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01 May 2008

Multimodal Solution for a Rectangular Waveguide Radiating into a Multimodal Solution for a Rectangular Waveguide Radiating into a

Multilayered Dielectric Structure and its Application for Dielectric Multilayered Dielectric Structure and its Application for Dielectric

Property and Thickness Evaluation Property and Thickness Evaluation

Mohammad Tayeb Ahmad Ghasr Missouri University of Science and Technology, mtg7w6@mst.edu

R. Zoughi Missouri University of Science and Technology, zoughi@mst.edu

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Recommended Citation Recommended Citation M. T. Ghasr and R. Zoughi, "Multimodal Solution for a Rectangular Waveguide Radiating into a Multilayered Dielectric Structure and its Application for Dielectric Property and Thickness Evaluation," Proceedings of the IEEE Instrumentation and Measurement Technology Conference, 2008 (2008, Victoria, BC, Canada), pp. 552-556, Institute of Electrical and Electronics Engineers (IEEE), May 2008. The definitive version is available at https://doi.org/10.1109/IMTC.2008.4547098

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I2MTC 2008 - IEEE International Instrumentation andMeasurement Technology ConferenceVictoria, Vancouver Island, Canada, May 12-15, 2008

Multimodal Solution for a Rectangular Waveguide Radiating into a Multilayered DielectricStructure and its Application for Dielectric Property and Thickness Evaluation

M.T. Ghasr, R. ZoughiApplied Microwave Nondestructive Testing Laboratory (amntl)

Electrical and Computer Engineering DepartmentMissouri University of Science and Technology (Missouri S&T)

(formerly University of Missouri-Rolla), Rolla, Missouri 65409, USAPhone: (573) 341-4728, Fax: (573) 341-6671

Abstract -Open-ended rectangular waveguides are widely used formicrowave and millimeter wave nondestructive testing applications.Applications have included detecting disbonds and delaminations inmultilayered composite structures, thickness evaluation of dielectricsheets and coatings on metal substrates, etc. When inspecting acomplex multilayered composite structure, made of generally lossydielectric layers with arbitrary thicknesses and backing, the dielectricproperties of a particular layer within the structure is ofparticularinterest, such being health monitoring ofstructures such as radomes.The same is also true where one may be interested in the thickness ormore importantly thickness variation ofa particular layer within suchstructures. An essential toolfor estimating the dielectric constant orthickness is an accurate modelfor simulating reflection coefficient atthe aperture of the probing open-ended waveguide. One issue ofinterest is that radiation from open-ended rectangular waveguidesinto layered dielectric structures has been considered only whenaccountingfor the dominant waveguide mode. However, when usingthese models for recalculating dielectric constant or thickness, theresults may not be accurate (depending on the measurementrequirements). To this end, this paper provides an accurate modelforthe reflection coefficient which also accounts for the effect of higher-order modes. Finally, the potential of this model for accuratelyestimating dielectric constant is shown.

Keywords - open-ended waveguide, stratified dielectric medium,higher-order modes, dielectric constan, thickness.

I. INRODUCTION

Open-ended rectangular waveguides are the most widelyused probes in near-field microwave and millimeter wavenondestructive testing (NDT) applications, imaging such asdielectric property measurement, thickness measurement of adielectric slab, crack detection, and porosity level estimation toname a few [1]. When operating in the near-field, spatialresolution is a function of probe dimensions and henceobtaining high-resolution images are readily possiblecompeting very well with other NDT methods [2]. In addition,when concerned with multilayered composites, thicknessvariation detection can be conducted with micrometer rangeaccuracy at relatively low microwave frequencies [1, 3].Furthermore, microwave NDT has the potential of providinginformation about electrical properties of dielectric structures.This is particularly important for inspecting structures such asradomes since the electrical properties of each layer in thestructure directly affects whether a radome is transparent toelectromagnetic radiation or not. Dielectric property

measurement using open-ended rectangular waveguides hasreceived significant attention both experimentally and from themodeling point of view. These works have included singlelayer or infinite half-space material evaluation [1, 4-6]. Anissue of concern in multilayered composite characterization iswhen the effect of a particular layer on the reflectioncoefficient is small and any small error in modeling ormeasurement causes substantial error in evaluating theproperties (thickness or dielectric) of another layer. In suchcases the modeling error is commonly associated with ignoringthe influence of higher-order modes while only taking thedominant mode into account. Subsequently, when the modelis used in an inverse manner (or forward iterative) formeasuring dielectric constant or thickness, slight to significanterrors may be experienced. Some of these studies have utilizedvariational method, which results in an approximate solution[3], and some utilize more rigorous formulations including theeffect of higher-order modes [4-6]. None of these studiesexcept [4] offers a solution for reflection from a multilayeredstructure while taking higher-order modes into account.Furthermore, no study has shown the contribution of higher-order modes as function of various structural features of amultilayered structure such as thicknesses, dielectric profiles,and when the composite is backed by a dielectric infinite half-space or a conducting plate. As this paper will show, the effectof higher-order modes is much more prominent whenanalyzing multilayered structures.

This paper gives an exact formulation based on Fourieranalysis for the reflection coefficient at the aperture of anopen-ended rectangular waveguide irradiating a stratifieddielectric structure. This formulation accounts for thecontribution of TE and TM higher-order modes.Subsequently, an analysis of the effect of higher-order modesper structure features (thickness and dielectric constant) andnumber of higher-order modes required for convergence willbe presented. Finally, the use of this model for extraction ofdielectric constant will be presented through measuringreflection coefficient of lossy rubber and low-loss plexiglasssheets.

II. FORMULATION AND ANALYSIS

The reflection coefficient seen by a waveguide irradiating astratified dielectric medium is formulated in a similar fashionto [5-6]. Yoshitomi et. al. [5] considered a waveguide with a

1-4244-1541-1/08/$25.00 C 2008 IEEE

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lossy flange radiating into free-space for the purpose ofexamining the influence of the flange on the radiation patternof open-ended waveguides, while and Bois et. al. [6] expandedthese formulations for application to accurate extraction of thedielectric constant of a generally lossy infinite half-space of adielectric material. This paper extends this formulation to ageneral case of a stratified dielectric medium possessing anynumber of layers (backed by an infinite half-space or aconductor) for the purpose of accurate extraction of dielectricconstant or thickness of any layer. Fig. 1 shows a schematic ofa waveguide radiating into a multilayered structure. Thewaveguide has a broad dimension of 2a and a narrowdimension of 2b. The waveguide has an infinite flange andradiating a multilayered dielectric structure backed by aconductor or an infinite half-space (IHS) of a dielectric. Each

layer is defined by its complex permittivity (Er =r -jEr ) and

permeability (,ur =,ur-j, ) as well as its thickness, d.The electric and magnetic fields in all regions may be

derived from Hertzian vectors. Inside the waveguide, for theincident field a magnetic Hertzian vector with a TElo modedistribution is used. On the other hand, the reflected waves arerepresented by electric and magnetic Hertzian vectors whichrepresent a summation of all possible TM and TE waveguidemodes. The definition for these Hertzian Vectors may befound in [5] and [6].

*yInfiniteGroundplane

Frl

di

Iz

-b

£rL

,trL

dL

Conductoror

/ IHS of Dielectric

zE0

aze

(X,y,z) =

4zT2K 2 ff [Ae (j, 77)e-I- 00 0

(1)+ Ae- (j, r)ej'z I e-j(jx+qy)djd77

.fJh (X,y,z) =

4z2K 2Z

f f[AI ( 77)e -+ Alh- (j 77)ejz ]e-j( x+' y)djd774

I oo oo

where K = EJ,I z vi,I =K2= , and

Ae,h (,) are the spectral domain wave functions representing

radiated electromagnetic fields from the waveguide. Thesefunctions are unknown and are found by applying theboundary conditions. Since there are discontinuities along thez-direction, forward and backward propagating waves arepresents in each layer which are donated by a + and-superscripts respectively. Furthermore, subscript 1 denotes thelayer number. Expressions for converting Hertzian vectorsinto electric and magnetic fields, and Fourier relationshipsmove the field between the spatial and spectral domains maybe found in [5] and [6].The solution is obtained by matching boundary conditions at

all interfaces. This process starts at the last layer ( z = ZL ). For

a conductor backed (CB) case, the tangential electric field iszero resulting in total reflection:

On the other hand, for the IHS case, the backward travelingwave is zero, since there is no mechanism for reflections,therefore:

AL =0, and AL = 0 (4)

Matching the boundary conditions at intermediate layersresults in the following relationships between the forward andbackward traveling waves:

A9

e.

cee2j~zP-2jSl-,zl-I I 1-I (I+Be)-e2jSlzj (1-Be )

-C, e jS-1(I-B )+e2j' (t+B;)Fig. 1. The geometry of the problem.

where,

In the multilayered dielectric structure, the electric andmagnetic Hertzian vectors are defined as follows:

e Zl_lK1;I_lBI ZIKI- lS

(6)

(2)

L ALe L ,and AL = LA+ e-2j (3)

(5)

Y4

cz

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Ce1l-1Ce-2jSd, 1+ B, e

-Cee2;jSjdj I e + I + Be

Similar expressions are obtained for the magnetic fieldcomponents:

Bh= 1IIZ, 1K1

hc1-

(7)

(8)

(9)

The Ce,h coefficients are obtained iteratively starting frome hthe last layer where CL = 1Ci=I for a CB case, and

Ch Ce 0 for an IHS case.

The boundary conditions at the waveguide aperture (z=0 ),dictate the continuity of the total electric and magnetic field, asfollows:

|x| < a, y| < b

elsewhere

Z [a.3 (m,n, p,q)+b4 (m,n,p,q)]K Ae

+ , [b 3 (m,n, p,q) - aI4 (m,n, p,q)]KOAmnm,n=0m=n#O

+ab[A>K0ap+Ah Kpqb (1+3 )]zo

= K0ai (1,0,p,q)A

(13)

forp=O,1, 2, 3, ..., q=], 2, 3, ....

In these equations A e and Ah are the coefficients of thereflected and/or generated TM and TE modes at the waveguideaperture am m b2 K K2 2 s the

2a 'n 2b ' mn=

amn pqishKronecker delta, and the integrals 1,234 are:

I (m, n, p, q) 2 f fm ()Sb ()Sp(/)Cb(j1)d%Lq (14)

I2 (m, n, p, q) 1 2 J )Cb() )Cb(1)d d1 (15)4fT

(16)(10) I, ( m, n, p, q ) = 2f f|3Cm (()4 (f

HXY =HY||<wgH,y Hx, lx.a, y.<b (1 1)

where the superscript wg denotes total fields in the waveguideand superscript 1 refers to the total fields in the first layer.Applying these boundary conditions and utilizing theorthogonal properties of the modes in the waveguide in similarfashion to [5] and [6] resulting in these two linear systems ofequations:

[afIM (m,n,p,q)+bb2 (m,n,p,q)]K YA'

+ Y [bjI, (m,n,p,q)-ajI2 (m,n,p,q)]K,Ajj

m=n#O (12)

+ab[A>K0b -APqKpq a (1+ dq)]Z

= Ko alI2 ( 1, O, p, q ) - 2KIo a, abZ, glpgOq ]A'zf

for p-i , 2, 3 q =O, 1, 2, 3 ,and

14(m, n, p,q) 12 S (_)C (_77)Ca(>)S>(17)dCdq (17)4T

The variables V,23 contain information about the properties ofthe multilayered dielectric structure and defined as:

K2g~21ghDe1,-,Z17DI (18)

VI K ( j2 + 172 ) etD (19)

K17 + 1 (20)

K 2 21 qD

1-Che21 (22)1 _C1 e

-2jsldl

e +e2D IeI

;Cl 2rd (22)

The integral in (14) to (17) may become singular, especiallywhen the total loss in the multilayered dielectric structure islow. When numerically evaluating these integrals, it isadvantageous to transform these integral to polar coordinatesas reported in [5]. However unlike the cases investigated in

h -2j.,d, h ) h )C, e (I+BI -(I-BI

-Clh e-2j.jdj (I-B h )+(I+B h )

[f.9-I X'yE =

X'y 0

_77)C a (.)S'(77)d.dqp q

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[5] and [6] where the structure is always an IHS and thesingularity location is known and analytically extracted, in amultilayered structure, the singularity of the integrals dependson profile of the structure. Therefore, analytically extractingthe singularities is not feasible and the integration is performedon a contour around singular points. Furthermore, aconversion test on these integrals is performed prior toobtaining a solution.

III. ANALYSIS

To verify the formulation, the complex reflection coefficient(F = IFr eo ) seen by an X-band (8.2 -12.4 GHz) waveguideradiating into free-space was compared to results obtainedusing HFSSTM which employs 3D electromagnetic simulationsusing finite element methods (FEM). The results are shown inFig. 2. These results show that while there is a majoradvantage in including higher order-modes in the solution,their contribution becomes minimal beyond the first fewmodes. In fact with only 6 modes, it is possible to approximatethe actual solution adequately as similarly reported in [6].Consequently, from now on all solutions that include higher-order modes will only utilize 6 modes.

In [6], it was shown that the error in TEIO reflectioncoefficient due to the exclusion of higher-order modesdecreases as permittivity and loss factor increase. However,that finding is only true when the waveguide is radiating intoinfinite half-space of dielectric. In the cases of a finitedielectric slab, or a multilayered case, the thickness of thedielectric layers has an effect on the contribution of higher-order modes as well. To investigate this phenomenon,simulations were performed for a slab of rubber with a relativedielectric constant of ( r = 7.3 - j 0.3) and a slab of Plexiglaswith a relative dielectric constant of ( r = 2.6 - j 0.01). Fig. 3shows the error in calculating TE-1 mode reflection coefficientin terms of the slab thickness (normalized to the wavelength inthe dielectric slab) due to the exclusion of higher-order modes.The results show that, as the thickness of the slab increases(moving toward infinite half-space), materials with higherpermittivity and loss factor (i.e. rubber) produce lesssignificant higher-order modes compared to material with lowpermittivity and loss factor (i.e. plexi glass) and it agrees withthe results reported in [6]. Conversely, when the thickness ofthe slab becomes small, the opposite occurs and thecontribution of higher-order modes becomes more significantfor high permittivity materials.

0.09

0.26r-----1-----T----- -----------.1

0.25

0.24 -(

L- 0.23

0.22

0.21

0.28

1 mode----- 6 modes

15 modesO HFSS

9 10 11Frequency (GHz)

(a)

----- RubberPlexi glass0.08

0.07

0.06

H-.: 0.05

LU0.04

0.03

12 130.02

0.01 L0 0.2 0.4 0.6 0.8 1 1.2 1.4

Normalized Slab Thickness (X)1.6 1.8 2

a)

L

-90

-9518 10 11

Frequency (GHz)12 13

(b)

Fig. 2. Reflection coefficient seen by an X-band open-ended waveguideradiating into free space; the effect of higher order modes (a)

magnitude, and (b) phase.

Fig. 3. Error in reflection coefficient calculation due to the exclusion ofhigher-order modes vs. normalized slab thickness.

IV. INVERSE PROBLEM

The main goal of this study is to extract the dielectricconstant (or thickness, not specifically shown here) of amaterial within a multilayered structure. To illustrate this,measurements were performed with an Agilent 8510C vectornetwork analyzer (VNA) using a sheet of 4.42 mm-thickrubber utilizing the setup shown in Fig. 4. The rubber sheethad an area of 15 x 15 mm2, large enough to contain theradiations of the open-ended waveguide.

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Fig. 4. Measurement setup.

The dielectric constant is found by minimizing theEuclidean distance, i.e. cost function (43), between themeasured and simulated reflection coefficients, 1m (usingVNA) and Fs (using the proposed formulation), respectively.

e= Fm -I7- s 2

f(43)

This Euclidean distance is iteratively minimized by varyingrelative dielectric constant r using the minimax optimizationalgorithm [7]. Table 1 shows a solution obtained for measuringdielectric constant of the rubber sheet, with and without takinginto account the effect of the higher-order modes. Theseresults are compared to measurements performed using loadedwaveguide technique [8-9]. When taking the higher-ordermodes into consideration, the estimated r is within less than1% of the actual r for the real part and within less than 10%for the imaginary part which is within the accuracy limits ofthe loaded waveguide technique [8]. These results also showthat not taking higher-order modes into account, significanterrors are encountered specially in estimating the loss factor.

Table 1. Measured dielectric constant of rubber.

V. SUMMARY

Results of formulating reflection properties of an open-

ended rectangular waveguide irradiating a stratified dielectricmedium were presented. This formulation utilizes Fourieranalysis which provides a complete non-approximate solutionincorporating higher-order modes. It is shown that thecontribution of higher-order modes is frequency and structuredependant. Most importantly, when measuring reflectioncoefficient from a dielectric slab, the effect of higher-ordermodes increases significantly when the slab is thin relative tothe wavelength. This formulation is used to measure thedielectric constant of thin slab of rubber and significantimprovement in measurement accuracy is obtained.Potentially, dielectric constant of a layer in a multilayeredstructure or the dielectric constants of multiple layers may beobtained using this formulation. Furthermore, if the thicknessof dielectric layers is of interest, it may also be accuratelyestimated. The results of applying this technique to thicknessevaluation will be presented in the full paper.

REFERENCES

[1] R. Zoughi, Microwave Non-Destructive Testing and Evaluation, KluwerAcademic Publishers, The Netherlands, 2000.

[2] M.A. Abou-Khousa, A. Ryley, S. Kharkovsky, R. Zoughi, D. Daniels, N.Kreitinger and G. Steffes, "Comparison of X-ray, millimeter wave,

shearography and through-transmission ultrasonic methods forinspection of honeycomb composites," in Review of Progress inQuantitative Nondestructive Evaluation Vol. 26B, Portland, Oregon, July31- August 5, 2006, AIP Conference Proceedings, edited by D.O.Thompson and D.E. Chimenti, vol. 894, AIP, Melville, NY, 2007, pp.

999-1006.[3] S. Bakhtiari, S. Ganchev and R. Zoughi, "Open-ended rectangular

waveguide for nondestructive thickness measurement and variationdetection of lossy dielectric slabs backed by a conducting plate," IEEETransactions on Instrumentation and Measurement, vol. 42, no. 1, pp.

19-24, February 1993.[4] V. Theodoris, T. Sphicopoulos, and F. E. Gardiol, "The reflection from

an open-ended waveguide terminated by a layered dielectric medium,"IEEE Transactions on Microwave Theory Techniques, vol. MTT-33, pp.

359-366, May 1985.[5] K. Yoshitomi and H. R. Sharobim, "Radiation from a rectangular

waveguide with a lossy flange," IEEE Transactions on Antennas andPropagation, vol. 42, pp. 1398-1403, October 1994.

[6] K. J. Bois, A. D. Benally, and R. Zoughi, "Multimode solution for thereflection properties of an open-ended rectangular waveguide radiatinginto a dielectric half-space: the forward and inverse problems." IEEETransactions on Instrumentation and Measurement, vol. 48, no. 6, pp.

1131- 1140, December 1999.[7] Brayton, R.K., S.W. Director, G.D. Hachtel, and L.Vidigal, "A New

Algorithm for Statistical Circuit Design Based on Quasi-NewtonMethods and Function Splitting," IEEE Transactions on Circuits andSystems, vol. CAS-26, pp. 784-794, September. 1979.

[8] J. Baker-Jarvis, M. D. Janezic, J. H. Grosvenor, and R. G. Geyer,"Transmission/reflection and short-circuit line method for measuringpermittivity and permeability," NIST Tech. Note 1355-R, U.S. Dept.Commerce, Boulder, CO, pp. 52-57, December 1993.

[9] Bois, K, L. Handjojo, A. Benally, K. Mubarak and R. Zoughi,"Dielectric Plug-Loaded Two-Port Transmission Line MeasurementTechnique for Dielectric Property Characterization of Granular andLiquid Materials," IEEE Transactions on Instrumentation andMeasurement, vol. 48, no. 6, pp. 1141-1148, December 1999.

Calibration,,/plane

IHS of Air

Rubber

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