Artificial Dielectrics
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FOCUSED ISSUE FEATURE Artificial Dielectrics NG I SH I NGRAM PUBL I Rajind Mendis and CENSED BY CENSED BY Daniel M. Mittleman I MAGE L MAGE I TOCK, TOCK, S C I RAPH G CENSED BY CENSED BY I IMAGE L IMAGE onsider an electromagnetic wave propa- though simply empty space, mimics the properties of gating in the empty space between two a dielectric medium with a wave velocity yp different parallel metal plates. This situation is from c, that of waves in vacuum. The waveguide there- often the first example of guided waves fore can be described as an artificial dielectric [1] with encountered by undergraduate students an effective refractive index nc= /yp different from Csince the solutions (the guided modes) are analytic and unity. In contrast to naturally occurring dielectrics easily obtained. And yet, this seemingly simple con- for which n 2 1, n can have a value less than unity. figuration can give rise to a number of interesting and These waveguide-based artificial dielectrics were first even counterintuitive phenomena. For the set of modes introduced in a small body of work by the microwave with the electric field pointing parallel to the surfaces community about half a century ago [2]–[5]. As dis- of the two metal plates, the region between the plates, cussed here [6]–[10], the wavelength scaling that results Rajind Mendis ([email protected]) and Daniel M. Mittleman ([email protected]) are with the Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, United States. Digital Object Identifier 10.1109/MMM.2014.2355696 Date of publication: 12 November 2014 34 1527-3342/14©2014IEEE November/December 2014 when moving from the microwave region to the tera- To extend the functionality to the hertz (THz) region results in more practical structural dimensions. This gives new life to these artificial di- third dimension (and be able to electrics for a variety of novel and exotic applications in handle large cross-sectional beam THz science and technology. sizes), one can construct a stacked set Figure 1 illustrates one implementation (a lens- antenna) of some of the historical work [2]. These early of thin metal plates (as in Figure 1). devices consisted of a stack of metal plates, with each plate shaped into a unique geometry. They behaved as an array of parallel-plate waveguides (PPWGs) oper- tially, resulting in an inhomogeneous medium. In fact, ating in tandem. In very general terms, these devices using the latter approach, one may create a desired function (or affect refraction) very similarly to a natu- refractive index profile (in two dimensions) simply rally occurring dielectric medium, i.e., by imparting a by choosing an appropriate plate separation. This change in the phase velocity of electromagnetic waves. approach is effective as long as the change in the plate Since the overall device consists of a stacked array of waveguides, we can understand the principle of opera- tion by looking at a single PPWG: a unit cell. When the electric field of the input wave is polarized parallel to the plates, only the transverse-electric (TE) modes are excited in a PPWG. Furthermore, it is possible to excite only (or primarily) the lowest-order TE mode (the TE1 mode) by matching the input beam size to the plate separation b. As shown in Figure 2(a), the phase velocity yp of the TE1 mode has a characteristic frequency dependence; it increases from the free-space value c to infinity as the frequency approaches the cutoff frequency fc . Using the well-known expression for yp [11], we can derive an effective refractive index n, given by 2 nc==//ypc1 - ff, (1) ^h where fcc = /(2b). This function is plotted in Fig- ure 2(b) for two different values of the plate separa- (a) tion, showing that n varies from a value close to unity at high frequencies to zero as the frequency reaches cutoff. Therefore, a wave propagating in the TE1 mode inside the PPWG experiences a medium with a refrac- tive index n varying between zero and unity. Using only a single PPWG, i.e., a single unit cell, would limit the functionality of these components to only two dimensions. To extend the functionality to the third dimension (and be able to handle large cross- sectional beam sizes), one can construct a stacked set of thin metal plates (as in Figure 1). However, even in this case, one does not achieve true three-dimensional (3-D) behavior since there cannot be any propagation in the direction perpendicular to the plates. Equation (1) also reveals that the effective refrac- tive index n depends on both the operating fre- (b) quency and the plate separation. This implies that n can be tuned between zero and unity, either by vary- Figure 1. Metal lens antennas using waveguide-based ing the frequency for a given plate separation or by artificial dielectrics. (a) A plano-concave geometry that varying the plate separation for a given frequency. operates as a convergent lens. The aperture diameter is 14 In the former case, at a given operating frequency, n wavelengths, and the focal length is 23 wavelengths. (b) A would be uniform throughout, resulting in a homoge- repeater antenna operating at 4 GHz. Note the structural neous medium. In the latter case, n would vary spa- size of the device compared to a human. (From [2].) November/December 2014 35 separation is adiabatic with respect to the wavelength mode, the plates would no longer be precisely parallel so that the single-mode nature of the input wave is to each other. preserved. In this case, even though the propagation In the following sections, we describe some exam- would be governed by the characteristics of the TE1 ples of the phenomena that can result from the use of these artificial dielectric structures and a few functional components that may be valuable in THz applications. 5 4 Refraction To demonstrate the refractive behavior of this artificial 3 dielectric medium, one can perform an experiment Vp using a 45° PPWG prism, fabricated using triangu- 2 Velocity (c) lar aluminum plates [6]. A schematic and the device Cutoff 1 geometry are shown in Figure 3(a) and (b), respec- Vg tively. As shown, a THz beam is coupled into the prism 0 0.0 0.2 0.4 0.6 0.8 1.0 at normal incidence with the electric field polarized parallel to the plate surfaces. The beam consists of a Frequency (THz) train of broadband THz pulses with a spectral content (a) that varies from about 0.05 THz to about 1 THz. The 1.0 beam propagates via the TE1 mode through the prism 0.8 and encounters the exit face at an angle of 45°. At this b = 1 mm face, if the artificial dielectric behaves as predicted, 0.6 the wave should experience a sudden change in index n Theory and, in keeping with Snell’s law, should change its 0.4 Experiment propagation direction. In fact, the output beam should 0.2 b = 2 mm bend toward the normal to the face, since it is traveling from a medium of low index ()n11 to free space. This 0.0 0.0 0.2 0.4 0.6 0.8 refractive effect is the opposite of that observed in a Frequency (THz) conventional dielectric, where the wave would travel from a high-index ()n21 medium to free space, bend- (b) ing away from the normal. A THz receiver is scanned along an arc to detect Figure 2. (a) The phase and group velocity of the TE1 mode with respect to the free-space value c, for a PPWG with the output signals at various angular positions given b1= mm. The PPWG shown in the inset can operate only by i [Figure 3(a)]. The electric field amplitude spec- in two dimensions as indicated by the two arrows. (b) The tra of the detected signals are shown in Figure 3(c) for effective refractive index for b1= mm and 2 mm, where b = 1 mm (corresponding to fc = 01. 5 THz). As i in- f0c = .15 THz and 0.075 THz, respectively. (From [6].) creases, the spectral content shifts to lower frequencies, 1.0 i = 3° THz 0.8 i = 10° 45° Rx i = 20° THz i i = 30° E 0.6 Beam i = 40° Amplitude 0.4 (a) eld Fi 0.2 b 0.0 0.1 0.2 0.3 0.4 0.5 Frequency (THz) (b) (c) Figure 3. (a) A schematic of the PPWG prism experiment. The width of the input face is 38.1 mm. (b) A 3-D rendition of the PPWG prism indicating the input beam. (c) Amplitude spectra of the detected signals at various receiver angular positions. Here, b1= mm and f0c = .15 THz. (From [6].) 36 November/December 2014 reaching frequencies near fc at 45°. This demonstrates To demonstrate the refractive the frequency-dependent beam refraction at the an- gled output facet. Using these spectra, one can derive behavior of this artificial dielectric the values of the refractive indices experienced by the medium, one can perform an beam inside the waveguide as a function of frequency. experiment using a 45° PPWG These values can be compared with the theoretical values derived from (1). This comparison, displayed in prism, fabricated using triangular Figure 2(b), shows very good agreement and demon- aluminum plates. strates that the space between the plates (which is just air) indeed acts as if it has a dispersive refractive index less than unity. curves given in Figure 2(b), for a plate separation of 1 mm, this should occur at a frequency of approxi- Total Internal Reflection mately 0.18 THz.