## Abstract

A new holographic data processing path for accurate quantitative tomographic reconstruction of 3D samples placed in a cylindrical capillary is proposed. The method considers strong unintentional focusing effects induced by the inner cylindrical boundary of the vessel: 1) introduction of cylindrical wave illumination of a sample, and 2) object wave deformation. The first issue is addressed by developing an arbitrary illumination tomographic reconstruction algorithm based on filtered backpropagation, while the second by a novel correction algorithm utilizing the optical rays analysis. Moreover, the processing path includes a novel holographic method for correction of spherical aberration related to refraction at a planar surface. Utility of the developed data processing path is proven with numerical simulations and experimental measurement of a specially prepared test sample.

© 2015 Optical Society of America

## 1. Introduction

Over the years, digital holographic microscopy (DHM) has proven its great potential in investigation of biological micro-objects [1–5]. The key properties of DHM that contribute to its success are possibility for quantitative imaging of phase changes, submicrometer resolution and noninvasive character of the measurement. DHM is a so-called 2½D imaging technique, which provides information about refractive index changes accumulated along path of optical rays. The exciting possibility of true 3D sample characterization is offered by optical diffraction tomography (ODT), which uses multiple DHM measurements captured for various sample perspectives. However, to maintain all the mentioned advantages of DHM, ODT requires devising a noninvasive method for alternation of the measurement views. In the case of living biomedical samples with high demands for stable cultivation conditions, this requirement proved to be very difficult to fulfill. For this reason, most researchers turned their attention to the illumination scanning configuration (ISC) of ODT, in which angular views are altered by scanning the illumination beam while keeping a sample and direction of observation fixed [6–10]. This solution provides high sample stability and high measurement speed. However, the major disadvantage of ISC is capturing measurements in a limited angle, which results in a distorted reconstruction and anisotropic resolution [11].

An alternative to ISC is a classical object rotation configuration (ORC) [12,13], in which angular scanning is achieved by rotating a sample under constant illumination and observation conditions. The ORC provides large and isotropic object frequency coverage, which translates into high quality tomographic reconstruction. To enable application of ORC to biomedical study, an approach using rotary capillary was proposed [14–16]. In this method a bio-object is introduced into a capillary filled with a cell culture medium and rotation of the vessel enables observation of a sample in a full angle of 360°. The capillary-supported ORC theoretically offers much higher quality of reconstruction than ISC. However, the full potential of this method is constrained by problems related to optical properties of the capillary. Firstly, the front cylindrical surface of the vessel introduces cylindrical wave illumination of a sample, while conventional ODT tomographic reconstruction algorithms work with plane wave illumination conditions. Secondly, the object wave is refracted by the rear surface of the capillary, resulting in a strong wavefront aberration [17]. The effect is similar to unintentional introduction of cylindrical lens into optical path of the object wave. Another problem is spherical aberration caused by refraction of the object wave at the planar bottom of the Petri dish, which is used in the capillary rotational module. The spherical aberration is, in this case, especially significant due to a strong cylindrical wave carrier produced by both surfaces of the capillary.

In all existing ODT systems, the capillary-related problems are addressed using thin element approximation (TEA). In this approach, the capillary correction boils down to direct subtraction of the phase distribution related to the capillary from the original phase measurements [14–16]. As it will be shown further in this paper, the TEA correction is insufficient. According to Ref. 17, the reminding problems, which are not corrected with this method, are spatially dependent transverse shift of the image and a change in magnification. Moreover, the TEA approach does not address the challenge of tomographic reconstruction with cylindrical illumination conditions. Instead, a conventional plane wave illumination algorithm is used. Furthermore, the TEA correction does not deal with the problem of spherical aberration introduced by refraction on the planar bottom wall of the Petri dish. In conclusion, the TEA approach is unable to provide the level of correction, which is necessary for reliable quantitative tomographic measurements.

In this paper we propose a new, accurate tomographic data processing path, which allows unlocking the full potential of the capillary-supported ORC. The proposed path addresses all of the mentioned problems related to the capillary holder. The first step of the processing path is holographic correction of object waves, which includes 1) correction of spherical aberration and 2) compensation of the object wave refraction at the rear cylindrical surface of the capillary. The first problem is solved with a novel, efficient method, which provides complete compensation of the aberration with two fast Fourier transformations. The second problem, i.e. aberration due to refraction at a cylindrical surface, is solved by employing local ray approximation (LRA) technique, which originally was developed for accurate phase-to-shape transition [18–20]. The second step of the proposed data processing method is 3D refractive index reconstruction with a novel tomographic algorithm, taking into account arbitrary illumination of a sample. The proposed algorithm is a generalization of an extended depth of field filtered backpropagation algorithm (EDOF-FBPP) [21], and maintains the key feature of EDOF-FBPP, i.e. it is capable of high accuracy evaluation in a large reconstruction volume.

The proposed data processing approach is developed for capillary-supported ODT systems, however, the individual steps of the proposed data processing may find many other interesting applications, e.g. the method for spherical aberration correction can provide substantial enhancement of the results in any holographic system, in which high NA wave is incident on a planar interface.

Utility of the proposed tomographic data processing path is demonstrated with numerical simulations and experimental measurement of a test sample – a set of PMMA microspheres of diameter 23.5μm and refractive index 1.489.

The paper is organized as follows: in Sec. 2 a tomographic measurement system with a rotary capillary module is presented. Section 3 outlines individual stages of the proposed tomographic data processing. In Sec. 4 the proposed approach is numerically tested and compared with the TEA method. Section 5 presents results of the experimental measurement. Finally, Sec. 6 summarizes the findings and draws conclusions.

## 2. Tomographic DHM system with a capillary rotation module

#### 2.1 The measurement system

The schematic of the capillary-supported ODT system is shown in Fig. 1. The setup is a digital holographic microscope based on Mach-Zehnder interferometer with a custom-made capillary rotation module [15]. In the module, a hollow-core fiber capillary (FC) (inner diameter 128μm, outer diameter 340μm, *n*_{He-Ne} = 1.4570) contains specimens suspended in a cell culture medium. The capillary is placed inside a Petri dish (PD) (iBidi GmbH, µDish 25mm) filled with a refractive index matching liquid to suppress refraction on the outer wall of the vessel. The walls of PD are drilled and syringe needles are used as guides for the capillary in order to minimize the run-out during rotation. More details on the capillary-module preparation can be found in Ref. 15.

In the DHM system, the beam (λ = 632.8nm), collimated by the lens CO (*f*’ = 100mm), is split into object and reference beam at the beam splitter BS1. The object beam passes through a sample introduced into a capillary rotation module. The object beam is imaged onto a CCD camera sensor (JAI BM-500GE, pixel size 3.45µm) with an afocal imaging system composed of the microscope objective MO (20x, NA = 0.4, infinity corrected) and the tube lens TL (*f*’ = 200mm). In the reference beam, a neutral density filter NDF is used to adjust the intensity of the beams and optimize the resulting fringe contrast. The L1-L2 beam expander adjusts dimensions of object and reference beams. During tomographic measurement, the capillary is rotated using a motorized fiber holder (Elliot Martock MDE235) enabling the registration of multiple angular measurements.

#### 2.1 Optical properties of the capillary rotation module

Figure 2 illustrates the capillary rotation module. The capillary (inner radius *R* = 64μm, refractive index *n*_{2} = 1.4570) is filled with a cell culture medium (*n*_{1}≈1.337), which ensures the required environment for living biological specimens. Since the choice of the cell culture medium is dictated by requirements of the samples, usually it is not possible to obtain a good match between refractive index of the medium *n*_{1} and the capillary (here ∆*n*_{inner} *= n*_{1}- *n*_{2} = −0.12). This creates a strong refraction effect on the inner boundary of the vessel. In fact, the inner boundary of the capillary can be considered as a set of two optical surfaces, whose focal lengths are *f* = *n*_{1}*R*/(*n*_{1}-*n*_{2})≈-713μm. To suppress refraction on the outer boundary of the capillary, the vessel is immersed in an index-matching liquid with *n* = 1.4590. Thus, the mismatch on the outer boundary is negligible (∆*n*_{outer} = −0.002).

In the tomographic measurement the capillary is illuminated with an on-axis plane wave *W*. The plane wave is refracted at the first boundary *B*_{1} of the vessel, generating a cylindrical outgoing wave. The cylindrical wave is scattered by a sample *S* and hits the second boundary *B*_{2} of the capillary. Refraction at the cylindrical boundary *B*_{2} generates aberration of the object wave. Afterwards, the aberrated wave with a strong, cylindrical carrier is incident on the planar bottom wall *B*_{3} of the Petri dish (glass-air interface ∆*n*≈0.45), which introduces another deformation (spherical aberration) of the wavefield. The aberrated object wave is imaged by the microscope imaging system onto CCD. The measurement process is repeated for multiple angular positions of the capillary, which provides a set of sample angular projections required for tomographic evaluation.

To sum up, the capillary holder introduces three important effects: 1) cylindrical illumination of a sample, 2) object wave aberrations due to refraction on the cylindrical surface *B*_{2}, 3) spherical aberration due to refraction on the planar surface *B*_{3}. As it will be proven with numerical simulations in Sec. 4 of this paper, each of these effects introduces errors to tomographic reconstruction, causing very low quality of the final 3D result. In this paper we propose the processing path, which incorporates the capillary-related effects 1-3 and enables an accurate and reliable tomographic measurement.

## 3. Data processing

This section presents individual steps of the tomographic data processing for the capillary-supported ODT. After registering a tomographic data series, the holograms are reconstructed with a suitable hologram reconstruction method. Next, the reconstructed object waves undergo holographic corrections, including removal of spherical aberration (algorithm described in Sec. 3.1) and the cylindrical boundary refraction compensation (Sec. 3.2). The last step of the processing is tomographic evaluation with the novel reconstruction algorithm considering cylindrical wave illumination conditions (Sec. 3.3).

#### 3.1 Compensation of aberration from planar interface

It is well known that a planar interface produces an aberrated image of a point source [22]. This effect is illustrated in Fig. 3, in which different optical rays emitted from the source *P* produce images *P*’ in various axial locations (spherical aberration). To remove this aberration, we propose a simple, holographic compensation procedure consisting of two operations of numerical propagation. First, the aberrated wavefront is propagated for a distance *L’ = Ln’/n* in a medium *n*’ using a numerical diffraction algorithm, e.g. the plane wave spectrum decomposition method [23]. The operation is equivalent to reaching the planar surface with the wavefield. Then, the wavefield is propagated backwards in *n* for a distance *L* to the actual location of a sample (Fig. 3).

However, the parameter *L* (distance between a sample and the planar interface) is usually large, e.g. in the capillary rotation module *L* = 1.562mm. Therefore, the propagation operations shall require an additional care [23], i.e. extending the signal using for example zero padding technique. To avoid this problem, we propose to incorporate two operations of propagation (forwards and backwards) into one effective propagation kernel:

*f*

_{x},

*f*

_{y}denotes spatial frequencies and

*k*

_{0}is a wave number in vacuum. The corrected wavefield is obtained via:where FT denotes Fourier transformation and

*u*

_{aberr}is the aberrated wavefield.

It is notable that the correction of spherical aberration is crucial only when a microscope objective of the ODT system works with the air. In this case, the difference of refractive indices at the Petri dish bottom wall is very high (∆*n*≈0.5). Otherwise, if the whole volume between the capillary and the objective is filled with an immersion liquid, like in Ref. 14, ∆*n* is much smaller and the spherical aberration compensation procedure can be omitted.

#### 3.2 Compensation of aberration from cylindrical interface

In the conventional approach, the aberration related to the object wave refraction at the rear cylindrical boundary of the capillary is corrected using the TEA method. However, the TEA is valid only for flat objects with slow variation of refractive index and these conditions are not met for the case of the capillary rotation module.

The accurate correction procedure, proposed in the paper, is based on tracking optical rays during the transmission through the capillary (LRA approach). Figure 4 shows a trace of an optical ray of the wavefield *u* = *A*exp(i*φ*), which passes the imaging plane π at a transverse coordinate *x*_{A}. The ray hits the rear boundary of the capillary at *x*_{B}, where, due to refraction, changes direction from **k _{1}** = [

*k*

_{0}

*n*

_{1}sin

*α*

_{1,}

*k*

_{0}

*n*

_{1}cos

*α*

_{1}] to

**k**= [

_{2}*k*

_{0}

*n*

_{2}sin

*α*

_{2,}

*k*

_{0}

*n*

_{2}cos

*α*

_{2}]. The microscope imaging system, looking at the object space, sees the outgoing ray

**k**, which forms an aberrated image of a sample point at

_{2}*x*

_{C}. The correction of an aberrated object wave

*u*

_{aberr}=

*A*

_{aberr}exp(i

*φ*

_{aberr}) can be achieved by retracing the optical rays back along the described path from point

**, through**

*C***, to**

*B***.**

*A*Unwrapped phases of the aberrated and non-aberrated wavefields satisfy the relation:

**k**of the aberrated wavefield can be readily obtained from the first derivative of

_{2}*φ*

_{aberr}:

**k**requires calculating the spatial coordinates of the refraction point

_{1}**:**

*B**R*is the inner radius of the capillary. The angular direction of the non-aberrated vector

**k**is found using Snell’s law of refraction:

_{1}*φ*

_{corr}, given by Eq. (8), is calculated at the shifted spatial coordinates

*x*

_{C}. To obtain the corrected optical field at the original coordinates

*x*

_{A}and thus correct for transverse shift of the rays, the phase distribution is remapped into new coordinates:

*A*

_{aberr}of the aberrated wavefield, is remapping

*A*

_{aberr}to the new coordinates

*x*

_{A}. For the sake of simplicity, the correction algorithm was presented for one dimensional case only, however, its extension to two dimensions is straightforward.

#### 3.3 Tomographic reconstruction with cylindrical illumination

All reported capillary-supported ODT systems share a set of important problems related to tomographic evaluation. Firstly, the TEA correction of object waves is usually followed by 3D refractive index reconstruction using a non-diffractive tomographic reconstruction algorithm, e.g. filtered backprojection. Moreover, the reconstruction is performed under an incorrect assumption of plane wave illumination of a sample. In this paper we propose a more accurate approach, which takes into account 1) diffraction; 2) cylindrical wave illumination of a sample. The proposed tomographic reconstruction method is a generalization of the filtered backpropagation algorithm FBPP [24] and its modification - the extended depth of field filtered backpropagation algorithm (EDOF-FBPP) [21].

For the sake of simplicity, we present the algorithm for the case of 2D object reconstruction. The geometry used in the algorithm is illustrated in Fig. 5. In the image, ∆*n*(*x,y*) is a variation of refractive index to be found, *n*_{0} is refractive index of the surrounding medium and coordinate system (*ξ*,*η*) relates to the measurement geometry at a given angle *θ* of illumination (rotation) of a sample.

In the proposed arbitrary illumination tomographic algorithm, called from here AI-FBPP, the reconstruction starts with extending each 1D optical field *U*^{θ}(*ξ*) into 2D backpropagated view Π^{θ}(*ξ*,*η*):

_{η}= (k

_{0}

^{2}n

_{0}

^{2}-k

_{ξ}

^{2})

^{½}. The extended 2D view Π

^{θ}contains a fast changing component related to the cylindrical carrier of the object wave, which has to be removed prior to the further evaluation. For this reason, the 2D view Π

_{ill}, related to an illuminating wave

*U*

_{ill}, is calculated by replacing

*U*

^{θ}with

*U*

_{ill}in Eq. (10). The normalized 2D view is obtained via:Afterwards, the Rytov phase is calculated according to:

*θ*are added up giving an object function (scattering potential):

The illuminating wave *U*_{ill}, required for the evaluation of the extended illumination image Π_{ill} [Eq. (11)], can be obtained with at least three different methods. It can be calculated using an additional, reference hologram of the empty vessel [25] or it can be extracted from the original hologram using regions, which are free of samples [26]. The third method involves computation of the plane wave transition through the front cylindrical boundary using a numerical model of the capillary.

The proposed AI-FBPP algorithm differs from the original FBPP in two aspects. Firstly, FBPP backpropagates a Rytov phase [24], while AI-FBPP backpropagates an original optical field [Eq. (10)]. This modification ensures enhanced accuracy of the backpropagation process and provides uniform, high quality of the measurement in the large reconstruction area, including parts of a sample, which are strongly defocused during the measurement. For details please refer to Ref. 21. Secondly, in AI-FBPP the backpropagation is performed on an original wavefield without removing its carrier wave (*U*_{ill}). In this method the removal of the cylindrical carrier is done after the propagation step using axially extended images Π^{θ} and Π_{ill} [Eq. (11)]. In contrast, in the classical FBPP the backpropagation is performed using an artificially ‘flattened’ phase, i.e. the normalization is performed prior to the backpropagation. This approach causes deformation of an extended image Π^{θ}.

The consequence of backpropagating an artificially ‘flattened’ wavefield is illustrated in Fig. 6. Figure 6a depicts cylindrical wave illuminating a sample *S* (such conditions can be found e.g. in the interior of the capillary). The effect of backpropagating a ‘flattened’ wavefield is shown in Fig. 6b. In this case, the image *S’* is shifted and rescaled. The proper extension of 1D optical field *U*^{θ}(*ξ*) into 2D contribution Π^{θ}(*ξ*,*η*) requires backpropagating the original wavefield [Fig. 6(c)]. The latter approach [Fig. 6(c)] is utilized in AI-FBPP, the former [Fig. 6(b)] - in the classical FBPP method.

## 4. Computational tests

This section is devoted to the numerical investigation of the proposed data processing path for tomographic reconstruction of a sample placed in a cylindrical vessel. We assume following parameters of the simulation: inner radius of the capillary *R* = 64μm, refractive index of the capillary *n*_{2} = 1.457, refractive index of the cell culture medium *n*_{1} = 1.337, distance between the capillary and planar bottom of the Petri dish *L* = 2mm, a microscope objective working with air, wavelength of light λ = 632.8nm. These parameters mimic a real experimental situation. The model of a biological sample is a circle with refractive index *n*_{s} = 1.37 and radius *R*_{s} = 10μm. The sample is located in a distance *r*_{s} = 20μm from the center of the capillary. The simulations are performed using three methods: algorithm based on Born expansion [27] for transition through the boundary of the capillary, angular spectrum method [23] for free-space propagation, and wave-propagation method [28] for transition through the sample.

Figure 7 shows refractive index reconstruction of the phantom cell obtained with a standard tomographic approach (without correction of any aberrations and with the standard FBPP algorithm). The result clearly demonstrates that the capillary is a source of significant reconstruction errors. The deformation is further investigated by analyzing the impact of individual elements of the rotation capillary module.

First, we investigate influence of the object wave refraction at the planar bottom of the Petri dish. Figure 8 shows refractive index reconstruction obtained with the application of LRA correction and AI-FBPP tomographic reconstruction algorithm, but without treating refraction on the Petri dish. As it appears, the planar surface has a profound impact on the reconstruction - the spherical aberration causes a strong ring artifact, which is concentric with the sample.

The next investigated issue is the object wave refraction at the cylindrical boundary of the capillary. Figure 9 presents results of tomographic reconstruction (Fig. 9(a)- reconstructed ∆*n*, Fig. 9(b)- error distribution) obtained without LRA correction, but with application of spherical aberration correction and AI-FBPP reconstruction algorithm. The results show clearly that the TEA approach generates significant errors. The reconstructed sample obtained with TEA is shrunk and its center is shifted towards the capillary center.

Impact of the last investigated element – cylindrical wave illumination introduced by the rear cylindrical boundary of the capillary – is presented in Fig. 10. The image shows a refractive index reconstruction obtained with full correction of the object wave aberrations, but without treating cylindrical illumination conditions (the reconstruction was obtained with a standard FBPP). The reconstructed image of the sample is blurred in the azimuthal direction, which indicates that AI-FBPP is the crucial factor to obtain accurate tomographic reconstruction.

Finally, in Fig. 11, we show a tomographic result (Fig. 11(a)- reconstructed ∆*n*, Fig. 11(b)- error distribution) obtained with the full data processing path proposed in the paper (spherical aberration and LRA correction, tomographic reconstruction with AI-FBPP). It can be seen that all the deformation visible in Fig. 7 has been removed. The remaining error is a minor blurring of the sample boundary.

It is intuitively obvious that the reconstruction errors related to refraction at the cylindrical
boundaries of the capillary may depend on the radial location of a sample in the vessel. To
investigate this issue quantitatively, we repeated the simulations presented in Figs. 8-11 for various radial positions
*r*_{s} of the sample. For each case we computed the error of the
reconstruction:

*n*and ∆

_{rec}*n*are reconstructed and modelled refractive index variations, respectively and

_{model}*N*denotes the number of investigated pixels. The computations were performed for the sample positions up to

*r*

_{s}= 40μm due to the limitations of the applied simulation methods. The results of the numerical experiment are presented in Fig. 12: plot A – lack of the spherical aberration correction procedure; plot B – lack of the LRA correction; plot C – lack of AI-FBPP; D – results obtained with the full data processing path; (see the table-legend in Fig. 12). The charts indicate that the largest source of errors is the object wave refraction on the cylindrical boundary of the capillary (chart B). Additionally, this error shows the strongest dependence on the sample position

*r*

_{s}. Another source of errors that grow with

*r*

_{s}is neglecting the cylindrical illumination conditions (chart C). The increasing character of the curve C is also partially caused by the inherent characteristics of FBPP, which does not maintain the same accuracy in the whole reconstruction area [21]. The smallest source of errors is related to spherical aberration (chart A), however, strong oscillatory character of the deformation disturbs 3D imaging capability at higher resolution hindering analysis of the reconstructed structure. Thus, spherical aberration compensation should not be omitted. Finally, the chart D in Fig. 12, which corresponds to the full data processing procedure proposed in the paper, demonstrates efficient reduction of the capillary-related errors up to the investigated limit

*r*

_{s}= 40μm.

## 5. Experimental results

In order to verify the proposed data processing path for capillary-supported ODT, a special test object consisting of a set of PMMA calibrated microspheres, fixed inside the capillary with an optical adhesive, was prepared. The choice of such sample enabled us to perform a quantitative test of accuracy of the proposed data processing approach. To prepare the sample, a mixture of non-polymerized optical adhesive (*n*_{He-Ne} = 1.517, NOA 65 Norland Products, Inc.) and PMMA microspheres (diameter 23.5µm, *n*_{He-Ne} = 1.489, Microparticles, GmbH) was introduced into a fused silica fiber capillary (inner radius 64µm, *n*_{He-Ne} = 1.457) using capillary action. During this process, the capillary was maintained at approx. 60 degrees angle to the vertical and rotated constantly in order to prevent microspheres from flowing towards the walls of the vessel. Then, the polymer in the capillary was cured with 380nm UV light for 60 seconds. The proposed sample preparation procedure allowed avoiding attaching the microspheres to the capillary wall. Additionally, fixing the microspheres in the adhesive excluded any uncertainty concerning movement of the objects relative to the capillary during the tomographic measurement. In the case of biological objects the procedure is different and it neither requires to apply adhesive nor constantly rotate the tube [15].

For the prepared test sample, 180 off-axis holograms were captured with an angular step of 2° using DHM system presented in Fig. 1. Then, the Fourier reconstruction method was applied to retrieve amplitude and phase of object waves (example results in Fig. 13).

In the next step, the spherical aberration procedure from Sec. 3.1 was performed with the distance *L* between the capillary and the Petri dish wall set to 1.562mm. The distance *L* was found by measuring an axial shift of the object that was required for changing focus of the imaging system from the central cross-section of the capillary to the bottom surface of the Petri dish. Then, the corrected amplitude and phase images were subject to LRA procedure for compensating the object wave refraction at the rear boundary of the capillary. The last step of the processing path was to reconstruct the refractive index with AI-FBPP tomographic algorithm. The reference measurement *U*^{θ}_{ill} of illuminating wave, which is required in AI-FBPP, was obtained from the original measurements *U*^{θ} using a fitting procedure.

The result of tomographic evaluation is presented in Fig. 14 as a 3D view. The obtained refractive index distribution shows good agreement with technical data (homogeneous spheres of diameter *d* = 23.5μm and refractive index *n* = 1.489). According to the simulation section, the largest reconstruction problems are expected for the third microsphere (counting from the top), which is located in the furthest location from the axis of the capillary (radial postion *r*_{s} = 18μm). For this sample, an average value of the measured refractive index is *n* = 1.4888 and standard deviation of *n* is *δ*_{n} = 0.0024. The diameter of the microsphere measured in *x*, *y* and *z* directions is *d*_{x} = 23.52μm, *d*_{y} = 23.86μm and *d*_{z} = 23.69μm, respectively.

Figure 15a shows a transverse cross-section through the reconstructed refractive index distribution at *y* = 40μm (central cross-section of the third microsphere). For comparison, in Fig. 15(b) the same slice obtained with TEA correction (and AI-FBPP algorithm) is shown. Figure 15c shows difference between the results obtained with LRA [Fig. 15(a)] and TEA [Fig. 15(b)]. The differential image in Fig. 15(c), which can be treated as a distribution of the TEA errors, corresponds well to the results obtained from the simulation[Fig. 9(b)]. In the case of experimental data, the error is smaller, which is related to smaller focusing power of the capillary filled with adhesive (*f* = *n*_{1}*R*/2/(*n*_{1}-*n*_{2})≈809μm) than the focusing power of the same capillary filled with cell culture medium (*f*≈-357μm). Moreover, the capillary with adhesive acts as a positive lens, while the tube filled with cell culture medium has negative focusing power, therefore, the character of the deformation for the experimental data is opposite to that obtained in the simulation section, i.e. the PMMA microsphere reconstruction obtained with TEA [Fig. 15(b)] is broadened and shifted away from the axis of the capillary, while the bio-object phantom reconstruction [Fig. 9(a), Sec. 4] is narrowed and shifted towards the center of the capillary. It is worth noting, that the object deformation obtained with TEA can be easily overlooked due to specific character of the changes (stretching of a sample).

Figure 16 shows a tomographic reconstruction obtained with the LRA correction and the conventional FBPP algorithm. As expected, in the case of off-axis rotation and cylindrical wave illumination, the conventional FBPP generates significant reconstruction errors. Vast improvement is achieved with AI-FBPP, which 1) incorporates cylindrical wave illumination conditions and 2) similarly like EDOF-FBPP [21] provides improved accuracy for reconstruction of the off-axis features.

Additionally, to visualize the importance of correction of the object wave refraction at the planar surface of the Petri dish, we present a slice of the LRA + AI-FBPP tomographic reconstruction obtained without the compensation of spherical aberration [Fig. 17]. The image shows a strong, characteristic ring artefact and it is in agreement with the results obtained from the simulation [Fig. 8].

## 6. Conclusions

In the paper, we proposed a set of novel numerical methods, which allow for accurate tomographic measurement of samples placed in a rotary capillary. Two crucial steps of the proposed data processing path are: 1) accurate correction of the object wave refraction at the cylindrical surface of the capillary and 2) refractive index reconstruction with a novel arbitrary-illumination tomographic algorithm. The correction of the object wave refraction is achieved using LRA approach, which is based on exact laws of geometrical optics and is therefore far more accurate than the TEA approach. The second step, i.e. 3D reconstruction, is done with a novel tomographic method based on FBPP, which incorporates cylindrical illumination conditions generated by a capillary and provides improved accuracy of off-axis reconstruction. Moreover, in the paper we proposed a novel, efficient method for numerical correction of spherical aberration introduced by the planar surface. We apply the method for compensating the object wave refraction at the glass bottom of a Petri dish.

The numerical simulations, presented in the paper, show that each of the three investigated elements: two cylindrical boundaries of the capillary and planar surface of the Petri dish, is a source of its own characteristic reconstruction errors. Furthermore, the simulations indicate that reconstruction errors related to the capillary are larger for samples, which are located far from the axis of the capillary. For these samples application of the proposed method is especially important. The simulations proved utility of the proposed correction and tomographic algorithms.

The proposed tomographic data processing path was also validated with an experimental measurement of a specially prepared test sample, i.e. a set of homogenous microspheres of diameter 23.5μm and refractive index *n* = 1.489 (refractive index variation ∆*n* = −0.028). The experimental results confirmed utility of the individual algorithms: LRA correction method, AI-FBPP tomographic reconstruction algorithm and the method for spherical aberration compensation.

On the final note, the algorithms proposed here were created with the aim of enabling accurate ODT measurements of biological samples, however, the potential scope of applicability of the methods is much wider. The example possible applications of the method are in 2D holographic microflow visualization and in novel arbitrary-illumination ODT systems.

## Acknowledgments

The research leading to the described results are realized within the program TEAM/2011-7/7 of Foundation for Polish Science, co-financed from the European Funds of Regional Development. The authors acknowledge also the support from the statutory funds of Warsaw University of Technology.

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