1. Introduction

Diffusing wave spectroscopy (DWS) is a group of techniques that use light transmitted through a scattering medium to extract the dynamics and the properties of the scattering medium [1–7]. DWS methods rely on detecting the transmitted speckle light field from the scattering medium and measuring the speckle decorrelation time as a quantifier of the scattering medium’s dynamics. The chief advantage of DWS lies in its non-invasiveness when used to probe biological samples. The technique is widely used in biological studies, such as measuring cerebral blood flow through tissues and bones [8–10]. In DWS, the speckle decorrelation time between the measured data is extracted as a metric to provide the dynamic information of the scattering medium, and the speckle decorrelation time is typically calculated in two different methods: temporal sampling and spatial ensemble.

Temporal sampling based DWS is also called diffuse correlation spectroscopy (DCS) [8–16]. In general, DCS applications, the laser light propagates through a dynamic scattering medium and the transmission is collected back by using a single photon avalanche diode (SPAD) or a photodetector. Typically, a single speckle or a small collection of speckles (constituting a single aggregate speckle) is detected at a sufficiently high frame rate such that the inverse of the frame rate is significantly shorter than the expected decorrelation time of the speckle [17–23]. The typical DCS algorithm then calculates the signal’s time correlation to measure the scattering medium’s dynamics. The autocorrelation function derived from photon’s arrival time series can be fitted to multiple scattering models where models used are determined by the motion type and the scattering regime of the scattering medium [6]. The intensity autocorrelation is fitted to the ${g_2}(\tau )$ function, and the field autocorrelation function ${g_1}(\tau )$ is then retrieved via the Siegert relation. In biological applications, DCS generally employs a fast (typically sampled in MHz range) sensor with high quantum efficiency and low detector noise characteristics, such as an ultrafast SPAD or other forms of fast time-of-flight photon detectors, to better detect the typically weak signals.

There are different and more advanced forms of DCS. In the interferometric DCS (iDCS) [18–25], a reference light field is introduced to enable interferometric detection of the optical signal. This provides the ability to reject stray light and enable interferometric suppression of the detector noise when measuring the optical signal. It is also possible to simultaneously detect multiple speckles with multiple high-speed detectors, such as a SPAD array [17–19,21]. Recently, a method that combines interferometry and ultrafast cameras for multiple speckle detection has also been reported that sought to combine both advantages to yield a stronger DCS signal [26–28].

The alternate DWS approach is based on the use of spatial ensemble. This class of method is generally referred to as the speckle visibility spectroscopy (SVS) [29–32]. Other approaches utilizing spatial ensembles of speckles include laser speckle contrast imaging (LSCI) [33–41]. Here, a high pixel count camera is used to simultaneously detect multiple speckles in the optical transmission. Generally, the camera exposure time is set to be substantially longer than the speckle decorrelation time (typically a magnitude or more). This results in multiple different speckle patterns summing up in a single camera frame within the exposure time and yielding a blurred or washed-out speckle pattern. The speckle decorrelation time can then be calculated from the degree of blurring, or more specifically, the contrast of the speckles over all the detected speckles in the whole frame. By the nature of its operation, SVS does not require high-speed detectors and can work with mature and commercially available video-rate sCMOS cameras. In addition, SVS can naturally collect more transmitted light as video-rate cameras generally have very high pixel counts – the photon count per pixel may still be low, but a higher aggregate signal is obtained by using more pixels. On the other hand, SVS must contend with the relatively high detector noise characteristics of these cameras. To mitigate this issue, an interferometric SVS (iSVS) system has recently been demonstrated and shown to work well at suppressing detector noise through the interferometric measurement of the optical signal [29].

For DWS methods in general (both DCS and SVS), the accuracy of the speckle decorrelation time is a key specification. Mathematically, this accuracy can be quantified as the ratio of the measured decorrelation time to the standard error of the measurement, and it can be interpreted as the signal-to-noise performance specification of DWS methods. While DCS and SVS employ different methods to determine the speckle decorrelation time, their SNR performance can ultimately be tied back to the number of independent observables (NIO) and the detected photon flux [30]. The NIO corresponds to the number of speckle decorrelation events measured for single speckle DCS methods. The NIO then scales proportionately with the speckle count if multiple speckles are simultaneously detected for multi-speckle DCS methods. For SVS methods, the NIO corresponds to the number of speckle grains observed per exposure. Reference [30] details the exact NIO relationships for both DCS and SVS methods. The reference also compared the application of both DCS and SVS methods for biological studies, and found that for blood flow measurements, it is generally easier for SVS method to achieve a higher NIO count and, therefore, better accuracy in decorrelation time measurement. In addition, SVS methods cannot quantify the exact shape of the time decorrelation function, while DCS methods are generally able to do so.

The mathematical treatment of SNR performances in Ref. [30] assumes that the camera/detector is noise-free or that the camera/detector noise variance is low enough to have little impact on the measurements. While such an assumption is reasonable for a broad range of DWS applications, it will generally fail for low-light situations where the low number of detected signal photons can easily be overwhelmed by the intrinsic camera or detector noise, especially in biological applications where the exposure and emission levels are restricted [42]. We recognize that James et al has established the analysis and optimization of SNR performance for holographic DCS, but detailed noise analysis has not been published for iSVS type systems yet [43].

This study aims to expand the mathematical model to account for detector noise. We will focus our analysis on an iSVS system, as such a system is specifically designed for low-light application uses, where interferometry is employed to enhance the detection of the signal light in the presence of significant detector noise. This advantage of iSVS over SVS has been experimentally demonstrated in previous studies [29]. It was shown that human cerebral blood flow could be determined with a high sampling rate of 100 Hz, even when the number of collected signal photons per pixel was low (i.e., below the camera noise variance) [29]. That promising result motivates our current analysis.

In this paper, extending from the iSVS technique first proposed by Xu et al in Ref. [29], we mathematically analyze and experimentally verify the signal-to-noise ratio (SNR) gains of iSVS over SVS in the presence of camera noise. We also show that with reasonable prior knowledge of the average signal intensity, the SNR of iSVS will always be higher or on par than SVS with a uniform reference field. Previous work used a uniform plane wave reference beam with a specific angle (off-axis interferometry) to enable cropping in the Fourier domain to select the signal [29]. We show that such requirements can be relaxed, and the results are nearly independent of the reference beam’s shape and angle, allowing one to use a non-uniform wave reference, on-axis interferometry setup with no angle between reference and source, or even an entirely spatially speckled reference to achieve a similar result. We also deployed standard calibration methods for iSVS to calibrate noise in experiments, which is necessary for the system to operate in a wider range of intensity. We demonstrated that iSVS has the advantage of retrieving more consistent decorrelation time, especially in signal limited situations where the number of photons collected are low compared to noise. Finally, we conclude that the properties discovered justify and simplify iSVS application implementation.

2. Theoretical derivations and results

In this section, we derive the SNR expressions associated with iSVS, then compare the theoretical performance of iSVS and SVS in the presence of camera noise. The SNR is defined by the ground truth decorrelation time $\tau $ and the standard error ($SE(\tau)$) of the measured $\tau $ as:

To obtain a reference-subtracted or reference-calibrated image, we subtract the captured image by the reference image. The reference-calibrated signal readout, in terms of photon counts, on pixel j from the CMOS sensor can be mathematically expressed as:

$K_{up}^2$ can be expressed as:

Assuming the negative exponential decay holds, we approximate the decorrelation function as follows:

By substituting Eq. (7) into Eq. (6), we can obtain:

$K_{down}^2$ can be expressed as the mean of the calibrated image, which only depends on the sample intensity as the reference is subtracted. By combining the denominator and numerator of the term ${K^2}$, we obtain its full relationship with ${\tau _s}$ and ${\tau _{field}}$:

The above analysis assumes ${I_{Rj}}$ is much greater than ${I_{Sj}}$. For situations where the reference field is uniform, the condition is well satisfied for all pixels if it is well satisfied for any given pixel. When the reference field intensity is spatially heterogeneous, this condition may not be satisfied for all pixels. In this case, the above speckle contrast expression is still valid, except we would need to exclude pixels where ${I_{Rj}}$ is low compared to ${I_{Sj}}$ from our count. All averages and expectation values would only take into account the pixels where the condition ${I_{Rj}} \gg {I_{Sj}}$ is satisfied. In experiments, it is generally possible to arrange $\overline {{I_R}} $ to be much higher than $\overline {{I_S}} $, so that the ${I_{Rj}} \gg {I_{Sj}}$ is satisfied for the majority of the pixels. In such a case, the class of pixels to be excluded is so small that including them will only lead to a negligible error. In our later experiments, this is proven to be true since the reference intensity can be tuned to an arbitrarily high level, as long as it does not saturate the camera. Even in situations where reference is a fully developed speckle with negative exponential distributed intensities, we believe the assumption that ${I_{Rj}} \gg {I_{Sj}}$ should still hold for the majority of the pixels because iSVS typical operates in photon-limited scenarios where ${I_{Sj}}$ is minimal in comparison. However, we do want to emphasize this assumption clearly for future designs.

We preserved the first term in the numerator of Eq. (10) to show the contributions of both ${\tau _s}$ and ${\mathrm{\tau }_{field}}$. In practice, we can further simplify the expression by noting that the first term is much smaller than the second and third numerator terms. If we set ${U_I} = \overline {{I_R}} /\overline {{I_S}} $, and define ${N_{ST}}$ as the average number of sample beam photons per camera pixel ${N_{ST}} = \alpha \overline {{I_S}} T$:

The speckle contrast can be exactly determined for an ideal sensor with an infinite number of pixels so that the ensemble averages in the above calculation mirror the spatial averages. However, since only a finite number of pixels is available in a camera, the measured speckle contrast squared $\widehat {{K^2}}$ will contain an error, as $\widehat {{K^2}}$ will vary from measurement to measurement. The error will depend on the spatial profile of the reference beam and the NIO. We define a metric $R = \frac{{\left\langle {I_{Rj}^2} \right\rangle }}{{{{\left\langle {{I_{Rj}}} \right\rangle }^2}}}$ to evaluate the uniformity of reference distribution. $R = 1$ represents a perfectly uniform reference intensity distribution, and a nonuniform reference pattern will have $R > 1$. As an example, a static speckled reference field where the speckle grain size is comparable or larger than the camera pixel will have an R-value approximately equal to 2. We also assume that the error of ${K^2}$ from measurement is only contributed by the numerator $K_{up}^2$, because $K_{down}^2 = N_{ST}^2$ can be accurately determined in experiments as a prior. Experimentally, we can determine $K_{down}^2 = N_{ST}^2$ exactly by blocking the reference beam and averaging arbitrarily many image frames.

This leads to a speckle contrast estimate given by:

Equation (13) can be rewritten to provide a mathematical expression for determining ${\tau _{field}}$ and its measurement-to-measurement variation from the measured speckle contrast:

The above analysis indicates that, even though iSVS is an interferometry method, the requirements on the reference field are significantly relaxed. While many interferometry methods require the reference field to be planar and have a spatially invariant phase (or a phase with a linear spatial ramp), iSVS can be performed with a reference field with a spatially varying phase and amplitude. The SNR of iSVS is not penalized by a spatially varying phase and is only mildly penalized by a factor of $\sqrt {\frac{2}{{3R - 1}}} $ for spatial heterogeneity. It should be noted that the number of independent observables (NIO) may vary under extreme heterogeneous reference cases, such as when some pixels have a reference intensity lower than the camera noise or the signal intensity, which breaks the assumption overlying the above analysis. In such cases, one should account for those pixels and modify the NIO appropriately. However, these pixels are generally an overwhelming small class when the average reference intensity is set much higher than the sample intensity. In the following iSVS experiments, the reference intensity is sufficiently high such that less than 0.1% of the pixels had a reference intensity lower than camera noise and corresponding $\overline {{I_S}} $, and therefore, the effect of scattered reference illumination on NIO was negligible in the experimental data presented in this paper.

Figure 1 shows the SNR as a function of the decorrelation time for different averaged sample beam intensities $\overline {{I_S}} $, plotted with the following parameters: ${\sigma _n}_{cam}^2 = 8$ ph/pxl, ${I_R} \approx 3000$ ph/(pxl.ms), $NIO = 2000, $ and $T = 200\; \mu s$. Camera noise variance is converted to keep consistency with the measured gray scale value. Note that the decorrelation time on the x-axis is the intensity decorrelation time. Also, the intensity units are simulated to match those in the experiments closely. Figure. 1(a) demonstrated the ability of both iSVS systems, uniform reference and non-uniform reference, to achieve significantly better SNR than the typical SVS system as it overcomes the camera noise variance. In Fig. 1(b), the performance of SVS and uniform iSVS system overlaps, while non-uniform reference iSVS systems has a lower SNR as a result of larger R value from the speckled reference. From Fig. 1(c), a glance of SNR performance over a wide range of sample intensity, one note that iSVS performs better than SVS at low signal intensities $\overline {{I_S}} $, but both uniform iSVS and SVS SNRs converge to the same performance as $\overline {{I_S}} $ is high enough to overwhelm the camera noise variance. The non-uniform reference penalizes the SNR, thus showing a lower SNR, but the advantage at low sample intensity levels still holds. To further validate the theoretical results, we numerically generated different speckle patterns with results shown in Supplement 1, and the numerical simulation agrees with the results shown in Fig. 1 [48].

 

Fig. 1. Theoretical comparison between iSVS and SVS by using the SNR and the derived speckle contrast equations. (a), (b) SNR as a function of the decorrelation time at different averaged intensities $\overline {{I_S}} $ of the sample laser light field at below or above the cameras noise variance. The corresponding sample intensity is labeled in the title. Decorrelation time is the intensity decorrelation time ${\tau _s}$. (c) average SNR as a function of $\overline {{I_S}} $ for ${\tau _s} = [{3,\; 20} ]\; \mu s$. As evident, iSVS performs better than SVS at low intensities $\overline {{I_S}} $, but both converge to same SNR as $\overline {{I_S}} $ overcome the camera noise variance. Parameters: $\sigma _{{n_{cam}}}^2 = 8 $ ph/pxl, ${I_R} = 3000\textrm{ph}/({\textrm{pxl}.\textrm{ms}} )$, $NIO = 2000$, and $T = 300\; \mu $s. Reference beam is uniform ($R = 1$) or speckled in iSVS non-uniform ($R = 2$).

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3. Experimental arrangement and results

To validate the theoretical and simulation results shown previously in sections 2 and in numerical simulations, we designed an experiment to measure the SNR utilizing a rotating diffuser and an interferometer. The rotating diffuser is controlled by a DC motor, whose rotation speed is digitally controlled; thus, the decorrelation time of the speckle images is controlled electronically. The experimental arrangement is shown in Fig. 2. It is based on a Mach-Zehnder interferometer, whose input is a highly coherent (single-mode, single-frequency) continuous wave 532 nm collimated laser beam [Spectra-Physics Excelsior-532-200-LDRH]. Note that similar results should be obtained for any wavelength as long as the camera’s quantum efficiency is reasonable at the illumination wavelength.

 

Fig. 2. Experimental arrangement for measuring the decorrelation time and SNR of a rotating diffuser, acting as moving sample. A diffuser 2 can be inserted to generate a distorted (randomly scattered) reference laser light beam in iSVS. In SVS, the reference laser beam is blocked.

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The input laser beam is first divided by a beam splitter (‘BS1’ in Fig. 2) into two light beams; one is the sample laser beam, and the other is the reference laser beam. The sample laser light beam propagates through a rotating diffuser and a static diffuser. The static diffuser counteracts the one-dimensional smearing effects from a single rotating diffuser, generating random speckles at the output. The reference laser light beam propagates in free space onto a second beam splitter (‘BS2’ in Fig. 2). A diffuser can be inserted in the path of the reference beam to generate a speckled reference laser light beam. We use tunable neutral density (ND) filters to control or to block (shut down) the intensities of the sample and the reference laser light beams. At the second beam splitter (‘Beam Splitter 2’ in Fig. 2), the two laser light beams interfere and form an interference pattern that is imaged to and recorded by a scientific CMOS camera sensor [PCO Edge 5.5] with an exposure time T.

To satisfy the prior of measuring $\left\langle {{N_j}} \right\rangle $ accurately, the reference laser light beam was blocked (at $t = 0$) by closing the reference arm shutter, creating a setup similar to an SVS setup, and $\left\langle {{N_j}} \right\rangle $ was estimated accurately from an average over $1000$ images. This measurement is utilized later in calculating speckle contrast K and ${K^2}$ for iSVS. Camera noise and its variance are also measured with both shutters closing, measuring only the camera background fluctuations.

Note that we are using an on-axis Mach–Zehnder interferometer setup, wherein there is no angle between the reference and the sample beams. This approach, as showed in previous works on laser Doppler holography for human blood flow imaging, presents both advantages and limitations [49,50]. On-axis design simplifies alignment and maximizes field of view, beneficial for working with lower sample beam intensities by leveraging the reference laser beam to mitigate both the camera and laser noise. However, on-axis lacks certain off-axis interferometry advantages like spatial Fourier filtering and frequency-shifted reference beam for selective frequency imaging [49,50]. Since we are implementing an on-axis setup, there is no need to enforce an angle between the reference and the source; the two beams can be aligned to overlap each other.

A benchmark process is established to compare the performance of iSVS and SVS. The benchmark includes rotating the diffuser at 16 different rotational velocities. The rotation speed starts at 0 percent, then goes from 25 percent to 95 percent in 5 percent increments. At 100 percent rotational speed, the motor operates at 30 rpm. Also, the motor’s rotational speed and the electronically controlled velocity percentage are believed to be linear across most of the range except for low rotational speed, for which we have intentionally skipped. The speeds are specifically chosen such that ${\tau _s} \ll T$ holds true. A set of reference ${\tau _s}$ is obtained by DCS experiment operating at a high signal level using an adjustable optical receiver [Newport 2051-FS] sampling at 1.25 MHz. Then, the temporal autocorrelation curves are fitted with a negative exponential function, and decorrelation time is extracted for each rotation speed. Based on this benchmarking DCS reference experiment, the decorrelation time associated with the rotating diffuser was related to the rotational velocity of the DC motor, see Supplement 1.

Furthermore, despite incorporating the framework of shot noise and camera noise in theory, there are other external factors during experiments that would impact the noise profiles. To mitigate this and calibrate noise accurately, noise is measured instead of calculated: by closing both shutters, camera noise can be retrieved by calculating the variance of multiple measurements. Reference noise and sample arm noise can also be measured by keeping the diffuser in the static position and making multiple measurements. A total of 100 calibration measurements are taken for each trial during experiments, and noise is measured and factored in for both SVS and iSVS. Details of the calibration are presented in Supplement 1.

We also observed additional noise terms with low spatial frequency characteristics in the experiment. These terms are likely attributable to the experimental spatial uniformity of the reference field and unaccounted laser mode instability. They introduce systematic errors into the experimental determination of ${\tau _{field}}$ and ${\tau _s}$. We addressed this by subtracting the empirically determined systematic error from the measured speckle contrast K prior to calculating the SNR. For the details regarding the offset calibration as well as results without calibration, please refer to Supplement 1.

We calculated the R values for the experimental reference beams. The distorted reference field pattern is speckled, but over 99.9% of the pixels are illuminated with a reference intensity larger than the camera noise variance, and therefore the effective NIO of the uniform reference iSVS and speckled reference iSVS can be considered equal. The uniform reference has a Gaussian spatial distribution with $R \approx 1.5$, while the speckled reference has $R \approx 1.9$. Therefore, based on Eq. (18a), the theoretical performance difference can be calculated as 1 − $\sqrt{\frac{R_{{speckle }}}{R_{{uniform }}}}$ ≈ 12%, which is a relatively minor difference. Given the minor impacts of both considerations, we do not expect the two profiles of the reference beams used to significantly impact the SNR.

For each rotational velocity, 200 images are captured with a frame rate of 150 FPS and exposure time $T = 300\; \mu s$. For each of the 200 captured images, speckle contrast is calculated for both iSVS and SVS according to Eq. (4). During SVS experiments, the reference laser beam path is blocked. SNR without noise calibration is demonstrated in Supplement 1, but in the main text, we prefer to focus on the calibrated result as it is based on the set of measurements that has good accuracy and are close to the ground truth.

The calibrated SNR results are shown in Fig. 3. The ground truth is obtained by using the high sample intensity data. As detailed in Supplement 1, a series of noise calibrations is executed to subtract noise measured empirically. As evident in the data, the experimental results shown in Fig. 3 coincide with the theoretical results in Fig. 1 and numerical results in Supplement 1. The overall trends in Fig. 3 match well with the expectation. Figure 3(a) shows that iSVS performs better than SVS at low intensities $\overline {{I_S}} $, where the SNR of the iSVS signal can still guarantee retrieval of decorrelation time while SVS signal failed. In this case, the non-uniform, or scattered reference performed slightly worse than uniform iSVS as predicted by the theory. Figure 3(b) shows a situation where both iSVS and SVS signals converge to the same SNR as $\overline {{I_S}} $ overcomes the camera noise variance, indicating no disadvantage present for performing interferometry. We also observed a slight decay in SNR in longer decorrelation times, and this may be due to a larger motor rotation speed discrepancies at these lower percentages. In this case, the scattered reference SNR is in line with the uniform SNR. We attribute the lack of difference in performance to the larger temporal fluctuation that may overwhelm the 15% difference caused by the reference field shape. We note that given the similar NIO and other parameters, the SNR performance is slightly lower than theoretical and simulation values [30]. This can be attributed to instability in the DC rotating motor speed at a high rotation speed. Laser noise in the form of mode hopping, laser intensity fluctuations, and/or environmental noise can also disrupt the coherence condition required for iSVS’s interferometry scheme. These factors lead to variations in the measurements [43].

 

Fig. 3. The iSVS and SVS experiment results using the measured speckle fields in the controlled experiment. (a), (b) SNR as a function of the decorrelation time at different averaged intensities $\overline {{I_S}} $ of the sample laser light beam at below or above the camera noise variance. X-axis ${\tau _s}$ derived DCS benchmark measurement associated with different rotation speeds. (c) Average SNR at various $\overline {{I_S}} $. As evident, iSVS performs better than SVS at low intensities $\overline {{I_S}} $, but both converge to same SNR as $\overline {{I_S}} $ overcome the camera noise variance (as in Fig. 2). Parameters: $\sigma _{{n_{cam}}}^2 = 8$ ph/pxl (converted from electron per pixel according to camera QE and conversion factor), $\overline {{I_R}} \approx 3000$ ph/(pxl.ms), $NIO\; \sim \; 2000, $ and $T = 300\; \mu $s.

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Figure 3(c) shows the SNR as a function of the average signal intensities $\overline {{I_S}} $ for the three cases: SVS, plane-wave uniform reference iSVS, and non-uniform speckled reference iSVS. Note that plane-wave reference iSVS and speckled reference iSVS exhibit an SNR index gain of at least two to three orders of magnitude higher at low signal situations than SVS. This allows the detection of the dynamics of the sample even if the signal level is below the camera noise. To achieve a similar SNR, iSVS and speckled iSVS both require 100 times fewer photons than SVS. Observe from Fig. 3(c) that there is no significant performance difference between the speckled reference iSVS and the uniform reference iSVS in the experiment. In low signal situations, the speckled non-uniform reference iSVS performed slightly worse, as shown in Fig. 3(a). However, this discrepancy diminishes as sample’s signal strength increases. Again, the reason is likely because laser noise generates temporal fluctuations in reference intensity that equally affected both reference fields, overwhelming the 15% performance difference that may be presented with an ideal stable laser. Note that under the ideal situations, as shown in theory, SNR can reach up to 30, but in experiments the average SNR is constrained to 20 due to the aforementioned instabilities.

4. Concluding remarks

In this study, we found that, by the merit of interferometry’s ability to suppress camera noise, iSVS can accurately measure field decorrelation time in situations where the signal would otherwise be overwhelmed by camera noise in an SVS setup. This is particularly important in biomedical applications where the signal intensity tends to be limited and may otherwise require expensive cooled cameras or low-noise detectors to detect DWS signals. We also found that iSVS and SVS performed similarly when the signal intensity is high. This indicates that SVS can be expected to be sufficient in high sample intensity applications. Also, iSVS is more robust to other noises and can be easily calibrated to obtain consistent decorrelation times predictions, while SVS requires a larger signal to overcome the noise. Moreover, the theory shows the potential ways to increase SNR, such as utilizing larger sensors with more pixels or to choose exposure time suitable for the particular target’s dynamics, but in most cases, the laser instability and the motion artifacts are the limiting factors that prevent experimental SNR from reaching the theoretical SNR. In this study, we additionally found that the reference field used in iSVS can be spatially heterogeneous (unlike many other interferometry applications), with the heterogeneity only leading to slight performance degradation. This flexibility can potentially benefit various iSVS system designs. For instance, specific interferometer designs of the optical systems can achieve simultaneous direct measurement of both field and intensity properties, as indicated in the theoretical derivation where both direct and cross terms are present in Eq. (10).

There is a potential advantage of iSVS that is not explored in this study – the ability to perform path length selection with a short coherence laser, which then can be applied to increase brain-to-scalp sensitivity in cerebral blood flow measurements. This capability can potentially be used in biomedical applications to select a sub-group of photon trajectories for detection. We anticipate that this is an area worth exploring in a follow-up study. Further investigation can also explore more in depth on different types of decorrelations and how it impacted the results in SVS. There are also potential to apply existing calibrations to further suppress the noise, either by measuring them more accurately or through filtering, increasing accuracy at various signal ranges.

In summary, we have demonstrated a unified approach to characterize the accuracy and SNR of iSVS method compared to SVS. As Ref. [30] has made a similar comparison between SVS and DCS methods, albeit without the presence of noise, we now have a unified analytical approach for comparing the major DWS methods.