Interferometric near-infrared spectroscopy ( iNIRS ) : performance tradeoffs and optimization

Interferometric near-infrared spectroscopy (iNIRS) is a time-of-flight(TOF-) resolved sensing modality for determining optical and dynamical properties of a turbid medium. iNIRS achieves this by measuring the interference spectrum of light traversing the medium with a rapidly tunable, or frequency-swept, light source. Thus, iNIRS system performance critically depends on the source and detection apparatus. Using a current-tuned 855 nm distributed feedback laser as the source, we experimentally characterize iNIRS system parameters, including speed, sensitivity, dynamic range, TOF resolution, and TOF range. We also employ a novel Mach-Zehnder interferometer variant with a multi-pass loop to monitor the laser instantaneous linewidth and TOF range at high tuning speeds. We identify and investigate tradeoffs between parameters, with the goal of optimizing performance. We also demonstrate a technique to combine forward and backward sweeps to double the effective speed. Combining these advances, we present in vivo TPSFs and autocorrelations from the mouse brain with TOF resolutions of 22-60 ps, 36-47 dB peaksidelobe dynamic range, 4-10 μs autocorrelation lag time resolution, a TOF range of nanoseconds or more, and nearly shot noise limited sensitivity. © 2017 Optical Society of America OCIS codes: (030.1640) Coherence; (140.3600) Lasers, tunable; (140.3490) Lasers, distributed-feedback; (160.4760) Optical properties; (170.7050) Turbid media; (170.4500) Optical coherence tomography; (290.4210) Multiple scattering; (170.6920) Time-resolved imaging; (170.6480) Spectroscopy, speckle. References and links 1. F. F. Jobsis, “Noninvasive, infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198(4323), 1264–1267 (1977). 2. A. Torricelli, D. Contini, A. Pifferi, M. Caffini, R. Re, L. Zucchelli, and L. Spinelli, “Time domain functional NIRS imaging for human brain mapping,” Neuroimage 85(Pt 1), 28–50 (2014). 3. 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Introduction
The field of near-infrared spectroscopy (NIRS) aims to non-invasively probe the physiology of highly scattering biological tissues with near-infrared light [1].Continuous wave (CW-) NIRS measures changes in the absorption coefficient at different wavelengths to assess oxyhemoglobin and deoxyhemoglobin concentration changes, but not baseline values.By adding additional dimensions to the measurement, time-domain (TD-) [2] and frequencydomain (FD-) [3] NIRS methods can quantify optical properties, enabling them to measure baseline hemoglobin concentration and saturation.However, oxygenation alone may be ambiguous without information about blood perfusion or flow [4].To evaluate blood flow, Diffuse Correlation Spectroscopy (DCS) [5,6] analyzes temporal autocorrelations of coherent light, multiply scattered through tissue, to infer red blood cell dynamics.Equipped with flow, hemoglobin concentration, and oxygen saturation, multimodality NIRS-DCS instruments can monitor tissue metabolic rate of oxygen non-invasively [7,8].Current challenges in the field of NIRS are 1) the large number of assumptions needed to quantify baseline flow and oxygenation from simple CW measurements, 2) the high cost of more accurate TD-NIRS and FD-NIRS methods, and 3) the need for complex multimodality instrumentation to perform flowmetry and oximetry simultaneously.A single NIRS modality to quantify both tissue optical properties and dynamics would both improve accuracy of NIRS and enhance its numerous applications [9][10][11].
Interferometric near-infrared spectroscopy (iNIRS) is a new time-of-flight (TOF-) resolved optical modality that can assess optical properties and dynamics with a single instrument.The theoretical iNIRS framework [12,13] and in vivo application [14] have been described previously.In this work, for the first time, we characterize and optimize the iNIRS setup with respect to speed, sensitivity, dynamic range, TOF resolution (related to laser tuning range), and TOF range (related to laser linewidth, sampling, and detection bandwidth).In doing so, we introduce a method for monitoring the instantaneous linewidth and coherence time of temporally coherent and rapidly tuned lasers, based on a novel multi-pass loop variant of a modified Mach-Zehnder interferometer.We show that both forward and backward sweeps nearly achieve the shot noise limit, and combine them to double the temporal sampling of the field autocorrelation.Finally, after accounting for tradeoffs to determine the system operating point, we show iNIRS TPSFs and autocorrelations in vivo at 2.3x higher TOF resolution and 5x higher autocorrelation time lag sampling than demonstrated previously [14].

iNIRS theory
The theory for iNIRS is based on the TOF-resolved field autocorrelation [14].The iNIRS method uses a tunable narrow-linewidth source to illuminate a turbid sample, and interferes the multiply scattered sample light with light traversing a reference path via a modified Mach-Zehnder interferometer (MZI), as shown in Fig. 1(a) [12].The source tunes over optical wavelength (λ) or frequency (ν).Fourier transformation of the spectral interference signal (resampled, if needed, to be uniformly spaced in ν) yields the mutual coherence function between sample and reference fields, Г rs (τ s , t d ), which depends on TOF (τ s ) and delay time (t d ).Incoherent averaging over t d yields the temporal point spread function (TPSF), I iNIRS (τ s ), where d 2 iNIRS s r s s d t The TPSF represents the intrinsic medium distribution of times-of-flight (DTOF), I(τ s ), convolved (*) in TOF (τ s ) with the instrument response function (IRF), IRF(τ s ): ( TPSFs are then fitted with A × [ (τ s )*IRF(τ s )], where A is a scaling factor, and (τ s ) is the TOF-resolved diffusion approximation of reflectance from a semi-infinite medium [15], to extract absorption and reduced scattering coefficients: μ a and μ s ′, respectively [14], by minimizing the norm-squared of the error:  Similarly, TOF-resolved, un-normalized iNIRS field autocorrelations, G 1 iNIRS (τ s , τ d ), are functions of both TOF (τ s ) and lag time (τ d ) [12,13]: iNIRS autocorrelations are related to the intrinsic medium field autocorrelation, G 1 (τ s , τ d ), by a convolution in TOF (τ s ) with the IRF: Thus, iNIRS provides the unique capability to measure TOF-resolved field autocorrelations.This capability enables more direct quantification of sample dynamics [12,14] than is possible with intensity autocorrelations [16].In the Diffusing Wave Spectroscopy (DWS) regime [17], the normalized TOF-resolved field autocorrelation g 1 (τ s , τ d ) is given by: [ ] where ξ(τ s ) = 2k 2 αD B μ s ′cτ s /n is the TOF-resolved decay rate, L = cτ s /n is the photon path length, k = 2πn/λ c is the medium wavenumber, and αD B = BFI is the blood flow index [5].Provided that μ s ′ can be determined from Eq. ( 3), BFI can be straightforwardly determined [12].Note that I iNIRS (τ s ) = G 1 iNIRS (τ s , 0) and I(τ s ) = G 1 (τ s , 0); hence Eq. ( 5) also implies Eq. ( 2).As convolution involves a smoothing along the TOF axis, iNIRS requires a fine TOF resolution to determine optical properties and TOF-resolved medium dynamics.

Tuning range determines time-of-flight resolution
The TOF resolution, or width of the IRF, is inversely related to the tuning range, or optical bandwidth, of the light source [12].A fine TOF resolution causes the TPSF to better approximate the DTOF in Eq. ( 2), and improves TOF selectivity for recovering dynamics from the field autocorrelation in Eq. (5).We assume that Δν and Δλ are the laser full-width at half-maximum (FWHM) bandwidths in optical frequency and wavelength respectively, c is the speed of light in vacuum, and λ c is the central wavelength.For a Gaussian spectrum, the theoretical FWHM TOF resolution, δτ s, is given by In reality, it is not possible to achieve a perfectly Gaussian spectrum as the tuning range is finite, and the spectrum must be truncated.In order to minimally degrade the resolution by truncating the spectrum, we specify that ΔΛ and ΔV (full end-to-end tuning ranges) are [18]: Since the iNIRS TOF resolution is inversely related to the tuning range or optical bandwidth, a temporally coherent λ c = 855 nm DFB laser (Eagleyard) with a large mode-hop free tuning range was chosen as the source.The DFB laser is tuned rapidly in wavelength by changing the drive current as illustrated in Fig. 1(b).In contrast to external cavity laser tuning via mechanical elements [19], current tuning is highly stable and repeatable.However, this tuning strategy induces modulation of output power as well, which limits the practical tuning range.The laser free spectral range is already large due to the short DFB cavity length, and the mode hop free tuning range is further extended to a few nanometers by the shift in grating reflectivity achieved by a DFB laser during current tuning.Practically, the current tuning range is limited by the difference between the maximum allowable forward current (conservatively set to 190 mA because 220 mA is the irreversible damage threshold) and the minimum current required for lasing (~30 mA).A maximal tuning range of ΔΛ~480 pm (ΔV~197 GHz) is then obtained simply from the product of the available current range and a typical wavelength tuning coefficient dλ/dI = 0.003 nm/mA for the DFB laser provided by the manufacturer.However, for the particular DFB laser used in this study, a wavelength tuning coefficient of ~0.002 nm/mA was measured, limiting the expected tuning range to ΔΛ~320 pm (or ΔV~131 GHz).
In iNIRS, a high tuning speed (F S ) is required to measure the electric field faster than the intrinsic decorrelation time scale of the sample.However, the tuning range of DFB lasers is diminished by the transition of the dominant tuning mechanism from thermal to carrier density modulation at fast modulation speeds [20].This reduces the wavelength tuning coefficient, in turn decreasing the tuning range.To characterize this effect, the tuning range ΔV was measured at different tuning speeds [Fig.2(a)] by analyzing the spectral interference pattern from an MZI with mismatched sample and reference paths.The tuning range is determined as follows:  Here, N fringes represents the number of fringes (obtained from the total deviation of the interference signal's unwrapped phase divided by 2π), n is the refractive index of the mismatch medium (n = 1.455 for fiber core), and ∆L = cΔτ s /n is the physical length of the path mismatch between arms.Due to possible filtering in the current controller electronics, input voltage to the controller may not be a reliable indicator of the actual laser drive current at high tuning speeds.Instead, laser output power was measured by a photodiode.By calibrating the change in photodiode signal to milliamps of current modulation, the photodiode signal was used, indirectly, to infer the drive current [Fig. 2 While a broader tuning range and higher TOF resolution can be achieved by increasing current modulation amplitude, several limitations arise.First, as the maximum forward current (I max in Fig. 1(b)) is limited, increasing modulation amplitude requires reducing average current, and therefore, reducing average output power.Second, as the modulation amplitude is increased, energy distributes more asymmetrically across wavelengths in the sweep.In particular, if the drive current approaches the lasing threshold (I min in Fig. 1(b)) at shorter wavelengths, power diminishes.Though asymmetries in the spectrum can be compensated numerically, this shaping process incurs a sensitivity penalty, as described in Section 2.5.To better assess this tradeoff between TOF resolution and sensitivity, we investigated both a high resolution mode (ΔΛ = 157 pm) and a low resolution mode (ΔΛ = 58.6 pm) at 50 kHz tuning speed.

Tuning speed determines autocorrelation lag time (ALT) resolution
According to DWS [Eq.( 6)], the autocorrelation decay rate increases with TOF.As a rule of thumb, we assume that, due to decorrelation during the sweep and sampling considerations, accurate measurement of the decay rate is possible up to TOF values where ξ(τ s ) ~1/δτ d , where δτ d is the autocorrelation lag time (ALT) resolution.In our previously reported results [12][13][14], the tuning speed was limited to F S = 50 kHz, and we employed only forward sweeps, with δτ d = 1/F S = 20 μs.In order for iNIRS to measure autocorrelations at longer TOFs (corresponding to larger penetration depths), both high sensitivity and lag time resolution (speed) are desired.For low resolution and high resolution mode at F S = 50 kHz, we introduced a novel method, described in Section 2.9, to combine forward and backward sweeps to achieve an ALT resolution of δτ d = 1/2F S = 10 μs, thus using detected photons more effectively.We also investigated a high speed mode (ΔΛ = 62.5 pm) at F S = 500 kHz, which achieved an ALT resolution of δτ d = 4 μs due to data transfer limitations, allowing us to investigate the tradeoffs between sensitivity and speed.

Signal processing
In our iNIRS setup, the DFB laser optical frequency is tuned by sinusoidal current modulation, generating both a forward (fwd) and a backward (bwd) sweep.However, optical frequency (ν) is the conjugate variable to TOF (τ s ), and the Discrete Fourier Transform requires uniform sampling [18].Hence a resampling or interpolation procedure was implemented.The analytic phase of the interference signal as a function of time (t) is given by φ(t) = 2πΔτ s ν(t) + φ 0 , where φ 0 is a phase offset, and Δτ s is the known TOF mismatch (Δτ s = 1.25 ns in Fig. 3).The phase is obtained from the analytic function via Hilbert transformation, a high order polynomial is fitted to it, and all subsequent fringe patterns are interpolated such that the samples are linearly spaced in φ (and hence ν as well).Note that two distinct solutions for the phase of the analytic function (φ and -φ) are possible.The solution with increasing / decreasing phase over time was chosen for the forward / backward sweep, respectively.Given the small tuning range (ΔΛ ≈0.1 nm), linearity in wavelength implies linearity in optical frequency, at least for Δτ s up to 100 ns (~22 m).Hence for TOF mismatches of interest, the nonlinearity to be corrected by resampling arises mainly from sinusoidal tuning, and not the nonlinear relationship between wavelength and frequency (ν = c/λ) [18].Finally, the interpolated fringe pattern is multiplied by a Hamming window, zeropadded, and inverse Fourier transformed to yield Γ rs .As

[e] [g] [f] [h] [i] [k] [j] [l] [m] [o] [n] [p] [q] [s] [r] [t] [u] [w] [v] [x]
nonlinear linearized ideal As shown in Fig. 3, interpolation linearizes the phase and dramatically improves TOF resolution (red) relative to no interpolation (blue).To assess whether phase nonlinearity is completely removed by interpolation, an "ideal" IRF can be constructed from the Fourier transform of the resampled and Gaussian shaped fringe envelope (black), thereby defining the TOF resolution achieved with no residual phase error (dotted black).Indeed, this ideal TOF resolution (FWHM ≈60.3 ps for low resolution mode, FWHM ≈22.6 ps for high resolution mode, and FWHM ≈56.7 ps for high speed mode), was achieved for both sweeps (compare red and dotted black lines).As a sanity check, these experimental results are roughly consistent with the theoretical TOF resolutions of δτ s ≈58.9 ps for low resolution mode, δτ s ≈22.0 ps for high resolution mode, and δτ s ≈55.2 ps for high speed mode, determined from Gaussian assumptions in Eqs. ( 7) and ( 8).Although the ideal IRF FWHM was achieved for all modes, sidelobes were slightly larger than predicted in high resolution mode [Fig.3(o-p)].We found that the sidelobes may result from wavelength tuning noise at frequencies above the drive frequency.This problem can be addressed in the future by reducing current controller bandwidth, or additional electronic filtering stages.

Sensitivity
If a single mode fiber is used for light collection in iNIRS, the detection system must use collected photons efficiently.The sensitivity is the minimum transmittance or reflectance that can be measured.As in Fourier domain Optical Coherence Tomography [21], iNIRS uses heterodyne gain to potentially achieve sensitivity limited by shot noise, not detector noise, even though photon counting detectors are not used.Since Fourier domain detection is used [21,22], all detected sample photon paths interfere with the reference path, yielding an efficiency advantage over time-gated NIRS techniques [2,23].
To derive the theoretical shot noise limit for the minimum measurable reflectance or transmittance, we assume that N S and N R are the number of photons from the sample and reference arms at the detector, respectively, and s 2 (ν) represents the source spectrum, or density of photons across optical frequency, ν [24].We further assume that s 2 (ν)dν = 1.If spectral shaping in the reference and sample arms is negligible, S R (ν) = N R s 2 (ν) and S S (ν) = N S s 2 (ν) represent the reference and sample spectral photon densities, respectively.To determine the shot noise limited sensitivity, we write the peak iNIRS signal as If N S << N R , the variance of the noise floor is given by since the detected noise is dominated by Poisson-distributed shot noise from the reference arm.Under these assumptions, the signal-to-noise ratio (SNR) is given by The number of sample photons at the detector is N S = αρ d N S,inc , where α is the sample reflectance or transmittance, ρ d accounts for detector quantum efficiency, and N S,inc is the number of incident photons.The sensitivity is the minimum measurable reflectance or transmittance, α min , where "measurable" is defined as having an SNR of 1 ≥ .Therefore, the shot noise limited sensitivity is Here, P S is the time-averaged illumination power over the full sweep in the sample arm, Δt is the acquisition time or sweep duration, and h is Planck's constant.A notable feature of this expression is that the shot noise limited sensitivity does not depend on the reference power.However, to reach this limit, the reference arm power must be increased so that the shot noise in Eq. ( 11) dominates detector noise.As previously discussed, current modulation for tuning in iNIRS also modulates the spectral shape.This undesirable modulation can be compensated digitally.In order to determine the effects of digital shaping or windowing on sensitivity, we can add an additional "window" function, w(ν) to Eqs. ( 10) and ( 11

low resolution high resolution high speed [a] [b] [c]
Finally, we can determine the sensitivity loss from shaping: w ν s ν dν sensitivity loss from shaping = .w ν s ν dν This can further be simplified to yield: where s w (ν) = s(ν)w(ν) describes the windowed, or the "shaped," spectrum.By the Cauchy-Schwartz inequality, and since s 2 (ν)dν = 1, with equality achieved if w(ν) is constant (no shaping).Thus Eq. ( 17) allows us to calculate the theoretical sensitivity loss from shaping alone, again, assuming the shot noise limit.In reality, the sensitivity loss from shaping may be worse in the presence of detector noise.
The calculated theoretical sensitivities are compared with experimental sensitivities (obtained by attenuating the sample power and measuring the signal-to-noise ratio in direct transmission) in Table 1 for each of the three modes (low resolution, high resolution, and high speed).A quantum efficiency of ρ d = 0.65 was estimated, and λ c = 855 nm was assumed.The acquisition times are Δt fwd = 8.01 μs and Δt bwd = 8.53 μs for low resolution mode; Δt fwd = 8.01 μs and Δt bwd = 8.53 μs for high resolution mode; and Δt fwd = 0.715 μs and Δt bwd = 0.770 μs for high speed mode.The sample power P S is determined for each sweep individually based on the measured average power and ratio of the forward/backward envelope integrals, and taking into account the exclusion of portions of the sweep where the wavelength is stationary.Experimental measurements are compared with theory for all three modes, both with and without windowing/shaping [Fig.4].The experimental sensitivity is calculated as the sum of two terms on a dB scale.The first is the maximum value of the intensity, divided by the variance (σ 2 noise,τd ) of the complex Γ rs , measured along delay time (τ d ) and averaged across a small TOF (τ s ) range at the noise floor.The second is the measured attenuation in the sample arm (calculated as the ratio of P S,in and P S,out , the sample powers before and after attenuation, respectively) in dB: In agreement with the theory, we found that for all modes, iNIRS sensitivity increases with reference arm power, approaching a plateau near the shot noise limit [Fig.4].Note that sensitivity values vary slightly between forward and backward sweeps due to small differences in energy between the two sweeps, possibly related to the current controller.The wide plateau in sensitivity at higher reference powers suggests that excess noise does not limit sensitivity.Experimental sensitivity values (measured with 53.3 dB attenuation in the sample arm for low resolution mode, 51.6 dB for high resolution mode, and 52.5 dB for high speed mode) are ~0-3 dB below theoretical values (Table 1).The slight discrepancies between experimental and theoretical sensitivities are attributed to additional system losses that are not accounted for by the detector quantum efficiency (ρ d ) alone in Eq. ( 13), or dynamic range limitations.Finally, we note that an additional 3 dB reduction in sensitivity would be expected from the random polarization of diffuse light from a tissue sample, but is not observed here with direct transmission.

Sampling and time-of-flight range
TOF ranges of nanoseconds or more are needed to measure diffusive DTOFs in iNIRS.The maximum measurable TOF without aliasing (τ s,max ) depends on the sampling interval in the optical frequency (ν) domain.In principle, the resampling step could decrease the sampling interval arbitrarily; however, aliasing during data acquisition would still set the maximum measurable TOF.The number of acquired samples in a sweep is N s = Δtf s , where f s is the digitizer sampling rate (not to be confused with F S , the sweep rate) and Δt is the sweep duration.The maximum measurable TOF is set by the Nyquist limit, given the number of samples (N s ) and the tuning range (ΔV).This maximum TOF for a linear sweep can then be related to the TOF resolution (δτ s ) as follows: In our setup, f s = 200 MS/s; thus, the theoretical maximum measurable TOF values are τ s,max = 33.4ns for low resolution mode, τ s,max = 12.4 ns for high resolution mode, τ s,max = 2.8 ns for high speed mode.In reality, however, Eq. ( 20) represents a rough guideline and aliasing may occur for smaller TOFs due to sweep nonlinearity.A more conservative TOF limit is obtained by stipulating that aliasing may not occur at any point during the nonlinear sinusoidal sweep: In deriving Eq. ( 21) we assumed that Δt is the time for half of a full sinusoidal oscillation in wavelength.This more conservative definition yields τ s,max = 21.3 ns for low resolution mode, τ s,max = 7.9 ns for high resolution mode, τ s,max = 1.8 ns for high speed mode.Note that if the electronic detection bandwidth is sufficiently large, ranging may be performed past the limits in Eqs. ( 20) and ( 21), if aliasing can be tolerated.

Dynamic range
The dynamic range is the ratio of the minimum and maximum signal levels that can be measured at once.One definition of the iNIRS dynamic range is the ratio between the maximum value of the intensity that can be obtained without saturating the detector, and the corresponding noise background variance σ 2 noise,τd , on a dB scale: The variance is once again measured along delay time (τ d ) and averaged across a small TOF (τ s ) range around the noise floor, as in Eq. (19).In order to quantify the available dynamic range when large and small signals are more closely separated, which is the case at small source-detector separations [25,26], the dynamic range is also quantified by the ratio of the maximum, or peak, intensity, and the intensity of the first sidelobe: Since discontinuities at the edges of the interference spectrum can cause sidelobes, apodization and spectral shaping methods can optimize the tradeoffs between sensitivity, TOF resolution, and peak-sidelobe ratio.To this end, in addition to the sensitivity results in Table 1, we demonstrate the impact of Hamming windowing and Gaussian shaping on the dynamic range and time-of-flight resolution [Fig.5 and Table 2].The results for both definitions of dynamic range are summarized in Table 2 for all three operating modes.
In practice, the reference power during the dynamic range measurement is increased until it approaches the shot noise limit as determined by Fig. 4, while the sample power is subsequently attenuated to reduce the interference signal to just below saturation.As a result, reference powers were ~4-5.5 mW and sample powers were on the order of 1 μW for all regimes.Note that the differential amplifier in the dual balanced detector (Newport 80 MHz Balanced Photoreceiver Model 1807-FC) reaches saturation when the difference in powers between the two inputs reaches 125 μW.To achieve a high dynamic range, it is thus imperative that the splitting ratio of the second fiber coupler, shown in Fig. 1(a), is as close as possible to 50:50.

Time-of-flight range
In iNIRS, the TOF range of the system must exceed the extent of the DTOF.Two important determinants of TOF range are the electronic detection bandwidth and dynamic coherence time.For rapidly tunable lasers, the dynamic coherence time is defined by analogy to the coherence time of a laser that is stationary in wavelength.For a wavelength-stationary laser, temporal phase fluctuations (φ n ) of the light field reduce the interference fringe visibility at large TOF mismatches.Similarly, for a wavelength-swept laser, random phase fluctuations (φ n ) during a frequency sweep φ (t) = 2πΔτ s ν (t) + φ n (t) + φ 0 , reduce the interference fringe visibility at large time delays.In order to characterize this effect, if present, we define the coherence time as the delay in the interferometer at which the visibility of the fringes is reduced to one-half of the visibility at zero delay: Here, τ c is the half-width at half-maximum (HWHM) TPSF measurement range or coherence time, and δν is the FWHM of the instantaneous laser linewidth, assumed to be Gaussian.Note that the HFHM coherence length is given by L c = cτ c .In swept source OCT, the coherence length is measured experimentally by obtaining interference signals at various path delays sequentially and plotting the point spread function height rolloff versus path mismatch [21].Alternatively, a method to retrieve the complex electric field of a swept laser was recently introduced, but requires a complicated interferometer setup with multiple detection channels [27].Here we introduce a simple multipass loop method to measure the rolloff of a tunable laser at multiple path mismatches simultaneously.Importantly, to exclude multi-pass losses, the method requires measuring once at a slow tuning rate (F S = 500 Hz in our experiment), where coherence rolloff and electronic bandwidth effects can be neglected.and 500 kHz (g) tuning rates.The TOF regions around each peak were summed, to mitigate TPSF broadening caused by resampling errors at large TOFs, for 50 kHz (e) and 500 kHz (h) tuning rates.Assuming that the rolloff at a very slow speed of 500 Hz, represented by the bold black line in (e) and (h), is only due to multi-pass losses, normalized rolloffs at higher speeds (50 kHz and 500 kHz) that exclude multi-pass losses can be estimated as the difference in summed rolloffs, (f and i, respectively).
A custom 850 nm, 99:1 fiber coupler with short arms (~16 cm each) was inserted in the reference path of a Mach-Zehnder interferometer.One input and one output FC/APC port of the coupler were connected to form a recirculating loop [Fig.6(a)].The multiple passes create a superposition of fringes, where the fringe frequency increases with the number of passes [Fig.6(b)].Resampling and Fourier transformation yields a series of peaks with increasing TOF, one for each fringe frequency, where the peak heights are proportional to the fringe amplitudes (corresponding colors in Fig. 6(c)) and peak locations are proportional to the time delay.If the same amount of light were transmitted on each pass, the fringe amplitude would be proportional to the fringe contrast, and the laser rolloff could be easily determined [Fig.6(c)].However, in practice, one must account for multi-pass losses.The output intensity after n passes through the loop, assuming a coupling ratio of α c and a loop transmission of β, which incorporates connector losses, is given by: ( ) Since rolloff may increase with increasing sweep speed due to linewidth broadening and limited electronic bandwidth, we provide measurements at F S = 50 kHz and F S = 500 kHz to illustrate tradeoffs at higher speed.Interferograms were acquired once without connecting the loop portion of the interferometer to set baseline IRFs, and then once more with the loop connected so the rolloff was sampled at TOF values equal to multiples of the loop delay.The processed loop signals are presented in Figs.6(d) and 6(g) for 50 and 500 kHz, respectively.At very large path delays, error in the linearization procedure and phase distortion caused by electronic filtering effects lead to IRF broadening.This is by summing the intensity in regions around each peak [Figs.6(e) and 6(h)].Importantly, a measurement at a very slow speed (F S = 500 Hz) is used to adjust for multipass losses.Assuming that laser tuning at F S = 500 Hz is quasi-static, coherence rolloff effects over the TOF range are negligible (the static DFB laser linewidth is ~2 MHz).Moreover, the electronic bandwidth rolloff should be insignificant for TOFs of interest at F S = 500 Hz.These assumptions are confirmed by fitting a line to the rolloff at 500 Hz (bold black line in Figs.6(e) and 6(h)) which yields a loss of 0.17 dB/ns, or 0.28 dB/pass, consistent with the expected FC/APC connector loss at C4 (β ~0.2 dB) and the coupling ratio (α c ~0.99 or ~-0.0436 dB).Finally, the summed intensity rolloffs are normalized by the rolloff fit at F S = 500 Hz.Thus, Figs.6(f) and 6(i) most accurately represent the rolloff of the iNIRS system itself, excluding losses in the multi-pass loop.
In addition to temporal coherence of the laser, filtering in the photodiode or subsequent electronics may further reduce the TOF range.The required electronic frequency bandwidth (Δf) for sinusoidal wavelength tuning is approximately The electronic bandwidth (Δf) of the dual balanced detector is 80 MHz, suggesting that increasing TOF resolution (increasing ΔΛ) will reduce the TOF mismatch Δτ s corresponding to the detection bandwidth limit.This TOF value is represented as a dashed line in Figs.6(d)-6(i) for various levels of δτ s to illustrate the tradeoff between TOF resolution and TOF range.Investigation of Figs.6(d)-6(i) suggests that the rolloff is determined by the electronic detection bandwidth, not coherence effects in the laser, at least under our experimental conditions.These results suggest that τ c ≈7 ns at the highest possible TOF resolution of δτ s = 22.2 ps at 50 kHz, which is much larger than the expected DTOF duration (typically a couple of nanoseconds [2]), and τ c ≈3 ns for a TOF resolution of δτ s = 80.6 ps at 500 kHz.Furthermore, the slope of the rolloff was always much lower than the asymptotic slope of a typical diffusive DTOF (~-25 dB/ns) [28].Finally, the rolloff can be compensated numerically, if known a priori, though the associated sensitivity loss cannot be recovered.Balanced detectors with higher bandwidth and a digitizer with a higher sampling rate will further improve the iNIRS rolloff in the future.

Combining forward and backward sweeps
As discussed above, the choice of iNIRS operating regime involves tradeoffs between various system parameters.Here we show that coherently combining forward and backward sweeps can improve autocorrelation lag time (ALT) resolution, with no other tradeoffs.To achieve this, the interference fringes for both sweeps, measured over time by the dual-balanced photodetector, must be assigned to the correct optical frequencies (ν) by the resampling procedure.Also, the spectral envelopes must be identical for both sweeps prior to inverse Fourier transformation.Provided these conditions are met, forward and backward sweeps can be successfully integrated in iNIRS analysis.In practice, the unwrapped phase (and optical frequency) during the sweep is obtained via the analytic function of a spectral interference pattern, usually acquired in transmission mode with a known path mismatch.As described in Section 2.4, we assign increasing phase (and frequency) to the forward sweep, and decreasing phase (and frequency) to the backward sweep.Though a sinusoidal drive is used, the actual laser tuning is slightly asymmetric, but highly repeatable.Hence the transition between sweeps was determined as the locally symmetric folding point of the fringe pattern.The unwrapped phase of the analytic function is inaccurate near the edges of the sweep where the phase and frequency are stationary.However, this error was determined through simulations to be <1 radian for TOF values of interest.

low resolution high resolution high speed [a] [b] [c] [e]
[f] bwd fwd After each point in both forward and backward sweeps was assigned to an optical frequency, both sweeps were interpolated to achieve linear sampling in ν.Next, both sweeps were digitally shaped to achieve the same spectral envelope, as shown in (τ s, τ d ) was estimated as described previously [13], treating data from both sweeps identically.This procedure has the advantage of being both simple and unbiased.However, as forward and backward sweeps had slightly different signal-to-noise ratios (Table 1), a weighted autocorrelation estimate may improve results in the future.

In vivo results
The iNIRS optimizations described in Section 2 were tested in vivo in the nude mouse brain.
After anesthesia induction, a male nude mouse (SKH-1E, Charles River) was immobilized in a stereotactic frame with a ventilating system supplying 1.2% v/v isoflurane in medical air.Contact-free measurements were performed at a source-detector (SD) separation of 7.6 mm and at null SD separation, for all three modes: low resolution, high resolution, and high speed.Note that the forward and backward sweeps overlap for all SD separations and modes, supporting the consistency of the processing method for forward and backward sweep TPSFs.

Mouse brain optical properties
As forward and backward sweeps yielded consistent TPSFs [Fig.8], they were averaged to improve signal-to-noise ratio, and re-normalized by the maximum value.Optical properties were then determined by fitting with diffusion theory.The forward and backward sweep TPSFs were determined individually and averaged [Eq.( 1)], and then fitted with A × [ (τ s )*IRF(τ s )] as described by Eq. (3); the fitting results are shown in Fig. 9 and Table 3.

Discussion
This work comprehensively describes the tradeoffs in an interferometric near-infrared spectroscopy (iNIRS) system that uses a current tuned distributed feedback (DFB) laser.
While some tradeoffs are fundamental to the iNIRS method, others are particular to the laser and tuning mechanism used in this study.Here, we summarize the major performance tradeoffs, comment on their relevance to other tunable laser technologies, and point the way towards future system improvements.

TOF resolution vs. tuning rate
Because current tuning of the DFB laser does not involve mechanical elements [Fig.1], tuning is stable and repeatable.However, current tuning is limited by the switch from thermal to carrier density modulation at fast modulation speeds [20].The resulting degradation of tuning coefficient with tuning rate results in the observed loss of TOF resolution [Fig.2], because, as per Eq. ( 7), TOF resolution is inversely proportional to the tuning range.While the dynamic tuning range approaches the quasi-static limit at slow tuning rates, the maximum tuning range decays at a rate of ~20 GHz/decade beyond a few kHz tuning rate [Fig.2(a)].Therefore, the achievable TOF resolution δτ s increases from 11.5 ps at 10 Hz to nearly 40 ps at 1 GHz [Fig.2(d)].Although the fitting procedure to determine μ a and μ s ′ [Eq.( 3)] does account for the finite TOF resolution, worse TOF resolution may increase uncertainty in the two fitted parameters in the presence of noise.Nevertheless, TOF resolutions achieved by iNIRS are competitive with or better than state-of-the-art time-domain systems [31].To explicitly use the non-diffusive information at early TOFs, higher TOF resolutions are desired.TOF resolution could be improved by a DFB laser with a higher current tuning coefficient, which can reach 0.02 nm/mA albeit with lower output power [32], or alternative lasers with different tuning mechanisms.

TOF resolution vs. sensitivity
We showed that iNIRS approaches the shot noise limit [Fig.4], where the sensitivity is proportional to the number of photons incident on the sample.As the sample power (photons incident per unit time) was limited by the laser output power in this study, the maximum achievable sensitivity was also limited by the maximal laser output power.When current tuning the DFB laser, variation of the injection current changes not only the optical wavelength, but also the gain, and therefore, the optical power.The tradeoff between TOF resolution and sensitivity arises from this concomitant modulation of the laser output power [Fig.1(b)].Low TOF resolution needs a low tuning range, ΔΛ, which requires only a small current modulation; thus the central current I c can be increased.In the limiting case where ΔΛ is infinitesimally small, the central current could be raised all the way to maximum current (I max ).Note that the additional digital spectral shaping required for a wider sweep incurs an additional sensitivity penalty, as per Eq. ( 17).Consequently, by going from a high to low TOF resolution, the shot noise limit described by Eq. ( 13) can be increased by >3 dB.The tradeoff between TOF resolution and sensitivity is summarized by Fig. 4 and Table 1.
Finally, the resolution-sensitivity tradeoff, arising from undesired power modulation of the laser, can be circumvented by other laser tuning mechanisms which decouple the tuning of wavelength selectivity from injection current to the active gain region [33,34].For tunable lasers where the output power is below the American National Standards Institute (ANSI) limit, optical amplification techniques can be considered as well.

Number of resolvable TOFs vs. tuning rate
A tradeoff between the number of resolvable TOFs (the TOF range τ s,max divided by the TOF resolution δτ s ) and tuning rate F S can arise from the limited sampling rate, limited detection bandwidth, and possible coherence effects in the laser.The sampling rate limitation is described in Eq. ( 21), which expresses the requirement that aliasing does not occur at any point during a sinusoidal sweep.The detection bandwidth limitation is described in Eq. ( 26) and was demonstrated experimentally in Fig. 6 using a multi-pass loop.Both equations can be summarized succinctly as (τ s,max /δτ s )F S ~fs (sampling rate), and (τ s,max /δτ s )F S ~Δf (bandwidth), where τ s,max /δτ s is the number of resolvable TOFs.These tradeoffs can be ameliorated by increasing the digitizer sampling rate f s and detection bandwidth Δf, respectively.Finally, though experimental rolloffs in Fig. 6 can be attributed to the limited 80 MHz detection bandwidth, our multi-pass loop approach can also be used to characterize degradation in instantaneous coherence length as well.This may be expected for faster wavelength tuning (i.e. more nanometers per second), or different tunable source technologies.

Tuning rate vs. sensitivity
We showed that shot noise limited sensitivity in iNIRS is nearly achieved for all modes [Fig.4], though higher loss at 500 kHz may suggest the presence of excess noise [Table 1].If the average incident sample power is limited by ANSI exposure limits [35], at higher tuning rates, the sensitivity for a single sweep decreases due to the decrease in sweep photon number.To partially offset this tradeoff, here we demonstrated combination of forward and backward sweeps, essentially doubling the autocorrelation lag time resolution and using detected photons more effectively, with no associated tradeoffs.It should also be noted that if coherent averaging of multiple shot noise limited sweeps is performed, the sensitivity can be theoretically recovered [36].

Dynamic range
System dynamic range is particularly important in time-resolved measurements at short source-detector separations where backscattered and few-scattered light reaches the detector [25,26].A high system dynamic range is needed so that these early paths do not obscure more deeply penetrating, diffuse light paths.Here we characterize the tradeoffs between dynamic range and sensitivity and TOF resolution via digital spectral shaping or windowing methods (Tables 1 and 2), achieving peak-sidelobe ratio dynamic ranges of ~36-47 dB.In particular, Gaussian shaping improves the peak-sidelobe ratio dynamic range, with a minor penalty in sensitivity and TOF resolution.iNIRS achieves considerably cleaner IRFs and better peaksidelobe dynamic ranges (up to >45 dB) than time-domain NIRS systems (~20-25 dB peaksidelobe dynamic range) [37][38][39], which are susceptible to afterpeaks and afterpulses (PMTs) [40] or diffusion tails (SPADs) [38,39].Such non-uniformities in the IRF severely contaminate information from deep structures [39].Yet, time-domain NIRS systems can achieve slightly better SNR dynamic ranges (~70 dB) [37] than iNIRS (~57-66 dB with shaping or windowing).

Comparison with existing techniques and future directions
This work explores optimization of an iNIRS system for measurements of tissue optical and dynamical properties.Currently, iNIRS fills a mesoscopic niche between microscopic (Optical Coherence Tomography) and macroscopic (time-domain NIRS) systems.Though single mode fibers are used in iNIRS, shot noise limited sensitivity is nearly achieved.Thus iNIRS can be expected to achieve similar effective photon count rates as diffuse correlation spectroscopy (DCS) with polarized detection [5], which also uses single mode fibers [41].
The TOF range of iNIRS is already sufficient to measure nanosecond-scale DTOFs [Fig.6].Therefore, it is appealing to speculate on applying iNIRS to the adult human brain.The autocorrelation lag time resolution of δτ d = 4 μs currently enables measuring an exponential autocorrelation decay rate of up to 1/δτ d = 1/4 μs −1 , which approaches values needed for the adult human brain.However, a lag time resolution of δτ d = 1 μs would be more desirable for deep brain sensitivity.This will be achievable at the laser tuning rate of F S = 500 kHz demonstrated here, with improved data acquisition.For iNIRS to be competitive with timedomain NIRS, further improvement in light collection by incorporating multiple detectors is needed [42].Additional improvement may also be achieved by reducing the source detector separation in the future, taking advantage of the demonstrated high TOF resolution and dynamic range to maintain depth sensitivity [25].

Conclusion
In summary, we have characterized key iNIRS system parameters including tuning rate, sensitivity, dynamic range, TOF resolution, and TOF range.We showed that our distributed feedback laser iNIRS system approaches the shot noise limit, and can achieve a dynamic range (peak-sidelobe ratio) of >45 dB.In addition, we introduced a novel multi-pass method to assess the TOF measurement range of our system, concluding that it is limited by electronic detection bandwidth, not intrinsic laser coherence effects, under typical operating conditions.Finally, we demonstrated a method to coherently combine forward and backward sweeps, doubling the temporal sampling and using detected photons more effectively, with no associated tradeoffs.We demonstrated the self-consistency of iNIRS TPSFs and autocorrelations in vivo at a variety of tuning rates and TOF resolutions.The methods of characterization and benchmarks presented here lay the groundwork for assessing the performance of future interferometric sensing systems that use rapidly tunable lasers.

Fig. 1 .
Fig. 1. iNIRS optical system and modulation scheme.a) The function generator (FG)sinusoidally modulates the injected drive current via the integrated current/temperature controller (I/T controller); the injected drive current in turn modulates the optical frequency of the distributed feedback laser (DFB).The laser output beam is collimated (L1), shaped with an anamorphic prism pair (APP), isolated (55 dB Thorlabs isolator), and finally coupled (L2) to a few mode fiber (SMF-28).M1-M4 are steering mirrors.The beam is split into reference (1%) and sample (99%) arms with a 99:1 fiber coupler, where the sample arm is collimated (L3) before illuminating the turbid medium.A single mode fiber coupler (L4) collects multiply scattered sample light, which is combined with the reference light by a 50:50 fiber coupler, before detection by a dual-balanced detector (DBD).Simultaneous reference power measurements are recorded with a photodiode (PD).Finally, iNIRS interference signals and reference power measurements are digitized and processed with a PC.b) This implementation of iNIRS relies on injection current modulation for wavelength tuning, resulting in concomitant modulation of the DFB laser output power as the wavelength is tuned.

Fig. 2 .
Fig. 2. Current tuning of a distributed feedback (DFB) laser incurs a reduction in tuning range and consequently, TOF resolution, with increased tuning speed.a) Maximal tuning range (ΔV) versus tuning speed.b) Peak-peak current during the sweep.c) Current tuning coefficient (dλ/dI) versus tuning speed.d) "Max", or best, achievable time-of-flight (TOF) resolution (δτ s ) based on measurements shown in (a) and Eqs.(7) and (8).
(b)].The wavelength tuning range divided by the current modulation yielded the wavelength tuning coefficient [Fig.2(c)].The decrease in tuning coefficient suggests that thermal tuning becomes less effective with increased tuning speed.As a result, the maximal achievable tuning range is reduced by approximately 53% at F S = 50 kHz, thence reducing the maximum achievable time-of-flight resolution from 11.5 ps at F S = 10 Hz tuning speed to nearly 21.7 ps at F S = 50 kHz tuning speed [Fig.2(d)].
Γ rs is related to the sample electric field, |Γ rs | 2 is related to sample photon number or intensity, and thus 10log 10 (|Γ rs | 2 ) = 20log 10 (|Γ rs |) must be presented on a dB scale.

Fig. 3 .
Fig. 3. Interpolation procedure optimizes time-of-flight (TOF) resolution by linearizing interference fringes.Raw mean-subtracted interference signals (a-b, i-j, and q-r).Phase of interference signals before (blue) and after (red) interpolation (c-d, k-l, and s-t).The interpolation procedure linearizes the fringes (e-f, m-n, and u-v) and improves the TOF resolution (g-h, o-p, and w-x), nearly achieving the "ideal" IRF (dotted black) set by the fringe envelopes.

Fig. 4 .
Fig. 4. The iNIRS sensitivity increases with increasing reference arm power and approaches the shot noise limit for both low (a) and high (b) resolution modes, as well as high speed mode (c).The input sample powers were P S,in = 35.6 mW for low resolution mode, P S,in = 23.2mW for high resolution mode, and P S,in = 29.3mW for high speed mode.Note that in (a), sensitivities are nearly identical for forward and backward sweeps.

Fig. 5 .
Fig. 5. Impact of Hamming windowing and Gaussian shaping on IRFs (a-d: low resolution mode, e-h: high resolution mode, i-l: high speed mode).Corresponding interference spectra (a-b, e-f, and i-j) and IRFs (c-d, g-h, and k-l) are shown, illustrating a tradeoff between dynamic range (peak-sidelobe ratio) and TOF resolution.

Fig. 6 .
Fig. 6. a) Multi-pass loop method for coherence time measurement of rapidly tunable lasers (DFB -distributed feedback laser, APP -anamorphic prism pair, L1/L2 -lenses, OI -optical isolator, FOPC -fiber optic polarization controller, C1-C4 -FC/APC connections, DBDdual balanced detector).b) Interference signals are generated for each pass through the loop.c) Without multi-pass losses, Fourier analysis of interference signals yields the rolloff, whose half-width at half maximum is the coherence time (τ c ) of the laser.Loop signals for 50 kHz (d)and 500 kHz (g) tuning rates.The TOF regions around each peak were summed, to mitigate TPSF broadening caused by resampling errors at large TOFs, for 50 kHz (e) and 500 kHz (h) tuning rates.Assuming that the rolloff at a very slow speed of 500 Hz, represented by the bold black line in (e) and (h), is only due to multi-pass losses, normalized rolloffs at higher speeds (50 kHz and 500 kHz) that exclude multi-pass losses can be estimated as the difference in summed rolloffs, (f and i, respectively).

Fig. 7 .
Fig. 7. a-c) Envelopes and fringe frequencies are nearly identical for forward and backward sweeps prior to inverse Fourier transformation.d-f) Resulting forward and backward sweep IRFs show good agreement.
Fig.s 7(a)-7(c).Finally, after inverse Fourier transformation, the residual phase shift between the forward and backward sweep was corrected, and the two were intercalated to form a new Γ rs (τ s, t d ) time series with twice the temporal (t d ) resolution.A comparison of IRFs derived from either forward sweeps or backward sweeps is shown in Figs.7(d)-7(f).The autocorrelation G 1 iNIRS

Fig. 8 .
Fig. 8. a) iNIRS was performed noninvasively in the nude mouse brain in vivo in reflectance mode.(b-d) In vivo TPSFs at null (dashed blue/red) and 7.6 mm (solid blue/red) sourcedetector separations.The solid black vertical line in b-d represents the zero TOF position as determined by the centroid of the null SD TPSF.

Figure 8 (
a) displays a cartoon of the reflectance geometry.The zero TOF position for TPSFs is approximately the centroid the null SD reflectance TPSF, with the caveat that the centroid of the null SD TPSF is slightly biased towards longer TOFs by scattering.Therefore, all TPSFs were delayed by an additional 6 ps (estimated by Monte Carlo simulation) to correct this bias.The IRF was measured in transmission geometry.The resulting null SD and 7.6 mm SD TPSFs are displayed in Figs.8(b)-8(d).

Fig. 10 .
Fig. 10.Normalized optical field autocorrelations for all modes approximately agree at a timeof-flight of τ s ≈100 ps.Note that autocorrelations with δτ d = 10 μs are derived from coherent comparisons of bidirectional (forward and backward) sweeps at 50 kHz, while autocorrelations with δτ d = 4 μs are derived from unidirectional sweep comparisons only.