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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 1 — Jan. 3, 2011
  • pp: 247–254
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Hologram transmission through multi-mode optical fibers

Roberto Di Leonardo and Silvio Bianchi  »View Author Affiliations


Optics Express, Vol. 19, Issue 1, pp. 247-254 (2011)
http://dx.doi.org/10.1364/OE.19.000247


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Abstract

We demonstrate that a structured light intensity pattern can be produced at the output of a multi-mode optical fiber by shaping the wavefront of the input beam with a spatial light modulator. We also find the useful property that, as in the case for free space propagation, output intensities can be easily superimposed by taking the argument of the complex superposition of corresponding phase-only holograms. An analytical expression is derived, relating output intensities ratios to hologram weights in the superposition.

© 2011 Optical Society of America

1. Introduction

Optical fibers can guide a light beam across long distances or through turbid media like biological tissues [1

1. Y. Wu, Y. Leng, J. Xi, and X. Li, “Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues,” Opt. Express 17, 7907–7915 (2009). [CrossRef] [PubMed]

]. The total intensity of the output light can be easily modulated on the input side when serial information has to travel along the fiber. However multimode fibers can propagate a light beam carrying a much larger information encoded in the complex coefficients of its expansion in the propagating modes. The main obstacle in using such a set of degrees of freedom comes from the fact that the phases of modes’ amplitudes are rapidly shuffled upon propagation [2

2. D. Z. Anderson, M. A. Bolshtyansky, and B. Y. Zel’dovich, “Stabilization of the speckle pattern of a multimode fiber undergoing bending,” Opt. Lett. 21, 785–787 (1996). [CrossRef] [PubMed]

, 3

3. A. Lucensoli and T. Rozzi , “Image transmission and radiation by truncated linearly polarized multimode fiber,” Appl. Opt. 46, 3031–3037 (2007). [CrossRef]

]. As a result, one always ends up having a random speckle pattern at a multimode fiber output. In this sense a multimode fiber can be thought of as a strongly aberrating optical element. Spatial light modulators (SLM) have been shown to be extremely useful for correcting aberrations both in weakly [4

4. K. D. Wulff, D. G. Cole, R. L. Clark, R. Di Leonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett , “Aberration correction in holographic optical tweezers,” Opt. Express 14, 4169–4174 (2006). [CrossRef] [PubMed]

6

6. A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express 15, 5801–5808 (2007). [CrossRef] [PubMed]

] and strongly [7

7. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007). [CrossRef] [PubMed]

, 8

8. T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010). [CrossRef]

] aberrated optical systems. In the case of LMA fibers, phase modulation has been already used to maximize the output signal of fiber lasers by coherently adding the light coming from a few monomodal cores [9

9. M. Paurisse, M. Hanna, F. Druon, and P. Georges, “Wavefront control of a multicore ytterbium-doped pulse fiber amplifier by digital holography,” Opt. Lett. 35, 1428–1430 (2010). [CrossRef] [PubMed]

, 10

10. C. Bellanger, A. Brignon, J. Colineau, and J. P. Huignard, “Coherent fiber combinig by digital holography,” Opt. Lett. 33, 2937–2939 (2008). [CrossRef] [PubMed]

] or a few different modes in the same core [11

11. M. Paurisse, M. Hanna, F. Droun, P. Georges, C. Bellanger, A. Brignon, and J. P. Huignard, “Phase and amplitude control of a multimode fiber beam by use of digital holography,” Opt. Express 17, 13000–13008 (2009). [CrossRef] [PubMed]

]. Phase modulation can be used to manipulate the spatial and spectral properties of high-harmonics in hollow fibers [12

12. D. Walter, T. Pfeifer, C. Winterfeldt, R. Kemmer, R. Spitzenpfeil, G. Gerber, and C. Spielmann, “Adaptive spatial control of fiber modes and their excitation for high-harmonic generation,” Opt. Express 14, 3433–3442 (2006). [CrossRef] [PubMed]

]. In this last paper, genetic algorithms have been used to modulate the output speckle pattern with large scale intensity masks.

However multimode fibers can propagate thousands of modes that when combined in random superposition give rise to a large number of speckles. It is natural then to ask whether such a structured noisy pattern could be shaped with a spatial resolution of a single speckle size. One or few, diffraction limited spots could be delivered at a fiber output and dinamically reconfigured. That possibility would be particularly relevant for in vivo biological applications, where multimode fibers could penetrate through highly turbid biological tissues to perform endomicroscopy [1

1. Y. Wu, Y. Leng, J. Xi, and X. Li, “Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues,” Opt. Express 17, 7907–7915 (2009). [CrossRef] [PubMed]

, 13

13. I. M. Vellekoop and C. M. Aegerter , “Scattered light fluorescence microscopy: imaging through turbid layers,” Opt. Lett. 35, 1245–1247 (2010). [CrossRef] [PubMed]

] or endo-micromanipulation [8

8. T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010). [CrossRef]

, 14

14. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a-fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993). [CrossRef] [PubMed]

, 15

15. P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fibre trap for Raman micro-spectroscopy of single cells,” Opt. Express 14, 5779–5791 (2006). [CrossRef] [PubMed]

].

In this paper we demonstrate that phase only modulation can be used to shape a light beam in such a way that, after propagation along a multimode fiber, most of the outgoing light will flow through one or few target spots having the size of a single speckle and arbitrarily located in space. We also show that a set of separate holograms, each producing a single target spot, can be combined in a complex superposition for quick multispot generation. Finally we derive and experimentally validate an analytical expression providing the actual fraction of power that falls on a given spot produced by a superposition hologram.

2. Results and discussion

A schematic view of our experimental setup is shown in Fig. 1. The laser beam (CrystaLaser CL-2000, 100 mW CW, λ =0.532 μm) is expanded to fill the entire SLM (Holoeye LCR-2500) active area. The modulated beam is then compressed by a telescope and focused onto the core of a multimodal fiber (Thorlabs AFS105/125Y, NA=0.22, length=2 m). Outcoming light is collected by a collimating lens and sent on a CMOS camera (Prosilica GC1280). Both lenses L5 and L6 have a numerical aperture of 0.25 that is slightly larger than that of the fiber. The SLM is located on the Fourier plane of the fiber input. The modulated beam is on the first diffraction order of a linear grating while unmodulated light, propagating on the zeroth order, is blocked by the diaphragm (D). Working on the first diffraction order will allow us to efficiently superimpose independently obtained holograms without taking into account interference with unmodulated light on the zeroth order. Our fiber doesn’t preserve polarization and we only detect the linearly polarized component emerging from the analyzer polaroid P. In the absence of the analyzer, two output beams with orthogonal polarizations have to be shaped simultaneously, which makes our task much harder, although still feasible in principle.

Fig. 1 Schematic view of the experimental setup. L1-L4 planoconvex lenses; L5, L6 fiber collimators; P polaroid; D iris diaphragm.

Fig. 2 Single spot optimization. Power on the target spot (normalized to total output power) is plotted as a function of iteration step for four optimization runs (colored solid lines). An average over the four runs is displayed as a black solid line and clearly evidences a convergence after 2000 iterations.

In Fig. 3 we show an optimized spot as compared to the speckle pattern obtained with an unmodulated beam. We observe a peak intensity which is about 35 times larger than the average nearby speckles in the unmodulated case (Fig. 3d). One might expect that for an optimal phase modulation, all the N modes contributing to the intensity I at the target spot will interfere constructively (IN2). On the other hand, when no modulation is applied, the same modes will sum up incoherently (IN). As a consequence, the brightness of the target spot is expected to increase roughly as the number of contributing modes. In particular, if the unmodulated complex amplitudes of the modes have a circular Gaussian distribution, the expected enhancement is found to be π (N – 1)/4 + 1 [7

7. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007). [CrossRef] [PubMed]

]. However, for each target spot, only a fraction of the available guided modes will be contributing to the total intensity. For example, when the target spot is located on the axis of an ideal fiber, only the l = 0 modes will contribute. In our fiber there will be 43 modes with l = 0 which gives an expected enhancement of 34, which compares surprisingly well with the observed factor 35. It is worth noting that our CMOS camera (Prosilica GC1280) has an intensity threshold such that light with an intensity lower than this threshold is not detected. For this reason, at a shutter speed that avoids saturation on the target spot, the background speckles are not detected. To overcome this issue we artificially increase the bit depth of our camera by collecting multiple frames at different shutter speeds and than merging the frames into a single picture.

Fig. 3 a) Random speckle pattern on the fiber output without phase modulation. b) Detail of the full 768x768 optimized phase modulation. The linear grating, shifting the beam away from the zero order, has been subtracted. c) When applying the optimized phase modulation a single spot appears on the fiber output. d) Intensity profile across the target spot with (black line) and without (blue line) phase modulation.

If we aim to an array of spots we will find out that their intensities depends on the number of targets as well as on their geometry. We qualitatively observe that, while the average target intensity obviously decreases when increasing the number of targets, the total light power flowing through all the spots increases slightly. We report in Fig. 4 the result of a simultaneous optimization of 17 targets arranged to form the letters “cnr”. Such multispot targets can also be displayed dynamically on the SLM to transmit a holographic movie across our two meters long multi mode fibers. Figure 5 shows some frames from a movie of a spinning square coming out of the fiber.

Fig. 4 Multiple target optimization. Incoming light is modulated so that the light that comes out of the multimode fiber is concentrated onto 17 spots arranged to form the letters “cnr”.
Fig. 5 An holographic movie of a spinning square can be delivered through a 2 meters long multimode fiber by modulating the incoming beam with a time sequence of phase masks.

Holograms that result in spatially separated target spots can be combined to get multispot arrays. For example, calling ϕA and ϕB the two phase only modulations corresponding to spots in points A and B respectively, we can build the complex modulation:
ujAB=xexp[iϕjA]+1xexp[iϕjB]
(2)
that would result in two simultaneous spots with a fraction x of total intensity going in A and the remaining 1 – x in B. However u will in general correspond to an amplitude and phase modulation while we can only apply a phase only modulation on the SLM. A similar problem is encountered in holographic optical tweezers where phase only holograms for a single trap are easily computed [21

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

, 22

22. G. C. Spalding, J. Courtial, and R. Di Leonardo, “Holographic optical tweezers,” in Structured Light Its Applications, D. L. Andrews, ed. (Academic Press, 2008) pp. 139–168. [CrossRef]

]. In that context it has been found that for multiple traps, by simply neglecting the amplitude modulation, one gets an intensity distribution that is close to the superposition of the separate hologram intensities [17

17. R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007). [CrossRef] [PubMed]

, 23

23. 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]

]. We might hope that this is also the case for propagation in multimode fibers and apply the phase only modulation ϕj=arg(ujAB) so that the field on the SLM plane will be uj=ujAB/|ujAB|. Indeed, such a phase only modulation results in a double spot output as shown in Fig. 6, where the intensity of the two spots as a function of x is reported with open circles. The shape of the two curves seems to be very clean and reproducible suggesting a robust statistical averaging. At our low power levels, the complex field on target spot A will be a linear combination of the complex field on SLM pixels:
vA=jGjAuj
(3)
where GjA is the light propagator from pixel j to target point A. If we iteratively find a phase modulation ϕjA that maximizes the light intensity through the target spot A, we might assume that a good approximation for the propagator will be given by GjAgjAexp[iϕjA]. Where gjA are unknown real amplitudes.

Fig. 6 Normalized intensities (vA,B(x)2/vA,B(0)2)on the two target spots A (red dots) and B (black dots) as a function of the weight x that the hologram ϕA has in the superposition. The theoretical prediction given by equation 6 is also shown as solid lines. Gray dashed line is the expectation for full phase and amplitude modulation.

Therefore we can anticipate that the complex field on target A can be obtained by:
vA(x)=jGjAujAB|ujAB|jgjAx+1xexp[iθj]|x+1xexp[iθj]|
(4)
where θj=ϕjBϕjA. The above expression is a sum of single pixel terms and therefore only depends on the distributions of θj and gjA but not on their particular spatial arrangement on the SLM. We experimentally found that the single spot holograms are characterized by a uniform distribution of phase values between 0 and 2π. The same will then hold for θj. If, in addition, we assume that amplitudes gjA and phases ϕjA are statistically uncorrelated, we can replace the summation in Eq. (4) with the averages:
vA(x)vA(0)12π02πa+bexp[iθ]|a+bexp[iθ]|dθ=1π0πa+bcosθa2+b2+2abcosθdθ=
(5)
=abaπE[4ab(ab)2]a+baπk[4ab(ab)2]
(6)
where a=x,b=1x, K[⋯] and E [⋯] are the complete elliptic functions of respectively first and second kind. The intensity of spot A is then obtained as vA(x)2, vB can be obtained from Eq. (6) by swapping A and B. The theoretical expression is plotted in Fig. 6 showing a remarkable good agreement with experimental data. It is worth noting here that the approximations involved in the derivation of Eq. (6) will hold exactly for the superposition of two single trap holograms in holographic tweezers, therefore expression Eq. (6) could be equally well used to choose the right weights for a desired intensity ratio between traps.

3. Conclusions

References and links

1.

Y. Wu, Y. Leng, J. Xi, and X. Li, “Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues,” Opt. Express 17, 7907–7915 (2009). [CrossRef] [PubMed]

2.

D. Z. Anderson, M. A. Bolshtyansky, and B. Y. Zel’dovich, “Stabilization of the speckle pattern of a multimode fiber undergoing bending,” Opt. Lett. 21, 785–787 (1996). [CrossRef] [PubMed]

3.

A. Lucensoli and T. Rozzi , “Image transmission and radiation by truncated linearly polarized multimode fiber,” Appl. Opt. 46, 3031–3037 (2007). [CrossRef]

4.

K. D. Wulff, D. G. Cole, R. L. Clark, R. Di Leonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett , “Aberration correction in holographic optical tweezers,” Opt. Express 14, 4169–4174 (2006). [CrossRef] [PubMed]

5.

Y. Roichman, A. Waldron, E. Gardel, and D. Grier, “Optical traps with geometric aberrations,” Appl. Opt. 45, 3425–3429 (2006). [CrossRef] [PubMed]

6.

A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express 15, 5801–5808 (2007). [CrossRef] [PubMed]

7.

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007). [CrossRef] [PubMed]

8.

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010). [CrossRef]

9.

M. Paurisse, M. Hanna, F. Druon, and P. Georges, “Wavefront control of a multicore ytterbium-doped pulse fiber amplifier by digital holography,” Opt. Lett. 35, 1428–1430 (2010). [CrossRef] [PubMed]

10.

C. Bellanger, A. Brignon, J. Colineau, and J. P. Huignard, “Coherent fiber combinig by digital holography,” Opt. Lett. 33, 2937–2939 (2008). [CrossRef] [PubMed]

11.

M. Paurisse, M. Hanna, F. Droun, P. Georges, C. Bellanger, A. Brignon, and J. P. Huignard, “Phase and amplitude control of a multimode fiber beam by use of digital holography,” Opt. Express 17, 13000–13008 (2009). [CrossRef] [PubMed]

12.

D. Walter, T. Pfeifer, C. Winterfeldt, R. Kemmer, R. Spitzenpfeil, G. Gerber, and C. Spielmann, “Adaptive spatial control of fiber modes and their excitation for high-harmonic generation,” Opt. Express 14, 3433–3442 (2006). [CrossRef] [PubMed]

13.

I. M. Vellekoop and C. M. Aegerter , “Scattered light fluorescence microscopy: imaging through turbid layers,” Opt. Lett. 35, 1245–1247 (2010). [CrossRef] [PubMed]

14.

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a-fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993). [CrossRef] [PubMed]

15.

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fibre trap for Raman micro-spectroscopy of single cells,” Opt. Express 14, 5779–5791 (2006). [CrossRef] [PubMed]

16.

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).

17.

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007). [CrossRef] [PubMed]

18.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley & Sons, 1991). [CrossRef]

19.

S. Bianchi and R. Di Leonardo , “Real-time optical micro-manipulation using optimized holograms generated on the GPU,” Comp. Phys. Commun. 181, 1444–1448 (2010). [CrossRef]

20.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010). [CrossRef]

21.

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

22.

G. C. Spalding, J. Courtial, and R. Di Leonardo, “Holographic optical tweezers,” in Structured Light Its Applications, D. L. Andrews, ed. (Academic Press, 2008) pp. 139–168. [CrossRef]

23.

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]

OCIS Codes
(060.2350) Fiber optics and optical communications : Fiber optics imaging
(090.1760) Holography : Computer holography
(230.6120) Optical devices : Spatial light modulators

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: October 7, 2010
Revised Manuscript: December 1, 2010
Manuscript Accepted: December 2, 2010
Published: December 22, 2010

Citation
Roberto Di Leonardo and Silvio Bianchi, "Hologram transmission through multi-mode optical fibers," Opt. Express 19, 247-254 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-247


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References

  1. Y. Wu, Y. Leng, J. Xi, and X. Li, "Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues," Opt. Express 17, 7907-7915 (2009). [CrossRef] [PubMed]
  2. D. Z. Anderson, M. A. Bolshtyansky, and B. Y. Zel’dovich, "Stabilization of the speckle pattern of a multimode fiber undergoing bending," Opt. Lett. 21, 785-787 (1996). [CrossRef] [PubMed]
  3. A. Lucensoli, and T. Rozzi, "Image transmission and radiation by truncated linearly polarized multimode fiber," Appl. Opt. 46, 3031-3037 (2007). [CrossRef]
  4. K. D. Wulff, D. G. Cole, R. L. Clark, R. Di Leonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett, "Aberration correction in holographic optical tweezers," Opt. Express 14, 4169-4174 (2006). [CrossRef] [PubMed]
  5. Y. Roichman, A. Waldron, E. Gardel, and D. Grier, "Optical traps with geometric aberrations," Appl. Opt. 45, 3425-3429 (2006). [CrossRef] [PubMed]
  6. A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Wavefront correction of spatial light modulators using an optical vortex image," Opt. Express 15, 5801-5808 (2007). [CrossRef] [PubMed]
  7. I. M. Vellekoop, and A. P. Mosk, "Focusing coherent light through opaque strongly scattering media," Opt. Lett. 32, 2309-2311 (2007). [CrossRef] [PubMed]
  8. T. Cizmar, M. Mazilu, and K. Dholakia, "In situ wavefront correction and its application to micromanipulation," Nat. Photonics 4, 388-394 (2010). [CrossRef]
  9. M. Paurisse, M. Hanna, F. Druon, and P. Georges, "Wavefront control of a multicore ytterbium-doped pulse fiber amplifier by digital holography," Opt. Lett. 35, 1428-1430 (2010). [CrossRef] [PubMed]
  10. C. Bellanger, A. Brignon, J. Colineau, and J. P. Huignard, "Coherent fiber combinig by digital holography," Opt. Lett. 33, 2937-2939 (2008). [CrossRef] [PubMed]
  11. M. Paurisse, M. Hanna, F. Droun, P. Georges, C. Bellanger, A. Brignon, and J. P. Huignard, "Phase and amplitude control of a multimode fiber beam by use of digital holography," Opt. Express 17, 13000-13008 (2009). [CrossRef] [PubMed]
  12. D. Walter, T. Pfeifer, C. Winterfeldt, R. Kemmer, R. Spitzenpfeil, G. Gerber, and C. Spielmann, "Adaptive spatial control of fiber modes and their excitation for high-harmonic generation," Opt. Express 14, 3433-3442 (2006). [CrossRef] [PubMed]
  13. I. M. Vellekoop, and C. M. Aegerter, "Scattered light fluorescence microscopy: imaging through turbid layers," Opt. Lett. 35, 1245-1247 (2010). [CrossRef] [PubMed]
  14. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, "Demonstration of a-fiber-optical light-force trap," Opt. Lett. 18, 1867-1869 (1993). [CrossRef] [PubMed]
  15. P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, "Dual beam fibre trap for Raman micro-spectroscopy of single cells," Opt. Express 14, 5779-5791 (2006). [CrossRef] [PubMed]
  16. R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 237-246 (1972).
  17. R. Di Leonardo, F. Ianni, and G. Ruocco, "Computer generation of optimal holograms for optical trap arrays," Opt. Express 15, 1913-1922 (2007). [CrossRef] [PubMed]
  18. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, (John Wiley & Sons, 1991). [CrossRef]
  19. S. Bianchi and R. Di Leonardo "Real-time optical micro-manipulation using optimized holograms generated on the GPU," Comput. Phys. Commun. 181, 1444-1448 (2010). [CrossRef]
  20. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, "Exploiting disorder for perfect focusing," Nat. Photonics 4, 320-322 (2010). [CrossRef]
  21. D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003). [CrossRef] [PubMed]
  22. G. C. Spalding, J. Courtial, and R. Di Leonardo, "Holographic optical tweezers," in Structured Light and Its Applications, D. L. Andrews, ed. (Academic Press, 2008) pp. 139-168. [CrossRef]
  23. 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]

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