1

Fast and Robust Characterization of Dielectric SlabsUsing Rectangular Waveguides

Xuchen Wang and Sergei A. TretyakovDepartment of Electronics and Nanoengineering, Aalto University, Espoo, Finland

xuchen.wang@aalto.fi

Abstract—Waveguide characterization of dielectric materials isa convenient and broadband approach for measuring dielectricconstant. In conventional microwave measurements, materialsamples are usually mechanically shaped to fit the waveguideopening and measured in closed waveguides. This method is notpractical for millimeter-wave and sub-millimeter-wave measure-ments where the waveguide openings become tiny, and it is ratherdifficult to shape the sample to exactly the same dimensions asthe waveguide cross-section. In this paper, we present a methodthat allows one to measure arbitrarily shaped dielectric slabsthat extend outside waveguides. In this method, the measuredsample is placed between two waveguide flanges, creating adiscontinuity. The measurement system is characterized as anequivalent Π-circuit, and the circuit elements of the Π-circuit areextracted from the scattering parameters. We have found thatthe equivalent shunt impedance of the measured sample is onlydetermined by the material permittivity and is rather insensitiveto the sample shape, position, sizes, and other structural detailsof the discontinuity. This feature can be leveraged for accuratemeasurements of permittivity. The proposed method is veryuseful for measuring the permittivity of medium-loss and high-loss dielectrics from microwave to sub-terahertz frequencies.

Index Terms—Permittivity measurement, millimeter-wave,sub-millimeter-wave, rectangular waveguide.

I. INTRODUCTION

Material characterization is an essential step in designingelectromagnetic devices. The recent fast developments of wire-less communications (5G and beyond) impose strong demandsfor the characterization of dielectric materials at millimeter-wave and even higher frequencies. In general, the methodsfor measuring dielectric constant can be divided into twogroups: one is based on resonating systems and the otheris utilizing non-resonant transmission-lines structures [1]. Inresonance-based methods, the sample is usually machined intoa dielectric resonator with a high Q-factor. The real part ofthe permittivity can be predicted from the resonant frequencyand the loss tangent is extracted from the Q-factor of themeasurement system [2], [3]. Alternatively, one can positionthe sample into a high-Q cavity and obtain the permittivityvalue by measuring the perturbation of the resonant frequencyand the Q-factor of the system before and after loading thecavity [4]. Generally speaking, resonance methods provide thebest accuracy in the estimation of both real and imaginaryparts of permittivity for low-loss dielectrics. The drawbackof this method is that the measured frequencies must bediscrete, corresponding to the resonant frequencies of thesystem. In addition, the dimensions of the resonators become

tiny at millimeter-wave or higher frequencies, which imposesconsiderable practical difficulties.

MUT

Alignment hole

Screw

Alignment pin

𝑊𝐿

𝑧

𝑦𝑥

Fig. 1. Measurement setup using millimeter-wave rectangular waveguides.The sample under test (SUT) is positioned in between two waveguide flanges.The actual setup is fastened by screws.

In the transmission-line-based method, the measured sampleis connected as a load or insertion in a waveguiding structure.By measuring the reflection and/or transmission coefficients(S-parameters) of the system, the dielectric properties of thematerial can be determined in a broad frequency range [5].Due to the absence of a setup resonance, the measured S-parameters are not so sensitive to the dielectric losses as inthe resonator method (especially for thin dielectric samples),and therefore the measurement accuracy for the loss tangentis generally worse than in the resonator method. For thisreason, the transmission-line method is most suitable for thecharacterization of medium-loss and high-loss dielectrics [1],[6]. Transmission-line structures can be formed by manystructures, such as metallic [7]–[9] or dielectric waveguides[2], [10], coplanar waveguides [11], microstrip lines [12],free space [13], and so on. One of the most commonly usedmethods is developed in [7], [8], which is well known as theNicolson–Ross–Weir method. In this method, the sample undertest (SUT) is embedded into a rectangular waveguide and fullyin contact with the waveguide walls. However, in practice,there are inevitable air gaps between the surfaces of the SUTand the waveguide walls. This is a considerable restrictionfor measurements at millimeter-wave and above frequencieswhere the waveguide dimensions are of the order of millimeteror smaller [14].

In order to avoid the problem of imperfect contact with thewaveguide walls, it is preferable and easier to test the sampleoutside waveguides [15]–[18]. For example, in [16], [17], thesamples are positioned between two waveguide flanges, creat-

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ing a discontinuity from which the electromagnetic energy isallowed to leak away. The measurement setup is then modeledin commercial simulation tools or using self-developed numer-ical algorithms to calculate the S-parameters. By fitting thenumerically simulated S-parameters with the measured values,the permittivity of the sample can be estimated. However, inthis method, it is necessary to ensure that the actual measure-ment setup is accurately represented in numerical modeling,e.g., the dimensions of the waveguide flanges and the testsample, since the S-parameters of the system are affected byall these details. Practical limitations on the modeling accuracyof all the setup details do not allow accurate millimeter-wavemeasurements, because the configuration of millimeter-wavewaveguide flanges is usually not planar and contains otherstructures such as holes, chokes, alignment pins, and screws,which are difficult to model accurately (see Fig. 1). Obviously,in the millimeter-wave range and above, it is more convenientto measure the sample outside waveguides, without caringabout the sample shape and positioning, the flange types, andother accessories.

In this paper, we propose such a method that can be usedto measure arbitrary-shaped dielectric slabs outside rectangularwaveguides (see Fig. 1). We qualitatively analyze the electro-magnetic fields in the discontinuity and use the understandingof field distribution to model the discontinuity as a Π-circuitwhere each circuit component can be extracted from themeasured S-parameters. We have found that the equivalentshunt impedance of the discontinuity is rather insensitive tothe sample shape and the structural details in the discontinuity,and it is only determined by the permittivity and thickness ofthe sample. This feature can be readily used for the extractionof dielectric permittivity if the sample thickness is known.

The paper is organized as follows: in Section II, we in-troduce the physical principle of the proposed method. InSections III and IV, we separately discuss the permittivity ex-traction methods for electrically thin and thick dielectric slabs.The measurement uncertainties are analyzed in Section V.

II. MEASUREMENT PRINCIPLE

In this section, we introduce the physical principle ofthe proposed measurement method. We start from the fieldanalysis for the measurement setup. Then, we model thediscontinuity as an equivalent circuit, and verify stability ofthe equivalent shunt impedance.

A. Field distribution in the discontinuity

Figure 1 shows the actual measurement setup basedon millimeter-range rectangular waveguides. An arbitrarilyshaped piece of a dielectric slab is positioned between twowaveguide flanges. The waveguide aperture is fully coveredby the dielectric sample. Although the structure of waveguideflanges contains many small details, for the following concep-tual analysis it is possible to simplify the flanges as planarmetallic walls. A cross-section of the discontinuity is shownin Fig. 2(a). Waves incident from Port 1 are partially reflectedand absorbed by the sample (shown as a black rectangle) andpartially leak to free space via the gap. The rest of the power

Port 1 Port 2

(a)

TM𝑛

+

TEM

TEM

TEM

Region I Region II

(b)

𝑧

𝑦

𝑥

Fig. 2. (a) Vectorial field distribution in the yz cross-section of themeasurement setup for excitation from Port 1. In the simulations, the operatingfrequency is 60 GHz, the waveguide aperture size is L = 3.76 mm andW = 1.88 mm (WR-15). Note that, throughout the paper, we use WR-15waveguides for all the numerical and experimental analyses. The dielectricslab has the permittivity of εr = 4(1−0.01j) and thickness of d = 625 µm.(b) Division of regions.

enters Port 2. The discontinuous junction is composed of aparallel-plate waveguide (PPWG) connected with a pair ofrectangular waveguides. As we know, if the waveguides arecontinuous, and the operating frequency is below the cutofffrequencies of higher-order modes, the rectangular waveguideonly supports the TE01 mode, and the PPWG only supportsthe TEM mode. However, in the junction, both rectangu-lar waveguides and PPWG are discontinuous. To adapt tothe configuration of the junction, higher-order modes of thewaveguides are excited. Therefore, the fields in the junctionhave a complicated composition, which is a combination ofwaveguide fundamental modes and many higher-order modes.The fields distribution in the junction region can be rigorouslycomputed using the mode-matching method [16]. Figure 2(a)shows the simulated electric field distribution in the waveguidejunction. As we can see, near the rectangular waveguideapertures, strong TMn modes (Ey 6= 0) of the PPWG areexcited, and these modes continue to propagate in the PPWGalong the y-direction. However, after the higher-order modesleave the junction region, they only see a continuous PPWG.Since the excitation frequency is below the cutoff frequenciesof these higher-order modes in the PPWG, these modes areevanescent and decay exponentially away from the junction.At some distance (see the top/bottom edge of the red dashedrectangle), the higher-order modes become negligible andonly the fundamental TEM mode continue propagation in thePPWG formed by two flanges.

B. Circuit modeling of the discontinuity

According to the field distribution in Fig. 2(a), the electro-magnetic environment of the discontinuity can be divided intotwo volumetric regions [see Fig. 2(b)]: Region I (highlightedin pink) encloses the volume where TMn modes survive;Region II (highlighted in green) includes the remaining volumeand all the surroundings outside the setup. Region I (Ey 6= 0)

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is directly connected with the waveguide ports and thereforeit can be viewed as a two-port system.

Any passive two-port electromagnetic system can be mod-eled by an equivalent Π-circuit. We model Region I as aΠ-circuit formed by two parallel admittances Yp (these twoadmittances are identical due to the structural symmetry)and one series impedance Za, as shown in Fig. 3. Forelectrically thin gaps, the shunt admittance is capacitive (aswill be numerically confirmed in the next section), becausethe vertical electrical field of TMn modes, Ey , representscapacitive coupling between the top and bottom edges ofthe waveguide apertures. Region II (Ey = 0) is an open-ended parallel-plate waveguide, which can be considered asa section of a transmission line terminated with an effectiveload impedance Zr as a model of the open end (Zr includesedge reactance and the radiation resistance of the open endof the PPWG). The input impedance of Region II (seen fromRegion I) is denoted as Zin which is a shunt connected toZa, as shown in Fig. 3. From Fig. 3, it is obvious that thewhole gap (including Regions I and II) can be characterizedas a unified Π-circuit, where the two shunt admittances Ypare inherited from Region I and the series impedance Zg isformed by a parallel connection of Zin and Za, denoted asZg = Za ‖ Zin.

Port 1 Port 2𝑌𝑝 𝑌𝑝

𝑍𝑎

𝑍in ∥ 𝑍𝑎 = 𝑍𝑔

𝑍𝑟

𝑍in

Region I

Region II

Fig. 3. Equivalent circuit of the measurement setup.

If the sample size is larger than Region I, modifications ofthe sample shape change the electromagnetic environment inRegion II and thus influence Zin. Moreover, for millimeterand sub-terahertz waveguides, the flanges walls are normallynot planar. Any additional passive structures in Region II, e.g.,fastening screws, choke grooves, tapped holes, and alignmentpins can be viewed as additional loadings of the PPWG, andthus they also affect the value of Zin. As a consequence,the total series impedance may significantly vary when theshape and size of the dielectric sample are changed. Evendifferent positioning of the same sample or different tightnessof fastening affects the series impedance. However, the shuntimpedance of the discontinuity will not be affected by thesestructural details in Region II, since it is only determined bythe capacitive coupling of the waveguide walls in Region Iwhich is very stable once the sample area is larger thanRegion I. For this reason, we can leverage the stable shuntimpedance to characterize slab samples with arbitrary shapespositioned between arbitrary flanges. Note that, the stability

of shunt impedance was noticed in our previous work [19],but at that time, we did not realize that it can be used forpermittivity extraction.

C. Stability of the shunt impedance

In the equivalent circuit of Fig. 3, the values of Zg andYp can be extracted from measured S-parameters. We use thetransfer matrix method. After expressing the circuit compo-nents Yp and Zg in terms of ABCD matrices, the total transfermatrix of the discontinuity can be calculated as the cascadedmultiplication of them:[

A BC D

]=

[1 0Yp 1

] [1 Zg0 1

] [1 0Yp 1

]

=

[1 + ZgYp ZgZgY

2p + 2Yp 1 + ZgYp

].

(1)

The matrix elements, A, B, C, and D, can be expressed asfunctions of S-parameters [20, § 4.4]. Therefore, we can relatethe circuit values with S-parameters. Parameter Zg can beexpressed as

Zg = B = Z0(1 + S11)2 − S2

21

2S21. (2)

Here, Z0 = ωµ0/√ω2µ0ε0 − ( πL )2 is the characteristic

impedance of the TE10 mode in the rectangular waveguide.Another equation can be written as

1 + ZgYp = A =1− S2

11 + S221

2S21. (3)

Solving Yp from (2) and (3), we obtain

Yp =A− 1

Zg=

1− S11 − S21

Z0(1 + S11 + S21). (4)

Next, we numerically demonstrate that Yp is insensitive tothe shape of the sample as well as to possible additional struc-tures inside the waveguide discontinuity. In the simulations,the values of the permittivity and the thickness of the dielectricslab are the same as assumed in Fig. 2(a). The measurementsetup is modeled in three different ways. In the first case,the sample covers the waveguide aperture with dimensionW × L and extends to the distance ∆s from the apertureedges, as illustrated in the first inset picture of Fig. 4 (top). Weincrease the extended size ∆s and extract the shunt impedanceZp = 1/Yp and series impedance Zg from the simulated S21

and S11 according to Eqs. (2) and (4). It can be seen thatas ∆s increases, the shunt impedance remains constant, whilethe series impedance is very unstable. This is because changesof the dielectric sample sizes modify the input impedance ofRegion II and thus change Zg dramatically. Notice that whenthe sample size is close to the waveguide aperture (∆s ≈ 0),the extracted shunt impedance becomes sensitive to the sizevariations. This is because in this case the dominating TM1

mode does not fully decay in the sample, and variations ofthe sample size affect the field distribution in Region I andtherefore change the shunt impedance. Obviously, there existsa critical extension size ∆scr, for which the amplitude of the

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TM1 mode decays to e−α (α is the decaying factor) of theoriginally excited amplitude when it propagates in PPWG.According to this criterion, ∆scr can be calculated as

∆scr =α√

(πd )2 − ω2εrε0µ0

(5)

Therefore, the size of the test slab should be larger than (L+∆scr)×(W +∆scr) to ensure that the higher-order modes arenegligible at the edges of the sample. The critical extensionsize for Case I in Fig. 4 is ∆scr = 0.69 mm for α = 3. Aswe can see from Fig. 4 (top panel), the shunt impedance ofthe gap does not change when ∆s > ∆scr.

In Case II, we choose an arbitrarily shaped dielectric slabthat is larger than the critical size. It is shown that the shuntimpedance still keeps unchanged. In Case III, the waveguideflanges are modified into a circular shape with actual screws,alignment pins, and choke grooves. We see that even withsuch a complicated gap environment, the shunt impedance Ypis still not affected at all.

𝑊

𝐿

Case I Case II Case III

Δ𝑠

Screw

Pin

Choke

𝑔 𝑔

𝑔

𝑝 𝑝

𝑝

Fig. 4. Extracted shunt (top) and series (bottom) impedances for differentsample dimensions. In Case I, the sample size is (L + ∆s) × (W + ∆s).In Case II, the sample shape is randomly chosen but it is larger than thecritical size. In Case III, the waveguide flanges are round with small accessorystructures.

The above numerical experiments fully verify the predic-tions based on circuit modeling, confirming that the shuntimpedance of the gap is insensitive to the gap environmentas well as to the shape and size of the sample. In the nextsection, we will show how to extract the permittivity from themeasured Yp.

III. CHARACTERIZATION OF ELECTRICALLY THINDIELECTRIC SLABS

Next, we discuss how Yp is related to the permittivity of thedielectric slab under test. Unlike the conventional waveguidecharacterization, here, the S-parameters have no explicit an-alytical relations with εr. Therefore, it is not straightforward

to find the permittivity from measured S-parameters and theshunt admittance Yp. In this section, we discuss the extractionmethods for electrically thin dielectric layers (d < λd/10),derive the extraction formulas, and show the measurementresults.

A. Extraction formula

For samples with ultra-subwavelength electrical thickness(d < λd/10), the fields in Region I are similar to the field ina material slab placed in a continuous rectangular waveguide.This is because very thin flange gaps have very large parallel-plate capacitances, allowing the currents on metal walls ofthe waveguides to pass through the gap. Therefore, the shuntimpedance of Region I can be approximated consideringthe same dielectric slab in a closed waveguide. To find theshunt impedance of a dielectric slab inside the waveguide, aconvenient way is to use the corresponding ABCD matrix.By equating the matrix elements with that in Eq. (1), we cansolve all the circuit components (Yp and Zg). The ABCDmatrix of a dielectric slab inside a continuous waveguide canbe expressed as[

A BC D

]=

[cos (βdd) jZd sin (βdd)

jYd sin (βdd) cos (βdd)

], (6)

where βd =√ω2µ0ε0εr − ( πL )2 is the propagation constant

in dielectric slab (TE01 mode), and Zd = 1/Yd = µ0ω/βd isthe corresponding characteristic impedance. After equating thematrix elements in Eq. (6) and Eq. (1), Yp can be analyticallysolved:

Yp =cos (βdd)− 1

jZd sin(βdd)=

1− S11 − S21

Z0(1 + S11 + S21). (7)

Once the S-parameters are measured, the above equationuniquely determines εr [note that in Eq. (7), βd is a functionof εr].

To examine the accuracy of extraction formula Eq. (7),let us consider a dielectric slab with d = 100 µm andεr = ε′r − jε′′r = 4 − j0.04. The electrical thickness isd = λd/25 at 60 GHz. We simulate the setup with theseassumed physical parameters and obtain the S-parametersfrom 50 GHz to 75 GHz. Using Eq. (7), we solve the complexpermittivity at each frequency point. The results are shownin Fig. 5. The retrieved permittivity perfectly agrees withthe value assumed in the simulation. It should be noted thatalthough for thin slabs only a very small amount of powerleaks away from the discontinuity, one cannot ignore it anduse the conventional Nicolson-Ross-Weir formulas (which arederived for closed waveguides) to extract the permittivity. InFig. 5, the extraction results using Eq. (7) and the Nicolson-Ross-Weir formulas [7] are compared. We see that even forsuch a thin gap, the Nicolson-Ross-Weir method does notwork due to the negligence of leaked power. The proposedmethod is, however, fully applicable, because it extracts thepermittivity via the shunt impedance, but not directly fromthe S-parameters.

It is important to stress that the permittivity extractionformula Eq. (7) is only accurate for ultra-thin dielectricmaterials, i.e., when |βd|d � 1. Under this condition, we

5

(a)

(b)

Fig. 5. (a) Real and (b) imaginary (bottom) parts of permittivity solved fromEq. (7), and using Nicolson–Ross–Weir (NRW) method. In the simulationsetup, the sample is rectangular shaped with ∆s = 1 mm.

can make the following additional approximations in Eq. (7):cos (βdd)−1→ −(βdd)2/2 and sin (βdd)→ βdd, and obtain

Yp ≈j

2

(ωε0εr −

π2

ωµ0L2

)d. (8)

Equation (8) implies that, at a fixed frequency, <(Yp) and=(Yp) are linearly dependent on ε′′r and ε′r, respectively.

B. Measurement results

The extraction formula Eq. (7) is suitable for thin-filmcharacterization at microwave and millimeter-wave frequen-cies, e.g., Polyethylene Naphthalate (PEN) and PolyethyleneTerephthalate (PET) films with the thickness around onehundred microns which is much smaller than the wavelength.It is important to mention that measuring extremely thin sheets(several tens of microns) requires more accurate mechanicalcontact between the sample and flange walls. With loosefastening, imperfection of contact can be a noticeable errorsource. To avoid this problem, one can stack several layersof thin film to increase the thickness of the measured sample,but the total thickness still should be much smaller than thewavelength.

Here, we measure the permittivity of common copy paperand experimentally show the stability of shunt impedance.We stack four layers of 80 µm thick STAPLES copy paper,forming a 320 µm thick sample (d = λd/10 at 60 GHz).The sample is cut into an arbitrary shape but larger than(L + ∆scr) × (W + ∆scr) to ensure the stability of shuntimpedance. The measurement comprises several steps:

1) Calibrate the system using Thru-Reflect-Line (TRL)method.

SUT

Fig. 6. Photo of the measurement setup. In this example, the SUT is fourlayers of stacked copy papers. The waveguides are connected to WR-15 VectorNetwork Analyzer Extender (black modules).

2) Embed the sample between the flanges (see Fig. 6). Notethat it is not necessary to use a sample holder. Onecan cut the sample into a long strip that covers thewaveguide aperture and hold it by hand when connectingthe waveguides. After the sample was placed, close thewaveguides and fasten the flanges using screws.

3) Measure the S-parameters of the setup. At this step,proper time gating can be applied to filter parasitic reflec-tions caused by waveguide misalignments. One shouldbe careful not to remove the harmless reflections fromthe sample edge and other structures inside the flanges,otherwise, the extraction results will instead become lessaccurate.

Fig. 7. Measured S-parameters of two paper samples of different shapes.

4) Record the S-parameters and use Eq. (7) to numericallyextract the permittivity.

Figure 7 shows the measured magnitudes of S-parameters fortwo samples cut in arbitrarily different shapes. One can seethat the measured S-parameters are obviously not the samefor the two samples, since the sample shapes and sizes inRegion II are different. The difference in the measured S-parameters will pass on to the extracted series impedance,which is different for different samples, as shown in Fig. 8(a).In contrast, the measured shunt impedance is very stable, asshown in Fig. 8(b).

The extracted complex permittivity [using Eq. (7)] of twopaper samples is shown in Fig. 9. The results for the twosamples are very close and stable in the studied frequencyranges, also agreeing with previously reported results [21]. The

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slight difference might result from different actual thickness ofthe paper samples caused by different tightness of the screws.

(a)

(b)

Fig. 8. Extracted (a) series and (b) shunt impedances of the discontinuitycreated by two different samples.

To further verify the accuracy of the method, we measurethe permittivity of Polyethylene Naphthalate (PEN) layers and,as a further validation check, of free space. PEN samples arestacked in two layers (the thickness of each layer is 125 µm),and the total thickness is about d ≈ λd/12 at 60 GHz. Theextracted permittivity is shown in the yellow curves in Fig. 9.The measured value is around εr = 3.05−j0.05, being in goodagreement with the previously reported values in [22], [23] atmillimeter-wave frequencies. We also measure the permittivityof air. The “air sample” is formed by opening an arbitraryshaped (but larger than the critical size) hole in an 400 µmthick FR4 laminates. The extraction results are shown in Fig. 9(purple curves), confirming good accuracy. In all the measuredsamples, it appears that the imaginary part of permittivitysuffers more perturbations than the real part. This is causedby uncertainties of measured S-parameters. The reason willbe explained Sec. V where the measurement uncertainties forboth real and imaginary parts of permittivity are analyzed indetail.

Importantly, one should remember that the extraction for-mula Eq. (7) is only accurate for electrically thin materials(d < λd/10). As the electrical thickness of SUT increases,the extraction formula gradually becomes inaccurate.

IV. CHARACTERIZATION OF THICK DIELECTRIC LAYERS

When the thickness of dielectric slabs increases (λd/10 <d < λd/2), the higher-order TM modes become more and

(a)

(b)

Fig. 9. Extracted (a) real and (b) imaginary parts of permittivity for differenttypes of materials.

more significant in Region I, and the field in Region I canbe obviously different from the field in the closed waveguide.Therefore, one cannot use a simple transmission-line sectionmodel for Region I, and the extraction formula Eq. (7)becomes inaccurate. Obviously, the relation between Yp andεr is not as straightforward as for thin samples.

A. Simulation-assisted extraction methodHere, we utilize numerical tools (Ansys HFSS), to find the

relation between Yp and εr. Numerical fitting is a commonmethod to extract material parameters from measured data. Bymodeling the measurement setup in numerical tools and fittingthe simulated S-parameters with the measured values, one canestimate the permittivity of the sample. In the conventionalnumerical fitting method, one should accurately model theactual measurement setup [10] since the modeling errors caninduce significant inaccuracy in the simulated S-parametersand thus result in erroneous estimations of permittivity. Toovercome this problem, instead of fitting the scattering param-eters, we fit the equivalent shunt impedance/admittance. As wedemonstrated in Sec. II-C, the shunt impedance/admittance isonly related to the thickness (which is easy to measure) andthe sample permittivity, and is not affected by details of thediscontinuity and external environment. In this way, one canavoid the need to accurately reproduce all the setup detailsin simulation tools. The measurement procedure compromisesthe following steps:

1) The sample is measured in a waveguide junction. Theshunt admittance is extracted from the measured S-parameters using Eq. (4).

2) The physical setup is modeled in the simulation tool. Notethat it is not necessary to accurately model the mea-

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surement setup in the simulation since the complicatedstructures of the waveguide junction and the shape of thesample (as long as it is larger than the critical size) doesnot affect the shunt admittance.

3) At the frequencies of interest, different values of permit-tivity (both real and imaginary parts) are assumed in thesimulation, and the shunt admittance is extracted fromnumerical results. The simulated shunt impedance is thencompared with the measured values. When the simulatedand measured values are identical, the permittivity as-sumed in the simulation is the actual permittivity of thematerial under test.

𝜖r = 6.58 − 𝑗0.6

(a)

𝜖r = 6.58 − 𝑗0.6

(b)

Fig. 10. (a) Real and (b) imaginary parts of shunt admittance at 75 GHz interms of real and imaginary parts of permittivity. The data is obtained fromnumerical simulation using HFSS.

For example, we assume the thickness of SUT is d =660 µm and the frequency of interest is f = 75 GHz. Wemodel the measurement setup in HFSS and vary ε′r and ε′′rwithin reasonable ranges. For each pair of ε′r and ε′′r , we cancalculate the corresponding shunt admittance using Eq. (4). Inthis way, we can plot <(Yp) and =(Yp) as functions of ε′r andε′′r , as shown in Fig. 10. The measured shunt admittance at75 GHz is Yp = 0.005 + j0.018. Then, we draw two contourcurves <(Yp) = 0.005 and =(Yp) = 0.018 in Fig. 10(a) and(b) (white solid curves), respectively. The intersection point ofthe two contour curves (εr = 6.58− j0.6) in Fig. 10(b) is themeasured value of permittivity at 75 GHz.

B. Measurement results

In this section, we measure the permittivity of a mobilephone screen glass (Corningr Gorillar Glass 6) with thethickness d = 660 µm. The shunt admittance is extractedfrom the measured S-parameters, as shown in Fig. 11(a).To extract the permittivity at all measured frequencies, it isnot efficient to fit the permittivity value at each frequencyone by one, following the procedure introduced at the end ofSec. IV-A. Here, we use a deep-learning technique to analyzethe simulation data and quickly extract the permittivity at allthe frequencies of interest.

In numerical simulations, we model the setup and per-form parametric studies in terms of f , ε′r, and ε′′r . Fromsimulations, we obtain more than 2000 sets of data,[f, ε′r, ε

′′r ,<(Yp),=(Yp)]simu. The task is to use the sim-

ulated dataset to find ε′r and ε′′r for a given set of

[f,<(Yp),=(Yp)]meas that is obtained from measurements.This is a multi-dimensional fitting problem. We use the NeuralNet Fitting app in MATLAB to train a fitting network. In themodel training, the input datasets are [f,<(Yp),=(Yp)]simu

and the output datasets are [ε′r, ε′′r ]simu. The Levenberg-

Marquardt Algorithm is chosen to train the neural network.For 2000 datasets, training can be completed within severalseconds. Once the fitting model is trained, the measureddatasets [f,<(Yp),=(Yp)]meas are fed to the model as inputs,and the output is the predicted permittivity. Figure 11(b) showsthe extracted permittivity in the measured frequency range.

The real part of permittivity is between 6.6 and 6.8, whichis in good agreement with the reference value (εr = 6.69 −j0.087 at f = 3 GHz, measured in [24]). At millimeter-wave frequencies, the material loss (the measured values0.4 < ε′′r < 0.66 over this frequency range) significantlyincreases as compared to the provided value at microwavefrequencies.

(a)

(b)

Fig. 11. Measured shunt impedance (a) and permittivity (b) of CorningrGorillar Glass 6 used for screens of mobile devices.

V. UNCERTAINTY ANALYSIS

The measurement uncertainty originates from inaccuraciesin the measurement of sample thickness, flanges alignments,imperfect contact between SUT and flange walls, and soon. The measurement errors caused by those factors can bereduced by using high-precision thickness characterizationdevices (e.g., profilometers) and careful assembling of themeasurement setup. Other important sources of measurementerrors include uncertainties of the measured S-parameters(both magnitude and phase), which are unavoidable and de-termined by the VNA device parameters.

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In this section, we study the measurement errors caused byuncertainties of the measured S-parameters. The uncertaintiesof S-parameters on both magnitude and phases are denoted as|∆Sα| and ∆θα (α = 11, 21). To analyze the impact of theseparameters on the extracted permittivity, we use the differentialmethod, where the dependent variable, εr, is differentiatedwith respect to each possible error parameter (the magnitudesand phases of the S-parameters involved in the extractionmethod) [19], [25]. Since each derivative can take positiveor negative values, the final error is calculated as a sum of thesquared values of all derivatives:

∆εr =

√√√√∑α

(∂εr∂|Sα|

∆|Sα|)2

+∑α

(∂εr∂θα

∆θα

)2

, (9)

where∂εr∂|Sα|

=∂εr∂Yp

∂Yp∂|Sα|

,∂εr∂θα

=∂εr∂Yp

∂Yp∂θα

. (10)

In Eq. (10),

∂Yp∂|Sα|

=−2

Z0F 2

Sα|Sα|

,∂Yp∂θα

=−2j

Z0F 2Sα, (11)

with F = 1+S11 +S21. Furthermore, in Eq. (10), ∂εr/∂Yp =2/ (jωε0d) according to Eq. (8) for thin dielectric samples. Fora thick layer, the linear dependence of Yp on εr does not hold,as seen from Fig. 10. Therefore, numerical fitting techniques,e.g., ‘lsqnonlin’ function in MATLAB, are needed to modelthe nonlinear relation between εr and Yp.

(a)

(b)

Fig. 12. Measurement uncertainties for (a) real and (b) imaginary partsof permittivity caused by each error source (|∆Sα| and θα) and their totaleffects.

Let us assume that the sample thickness is d = 280 µmand the testing frequency is f = 60 GHz. In the first analysis,we evaluate the uncertainty of the real part of permittivity. Todo this, we fix tan δ = 0.01 and sweep ε′r from ε′r = 1 to

ε′r = 30 in simulation, and obtain the corresponding Sα foreach permittivity value. In real measurements, the uncertaintiesof |∆Sα| and ∆θα depend on the magnitude of Sα, andthe dependence can be obtained from Keysight UncertaintyCalculator for a specific vector analyzer [26]. In this work,we choose E8361C Vector Network Analyser with V11644ACalibration Kit in the calculator. Therefore, for each set of ε′rand ε′′r , we can calculate the uncertainties of ∆ε′r and ∆ε′′rusing Eq. (9), where ∂εr/∂|Sα| and ∂εr/∂|θα| are obtainedfrom the simulation, and, ∆Sα and ∆θα are provided by VNAmanufacturer.

The uncertainty of ε′r is plotted in Fig. 12(a). It can beseen that the measurement uncertainty remains relatively low(below 5%) for ε′r ranging from ε′r = 1 to ε′r = 20. Theuncertainty reaches its minimum (1.5%) for ε′r ≈ 4. For largerε′r, ∆ε′r increases. This is because as the slab becomes morereflective, the uncertainty of |S11| increases as its magnitudeincreases, which results in a decrease in the measurementaccuracy.

In the second example analysis, the uncertainty of theimaginary permittivity is evaluated. In this case, we fix ε′r = 5and vary ε′′r from ε′′r = 0 to ε′′r = 1 in simulations.We can see in Fig. 12(b) that, for high-loss dielectrics, themeasurement errors of ε′′r are small. However, for low-lossdielectrics, the relative uncertainty significantly increases. Thisis because when the wave goes through a low-loss dielectricslab, the attenuation cannot be sufficiently accumulated, andthe changes of S-parameters caused by material losses arenot evident. In this case, uncertainties in S-parameters caneasily cause inaccurate estimations of the loss tangent. Thisis the common shortcoming of the transmission/reflectionmethod for measuring thin low-loss material samples [25].For lossy dielectrics, the influences of material loss on themeasured S-parameters are observable, and the imaginary partof permittivity can be estimated accurately.

Finally, we should note that the S-parameter uncertaintiesprovided by the VNA manufacturer are their worst values. Inreality, the perturbations of S-parameters are not so strong, andthe uncertainties shown in Fig. 12 might be overestimated. Agood evidence is Fig. 5(b) where the extracted imaginary partof permittivity does not fluctuate as strong as estimated.

VI. CONCLUSION

To summarize, this paper reports a fast and robust method tomeasure dielectric slabs in a rectangular waveguide junction.The method does not require meticulous control of sampleshape and position, which is particularly useful for millimeter-wave and sub-terahertz-wave measurements. The physics be-hind this method is that, the equivalent shunt impedance ofthe waveguide junction is only related to the permittivity ofmeasured material if the thickness of the sample is knownin advance. We develop an analytical formula to extract thepermittivity of electrically thin materials (d < λd/10). Forthick dielectrics (λd/10 < d < λd/2), numerical tool isneeded to extract the permittivity. The method can accuratelyretrieve the real part of permittivity, while the prediction of theimaginary part is accurate only for medium-loss and high-lossmaterials.

9

VII. ACKNOWLEDGEMENTS

The authors would like to thank Francisco Cuesta for hishelp in glass sample measurements.

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