Stratified dispersive model for material characterization using terahertz time-domain spectroscopy
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l Stratified dispersive model for material characterization usingTHz time-domain spectroscopy
J.L.M. van Mechelen, , ∗ A.B. Kuzmenko, and H. Merbold ABB Corporate Research, Segelhofstrasse 1K, 5405 Baden-D¨attwil, Switzerland D´epartement de Physique de la Mati`ere Condens´ee, Universit´e de Gen`eve, Gen`eve, Switzerland compiled: September 12, 2018We propose a novel THz material analysis approach which provides highly accurate material parameters andcan be used for industrial quality control. The method treats the inspected material within its environmentlocally as a stratified system and describes the light-matter interaction of each layer in a realistic way. Theapproach is illustrated in the time- and frequency-domain for two potential fields of implementation of THztechnology: quality control of (coated) paper sheets and car paint multilayers, both measured in humid air.
OCIS codes: (300.6495); (120.4825); (120.4290); (120.4630); (070.4790)http://dx.doi.org/10.1364/XX.99.099999
The last decade has seen a significant developmentin the maturity of THz technology which is paving theroad towards industrial applications. THz technologyhas the potential to become a real differentiator givenits unusual properties ranging from the sensitivity todepth information to providing complex optical mate-rial functions while being innocuous for human tissue.A field of application which would substantially bene-fit from these aspects is material quality control in au-tomatized processes for detecting e.g. in-depth failures.Recently, THz spectroscopy based quality control hasbeen explored on materials such as paper [1], food, plas-tics, semiconductors, and biological tissues.[2] Commonmethods to obtain the material properties range fromtime-domain peak subtraction [1, 3] to inversion of thetransfer function [4, 5]. Both methods require severalinternal reflections and typically apply to single layeredmaterials which are assumed to have low absorption andlittle dispersion. Recently, a time-domain fitting proce-dure has been proposed for multilayers,[6] although it re-quires prior knowledge of the optical material propertiesand only provides the thicknesses. Despite the fact thatthese methods may give satisfactory material parame-ters in some situations, they break down or lack accu-racy when applied to industrial quality control where theinspected material and its environment form a complexstructure with properties that may change over time.In this Letter we describe a novel THz material analy-sis approach allowing for high precision material param-eter determination of complex structures, consisting of(i) a stratified dispersive model, (ii) an appropriate mea- ∗ Corresponding author: [email protected] surement configuration and (iii) a time-domain based fit-ting procedure. The underlying concept is to model themeasurement configuration as a stratified system wherefor each layer the physical processes that occur uponthe light-matter interaction are described in a realisticway. Subsequently, the light propagation through themultilayer system is calculated and fitted to the experi-mental data in the time-domain, thereby optimizing allmaterial parameters together. The method is demon-strated for two prominent industrial examples, inspec-tion of (coated) paper sheets and car paint multilayers.It is in particular shown that the method is applicablein ambient air with variable humidity and irrespective ofthe reference position in reflection geometry, two of themajor obstacles encountered in today’s THz analysis.The measurement configuration in common optical se-tups consists of the material under investigation posi-tioned in a certain environment, e.g. ambient air. Inindustrial circumstances, a reflection geometry is of-
Fig. 1. Schematic representation of a light ray incident atangle θ i interacting with a double layer structure (a) on areflector and (b) suspended above a reflector, in ambient air. ten preferable to a transmission arrangement due tomore convenient accessibility of the sample. The over-all strength of the reflected signal can be increased bypositioning a (metal) reflector behind the material [1].A central principle of the proposed analysis approach isto locally consider the measurement configuration, com-posed of the inspected material, the surrounding envi-ronment and the reflector, as a stratified system. Theadvantage is that the light propagation through this sys-tem can be analytically described by the Fresnel equa-tions. For the case of a bilayer material system shownin Fig. 1a with the angle of incidence θ i = 0, the re-flected electric field E r can be calculated using the inci-dent electric field E r, and the transfer function of theentire multilayer structure T through E r ( ω ) = T ( ω ) E r, ( ω ) T ( ω ) = r + t r t e − i β + t r r r t e − i β + t t r t t e − i β + β ) + . . . (1)where β k = ωn k d k /c is the phase shift accumulated inlayer k , ω is the frequency of the radiation, n k is thecomplex index of refraction of layer k , d k is the thicknessof layer k , c is the speed of light in vacuum, and thetransmission t ij and reflection r ij coefficients are t ij = 2 n i n i + n j , r ij = n i − n j n i + n j . (2)The full expression of T ( ω ) as well as the case of obliqueincidence can be found in standard optics literature [7].The stratified model (Eqs. 1-2) contains d k and n k ofeach individual layer, opposite to simple analysis meth-ods which only consider d and n of the material layer,and furthermore assume n being real and frequency in-dependent. We assume that d k and n k are not known,and aim at a realistic description of n k by addressing theindividual physical processes in each layer which resultsin n k being complex and dispersive. Although academicstudies have shown that many exotic materials, such assemi- or superconductors, are characterized by diverselight induced processes, materials encountered in indus-trial inspection applications are often much simpler. Insuch common materials the interaction with THz radi-ation is typically limited to lattice vibrations and (freeand/or collective) electron oscillations which can oftenbe well described by Lorentzian line shapes in n k ( ω ).However, the analysis method is general in a sense thatinteractions may also be described by other line shapessuch as Gaussian, Fano or Tauc-Lorentz. An appropri-ate model for n k ( ω ) is thus a summation of oscillators,each representing a specific excitation. For the case of aseries of Lorentzians, this model is commonly known asthe Drude-Lorentz parameterization and the dielectricfunction ǫ ( ω ) = n ( ω ) of each layer k is given by ǫ ( ω ) = ǫ ∞ + m X ℓ =1 ω p,ℓ ω ,ℓ − ω − iγ ℓ ω (3)where ǫ ∞ is the high frequency limit of ǫ ( ω ), ω p,ℓ theplasma frequency, ω ,ℓ the characteristic frequency and γ ℓ the relaxation rate of excitation ℓ in layer k . For ameasurement configuration as shown in Fig. 1b the twomaterial layers 2 and 3 are parameterized in this way.The thin air layers 1 and 4, on the other hand, can sim-ply be modeled by fixing n air = 1 + 0 i through setting ǫ ∞ , air = 1 without further Drude-Lorentz parameters.For the metallic reflector, layer 5, a single fixed Drudeoscillator is used which accounts for the reflectivity ofthe metal in the THz range. In this way n k describesthe interaction with each layer k in a realistic way. Sub-stitution of n k into Eqs. 1-2 now fully describes E r of theexamined material in its measurement configuration.From an experimental perspective, quantitative infor-mation of a material can be obtained by performing botha measurement of the sample structure and of a knownreference, leading to a data set consisting of | E ( ω ) | and | E ( ω ) | , respectively. The goal of the analysis is to ob-tain d k and n k ( ω ) of the probed material layer(s). Modelfree analytical inversion of Eq. 1 is often applied for sin-gle or double layers [6], but structures as shown in Fig. 1are from a practical point of view too complicated forthis approach. The commonly used alternative is fitting T of Eq. 1 to the reflectivity | r ( ω ) | = | E r ( ω ) /E r, ( ω ) | .However, multiple internal reflections occurring in struc-tures as shown in Fig. 1 make | r ( ω ) | rarely easy to un-derstand. The more comprehensive data set is composedby the time-domain functions E r ( t ) and E r, ( t ) whichoften clearly reveal the partial reflections from the vari-ous interfaces. Central to the proposed analysis methodis to perform a time-domain fitting procedure of E r ( t )to the experimentally determined E exp r ( t ) using a leastsquares algorithm. Hereto, the transfer function T ofEq. 1 has to be Fourier transformed to the time domainand subsequently convolved with E r, ( t ). It can nowbe seen that the purpose of having a thin air layer 1 inFig. 1, modeled as n air = 1, is to account for the possi-ble mismatch between the sample and reference position.Namely, the fitting parameter d air corresponds uniquelyto a temporal displacement of E r ( t ), and thus to therelative position of sample and reference. Another ma-jor advantage of the approach is that ambient humiditypresent in the reference E r, ( t ) is inherently (see Eq. 1)also present in E r ( t ) which allows analysis in ambientair without loss of accuracy.In the following we will illustrate the robustness of theanalysis method by giving two experimental examples.The first one is the quality control of paper for which theindustry requires sensitivity to a variety of parameterson which it imposes strict accuracy standards. We applythe proposed method to obtain the thickness and the ash(filler) concentration of various kinds of paper sheets. Fig. 2. E exp r ( t ) of (a-c) uncoated and (d) coated paper sheets, E exp r, ( t ) of a copper reflector, and fits E r ( t ), using s-polarizedincident radiation, θ i = 13 ◦ , 26 ± ◦ C and 22 ± copy paper,(c) 200 g/m copy paper, (d) coated board. The inset showsthe total thickness of the samples (a-d) obtained from thefitting procedure as compared to mechanical caliper values. Fig. 2 shows the experimental E exp r ( t ) of three un-coated and one coated papers sheet with varying thick-ness, and E exp r, ( t ) of an optically polished copper ref-erence measured by THz time-domain spectroscopy(TAS 7500, Advantest Inc.) in the range 0.1-4 THz,recorded with 512 averages at 125 Hz. The measurementconfiguration consists of air-(coating)-paper-air-copper(see Fig. 1b) and can thus be modeled as a five (four)layer system for (un)coated paper. In order to perform atime-domain fit to E exp r ( t ), we need to determine n k for Table 1. THz thickness values of various paper sheets as com-pared to values from rated mechanical techniques (in µ m).paper sample rated thickness THz thicknesstissue 37 38.7100 g/m copy paper 106 ± copy paper 200 ± ± each layer k of the system. Setting n k of air and copperas discussed before, the question arises how the interac-tion of THz radiation with paper can be described. Theusage of too many oscillators could provide a satisfac-tory fit, but their parameters may not have any physi-cal meaning. It turns out that the fibrous structure ofpaper slightly modifies n from being frequency indepen-dent. Paper may further contain so-called ash which is amineral that fills the spaces between the fibers and oftenfeatures a characteristic phonon absorption in the THzrange. This means that paper can typically be modeledwith one or two oscillators, one for the fibrous structurewith ω ≈ −
200 THz and in case it also containsash, like copy paper, a second one with ω <
10 THz.Tissue paper just contains fibers, whereas the coating ofthe board consists of only ash. Based on the experimen-tal data, we choose the oscillators to be Lorentzians andparameterize them as given by Eq. 3. At this stage, wehave fully described the optical properties of the entiremultilayer configuration at THz frequencies, which al-lows to calculate E r ( t ) using Eq. 1 and fit it to E exp r ( t ).The fitting parameters are d k , and ǫ ∞ , ω , ω p , γ whichmodel the paper sheet and coating layer. The resultof the fitting procedure is shown in Fig. 2 and the ob-tained thicknesses are shown in Table 1 where they arecompared to rated values measured by standard mechan-ical techniques (see also inset of Fig. 2). The excellentfit results together with the close match of the thick-ness values demonstrates the robustness of the approach.Moreover, the fit accurately describes the ambient mois-ture present in E exp r ( t ) and accounts for the mismatch ofthe reference position with respect to the sample (herearound 0.5 mm), both without compromising the ac-curacy. Note that although E exp r ( t ) has been recordedin 4 s, industrial paper quality control typically allowsmuch shorter integration times. This will add noise onthe amplitude of E exp r ( t ), but affects less the temporalinformation, which is of main importance for the thick-ness determination using the proposed method.The filler concentration, on the other hand, can be de-termined by the strength of the ash’s phonon absorptionas manifested in n k ( ω ) of the paper sheet. Fig. 3 showsthe real and imaginary part of n ( ω ) of several papersamples containing CaCO ash, obtained from a time-domain fitting analysis as outlined above. For compar-ison, tissue paper which contains no ash is also shown.For all ash-containing samples Im n ( ω ) shows a strongincrease around 3 THz which corresponds well to theliterature value of the CaCO lattice vibration [8]. Thespectral weight of the absorption is proportional to ω p which is directly obtained from the analysis procedure.The correlation of ω p with the rated CaCO content (asstated by the paper manufacturer) suggests a linear be-havior in the measured ash range (see inset of Fig. 3). Inaddition to the thickness, the analysis method can thusalso provide information on the consistency of paper.In a second example, we have applied the analysismethod to automotive paint layers, another widely sug- Table 2. THz thickness values (in µ m) of a triple paint layeron steel and silicon (see text) compared to rated techniques.sample THz individual THz total mechanical magneticsteel 42.5, 22.3, 31.9 96.7 98 ± ± ± ± gested industrial application of THz technology [3]. Dif-ferent kinds of Glasurit BASF car paints were used in or-der to make single, double and triple layer structures ona substrate as employed in the automotive industry. Al-though up to now we have solely dealt with reflection ge-ometries, in certain cases industrial inspection may alsorequire transmission geometries, for which the stratifieddispersive model (cf. Eqs. 1-2) can be straightforwardlyextended. In order to demonstrate both configurations,we have used substrates of both steel and silicon for re-flection and transmission measurements, respectively.Fig. 4a,b shows E exp r ( t ) and E exp t ( t ) of a triple paintlayer consisting of white primer, blue waterborne basecoat and clear coat on steel and silicon, respectively, asmeasured by THz time-domain spectroscopy (TPI 1000,TeraView Ltd.) in the range 0.03-3 THz. As comparedto Fig. 2, E exp r ( t ) seems less complicated and appears assingle cycle pulses with little structure. This is, however,deceptive since the small layer thicknesses and the ab-sence of an air gap between paint and substrate leads toan overlap of the multiple internal reflections. Fig. 4c,dshows the corresponding | r exp ( ω ) | and | t exp ( ω ) | in thefrequency-domain. In this example, we have appliedthe proposed analysis method in both the time- andfrequency-domain simultaneously (see Fig. 4). This en-hances the accuracy of the fit parameters in case | r ( ω ) | and/or | t ( ω ) | is strongly frequency dependent, as in thepresent example. The thicknesses resulting from the fitare shown in Table 2 for each individual layer as well Fig. 3. Real and imaginary part of n ( ω ) of (a) tissue paper,(b-c) copy paper, (d) magazine paper. The inset shows ω p vs. the rated CaCO content. The error bar gives the spreadin ω p of 5 sheets of different basis weight of the same grade.Note that as compared to samples (c) and (d), copy paper(b) requires an additional oscillator around 1.8 THz. Fig. 4. (a) E exp r ( t ) and E exp r, ( t ), (b) E exp t ( t ) and E exp t, ( t ),(c) the corresponding | r exp ( ω ) | and (d) | t exp ( ω ) | of a triplepaint layer on (a,c) steel and (b,d) silicon, recorded using 200averages at 30 Hz, at 26 ± ◦ C and 55 ± as for the total stack, and are compared to rated valuesfrom mechanical caliper and magnetic induction tech-niques. The inset of Fig. 4c shows the total thicknessvalues of a larger set of single and multilayer samples ver-sus the rated values. In all cases the comparison is veryclose which proves the accuracy of the analysis method.In conclusion, we have proposed a widely applicablemethod to perform and analyze THz measurements ofcomplex structures which is able to provide accuratematerial parameters. The approach treats the inspectedmaterial and its environment as a stratified system andfor each layer parameterizes the material properties us-ing a set of oscillators. The method has been demon-strated for paper and paint multilayer samples in boththe time- and frequency-domain, for reflection and trans-mission geometries in humid ambient air.We would like to thank Dirk van der Marel for the useof the TeraView spectrometer and Xun Gu for a criticalreading of the manuscript. References [1] J. White, J. Morgan, J. Riccardi, M. Friese, and I. Duling,in
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