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Abstract
We present the numerical analysis of the effect of the temporarily partially coherent illumination on the phase measurement accuracy in digital holography microscopy (DHM) and optical diffraction tomography (ODT), as reconstruction algorithms tend to assume purely monochromatic conditions. In the regime of reduced temporal coherence, we simulate the hologram formation in two different optical setups, representing classical off-axis two-beam and grating common-path configurations. We consider two ODT variants: with sample rotation and angle-scanning of illumination. Besides the coherence degree of illumination, our simulation considers the influence of the sample normal dispersion, shape of the light spectrum, and optical parameters of the imaging setup. As reconstruction algorithms we employ Fourier hologram method and first-order Rytov approximation with direct inversion and nonnegativity constraints. Quantitative evaluation of the measurement results deviations introduced by the mentioned error sources is comprehensively analyzed, for the first time to the best of our knowledge. Obtained outcomes indicate low final DHM/ODT reconstruction errors for the grating-assisted common-path configuration. Nevertheless, dispersion and asymmetric spectrum introduce non-negligible overestimated refractive index values and noise, and should be thus carefully considered within experimental frameworks.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recent advances in optical imaging techniques prove great potential and demand for label-free, non-invasive quantitative phase imaging (QPI) methods [1]. A prime example is digital holography microscopy (DHM), which is based on endogenous contrast, i.e., refractive index (RI) distribution of a sample. However, DHM, on its own, provides only combined information about local RI and the sample geometry. Optical diffraction tomography (ODT) combines DHM optical setup, multiple views acquisition and tomographic reconstruction algorithms for obtaining quantitative 3D RI distribution in a semi-transparent microsample. State-of-the-art ODT solutions based on synergistic combination of hardware and software enable highly accurate reconstruction of complex 3D structures, both biological [1,2] and technical [3,4]. The most commonly used ODT setups apply mechanical sample rotation [5–7] or scanning of an illumination beam [8–10]. The former type is typically applied to nonbiological samples, e.g., optical fiber elements, while the latter gives a possibility of in-vivo investigation of biological samples, which makes this configuration more versatile.
Statistical, coherence-related properties of an illuminating source are of particular importance in QPI techniques performed in various DHM and ODT configurations [11–13]. Coherence alteration allows for increasing measurement capabilities, see [14] and references therein, therefore it is crucial to understand its possible effects on DHM and ODT accuracy. Highly coherent sources such as single-mode stabilized lasers greatly simplify the process of phase retrieval via fringe pattern generation, recording and demodulation, nonetheless simultaneously leading to a number of undesirable coherent noises [5,15,16]. High coherence of the illumination source implies increased sensitivity for random phase modulation, resulting in the speckle noise. Furthermore, back-reflections from various optical elements, e.g., microscope slides or camera cover glass, give rise to parasitic fringe families. Similarly, diffraction on dust particles causes formation of local in-line Gabor hologram fringe patterns.
A number of methods for coherent noise reduction were proposed, both for DHM [17–19] and ODT, e.g., off-axis sample placement [5,20] and reconstruction with a large number of projections [20]. Alternatively, digital image processing techniques showed their promising denoising capabilities with deep learning [21], algorithmic phase discontinuity regions localization [22] or adaptive regularization [23]. The outlined methods enable minimizing the influence of the coherent noise on the measurements; however, they are geared towards compensation of the effect, not the physical cause itself (highly coherent illumination source).
Lowering degree of temporal coherence was recently found to be growing in popularity as an effective remedy for the coherent noises [24] both in DHM [11–13] and ODT based on conventional [8,9,15,25] as well as alternative 3D microscopy approaches [26–28], including fringe-less methods, e.g., 3D transport of intensity [29], 3D Fourier ptychographic microscopy [30,31], intensity diffraction tomography [32]. Most illumination-scanning ODT configurations suffer from well-known limitation, i.e., the missing cone problem, which occurs due to bounded cone of illumination angles. Partially coherent illumination emphasizes the problem, since low-coherence full-field ODT with steep illumination angle is not easily achievable in the classical interferometric setups, however several effective approaches have been proposed [9,15,27,33]. Park et al. demonstrated two successful modifications of wide-spectrum, angle-scanning Mach-Zehnder interferometry: one based on diffractive tilting with a digital micromirror device [9], and another utilizing binary phase modulation using a ferroelectric liquid crystal spatial light modulator with additional dispersion compensation unit [15]. Shaked et al., proposed a compact, portable interferometric module for quantitative phase measurement with high spatial and low temporal coherence, achieving coherent noise reduction and maintaining marginal level of spatial incoherence artifacts [34]. Chowdhury et al., introduced a diffractively formed, structured, broadband illumination to widefield microscopy, proving its capability for 3D specimen visualization by multiplexing plane-wave tilts [25]. Furthermore, axial-scanning configurations, using phase contrast microscopes with a broadband light have been investigated for tomographic RI reconstruction [26–28,35]. These methods employ deconvolution theorem, requiring additional knowledge about the optical transfer function of the setup [26–28], or Wolf equations with determined temporal spectrum of illumination and its autocorrelation properties [35]. However, besides significant reduction of coherent noises, these optical setups are relatively complicated in architecture and require advanced, high-end hardware components. On the other hand, previously proposed ODT configuration, based on the grating deployed total-shear 3-beam interferometer, proved achievability of coherent noise reduction with partially-coherent illumination, simultaneously maintaining the simplicity of the common-path optical setup [8,36]. In this layout, temporally partial-coherent illumination (SLED) was introduced to compact, common-path DHM microscope, showing the performance improvement of the tomographic measurement [8] in comparison with the laser source. Crucially for this paper, the listed, diverse solutions show clearly a high interest of optical community for temporal coherence reduction in (simpler) DHM and (more advanced) ODT systems.
Importantly, besides temporal properties of the source, the spatial coherence also have unquestionable influence over the large range of optical measurement methods. Decreased spatial coherence might be found both as a remedy for coherent noises, e. g., rotating diffuser [37–41] or LED illumination [42–44] and a difficulty to handle, e. g., halo effect [13]. Both coherence aspects should be carefully considered and investigated; however, in this work we focus on the temporal coherence exclusively as it is a more popular solution and it does not need any external additional elements.
The most popular reconstruction algorithm for ODT is a direct inversion method (DI) [45], which is based on a direct mapping of the 2D object information on Ewald spheres in the 3D object spectrum. The algorithm is versatile, i.e., it can be applied for various data acquisition configurations, fast and relatively easy to understand and implement. Moreover, it allows for addressing the missing cone problem [46,47]. However, the algorithm is based on the assumption of a monochromatic light illumination. The primary goal of this work is to investigate robustness of integrated (2D projection) and tomographic (3D) RI reconstruction process and its reliability under the terms of reduced temporal coherence. The impact of a wider spectrum of illumination still leaves unanswered questions about the limitations and perspective accuracy in DHM and ODT measurements. Motivated by the outcome of the experimental works [8,36], we present numerical analysis of the effect of reduced temporal coherence in refractive index microscopy and tomography.
2. Theoretical background and methods
2.1 DHM and ODT
DHM is a potent tool, which enables QPI measurements by recording the holographic representation of a sample. Amplitude and phase of an illuminating beam, that was modulated by a sample, is magnified and projected onto a detection plane with a microscope system. The holographic information is obtained by registration of an interference pattern, being a superposition of an object beam and a reference beam. This may be achieved in two configurations: separate-arms (Fig. 1(a)) and common-path (Fig. 1(b)) [1,2]. In both cases usually the spatial frequency carrier is introduced (by a tilted mirror diffraction grating), which results in creation of a linear carrier fringe pattern. Owning to that, specimen information can be extracted from a single holographic frame, e.g., by Fourier analysis in the spatial frequency domain. The most straightforward and commonly used hologram reconstruction method isolates the proper object peak in Fourier spectrum and the object wave is obtained as the complex field calculated by the inverse Fourier transformation of the filtered spectrum.
Fig. 1. DHM setups applying (a) Mach-Zehnder, (b) grating deployed total-shear 3-beam interferometric architectures: IS – illumination source; CL – collimator lens; SP – specimen plane; MO – microscope objective; TL – tube lens; G – diffraction grating; OP – output plane; BS1/2 – beam splitters; M1/2 – mirrors.
In this paper we focus on two DHM optical setups (Fig. 1) that are representatives of two classes of interferometric schemes. Our goal is to quantitatively (RI errors) investigate their robustness to decreased temporal coherence and thus evaluate their potential use for ODT with broadband light sources. Former configuration (Fig. 1(a)) is a well-known Mach-Zehnder type. Object beam modulated by specimen S interferes with a tilted reference beam in output plane OP. Assuming that the input beam is a monochromatic plane wave, interference of object and reference beam is described by:
The latter holographic system, which we would like to focus on, is the grating deployed total-shear 3-beam configuration (Fig. 1(b)) with Ronchi diffraction grating, described in [8,36]. The system is an example of a broader class of achromatic, grating-based setups [48–51], which recently gained popularity in ODT [15,25]. In our system, the object beam is incident on a diffraction grating. The interference between 0, +1 and -1 diffraction orders of the grating in the Fresnel regime is given by:
From Eq. (4), two conclusions are fundamental: (1) specimen information is encoded in both first and second harmonic of the hologram spectrum, (2) second harmonic demodulation of the interference pattern is achromatic, that is, the period of the carrier frequency is independent of the light wavelength. This property is of great importance as it provides the theoretical possibility to achieve successful interference encoding using the light with reduced temporal coherence. The detailed analysis of this hypothesis will be presented in the following sections.
Since ODT is aimed at retrieval of spatial RI distribution, the problem of reconstructing a 3D complex specimen function, i.e., the scattering potential, must be resolved. This is commonly and efficiently done by employing Fourier diffraction theorem [52] under the first order Rytov approximation, thanks to weakly-scattering assumption [53,54], which analytically relates 2D holographic views of a sample with its 3D scattering potential. The method can be implemented in the space domain, which results in filtered backpropagation algorithm [55], as well as in the spatial frequency domain, yielding DI [45]. The latter approach is however much more popular due to its computational efficiency and versality. In DI the 2D holographic views spectra are mapped onto suitable Ewald spheres that lie in the spectrum of the 3D scattering potential. Then, the scattering potential is reconstructed by inverse 3D Fourier transformation and further proceed to obtain 3D RI distribution [54].
Crucially, the above operations exploit a priori knowledge about the sample illumination, which is assumed to be perfectly coherent, both spatially and temporally. The major goal of this contribution is to investigate the consequences of departure of this assumption from the reality due to the use of non-monochromatic sources in ODT systems. In our study we will analyze the effect of the width and shape of the illuminating light spectrum on the accuracy of the ODT 3D RI reconstructions.
2.2 Outline of numerical simulation
The designed algorithm enables numerical simulation of the complete data acquisition process in ODT, i.e., from the illumination beam propagation through the sample to the interference pattern formation at the detector. The core of the performed numerical simulations that is the object beam propagation in a sample, is common for both analyzed interferometric configurations. In the beginning, let us assume that the illumination beam is a monochromatic plane wave. First, this synthetic optical field is numerically propagated through the sample, which is represented as a 3D array of discrete RI values, as presented in Fig. 2 .
Scattering simulation is based on wave propagation method [3,56], which accounts for multiple scattering, and thus provides high accuracy and reliability. Then, the effect of imaging optics is introduced by accounting for numerical aperture, magnification and optical conjunction of the object plane with the detection plane. The last step is done by computing backpropagation of the resultant scattered optical field, in the free space at desired imaging depth, which we define precisely as a center of the specimen. As the backpropagation is done in homogenous medium and the propagation distance is relatively small (here about 5 µm), the angular spectrum method is utilized [57,58]. It should be noted that space-bandwidth product of propagated optical field is kept constant along whole process and sampling of spectra is changed only in the final part when the magnification is applied.
At this point the interference process is implemented, which is done differently depending on the considered interferometric configuration. In the case of Mach-Zehnder, the reference beam is defined as a tilted plane wave with the optical frequency carrier depended on the mirror tilt and the interference is computed as its superposition with the object beam.
Elucidations are slightly more complicated in the case of grating-deployed 3-beam total-shear configuration, where object beam interferes with its inclined replicas. Normally, the interference pattern consists of two components: the cosine product, which encodes the object information in the first harmonic and the latter cosine which carries same data in the achromatic second harmonic (see Eq. (4)). However, as it was described in [8,36], OPD between zero and first order is usually larger than the coherence length once the temporal coherence is limited, thus first harmonic fringes vanish. Besides this phenomena, undesired diffraction orders can be filtered out with the additional 4f system or simply filtered in the Fourier spectrum (ensuring proper separation of all terms in the hologram spectrum). Having regard to the above, in our simulation we considered presence of +1 and -1 diffraction orders only:
With the above assumptions the interference simplifies to two-beam case giving:
Importantly, even though Eq. (5)–(6) where derived for the Fresnel regime, in the simulation no such assumptions were employed.
The above considerations were done assuming monochromatic light, however broader spectra is also often represented with a single wavelength, which raises metrological concerns. In this work we investigate the effect of reduced temporal coherence of illuminating beam so that the statistical properties of light must be considered. The theoretical analysis of partially-coherent illumination supported with 2D interference-pattern-based QPI experimental study was reported in detail by Park et al. [12]. On the basis of their work, it can be stated that the temporarily partially coherent illumination can be perceived as a set of fully-coherent optical fields with the same directions and different lengths of wave vectors. As shown in [12], this approach, in the regime of time-averaging intensity measurement, evolves to the sum of intensities detected for different wavelengths. Therefore, defining the spectrum of illumination beam as a set of spectral lines ${\lambda _1}$, ${\lambda _2} \ldots ,\,{\lambda _N}$ with amplitudes A, ${A_2} \ldots ,\,{A_N}$, respectively, allows to represent the resultant interference pattern obtained for temporarily partially coherent illumination as:
where ${I_n}({x,\; y,\; {\lambda_n},\; {A_n}} )$ is the interference pattern obtained for n-th spectral line.For most tests in our study, we assume Gaussian shape of the spectrum, where the width of the spectrum is characterized with full width at half maximum (FWHM) parameter. This could be a good approximation for relatively narrow spectra, however, as reported in [11], for a broadband light, especially when the bandpass filter is introduced, shape of the light spectrum, may be far from perfect symmetry in relation to dominant peak. In classical ODT, obtaining values of RI inevitably relies on choosing a specific spectral line for the reconstruction, ergo, we address this problem by simulating also the aberrated Gaussian spectrum as shown in Fig. 3(a).
Fig. 3. Investigated spectra of the illuminating beams (a); RI dispersion for simulated media, approximated from experimental data with Cauchy’s method with coefficients A = 1.457, B = 4.589 · 10−3, C = 2.28 · 10−4 and A = 1.471, B = 4.952 · 10−3, C = - 5 .089 · 10−5 for immersion oil and PMMA, respectively (b).
The last analyzed factor is the dispersion of the refractive index. In the simulation we implemented two types of optical media in the sample region. The dispersions of a sample and its surrounding environment are typically different, which potentially may affect the tomographic reconstruction. We address this issue in Sec. 3.3 by generating a combined hologram consisting of a set of interference patterns obtained for various wavelengths and properly dispersed RI values. To model the dispersion, we assumed that the sample and its environment, i.e., immersion, have experimental characteristics of Tomson PMMA [59] and Cargille Immersion oil type A, respectively. Based on empirically obtained RI discrete probes, we derive the formula for continuous wavelength-dependent RI distribution by employing Cauchy’s approximation formula:
where A, B, C are the Cauchy’s coefficients (Fig. 3(b)). Schematic representation of the synthetic interference pattern simulation is presented in Fig. 4.3. Simulations
3.1 Method verification
As an introduction to the simulation, we present a verification of the implemented numerical methods in the form of a comparison of the interference result obtained for temporally partially coherent illumination in Mach-Zehnder configuration with its theoretical prediction. Reference beam tilt, effectuated by an inclined virtual mirror, introduces the linearly changing OPD between interfering beams. Since the temporal coherence degree of light is considered, the satisfying condition for successful interference detection is that the OPD is much smaller than the coherence length $({{l_c}} )$ of light [60]. Assuming a Gaussian spectrum with the frequency bandwidth (Δν) expressed with FWHMν, the coherence time is given by [60]:
where c denotes the speed of light in a vacuum. From the above equation one can easily obtain ${l_c}$. In Fig. 5, we marked the theoretical calculations of a coherence area in the detection plane, which should be here understood as the interference region with OPD < ${l_c}$, for λ = 500 nm and FWHMν = 0.43 THz (FWHMλ = 0.36 nm). It is clearly visible that the contrast of the fringes decreases dramatically in the regions with OPD > ${l_c}$ which limits the effective field of view and complies with our predictions.Fig. 5. Influence of the reduced temporal coherence in Mach-Zehnder configuration for $\textrm{FWH}{\textrm{M}_\nu }$ = 0.43 THz ($\textrm{FWH}{\textrm{M}_\lambda }$= 0.36 nm) : OPD distribution (a), obtained interferogram with the contrast decrease due to increasing OPD (b).
3.2 2D Quantitative evaluation of the phase measurement accuracy
In this section we apply the created simulation engine to investigate influence of temporal coherence on accuracy of the phase measurement in QPI. The tomographic measurement accuracy will be discussed in Sec. 3.3. To quantitively evaluate the impact of partially-coherent illumination, we simulated the phase retrieval process in two considered interferometric configurations using the same parameters: NA = 1.3, magnification 90× and central wavelength λ = 500 nm. In both cases we used the same object under study, i.e., a sphere made of PMMA (RI = 1.49021; radius = 5 µm) and surrounded with the immersion oil (RI = 1.4802). Simulations were computed for the array dimensions of 1100 ${\times} \; $ 1100 pixels and the pixel size of approx. 0.027 µm. The object phase was obtained from the interference pattern via Fourier transform method.
In Fig. 6 we present the comparison of the resultant phase distributions along with the interference patterns and phase error maps, calculated as a difference between the obtained, partial-coherent phase measurement and the corresponding phase measurement obtained for fully coherent illumination.
Fig. 6. Phase retrieval with reduced temporal coherence. In each row - interferogram with the contrast cross-section, retrieved phase and phase error with respect to the coherent phase measurement. Results calculated for Mach-Zehnder (a)-(f) and grating-assisted interferometer (g)-(l) for FWHMλ = 0.3 nm and FWHMλ = 0.5 nm.
As it was shown in Fig. 5 (paragraph 3.1), non-zero OPD between the object and reference beam strongly affects the fringe contrast in Mach-Zehnder configuration, which yields significant loss of object information, even for relatively small spectral width. Meanwhile, in the case of the diffraction grating-assisted setup, this effect is not observed, as the chromatic OPD nominally remains constant across the entire field of view. The reported contrast decrease may be explained by the fringe pattern generation mechanism, since in Mach-Zehnder setup, the fringe period depends on the light wavelength, yet in the grating-assisted configuration it does not (it is rather defined by the period of the diffraction grating) [9]. In Mach-Zehnder interferometry the resultant interference pattern obtained for broadband source is therefore the superposition of sinusoidal intensity distributions with various periods, leading to the contrast decrease. Quantitative comparison of the phase measurement errors, for various spectrum widths, is presented in Fig. 7. For our measurements, we adopted the root mean square error (RMS), calculated from the difference between any reconstruction of interest and the appropriate reference one. As it can be seen in Fig. 7 (a), the phase error of Mach-Zehnder setup increases dramatically for FWHMν larger than 0.35 nm, which may be defined as a threshold for measurement with phase inaccuracy under 0.03 rad (of RMS error).
Fig. 7. Interferometric setups robustness comparison – RMS phase errors due to reduced temporal coherence: in logarithmic scale for both considered setups (a), in normal scale for grating-assisted DHM (b).
3.3 Quantitative evaluation of the measurement errors in optical diffraction tomography with specimen rotation
The previous section demonstrated robustness of the grating-assisted interferometric configuration to decreased temporal coherence. This suggests that the system may be potentially used for potent tomographic measurements in ODT systems with broadband illumination. However, the analysis from Sec. 3.2 focuses solely on the phase retrieval algorithm. The remaining question considerers immunity of the tomographic reconstruction algorithms to the partially coherent illumination. Nonetheless, at this stage, it is already clear that basic arrangement of Mach-Zehnder configuration is not suitable for low-coherence ODT. Therefore, in the following paragraphs we focus on the grating-assisted setup exclusively. As a method of tomographic reconstruction, we employ the direct inversion algorithm (DI) with the first-order Rytov approximation. This approach to RI retrieval is well-known and widely used in practice, thus we find it important to evaluate it in this context. The first simulated tomographic configuration is ODT with the specimen rotation. In our calculations, we applied exactly the same optical setup parameters as in Sec. 3.2, except for pixel size (approx. 0.025 µm), with additional feature of the specimen rotation by full angle around y-axis (360 projections with a constant angular step of 1°). The imaged sample was a concentric spheres sample (Fig. 2) for which the coherent ODT reconstruction is illustrated in Fig. 8.
Fig. 8. Coherent ODT measurement results with the specimen rotation simulated for 360 evenly spaced projections: sample k-space (a) – (c), retrieved RI distribution cross-sections (d) – (f)
In the course of the simulations process, we tested the tendency of the reconstructed RI errors for a number of symmetric spectral distributions with various widths, up to FWHMλ = 50 nm – Fig. 9.
Fig. 9. ODT reconstruction with reduced temporal coherence (FWHMλ = 50 nm): RI distribution cross-sections (a) – (c) corresponding RI error maps with respect to the coherent reconstruction (d) – (f).
Apart from the pure spectral width of illumination, additional factors were added to the simulation process. As described in Sec. 2.2, we expanded our numerical analysis and included the presence of optical medium normal dispersion and spectral distribution aberration, i.e., nonsymmetric spectrum. In Fig. 10, we presented the comparison of these temporal coherence alteration effects for $\textrm{FWH}{\textrm{M}_\lambda }$ = 50 nm.
Fig. 10. Analysis of additional temporal coherence effects. RMS of RI reconstruction errors calculated with respect to the coherent reconstruction (i.e. generated and reconstructed for single wavelength λ = 500 nm), in the function of FWHMλ for dispersive and nondispersive media; RMS value was computed in the volume of 12 × 12 × 11 µm centered around the sample (a). RI error maps calculated with respect to the coherent reconstruction, caused by: reduced temporal coherence with FWHMλ = 50 nm (b) – (c), the same reduced temporal coherence and, additionally, dispersion (d) – (e), the same reduced temporal coherence and, additionally, aberrated spectrum distribution (f) – (g). In this figure RI error cross-sections are organized column-wise.
The purely spectral factor (no dispersion, symmetric spectrum) causes only marginal RI errors (Fig. 10(b) - (c)), which have their maxima located along boundary between optical media. As it appeared, this does not affect the shape and visibility of the sample, even for large spectral width of FWHMλ = 50 nm (FWHMν = 59.96 THz) – Fig. 9, which may be interpreted as a robustness of the reconstruction algorithm for such defined reduced temporal coherence. Though, one can see that both the spectrum asymmetric shape and dispersion, causes an observable increase in the RI error. Different character of chromatic RI changes in the PMMA and the immersion causes different RI scaling and thus the observed value shift (Fig. 10(d) - (e)). The effect related to dispersion is however strongly determined by the individual characteristics of the investigated materials. Similarly, the effect of aberrated spectrum introduced a relatively large RI error (Fig. 10(f) - (g)), which implies the value mismatch in the sample area, with respect to the ground-truth coherent reference. Notably, the consequence of the asymmetric spectrum may be different depending on the choice of a wavelength that is used for reconstruction. In our case, we used maximal spectrum peak; however, the issue is not straightforward, e.g., the center of mass of the spectrum could be also considered.
3.4 Quantitative evaluation of the measurement errors in optical diffraction tomography with scanning of illumination
In this section we analyze the influence of broadband sources on tomographic measurements in ODT setups with scanning of illumination beam. For our simulation we used the same optical parameters as in the previous paragraphs. We utilized the cone illumination scanning scenario [8] with the cone half angle of 30° (to the optical (Z) axis) and 45 uniformly distributed projections (azimuthal step of 8°). The coherent reconstruction of the sample with its spectra mapped on the Ewald spheres is presented in Fig. 11.
Fig. 11. Coherent ODT measurement with the cone illumination: sample k-space (a) – (c), retrieved RI distribution cross-sections (d) – (f).
For the cone illumination, we investigated the impact of varying spectral width, similarly to Sec. 3.3. For applied values of FWHMλ in the range 0–50 nm, none of the RI reconstructions revealed noticeable undesired effect. The results show that even a wide spectrum of 50 nm can give satisfactory performance (Fig. 12). The full plot of the RMS reconstruction errors for various $\textrm{FWH}{\textrm{M}_\lambda }$ can be seen in Fig. 13(a).
Fig. 12. ODT reconstruction with reduced temporal coherence ($\textrm{FWH}{\textrm{M}_\lambda }$ = 50nm): RI distribution (a) – (c), RI error maps with respect to the coherent reconstruction (d) – (f).
The presented above results were obtained with DI reconstruction algorithm, which, by its own, does not address the major challenge of the illumination scanning tomographic configuration, that is the limited angle problem. One of the most popular numerical methods of suppressing the missing cone artifacts utilizes nonnegativity constraint regularization (NNC). We applied the algorithm to rate the performance in the regime of decreased temporal coherence. The numerical calculations outline the successful RI reconstruction improvement with respect to the ground truth, achieving a stable quantitative efficiency regardless of the coherence degree (Fig. 13(b)). The exemplary simulation of temporarily partially coherent ODT reconstruction (FWHMλ = 20 nm) with NNC is shown in Fig. 13(c) - (h).
Fig. 13. The effect of reduced temporal coherence in the tomographic configuration with the limited angle illumination. RMS error of partially coherent DI (no NNC) reconstruction calculated with respect to the analogical coherent reconstruction (i.e. generated and reconstructed for single wavelength λ = 500 nm), in the function of FWHMλ (a); RMS error of RI reconstructions calculated with respect to the ideal RI distribution of the sample (ground-truth model) with and without NNC (b); k-space resolved with 30 NNC iterations for partially coherent illumination (FWHMλ = 20 nm) (c) – (e) and corresponding RI reconstruction cross-sections (f) – (h). RMS errors were computed in the volume of 12 × 12 × 11 µm centered around the sample.
In this section, we limited our simulations to the investigation of symmetric spectrum, without sample dispersion, since the computations of ODT projections are highly time consuming and the general error tendency is expected to remain the same as in Sec. 3.3.
4. Conclusion
In this work we performed numerical examination of the effect of reduced temporal coherence in DHM and ODT measurements, including sample scattering, dispersion and spectrum shape. Results proved a promising level of achromaticity of phase and RI measurements in the 3-beam total-shear diffraction grating interferometer. The achromatic character of interferometric fringes obtained in this kind of configurations proved its advantage over widely known and exploited interferometric setups, such as Mach-Zehnder system, where the effective field of view is strictly limited by the coherence area.
For the grating-assisted setup, as far as the spectral width of illumination is limited to temporal parameters typical for standard coherent or partially coherent sources such as LASER, SLED, PTLS etc. (FWHMλ < 10 nm), the RI and phase reconstruction errors appear to be almost totally achromatic, considering simulated phenomena. This observation corroborates the potential of the setup to increase the phase and as well as RI imaging quality by lowering the degree of temporal coherence. The overall performance of the exploited reconstruction algorithms, i.e., Fourier hologram reconstruction method, DI and NNC, proved their applicability even for significantly lowered temporal coherence, which is highly advantageous within the process of coherent noise reduction. However, for substantial decrease of temporal coherence corresponding to FWHMλ larger than 30–40 nm, quantitative non-linear errors are observed. Particularly, when the spectrum is found to be irregular or simply asymmetric, the reconstructed RI errors may be noticeable (around ${10^{ - 4}})$. Aforementioned could be perceived as a potential challenge for the high accuracy measurements in wide-spectrum tomography yet in investigated case it did not reveal its crucial adverse impact on visual specimen inspection. The results presented herein prove a possibility of successful and relatively accurate ODT measurement even with significantly lowered temporal coherence of illumination, when the properly achromatic optical setup is employed. Referring to the content of this manuscript, the class of common-path grating-assisted DHM and ODT configurations has prospective ability of noiseless and easily achievable sample inspection.
In sum, we believe that the presented analysis paves a way to systematic identification of quantitative limitations and error sources in DHM and ODT measurements in general, which in further perspective may lead to overall performance improvement. Notably, in this paper we took only the temporal coherence phenomena into consideration. In our future works, we aim at the comprehensive analysis of the spatiotemporal coherence effect on DHM and ODT to achieve the highly faithful numerical model of the optical measurement with a potential for exact accuracy prediction.
Funding
Narodowe Centrum Nauki (OPUS 2020/37/B/ST7/03629).
Disclosures
Authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are to be made available upon reasonable request.
References
1. Y. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018). [CrossRef]
2. D. Jin, R. Zhou, Z. Yaqoob, and P. T. C. So, “Tomographic phase microscopy: principles and applications in bioimaging,” J. Opt. Soc. Am. B 34(5), B64 (2017). [CrossRef]
3. X. Ma, W. Xiao, and F. Pan, “Optical tomographic reconstruction based on multi-slice wave propagation method,” Opt. Express 25(19), 22595 (2017). [CrossRef]
4. J. Kostencka, T. Kozacki, and M. Józwik, “Holographic tomography with object rotation and two-directional off-axis illumination,” Opt. Express 25(20), 23920 (2017). [CrossRef]
5. J. Kostencka, T. Kozacki, M. Dudek, and M. Kujawińska, “Noise suppressed optical diffraction tomography with autofocus correction,” Opt. Express 22(5), 5731 (2014). [CrossRef]
6. M. Habaza, B. Gilboa, Y. Roichman, and N. T. Shaked, “Tomographic phase microscopy with 180° rotation of live cells in suspension by holographic optical tweezers,” Opt. Lett. 40(8), 1881 (2015). [CrossRef]
7. W. Gorski and W. Osten, “Tomographic imaging of photonic crystal fibers,” Opt. Lett. 32(14), 1977 (2007). [CrossRef]
8. P. Zdańkowski, J. Winnik, K. Patorski, P. Gocłowski, M. Ziemczonok, M. Józwik, M. Kujawińska, and M. Trusiak, “Common-path intrinsically achromatic optical diffraction tomography,” Biomed. Opt. Express 12(7), 4219 (2021). [CrossRef]
9. K. Lee, S. Shin, Zahid Yaqoob, P. T. C. So, and Y. Park, “Low-coherent optical diffraction tomography by angle-scanning illumination,” J. Biophotonics 12(5), 1 (2019). [CrossRef]
10. A. Kuś, W. Krauze, P. L. Makowski, and M. Kujawińska, “Holographic tomography: hardware and software solutions for 3D quantitative biomedical imaging,” ETRI Journal 41(1), 61–72 (2019). [CrossRef]
11. P. Bouchal, R. Chmelík, and Z. Bouchal, “Phase of white light and its compatibility to the optical path,” Opt. Express 29(8), 12398 (2021). [CrossRef]
12. S. Shin, K. Kim, K. Lee, S. Lee, and Y. Park, “Effects of spatiotemporal coherence on interferometric microscopy,” Opt. Express 25(7), 8085 (2017). [CrossRef]
13. T. H. Nguyen, C. Edwards, L. L. Goddard, and G. Popescu, “Quantitative phase imaging with partially coherent illumination,” Opt. Lett. 39(19), 5511 (2014). [CrossRef]
14. Y. Choi, P. Hosseini, J. W. Kang, S. Kang, T. D. Yang, M. G. Hyeon, B.-M. Kim, P. T. C. So, and Z. Yaqoob, “Reflection phase microscopy using spatio-temporal coherence of light,” Optica 5(11), 1468 (2018). [CrossRef]
15. C. Park, K. Lee, Y. Baek, and Y. Park, “Low-coherence optical diffraction tomography using a ferroelectric liquid crystal spatial light modulator,” Opt. Express 28(26), 39649 (2020). [CrossRef]
16. W. Krauze, A. Kuś, D. Śladowski, E. Skrzypek, and M. Kujawińska, “Reconstruction method for extended depth-of-field optical diffraction tomography,” Methods 136, 40–49 (2018). [CrossRef]
17. F. Pan, L. Yang, and W. Xiao, “Coherent noise reduction in digital holographic microscopy by averaging multiple holograms recorded with a multimode laser,” Opt. Express 25(18), 21815 (2017). [CrossRef]
18. C. Buitrago-Duque and J. Garcia-Sucerquia, “Physical pupil manipulation for speckle reduction in digital holographic microscopy,” Heliyon 7(1), e06098 (2021). [CrossRef]
19. F. Pan, W. Xiao, S. Liu, F. Wang, L. Rong, and R. Li, “Coherent noise reduction in digital holographic phase contrast microscopy by slightly shifting object,” Opt. Express 19(5), 3862 (2011). [CrossRef]
20. J. van Rooij and J. Kalkman, “Large-scale high-sensitivity optical diffraction tomography of zebrafish,” Biomed. Opt. Express 10(4), 1782 (2019). [CrossRef]
21. G. Choi, D. Ryu, Y. Jo, Y. S. Kim, W. Park, H. Min, and Y. Park, “Cycle-consistent deep learning approach to coherent noise reduction in optical diffraction tomography,” Opt. Express 27(4), 4927 (2019). [CrossRef]
22. X. Ma, W. Xiao, and F. Pan, “Reconstruction method for samples with refractive index discontinuities in optical diffraction tomography,” Opt. Lasers Eng. 94, 58–62 (2017). [CrossRef]
23. T. Pham, E. Soubies, A. Ayoub, D. Psaltis, and M. Unser, “Adaptive Regularization for Three-Dimensional Optical Diffraction Tomography,” in 2020 IEEE 17th International Symposium on Biomedical Imaging (ISBI) (IEEE, 2020), pp. 182–186.
24. C. Larivière-Loiselle, E. Bélanger, and P. Marquet, “Polychromatic digital holographic microscopy: a quasicoherent-noise-free imaging technique to explore the connectivity of living neuronal networks,” Neurophotonics 7(04), 040501 (2020). [CrossRef]
25. S. Chowdhury, W. J. Eldridge, A. Wax, and J. Izatt, “Refractive index tomography with structured illumination,” Optica 4(5), 537 (2017). [CrossRef]
26. T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014). [CrossRef]
27. M. Chen, L. Tian, and L. Waller, “3D differential phase contrast microscopy,” Biomed. Opt. Express 7(10), 3940–3950 (2016). [CrossRef]
28. J. Li, Q. Chen, J. Sun, J. Ding, and C. Zuo, “Three-dimensional tomographic microscopy technique with multi-frequency combination with partially coherent illuminations,” Biomed. Opt. Express 9(6), 2526–2542 (2018). [CrossRef]
29. C. Zuo, J. Li, J. Sun, Y. Fan, J. Zhang, L. Lu, R. Zhang, B. Wang, L. Huang, and Q. Chen, “Transport of intensity equation: a tutorial,” Opt. Lasers. Eng. 135, 106187 (2020). [CrossRef]
30. C. Zuo, J. Sun, J. Li, A. Asundi, and Q. Chen, “Wide-field high-resolution 3D microscopy with Fourier ptychographic diffraction tomography,” Opt. Lasers. Eng. 128, 106003 (2020). [CrossRef]
31. R. Horstmeyer, J. Chung, X. Ou, G. Zheng, and C. Yang, “Diffraction tomography with Fourier ptychography,” Optica 3(8), 827 (2016). [CrossRef]
32. J. Li, A. Matlock, Y. Li, Q. Chen, C. Zuo, and L. Tian, “High-speed in vitro intensity diffraction tomography,” Adv. Photon. 1(06), 1 (2019). [CrossRef]
33. J. M. Soto, J. A. Rodrigo, and T. Alieva, “Optical diffraction tomography with fully and partially coherent illumination in high numerical aperture label-free microscopy,” Appl. Opt. 57(1), A205 (2018). [CrossRef]
34. P. Girshovitz and N. T. Shaked, “Compact and portable low-coherence interferometer with off-axis geometry for quantitative phase microscopy and nanoscopy,” Opt. Express 21(5), 5701 (2013). [CrossRef]
35. X. Chen, M. E. Kandel, C. Hu, Y. J. Lee, and G. Popescu, “Wolf phase tomography (WPT) of transparent structures using partially coherent illumination,” Light: Sci. Appl. 9(1), 142 (2020). [CrossRef]
36. K. Patorski, P. Zdańkowski, and M. Trusiak, “Grating deployed total-shear 3-beam interference microscopy with reduced temporal coherence,” Opt. Express 28(5), 6893 (2020). [CrossRef]
37. V. Dubey, D. Popova, A. Ahmad, G. Acharya, P. Basnet, D. S. Mehta, and B. S. Ahluwalia, “Partially spatially coherent digital holographic microscopy and machine learning for quantitative analysis of human spermatozoa under oxidative stress condition,” Sci. Rep. 9(1), 3564 (2019). [CrossRef]
38. A. Nativ and N. T. Shaked, “Compact interferometric module for full-field interferometric phase microscopy with low spatial coherence illumination,” Opt. Lett. 42(8), 1492 (2017). [CrossRef]
39. K. Zhuo, Y. Wang, Y. Wang, K. Wen, M. Liu, Y. Ma, J. Zheng, and P. Gao, “Partially Coherent Illumination Based Point-Diffraction Digital Holographic Microscopy Study Dynamics of Live Cells,” Front. Phys. 9, 796935 (2021). [CrossRef]
40. F. Dubois, M.-L. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. 43(5), 1131 (2004). [CrossRef]
41. A. El Mallahi, C. Minetti, and F. Dubois, “Automated three-dimensional detection and classification of living organisms using digital holographic microscopy with partial spatial coherent source: application to the monitoring of drinking water resources,” Appl. Opt. 52(1), A68 (2013). [CrossRef]
42. F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38(34), 7085 (1999). [CrossRef]
43. F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, J.-C. Legros, and P. Kischel, “Pattern recognition with a digital holographic microscope working in partially coherent illumination,” Appl. Opt. 41(20), 4108 (2002). [CrossRef]
44. J. M. Soto, J. A. Rodrigo, and T. Alieva, “Partially Coherent Optical Diffraction Tomography Toward Practical Cell Study,” Front. Phys. 9, 666256 (2021). [CrossRef]
45. S. Pan and A. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31(5), 1262–1275 (1983). [CrossRef]
46. J. Lim, K. Lee, K. H. Jin, S. Shin, S. Lee, Y. Park, and J. C. Ye, “Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography,” Opt. Express 23(13), 16933 (2015). [CrossRef]
47. W. Krauze, P. Makowski, M. Kujawińska, and A. Kuś, “Generalized total variation iterative constraint strategy in limited angle optical diffraction tomography,” Opt. Express 24(5), 4924 (2016). [CrossRef]
48. P. Kolman and R. Chmelík, “Coherence-controlled holographic microscope,” Opt. Express 18(21), 21990 (2010). [CrossRef]
49. E. N. Leith and G. J. Swanson, “Achromatic interferometers for white light optical processing and holography,” Appl. Opt. 19(4), 638 (1980). [CrossRef]
50. R. Chmelik, “Coherent imaging with incoherent light in digital holographic microscopy,” in J. Peřina, L. Nozka, M. Hrabovský, D. Senderáková, W. Urbańczyk, and O. Haderka, eds. (2012), p. 869711.
51. T. Slabý, P. Kolman, Z. Dostál, M. Antoš, M. Lošťák, and R. Chmelík, “Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic microscope,” Opt. Express 21(12), 14747 (2013). [CrossRef]
52. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969). [CrossRef]
53. A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6(8), 374 (1981). [CrossRef]
54. Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009). [CrossRef]
55. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982). [CrossRef]
56. D. Suski, J. Winnik, and T. Kozacki, “Fast multiple-scattering holographic tomography based on the wave propagation method,” Appl. Opt. 59(5), 1397 (2020). [CrossRef]
57. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (Vol. 164 of McGraw-Hill Physical and Quantum Electronics Series, 1996).
58. T. Kozacki, “Numerical errors of diffraction computing using plane wave spectrum decomposition,” Opt. Commun. 281(17), 4219–4223 (2008). [CrossRef]
59. X. Zhang, J. Qiu, X. Li, J. Zhao, and L. Liu, “Complex refractive indices measurements of polymers in visible and near-infrared bands,” Appl. Opt. 59(8), 2337 (2020). [CrossRef]
60. Bahaa E. A. Saleh and Malvin C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley Series in Pure and Applied Optics, 2019).
