Beating temporal phase sensitivity limit in off-axis interferometry based quantitative phase microscopy

Measuring morphological changes in time is important for studying cellular activities and material processes. Such changes can be quantified through quantitative phase microscopy (QPM). As a label-free imaging method, QPM precisely maps the optical path length (OPL) through interferometry or computation from intensity measurements.^1–4 However, measuring intrinsically weak nanoscale changes (e.g., protein aggregation,⁵ virus particle dynamics,⁶ and spiking-induced membrane fluctuations⁷) without exogenous labeling is very challenging with conventional QPM methods due to the low signal-to-noise ratio (SNR). Many factors can contribute to noise during phase measurements, such as photon shot noise, electronic source (e.g., device instability, camera noise, and quantization errors), light source instability, light source coherence properties, and other environmental factors (e.g., mechanical vibrations and air-density variations).^8–12 

Realizing the influence of different noise factors, over the past 15 years, much effort has been made to enhance the phase sensitivity in off-axis interferometry-based QPM systems (off-axis QPM in short), especially the temporal phase sensitivity. Chen et al.¹³ derived the best achievable phase sensitivity using the Fourier transform-based algorithm in off-axis digital holography. Shaked et al.¹⁴ reported placing of the interferometer in a vacuum-sealed enclosure to avoid the influence of the air flow and used a floating optical table to dampen the device oscillations to some extent. Zhou et al.¹⁵ explored the use of high stability laser sources (i.e., temperature and current controlled semiconductor lasers) and broadband light sources to minimize the phase noise. Popescu et al.¹⁶ developed diffraction phase microscopy (DPM) with a common-path off-axis interferometry geometry to diminish the mechanical vibrations. By introducing white-light illumination sources, both temporal and spatial sensitivities have been significantly improved in DPM due to the reduction of speckle noise inherent to laser sources.⁹ Considering the coherence properties of light, Shin et al.¹⁷ showed that the speckle noise can be minimized by reducing both spatial and temporal coherence lengths of the illumination. Slabý et al.¹⁸ proved that coherent noise in phase images can be suppressed by exploring the coherence gating effect. Majeed et al.¹⁹ demonstrated that applying spatiotemporal filtering can push the temporal phase sensitivity to 5 pm. Despite efforts using low noise cameras, high stability light sources, white-light sources, common-path interferometry geometries, and digital filtering schemes to reduce the phase noise, the photon shot noise plays a dominant role in limiting the phase sensitivity. On the other hand, the noise due to environmental factors is often difficult to control. Hosseini et al.²⁰ reported that using a high electron-well-capacity camera in a common-path off-axis QPM system results in a much higher temporal sensitivity than using a normal camera. Ling et al.²¹ achieved an even higher phase sensitivity of less than 4 pm by applying a spike-triggered averaging method with an ultra-fast speed camera. To observe the roadmap in improving the temporal phase sensitivity in off-axis QPM, we highlighted several representative works and extracted their reported temporal phase sensitivity in OPL values (Fig. 1). To alternatively characterize phase sensitivity, we define a new phase SNR metric as $SNR=10log10(2π/φ)=10log10λ/OPL$⁠, where φ is the phase noise value in radians, OPL is related to φ through OPL = φ · λ/2π, and λ is the central wavelength of the light source in free space. Note that in Refs. 20 and 21, where the phase SNR values are calculated to be over 45 dB, the systems were claimed to be operating under the photon shot noise limit. Apart from off-axis QPM methods, Wang et al.²² reported spatial light interference microscopy, based on the phase-shifting technique, that achieved both high temporal and spatial sensitivities. Chen et al.⁸ proposed a three-level framework for sensitivity evaluation in wavelength shifting interferometry. Like many QPM methods using laser interferometry, an extremely high path length sensitivity, on the order of 10⁻⁹ nm–10⁻¹² nm, was achieved by using the Laser Interferometer Gravitational-Wave Observatory (LIGO) gravitational wave detector [Ref. 10 (p. 137) and Ref. 23].

Based on the previous experimental observations and theories, in this work, we mainly explore how mechanical vibrations from different environmental sources and photon shot noise limit the temporal phase sensitivity in off-axis QPM with laser illumination. We also propose and demonstrate several methods or in combination to push the temporal phase sensitivity limit in this type of system. The temporal phase sensitivity limit is first explored and compared in common-path (i.e., DPM configuration) and non-common-path interferometry based off-axis QPM systems when operating under different environmental conditions. A spatiotemporal filtering method and a frame summing method are then used to further push the temporal phase sensitivity to around 2 pm, which, to the best of our knowledge, is better than any reported results for off-axis QPM.

To first explore how the mechanical vibration from the environment affects the phase sensitivity, we constructed two different QPM systems schematically shown in Figs. 2(a)–2(c). Figures 2(a) and 2(b) are configured to form a QPM design with a common-path Mach–Zehnder interferometer geometry (i.e., a DPM system), while Figs. 2(a) and 2(c) are configured to form a QPM design with a regular non-common-path Mach–Zehnder interferometer geometry. In both QPM systems, a single-mode fiber coupled laser with a central wavelength of 532 nm is used as the illumination source (MGL-FN-532, CNI laser). After the laser beam is collimated by lens L2, a 4f system, consisting of FL1 and object lens OL1 (EC Plan-Neofluar 10×/0.3 Ph1 M27, Zeiss), is used to bring the beam to the sample plane with a relatively uniform intensity distribution. The scattered light from the sample is collected by an imaging objective OL2 (EC Plan-Neofluar 40×/0.75 M27, Zeiss), and an intermediate image is formed after the tube lens TL1 (TL1 and OL2 forms a 4f system). From the intermediate image plane, an off-axis interferometer is constructed for retrieving the image field phase. For the common-path design [Fig. 2(b)], a diffraction grating (DG) is placed precisely at the intermediate image plane to divide the image field into multiple orders with each order containing the same image information (note that only the zeroth order and first order beams are used later). At the Fourier plane after L4, a 10 µm diameter pinhole (PH) is used as a low-pass filter to convert the zeroth order beam into a reference beam that does not contain sample information, while the first order beam remains unfiltered. Lenses L4 and L5 form another 4f system to relay the sample to the final image plane, where interferograms are finally formed on a camera. For the non-common-path design [Fig. 2(c)], the illumination beam is separated into two by using a 1 × 2 fiber coupler. The reference beam is collimated by lens L3 and then combined with the sample beam through a beam splitter (BS). P1 and P2 are the polarizers for adjusting the intensities of the two beams to achieve the best fringe contrast. In this study, a high electron-well-capacity camera with a full electron-well-capacity of 2 × 10⁶ electrons (Q-2A750/CXP, Adimec) is used to capture interferograms in both QPM systems. The whole system is sealed in a black box to minimize the airflow and prevent ambient light entering the camera. The system was constructed on a thick isolating optical table.

To quantify temporal phase sensitivity and spatial phase sensitivity of both QPM systems, we measured sample-free interferograms under four different environmental conditions: (1) during day and air conditioner on, (2) day and air conditioner off, (3) night and air conditioner on, and (4) night and air conditioner off. Under each condition, 25 sample-free interferogram stacks were recorded at 500 frames per second (fps) with an exposure time of 1337 µs. Each stack contains 601 interferograms, and the image size is 1024 × 1024 pixels. Different stacks of data were recorded over a long period of time, i.e., >2 h when recording 25 stacks, and each stack took around 3 min–5 min to save to the hard drive (note that at 500 fps, the maximal recording time for the camera is 20 s, i.e., 10 000 frames in total as limited by the camera buffer). In each stack, the first frame is used for calibrating the phase map retrieved using a Fourier transform based algorithm.²⁴ After obtaining the phase frames in each stack, we computed the frequency spectrum using the mean phase values of each phase frame. The averaged frequency spectra of 25 stacks are shown in Figs. 2(d) and 2(e) for common-path and non-common-path systems, respectively. We found the environmental disturbance (mainly the mechanical vibrations of the building structure) is prominent at the 15 Hz–30 Hz frequency band. In both the low frequency region and the high frequency region, spectrum peaks are observed in the non-common-path system. From the spectrum analysis, we conclude that the common-path design can achieve better phase sensitivity through isolating both high frequency and low frequency noise induced by mechanical vibrations from the environment.

To quantify the phase noise under each experimental condition, we calculated the phase sensitivity values in terms of OPL values and summarized them in Table I (OPL_t—temporal phase sensitivity and OPL_s—spatial phase sensitivity; these values are obtained from the 25 stacks and both mean and standard deviation values are included). OPL_t is defined as the mean value of the standard deviation map of all the points in the frame, while the standard deviation map is obtained by computing the standard deviation at each pixel in the time series of N measurements. OPL_s is defined as the mean value of standard deviation maps corresponding to the N measurements. At condition 4 (night and air conditioner off), when environmental disturbance is minimized, the lowest phase noise occurred, and the best temporal phase sensitivity was around 0.077 nm in average. The results also indicate that the non-common-path system is more sensitive to environmental disturbance as expected. This specifically designed experiment serves as a confirmation of the common-path QPM design in isolating environmental vibrations. When environment factors are minimized, the temporal phase sensitivity is mainly determined by the photon shot noise, which originates from the discrete nature of photons. In off-axis QPM, the temporal phase sensitivity is related to the effective electron-well-capacity (N_eff) of the camera as $φ≈1/Neff$⁠.²⁰ N_eff can be derived from the histogram of a typical interferogram, as shown in Fig. 3(b). For our common-path system, we estimate $Neff=N2−N1≈500000$ electrons. Note that in off-axis QPM, the phase is interpreted from fringes and the phase value at each pixel is affected by the whole diffraction spot area. Therefore, the phase sensitivity needs to be weighted over all the pixels within one diffraction spot, i.e., $φ≈1/m⋅Neff$⁠, where m is the effective number of pixels.²⁵ A detailed derivation on the theoretical temporal phase sensitivity limit (also considering the frame summing method in Sec. III A) is provided in Sec. I of the supplementary material. In our QPM system, there are around 4.3 pixels for each fringe period and 12 pixels for each diffraction-limited spot, and we determined m to be 14. Then, the theoretical temporal phase sensitivity value, OPL_t, is calculated to be around 0.032 nm for our system. From the values in Table I, we found that the best matched result is from the common-path system that has the minimum noise contribution from the environmental vibrations. The smallest OPL_t among the 25 OPL_t values computed from all stacks is 0.044 nm. We also compared the temporal phase sensitivity between a normal camera and the high electron-well-capacity camera in Sec. II of the supplementary material. The results were found to follow the theoretical model.

To improve the phase sensitivity in common-path QPM, we explore a frame summing method that effectively increases the electron well capacity. The procedure of this method is illustrated in Fig. 3(a). We initially subtract the mean value of the entire frame from each pixel and normalize the intensity of the raw interferograms to 0–1. The normalized n · T raw interferograms are divided into n groups with each group containing T frames. After that, we sum the images in each group pixel by pixel to generate a summed interferogram stack (P = {I_s1, I_s2, …, I_sn}). Following that, we use the first summed interferogram I_s1 (indicated by the blue dotted box) as the reference for calibrating the phase images. After obtaining the phase stack (Q = {p_s1, p_s2, …, p_s(n−1)}), each phase map is calibrated by subtracting the frame mean value. Then, the standard deviation of each pixel is calculated for the calibrated phase map sequences. Finally, the sensitivity value is obtained by averaging the standard deviation map. In experiments, a stack of 600 sample-free interferograms were acquired at 500 fps with an exposure time of 1337 µs from the DPM system. As shown in Table II, improvement of more than four times in OPL_t can be achieved to around 8 pm by summing 100 frames. The result is smaller than the theoretical improvement of ten times, which is likely due to the residual environmental vibration-induced noise. Note that as temporal phase sensitivity improves, spatial phase sensitivity also improves. A further study on the influence factors in the frame summing method, including bit depth, frame rate, effective electron-well-capacity, and number of frames recorded, is provided in Sec. III of the supplementary material.

To further beat the temporal phase sensitivity limit, we applied a temporal and spatial bandwidth (BW) filtering method (or spatiotemporal filtering method in short¹⁹) to diminish the vibrational noise. To determine how the spatial and temporal scales affect the sensitivity, we draw a 3D spatiotemporal map by taking the Fourier transform along x, y, and t of the selected image stack, as shown in Fig. 4(a). The spatial bands selected are circles centered at the origin with radius of 2π, π, and π/2. When selecting the low frequency BW [15 Hz–55 Hz band with noticeable vibration-induced noise according to Fig. 2(d)], the OPL_t values are reduced to 5.6 pm (2π), 4.4 pm (π), and 3.4 pm (π/2) as we reduce the spatial BW. This indicates that the temporal phase sensitivity can be improved through sacrificing the spatial resolution for a selected temporal frequency range. When selecting the high temporal frequency BW (180 Hz–220 Hz band with negligible vibration-induced noise), the best OPL_t value is 1.9 pm when the spatial BW is π/2. This analysis indicates that if there is flexibility in selecting the spatial and temporal frequency band, one can beat the phase sensitivity limit.

To demonstrate the feasibility of the proposed methods for imaging biological specimens, we measured the membrane displacement of live human red blood cells (RBCs) suspended in a phosphate-buffered saline (PBS) solution on glass [Fig. 5(a)]. For the selected RBC [Fig. 5(b)], the diameter was ∼7.8 µm and the average heights of the rim and the dimple areas were 2.2 µm and 1.4 µm, respectively. The morphological parameters are consistent with the literature.²⁶ We recorded 500 sample frames at 500 fps with 1337 µs exposure time and obtained the OPL displacement map. We selected eight representative regions with 30 × 30 pixels for comparing the phase sensitivity improvement. A higher displacement is observed at the rim of the RBC than in the dimple region. The original OPL_t values of region 1 and 2 are 3.1 nm and 2.9 nm, respectively. By dividing the refractive index difference of 0.063 between the RBC and medium, we obtain membrane displacements of 48.6 nm and 45.7 nm that are comparable to the values in the literature.²⁷ Referring to Ref. 28, we selected a 0 Hz–25 Hz temporal BW and a 3π radius spatial BW central circle for spatiotemporal filtering. A 100-frame summing method is then applied to the reconstructed phase stack. At background regions 7 and 8, improvement of more than three times in temporal phase sensitivity has been achieved to 0.13 nm and 0.15 nm, respectively. The sensitivity enhancement is not close to the sample-free theoretical limit, which might be due to the following two reasons: (1) we did not eliminate the frequency at around 20 Hz where there is vibration-induced noise, and (2) the liquid medium and sample motions prevented us to achieve temporal phase sensitivity close to the theoretical limit.

We added a further study on the liquid medium effect in Sec. IV of the supplementary material, where we compared the phase sensitivity of different mounting schemes, including air, glass, PBS on glass, and cell culture medium on glass. We found that PBS does not have a significant influence on the temporal sensitivity measurement, while the cell culture medium largely deteriorates both temporal and spatial phase sensitivities. Next, we explored the sample motion effect on phase sensitivity. We measured a fabricated phase target on glass (quantitative phase target, Benchmark Technologies) [Fig. 5(e)] and live human embryonic kidney (HEK) 293T cells immersed in PBS on glass [Fig. 5(f)]. Note that the cell culture medium was replaced with PBS to minimize the medium motion effect. We recorded 500 frames at 500 fps with an exposure time of 1337 µs for each specimen. Then, we applied the same sensitivity enhancement procedure as used in RBC imaging and selected two 30 × 30 region of interest (ROI) in the background. The temporal phase sensitivity values at region 10 (phase target) have improved from 0.051 nm to 0.0097 nm, while at region 11 (HEK 293T cell), the values have improved from 0.13 nm to 0.077 nm. The result from the phase target is close to the sample-free result in Table II. For the HEK cell imaging, the temporal phase sensitivity value is larger than the value recorded with PBS only (i.e., sample-free). From those additional experiments, we conclude that the lower phase sensitivity achieved in RBC imaging is likely to be affected by both the motion of the RBCs themselves and the mounting medium.

In this letter, we have quantified the temporal phase sensitivity of both common-path and non-common-path off-axis QPM systems under different environmental conditions, demonstrating that the non-common-path system is more sensitive to environmental disturbance. We applied the frame summing method and spatiotemporal filtering method to push the temporal phase sensitivity to a certain extent while compromising the temporal and spatial resolution. Although we have demonstrated a potential to achieve less than 2 pm of temporal phase sensitivity, there are still issues to be solved when applying our method for nanoscale dynamics observation in practical applications. Both the liquid medium and sample motion can affect the background sensitivity. A further exploration on minimizing the effect of liquid medium will be carried out by us. If one can achieve an ultra-high temporal phase sensitivity, it will open new avenues in applications such as mapping single-layer material topography, measuring spiking-induced membrane fluctuation, and observing protein aggregation. Note that the spatial phase noise will also limit the temporal phase noise enhancement performance and the overall imaging performance. To address this issue, we can explore controlling the spatial and temporal coherence properties of light illumination sources. Note that part of this work was presented at SPIE Photonics West 2020 and published in a different format in a conference proceeding.²⁵ 

See the supplementary material for a detailed description of temporal phase sensitivity theory, phase sensitivity comparison when using cameras of different electron-well-capacity levels, analysis of influence factors in the frame summing method, and analysis of the liquid medium effect.

This work was supported by the Croucher Foundation (Grant No. CM/CT/CF/CIA/0688/19ay), the Shun Hing Institute of Advanced Engineering (Grant No. BME-p3-18), and the Innovation and Technology Commission, Hong Kong (Grant Nos. ITS/098/18FP and ITS/394/17).

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