In this work, an asymptotic analysis is presented for bound-mode and free-space residual-wave currents, with the aim of obtaining an analytical representation of the continuous-spectrum current excited by a delta-gap source on a microstrip line. The proposed approach is valid in a frequency region which extends through the entire spectral gap of the dominant leaky mode. The analysis proves in a rigorous way the different asymptotic behaviors of the TM0 bound-mode and free-space residual-wave currents. The relevance of the nature of the involved branch points to these asymptotic behaviors is also discussed. An explicit expression for the numerical coefficient arising in the asymptotic expansion obtained through Watson's Lemma is provided in a simple closed form for both types of residual-wave current. Analytical details are supplied, together with numerical results which confirm the predicted asymptotic behaviors and the accuracy of the closed-form representation also in proximity of the source, if the presence of singularities of the relevant spectral integrands is properly taken into account.
 In this work, the problem of approximating the continuous-spectrum (CS) current excited by a finite source on a microstrip line is addressed, on the basis of an asymptotic evaluation of the relevant spectral-domain integral representation. As is well known, the CS current is responsible for spurious effects (i.e., attenuation and interference) in microwave planar circuits, whose operation is based on the excitation of a single bound mode [Freire et al., 1999; Mesa and Jackson, 2002]; moreover, undesired crosstalk between neighboring circuits may also be produced [Bernal et al., 2003]. On the other hand, accurate knowledge of the CS current is necessary in order to evaluate the radiated field in planar traveling-wave antennas, e.g., leaky-wave antennas [Tamir and Oliner, 1963; Oliner, 1987].
 In general, the total current excited on a planar structure can be represented as an inverse Fourier transform with respect to the spectral variable kz, associated with the longitudinal z direction of the waveguide under analysis [Di Nallo et al., 1998]. We refer here to the basic microstrip structure shown in Figure 1. The relevant spectral integrand has branch points and pole singularities [Mesa et al., 1999] (see Figure 2); by deforming the integration path along the Sommerfeld branch cuts, the total current can be represented as the sum of the residue contributions of a finite number of poles, associated to the bound modes of the structure, and the contribution of the integral around the branch cuts, which constitutes the CS current [Mesa et al., 2001].
 By means of a further contour deformation along the vertical steepest-descent paths (SDPs) through the branch points, the CS can be expressed as the sum of the residue contribution of the captured (i.e., physical) leaky-wave (LW) poles, if any, and the contribution of the integrals along the SDPs, which is known as the residual wave (RW) [Mesa et al., 2001]. Such a representation is particularly useful in those cases when the contribution of a single LW pole allows us to accurately represent the CS. However, there are frequency regions in which the LW poles are not physical (spectral-gap regions [Baccarelli et al., 2002, 2004]), and therefore the RW is the only contribution to the CS. In any case, since the LW has an exponential decay while the RW has an algebraic asymptotic decay with the longitudinal z coordinate [Mesa et al., 2001], the latter is the dominant contribution to the CS at large distances from the source.
 In the case of a microstrip line, it has been shown in Mesa et al.  that a logarithmic-type branch point is located in the kz plane at the k0 free-space wavenumber, while square-root-type branch points are located at the wavenumbers of the TM and TE modes of the background structure; these TE and TM modes can be either above-cutoff modes, with a real propagation constant, or below-cutoff modes, which may be leaky waves with a complex propagation constant. Consequently, the RW current excited on a microstrip line can be expressed as the sum of a finite number of distinct components: a free-space RW current (FSRWC), associated to the k0 branch point; bound-mode RW currents (BMRWC), associated to the above-cutoff TM and TE modes of the background structure; finally, leaky-mode RW currents (LMRWC), associated to the physical leaky modes of the background structure, whose contribution decays exponentially with z and therefore can typically be neglected.
 The aim of the present work is to provide a rigorous asymptotic study of both the TM0-BMRWC and the FSRWC for a microstrip line excited by a delta-gap source in a frequency region that extends through the entire spectral gap of the dominant leaky mode, in order to obtain an analytical representation of the CS current. In particular, by means of suitable integral evaluations in the complex plane of the transverse spectral variable kx via Cauchy Integral Theorem, it will rigorously be shown that both TM0-BMRWC and FSRWC have an algebraic asymptotic behavior for large values of the longitudinal z coordinate: the former as z−3/2 and the latter as z−2 [Jackson et al., 2000b; Mesa et al., 2001]. Moreover, as it will be made clear in the analysis, such different behaviors are related to the square-root or logarithmic nature of the involved branch points. An explicit expression for the numerical coefficient arising in the asymptotic expansion obtained through Watson's Lemma will be provided in a simple closed form for both the TM0-BMRWC and the FSRWC. In order to obtain a good agreement also very close to the source, a more accurate formulation, which takes into account the presence of one pole singularity, will also be derived for the TM0-BMRWC.
 The paper is organized as follows. Analytical derivations are reported in section 2 for both TM0-BMRWC and FSRWC, on the basis of a simple spectral-domain representation of the microstrip current obtained through the method of moments [Di Nallo et al., 1998]. A discussion on the proposed formulations is given in section 3. Numerical results for two different microstrip structures at frequencies inside the spectral gap of the dominant leaky-mode are presented in section 4. Some concluding remarks are provided in section 5.
2.1. Problem Statement and Background
 The reference structure is a microstrip line of width w on a homogeneous isotropic lossless grounded dielectric slab of thickness h and relative permittivity ɛr (see Figure 1). A delta-gap source is assumed, with a longitudinal impressed electric field Ezgap with transverse profile η(x) and longitudinal profile Lg(z) (ideally, Lg(z) = δ(z), although in practice it can be chosen as a narrow pulse function). By virtue of the symmetry of the excitation and of the structure, an even electromagnetic-field configuration will be excited, with the x = 0 plane being a perfect magnetic conductor.
 An Electric-Field Integral Equation (EFIE) can be written, by assuming a longitudinal current density Jz on the strip (accurate for a narrow strip) with transverse profile η(x) [Di Nallo et al., 1998]. The consideration of a single basis function in a Galerkin discretization of the EFIE through the method of moments in the spectral domain is sufficient to achieve an accurate asymptotic representation, as shown in Jackson et al. [2000a]. The resulting total current I(z) (obtained by integrating Jz across the strip width) can then be written as an inverse Fourier transform as:
Here ��x is an appropriate integration path in the transverse kx plane and zz(kx, kz) is the spectral-domain Green's function of the background grounded dielectric slab for the z component of the electric field due to a z-directed electric dipole, which is known in a simple closed form:
where kt = , and ViTE(kt), ViTM(kt) are the voltage waves excited by a one-Amp current generator on the transverse equivalent transmission line along y associated to TE and TM waves, respectively [Itoh and Menzel, 1981]:
where ζ0 is the characteristic impedance of free space. The ��x integration path may or may not detour around the different singularities of the spectral-domain Green's function; these are poles at k = ± (where kbs is the wavenumber of the modes of the background structure, i.e., the grounded dielectric slab) and branch points at k = ±. The nonuniqueness of the choice of the ��x integration path implies that (kz) is a multivalued function of its argument. A deep discussion about the nature of its singularities is carried out in Mesa et al. , which shows that different types of branch points exist in the kz plane. In particular, logarithmic-type branch points are located at ±k0, while algebraic square-root-type branch points are located at the wavenumbers ±kbs of the TM and TE modes of the background structure.
 Assuming now that, for given structure parameters and frequency, only the TM0 mode of the background structure is above cutoff, the integral contour in equation (1) can be deformed as shown in Figure 2 to the SDPs through the branch points k and k0. The residual-wave currents are then expressed as:
where SDP stands for SDP or SDP.
2.2. TM0 Bound-Mode Residual-Wave Current
 Let us consider the TM0-BMRWC, that is given by equation (6) with SDP being the steepest-descent path SDP through the k branch point in the kz plane (see Figure 2). By letting kz = k − js, equation (6) can be written as:
where F(s) is given by
and D0 and D are defined as:
is the Fourier transform of a rectangular profile function η(x). In equation (9), according to Figure 3, ��0 is the integration path in the kx plane along the real axis (which corresponds to evaluating the (kz) function on the top sheet with respect to all the branch points), while �� is the integration path in the kx plane which detours around the two poles at kx = ±k = ± (which corresponds to evaluate the (kz) function on the bottom sheet with respect to the k branch point and on the top sheet with respect to all the other branch points).
 The F(s) function can then be written as:
The numerator D(s) − D0(s) in equation (11) is equal to the sum of the residue contributions at the poles ±k. By virtue of the parity properties:
The problem now is to approximate the D0(s) and D(s) functions for s → 0 and, in view of equations (13) and (15), this task is reduced to the evaluation of the D0(s) (or the D) function only. As shown in Baccarelli et al. , this can be achieved by means of Cauchy Integral Theorem once the D0(s) function is cast in the form:
by a proper rearrangement of the 2(kx) factor. In fact, the integral in equation (16) can be evaluated by means of the contour shown in Figure 4, and it turns out to be expressed as the sum of four distinct contributions:
The first contribution, I, is due to the pole at kx = k; the second, I, is due to the integral along the semicircular path γr around the pole in kx = 0; the third, I, is due to the integral along the semicircular path ΓR; finally, the fourth, Ibc, is due to the integral along the Sommerfeld branch cut Γbc through the branch point k = . The contribution I has been asymptotically evaluated in Baccarelli et al. , and the resulting expression is
while the contribution I tends to zero in the limit R → +∞. By virtue of equation (3), Iγ can thus be expressed as:
In the limit s → 0, the ViTM function can be approximated by an asymptotic power series in s, whose dominant contributions are given by the singular part (∼s−1) associated to the poles in kt = ±kand a regular term Cγ (∼s0):
The contribution Ibc has to be evaluated numerically, taking into account that, in the limit s → 0, the branch point k tends to the finite value j and the integration path Γbc simply lies along the imaginary axis through the branch point k. The final result Ibc(0) is seen to be a numerical constant independent of s.
is the pole of the approximate F(s) function, which can be considered an approximation of the actual pole of the exact F(s) function.
 If other modes of the background structure with wavenumber kbs are above cutoff, the analysis remains the same: the only change is in the definition of the coefficient A2, which has to take into account the finite contribution of the residues in the additional poles k = j in the kx plane.
2.3. Free-Space Residual-Wave Current
 We consider now the FSRWC, that is given by equation (6) with SDP being the steepest-descent path SDP through the k0 branch point in the kz plane (see Figure 2). By letting kz = k0 − js, we get:
where, assuming that the only TM0 mode is above cutoff, F(s) is given by
where D and D are defined as:
where �� is, according to Figure 3, the integration path in the kx plane which detours around both the poles in kx = ±k00486604 = ± and the branch points in kx = ±k= ± = ± (which corresponds to evaluating the (kz) function on the Riemann sheet S−1 with respect to the k0 branch point [Mesa et al., 1999]). The F(s) function can then be written as:
In this case the numerator D (s) − D(s) in equation (34) is equal to an integral along a closed path ℒ that joins the two branch points in kx = ±k and lies partly on the top and partly on the bottom sheet of the kx plane (see Figure 5):
By joining the branch points through line segments as in Figure 5, we can parameterize the integration path ℒ as follows:
and, in the limit s → 0, we obtain:
where zzT(s, t) and zzB(s, t) are the spectral Green's functions, written as functions of the variables s and t, evaluated on the top and on the bottom sheet of the kx plane, respectively. From equation (3), after lengthy calculations, the following simple result can be obtained:
where ζ0 is the characteristic impedance of free space, so that, from equation (37):
As a first result, from equation (34), we can then express the F(s) function as:
Now we have to evaluate D(s) in the limit s → 0; D(s) can then be obtained from equation (35). With reference to Figure 3, by closing the �� path at infinity, as it was done in Figure 4 for the ��0 path, D(s) can be expressed as:
The first contribution, I, is due to the pole in kx = −k and can be asymptotically evaluated with the usual procedure that leads to
The other terms in equation (42) have the same meaning as in equation (17). The contribution I tends to zero in the limit R → +∞, while I(s), in the limit r → 0, can be written as:
By virtue of equation (3), in the limit s → 0, it can be obtained:
Finally, we have to evaluate Ibc(s). By letting t = and kz = k0 − js, it follows:
and, by virtue of equation (3) and in the limit s → 0, we obtain:
that does not depend on s and can simply be evaluated numerically (it is to be noted that the integrand is regular everywhere, since at t = 0 and at t = ±k0 removable singularities occur).
and from equation (41), again in the limit s → 0, we can conclude:
By means of Watson's Lemma, we finally obtain the asymptotic behavior of I as [Mesa et al., 2001]:
If other modes of the background structure with wavenumber kbs are above cutoff, the analysis remains the same: the only change is in the definition of the coefficient A0, which has to take into account the finite contribution of the residues in the additional poles k= j of the kx plane.
 From the analysis reported in Section 2, several remarks can be pointed out. As a first result, the dominant asymptotic representation of the RW currents on a microstrip line has been derived in a closed form. In particular, explicit analytical expressions for the relevant Watson coefficients have been given in equations (28) and (53). In this connection, it is to be noted that the algebraic decays of the RW currents in a microstrip line expressed by equations (27) and (55) are equal to those reported in Mesa et al.  without derivation. The asymptotic behaviors of the RW currents have now been proved rigorously for the case of a delta-gap source: two different algebraic decays can be observed, i.e., z−3/2 for the TM0-BMRWC and z−2 for the FSRWC. Our analysis has further revealed that the Watson coefficient for the TM0-BMRWC does not depend on the strip width w, while the one for the FSRWC depends explicitly on it through the term A0 in equation (54) which depends on the term I in equation (43).
 It is also worth noting that all the analytical derivations have been carried out by assuming that the RW currents can accurately be represented by means of just one longitudinal rectangular basis function: such an assumption is expected to be correct in a neighborhood of the spectral-gap region of the dominant leaky mode, whose current configuration resembles that of the dominant bound EH0 mode. In fact, in this frequency range, the character of the RW currents is mainly influenced by the closeness of the corresponding leaky pole to the k or k0 branch point. However, at very high frequencies, the dominant leaky mode may evolve into a different mode by changing its spectral character and therefore its current configuration; in addition, other leaky poles (which correspond to higher-order modes) can approach the involved branch points, as shown in Mesa and Jackson . This may cause the RW current to assume a more complex character, not accurately described by just one basis function, thus affecting the results for the Watson coefficients. Instead, the conclusions for the algebraic asymptotic decays remain the same: in fact, the higher-order basis functions do not directly influence the behavior of the spectral integrand for s → 0 [Jackson et al., 2000a], but their presence may affect the numerical coefficient of the first longitudinal basis function, which determines the Watson coefficient.
 Another interesting point arises in connection with the nature (algebraic or logarithmic) of the k and k0 branch points and the corresponding asymptotic behavior of the TM0-BMRWC and the FSRWC. The asymptotic behavior, for large z, of the FSRWC is related to the behavior, for s → 0, of the relevant integrand. By examining equations (34), (49), and (52), it turns out that the dominant contribution in the neighborhood of s = 0 is due to the closed path ℒ in the transverse plane and is given by equation (39). This contribution is not present in the corresponding integrand of the TM0-BMRWC in equation (11), and it concurs to determine the different algebraic decays of the two kinds of residual-wave currents. Since the presence of the k branch point in the transverse complex plane determines the logarithmic character of the k0 branch point in the longitudinal kz plane [Mesa et al., 1999], it can be concluded that, in the considered microstrip structure and excitation, the different nature of the k and k0 branch points is directly related to the asymptotic behavior of the corresponding residual-wave currents. However, it should be added that, in general, i.e., by considering an arbitrary analytic function of one complex variable, the logarithmic nature of a branch point does not determine in itself the behavior of the function in the neighborhood of the branch point [Markushevich, 1965].
 Finally, the asymptotic analysis has been further refined for the case of the TM0-BMRWC, by taking into account the presence of a pole singularity in the approximate integrand. This uniform formulation is expected to provide more accurate results in the lower frequency range of the involved spectral-gap region, where one pole singularity becomes arbitrarily close to the k branch point and the simple Watson formulation is accurate only for extremely large distances from the source. A similar formulation has not been carried out for the case of the FSRWC, because of the extremely cumbersome analytical calculations involved.
4. Numerical Results
 In order to validate the proposed formulations for both TM0-BMRWC and FSRWC, two reference microstrip lines will be considered, with different strip widths, excited by a longitudinal delta-gap source. In both cases the relative permittivity is ɛr = 2.2 and the substrate height is h = 1 mm (see Figure 1). For numerical purposes [Mesa et al., 2001], the width Δ of the gap source is taken finite: in particular, we consider a rectangular profile for the impressed gap source Lg(z) = 1/Δ for ∣z∣ ≤ Δ/2, with Δ = 0.1λ0, so that its Fourier transform is g(kz) = sinc (kz Δ/2).
4.1. Large Strip Width: w/h = 3
 We consider first a large strip width w = 3 mm for which the choice of a single, rectangular longitudinal basis function is known to be appropriate [Jackson et al., 2000a]. The relevant dispersion curves are reported in Figure 6 for both the fundamental bound mode EH0 and the dominant leaky mode. The spectral-gap region of the dominant leaky mode begins at zero frequency and ends at the frequency for which the phase constant of the leaky mode is equal to the phase constant of the TM0 mode of the background structure (this occurs at about f = 27.5 GHz). At low frequencies two distinct real improper poles exist, which then merge to become a complex conjugate pair at higher frequencies. Outside the spectral gap, a physical leaky pole exists in the frequency range between about 27.5 and 33 GHz and its behavior was first described in Nghiem et al. .
 In Figure 7, a comparison is shown among three different formulations for the TM0-BMRWC inside the spectral gap at f = 25 GHz, where the poles are a complex conjugate pair: in particular, the absolute value of the TM0-BMRWC is reported as a function of the normalized longitudinal distance z/λ0 (where λ0 is the free-space wavelength). The exact residual-wave current ERW (3 + 2 BF) has been obtained by a full-wave spectral-domain formulation [Mesa and Jackson, 2002] with three longitudinal and two transverse basis functions for the current density on the strip. The adopted basis functions are the customary Chebyshev polynomials weighted by suitable factors which take into account the expected edge behavior [Collin, 1991]. The exact residual-wave current ERW (Rect BF) has been obtained instead by means of just one longitudinal rectangular basis function as the one considered in Section 2 for the analytical derivations. Finally, the asymptotic residual-wave current ARW has been calculated according to equation (27). As it can be seen, the analytical formulation ARW predicts the correct asymptotic behavior for large z/λ0 values: in particular, as it can be observed in the inset, the z−3/2 algebraic decay is confirmed and the Watson coefficient obtained in equation (28) is correct. It is worth noting that the use of just one basis function does not affect the accuracy of the results, due to the dominant-like character of the TM0-BMRWC in the considered frequency range.
 In Figure 8 a comparison is shown, again at f = 25 GHz, among three different formulations for the FSRWC. The exact formulations ERW (3 + 2 BF) and ERW (Rect BF) have been computed as the corresponding ones in Figure 7, while the asymptotic residual-wave current FS-ARW represents the FSRWC calculated according to equation (55). Also in this case the algebraic asymptotic decay as z−2 is correctly predicted; moreover, the Watson coefficient obtained in equation (53) is the correct one for the ERW (Rect BF) formulation, and it is in a very good agreement with the one arising from the full-wave ERW (3 + 2 BF) formulation. In fact, in this case a slight discrepancy can be observed between the exact results obtained with one basis function and those obtained with multiple basis functions.
 By lowering the frequency, thus entering the spectral-gap region where both poles are improper real, the ARW formulation for the TM0-BMRWC becomes less and less accurate, as it can be seen in Figure 9, corresponding to f = 10 GHz. In fact, the ARW formulation overestimates both the ERW formulations by one or two orders of magnitude. However, the asymptotic z−3/2 algebraic decay and the Watson coefficient predicted by the ARW formulation are the correct ones, as it can be observed in the inset. On the other hand, the uniform asymptotic residual wave current UARW, computed by means of equation (29) which takes into account the presence and the influence of a pole singularity located near the branch point k, is in excellent agreement with the ERW formulations also very close to the source, down to z ≃ λ0, confirming its uniform character.
 The FSRWC at f = 10 GHz is shown in Figure 10: at this frequency, the FS-ARW does not provide an accurate representation of the exact residual-wave current in the first tens of wavelengths from the source, due to the presence of singularities in the neighborhood of the k0 branch point. However, far from the source, both the z−2 algebraic decay and the Watson coefficient are again correctly reproduced. Moreover, it is worth noting that at this frequency the ERW formulations are almost superimposed.
4.2. Moderate Strip Width: w/h = 1
 We consider now the case of a microstrip line with a moderate strip width, i.e., w = 1 mm; the other parameters are the same of Figure 6. In Figure 11 the dispersion curves are reported for both the fundamental bound mode EH0 and the dominant leaky mode. As it can be seen, the spectral gap region is now larger with respect to that of Figure 6, as a consequence of the narrower metal strip. Three frequencies inside the spectral gap will be considered in this case, i.e., f = 30 GHz (corresponding to the presence of two complex-conjugate improper poles), f = 20 GHz, and f = 10 GHz (corresponding to the presence of two real improper poles).
 In Figures 12 and 13, results at f = 30 GHz are shown for the TM0-BMRWC and the FSRWC, respectively. Both ARW and FS-ARW formulations are in very good agreement with the exact results, confirming the expected asymptotic trends and the accuracy of the calculated Watson coefficients.
 In Figure 14, results at f = 20 GHz are shown for the TM0-BMRWC. In this case, the ARW formulation fails close to the source. In fact, while in the previous case at f = 30 GHz both the poles were sufficiently far from the k branch point to guarantee the correctness of the ARW formulation not only for large z but also relatively near the source, at f = 20 GHz one improper real pole is close to the k branch point and it influences the behavior of the residual wave: therefore the ARW is a good representation of the TM0-BMRWC only far from the source. Moreover, the UARW, which takes into account the presence of a pole singularity, begins to provide a better approximation.
 In Figure 15 results at f = 20 GHz are shown for the FSRWC. Again, the asymptotic FS-ARW formulation is a good representation of the exact residual-wave current from z ≃ 5 λ0, giving the correct asymptotic z−2 algebraic decay and the exact Watson coefficient.
 Finally, in Figures 16 and 17, results at f = 10 GHz are shown for the TM0-BMRWC and the FSRWC, respectively. Again, the correctness of the Watson formulations, i.e., ARW in Figure 16 and FS-ARW in Figure 17, is confirmed for large values of z/λ0. In this case, one pole is very close to both the k and k0 branch points and it strongly influences both the relevant residual-wave currents: therefore the ARW in Figure 16 is still more than one order of magnitude above the exact results in the first hundreds of wavelengths of distance from the source, and the FS-ARW in Figure 17 begins to be accurate from z ≃ 40 λ0. However, for the TM0-BMRWC the uniform representation UARW in Figure 16 allows us to achieve accurate results down to a fraction of wavelength of distance from the source.
 In this work, a rigorous asymptotic analysis has been presented for the TM0 bound-mode residual-wave current and the free-space residual wave current of a microstrip line excited by a delta-gap source.
 By means of suitable approximations of the involved spectral integrands in the neighborhood of the relevant branch points, Watson's Lemma could be used to determine the order of the algebraic decays of the residual-wave currents with the longitudinal distance from the source, and to obtain the corresponding Watson coefficients in a simple closed form.
 In addition, the analysis has made clear which is the role of the different nature of the involved branch points (i.e., square-root type versus logarithmic type), showing that, for the considered microstrip structure and excitation, such a different nature determines the asymptotic behavior of the corresponding residual-wave currents.
 For the case of the TM0 bound-mode residual-wave current, a uniform asymptotic formulation has also been derived, which takes into account the presence of one pole singularity in the spectral integrand and is therefore extremely more accurate at low frequencies. Various numerical results have been provided, which confirm the correctness and the accuracy of the proposed formulations.
 The authors wish to acknowledge Prof. D. R. Jackson of the University of Houston, Texas, for stimulating discussions on this topic, and Prof. F. Mesa of the University of Seville, Spain, for his helpful suggestions.