Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography

Christoph K. Hitzenberger; Erich Götzinger; Markus Sticker; Michael Pircher; Adolf F. Fercher

doi:10.1364/OE.9.000780

Optics Express
Vol. 9,
Issue 13,
pp. 780-790
(2001)
•https://doi.org/10.1364/OE.9.000780

Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography

Christoph K. Hitzenberger, Erich Götzinger, Markus Sticker, Michael Pircher, and Adolf F. Fercher

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Christoph K. Hitzenberger, Erich Götzinger, Markus Sticker, Michael Pircher, and Adolf F. Fercher, "Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography," Opt. Express 9, 780-790 (2001)

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Abstract

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.

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Fig. 1. Sketch of instrument. BS, beam splitter; QWP, quarter wave plate.

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Fig. 2. Measured versus set retardation. (a) Plot of measured retardation (data points) and standard deviation (error bars) as a function of set retardation for a fast axis orientation of 40°. For better comparison, the expected (set) retardation value is indicated as solid line. (b) Polar plot of measured retardation versus set retardation for several fixed values of fast axis orientation (indicated along circumference of the plot). The color of a data point indicates the set value of retardation, the radial distance from the half-circle center indicates the corresponding measured value. Ideally, the data points should lie on the corresponding half-circle.

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Fig. 3. Measured versus set fast axis orientation. (a) Plot of measured axis orientation (data points) and standard deviation (error bars) as a function of set fast axis for a retardation of 30°. For better comparison, the expected (set) axis orientation is indicated as solid line. (b) Polar plot of measured axis orientation versus set fast axis for several fixed values of retardation (indicated along circumference of the plot). The color of a data point indicates the set value of axis orientation, the radial distance from the quarter-circle center indicates the corresponding measured value. Ideally, the data points should lie on the corresponding quarter-circle.

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Fig. 4. OCT images recorded in a chicken myocardium in vitro. Dimensions are indicated in mm (the ordinate shows optical distance). (a) Intensity image (color bar: logarithmic intensity scale); (b) phase retardation image (color bar: retardation [deg]); (c) image of fast axis distribution; interpretation: see text (color bar: axis orientation [deg]).

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Equations (10)

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\vec{E} = E_{0} (\begin{matrix} 0 \\ 1 \end{matrix})

M (δ, θ) = [\begin{matrix} \cos^{2} (θ) + \sin^{2} (θ) \cdot \exp (- i δ) & \cos (θ) \cdot \sin (θ) \cdot (1 - \exp (- i δ)) \\ \cos (θ) \cdot \sin (θ) \cdot (1 - \exp (- i δ)) & \cos^{2} (θ) \cdot \exp (- i δ) + \sin^{2} (θ) \end{matrix}]

E_{r} = \frac{1}{2} M_{QWP 2} \cdot M_{QWP 2} \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) = \frac{1}{2 \sqrt{2}} (\begin{matrix} 1 \\ - 1 \end{matrix}) .

E_{s} = \frac{1}{2} M_{QWP 1} \cdot M_{sample} (δ, θ) \cdot \sqrt{R} \cdot M_{sample} (δ, θ) \cdot M_{QWP 1} \cdot (\begin{matrix} 0 \\ 1 \end{matrix})

= \frac{\sqrt{R}}{2} (\begin{matrix} \cos (δ) \exp (- i δ) \\ \sin (δ) \exp (i (π - δ - 2 θ)) \end{matrix})

I_{k} (z) = I_{r, k} + I_{s, k} + 2 \sqrt{I_{r, k} I_{s, k}} \cdot ∣ γ (z - z_{0}) ∣ \cdot \cos (Φ_{k}) .

{\tilde{A}}_{k} (z) = I_{k} (z) + i \cdot H {I_{k} (z)} = A_{k} (z) \cdot \exp [i \cdot Φ_{k} (z)]

R (z) \sim A_{1} {(z)}^{2} + A_{2} {(z)}^{2}

δ (z) = \arctan (\frac{A_{2} (z)}{A_{1} (z)}) .

θ = (180^{o} - △ Φ) / 2 .

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Equations (10)

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