**Impulse-Regime Analysis of Novel Optically-Inspired Phenomena at Microwaves**

J. Sebastian Gomez-Diaz1, Alejandro Alvarez-Melcon1, Shulabh Gupta2 and Christophe Caloz2 <sup>1</sup>*Universidad Politécnica de Cartagena* <sup>2</sup>*École Polytechnique de Montréal* <sup>1</sup>*Spain* <sup>2</sup>*Canada*

#### **1. Introduction**

26 Fourier Transform Applications

Yang, T.; Bayram, Y. & Volakis, J. L. (2010). Hybrid Analysis of Electromagnetic Interference

(Aug. 2010), Vol.52, No.3, pp. 745-748, ISSN 0018-9375

Effects on Microwave Active Circuits Within Cavity Enclosures. *IEEE Trans. EMC*,

The ever increasing of needs for high data-rate wireless system is currently producing a shift from narrow-band radio towards ultra-wideband (UWB) radio operation [see Ghavami et al. (2007)]. Novel microwave tools, concepts, phenomena and direct applications must be developed to meet this demand. While the past decades have been focused on the "magnitude engineering" and filter design [see Pozar (2005)], there is a renewed interest in the "dispersion engineering". In the dispersion engineering approach, the phase is engineered to met various specifications within a given frequency range, so as to process signals in real time.

In this context, the development of electromagnetic metamaterials over the last years [see Caloz & Itoh (2006) or Marques et al. (2008)], with their intrinsically dispersive nature and subsequent impulse-regime properties, may provide novel and original solutions (see Fig. 1). Metamaterials can easily be synthesized in planar technology under the form of composite right/left-handed (CRLH) transmission lines (TLs), using non-resonant [see Caloz & Itoh (2006)] or resonant [see Duran-Sindreu et al. (2009)] approaches. These structures have provided novel and exciting applications, such as multi-band components, diplexers, couplers, phase-shifters, power-dividers or antennas with enhanced features, to mention just a few [see Caloz (2009) or Eleftheriades (2009) for a recent review]. However, CRLH structures have mostly been analyzed in the harmonic regime to date, and therefore only a few impulse-regime components and systems have been proposed so far. An example of these applications is the tunable pulse delay line presented in in Abielmona et al. (2007).

In this chapter, we present recent advances based on Fourier transformation techniques to model dispersive UWB phenomena and far-field radiation from complex CRLH structures. Section 2 first employs inverse Fourier transforms to study pulse propagation along this type of medium. Then, a Fourier transform approach is applied to the current which flows along the CRLH line, accurately retrieving the time-domain far-field radiation of the structure [which behaves as a leaky-wave antenna, (LWA)]. The main advantages of the proposed techniques are the easy treatment of complex CRLH structures, a deep insight into the physics of the phenomena, and an accurate and a fast computation, which avoids the time-consuming analysis required by completely numerical simulations.

In this section, a general time-domain Green's function approach is presented for the analysis of pulse propagation in electrically thin CRLH TLs. This method is based on the transient analysis of 1D transmission lines [see Paul (2007)] combined for the first time with CRLH TL concepts introduced in Caloz & Itoh (2006). With this equivalent transmission line simplification of the geometry, the Green's functions are available in closed-form, and directly correspond to the voltages and currents along the transmission line. The main advantages of this approach are the unconditional stability and fast computation, due to the continuous treatment of time, and the insight into the physical phenomena provided by the Green's functions. Subsequently, the method is extended to analyze impulse-regime CRLH leaky-wave antennas. The approach is based on the use of the time-domain current which flows along the structure to compute the far-field radiation of the antenna. This technique is especially appropriate to characterize complex radiated-wave UWB phenomena and devices,

Impulse-Regime Analysis of Novel Optically-Inspired Phenomena at Microwaves 29

The introduction of metamaterials [see Caloz & Itoh (2006) or Marques et al. (2008)] in the last decade has paved the road to the development of new devices and applications based on the novel fundamental features and phenomena associated to this type of media. Among the most useful metamaterials, one can find the CRLH transmission lines [see Caloz & Itoh (2006)]. This type of transmission lines, which are inherently nonresonant and low-loss, can be easily implemented in planar technology (such as microstrip or coplanar waveguide, for instance) and provides a practical realization of electromagnetic metamaterials. As any metamaterial, CRLH TLs are generally periodic structures formed by the repetition of unit-cells (an example of this type of cells is shown in Fig. 2) whose size, *p*, must fulfill the condition *p λ<sup>g</sup>* (where *λg* is the guided wavelength) ir order to emulate an effectively homogeneous material. A powerful method to analyze these metamaterial lines is the TL approach, presented in Caloz & Itoh (2006), which employs an ABCD-matrix technique of periodically arranged unit-cells to model the artificial transmission line (see Fig. 3) and to determine its wave propagation characteristics (such as propagation constant or Bloch impedances, see Fig. 4). There are many examples of interesting and groundbreaking applications of planar TL metamaterials at microwaves, such as for instance multi-band components, filters and diplexers, couplers, power-dividers, phase-shifters, lenses, or backfire to endfire leaky-wave antennas. A review of these and much more applications and devices can be found in metamaterials textbooks, such as in Eleftheriades & Balmain (2005), in Caloz & Itoh (2006), or in Marques et al. (2008). Note that all previously mentioned and most of metamaterials applications operate in the harmonic regime up to now, and they have been designed for narrow-band components and systems (even though some of them may support a multi-band

Many media, ranging from traditional purely right-handed materials to recent CRLH metamaterials [see Caloz & Itoh (2006)], can be advantageously analyzed by the transmission line theory described in Pozar (2005). So far, this theory has been applied mostly in the *harmonic* regime, where Green's functions for both the voltage and the current along the line are available. However, it may also be employed in the time-domain, where the Green's function approach provides an efficient tool to analyze *impulse regime* signals along

as it will be demonstrated in Section 3.

operation).

**2.1 Composite right/left-handed structures**

**2.2 Impulse regime analysis of CRLH transmission lines**

Fig. 1. Illustration of the dispersion engineering concept using metamaterial and CRLH structures. Reprinted with permission from Abielmona et al. (2008). Copyright 2008, URSI.

Section 3 applies the previously derived theory to study the impulse-regime phenomenology of CRLH structures, and subsequently demonstrates several optically-inspired phenomena and applications at microwaves in both the guided and the radiative regime. This study is divided into two main groups, according to the *guided-wave* or *radiative-wave* natures of the proposed phenomena and applications.


In all cases, the proposed theory and phenomena will be validated by using simulation results from full-wave commercial softwares and measurements, from fabricated prototypes. Therefore, the usefulness of the Fourier transform approach will be fully demonstrated as an essential mathematical tool for the fast and accurate modeling of very complex UWB structures and phenomena.
