1. Introduction

Coherence is a fundamental notion in the science of light and offers a central degree of freedom to manipulate electromagnetic radiation of diverse physical character [1]. The concept of spatial coherence, in particular, has had a profound impact on the foundations and development of modern optics [2], and it is manifested in countless technological applications, e.g., in imaging [3,4], tomography [5–7], beam propagation [8], nanophotonics [9,10], trapping [11–14], and free-space optical communications [15,16]. Nevertheless, the control of spatial coherence is not yet exploited to its full extent. Researchers have developed several methods to control this coherence property, either via active devices such as spinning phase diffusers [17], spatial light modulators (SLMs) [18] and digital micromirror devices (DMDs) [19], or via passive methods such as fine tuning of the optical path [20–22]. Yet, all these techniques have limitations on the attainable speed or are limited to a single pair of fields.

Recently, a new class of tools to control light has emerged that exploits the properties of optical complex media when combined with wavefront shaping devices. A complex medium is an optical system that mixes the spatial and temporal degrees of freedom of an impinging field, resulting in a scrambled intensity distribution at its output [23]. The extremely large number of internal degrees of freedom of a complex medium makes the output intensity pattern disordered, yet it remains deterministic. Therefore, it is possible to fully characterize the effect of the propagation through the medium on an incident field with a linear transmission matrix (TM) [24]. Knowledge of the TM allows complex media to be used to perform a variety of tasks, once combined with programmable modulators. Applications include the control of different properties of light, e.g., intensity [25–27], polarization [28,29] and spectrum [30–33]. In particular, complex media have been proposed as a compact, highly-dimensional multi-port device [34], e.g., to perform quantum operations [35,36]. The use of these media in combination with wavefront shaping presents a potential alternative to other platforms, in free space [37] or integrated optics [38–40], which suffer from scalability issues. Interestingly, even though both random diffusers and wavefront shaping devices, such as SLMs and DMDs, have been used for the control of the spatial coherence [17–19], they have not been employed in combination to overcome the previous limitations.

In this work, we propose a technique to control the correlations between an arbitrary number of field pairs in a single-shot fashion, based on a linear transformation applied to $n$ mutually incoherent input fields. Employing optical coherence theory, we first derive a general expression for the linear transformation that yields access to such coherence control. The linear transformation is then experimentally implemented with a complex medium in combination with SLM-based wavefront shaping. As a proof of principle, we realize a $3 \times 3-$port device and show that it generates any combination of mutual correlations, within the technical limitations.

2. Theory

In this section, after providing some background concepts, we show how to generate a set of $n$ fields with precisely controlled mutual correlations by applying an $n \times n$ linear transformation to a set of $n$ mutually incoherent input fields.

2.1 Basic definitions

Let the complex analytic signals $E_1,\ldots,E_n$ represent random, statistically stationary, quasimonochromatic electric fields at $n$ different points in space. The spatial correlation between two fields $E_i$ and $E_j$, with $i,j\in \{1,\ldots,n\}$, can be characterized via the mutual (or complex) degree of coherence [41]

To characterize the overall spatial coherence of the system, we employ the measure [43]

2.2 Coherence control with linear transformation

Let us now consider a vector $\boldsymbol {E}_{\mathrm {in}}$ that represents $n$ mutually incoherent input fields. Since in this case all mutual degrees of coherence are zero, whereupon the overall spatial coherence of the whole system is also zero, the input coherence matrix obeys $\mathbb{K}_{\mathrm {in}} = \langle \boldsymbol {E}_{\mathrm {in}} \boldsymbol {E}_{\mathrm {in}}^\dagger \rangle = \mathbb{I}$. We combine the input fields via a linear transformation $\hat {T}$, according to

Using the fact that $\hat {T}$ is deterministic and the inputs are mutually incoherent, we obtain

Therefore, it is possible to generate an arbitrary output coherence matrix upon choosing a linear transformation which fulfills

We observe from Eq. (6) that under unitary transformations, $\hat {T}\hat {T}^\dagger = \hat {T}^\dagger \hat {T} = \mathbb{I}$, the output coherence matrix always obeys $\mathbb{K}_{\mathrm {out}}=\mathbb{K}_{\mathrm {in}}=\mathbb{I}$ for an incoherent input system. The control of the output coherence thus relies on the nonunitary character of the chosen transformation. In practice, we implement the desired transformation using a multi-port linear optical device, as we describe in the following section.

3. Implementation

We implement a multi-port linear optical device by transmitting wavefront-shaped light through a complex medium. To use the medium to perform the desired transformation, we need to characterize its transmission matrix. To do that, we use a phase-only SLM to inject light into the medium in a well defined input basis, and we measure the output speckle pattern for each vector of the basis [24]. The number of degrees of freedom of the scattering layer (given by the number of the propagating modes supported by the material) is practically unlimited, hence the dimensionality of the TM is only limited by the number of pixels of the SLM and of the camera [44], both of which are typically on the order of one million. From the knowledge of this large fixed random matrix, we can employ the SLM in combination with the complex medium to implement a smaller but reconfigurable linear transformation [34]. Importantly, since the whole scattering matrix is too large to be fully characterized with the limited number of pixels of the SLM, all the uncontrolled modes can be considered lossy channels. These channels enable the implementation of non-unitary transformations with our compact system, comprising the SLM and the complex medium, hence allowing the control of the coherence matrix of the output fields. We note that, in integrated or standard free-space optics, realizing non-unitary transformations requires a large number of optical components, many of which are employed only to implement lossy channels [45].

3.1 Multi-port linear device

Here, we describe how to obtain a programmable linear transformation for coherence control using the system of SLM and complex medium. For illustration purposes, we focus on the simple case of a linear $3\times 3-$port device, which we implement experimentally.

Let us consider three mutually incoherent light beams $E^{\mathrm {in}}_1$, $E^{\mathrm {in}}_2$ and $E^{\mathrm {in}}_3$. The three non-overlapping beams reach different regions of a phase-only SLM. For each region, $N$ segments of the SLM modulate the field locally, effectively generating $N$ spatially separated modes with controlled phase. Thus, each input $E^{\mathrm {in}}_i$ undergoes a transformation $\hat {T}_{\mathrm {SLM}, i}$ with dimensions $N \times 1$. Next, the three sets of $N$ modes enter the complex medium. The output intensity emanating from a complex medium typically forms a disordered interference pattern, known as speckle pattern, resulting from the mixing of the input modes. In the case under analysis, the three sets of modes entering the medium are mutually incoherent, so they do not interfere, thus leading to the sum of three independent speckle patterns. Hence, we characterize the effect of the medium with three distinct TMs, which we denote as $\hat {T}_{\mathrm {CM}, i}$. The dimension of each $\hat {T}_{\mathrm {CM}, i}$ is $M \times N$, since they connect each set of $N$ input modes to $M$ output speckles. We want to control only three modes out of the many ones at the output. These three modes should then enclose the largest amount of the output power. Therefore, we apply a projection $\hat {P}$ (of size $3 \times M$) to each $\hat {T}_{\mathrm {CM}, i}$ to select only the output speckles we are interested in, while zeroing out the intensity of the rest.

The overall operation transforms each input beam into three outputs, according to the relation $[ E^{\mathrm {out}}_{1,i}, E^{\mathrm {out}}_{2,i}, E^{\mathrm {out}}_{3,i} ]^\intercal = \hat {P}\hat {T}_{\mathrm {CM}, i} \hat {T}_{\mathrm {SLM}, i} E^{\mathrm {in}}_i$, for $i = 1,2,3$. We then sum the independent speckle patterns to get the final result:

Since we know from Eq. (7) the target coefficients $t_{ij}$ to obtain the desired output coherence matrix, we only need to invert Eq. (9) to find the transformation to be implemented with the SLM. In practice, we apply a phase conjugation, which has already been proven successful to focus light into few speckles [46]. Finally, we get the following relation:

4. Experiment

In Fig. 1 we show the experimental setup, which comprises two main blocks. The first one (preparation) generates three fields characterized by a programmed coherence matrix, while the second block (verification) verifies that the encoded degrees of coherence correspond to the desired ones.

We use two $512\times 512$ pixel spatial light modulators (Meadowlark Optics P512). In the preparation stage, we use a first SLM (SLM1) to modulate three mutually incoherent input lasers (Thorlabs HRP050 and Meredith Instruments $633\,\mathrm {nm}$ HeNe lasers, and $\approx 650\,\mathrm {nm}$ FOSCO BOB-VFL650-10, see Supplement 1 for more details). Next, we focus them onto an optical complex medium (ground glass diffuser, Thorlabs DG10-1500). The SLM1 and the scattering medium together form the programmable multi-port linear device. Through wavefront shaping, the light scattered by the medium and collected by a lens forms three output beams.

In the verification stage, we use a second SLM (SLM2) to modulate the phase of the beams before a lens. This allows us to control which beam is focused onto the camera plane and to which location. From the interference patterns measured with the camera (Basler acA640-750um), we reconstruct the mutual degree of coherence. In the following sections, we describe the procedures used to encode and measure the coherence matrix of the output fields.

 

Fig. 1. Experimental setup. We employ three different lasers as mutually incoherent inputs. The three lasers are modulated by a phase-only SLM, then they are focused onto a complex medium (ground glass diffuser) by a lens. The propagating beams are mixed by the complex medium and then collected by another lens. Through wavefront shaping, we obtain three output beams with the desired coherence matrix. The three output beams are focused by a third lens and interfere in the camera plane. A second SLM is used to characterize the degrees of coherence from the interference patterns. $L_1$, $L_2$, $L_3$: lenses; $\textrm {CM}$: complex medium; $\textrm {CAM}$: camera.

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4.1 Preparation

The transmission matrix of the complex medium must be characterized to employ it as a part of the reconfigurable multi-port linear device. Each element of the TM connects the field modulated by the $n$th pixel of SLM1 to the complex field of the $m$th output mode (a camera pixel used for TM characterization). We reconstruct the TM by configuring the SLM with each vector of a complete basis of the input modes, and measuring the corresponding complex fields at the output camera. Having at our disposal phase-only SLMs, we choose the Hadamard basis, which maximizes the measured intensity and increases the signal-to-noise ratio of the reconstruction [24]. Moreover, we measure the phase of the outputs with an interference measurement, employing part of the SLM to provide a static reference field [24,47].

As discussed in Sec. 3.1, the speckle patterns generated by each input field are mutually incoherent, thus they do not interfere. Therefore, we assign a different transmission matrix $\hat {T}_{\mathrm {CM}, i}$ to each of the three non-overlapping input beams. Each laser is spatially phase modulated by a different quadrant consisting of $256\times 256$ pixels of SLM1, out of a total of $512\times 512$ pixels. The outer part of each quadrant is used as a static reference for the interference measurement, while we use an area of $128\times 128$ pixels (divided into $4096$ square segments of 4 pixels each) to encode the Hadamard basis employed in the TM reconstruction. Once we have reconstructed the TM for each input laser, we can implement any desired linear transformation according to Eq. (10). We refer the reader to Supplement 1 for a thorough characterization of the multi-port linear device.

4.2 Verification

The multi-port linear device described above is able to encode any desired coherence matrix. To verify the correctness of the encoding, we measure each entry of the coherence matrix, that is the mutual degree of coherence $\gamma _{ij}$ of each field pair. The measurement of $|\gamma _{ij}|$ can be carried out from the relations [41]

In Fig. 2, we summarize the procedure employed to measure the degree of coherence. Firstly, we use SLM2 to apply a linear phase grating to two of the three output beams [48]. In the focal plane, which corresponds to the camera plane, the phase grating spatially displaces the two beams, allowing us to measure the intensity of the remaining one. In Fig. 2(a) and Fig. 2(b), we show the intensity distribution of the first and the second beam, respectively, when the other two are displaced. We then use the phase grating to displace only one beam, and let the other two interfere, leading to the typical sinusoidal modulation across the area of the camera, as shown in Fig. 2(c) and Fig. 2(d). In particular, Fig. 2(c) shows the intensity distribution when the mutual degree of coherence of the two interfering beams is low ($\gamma _{ij} = 0.2$), while Fig. 2(d) shows the case of high degree of coherence ($\gamma _{ij} = 0.8$). Figure 2(e) shows a cross-section of the interference fringes for three degrees of coherence ($\gamma _{ij} = 0.2, 0.5, 0.8$). The modulation depth increases for higher degrees of coherence, as expected. Next, we modulate the phase of one of the beams (termed reference phase later) from $0$ to $2\pi$. This modulation results in a spatial shift of the interference fringes. Thus, we are able to measure the visibility at each pixel of the camera. Figure 2(f) shows examples of the intensity at a camera pixel with respect to the reference phase for three different degrees of coherence ($\gamma _{ij} = 0.2, 0.5, 0.8$). From the visibility and the intensities of the single beams, we reconstruct the degree of coherence at a specific location, according to Eq. (11). As the reconstruction of the degree of coherence is noisier for regions of low intensities, we choose to only consider pixels where both the single-beam intensities are above 60% of their maximum value. We repeat the previous procedure for all the considered pixels, and average the results to obtain the reconstructed degree of coherence $\gamma _{ij}$.

 

Fig. 2. Reconstruction of the degree of coherence. (a, b) Intensity $I$ of the (a) first and the (b) second beam. (c, d) Interference patterns for the degree of coherence ($\gamma$) equal to (c) 0.2 and (d) 0.8. (e) Cross-sections of the intensity distributions for different values of $\gamma$. (f) Normalized intensity distribution at a fixed pixel as a function of the phase of one of the interfering beams (reference phase), which is swept from 0 to $2\pi$. The different coloured dots correspond to measurements taken for different degrees of coherence. The solid lines are the cosine fits of the data points. (g) Example of degree of coherence control. We show the reconstructed degrees of coherence ($\gamma _{\mathrm {rec}}$) with respect to the encoded ones ($\gamma _{\mathrm {enc}}$). We choose the degrees of coherence of the three pairs of fields ($\gamma _{12}$, $\gamma _{13}$ and $\gamma _{23}$) to be equal.

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Inevitably, the encoding of a chosen coherence matrix is subject to errors, leading to a discrepancy between the encoded and the reconstructed $\gamma _{ij}$. To minimize this discrepancy, we implement a gradient descent algorithm to optimize the multi-port linear device for minimum error (see Supplement 1 for details). The gradient descent is performed a single time to achieve better calibration of the linear port. The phase masks obtained with this final procedure can be then employed without further calibration as long as the complex medium is stable (which in the case of ground glass diffusers is only limited by the pointing stability of the lasers [34]).

Figure 2(g) shows an example of the achieved precision in the control of the degree of coherence. Here, we report the reconstructed degrees of coherence $\gamma _{\mathrm {rec}}$, with respect to the encoded values $\gamma _{\mathrm {enc}}$. We encoded coherence matrices with identical degrees of coherence between each field pair, i.e., $\gamma ^{\mathrm {enc}}_{12}=\gamma ^{\mathrm {enc}}_{13}=\gamma ^{\mathrm {enc}}_{23}$, ranging from 0 to 1. The reconstructed and the encoded degrees of coherence agree to within an average error of 0.004 in the region between 0.2 and 0.8. Outside this range, we observe deviations from the expected behaviour. For low coherence, the measurement of $\gamma _{ij}$ is affected by the background noise caused by the uncontrolled modes of the complex medium, whereas for high coherence we are limited by the self-coherence of the input lasers (see Supplement 1).

5. Results

We now show the level of control over the coherence matrix achievable with the presented implementation. The coherence matrix is completely defined by its off-diagonal values, which are the mutual degrees of coherence of the field-field pairs. We can therefore assign to each coherence matrix a vector $\pmb {\gamma } = [\gamma _{12}, \gamma _{13}, \gamma _{23}]$, and visualize all the possible vectors in a space where the axes are the magnitudes of the mutual degrees of coherence. Note that the positive semi-definiteness of the coherence matrix bounds the domain of allowed vectors (see Supplement 1 for details). We discretize the three-dimensional space in a cubic grid with a step size of 0.1. In Fig. 3(a), we show the experimentally achieved coherence matrices. The blue (red) dots represent the measured (encoded) vectors. We restrict ourselves to the region of degrees of coherence between 0.2 and 0.8, where the reconstructed coherence matrices do not have a large deviation with respect to the encoded ones, caused by technical limitations (see Fig. 2(g) and Supplement 1). To graphically make more evident the typical distance between encoded and reconstructed coherence matrices, we show in Fig. 3(b) one of the orthographic projections of the three-dimensional space. The blue dots represent the reconstructed vectors, while the red circles with a radius of 0.025 are centered on the encoded matrices, and provide a visual reference.

As a quantitative measure for the accuracy of our coherence matrix control scheme, we define, for each measured point, the error $\varepsilon$ as the root-mean-square distance between the encoded and the reconstructed vector in the space of the coherence matrices:

Here the subscript $i$ indicates the field-field pairs $i = \{1,2\}, \{1,3\}, \{2,3\}$. Figure 3(c) shows the histogram of the errors. For the majority of the coherence matrices, the error is below 0.01, which is the threshold value set in the gradient descent optimization. The mean value of the error is 0.01.

 

Fig. 3. Coherence matrix control. (a) Coherence matrix space. Each point of the space represents a different coherence matrix, for which the off-diagonal elements are given by the three coordinates of the point $[\gamma _{12}, \gamma _{13}, \gamma _{23}]$. The blue and the red dots are the encoded and reconstructed coherence matrices, respectively. The encoded degrees of coherence range from 0.2 to 0.8. (b) One of the 2-dimensional orthographic projections of the three-dimensional space. The blue dots represent the measured vectors, while the red circles with radius 0.025 are centered on the encoded vectors. (c) Histogram of the error $\varepsilon$, i.e., the distance between the reconstructed and the encoded values. (d) Histogram of the statistical error $\sigma$. For each encoded coherence matrix, we repeat the reconstruction of the three degrees of coherence 10 times. The statistical error $\sigma$ is calculated as the standard deviation of each ensemble. (e) Encoded overall coherence $\mathcal {S}_{\mathrm {enc}}$ vs. reconstructed one $\mathcal {S}_{\mathrm {rec}}$. The blue dots are the measured values, while the dashed line represents the ideal relation.

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Next, we characterize the statistical error associated with the measured $\pmb {\gamma }$. To do so, we repeat the previously described reconstruction procedure 10 times for each vector of the space. We then estimate the statistical error $\sigma$ as the standard deviation of the measured ensemble. In Fig. 3(d) we plot the histogram of $\sigma$. The average statistical error is 0.008. This justifies the chosen threshold in the optimization algorithm.

Finally, we characterize the whole system of fields with a single quantity, i.e., the overall coherence. The overall coherence $\mathcal {S}$ is a real number ranging from 0 (full incoherence) to 1 (full coherence) and measures the coherence of the whole system, independently of how it is shared between the degrees of coherence (see Sec. 2.1). In Fig. 3(e), we plot the reconstructed ($\mathcal {S}_{\mathrm {rec}}$) versus encoded ($\mathcal {S}_{\mathrm {enc}}$) overall coherence, computed from all the measured vectors shown in Fig. 3(a). The average error (defined as $\left | \mathcal {S}_{\mathrm {enc}} - \mathcal {S}_{\mathrm {rec}} \right |$) over all the measurements is 0.003.

6. Discussion

In summary, we have presented a technique to program the coherence matrix of a set of spatially separated fields through the use of a multi-port linear optical device. We have experimentally realized a $3 \times 3$-port system, based on wavefront shaping of mutually incoherent inputs that propagate through a complex medium. By sampling the set of allowed coherence matrices, we have shown that we can encode and successfully retrieve the majority of the matrices within an average error of 0.01. Configurations with overall coherence close to 0 and 1 are not obtainable with our implementation. Nevertheless, we point out that those are cases in which our scheme has no advantages compared to traditional approaches. In fact, for low coherence one can directly employ the input fields. For high coherence, instead, one could use a single laser and split it with beam splitters.

Remarkably, to our best knowledge this is the first time that the spatial coherence of multiple fields is controlled in a single-shot fashion. Single-shot means that, once the correct phase mask is programmed into the SLM, the spatial coherence modulation occurs after a single propagation through the system, in contrast with previous works which rely on the collection of a large ensemble of phase masks introduced by some active device [17–19]. Moreover, our complex medium-based device can operate on many ports. In fact, in order to increase the number of controlled fields with the same performance in terms of coherence control, we only need to keep a constant background, which allows us to retain the same minimum coherence (see Supplement 1). We can achieve this by maintaining a constant number of SLM pixels for each input laser [46], for instance by employing a larger SLM.

Our work adds an important tool to the available methods for controlling the various attributes of light. Among the various potential applications [49], our work is of considerable interest for free-space optical communications, where beams with partial spatial coherence are more robust to atmospheric turbulence compared to the coherent counterpart [15,16]. Moreover, one could use the mutual degree of coherence as a physical bit, leading to a favorable quadratic scaling of the number of communication channels with the number of input fields.