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Second-class postage paid at New York, NY and at additional mailing offices. %stmm.ten Send Address changes to IEEE, 445 Hoes Lane, Piscataway, NJ 08354. IEEETRANSACTIONSON MICROWAVE THEORYANDTECHNIQUES, VOL. MTT-31, NO. 3, MARCH 1983 249 A Fast Computational Technique for Accurate Permittivity Determination Using Transmission Line Methods LEO P. LIGTHART Abstract—Afast analyticalmethodisgiven for determining perrnittivity measured with sufficient accuracy by a slotted line, or by a characteristics at microwave frequencies. The experimental setup uses a calibrated network analyzer (type HP 8542B) at different single-moded cyfindrfcaf waveguide filled with dielectric and followed by a positions of the moving short. For the use of the moving load or by a moving short. In this way, transmission-reflection and short in this measurement technique, one is referred to short-circuited line methods are compared. By includhrg the uncertainties in length and in the reflection and transmission parameters, the permittfvity Deschamps [2]. The coupled scattering coefficients and uncertainty region is determined. It is shown that for optimum accuracy of their mutual uncertainties are calculated from these mea- the pernrittfvity, specific lengths in combination with a moving short are surements. The computations for the scattering coefficients needed. and the permittivity, including their uncertainties, can be I. INTRODUCTION performed within a negligible computing time by using the analytical expressims derived in Sections II and III. T HE MOST ACCURATE determination of the per- It is claimed that this method is useful for accurate rnittivity of dielectrics at high frequencies can be determination of the permittivity over a wide range of the obtained by using high Q resonant circuits. The main frequency, the dielectric constant, and the loss factor. This disadvantages of this method are that it can be applied permittivity measuring technique fills up the gap between only in a narrow frequency range, and that it is necessary the existing transmission-line and resonator-measuring to design the resonator. Permittivity measurements over a techniques. Further, it allows a reconsideration of the wide range of frequencies can be done, with reduced accu- existing methods in view of the possible known inaccu- racy, when transmission line methods are used. A homoge- racies. The computer programs are simple and can be run neous dielectric with specific sample length is positioned in on a 16-K minicomputer. the terminated transmission line, and the permittivity can be derived from the measured transmission and reflection II. THE MUTUAL DEPENDENCIESOFTHE SCATTERING coefficient. In the case of finite measurement uncertainties, COEFFICIENTSAND THE PERMITTIVITY Stuchly and Matuszewski [1] found a considerable permit- In this section, we derive, under very strict model as- tivity uncertainty. This uncertainty can be reduced by sumptions, two independent equations, both of which give using the mutual relationship between the scattering coeffi- the permittivity as a function of the two measured scatter- cients. The method described here takes this relationship ing coefficients. Starting from a given uncertainty limit of into account, and gives optimal results for a transmission one of the scattering coefficients, we compute the accom- line setup as found via the uncertainty analysis (Section panying scattering coefficient and vice versa. Only those III). The advantages of the present method are: a) the scattering coefficients within uncertainty limits are used to computation times required are small (a few milliseconds compute the permittivity. For this purpose, the average per permittivity uncertainty region); and b) the analytical values and the extreme variations of the scattering coeffi- expressions allow an optimization of the measurement cients are selected. configuration. Attention is given to model imperfections, and at the The disadvantages are: a) the limitation of small mea- end of this section the measuring method with the moving surement uncertain y regions; and b) the complicated anal- short is shown to obtain the best measurement accuracy. ysis. To derive the theory, the cross section of a cylindrical This method is based on small measurement uncertain- single-moded waveguide is shown in Fig. 1 filled over a ties and a single-moded cylindrical waveguide of specific length 1with a dielectric having unknown permittivity c. length filled with dielectric and terminated with both a The propagation constants are yOin the empty waveguide load and a moving short. For a set of frequencies, the and y in the filled waveguide, and depend on the mode scattering parameters at the waveguide reference planes are number m, the waveguide dimensions, the radian frequency u, and the dielectric constant n’= &. Manuscript received February 9, 1982; revised November 2, 1982. We know that The author is with the Delft University of Technology, Department of Electrical Engineering, Microwave Laboratory, P.O. Box 5031, Delft, the Yo= F’k,–ko (1) Netherlands. 0018-9480/83/0300-0249$01.00 01983 IEEE 250 IEEETRANSACTIONSONMICROWAVE THZORY ANDTECHNIQUEVSO,L.MTT-31, NO. 3, MARCH 1983 reference plane 1 reference plane 2 I I t I , i ‘Re(S211 Fig. 1. Cross section of waveguide filled with dielectric. where kC= wave number at the cutoff frequency, depend- ing on m and the waveguide dimension (e.g., kC= r/a for the TEO1 mode where a = width of the waveguide and kC= Ofor the TEM mode); and kO= wave number in free — space = o= where E. and POare the permittivity and Fig. 2. Uncertainty regions around $1, and S21.S1,,., S11,b, SII.,, and S,l, dare the uncertainty limits for S1l and S21,.; S~l,b, S21,.. and S21,d permeability in free space, respectively; and that for %2,. -(nko)2. y=@: (2) where the positive root can be taken in yl because in (9) The scattering matrix coefficients at the reference planes only cosh(yl) is present. Consequently, not only the mea- of this symmetrical two-port become [1] surement accuracy, but also the relationship just calculated between S1, and S21,influences the uncertainties. This can sl, =s22=–p(l –w2)/(1–p*l’v’) (3) be proved by analyzing the limits in the uncertainty regions s2, =s,2=w(l –p’)/(1–p’w2) (4) which are due to measurement inaccuracies. The uncer- tainty knits S1~,~, S1~,~, S1,,~, and S1~~ around the mea- where sured S1l and those of S21,~, S’l,b, S21,=, and S21,d around P = (Y/Yo – 1)/(Y/Yo + 1) the measured ~21 can be visualized as in Fig. 2. The and different limits are substituted successively in (9) to com- pute the corresponding scattering coefficient. The proce- w=exp(–yl). dure is asfollows. From (3) and (4), the reflection coefficient p and the Starting from S1~= S1~,~, we compute the corresponding propagation factor w can be derived as follows: Sj by assuming (see Fig. 2) -(l+s;, -s; )i-/(l+s~, -s;,) 2-4s:, S;l = S*,+ 8;, (lo) p= (5) 2s,, Y;,l=YOZ ((h%)J22)-/({S(jl,+s1,,a)2-(g, )2} . (11) To find ii;l, we approximate (9) by the first terms of the Taylor expansion which means that (Y/Yo)2= {(1- s,,)2-s:,}/{(1+ ~,,)2- ~:,} (7) O*= {(1 – S21)2– Sfl}/{(l + S21)2– Sfl} (8) (&,)2–2$lcosh(y;lz) 1–(s1,,a)2+ Sjl = where o = tgh(yl/2). l–(sll,. )2+(31)2 Equations (7) and (8) relate the propagation constant –2~2, + 32, and the propagation factor to the scattering coefficients. To determine y, we have two different equations, namely (7) and (8), in which y is related to the scattering coeffi- +2 S21sinh(y~11) - (12) 21 cients. With the exact scattering coefficients S,, and S21 and a given length 1,we are able to compute the permittiv- ~(Y$lz) = _y;,l 1 ity by using either (7) or (8). This means that by elimina- a~2, tion of y, a mutual dependency h(S1,, S21) between S1~and { (1-%.) 2-(s,)2 S21can be derived 1 q. (13) 1– s:, + S;l ~(511,521)= s –2cosh(yl) = O - (1+s,,,=)2-(s,)2 } 21 We follow the same procedure for S1~= S1~,~, S1,,~, and Yl=yol {(1–s,*)2– s;,}/{ (l+s,, )2–s; J 511,~ to compute ~~1, ~~1, and 8il, respectively, whale for S*, = S’,,o, szl,~, %,., and %,d we compute 8;1, 13~1,~fl (9) and i3~1,respectively, by a set of equations as listed below LIGTHARTT:ECHNIQUFEORACCURATPEERMHTIVITYDETERMINATION 251 within the sample; and f) the nonplanar ends of the sample at the reference planes. The last four model imperfections are taken into account yf~l=yJ {(1–s1,)2–(s2,,=)2)/{(1+s,, )2–(s,,,. )2) by the ~nlargements of the uncertainty regions of ~1, / and/or S21with the measured differences between ~21and (15) ~12, and between modulus (~1,) and modulus (~22). It is l–(S,1)2+ a)2–2s21, clear that, for increased measurement accuracy, the com- (s21, acosh(yflz) mon S1, and S21 areas become smaller so that stronger sf, = (16) d(y:ll) model conditions have to be fulfilled, The measurement 2F,1 +2 S21,asinh(y;lZ) or accuracy in S1, and S21, when measured as transmission 11 and reflection coefficients with the HP 8542 B calibrated C?(y:,l) l–s,l network analyzer, is sufficient for rather lossy materials if — .——. only (7) is used [1]. For low-loss materials with optimum W,, ‘:’1 ( (1- F,,)2-(S21,J2 results, the approach formulated in (7)–( 18) has to be used. Examples of experimental results obtained with a wave- 1+s,, + . (17) guide slotted line and with the calibrated network analyzer (1+%)2 -(s2,,.)2 1 are given in Section IV. The random errors in the measurements are reduced To get a y region solution, the S1, region determined by with the reflection coefficient measurement method as the four 81~ points needs an area in common with the described by Deschamps [2], The method uses a load and a uncertain y region of S1~, and can therefore be char- moving short behind the sample of Fig. 1 and was origi- acterized by at least three points and maximally eight nally used for perrnittivity measurements by Altschuler [3]. points. At the same time, the S21region formed by the four When accurate scattering coefficient measurements of ~al points needs an area in common with the uncertainty S1,, SZ2, S21, and S12 are desired, eight or even sixteen region of S21and can also be characterized by at least three short-circuit positions spaced over half a wavelength can points and maximally eight points. If the measurement be used, The appeal of this method over the transmission accuracy is worse than specified, or the model assumptions and reflection method is that the mutual dependencies are incorrect, it is possible that no common area is found, between the scattering coefficients are implicitly taken into rendering an accurate determination of y impossible. From account. For the ‘calculation of the scattering coefficients, the common area with minimally three and maximally one is referred to [2]. eight extrema in S1,, the corresponding S2, values have to be found, and likewise with the extrema in Szl, the corre- HI. THE OPTIMAL MEASUREMENT CONFIGURATION sponding S1, values. These extrema are used to compute FORCOMPUTING THEPERMITTIVITY the average value Sfl with corresponding S21, and the In Section 11,we have proved that there exists a mutual average S~: with corresponding ~1,. From the same data, dependency between S1~and S21.If we assume that S21is a the differences dSfl, d~l,, dj21, and dS~~ are derived as function of S1,, the error d~,, in c due to a sample’ length follows: difference 81and to a difference dS1, (this can be dSfl or dSfl = extrema in S1, within common S1, area – Sfl dS1,) becomes d~21= corresponding S21– $Tl ed =~ 81+ ik “dS’” forc=c(S1,,l) (19) “ 11 al 8s,, dS;: = extrema in Szl within common SZLarea – S$; and for S1, as a function of S21 cd %= +la ae The combinations Sfl, ~21, and ~1,, S~~ are used to forc=t(S21,1). (20) 2’ al “ 13S21“‘s21‘ compute y twice by substituting these S parameters into (8). The differences in (18) are then used to compute the By using (l)–(4), (6), (7), and (9) we derive differences in y according to the theory given in Section III. If the four areas formed by these differences in y do (21) not contain the difference in the two y solutions, this indicates a nonlinearity in the method. –ik () 21 (1-p%2)2 To study the model imperfections we distinguish: a) a length inaccuracy which can be taken into account as in as,, ‘4 : l–pz 4pw2yl+ (l+pW)(l-w2) Section III; b) a displacement of the reference planes, (22) resulting in differences in the arguments of the measured ~1, and ~2zwhich can be taken into account by averaging (?E 2y21 (1-p2w2)2 (23) these arguments; c) an air gap between the sample and the 8s2, = G ()~ l–p’2 p(l–wz)+(l +pzwz)yl” waveguide walls; d) the inhomogeneities within the sample; e) the excitation and propagation of higher order modes From (21) we see that for length uncertainties 252 IEEETRANSACTIONSONMCROWAVE THEORYAND TECHNIQIJSS, VOL. MTT-31, NO. 3, MARCH 1983 =lcm I&l I c, = 8.79 c“ = 0.00 40 — kclko = O 35 _ k=/kO = 0.9 30 =2CUI 25 20 15 =4cm 10 =8cn 5 o 0 0 T 2T 311 —411 5n Iia(yt) Fig. 3. The real part of (21) asa function of c’ for different lengths and kC/ko = 0.59. Fig. 5. The modulus of (22) asafunction of Im(y[) for different kc/ko. .lcm c’ = 8.79 l–3El t ] E“ = 0.00 aszl — kc/ko = O 40 =2cm =4CM =8crn 0 o 5 - 10 E“ o II 27r 31T -..% ST Im(yL) Fig. 4. The imaginary part of (21) as a function of c“ for different lengths. Fig. 6. The modulus of (23) asafunction of Im(y/) for different kC/ko. Re( d~/dl) = real part of 6’(/81 depends on Re(c) = c’ and where larger k gives more accurate d values but at the becomes minimal for maximum length and for measure- same time implies more stringent considerations concern- ment wavelengths near cutoff. Im( A / dl ) = imaginary part ing the model assumptions. Increasing p can be reached by of 13c/al is a function of Im(~ ) = c“, and becomes minimal using the empty waveguide near cutoff. The most attractive for maximum length, but does not depend on the empty feature of the method with the automatic network analyzer, transmission line cutoff wavelength. For that reason, the however, is broad-bandedness. This means that the near real and imaginary parts of(21 ) are shown in Figs. 3 and 4. cutoff waveguide option would not be generally available. To analyze the influences of scattering coefficient uncer- A second advantage of the sample lengths given by (25) tainties for different sample lengths, first the limitation of is that for this idealized case ]Sl, Ibecomes O.This means dielectric materials without losses is considered. This means that IdS1~1,and also ldSzl 1,because of the mutual dependen- cy between SI, and Szl, become minimal. “=()- Im(p)=Re(y)=O. (24) e If losses are taken into account, ~“ * O,and so Re(y) = a In that case, the absolute values of (22) and (23) have * O.Because IS1~I* O,and thus ldS1,]increases when com- been visualized in Figs. 5 and 6, e.g., for c’= 8.79. The pared to the lossless case, the permittivity results become reason why we take the absolute values is because dS1, and less accurate for optimum sample lengths given by (25). tiSzl have complex vahtes around Sfl ~ith corresponding From (22) and (23) we seethat for lossy materials the Szl Szl, or around S~~with corresponding S1,. Contrary to the measurements largely determine the accuracy in the per- length uncertainty, we therefore assume equal influences mittivity results for sample lengths very large in relation to on dc’ and dc” due to uncertainties in dS1, and dSzl. From the wavelength because only the minima of (23) and ISZII Figs. 5 and 6, we conclude that for optimum computation decrease with increasing sample length. This is in agree- of the permittivity, p has to be aslarge aspossible and ment with [4], where the influence of the sample length for Im(yl) = km, k=0,1,2 ... (25) high d ferroelectric materials in a TE measurement config- LIGTHARTT:ECHNIQUFEORACCURATPEERMITTIVITDYETERMINATION 253 uration has been studied by using the amplitude and phase -c,, 1.0 Aluminhm Oxide of Szl only. For an arbitrary sample length, both (22) and I ‘o 1made – —... -TEMImde (23) have to be used to achieve the most accurate results. ~%.,. . IV. RESULTS 0.4 / , .< . .. -. , ... Firstly, we show the influence on the accuracy of c= c’+ 0.2 ----/.. . ‘.. ,-.\ jc” measurements at frequencies near the cutoff frequency 1<\1l... “85 8.3 8.75 9.0 9.25 of the empty waveguide. We take the example of the TEO1 mode propagating in a rectangular waveguide (inner di- c’ mensions 25.8 mm X51.6 mm), and homogeneously filled Fig. 7. Uncertainty limits of c, predicted from uncertainty limits of S,, and S2,, asobtained for aquarter-wave sample in aTEM transmission over a length L = 8.92 mm with aluminum oxide corre- line andaTEn, waveguidenearcutoff. (Thelowerboundaryof d’ has sponding to a quarter wavelength inside this medium at a not beenshow-nbeca~seit correspondswith activematerials.) frequency of 3 GHz. The scattering coefficients ~1, and & and the uncertainty limits as suggested by the manufac- I Aluminim oxide mode turer of the HP calibrated network analyzer type HP 8542 -e“ ~._. -. -, —------ 1 ----- ‘mon1mcde B become 0.095 . ! I I ~= 3 GHz, length =8.92 mm+Omm 0.090 0.088 . Re(S1l) = –0.983+0.030, Im(S1l) =0+0.030 0.085 ~D~ p2,1=o.177*o.oo3, arg(s21) = – 1.57+0.040. i i 0.080 i-, _._. -.-. -._._i 1 The uncertainty limits are checked by turning the sample so that Szzand S12are measured, and further by computing v) 8.78 8.79 8.80 — 821((12)), under the assumption of an infinitesimal uncer- E, Fig. 8. Uncertainty limits of e, predicted from uncertainty limits of S1, tainty region of S1,, and by computing 8,~((16)), under the and S.,. as obtained for half awavelength sanmle in aTEM transmis- assumption of an infinitesimal uncertainty region of S21.If sion~;e andaTEO1waveguidenearcu;off. ‘ the differences between ~1, and ~zzor between ~zl and &, or if 8Z1or 8,, are larger than the HP uncertainty limits due II Plexi-glass to incorrect model considerations, including length uncer- .,. .~ = ;.+ j;,, -E” . tainties, the uncertainty region has to be enlarged. By using 0.017 1 a (10)-(17), and (l), (2), and (18)-(23), the common c area x .:! has been calculated. In Fig. 7 the common t area has been 0.015 ‘, . . .“ drawn with solid lines; for comparison, the broken lines 0.013 t indicate the common c area as obtained in a TEM config- uration with the same electrical length of the sample. For a sample length of 17.83 mm, corresponding to the optimum electrical length of half a wavelength, the scatter- Fig. 9. Uncertainty fimits of c predicted from computed uncertainty limits of S1,,S21(solidlines)[2],andsixteencomputedS1,,S21values ing coefficients with their uncertainties become (dots). j= 3 GHz, length= 17.83 mm+o mm Re(S,l) = –0.089Y0.003, Im(Sll) = 0.001 ~0.003 The results (solid lines) are given in Fig. 9. If we use the sixteen measurements separately to com- IS*,I = 0.91+0.017, arg(Szl) = 3.142 +0.027, pute S2, via For this optimum length, the common c area is shown in Fig. 8 for the TEOI mode and is compared to the TEM s,, = (rj-s,,)(l-s,,r,) mode. 2 rk From Figs. 7 and 8, we see the advantage of using the TEOI mode near cutoff for sample lengths which satisfy where r~ = complex reflection coefficient of the moving (25). short at the output reference plane, and r; = measured The method with the moving short has been used for the reflection coefficient at the input reference plane, we are ( determination of plexiglass in the case of nonoptimum able to compute sixteen different t‘s ((8)), indicated by sample length. From sixteen measurements with the mov- dots in Fig. 9, resulting in ing short, and one S1, measurement with a load, we have c’= average in c’= 2,5793, SC.= spread in e’= 0.0020 derived from Deschamps [2] the scattering coefficients with i“ = average in C“= – 0.0156, S,,,= spread in d’ = 0.0006. their uncertainties for f = 9.814 GHz, length = 74.5 mm~O mm. Comparison between the dots and the common carea in Fig. 9 indicates an accuracy in C’and C“ better than 0.001. Re(S1l) = –0.142*0.019, Im(S1l) = –0.186&0.019 The fact that the dots follow a smooth contour rather than ISZII= 0.881+0.021, arg(Szl) = 2.696&0.031. scatter randomly suggests that there is some systematic 254 IEEETRANSACTIONOSNMICROWAVTEHEORYANDTECHNIQUEVSO,L.MTT-31, NO. 3, MARCH 1983 error in S1~and S21.Minimizing the area of the contour by ACKNOWLEDGMENT varying SI, within the uncertain y limits can further im- Thanks are due to M. K. Smit, who thoroughly discussed prove the c accuracy. this subject with me, and to M. de Kok, who did the measurements with and the programming for the auto- V. CONCLUSIONS matic and calibrated network analyzer. The analytical approach to computing c from reflection REFERENCES and/or transmission coefficients as derived in Section II [1] S.S.Stuchlyand M. Matuszewski,“A combined totaf reflection- has the advantage that the programming is simple and can transmissionmethodinapplicationtodielectricspectroscopy,”IEEE be performed within a negligible computing time in the Trans. Instrum. Mess., vol. IM-27, no. 3, pp. 285-288, Sept. 1978. minicomputer which is part of the automatic and calibrated [2] G. A. Deschamps, “Determination of reflection coefficients and insertion loss of awaveguide junction,” J. ofAppL Phys., vol. 24, no. network analyzer. In Section III, the ~uncertain y region is 8, pp. 1046-1051, Aug. 1953. related to the sample length uncertainty, and to the scatter- [3] H. M. Aftschuler,Handbook of Microwave Measurements, M. Sucher ing coefficient uncertainties. For low-loss materials, accu- andJ.Fox,Eds.vol.2. NewYork: Wiley,section9.08,1963. [4] J.B.Horton andG.A.Burdick,”Measurementofdielectricconstant rate t results are obtained by measuring S11 for long andloss tamzent in materials havinx lame dielectric constants.” IEEE sample lengths corresponding to an integer number times Trans. Micr;waoe Theo~ Tech,, ~ol. ‘M’fT- 16, pp. 873-875, Oct. half a wavelength in the medium, especially when a rectan- 1968. gular waveguide near cutoff is used. However, the longer the sample lengths are, the more sensitive the c computa- tions are to failures in the model, For higher losses, the influence of Szl becomes more important and both minima Leo P. L&thart was born in Rotterdarn, The of (22) and (23) have to be used to get the most accurate Netherlands, on September 15, 1946. He results. Under the assumption of sufficient measurement graduated with distinction in 1969 and received the M.S. degree in electrical engineering from sensitivity, the method with the moving short has the Delft University of Technology. advantage that only reflection coefficients have to be mea- Since 1969 he has been with the Microwave sured. From these measurements, the scattering coefficients Laboratory of the Delft University of Tech- nology. In 1974 he became Senior Lecturer teach- and their uncertainties are derived. A variant makes use of ing undergraduate courses in Transmission Line the relation between the scattering coefficients via the Theory, Antennas, and Propagation. During measured reflection coefficients. The different c results the 1976-1977 academic year he was on leave at the Chalmers University of Technology, Gothenborg, Sweden. Since 1977 found in this manner yield comparable results and can be he has been a Senior Staff Engineer working on antennas and tropo- analyzed statistically. spheric radar. IEEETRANSACTIONOSNMICROWAVTEHJ30RAYNDTECHNIQUEVSO, L.M’tT-31, NO. 3, MARCH 1983 255 Nondestructive Measurement of a Dielectric Layer Using Surface Electromagnetic Waves WEIMING OU, MEMBER,IEEE,C. GERALD GARDNER, AND STUART A. LONG, SENIORMEMBER,IEEE Abstract—In thisinvestigation,thepossibilityof nondestructively mea- For our work, we are interested in layers of dielectric suring the thickness and dielectric constant of alayer of dielectric material material on conductive substrates. These layers are rela- on a conducting substrate by surface electromagnetic waves (SEW) has tively thick compared to the thin films, and are not usually been demonstrated. The theoretical approximate dispersion relations near optically transparent. Therefore, microwaves or millimeter cutoff were derived for both the TE and TM modes and fonnd to be finear functions of frequency, The thickness and dielectric constant were then waves are more appropriate for our study. Thus, it would calculated as simple idgebraic functions of the slope and intereept of the seemingly be desirable to develop a single-mode technique dispersion curve. An experimental apparatus utilizhg a prism-conpler was which could calculate the dielectric constant and thickness constructed to excite snrfaee electromagnetic waves in a d]eleetric layer of the layer independently without the use of lengthy whose characteristics were known. By suitable measurements of the computer analysis. The microwave system also allows opti- freqnency and the conpfing angle of the source, the dispersion curve was determined experimentally and the resulting dielectric constant and thick- cally nontransparent materials to be measured and does ness of the layer calculated. not require recording film as a detector, making it more suitable for real-time applications. I. INTRODUCTION s Our study shows that a multifrequency method of mea- URFACE electromagnetic waves (SEW) have been used suring both the cutoff frequency and the slope of the in the past in the field of nondestructive evaluation. dispersion curve near cutoff allows a pair of simple simul- Previously, Alexander and Bell [1] used this technique to taneous equations to be found which can then be solved study planar optical waveguides in which a thin film of for both the dielectric constant and the thickness of the optically transparent material was coated on a conductive layer. This method is seen to be suitable for any TE mode or semiconductive substrate. Later, Ulrich and Torge [2] and all TM modes, except the TMO which has a zero cutoff published a paper which discussed measuring the thickness frequency. and refractive index of a light-guiding film using a prism- A prism coupling method developed by Iogansen [5]-[7] coupling technique. There are unfortunately, a number of was then applied to an experimental test setup consisting disadvantages in using their method: 1) the fihn must be of layers of plastic over a large sheet of aluminum. Results thick enough so that at least two modes can propagate; 2) were then compared with direct measurements made of the the method is most often destructive in nature since the thickness and dielectric constant. film must be mechanically pressed against the prism; 3) Several potential applications of this method can be computer processing was required to fit the experimental readily visualized. Thin protective coatings on components data to the theoretical curves; 4) the method employs a of jet engines, magnetohydrodynamic generators, or coal laser as a source so only light-transparent materials can be combustion chambers could be evaluated. Similarly, one measured. Later, Walter [3] used both TE and TM modes might examine polymeric coatings for environmental pro- synchronously, resulting in a pair of transcendental equa- tection or the surfaces of metals prepared for adhesive tions for the thickness and refraction index of the layer. bonding, where the strength attained by the bond is known This method was based on the previous theory, so the same to be sensitive to the condition of the adherent surfaces. basic disadvantages were still present. Chung [4] then dis- Since the measurements must be taken at frequencies above cussed using a single-mode, multiwavelength method, but cutoff for the lowest order mode, the thickness of the required knowledge of the dispersion curve of the layer in dielectric layer must be greater than some minimum value advance. which is determined by the frequency of operation and the dielectric constant of the material (e.g., for a frequency of Manuscript received June 17, 1982; revised September 2, 1982. This 100 GHz and c,= 10, a minimum thickness of 80 pm is work was supported in part by the U.S. Air Force Office of Scientific required). Thus, for thin coatings, frequencies well into the Research under Grant 77-3457, and in part by the Energy Laboratory of millimeter-wave region may be required. The experimental the University of Houston. W. Ou is with the Varian Solid-State Microwave Division, Santa Clara, measurements carried out in this work were made in the CA. 8– 12-GHz band with dielectric constants near 2 and, there- C. G. Gardner iswith the Welex Division of Haltiburton, Houston, TX. fore, the thickness of the resultant layers was in the centi- S.A. Long is with the Department of Electrical Engineering, University of Houston, Houston, TX 77004. meter range. 0018-9480/83/0300-0255$01.00 01983 IEEE 256 IEEETRANSACTIONOSNMICROWAVTEHEORYANDTECHMQUESV,OL.M~-31, NO. 3, MARCH 1983 DIELECTRIC ? LAYER t + ,. —x T CONDUCTING SUBSTRP.TE Fig. 1. Geometry and coordinate system for dielectric layer over a conducting substrate. II. THEORY The geometry of the problem is shown in Fig. 1 and consists of a dielectric layer with relative permittivity t-, and thickness 1over an infinitely conducting plane. For modes TM to i, the fields in the air region y >1 are of the form EX, fiY, Hz - ~-jckoxx+koyj, y > 1; Fig. 2. Graphical solution of dispersion relation. with k; = k& + k~Y= ti2poCo (1) and in the dielectric material persion relation about the cutoff point and neglecting all EX, EY, Hz - e‘J(k@+%YJ O<y<l (2) but first-order terms, the relationship between kOx and kO can be shown to be where k; =k;x + kfY = 02po~1 =c,k:. Using the appropriate boundary conditions at y = 1and with k. given by (5). y = O,the following equations can be derived in the manner It should be noted that a linear relation will result if the of Barrington [8]: quantity klYltan(klYl) = c,@ (3) [kH2 11/2 ox and k. ‘1 (k,yl)2+(Pl)2 = ((, - l)(kOl)2 (4) is plotted versus k.. Its intercept will be where k~Yis the wavenumber in the dielectric in the ; k,= ‘~ direction, and B = jkoY accounts for the decay in air nor- 1(6, –1)”2 mal to the surface for the pure surface wave. This pair of equations cannot be solved exactly. The usual graphical and its slope will be solution is shown in Fig. 2 in the k,yl–~l plane. Equation (Er-l)l (4) is a circle of radius (c, – 1)1/2kOl, and (3) is represented Cr by the tangent-like curves. The points of intersection be- tween these two curves determines the solution for kl y and Thus, if the value of koX is found for several frequencies ~ as a function of thickness 1and dielectric constant c,. It just above ~,, the slope and intercept of the above curve can be seen from the figure that the TMO mode will be able can be found, and the quantities ~, and 1can be determined to propagate for any frequency, but all higher order modes from kCand the slope S have a finite cutoff frequency. The cutoff wavenumber of +4(%’1”2 each mode is given by , — (7, r SkC 2 kC= ‘v n=o, 1,2, ... . (5) 2— 1(6, –1)”2’ ()n7r This one relation between C,and 1can be determined (for (8) all but the TMO mode) by simply measuring the cutoff frequency. (The equation for C,is a double valued function and thus If another measurable parameter can be found that is a some approximate value of Crmust be known from another different function of c, and 1, then both c, and 1could be source.) found independently from the two resulting equations. To It should be noted that for the TMO mode, (5) does not this end, the dispersion relation of these TM modes can be provide one of the required relationships between ~r and 1, examined more closely near cutoff. Expansion of the dis- and thus (7) and (8) cannot be used. Equation (6) can still