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Holography and phase retrieval in terahertz imaging

Nikolay V. Petrova, Andrei A. Gorodetskyb, Victor G. Bespalova

aDepartment of photonics and optical informatics, National Research Univ. of InformationTechnologies, Mechanics and Optics,

Kadetskaya Linia 3, St. Petersburg, Russia, 199004.bInstitute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas

(Greece).

ABSTRACT

In this paper, we present review and latest results obtained in the scope of terahertz holographic and othermethods for phase retrieval in terahertz imaging. Not only accurate change of amplitude, but also rigorousphase retrieval is essential for precise calculation of optical parameters of the samples in terahertz range. Pulseterahertz holography introduced some years ago shows itself as perfect method for overall-object phase retrievaltechnique, but in the same time it allows measurement with low signal to noise that leads to less precise derivationof sample optical parameters. And certainly just point-by-point terahertz time-domain spectroscopy provides themost precise information of sample phase, but it is rather time consuming and has low spatial resolution as well.The other possible way assumes, in contrary to pulse terahertz holography and spectroscopy, using narrow-bandcontinuous terahertz source, which tunability might also make the measurement process easier. And diffractionpatterns registered with microbolometer array or any other terahertz intensity sensor placed at several differentdistances from the object and/or taken for several different terahertz frequencies are used for phase retrieval inthis case. We present both numerical predictions and experimental results for the proposed methods, estimatethe achievable spatial and other limits of the techniques and compare them to the others used in different spectralranges.

Keywords: terahertz digital holography, terahertz imaging, phase retrieval, phase and amplitude recovery,phase imaging, wavefront reconstruction, time-domain imaging, inverse problem.

1. INTRODUCTION

Terahertz (THz) imaging proposes emerging number of its practical applications due to the ability of THzradiation to penetrate various non-metal materials, that are opaque for visible light.1 From the numerous paperson THz imaging,2–4 several main methods of obtaining 3-dimensional inner structure of an object are found. Thefirst approach is based on tomographic object reconstruction, found in different experimental decisions: time offlight in reflection geometry,5 classic computer tomography involving inverse Radon transform,6,7 Fresnel binarylens method.8 Investigation of mentioned approaches was presented in.9 And the second approach comes fromoptical interferometry and classical digital holography.10 Up to now, several 3D imaging methods in THz regimebased on interferometric and holographic techniques have been demonstrated: classical interferometric scheme,11the same scheme with multi-wavelength phase unwrapping12 was introduced earlier, phase-shifting methods13,14and classical holographic recording.15 These methods allow phase retrieval, thus allowing to convert time-domaindata into third spatial dimension and hence obtain object’s optical relief information.

In this paper, we review terahertz pulse time-domain holography (THz PTDH) method proposed by authorsmore than 5 years ago,16 that is based on direct time domain field profile recording followed by numerical wave-front inversion for object amplitude-phase characteristics reconstruction. This method is the most experimentallysimple among the previously mentioned, since it’s just slightly upgraded version of classical THz time-domainspectroscopic (THz TDS) setup and avoids reference THz beam, but at the same time has relatively low signal-to-noise ratio. However, it should be mentioned here, that in majority of previously cited works, more expensiveand less widespread high-power THz sources were used.

E-mail: [email protected]

Invited Paper

Terahertz Emitters, Receivers, and Applications IV, edited by Manijeh Razeghi, Alexei N. Baranov, John M. Zavada,Proc. of SPIE Vol. 8846, 88460S · © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2023918

Proc. of SPIE Vol. 8846 88460S-1

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/26/2013 Terms of Use: http://spiedl.org/terms

For the case of single (fixed or tunable) wavelength high-power THz source the other method that we haveproposed recently and review here in the second part of the paper, can be used. This method utilizes iterativephase retrieval, currently introduced in optical imaging and is also experimentally relatively simple, as the onethat does not require neither reference beams, non direct field measurements, putting all complexity into the dataprocessing algorithm. This method allows to recover full wavefront picture after measuring of spatial intensitydistributions at several frequencies (for example, as in12) and/or at different planes of registration and then useiterative procedure for wavefront restoration.

2. THE PRINCIPLES OF TERAHERTZ PULSE TIME-DOMAIN HOLOGRAPHY

Terahertz pulse time-domain holography method got it’s name because the full electric field can be recordedwith electro-optic (EO) sampling. There’s no need in reference source used in optical holography to record phaseinformation, full broadband THz registered wavefront can be easily inverted numerically.

In our earlier paper17 this technique was studied in detail - namely, its spatial resolution, dynamic diapason,and noise affect on reconstruction. Later,18 it was suggested to use virtual reference source, which radiation isnumerically added to the registered diffraction pattern, and reconstruction can be performed both numericallyand optically, if hologram mask is properly scaled to fit the optical wavelength. Methods comparison can alsobe found in previously mentioned reference.18 Experimental results on THz pulse wavefront inversion objectreconstruction were published later.19

2.1 Scheme for recording THz broadband hologramsDirect time-domain THz field distribution is recorded with upgraded classic THz TDS setup (fig. 1, a). Fem-tosecond IR radiation is generated by Solar FL-1 Yb:KYW femtosecond laser, delivering more than 1 W averagepower, in pulses of around 150 fs in duration and 70 MHz repetition rate at 1040 nm wavelength. Then, radiationis split in the ratio of 99 : 1 into pump (99%) and probe beams. Pump beam comes through delay stage forcontrol of its relative time position and falls onto InAs semiconductor crystal that serves as THz generator, placedinto constant magnet field of 2 T for THz generation enhancement.20 Produced THz radiation is collimated witha parabolic mirror and then coupled back with a probe beam by high-resistive silicon beam splitter. Probe IRbeam passes the EO CdTe crystal together with diffracted THz beam. THz electric field induces birefringence,introducing rotation of probe beam polarization map proportional to the field amplitude. To record this map,analyzing polarizer and fast CCD camera are used, to follow the time-domain profile, the delay line in THz armof the setup is used. Changing the relative path of THz and probe beam, the full time-domain picture is recordedfor every point of the diffraction pattern. The same registration method for pulse THz holograms is describedelsewhere.21

Experimental recording and reconstruction of the “K”-shaped object opaque in THz frequency range are shownin fig. 1. The size of the studied object area was about 2× 2 cm. Spatial resolution corresponds to the samplingused in measurement, and sampling is selected in accordance with THz wavelength of maximum intensity inspectrum. One should notice, that mathematical model excludes virtual image at reconstruction, and absenceof reference source cancels the problems raising from zero-order diffraction at the image, unlike classical opticalschemes.22

2.2 Mathematical model of image reconstructionIn our first paper on THz PTDH,16 our approach to numerical modeling of reference-free holographic recordingand reconstruction is described. For modeling the field propagation, equations of scalar diffraction theory isused.

For solving direct diffraction problem, the whole broadband THz spectrum in the object plane (x′, y′) is splitinto monochromatic components ωi, and then, we propagate these waves separately to the registration plane(x, y), thus obtaining amplitude |U(x, y)| and phase Φ(x, y) for each frequencies ωi at the registration plane.In practice, the data obtained from measured spatial-temporary distribution of electromagnetic field by Fouriertransform F :

Uω(x, y) = F [Ut(x, y)]. (1)



FLBS

Si-BS

IPM I

P

uO CdTe P

>L

CCD

I

(a)

(b) (c) (d)Figure 1. (a) Setup for THz PTDH: FL – femtosecond hi-power Yb:KYW laser, BS – beam splitter, M – mirrors, THz– THz generator, PM – off-axis parabolic mirror, O – object under study, Si-BS – silicon beam splitter, L – lenses, P –polarisers, CdTe – EO crystal, CCD – camera; and (b-d) experimental results of THz pulse holographic reconstructionof the opaque object. (b) – photo of the object, (c) – total intensity of the reordered diffraction pattern (500 µm pointsize), (d) – reconstructed object.

Possessing a set of data on the spatial distribution of the complex amplitude of electromagnetic field in theregistration plane, we can calculate the field uωi

(x,′ y′) in the object plane for selected monochromatic componentsωi. At first, we suggested to use for that Rayleigh-Sommerfeld equation, without paraxial approximation, sincethe setup we were using did not fulfill the paraxial approximation limitations for all the broadband spectrumof the terahertz pulse radiation. However, the direct Rayleigh-Sommerfeld integral calculation is very time-consuming, so for practical purposes angular spectrum algorithm based on representation of the electromagneticfield by angular spectrum of plane waves23 can be used:

u(x′, y′) = F−1{H(fx′ , fy′ ,−l)· F [U(x, y)]

}, (2)



where H(fx′ , fy′ ,−l) is the transfer function :

H(fx′ , fy′ ,−l) =

{exp

[i 2πλ (−l)

√1− λ2(f2x′ + f2y′)

]if f2x′ + f2y′ <

1λ2 ;

0 if f2x′ + f2y′ >1λ2 ;

(3)

Here l is the distance from the object to the registration plane, F−1 is operator of inverse Fourier transform, fx′

and fy′ are spatial frequencies, λ = c/nairω is the wavelength of selected for reconstruction spectral component,c is the speed of light, nair is the refractive index of air. The critical range of angular spectrum algorithm islimited by inequality:

l 6 lmax =∆xD

λ, (4)

if pixel size at the object plane ∆x′ is equal to pixel size at the registration plane ∆x. Here D is the typical linearsize of object. For broadband THz radiation, this limitation affects only low-frequency part of the spectrum. Inorder to increase the critical range, object zero-padding or pixel merging can be used. Resolution in pixel mergingwill not be affected, because diffraction limit for low-frequency components is more low. Another strategy couldbe using another wave propagation equation for these monochromatic components, which usage is not limited onthe distances bigger than lmax. It can be two-dimensional Fresnel transform, or its numerically optimized analogFresnel-Bluestein transform.24 In that case paraxial limitation

l� lpar =1

2

3

√D4

λ(5)

can be neglected, as shown in our previous work,25 because it does not bring any affect into diffractionpattern.

In principle, for broadband holographic modeling in THz frequency range, other mathematical models can beadapted, like convolution algorithm, derived from Rayleigh-Sommerfeld diffraction integral formula by linearitysystem theory,26 or even spectral approach.27

3. THE CONCEPT OF PHASE RETRIEVAL IN TERAHERTZ IMAGING

At present, phase retrieval methods have been successfully used in the visible spectrum for a wide range of tasks.It allow one to obtain a set of real field amplitudes, using which in an iterative procedure makes it possible torecover the phase information that was lost during the registration of the intensity distributions.

General concept of phase problem solution in the case of recorded intensity patterns is based on using addi-tional data sets, that compensate the lack of phase information. Usually, several additional intensity patterns areused, recorded for different sets of experimental parameters. For effective phase retrieval we need mathematicalmodel describing the whole process of radiation propagation from object to registration plane and back andtaking all the variable parameters into account.

In the discussion on transition a phase retrieval methods from optical to THz frequency range, one shouldmention, that as variables the same parameters as in optical methods can be used. Very good results werereported for changing the wavelength of radiation,28,29 and/or distance from the object to the registrationplane.30 Problems in that case arise from the increase of the wavelength by three orders of magnitude, whilegeometric dimensions of the individual elements traditionally used in setups remain practically invariable. Thisleads to a necessity to work in the near-field diffraction, which, in turn, requires an adjustment in the usedmathematical models of optimization of the parameters of the optical system. In the THz region we examinedthe configuration of the phase retrieval methods, based on the iterative procedure of solving the wave propagationequation (specifically - the Fresnel transform), using several spatial intensity distributions25 and free space asthe phase analyzer. In the mentioned work, generalized iteration algorithm allowing to use both wavelength anddistance as variable parameters is described in detail.

As shown by numerical simulations, the main restriction on the phase reconstruction using this configurationis applied by the sampling theorem. In accordance with it, sampling the signal while recording the intensity



distributions forbids the iterative procedure using Fresnel transform in the very near zone, where fine parts ofthe object are reconstructed due to the presence of high spatial frequencies. Also, we have investigated thealternative configuration of the method based on angular spectrum propagator.31 The longitudinal intensitydistributions was analyzed and the selection criteria for the data in case of using plane waves was determined.

3.1 Comparison of the THz PTDH and phase retrieval methodsTo show the ability of both techniques to reconstruct 3D objects, we demonstrate, yet numerically, the recon-struction of purely flat phase object. As an object we take Teflon, that is transparent in THz and has the flatrefractive index of nobj = 1.46 in the whole available THz spectral range,32 thus allowing to call it purely phaseobject (fig. 2, a). For dispersive objects refractive index can be measured by THz TDS at first.

(a) (b) (c)

(d) (e) (f)Figure 2. 3D model of the sample (a), phase distribution brought by the object to 200 µm (b) and 300 µm (d) spectralcomponents, phase reconstructed by THz PTDH at 200 µm (c) and 300 µm (e), phase reconstructed by iterative algorithmusing 300 µm and 10 different intensity patterns at different distances (f).

The size of the object is 6, 4× 6, 4 mm, its thickness is changed from 1 to 2 mm. When passing through thephase object wavefront undergoes the following phase retardance:

ϕ(x′, y′) =2π

λh(x′, y′) (nobj − nair) . (6)

Here h(x′, y′) is the object profile. On the fig. 2, b, d the phase distribution after passing through the objectspectral component of THz radiation with λ = 200 µm and λ = 300 µm is presented. At fig. 2, c, e, and f phasereconstruction results are shown, Image “c” and “e” are obtained by wavefront inversion, and “f” is obtained byusing iterative algorithm.



4. CONCLUSIONTerahertz holographic and phase retrieval techniques are very promising tool for security, biological and otherapplications, because they allow to reconstruct not only amplitude and spectrum, but also its phase informationthus allowing to calculate numerous optical parameters of the sample. Holographic imaging coupled with fastdelay line assumes single shot measurements, thus allowing to work with living and fast changing samples, andalso follow the system dynamics. Iterative phase retrieval method requires only spatial intensity measurementsand can be used with high power CW or pulsed sources of THz radiation with matrix detector. Moreover, therecording scheme is considerably simplified in comparison with analogous methods using reference wave. In bothmethods, there are no virtual image and zero-order diffraction due to their reference-free nature. The results ofnumerical experiments show us that the resolution of the methods is limited by the wavelength of the radiation.We believe that this imaging techniques have great potential.

5. ACKNOWLEDGMENTSThe work was performed under the Federal Target Program ”Scientists and Science Educators of InnovativeRussia” in 2009–2013 (Agreement No. 14.B37.21.1561) and supported by Russian Fund Basic Research (12-02-09550-Mob z).

A.G. thanks Magicplot Systems, LLC, for providing MagicPlot Pro plotting software used for fig. 1 prepara-tion.

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