## Abstract

We experimentally demonstrate the use of the transmission matrix (TM) to quantitatively control the amplitude and phase of the light transmitted through highly scattering media. This is achieved by measuring the absolute value of the TM elements. We also use the fact that the cross-correlations between the contributions of different input channels at the observation plane is important in describing the transmitted optical field. In addition, we demonstrate both quantitative control of the intensity at multiple output spatial modes, each with a different intensity, as well as a “dark” area of low intensity. Our experiments are carried out using a low cost (less than US$600) spatial binary amplitude modulator that we modify for phase-only operation, as well as a novel optical setup that enables independent control of a reference and control signal while maintaining interferometric stability. The optical implementation used in this paper will make such experiments widely accessible to many researchers. Furthermore, the results presented could serve as the foundation for many useful potential applications ranging from the biomedical sciences to optical communications.

© 2013 OSA

## 1. Introduction

The ability to control the transmission of light through highly scattering media has several potential applications ranging from imaging and actuation of thick biological systems to fabrication and characterization of three dimensional structures. Recently, these and other potential applications have led to several interesting studies [1–5]. For example, Vellekoop and Mosk [1] showed in 2007 that light can be focused through a scattering medium by iteratively modifying the light’s incident wavefront. Later it was shown that, with an appropriate feedback mechanism, light can also be focused inside the scattering medium [2]. A key insight in this work is that in the absence of any time-varying processes, scattering is deterministic. As a result, the contribution of each point from the incident field to the field measured at each point in the observation plane would also be independent of time [1, 3]. These contributions can be measured and arranged in what is known as the transmission matrix (TM) [3]. The first experimental measurement of the TM was reported in 2010 [3]. In this case, the measured elements of the TM were used to design the appropriate input field profiles necessary to focus light through a scattering medium [3]. Although a dynamic scattering medium has a time-dependent TM, recently it has been shown that when the measurement process is fast compared to the time constant of the scattering medium, the TM can be used to focus light through a dynamic scattering medium as well [4]. A goal of these and other similar studies have been to maximize the amount of light transmitted to the desired output spatial modes. However, it is well established from the scalar-wave model of light that intensity (or amplitude) and phase are two of the intrinsic properties that are required to completely describe the state of light [6]. In addition, for some applications it is necessary to exercise precise control over the intensity and/or phase of the light. For example, in photodynamic therapy, the effectiveness and side effects of the treatment depend both on the light dose and fluence rate [7]. Another example is with generating arbitrary states of polarization which requires controlling the relative complex weights of the constituent eigenpolarizations. Similarly, exotic optical field states like vector beams and optical vortexes [8–10], which have been shown to possess several interesting properties and applications [10–13], also require full control of the amplitude and phase of the input light.

In this paper, we show that the TM can be used to design input optical field profiles to control both the amplitude and phase of the light transmitted through a scattering medium. We find that doing so requires the use of the absolute value of the TM elements [5] rather than a scaled version of the TM elements [3]. In a previous report, we presented a technique for measuring the absolute value of the TM elements in [5]. However, the approach required that a separate reference beam path be used in the measurement process. In contrast, the optical setup we present in this paper allows measurement of the absolute value of the TM elements using a single incident beam path, thereby both reducing the complexity of the optical setup and improving interferometric stability. We also find that it is necessary to account for the partially coherent nature of the light transmitted through the scattering medium when quantitative control over the transmitted light is to be exercised.

The article is organized as follows. In Section 2 we describe our approach for measuring the TM elements, whereas in Section 3 we discuss the calculation of the phase images for controlling the intensity and phase of the transmitted light. We present our results in Section 4 and our conclusions in Section 5.

## 2. Measurement of the transmission matrix

Figure 1 is a schematic of our experimental setup for measuring the TM elements. A collimated, 808-nm diode laser beam [Power Technology, Inc IQ2C(808-150)] is incident on a deformable mirror device micro-mirror array (DMD-MMA) [14]. This DMD-MMA, acquired as part of the DLP LightCrafter DMD (Texas Instruments), costs less than US$600, and imparts only a binary amplitude modulation. However, measuring the TM elements requires a spatially dependent phase modulation [3–5]. To address this challenge, we use Lee’s synthetic binary holograms [15], which can be implemented through binary amplitude modulation and have been shown [4, 15] to be able to provide spatially dependent phase modulation. Thus, for a desired phase profile *g*(*x*, *y*), we implement Lee’s synthetic binary hologram *f* (*x*, *y*) using the following expression [15]

*T*is the period of the grating in the hologram. The parameter

*q*defines the duty cycle of the grating [15] and is set to 0.5 in our case. Since the DMD-MMA is a pixelated device, we calculate the hologram function

*f*(

*x*,

*y*) only at the center of each of the available micro-mirrors. The calculated binary hologram is then displayed on the DMD-MMA which results in several diffraction orders. The desired phase modulation is in the first diffraction order, which is selected by using an iris to obstruct the other orders at the Fourier plane of lens L1. The selected order is then collimated by lens L2, wherein the desired phase modulation is observed at its back focal plane. This field is then focused onto the sample S by an infinity corrected, 10X microscope objective OBJ1 (Spencer) with numerical aperture (NA) 0.25. Part of the scattered light is then collected by a second, infinity corrected objective OBJ2 (Reichert) with magnification 45X and NA of 0.66, and passed through an analyzer P (Thorlabs LPNIR100-MP) before finally being recorded by a CMOS camera (Thorlabs DCC1545M). The scattering samples were prepared by depositing a mixture of ZnO and ethanol onto the standard microscope slides which resulted in ZnO films with average thickness of 100

*μm*. Similarly deposited ZnO films have been reported in the literature to have an average mean free path of 6

*μm*[16].

To measure the TM elements, we divide the DMD-MMA into two areas, the central control area and the peripheral reference area, such that the measured matrix elements relate the input field corresponding to the control area to the field observed on the camera. The input field corresponding to the peripheral area provides the reference signal required in the four-point phase-shifting interferometry used in the measurement process. Figure 2(a) shows an example hologram that extends over the whole DMD-MMA, whereas Figs. 2(b) and 2(c) show holograms that cover only the control and reference areas, respectively. The control area is divided into 64 independently controllable segments. Since the contribution of each control segment at the input to an observation point at the output is weak, measuring the TM elements in the canonical basis results in a low signal-to-noise ratio (SNR); therefore we measure the TM elements using a Hadamard basis at the input [3, 5]. The measured TM elements are of the form

where ${I}_{n,m}^{C}$ and ${I}_{n,m}^{R}$ are the intensities of the control and reference signals at the*n*th observation point corresponding to the

*m*th Hadamard basis element input. Similarly,

*φ*is the phase difference between the control and reference signals. The cross-correlation term [6] |

_{n,m}*g*| defines the correlation between the reference signal and the control signal at the

_{nm}*n*th observation point corresponding to the

*m*th Hadamard basis element input. In the previous studies [3–5], this term was not considered. However, we find its inclusion to be crucial in exercising quantitative control over the intensity and phase of the transmitted light. We note that in our experiments each observation point or region, unless otherwise specifically stated, is square-shaped with a side length of 10.4

*μm*(equivalent to two physical pixels of DCC1545M on each side) and is smaller than the speckle size.

Ideally, the TM elements should not depend on the reference signal. However, Eq. (2) shows that the measured TM elements depend on both the reference and control signals. To filter out the effect of the reference signal, we measure its strength at each of the observation points and divide the TM elements obtained from Eq. (2) by the square root of the measured reference signal. Measurement of the strength of the reference signal is achieved by turning off the control area on the DMD-MMA using the hologram shown in Fig. 2(c) and subsequently measuring the intensity of the transmitted optical field. This allows us to measure the absolute values of the TM elements

which now does not depend upon the reference signal. Similarly, we also calculate the amplitude of the cross-correlation by displaying the hologram shown in Fig. 2(b) on the DMD-MMA, which turns off the reference area, and subsequently measuring the intensity of the transmitted field; this gives us the strength of the control signal. From this |*g*| is calculated as This process of turning on and off the signal in the way described above is not possible using conventional spatial light modulators that are based on nematic liquid crystal materials and is an advantage of the DMD-MMA.

_{nm}Further, we find that determining *t _{n,m}* through curve fitting consistently provides better results. Here, the intensity at the

*n*th observation point corresponding to the

*m*th Hadamard basis element is phase modulated by

*α*such that

*g*| and

_{nm}*φ*as free parameters, ${I}_{n,m}^{C}$ and ${I}_{n,m}^{R}$ as the problem dependent parameters, and

_{n,m}*α*and

*I*as the independent and dependent variables, respectively.

_{n,m}*I*is measured for phase shifts corresponding to

_{n,m}*α*= 0,

*π*/2,

*π*, and 3

*π*/2. In the standard approach to measuring the TM elements, 4

*M*phase profiles are required, where

*M*corresponds to the number of input channels [3, 5]; in contrast, using our approach, as discussed above, 5

*M*+ 1 phase images are required. However, our approach provides information that is not captured in the standard process, namely, the absolute value of the TM elements as well as the magnitude of the cross-correlation between the reference and the control signals. Note that although the TM elements are measured for a Hadamard basis input, we convert them to the canonical basis input

*T*using a standard Hadamard to canonical conversion [17] before using them in designing of the input phase profiles.

_{n,m}## 3. Input phase profile calculation

To control the intensity and/or phase of the transmitted light using the TM, one needs to design and implement an input field profile that would result in the desired transmitted field. In our experiments, this is achieved using computational optimization techniques. For example, an input phase profile that would result in the desired intensities
${I}_{k}^{D}$s at the chosen observation points *k* = *k*_{1}, *k*_{2}, ··· ,*k _{N′}* is calculated by solving the optimization problem of the form

*ϕ*is the phase modulation to be applied to the

_{m}*m*th control segment and

*N′*is the number of points to be optimized. The predicted intensity ${I}_{k}^{P}$ needed in the optimization process can be calculated as

*m*th and

*m′*th control segments to the

*k*th observation point, whereas ${\delta}_{m,{m}^{\prime}}^{k}$ is the phase difference between those contributions. For

*M*control segments there are

_{M}C_{2}cross-correlation terms, where the symbol

_{M}C_{2}is used in combinatorics to refer to the combination of

*M*elements taken 2 at a time without repetition. To use Eq. (7) these cross-correlation terms need to be experimentally measured. One approach to measuring these values could be to perform pairwise interference measurements between the contributions of all the available control segments. However, these measurements in canonical basis could lead to a low SNR. An alternative approach would be to measure the transmitted intensities corresponding to at least

_{M}C_{2}distinct input phase profiles and to solve the resulting system of linear equations. However, if one is to limit interest to regions where the measured values of |

*g*| are relatively large, results with sufficient accuracy can be expected with a simpler relation of the form with ${I}_{k}^{P}={\left|{u}_{k}^{P}\right|}^{2}$. This approach, which we follow in our experiments, obviates a need to measure the aforementioned

_{nm}

_{M}C_{2}cross-correlation terms. Once a phase profile is designed, corresponding holograms are generated using Eq. (1) and then displayed on the DMD-MMA.

To deliver an optical field of intensity ${I}_{k}^{D}$ at a phase of ${\varphi}_{k}^{D}$, one needs to solve the optimization problem

## 4. Results and discussion

In Fig. 3 we show the magnitude of the cross-correlations between the control signal, corresponding to each of the Hadamard basis inputs, and the reference signal at a region on the observation plane. Contrary to what has been assumed in previous studies [3–5], we observe that the cross-correlations values are less than one. The magnitude of these cross-correlations depend upon the temporal and spatial coherence of the input field as well as the distribution of optical path lengths traversed by the incident photons [19]. Further, it also shows that each control segment contributes a partially coherent field to the region of interest.

Figure 4 are results demonstrating quantitative control of the transmitted intensity using our aforementioned approach. Here, we first measure the TM elements. Next, we select the regions to be studied. These regions are randomly selected under the constraint that at least 50 of the 64 transmission matrix elements of the selected region have a measured |*g _{nm}*| between 0.7 and 1. For each selected region, we calculate 25 phase profiles, each predicted to generate one of the 25 targeted intensities at that region. The 25 targeted intensities range from grayscale intensity level of digital number (DN) = 10 to 250 in steps of 10. This range is chosen since the camera used in our experiment is limited to an 8-bit output with possible outputs ranging from DN = 0 to 255. Then, we generate input optical fields with the prescribed phase profiles using the binary holograms calculated according to Eq. (1). Finally, we measure the respective transmitted intensities at the region of interest and compare the measured intensities with the targeted ones. From the Fig. 4, we see that the measured values follow the targeted ones with a maximum standard deviation of less than 30 DN and a mean standard deviation of 23.8 DN. A trend in the observed intensity values to be larger than the targeted intensities for smaller targeted intensities and smaller than the targeted intensities for larger intensities is consistent with our using Eq. (8) in the calculation of the input phase profile. Specifically, when the magnitude of the cross-correlations is not one, Eq. (8) underestimates the intensities for lower intensities and overestimates for larger intensities. Figure 4 demonstrates that even in the presence of multiple scattering events, it is possible to transmit light of a desired intensity through the scattering medium. It is expected that using Eq. (7) in the optimization process compared to Eq. (8) that is currently used, the quality of the control can further be improved. We also point out that the ability to modulate intensity is dependent upon the magnitude of the cross-correlation values. The maximum range can be expected when the magnitude of the cross-correlations is unity and the range decreases progressively with decreasing magnitudes. In the limit, when all the cross-correlation values are zero, no intensity modulation can be expected.

The ability to simultaneously maximize the intensity at multiple points using the transmission matrix has previously been demonstrated [3]. However, controlling each of the multiple points to a specified intensity value has not yet been demonstrated. In Fig. 5, we demonstrate such control. Figure 5(a) is captured by modulating the control area with a phase profile designed to make the region on the right brighter, whereas Fig. 5(b) uses a phase profile calculated to turn the left area brighter. Both of these phase profiles are calculated by taking *N′* = 2 in Eq. (6). The ability to independently modulate the intensity at multiple points demonstrated in this paper should be useful in projecting grayscale images through highly scattering media. This is in contrast to transmission of phase images through such media as demonstrated in [20], and projection of binary intensity images as reported in [21].

We also use our ability to control the intensity profile to generate a “homogeneously” low intensity speckle field, and show such a field in Fig. 6. In Fig. 6, an area of approximately 218 *μm*^{2} is targeted to have a low intensity whereas no control is exercised on the rest of the observation plane. The maximum value of intensity in all the areas is 111.5 DN whereas the maximum intensity value within the area of interest (AOI) is only 19.5 DN; note that this further decreases to 12.5 DN when only the central portion of the AOI which is half the size of the AOI is considered. This type of control could lead to interesting applications such as encoding of information in highly scattering channels.

Figure 7 shows our experimental results for controlling the phase of the light transmitted through the scattering medium. In this case, we also restrict ourselves to the regions on the observation plane that have at least 50 of the 64 measured values of |*g _{nm}*| larger than 0.7. For each selected region of interest, we calculate the initial input phase profiles necessary to generate a constant targeted intensity of 190 DN, and a concomitant range of phase values from 0 to 330° at an interval of 30° using the approach outlined in Section 3. From these phase profiles we calculate four new phase profiles shifted from the original by 0°, 90°, 180° and 270°. Next, we determine the reference and control signals by judiciously using the area of the DMD as discussed above, and shown in Fig. 2, and measure the corresponding transmitted intensities.

Finally, we use the measured intensities to calculate the phase angle *φ* of the transmitted signal using the curve fitting approach outlined in Section 2. Since different regions have different reference signals, to filter out the influence of the reference signal on the measured values, for each region we take the phase value measured corresponding to a targeted phase of 0° as offset and subtract that value from the rest of the measured phase values for that observation region. From the graph we find that, unlike the case of the intensity control, all the targeted phase values show a small standard deviation of about 10°. Further, the mean values of the measured phases are close to the targeted values. It shows that the phase angles are less strongly affected by the amplitude of the cross-correlations compared to the magnitude. It can be easily seen for two channels by looking at the expression for the relative phase in four point phase shifting interferometry which is
${\text{tan}}^{-1}\frac{{I}^{\alpha =3\pi /2}-{I}^{\alpha =\pi /2}}{{I}^{\alpha =0}-{I}^{\alpha =\pi}}$, where *I*^{α=x} is the resultant intensity with a phase modulation *α* of *x* radians. Although the intensities are dependent upon the amplitude of the cross-correlation, the relative phase is not as the amplitude of the cross-correlations in the numerator and the denominator cancel out.

## 5. Conclusion

In this paper, we experimentally demonstrated that by measuring the absolute value of the transmission matrix elements it is possible to quantitatively control the intensity and phase of light transmitted through a scattering medium. To achieve this, we used a novel, phase-stable, interferometric optical setup that comprised a low cost, micro-mirror array that was modified to behave as a phase-only spatial light modulator. Our setup was used to generate areas of low intensity in the speckle field of the transmitted light, as well as to quantitatively control the intensity transmitted to multiple observation points. The quantitative control reported in the paper has the potential to be useful in varied applications ranging from biomedical imaging to optical communications. Moreover, the low cost, optical setup employed will make these studies more (experimentally) accessible to many optical researchers.

## Acknowledgments

K. C. T. acknowledges partial support from an NSF CAREER award (NSF DBI 09-54155).

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