Speckle-dependent accuracy in phase-sensitive optical coherence tomography: supplement

Phase-sensitive optical coherence tomography (OCT) is used to measure motion in a range of techniques, such as Doppler OCT and optical coherence elastography (OCE). In phase-sensitive OCT, motion is typically estimated using a model of the OCT signal derived from a single reflector. However, this approach is not representative of turbid samples, such as tissue, which exhibit speckle. In this study, for the first time, we demonstrate, through theory and experiment that speckle significantly lowers the accuracy of phase-sensitive OCT in a manner not accounted for by the OCT signal-to-noise ratio (SNR). We describe how the inaccuracy in speckle reduces phase difference sensitivity and introduce a new metric, speckle brightness, to quantify the amount of constructive interference at a given location in an OCT image. Experimental measurements show an almost three-fold degradation in sensitivity between regions of high and low speckle brightness at a constant OCT SNR. Finally, we apply these new results in compression OCE to demonstrate a ten-fold improvement in strain sensitivity, and a five-fold improvement in contrast-to-noise by incorporating independent speckle realizations. Our results show that speckle introduces a limit to the accuracy of phase-sensitive OCT and that speckle brightness should be considered to avoid erroneous interpretation of experimental data.


Speckle bias characterization
In Fig. S1, we demonstrate that the inaccuracy in phase difference from speckle is unbiased by comparing the mean value of many experimental measurements between a specular reflector and a scattering phantom. As described in Section 2.1, speckle introduces a fundamental inaccuracy to phase difference measurements. Importantly, this inaccuracy is dependent on the degree of local interference. In practice, whilst the distribution of scatterers in a turbid sample is deterministic, the location of each scatterer can be considered randomly distributed. As described in Section 2.1, consistent with existing literature, phase difference measured in a specular reflector is unbiased [1,2]. Therefore, we use the mean value of many phase difference measurements acquired in a specular reflector (provided by a polished mirror) with a high OCT SNR as an approximation to the true phase difference. For an equivalent translation between a specular reflector and speckle, this true phase difference can be used to analyze whether phase difference measured in speckle is unbiased.
The experimental setup is described in Section 3 of the main paper. To ensure the translation is equivalent between the two cases, we first set the voltage applied to the actuator and acquire an (x,z) complex OCT B-scan pair of the mirror. Without changing the system, the mirror was removed and replaced with the scattering phantom. An (x,z) complex OCT Bscan pair of the scattering phantom was then acquired. We applied an arbitrary translation of 0.325 μm (equivalent to a π radians phase shift) in both cases. For both cases, we acquire one B-scan pair and calculate the phase difference between B-scans acquired at different translation levels in each pair. Phase difference is calculated between pixels at the same location. Regions of the OCT SNR and phase difference (x,z) B-scan of the specular reflector that were analyzed in this section are shown in Figs. S1(a) and S1(b), respectively. The mean of 200 phase difference measurements, μ Δϕ , acquired at a constant depth along the xdimension in a region of high OCT SNR in the specular reflector was ~2.406 radians. The effective SNR [3] in this case was ~43 dB. As the system was unchanged between both samples, this value is taken to be the true phase difference applied in each case.
Regions of the OCT SNR and phase difference (x,z) B-scans of the scattering phantom that were analyzed are shown in Figs. S1(c) and S1(d), respectively. In Fig. S1(c), speckle is clearly visible in the OCT SNR. A histogram of 200 phase difference measurements acquired in independent speckle realizations at a constant depth along the x-dimension in the scattering phantom is shown in Fig. S1(e). The effective SNR in this case was ~21 dB. The mean of these phase difference measurements was ~2.435 radians. Here, the difference in translation measured between the two cases is approximately 3 nm, which is below the displacement sensitivity of the system at 20 dB (~10 nm). These results demonstrate that the phase difference measured across multiple speckle realizations, i.e., the inaccuracy in speckle, is unbiased.

Speckle brightness metrics
In this section, we describe two different methods of quantifying the amount of constructive interference of the OCT signal in speckle. As described in Section 2, speckle brightness, i.e., the amount of constructive interference in the OCT signal, influences phase difference accuracy and sensitivity. As described in Section 3.3, an intuitive approach to measure speckle brightness is to compare the measured OCT amplitude to the maximum possible amplitude at each spatial location. This is illustrated in Fig. S2(a) for the case of a onedimensional (1-D) simulation of an OCT A-scan. Details of the simulation are provided in Sections 2.1 and 3.1 of the main paper. Here, to demonstrate the principle of measuring speckle brightness, we analyze the OCT amplitude in the absence of optical noise. Figure  S2(a) shows an arbitrary distribution of scatterers in depth. We simulate an OCT A-scan by convolving the distribution of scatterers with the complex point-spread function (PSF), where the real (blue line) and imaginary (green line) components of the complex OCT signal at each depth are shown. In addition, we plot the envelope (red line), equivalent to the OCT amplitude, of the complex OCT signal. As indicated by the arrows in Fig. 2(a), regions of constructive interference (bright speckle), and destructive interference (dark speckle) are clearly visible in this A-scan. The maximum OCT amplitude is determined in simulation by convolving the scatterers with the OCT point-spread function (PSF) envelope (dotted black line), i.e., by ignoring the phase of the constituent responses.
Here, speckle brightness is quantified by dividing the measured OCT amplitude by the maximum possible amplitude which yields a measure from 0 (darkest speckle) to 1 (brightest speckle). We term this speckle brightness metric, metric one (M1). The corresponding values of M1 at each location for the simulation in Fig. S2(a) are shown in Fig. S2(b). Whilst M1 provides a direct measure of speckle brightness, it is impractical in experiment as it requires prior knowledge of the precise location and reflectivity of each sub-resolution scatterer. Therefore, to quantify the degree of interference in experiment, we instead propose a metric based on the normalized Rayleigh distribution which is described in Section 3.3 of the main paper. We term the speckle brightness metric based on the Rayleigh distribution, metric two (M2). Whilst M2 does not provide a direct measure of speckle brightness, it has the advantage of only requiring the total OCT signal, and hence can be applied in experiment where the exact locations and reflectivity of the sample scatterers are unknown. Importantly, both M1 and M2, can be applied in simulation, which can be used to assess the validity of M2. In Fig. S3, we compare the relationship between phase difference sensitivity and speckle brightness using both metrics in simulation. 255 million phase difference measurements are analyzed in Fig. S3 and Fig. S4. In Fig. S3, we translate the homogenous phantom 1.3 µm, corresponding to a 4π phase difference and compare points with the same OCT SNR and different values of speckle brightness. Note that M1 is in the range [0, 1], whereas M2 is theoretically in the range [0, ∞], and practically [0, 5]. Details of the simulation parameters, including the values used for optical noise and attenuation, are provided in Sections 2.1 and 3.1. In Figs. S3(a), S3(b), and S3(c), we present histograms of phase difference versus M1 for points with an SNR of 15±1 dB, 20±1 dB, and 25±1 dB, respectively. Similarly, in Figs. S3(d), S3(e), and S3(f), we present histograms of phase difference versus M2 for points with an SNR of 15±1 dB, 20±1 dB, and 25±1 dB, respectively. The color scale in Fig. S3 represents the number of measurements in each histogram bin divided by the total number of measurements. Here, both metrics are in close agreement, and in each case, the distribution of phase difference decreases in width for increasing speckle brightness. Furthermore, in Fig. S4, we show the correlation between M1 and M2 by plotting the speckle brightness measured at each corresponding location using both metrics. The 45degree line (dotted black line) represents a 1:1 correlation. The color scale in Fig. S4 represents the number of measurements in each histogram bin divided by the total number of measurements. Figure S4 shows a positive correlation (correlation coefficient of ~0.75) between M1 and M2, which along with the results in Fig. S3, demonstrates that both metrics can be used to characterize the relationship between speckle brightness and phase difference sensitivity.

Averaging method
Here, we describe the method used to compute the mean OCT amplitude, OCT SNR and phase difference B-scans presented in Section 4. As described in Section 3.2 of the main paper, we acquire N B-scan pairs at each lateral y-location. Each B-scan pair consists of one complex OCT amplitude B-scan acquired with the sample under the static preload (referred to as the unloaded scan), and one complex OCT amplitude B scan acquired with an additional microscale compression loading (referred to as the loaded scan). For temporal averaging, we acquire N B-scan pairs at one lateral y-location. For spatial averaging, we acquire one B-scan pair at N lateral y-locations. In Sections 4.3 and 4.4, N = 100, and each B-scan pair in the spatially averaging case was acquired 10 µm apart. We use the same method to compute the mean amplitude, OCT SNR, and phase difference in both the temporal and spatial averaging cases which is described below.
Firstly, we separate the unloaded and loaded complex OCT amplitude B-scans, A, from each B-scan pair. The complex intensity, I, for the both the unloaded and loaded cases is computed by: We compute the mean OCT intensity, μ I , in the complex domain: The mean complex OCT intensity is converted back to mean complex OCT amplitude, μ A , by: The mean OCT amplitude is found by taking the absolute value of Eq. (S3). Speckle brightness in Section 4 is computed using the unloaded mean OCT amplitude.
To determine the OCT SNR for a given B-scan, firstly, the standard deviation, σ, inside a region containing no signal in the mean complex OCT amplitude is computed for both cases. The linear OCT SNR (SNR linear ) is then computed by: The OCT SNR in decibels (dB) (SNR dB ) is given by: We present the unloaded OCT SNR in dB in Section 4. To compute the mean phase difference, we first compute the phase difference between the unloaded (A UL ) and loaded (A L ) complex OCT amplitude B-scans in each B-scan pair. We then compute the mean in the complex domain, where the mean phase difference (μΔϕ) B-scan is given by: where A represents the complex conjugate of A.
In Section 4, the corresponding strain B-scans are computed from the mean phase difference B-scans given by Eq. (S6).