Radiation properties of leaky modes near the spectral gap region for semi-infinite printed-circuit lines

Transcription

1 RADIO SCIENCE, VOL. 38, NO. 3, 151, doi:1.129/22rs2777, 23 Radiation roerties of leaky modes near the sectral ga region for semi-infinite rinted-circuit lines Frank J. Villegas, 1 David R. Jackson, and Jeffery T. Williams Alied Electromagnetics Laboratory, Deartment of Electrical and Comuter Engineering, University of Houston, Houston, Texas, USA Arthur A. Oliner Deartment of Electrical Engineering, Polytechnic University, Brooklyn, New York, USA Received 14 Setember 22; revised 24 January 23; acceted 14 March 23; ublished 7 June 23. [1] The radiation field from a leaky mode on a semi-infinite rinted-circuit line is investigated. Asymtotic formulas for the radiation field are resented for a mode that is either in the hysical region (a fast wave with resect to the substrate mode into which leakage occurs), or in the sectral-ga region (a slow wave). These formulas, as well as a numerically exact reresentation of the leaky-mode radiation field, are used to examine the nature of the radiation field and to study how it changes as the leaky mode transitions from the hysical region to the sectral-ga region. A comarison is made between the field radiated by the leaky-mode current and that radiated by a ractical robe feed that launches the semi-infinite leaky mode current. Another comarison is made between the field radiated by the leaky-mode current and the exact field radiated by the total current that is excited by a ractical ga voltage source on an infinite line. INDEX TERMS: 624 Electromagnetics: Guided waves; 689 Electromagnetics: Wave roagation (4275); 619 Electromagnetics: Electromagnetic theory; KEYWORDS: leaky mode, rinted circuit, striline Citation: Villegas, F. J., D. R. Jackson, J. T. Williams, and A. A. Oliner, Radiation roerties of leaky modes near the sectral ga region for semi-infinite rinted-circuit lines, Radio Sci., 38(3), 151, doi:1.129/22rs2777, Introduction [2] In the recent ast, extensive investigations have led to an increased understanding of the ower leakage from leaky modes on rinted-circuit lines (Das and Pozar [1991] through Villegas et al. [1999]). For covered structures having a sufficiently small ground lane searation, leakage occurs into the TM arallel-late mode of the substrate structure that surrounds the rintedcircuit line. Examles of covered rinted-circuit lines that suort leaky modes include the striline with an air ga (Figure 1a)[Nghiem et al., 1993, 1995] and the covered microstri (Figure 1b) [Langston et al., 21; Mesa and Marques, 1996]. For oen structures radiation may occur into the TM surface-wave mode as well as directly into sace. 1 Now at Raytheon Comany, El Segundo, California, USA. Coyright 23 by the American Geohysical Union /3/22RS [3] Initial investigations in this area focused on the roagation and radiation characteristics of leaky modes on infinite transmission lines. Recently, however, a study was made of the radiation characteristics of semi-infinite transmission lines [Villegas et al., 1999]. This rovides a canonical model for leakage on a certain class of ractical circuits for which a leaky mode is launched from a feed oint and then roagates in one direction along the line from the feed. [4] In the resent study, the investigation of radiation into the TM mode from leaky modes on semi-infinite rinted-circuit lines is extended to examine the nature of the radiated fields for leaky modes in the sectral-ga (SG) region, where b z > k TM. This extension is imortant because it directly aids in the understanding of the hysical significance (or lack thereof) for leaky modes in the SG region. It also verifies how feed radiation may become very significant relative to the leaky-mode radiation for this tye of structure, when the leaky mode is within the SG region. The analysis also shows how the field from a leaky mode in the sectral ga may be interreted hysically as the sum of an exonentially

2 17-2 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES arallel-late mode). The amlitude of the leakage field will, however, deend on the articular structure. 2. Radiation From a Semi-Infinite Stri Current [6] Figure 2 shows a to view of a semi-infinite rinted-circuit line that is inside of a covered structure (such as one of the structures shown in Figure 1) and is fed by a vertical robe. The stri begins at the feed location (z = ), and extends to infinity. A robe current of 1 A is assumed, which launches a traveling-wave current on the stri of the form Iz ð Þ ¼ e jk zz ; ð1þ Figure 1. (a) End view of the air-ga striline structure. (b) End view of the covered microstri structure. These are two different covered structures that suort leaky modes (modes that leak into the arallel-late mode of the background structure). decaying field that would be excited by the leaky mode on an infinite line, lus a radiation term that arises from the discontinuity at the beginning of the line. Hence, hysical insight into leaky-mode radiation is obtained from the analysis. The results are alicable to all covered structures where radiation occurs into only the TM arallel-late mode, and also to oen structures, rovided that only the radiation into the TM surfacewave mode is considered. [5] Figure 2 deicts a to view of a semi-infinite rinted-circuit line fed by a robe at z =, which will be used for the resent analysis. Numerical results will be resented for the air-ga striline case of Figure 1a, although the conclusions regarding the shae of the leakage field will be valid for any covered structure, since the shae of the leakage field is only a function of the normalized wavenumber of the leaky mode (the wavenumber of the leaky mode relative to that of the where k z = b z ja z is the comlex roagation constant of the leaky mode. Although the analysis will be alicable to a semi-infinite rinted stri in any covered structure, results will be resented later for one articular tye of structure, the air-ga striline structure shown in Figure 1a [Nghiem et al., 1993, 1995]. The dielectric structure is assumed to be lossless, so that the wavenumber of the dominant arallel-late mode k TM is urely real. The loss tangent of the dielectric should have relatively little effect if it is small comared to the attenuation constant of the leaky mode. [7] Even though no junction effects are accounted for at the attachment oint of the vertical robe and the stri, the current in equation (1) is nevertheless consistent with the 1 A current on the vertical robe that is assumed. Hence, a Kirchhoff condition is correctly satisfied in the sense that there is continuity of total current going from the vertical feed robe to the horizontal stri. Of course, if a filamentary robe meets a stri of finite width, there would be a small region of radially sreading current near the attachment oint. However, this region should be very small comared to a wavelength (assuming a narrow stri). Furthermore, the radiation from the different arts of the radially sreading current would tend to cancel, by symmetry. For both of these reasons, the Figure 2. To view of a semi-infinite rinted-circuit line fed by a vertical robe feed, with the coordinate system and geometrical arameters that are used in the field calculation labeled.

3 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES 17-3 radiation from the radially sreading current would be very small. [8] Alternatively, another hysical interretation of the assumed feed model is that of a vertical flat-stri feed robe, having the same width as the line stri, which makes a right-angle bend to form the horizontal line stri. In this case the current on the robe would simly bend over to form the stri current, and there would be no attachment oint or radial current. From this oint of view, the calculation of the field from the robe (discussed later) using a filamentary model of the robe, as shown in Figure 2, is an aroximation; however, it should be an excellent aroximation rovided the width of the stri is small comared to a wavelength. [9] Although the current on the stri (and the feed robe) has been assumed to be 1 A for simlicity, the actual current on the feed robe and stri would be determined by the voltage at the feed, and the characteristic imedance of the line. That is, I() = V/Z, where V is the voltage at the feed (the voltage at the coaxial connector for a ractical feed). This would simly scale all of the results by a factor of I(), and would not change any of the conclusions based on the resent analysis. [1] Assuming that the TM mode is the only substrate mode above cutoff, and that the stri width is small comared to a wavelength, the vertical comonent of the arallel-late leakage field, E y, radiated by the semiinfinite stri current source is given by E y ðx; y; zþ ¼ A Z 1 H ð2þ 1 ðk TM r Þcosðf ÞIz ð Þdz ; ð2þ where A is an excitation coefficient, giving the amlitude of the arallel-late TM field excited by a unitamlitude horizontal electric diole in the ^z direction (essentially the amlitude of the Green function of the background structure). The value of A, which differs with each rinted-circuit structure, is not imortant for examining the nature of the leakage field radiated by the line, but becomes imortant when the field radiated by the line current is comared to that radiated by the robe feed. The value of A for the air-ga striline structure of Figure 1a is given in Aendix A. The angle f is measured from the stri axis, and is given by f ¼ tan 1 x z z ; ð3þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x 2 þ ðz z Þ 2 is the radial distance to the observation oint measured from the source location at z, as shown in Figure 2. [11] The integral in (2) is analogous to the familiar hysical otics (PO) integration encountered in diffraction theory. An asymtotic evaluation of this PO integral using the stationary-hase method (or the saddle-oint Figure 3. The s lane used for the integration to calculate the radiation field of the semi-infinite line, showing the branch cuts associated with the integrand. The location of the stationary-hase oint (SPP) is shown for three different cases: the hysical-lit case (oint a), the hysical-dark case (oint b), and the sectral-ga case (oint c). method) [Bleistein and Handelsman, 1975] can be obtained in the limit that z/l!1while the arameter a z z remains finite. This is essentially equivalent to having the semi-infinite stri current aroach an infinite electrical extent in the +z direction. The integral in (2) can be ut into a more convenient normalized form for the asymtotic evaluation by defining a normalized distance variable s =(z z)/z and relacing the Hankel function by its large-argument aroximation (k TM z 1). This results in y ¼ B ffiffiffiffiffiffi k z E where Z 1 e jk z z 1 s ðs 2 þ tan 2 fþ 3=4 ð Þ 1=2 ds e s e j b zzs þ k TM z s 2 þtan 2 f sffiffiffiffiffiffiffiffiffiffiffi B ¼ 1 2j ffiffiffiffi ja k k TM ð4þ ð5þ and tan f = x/z. The integrand in (4) above has branch oints at s =±j tan f as deicted in Figure 3. The to sheet of the Riemann surface in Figure 3 is defined as the analytic continuation from the real axis, where the terms (s 2 +tan 2 f) 1/2 and (s 2 +tan 2 f) 3/4 are ositive real numbers. The integration ath C :[ 1, + 1 ) along the real axis is also shown. The stationary-hase oint (SPP) for this integral can be determined in a straightforward

4 17-4 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES manner. We first identify the oscillatory ortion of the integrand, denoted as ex [ jvg(s)], where gs ðþ¼ bs þ s 2 þ tan 2 1=2 f ; ð6þ with b b z =k TM, and V k TM z ð7þ Table 1. Regions Corresonding to SPP Locations in the s Plane Region Conditions (a) hysical lit b <1,jfj < cos 1 b (b) hysical dark b <1,jfj > cos 1 b (c) sectral ga b >1 is a large arameter. Setting g (s ) = yields the equation b s 2 þ tan 2 1=2þ f s ¼ ; ð8þ which then gives the solution for the SPP s as s bjtan fj b; f ¼ 1 b 2 1=2 : ð9þ The square root in this equation is interreted as a ositive real number when b < 1, since s must be a negative number in this case (this follows from an insection of the SPP exression in (8)). When b > 1the SPP is located on the imaginary axis and is more roerly called a saddle oint, since the terminology of the stationary-hase method is customarily restricted to integrations along the real axis. For simlicity, however, the term SPP is used in Figure 3 to include both cases. 3. Asymtotic Evaluation of Radiation Integral [12] It is seen from (9) that s is a function of b as well as f. Hence, in treating the PO integral (4) asymtotically, the ossible locations of the SPP must be determined as b and f vary. This is imortant for both roerly evaluating the integral as well as hysically classifying the leaky mode. In articular, we note from classical leaky-mode theory [Tamir and Oliner, 1963] that b < 1 corresonds to a hysically meaningful imroer (leaky) modal solution, in which the leaky mode exhibits an exonential behavior within an angular leakage region of sace, denoted here as the hysical-lit region, given aroximately by jfj < cos 1 b. For a hysical leaky mode, the field is exected to decrease monotonically in the region jfj > cos 1 b, and in analogy with otics, this region in sace is referred to here as the hysical-dark region. The boundary between these two regions, called the leakage shadow boundary, occurs when b ¼ cos f, which corresonds to s = 1, from (9). [13] The SG region is the frequency region for which a leaky mode has the roerty b > 1. The radiation characteristics of the leaky mode inside the SG, as well as the behavior near the hysical/sg boundary (i.e. b ¼ 1) is a rimary focus of the resent aer. [14] In total, three distinct cases arise, with the location of the SPP, and hence the nature of the asymtotic evaluation, deending on the case. The three cases are summarized in Table 1. [15] Figure 3 shows the locus of the SPP in the s lane for the three different cases. In the hysical region (cases (a) and (b) above), the SPP on the to sheet is located along the negative real axis. (There is another SPP symmetrically located along the ositive real axis on a lower sheet, which is inconsequential in our resent discussion since it has no effect on the integration ath C, which resides on the to sheet.) At the hysical/sg boundary, the SPP is located on a circle at infinity as shown in the figure. In the SG region (case (c) above), the SPP is located along the negative imaginary axis, as seen from (9). Its location is just to the left of the branch cut on the to sheet, which may be verified by an insection of (8). As mentioned reviously, the SPP is more roerly denoted as a saddle oint in this case, although the term SPP is used in Table 1 for simlicity. [16] In the following subsections, closed-form asymtotic exressions for the radiated leaky-mode field in each of the three above-mentioned regions will be resented. Those for the first two cases, i.e., the hysical-lit and hysical-dark regions, will be given in summary only since their detailed derivation can be found in Villegas et al. [1999] Physical Region [17] In the hysical-lit region, b < 1 and jfj < cos 1 b, which imlies that s > 1 from (9). From Figure 3, it is seen that in this articular case the SPP (oint a) is encountered by the integration ath C. Alication of the stationary-hase method to the PO integral in (4) yields a geometrical otics (GO) field, in analogy with diffraction theory. Foregoing the details of the calculation, the final exression is given as follows: Ey j2a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cot q e jk zz jk TM x o e 2 þ ð z o zþ2 ; ð1þ k TM where q = cos 1 (b z /k TM ) is the leakage angle (subject to the conditions a z 1 and b z a z ), and z o is the

5 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES 17-5 hysical location on the stri corresonding to the stationary-hase oint s, defined by the relation s ¼ z z =z: ð11þ The oint z is interreted as the location along the stri axis from which a ray emanating at the leakage angle q reaches the observation oint. At z, the field has an amlitude which is roortional to ex ( a z z ) as is evident from (1). At this oint, the ray roagates toward the observation oint with the wavenumber of the substrate mode, k TM. Hence, (1) imlies that in the hysical-lit region, the majority of the radiated field at a given observation oint is essentially coming from a single oint along the stri, and is emanating from this oint at the leakage angle. This in turn imlies an exonentially increasing field in the lit region. When the observation oint is located outside of the leakage (lit) region, i.e., when b < 1 and jfj > cos 1 b the SPP is located at oint b in Figure 2. In this case the SPP is not encountered by the integration ath C. Therefore, a simle alication of integration-by-arts (IP) rovides an asymtotic exression for the radiated field. Once again, omitting the details of the calculation [see Villegas et al., 1999], the result is given as!! Ey jbe jk 1 1 zz ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi e cos f k TM r b cos f k TM e jv ð b sec fþ ; ð12þ where B is given by (5), ktm = k TM /k, and r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ z 2. In contrast with (1), where it was convenient to exress field in terms of x and z, itis more useful here to leave (12) in terms of the angle f measured from the stri axis. There are several imortant features inherent in this result. First and foremost, we realize that since the SPP is not encountered by the integration ath when the leaky mode is in the hysical-dark region, there is no direct radiation from the leaky mode in the GO sense. Furthermore, the radiation field of the leaky mode in the hysical-dark region may be interreted as being the result of an equivalent source discontinuity (ESD) located at the stri origin (z = ), since the hase center of the radiation is at the origin. This follows from the last term in (12), which may be rewritten as e jv sec f ¼ e jk TM r : ð13þ Also, note that (12) is singular at the shadow boundary (i.e. b ¼ cos f or equivalently f = q ). The GO equation (1) is not singular when corresondingly evaluated at x = z tan q. However, (1) still loses accuracy at the leakage shadow boundary, due to the Figure 4. The s lane showing the original ath of integration deformed to two new integration aths, a vertical ath C d that descends from the oint s = 1, and the ath C b that goes around the lower branch cut. This new set of aths is used to asymtotically evaluate the radiation field for a leaky mode in the sectral-ga region. A saddle oint (SP) is shown on the left side of the branch cut. fact that the SPP is located at the limit of integration (s = 1). (In fact, (1) will be off by aroximately a factor of 2 at the shadow boundary). More accurate exressions for the radiated field in the hysical-lit and hysical-dark regions may be obtained by emloying a uniform asymtotic exansion (UAE). Nevertheless, the UAE formulas lose the simle hysical interretation rovided by (1) and (12). The details of the UAE formulation are resented by Villegas et al. [1999] and are omitted here Sectral Ga Region [18] In the SG region, b > 1. The SPP, which is now a saddle oint (SP), is urely imaginary, and is located as shown in Figure 3. In this case it is most convenient to deform the ath of integration C in (4) from ( 1, 1)into a sum of two aths, C b and C d, as shown in Figure 4, so that Ey ¼ B ffiffiffiffiffiffi k z e jk z z ði b þ I d Þ: ð14þ This deformation is ossible in the SG case because the integrand in (4) decays exonentially along both sides of the branch cut in the lower half lane (in the hysical region, where b z < k TM, there is exonential decay on the right side of the branch cut in the lower half lane, but exonential growth on the left side). The asymtotic field of the branch-cut contribution, I b, can be conveniently found by first emloying the change of variables s = jy.

6 17-6 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES Combining the contributions from the integration along both sides of the branch cut results in Z " # 1 y I b ¼ e byv e jy tan f ðy 2 tan 2 fþ 3=4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi e j3=4 y e 2 tan 2 f V e j3=4 e y 2 tan 2 f V dy : ð15þ The integration arising from the first term in arenthesis in (15), corresonding to the integration along the left side of the branch cut, has a saddle oint at y ¼ q b ffiffiffiffiffiffiffiffiffiffiffiffiffi tan f : ð16þ b 2 1 Thus, the integral I b may be asymtotically evaluated by virtue of its saddle-oint contribution. In articular, the integral that is associated with the first term in (15) is of the form I b1 ¼ Z 1 a fðy Vgðy Þe Þ dy ; ð17þ which can be conveniently evaluated asymtotically by using Lalace s method (secialization of the steeestdescent method to the real axis [Bleistein and Handelsman, 1975]). The resulting closed-form asymtotic exression is given by 1 sffiffiffiffiffiffiffiffiffi I b 1 2 ffiffiffiffiffiffi e j3=4 B b qffiffiffiffiffiffiffiffiffiffiffiffiffiae axx ; ð18þ k z k TM b 2 1 where B is again given by (5). The transverse decay constant in this equation is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a x ¼ b 2 ktm 2 : ð19þ [19] Equation (18) has an interesting hysical interretation. It may essentially be regarded as the TM substrate-mode field radiated by an infinite line current ex ( jk z z) on the rinted-circuit structure. However, in assigning this interretation, care must be taken when defining recisely what is meant by the field of an infinite line current. Because the stri current has a comlex roagation constant, it exonentially grows to infinity in magnitude as z! 1, thus violating the radiation condition at (minus) infinity. Because of this, the radiation field from such a current on an infinite stri is not uniquely defined (as it is for the semi-infinite line). In articular, there are two ossible solutions for the radiation field of an infinite line source, if one is willing to abandon the radiation condition. One has a transverse decay as ex ( a x x) and the other has an exonential increase as ex (+a x x). The first corresonds to a bound (roer) modal field, and the second corresonds to an imroer surface-wave tye of field. Both fields are valid mathematically, in the absence of a radiation condition. Generally, it is believed that an imroer surface wave solution lacks hysical meaning [Lamariello et al., 199; Shigesawa et al., 1993]. That is, when b z > k TM, the hysical modal field of the infinite line source should be that of a bound mode, not an imroer mode. It is therefore quite satisfying that the roer modal field behavior is the one that is observed in the field of the semi-infinite line, as seen from (18). This rovides conclusive evidence that an imroer surface-wave solution on a rinted-circuit line is not hysically meaningful. Because an imroer leaky mode in the SG region becomes an imroer surface wave mode in the asymtotic limit (where we assume a z! ), it can corresondingly be stated that in the SG an imroer leaky modal solution has little hysical significance, at least in the asymtotic limit. For finite values of a z, the issue of hysical meaning becomes qualitative, because the leaky mode changes in nature gradually as it enters the SG. In articular, as a z increases, the transition from a hysical to a nonhysical solution occurs more gradually. This will be demonstrated later when results are resented for the radiation field of a leaky mode in the SG region, for different values of a z (or equivalently, different values of z/l, with = a z z fixed). [2] The above conclusion regarding the hysical validity of a leaky mode is in agreement with that reached by Di Nallo et al. [1998], where the current excited on an infinite rinted-circuit line by a finite source was studied. [21] The integral I d along the ath C d in Figure 4 does not encounter any saddle oint along the ath of integration, and may therefore be asymtotically evaluated using integration by arts, as was done for the hysicaldark case. The result is 1 1 B 1 C I d j qffiffiffiffiffiffiffiffiffia k z k TM! 1 ffiffiffiffiffiffiffiffiffiffiffi k TM r e cos f b cos f jv b sec f e ð Þ : ð2þ The above asymtotic result (2) for the integral I d is always fairly accurate, since the saddle oint (on the imaginary axis) never aroaches the endoint of integration (s = 1) for the integral I d. Therefore, no UAE formulation is necessary to imrove the accuracy, unlike the hysical-dark case. [22] The result in (18) is likewise always accurate, unless f!. In this case, the saddle oint aroaches

7 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES 17-7 the branch oint at s = j tan f. This corresonds to the observation oint aroaching the stri axis. Although a more accurate asymtotic evaluation could in rincile be erformed in this limit, the degree of imrovement in the overall radiated field would be questionable, since a fundamental aroximation carried throughout all of the derivation has been that only the leakage field of the dominant TM substrate mode is areciable. In fact, higher-order arallel-late modes would indeed be excited by the current source in the structure of Figure 2a, which would decay raidly away from the stri transversely. Nevertheless, close to the stri these higherorder modes could in rincile become imortant. [23] The revious results show that in the SG region, the field radiated by a leaky mode on a semi-infinite stri is comosed of two arts: an exonentially decaying field that may be interreted as the bound-mode field of an infinite line source (I b ), and an equivalent source discontinuity (ESD) term (I d ), which acts as a radiating source at the beginning of the line. Using (18) and (2) in (14), and simlifying (2) slightly, the total field has the form E y sffiffiffiffiffiffiffiffiffi 2 B e j3=4 k TM B 1 C þ jb@ qffiffiffiffiffiffiffiffiffia k TM 1 1 B b qffiffiffiffiffiffiffiffiffiffiffiffiffiae jkzz e a xx b 2 1! 1 cos f ffiffiffiffiffiffiffiffiffiffiffi e jk TM r : ð21þ k TM r b cos f Unless the observation oint is extremely close to the stri, the field of the infinite line source (first term in the above equation) is usually negligible due to the exonentially decreasing behavior. This turns out to indeed be the case as results resented in Section V show. The remaining field, the ESD field, is a monotonically decreasing function of f, which never exhibits any exonential growth. Another interesting observation is that in the vicinity of hysical/sg boundary, i.e., b 1, the field is changing very raidly as f!. In fact, the asymtotic result (21) becomes singular at the hysical/ SG boundary, when f =. The overall field is thus exected to be a monotonically decreasing function of f, which is sharly eaked at f =, when b = k TM. 4. Radiation From Feed Current [24] In this section we resent an exression for the radiation from the feed current, derived using standard sectral-domain techniques [Itoh, 1989]. For modeling uroses, we assume a vertically-oriented filamentary robe carrying 1 A of current between the lower ground lane and the stri as our model for the robe feed. This would be an accurate model for a coaxial robe-fed rinted-circuit structure. For simlicity, this robe model is further simlified by assuming an equivalent diole in the middle of the substrate, for the uroses of calculating the radiation from the robe. The resulting aroximate exression resented below is secific to one articular structure (air-ga striline of Figure 1a). Neglecting the robe to stri interface in our model is inconsequential since we are rimarily concerned with comaring the relative amlitudes of the semi-infinite stri and robe radiation comonents, and not the total field itself. In addition, the asymtotic observation distances recludes the need to account for the localized field erturbation introduced by the robe/stri discontinuity. The ratio of robe to line radiation will of course deend on the tye of structure in question. However, the results resented in the next section are reresentative of how significant the direct feed radiation can be, relative to the field radiated by the semi-infinite line current. [25] The final result for the robe radiation field at y = (the ground lane), derived in Aendix B, is rffiffiffiffiffi " # ho k 2 I hk 3 e jk TM r E robe y ðþj r j 2 r TM ffiffiffiffiffiffiffiffiffiffiffi k TM r Residue; ð22þ where the normalized residue term Residue is defined as Residue ¼ lim k t k TM I TM V k t ; ; h ; ð23þ k t!k TM 2 with the function IV TMðk t; ; h 2Þ defined as the current at y = (ground-lane location) on the TM y transmissionline model of the layered structure [Michalski and Zhang, 199] for a normalized wavenumber k t ¼ k t =k, due to a 1 V series source at y = h/2. [26] The main conclusion from (22) above is that the feed radiation has a satial sreading factor of 1= ffiffi r,the same as that of the ESD that is resonsible for the field in the hysical-dark region of sace, as well as the field when the leaky mode is in the SG region. 5. Results [27] In this section, results based on the reviously derived closed-form asymtotic exressions and a numerical evaluation of the exact integral in (2) are given, as well as a comarison of the radiation from the stri and robe-feed currents. The characteristics of the field within the hysical and SG regions are illustrated, and close attention is given to the transition between these two regions. Unless otherwise noted, the results are for the amlitude of the electric field E y measured at the ground lane of the air-ga striline structure in Figure 1a, due to a leaky mode on a semi-

8 17-8 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES Figure 5. A lot of the exact radiation field versus angle for the structure of Figure 1a, for various values of b ¼ bz =k TM that are less than unity (hysical leaky modes). The observation distance is z/l = 1. The radiation field is calculated by using a numerical integration in (2). infinite stri as shown in Figure 2. The results are general, however, in the sense that the shae of the leakage field for any covered structure only deends on the value of b b z =k TM. As the structure changes, the amlitude of the field will simly change by a multilicative factor due to a different A constant (see (2)). The calculation uses the secific values r = 2.2, w/h = 1., d/h =.1, and h/l =.133. The hase constant of the TM mode for this case is k TM /k = The results have been normalized by multilying E y by the substrate height h, so that (to a good aroximation) the voltage across the substrate is lotted. Unless otherwise noted, all of the results corresond to the case = a z z =2(a z /k ) (z/l )=. [28] Figure 5 first shows the exact leakage field of the air-ga striline for leaky modes that are within the hysical region b < 1 at an observation distance of z/l = 1, for various values of b b z =k TM. As exected, the field is that of an imroer leaky mode with an exonential increase within the lit region, and oscillations due to the diffractive nature of the source discontinuity. The sloe of the exonential behavior is consistent with that redicted by (1). The oscillations are not redicted by (1), however, but are catured with fidelity here due to the exact nature of the PO integral in (2). The field eaks at values of f which are fairly consistent with the exected leakage angle q ¼ cos 1 b, and subsequently exhibits a monotonically decreasing behavior in the dark region, consistent with earlier comments made after (21). [29] Figure 6 deicts the characteristics of the radiation field for the same case as Figure 5, but for values of b that transition from below unity to above unity (within the sectral ga). This allows an examination of the nature of the field transition as the leaky mode enters the sectral ga region. The radiated field was calculated using a numerical integration of 2. The first obvious feature of the field when the mode is within sectral ga is the lack of any exonential increase in the amlitude, as well as the lack of any oscillations, both of which were seen in the hysical-lit region of the revious figure. The field in the SG region is a monotonically decreasing function of f, as redicted by (2) and (21). Another feature, which is redicted by the asymtotic formula (1), is that the eak amlitude increases as b! 1. This behavior is also redicted by (2), which actually redicts a singular field behavior at f = when b ¼ 1. (The actual leakage field is not singular near the stri axis, since (2) loses accuracy as x aroaches zero for a fixed value of z. Nevertheless, for any fixed angle f, (2) becomes more accurate as z increases.) [3] The leakage field in the SG region is tyically much smaller than that for a mode in the hysical region, rovided b is not too close to unity. This difference in field level becomes more ronounced as z becomes larger (for a fixed value of = a z z). ffiffiffiffiffiffiffiffiffiffiffi This is because the field in the SG region varies as 1= k TM r, from (14) and (2), whereas the field in the hysical-lit region does not (see (1)). [31] Another observation is that the eak field level in the SG region (at f = ) decreases raidly as b increases Figure 6. A lot of the exact radiation field versus angle for the structure of Figure 1a, for various values of b ¼ bz =k TM, including modes that are in the hysical region and in the sectral-ga region. The observation distance is z/l = 1. The radiation field is calculated by using a numerical integration in (2).

9 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES 17-9 Figure 7. A lot of the radiation field versus angle for the structure of Figure 1a, for various values of b ¼ bz =k TM that are greater than unity (modes that are in the sectral-ga region). The observation distance is z/l = 1. Results from the new asymtotic formula (21) are comared with the exact results, obtained by a numerical integration in (2). beyond unity, and more so as z/l becomes larger. This is due to the fact that the asymtotic result (2) becomes increasingly accurate as z/l increases. This result redicts a field that varies inversely roortional to b 1, and hence a eak amlitude that is infinite (and varying infinitely fast) when b ¼ 1. [32] To verify the accuracy of (21), Figure 7 comares the field obtained from (21) with that obtained by numerical integration of (2) for several different values of b, with z/l = 1. As can be seen, the agreement is quite good for most angles of observation throughout. The main difference is that at the SG boundary (b ¼ 1) the asymtotic formula redicts a singular field, whereas the actual field is finite. [33] Equation (21) shows that the total field in the SG region is the sum of an equivalent source discontinuity (ESD) field and a term that is interreted as the field of an infinite line source (ILS). To exlore this further, the field of the ILS is shown in Figure 8. This field has roughly the same order of magnitude as the total field of Figure 6 at the stri edge, but it becomes extremely small as the observation oint moves transversely away from the stri, and the rate of exonential decay increases as the hase constant increases. We can thus conclude that as a direct consequence of the exonentially decaying nature of the ILS field in the SG, the contribution of the ILS to the total leakage field is negligible for most observation angles away from the stri. Thus, while the radiated field for a mode in the SG region can be interreted as the interference between two distinct field Figure 8. The field of an infinite line source, calculated by the asymtotic formula for this field, given by (18). The fields are lotted versus normalized transverse distance x/l for the structure of Figure 1a, for various values of b ¼ b z =k TM for modes in the sectral-ga region. The observation distance is z/l = 1. For b/k TM = 1.1, a comarison is also made with the exact field of the infinite line source, defined as the integral around the branch cut in Figure 3. contributions, the ILS contribution is almost always negligible, leaving the ESD as the dominant contributor. For the sake of verification, a comarison is also made in this figure between the asymtotic evaluation of the ILS field (18) and the exact integration in (15), for one reresentative value of b ¼ 1:1. Note that the two give essentially identical results. [34] Figure 9 shows lots of the leakage field for the case of z/l = 1, as the leaky mode transitions into the Figure 9. The same lot as Figure 6, excet that z/l = 1.

10 17-1 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES Figure 1. A lot of the radiation field versus angle for the structure of Figure 1a, for various values of b ¼ bz =k TM, including modes that are in the hysical region, in the sectral-ga region, and at the hysical/ sectral-ga boundary. The radiation field from the robe feed is also included for comarison. The observation distance is z/l = 1. The radiation fields are calculated using a numerical integration of (2). SG region. Interestingly, in this case the distinction between the hysical and SG lots is much harder to make. This verifies that the radiation field changes more raidly for larger values of z (rovided = a z z is fixed) as the mode transitions from the hysical to the SG region. For relatively small values of z/l, the field changes quite smoothly in the transition region. Therefore, one imortant conclusion is that there is no shar boundary between a hysically meaningful leaky mode field and a nonhysical modal field for small values of z/l. A similar conclusion is that the field transition from a hysical to a nonhysical leaky mode becomes less shar as the attenuation constant of the leaky mode increases. [35] In Figure 1, the radiation field is shown well within each relevant region for comarison uroses, calculated via numerical evaluation of (2) in the hysical region and (21) in the SG region, with z/l = 1. Also shown is the field at the hysical/sg boundary (b = k TM ), as well as the field due to the robe feed current, calculated using (22). An imortant characteristic is evident in this figure, namely that the amlitude of the radiation from the robe feed is greater than that of the line radiation for a mode in the SG region. Additionally, although not illustrated in the resent work in the interest of sace, it has also been determined that the ratio of the field amlitude radiated by the line and the robe feed is indeendent of z/l. This will always be the case for any tye of rinted-circuit structure, although the secific value of the ratio will deend on the articular structure being investigated. This is because the robe field and the ESD field in the SG are qualitatively similar, with both fields decaying as 1= ffiffi r, with a hase center at the beginning of the stri. The only difference is that the robe field amlitude is omnidirectional in f, for a fixed value of r, while the ESD field is not. From (2), the f deendence of the ESD field is seen to be (for a fixed r) cos f= b cos f. This imlies that the feed radiation cannot be ignored when examining the radiated fields from a leaky mode in the SG region. [36] When the leaky mode is well within the hysical region, and the leaky-mode current is dominant on the stri, the leaky-mode radiation is a good redictor of the actual field radiated by a ractical source on the rintedcircuit line. To illustrate this, the structure of Figure 11 is considered, which consists of an infinite stri excited by a 1 V ga source at z =. For this structure the exact current on the stri can be calculated using a semianalytical rocedure [Di Nallo et al., 1998]. This rocedure also redicts the amlitude of the leaky-mode current that is launched by the ga source [Langston et al., 21]. Hence, a direct comarison between the field of the leaky mode and the actual field from the total stri current can be calculated. The field due to the leaky mode can be calculated in a similar manner as in (2), excet that the integration extends from minus infinity to infinity, with an even symmetry in the leaky-mode current. In addition, the result from (2) must be scaled by the amlitude of the leaky-mode current launched by the 1 V source. The total field is calculated by integrating over the total current that is roduced by the 1 V ga source. [37] Figure 12 shows a comarison of the field radiated by the total current on the infinite stri with the field radiated by the leaky-mode current. The field of the leaky mode is calculated in two different ways: by using the current on the infinite stri (which has an even symmetry about the origin) and by using the current in the semi-infinite region z > as in (2). The comarison is made for the covered microstri structure of Figure 1b, using r = 2.2, h c /h =.455, w/h = 1, and h/l =.2. Figure 11. To view of an infinite rinted-circuit stri excited by a 1 V ga voltage source.

11 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES it is weakly excited, or that it is a nonhysical mode that is within the sectral ga. Figure 1 illustrates that in the latter case the feed radiation may be larger than the radiation from the leaky-mode current. Figure 12. The normalized substrate field at a fixed distance of x =4.l from the stri, for the covered microstri structure of Figure 1b, using an infinite stri excited by a 1 V ga source at z =, as shown in Figure 11. This structure is one that has been studied by Mesa and Marques [1996] and Langston et al. [21], where it was demonstrated that a dominant leaky mode is excited by the ga source. For this structure the hase constant of the TM mode is k TM /k = and the wavenumber of the leaky mode is k z /k = b z ja z = j (.315) (hence b ¼ :8292). The amlitude of the leaky-mode current at z = is.67 A. [38] Because this case is one for which the leaky mode is fairly dominant, the agreement between the exact field and the field radiated by the leaky mode on the infinite line is quite good. The disagreement is due to the fact that the exact result includes radiation from the bound-mode current (which radiates at the source discontinuity) and also radiation from a high-order current known as the residual-wave current [Langston et al., 21]. [39] In the lotting range shown, there is negligible difference in the fields radiated by the semi-infinite and the infinite leaky-mode currents. This is because the fields are lotted only in the z > region, and radiation from the leaky-mode current on the z < art of the infinite stri (which radiates mainly in the backward direction) is very small in the z > region. [4] If a case were icked for which the leaky-mode current on the stri was not very dominant, then good agreement between the exact and the leaky-mode fields would not be exected, since the radiation from the bound-mode and residual-wave currents would be areciable relative to that from the leaky-mode current. In such a case the nature of the source and the layout of the stri (semi-infinite or infinite) would have an imortant influence on the shae of the total radiated field. The lack of dominance of a leaky mode may be due to the fact that 6. Conclusions [41] In this aer, the radiation from leaky modes on semi-infinite rinted-circuit lines has been investigated, and in articular, the nature of the radiation field as the leaky mode transitions from a hysical leaky mode (b z < k TM ) to a mode in the sectral ga (SG) region (b z > k TM ) has been investigated. Asymtotic formulas for the radiation field were resented for a leaky mode in the hysical region or the SG region. These formulas become accurate in the limit that z/l becomes large while the roduct = a z z remains finite. The formulas for the hysical region were derived reviously by Villegas et al. [1999], and were only resented in summary here. The formulas for the SG region are new, and the derivation was resented in some detail. A derivation for the radiation field of a vertical robe, modeling a coaxial feed at the beginning of the semi-infinite line, was also resented. [42] For a leaky mode in the SG region, the radiated field was shown to consist of the sum of two arts: the exonentially decaying field of an infinite line source, and an equivalent source discontinuity (ESD) tye of field that corresonds to radiation from an equivalent source at the beginning of the line. This allows for an imortant conclusion, namely, that the field of an imroer leaky (i.e. imroer surface-wave) tye of mode is never hysically meaningful in the asymtotic limit of z!1with a z z finite. In this asymtotic limit the field changes very raidly as the mode on the semiinfinite stri changes from a hysical leaky mode to a mode in the SG region. However, for a moderate value of z/l or a z, the transition from a hysical leaky mode field to a nonhysical modal field occurs gradually as b z! k TM. When lotted as a function of angle from the stri axis, the field of the infinite line source becomes negligible due to the raid rate of decay, and hence the field in the SG region is essentially that of an ESD radiation field. [43] The asymtotic formulas and an exact calculation of the radiation field were used to study how the field changes as a mode transitions from the hysical to the SG regions. As b ¼ b z =k TM aroaches unity from below, the eak of the radiated field (which occurs at aroximately f = q, where f is the angle measured from the stri axis and q is the leakage angle) moves toward the stri axis and becomes larger in amlitude. Exactly at the SG boundary, b ¼ 1:, the field is a sharly eaked monotonically decreasing function of angle f. Asb increases beyond unity, the field remains

12 17-12 VILLEGAS ET AL.: RADIATION PROPERTIES OF LEAKY MODES a monotonically decreasing function of angle, but the eak field amlitude decreases raidly with increasing b. [44] A comarison of the field radiated by the stri current and that radiated by the robe feed shows that the direct feed radiation cannot be neglected when the mode is in the SG region, regardless of the observation distance z. [45] Finally, results were resented to comare the leaky-mode radiation and the exact radiation field for an infinite rinted-circuit line excited by a ga voltage source. These results confirm that leaky-mode radiation on a semi-infinite line is a good aroximation to the total radiation field when a dominant leaky mode is excited by the source. Aendix A [46] Standard sectral-domain methods may be used to find the field radiated by a horizontal electric diole. The residue of the sectral integral at the TM y ole location k TM gives the launching amlitude A, which determines the fundamental TM y substrate-mode field (arallel-late field) radiated by the diole. Omitting the details, the final formula for the constant A for a diole at the to of the lower substrate in Figure 1a is ktm 2 A ¼ Res V TM i ðk TM Þ ; ða1þ j2k y sin k y h where Res( f (k TM )) denotes the residue of the function f (k t ) at the ole k TM and V i TM (k t ) is the sectral-domain voltage function [Michalski and Zhang, 199], which gives the voltage in the TM y transmission-line model at y = (where the field is comuted) due to a 1 A arallel current source in the model, at the location of the stri current. The wavenumber k y is the vertical wavenumber in the substrate, corresonding to the transverse wavenumber k t = k TM. Aendix B [47] Formula (22) for the radiation field of the vertical robe feed in the structure of Figure 1a is derived here. First, the vertical (y) comonent of the Fourier-transform of the field E VED y (r, y, y ) from a vertical electric diole in the middle of the lower substrate (y = h/2) is calculated by standard sectral-domain methods, and is exressed as ~E y VED k t ; ; h 2 ¼ k 2 t IV TM k t ; ; h ; ðb1þ w 2 where y = has been assumed (observation oint on the ground lane). The function I v TM is the current in the sectral-domain transmission-line model (at y = ) due to a series 1 V source at y = h/2 [Michalski and Zhang, 199]. Taking the inverse Fourier transform (using a Hankel transform integral in olar coordinates), the normalized electric field of the robe is then exressed as an integral in the normalized wavenumber lane as E robe y ðþ¼ r I h ð4þ h o k 2 Z 1 r k 3 t I V TM k t ; ; h 1 2 H ð2þ ðk t rþdk t ; ðb2þ where k t = k t /k. The arallel-late excitation amlitude of the robe is then equal to the residue contribution from the ole at k t = k TM, so that rffiffiffiffiffi j ho k 2 ðþj r I hk 3 2 r Res IV TM k TM ; ; h 2 E robe y TM " # e jk TM r ffiffiffiffiffiffiffiffiffiffiffi k TM r ; ðb3þ where Res( f (k TM )) denotes the residue of the function f (k t ) at the normalized ole location k TM and use has been made of the large argument aroximation of the Hankel function in (B2). References Bleistein, N., and R. A. Handelsman, Asymtotic Exansion of Integrals, Holt, Rinehart, and Winston, Fort Worth, Tex., Das, N. K., and D. M. Pozar, Full-wave sectral-domain comutation of material, radiation and guided-wave losses in infinite multilayered rinted transmission lines, IEEE Trans. Microwave Theory Tech., 39, 54 63, Di Nallo, C., F. Mesa, and D. R. Jackson, Excitation of leaky modes on multilayer striline structures, IEEE Trans. Microwave Theory Tech., 46, , Itoh, T., Numerical Techniques for Microwave and Millimeter- Wave Passive Structures, John Wiley, New York, Lamariello, P., F. Frezza, and A. A. Oliner, The transition region between bound-wave and leaky-wave ranges for a artially dielectric-loaded oen guiding structure, IEEE Trans. Microwave Theory Tech., 38, , 199. Langston, W. L., J. T. Williams, D. R. Jackson, and F. Mesa, Surious radiation from a ractical source on a leaky covered microstri line, IEEE Trans. Microwave Theory Tech., 49, , 21. Mesa, F., and R. Marques, Low-frequency regime in covered multilayered strilines, IEEE Trans. Microwave Theory Tech., 44, , Michalski, K. A., and D. Zhang, Electromagnetic scattering and radiation by surfaces of arbitrary shae in layered media. I. Theory, IEEE Trans. Antennas Proag., 38, , 199.