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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 17 — Aug. 21, 2006
  • pp: 7623–7629
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Design of Talbot array illuminators for three-dimensional intensity distributions

Markus Testorf, Thomas J. Suleski, and Yi-Chen Chuang  »View Author Affiliations


Optics Express, Vol. 14, Issue 17, pp. 7623-7629 (2006)
http://dx.doi.org/10.1364/OE.14.007623


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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

The design of phase-only diffractive optical elements (DOEs) which generate a desired intensity distribution in a single specified diffraction plane has received a great deal of attention [1

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

]. Early work on computer generated holography has been used to generate specific three-dimensional (3-D) intensity distributions [2

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

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

, 4

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]

] for applications including holographic displays.

This study was inspired by recent interest in interference lithography for fabricating volume structures, such as photonic crystals, where the 3-D intensity distribution behind a diffraction grating is recorded in a photosensitive material [5

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]

]. Better control over the propagating wave-field via the design of the diffraction gratings provides equal control over the properties of the synthesized volume structure. In contrast to computer holograms for display purposes the 3-D intensity structure which is desired does not necessarily comply with the wave equation. This means, the design of suitable diffraction screens typically involves nonlinear optimization to determine the best compromise between several competing properties of the resulting intensity profile. The design of DOEs for generating 3-D intensity distributions has been investigated before, for instance in the context of non-diffracting beams [6

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

, 7

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

, 8

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

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]

] and self-imaging [10

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

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]

].

Our contribution describes an alternative approach to design DOE’s for generating periodic intensity distributions. Based on the fractional Talbot effect [12

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]

] and the concept of Talbot array illuminators (TAIs) [13

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

], we employ the matrix formalism of the fractional Talbot effect [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]

]. This formalism was shown to provide a convenient framework for implementing numerical design algorithms if the output is specified in a single diffraction plane [15

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]

].

We show that it is straightforward to implement a similar algorithm for the design of volume intensity distributions. In particular the simulated annealing algorithm described in [15

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]

] can be used with little modifications to design binary optics DOEs with a small number of discrete phase levels. However, the analysis of the degrees of freedom which can be exploited defines a non-trivial problem, since the propagating wavefield is governed by the wave equation. We discuss how constraints imposed by the wave equation translate into inherent symmetries of periodic wavefields. The performance of the design algorithm then critically depends on the specified intensity to be compatible with these symmetries. Features as well as limitations of the design problem in 3-D are highlighted based on results obtained for a particular example, i.e. the design of TAIs with extended focal depth.

Fig. 1. Basic geometry of the optical setup.

2. Degrees of freedom

For this study we restrict the discussion to a two-dimensional configuration, i.e. to optical signals which are functions of one transverse coordinate and the longitudinal position. For separable problems this automatically yields the solution to the 3D problem. For non-separable configurations the proposed procedure can be generalized accordingly. In addition, we constrain our discussion to the domain of paraxial optical signals. Figure 1 illustrates the geometry we consider. A periodic binary optics DOE with 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 zT = 2d 2/λ, where l is the wavelength of the incident wave. This is the essence of the Talbot effect [16

16. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed., vol. XXVII, pp. 2–108 ( Elsevier, Amsterdam,1989).

]. For our discussion we consider so-called fractional Talbot planes zN,M = zT 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,zN,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]

], i.e.

u(x,zN,M)=q=0Q1cqg(xqdQ).
(1)

un=q=0Q1Dn,qgq,
(2)

where elements Dn,q are the Talbot coefficients ck , 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]

]

cq=1Qexp[(q2Q14)]
(3)

For 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 zN,M provides us with Q equations,

un2=q,q′=0Q1Dn,q,Dn,q′*gqgq′*,
(4)

which can be used to determine the set of Q 2 unknowns gqgq*. This might imply that the intensity can be specified for Q out of TV planes before exhausting the available degrees of freedom. However, the gqgq* are not independent variables, which becomes obvious if we re-write Eq. (4) by substituting gqgq* = 2 cos(Δϕ q,q′). Here, Δϕ q,q′ is the phase difference between pixels gq′ . and gq . Since cases (q, q′) and (q′,q) result in equal intensities and terms q = q′ do not correspond to unknowns, we can specify at the most Q/2 - 1 diffraction planes. We note that Eq. (4) constitutes a nonlinear relationship between the phase only distribution of the DOE and the output signal. This makes the calculation of the number of independent parameters difficult. We therefore emphasize that our discussion only provides an estimate which should not be regarded as a strict condition. Additional constraints are imposed by the physics of wave propagation which limit the variety of intensity distributions which can be implemented. The matrix formalism of the fractional Talbot effect again is the convenient tool to investigate these constraints. In particular it is possible to analyze inherent symmetries of the periodic diffraction amplitude. For diffraction planes where N and M share a common denominator only a subset of matrix elements Dn,q is non-zero, i.e. the number of shifted and modulate copies of g(x) in Eq. (1) is smaller than Q. For instance, the complex amplitude at the plane M = N/2 is described by a single copy of the grating profile shifted by half the period d. This has two consequences. Firstly, as illustrated in Fig. 1, the intensity distributions within the first and the second half of the Talbot length are identical, except for a transverse shift of d/2, and cannot be specified independently. Secondly, at z = zT /2 the complex amplitude again is phase-only regardless of the structure of the DOE. Obviously, no DOE can be designed to generate the specified 3D intensity distribution with high accuracy if the desired intensity is not consistent with these fundamental properties of the propagating wavefield. Similarly, we find that the wavefield at z = zT /4 is described by two copies of the grating profile laterally shifted by d/2 and d/4, respectively. This implies that the intensity in this plane cannot exceed twice that of the incident field, no matter how the DOE is designed. This in turn means that we cannot hope to confine all of the intensity to a cross section smaller than d/2. Similar relationships can be deduced for 1/8,1/16,.... of the Talbot distance as well as for multiples of these fractional Talbot planes for which N and M are coprime.

3. Nonlinear optimization

Fig. 2. TAIs with extended focal depth. The intensity is specified in (a) one, (b) two, and (c) three diffraction planes. For the optimum DOE one period of the intensity in all specified planes (upper part) and the intensity distribution in x and z (lower part) is shown.

A quantitative evaluation of the confined intensity as a function of the propagation distance is shown in Fig. 3. The average intensity within the designated quarter of the transverse period is plotted in Fig. 3(a). While specifying the the desired intensity in more than one plane reduces the maximum intensity by little more than 6%, the half width of the intensity distribution is increased about five times. The ideal plateau-like shape is not reproduced and part of the total intensity is confined with the designated volume. In part, the deviations from the specified distribution are caused by the phase quantization and the spatial quantization of the DOE. Similar to DOEs which are design for the Fraunhofer domain [20

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

] either the number of pixels per period 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.

Fig. 3. Average intensity confined to (a) 1/4, (b) 1/2 of the transverse period. Solid lines refer to DOE with L = 8 discrete phase levels, the dashed green line corresponds to L = 16.

4. Discussion and Summary

The example used in Figs. (2) and (3) primarily serves as an illustration of the behavior we observed with a larger set of simulations. The algorithm typically converges to an acceptable solution if for optimization in a single diffraction plane a perfect or near perfect design can be obtained. Consistent with our expectations the algorithm invariably provides poor results if the specified intensities do not match the inherent properties of periodic propagating waves. The nonlinear optimization procedure cannot be applied blindly, but requires a substantial amount of guidance, to define the optimization problem in accordance with the wave equation.

We also point to the behavior of the intensity in Fig. 2 for diffraction planes z > zT /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 < zT /4 and a better confinement for planes z > zT /4. However, the resulting design is governed by the diffraction pattern at plane zT /4 where the maximum compression ratio cannot exceed a factor of two.

Finally, we would like to emphasize that the assumptions we have used for this study do not limit the generality of this approach. This includes the strict periodicity of the intensity distribution. For sufficiently large 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

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

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]

] into the framework of the fractional Talbot effect. However, while we believe that a thorough investigation of this relationship would be beneficial, it is beyond the scope of this contribution.

Acknowledgment

This work was financially supported by DARPA grant W911NF-04-1-0319.

References and links

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]

13.

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

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]

16.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed., vol. XXVII, pp. 2–108 ( Elsevier, Amsterdam,1989).

17.

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).

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]

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]

20.

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

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

  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. 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]
  5. 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]
  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. 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]
  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]
  13. A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik 79, 41-45 (1988).
  14. 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]
  15. 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]
  16. K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, E.Wolf, ed., vol. XXVII, pp. 2-108 (Elsevier, Amsterdam, 1989).
  17. 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).
  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]
  19. V. Arrizon, E. Lopez-Olazagasti, and A. Serrano-Heredia, "Talbot array illuminators with optimum compression ratio," Opt. Lett. 21, 233-235 (1996). [CrossRef] [PubMed]
  20. 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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