Electromagnetic Characterisation of Materials by Using Transmission/Reﬂection (T/R) Devices

electronics

Review Electromagnetic Characterisation of Materials by Using Transmission/Reﬂection (T/R) Devices

Filippo Costa * ID , Michele Borgese ID , Marco Degiorgi and Agostino Monorchio

Dipartimento di Ingegneria dell’Informazione, Università di Pisa, 56126 Pisa, Italy; [email protected] (M.B.); [email protected] (M.D.); [email protected] (A.M.) * Correspondence: ﬁ[email protected]; Tel.: +39-050-2217577

Received: 1 October 2017; Accepted: 7 November 2017; Published: 9 November 2017

Abstract: An overview of transmission/reﬂection-based methods for the electromagnetic characterisation of materials is presented. The paper initially describes the most popular approaches for the characterisation of bulk materials in terms of dielectric permittivity and magnetic permeability. Subsequently, the limitations and the methods aimed at removing the ambiguities deriving from the application of the classical Nicolson–Ross–Weir direct inversion are discussed. The second part of the paper is focused on the characterisation of partially conductive thin sheets in terms of surface impedance via waveguide setups. All the presented measurement techniques are applicable to conventional transmission reﬂection devices such as coaxial cables or waveguides.

Keywords: dielectric permittivity measurement; magnetic permeability measurement; surface impedance measurement

1. Introduction Recent technological advances have fostered the use of new materials and composites in several different fields spanning from electronics to mechanics, agriculture, food engineering, medicine, etc. Two-dimensional materials [1] have attracted considerable interest from the electronic devices community. Graphene is probably the foremost representative of that class of materials and promises a revolution in many applications. It has also aroused the interest of the scientific community towards other two-dimensional, or quasi-2D, materials, which are receiving increasing attention thanks to the development of printable electronics technology, which allows fast and low-cost prototyping and fabrication of device like antennas [2–4], periodic surfaces [5,6], or sensors and tags [7–9]. Those kinds of application require highly conductive inks that may exhibit high losses, which, in turn, can be exploited for the design of absorbing materials [10–15] and for electromagnetic signature control [16]. The applicability of these techniques has been recently pushed to above 100 GHz using reverse offset (RO) printing, which allows high line-resolution and increases printing speed [17,18]. Microwave properties of materials are also of crucial importance for the development of electronic and flexible circuits working at microwave frequencies: flex circuits [19–22] are often adopted in computer peripherals, cellular phones, cameras, printers, and LCD fabrication. The film deposited on top of the substrate exhibits a thickness of a few micrometres. That plethora of new materials and their innovative applications represent a driver for the development of new and more accurate techniques to characterise their electromagnetic properties. However, the measurement of thin materials is still a challenge. Unambiguous and precise characterisation of specimens over a very wide band is increasingly important. Non-destructive testing (NDT), which is fundamental in science and technology for the screening of components or systems without damaging or permanently altering the sample, can find a natural solution in the electromagnetic techniques for the material’s properties measurement. Free-space reflection

Electronics 2017, 6, 95; doi:10.3390/electronics6040095 www.mdpi.com/journal/electronics Electronics 2017, 6, 95 2 of 27 and open-ended probe techniques are typical transmission/reflection or reflection-only NDT methods [23–28]: they are indeed attractive because they maintain material integrity and are increasingly used in various fields (mechanics, constructions, medicine, and so on), but they usually pay a price in terms of accuracy and mathematical model complexity with respect to the guided T/R methods. In this context, it is fundamental to have access to methodologies able to accurately characterise the properties of materials. A number of those techniques are available in the scientific literature [29,30]. They can be substantially divided into two categories: resonant methods and transmission/reflection (T/R) methods. In the former case, very accurate electromagnetic (EM) characterisation is achievable but requires the precise fabrication of both the measured sample and the designed resonator [29]. In addition, these methodologies are often customised for a reduced set of materials (range of measured dielectric permittivity and magnetic permeability, size and physical state of the sample). Unfortunately, this approach can be impractical and expensive for research laboratories, where, usually, the characterisation of materials is not the focus of the working activity but a preliminary step towards the design of an electronic or electromagnetic device. In the latter case, commercial T/R devices, such as coaxial cables, waveguides or even free space propagation, can be employed to perform the characterisation of materials. Unlike the methods, which are based on single-frequency or multiple-frequencies measurements, the T/R characterisation is intrinsically wideband. The frequency band in which the characterisation is valid depends on the specific T/R device and on the employed equipment. In both cases of coaxial cables and waveguides, the material is characterised in the unimodal propagation band of the device. In addition, coaxial cables offer a wider bandwidth and allow one performing (at least in theory) the measurement at low frequency since they do not have a cut-off band. However, since this measurement technique requires that the shape of the material be adapted to the concentric corona of the cable, the manufacturing of the sample is not straightforward. On the contrary, the preparation of the sample for waveguides is much simpler, thus justifying the success of waveguides over the coaxial cables even at the cost of a smaller allowed bandwidth. The purpose of this paper is to present an overview of T/R-based methods for the characterisation of a large set of materials by employing standard laboratory measurement equipment. Electric and magnetic polarisations effects of materials are macroscopically described by two well-known parameters, namely the dielectric permittivity and the magnetic permeability. A summary of the methods presented in the paper is displayed in Figure1. In the second section, the most well-known T/R characterisation procedure for retrieving dielectric permittivity and magnetic permeability of materials, the Nicholson–Ross–Weir [31,32] method (NRW), is presented. The method is easy to use as it allows for retrieving the dielectric permittivity and magnetic permeability from the measured scattering parameters of the unknown sample through a simple analytic procedure. However, the NRW approach presents a number of limitations that have been extensively discussed in the literature [33–35], and a number of alternative procedures have been proposed throughout the years [35–39]. The most relevant limitations of the method and some of the most remarkable approaches aimed at overcoming the NRW drawbacks will be presented, with examples. Specifically, particular attention will be dedicated to the possible ambiguities in the determination of the dielectric permittivity and magnetic permeability, which arise in the direct inversion procedure typical of the NRW algorithm [40–42]. It will be shown that these limitations can be circumvented by treating the inversion problem as a global optimisation procedure, where only solutions obeying to causal models of materials (Lorentz or Debye models) are searched by using an iterative procedure. This approach will be called Baker–Jarvis (BJ) method [36]. Another useful technique to characterize the dielectric properties of thin low-dielectric permittivity materials based on resonant FSS filters in waveguide environment will also be discussed [43]. The third section is dedicated to the characterisation of thin two-dimensional materials with non-negligible conductivity. Moreover, the cases where the sheet impedance approximation is valid will be discussed. If the sheet impedance condition is verified, the 2D material can be characterised in terms of complex sheet impedance instead of using bulk parameters Electronics 2017, 6, 95 3 of 27 Electronics 2017, 6, 95 3 of 28 such as the complex dielectric permittivity. Afterwards, a characterisation approach employing the permittivity. Afterwards, a characterisation approach employing the same waveguide experimental same waveguide experimental setup [44–46] to derive the surface impedance of the sample is presented. setup [44–46] to derive the surface impedance of the sample is presented.

Figure 1. ClassiﬁcationClassification of the T/R methods methods fo forr EM EM characterisation characterisation of of materials. materials.

2. Measuring Measuring Properties of Dielectric Materials The electromagnetic characterisati characterisationon of dielectric materials is a relevant aspect in numerous practical applications.applications. Over Over the the years, years, several several experimental experimental techniques techniques aimed ataimed extracting at extracting the dielectric the dielectricproperties ofproperties a materials of havea materials been investigated have been [29, 30investigated]. The choice of[29,30]. the EM The characterisation choice of the method EM characterisationdepends on the EMmethod parameters depends such on as the the EM frequency parameters band andsuch the as required the frequency accuracy band of the and results the requiredand, more accuracy remarkably, of the also results on the physicaland, more and remarkably, mechanical parametersalso on the of thephysical investigated and mechanical specimen parameterssuch as its physical of the investigated state (solid, specimen liquid), its such size andas its shape. physical In addition, state (solid, a priori liquid), information its size and regarding shape. Inthe addition, nature of thea priori analysed information material isregarding needed. Forthe example,nature of resonant the analysed methods material provide is reliable needed. results For example,for dielectric resonant materials methods with low provide losses. reliable These methods, results for which dielectric employ materials resonant structureswith low suchlosses. as These cavity methods,resonators, which rely on employ the measure resonant of the structures Q-factor such and the as resonancecavity resonators, frequency rely of theon cavity.the measure Alternatively, of the Q-factornon-resonant and methods,the resonance which frequency employ transmission of the cavi lines,ty. Alternatively, rely on the measure non-resonant of the signal methods, reflected which and employtransmitted transmission by the specimen lines, underrely on test. the While measur resonante of the methods signal are reflected able to accuratelyand transmitted characterise by the specimenmaterial over under a discrete test. While number resonant of frequencies, methods are non-resonant able to accurately methods characterise inherently providethe material broadband over a discreteresults. Innumber addition, of non-resonantfrequencies, non-resonant methods can methods be applied inherently to different provide measuring broadband setups. results. Although In addition,coaxial line non-resonant probes require methods a very can accurate be applied shaping to different of the sample, measuring they setups. can measure Although EM coaxial properties line probesof the materialrequire overa very a veryaccurate wide shaping band and, of inthe theory, sample, at lowthey frequencies. can measure Waveguide EM properties methods of the are materialusually moreover populara very wide since band circular and, orin rectangulartheory, at low samples frequencies. are easier Waveguide to produce methods than coaxial are usually ones. moreHowever, popular waveguide-based since circular estimations or rectangular are only samp validles overare aeasier limited to frequencyproduce range.than coaxial On the ones. other However,hand, free-space waveguide-based techniques estimations not only circumvent are only vali thed problemover a limited of the frequency sample fit range. precision On the but other also hand,preserve free-space the integrity techniques of the sample not only (non-destructive). circumvent the A problem major disadvantage of the sample of the fit free-space precision method but also is preservethat, in order the tointegrity avoid diffraction of the sample effects due(non-destructive). to sample edges, A the major wireless disadvantage measurement of ofthe the free-space scattering methodparameters is that, through in order antennas to avoid requires diffraction a sample effects of several due to wavelengths. sample edges, the wireless measurement of the scattering parameters through antennas requires a sample of several wavelengths. 2.1. Nicolson–Ross–Weir Procedure 2.1. Nicolson–Ross–Weir Procedure One of the most popular procedures for the EM characterisation of materials is the Nicolson– Ross–WeirOne of (NRW) the most method, popular based procedures on T/R measurements for the EM ch ofaracterisation the specimen of under materials test. Theis the properties Nicolson– of Ross–Weirmaterials are (NRW) retrieved method, from theirbased impedance on T/R measurements and the wave of velocities the specimen in the under materials. test. It The is well properties known ofthat materials the main are drawback retrieved of thisfrom method their impedance is related to and the electricalthe wave thickness velocities of in the the analysed materials. material It is well [40]. knownIn particular, that the when main the drawback thickness of ofthis the method specimen is related is an integer to the multipleelectrical ofthickness half wavelength of the analysed in the materialmaterial, the[40]. method In particular, produces when ambiguous the thickness results due of tothe the specimen 2π periodicity is an of theinteger phase multiple of the EM of wave. half wavelength in the material, the method produces ambiguous results due to the 2π periodicity of the phase of the EM wave. In principle, from a priori information regarding the specimen, it is possible

Electronics 2017, 6, 95 4 of 27 Electronics 2017, 6, 95 4 of 28

Into principle, achieve unambiguous from a priori information results. In addition, regarding as the wi specimen,ll be presented it is possible in this to section, achieve because unambiguous of the results.inversion In addition,procedure as adopted will be presented for the extraction in this section, of the because EM parameters, of the inversion the NRW procedure method adopted is not foraccurate the extraction in the case of the of EMlow parameters,reflecting materials the NRW or method thin samples. is not accurate Two examples in the case of transmission of low reﬂecting lines materialsfor the material or thin characterisation samples. Two examples are reported of transmission in Figure 2. linesThe sample for the materialunder test characterisation with thickness ared is reportedlocated inside in Figure a two-port2. The sample microwave under testdevice with such thickness as a coaxiald is located cable inside (Figure a two-port 2a) or a microwavewaveguide device(Figure such 2b). as a coaxial cable (Figure2a) or a waveguide (Figure2b).

(a) (b)

Figure 2. Transmitting/reflecting devices with the material sample to be characterised: (a) coaxial Figure 2. Transmitting/reﬂecting devices with the material sample to be characterised: (a) coaxial cable; (b) waveguide. cable; (b) waveguide.

Due to the presence of a sample of thickness d, the phase factor (T) due to phase delay, (T) can d T T be definedDue to theas: presence of a sample of thickness , the phase factor ( ) due to phase delay, ( ) can be deﬁned as: −−j kd γ d TTe===e−j k d = e− eγ d.. (1)

TheThe reflection reflection coefficient coefficient at at the the first first interface interface between between the the transmission transmission line’s line’s sections sections with with and and withoutwithout the the specimen specimen is is defined defined as: as: Z − Z Γ = 1 − 0 , (2) ZZ10+ ZZ Γ= 1 0 , (2) Z + Z where Z0 and Z1 are the transmission line’s characteristic10 impedances in absence and in the presence of the sample, respectively. The expression of Z can be straightforwardly derived from Equation (2) where Z0 and Z1 are the transmission line’s characteristic1 impedances in absence and in the presence as follows: of the sample, respectively. The expression of Z1 can1 + beΓ straightforwardly derived from Equation (2) Z = Z . (3) as follows: 1 0 1 − Γ +Γ According to the theory of multiple reflections= [311 ], the scattering parameters of the finite length ZZ10 . (3) sample (S11 and S21) can be written as a function1 of−Γ the reflection coefficient at the interface of two infinite media Γ and as a function of the phase factor T: According to the theory of multiple reflections [31], the scattering parameters of the finite length sample (S11 and S21) can be written Γas(1 a− functionT2) of the(1 reflection− Γ2)T coefficient at the interface of S11 = ΓIN = , S21 = . (4) two infinite media Γ and as a function of the1 − phaseΓ2T2 factor T: 1 − Γ2T2 22 At this point, the reflection coefficient ΓΓ−, which(1TT contains ) the (1 −Γ impedance ) of the unknown medium, SS=Γ =, = . (4) and the phase factor T, which contains11 IN the11 propagation−Γ22TT 21 constant −Γ 22of the unknown medium, can be expressed as a function of the scattering parameters of the sample: At this point, the reflection coefficient Γ, which contains the impedance of the unknown ( √ medium, and the phase factor T, which containsΓ = K ± theK propagation2 − 1 constant of the unknown medium, . (5) can be expressed as a function of the scatteringT = Sparameters11+S21−Γ of the sample: 1−(S11+S21)Γ Γ=KK ±2 −1   SS+−Γ. (5) T = 11 21  −+Γ  1(SS11 21 )

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where the auxiliary parameter K is a function of the scattering parameters S11 and S21:

S2 − S2 + 1 K = 11 21 . (6) 2S11

As it is apparent in Equation (5), the reflection coefficient Γ is characterised by a sign ambiguity. The sign should be properly chosen to obtain the correct impedance and propagation constant of the medium [47,48] (passivity condition), which can be verified at the end of the inversion procedure. However, if one needs to estimate only the dielectric permittivity and the magnetic permeability of the medium, both signs can be arbitrarily chosen. Indeed, since the dielectric permittivity and the magnetic permeability are a product of the impedance of the medium and its propagation constant, the correct sign is always restored [40]. The phase factor T of relation (1) can be expressed with its amplitude |T| and a phase term φ as follows: T = e−γ d = |T|e−jφ. (7)

Consequently, the propagation constant γ can be obtained by applying the natural logarithm of the complex variable T. The natural logarithm of the complex variable T will be complex. Its real part can be expressed as the logarithm of its amplitude and the imaginary part will be the phase term φ plus a 2π phase ambiguity:

1 γ = jk = [− ln(|T|) − jφ + j2πn ], n = ... − 2, −1, 0, 1, 2 . . . (8) d It is evident from Equation (8) that there is an ambiguity in the calculation of γ because of the term 2πn. The existence of an inﬁnite number of valid solutions is called branching. Therefore, in order to solve this ambiguity, the NRW method requires a thin specimen material (usually a quarter of a wavelength) so that n = 0 at the analysed frequencies. Once γ is calculated, it is possible to estimate the electrical permittivity and the magnetic permeability:  γ  ε∗ = 1−Γ γ0 1+Γ = γ with n 0; (9)  µ∗ = 1+Γ γ0 1−Γ In a waveguide environment [32], the magnetic permeability can be similarly calculated by using the equivalent transmission line circuit of the waveguide: q q TE 2 2 2 2 γ Z k0εrµr − kt ωµ µ k0 − kt ∗ = = 0 r = µ TE q q µr, (10) γ0 Z 2 2 2 2 ωµ0 0 k0 − kt k0εrµr − kt where k0 is the propagation constant in free space and kt is the cutoff wavenumber of the waveguide. On the contrary, the dielectric permittivity should be derived from the propagation constant q 2 (k = k0εrµr − kt ) after the derivation of the magnetic permeability:

k2 + k2 = t εr 2 ∗ . (11) k0µ

An example of an application of the NRW procedure for the characterisation of a dielectric material is reported hereinafter. A material sample with thickness equal to 8 mm and εr = 9 − j0, µr = 1 − j0 is considered. The transmission and reﬂection coefﬁcient simulated with a transmission line model are reported in Figure3. Electronics 2017, 6, 95 6 of 27 Electronics 2017, 6, 95 6 of 28

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λ g 2 resonance λ g 2 resonance

(a) (b)

Figure 3. Transmission and reflection coefficient: (a) amplitude; (b) phase of a rectangular sample of Figure 3. Transmission(a and) reﬂection coefﬁcient: (a) amplitude; (b) phase of(b a) rectangular sample of 8 mm thickness with εr = 9 − j0, µr = 1 − j0. 8 mmFigure thickness 3. Transmission with εr = and 9 − reflectionj0, µr = 1 coefficient:− j0. (a) amplitude; (b) phase of a rectangular sample of

8 mm thickness with εr = 9 − j0, µr = 1 − j0. The application of the NRW with n = 0 provides the results reported in Figure 4a,b. As is evident fromThe applicationthe graphs ofdisplayed the NRW in with Figuren = 04, provides when th thee thickness results reported of the insample Figure is4 a,b.equal As isto evidenthalf a The application of the NRW with n = 0 provides the results reported in Figure 4a,b. As is evident fromwavelength the graphs in displayed the dielectric in Figure media4, when (λg/2) the the thickness NRW method of the produces sample is ambiguous equal to half results. a wavelength In this from the graphs displayed in Figure 4, when the thickness of the sample is equal to half a in theparticular dielectric case, media the thickness (λg/2) the of NRWthe sample method is equal produces to λg/2 ambiguous at 6.4 GHz: results. In this particular case, wavelength in the dielectric media (λg/2) the NRW method produces ambiguous results. In this the thickness of the sample is equal to λg/2 at 6.4 GHz: π particular case, the thickness ofλ the() sample== is equal to λg/22 at 6.4 GHz: = g f 6.4GHz 16mm . (12) ε2πkk22− λg(()ff===6.46.4GHz GHz) = q rt0 = 16mm16 mm. (12) g 2 2 . (12) εεrk22− k rtkk00 t

λ g λ ambiguity g 2 2 ambiguity λ g λ 2 ambiguity g ambiguity n=0 n=0 2 n=0 n=0

(a) (b)

n=1 n=1 n=1 n=1

Figure 4. Estimated( celectric) permittivity and magnetic permeability for a dielectric(d) sample of 8 mm with εr = 9 − j0, µr = 1 − j0 in the case of n = 0 (a,b) and n = 1 (c,d). Figure 4. Estimated electric permittivity and magnetic permeability for a dielectric sample of 8 mm Figure 4. Estimated electric permittivity and magnetic permeability for a dielectric sample of 8 mm with εr = 9 − j0, µr = 1 − j0 in the case of n = 0 (a,b) and n = 1 (c,d). withAsεr =reported 9 − j0, µ rin= Figure 1 − j0 in 4, the the case results of n = obtained 0 (a,b) and withn = 1n (=c, d0). are valid up to 6.4 GHz, where the estimated parameters are constant. Similarly, with n = 1, the results are valid starting from 6.4 GHz. As reported in Figure 4, the results obtained with n = 0 are valid up to 6.4 GHz, where the Therefore, it is evident that with this formulation of the NRW method the user needs to choose the estimatedAs reported parameters in Figure are4 ,constant. the results Similarly, obtained with with n =n 1,= 0the are results valid upare to valid 6.4 GHz,starting where from the 6.4 estimated GHz. valid results depending on the different values of n. In order to overcome the limitations of the parametersTherefore, areit is constant. evident that Similarly, with this with formulation = 1, then resultsof the NRW are valid method starting the user from needs 6.4 GHz. to choose Therefore, the it isvalid evident results that depending with this formulation on the different of the values NRW of method n. In order the user to overcome needs to choosethe limitations the valid of resultsthe

Electronics 2017, 6, 95 7 of 27 Electronics 2017, 6, 95 7 of 28 dependingbranch point on theselection, different a valuesstepwise of nscheme. In order of the to overcomeNRW method the limitations has been ofproposed the branch [38]. point The selection,procedure a stepwiseis based schemeon the fact of the that NRW the measured method has sample been proposedis thinner [than38]. Thehalf-guided procedure wavelength is based on at thethe fact lowest that frequency. the measured sample is thinner than half-guided wavelength at the lowest frequency. Therefore,Therefore, the then n= = 0 0 branch branch is is selected selected at at the the lowest lowest measured measured frequency: frequency: 11 γφγ=Τ−+= [ln(()1/ 1/|T|) − jjφ + 2π2πn]nwith withn = nff0 at==0f at= fmin; ; (13)(13) ddmin The argument of the propagation constant for the subsequent frequency point p is selected using The argument of the propagation constant for the subsequent frequency point p is selected a step-by-step increment, which avoids the selection of the branch value n: using a step-by-step increment, which avoids the selection of the branch value n: n jk d ! T(Tpp)() n e Mjk,pMp, d φ(φφp)()= φ=−+(p − (1) + )arg = φ =( φ0)()+ ∑ +arg e . (14) pp1argT(p − 1) 0 argjkM,jkp−1d d . (14) ()− p=1 eMp,1− Tp1 p=1 e

TheThe real real and and imaginary imaginary parts parts of theof the propagation propagation constant constant characterised characterised with with the NRW the NRW (n = 0) (n and = 0) stepwiseand stepwise method method are reported are reported in Figure in Figure5a,b, respectively. 5a,b, respectively.

Stepwise method NR, n=0

(a) (b)

Figure 5. Propagation constant of the medium with NRW with n = 0 (a) and with the stepwise Figure 5. Propagation constant of the medium with NRW with n = 0 (a) and with the stepwise method (b). method (b).

If the sample is not thinner than a half-guided wavelength at the lowest frequency, unphysical solutionsIf the sampleare also is obtained not thinner using than the a half-guided stepwise method. wavelength This atcase the can lowest be analysed frequency, considering unphysical a solutions are also obtained using the stepwise method. This case can be analysed considering a sample sample with thickness d = 8 mm and εr = 30 − j0.2, µr = 1 − j0. The scattering parameters S11 and S21 and d − − S S withthe real thickness and imaginary= 8 mm part and εofr =the 30 dielectricj0.2, µr perm= 1 ittivityj0. The estimated scattering with parameters the stepwise11 and method21 and when the n n real= 0 andand imaginary n = 1 are partreported of the in dielectric Figure 6a,b, permittivity respectively. estimated As is with evident the stepwise from Figure method 6b, when unphysical= 0 n andvalues= 1of are the reported dielectric in permittivity Figure6a,b, respectively.are provided As by isthe evident stepwise from method Figure when6b, unphysical n = 0. values of the dielectric permittivity are provided by the stepwise method when n = 0. The NRW direct inversion procedure is sensitive to the inaccuracies of the S-parameters at 0 the N*λg/2 resonances, with N entire number.S Indeed, at that frequency, the sample becomes 11 S a half-wavelength window and the reflection21 coefficient tends to zero. Consequently, it is not possible -5 to apply the direct inversion procedure. Moreover, in the case of noisy measurements, this method will induce a high uncertainty in the extracted parameters of the material and may cause problems in -10 finding (dB) the correct branch in the vicinity of these frequencies [49]. 21

As, s an example, a material characterised by a constant dielectric permittivity (ε = 9 − j0.2) 11 -15 r and as thickness of 9 mm simulated in a waveguide environment (WR137 waveguide) is analysed.

The synthetic-20 scattering parameters are perturbed with Additive White Gaussian Noise (AWGN) characterised by a mean value (mv) equal to zero and standard deviation (σ) equal to 0.01. As is shown in Figure-25 7, the application of the NRW formulas leads to high inaccuracies around 6.4 GHz where 4.5 5 5.5 6 6.5 7 7.5 8 8.5 the reﬂection coefﬁcientFrequenc of they (GHz samples) is very low. The estimation error depends on the accuracy of the available scattering(a parameters.) In [50,51], the Cramér–Rao lower bounds,(b) that is, the minim

Figure 6. (a) Scattering parameters for a sample of εr = 30 − j0.2, µr = 1 − j0 with d = 8 mm; (b) estimated dielectric permittivity for n = 0 and n = 1.

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branch point selection, a stepwise scheme of the NRW method has been proposed [38]. The procedure is based on the fact that the measured sample is thinner than half-guided wavelength at the lowest frequency. Therefore, the n = 0 branch is selected at the lowest measured frequency: 1 γφ=Τ−+ln() 1/j 2π n with nff==0 at ; (13) d min The argument of the propagation constant for the subsequent frequency point p is selected using a step-by-step increment, which avoids the selection of the branch value n:

Tp() n e jkMp, d φφ()pp=−+ (1arg )  = φ() 0 + arg. (14) ()−  jkMp,1− d Tp1 p=1 e

The real and imaginary parts of the propagation constant characterised with the NRW (n = 0) and stepwise method are reported in Figure 5a,b, respectively.

Stepwise method NR, n=0

(a) (b)

Figure 5. Propagation constant of the medium with NRW with n = 0 (a) and with the stepwise method (b).

ElectronicsIf2017 the, 6sample, 95 is not thinner than a half-guided wavelength at the lowest frequency, unphysical8 of 27 solutions are also obtained using the stepwise method. This case can be analysed considering a sample with thickness d = 8 mm and εr = 30 − j0.2, µr = 1 − j0. The scattering parameters S11 and S21 and variance achievable with a ﬁxed SNR of the measured scattering parameters, are derived. Using the real and imaginary part of the dielectric permittivity estimated with the stepwise method when n these bounds, it is possible to estimate the minimum Signal to Noise Ratio (SNR) required to estimate = 0 and n = 1 are reported in Figure 6a,b, respectively. As is evident from Figure 6b, unphysical thevalues material of the parameters dielectric with permittivity a given ar accuracy.e provided by the stepwise method when n = 0. Electronics 2017, 6, 95 8 of 28

The0 NRW direct inversion procedure is sensitive to the inaccuracies of the S-parameters at the S 11 N*λg/2 resonances, with N entire number.S Indeed, at that frequency, the sample becomes a 21 half-wavelength-5 window and the reflection coefficient tends to zero. Consequently, it is not possible to apply the direct inversion procedure. Moreover, in the case of noisy measurements, this method -10

will (dB) induce a high uncertainty in the extracted parameters of the material and may cause problems 21

in finding, s the correct branch in the vicinity of these frequencies [49].

11 -15 s As an example, a material characterised by a constant dielectric permittivity (εr = 9 − j0.2) and a thickness of 9 mm simulated in a waveguide environment (WR137 waveguide) is analysed. The -20 synthetic scattering parameters are perturbed with Additive White Gaussian Noise (AWGN) characterised by a mean value (mv) equal to zero and standard deviation (σ) equal to 0.01. As is -25 shown4.5 in Figure 5 5.5 7, the 6 application 6.5 7 7.5of the 8 NRW 8.5 formulas leads to high inaccuracies around 6.4 GHz Frequency (GHz) where the reflection coefficient of the samples is very low. The estimation error depends on the (a) (b) accuracy of the available scattering parameters. In [50,51], the Cramér–Rao lower bounds, that is, the minimFigure variance 6. (a) achievableScattering paramete with a fixedrs for SNRa sample of the of εmeasuredr = 30 − j0.2, scattering µr = 1 − j0parameters, with d = 8 aremm; derived. (b) Figure 6. (a) Scattering parameters for a sample of εr = 30 − j0.2, µr = 1 − j0 with d = 8 mm; (b) estimated estimated dielectric permittivity for n = 0 and n = 1. Usingdielectric these permittivity bounds, it foris possiblen = 0 and ton estimate= 1. the minimum Signal to Noise Ratio (SNR) required to estimate the material parameters with a given accuracy. (dB) (deg) 21 21 , s , S 11 11 s S

(a) (b) r

Figure 7. Scattering parameters for a sample of εr = 9 − j0.2, µr = 1 − j0 with d = 8 mm perturbed by Figure 7. Scattering parameters for a sample of εr = 9 − j0.2, µr = 1 − j0 with d = 8 mm perturbed by Additive White Gaussian Noise (AWGN) with mean value mυ = 0 and standard deviation σ = 0.01 Additive White Gaussian Noise (AWGN) with mean value mυ = 0 and standard deviation σ = 0.01 (a,b); estimated dielectric permittivity and magnetic permeability for n = 0 for the first frequency (a,b); estimated dielectric permittivity and magnetic permeability for n = 0 for the ﬁrst frequency point point and stepwise algorithm (c,d). and stepwise algorithm (c,d). Finally, it is worth mentioning that several variations of the original formulation of the NRW method have been proposed in the literature. Several approaches to improve the stability of the results with “noisy” or not enough accurate data when the sample is thin, or the material exhibits a low reflectivity have been presented [52–54]. Moreover, formulations able to make the characterisation of the material independent of the calibration planes have been also studied [54,55].

Electronics 2017, 6, 95 9 of 27

Finally, it is worth mentioning that several variations of the original formulation of the NRW method have been proposed in the literature. Several approaches to improve the stability of the results with “noisy” or not enough accurate data when the sample is thin, or the material exhibits a low reflectivity have been presented [52–54]. Moreover, formulations able to make the characterisation of the material independent of the calibration planes have been also studied [54,55]. Finally, some techniques based only on measured reflection or transmission amplitude have been also investigated [56,57].

2.2. Iterative Non-Ambiguous Estimation of Dielectric Permittivity from Broadband Transmission/Reflection Measurements The extraction of the dielectric constants from T/R measurements can be ambiguous when the S-parameter inversion is performed frequency-by-frequency. The reason is the impossibility of determining the correct branch of the solution with any initial guess of the material properties. High values of permittivity or permeability might lead to a sample with an electrical thickness of several multiples of half-wavelengths, even at the lowest measurement frequency. Consequently, the choice of the ambiguity integer n in (13) is not straightforward. This is particularly true when the NRW inversion algorithm is applied to metamaterials where the permittivity and permeability tensor values are dispersive with frequency and they can rapidly increase in certain frequency bands around the resonances [42,47,49,58–60]. One of the main problems of the NRW direct inversion procedure is that it admits unphysical non-causal solutions [41,49]. An alternative approach that exploits the correlation among contiguous frequency points, thus circumventing the ambiguity and discontinuity problems that plague the NRW method, was presented by Baker Jarvis in 1992 [36]. The best estimation of the dielectric permittivity can be obtained with an iterative fitting of an objective function, which minimises the quadratic errors on the magnitude and phase of S-parameters among the predicted and measured data. The method does not require any initial guess on the permittivity and may search the best estimation on the whole complex permittivity domain. Moreover, the algorithm can be successfully employed both with T/R or R measurements [61]. The idea of the iterative method is to search only physical solutions analysing the scattering parameter over all the available frequency points at the same time. In this way, it is possible to exploit the correlation among frequency points, which is instead neglected by the NRW method. The scattering parameters of a sample transversally placed in a T/R device can be analytically retrieved by using a simple transmission line model [44,62,63]. Therefore, the unknown permittivity and permeability profiles can be estimated by computing the discrepancy between the measured scattering parameters and the analytically estimated ones based on physical permittivity models. The discrepancy can be estimated by using the following cost function: ˆ 2 ˆ 2 FC(ε) = ∑ S11,M( fn) − S11( fn) + ∑ S21,M( fn) − S21( fn) , (15) n n where ε = [εR, ε I ], Sxy,M are the measured scattering parameters at the frequencies of interest fn and Sˆxy are the S parameters estimated according to the transmission line model. The cost function FC, which is reported in terms of real and imaginary part in the Formula (16) equation reference goes here, can also be expressed in terms of amplitude and phase, as follows:

ˆ 1 ˆ FC = ∑ |S11,M( fn)| − S11( fn) + 2π ∑ ]S11,M( fn) − ]S11,M( fn) + n n ˆ 1 ˆ . (16) ∑ |S21,M( fn)| − S21( fn) + 2π ∑ ]S21,M( fn) − ]S21,M( fn) n n

In the simplest case of a dielectric sample with constant permittivity parameters, only the real and the imaginary part of the dielectric constant needs to be estimated and a two-dimensional space is analysed. However, in the presence of noise, the cost function is not convex and exhibits local minima. Consequently, a single starting point is not sufﬁcient for the analysis, especially when the number of unknown parameters increases. Unfortunately, without an initial guess of the sample, Electronics 2017, 6, 95 10 of 27 a very large range of possible values of the unknown parameters needs to be explored. For this reason, a certain number of initial values have to be chosen initially. It is obvious that, in order to explore the two-dimensional space with a reasonable computation time, a coarse sampling is recommended. Let us select a set of initial points ε(0i) and apply an iterative line-search gradient algorithm for every initial point in order to exhaustively explore the surrounding space. Given a starting point ε(0), the algorithm reﬁnes the permittivity values by moving on the steepest descent path:

εm = εm−1 − αm∇FC(εm−1), (17) where the subscript m individuates the convergence step and:

1 αm = . (18) |∇FC(εm−1)|

If the new computed cost function respects the condition FC(εm) ≤ FC(εm−1), the next value of the complex dielectric permittivity is retained. Otherwise, the multiplying coefﬁcient is halved and the iteration is repeated as follows:

αm F (ε ) > F (ε − ) → α = . (19) C m C m 1 m 2 The algorithm converges when the following condition is fulﬁlled:

−6 |εm − εm−1| < 10 . (20)

The algorithm for the simple 2D space is very fast since the formulation is based on an analytic solver. In case of good estimation, the cost function will decrease towards zero. Moreover, for a further speed-up of the algorithm, the measured data can be under-sampled. In addition, the estimation of the magnetic permeability can be easily included in the method by enlarging the initial explored space up to four dimensions. The presented iterative procedure will be applied to different dielectric materials hereinafter. The first numerical example is a specimen of 0.1 m with a relative dielectric permittivity εR = 20 − j10. The sample is considered transversally accommodated in coaxial cable and measured in the band 100–500 MHz over 50 frequency samples. In this preliminary analysis, the target scattering parameters are synthetically generated by using the equivalent transmission line model. The complex space of the dielectric permittivity is sampled over nine points uniformly distributed between 0 and 50 for the real part and over 0 and 250 for the imaginary part. The initial and final positions provided by the iterative algorithm for each starting point are shown in Figure8a. As is evident from Figure8a, the algorithm converges to the same value for every starting point. If the same approach is applied to a sample of 0.2 m characterised by a relative dielectric permittivity εR = 6.15 − j0.0184 in the frequency range 100–500 MHz, not all the starting points of the complex space converge to the correct solution (Figure8b). Finally, it is worth underlining that the proposed method does not suffer the ambiguities of phase extraction from transmission coefficient that affect the classical NRW algorithm. In fact, if the NRW inversion procedure is applied to the synthetic data of the second example, divergent solutions are obtained when the sample length is equal to or longer than a half wavelength. On the contrary, the proposed method is stable on the whole bandwidth, thus proving its broadband applicability. It is worth highlighting that, if general and realistic dielectric permittivity and magnetic permeability models are selected, the number of parameters to be found increases with respect to the case of constant values. On the contrary, by recurring to causal models for the sample under testing, the dielectric permittivity and the magnetic permeability of a material can be characterised using only a few parameters. The most important models, widely adopted to describe the dielectric behaviour of materials, are provided by Lorentz and Debye. In this case, the iterative search consists of optimising the parameters of the causal models by matching the measured scattering parameters with Electronics 2017, 6, 95 11 of 27 those obtained from the simulations of the material. We will address this method as Backer–Jarvis (BJ) Electronics 2017, 6, 95 11 of 28 since a similar approach was presented by the author [36].

(a) (b)

Figure 8. Representation of the convergence of the algorithm on the complex space of the dielectric Figure 8. Representation of the convergence of the algorithm on the complex space of the dielectric permittivity. The searched values are (a) εR = 20 − j10 and (b).εR = 6.15 − j0.0184. permittivity. The searched values are (a) εR = 20 − j10 and (b) εR = 6.15 − j0.0184. It is worth highlighting that, if general and realistic dielectric permittivity and magnetic The definition of the employed causals models is provided hereinafter. According to the Lorentz permeability models are selected, the number of parameters to be found increases with respect to the model, the electrons are considered as damped harmonically bound particles subject to the external case of constant values. On the contrary, by recurring to causal models for the sample under testing, field. Lorentz’s model relies on the fact that the mass of the nucleus of the atom is much higher than the dielectric permittivity and the magnetic permeability of a material can be characterised using the mass of the electron. Consequently, the problem can be treated as an infinite mass connected to an only a few parameters. The most important models, widely adopted to describe the dielectric electron-spring system. In addition, the hypothesis that the biding force can be modelled as a spring, behaviour of materials, are provided by Lorentz and Debye. In this case, the iterative search consists can be considered valid for any kind of binding because of the small displacement. The permittivity of optimising the parameters of the causal models by matching the measured scattering parameters provided by Lorentz’s model can be written as follows: with those obtained from the simulations of the material. We will address this method as Backer–

Jarvis (BJ) since a similar approach was presentedεs − byε∞ the author [36].σ ε( f ) = ε∞ + − j . (21) The definition of the employed causals ∆modelsf isf provided2 2π f εhereinafter.0 According to the 1 + j 2 f − f Lorentz model, the electrons are considered as fda0 mped harmonically0 bound particles subject to the external field. Lorentz’s model relies on the fact that the mass of the nucleus of the atom is much higherIn thisthan formula,the massε sof andthe electron.ε∞ are the Consequent static dielectricly, the constantproblem andcan be the treated optic region as an infinite permittivity, mass respectively.connected to The an parameterelectron-springf 0 is the system. resonance In addition frequency,, the∆ fhypothesisis the width that of thethe Lorentzian biding force resonance can be 2 linemodelled at the –3as dBa level,spring, and can the be term considered∆ f / f0 is thevalid relaxation for any frequencykind of binding parameter. because of the small displacement.The Debye The model permittivity describes provided the properties by Lorentz’s of polar model dielectric can be materials. written as It canfollows: be interpreted as a special case of the Lorentz model when the displacement vector is replaced by the dipole rotation εε− σ s ∞ angle θ (i.e., the angle betweenεε the()fj dipoles=+∞ and the excitation E −field). . 2 πε The rate at which polarisation can occur isΔff limited: as the2 frequencyf 0 of the external EM field(21) 1+−jf 2  increases, the dipole moments are not able to orientf0 themselvesf0 fast enough to maintain alignment with the excitation and the total polarizability decreases. This fall, with its related reduction of permittivity and theIn this occurrence formula, of energy and absorption, are the isstatic referred dielectric to as dielectricconstant and relaxations the optic or region dispersions. permittivity, respectively.The relaxation The parameter mechanism f0 is in the dipolar resonance dielectric frequency, materials Δf is is the described width of by the the Lorentzian Debye model: resonance line at the –3 dB level, and the term ∆⁄ is the relaxation frequency parameter. The Debye model describes the properties εofs − polarε∞ dielectricσ materials. It can be interpreted as a ε( f ) = ε∞ + − j . (22) special case of the Lorentz model when the displace1 + j2π mentf τ vector2π f ε0 is replaced by the dipole rotation angle (i.e., the angle between the dipoles and the excitation E field). The real and imaginary part as a function of the frequency of the electric permittivity in the case of The rate at which polarisation can occur is limited: as the frequency of the external EM field the Debye (a) and Lorentz (b) models are reported in Figure9a,b, respectively. It is worth underling that, increases, the dipole moments are not able to orient themselves fast enough to maintain alignment in the iterative process, five parameters need to be estimated for Lorentz’s model. These parameters with the excitation and the total polarizability decreases. This fall, with its related reduction of are highlighted in Figure9b. Similarly, the estimation of four parameters is required for Debye’s model, permittivity and the occurrence of energy absorption, is referred to as dielectric relaxations or as reported in Figure9a. dispersions. The relaxation mechanism in dipolar dielectric materials is described by the Debye model:

Electronics 2017, 6, 95 12 of 28

εε− σ εε()fj=+s ∞ − ∞ + πτ πε. (22) 12jf 2 f0

The real and imaginary part as a function of the frequency of the electric permittivity in the case of the Debye (a) and Lorentz (b) models are reported in Figure 9a,b, respectively. It is worth underling that, in the iterative process, five parameters need to be estimated for Lorentz’s model. ElectronicsThese2017 ,parameters6, 95 are highlighted in Figure 9b. Similarly, the estimation of four parameters12 of 27 is required for Debye’s model, as reported in Figure 9a.

ε 1 Δf s f = trans πτ ε 2 ε s ∞ ε∞ 15 f0 Re( ) r Im( ) 10 r

0 r

-5 σ -10 − j σ 2πεf − j 0 πε -15 2 f 0

-20 0 5 10 15 20

(a)(b)

Figure 9. Real and imaginary parts as a function of the frequency of the electric permittivity in the Figure 9. Real and imaginary parts as a function of the frequency of the electric permittivity in the case case of the Debye (a) and Lorentz (b) models. of the Debye (a) and Lorentz (b) models.

Real materials might exhibit several resonances, which can be characterised as a combination of Realmultiple materials Lorentz might and exhibit Debye several terms as resonances, follows: which can be characterised as a combination of multiple Lorentz and Debye terms as follows: NMεε−− εε sn,,∞∞ sn σ εε()fj=+∞ + − . N − M2 +− πτ πε nm==11εs,n Δεf∞ εs,12n jfε∞ n σ 2 f0 ε( f ) = ε∞ + +−n +f − j . (23) (23) ∑ 1 jf 2 ∑ 1 + j2π f τ 2π f ε n=1 ∆ fn f 2 f f m=1 n 0 1 + j 2 f −0,n f 0,n f0,n 0,n As is evident from Equation (23), if a material can be modelled with N Lorentz resonances and As is evident from Equation (23), if a material can be modelled with N Lorentz resonances and M M Debye resonances, the total number of parameters (Tp) to estimate is equal to: Debye resonances, the total number of parameters (Tp) to estimate is equal to: =+⋅+⋅ TNMp 23 2 . (24) Tp = 2 + 3 · N + 2 · M. (24) In order to improve the convergence of the algorithm, since the dimension of the solutions Inspace order increases to improve with the convergencethe complexity of the of algorithm, the employed since the model, dimension the ofiterative the solutions method space can be increasesimplemented with the complexity with a genetic of the algorithm employed (GA) model, instead the iterativeof the iterative method gradient can be implementedtechnique. The with iterative a geneticapproach algorithm employing (GA) instead the Debye of the model iterative is compared gradient technique. with the NRW The iterative method approachby analysing employing the case of a the Debyedielectric model sample is compared of thickness with equal the NRW to 10 method cm with by µ analysingr = 1. The thegeometrical case of a and dielectric electrical sample parameters of thicknessare summarised equal to 10 cmin Table with µ1.r = 1. The geometrical and electrical parameters are summarised in Table1. Table 1. Electrical and geometrical parameters of the material characterised by a Debye dispersion. Table 1. Electrical and geometrical parameters of the material characterised by a Debye dispersion. εs ε ftrans [MHz] σ µ t [mm] Actual values εs 100ε f 2trans [MHz] 300 σ 0.5 µ 1 t [mm] 100 ActualBest values values (3 runs) 100 87 2 1.43 300 305 0.5 0.64 1 1.01 100 100 Best valuesActual (3 runs) values 87 1.43100 2 305 300 0.64 0.5 1.01 1 100 150 ActualBest values values (6 runs) 100 97.9 2 1.1 300 305 0.5 0.49 1 1.0 150 150 Best values (6 runs) 97.9 1.1 305 0.49 1.0 150 GA searchGA space search (min) space (min) 1 1 1 1 1 0 0 1 1 GA searchGA space search (max) space (max) 200 200 50 50 1000 1000 10 10 5 5

As is evident from Figure 10a, both the NRW and the iterative method accurately estimate the real and imaginary parts of the dielectric permittivity (continuous line). Conversely, when the thickness of the same dielectric sample is 15 cm, the NRW method is not able to correctly estimate the dielectric permittivity (Figure 10b). Indeed, the NRW method fails to retrieve the dielectric permittivity and magnetic permeability if n = 0 is not suitable for the ﬁrst point of the frequency window. On the other Electronics 2017, 6, 95 13 of 28 Electronics 2017, 6, 95 13 of 28

As is evident from Figure 10a, both the NRW and the iterative method accurately estimate the As is evident from Figure 10a, both the NRW and the iterative method accurately estimate the real and imaginary parts of the dielectric permittivity (continuous line). Conversely, when the real and imaginary parts of the dielectric permittivity (continuous line). Conversely, when the thickness of the same dielectric sample is 15 cm, the NRW method is not able to correctly estimate thickness of the same dielectric sample is 15 cm, the NRW method is not able to correctly estimate the dielectric permittivity (Figure 10b). Indeed, the NRW method fails to retrieve the dielectric Electronicsthe dielectric2017, 6 ,permittivity 95 (Figure 10b). Indeed, the NRW method fails to retrieve the dielectric13 of 27 permittivity and magnetic permeability if n = 0 is not suitable for the first point of the frequency permittivity and magnetic permeability if n = 0 is not suitable for the first point of the frequency window. On the other hand, the iterative method accurately estimates both the real and imaginary window. On the other hand, the iterative method accurately estimates both the real and imaginary parts of the dielectric permittivity. The optimisation is performed using a GA. Clearly, the smaller hand,parts of the the iterative dielectric method permittivity. accurately The estimates optimisati bothon is the performed real and imaginaryusing a GA. parts Clearly, of the the dielectric smaller the space where the solution is searched, the closer will be the found solution. In the case reported in permittivity.the space where The the optimisation solution is searched, is performed the closer using wi all GA. be the Clearly, found the solution. smaller In the thespace case reported where the in Figure 10, the GA is run three times before selecting the best solution. Every run of the GA lasts 20 s, solutionFigure 10, is the searched, GA is run the closerthree time wills be before the found selecting solution. the best In thesolution. case reported Every run in Figureof the GA 10, thelasts GA 20 iss, thus obtaining an overall optimisation time equal to one minute. The results provided by the NRW runthus three obtaining times an before overall selecting optimisation the best time solution. equal to Every one runminute. of the The GA results lasts 20provided s, thus obtainingby the NRW an and iterative method have also been analysed for the case of noisy input data. In particular, S11 and overalland iterative optimisation method time have equal also tobeen one analysed minute. Thefor resultsthe case provided of noisy byinput the NRWdata. In and particular, iterative methodS11 and S21 obtained from the measurement of a dielectric sample with µr = 1 and thickness equal to 10 cm haveS21 obtained also been from analysed the measurement for the case of noisya dielectric input sample data. In with particular, µr = 1 Sand11 and thicknessS21 obtained equal to from 10 thecm have been corrupted with Additive White Gaussian Noise (AWGN) with a mean value (mv) equal to measurementhave been corrupted of a dielectric with Additive sample White with µGar =ussian 1 and Noise thickness (AWGN) equal with to 10 a mean cm have value been (mv corrupted) equal to zero and standard deviation (σ) equal to 0.01. The measured and noisy S-parameters are reported in withzero and Additive standard White deviation Gaussian (σ) Noiseequal (AWGN)to 0.01. The with measured a mean and value noisy (mv )S-parameters equal to zero are and reported standard in Figure 11a. As it is apparent from Figure 11b, the Baker–Jarvis (BJ) iterative method is robust with deviationFigure 11a. (σ As) equal it is toapparent 0.01. The from measured Figure and11b, noisythe Baker–Jarvis S-parameters (BJ) are iterative reported method in Figure is robust11a. As with it is respect to the AWGN, thus providing an accurate estimation of the real and imaginary part of the apparentrespect to from the AWGN, Figure 11 thusb, the providin Baker–Jarvisg an accurate (BJ) iterative estimation method of is the robust real withand imaginary respect to thepart AWGN, of the dielectric permittivity. thusdielectric providing permittivity. an accurate estimation of the real and imaginary part of the dielectric permittivity.

100 100 100 100 Re( ) Re( r ) Im( )r Im( r ) 50 50 Re( r)-BJ 50 50 Re( r )-BJ Im( )-BJr Im( r )-BJ Re( r)-NR 0 0 Re( r )-NR 0 0 Im( )-NRr Im( r )-NR r r r Re( ) -50 Re( r ) -50 -50 Im( )r -50 Im( r ) Re( r)-BJ Re( r )-BJ -100 Im( )-BJr -100 Im( r )-BJ -100 Re( r)-NR -100 Re( r )-NR Im( )-NRr Im( r )-NR -150 r -150 -150100 200 300 400 500 600 700 -150100 200 300 400 500 600 700 100 200 300 400 500 600 700 100 200 300 400 500 600 700 Frequency (MHz) Frequency (MHz) Frequency (MHz) Frequency (MHz) (a) (b) (a) (b) Figure 10. Real and imaginary parts of the dielectric permittivity estimated with NRW and the Figure 10.10. RealReal and and imaginary imaginary parts parts of theof dielectricthe dielectric permittivity permittivity estimated estimated with NRW with and NRW the iterativeand the iterative method employing the Debye dispersion model with µr = 1 for the case of a dielectric sample iterative method employing the Debye dispersion model with µr = 1 for the case of a dielectric sample withmethod thickness employing equal the to 10 Debye cm ( dispersiona) and 15 cm model (b). In with thisµ rcase,= 1 forthe theprocessed case of data a dielectric are not sample affected with by with thickness equal to 10 cm (a) and 15 cm (b). In this case, the processed data are not affected by noise.thickness equal to 10 cm (a) and 15 cm (b). In this case, the processed data are not affected by noise. noise.

0 0 S S11 -10 S 11 -10 S21 S 21-downsample S11 -downsample -20 S 11-downsample -20 S21 -downsample S 21-BJ S11 -BJ -30 S 11-BJ (dB) S21 -BJ -30 r

(dB) 21 21 r 21 , s -40 , s 11 -40 s 11 s -50 -50

-60 -60

-70 -70100 200 300 400 500 600 700 100 200 300 400 500 600 700 Frequency (MHz) Frequency (MHz) (a) (b) (a) (b) Figure 11. Comparison between the NRW and iterative method (BJ) employing the Debye dispersion Figure 11. Comparison between the NRW and iterative methodmethod (BJ) employing the Debye dispersion model for the estimation of the dielectric permittivity with µr = 1. (a) Measured, downsampled and model for the the estimation estimation of of the the dielectric dielectric permittivity permittivity with with µr µ=r 1.= ( 1.a) Measured, (a) Measured, downsampled downsampled and GA-estimated S-parameters for the case of a dielectric sample with thickness equal to 10 cm; (b) andGA-estimated GA-estimated S-parameters S-parameters for the for case the caseof a ofdiel aectric dielectric sample sample with with thickness thickness equal equal to 10 to cm; 10 cm;(b) estimated dielectric permittivity with the two methods. In this case the processed data are affected by (estimatedb) estimated dielectric dielectric permittivity permittivity with with the thetwo two methods. methods. In this In this case case the theprocessed processed data data are areaffected affected by AWGN noise with mv = 0 and σ = 0.01. byAWGN AWGN noise noise with with mvmv = 0= and 0 and σ =σ 0.01.= 0.01.

The NRW and the BJ iterative methods have also been compared in the case of noisy data for the estimation of both dielectric permittivity and magnetic permeability. The input S-parameters are obtained from a material sample of 10 cm characterised by the same electrical parameters summarised Electronics 2017, 6, 95 14 of 28

ElectronicsThe 2017NRW, 6, 95and the BJ iterative methods have also been compared in the case of noisy data14 offor 27 the estimation of both dielectric permittivity and magnetic permeability. The input S-parameters are obtained from a material sample of 10 cm characterised by the same electrical parameters in Table1, except that µ = 3. The S-parameters have been corrupted with AWGN (mv = 0, σ = 0.01). summarised in Table 1, exceptr that µr = 3. The S-parameters have been corrupted with AWGN (mv = 0,The σ = real 0.01). and The imaginary real and partsimaginary of the parts dielectric of the permittivitydielectric permittivity estimated estimated with NRW with and NRW the iterativeand the iterativemethod employingmethod employing the Debye the dispersion Debye disper modelsion are model reported are reported in Figure in 12 Figurea. 12a.

100

0 r r Re( ) r -50 Im( ) r Re( )-BJ r Im( )-BJ r -100 Re( )-NR r Im( )-NR r

-150 100 200 300 400 500 600 700 Frequency (MHz) (a) (b)

Figure 12. Real and imaginary parts of the (a) dielectric permittivity and (b) magnetic permittivity Figure 12. Real and imaginary parts of the (a) dielectric permittivity and (b) magnetic permittivity estimated with NRW and the iterative method (BJ) employing the Debye dispersion model with µr = estimated with NRW and the iterative method (BJ) employing the Debye dispersion model with µr = 3 3 for the case of a dielectric sample with thickness equal to 10 cm. In this case, the processed data are for the case of a dielectric sample with thickness equal to 10 cm. In this case, the processed data are affected by AWGN noise with mv = 0 and σ = 0.01. affected by AWGN noise with mv = 0 and σ = 0.01.

From Figure 12a, we note that the iterative method is sufficiently accurate whereas the NRW methodFrom provides Figure 12unphysicala, we note results that the since iterative the methodelectrical is sufﬁcientlylength of the accurate sample whereas exceeds the a NRWhalf wavelengthmethod provides at the unphysical lowest investigated results since frequency. the electrical Conversely, length of for the the sample NRW exceeds method a half the wavelengthestimation ofat the lowestmagnetic investigated permeability frequency. is critical Conversely, (Figure 12 forb), whereas the NRW the method iterative the method estimation provides of the magneticaccurate results.permeability is critical (Figure 12b), whereas the iterative method provides accurate results. The main advantages and disadvantages of the tw twoo approaches for retrieving the dielectric and magnetic properties of the mate materialsrials are summarised in Table2 2..

Table 2. Advantages and disadvantages of the direct inversion procedure based on the Table 2. Advantages and disadvantages of the direct inversion procedure based on the Nicolson-Ross-Weir algorithm and the Baker–Jarvis iterative search based on the S-parameters Nicolson-Ross-Weir algorithm and the Baker–Jarvis iterative search based on the S-parameters cost cost function. function.

ComputationComputation AdvantagesAdvantages Disadvantages Disadvantages TimeTime -- AmbiguitiesAmbiguities problems (when(when the the - - WidebandWideband extraction extraction of of both both samplesample thickness isis multiplemultiple of of Nicolson Nicolson Ross dielectricdielectric permittivity permittivity and and magnetic half-wavelengths).half-wavelengths). Ross Weir Weir direct permeability.magnetic permeability. - - HighHigh dielectric dielectric permittivitypermittivity can can be be ImmediateImmediate direct hardly measured. inversion - - DielectricDielectric properties properties of ofdispersive dispersive hardly measured. inversion materials (including metamaterials). - Unphysical solutions are admitted materials (including metamaterials). - Unphysical solutions are admitted - Sensible to noise. - Sensible to noise. - Wideband extraction of both - Widebanddielectric extraction permittivity of andboth - Evolutionary optimisation is required. Baker-Jarvis dielectricmagnetic permittivity permeability and magnetic - - EvolutionaryThe search space optimisation can be large is ∼1 min Baker-Jarvisiterative search - permeabilityDielectric properties of dispersive required.but limited. iterative - Dielectricmaterials properties (including of metamaterials) dispersive ∼1 min - The search space can be large but search - materialsOnly physical (including solutions metamaterials) are admitted. limited. - Only physical solutions are - Characterisation limited to - Very thin samples. Resonant admitted. dielectric permittivity. - Accurate estimation of Immediate Methods -- CharacterisationPresence of air gaps limited may degradeto dielectric the - Verydielectric thin samples. losses Resonant permittivity.accuracy of the estimation. - Accurate estimation of dielectric Immediate Methods - Presence of air gaps may degrade the losses accuracy of the estimation.

Electronics 2017, 6, 95 15 of 27 Electronics 2017, 6, 95 15 of 28

2.3.2.3. FSS-Based FSS-Based Waveguide Waveguide Resonant Resonant Method Method ResonantResonant methodsmethods usually usually exhibit exhibi highert higher accuracies accuracies and sensitivitiesand sensitivities than non-resonant than non-resonant methods, methods,and they areand more they suitableare more for suitable low-loss for samples. low-loss Resonant samples. methods Resonant are methods based on are the based property on thatthe propertythe resonant that frequency the resonant and quality frequency factor ofand a dielectric quality resonatorfactor of with a dielectric given dimensions resonator are with determined given dimensionsby its permittivity are determined and permeability. by its Thesepermittivity methods and are permeability. usually employed These for methods the characterizsation are usually employed for the characterizsation of low-loss dielectrics with permeability equal to µ0. There are a of low-loss dielectrics with permeability equal to µ0. There are a large number of resonant methods largefor the number estimation of resonant of the dielectric methods properties for the estimation of materials of the [29 dielectric] but they properti usuallyes require of materials ad hoc [29] cavities but theyand ausually direct etchingrequire ofad the hoc resonator cavities and on the a direct unknown etching sample. of the The resonator preparation on the of unknown the sample sample. under Thetest canpreparation be a difficult of the task sample in cavity under perturbation test can be techniques, a difficult task as the in samplecavity perturbation requires a regular techniques, geometry. as theIn resonantsample requires methods, a theregular sample geometry. permittivity In resonant is evaluated methods, from the the sample shift ofpermittivity the resonant is evaluated frequency. fromAll the the resonant shift of methodsthe resonant are limitedfrequency. to either All the a specificresonant frequency methods range,are limited some to kinds either of a materials, specific frequencyor specific range, applications. some kinds of materials, or specific applications. AA simple simple resonant resonant method method employable employable in in a awa waveguideveguide environment environment was was proposed proposed in [43]. in [43 It]. adoptsIt adopts a waveguide—which a waveguide—which is isa acommon common transmission/reflection transmission/reflection device—in device—in conjunction conjunction with with a resonanta resonant device, device, a aplanar planar Frequency Frequency Selective Selective Surf Surfaceace (FSS) (FSS) [10,64] [10,64 ]inserted inserted in in the the waveguide. waveguide. When When thethe dielectricdielectric samplesample is is accommodated accommodated close close to the to FSS,the FSS, the resonant the resonant frequencies frequencies of the transfer of the functiontransfer functionof the resulting of the cascadedresulting structure cascaded (FSS structure and sample) (FSS areand shifted sample) as aare function shifted of as the a thicknessfunction andof the the thicknessdielectric permittivityand the dielectric of the permittivity sample. The advantageof the sample. of the The proposed advantage technique of the isproposed the cost-effectiveness technique is theof thecost-effectiveness printed FSS and of the its simplicity,printed FSS because and its simp it relieslicity, only because on the it measurementrelies only on the of themeasurement amplitude ofwithout the amplitude measuring without the phase. measuring The the technique phase. The allows technique the estimation allows the of estimation the dielectric of the permittivity dielectric permittivitywithout employing without any employing ad hoc device any ifad the hoc sample device is correctly if the sample pressed is together correctly withthe pressed resonant together FSS. withtheThe procedure resonant is FSS. particularly The procedure suitable is for particularly the characterisation suitable for of the thin characterisation dielectric slabs of where, thin dielectric instead, slabswideband where, approaches instead, wideband presented approaches in the previous presen sectionted in are the not previous suitable section for thin are materials. not suitable Indeed, for thinthin samplesmaterials. are Indeed, not suitable thin forsamples the transverse are not positioning suitable for inside the atransverse waveguide positioning without a mechanical inside a waveguidesupport because without they a mechanical exhibit a low suppo reflectionrt because coefficient, they exhibit typically a low leading reflection to lowcoefficient, accurate typically results. leadingThe measurement to low accurate setup results. for waveguide The measurement resonant setup method for waveguide is shown in resonant Figure 13method. The presenceis shown in of Figurean additional 13. The dielectricpresence of substrate an additional close todielectric the FSS substrate determines close the to shift the ofFSS the determines resonance the frequency shift of thetowards resonance lower frequency values depending towards on lower the slab values thickness. dependi Ifng the on dielectric the slab thickness thickness. is lowerIf the than dielectric a half thicknessFSS periodicity, is lower its than effect a half can FSS be describedperiodicity, by its an effect effective can be permittivity described by that an dependseffective onpermittivity dielectric thatthickness. depends The on fabrication dielectric of thicknes the FSSs. filter The mustfabrication guarantee of the the FSS continuity filter must of the guarantee waveguide the through continuity the ofsubstrate the waveguide of the filter. through This isthe accomplished substrate of bythe putting filter. This a set is of accomplished closely spaced by vias putting across a the set supporting of closely spacedsubstrate vias [43 across]. the supporting substrate [43].

(a) (b)

FigureFigure 13.13. ((aa)) Resonant Resonant measurement measurement setup setup in a in waveguide a waveguide based based on FSS on filters; FSS (bfilters;) estimation (b) estimation principle. principle.

The resonant technique is aimed at retrieving the unknown permittivity of a dielectric of arbitrary thickness that is stacked to the free side of the printed FSS. The procedure can be summarised in the following steps:

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• The transmission/reflectiontransmission/reflection response response of theof unloadedthe unloaded filter isfilter initially is measured,initially measured, thus identifying thus identifyingthe resonant the peak; resonant peak; • The additional additional dielectric dielectric is is placed placed close close to tothe the FSS, FSS, measuring measuring the the new new response response and and hence hence the frequencythe frequency shift shift of the of theresonant resonant peak; peak; • The frequency shift is mapped into the unknown permittivity through a shift–permittivity map that is pre-defined pre-defined through anan iterativeiterative PeriodicPeriodic MethodMethod ofof MomentsMoments (PMM)(PMM) simulationsimulation [[65].65].

The frequency shift versus the permittivity ma mapp is computed by running iterative simulations of the filterfilter with the PMM code. The The frequency frequency shift shift as as a a function function of of the permittivity, which is determined for a specific specific FSS, allowsallows a straightforward determination of the unknown permittivity from thethe measuredmeasured frequency frequency shift. shift. Moreover, Moreover, similar similar behaviour behaviour is observed is observed for different for different FSS elements FSS elementssince the shiftsince of the the shift resonance of the resonance is due to the is due characteristics to the characteristics of the dielectrics of the anddielectrics it is weakly and it dependent is weakly dependenton the shape on of the the shape printed of the pattern. printed The pattern. mentioned The me map,ntioned for a specificmap, for example a specific of example an FSS filter of an tuned FSS filterin a WR90 tuned waveguide, in a WR90 waveguide, is reported inis Figurereported 14 ain [ 43Figure]. 14a [43].

(a) (b)

Figure 14. (a) Shift-permittivity map: unkn unknownown dielectric dielectric permittivity permittivity as as a a function function of of the the frequency frequency shift of the resonant peak;peak; ((bb)) maximummaximum value value of of the the measurable measurable dielectric dielectric permittivity permittivity as as a functiona function of ofthe the sample sample thickness. thickness.

When the thickness of the unknown dielectric is small in comparison to the cell periodicity, the When the thickness of the unknown dielectric is small in comparison to the cell periodicity, shift of the resonance frequency as a function of the sample permittivity is small as well. the shift of the resonance frequency as a function of the sample permittivity is small as well. Consequently, the determination of the unknown parameter is unambiguous even for high values of Consequently, the determination of the unknown parameter is unambiguous even for high values of εr. εr. For this reason, the method is particularly suitable for a precise estimate of the dielectric For this reason, the method is particularly suitable for a precise estimate of the dielectric permittivity of permittivity of thin slabs. On the contrary, a thickness larger than 1/10 of the cell periodicity thin slabs. On the contrary, a thickness larger than 1/10 of the cell periodicity determines a considerable determines a considerable shift of the resonance frequency with respect to the thin dielectric case. shift of the resonance frequency with respect to the thin dielectric case. When the thickness of the When the thickness of the dielectrics surrounding the FSS is larger than half cell periodicity, the shift dielectrics surrounding the FSS is larger than half cell periodicity, the shift of resonance frequency of resonance frequency becomes independent from the thickness since the FSS can be considered becomes independent from the thickness since the FSS can be considered embedded in a homogeneous embedded in a homogeneous dielectric [63,64]. dielectric [63,64]. If the considered resonant peak shifts below the cut-off frequency of the waveguide, a second If the considered resonant peak shifts below the cut-off frequency of the waveguide, a second resonant peak appears in the upper frequency range of the waveguide. The variation from the first to resonant peak appears in the upper frequency range of the waveguide. The variation from the ﬁrst the second resonant peak determines an abrupt transition in the map as reported in Figure 14a. In to the second resonant peak determines an abrupt transition in the map as reported in Figure 14a. this situation, the procedure generates an ambiguity in the determination of the permittivity. The In this situation, the procedure generates an ambiguity in the determination of the permittivity. ambiguity might be solved by observing the characteristics of the resonant peak. In particular, the peak is sharp when a low dielectric loading is applied to the FSS and it becomes shallow when the dielectric permittivity of the loading sample is high. In addition, another possible source of inaccuracy is the air gap between the FSS and the unknown sample. In order to avoid the mentioned inaccuracies, the unknown sample should be in direct contact with the FSS.

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The ambiguity might be solved by observing the characteristics of the resonant peak. In particular, the peak is sharp when a low dielectric loading is applied to the FSS and it becomes shallow when theElectronics dielectric 2017, 6 permittivity, 95 of the loading sample is high. In addition, another possible source17 of of28 inaccuracy is the air gap between the FSS and the unknown sample. In order to avoid the mentioned inaccuracies,The maximum the unknown frequency sample shift should for avoiding be in direct the contact ambiguity with theis determined FSS. by the difference betweenThe maximumthe frequency frequency where shift the forinvestigated avoiding thepeak ambiguity is located, is determined f0 (i.e., 13 GHz by the indifference our case) betweenand the lower limit of the waveguide flow [66]. The frequency shift between the initial peak and the shifted the frequency where the investigated peak is located, f 0 (i.e., 13 GHz in our case) and the lower limit ofone the is waveguidetherefore dueflow to:[66 ]. The frequency shift between the initial peak and the shifted one is therefore due to: f0 high low −=−f0 high fff−0..= waveg− low waveg , (25) f0 q ε eff fwaveg. fwaveg., (25) e fr f εr where f0 is the initial resonance frequency, e f f is the effective permittivity of the unknown sample where f0 is the initial resonance frequency, εr is the effective permittivity of the unknown sample and the FSS filter substrate andhigh andlow represent the maximum and the minimum and the FSS ﬁlter substrate and fwaveg and fwaveg represent the maximum and the minimum working frequencyworking frequency of the waveguide. of the waveguide. As an example, As an theexample, measurable the measurable values of permittivityvalues of permittivity for a sample for ina asample WR90 in waveguide, a WR90 waveguide, where the maximum where the frequency maximum shift frequency for avoiding shift theforambiguity avoiding isthe determined ambiguity byis high thedetermined difference by between the difference the frequency between where the frequency the investigated where the peak investigated is located, peakfhigh (i.e., is located, 13 GHz f in (i.e., our low case),13 GHz and in theour lowercase), limitand the of the lower waveguide limit of theflow waveguide(i.e., 8 GHz), f is (i.e., shown 8 GHz), in Figure is shown 14b. in Figure 14b.

3. Measuring the Surface Impedance of Thin Sheets Controlled resistance coatings coatings are are widely widely adopted adopted in inseveral several electronic electronic applications applications such such as the as thefabrication fabrication of a of thin-film a thin-ﬁlm transistor transistor [67], [67 ],transparent transparent electrodes electrodes [68] [68 ]and and printed printed resistors resistors [69]. [69]. Consequently, the EM estimationestimation of thethe surfacesurface impedanceimpedance of thin sheets is desirabledesirable in severalseveral applications. A A thin thin sheet sheet of of material material can can be be repres representedented by by a ashort short piece piece of of transmission transmission line line with with a acertain certain impedance impedance ζ1ζ, 1as, as depicted depicted in in Figure Figure 15a. 15a. However, However, under under certain certain conditions, conditions, the thin sheet can alsoalso be be represented represented by aby lumped a lumped component component with certain with impedancecertain impedanceZs, named Z surfaces, named impedance, surface asimpedance, shown in as Figure shown 15 b.in Figure 15b.

(a) (b)

Figure 15. (a) Transmission line model of a thin dielectric slab with characteristic impedance ζ ; (b) Figure 15. (a) Transmission line model of a thin dielectric slab with characteristic impedance1 ζ1; (transmissionb) transmission line line model model of an of infinitesimal an inﬁnitesimal sheet sheet characterised characterised by bya surface a surface impedance impedance Zs. Zs.

With referencereference to to the the model model depicted depicted in Figurein Figure2a, the2a, inputthe input impedance impedance of a dielectric of a dielectric slab reads: slab reads: ζζζ0 + jζ1 tan()(kd) = 01jkdtan ZV1 = ς 1 . (26) ZV 1 ς + jζ tan(kd). 1 ςζ1+ 0 () (26) 10jkdtan

Differently, the input impedance calculated by using the model of Figure 2b is: Z ζ Z = s 0 . V2 +ζ (27) Zs 0

Electronics 2017, 6, 95 18 of 28 Electronics 2017, 6, 95 18 of 27 By imposing the equation of the two input impedances ZZ= , the equivalent sheet VV12 impedanceDifferently, necessary the input to satisfy impedance the equation, calculated can bybe usingfound the as follows: model of Figure2b is: ζ 0ZV = Zsζ02 ZZs = . (28) (27) V2 ζZ −+Zζ 0s V02 = LetBy imposingus now theassume equation that ofthe the diel twoectric input slab impedances is sufficientlyZV1 thinZV2 , theand equivalent that the good sheet impedanceconductor necessary to satisfy the equation, can be found as follows: condition is satisfied (kd << 1 and σ/ωε0εr >> 1); the expression of Z can be simplified as follows: V1 ζ0ZV ()1+ j Z = 2 d (28) ζ + s ζ − Zζ + 2 j ()++0 jkd()0 V20 2 ζ + ωμ 11jjσδ σδ 00jd Z ≈= =, (29) V1 σδ()11++jj σδ () d 1+ζ dσ Let us now assume that the dielectric++−jkdζζ slab is sufficiently thin j() and1 j that the good0 conductor condition σ σδ00 σδ δ is satisfied (kd << 1 and /ωε0εr >> 1); the expression of ZV1 can be simplified as follows:

ζ ≈+()11 ≈−()(1+j) δω= μ σ d where 1 1,1,jkj(1 + j) ζ0 + j withkd (1 +2j) 0 ζare0 + used.2j 2 ζ + jdωµ Z σδ≈ δ σδ = σδ = 0 0 , (29) V1 (1+j) (1+j) σδ σδ d 1 + ζ0dσ By using Equation (28), theσδ equivalent+ jζ0kd sheet impedanceσδ + jζ 0under(1 − j) theδ initial hypotheses assumes the following expression: 1 1 p where ζ1 ≈ (1 + j) σδ , k ≈ (1 − j) δ , with δ = 2/ωµ0σ are used. 2 By using Equation (28), the equivalent sheetζζ+ impedancejdωμ under the initial hypotheses assumes Z ≈ 000. (30) the following expression: s ζ 2 σωμ− 00djd ζ2 + jζ dωµ Z ≈ 0 0 0 Under the initial hypothesis of goods conductor2 and thin. sheet, the expression can be further (30) ζ0dσ − jdωµ0 simplified as the imaginary parts of the numerator and of the denominator of Equation (30) are Under the initial hypothesis of good conductor and thin sheet, the expression can be further negligible with respect to the real part: ζ 2σωμζ,12ddωμ → λπ . Once this simplified as the imaginary parts of the numerator0000 and of the denominator of Equation (30) condition is verified, it is easy to demonstrate th2 at the sheet resistance approaches the classical DC are negligible with respect to the real part: ζ0σ ωµ0 , ζ0 dωµ0 → d/λ 1/2π . Once this resistancecondition isvalue: verified, it is easy to demonstrate that the sheet resistance approaches the classical DC resistance value: 1 ≈ 1 ZZs ≈ .. (31) (31) s σσdd The derivations cancan alsoalso be be numerically numerically verified. verified It. It is is possible possible to considerto consider a thin a thin layer layer of d of= 10d =µ 10m, µm,and σand= 1000 σ = S/m. 1000 TheS/m. equivalent The equivalent sheet impedance sheet impedance satisfying satisfying the general the Equation general (25) Equation is represented (25) is representedin Figure 16. Asin isFigure evident, 16. the As equivalent is evident, sheet the impedance equivalent approaches sheet impedance the DC resistance, approaches as predicted the DC resistance,by Equation as (30), predicted below by 100 Equation GHz. (30), below 100 GHz.

200 real(Z ) s imag(Z ) s Z 150 DC /sq) 6 10 ) r

100 kd 0 /(

50 Sheet impendance (

0 10-2 100 102 104 Frequency (GHz) (a) (b)

Figure 16. (a) Equivalent surface impedance to verify the equivalence of the thin sheet with a finite Figure 16. (a) Equivalent surface impedance to verify the equivalence of the thin sheet with a ﬁnite dielectric conductivity (Figure 15b) and the infinitesimally thin sheet characterised by a lumped dielectric conductivity (Figure 15b) and the inﬁnitesimally thin sheet characterised by a lumped impedance (Figure 15a). (b) Behaviour of the kd term and the good conductor condition as a function impedance (Figure 15a). (b) Behaviour of the kd term and the good conductor condition as a function of frequency. of frequency.

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For those materials that meet these conditions, the standard dielectric permittivity and magnetic permeability values can be replaced by the sheet impedance. For thin sheets, where the thickness of the layer is much lower than the skin depth, the sheet impedance can be used in both reflection and transmission measurements. Consequently, the thin dielectric layer can be replaced by a resistive sheet that is completely characterised only by the measurable quantity Zs (Ω/sq). A slab of lossy material with thickness d and conductivity σ can be analysed as an infinitesimal sheet and the tangential fields satisfy the impedance boundary condition:

+ − Et = Zs Ht − Ht . (32)

Equation (32) implies that the electric field is continuous (1 + S11 = S21) at the deposition interface. This hypothesis can be validated with measurements of the scattering parameters. The fabrication of thin conductive and partially conductive coatings is a common practice in a number of engineering areas. Their employment spans from electronic circuitry for the fabrication of thin-film transistors [67], printed resistors [69,70] or transparent electrodes [68] to shielding or absorbing materials [11,12,16] for applications in Electro-Magnetic Interference (EMI) [71] and Electrostatic Discharge (ESD). Non-perfect conductive thin films can be synthesised through a wide set of materials. One of the most popular for its peculiar properties of optical transparency and high conductivity is Tin Indium Oxide (ITO) [72]. Moreover, graphene [46,73] or commercial carbon-based resistive inks allow controlled value of surface resistances. While conductive coatings require sheet resistances as low as possible, a partially conductive film requires a controlled value of the sheet resistance. It is worth noticing that the desired impedance value is only achievable with an accurate control of the thickness of the deposition. Vacuum deposition techniques, such as Physical Vapor Deposition (PVD), provide an adequate control of the film thickness but their employment produces high fabrication costs and, furthermore, they are not suitable for the fabrication of large samples. Nowadays, there is strong interest in fabricating conductive or resistive coatings by using screen printing, ink-jet printing or roll to roll technologies on Kapton films, FR4, silicon or ceramic substrates [74]. A reliable and simple technique to measure the complex surface impedance of thin film depositions within the microwave range is useful for establishing a closed loop between the manufacturing and design stages. The most common technique to measure the DC surface resistance is the Van Der Pauw method, based on four-point probes [75]. The measurement of the surface impedance at microwave frequencies can be accurately carried out through ad hoc resonant cavities [76,77], which may be expensive and narrowband. A wideband estimation of the surface impedance can be performed with transmission/reflection measurements in free space or in a guided device [78–83]. In contrast to the CPW-based techniques, the T/R method does not require any patterning of the sample or any electrical contact, thus providing the estimation of the surface impedance regardless of contact resistance considerations [46]. In the following section, we will discuss the T/R radio frequency approach to characterise thin sheets as a versatile, simple and very accurate approach that can be simply implemented with standard laboratory equipment. See [29] for a complete discussion of alternative methods to characterise thin sheets.

3.1. Inversion Procedure in a Closed Waveguide A two-dimensional sketch of the measurement setup and its equivalent circuit model are shown in Figure 17a,b, respectively. The thin resistive sheet is represented by a lumped resistor in the equivalent network, while the supporting substrate is represented as a short transmission line section. Electronics 2017, 6, 95 20 of 27 Electronics 2017, 6, 95 20 of 28

(a) (b)

FigureFigure 17. 17. (a(a) )Measurement Measurement setup setup and and (b (b) )simplified simpliﬁed transmission transmission line line model. model.

The impedance sheet and the substrate can be described as a shunt connection of a transmission The impedance sheet and the substrate can be described as a shunt connection of a transmission line and a lumped load. By using the ABCD description of the cascade system, the explicit line and a lumped load. By using the ABCD description of the cascade system, the explicit expression expression of the surface impedance as a function of the transmission coefficient S21 can be derived of the surface impedance as a function of the transmission coefﬁcient S can be derived [44]: [44]: 21 TE TE TE TE −S TEZ TEZ TEZ cos(k d) + jZ TE sin(k d) T = −+SZ21 0 Zm() Z0 cos() kdz jZm sin () kdz TZs 21 00mmzz , (33) Z = TE TE( ( ) − ) + TE2 + TE2 ( ) , s 2Z Zm S21 cos kzd 1 j Zm 22Z S21 sin kzd (33) 2cos1ZZTE0 TE() S() kd−+ jZ(()() TE + Z0 TE ) S sin() kd 0 mm21zz0 21 q TE 2 2 √ where Zm = ωµ0/kz, kt = π/a, kz = km − kt , km = k0 εrµr, a is the wider waveguide TE ===−=ωμ π22 ε μ wheredimension, Zkkakkkkkmztzmtmrrk0 is00 the free, space propagation, constant, , and kt is, the a is transversethe wider waveguide wavevector dimension, of the TE10 kpropagation0 is the free space mode. propagation constant, and kt is the transverse wavevector of the TE10 propagation mode.If the thickness of the dielectric substrate is negligible, Equation (33) simpliﬁes to:

If the thickness of the dielectric substrate is negligible,TE Equation (33) simplifies to: S21Z ZT ' 0 , i f d λ. (34) s ( −TE ) T 2SZ121 S21 Zifd0 , λ . (34) s 21()− S 3.2. Sample Positioning 21 A significant difficulty in the measurement of the scattering parameters is the positioning of 3.2.the Sample sample. Positioning In order to address this issue, two different measurement setups can be considered. Most likely,A significant the most intuitive difficulty way in the to placemeasurement the sample of the in thescattering waveguide parameters is to cut is the a sample positioning (impedance of the sample.sheet and In substrate)order to address to the same this issue, dimensions two diff aserent the waveguide measurement at the setups most can reachable be considered. tolerances Most and, likely,thesubsequently, most accommodateintuitive way to the place sample the insidesample the in waveguidethe waveguide (Figure is to 18 cutb—setup a sample 1). (impedance The second sheetoption and is tosubstrate) cut the sample to the same larger dimensions than the waveguide as the waveguide flanges and at the then most press reachable it against tolerances the waveguide and, subsequently,flanges (Figure accommodate 18c—setup 2). the The sample two setups inside have the beenwaveguide experimentally (Figure tested18b—setup using 1). a resistive The second sheet option(nominally is to ofcut 20 the ohm/sq) sample deposited larger than on a the 1.6 mmwaveguide thick FR4 flanges substrate. and then The scatteringpress it against parameters the waveguidehave been measuredflanges (Figure with both 18c—setup setups and2). The the inversiontwo setups procedure have been in Equationexperimentally (33) has tested been using applied. a resistiveThe surface sheet impedance (nominally values of 20 obtained ohm/sq) after deposited the inversion on a 1.6 procedure mm thick and FR4 the substrate. two investigated The scattering setups parametersare reported have in Figurebeen measured 18a. Although with both setup setups 1 is the and most the inversion intuitive, procedure it leads to in a strongEquation unexpected (33) has beenimaginary applied. part The when surface the impedanceimpedance isvalues retrieved. obta Theined impedanceafter the inversion curve follows procedure that of and a capacitive the two investigatedelement (1/ω setupsC) in seriesare reported with the in real Figure part 18a. of the Although surface impedance.setup 1 is the This most effect intuitive, is due it to leads the small to a stronggaps formed unexpected between imaginary the measured part when sample the and impeda the topnce andis retrieved. bottom waveguide The impedance walls. curve Even iffollows it does thatnot of ensure a capacitive the electric element continuity (1/ωC) ofin theseries waveguide with the real across part a shortof the sectionsurface of impedance. the waveguide, This effect setup is 2 dueprovides to the two small main gaps advantages. formed between Firstly, the it eliminates measured the sample strong and capacitive the top behaviourand bottom achieved waveguide with walls.the previous Even if setupit does and, not inensure addition, the electric it allows cont aninuity accurate of the orthogonal waveguide positioning across a short of the section sample of with the waveguide,respect to the setup waveguide. 2 provides two main advantages. Firstly, it eliminates the strong capacitive behaviour achieved with the previous setup and, in addition, it allows an accurate orthogonal positioning of the sample with respect to the waveguide.

Electronics 2017, 6, 95 21 of 27 Electronics 2017, 6, 95 21 of 28 Electronics 2017, 6, 95 21 of 28

(b) (b)

(a) (c) (a) (c) Figure 18. Estimated surface resistance of an ink deposition with a DC surface resistance of 20 Figure 18.18. EstimatedEstimated surface surface resistance resistance of anof inkan depositionink deposition with awith DC surfacea DC surface resistance resistance of 20 ohm/sq of 20 ohm/sq (a). The substrate is FR4 with a thickness of 1.6 mm. The results are obtained by using setup 1 (ohm/sqa). The ( substratea). The substrate is FR4 with is FR4 a thickness with a thickness of 1.6 mm. of 1. The6 mm. results The areresults obtained are obtained by using by setup using 1 (setupb) and 1 (b) and setup 2 (c). setup(b) and 2 (setupc). 2 (c). 3.3. Calibration Procedure of the Open Junction to Improve Estimation 3.3. Calibration Procedure of the Open Junction to Improve EstimationEstimation As demonstrated in the previous section, setup 2 is the most suitable for impedance As demonstrated in the previous section, se setuptup 2 is the most suitablesuitable for impedanceimpedance characterisation. However, some residual errors are introduced by the presence of a small part of the characterisation. However, However, some some residual residual errors errors are are introduced introduced by the by presence the presence of a small of a smallpart of part the waveguide, which is unshielded. Moreover, the thinner the substrate thickness the lower the ofwaveguide, the waveguide, which whichis unshielded. is unshielded. Moreover, Moreover, the thinner the thinner the substrate the substrate thickness thickness the lower the lower the introduced measurement error. However, as demonstrated in [45], this problem can be solved with a theintroduced introduced measurement measurement error. error. However, However, as demonstrated as demonstrated in [45], in this [45 ],problem this problem can becan solved be solvedwith a preliminary calibration procedure. According to [45], the partially open junction formed by the withpreliminary a preliminary calibration calibration procedure. procedure. According According to [45], to the [45 ],partially the partially open openjunction junction formed formed by the by waveguide flanges and the dielectric substrate can be characterised by a T-type circuit formed by a thewaveguide waveguide flanges ﬂanges and andthe dielectric the dielectric substrate substrate can be can characterised be characterised by a byT-type a T-type circuit circuit formed formed by a series impedance, Zg3, and two parallel admittances, YG1 and YG2. The equivalent transmission line byseries a series impedance, impedance, Zg3, andZg3, two and twoparallel parallel admittances, admittances, YG1 YandG1 andYG2. YTheG2. equivalent The equivalent transmission transmission line circuit of the junction in the absence and presence of the impedance sheet, respectively, is shown in linecircuit circuit of the of junction the junction in the in theabsence absence and and presence presence of the of theimpedance impedance sheet, sheet, respectively, respectively, is shown is shown in Figure 19a,b. The values of these terms can be derived by a preliminary measurement aimed at the inFigure Figure 19a,b. 19a,b. The The values values of these of these terms terms can can be derived be derived by a by preliminary a preliminary measurement measurement aimed aimed at the at characterisation of the junction. thecharacterisation characterisation of the of thejunction. junction.

(a) (b) (a) (b) Figure 19. Equivalent circuit of the waveguide junction in the absence (a) or presence (b) of the Figure 19. Equivalent circuit of the waveguide junction in the absence (a) or presence (b) of the Figureimpedance 19. Equivalentsheet. circuit of the waveguide junction in the absence (a) or presence (b) of theimpedance impedance sheet. sheet. According to [45], the values of the lumped components describing the junction can be According to [45], the values of the lumped components describing the junction can be computed by using the scattering parameters of the junction S without the impedance sheet (Figure computedAccording by using to [45 the], the scattering values of parameters the lumped of components the junction describing S without thethe junctionimpedance can sheet be computed (Figure 19a) as follows: by19a) using as follows: the scattering parameters of the junction S without the impedance sheet (Figure 19a) as follows: ()+−2 2 TE 1 SS112 2 21 ZZ= TE (()11++−SSS11) − S21 (35) Z g=3 Z TE0 11 21 (35) ZZg3 = 0 2S (35) g3 0 2S2121 2S21

Electronics 2017, 6, 95 22 of 28

Electronics 2017, 6, 95 22 of 27 ()1−−SS2 2 −± − 11 21 244Zg3 TE 2SZ21 (36) Y = 0 g r . 2Z (1−S )2−S2 − ± − Zg3 11 21 2 4 4 g3 2S ZTE 21 0 It is worth underlining thatYg only= a solution of Equation (36). with a positive real part (36) is 2Zg3 acceptable. After the derivation of the calibration parameters, the scattering matrix in presence of the sheetIt is worth S' (and underlining the same that dielectric only a solution stack-up) of Equation is obtained (36) with through a positive a second real part measurement. is acceptable. After the derivation of the calibration parameters, the scattering matrix in presence of the sheet S0 Consequently, the sheet impedance Zs can be obtained as: (and the same dielectric stack-up) is obtained through a second measurement. Consequently, the sheet ' TE TE impedance Z can be obtained as: SZZYZ()++ Z s 2100gg 3 g 3 Z = . s ' TE TE TE (37) 2222−++++SZYZY()0 () TE Z ZTE YZ 21gg 3S Z000 gZg3Yg + Z gg 33 + Z g Z = 21 0 0 s 0 TE TE TE . (37) 2 − S Zg3Yg Z Yg + 2 + Zg3/Z + 2YgZ + 2 Moreover, the authors of [45]21 provided0 a more accurate expression0 0of the impedance that relies on theMoreover, measurement the authors of the of [45S22].provided In addition, a more the accurate following expression expression of the corrects impedance some that residual relies on perturbationsthe measurement of the of thecalibrationS22. In addition, parameters the intr followingoduced expression by the presence corrects of some the impedance residual perturbations sheet: of the calibration parameters introduced by the presenceSZ' TE of the impedance sheet: = 21 0 Zs . (38) −−''()()TES0 ++ZTE TE + TE + 2222SSZYZY22 21gg 3 g21 0 ZZ g 3 YZ g Zs = 000. (38) − S0 − S0 Z Y ZTEY + + Z ZTE + Y ZTE + 2 22 21 g3 g 0 g 2 g3/ 0 2 g 0 2 This second solution, which provides the most accurate results, will be considered in the followingThis secondexample. solution, A resistive which sheet provides with the a mostsurfac accuratee impedance results, equal will beto considered21 Ohm/sq in is the considered. following Theexample. substrate A resistive of the resistive sheet with sheet a surface is a FR4 impedance slab of thickness equal to 21 d. Ohm/sqIn order isto considered. analyse the The effect substrate of the gapof the size, resistive three sheetdifferent is a thicknesses FR4 slab of of thickness substrat d.e d Inare order considered. to analyse The the simulations effect of the have gap been size, performedthree different by using thicknesses Ansys ofElectronics substrate dv.16. are Two considered. different The sizes simulations of sample have (size been 1 and performed size 2) and by threeusing different Ansys Electronics thicknesses v.16. (d Two = 0.5 different mm, d sizes = 1.5 of samplemm and (size d 1= and2.5 sizemm) 2) are and considered. three different A two-dimensionalthicknesses (d = 0.5 representation mm, d = 1.5 mm of and thed =simulated 2.5 mm) aresetup considered. is reported A two-dimensional in Figure 20. representationAccording to Equationof the simulated (33), the setup inversion is reported procedure in Figure requires 20. According a single to Equation simulation, (33), which the inversion is necessary procedure to pre-emptivelyrequires a single estimate simulation, the whichdielectric is necessary permittivity to pre-emptively of the substr estimateate. The the second dielectric approach, permittivity which of includesthe substrate. the calibration The second of approach, the open which junction, includes requires the calibrationa first simulation of the open without junction, the resistive requires asheet ﬁrst andsimulation a second without simulation the resistive with the sheet resistive and asheet. second The simulation scattering with parameters the resistive are combined sheet. The according scattering toparameters Equations are (33) combined and (38) accordingto estimate to the Equations surface (33)impedance and (38) of to the estimate sample. the The surface estimated impedance surface of impedancethe sample. (real Theestimated and imaginary surface part) impedance of the (real sample and imaginaryaccommodated part) of on the the sample different accommodated the three differenton the different substrate the thicknesses three different is shown substrate in Figu thicknessesre 21.While is a shown substrate in Figurethickness 21.While lower than a substrate 1 mm doesthickness not create lower relevant than 1 mmproblems, does not a thickness create relevant of 1.5 mm problems, and 2.5 a mm thickness can significantly of 1.5 mm andperturb 2.5 mmthe valuecan signiﬁcantly of the estimated perturb parameter. the value However, of the estimated the ap parameter.proach including However, the thecalibration approach of including the junction the restorescalibration the ofexpected the junction values restores also in the the expected presence values of non-negligible also in the presence gaps. of non-negligible gaps.

(a) (b)

Figure 20. Measurement setup simulated with Ansys Electronics v. 16: (a) side view; (b) front view. Figure 20. Measurement setup simulated with Ansys Electronics v. 16: (a) side view; (b) front view.

Electronics 2017, 6, 95 23 of 27 Electronics 2017, 6, 95 23 of 28

30 closed Acc. to (33) - size1 28 Acc. to (33) - size1 Acc. to (38) - size1 26 Acc. to (38) - size2 24

20 (ohm/sq) (ohm/sq) s s Z Z 18

55.566.577.58 Frequency (GHz) (a) (b) (ohm/sq) (ohm/sq) s s Z Z

30 closed Acc. to (33) - size1 28 Acc. to (33) - size1 Acc. to (38) - size1 26 Acc. to (38) - size2 24

20 (ohm/sq) (ohm/sq) s s Z 18 Z

55.566.577.58 Frequency (GHz) (e) (f)

Figure 21. Estimated surface impedance (real and imaginary part) for three different thicknesses of Figure 21. Estimated surface impedance (real and imaginary part) for three different thicknesses of the substrate: Zs was set to 20 + j0 in the simulations performed with Ansys Electronics v. 16. Closed: the substrate: Zs was set to 20 + j0 in the simulations performed with Ansys Electronics v. 16. Closed: sample entirely contained in a closed waveguide. Size 1: Δx = 2 mm, Δy = 2 mm, Size 2: Δx = 6 mm, sample entirely contained in a closed waveguide. Size 1: ∆x = 2 mm, ∆y = 2 mm, Size 2: ∆x = 6 mm, ∆Δyy = = 4 4 mm. mm. ( a(a,b,b))d d= = 0.5 0.5 mm, mm, ( c(c,d,d))d d= = 1.5 1.5 mm, mm, ( e(e,f,)f)d d= = 2.5 2.5 mm. mm.

4. Conclusions 4. Conclusions A comprehensive overview of material characterisations using conventional A comprehensive overview of material characterisations using conventional transmission/reflection transmission/reflection devices is presented. At first, the classical Nicholson–Ross–Weir inversion devices is presented. At first, the classical Nicholson–Ross–Weir inversion procedure is described and procedure is described and its limitations in terms of ambiguities and unphysical solutions are its limitations in terms of ambiguities and unphysical solutions are discussed. An iterative procedure, discussed. An iterative procedure, inspired by the Baker–Jarvis method, which admits only causal inspired by the Baker–Jarvis method, which admits only causal solutions, has been discussed as a valid solutions, has been discussed as a valid alternative to characterise high-permittivity substrates. The alternative to characterise high-permittivity substrates. The solution is found by optimising the unknown solution is found by optimising the unknown parameters of classical dispersion models such us parameters of classical dispersion models such us Debye and Lorentz. Finally, a resonant waveguide Debye and Lorentz. Finally, a resonant waveguide method employing FSS filters in a waveguide is reported for the characterisation of thin samples, that are problematic with classical wideband approaches.

Electronics 2017, 6, 95 24 of 27 method employing FSS filters in a waveguide is reported for the characterisation of thin samples, that are problematic with classical wideband approaches. The second part of the paper has been focused to the characterisation of thin two-dimensional materials characterised by a relevant conductivity. The conditions required to replace the thin sample with an infinitesimally thin sheet characterised by a finite value of surface impedance have been analysed. A simple technique to characterise the surface impedance of thin samples based on a waveguide setup has been described. The correct positioning of the sample in the waveguide has been verified and a calibration procedure, aimed at removing residual errors due to parts of unshielded waveguide because of the presence of a dielectric substrate, has been illustrated via numerical examples.

Author Contributions: F.C. and A.M. conceived the work, M.D. and F.C. prepared the codes and performed simulations; F.C. and M.B. prepared the manuscript; F.C., M.B. and M.D. analysed the data; M.D. and A.M. revised the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

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