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Abstract
A new quantitative phase imaging approach is proposed based on self-reference holography. Three on-axis interferograms with different values of the phase filter are superposed. The superposition yields a more accurate phase map of the wavefront emerging from the object, compared with standard off-axis interferometry. Reduced temporal noise levels in the measured phase map and efficient phase recovery process for optically thin and thick transmissive phase objects highlight the applicability of the suggested framework for various fields ranging from metrology to bio-imaging. Qualitative phase imaging is also done online without altering the optical configuration. Qualitative phase detections of multiple planes of interest are converted to quantitative phase maps of the multiplane scene by a rapid phase contrast-based phase retrieval algorithm, from a single camera exposure and with no moving parts in the system.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetic radiation is, among others, the natural choice for propagating and storing spatial information, as evident by the synergy between the human eye and light. In the optical regime, electromagnetic radiation oscillates at a relatively high frequency that prevents direct detection of the phase information of the electromagnetic wave. Quantitative phase imaging (QPI) is a methodology that involves recovering the spatial phase distribution that is lost during the imaging process [1]. Restoring the phase information has shown to be beneficial in the life sciences, material engineering, and many technology-related areas. Capturing the dynamics of biological cells [2,3], measuring their refractive index [4,5], analyzing the kinetic fluid flow [6,7], and characterizing optical elements [8,9] become possible by the instantaneous phase measurement of the probing wave using QPI. Over the past two decades, several QPI methodologies have been suggested and demonstrated. For instance, digital holographic microscopy [10–12], lateral shearing interferometry [13–15], and diffraction phase microscopy [16,17] all belong to the classical interferometric approach, in which a wave carrying the phase information is interfered with a reference wave to create a digital hologram. More recently, non-interferometric methods have been emerged such as Fourier ptychographic microscopy [18–20], methods of transport-of-intensity equation [8,21–24] and other phase retrieval algorithms [25–28]. These techniques do not suffer from constraints of sensitivity and stability typical to interferometers. Recently, we proposed a method that significantly shortens the phase recovery process using a phase retrieval algorithm fed by a single-phase contrast image [29]. However, this method recovers the phase of only a single axial plane, and it is limited to objects having an optical thickness (OT) below the illumination wavelength. These limitations are addressed throughout the current paper.
In the present study, we demonstrate a QPI method that relies on the self-reference principle, in which a separate reference wave is not required. Additionally, the system enables microscopic phase detection by on-axis holography. Thus, we refer to this method as self-reference on-axis quantitative phase imaging (SO-QPI). The self-reference on-axis interference significantly improves the accuracy of the quantitatively imaged phase and reduces the measurement uncertainty, as is demonstrated in the following. Moreover, in comparison to off-axis, on-axis holography has a larger field-of-view (FOV) in the case of Fourier holography [30] or higher spatial bandwidth in the case of Fresnel holography [31]. QPI based on common-path configuration [32,33] and self-reference approach [34,35] has been demonstrated in the past. However, these previous implementations include movement of elements within the system, complex illumination modules, or only partial overlap between the self-reference beam and the sample beam, which eventually limit their use and accuracy. The proposed SO-QPI integrates the common-path and self-reference features under a simple optical apparatus that provides motionless and accurate extraction of the phase information. Furthermore, SO-QPI is implemented by a compact design that is also capable of prompt qualitative detection of phase objects from multiple planes of interest in a motionless fashion. This bimodality, alongside the improved phase sensitivity, distinguishes the proposed quantitative phase microscope from other QPI techniques. To quantify the contribution of the suggested framework, we statistically compare the accuracy and reproducibility of the SO-QPI with other QPI approaches. Our findings support the claim that SO-QPI can be classified as a useful bimodal diagnostic tool for multiplane phase imaging tasks in various fields that require high stability, sensitivity, and accuracy.
2. Self-reference on-axis quantitative phase imaging
2.1 System theory
One of the pioneering methods to observe transparent microscopic objects in the optical wavelength regime is the phase contrast technique invented by Frits Zernike almost nine decades ago [36]. This technique relies on shifting the phase between the non-constant phase signal and its constant background. Thus, the previously invisible features of the tested phase object now seem brighter (or darker) in proportion to their optical thickness. Nonetheless, this is also the main drawback of this technique, which is its inability to quantitatively extract the OT of the tested object, but only the qualitative estimations of this thickness. In the SO-QPI, we propose a way to extract the phase information quantitatively from three different phase-shifting measurements of the same object. Inspired by Zernike's technique, the phase contrast image in our system is formed by introducing a known phase delay of $\xi$ radians between the zero-order and the higher-order components in the spatial spectrum domain of the tested object. After inversely Fourier transforming the modified spatial spectrum, the emerging intensity pattern at the image plane, is given by,
In the second step, the complex amplitude of the original phase object can be computed by dividing Eq. (3) by 4i and adding unity to the result. At this point, the phase of the examined object can be extracted with no further calibration or post-processing to account for background phase noise, which might emerge in cases of interference with a tilted reference wave.
2.2 Optical configuration and demonstration
The optical implementation of the proposed SO-QPI scheme relies on a 4f spatial filtering system with a phase plate positioned at the spatial frequency domain to introduce the desired phase delay between the zero-order and the higher orders of the spectrum. Microscopic imaging with the SO-QPI can be achieved either by attaching the 4f system as an add-on module at the image plane of a given microscope or by choosing the first lens in the 4f configuration to be a microscope objective. In the current study, we adopt the latter alternative as illustrated in Fig. 1(a), where the focal length f1 of the first lens L1 is much shorter in comparison to the focal length f2 of the second lens L2. The phase modulation in the proposed scheme is done by a pure phase spatial light modulator (SLM) (Holoeye, PLUTO-2, 1920 × 1080 pixels, 8μm pixel pitch, phase-only reflective modulation). Since the used SLM operates as a reflective modulator, the actual experimental system had to undergo two modifications shown in Fig. 1(b). After being filtered and expanded, a coherent beam from HeNe laser (AEROTECH, a maximum output power of 25 mW @ λ = 632.8 nm) is transmitted through a transparent phase object positioned at the object plane of the system. The transmitted signal is collected by a microscope objective (MO, Olympus PLN 10X, NA = 0.25), so that the object spatial frequency spectrum is formed immediately after the objective and is relayed towards the SLM by using a unit-magnification 4f relay lens assembly composed of two identical lenses Lr1 and Lr2 (f = 50 mm, D = 25.4 mm). On the SLM, a phase-pinhole mask is displayed according to the required phase delay between the zero-order and the higher orders. The mask consists of several central pixels having phase modulation of ξ radians surrounded by the entire SLM pixels having no phase modulation. The number of the modulating central pixels in the phase-pinhole mask was determined according to the Rayleigh criterion [37] to be 6 × 6 pixels. After mechanical alignment of all the system apertures’ centers with the optical axis, these central pixels might have to be slightly translated electronically to achieve the correct modulation implied by the quality of the phase contrast image. This calibration ensures that the phase modulation is only applied to the zero-order of the object spectrum. Both, the modulated (zero-order) and the unmodulated (higher orders) signals of the spectrum domain, are transformed back to the spatial domain by a lens LF (f = 500 mm, D = 50.8 mm). The resulting intensity, according to Eq. (1), is recorded by the camera (Thorlabs 8051-M-USB, 3296×2472 pixels, 5.5 μm pixel pitch). Obviously, in the case of using a transmissive phase plate, the additional 4f relay system and the beam-splitter in front of the SLM are not necessary. Therefore, the suggested optical configuration can be considered as a single-channel, inline setup, in which the phase object and the reference signals overlap along their entire route. As explained in the next section, this feature has major implications on the apparatus performance in terms of accuracy and stability. It is important to note that the SLM modulation is not perfect, and each SLM pixel has a non-modulating area. This dead-zone gives rise to a slightly different intensity pattern than is given in Eq. (1). This modified recorded intensity should be considered when computing the object phase. Further details regarding the correction due to the imperfect SLM modulation can be found in the Appendix.
Fig. 1. (a) Schematic of the suggested SO-QPI system and (b) the implemented experimental setup. BE, beam expander; AS, aperture stop; MO, microscope objective; BS, beam-splitter; SLM, spatial light modulator.
Figure 2 depicts the working flow of the SO-QPI, starting from the three phase-shifting images of a transparent sample, and continuing with the digital reconstruction of its complex amplitude $\exp [{i\varphi ({\boldsymbol r})} ].$ The first row shows the phase-shifting images captured by the image sensor for a binary phase target, where different contrast levels of the transparent object are shown according to the value of the phase-pinhole in the phase plate. Once the three images are recorded, the digital reconstruction process is done by superposing the images, according to Eq. (2), into the complex phase hologram $h({\boldsymbol r})$ shown in the second row of Fig. 2. Finally, the complex amplitude of the sample is accurately recovered by normalizing $h({\boldsymbol r})$ to the background intensity (see Appendix) and adding a constant matrix of unity. The phase distribution of the sample can be readily extracted as the argument of the complex result, as illustrated in the final step of Fig. 2. The binary phase target used in this demonstration is a phase-only USAF 1951 resolution chart (Quantitative Phase Microscopy Target, Benchmark Technologies) of 300 nm thickness, in which the elements are deposited on a glass substrate. The substrate and elements have their refractive indices matched at 1.52. According to the manufacturing data, the height difference results in a phase delay of 1.55 radians between the two parts of the wave, one that travels through the elements and the other which travels through the substrate only. Quantitative assessment of the recovered phase in Fig. 2 is obtained by evaluating the phase difference between the square phase element (dashed yellow square) and the substrate (dashed red rectangle). The mean phase of the square is 0.94 radians, whereas the mean phase of the substrate is -0.63 radians, lead to a phase difference of 1.57 radians. In the subsequent sections, the accuracy, sensitivity, and stability of the SO-QPI technique are studied statistically and compared to other QPI techniques.
Fig. 2. The process of the quantitative phase imaging of a binary phase-only resolution chart (see text for details) using the SO-QPI method. The first row illustrates the captured phase-shifting images by the camera, and the second row describes the digital steps to recover the phase of the sample. Grayscale bar units are radians.
2.3 Evaluation of the system sensitivity and stability
In this section, we evaluate the measurement sensitivity of SO-QPI, i.e., the minimal detectable phase change by a single measurement. We also estimate the stability of SO-QPI during several measurements captured over time. To put this evaluation in perspective, we compare the results with two other QPI techniques, namely with off-axis holography [38] implemented by the well-known Mach-Zehnder interferometer (MZI) [39,40] and with the phase contrast-based phase retrieval (PCPR) method recently proposed by us [29]. Conventionally, noise evaluation of QPI systems is carried out on sample-free measurements [15,41], so that the expected value for the standard deviation should ideally be zero. However, due to different noise mechanisms such as ambient fluctuations and camera shot noise, the system sensitivity is greater than zero, and the stability is limited. To evaluate the noise levels in the suggested SO-QPI system, we continuously recorded phase-shifting images and recovered the phase maps 100 times (total of 300 phase-shifting images) over 30 minutes without any sample at the object plane. For comparison with the noise levels in the MZI, we recorded 100 sample-free off-axis holograms and extracted the phase using the same optical configuration and procedure described in [42]. For the phase reconstructions using PCPR, we used a phase contrast image from the data captured for the SO-QPI (the one with $\xi = \pi /2$) and fed it into the digital algorithm of PCPR described in [29]. In the SO-QPI case, the digital procedure described in Fig. 2 was repeated 100 times for each 3-batch phase-shifting images from the 300 acquired images. In the three different QPI techniques, the standard deviation of each of the 100 recovered phase maps was calculated and considered as the minimum detectable phase change by the measurement, also defined as the spatial sensitivity. The results are shown in Fig. 3(a). Although MZI shows the lowest average standard deviation of 1.9 mRad, it can be seen that the standard deviation over time in these measurements is 6.66 times higher compared to the SO-QPI and PCPR performance, while the average value of the SO-QPI is 2.6 mRad, only 1.37 times higher than the MZI standard deviation. The PCPR exhibits a slightly lower phase sensitivity of 4 mRad with the same stability as the SO-QPI.
Fig. 3. Comparative evaluation of noise levels in the recovered phase map using SO-QPI, PCPR, and MZI. (a) Spatial sensitivity and (b) temporal stability of the recovered phase.
For further temporal stability evaluation, 40,000 pixels from each phase map were randomly chosen, and the temporal standard deviation of each pixel was calculated over the 100 recovered phase maps. A histogram of the emerging values is shown in Fig. 3(b), which illustrates the temporal stability performance of each technique. The number of bins used in each histogram is automatically chosen by the plotting software (MATLAB R2018b) according to the measured data to cover the entire data range and reveal the shape of the underlying distribution. One can see the superiority of the inline, single-channel methods with mean temporal stability of 2.8 and 2.1 mRad for SO-QPI and PCPR, respectively, while MZI shows a value of 9.5 mRad. The relatively large spread of the MZI histogram indicates poor temporal stability in comparison to the on-axis methods over the entire image area. The measured noise levels for each method presented in this section are summarized in Table 1.
Table 1. Spatial sensitivity and temporal stability of the phase measurement
3. Experimental results
In this section, we describe the performance of the SO-QPI in various tasks of measuring the quantitative phase information of unstained transparent samples. This includes statistical evaluation of thin (OT<λ) binary phase objects and thick (OT>λ) phase objects. Comparison with MZI and PCPR is also provided. Furthermore, the inherent multiplane QPI capability of the SO-QPI is demonstrated. Finally, a new technique to extend the PCPR method toward multiplane QPI, despite its restriction of imaging phase modulation below 2π, is proposed and demonstrated.
3.1 Thin binary phase objects
One of the main goals of QPI is to measure the thickness of transparent objects as accurately as possible. Given that the refractive index of the examined homogeneous material is known, the thickness of objects can be calculated from the quantitative phase map. In this demonstration, it is shown that SO-QPI provides high-accuracy thickness measurement, in a nanoscale, of transparent thin films. We measured the phase delay of phase-only USAF 1951 resolution charts of different thicknesses, varying from 50 to 300 nm, and by knowing the refractive index of the film, we evaluated their respective thickness. To diminish measurement errors and reduce the possibility of false examination, we collected sufficient data for the reconstruction of 100 phase maps using SO-QPI, as outlined in section 2.3. Figures 4(a)–4(d) illustrate the reconstructed phase of the transparent resolution charts using SO-QPI for elements thickness of 50, 100, 200, and 300 nm, respectively. To evaluate the object thickness based on its measured phase delay relative to the substrate, the same approach described in section 2.2 was adopted, and the mean value of the 100 phase reconstructions was considered as the object thickness. The estimated thicknesses using SO-QPI of 48, 94.7, 199, and 300 nm are remarkably close to the manufacturing data. To further characterize the precision of the SO-QPI, we measured the thicknesses of the same charts using the MZI and the PCPR by the same procedures described in section 2.3. The 100 estimated thicknesses for each chart using the three described QPI methods are plotted in Fig. 5, whereas the mean and variance values are summarized in Table 2. Among the three methods, the SO-QPI superiority is prominent with a more accurate estimation of the thickness for all the measured samples. Although the MZI variance is slightly smaller than that of SO-QPI for some of the samples, the MZI estimation fails to give the precise value for the thickness. In this experiment, PCPR performed the least with the relatively highest deviations from the expected thickness for most of the samples.
Fig. 4. Quantitative phase imaging of binary phase-only USAF 1951 resolution charts using SO-QPI. Chart elements thickness upon the substrate is (a) 50, (b) 100, (c) 200 and (d) 300 nm. The white scale bar is equivalent to 15 μm. Grayscale bar units are radians.
Fig. 5. Thickness estimations of the USAF 1951 resolution charts based on 100 quantitative phase measurements using the three different QPI methods described in the text.
Table 2. Mean measured thickness of binary phase object using the different QPI methods
3.2 Thick phase objects
Many real-world applications of QPI are associated with measuring the phase of thick and/or dense samples, in which the largest phase retardation across the sample exceeds 2π radians. Additionally, there are phase objects with no clear background phase or cases in which the bias term is lower than some of the other terms. To demonstrate the SO-QPI ability to accurately capture the object phase in such conditions, we used a set of three spherical plano-convex refractive lenses of different focal lengths and recovered their respective phase distributions. Thereafter, a phase-unwrapping procedure of the imaged phases was implemented, and the results were compared with a theoretical lens function of the same focal length. In this demonstration, the Goldstein branch cut method was used in the phase-unwrapping process [43]. It should be noted that, since the examined objects in this section are lenses with a transmission of the quadratic phase function, most of the signal energy is not in the bias term. Therefore, normalizing $h({\boldsymbol r})$ of Eq. (3) and adding a constant is unnecessary. The examined phase can be readily recovered by taking the argument of $h({\boldsymbol r})$ given in Eq. (3). Figure 6 illustrates the results obtained in this experiment, along with a comparison to the theoretical phase values expected with the imaged lens parameters. Figure 6(a) is the theoretical wrapped phase for a 15 mm focal length lens, and Fig. 6(b) is the wrapped phase measured using SO-QPI. Figures 6(c) and 6(d) are the theoretical and measured unwrapped phase maps of the lens, respectively. Figure 6(e) plots theoretical and measured unwrapped phase for the 15 mm focal length lens along the red diagonal shown in Fig. 6(d). The phase maps of two additional lenses, 25 mm and 50 mm focal length, were measured using the SO-QPI, and the obtained values are compared with the theoretical values in Figs. 6(f) and 6(g), respectively. To further verify the matching between the expected and the measured phase values, the root-mean-square error (RMSE) between the two was calculated and normalized by the maximal unwrapped phase delay for each of the three examined lenses. This gives the percentage merit of the error in the unwrapped phase measurement per pixel. The normalized RMSE values for the 15 mm, 25 mm, and 50 mm focal length lenses were calculated to be 1.6%, 2.0%, and 2.4%, respectively. These values emphasize the SO-QPI accuracy in measuring the unwrapped phase of optically thick objects, with no more than 2.5% mean error per pixel in comparison with the theoretical values.
Fig. 6. Phase measurement of spherical plano-convex refractive lenses using SO-QPI in comparison to theoretical curves. (a) Theoretical and (b) measured wrapped phase of a 15 mm focal length lens. The corresponding unwrapped phase maps are shown in (c) and (d), respectively. Cross-section plots of the unwrapped phase map along the principal diagonal [shown in red in (d)] for (e) 15 mm, (f) 25 mm, and (g) 50 mm focal length lens. Grayscale bar values are in radians. The white scale bar in (a)-(d) is equivalent to 25 μm.
As an additional demonstration of the SO-QPI applicability, we measured the phase of cotton pollen which can be regarded as a thick biological sample. Figures 7(a) and 7(b) illustrate the wrapped and unwrapped phases of an ensemble of cotton pollens, respectively, where their OT is clearly beyond the illumination wavelength. Capturing the true phase distribution of the examined cotton pollens enables the observation of the spikes on the pollen shells, which are optically thicker than the shell itself.
Fig. 7. (a) Wrapped and (b) unwrapped phase map of cotton pollens measured by using the SO-QPI method. Phase values of the grayscale bars are in radians. The white scale bar is 20 μm.
3.3 Quantitative phase imaging of multiplane phase objects
Capturing the phase of a multiplane scene with several transparent samples positioned at different transverse planes is a mission encountered occasionally in various fields of QPI [44]. For any scene with multiple specimens positioned at different z distances, it is important to have a technique that can quantitatively extract the phase information of these different specimens. In this section, the capability to recover the phase of a multiplane scene at the desired axial location by the two different inline, single-channel QPI methods (SO-QPI and PCPR) is evaluated and compared. In SO-QPI, the phase at the object plane is captured three times, and the corresponding phase-shifting images are superposed to create a complex signal of the wavefront at a specific transverse plane. Therefore, SO-QPI is inherently capable of recovering the phase at neighboring planes by Fresnel propagation of the complex amplitude from the measured plane. On the other hand, in PCPR, a recorded phase contrast image is fed into a phase retrieval algorithm so that the phase at the object plane is digitally recovered. However, our investigations indicate that true phase distribution is obtained by the PCPR only for thin phase objects. Thus, PCPR, in general, is not capable of recovering phase information from neighboring planes since the unwrapped phase distribution over the transverse plane might exceed the limit of 2π radians. Fortunately, by introducing a diffractive lens around the phase-pinhole of the phase plate, a refocusing algorithm can be applied to capture the phase contrast image at neighboring axial planes. This image is served as the input of the phase retrieval algorithm, and therefore, the phase distribution of the neighboring planes can also be recovered quantitatively by the PCPR method. To the best of our knowledge, this approach for multiplane QPI with PCPR is proposed here for the first time, and the current study is its first demonstration.
For the experiments presented in this section, we attached two different slides of Polystyrene beads (Focal Check, 6 μm and 15 μm bead diameter) and placed them at the input plane of the system. In this way, a two-plane phase object was constructed with different beads diameter in each plane. The separation between the two planes is about 1mm, determined by the thickness of the microscope slide. Starting with the SO-QPI, Fig. 8(a) shows the recovered phase of the sample plane containing the 6 μm beads, where three different beads can be observed. As mentioned above, this recovered phase can be propagated to the other plane of interest, which contains the 15 μm beads. Figure 8(b) shows the phase after Fresnel propagation to the best focus plane of the three other beads at a distance of 1 mm, which is approximately the distance between the two different specimens. Figures 8(c) and 8(d) are the PCPR equivalents of Figs. 8(a) and 8(b), respectively. Here, the reconstruction of the phase information from the out-of-focus plane is done by an appropriate diffractive lens displayed on the SLM around the phase-pinhole central pixels.
Fig. 8. Quantitative phase measurements of multiplane phase object. (a)-(d) The recovered phase for 6 μm and 15 μm diameter beads using SO-QPI [(a) and (b)] and PCPR [(c) and (d)], respectively. The 6 μm diameter beads are indicated by the red and blue arrows in (a) and (c). The white scale bar is equivalent to 15 μm. (e)-(h) Cross-sections of the squared measured phase and the corresponding theoretical fittings of a single bead from (a)-(d) indicated by blue arrows, respectively. Curve fittings were used to evaluate the bead diameters to verify the reliability of the quantitative phase imaging (see text for details).
To assess the QPI performance of SO-QPI and PCPR in the present case of multiplane phase objects, we calculated each bead diameter solely based on the phase profiles of Figs. 8(a)–8(d) as follows. Assuming the refractive index of the bead is constant, and the light propagates along the z-axis, the transversal phase values of the wavefront after it passes the bead are proportional to the distance the light passes through the bead, i.e., φ(r)=αz(r). Thus, by the knowledge of the bead shape, α can be evaluated from the measured phase profiles φ(r) of each bead. Once the proportionality factor α is known, each bead diameter d can be evaluated by dividing the highest phase delay across the bead by α, i.e., $d = {{\varphi (\boldsymbol{r} = 0)} / \alpha }$. From the spherical shape of the bead, it follows that the distance covered by the illumination inside the bead is $z({\boldsymbol r}) ={\pm} \sqrt {{R^2} - {{\boldsymbol r}^2}}$, where R is the bead’s radius. Therefore, the squared accumulated phase of the wavefront after it passes through the bead is,
Based on this relation, the squared measured phase is fitted to a second-order polynomial of r, and α is accordingly evaluated. Figures 8(e)–8(h) show the horizontal cross-sections of the squared measured phase, where r = (x,0), along with the theoretical second-order polynomial fitting of a single bead from Figs. 8(a)–8(d) marked by the blue arrow, respectively. By using this kind of fitting for each bead, $\alpha $ was determined for each phase profile, and the corresponding diameters of the beads were calculated. The average deviations from the expected diameters 6 μm and 15 μm, based on the recovered phase profiles by each method, were calculated to be 0.76 μm and 2.35 μm using PCPR and 0.98 μm and 2.03 μm using SO-QPI, respectively.
4. Discussion
Comparing the suggested SO-QPI with other established methods is vital to objectively evaluate the method’s applicability and ability. For this end, MZI and PCPR were chosen since the former belongs to the ad-hoc methodologies of interferometric phase measurements, while the latter shares similar properties with SO-QPI, but at the same time, it is classified differently as an iterative phase retrieval scheme. By this comparison, we seek to estimate the contribution of SO-QPI to the field of QPI among the various approaches and modalities. From our first set of experiments, it can be deduced that SO-QPI outperforms the other two methods in terms of accuracy in phase extraction of an examined static object, with more accurate estimations for the object thickness and lower degree of uncertainty. Although MZI shows lower uncertainty in three out of the four measured objects in section 3.1, the true thickness (given by the sample manufacturer) does not fall within the measurement error range. Thus, MZI is less accurate than SO-QPI. We believe that the main reason for the better performance of SO-QPI is the tripled sampling of the object phase. Superposing three different phase-shifting images into a single complex hologram reduces the effect of unwanted noise sources by a type of averaging. Naturally, the high sensitivity and stability of SO-QPI demonstrated in section 2.3, which mainly stem from the single-channel common-path configuration, also contribute to its superior accuracy. An additional benefit of the tripled phase sampling and superposing in SO-QPI is the ability to recover the phase of thick objects, which have phase modulations higher than 2π radians. This ability is not applicable in the single-shot PCPR simply because the digital phase retrieval algorithm cannot capture the cyclic nature of the phase in a single-phase contrast intensity image. In addition, it should be noted that the MZI low uncertainty compared with the on-axis methods is probably due to the filtering operation of extracting the desired information from the off-axis hologram. This process decreases the ambient noise of the reconstructed phase and thus increases the method’s sensitivity, albeit in the vicinity of inaccurate mean phase value.
In this study, we introduce a modified version of the original PCPR that can recover phase information from multiple planes of interest without scanning or moving any part of the optical system. As shown, the multiplane QPI capability is inherent in SO-QPI, and a comparison between the two methods in recovering the phase of Polystyrene beads placed at different axial planes finds they provide comparable results. This result is not inconsistent with the previous findings in this study regarding the SO-QPI superiority in terms of accuracy. We found SO-QPI to be more accurate than PCPR in the phase range of milliradian. This translates to the physical thickness of the beads in the nanometric region – clearly, an insignificant value for bead diameters of several microns as in the present case of the multiplane phase object. Moreover, in SO-QPI, only three camera shots are needed for reconstructing the entire transverse planes in the volume, whereas in PCPR, a single shot is needed for every investigated plane. Obviously, for a volume with more than three planes of interest, SO-QPI is faster than PCPR. Nonetheless, it is still attractive to have the two methods for multiplane QPI in a single optical configuration. While SO-QPI provides information about the out-of-focus object only after digital processing, PCPR can provide a qualitative map of the out-of-focus object already during the acquisition stage. The better-suited method can be selected for optimal operation – whether it is the rapid acquisition of the complex hologram using SO-QPI or it is the qualitative examination of the object before the acquisition of the phase contrast image by PCPR.
5. Conclusions
In this study, we established an optical framework that converts qualitative phase-shifting images to their corresponding quantitative phase image while preserving the benefits of a single-channel and common-path configuration. Due to the self-reference property, the new approach exhibits phase sensitivity, temporal stability, and higher accuracy in comparison to a classic interferometric device and PCPR [29]. Unlike the PCPR, in the proposed SO-QPI, the phase recovery process is non-iterative, and the phase component of the wavefront is measured unambiguously. This grants SO-QPI two important advantages: the inherent multiplane imaging capability and the ability to capture phase object retardations beyond 2π radians. The experimental demonstrations presented here for different kinds of phase objects, and the implementation by a simplified optical configuration, turn SO-QPI into an attractive apparatus for various tasks of QPI. SO-QPI should not be confused with another scheme for phase imaging tasks that is based on a similar phase pinhole filter positioned at the Fourier plane [45]. While [45] uses a three-wave interference modality, SO-QPI employs a simplified two-wave interference output which significantly reduces the number of camera shots needed. Therefore, SO-QPI can be jointly implemented with PCPR on the same instrument offering two types of multiplane imaging. In a broader perspective, the contribution of the present study is twofold. First, an accurate single-channel common-path quantitative phase imager is demonstrated; and second, its technical capabilities are characterized in comparison to two other methods to show the strengths of the SO-QPI. The results presented in this paper draw clear lines between the different imaging scenarios in terms of the optimal phase microscopy approach to be used. Whenever inspection of unknown transparent and static objects is involved, the robust SO-QPI technique is preferred, offering two modes of operation (single plane or multiplane QPI) and higher phase accuracy. However, for the rapid analysis of phase objects, it is recommended to use the well-known off-axis interferometric approach at the cost of reduced spatial bandwidth or FOV. In case both rapid operation and full spatial bandwidth or FOV coverage are required, and the target OT is smaller than the illumination wavelength, the PCPR can be adopted for optimal QPI qualities. Of course, using PCPR comes at the price of higher computational resources, convergence duration of the digital algorithm, and some reduction in the measurement accuracy. We hope this research will pave the road for more complete versions of QPI frameworks, which will offer both high accuracy and sensitivity of on-axis devices, along with a single shot data acquisition for capturing dynamic events. Meanwhile, the SO-QPI suggested in this study can be integrated with its closely related phase retrieval method [29] to achieve optimal imaging capabilities of phase measurements within a single optical apparatus.
Appendix
Considering the non-ideal SLM device, its pixels do not modulate the entire incident wave. Since we used only a few of the central SLM pixels to create the phase plate, and all the other pixels are switched off, we get an uneven reflection for the two signals arriving at the sensor plane. Therefore, the recorded intensity at the sensor plane from Eq. (1) is slightly modified as follows,
To recover the desired phase, one must divide the phase hologram of Eq. (6) by the factor 4i·ab and not by 4i as in the ideal case of a = b=1. This will isolate the n>0 terms in the Maclaurin expansion of the examined phase object, from which its quantitative values can be recovered by taking the argument of these terms plus unity. Accessing the normalization factor ab does not require any additional measurement and can be achieved for each measurement as the following. Adding of the already recorded phase-shifting images $I({\mathbf r};\xi )$ for $\xi $ values of π/2 and 3π/2 yields,
Finally, the quantitative phase values in the case of thin or sparse phase objects can be extracted by using Eqs. (6) and (8) as follows,
Funding
Israel Science Foundation (1669/16).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this study are not publicly available at this time but may be obtained from the authors upon a reasonable request.
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