Full-range spectral domain Jones matrix optical coherence tomography using a single spectral camera

Chuanmao Fan; Gang Yao

doi:10.1364/OE.20.022360

1. Introduction

Polarization sensitive optical coherence tomography (PSOCT) [1,2] extends conventional optical coherence tomography (OCT) [3] 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 [10], 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 [25]. 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. Methods

2.1 Jones matrix PSOCT system

The Jones matrix PSOCT system (Fig. 1 ) was designed based on a conventional single camera PSOCT reported previously [25]. 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.

Fig. 1 Schematics of the proposed Jones matrix PSOCT system. SLD: Superluminescent diode; C: collimator, P: a polarizer for generating vertically polarized light; EOM: electronic optical modulator; BS: non polarization beam splitter; ND: neutral density filter; PBS: polarization beam splitter; M1, M2: reference mirrors for the horizontal and vertical polarization components; L1: achromatic collimation lens (f = 30 mm); L2 achromatic focusing lens (f = 120 mm); OL: objective lens (f = 60 mm); VPHTG: volume phase holography transmission grating (1200lines/mm).

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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 [25]. 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 [25].

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.

Fig. 2 An illustration of the system synchronization sequence. Ch 0 is the trigger sequence for CCD A-line acquisition. Ch 1 is the drive signal for the EOM where a high voltage generates a left-circularly polarized light, and a low voltage generates a right-circularly polarized light. Ch 2 triggers the driving signals for the B- and C-scanner.

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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:

\begin{matrix} (\begin{matrix} H_{s}^{L} (z, x) \\ V_{s}^{L} (z, x) \end{matrix}) = J_{\det} J (z, x) \frac{\sqrt{2}}{2} (\begin{matrix} 1 \\ + i \end{matrix}) \\ (\begin{matrix} H_{s}^{R} (z, x) \\ V_{s}^{R} (z, x) \end{matrix}) = J_{\det} J (z, x) \frac{\sqrt{2}}{2} (\begin{matrix} 1 \\ - i \end{matrix}) \end{matrix},

where J(z, x) is the round-trip Jones matrix of the sample at depth z and lateral scanning position x. For convenience, the depth z is represented in “round-trip” optical pathlength which is the product of refractive index and physical distance. The

{[H_{s}^{L} (z, x), V_{s}^{L} (z, x)]}^{T}

and

{[H_{s}^{R} (z, x), V_{s}^{R} (z, x)]}^{T}

are the backscattered complex Jones vectors for LC and RC polarized incident light, respectively. J_det represents the system Jones matrix from the sample to the detector and should be calibrated for a particular system. 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:

J (z, x) = \sqrt{2} J_{\det}^{- 1} [\begin{matrix} H_{s}^{L} (z,x) & H_{s}^{R} (z,x) \\ V_{s}^{L} (z,x) & V_{s}^{R} (z,x) \end{matrix}] {[\begin{matrix} 1 & 1 \\ + i & - i \end{matrix}]}^{- 1} .

To construct the Jones vectors of the backscattered sample light, we consider the detected OCT interference signal from the reference and sample light:

\begin{matrix} I (k, x) = S (k) | \int_{- \infty}^{+ \infty} E_{r}^{H} δ (z - z_{H}) e^{i (k z + θ_{H})} d z + \int_{- \infty}^{+ \infty} E_{r}^{V} δ (z - z_{V}) e^{i (k z + θ_{V})} d z + \\ {\int_{- \infty}^{+ \infty} E_{s}^{H} (z, x) e^{i (k z + ϕ_{H} + f_{m} x)} d z + \int_{- \infty}^{+ \infty} E_{s}^{V} (z, x) e^{i (k z + ϕ_{V} + f_{m} x)} d z |}^{2}, \end{matrix}

where I(k, x) represents the signal measured at wave vector k and B scan position x. The symbol E represents the real amplitude of the light. S(k) is the spectrum of the light source; the z_H and z_V are positions of the H- and V-polarized reference beam, respectively.

ϕ_{H}

and

ϕ_{V}

are polarization- and depth-dependent phases associated with the backscattered light. The

θ_{H}

and

θ_{V}

are phases of the reference light. For LC incidence,

θ_{H} = α

and

θ_{V} = π / 2

; whereas

θ_{H} = α

and

θ_{V} = - π / 2

for RC incidence.

\begin{matrix} \begin{matrix} LC & incidence : & θ_{H} \end{matrix} = α, θ_{V} = π / 2 \\ \begin{matrix} RC & incidence : & θ_{H} \end{matrix} = α, θ_{V} = - π / 2 \end{matrix} .

There are two potential sources for the phase shift α. First, a phase shift can be induced by the reflection at the beam splitter [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

e^{i f_{m} x}

in Eq. (3) is a linear phase modulation induced by the off-axis B-scanning that was implemented to achieve full-range imaging [25]. The DC intensities from the reference can be combined as

I_{r} = {| E_{r}^{H} |}^{2} + {| E_{r}^{V} |}^{2}

and the interference signal becomes

\begin{matrix} I (k, x) = S (k) (I_{r} + I_{s}) + 2 S (k) \int_{- \infty}^{+ \infty} E_{r}^{H} E_{s}^{H} (z, x) \cos [k (z - z_{H}) + ϕ_{H} - θ_{H} + f_{m} x] d z, \\ + 2 S (k) \int_{- \infty}^{+ \infty} E_{r}^{V} E_{s}^{V} (z, x) \cos [k (z - z_{V}) + ϕ_{V} - θ_{V} + f_{m} x] d z \end{matrix}

where I_s represents the self-interference of the backscattered sample light:

I_{s} = 2 \iint_{z, z'} [E_{s}^{H} (z, x) E_{s}^{H} (z', x) + E_{s}^{V} (z, x) E_{s}^{V} (z', x)] \cos [k (z - z')] d z d z' .

Fourier transform was then applied on the x variable. After band-pass filtering to remove the negative frequencies [25] and the slow-changing term of I_s, the resulted signal contains only the interference terms:

\begin{matrix} I (k, u) = \int_{- \infty}^{+ \infty} {\hat{F}}_{x} [2 S (k) E_{r}^{H} e^{- i θ_{H}} H_{s} (z, x)] \otimes [\frac{\sqrt{2 π}}{2} e^{i [k (z - z_{H})] \frac{u}{f_{m}}} δ (u - f_{m})] d z \\ + \int_{- \infty}^{+ \infty} {\hat{F}}_{x} [2 S (k) E_{r}^{V} e^{- i θ_{V}} V_{s} (z, x)] \otimes [\frac{\sqrt{2 π}}{2} e^{i [k (z - z_{V})] \frac{u}{f_{m}}} δ (u - f_{m})] d z, \end{matrix}

where

H_{s} = E_{s}^{H} e^{i ϕ_{H}}, V_{s} = E_{s}^{V} e^{i ϕ_{V}}

are complex amplitudes of the backscattered light. Inverse Fourier transform was then followed to construct the complex interference signal as:

\begin{matrix} \tilde{I} (k, x) = {\hat{F}}_{u}^{- 1} [I (k, u)] = \int_{- \infty}^{+ \infty} S (k) E_{r}^{H} e^{- i θ_{H}} H_{s} (z, x) e^{i [k (z - z_{H}) + f_{m} x]} d z \\ + \int_{- \infty}^{+ \infty} S (k) E_{r}^{V} e^{- i θ_{V}} V_{s} (z, x) e^{i [k (z - z_{V}) + f_{m} x]} d z . \end{matrix}

The above equation can be reformatted by using inverse Fourier transform over depth z:

\begin{matrix} \tilde{I} (k, x) = \sqrt{2 π} S (k) {F_{z}^{- 1} [E_{r}^{H} e^{- i θ_{H}} H_{s} (z + z_{H}, x) e^{i f_{m} x}] \\ + F_{z}^{- 1} [E_{r}^{V} e^{- i θ_{V}} V_{s} (z + z_{V}, x) e^{i f_{m} x}]} . \end{matrix}

A second Fourier transform was applied in k-space to obtain the complex depth-resolved images:

\tilde{I} (z, x) = {\hat{F}}_{k} [\tilde{I} (k, x)] = \sqrt{2 π} {\hat{F}}_{k} [S (k)] \otimes E_{r}^{V} e^{i f_{m} x} [\frac{E_{r}^{H}}{E_{r}^{V}} e^{- i θ_{H}} H_{s} (z + z_{H}, x) + e^{- i θ_{V}} V_{s} (z + z_{V}, x)] .

The term

{\hat{F}}_{k} [S (k)]

determines the spatial resolution of the OCT system, i.e. the size of each pixel where the OCT measurements are performed. The constant term

\sqrt{2 π} E_{r}^{V} e^{i f_{m} x}

can be eliminated from subsequent calculations. From Eq. (10), the horizontal component H_s and vertical component V_s are mapped into different depth locations separated by a distance ∆z = z_H-z_V. Therefore the Jones vector of the backscattered sample light can be obtained by extracting data from different locations in the complex image:

[\begin{matrix} H_{s} (z, x) \\ V_{s} (z, x) \end{matrix}] = [\begin{matrix} \tilde{I} (z - z_{H}, x) e^{i θ_{H}} \frac{E_{r}^{V}}{E_{r}^{H}} \\ \tilde{I} (z - z_{V}, x) e^{i θ_{V}} \end{matrix}] .

Using the phase information from Eq. (4), the above equation can be derived as:

[\begin{matrix} H_{s}^{L} (z, x) & H_{s}^{R} (z, x) \\ V_{s}^{L} (z, x) & V_{s}^{R} (z, x) \end{matrix}] = [\begin{matrix} e^{i α} {\tilde{I}}_{L} (z - z_{H}, x) \frac{E_{r}^{V}}{E_{r}^{H}} & e^{i α} {\tilde{I}}_{R} (z - z_{H}, x) \frac{E_{r}^{V}}{E_{r}^{H}} \\ + i {\tilde{I}}_{L} (z - z_{V}, x) & - i {\tilde{I}}_{R} (z - z_{V}, x) \end{matrix}] .

From Eq. (2), the sample Jones matrix can then be calculated as:

\begin{matrix} J (z, x) = \sqrt{2} J_{\det}^{- 1} [\begin{matrix} e^{i α} & 0 \\ 0 & i \end{matrix}] [\begin{matrix} E_{r}^{V} / E_{r}^{H} & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} {\tilde{I}}_{L} (z - z_{H}, x) & {\tilde{I}}_{R} (z - z_{H}, x) \\ + {\tilde{I}}_{L} (z - z_{V}, x) & - {\tilde{I}}_{R} (z - z_{V}, x) \end{matrix}] {[\begin{matrix} 1 & 1 \\ + i & - i \end{matrix}]}^{- 1} \\ = J_{c a l} [\begin{matrix} E_{r}^{V} / E_{r}^{H} & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} {\tilde{I}}_{L} (z - z_{H}, x) & {\tilde{I}}_{R} (z - z_{H}, x) \\ + {\tilde{I}}_{L} (z - z_{V}, x) & - {\tilde{I}}_{R} (z - z_{V}, x) \end{matrix}] {[\begin{matrix} 1 & 1 \\ + i & - i \end{matrix}]}^{- 1} . \end{matrix}

The effective calibration Jones matrix J_cal can be represented by a general retarder that consists of a retarder and a rotator:

J_{c a l} = (\begin{matrix} \cos θ_{c} & - \sin θ_{c} \\ \sin θ_{c} & \cos θ_{c} \end{matrix}) (\begin{matrix} e^{i δ_{c} / 2} & 0 \\ 0 & e^{- i δ_{c} / 2} \end{matrix}) (\begin{matrix} \cos θ_{c} & \sin θ_{c} \\ - \sin θ_{c} & \cos θ_{c} \end{matrix}) (\begin{matrix} \cos ϕ_{c} & \sin ϕ_{c} \\ - \sin ϕ_{c} & \cos ϕ_{c} \end{matrix}) .

The three variables θ_c, δ_c and ϕ_c in J_cal can be obtained during system calibration by using standard polarization component whose Jones matrix J(z, x) is known. Such calibration was implemented in this study using optimization method by searching for the best J_cal that resulted with the minimal least square errors between experimental results and calculated results.

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 [25] 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 [25]. 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 = z_H-z_V. 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:

I = 10 \log_{10} [({| J(1,1) |}^{2} + {| J(2,1) |}^{2} + {| J(1,2) |}^{2} + {| J(2,2) |}^{2}) / 2] .

Eigendecomposition was applied to calculate the retardance and relative attenuation images:

J = V Λ V^{- 1} = [\begin{matrix} ν_{11} & ν_{21} \\ ν_{12} & ν_{21} \end{matrix}] [\begin{matrix} λ_{1} & 0 \\ 0 & λ_{2} \end{matrix}] {[\begin{matrix} ν_{11} & ν_{21} \\ ν_{12} & ν_{21} \end{matrix}]}^{- 1},

where 𝚲 is a diagonal matrix formed from the eigenvalues λ₁ and λ₂ of J. The two columns  [

v_{11}

,

v_{12}

]^T and [

v_{21}

,

v_{22}

]^T of V are the corresponding eigenvectors of J. The two eigenvalues can be represented in term of retardance and relative attenuation as:

{\begin{cases} λ_{1} = e^{i δ + σ} = e^{i ρ} \\ λ_{2} = e^{- (i δ + σ)} = e^{- i ρ} \end{cases},

where ρ = σ + iδ is the complex retardance with δ being the sample retardance and σ being the relative attenuation coefficient. Therefore the single-trip retardance [27,28] can be calculated from the two eigenvalues as:

δ = \frac{1}{2} | \tan^{- 1} \frac{Im (λ_{1} \times λ_{2}^{*})}{Re (λ_{1} \times λ_{2}^{*})} | .

The optical axis can be calculated from the eigenvector corresponding to the fast eigenvalue λ₂ [29]

θ = \frac{1}{2} \tan^{- 1} (\frac{2 | v_{21} v_{22} | \cos τ}{{| v_{21} |}^{2} - {| v_{22} |}^{2}}),

where τ is the phase difference between v₁₂ and v_11. The single-trip relative attenuation is then calculated as [28]:

σ = \tan h^{- 1} (| \frac{{| λ_{1} |}^{2} - {| λ_{2} |}^{2}}{{| λ_{1} |}^{2} + {| λ_{2} |}^{2}} |), [0, \infty] .

3. Results

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 J_cal, 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 (θ_c, δ_c, ϕ_c) were determined by searching the variable space (θ_c ∊ [0, π], δ_c ∊ [0, π], ϕ_c ∊ [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 θ_c = 2.39 rad, δ_c = 0.70 rad and ϕ_c = 2.50 rad. The LSE error as a function of parameters δ_c and ϕ_c at θ_c = 2.39 rad was shown in Fig. 3 .

Fig. 3 An illustration of the error distribution as a function of the calibration parameters δ_c and ϕ_c with θ_c set at 2.39. (a) The LSE shown in a pseudo color image with black indicating small error. (b) The LSE profiles as a function of δ_c and ϕ_c along the lines shown in (a).

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The obtained calibration parameters (θ_c, δ_c, ϕ_c) 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 J_cal. 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.

Fig. 4 The effect of calibration matrix on the measured retardance and optical axis of a variable waveplate whose retardance was set at π/2 (in a and b) and π/4 (in c and d).

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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:

J = J_{QWP}^{T} J_{QWP} = [\begin{matrix} \cos (2 θ) & \sin (2 θ) \\ \sin (2 θ) & - \cos (2 θ) \end{matrix}] .

As shown in 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.

Fig. 5 (a) The measured amplitudes of Jones matrix elements J(1,1) and J(2,1) of the quarter wave plate. Solid lines are calculation results from of |cos(2θ)| and |sin(2θ)|. (b) The measured retardance, optical axis and relative attenuation of a quarter waveplate. The axis of the quarter waveplate was rotated from –π/2 to π/2 during the test.

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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 [32], 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)).

Fig. 6 Depth-resolve polarization imaging of a piece of chicken tendon sample. (a)Intensity; (b)retardance; (c)optical axis; and (d) relative attenuation.

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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 [12]. 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 [28] 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 [12].

Fig. 7 Example A-line profiles of retardance, optical axis and relative attenuation extracted from the dashed lines marked in Fig. 6. The line shown in (b) was a linear regression fitting of the experimental data (R² = 0.86).

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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).

Fig. 8 Depth-resolve polarization imaging of a finger bed in vivo. Example cross-sectional images are shown in the first row: (a) intensity (Media 1), (b) retardance (Media 2), (c) optical axis (Media 3), and (d) relative attenuation (Media 4). Shown in the 2nd row are enface images extracted at the position of the dashed line in (a): (e) intensity, (f) retardance, (g) optical axis, and (h) relative attenuation. The labels shown indicate different structures: “e” epidermis, “d” dermis, “c” cuticle, “p” nail plate, “b” nail bed, “m” nail matrix.

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4. Conclusion

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.

Acknowledgment

This project is supported in part by the National Science Foundation (NSF) under grant award CBET-0643190.

References and links

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Full-range spectral domain Jones matrix optical coherence tomography using a single spectral camera

Abstract

1. Introduction

2. Methods

2.1 Jones matrix PSOCT system

2.2 Jones matrix construction

2.3 Signal processing

2.4 Determining polarization properties

3. Results

3.1 System calibration

3.2 Imaging of biological tissues

4. Conclusion

Acknowledgment

References and links

Supplementary Material (4)

Cited By

Figures (8)

Equations (21)

Optics Express