## Design of Talbot array illuminators for three-dimensional intensity distributions

Optics Express, Vol. 14, Issue 17, pp. 7623-7629 (2006)

http://dx.doi.org/10.1364/OE.14.007623

Acrobat PDF (299 KB)

### Abstract

The self-imaging phenomenon is investigated as the basis for designing diffractive optical elements to generate three-dimensional diffraction patterns. The phase-only diffractive element is related to the intensity distribution at a finite and discrete set of Fresnel diffraction planes by use of the matrix formalism of the fractional Talbot effect. This description provides a framework to determine the degrees of freedom which can be exploited for design. It also helps to identify inherent symmetries of periodic wavefronts, which limit the set of intensity patterns that can be implemented. A simulated annealing algorithm is used to exploit the design freedom. Our discussion includes an example to illustrate observations applicable to a more general class of design problems.

© 2006 Optical Society of America

## 1. Introduction

1. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A **12**, 2145–2158 (1995). [CrossRef]

2. M. C. King, A. M. Noll, and D. H. Berry, “A new approach to computer generated holograms,” Appl. Opt. **9**, 471–475 (1970). [CrossRef] [PubMed]

3. T. Yatagai, “Steroscopic approach to 3-D display using computer generated holograms,” Appl. Opt. **15**, 2722–2729 (1976). [CrossRef] [PubMed]

4. C. Frére, D. Leseberg, and O. Bryngdahl, “Computer-generated holograms of three-dimensional objects composed of line segments,” J. Opt. Soc. Am. A **3**, 726–730 (1986). [CrossRef]

5. S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, “Three-dimensional nanofab-rication with rubber stamps and conformable photomasks,” Adv. Mater. **16(15)**, 1369–1373 (2004). [CrossRef]

6. R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. **19**, 771–773 (1994). [CrossRef] [PubMed]

7. R. Piestun, B. Spektor, and J. Shamir, “Pattern generation with an extended focal depth,” Appl. Opt. **37**, 5394–5398 (1998). [CrossRef]

8. R. Liu, B.-Z. Dong, G.-Z. Yang, and B.-Y. Gu, “Generation of pseudo-nondiffractive beams with use of diffractive phase elments designed by the conjugate gradient method,” J. Opt. Soc. Am. A **15**, 144–151 (1998). [CrossRef]

9. U. Levy, D. Mendlovic, Z. Zalevsky, G. Shabtay, and E. Marom, “Iterative algorithm for determining optimal beam profiles in a three-dimensional space,” Appl. Opt. **38**, 6732–6736 (1999). [CrossRef]

10. N. Guérineau, B. Harchaoui, J. Primot, and K. Heggarty, “Generation of achromatic and propagation invariant spot arrays by us of continuous self-imaging gratings,” Opt. Lett. **26**, 411–413 (2001). [CrossRef]

11. J. Courtial, G. Whyte, Z. Bouchal, and J. Wagner, “Iterative algorithm for holographic shaping of non-diffracting and self-imaging light beams,” Opt. Express **14**, 2108–2116 (2006). [CrossRef] [PubMed]

12. 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]

14. V. Arrizón, J. G. Ibarra, and J. Ojeda-Castan˜eda, “Matrix formulation of the Fresnel transform of complex trans-mittance gratings,” J. Opt. Soc. Am. A **13**, 2414–2422 (1996). [CrossRef]

15. M. Testorf, V. Arrizón, and J. Ojeda-Castan˜eda, “Numerical optimization of phase-only elements based on the fractional Talbot effect,” J. Opt. Soc. Am. A **16**, 97–105 (1999). [CrossRef]

15. M. Testorf, V. Arrizón, and J. Ojeda-Castan˜eda, “Numerical optimization of phase-only elements based on the fractional Talbot effect,” J. Opt. Soc. Am. A **16**, 97–105 (1999). [CrossRef]

## 2. Degrees of freedom

*Q*pixels of constant phase per period

*d*is illuminated with a coherent plane wave. Transverse periodicity results in a longitudinal periodicity of the complex amplitude distribution with period

*z*

_{T}= 2

*d*

^{2}/

*λ*, where l is the wavelength of the incident wave. This is the essence of the Talbot effect [16]. For our discussion we consider so-called fractional Talbot planes

*z*

_{N,M}=

*z*

_{T}

*M*/

*N*specified by integer numbers

*M*and

*N*, and we limit our attention to the case of even numbers

*Q*=

*N*/2. It can be shown that the complex amplitude

*u*(

*x*,

*z*

_{N,M}) can be expressed as a superposition of

*Q*copies of the complex amplitude

*g*(

*x*) in the grating plane

*z*= 0, where each copy is shifted and modulated [14

14. V. Arrizón, J. G. Ibarra, and J. Ojeda-Castan˜eda, “Matrix formulation of the Fresnel transform of complex trans-mittance gratings,” J. Opt. Soc. Am. A **13**, 2414–2422 (1996). [CrossRef]

*D*

_{n,q}are the Talbot coefficients

*c*

_{k}, with

*k*= (

*n*-

*q*) mod

*Q*. For all planes (

*M*,

*N*) the matrix

**D**is unitary and symmetric. In other words, the diffraction planes associated with these linear transforms are those where the number of equidistant samples necessary for uniquely describing the complex amplitude corresponds to the degrees of freedom of the optical signal and the diffraction grating. For

*M*= 1 the Talbot coefficients can be calculated as [14

14. V. Arrizón, J. G. Ibarra, and J. Ojeda-Castan˜eda, “Matrix formulation of the Fresnel transform of complex trans-mittance gratings,” J. Opt. Soc. Am. A **13**, 2414–2422 (1996). [CrossRef]

*M*> 1 the Talbot coefficients can conveniently be determined with the help of Eqs. (1) and (2), by computing (

**D**)

^{M}. Now we are prepared to estimate the degrees of freedom of a volume distribution of the intensity which is controlled with the DOE. From Eq. (1) we find that specifying the intensity distribution in any plane

*z*

_{N,M}provides us with

*Q*equations,

## 3. Nonlinear optimization

15. M. Testorf, V. Arrizón, and J. Ojeda-Castan˜eda, “Numerical optimization of phase-only elements based on the fractional Talbot effect,” J. Opt. Soc. Am. A **16**, 97–105 (1999). [CrossRef]

*L*discrete phase values. The set of discrete phase levels can conveniently be interpreted as a state of the thermodynamic system which is annealed. We use a standard implementation of simulated annealing [17, 18

18. M. S. Kim, M. R. Feldman, and C. C. Guest, “Optimum coding of binary phase-only filters with a simulated annealing algorithm,” Opt. Lett. **14**, 545–547 (1989). [CrossRef] [PubMed]

*d*. In accordance with the results of the previous section we selected (

*N*= 16,

*M*= 2) for the location of the spot array, i.e. the associated diffraction plane is located at

*z*

_{T}/8. If the intensity is specified for this diffraction plane only, the solution of the design problem can be obtained analytically [19

19. V. Arrizón, E. López-Olazagasti, and A. Serrano-Heredia, “Talbot array illuminators with optimum compression ratio,” Opt. Lett. **21**, 233–235 (1996). [CrossRef] [PubMed]

*L*= 8 discrete phase levels the DOE can compress 100% of the intensity into one quarter of each period (see upper part of Fig. 2(a)). The intensity distribution along the

*z*axis (lower part of Fig. 2(a)) reveals a sharp set of periodic spots at the specified diffraction plane marked with white triangles. The intensity of the propagating wavefront was obtained with a paraxial angular spectrum propagation method. The input vector contained four periods of the DOE where each pixel of the DOE was sampled with four points. If the same intensity distribution is used to specify the intensity in two planes

*M*= (1,2), and in three planes

*M*= (1,2,3) the increase of the longitudinal extension of the spots is clearly recognizable. Figs. 2(b) and (c) show the respective intensity distributions in all specified diffraction planes as well as the intensity distributions of the propagating wavefront.

20. V. Arrizón and M. Testorf, “Efficieny limit of spatially quantized Fourier array illuminators,” Opt. Lett. **22**, 197–199 (1997). [CrossRef] [PubMed]

*Q*or the number of phase levels

*L*can be increased to improve primarily the confined portion of intensity and to some extent the shape of the intensity distribution. Figure 3 also contains the result for three diffraction planes and

*L*= 16 phase levels which results in a small, but distinct improvement for diffraction planes between

*M*= 2 and

*M*= 4. For comparison we also evaluated the part of the longitudinal intensity distribution which is confined to one half of the period (Fig. 3(b)). The behavior is similar to that in Fig. 3(a), however a slightly more distinct plateau shape can be observed. This result can be generalized in that the optimization can be performed for a high compression ratio in order to obtain acceptable results for the design goal with smaller compression ratio.

## 4. Discussion and Summary

*z*>

*z*

_{T}/4. From Fig. 2(c) we find that the intensity is also confined to some extent. The intensity can be specified for additional planes in this range as well. The algorithm then determines the optimum configuration which is a reduced intensity confinement for planes

*z*<

*z*

_{T}/4 and a better confinement for planes

*z*>

*z*

_{T}/4. However, the resulting design is governed by the diffraction pattern at plane

*z*

_{T}/4 where the maximum compression ratio cannot exceed a factor of two.

*Q*it is possible to investigate the case, where the intensity distributions of adjacent periods are essentially isolated from one another. This in effect allows treatment of non-periodic problems within the framework of the fractional Talbot effect. In particular, this allows us to recast the design problem of non-diffracting beams [7

7. R. Piestun, B. Spektor, and J. Shamir, “Pattern generation with an extended focal depth,” Appl. Opt. **37**, 5394–5398 (1998). [CrossRef]

8. R. Liu, B.-Z. Dong, G.-Z. Yang, and B.-Y. Gu, “Generation of pseudo-nondiffractive beams with use of diffractive phase elments designed by the conjugate gradient method,” J. Opt. Soc. Am. A **15**, 144–151 (1998). [CrossRef]

9. U. Levy, D. Mendlovic, Z. Zalevsky, G. Shabtay, and E. Marom, “Iterative algorithm for determining optimal beam profiles in a three-dimensional space,” Appl. Opt. **38**, 6732–6736 (1999). [CrossRef]

## Acknowledgment

## References and links

1. | J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A |

2. | M. C. King, A. M. Noll, and D. H. Berry, “A new approach to computer generated holograms,” Appl. Opt. |

3. | T. Yatagai, “Steroscopic approach to 3-D display using computer generated holograms,” Appl. Opt. |

4. | C. Frére, D. Leseberg, and O. Bryngdahl, “Computer-generated holograms of three-dimensional objects composed of line segments,” J. Opt. Soc. Am. A |

5. | S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, “Three-dimensional nanofab-rication with rubber stamps and conformable photomasks,” Adv. Mater. |

6. | R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. |

7. | R. Piestun, B. Spektor, and J. Shamir, “Pattern generation with an extended focal depth,” Appl. Opt. |

8. | R. Liu, B.-Z. Dong, G.-Z. Yang, and B.-Y. Gu, “Generation of pseudo-nondiffractive beams with use of diffractive phase elments designed by the conjugate gradient method,” J. Opt. Soc. Am. A |

9. | U. Levy, D. Mendlovic, Z. Zalevsky, G. Shabtay, and E. Marom, “Iterative algorithm for determining optimal beam profiles in a three-dimensional space,” Appl. Opt. |

10. | N. Guérineau, B. Harchaoui, J. Primot, and K. Heggarty, “Generation of achromatic and propagation invariant spot arrays by us of continuous self-imaging gratings,” Opt. Lett. |

11. | J. Courtial, G. Whyte, Z. Bouchal, and J. Wagner, “Iterative algorithm for holographic shaping of non-diffracting and self-imaging light beams,” Opt. Express |

12. | J. T. Winthrop and C. R. Worthington, “Theory of Fresnel Images. I. Plane Periodic Objects in Monochromatic Light,” J. Opt. Soc. Am. |

13. | A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik |

14. | V. Arrizón, J. G. Ibarra, and J. Ojeda-Castan˜eda, “Matrix formulation of the Fresnel transform of complex trans-mittance gratings,” J. Opt. Soc. Am. A |

15. | M. Testorf, V. Arrizón, and J. Ojeda-Castan˜eda, “Numerical optimization of phase-only elements based on the fractional Talbot effect,” J. Opt. Soc. Am. A |

16. | K. Patorski, “The self-imaging phenomenon and its applications,” in |

17. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

18. | M. S. Kim, M. R. Feldman, and C. C. Guest, “Optimum coding of binary phase-only filters with a simulated annealing algorithm,” Opt. Lett. |

19. | V. Arrizón, E. López-Olazagasti, and A. Serrano-Heredia, “Talbot array illuminators with optimum compression ratio,” Opt. Lett. |

20. | V. Arrizón and M. Testorf, “Efficieny limit of spatially quantized Fourier array illuminators,” Opt. Lett. |

**OCIS Codes**

(050.1380) Diffraction and gratings : Binary optics

(050.1970) Diffraction and gratings : Diffractive optics

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

**ToC Category:**

Fourier Optics and Optical Signal Processing

**History**

Original Manuscript: May 23, 2006

Revised Manuscript: July 18, 2006

Manuscript Accepted: July 22, 2006

Published: August 21, 2006

**Citation**

Markus Testorf, Thomas J. Suleski, and Yi-Chen Chuang, "Design of Talbot array illuminators for three-dimensional intensity distributions," Opt. Express **14**, 7623-7629 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7623

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### References

- J. N. Mait, "Understanding diffractive optic design in the scalar domain," J. Opt. Soc. Am. A 12, 2145-2158 (1995). [CrossRef]
- M. C. King, A. M. Noll, and D. H. Berry, "A new approach to computer generated holograms," Appl. Opt. 9, 471-475 (1970). [CrossRef] [PubMed]
- T. Yatagai, "Steroscopic approach to 3-D display using computer generated holograms," Appl. Opt. 15, 2722-2729 (1976). [CrossRef] [PubMed]
- C. Frere, D. Leseberg, and O. Bryngdahl, "Computer-generated holograms of three-dimensional objects composed of line segments," J. Opt. Soc. Am. A 3, 726-730 (1986). [CrossRef]
- S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004). [CrossRef]
- R. Piestun and J. Shamir, "Control of wave-front propagation with diffractive elements," Opt. Lett. 19, 771-773 (1994). [CrossRef] [PubMed]
- R. Piestun, B. Spektor, and J. Shamir, "Pattern generation with an extended focal depth," Appl. Opt. 37, 5394-5398 (1998). [CrossRef]
- R. Liu, B.-Z. Dong, G.-Z. Yang, and B.-Y. Gu, "Generation of pseudo-nondiffractive beams with use of diffractive phase elments designed by the conjugate gradient method," J. Opt. Soc. Am. A 15, 144-151 (1998). [CrossRef]
- U. Levy, D. Mendlovic, Z. Zalevsky, G. Shabtay, and E. Marom, "Iterative algorithm for determining optimal beam profiles in a three-dimensional space," Appl. Opt. 38, 6732-6736 (1999). [CrossRef]
- N. Guerineau, B. Harchaoui, J. Primot, and K. Heggarty, "Generation of achromatic and propagation invariant spot arrays by us of continuous self-imaging gratings," Opt. Lett. 26, 411-413 (2001). [CrossRef]
- J. Courtial, G. Whyte, Z. Bouchal, and J. Wagner, "Iterative algorithm for holographic shaping of non-diffracting and self-imaging light beams," Opt. Express 14, 2108-2116 (2006). [CrossRef] [PubMed]
- 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]
- A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik 79, 41-45 (1988).
- V. Arrizon, J. G. Ibarra, and J. Ojeda-Castaneda, "Matrix formulation of the Fresnel transform of complex transmittance gratings," J. Opt. Soc. Am. A 13, 2414-2422 (1996). [CrossRef]
- M. Testorf, V. Arrizon, and J. Ojeda-Castaneda, "Numerical optimization of phase-only elements based on the fractional Talbot effect," J. Opt. Soc. Am. A 16, 97-105 (1999). [CrossRef]
- K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, E.Wolf, ed., vol. XXVII, pp. 2-108 (Elsevier, Amsterdam, 1989).
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, chap. 10.9, pp. 444-455, 2nd ed. (Cambridge University Press, New York, 1992).
- M. S. Kim, M. R. Feldman, and C. C. Guest, "Optimum coding of binary phase-only filters with a simulated annealing algorithm," Opt. Lett. 14, 545-547 (1989). [CrossRef] [PubMed]
- V. Arrizon, E. Lopez-Olazagasti, and A. Serrano-Heredia, "Talbot array illuminators with optimum compression ratio," Opt. Lett. 21, 233-235 (1996). [CrossRef] [PubMed]
- V. Arrizon and M. Testorf, "Efficieny limit of spatially quantized Fourier array illuminators," Opt. Lett. 22, 197-199 (1997). [CrossRef] [PubMed]

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