Investigations and improvements of digital holographic tomography applied for 3D studies of transmissive photonics microelements

Investigations and improvements of digital holographic tomography applied for 3D studies of transmissive photonics microelements

Malgorzata Kujawinska*, Agata Jozwicka, Tomasz Kozacki

Institute of Micromechanics and Photonics Warsaw University of Technology

8 Św. A. Boboli St., 02-525 Warsaw, POLAND

ABSTRACT

In order to control performance of photonics microelements it is necessary to receive 3D information about their amplitude and phase distributions. To perform this task we propose to apply tomography based on projections gather by digital holography (DH). Specifically the DH capability to register several angular views of the object during a single hologram capture is employed, which may in future shorten significantly the measurement time or even allow for tomographic analysis of dynamic media. However such a new approach brings a lot of new issues to be considered. Therefore, in this paper the method limitations, with special emphasis on holographic reconstruction process, are investigated through extensive numerical experiments with special focus on 3D refractive index distribution determination.. The main errors and means of their elimination are presented. The possibility of 3D refractive index distribution determination by means of DHT is proved numerically and experimentally. Keywords: digital holographic tomography, digital holography, interferometric tomography, phase microelements.

1. INTRODUCTION The progress in microtechnology and rapid development of a variety of amplitude-phase microelements including 3D

waveguides, photonics fibres, gradient refractive index optics requires development of new instruments to determine their basic physical parameters in 3D. In order to control performance of these elements it is necessary to receive quantitative information about amplitude and phase distribution of the field at each point (x,y,z) of the microelement under investigation. Various methods are already developed to determine these material parameters. The group of destructive methods [1, 2] using microtome measurement is the most known and enabling high measurement accuracy and sensitivity. However their invasiveness makes them unpractical for industrial application. From this point of view the methods based on tomographic algorithms, such as absorption tomography [3], interferometric tomography [4] and optical coherence tomography [5] are much more attractive, however they also do not fulfill all measurement requirements. The measurement time and limitations of application connected with some classes of object, measurement setup parameters and measurement conditions are still a crucial issues. The analysis of possibilities of interferometric tomography and digital holography have demonstrated that the combination of these two methods could be an effective tool for measurement of transmissive amplitude-phase microelements and also would permit to avoid some disadvantages of these methods when investigating amplitude or phase elements only [6,7,8]. In the paper at first we present the principle of the proposed digital holographic tomography method and its component parts are described. Next we perform the numerical analysis of DHT method with a special emphasis given to numerical errors received due to holographic reconstruction algorithm. Moreover, the advantages and limitations of the method are discussed. Finally, the results of experimental implementation of DHT method are presented at the example of 3D refractive index distribution determination in an optical fibre.

*)[email protected]

Invited Paper

Interferometry XIV: Techniques and Analysis, edited by Joanna Schmit, Katherine Creath, Catherine E. Towers, Proc. of SPIE Vol. 7063, 70630F, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.798314

Proc. of SPIE Vol. 7063 70630F-12008 SPIE Digital Library -- Subscriber Archive Copy

1. THE PRINCIPLES OF DIGITAL HOLOGRAPHIC TOMOGRAPHY

Digital holographic tomography is a combination of two measurement methods: interferometric tomography and digital holography [8]. Its basic advantage is the capability of registration of several angular views of the object during one registration. This is performed by a multiple pass arrangement (Fig.1a), which decreases significantly the number of registered images used for determination of 3D phase distribution in measurement volume, shortens significantly the measurement time and decreases the volume of captured data. DHT consists of two steps (Fig.1b): - holographic registration and numerical reconstruction process: information about 2D distribution of integrated phase function obtained at different reconstruction distances which are the analogue of sequential angular positions of an object in classical tomography,

- tomographic (numerical) reconstruction process: a set of the phase maps reconstructed from the hologram is treated as input data for tomographic reconstruction algorithm. The concept of measurement setup for hologram registration is shown in Fig.1a. The light from the He-Ne laser is introduced into optical fiber FO. It is split into two beams using FO coupler C. Collimators K1 and K2 form reference and object beams. The object beam (ΣO) is directed by the cuvette’s facets with reflective coating M1-M16, so that it passes, in this case nine times, through the object. The reference beam (ΣR) impinges at CCD camera after its redirecting by the beam splitter BS. The result of the interference between the object beam and the reference one is registered by CCD camera and the digital hologram is stored in computer memory. For some classes of the object it is advantageous to use imaging system in order to magnify objects at the hologram plane. In the paper we assume that the measurement volume is much bigger than the object. This allows the arrangement in which the object is situated at different part of the object beam during each pass through the object (i.e. the information about the object coded into the object beam at different distances is separated. The data captured in the system presented above contains nine out of focus images of an object, all in different planes. If imaging system is used it can not produce focused images within one image. Therefore the application of numerical refocusing, included in digital hologram reconstruction algorithm, is necessary. In such case prior to application of numerical refocusing the captured hologram can be processed with one of fringe pattern analysis methods [9] producing complex out of focus images. In the next step, numerical reconstruction at nine different distances should be performed. Usually, for that purpose, in digital holography the Fresnel convolution free space propagation method is applied [7]. However, in our case the propagation distances are relatively small. The study of Rayleigh-Sommerfeld free space propagation algorithm had shown that the convolution method gives meaningful propagation errors for small distances, therefore in DHT method, we propose to apply plane wave spectrum decomposition algorithm with frequency sampling adjusted according to propagation distance [10] for numerical holographic reconstruction process. Subsequently, the reconstructed results are post-processed, including data extraction and centring, in order to receive series of 2D phase projections for different angular position of the object as initial data for a sinogram. From the sinogram a 3D refractive index distribution is calculated by means of tomographic reconstruction. Two algorithms are the most widely used for that purpose. First one uses assumption that optical radiation propagates nearly along straight rays. Numerically it was proven that in this case the most accurate reconstructions are obtained, if instead of scattered field the projection image of the object centre is measured [11, 12]. Second reconstruction algorithm (that accounts for diffraction using either Born or Rytov weak-scattering approximations) is the filtered backpropagation algorithm [13]. The most accurate reconstruction can be received by hybrid filtered backpropagation algorithm, in which prior to application of this algorithm, an optical field scattered at an object is numerically propagated back to the object center [12, 14, 15]. However through numerical simulations and experiments it was shown that the both above mentioned reconstruction algorithms give results with comparable accuracy [12].

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Acquisition of

a serie of

holograms

Acquisition of

a serie of

holograms

Serie of phase/

amplitude

projections

Serie of phase/

amplitude

projections

Tomographic

recostruction

Tomographic

recostruction

3D distribution

of

Phase or

amplitude

3D distribution

of

Phase or

amplitude

ScalingScaling

3D distribution.

Of refractive

Index or

absorption

3D distribution.

Of refractive

Index or

absorption

Algorithm

Holographic

rekonstruction

methods

Holographic

rekonstruction

methods

Tomographic

rekonstruction

method

Tomographic

rekonstruction

method

Processing modules

b) o

Fig. 1. The concept of digital holographic tomograph (DHT): (a) setup for hologram registration and (b) the processing scheme. FO – fiber optics, C – fiber optics coupler, K1, K2 – collimators, M1…M16 – cuvette’s facets with reflective coating, BS – beam splitter, DH- digital hologram plane.

2. NUMERICAL VERIFICATION In the paper we have proposed to use a new DHT method for 3D refractive index distribution determination, which consists of two parts: holographic and tomographic process. As it was already mentioned, tomographic process has been previously extensively analyzed and the best suited reconstruction algorithms have been proposed [15]. However, in order to achieve high accuracy of tomographic reconstruction it is necessary to determine optimum conditions for 2D phase projections reconstruction giving the best precision of results obtained within holographic process. Therefore there is a need of advanced numerical analysis of errors arising from DH reconstruction. Moreover, the method limitations due to an object dimensions as well as measurement setup parameters (optimum magnification conditions and spatial separation of objects in axis x) should be also studied in details. For numerical analysis of DHT method a cylindrical object (Fig. 2a) corresponding to typical optical fiber is chosen. For simplicity of further analysis only three passes of the object beam through the object are considered (Fig.2b). Moreover, the sequential passes through the object are modeled by the linear configuration (three virtual cylindrical objects with different location from the hologram are analyzed), which is equivalent to DHT multipass system presented in Fig. 1a.

(a) (b)

Fig. 2. The configuration taken for computer simulation: (a) the phase object applied and linear configuration presenting the multipass arrangement taken for numerical registration and reconstruction of the model objects.

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(a) (b) (c)

There are many optical modeling methods based on Maxwell equation, wave optics or ray optics. Unfortunately none of them could be used to simulate complete holographic registration process. These methods either require too much computer power or simply would produce inaccurate results. Therefore to simulate the complete holographic process in a reliable way we use combination of two methods applied for different system region: wave propagation method (WPM) [16] and plane wave spectrum decomposition method (PWS) [17]. The WPM is numerically very efficient tool to study slowly varying refractive index structures and it is applied within the object volume. At all the other system regions the PWS method is applied. As a result of full holographic process we receive three 2D angular views of the object reconstructed at three different distances from a single hologram. These projections are used as the initial data for tomographic reconstruction based on filtered backprojection method. The previous analysis [18] have demonstrated that when the pure PWS algorithm for free space propagation is applied the reconstructed objects with off-axes location suffer significant asymmetry (Fig. 3), moreover the reconstructed phase value decreases significantly. That is due to an improper sampling of a phase part of the PWS kernel which is a major source of the PWS algorithm error. The consecutive angular views of the object captured in DH system have significant off-axis location, therefore additional correction algorithms in the holographic reconstruction process had to be implemented.

Fig. 3. Holographic reconstruction of the object located at (a) left side of the axis, (b) on-axis and (c) right side of the axis for ro=0.025mm and

|ncl-nco|=0.01.

Fig. 4. Holographic reconstruction of the object phase with proper sampling algorithm implementation. Object located at (a) left side of the axis, (b) on-axis

and (c) right side of the axis for ro=0.025mm, |ncl-nco|=0.01.

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nLJ

Elimination of the PWS algorithm error is possible by changing the original sampling. The phase part of the PWS kernel is a phase function with a constant amplitude containing high frequency information. To be properly sampled the phase difference of the PWS kernel between two next samples can not exceed 2π. Therefore it was proven [10] that in order to avoid PWS kernel aliasing error the following condition has to be fulfilled through the signal data extension

224

2

λ

λ

−∆∆

≥

xx

x

zN (1)

where Nx - number of samples, ∆x - distance between the samples and z – propagation distance. However, such an extension can present practical difficulty due to large memory requirements. In that case the multi Fourier transform PWS method (MPWS) [10] has to be applied. The effect of proper sampling algorithm implementation is shown in Fig. 4. The algorithm was used in phase reconstruction of the same object that in case presented in Fig.3. The problem of asymmetry and decrease of phase value was eliminated. Next, the full DHT process was numerically simulated for the object with parameters similar to multimode fiber (nco=1.48, ncl= 1.47, ro=0.05 mm) and the distance between objects x=x1=x2=0.8 mm. The tomographic reconstruction was calculated based on 180 projections (homogenously distributed within 180°) reconstructed from 90 holograms (each with 3 angular views of the object). The exemplary integrated phase projection reconstructed from a single hologram and 3D refractive index distribution calculated using tomographic algorithm are shown in Fig. 5.

Fig. 5. (a) One of holographic reconstruction of the object phase with proper sampling algorithm implementation and tomographic reconstruction: b) an exemplary layer and c) a central cross-section of 3D refractive index distribution. The calculations performed for the object with |ncl-nco|=0.01 and ro=0.025 mm and for projections calculated for objects separated by distances x=x1=x2= 0.8mm located at d2=100 mm (left side of the axis), d3= 50 mm (on-axis)

and d1=150 mm ( right side of the axis) from DH plane. Computer simulations presented above have proven possibility of reconstruction of 3D refractive index distribution using DHT method. However, in order to optimize our system consecutive angular views of the object, as referred to hologram plane, have to be located as close to each other as possible. Therefore, additional simulations have been performed to verify the influence of the spatial separation of objects in axis x on the 2D phase reconstruction. It was shown (see in Fig. 6) that when the objects are located close to each other severe diffraction effects appear. The error increases when the distance x=x1=x2 between objects decreases and in future there is a need to elaborate algorithms based on iterative methods in order to eliminate this problem.

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I \ nwL— fl<:-a 1_i(b)

Fig. 6. Cross-sections of the integrated phase holographically reconstructed at d1=150 mm and adequate tomographic reconstruction results for distances x=x1=x2 equal to: (a) 0.5 mm, (b) 0.4 mm and (c) 0.3 mm.

3. EXPERIMENTAL VERIFICATION

In the previous section the numerical analysis of DHT process have been presented, also the crucial problems of the method have been determined. The computer simulations of measurement process, in which the beam passes three times through the object, have shown that it is possible to reconstruct refractive index distribution in microobjects. However the problem of influence of the diffraction due to too small spatial separation of objects in axis x is still unsolved issue. In order to avoid that problem in the experiment a small magnification (telescopic system L1-L2 with magnification 3x) and double pass of the object beam through the object only have been implemented, as it is shown in Fig.7a. In this configuration the object beam (Σo) is directed by the mirror M so that it passes two times through the cuvette with immersion liquid in which the measured element is placed. Next, it interferes with the reference beam (Σo) and the resulting hologram is numerically recorded at CCD camera. The data captured in our system contains two out of focus images of an object, both in different planes. Therefore, as it was mentioned in section 2, the application of numerical refocusing, included in digital hologram reconstruction algorithm, is necessary. First, the captured hologram was processed with Fourier Transform fringe pattern analysis method [9] producing complex out of focus phase images. For free space propagation we have used plane wave spectrum decomposition algorithm with frequency sampling adjusted according to the propagation distance. Such a composed holographic reconstruction

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FOfl ER

L2 LI cuvette with

I j immersion; I EliLaser — .

process gives two focused object integrated phase projections for two different object angular orientations (Fig. 7b). In our configuration setup the angular separation of projections was 9°. The object was rotated sequentially and the holographic process was repeated. Such a process gives several projections homogeneously distributed within 180° which is necessary for application of the filtered backprojection algorithm.

(a) (b)

Fig. 7. The experimental setup for registration at hologram the beam with double passing through an object (a) and the experimentally registered hologram and 2D angular views of the multimode fiber numerically reconstructed at two different distances (b). FO – fiber optics, C – fiber optics coupler, K1-K2 –

collimators, M – mirror, BS – beam splitter, L1-L2 – magnifying assembly, CCD – digital camera with DH plane.

Fig. 8. DHT: 3D refractive index distribution in multimode fiber numerically reconstructed using tomographic algorithms. nimm – refractive index of

immersion liquid, nco – refractive index of FO core, ncl – immersion liquid of FO cladding.

The experimental results are shown in Fig. 7b and Fig. 8. 2D phase projections were correctly reconstructed at two different distances from a single digital hologram. Therefore, based on the set of 2D phase projections, it was possible to reconstruct 3D refractive index profile correctly. The value difference between refractive index of immersion liquid (nimm) and of fiber’s cladding (ncl) results from accuracy of matching between these two indices and it does not introduce errors at this stage of the work.

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4. CONCLUSIONS

It was proven numerically and experimentally that it is possible to reconstruct 3D refractive index distribution using digital holographic tomography method (DHT). Digital holography allows to reconstruct correctly phase objects at different distances and the 2D phase projections obtained can be used for 3D phase tomographic reconstruction. The problems in holographic reconstruction connected with size of an object and object localization reported in the previous papers have been resolved by implementation of new correction algorithm. However, the influence of the diffraction resulting from small spatial separation of objects in axis x is still a crucial error. In future there is a need to elaborate algorithms in order to eliminate this problem.

AKNOWLEDGEMENTS

The authors acknowledge the financial support of Ministry of Education and Science within the project no. 3 T10C 015 29 and N505 008 31/1374. The project was also supported by European Union within the Network of Excellence for Micro-Optics NEMO.

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