## Abstract

We propose a novel tomographic measurement approach that enables a noise suppressed characterization of microstructures. The idea of this work is based on a finding that coherent noise in the input phase data generates an artificial circular structure whose magnitude is the highest at the centre of tomographic reconstruction. This method decreases the noise level by applying an unconventional tomographic measurement configuration with an object deliberately shifted with respect to the rotation axis. This enables a spatial separation between the reconstructed sample structure and the area of the largest refractive index perturbations. The input phase data defocusing that is a by-product of the introduced modification is numerically corrected with an automatic focus correction algorithm. The proposed method is validated with simulations and experimental measurements of an optical microtip.

© 2014 Optical Society of America

## 1. Introduction

The recent rapid development in the field of micro-optics triggers the need to develop a reliable method for inspection of technological elements with a micro-sized three dimensional structure, such as single mode and photonic crystal fibres, microbridges, polymer microtips and GRIN microlenses. A method that has capability to meet these requirements is optical diffraction tomography (ODT) [1–6]. In addition, ODT offers a number of other advantages such as a non-destructive and quantitative character. When using this technique the inner structure of a sample is characterized by providing its 3D refractive index distribution. This is achieved by using a series of 2D measurements of light transmitted through the sample for various illumination directions, which are captured by means of laser interferometry, including digital holography (DH) [7–10]. The scattered data is numerically processed using e.g. filtered back projection algorithm (FBPJ) [11] giving a 3D image of the refractive index variations in the investigated volume.

In recent years intensive research effort has been focused on improving performance of ODT by numerical refocusing to the best focus plane [12–15] and by minimizing the radial run-out [13,16]. However, another important factor severely affecting quality of ODT reconstructions is high level of noise. The poor signal-to-noise ratio originates from the inherent characteristic of high coherence light used in ODT systems to obtain scattered data. This includes unwanted interference between light reflected from random scatterers within the measurement system, e.g. minor roughness, scratches, dust particles. The resulting granular coherent noise pattern, the so-called speckle pattern, is even more strengthened via a tomographic reconstruction algorithm. In the common tomographic measurement configuration when the relative angle of observation is varied by rotating the object under fixed illumination and detection directions, all the 2D measurements contain the same, or at least very similar, speckle patterns. As it will be shown in the following sections, this effect causes a single speckle to be interpreted by a tomographic reconstruction algorithm as a semi-circular structure with the centre of curvature located in the rotation axis and with a magnitude inversely proportional to its radius.

The adverse effect related to coherent noise severely restricts capability of ODT to visualize object features of size comparable with size of the speckles, and thus, prohibits accurate characterization of micro-optical elements. The most straightforward way to reduce this undesirable effect is to exploit the fact that the magnitude of noise is the largest in the central area of the reconstruction and decreases rapidly with increasing distance from the rotation axis. Therefore, the presence of a substantial distance between the object and the rotation axis (the so-called radial run-out) should potentially improve the signal-to-noise ratio. However, in off-axis rotation configuration the sample travels along a vast circular path in the plane perpendicular to the rotation axis and the axial component of this trajectory (i.e. the component parallel to the optical axis) may potentially cause a considerable degree of defocusing of the captured data. This undesirable effect, which is especially strong for object sizes in the order of several wavelengths, leads to violation of the assumption of a straight light propagation through the sample, which is a basis for FBPJ algorithm. Hence, the quality of the reconstruction is significantly reduced [17]. For this reason, the noise suppression approach based on deliberate introduction of radial run-out has not been so far realized. In this paper, we propose a method that allows for a numerical correction of the defocus errors and thus enables implementation of the proposed noise suppression methodology. The proposed numerical defocus correction procedure is fully automatic. In this approach a holographic autofocusing technique [18–21] is employed to bring all the angular sample measurements to the focus before the application of FBPJ algorithm.

Although the investigation of the noise performance of ODT is conducted here using FBPJ algorithm, its findings regarding character of the noise reconstruction are general, i.e. they apply to any tomographic reconstruction method, e.g. algebraic reconstruction technique (ART) [22,23] and filtered backpropagation method (FBPP) [24,25]. Therefore, the measurement in the off-axis rotation configuration should potentially improve results of ODT, regardless of the tomographic reconstruction algorithm used to process 2D data. In the case of FBPP algorithm, which unlike FBPJ accounts for diffraction, reconstructions of the object rotated with a minor run-out (of the order of a few micrometres) may not necessary demand the prior defocus correction. However, in the case of larger run-out (30μm and more) it is advisable to perform the run-out correction in order to reduce the errors due to inaccurate approximated numerical propagation formulas used in FBPP [12].

This paper is organized as follows. The principle of ODT is explained in Sec. 2. Section 3 provides a detailed analysis of the influence of coherent noise on tomographic reconstruction. In Sec. 4 the proposed noise suppression method is presented, including description of the run-out correction procedure [Sec. 4.2] and discussion of applicability and potential gain of the method [Sec. 4.3]. In Sec. 5 the experimental results obtained for the case of an optical microtip are shown. Finally, discussion and conclusions are presented in Sec. 6.

## 2. Principles of optical diffraction tomography

#### 2.1 Acquisition of scattered fields

The basic principle of tomography is to record multiple images for various illumination directions with respect to the sample and use these data to recreate the internal 3D structure of the investigated object. In ODT the individual scattered fields required for tomographic evaluation are obtained by means of digital holographic microscopy, usually using an experimental setup based on Mach-Zehnder interferometer configuration as e.g. the one depicted in Fig. 1.In this setup a beam of coherent light (He-Ne laser, λ = 632.8nm) is collimated by a beam expander system consisting of a spatial filter SF and a collimator lens C. The expanded beam is divided by a beam splitter BS into an object and a reference waves. The object wave illuminates a sample MS mounted on a motorized rotary stage enabling alteration of the relative illumination direction with respect to the sample. In order to minimize refraction on the external surface of the specimen, the object is placed in a cuvette IC filled with a properly matched immersion liquid. The optical field scattered by the sample is imaged by an optical system consisting of a microscope objective MO1 (NA = 0.4) and an imaging lens IL (f = 200mm) on a CCD detector (Scorpion SCOR-20SO by Point Grey, resolution 1600x1200, pixel pitch 4.4μm) providing 20-fold magnification. The imaging system utilizes an afocal imaging condition [26], which ensures a constant lateral magnification that is independent of the object position. In the reference path, the light is reflected by a mirror attached to a piezoelectric transducer (PZT-M) used to obtain precise phase changes, required for temporal phase shifting reconstruction of an on-axis hologram, and then passes through the microscope objective MO2 of the same type as MO1 to partially compensate for aberration introduced to the object beam. At the CCD plane the object wave is superimposed with a reference plane wave and the resulting interference intensity patterns forms an on-axis digital hologram.

#### 2.2 Tomographic processing

A single digital hologram obtained with the system described in Sec. 2.1 does not contain sufficient information to reconstruct 3D structure of the sample. If volumetric information is required, a series of 2D measurements has to be performed for various relative angles of illumination α with respect to the sample. Afterwards, the complex scattered waves encoded in the holograms are recovered using e.g. temporal phase shifting (TPS) [27] or spatial carrier phase shifting (SCPS) [4] techniques.

Provided that an object under study features small deviations of the refractive index [17], the phase *φ _{α}*(

*x*,

*y*) of the scattered fields can be interpreted as object projections, i.e. integrals of the refractive index variations ∆

*n*,

_{α}(x*y*,

*z*), which is a difference between refractive index of the object

*n*,

_{α}(x*y*,

*z*) and the background medium

*n*:

_{o}*z*:

If this condition is met, the 3D structure Φ*(x*,*y*,*z*) of the object can be efficiently reconstructed using the FBPJ algorithm. Assuming that the relative angle of illumination is altered by rotating the sample around the vertical axis, the reconstruction is performed separately for each transverse slice of the measured volume. The computations start with preparation of a sinogram *S*(*x*,*α*), which is a 2-D array containing all the unwrapped phase projections *φ _{α}*(

*x*,

*y*=

*y*) of a current slice. Then all projections are high-pass filtered by multiplication by a ramp function in the Fourier domain:

_{o}*f*is a spatial frequency and

*FT*denotes the inverse Fourier transformation. After this preparation the individual projections, corresponding to different sample views, are backprojected through the image along directions in which they were originally captured and the final cross-section of the object is obtained by summing these backprojected views:

^{−1}The particular stages of the reconstruction process, from acquisition of projections, through the sinogram processing, to the backprojected views assembly, are schematically presented in Fig. 2.

The complete 3D object structure is finally obtained by successive stacking the reconstructed transverse slices. The last step within the tomographic evaluation is the scaling of the phase data Φ*(x*,*y*,*z*) to the refractive index values:

*denotes a pixel pitch of the CCD detector divided by transverse magnification of the imaging system.*

_{x}## 3. Impact of coherent noise on reconstruction

Under ideal tomographic measurement conditions (when the sample is rotated around the well-defined axis with a strictly controlled angular step and in absence of any other sources of motion, such as mechanical vibrations) all points of a rigid object move along circular trajectories around the rotation axis. Consequently, each point of the object appears in a sinogram as a sine wave with amplitude equal to a distance from this point to the rotation axis and the initial phase corresponding to its angular position [Fig. 3]. Within the tomographic reconstruction all these sinusoidal traces are properly converted into the object points.

However, under realistic conditions of ODT measurement, the phase images captured with a DH
system contain also structures other than those associated with the rotating sample. In
particular, the phase images are corrupted by adverse phenomena due to high coherence of light
used in DH setup, such as speckles and parasitic interference fringes. These structures are
usually not related to the rotating sample and change very slowly in time. Thus, they appear in
the sinogram as a set of parallel lines [Fig. 4(a)]. After application of the FBPJ algorithm, these parallel lines are reconstructed into
fictitious circular structures composed of two concentric arcs with opposite signs of refractive
index variations [Fig. 4(b)]. The centre of the curvature
of the arc structure coincides with the centre of the reconstruction and its radius
*r* is equal to a distance between the corresponding line in the sinogram and
the sinogram central axis $\alpha .$

From the fact that the integral over a single phase projection [Eq. (2)] is equal to a total sum of all refractive index variations in the object:

*r*, the magnitude of the refractive index is proportional to ${r}^{-1}$ [Fig. 5].

In order to verify the predictions made using the previous consideration for the coherent
noise reconstruction, we set up an experiment involving registration of 90 holograms of an empty
field (without a sample), which represents a typical background of images captured within a
tomographic measurement series. The holograms were registered in the Mach-Zehnder DH microscope
setup [Fig. 1] and processed with TPS algorithm. The
obtained phase images were reconstructed using the FBPJ tomographic reconstruction algorithm.
Figure 6 shows results of the reconstruction obtained
for one cross-section: (a) sinogram with a clearly visible linear pattern; (b) reconstructed
refractive index distribution Δ*n* with the characteristic circular
structure; (c) a section through the middle row of Δ*n*. As expected, the
refractive index perturbation is the strongest in the central area of the reconstruction, which
typically is occupied by the image of an investigated sample.

## 4. Noise suppression tomographic method

#### 4.1 The noise suppression methodology

The analysis in Sec. 3 shows that phase images captured in the actual conditions of ODT measurement form sinograms that are composed of two families of traces: 1) a collection of sine functions associated with an object and 2) a set of parallel lines related to noise. After tomographic evaluation, the first family of traces gives a proper reconstruction of the object whereas the second family generates a fictitious pattern consisting of multiple concentric arcs, whose magnitude is the strongest in the central area of the reconstruction. This circular pattern, referred to as a ring artefact, has an adverse effect on the reconstruction, because it obscures the actual sample structure and hinders resolving of fine details, especially those of size comparable with size of the speckles.

Given the fact that the magnitude of the ring artefact is the largest in the centre of the 3D image and decreases rapidly with increasing distance from the rotation axis, we have drawn a conclusion that a significant improvement of the reconstruction quality can be achieved by moving the position of a sample from the typical location on the rotation axis to an off-axis position. It is important to note, that the proposed modification, although advantageous in terms of noise, stays in a complete opposition to a typical experimenters’ intention: minimization of run-out in order to simplify the reconstruction process and to reduce errors of FBPJ algorithm due to defocusing of phase projections.

As it will be shown in the following sections by simulations and experimental measurements, the off-axis configuration allows for a significant reduction of the noise level. However, in this case to obtain the correct reconstruction it is necessary to address the mentioned issue of tomographic data defocusing.

#### 4.2 Run-out related errors and the correction algorithm

The detailed explanation of the run-out-related problems will be provided based on a schematic
diagram in Fig. 7.The diagram shows a trajectory of a sample during the off-axis rotation. As shown, at the
beginning of the measurement the sample is placed exactly in the plane optically conjugated
with the image sensor (the imagining plane *π _{o}*), which
ensures acquisition of properly focused data. Nevertheless, after rotation by an angle

*α*the object centre shifts in both lateral (

*x*) and axial (

*z*) direction. If the axial displacement

*dz*is sufficiently large, the object is set out of depth of focus (DOF) of the imaging system. This causes a defocusing of phase data and consequently leads to the erroneous reconstruction.

It is important to note, that in the case of the proposed noise suppression measurement the described effect of phase defocusing is especially severe. This is due to two major factors. Firstly, in this case the magnitude of run-out is relatively large, and secondly, micro-optical elements for which the proposed method is intended produce strong diffraction changes and are thus very sensitive to defocusing.

Therefore, in order to truly benefit from the noise reduction of off-axis reconstructions and maintain a high accuracy of the refractive index determination, it is necessary to correct the mentioned degradation of phase measurements due to defocused imaging conditions. This is achieved by applying a post-processing correction procedure based on a digital holographic autofocusing technique, which enables numerical refocusing of the data from the registration plane *π _{o}* to the in-focus position

*π*. The consecutive steps of the modified tomographic algorithm are presented below:

_{α}- The individual holograms captured for various angular views of the sample are reconstructed giving a series of complex scattered waves in the plane
*π*._{o} - To remedy potential defocus of the measurements, each complex wave is numerically propagated to multiple axial locations using an accurate plane wave spectrum decomposition method [28]. For each position, the focusing condition of the optical field is evaluated using a focus criterion suitable for non-absorbing samples (phase objects); for each location a variance of the amplitude distribution is calculated and the position with the minimal variance, and thus minimal diffraction changes, is identified as the proper in-focus location.
- The phase distributions of the optical fields at the in-focus positions are unwrapped and undergo further tomographic evaluation according to the description in Sec. 2.2.

The second component of the sample displacement, i.e. the transverse shift *dx* basically does not introduce any errors to the reconstruction, provided that the axis of the rotation is properly aligned with the central column of the captured phase images. However, larger transverse shifts require an extension of the 3D reconstruction matrix, which considerably increases the computational cost of tomographic evaluation. Therefore, before applying the backprojection procedure according to Eqs. (3)-(4) the transverse displacement in the individual phase projections should be determined [1,3–5,16] and compensated by transversally shifting the images by the appropriate values *x _{shift} = -dx*.

#### 4.3 *Range of applicability and the prospective gain*

In this section the ability of the proposed noise suppression methodology is studied via
numerical simulations. Figures 8(a) and (b) show noisy
reconstructions of cylinders with diameter Φ 5λ and 20λ, respectively,
obtained with the FBPJ algorithm for the case of on-axis configuration (radial run-out
*R* = 0μm). The value of refractive index variation
Δ*n* of the cylindrical objects is 0.01. The optical fields scattered by
the cylinders, required for tomographic evaluation, were computed using a wave-plane
propagation method (WPM) [29]. To account for the noisy
reconstruction, the obtained phase images were superimposed with the experimentally measured
background images [Sec. 3, Fig. 6]. The simulation was
carried out with λ = 0.5μm and *n _{o}* = 1. As it can be
seen from Figs. 8(a),(b), the strong refractive index
perturbation in the central area of the images prohibits proper reconstruction of the smaller
cylinder (Φ = 5λ = 2.5μm) and considerably reduces quality of
reconstruction of the larger structure (Φ = 20λ = 10μm). However, when
using the modified measurement methodology with an off-axis rotation (

*R*= 20μm) and defocus correction, one can observe a significant improvement of the results [Figs. 8(c),(d)].

To quantify the gain from the proposed noise suppression methodology we define an error
$E$of the tomographic reconstruction
Δ*n*:

*n*. With this measure we established that the proposed noise suppression methodology (off-axis measurement and defocus correction) reduces the error of noisy reconstruction ∆

_{ref}*n*, which is calculated here with respect to the noiseless case ∆

_{noisy}*n*, from

_{noiseless}*E1*=

*E*(∆

*n*|∆

_{noisy}*n*) = 1.02 to

_{noiseless}*E1*= 0.15 for the cylinder Φ = 5λ, and from

*E1*= 0.50 to

*E1*= 0.15 for the cylinder Φ = 20λ, which corresponds to the improvement of the results by a factor of 6.8 and 3.3, respectively. Analogous data, but obtained for a number of cylinder diameters and values of radial run-out, are presented in Fig. 9(a) (error

*E1*of the noisy reconstruction calculated with respect to the noiseless reconstructions, the results obtained with the defocus correction) and in Fig. 9(b) (reduction of the error

*E1*relative to the on-axis measurement).

Furthermore, as proof of importance of the defocus correction, in Fig. 9(c) we present an error $E2=E(\Delta {n}_{off-axis}^{no\text{\hspace{0.17em}}corr}|\Delta {n}_{on-axis})$ of the off-axis reconstruction $\Delta {n}_{off-axis}^{no\text{\hspace{0.17em}}corr}$ obtained without defocus correction, calculated with respect to the on-axis reconstruction ∆*n _{on-axis}* (both results obtain from the noiseless data). As it can be seen from the charts, defocusing of phase images in the off-axis configuration leads to a significant increase of the reconstruction errors. These errors, however, are fully compensated with application of the defocus correction procedure. This can be seen from Fig. 9(d) showing an error $E3=E(\Delta {n}_{off-axis}^{corr}|\Delta {n}_{ideal})$ of the off-axis defocus corrected reconstruction $\Delta {n}_{off-axis}^{corr}$ calculated with respect to the ideal refractive index distribution ∆

*n*(perfect cylinder). The constant value of

_{ideal}*E3*for a given diameter is due to intrinsic errors of FBPJ algorithm.

For the purpose of clarifying the applicability and the potential gain of the noise suppression
method, the computation of error *E1* =
*E*(∆*n _{noisy}*
|∆

*n*) was carried out for multiple cylinders of diameters Φ ranging from 2λ to 100λ, ∆

_{noiseless}*n*from 0.001 to 0.05, and for various values of run-out $R$ = {0μm, 5μm, 20μm} [Fig. 10]. From the error distribution for the on-axis case [Fig. 10(a)] one can conclude that two major indications for the noise suppression method are small dimensions and small refractive index values of a sample (the largest values of

*E1*). Furthermore, when comparing the on-axis error to the off-axis error, it becomes clear that the most significant improvement of the results can be achieved for micro-sized structures.

## 5. Experimental results

The feasibility of the proposed noise suppression method was experimentally demonstrated
by the measurement of a polymer microtip [Fig. 11]
manufactured at the extremity of a single mode optical fibre [3,30]. For this specimen two tomographic
measurement series, consisting of 90 sets of phase-shifted holograms, were captured using the DH
microscope setup described in Sec. 2.1. The first series was performed in a conventional on-axis
configuration (*R$\approx $*0μm) and the second series had a relatively large
run-out (*R* = 17μm). The measurement in the off-axis configuration was
performed in order to suppress the noise (see Fig. 11),
which is a crucial factor reducing the effective resolution with respect to the diffraction
limit δx = λ/NA≈1.5μm associated with the resolving power of the
imaging system.

Prior to the tomographic reconstruction, the off-axis measurement series was subjected to the full run-out correction procedure (compensation of both transverse *dx* and axial *dz* displacement). The numerical defocusing correction was performed according to the description in Sec. 4.2. For this purpose, the object waves reconstructed from the individual holograms were numerically propagated to 20 axial locations distributed with a step Δ*z* = 1μm. In each plane the focus measure of the optical field (variance of the amplitude distribution) was calculated using the region of the hologram related to the image of the tip structure only. The minimum of the calculated variance indicated the optimal focus position.

The results of the autofocusing algorithm, obtained for an exemplary sample view (α
= 60), are presented in Fig. 12, where the clearly
visible defocus of the original object wave ((a) amplitude and (b) unwrapped phase) is fully
compensated by propagation to the in-focus position *z* = −12μm
[Fig. 12(c),(d)].

After application of the autofocusing algorithm to all angular measurements, the corrected phase images were unwrapped. In the subsequent step, the transverse sample displacement *dx* was compensated by aligning boundaries of the microtip in the successive unwrapped images. This was achieved with an automatic recentring method based on detection of the maximum of cross-correlation between two successive images [16]. After completion of both stages of the run-out correction procedure, the obtained phase images were processed with the conventional FBPJ tomographic reconstruction algorithm.

The final result of tomographic evaluation is displayed in Fig. 13(a) (3D representation) and in Figs. 13(b)-(d) (central cross-sections). Additionally, in order to underline importance of the defocus compensation, the images in Figs. 13(b)-(d) are compared with the corresponding slices of the 3D reconstruction obtained from the same measurement data but applying no numerical autofocusing [Figs. 13(e)-(g)]. As it can be seen, the refractive index distribution reconstructed without compensation of the axial displacement contains diffraction induced errors, which may be misinterpreted as a double core of the microtip; however, this artefact is fully removed via the defocus correction procedure. Moreover, the microtip structure obtained with defocus correction is much more regular, in particular the cross-section in Fig. 13(b) has much more circular shape.

Additionally, to visualize the gain due to the proposed noise suppression method, the results in Figs. 13(b)-(d) are compared to the reconstruction obtained for the case of the minimal run-out (*R$\approx $*0μm) [Figs. 13(h)-(j)], where strong fluctuations in the central area of the ring artefact severely reduces quality of the reconstruction.

## 6. Summary

The speckle phenomenon is a common problem in many coherent imaging techniques, including optical tomography. However, in ODT the effect of coherent noise on the final image is very specific. In this technique the speckle pattern present in the input phase images is transformed by the tomographic reconstruction algorithm into an artificial circular structure, whose magnitude decreases rapidly with increasing distance from the rotation axis. This implies that, uniquely for ODT, the signal-to-noise ratio of the final reconstruction can be adjusted by changing relative location of the sample with respect to the rotation axis. Therefore, in contrary to the conventional tomographic configuration, where minimization of the radial run-out results in high level of noise, in this paper we propose an alternative approach that employs measurements in the off-axis configuration. This modification enables the spatial separation between the sample structure and the central area of the reconstruction which inhibits the highest level of noise. However, the substantial gain in the noise performance comes at the cost of defocus degradation of phase images, which is caused by an inevitable axial displacement of the sample during the off-axis rotation. To account for this effect we employ a digital holographic technique to numerically refocus the phase data to the proper in-focus position the precise location of whose is determined with a standard autofocusing algorithm.

The feasibility of the proposed noise suppression method has been confirmed via numerical simulations, which show an up to sevenfold decrease of the reconstruction error. The full noise suppression measurement methodology (off-axis rotation and numerical defocus correction) has been successfully applied to tomographic reconstruction of an optical microtip that was measured with a radial run-out of *R* = 17μm. The obtained results have proved, firstly, a superiority of the off-axis configuration compared to the classical approach in term of noise and, and secondly, a possibility of an effective post-processing correction of an error due to defocusing of phase projections.

## Acknowledgments

This work was realized within TEAM program of Foundation for Polish Science which is funded from the European Regional Development Fund under the Agreement No TEAM/2011-7/7. The support of Warsaw University of Technology within the statutory funds is also acknowledged.

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