1. Introduction

When waves travel through disordered media such as ground glass and skin tissues, they are scattered multiple times. Large fraction of the incoming energy bounces back as a consequence, which leads to the attenuation of transmission. Considering that waves have to penetrate through scattering layers for the efficient sensing and manipulation of the targets embedded in them, this attenuation in transmission has been a limiting factor for the working depth of various optical techniques [1–4]. For example, poor light penetration through skin tissues is a major barrier to the improvement of treatment depth in phototherapy [1] and to the imaging depth of various imaging techniques including photoacoustic tomography [2], optical projection tomography [3], and diffuse optical tomography [4].

In recent studies, much effort [5–23] has been devoted to measuring scattered waves from a disordered medium and controlling wave propagation using wavefront shaping. In particular, advances have been made for enhancing light transmission through scattering layers [10, 15]. The main strategy of these previous studies has been to find the specific shape of incident wave that induces strong constructive interference at the far side of the disordered medium. This approach can be so effective that even the complete elimination of the attenuation is possible in theory [24–28]. However, all such previous efforts have required detectors positioned on the opposite side of a scattering medium to the incident wave for the direct monitoring of the transmitted wave. This makes them unsuitable for realistic applications in which detectors can only be positioned on the same side as the incident wave and the monitoring of reflected wave is the only possible mode of detection.

In this study, we measured the reflection matrix of a disordered medium with wide angular coverage for each orthogonal polarization states. From the reflection matrix, we identified reflection eigenchannels of the medium. And we demonstrated that shaping of an incident wave into the reflection eigenchannel with smallest eigenvalue, which we call anti-reflection mode, reduces reflectance, and that, due to the energy conservation, the reduced reflection energy is transferred to the opposite side of the disordered media. In essence, we interrogated reflected waves so as to find specific incident wave patterns that induce the destructive interference of reflected waves. We observed more than a factor of 3 enhancement in light transmission after the coupling of the incident waves to the anti-reflection modes. Our approach of delivering light deeper into the scattering media will contribute to enhancing the sensitivity of detecting objects hidden under scattering layers, which is universal problem ranging from geology to life science.

2. Experimental setup

In order to identify the anti-reflection modes of a disordered medium, we implemented an experimental setup (Fig. 1) and measured a reflection matrix of the medium. Specifically, we sent a clean planar wave from a He-Ne laser to the medium at various incident angles covering two orthogonal polarizations. Using an interferometric microscope [6], we recorded the complex field map of reflected waves from the medium. The laser beam having ahorizontal polarization was split into two and one of the beams reflected at the beam splitter (BS1) was guided through free space to the camera to serve as a reference wave. A waveplate (WP1) installed in the beam path was used to set the polarization of the reference wave as either vertical or horizontal. The other beam was sent through a waveplate (WP2) to rotate the direction of polarization by 45 degrees. The beam was divided into two by a polarizing beam splitter (PBS1). The transmitted wave at the PBS1 having horizontal polarization was sent to SLM I via BS2. Then, the beam was reflected by the SLM I while its polarization unchanged and subsequently reflected by the BS2. Afterwards, it was sent through PBS2 and then guided through a condenser lens (C) to a disordered medium (S). Therefore, SLM I controls the horizontal polarization component. The reflected wave at PBS1 having vertical polarization was controlled by the SLM II and then overlapped with the beam from SLM I at the PBS2. The two SLMs were positioned at the conjugate planes of the disordered medium (S). The wave reflected by the disordered medium went through the condenser lens and then reflected at BS4. It was relayed to the camera through BS5 and BS6. In the case of transmitted wave through the disordered medium, it was captured by an objective lens (OL) and relayed to the camera through BS5 and BS6.

 

Fig. 1 Experimental schematic for the recording of a reflection matrix and implementation of reflection eigenchannels. WP1, 2: waveplates, BS1-6: beam splitters, SLM I, II: spatial light modulators, PBS1, 2: polarizing beam splitters, C: condenser lens (Nikon, 1.4 NA), OL: objective lens (Olympus, UPLSAPO, 100XO, 1.4 NA), S: scattering sample, BB1, 2: beam blocks.

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3. Measurement of reflection matrix

The procedure of recording a reflection matrix is as follows. A beam block plate (BB1) was inserted to block the transmitted wave through the disordered medium and the other beam block (BB2) was removed. And then we controlled SLM I to generate a horizontally polarized plane wave with an incident angle (θ_x, θ_y) at the disordered medium. For each incident angle with a certain polarization state (θ_x, θ_y, p), where (θ_x, θ_y) indicates illumination angle and p stands for either horizontal (H) or vertical (V) polarization state, we took images of the H and V components of the reflected waves. The acquired image is a complex field map, E(x, y, p; θ_x, θ_y, p), where (x, y) is a position vector in the sample plane.

In order to record both horizontal and vertical polarization components of the reflected wave, WP1 was first set to make the reference wave horizontally polarized and an interference image was captured. And then WP1 was rotated to make the reference wave to be vertically polarized, and another interference image was recorded. Each interference image was processed to obtain both amplitude and phase maps of the reflected wave. The same procedure was repeated for the SLM II that controls the vertical polarization of incident wave.

The illumination area was set to be 11 × 11 μm² and the angular range covered was up to a numerical aperture of 0.8. The minimum required number of incident angles for the complete coverage of the given area and numerical aperture is 640. Here we used 800 angles of incident wave, which is a slight oversampling, for enhancing the signal to noise in the matrix construction. The collection area (32 × 32 μm ²) for the reflected wave was set larger than the illumination area to capture waves that laterally diffuse away from the illumination area. The angular coverage of collection was set to the numerical aperture of 1.0 to capture all the angularly scattered waves. As two polarization states were covered for both input and output waves, in total 3,200 images of reflected waves were recorded. As a disordered medium, we used randomly distributed TiO₂ particles on a coverslip, whose average transmittance ranges between 10% and 15%. The thickness of the medium is about 8 ± 2 μm and the transport mean free path is 0.5 ± 0.2 μm such that the transmitted light is scattered hundreds of times on average.

We then processed the 3,200 images to construct a reflection matrix R. The input and output bases of this matrix are n = (θ_x, θ_y, p) and m = (x, y, p), respectively. The matrix element r_mn connects the n^th coordinate in the input to the complex amplitude of the wave at the m^th coordinate in the output. Below is the procedure of the matrix construction. For each input coordinate, (θ_x, θ_y, p), there are two sets of reflection images, E(x, y, H; θ_x, θ_y, p) and E(x, y, V; θ_x, θ_y, p). We converted each image into a vector by appending each column in the reflected image to the end of the preceding column. We then appended the image vector for V polarization to the one for H polarization. The resulting merged reflection image vector was assigned to a column of the reflection matrix. By repeating the same procedure for all input coordinates, the reflection matrix R was constructed (see Section A1 for the constructed reflection matrix).

4. Implementation of the Anti-Reflection Modes and Transmission Enhancement

From the constructed reflection matrix, we extracted anti-reflection modes of a disordered medium. Based on the random matrix theory (RMT), we factorized the reflection matrix into R = USV* by the singular value decomposition [24]. Here S is a rectangular diagonal matrix with nonnegative real numbers on the diagonal called singular values, and V and U are the unitary matrices whose columns are the reflection eigenchannels for the input and output, respectively. The V* denotes the conjugate transpose of the matrix V. The singular values in S were sorted in the descending order. We assigned eigenchannel index from 1 to N for the sorted singular values, where N = 1280 is the total number of eigenchannels. Therefore, the i^th column in V is the i^th reflection eigenchannel on the input plane, and the individual element in the column is the complex amplitude of the corresponding (θ_x, θ_y, p). The square of the singular value is the eigenvalue of the R*R, which is the expected reflectance of the corresponding eigenchannel. We define the eigenchannel with smallest eigenvalue, whose eigenchannel index corresponds to 1280, as the anti-reflection mode of the medium.

We physically shaped an incident wave into a reflection eigenchannel, i.e. each column of V. SLM I and SLM II, which operate phase-only control mode, were used to shape the H and V polarization components of individual eigenchannel. They were carefully aligned, both laterally and axially, to be overlapped at the input plane of the sample with a positioning accuracy of 150 nm. In particular, we experimentally measured the relative phase fluctuation between two SLMs and corrected it when we implemented eigenchannels. From the measured reflection matrix covering two polarization states, we identified separate incident waves for horizontal (SLM I) and vertical (SLM II) polarizations that can generate a single focused spot. We then measured the intensity of the spot by varying the overall phase of SLM II. We found and added the phase that makes the intensity maximum.” Figs. 2(a) and 2(b) show the intensity maps of the incident waves for the first eigenchannel and the anti-reflection mode, respectively. We then measured the reflected wave images for each case (Figs. 2(d) and 2(e)). Total reflection intensity of the first eigenchannel is much higher than that of the anti-reflection mode, and the reflection intensity of the anti-reflection mode, the least reflecting mode of the medium, was highly attenuated. We measured the intensity map of the transmitted waves for each reflection eigenchannel on the opposite side of the medium (Figs. 2(g) and 2(h)) and observed that the total transmission intensity of the anti-reflection mode is significantly higher than that of the first eigenchannel. As a point of reference, we sent a planar wave of normal incidence to the medium (Fig. 2(c)) and recorded the intensity maps of the reflected (Fig. 2(f)) and transmitted (Fig. 2(i)) waves. Compared with the transmittance of this normally incident plane wave, transmittance of the anti-reflection mode was enhanced by a factor of 3. The transmission enhancement was highly reproducible. Typically, the range of enhancement factor varies from 2 to 3 for 10 different samples. This result proves that we have enhanced wave penetration through a highly scattering medium with the measurement of reflected waves, not the transmitted waves.

 

Fig. 2 Shaping incident wave into reflection eigenchannels. (a), (b) and (c): Intensity maps of experimentally shaped incident waves for the first eigenchannel, 1280th eigenchannel and normally incident plane wave, respectively. They correspond to the highest, lowest, and average reflectance modes, respectively. The H and V polarization components were superposed at the incident plane. (d), (e) and (f): Experimentally recorded intensity maps of reflected waves for the cases of (a), (b) and (c), respectively. (g), (h) and (i): Measured intensity maps of transmitted waves for the case of (a), (b) and (c), respectively. Scale bar, 5 μm. The color bar next to (c) indicates intensity in an arbitrary unit and applies also to (a) and (b). The other two color bars follow the same rule.

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Green square dots in Fig. 3(a) show the eigenvalue distribution calculated from the measured reflection matrix where we observed monotonic decrease in the eigenvalue. We recorded the reflectance for many of the representative eigenchannels (see Section A2 and Media 1). The reflectance monotonically decreased as the eigenchannel index was increased (blue dots in Fig. 3(a)). The measured reflectance for the anti-reflection mode was 35.7%, which is far smaller than the reflectance of uncontrolled input wave of 71.1%. As a consequence of the net decrease of reflectance by 35.6%, the transmission was increased from an average transmittance of 12.8% to 38.8% (Fig. 3(b)). We found that not all of the reduced reflection energy was converted to the transmission energy. In fact, there was an approximate 9.6% loss due to the use of slab geometry, rather than the ideal waveguide geometry, which allowed some of the scattered waves to leak out of collection range. This limitation is consistent with real applications such as the interrogation of targets under biological skin tissues because the scattering layers are in slab geometry by nature. According to our numerical analysis based on RMT (see Section A3), the expected enhancement factor was around 4 after accounting for the effect of this leakage and limited angular coverage, which is close to the experimentally achieved enhancement factor of 3. This means that the transmission enhancement by the experiment nearly reached to 75% of the theoretical limit. The residual discrepancy was mainly due to the imperfect wavefront shaping of the incident eigenchannels. If our wavefront shaping were perfect, the reflectance of the eigenchannels would have followed green dots in Fig. 3(a) rather than the blue dots. Then the reduced reflectance of the closed eigenchannels would amount to 50%. Considering that about 75% of the reduced reflectance is converted to the transmittance, the transmission enhancement factor of the closed eigenchannels is estimated to be about 4. This is almost the same as the estimation by RMT shown in (see Section A3).

 

Fig. 3 Reflectance and transmittance of various reflection eigenchannels. (a) Reflectance of experimentally implemented reflection eigenchannels (blue circular dots) and those predicted from the eigenvalues of the measured reflection matrix (green square dots). Black dots indicate mean reflectance of the medium. (b) Transmittance of the same reflection eigenchannels (red circular dots) implemented in (a). Black dots indicate mean transmittance of the medium. (Media 1: On the left are the individual incident eigenchannels experimentally generated by the two spatial light modulators. For each incident eigenchannels, we measured reflected images (middle) and their transmitted images (right). The size of each image is 32 × 32 μm². The number at the top of each image is the eigenchannel index.) (c) Relation between reflection and transmission eigenchannels. Reflection and transmission matrices were measured for the same disordered medium, and their respective eigenchannels were obtained by performing the singular value decomposition. Shown here is the absolute square of normalized cross-correlation between the first reflection eigenchannel and all the transmission eigenchannels.

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The increase in transmission through the wave coupling into the anti-reflection mode implies that the incident wave has preferentially coupled to those transmission eigenchannels with large transmittance. We explored the explicit connection between reflection eigenchannels and their transmission counterparts. We recorded a transmission matrix for the same medium by turning BB1 off and BB2 on (see Section A4), and obtained the transmission eigenchannels. By calculating the cross-correlation between the reflection eigenchannel of minimum eigenvalue and each transmission eigenchannel, we found that the transmission eigenchannels with larger transmission eigenvalues (i.e. of smaller transmission eigenchannel indices) preferentially contribute to the anti-reflection mode (Fig. 3(c)).

5. Conclusion

In conclusion, we have demonstrated the enhancement of wave penetration to the scattering medium by coupling light into anti-reflection modes of the medium. For the sensing, imaging and manipulation of targets obscured by highly scattering layers, it is essential to send interrogating waves with sufficient intensity to the target sites. Therefore, our study will lay a foundation for advancing many practical techniques for which reflection loss caused by the scattering layers is a limiting factor. For example, our method can enhance the working depth of photoacoustic imaging, diffuse optical tomography, and in vivo glucose sensing [29]. Similarly, by administering an intense light source and minimizing loss in this way, the treatment depth of various phototherapies can be increased, and the optical manipulation of cells embedded in the tissues can be made efficient.

A1. Experimentally measured reflection matrix

Figure 4 is the experimentally measured reflection matrix. Figures 4(a) and 4(b) are the amplitude part and the phase part of the reflection matrix, respectively.

 

Fig. 4 Experimentally recorded reflection matrix of a disordered medium. a, Amplitude part of the reflection matrix. Column indices correspond to input coordinates, and were assigned with an ascending order of the azimuthal angle for H polarization and then the same order of angle for V polarization. Row indices indicate spatial coordinate (x,y) for both polarization states of reflected waves. Color bar indicates amplitude in an arbitrary unit. b, The phase part of the reflection matrix. Color bar indicates phase in radians. Yellow and black dashed lines were drawn for visual guidance.

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A2. Representative reflection eigenchannels

In order to supplement Fig. 2 in the main text, the intensity maps of a few representative reflection eigenchannels and their respective reflection images are shown in Fig. 5. Figures 5(a)-(e) show the intensity maps of reflection eigenchannels at the input plane for the eigenchannel indices of 47, 239, 551, 863, and 1187, respectively. And Figs. 5(f)-(j) show the reflection images when Figs. 5(a)-(e) were used as input waves, respectively.

 

Fig. 5 Representative reflection eigenchannels incident to and reflected from the disordered medium. Scale bar, 5 μm. Color bars indicate intensity in arbitrary units.

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A3. Random matrix theory for the limited channel coverage

Based on the random matrix theory (RMT), we assess the enhancement factor of transmission when incident wave couples to the anti-reflection mode. For ideal waveguide geometry, the RMT predicts the distributions of transmission and reflection eigenvalues. By employing random unitary matrices V and U for the input and output channels, respectively, we can generate both transmission and reflection matrices. Figures 6(a) and 6(b) show the reflection matrix and its associated transmission matrix for a disordered waveguide equivalent to having 20% average transmittance and 80% average reflectance, respectively. For both matrices, the column index corresponds to the input channel. The row index in the reflection matrix corresponds to the output channel on the reflection side while that in the transmission matrix indicates the output channel on the transmission side.

 

Fig. 6 Reflection a, and transmission b, matrices calculated by RMT. c, Blue square dots indicate reflection eigenvalues obtained from the reduced reflection matrix. Black dots indicate the mean reflectance of the medium. d, Red circular dots indicate the expected transmittance for the reflection eigenchannels obtained in (c). Black dots indicate the mean transmittance of the medium.

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In the experiment, we have used a sample with slab geometry in which there exists inevitable loss in collecting scattered waves. For the sample used for Fig. 3 in the main text, the measured transmittance and reflectance by the camera were 13% and 71%, respectively. Therefore, there was 14% loss, which is mainly due to the leakage of some of the multiply scattered waves from the camera view field. In order to confirm this, we measured transmittance and reflectance using detectors that cover wider area than the camera can capture. In this measurement, we found that average transmittance and reflectance were 20% and 80%, respectively. Therefore, the finite size of the camera sensor was the source of loss. In order to account for this loss in the RMT, we reduce the number of output channels from the ideal matrix. Specifically, those output channels located between two yellow horizontal lines in both Figs. 6(a) and 6(b) were selected for further analysis. Note that larger fractional loss in the transmission than the reflection was considered in determining the width of window.

Another experimental factor was that we covered the input solid angles up to about 0.8 NA. This was due to the limited number of pixels in SLM with which we need to cover both area and angular range. This factor can be considered as the loss of the input channels in the given matrix. In order to account for this experimental limitation, we chose those input channels located within two vertical yellow lines in Figs.6(a) and 6(b) for the further analysis. In the end, we consider the reduced matrices defined by 4 yellow lines as the transmission and reflection matrices comparable to those acquired in the experiment.

By performing the singular value decomposition for the reduced reflection matrix, we obtain reflection eigenvalues (Fig. 6(c)) and their respective reflection eigenchannels. Next, we predict the transmittance of each reflection eigenchannels by multiplying the reduced transmission matrix to the reflection eigenchannels (red circular dots in Fig. 6(d)). From this result, we found that the expected enhancement in transmission by the coupling light into the minimum reflection eigenchannel is a factor of 4. The experimentally observed enhancement factor of 3 is close to this theoretical expectation. The residual discrepancies can be attributed to the imperfect wavefront shaping and mechanical perturbation during the measurements.

A4. Transmission matrix and transmission eigenvalues of a disordered medium

For the disordered medium used for Fig. 3 in the main text, we recorded a transmission matrix and computed its eigenvalues. Figures 7(a) and 7(b) show the amplitude and phase parts of the transmission matrix. Here (x, y) indicates spatial coordinate at the transmission side of the medium. Color bar in Fig. 7(a) indicates amplitude in an arbitrary unit and that in Fig. 7(b) shows phase in radians. From the measured transmission matrix, we obtained transmission eigenvalues (Fig. 7(c)) using the singular value decomposition. Here the eigenchannel index was assigned after sorting the transmission eigenvalues in the descending order.

 

Fig. 7 Experimentally measured transmission matrix and its eigenvalue distribution. a, The amplitude part and b, the phase part of the transmission matrix. c, Blue square dots represent the transmission eigenvalues. Black dots indicate the mean transmittance of the medium

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A5. Sample preparation

In this experiment, TiO₂ particle (Sigma Aldrich) layers were used as disordered media. TiO₂ particles have high refractive index (2.58) and are almost free from absorption for the light source with 633 nm wavelength. In order to prepare relatively uniform layers, we made a solution of TiO₂ in ethanol and spread the layer on the slide glass (or cover glass) by using air spray. The thickness of the TiO₂ layer measured by atomic force microscopy was 8 ± 2 μm and the transport mean free path found from the relation between the transmission and the thickness of the layer was 0.5 ± 0.2 μm.

Acknowledgments

This research was supported by the IT R&D Program (R2013080003), the Global Frontier Program (2014M3A6B3063710), IBS-R023-D1-2015-a00, the Basic Science Research Program (2013R1A1A2062560, 2013R1A1A2062808) and the Nano-Material Technology Development Program (2011-0020205) through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning. It was also supported by the Korea Health Technology R&D Project (HI14C0748) funded by the Ministry of Health & Welfare, South Korea.

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