Complex polarization ratio to determine polarization properties of anisotropic tissue using polarization-sensitive optical coherence tomography

Abstract

Complex polarization ratio (CPR) in materials with birefringence and biattenuance is shown as a logarithmic spiral in the complex plane. A multi-state Levenberg-Marquardt nonlinear fitting algorithm using the CPR trajectory collected by polarization sensitive optical coherence tomography (PS-OCT) was developed to determine polarization properties of an anisotropic scattering medium. The Levenberg-Marquardt nonlinear fitting algorithm using the CPR trajectory is verified using simulated PS-OCT data with speckle noise. Birefringence and biattenuance of a birefringent film, ex-vivo rodent tail tendon and in-vivo primate retinal nerve fiber layer were determined using measured CPR trajectories and the Levenberg-Marquardt nonlinear fitting algorithm.

Figures

Fig. 1

(a) Assignment of polarization states in the (h, v) basis Cartesian complex plane (L = 0°: Linearly horizontal, L = 45°: Linearly 45°, L = –45°: Linearly –45°, R.C.: right circular, L.C.: left circular polarization states) (b) Loci of polarization states of constant relative magnitude (| Cvh |=| Ev | / | Eh |) and phase (∠Cvh=θv−θh). (c) Relationship between (h, v) basis complex plane and Poincaré sphere. The Poincaré sphere of unit diameter is transformed to the complex plane by stereographic projection.

Fig. 2

CPR trajectories in the complex plane. (a) Trajectory with only phase retardation (δ(z) = 360°) uniformly rotates around the origin (black dot) (b) Trajectory with phase retardation (δ(z) = 360°) and amplitude attenuation (ε(z) = 36°) is a logarithmic spiral converging toward the origin. (Red and blue dots represent the first and last CPRs, respectively).

Fig. 3

Trajectories of CPRs in the (h, v) and (x, y) basis complex planes. (a) Trajectory with double-pass phase retardation (2δ(z) = 360°) and an optic axis (cvh _ oa = 0.3exp(j45°), black dot) in the (h, v) basis complex plane (b) Trajectory in the (x, y) basis complex plane. The optic axis (cyx _ oa) in (x, y) basis complex plane is zero and at the origin of trajectory (Red and blue dots represent the first and last CPRs, respectively).

Fig. 4

Multi-state trajectories of CPRs in the (h, v) and (x, y) basis complex planes (a) Trajectories with 2δ(z) = 60°, 2ε(z) = 6.0° and an optic axis (cvh _ oa = 0.3exp(j45°), black dot) in the (h, v) basis complex plane (b) Trajectories in the (x, y) basis complex plane. Identical δ(z) and ε(z) are visually observed (Red and blue dots represent the first and last CPRs, respectively).

Fig. 5

CPRs of simulated PS-OCT data in the (h, v) basis complex plane. The CPR of optic axis (black dot) and noise-free polarization arcs (black trajectories) were estimated from speckle-noise CPRs (colored trajectories)(Red dot behind the black dot represents true CPR of optic axis).

Fig. 6

Trajectory of CPRs by ex-vivo rodent tail tendon specimen in the (h, v) basis complex plane. The CPR of optic axis (black dot) and noise-free polarization arcs (black trajectory) are estimated from speckle-noise CPRs (red trajectory) (Red and blue dots represents the first and last CPRs, respectively).

Fig. 7

Multi-states trajectories of CPRs by in-vivo primate RNFL in the (h, v) basis complex planes (a) Speckle noise-corrupted and fitted (black) trajectories in an inferior region (thickness z = 150.0μm). Phase retardation is δRNFL(z = 150.0μm) = 28.2° (b) Speckle noise-corrupted and fitted (black) trajectories in a nasal region (thickness z = 53.5μm). Phase retardation is δRNFL(z = 53.5μm) = 2.78° (Black dots represents the CPR of optic axis).

Fig. 8

Relative processing times between the two nonlinear fitting algorithms using CPRs and Stokes vectors. The relative processing times were measured by 100 independent estimate sets using simulated PS-OCT data (a) Relative processing times with different phase retardation (δ(z) = 10°, 30°, 50°, 70° and 90°) at a high speckle noise with standard deviation (σ = 5°) (b) Difference of relative processing times computed from (a) (c) Relative processing times with speckle noise (σ = 1°, 2°, 3°, 4° and 5°) at a fixed phase retardation (δ(z) = 30°) (d) Difference of relative processing times computed from (c).

Fig. 9

Estimated retardation and 95% confidence intervals by simulated PS-OCT data in two Levenberg-Marquardt nonlinear fitting algorithms using CPRs and Stokes vectors (a) Confidence intervals with different phase retardation (δ(z) = 10°, 30°, 50°, 70° and 90°) at a fixed optic axis and high speckle noise with standard deviation (σ = 5°) (b) Estimated retardation and 95% confidence intervals (green error bar) from (a) (c) Confidence intervals with speckle noise (σ = 1°, 2°, 3°, 4° and 5°) at a true phase retardation (δ(z) = 30°) (d) Estimated retardation and 95% confidence intervals (green error bar) from (c).