Radiation Model for Terahertz Transmission-Line Metamaterial

IEEE TRANSACTIONS ON TERAHERTZ SCIENCE AND TECHNOLOGY, VOL. 2, NO. 3, MAY 2012 323

Radiation Model for Terahertz Transmission-LineMetamaterial Quantum-Cascade Lasers

Philip W. C. Hon, Student Member, IEEE, Amir A. Tavallaee, Student Member, IEEE, Qi-Sheng Chen,Benjamin S. Williams, Senior Member, IEEE, and Tatsuo Itoh, Life Fellow, IEEE

Abstract—We present the use of the cavity antenna model inpredicting the radiative loss, far-field polarization and far-fieldbeam patterns of terahertz quantum-cascade (QC) lasers.Metal–metal waveguide QC-lasers, transmission-line metama-terial QC-lasers, and leaky-wave metamaterial antennas areconsidered. Comparison of the fundamental and first higher orderlateral mode in a metal-metal waveguide QC-laser shows distinctdifferences in the radiation characteristics. Full-wave finite-ele-ment simulations, cavity model predictions and measurementsof far-field beam patterns are compared for a one-dimensionalleaky-wave antenna. Lastly, an active leaky-wave metamaterialantenna with full backward to forward wave beam steering isproposed and analyzed using the cavity antenna model.

Index Terms—Cavity model, composite right/left-handedtransmission line, leaky-wave antenna, quantum-cascade (QC)lasers, terahertz active metamaterials.

I. INTRODUCTION

T ERAHERTZ quantum-cascade (QC) lasers are attractivecandidates as sources for many applications in terahertz

imaging, sensing, and spectroscopy, and have been demon-strated to operate at frequencies between 1.2 and 5.0 THz(without the assistance of a magnetic field) [1]–[3]. The THzQC-lasers that exhibit the best high-temperature operation arebased upon the so-called metal–metal (MM) waveguide, inwhich the multiple-quantum-well active region is sandwichedbetween two metal cladding layers, typically separated by2–10 m [4]–[6]. This waveguide is characterized by low lossand a strong overlap of the mode with the active region, evenwhen the transverse waveguide dimensions are scaled far belowthe wavelength. Reducing the dimensions in this way is oftendesirable to reduce power dissipation as well as to increase thelight-matter interaction strength [7]–[9]. Moreover, because ofits strong resemblance to a microstrip transmission line, it hasrecently been proposed to use MM waveguides as a platform

Manuscript received September 02, 2011; revised March 02, 2012; acceptedMarch 04, 2012. Date of publication April 30, 2012; date of current version May08, 2012. This work was supported by the National Science Foundation underNSF Grant ECCS-0901827.

P. W. C. Hon and T. Itoh are with the Electrical Engineering Department,University of California, Los Angeles, CA 90095 USA (e-mail: [email protected], [email protected]).

A. A. Tavallaee, and B. S. Williams are with the Electrical Engineering De-partment, University of California, Los Angeles, CA 90095 USA, and also withthe California NanoSystems Institute (CNSI), University of California, Los An-geles, CA 90095 USA (e-mail: [email protected], [email protected]).

Q.-S. Chen is with Northrop Grumman Aerospace Systems, Redondo Beach,CA 90278 USA.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TTHZ.2012.2191023

for the development of planar composite right/left-handed(CRLH) transmission-line metamaterials (MTMs) [10], [11].

Soon after the demonstration of MM-waveguide QC-lasers,it was recognized that the beam pattern from a conventionalcleaved-facet Fabry–Pérot (FP) ridge cavity produced a highlydivergent beam pattern, characterized by concentric rings in thefar-field [12]. This beam pattern was qualitatively explainedusing an antenna model of a so-called wire laser [13]. Thismodel applies when the MM waveguide exhibits transverse di-mensions much less than a wavelength, which models the wirelaser as a linear one-dimensional (1-D) array of isotropic radia-tors similar to the analysis of a dielectric traveling wave antenna[14]. However, because the antenna model as implemented in[13] neglects the details of the mode profile, as well as radi-ation from the ends of the “wire” cavity (i.e. the facets), thismodel does not provide useful information such as the field po-larization, the relationship of the far-field beam pattern to thelateral modes, or quantitative estimates of radiative losses. Thiswas pointed out by Gellie et al., who used full-wave three-di-mensional (3-D) electromagnetic simulations to show that asexpected a first higher-order lateral mode resulted in afar-field beam pattern that mirrored the lateral mode symmetryfor a 130 m wide ridge [15]. Namely, for the mode, anull is present in the center of the far-field beam pattern alongthe axis of the ridge. Nonetheless, the use of 3-D full-wave sim-ulations is computationally intensive, prohibitive for large struc-tures, and provides limited physical insight.

In this work, we present the use of the cavity antenna modelfor modelling the radiative properties of cavities, leaky-waveantennas, and transmission-line MTMs that are based uponMM waveguides. The cavity model is widely used to modelmicrostrip patch antennas in the microwave frequency range[16]. It is based upon application of the field equivalence prin-ciple, where effective magnetic currents on the outer surface ofthe cavity act as sources for the far-field radiation. Combinedwith antenna array theory, we are able to predict far-fieldbeam patterns and polarizations, approximate cavity qualityfactors, and associate these properties with individual surfacesor structures of the waveguide.

The paper is organized in the following manner. In Section II,a description of the cavity model and its implementation inMM-waveguide QC-lasers is covered. A quantitative compar-ison for predictions for MM-waveguide resonator radiativelosses and beam patterns is made between full-wave finite-el-ement simulations (Ansys’ HFSS) and the cavity model forthe lateral mode in Section III. In Section IV, the

mode of a MM-waveguide QC-laser is also consid-ered and shown to have desirable radiative properties as aleaky-wave antenna. The application of the cavity model to the

2156-342X/$31.00 © 2012 IEEE

324 IEEE TRANSACTIONS ON TERAHERTZ SCIENCE AND TECHNOLOGY, VOL. 2, NO. 3, MAY 2012

Fig. 1. A perspective and top view of the MM waveguide with its completelytangential electric field profile and equivalent magnetic current sources �� ,represented by double-headed arrows for: (a), (b) �� and (c), (d) ��lateral modes.

MM-waveguide QC-laser operating in its lateral mode asa resonator and leaky-wave antenna is discussed. Comparisonsof its radiative loss (resonator) and beam patterns (leaky-waveantenna) are presented with simulations and backed by mea-surements. Finally, in Section V, the cavity model is appliedto a proposed MTM antenna with backward-to-forward beamscanning capability.

II. CAVITY MODEL

The cavity antenna model is based upon the use of Huygen’sprinciple to formulate a simplified equivalent problem. The prin-ciple states, for a closed surface around the structure of interest,the fields at a given observation point outside of the closedsurface can be found by considering radiation from equivalentsources represented by the surface electric and magnetic surfacecurrent densities, and respectively, on the closed surface.From the uniqueness theorem, knowledge of the tangential fieldcomponents on the closed surface gives the equivalent sourcesaccording to

(1)

(2)

where is the surface normal. One can find the exact radiatedfar-field of the structure by using the far-field integral expres-sions (8)–(14) located in the Appendix.

In analogy to the application of the cavity model to the patchantenna, we make some approximations of the near-field com-ponents of the QC-laser. Specifically, we assume our structureto have a large width-to-height ratio and that the height is muchless than the free space wavelength [16]. For this ideal situation,fringing fields are negligible, the surface electric fields are tan-gential to the sidewalls and magnetic fields are normal to the sur-face so . We assume an infinite ground plane, so that onecan invoke image theory which doubles the equivalent magneticcurrent source. Also, any electric surface current that is presenton the top surface of the metallic plate will not radiate efficientlysince it will cancel with its image. With these approximations,the far-field beam pattern is given by only the equivalent mag-netic currents.

The structure of interest is then modelled as an array of radi-ating equivalent magnetic current elements (see Fig. 1), wherethe array factor is a function of the structure’s dispersion char-acteristics. For example, in a MM-waveguide FP QC-laser, themodelled standing wave is a sinusoidal function with a guidedwavelength where is the dis-persive propagation constant. The array factor is then modulatedby this sinusoid. For the case of a leaky-wave antenna, the si-nusoidal varying field profile also experiences a decaying fieldfactor given by its power attenuation coefficient , which thendictates the envelope of the array factor along the direction ofmode propagation. For example, considering only the two side-walls of a MM waveguide operating in its mode as aleaky-wave antenna [see Fig. 1(c), (d)], first an array factor forone sidewall is calculated (along the -direction) using

(3)

where

(4)

(5)

where is the normalized element amplitude on the sidewall,and are the far-field observation angles, is the element

to element distance in the y direction, is the power atten-uation coefficient, is the propagation constant along theleaky-wave antenna, is the element index, is the mainbeam scan angle, and is the free-space wave vector. Thissidewall is then treated as one radiating source element modifiedby the approriate array factor applied in the transverse direction( -direction) to account for the other sidewall.

Using the cavity model’s calculated total radiated power andthe structure’s stored energy, the quality factor (radiative loss)can be quantified as shown in the Appendix (22). The structure’stime average stored energy is calculated by integrating the elec-tric field within the boundaries of the structure. Of course, this isan approximation since the stored near-field energy is neglectedwith this method. However, this method does provide a goodqualitative understanding of design parameter effects on radia-tive losses and its accuracy does improve when the structure’s

ratio is large and height is much less than its free spacewavelength.

Our assumption of an infinite ground plane means that ourantenna model is most accurate for lasers with dry etched facets.However, despite this limitation, we can still obtain insight forTHz QC-lasers with cleaved facets provided we consider thefollowing. First, the absence of a ground plane will reduce theeffective magnetic image current, which will decrease the radi-ated power from the facet. Second, the beam pattern will changesomewhat as radiation into the lower half-space (previouslyscreened by the ground plane) is now possible. However, as canbe seen from data reported in the literature ([12], [15]) the beampattern in the upper half space still exhibits the characteristicfringe patterns that we will show is well described by the cavitymodel. The cavity model is particularly useful for the emergingdesigns for THz QC-lasers in MM waveguides with directivebeam patterns (such as leaky-wave antennas [17], second and

HON et al.: RADIATION MODEL FOR TERAHERTZ TRANSMISSION-LINE MTM QC LASERS 325

third-order distributed feedback lasers [18], [19], and 2-Dphotonic crystals [20]), which do not possess cleaved facets.

III. MODELING OF MM-WAVEGUIDE RESONATORS:FUNDAMENTAL LATERAL MODE

We begin by applying the cavity model to a MM-waveguideFP cavity oscillating in an axial mode with index .We first consider the radiative quality factor Q for themode as a function of cavity length using approximated analyticexpressions, the cavity model and a full-wave 3-D finite-ele-ment solution. The 3-D finite-element simulations for this paperwere conducted with HFSS for MM-waveguide QC-laser cav-ities with a 10 m or 5 m thick active region, 0.2 m thickupper metal, 15 m ridge width, and an infinite ground plane.A 15 m ridge width is selected in this study since itsmode’s cutoff frequency ( 2.7 THz with ) correspondsto a frequency within the terahertz spectral range and is dis-cussed in greater depth in a later section.

In order to obtain the appropriate magnitude of the sidewallmagnetic currents, an infinitely long structure is simulated usingthe finite-element method to obtain the transverse waveguidemode profile for the and modes in the geometriesof interest. In this way, the value of the transverse E-field at thesidewalls can be related to the stored energy of the cavity. Apolynomial fit of the lateral field profile was used in the calcu-lations for the equivalent magnetic current sources. The samefitted lateral field profile was also used to calculate the storedenergy of the MM-waveguide resonator.

Additionally, for the MM waveguide cavities pictured inFig. 1, the termination is ideally open, and the E-field exhibitsan anti-node (maximum) at the facet. In realistic structures, thetermination deviates slightly from this condition resulting in aslightly smaller facet field amplitude [21]. This is accountedfor by extracting the complex facet reflectivity from a 3-Dfinite-element simulation and incorporated into the far-fieldbeam pattern calculations using the cavity model.

A. Radiative Losses

HFSS eigenmode simulations for MM-waveguide QC-lasersresonating (cold cavity) in its mode at 2.7 THz were con-ducted for structures up to only an axial mode index of ,

mm, due to simulation size constraints—a limita-tion that does not exist for the implemented cavity model. Sim-ulations were conducted with a lossless active region, and themetallization was considered to be a perfect electric conductor,therefore, the simulated quality factor reflects only loss associ-ated with radiation. The facet reflectivity was calculated to be0.84 and the confinement factor 0.79 from finite element simu-lations.

A nice feature of the cavity model is that it allows one tostudy the contributions of each radiating component (sidewalls,facets) separately. Fig. 2 illustrates this with the facet and side-wall radiative losses shown separately for a structure with aof m m. The radiative losses from the sidewalls are atleast three orders of magnitude less than that of the facets—aclear indication that even for MM-waveguide lasers with sub-wavelength dimensions and substantial evanescent field in theair, for the mode, the facets remain the primary source oflaser emission. Hence for the mode, the far-field rings in

Fig. 2. Quality factor of a MM-waveguide FP cavity with a � � �� m and� � �� m operating at 2.7 THz in its �� mode as calculated using thecavity model. Analytic calculations (16) and (18) provided in the Appendixshow close agreement with the cavity model and HFSS simulations.

the beam pattern can be considered to originate from the inter-ference of two dipole sources separated by distance . The strongsuppression of sidewall radiation is due to two effects: the par-tial near-field cancellation of the oppositely directed equivalentmagnetic current sources along the sidewalls and the far-fieldcancellation of the rapidly oscillating that occurs for

, when . Adjacent data points correspond to a differencein MM-waveguide length of . Oscillations in the qualityfactor with a period of three axial mode indices (correspondingto a total distance of since the ) are thereforedue to the constructive and destructive interference of the facets.Lastly, as shown in the Appendix, we can also apply the cavitymodel to derive analytic expressions for radiative Q in the limitswhere and for the mode (facets areapproximated to be ideal opens and an uniform lateral field pro-file is assumed). This allows us to identify scaling trends withdimensions. The small discrepancy between the cavity modeland the analytic expressions is due to the nominally more re-alistic incorporation of the simulated lateral field profile andfacet reflectivities. The cavity model predicted Q is in goodgeneral agreement with the full-wave 3-D simulations althoughthe cavity model consistently predicts a higher value for thequality factor. This discrepancy results from the assumption thatfringing fields can be neglected. Indeed, our calculations showa convergence between HFSS and cavity model simulations forlarger ratios. For example for m m (datanot shown) the Q values calculated by HFSS are 80% of thosecalculated by the cavity model, compared with 50% for the

m m case shown in Fig. 2. Thus, while thecavity model predictions for are at best semi-quantitative for

, they still provide qualitative insight. Furthermore,THz QC-lasers with an active region thickness of 1.75 m haverecently been demonstrated [6], for which the cavity model willbe increasingly accurate for calculating radiative losses.

B. Beam Patterns

As a representative case, a comparison between the cavitymodel predicted and simulated (using a HFSS driven modal sim-ulation) far-field beam patterns for a MM-waveguide QC-laser


Fig. 3. A comparison of predicted far-field intensities by the cavity model andHFSS simulation for a MM-waveguide with � � �� m and � � �� moperating at 2.7 THz in its �� mode. (a) Cavity model: radiation due onlyto the two sidewalls. Far-field intensity due to the sidewalls is three orders ofmagnitude less than that of the two facets. (b) Cavity model: radiation due onlyto the two facets. (c) Cavity model: radiation due to two facets and sidewalls. (d)HFSS simulation: radiation due to two facets and sidewalls. Field is primarilypolarized in � direction.

with a m m and an axial mode index ofis shown in Fig. 3. We see that the radiation from the facets is

the dominant contribution—an observation which is consistentwith the cavity model’s prediction ofshown in Fig. 2, and is supported by the decomposed beampatterns. Namely, the radiation pattern from the sidewalls [seeFig. 3(a)] is expected to give a null along the longitudinal cutof the structure since the equivalent magnetic currents along thetwo sidewalls are in opposite directions, which leads to destruc-tive interference of their far-field beam patterns directly abovethe laser ridge (at ). However, this is not seen in theoverall beam pattern, which is dominated by interference of ra-diation from the two facets, and exhibits strong polariza-tion. The number of rings (fringe pattern) is directly related tothe length of the structure [13]. The radiation pattern predictedby the cavity model matches very well with the full-wave fi-nite-element simulation even though the ratio is close toone, illustrating that the far-field calculations are quite robust.

IV. HIGHER ORDER LATERAL MODE OPERATING AS A

LEAKY-WAVE ANTENNA

Unlike the fundamental mode, the mode ex-hibits a cut-off frequency at which . For certainfrequencies close to its cutoff, this mode propagates with aneffective index , and can be made to couple radia-tion into a directive beam according to the leaky-wave mech-anism. To examine the radiative mechanisms of this in detail,we first consider a MM-waveguide finite length resonator op-erating in its mode. As a representative case, the radia-tion from the facets and sidewalls of a resonator with a lengthof 250 m is analyzed (Fig. 4). For operation within the

Fig. 4. Quality factor versus effective index for MM-waveguide in its ��mode with� 250 �m length. Various effective indices are obtained by varyingridge widths from 15 �m to 45 �m while keeping � � � �m. Operating fre-quency is 2.7 THz. Field profile along facet and sidewalls assumed to be idealsinusoids with a normalized electric field amplitude of 1 V/m. Analytic calcu-lations provided in (17) and (19) in the Appendix are derived with the sameapproximations. Insets show the electric field profile in the lateral direction forvarious ridge widths.

light cone , the cavity model predicts that the con-structive radiation of the sidewall is orders of magnitude greaterthan that of the facets due to the in-phase contributions fromthe two sidewalls (when ). Only for effective indices

does the facet contribution become greater than thatof the sidewalls. This is a somewhat unexpected result. Sincethe magnetic currents from the two sidewalls are in phase fora mode, the far-field beam pattern will have even sym-metry without a broadside null. However, for structures with

, the far-field beam pattern is expected to have a nullsince the facets are the dominant radiating components and haveequivalent magnetic currents in opposite directions, as observedin [15] for 130 m wide MM waveguides. This result indicatesthat examination of the symmetry of the far-field beam patternalone is not sufficient to identify which lateral mode is lasing.We will see in a later part of this section, the polarization of thefar-field beam pattern can be used instead if is unknown.We, therefore, propose the use of a MM waveguide in itslateral mode as a leaky-wave antenna since it is associated withdispersive characteristics not achievable with the modesuch as strong radiative loss, which can be made to exceed un-wanted ohmic and material losses for more efficient radiativecoupling.

A. Radiative Losses

The waveguide radiation loss is quite large for the MM wave-guide operating in its mode; while this leads to more ef-ficient radiative coupling, one would often like to reduce foruse as a laser or leaky-wave antenna. An obvious strategy is todecrease the height to reduce the area of the emitting aper-ture, hence a m was chosen for this analysis. A secondstrategy suggested by the cavity model is to introduce holes witha sub-wavelength periodicity, , into the upper metallization.Then the MM-waveguide structure in its mode can be an-alyzed as repeated unit cells of size m (see Fig. 5). The


Fig. 5. Dispersion curve for a MM-waveguide operating in its �� mode.Two cases are considered; namely, a unit cell with and without holes. In the lattercase, a transverse �� and longitudinal �� hole dimension of � � � �mand � � � �m is introduced into the unit cell �� m� to bring theshunt resonant frequency from 2.75 to 2.63 THz. Eigenfrequencies are obtainedfrom unit cell eigenmode simulations with periodic boundary conditions corre-sponding to an infinitely long structure in HFSS. The circuit model for the casewithout the hole (� � � pH, � � �� pH, � � �� fF) and with thehole (� � � pH, � � �� pH, � � �� fF) are in close agreement withthe simulations.

addition of the holes has two effects. First, the dispersion rela-tion is altered as illustrated in Fig. 5. Without the addition of theholes, the MM waveguide’s mode has a cut-off frequencyof 2.7 THz, or equivalently, a shunt resonant frequency givenby the transmission-line circuit model [10]

(6)

In the language of transmission-line MTMs, increasing thehole size narrows the inductive path leading to a larger shuntinductance , and reduces the shunt resonant frequency[17]. The change in the shunt capacitance is not substantialsince the electric field has a null in the center of the ridge[see Fig. 6(a)].

In addition to controlling the shunt frequency, the hole sizealso gives the designer a parameter to control the radiativelosses. This can be explained qualitatively using the cavitymodel by considering additional equivalent magnetic currentsat the sidewalls of the holes [see Fig. 6(b)]. Components labeled

are the dominant in-phase components along the sidewallsof the structure and contribute most to the radiated power. The

components have a negligible contribution due to theirquadrupole arrangement. As the hole size in the longitudinaldirection is made larger, larger components are generated,which leads to a destructive near- and far-field contributionrelative to that from the components; as a result the qualityfactor should increase. Lastly, the components are onlypresent at the ends of the structure since they correspond withthe facets and have negligible contribution when operatingwithin the fast-wave region (see Fig. 4). A comparison of theradiative losses for an 80 m long MM-waveguide cavitywith varying hole sizes is presented in Table I. The calculatedquality factors confirm the trend in radiative losses qualitativelyexplained by the cavity model. It is important to note that for

TABLE IRADIATIVE LOSSES FOR THE �� MODE IN ITS � � (ZERO-INDEX)

RESONANCE WITH DIFFERENT LONGITUDINAL HOLE SIZES

the finite length structure reported in Table I there will be aslight shift in shunt frequency compared to that of an infinitelylong structure due to the loading effect of radiation from thefacets.

B. Beam Patterns

Operation within the leaky-wave bandwidth of themode is accompanied by a directive beam launched by theantenna due to the small phase variation between radiatingelements as suggested by (3); similar microwave microstripline versions have previously been demonstrated [22]. As anexample, we compare cavity model predictions against anexperimentally measured beam pattern from a leaky-waveantenna presented in [17], namely, with m, m,

m, and 3 m 3 m holes. Similar to Section III, fromthe unit cell simulation, the field profile is extracted forcalculation of the equivalent magnetic currents. Instead of astanding wave pattern, the fields in the leaky-wave antennaare modelled according to (3). The leakage/power attenuationcoefficient is obtained from HFSS scattering parameter simu-lations of the leaky-wave structure, in this case, 51 unit cellslong. Having these parameters, the radiation pattern for anarbitrarily long device can be constructed with fairly good ac-curacy (Fig. 7) and as expected a directive, single beam far-fieldpattern is observed. As the angle of emission is dictated by thedispersion characteristic, the beam can in principle be steeredeither by varying the frequency or transmission-line character-istics—for instance, by changing the hole size. The inset showsthe predicted scan angle as a function of frequency along withthe corresponding measured and cavity model prediction. Thecavity model also gives qualitative insight for polarization ofthe far-field beam pattern. For the mode, the far-fieldpattern along the top of the ridge is expected to have primarily

components (transverse to the length of the ridge) dueto the in-phase equivalent magnetic current sources. Cavitymodel predictions and HFSS simulations (Fig. 8 insets) confirmthis analysis and is matched by the experimentally measuredfar-field polarization. Fig. 9 shows a longitudinal cut of thebeam pattern for the leaky-wave antenna when fed by a masteroscillator THz QC-laser operating in the (at 2.50 THz)mode and the (at 2.81 THz) mode. For the mode,the detected radiation is dominantly polarized in the directionperpendicular to the antenna axis, which corresponds tocomponents as mentioned above. The mode, on theother hand, exhibited a non-directional beam pattern withmany fringes in the far-field and is polarized primarily along


Fig. 6. (a) Electric field within a �� m unit cell resonating at � with hole dimensions � � � �m and � � � �m and (b) a top-down view of theunit cell with its equivalent magnetic currents sources.

Fig. 7. Comparison of the longitudinal cut of the far-field beam pattern at 2.74THz. Inset shows main beam angle predictions from HFSS simulations, thecavity model and measurements for 2.61, 2.74, 2.81 THz, where the leaky-waveantenna is passive and is excited by a master oscillator.

Fig. 8. Total far-field intensity of leaky-wave antenna at 2.81 THz predictedby: (a) the cavity model and (b) HFSS. (Insets show theta and phi components.)

the axis of the waveguide, corresponding to the predictedcomponents, and is an order of magnitude lower in intensity.

V. BALANCED MTM TRAVELING-WAVE ANTENNA

The structure presented in Section IV can serve as a plat-form for development of a THz CRLH transmission-line meta-material [11]. We now propose a nearly balanced, CRLH MTMleaky-wave antenna based on the lateral mode MM wave-guide of Section IV. This can be achieved with the periodic

Fig. 9. Measured far-field beam patterns and spectra under two different biasconditions corresponding to antenna operation in its�� mode (at 2.50 THz)outside the light cone and �� mode (at 2.81 THz) within the leaky-wavebandwidth. Data was collected at 77 K with the master oscillator QC-laser bi-ased in pulsed mode (5 �s pulses repeated at 10 kHz) and the antenna passive.

m addition of series gap capacitors in the upper met-allization as shown in Fig. 10(a). A MTM structure comprisedof such unit cells exhibits CRLH properties such as left-handed(backward waves) propagation. Compared to the design pro-posed in [11], where series capacitance and shunt inductance el-ements are loaded into a MM waveguide operating in itsmode, this design exhibits greater radiation loss. In this section,we present simulations and cavity model predictions for the ra-diation pattern and radiative losses of a structure composed ofbalanced CRLH unit cells.

The unit cell in Fig. 10(a), however, does not meet two prac-tical design necessities for active QC-laser structures, namely,the ability to apply an electrical DC bias to the upper metal con-tact and the ability to achieve a balanced MTM design (a largeseries capacitance is needed corresponding to an unreasonablysmall gap size). However, with a slight change in design, bothconcerns can be addressed [see Fig. 10(b)] with alternating gapcapacitors of 238 nm incorporated into the upper metallization,leading to a meander-type structure. Due to its continuous DCconnectivity, the device can operate as an active QC-laser as wella purely passive MTM leaky-wave antenna. This, however, in-creases the overall unit cell length to 16 m, which translatesto a larger series inductance helping achieve a nearly balancedcondition with THz, THz ( 10 GHzbandgap reported by HFSS eigenmode simulation). Under the


Fig. 10. (a) MTM unit cell with symmetric series gap capacitors. (b) MeanderMTM unit cell with alternating series gap capacitors and its (c) circuit modelalong with its dispersion relation obtained via unit cell analysis in HFSS. Leastsquares fit obtained circuit parameters are � � �� pH, � � �� pH, � �� fF, and � � �� fF.

perfectly balanced case, the transition, shunt and series frequen-cies would be the same , where

(7)

The simple circuit model is qualitatively a good match withthe HFSS calculation near the transition point, although thereis some discrepancy in the left-handed branch near the lightline [see Fig. 10(c)]. The light cone spans from 2.4 to 3.1 THz,within which the CRLH waveguide can be used as a leaky-waveantenna.

We have calculated the radiative and the associated radia-tive loss coefficient [given by (22)] for the CRLH structureusing both HFSS and the cavity model, as shown in Fig. 11. Thesimulated is obtained using HFSS’s eigenmode solver withperiodic boundary conditions, which estimates radiation froman infinitely long structure. The cavity model is applied to findthe along with a perfectly balanced analytic dispersion rela-tion obtained from the circuit model (as shown in Fig. 10). Thiscalculation is very similar to that shown in Fig. 4 and Table Ifor a leaky-wave mode, except modified to allow for theinclusion of the series capacitor in the stored cavity energyfor Q calculations. In this CRLH structure, electric field energycan be stored in the shunt capacitor , which is associatedwith strong radiation from the sidewalls, and/or , which con-tributes negligible radiation due to the antisymmetric con-tributions within the metallization gaps.

Fig. 11 shows generally good agreement between the cavitymodel and HFSS calculations of the radiative loss coefficient.

Fig. 11. Power attenuation coefficient (radiative loss) within the leaky-wavebandwidth for the nearly balanced MTM structure calculated from HFSS and thecavity model for a balanced CRLH waveguide and the RH only leaky-wave an-tenna from Section IV. �� THz. The purely right handed leaky-waveantenna of Section IV exhibits much greater loss near its band edge comparedto the nearly balanced MTM case. Inset: radiative Qs from HFSS’s eigenmodesolver and the cavity model (� 1000 �m long structure).

However, near 2.6 THz, the HFSS calculation shows a discon-tinuity in and due to the fact that in the numerical simula-tion the CRLH structure is not perfectly balanced, and a residual10 GHz bandgap remains. In the cavity model simulation, aperfectly balanced circuit model was used which exhibits nobandgap, and thus the radiative loss characteristic is smooth.For comparison, the loss coefficient is also plotted for the righthanded (RH) only leaky-wave structure whose dispersion liesbetween the two cases given in Fig. 5 (15 m wide waveguidewith 3 3 m holes). This structure exhibits a power loss coef-ficient that diverges as the cutoff frequency is approached,due to a group velocity that approaches zero. The CRLH struc-ture, on the other hand, maintains a nonzero group velocity evenat , and hence there is no divergence in loss. This makesthis structure highly attractive for use as a leaky-wave antenna.

Using the HFSS derived dispersion relation, the radiationpattern for an arbitrarily long device is constructed with fairlygood accuracy using the cavity model. The backward to forwardscanning of the main beam for the balanced MTM structure of10 unit cells (160 m) is observed [see Fig. 12(a)–(c)]. Thereis good agreement between the simulated and cavity modelpredicted beam patterns along the longitudinal direction ofthe structure for all scan angles. The largest deviation occursaround the transition frequency where the dispersion charac-teristic of a simulated finite length structure will differ slightlyfrom that of an infinitely long one, which is how the dispersionrelation was obtained; this could also be associated with largeruncertainty in the vicinity of . For observation angles farfrom the longitudinal cut of the device, slight deviations areseen compared to the simulated radiation patterns from HFSS(Fig. 12 insets). The deviation is due to the fact that themode is not highly confined and hence the effective transversedimension is effectively larger than the physical size. In ourcavity model, the distance separating the equivalent magneticcurrent elements along the sidewalls is modelled to be the


Fig. 12. HFSS and cavity model longitudinal far-field beam pattern cuts fordifferent regions of operation: (a) left-handed (2.5 THz), (b) transition frequency(�2.61 THz), and (c) right-handed (2.80 THz). Insets: HFSS and cavity modelrectangular contour plots of the far-field intensity.

same distance as the physical ridge width (in this case 15 m).Indeed, in [22], Menzel also considers an effective width due tofringing fields, which we have neglected in order to simplify themodel. By separating the equivalent magnetic current sourceswith a greater element spacing (i.e., a wider equivalent ridgewidth), the cavity model resulted in a beam pattern that wasless divergent in the transverse dimension, closely matchingthat predicted by HFSS (not shown here).

VI. CONCLUSION

In conclusion, we have proposed the application of themicrowave patch antenna cavity model as a general tool foranalyzing the radiative properties and beam patterns of tera-hertz MM-waveguide structures. The method is computationalresource non-intensive, allowing for a quick assessment ofdesigns with physical intuition. Its implementation is also notconstrained to very narrow wire lasers as in [13], and as suchcan account for the effects of transverse mode structure, anddescribes the far-field polarization. As a benchmark, compar-isons between full-wave 3-D finite-element method simulationsand the cavity model were performed for andFP cavity QC-lasers, MM waveguides operating as

leaky-wave antennas, and a balanced CRLH leaky-wave an-tenna. Generally speaking, excellent agreement between thecavity model and full-wave simulations was seen for observedbeam patterns, and a good qualitative agreement was observedfor the radiative quality factors. Disagreements in calculated

-factors generally originated from the approximations in-voked to make the cavity model tractable, and were greatlyreduced for larger width-to-height ratios that reduced fringingfields. Given the flexibility of the cavity model, it is possiblefor it to be used for other sorts of terahertz radiating structures.

Notable predictions include the relative roles of sidewall andfacet radiative contributions for narrow MM waveguides oper-ating in their and modes. While radiation isdominated by the facet contribution (even for highly subwave-length dimensions), the mode exhibits a switchover fromsidewall to facet radiation for , which is characterizedby a change in the far-field beam pattern symmetry from evento odd, as well as a polarization change. A MM-waveguide inits mode can operate within the light cone and be usedas a leaky-wave antenna [17]. Due to its strong resemblance toa waveguiding microstrip transmission line, a circuit model isused to model the MM waveguide’s dispersion relation, whichgives the designer physical insight in changing the structure’sgeometry in order to achieve a desired dispersion relation. Thepredicted cavity model beam patterns are directive, and com-pare well with full-wave simulations and experimentally mea-sured beam patterns and angles.

With only the addition of series capacitance to a MM-wave-guide operating in its mode, a balanced 1-D CRLHtransmission-line MTM can be produced. Recently, as aproof of concept demonstration, a similar structure to theone mentioned in Section V., was fabricated using spin-coatedBenzocyclobutene, which showed left-handed and right-handedoperation [23]. The proposed meander structure in Section Voperates as a leaky-wave antenna with a directive beam inthe axial direction and favorable single lobe beams due to thein-phase addition of equivalent radiating magnetic sidewallcurrents. This structure is a candidate for full scan THz an-tennas—in this design a frequency dependent steering of thefar-field main beam from backward to broadside to forward overa bandwidth of 0.7 THz can be achieved with nearly no stop-band. Furthermore, the CRLH structure maintains a finite groupvelocity at all frequencies within the leaky-wave bandwidth,which prevents divergence of the radiative loss coefficient.Lastly, the meander design allows an electrically connectedcontact suitable for implementation of active QC-laser MTMcavities, waveguides, and antennas [11].

APPENDIX

A. Far-Field Expressions

Following the treatment of Balanis [16], the far-field integralexpressions are given as

(8)

(9)


(10)

(11)

where we have defined our closed surface to extend conformallyover the surface of the MM-waveguide cavity and ground plane.

is the angle between the vector and . is the magneticvector potential induced by an electric current and is theelectric vector potential induced by a magnetic current . TheE field components in the far-field can then be written as:

(12)

(13)

(14)

B. Analytic Quality Factor Expressions

The cavity model can be used to derive analytic expressionsfor the radiative quality factor for FP modes in a MM-waveguidecavity under certain limiting conditions. For a cavity of width

, height and length , as shown in Fig. 1, we consider theradiation from the mode when , , and .As shown in detail in [21], and when , the edges ofthe MM-waveguide ridge act as nearly ideal open terminations.Additionally, a uniform or perfect half sinusoid is assumed forthe and lateral field profiles, respectively. Underthese conditions the necessary integrals simplify.

The tangential electric fields at the surface are used to obtainthe equivalent magnetic currents which in turnenters into the expressions for given in (11), and in turn isused to calculate the far-field radiated power . We calculatethe radiative quality factor by considering the ratio of the storedelectromagnetic energy within the cavity to the total radiatedpower integrated over the far-field half-space over the groundplane:

(15)

We use this to obtain expressions for the quality factor dueto radiation separately from the facets or sidewalls, in either the

or lateral modes:

(16)

(17)

(18)

(19)

where

(20)

and

(21)

In these expressions, is the relative permittivity of the cavitydielectric, and is the effective index of the propagatingmode. It should be noted that the expressions for givenabove are only valid when .

For a Fabry–Pérot cavity characterized by counterpropa-gating modes with group velocity , we can relate the radiativeQ to the power attenuation coefficient using the relation

(22)

REFERENCES

[1] R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. Harvey, E. H. Lin-field, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertzsemiconductor-heterostructure laser,” Nature, vol. 417, pp. 156–159,May 2002.

[2] C. Walther, G. Scalari, J. Faist, H. Beere, and D. Ritchie, “Low fre-quency terahertz quantum cascade laser operating from 1.6 to 1.8 THz,”Appl. Phys. Lett., vol. 89, p. 231121, Dec. 2006.

[3] B. S. Williams, “Terahertz quantum-cascade lasers,” Nature Photonics,vol. 1, pp. 517–525, Sep. 2007.

[4] B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Ter-ahertz quantum-cascade laser at � � �� m using metal waveguidefor mode confinement,” Appl. Phys. Lett., vol. 83, pp. 2124–2126, Sep.2003.

[5] S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertzquantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett.,vol. 94, p. 131105, Apr. 2009.

[6] E. Strupiechonski, D. Grassani, D. Fowler, F. H. Julien, S. P. Khanna,L. Li, E. H. Linfield, A. G. Davies, A. B. Krysa, and R. Colombelli,“Vertical subwavelength mode confinement in terahertz and mid-in-frared quantum cascade lasers,” Appl. Phys. Lett., vol. 98, p. 101101,2011.

[7] B. Williams, S. Kumar, Q. Hu, and J. Reno, “Operation of terahertzquantum-cascade lasers at 164 K in pulsed mode and at 117 K in con-tinuous-wave mode,” Opt. Express, vol. 13, pp. 3331–3339, May 2005.

[8] Y. Todorov, I. Sagnes, I. Abram, and C. Minot, “Purcell enhancementof spontaneous emission from quantum cascades inside mirror-gratingmetal cavities at THz frequencies,” Phys. Rev. Lett., vol. 99, p. 223603,Nov. 2007.

[9] C. Walther, G. Scalari, M. Beck, and J. Faist, “Purcell effect in theinductor-capacitor laser,” Opt. Lett., vol. 36, pp. 2623–2625, Jul. 2011.

[10] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmis-sion Line Theory and Microwave Applications. Hoboken, NJ:Wiley-IEEE Press, 2005.

[11] A. A. Tavallaee, P. W. C. Hon, K. Mehta, T. Itoh, and B. S. Williams,“Zero-index terahertz quantum-cascade metamaterial lasers,” IEEE J.Quantum Electron., vol. 46, no. 7, pp. 1091–1098, Jul. 2010.

[12] A. J. L. Adam, I. Kasalynas, J. N. Hovenier, T. O. Klaassen, J. R. Gao,E. E. Orlova, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Beampatterns of terahertz quantum cascade lasers with subwavelength cavitydimensions,” Appl. Phys. Lett., vol. 88, p. 151105, 2006.

[13] E. E. Orlova, J. N. Hovenier, T. O. Klaassen, I. Kasalynas, A. J. L.Adam, J. R. Gao, T. M. Klapwijk, B. S. Williams, S. Kumar, Q. Hu,and J. L. Reno, “Antenna model for wire lasers,” Phys. Rev. Lett., vol.96, p. 173904, May 2006.

[14] H. Jasik, Antenna Engineering Handbook. New York: McGraw-Hill,1961.

[15] P. Gellie, W. Maineult, A. Andronico, G. Leo, C. Sirtori, S. Barbieri,Y. Chassagneux, J. R. Coudevylle, R. Colombelli, S. P. Khanna, E. H.Linfield, and A. G. Davies, “Effect of transverse mode structure on thefar field pattern of metal-metal terahertz quantum cascade lasers,” J.Appl. Phys., vol. 104, p. 124513, 2008.

[16] C. A. Balanis, Antenna Theory. Hoboken, NJ: Wiley-Interscience,2005.


[17] A. A. Tavallaee, B. S. Williams, P. W. C. Hon, T. Itoh, and Q.-S.Chen, “Terahertz quantum-cascade laser with active leaky-wave an-tenna,” Appl. Phys. Lett., vol. 99, p. 141115, 2011.

[18] S. Kumar, B. S. Williams, Q. Qin, A. W. Lee, Q. Hu, and J. L. Reno,“Surface-emitting distributed feedback terahertz quantum-cascadelasers in metal-metal waveguides,” Opt. Express, vol. 15, pp. 113–128,Jan. 2007.

[19] M. I. Amanti, M. Fischer, G. Scalari, M. Beck, and J. Faist, “Low diver-gence single-mode terahertz quantum cascade laser,” Nature Photon.,vol. 3, pp. 586–590, Oct. 2009.

[20] Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere,D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electri-cally pumped photonic-crystal terahertz lasers controlled by boundaryconditions,” Nature, vol. 457, pp. 174–178, 2009.

[21] Y. Todorov, L. Tosetto, J. Teissier, A. M. Andrews, P. Klang, R.Colombelli, I. Sagnes, G. Strasser, and C. Sirtori, “Optical propertiesof metal-dielectric-metal microcavities in the Thz frequency range,”Opt. Express, vol. 18, pp. 13 886–13 907, Jun. 2010.

[22] W. Menzel, “A new travelling wave antenna in microstrip,” in Mi-crowave Conference, 1978, 8th European, Sept. 1978, pp. 302–306.

[23] Z. Liu, P. W. C. Hon, A. A. Tavallaee, T. Itoh, and B. S. Williams,“Terahertz composite right-left handed transmission-line metamaterialwaveguides,” Appl. Phys. Lett., vol. 100, p. 071101, 2012.

Philip W. C. Hon (S’08) was born in New YorkCity in 1982. He received the B.S. and M.S. degreein electrical engineering from the University ofCalifornia, Los Angeles (UCLA), in 2004 and2007, respectively, and is currently working towardsthe Ph.D. degree in electromagnetics at the sameuniversity.

Since 2005, he has worked in industry program-ming FPGAs, designing MMICs and microwaveantennas. His research interest include metamate-rial, plasmonic, reconfigurable, and phased array

antennas.

Amir Ali Tavallaee (S’07) was born in Mashhad,Iran, in 1981. He received the B.Sc. degree fromSharif University of Technology, Tehran, Iran, in2003, the M.Eng. degree (Hons.) from McGillUniversity, Montreal, Canada, in 2006, both inelectrical engineering, and is currently workingtowards the Ph.D. degree in electrical engineering atthe University of California, Los Angeles.

His research interests include terahertzquantum-cascade lasers, active transmission linemetamaterials, and sub-wavelength plasmonics.

Mr. Tavallaee was a recipient of the URSI Student Fellowship Grant Awardin 2007.

Qi-Sheng Chen, photograph and biography not available at time of publication.

Benjamin S. Williams (S’02–M’03–SM’10) wasborn in Syracuse, NY, in 1974. He received the Ph.D.degree from the Massachusetts Institute of Tech-nology (MIT), Cambridge, in electrical engineeringand computer science in 2003.

He was a Postdoctoral Associate at the ResearchLaboratory of Electronics at MIT from 2003 to 2006.In 2007, he joined the Electrical Engineering Depart-ment at the University of California, Los Angeles,where he is currently an Assistant Professor and aHenry Samueli School of Engineering and Applied

Sciences Fellow. His research interests include quantum-cascade lasers, inter-subband and intersublevel devices in semiconductor nanostructures, and tera-hertz metamaterials and sub-wavelength plasmonics.

Tatuso Itoh (S’69–M’69–SM’74–F’82–LF’06)received the Ph.D. Degree in electrical engineeringfrom the University of Illinois, Urbana, in 1969.

After working for University of Illinois, SRI andUniversity of Kentucky, he joined the faculty atThe University of Texas at Austin in 1978, wherehe became a Professor of Electrical Engineeringin 1981. In September 1983, he was selected tohold the Hayden Head Centennial Professorship ofEngineering at The University of Texas. In January1991, he joined the University of California, Los An-

geles as Professor of Electrical Engineering and holder of the TRW EndowedChair in Microwave and Millimeter Wave Electronics (currently NorthropGrumman Endowed Chair). He has 400 journal publications, 820 refereedconference presentations and has written 48 books/book chapters in the area ofmicrowaves, millimeter-waves, antennas and numerical electromagnetics. Hehas supervised up to 73 Ph.D. students.

Dr. Itoh received several awards, including IEEE Third Millennium Medal in2000, and IEEE MTT Distinguished Educator Award in 2000. He was electedto a member of National Academy of Engineering in 2003. In 2011, he receivedMicrowave Career Award from IEEE MTT Society. He is a member of the Insti-tute of Electronics and Communication Engineers of Japan, and CommissionsB and D of USNC/URSI. He served as the Editor of IEEE TRANSACTIONS ON

MICROWAVE THEORY AND TECHNIQUES for 1983–1985. He was President of theIEEE Microwave Theory and Techniques Society in 1990. He was the Editor-in-Chief of IEEE MICROWAVE AND GUIDED WAVE LETTERS from 1991 through1994. He was elected as an Honorary Life Member of MTT Society in 1994.He was the Chairman of Commission D of International URSI for 1993–1996.the Chairman of Commission D of International URSI for 1993–1996. He serveson advisory boards and committees of a number of organizations. He served asDistinguished Microwave Lecturer on Microwave Applications of MetamaterialStructures of IEEE MTT-S for 2004–2006.

Documents

Radiation Model for Terahertz Transmission-Line Metamaterial