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Abstract
Tomographic diffractive microscopy (TDM) is increasingly gaining attention, owing to its high-resolution, label-free imaging capability. Fast acquisitions necessitate limiting the number of holograms to be recorded. Reconstructions then rely on optimal Fourier space filling to retain image quality and resolution, that is, they rely on optimal scanning of the tomographic illuminations. In this work, we theoretically study reflection TDM, and then the 4Pi TDM, a combination of transmission and reflection systems. Image simulations are conducted to determine optimal angular sweeping. We found that three-dimensional uniform scanning fills Fourier space the best for both reflection and 4Pi configurations, providing a better refractive index estimation for the observed sample.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION
Tomographic diffractive microscopy (TDM) has increasingly been used as a non-invasive and label-free imaging technique to provide three-dimensional (3D) complex refractive index (RI) distribution of quasi-transparent samples with a sub-wavelength resolution [1–5]. It is based on holographic recording of many diffracted fields from the object under varying illuminations, and their numerical recombination to compute 3D quantitative information.
The TDM can be implemented in various configurations. The most common is transmission TDM [6–12]. In the classical transmission TDM, where the illumination angles are varied while the sample is kept fixed, the frequency components along the optical axis are inaccessible [2,6,13]. Hence, the axial resolution is poor compared to transverse resolution [3,14].
Reflection TDM is preferable for samples that are mostly reflecting and quasi-flat [15,16]. However, such configuration has two main limitations. First, it lacks collecting low-frequency components [13], which prevents quantifying the smoother RI variations properly, as long as the first-order Born or Rytov approximations are used. Second, the asymmetric optical transfer function (OTF) in reflection TDM could lead to wrong estimation of the complex RI, by intermixing the refraction and absorption components [17,18].
Tomographic diffractive microscopy with sample rotation delivers isotropic resolution images, for which non-contact rotation techniques, such as optical tweezers [19,20] or dielectrophoretic rotation [21], simplify sample preparation/manipulation. Another approach is multimodal TDM-fluorescence microscopy, in which the fluorescence maps serve as constraint to the tomographic reconstruction [22].
However, for best results in terms of 3D resolution and image quality, one has to combine sample and illumination rotations [23,24]. Still, while successful [25–28], this approach is limited to free-standing samples (such as pollen grains, blood cells, optical fibers), which can be manipulated using either mechanically induced rotation or optical tweezers.
For more general specimens, one can consider 4Pi tomography, a configuration proposed by Lauer in 2002 [2], but which has not yet been implemented, to the best of our knowledge. An alternate configuration is mirror-assisted tomography, introduced by Mudry et al. [29], which has the advantage of being easier to implement, starting from a reflection system. In its full configuration, mirror-assisted TDM is equivalent to a 4Pi setup, and for mostly forward-diffracting samples, it can be simplified to constitute an alternative to a conventional transmission system [17].
For all configurations, an optimal sample illumination pattern is critical for fast imaging, requiring a low number of sequential illumination. For transmission TDM, various classes of illumination scanning have been studied [18], showing that optimal patterns can indeed provide reconstructed images of better quality. In this work, we consider reflection TDM and then extend our investigations to 4Pi microscopy, which necessitates combining transmission and reflection modalities. Fourier space filling factor and phantom reconstruction simulations are used to evaluate the efficiency of illumination scanning.
Fig. 1. Synthetic aperture process for reflection, transmission, and 4Pi TDM: (a) 2D synthetic aperture process in the $({k_y} - {k_z})$ plane and (b) its extended effective OTF in 3D for reflection TDM; (c) 2D synthetic aperture process in the $({k_y} - {k_z})$ plane and (d) the corresponding extended effective OTF in 3D for transmission TDM; (e)–(g) 2D central slice of the effective OTF computation process for a 4Pi configuration from reflection and transmission measurements.
2. SUMMARY OF THE TDM DATA ACQUISITION SCHEME
In TDM, the 3D RI map of an object is computed through numerical combination of two-dimensional (2D) holograms acquired from multiple illumination angles (see Refs. [2] through [30]). The simplest, fastest reconstruction algorithms in TDM use Born or Rytov approximations assuming weakly scattering samples, for which we have shown that reconstructed image quality is dependent on the angular sweeping scheme [18]. Within this framework (weak diffraction/absorption), the object wavevectors support ${{\boldsymbol k}_{o}}$ is obtained from the cap of sphere, which depicts diffracted wavevectors ${{\boldsymbol k}_d}$ collected by the objective and translated by the incident wavevector ${{\boldsymbol k}_i}$ given by
Thus, in tomography, by sequentially using $N$ illuminations, one records $N$ ${{\boldsymbol k}_d}$ sets, to be properly relocated in Fourier space using their respective ${{\boldsymbol k}_i}$ to obtain N ${{\boldsymbol k}_o}$ vector sets. As microscope objectives are limited in numerical aperture (NA), only cap of spheres are collected, but anyway, an extended optical transfer function (OTF) can be syntetized, as depicted Fig. 1, for transmission, reflection, and 4Pi configurations.
In reflection, the final OTF takes the shape of a “cap of ball,” as illustrated in Figs. 1(a) and 1(b), shifted toward high longitudinal frequencies, but which does not allow for registering low-frequency components. In transmission, the OTF takes the shape of an angularly limited horn torus (so-called doughnut [24]), as illustrated in Figs. 1(c) and 1(d), and it is characterized by a so-called missing cone [31] of non-recorded frequencies along the optical axis.
A 4Pi TDM system, as originally proposed by Lauer [2], uses a two-facing-objective configuration, as in 4Pi fluorescence microscopy [32], in view of combining reflection and transmission configurations, as shown in Fig. 1(g). When illuminating through the first objective, and detecting reflection through that objective and transmission through the facing objective, one obtains the doughnut and a first cap of ball, as shown in Fig. 1(e). When scanning illumination through the second objective, one records again the doughnut, but in a different way, plus the complementary cap of ball, as shown in Fig. 1(f). Recombining the four data subsetsleads to the final 4Pi OTF, as shown Fig. 1(g). Note that the data acquired in transmission mode from both directions of illumination are not identical but combine to obtain a better filling of the central doughnut.
A. Scanning Patterns
As for transmission TDM, efficient data collection in reflection, and therefore also 4Pi TDM, requires diminishing as much as possible the necessary number of acquisitions, while trying to keep the final image quality necessary for proper sample investigations. Requirements may change whether ultimate 3D resolution is foreseen, or if only general shape of the observed sample is needed.
This work therefore investigates the influence of the illumination scanning scheme in reflection TDM, as well as in the 4Pi setup. The considered scanning patterns are star-like, grid, circle, spiral, flower, Fermat’s spiral with golden angle, 3D uniform distribution on a cap of sphere (3D UDCS), and 3D uniform distribution on a hemisphere (3D UDHS) scanning patterns. Their mathematical descriptions and implementations have been detailed in [18], in the case of transmission TDM.
Fig. 2. 2D (${k_x} - {k_y}$) and (${k_y} - {k_z}$) representations of OTFs for 4Pi TDM. (a) Theoretically, fully filled OTF; (b)–(g) OTFs corresponding to 600 illumination angles: (b) three-axes star; (c) rectangular grid; (d) annular pattern with four concentric circles; (e) double spiral with six turns and regular spacing of points; (f) eight-petal flower; (g) Fermat’s spiral; (h) 3D UDCS; and (i) 3D UDHS.
Note that star-like, grid, circle, spiral, and flower patterns are the families of patterns depending not only on the number of illuminations to be considered, but also on some parameters such as the number of branches for star-like illumination. In contrast, Fermat’s spiral, 3D UDCS, and 3D UDHS patterns are governed by equations involving only the number of points to be distributed [18].
B. Evaluation Criteria
The filling factor of the Fourier space is used as a metric to compare the several illumination patterns, defined as the ratio of the volume ${V_s}$ (in pixel counts) effectively filled by the cap of spheres when using the scanning (Fig. 2) to that of the theoretical OTF volume ${V_T}$ of the cap of ball or of the full 4Pi OTF:
For quantifying image reconstruction quality when simulating TDM acquisitions using the considered scanning patterns, the root mean square error (RMSE) of those reconstructed images compared to the test objects is used:
3. 3D OPTICAL TRANSFER FUNCTION
A. OTFs Construction
Simulations of the corresponding OTFs when using the considered illumination schemes are shown in Figs. 2(b)–2(i), using 600 illumination angles in each case. We consider oil immersion (${n_{{\rm oil}}} = 1.515$) objectives with NA = 1.4. Figures 2(b)–2(i) illustrate 4Pi TDM OTFs, out of which reflection TDM OTF can be obtained simply by only considering the outer parts at high ${k_z}$ values.
Note that scanning with three-axes star pattern [Fig. 2(b)] and non-overlapping flower petals [Fig. 2(f)] shows poor (${k_y} - {k_z}$) frequency coverage, leaving large gaps in the reflection OTFs. Such large gaps are not observed in the corresponding transmission OTF. Using grid [Fig. 2(c)], spiral [Fig. 2(e)], and Fermat’s spiral [Fig. 2(g)] illuminations allows for better evenly distributed spatial frequency components coverage, but the grid scanning results in bad frequency coverage along deprived directions in the (${k_x} - {k_y}$) plane [18]. On the contrary, scanning schemes such as 3D UDCS [Fig. 2(h)] and 3D UDHS [Fig. 2(i)] patterns lead to OTFs characterized by string of holes along the optical axis, while annular scanning leaves few, but large holes along the ${k_z}$ direction, Fig. 2(d).
Finally, comparing Fig. 2 with Fig. 3 in Ref. [18], note that in the 4Pi configuration, the central doughnut is symmetrical and better filled than in conventional transmission TDM, thanks to the two successive illuminations via both objectives.
B. Filling Factor
The filling factor ${\rm FF}(\%)$, computed as a function of the number of illuminations (from 1 to 600) is plotted in Fig. 3. For simplicity, Fig. 3(a) depicts the curves obtained for reflection TDM, and only for those patterns giving the best results for Fourier space filling in their respective families: the star pattern with three axes, flower pattern with eight petals, regularly sampled double-spiral along the curvilinear abscissa with six turns, and regular azimuthal spacing annular pattern with three concentric circles (in that case, different from transmission for which the optimum was attained with four concentric circles [18]).
Fig. 3. OTF ${\rm FF}(\%)$ plots for reflection and 4Pi TDM: (a) ${\rm FF}(\%)$ values for reflection, and (b) mean ${\rm FF}(\%)$ with maximum variation of the parameters in the illumination patterns; (c) ${\rm FF}(\%)$ values for 4Pi configuration.
Note that for low numbers of illuminations, ${\rm FF}(\%)$ is very similar for each scanning pattern, but curves plot further apart when the number of illumination angles increases. These curves follow the same trend as for transmission TDM [18], but with two main differences. First, the annular pattern now gives results similar as the 3D UDHS scanning, and spiral pattern gives much higher filling compared to the star illumination pattern (compare with Fig. 4(a) of Ref. [18]. Second, the dispersion of the curves is higher for reflection TDM. Flower, Fermats’s spiral and grid scanning deliver very similar results in terms of Fourier space filling, while star scanning appears to be the least efficient. Surprisingly, scanning along a single circle at maximum illumination NA in fact delivers a higher filling factor than using a star pattern, although it only fills the lower part of the reflection OTF, which then takes the shape of half the doughnut of transmission OTF, but shifted to higher ${k_z}$.
The star, spiral, annular, and flower scanning in fact are families of patterns, which exact shape depends on various parameters. Figure 3(b) therefore plots the mean value of each family, with error bars around representing the maximum and minimum values obtained when computing a ${\rm FF}(\%)$ for a given family.
The spiral pattern is characterized by the largest variations as a function of its parameters (number of turns, simple or double spiral, type of point separation [18]). As for transmission TDM, two-turn non-regularly sampled single spiral delivers the lowest ${\rm FF}(\%)$, while six-turn regularly sampled double spiral delivers the highest. Star patterns are characterized by low dispersion of the filling factor. For flower patterns family, increasing the number of petals results in more evenly distributed illumination angles, which in turn results in a better filling factor, thanks to reduced redundant information. Maximum and minimum filling are obtained with eight and three petals, respectively. Annular scanning with concentric circles gives best results with three circles, with even azimuthal spacing.
Finally, Fig. 3(c) plots ${\rm FF}(\%)$ for full 4Pi configuration. Curves exhibit the same trend as for reflection Fig. 3(a) and transmission Fig. 4 from Ref. [18] setups. There are, however, noticeable differences.
First, the filling factor systematically increases, compared to separate transmission and reflection cases, reaching a maximum close to 80% is the best case scenario. This is a benefit of the symmetric 4Pi configuration, which, with the same number of illumination angles allows for recording two transmission OTFs, resulting in a better filling factor of that part of the final 4Pi OTF.
Then, the combination of reflection and transmission results in families of curves, with the 3D UDHS, 3D UDCS, and annular scanning giving best results, and star patterns being noticeably less efficient than the flower, Fermat, spiral, and grid ones, which deliver very similar results.
Finally, we considered a special case of annular scanning: annular illumination at maximum NA, which consists of a single annular illumination, performed at maximum illumination angle, which in a 4Pi configuration is automatically the same as the maximum detection angle, as same objectives are used in a symmetric configuration for illumination and detection (see also Fig. 2 of Ref. [18]). Such scanning presents peculiar properties when considering mirror-assisted tomography [17,29], which can be viewed as a special case of 4Pi tomography. This peculiar scanning scheme delivers the same overall filling factor as the flower, Fermat, spiral, and grid scanning. This is the result of annular illumination at maximum NA being the most efficient scanning to fill half a doughnut of the transmission OTF. In a simple transmission TDM, this results in a 50% ${\rm FF}(\%)$ at best [18], which is lower than all other scanning patterns. However, for some specific configurations, this peculiar property becomes an advantage. If the sample presents geometrical symmetries [33], scanning can be greatly simplified. For non-absorbing specimens, the Hermitian symmetry of the sample spectrum can be involved, which also leads to a better angular scanning [18]. Similarly, for mirror-assisted TDM [29], annular illumination at maximum NA would also be the best scanning scheme if the sample is mostly forward diffracting [17]. Here, symmetrization results from the 4Pi configuration itself, leading to an excellent filling factor of the transmission part of the 4Pi OTF. This high filling factor even compensates for lower filling factor of the reflection parts of the 4Pi OTFs, as presented in Fig. 3(a). However, the reflection parts of the 4Pi OTF with annular illumination at maximum NA do not allow recording of high ${k_z}$ frequencies; thus, the overall imaging quality is expected to be lower along the optical axis.
In summary, in terms of Fourier space filling, the 3D UDHS and annular scanning patterns deliver the best overall results. The 3D UDHS giving best filling factor for both transmission and reflection has interesting implications in view of practical implementation of 4Pi tomography (see Section 5).
4. IMAGE SIMULATIONS
A. Shepp–Logan Phantom
We first consider a Shepp–Logan phantom with constrast corresponding to pure index of refraction variations [34] (no absorption). Here the background medium is set as RI $n_{0}=n_{0}^{\prime}+in_{0}^{\prime\prime}=1.51+i0$, with maximum index of refraction departure at 0.04, and the phantom is $5\;\unicode{x00B5}{\rm m}$ wide along the optical axis. Figures 4(a)–4(d) show $y - z$ cuts of the original phantom, its reconstruction with transmission TDM, reflection TDM, and 4Pi TDM, respectively. The 3D UDHS with 600 angles is considered at $\lambda = 633\,\,\rm nm$, and considering NA = 1.4 oil immersion objectives (${n_{{\rm oil}}} = 1.515$). We focus on longitudinal sections, for which the 4Pi TDM is expected to deliver improved results, its lateral resolution being the same as transmission or reflection TDM [2,13,25]. As for our previous work [18], images are simulated assuming Born approximation, by filtering the object’s Fourier spectrum of the OTFs generated by the considered scanning pattern. This limits present investigations to small-thickness samples with a low index contrast, such as single isolated cells, and not tissue or strongly scattering samples, for which more advanced diffracted beam simulations, as well as image reconstruction approaches, should be used.
Fig. 4. TDM simulations: (a) a modified Shepp–Logan phantom $({k_y} - {k_z})$. (b)–(d) Reconstructions using (b) transmission, (c) reflection, and (d) 4Pi TDM. The scale bar is $1\,\,\unicode{x00B5}{\rm m}$ (Visualization 1).
Figure 4(b) depicts the characteristic elongation along the optical axis of transmission microscopes. Note how the small inclusions are smeared out (single-head arrow), so that the two near the large one along the $z$-axis are not distinguishable anymore (double-head arrow). In Fig. 4(c), only edges are visible, which is easily understood as the pure reflection TDM only captures sample’s high spatial frequencies (see Fig. 2), and therefore acts as a high-pass filter. Only surfaces with edges from abrupt RI changes can be reconstructed. Combining both techniques into a 4Pi method gives better reconstruction and RI estimation, as seen from Fig. 4(d). Note however that oscillations along the ${z}$ direction remain. They are consequences of the 4Pi OTF not being continuous: gaps remain between the central, transmission “doughnut,” and both reflection parts. Anyway, morphological reconstruction and RI estimation are of better quality, even considering the simplest Born approximation (Visualization 1).
This modified 3D Shepp–Logan phantom present complex features, which make it ideal to test the influence of scanning patterns in 4Pi TDM, especially when the number of illumination angles is low. This makes it useful if one wants to speed up data acquisitions. Figures 5(a)–5(d) show $(y - z)$ cross-sections of the phantom, and reconstructions using the star, 3D UDHS, and annular at maximum NA illuminations, now considering only 60 illuminations. The zoomed area emphasizes the ability of the various scanning patterns to reconstruct smaller features, especially along the optical axis, which is the direction for which the 4Pi TDM is expected to improve results with respect to transmission TDM only.
Fig. 5. 4Pi TDM simulations: (a) a modified Shepp–Logan phantom. Reconstructed images using (b) star, (c) 3D UDHS, and (d) annular at max. NA. The scale bar is $1\,\,\unicode{x00B5}{\rm m}$. The zoomed area below shows the smaller details inside the phantom (Visualization 2).
The 3D UDHS illumination pattern gives better size and shape estimation of the three small structures visible on the left side of the zoomed image, by opposition to star pattern scanning, which fails at reconstructing them and is plagued by more pronounced artefacts. Annular illumination at maximum NA identifies the smaller structures on the left as 3D UDHS, but it has lower performances along the $z$-axis. This is due, as already mentioned, to the peculiar shape of the 4Pi OTF with this scanning scheme, unable to record high ${k_z}$ object frequencies. It may be an efficient scanning in transmission TDM, but should therefore be avoided for efficient 4Pi scanning (see also Visualization 2).
B. Refracting Bead with Absorbing Inclusion
The above described Shepp–Logan phantom is purely refracting. The TDM is indeed capable of measuring absorption, which is also an interesting [3,14,18,35–38], albeit often neglected contrast mechanism. Worst, absorption and refractive components can mix, distorting measurements [17,18]. We therefore also consider an object, suspended in a background medium of RI ${n_0=n_0^{\prime}+in_0^{\prime\prime}=1.51+0i}$ and consisting into a microsphere of diameter $2\,\,\unicode{x00B5}{\rm m}$ and refraction index ${n_1} = 1.48 + 0i$, with two absorptive inclusions of a $0.8\,\,\unicode{x00B5}{\rm m}$ diameter, with a complex RI ${n_2} = {n_3} = 1.48 + 0.02i$, and located along the optical axis, at $z = \pm 0.5\,\,\unicode{x00B5}{\rm m}$, respectively.
Figures 6(a) and 6(f) depict the $(y - z)$ cross sections within reconstructions of this synthetic object, considering the 3D UDHS, Fermat, and star scanning, applying the same wavelength and NA as in previous section, and using 60 illuminations. Because two images of the sample are to be reconstructed (refraction and absorption), we adopt the same presentation as for Fig. 5 in Ref. [18]. This highlights another advantage of the 4Pi configuration: the resulting OTF is not only better filled, but it is also much more symmetrical than in transmission-only TDM, as can be seen when comparing Fig. 2 with Fig. 3 of Ref. [18]. This results in no (or extremely low) intermixing of real and imaginary components, as can be seen when comparing Fig. 6 with Fig. 5 of Ref. [18], in which for annular scanning at maximum NA, absorptive inclusions are clearly visible in the refraction image.
Fig. 6. 4Pi TDM simulations of an absorbing phantom, $(y - z)$ cross sections: (a) phantom refractive bead, and (e) phantom absorptive inclusions; (b)–(d) and (f)–(h) represent the corresponding reconstructed images using (b), (f) star, (c), (g) Fermat’s, and (d), (h) 3D UDHS. The scale bar is $1\,\,\unicode{x00B5}{\rm m}$.
For such a small number of illuminations, the filling factor FF(%) is rather low and very similar for each scanning pattern. As a result, image reconstruction is of the same quality, but with slightly better separation of the inclusions with 3D UDHS, than with Fermat and star scanning, with a slightly better index estimate.
This behavior is confirmed by analysing lateral and longitudinal profiles, depicted in Figs. 7(a)–7(c). For RI $n^\prime $ in Figs. 7(a) and 7(b), along both $y$- and $z$-axes, the best profile estimations are obtained for the 3D UDHS scanning, with the star scanning delivering the lowest, and Fermat’s spiral being in between.
Fig. 7. Profile plots of the bead in Fig. 6 for the RI (a) in the $y$-axis, (b) in the $z$-axis, and (c) for the absorption in the $z$-axis.
As for absorptive part profiles for both inclusions, same conclusion holds (see Fig. 7(c)). In particular, considering the small separation between the two inclusions ($\approx 200\,\,\rm nm$), we focus on how well each scanning reconstructs this separation, giving an insight about the $z$-axis resolution: the 3D UDHS illumination pattern more clearly separates the inclusion than Fermat’s spiral, and finally star patterns. In all cases, the Fermat scanning results in smoother profiles, with less oscillations, a feature also noticeable in transmission TDM [18] (note that the Fermat scanning also presents interesting advantages in magnetic resonance imaging scanning [39,40]).
To evaluate the accuracy of reconstructions when using the star, Fermat, and 3D UDHS scanning patterns, we also compute the RMSE [Eq. (3)] in a $25 \times 25 \times 25\,\,\unicode{x00B5}{\rm m^3}$ cubic volume around the object. Table 1 displays results for the RI and for the absorption. The 3D UDHS scanning gives the overall (sum of RI and absorption RMSE) lowest RMSE, with star scanning very close and Fermat scanning slightly higher.
Table 1. RMSE of Reconstructed Images for Refractive Index and Absorption in a $25 \times 25 \times 25\,\,$µ${{\rm m}^3}$ Volume Around the $2\,\,$µm Bead
5. EXPERIMENTAL CONSIDERATIONS
Since its original proposal [2], the 4Pi TDM has not been implemented, to the best of our knowledge. 4Pi in fluorescence microscopy [32] is very difficult to set up, mainly because of the short coherence length of fluorescence, which requires very tight balancing of the 4Pi interferometer arms. However, in 4Pi TDM, this problem would be greatly leveraged, thanks to the coherence length of the lasers that can be used, and because the synthetic aperture process is performed numerically, after separate detection of the diffracted field by both objective, and not physically as in 4Pi florescence, in which detected fields must interfere.
Other difficulties are however foreseeable. Sample preparation must be made between two cover glasses, to be observed through both objectives. It renders preparations more fragile, but has been proven feasible by several groups using TDM setups built around two high-NA objectives, one acting as a condenser and other as an objective [4,8,18,25,26,38,41–43].
Compared to these systems, a full 4Pi TDM microscope could be built by symmetrizing illumination and detection, as illustrated in Fig. 8. As 3D UDHS delivers best results in both transmission and reflection, it optimizes 4Pi scanning, which, possibly combined with multiplexed acquisition [44], would help to fasten acquisitions. With two successive illuminations, two transmission OTFs, and two reflection OTFs can be recorded, to be registered to form a final 4Pi OTF as depicted in Figs. 1 and 2.
Fig. 8. Sketch of a possible 4Pi TDM setup. BS, beam splitter; FM, flip mirror; TTM, tip-tilt mirror; RC, recombination cube.
A difficulty however will be registration of each sub-part of the OTF with respect to the others. As central doughnut in transmission and both reflection cap of ball are recorded in different experiments, their relative amplitude and/or phase may not be properly normalized, which is unfavourable for Fourier inversion. With an NA high enough, these three sub-parts of the 4PI OTF overlap, as depicted in Fig. 11 of Ref. [2]. However, that happens only for aperture angles greater then 70.5°, or NA that is 1.43 in oil (oil immersion objectives with NA = 1.45 exist), or a NA that is 1.25 in water (water immersion objectives with NA = 1.27 exist).
An alternative could be to use sample rotation as in [25–28], but within a 4Pi TDM system. A large overlap would then exist between the recorded 4Pi OTFs, which would facilitate registering, and finally completely filling in Fourier space with a ball having a radius $2{k_i}$. This would also result in an isotropic Nyquist resolution of $\lambda /4\,{\rm n}$ (${n}$ being the average index of the sample refraction) [2], higher than a combined sample/illumination TDM, which is limited to a Nyquist resolution of $\lambda /4\,{\rm NA}$ [24,25]. Alternately, advanced, constrained reconstructions or learning approaches could be used, which could also contribute to at least partially fill the remaining gaps in the 4Pi OTF and bolster the image reconstruction quality [34,45–51]. They could also help in reducing scanning constraints [45].
Another potential hurdle could be the difficulty to record the field, which is scattered back by the sample, as it may be very dim compared to the transmitted diffracted field. Note that this field is detected on a dark background, contrary to transmitted field, which also contains the original illumination beam: this will render detection of the weak back-scattered component easier. A solution could then be to use different exposure times for transmission and reflection cameras. The speed of the acquisition process would be then determined by the reflection camera acquisition time. If not sufficient, a high dynamic range approach could be used, recording transmitted components with a low illumination to not saturate the camera, and then the back-scattered would be recorded with a higher intensity illumination. This would slow down the process (roughly by another Factor 2), but permit higher dynamic detection for the back-scattered components of the diffracted field.
6. CONCLUSION
In this work, we studied the influence of illumination scanning in reflection TDM, which, combined with transmission, leads to a 4Pi TDM configuration, able to deliver an almost isotropic resolution in 3D. In particular, we found that the 3D UDHS illumination pattern best fills the Fourier space in reflection, like in previously studied transmission [18]. Simulations using a refractive-only Shepp–Logan phantom and a bead phantom with absorptive inclusions also show that image reconstruction quality is higher using this peculiar scanning. Thus, the 3D UDHS forms an optimal scanning for 4Pi TDM configurations, which will help to simplify and speed up image acquisition in an actual setup, and we discussed practical implementation hurdles and possible solutions to those.
Funding
Région Grand Est (FRCR 18P-07855, FRCR 19P-10656); Agence Nationale de la Recherche (ANR-18-CE45-0010, ANR-19-CE42-0004).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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