wave modes in multilayered planar open parallelâ - Wiley Online Library

A Full-Wave Space-Domain Method for the Analysis of Leaky-Wave Modes in Multilayered Planar Open Parallel-Plate Waveguides Jose´ Luis Go´mez-Tornero, Fernando Daniel Quesada-Pereira, Alejandro A´lvarez-Melco´n Technical University of Cartagena, Cartagena, Murcia, Muralla del Mar s/n 30202, Spain Received 9 November 2003; accepted 22 May 2004

ABSTRACT: This work presents a novel numerical technique for the analysis of leaky-wave modes in laterally shielded, top-open multilayered planar transmission lines. The method develops the multilayered-media boxed Green’s functions for this type of electromagnetic devices, and solves the characteristic electric-field integral equation (EFIE) by using the method of moments (MoM). The analytical expressions used for the computation of the kernel of this technique are derived in this article, and results for leaky-wave modes in different practical structures are presented. The strong analytical nature of the proposed approach allows for a low-cost computational code without losing its full-wave accuracy. Comparisons with previous works are presented, showing the advantages of this new method for the analysis and design of practical millimeter-band leaky-wave antennas. © 2004 Wiley Periodicals, Inc. Int J RF and Microwave CAE 15: 128 –139, 2005.

Keywords: leaky-wave modes; multilayered Green’s functions; modal expansion; method of moments

I. INTRODUCTION The study of leaky-wave modes in practical printed structures has been a topic of much interest for many years [1–3]. This type of complex mode can accurately characterize the radiation behavior of open transmission lines. In this way, it is possible to avoid undesired cross-coupling effects [4]. Also, the study of the radiation characteristics of leaky modes allows for the design of versatile and novel leaky-wave antennas for the millimeter-wave band [5, 6]. Many different methods have been used to analyze such complex modes in open microwave devices. For Correspondence to: J. L. Go´mez-Tornero; email: josel. [email protected]. DOI 10.1002/mmce.20059 Published online 3 December 2004 in Wiley InterScience (www.interscience.wiley.com). © 2004 Wiley Periodicals, Inc.

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totally open planar transmission lines, the preferred technique in past works used a formulation of the Green’s functions in the spectral domain (SD), both for 2D [7, 8] and 3D [9, 10] structures, due to its versatility. However, much numerical effort needs to be done to solve the required Fourier integrals, and much care must be taken to deal with the branch cuts and integration contours [11]. When analyzing laterally shielded top-open planar transmission lines (LShPTL), such as the microstrip or the slot-line leaky-wave antennas shown in Figure 1, a method directly derived in the space domain (SpD) can be formulated. This is due to the fact that the side walls can be viewed as a parallel-plate waveguide (PPW), which allows for the existence of PPW modes. Therefore, this set of PPW modes can be used to expand the possible electromagnetic solutions that can exist in the open transmission line, namely, the LShPTL modes. According to the reference axes

Leaky-Wave Modes in Multilayered Waveguides

129

orthogonal PPW modes, and not only with the main PPW mode. All the integrals of the MoM matrix are analytical and the summation of the PPW modes is rapidly convergent. The numerical search of the complex leaky-wave roots has been improved by using a novel iterative algorithm. All these features will be described throughout this article to show the advantages of the proposed method.

Figure 1. Microstrip and slot-line leaky-wave antennas.

shown in Figure 1, leaky-wave LShPTL modes are characterized by their complex propagation constant in the longitudinal yˆ-axis of the open waveguide, given by k yLShPTL ⫽ ␤ y ⫺ j ␣ y.

(1)

Two main different approaches have been used to study these complex modes. The first is based on an equivalent network obtained for the main PPW mode, from which a transverse resonance equation (TRE) is derived to find the permitted propagation modes. This method was used for the analysis and design of a slot-line leaky-wave antenna [12], exhibiting fast and analytical computation. However, the approximated nature of the equivalent network used to model the metal-strip discontinuity made this approach accurate only for large strip values. The second possible choice is based on a more accurate full-wave approach that makes use not only of the main PPW mode, but of the entire PPW mode spectrum. In this way, the modematching procedure was used in [13] to analyze various types of laterally shielded planar transmission lines, such as the conductor-backed coplanar line, the conductor-backed slot line, and the microstrip line. This method proved to be as accurate as the SD approach, as demonstrated in [14]. However, the computation cost rises due to the necessity of computing the mode-matching integrals in the different discontinuity interfaces between many PPW modes (on the order of 100). Also, the relative-convergence effect is a problem often associated with the mode-matching technique. In this work, we present a novel method based on the space-domain Green’s functions developed for laterally shielded multilayered planar transmission lines. This technique does not need the computation of complicated Fourier integrals, since it is directly developed in the SpD for a laterally shielded structure. Also, it is accurate due to its full-wave nature, since the LShPTL fields are expanded by a series of

II. SPACE-DOMAIN GREEN’S FUNCTIONS IN A LATERALLY SHIELDED MULTILAYERED MEDIA AND EQUIVALENT NETWORK The Green’s functions for an elementary infinite wire of electric current located at a given point (x⬘, z⬘ ⫽ 0) inside the LShPTL structures shown in Figure 1, can be analytically derived by expanding the LShPTL mode fields by a set of PPW modes: ញ EtJ共 x⬘, z⬘ ⫽ 0; x, y, z兲 G

冘冘冋 2

⫽

M

p⫽1 m⫽0

eជ m共 p兲*共 x⬘兲 ⫺ C m 䡠 eជ m共q兲*共 x⬘兲 1 ⫺ C m2 ⫻ Vˆm共 p兲 共z兲 䡠 eជm共 p兲 共x兲 䡠 e⫺jk y

册

LShPTL

y

,

(2)

y

,

(3)

y

,

(4)

,

(5)

ញ HtJ共 x⬘, z⬘ ⫽ 0; x, y, z兲 G

冘冘冋 2

⫽

M

p⫽1 m⫽0

eជ m共 p兲*共 x⬘兲 ⫺ C m 䡠 eជ m共q兲*共 x⬘兲 1 ⫺ C m2 ⫻ ˆIm共 p兲 共z兲 䡠 hជ m共 p兲 共x兲 䡠 e⫺jk y

册

LShPTL

ញ EzJ共 x⬘, z⬘ ⫽ 0; x, y, z兲 G

冘冋

eជ mTM*共 x⬘兲 ⫺ C m 䡠 eជ mTE*共 x⬘兲 1 䡠 ⫽ j ␻ ␧ m⫽0 1 ⫺ C m2 M

2 ជ mTM 共x兲 䡠 e⫺jk y ⫻ ˆImTM 共z兲 䡠 kcm 䡠␾

册

LShPTL

ញ HzJ共 x⬘, z⬘ ⫽ 0; x, y, z兲 G

冘冋

eជ mTE*共 x⬘兲 ⫺ C m 䡠 eជ mTM*共 x⬘兲 1 䡠 ⫽ j ␻␮ m⫽0 1 ⫺ C m2 M

2 ជ mTE 共x兲 䡠 e⫺jk y ⫻ VˆmTE 共z兲 䡠 kcm 䡠␾

册

LShPTL

y

where the asterisk denotes complex conjugate. In eqs. (2)–(5), eជm共 p兲 and hជ m共 p兲 共x兲 are the customarily known PPW mode transverse (xˆ- and yˆ-directed) vector

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Figure 2. Equivalent model for the open-waveguide cross section and parameters for the analyzed microstrip line.

modal functions, but they are modified to allow for a ជ mTM and ␾ ជ mTE are the propagating vector in 3D, while ␾ longitudinal (zˆ-directed) PPW fields, given by eជ mTE共 x兲 ⫽

⫺jky cos共kxm x兲xˆ ⫹ kxm sin共kxm x兲y` , Nm

(6)

eជ mTM共 x兲 ⫽

⫹kxm cos共kxm x兲xˆ ⫺ jky sin共kxm x兲y` , Nm

(7)

hជ m共 p兲共 x兲 ⫽ zˆ ⫻ eជ m共 p兲共 x兲

m ⫽ 0, 1 . . . M and p ⫽ TEZ or TMZ ,

(8)

ជ mTE共 x兲 ⫽ ␾

⫺kcm cos共kxm x兲 zˆ, Nm

(9)

ជ mTM共 x兲 ⫽ ␾

⫺kcm sin共kxm x兲 zˆ. Nm

(10)

These modes are defined by the index m, which establishes the order of the PPW mode, and therefore the harmonic variation of the fields between the side metallic walls in the x-axis as follows: k xm ⫽ m 䡠

冉 冊

␲ rad a m

(p) (p) ˆm Moreover, the scalar functions V (z) and Iˆm (z) respectively represent the modal voltage and current in the equivalent transmission line shown in Figure 2, which was studied in more detail in [16]. Of particular relevance is the coupling coefficient Cm, which models the energy transfer from a TEZ PPW mode to a TMZ PPW mode. This coupling occurs for complex modes, such as the leaky-wave modes studied in this work. This modal-coupling coefficient can be computed using the following expression:

m ⫽ 0, 1, 2 . . . M.

(11)

The indexes p and q make the distinction between TEZ (p or q ⫽ 1) and TMZ (p or q ⫽ 2) PPW modes, since a transverse-longitudinal notation with respect to the z-axis has been used to facilitate the derivation of an equivalent transmission-line model [15].

Cm ⫽

⫹kxm 䡠 2␣y a 䡠 ␦m 兩Nm 兩2 2

再

2; m ⫽ 0 ␦m ⫽ 1; m ⫽ 0. (12)

The authors refer to [16] for further details about the derivation of the coefficient Cm and its physical meaning. The results presented in [16] show that it is important to take this derivation into account in order to obtain accurate results, especially when the ky complex solution has a large imaginary part ␣y. Also, [16] describes how to introduce the coefficient Cm in the space-domain Green’s function to obtain eqs. (2)– (5). Finally, Nm denotes the PPW-mode normalization constants, and kcm is the transverse wavenumber, given by 2 2 k cm ⫽ 冑k xm ⫹ k yLShPTL

(13)

Also, the complex wavenumber in the z-direction, kzm, is needed in order to obtain the equivalent transmission line in the multilayered-media stratification direction. This propagation constant can be computed for each PPW mode via the usual relation:

Leaky-Wave Modes in Multilayered Waveguides 2 2 k zm ⫽ 冑k 02 ⫺ k xm ⫺ k yLShPTL ⫽ ␤ zm ⫹ j ␣ zm (14)

where eq. (1) has been used, thus leading to a complex wavenumber in the z-axis with a positive imaginary part. This imaginary part indicates the increase of power of the leaky wave as we move far from the source. This annoying phenomenon is well explained in the literature [1], and does not cause any trouble in our formalism when computing the leaky fields, as will be shown. In order to obtain the leaky-wavemode fields inside the waveguide, radiation impedRAD ance Zm , derived in [17] for higher-order PPW modes, has been used to model the top aperture, as can be seen in the equivalent network of Figure 2. In order to extend the Marcuvitz radiation impedance for higher-order PPW modes, a modification has been introduced to the expressions presented in [17]. Basically, we have considered the propagation direction of the inhomogeneous plane-wave associated with each PPW mode. The expression of the electromagnetic fields for these PPW plane waves, and the angle in which they propagate inside the PPW, are derived in [18]. When these quantities are introduced in the expressions of Marcuvitz (see [17]), the radiation impedance can be computed as follows: Z m共 p兲 RAD ⫽

1 1 ⫽ Y m共 p兲 RAD G m共 p兲 RAD ⫹ jB m共 p兲 RAD

m ⫽ 0, 1 . . . M and p ⫽ TEZ or TMZ ,

G m共 p兲 RAD ⫽

1 共 p兲 䡠 Z 0m

共 p兲 RAD m

B

(15)

冉 冊 冉 冊 sinh ␲

cosh ␲

1 ⫽ 共 p兲 䡠 Z0m

a ␭Zm

a ⫹ cos ␹m共 p兲 ␭Zm

sin ␹m共 p兲 , a cosh ␲ ⫹ cos ␹m共 p兲 ␭Zm

冉 冊

(16)

冉 冊 冘冋 冉 冊 ⬁

␹ m共 p兲 ⫽ 2 ⫺

2e a a 䡠 ln ␭Zm ⫺ 2 䡠 arcsin ␭ Zm a␥ ␭Zm 䡠 i i⫽1

册

a ; ␭Zm 䡠 i

e ⫽ 2.718

␥ ⫽ 1.781, (17)

where the wavelength in the z-direction is computed from the real part of the complex propagation factor in that direction, as expressed in eq. (14):

␭ Zm ⫽

2␲ . ␤ zm

(18)

131

As commented upon, this last expression takes into account the angle in which each PPW mode travels inside the PPW, therefore extending the radiation impedance of Marcuvitz for higher-order PPW modes. From this modal-radiation impedance, the reflection coefficient at the aperture for each PPW mode can be easily computed:

␳ m共 p兲 ⫽

共 p兲 Z mRAD ⫺ Z 0m 共 p兲 Z mRAD ⫹ Z 0m

m ⫽ 0, 1 . . . M and p ⫽ TEZ or TMZ ,

(19)

where the modal characteristic impedance of the equivalent transmission line shown in Figure 2, is computed for TEZ and TMZ PPW modes using the following well-known expressions: TE Z 0m ⫽

␻␮ , k zm

(20)

TM Z 0m ⫽

k zm , ␻␧

(21)

In this way, the outgoing leaky-wave PPW modes suffer a partial reflection at the aperture, thus creating a reflected wave that reflects back toward the source. As demonstrated in [18], the modal reflection coefficient at the aperture tends to one, as the order of the PPW modes increases. This causes interference between the incident and reflected PPW waves, thus creating a standing-wave pattern, whose shape for higher-order PPW modes provides a decaying field toward the aperture, as would correspond to higherorder modes below cutoff. In this way, the fields associated with higher-order modes are not amplified at the aperture, and their interference on the open end can be treated without any difficulty. In previous works in the SD, some authors have used an infinite stub, as in [13], and others have used the radiation impedance only for the main PPW mode, as in [12, 19]. The stub’s finite length, however, must be taken into account for a practical antenna design, since it introduces important phenomena. In [19], Shigesawa studied the coupling with channel-guide leaky modes, which occurs for long-stub structures. Our method is able to predict this unwanted phenomenon, since the finite length of the upper stub is taken into account, as will be shown in the last section of the article. Moreover, for short-stub antennas, several evanescent modes can reach the top aperture. This multimode interaction has never been taken into account before, since the radiation impedance has been applied only

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Figure 3. Iterative search of zeros of ⌿(␤y, ␣y) in the complex ky plane for the structure shown in Fig. 2.

for the main PPW mode, expecting higher-order modes to have decayed to negligible values at the open end. Since Eqs. (15–19) are computed for the full-wave spectra of PPW modes, the proposed fullwave method predicts the influence of higher-order PPW modes at the aperture, as will be shown in the last section of this article.

III. SEARCHING FOR COMPLEX LEAKY-WAVE MODES Once the Green’s functions are formulated in this way, and the EFIE is set up and solved via the method of moments (Galerkin procedure). The resulting basis and test integrals have direct analytical expressions, since they involve the product of sinusoidal functions, as established by the Kernel shown in eqs. (2)–(10). In order to ensure a nontrivial solution for the characteristic homogeneous system, the determinant of the MoM matrix P must be equal to zero. In general, this determinant is a complex function which depends on the unknown complex variable kyLShPTL, as expressed in eq. (1): det共P兲 ⫽ ⌿共ky LShPTL兲 ⫽ ⌿共␤y , ␣y 兲.

(22)

For the case in which the LShPTL mode is a leakywave mode, a numerical search of the zeroes of the complex function ⌿ (␤y, ␣y), which depends on two variables, must be performed in the complex plane ␤y ⫹ j␣y. This numerical search in two variables can be very difficult if we do not have a good starting point to seed the search algorithm [20]. For this purpose, we have applied a novel iterative algorithm in which we use as a starting point the solution for the

completely closed structure, that is, when the top aperture is covered with a metallic wall. With this strategy, it is easy to find this starting point, since the solutions must be in the real ␤y axis, as the mode cannot radiate and the medium is lossless. After that, the top-wall impedance is gradually changed from that of a perfect metallic wall to the final multimodal (P)RAD Marcuvitz radiation impedance, Zm . This transformation is carried out in small steps in order to provide a smooth transition of the solution. In this way, we are able to find the final solution in the complex plane in a few steps, as shown in Figure 3, where two- and ten-step search mechanisms are applied.

IV. VALIDATION AND POSTPROCESSING RESULTS In order to validate the proposed method, Figure 4 shows comparisons between our full-wave SpD method, a single-mode technique based on the TRE [12], and a full-wave method which employs the Green’s functions in the SD [14]. As a measurement of accuracy, the normalized phase and attenuation constants are represented as a function of the ratio strip width to the PPW width (W/a), for a leaky-wave mode in a microstrip line. As commented upon, the equivalent network used in the TRE is less accurate as the strip becomes narrower, as illustrated by the inaccurate values of ␤ for W/a ⬍ 0.4 in Figure 4. On the other hand, our method agrees very well with the accurate SD technique due to its full-wave nature, but it is entirely formulated in the SpD, thus avoiding complicated transformations in the SD [11].

Leaky-Wave Modes in Multilayered Waveguides

133

Figure 4. Comparison of the accuracy between our implementation and the other two techniques.

Figure 5 shows the convergence behavior of this method for the microstrip structure shown in Figure 2. The formulation depends on two infinite summations, which are truncated by two limits. The first limit sets the number of PPW modes considered in the fields expansion, namely, M, as in eqs. (2)–(5). The second limit is N, which establishes the number of basis and test functions and therefore the size of the MoM matrix P. It can be seen from Figure 5(a) that, with M ⫽ 100 PPW modes and N ⫽ 4, the solution has converged. The value of M needed for convergence depends on the ratio strip width to PPW width (W/a), and also on the number of basis functions N, as depicted in Figure 5(b). Much care must be taken with the relative convergence phenomenon in the MoM

procedure [21], as illustrated in Figure 5(b) for the results N ⫽ 2 and N ⫽ 8. As the number of basis functions N is increased, a higher amount of PPW modes M is needed to account for the higher-harmonic variations in the currents. Fortunately, with a low number of basis functions (N ⫽ 4) it is sufficient to obtain a stable solution, as depicted from the summary table shown in Figure 5(c). All these results have been obtained for the leaky-wave mode of the microstrip structure analyzed with the parameters shown in Figure 2. The results show a good numerical stability. Once the complex solution has been obtained, the electromagnetic fields of the LShPTL leaky-wave mode can also be computed, both inside and outside

Figure 5. Convergence of the proposed method for the structure shown in Fig. 2: (a) with number of PPW modes M; (b) with number of basis functions N and according to strip width W/a; (c) summary table showing the dependence between M and N.

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Figure 6. Electromagnetic fields of leaky-wave mode in microstrip, ky ⫽ 665.3 ⫺ j34.8 (m⫺1): (a) electric-field components at the air-dielectric interface; (b) transverse magnetic-field boundary condition at the strip and air-dielectric slab regions; (c) currents from the MoM expansion.

the waveguide. In the inside region, the following expressions are derived for the retrieval of the electric and magnetic fields, respectively:

Eជ 共 x, y, z兲 ⫽

⫻ f 共x⬘兲 xˆ 䡠 ⭸x⬘ ⫹

M

A m共 p兲 䡠 Vˆ m共 p兲共 z兲 䡠 eជ m共 p兲共 x兲 䡠 e ⫺jky

y

LShPTL

Y n

n⫽0

⫻

冘冘 2

冘␣ N

X n

冕

X2

x⬘⫽X1

eជm共 p兲* 共x⬘兲 ⫺ Cm 䡠 eជm共q兲* 共x⬘兲 Y 䡠 f n 共x⬘兲 yˆ 䡠 ⭸x⬘ 1 ⫺ Cm2

(25)

p⫽1 m⫽0

冘 ⬁

1 ⫹ 䡠 A TM 䡠 Iˆ mTM共 z兲 䡠 k cm j ␻ ␧ m⫽0 m ⫻ ␾mTM 共x兲 䡠 e⫺jk y

冘冘A 2

ជ 共 x, y, z兲 ⫽ H

LShPTL

y

zˆ,

(23)

M

共 p兲 m

䡠 ˆI m共 p兲共 z兲 䡠 hជ m共 p兲共 x兲 䡠 e ⫺jky

y

LShPTL

p⫽1 m⫽0

冘 ⬁

⫹

1 䡠 A TE 䡠 Vˆ mTE共 z兲 j ␻␮ m⫽0 m ⫻ kcm 䡠 ␾mTE 共x兲 䡠 e⫺jk y

LShPTL

y

zˆ,

(24)

(P) where the coefficients Am are computed for each PPW mode as follows (with ␣n the current coefficients obtained via the MoM):

冘 N

A m共 p兲 ⫽

n⫽0

␣ nX

冕

X2

x⬘⫽X1

eជ m共 p兲*共 x⬘兲 ⫺ C m 䡠 eជ m共q兲*共 x⬘兲 1 ⫺ C m2

These coefficients Am are studied in more detail in [18], where the authors derive from them the discrete propagation and radiation spectra of a laterally shielded open waveguide. From these spectra, analytical expressions can be obtained to measure the propagation and radiation characteristics (radiation efficiency, cutoff zone, polarization purity, and coupling with undesired channel-guide modes) of a laterally shielded leaky-wave antenna. Nevertheless, it can be observed from these last expressions that all the functions are analytical, and therefore the computation of the fields is direct. Of special interest is the representation of the electromagnetic fields at the metal-dielectric interface, since it can be used for physicalchecking purposes. At this coordinate, the transverse electric field (x- and y-directed) must be zero on the metal strip, as imposed by the EFIE. In addition to the condition imposed by the EFIE, the magnetic field must fulfill the boundary conditions on the strip and outside the strip. In Figure 6(a), the 1D plot of the electric field is shown, together with the difference of the magnetic fields at the interface [Fig. 6(b)]. It can be seen that the EFIE is fulfilled and the currents induced by the discontinuity of the magnetic fields perfectly agree with the currents obtained from the

Leaky-Wave Modes in Multilayered Waveguides

135

Figure 7. (a) Current distribution and (b) radiation pattern and transverse power flux inside the waveguide for the leaky-wave mode of the microstrip antenna shown in Fig. 2.

MoM expansion [Fig. 6(c)]. Also, the transverse magnetic field is continuous at the dielectric-slab–air region, thus leading to a null induced-current density outside the strip, as physically expected. Finally, a 2D plot of the fields can provide useful information about the leaky-wave mode. Figure 7(a) presents the current distribution of the studied leaky-wave mode, showing the phase change due to the real part of the complex ky solution (a wave propagating along the y-direction with ␤y ⫽ 665.3 rad/m) and the radiation-loss rate due to the imaginary part (an exponential decaying behavior with attenuation constant ␣y ⫽ 34.8 Np/m). Of much interest is the transverse Poynting vector inside the open waveguide, which is plotted in Figure 7(b). When the waveguide is completely shielded, this transverse power flux is zero, since all the energy is reflected in the metal walls surrounding the structure. However, when the top wall is open, some power will leak to the outside, leading to a net outgoing power flux, as shown in Figure 7(b). The radiation pattern for a given LShPTL leaky-wave mode can also be easily obtained. For this purpose, the equivalent magnetic currents at the top aperture are computed, and from them the radiated fields can be obtained by calculating a few analytical integrals [18]. In Figure 7(b), the radiation pattern is plotted above the open waveguide for the same leaky-wave mode studied in previous figures.

In addition, it is very interesting to compute the dispersion curves for ␤y and ␣y as a function of frequency or other geometrical parameters of the structure, since this gives important information about the electrical performance of the antenna. Particularly, ␤y gives information on the main beam direction of the leaky wave, while ␣y is directly related to the beamwidth [6, 12]. Therefore, it is necessary to know how each physical parameter of the structure affects the propagation constant of the excited leaky-wave mode. In particular, the PPW stub is needed to reduce cross polarization and grating lobes [12]. In some past works such as [13], the stub length L was assumed to be infinite so that the analysis was much simpler. However, a practical antenna has a finite height, which can introduce unexpected phenomena. Of much interest is the appearance of so-called channel-guide leaky modes. These modes are not desired for radiation, since their propagation and radiation features cannot be controlled with the strip. Consequently, it is important to take into account the finite length of the stub in order to avoid undesired couplings, which might spoil the performance of the antenna. Coupling with channel-guide leaky-modes in stubloaded leaky-wave antennas was studied in much detail in [19]. However, the effects of higher-order evanescent PPW modes for short-stub structures have never been taken into account. As described in

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Figure 8. Normalized ␤ y and ␣y for different leaky-wave modes of a slot-line LWA at 50 GHz as a function of stub length L (a ⫽ 2.2 mm, D ⫽ 1.59 mm, X1 ⫽ 1 mm, X2 ⫽ 2.1 mm, ␧r ⫽ 2.56).

section II, the higher-order PPW modes were always supposed to have decayed to negligible values at the radiating aperture [12, 19]. They are therefore not needed to model the multimode interaction on the open waveguide. This assumption is no longer necessary with our theory. Figure 8 shows the dependence of the real and imaginary parts of the complex longitudinal wavenumber k y with the value of the stub length L, as obtained with the approach derived in this article for the slot leakywave antenna studied in [12]. The analysis is carried out at a fixed frequency of 50 GHz. The dashed line (DL-m ⫽ 0) represents the desired leaky-mode complex solution obtained when only the main PPW mode (m ⫽ 0) is taken into account in the top aperture, while the continuous line (DL-@m) is for the desired leaky-mode solution found with our full-wave method. Also, the results for an infinite stub analysis are plotted (labeled INFINITE in Fig. 8). The variation with the stub length L should cause the complex wavenumber to vary in a quasi-periodical fashion around the infinite solution, as it would correspond to the standing-wave variation produced by the reflection of the m ⫽ 0 PPW mode when reaching the top-aperture discontinuity. However, for L ⬎ 5 mm, the desired leaky-wave mode couples to a channel-guide mode (labeled CH#2), which has a totally different nonperiodical dependence with L. Also, another channel-guide mode solution (CH#1) is plotted to show the typical shape of the variation of the ␤y and ␣y solutions of a channel mode with channel length L. Completely original results are obtained for L ⬍ 1 mm, where it can be seen that the solutions become different between the monomodal (DL-m ⫽ 0) and multimodal (DL-@m) approaches. This is due to the interactions of higher-order modes at the top aper-

ture. Even more interesting than the variation in the complex propagation constant of the desired leakywave mode is the variation of its radiation pattern. Since higher-order PPW modes can reach the radiating aperture for short stubs, they can contribute to radiation. Figure 9(a) shows the transverse power flux inside the slitted leaky-wave antenna for three different values of L (0.2, 0.8, and 3 mm). It can be seen that higher-order PPW modes have decayed to negligible values above L ⫽ 1 mm, as seen in the uniform-field region for L ⬎ 1 mm shown in Figure 9(a). Figure 9(b) shows the relative-power illumination at the aperture for the three stub lengths, illustrating that a uniform illumination is only achieved for the case L ⫽ 3 mm. The nonuniform illuminations obtained for L ⬍ 1 mm cause the cross section (E-plane) radiated beams to be less directive, as can be seen in Figure 9(c). Moreover, the beams suffers a tilt in its maximum radiation angle ␾. Another observable higher-order effect is the appearance of cross-polarization for short stubs, as can be seen in Figure 9(d). This is due to higherorder PPW-mode excitation, which contributes to both E␾ and EY polarizations, while the main PPW mode (m ⫽ 0) is purely E␾ polarized. The interaction of evanescent higher-order PPWs at the top aperture must be taken into account, since the radiation properties of a stub-loaded leaky-wave antenna can be strongly perturbed. The full-wave method presented in this article permits us to obtain the dispersion analysis of ␤y and ␣y as a function of any antenna parameter accurately and rapidly, and to take into account all higher-order effects. These features make this tool very appropriate for the automated design and optimization of practical leakywave antennas.

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Figure 9. Relative transverse power density for slot-line LWA: (a) power pattern inside the open waveguide with L ⫽ 0.2, 0.8, and 3 mm; (b) aperture illumination; cross section (E-plane) radiation diagrams in (c) E␾ and (d) EY polarizations.

V. CONCLUSION This work has presented a new method, based on the Green’s functions in multilayered media inside a parallel-plate waveguide (PPW), which is very well suited for the analysis of leaky modes in a broad class of millimeter-wave line-source leaky-wave antennas. The method is entirely formulated in the space domain (SD) due to the possibility of expanding the fields in an orthonormal set of PPW modes. The analytical nature of this approach allows very fast computations, while maintaining accurate results, as checked via comparisons with other techniques. Both the planar discontinuities and the waveguide aperture are treated in a multimodal fashion. The method also overcomes some problems, such as the search for leaky-wave poles or the treatment of higher-order PPW modes, thus leading to a robust full-wave tool

for the analysis and design of laterally shielded multilayered planar antennas.

ACKNOWLEDGMENTS This work has been supported by Spanish National projects ESP2001-4546-PE, TIC2000-0591-C03-03 and Regional Seneca project 2002 PB/4/FS/02 and PMPDI-UPCT-2004. The authors also want to thank EPSON-Ibe´rica foundation for its stimulating support.

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13. Z. Ma and E. Yamashita, Space-wave leakage from higher-order modes on various planar transmissionlines structures, IEEE MTT-S Int Microwave Symp Dig 2 (1994), 1033–1036. 14. P. Baccarelli, P. Burghignoli, C. Di Nallo, F. Frezza, A. Galli, P. Lampariello, and G. Ruggieri, Full-wave analysis of printed leaky-wave phase arrays, Int J RF and Microwave CAE 12 (2002), 272–285, 15. S.G. Pan and I. Wolff, Scalarization of dyadic spectral Green’s functions and network formalism for threedimensional full-wave analysis of planar lines and antennas, IEEE Trans Microwave Theory Tech 42 (1994), 2118 –2127. 16. J.L. Go´mez and A.A. Melco´n, Nonorthogonality relations between complex-hybrid-modes: An application for the leaky-wave analysis of laterally shielded topopen planar transmission lines, IEEE Trans Microwave Theory Tech 52 (2004), 760 –767. 17. N. Marcuvitz, Waveguide handbook, McGraw-Hill, New York, 1951, pp, 179 –181. 18. J.L. Go´mez and A.A. Melco´n, Radiation analysis in the space domain of laterally shielded planar transmission lines, Pt I: Theory, Radio Sci 39 (2004), 1–11. 19. H. Shigesawa, M. Tsuji, P. Lampariello, F. Frezza, and A. A. Oliner, Coupling between different leaky-mode types in stub-loaded leaky waveguides, IEEE Trans Microwave Theory Tech 42 (1994), 1548 –1560. 20. C.G. Hsu, R.F. Harrington, J.R. Mautz, and T.K. Sarkar, On the location of leaky wave poles for a grounded dielectric slab, IEEE Trans MTT 39 (1991), 346 –349. 21. R. Mittra, T. Itoh, and Ti-Shu Li, Analytical and numerical studies of the relative convergence phenomenon arising in the solution of an integral equation by the moment eethod, IEEE Trans Microwave Theory Tech 20 (1972), 96 –104.

BIOGRAPHIES Jose´ Luis Go´mez Tornero was born in Murcia, Spain, in 1977. He received his Telecommunications Engineer degree from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 2001, and is currently working toward his Ph.D. degree at the Technical University of Cartagena, Cartagena, Spain. In 1999, he joined the Radiocommunications Department of UPV as a Research Student, where he was involved in the development of analytical and numerical tools for the study and automated design of microwave filters in waveguide technology for space applications. In 2000, he joined the Radio Frequency Division, Industry Alcatel Espacio, Madrid, Spain, where he was involved with the development of microwave active circuits for telemetry, tracking and control (TTC) transponders implicated in many different spatial missions for the European Space Agency (ESA), National Aero-

nautics Space Administration (NASA), and other Space Agencies. In 2001, he joined the Technical University of Cartagena as Assistant Professor, and he is currently developing his teaching activities there. His scientific research is focused on the analysis and design of leaky-wave antennas for millimeter-wave-band applications and the development of numerical methods for the analysis of novel passive radiating structures in planar and waveguide technologies. His scientific interests also include the study of active devices for microwave and millimeter-wave bands, such as oscillators and active antennas. In July 2004, Jose´ Luis Go´mez received the second national award from the foundation EPSON-Ibe´rica for the best Ph.D. project in the field of Technology of Information and Communications (TIC).

Leaky-Wave Modes in Multilayered Waveguides Fernando Daniel Quesada Pereira was born in Murcia, Spain, in 1974. He received his Telecommunications Engineer degree from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 2000, and is currently working toward his Ph.D. degree at the Technical University of Cartagena, Cartagena, Spain. In 1999, he joined the Radiocommunications Department of UPV as a Research Assistant, where he was involved in the development of numerical methods for the analysis of anechoic chambers and tag antennas. In 2001, he joined the Technical University of Cartagena, Cartagena, Spain, first as a Research Assistant and then as an Assistant Professor. His current scientific interests involve integral-equation numerical methods for the analysis of antennas and microwave devices. Alejandro Alvarez Melco´n was born in Madrid, Spain, in 1965. He received his Telecommunications Engineer degree from the Polytechnic University of Madrid (UPM), Madrid, Spain, in 1991, and his Ph.D. degree in electrical engineering from the Swiss Federal Institute of Technology, Lausanne, Switzerland, in 1998. In 1988, he joined the Signal, Systems and Radiocommunications Department of UPM as a Research Student, where he was involved in the design, testing, and measurement

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of broadband spiral antennas for electromagnetic measurements support (EMS) equipment. From 1991 to 1993, he worked in the Radio Frequency Systems Division of the European Space Agency (ESA/ESTEC), Noordwijk, The Netherlands, where he was involved in the development of analytical and numerical tools for the study of waveguide discontinuities, planar transmission lines, and microwave filters. From 1993 to 1995, he joined the Space Division of Industry Alcatel Espacio, Madrid, Spain, and he worked at the ESA, where he collaborated in several ESA/ESTEC contracts. From 1995 to 1999, he joined the Swiss Federal Institute of Technology, E´cole Polytechnique Fe´de´rale de Lausanne, Lausanne, Switzerland, where he worked in the field of microstrip antennas and printed circuits for space applications. In 2000, he joined the Technical University of Cartagena, Cartagena, Spain, where he is currently developing his teaching and research activities. He received the JINA (Journe´e Internationales de Nice sur les Antennes) Best Paper Award for the best contribution to the JINA’98 International Symposium on Antennas, and the COIT/AEIT (Colegio Oficial de Ingenieros de Telecomunicacio´n) award for the best Ph.D. thesis in basic information and communication technologies.