Digital calibration method to enable depth-resolved all-fiber polarization sensitive optical coherence tomography with an arbitrary input polarization state

We present a fully integrated depth-resolved all fiber-based polarization sensitive optical coherence tomography (PSOCT). In contrast to conventional fiber-based PSOCT systems, which require additional modules to generate two or more input polarization states, or a pre-adjustment procedure to generate a circularly polarized light, the proposed all-fiber PSOCT system can provide depth-resolved birefringent imaging using an arbitrary single input polarization state. Utilizing the discrete differential geometry (DDG)-based polarization state tracing (PST) method, combined with several geometric rotations and transformations in the Stokes space, two problems induced by the optical fibers can be mitigated: 1) The change in the polarization state introduced by the optical fibers can be effectively compensated using a calibration target at the distal end of the probe, and the computations of the local axis orientation and local phase retardation can be achieved with a single arbitrary input polarization state, eliminating the need for a pre-defined input polarization state, allowing a flexible system design and user-friendly experimental procedure; 2) The polarization mode dispersion (PMD) induced by the optical fibers can be compensated digitally without the requirement of additional input polarization states, providing an accurate PSOCT imaging result. To demonstrate the performance of the proposed method, the depth resolved PSOCT results of a plastic phantom and in vivo skin imaging are obtained using the proposed all-fiber PSOCT system.


Introduction
Polarization sensitive optical coherence tomography (PSOCT) is capable of non-invasive, threedimensional (3D) imaging of anisotropic biological tissues such as tendon, muscle, mineralized tissue, retinal nerve fiber tissue, and scar [1][2][3][4][5][6][7][8][9][10].These anisotropic biological tissues are related to several important clinical conditions, for example dental caries [11,12], myocardial infarction [13], glaucoma [14], scar formation [15,16] and lung diseases [17,18].Given that polarization detection necessitates a more intricate configuration compared to traditional OCT systems, efforts have been dedicated to simplifying PSOCT design and procedures, aiming for seamless clinical translations.The original PSOCT system is based on free space bulk-optic system setup, permitting precise control over the polarization states of light.However, the large volume of free space system setup is impractical in clinical settings.Compared with the free space setup, fiber-based PSOCT systems [5,10] offer distinct advantages in terms of system alignment and handling, desirable for clinical translations.However, optical fibers employed in the system can introduce two problems in the polarization measurements: 1) unknown changes in the polarization states of the probing light during propagation in the fibers due to the non-perfect circular symmetry of the core, external perturbations and stress; 2) polarization mode dispersion (PMD) effect due to the nature of broadband light source employed, distorting the final measurements of sample birefringent properties [10]; 3) restricted to only relative optic axis measurements rather than absolute optic axis due to the implementation with single mode fiber in the system [19].
To circumvent the necessity of a predefined polarization state of probing light incident on the sample, PSOCT techniques that probe the sample with two input polarization states, either by sequential illumination [10] or multiplexing [5], were developed to achieve the depth-resolved birefringent imaging.Notably, spectral binning technique was developed [10,20] using tomogram reconstructed with Jones matrix formalism and then cast to Stokes space to mitigate the PMD artifacts.However, additional modules such as the polarization delay unit or electro-optic modulator are required in the system to generate two input polarization states.Therefore, it not only increases the system complexity, but also slows down the data acquisition rate and increases the cost of computation, significantly inhibiting the clinical translation of PSOCT systems.Recently, a signal processing method that utilizes the residual PMD demonstrating a cumulative retardance estimation error of 13.2 o was developed for the catheter-based PSOCT [21].It is known that single input polarization PSOCT can suffer from intrinsic measurement ambiguities when the input polarization state is aligned with the optic axis of the sample [19,21].To avoid the measurement ambiguity, a polarization symmetry was utilized along with the residual spectral variation of the incident polarization states.The method in [21] relies on a Jones to Stokes transformation and utilizes the weak birefringence of the sheath of the probe as a calibration reference to both estimate the absolute local axis orientation and compute the PMD.
Another method that requires a manual calibration procedure to generate a predefined single input polarization state was proposed for the quantitative PSOCT measurements in a single-mode fiber based PSOCT system [22].Although the accumulated PSOCT results can be quantified in this method, depth-resolved birefringence information with PMD compensation is yet to be obtained.More importantly, the pre-calibration method relies on manual adjustment of the polarization controller (PC).When the fibers are disturbed, which occur constantly and inevitably in the handheld probe system, manual calibration is required before each experiment, which becomes cumbersome in clinical settings.
In this paper, we propose a fully integrated depth-resolved all fiber-based PSOCT system with a single state of input polarization.A digital post-processing method using a calibration target located at the distal end of the probe, combined with the discrete differential geometry (DDG) based polarization state tracing method [13], is proposed to enable absolute and depthresolved axis orientation and phase retardation imaging without the requirement of pre-defined input polarization state.Furthermore, the PMD induced by the optical fibers can be digitally compensated directly in the Stokes space following the computation of birefringent parameters and the calibration step.By using our proposed method, a new depth-resolved PMD-compensated PSOCT imaging system with a portable handheld probe can be achieved without introducing additional hardware components.
In this study, the system and method was tested by a ring-like polylactic acid (PLA) 3D printer filament phantom with its fast optic axis oriented circumferentially.The filament was illuminated by arbitrarily varied input polarization states and imaged by the proposed PSOCT system, respectively.After calibration, the results obtained from different measurements with perturbation of fibers were shown consistent, demonstrating that the change in the polarization state induced by the optical fibers can be mitigated using the proposed method.Finally, the forehead skin of a healthy human was imaged in-vivo and the corresponding local birefringence information was presented, showing that the proposed method can mitigate the artifacts induced by the PMD and provide more accurate and consistent results.

Fiber-based PSOCT system setup
Figure 1 shows the schematic diagram of the all fiber-based PSOCT system.The system was constructed with single mode fiber rather than polarization maintaining fiber (PMF) and is based on a typical Mach-Zehnder interferometer configuration.This choice was made for several reasons: 1) only a single input polarization state is needed in this method, eliminating the need for a long PMF to generate two input polarization states; 2) ghost artifacts, which can be introduced by the PMF, can be mitigated by using single mode fibers [23]; 3) single-mode fibers are more cost effective than PMF; and 4) the proposed digital method eliminates the requirement for maintaining the state of polarization in PSOCT systems.
based on a typical Mach-Zehnder interferometer configuration.This choice was made for several reasons: 1) only a single input polarization state is needed in this method, eliminating the need for a long PMF to generate two input polarization states; 2) ghost artifacts, which can be introduced by the PMF, can be mitigated by using single mode fibers [23]; 3) single-mode fibers are more cost effective than PMF; and 4) the proposed digital method eliminates the requirement for maintaining the state of polarization in PSOCT systems.The light source was a 200 kHz swept laser source (Thorlabs SL132120), providing a central wavelength of 1310 nm and a spectral tuning range of 100 nm.The output of the laser source was split by a 25/75 coupler and sent to the reference arm (25%) and the sample arm (75%), respectively.In the sample arm, a commercial handheld probe (Thorlabs OCTG-1300NR-SP1) was used with a calibration target integrated at the distal end between the sample and the handheld probe.The calibration target (CA in Fig. 1) was made by a common polylactic acid (PLA) material 3D printed (formlabs 3D printer, model Form 3+) with a series of ringlike filaments.It had a central opening of 8 mm in diameter and a thickness of 2 mm.The target with its fast optical axis oriented circumferentially was situated within the field of view (FoV) when imaging.In this case, the sample of imaging interest and the calibration target were imaged in parallel, with the sample image situated in the middle of the calibration target ring, enabling a fast digital calibration and PMD compensation.The back-reflected light from the sample and the reference arms were recombined by a 50/50 coupler and were sent to the polarization detection unit, where the coherent interference signals were split into orthogonal linear polarization components using two polarization beam splitters (PBS) and were collected by the photo-diode detectors (Thorlabs PDB480C-AC), followed by a digitizer (ATS9360, AlazarTech, Inc.) to convert the analog signals to digital domain.The polarization in the reference arm was adjusted to have equal power in the two polarization channels by using the polarization controllers PC1, PC3 and PC4.Because the proposed system can be calibrated by a digital method combined with a standard calibration target, there is no need to ensure the The light source was a 200 kHz swept laser source (Thorlabs SL132120), providing a central wavelength of 1310 nm and a spectral tuning range of 100 nm.The output of the laser source was split by a 25/75 coupler and sent to the reference arm (25%) and the sample arm (75%), respectively.In the sample arm, a commercial handheld probe (Thorlabs OCTG-1300NR-SP1) was used with a calibration target integrated at the distal end between the sample and the handheld probe.The calibration target (CA in Fig. 1) was made by a common polylactic acid (PLA) material 3D printed (formlabs 3D printer, model Form 3+) with a series of ringlike filaments.It had a central opening of 8 mm in diameter and a thickness of 2 mm.The target with its fast optical axis oriented circumferentially was situated within the field of view (FoV) when imaging.In this case, the sample of imaging interest and the calibration target were imaged in parallel, with the sample image situated in the middle of the calibration target ring, enabling a fast digital calibration and PMD compensation.The back-reflected light from the sample and the reference arms were recombined by a 50/50 coupler and were sent to the polarization detection unit, where the coherent interference signals were split into orthogonal linear polarization components using two polarization beam splitters (PBS) and were collected by the photo-diode detectors (Thorlabs PDB480C-AC), followed by a digitizer (ATS9360, AlazarTech, Inc.) to convert the analog signals to digital domain.The polarization in the reference arm was adjusted to have equal power in the two polarization channels by using the polarization controllers PC1, PC3 and PC4.Because the proposed system can be calibrated by a digital method combined with a standard calibration target, there is no need to ensure the input polarization state to be a certain known state.Taking advantage of this property, we installed a PC2 in the sample arm and adjusted it to generate an input polarization state that provides best contrast during the measurements.For the results presented in Section 3, all the imaging was conducted with a 10mm x 10mm scan pattern.This scan pattern consisted of 500 A-scans per B-scan at 500 B-scan positions, resulting in a uniform spacing of 20µm between adjacent A-scans in a 10mm x 10mm field of view.

Theoretical considerations
The theoretical considerations and proofs for the proposed calibration method are presented in this section.Note that the theory and the experimental results for PMD compensation will be presented in Section 4. According to the model presents in our previous work [13], the state of polarization for the light scattered back from each depth of the sample can be considered as a polarization trajectory at the Poincaré sphere.In this work, all the theoretical discussions are based on this Poincaré sphere model (Fig. 2).To present the methodology systematically, the optical path in the fiber-based PSOCT system is divided into two parts for discussion.The first part of the optical path is from the laser to the sample, and the second part is from the sample to the detectors.In the first path, the fibers may alter the polarization state of the light beam due to the birefringent property of the single-mode optical fibers.Hence, the input polarization state that illuminates the sample is unknown and can be arbitrary.However, for any input polarization state traveling through the same sample, each of their corresponding trajectories on the Poincaré sphere should be parallel to each other, because they share the same set of the binormal vectors that are only determined by the local axis orientation and local phase retardation of the sample [13].Since the DDG-based method [13] derives the local axis orientation and phase retardation based on the computation of the binormal vectors, which are independent from the input polarization state, the fibers in the path from laser to sample would not impact the final computation results.Thus, it is not necessary to compensate for the change introduced in this path.In other words, if one can directly detect the output polarization states that are back-scattered from the sample, the DDG-based method would be sufficient to derive the absolute birefringent information of the sample.
However, the lights that are backscattered from the sample must route through the fibers in the detection path for detection.This process leads to an unknown change in all the output polarization states scattered back from the sample.The situation can get worse if the optical fiber introduces random polarization perturbations when the light is propagating with this path before the signals are finally received by the detectors (PD1 and PD2 in Fig. 1), which is unfortunately true for in vivo imaging applications.Both elliptically and linearly polarized components can add to the output polarization states of interest.The resulting polarization state can be presented as a 3D rotation of the polarization trajectories of the detected signal scattered back from the sample in the Poincaré sphere, shown in Fig. 2(A), leading to a change of the axis orientation of interest.If left uncorrected, incorrect computation of the birefringent parameters would result.Therefore, calibration is crucial.
To compensate for the 3D rotation introduced by the optical fibers and calibrate the axis orientation, a stepwise compensation method is proposed.The method compensates for the elliptically and linearly polarized components separately by two steps.The first step is to compensate for the elliptically polarized component introduced by the fibers.Without the effect of the fibers, PSOCT would not detect any elliptically polarized component of the sample due to the round-trip measurement, and the measured optic axis orientation should be constrained Fig. 2 Digital calibration from measured polarization state trajectories to absolute optical axis using 3D rotations determined by the calibration target.(A) shows the main steps of the algorithm to obtain the absolute optical axis from measured polarization state trajectories of the calibration ring:   (  ,  ,  ) is the measured polarization state trajectories for calibration ring;   are the computed 3D vectors from   (  ,  ,  ) ;   are the computed binormal vectors;  are the computed phase retardations;   ( -  ;  ) is the 3D rotation from   (  ,  ,  ) to   (  ,  ,  ) , as shown in Eq.2;   are the absolute and calibrated optical axis.(B) Poincaré sphere representations of the DDG method and the 3D rotations performed to obtain the absolute and calibrated optical axis   from the measured polarization state trajectories   (  ,  ,  ) (black dotted curve):  1(black arrow) is the computed optical axis for the 1 st layer of   (  ,  ,  ) ;  2 (blue arrow) is the binormal vector and note that  1 =  1according to the DDG-based PST method;  2 (green arrow) is the true optical axis computed by rotating  2 around  1 by 3D rotation matrix shown in Eq.3;  (green arrows) are the optical axes computed by repeating the DDG method for n times from the first to n th depth-layer of the sample, and   (pink arrows) denote the relative optical axes by rotating the plane of   back to the QU plane;   (yellow arrows) are the absolute calibrated optical axes.
The process flow chart is shown in Fig. 2A, together with their corresponding Poincaré sphere representations at each step (Fig. 2B).By applying the DDG-based PST method on the measured polarization state trajectories  (,,) of the calibration ring shown in Fig. 2B, the 3D vectors     and   can be computed using the Frenet-frames method [13,25], where the plane composed by   can be obtained.Then we apply the elliptically-linearly polarization stepwise compensation method to compensate for the induced polarization changes by the optical fibers.The first step is to rotate the plane composed by   to the QU-plane and the elliptically polarized component introduced by the system can be compensated.The second step is to rotate the plane about V-axis to calibrate the linearly polarized component according to the absolute axis orientation information of the target ring.Importantly, because the calibration and the sample data are acquired in parallel using a single input state of polarization, the local birefringence parameters (local optical axis and phase retardation) of the sample can be computed together with the calibration procedure within the same measurement.

Experimental results for absolute axis orientation calibration
To demonstrate that the proposed fiber-based PSOCT enables the local birefringence imaging with arbitrary input polarization state, three PSOCT measurement trials of the calibration target of PLA ring were taken (shown in Fig. 1 insert).In each trial, we purposely varied input polarization state of the probing light incident upon the sample, by randomly twisting the optic fibers in the system with all other experimental conditions remained unchanged.The resulting en-face polarization state images measured from the target are shown in Figs.3A-3C, where the effect of optic fibers in the system on the results is apparent, rendering the interpretation of  2); A n are the absolute and calibrated optical axis.(B) Poincaré sphere representations of the DDG method and the 3D rotations performed to obtain the absolute and calibrated optical axis A n from the measured polarization state trajectories within the QU-plane.Hence, any elliptically polarized component in the detected polarization states should belong to the system itself, not by the sample.To compensate for this elliptically polarized component, the planes for all the measured local axis orientations (the details will be presented below) are first determined, and then rotated back to the QU-plane.This procedure eliminates the elliptically polarized component introduced by the system (fibers), giving rise to a relative local axis orientation representative of the sample.To compensate for the linearly polarized component and obtain the absolute axis orientation, the PLA reference ring is utilized.Since the axis orientation of the standard ring is known, the absolute axis orientation of both the sample and the calibration ring target can be obtained by rotating the plane of the relative local axis orientation from the first step about the V-axis.Upon completion of these two steps, both the elliptically and linearly polarized components of the fibers are removed from the measurements, resulting in the absolute axis orientation of the sample.
The plane of the measured local axis orientation which includes the birefringent information of the fibers can be obtained by applying the DDG-based PST method to the trajectories of the rotated output polarization states directly measured by the detectors.The verification of this statement is presented below.Firstly, by applying the DDG to the trajectories of the output polarization states detected by the system, a set of tangent vectors, normal vectors and binormal vectors (i.e., TNB frames) would be obtained.Without the effect of the fibers, the relationship between the binormal vectors, the local axis orientation and local phase retardation of the sample has been demonstrated in the previous study [13], which can be expressed as the following, It was shown that the local axis orientation A n can be obtained by applying a set of 3D rotations (R n−1 R n−2 . . .R 1 ) to B n .B n is the binormal vector representing the cross-product between the tangential and normal vectors, with n representing the n-th layer along the depth of the sample.B n is orthogonal to the osculating plane and thus includes information related to the local axis orientation.δ n−1 δ n−2 . . .δ 1 represents the local phase retardation along the depth of the sample.For instance, R 1 (−δ 1 ; A 1 ) is the 3D rotation matrix determined by A 1 , which is the local axis orientation in the first layer of the sample.Again, since the local axis orientation and local phase retardation are the intrinsic properties of the sample and are independent from the input polarization states, the change in the polarization state introduced by the optical path from the light source to the sample should not affect the computation of the local birefringence parameters by using the DDG-based PST method.However, this is not true for the detection path as we discussed above.
Here, we demonstrate that the DDG-based PST method combined with the digital geometric rotation to analyze the detected signals can recover the local birefringence parameters.The polarization states of the light that are back-scattered from the sample and emerged at the sample surface can be expressed as P n .Due to the optical fibers in the detection path, P n is altered as the light propagates in the fiber, it cannot be detected directly.Because the change in the polarization states in the detection path can be considered as an overall three-dimensional (3D) rotation in the Stokes space (i.e., Q, U, V coordinate system) on the Poincaré sphere [24], the polarization states F n that are finally detected at the detectors can be expressed as where is the 3D rotation matrix determined by the optical fibers in the detection path, ) and δ d is the rotation angle δ d (subscript d denotes the detection path)).As mentioned above, by applying the DDG-based PST method to the back-scattered polarization trajectory P n (Q n , U n , V n ), the local birefringence parameters of the sample can be obtained.However, due to the optical fibers in the detection path is random.Hence, we need to compute the local birefringence parameters of the sample by utilizing the detected signal When applying the DDG-based PST method to the rotated polarization trajectory of ) and phase angle δδ n , which satisfy Eq. ( 1) and can be written as, Next, we derive the local optical axis A n (Q n , U n , V n ) and the local phase retardation δ n of the sample.As proved in the previous study [13,27], the overall rotation operation to the trajectory would not change the rotation angle between the adjacent points about the corresponding binormal vector.Thus, δδ n = δ n .We can obtain, where B n is the binormal vector of P n .According to the DDG-based PST method, . By substituting these two equations into Eq.( 2), we can obtain the relationship between the local optical axis A 1 and the computed axis For the computed axis AA 2 (Q 2 , U 2 , V 2 ), it can be expressed as: where δ 1 = δδ 1 as shown above.Under the (Q, U, V) coordinate system, it is hard to determine the relationship between AA 2 and A 2 based on Eq. ( 3).Hence, we consider Eq. ( 3) in a new coordinate system (Q', U', V'), which can be obtained by rotating the original coordinate system (Q, U, V) by the rotation matrix R d .In the new coordinate system (Q', U', V'), the value of F n is totally the same as the value of P n in the original coordinate system (Q, U, V) because of Eq. ( 2).Therefore, the value of AA n in the new coordinate system is the same as the value of A n of the original coordinate system.Hence, by describing AA n in the new coordinate system, A n can be obtained as below: As can be seen in Eq. ( 7), there is a 3D rotation relationship described by

the final measurements and the true local axis orientation
due to the detection path is known during imaging, the local optic axis and phase retardation of the sample can be recovered by rotations, in reverse direction , to the measured states of polarizations at the photodiode detectors.In our proposed approach, this requirement is met by the calibration target-ring situated at the distal end of the OCT probe, where the target ring and the sample are imaged in parallel within the same FoV with the sample located at the central 8mm circular region and the calibration ring located in the periphery of the FoV.Before the local optical axis and phase retardation of the sample are computed, R d (δ d ; A d (Q d , U d , V d )) due to the detection path is first computed from the measurements received from the peripheral target ring in the imaging FoV (where the optic axis of the target ring is known in prior).This process is referred here to as the digital calibration by using DDG-based PST method combined with the digital geometric rotation for deriving the local birefringent parameters of the sample.
The process flow chart is shown in Fig. 2(A), together with their corresponding Poincaré sphere representations at each step (Fig. 2(B)).By applying the DDG-based PST method on the measured polarization state trajectories F n (Q n , U n , V n ) of the calibration ring shown in Fig. 2(B), the 3D vectors AA n , BB n , and δδ n can be computed using the Frenet-frames method [13,25], where the plane composed by AA n can be obtained.Then we apply the elliptically-linearly polarization stepwise compensation method to compensate for the induced polarization changes by the optical fibers.The first step is to rotate the plane composed by AA n to the QU-plane and the elliptically polarized component introduced by the system can be compensated.The second step is to rotate the plane about V-axis to calibrate the linearly polarized component according to the absolute axis orientation information of the target ring.Importantly, because the calibration and the sample data are acquired in parallel using a single input state of polarization, the local birefringence parameters (local optical axis and phase retardation) of the sample can be computed together with the calibration procedure within the same measurement.

Experimental results for absolute axis orientation calibration
To demonstrate that the proposed fiber-based PSOCT enables the local birefringence imaging with arbitrary input polarization state, three PSOCT measurement trials of the calibration target of PLA ring were taken (shown in Fig. 1 insert).In each trial, we purposely varied input polarization state of the probing light incident upon the sample, by randomly twisting the optic fibers in the system with all other experimental conditions remained unchanged.The resulting en-face polarization state images measured from the target are shown in Figs.3(A)-3(C), where the effect of optic fibers in the system on the results is apparent, rendering the interpretation of PS-OCT images difficult.We selected three locations at the PLA ring with differed optic axis orientations (P1, P2 and P3 in Figs.3(A PS-OCT images difficult.We selected three locations at the PLA ring with differed optic axis orientations (P1, P2 and P3 in Figs.3A-3C) and presented the corresponding trajectories of the measured output polarization states on Poincare sphere in Figs.3J-3L, where the starting points of the trajectories (i.e., the averaged point of the start points at 3 locations, pointed by the black arrows) are different in the 3 trials as expected, suggesting that the input polarization states sent to the sample differ.Further, the trajectories of the output polarization states of the same point (P1, or P2, or P3) in the trials appear totally different and non-parallel, demonstrating the optical fibers in the input path and output path both caused the changes in the polarization states of the probing light.By applying the DDG-based PST method to each of the trajectories, the computed axes   along the depth of the target ring were obtained using Eq. 3, with the results shown in Figs.3M-3O.For each trial, all the computed axes   are constrained within a plane.And there is a 3D rotation relationship between each plane in different groups.
The computed axes   along the depth were averaged to become the dominate vectors at points P1, P2 and P3 (as indicated by the black, red and blue arrows respectively in Figs.3M-3O).The values of the dominate vectors and the relative angles between the dominate vectors: φ1 (between vectors of P1 and P2), φ2 (between vectors of P2 and P3) and φ3 (between vectors of P1 and P3) are tabulated in Table 1, respectively.The values of the angles φ1, φ2 By applying the DDG-based PST method to each of the trajectories, the computed axes AA n along the depth of the target ring were obtained using Eq. ( 3), with the results shown in Figs.3(M)-3(O).For each trial, all the computed axes AA n are constrained within a plane.And there is a 3D rotation relationship between each plane in different groups.
The computed axes AA n along the depth were averaged to become the dominate vectors at points P1, P2 and P3 (as indicated by the black, red and blue arrows respectively in Figs. 3

(M)-3(O)).
The values of the dominate vectors and the relative angles between the dominate vectors: φ1 (between vectors of P1 and P2), φ2 (between vectors of P2 and P3) and φ3 (between vectors of P1 and P3) are tabulated in Table 1, respectively.The values of the angles φ1, φ2 and φ3 in different measurement trials are similar, demonstrating the proposed DDG-based PST method can retrieve the relative orientation of the PLA sample when the input polarization state of the probing light is unknown.This result is consistent with the theoretical analysis discussed in Section 3.1.By rotating the plane back to the QU-plane, the relative local axis orientation map is obtained (Figs.3(D)-3(F)).It should be noted that because the optical fibers can introduce an arbitrary 3D rotation to the trajectories of the polarization states for the light that are back-scattered from the sample, it is possible that the plane of all the possible optic axes can be flipped over by the optical fibers.Fortunately, as will be demonstrated later, the ring calibration target can help us to identify if the plane of optical axes is flipped or not.Next, by rotating the relative axes about the V axis, the absolute local axis orientation A n of the sample can be obtained (Figs.3(G)-3(I)), where they show substantial equivalence with the three arbitrary chosen input polarization states.These results of the 3D printed PLA ring demonstrate that the local axis orientation can be retrieved in the all fiber-based PSOCT system with arbitrary input polarization state.

Table 1. Calculated values of the dominate vectors and angles at the point P1, P2 and P3, and the
relative angles (φ1, φ2, andϕ 3) between the dominate vectors for the 3 measurement trials, respectively, as described in Fig. 3.
and φ3 in different measurement trials are similar, demonstrating the proposed DDG-based PST method can retrieve the relative orientation of the PLA sample when the input polarization state of the probing light is unknown.This result is consistent with the theoretical analysis discussed in Section 3.1.By rotating the plane back to the QU-plane, the relative local axis orientation map is obtained (Figs.3D-3F).It should be noted that because the optical fibers can introduce an arbitrary 3D rotation to the trajectories of the polarization states for the light that are back-scattered from the sample, it is possible that the plane of all the possible optic axes can be flipped over by the optical fibers.Fortunately, as will be demonstrated later, the ring calibration target can help us to identify if the plane of optical axes is flipped or not.Next, by rotating the relative axes about the V axis, the absolute local axis orientation   of the sample can be obtained (Figs.3G-3I), where they show substantial equivalence with the three arbitrary chosen input polarization states.These results of the 3D printed PLA ring demonstrate that the local axis orientation can be retrieved in the all fiber-based PSOCT system with arbitrary input polarization state.
Table 1.Calculated values of the dominate vectors and angles at the point P1, P2 and P3, and the relative angles ( φ1 , φ2, and φ3 ) between the dominate vectors for the 3 measurement trials, respectively, as described in Fig. 3.

PMD compensation for fiber-based PSOCT with digital calibration
Due to the broad spectral band nature of the light source, the optical fibers in the system can also introduce PMD, meaning that the optical fibers can introduce wavelength/wavenumberdependent changes in the polarization state of the propagating light.Similar to calibration described in the last Section, the PMD generated by the system can be classified into two parts: (1) the PMD introduced to the signal by the optical propagation from the source to the sample; (2) the PMD introduced to the signal by the optical propagation from the sample to the detectors.The fibers in the optical path from the source to the sample can result in wavenumber-dependent input polarization states (relative to the desired ones) to illuminate the sample, generating parallel trajectories    (   ,   ,   ) in the Poincaré sphere when the broadband light transmits through the sample, where superscript k denotes wavenumber.As mentioned above, different input polarization states would not affect the computation of the local birefringence parameters when utilizing the binormal vectors of the trajectory of the output polarization states to derive the local optic axis and local phase retardation.Hence, by leveraging the method of spectral binning concept proposed by Villiger et al [10], the PMD introduced by the optical path from the source to the sample can be mitigated by computing and averaging the binormal vectors of each trajectory    (   ,   ,   ) obtained from each wavelength bin and utilizing the averaged binormal vectors to compute the final local birefringence parameters.Additionally, in a recently published work [21], a technique was demonstrated for a calibrated, PMD-compensated, single input polarization PSOCT for a catheter-based system.However, their local birefringent parameters were computed using the conventional dual-input methodology presented in [26], while our proposed work calibrated for the absolute optic axis and phase retardation, and mitigated PMD using the DDG-based PST method [13,27].

PMD compensation for fiber-based PSOCT with digital calibration
Due to the broad spectral band nature of the light source, the optical fibers in the system can also introduce PMD, meaning that the optical fibers can introduce wavelength/wavenumber-dependent changes in the polarization state of the propagating light.Similar to calibration described in the last Section, the PMD generated by the system can be classified into two parts: (1) the PMD introduced to the signal by the optical propagation from the source to the sample; (2) the PMD introduced to the signal by the optical propagation from the sample to the detectors.The fibers in the optical path from the source to the sample can result in wavenumber-dependent input polarization states (relative to the desired ones) to illuminate the sample, generating parallel trajectories in the Poincaré sphere when the broadband light transmits through the sample, where superscript k denotes wavenumber.As mentioned above, different input polarization states would not affect the computation of the local birefringence parameters when utilizing the binormal vectors of the trajectory of the output polarization states to derive the local optic axis and local phase retardation.Hence, by leveraging the method of spectral binning concept proposed by Villiger et al [10], the PMD introduced by the optical path from the source to the sample can be mitigated by computing and averaging the binormal vectors of each trajectory P k n (Q k n , U k n , V k n ) obtained from each wavelength bin and utilizing the averaged binormal vectors to compute the final local birefringence parameters.Additionally, in a recently published work [21], a technique was demonstrated for a calibrated, PMD-compensated, single input polarization PSOCT for a catheter-based system.However, their local birefringent parameters were computed using the conventional dual-input methodology presented in [26], while our proposed work calibrated for the absolute optic axis and phase retardation, and mitigated PMD using the DDG-based PST method [13,27].
When the light that are back-scattered from the sample and passes through the detection path, these parallel trajectories would be rotated by wavelength-dependent rotation matrix R k d determined by the optical fibers in the detection path.By separating the spectrum into several wavelength bins and reconstructing the depth dependent Stokes parameters from each bin, the trajectories over depth with different wavelengths can be obtained as Since the rotation matrix R k d is wavelength-dependent, the trajectories F k n that can be obtained directly from the spectral binning are no longer parallel.Hence, by rotating the trajectories with different wavelength to make all the trajectories parallel to that of the central wavelength , the PMD introduced by the optical fibers in the detection path can be mitigated.This strategy can compensate the two parts of the PMD in the fiber-based system.Unlike the previous study that utilized two input polarization states to compensate the PMD in the Jones matrix formulism [10,20], the proposed PST-based method only uses a single and arbitrary input polarization state measured in the Jones space with the remaining steps including digital calibrations, birefringent parameter reconstruction and PMD compensations, being operated in the Stokes space, making the method more straightforward, a useful attribute for clinical implementations.

Experimental results for PMD compensation and digital calibration
To demonstrate the PMD compensation method by using a single input polarization state in the fiber-based PSOCT system, a B-scan of the PLA target ring was selected for analysis (the location is indicated by the black dash line in Fig. 3(G)).The OCT spectrum of the B-scan was split into 5 bins evenly spread across the 100 nm bandwidth of the light source, with the middle bin having a central wavelength at 1300nm.Then the cross-sectional PSOCT images of each bin were reconstructed as shown in Fig. 4. The output polarization states vary from one bin to another (Fig. 4(A)) and the trajectories of the output polarization states in an A-scan (indicated by the dash line in Fig. 4(A)) in different bins do not overlap, neither parallel to each other (Fig. 4(I)), demonstrating the existence of the PMD in the fiber-based PSOCT system.Hence, the corresponding computed local axis orientation maps of different bins are not consistent (Fig. 4(B)).Note that the colored lines in the Poincaré sphere shown in Fig. 4(I) represent the local axis orientations for each polarization trajectory of the 5-bins.By aligning the trajectories of each bin to that of the central bin 3 as shown in Fig. 4(J), the local axis orientation maps of each bin become more consistent with and parallel to each other.We averaged each of the 5 binned B-scans over depth (the location of the B-scan is indicated by the black dash line in Fig. 3(G)) in Fig. 4(B) and 4(C) to obtain the averaged axis orientation angle of the ring target.The horizontal axes in Fig. 4(K) and 4(L) represent the pixel index in the B-scan, while the vertical axes represent the mean optic axis orientation angle in unit of degrees for the ring target.As the pixel index in the B-scan increases (i.e., along the B-scan fast axis), the mean optic axis orientation of the ring changes gradually, as expected, because the B-scan selected is at the ring located across a varying angle of optic axis, indicated by the dash line in Fig. 3(G).Without the PMD compensation, the curves of the mean axis orientation values of different bins are deviated from each other.With the PMD compensation, the curves obtained by different bins overlapped, demonstrating that the proposed method compensated for the PMD and the results are calibrated.In addition, the OCT structure image (Fig. 4(H  Compared with the local axis orientation map without PMD compensation (Fig. 4E), where the band-like PMD artifacts (discussed in details in the previous studies [10]) are seen as apparent regional birefringent structures (indicated by the red arrows), the averaged local axis orientation map with PMD compensation shows a more uniform color throughout the depth (Fig. 4D).In the averaged local phase retardation image with PMD compensation (Fig. 4F), the band-like PMD artifacts (indicated by the yellow arrows), which are visible in the local phase retardation image without the PMD compensation (Fig. 4G), are also mitigated.It should be noted that, aside from the PMD artifacts indicated by the yellow arrows, there are other small band-like artifacts in the phase retardation images (Figs.4F and 4G).These artifacts are likely introduced by the gaps between each layer of the PLA sample, as seen in Fig. 4H.These results demonstrate that the proposed method is efficient in compensating for the PMD introduced by the optical fibers and provides a more consistent result.

Demonstration of the proposed system for in vivo imaging
To showcase the performance of the proposed fiber-based PMD-compensated PSOCT system, we conducted in vivo imaging of the forehead skin on a healthy volunteer, and the corresponding PSOCT results are presented in Fig. 5. Two-measurement trials were performed at the same area using two randomly selected input polarization states.The FoV of each image was cropped down to 7 mm  7 mm from the original scan pattern, with a pixel density of 500  500 pixels.
Figures 5A and 5E show the polarization state images of the sample using randomly selected input 1 and input 2 as the input polarization states, respectively.Different colors are observed when imaging the same area with different input polarization states, suggesting that polarization Compared with the local axis orientation map without PMD compensation (Fig. 4(E)), where the band-like PMD artifacts (discussed in details in the previous studies [10]) are seen as apparent regional birefringent structures (indicated by the red arrows), the averaged local axis orientation map with PMD compensation shows a more uniform color throughout the depth (Fig. 4(D)).In the averaged local phase retardation image with PMD compensation (Fig. 4(F)), the band-like PMD artifacts (indicated by the yellow arrows), which are visible in the local phase retardation image without the PMD compensation (Fig. 4(G)), are also mitigated.It should be noted that, aside from the PMD artifacts indicated by the yellow arrows, there are other small band-like artifacts in the phase retardation images (Figs.4(F) and 4(G)).These artifacts are likely introduced by the gaps between each layer of the PLA sample, as seen in Fig. 4(H).These results demonstrate that the proposed method is efficient in compensating for the PMD introduced by the optical fibers and provides a more consistent result.

Demonstration of the proposed system for in vivo imaging
To showcase the performance of the proposed fiber-based PMD-compensated PSOCT system, we conducted in vivo imaging of the forehead skin on a healthy volunteer, and the corresponding PSOCT results are presented in Fig. 5. Two-measurement trials were performed at the same area using two randomly selected input polarization states.The FoV of each image was cropped down to 7 mm × 7 mm from the original scan pattern, with a pixel density of 500 × 500 pixels.
state image results are dependent on the input polarization state.Figs.5B and 5F display the local axis orientation results without utilizing the calibration and PMD compensation methods.Without the calibration described in Section 3, the coordinate systems of Figs.5B and 5F differ, as expected, illustrating the unpredictable effects due to light propagation within the optical fibers.It is noteworthy that the mean error in local axis orientation was calculated to be approximately 17.82 o with a standard deviation of 8.99 o .Figures 5C and 5G present the local axis orientation results with calibration but without PMD compensation.Using the digital calibration method, the local axis orientation images with two different input polarization states are calibrated to the same coordinate system, resulting in a mean error in local axis orientation of 8.96 o with a standard deviation of 3.51 o .This indicates the improved consistency between polarization measurements due to variations of the input polarization states.However, without PMD compensation, the imaging quality is significantly degraded by the polarization dispersion effect and noises, as evident in Figs.5C and 5G.
With PMD compensation, the collagen organization can be visualized clearly in Figs.5D  and 5H, establishing substantial equivalence between the two results.The quantitative evaluation yielded a mean error of only 2.26 o and a standard deviation of 1.47 o .The residual small mean error may be attributed to other factors including system noise propagations and the omission of diattenuation in our framework.Overall, these findings demonstrate that PMDcompensated local axis orientation can be achieved with an arbitrary input polarization state using digital calibration and PMD compensation, enabling a straightforward and robust PSOCT imaging modality.
Figure 6 shows the birefringent measurements by putting OCT structural information into the context.To make it easier for scrutinization, the same polarization state images of the sample as shown in Figs.5A and 5E are given in Figs.6A and 6E, respectively.Figs.6B and 6F display the corresponding structural en-face images.Although the collagen organizations Figures 5(A) and 5(E) show the polarization state images of the sample using randomly selected input 1 and input 2 as the input polarization states, respectively.Different colors are observed when imaging the same area with different input polarization states, suggesting that polarization state image results are dependent on the input polarization state.Figures 5(B) and 5(F) display the local axis orientation results without utilizing the calibration and PMD compensation methods.Without the calibration described in Section 3, the coordinate systems of Figs.5(B) and 5(F) differ, as expected, illustrating the unpredictable effects due to light propagation within the optical fibers.It is noteworthy that the mean error in local axis orientation was calculated to be approximately 17.82 o with a standard deviation of 8.99 o .
Figures 5(C) and 5(G) present the local axis orientation results with calibration but without PMD compensation.Using the digital calibration method, the local axis orientation images with two different input polarization states are calibrated to the same coordinate system, resulting in a mean error in local axis orientation of 8.96 o with a standard deviation of 3.51 o .This indicates the improved consistency between polarization measurements due to variations of the input polarization states.However, without PMD compensation, the imaging quality is significantly degraded by the polarization dispersion effect and noises, as evident in Figs. 5

(C) and 5(G).
With PMD compensation, the collagen organization can be visualized clearly in Figs.5(D) and 5(H), establishing substantial equivalence between the two results.The quantitative evaluation yielded a mean error of only 2.26 o and a standard deviation of 1.47 o .The residual small mean error may be attributed to other factors including system noise propagations and the omission of diattenuation in our framework.Overall, these findings demonstrate that PMD-compensated local axis orientation can be achieved with an arbitrary input polarization state using digital calibration and PMD compensation, enabling a straightforward and robust PSOCT imaging modality.
can be observed, the structural contrast is relatively low and without clear information about the orientation.Fig. 6C and 6G present the local phase retardation results without PMD compensation.The contrast in phase retardation of the collagen organization is improved compared with the other components within the tissues.However, a reasonable amount of disconnection of the collagen fibers is observed in some regions due to the PMD introduced by the optical fibers.The computed mean error between the results from input 1 and input 2 (e.g., Figs.6C and 6G) is approximately 8.55 o with a standard deviation of 2.89 o .With PMD compensation, more collagen structures can be visualized and the collagen organization can be seen clearly with good continuities, as displayed in Figs.6D and 6H, giving the mean error of approximately 2.34 o with a standard deviation of 1.12 o .Overall, these findings including both local phase retardation and local axis orientations demonstrate that digitally calibrated and PMD-compensated birefringent information can provide more reliable in-depth information of the collagen including its structure and organization, promising a straightforward and robust PSOCT imaging modality.

Conclusion
We proposed a fully integrated all fiber-based PSOCT system with a handheld probe and a calibration target in the distal end to enable accurate and consistent measurements of depthresolved optic axis and phase retardation within the sample of interest.Using only a single input polarization state, we addressed two polarization-related issues induced by the optical fibers used in the system, i.e., the issue of the changes in the polarization state and the issue of the polarization mode dispersion.With the proposed methods and the fiber-based system by using the DDG-based PST method combined with several geometric transformations of the trajectories of the output polarization states in the Stokes space, the results from PLA phantom and the in vivo forehead skin demonstrated the excellent performance.Figure 6 shows the birefringent measurements by putting OCT structural information into the context.To make it easier for scrutinization, the same polarization state images of the sample as shown in Figs.5(A) and 5(E) are given in Figs.6(A) and 6(E), respectively.Figures 6(B) and 6(F) display the corresponding structural en-face images.Although the collagen organizations can be observed, the structural contrast is relatively low and without clear information about the orientation.Figure 6(C) and 6(G) present the local phase retardation results without PMD compensation.The contrast in phase retardation of the collagen organization is improved compared with the other components within the tissues.However, a reasonable amount of disconnection of the collagen fibers is observed in some regions due to the PMD introduced by the optical fibers.The computed mean error between the results from input 1 and input 2 (e.g., Figs.6(C) and 6(G)) is approximately 8.55 o with a standard deviation of 2.89 o .With PMD compensation, more collagen structures can be visualized and the collagen organization can be seen clearly with good continuities, as displayed in Figs.6(D) and 6(H), giving the mean error of approximately 2.34 o with a standard deviation of 1.12 o .Overall, these findings including both local phase retardation and local axis orientations demonstrate that digitally calibrated and PMD-compensated birefringent information can provide more reliable in-depth information of the collagen including its structure and organization, promising a straightforward and robust PSOCT imaging modality.

Conclusion
We proposed a fully integrated all fiber-based PSOCT system with a handheld probe and a calibration target in the distal end to enable accurate and consistent measurements of depthresolved optic axis and phase retardation within the sample of interest.Using only a single input polarization state, we addressed two polarization-related issues induced by the optical fibers used in the system, i.e., the issue of the changes in the polarization state and the issue of the polarization mode dispersion.With the proposed methods and the fiber-based system by using the DDG-based PST method combined with several geometric transformations of the trajectories of the output polarization states in the Stokes space, the results from PLA phantom and the in vivo forehead skin demonstrated the excellent performance.
However, as a trade-off, the step of the spectral binning for the PMD compensation can sacrifice the axial resolution.Without the spectral binning step, the axial resolution can be retained.The PMD compensation step using spectral binning can also increase computational cost to the system.To mitigate this issue, a further study to compensate the PMD in the raw spectrum is desirable.Overall, the proposed fiber-based PSOCT system and the method of reconstruction of the depth-resolved PSOCT images promise a step forward to clinical translation of PSOCT imaging.

Fig. 1
Fig. 1 Schematic diagram of the all fiber-based PSOCT system.SS: swept source; PC1, PC2, PC3 and PC4: polarization controllers; OC1 and OC2: optical circulators; PBS1 and PBS2: polarization beam splitters; PD1 and PD2: photodiode detectors; M: mirror; L: focusing lens; C: collimator lens; GS: galvo scanner.Insert images show the photograph of the fiber-based PSOCT system and the handheld probe; CA: calibration target at the distal end integrated with handheld probe right before the sample, with the black arrow indicating the direction of optical axis varying azimuthally along the circle.

Fig. 1 .
Fig. 1.Schematic diagram of the all fiber-based PSOCT system.SS: swept source; PC1, PC2, PC3 and PC4: polarization controllers; OC1 and OC2: optical circulators; PBS1 and PBS2: polarization beam splitters; PD1 and PD2: photodiode detectors; M: mirror; L: focusing lens; C: collimator lens; GS: galvo scanner.Insert images show the photograph of the fiber-based PSOCT system and the handheld probe; CA: calibration target at the distal end integrated with handheld probe right before the sample, with the black arrow indicating the direction of optical axis varying azimuthally along the circle.

Fig. 2 .
Fig. 2. Digital calibration from measured polarization state trajectories to absolute optical axis using 3D rotations determined by the calibration target.(A) shows the main steps of the algorithm to obtain the absolute optical axis from measured polarization state trajectories of the calibration ring: F n (Q n , U n , V n ) is the measured polarization state trajectories for calibration ring; AA n are the computed 3D vectors from F n (Q n , U n , V n ); BB n are the computed binormal vectors; n are the computed phase retardations; R d (−δ d ; A d ) is the 3D rotation from F n (Q n , U n , V n ) to P n (Q n , U n , V n ),as shown in Eq. (2); A n are the absolute and calibrated optical axis.(B) Poincaré sphere representations of the DDG method and the 3D rotations performed to obtain the absolute and calibrated optical axis A n from the measured polarization state trajectories F n (Q n , U n , V n ) (black dotted curve): AA 1 (black arrow) is the computed optical axis for the 1 st layer of F n (Q n , U n , V n ); BB 2 (blue arrow) is the binormal vector and note that BB 1 = AA 1 according to the DDG-based PST method; AA 2 (green arrow) is the true optical axis computed by rotating BB 2 around AA 1 by 3D rotation matrix shown in Eq. (3); AA n (green arrows) are the optical axes computed by repeating the DDG method for n times from the first to n th depth-layer of the sample, and A rel (pink arrows) denote the relative optical axes by rotating the plane of AA n back to the QU plane; A n (yellow arrows) are the absolute calibrated optical axes.
Fig. 2. Digital calibration from measured polarization state trajectories to absolute optical axis using 3D rotations determined by the calibration target.(A) shows the main steps of the algorithm to obtain the absolute optical axis from measured polarization state trajectories of the calibration ring: F n (Q n , U n , V n ) is the measured polarization state trajectories for calibration ring; AA n are the computed 3D vectors from F n (Q n , U n , V n ); BB n are the computed binormal vectors; n are the computed phase retardations; R d (−δ d ; A d ) is the 3D rotation from F n (Q n , U n , V n ) to P n (Q n , U n , V n ),as shown in Eq. (2); A n are the absolute and calibrated optical axis.(B) Poincaré sphere representations of the DDG method and the 3D rotations performed to obtain the absolute and calibrated optical axis A n from the measured polarization state trajectories F n (Q n , U n , V n ) (black dotted curve): AA 1 (black arrow) is the computed optical axis for the 1 st layer of F n (Q n , U n , V n ); BB 2 (blue arrow) is the binormal vector and note that BB 1 = AA 1 according to the DDG-based PST method; AA 2 (green arrow) is the true optical axis computed by rotating BB 2 around AA 1 by 3D rotation matrix shown in Eq. (3); AA n (green arrows) are the optical axes computed by repeating the DDG method for n times from the first to n th depth-layer of the sample, and A rel (pink arrows) denote the relative optical axes by rotating the plane of AA n back to the QU plane; A n (yellow arrows) are the absolute calibrated optical axes.
)-3(C)) and presented the corresponding trajectories of the measured output polarization states on Poincare sphere in Figs.3(J)-3(L), where the starting points of the trajectories (i.e., the averaged point of the start points at 3 locations, pointed by the black arrows) are different in the 3 trials as expected, suggesting that the input polarization states sent to the sample differ.Further, the trajectories of the output polarization states of the same point (P1, or P2, or P3) in the trials appear totally different and non-parallel, demonstrating the optical fibers in the input path and output path both caused the changes in the polarization states of the probing light.

Fig. 3
Fig. 3 PSOCT results of the PLA circle ring phantom obtained from 3 PSOCT measurement trials with each trial having differed input polarization state.The results are arranged in rows with the first row (A, D, G, J, M), the second row (B, E, H, K, N) and the third row (C, F, I, L, O) corresponding to the 1 st , 2 nd and 3 rd trial, respectively.(A-C) The en-face polarization state images of "presumably" the surface of the phantom using the inputs 1, 2 and 3 as the input polarization states, respectively.(D-F), The en-face relative local axis orientation images of the PLA phantom obtained from 3 trials.(G-I), The calibrated local axis orientation images of the PLA phantom using the DDG-based PST method combined with the calibration rotation method, showing the calibrated local axis orientation images are consistent.(J-L), The trajectories of measured output polarization states of the 3 selected points P1, P2 and P3 represented on the Poincaré sphere in (A-C).The black arrows indicate the start points of the trajectories.(M-O), The measured axes   of P1, P2 and P3 over depth from 3 trials.For each trial, the measured axes are constrained with a plane.The relative angles among the points are the same in the 3 trials.The polar form color bar for LOA is added to the right hand bottom next to (G).The white scale bar is 1 mm.

Fig. 3 .
Fig. 3. PSOCT results of the PLA circle ring phantom obtained from 3 PSOCT measurement trials with each trial having differed input polarization state.The results are arranged in rows with the first row (A, D, G, J, M), the second row (B, E, H, K, N) and the third row (C, F, I, L, O) corresponding to the 1 st , 2 nd and 3 rd trial, respectively.(A-C) The en-face polarization state images of "presumably" the surface of the phantom using the inputs 1, 2 and 3 as the input polarization states, respectively.(D-F), The en-face relative local axis orientation images of the PLA phantom obtained from 3 trials.(G-I), The calibrated local axis orientation images of the PLA phantom using the DDG-based PST method combined with the calibration rotation method, showing the calibrated local axis orientation images are consistent.(J-L), The trajectories of measured output polarization states of the 3 selected points P1, P2 and P3 represented on the Poincaré sphere in (A-C).The black arrows indicate the start points of the trajectories.(M-O), The measured axes AA n of P1, P2 and P3 over depth from 3 trials.For each trial, the measured axes are constrained with a plane.The relative angles among the points are the same in the 3 trials.The polar form color bar for LOA is added to the right hand bottom next to (G).The white scale bar is 1 mm.
)) shows that the PLA circle is not a perfect homogenous phantom because the scattering signals are not uniform.When the scattering signals are weak, the PSOCT signals are more sensitive to noise.Hence, there are some noises in the local axis orientation maps in Figs.4(B) and 4(C) as expected.

Fig. 4
Fig. 4 Cross-sectional PSOCT results of the PLA phantom with and without the PMD compensation.(A) Cross-sectional polarization state images resulting from 5 spectral bins with central bin centered at 1310nm.(B, C), Cross-sectional local axis orientation images (B) without the PMD compensation and (C) with the PMD compensation for 5 spectral bins, respectively.(D, E), Averaged-local axis orientation image with and without the PMD compensation, respectively.(F, G), Averaged local phase retardation with and without the PMD compensation, respectively.(H), Corresponding OCT structural image.(I), The trajectories of the depth extend marked by the black dashed lines in (A) in different spectral bins at the Poincaré sphere, and (J), the corresponding trajectories after the PMD compensation.(K, L), Plots of the orientations across a B-scan at the positions masked as the dashed white line in (B) without (K) and with (L) PMD compensations, respectively.

Fig. 4 .
Fig. 4. Cross-sectional PSOCT results of the PLA phantom with and without the PMD compensation.(A) Cross-sectional polarization state images resulting from 5 spectral bins with central bin centered at 1310nm.(B, C), Cross-sectional local axis orientation images (B) without the PMD compensation and (C) with the PMD compensation for 5 spectral bins, respectively.(D, E), Averaged-local axis orientation image with and without the PMD compensation, respectively.(F, G), Averaged local phase retardation with and without the PMD compensation, respectively.(H), Corresponding OCT structural image.(I), The trajectories of the depth extend marked by the black dashed lines in (A) in different spectral bins at the Poincaré sphere, and (J), the corresponding trajectories after the PMD compensation.(K, L), Plots of the orientations across a B-scan at the positions masked as the dashed white line in (B) without (K) and with (L) PMD compensations, respectively.

Fig. 5
Fig. 5 En-face PSOCT images of the forehead skin of a healthy volunteer at depth of 120 μm below the surface of the skin.(A) and (E) The polarization state images using (A) input 1 and (E) input 2 as the input polarization states, respectively.(B) and (F) The local axis orientation images without the calibration and without the PMD compensation using (B) input 1 and (F) input 2 as the input polarization states respectively.(C) and (G) the local axis orientation images with the calibration and without the PMD compensation using (C) input 1 and (G) input 2 as the input polarization states, respectively.(D) and (H) the local axis orientation images with the calibration and with the PMD compensation using (D) input 1 and (H) input 2 as the input polarization states, respectively.With different input polarization states, the proposed method can provide substantially equivalent results (D) and (H).The scale bar is 1 mm.

Fig. 5 .
Fig. 5. En-face PSOCT images of the forehead skin of a healthy volunteer at depth of 120 µm below the surface of the skin.(A) and (E) The polarization state images using (A) input 1 and (E) input 2 as the input polarization states, respectively.(B) and (F) The local axis orientation images without the calibration and without the PMD compensation using (B) input 1 and (F) input 2 as the input polarization states respectively.(C) and (G) the local axis orientation images with the calibration and without the PMD compensation using (C) input 1 and (G) input 2 as the input polarization states, respectively.(D) and (H) the local axis orientation images with the calibration and with the PMD compensation using (D) input 1 and (H) input 2 as the input polarization states, respectively.With different input polarization states, the proposed method can provide substantially equivalent results (D) and (H).The scale bar is 1 mm.

Fig. 6 .
Fig. 6.The same as Fig. 5 but presented with OCT structure and phase retardation images of the forehead skin of a healthy volunteer at depth of 120 μm below the surface of the skin.(A) and (E) The polarization state images using (A) input 1 and (E) input 2 as the input polarization states, respectively.(B) and (F) The structure images compensation using (B) input 1 and (F) input 2 as the input polarization states respectively.(C) and (G) the local phase retardation images without the PMD compensation using (C) input 1 and (G) input 2 as the input polarization states, respectively.(D) and (H) the local phase retardation images with the PMD compensation using (D) input 1 and (H) input 2 as the input polarization states, respectively.The scale bar is 1 mm.The color range for C, D, G and H is [0, 40] degrees.

Fig. 6 .
Fig. 6.The same as Fig. 5 but presented with OCT structure and phase retardation images of the forehead skin of a healthy volunteer at depth of 120 µm below the surface of the skin.(A) and (E) The polarization state images using (A) input 1 and (E) input 2 as the input polarization states, respectively.(B) and (F) The structure images compensation using (B) input 1 and (F) input 2 as the input polarization states respectively.(C) and (G) the local phase retardation images without the PMD compensation using (C) input 1 and (G) input 2 as the input polarization states, respectively.(D) and (H) the local phase retardation images with the PMD compensation using (D) input 1 and (H) input 2 as the input polarization states, respectively.The scale bar is 1 mm.The color range for C, D, G and H is [0, 40] degrees.