## Computer generation of optimal holograms for optical trap arrays

Optics Express, Vol. 15, Issue 4, pp. 1913-1922 (2007)

http://dx.doi.org/10.1364/OE.15.001913

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### Abstract

We propose a new iterative algorithm for obtaining optimal holograms targeted to the generation of arbitrary three dimensional structures of optical traps. The algorithm basic idea and performance are discussed in conjunction to other available algorithms. We show that all algorithms lead to a phase distribution maximizing a specific performance quantifier, expressed as a function of the trap intensities. In this scheme we go a step further by introducing a new quantifier and the associated algorithm leading to unprecedented efficiency and uniformity in trap light distributions. The algorithms performances are investigated both numerically and experimentally.

© 2007 Optical Society of America

## 1. Introduction

1. A. Ashkin, J. M. Dziedzic, and J. E. Bjorkholm, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**,288–290 (1986). [CrossRef] [PubMed]

2. M. Reicherter, T. Haist, E.U. Wagemann, and H.J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. **24**,608–610 (1999). [CrossRef]

3. J. Liesener, M. Reicherter, T. Haist, and H.J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. **185**,77–82 (2000). [CrossRef]

4. E.R. Dufresne, G.C. Spalding, M.T. Dearing, S.A. Sheets, and D.G. Grier, “Computer-generated holographic optical tweezers arrays,” Rev. Sci. Instrum. **72**,1810–1816 (2001). [CrossRef]

5. J. Curtis, B.A. Koss, and D.G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. **207**,169–175, (2002). [CrossRef]

6. J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express **12**,220–226 (2004). [CrossRef] [PubMed]

7. D.G. Grier, “A revolution in optical manipulation,” Nature **424**,810–816 (2003). [CrossRef] [PubMed]

## 2. Algorithm description and performance

*u*= |

_{j}*u*|exp(

*iϕ*) the complex amplitude of electric field reflecting off the

_{j}*j*th pixel, where

*ϕ*is the corresponding phase shift. The total energy flux through the SLM is given by

_{j}*W*

_{0}=

*cε*

_{0}

*N*|

*u*|

^{2}

*d*

^{2}/2 where N is the total number of pixels and

*d*

^{2}is the pixel’s surface area. We can use scalar diffraction theory to propagate the electric field complex amplitude from the

*j*th pixel surface to the location of the

*m*th trap in image space [9]. Summing up the contributions from all the

*N*pixels we obtain the complex amplitude

*v*of electric field on trap

_{m}*m*:

*x*,

_{j}*y*are the

_{j}*j*th pixel’s coordinates on the back focal plane (SLM) and

*x*,

_{m}*y*,

_{m}*z*are the

_{m}*mth*trap coordinates referred to the front focal plane (Fig. 1). We can easily generalize the Δ

^{m}

_{j}to add orbital angular momentum to trapping beams [5

5. J. Curtis, B.A. Koss, and D.G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. **207**,169–175, (2002). [CrossRef]

*V*:

_{m}*I*= |

_{m}*V*|

_{m}^{2}measures the energy flux in units of

*W*

_{0}flowing through an area

*f*

^{2}λ

^{2}/(

*Nd*) (the area of a diffraction peak) centered at the

*m*th trap site. For

*z*= 0

_{m}*V*, corresponds to the discrete Fourier transform of

_{m}*e*evaluated at the spatial frequencies (

^{iϕj}*x*/λ

_{m}*f*,

*y*/λ

_{m}*f*).

^{m}

_{j}, for the best choice of

*ϕ*s to maximize the modulus of

_{j}*V*on all traps. We will use as a benchmark to compare different strategies, the task of computing an

_{m}*N*= 768×768, 8 bit hologram aiming at a target intensity of

*M*= 100 traps arranged on a 10×10 square lattice located in the Fourier (

*z*= 0) plane. The performance of different strategies is quantified by three parameters: efficiency (

_{m}*e*), uniformity (

*u*) and percent standard deviation (σ)

*m*.

*M*= 1. The choice here is easily found by setting

*ϕ*= Δ

_{j}_{j}

^{1}which makes all terms in the sum (3) real and equal to 1/

*N*, thus giving |

*V*

_{1}|

^{2}= 1. When

*M*> 1 we have to seek for a compromise between the

*M*different choices

*ϕ*= Δ

_{j}^{m}

_{j}(one choice for each

*m*value) that would divert all energy on trap

*m*. One of the fastest routes is the random mask encoding technique (RM) [17

17. M. Montes-Usategui, E. Pleguezuelos, J. Andilla, and E. Maríin-Badosa, “Fast generation of holographic optical tweezers by random mask encoding of Fourier components,” Opt. Express **14**,2101–2107, (2006). [CrossRef] [PubMed]

*m*is a number between 1 and

_{j}*M*randomly chosen for each

*j*. The technique is very fast, and performs remarkably good as far as uniformity is concerned. However the overall efficiency can be very low when

*M*is large. In fact, on average, for each

*m*only

*N*/

*M*pixels will interfere constructively, all the others giving a vanishing contribution. Therefore |

*V*|

_{m}^{2}≃ 1/

*M*

^{2}and

*e*≃ 1/

*M*which can be significantly smaller than one when

*M*is large. In the present case, where

*M*= 100, we numerically obtained

*u*= 0.58 but

*e*= 0.01 = 1/

*M*.

*V*real and equal to

_{m}*M*= 1 case but we can try to maximize the real part of ∑

_{m}

*V*with respect to

_{m}*ϕ*. The stationary points are easily obtained imposing the condition of a vanishing gradient:

_{j}*n*are set to 0 and therefore:

_{j}2. M. Reicherter, T. Haist, E.U. Wagemann, and H.J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. **24**,608–610 (1999). [CrossRef]

3. J. Liesener, M. Reicherter, T. Haist, and H.J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. **185**,77–82 (2000). [CrossRef]

*e*= 0.29 uniformity is only

*u*= 0.01. Moreover, when, as for the square lattice, highly symmetrical trap geometries are sought, a consistent part of energy is diverted to unwanted ghost traps.

*V*projected on randomly chosen directions in complex plane. In other words we seek a maximum of ∑

_{m}_{m}Re{

*V*exp(-

_{m}*iθ*)} where

_{m}*θ*are random numbers uniformly distributed in [0,2

_{m}*π*]. In this case we obtain:

10. L.B. Lesem, P.M. Hirsch, and J.A. Jordan “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. **13**,150–155 (1969). [CrossRef]

*e*= 0.69,

*u*= 0.01).

11. J.E. Curtis, C.H.J. Schmitz, and J.P. Spatz, “Symmetry dependence of holograms for optical trapping,” Opt. Lett. **30**,2086–2088 (2005). [CrossRef] [PubMed]

*θ*and try to maximize ∑

_{m}_{m}|

*V*| allowing for any possible value for

_{m}*θ*. Again differentiating with respect to

_{m}*ϕ*we obtain the stationary points:

_{j}*N*smaller than the diagonal ones. It can be shown that such a perturbation will only affect the sign of one eigenvalue at most [12

12. L. Angelani, L. Casetti, M. Pettini, G. Ruocco, and F. Zamponi, “Topological signature of first-order phase transitions in a mean-field model,” Europhys. Lett. **6**,775–781 (2003). [CrossRef]

*N*is very large we can neglect this eventuality and call the stationary point

*ϕ*are obtained as the phase of the linear superposition of single trap holograms with coefficients of unit modulus and a phase given by the phase of

_{j}*V*, that is the field produced by the

_{m}*ϕ*themselves on trap site

_{j}*m*. It’s now impossible to have the

*ϕ*in an explicit form given the implicit dependence of

_{j}*V*on

_{m}*ϕ*. A possible route is that of starting with a guess for

_{j}*ϕ*, i.e. the one obtained from SR, and use (14) in an iterative procedure. This algorithm is called Gerchberg-Saxton (GS)[13

_{j}13. 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]

14. G. Sinclair, J. Leach, P. Jordan, G. Gibson, E. Yao, Z. Laczik, M. J. Padgett, and J. Courtial, “Interactive application in holographic optical tweezers of a multi-plane Gerchberg-Saxton algorithm for three-dimensional light shaping,” Opt. Express **12**,1665–1670 (2004). [CrossRef] [PubMed]

*e*= 0.94, and

*u*= 0.60.

*V*, having no bias towards uniformity. Such a bias is present when we seek for a maximum in a quantity like ∏

_{m}_{m}|

*V*| or equivalently ∑

_{m}_{m}log|

*V*|. By differentiating the biased function with respect to

_{m}*ϕ*we obtain:

_{j}5. J. Curtis, B.A. Koss, and D.G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. **207**,169–175, (2002). [CrossRef]

*u*= 0.79 with the same efficiency

*e*= 0.93 as GS.

*P*= 256 gray levels looking for an improvement (increase) in the gain function:

15. M. Meister and R. J. Winfield, “Novel approaches to direct search algorithms for the design of diffractive optical elements,” Opt. Commun. **203**,3949 (2002). [CrossRef]

16. M. Polin, K. Ladavac, S.H. Lee, Y. Roichman, and D. Grier, “Optimized holographic optical traps,” Opt. Express **13**,5831–5845, (2005). [CrossRef] [PubMed]

16. M. Polin, K. Ladavac, S.H. Lee, Y. Roichman, and D. Grier, “Optimized holographic optical traps,” Opt. Express **13**,5831–5845, (2005). [CrossRef] [PubMed]

*f*= 0.5, the algorithm achieves a perfect uniformity (

*u*= 1.00) after 1.3 ×

*N*steps of computational cost scaling as

*M*×

*P*, though the overall efficiency is diminished to

*e*= 0.68. Better holograms can be obtained by giving more bias to efficiency (

*f*=0.25) and waiting for a substantially longer time (∼ 10

*N*steps that is about a hundred times longer than GS). However, we observed that reducing the number of gray-levels

*P*to just 8 can significatively reduce (by a factor 32) the computational cost without affecting performance too much (see [15

15. M. Meister and R. J. Winfield, “Novel approaches to direct search algorithms for the design of diffractive optical elements,” Opt. Commun. **203**,3949 (2002). [CrossRef]

*e*= 0.84 and

*u*= 1.00 after 7

*N*steps that is still about 3 times longer than GS. A this point the whole hologram has been reduced to 3 bit and a comparison with other algorithms working with full 8 bit depth is out of purpose. So far, we reviewed the most common strategies for generating phase only holograms for optical tweezers applications. Each one of the presented methods is not fully satisfying regarding at least one parameter between efficiency and uniformity.

*M*extra degrees of freedom

*w*, that maximize the weighted sum ∑

_{m}_{m}

*w*|

_{m}*V*| with the constrain that |

_{m}*V*| are all equal. By differentiating with respect to

_{m}*ϕ*, we obtain the maximum condition:

_{j}*ϕ*in an implicit form, this time containing also the unknown weights

_{j}*w*. Starting from a

_{m}*SR*guess for

*ϕ*and setting

_{j}*w*= 1, the iteration proceeds as follows:

_{m}*w*in such a way to reduce |

_{m}*V*| deviations from the average 〈|

_{m}*V*|〉. The above procedure converges, with a speed typical of GS and GAA, to a hologram having the almost optimal performance

*e*= 0.93,

*u*= 0.99. We will refer to this new algorithm as weighted Gerchberg-Saxton or GSW. The optimization progress for GSW is reported in detail in Fig. 2 and shows how we can efficiently use it to obtain above 90% efficiency and uniformity in only 10 steps. The performance of our algorithm remains the highest when we turn to three dimensional lattices. For example a 3 × 3 × 3 simple cubic lattice can be generated with an efficiency of

*e*= 0.91 and a

*u*= 0.99 uniformity. The idea of aiming at a slightly modified target intensity distribution in GS optimizations has been also proposed in a different algorithm targeted to two dimensional beam shaping tasks [18

18. J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. **27**,1463–1465, (2002). [CrossRef]

## 3. Experimental results

*P*= -4°,

*A*= 20°, where angles are measured from the vertical direction and rotating clockwise when viewing from the beam direction). The phase modulated wavefront is imaged onto the exit pupil of a 100x NA 1.4 objective lens mounted in a Nikon TE2000-U inverted optical microscope.

*μm*. We obtain experimental values for

*I*by summing the values of pixels inside a 1

_{m}*μ*m × 1

*μ*m square area centered on the peak center of mass. Such a determination of

*I*is affected by many sources of uncertainty: i) the coverglass plane might not be exactly parallel to the traps plane, ii) the diffusing power of the interface might be non uniform, ii) the imaging with coherent and polarized light might be non uniform. To correct for these uncertainties, we measured an ensemble of

_{m}*I*obtained from high uniformity algorithms (GSW, DS) and calculated starting from independent SR holograms. For each

_{m}*m*, we thus obtained a normalization factor as the average of the measured

*I*s. The corrected

_{m}*I*for the algorithms described in section 2 are reported, together with the raw image data, in Fig. 4.

_{m}*e*,

*u*,

*σ*in Table 2. Efficiencies have been multiplied by a constant factor chosen to give the theoretically expected value for GSW.

*u*≃ 1), probably for some residual error in the determination of

*I*.

_{m}## 4. Conclusion

*I*. In this scheme, we went a step further by introducing a new algorithm leading to unprecedented efficiency and uniformity in trap light distributions. The algorithm converges in a few tens of iterations of

_{m}*N*×

*M*computational cost. We compared, both numerically and experimentally, the performances of investigated algorithms in producing a 10×10 square grid target. The obtained results demonstrate that the proposed algorithm allows to achieve almost perfect efficiencies and uniformities using phase only holograms and a modest computational time.

## References and links

1. | A. Ashkin, J. M. Dziedzic, and J. E. Bjorkholm, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

2. | M. Reicherter, T. Haist, E.U. Wagemann, and H.J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. |

3. | J. Liesener, M. Reicherter, T. Haist, and H.J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. |

4. | E.R. Dufresne, G.C. Spalding, M.T. Dearing, S.A. Sheets, and D.G. Grier, “Computer-generated holographic optical tweezers arrays,” Rev. Sci. Instrum. |

5. | J. Curtis, B.A. Koss, and D.G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. |

6. | J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express |

7. | D.G. Grier, “A revolution in optical manipulation,” Nature |

8. | E. Martn-Badosa, A. Carnicer, I. Juvells, and S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. |

9. | J.W. Goodman, “Introduction to Fourier Optics,” McGraw-Hill (1996). |

10. | L.B. Lesem, P.M. Hirsch, and J.A. Jordan “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. |

11. | J.E. Curtis, C.H.J. Schmitz, and J.P. Spatz, “Symmetry dependence of holograms for optical trapping,” Opt. Lett. |

12. | L. Angelani, L. Casetti, M. Pettini, G. Ruocco, and F. Zamponi, “Topological signature of first-order phase transitions in a mean-field model,” Europhys. Lett. |

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

14. | G. Sinclair, J. Leach, P. Jordan, G. Gibson, E. Yao, Z. Laczik, M. J. Padgett, and J. Courtial, “Interactive application in holographic optical tweezers of a multi-plane Gerchberg-Saxton algorithm for three-dimensional light shaping,” Opt. Express |

15. | M. Meister and R. J. Winfield, “Novel approaches to direct search algorithms for the design of diffractive optical elements,” Opt. Commun. |

16. | M. Polin, K. Ladavac, S.H. Lee, Y. Roichman, and D. Grier, “Optimized holographic optical traps,” Opt. Express |

17. | M. Montes-Usategui, E. Pleguezuelos, J. Andilla, and E. Maríin-Badosa, “Fast generation of holographic optical tweezers by random mask encoding of Fourier components,” Opt. Express |

18. | J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(090.1760) Holography : Computer holography

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Trapping

**History**

Original Manuscript: October 13, 2006

Revised Manuscript: December 13, 2006

Manuscript Accepted: December 16, 2006

Published: February 19, 2007

**Virtual Issues**

Vol. 2, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Roberto Di Leonardo, Francesca Ianni, and Giancarlo Ruocco, "Computer generation of optimal holograms for optical trap arrays," Opt. Express **15**, 1913-1922 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1913

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### References

- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986). [CrossRef] [PubMed]
- M. Reicherter, T. Haist, E.U. Wagemann, H.J. Tiziani, "Optical particle trapping with computer-generated holograms written on a liquid-crystal display," Opt. Lett. 24, 608-610 (1999). [CrossRef]
- J. Liesener, M. Reicherter, T. Haist, H.J. Tiziani, "Multi-functional optical tweezers using computer-generated holograms," Opt. Commun. 185, 77-82 (2000). [CrossRef]
- E.R. Dufresne, G.C. Spalding, M.T. Dearing, S.A. Sheets, D.G. Grier, "Computer-generated holographic optical tweezers arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001). [CrossRef]
- J. Curtis, B.A. Koss, D.G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175, (2002). [CrossRef]
- J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, "3D manipulation of particles into crystal structures using holographic optical tweezers," Opt. Express 12, 220-226 (2004). [CrossRef] [PubMed]
- D.G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003). [CrossRef] [PubMed]
- E. Martn-Badosa, A. Carnicer, I. Juvells, and S. Vallmitjana, "Complex modulation characterization of liquid crystal devices by interferometric data correlation," Meas. Sci. Technol. 8, 764-772 (1997). [CrossRef]
- J.W. Goodman, "Introduction to Fourier Optics," McGraw-Hill (1996).
- L.B. Lesem, P.M. Hirsch, J.A. Jordan "The kinoform: a new wavefront reconstruction device," IBM J. Res. Dev. 13, 150-155 (1969). [CrossRef]
- J.E. Curtis, C.H.J. Schmitz, J.P. Spatz, "Symmetry dependence of holograms for optical trapping," Opt. Lett. 30, 2086-2088 (2005). [CrossRef] [PubMed]
- L. Angelani, L. Casetti,M. Pettini, G. Ruocco, F. Zamponi, "Topological signature of first-order phase transitions in a mean-field model," Europhys. Lett. 6, 775-781 (2003). [CrossRef]
- T. Haist, M. Schönleber, H.J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997). [CrossRef]
- G. Sinclair, J. Leach, P. Jordan, G. Gibson, E. Yao, Z. Laczik, M. J. Padgett, and J. Courtial, "Interactive application in holographic optical tweezers of a multi-plane Gerchberg-Saxton algorithm for three-dimensional light shaping," Opt. Express 12, 1665-1670 (2004). [CrossRef] [PubMed]
- M. Meister and R. J. Winfield, "Novel approaches to direct search algorithms for the design of diffractive optical elements," Opt. Commun. 203, 3949 (2002). [CrossRef]
- M. Polin, K. Ladavac, S.H. Lee, Y. Roichman, D. Grier, "Optimized holographic optical traps," Opt. Express 13, 5831-5845, (2005). [CrossRef] [PubMed]
- M. Montes-Usategui, E. Pleguezuelos, J. Andilla, E. Mart ýn-Badosa, "Fast generation of holographic optical tweezers by random mask encoding of Fourier components," Opt. Express 14, 2101-2107, (2006). [CrossRef] [PubMed]
- J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27, 1463-1465, (2002). [CrossRef]

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