Ординатура / Офтальмология / Английские материалы / Handbook of Optical Coherence Tomography_Bouma, Tearney_2002

The Stokes vector measured in the laboratory frame can be transformed to a new frame, which we will call the sample frame, according to the matrix in Eq. (39).

where C ¼ cos 2; S ¼ sin 2 , with the angle of the optic axis with the horizontal and the retardance. Equation (40) is the Mueller matrix representation of the linear retarder described by the Jones matrix in Eq. (11), with ¼ k0z n. Upon specular reflection inside the sample, the Stokes parameters U and V change sign. After the return pass through the retarder, the Mueller matrix is given by

with ¼ 2k0z n. For example, when right circularly polarized light is incident on a sample with linear retardance, the reflected light polarization state can be defined by the product of the Stokes vector (1,0,0,1) and the matrix of Eq. (41). The reflected light Stokes vector is

Vcos 2k0z n

Using Eq. (39) to transform the reflected light Stokes vector into a rotated reference frame, we can show that the Q 0 parameter equals 0 if the rotation angle equals . Therefore, the angle of the optic axis is found by determining the angle of rotation of the reference frame that minimizes the amplitude of oscillations with depth in the Q parameter. This angle defines a rotation of the laboratory frame to a sample frame whose basis vectors are aligned with the optic axes of the sample.

9.2.8Depth-Resolved Imaging of Stokes Parameters

Rodent muscle was mounted in a chamber filled with saline solution and covered with a thin glass slip to avoid dehydration during measurements. Figure 5 shows images of the four Stokes parameters in the sample frame for right circularly polarized incident light. Several periods of normalized U and V, cycling back and forth between 1 and 1, are observed in muscle, indicating that the sample is birefringent, further demonstrated by the averages of the Stokes parameters over all depth profiles in Fig. 6a. To verify experimentally the orientation of the optic axis computed by rotating the reference frame so as to minimize the amplitude of oscillation in Q, three measurements were performed by replacing the QWP in the sample arm by a halfwave retarder. Light incident on the sample was prepared in three linear polarization

9.2.9Polarization Diversity Detection and Speckle Averaging

Figure 8 shows six images reconstructed from a single scan of rodent skin, 3 weeks after exposure to a 100 C brass rod for 20 s. The six images display, respectively, in the left column from top to bottom, the sum of the reflected intensity detected by the detectors, the vertically polarized reflected intensity (detector 1), and the horizontally polarized reflected intensity (detector 2), and in the right column from top to bottom, the normalized Stokes parameters Q, U, and V. The Stokes parameters were calculated as described previously and give the polarization state of light reflected from the sample before the return pass through the QWP. Images display the Stokes parameters normalized on the intensity and gray-scale-coded from 1 to 1. Contour lines indicating 1/3 and 1=3 values were calculated after low-pass filtering by convolving the images with a Gaussian filter of 4 4 pixels and overlaid with the original image. The scan was made from normal skin (left) into thermally damaged skin with scar formation (right).

Two interesting observations can be made. First, the images recorded by a single detector show dark areas indicating low tissue reflectivity that are completely absent from the summed image. These features are solely due to polarization changes in the tissue and are not correlated with tissue reflectivity. Images of the Stokes parameters show the polarization state changing from circular to linear and back as a function of increasing depth.

Figure 8 (a) OCT and (b) PS-OCT images generated from a single scan of rodent skin, 3 weeks postexposure to a 100 C brass rod for 20 s. Image size is 4 mm 1 mm. The six images display, respectively, in the left column from top to bottom the sum of the detected reflected intensity, the vertically polarized reflected intensity (detector 1), and the horizontally polarized reflected intensity (detector 2) gray-scale coded on a logarithmic scale, and, in the right column from top to bottom, the normalized Stokes parameters Q, U, and V gray-scale coded from 1 to 1. Incident light was right circularly polarized (V ¼ 1). The Stokes parameters Q, U, and V represent the polarization state reflected from the sample before the return pass through the quarter-wave plate. White lines are contours at 1/3 (white to gray transition) and 1=3 (gray to black transition) values. The scan was made from normal (left) into thermally damaged skin with scar formation (right). Punch biopsy and histological evaluation of the imaged location indicate that the banded structure in the lower right half of the summed intensity image is muscle tissue. (Reprinted from Ref. 17 with permission of the Institute of Electrical and Electronics Engineers, Inc.)

Second, the summed image has a smoother appearance, which suggests that speckle noise is averaged. Presently, speckle-averaging algorithms in OCT use four detectors recording light that has traveled over a slightly different path through the sample to the focal point and back [18]. Wave front distortions by tissue inhomogeneities create a different speckle pattern at each detector. However, implementation of this approach in a single-mode fiber is difficult. Because light has two degrees of freedom, at each spatial location two (partially) independent speckle patterns are present, one in each polarization channel. Summing the images recorded in orthogonal polarization channels provides an alternative approach to speckle averaging that is simple to implement in a single-mode fiber. The number of images with (partially) independent speckle patterns can be further increased by modulating the polarization incident on the sample over orthogonal states. Two additional images can be recorded, giving four interference signals available for speckle averaging, which would increase the signal-to-noise ratio (SNR) by a factor of 2. However, the actual increase in SNR that can be realized may be considerably less, because the speckle patterns in orthogonally polarized channels need not be completely independent due to the low-order scattering of OCT signals [19]. To summarize, PS-OCT is important not only for measuring birefringence but also for accurate interpretation of OCT images. Most fibrous structures in tissue (e.g., muscle, nerve fibers) are birefringent owing to their anisotropy. Single-detector OCT systems can generate images that show structural properties by a reduction in tissue reflectivity, solely due to polarization effects. In addition, polarization diversity detection and polarization modulation in the sample arm could be interesting approaches to speckle averaging in OCT systems based on single-mode fibers.

So far, the source light was assumed to be polarized. The presented analysis is easily extended to include unpolarized sources. An unpolarized source can be described by the addition of two orthogonally polarized sources that are mutually incoherent. The interference fringes at the detector(s) need to be analyzed separately for the two pure polarized states, and the total interference fringe pattern is given by the sum of the fringe patterns of the individual polarized states. An OCT system with an unpolarized source and a single detector does not imply polarization diversity detection. On the contrary, this system is even more sensitive to polarization effects than a system with a polarized source. Consider polarized source light incident on a birefringent sample acting as a linear retarder with its optic axis at 45 with respect to the incident light polarization axis. The polarization state of reflected light that has undergone a phase retardation is orthogonal to the incident polarization state. Because orthogonally polarized states cannot interfere, light from the sample and reference arms does not produce interference fringes. The same holds for each of the orthogonally polarized states in the decomposition of an unpolarized source into linear states at 45 and 45 to the optic axis. Therefore, for the unpolarized source no interference fringes will be detected either. Suppose now that the decomposition of the unpolarized source is chosen differently for the above-mentioned birefringent sample, such that the two orthogonal linearly polarized mutually incoherent states are along and perpendicular to the optic axis. Both orthogonal polarization states reflected from the sample are unaltered by the birefringence and will produce interference fringes with the reference arm light. However, the interference fringes for orthogonal polarization states are exactly out of phase and cancel after summation, and no interference fringe pattern is present at the detector. Thus, in this

example, the unpolarized source will not produce interference fringes regardless of the orientation of the optic axis (as is expected from symmetry arguments). In contrast, a polarized source would produce interference fringes if the polarization state is (partially) along the optic axis.

9.2.10Accuracy and Noise: Birefringence, Dichroism, and Scattering

Everett et al. [9] presented an analysis of the systematic error in the phase retardation due to background noise for the incoherent PS-OCT detection scheme described in Section 9.2.3. They showed that for phase retardations close to 0 or 90 the background noise on the detectors introduces a significant and systematic error of, e.g., 15 at a signal-to-noise ratio of 10 dB. The coherent detection scheme described in Section 9.2.5, which calculates the Stokes parameters, has better immunity to this systematic error. A closer look at Eqs. (25), (34), or (36) reveals that in the calculation of the Q parameter (also denoted by s1) the spectral density in one polarization channel is subtracted from the spectral density in the orthogonal polarization channel, thus eliminating constant background noise terms, and the U (s2) and V (s3) parameters are calculated from the cross-correlation between fringes in orthogonally polarized channels, eliminating autocorrelation noise. Noise will decrease the degree of polarization P, because it will be present as autocorrelation noise in the Stokes parameter I. The better noise immunity can be illustrated with the help of Fig. 6a, which shows the Stokes parameters for rodent muscle. In the incoherent detection scheme only V (s3 in the figure) is measured, and the error in the phase retardation is introduced by the decrease of the amplitude of oscillations with increasing depth. In the coherent detection scheme, the Stokes parameters Q, U, and V can be renormalized on P, restoring the amplitude of the oscillations and thus eliminating the systematic error.

Schoenenberger et al. [11] went further and also analyzed system errors introduced by the extinction ratio of polarizing optics and chromatic dependence of wave retarders, and errors due to dichroism, i.e., the differences in the absorption and scattering coefficients for polarized light in tissue. System errors can be kept small by careful design of the system with achromatic elements but can never be completely eliminated. Dichroism can possess a more serious problem when the results are interpreted as solely due to birefringence. However, Mueller matrix ellipsometric measurements have shown that the error due to dichroism in the eye is relatively small [20,21], and Fig. 6c shows that dichroism is of minor importance in rodent muscle. More research is necessary to determine the importance of dichroism in other types of tissue.

We believe that the dominant noise sources are related to multiple scattering and speckle. As argued earlier, multiple scattering will scramble the polarization mainly in a random manner. This offers some means to distinguish it from birefringence. Reported birefringence values for cornea, tendon, and muscle are on the order of n ¼ 10 3 [8,22–24] which will give a 90 phase retardation at a depth on the order of several hundreds of micrometers. Thus, birefringence-induced changes are relatively slow, and the Stokes parameters change according to the Mueller matrix of a linear retarder. However, an optic axis that varies with depth will give changes in the polarization state that will be difficult to distinguish from the randomness of

multiple scattering. Measurement of the full Mueller matrix of the sample by varying the incident light over four different polarization states, as recently demonstrated by Yao and Wang [25], will provide additional information that could aid the analysis of PS-OCT signals. More research is necessary on this complex problem.

Speckle introduces noise on the Stokes parameters by the large fluctuations in the interference fringes that could be uncorrelated in the orthogonal detection channels. Speckle-averaging techniques demonstrated by Schmitt et al. [18] will reduce this noise, as will averaging the Stokes parameters over distances greater than the coherence length. Averaging images with different input polarization states will reduce speckle noise also. Speckle noise remains one of the main problems that need to be addressed in OCT.

9.3IMAGING OF THERMAL DAMAGE WITH PS-OCT

9.3.1Introduction

Collagen is the dominant structure in human skin. Its birefringence has two components. Form birefringence results from ordered proteins surrounded by a ground substance with a different refractive index; intrinsic birefringence results from chemical groups arranged in an ordered configuration with anisotropic optical retardance. Thermal damage, which starts to take place between 56 C and 65 C, reduces both the form and intrinsic birefringence by changing the collagen from a rodlike to a random coil structure. The reduction of collagen birefringence can be used to determine burn injury, because partial loss of birefringence is known to be an indication of tissue thermal damage [26].

9.3.2Laser-Irradiated Bovine Tendon

To demonstrate an application of PS-OCT, we present 1 mm and 1.5 mm wide by 700 m deep images of bovine tendon birefringence before and after pulsed laser irradiation. For comparison, reflection OCT images are shown that were formed by the sum of detected signals in orthogonal polarization channels. A detailed description of the experimental configuration used to obtain Figs. 9–12 is given in de Boer et al. [8]. Measurements on 1 cm 2 cm samples at least 1 cm thick were done within 48 h of bovine death. In Fig. 9 conventional and birefringence images of fresh bovine tendon are shown. The banded structure in Fig. 9b, indicative of birefringence, is clearly visible up to a physical depth of 700 m, whereas this structure is completely absent from the conventional reflection image in Fig. 9a. This shows that the banded structure in the birefringence image is not an anatomical artifact in the tendon but is due solely to its birefringence. By measuring the optical versus physical thickness of a thin slice [28], we found the average refractive index of the tendon to be n ¼ 1:42 0:03. We determined the birefringence by the average distance between the first and second dark bands from the top of Fig. 9b over the full width (100 lateral scans). The average distance z ¼ 116 13 m corresponded to a polarization rotation ’ ¼ k0z ¼ in Eq. (19). The experimentally determined birefringence ¼ ð3:7 0:4Þ 10 3 of bovine tendon (predominantly type I collagen) is in agreement with reported values of ð3:0 0:6Þ 10 3 [24] and (2.8–3:0Þ 10 3 [22,23] for type I collagen. Fitting e 2z= at z ¼ 150–600 m depth to the total backscattered intensity IT [Eq. (18)] in the sample, averaged over the image in Fig. 9a (100 lateral scans) gave

Figure 9 Images of fresh bovine tendon 1 mm wide by 700 m deep, constructed from the same measurement. (a) Conventional reflection image generated by computing 10 log½IT &. The gray scale to the right gives the magnitude of the signals. (b) Birefringence image generated by computing the phase retardation ’ in Eq. (19). The gray scale at the right gives the angle ’. The banded structure, indicative of the birefringence, is clearly visible. Each pixel represents a 10 m 10 m area. The dynamic range within the image was 48 dB. (Reprinted from Ref. 27 with permission of the Society of Photo-Optical Instrumentation Engineers.)

0:2 mm. Decay of total backscattered light intensity with depth depends on several factors, among them attenuation of the coherent beam by scattering and the geometry of the collection optics. In Fig. 10 conventional OCT and birefringence images of laser-irradiated bovine tendon are presented. A decrease in the birefringence at the center of the irradiation zone, extending into the tendon over the full depth of the image (700 m), is clearly seen. Furthermore, the direction of incoming laser light (from the upper left corner, at an angle of 35 with the normal to the surface) is observed. The surface temperature of the tendon was monitored during laser irradiation by infrared radiometry. For comparison, Fig. 10a shows an OCT image of the total backscattered intensity IT . Although less backscattered light from

Figure 10 Images of fresh bovine tendon 1 mm wide by 700 m deep, constructed from the same measurement. (a) Conventional reflection image generated by computing 10 log½IT &. The gray scale to the right gives the magnitude of the signals. (b) Birefringence image generated by computing the phase retardation ’ in Eq. (19). The gray scale at the right gives the angle ’. The bovine tendon was exposed to three consecutive 1 J, 150 s laser pulses ( ¼ 1:32 m) spaced by 10 ms, incident from the upper left at an angle of 35 with respect to the surface normal. The beam diameter was 2 mm. Initial surface temperature after laser irradiation was 77 C, dropping to 61 C after 0.25 s. The displacement of the banded structure in the image indicates the loss of birefringence due to the thermal damage in the irradiated zone. Each pixel represents a 10 m 10 m area. (Reprinted from Ref. 27 with permission of the Society of Photo-Optical Instrumentation Engineers.)

the irradiated area can be observed, the polarization-sensitive image (Fig. 10b) reveals important structural information not evident in Fig. 10a. In Figs. 11 and 12, conventional and birefringence images 1.5 mm wide by 700 m deep of laserirradiated tendon with and without surface cooling are presented. Because loss of birefringence is attributed to thermal damage, comparison of the two figures show the effect of surface cooling on laser-mediated thermal damage. Application of cooling clearly reduces the birefringence loss near the surface in Fig. 12 compared to Fig. 11.

9.3.3Slowly Heated Porcine Tendon

To investigate the effect of temperature on collagen birefringence, porcine tendon was mounted in a Rose chamber filled with 5% saline solution. A thermocouple monitored the temperature inside the chamber. A heating device was mounted outside the chamber that could increase the temperature inside to 77 C. Lateral scan velocity v (x direction) was 100 m=s, axial number of scans was 200 at 2 m increments, giving an image size of 200 m 400 m and a pixel size of 1 m 2 m. Figure 13 shows the OCT and PS-OCT images of slowly heated tendon at 25, 45, 55, 60, 70, and 77 C, respectively. Image acquisition was started after the Rose chamber had reached the target temperature (15–30 min between consecutive scans). The last scan was taken after the sample was heated for 5 h at 77 C.

The images at 25 C, 45 C, and 55 C (Figs. 13a, 13b, and 13c, respectively) showed no change in the birefringence. The more closely spaced the banded structure, the greater the birefringence, because the accumulated phase retardation is the product of depth z and birefringence n. At 60 C (Fig. 13d), a reduction in the birefringence can be observed. The tendon shrank considerably along the axis of the fibers during the scan, which contributed to motion artifacts in the image. At 70 C (Fig. 13e) the birefringence was reduced further, and after 5 h at 77 C (Fig. 13f) the

Figure 11 Images of fresh bovine tendon 1.5 mm wide by 700 m deep, constructed from the same measurement. (a) Conventional reflection image generated by computing 10 log½IT &. The gray scale to the right gives the magnitude of the signals. (b) Birefringence image generated by computing the phase retardation ’ in Eq. (19). The gray scale at the right gives the angle ’. The bovine tendon was exposed to three consecutive 1.4 J, 150 s laser pulses ( ¼ 1:32 m) spaced by 10 ms, incident from the upper left at an angle of 35 with respect to the surface normal. The beam diameter was 2 mm. Initial surface temperature after laser irradiation was 88 C, dropping to 70 C after 0.25 s. The displacement of the banded structure in the image indicates the loss of birefringence due to thermal damage in the irradiated zone. Each pixel represents a 10 m 10 m area. (Reprinted from Ref. 27 with permission of the Society of Photo-Optical Instrumentation Engineers.)