IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 15, NO. 2, FEBRUARY 2005
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Growing Evanescent Waves in Continuous Transmission-Line Grid Media Trevor Andrade, Anthony Grbic, Member, IEEE, and George V. Eleftheriades, Senior Member, IEEE
Abstract—We demonstrate the growth of evanescent waves in a backward-wave medium which consists of a printed continuous transmission-line (TL) grid without any embedded lumped elements (chip or printed) or vias. This proposed medium is referred to as a backward-wave transmission-line (BWTL) grid. When such a BWTL medium is sandwiched between two complementary forward-wave transmission-line (FWTL) grids, growing evanescent waves are supported in the BWTL medium. Analysis and microwave simulations verify the growth of evanescent waves including in the realistic case of lossy TLs. The proposed BWTL medium is easy to construct and inherently scalable with frequency.
Fig. 1. Transmission-line (TL) unit cells. (a) Unit cell of the FWTL medium. (b) Unit cell of the BWTL medium.
Index Terms—Backward waves, evanescent waves, left-handed metamaterials, negative refraction, periodic sturctures, plasmons.
I. INTRODUCTION
T
HE feasibility of media with negative permeability and permittivity, and hence a negative refractive index (NRI), was proposed by Veselago [1] in the 1960’s. More recently Pendry [2] proposed that a slab of such a NRI medium could be used as a “perfect” lens because it could both phase correct propagating waves and “amplify” evanescent waves, thus undoing the effect of propagation in free space. Shelby et al. have constructed a bulk NRI medium with thin wire strips and split-ring resonators which has successfully demonstrated negative refraction [3]. NRI media have also been realized using a periodic two-dimensional (2-D) array of transmission-lines (TL) loaded in a dual (high-pass) configuration with inductors and capacitors. These dual TL structures have been used to experimentally demonstrate focussing [4]. Analysis [5], simulation [6], and experiments have confirmed that a dual TL lens is capable of sub-wavelength focussing by means of growing evanescent waves [7]. Other useful microwave devices based on dual TLs are described in [8]. In this letter, we propose a simple medium, capable of supporting negative refraction and growing evanescent waves, consisting of a continuous 2-D grid of transmission-lines without any embedded elements (chip or printed) or vias as depicted in Fig. 1. The dimensions of each unit cell are on the order of a wavelength, thus it cannot be considered a homogeneous medium. Hence, it is not possible to define an effective permeability and permittivity. However, we will demonstrate through
Manuscript received June 1, 2004; revised October 26, 2004. This work was supported by the Ontrario Government’s Centre of Excellence on Information Technology (CITO). The authors are with The Edward S. Rogers, Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5G 1G6, Canada (e-mail:
[email protected]). Digital Object Identifier 10.1109/LMWC.2004.842864
analysis and simulation that this structure is capable of supporting growing evanescent waves. This grid comes in two flavors: a backward-wave transmission-line (BWTL) grid and a forward-wave transmission-line (FWTL) grid. The unit cell of the FWTL grid is depicted in Fig. 1(a) and is just a square cross . of connected transmission-lines with a node spacing of The BWTL grid is the same except its unit cells have a dimen[see Fig. 1(b)]. In the rest of this letter, we will only sion of consider complementary BWTL and FWTL grids that have the pair of dimensions stated above. II. ANALYSIS OF FWTL AND BWTL MEDIA The dispersion relations and Bloch impedances for a general grid of loaded 2-D transmission-lines have been derived in [9]. This analysis is directly applicable to the proposed TL grids. Accordingly the dispersion relation, -directed Bloch impedance and are and relation between the voltages to ground, (1)
(2) (3) where and are the Bloch wavenumbers and is the intrinsic propagation constant on the transmission-lines. These equations apply to both the FWTL and BWTL grids with the and , respectively. Note that substitutions the factor of 2 in the right-hand-side of (1) is necessary to properly account for the scattering at the cell boundaries in these 2-D grids. This complies with earlier work on the dispersion of dense periodic metallic grids over ground (e.g., see equation (7) in [10]).
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IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 15, NO. 2, FEBRUARY 2005
Fig. 2. Brillouin diagram for proposed TL grid.
Fig. 3. BWTL grid sandwiched between two semi-infinite FWTL grids with infinitely long interfaces in the y direction.
The easiest way to see that the BWTL grid is indeed a backward-wave medium is to examine Fig. 2 which plots the dislies between persion relation (1). It can be seen that when and the gradient of the dispersion curve, which indicates the direction of power flow, points along the opposite direction , must be negative in of the k vector. Thus when . This condition the BWTL medium whenever is less than implies that the BWTL medium must operate in the second frequency band shown in Fig. 2. Hence, at the interface between a FWTL and a BWTL medium, negative refraction effects will take place. Notomi has also shown that higher frequency bands can be exploited to synthesize NRI photonic crystals [11]. Now consider such a BWTL medium sandwiched between two semi-infinite FWTL media as shown in Fig. 3. An evanescent wave could be launched using an infinite array of voltage sources with a progressive phase to excite the leftmost edge of the FWTL medium (boundary “a” in Fig. 3) [6]. To excite the evanescent wave we set the progressive phase per unit cell of the , to be greater than the magnitude of the involtage sources, trinsic Bloch wavenumber of the FWTL medium k, defined by (4) an evanescent wave, is launched in the longiudWhen inal direction since the only way to satisfy (1) is to have an imaginary value for . This situation of a BWTL grid embedded in between two FWTL grids is similar to that found in [5] (also see Erratum [12]) except that in this case the proposed BWTL grid is used instead of a dual TL structure. The analysis of [5], which demonstrates analytically the growth of evanescent waves in a NRI and material, requires that for evanescent waves
, where is the complex conjugate of the Bloch impedance in the backward-wave medium; the BW and FW superscripts indicate quantities in the forward and backward wave media, respectively. According to the analysis of [5], in order to satisfy the radiation condition, the (imaginary) longiand for evanescent waves must tudinal wavevectors have the same sign (negative) in the FWTL and BWTL media. Moreover, these are inhomogeneous waves in which the transverse wavenumbers are real and equal due to phase matching . Hence, in this at the interfaces of Fig. 3, i.e., case, the longitudinal wavenumbers are also equal in order to (note that satisfy the dispersion equation (1), i.e., given the complementary dimensions of Fig. 1, the right-hand side of (1) is identical for the two media). This latter condition and the fact that the dimensions of the unit cells in the BWTL and FWTL media are complementary (see Fig. 1) when applied to (2) imply that the longitudinal Bloch impedances of the BWTL and FWTL media are imaginary, equal in magnitude and have opposite signs. Specifically the FWTL medium is inductive whereas the BWTL medium is capacitive and conjugately 0. matched for evanescent waves, i.e., The above permits a simple explanation for the growth of evanescent waves. Interface 2 in Fig. 3 corresponds to an inwhereas interface 1 is finite reflection coefficient matched. Therefore, an evanescent wave injected from the leftmost FWTL grid would attenuate exponentially in that medium (since it does not experience any reflection at interface 1) as well as in the FWTL medium right of interface 2 (assumed infinite or matched). On the other hand, in the BWTL region it is the exponentially growing solution to the wave equation that dominates, due to the infinite reflection coefficient at interface 2. Indeed at the second interface (5) Furthermore, the reflection coefficient just on the right of interface 1 is also infinite since where is the width of the BWTL grid. Hence, the input impedance just on the left of interface 1 is indeed matched (6) Under these circumstances, the voltage at interface 2 grows exponentially with respect to the voltage at interface 1 (7) Note that the voltages at both interfaces remain finite. Moreover, (7) remains valid for every transverse wavenumber, which is essential in the “perfect lens” concept of [2]. III. RESULTS: GROWTH OF EVANESCENT WAVES To verify the growth of evanescent waves in the BWTL medium we used Agilent’s Advanced Design System (ADS). In the ADS simulation, the array of voltage sources in the FWTL medium was placed three unit cells before interface 1 in Fig. 3. The voltage generators and boundaries were terminated in the appropriate Bloch impedances. The width of the
ANDRADE et al.: GROWING EVANESCENT WAVES IN CONTINUOUS TRANSMISSION-LINE GRID MEDIA
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TABLE I ELECTRICAL PARAMETERS FOR THE FWTL AND BWTL MEDIA
TABLE II ELECTRICAL PARAMETERS FOR THE FWTL AND BWTL MEDIA
Fig. 5. Simulated voltage magnitudes to ground (in the y direction) at 1 GHz; lossy case.
transmission-line (TL) grid without any embedded lumped elements (chip or printed) or vias. The proposed BWTL medium is scalable with frequency and easy to construct. Complementary anisotropic grids have been experimentally demonstrated in [13]. REFERENCES Fig. 4. Simulated voltage magnitudes to ground (in the y direction) at 1 GHz; lossless case.
BWTL medium in the direction was four unit cells. In order to compare results with [6], the same values were used for attenuation per unit cell, characteristic impedance and intrinsic wavenumber. Using these values, it is possible to calculate (which is analogous to ), , and the Bloch impedances using (4), (1), and (2), respectively. All the relevant quantities are shown in Tables I and II. These parameters are essentially the same as those found in [6] and they result in the same values for the Bloch impedances and wavenumbers. Fig. 4 shows the voltage magnitudes for a finite structure 10 cells wide and 14 cells long. These results are indeed identical to Fig. 4 in [6]. It was necessary to place voltage generators along boundaries a and b, as was done in [6], in order to simulate an infinite array of voltage sources. The same edge effects observed in [6] can be seen in Fig. 4. Fig. 5 shows the effect of simulating the transmission-lines with the losses of a practical microstrip implementation. In particular, the lines were simulated with the conductivity of copper 5.8 10 Siemens/m. The losses result in a significant distortion of the evanescent waves. However, as Fig. 5 shows, it is still possible to observe the growth of evanescent waves in the BWTL medium. IV. CONCLUSION We demonstrate the growth of evanescent waves in a backward-wave medium which consists of a printed continuous
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