IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 12, NO. 6, NOVEMBER 2013 1191

Tunable Subwavelength Terahertz PlasmonicStub Waveguide FiltersJin Tao, Bin Hu, Xiao Yong He, and Qi Jie Wang

Abstract—Tunable subwavelength terahertz (THz) plasmonicstub waveguide filters based on indium antimonide are proposedand numerically investigated for the first time. The transmissionline theory and the finite different time domain simulation resultsreveal that the single-stub waveguide structure can realize a stop-band filtering function and the central wavelength of the notch islinearly dependent on the stub length while nonlinearly dependenton the stub width. The central wavelength of the notch can be ac-tively controlled by tuning the temperature. As an extension to thesingle-stub structure, multiple-stub InSb slot waveguide structuresare also proposed and used to realize a wide stop-band filteringfunction. The proposed filters may have applications in THz highlyintegrated plasmonic circuits.

Index Terms—Filters, plasmonics, terahertz (THz), waveguides.

I. INTRODUCTION

T ERAHERTZ (THz) technology is now drawing extensiveattention because of its potential applications in biochemi-

cal sensing, spectroscopy, and high-speed communication. Withthe rapid development of THz sources [1]–[5] and detectors [6],there is a high demand for THz components, such as waveguides,polarizers, filters, and collimators [7], [8]. Waveguide filter isprobably the most basic component of all. Various waveguidefilters have been demonstrated within the last few years. Most ofthese are based on conventional guiding structures, such as metaltubes [9], [10], metal wires [11], [12], and dielectric waveg-uides [13], [14]. Confining and manipulating electromagneticwaves at dimensions much smaller than the wavelength are ofgreat importance for miniaturized integrated photonic circuits.

Manuscript received March 13, 2013; accepted October 3, 2013. Date ofpublication October 9, 2013; date of current version November 6, 2013. Thiswork was supported by the grant (grant number M58040017) from NanyangTechnological University (NTU), and the grant (MOE2011-T3-1-005) fundedby Ministry of Education, Singapore, (MOE2011-T2-2-147) Singapore. Thereview of this paper was arranged by Associate Editor N. G. Ing.

J. Tao is with the OPTIMUS, Photonics Centre of Excellence, Schoolof Electrical and Electronic Engineering, Nanyang Technological University,Singapore 639798, Singapore and also with the CINTRA CNRS/NTU/THALES, UMI 3288, Singapore 637553, Singapore (e-mail: jtao2@e.ntu.edu.sg).

B. Hu and X. Y. He are with the OPTIMUS, Photonics Centre of Excel-lence, School of Electrical and Electronic Engineering, Nanyang Technolog-ical University, Singapore 639798, Singapore (e-mail: HUBIN@ntu.edu.sg;xyhethz@hotmail.com).

Q. J. Wang is with the OPTIMUS, Photonics Centre of Excellence, Schoolof Electrical and Electronic Engineering, and CDPT, Centre for Disrup-tive Photonic Technologies, School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore 639798, Singapore (e-mail:qjwang@ntu.edu.sg).

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

Digital Object Identifier 10.1109/TNANO.2013.2285127

The minimum confinement of a guided mode in conventionalwaveguides is limited by the diffraction limit.

Plasmonic devices, based on surface plasmon-polaritons(SPPs) propagating at metal-dielectric interface, have showngreat potential to guide and manipulate light by metallic nanos-tructures at subwavelength scales [15]. Most of the efforts inplasmonic devices have been concentrated at optical and near-infrared frequencies [16]–[18], and the tuning mechanisms ofactive plasmonic device are based on changing the permittiv-ity of the dielectric materials [19] through thermo-optic [20],electro-optic [21], and nonlinear optical effects [22], [23] ratherthan changing permittivity of the metal because of the largecharge carrier concentration (∼1022 cm−3 of metals), whichmakes it difficult to change its permittivity by varying the car-rier density. In the THz region, both the real and imaginaryvalues of the metal permittivity are huge, with typical absolutevalues to the order of 105 . These values of permittivity give riseto weak coupling of the electromagnetic field to the electrons inthe metal. Actually, the metal resembles a perfect electric con-ductor, where the electromagnetic waves hardly penetrate intothe metal, instead of hundreds of wavelength into the dielectricmedium [24]. In this regard, surface plasmons (SPs) in the realsense do not exist on planar metal/dielectric interface in THz.Spoof SPs based on corrugated metallic structures can provideTHz wave localization and slow light localization by introduc-ing localized electromagnetic modes strongly interacting withstructured surfaces [25]. A dynamic THz SPP switch based onresonance and absorption has been proposed [26]. Alternativematerials for exciting low-frequency SPs on planar structuresare doped semiconductors. Some semiconductors have a per-mittivity at THz range close to that of metal at optical range.In addition, the permittivity of semiconductors can be modi-fied [27], [28] by varying temperature, dopant concentrations,and magnetic fields which can be used to realize the tunabilityof plasmonic devices.

In this paper, we propose planar tunable subwavelength plas-monic stub waveguide filters in THz range using indium anti-monide (InSb) as the “metal” material. Characteristics of deep-subwavelength InSb plasmonic slot waveguide are first inves-tigated. The single-stub slot InSb stub waveguide filters areproposed and analyzed by the transmission line theory andthe finite different time domain (FDTD) simulation [29]. Theresults obtained from both the methods agree well. As anextension to the tunable single-stub filter, a multiple-stub struc-ture is also demonstrated with more functionalities. Our struc-tures may have great potential for ultracompact THz integratedcircuits.

1536-125X © 2013 IEEE

1192 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 12, NO. 6, NOVEMBER 2013

(b)

(a)

Fig. 1. (a) Real part of the effective index of InSb slot waveguide as a functionof slot widths at different temperatures for incident frequency at 1 THz. Inset:schematic structure of an InSb slot waveguide. (b) Imaginary part of the effectiveindex of InSb slot waveguide.

II. CHARACTERISTICS OF SUBWAVELENGTH INSB PLASMONIC

SLOT WAVEGUIDE

The inset of Fig. 1(a) shows the InSb slot waveguide structurecomposed of two parallel InSb plates with a dielectric core. Thedispersion relation of the fundamental SP mode in the InSb slotwaveguide is given by [30]

tanh(√

n2eff − εdk0w/2

)=

−εd

√n2

eff − εinsb(ω, T )εinsb(ω, T )

√n2

eff − εd

(1)

where neff is the effective refractive index of the SP wave inthe waveguide and k0 = 2π/λ0 is the free-space wave vector.εd denotes the permittivity of medium in the slot guide regionand the permittivity εinsb (ω, T ) of InSb can be described bythe Drude model [31], [32] approximation,

εinsb(ω, T ) = ε∞ −ω2

p (T )ω[ω + iΓ(T )]

(2)

where ω is the angular frequency of the incident electromagneticradiation, and ω∞ = 15.75 is the high-frequency permittivity;ωp(T ) and Γ(T ) are the plasma frequency and collision rate ofthe charge carriers, respectively. The plasma frequency ωp (T ) =√

Ne2/ε∞ε0m∗ depends on the density of carriers, where e isthe effective charge unit, ε0 is the permittivity in vacuum, andm∗ is the effective mass of electrons. Different from metal, theplasma frequency of the InSb significantly depends on the tem-perature T . The intrinsic concentration in the InSb is described

by the formula N = 5.76 × 1014T 1.5 exp(−0.129/KBT )cm−3

[33], where KB and T are the Boltzmann constant and tempera-ture, respectively. At 295 K, the permittivity of InSb at 1 THz is−44.19 + 15.58i, which is similar to that of metals at optical fre-quencies. The InSb permittivity is changed to −26.69 + 10.18iat temperature of 280 K. The dielectric constant ε(ω, T ) of InSbmaterial is affected by the variation of the temperature in theTHz range because of the plasma frequency and collision rateof the charge carriers are both dependent on the temperature.Therefore, tunable plasmonic filters based on thermal tuningcan be expected.

In the InSb slot waveguide, the fundamental transverse mag-netic mode (TM0) always exists regardless of the waveguidewidth, while other high-order modes have a cutoff wave-guide width. To satisfy the single mode propagation condi-tion [34], the width of the waveguide must be smaller thanλ0arc tan(

√−Re(εinsb)εd/π

√εd). In this paper, the dielectric

in the guided region is assumed to be air with a permittivityεd = 1. Fig. 1(a) and (b) shows the dependence of the real partand imaginary part, respectively, of the effective refractive indexof InSb slot waveguide on the slot widths at different tempera-tures for an incident light at 1 THz. From Fig. 1(a), one can seethat the real part of effective refractive index of the waveguidedecreases with increasing w at a certain temperature, and for agiven width it decreases as the temperature rises. The waveguideintroduces higher losses for smaller slot width, as indicated bythe imaginary part of waveguide refractive indices in Fig. 1(b).So there is a tradeoff between strong mode confinement and highattenuation for the slot waveguide. And for the slot waveguidewith a given width, the imaginary part of the effective refractiveindex increases as the temperature decreases, which means thepropagation loss is higher at a lower temperature.

For the InSb slot waveguide, the Hy field component distri-bution can be written as [35]

Hy =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(ε

1/2insb(T )k0c

ω

)(eik1 w/2 + e−ik1 w/2)e−ik2 (x−w/2)

x > w/2(εdk0c

ω

)(eik1 x + eik1 x) |x| ≤ w/2

(ε

1/2insb(T )k0c

ω

)(eik1 w/2 + e−ik1 w/2)e−ik2 (x−w/2)

x < −w/2(3)

with

k21 = β2 − εinsb(T )k2

0 , k22 = εdk

20 − β2 . (4)

Here [Re(k1)]−1 and [Re(k2)]−1 determine the skin depthsof SPs in the dielectric guided region and the InSb region, re-spectively; β is the propagation constant of SP wave that canbe represented by β = k0neff . Fig. 2 shows the normalized Hy

profiles as a function of the distance from the waveguide me-dian for the incident wave at 1 THz. One can see from the figurethat the majority of the SPs energy is localized inside the di-electric guided region. The Hy profile has nearly a parabolicshape in the slot, reaching a maximum value at the interface

TAO et al.: TUNABLE SUBWAVELENGTH TERAHERTZ PLASMONIC STUB WAVEGUIDE FILTERS 1193

ω

Fig. 2. Normalized Hy (y component of the magnetic field) profiles for theInSb-Air-InSb waveguide with slot with of 50 μm at different temperatures forincident frequency of 1 THz.

between the slot and the InSb material. Outside the guided re-gion, Hy field decays exponentially. It should be noted that thepenetration depth of the electromagnetic wave in InSb decreaseswhen the temperature rises which means bigger part of the THzwave propagates in the dielectric slot region than in the InSbmaterial. Therefore, the propagation constant decreases (movesclose to the propagation constant in air) with the increase of thetemperature. As a consequence to the smaller skin depth, SPPsare less dissipated, which gives rise to a larger propagationdistance.

III. ANALYSIS OF INSB PLASMONIC SLOT STUB

WAVEGUIDE FILTERS

A. Single-Stub Structure Filter

The InSb plasmonic waveguide filter with a single-stub struc-ture is shown in Fig. 3(a). The plasmonic waveguide can bemodeled as a transmission line with characteristic impedance.Fig. 3(b) shows the transmission line equivalent circuit of thesingle-stub structure. Z0 is the characteristic impedance ofthe waveguide. The characteristic susceptance of the single-stub structure is given Ys = −jY0 tan (β(ws)L), where j =√−1, Y0 = 1/Z0 , β(ws), and β(w) are the propagation con-

stant of the stub and input waveguide [36]. The transmittance ofthe single-stub structure can be expressed as follows:

Ts =4

4 +(

β (w )wβ (ws )ws

)2tan2

(2πneff L/λ

) exp(− D

LSPP

)

(5)where LSPP = (2Imβ)−1 is the characteristic propagationlength of the SP mode. The equation above exhibits a clearphysical process: the first part accounts for interference be-tween the incident wave and the wave reflected from the stub.The exponential factor describes the attenuation of the SP wave.When the phase delay satisfies 2πneff L/λ = (m + 1/2)π(m =0, 1, 2,...), the transmittance Ts reaches the minimum. There-fore, the wavelength λm of the notch of the transmittance is

(a)

(b)

Fig. 3. (a) Schematic structure of an InSb plasmonic slot waveguide filterwith the single-stub structure. (b) Transmission line equivalent circuit of thesingle-stub structure.

determined as follows:

λm =2neff L

m + 1/2. (6)

It can be seen that the wavelength λm is linear to the stublength L and the effective refractive index of the stub.

It should be noted that the transmission model cannot be veryaccurate as the phase changes at the junction of the stub andincident waveguide are not included in the model. In the fol-lowing part, the FDTD method is used to investigate the trans-mission response of the stub filters. The perfectly matched layerabsorbing boundary conditions is set at all boundaries of thesimulation domain. Since the width of the InSb slot waveguideis much smaller than the operating wavelength in the structure,only fundamental waveguide mode is supported. The incidentwave for excitation of SP mode is a TM-polarized (the magneticfield is parallel to y-axis) fundamental mode. The width w ofthe incident waveguide is set to be 50 μm and the distance L isfixed to 120 μm. Two power monitors are, respectively, set atthe positions of P and Q to detect the power of the incident andthe transmission field.

The transmission spectra of the InSb single-stub filter areshown in Fig. 4(a), which are obtained by the transmission linemodel (red dash curve) and FDTD (black solid curve and blackdot curve correspond to the lossy and lossless case, respectively).For the lossy case, one can see that there are two transmis-sion notches at the frequencies of 0.213 THz (corresponding tom = 0) and 0.642 THz (corresponding to m = 1), with min-imum transmittance of 2 and 3%, respectively. The maximumtransmittance is ∼80% due to the Ohmic loss of InSb, corre-sponding to an insertion loss of 10.5 dB and the full width halfmaximum (FWHM) are 0.136 and 0.137 THz, respectively, forthe notch frequencies of 0.213 and 0.642 THz. While for thelossless case, the maximum transmittance is 1, and FWHMs attwo notch frequencies is 0.127 and 0.118 THz, which are smaller

1194 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 12, NO. 6, NOVEMBER 2013

(a)

(b)

Fig. 4. (a) Transmission spectra of the InSb single-stub filter with the wave-guide width w = 50 μm, the stub width ws = 50 μm, and the length L =300 μm at temperature of 295 K. (b) Corresponding simulated magnetic field(Hy ) distributions for frequencies at 0.213 THz [one notch in (a)] and 0.43 THz[one peak in (a)], respectively.

than those of lossy case. It is found that the FDTD results almostagree well with those calculated by the transmission line model.There is a small deviation at the positions of the transmissionnotch because the phase changes at the junction of the stub andincident waveguide are not considered in the transmission linemodel. Submitting ω = 0.213 THz into (6) gives a total phasechange Δϕ = 0.019π for the first notch with the stub lengthL = 300 μm, and the effective index neff = 1.1302. The sec-ond resonance frequency can be approximately calculated as0.661 THz by the formula neglecting the frequency dependenceon the phase change. The magnetic fields (Hy ) at the frequen-cies of 0.213 and 0.43 THz corresponding to the transmissionnotch and peak in Fig. 4(a) are displayed in Fig. 4(b).

Fig. 5(a) shows the transmission spectra of the single-stubInSb slot waveguide structure functioning as a notch filter withdifferent stub lengths L. It reveals that the central wavelengthmoves to a longer wavelength with the increase of the stub lengthL. Fig. 5(b) shows that the central wavelength of the notch has alinear relationship with the length of the stub as expected from(6).

Fig. 6 shows the central wavelength (frequency) of the firsttransmission notch versus the stub width of ws for stub lengthof 250 μm at temperature of 295 K. It can be seen that theinitial notch of the transmission moves quickly to the shorterwavelength (blue-shift) with the increase of ws for ws < 30 μm,

(a)

(b)

Fig. 5. (a) Transmission spectra of the single-stub InSb slot waveguide filterwith different stub lengths L. (b) Relationship between the frequency (wave-length) of the transmission notch and the stub length. The width of the waveguideis w = 50 μm, and the stub width is ws = 50 μm at temperature of 295 K.

Fig. 6. Relationship between the frequency (wavelength) of the first transmis-sion notch and the stub width. The width of the waveguide is w = 50 μm, andthe stub length L = 250 μm at temperature of 295 K.

and the shift becomes small after ws > 30 μm. As revealedin (6), the above relationship between the notch position andws is mainly due to the contribution of inverse-proportion-likedependence of the effective index neff on the stub width wt asshown in Fig. 1(a). Therefore, one can realize filtering at variousrequired frequencies by properly choosing the width and/or thelength of the stub structure.

TAO et al.: TUNABLE SUBWAVELENGTH TERAHERTZ PLASMONIC STUB WAVEGUIDE FILTERS 1195

(a)

(b)

Fig. 7. (a) Transmission spectra of the InSb slot single stub waveguide filterat the temperatures of 225 and 325 K with w = 50 μm, ws = 30 μm, andL = 200 μm. (b) Relationship between the central frequency of the transmissionnotch and the temperature.

B. Actively Controlled SPs in Single-Stub Structure

As discussed in Section II, the permittivity of the InSb andthe effective index of the InSb slot waveguide are affected bythe variation of the temperature at the THz range. One canexpect that the filtering characteristics of the InSb single-stubwaveguide filter can be controlled by tuning the temperature.Fig. 7(a) shows the transmission spectra of the InSb single-stubwaveguide filter at temperatures of 225 and 325 K, respectively.The other parameters of the structure are set as: slot waveguidewidth w = 50 μm, stub width ws = 30 μm, stub length L =200 μm. From Fig. 7(a), it can be seen that the central frequencyof the transmission notch shifts from 0.231 to 0.305 THz as tem-perature increases from 225 to 325 K. The transmission of thepass band at 225 K is lower than that at 325 K because theimaginary part of the effective index the InSb slot waveguideis higher at lower temperatures [as shown in Fig. 2(b)] whichcauses higher transmission losses in the waveguide. Fig. 7(b)shows the relationship between the central frequency of thetransmission notch and the temperature. One can see that thecentral frequency of the transmission notch increases with theincrease of the temperature. This is because the wavelength ofthe transmission notch has a linear relationship with the effec-tive index of the stub which decreases with the increase of thetemperature [as shown in Fig. 1(a)]. Therefore, active control of

(b)

(a)

Fig. 8. (a) Schematic structure of a multiple stubs InSb slot waveguide.(b) Transmission spectra of the multiple stubs InSb slot waveguide with dif-ferent period numbers with the slot width w = 50 μm, the stub width ws =50 μm, the length L = 300 μm, the distance d = 100 μm, period length Λ =100 μm, and the temperature T = 295 K.

SP wave in InSb slot stub waveguide structure can be realizedby tuning the temperature.

C. Periodic Multiple-Stubs Structure Filter

It is of interest to expand the plasmonic single-stub wave-guide structure to periodic multiple-stubs structure, as shown inFig. 8(a). The InSb slot waveguide width w and the distance d arefixed to be 50 and 100 μm. Λ and P are the period length and thenumber of stubs, respectively. Fig. 8(b) shows the transmissionspectra of the multiple stubs InSb slot structure for different pe-riod numbers with w = 50 μm, ws = 50 μm, length L = 300 μm,and the distance d = 100 μm at temperature T = 295 K, whichis obtained by the FDTD method. Two wide stop bands occuraround 0.221 THz (m = 0) with a bandwidth of 0.08 THz (de-fined as the difference between the two frequencies at each ofwhich the transmittance is equal to 1%) and 0.646 THz (m =1) with bandwidth 0.09 THz for a period number P = 3. Thesecond stop band appears with a better filtering performance. Itcan be seen that the width of the stop band broadens with theincrease of the period number. The bandwidths are 0.108 and0.057 THz for P = 4 and P = 2, respectively. It should alsobe noted that large period induces a high transmission losses.The maximum transmittance of pass band decreases from 69to 52% as the period number increases from 2 to 4. There-fore, there is a tradeoff between the bandwidth and the periodnumber. For the multiple stub structure with the parameters ofRe(neff ,stub) = 1.1094 for the width of D = L + w = 350 μm,

1196 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 12, NO. 6, NOVEMBER 2013

Fig. 9. Relationship between the central frequency of the second stop band(corresponding to m = 1) and the stub length for the input waveguide widthw = 50 μm, the stub width ws = 50 μm, the distance d = 100 μm, the periodnumber N = 3, and the temperature T = 295 K.

Fig. 10. Relationship between the central frequency of the second transmis-sion stop band and the temperature. The parameters include the waveguide widthw = 50 μm, the stub width ws = 50 μm, the distance d = 100 μm, and theperiod number N = 3.

Re(neff ,w ) = 1.1222 for w = 50 μm, and the stub period ofΛ = 100 μm in the z-axis direction (see Fig. 8). One can seethat the ws Re(neff ,stub) + (Λ− ws ) Re(neff ,w ) ≈ 111.58 μm< 464.4 μm/2 (Bragg wavelength λbragg ). Thus, the structuredoes not follow the Bragg condition in the z-axis direction. Thefiltering feature of the stub structures can be attributed to themultiple-interference the reflected and transmitted fields fromeach of the three stubs structures.

Fig. 9 reveals that the central wavelength of the second stopband (corresponding to m = 1) has a linear relationship withthe length of the stub as expected from (6). The parameters ofthe structure are set as: input waveguide width w = 50 μm,stub width ws = 50 μm, distance d = 100 μm, period numberN = 3, and temperature T = 295 K. The central wavelength ofthe stop band moves to a longer wavelength with the increaseof the stub length L. In addition, similar to the InSb slot singlestub waveguide structure, the central wavelength of the stopband of multiple-stubs structures can be tuned by temperature.Fig. 10 shows the relationship between the central wavelengthof the second transmission stop band and the temperature forthe incident waveguide width w = 50 μm, stub width ws =50 μm, distance d = 100 μm, and period number N = 3. Thecentral frequency of the transmission stop band increases with

the increase of the temperature. The frequency tuning rangeis 0.13 THz for temperature from 225 to 325 K. Therefore,one can realize wide stop-band filtering function at the desiredTHz range in multiple-stubs InSb slot waveguide structures byproperly choosing the structure parameters, such as the stublength and width.

Plasmon dissipation is a critical issue for designing the plas-monic waveguide filter. A possible solution to improve the per-formance of the filter is to combine SPs with electrically andoptically pumped gain media such as semiconductor quantumwell [3] embedded into the structure. The emergence technologysuch as graphene technology is also to provide loss compensa-tion for THz spectral range [37], [38].

IV. CONCLUSION

In summary, we propose planar tunable subwavelength plas-monic stub waveguide filters in THz range. The InSb plasmonicslot waveguide has a strong confinement and guides THz wavesbeyond the diffraction limit. The single-stub InSb plasmonic slotwaveguide structure can operate as a notch filter with its centralwavelength designed by choosing the width and/or the length ofthe stub structure. As an extension to the single-stub structure,the multiple-stubs InSb slot waveguide structure can realize awide stop-band filtering function. These filtering characteristicsof the stub structures can be actively controlled through tem-perature tuning. The proposed structures open a possibility forfuture applications in ultracompact THz integration circuits.

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