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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 6 — Mar. 20, 2006
  • pp: 2108–2116
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Iterative algorithms for holographic shaping of non-diffracting and self-imaging light beams

Johannes Courtial, Graeme Whyte, Zdeněk Bouchal, and Jaroslav Wagner  »View Author Affiliations


Optics Express, Vol. 14, Issue 6, pp. 2108-2116 (2006)
http://dx.doi.org/10.1364/OE.14.002108


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Abstract

We have developed iterative algorithms for the calculation of holograms for non-diffracting or self-imaging light beams. Our methods make use of the special Fourier-space properties of the target beams. We demonstrate experimentally the holographic generation of perhaps the most challenging type of beam: a self-imaging beam shaped in more than one plane. Potential applications include the generation of light “crystals” for optical trapping and atomic diffraction studies.

© 2006 Optical Society of America

1. Introduction

Since their discovery [1

1. J. Durnin, J. J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

], non-diffracting (ND) light beams have enjoyed great interest, for example in the areas of optical micro-manipulation [2

2. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001). [CrossRef]

, 3

3. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002). [CrossRef] [PubMed]

, 4

4. D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003). [CrossRef] [PubMed]

] and medical imaging [5

5. E. Goldfain, “Optical biopsy with long-range nondiffracting beams,” in Optical Biopsy III, R. R. Alfano, ed., Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), pp. 119–127 (2000).

].

It is easy to understand why ND beams do not change shape on propagation [1

1. J. Durnin, J. J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

]. Like all light beams, they can be understood as superpositions of plane waves. On propagation through a distance z, each individual plane-wave component changes phase by kzz (kz is the wave number in the z direction). Light beams usually change shape on propagation because they consist of plane-wave components with different values of kz, and which correspondingly change phase relative to one another on propagation. ND beams consist of plane-wave components that all have the same kz; on propagation, they change phase in exactly the same way, and therefore retain their relative phase, which in turn means that their interference pattern – the beam – does not change.

kx2+ky2+kz2=k2.
(1)

For monochromatic beams (to which we restrict our discussion in this paper) a constant value of kz is associated with transverse wave vector components that satisfy the equation

kx2+ky2=k2kz2=const.
(2)

Equation (2) describes a circle in the kx-ky plane. This can be used to create ND light beams as follows (Fig. 1): illuminate a thin ring aperture (an approximation to a circle) in the front focal plane of a lens (we refer to this plane as the Fourier plane), and behind the lens (specifically in the back focal plane, where the beam’s amplitude is the Fourier transform of the amplitude in the Fourier plane) the light will be approximately non-diffracting. (Note that experimentally created light beams are never perfectly non-diffracting; in the setup discussed here this is due to the fact that the intensity in the Fourier plane is a ring of finite width instead of a circle, and also because the aperture of any real Fourier lens is of finite size [6

6. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]

]). This method has been used in various experiments to create different ND beams [7

7. S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt. 4, S52–S57 (2002). [CrossRef]

, 8

8. Z. Bouchal and J. Kyvalský, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004). [CrossRef]

].

An interesting generalisation of ND beams are self-imaging (SI) beams [9

9. K. Patorski, “The self-imaging phenomenon and its applications,” Progr. Opt. XXVII, 3–108 (1989).

]. If a second bright ring is present in the Fourier plane, it creates a second ND beam whose phase changes linearly with z relative to that of the first beam. The beams periodically go in and out of phase – they beat – and the sum of the two beams changes shape periodically. This can be generalised to more than two ND components: all ND components whose longitudinal wave vectors, kz, satisfy the equation

kzΔzmod2π=Φ0
(3)
Fig. 1. Experimental generation of ND beams. A laser beam illuminates a thin ring aperture in the front focal plane (plane A) of the Fourier lens. Behind the lens, the beam is non-diffracting. The beam amplitude in the back focal plane of the Fourier lens (plane B) can be easily calculated as the Fourier transform (FT) of the amplitude in plane A. The intensity distributions on the far left and right illustrate the case of a single bright ring (uniform intensity, planar phase) at A, which leads to the simplest Bessel beam behind the lens.

suffer the same phase change, Φ0, during propagation through the distance ∆z. A light beam consisting exclusively of such components will be self-imaging with a period ∆z. SI beams have also been used in optical micro-manipulation [10

10. E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express 13, 3777–3786 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3777. [CrossRef] [PubMed]

].

ND and SI beams have been shaped before: for example, the projections-onto-convex-sets algorithm was used to construct ND beams with a cross-section of arbitrarily positioned bright points [11

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

]; diffractive optical elements for the shaping of longitudinally periodic beams were calculated iteratively [12

12. V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Algorithm for the Generation of Non-diffracting Bessel Modes,” J. Mod. Opt. 42, 1231–1239 (1995). [CrossRef]

, 13

13. V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997). [CrossRef]

]; SI beams that contain braided vortex lines were constructed mathematically [14

14. M. R. Dennis, “Braided nodal lines in wave superpositions,” New J. Phys. 5, 134.1–134.8 (2003). [CrossRef]

]; and the intensity of ND [15

15. Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. 27, 1376–1378 (2002). [CrossRef]

] and SI [16

16. G. Indebetouw, “Quasi-self-imaging using aperiodic sequences,” J. Opt. Soc. Am. A 9, 549–558 (1992). [CrossRef]

, 8

8. Z. Bouchal and J. Kyvalský, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004). [CrossRef]

] beams was made to take on the shape of an aperture in a double-Fourier-lens setup.

2. Gerchberg-Saxton (GS) algorithm

In this section we show how to use the computationally very efficient Gerchberg-Saxton (GS) algorithm [17

17. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

] to construct non-diffracting or self-imaging light beams.

Fig. 2. Simulation of ND (a) and SI (b) light beams, calculated with the GS method, whose intensity cross-section has been shaped to resemble the target intensity shown in (a). In both cases the focal length of the Fourier lens is f = 1m; the radii of the Fourier-space rings are r =10mm (a) and r =6.06mm, 10mm, 12.77mm and 15.05mm (b), resulting in a self-imaging period of 20mm. Shown are intensity cross-sections over a 1mm×1mm area in different transverse planes. The finite width of each ring (the profile is Gaussian with a width of 300μm) limits the distance over which the beams are non-diffracting or diffraction-free to a finite distance; this can be seen in (c).

Figure 2 demonstrates the simulated propagation of a ND and a SI beam calculated with this method. The beams’ intensity cross-sections (in all planes in the case of the ND beam, in planes z = 0mm and 20mm in the case of the SI beam) are clearly similar to the target pattern. The intensity cross-sections of the SI beam are noticeably more similar to the target pattern than those of the ND beam; this is due to the larger number of rings in k-space representation of the SI beam, which corresponds to more parameters that can be varied to improve the beam’s intensity cross-section, and the fact that in the particular example shown in Fig. 2 the largest k-space rings of the SI beam are larger than that of the ND beam, which means that higher spatial frequencies and correspondingly more small-scale details are present in the SI beam. The intensity cross-sections of the ND beam are shown over a relatively large propagation range to demonstrate the fact that the beam is ND over a finite range only. On a 600MHz Apple G3 laptop, our program – written in LabVIEW and using a grid of 512×512 pixels – performs one iteration in approximately 5 seconds and takes about 30 iterations to reach good solutions.

A possible extension of this method is the use of the multi-plane GS algorithm [20

20. T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997). [CrossRef]

] or even a full 3D GS algorithm [21

21. G. Shabtay, “Three-dimensional beam forming and Ewald’s surfaces,” Opt. Commun. 226, 33–37 (2003). [CrossRef]

, 22

22. G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm,” New J. Phys. 7, 117 (2005). [CrossRef]

], which would allow SI beams to be shaped in multiple planes or even over the entire volume of one period.

3. Direct-Search (DS) algorithm

DS algorithms [18

18. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987). [CrossRef] [PubMed]

] provide perhaps the simplest way to design light beams. The basic DS algorithm can be summarised as iteration of the following procedure. Make a random change to the light beam in one plane. Model the effect this has on the beam and evaluate its “merit”, which has to be defined such that it indicates to which degree the new light beam has the desired properties. For example, the merit function could be defined such that it is greatest if the new light beam’s intensity profile resembles most closely a pre-defined target profile. If the change has resulted in a better beam keep the change, otherwise discard it. As the merit can only increase, this algorithm has to converge. However, it does not always converge to the global merit maximum, but to a local merit maximum. Fast computers have enabled the useful application of this algorithm.

We apply this algorithm by randomly altering the phase and/or intensity at a fixed number of “source points” on one or more centered circles in the Fourier plane. The remainder of the beam in that plane is dark; as before, this ensures that the resulting beam is non-diffracting (one circle) or self-imaging (several circles). The beam is evaluated at a fixed number of “merit points”, which are points at which the intensity is of particular interest. In optical micro-manipulation, for example, the aim of light shaping is often to achieve high intensity at the positions where particles are to be trapped (and low intensity everywhere else). In this case the trapping positions could be the merit points, and the merit function could be defined such that it is increases if the intensity in the merit points increases.

Here we define the merit of a beam in terms of the intensities, Ii, at the merit points as a fraction of the overall power in the beam, P, as

i(niIiP+ε)1,
(4)

where ni is the number of merit points in the same plane and ε is a constant much smaller than the other terms that prevents individual terms from becoming -∞. We choose the form (4) because it prefers an equal distribution of the intensity between all the merit points to simply putting all the intensity into a few or even one merit point (an infinitesimal increase in the intensity leads to a merit increase proportional to the increase divided by the square of the intensity). Each term (niIi)/P describes the fraction of the power ideally in a point (P/ni) and the intensity there. In this way we preferentially brighten up the darkest merit points in each plane. Simple modifications of the merit function could lead to arbitrary relative merit-point intensities, or even a preference for darkness (no intensity) at some points. Other merit functions could be used to shape the beam in whole areas or volumes, for example by densely (of the order of the wavelength apart) covering the areas or volumes with merit points.

The intensity at the merit points due to the field at the source points can be calculated in a number of ways. We use here the fact that each source point (like every point in the Fourier plane) gives rise to a uniform plane wave behind the Fourier lens, whose complex amplitude is proportional to the complex amplitude at the source point. The overall complex amplitude at a merit point is simply the sum of the complex amplitudes due to all the source points. As only one source point is modified during each iteration, we calculate only the change of the merit-point fields due to the modification of the point source (for each merit point, we calculate the field due to the altered source point alone twice, respectively using the old and new source amplitude; the field change is then the difference between new and old field value). Running on the same 600MHz Apple G3 laptop, our program (again written in LabVIEW), when using 10 merit points, performs about 270 iterations per second, whereby the time per iteration is roughly proportional to the number of merit points. It takes typically of the order of 50 “visits” to each source pixel to arrive at a good hologram, and as we typically use around 500 source pixels this takes roughly 1.5 minutes.

Fig. 3. Simulated shaped ND and SI beams calculated with our DS method. (a), (b): intensity cross-sections of ND beams shaped into the constellation “Orion” (a diagram of the star position is inset). (c), (d): intensity cross-sections over two self-imaging periods (∆z = 40mm) of SI beams shaped into stretched unit cells of face-centred cubic (fcc, (c)) and body-centred cubic (bcc, (d)) “crystals” of bright spots; the merit points were located in the planes z = 0 and z = 10mm. In (a) both phase and intensity were modulated, in (b) and (d) only the phase (uniform intensity), in (c) only the intensity (uniform phase). In (a) and (b) r = 25mm (1 circle in the Fourier plane), in (c) and (d) r = 6.06mm, 10mm, 12.77mm and 15.05mm (4 circles); each circle contained 512 source points. In all cases f = 1m and λ = 633nm. All intensity cross-sections represent an area of approximately 0.5mm × 0.5mm. The additional multimedia material contains movies showing a computer rendition of the three-dimensional intensity distributions over slightly more than one self-imaging period of the fcc beam (308K) and the bcc beam (348K) as seen from different viewing angles. [Media 1] [Media 2]

Figure 3 shows some examples of intensity distributions calculated with our DS algorithm. In two examples the algorithm modulated only the phase of source points, in one example it modulated only the intensity, and in another example both. These examples demonstrate the creation of arbitrary non-diffracting point patterns, in our case in the shape of the stars in the constellation Orion, and “light crystals”: periodic light distributions in the shape of a series of simple crystallographic unit cells. The detailed intensity is not exactly the desired point pattern, but this can be improved with larger and/or more rings in Fourier space.

4. Experiment

To convert one laser beam into another both phase and intensity of the original beam has to be altered (unless either the intensity or phase structure of the original beam happens to be that of the desired beam, which is not the case here). Our experiment can be understood by thinking of the original beam as the sum of the desired beam (travelling at an angle α with respect to the original beam, which is taken to travel along the z axis) and a rest; our setup simply subtracts this rest from the original beam. Specifically, we use essentially a uniform plane wave (in fact a widened and collimated Gaussian beam from a commercial HeNe laser) as our original beam, and pass it through an intensity hologram (in the form of an SLM), which absorbs one part of the rest and transmits the sum of three beams: u(x,y)exp(iksin(α)x), the desired beam (u(x,y)), travelling at an angle α in the xz plane with respect to the original beam; u *(x,y)exp(-iksin(α)x), a beam (specifically the complex conjugate of the desired beam) travelling at an angle -α with respect to the original beam in the xz plane; and u 0 = minx,y(u(x,y)exp(iksin(α)x) + u *(x,y)exp(-iksin(α)x)), a uniform plane wave travelling in the z direction. This particular sum of beams is chosen because it has planar phase fronts, and its generation from a beam with planar phase fronts (like our collimated Gaussian) therefore only requires intensity modulation into the form

I(x,y)=[u(x,y)exp(iksin(α)x)+u*(x,y)exp(iksin(α)x)+u0]2.
(5)

The exp(±iksin(α)x) terms give the intensity modulation some characteristics of an intensity grating: the three transmitted beams at angles +α, 0 and -α with respect to the original beam can be seen as the +1st, 0th and -1st diffraction orders in the Fourier plane, respectively. The 0th and -1st orders are subsequently filtered out (Fig. 4(a)).

As the desired beam already has to be present in the illuminating beam, the holographic conversion works most efficiently in a plane where the intensity of the original and desired beams (the latter at an angle) overlap most. To achieve a good intensity overlap with the uniform illumination beam, we avoid generating the desired beam in a plane where its intensity distribution is concentrated in only a few bright rings (like in the image of the hologram plane) or spots, like for example in the planes z = 0 and z = 10mm in Fig. 3(d), and instead generate it in a plane half-way between these two planes, that is by choosing u(x,y) to be the amplitude distribution of the desired beam at z = 5mm. The corresponding intensity-hologram pattern, calculated with equation (5), is also shown in Fig. 4(a).

Fig. 4. Experimental setup for holographic generation of shaped self-imaging beams (a) and observed intensity distributions over an area of 3.2mm×3.2mm ((b); the corresponding modelled intensity distributions are those shown in Fig. 3(d)). Insets in (a) show the intensity hologram (corresponding to an area of 8.5mm×8.5mm) and the resulting diffraction orders in the Fourier plane, A; z 0 = 100mm is the z coordinate of plane B, into which the hologram plane, H, is imaged (after Fourier filtering). We use f 1 = 119mm and f 2 = 550mm. L 2 is the Fourier lens from Fig. 1.

Figure 4(b) shows the intensity distributions recorded over one self-imaging period. These experimental intensity distributions are not perfect reproductions of the modelled ones shown in Fig. 3(d); we believe this is mainly due to the difficulty of our specific SLM [23

23. CRL Opto Ltd., 1024 × 768 pixels, 13.9mm × 8.5mm active area.

] to display high spatial frequencies. However, the intensity distributions clearly show the desired stretched body-centred cubic (bcc) structure.

5. Conclusions

We have demonstrated that the special Fourier-space properties of ND and SI beams can be utilised in well-known algorithms to design such beams. Specifically we demonstrate, by computer-modelling and experimentally, the generation of SI beams shaped in more than one plane. Because they allow the creation of optical lattices with a wide range of symmetries, we believe that such beams offer interesting new possibilities in research areas such as study of cold atomic gases in optical lattices [24

24. L. Santos, “Introduction to Focus Issue: Cold Atomic Gases in Optical Lattices,” Optics Express 12, 2–3 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-2. [CrossRef] [PubMed]

].

Future work could include a more detailed analysis. Specifically, the two algorithms should be applied to the same examples and compared in terms of the merit of their output, which could for example be measured in terms of merit function such as the one given in equation (4) or simply the combined power in all the target points, but also in terms of their speed and suitability to various design problems. The dependence on the number of rings could also be quantified.

Acknowledgments

J.C. is a Royal Society University Research Fellow and gratefully acknowledges the generous hospitality of the Department of Optics, Palacký University. Z.B. and J.W. acknowledge support from the Ministry of Education of the Czech Republic (projects LN00A015 and CEZ J14/98).

References and links

1.

J. Durnin, J. J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

2.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001). [CrossRef]

3.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002). [CrossRef] [PubMed]

4.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003). [CrossRef] [PubMed]

5.

E. Goldfain, “Optical biopsy with long-range nondiffracting beams,” in Optical Biopsy III, R. R. Alfano, ed., Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), pp. 119–127 (2000).

6.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]

7.

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt. 4, S52–S57 (2002). [CrossRef]

8.

Z. Bouchal and J. Kyvalský, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004). [CrossRef]

9.

K. Patorski, “The self-imaging phenomenon and its applications,” Progr. Opt. XXVII, 3–108 (1989).

10.

E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express 13, 3777–3786 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3777. [CrossRef] [PubMed]

11.

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

12.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Algorithm for the Generation of Non-diffracting Bessel Modes,” J. Mod. Opt. 42, 1231–1239 (1995). [CrossRef]

13.

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997). [CrossRef]

14.

M. R. Dennis, “Braided nodal lines in wave superpositions,” New J. Phys. 5, 134.1–134.8 (2003). [CrossRef]

15.

Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. 27, 1376–1378 (2002). [CrossRef]

16.

G. Indebetouw, “Quasi-self-imaging using aperiodic sequences,” J. Opt. Soc. Am. A 9, 549–558 (1992). [CrossRef]

17.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

18.

M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987). [CrossRef] [PubMed]

19.

V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd, London, 1997).

20.

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997). [CrossRef]

21.

G. Shabtay, “Three-dimensional beam forming and Ewald’s surfaces,” Opt. Commun. 226, 33–37 (2003). [CrossRef]

22.

G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm,” New J. Phys. 7, 117 (2005). [CrossRef]

23.

CRL Opto Ltd., 1024 × 768 pixels, 13.9mm × 8.5mm active area.

24.

L. Santos, “Introduction to Focus Issue: Cold Atomic Gases in Optical Lattices,” Optics Express 12, 2–3 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-2. [CrossRef] [PubMed]

OCIS Codes
(090.1760) Holography : Computer holography
(110.6760) Imaging systems : Talbot and self-imaging effects
(140.3300) Lasers and laser optics : Laser beam shaping

ToC Category:
Holography

History
Original Manuscript: December 21, 2005
Revised Manuscript: March 14, 2006
Manuscript Accepted: March 14, 2006
Published: March 20, 2006

Citation
Johannes Courtial, Graeme Whyte, Zdeniek Bouchal, and Jaroslav Wagner, "Iterative algorithms for holographic shaping of non-diffracting and self-imaging light beams," Opt. Express 14, 2108-2116 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2108


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References

  1. J. Durnin, J. J. J. Miceli, and J. H. Eberly, "Diffraction-free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
  2. J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001). [CrossRef]
  3. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002). [CrossRef] [PubMed]
  4. D. McGloin, V. Garcés-Chávez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003). [CrossRef] [PubMed]
  5. E. Goldfain, "Optical biopsy with long-range nondiffracting beams," in Optical Biopsy III, R. R. Alfano, ed., Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), pp. 119-127 (2000).
  6. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987). [CrossRef]
  7. S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002). [CrossRef]
  8. Z. Bouchal and J. Kyvalský, "Controllable 3D spatial localization of light fields synthesized by non-diffracting modes," J. Mod. Opt. 51, 157-176 (2004). [CrossRef]
  9. K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).
  10. E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, "3D interferometric optical tweezers using a single spatial light modulator," Opt. Express 13, 3777-3786 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3777. [CrossRef] [PubMed]
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