Effect of Relative Phase Delay Error of Quarter-Wave Plate on Measurement Accuracy of Birefringence Parameters in Polarization-Sensitive Full-Field Optical Coherence Tomography System

Shuanglu Zou; Wanrong Gao; Siyu Liu

doi:10.3788/LOP202259.1617003

Journals >Laser & Optoelectronics Progress >Volume 59 >Issue 16 >Page 1617003 > Article

Laser & Optoelectronics Progress
Vol. 59, Issue 16, 1617003 (2022)

Effect of Relative Phase Delay Error of Quarter-Wave Plate on Measurement Accuracy of Birefringence Parameters in Polarization-Sensitive Full-Field Optical Coherence Tomography System

Shuanglu Zou, Wanrong Gao^*, and Siyu Liu

Author Affiliations

School of Electronic and Optical Engineering, Nanjing University of Science & Technology, Nanjing 210094, Jiangsu , China

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DOI: 10.3788/LOP202259.1617003 Cite this Article Set citation alerts

Shuanglu Zou, Wanrong Gao, Siyu Liu. Effect of Relative Phase Delay Error of Quarter-Wave Plate on Measurement Accuracy of Birefringence Parameters in Polarization-Sensitive Full-Field Optical Coherence Tomography System[J]. Laser & Optoelectronics Progress, 2022, 59(16): 1617003 Copy Citation Text

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Abstract

Polarization-sensitive full-field optical coherence tomography (PS-FFOCT) system can produce the depth-resolved en face images of the birefringence and diattenuation of biological tissue samples. One of the important problems in the birefringence measurement of biological tissue using PS-FFOCT system is the analysis of the effects of relative phase delay error of quarter-wave plate on the measurement accuracy. A PS-FFOCT system was set up in this study. First, the Jones matrix model of the system was derived, and the effects of relative phase delay errors of quarter-wave plate on the measurement accuracy of birefringence were then analyzed. Finally, the results simulated using Matlab software were presented to reveal the specific behaviors. The research results can help the design of PS-FFOCT system and the accurate measurement of birefringence characteristics of biological tissues.

Keywords

birefringence characteristic polarization optical coherence tomography quarter-wave plate relative phase delay error

[\begin{matrix} I^{⊥} \\ I^{‖} \end{matrix}] = [\begin{matrix} {\bar{I}}^{⊥} + A c o s φ c o s ϕ \\ {\bar{I}}^{‖} + A s i n φ c o s (ϕ + 2 β) \end{matrix}]

，（1）

\bar{ϕ_{s a m} - ϕ_{r e f}} = Ψ s i n (ω t + θ)

，（2）

[\begin{matrix} I^{⊥} \\ I^{‖} \end{matrix}] = [\begin{matrix} {\bar{I}}^{⊥} + A c o s φ c o s [ϕ + Ψ s i n (ω t + θ)] \\ {\bar{I}}^{‖} + A s i n φ c o s [ϕ + 2 β + Ψ s i n (ω t + θ)] \end{matrix}]

。（3）

[\begin{matrix} {E^{⊥}}_{k} \\ {E^{‖}}_{k} \end{matrix}] = [\begin{matrix} \int_{(k - 1) T / 4}^{k T / 4} \{{\bar{I}}^{⊥} + A c o s φ c o s [ϕ + Ψ s i n (ω t + θ)]\} d t \\ \int_{(k - 1) T / 4}^{k T / 4} \{{\bar{I}}^{‖} + A s i n φ c o s [ϕ + 2 β + Ψ s i n (ω t + θ)]\} d t \end{matrix}]

，（4）

\{\begin{array}{l} Σ_{S}^{⊥} = - E_{1}^{⊥} + E_{2}^{⊥} + E_{3}^{⊥} - E_{4}^{⊥} = (4 T / π) Γ_{S} A c o s φ s i n ϕ \\ Σ_{C}^{⊥} = - E_{1}^{⊥} + E_{2}^{⊥} - E_{3}^{⊥} + E_{4}^{⊥} = (4 T / π) Γ_{C} A c o s φ c o s ϕ \\ Σ_{S}^{⊥} = - E_{1}^{‖} + E_{2}^{‖} + E_{3}^{‖} - E_{4}^{‖} = (4 T / π) Γ_{S} A s i n φ s i n (ϕ + 2 β) \\ Σ_{C}^{‖} = - E_{1}^{‖} + E_{2}^{‖} - E_{3}^{‖} + E_{4}^{‖} = (4 T / π) Γ_{C} A s i n φ c o s (ϕ + 2 β) \end{array}

，（5）

\{\begin{array}{l} Γ_{S} = \sum_{n = 0}^{+ \infty} {(- 1)}^{n} \frac{J_{2 n + 1} (Ψ)}{2 n + 1} s i n [(2 n + 1) θ] \\ Γ_{C} = \sum_{n = 0}^{+ \infty} \frac{J_{4 n + 2} (Ψ)}{2 n + 1} s i n [2 (2 n + 1) θ] \end{array}

。（6）

t a n^{2} φ = \frac{(Σ_{S}^{⊥})^{2} + (Σ_{C}^{⊥})^{2}}{(Σ_{S}^{⊥})^{2} + (Σ_{C}^{‖})^{2}}

。（7）

S = (Σ_{S}^{‖})^{2} + (Σ_{C}^{‖})^{2} + (Σ_{S}^{⊥})^{2} + (Σ_{C}^{⊥})^{2} = {[(\frac{4 T}{π}) Γ A]}^{2}

，（8）

Δ z = \frac{2 \sqrt[]{2} l n 2}{n π} \times \frac{λ_{0}^{2}}{Δ λ}

，（9）

Δ z = \frac{0.5 λ_{0}}{N A}

，（10）

\begin{array}{l} E = E_{x} x_{0} + E_{y} y_{0} = x_{0} a_{1} e x p [i (α_{1} - ω t)] + \\ y_{0} a_{2} e x p [i (α_{2} - ω t)] \end{array}

，（11）

\{\begin{array}{l} E_{x} = a_{1} e x p [i (α_{1} - ω t)] \\ E_{y} = a_{2} e x p [i (α_{2} - ω t)] \end{array}

。（12）

\{\begin{array}{l} E_{x} = a_{1} e x p (i α_{1}) \\ E_{y} = a_{2} e x p (i α_{2}) \end{array}

，（13）

E = [\begin{matrix} E_{x} \\ E_{y} \end{matrix}] = [\begin{matrix} a_{1} e x p (i α_{1}) \\ a_{2} e x p (i α_{2}) \end{matrix}]

。（14）

\{\begin{array}{l} A_{1} = g_{11} A_{1} + g_{12} B_{1} \\ B_{2} = g_{21} A_{1} + g_{22} B_{1} \end{array}

，（15）

[\begin{matrix} A_{2} \\ B_{2} \end{matrix}] = [\begin{matrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{matrix}] [\begin{matrix} A_{1} \\ B_{1} \end{matrix}]

。（16）

E_{t} = J E_{i}

，（17）

J = [\begin{matrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{matrix}]

。（18）

E_{t} = J_{N} \dots J_{2} J_{1} E_{i}

。（19）

J_{S} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]

，（20）

J_{P} = [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]

。（21）

J_{Q} = e x p (i \frac{π}{4}) [\begin{matrix} 1 & 0 \\ 0 & - i \end{matrix}]

。（22）

J_{θ} = [\begin{matrix} c o s θ & s i n θ \\ - s i n θ & c o s θ \end{matrix}]

。（23）

J {}_{Q_{θ}}= J_{- θ} J_{Q} J_{θ} = e x p (i \frac{π}{4}) [\begin{matrix} c o s^{2} θ - i s i n^{2} θ & c o s θ s i n [θ (1 + i)] \\ c o s θ s i n [θ (1 + i)] & s i n^{2} θ - i c o s^{2} θ \end{matrix}]

。（24）

J_{Q_{45}} = \frac{\sqrt[]{2}}{2} [\begin{matrix} 1 & i \\ i & 1 \end{matrix}]

。（25）

J_{R} = r [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]

，（26）

J_{B} = r [\begin{matrix} e x p [- i (\frac{ϕ_{B}}{2})] & 0 \\ 0 & e x p [+ i (\frac{ϕ_{B}}{2})] \end{matrix}]

，（27）

J_{B_{β}} = r \times [\begin{matrix} \begin{array}{l} c o s^{2} β e x p [- i (\frac{ϕ_{B}}{2})] + \\ s i n^{2} β e x p [+ i (\frac{ϕ_{B}}{2})] \end{array} & - i s i n 2 β s i n \frac{ϕ_{B}}{2} \\ - i s i n 2 β s i n \frac{ϕ_{B}}{2} & \begin{array}{l} c o s^{2} β e x p [+ i (\frac{ϕ_{B}}{2})] + \\ s i n^{2} β e x p [- i (\frac{ϕ_{B}}{2})] \end{array} \end{matrix}]

。（28）

J_{b s R} = r_{b s} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

，（29）

J_{b s T} = t_{b s} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

，（30）

J_{r e f} = J_{b s T} J_{Q_{θ}} J {}_{R}J J_{Q_{θ}} J_{b s R}

，（31）

J_{r e f} = [\begin{matrix} c o s 2 θ & s i n 2 θ \\ s i n 2 θ & - c o s 2 θ \end{matrix}] r_{b s} t_{b s} r_{r e f} e x p (i φ_{r e f})

，（32）

J_{B_{β}} = r \times [\begin{matrix} \begin{array}{l} c o s^{2} β e x p [- i (\frac{ϕ_{B}}{2})] + \\ s i n^{2} β e x p [+ i (\frac{ϕ_{B}}{2})] \end{array} & - i s i n 2 β s i n \frac{ϕ_{B}}{2} \\ - i s i n 2 β s i n \frac{ϕ_{B}}{2} & \begin{array}{l} c o s^{2} β e x p [+ i (\frac{ϕ_{B}}{2})] + \\ s i n^{2} β e x p [- i (\frac{ϕ_{B}}{2})] \end{array} \end{matrix}]

。（33）

J_{s a m} = J_{b s R} J_{Q_{θ}} J_{S_{β}} J_{Q_{θ}} J_{b s T}

。（34）

J_{s a m} = i [\begin{matrix} - s i n ϕ {}_{B}e x p (2 i β) & c o s ϕ {}_{B} \\ c o s ϕ {}_{B} & s i n ϕ_{B} e x p (- 2 i β) \end{matrix}] t_{b s} b_{b s} r e x p (i φ_{s a m})

。（35）

E_{P} = \sqrt[]{I} [\begin{matrix} 0 \\ 1 \end{matrix}]

。（36）

{E_{d e t}}^{(P)} = J_{P} (J_{r e f} + J_{s a m}) E_{P}

，（37）

{E_{d e t}}^{(P)} = \frac{\sqrt[]{I}}{2} [- r_{r e f} c o s 2 θ e x p (i φ_{r e f}) + i r_{s a m} s i n ϕ_{B} e x p (- 2 i β) e x p (i φ_{s a m})]

，（38）

I_{d e t}^{(P)} = \frac{I}{4} [r_{r e f}^{2} c o s^{2} 2 θ + r_{s a m}^{2} s i n^{2} ϕ_{B} + 2 r_{r e f} r_{s a m} c o s 2 θ s i n ϕ_{B} s i n (φ - 2 β)]

，（39）

E_{d e t}^{(S)} = \frac{\sqrt[]{I}}{2} [r_{r e f} s i n 2 θ e x p (i φ_{r e f}) + i r_{s a m} c o s ϕ_{B} e x p (i φ_{s a m})]

，（40）

I_{d e t}^{(S)} = \frac{I}{4} [r_{r e f}^{2} s i n^{2} 2 θ + r_{s a m}^{2} c o s^{2} ϕ_{B} - 2 r_{r e f} r_{s a m} s i n 2 θ c o s ϕ_{B} s i n φ]

。（41）

I_{d e t}^{(P)} = I_{0}^{(P)} + A^{(P)} c o s (\bar{φ_{s a m} - φ_{r e f}}) + B^{(P)} s i n (\bar{φ_{s a m} - φ_{r e f}})

，（42）

I_{d e t}^{(S)} = I_{0}^{(S)} + A^{(S)} c o s (\bar{φ_{s a m} - φ_{r e f}}) + B^{(S)} s i n (\bar{φ_{s a m} - φ_{r e f}})

，（43）

\{\begin{array}{l} I_{0}^{(P)} = \frac{I}{4} [{(r_{r e f} c o s 2 θ)}^{2} + {(r_{s a m} s i n ϕ_{B})}^{2}] \\ A^{(P)} = \frac{I}{2} r_{r e f} r_{s a m} c o s 2 θ s i n ϕ_{B} s i n (ϕ_{z} - 2 β) \\ B^{(P)} = \frac{I}{2} r_{r e f} r_{s a m} c o s 2 θ s i n ϕ_{B} c o s (ϕ_{z} - 2 β) \\ I_{0}^{(S)} = \frac{I}{4} [{(r_{r e f} s i n 2 θ)}^{2} + {(r_{s a m} c o s ϕ_{B})}^{2}] \\ A^{(S)} = - \frac{I}{2} r_{r e f} r_{s a m} s i n 2 θ c o s ϕ_{B} s i n ϕ_{z} \\ B^{(S)} = - \frac{I}{2} r_{r e f} r_{s a m} s i n 2 θ c o s ϕ_{B} c o s ϕ_{z} \end{array}

。（44）

t a n^{2} ϕ_{B} t a n^{2} (2 θ) = \frac{A^{(P) 2} + B^{(P) 2}}{A^{(S) 2} + B^{(S) 2}}

。（45）

t a n ϕ_{z} = \frac{A^{(S)}}{B^{(S)}}

。（46）

t a n (2 β) = \frac{A^{(S)} B^{(P)} - A^{(P)} B^{(S)}}{B^{(S)} B^{(P)} + A^{(S)} A^{(P)}}

。（47）

J_{Q}^{’} = e x p (i \frac{π}{4}) [\begin{matrix} 1 & 0 \\ 0 & i e x p (- i δ) \end{matrix}]

，（48）

J_{Q_{θ} δ} J_{- θ} J_{Q}^{’} J_{θ} = e x p (i \frac{π}{4}) [\begin{matrix} c o s^{2} θ - i s i n^{2} θ e x p (- i δ) & s i n θ c o s θ [1 + i e x p (- i δ)] \\ s i n θ c o s θ [1 + i e x p (- i δ)] & s i n^{2} θ - i c o s^{2} θ e x p (- i δ) \end{matrix}]

。（49）

J_{r e f}^{’} = J_{b s T} J_{Q_{θ}}^{’} J {}_{R}J J_{Q_{θ}}^{’} J_{b s R}

，（50）

J_{r e f}^{’} = [\begin{matrix} c o s^{2} θ - s i n^{2} θ e x p (- 2 i δ) & s i n θ c o s θ [1 + e x p (- 2 i δ)] \\ s i n θ c o s θ [1 + e x p (- 2 i δ)] & s i n^{2} θ - c o s^{2} θ e x p (- 2 i δ) \end{matrix}] \cdot r_{b s} t_{b s} r_{r e f} e x p (i φ_{r e f})

。（51）

J_{s a m}^{’} = J_{b s R} J_{Q_{θ}}^{’} J_{S_{β}} J_{Q_{θ}}^{’} J_{b s T}

。（52）

J_{s a m}^{’} = i [\begin{matrix} \begin{array}{l} - s i n ϕ_{B} c o s 2 β \\ - i s i n ϕ_{B} s i n 2 β e x p (- i δ) \end{array} & c o s ϕ_{B} \\ c o s ϕ_{B} & \begin{array}{l} s i n ϕ_{B} c o s 2 β \\ - i s i n ϕ_{B} s i n 2 β e x p (- i δ) \end{array} \end{matrix}] \cdot r_{b s} t_{b s} r e x p (i φ_{s a m})

。（53）

E_{P} = \sqrt[]{I} [\begin{matrix} 0 \\ 1 \end{matrix}]

。（54）

E_{d e t}^{(P)} = J_{P} (J_{r e f}^{’} + J_{s a m}^{’}) E_{P}

。（55）

E_{d e t}^{(P) ’} = \frac{\sqrt[]{I}}{2} i s i n ϕ_{B} [c o s 2 β - i s i n 2 β e x p (- i δ)] r_{s a m} e x p (i φ_{s a m}) + \frac{\sqrt[]{I}}{2} [s i n^{2} θ - c o s^{2} θ e x p (- 2 i δ)] r_{r e f} e x p (i φ_{r e f})

。（56）

\begin{array}{l} I_{d e t}^{(P) ’} = \frac{I}{4} \{s i n^{2} ϕ_{B} {[c o s 2 β - i s i n 2 β e x p (- i δ)]}^{2} r_{s a m}^{2} + {[c o s^{2} θ e x p (- 2 i δ) - s i n^{2} θ]}^{2} r_{r e f}^{2} + 2 s i n ϕ_{B} r_{s a m} r_{r e f} c o s 2 θ s i n (φ - 2 β)\} - \\ \frac{I}{4} \{2 s i n ϕ_{B} r_{s a m} r_{r e f} {c o s^{2} θ [1 - e x p (- 2 i δ)] \times [c o s 2 β - i s i n 2 β e x p (- i δ)] - i s i n 2 β c o s 2 θ [1 - e x p (- i δ)]\} s i n φ\} \end{array}

。（57）

E_{d e t}^{(S) ’} = \frac{\sqrt[]{I}}{2} \{r_{r e f} s i n θ c o s θ [1 + e x p (- 2 i δ)] e x p (i φ_{r e f}) + i r_{s a m} c o s ϕ_{B} e x p (i φ_{s a m})\}

。（58）

\begin{array}{l} I_{d e t}^{(S) ’} = \frac{I}{4} \{{\{s i n θ c o s θ [1 + e x p (- 2 i δ)]\}}^{2} r_{r e f}^{2} + c o s^{2} ϕ_{B} r_{s a m}^{2} - 2 r_{r e f} r_{s a m} s i n 2 θ c o s ϕ_{B} s i n φ\} + \\ \frac{I}{4} \{2 r_{r e f} r_{s a m} c o s ϕ_{B} s i n θ c o s θ [1 - e x p (- 2 i δ)] s i n φ\} \end{array}

。（59）

\begin{array}{l} I_{d e t}^{(P) ’} = I_{0}^{(P) ’} + A^{(P)} c o s (\bar{φ_{s a m} - φ_{r e f}}) + B^{(P)} s i n (\bar{φ_{s a m} - φ_{r e f}}) + \frac{I}{2} s i n ϕ_{B} r_{s a m} r_{r e f} \{c o s^{2} θ [1 - e x p (- 2 i δ)] \times \\ [c o s 2 β - i s i n 2 β e x p (- i δ)] - i s i n 2 β c o s 2 θ [1 - e x p (- i δ)]\} \times \\ [c o s (\bar{φ_{s a m} - φ_{r e f}}) s i n ϕ_{z} + s i n (\bar{φ_{s a m} - φ_{r e f}}) c o s ϕ_{z}] \end{array}

，（60）

\begin{array}{l} I_{d e t}^{(S) ’} = I_{0}^{(S) ’} + A^{(S)} c o s (\bar{φ_{s a m} - φ_{r e f}}) + B^{(S)} s i n (\bar{φ_{s a m} - φ_{r e f}}) + \frac{I}{2} r_{r e f} r_{s a m} c o s ϕ_{B} s i n θ c o s θ [1 - e x p (- 2 i δ)] \times \\ [c o s (\bar{φ_{s a m} - φ_{r e f}}) s i n ϕ_{z} + s i n (\bar{φ_{s a m} - φ_{r e f}}) c o s ϕ_{z}] \end{array}

，（61）

I_{d e t}^{(P) ’} = I_{0}^{(P) ’} + A^{(P)} c o s (\bar{φ_{s a m} - φ_{r e f}}) + B^{(P)} s i n (\bar{φ_{s a m} - φ_{r e f}}) + C^{(P)} c o s (\bar{φ_{s a m} - φ_{r e f}}) + D^{(P)} s i n (\bar{φ_{s a m} - φ_{r e f}})

，（62）

I_{d e t}^{(S) ’} = I_{0}^{(S) ’} + A^{(S)} c o s (\bar{φ_{s a m} - φ_{r e f}}) + B^{(S)} s i n (\bar{φ_{s a m} - φ_{r e f}}) + C^{(S)} c o s (\bar{φ_{s a m} - φ_{r e f}}) + D^{(S)} s i n (\bar{φ_{s a m} - φ_{r e f}})

，（63）

\{\begin{array}{l} I_{0}^{(P) ’} = \frac{I}{4} \{{\{r_{r e f} [s i n^{2} θ - c o s^{2} θ e x p (- 2 i δ)]\}}^{2} + {\{r {}_{s a m}s i n ϕ_{B} [c o s 2 β - i s i n 2 β e x p (- i δ)]\}}^{2}\} \\ A^{(P)} = \frac{I}{2} r_{r e f} r_{s a m} c o s 2 θ s i n ϕ_{B} s i n (ϕ_{z} - 2 β) \\ B^{(P)} = \frac{I}{2} r_{r e f} r_{s a m} c o s 2 θ s i n ϕ_{B} c o s (ϕ_{z} - 2 β) \\ \begin{array}{l} C^{(P)} = - \frac{I}{2} s i n ϕ_{B} r_{s a m} r_{r e f} \{c o s^{2} θ [1 - e x p (- 2 i δ)] \times [c o s 2 β - i s i n 2 β e x p (- i δ)]\} - \\ \frac{I}{2} \{i s i n 2 β c o s 2 θ [1 - e x p (- i δ)]\} s i n ϕ_{z} \end{array} \\ \begin{array}{l} D^{(P)} = - \frac{I}{2} s i n ϕ_{B} r_{s a m} r_{r e f} \{c o s^{2} θ [1 - e x p (- 2 i δ)] \times [c o s 2 β - i s i n 2 β e x p (- i δ)]\} - \\ \frac{I}{2} \{i s i n 2 β c o s 2 θ [1 - e x p (- i δ)]\} c o s ϕ_{z} \end{array} \end{array}

，（64）

\{\begin{array}{l} I_{0}^{(S) ’} = \frac{I}{4} \{{\{s i n θ c o s θ [1 + e x p (- 2 i δ)] r_{r e f}^{}\}}^{2} + {(c o s ϕ_{B} r_{s a m}^{})}^{2}\} \\ A^{(S)} = - \frac{I}{2} r_{r e f} r_{s a m} s i n 2 θ c o s ϕ_{B} s i n ϕ_{z} \\ B^{(S)} = - \frac{I}{2} r_{r e f} r_{s a m} s i n 2 θ c o s ϕ_{B} c o s ϕ_{z} \\ C^{(S)} = \frac{I}{2} r_{r e f} r_{s a m} c o s ϕ_{B} s i n θ c o s θ [1 - e x p (- 2 i δ)] s i n ϕ_{z} \\ D^{(S)} = \frac{I}{2} r_{r e f} r_{s a m} c o s ϕ_{B} s i n θ c o s θ [1 - e x p (- 2 i δ)] c o s ϕ_{z} \end{array}

。（65）

t a n^{2} ϕ_{B}^{’} t a n^{2} (2 θ) = \frac{{(A^{(P)} + C^{(P)})}^{2} + {(B^{(P)} + D^{(P)})}^{2}}{{(A^{(S)} + C^{(S)})}^{2} + {(B^{(S)} + D^{(S)})}^{2}}

。（66）

t a n ϕ_{z}^{’} = \frac{A^{(S)} + C^{(S)}}{B^{(S)} + D^{(S)}}

。（67）

t a n (2 β^{’}) = \frac{(A^{(S)} + C^{(S)}) (B^{(P)} + D^{(P)}) - (A^{(P)} + C^{(P)}) (B^{(S)} + D^{(S)})}{(B^{(S)} + D^{(S)}) (B^{(P)} + D^{(P)}) + (A^{(S)} + C^{(S)}) (A^{(P)} + C^{(P)})}

。（68）

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