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Abstract
This paper proposes a flexible and accurate dynamic quantitative phase imaging (QPI) method using single-shot transport of intensity equation (TIE) phase retrieval achieved by division of focal plane (DoFP) polarization imaging technique. By exploiting the polarization property of the liquid crystal spatial light modulator (LC-SLM), two intensity images of different defocus distances contained in orthogonal polarization directions can be generated simultaneously. Then, with the help of the DoFP polarization imaging, these images can be captured with single exposure, enabling accurate dynamic QPI by solving the TIE. In addition, our approach gains great flexibility in defocus distance adjustment by adjusting the pattern loaded on the LC-SLM. Experiments on microlens array, phase plate, and living human gastric cancer cells demonstrate the accuracy, flexibility, and dynamic measurement performance for various objects. The proposed method provides a simple, flexible, and accurate approach for real-time QPI without sacrificing the field of view.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Phase contains lots of important information of light filed, such as surface morphology and internal structure of the object. To measuring it accurately, quantitative phase imaging (QPI) [1], which computes phase distribution by convenient-recorded intensity images, has been extensively studied in recent decades. With the demand of industrial real-time measurement and biological label-free living detection [2,3], dynamic QPI becomes the current research focus in this field. Among them, interferometric methods [4–7] get limited due to their susceptibility to disturbances, coherent noise leaded by light source and complex phase retrieval algorithms. The means of wavefront detection suffer from the problem of poor spatial resolution caused by processing of optical devices. Iterative-based non-interferometric approaches [8–10] are not suitable for dynamic imaging because the large amount of data and computation required. In contrast, transport of intensity equation (TIE), as a deterministic non-interferometric method, has been well applied in dynamic phase measurement and imaging [11–13] due to its simple system, fast and efficient algorithm, robustness to environmental disturbances and applicability to partially coherent illumination.
TIE retrieves phase information by axial intensity changes. Normally, a good phase result can be obtained by solving TIE with two intensity maps at different positions of the optical axis. Obviously, to capture intensity images, camera needs to be moved mechanically along the axis, preventing its application in dynamic QPI. To address this problem, initially, researchers utilized electronically controlled zoom devices [14–16] or other ways [17–19] to quickly acquire multiple defocus images to achieve quasi dynamic measurement. Further, some single-shot methods (as shown in Table 1) have been proposed successively to realize real-time QPI. Essentially, those approaches accomplish simultaneous capture two defocus intensity images with single exposure, including the scheme of field of view (FOV) multiplexing, the way of wavelength multiplexing and deep learning algorithm. Among them, the FOV multiplexing technique is the simplest and easy to implement, which achieves single-shot QPI by lateral beam separation, using various optical configurations and devices, such as volume hologram [20], tilted mirror [21–23], microscopy [24], beam splitter [25,26], SLM [27,28], diffractive optical elements [29–31] and metasurface [32,33]. However, these methods are limited by sacrificing FOV and image registration. Furthermore, the strategy of wavelength multiplexing avoids these problems by exploiting the chromatic aberration (CA) of the lens [34] or phase characteristics of the SLM [35,36], but it suffers from the color coupling caused by color camera, and is only suitable for non-dispersive samples testing. Alternatively, some researchers have tried to implement dynamic QPI with the help of deep learning algorithm [37–42], while it still plagued by generalization issues.
Table 1. Comparison of current dynamic TIE-based QPI methods
To bypass the limitations of the above methods, in this work, DoFP polarization imaging technique [43,44] and synchronous multiplexing of polarization are adopted to achieve dynamic QPI. As the information carrier, the polarization obtains excellent anti-interference ability. Therefore, the single-shot QPI can be achieved by modulating different defocus beams with different polarization states and recording them simultaneously. To capture those polarized images, the traditional division of time polarimeter [45], division of amplitude polarimeter [46], and division of aperture polarimeter [47] ways are still limited by inability of single-shot imaging, image registration and sacrifice of FOV, respectively. Differently, DoFP polarization imaging [48] acquires multiple polarization direction images by a micro polarizer array, which provides a sub-pixel polarization imaging capability without the above issues. Although this cause down sampling issue compared with monochrome camera, the resolution can be effectively improved by advanced interpolation algorithm.
Specifically, our method exploits the property of SLM to produce pure phase modulation for only one polarization direction. Thus, two intensity images of different defocus can be obtained by a pair of orthogonal polarized incident light. Next, by the DoFP polarization imaging, those defocus images of different polarization directions can be simultaneously acquired without FOV sacrifice, enabling a single-shot full-field QPI by solving the TIE. And the defocus distance can be adjusted by changing the Fresnel lens pattern, providing great flexibility for TIE-based QPI. In this manner, the proposed method preserves both the simplicity of the FOV multiplexing scheme and the advantage of the wavelength multiplexing methods without FOV sacrifice and image registration. Moreover, the polarization channel of DoFP owns excellent anti-coupling ability, which ensures the accuracy of the phase retrieved result.
2. Theory
Generally, the QPI of TIE requires two intensity images acquisition (one in-focus and the other defocus). To achieve dynamic measurement, the experimental setup of our method is designed as shown in Fig. 1, for the simultaneous capture of two images with different defocus distance by the synchronous multiplexing of polarization strategy using the polarization property of LC-SLM. Specifically, the object is illuminated by a parallel partially coherent light polarized at 45, after passing through the lens, the light incident the LC-SLM in frequency domain, where the 0 component (the working polarization direction of LC-SLM) is modulated by the loaded Fresnel lens to produce defocus, while the 90 component stays in-focus due to being unmodulated. Then, after the imaging lens, the light of two polarization states can be simultaneously captured by the DoFP polarization imaging camera without FOV sacrifice. Moreover, the polarization channels are excellent independent and not affected by the channel coupling. Finally, an accurate phase distribution can be reconstructed by TIE with those two images. The detailed mathematical derivation is provided in the following.
Fig. 1. Schematic diagram of our method. (a) The principle of DoFP polarization. NPBS, nonpolarizing beam splitter.
2.1 Synchronous multiplexing of polarization by DoFP polarization imaging
As a convenient and precise optical filed modulator, LC-SLM works in a specific polarization state (alignment direction of LC molecules) to produce pure phase modulation [49], i.e., when a beam with orthogonal polarization directions is incident on the LC-SLM, one remains unchanged, while the phase of the other is modulated as:
where $\Phi $ denotes the phase delay, $\lambda$ represents the wavelength of incident light, $\Delta \textrm{n}$ means the birefringence index of LC molecule, h stands for the thickness of LC cavity and $\theta$ is the deflection angle of LC molecule, which is controlled by the pattern loaded on LC-SLM. Therefore, by axial modulation and multiplexing the information of these two polarization states, two intensity images with different defocus distances can be simultaneously generated. By introducing a Fresnel lens ($\Phi ={-} \frac{\pi }{{\lambda \textrm{f}}}({\textrm{x}^2} + {\textrm{y}^2})$, where f denotes the focal length) in frequency domain, combining Eq. (1) and defocusing formula of LC-SLM [50] ($\textrm{d} = \frac{{{\textrm{f}_\textrm{L}}^2}}{\textrm{f}}$), the ray with 0 polarization ${\textrm{I}_0}$ is defocused with a distance ofAfter simultaneously producing two defocus maps, generally, these images are recorded with the means of FOV separation [27,28,31]. However, they require complex image registration and FOV sacrifice, which limit their application. [35] presents a way to acquire these images without FOV sacrifice by wavelength multiplexing, but its measurement accuracy is affected by color channel coupling.
Therefore, it is of great significance to capture them without FOV sacrifice and channel coupling. DoFP polarization imaging technique is adopted to accomplish it. As shown in Fig. 1(a), the image sensor of polarization camera consists of three layers, where the microlens array is at the first layer, followed by polarizer array and imaging array. With this arrangement, the microlens array maximizes light collection and minimizes the pixel crosstalk, which avoids the channel coupling issue. And the microarrayed polarizers are closely configured to provide pixel-level polarization imaging capability, which addresses the problems of FOV sacrifice and image registration. Further, with the advanced interpolation algorithm [49,51–55], the imaging resolution can be effectively improved. As a result, the pair of orthogonally polarized light is captured by DoFP-polarization-imaging-based polarization camera with single exposure.
2.2 Single-shot QPI with TIE solution
After obtaining the two intensity maps at two different axial positions, the phase can be solved by TIE [56], which is given in the form of
FFT-TIE is simple and effective, but has limitations in some respects, such as phase discrepancy error [58,59] caused by auxiliary function and phase singularity issue [60,61] leaded by zeros in intensity term. To address those problems, an improved iterative TIE solution [62] is employed in our method, which computes phase through establishing iteration by reconstructing TIE into two solvable equations and setting ${\textrm{I}_{\max }}$ with the maximum value of I
Easily, Eq. (4) and (5) can be numerically solved respectively by
3. Verification experiments
To demonstrate the accuracy of our method, the experiment on microlens array (LBtex, MLAS10-F15-P300, 300µm pitch) was conducted. The experimental setup is shown in Fig. 2(a), where the center wavelength of the LED source (Daheng, GCI0604) is 620 nm with 10 nm full-width-at-half-maximum, the 4f system composed of two lenses with 150 mm focal length, a beam splitter is for imaging, and the DoFP polarization camera (Baumer VCXU.2-50MP, Sony IMX250 Gen2 COMS image sensor, 76fps full frame) is 2448 × 2048 resolution with 3.45 × 3.45µm pixel size. According to Eq. (2), the Fresnel lens pattern is designed and loaded on the LC-SLM (UPOLabs HDSLM80R, 1920 × 1200, > 95% filling factor) to produce 0.5 mm defocus [63–67] for 0 polarization light. Due to the high filling factor of the LC-SLM used, zero-order diffractive beam is not filtered in this experiment. If a high- accurate phase reconstruction is required, the error in unmodulated areas caused by fill factor can be effectively solved by zero-order diffractive light filtering [68]. The raw intensity image with single-shot capture is shown in Fig. 2(b), from which the in-focus and defocus maps can be obtained by down sampling and bicubic interpolation [52]. As a comparison, under the same experiment, the result of conventional two-plane FFT-solved TIE is presented, with the camera moved by the same defocus distance. And object is also measured by the digital holography (DH) as reference.
Fig. 2. The experimental results of microlens array. (a) Experimental setup. (b) The raw image recorded by polarization camera with single exposure. (b1) In-focus and (b2) defocus images obtained from (b) by process of (c). The retrieved phase maps by (d) conventional TIE solution, (e) our method and (f) DH. (g) The height profiles indicated by dotted lines in (d)–(f).
Obviously, both our method and traditional TIE can provide a good phase reconstruction result. However, looking further at the details of the retrieved phases (see the height profiles in Fig. 2(g)), the lens shape recovered by conventional TIE deviates slightly from the DH result (see the enlarged part in red box), while our method remains consistent with the ground truth. This is because, in conventional TIE, errors introduced by mechanical movement and inconsistencies of image frame-to-frame noise caused by multi-frame acquisition affect the measurement result. In contrast, our approach captures images in single-shot way from the polarization camera without moving, enabling an accurate and robust QPI.
To further illustrate the performance of our method on more complex phase types, a phase-only SLM (UPOLabs HDSLM80R) is employed as tested object to flexibly generate different phases (as shown in Fig. 3(a)). Since the actual phase distribution (see Fig. 3(c), ‘pollen’) is known, the residual error (RE) maps and root mean square error (RMSE) can be calculated to evaluate the accuracy and quality of the reconstructed phase.
Fig. 3. The experiment for phase object produced by SLM. (a) Experimental setup. (b) The single shot captured raw image. (b1) In-focus and (b2) 2mm-defocus maps obtained by (b). (c) Actual phase map. The retrieved phase by (c1) conventional TIE and (c2) our method. (d1) and (d2) The respective RE maps. (e) The height profiles indicated by the red lines in (c), (c1), and (c2).
It can be observed in Fig. 3(c1) and (c2) that both the mechanical movement-based conventional TIE and our method recover the morphology of pollen, but our method obtains a higher accuracy, especially in reconstruction of the background (see Fig. 3(d1) and (d2)). The reason is the same with the previous experiment. Comprehensively, the details of the object are further analyzed. As shown in the enlarged view in red box, the grainy structures on the pollen surface are basically recovered, but it has to be said that the results are slightly blurred. This is clearly shown in Fig. 3(e), compared to the actual value, the profile of our method is smoothed. In fact, this is caused by the limitation of the aperture of optical system. The beam modulated by high-frequency part has a larger propagation angle, which cannot be fully captured by the optical system. As a result, some details are lost in reconstructed phase map. Fortunately, this problem can be well solved by optimizing the pattern of the illumination, which is closely discussed in section 4.
Since the merit of our method is the dynamic measurement capability, the experiment on living human gastric cancer cells was implemented with a vertical optical system (see Fig. 4(a)). In the experiment, an achromatic objective (Leica N PLAN EPI 50x/0.75 NA) and a tube lens (focal length of 120 mm) are arranged in front of the sample, which provides a lateral magnification of 34x and resolution of 505 nm. The sample was prepared by diluting directly from the culture medium (Refractive index, RI = 1.34) to a low concentration, allowing for better observation of individual cell. Notably, this phase (1200 × 1200 resolution) frame rate is 10fps, which is computed in MATLAB on a computer with an Inter Core i7 processor with a clock rate of 3.3 G Hz and a memory size of 16GB without GPU acceleration.
Fig. 4. Dynamic experimental phase reconstruction results for living human gastric cancer cells (Visualization 1). (a) Setup. (b) Raw image. (c) Intensity derivative. (d) Retrieved phase map. (e) Color-coded 3D phase distribution. (f) Enlarged view of the selected cell in red box and its nucleus segmentation result. (g) Cell thickness profile indicated by the green line in (f).
Totally, 1800 images are captured over a 3-minute period. The first raw image captured with single exposure is shown in Fig. 4(b), from which the in-focus and 1.5mm-defocus (a proper distance for cancer cells [36]) intensity maps can be obtained. And the intensity derivative (see Fig. 4(c)) estimated by the in-focus and defocus maps shows the phase contrast effect, because it is directly related to the phase gradient. Further, the reconstructed phase and its 3D color-coded distribution are shown in Fig. 4(d) and (e) respectively, from which we can clearly observe the morphology of cells in the entire FOV (74 × 74µm2). To better demonstrate the recovered result, the cell in the bottom right corner of the FOV is selected and enlarged in Fig. 4(f). Obviously, the cancer cell is huge (approximately 10µm diameter) and acquires a typical wrinkle surface and large nucleus. This can be further illustrated by the thickness distribution shown in Fig. 4(g), where the profile changes of the cell versus the background and the nucleus versus the cytoplasm are evident. And it can be seen that there is an extremely prominent part in the nucleus, which is be deemed to the aggregation of a large amount of DNA and other substances due to the cancer cell rapid proliferation [69]. Quantitatively, the nuclear-cytoplasmic ratio (NC ratio) can be computed by nucleus segmentation [70] (see Fig. 4(f)). By counting the number of pixels occupied, the NC ratio of the cell is 0.55, which is greater than that of the normal gastric cells [71,72].
4. Discussion
In this section, the effect of illumination pattern on TIE-based QPI is comprehensively studied. With the optimized annular illumination (AI), a high-solution and high-accurate phase retrieval result can be achieved. Meanwhile, the limitations of this approach are be discussed.
Compared with coherent light, partially coherent illumination can effectively improve the imaging quality and reduce the coherent noise [11,73,74]. With the establishment of the theory of TIE under partially coherent illumination [57,75,76], researchers further find that the imaging spatial resolution can be improved by changes the coherence parameters and the shape of light source [74,77–81], which provides a solution to the problem of low resolution for high-frequency information. Although these approaches does not push the diffractive limit compared with super-resolution means [82], they does not require multiple illumination and acquisitions, enabling resolution improvement with single-shot imaging. Among them, Li et al. [79] develop the optimal illumination pattern by a programmable LED array for the best imaging resolution and image contrast, which requires the light source to be a thin annular pattern matching the numerical aperture (NA) of the system. Thus, by updating the light source in setup with the programmable LED array and setting it as a thin AI (as shown in Fig. 5(a)), our approach enables high-resolution imaging, which is close to the diffraction limit [74] (414 nm spatial resolution with our experimental configurations (620 nm light source, 0.75 NA objective)). However, it should be pointed that there are two limitations in this method. Firstly, the cost of the use of AI is a significant decrease in phase transfer function response [11,76], resulting in a weak phase contrast and further causing a low signal-to-noise ratio (SNR) of the recovered phase distribution. Moreover, the scheme of AI is generally suitable for tiny objects, not for large ones. This is because compared with the miniature sample, the angle change leaded by AI is very small for the large ones due to the optical aperture limitation.
Fig. 5. The comparison for different illumination pattern and object size. (a) AI produced by programable LED array. (b) Raw image captured under AI. (c) Actual phase. The phase retrieved by (c1) LED point source, (c2) AI and (c3) AI for a large size object. (d1), (d2), and (d3) The respective RE maps. (e) The height profiles indicated by the red lines in (c), (c1), (c2), and (c3).
To well illustrate this, comparison experiments are implemented on the same object of Fig. 3(b). A 64 × 64 programmable LED array with the size of luminous die of 150µm and intervals of 1.25 mm between every adjacent LED element is employed for illumination. To match the aperture of the optical system, only 20 × 20 (coverage of 25 × 25mm2) LEDs are actually used. The reconstructed phase results by our method under LED light point source and annular LED illumination (with the same 2mm-defocus distance) are presented in Fig. 5(c1) and (c2) respectively. In addition, the retrieved phase of the object with a larger size (16 times larger than (c2)) under AI is also provided in Fig. 5(c3). Obviously, the result under AI shows higher resolution, which is mainly reflected in the recovery of object detail information (see Fig. 5(d2)). This is as we expected, multi-angle lighting leaded by AI provides more high-frequency information for phase retrieval, which makes up for the lack of the optical aperture, resulting in a high-solution and high-accurate result. This is clearly shown in Fig. 5(e), the microstructure of the pollen surface that was blur in the previous phase reconstruction (Fig. 3) is greatly improved by AI, which is almost the same as the true value. However, there is slight increase in background noise due to the reduced SNR. On the other hand, as the size of the object becomes larger, the resolution improvement leaded by the AI gradually weakens, as shown in Fig. 5(c3). Details in the pollen get misty again, and the background noise still exist. It can be seen from Fig. 5(e) that its resolution for pollen details is almost the same as that of (c1).
Therefore, by replacing the light source as the optimized AI, a high-resolution phase reconstruction result can be effectively realized, while it will reduce the SNR of the image to a certain extent. And it should be noted that the resolution improvement for large object is limited.
5. Conclusion
In conclusion, a flexible and accurate dynamic QPI method is presented in this work based on DoFP polarization imaging technique. Using the polarization property of the LC-SLM, intensity images at two different axial positions can be produced and contained in a beam of orthogonal polarization states. Next, with the help of DoFP polarization camera, those maps can be captured with single exposure, without FOV sacrifice and channel coupling. Finally, an accurate and dynamic phase distribution can be reconstructed by TIE using these two intensity images (one in-focus and the other defocus). In addition, by optimizing the illumination pattern, a high-resolution result can be provided. In this manner, on the one hand, our method gains the advantage of simple and high-resolution measurement due to the DoFP polarization imaging and polarization multiplexing strategy. On the other hand, our approach obtains great applicability to various objects benefitted from the flexibility of defocus distance adjustment. Experiments on static and dynamic samples demonstrate the accuracy, flexibility and dynamic measurement capability of our method. In practice, the proposed method can be well applied to label-free biomedical imaging and real-time industrial testing.
Funding
National Natural Science Foundation of China (52175516, 61975161); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2022410).
Acknowledgments
National Natural Science Foundation of China (No. 61975161, No. 52175516), Youth Innovation Promotion Association, Chinese Academy of Sciences, CAS (2022410).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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