Abstract
 Top of page
 Abstract
 1. Introduction
 2. Formulation
 3. Discussion and Results
 4. Conclusion
 Appendix A: Continuous Spectrum Contribution
 Appendix B: SemiInfinite Interconnect
 References
 Supporting Information
[1] Leaky wave excitation on threedimensional, viafed single and coupled microstrip interconnects is studied. Closedform asymptotic expressions for the fields associated with the interconnect are derived and are applied in the traveling/standing wave and leaky wave regimes, both of which lead to radiation. The leaky wave beam angle is found to correspond to the usual twodimensional ray optics leakage angle for long interconnects, as expected, and depends on interconnect length, spacing, and excitation for shorter interconnects. Comparisons with fullwave results are shown for the case of viafed coupled interconnects.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Formulation
 3. Discussion and Results
 4. Conclusion
 Appendix A: Continuous Spectrum Contribution
 Appendix B: SemiInfinite Interconnect
 References
 Supporting Information
[2] Leaky waves on printedcircuit transmission lines have recently received considerable attention. Radiation into space waves, surface waves, and both wave types have been investigated for a variety of twodimensional microstriplike lines, coplanar lines, and various printed transmission line modifications [Michalski and Zheng, 1989; Shigesawa et al., 1991, 1995, 1996; Carin and Das, 1992; Bagby et al., 1993; Nghiem et al., 1995, 1996; Yakovlev and Hanson, 1997; Nyquist et al., 1997]. Experience gained in the study of leaky wave open boundary waveguides has resulted in the development of novel microwave and millimeter wave devices and components, such as leaky wave couplers [Niu et al., 1993] and leaky wave antennas, although many studies have concentrated on the potentially harmful effects of unintended crosstalk, coupling, and radiation due to leaky waves on integrated circuits.
[3] In many practical applications, circuit board and onchip interconnects are constructed from finite sections of transmission lines. The possibility of leaky wave excitation on these threedimensional interconnects has received relatively little attention [Das, 1996; Carin et al., 1998], compared to twodimensional structures, although semiinfinite sourceexcited lines have been somewhat more thoroughly studied [Villegas et al., 1999; Mesa et al., 1999, 2001; Jackson et al., 2000; Mesa and Jackson, 2002; Langston et al., 2003]. While it is clear that the simpler twodimensional transmission line model can be used as a starting point for considering leaky waves on threedimensional structures, the degree to which twodimensional results predict threedimensional phenomena is still an open question. For example, Das [1996] reported that the onset of leakage on certain threedimensional transmission line structures seems to occur at a different point than that predicted by twodimensional theory. Furthermore, in previous studies simplified excitations (generally, delta gaps) were considered, or the influence of the feed was ignored.
[4] In this paper we investigate threedimensional microstrip interconnect geometries, with realistic excitations, and derive asymptotic closedform expressions for the field radiated by such structures. The field along the airdielectric interface is considered, since potential crosstalk with other devices may depend on the field at this interface. The results are compared to a fullwave, threedimensional simulation in both the traveling/standing wave (TSW) regime (analogous to the bound regime on twodimensional transmission lines; see section 3) and leaky wave (LW) regime, verifying the developed expressions. It is shown that in the leaky wave regime the angle of leakage on a threedimensional interconnect corresponds with the usual twodimensional ray optics result if the interconnect is sufficiently long, even for coupled interconnects. For shorter interconnect strips, the influence of interconnect length, spacing, and of the via is important. This paper is a considerably expanded version of Hanson et al. [2004] and Hanson and Yakovlev [2004].
2. Formulation
 Top of page
 Abstract
 1. Introduction
 2. Formulation
 3. Discussion and Results
 4. Conclusion
 Appendix A: Continuous Spectrum Contribution
 Appendix B: SemiInfinite Interconnect
 References
 Supporting Information
[5] A threedimensional microstrip interconnect, fed and terminated by cylindrical vias, is depicted in Figure 1 (notation associated with the grounded slab was chosen to be consistent with Hanson [2005]).
[6] For the laterally infinite grounded dielectric slab, the Hertzian potential dyadic Green's function can be written in eigenfunction form as [Hanson, 2005]
where the overbar indicates complex conjugation, n_{e} and n_{h} indicate E (transverse magnetic (TM)) and H (transverse electric (TE)) discrete surface wave modes, respectively (and similarly for the TM and TE continuous spectrum over ξ_{e} and ξ_{h}), N_{e} and N_{h} are the number of above cutoff TM and TE surface waves, respectively, k_{ρ}^{2} = k_{y}^{2} + k_{z}^{2} is the radial wave number, β_{sw,n} is the propagation constant of the nth surface waveguided by the grounded dielectric slab, and D^{xx} = , D^{αα} = + for α = y, z. The superscripts d and c indicate the discrete and continuous contributions to the Green's dyadic, respectively. The eigenfunctions u and adjoint eigenfunctions v are defined by Hanson [2005]. An alternative form for the Green's function is
where ρ = ((y − y′)^{2} + (z − z′)^{2})^{1/2}. The pure spectral form (1) will be used to obtain the field due to horizontal currents on the microstrip interconnect, whereas the modal form (2) will be used to obtain the field from the vertical via feeds.
[7] The electric and magnetic fields are
where
[8] We will concentrate on the fields excited along the surface of the slab (at x = 0) for ρ ≫ 1. In this case the continuous spectrum contribution g^{c} is very small compared to the discrete spectrum contribution g^{d}. In particular, as shown in the following, the discrete contribution varies as 1/. As shown in Appendix A, the continuous spectrum contribution varies as 1/ρ^{2}. Therefore the discrete contribution is dominant for ρ ≫ 1, and so in what follows we will consider only the discrete contribution to g^{d}, leading to ^{d}.
2.1. Field Due to a Vertical Via Source
[9] We consider a microstrip interconnect fed by a vertical cylindrical via. The via is infinitely thin, and carries the current = J_{x},
for x ∈ [−d, 0], where f(x) is a given current profile. The total current on the via at x = 0 (the viastrip intersection) is
We will assume f(0) = 1. The resulting potential is, from (4),
where
(since a vertical current will only excite TM modes) with ρ_{p} = ((y − y_{0})^{2} + (z − z_{0})^{2})^{1/2}. Upon defining
assuming ≫ 1 and approximating the Hankel function using its largeargument form, and using the eigenfunctions (56) and (63)–(64) of Hanson [2005], the electric field along the surface of the interface (x = 0) due to a via is obtained as
where
and where
with γ_{j}^{2} = β_{sw,n}^{2} − k_{j}^{2}, j = 2,3.
[10] The fields maintained by the via current are completely determined subject to specifying the current profile f(x) in (9), leading to h(). The via probes are not assumed to be matched to the interconnect, although reasonable choices for via radius and position are assumed. Since the substrate thickness d is not necessarily small compared to the wavelength in the dielectric, the assumption of constant current on the via is inappropriate. Because of the connection between the via and the strip at x = 0, and the via feed at x = −d, the current should be nonzero at the top and bottom of the via. At frequencies considered in this study (f = 6.5 GHz, 9 GHz, and 13 GHz; see section 3), d is approximately λ/3, λ/2, and 2λ/3, respectively, where λ is the wavelength in the dielectric. Therefore it would be expected that the current would become zero somewhere along the via's length (e.g., a null point is forced to be near the midpoint of the via since the current is nonzero at the via ends). Fullwave simulation using Ansoft Ensemble verifies this fact, and so we assume the via current profile to be
for x ∈ [−d, 0], which is the current on a traditional halfwave dipole shifted to put the null at the midpoint (x = −d/2) of the via and current maximums at the strip and feed connections, leading to
[11] It was found that while the magnitude of the via field is somewhat insensitive to the choice of f(x), the phase is not, and therefore the total field due to the sum of the interconnect strip current and via current is sensitive to the choice of via current profile. In particular, proper via modeling is important for correct prediction of the leakage angle for shorter interconnects, as described later.
2.2. Potential Due to a Traveling Wave Interconnect Current
[12] We now model the field due to the horizontal interconnect current. Assume a perfectly conducting printed strip at the airdielectric interface x = 0 having length L (extending from z = −L_{1} to z = L_{2}, L_{1,2} ≥ 0, L_{2} − L_{1} = L), width 2w, and centered at y = y_{0}. Assume the traveling wave strip current = J_{z},
where γ_{ms} is a complex propagation constant and T_{n} is the nth order Chebyshev polynomial. For bound microstrip modes γ_{ms} = β_{ms} is real valued and β_{ms} > β_{sw}. For leaky microstrip modes γ_{ms} = β_{ms} − jα is complex valued and β_{ms} < β_{sw}. The total current on the strip at the viastrip junction (x = 0, z = z_{0}) is
Since the current on the via at the viastrip junction (6) is assumed to be I_{0}^{p}, current continuity then demands that
[13] The associated potential is given by (4) where the necessary Green's components are
with
for α = x, z. Defining
and
assuming n = 0 and that only the TM_{0} mode is above cutoff (in which case u_{zz}^{(3)} = 0), and denoting β_{sw} = β_{sw,0} leads to
where
2.2.1. Asymptotic Evaluation of I(y, z, β_{sw}) by the Method of Steepest Descent
[14] To evaluate (27), note that
where the integral over k_{y} was evaluated using the residue theorem, and where
With ρ_{s} = ((y − y_{0}) ^{2} + z^{2})^{1/2},
(28) becomes
where
The path of integration C is the path k_{z} ∈ (−∞, ∞) mapped to the steepest descent plane, as shown in Figure 2. In Figure 2, P indicates the mapping of the proper sheet in the k_{z} plane (where ℜ (≷) 0 for y(≷)0) into the steepest descent plane, and I indicates the mapping of the improper sheet.
[15] Deforming the integration contour C to the steepest descent contour SDC leads to
where S represents any singularities encountered in the deformation of the original integration contour C to the steepest descent contour SDC. Here we consider a finite length interconnect, and in Appendix B a semiinfinite interconnect (which leads to quite different results).
[16] Assuming L_{1}, L_{2} < ∞, there are no singularities in the steepest descent plane. In particular, in the following we will consider the case L_{1} = L_{2} = L/2. In the integral (33) a stationary phase point occurs at θ_{s} = ϕ, leading to
where
For a narrow strip (wβ_{sw} ≪ 1) the Bessel function does not contribute significantly to the angular dependence, and from (37) it can be seen that as L increases the width of the radiation beam will decrease.
2.2.2. Strip Fields
[17] The electric field along the interface due to a single strip interconnect is obtained from the potential (26), using (3), as
where
Since
we obtain
where, using (53), (54), (60), and (65) from Hanson [2005],
and where C_{n0} is given by (16).
[18] The angular dependence of the field is given approximately by the angular dependence of the term f_{sd}, except if ϕ ≃ 0, π/2. Defining ψ = (γ_{ms} − β_{sw} sin ϕ) and ignoring the Bessel function contribution in (37) we obtain
A beam maximum occurs when ψ = 0, leading to
which is the usual twodimensional leakage angle for β_{ms} < β_{sw} (we assume γ_{ms} = β_{ms} − jα ≃ β_{ms}). For β_{ms} > β_{sw} there is no real angle for leakage, although radiation maxima occur approximately when the numerator of (53) is unity, leading to
n = 1, 2, 3, .N_{Lobes}^{TSW}, where
is the number of forward lobes in the range 0 ≤ ϕ ≤ π/2. More accurate angles can be obtained by replacing (2n − 1) π in (55) with 2x_{0}, where x_{0} are the maxima of the sin(x)/x function (x_{0} = 4.493, 7.725, 10.904, etc.). Note that (56) is a generalization of the formula for the number of lobes of a traveling wave antenna in free space, N = L/λ [Stutzman and Thiele, 1998]. Pattern nulls occur when sin ψ = 0, leading to
for n = 1, 2, 3, .N_{Lobes}^{TSW}. The results (54)–(57) hold for a single strip in the absence of a via. The presence of the trigonometric functions multiplying f_{sd} in (45)–(47) will modify the above angles somewhat.
2.2.3. Single Strip With Standing Wave Current
[19] If, instead of (19) we assume a general traveling wave–standing wave current,
for ∣y − y_{0}∣ < w and −L_{1} < z < L_{2}, where Γ is a reflection coefficient,
then (21), because of enforcement of current continuity, is replaced by
As with the previous case, we take n = 0 in (58). The results (45)–(47) are applicable, with f_{sd}(37) replaced by f_{sd}^{stw}, where
2.2.4. Coupled Strips
[20] Assume that, instead of a single strip, we have two coupled strips as shown in Figure 3. If the strips are centered at y = ±y_{0}, we have
for ∣y ± y_{0}∣ < w and −L_{1} < z < L_{2}, where Γ_{1,2} are the reflection coefficients at the load of the strips (each has the form (59)) and we will again use n = 0. The resulting fields due to coupled strips are
where
and
[21] For coupled strips (ignoring the vias) the beam maximum and minimum can be obtained numerically from (64)–(66), or from a simpler approximate expression as derived below. Using
where ρ = (y^{2} + z^{2})^{1/2}, then, for example,
If a_{1} = −a_{2} = −1 (leading to the odd mode, which is the case of most interest for leakage on coupled strips), then
The presence of reflected waves is found to have a minor effect on beam angles (see section 3, and so, setting Γ = 0 in f_{sd}^{stw}, beam maxima and nulls for coupled strips can be found numerically from the relatively simple expression
where the second expression is obtained assuming βy_{0} ≪ 1. Beam maxima obtained from a numerical search of (73) will be denoted as ϕ_{max}^{BE}, which obviously includes strip coupling but not the via fields (and holds in both the TSW and LW regimes). Comparing (73) and (53) it is clear that the number of beams for coupled strips in the TSW regime, and the angles for beam nulls, will be approximately the same as for single strips (the term sin (βy_{0}cos (ϕ)) will not contribute additional beam nulls if βy_{0} is reasonably small). Thus, for coupled strips,
for n = 1,2,3,.N_{Lobes}^{TSW}. However, the angles of beam maxima will be somewhat different from (54) and (55). In particular, for sufficiently large L the sine function in (73) is rapidly varying compared to the term sin (2ϕ), and thus (54) and (55) will hold approximately for sufficiently long coupled strips. For small L the sinc function is smoothly varying, and the function sin (2ϕ) will play an important role in determining the beam maxima. The influence of strip length is further discussed in section 3.
[22] Note that mutual coupling effects are implicitly accounted for, since the fullwave complex propagation constant of two coupled strips is used in the asymptotic formula. The current profiles, (62) and (63), do not incorporate proximity effects (since only the dominant term of the current expansion in terms of Chebyshev polynomial is used), although it is well known that the far field of an antenna is somewhat insensitive to the antenna's current. Therefore the error due to neglecting the proximity effect in (62) and (63) is expected to be small.
2.2.5. Total Fields
[23] The field due to strip interconnects and vias is given by
where E^{s} is the field due to the strips, given by (45)–(47) for a single strip and (64)–(66) for coupled strips, N_{V} is the number of vias, and for each via E_{n}^{v} is given by (10)(11)–(12). Beam maxima obtained from a numerical search of (76) will be denoted as ϕ_{max}^{ASY}.
3. Discussion and Results
 Top of page
 Abstract
 1. Introduction
 2. Formulation
 3. Discussion and Results
 4. Conclusion
 Appendix A: Continuous Spectrum Contribution
 Appendix B: SemiInfinite Interconnect
 References
 Supporting Information
[24] Before presenting results it is worthwhile to emphasize some differences between two and threedimensional transmission line/interconnect structures. In the usual twodimensional transmission line theory leading to the generation of dispersion curves, one studies a sourcefree infinite structure. For perfectly conducting lines immersed in lossless dielectrics, this leads to the concept of bound modes characterized by realvalued propagation constants, and “leaky modes” characterized by complexvalued propagation constants. The bound modes do not radiate, but carry energy along the line. The leaky modes are not part of the proper spectrum of the integrated transmission line (i.e., their fields do not obey the usual radiation condition at infinity in certain directions normal to the line; the leaky mode field is exponentially growing in these directions), but can be useful as an approximation of the continuous spectrum of the structure in restricted areas of space. If radiation takes place, it is represented by the continuous spectrum, or its leaky mode approximation.
[25] For a threedimensional interconnect the situation is quite different. An excitation such as a via may result in currents on the finite length transmission line section that resemble various bound and/or leaky modes corresponding to an infinite line, but with a different interpretation in terms of radiation. For example, on a finitelength transmission line a general current such as I_{0}(e^{−jβz} − Γe^{j}β^{z}), with 0 ≤ ∣Γ∣ ≤ 1, will radiate energy into space and surface waves, even though β is real valued (e.g., consider a printed patch antenna, which typically has a standing wave current given by setting Γ = 1). This will be the case even if β is equal to a modal propagation constant on an infinite line. A traveling wave antenna [Johnson, 1993; Stutzman and Thiele, 1998] results from setting Γ = 0. We refer to this general situation as the traveling/standing wave (TSW) regime. For the printed interconnect in the TSW regime there will be N_{Lobes}^{TSW} = Lβ_{ms}/2π (see (56) or (74)) forward lobes in the angular range 0 ≤ ϕ ≤ 90° for a traveling wave antenna, and, in the event of a standing wave pattern, an equal number of backward lobes in the range −90 ≤ ϕ ≤ 0°.
[26] If, however, the current on the line resembles a leaky wave current, then the beam angle should correspond to a leaky wave angle (β_{ms} represents a fast wave with respect to β_{sw}). If the leakage constant is sufficiently large, in the leaky wave regime there will be one forward lobe in the angular range 0 ≤ ϕ ≤ 90° (N_{Lobes}^{LW} = 1). This is a principal difference between the TSW and LW regimes on a threedimensional interconnect; in the former γ_{ms} ≃ β_{ms} > β_{sw} (α_{ms} ≪ 1), resulting in a multilobe pattern, and in the latter, α_{ms} becomes larger, resulting in essentially singlelobe radiation.
[27] Furthermore, on an infinite twodimensional structure the leakage field increases exponentially in the leakage direction, and on a sourceexcited infinite or semiinfinite line the leakage field increases up to a point called the leakage boundary [Villegas et al., 1999]. On a threedimensional line, radiation into a surface wave will be in the form of beams or lobes, regardless of whether the current is a standing wave (such as when ∣Γ∣ = 1), a traveling wave (as in a conventional traveling wave antenna), or a leaky wave. In this study TSW and leaky wave excitation will be identified by the correspondence between the predicted and observed number of lobes, and angle of maximum radiation.
[28] A plot of normalized E_{x} (vertical electric field) is shown in Figures 4, 5, 6, 7, 8, and 9 for two coupled microstrip interconnects on a grounded slab having dielectric constant ɛ_{2} = 2.25ɛ_{0}, ɛ_{3} = ɛ_{0}, and dielectric thickness d = 1 cm (see Figure 3). Coupled interconnects are considered since the odd mode becomes leaky above a certain frequency [see, e.g., Shigesawa et al., 1995; Yakovlev and Hanson, 1997], and hence one can examine field behavior in what should be (via twodimensional predictions) leaky and nonleaky regimes. The interconnects are perfectly conducting, and have width 0.25 cm and centertocenter separation 0.5 cm. Each interconnect is fed by a via on one end and open circuited at the other end. In Figures 4–6 long interconnects are considered, L = 40 cm (as an aid in comparing with twodimensional leaky wave theory), and in Figures 7–9 shorter interconnects are considered, L = 5 cm. In each figure the computational space is 150 × 150 cm^{2}, and the interconnects are located in the center of the computation area.
[29] In Figures 4–9, part a shows the closedform analytical result (76), which is the sum of the fields due to two vias (each carrying opposing ±1 A currents so as to excite the odd strip mode), given by two terms of the form (10), and horizontal strip fields given by (64) using a_{1} = I_{0}^{s}, a_{2} = −I_{0}^{s}. Since the load end of the strips are open circuited, we assume Γ_{L} = 1 in the reflection coefficients for each strip. The angle of maximum radiation for the asymptotic result, ϕ_{max}^{ASY}, was obtained by numerical root search of the total field (76), then rounded to the nearest degree. In part b the field E_{x} obtained by fullwave (FW) simulation (Ansoft Ensemble) is shown. For the fullwave result the radiation angle ϕ_{max}^{FW} was obtained from the plot. In obtaining radiation angles from the plots, errors of several degrees can be expected, primarily due to ambiguity in locating the beam peak. Plot legends are not included since the asymptotic result assumes ±1 A on each via, and the fullwave result places ±1 volt at the base of each via, and so numerical comparison of the field amplitudes is not possible.
[30] The fullwave dispersion curves for the infinitelength coupled microstrip transmission line case are shown in Figure 1 of Yakovlev and Hanson [1997], from which the propagation constant values utilized below were obtained. From these plots it is evident that for f ≲ 8 GHz the odd mode is bound, whereas above 8 GHz the odd mode is leaky. In this study we consider three frequencies, f = 6.5 GHz (bound regime), f = 9 GHz, and 13 GHz (leaky regime).
[31] The case of a very long interconnect (L = 40 cm) is considered in Figures 4–6. In Figure 4 the frequency of operation is 6.5 GHz, at which point the odd mode propagation constant on an infinite pair of strips is γ_{ms} ≃ 1.29k_{0}, whereas β_{sw} ≃ 1.25k_{0}. Since β_{ms} > β_{sw} we would expect that the mode is not leaky. The vertical electric field at the airdielectric surface shows a fairly symmetric pattern, consistent with standing current waves, and does not seem to exhibit classical leakage. Radiation exists, in the same way that a printed patch antenna or strip dipole antenna radiates into space and surface waves. The number of radiation lobes predicted by the traveling/standing wave theory (74) is N_{Lobes}^{TSW} = 11, which is the same number observed in the fullwave and asymptotic plots (although in the asymptotic plots some lobes are difficult to see because of plot contrast). It is clear that the traveling/standing wave theory accounts for the observed field behavior.
[32] In Figure 5 the frequency of operation is 9 GHz, where γ_{ms} ≃ (1.31 − j0.019) k_{0} and β_{sw} ≃ 1.34k_{0}, and, since ℜ{γ_{ms}} < β_{sw}, one expects the mode to be leaky. The asymptotic result predicts a leakage angle of ϕ_{max}^{ASY} ≃ 74°, which agrees with the fullwave result, and the twodimensional single strip leakage angle is ϕ_{max}^{LW,2D} = 77° from (54). The number of observed radiation lobes (N_{Lobes}^{FW} ≃ 16) is the same as the value predicted by the traveling/standing wave theory. Thus, although the leakage angle is close to the observed radiation angle, the presence of multiple lobes, close to the TSW prediction, indicates that the TSW theory seems to account for the interconnect radiation. However, it seems that leakage is starting to emerge.
[33] In Figure 6, results are shown for f = 13 GHz, in which case γ_{ms} ≃ (1.30 − j0.03) k_{0} and β_{sw} ≃ 1.42k_{0}. Therefore, on the basis of propagation constant values one expects the mode to be leaky. The radiation angle predicted by the asymptotic theory is ϕ _{max}^{ASY} ≃ 67°, which is approximately the same angle obtained from the fullwave plot, and, also, ϕ_{max}^{LW,2D} = 67°. In this case one lobe is observed, which, together with the correspondence between the beam angles, indicates that a leaky wave is providing the dominant effect.
[34] Results are summarized in Table 1. Note that the angle ϕ_{max}^{ASY}, arising from (76), accounts for via excitation, whereas ϕ_{max}^{BE}, arising from (73), assumes close strip spacing and ignores the via excitation. For the L = 40 cm case they provide the same result.
Table 1. Radiation Parameters for L = 40 cm Interconnect^{a}f, GHz  N_{Lobes}^{TSW}  N_{Lobes}^{FW}  ϕ_{max}^{FW}, deg  ϕ_{max}^{ASY/BE}, deg  ϕ_{max}^{LW,2D}, deg 


6.5  11  11  79  79/79  – 
9.0  16  16  74  74/74  77 
13  –  1  67  67^{/}67  67 
[35] In Figures 7–9 coupled interconnects having L = 5 cm are considered at f = 6.5 GHz, 9 GHz, and 13 GHz, respectively. Generally, the above described conclusions for the L = 40 cm interconnect seem to apply to the L = 5 cm interconnect, and results are summarized in Table 2. However, unlike the L = 40 cm case, upon comparing ϕ_{max}^{ASY} and ϕ_{max}^{BE} it can be seen that for the L = 5 cm interconnects the presence of the via has some influence on the beam angle. Results at other frequencies yielded good agreement as well, but will be omitted because of space limitations. In all cases the asymptotic theory agrees well with the fullwave results, in both the TSW and LW regimes.
Table 2. Radiation Parameters for L = 5 cm Interconnect^{a}f, GHz  N_{Lobes}^{TSW}  N_{Lobes}^{FW}  ϕ_{max}^{FW}, deg  ϕ_{max}^{ASY/BE}, deg 


6.5  1  1  53  54/56 
9.0  2  2  56  56/58 
13  –  1  56  56/59 
[36] The influence of reflected waves was also more important on the shorter interconnect. If, in the asymptotic model, Γ = 0 was used, the radiation angle changed approximately one degree for the L = 5 cm interconnect, although there was no change for the L = 40 cm interconnect.
[37] Of more importance, at least for shorter interconnects, is strip coupling (as explained in the text after (73)). For example, at f = 13 GHz the L = 5 cm (L ≃ 2.1λ_{0}) coupled interconnects leak at ϕ_{max}^{FW} ≃ 56°, and the L = 40 cm (L ≃ 17.3λ_{0}) interconnects leak at ϕ_{max}^{FW} ≃ 67°, which is also the twodimensional, single strip leakage angle. For small L the observed difference in leakage angles is explained by the presence of two strips, as indicated by a careful consideration of (73). As further confirmation of this, we simply removed one strip and via from the asymptotic model, leading to a single strip (SS) and via (keeping the same value of γ_{ms} as for the coupled strips). The result was that for the L = 40 cm interconnect the leakage angle was 67° for both the single and coupled strip cases. However, for the L = 5 cm interconnect the leakage angles were different, with the single strip case leakage angle close to the simple twodimensional result, as shown in Table 3. The ability to include or not include various factors such as the via excitation, reflections, and strip coupling is one reason the asymptotic development is particularly useful.
Table 3. Influence of the Presence of Coupled Versus Single Strips on Leakage Angle (Keeping the Propagation Constant γ_{ms} Constant)^{a}  L = 5 cm  L = 40 cm 


f, GHz  13  13 
ϕ _{max}^{ASY,CS}, deg  56  67 
ϕ _{max}^{ASY,SS}, deg  65  67 
[38] In Figure 10 the magnitude of the vertical electric field versus position z (i.e., parallel to the interconnect), at three distances perpendicularly away from two coupled long interconnects (y = 10, 25, 40 cm) is shown at 13 GHz (I_{0}^{p} = 1 A). The geometry is the same as in Figure 6 (L = 40 cm), and the field shown in Figure 10 is simply a horizontal cut from Figure 6a. Results are calculated from the asymptotic formula, and the presence of a strong leakage beam is evident. Figure 11 shows the same result for L = 5 cm interconnects (the field shown is simply a horizontal cut from Figure 9a). It can be appreciated that the field is stronger for the longer interconnects, since they have sufficient length for a strong leakage beam to build up.
[39] On the basis of results of the asymptotic theory, verified by fullwave simulation, the following observations can be made.
[40] 1. At frequencies well below and well above the onset of surface wave leakage on twodimensional structures (f = 6.5 GHz and 13 GHz, respectively for the structure considered here, with f ≃ 8 GHz being the leakage cutoff point), the presence or absence of leakage into surface waves on finite viafed interconnects did seem to correspond with the regimes predicted by twodimensional theory. Closer to the leakage cutoff point the situation is less clear; however, no attempt was made in this study to examine the emergence of leakage as a function of frequency.
[41] 2. On a finitelength interconnect with a nonideal load termination, the presence of reflected current waves seems to merely produce backward radiation beams, in both the traveling/standing wave and leaky wave regimes (as might be expected).
[42] 3. On relatively long coupled interconnects the leakage angle is well predicted by simple twodimensional ray optic theory, although on shorter interconnects the interconnect length L plays a significant role in determining the leakage angle, mainly due to strip coupling, but also to the interaction between the via and strip fields, and to the presence of reflected waves.