## Iterative algorithms for holographic shaping of non-diffracting and self-imaging light beams

Optics Express, Vol. 14, Issue 6, pp. 2108-2116 (2006)

http://dx.doi.org/10.1364/OE.14.002108

Acrobat PDF (378 KB)

### 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

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]

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

*z*, each individual plane-wave component changes phase by

*k*∙

_{z}*z*(

*k*is the wave number in the

_{z}*z*direction). Light beams usually change shape on propagation because they consist of plane-wave components with different values of

*k*, and which correspondingly change phase relative to one another on propagation. ND beams consist of plane-wave components that all have the same

_{z}*k*; 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.

_{z}*k*is associated with transverse wave vector components that satisfy the equation

_{z}*k*-

_{x}*k*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

_{y}*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]

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]

*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,

*k*, satisfy the equation

_{z}_{0}, 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]

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]

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]

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]

## 2. Gerchberg-Saxton (GS) algorithm

*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.

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]

## 3. Direct-Search (DS) algorithm

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]

*I*, at the merit points as a fraction of the overall power in the beam,

_{i}*P*, as

*n*is the number of merit points in the same plane and

_{i}*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 (

*n*)/

_{i}I_{i}*P*describes the fraction of the power ideally in a point (

*P*/

*n*) 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.

_{i}*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

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

*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(

*ik*sin(

*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(-

*ik*sin(

*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}= min

_{x,y}(

*u*(

*x*,

*y*)exp(

*ik*sin(

*α*)

*x*) +

*u*

^{*}(

*x*,

*y*)exp(-

*ik*sin(

*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

*ik*sin(

*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)).

*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).

## 5. Conclusions

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]

## Acknowledgments

## References and links

1. | J. Durnin, J. J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. |

2. | J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. |

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 |

4. | D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. |

5. | E. Goldfain, “Optical biopsy with long-range nondiffracting beams,” in |

6. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

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. |

8. | Z. Bouchal and J. Kyvalský, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. |

9. | K. Patorski, “The self-imaging phenomenon and its applications,” Progr. Opt. |

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 |

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

12. | V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Algorithm for the Generation of Non-diffracting Bessel Modes,” J. Mod. Opt. |

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. |

14. | M. R. Dennis, “Braided nodal lines in wave superpositions,” New J. Phys. |

15. | Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. |

16. | G. Indebetouw, “Quasi-self-imaging using aperiodic sequences,” J. Opt. Soc. Am. A |

17. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik |

18. | M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. |

19. | V. Soifer, V. Kotlyar, and L. Doskolovich, |

20. | T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. |

21. | G. Shabtay, “Three-dimensional beam forming and Ewald’s surfaces,” Opt. Commun. |

22. | G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm,” New J. Phys. |

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 |

**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

- J. Durnin, J. J. J. Miceli, and J. H. Eberly, "Diffraction-free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001). [CrossRef]
- 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]
- D. McGloin, V. Garcés-Chávez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003). [CrossRef] [PubMed]
- 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).
- J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987). [CrossRef]
- 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]
- 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]
- K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).
- 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]
- R. Piestun and J. Shamir, "Control of wave-front propagation with diffractive elements," Opt. Lett. 19, 771-773 (1994). [CrossRef] [PubMed]
- 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]
- 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]
- M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5, 1341-1348 (2003). [CrossRef]
- Z. Bouchal, "Controlled spatial shaping of nondiffracting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002). [CrossRef]
- G. Indebetouw, "Quasi-self-imaging using aperiodic sequences," J. Opt. Soc. Am. A 9, 549-558 (1992). [CrossRef]
- 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).
- 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]
- V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd, London, 1997).
- T. Haist, M. Schönleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997). [CrossRef]
- G. Shabtay, "Three-dimensional beam forming and Ewald’s surfaces," Opt. Commun. 226, 33-37 (2003). [CrossRef]
- 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]
- CRL Opto Ltd., 1024 × 768 pixels, 13.9mm × 8.5mm active area.
- 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]

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