Abstract: We demonstrate that a structured light intensity pattern canbe produced at the output of a multi-mode optical fiber by shaping thewavefront of the input beam with a spatial light modulator. We also findthe useful property that, as in the case for free space propagation, outputintensities can be easily superimposed by taking the argument of the com-plex superposition of corresponding phase-only holograms. An analyticalexpression is derived, relating output intensities ratios to hologram weightsin the superposition.
References and links1. 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).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).3. A. Lucensoli and T. Rozzi “Image transmission and radiation by truncated linearly polarized multimode fiber,”
Appl. Opt. 46, 3031–3037 (2007).4. K. D. Wulff, D. G. Cole, R. L. Clark, R. Di Leonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett “Aber-
ration correction in holographic optical tweezers,” Opt. Express 14, 4169–4174 (2006).5. Y. Roichman, A. Waldron, E. Gardel, and D. Grier, “Optical traps with geometric aberrations,” Appl. Opt. 45,
3425–3429 (2006).6. A. Jesacher, A. Schwaighofer, S. Furhapter, 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).7. I. M. Vellekoop and A. P. Mosk “Focusing coherent light through opaque strongly scattering media,” Opt. Lett.
32, 2309–2311 (2007).8. T. Cizmar, M. Mazilu, and K. Dholakia “In situ wavefront correction and its application to micromanipulation,”
Nat. Photonics 4, 388–394 (2010).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).10. C. Bellanger, A. Brignon, J. Colineau, and J. P. Huignard “Coherent fiber combinig by digital holography,” Opt.
Lett. 33, 2937–2939 (2008).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).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).13. I. M. Vellekoop and C. M. Aegerter “Scattered light fluorescence microscopy: imaging through turbid layers,”
Opt. Lett. 35, 1245–1247 (2010).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).
#135990 - $15.00 USD Received 7 Oct 2010; revised 1 Dec 2010; accepted 2 Dec 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 247
15. P. R. T. Jess, V. Garces-Chavez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, andK. Dholakia “Dual beam fibre trap for Raman micro-spectroscopy of single cells,” Opt. Express 14, 5779–5791(2006).
16. R. W. Gerchberg and W. O. Saxton “A practical algorithm for the determination of the phase from image anddiffraction 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).
18. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley & Sons, 1991).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).20. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk “Exploiting disorder for perfect focusing,” Nat. Photonics 4,
320–322 (2010).21. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).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. 139–168.23. 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).
1. Introduction
Optical fibers can guide a light beam across long distances or through turbid media like bio-logical tissues [1]. The total intensity of the output light can be easily modulated on the inputside when serial information has to travel along the fiber. However multimode fibers can prop-agate a light beam carrying a much larger information encoded in the complex coefficients ofits expansion in the propagating modes. The main obstacle in using such a set of degrees offreedom comes from the fact that the phases of modes’ amplitudes are rapidly shuffled uponpropagation [2, 3]. As a result, one always ends up having a random speckle pattern at a multi-mode fiber output. In this sense a multimode fiber can be thought of as a strongly aberratingoptical element. Spatial light modulators (SLM) have been shown to be extremely useful forcorrecting aberrations both in weakly [4–6] and strongly [7,8] aberrated optical systems. In thecase of LMA fibers, phase modulation has been already used to maximize the output signal offiber lasers by coherently adding the light coming from a few monomodal cores [9,10] or a fewdifferent modes in the same core [11]. Phase modulation can be used to manipulate the spa-tial and spectral properties of high-harmonics in hollow fibers [12]. In this last paper, geneticalgorithms have been used to modulate the output speckle pattern with large scale intensitymasks.
However multimode fibers can propagate thousands of modes that when combined in ran-dom superposition give rise to a large number of speckles. It is natural then to ask whethersuch a structured noisy pattern could be shaped with a spatial resolution of a single specklesize. One or few, diffraction limited spots could be delivered at a fiber output and dinamicallyreconfigured. That possibility would be particularly relevant for in vivo biological applications,where multimode fibers could penetrate through highly turbid biological tissues to performendomicroscopy [1, 13] or endo-micromanipulation [8, 14, 15].
In this paper we demonstrate that phase only modulation can be used to shape a light beamin such a way that, after propagation along a multimode fiber, most of the outgoing light willflow through one or few target spots having the size of a single speckle and arbitrarily locatedin 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 deriveand experimentally validate an analytical expression providing the actual fraction of power thatfalls on a given spot produced by a superposition hologram.
#135990 - $15.00 USD Received 7 Oct 2010; revised 1 Dec 2010; accepted 2 Dec 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 248
532 nm
SLM
CMOS
L1L2
L3L4 L5
L6P
D
fiber
Fig. 1. Schematic view of the experimental setup. L1-L4 planoconvex lenses; L5, L6 fibercollimators; P polaroid; D iris diaphragm.
2. Results and discussion
A schematic view of our experimental setup is shown in Fig. 1. The laser beam (CrystaLaserCL-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 coreof a multimodal fiber (Thorlabs AFS105/125Y, NA=0.22, length=2 m). Outcoming light is col-lected by a collimating lens and sent on a CMOS camera (Prosilica GC1280). Both lensesL5 and L6 have a numerical aperture of 0.25 that is slightly larger than that of the fiber. TheSLM is located on the Fourier plane of the fiber input. The modulated beam is on the firstdiffraction order of a linear grating while unmodulated light, propagating on the zeroth order, isblocked by the diaphragm (D). Working on the first diffraction order will allow us to efficientlysuperimpose independently obtained holograms without taking into account interference withunmodulated light on the zeroth order. Our fiber doesn’t preserve polarization and we only de-tect the linearly polarized component emerging from the analyzer polaroid P. In the absence ofthe analyzer, two output beams with orthogonal polarizations have to be shaped simultaneously,which makes our task much harder, although still feasible in principle.
Our task is to find the best phase mask on the input plane (SLM) so that the light on theoutput plane (camera) is mostly delivered onto an array of chosen target spots. A Gerchberg-Saxton-like algorithm would be a good choice but it requires the knowledge of the propagationkernel between input and output, as in the case of free space propagation [16, 17]. However,light propagation in a multimodal fiber is so sensitive to tiny external perturbations that anyestimate of the propagated output field would be impractical. A direct search algorithm, whereone explores the space of phase modulations guided by some experimentally measured meritfunction, seems to be a straightforward way to find the optimal phase mask. We choose toproceed by a Monte-Carlo search in the reciprocal space of our SLM. Calling φ j the phase shiftapplied on the jth SLM pixel, a the fiber core size and f = f3 f5/ f4 the equivalent focal lengthof lenses L3-L4-L5, only those Fourier components in exp[iφ j], with a wave vector k smallerthan πa/λ f , will be focused within the fiber core. At iteration step n we randomly choose k
#135990 - $15.00 USD Received 7 Oct 2010; revised 1 Dec 2010; accepted 2 Dec 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 249
Fig. 2. Single spot optimization. Power on the target spot (normalized to total outputpower) is plotted as a function of iteration step for four optimization runs (colored solidlines). An average over the four runs is displayed as a black solid line and clearly evidencesa convergence after 2000 iterations.
within that circle of allowed k vectors and compute the trial phase modulation:
φ ∗j = arg
[(1−ξ )exp[iφ n−1
j ]+ξ exp[i(k · r j +θ)]]
(1)
where ξ is a random number in the interval [0,1/2] and θ a random phase. The trial phasemodulation is then displayed on the SLM and the intensity distribution emerging from theother side of the fiber is recorded on the camera. The performance of the new hologram φ∗
j isevaluated as the product ∏m Im of target spots intensities. Finally, we update φn
j to φ ∗j if the
performance is improved or otherwise set φnj = φ n−1
j . By maximizing the product of intensitiesour algorithm converges towards a bright and also uniform distribution of spots. The algorithmconverges after about 2500 iterations which is about the number of propagating modes in ourfiber (∼ 2V 2/π2 = 3770 [18]). The overall optimization time is 10 minutes and it’s dominatedby data acquisition on the camera. Equation (1) is computationally heavy but easy to parallelizeon a commercial graphics card GPU [19]. Figure 2 shows the fraction of power coming out ofthe fiber that falls on a single target spot as a function of iteration step. The power on the targetspot is 14% of the total output power. An estimate of the best performance we could hope toget by phase-only modulation can be obtained by numerical simulation. In particular we placea point light source on the output plane of our model optical system whose parameters (focallengths, fiber dimensions and refractive indices, etc.) are chosen as the actual experimentalvalues. We then propagate the light back onto the SLM plane. That amounts to performing asequence of: i) one FFT to propagate from the camera plane to the fiber output face through lensL6; ii) end to end fiber propagation; iii) one last FFT to propagate through the combined lenssystem L3-L4-L5 when going from the input fiber face to the SLM plane. There we replacethe amplitude with a constant value and leave the phase modulation unchanged. Finally wepropagate the beam forward up to the camera plane. As a result we find that the fraction ofpower on the target spot depends on its location within the camera plane and is always in the
#135990 - $15.00 USD Received 7 Oct 2010; revised 1 Dec 2010; accepted 2 Dec 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 250
Fig. 3. a) Random speckle pattern on the fiber output without phase modulation. b) Detailof the full 768x768 optimized phase modulation. The linear grating, shifting the beam awayfrom the zero order, has been subtracted. c) When applying the optimized phase modulationa single spot appears on the fiber output. d) Intensity profile across the target spot with(black line) and without (blue line) phase modulation.
range of a few tens of percent. Those values compare pretty well with the observed efficiencies.Other direct search algorithms have been used to focus light through turbid media or correctoptical system aberrations [7,8,13,20]. In these algorithms the SLM array is divided into squaresub-matrices with a constant phase shift. The optimal phase mask is obtained by scanning thephase of each square until constructive interference with a reference square is found on a targetspot.
In Fig. 3 we show an optimized spot as compared to the speckle pattern obtained with anunmodulated beam. We observe a peak intensity which is about 35 times larger than the aver-age nearby speckles in the unmodulated case (Fig. 3d). One might expect that for an optimalphase modulation, all the N modes contributing to the intensity I at the target spot will interfereconstructively (I ∝ N2). On the other hand, when no modulation is applied, the same modes willsum up incoherently (I ∝ N). As a consequence, the brightness of the target spot is expected toincrease roughly as the number of contributing modes. In particular, if the unmodulated com-plex amplitudes of the modes have a circular Gaussian distribution, the expected enhancementis found to be π(N−1)/4+1 [7]. However, for each target spot, only a fraction of the availableguided modes will be contributing to the total intensity. For example, when the target spot islocated on the axis of an ideal fiber, only the l = 0 modes will contribute. In our fiber therewill be 43 modes with l = 0 which gives an expected enhancement of 34, which compares sur-
#135990 - $15.00 USD Received 7 Oct 2010; revised 1 Dec 2010; accepted 2 Dec 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 251
Fig. 4. Multiple target optimization. Incoming light is modulated so that the light thatcomes 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 longmultimode fiber by modulating the incoming beam with a time sequence of phase masks.
prisingly well with the observed factor 35. It is worth noting that our CMOS camera (ProsilicaGC1280) has an intensity threshold such that light with an intensity lower than this thresholdis not detected. For this reason, at a shutter speed that avoids saturation on the target spot, thebackground speckles are not detected. To overcome this issue we artificially increase the bitdepth of our camera by collecting multiple frames at different shutter speeds and than mergingthe frames into a single picture.
If we aim to an array of spots we will find out that their intensities depends on the numberof targets as well as on their geometry. We qualitatively observe that, while the average tar-get intensity obviously decreases when increasing the number of targets, the total light powerflowing through all the spots increases slightly. We report in Fig. 4 the result of a simultaneousoptimization of 17 targets arranged to form the letters ”cnr”. Such multispot targets can also bedisplayed dynamically on the SLM to transmit a holographic movie across our two meters longmulti mode fibers. Figure 5 shows some frames from a movie of a spinning square coming outof the fiber.
Holograms that result in spatially separated target spots can be combined to get multispotarrays. For example, calling φA and φB the two phase only modulations corresponding to spotsin points A and B respectively, we can build the complex modulation:
uABj =
√xexp[iφA
j ]+√
1− xexp[iφBj ] (2)
that would result in two simultaneous spots with a fraction x of total intensity going in Aand the remaining 1− x in B. However u will in general correspond to an amplitude and phasemodulation while we can only apply a phase only modulation on the SLM. A similar problem
#135990 - $15.00 USD Received 7 Oct 2010; revised 1 Dec 2010; accepted 2 Dec 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 252
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 dashedline is the expectation for full phase and amplitude modulation.
is encountered in holographic optical tweezers where phase only holograms for a single trapare easily computed [21, 22]. In that context it has been found that for multiple traps, by sim-ply neglecting the amplitude modulation, one gets an intensity distribution that is close to thesuperposition of the separate hologram intensities [17, 23]. We might hope that this is also thecase for propagation in multimode fibers and apply the phase only modulation φ j = arg(uAB
j )
so that the field on the SLM plane will be uj = uABj /|uAB
j |. Indeed, such a phase only modu-lation results in a double spot output as shown in Fig. 6, where the intensity of the two spotsas a function of x is reported with open circles. The shape of the two curves seems to be veryclean and reproducible suggesting a robust statistical averaging. At our low power levels, thecomplex field on target spot A will be a linear combination of the complex field on SLM pixels:
vA = ∑j
GAj u j (3)
where GAj is the light propagator from pixel j to target point A. If we iteratively find a phase
modulation φAj that maximizes the light intensity through the target spot A, we might assume
that a good approximation for the propagator will be given by GAj ∼ gA
j exp[−iφAj ]. Where gA
jare unknown real amplitudes.
Therefore we can anticipate that the complex field on target A can be obtained by:
vA(x) = ∑j
GAj
uABj
|uABj | ∼ ∑
jgA
j
√x+
√1− xexp[iθ j]
|√x+√
1− xexp[iθ j]|(4)
where θ j = φBj − φA
j . The above expression is a sum of single pixel terms and therefore onlydepends on the distributions of θ j and gA
j but not on their particular spatial arrangement on the
#135990 - $15.00 USD Received 7 Oct 2010; revised 1 Dec 2010; accepted 2 Dec 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 253
SLM. We experimentally found that the single spot holograms are characterized by a uniformdistribution of phase values between 0 and 2π . The same will then hold for θ j. If, in addition,we assume that amplitudes gA
j and phases φAj are statistically uncorrelated, we can replace the
summation in Eq. (4) with the averages:
vA(x)vA(0)
∼ 12π
∫ 2π
0
a+bexp[iθ ]|a+bexp[iθ ]|dθ =
1π
∫ π
0
a+bcosθ√a2 +b2 +2abcosθ
dθ = (5)
= −a−baπ
E
[− 4ab(a−b)2
]− a+b
aπK
[− 4ab(a−b)2
](6)
where a =√
x, b =√
1− x, K[· · · ] and E[· · · ] are the complete elliptic functions of respec-tively first and second kind. The intensity of spot A is then obtained as vA(x)2, vB can be ob-tained from Eq. (6) by swapping A and B. The theoretical expression is plotted in Fig. 6 showinga remarkable good agreement with experimental data. It is worth noting here that the approx-imations involved in the derivation of Eq. (6) will hold exactly for the superposition of twosingle trap holograms in holographic tweezers, therefore expression Eq. (6) could be equallywell used to choose the right weights for a desired intensity ratio between traps.
3. Conclusions
We have demonstrated that phase only modulation can be used to shape a light beam in sucha way that when coupled to a multimode fiber, the output light is distributed over an array oftarget spots having the size of a single speckle and arbitrary located in space. Our direct searchstrategy has the limit of being very sensitive to fiber stresses, but even our two meter fiber, whenheld stationary, can preserve the optimized target output for hours. Once a fiber is enclosedin a thin rigid needle, our observations could open the way towards new strategies for endo-microscopy or endo-micromanipulation. This work was supported by IIT-SEED BACTMOBILproject and MIUR-FIRB project RBFR08WDBE.
#135990 - $15.00 USD Received 7 Oct 2010; revised 1 Dec 2010; accepted 2 Dec 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 254
