IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER

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1 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER Transmission Line Modeling Asymptotic Formulas for Periodic Leaky-Wave Antennas Scanning Through Broadside Simon Otto, Student Member, IEEE, Andreas Rennings, Member, IEEE, Klaus Solbach, Christophe Caloz, Fellow, IEEE Abstract It is shown, using three specific examples a series fed patch (SFP) array, a phase reversal (PR) array a composite right/left-hed (CRLH) antenna that one-dimensional periodic leaky-wave antennas scanning through broadside build a class of leaky-wave antennas sharing qualitatively similar quantitatively distinct dispersion radiation characteristics. Based on an equivalent transmission line (TL) model using linearized series shunt immittances to approximate the periodic (Bloch) antenna structure, asymptotic TL formulas for the characteristic propagation constant, impedance, energy, power quality factor are derived for two fundamentally different near off-broadside radiation regimes. Based on these formulas, it is established that the total powers in the series shunt elements are always equal at broadside, which constitutes one of the central results of this contribution. This equal power splitting implies a severe degradation of broadside radiation when only one of the two elements series or shunt efficiently contributes to radiation the other is mainly dissipative. A condition for optimum broadside radiation is subsequently established shown to be identical to the Heaviside condition for distortionless propagation in TL theory. Closed-form expressions are derived for the constitutive (LCRG) parameters of the TL model for the specific SFP, PR CRLH antenna circuit models, quantitative information on the validity range of the TL model is subsequently provided. Finally, full-wave simulation measurement LCRG parameter extraction methods are proposed validated. Index Terms Bloch-Floquet theorem, broadside radiation, composite right/left-hed (CRLH) metamaterial, Heaviside condition, leaky-wave antennas, periodic structures, phase-reversal (PR) array, quality factor, series-fed patch (SFP) array, transmission line (TL) theory. I. INTRODUCTION P ERIODIC leaky-wave antennas have been extensively studied widely used for over five decades [1], [2]. Particularly, planar leaky-wave antennas configurations, with their advantages of low cost, light weight, simple fabrication integrability with electronic components have found Manuscript received November 11, 2010; revised February 24, 2011; accepted April 07, Date of publication August 04, 2011; date of current version October 05, S. Otto, A. Rennings, K. Solbach are with the University of Duisburg- Essen, Duisburg, Germany ( simon.otto@uni-due.de). C. Caloz is with the École Polytechnique de Montréal, 2500, ch. de Polytechnique, H3T 1J4, Montréal, Quebec, Canada. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TAP applications in various radar communication systems at microwave frequencies. While the first planar periodic leaky-wave antennas were reported back in the 1980s, they experience a regain of interest nowadays following recent contributions on novel broadside optimum designs [3] [5] on designs with novel features or system functionalities [6] [12]. An overview of recent advances in this field is available in [13]. This paper presents a lattice-network based TL model of periodic leaky-wave antennas characterized by continuous backward to forward radiation through broadside derives subsequent asymptotic formulas for the regimes of near-broadside off-broadside radiation either toward forward or backward directions. The proposed approach applies to all leaky-wave antennas of this class. It is illustrated by three examples, a series fed patch (SFP) array, a phase reversal (PR) array a composite right/left-hed (CRLH) antenna, supported by full-wave experimental results. The asymptotic formulas are the central result of the paper. They provide physical insight electromagnetic understing of the behavior of leaky-wave antennas around the broadside frequency. They also explain the issue of inefficient radiation of one-dimensional periodic leaky-wave antennas at broadside. When the unit cell is symmetric with respect to its transverse axis, as assumed in this work, the radiative modes decouple at the broadside radiation frequency [1] it may be designed to exhibit a seamless transition between the forward backward frequency ranges by closure of the open stopb. Transversally asymmetrical unit cells, which exhibit an open stopb as a result of mode coupling [1], [2], [4], [13], form a different class of antennas are out of scope of the present paper. In this work we show that two conditions are required to overcome the broadside radiation issue. In addition to frequency-balancing for closing the b gap, Q-balancing, which is based on the Heaviside condition (distortionless TL), is also required for optimum broadside radiation. The paper is organized as follows. Section II defines the class of antennas under consideration describes their common qualitative features their quantitative differences. Section III derives the transmission line (TL) model, based on linearized lattice-network series shunt immittances, for this class of antennas. Section IV develops a simple equivalent circuit for each of the considered antennas (SFP, PR CRLH) compares the periodic (Bloch) propagation constant the impedance with the TL model results for the validation estimation of the frequency validity range of the latter. Section V proposes LCRG parameter extraction methods validates the TL model by X/$ IEEE

2 3696 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011 Fig. 1. Frequency scanning periodic leaky-wave antennas under consideration in this paper. (a) Series-fed patch (SFP) array antenna. (b) Phase reversal (PR) array antenna. (c) Composite right/left-hed (CRLH) antenna with shunt stub inductors different series capacitors (left: interdigital, right: metal-insulatormetal). full-wave simulation experimental results for the SFP case. Upon the basis of Section III to V, Section VI derives asymptotic formulas for the phase constant, leakage factor, impedance, quality factor, energy power. Finally, conclusions are given in Section VII. II. CLASS OF ANTENNAS CONSIDERED The class of antennas considered in this paper covers a wide range of one-dimensional periodic 1 leaky-wave antennas [13] characterized by a full or partial frequency beam scanning response from backward to forward angles including broadside. Fig. 1 depicts three antennas which belong to this class which will be specifically analyzed: a series fed patch (SFP) array antenna [14] [16], a phase reversal (PR) array antenna [17] [19] a composite right/left-hed (CRLH) antenna [6], [20], [21]. While such antenna structures maybe short-ended or open-ended to operate in a sting-wave regime as resonant antennas, with fixed radiation beam, we consider here the case where the antenna structures are terminated by matched loads, so as to operate in a traveling-wave regime as leaky-wave antennas, following the stard frequency-angle scanning law [2], [13] (1a) 1 In fact, although only periodic structures are discussed in this paper, this class could be extended to uniform (non-periodic) antennas, such as the ferrite-loaded open waveguide CRLH leaky-wave antenna recently reported in [10]. Fig. 2. Typical dispersion diagrams for the three antennas shown in Fig. 1 obtained by full-wave analysis using periodic boundary conditions (FEM Ansoft-HFSS). (a) SFP antenna [Fig. 1(a)]. (b) PR antenna [Fig. 1(b)]. (c) CRLH antenna [Fig. 1(c)]. Only the lowest few modes are shown the diagram is restricted to the first Brillouin zone (j8j = jjp < ). The angular frequency! is normalized with respect to the frequency of broadside radiation, i.e.! =!(8 = p =0). The white areas indicate the slow-wave regions (v =!= < c =!=k ), where radiation may potentially exist in the substrate [22] but not in free space, while the shaded regions correspond to the fast-wave leaky-wave region (v =!= > c =!=k ) of interest. (d) Superposition of the (a) (c) dispersion curves of interest, i.e. the positive group velocity (v > 0) leaky-wave curves around the broadside frequency region [thicker parts in (a) (c)], corresponding to a specific (n ) space harmonic. with (1b) where represents the main-beam radiation angle, denotes the space harmonic, is the free-space wavenumber, : angular frequency, : speed of light in free space is the period of the structure [1]. Despite their seemingly very different configurations, the three antennas of Fig. 1, as well as all other antennas belonging to the aforementioned class, exhibit a qualitatively similar leaky-wave behavior following from their qualitatively similar leaky dispersion responses [(1)]. Fig. 2 shows the exact dispersion diagrams computed by full-wave analysis for these three antennas, where each antenna is designed to have a closed b gap at. While the overall dispersion diagrams substantially differ between the three cases, in particular in the guided wave regions, the leaky-wave curves around the broadside frequency, occurring at, are qualitatively identical, they all display a seamless quasi-linear transition from negative to positive values of, allowing frequency scanning from backward to forward angles according to (1) [13]. All three antennas have been designed to exhibit a

3 OTTO et al.: TRANSMISSION LINE MODELING AND ASYMPTOTIC FORMULAS FOR PERIODIC LEAKY-WAVE ANTENNAS 3697 TABLE I ELECTRICAL PERIODS (p), PHASE SLOPES ( _ 8 ) SCANNING SENSITIVITIES (s) FOR THE THREE ANTENNAS IN FIG. 1 single radiating space harmonic so as to provide single-beam full-space scanning [1]. Beyond their aforementioned fundamental similarities, the three antennas also exhibit some important differences. The SFP antenna uses its space harmonic to radiate [2]. For the PR antenna, the radiating space harmonic may be considered either to be the or the [19]. The CRLH antenna radiates in the fundamental space harmonic, 2 [21]. Moreover, the three antennas have very different quantitative parameters, as shown in Table I, which has an important consequence on the frequency range validity of the derivations performed in the reminder of the paper, as will be shown next. The first column of Table I shows that the electrical periods,, where is the free space wavelength at, vary by a 3.4 factor between SFP CRLH antennas while the PR antenna has an intermediate value of. For radiation angles near broadside, the inverse sinus in (1) may be approximated by its argument, i.e., where may itself be approximated by its dominant Taylor expansion term, where is the phase slope at. Therefore, from which it follows that the scanning sensitivity of the antennas may be written as Since is inversely proportional to the frequency bwidth over which the antenna s main beam scans from broadside to backfire endfire since these two extreme directions approximately correspond to the grounded-slab mode (following the light line at lower frequencies), the antenna with the largest will have the smallest frequency bwidth immune of coupling to this mode. Table I shows that the phase slopes for the PR CRLH antennas are fairly close, while the phase slope for the SFP is twice larger. Incorporating the electrical periods along with the slopes into (4), yields the scanning sensitivities, which are listed in the third column of the table. Thus, the CRLH antenna exhibits the highest scanning sensitivity, as a result of its extremely small electrical period. As a consequence, it exhibits 2 The mode starting from! =0in Fig. 2(c) is the light line mode (TM ), so that the mode around! is the fundamental mode (n =0)of the CRLH structure. (2) (3) (4) Fig. 3. Two-port network modeling of a periodic leaky-wave structure in terms of the ABCD parameters of its unit cell application of Bloch-Floquet theorem to relate its input output currents voltages. the smallest frequency range immune of perturbation related to coupling to the -mode, i.e. the smallest frequency range with a linear dispersion curve, as seen in Fig. 2(c) Fig. 2(d). The PR antenna, with its relatively large small has the lowest scanning sensitivity, therefore displays the broadest range of linear phase-frequency response. The modeling formulas of the next sections will not account for coupling to the mode, will therefore be restricted to the regions of nearly constant group velocity corresponding to a quasi linear phase in Fig. 2(d). Nevertheless, the proposed analysis will prove to be very efficient within a reasonable bwidth for all the cases, including the CRLH case. III. PERIODIC NETWORK MODELING AND EQUIVALENT TRANSMISSION LINE MODEL Due to the qualitative similarity of their leaky-wave dispersive responses, the three antennas of Fig. 1 can be described by a common network model. This model will be derived as an approximation of the periodic or Bloch solution, characterized by the propagation constant impedance, will be referred to as the transmission line (TL) model, with characteristic propagation constant impedance. This section establishes the TL model. It will serve as a foundation for the derivation of asymptotic formulas in Section VI. A. Bloch Model, Propagation Constant Impedance The periodic antenna structures depicted in Fig. 1 may be modeled by their unit cell equivalent two-port network, which is shown in Fig. 3. Applying Bloch-Floquet theorem [23] to the terminals of this two-port network, described in terms of its ABCD matrix, assuming that the unit cell is symmetrical, i.e., yields the dispersion relation for the Bloch propagation constant [24] where is the phase constant, which determines the angle of main beam radiation according to (1) (replacing by ), while is the attenuation constant [2], which includes a radiation leakage contribution, determining the directivity of the antenna, a dissipation contribution. By the same token, the Bloch impedance is obtained for the symmetrical case as (5) (6)

4 3698 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011 Fig. 5. Specific lattice equivalent circuits corresponding to the general lattice of Fig. 4(a). (a) Lattice incorporating the equivalent lumped elements given by (13) (14). (b) Approximate equivalent lattice obtained from the linearized immittances Z Y as defined in (15). Fig. 4. Lattice type equivalent circuit model for the unit cell of periodic leakywave antennas. (a) General circuit. (b) Equivalent circuit under odd input-output excitation. (c) Equivalent circuit under even input-output excitation. Inside the black box equivalent network of Fig. 3, the physical unit cell of the periodic structure is modeled by the lattice circuit [26] shown in Fig. 4(a). This topology offers the advantage of decoupling the series impedance shunt admittance under odd even input-output termination conditions [27], respectively, as shown in Fig. 4(b) Fig. 4(c). While the lattice topology of Fig. 4(a) was actually implemented by distributed lumped elements to exploit the broadb all-pass nature of this circuit in [28], it is used here as a generic model for the antennas under consideration in Section II. The impedance matrix parameters of the lattice circuit of Fig. 4(a), which is assumed to be symmetric, are given by [29] The difference of these two expressions yields (7a) (7b) (8a) while the sum yields, which leads, using stard network transformation formulas [24], to (8b) We now seek an expression for the Bloch propagation constant in (5) as a function of the series impedance shunt admittance. For this purpose, we write the parameters in (8) in terms of the ABCD parameters, using again stard network conversion formulas, i.e.,, [24], which leads to (9) Using (5) assuming reciprocity in addition to symmetry ( ), we obtain the following relation for the Bloch propagation constant in terms of (10) Similarly, we seek an expression for the Bloch impedance. For this purpose, we first write the parameters in (6) in terms of the matrix elements using stard network conversion formulas, i.e. [24], which yields. Inserting next the conversion formulas into (8b) yields, from which we find B. Transmission Line Characteristic Propagation Constant Impedance Following From Immittance Linearization (11) The formulas for the Bloch propagation constant the Bloch impedance given by (10) (11) may be simplified by modeling as a series resonance circuit with resonance frequency as a parallel resonance circuit with resonance frequency, respectively. In a leaky-wave antenna, these resonances are typically designed to be equal, i.e., so as to provide seamless beam scanning from backward to forward angles across broadside [20]. In the remainder of this paper, we will consider that this equality, called frequency-balancing, 3 always holds. Fig. 5(a) depicts the lattice circuit with its explicit series shunt resonators. The series resonance frequency the shunt resonance frequency are defined by The corresponding lumped elements are determined by 3 as opposed to Q-balancing which is addressed in Section VI-D. (12) (13a) (13b) (13c)

5 OTTO et al.: TRANSMISSION LINE MODELING AND ASYMPTOTIC FORMULAS FOR PERIODIC LEAKY-WAVE ANTENNAS 3699 where by, for the series resonance circuit TABLE II SUMMARY OF THE RESULTS DERIVED IN SECTION IV-A TO SECTION IV-C (14a) (14b) (14c) where, for the shunt resonance circuit, respectively. The immittances may next be linearized around their respective resonances (15a) (15b) where it will be assumed that, as previously announced. Here, the LCRG parameters (without any subscripts) have emerged as the parameters of the equivalent linearized model, which is represented in Fig. 5(b). The LCRG parameter can be calculated analytically based on the specific circuit model of the SFP, PR or CRLH unit cell [Section IV-A to Section IV-C] or they can be extracted from full-wave simulation or measurement data [Section V]. The linearized lattice model of Fig. 5(b) will be used as the TL model in the forthcoming evaluations derivations. Around the broadside frequency, where, it exhibits simple characteristic parameters. By approximating both sides of (10) with a Taylor expansion with respect to, respectively, truncating higher order terms one obtains (16) from which the linearized-immittance propagation constant is defined as (17) with the parameters given by (15). On the other h, the linearized-immittance impedance is directly (without approximation) obtained from (11) as (18) Due to their similarity with TL expressions, in (17) in (18) will be subsequently referred to as the characteristic propagation constant the characteristic impedance (hence the subscript c ) of the antenna structures, respectively. These quantities may be interpreted as the characteristic parameters of a uniform TL approximating the actual periodic structure with the characteristic propagation constant the characteristic impedance. IV. VALIDATION OF THE TL MODEL WITH SFP, PR AND CRLH CIRCUIT MODELS This section validates the TL model by broadb periodic (Bloch) distributed/lumped circuit models specific to the SFP, PR CRLH antennas, thereby also determines the frequency validity range of this model. We introduce a perturbational approach for the immittances yielding simple closed-form expressions for,,. For each unit cell circuit, the TL model is validated by comparing the characteristic propagation constant characteristic impedance [(17) (18)] with Bloch parameters [(5) (6)]. The following procedure is used: 1) Compute the lossless 4 overall matrix of the unit cell by cascading the sub-matrices of its successive equivalent elements ignoring the lossy elements. (e.g. SFP unit cell in Fig. 6(a): ). 2) Compute the lossless matrices corresponding to the matrix obtained in 1) using stard network conversion formulas [24]. 3) Insert the appropriate terms of these matrices into (8a) (8b) to determine, respectively. 4) Compute the frequency derivatives of using (13b) using (14b), insert the resulting expressions, evaluated at, into (15) to obtain the linearized immittances, which are purely imaginary since the losses are not taken into account so far. 5) Determine the, from (15), as they are the coefficients in. 6) Introduce the lossy elements in the respective circuit model apply odd [Fig. 4(b)] even [Fig. 4(c)] excitations at the resonance frequency to isolate the resistance conductance, respectively. 7) With found in 5) found in 6), insert the expressions obtained for into (17) (18) to compute. The LCRG formulas obtained from this seven-step procedure based on the forthcoming SFP, PR CRLH distributed/ lumped element equivalent circuits, are given in Table II. For each unit cell circuit, a numerical example comparing Bloch TL model is presented next. 4 This initial lossless approximation is reasonable, as will be shown, because the slopes of the immittance parameters are not significantly affected by loss radiation.

6 3700 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011 Fig. 6. SFP antenna: (a) Layout of a microstrip SFP unit cell corresponding circuit model with normalized electrical period p =0:54. Comparison Bloch model versus TL model, (b) propagation constant ; (the attenuation constant Re(p) =p is multiplied by a factor of 5 for better visualization) (c) impedance Z ;Z. Fig. 7. PR antenna: (a) Layout of a microstrip PR unit cell the corresponding circuit model with normalized electrical period p =0:36. Comparison Bloch model versus TL model, (b) propagation constant ; (the attenuation constant Re(p) = p is multiplied by a factor of 5 for better visualization) (c) impedance Z ;Z. A. Series Fed Patch (SFP) Antenna Fig. 6(a) shows the layout the equivalent circuit model for the SFP antenna [Fig. 1(a)] unit cell [15], [16]. The microstrip patch, of length, is modeled by a low-impedance TL section of impedance, while the interconnecting microstrip lines on both sides of the patch, of length, are modeled by high-impedance TL sections of impedance. The radiating edges of the patch are modeled by the conductances. Distributed losses i.e. conductor loss dielectric loss are lumped into. The electrical line lengths of the connection line the patch TL have to be at the broadside frequency. Therefore, we write the electrical length as follows where. A set of realistic parameters,, has been selected to generate Fig. 6(b) Fig. 6(c), which will be described in Section IV-D. B. Phase Reversal (PR) Antenna Fig. 7(a) shows the layout the equivalent circuit model for the PR antenna [Fig. 1(b)] unit cell. The black gray traces represent the top bottom metal layers at either side of the substrate, respectively, radiation occurs from the cross-over region [18], which is modeled by an ideal transformer with a transformation ratio to provide the 180 phase reversal. In addition, two independent resistors,, are used to model radiation dissipation loss (conductor dielectric loss), respectively. The overall length of the unit cell is. The electrical length of the unit cell is at the broadside frequency with The following set of parameters has been chosen for validation:,. The TL-Bloch comparison is shown in Figs. 7(b) 7(c) will be discussed in Section IV-D. Fig. 8. CRLH antenna: (a) Layout of a microstrip CRLH unit cell (left: MIM, right: interdigital) corresponding circuit model with normalized electrical period p =0:16. Comparison Bloch model versus TL model, (b) propagation constant ; (the attenuation constant Re(p) = p is multiplied by a factor of 5 for better visualization) (c) impedance Z ;Z. C. Composite Right/Left-Hed (CRLH) Antenna Fig. 8(a) shows the layout the equivalent circuit model for the CRLH antenna [Fig. 1(c)] unit cell, where both the MIM interdigital implementations are considered. The length of the unit cell is. The set of parameters,,,, has been selected for the comparison in Figs. 8(b) 8(c). This example will be further discussed in Section IV-D. D. Comparison Validity Range The circuit models for the SFP, PR CRLH antennas in Figs. 6(a), 7(a) 8(a), respectively are inherently broadb,

7 OTTO et al.: TRANSMISSION LINE MODELING AND ASYMPTOTIC FORMULAS FOR PERIODIC LEAKY-WAVE ANTENNAS 3701 since they are based on the exact physical nature of the corresponding structures. In contrast, the TL model is a much simpler ( therefore narrower b) model; however, this model has the fundamental interest to be common to all one-dimensional periodic leaky-wave antennas scanning through broadside while still being very accurate around the broadside radiation frequency. The degree of accuracy of the TL model away from the broadside frequency strongly depends on the particular antenna structure. The TL model for the SFP antenna, with its large electrical unit cell period, has the highest deviation in the propagation constant [Fig. 6(b)] impedance [Fig. 6(c)] compared to the PR antenna, which is in excellent agreement with the TL model [Figs. 7(b) 7(c)]. This excellent agreement is due to the homogeneous TL nature of the PR antenna, which is essentially composed of TL sections of constant impedance. The approximation accuracy of the CRLH circuit antenna [Fig. 8(a)] is in between the two former cases, with a good agreement for the impedance a reasonable agreement for the propagation constant. Another factor limiting the frequency range validity, quite far from the broadside frequency region, is the coupling to the light line (Fig. 2), not taken into account in the model. The shaded areas in the dispersion diagrams of Figs. 6(b), 7(b) 8(b) indicate the fast-wave regions, where the model is applicable. The formulas presented in Table II provide design guidance insight into the influence of the different structural parameters for the development of the specific leaky-wave antenna of interest. For instance, they provide, via Table II, parametric expressions for the Bloch impedance near broadside off broadside for each of the three discussed antennas using the formulas, respectively, which will be derived in Section V. Finally, it is important to note that the results in Table II are based on a perturbational (lossless) approach (step 4 above) for the series shunt immittances. Such an approach can clearly not be applied to the propagation constant or impedance directly, as is done in TL theory as will be exploited in Section V, since this would not capture the dominant resistive effects at the broadside frequency. V. FULL-WAVE AND EXPERIMENTAL PARAMETER EXTRACTION AND VALIDATION This section validates the formulas derived in Sections III IV by full-wave measurement results. The SFP antenna is selected for this purpose. This is essentially an arbitrary choice, except for the fact that this antenna is particularly simple to fabricate test. In the full-wave case, two completely different methods are used in parallel for the extraction of the LCRG circuit model parameters, a driven-mode network method using immittance slope parameters an eigenmode electromagnetic method using energies powers computed by integrating electromagnetic fields, while only the former is applicable applied in the experimental case. Fig. 9 shows the layout dimensions of the test SFP antenna, which was designed to operate at. Fig. 9. Layout dimensions (unit cell) of the selected SFP antenna. The substrate has a relative permittivity of " =2:2 a height of h =1:5 mm. Fig. 10. Simulation setup of the 10-cell structure with microstrip TL ports in EMPIRE XCcel. The structure is simulated with a finite ground plane is fully enclosed by absorbing PML boundaries. A. Driven-Mode Network LCRG Parameter Extraction The network LCRG parameter extraction consists in the following simulation steps: 1) set up a parameterized (in ) antenna structure composed of a sufficient number of unit cells to effectively take into account mutual coupling effects [4], [30], [31] to sufficiently dilute edge (termination) aperiodic effects (here is set to 10); 2) compute the transmission matrix for this -cell structure; 3) take the -root of this matrix,, to obtain the corresponding matrix for one unit cell,, also evaluating a different length (e.g. ) or using an estimation of the broadside frequencies in order to select the correct physical solution among solutions [31]; 4) use stard conversion formulas to compute the corresponding (one unit cell) impedance matrix admittance matrix ; 5) insert the appropriate elements of the resulting matrices into (8a) (8b) to determine, respectively, apply (12) to find initial (approximate) values for ; 6) optimize the parameters to obtain exactly in (12); 7) insert the optimized impedance admittance parameters into (8a) (8b) to obtain the exact, respectively, compare these results with (15) to obtain the final LCRG parameters. The corresponding structure (FDTD EMPIRE XCcel) used in this procedure for the forthcoming results is shown in Fig. 10. B. Eigenmode Electromagnetic LCRG Parameter Extraction The electromagnetic LCRG parameter extraction was developed applied to a CRLH antenna in [32] is recalled here for self-consistency. It employs an eigenmode solver (here FEM Ansoft-HFSS) with periodic boundaries along the axis of

8 3702 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011 Fig. 12. Equivalent circuit models of Fig. 5(a) of the structure at 8 =0. (a) PEC-wall (short-circuited) mode!, where the series impedance is isolated. (b) PMC-wall (open-circuited) mode!, where the shunt admittance is isolated. Fig. 11. Simulation setup for parameter extraction. (a) Periodic boundaries with a constant null phase shift (8 =0)are applied at both sides of the unit cell along the axis of the antenna structure (x direction). The sides of the computational domains are terminated by PML boundaries in the transverse direction (y direction) as well as the opening to free space on the top (z direction). While the unit cell shown is that of the SFP antenna, the same setup can be used for any antenna of the class considered in the paper. (b) Voltage current integration paths for (25) (26), respectively. the antenna structure, set to a constant null periodic phase shift,, determines the circuit parameters for the circuit model of Fig. 5(a) from energies powers computed by integration of the electric magnetic fields over appropriate domains. Fig. 11 shows the simulation setup used for this purpose. The required electromagnetic quantities are [24] (19) (20) : one corresponding to PEC walls another one corresponding to PMC walls at the positions of the periodic boundaries 5 [32]. With such walls, the lattice circuit model of Fig. 5(a) splits in the two simplified circuits shown in Fig. 12, which allows to separate the elements of the series impedance of the shunt admittance. The corresponding current voltage are obtained from field quantities as (25) (26) where the integration contours are indicated in Fig. 11(b). The reactive parameter values ( ) for the circuit components of each resonator are obtained by relating the stored magnetic or electric energy to the current or voltage for the corresponding mode, while the resistive parameter values ( ) are computed by relating the power loss for each type of loss to the terminal current or voltage. The formulas for the series resonator, corresponding to the circuit model of Fig. 12(a) read (21) (22) (23) (27) while the formulas for the shunt resonator, corresponding to the circuit model of Fig. 12(b) read where represent energies powers, respectively, the subscripts refer to electric magnetic quantities, respectively. The subscripts, refer to radiation, dielectric conductor losses. The vector quantities, represent the electric field, the magnetic field the conduction current density, respectively. Fig. 11 indicates the appropriate integration domains,,. The overall quality factor of the unit cell can be expressed in terms of the above quantities as (24) where represents to total power loss. Assuming that the unit cell is symmetrical ( direction as symmetry axis), which is the case in all the antenna structures considered here, two modes are supported by the structure at (28) The field calculator in HFSS provides a simple interface for the computation of the integral expression (19) (23) the current voltage path integration (25) (26). C. Prototype Measured LCRG Parameters Fig. 13 shows the measurement setup with the 10-cell SFP antenna prototype corresponding to the layout design parameters of Fig. 9. The LCRG parameters are experimentally determined by using the same driven-mode network method as in the full-wave analysis case (Section V-A), with a careful setup 5 Therefore, the problem could also be analyzed without periodic boundaries by alternatively using PEC PMC walls in two distinct simulations.

9 OTTO et al.: TRANSMISSION LINE MODELING AND ASYMPTOTIC FORMULAS FOR PERIODIC LEAKY-WAVE ANTENNAS 3703 Fig. 13. Measurement setup with the 10-cell SFP antenna prototype based on the layout design parameters of Fig. 9. TABLE III COMPARISON OF EXTRACTED CIRCUIT PARAMETERS FOR THE SFP UNIT CELL Fig. 14. Compared Bloch propagation constant characteristic (TL) propagation constant [(17)] with the LCRG parameters taken from Table III. (a) FDTD driven mode, following Section V-A. (b) Measurement, following Section V-C. The attenuation constant Re(p) =p is multiplied by a factor of 5 for better visualization. Fig. 15. Compared Bloch impedance Z characteristic (TL) impedance Z [(18)] with the LCRG parameters taken from Table III. (a) FDTD driven mode, following Section V-A. (b) Measurement, following Section V-C. of the measurement reference planes using the built-in port extension function of the vector network analyzer Agilent PNA E8363B. D. Comparison of the Extraction Methods Validation of the Transmission Line Model The LCRG circuit parameters extracted by the three approaches described above are listed in Table III. All the parameters are in close agreement, except for the shunt quality factor corresponding conductance. This discrepancy may be due to the higher sensitivity of the shunt resonator following from its lower loss compared to the series resonator. The Bloch analysis (determination of ) in the driven FDTD simulation the measurement is based on a finite number of cells yielding a close approximation of the exact Bloch solution in the limit of an infinite number of cells. Fig. 14(a) compares FDTD full-wave simulation results for the real imaginary parts of the Bloch propagation constant of the characteristic propagation constant resulting from the TL modeling of Section III. The Bloch propagation constant is computed by (5) using the parameter of the unit cell transmission matrix obtained from the first four steps of the procedure given in Section V-A from FDTD simulations. The characteristic constant is computed by (17) using the LCRG parameters following all the seven steps of procedure of Section V-A leading to the parameters listed in the first column of Table III. Excellent agreement is observed between the two types of results over the frequency range from 4.8 to 6.8 GHz for over the frequency range from 4.8 to 6.4 GHz for. Similarly, Fig. 14(b) compares measurement results for the Bloch propagation constant for the characteristic propagation constant, using the experimentally determined parameter in the former case LCRG parameters in the second column of Table III for the latter case. The experimental dip at the broadside frequency [Fig. 14(b)] is less pronounced than the simulated one [Fig. 14(a)], due to the already mentioned higher shunt loss encountered in the measurement, but very good agreement between Bloch characteristics results are again observed. Figs. 15(a) 15(b) provide the same comparisons as Figs. 14(a) 14(b) but for the Bloch characteristic impedances, where (5) (17) are replaced by (6) (18), respectively. As for the propagation constant, good agreement is observed in both graphs over the frequency range from 4.8 GHz to 6.8 GHz. The ripples in the Bloch results are related to the finite number of cascaded unit cells correspond to the frequencies where the overall phase across the 10-cell structure is a multiple of is thus an effect of the finite number of unit cells. Around the broadside frequency, the maximum Bloch impedance (real part) is higher in the simulation [Fig. 15(a)] than in the measurement [Fig. 15(b)]. This is again due to the different shunt losses in Table III.

10 3704 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011 Fig. 16. Equivalent TL model for an N -cells periodic leaky-wave antenna, in particular for the 10-cell prototype shown in Fig. 13. TL analysis based on the propagation constant the characteristic impedance Z is applied to compute compare the Bloch characteristic scattering matrices. of approximately 15% around, with an almost perfect accuracy near the broadside frequency, where the immittances were linearized. In conclusion, the TL model developed in Section III has been fully-confirmed by full-wave experimental results, beyond the circuit based validations provided in Sections IV-A to IV-C. Consequently, the TL model derived in Section III validated in Section V, is really applicable for all the leaky-wave antennas belonging to the class described in Section II. VI. ASYMPTOTIC TL FORMULAS Based on the TL model established in Section III, this section derives analytical formulas describing the asymptotic response of the considered antennas in two regimes, the regime near the broadside frequency the regime off the broadside frequency. In all the subsequent formulas, the LCRG parameters may be substituted by the analytical expressions given in Table II, or may be extracted from full-wave simulation, as described in Sections V-A V-B or from measurement as presented in Section V-C. Fig. 17. Comparison of the S parameters obtained by the TL model [(29)] by direct FDTD simulation measurement for the N =10cells antenna of Fig. 9. (a) FDTD TL (using Section V-A LCRG parameter extraction for the latter). (b) Experiment TL (using Section V-C LCRG parameter extraction for latter). In order to obtain a final validation immune of any possible artifacts, the TL model is compared in terms of raw scattering data with full-wave experimental results for the 10-cell antenna of Fig. 13. From TL theory, the scattering matrix for the TL model is given in terms of by (29) where the -referenced TL scattering matrix has been re-normalized to the measurement impedance reference using stard conversion formulas. The LCRG parameters needed in the computation of are determined from the simulated (Section V-A) or measured (Section V-C) scattering matrices, respectively. Fig. 16 shows the TL model for the reference. The range of validity of the TL model is shown in Fig. 17, where the scattering parameters of the model are compared with full-wave measurement data. The antenna is poorly matched to, due to the relatively high impedance (, see Fig. 15). This high value of may be understood from the SFP relations of Table II, corresponding to the circuit model of Fig. 6(a), which show that the SFP lossless characteristic impedance is, typically yielding due to the alternating high-impedance low-impedance TL sections of the structure. This strong mismatch is beneficial to the present validation purpose as it leads to very clear ripples in the scattering parameters, allowing optimal comparison. The proposed TL model is validated by both full-wave measurement results over a substantial frequency range A. General Case For the case of a frequency-balanced antenna unit cell with the angular frequency is written as (30) (31) where spans a relatively narrow frequency range around the broadside frequency with. We rewrite (15) as using the general quality factor definition [23] (32a) (32b) (33) the corresponding specific series shunt quality factors (34a) (34b) for the equivalent series shunt resonators, with corresponding stored energies, power losses, [24]. Inserting these expressions into (17), the characteristic propagation constant is given in terms of the LCRG parameters the quality factors with

11 OTTO et al.: TRANSMISSION LINE MODELING AND ASYMPTOTIC FORMULAS FOR PERIODIC LEAKY-WAVE ANTENNAS 3705 (35) The corresponding TL quality factor may be computed [23] from this relation as (36) Finally, the characteristic impedance is obtained by insertion of the same expressions into (18) as (39) Finally, the off-broadside characteristic impedance is obtained by neglecting the terms in (37), by reusing the aforementioned Taylor approximation, which yields (40) Assume, so to fully neglect the loss contributions in (32), (40) further simplifies to (41) (37) where, are the TL voltage current, respectively. Rearranging (41), we obtain the relation (42) B. Off-Broadside Regime Consider the frequency range (still assuming ), which extends on both sides of, excluding a small region around. Note that the condition may be satisfied even very close to, because the product represents a very small quantity, essentially modeling the radiation loss per unit length of the leaky-wave antenna. In this range, the general propagation constant in (35) reduces to showing that the stored energies are equal. Using (33) along with (34a) (34b), we obtain for the off-broadside quality factor (43) which is identical to (39). However, the energy power based derivation which lead to (43) provides deeper physical insight than the other one. with (38a) C. Near-Broadside Regime Now consider the complementary frequency range (still assuming ), which corresponds to an extremely narrow frequency region centered at, where according to (31). In this range, the general propagation constant of (35) becomes (38b) (38c) where the subscript has been introduced to indicate the offbroadside regime. The last approximate result in (38a) was obtained using the Taylor approximation for, with. The condition safely holds here due to the aforementioned low per-unit-length leaky-wave radiation loss. An explicit expression for the off-broadside TL quality factor is then obtained by inserting (38b) (38c) into (36), which yields with as presented in [4] (44a) (44b) (44c) where the subscript 0 has been introduced to indicate the nearbroadside regime.

12 3706 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011 An explicit expression for the near-broadside TL quality factor is then obtained by inserting (44b) (44c) into (36), which yields (45) near- leads to identical formulas for the off-broadside broadside (0) regimes, (51) (52) Finally, the near-broadside characteristic impedance is obtained by neglecting the term in the nominator denominator of (37), by reusing the aforementioned Taylor approximation, which yields The characteristic impedance of (46) may be simplified to (46) (47) now assuming, so as to neglect the reactive contributions in (32). Rearranging (47), we obtain the relation (48) revealing that the powers dissipated in the series shunt resonators are equal. Finally, using (33) along with (34a) (34b), we obtain the near-broadside (0) quality factor (49) which is identical to (45). The power equality in (48) together with the quality factor of (49), their implications in broadside radiating periodic leaky-wave antennas, are the major result of this work. At broadside, the wave propagation radiation mechanism is essentially resistive follow these unusual formulas. The TL theory based on the conventional propagation constant line impedance fails to model this phenomenon, whereas the followed TL approach based on the derived parameters fully captures the dominant resistive effects explains the physics of the phenomenon. D. Q-Balancing An interesting situation arises when the series shunt quality factors are equal (50) which in a conventional homogeneous TL is referred to as the Heaviside condition, corresponding to distortionless transmission [24]. In the present context, this condition simplifies the general formulas for the propagation constant (35), the quality factor (36) the characteristic impedance (37) in a way that (53) (54) as may be easily verified by substituting (50) into (35) to (37). Furthermore, the following energy power relations hold for the Q-balanced case in both off- near-broadside regimes: (55) (56) Thus, in the Q-balanced case, the wave propagation in the off near-broadside regimes is distortionless the leaky-wave structure is modeled by the same set of equations [(51) to (56)]. In [4] a CRLH antenna has been optimized for broadside radiation by using a single-sided open-ended stub producing an orthogonally polarized radiation from the shunt resonator. The condition of Q-balancing in (50) will ultimately lead to a distortionless propagation in longitudinal antenna direction with a constant leakage factor a constant impedance over frequency hence results in an antenna gain which is constant while the beam is scanned across broadside. It may be deduced from the above developments that the empirically designed antenna in [4] fairly well meets the condition (50). E. Summary Validation Example Table IV summarizes the off-broadside near-broadside asymptotic formulas derived in this section. At the broadside frequency the Bloch wave propagation forces the overall power dissipation (radiation together with power loss) to be exactly the same in the series resonator in the shunt resonator, which has a major impact on the antenna performance it has, to the authors best knowledge, not been discovered so far. If a periodic leaky-wave antenna has one resonator, either series or shunt with a poor radiation efficiency then the antenna performance will be strongly degraded at broadside due to the equal power split into the two resonators. In case one resonator is not radiating at all only dissipating power, the radiation efficiency cannot exceed more than 50% at broadside, independent of the amount of power dissipated in the non-radiating resonator. In the off broadside regime, where the energies in the two resonators are equal (42), the overall radiation efficiency depends on the amount of power loss in the dissipative, non-radiating resonator, so with its quality factor approaching infinity a radiation efficiency of ideally 100% can theoretically be achieved. A numerical example is given to verify the formulas derived in Section VI-B Section VI-C to visualize their asymptotic behavior. Figs. 18, compare these formulas with

13 OTTO et al.: TRANSMISSION LINE MODELING AND ASYMPTOTIC FORMULAS FOR PERIODIC LEAKY-WAVE ANTENNAS 3707 TABLE IV SUMMARY OF THE ASYMPTOTIC OFF- AND NEAR-BROADSIDE TL FORMULAS DERIVED IN SECTION VI Fig. 18. Complex propagation constant comparison between the exact formula of the TL model (35) the asymptotic formulas of (38) (44). Fig. 20. Characteristic impedance comparison between the exact TL model formula of (37) the asymptotic formulas of (40) (46). Fig. 19. Quality factor comparison between the exact formula of the TL model (36) the asymptotic formulas of (39) (45). their exact counterparts given by (35) to (37) for the following set of parameters:,,,. In all the cases, the asymptotic formulas are found to accurately model the off-broadside near-broadside behaviors of the structures. VII. CONCLUSION Network transmission line analyzes of periodic leaky-wave antennas scanning through broadside with frequency have been performed novel asymptotic formulas have been subsequently derived. It has been shown that, despite seemingly very different configurations, these antennas exhibit a qualitatively similar response, therefore can be described by the same transmission line model. This has been specifically demonstrated for the examples of a series fed patch array antenna, a phase reversal array antenna a composite right/left-hed antenna. Based on the theory of periodic structures, a simple lattice network model has been proposed a TL model with linearized immittances has been subsequently developed. This model has been validated by comparison with periodic Bloch results for the three aforementioned antennas, confirmed by full-wave experimental results in the particular case of the series fed patch array. The asymptotic formulas, following from this simple model corresponding to broadside off-broadside radiation regimes, are the central result of the paper. These formulas provide physical insight electromagnetic understing of the behavior of leaky-wave antennas around the broadside frequency. An important result is that

14 3708 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 10, OCTOBER 2011 the total power radiated from or dissipated in the unit cell at broadside equally splits between the series shunt paths, which explains poor broadside radiation when only one of the two paths efficiently contributes to radiation. REFERENCES [1] A. Hessel, Antenna Theory, Part II, R. E. Collin R. F. Zucker, Eds. New York: McGraw-Hill, 1969, ch. 19. [2] A. A. Oliner D. R. Jackson, Antenna Engineering Hbook, J. Volakis, Ed., 4th ed. New York: McGraw-Hill, [3] P. Burghignoli, G. Lovat, D. R. Jackson, Analysis optimization of leaky-wave radiation at broadside from a class of 1-D periodic structures, IEEE Trans. Antennas Propag., vol. 54, no. 9, pp , Sep [4] S. Paulotto, P. Baccarelli, F. Frezza, D. R. Jackson, Full-wave modal dispersion analysis broadside optimization for a class of microstrip CRLH leaky-wave antennas, IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp , Dec [5] S. Paulotto, P. Baccarelli, F. Frezza, D. R. Jackson, A novel technique for open-stopb suppression in 1-D periodic printed leakywave antennas, IEEE Trans. Antennas Propag., vol. 57, no. 7, pp , Jul [6] L. Liu, C. Caloz, T. Itoh, Dominant mode (DM) leaky-wave antenna with backfire-to-endfire scanning capability, Electron. Lett., vol. 38, no. 23, pp , Nov [7] S. Lim, C. Caloz, T. Itoh, A reflecto-directive system using a composite right/left-hed (CRLH) leaky-wave antenna heterodyne mixing, IEEE Microwave Wireless Compon. Lett., vol. 14, no. 4, pp , [8] S. Lim, C. Caloz, T. Itoh, Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle beamwidth, IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp , Jan [9] F. P. Casares-Mira, C. Camacho-Peñalosa, C. Caloz, High-gain active composite right/left-hed leaky-wave antenna, IEEE Trans. Antennas Propag., vol. 54, no. 8, pp , Aug [10] T. Kodera C. Caloz, Uniform ferrite-loaded open waveguide structure with CRLH response its application to a novel backfire-to-endfire leaky-wave antenna, IEEE Trans. Microwave Theory Tech., vol. 57, no. 4, pp , April [11] T. Kodera C. Caloz, Integrated leaky-wave antenna Duplexer/ diplexer using CRLH uniform ferrite-loaded open waveguide, IEEE Trans. Microwave Theory Tech., vol. 58, no. 8, pp , Aug [12] T. Kodera C. Caloz, Low-profile leaky-wave electric monopole loop antenna using the =0regime of a ferrite-loaded open waveguide, IEEE Trans. Antennas Propag., vol. 58, no. 10, pp , Oct [13] C. Caloz, D. R. Jackson, T. Itoh, Frontiers in Antennas, F. Gross, Ed. New York: McGraw-Hill, [14] J. James, P. Hall, C. Wood, Microstrip Antenna Theory Design (Electromagnetic Waves). London, U.K.: Institution of Engineering Technology, [15] M. Danielsen R. Jorgensen, Frequency scanning microstrip antennas, IEEE Trans. Antennas Propag., vol. 27, no. 2, pp , Mar [16] A. Derneryd, Linearly polarized microstrip antennas, IEEE Trans. Antennas Propag., vol. 24, no. 6, pp , Nov [17] N. Yang, C. Caloz, K. Wu, Fixed-beam frequency-tunable phase-reversal coplanar stripline antenna array, IEEE Trans. Antennas Propag., vol. 57, no. 3, pp , Mar [18] N. Yang, C. Caloz, K. Wu, Wideb phase-reversal antenna using a novel bwidth enhancement technique, IEEE Trans. Antennas Propag., vol. 58, no. 9, pp , Sep [19] N. Yang, C. Caloz, K. Wu-, Full-space scanning periodic phasereversal leaky-wave antenna, IEEE Trans. Microw. Theory Tech., vol. 58, no. 10, p. 1, Oct [20] C. Caloz T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory Microwave Applications. Piscataway,NJ: Wiley-IEEE Press, [21] C. Caloz, T. Itoh, A. Rennings, CRLH metamaterial leaky-wave resonant antennas, IEEE Antennas Propag. Magn., vol. 50, no. 5, pp , Oct [22] P. Baccarelli, S. Paulotto, D. R. Jackson, A. A. Oliner, A new Brillouin dispersion diagram for 1-D periodic printed structures, IEEE Trans. Microw. Theory Tech., vol. 55, no. 7, pp , [23] R. E. Collin, Field Theory of Guided Waves. New York: IEEE Press, [24] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, [25] R. E. Collin, Foundations for Microwave Engineering. Piscataway, NJ: Wiley-IEEE Press, [26] F. E. Terman, Network theory, filters, equalizers, Proc. IRE, vol. 31, no. 5, pp , [27] S. Wane D. Bajon, Broadb equivalent circuit derivation for multi-port circuits based on eigen-state formulation, in Proc. IEEE MTT-S Int. Microwave Symp. Digest MTT 09, 2009, pp [28] F. Bongard, J. Perruisseau-Carrier, J. R. Mosig, Enhanced CRLH transmission line performances using a lattice network unit cell, IEEE Microw. Wireless Compon. Lett., vol. 19, no. 7, pp , Jul [29] J.-S. G. Hong M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley-Interscience, [30] P. Baccarelli, C. Di Nallo, S. Paulotto, D. R. Jackson, A full-wave numerical approach for modal analysis of 1-D periodic microstrip structures, IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp , Jun [31] T. Liebig, S. Held, A. Rennings, D. Erni, Accurate parameter extraction of lossy composite right/left-hed (CRLH) transmission lines for planar antenna applications, in Proc. Metamaterials, Karlsruhe, Germany, Sep. 2010, pp [32] S. Otto, A. Rennings, T. Liebig, C. Caloz, K. Solbach, An energy-based circuit parameter extraction method for CRLH leaky-wave antennas, presented at the EuCAP, Barcelona, Spain, Apr Simon Otto (S 10) received the Diplom-Ingenieur degree from Duisburg-Essen University in 2004, where he is working toward the Ph.D. degree. He was a visiting student at the Microwave Electronics Laboratory of the University of California at Los Angeles (UCLA). Currently, he is with the Antenna EM Modeling Department at IMST in Kamp-Lintfort, Germany. He has authored or coauthored more than 25 conference journal papers related to antennas, filter designs, RF components, simulation techniques, magnetic-resonance imaging (MRI) systems filed three patents. His research interests include array antennas, transmission line metamaterials, EM-theory numerical modeling. Mr. Otto received the VDE prize (Verb der Elektrotechnik Elektronik Informationstechnik e.v.) for his Diplom-Ingenieur thesis the second prize of the Antenna Propagation Symposium (AP-S) student paper award 2005 in Washington. Andreas Rennings (M 08) studied electrical engineering at the University of Duisburg-Essen, Germany. He carried out his diploma work at the Microwave Electronics Laboratory of the University of California at Los Angeles (UCLA). He received the Dipl.-Ing. the Dr.-Ing. degrees from the University of Duisburg-Essen, in , respectively. From 2006 to 2008, he was with IMST GmbH in Kamp-Lintfort, where he worked as an RF Engineer. Since then he is a Scientist in the Department of Theoretical Electrical Engineering (ATE) of the University of Duisburg-Essen, where he leads the Bio-Electromagnetics/Med-Tech group. His general research interests include all aspects of theoretical applied electromagnetics, currently with a focus on medical applications. He has authored coauthored over 50 conference journal papers, one book chapter, filed six patents. Dr. Rennings has received several awards, including the second prize at the student paper competition of the 2005 IEEE Antennas Propagation Society (AP-S) International Symposium the VDE-Promotionspreis 2009 for his doctoral thesis.

15 OTTO et al.: TRANSMISSION LINE MODELING AND ASYMPTOTIC FORMULAS FOR PERIODIC LEAKY-WAVE ANTENNAS 3709 Klaus Solbach was employed at the University of Duisburg, from 1975 to 1980, as a Junior Researcher in the field of integrated dielectric image line circuits. In 1981, he joined the Millimeterwave Research Laboratory at AEG-Telefunken in Ulm in 1984 changed to the Radar Systems Group of Daimler-Benz Aerospace (now part of EADS) where he engaged in the design production of microwave-subsystems for ground based airborne Radar, EW communication systems including phased array active phased array antenna systems. His last position was manager of the RF--Antenna-Subsystems Department. In 1997 he joined the faculty of the University of Duisburg as Chair of RF Microwave Technology. He has authored coauthored more than 200 national international papers, conference contributions, book chapters patent applications. Prof. Solbach was Chairman of the VDE-ITG Fachausschuss Antennen, Executive Secretary of the Institut für Mikrowellen-und Antennentechnik (IMA), Chair of the IEEE Germany AP/MTT Joint Chapter. He was General Chair of the International ITG-Conference on Antennas INICA2007 in Munich the General Chair of the European Conference on Antennas Propagation EuCAP2009 in Berlin. Christophe Caloz (F 10) received the Diplôme d Ingénieur en Électricité the Ph.D. degrees from École Polytechnique Fédérale de Lausanne (EPFL), Switzerl, in , respectively. From 2001 to 2004, he was a Postdoctoral Research Engineer at the Microwave Electronics Laboratory of University of California at Los Angeles (UCLA). In June 2004, he joined École Polytechnique of Montréal, where he is now a Full Professor, a member of the Poly-Grames Microwave Research Center, the holder of a Canada Research Chair (CRC). He has authored coauthored over 380 technical conference, letter journal papers, 3 books 8 book chapters, he holds several patents. His research interests include all fields of theoretical, computational technological electromagnetics engineering, with strong emphasis on emergent multidisciplinary topics. Dr. Caloz is a Member of the Microwave Theory Techniques Society (MTT-S) Technical Committees MTT-15 (Microwave Field Theory) MTT-25 (RF Nanotechnology), a Speaker of the MTT-15 Speaker Bureau, the Chair of the Commission D (Electronics Photonics) of the Canadian Union de Radio Science Internationale (URSI). He is a member of the Editorial Board of the International Journal of Numerical Modelling (IJNM), of the International Journal of RF Microwave Computer-Aided Engineering (RFMiCAE), of the International Journal of Antennas Propagation (IJAP), of the journal Metamaterials of the Metamorphose Network of Excellence. He received several awards, including UCLA Chancellor s Award for Post-doctoral Research in 2004 the MTT-S Outsting Young Engineer Award in He is an IEEE Fellow.