We present an improved method of polarization sensitive optical coherence tomography that enables measurement and imaging of backscattered intensity, birefringence, and fast optic axis orientation simultaneously with only one single A-scan per transverse measurement location. While intensity and birefringence data are obtained in a conventional way, the optic axis orientation is determined from the phase difference recorded in two orthogonal polarization channels. We report on accuracy and precision of the method by measuring birefringence and optic axis orientation of well defined polarization states in a technical object and present maps of birefringence and, what we believe for the first time, of optic axis orientation in biological tissue.
©2001 Optical Society of America
Optical coherence tomography (OCT) is an emerging technology for high-resolution, non-contact imaging in transparent and translucent structures [1–3]. Originally developed for imaging of the human retina, the application fields of OCT have been extended to a wide variety of tissues and also to imaging of non-biologic structures. Conventional OCT is based on the intensity of backreflected or -scattered optical radiation. However, as is well known from microscopy, many samples show only poor contrast if they are imaged on a pure intensity basis. Therefore, OCT has been extended to exploit additional properties of light to improve the image contrast and to perform quantitative measurements. One of the most promising of the extended OCT techniques is polarization sensitive (PS)-OCT. Several reports on PS-OCT have demonstrated its ability to measure and image birefringence in different tissues [4–8]. Among the suggested applications of PS-OCT are burn depth estimation, glaucoma diagnosis, and early detection of carious lesions.
Most of the work on PS-OCT reported so far is based on the PS low coherence interferometry method demonstrated first by Hee et al. . With this technique, the sample is illuminated by a circular polarized beam; the sample birefringence is obtained by measuring the envelope of the interferometric signal in two separate detection channels of orthogonal polarization state. From the ratio of the two measured envelope amplitudes, birefringence is obtained as a function of depth, independent of the sample optical axis. Other physical parameters that change the polarization state of the probing light, e.g., dichroism and optic axis orientation, cannot be determined by this method.
Recently, more advanced PS-OCT techniques enabling the recording of full Stokes vectors and Müller matrices were reported [10–12]. While these techniques are able to obtain more information on the polarizing properties of a sample (the full information is contained in the Müller matrix), they usually require more than one A-scan at a given sample location [11,12] (up to 16 for measuring the full Müller matrix). On the other hand, the information contained in the individual Müller matrix elements is difficult to interpret on the basis of simple, physically meaningful parameters.
One of these parameters is the orientation of the optic axis of the sample at the measured position. In the context of tissue imaging, the orientation of the optic axis is a measure of the orientation of fibrous structures, as muscle fibers or axons in the retinal nerve fiber layer. So far, only two reports on determining the optic axis of a sample by OCT have been presented: De Boer et al.  determined an overall orientation of the sample optical axis by computing the angle of rotation that minimizes the amplitude of oscillations with depth of the Stokes parameters S1 or S2. No detailed information on the procedure and number of A-scans needed was provided. Furthermore, this method seems to provide only an overall orientation of the optical axis, no information on the local distribution (i.e., mapping) of the optic axis has been reported. Roth et al.  report on an explicit measurement of the optic axis of birefringent structures by OCT. This method, however, requires three subsequent A-scans with different input polarization states.
We report on a method to measure backscattered intensity, birefringence, and fast optic axis orientation simultaneously with only one single A-scan. The method employs a standard two-channel PS-OCT setup  in combination with a phase sensitive recording of the interferometric signals in the two orthogonally polarized detection channels. We use an algorithm previously reported in the context of Doppler OCT  and differential phase contrast OCT  to extract amplitude and phase information contained in the interferometric signals. While the birefringence information is obtained from the signal amplitudes, as usual in PS-OCT, a careful analysis of the propagating beams by the Jones calculus reveals, that the information on the fast axis orientation is encoded entirely in the phase difference of the interferometric signals . We demonstrate our method and report on accuracy and precision of birefringence and fast axis measurements. Finally, we present PS-OCT maps of birefringence, and, what we believe for the first time, of transversal distribution of fast axis orientation recorded in biological tissue.
2.1 Experimental setup
Figure 1 shows a sketch of the instrument. A broadband fiber optic light source (AFC Technologies, Canada) emits a beam (output power ≅20 mW) with center wavelength λ0=1310 nm and a bandwidth of Δλ≅55 nm, corresponding to a round trip coherence length  of lc≅13.5 µm. This beam illuminates, after being vertically polarized, a Michelson interferometer where it is split by a non-polarizing beam splitter into a reference and a sample beam. The reference beam transmits quarter wave plate QWP2, oriented with its fast axis at 22.5° to the horizontal, and is reflected at the movable reference mirror. After backpropagating through QWP2, the orientation of the beam polarization plane is at 45° to the horizontal, providing equal reference power in two orthogonal polarization states. The quarter wave plate QWP1 in the sample arm is oriented at 45° to provide circularly polarized light to the sample. After reflection at the sample and propagating back through QWP1, the light in the sample arm is in an elliptical polarization state containing information on the birefringence and optic axis orientation of the sample. After recombination with the reference beam at the interferometer beam splitter, both beam components are directed towards a polarization sensitive two-channel detection unit. The HeNe laser and the infrared camera are for alignment purposes only.
OCT scanning is performed in the usual way, i.e., A-scans are recorded by shifting the reference mirror, transversal information is obtained by a lateral shift of the sample between individual A-scans. Contrary to standard OCT and earlier versions of PS-OCT, not only the envelopes of the interferometric signals are recorded. Instead, for each detection channel an independent detector/amplifier/AD-converter combination records the whole interference signal Ik(z) (channel number k=1, 2). The axial resolution is equal to the round trip coherence length lc. The transversal resolution is determined by the spot size of the focused sample beam at the object and by the step increments in transverse direction. Using a beam diameter of ~1 mm and a focal length f=30 mm, we obtain an FWHM focus diameter of ≅17 µm. This is lower than the transverse step size of 25 µm between the individual A-scans, so our transverse resolution is 25 µm.
2.2 Calculation of the Polarization States
The polarization state of a fully polarized light beam traversing polarizing optical elements can be calculated by the Jones formalism [17,18]. The beam incident on the Michelson interferometer, after passing the polarizer, is in a linear, vertical polarization state. Its Jones vector can be written as:
with E0=A0·exp(iωt). E⃗ is the electric field vector, E0 the scalar electric field, A0 the field amplitude, ω the angular frequency, and t the time. The upper and lower components of the vector in equation (1) correspond to the horizontal and vertical components of the electric field vector. For the sake of simplicity, we ignore the oscillating term exp(iωt) and set A0=1 in the following.
The Jones vector of a beam traversing an optical element can be found by multiplying the Jones vector of the incident beam by the Jones matrix corresponding to the optical element. The Jones matrix is a 2×2 matrix consisting of - usually complex - elements. If more than one optical element is traversed, the input Jones vector has to be multiplied by the Jones matrices of each element, in the order they are transmitted by the beam.
The reference beam passes both, the beam splitter and the quarter wave plate QWP2, twice. The effect of the beam splitter is simply to reduce the intensity of the reference beam by a factor of 2 at every pass, leading to a total intensity reduction of a factor of 4, equivalent to a reduction in field amplitude by a factor of 2. The Jones matrix of a general retarder of retardation δ and fast axis orientation θ is :
For the reference beam, δ=π/2 and θ=π/8. The Jones vector of the reference beam, after double passing QWP2 (and the beam splitter) is:
This is a linear polarized beam with its polarization axis oriented at -45°. This beam provides equal reference intensity for both, the horizontal and the vertical polarization component, which are separated by the polarizing beam splitter PBS of the detection unit. Furthermore, no phase shift occurs between the two polarization components (this would cause at least one of the vector components to have an imaginary part). Therefore, the reference beam influences neither the intensity ratio nor the phase of the interference signals recorded in the two detection channels.
The sample beam passes the beam splitter, QWP1, and the sample (reflectivity: R) twice each. Again, the effect of the beam splitter is simply to reduce the field amplitude by a factor of 2. QWP1 has retardance and axis values of δ=π/2 and θ=π/4, respectively. The change of polarization state caused by the sample is described by Msample, a general retardation matrix of the form given by equation (2). The Jones vector of the sample beam, after exiting the interferometer, is:
A similar dependence of the Jones vector components on optic axis orientation θ has already been derived earlier .
2.3 Determination of sample reflectivity, birefringence, and optic axis orientation
As is well known from the theory of partial coherence interferometry, a sample interface located at (optical) depth position z0, and with reflectivity R, gives rise to an intensity Ik(z) recorded by the detector in channel k :
Ir,k and Is,k are the intensities that would be recorded at the detector k from the reference mirror and the sample interface, respectively, without interference; k=1,2 for the horizontal and vertical polarization state, respectively, Φk=2π(z-z0)/λ0 is the corresponding phase term of the oscillating interference signal, and |γ(z-z0)| is the modulus of the complex degree of coherence.
With equations (4) and (5), and with Ir,k∝ and Is,k∝, (Er,k and Es,k are given by equations (3) and (4)), we can derive the physical parameters: reflectivity R, birefringence δ, and fast optic axis orientation θ. To determine reflectivity and birefringence, we need only the envelope of the AC term of equation (5) . This is usually obtained by AC coupling the detector signal and subsequently rectifying and low pass filtering the amplified signal [4–8]. However, to measure the optic axis orientation, we need the phase difference ΔΦ=Φ2-Φ1 occurring between the two signals. Therefore, we use a different approach. After AC coupling, the full interferometric signal is recorded. An algorithm previously reported in the context of Doppler OCT  and differential phase contrast OCT  is used to determine the amplitude Ak(z) and the phase Φk(z) of the oscillating term of each interference signal at any depth. These quantities can be extracted from the complex function:
which is determined by analytic continuation of the measured interference signal Ik(z) by use of the Hilbert transform H.
Furthermore, equation (4) indicates that only the signal corresponding to the vertical polarization state (channel 2) depends on the sample optic axis orientation θ, while both signals oscillate synchronously with retardation δ. Therefore, the axis orientation is directly encoded in the phase difference ΔΦ=Φ2-Φ1 which can be obtained with the help of eq. (6):
To evaluate our method and to test its precision, we used a Berek’s polarization compensator (New Focus Model 5540) in front of a mirror as the sample. This device allows an arbitrary and independent setting of optical axis and retardation for a wide range of wavelengths. We performed two series of measurements.
At first, we held the optic axis at fixed values of 0°, 15°, …180° and varied the retardation from 0° to 90° in steps of 15°; at each set value we measured the retardation with our method. Figure 2 shows the results. In fig. 2a, measured values of δ are plotted versus the retardation adjusted at the Berek’s compensator (optical axis orientation set at 40°). The measured data points are in good agreement with the expected values (solid line). The largest deviations of ~3–4° occur near δ=0°, 90°, and 180°. These systematic deviations are probably caused by imperfect polarization optics. The error bars indicate the standard deviation (SD) of 5 repeated measurements. The mean SD, averaged over the whole data set, is SDδ=0.5°. Fig. 2b summarizes the results of the recorded data in a polar diagram. The optic axis set at the Berek’s compensator is indicated along the circumference of the plot, the half-circles represent the test plate retardation settings, the data points represent the measured retardation. In most cases, the deviation between set and measured retardation values is small, again the largest deviations occur near δ=0° and 90°.
In a second measurement series, the retardation was set at fixed values of 15°, 30°, …75°, while the optic axis was varied from -90° to +90° in steps of 10°. Fig. 3 shows the results. In fig. 3a, measured values of θ are plotted versus those set at the Berek’s compensator (retardation set at 30°). The measured data points (error bars: SD) show a systematic deviation from the expected values (solid line) which might be caused by incorrect calibration of the compensator. The mean SDθ=1.8°. Fig. 3b again summarizes the data in a polar diagram. In this case, the retardation set at the Berek’s compensator is indicated along the circumference of the plot, the quarter-circles represent the fast axis set at the compensator, the data points represent the measured axis orientation. The mean absolute deviation between set and measured axis orientation is ~9.4°, the maximum deviation ~15.3°. The red data points indicate measured data points where the axis orientation was set at +90°. The systematic deviation leads to θ — values larger than +90° in this case. Our algorithm maps all the data between -90° and +90°. This corresponds to a subtraction of 180° from any θ — value larger than +90°. Therefore, these (red) data points are wrapped to slightly over -90° (in fig. 3a, the corresponding data point was unwrapped for better readability of the figure). At retardation values of 0° and 90°, no measurements of optic axis orientation could be obtained because in these cases either detection channel 1 or 2 obtained no signal (because of crossed polarization states).
To demonstrate the performance of our system in scattering tissue, we recorded PS-OCT cross sectional images in a chicken myocardium in vitro. The heart was sectioned perpendicular to its longitudinal axis into two halves, OCT images were recorded perpendicular to the section plane (i.e., parallel to the longitudinal axis), approximately across the center of the myocardium, covering ventricular muscle tissue on the left and right hand side of the tomograms, and tissue corresponding probably to the septum interventriculare near the central image part . 560 A-scans were recorded with lateral step increments of 25 µm. Scanning speed was ~6 mm/s; with an A-scan length of ~6 mm, the total scanning time was ~20 min (unidirectional scanning).
Figure 4 shows the result. Fig. 4a shows the intensity image, fig. 4b the phase retardation, and fig. 4c the fast axis distribution in the sample. The intensity image shows very little structure and contains the least information. The retardation image shows rather constant color with depth near the (transversal) central third of the image, and more than one full oscillation (i.e., from blue to red to blue), indicating a phase shift of > 180°, in the right and left thirds of the image. This is probably due to birefringence caused by the fibrous structure of the muscle tissue. The parts of the image shown in gray correspond to regions where the signal intensity is not significantly above the noise level (less than 10 dB). No birefringence (and optic axis orientation) was calculated in these areas because the noise would severely influence the result . The low birefringence in the central area of the image is probably caused by a steeper orientation of the muscle fibers in this area. Apart from a different surface relief, no difference is observed between left and right hand parts of the image. Especially the amount of birefringence, indicated by the oscillation length, is approximately equal. The map showing the optic axis orientation (fig. 4c) shows information complementary to that of the retardation image. Clearly, different orientations of fast optic axis can be observed in the right and left hand parts of the tomogram, respectively, indicating different orientations of muscle fibers (the 90° color change in axial direction at depth positions where the retardation (fig. 4b) passes the 90° and 180° values are caused by the data processing algorithm and do not indicate layers of different orientation).
While intensity and birefringence measurements by OCT have been reported in several previous papers, only two reports on optic axis determination have been presented [10,13]. Only one of these papers  reports on explicit measurement of optic axis, however, this method requires three subsequent A-scans per transverse measurement location with different input polarization states. While this method was demonstrated successfully for optic axis orientation on a single location in a technical object (a Berek’s compensator), no application of spatially resolved axis measurement or in biological tissue has been reported. We have developed a new method of PS-OCT that allows to measure and image backscattered intensity, birefringence, and fast optic axis orientation simultaneously with only one single A-scan per transverse measurement position. We have demonstrated our technique in the case of a Berek’s compensator, and, for the first time in PS-OCT imaging, for recording transversally resolved axis orientation in a scattering tissue.
Our measurements revealed a systematic deviation between measured and expected optic axis orientation. The mean absolute deviation was ~9°, with a systematic decrease of the deviation from θ=-90° to θ=+90°. Similar systematic deviations were found in  where they were attributed to incorrect calibration of the Berek’s compensator. We assume a similar reason. Another contribution to the offset might originate from imperfect polarization optics (see below). The latter deviation, caused by the instrument, can be corrected for by calibration.
The interpretation of axis orientation maps as that of fig. 4c deserves some care:
(i) Very close to the entrance surface of the sample beam, a thin layer (~50–100 µm) is observed whose color indicates an optic axis orientation of near 90°. This is an artifact caused by imperfect polarization optics. Close to the surface, the total retardation is still very low, ideally yielding no signal in channel 2 (because of crossed polarization states). However, the interferometer beam splitter is not perfectly non-polarizing but causes a small retardance (~3°) in the sample beam (cf. fig. 2a). This causes a small but measurable residual signal in channel 2 which is independent from the sample. The phase difference of this residual signal to that of channel 1 is constant and corresponds to an optic axis orientation of 73°. Once a signal of sufficient intensity is built up in channel 2 (caused by increased retardation with depth), the influence of this residual signal vanishes.
(ii) Equations (4) and (9), which were used to derive the optic axis orientation as a function of ΔΦ, are based on the assumption that the sample consists of only one birefringent layer of constant axis orientation θ. If the axis orientation changes with depth, an accumulated effect occurs, yielding a deviation of the computed axis orientation from the true value. Therefore, the true axis orientation can only be determined for the first birefringent layer in the sample. This layer’s axis orientation can be mapped in transversal direction. If a second birefringent layer of different axis orientation is stacked beneath the first, its boundary can be recognized by a color change, however, its orientation cannot be exactly determined by the present algorithm. A more sophisticated method employing the birefringence information derived from the first layer has to be used.
A possible disadvantage of our method is that it would be more difficult to implement in fiber optics since the polarization state of the light has to be maintained in the sample arm. The main advantage of our method is that only a single A-scan per transverse location is required. This can reduce measurement time and sensitivity to sample movements. Furthermore, the unambiguous range of optic axis determination of our method (-90° to +90°) is twice as large as that described in  (-45° to +45°).
We have demonstrated our technique for imaging of birefringence and axis orientation in myocardial tissue. In this case, as well as in other types of muscle tissue, the axis orientation images provide additional information on the orientation of muscle fibers. Our technique is of special interest for tissues consisting of only one birefringent layer. Such tissues can be found, e.g., in the eye. The cornea is known to be birefringent, the optic axis orientation varies between individuals. The knowledge of this parameter might improve nerve fiber polarimetry for glaucoma diagnostics . In the retina, the orientation of the nerve fiber bundles around the optic disk contains information about the retinal region its nerve fibers are connected to. Imaging of nerve fiber loss as a function of nerve fiber orientation might provide additional information on the process and degree of glaucomatous damage.
The authors wish to thank Mr. H. Sattmann and Mr. L. Schachinger for technical assistance. Financial assistance from the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (FWF grant P14103-MED) is acknowledged.
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