Hologram transmission through multi-mode optical fibers

Roberto Di Leonardo; Silvio Bianchi

doi:10.1364/OE.19.000247

1. Introduction

Optical fibers can guide a light beam across long distances or through turbid media like biological tissues [1]. 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, 3]. 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 [4–6] and strongly [7, 8] 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, 10] or a few different modes in the same core [11]. Phase modulation can be used to manipulate the spatial and spectral properties of high-harmonics in hollow fibers [12]. 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, 13] or endo-micromanipulation [8, 14, 15].

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.

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Our task is to find the best phase mask on the input plane (SLM) so that the light on the output 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 propagation kernel 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 any estimate of the propagated output field would be impractical. A direct search algorithm, where one explores the space of phase modulations guided by some experimentally measured merit function, seems to be a straightforward way to find the optimal phase mask. We choose to proceed by a Monte-Carlo search in the reciprocal space of our SLM. Calling ϕ_j the phase shift applied on the jth SLM pixel, a the fiber core size and f = f₃f₅/f₄ the equivalent focal length of lenses L3-L4-L5, only those Fourier components in exp[iϕ_j], with a wave vector k smaller than πa/λf, will be focused within the fiber core. At iteration step n we randomly choose k within that circle of allowed k vectors and compute the trial phase modulation:

ϕ_{j}^{*} = arg [(1 - ξ) exp [i ϕ_{j}^{n - 1}] + ξ exp [i (k \cdot r_{j} + θ)]]

where ξ is a random number in the interval [0, 1/2] and θ a random phase. The trial phase modulation is then displayed on the SLM and the intensity distribution emerging from the other side of the fiber is recorded on the camera. The performance of the new hologram

ϕ_{j}^{*}

is evaluated as the product Π_m I_m of target spots intensities. Finally, we update

ϕ_{j}^{n}

to

ϕ_{j}^{*}

if the performance is improved or otherwise set

ϕ_{j}^{n} = ϕ_{j}^{n - 1}

. By maximizing the product of intensities our algorithm converges towards a bright and also uniform distribution of spots. The algorithm converges after about 2500 iterations which is about the number of propagating modes in our fiber (∼ 2V²/π² = 3770 [18]). The overall optimization time is 10 minutes and it’s dominated by data acquisition on the camera. Equation (1) is computationally heavy but easy to parallelize on a commercial graphics card GPU [19]. Figure 2 shows the fraction of power coming out of the fiber that falls on a single target spot as a function of iteration step. The power on the target spot is 14% of the total output power. An estimate of the best performance we could hope to get by phase-only modulation can be obtained by numerical simulation. In particular we place a point light source on the output plane of our model optical system whose parameters (focal lengths, fiber dimensions and refractive indices, etc.) are chosen as the actual experimental values. We then propagate the light back onto the SLM plane. That amounts to performing a sequence of: i) one FFT to propagate from the camera plane to the fiber output face through lens L6; ii) end to end fiber propagation; iii) one last FFT to propagate through the combined lens system L3-L4-L5 when going from the input fiber face to the SLM plane. There we replace the amplitude with a constant value and leave the phase modulation unchanged. Finally we propagate the beam forward up to the camera plane. As a result we find that the fraction of power on the target spot depends on its location within the camera plane and is always in the 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 correct optical system aberrations [7,8,13,20]. In these algorithms the SLM array is divided into square sub-matrices with a constant phase shift. The optimal phase mask is obtained by scanning the phase of each square until constructive interference with a reference square is found on a target spot.

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.

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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 (I ∝ N²). On the other hand, when no modulation is applied, the same modes will sum up incoherently (I ∝ N). 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]. 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.

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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”.

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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.

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Holograms that result in spatially separated target spots can be combined to get multispot arrays. For example, calling ϕ^A and ϕ^B the two phase only modulations corresponding to spots in points A and B respectively, we can build the complex modulation:

u_{j}^{AB} = \sqrt{x} exp [i ϕ_{j}^{A}] + \sqrt{1 - x} exp [i ϕ_{j}^{B}]

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, 22]. 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, 23]. We might hope that this is also the case for propagation in multimode fibers and apply the phase only modulation

ϕ_{j} = arg (u_{j}^{AB})

so that the field on the SLM plane will be

u_{j} = u_{j}^{A B} / | u_{j}^{A B} |

. 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:

v^{A} = \sum_{j} G_{j}^{A} u_{j}

where

G_{j}^{A}

is the light propagator from pixel j to target point A. If we iteratively find a phase modulation

ϕ_{j}^{A}

that maximizes the light intensity through the target spot A, we might assume that a good approximation for the propagator will be given by

G_{j}^{A} \sim g_{j}^{A} exp [- i ϕ_{j}^{A}]

. Where

g_{j}^{A}

are unknown real amplitudes.

Fig. 6 Normalized intensities (v^A,B(x)²/v^A,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 ϕ^A 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.

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Therefore we can anticipate that the complex field on target A can be obtained by:

v^{A} (x) = \sum_{j} G_{j}^{A} \frac{u_{j}^{A B}}{| u_{j}^{A B} |} \sim \sum_{j} g_{j}^{A} \frac{\sqrt{x} + \sqrt{1 - x} exp [i θ j]}{| \sqrt{x} + \sqrt{1 - x} exp [i θ j] |}

where

θ_{j} = ϕ_{j}^{B} - ϕ_{j}^{A}

. The above expression is a sum of single pixel terms and therefore only depends on the distributions of θ_j and

g_{j}^{A}

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 θ_j. If, in addition, we assume that amplitudes

g_{j}^{A}

and phases

ϕ_{j}^{A}

are statistically uncorrelated, we can replace the summation in Eq. (4) with the averages:

\frac{v^{A} (x)}{v^{A} (0)} \sim \frac{1}{2 π} \int_{0}^{2 π} \frac{a + b exp [i θ]}{| a + b exp [i θ] |} d θ = \frac{1}{π} \int_{0}^{π} \frac{a + b cos θ}{\sqrt{a^{2} + b^{2} + 2 a b cos θ}} d θ =

= - \frac{a - b}{a π} E [- \frac{4 a b}{{(a - b)}^{2}}] - \frac{a + b}{a π} k [- \frac{4 a b}{{(a - b)}^{2}}]

where

a = \sqrt{x}, b = \sqrt{1 - x}

, K[⋯] and E [⋯] are the complete elliptic functions of respectively first and second kind. The intensity of spot A is then obtained as v^A(x)², v^B 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

We have demonstrated that phase only modulation can be used to shape a light beam in such a way that when coupled to a multimode fiber, the output light is distributed over an array of target spots having the size of a single speckle and arbitrary located in space. Our direct search strategy has the limit of being very sensitive to fiber stresses, but even our two meter fiber, when held stationary, can preserve the optimized target output for hours. Once a fiber is enclosed in 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 BACTMOBIL project and MIUR-FIRB project RBFR08WDBE.

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Hologram transmission through multi-mode optical fibers

Abstract

1. Introduction

2. Results and discussion

3. Conclusions

References and links

Cited By

Figures (6)

Equations (6)

Optics Express