Abstract

The problem of designing Talbot array illuminators is revisited in the context of phase-space optics. It is shown that for Talbot array illuminators with optimum compression ratio the construction of phase-only grating profiles can be simplified significantly by using phase-space representations of optical signals. Based on the Wigner distribution function a graphical procedure is derived for obtaining the complete design of the array generator for a given compression ratio. The application of phase-space optics to other classes of Talbot array illuminators, and its use as part of numerical optimization algorithms, is considered as well.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik (Stuttgart) 79, 41-45 (1988).
  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. J. R. Leger and G. J. Swanson, "Efficient array illuminator using binary-optics phase plates at fractional Talbot planes," Opt. Lett. 15, 288-290 (1990).
    [CrossRef] [PubMed]
  5. C. Zhou and L. Liu, "Simple equations for the calculation of multilevel phase gratings for Talbot array illuminations," Opt. Commun. 115, 40-44 (1995).
    [CrossRef]
  6. H. Hamam, "Design of Talbot array illuminators," Opt. Commun. 131, 359-370 (1996).
    [CrossRef]
  7. T. J. Suleski, "Generation of Lohmann images from binary-phase Talbot array illuminators," Appl. Opt. 36, 4686-4691 (1997).
    [CrossRef] [PubMed]
  8. W. Klaus, Y. Arimoto, and K. Kodate, "High performance Talbot array illuminators," Appl. Opt. 37, 4357-4365 (1998).
    [CrossRef]
  9. M. Testorf, V. Arrizón, and J. Ojeda-Castañeda, "Numerical optimization of phase-only elements based on the fractional Talbot effect," J. Opt. Soc. Am. A 16, 97-105 (1999).
    [CrossRef]
  10. V. Arrizón and J. Ojeda-Castañeda, "Fresnel diffraction of substructured gratings: matrix description," Opt. Lett. 20, 118-120 (1995).
    [CrossRef] [PubMed]
  11. V. Arrizón, J. G. Ibarra, and J. Ojeda-Castañeda, "Matrix formulation of the Fresnel transform of complex transmittance gratings," J. Opt. Soc. Am. A 13, 2414-2422 (1996).
    [CrossRef]
  12. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001), Chap. 8, pp. 265-318.
  13. M. J. Bastiaans, "Application of the Wigner distribution function in optics," in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbräuker and F.Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.
  14. J. Ojeda-Castañeda and E. E. Sicre, "Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields," Opt. Acta 32, 17-26 (1985).
    [CrossRef]
  15. V. Arrizón and J. Ojeda-Castañeda, "Irradiance at Fresnel planes of a phase grating," J. Opt. Soc. Am. A 9, 1801-1806 (1992).
    [CrossRef]
  16. M. Testorf and J. Ojeda-Castañeda, "Fractional Talbot effect: analysis in phase space," J. Opt. Soc. Am. A 13, 119-125 (1996).
    [CrossRef]
  17. M. Testorf, "Phase-space optics applied to the design of Talbot array illuminators," in Diffractive Optics and Micro-optics, OSA Technical Digest (Optical Society of America, 2004), paper DMC1.
  18. V. Arrizón, E. López-Olazagasti, and A. Serrano-Heredia, "Talbot array illuminators with optimum compression ratio," Opt. Lett. 21, 233-235 (1996).
    [CrossRef] [PubMed]
  19. M. Stern, "Binary optics fabrication," in Micro-optics, H.P.Herzig, ed. (Taylor & Francis, 1997), pp. 53-85.
  20. O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. (Bellingham) 43, 2549-2556 (2004).
    [CrossRef]

2004 (1)

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. (Bellingham) 43, 2549-2556 (2004).
[CrossRef]

1999 (1)

1998 (1)

1997 (1)

1996 (4)

1995 (2)

V. Arrizón and J. Ojeda-Castañeda, "Fresnel diffraction of substructured gratings: matrix description," Opt. Lett. 20, 118-120 (1995).
[CrossRef] [PubMed]

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

1992 (1)

1990 (2)

1988 (1)

A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik (Stuttgart) 79, 41-45 (1988).

1985 (1)

J. Ojeda-Castañeda and E. E. Sicre, "Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields," Opt. Acta 32, 17-26 (1985).
[CrossRef]

1965 (1)

Arimoto, Y.

Arrizón, V.

Bastiaans, M. J.

M. J. Bastiaans, "Application of the Wigner distribution function in optics," in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbräuker and F.Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.

Hamam, H.

H. Hamam, "Design of Talbot array illuminators," Opt. Commun. 131, 359-370 (1996).
[CrossRef]

Herzig, H.-P.

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. (Bellingham) 43, 2549-2556 (2004).
[CrossRef]

Ibarra, J. G.

Kettunen, V.

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. (Bellingham) 43, 2549-2556 (2004).
[CrossRef]

Klaus, W.

Kodate, K.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001), Chap. 8, pp. 265-318.

Leger, J. R.

Liu, L.

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

Lohmann, A. W.

A. W. Lohmann and J. A. Thomas, "Making an array illuminator based on the Talbot effect," Appl. Opt. 29, 4337-4340 (1990).
[CrossRef] [PubMed]

A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik (Stuttgart) 79, 41-45 (1988).

López-Olazagasti, E.

Ojeda-Castañeda, J.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001), Chap. 8, pp. 265-318.

Ripoll, O.

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. (Bellingham) 43, 2549-2556 (2004).
[CrossRef]

Serrano-Heredia, A.

Sicre, E. E.

J. Ojeda-Castañeda and E. E. Sicre, "Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields," Opt. Acta 32, 17-26 (1985).
[CrossRef]

Stern, M.

M. Stern, "Binary optics fabrication," in Micro-optics, H.P.Herzig, ed. (Taylor & Francis, 1997), pp. 53-85.

Suleski, T. J.

Swanson, G. J.

Testorf, M.

Thomas, J. A.

Winthrop, J. T.

Worthington, C. R.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001), Chap. 8, pp. 265-318.

Zhou, C.

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

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

J. Ojeda-Castañeda and E. E. Sicre, "Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields," Opt. Acta 32, 17-26 (1985).
[CrossRef]

Opt. Commun. (2)

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

H. Hamam, "Design of Talbot array illuminators," Opt. Commun. 131, 359-370 (1996).
[CrossRef]

Opt. Eng. (Bellingham) (1)

O. Ripoll, V. Kettunen, and H.-P. Herzig, "Review of iterative Fourier transform algorithms for beam shaping application," Opt. Eng. (Bellingham) 43, 2549-2556 (2004).
[CrossRef]

Opt. Lett. (3)

Optik (Stuttgart) (1)

A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik (Stuttgart) 79, 41-45 (1988).

Other (4)

M. Testorf, "Phase-space optics applied to the design of Talbot array illuminators," in Diffractive Optics and Micro-optics, OSA Technical Digest (Optical Society of America, 2004), paper DMC1.

M. Stern, "Binary optics fabrication," in Micro-optics, H.P.Herzig, ed. (Taylor & Francis, 1997), pp. 53-85.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001), Chap. 8, pp. 265-318.

M. J. Bastiaans, "Application of the Wigner distribution function in optics," in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbräuker and F.Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.

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 (5)

Fig. 1
Fig. 1

Model system for implementing TAIs.

Fig. 2
Fig. 2

Desired Fresnel diffraction intensity pattern.

Fig. 3
Fig. 3

WDF of a delta comb function. Symbols (+) and (−) refer to delta functions with positive and negative sign, respectively.

Fig. 4
Fig. 4

WDF of a comb function after horizontal shear; Q = 4 (even compression ratio).

Fig. 5
Fig. 5

WDF of a comb function after horizontal shear; Q = 5 (odd compression ratio).

Equations (22)

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

u out ( x ) = Q rect ( x d ) n = δ ( x n p ) ,
u Fr * ( x ) = exp ( i k z Fr + i π 4 ) λ z Fr exp ( i π λ z Fr x 2 ) .
u T A I = Q rect ( x d ) n = δ ( x n p ) u Fr * ( x ) .
W ( x , ν ) = u ( x + x 2 ) u * ( x x 2 ) exp ( i 2 π ν x ) d x .
W comb ( x , ν ) = 1 2 p n , n = ( 1 ) n n δ ( x n p 2 ) δ ( ν n 2 p ) .
I ( x ) = u ( x ) 2 = W ( x , ν ) d ν ,
W ( x , ν ) W ( x Δ z λ ν , ν ) .
z Fr = 1 2 Q z T ,
u ( x ) u * ( 0 ) = W ( x 2 , ν ) exp ( i 2 π ν x ) d ν ,
u L ( x ) = exp [ i ϕ L ( x ) ] = exp ( i π λ f Fr x 2 )
W ( x , ν ) W [ x , ν + x ( λ f Fr ) ] .
ϕ L ( x ) = π Q p 2 x 2 .
ϕ n = π n 2 Q .
ϕ n = π ( n 2 Q + n Q ) ,
u δ δ ( x ) = a 1 δ ( x x 0 ) + a 2 δ ( x + x 0 ) ,
W δ δ ( x , ν ) = a 1 2 δ ( x x 0 ) + a 2 2 δ ( x + x 0 ) + 2 a 1 a 2 cos ( 4 π ν x 0 + Δ ϕ ) δ ( x ) ,
W t c ( x , ν ) = 1 2 p n , n = A n , n δ ( x n p 4 ) δ ( ν n 2 p ) ,
A n , n = { ( 1 ) n n 2 2 a 1 a 2 cos ( π n 2 + Δ ϕ ) , n = even ( 1 ) ( n 1 ) n 2 a 1 2 + ( 1 ) ( n + 1 ) n 2 a 2 2 , n = odd } .
I k = 1 2 p n = 0 3 A n , n ,
I k = 1 Q [ a 1 2 + a 2 2 + ( 1 ) k 2 2 a 1 a 2 cos ( Δ ϕ ) ] ,
u p ( x ) = k = 0 Q 1 a k δ ( x k d ) .
W g ( x , ν ) = 1 2 p n , n = ( 1 ) n n k , k = 0 Q 1 a k a k * exp [ i π ( k k ) n Q ] δ ( x m d 2 ) δ ( ν n 2 p ) ,

Metrics