Jones matrix optical coherence tomography can fully characterize depth-resolved polarization properties in tissue. In this report, we described a simple single-camera based implementation of full-range spectral domain Jones matrix optical coherence tomography. The Jones matrix reconstruction algorithm was described in detail and system calibration was demonstrated with comprehensive examples. In addition to the conventional structural image, the images of retardance, optical axis and relative attenuation can be obtained from the measured Jones matrix image. Both in vitro and in vivo image examples were presented to demonstrate the polarization imaging ability of the system.
© 2012 OSA
Polarization sensitive optical coherence tomography (PSOCT) [1,2] extends conventional optical coherence tomography (OCT)  by providing polarization sensitive measurements. It acquires not only structural images but also polarization contrast images which are useful in many biomedical applications [4,5] and material characterization [6,7]. For birefringence samples, a simple PSOCT implementation is to use a circularly polarized incident light and detect the two orthogonal horizontal and vertical polarized backscattered components from the sample [2,8]. To completely characterize all sample polarization properties, both intensity-based Mueller matrix PSOCT [9,10] and phase sensitive Jones matrix PSOCT [11,12] were developed. Because of the coherent detection in OCT , the obtained Mueller matrix and Jones matrix can be converted to each other. In recent years, spectral domain OCT has become popular due to its superior signal-to-noise and faster imaging speed than the time domain OCT [13,14]. Since phase information is readily available during the image reconstruction in spectral domain OCT, Jones matrix based PSOCT can be conveniently implemented.
Several spectral domain Jones matrix PSOCT systems have been reported [15–24]. A common feature of existing PSOCT systems is the detection of the two orthogonal polarization components of the backscattered signal. When a swept light source is used in a spectral domain system, the two orthogonal signal channels can be easily detected using two individual detectors as in time domain system [15,17]. Spectral camera based systems are more widely adopted due to their relatively lower cost than the swept source based system. In such a system, a straightforward approach is to split the horizontally and vertically polarized components in space and project each to a separate linear CCD array [18,19]. Alternatively, the two components can be aligned and projected to adjacent space of a single planar CCD or multi-line CCD [20,22]; or two orthogonal polarized components can be realigned and projected to adjacent space in a single linear CCD [23,24]. However, it is challenging to achieve spectrum alignment in such systems in addition to increased system complexity and cost.
In this paper, we describe a simple implementation of spectral domain Jones matrix PSOCT system by using a single spectral camera. Instead of separating the two orthogonal polarization channels in the detection arm, two reference beams with different pathlengths were used to separate the two orthogonal detection channels in the depth direction . The procedures for Jones matrix reconstruction and system calibration are described in detail. Imaging in vitro and in vivo tissue samples were demonstrated. This system provides a simple alternative implementation of Jones matrix PSOCT system for fully characterize of sample polarization properties including retardance, optical axis and diattenuation.
2.1 Jones matrix PSOCT system
The Jones matrix PSOCT system (Fig. 1 ) was designed based on a conventional single camera PSOCT reported previously . A superluminescence diode (SLD) with a central wavelength λ0 of 844 nm and 46.8 nm spectral bandwidth was used as the light source. The calculated axial resolution was 6.7 μm in air. Light emitting from the SLD was vertically polarized by using a polarizer (P) after passing through an isolator and collimated by a collimator (C). The polarization state of the light source was modulated by using an electro-optical modulator (EO-AM-NR-C1, Thorlabs Inc.) to achieve alternating left-circularly (LC) and right-circularly (RC) polarized light. The modulated light was then split by a beam splitter BS into a reference arm and sample arm with a ratio of 50:50.
In the sample arm, the modulated incident light was reflected by a galvanometer scanner (B-scan). The reflection position of the incident beam was slightly deviated from the pivot axis to implement full range measurement . Then the reflected light was guided onto an objective lens (OL) via another galvanometer scanner (C-scan) and focused into the sample. The reflection position on the C-scanner was centered at the pivot axis. The beam waist on the sample surface was 15 μm. The incident power on the sample surface was 1.56 mW.
In the reference arm, a polarization beam splitter (PBS) split the light into two beams of orthogonal polarization states: horizontal linear (H) and vertical linear (V). Two reference mirrors (M1 and M2) were used to form two co-axial reference beams with different delays, which were essential to map the horizontally and vertically polarized components of the backscattered sample light to two separate depth positions in the reconstructed image .
The recombined reference light and sample light were coupled into a custom-made spectrometer through a single mode fiber. The coherence spectra were captured by a line scan CCD camera with 1024 pixels and a pixel size of 14 μm (AVIIVA SM2, e2v, France). The spectral images were acquired via a frame grabber (PCIe-1427, National Instruments). The image acquisition speed was 52k A-line/s.
The spectral CCD, EOM and scanners were synchronized using signals generated from a DAQ board (PCI-6733, National Instrument Inc.). As shown in Fig. 2 , Channel 0 provided a TTL signal (52 kHz) to trigger each A-line acquisition. A square wave from Channel 1 was used to drive EOM at 26 kHz. The 26 kHz TTL signal from Channel 2 was used to trigger the output of two analog pulse trains to control the B and C galvanometer scanners. At each scanning position, a pair of high and low voltages were applied to EOM to generate L and R circularly polarized light; at the same time two A-lines signals were acquired sequentially. The CCD trigger signal (ch 0, 52 kHz) and the EOM driving signal (ch 1, 26 kHz) were synchronized at the rising edge. The CCD exposure time for each A-scan was 18 μs which was slightly smaller than the duration (19.2 μs) of the high or low voltage of the EOM driving signal. The system sensitivity measured at zero delay line of the image is 106 dB.
2.2 Jones matrix construction
The Jones vectors of the backscattered sample light for left-circularly (LC) and right-circularly (RC) polarized incident light can be represented as:Equation (1) can be rearranged using a matrix format to derive the true sample Jones matrix J(z,x) from the two measured Jones vectors:26]. Secondly, because the position difference of the two reference mirrors M1 and M2 was not an integer number of the depth resolution (the pixel size), quantization errors were generated in the computation and resulted in a phase shift between the two channels. The term in Eq. (3) is a linear phase modulation induced by the off-axis B-scanning that was implemented to achieve full-range imaging . The DC intensities from the reference can be combined as and the interference signal becomes25] and the slow-changing term of Is, the resulted signal contains only the interference terms:Eq. (10), the horizontal component Hs and vertical component Vs are mapped into different depth locations separated by a distance ∆z = zH-zV. Therefore the Jones vector of the backscattered sample light can be obtained by extracting data from different locations in the complex image:Eq. (4), the above equation can be derived as:Eq. (2), the sample Jones matrix can then be calculated as:
2.3 Signal processing
To facilitate the discussions, we use the coordinate system shown in Fig. 1. The z-axis represents the depth direction in the sample; x- and y-axis represent the B-scan and C-scan directions at the sample surface. In addition, we use the λ-space to denote the wavelength dimension in the acquired raw spectrum. Due to the detection sensitivity fall-off over depth, each A-line signal was corrected by dividing the sensitivity fall-off curve obtained during system calibration. The acquired one B-scan data consisted of 2000 interference spectra of 1024 pixels each. Such B-scan data in the (λ, x) space were first split in the x-dimension into the “odd” and “even” scans corresponding to the LC and RC polarized incident light (Fig. 2). Each frame had the same size 1024 × 1000 pixels and was processed following the same procedures  described below to extract the horizontal and vertical polarization components.
First, every spectral line (1024 pixels) along the original λ-space was converted to a uniform wave number k-space by using linear interpolation. Fourier transform, band pass windowing and inverse Fourier transform were then applied on each B-scan (along the x dimension) to obtain the analytic complex coherence signal . Fourier transform was then applied on each A-line (k-space) to get full range images (in z dimension) where the horizontal (H) and vertical (V) components were separated by ∆z = zH-zV. Another image was created by circ-shifting down the obtained 2D image by ∆z to match the H and V images. The Jones matrix images were then computed by using Eq. (13).
2.4 Determining polarization properties
Once Jones matrix J is obtained, the intensity image I (in dB) was calculated as:27,28] can be calculated from the two eigenvalues as:29]28]:
3.1 System calibration
The extinction ratio of the H and V polarized reference components were 32.8 dB and 19.2 dB as characterized by using a polarizer. The polarization purity of the LC and RC incident light were measured as 48.6 dB and 42.6 dB, respectively by using a rotating polarizer. To obtain calibration matrix Jcal, a variable waveplate (VWP) with retardance set at π/2 was placed in the sample arm. Its Jones matrix was then measured when rotating its axis from –π/2 to π/2. The optimal set of calibration parameters (θc, δc, ϕc) were determined by searching the variable space (θc ∊ [0, π], δc ∊ [0, π], ϕc ∊ [0, π]) to determine the set of parameters resulting the minimal least square error (LSE) between calculated retardance/axis and their corresponding true values. The obtained calibration parameters for our particular system were θc = 2.39 rad, δc = 0.70 rad and ϕc = 2.50 rad. The LSE error as a function of parameters δc and ϕc at θc = 2.39 rad was shown in Fig. 3 .
The obtained calibration parameters (θc, δc, ϕc) were applied to calculate the true sample Jones matrix from Eq. (13) and extract the retardance and optical axis from Eqs. (18) and (19). As shown in Figs. 4(a) and 4(b), the calculated retardance and axis of this π /2 waveplate were incorrect without applying the calibration matrix Jcal. However, after applying the calibration matrix, the correct retardance and optical axis were obtained. The obtained average retardance was 1.552 ± 0.016 rad. Figures 4(c) and 4(d) show the measurements when the retardance of the same variable waveplate was changed to π/4. Again the correct retardance and optical axis were obtained when the same calibration matrix were used. The measured average retardance at various angles was 0.775 ± 0.014 rad.
To further verify the system calibration, the Jones matrix of another quarter waveplate (QWP) was measured. The axis orientation of the waveplate θ was rotated from –π/2 to π/2. The theoretical round-trip Jones matrix of quarter wave plate is:Fig. 5(a) , the measured amplitudes |J(1,1)| and |J(2,1)| were in good agreement with the theoretical predications of |cos(2θ)| and |sin(2θ)|. Figure 5(b) shows the obtained retardance, optical axis, and relative attenuation. The measured single-trip retardance was 1.516 ± 0.028 rad. The error between measured optical axis and the true (pre-set) axis orientation was within [-0.061 rad, 0.059 rad]. As expected, the measured relative attenuation was close to zero (0.046 ± 0.019) for this quarter waveplate.
3.2 Imaging of biological tissues
Figure 6 shows the imaging results obtained in a piece of chicken tendon sample. To improve imaging depth, a thin film of 6% glycerin was applied on the sample surface before the imaging. Some vertical crimp patterns appeared in the structure image (Fig. 6(a)) similar to those reported previously [30,31]. The horizontal “band” pattern shown in the retardance image (Fig. 6(b)) was a typical representation of organized collagen structures in tendon. Each band of the pattern covered a range of π in phase retardance with phase wrapping at π/2. The relatively uniform distributed band pattern indicated a uniform birefringent structure, which suggested homogeneous distribution of collagen fibers with depth. The band structure shown in optical axis image (Fig. 6(c)) was due to the π/2 phase wrapping in retardance calculation , and thus the “banded” pattern was identical in the retardance image (Fig. 6(b)) and optical axis image (Fig. 6(c)). The relative attenuation image was relatively uniform and appeared to increase with depth (Fig. 6(d)).
For a quantitatively study, example A-line profiles of the retardance, optical axis and relative attenuation extracted along the dashed lines in Fig. 6 were shown in Fig. 7 . The retardance changed with depth in a sawtooth waveform. The average retardance calculated from the slope of the linear portion of the A-line (Fig. 7(a)) was about 21.3 rad/mm, similar to the previously reported value of 20.67 rad/mm . The optical axis curve appeared to be rectangular (Fig. 7(a)). Without the phase wrapping, the optical axis was relatively flat within each line segment, suggesting roughly parallel collagen fiber arrangement. The relative attenuation (Fig. 7(b)) increased linearly with depth. The slope of the linear regression was 0.54 mm−1. The relative attenuation value can be converted to conventional single trip diattenuation  value of tanh(σ) = tanh(0.54)≈0.49 mm−1, which was similar with a previously reported diattenuation value of 0.39 mm−1 .
This system was also applied to acquire 3D images of a human finger bed in vivo. The 3D scans consisted of 300 frames of 1024 × 1000 2D images and were acquired in around 12 seconds. A sample cross-sectional image and enface image were show in Fig. 8 . The images were filtered for display by using a 5 × 5 median filter. In the intensity image (Fig. 8(a)), the epidermis (“e”) can be discriminated from the dermis (“d”). The eponychium or cuticle (“c”) is located between the dermis and the nail plate [33,34]. The boundary between the nail plate and the nail bed was not clear because the gradual change in the intensity signals. In the polarization images, the band structure in the nail bed was significantly denser than that in the nail plate, i.e. there were more bands in the nail bed than the nail plate. This indicated higher retardance in the nail bed, likely due to higher collagen content. Nail matrix underneath the nail bed showed a slower changing band pattern in retardance. The retardance also showed depth-dependent changes in dermis likely due to the rich collagen in dermis. As expected, the band patterns in optical axis were similar to that in the retardance image. Similarly structures were observed in the corresponding enface images retrieved at the depth marked in Fig. 8(a).
In summary, we demonstrated a simple Jones matrix PSOCT imaging system using a single spectral camera. Instead of using separate detection arms in other reported systems, the horizontally and vertically polarized components of the backscattered light were mapped into different position in the full range image space by using spatially separated double reference arms. The alternating left- and right- circularly polarized incident lights were generated by an electro-optical modulator. By using a single spectral camera, the system synchronization and alignment were considerably simplified. The Jones matrix reconstruction algorithm and system calibration were described in detail. Both in vitro and in vivo tests were conducted to demonstrate the capability of this system for imaging retardance, optical axis and relative attenuation distributions in tissue samples.
This project is supported in part by the National Science Foundation (NSF) under grant award CBET-0643190.
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