1. Introduction

Optical microscopy is a powerful tool for detecting tiny structures of samples, and therefore, it is of great importance to many fields. However, traditional optical microscopy can only obtain the intensity information of the samples, and thus it is inaccessible to transparent samples, such as live cells. Fluorescence microscopy can overcome the shortage of low imaging contrast [1], despite the photobleaching and phototoxicity involved, but is troublesome for long-term observation of live samples. Alternatively, quantitative phase imaging (QPI) [2–4], as a label-free microscopic technique, can not only observe transparent samples with high contrast but also quantitatively obtain the intrinsic information of the samples. Therefore, it is very suitable for nondestructively imaging live samples.

There are many single-beam approaches that enable us to perform quantitative phase imaging on transparent samples. First, wavefront sensing, employing a Shack-Hartmann sensor [5] or a pyramid sensor [6], is a simple quantitative phase imaging approach. Second, phase-contrast microscopy [7–11] enables us to convert the phase distribution to the intensity distribution by phase retarding the DC term of the object wave. Third, differential interference phase-contrast (DIC) microscopy [12] is capable of producing high contrast images of transparent samples, while the contrast reflects the phase derivative of the measured sample rather than the phase. Fourth, beam-propagation-based phase retrieval approaches can also retrieve the phase distribution by recording a series of diffraction patterns under different conditions or applying different perturbations [13–16]. Recently, Fourier ptychographic microscopy (FPM) was proposed to reproduce the sample’s information by combining phase recovery and synthetic aperture. It can obtain the amplitude and phase information of a sample with a large field of view and high spatial resolution by recording low-resolution images at different angles and utilizing an iterative algorithm [17].

The single-beam phase retrieval approaches mentioned above have a simpler configuration but a moderately-complicated recovery process. In contrast, interferometric approaches have much higher phase measurement accuracy through interference between an object wave and a reference wave to sensor the phase [18–20]. However, in interferometric approaches, the object and reference beams follow widely separated paths and are, therefore, differently affected by mechanical shocks and temperature fluctuations. Thus, the fringe pattern in the observation plane is sensitive to environmental disturbances. To overcome this drawback, Medecki [21] proposed a point diffraction interferometer in 1996, and then Naulleau applied it to investigating wave-front aberrations [22]. Popescu [23] integrated a point-diffraction interferometer and microscopy into diffraction phase microscopy (DPM) to perform quantitative phase imaging on microscopic samples. In DPM, a diffraction grating was used to divide the object wave into two parts, one of which was converted into a reference wave after being pinhole-filtered on the Fourier plane, and the other part was still used as the object wave. The amplitude and phase information of the sample were reconstructed from the recorded off-axis hologram. In such a DPM configuration, the object and reference waves propagate along a common optical path, commonly affected by environmental disturbances. Therefore, fast, stable, and accurate phase imaging of a sample can be realized. In 2010, our group [24] proposed a phase-shifting in-line DPM based on a pair of gratings, and later incorporated it with a parallel phase-shifting module for real-time phase imaging [25]. Compared with traditional off-axis DPM, this method could make full use of the spatial bandwidth product of the camera. In general, all existing DPM techniques generate the reference wave by the pinhole-filtering of a copied object wave. Therefore, the intensity of the reference wave is substantially dependent on samples, making it challenging to have high-contrast interference patterns which are necessary for a phase reconstruction with a high-accuracy and high signal to noise ratio (SNR).

In this paper, we propose a polarization grating based diffraction phase microscopy (PG-DPM). PG-DPM retains the characteristic of conventional DPM, i.e., real-time measurement, high-stability, high phase-measurement accuracy. Furthermore, in PG-DPM the fringe contrast is adjustable by combining a quarter-wave plate (QW) and a polarization diffraction grating. Moreover, we apply PG-DPM to quantitative phase imaging of live paramecia for the first time, investigating their self-helical forward motion characteristics and digestion-related dynamics.

2. Methods

The schematic diagram of the PG-DPM system is shown in Fig. 1(a), and 3D view of the mechanical configuration in Fig. 1(b). A 532-nm solid-state crystal laser (1875-532L, Laserland, Wuhan, China) is used as the illumination source. The emitted laser beam is linearly polarized along the horizontal direction, and its polarization state is denoted with $\left[ {\begin{array}{c} \textrm{1}\\ \textrm{0} \end{array}} \right]$. A QW plate, of which the principal axis has an angle θ with respect to the horizontal direction, is placed on the beam path. After passing through the QW, the linearly polarized beam is turned into an elliptically polarized beam, denoted with $\left[ {\begin{array}{c} {\textrm{cos}\mathrm{\theta }}\\ {{i \cdot \textrm{sin}}\mathrm{\theta }} \end{array}} \right]$. After being expanded by a beam expander composed of lens L₁ and L₂, the beam illuminates perpendicularly the sample that is placed on the front focal plane of the telescope system consisting of a microscopic objective (10X/0.25, Nanjing Yingxing Optical Instrument Co., Ltd., Nanjing, China) and an achromatic lens L₃. The image of the sample appears in the rear focal plane of L₃, where a polarization grating G (#12-677, Edmund Optics, Barrington, New Jersey, America) is located and splits the beam into different copies along diffraction orders. The ±1^st diffraction orders have more than 42.5% total energy for each. The +1^st order is used as the object wave, while the -1^st order is used for the reference wave after being filtered on the middle focal plane of a telescope system L₄-L₅, as shown in the second inset of Fig. 1(a). Specifically, the +1^st order passes through a large hole on the filter mask with its spectrum not being affected, while the -1^st order is filtered by a pinhole (GCO-01100, Daheng Optics, Beijing, China) and used as the reference wave. It should be noted that it is preferable to choose a pinhole with its diameter close to 1 airy unit (AU) to generate a plane wave in the field of view of the camera. After passing through a linear polarizer P with its polarization orientation having an angle of 45° with respect to the horizontal direction, the object and reference waves become linearly polarized and interfere with each other on the plane of a camera (4000×3000 pixels, pixel size 1.85 μm, DMK 33UX226, The Imaging Source Asia Co., Ltd., China). The object wave and the reference wave at the camera plane (see Appendix A) can be expressed as

 

Fig. 1. The schematic and 3D view of PG-DPM. (a) The schematic diagram of the proposed PG-DPM system. (b) The 3D structure of the PG-DPM mechanical configuration with a compact size of 40 cm × 10 cm × 25 cm. F, filter; G, polarization grating; L₁-L₅, lenses; M, mirror; P, polarizer; QW, quarter-wave plate.

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3. Performances of PG-DPM

3.1 Adjustable fringe contrast

The most prominent advantage of the proposed system is stripe-contrast adjustable. In conventional DPM, the reference wave intensity varies with specific samples (see Appendix B), and therefore, the fringe contrast is difficult to maximize. In PG-DPM, a polarization grating which has a unique polarization-dependent diffraction characteristic, is used to overcome this disadvantage. To be specific, upon the incidence of a left (right) circularly polarized beam, the majority of the diffracted intensity will go to the +1^st (-1^st) order, as shown in Figs. 2(a) and 2(b). Moreover, the intensities of the ±1 orders vary oppositely when rotating the QW (Fig. 2(c)), i.e., the intensities have the relation of cos²θ and sin²θ with respect to the azimuth of the QW (Fig. 2(e)). Therefore, the stripe contrast of the hologram, and eventually the signal to noise ratio of the reconstruction, can be maximized (Fig. 2(f)) by rotating the QW to balance the intensities of the object wave and the reference wave (see Fig. 2(d)).

 

Fig. 2. The polarization-dependent grating and its application to stripe-contrast adjustment in PG-DPM. The majority of the intensity is diffracted to the +1^st order and the -1^st order upon the incidence of a left (a) and right (b), respectively. The intensities of the ±1^st orders (after pinhole filtering on the -1^st order) can be balanced by the incidence of an elliptically polarized beam (d), of which the principal-axis azimuth θ can be varied by a QW (c). (e) The intensities of the object and reference waves vary oppositely with the θ of the QW. (f) The stripe contrast and SNR of the reconstructed phase image versus the azimuth θ of the QW. In the SNR calculation, the real value of the phase step (with a ground-truth phase value of 2π/3 [25]) was used as the effective signal, and the deviation against the real value within the phase-step region was used as the noise. (g) The intensity distributions of the object wave, the reference wave, the hologram, and the reconstructed phase images obtained with θ = 20°, 50°, 80°, 110°, respectively. CPL, circularly polarized light; LPL, linearly polarized light; EPL, elliptically polarized light; QW, quarter-wave plate; G, grating; O, objective wave; R, reference wave; I, recorded hologram; φ, reconstructed phase.

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3.2 High stability

Another important characteristic of the proposed PG-DPM system is high stability due to its common-path configuration. Often, high stability is essential to detect minute optical path length (OPL) variations, for example, induced by nanoscale thermal fluctuations of red blood cells of live blood cells [26]. To assess the stability of the proposed setup against the environmental disturbance, we performed a continuous phase measurement in the absence of any specimen for 180 mins at the interval of 20 s using both traditional DHM and the proposed PG-DPM system. The phase value of a randomly-selected point (indicated with the rectangle in Fig. 3(a)) over the 180 mins is calculated for both systems and shown with red and blue dots in Fig. 3(b), respectively. The statistics tell that the phase of the traditional DHM system floats randomly between 0 and 2π. In contrast, the phase of PG-DPM fluctuates in a small region with a full-width of 0.3 rad, implying that the proposed PG-DPM setup has 20-fold higher stability.

 

Fig. 3. The stability test of the PG-DPM system. (a) The phase distribution obtained at a particular time point without samples. The red curves indicate the contour lines with an interval of 1 rad. (b) comparison of phase stability of the traditional DHM and PG-DPM. The red and blue dots indicate the phase of a randomly-selected point (indicated with the black rectangle in (a)) measured at different time using traditional DHM and PG-DPM, respectively. Scale bar in (a), 10 μm.

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4. Experimental results

4.1 Quantitative phase imaging of lily anther

PG-DPM was firstly demonstrated with the imaging of lily anther (LIOO Optics, Beijing JinghaoYongcheng Trading Co., Ltd., Beijing, China), which was sandwiched between two cover-slips. The recorded hologram I($\vec{r}$) of the lily anther is shown in Fig. 4(a), while the spectrum of I·R_D($\vec{r}$) is shown in Fig. 4(b). The amplitude and phase images were reconstructed using Eq. (3) with d₀=0, and shown in Figs. 4(c) and 4(d), respectively. There are almost no structures visible in the amplitude image due to its transparency. By contrast, the detailed cell wall structures of the anther become apparent in the phase image. The comparison implies that PG-DPM is capable of extracting the fine structures of transparent samples.

 

Fig. 4. PG-DPM imaging on lily anther. (a) The recorded hologram. (b) The spectrum of IR_D and the selection of a window function ${\hat{\textrm W}(}\mathrm{\xi }{,\eta )}$ (shown as a yellow circle). (c)-(d) The reconstructed amplitude (a.u.) and phase (unit: rad) images, respectively.

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4.2 Quantitative phase imaging of live paramecia

PG-DPM has then been applied to image live paramecia in the water-glue mixture (1:1, v/v), and target their helical motion and digestion-related morphological dynamics. Paramecia are unicellular protists that are naturally found in aquatic habitats. The body of a paramecium is with thousands of fine cilia that facilitate the movement of paramecia. Often, paramecia perform helical motion when encountered with preys or predators, as well as swimming in the confined or viscous environment [27–34]. Compared with conventional microscopy, quantitative phase microscopy is more suitable to observe the helical motion of paramecia, since the latter one is capable of resolving the thickness variation during the helical motion.

In this experiment, paramecia, incubated with water, are confined in a micro-channel between two cover-slips. After placing the micro-channel in the PG-DPM setup, a hologram series were recorded in sequence with a time interval of one second. Then, the amplitude and phase images were reconstructed with Eq. (3). Figures 5(a)–5(c) shows the holograms, the reconstructed amplitude images, and phase images at different time with intervals of 5 seconds, respectively. We can find that, compared to the amplitude images, the phase images in Fig. 5(c) and especially those in Visualization 1 clearly reveal the helical-motion related morphological change of the paramecium. Furthermore, we have also conducted statistics on the self-helical direction for the forward-motion with 60 paramecia. The statistics in Fig. 6(a) tells the clockwise and anti-clockwise helical rotation-directions for the forward-movement (when facing the forward movement direction) have the proportions 77% and 23%, respectively.

 

Fig. 5. PG-DPM imaging on live paramecia. The helical motion of a paramecium in the mixed water-glue solution (1:1, v/v). The holograms (a), the reconstructed amplitude images (b), and phase images (unit, rad) of a paramecium (c) at different time points with a time interval of 5 s. The green scale bar in (b) and (c), 60μm. The red arrow in (c) positions the oral groove of the paramecium. The attached Visualization 1 and Visualization 2 display the helical motion and the morphological dynamics of a paramecium, respectively. The displaying frame rates of the two visualizations are 7 Hz and 20 Hz, respectively.

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Fig. 6. The statistics on helical motion and shapes of paramecia. The self-helical direction (a) and the shape parameters (b) (the aspect ratio along the x-axis and area-to-perimeter ratio along the y-axis) of 60 paramecia.

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Moreover, the digestion-related morphological dynamics of paramecia have also been investigated with PG-DPM. It is known that paramecia live with microorganisms, such as bacteria and single-celled algae, as their main food. The digestion of these microorganisms follows an inner-circulation along a certain route in the bodies of paramecia. Such digestion-orientated dynamics was observed with PG-DPM and shown in Visualization 2. Interestingly, we found from the phase images in the movie that several components on the digestion path were repeatedly twinkling with time, but they are not observable under the amplitude images due to their transparency. However, the nature of the twinkling components is still in need of exploration later on.

Lastly, we have also conducted statistics on the geometrical sizes of 60 paramecia imaged with our PG-DPM system. For simplicity, the shape of a paramecium is modeled as an ellipse. The statistics in Fig. 6(b) reveals that the aspect ratio (defined with the ratio of the long axis to the short axis) is 4.3 ± 0.4 (mean ± standard deviation), and the area-to-perimeter ratio (defined with the cross-sectional area of the ellipse to the perimeter) is 16.3 ± 1.3 (mean ± standard deviation). These two characteristic parameters are essential to automatically identify the types of paramecia. The above applications of PG-DPM indicate that PG-DPM has the capability to image transparent samples with high-contrast and in a quantitative manner.

5. Conclusion

In this study, we have developed a polarization-grating-based point-diffraction interferometric microscopy (PG-DPM) for quantitative phase measurement. A polarization grating is used to diffract the object wave into several copies, and the ±1^st diffraction orders are used as the object wave and the reference wave after pinhole-filtering on the -1^st order. The relative intensity of the object and reference waves can be freely adjusted by rotating a quarter-wave plate and therefore, the stripe contrast of the recorded holograms can be maximized. PG-DPM, featuring with a common-path configuration and adjustable stripe-contrast, has the advantages of high stability, high accuracy, and real-time amplitude/phase imaging. It is also worthy to mention that when the PG-DPM system is used to image samples with intrinsic birefringence, the fringe contrast and the measured phase will be influenced or complicated by the birefringence. Nevertheless, the birefringence of the sample can be characterized, once the phase of the sample is measured with different polarizations of the illumination wave.

Using the proposed system, we have captured the internal structures of fixed lily anther and live paramecia, verifying the feasibility of PG-DPM for real-time quantitative phase imaging. Of note, it is found that quantitative phase imaging via PG-DPM is more suitable for investigating the helical movement of paramecia. Eventually, we report, as the first time, that paramecia have a majority helical rotation-direction along the clockwise direction when performing forward-motion. In the future, with PG-DPM we will conduct a more systematic study on the motion-orientated features of paramecia, as well as their inner-relations.

Appendix A: Theoretical derivation of the polarization states

In this section, we demonstrate that the fringe contrast of the hologram can be adjusted by rotating the quarter-wave (QW) plate in Fig. 1(a).

The light emitted by the laser is linearly polarized along the horizontal direction with its polarization state recorded as $\left[ {\begin{array}{c} \textrm{1}\\ \textrm{0} \end{array}} \right]$. After passing through a quarter-wave plate (QW) of which the principal axis has an azimuth θ with respect to the horizontal direction, the illumination light is turned into an elliptically polarized light and its polarization state can be expressed as$\; \left[ {\begin{array}{c} {{cos}\mathrm{\theta }}\\ {{i \cdot sin}\mathrm{\theta }} \end{array}} \right]$.

Then, the illumination light passes through a sample, and imaged by a telescope system MO-L₃ to a polarization grating G. The polarization grating G divides the object wave into several parts, and propagate along the directions of different diffraction orders, respectively. Among them, the ±1^st diffraction orders are respectively used as the object wave and the reference wave after filtering the -1^st order by a pinhole at the Fourier plane. Passing through a linear polarizer P located before the camera, the object wave and the reference wave become linearly polarized with the same polarization direction, and then interfere with each other on the camera plane. Here, we theoretically analyze the polarization diffraction process and the complex transmittance of the polarization grating can be expressed as

(4)$$\tilde{T}\textrm{(}x,\textrm{y)} = {e^{i{k_0}{n_0}d}}\left[ {\begin{array}{cc} {{\tau_e}{\textrm{e}^{i{k_0}\Delta {n_\textrm{e}}d}}{{\cos }^2}{\psi_e} + {\tau_o}{\textrm{e}^{i{k_0}\Delta {n_\textrm{o}}d}}{{\sin }^2}{\psi_e}}&{({\tau_e}{\textrm{e}^{i{k_0}\Delta {n_\textrm{e}}d}} - {\tau_o}{\textrm{e}^{i{k_0}\Delta {n_\textrm{o}}d}})\sin {\psi_e}\cos {\psi_e}}\\ {({\tau_e}{\textrm{e}^{i{k_0}\Delta {n_\textrm{e}}d}} - {\tau_o}{\textrm{e}^{i{k_0}\Delta {n_\textrm{o}}d}})\sin {\psi_e}\cos {\psi_e}}&{{\tau_e}{\textrm{e}^{i{k_0}\Delta {n_\textrm{e}}d}}{{\sin }^2}{\psi_e} + {\tau_o}{\textrm{e}^{i{k_0}\Delta {n_\textrm{o}}d}}{{\cos }^2}{\psi_e}} \end{array}} \right].$$

Here, τ_o and τ_e represent the transmittance of o light and e light, respectively. Δn_o and Δn_e denote the refractive index of the grating for the o light and e light, respectively. d(x,y) represents the grating thickness distribution. φ_o(x, y)=k₀Δn_od and φ_e(x, y)=k₀Δn_ed are the spatial phase modulation functions of the polarization grating for the o light and e light, respectively, and k₀=2π/λ. Ψ_e(x,y) denotes the direction of the principal axis of the anisotropy changing with space. We can expand the polarization components ${{\tilde{\textrm T}}_{\textrm{11}}}$, ${{\tilde{\textrm T}}_{\textrm{12}}}$, ${{\tilde{\textrm T}}_{\textrm{22}}}$ of ${\tilde{\textrm T}(t,\; x,}\mathrm{\gamma }\textrm{)}$ into Fourier series as ${\textrm{T}_{\textrm{ij}}}\textrm{(x,y) = }\sum \textrm{T}_{\textrm{ij}}^\textrm{l} \cdot \textrm{exp}({\textrm{ilKx}} )$. Here, the values of i and j are 1 and 2 while l=0,±1, ±1…representing the diffraction order. K=2π/Λ is the grating vector with Λ the period of the grating. Then, the transmission matrix for the light wave propagating along the +1^st order of the grating can be written as

(5)$$\begin{array}{l} \tilde{T}_1^{ + 1}\textrm{(}x,y\textrm{)} = \left[ {\begin{array}{cc} {T_{11}^{ + 1}}&{T_{12}^{ + 1}}\\ {T_{12}^{ + 1}}&{T_{22}^{ + 1}} \end{array}} \right],\\ T_{11}^{ + 1} = \frac{1}{{4\pi }}\int_0^{2\pi } {[{\tau _e}\textrm{exp(}i{k_0}\Delta {n_e}d){{\cos }^2}{\psi _e} + {\tau _o}\textrm{exp(}i{k_0}\Delta {n_o}d){{\sin }^2}{\psi _e}]\exp ( - iKx)} d(Kx),\\ T_{12}^{ + 1} = \frac{1}{{4\pi }}\int_0^{2\pi } {[{\tau _e}\textrm{exp(}i{k_0}\Delta {n_e}d) - {\tau _o}\textrm{exp(}i{k_0}\Delta {n_o}d)]\sin (2{\psi _e})\exp ( - iKx)} d(Kx),\\ T_{22}^{ + 1} = \frac{1}{{4\pi }}\int_0^{2\pi } {[{\tau _e}\textrm{exp(}i{k_0}\Delta {n_e}d){{\sin }^2}{\psi _e} + {\tau _o}\textrm{exp(}i{k_0}\Delta {n_o}d){{\cos }^2}{\psi _e}]\exp ( - iKx)} d(Kx). \end{array}$$

The transmission matrix for the light wave propagating along the -1^st diffraction order of the grating can be expressed as

(6)$$\begin{array}{l} \tilde{T}_1^{ - 1}\textrm{(}x,y\textrm{)} = \left[ {\begin{array}{cc} {T_{11}^{ - 1}}&{T_{12}^{ - 1}}\\ {T_{12}^{ - 1}}&{T_{22}^{ - 1}} \end{array}} \right],\\ T_{11}^{ - 1} = \frac{1}{{4\pi }}\int_0^{2\pi } {[{\tau _e}\textrm{exp(}i{k_0}\Delta {n_e}d){{\cos }^2}{\psi _e} + {\tau _o}\textrm{exp(}i{k_0}\Delta {n_o}d){{\sin }^2}{\psi _e}]\exp ( - iKx)} d(Kx),\\ T_{12}^{ - 1} = \frac{1}{{4\pi }}\int_0^{2\pi } {[{\tau _e}\textrm{exp(}i{k_0}\Delta {n_e}d) - {\tau _o}\textrm{exp(}i{k_0}\Delta {n_o}d)]\sin (2{\psi _e})\exp ( - iKx)} d(Kx),\\ T_{22}^{ - 1} = \frac{1}{{4\pi }}\int_0^{2\pi } {[{\tau _e}\textrm{exp(}i{k_0}\Delta {n_e}d){{\sin }^2}{\psi _e} + {\tau _o}\textrm{exp(}i{k_0}\Delta {n_o}d){{\cos }^2}{\psi _e}]\exp ( - iKx)} d(Kx). \end{array}$$

After passing through a sample, the incidence beam forms the object wave O(x,y) and then diffracted by the grating, the polarization state of ±1^st diffraction orders can be expressed as

(7)$$\begin{array}{c} {{D_1} = \left[ {\begin{array}{c} {T_{11}^{ + 1} \cdot \textrm{cos}\mathrm{\theta } + i \cdot T_{12}^{ + 1} \cdot \textrm{sin}\mathrm{\theta }}\\ {T_{12}^{ + 1} \cdot \textrm{cos}\mathrm{\theta } + i \cdot T_{22}^{ + 1} \cdot \textrm{sin}\mathrm{\theta }} \end{array}} \right],}\\ {{D_{ - 1}} = \left[ {\begin{array}{c} {T_{11}^{ - 1} \cdot \textrm{cos}\mathrm{\theta } + i \cdot T_{12}^{ - 1} \cdot \textrm{sin}\mathrm{\theta }}\\ {T_{12}^{ - 1} \cdot \textrm{cos}\mathrm{\theta } + i \cdot T_{22}^{ - 1} \cdot \textrm{sin}\mathrm{\theta }} \end{array}} \right].} \end{array}$$

The constant term $\textrm{C}{\textrm{e}^{\textrm{i}{\textrm{k}_\textrm{0}}{\textrm{n}_\textrm{0}}\textrm{d}}}$ is ignored here. After the diffraction of the grating, the object wave is split into different copies. The copy along the +1^st order with the polarization state of D₁ is used as the object wave, while the -1^st diffraction order with the polarization state of D_-1 is filtered by the pinhole to form the reference wave. After passing through the polarizer P, the object and reference waves become linearly polarized with the same polarization direction, and interfere with each other at the camera plane. Specifically, when the polarization direction of the polarizer P is 45° relative to the horizontal direction, its transmittance function can be written as

(8)$${\tilde{t}_P}({t,x} )\textrm{ = }\frac{1}{2} \cdot \left[{\begin{array}{cc} 1&1\\ 1&1 \end{array}}\right].$$

Thus, the complex amplitude of the object and reference waves after passing through the polarizer P can be expressed as

(9)$$\begin{array}{c} {\tilde{O} = \frac{1}{2} \cdot [{({T_{11}^{ + 1} + T_{12}^{ + 1}} )\cdot \cos \theta + i \cdot ({T_{12}^{ + 1} + T_{22}^{ + 1}} )\cdot \sin \theta } ]\cdot \left[ {\begin{array}{c} 1\\ 1 \end{array}} \right],}\\ {\tilde{R} = \frac{1}{2} \cdot [{({T_{11}^{ - 1} + T_{12}^{ - 1}} )\cdot \cos \theta + i \cdot ({T_{12}^{ - 1} + T_{22}^{ - 1}} )\cdot \sin \theta } ]\cdot \left[ {\begin{array}{c} 1\\ 1 \end{array}} \right].} \end{array}$$

As $(T_{11}^{ + 1} + T_{12}^{ + 1}) > > (T_{12}^{ + 1} + T_{22}^{ + 1})$ and $(T_{12}^{ - 1} + T_{22}^{ - 1}) > > (T_{11}^{ - 1} + T_{12}^{ - 1})$ for the polarization grating we used, thus Eq. (9) can be re-written as Eq. (1). It can be seen from Eq. (1) that the relative intensity of $\tilde{O}$ and $\tilde{R}$ is related to the polarization state of the illumination light that is in turn determined by the azimuth of the principal axis of the QW relative to the horizontal direction. Therefore, the stripe contrast of the hologram can be maximized (Fig. 2(f)) by rotating the QW to balance the intensities of the object wave and the reference wave.

Appendix B: The contrast of interference stripes is sample-dependent

In this section, we demonstrate that the intensity of the reference wave and as well as the fringe contrast of the hologram is sample-dependent. In traditional DPM, the reference wave is generated by pinhole-filtering one copied object wave to eliminate the sample-related information. Also, there is the fact that the illumination beam is divided into the scattered wave and unscattered wave after passing through the sample, whose energy proportions are significantly impacted by the scattering degree of the sample. In addition, the role of the pinhole is to reject the scattered wave and transfer the unscattered wave to form a uniform reference wave. Therefore, the intensity of the reference wave reaching the camera plane is dramatically influenced by the measured sample. In order to verify this feature, we measured the intensity of the object wave and the reference wave before and after placing a scattered sample, and the results are shown in Fig. 7. The images in the first row of Fig. 7 are the intensity distributions of the object wave (O, left) and the reference wave (R, right) before placing a scattered sample. The second row is the corresponding intensity distributions after placing a sample of the mature lily anther. Comparing the images of the upper row and the lower one, we can clearly find that the intensity of the reference wave generated by the pinhole filtering is relatively small when the scattering of the sample is stronger. This is due to the fact that when the sample structure is relatively complex, the energy of the sample is mostly distributed in high-frequency regions, and has no contributions to the reference wave during the pinhole-filtering operation. To summarize, in the traditional DPM, the contrast of the interference stripe is sample dependent, which can not guarantee high-contrast interference patterns and eventually a high-quality reconstruction for all the samples. Therefore, it is of great importance to make the fringe-contrast adjustable in DPM techniques.

 

Fig. 7. The effect of the sample on the intensity of the reference wave generated by pinhole-filtering of the object wave.

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Appendix C: The stability test of a traditional DHM

For comparison of the stability of the PG-DPM and the traditional DHM systems, we conducted a stability test with both the traditional DHM system (home-built, see Figs. 8(a) and 8(b)) and the PG-DPM system. In the traditional DHM system, a 473-nm diode laser (MLL-U-473, Changchun New Industries Optoelectronics Technology Co., Ltd., China) is coupled into a 1 × 2 fiber splitter and used for the illumination source. The light from one end of the fiber splitter is collimated by a lens L₂, demagnified by the telescope system L₃-MO₁, and illuminates a sample from the bottom to the top. After passing through the sample, the object wave is magnified by the second telescope system MO₂-L₄, and the image appears at the CCD plane. Meanwhile, the light from the other end of the fiber splitter is collimated by a lens L₁ and is used as the reference wave. The object and reference waves are superimposed with each other after passing through a beamsplitter BS and interference with each other. A CCD camera (DMK 23U274, The Imaging Source Asia Co., Ltd., China) records the generated hologram, from which both the amplitude and phase distributions of the sample can be obtained [23].

 

Fig. 8. Stability test of the traditional DHM and the PG-DPM system. (a) the schematic diagram and (b) the picture of our home-built DHM system, which has separate object and reference waves. (c) the phase of a randomly-selected point fluctuates with time for both the traditional DHM and the PG-DPM systems. (d) 5-s phase fluctuation of the selected point in the traditional DHM. (e) the statistics of the phases in (c) for the traditional DHM and the PG-DPM system. The attached Visualization 3 displays the fringe vibration in a traditional DHM hologram, of which the displaying rate is ten frames per second.

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As described in Section 3.2, we performed quantitative phase measurement in the absence of any specimen for 180 mins at the interval of 20 s using both the traditional DHM and PG-DPM. The phase of a randomly-selected point in the field of view fluctuates with time for both the traditional DHM and the PG-DPM system, and the results are shown with blue and red dots in Fig. 8(c), respectively. In addition, Fig. 8(d) shows a short-term (5 s) phase fluctuation in the traditional DHM system. Furthermore, the statistics in Fig. 8(e) tell that the phase of the traditional DHM system floats randomly between 0 and 2π with a standard deviation of 1.80 rad. By contrast, the phase of PG-DPM fluctuates around a static value of 0.7 rad with a standard deviation of 0.08 rad, indicating that the proposed setup has around 20-fold higher stability.

Funding

Natural Science Foundation of Shaanxi Province (2020JM-193, 2020JQ-324); China Postdoctoral Science Foundation (2017M610623); Higher Education Discipline Innovation Project; State Key Laboratory of Transient Optics and Photonics (SKLST201804); Fundamental Research Funds for the Central Universities (XJS190508).

Disclosures

The authors declare no conflicts of interest.

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