Multiple scattering is one of the main degrading influences in optical coherence tomography, but to date its presence in an image can only be indirectly inferred. We present a polarization-sensitive method that shows the potential to detect it more directly, based on the degree to which the detected polarization state at any given image point is correlated with the mean state over the surrounding region. We report the validation of the method in microsphere suspensions, showing a strong dependence of the degree of correlation upon the extent to which multiply scattered light is coherently detected. We demonstrate the method’s utility in various tissues, including chicken breast ex vivo and human skin and nailfold in vivo.
©2007 Optical Society of America
In optical coherence tomography (OCT) of turbid media (such as biological tissue), the maximum sample penetration depth for which a meaningful image may be formed is limited by multiple scattering [1, 2]. The collected fraction of multiply scattered, relative to singly scattered, light increases with depth, reducing image contrast  and degrading the axial and lateral resolutions [2,4–6]. The reduced contrast and resolution mask the real structure in a turbid sample, and it has not generally been possible to determine without a priori knowledge the extent to which such corruption is present in an OCT image. As well as degrading image quality, the presence of multiple scattering has been shown to deleteriously affect quantitative measurements of tissue optical properties [4,7–9]. Dynamic light scattering methods, implemented with low-coherence interferometry, have been used to study its effects in liquid samples [10,11] with success, but the continued inability to directly detect multiple scattering is a general impediment in advancing OCT imaging.
Multiple scattering is known to cause change in not only the propagation direction of light in a turbid medium, but also to randomize its (in general, elliptical) polarization state and to decrease its degree of polarization (DOP), phenomena which are commonly referred to as depolarization [12–18]. These depolarization effects provide a means for polarization gating to discriminate shallow from deep backscattering in biological tissues [19–22].
In this paper, we utilize polarization-sensitive (PS)-OCT [23, 24] to measure the detected polarization states at different system axial and lateral scan positions, and determine the extent to which they are correlated. We note the instantaneous OCT signal is not directly sensitive to reductions in the DOP of the backscattered light , because the detected interferometric signal arises only from the component of the backscattered light that is capable of being coherently mixed with the polarized reference beam. This component is necessarily fully polarized, although its polarization state is free to vary.
We hypothesize that when the detected signal at any position is composed mainly of singly scattered light, its associated polarization state will be highly correlated with those in its vicinity. This is justified by noting that in a number of important special cases, the light directly backscattered from a single sample scatterer retains its incident polarization state (subject to a coordinate-system reflection transformation). This holds true for the special cases of: a homogeneous spherical scatterer (Mie theory); an arbitrarily shaped scatterer whose relative refractive index (with respect to the background) is near-unity (Rayleigh-Debye scattering); an extremely small homogeneous (Rayleigh) scatterer; or an extremely large arbitrarily shaped scatterer (WKB interior wave number approximation) . More generally, the statement must be qualified; for example, light depolarization will occur for light backscattered randomly oriented aggregates of ellipsoidal particles of size comparable to the illumination wavelength . Even in this case, the degree to which depolarization occurs will be fairly moderate over a broad range of particle size and ellipsoid aspect ratio conditions. In contrast, when predominantly multiply scattered light is detected, its associated polarization states within small scan regions will be highly uncorrelated , because wide-angled scattering in general is highly polarization-dependent. (For example, independent of its incident polarization state, light scattered at 90° from a Rayleigh scatterer will be linearly polarized in the direction orthogonal to the scattering plane.) The measurement of a strong “cross-polarized” backscattered component in OCT  was attributed to both multiple scattering and single backscattering from non-spherical particles.
Sample birefringence tends to vary the polarization state of the incident light as it propagates through the tissue, both prior and subsequent to a backscattering event. However, the variations in many samples (such as linearly oriented fibrous tissues) tend to be systematic (not random), over length scales hundreds of micrometers or more, much greater than the coherence length of the light source . As such, these variations should be readily distinguishable from those due to multiple scattering . This statement may hold even for tissue types with randomly oriented fibers, such as collagen networks. Although the length scale of the sample birefringence variations (for example, fluctuations in the optic axis direction) may be small, compared to the coherence length, the magnitude of the birefringence will be sufficiently low that the polarization state of the light will remain constant over relatively large propagation distances.
We propose a measure of the polarization-state correlation within a small spatial kernel, based on the variation of the reduced Stokes vectors detected within it, and demonstrate that it is correlated with the presence of multiple scattering in OCT images. In Section 2 we describe the theoretical framework and method for analyzing the OCT signals; we provide a detailed description of the experimental setup in Section 3. The results of experiments performed on aqueous microsphere suspensions and tissue samples are presented in Section 4. The discussion of Section 5 concentrates on the impact of the sample and system characteristics upon the results, and in Section 6, we draw together the main conclusions.
2. Theoretical framework and method
2.1. Acquisition of Stokes vectors at each sample location
We consider a single-mode optical fiber-based PS-OCT system. In conventional PS-OCT optical setups , it is generally desired to illuminate the sample with circularly polarized light. This is because many, although not all, birefringent samples of interest are linearly birefringent [24, 28–32]; circularly polarized light, therefore, contains equal components of both sample eigenpolarization states, ensuring maximum variation in polarization state as the beam propagates through the sample . For our purposes, illumination with circularly polarized light is unnecessary, since our technique does not require that the system be sensitive to sample birefringence. With reference to the schematic shown in Fig. 1, which will be further described in Section 3, we utilize the polarization controllers PCS and PCD to impose two conditions on the returning light incident upon the polarizing beam splitter (PBS). The first, PCS, ensures that the elliptical polarization state of singly backscattered (or specularly reflected) light from the sample arm is identical to that from the reference arm after the signals have been recombined at the output of the 50/50 coupler. For general samples, the scattered light will have undergone a polarization state change, which may be described as a superposition of a large component in this state and an additional (usually smaller) component in the orthogonal state. This is indicated in Fig. 1, by denoting the light returning from the sample arm with two distinct ellipses, corresponding to these co- and cross-polarized states. The second polarization controller, PC D, ensures that the singly backscattered component of the sample light is equally split between the two outputs of the PBS. Equal splitting will occur provided that the major axis of the ellipse describing the reference polarization state at the PBS input is oriented at ±45° to the PBS axes. Equal splitting will occur provided that the major axis of the ellipse describing the reference polarization state at the PBS input is oriented at ±45° to the PBS axes. The reference and sample electric fields may be expressed in terms of their analytic-signal (complex, time-dependent) vector representations with respect to these eigenaxes, E R=[E R1(t) E R2(t)]T and E S=[E S1(t) E S2(t)]T, where the parameter t represents time, and the superscript T represents matrix transpose. The detected complex interferometric signals at each output (following full-fringe detection) can be written (up to an unimportant scaling constant) as:
where the triangle brackets indicate a long time average (over the photoreceiver response time). If, as is expected for singly scattered sample light, the detected reference and sample polarization states are matched, so that E S and E R are related by a complex scaling factor, then Ĩ1=Ĩ2, since the light is split equally between the orthogonal detectors. If the states are unmatched, then this equality will not hold.
The signals Ĩ1 and Ĩ2 are functions of reference-arm position z (which corresponds to a particular optical depth in the sample). They can be represented in terms of the real Stokes vector  parameters:
where δ=Arg(Ĩ2/Ĩ1), and for which I 2=Q 2+U 2+V 2, a consequence of the fact that our system detects a fully polarized signal. (These Stokes vector parameters, we reiterate, do not describe the polarization state of light scattered from the sample; instead, they describe the relative amplitudes of, and the phase difference between, the detected interferometric signals.) The normalized reduced Stokes vector (hereafter, Stokes vector) Ŝ=[Q̂ÛV̂]T, where Q̂=Q/I, Û=U/I, V̂=V/I, when plotted in 3-dimensional space, lies on a unit Poincaré sphere. The condition Ĩ1=Ĩ2 corresponds to the Stokes vector Ŝ0=[0 1 0]T.
2.2. Determination of the mean Stokes vector
As noted in the introduction, the presence of multiple scattering will tend to randomize the polarization state of the detected scattered light, whereas a single backscattering event will tend to preserve the incident state. Within the recorded two-dimensional B-scans, let the z direction denote optical depth (with the origin situated at the sample interface), and the x direction denote lateral displacement. Then the Stokes vector associated with the sample location (x, z) may be denoted Ŝ(x, z). Thus, if the signal at (x, z) is primarily associated with singly backscattered light, then in the absence of sample-induced systematic polarization transformations, Ŝ(x, z)=Ŝ0. If such polarization transformations do occur, principally due to systematic birefringence (and biattenuance) , then Ŝ(x, z) may not be equal to Ŝ0, but should be highly predictable based on knowledge of the Stokes vectors Ŝ associated with backscattering from the surrounding region. This statement will not hold for multiply scattered light, for which the detected polarization state at a given location will be, in general, nearly uncorrelated with the values of Ŝ in the surrounding region.
We apply a (scalar) averaging kernel K 1(x, z) to the map of Ŝ(x, z), to obtain the mean Stokes vector in the vicinity of a location (x, z),
where ⊗ denotes the (two-dimensional) convolution operation, acting on all three elements of the vectors Ŝ, and ‖.‖ denotes vector norm. (The normalization factor in the denominator ensures that this quantity is a unit vector.) We now consider the singly backscattered detected signal from a sample of uniform birefringence. If the quantity ∫∞-∞K 1(x, z)dx, taken as a function of z, is symmetric about the origin, for which a sufficient condition is that the x-axis is an axis of symmetry of K 1(x, z), then the vector will be an almost unbiased predictor of Ŝ. This is because the detected polarization state path over each axial scan will trace out a circle on the Poincaré sphere, the axis of which is parallel to the birefringent eigenaxes . The stated condition would ensure that equal contributions from both sides of the polarization state at the center of the kernel contribute to the mean state. The birefringent artifacts from these contributions tend to cancel, thereby rendering the mean state relatively immune to such artifacts. (There will be a slight systematic difference between Ŝ(x, z) and (x, z) when the polarization state path is not a great circle of the Poincaré sphere; in practice, this effect is negligible.)
In the case of samples with random variations in birefringence on a small scale, such as fluctuations in the direction of the optic axis, then the polarization state path will not be a circle on the Poincaré sphere; instead, it will resemble a random walk. Provided that the magnitude of the birefringence is low, then the random walk will traverse only a short distance on the sphere over the axial length of the kernel K 1, a few times the coherence length of the light source. (This will be the case for virtually all biological tissues, for which the maximum birefringence corresponds to a beat length on the hundreds of micrometers to millimeter scale [24, 33], much greater than the source coherence length.) The mean polarization state would remain an excellent predictor of the state at the center of the path. However, this argument only applies to variations in detected polarization state along an axial path. The signals from the same depth in successive axial scans, that is, from locations displaced laterally, are generated from light which has undergone different propagation paths through the sample. Therefore, the accumulated polarization state variations may be significantly different, and seemingly random, for such signals. (The issue may be of significance even for samples of apparently uniform birefringence.) Quantifying this effect requires further investigation. It suffices to note that the method we propose in this paper will still be applicable to samples for which it is applicable, provided that the kernel K 1 is chosen to be narrow in the lateral direction (and perhaps correspondingly broader in the axial direction).
We make the distinction between sample birefringence (polarization-state-dependent refractive index), which affects the input beam polarization state along the propagation path to and from the backscatterer, and polarization-randomizing backscattering, as from non-spherical particles. The arguments of the preceding paragraphs demonstrate that the approach we adopt in this paper, comparing detected Stokes vectors to their corresponding mean Stokes vectors, is relatively immune to sample birefringence. However, the extent to which the incident polarization state is randomly transformed by backscattering from non-spherical scatterers of arbitrary orientation is an impediment to our method.
2.3. Quantification of random polarization state variations
At each (x, z) location in the two-dimensional map, a parameter ζ is defined that measures the variation between the unit vectors Ŝ(x, z) and (x, z), specifically,
where • represents vector inner (dot) product. The parameter ζ can range between -1 and 1. If the detected polarization state at a location is identical to the mean state associated with that location, which is to be expected in the case of single scattering (in the presence of birefringence or not), then ζ will be close to 1. In the presence of strong multiple scattering, the state Ŝ(x, z) will be random, and virtually independent of (x, z), so that ζ may take any value from -1 to 1. Indeed, it will be uniformly distributed between these quantities, with expected value zero.
To effectively quantify the degree of multiple scattering detected within a region, it is useful to take a weighted average of ζ in the vicinity of a single point, that is, we define:
where K 2 is an averaging kernel, which has been appropriately normalized so that ∫∫∞ -∞K 2(x, z)dxdz=1. The parameter ζ̄ is therefore limited to values between -1 and 1, although extreme negative values will be very rare, and the most extreme, mathematically impossible. We should expect that will take values near 1 in the presence of single scattering, and values near 0 in the presence of strong multiple scattering.
At this point, we consider the relationship between the two envelope speckle patterns obtained from the orthogonal detection channels, particularly the extent to which it is relevant to our method. When the detected signal is dominated by singly scattered light, the two speckle patterns will be highly correlated. This follows from our assumption that singly backscattered light retains its incident polarization state. A given point of a speckle pattern can be described as a random sum of complex phasors, each of which is associated with an individual sample backscatterer [34, 35]. If the incident-light polarization state transformation corresponding to each is the same (a more general condition than the stated assumption), or nearly so, then the detected sample (scalar) fields in the orthogonal detection channels will be equal, up to a complex constant which is independent of scan position. Their envelope speckle patterns will be identical. However, multiply scattered light, which randomizes the polarization state of the incident light, will likewise decorrelate the two directly detected speckle patterns. Thus, the degree of correlation between the detected speckle patterns is closely related to our parameter ; alternatively, we interpret polarimetric speckle noise  is an indicator of the presence of multiple scattering.
We also draw a comparison between ζ and the “degree of polarization” (DOP) parameter defined in Ref. . It was noted in the Introduction that the detected light in PS-OCT is fully polarized; this condition applies to the instantaneous signal, detected at any point of an A-scan. However, if multiple (non-reduced) Stokes vectors are added “incoherently” , then the result will, in general, have a DOP of less than 1. In Refs. [24, 37], such a summation is performed by Fourier-transforming a small segment of the OCT A-scan, calculating the non-reduced Stokes vector for each spectral component, and integrating each element of the vector over the source spectrum. There is a trade-off between spectral resolution (and, thus, the effective number of independent Stokes vectors in the integral) and the length of the segment (spatial resolution). Likewise, the use of the spatially resolved parameter ζ involves a loss of spatial resolution, corresponding to the size of the kernel K 1. Both parameters are sensitive to variations between the orthogonally detected interference signals; for a DOP of near 1 over an A-scan segment, the corresponding speckle patterns from the two detection channels must be the same (up to a constant factor). In Ref. , reduction in DOP, calculated this way, was attributed in part to multiple scattering. The detection of its presence is the objective of the current study.
The following section describes in detail the experimental setup used to generate maps of ζ and versus position, for a number of different sample types.
3. Experimental setup and data analysis
A schematic of the fiber-based PS-OCT experimental setup is shown in Fig. 1. A 7.75-mW polarized broadband source, with a center wavelength of 1330 nm and a bandwidth of 43 nm, was split by a 50/50 coupler into the sample and reference arms of the interferometer. Reference arm scanning, indicated schematically with a translating mirror, was performed with a frequency-domain optical delay line in the off-axis configuration . The sample arm optics consisted of a collimator outputting a 2.76-mm (1/e 2 width) Gaussian beam focused onto the sample with a Steinheil triplet lens (f=20 mm). The resulting measured axial full-width-at-half- maximum (FWHM) resolution was 18 µm, and the transverse resolution, as described by the 1/e 2 beam width, was equal to 12 µm. The depth at which the beam was focused varied between the experiments.
The detection arm consisted of a fiber-based polarizing beamsplitter (PBS) with single-mode fiber at the input and polarization-maintaining fiber at the two outputs. Light from the output was fiber-coupled into two DC-coupled, equal-gain photoreceivers. Signals from the photoreceivers were amplified and bandpass filtered prior to digitization, which was performed with a data acquisition board operating at 625 kSamples per second.
The system polarization states within the optical fiber were transformed at several different locations in the optical path using fiber-based polarization controllers. The polarization controller PCR (see Fig. 1) was adjusted to select the polarization state launched into the optical delay line which minimized its polarization-dependent loss. A specular reflector in the sample arm was then used to match the reference and sample polarization states; the polarization controller PCS was adjusted to optimize the interference visibility by placing a photoreceiver prior to the PBS. Finally, the PBS was re-connected and the polarization controller PCD was utilized to ensure the major and minor axes of the elliptical detected state were aligned at 45 ° to the PBS axes, according to the requirements outlined in Sub-section 2.1. This was achieved by equalizing the interference signal amplitudes in the two orthogonal photoreceiver outputs.
4.1. Microsphere solutions
Measurements were performed on 0.51 µm-diameter polystyrene microsphere aqueous suspensions, with seven concentrations ranging from 0.034 µm-3 to 0.69 µm-3, and the beam focus located at an optical depth of 0.6 mm. For each concentration, the scattering coefficient µs of the suspension (at the center optical wavelength) was calculated using Mie theory. It was equal to µs=0.99 mm-1 for the lowest concentration of 0.034 µm-3, and, assuming that correlated scattering effects can be discounted, was proportional to concentration. Based on the same assumption, the anisotropy of each suspension was 0.47.
The variation of Stokes vectors with depth is illustrated in the movie of Fig. 2. Each frame corresponds to a particular optical depth, which is indicated by the advancing magenta vertical line in the OCT envelope image on the left. This envelope image corresponds to a concentration of 0.41 µm-3; it is the Pythagorean sum of the envelope signals from the individual detectors. For this concentration (green), and the lowest concentration (0.034 µm-3, dark blue), the detected Stokes vectors at each depth are plotted on the Poincaré sphere on the right of Fig. 2. It is clear that the variation in the detected polarization states at the higher concentration is uniformly greater than that at the lower concentration. Indeed, at the lower concentration, the detected states remain clustered around the Û -axis, that is, the vector Ŝ0, over the entire scan range. (Large variations in the mean of the detected states near the interface are artifacts due to ringing from the strong specular reflection.) For the purposes of greater statistical sampling, five separate recordings (each corresponding to a lateral length of 2.5 mm) were recorded and appended together, yielding the artificial lateral range of over 12 mm displayed on the figure.
To investigate this phenomenon using the theory of Section 2, we present in Fig. 3(a) the OCT B-scan envelope image from the mid-range sphere concentration, 0.28 µm-3, obtained in the same way as the corresponding image in Fig. 2. (In this paper, for the purpose of conveying information over a large dynamic range, false color scales are used for all envelope images that are not overlaid with a separate color map.) Maps of ζ(x, z) and (x, z) are plotted in parts (b) and (c), and the kernels K 1 and K 2 that were used to generate them are displayed graphically. Both kernels are symmetrical, and separable when expressed as functions of axial and lateral coordinates. The kernel K 2 exhibits a Gaussian profile in both directions, with lateral FWHM equal to three times the system lateral resolution, and axial FWHM equal to twice the system axial resolution. (Different factors were chosen in the axial and lateral directions due principally to the larger sampling spatial period in the lateral direction.) The kernel K 1 is the difference between such a Gaussian kernel with lateral and axial FWHMs equal to 5 and 4 times the respective system resolutions, and one with lateral and axial FWHMs equal to 3 and 2.5 times these quantities. (The reason for this choice of K 1 is to ensure that the “mean” polarization state Ŝ is not heavily biased by the center state Ŝ, with which it must be compared. For the same reason, the size of K 1 is larger than that of K 2.) These kernel specifications apply to all results presented in this paper.
Figure 4(a) shows the average A-scans for the seven microsphere concentrations. Shown in part (b) are plots of the mean parameter versus optical distance z (averaged over all lateral scan positions). To effectively compare the latter curves for different sphere concentrations, the sample depth parameter for each was converted to “number of mean-free paths”, where one mean-free path is equal to the inverse of the scattering coefficient µs. (This is a physical distance, and the refractive index n=1.32 was used at the center wavelength of 1330nm in order to convert from optical to physical depth.) Limited regions of these scaled curves (corresponding to depths surrounding the focal point) are plotted in part (c) of the figure.
In Fig. 4(a), at the highest concentrations, a reduction in slope (on the logarithmic scale) is observed in the A-scans, as depth increases. This has previously been interpreted as signaling the onset of the multiple scattering regime [3, 10, 39, 40]. At lower concentrations, the axial confocal sectioning effect produces the distinct peak at approximately 0.6 mm, but with slightly different apparent attenuation coefficients either side of the confocal waist, due to the increase in multiple scattering with depth, especially for the intermediate concentrations. The rapidity at which multiple scattering besets the signal immediately following the beam focus can be explained by noting that multiply scattered light that is (preferentially) accepted by the confocal gate will have an increased apparent path length and, therefore, appear to be reflected from a position in the sample beyond the focal point. The system noise floor corresponds to a signal level of less than -55 dB, at least 8 dB below the minimum mean envelope signal for any of the sample concentrations. Note that the sidelobes corresponding to the signal from the glass interface at negative optical depths are clearly discernible, so they have not been overshadowed by system noise. (In this figure, the decibel scale is referenced to the mean signal at the axial location corresponding to the peak, to facilitate the comparison between the seven samples. In all other figures, it is referenced to the maximum envelope level over the two-dimensional image.)
The curves of Fig. 4(b) exhibit the near-uniform trend that the mean of decreases with increasing sphere concentration, at all depths. The maximum value (following the sample interface) of nearly 0.9 is consistent with the notion that single scattering dominates for the lower concentrations, over the entire length of the scan. As concentration and multiple scattering increase, the baseline value (of about 0.25–0.3), is reached at progressively shallower optical depths.
The rescaled curves of Fig. 4(c) highlight the fact that, given the optical depth at which it is measured, the parameter does not depend strongly on the specific sphere concentration, at least in the vicinity of the focal plane. For the current value of anisotropy (and system characteristics), extinction of single scattering (when reaches its baseline) occurs at about 9 mean-free paths.
4.2. Chicken breast
Since birefringence causes the polarization state to vary systematically with depth, we investigate its influence on the parameter in a homogeneous birefringent sample - chicken breast. The polarization controller PCS was adjusted to ensure that linearly polarized light was incident upon the sample arm. (An additional polarizer was utilized to verify this, and then removed.) Measurements of the chicken breast were then obtained at various sample rotation angles (in steps of about 22.5°). The movie of Fig. 5 corresponds to the data set exhibiting the most prominent bands. As in Fig. 2, the variation in detected polarization state tends to increase with depth, but for this birefringent sample, the mean state (of those plotted in each frame) does not remain constant. Instead, it traces large circles on the Poincaré sphere.
Figure 6(a), (b), and (c) present the scaled Q̂, Û, and V̂ Stokes parameters versus location, for the same data set, showing relatively homogeneous morphology in the lateral dimension and a banding pattern with depth corresponding to a double-pass phase retardation rate of approximately 0.88°/µm (assuming the refractive index n=1.4). The fact that the banding pattern is clearly visible throughout the entire image indicates that a high proportion of the light detected from all imaging depths was singly backscattered. However, it is also clear that the banding pattern tends to be more corrupted at greater depths within the sample. The envelope signal and parameter are displayed in the second row of the figure (parts (d) and (e)). The high value of the latter quantity confirms the observation that single scattering dominates the detected signal at all depths. Even so, there are small contiguous regions for which the polarization state seems to be highly randomized (corresponding to low values of ). Interestingly, they correspond to one particular mean polarization state, because they recur at the period of the banding pattern (or beat length), approximately at regions for which the parameters Q̂ and V̂ are greatest. This effect is displayed even more prominently in Fig. 6(g), the plot of the mean versus position. This parameter is relatively high for most of the scan, but a prominent highly fluctuating periodic component is also visible. The existence of this fluctuating or banding effect does not contradict with the previous contention that the parameter ζ is insensitive to strong uniform birefringence. This is because, upon close inspection, the regions of the Q̂, Û, and V̂ maps for which is low clearly show a grainy pattern, indicative of a highly variable detected polarization state, not birefringence “artifacts.” However, the correlation between the magnitude of and the detected (mean) polarization state requires an explanation. One possibility is that the degree to which the backscattered light’s polarization state is randomized depends on the polarization state of the light incident upon the scatterer. Alternatively, the amount of backscattering from a region, and thus, the singly scattered fraction of its associated detected signal, may depend on the incident-light polarization state. Further investigation is required to confirm or refute these suggestions. (We note that the DOP parameter of Ref.  does not appear to show this periodic effect when plotted for a chicken breast sample.)
The mean signal level is displayed in Fig. 6(f); it shows a gradual decrease in detected signal following the beam focal plane. (The modest fluctuations in this signal provide only marginal evidence for the “alternative” explanation suggested at the end of the previous paragraph.) However, the high values of at the large depths in the chicken breast sample suggest that the penetration depth of this measurement is not strongly limited by multiple scattering and may be increased by illuminating the sample with greater light power, or employing longer OCT acquisition times.
4.3. Human skin
The measurements in human skin were conducted to examine the behavior of the parameter in tissue that is not uniformly birefringent. In vivo OCT images were taken of the anterior surface of the distal forearm (near the wrist) of a male Caucasian volunteer using glycerol for index matching to the sample-arm glass window. The top row of Fig. 7 shows the envelope signals of the two orthogonal OCT channels corresponding to an OCT B-scan. Skin tissue is a stratified medium, and its distinct layers are identifiable in these images. Based on inspection of the figure, the first layer, the epidermis, extends from the interface (z=0) to an optical depth of z=0.15 mm. The subsequent highly scattering layer from z=0.15 mm to z=0.4 mm we attribute to the papillary dermis. Below this layer is the reticular dermis, which accounts for the remainder of the scan depth.
The closely packed spheroidal cells of the epidermis are not expected to display any birefringence. In this region, the signal envelopes corresponding to the two orthogonal polarization states are of similar magnitude, consistent with the contention that the incident polarization state remains largely unchanged upon backscattering. There are strong differential scattering effects in the dermal layers (at optical depth 0.25 mm), that is, clearly different envelope signals recorded in the two detection channels. These effects possibly indicate the presence of cylindrical scattering structures such as clusters of collagen fibrils with polarization-dependent scattering cross-sections. The sharp drop in the mean of throughout the papillary dermis, evident in Fig. 7(d), suggests a possible increase in the contribution of polarization-randomizing scattering to the detected signal. However, despite this apparent dependence on tissue composition, the mean of consistently decreases with depth throughout the sample, well into the reticular dermis, implying that the signal is increasingly being affected by multiple scattering.
4.4. Human nailfold
In the final experiment, we investigated the behavior of the parameter in heterogeneous tissue - the human nailfold [4, 33]. It is a longstanding example of a biological sample thought to be influenced by multiple scattering (see Fig. 1 of Ref. ). In vivo OCT images were taken of the nailfold of a male Caucasian volunteer using glycerol to effect index matching with respect to the glass interface. The OCT B-scan envelope image of Fig. 8(a) shows heterogeneous structure corresponding to the anatomical features  of the exposed nail, nail covered by the proximal nailfold (or eponychium), and the cuticle at the tip of the eponychium. The cuticle is composed of epidermis which appears to be a continuation of the superficial epidermal layer of the eponychium. Sandwiched between this epidermal layer and the nail is a layer of “soft tissue,” which, like the papillary dermis of thin skin, is fairly homogeneous. The signal from the nail underlying the eponychium is significantly lower than that for the exposed nail, which is consistent with previous results [4, 33].
The map of versus position in Fig. 8(b) shows that even at the greatest depths (z~1.5mm), the value of remains moderately high (greater than 0.5), well above the multiple scattering baseline (~0.3) observed for the microsphere solutions. Its overlay onto the envelope signal in Fig. 8(c) highlights the strong dependence of local values of upon the sample layer in which they are observed, not on the amount of overlying tissue. This is consistent with our previous observation that the OCT signal contains a strong singly scattered component at all depths; variations in are likely due primarily to polarization-randomizing scattering structures. Further evidence for this interpretation is found in the high value of at the location of the soft tissue below the exposed nail (bottom-left of Fig. 8(b)), suggesting that a large fraction of the collected light at this scan position was singly backscattered. Since the effects of multiple scattering tend to accumulate with depth, this implies a high contribution of singly scattered light to the signal at shallower depths along the A-scan. Yadlowsky et al.  suggest that the absence of signal from the unexposed nail (which is a distinct feature of their image) is due to complicated multiple scattering. A key difference between this measurement and our own, in which the unexposed nail and the layers beneath it are clearly visible, is that we used glycerol to index-match the tissue to a glass window in the sample arm. Light propagating through the tissue-glass interface, though attenuated, is not subject to significant scattering or wavefront distortion.
The experiments with microsphere solutions confirm the basic hypothesis advanced in this paper, namely, that the local degree of polarization state correlation characterizes the relative predominance of single and multiple scattering. The transition between predominantly single scattering, and predominantly multiple scattering, was found to occur at around 9 scattering mean-free paths, for anisotropy g=0.47. This is greater than the OCT contrast limits reported by Pan et al.  of approximately 6 mean-free paths for g=0.8. An increase in penetration depth is consistent with the dependence of this quantity upon g reported by Fercher et al. ; the variation is also likely to be due to the differences in other system parameters (such as numerical aperture, NA) and the definitions of penetration depth. Indeed, our results support the suggestion of Ref.  that a low objective NA be used in suppressing multiple scattering from an isotropic medium. For our purposes, using a low NA will also ensure that the detected singly scattered light has been nearly directly backscattered, as required to preserve the incident polarization state for spherical particles.
Additional work with tissue phantoms may be useful for demonstrating further the relationship between the polarization-based multiple-scattering quantifiers and OCT contrast limits. Since OCT resolution is known to be degraded by multiple scattering, the relationship between the polarization-based quantifiers and resolution degradationwarrants further investigation. The influence of anisotropy g and system NA on our technique, and the relationship between the rate of depolarization and rate of delay accumulation for multiply highly forward scattered light, are other intriguing subjects for future research.
Taken as a whole, the results on tissues, shown in Figs. 5–8, confirm the potential of the approach to identify sample regions in which single scattering predominates. In the reticular dermal layer of the skin sample, the mean of appears to be affected by multiple scattering, and in the nailfold sample, the detection of singly backscattered light from structure below the exposed and unexposed nail was confirmed. Since the technique is expected to be immune to (at least uniform) birefringence, the measured modulation of the mean of in the chicken breast tissue requires further investigation, especially the dependence of the backscattered signal upon the incident polarization state.
In biological samples, the value of is affected by polarization-randomizing backscattering from non-spherical scatterers  with the spatial scale of polarization-state variation comparable to or below the size of the kernel K 1. Polarization-randomizing scattering is likely to have influenced the values of observed in the reticular dermis of the skin (Fig. 7), and be the cause of low values of this quantity apparent in the exposed nail in Fig. 8.
It may be possible to use smaller kernels K 1 and K 2, by employing a smaller coherence length light source and higher system NA, or by three-dimensional acquisition. This would permit the measurement of a statistically significant number of independent resolution elements (or speckle states) whilst improving the resolution of the maps of (and increasing the likelihood that it is less than the scale of the systematic polarization variations in the sample). Another approach for obtaining a statistically significant number of speckle states could be the use of custom shaped kernels, deformed so that they sample large areas (or volumes) corresponding to contiguous regions of homogeneous structure (such as tissue layers).
A factor that potentially influences the parameter is system noise . Noise sources can be correlated between the detection channels (for example, source intensity noise), or uncorrelated (for example, shot noise or electrical noise). In the former case, the parameter may be artificially boosted, since a favored background polarization state is constantly present in the signal noise; in the latter case, the parameter will be artificially decreased, due to the additional randomness inherent in the noise. Correlated source intensity noise, in particular, may partially account for the fact that the baseline value of was consistently greater than zero. (This correlation may affect even when the total signal strength is much greater than the noise level.) In addition to system noise contributions, sidelobes in the coherence function of the source could artificially boost the value of , particularly when strong reflections are present shallow in the sample, such as the glass interface in our results. The influence of the coherence function on can be discounted by verifying that the dynamic range of the OCT signal is not limited by the coherence function baseline.
In general, caution must be exercised when interpreting low values of , since they do not necessarily imply the detection of a strong multiply scattered signal component. The preceding paragraphs highlighted system noise and polarization-randomizing backscattering as the most likely alternative causes of low . A high proportion of single scattering in the detected signal can still be inferred from a particular spatial location if is high at its location. However, as noted in the discussion of the nailfold sample, this is often a valid inference even if is merely high at greater depths in the A-scan or OCT image than the location of interest. Since multiple scattering has the property of masking or blurring sample structure, the visibility of distinct sample structures in the OCT images automatically implies that the detected signal is a result of single scattering. It is in regions where the sample appears homogeneous that polarization-sensitive multiple-scattering quantifiers are most needed.
We have introduced a polarization-based method that uses the spatial variation of polarization states to detect multiple scattering in OCT images, and presented maps displaying the areas for which the signal is dominated by single backscattering in images of solid biological tissue. The behavior of the parameter was studied in a range of samples, from homogeneous microsphere solutions to heterogeneous biological tissue, and confirmed the potential of the approach.
Polarization-based analysis of multiple scattering may be useful in studying the penetration depth limits of OCT in various solid biological samples, and the effects of optical clearing agents. In general, the penetration depth of OCT measurements is limited by SNR or multiple scattering. Our human nailfold measurements indicate that the low visibility of the unexposed nail is not due to the collection of a high fraction of multiply scattered light at its depth. Indeed, high values of were observed at appreciable depths in both our human nailfold and chicken breast results. This suggests that the maximum imaging depth could be improved with an increase in optical power or acquisition time.
The authors are grateful to Dirk Schneiderheinze for useful discussions.
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