Polarization-sensitive optical coherence tomography (PS-OCT) is a non-invasive imaging modality that provides depth-resolved polarimetric information with high-resolution in anisotropic tissues [1
]. Measurement of depth-resolved polarimetric information by PS-OCT is widely investigated for diagnosis of various pathological conditions or trauma by observing the variation of polarimetric properties between normal and abnormal tissues [3
In general PS-OCT, horizontal
interference fringe magnitudes and relative phase
as a function of depth (z
) are recorded by controlling polarization elements in source, sample and reference paths of time- or frequency-domain OCT instrumentation [9
]. Diverse analytic techniques have been used to extract polarimetric information such as depth-resolved phase retardation (δ(z)
), birefringence (Δn
), amplitude attenuation (ε(z)
) (or diattenuation (d(z)
)), biattenuance (Δχ
), and optic axis orientation (α
]. In early PS-OCT studies, the phase retardation was computed from the arc tangent of the ratio of the horizontal and vertical fringe magnitudes
]. Although this technique was simple and appropriate for a coarse two-dimensional image, polarimetric values in a target region (e.g. retinal nerve fiber layer in retina) were not accurately computed due to background noise including speckle [9
A Mueller matrix formalism was applied to time- and frequency-domain PS-OCT to obtain the polarimetric properties of biological tissue [12
]. A two-dimensional depth-resolved 4 × 4 Mueller matrix of the tissue sample was measured by open-air and fiber-based PS-OCT systems. Birefringence, diattenuation and optic axis orientation were extracted from the measured Mueller matrix. Although the Mueller matrix can provide the complete polarimetric transformation of the tissue specimen, each element of the Mueller matrix is not easily interpreted.
A Jones matrix formalism was applied to analyze polarization state of light backscattered from tissue and recorded by PS-OCT [15
]. Light propagation into many components (including tissue specimen) was described by products of Jones matrices in time-domain fiber-based PS-OCT. Birefringence, diattenuation and optic axis orientation of the samples were determined by calculation of the Jones matrices [15
]. Similar to Mueller matrix images using PS-OCT, two-dimensional depth-resolved Jones matrix images were demonstrated and local polarization properties such as phase retardation, diattenuation, and optic axis orientation were computed in open-air [16
] and fiber-based [18
] Fourier-domain PS-OCT.
Stokes parameters are also utilized to analyze polarimetric properties of biological tissue using PS-OCT [20
]. In early studies using Stokes parameters, two-dimensional images of depth-resolved Stokes parameters were generated and compared to intensity images for identifying tissue characteristics [20
]. As the Stokes parameters are represented geometrically as a three-dimensional vector (Stokes vector), polarization properties of tissue in PS-OCT can be visually interpreted compared with other polarization analysis techniques such as Jones vectors. Depth-resolved Stokes vectors on the Poincaré sphere allow visualization of the polarization state of light backscattered by a tissue specimen. Birefringence and optic axis orientation were obtained by vector calculation using the depth-resolved Stokes vectors in a fiber-based PS-OCT instrument [22
]. Trajectory of the depth-resolved Stokes vectors on the Poincaré sphere was theoretically and experimentally investigated corresponding to light propagation in anisotropic tissues. Numerical expressions of the trajectory and associated differential geometry were derived for materials with arbitrary birefringence, biattenuance and optic axis orientation [24
]. A Levenberg-Marquardt nonlinear fitting algorithm with multi-incident polarization states of light were applied to analyze the depth-resolved Stokes parameters of backscattered light from tissue specimens recorded by an open-air PS-OCT instrument. Tissue birefringence, biattenuance, and optic axis orientation were determined by estimating Stokes parameters from speckle-noise corrupted PS-OCT data. Multiple incident polarization states of light were used to suppress noise and increase polarimetric signal to noise ratio (PSNR) of PS-OCT data [25
Complex valued analytic signals are utilized to solve a variety of problems arising in science and engineering. Even though real numbers are natural for representing recorded data, complex numbers provide a useful approach to analyze many engineering problems. In control theory, systems are transformed from time-domain to frequency-domain, and vice versa using Laplace or Z-transforms. Characteristics of systems are analyzed by poles and zeros using signals represented as complex numbers in the complex plane [28
]. In other fields such as fluid dynamics, quantum mechanics, relativity and applied mathematics [29
], complex numbers are routinely employed to represent real phenomena.
We demonstrate a new approach which analyzes the trajectory of the complex polarization ratio (CPR) in the complex plane of PS-OCT data to determine polarimetric properties of biological tissues. The technique using CPR combines the advantages of matrix formalisms (Jones and Mueller matrices) and Stokes parameters. Similar to Jones and Mueller matrices, CPR can mathematically express the polarization state of light backscattered from tissue as a single complex number. Computations relating to the polarization state of light are simplified, however, because the CPR is a single number. Similar to Stokes vectors on the Poincaré sphere, CPR may be displayed geometrically to visualize the polarization state of light backscattered from tissue on a complex plane. Moreover, two-dimensional display of CPR on the flat complex plane is more convenient than three-dimensional display of Stokes vectors on the Poincaré sphere.
We utilized a Levenberg-Marquardt nonlinear fitting algorithm to determine the polarization properties of a tissue specimen from CPR trajectories in the complex plane. The algorithm was verified using CPR trajectories of simulated PS-OCT data corrupted with polarimetric speckle noise. In addition, the CPR algorithm was applied to PS-OCT data recorded from a birefringent film, ex-vivo rodent tail tendon and in-vivo primate retinal nerve fiber layer (RNFL).