Mode conversion via wavefront shaping

Anat Daniel; Xin Bing Song; Dan Oron; Yaron Silberberg

doi:10.1364/OE.26.022208

1. Introduction

When coherent light passes through a medium that exhibits an inhomogeneous index of refraction, interference between scattered-light components occurs and is manifested as random speckle patterns with an intensity distribution that appears highly disordered and stochastic. However, these complex random interferences are completely deterministic and can, therefore, be manipulated by controlling the input mode of the field coherently.

The availability of Spatial Light Modulators (SLMs) has allowed controlling this type of optical fields with many degrees of freedom and opened the way to a class of experiments and applications that deals with complex optical fields [1–12]. Exploiting the deterministic nature of light scattering, Vellekoop and Mosk [1] demonstrated coherent control of multiple scattering to generate a focus behind the turbid medium. In their pioneering work, they used an adaptive algorithm and a SLM with 1,000 degrees of control, to achieve a ~1,000-fold enhancement in the focused brightness. By adjusting the target function used as feedback, they have shown that not only were they able to enhance a single focal point in the target plane but also multiple foci simultaneously. In contrast with the more common application of wave front shaping, where the goal is to enhance one speckle grain or more [12–15], it has been recently shown that adaptive optimization of the incident wave front can also practically eliminate the intensity of light that scatters into some aperture, for example the light that is scattered into a camera, as long as the aperture is small enough [16].

Enhancing and reducing intensity by wave front shaping through scattering media over a broad area has been also demonstrated in [17–19], and while this subject is being thoroughly studied, there is also an increasing interest in manipulating the complex speckle field into other spatial modes [20,21]. One such example is the generation of reconfigurable Bessel-like beams at user defined positions through scattering media by exploiting adaptive focusing [22,23]. Another example is generating spatial modes prior to the scattering media and then using a control-SLM for sorting the input light modes into different output channels [24].

While the limits to the efficiency of focusing light into an isolated focal spot are well understood, the question of how effective the conversion of scattered field into some general mode has not been thoroughly explored. Here we generalize the concept and study the enhancement of several desired target fields of light over a region behind a scattering media, so as to examine how the achievable conversion efficiency depends on the distribution of the target field and on its size. We focus here on target fields with uniform amplitude yet with some desirable phase structure. In order to study a general mode conversion, we choose three types of phase structures that exhibit very different properties from one another: homogenous (constant phase), completely inhomogeneous (complex phase structure that is assigned according to random distribution) and slowly varying (spiral phase). We choose to work at the limit of a thin diffuser, as this allows a detailed comparison between experiments, simulations and analytical or semi-analytical derivations, which in turn provide some intuition relating to the origin of the dependence of the conversion efficiency on the desired mode.

2. Experiment

Figure 1(a) illustrates an experiment that demonstrates the enhancement of a general target field: A telescope expands the laser beam which then reflected off the SLM. The intensity of the beam on the SLM has a Gaussian distribution; hence each pixel on the SLM contributes differently to the enhancement process. The SLM which is divided into 15X20 effective blocks is then imaged on the diffuser with 5x de-magnification imaging set up. Light scattered by the diffuser creates a speckle pattern that propagates and encounter a phase mask (PP) at the far field (at the Fourier plane of a lens L1). The speckle size at the target plane is approximately 40 µm in diameter, as determined by the intensity autocorrelation of speckle pattern (Fig. 1(b)). A variable aperture (A) attached to a phase mask is positioned at the target plane and it is yet another (amplitude) degree of freedom to determine the size of the spatial mode we wish to enhance. The phase mask is altered between constant phase (no mask), spiral phase (with an angular momentum charge of 2) and random phase (a thin diffusing plate - Light Shaping Diffuser, angle: 5°, Newport). The light that passes through the phase mask is then focused by a lens onto a CCD camera (Andor Luca S.)

Fig. 1 (a) Schematic of the optical setup used for enhancing a particular mode of a light by wave front shaping (a helical mode in this case). A speckle field propagates and encounters a helical phase plate (PP) at the far field (or at the Fourier plane of lens L1, as shown). The resulting field is then focused on a CCD camera by lens L2. Using a feedback algorithm, the phases on the SLM are adjusted to maximize the focus, which corresponds to a uniform phase distribution at the output of the phase mask. A variable size aperture (A), attached to the mask, enables the control the size of the target mode. (b) Intensity autocorrelation of the speckle pattern: the size of the autocorrelation spot is approximately 12 pixels, each pixel is 5.3 µm. Therefore the full width is ~60 µm. Since we take the FWHM of the distribution as the speckle size, the diameter is smaller than the full width by a factor of about 1.5, namely ~40 µm. (c) The intensity distribution (to scale) of the scattered light before and after optimization for a particular aperture size as observed at the aperture plane (target plane). A doughnut shape which is associated with the helical phase distribution is obtained.

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A feedback algorithm [25, 26] is used to find the SLM phase pattern that maximizes the intensity at a target pixel on the CCD, that is, the pattern that maximizes a plane wave at the back surface of the phase mask. The field at the front surface of the phase mask correspondingly obtains the phase conjugate distribution of the mask. Since the SLM is based on a single liquid crystal device, its operation is limited to a single polarization. Therefore all experiments are performed with a linearly polarized input beam. Notably, this is not an intrinsic limitation of the method. Figure 1(c) shows that after the optimization process, the intensity distribution of the scattered light acquired a doughnut shape which is consistent with the helical phase structure of the spiral phase plate (with topological charge of 2) used in the experiment.

We studied both the dependence of the enhancement on the aperture size and, more importantly, on the shape of the desired mode.

Figure 2 shows the maximal enhancement factor obtained experimentally as a function of the aperture diameter for three target fields. The enhancement factor is defined as the ratio between the optimized intensity on the target CCD pixel and the initial average intensity before optimization. Every data point is taken as the average value of 4 different realizations of the scattering medium. The optimization procedure was performed using the MATLAB genetic algorithm toolbox, with the target of increasing the total intensity at a small region (typically about the size of a single speckle) on the CCD [25].

Fig. 2 Experimentally measured field enhancement (enhancement factor is defined as the ratio between the optimized intensity on the target CCD pixel and the initial average intensity before optimization) for different target modes. The enhancement factor is plotted as a function of aperture diameter for no mask (blue) helical phase mask of topological charge 2 (red) and a random phase mask (black). The upper axis indicates the aperture diameter in terms of average speckle diameter. Error bars were determined from multiple realizations of the scattering medium. Note that for helical phase, the minimum pinhole size is 150 μm, which is limited by the finite singularity dimension of the phase mask. For the constant and the random masks, the minimum pinhole size is 25 μm, which is approximately half the size of a single speckle at the pinhole plane.

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The plots exhibit the following trends: For a constant phase (i.e. no mask at all), the enhancement factor decays with aperture diameter. For helical phase, the enhancement factor increases at first, and then decreases with the aperture size. Notably, for apertures larger than 200 μm the optimization process has reached a higher value of the enhancement than that of the constant phase, and remains above it for wider apertures. The enhancement factor for a target field with quasi-random phase distribution exhibits an approximately constant value, higher than those obtained for the other phase structures.

As is clearly shown by the experiments, the attainable enhancement strongly depends on the shape of the desired output mode, as well as on the size of the optimized field. These somewhat surprising results call for a more detailed analysis of the origin of this behavior.

For further understanding of the experimental results we performed computer simulations and derived theoretical estimates of the optimal enhancements for the same three types of phase masks. The simulations are performed by averaging 50 realizations of speckle fields produced by a 64X64 random phase matrix padded with zeros to form a 256X256 matrix. This leads in the far field to a speckle field (4096 speckles in total, each about 4 cells wide). The maximal enhancement is estimated assuming 256 control elements in the SLM plane, and calculating the optimal control phases by first estimating the phase each SLM pixel contribute to the target field, and applying the conjugate phase to the appropriate control element to assure constructive interference of all contributions. The enhanced value is compared to the value without control (averaged over 100 realizations) to extract the enhancement factor.

Simulation results are shown in Fig. 3(a); they reproduce qualitatively the trends observed in the experimental measurements. Quantitatively, the experimentally observed enhancements were lower than the theoretically predicted values. One possible reason could be the non-uniform Gaussian intensity distribution of the illuminating beam on the SLM. As a result, the control elements do not contribute equally to the enhancement.

Fig. 3 Simulations (a) and theoretical estimates (b) showing the optimized enhancements for a flat field with no mask (red), a field with spiral phase of charge 2 (green) and a field with random phase structure (gray), as a function of aperture diameter S (normalized by the speckle size d). (The derivation of the random phase is not valid for small apertures (details in theoretical derivation section). The enhancement factor is defined as the ratio between the optimized intensity on the target pixel and the initial average intensity before optimization.

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3. Theoretical derivation

The experiments utilize thin diffusers. This enables us not only to simulate the system response but also to derive an analytical expression to estimate the expected enhancement. These estimates are based on Fourier analysis of the propagation of the speckle field, as detailed below, and the results are shown in Fig. 3(b). As can be seen, the analytical model agrees well with the simulation results. Notably, the model captures the non-trivial dependence of the maximal enhancement on the aperture size for the three different targets, and it can help us to get some insight into this behavior.

We consider an SLM with $N_{c}$ control elements in front of a diffuser with aperture D, as shown in Fig. 4(a). We wish to control the field shape across an aperture, assumed here circular with diameter S, at a distance L. The target field is phase conjugated by a phase mask M and the goal is to maximize the DC component (that is, the light at the focal point P of a Fourier Transform lens as shown).

Fig. 4 (a) The simplified geometry for estimating the maximal possible enhancement. (b) To estimate the number of effective control elements and their contributions, we consider a wave propagating backwards from the target to the SLM plane. The distribution of light on the SLM is determined by the convolution of the phase mask and the aperture. For a constant phase, as illustrated here, as the aperture is closed the diffracted beam broadens, covering more pixels on the SLM, enabling higher enhancement.

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Field with a flat phase – no mask

In the simplest case, we wish to maximize a uniform field over the aperture S (that is, no mask). When S is completely open, the system images one point on the scatterer to the focal point, hence only one element of the SLM is effective, and no control is possible. More control is possible for smaller apertures. This is easier to see if we consider a wave propagating backwards from P towards the scatterer, as shown in Fig. 4(b). As the aperture S is closed, the diffraction point on the scatterer increases, so that more elements get into action, until, for a very small aperture, all elements are uniformly illuminated and we should regain the full Vellekoop-Mosk factor $F_{V M} = \frac{π}{4} N_{c}$ , where the $\frac{π}{4}$ originates from the complex- Gaussian statistics of amplitude [1,27].

To quantify this, consider the diffraction of the backward- propagating wave. This wave diffracts at the aperture to form the following field and intensity distribution at the scatterer/SLM plane:

\begin{matrix} E (q) = F {c i r c (\frac{2 r}{S})} = \sqrt{I_{0}} [\frac{2 J_{1} (x)}{x}]; \\ I (q) = I_{0} {[\frac{2 J_{1} (x)}{x}]}^{2} . \end{matrix}

With $x = π S q / λ L$ , r and q are the radial coordinates at the aperture and the SLM planes, respectively.

To estimate the maximal possible enhancement we need to integrate the fields arriving from all control elements to P. Without control, these fields arrive with random phases, while proper control will add them all in-phase. We do that by counting the elements included in each ring around the optical axis, weighted by the field at that radius. Assuming, for simplicity a circular SLM with radius $q_{m} = D / 2$ , the density of control elements is $N_{c} / π q_{m}^{2} .$ Without optimization, the intensity is the result of addition of phasors with random phases yielding:

\begin{matrix} I_{r p} = \frac{4}{π} \frac{N_{c}}{π q_{m}^{2}} 2 π \int_{0}^{q_{m}} q I (q) d q \\ = \frac{4}{π} \frac{N_{c}}{π q_{m}^{2}} 4 π {(\frac{λ L}{π S})}^{2} [1 - J_{0}^{2} (\frac{π S q_{m}}{λ L}) - J_{1}^{2} (\frac{π S q_{m}}{λ L})] \\ = \frac{4}{π} 4 N_{c} {(\frac{d}{S})}^{2} [1 - J_{0}^{2} (\frac{S}{d}) - J_{1}^{2} (\frac{S}{d})] . \end{matrix}

We have used $J_{0}^{'} = - J_{1}$ and $J_{1}^{'} = J_{0} - J_{1} / x$ , and defined $d = λ L / (π q_{m})$ which is approximately the average speckle size at the aperture plane. The optimal enhanced field is obtained by aligning all phasors:

\begin{matrix} E_{o p t} = \frac{N_{c}}{π q_{m}^{2}} 2 π \int_{0}^{q_{m}} q | E (q) | d q \\ = 4 N_{c} {(\frac{d}{S})}^{2} [1 - 2 J_{0} (z_{1}) + 2 J_{0} (z_{2}) - \dots J_{0} (\frac{S}{d})] . \end{matrix}

Here $z_{1}, z_{2} \dots$ are the zeros of $J_{1} (x)$ and the sum is over all zeros smaller than $S / d$ . We then get for the enhancement with optimal phases:

F_{f} = \frac{{| E_{o p t} |}^{2}}{I_{r p}} = \frac{π}{4} 4 N_{c} {(\frac{d}{S})}^{2} \frac{{[1 - 2 J_{0} (z_{1}) + 2 J_{0} (z_{2}) - \dots J_{0} (\frac{S}{d})]}^{2}}{[1 - J_{0}^{2} (\frac{S}{d}) - J_{1}^{2} (\frac{S}{d})]} .

This is shown as the red line in Fig. 3(b). We note that for small apertures ( $S ≪ d)$ one gets $F_{f} = F_{V M}$ . This happens when the aperture is smaller than a speckle at the target plane, and then the optimum is achieved by simply enhancing the intensity in this single speckle, which reproduces the Vellekoop-Mosk factor.

Field with a spiral phase structure

Consider now a mask with phase vortex of charge 2, i.e. $\exp (2 i θ)$ . The field on the SLM is now

E (q, φ) = F {T (r, θ)} = F {c i r c (\frac{2 r}{S}) \exp (i 2 θ)} .

Since the expansion of the integrand $T (r, θ) = \sum^{} f_{n} (r) \exp (- i n θ)$ contains in this case only one non zero term n = 2, we get [28]

E (q, φ) = - 2 π \exp (2 i φ) \int_{0}^{S / 2} J_{2} (\frac{2 π r q}{λ L}) r d r .

Using

\int J_{2} (x) x d x = - 2 J_{0} - x J_{1},

we get:

E (q, φ) = - 2 π \exp (i 2 φ) {(\frac{λ L}{2 π q})}^{2} [2 - 2 J_{0} (\frac{π s q}{λ L}) - \frac{π s q}{λ L} J_{1} (\frac{π s q}{λ L})],

hence,

\begin{matrix} E_{o p t} = \frac{N_{c}}{π q_{m}^{2}} 2 π \int_{0}^{q_{m}} q | E (q) | d q \\ = \frac{N_{c}}{π q_{m}^{2}} {(2 π)}^{2} {(\frac{λ L}{2 π})}^{2} \int_{0}^{S / d} x^{- 1} | 2 - 2 J_{0} (x) - x J_{1} (x) | d x . \end{matrix}

While:

\begin{array}{l} I_{r p} = \frac{4}{π} \frac{N_{c}}{π q_{m}^{2}} 2 π \int_{0}^{q_{m}} q I (q) d q = \\ = \frac{4}{π} \frac{N_{c}}{π q_{m}^{2}} 8 π^{3} {(\frac{λ L}{2 π})}^{4} {(\frac{d}{S})}^{2} \int_{0}^{S / d} x^{- 3} {[2 - 2 J_{0} (x) - x J_{1} (x)]}^{2} d x, \end{array}

the enhancement factor in this case is then:

F_{s} = \frac{{| E_{o p t} |}^{2}}{I_{r p}} = \frac{π}{4} 2 N_{c} {(\frac{d}{S})}^{2} \frac{{[\int_{0}^{S / d} x^{- 1} | 2 - 2 J_{0} (x) - x J_{1} (x) | d x]}^{2}}{\int_{0}^{S / d} x^{- 3} {| 2 - 2 J_{0} (x) - x J_{1} (x) |}^{2} d x} .

We note that for small apertures (

S ≪ d)

one gets

F_{s} = \frac{π}{4} \frac{3 N_{c}}{4}

(i.e. ¾ of the Vellekoop-Mosk factor). The enhancement

F_{s}

is shown as the green line in Fig. 3(b). Experimental corroboration of this limit is, however, inaccessible, since the intensity at the node of the spiral phase rapidly decreases for small apertures.

Fields with random phase structures

If the target field has a random structure of phases, evenly distributed between 0 and $2 π$ , we can use the same approach to find the average enhancement. The back-propagating field on the SLM can be estimated then to be a speckle field with intensity and field probability distributions of:

\begin{array}{l} P (I) = \frac{1}{2 σ^{2}} e x p (- \frac{I}{2 σ^{2}}), \\ P (| E |) = \frac{| E |}{σ^{2}} e x p (- \frac{E^{2}}{2 σ^{2}}), \end{array}

where

σ = \sqrt{〈 I 〉 / 2}

and

〈 \cdot 〉

stands for averaging over statistical realizations. Hence

〈 E_{o p t} 〉 = 〈 2 π \frac{N_{c}}{π q_{m}^{2}} \int^{​} | E (q, φ) | q d q 〉 = N_{c} 〈 E 〉 = N_{c} \int^{​} E P (E) d E = \frac{N_{c} σ}{2} \sqrt{2 π} .

While for un-optimized SLM we get:

I_{r p} = 〈 2 π \frac{4}{π} \frac{N_{c}}{π q_{m}^{2}} \int^{​} I (q, φ) q d q 〉 = \frac{4}{π} N_{c} 〈 I 〉 = \frac{4}{π} N_{c} \int^{​} I P (I) d I = \frac{4}{π} N_{c} 2 σ^{2} .

From these we get the enhancement factor:

F_{r} = \frac{{| E_{o p t} |}^{2}}{I_{r p}} = {(\frac{π}{4})}^{2} N_{c} .

Note that in this case, the enhancement is predicted to be independent of the aperture size, as long as the aperture is significantly larger than the phase correlation length of the speckle field, corresponding roughly to the size of a single speckle grain.

4. Discussions and conclusions

It is known that the attainable enhancement depends on the number of control elements that take part in the optimization. In the simple situation considered by Vellekoop and Mosk [1], $N_{c}$ control elements, each contributing equally to the enhanced field, lead to a maximal enhancement factor of $F_{V M} = π N_{c} / 4$ . As we have discussed above, when the target is to match a particular mode over some area, the number of contributing control elements and their relative contributions depend on the target field and its size. For the case of a target plane-wave with a flat phase, the field on the SLM is simply the backward-diffraction pattern of the aperture as demonstrated in Fig. 4(b).

Similar convolution-diffraction reasoning have been applied for other modal structures. It is clear, for example, why it is easier to optimize a field with random patterns (e.g. a randomly changing phase pattern) than a plane wave. The light from most of the SLM elements reaches the target pixel, regardless the aperture size; therefore, the enhancement factor is nearly constant because the same numbers of control elements are contributing to the optimizations process. This description breaks down when the aperture size is of the order of the speckle grain size or smaller. In that case, the optimum is obtained by maximizing one speckle onto the aperture, as in the original Vellekoop-Mosk setup. In our experiment, the speckle size was about 40 μm (see Fig. 1(b) Note that this is not the case for a spiral phase that has a singularity at the origin which affects the system at any aperture size, as detailed in the theory section above.

It follows from the above discussion that faster phase changes in the target field should lead to improved enhancement factor. To demonstrate this point, we examine how the enhancement of a target with a spiral phase depends on its topological order. Figure 5 shows simulation results for topological charge of 1 (red), 2 (green) and 4 (blue). We note that for all topological charges (namely m = 1, m = 2, and so forth), the spatial shape of the modes has a doughnut shape, and the only difference is in the radius of the inner low intensity “core”. Specifically, the radius of the inner core increases with the increasing of the topological charge value. There is only one spatial position where there is a singularity in the phase mask (at r = 0). As the topological charge increases, the phase structure changes faster, more SLM elements contribute to the optimized field, and indeed the enhancement factor increases for apertures that are larger than a few speckles. We would expect that with increasing topological charge, the Enhancement vs. Aperture plot will approach that of a random phase structure.

Fig. 5 Optimal enhancement of phase-singular fields vs aperture radius for m = 1 (red), m = 2 (green), m = 4 (blue).

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It is important to note that the experimental method used for optimization procedure used here employs a mask which imprints the inverse of the desired phase pattern. Without detailed knowledge of the transfer matrix of the scattering medium (as was done in [12]) it is generally not possible to optimize a phase pattern using only the intensity pattern at the aperture plane. One notable exception is the case of non-diffracting beams, where optimization by imaging the (same) intensity pattern at two adjacent axially shifted planes is perhaps possible.

Our experiments and analysis used thin diffuser as the scattering medium. It is therefore reasonable to ask to which extent these results will be valid for thick diffuser. Using the same arguments as in our theoretical analysis, we can reason that these conclusions should hold as long as (1) the SLM is imaged on the input plane of the scatterer and (2) the light at the output of the diffuser stays reasonably localized, that is, it will not spread much beyond the aperture of a single SLM pixel. In our experiment, we used 15X20 pixels, which were demagnified by X5 on the scatterer. This suggests that we would have obtained similar results as long as the scatterer was thinner than about 100 micrometers.

To conclude, we have investigated the possibility of using wavefront shaping for enhancing a particular spatial mode over a given target area. We have found that the enhancement depends on both the target area size and the structure of the mode. In particular, it is harder to enhance a flat-phase field over a large aperture, as compared with fields with more complex phase structures. These observations and insights could be useful in order to expand the reach of wave front shaping and apply it to achieve more complex field shapes for various applications.

Funding

I-CORE (Israeli Centers for Research Excellence “Circle of Light” program) of the Israel Science Foundation (ISF); Weizmann Institute of Science, Crown Photonics Center; Israeli Ministry of Science, Technology, and Space (Grant No. 712845).

Acknowledgments

The authors thank Uri Rossman for helpful discussions.

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Mode conversion via wavefront shaping

Abstract

1. Introduction

2. Experiment

3. Theoretical derivation

Field with a flat phase – no mask

Field with a spiral phase structure

Fields with random phase structures

4. Discussions and conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Equations (15)

Optics Express