2.1.

Heterodyne Gain

In iDWS, heterodyne gain enables inexpensive sensors to measure weak sample light levels. Here we adopt the definition of “heterodyne” from dynamic light scattering,⁴⁵ which differs from the definition in optical communications. In heterodyne DCS/DWS methods, a strong reference light field ${\mathbf{E}}_R$ is interfered with a weak diffuse (sample) light field(s) ${\mathbf{E}}_S(t)$:

For a simple signal comparison between homodyne and heterodyne detection, we assume that (1) all powers are given in units of photoelectrons per second; (2) single detector/pixel for single speckle, indicating homodyne coherence factor³ $\beta$ and heterodyne mutual coherence degree¹² $\overline{\gamma }$ are both equal to 1; and (3) the detector exposure time $T$ is much shorter than the decorrelation time of ${\mathbf{E}}_S(t)$, ${\tau }_c$ (i.e., $T\ll {\tau }_c$), ensuring no decorrelation during the exposure time.¹⁵ Thus the heterodyne gain is the heterodyne to homodyne signal ratio of $2{\overline{P}}_R/{\overline{P}}_S$ (see Table 1).

Table 1

Signal and SNR for heterodyne and homodyne detection.

Heterodyne gain or amplification is particularly beneficial for lower sample powers (Fig. 3). With an appropriate detector, we can achieve ${\overline{P}}_R/{\overline{P}}_S\gg 1$. Boosting the sample signal in this manner has three important advantages. First, it enables digitizing the signal of interest with more bits and thus higher fidelity [Fig. 3(b)]. Second, it boosts the associated signal and shot noise above the detector noise [Fig. 3(c)], so it is possible to reach the shot noise limit, as described in the next section. Third, it entails measuring the sample field, or at least its in-phase part, as opposed to the sample power. Thus the heterodyne method is sensitive to sample intensity and phase fluctuations, whereas the homodyne method is just sensitive to intensity fluctuations.⁴⁶

Fig. 3

Strong reference light boosts weak sample signal in heterodyne detection. (a) Comparison between homodyne and heterodyne setups, with MMF and 2D camera collecting and detecting diffuse light, respectively. MMF, multimode fiber; FC, fiber coupler; BS, beamsplitter; $\rho$, S–C separation. (b) Simulated images of multimode interference between strong reference light, with uniform intensity pattern, and a weak sample light, with MMF speckle pattern. (c) Simulated 12-bit images with additive shot noise and camera noise under real measurement conditions, corresponding to images in (b). One digital number level is assumed to be $\sim 3$ photoelectrons. The MMF speckle pattern is dominated by camera noise in the homodyne approach (sample only), but it is clearly visualized in the shot noise-limited heterodyne image.

2.2.

Shot-Noise Limited SNR

Though heterodyne gain is a useful concept, it does not account for noise. For instance, a large reference power increases both the signal and the shot noise. For this reason, SNR is a more appropriate metric to compare detection approaches. First, we consider a noiseless detector in which the only noise source is shot noise (${\sigma }_{\text{shot}}^2$). If ${\overline{P}}_{R}T\gg {\overline{P}}_{S}T$ for weak sample light, we find that ${\mathrm{SNR}}_{\text{shot}}=2{\overline{P}}_{S}T$ for heterodyne detection (Table 1), double that of homodyne detection. The factor of 2 in ${\mathrm{SNR}}_{\text{shot}}$ arises from the heterodyne signal term in Eq. (1).

Next, we consider detector or camera noise (i.e., total noise given by ${\sigma }_{\mathrm{tot}}^2={\sigma }_{\text{shot}}^2+{\sigma }_{\mathrm{cam}}^2$). If ${\overline{P}}_{R}T\gg {\sigma }_{\mathrm{cam}}^2\gg {\overline{P}}_{S}T$ for weak sample light detected by a noisy sensor, we find that ${\mathrm{SNR}}_{\mathrm{tot}}\approx 2{\overline{P}}_{S}T$ is still obtained for heterodyne detection (Table 1), whereas ${\mathrm{SNR}}_{\mathrm{tot}}\ll {\overline{P}}_{S}T$ for homodyne detection, much lower than that of case 1. This is because heterodyne detection can achieve the shot noise limit even for a noisy camera/detector (i.e., ${\sigma }_{\text{shot}}^2\approx {\overline{P}}_{R}T\gg {\sigma }_{\mathrm{cam}}^2$), enabling us to “see” a small sample signal ${\overline{P}}_S$, which is otherwise buried in the camera noise [compare “heterodyne” and “sample only” in Fig. 3(c)].

2.3.

Choice of Reference Power

The above considerations suggest that a reference count that is much larger than the sample count is beneficial. This ensures that the homodyne autocorrelation term is much smaller than the heterodyne autocorrelation term, which simplifies the interpretation of data. An ideal reference power should enable all pixels to reach the shot noise limit^12,15,36 but, obviously, not cause the detector to saturate. A high reference power also places stringent requirements on stability so as not to induce additional spurious decorrelation dynamics from reference fluctuations. We have not observed excess noise that would preclude high reference powers in our experiments³⁶ with isolated and temperature-controlled DFB and DBR lasers. Others have proposed signal processing strategies to deal with noise from mode hopping.⁴⁷

Despite the benefits of a large reference count, several works have operated in a regime in which the reference count is comparable to the sample count.⁴⁸ Whether this is a necessary consequence of using detectors with a low saturation level at short S–C separation, or whether there are actually advantages to working in this regime, remains unclear. Early work suggested the choice of an optimal reference power approximately equal to the sample power for measuring low-correlation levels,⁴⁹ but a high reference power was later shown to be optimal for large S–C separations over much of the autocorrelation decay;⁴² it was demonstrated that relative variability of fitted BFIs monotonically decreased by up to 80% for a reference to total power ratio, $\alpha ={\overline{P}}_R/({\overline{P}}_R+{\overline{P}}_S)$, increasing from 0 to $\sim 1$. Data analysis in the case of comparable reference and sample powers can be handled by a modified Siegert relationship that includes a static term,^12,42,46,48 which is expressed as

2.4.

Coherence Gating

In homodyne DCS, all optical paths must be coherent with each other to maximize the available fluctuation signal (i.e., $\beta$ of intensity autocorrelation). Reduced coherence superposes independent speckles and effectively reduces $\beta$ [Figs. 4(a) and 4(c)]. Low coherence is undesirable in CW DCS and is unavoidable in time-domain DCS.^10,50 In iDWS with a large heterodyne gain, the predominant signal is formed by the interference of sample paths with the reference path. A low-coherence light source can, therefore, create a TOF filter, attenuating paths that do not fall within the coherence function [Fig. 4(b)].⁵¹ TOF filtering reduces the height of $G_1$, but is beneficial, on balance, as superficial paths can be preferentially attenuated.⁵² Thus coherence reduction is a simple approach to achieving TOF filtering or discrimination, which can attenuate scalp photons to enhance brain sensitivity at small S–C separations [Fig. 4(d)].³⁹

Fig. 4

Source coherence affects homodyne and heterodyne techniques differently. Photon DTOFs, with coherence lengths from $l_{c,1}$ to $l_{c,3}$ for (a) homodyne DCS and (b) heterodyne iDWS. DTOF, distribution of times-of-flight; CF, coherence function; and $l_{c,1-3}$, coherence lengths of source light. (c) Reduced coherence decreases $g_2$ amplitude ($\beta$) for DCS. (d) Reduced coherence creates a TOF filter, affecting decorrelation time but not the $g_1$ amplitude for iDWS. $g_2$, normalized intensity autocorrelation; $g_1$, normalized field autocorrelation; and ${\tau }_c$, $1/e$ decorrelation time of $g_1$ (i.e., ${\tau }_c/2$ for $g_2$).

2.5.

TOF-Resolved Interferometric Diffuse Optics

Going beyond mere TOF discrimination, TOF resolution aims for a full temporal point spread function (TPSF)⁵³ and concordant TOF-resolved dynamics.^10,24 iNIRS is an interferometric diffuse optical approach that measures superposed interference spectra, which are Fourier transformed to yield TOF-resolved complex optical field time courses, from which TOF-resolved intensity and field autocorrelations are determined (Fig. 5). iNIRS can separate static and dynamic components in the TPSF,⁴¹ providing rich information about multiple scattered light paths. TPSFs provide optical properties, which facilitate recovering absolute BFI.^3,23,35 Admittedly, any technique providing TOF-resolved CLFs^10,40,51 can also measure TPSFs and, in principle, with sufficient TOF resolution, quantify optical properties. However, unlike TD-DCS, iNIRS does not suffer from the tradeoff between coherence and TOF resolution⁵⁴ and can achieve TOF resolutions of $<25$ picoseconds⁴¹ with a high dynamic range through spectral shaping.⁵⁴ The major drawback of iNIRS is its intrinsically low-light throughput, resulting from the high cost of channels digitized at hundreds of MHz. As argued elsewhere, TOF-discrimination is compatible with highly parallel detection by less expensive sensor arrays.

Fig. 5

iNIRS measures intensity and dynamics with TOF resolution (a) by encoding TOF information in interference spectra acquired by frequency tuning of a laser source. FC1 and FC2 indicate single-mode fiber couplers. (b) Temporal fluctuations of the mutual coherence function ${\mathrm{\Gamma }}_{rs}({\tau }_s,t_d)$ obtained by inverse Fourier transformation of interference spectra. The autocorrelation of ${\mathrm{\Gamma }}_{rs}({\tau }_s,t_d)$ yields $G_1({\tau }_s,{\tau }_d)$, which encapsulates both the (c) TPSF and (d) TOF-resolved field autocorrelations. $G_1({\tau }_s,{\tau }_d)$ represents an information-rich two-dimensional data set from which optical properties and BFI can be derived. Red, blue, and green denote short, medium, and long TOFs in (a)–(d), respectively.

2.6.

2D Camera Interferometric Approaches

Recently, interferometric diffuse optical approaches have been adapted for low frame rate 2D cameras. Conventional iDWS uses a fast camera with a short exposure time $T$ relative to the decorrelation time ${\tau }_c$.¹² MiDWS relaxes this requirement, instead probing the autocorrelation indirectly by varying exposure time of a 2D camera with low frame rate.¹⁵ Other 2D approaches probe multiple heterodyne frequencies¹³ or even use a single exposure and measure the sample power separately.¹⁴ 2D approaches can theoretically achieve orders of magnitude lower cost per speckle than iDWS due to the massive number of available pixels (${10}^5$ to ${10}^7$) on 2D cameras. In benefitting from 2D cameras, these approaches are akin to SCOS/DSCA (diffuse speckle contrast analysis), which measures BFI from homodyne speckle contrast(s).^11,55–58 MiDWS and SCOS/DSCA are compared for human brain BFI measurements in Table 2.

Table 2

Comparisons of key features between SCOS/DSCA and MiDWS/iSVS for the human brain.

Is it possible to obtain TOF information while benefitting from the scalability of 2D interferometric approaches? A brute force approach that borrows concepts from iNIRS and iDWS is to employ a high-speed and expensive camera for parallelization of iNIRS.⁵⁹ However, to achieve sufficient sampling of the interference spectrum for reconstruction of a full TPSF⁴¹ with this approach, even at frame rates of MHz, maximal sweep rates are, at best, in the tens of kHz range. An alternative strategy is to forego TOF resolution in favor TOF discrimination,³⁹ which relaxes the frame rate requirements and is (in theory, at least) compatible with highly parallel multiexposure approaches.

3.1.

Motion Artifacts

Interferometric detection, which depends on fluctuations of both light intensity and phase, is more susceptible to motion artifacts than conventional intensity-based DCS/DWS. Moreover, many iDWS systems collect light with MMFs, the motion or vibration of which could induce dynamics beyond those intrinsic to the sample. We generally find that iDWS measurements are not that susceptible to moderate motion of a few meter collection MMF, given that the motional decorrelation dynamics are much slower than intrinsic sample decorrelation dynamics.³⁶ To further mitigate MMF motion artifacts, an MMF bundle with a smaller core size and NA could be employed,⁶⁰ as suggested by related work.¹⁴ Another issue for noncontact interferometric detection^41,43 is axial motion between the subject and probe, which causes Doppler phase drifts resulting in overestimated sample dynamics. The iNIRS method solves this issue by measuring TOF-resolved complex optical fields, from which the Doppler shift can be quantified from static light at short TOFs, and such artifacts can be corrected.⁴¹

3.2.

SNR or Brain Specificity

Even with recent advances, state-of-the-art iDWS has not yet reached the light throughput of widely used CW-NIRS, and measurements at large S–C separations ($\ge 4\mathrm{cm}$), at late TOFs, or in regions with overlying hair remain challenging.³⁶ How do we further improve SNR by additional 1 to 2 orders-of-magnitude and better achieve theoretical advantages of BFI for brain specificity (Fig. 2)? Here we describe three potential solutions, not mutually exclusive, that are being investigated: first, enhancing SNR by further increasing the speckle number, based on parallelizing multiple cameras for iDWS/MiDWS, is a straightforward way to achieve larger S–C separation (i.e., higher brain specificity); second, combining iDWS with coherence gating, to filter out extracerebral photons with short TOFs at a small S–C separation, can improve both SNR (more cerebral photons) and brain specificity;^39,41,43 third, the intrinsic advantages of SNR and high source power for DCS at 1064 nm can be carried over to interferometric detection,^61,62 in which extremely costly 1064 nm photon counting detectors⁶³ can be replaced by InGaAs cameras,⁶⁴ which are less costly.

3.3.

Light Source Cost

With detector technologies supported by economies of scale, we project that source cost will soon become a limiting factor in scaling interferometric diffuse optics. Benefiting from the high pixel counts of CMOS sensors, interferometric detection has reduced the cost of a single detected speckle by orders of magnitude³⁶ compared with conventional multispeckle DCS/DWS.⁴⁴ This trend is likely to continue. For instance, the 2D CMOS camera used in Fourier domain DCS^13,47 only costs hundreds of dollars, which is dozens of times less expensive than the source. State-of-the-art iDWS³⁶ has used similar light sources as state-of-the-art DCS/DWS⁴⁴ to date. We believe that more systematic iDWS investigations with lower cost laser sources⁶⁵ are warranted. Indeed, we have found that a lower temporal coherence, characteristic of lower cost sources, can actually be helpful to reducing the impact of unwanted system reflections in iDWS. Finally, although iDWS at 1064 nm can also reduce the laser source cost,⁶⁴ InGaAs sensors unfortunately are expensive. Though not strictly a limitation, we project that lowering the light source cost, perhaps in conjunction with noise mitigation strategies,⁴⁷ will be important to enabling new applications in the coming years.