Absolute Measurement of Coherent Backscattering Using a Spatial Light Modulator for Coherence Modification

Abstract

Coherent backscattering is an interference phenomenon that occurs in the backwards direction of the incident illumination. It arises from photons traveling the same path in opposite directions within a scattering medium. Accurately determining the background signal for normalization can be challenging in such measurements. This study investigates the use of a spatial light modulator to control spatial coherence, effectively switching the interference on and off. This approach enables independent, absolute measurements of both the signal and background across the full angular detection range without modifying the experimental setup. We demonstrate this method experimentally, highlighting the importance of accurate background determination in coherent backscattering measurements using a Fourier-based setup. Additionally, we demonstrate that measurements normalized to the correct background closely match Monte Carlo simulations of the coherent backscattering signal.

1. Introduction

The interference effect of coherent backscattering (CBS) is well described in the literature and has been shown experimentally as well as theoretically for a variety of specific cases. Experimental research was carried out investigating disordered media in general [1,2], emulsion [3], cold atom [4,5], and biological tissue [6] using different setup styles. Furthermore, it was shown that Raman scattering can also exhibit a CBS signal [7]. The topic was investigated considering both the polarization [8] and the coherence properties of the light source, as well as their influence on the interference cone [9,10]. Simulations were performed for pseudo two-dimensional samples [11] and for three-dimensional finite and semi-infinite samples [12,13], showing good consistency with the theoretical expectations when altering the optical parameters [14]. In theoretical calculations, the signal can be divided into an interference term and a background term, which is used to normalize the complete signal [15]. However, in the experiment, the background is also influenced by the setup’s transfer function and stray light. It is challenging to measure the correct background for normalization. By using an incoherent light source, it would be possible to measure the background. The main problem is that changing a coherent light source to an incoherent one influences the system transfer function and requires further intensity adjustment.

In this study, we investigate a new method to determine the background using a phase-only spatial light modulator (SLM) for the modification of the spatial coherence of a collimated beam, resulting in suppressing nonrandom interference, hence enabling the absolute comparison of signal and background. The methodology of suppressing nonrandom interference via wavefront shaping (WFS) is based on the idea of light manipulation presented in [16], where the phase of light incident on a scattering sample is adjusted to force an enhancement in the constructive interference at an arbitrary location behind the medium. In addition to its application in CBS, this technique has multiple uses in various imaging modalities, material processing, and communications [17]. Consequently, this method enables absolute comparisons to be made between signal and background measurements, which, to the knowledge of the authors, has not been demonstrated before. The newly developed method of measuring the CBS background enables a correct comparison to be made between the measurement and the corresponding simulation for this type of setup.

2. Materials and Methods

2.1. Measurement Setup

The measurement setup is a system to acquire the CBS signal using a Fourier arrangement modified by implementing an SLM. The Fourier arrangement transfers the angular distribution of the reflected light into a spatial distribution in the detection plane. Due to the dimensions of the sensor, only small angles can be imaged. The system is shown in Figure 1. Monochromatic light emitted by a 532 $\mathrm{n}$$\mathrm{m}$ laser (LS) (LaserBoxx LCX-532S, Oxxius Simply Light, 4 Rue Louis de Broglie, 22300 Lannion, France) is coupled into the system via a single-mode fiber (SF) with a core diameter of 4 $\mathrm{\mu }$$\mathrm{m}$. A linear polarizer (LP1) selects the suitable polarization plane defined by the SLM. A collimation lens (CL), with a focal length of 150 $\mathrm{m}$$\mathrm{m}$, collimates the beam, which is laterally limited by an adjustable aperture (A1) to a size of 8 $\mathrm{m}$$\mathrm{m}$. After this beam modeling, the light hits the phase-only SLM (GAEA2.1, HOLOEYE Photonics AG, Volmerstrasse 1, 12489 Berlin, Germany), where it is deflected due to diffraction by a blazed phase grating on the SLM. To filter the diffraction orders and select the order of interest, a telescope is used consisting of a focus lens (FL1) with a focal length of 60 $\mathrm{m}$$\mathrm{m}$, an adjustable aperture (A2), and another collimation lens (CL) with a focal length of 60 $\mathrm{m}$$\mathrm{m}$. The aperture functions as a spatial filter, and the correct pupil diameter is crucial for the correct absolute measurement of signal and background. The light then propagates towards a beam splitter (BS) and is redirected onto the sample (S), which is mounted on a vibration table (VT) for suppressing speckles. To reduce stray light in the casing, the light transmitted through the beam splitter at the first interaction is guided towards a beam trap (BT). The backward scattered light from the sample propagates through the beam splitter; is filtered by a linear polarizer (LP2), which, in the case of the measurements shown in this study, selects the polarization parallel to the illuminating linear polarizer (LP1); and is imaged by a Fourier lens (FL2) with a focal length of 500 $\mathrm{m}$$\mathrm{m}$ onto a CCD-sensor (CCD) (Pixis512, Teledyne Princeton Instruments, 3660 Quakerbridge Road, Trenton, NJ 08619, USA). resulting in a detectable angular range of ±0.65° around the backward direction.

The general measurement procedure is as follows. At first, the CBS signal is measured, followed by a measurement of the absolute background; the corresponding dark measurements are acquired in the same order. These compensate for all contributions that are not caused by interaction with the samples.

2.2. Implementation of a Spatial Light Modulator for Coherence Manipulation

By manipulating the spatial coherence, the SLM can switch the CBS signal on and off by suppressing non-random interference. As explained above, this opens up the possibility of making an absolute comparison between the signal and the background, without having to change the experimental setup. The steps necessary for implementing the SLM are discussed below.

The SLM’s pixel fill factor equals 90%, with a pixel size of 3.74 $\mathrm{\mu }$$\mathrm{m}$, meaning the diffraction of the SLM electrode grating on the front surface must be considered. This grating leads to a large amount of light being diffracted without undergoing phase manipulation. In order to use only the light that has undergone phase manipulation, a vertically aligned blazed grating is displayed on the SLM pixels, with a consistent period throughout the experiments. The blaze grating period, determined to be 11.22 $\mathrm{\mu }$$\mathrm{m}$, is selected so that its diffraction orders are distinguishable from those originating from the front surface grating, and the experimental setup is adjusted accordingly.

When measuring the absolute background, a random phase pattern with uniformly distributed phase values and fixed-size pixel clusters is added to the blazed grating. This suppresses the non-random interference, which leads to the CBS signal, by significantly reducing the spatial coherence. Photons that propagate along the same path but in opposite directions within the medium now have different randomly chosen phases. Hereby, several cluster sizes were tested, and the best results were achieved using a 7 by 7 pixel cluster. A trade-off must be made between sufficient randomization to attenuate the CBS signal and the amount of light transferred to the diffraction order determined by the blazed grating.

Depending on the cluster size, additional diffraction caused by the random pattern diminishes the intensity and enlarges the illumination spot on the sample. When measuring the CBS signal, no additional random pattern is used, and the measurements are unaffected by this. To match the intensity in both scenarios and enable an absolute comparison of the signal and the background, a two-dimensional binary diffraction grating is added to the blazed grating before measuring the CBS signal. Matching the clustering of the random pattern, the period of the binary grating must be twice the cluster size, as a grid period must always contain two pixel clusters that have the specified contrast to each other, so the spatial frequency is matched. Furthermore, the phase contrast $C_{\mathrm{\Phi }}$ of the binary grating must be matched to that of a random phase pattern. It is defined as the ratio of the square root of the phase variance, ${\sigma }_{\mathrm{\Phi }}^2$, to the average phase value $\overline{\mathrm{\Phi }}$ [18]. Accordingly, the phase contrast of uniformly distributed phases on the interval from 0 to 2 $\pi$ is given by $C_{\mathrm{\Phi }}=1/\sqrt{3}$. To transfer the change in phase contrast to the SLM, the difference in phase values for the binary grid needs to be adjusted by a factor of $1/\sqrt{3}$ based on the aforementioned calculation. Excerpts of the phase patterns utilized for the generation of the spatially coherent and randomized signal and their respective composition are shown in Figure 2.

Calibration of the SLM is performed by the manufacturer, and all patterns are displayed in combination with the wavelength dependent calibration phase pattern.

In conclusion, an absolute comparison of the signal and the background requires joint use of the blazed grating in combination with a contrast-adjusted, two-dimensional binary grating for signal determination. The same blazed grating in combination with a random pattern featuring uniformly distributed phases and a cluster size equal to half the binary grating’s period size is used for background measurement.

2.3. Quartz Glass Samples

To verify the measured CBS signal, quartz glass-based phantoms, with spherical air-filled inclusions, were used. The optical properties of these phantoms were determined using integrating sphere measurements. It was also possible to determine the size of the scatterers and their standard deviation individually for each phantom by fitting calculations based on Mie theory. Three phantoms with different scatterer concentrations were used: Phantom 1, with a scatterer diameter of d = 3.38 $\mathrm{\mu }$$\mathrm{m}$, a standard deviation of std = 0.07 $\mathrm{\mu }$$\mathrm{m}$, and a relative scatterer volume concentration of cv = 0.042, with a sample thickness of 1 $\mathrm{m}$$\mathrm{m}$ and a sample diameter of 50 $\mathrm{m}$$\mathrm{m}$; Phantom 2, with a scatterer diameter of d = 4.0 $\mathrm{\mu }$$\mathrm{m}$, a standard deviation of std = 0.07 $\mathrm{\mu }$$\mathrm{m}$, a relative scatterer volume concentration of cv = 0.039, and two samples with smooth and rough surfaces with a thickness of 1 $\mathrm{m}$$\mathrm{m}$ and a sample diameter of 50 $\mathrm{m}$$\mathrm{m}$; Phantom 3, with a scatterer diameter of d = 4.55 $\mathrm{\mu }$$\mathrm{m}$, a standard deviation of std = 0.07 $\mathrm{\mu }$$\mathrm{m}$, a relative scatterer volume concentration of cv = 0.030, and two samples with smooth and rough surfaces with a thickness of 2 $\mathrm{m}$$\mathrm{m}$ and a sample diameter of 50 $\mathrm{m}$$\mathrm{m}$. The spherical shape and relatively good monodispersity of the scatterers enable an accurate calculation of the scattering phase function, which leads to precise simulations of the signal for comparison with measurements. All phantoms show negligible absorption in the analyzed spectral range.

2.4. Monte Carlo Simulation

The measured data is compared to results obtained via an electric field Monte Carlo simulation [19,20] (MC simulation). This method allows for taking both the light’s polarization and phase into account; therefore, it is well suited to simulate an interference effect like CBS. The MC simulation used was developed in-house, and the basic algorithms have been published [11,21]. It has been verified by comparison with analytic solutions of the vectorial radiative transfer equation, also derived by us [11,12].

In an MC simulation of light propagation, light is simulated by tracing many energy packets through the given system. Absorption and scattering are treated stochastically. The turbid medium is described by the scattering and absorption coefficients, ${\mu }_{\mathrm{s}}$ and ${\mu }_{\mathrm{a}}$, and the polarization-dependent amplitude scattering matrix, i.e., the probability distribution for light being scattered in a given direction. The polarization and phase of an energy packet are described by the combination of a Jones vector $E\in {\mathbb{C}}^2$ and a local frame. In this work, spherical scatterers describing the air bubbles are taken into account. Hence, the matrix-valued amplitude scattering function $S:\left[0,\pi \right]\to {\mathbb{C}}^{2\times 2}$ can be obtained from the Mie solutions of Maxwell’s equations, being diagonal.

CBS arises from the interference of partial waves that travel along the same path through the medium, yet in opposite directions. The electric field of an energy packet after N scattering events can be described by the multiplication of the initial Jones vector with the matrix

$$M^{\mathrm{F}}=R\left({\varphi }_{N+1}\right)\left(S\left({\theta }_N\right)R\left({\varphi }_N\right)\cdots S\left({\theta }_1\right)R\left({\varphi }_1\right)\right),$$

(1)

where ${\theta }_k$ are the scattering angles, ${\varphi }_k$ the rotation angles to transform the Jones vector from the previous local frame into the next one, and $R\left({\varphi }_k\right)\in {\mathbb{R}}^{2\times 2}$ the rotation matrices. $R\left({\varphi }_{N+1}\right)$ is the rotation matrix that transforms into the meridian plane spanned by the medium’s surface normal and the energy packet’s propagation direction. Similarly, the electric field of an energy packet after traveling along the reverse path can be obtained from

$$M^{\mathrm{R}}=R\left({\varphi }_0^{\prime }\right)S\left({\theta }_1^{\prime }\right)R\left({\varphi }_1^{\prime }\right)\left(R\left({\varphi }_2\right)S\left({\theta }_2\right)\cdots R\left({\varphi }_{N-1}\right)S\left({\theta }_{N-1}\right)\right)R\left({\varphi }_{N-1}^{\prime }\right)S\left({\theta }_N^{\prime }\right)R\left({\varphi }_N^{\prime }\right),$$

(2)

where angles marked with a prime differ from the ones used in Equation (1) in order to match all local frames.

In order to reduce calculation times, the last flight method [21,22,23,24,25,26] has been used for variance reduction. The MC simulation itself is implemented in OpenCL C, and calculations are performed on GPUs.

Due to the long focal length of the collimating lens and the small diameter of aperture A1, only a small spatial range of the Gaussian beam is used. Hence, the truncated section of the Gaussian profile is well approximated as a plane wave. In the simulation, the illumination, however, is assumed to be a perfect plane wave. This can lead to small deviations. However, this error is expected to play a subordinate role.

3. Results and Discussion

As discussed above, modifying the spatial coherence without altering the experimental environment enables an absolute comparison of the signal and the background. However, despite subtracting a dark measurement from each of the two measurements, an offset remains. This offset is caused by stray light originating from optical components and by light leaving the sample in an angular range that falls outside the detection range, which is then led back onto the sensor via interaction with the measurement setup. Compensating for this background light allows for a comparison of the measurements with the results acquired via the aforementioned MC simulations. The experimental data and the simulations are presented as radial means over the rotation angle $\varphi$ to reduce noise and improve comparability. This is carried out even though a polarization-induced anisotropy of the signal is present, which is valid because the averaging is performed equally for both the simulations and the measurements.

3.1. Scattering and Stray Light Compensation

In addition to correcting the results using a dark measurement, further corrections are required. An additional intensity offset, which is constant versus the angle, must be compensated for when comparing the measurements with the MC simulations. To investigate this offset further, we conducted additional simulations, evaluating the integrated detector intensity in relation to the absolute number of photons passing through the Fourier lens (FL2 in Figure 1), as an indicator for light entering the detection system. In the simulation, the detector intensity, normalized to the number of photons on the lens, decreases with the scattering coefficient. However, the ratio of the number of photons hitting the lens to the number of simulated photons stays similar. Nevertheless, the measurements show similar detected intensities for all samples.

Additionally, the influence of surface roughness must be considered, as this leads to increased diffuse surface scattering and a larger amount of stray light. This is not represented in the MC simulation, but it does increase the offset. To the knowledge of the authors, this has not yet been demonstrated systematically in the literature. To further investigate the influence of surface roughness, goniometric measurements were performed, analyzing the intensity distribution of the reflected light. Figure 3 shows these measurements, with the intensity depicted versus the detection angle. In this case, the sample was illuminated at an angle of 8.2° referenced to the surface normal, with the detection centered around the corresponding reflection direction at 16.4°. In Figure 3, the specular reflections of Phantom 2 (red lines) and Phantom 3 (blue lines) for smooth (solid lines) and rough (dashed lines) surfaces are shown. The broadening of the measurements for smooth surfaces is expected to be caused by the system transfer function. Measurements for rough surfaces show a distinct decrease in reflected intensity and signal broadening. Phantom 3 exhibits this behaviour to an even greater extent, caused by greater surface roughness.

Figure 4 shows measurements of the CBS signal versus the detection angle ${\theta }_{det}$ for Phantom 2 (left graph) and Phantom 3 (right graph), where all measurements have been corrected with the corresponding background. For both phantoms, the measurements using samples with smooth (solid blue lines) and rough (solid red lines) surfaces are depicted. In both cases, an increased surface roughness leads to a decrease in the achievable enhancement, as expected following the results of the goniometric measurements. Furthermore, the influence of the rough surface is more pronounced for Phantom 3. Hence, the impact of the surface roughness on the signal to background ratio increases for a decreasing reduced scattering coefficient.

These observations suggest that the measurements for lower scattering samples with an increased surface roughness contain a higher stray light component, which ultimately increases the offset. Furthermore, the amount of stray light is also dependent on the sample-specific angular reflection distribution. In Figure 5, the impact of this offset is shown. However, these circumstances are only partially included in the MC simulation, which does not, for example, take into account the lateral extension of the samples or the light exiting the sides of the phantoms. The MC simulation used does not take all optical elements and system-relevant mechanical structures, where light can be scattered, into account. Expanding the simulation to include all these components is highly complicated, and the calculation time would increase significantly.

The correction factor for this additional offset, which was assumed to be constant for all angles, is determined experimentally by comparing the measured normalized enhancement with the theoretical curve, acquired via the MC simulation, as shown for the Phantom 1 sample on the left graph of Figure 5. When the measurement (dashed green line) is adjusted by an offset of 26%, the theoretical curve (solid blue line) and the resulting corrected curve (dashed red line) match very well. Phantom 2 with a smooth surface has the same relative stray light offset to match the simulated signal. This leads to the conclusion that the remaining offset for the samples of Phantom 1 and Phantom 2 is caused by scattering at optical components. These two samples suppress long propagation paths sufficiently so that the additional offset due to light exiting the phantom outside the illumination is negligible.

When changing the integration time, the offset remains the same relative to the overall intensity of the background, as shown in the bottom-right graph of Figure 5. The top right of Figure 5 shows an example of the corrected enhancement for measurements of the smooth sample of Phantom 1. The curves for 10 $\mathrm{s}$ (solid blue line), 20 $\mathrm{s}$ (dashed red line), and 30 $\mathrm{s}$ (dashed green line) match when corrected with their respective offsets. These are shown in the bottom right graphic. Similar measurements and evaluations were performed on all three Phantoms.

Measurements with a detection polarizer (PL2) rotated through 90° exhibit the same relative stray light compensation for each phantom as the aforementioned measurements with parallel polarizers (PL1 and PL2).

3.2. Absolute Measurement

Figure 6 shows an example of using a combination of a blazed phase grating and a two-dimensional binary grating with adjusted phase contrast to measure the CBS signal and the same blazed phase grating with an additional clustered random phase pattern to determine the background. As an example, an absolute measurement for Phantom 1 is shown, where the stray light correction discussed in Section 3.1 is subtracted. The data on both sides of Figure 6 is averaged versus the rotation angle $\varphi$ and plotted against the detection angle ${\theta }_{det}$.

On the left graph of Figure 6, the absolute values of the CBS signal (dashed red line) and the background (solid blue line) are compared. As well as showing very good agreement outside the angular range in which the CBS cone is prominent, the comparison reveals that the background is not flat. This non-flat background is inherent to the system and reflects the influence of the experimental setup on the propagation of light, due to the system’s transfer function and the reflection cone of the samples. Consequently, calculating the intensity enhancement by normalizing to the intensity at large angles yields an incorrect result, as it often does in the literature [6]. The innovation presented here, which uses an SLM to suppress the coherent signal component, now enables correct normalization and eliminates the need for further assumptions. As can be seen on the right graph of Figure 6 (dashed orange line), this results in the enhancement curve broadening and decaying more slowly at larger angles. Conversely, when the full background is used for normalization (solid green line), the CBS cone’s enhancement is smaller. As expected by theory, the graph decays to a flat curve progression slightly above one. This demonstrates the importance of normalizing with the correct background to obtain the right curvature.

3.3. Comparison of the Measurement Results with Monte Carlo Simulations

To validate the measurements, the CBS signal was compared to the MC simulations described above. Five measurements for each sample have been performed in the represented series to ensure sufficient averaging, where the standard deviation is sufficiently small (mean standard deviation ${\mathrm{std}}_m$ = 0.012 arb. un.) and not shown, to ensure better visualization of the results. The averaging over the rotation angle ${\varphi }_k$ was carried out after correction by the homogeneous offset for each pixel. All three samples show very good agreement with the simulated values. In Figure 7, this is shown exemplarily for Phantom 1 and Phantom 3. The results for Phantom 2 are not shown here, as they fall between those for Phantom 1 and Phantom 3, which would be detrimental to clarity. However, the results show a comparable level of agreement. The upper graph shows the mean value of the measurement for Phantom 1 (dashed orange line), compared with its simulation (solid green line) and the mean value of the measurement for Phantom 3 (dashed red line), compared with the respective simulation (solid blue line). The lower graph shows the relative deviations of each measurement and the corresponding simulation, for Phantom 3 (solid blue line) and Phantom 1 (dashed green line). Both are plotted versus the chosen angular spectrum, which was limited to 0.25° to better emphasize the CBS enhancement cone. Both measurements show less than ±5% deviation over the whole angular spectrum. In the literature, simulation results are often fitted iteratively to preceding measurements, with the optical parameters being freely changed [7,27,28]. However, it is not always possible to guarantee that normalization to a potentially incorrect background is not compensated for by assuming altered optical parameters in the fit, as these may differ from the sample’s actual optical properties. In contrast, the excellent characterization of the phantoms used in this study, the use of an SLM to suppress the coherent signal component to acquire the correct background, and the verified MC simulation allow for excellent agreement between the measurement and simulation results, without altering the optical properties. This methodology does not require fitting and represents an innovation. A slight decrease in the curvature of the slope around the exact backwards direction is noticeable. This could be the result of errors made by the discretization in the MC simulation and the limited resolution of the detector.

4. Conclusions

The application of an SLM to modify the spatial coherence of the incident light beam has proven useful and applicable to determine the background of a CBS measurement. Using a setup with an SLM to measure the CBS signal enabled us to suppress the interference peak without changing the experimental environment. This enables the background and the CBS signal to be compared absolutely, allowing for correct normalization and facilitating comparison with a simulation. To the knowledge of the authors, this has been achieved for the first time. However, to accomplish the latter, offset compensation was necessary, whereby volume scattering, stray light, and surface roughness must be taken into account. Using this technique, we achieved a high level of agreement between the measurements and the MC simulations. For all measured samples, the deviation from the respective simulations across the entire angular spectrum is less than ±5%. In order to make a better comparison between the simulation and measurement results, it would be beneficial to develop a more complex MC simulation. In conclusion, the spatial coherence-modifying characteristics of an SLM are useful in this context and can be transferred to other applications.