Cells, biological tissues, optical lenses, and the like are typical phase objects [1,2]. They are colorless and transparent and do not change the amplitude of the incident light, thus making it impossible to obtain high-contrast images under a traditional bright-field microscope [3]. Currently, the observation of cells often requires exogenous fluorescent dyes to label them, which can affect cellular physiological processes. Additionally, photobleaching and phototoxicity of fluorescent dyes hinder the long-time observations of live cells [4,5]. Phase delay, serving as an endogenous label, can reflect the morphology of cells and the surface features or defects of optical elements. The quantitative measurement of phase information finds wide applications in the fields of biomedical photonics and optical detection [6]. Interferometry is the most classic quantitative phase imaging (QPI) technique, but it requires a complex and precise interferometric setup and a stable measurement environment. Furthermore, its resolution is limited by the coherent diffraction limit, and speckle noise can lead to image quality degradation. These drawbacks limit its application scenarios [7–11].

Partially coherent QPI (PC-QPI) plays an increasingly important role in various fields such as biomedicine, optical measurement, and material physics, owing to its high signal-to-noise ratio, simple imaging setup, and higher resolution, which is twice the interference systems [12–14]. In recent years, PC-QPI methods, represented by Fourier ptychographic microscopy (FPM) [15–17], transport-of-intensity equation (TIE) [18–20], and differential phase contrast (DPC) imaging [21–23], have made significant progress. FPM is a representative method employing iterative reconstruction for phase retrieval, which typically requires a large number of intensity images to ensure algorithm convergence. The need for multiple iterations leads to excessive computational costs and reconstruction time, making it unsuitable for applications that require real-time processing. Moreover, iterative phase retrieval methods rely on the strict adherence of the light field to scalar diffraction laws, which limits their application to fully coherent light fields. The accuracy of the reconstruction heavily depends on the coherence of the light source [24].

Another class of propagation-based QPI can directly recover the phase in a non-iterative manner. TIE obtains multiple defocused images along the axial plane and recovers the phase by directly solving the phase-intensity equation [25]. Obtaining accurate axial intensity derivative is crucial for the precise solution of TIE. Teague first proposed the two-plane method to estimate axial intensity derivative [26]. However, a too small defocus distance can cause phase contrast to be overwhelmed by noise, while a too large defocus distance can introduce significant nonlinear errors [27]. To address the shortcomings of the two-plane method, Waller et al. proposed a higher-order finite difference method, which approximates the higher-order terms in the Taylor expansion of the axial intensity derivative by increasing intensity measurements [28]. Zuo et al. proposed a unified framework theory based on the Savitzky–Golay differential filter, incorporating all axial intensity derivative methods under the Savitzky–Golay differential filter framework [29]. Although multi-plane methods significantly improve the accuracy of axial intensity derivative estimation, they also greatly increase data acquisition time. Additionally, the effectiveness of TIE is limited by the paraxial approximation, making it applicable only to imaging with low numerical aperture (NA). It is true that increasing the illumination NA can enhance resolution; it also introduces severe high-frequency blurring, restricting its application under high NA [13,30].

Different from TIE, DPC generates phase contrast information through asymmetric illumination. The quantitative phase distribution of the sample is obtained by deconvolving the DPC image with the phase transfer function (PTF). As a deterministic solving method, it does not require any iterative processes. However, the accuracy of the recovered phase highly depends on the precision of the PTF. The transmission cross-coefficient (TCC) model can perfectly describe the complex amplitude transmission by establishing a four-dimensional transfer model for the spatial frequency of the sample in the spatial frequency domain [31,32]. Yet, accurately obtaining this high-dimensional representation is challenging and requires a substantial amount of data. Current implementations of QPI often simplify the bilinear transfer model to a linear process. The phase retrieval of weak scattering objects can be solved by using the Born approximation [33] and Rytov approximation [34], etc. The split-step non-paraxial (SSNP) model [35] and multi-slice beam-propagation (MSBP) forward model [36] extend the application of phase retrieval to thick objects and strongly scattering specimens by employing more complex transfer models. For two-dimensional DPC, the utilization of weak object approximation (WOA) to linearize the phase-to-intensity transfer process often leads to discrepancies between the reconstructed quantitative phase and the actual phase. Recently, a strict definition of the WOA has been proposed, and accurate quantitative phase results can be obtained through a one-step deconvolution approach only for samples with phase delay less than 0.5 rad [37]. This makes DPC unable to accurately reconstruct large phase objects in a non-iterative manner, and it loses its quantitative features.

Another challenging issue in the DPC is that the NA of the illumination needs to strictly match the NA of the objective lens (the NA of the illumination should be equal to the NA of the objective lens) [38]. For asymmetric illumination condition, the PTF contains two shifted circular pupils. Only when the illumination is matched can these two circular pupils be tangent at zero frequency, allowing low frequencies to be fully transmitted to the intensity image [38–40]. Non-matched-illumination can lead to the loss of low-frequency information, resulting in a recovered phase object that is incomplete and unable to reflect the true morphology of the sample. However, achieving precisely matched-illumination requires precise optical adjustments, and implementing high NA illumination is often challenging and costly. Recent research has discovered that when projecting the two-dimensional pupil function into the three-dimensional (3D) Fourier space, it forms two back-to-back Ewald spheres. Even if the illumination is not matched, these two Ewald spheres are completely separated at low frequencies and do not cancel each other out [41,42]. Expanding the two-dimensional measurement to 3D space eliminates the strict requirement for illumination matching, but this necessitates a significant number of scans along the Z axis [43]. The transport-of-intensity effect introduces an imaginary component in the transfer function, which inspires us to use intensity transmission to compensate for the missing low-frequency information [44]. By solving the TIE, the low-frequency recovered phase results can be obtained, requiring only one additional intensity image. And regardless of the total phase delay, TIE can always sufficiently recover a smooth phase.

In this paper, the relationship between the intensity recorded by a forward model using WOA and the actual intensity under different illumination NA is explored. The applicability of the WOA condition requires considering both the sample phase distribution and the NA of the illumination. Smaller NA illumination can expand the applicability of the WOA, and missing low-frequency information due to mismatched NA can be recovered through transport of intensity. The proposed transport-of-intensity differential phase contrast (TI-DPC) method in this paper does not require iteration as a deterministic solving method; it accurately recovers large phase targets while preserving high-frequency details. It defies the WOA condition and the matched-illumination condition. Phase retrieval was performed on a customized artifact with fine structure and a microlens array (MLA) with a large phase, achieving accurate phase retrieval consistent with the nominal results. Additionally, tongue slices and oral epithelial cells were used as samples, and the precise phase retrieval accurately reflected the actual thickness and morphology of the samples. This demonstrates the tremendous potential of TI-DPC for applications in the biomedical field and precision measurement fields.

DPC employs asymmetric partially coherent illumination to generate phase contrast information. The intensity distribution in the image plane follows the Abbe diffraction theory and can be viewed as the incoherent superposition of images formed by coherent light illuminating the object [45]. Due to the cross propagation and superposition of each point, the measured intensity is bilinearly related to the absorption and phase of the sample (the complete forward model of DPC is given in Appendix A) [31,32]: I(rc)=∬∬∬E(u)O(u1)O*(u2)H(u+u1)⁢H*(u+u2)ei2πr(u1−u2)d2u1d2u2d2u,(1)where u represents spatial frequency coordinates, rc represents the scaled coordinates at the image sensor, o(r) is the complex amplitude of the object and O(u)=o^(r), r represents the spatial coordinates, ·^ denotes the Fourier transform, E(u) is the illumination function, and H(u) is coherent transfer function. The component associated with the imaging system is defined as the TCC, TCC=∬E(u)H(u+u1)H*(u+u2)d2u [37]. TCC establishes a quantitative relationship between the absorption and phase of the sample and the measured intensity. However, it can be observed from Eq. (1) that the contributions of absorption and phase to the final measured intensity are not clearly expressed, making the phase retrieval from the measured intensity highly complex. To make phase retrieval feasible, the WOA is often employed to linearize the problem [46–48], o(r)=eiϕ(r)≈1+iϕ(r), where ϕ(r) represents the phase of the object. Ignoring higher-order terms between scattered light, the intensity distribution under the WOA can be obtained as [44] I(rc)=TCC(0,0)+Re{∬TCC(u,0)[iϕ^(r)]ei2πrud2u}.(2)

After introducing the WOA, the intensity becomes a linear function of the phase, where TCC(0,0) represents the background component of the system, and TCC(u,0) is the weak phase transfer function (WPTF) of the system. The frequency response of the phase is then described by the PTF [23], Tph(u)=TCC*(u,0)−TCC(u,0).(3)

By employing asymmetric illumination to achieve an odd-symmetric PTF and an even-symmetric absorption transfer function (APF), the phase and absorption are decoupled by utilizing differentiation [21], ImDPC(rc)=IDm,1−IDm,2IDm,1+IDm,2,(4)where m=1,2, {D1,1,D1,2}={up,down}, {D2,1,D2,2}={left,right}, and ImDPC(rc) represent DPC images in two directions. The Fourier spectrum of the DPC images can be expressed as [23] I^DPC(rc)=Tph(u)TCC(0,0)ϕ^(r),(5)where T(u)=Tph(u)/TCC(0,0). According to this principle, the quantitative phase of the sample can be obtained by the deconvolution process. After deconvoluting the I^DPC(rc) and the PTF that is equivalent to Tikhonov regularization, the phase information of an object can be expressed as [49] ϕtik(r)=F−1{∑jTj*(u)·I^jDPC(rc)∑j|Tj(u)|2+α},(6)where F−1 is the inverse Fourier transform, j is the index of DPC image measurement (up–down and left–right), and α is the regularization parameter, which is employed to mitigate the reconstruction error that arises from the excessive amplification of extremely small values in the PTF during the inversion process.

For accurate phase retrieval, it is imperative that the sample strictly adheres to the conditions of the WOA. Otherwise, significant errors may arise, jeopardizing the crucial “quantitative” characteristic. The deconvolution process involves two approximations: first, neglecting higher-order terms in the Taylor expansion of the sample function to linearize it and, second, ignoring interference terms between scattered light when deriving Eq. (2). However, when using Eq. (4) to compute the DPC image, the second error is eliminated. Therefore, phase errors primarily stem from the discrepancy between the ideal sample function eiϕ(r) and the linearized sample function 1+iϕ(r) [37].

In Fig. 1(a), the errors between eiϕ(r) and 1+iϕ(r) are illustrated. When the phase is less than 0.5 rad, the sample can satisfy the WOA condition [37]. As the phase shift increases, the error between eiϕ(r) and 1+iϕ(r) grows exponentially, and the sample no longer satisfies the WOA, at which point the phase retrieval process will experience severe nonlinear errors. To explore the effect of the linearized model in the intensity image, the intensity difference between the DPC images generated by actual sample distribution eiϕ(r) and the linearized sample distribution 1+iϕ(r) under different coherence parameters S (S=NAill/NAobj, where NAill is the illumination NA, and NAobj is the objective NA) was simulated. The simulated pixel size is 2.4 μm, the illumination wavelength is 520 nm, the magnification is 20×, and NAobj=0.4. Throughout this study, the same parameters are used for simulations. As shown in Fig. 1(b), the intensity differences and root mean square error (RMSE) of the DPC images generated using these two different imaging models were compared. When S=1, there is a significant difference between the DPC images generated by the two models. This leads to inaccurate phase recovery when WOA is not satisfied in conventional DPC imaging (S=1). As the illumination NA decreases, the RMSE gradually decreases. When S is less than 0.7, the RMSE is less than 0.017. In other words, as S decreases, the intensity difference brought by the two imaging models will decrease and the nonlinear error introduced due to WOA is canceled out. This inspires us to use a smaller NAill to defy the WOA condition. The resolution of the microscopic system is defined as R=λ/(NAill+NAobj), where λ is the wavelength. Therefore, it is not advisable to arbitrarily reduce NAill, which will cause a significant reduction in resolution. In this sense, illumination with a coherence parameter of 0.6–0.7 is used to achieve accurate quantitative phase imaging without significant resolution lost.

The derivation of the DPC imaging forward model in Section 2.A reveals the process of converting quantitative phase information into intensity information. Equation (3) indicates that the PTF is the difference between two shifted pupil functions. This theory predicts the loss of low-frequency information within (1−S)NAobj/λ, allowing the highest spatial frequency transmitted by (1+S)NAobj/λ [23]. This implies that when S<1, low-frequency phase cannot be effectively transmitted and recorded by the image sensor, resulting in the inability to quantitatively recover low-frequency phase. Only when S=1 can both high-frequency and low-frequency information be completely covered. Figure 2 shows the intensity distribution, absolute value of the transfer function, and intensity spectrum at different NAill and different focal planes of a simulated target. As shown in Figs. 2(a1)–2(a3), under the condition of matched-illumination, the two shifted pupils are tangent, allowing complete transmission of phase information. When the illumination angle decreases [Figs. 2(b1)–2(b3), S=0.65], the two shifted pupils cancel out in the low-frequency part, preventing the transmission of low-frequency phase. Therefore, when attempting to overcome the drawbacks of the WOA by using low NA illumination, the loss of low-frequency information hinders the quantitative phase retrieval. However, it is important to note that non-matched-illumination can occur not only in the proposed TI-DPC (where it is intentionally introduced) but also in traditional DPC, leading to the inability to achieve high quality phase retrieval. Therefore, defying the matched-illumination condition is crucial. In addition to using asymmetric illumination, TI-DPC also encodes phase information into the intensity image through defocus modulation. When defocus Δz, an imaginary part eikΔz1−λ2|u2| is introduced into the H(u), H(u)=P(u)eikΔz1−λ2|u2|, making PTF no longer a purely real function, where k represents the wave number and P(u) is the pupil function [50]. The imaginary component introduced by defocusing is well-suited to compensate for the low-frequency loss in DPC when S<1. As shown in Figs. 2(c1)–2(c3), when we defocused the intensity image by 4 μm, the low-frequency response of the transfer function reappeared. The distribution of the intensity spectrum also confirms the above analysis.

This means that the low-frequency phase can be encoded into the intensity image by defocusing. By combining the extension of WOA applicability under non-matched-illumination conditions and the low-frequency information carried by additional defocused images, TI-DPC can accurately recover the phase without degrading image quality and overcome the WOA and illumination matching conditions. Based on this, we propose a microscopic imaging method and conduct theoretical verification and experimental verification in Sections 3 and 4, respectively.

Due to the use of partially coherent illumination, the system can be conveniently implemented on a commercial inverted fluorescence microscope Ti2-U. Figure 3(a) shows the experimental setup. It only requires the replacement of the Köhler illumination with a circular programmable LED array which is capable of providing illumination at different angles and patterns. The left part of Fig. 3(a) shows the actual image of the LED array (241D-RGB-5 V, Shenzhen Gengyuan Technology Co., Ltd.), and the typical illumination patterns used in the experiment are generated by Arduino UNO programming. The LED array consists of eight concentric LED rings and one central LED. The diameter of the outermost LED ring is 172 mm, and it decreases by 20 mm from the outside to the inside ring by ring. The width of each LED ring board is 8 mm, and each LED has a side length of 5 mm. The number of LEDs on each ring from the outside to the inside is 60, 48, 40, 32, 24, 16, 12, and 8. A 3D manual stage is used to adjust the position of the LED array to make it perpendicular to the optical axis, and the central LED is across the optical axis. As shown in Fig. 3(b), the diameter of the LED array is DLED and the vertical distance between the LED array and the sample is zLED. Then the NAill achieved by each LED is NAill=(DLED/2)/(DLED/2)2+zLED2. Different NAill can be achieved by adjusting DLED and zLED. Since the emitting area of the LED is small enough and the distance from the sample is large enough, each LED can be considered to emit quasi-coherent light with a wavelength of 520 nm and a bandwidth of 25 nm. First, four in-focus images under asymmetric illumination conditions (up, down, left, and right) are acquired. Then, a motorized stage (ETS-50RG, Sanying MotionControl Instruments Ltd.) is used to move the sample and obtain one defocused image under symmetric illumination conditions. The defocus distance used to recover the phase of the customized artifact, MLA, and biological samples were 3 μm, 20 μm, and 2 μm, respectively. Intensity information transmitted through the different objective lenses (CFI Plan Fluor 10×, NA=0.3, CFI S Plan Fluor ELWD 20×, NA=0.45, or CFI Super Plan Fluor ELWD ADM 40×, NA=0.6) is recorded using a Nikon DS-Fi3 camera (pixel size=2.4μm).

The phase gradient transfer function (PGTF) is defined as the ratio of the measured phase gradient ∇rϕ1(r) in the image plane to the ideal phase gradient ∇rϕ2(r) of the object [44, 51,52], PGTF=∇rϕ1(r)/∇rϕ2(r). Figure 3(c) illustrates the PGTF under different coherence parameters S; the horizontal axis is the normalized spatial frequency [44]. When S=1, the recovered phase gradient is always half of the ideal gradient. When S<1, the phase gradient within 1−S is consistent with the true value, and phase gradients higher than 1−S are underestimated. The above analysis indicates that TIE tends to blur high-frequency information, but it is still capable of accurately reconstructing the overall smooth variations in the phase, even when the total phase shift is large [44]. On the other hand, the PTF of DPC imaging only needs to satisfy the WOA, preserving high-frequency information effectively. Therefore, combining the phase results recovered by TIE and DPC ensures the accuracy of low-frequency information while maintaining high-frequency details.

Figure 4 illustrates the data processing steps of TI-DPC using oral epithelial cells as an example. For the first step, four in-focus intensity images [Fig. 4(a)] are acquired under asymmetric illumination conditions of S=0.65, and then the sample is moved to obtain a defocus intensity image [Fig. 4(b)] under symmetric illumination conditions with the same S. All images were normalized before used to remove background components. Step 2 is to obtain the initial phase by solving the acquired image using DPC and TIE (detailed algorithms can be found in Appendices A and B), respectively. As shown in Fig. 4(c), using the Tikhonov regularization process denoted by Eq. (6) for the four in-focus images, a poor phase retrieval result can be obtained. From the intensity spectrum distribution in Fig. 4(d), it can be demonstrated that due to the cancellation of the low-frequency region of the PTF, all the low-frequency information of the cells is lost, and only limited high-frequency information can be restored. To compensate for the missing low-frequency information of the cell, it is necessary to use intensity transmission to correct the error. According to the Abbe’s superposition method, overlaying the intensity images obtained under asymmetric illumination conditions will result in an in-focus image under symmetric illumination condition. Therefore, the in-focus image under symmetric illumination can be obtained by overlaying the images within the blue box or the green box in Fig. 4(a). By solving the TIE using both in-focus and defocus images, the intensity spectrum and phase distribution shown in Figs. 4(e) and 4(f) can be obtained. As shown in Fig. 4(e), low-frequency information is fully retained in the results. However, the TIE solving process utilizing finite difference approximation and virtual boundary conditions [28,42] results in the loss of high-frequency information. In step 3, based on this, a complementary filter is designed accordingly in this paper. The threshold of the complementary filter is set to (1−S)NAobj, which is the threshold of the missing low frequency of DPC, and the threshold of PGTF less than 1. The complementary filter consists of a high-pass filter to collect the high-frequency information [from (1−S)NAobj to 2NAobj] of the phase retrieved by DPC and a low-pass filter to collect the low-frequency information [from 0 to (1−S)NAobj] of the phase retrieved by TIE. The high-frequency information is then added to the low frequency to obtain the mixed spectrum (detailed algorithms can be found in Appendix C). The mixed spectrum is shown in Fig. 4(g). Finally, by performing an inverse Fourier transform on the mixed spectrum, the ultimate phase distribution as shown in Fig. 4(h) is obtained. By mixing the spectrum in the frequency domain, the target phase can be accurately retrieved under non-matched-illumination. TI-DPC exhibits more pronounced advantages when recovering large phase objects using a high NA objective.

To demonstrate the feasibility of our proposed method, as shown in Fig. 5(a), we conducted simulations on a pure phase target composed of the NKU logo (1 rad, representing high-frequency information) overlaid with a large phase lens (0–3.5 rad, representing low-frequency information). Under a 20×, NA=0.4 objective, the intensity distributions under asymmetric illumination at the focal plane and symmetric illumination in the defocused position were simulated based on the Abbe’s theory of microscopy. Following the given reconstruction principles of TI-DPC, the simulated phase target was reconstructed. Figure 5(b) shows the phase retrieval results, spectrum distribution, and phase error for matched-illumination DPC (matched-illum DPC, S=1), non-matched-illumination DPC (non-matched-illum DPC, S=0.65), TIE (S=0.65), and proposed TI-DPC (S=0.65), respectively. Structural similarity index measure (SSIM) was used to quantify the degradation in image quality due to the loss of high-frequency information, and RMSE was used to quantify the absolute difference between the true and reconstructed phases.

Consistent with theoretical predictions, matched-illum DPC exhibited the highest SSIM (SSIM=0.8053) because of its maximum illumination NA, providing the broadest frequency coverage [as shown in the second row of Fig. 5(b)]. However, since the simulation target is no longer adhering to the WOA condition [37], its RMSE (RMSE=0.7919) was relatively high, leading to a loss of quantitative characteristics. Non-matched-illum DPC, with the cancellation of low-frequency components and the narrower passband of the PTF, resulted in the poorest reconstruction (SSIM=0.0062,RMSE=2.1480). TIE, due to the introduction of paraxial condition and nonlinearity error introduced by defocusing, led to high-frequency blurring (SSIM=0.6602) but demonstrated good low-frequency retrieval (RMSE=0.2645). The aforementioned three methods are limited by WOA, paraxial approximation, small defocus approximation, and illumination matching, making it impossible to accurately recover the phase. However, TI-DPC preserves as much high-frequency information as possible while recovering low-frequency information through intensity transmission. The TI-DPC achieved an SSIM of 0.7668 and an RMSE of 0.2217. From the phase error, it is also evident that the TI-DPC has significantly accurate reconstruction results.

As shown in Figs. 5(c) and 5(d), the imaging results of non-matched-illum DPC, TIE, and TI-DPC under different S (S=0.1, S=0.3, S=0.5, S=0.7, and S=0.9) are compared to demonstrate the excellent generalization ability and robustness of TI-DPC. Comparing the results under the same S, TI-DPC consistently achieves the smallest RMSE and the largest SSIM, which proves that it improves the imaging quality under all S. As shown in Fig. 5(c), the non-matched-illum DPC cannot accurately recover the low-frequency information due to the mismatch of NAill, which has the largest RMSE. The RMSE of the TI-DPC decreases with the increase of S. When S exceeds 0.7, the RMSE begins to increase. When S=0.9, the RMSE increases significantly because TIE requires a certain level of spatial coherence to accurately recover the phase. Figure 5(d) compares the SSIM of non-matched-illum DPC, TIE, and the proposed TI-DPC under different S. Similarly, the non-matched-illum DPC has the worst imaging performance due to the mismatch of NAill. As S increases, the frequency range recovered by DPC gradually expands, resulting in progressive increases in SSIM of TI-DPC. However, its SSIM decreases when S=0.9, because the coherence of the illumination is too poor and TIE cannot effectively recover the phase. Although TI-DPC has excellent generalization ability and robustness to significantly improve the image quality across all values of S, our simulations indicate that TI-DPC can better balance high-frequency and low-frequency information to achieve the best image quality when S is in the range between 0.6 and 0.7. This approach overcomes the phase underestimation caused by the target not satisfying the WOA condition, eliminates the need for strict NA matching, reduces hardware and optical alignment costs, and expands the system’s applicability.

As the proposed TI-DPC microscopy imaging method is preliminarily verified by the simulation in Section 3.B, further experimental studies need to be conducted to confirm the phase retrieval accuracy and verify the related imaging performance in the full detected frequency domain. We verified the accuracy of phase retrieval using customized artifact and further validated the performance of this method in the field of surface measurement through MLA samples. Finally, we confirmed the imaging capability of this method in the field of biological imaging through tongue slices and oral epithelial cells. TI-DPC achieves phase retrieval based on fast Fourier transform, resulting in high computational efficiency. On a laptop (Intel Core i7-9750H CPU @ 2.60 GHz) equipped with MATLAB 2021b, TI-DPC takes 0.458 s to reconstruct a 512×512pixels image.

To demonstrate that our proposed method can accurately recover the phase, we use RMSE to conduct quantitative analysis through experiments on a customized phase plate. The phase target was fabricated on a fused silica glass substrate with a refractive index of 1.45 and contains two structures: one is a set of rectangular steps with side lengths of 5 μm and heights of 70 nm, 120 nm, 170 nm, 220 nm, and 270 nm; the other is a USAF resolution test target from group 8 to group 10 with a height of 180 nm. The phase of the rectangular steps ranges from 0.38 rad to 1.47 rad, covering different conditions from those that satisfy the WOA to those that do not satisfy it, which makes it an ideal sample for validating the imaging capability of the proposed method. A 20× objective with an NA of 0.45 was used to collect intensity information. Figure 6(a) shows the reconstruction results of the step samples by non-matched-illum DPC (S=0.65), matched-illum DPC (S=1), TIE (S=0.65), and TI-DPC (S=0.65). Non-matched-illum DPC could hardly reconstruct the step structure due to the lack of low-frequency information. Although the other three methods could reconstruct the step structure, matched-illum DPC significantly underestimated the phase because some steps do not satisfy the WOA. To compare the phase retrieval results more clearly, the phase distribution at the green, red, blue, and yellow rectangular slices in Fig. 6(a) was extracted and is plotted in Fig. 6(b). Even though the shortest step was only 0.38 rad, which meets the WOA, the phase still could not be accurately recovered by matched-illum DPC. This is because the phase contrast produced by high-angle illumination is partially counteracted by the low-angle illumination, leading to the inaccurate phase reconstruction. As the step height increased, the phase error recovered by matched-illum DPC also increased. However, the RMSE of TI-DPC at 0.119 is much lower than the matched-illum DPC at 0.285 and non-matched-illum DPC at 0.540. Meanwhile, it is understandable that the improvement of TI-DPC over TIE is small, since the step sample mainly contains low-frequency information and TI-DPC recovers high-frequency information more efficiently than TIE. On the other hand, our proposed TI-DPC has a more significant advantage over TIE when it comes to phase retrieval of samples with fine structures. Therefore, we have presented and compared the phase retrieval results for the USAF resolution test target that is illustrated in Fig. 7(a). To more intuitively demonstrate the resolution achieved by the system, we printed the group number and element number of the resolution target onto the phase reconstruction results. The result for non-matched-illum DPC was the worst, which is only able to outline the contours of the resolution target slightly. To clearly demonstrate the differences in the reconstruction results, we extracted the phase distributions at the positions of the green, red, blue, and yellow solid lines and plotted them in Fig. 7(b). Since the phase of the resolution target (0.978 rad) exceeded the limit of the WOA, matched-illum DPC significantly underestimated the actual phase. TIE, due to its use of the paraxial approximation, loses high-frequency information and thus cannot fully resolve the elements in group 9. Only the proposed TI-DPC resolved all elements in group 9 and reflected the actual phase values. To compare the maximum resolution of TI-DPC and matched-illum DPC, we extracted the profile at the white dashed line (group 10). TI-DPC can resolve the third element of group 10 with a resolution of 775 nm, and matched-illum DPC can resolve the fifth element of group 10 with a resolution of 615 nm. The NAill of the TI-DPC is smaller than the NAill of the matched-illum DPC, so its resolution will be slightly lower than that of the matched-illum DPC. All the above experimental results strongly verified that the proposed method can overcome the limitations of the WOA and accurately recover phase values greater than 0.5 rad, and it does not require strict matching of the illumination NA. This enables TI-DPC to be qualified for scenarios requiring precise phase measurements, even when changing objectives with different NAs, allowing direct measurement without additional illumination adjustments.

To quantitatively validate the accuracy of TI-DPC in phase retrieval, an MLA (RPC Photonics, MLA-S100-f28-15010-A-2S, refractive index=1.56) with a pitch of 100 μm and a height of 754 nm was used as the sample. The MLA was positioned in air, and its maximum phase exceeded 5.1 rad, which does not satisfy the WOA condition. Intensity information was collected through a 10× objective. Figure 8(a) shows the actual image of MLA and its reference data. The vector height of the MLA was first obtained using a calibrated optical profilometer (NewView 9000, Zygo Corporation), and then the ground-truth phase was calculated using the equation ϕground_truth=2πlΔn/λ, which connects phase to height, where l is the vector height of the MLA and Δn is the difference between the refractive index of the MLA and air. As shown in Figs. 8(b)–8(e), the phase of MLA was reconstructed using non-matched-illum DPC (S=0.7), matched-illum DPC (S=1), TIE (S=0.7), and TI-DPC (S=0.7), respectively, and then the iterative closest point (ICP) algorithm was used to perform registration process of the 3D point cloud of the ground truth and the reconstructed phase. The red dots represent the 3D point cloud of the ground truth, and the blue dots represent the 3D point cloud of the reconstructed phase. The RMSE of the 3D point cloud matching results is used to assess the accuracy of the reconstructed surface profile. As a smooth phase object dominated by low-frequency components, the MLA posed challenges for phase reconstruction, especially when the illumination NA did not match. In such cases, non-matched-illum DPC failed to reconstruct the phase due to the absence of low-frequency information (has the largest RMSE=2.288 and the worst point cloud matching results). Even with a matched-illumination NA, the phase of the MLA was significantly underestimated due to its departure from the WOA condition. It also can be seen from the 3D point cloud that the reconstructed surface profile’s vertex height is significantly lower than the reference value (RMSE=1.304). Due to the dominance of low-frequency information in MLA, TIE has an advantage over the previous two methods, achieving a smaller error (RMSE=0.329). While the proposed TI-DPC is not constrained by WOA and can preserve high-frequency information as much as possible, it achieves twice the reconstruction accuracy of TIE, with a minimized RMSE (RMSE=0.144). To compare the phase retrieval results of MLA by these four methods more intuitively, we extracted the phase distribution at the position of the white dashed line in Fig. 8(a) and plotted it in Fig. 8(f). To show the results more clearly, the profile in the green box is enlarged in Fig. 8(f). Consistent with the previous analysis, non-matched-illum DPC fails to recover the phase of the MLA. Matched-illum DPC suffers from significant phase reconstruction noise and fails to accurately reflect the actual shape of the individual lens. Conversely, TI-DPC demonstrates superior signal-to-noise ratio and achieves accurate phase reconstruction. The associated reconstructed results are consistent with the surface profile data measured by the profilometer. The precise phase recovery of TI-DPC provides a new method for application scenarios requiring quantitative measurements such as lens surface profiling.

To validate the application of TI-DPC in the biomedical field, we performed quantitative phase imaging on unstained tongue slices and oral epithelial cells using a 40× objective with NA=0.6. Figure 9(a) presents a comparison of results among three traditional quantitative phase imaging techniques and the proposed TI-DPC, where matched-illum DPC was conducted under the condition of S=1, and the other three methods were under S=0.65. Figures 9(b1)–9(b4) provide a magnified comparison of the recovered results from the four phase imaging techniques at ROI 1. We extract the phase distribution profile of the nucleus (yellow curve) at the position of the white dashed line and the phase distribution profile of the fibrillar structure (blue curve) at the position of the white solid line. Since the phase delay of the nucleus is large and exceeds the WOA condition, matched-illum DPC underestimates the phase of the nucleus. The phase distribution of the cell nucleus is relatively smooth, mainly consisting of low-frequency information. Non-matched DPC cannot effectively reconstruct the cell nucleus due to the lack of low-frequency information. However, TI-DPC overcomes the drawback of inability to recover low frequency (phase distribution of the nucleus) under non-matched-illumination conditions and obtains a more accurate phase similar to TIE. Moreover, compared to the TIE, the proposed method recovers high-frequency information (fibrillar structure phase distribution) more effectively, achieving reconstruction results of high-frequency details comparable to matched-illum DPC. Meanwhile, Fig. 9(c) presents the quantitative phase of oral epithelial cells reconstructed by TI-DPC under a 40× objective (S=0.65). Figure 9(d) shows the magnified phase distributions at ROI 2 reconstructed using TI-DPC and matched-illum DPC. Figure 9(e) is the phase profile at the white dotted line position in Fig. 9(d), comparing the phase retrieval results between TI-DPC and matched-illum DPC. For example, in the region of the cell nucleus, the sample no longer satisfies the WOA condition, and the matched-illum DPC severely underestimates the phase value, while TI-DPC accurately recovers the true phase and thickness of cell contours and internal subcellular structures, representing the true morphology of the cells. Figures 9(f1)–9(f3) show the magnified images of the phase recovered at ROI 3 by matched-illum DPC, TIE, and TI-DPC. By observing the position indicated by the white ellipse, it is interesting to note that TIE cannot recover high-frequency information, resulting in the inability to reconstruct the organelles in the cells. The proposed TI-DPC, like matched-illum DPC, can accurately recover the phase distribution of the organelles. While matched-illum DPC significantly underestimates the phase for the nucleus, TI-DPC accurately recovers it. Due to the limitation of the NA of the illumination, the ultimate resolution of TI-DPC will be slightly lower than that of matched-illum DPC. It can be found that TI-DPC can recover the phase distribution of cells more accurately, providing more precise data for morphological analysis and dry mass analysis of cells.

This paper explores the relationship between different illumination NA and the applicability of WOA conditions, proposing a method to defy WOA using non-matched-illumination. Through intensity transmission and additional defocused intensity images, the proposed TI-DPC enhances the transmission of low-frequency phase, which realizes the precise retrieval of low-frequency information lost due to non-matched-illumination and accurately reconstructed large phase objects. Meanwhile, as a deterministic method without iteration, TI-DPC can directly recover quantitative phase, eliminating the need for strict illumination matching conditions in DPC imaging and overcoming the limitations of the WOA condition. First, the accuracy of TI-DPC was theoretically demonstrated by simulating the reconstruction process of a pure phase object with both large phase delays and high-frequency details. Moreover, the reconstruction precision of TI-DPC and its excellent transmission capability in the full detection frequency domain were validated through the reconstruction of customized artifact with fine structures and MLA with large phase delays. Experimental results on tongue slices and oral epithelial cells show that the proposed method reflects the true phase, morphology, and thickness of cells, providing precise data for subsequent analyses of cell dry mass and cell physiology. Despite reducing the NA of illumination, the method still provides high-frequency reconstruction capabilities comparable to DPC under matched-illumination conditions, with more accurate phase values at large phase locations (such as the cell nucleus). Additionally, as a PC-QPI, it only requires replacing the Köhler illumination on a commercial microscope with a programmable LED array, which means no additional illumination adjustment is needed when imaging the samples with objective lenses of different NAs—not to mention that manufacturing condenser lenses with large NA for high NA illumination is usually costly. In summary, this method opens up a new way for label-free imaging in biomedical field and precise measurement in industrial manufacturing. Due to the use of partially coherent illumination, the phase contrast produced by high-angle illumination can be counteracted by low-angle illumination, leading to reduced phase sensitivity. In the future, we will consider using annular illumination and deep learning to optimize the transfer function for more accurate phase reconstruction [22,50]. Meanwhile, the solution to the TIE currently uses approximate periodic boundary conditions, which can result in phase errors when objects are present at the field of view edges [44]. We will consider adding a hard aperture at the image plane to generate the Neumann boundary conditions required for solving the TIE as the next step, in order to achieve accurate phase solutions. An iterative approach is used to incorporate the information from the darkfield illumination into the phase results recovered by the DPC, which is able to break through the incoherent diffraction limit [30]. It is worthy of continued research in the future. Through the above improvements, TI-DPC is expected to achieve more accurate phase reconstruction, meeting more complicated demands in biomedical and precision measurement fields.

Acknowledgment. We thank Yanmei Liang from the Institute of Modern Optics for providing tongue slice and Wei Wu from the School of Physics for making the customized artifact.