ( x , y , z T / 4 ) exp ( i k z T / 4 )
I A ( x , y , z T / 4 ) = | U ( x , y , z T / 4 ) | 2 = ¼ | A ( x , y ) exp [ i ϕ ( x , y ) ] + A ( x + d / 2 , y ) × exp [ i ϕ ( x + d / 2 , y ) + i π / 2 ] + A ( x , y + d / 2 ) exp [ i ϕ ( x , y + d / 2 ) + i π / 2 ] + A ( x + d / 2 , y + d / 2 ) × exp [ i ϕ ( x + d / 2 , y + d / 2 ) + i π ] | 2 .
I ( x , y , z T / 4 ) = ¼ | exp [ i ϕ ( x , y ) ] + exp [ i ϕ ( x + d / 2 , y ) + i π / 2 ] + exp [ i ϕ ( x , y + d / 2 ) + i π / 2 ] + exp [ i ϕ ( x + d / 2 , y + d / 2 ) + i π ] | 2 ,
| u = 1 N exp ( i β u ) | 2 = N + 2 1 u < υ N cos ( β u β υ ) .
I ( x , y , z T / 4 ) = 1 + ½ { sin [ ϕ ( x , y ) ϕ ( x + d / 2 , y ) ] + sin [ ϕ ( x , y ) ϕ ( x , y + d / 2 ) ] cos [ ϕ ( x , y ) ϕ ( x + d / 2 , y + d / 2 ) ] + cos [ ϕ ( x , y + d / 2 ) ϕ ( x + d / 2 , y ) ] + sin [ ϕ ( x + d / 2 , y ) ϕ ( x + d / 2 , y + d / 2 ) ] + sin [ ϕ ( x , y + d / 2 ) ϕ ( x + d / 2 , y + d / 2 ) ] } .
I I ( x , y , z T / 4 ) + I ( x + d / 2 , y , z T / 4 ) + I ( x , y + d / 2 , z T / 4 ) + I ( x + d / 2 , y + d / 2 , z T / 4 ) .
I ( x + d / 2 , y , z T / 4 ) = ¼ | exp [ i ϕ ( x + d / 2 , y ) ] + exp [ i ϕ ( x , y ) + i π / 2 ] + exp [ i ϕ ( x + d / 2 , y + d / 2 ) + i π / 2 ] + exp [ i ϕ ( x , y + d / 2 ) + i π ] | 2 .
cos ( α π / 2 ) + cos ( α π / 2 ) + cos ( α π / 2 ) + cos ( α π / 2 ) .
I ( x , y , z T / 4 ) + I ( x + d / 2 , y , z T / 4 ) + I ( x , y + d / 2 , z T / 4 ) + I ( x + d / 2 , y + d / 2 , z T / 4 ) = 4 .
U ( x , y > 0 ) = exp ( ikz ) m T m exp ( 2 π imz / d ) × exp ( 2 π i m 2 z / z T ) .
P α f ( x ) = f ( x + α ) .
S = 1 r = 0 2 q 1 exp ( i 2 π r 2 / 4 q ) s = 0 2 q 1 exp ( i 2 π s 2 / 4 q ) P s d / 2 q
S m = exp ( 2 π i m 2 / 4 q ) s = 0 2 q 1 exp [ i 2 π ( s + m ) 2 / 4 q ] r = 0 2 q 1 exp ( i 2 π r 2 / 4 q ) .
s = 0 2 q 1 exp ( i 2 π s 2 / 4 q ) = exp ( i π / 4 ) 2 q .
U ( x , z T / 4 q ) = exp ( i k z T / 4 q ) exp ( i π / 4 ) 2 q × s = 0 2 q 1 exp ( i π s 2 / 2 q ) P s d / 2 q U ( x , z = 0 ) .
T m n exp ( 2 π imz / d ) exp ( 2 π i m 2 z / z T ) exp ( 2 π i n y / d ) × exp ( 2 π i n 2 z / z T ) .
U ( x , y , z T / 4 q ) = exp ( i k z T / 4 q ) exp ( i π / 2 ) 2 q × s = 0 2 q 1 exp ( i s 2 π / 2 q ) P s d / 2 q , 0 × r = 0 2 q 1 exp ( i r 2 π / 2 q ) P 0 , r d / 2 q U ( x , y , z = 0 ) = exp ( i k z T / 4 q ) exp ( i π / 2 ) 2 q × r , s = 0 2 q 1 exp [ i ( s 2 + r 2 ) π / 2 q ] × P s d / 2 q , r d / 2 q U ( x , y , z = 0 ) ,
U ( x , y , z T / 4 ) = exp ( i k z T / 4 ) i 2 ( 1 + i P d / 2 , 0 ) × ( 1 + i P 0 , d / 2 ) U ( x , y , z = 0 ) ,
s = 0 2 q 1 r = 0 2 q 1 P s d / 2 q , r d / 2 q I ( x , y , z T / 4 q ) = ( 2 q ) 2 ,
U ( x , z T / 3 ) = exp ( i k z T / 3 ) r = 0 2 exp(2 π i r 2 / 3 ) × s = 0 2 exp ( 2 π i s 2 / 3 ) P s d / 3 U ( x , z = 0 ) ,
U ( x , z T / 5 ) = exp ( i k z T / 5 ) r = 0 4 exp ( 2 π i r 2 / 5 ) × s = 0 4 exp ( 2 π i s 2 / 5 ) P s d / 5 U ( x , z = 0 ) ,
U ( x , z T / 6 ) = exp ( i k z T / 6 ) r = 0 2 exp ( 2 π i r 2 / 3 ) × s = 0 2 exp ( 2 π i s 2 / 3 ) P s d / 3 + d / 2 U ( x , z = 0 ) ,
0 I ( x , y ) + I ( x + d / 2 , y ) + I ( x , y + d / 2 ) 4 , I ( x + d / 2 , y + d / 2 ) = 4 [ I ( x , y ) + I ( x + d / 2 , y ) + I ( x , y + d / 2 ) ] .
sin ( ϕ 2 ) + sin ( ϕ 1 ϕ 3 ) = I ( x , y ) + I ( x + d / 2 , y ) 2 , sin ( ϕ 1 ϕ 2 ) + sin ( ϕ 3 ) = I ( x , y ) + I ( x , y + d / 2 ) 2 , cos ( ϕ 1 ) cos ( ϕ 2 ϕ 3 ) = I ( x + d / 2 , y ) + I ( x , y + d / 2 ) 2 .
ϕ ( x , y ) = π , ϕ ( x , y + d / 2 ) = ϕ ( x + d / 2 , y ) = arcsin [ ½ I ( x , y ) 1 ] .
ϕ ( x , y ) = arccos [ ½ I ( x + d / 2 , y ) 1 ] , ϕ ( x + d / 2 , y ) = π / 2 , ϕ ( x , y + d / 2 ) = ϕ ( x , y ) π / 2 .
ϕ ( x + s d / 2 q , y + r d / 2 q ) = ( s 2 + r 2 ) π / 2 q .
t ( x , y ) = exp [ i π λ F ( x d / 2 ) 2 ] × exp [ i π λ F ( y d / 2 ) 2 ] ,
I ( x , y , z T / 4 q ) = ( 2 q ) 2 1 r < s 2 q ( 1 ) s r + 1 × sin 2 [ π ( s r ) x / d ] 1 u < υ 2 q ( 1 ) υ u + 1 × sin 2 [ π ( υ u ) x / d ] .
I ( x , y , z T / 4 q ) = ( 2 q ) 2 sinc 2 [ 2 q ( x d / 2 ) / d ] × sinc 2 [ 2 q ( y d / 2 ) / d ] ,
B ( m , q ) = s = 0 2 q 1 exp [ i π ( s + m ) 2 / 2 q ] ,
B ( m + 1 , q ) = s = 1 2 q exp [ i π ( s + m ) 2 / 2 q ] = s = 0 2 q 1 exp [ i π ( s + m ) 2 / 2 q ] exp ( i π m 2 / 2 q ) + exp [ i π ( 2 q + m ) 2 / 2 q ] = B ( m , q ) + exp ( i π m 2 / 2 q ) × [ 1 + exp ( i π 2 m ) exp ( i π 2 q ) ] = B ( m , q ) ,
2 s = 0 2 q 1 cos ( π s 2 / 2 q ) = s = 0 2 q 1 cos ( π s 2 / 2 q ) + s = 2 q 4 q 1 cos [ π ( s 2 q ) 2 / 2 q ] = s = 0 4 q 1 cos ( π s 2 / 2 q ) .
B ( q ) = 1 2 s = 0 4 q 1 cos ( 2 π s 2 / 4 q ) + i 1 2 s = 0 4 q 1 sin ( 2 π s 2 / 4 q ) = 4 q 4 [ 1 + cos ( 4 q π / 2 ) + sin ( 4 q π / 2 ) ] + i 4 q 4 [ 1 + cos ( 4 q π / 2 ) sin ( 4 q π / 2 ) ] .
B ( q ) = 1 2 ( 1 + i ) 2 q ,
I = s = 0 2 q 1 P s d / 2 q I ( x , z T / 4 q ) = s = 0 2 q 1 P s d / 2 q 1 2 q × | r = 0 2 n 1 exp ( i π r 2 / 2 q ) exp [ i ϕ ( x + r d / 2 q ) ] | 2 .
| u = 0 N 1 exp ( i β u ) | 2 = 0 u , υ N 1 cos ( β u β υ ) = N + 0 u υ N 1 cos ( β u β υ ) ,
I = 1 2 q s = 0 2 q 1 P s d / 2 q u , υ = 0 2 q 1 cos [ π u 2 / 2 q + ϕ ( x + u d / 2 q ) π υ 2 / 2 q ϕ ( x + υ d / 2 q ) ] = 2 q + 1 2 q s = 0 2 q 1 u υ = 0 2 q 1 cos { π ( u 2 u 2 ) / 2 q + ϕ [ x + ( s + u ) d / 2 q ] ϕ [ x + ( s + υ ) d / 2 q ] } .
cos [ ( u + 2 q ) 2 π / 2 q + γ ] = cos [ u 2 π / 2 q + γ ] .
u 2 υ 2 = 2 s Δ υ u + α 2 β 2 ,
s = 0 2 q 1 cos ( 2 s Δ υ u π / 2 q + c ) = cos ( c ) s = 0 2 q 1 cos ( s Δ υ u π / q ) sin ( c ) s = 0 2 q 1 sin ( s Δ υ u π / q ) ,
s = 0 2 q 1 cos ( s Δ υ u π / q ) = sin ( Δ υ u π ) cos [ ( 2 q 1 ) Δ υ u π / 2 q ] / sin ( Δ υ u π / 2 q ) , s = 0 2 q 1 sin ( s Δ υ u π / q ) = sin ( Δ υ u π ) sin [ ( 2 q 1 ) Δ υ u π / 2 q ] / sin ( Δ υ u π / 2 q ) .
I = s = 0 2 q 1 r = 0 2 q 1 P s d / 2 q , r d / 2 q I ( x , y , z T / 4 q ) = r = 0 2 q 1 s = 0 2 q 1 u = υ = 0 2 q 1 1 ( 2 q ) 2 = ( 2 q ) 2 ,

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