Practical limitations of subwavelength resolution using ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 10, OCTOBER 2005

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Practical Limitations of Subwavelength Resolution Using Negative-Refractive-Index Transmission-Line Lenses Anthony Grbic, Student Member, IEEE, and George V. Eleftheriades, Senior Member, IEEE

Abstract—Previously, we have demonstrated both analytically and experimentally subwavelength imaging using a negative-reloaded transmisfractive-index lens made of a periodically , sion line (TL) network. This loaded transmission line network has been referred to as the dual TL lens. Here, we consider the limitations on subwavelength imaging imposed by impedance mismatches and the component losses of a practical dual TL lens. Simple expressions for estimating the resolving capability of a dual TL lens are given. It is found that the resolution enhancement of the dual lens is proportional to the quality factor of the series loading capacitors divided by the electrical thickness of the lens. The effective material parameters of the dual TL lens are also derived so that these expressions can be directly related to those of previous studies considering uniform and isotropic left-handed lenses. Finally, the resolving capability of an experimental lens that achieves subwavelength imaging is theoretically predicted. These theoretical predictions are then directly compared to previously reported experimental results. Index Terms—Diffraction limit, left-handed, metamaterials, negative refractive index, plasmons, superresolution, transmission lines (TLs).

I. INTRODUCTION

M

ATERIALS possessing negative material parameters ( , ) were first investigated by Veselago in the 1960s and shown to exhibit a negative refractive index (NRI) [1]. In his paper, Veselago theorized that a flat slab of left-handed material acts as a unique lens that focuses propagating waves (rays) from one side of the slab to the other. Recently, Pendry predicted that, in addition to focusing propagating waves, the left-handed slab also restores the amplitude of evanescent waves at the focal plane, thus allowing perfect imaging [2]. Such a lens could theoretically overcome the diffraction limit inherent to conventional lenses. It should be pointed out that this type of lens is a peculiar one since it does not have an optical axis and there is no associated magnification. The first left-handed medium was implemented using periodic arrays of wires to achieve negative permittivity and split-ring resonators to achieve negative permeability [3]. This composite structure was used to experimentally demonstrate negative refraction and paved the way for further experimental work. Others confirmed these initial experiments and demonstrated focusing using the wire/split-ring resonator medium Manuscript received April 23, 2004; revised April 1, 2005. The authors are with The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2005.856316

Fig. 1. mesh.

TL unit cells (including losses). Unit cell of (a) dual TL and (b) TL

[4], [5]. An alternative approach to synthesizing left-handed media was proposed in [6] by loading a two-dimensional (2-D) network of transmission lines with series capacitors and shunt inductors as shown in Fig. 1(a). This loaded transmission line (TL) network has been referred to as the dual TL in previous work. One-dimensional dual TLs, which have been known for some time [7], were revived in [8]–[10] and led to interesting left-handed structures and circuits. Experimental focusing by a left-handed medium was demonstrated for the first time using the 2-D dual TL in [6] and [11]. These initial experiments focused electromagnetic waves within the left-handed medium (dual TL) from a conventional dielectric (parallel-plate waveguide). Further work led to the development of dual TL lenses that focused electromagnetic waves from one dielectric to another. These lenses will be referred to as the three-region lens arrangement, where the three regions encompass the lens and a dielectric on either side of the lens. The restoration of evanescent waves and imaging beyond the diffraction limit were shown analytically and through simulation using a three-region lens arrangement in [12]–[14]. More recently, subwavelength imaging with three times the resolution of a conventional lens has been observed experimentally [15]. Diffraction-limited imaging with mushroom-like dual TL lenses resembling the structures in [16] and [17] has also been reported [18]. The sensitivity of subwavelength imaging to absorption and deviations in electric permittivity and magnetic permeability in uniform isotropic left-handed lenses was examined extensively in the past [2], [19]–[24]. In this paper, the periodic dual TL lens is considered and the limitations on image resolution imposed by impedance mismatches and losses are explored. The analytical formulation of imaging using a dual TL lens presented in [13] is expanded to include losses in order to study these effects. Simple expressions are derived for estimating the resolving capability of a dual TL lens and compared to those of previous

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studies considering uniform and isotropic left-handed lenses. The effective material parameters of the dual TL lens are found so that these comparisons can be made. Finally, the performance of the lens reported in [15], which experimentally achieves subwavelength focusing, is predicted and the theoretical predictions are directly compared to experimental results. II. THE LOSSY DUAL TL LENS A lossless version of the periodic two-dimensional dual TL depicted in Fig. 1(a) was shown to act as an isotropic left-handed material which possesses a negative index of refraction within a wide frequency range [6], [11]–[14], [17], [25]. Its unloaded counterpart, the TL mesh depicted in Fig. 1(b), was also shown to behave as an isotropic “right handed” medium that has a positive index of refraction. Imaging of a monochromatic current source by a lossy dual TL lens, in the three-region lens arrangement, is shown in Fig. 2 [13]. The current source is imaged from one TL mesh acting as a parallel-plate waveguide to the other. The . The dual current source is located at the origin ( in Fig. 2) to ( is 7.5 TL lens extends from in Fig. 2) and the focal plane on the far side of the lens is located at ( in Fig. 2). In order to characterize the lossy three-region lens arrangement, the propagation characteristics in both the lossy TL mesh and dual TL first need to be examined.

Fig. 2. Imaging using a dual TL lens.

A. Propagation Characteristics of the Lossy Dual TL and TL Mesh The dispersion relations and Bloch impedances for the lossless TL mesh and dual TL were derived in [25]. In this section, these dispersion relations and Bloch impedances are extended to include losses so that the lossy dual TL lens can be characterized in the following sections. A conductance 2 is added in parallel with the series capacitor 2 and a resistance is added in series with the shunt inductance in the dual TL unit cell shown in Fig. 1(a). These dissipative elements account for the losses inherent to practical capacitors and inductors and are for the incommonly expressed in terms of quality factors for the capacitor ductor and (1)

(4) and are the wavenumwhere is the angular frequency, and are bers in the and directions in the TL mesh, the wavenumbers in the dual TL, is the unit cell dimension, is the propagation constant of the interconnecting TL sections, is their characteristic impedance. and Similarly, the Bloch impedance expressions for the lossy TL mesh and dual TL can also be found. The Bloch impedances and represent the and directed Bloch impedances and are those in the dual TL in the TL mesh, while

Accounting for the lossy components simply means replacing 2 with 2 and with in the dispersion relation of the lossless dual TL in [25]. In addition, the loss in the interconnecting TLs can be accounted for by a complex . This loss can also be propagation constant expressed in terms of a TL quality factor

(5)

(2)

(6)

The dispersion relations for the lossy TL mesh and dual TL are therefore given by (3) and (4), respectively

Now that the lossy Bloch impedances and dispersion relations have been defined, the conditions for “perfect imaging” using the three-region lens depicted in Fig. 2 can be expressed in terms of the , components and TL parameters. In order to achieve “perfect imaging,” the TL mesh and dual TL must be lossless ) and impedance matched, and their ( relative refractive index must be 1 with respect to each other

(3)

GRBIC AND ELEFTHERIADES: PRACTICAL LIMITATIONS OF SUBWAVELENGTH RESOLUTION

[2], [13]. The requirement of a relative refractive index equal to 1 can be satisfied by equating the right-hand sides of (3) and (4). The third requirement of an impedance match between the TL mesh regions and dual TL lens can be satisfied by setting the numerator in (5) equal to the negative of the numerator in (6). With some algebraic manipulation, these two requirements of perfect focusing reduce to the following two simple equations: (7)

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can be related to the effective The isotropic impedances , of the dual TL and TL mesh, respecwave impedances , tively, by a geometrical factor (11) for planar geometries that have unit cells with where . As a result, the effective material padimensions rameters of the dual TL ( , ) and TL mesh ( , ) are

(8) It is worth mentioning that (7) also corresponds to the condition required to close the stopband that may exist between the “left-handed” propagation band and the next higher “righthanded” band in the dual TL, as was noted in [11]. Closing this stopband allows for a continuous transition between backward-wave propagation and forward-wave propagation as the frequency is increased. Moreover, it should be pointed out that a dispersion analysis for a one-dimensional ideal backward-wave line (i.e., without any interconnecting transmission lines) having the lossy elements arranged as shown in Fig. 1(a) has been carried out in [26]. B. Effective Medium Theory for the Lossy Dual TL and TL Mesh In this section, the effective material parameters ( , ) of the lossy dual TL and TL mesh are derived so that comparisons can be made between the three-region lens arrangement shown in Fig. 2 and the three-region isotropic uniform left-handed lens assumed in most theoretical studies [2], [19]–[24]. This derivation allows the dual TL and wire/split-ring resonator medium to be directly compared and may shed light on optimizing the dual TL medium as well as improving the wire/split-ring resonator medium which has shown to be lossy [5]. In order to derive the effective material parameters, some unit cell dimensions must be assumed. The unit cell dimensions are assumed to be in the and directions with a thickness of in the direction. For microstrip implementations of the dual TL and TL mesh, represents the height of the substrate on which the transmission lines are printed. In addition, the width of the transmission lines is assumed to be much smaller than the unit cell dimension . Effective material parameters can be derived for frequencies of isotopic and homogeneous propagation. Such conditions exist when the right-hand sides of (3) and (4) are much less than are much smaller one, and the unit cell dimensions and than the wavelength of operation. At frequencies of isotropic propagation, an isotropic wavenumber and impedance can be and defined for the TL mesh and dual TL. The variables will denote the isotropic wavenumbers in the dual TL and and will denote the TL mesh, respectively. Similarly, isotropic impedances. These values can be found by simply considering propagation along one of the principal ( or ) ) is used axes. Here, propagation along the axis ( (9) (10)

(12)

(13) The dual TL can therefore be approximated as a parallel-plate waveguide of thickness filled with a dielectric with material and . Likewise, the TL mesh can be approxiparameters mated as a parallel-plate waveguide filled with a dielectric with and . material parameters III. RESOLVING CAPABILITY OF THE LOSSY DUAL TL LENS that According to Fourier optics, the minimum feature can be resolved by a lens is related to the maximum transverse contributing to the image, by the wavenumber . When imaging Fourier transform relationship using conventional lenses, evanescent waves are lost due to the attenuation they experience from the source/object to the focal is bounded by the wavenumber in the plane. Therefore, remains on the order of a wavesurrounding medium and . Lenses with numerical apertures less length: than one are further limited by the fact that not all the propaare captured by the lens. A left-handed gating wavenumbers lens, however, supports growing evanescent waves which restore some of the object’s evanescent spectrum at the focal plane and the minimum resolvable [2]. This results in becomes smaller than a wavelength feature , which has been referred [27]. Hence, the ratio to as the resolution enhancement, can be used as a figure of merit for lenses [24], [28]. A conventional lens cannot surpass a resand a perfect lens has a , olution enhancement for a practical/imperfect left-handed lens lies somewhile . where in the range Having derived effective permittivities and permeabilities in the previous section, the approximate resolution enhancement of a dual TL lens can be computed using expressions derived in [24] for uniform left-handed lenses ( -polarized/perpendicularly polarized waves). In [24], an approximate expression is derived for small mismatches in permeability between the lens and the surrounding medium (14)

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where , is the thickness of the left-handed slab and is the wavelength of radiation. Alternatively, the optical transfer function (OTF) of the dual TL lens can be derived and the resolution enhancement found in a more direct and accurate manner. The OTF is the transmission coefficient from the source plane to the focal plane. It describes the spatial frequency response of the lens. Before proceeding, a few terms need to be defined. The spatial spectrum of the node voltages along the external focal plane will be referred to as the Bloch voltage spectrum of the image. Similarly, the spatial spectrum of the node voltwill be referred to ages incident along the source plane as the Bloch voltage spectrum of the source. The OTF of the dual TL lens is simply the ratio of the Bloch voltage spectrum of the image to the Bloch voltage spectrum of the source (15) where and an time-harmonic progression is asand are the Fresnel reflection cosumed. The variables efficients at the first and second interfaces, respectively, while and are the corresponding Fresnel transmission coefficients

has an inductive reactance and a capacitive reactance for evanescent waves [see (5) and (6)] [29]. In summary, the following conditions are satisfied for lossless “perfect imaging”: for propagating Bloch waves (19) for evanescent Bloch waves (20) For propagating waves, the conditions above imply that and so that the overall transmission coefficient . Therefore, all the propagating Fourier comporemains nents of the source are perfectly transmitted to the external focal plane of the lens. In the case of evanescent waves, the Fresnel ) but the overall transcoefficients diverge ( . As a result, all Fourier commission coefficient remains ponents are perfectly transmitted to the focal plane. It is interesting to note that, for evanescent waves, the -directed Bloch impedances are complex conjugates of each other ) under the conditions of “perfect imaging” ( [29]. This fact highlights that the growth of evanescent waves is in fact a resonant phenomenon. In addition, the condition is equivalent to the condition for the existence of a surface plasmon ( -polarized/perpendicularly polarized) at the interface between uniform NRI and PRI semi-infinite half-spaces [30] (21)

(16) The node voltages at the focal plane ( ) of the lens depicted in Fig. 2 are then given by the inverse discrete-space Fourier transform of the OTF multiplied by the Bloch voltage spectrum of the source (17) where is the Bloch voltage spectrum of the current source excitation given by the following expression [13]: (18) Note that the source spectrum is slightly different from that 2) since the source in this paper is located at the in [13] ( central node of a unit cell rather than at the unit cell terminals. The signs ( )of the Bloch impedances and wavenumbers in the OTF are determined by the radiation condition. The radiation condition stipulates that propagating waves carry power toward infinity and evanescent waves decay in amplitude toward infinity. For propagating waves, it is therefore assumed that the to Bloch impedances are positive quantities. This restricts to have a positive real part [achave a negative real part and cording to (5) and (6)] for frequencies of isotropic propagation. In other words, the TL mesh supports forward-wave propagation and the dual TL backward-wave propagation. Evanescent and are required waves are assumed to decay, therefore to have negative imaginary parts: , . As a result,

This can be shown by substituting the definitions of , for the TL mesh and dual TL into (21). The evanescent waves decaying from the source couple to the surface plasmon resonances at the interfaces of the dual TL lens. This interaction between the surface plasmons and the decaying incident wave is in fact what produces the growing evanescent wave within the lens [20], [31]. Under the conditions of “perfect imaging,” the resolution enhancement of the lossless dual TL lens is limited only by the periodicity of the lens. The periodicity restricts the maximum resulting in a restransverse wavenumber to [13]. In practical dual olution enhancement of TL lenses, however, impedance mismatches and losses further limit the resolution enhancement. The effect of these two resolution-limiting mechanisms can be studied by plotting vresus for variations in TL parameters ( , ), quality factors , ), and , component values. In general, losses ( , and impedance mismatches remove the higher spatial frequen(evanescent components) from the image that capture cies the subwavelength features of the source/object. As was shown in [20] and [24], the resolution enhancement of an imperfect lens can be estimated by examining the denominator of the OTF (15). Let us first consider a lossless impedance mismatch between the dual TL lens and TL mesh on either side. A lossless impedance mismatch refers to the situation where the Fresnel reflection coefficient is a real number. This type of mismatch term introduces a pole in the OTF (15) of the lens, since the in the denominator of the OTF is a positive real number. For small impedance mismatches (good dual TL lens designs),


this pole occurs for an evanescent component having a large and therefore . transverse wavenumber Setting the denominator of (15) equal to zero yields

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OTF (15). Nevertheless, it still attenuates the higher transverse wavenumbers. Equating the two terms in the denominator of the OTF (15) provides a conservative estimate of the transverse at which exponentially drops in magnitude wavenumber

(22)

(27)

where is the thickness of the dual TL lens is related to by dispersion (4). This and pole disproportionately augments the evanescent components in its vicinity and as a result distorts the image. Additionally, increases beyond the vicinity of the pole, the first term as in the denominator of the OTF (15) begins to dominate and exponentially drops in magnitude. Therefore, the transverse wavenumber identifying the pole can be used as a conservative , the maximum transverse wavenumber conestimate of tributing to the image. Such pole resonances have been associated with the excitation of polaritons of a uniform and isotropic left-handed slab [20], [32]. Equation (22) is simply the transverse resonance condition for guided modes that have an evanescent profile (polaritons), , that is, modes with imaginary -directed wavenumbers ( ) both inside and outside the dual TL lens. Therefore, the pole represents the excitation of a polariton guided by the dual TL lens. The surface plasmons at the two interfaces of the NRI ] couple and give rise to these TL lens [given by polaritons. The incident evanescent Bloch wave with transverse pumps a guided polariton and eventuwavenumber ally drives the fields of the mode to infinity under lossless conditions, effectively drowning out the source and its image. By taking the square roots of (22), the dispersion relation for the symmetric and asymmetric polaritons of the dual TL lens can be derived

This transverse wavenumber is approximately the 6 dB point in the OTF’s low-pass response when is plotted versus . As before, it can be used as an estimate for . There, whether the fore, a general formula for estimating is limited by loss or by a lossless resolution enhancement impedance mismatch, is

(23)

Equation (30) shows that the approximation is in fact a magnetostatic approximation [20]. It neglects the smaller effect of mismatches caused by the shunt inductor , which represents the permittivity of the dual TL lens. The larger effect of mismatches due to ,which represents the permeability of the dual TL lens, is only considered. Equations (28)–(30) apply to both lossless impedance mismatches and mismatches due to loss. For the latter, the TL mesh and dual TL lens are mismatched only due to component losses , ), such that (7) and (8) are still satisfied. Substi( , tuting (8) (2 ) into (30) reduces to the following simple expression:

(24) By expressing and in the above dispersion relations and , the dispersion relations become the in terms of same as those for polaritons of uniform left handed slab for perpendicular/ -polarization [20], [32] (25)

(28)

(

Assuming (28) occurs for a large transverse wavenumber ) such that , the resolution enhancement of the dual TL lens simplifies to [24]

(29) The above equation is equivalent to (14) given the expressions and and the fact that . Under isotropic and for homogeneous conditions ( , , ), the resolution enhancement further reduces to

(30)

(26) (31) As the electrical thickness of the lens is increased, the polariton cutoff wavenumber (the pole in the OTF) moves to a lower atten) corresponding to a lower . This reuation constant ( with increasing sults in a decreasing resolution enhancement lens thickness, given a fixed lossless . The effect of losses on the OTF is now considered in detail. In is an imaginary number resulting from the added this case, is negative. Therelosses in the dual TL. This implies that fore, this mismatch due to loss does not introduce a pole in the

In the magnetostatic approximation, the expression can be thought of as the resolution quality factor of the dual TL lens. Typically, the capacitor quality is smaller than 2 . Therefore (31) can be further factor approximated (32)

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This approximate expression indicates that the resolution enof a lossy dual TL lens is equal to the logarithm of hancement divided by the electrical thickthe capacitor quality factor ness (in radians) of the lens. The logarithmic dependence on the quality factor places severe constraints on losses. For example, of a dual TL in order to double the resolution enhancement lens with a given electrical thickness, one must square the overall resolution quality factor of the lens. In addition, the inverse relaindicates that electrically thin lenses perform tionship with better than thicker ones of the same composite lossy material. At a certain lens thickness, the losses inherent to a dual TL lens overcome the growth the evanescent wave experiences and its contribution to the image is lost. These losses prevent the amplitude of an evanescent wave (with finite transverse wavenumber ) from diverging as the lens thickness approaches infinity. In terms of the uniform left-handed lens, represents the imaginary part of the lens’ permeability [see (12)], which gives rise to an imaginary Fresnel reflection coefficient that limits reso, which represents the imaginary part of lution. The effect of the lens’ permittivity, on image resolution is much smaller than and therefore is neglected. The same that associated with on deviations in perdependence of resolution enhancement meability has been observed for the uniform left-handed lens for -polarized/perpendicularly polarized waves in [20] and [24].

IV. CHARACTERIZATION OF AN EXPERIMENTAL DUAL TL LENS In this section, the experimental microstrip dual TL lens presented in [15] is characterized in terms of TL circuit parameters as well as effective material parameters ( , , , ) so that its resolution enhancement can be predicted. A photograph of the experimental dual TL lens is shown in Fig. 3. As in the schematic of Fig. 2, the experimental setup consists of a lens made of a dual TL lens sandwiched between two TL meshes. The location of the source, image, and interfaces of the experiand ). The mental lens are as shown in Fig. 2 ( experimental dual TL lens in Fig. 3 extends five cells in the direction and 19 cells in the direction. The two TL meshes on either side of the lens extend 12 cells in the direction and 19 cells in the direction. Therefore, the overall experimental setup depicted in Fig. 3 is 29 12 cells. Both the TL mesh and dual TL unit cells were simulated using Ansoft’s High Frequency Structure Simulator (HFSS). In pF and the HFSS simulations, ideal capacitances of pF were used to model the chip capacitors and an ideal inductance of 20.2125 nH to model the chip inductors. These ideal capacitance and inductance values were extracted at 1 GHz from the -parameter files provided by the respective manufacturers. Using Ansoft HFSS, the Bloch impedances and propagation constants of the TL mesh and dual TL lens were matched at the design frequency of 1.00 GHz: and rad. Since the same TL parameters ( , ) were used in the TL mesh and dual TL, the parame, , could be extracted from , , , by ters , employing (3)–(6). These extracted values are listed in Table I. They take into account any mutual coupling interactions within and between the unit cells as well as parasitics introduced by the

Fig. 3. Experimental dual TL lens.

TABLE I EXTRACTED TL AND CIRCUIT PARAMETERS FOR THE DUAL TL LENS

TABLE II ISOTROPIC WAVENUMBERS AND IMPEDANCES FOR THE DUAL TL LENS AT 1 GHz

junctions in the microstrip TLs, since the cells were simulated with periodic boundary conditions in HFSS. The manufacturer’s quality factors were then used to account for the losses in the , components used. The quality factors quoted by the manufacturers are and at 1.0 GHz for the capacitors and inductors, respectively. In addition, the losses in the interconnecting microstrip transmission lines at 1 GHz. were estimated to be Using the above stated quality factors and the parameters given in Table I, the lossy , , , parameters were recomputed using (9) and (10) and are listed in Table II. With the use of (12) and (13), the effective material parameters of the TL mesh and dual TL lens are approximated to be (33) (34) From the effective material parameters, the refractive indexes and of the dual TL lens and TL mesh, respectively, can also be found (35) From the imaginary parts of the refractive indexes, it is evident that losses are significantly higher in the dual TL lens than the


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TL mesh. This is due to the lossy , components used. Nevertheless, the dual TL is still a relatively low-loss composite . medium with a loss tangent of only V. RESOLVING CAPABILITY OF THE EXPERIMENTAL DUAL TL LENS Now that the dual TL lens has been theoretically characterized, its performance and resolution enhancement can be predicted and directly compared to the experimental results reported in [15]. Instead of measuring the nodal voltages in the dual TL lens and the surrounding TL meshes, the vertical electric field (which is proportional to the nodal voltages) was detected above the experimental structure shown in Fig. 3. This vertical electric field will be compared to the theoretically predicted node voltages in order to analyze image resolution and lens performance. The resolution enhancement of the experimental lens can be estimated using either (14) or (31), depending on whether one views the three-region lens arrangement shown in Fig. 2 as an derived electrical network or as an effective medium. The for uniform isotropic left-handed slabs (14) can be used given that the slab thickness of the dual TL lens is cm cm, and the operating wavelength is cm. Substituting these and into two values and the effective material parameters (14) results in . Alternately, the resolution enhanceof the experimental lens can be estimated from the ment quality factors and the electrical length of the lens ( rad) using (31). The estimated resolution enhancement is therefore , which is the same as the previous estimate based on material parameters. These estimates can be verified by finding the spectral width of the two theoretically predicted OTFs shown in Fig. 4 at a 1 GHz frequency of operation. Theoretical OTF A is computed , ) and paramusing (15) given the quality factors ( , eters listed in Table I that model the experimental lens. A second theoretical OTF, theoretical OTF B, is also shown in Fig. 4. This OTF is computed by finding the fast Fourier transform (FFT) of to ) the 19 theoretically predicted node voltages ( along the focal plane and source plane, and then taking their ratio. This OTF accounts for the fact that there is only a finite number of node voltages to work with in a practical lens, where the transverse dimension of the lens is finite. Fig. 4 indicates that the magnitude of the theoretical OTFs drops to 0.5 ( 6 dB) at , justifying both of the preliminary approximately estimates: one based on the effective material parameters (14) (31). and the other solely based on The approximate OTF of the experimental NRI TL lens at 1.057 GHz is also shown in Fig. 4. The best experimental focusing was observed at 1.057 GHz; therefore experimental results at this frequency will be used when making comparisons to the theoretical results at 1.0 GHz. This shift in frequency can be attributed to the variation in chip inductors and capacitors from their nominal values, as well as fabrication tolerances in etching the microstrip lines. The experimental OTF was obtained by dividing the electric field spectrum at the focal plane by the electric field spectrum at the source plane. The electric

Fig. 4. Theoretical and experimental optical transfer functions.

Fig. 5.

Source and focal plane vertical electric field patterns.

field spectrum was found by computing the FFT of the electric field detected above the 19 unit cells along the source and focal planes. The experimental OTF is only approximate since it includes both incident and reflected waves present at the source plane, whereas the theoretical OTF only takes into account the incident wave. The experimental OTF shows a sharp cutoff at , corroborating our theapproximately oretical prediction of the lens’ resolution enhancement. Two spikes are also evident in the experimental OTF at approxi. These spikes are caused by a standing mately wave along the source plane, between the source and the edges of TL mesh. The standing wave results from reflections at the edges of the TL mesh and shows up as sidelobes on either side of the experimental source shown in Fig. 5. The sidelobe on the right is larger than the one on the left, which results in a larger than at . The notion that the spike at spikes are caused by a standing wave is supported by the fact that , the transverse dimension of the lens is 19 cells or 0.97

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which corresponds to a resonant length of 2 for waves with ( ). This is the dominant standing wave in the transverse direction, since the distance from the source to 4. the TL mesh edge is Next, the theoretically predicted voltage along the focal plane is compared to the normalized vertical electric field detected (above the focal plane) in the experimental setup. Both the experimental and theoretical images are plotted in Fig. 5 and show close agreement. Therefore, the expanded analytical formulation that includes losses predicts the performance of the experimental lens well. The theoretical diffraction-limited pattern with is also plotted in Fig. 5 to a resolution enhancement of highlight the fact that the experimental image is in fact subwavelength. The beamwidth of the source (shown in Fig. 5) is limited . The broadening of the predominantly by the periodicity image beyond that of the source results from the mismatch between the TL mesh and dual TL due to added losses in the dual TL. The added losses are introduced by the quality factors of the of the experimental components used and further limit the lens beyond that imposed by periodicity: . Even small losses in the dual TL lens degrade its resolution enhancement , as is evident from (31). VI. CONCLUSION The negative-refractive-index lens made of a dual TL has been characterized as both an electrical network and an effective medium. The effective material parameters as well as dispersion relations and Bloch impedance expressions for the lossy TL mesh and dual TL have been derived, taking into account both component and transmission line losses. Using these expressions, the sensitivity of subwavelength imaging to impedance mismatches and losses was explored and simple of the equations for estimating the resolution enhancement lens have been given. The resolution enhancement is the ratio of the largest transverse wavenumber reaching the image to the isotropic wavenumber of the medium. It was found that the of the dual lens is proportional to the quality factor of the series loading capacitors divided by the electrical thickness (in radians) of the lens. Finally, the resolution enhancement and lens performance of an experimental lens were predicted and compared to experiments. Good agreement between theory and experiment has been shown. REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of and ,” Sov. Phys. Usp., vol. 10, pp. 509–514, Jan.-Feb. 1968. [2] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, pp. 3966–3969, Oct. 2000. [3] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [4] C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett., vol. 90, p. 107 401, Mar. 2003. [5] A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett., vol. 90, p. 137 401, Mar. 2003.

[6] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index metamaterials supporting 2-D wave propagation,” in IEEE MTT-S Int. Microwave Symp. Dig., vol. 2, Seattle, WA, Jun. 2–7, 2002, pp. 1067–1070. [7] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. New York: Wiley, 1994, ch. 5. [8] A. Grbic and G. V. Eleftheriades, “A backward-wave antenna based on negative refractive index L-C networks,” in IEEE Int. Symp. Antennas Propag., vol. 4, San Antonio, TX, Jun. 16–21, 2002, pp. 340–343. [9] C. Caloz, H. Okabe, H. Iwai, and T. Itoh, “Transmission line approach of left-handed materials,” presented at the USNC/URSI Nat. Radio Science Meeting, San Antonio, TX, Jun. 16–21, 2002. [10] A. A. Oliner, “A periodic-structure negative-refractive-index medium without resonant elements,” presented at the IEEE AP-S/URSI Int. Symp., San Antonio, TX, Jun. 16–21, 2002. [11] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [12] A. Grbic and G. V. Eleftheriades, “Growing evanescent waves in negative refractive index,” Appl. Phys. Lett., vol. 82, no. 12, pp. 1815–1817, Mar. 2003. , “Negative refraction, growing evanescent waves, and sub-diffrac[13] tion imaging in loaded transmission-line metamaterials,” IEEE Trans. Microwave Theory Tech., vol. 51, no. 12, pp. 2297–2305, Dec. 2003. , “Subwavelength focusing using a negative-refractive-index trans[14] mission line lens,” Antennas Wireless Propag. Lett., vol. 2, pp. 186–189, 2003. , “Overcoming the diffraction limit with a planar left-handed trans[15] mission-line lens,” Phys. Rev. Lett., vol. 92, p. 117 403, Mar. 2004. [16] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [17] A. Grbic and G. V. Eleftheriades, “Dispersion analysis of a microstripbased negative refractive index periodic structure,” Microwave Wireless Comp. Lett., vol. 13, pp. 155–157, Apr. 2003. [18] A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 4, pp. 1252–1263, Apr. 2004. [19] R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E, vol. 64, p. 055 625, Oct. 2001. [20] S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Modern Opt., vol. 49, pp. 1747–1762, 2002. [21] J. T. Shen and P. M. Platzman, “Near field imaging with negative dielectric constant lenses,” Appl. Phys. Lett., vol. 80, pp. 3286–3288, May 2002. [22] F. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett., vol. 82, pp. 161–163, Jan. 2003. [23] S. A. Ramakrishna and J. B. Pendry, “Imaging the near field,” J. Modern Opt., vol. 50, pp. 1419–1430, 2003. [24] D. R. Smith, D. Schurig, R. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett., vol. 82, pp. 1506–108, Mar. 2003. [25] A. Grbic and G. V. Eleftheriades, “Periodic analysis of a 2-D negative refractive index transmission line structure,” Trans. Antennas Propag., vol. 51, no. 10, pp. 2604–2611, Oct. 2003. [26] C. Caloz and T. Itoh, “Transmission line approach of left-handed structures and microstrip realization of a low-loss broadband LH filter,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1159–1166, May 2004. [27] K. Iizuka, Elements of Photonics: In Free Space and Special Media. Toronto, ON, Canada: Wiley, 2002, vol. 1, ch. 2. [28] J. B. Pendry and S. A. Ramakrishna, “Near-field lenses in two dimensions,” J. Phys. Condens. Matter, vol. 14, pp. 8463–8479, 2002. [29] A. Alu and N. Engheta, “Pairing an Epsilon-negative slab with a Mu-negative slab: Resonance, tunneling and transparency,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2558–2571, Oct. 2003. [30] R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A, vol. 277, pp. 61–64, 2000. [31] J. B. Pendry and S. A. Ramakrishna, “Refining the perfect lens,” Physica B, vol. 338, pp. 329–332, 2003. [32] R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys. Condens. Matter, vol. 13, pp. 1811–1819, 2001.


Anthony Grbic (S’00) received the B.A.Sc. and M.A.Sc. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 1998 and 2001, respectively, where he is currently pursuing the Ph.D. degree. His research interests include printed antennas, microwave circuits, negative-refractive-index metamaterials, and periodic structures. Mr. Grbic received the Best Student Paper Award at the 2000 Antenna Technology and Applied Electromagnetics Symposium. He received an IEEE Microwave Theory and Techniques Society Graduate Fellowship in 2001.

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George V. Eleftheriades (S’86–M’88–SM’02) received the diploma (with distinction) from the National Technical University of Athens, Greece, in 1988 and the M.S.E.E. and Ph.D. degrees from the University of Michigan, Ann Arbor, in 1993 and 1989, respectively, all in electrical engineering. During 1994–1997, he was with the Swiss Federal Institute of Technology, Lausanne, where he developed millimeter- and sub-millimeter-wave receiver technology for the European Space Agency and fast computer-aided design tools for planar packaged microwave circuits. In 1997, he joined the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada, where he is now a Professor. He is leading a group of 15 graduate students in the areas of negative-refraction metamaterials and their microwave applications, integrated antennas and components for broadband wireless telecommunications, novel antenna beam-steering techniques, low-loss silicon micromachined components, sub-millimeter-wave radiometric receivers, and electromagnetic design for high-speed digital circuits. Prof. Eleftheriades received the Gordon Slemon Award (teaching of design) from the University of Toronto and the Ontario Premier’s Research Excellence Award, both in 2001. He received an E.W.R. Steacie Memorial Fellowship from the Natural Sciences and Engineering Research Council of Canada in 2004. Presently, he is an IEEE Distinguished Lecturer for the Antennas and Propagation Society.