Harnessing randomness to control the polarization of light transmitted through highly scattering media

Santosh Tripathi; Kimani C. Toussaint

doi:10.1364/OE.22.004412

1. Introduction

An optical field transmitting through a highly scattering medium undergoes multiple scattering events [1–8], which randomizes the field’s intensity, phase, and polarization [1]. This ultimately limits the utility of the transmitted field in applications such as deep tissue imaging [4] and information transmission [5], where often, an obscuring scattering medium is situated between the optical source and a detector. However, several recent studies [2–9] have shown that careful phase-shaping of the input optical field could mitigate the aforementioned effects of scattering-induced randomization. In 2007, the ability to focus light through a highly scattering medium was demonstrated by iteratively changing the phase profile of the input optical field [2], whereas similar focusing results were demonstrated through the use of the transmission matrix in 2010 [9]. In 2011, it was shown that multiple scattering of light in random media can be used to achieve both temporal and spatial focusing of ultrashort optical pulses [6, 10]. Later, in 2012, we experimentally demonstrated that such multiple scattering events can be used to control the amplitude of the polarization ratio of the light transmitted through such media [7]. Specifically, we introduced the concept of the vector transmission matrix (VTM) and showed that it can be used to control both the amplitude and phase of the light transmitted through scattering media; however, our efforts in experimentally controlling the phase was limited by interferometric stability. Subsequently, in the same year, it was shown that it is possible to focus light through a combination of a random medium and standard polarization components [8]. Here, the control of polarization is the result of utilizing external polarization components. This is in contrast to the approach presented in [7], where the polarization diversity introduced into the optical field by the multiple scattering from the sample is exploited to achieve control of the transmitted polarization state, independent of using any external polarization components. The use of the VTM has several advantages. First, it enables one to shape the input optical field such that any polarization state at any desired region of interest (ROI) in the observation plane can be generated without making any new physical measurements once the VTM is measured; this feature could be useful in projecting dynamic images through highly scattering media. Further, it allows for different spatial locations at the output of a random medium to have different states of polarization, which can be useful in the study of exotic polarization states known as vector beams and vector fields [11–13], which have been shown to have many promising applications [13–16]. Moreover, an ability to control polarization without external polarization components should be useful in polarization sensitive deep tissue imaging techniques where the use of external polarization components could be inconvenient.

In this paper, we experimentally demonstrate control of both the amplitude and phase of the polarization ratio, hence the state of polarization (SOP), of light transmitted through random media by utilizing the multiple scattering events taking place in such media. Our control builds on the work presented in [7] and the improvement in the experimental capability comes from the optical setup that permits measurement of the absolute value of the VTM elements with a single beam incident upon the media. This is made possible by the use of a deformable mirror device micro-mirror array (DMD-MMA) [17] adapted to work as a phase-only spatial light modulator. An added advantage of the technique is that it is low cost, as the DMD-MMA used in our experiments costs less than US$600.

This paper is organized as follows. In Section 2, we describe the method used in the measurement of the VTM elements. In Section 3 we discuss our algorithm for wavefront shaping. This is followed by results and discussion in Section 4 and conclusions in Section 5.

2. Measurement of the elements of the vector transmission matrix

The concept of the VTM was introduced in [7] to describe the amplitude, phase, and polarization changing behavior of scattering media. In this description, the input optical field is divided into M spatial segments. Similarly, the observation plane for the transmitted light is divided into N spatial segments. An element of the VTM, $t_{n, m}^{k l}$ , then represents the complex amplitude of the l-polarized component of the transmitted field at the nth observation region resulting from the k-polarized component of the field at the mth input spatial segment. Although in [7] the VTM elements corresponding to both the X- and Y -polarization components were measured, it turns out that, for controlling the polarization of the transmitted light, the randomness of the scattering medium can provide sufficient polarization diversity that measurement of both polarization components is not necessary. Thus, in this present study, we measure only the VTM elements corresponding to the Y -polarization input.

Figure 1 is a schematic of our experimental setup for measuring the VTM elements. A detailed description of a similar setup can be found in [18] where we recently demonstrate quantitative control over the intensity and phase of the light transmitted through a highly scattering medium. Briefly, the setup consists of an 808-nm diode laser [Power Technology, Inc IQ2C(808-150)] (not shown) whose output is spatially filtered and collimated before being incident on a DMD-MMA which, in our experiments, was acquired as part of a LightCrafter projector evaluation kit [17]. The binary amplitude modulation capabilty of the DMD-MMA is used to generate phase-only modulation through the use of Lee’s synthetic binary holograms [19] calculated as

f (x, y) = {\begin{array}{l} 1 & if \cos {g (x, y) + \frac{2 π x}{T}} > \cos (π q) \\ 0 & Otherwise \end{array},

where f(x, y), g(x, y), T, and q are the calculated hologram, desired phase profile, period of the grating in the hologram, and the duty cycle of the grating [19], respectively. When displayed on the DMD-MMA, the calculated binary hologram results in several diffraction orders. The desired phase modulation, which is present in the first diffraction order, is selected by using an iris to obstruct the other orders at the Fourier plane of lens L1. Lens L2 collimates the selected order and the desired phase modulation is observed at its back focal plane. An infinity corrected, 10X microscope objective OBJ1 (Spencer) with numerical aperture (NA) 0.25 then focuses the phase modulated field onto the sample S. A second, infinity corrected objective OBJ2 (Reichert) with magnification 45X and NA of 0.66 then collects the scattered light. The light collected by OBJ2 is passed through variable polarization components and then recorded by a CMOS camera (Thorlabs DCC1545M). To measure the VTM elements, an analyzer P (Thorlabs LPNIR100-MP) is placed between OBJ2 and CMOS, whereas to measure the state of polarization of the transmitted light, a quarter waveplate QWP is included in addition to the analyzer P in the beam path.

Fig. 1 Schematic of the experimental setup.See text for details.

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As in [18], we divide the DMD-MMA into a central control area and a peripheral reference area. The peripheral area is not phase modulated and provides the reference signal required in the interferometric measurements used in measuring the VTM elements. The control area is divided into 64 independently controllable segments. As pointed out in several previous studies [7, 9, 18], measuring the VTM elements in the canonical basis results in a low signal-to-noise ratio (SNR) so we also carry out our measurements in the Hadamard basis. Following the four point phase shifting interferometry, the measured VTM elements are of the form [7, 9, 18]

T_{n, m}^{k Y} = {| g_{n m} |}^{k} \sqrt{I_{n, m}^{R k} I_{n, m}^{S k}} e^{i Δ ϕ_{n, k}},

where

I_{n, m}^{R k}

and

I_{n, m}^{S k}

are the k– polarization components of the reference and control signals at the nth observation region, respectively. Similarly, Δϕ_n,k and |g_nm| are the phase difference and cross correlation between the reference and control signals, respectively, at the nth observation region corresponding to the mth Hadamard basis element input. To obtain the absolute value of the VTM elements, we measure the strength of the reference signal. This is achieved by turning off the grating for the portion of the DMD-MMA that represents the control area [18] and measuring the corresponding transmitted intensity. Further to measure the value of the magnitude of the cross-correlation term, the strength of the control signal is measured by, similarly, turning off the area on the DMD-MMA representing the reference area.

The phase angle of the VTM elements of the form $T_{n, m}^{X Y}$ are measured with respect to the phase of the X component of the reference signal, $E_{n, m}^{R X}$ , whereas the phase angle of the VTM elements of the form $T_{n, m}^{Y Y}$ are measured with respect to the phase of the Y component of the reference signal, $E_{n, m}^{R Y}$ [7, 18]. To correct for this phase offset, we measure the Stokes vector of the total reference signal using the rotating quarter waveplate technique [20], where the transmitted intensity is measured as the polarizer is oriented with its transmission axis along the X axis and the quarter waveplate preceding it is rotated. The measured intensities are related to the Stokes vector components through [20]

I (θ) = \frac{1}{2} {S_{0}^{R, n} + S_{1}^{R, n} \cos^{2} (2 θ) + S_{2}^{R, n} \cos (2 θ) \sin (2 θ) - S_{3}^{R, n} \sin (2 θ)},

where

S_{0}^{R, n}

,

S_{1}^{R, n}

,

S_{2}^{R, n}

, and

S_{3}^{R, n}

are the Stokes vector components of the reference signal at the nth observation region, and θ is the angle that the fast axis of the quarter waveplate makes with the X axis. The Stokes vector is calculated by fitting the experimentally measured values to Eq. (3). The relative phase between the two polarization components of the reference signal is then given by [7, 21]

φ^{n} = \frac{1}{2} ∠ (S_{2}^{R, n} + i S_{3}^{R, n}) .

Using these values, the absolute value of the VTM elements corrected for the phase offset between reference signals are calculated as

T_{n, m}^{X Y} = {| g_{n m} |}^{X} \sqrt{I_{n, m}^{S, X}} e^{i Δ ϕ_{n, X}}

and

T_{n, m}^{Y Y} = {| g_{n m} |}^{Y} \sqrt{I_{n, m}^{S, Y}} e^{i (Δ ϕ_{n, Y} + φ^{n})} .

As pointed out in [18], curve fitting yields better results in calculating the transmission matrix elements. Employing this approach, the transmission matrix elements are calculated by curve fitting to the expression

I_{n, m}^{k} = I_{n, m}^{S, k} + I_{n, m}^{R, k} + 2 \sqrt{I_{n, m}^{S, k} I_{n, m}^{R, k}} {| g_{n m} |}^{k} \cos (φ_{n, m}^{k} + α),

where

I_{n, m}^{k}

is the k polarized intensity at the nth observation region when the mth Hadamard basis element is phase modulated by α. The free parameters in the curve fitting are |g_nm|^k and

φ_{n, m}^{k}

. Similarly, the problem dependent parameters are

I_{n, m}^{S, k}

and

I_{n, m}^{R, k}

, and α and

I_{n, m}^{k}

are the independent and dependent variables, respectively. We measure

I_{n, m}^{k}

corresponding to the phase shifts of α = 0, π/2, π, and 3π/2.

As pointed out earlier, the VTM elements are measured in the Hadamard basis input. Before using them for the polarization control we convert them to the canonical basis input $t_{n, m}^{k Y}$ using a standard Hadamard to canonical conversion [22].

3. Input phase profile calculation

We use computational optimization to calculate the phase profile of the field incident upon the random media that would result in the desired state of polarization at the ROI. The optimization problem is of the form:

\begin{array}{l} Minimize \\ φ_{m} \end{array} {(I_{n}^{P} - I_{n}^{D})}^{2} subject to {\begin{matrix} | R^{2} - \frac{{| \sum u_{n}^{Y (P)} |}^{2}}{{| \sum u_{n}^{X (P)} |}^{2}} | < ε_{r} \\ (ϕ - ∠ \sum u_{n}^{X (P)} + ∠ \sum u_{n}^{Y (P)}) mod 2 π < ε_{p} \end{matrix},

where

I_{n}^{D}

is the desired intensity, whereas

u_{n}^{Y (P)}

,

u_{n}^{X (P)}

, and

I_{n}^{P}

are the respective Y and X components of the field and the total intensity predicted based on the VTM elements for an input phase profile ϕ_m.

In the general case, calculation of the predicted value of the intensity and field requires tedious measurements [18]; however, when the focus is limited to only those regions of interest where the cross-correlation values are large, simplified expressions of the following form can be used.

u_{n}^{k (P)} = \sum_{m} t_{k, m}^{k Y} e^{i ϕ_{m}},

and

I_{n}^{P} = {| u_{n}^{X (P)} |}^{2} + {| u_{n}^{Y (P)} |}^{2} .

The optimization problem in Eq. (8) is solved using the KNITRO optimization package [23] called from the MATLAB computational environment.

4. Results and discussion

We start by analyzing the control of the SOP at an ROI. Figures 2(a) and 2(b) show the X-and Y -polarized component of the speckle pattern generated by the light transmitted through the scattering sample for a phase unmodulated input optical field. These speckle patterns, as expected due to multiple scatterings, are uncorrelated with a correlation coefficient of less than 0.05. The ROI at which we consider the optimization of the polarization is is demarcated by a square box in Figs. 2(a) and 2(b), with 52-μm side length corresponding to 10 pixels on the camera on each side. Each of the M spatial segments of the field incident on the scattering sample contribute to both the X- and Y -polarized fields at the ROI. Figures 2(c) and 2(d) show the magnitude of the VTM elements governing the X-polarized and Y -polarized contributions at the ROI. The corresponding phase of these VTM elements are shown in Figs. 2(e) and 2(f). It is worthwhile to note here that the range of the magnitude of the VTM elements obtained from Eq. (2) for X- and Y -polarized contributions to the ROI are widely different and depend upon the reference signal. However, when the scaling due to the reference signal is removed, the magnitude of the VTM elements come to have similar range, as one would expect from the multiple scattering events taking place in the scattering medium. For example, before removing the scaling due to the reference signal, the range of magnitudes of the VTM elements contributing X-polarized field at the ROI shown in Fig. 2 is 7.4, whereas the range of the magnitudes of the VTM elements contributing Y -polarized field at the ROI is 20.8. This disparity stems from the difference in the strength of the X- and Y -polarized components of the reference signal at the ROI which are 3.4 DN and 20.5 DN, respectively. Once scaling due to the reference signal is removed, the amplitude of the VTM elements governing the X- and Y -polarized field at the ROI have a more comparable range of 4 and 4.5, respectively. It is crucial to apply this correction whenever quantitative controls are to be achieved through the use of the VTM or transmission matrix [18]. We also want to point out that the phase angles of the VTM elements corresponding to Fig. 2(f) have been corrected for the phase offset between X and Y components of the reference signal at the ROI. In this particular case the phase offset was calculated to be 26.5°.

Fig. 2 (a) and (b) show the X- and Y -polarized components, respectively, of the speckle pattern at the output of the scattering sample. The square box shown demarcates an ROI. The amplitude of the VTM elements governing the X- and Y -polarized optical field at the ROI are shown in (c) and (d), respectively. Similarly, the phase of the VTM elements governing the X- and Y -polarized optical field at the ROI are shown in (e) and (f), respectively.

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We used the VTM elements shown in Fig. 2 to design several different input phase profiles to achieve different SOPs at the ROI using the optimization process described in Section 3. We show five SOPs in Fig. 3. In the figure, the column on the left shows the polarization ellipses, where the red ellipses represent the targeted SOPs and the blue ellipses represent the SOPs experimentally measured. Note that in the approach outlined in this paper, the degree of polarization cannot be independently controlled. In our experiments we find it to be less than one but more than 0.9 in most of the cases. The parameters for the experimentally measured ellipses are derived from the polarized component of the optical field. The right column in Fig. 3 shows the intensity as a function of the quarter waveplate orientation using the Stokes vector measurement technique referred to in Section 2. The blue markers in these plots represent the experimentally measured intensities, whereas the blue curves represent the curves fitted to the measured intensities with Eq. (3). Finally, the red dashed curves are the expected intensity profiles corresponding to the targeted SOP and experimentally measured degree of polarization. The figure demonstrates that the method presented in the paper can be used to generate a variety of polarization states involving different ratios of the intensity of Y and X and different relative phases between these components. For example, the ratio of the intensity of Y and X in Figs. 3(a)–3(e) is 0.17, 5.83, 1, 1, and 1. Similarly, the relative phase between the Y and X components is −90, −90, 90, −90, and 0°.

Fig. 3 Demonstration of polarization control. The left column shows the polarization ellipses corresponding to five targeted (red) and experimentally measured (blue) SOPs at the ROI shown in Fig. 2. The right column shows the corresponding expected (red) and experimentally measured (blue) intensities using the rotating quarter-waveplate technique of measuring the Stokes vector components.

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This demonstration of polarization control is repeated at multiple ROIs for different SOPs. Table 1 lists the various SOPs attempted. In the table, 2χ and 2ψ are the latitude and longitude on the Poincaré sphere, whereas R² and ϕ are the square of the amplitude, and phase of the polarization ratios. These SOPs are represented on a Poincaré sphere in Fig. 4(a). Our experimental results are shown in Figs. 4(b)–4(d). In Fig. 4(b), we show the results corresponding to the polarization states with a constant targeted latitude of 45°. The plot on the left shows the measured azimuth angles versus the targeted azimuth angles, whereas the plot on the right shows the experimentally observed latitude angles corresponding to each of the targeted azimuth angles. In each case, the solid (red) line represents the ideal values. The results for constant targeted latitude angles of 0° and −45° are similarly shown in Fig. 4(c) and 4(d), respectively. The results show that the experimental values follow the targeted values closely. The results can be improved still further by measuring the cross-correlation terms between the contributions of different input channels. Currently, we do not measure these values and in designing the input phase profiles, take them to be unity. This approximation introduces errors in the strength of the X and Y components at the ROI resulting in a deviation of the experimentally observed SOP from the targeted one. Here, we point out that the relative phase between the X and Y components is not as much affected by the approximation related to the magnitude of the cross-correlation terms [18].

Table 1. The parameters of the SOPs generated.

View Table

Fig. 4 A Poincaré sphere showing targeted SOPs is shown in (a). Experimentally observed parameters are shown in (b)–(d). See text for details.

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In earlier studies on polarization control [7, 8] the goal was to achieve maximum intensity and the desired polarization at the ROI. However, treating intensity and polarization independently opens up many new opportunities for potential applications. For example, vector beams and vector fields are described by position dependent intensity and polarization profiles and generating them would require independent control over the intensity and phase. As a result, as can be seen from Eq. (8), we do not require the region of interest to be the brightest area in its vicinity. This is illustrated in Fig. 5. Here, we show the intensity profile at the observation plane for each position of the quarter waveplate corresponding to the state of polarization shown in Fig. 3(a). As can be seen, the region of interest, demarcated by a red square box, is not the brightest area in its vicinity.

Fig. 5 Intensity profile at the observation plane measured as a function of the quarter wave-plate orientation corresponding to an input optical field designed to generate the SOP shown in Fig. 3(a) at the ROI (red square). The ROI is demarcated by a red box.

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Finally, we look at the polarization distribution in the vicinity of the ROI as it helps us understand the type of control that is achieved. In Fig. 6 we show the polarization distribution in the observation plane in the vicinity of the ROI that was shown in Fig. 3(a). From the figure, it can be seen that although the input phase profile is designed to control the SOP at the ROI, the SOPs around the ROI do not change abruptly. This places a constraint on how fast (spatially) the SOPs can be changed through scattering media.

Fig. 6 Experimentally measured polarization distribution over the observation plane corresponding to an input optical field designed to generate the SOP shown in Fig. 3(a) at the ROI. The ROI is demarcated by a white box.

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5. Conclusion

In this paper we demonstrated that the multiple scattering events taking place inside a highly scattering medium can be used to control the polarization of the light transmitted through it. This control was achieved through the use of the vector transmission matrix for the scattering sample which we measured by using an interferometrically stable optical setup. This control has potential applications in many areas such as fundamental and application driven studies of inhomogeneously polarized optical fields such as vector beams and optical vortices. Similarly, it can also be useful in polarization sensitive characterization of optically random samples such as the sub-surface imaging of three dimensional integrated circuits and biological tissues.

References and links

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Harnessing randomness to control the polarization of light transmitted through highly scattering media

Abstract

1. Introduction

2. Measurement of the elements of the vector transmission matrix

3. Input phase profile calculation

4. Results and discussion

5. Conclusion

References and links

Cited By

Figures (6)

Tables (1)

Equations (10)

Optics Express

2χ (°)	2ψ (°)	R²	ϕ (°)

−45	0	0.17	−90
−45	90	1	−45
−45	180	5.83	−90
−45	270	1	−135

0	11.4	0.01	0
0	90	1	0
0	168.6	100	0
0	270	1	180

45	0	0.17	−90
45	90	1	−45
45	180	5.83	−90
45	270	1	−135