We present single-shot quantitative phase imaging with polarization differential interference contrast (PDIC) for a differential interference contrast microscope which records the unfiltered Stokes vector of the differential interference pattern with a polarization camera. PDIC enables high spatial resolution phase imaging in real-time, applicable to either absorptive or transparent samples and integrates simply with epi-fluorescence imaging. An algorithm with total variation regularization is further introduced to solve the quantitative phase map from the partial derivative along one single axis, improving the accuracy and the image quality from the Fourier transform solution. After quantifying the accuracy of PDIC phase imaging with numerical simulations and phantom measurements, we demonstrate the biomedical applications by imaging the quantitative phase of both stained and unstained histological tissue sections and visualizing the fission yeast Schizosaccharomyces pombe's cytokinesis.

Quantitative phase imaging (QPI) has emerged as a powerful tool for measuring biomass distribution and time-lapse imaging of cellular dynamics with recent advances in the spatial resolution,1 the imaging speed,2–5 high content imaging,6 and 3D imaging.7–10 Typical QPI, however, has a much poorer axial resolution (∼560 nm) than the lateral resolution (∼350 nm).9 In contrast, differential interference contrast (DIC) microscope, which images the interference fringes and is only qualitative, can achieve excellent lateral resolution and axial discrimination. Aided by computation, quantitative DIC has emerged as one attractive quantitative phase imaging modality,11–16 particularly the seminal work by Shribak et al.17–19 Its notable advantages include no need for phase unwrapping, tolerance to scattering,20 and the optical sectioning much sharper than other QPI methods.18,21

In a conventional Nomarski DIC microscope, the incident wavefront and its replica with orthogonal polarization directions pass through the same sample with an offset determined by the DIC shear s, recombine, and interfere. The sample phase profile is qualified by the resulting interference pattern typically after being filtered by a polarization analyzer to enhance the contrast. The recent introduction of the polarization camera has allowed the fast acquisition of the interference Stokes vector in a single snapshot without filtering (i.e., the DIC analyzer is disengaged). In a well-aligned polarization differential interference contrast (PDIC) microscope (see Fig. 1), the incident wavefront E1 and its replica E2 have equal amplitudes |E1|=|E2|=E0 and their polarization directions are ±π/4, respectively, away from the DIC polarizer orientation (y-axis). Both fields pass through the same specimen at an offset s. After passing through the sample and combining by a second Wollaston prism, E2 accumulates a phase difference Δ=s·ϕ+ϕb over E1 due to both the specimen of a phase profile ϕ and the DIC bias ϕb. The interference of these two fields, however, is not directly imaged inside a typical microscope. A mirror further reflects their interference E1+E2exp(iΔ) in our microscope to a polarization camera attached on a side port. The p-polarization component reflected by the mirror experiences an extra phase delay δ and an unequal reflectivity (the reflectivity ratio is r) than the s-polarization component.

FIG. 1.

(a) A schematic diagram and (b) orientations of the directions for a PDIC microscope. The first Wollaston prism splits the incident beam linearly polarized along the polarizer direction (the y-axis) into in-phase E1 and E2 polarized in the directions, ±π4 away from the y-axis, respectively. The DIC shear s is along the E1 direction. Es and Ep are the s- and p-polarization components of the electric field reflected by the mirror. The emerging wavefronts Es and Ep interfere and form an image at the side port recorded by a polarization camera. The epi-fluorescence optical path for simultaneous measurement is also shown.

FIG. 1.

(a) A schematic diagram and (b) orientations of the directions for a PDIC microscope. The first Wollaston prism splits the incident beam linearly polarized along the polarizer direction (the y-axis) into in-phase E1 and E2 polarized in the directions, ±π4 away from the y-axis, respectively. The DIC shear s is along the E1 direction. Es and Ep are the s- and p-polarization components of the electric field reflected by the mirror. The emerging wavefronts Es and Ep interfere and form an image at the side port recorded by a polarization camera. The epi-fluorescence optical path for simultaneous measurement is also shown.

Close modal

Inside the reference frame (viewed facing the impugning beam) of the polarization camera, the net electric field is expressed as

x̂EsŷEp=x̂E01+exp(iΔ)2ŷrE01exp(iΔ)2exp(iδ),
(1)

where x̂ and ŷ are the unit vectors along the x and y directions. The unfiltered interference pattern produced by Eq. (1) is recorded, yielding I=(I0°,I45°,I90°,I135°)T for the intensity of light linearly polarized along 0o,45o,90o, and 135o directions, respectively, and a spatially resolved Stokes vector for the specimen given by

S(ρ)=(S0(ρ)S1(ρ)S2(ρ))=(I0°(ρ)+I90°(ρ)I0°(ρ)I90°(ρ)I45°(ρ)I135°(ρ))
(2)

in a single snapshot where ρ=(x,y) are the pixel coordinates.

This provides a straightforward expression for Δ=atan2(sinΔ,cosΔ) from

S0=2r2I0,S1=2r2I0cosΔ,S2=2rI0sinδsinΔ,
(3)

where I0|E0|2 and S0,1[(1+r2)S0,1(1r2)S1,0]/2 according to Eq. (1). The results are particularly simple in the limit of r =1. In this case, S reduces to 2I0(1,cosΔ,sinδsinΔ)T and the variance of Δ is given by

Var(Δ)=12γI0(1sin2δcos2ϕb+sin2ϕb),
(4)

where the measured light intensities Ii (i=0o,45o,90o, and 135o) are assumed to be dominated by the shot noise and specified by their variances equaling to Var(Ii)=Ii/γ. The least error Var(Δ) is reached at ϕb=π/2.

To shed more light to the above result, the image formation is also analyzed with the weak object transfer function (WOTF) for a telecentric microscopic imaging system under Köhler illumination.22 Consider an in-focus thin sample whose complex transmission profile a(ρ)exp[iϕ(ρ)]=a0+Δa(ρ)+ia0ϕ(ρ), where Δa and ϕ are the amplitude and phase variations. The phase delay from the background has been assumed to be zero without losing generality. It is straightforward to show

S0=2ã2r2I0,S1=2ã2r2I0cos[ϕb+s·ϕ̃(ρ)],S2=2ã2rI0sinδsin[ϕb+s·ϕ̃(ρ)]+2a0rI0cos(δ)s·ã
(5)

to the first order, where ã=f1[g(ν)cos(πν·s)f[a]], ã=f1[g(ν)sinc(πν·s)f[a]], and ϕ̃=f1[g(ν)sinc(πν·s)f[ϕ]] are the filtered amplitude and phase, f and f1 denote Fourier transform and inverse Fourier transform, sinc(x)sin(x)/x is the sinc function, ν is the spatial frequency, and g(ν)CBF(ν,0)/CBF(0,0) is the normalized partially coherent transfer function for a brightfield microscope.23 According to Eq. (5), the setting of ϕb=π/2 is optimal as S1 gains the highest sensitivity to s·ϕ̃ and the influence of the unwanted second term in S2 due to absorption is minimized. The diffraction theory result (5) agrees with (3) from ray optics provided that the phase in (3) should be understood as the filtered ϕ̃ and its validity requires smooth variations in absorption (|s·ã|a0|tanδ|).

The filtered phase map ϕ̃ can then be obtained from s·ϕ̃=Δϕb. Fourier transform has been used before to find ϕ̃ from its partial derivative.24 An iterative approach based on total variance (TV) regularization is presented below to improve the above solution, solving the phase profile u(x, y) from xuu/x=ϕx(x,y) given the measured phase gradient ϕx assuming s is along the x-axis. Specifically, we formulate it as the following optimization problem:

minudρ[μ2|xuϕx|2+|yu|],
(6)

where μ is a positive constant and controls the degree of TV regularization. Equation (6) can be solved using an augmented Lagrangian function25,26

LA=dρ[|w|+β2|wyu|2λ(wyu)+μ2|xuϕx|2]
(7)

with the alternative direction method (ADM),

w(k+1)=(v(k)1βv(k)|v(k)|)H(|v(k)|1β),ũ(k+1)=iμkxϕx̃+βkyw̃(k+1)kyλ̃(k)μkx2+βky2+με,λ(k+1)=λ(k)β(w(k+1)yu(k+1)).
(8)

Here, v(k)yu(k)+λ(k)β,H is the Heaviside function, the superscript (k) means the quantities at the k-th iteration, ϕx̃,ũ,w̃, and λ̃ are the Fourier transforms of ϕx, u, w, and λ, respectively, kx and ky are the x- and y-components of the wavenumber inside the Fourier space, and ε=106 is introduced to avoid the division by zero. We initiate the iterative solution by setting

u(0)=f1[ikxϕx̃kx2+ε],λ(0)=0,β=10,μ=0.5/σ2,
(9)

where 0 is a zero matrix of the same shape as ϕx, and 1/σ2=I02/(I0/γ)=γI0 is a parameter representing the signal to noise ratio of the measurement. The initial solution u(0) is provided by the Fourier transform solution to the partial derivative.24 The final solution is reached when the relative change in u is less than a threshold (103), or the image quality (for example, sharpness) drops. Comparing to the Fourier transform solution, this TV regularization strategy improves the phase image quality by alleviating the imbalance in sensitivities along the DIC shear and its orthogonal directions for phase retrieval from the partial derivative along one single axis. The artifacts in the former4,24 largely disappear with the TV regularization strategy (see Fig. 3). The genuine phase map ϕ is obtained from ϕ̃ by deconvolution. The error of the recovered phase is approximately the same as the variance of ϕx, i.e., Var(u)Var(ϕx)(2γI0)1.

We then perform simulations and experiments to quantify the accuracy of PDIC phase imaging and demonstrate the biomedical applications. Experiments were performed on an inverted epifluorescence DIC microscope (IX73, Olympus). The light source is a Halogen 100 W lamp filtered by a Green filter (central wavelength 0.545μm). The numerical apertures for the condenser and the water immersion UPLANSAPO 60× objective are 0.55 and 1.2, respectively. The microscope is equipped with a focus drive and motorized scan system (ASI) and a dual-camera adaptor (TwinCam, Cairn). The PDIC and fluorescence images are recorded, respectively, by a polarization camera (Blackfly S Polarized, FLIR) and a digital color camera (Blackfly S Color, FLIR). PDIC and fluorescence images are co-registered and can be captured simultaneously. The exposure time is set at 10ms. The system parameters have been measured separately with r =0.981, δ=63.2o, s=0.22μm, and the pixel size 0.0564μm from a set of calibration experiments. To correct the potential phase artifacts introduced by optical elements inside the microscope, we measure Δbg for a uniform background under the identical condition and obtain s·ϕ̃=ΔΔbg. The total computation time for one phase image is ∼10 s in Matlab using one single 2.2 GHz CPU.

First simulations were performed using microlith,27 which accurately predicts images of a thin specimen observed with any partially coherent imaging system and is modified for PDIC. Figure 2(a) shows the simulated light intensities polarized along 0o,45o,90o, and 135o directions for an etched SiO2 phase pattern of thickness 300nm imaged under a PDIC configuration (λ=0.532μm in vacuum, numerical apertures NAo=1.2 and NAc=0.55 for a water immersion objective and a condenser, respectively, and ϕb=π/2). The phase pattern contains a set of phase rectangles of width 1,0.5, 0.5, 0.25, 0.25, 0.15, and 0.15μm. The original phase pattern and the recovered phase patterns under low noise (0.5% noise), inhomogeneous absorption and low noise (mean amplitude reduction 25% with a standard deviation 11% and 0.5% noise), and moderate noise (3% noise) conditions are shown in Figs. 2(b)–2(e) with their horizontal profiles displayed in Fig. 2(f). The recovered phase value agrees very well with the theoretical one (0.46 rad). The two smallest rectangles with a separation of 0.15μm are also resolved, confirming the spatial resolution limit, λ/(NAo+NAc)=0.3μm. The performance of the phase recovery is tolerant to both moderate inhomogeneous absorption and noise, whereas the accuracy of the recovered phase deteriorates for the structures of size smaller than the spatial resolution limit.

FIG. 2.

(a) The measured light intensities polarized along 0o,45o,90o, and 135o directions for a SiO2 phase pattern of thickness 300nm. I135o is multiplied by five for clarity. (b) The original phase pattern and the recovered phase patterns under (c) low noise (0.5% noise), (d) inhomogeneous absorption and low noise (mean amplitude reduction 25% with a standard deviation 11% and 0.5% noise), and (e) moderate noise (3% noise) conditions. (f) Their horizontal profiles [along the red line in (b)].

FIG. 2.

(a) The measured light intensities polarized along 0o,45o,90o, and 135o directions for a SiO2 phase pattern of thickness 300nm. I135o is multiplied by five for clarity. (b) The original phase pattern and the recovered phase patterns under (c) low noise (0.5% noise), (d) inhomogeneous absorption and low noise (mean amplitude reduction 25% with a standard deviation 11% and 0.5% noise), and (e) moderate noise (3% noise) conditions. (f) Their horizontal profiles [along the red line in (b)].

Close modal

Silica spheres (Bangs Laboratories, Catalog No. SS05002, refractive index: 1.43)28 embedded in glycerol (refractive index: 1.47) and a transmission grating (GT25–03, Thorlabs) of 300 grooves/mm and 17.5o groove angle were then imaged with the PDIC microscope under the water immersion 60× objective. Figure 3 shows the recovered negative phase images for one typical silica sphere (size: 3.85μm) by the Fourier transform solution and the TV regularization method. The latter significantly improves the image quality, mostly avoiding the halo effect and phase underestimation observed in the former. Figure 4 shows the recovered phase image and physical geometry for the transmission grating. The recovered maximum phase (1.84 ± 0.12 rad) is in excellent agreement with the nominal value (2.08 rad). The physical geometry of the grating is then deduced from the phase profile (the refractive index of the glass is n =1.52) with a groove angle 19.7°±0.8° and a pitch of 3.30±0.08μm.

FIG. 3.

The negative phase images of a silica sphere embedded in glycerol recovered by the Fourier transform solution in [(a), (c)] and the TV regulation method in (b) and (d). The horizontal (DIC shear direction) and vertical profiles along the red and green dashed lines in (a) and (c) are shown in (b) and (d) together with the ground truth (black line). The size of the sphere is 3.85μm.

FIG. 3.

The negative phase images of a silica sphere embedded in glycerol recovered by the Fourier transform solution in [(a), (c)] and the TV regulation method in (b) and (d). The horizontal (DIC shear direction) and vertical profiles along the red and green dashed lines in (a) and (c) are shown in (b) and (d) together with the ground truth (black line). The size of the sphere is 3.85μm.

Close modal
FIG. 4.

The GT25-03 transmission grating: recovered phase image (top) and physical geometry deduced from the phase profile along the dashed line (bottom).

FIG. 4.

The GT25-03 transmission grating: recovered phase image (top) and physical geometry deduced from the phase profile along the dashed line (bottom).

Close modal

After quantifying the accuracy of PDIC phase imaging with numerical simulations and phantom measurements, we demonstrate two biomedical applications for mapping the quantitative phase of histological tissue sections and visualizing live cells coregistered with fluorescence. Figure 5 shows the measured phase images for the hematoxylin and eosin (H&E) stained and unstained serial cuts of normal prostate tissue. The phase images for the stained and unstained serial cuts are overall similar. However, slight elevation of the phase is noticeable for the H&E stained section, especially in the areas surrounding some stromal regions.

FIG. 5.

(Left) The H&E stained histologic image, the measured phase images for the H&E stained and unstained serial cuts of normal prostate tissue. (Right) The magnified images for the region highlighted by the white rectangle. Scale bar: 10μm.

FIG. 5.

(Left) The H&E stained histologic image, the measured phase images for the H&E stained and unstained serial cuts of normal prostate tissue. (Right) The magnified images for the region highlighted by the white rectangle. Scale bar: 10μm.

Close modal

Live cell quantitative phase imaging of the fission yeast Schizosaccharomyces pombe coregistered with fluorescence is shown in Fig. 6 (Multimedia view). Cells were immobilized by mounting onto an agar pad containing essential nutrients. The fission yeast strain (nup211-GFP::ura4+ and ura4-D18 leu1–32 ade6) expresses the nuclear pore protein nup211 fused with the green fluorescent protein (nup211-GFP),29 which marks the position of the nucleus. The fluorescence images were cleaned by the iterative Poisson denoising with variance-stabilizing transformations (VST)30 and overlaid on the phase map. The cytokinesis process of cell No. 4 is clearly shown, including the formation and cleavage of the septum. We also observed the fast movements of structures in the cytoplasm. Further investigations are needed to identify these subcellular structures and elucidate their functions.

FIG. 6.

Live cell imaging of the fission yeast Schizosaccharomyces pombe at 3, 11, 22, 27, 30, and 34 min in (a)–(f). This strain expresses the nuclear pore protein nup211 fused with the green fluorescent protein (nup211-GFP), marking the position of the nucleus. Arrowheads point to the septum where cytokinesis occurs. Scale bar: 5μm. Multimedia view: https://doi.org/10.1063/5.0065129.1

FIG. 6.

Live cell imaging of the fission yeast Schizosaccharomyces pombe at 3, 11, 22, 27, 30, and 34 min in (a)–(f). This strain expresses the nuclear pore protein nup211 fused with the green fluorescent protein (nup211-GFP), marking the position of the nucleus. Arrowheads point to the septum where cytokinesis occurs. Scale bar: 5μm. Multimedia view: https://doi.org/10.1063/5.0065129.1

Close modal

The QPI system is defined by the compromise of four figures of merit: space–bandwidth product, time-bandwidth product, spatial phase sensitivity, and temporal phase sensitivity.31 While it is difficult to maximize each of these parameters in one single system, the PDIC microscope performs well in all four figures of merit with single snapshot broadband illumination and common-path geometry. The phase sensitivity can be estimated to be Var(ϕ)0.012 and the optical path length sensitivity 1.0nm at λ=0.545μm from I020000 (γ=0.176 for the polarization camera), agreeing well with the experimentally measured spatial and temporal sensitivity of 1.15±0.05nm from 250 time lapse phase images of a blank slide (data not shown). The phase sensitivity can be significantly enhanced when using a camera of an extremely deep well and/or by averaging over multiple images and performing spatially low pass filtering,32,33 provided that the phase error is dominated by photon noise rather than other sources, such as spatial or temporal fluctuations of the microscope system. The phase difference measured by PDIC is inherently between two polarization states (E1 and E2) and, hence, affected by birefringence if present. Cautious interpretation is needed for strongly birefringent specimens although birefringence is typically weak for biological samples.34 Compared to earlier studies,4 this work presented the PDIC quantitative phase imaging of both absorptive and transparent samples inside an inverted microscope and an iterative method to improve the quality of the recovered phase map. The polarization alteration due to the mirror reflection omitted in [4] is explicitly considered, critical for an accurate phase recovery. We derive the formalism for the PDIC microscope using both the geometrical optics and the weak object transfer function for a telecentric microscopic imaging system. The latter reduces to the former exactly and reveals its validity condition that the phase is the filtered version and the variation in sample absorption is spatially smooth. We note the geometrical optics formalism provides a simple interpretation for PDIC and is easy to extend to account for modifications in the optical system. Finally, we note that the current PDIC implementation is limited to measure the phase gradient along one single axis (the shear direction) and TV regularization plays an important role in phase recovery. Future studies are warranted to investigate the optimal regularization for a complex phase distribution.

In summary, we have presented single-shot quantitative phase imaging with polarization differential interference contrast (PDIC). It enables real-time quantitative phase imaging of either absorptive or transparent samples with high spatial resolution at a speed limited by the camera frame rate alone and integrates simply with epi-fluorescence imaging for coregistered simultaneous measurements. We have demonstrated the accuracy of PDIC phase imaging and the biomedical applications in imaging the quantitative phase of histological tissue sections and visualizing the cytokinesis process of the fission yeast. PDIC may emerge as an attractive quantitative phase imaging method for biology and medicine.

The research is funded by National Science Foundation, USA (No. 1607664), and The Immune Technology, Corp., USA.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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