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Abstract
Based on synchronous phase shift determination, we propose a differential phase measurement method for differential interference contrast (DIC) microscopy. An on-line phase shift measurement device is used to generate carrier interferograms and determine the phase shift of DIC images. Then the differential phase can be extracted with the least-squares phase-shifting algorithm. In addition to realizing on-line, dynamic, real-time, synchronous and high precision phase shift measurement, the proposed method also can reconstruct the phase of the specimen by using the phase-integral algorithm. The differential phase measurement method reveals obvious advantages in error compensation, anti-interference, and noise suppression. Both simulation analysis and experimental result demonstrate that using the proposed method, the accuracy of phase shift measurement is higher than 0.007 rad. Very accurate phase reconstructions were obtained with both polystyrene microspheres and human vascular endothelial.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Three-dimensional imaging of cell or tissue without labeling is essential for their biochemical and pathological investigation. However, imaging living cell with ordinary optical microscopes has low contrast due to the weak absorption of visible light [1]. Differential interference phase contrast (DIC) microscopy is a label-free imaging method based on polarized light shearing interference [2,3] that can transform the phase gradient of transparent specimen into the amplitude difference to achieve in situ high contrast and high-resolution imaging. This is an effective method for studying cell morphology and intracellular structures [4,5]. However, due to the nonlinear relationship between the captured intensity and the specimen inherent phase information, it is difficult to achieve quantitative phase imaging with a DIC microscope [6].
Recently, benefiting from the imaging theory development [7–9], several quantitative phase imaging (QPI) methods based on DIC microscopy have been proposed. Holmes described the imaging process of DIC microscopy using Fourier analysis and obtained the quantitative phase information of the specimen by using the iterative Gerchberg-Saxton algorithm [10]. Kou introduced the transport of intensity equation into DIC microscopy, and realized the phase reconstruction of human cheek cells via non-iterative algorithm [11], but it requires accurate calibration of axial defocus and accurate boundary condition, which inevitably affects the quantitative performance and accuracy. To improve the accuracy of phase reconstruction, the concept of direct quantitative phase in DIC is proposed, in which the DIC image is regarded as a kind of interferogram, and the gradient phase retrieval is performed by using the phase-shifting algorithm. The phase retrieval process via the phase-shifting algorithm requires high phase-shifting accuracy, which is limited by the inherent property of DIC image (i.e., the background is relatively uniform and there are no periodically distributed interference fringes), so many high-precision phase-shift estimation algorithms cannot be directly used [12,13]. 4-step phase-shifting algorithm has been widely applied in phase reconstruction of DIC images [14,15]. Phase-shifting methods such as Nomarski prism moving or wave plates rotating are commonly used to achieve phase-shifting DIC images [16], which is susceptible to mechanical adjustment limitations and then introduces large phase shift errors. Spatial light modulator based quantitative DIC microscopy can break through this limitation [17] and significantly improve phase shift accuracy, but it increases the cost. Moreover, liquid crystal variable retarder (LCVR) can realize the phase shift for DIC microscope [18,19], but the liquid crystal molecules are very sensitive to temperature as well as the incident light direction while the phase shifts have been calibrated in advance, so the phase shift accuracy is affected by environmental disturbance. To improve the temporal resolution and reduce the effect of environmental disturbance on the phase shift accuracy, the pixelated polarization cameras are applied to interferometry [20,21]. Four-frame phase-shifting interferograms achieved by the spatial phase-shifting approach are used for phase retrieval, accurate calibration is necessary to minimize the measurement errors induced by the polarization cameras [22]. Recently, the principal component analysis based on random phase shift is used in quantitative phase imaging of DIC microscopy [23]. It can rapidly realize high precision phase reconstruction of the specimen without prior knowledge of the phase shift. To ensure the accuracy of phase reconstruction, 4-20 frames DIC images with phase shifts uniformly distributed in [0, 2π] are required, which also increases the burden of the of acquisition.
Due to the small fringes number in DIC images, the small phase variation, and high precision measurement requirements, it is necessary to utilize the phase-shifting method for differential phase measurement, and then eliminate the influence of phase shift deviations on the accuracy at the same time. In this study, we propose a synchronous phase shift determination based differential phase measurement method for DIC microscopy. By using the polarizing beam splitting property of the Wollaston prism, we add an on-line phase shift determination device to generate carrier interferograms, which share the same interference field with DIC images, so the phase shift of DIC images can be determined. Consequently, the differential phase of the specimen can be reconstructed by using such phase shift. The proposed method can realize real-time, dynamic, and online phase shift determination whether the phase shifting is implemented by LCVR or Nomaski prism moving as phase shifters in DIC microscope, showing obvious advantages in accuracy, stability, and anti-interference ability.
2. Principle
2.1 DIC image formation and reconstruction
In the DIC microscope, the light is sheared by a crystal into two orthogonally polarized beams laterally displaced by Δr. The complex amplitude of the two orthogonally polarized beams can be written as:
The two orthogonally polarized beams in Eq. (1) can be combined and after passing them thorough a polarizer high contrast DIC image is achieved, the corresponding intensity distribution can be expressed as:
2.2 Phase shift measurement
As shown in Fig. 1(a), the proposed differential phase measurement method is implemented by adding an on-line phase shift determination device to generate spatial carrier interferograms for determining the phase shift of DIC images. The orthogonally polarized light from the DIC microscope is divided into phase shift measurement beam and DIC imaging beam by a beam splitter. DIC images are captured by Camera1. The phase shift measurement beam is converted by the Wollaston prism into two orthogonal linearly polarized beams having a certain angle in the propagation direction. These beams are focused by the lens L and pass through the polarizer P2 to generate a phase-shifting carrier interferogram on Camera2. Its intensity of the interference pattern can be expressed as:
Fig. 1. Schematic of the proposed differential phase measurement method. (a) optical setup; (b) flow chart of image acquisition and data processing. BS, beam splitter; P1, P2, polarizer; W, Wollaston prism; L, lens.
Note that the interferometric field of the spatial carrier interferograms used for the phase shift determination is the same as that of the DIC images, thus both the phase shift and the interference field suffer from the same external disturbance. As long as the acquisition of the two cameras is synchronous, the phase shift can be determined. Consequently, we can extract accurate differential phase by using the least-square phase-shifting algorithm whether the phase shift is an integral multiple of π/2.
2.3 Numerical simulation
The validity and accuracy of the proposed method are analyzed based on numerical simulation. As shown in Fig. 2(a), 201-frame simulated phase-shifting carrier interferograms are generated, the size is 400×400 pixels with pixel size of 5µm. The corresponding spatial carrier frequency in both x and y directions is 0.25 π/pixel, and the phase shift is increased from 0 to 2π rad with steps of 0.01π rad. The random noise with standard deviation of 2 gray value is added into the carrier interferogram. To compare the accuracy of the phase shift determination and the calculation time for different size interferograms, as shown in Fig. 2(a), we select three regions (200×200 pixels, 100×100 pixels, and 50×50 pixels) of an interferogram for calculation. These areas are marked as S1, S2, and S3.
Fig. 2. Simulation results. (a) carrier interference pattern; (b) the error between the calculated and expected phase shift.
Equation (5) is used for the phase shift determination in different regions. A computer with Intel i7 CPU, 2.6 GHz is utilized for the error analysis and the time of processing evaluation. The error between the calculated and expected phase shift is shown in Fig. 2(b). The accuracy of phase shift determination decreases with by decreasing the selected area. The proposed method use only one Fourier transform to determine the phase shift while the methods reported in Refs. [25,26] need to determine the phase distribution by inverse Fourier transform before the phase shift determination. Thus, the proposed method not only significantly improves the signal-to-noise ratio and achieves accurate phase shift, but also greatly reduces the calculation time.
The root-mean-square error (RMSE) and calculation time of different areas are reported in Table 1. It can be seen that the accuracy of the phase shift extracted from different regions is less than 10−3 rad, and the calculation time is in the range 0.2-1.2 ms, which fully satisfy the requirement for practical application. Note that it takes only 0.16 ms for phase shift extraction when the calculation area is 50×50 pixels.
Table 1. RMSE and calculation time achieved with different areas
3. Experiment
3.1 Experimental setup
The experimental setup is shown in Fig. 3. Both the DIC imaging acquisition and the added on-line phase shift determination device are placed at the exit of the inverted DICM (Olympus, IX73). The illumination light is a LED with a central wavelength λ=650 nm and a bandwidth Δλ=40 nm. A polarized beam arising from the DICM pass through the LCVR (Thorlabs, LCC1413-A) and is divided into the transmission beam and reflection beam by a beam splitter. The transmitted light is modulated by P1 and then forms DIC images on COMS1 camera (FLIR, BFS-U3-16S2M-CS). The reflected light passing through the phase shift measurement device is separated and deflected by the Wollaston prism, focused by a lens and modulated by P2 in order to generate a spatial carrier interferogram, which is recorded by COMS2 camera. The phase shift is implemented by a voltage-driven LCVR or Nomarski prism moving in a DICM along the direction perpendicular to its optical axis.
Fig. 3. Experimental setup of differential phase measurement based on synchronous phase shift determination for differential interference contrast microscope (DICM); LCVR, liquid crystal variable retarder; BS, beam splitter; P1, P2, polarizer; W, Wollaston prism (separation angle: Δβ=1° at λ=650 nm); L, lens with focal lengths f = 30 mm; COMS1, COMS2, complementary metal-oxide semiconductor camera.
3.2 Result of phase shift determination
To evaluate the effect of phase shift determination by the proposed method, we implemented the phase shift operation by two approaches: One is to change the driving voltage of LCVR, and the other one is to move the Nomarski prism of DICM perpendicularly to the optical axis, which will generate the phase shift between the two orthogonally polarized beams. Figures 4(a) and 4(b) show one captured phase-shifting spatial carrier interferogram and its local magnification, the corresponding phase shifts determined by the above two phase-shifting approaches are presented in Figs. 4(c) and 4(d), respectively.
Fig. 4. Experimental result of phase shift determination. (a) spatial carrier interferogram; (b) magnification of the marked red dotted box in (a); (c) phase shifts achieved by moving Nomarski prism of DIC microscope; (d) changing the driving voltage of LCVR.
As shown in Fig. 4(c), the variation curve of the phase shift determined with the Nomarski prism is not smooth due to the large and uneven spacing implemented by the manual adjustment of nuts, in spite of them accurate phase shifts can be gained. In Fig. 4(d), due to the small step length (0.005 V/step) and short time (<10 ms/step) implemented by voltage-driven LCVR, the curve variation of the phase shifts with the driving voltage is very smooth compared with Fig. 4(c), so we can achieve the desired phase shifts required for high precision differential phase measurement.
3.3 Accuracy of the phase shift determination
In order to verify the reliability of the phase shift determination by the online phase shift determination device, we replace the P1 and COMS1 used for DIC imaging in Fig. 3 by an online phase shift determination device. Two cameras synchronously capture the phase-shifting carrier frequency interferograms during the increasing of LCVR driving voltage, (the time accuracy of synchronization is less than 50 µs). Figure 5 shows the phase shift result determined by two channels interferograms. The phase shifts calculated from the two channels is consistent (Fig. 5(a), and the root-mean-square error (RMSE) of the phase shift deviation in the two channels is only 0.006 rad (Fig. 5(b)). In addition, we carry out multiple measurements (N = 4) at different times to evaluate the stability of the DICM system, as shown in Fig. 5(c), the corresponding RMSE of the phase shift deviation in the two channels is in the range 0.006-0.007 rad with maximum deviation of 0.025 rad, indicating high precision and excellent stability. It has the advantages in dynamic, synchronous and high-precision phase shift determination of the phase-shifting interferograms, and also reveals good capabilities in error compensation, anti-interference and noise suppression providing a powerful tool for the phase measurement system based on polarization phase-shifting interferometry.
Fig. 5. Accuracy and stability of the phase shift measurement. (a) relationship between phase shift and driving voltage in beam splitter transmitted and reflected optical paths; (b) difference between the phase shift curves of two channels in (a); (c) results of multiple measurements (N = 4) at different times.
4. Results and discussion
4.1 Quantitative phase imaging of microspheres
Using the proposed method, we measure the differential phase of two polystyrene microspheres with 7 µm diameter, and then reconstruct the phase with the phase-integral algorithm. The microspheres (n = 1.605) immersed in a UV curable adhesive (n = 1.56) are sandwiched between two cover glasses. In this experiment, a 20× objective with numerical aperture of 0.45 is used and the phase shift is implemented by the LCVR. Figure 6(a) shows 4-frame phase-shifting DIC images of the microspheres, in which the phase shifts determined by the proposed method are 0, π/2, π, and 3π/2, respectively. Figure 6(b) shows the differential phase achieved by the 4-step phase-shifting algorithm. If the phase shift is not exactly π/2 (e.g.: 0, 1.84 rad, 3.39 rad, and 4.92 rad), we also can gain accurate phase shifts by the proposed method, and the accurate differential phase by the least-squares algorithm, as shown in Fig. 6(c). Additionally, the profiles of differential phases (red dashed lines) in Figs. 6(b) and 6(c) are compared in Fig. 6(d), and the RMSE of the differential phase calculated by the two phase-shifting algorithm is only 0.011 rad. We also shows the phase reconstructed by the phase integration algorithm according to the shear direction and shear distance [27–28]. By combining the phase distribution and the refractive index information it is possible to obtain the height distribution of the two microspheres (see Fig. 6(e)). The cross profile of the centre line (see Fig. 6(f)) has the maximum height of 6.927 µm which is slightly different from the nominal height of 0.073 µm, suggesting that the main reason of this error might come from the phase integration algorithm [21,29]. In further work, we will focus on the effect of the phase-integral algorithm on the precision of the phase reconstruction in DICM.
Fig. 6. Proof-of-principle demonstration with polystyrene microspheres. (a) 4-frame phase- shifting DIC images; the differential phase extracted by (b) 4-step phase-shifting algorithm(PSA); (c) the least-squares phase-shifting algorithm when the phase shifts of DIC images are 0 rad, 1.84 rad, 3.39 rad, and 4.92 rad, respectively; (d) the comparison of the differential phase profiles in (b) and (c); (e) the reconstructed height map of two microspheres; (f) the nominal height (black dotted line) and the cross-sections of the retrieved height.
4.2 Quantitative phase imaging of cells
We measured the differential phase of human vascular endothelial cells (VE cells) and reconstructed their phase distribution to verify the applicability of the proposed method for the investigation of cells. The VE cells, provided by the BeNa Culture Collection (Beijing, China), are seeded on coverslips, and cultured for 24 hours before their use. The cells are fixed with 4% paraformaldehyde for 20 min, rinsed with PBS (pH =7.4) buffer and deionized water for three times, and then placed on the stage for measurement. In this experiment, a 20× objective with numerical aperture of 0.45 is used. Figure 7 shows the experimental result of intact VE cells and in their late stage of mitosis, respectively. Figures 7(a)–7(d) show 4-frame phase-shifting DIC images of VE cells. The right half and left half of each subgraph in Fig. 7 denote two intact VE cells and their late stage of mitosis, respectively. The phase shift step length is exactly π/2, this is achieved by the online phase shift measurement device. The differential phase of the VE cells is calculated by the least-squares algorithm, and their phase distribution is reconstructed by the phase-integral algorithm, as shown in Fig. 7(e). The intracellular structure details and the transition edge during VE cells division are visible in the achieved phase distribution, and the local magnification clearly reveals the cellular pseudopodia and cellular nucleus (Figs. 7(f)– 7(g)). These results demonstrate that the proposed method can be used for acquiring real time on-line high-resolution phase information during cell growth, which will provide a powerful tool for intracellular structure imaging during cell growth or pathological analysis.
Fig. 7. Experimental result obtained using VE cells. (a-d) 4-frame phase-shifted DIC images; (b) phase distribution and its local magnification.
5. Conclusion
In summary, we proposed a differential phase measurement method based on synchronous phase shift determination for high-precision quantitative DIC imaging. Due to the synchronous, real-time, dynamic, online, and high-precision phase shift determination for arbitrary phase shifts of DIC images, it can be used for high-quality phase reconstruction of the specimen. It has no limitation due to the type of phase shifter or its accuracy and does not need to estimate the phase shift from the interferograms. Thus, it can be used for easily implementing phase shift determination in laboratory or commercial system. Both numerical simulation and experiment indicate the advantages of the proposed method in precision (RMSE> 0.007 rad), stability and anti-interference capability. We have shown excellent performance of the method by high precision differential phase imaging of polystyrene microspheres and human VE cells. The method also can be applied to other phase-shifting interferometry systems.
Funding
National Natural Science Foundation of China (61727814, 61875059, 62175041).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Date underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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