Dispersion-Engineered Metamaterial Devices for Impulse-Regime

Samer Abielmona, Shulabh Gupta, Hoang Van Nguyen, and Christophe Caloz Poly-Grames, Ecole´ Polytechnique de Montreal,´ Quebec,´ Canada. Email: [email protected]

Abstract In this review, the authors present their research and development of an assortment of metamaterial devices applied to various applications. These devices, such as wide-band tunable delay lines and pulse position modulators, are apt for broad-band applications in impulse-regime systems. Recent advances as well future outlooks will also be outlined.

1 Introduction

In recent years, the “personal communicator” is constantly being redefined, driven mostly by market demands for consumer services. Among the many currently available trends, bandwidth-hungry and high data rate services and applications, have so far emerged as being the most functional, popular, and profitable [1]. These applications re- quire broad spectra, and thus a flurry of standards have been introduced: ultra-wide band (UWB), WiMAX, WiLAN. The emergence of these broadband technologies have presented designers with notable challenges and difficulties (dilemma/quandaries), eventually requiring new and innovative solutions. The recent introduction of metamaterials have allowed the utilization of dispersion-engineering, based on phase shap- ing as opposed to magnitude control, as an atypical approach to providing novel and innovative solutions [2], as shown in Fig. 1. Resonant-type and transmission line (TL) metamaterials have already enabled many narrow-band applications. However, TL metamaterials are garnishing more interest than resonant-type metamaterials due to their transmissive nature as well as their broad-band behavior. In this paper, the composite right/left-handed (CRLH) [3] TL’s affinity to broad-band applications will be described and demonstrated in various impulse-regime devices, along with the authors’prospective outlook.

2 Dispersion Engineering

The CRLH, an artificial TL composed of right-handed elements (LR,CR) and left-handed elements (LL,CL) is char- acterized by the following dispersion relation [3] µ ¶ ³ ´ 2 ³ ´2 1 −1 χ ω ωL 2 β(ω) = cos 1 − , with χ = + − κωc , and κ = LRCL + LLCR, (1) p 2 ωR ω √ √ where ωR = 1/ LRCR, ωL = 1/ LLCL, and p is the unit cell size. This dispersion relation is shown in Fig. 2 (a). As can be seen from Eq. (1), the CRLH TL offers an unprecedented level of dispersion (phase) control by simply varying the various CRLH parameters (LR,CR,LL,CL). As a direct result of this principle, many multi-band (dual- band [Fig. 1], tri-band, quad-band) devices have been devised for narrow-band applications. Some of the demonstrated examples include quarter-wavelength TLs and stubs, quadrature hybrids and power dividers [3]. Another advantage of dispersion engineering is bandwidth enhancement. The combination of right-handed TLs with linear phase and CRLH TLs with hyperbolic-linear phase yields a significant increase in the bandwidth response of several devices such as TL phase shifters (900, 1800). In both of the aforementioned benefits, the manipulation of the CRLH’s dispersion relation provides additional functionalities in narrow-band devices.

3 Metamaterials for Impulse-Regime Operation

As previously mentioned, most of the applications in today’s communication devices are converging towards broad bandwidth. Metamaterials are currently addressing some of the challenges presented by broad-band applications. The dispersion relation of the CRLH TL in Eq. (1) can be expanded in a Taylor series as follows [4]: µ ¶ ¯ n ¯ BALANCED CRLH ω ωL 1 2 d β(ω)¯ β(ω) = − ≈ β0 + β1(ω − ω0) + β2(ω − ω0) , where βn = n ¯ , (2) (LRCL=LLCR) ω ω 2 dω R ω=ω0 z

x y superluminal

DISPERSION leaky RH prop. g guided frozen

LH dual-band

NONLINEARITY

shock waves solitons

Figure 1: Authors’vision and conceptualization of dispersion-engineered metamaterials

µ ¶ µ ¶ ω0 ωL 1 ωL 2ωL with β0 = − , β1 = − 2 , β2 = − 3 , (3) ωR ω0 ωR ω0 ω0

where ω0 represents the carrier frequency for Taylor series expansion. Eq. (2) holds for a CRLH TL employed in devices requiring bandwidth (∆ω = ω − ω0) suitable for most broad-band applications. As well, it can be seen from Fig. 2 (b) that the CRLH TL exhibits a broad-band characteristic impedance, directly suitable for impulse-regime applications. The various Taylor series coefficients of Eq. (2) are given in Eq. (3), where β0, β1, and β2 represent the −1 phase velocity parameter, the group velocity parameter, and the group velocity dispersion, respectively. β1 is the propagation velocity of each carrier frequency (ω0), while β2 is a measure of dispersion strength of all frequencies.

4 First-Order (β1) Dispersion Applications

−1 st In a dispersive medium, the group velocity, vg = β1 , is frequency-dependent, where β1 is 1 order dispersion shown in Eq. (3). As a direct result, a recently implemented tunable pulse delay system has shown the ability to control the delay of pulses, over a broad band, by varying the carrier frequency f0 = ω0/2π [5]. Thus, a pulse is first modulated onto a carrier frequency (f0) via a mixer, then propagates through the CRLH TL acquiring a delay τg(f0), and is finally demodulated by f0 via another mixer at the CRLH’s output. The advantages of such a delay system are: continuous delay, frequency scalability, and immunity of pulses from the varying delay. Fig. 2 (c) shows the resulting τg(f0) and Fig. 3 (a) shows the prototype. A natural use of the tunable pulse delay system is in pulse-position modulation transceivers. A novel UWB CRLH delay line PPM transmitter is implemented in [6], where digital data is encoded by varying the carrier frequency of the transmitted pulses, thus controlling its delay and subsequent position. The advantages of this transmitter, besides the ones outlines above, are: no DC power consumption due to its purely passive implementation, and capability to support arbitrary M−ary PPM. Fig. 3 (b) shows the prototype. Another utilization of the pulse delay system is in antenna arrays as a time delayer for beam scanning while suppressing beam-squinting (∂θ/∂ω|θ0 = 0) [7]. Time delayers provide a phase shift proportional to frequency, eliminating the effect of beam-squinting in scanned antenna arrays if phase-shifters are employed. By utilizing an array of pulse delay systems, the RF pulses experience different delays before reaching the antennas in the same approach as outlined above, where each RF pulse is modulated onto a different f0 for beam scanning. In this manner, the array’s element- to-element delay is independent of the RF frequency, i.e. ∆τg = ∆τg(∆f0) (scanning) and ∆τg(∆fRF ) = 0 (no squinting). Fig. 3 (c) shows the prototype. 10 10 5.5 8

9 9 5.25 7.5 7 8 8 5 Real 6.5 7 7 Imag. 4.75 6 6 6 4.5 5.5 5 5 4.25 5 4.5

Frequency (GHz) Frequency Frequency (GHz) 4 4 4 Frequency (GHz) 4 3 3 Carrier frequency (GHz) 3.75 3.5 2 2 3.5 3 1 1 3.25 2.5 0 0 3 2 −π 0 +π 0 1 2 3 4 5 2.25 2.5 2.75 3 3.25 -90-60-30 0 30 60 90 βp τ (ns) 0 Z0/ZL g Main Beam θΜΒ (a) (b) (c) (d)

Figure 2: Various CRLH characteristics: (a) β(ω) from Eq. (1). (b) Characteristic impedance Z0 with ZL = 50. (c) o Group delay τg(ω). (d) Main beam leaky-wave scanning angle θ .

5 Second-Order (β2) Dispersion Applications

nd The dispersion strength in any dispersive medium is characterized by 2 order dispersion, β2, given in Eq. (3) for the CRLH TL. This leads to several interesting applications: the Talbot effect, real-time Fourier transformers, and compressive receivers. The Talbot effect generates repetitive constructive interference patterns at various locations of the CRLH TL, leading to novel pulse train generators [8]. Another application is the real-time Fourier transformer (RTFT) [9]. Due to the CRLH’s dispersion, a time domain signal is observed to be the Fourier transform of the in- put signal at a particular distance (Fraunhofer distance). A variation on the latter is the compressive receiver, which behaves as a frequency discriminator as well as an RTFT with a shorter Fraunhofer distance [10]. The compressive receiver compensates for the CRLH’s down-chirp with an equivalent but oppositely-sloped chirp, i.e., up-chirp, ef- fectively cancelling the CRLH’s dispersion in real-time, and achieving “pulse compression”. The above mentioned devices present attractive advantages which may lead to high-frequency, flexible and reconfigurable, and low-cost analog signal processors. Finally, for radiative applications, the CRLH TL can be exploited as a scanning leaky-wave antenna having a frequency- −1 dependent main radiation beam θMB = sin [cβ(ω)/ω] with β(ω) given in Eq. 2. Fig. 2 (d) shows the scanning law for the CRLH leaky-wave antenna. From this basis, a novel space-based real-time spectrum analyzer, capable of characterizing time-domain signals in the frequency domain, is implemented utilizing the CRLH TL as a leaky-wave antenna [11]. This device radiates the frequency content of any signal towards a particular spatial direction given by θMB(ω), where detectors at each location collect the intensities of the incoming waves from a specific direction. In this manner, the spectrum of the input signal is obtained without any FFT computations. Thus, a distinct advantage is gained by removing the time-frequency resolution constraint that FFT-based spectrum analyzers typically suffer from [12]. This may lead to a more accurate spectrum analyzer with increased resolution to measure and characterize complex pulses and digital signals. Fig. 3 (d) shows an illustration of this system.

6 Conclusions

Several novel and practical metamaterial devices utilizing the CRLH TL have been reviewed in this work. The CRLH’s broad-band characteristics are exploited to design devices operating in the impulse-regime for wide-band systems and applications. It is expected that these devices will be integrated in future systems. As well, design improvements in 1st order dispersion will lead to enhanced performance in time delayers and Talbot effect, while improvements in 2nd order dispersion can enhance the performance of PPM, compressive receiver, and RTFT systems. (a) (b)

(c) (d)

Figure 3: Prototype of impulse-regime CRLH TL devices. (a) Tunable pulse delay system. (b) UWB PPM transmitter. (c) Antenna array time delayer. (d) Space-based real-time spectrum analyzer

References

[1] www.plunkettresearch.com, “Wireless, Cellular, and RFID Industry Trends,” Plunkett Research, Ltd., accessed Feb. 2008. [2] S. Gupta and C. Caloz, “Dispersion and Nonlinearity Engineered Metamaterial Devices,” in Proc. Metamaterials 2007, Rome, Italy, March 2005, pp. 627–630.

[3] C. Caloz and T. Itoh, Electromagnetic metamaterials, transmission line theory and microwave applications, Wiley & IEEE Press, 2006. [4] G. P. Agarwal, Nonlinear fiber optics, Academic Press, 2005.

[5] S. Abielmona, S. Gupta, and C. Caloz, “Experimental demonstration and characterization of a tunable CRLH delay line system for impulse/continuous wave,” IEEE Microwave Wireless Compon. Lett., vol. 17, no. 12, Dec. 2007, pp. 864–866.

[6] H. V. Nguyen and C. Caloz, “CRLH delay line pulse position modulation transmitter,” IEEE Microwave Wireless Compon. Lett., submitted. [7] H. V. Nguyen, S. Abielmona, and C. Caloz, “Analog dispersive time delayer for beam-scanning phased array without beam-squinting,” in Proc. Antennas Propaga. Symp., San Diego, CA, July 2008, submitted. [8] S. Gupta and C. Caloz, “Temporal Talbot effect in left-handed metamaterial transmission lines,” in Proc. Interna- tional Symposium on Electromagnetic Theory (EMTS), Ottawa, ON, Canada, July 2007, CD-ROM. [9] T. Janson, “Real-time Fourier transformation in dispersive optical fibers,” Opt. Lett., vol. 8, pp. 232–234, 1983.

[10] S. Abielmona and C. Caloz, “Compressive receiver for frequency discrimination based on dispersion-engineered metamaterial delay line,” in Proc. CNC/USNC URSI National Radio Science Meeting, Ottawa, ON, Canada, July 2007, CD-ROM.

[11] S. Gupta, S. Abielmona, and C. Caloz, “Leaky-wave based spectrum analyzer with unrestricted time-frequency resolution,” in Proc. IEEE MTT-S Int. Microwave Symp. Dig., Atlanta, GA, June 2008. To be published. [12] L. Cohen, “Time-frequency distributions-a review,” Electron. Lett., Proc. of IEEE, vol. 77, no. 7, pp. 941–981.