Axially-offset differential interference contrast microscopy via polarization wavefront shaping

Changqin Ding; Chen Li; Fengyuan Deng; Garth J. Simpson

doi:10.1364/OE.27.003837

Optics Express
Vol. 27,
Issue 4,
pp. 3837-3850
(2019)
•https://doi.org/10.1364/OE.27.003837

Axially-offset differential interference contrast microscopy via polarization wavefront shaping

Changqin Ding, Chen Li, Fengyuan Deng, and Garth J. Simpson

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Changqin Ding, Chen Li, Fengyuan Deng, and Garth J. Simpson, "Axially-offset differential interference contrast microscopy via polarization wavefront shaping," Opt. Express 27, 3837-3850 (2019)

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Abstract

Sample-scan phase contrast imaging was demonstrated by producing and coherently recombining light from a pair of axially offset focal planes. Placing a homogeneous medium in one of the two focal planes enables quantitative phase imaging using only common-path optics, recovering absolute phase without halo or oblique-illumination artifacts. Axially offset foci separated by 70 μm with a 10x objective were produced through polarization wavefront shaping using a matched pair of custom-designed microretarder arrays, compatible with retrofitting into conventional commercial microscopes. Quantitative phase imaging was achieved by two complementary approaches: i) rotation of a half wave plate, and ii) 50 kHz polarization modulation with lock-in amplification for detection.

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Supplementary Material (1)

Name	Description
Visualization 1	The whole sets of images acquired with the half wave plate at different angles.

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Fig. 1 (A) The design of µRA as half-wave retardance with spatially varied azimuthal orientation of the fast-axis targeted for 532 nm light. Scale bar: 500 µm. Bottom: part of the measured different intensity distribution with horizontal (H) and vertical (V) polarization detection when horizontally polarized light passing through the µRA. (B) The working principle of traditional Nomarski phase contrast microscope. (C) The working principle of ADIC microscope. L1 and L2: lens; RP: reference plane; SP: sample plane.

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Fig. 2 Experiment set-up for QPI with a 10x objective to recover both bright field images and QP images. Blue circled optics: add-in parts for LIA detection.

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Fig. 3 Measured point spread functions in the x-z plane with (A) and without (B) the μRA installed in the beam path.

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Fig. 4 Transmittance image (A) and QP images (B) recovered with half wave rotation measurement of a single FoV of mouse tail section. (C) Overlay of the measured intensity of background (dots) with its nonlinear fit result (solid line). (D) Overlay of the measured intensity of random pixels (dots) with its nonlinear fit result (solid line) to recover transmittance image and phase contrast image. Scale bar: 50 μm.

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Fig. 5 Images measured from LIA detection with 1f (A, B) and 2f (C, D) as reference, and the recovered transmittance bright field image (E) and quantitative phase contrast image (F) of a single FoV of mouse tail section. Color bar unit: (E) transmittance percentage, (F) phase shift in radian. Scale bar: 50 μm.

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Fig. 6 Transmittance and quantitative phase contrast images recovered from HWP rotation (A, B) and LIA detection (C) strategies of a single FoV of mouse tail section. (D) Differences images of the phase shift calculated from the two strategies. Color bar unit: (A) transmittance percentage, (B-D) phase shift in radian. Scale bar: 50 μm.

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Fig. 7 Quantitative phase contrast images of the same FoV of 8 µm silica beads recovered from both HWP rotation (A) and LIA detection (B) strategies. Color bar unit: phase shift in radian. Scale bar: 50 μm. Inserts: zoom-in for one single bead. (C) Phase shift line profiles of the cross line in the insets retrieved from the HWP rotation (blue dots) and LIA detection (orange squares) approach.

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Equations (14)

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(\begin{matrix} \cos (γ) \\ \sin (γ) \end{matrix}) = \frac{1}{2} \cdot e^{- i \cdot γ} \cdot (\begin{matrix} 1 \\ i \end{matrix}) + \frac{1}{2} \cdot e^{i γ} \cdot (\begin{matrix} 1 \\ - i \end{matrix})

\overset{⇀}{e^{0}} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) \cdot \overset{⇀}{e^{0}} = \frac{1}{2} {[(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) + (\begin{matrix} 0 & i \\ - i & 0 \end{matrix})] + [(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) + (\begin{matrix} 0 & - i \\ i & 0 \end{matrix})]} \cdot \overset{⇀}{e^{0}}

\overset{⇀}{e^{t o t}} = \frac{1}{2} (| t^{+} | \cdot [\begin{matrix} 1 & i \\ - i & 1 \end{matrix}] \cdot e^{i \frac{δ}{2}} + | t^{-} | \cdot [\begin{matrix} 1 & - i \\ i & 1 \end{matrix}] \cdot e^{- i \frac{δ}{2}}) \cdot \overset{⇀}{e^{0}}

\overset{⇀}{e^{0} (γ)} = [\begin{matrix} \cos 2 γ & \sin 2 γ \\ \sin 2 γ & - \cos 2 γ \end{matrix}] \cdot [\begin{matrix} 1 \\ 0 \end{matrix}]

\overset{⇀}{e^{d e t} (γ)} = [\begin{matrix} \cos ϕ_{p o l} & - \sin ϕ_{p o l} \\ \sin ϕ_{p o l} & \cos ϕ_{p o l} \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} \cos ϕ_{p o l} & \sin ϕ_{p o l} \\ - \sin ϕ_{p o l} & \cos ϕ_{p o l} \end{matrix}] \cdot \overset{⇀}{e^{t o t} (γ)}

I (ϕ_{p o l}, γ) \propto {| \overset{⇀}{e^{d e t} (γ)} |}^{2}

I (γ) \propto {| t^{+} |}_{}^{2} + {| t^{-} |}_{}^{2} + 2 | t^{+} | | t^{-} | \cdot \cos (δ + 4 γ)

\overset{⇀}{e^{0} (τ)} \propto [\begin{matrix} 1 & - i \\ - i & 1 \end{matrix}] \cdot [\begin{matrix} e^{\frac{- i Δ (τ)}{2}} & 0 \\ 0 & e^{\frac{i Δ (τ)}{2}} \end{matrix}] \cdot [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}] \cdot [\begin{matrix} 1 \\ 0 \end{matrix}] = \sqrt{2} (1 - i) [\begin{matrix} \sin (\frac{Δ (τ)}{2} + \frac{π}{4}) \\ \cos (\frac{Δ (τ)}{2} + \frac{π}{4}) \end{matrix}]

Δ (τ) = 2 A \cdot \sin (2 π f τ)

I (τ) \propto {| t^{+} |}_{}^{2} + {| t^{-} |}_{}^{2} + 2 | t^{+} | | t^{-} | \cdot \sin (Δ (τ) - δ)

I (τ) \propto 2 ({| t^{+} |}_{}^{2} + {| t^{-} |}_{}^{2}) + 2 | t^{+} | | t^{-} | \cdot {\begin{cases} [(2 A - A^{3} + \frac{A^{5}}{6} - \frac{A^{7}}{72} + \dots) \cdot \sin (2 π f τ) + (\frac{A^{3}}{3} - \frac{A^{5}}{12} + \frac{A^{7}}{120} + \dots) \cdot \sin (3 \cdot 2 π f τ) + \dots] \cdot \cos δ \\ - [(1 - A^{2} + \frac{A^{4}}{4} - \frac{A^{6}}{36} + \dots) + (A^{2} - \frac{A^{4}}{3} + \frac{A^{6}}{24} + \dots) \cdot \cos (2 \cdot 2 π f τ) + (\frac{A^{4}}{12} - \frac{A^{6}}{60} + \dots) \cdot \cos (4 \cdot 2 π f τ) + \dots] \cdot \sin δ \end{cases}}

1 f_{Y} \approx 2 | t^{+} | | t^{-} | \cdot (2 A - A^{3} + \frac{A^{5}}{6} - \frac{A^{7}}{72}) \cdot \cos δ

2 f_{X} \approx - 2 | t^{+} | | t^{-} | \cdot (A^{2} - {\frac{A}{3}}^{4} + \frac{A^{6}}{24}) \cdot \sin δ

δ = Im [In (\cos δ + i \sin δ)]

Abstract

Supplementary Material (1)

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Figures (7)

Equations (14)

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