Analog signal processing in the terahertz communication links using waveguide Bragg gratings: example of dispersion compensation

Transcription

1 Vol. 5, No May 17 OPTICS EXPRESS 119 Analog signal proessing in the terahertz ommuniation links using waveguide Bragg gratings: example of dispersion ompensation TIAN MA, KATHIRVEL NALLAPAN, HICHEM GUERBOUKHA, AND MAKSIM SKOROBOGATIY* Department of Engineering Physis, Éole Polytehnique de Montréal, Montreal, Québe, H3T 1J4, Canada *maksim.skorobogatiy@polymtl.a Abstrat: We study the possibility of analog signal proessing for the upoming terahertz (THz) high-bitrate ommuniations using as an example the problem of dispersion ompensation in the THz ommuniation links. In partiular, two Waveguide Bragg Grating devies (WBGs) operating in the transmission mode are detailed. WBGs are designed by introduing periodi orrugation onto the inner surfae of the metalized tubes. The resultant devies operate in a single mode regime either in the viinity of the modal utoff or in the viinity of a bandgap edge, featuring large negative group veloity dispersions (GVD). We fabriate the proposed WBGs using 3D stereolithography, and metallize them using wet hemistry. Optial properties of the fabriated WBGs are investigated both theoretially and experimentally. The results onfirm single mode guidane, relatively high oupling effiieny, as well as large negative group veloity dispersions in the range of several -1 s ps/(thz m) in the viinity of.14thz. This makes the short setions of proposed WBGs suitable for ompensating positive dispersion inurred in the THz wireless links or fiberassisted THz interonnets for signals of several-ghz bandwidth. Finally, we omment on the hallenges assoiated with the analog signal proessing in the THz spetral range. 17 Optial Soiety of Ameria OCIS odes: (4.35) Far infrared or terahertz; (6.31) Fiber optis; (6.3735) Fiber Bragg gratings; (35.438) Nanophotonis and photoni rystals. Referenes and links 1. I. F. Akyildiz, J. M. Jornet, and C. Han, Terahertz band: Next frontier for wireless ommuniations, Phys. Commun. J. 1, 16 3 (14).. T. Nagatsuma, S. Horiguhi, Y. Minamikata, Y. Yoshimizu, S. Hisatake, S. Kuwano, N. Yoshimoto, J. Terada, and H. Takahashi, Terahertz wireless ommuniations based on photonis tehnologies, Opt. Express 1(), (13). 3. S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Shmogrow, D. Hillerkuss, R. Palmer, T. Zwik, C. Koos, W. Freude, O. Ambaher, J. Leuthold, and I. Kallfass, Wireless subthz ommuniation system with high data rate, Nat. Photonis 7(1), (13). 4. T. Nagatsuma, G. Duournau, and C. C. Renaud, Advanes in terahertz ommuniations aelerated by photonis, Nat. Photonis 1(6), (16). 5. L. J. Chen, H. W. Chen, T. F. Kao, J. Y. Lu, and C. K. Sun, Low-loss subwavelength plasti fiber for terahertz waveguiding, Opt. Lett. 31(3), (6). 6. A. Hassani, A. Dupuis, and M. Skorobogatiy, Low loss porous terahertz fibers ontaining multiple subwavelength holes, Appl. Phys. Lett. 9(7), 7111 (8). 7. A. Hassani, A. Dupuis, and M. Skorobogatiy, Porous polymer fibers for low-loss Terahertz guiding, Opt. Express 16(9), (8). 8. T. Ma, A. Markov, L. Wang, and M. Skorobogatiy, Graded index porous optial fibers dispersion management in terahertz range, Opt. Express 3(6), (15). 9. L. Vinetti, Hollow ore photoni band gap fiber for THz appliations, Mirow. Opt. Tehnol. Lett. 51(7), (9). 1. K. Nielsen, H. K. Rasmussen, P. U. Jepsen, and O. Bang, Porous-ore honeyomb bandgap THz fiber, Opt. Lett. 36(5), (11). 11. M. Skorobogatiy and A. Dupuis, Ferroeletri all-polymer hollow Bragg fibers for terahertz guidane, Appl. Phys. Lett. 9(11), (7). #86951 Journal 17 Reeived 16 Feb 17; revised Apr 17; aepted 8 Apr 17; published 3 May 17

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3 Vol. 5, No May 17 OPTICS EXPRESS Introdution With the demand of wider bandwidths and higher bit rates, using terahertz frequenies for wireless ommuniations is experiening a surge of attention in reent years. Various THz wireless ommuniation systems with arrier frequenies as high as.6thz and transmission data rates of up to 1Gbit/s have been developed and investigated [1 4]. However, appliations of these THz ommuniation systems are still limited due to inherent hallenges posed by the free spae propagation (FSP) modality, suh as strong dependene on atmospheri onditions, rapid divergene of the THz beams espeially at the lower arrier frequenies, as well as line-of-sight nature of the links. Partiularly, due to self-diffration and THz relatively long wavelength ~1mm, realizing low divergent THz beams requires the use of large beam diameters and fousing optis that an be as big as several tens of entimeter in diameter for even short ommuniation links of several tens of meters. Additionally, due to strong diretionality of the THz beams, wireless ommuniations aess to partially bloked areas an be problemati, thus, requiring additional THz steering solutions for reliable ommuniations. At the same time, low-loss terahertz fibers and waveguides offer good solutions to some of the limitations aused by the free spae THz propagation, and an have their own nihe appliations in THz ommuniations. There are some lear advantages of using THz waveguides and fibers. Partiularly, as light propagates through sealed THz fibers, influene of the atmospheri onditions on the ommuniation link quality is minimized. Additionally, THz fibers are flexible and of small diameters, hene, allowing aess to otherwise physially obstruted areas. Finally, THz fiber size is typially omparable to the wavelength of light, thus enabling ompat several-mm in diameter ommuniation links with very small footprint and low signal leakage outside of the fibers, whih is of speial interest for the ultra-high bandwidth on-hip interonnets. In the past deade, various THz fibers with low transmission losses (<.1m 1 ) have been proposed and demonstrated. Generally, suh waveguides maximize the fration of the guided power in the low-loss gas [5 15]. Among these designs, one generally distinguishes subwavelength THz fibers that guide using total internal refletion (TIR) mehanism [5 8] and hollow ore fibers that guide using either photoni bandgap onfinement [9 13] or anti-resonant refletion (ARROW) [14,15]. While loss redution in THz fibers an be onsidered as a solved problem, fiber dispersion management remains generally an unsolved problem. In free spae systems, Group Veloity Dispersion (GVD) of transmission media results in the temporal broadening of the THz pulses that are typially generated using THz-TDS (Time Domain Spetrosopy) systems. In the fiber-based ommuniation systems, modal GVD similarly leads to pulse broadening, and hene redution in the maximal bit rate supported by the fiber ommuniation link of fixed length. This happens beause adjaent bits of temporal size Δ t in the pulse train broaden and beome Δ t after propagation over length L in the fibers with GVD D, as Δt L D/ Δt. One onsiders that the pulse train gets srambled when adjaent bits overlap whih happens for the bits of temporal duration shorter than Δt D L. As detailed in Ref [16], the dispersion-limited maximum bitrate an be given as Bm = 1/(4 D L). For example, onsidering a typial THz fiber dispersion of D 1 ps/(thz m), in order to obtain even a moderate bit rate of 1 Gbit / s, the maximum transmission length is limited to ~6 m. On the other hand, transmission rates of 1 Gbit / s would limit transmission links to only 6 m with standard THz waveguides. Note that even in the wireless links, where dispersion of dry air at THz frequenies is relatively small D ps/(thz m) [17], wireless link length will still be limited due to pulse dispersion. For example, for a wireless link of length 5 m, the maximal bitrate supported by suh a link without dispersion ompensation will be 1 Gbit/s. In addition to

4 Vol. 5, No May 17 OPTICS EXPRESS 111 pulse distortion due to group veloity dispersion, frequeny dependent absorption loss due to water vapor will further ontribute to pulse distortion, and in some ases one might prefer using fiber-based THz links over free spae THz links as a bakup option for reliability onsiderations (for example, when onsistent performane is required at any weather onditions). Overall, while effets of dispersion on signal degradation appear modest in the free spae links, they are a major limiting fator in the waveguide-based interonnets, and therefore should be further addressed and mitigated. A straightforward way of reduing the GVD of a THz waveguide link is via a judiial design of the waveguide geometry. For example, in our previous work [8], we have demonstrated an effetively graded index porous THz fiber inorporating an air-hole array of variable air-hole diameters and inter-hole spaing. Compared to the porous THz fiber featuring regular air-hole array, the graded index fiber suessfully redued both the individual modal and intermodal dispersions. Further redution of the GVD below.1ps/(thz m), however, proved to be hallenging and there are only few examples in the literature of suh THz waveguides [18]. In order to address the issue of pulse dispersion, one an pursue an alternative approah. Thus, instead of reduing GVD of a transmission link, one an rather use dispersion ompensation at the link end, or dispersion pre-ompensation at the beginning of the link. There are several requirements for these effetive dispersion ompensation systems. First is effiient oupling between the transmission fiber and the dispersion ompensating waveguide. Seond is a single mode operation of both the transmission fiber and dispersion ompensating waveguide, in order to avoid multimode interferene effets and intermodal dispersion. Third is the requirement of the high negative dispersion of the dispersion ompensating waveguide D DCW in order to ompensate with a short devie length L DCW for the positive dispersion D L inurred over a transmission link of length L L so that DDCW LDCW + DL LL =, and L DCW L L [19]. Finally, dispersion ompensation ondition has to be satisfied over a wide enough frequeny range omparable to the ommuniation signal bandwidth. We note that, several on-hip devies based on parallel plate THz waveguides or spoofplasmon modes propagating between planar interfaes have been reported for the dispersion ontrol in the THz domain. For example, T. Fobbe et al. [] proposed a hybrid waveguide made by onneting a dieletri-metal and a metal-metal parallel plate waveguides. This resulted in a broadband tunable THz devie offering a small negative GVD on the order of tens of ps /( THz m). Meanwhile, Gao et al. [1] used a graded plasmoni resonator hain for the dispersion ontrol in the mirowave range. The authors theoretially predited that loalized spoof surfae plasmon polaritons in the resonator hain provide both positive and negative dispersion that an be easily tailored by tuning the strutural parameters. In the abovementioned works, however, the authors did not onsider the tradeoff between the bandwidth and the value of the GVD, whih is important and indispensable for pratial appliations. Moreover, for both proposed waveguides, the oupling effiienies with the external THz beam are not studied. Indeed, the use of a planar geometry suggests low oupling effiienies in the absene of mode onverters, whih would further limit pratial appliations of suh devies. In our paper, we pursue a different approah. Partiularly, we aim at designing a waveguide-based dispersion ompensator with the highest negative dispersion possible, while keeping its operational bandwidth large enough to support THz wireless ommuniations (5-1GHz). This design approah leads to shorter devies, and, therefore, lower propagation losses are expeted ompared to devies with smaller GVDs. Being hollow and irularly symmetri, our waveguides are inherently suitable for operation with the free spae beams. Thus, we expet muh higher oupling effiienies for our devies and lower modal propagation losses ompared to those of devies based on planar geometries. Furthermore, we

5 Vol. 5, No May 17 OPTICS EXPRESS 1113 present a disussion of the tradeoff between the waveguide GVD value and the devie bandwidth. Finally, we present several strategies aimed at inreasing the exitation effiieny of our waveguides with a free-spae Gaussian beam, while keeping a single mode operation. While fousing on the problem of dispersion ompensation in the THz ommuniation links, we reognize that this is a part of a muh broader researh area of analog signal proessing in the THz spetral range. Analog signal proessing is, in fat, a very ative researh area in the near-ir spetral range where most of the opti ommuniation systems are developed, while in THz this subjet is steadily gaining its reognition. In the near-ir one of the key enabling devies for analog signal proessing is a hirped Fiber Bragg Grating (FBG) that operates in the refletion mode. Used in ombination with fiber ouplers and fiber opti irulators, hirped FBGs an be used to perform a variety of signal proessing operations inluding dispersion ompensation [], signal modulation [3], wavelength division multiplexing [4], to name a few. Unfortunately, in the THz spetral range, there are only few examples to date of the advaned waveguide omponents that are required for building analog signal proessing devies using waveguide Bragg gratings (WBGs) in the refletion mode. While, some of the THz omponents have been demonstrated suh as ouplers [5] and FBGs [6], others like irulators have only being partially developed [7]. Therefore, in our study of the dispersion ompensation in the THz range we resort to the use of waveguide Bragg gratings operating in the transmission mode (rather than refletion mode), whih are onsiderably simpler to integrate into a ommuniation link. Moreover, in order to mitigate relatively high losses of the materials in the THz spetral range, we use hollow ore waveguides in our designs of the dispersion ompensating WBG devies. In this paper, we explore theoretially and experimentally two novel waveguide Bragg gratings (WBGs) operating in the transmission mode. The goal is to demonstrate the WBGbased devies apable of dispersion ompensation in THz ommuniation links that operate in a single mode regime, have high negative dispersions, and show high oupling effiieny to the inoming linearly polarized Gaussian beam. The proposed WBGs were first fabriated from photosensitive resins using 3D stereolithography, whih is a highly versatile and robust tehnique for the fabriation of optial devies at terahertz frequenies [13, 8 3]. Subsequently, the printed devies were metalized with silver nanolayers using the wet hemistry approah. Optial properties of the fabriated WBGs were then investigated numerially using the finite element method, as well as experimentally using both the ontinuous-wave terahertz (CW-THz) spetrosopy and terahertz time-domain spetrosopy (THz-TDS) imaging. The results onfirm both single mode operation and large negative dispersion for both fabriated WBGs near 14GHz. Speifially, for the first WBG, the single mode operation is attainable over the frequeny range of GHz, where the dispersion varies between 467 and 191 ps / ( THz m). Meanwhile, for the seond WBG, we ahieve a single mode operation range between 137 and 14GHz, where the dispersion is in the range of 33 and 138 ps / ( THz m).. Dispersion ompensation strategies using modes of a hollow ore waveguide As disussed in the introdution, for the effiient dispersion ompensation, several requirements should be satisfied by the dispersion ompensating devie, inluding effiient exitation, single mode operation, large negative dispersion, and sizable bandwidth. At THz wavelengths, some of these onditions an be satisfied using the large diameter metalli tubes that an show exellent oupling with a foused THz beam and high value of the negative GVD near the modal ut-off frequenies. In Fig. 1(a), we shematially show dispersion relations of the fundamental mode and the first higher order mode of a irular metalli tube, where ω and ω are the utoff frequenies for the fundamental mode (red line) and the first 1

6 Vol. 5, No May 17 OPTICS EXPRESS 1114 higher order mode (blue line). For frequenies between 1 ω and ω, metalli tube supports a single mode guidane with negative dispersion (gray region in Fig. 1(a)). As optial harateristis of the metalli tubes are strongly dependent on their ore size, one should onsider trade-offs between the ore size and the value of the negative dispersion, as well as single mode versus multimode operation, when using suh waveguides for dispersion ompensation. For example, the dispersion relation of guided modes in a metal tube of radius r is given by [31]: 1 mn ( ) βmn = ω ω (1) mn where ω = Xm n/ r is the modal utoff frequeny, and X mn is the n-th nonvanishing root of the m-th order seond kind Bessel funtion Ym ( ω ) (for TE modes) or the m-th order first kind Bessel funtion Jm ( ω ) (for TM modes). If λ is the operational wavelength, single mode regime is realized for the tube radii of.9λ < r <.38λ. A typial beam waist size of the foused diffration limited THz Gaussian beam is w ~4 λ f / ( π D ) [3], where f and D are the foal distane and diameter of the fousing lens, respetively, with field intensity dependent as E exp r / w. In our system, we use f = D = 1m, thus resulting in the Gaussian beam waste size of w ~ λ. Moreover, we note that in general, high quality THz paraboli mirrors and lenses that are urrently available ommerially feature f / D 1 ratios, whih is inherent to the design of lassial fousing optis. Therefore, resultant Gaussian waist size is typially larger than 1 λ for any hoie of the standard fousing optis. We note, that in order to ahieve effiient oupling to the fundamental HE 11 mode, the tube size has to be larger than the Gaussian beam size, namely r 1.5w [33]. However, when trying to operate near the first utoff frequeny (region of high negative dispersion), the tube radius should be r.9λ w λ, whih leads to ineffiient oupling between the foused Gaussian beam and the tube fundamental mode near its utoff frequeny. In order to inrease the oupling effiieny between the foused Gaussian beam and the fundamental mode of a tube, one needs either to inrease the tube diameter or, equivalently, inrease the operational frequeny. For instane, in the ase of a Gaussian beam with a waist of λ, the oupling effiieny is optimal and equal to (96%) if the tube size is r =.93λ. However, at this frequeny, the number of m = 1 modes supported by the tube will be 1 and the propagation regime is multimode. Fig. 1. Shematis of the dispersion relations for the few lowest order modes in (a) metal tube waveguide, (b) and () metal tubes waveguide with periodi orrugations. In gray, we show spetral regions of single mode operation and negative modal dispersion. Blak line: light line of air. Red urve: the fundamental mode. Blue urve: a higher order mode.

7 Vol. 5, No May 17 OPTICS EXPRESS 1115 In order to promote higher oupling effiieny, while maintaining high negative dispersion and single mode operation, we pursue two omplimentary approahes that are similar in spirit to those desribed in [34, 35]. One is to use a higher order mode that is ompatible in symmetry with a linearly polarized Gaussian beam (HE 1, for example), while operating in the viinity of its ut-off frequeny ω. At the same time, in order to guarantee single mode operation, we have to suppress oupling to the fundamental HE 11 mode. This an be ahieved by introduing periodi orrugation into the tube inner ore, while opening a bandgap (frequeny range from 1 1 ω bl to ω bu shown in gray in Fig. 1(b)) for the fundamental HE 11 mode in the viinity of the HE 1 ut-off frequeny ω. Alternatively, one an open a bandgap for the higher order mode HE 1 [frequeny range from ω bl to ω bu shown in gray in Fig. 1()], while operating in the viinity of the upper bandgap edge of the fundamental mode 1 ω bu. In this ase, the waveguide is again single mode, operation at higher frequenies allows better mode mathing between the foused Gaussian beam and the fundamental HE 11 mode, and finally, strong negative dispersion is ahieved due to operation lose to the upper bandgap edge of the fundamental mode. Partiularly, in the first WBG1, we implement the senario highlighted in Fig. 1(b), where single mode operation is using the HE 1 mode operating in the viinity of it utoff frequeny, while guidane in the fundamental HE11 mode is suppressed due to the bandgap effet. In the seond WBG we realize the senario highlighted in Fig. 1(), where single mode operation is using the fundamental HE 11 mode operating in the viinity of the orresponding upper bandgap edge, while guidane in the fundamental HE 1 mode is suppressed due to the bandgap effet. Both WBGs are designed and optimized for the dispersion ompensation in the viinity of the operation frequeny of ~14GHz. This partiular hoie for the operation frequeny is ditated by the relatively low atmospheri absorption at suh frequenies (atmospheri transpareny window), as well as ommerial availability of the high-power THz soures and detetors that will be used in the first-generation THz ommuniation systems [36]. Suessful appliation of the dispersion ompensation fibers in the THz ommuniation links also requires that the GVD of the dispersion ompensation fiber remains onstant over the signal bandwidth. Otherwise, problems aused by the higher order dispersion an be pronouned [3]. Here, we explore briefly the relation between the value of GVD and the operation bandwidth for the dispersion ompensating fibers operating either in the viinity of the modal utoff frequeny (WBG1) or in the viinity of the bandgap edge (WBG). Fig.. Shematis of the dispersion relations for (a) the mode operating lose to its ut-off frequeny ω and (b) the mode operating lose to its upper bandgap edge ω bu. () Shemati of the orresponding modal GVD. In gray: the spetral bands used for the dispersion ompensation, they are entered at ω and have a bandwidth of Δ ω. For the first waveguide Bragg grating (WBG1), we use the first higher order mode HE 1 ompatible in symmetry with the linearly polarized Gaussian beam for the dispersion

8 Vol. 5, No May 17 OPTICS EXPRESS 1116 ompensation, and operate in the frequeny range lose to the HE 1 ut-off frequeny [shown in gray in Fig. (a)]. The GVD of the HE 1 mode an be derived from its modal dispersion relationship (Eq. (1) as: β 1 ω DWBG1 = = () 3/ ω ω ω On the other hand, the seond waveguide Bragg grating (WBG) is operated in the frequeny range lose to the upper bandgap edge of the fundamental (HE 11 ) mode [shown in gray in Fig. (b)]. In this frequeny range, the dispersion relation of the mode an be approximated as π 1 β = + ω ωbu, where Λ is the periodiity of orrugation and γ is a ertain Λ γ oeffiient. The GVD of the fundamental HE 11 mode an then be approximated as: D β ω 1 bu WBG = = 3/ ω κ ω ω bu (3) where ω b is the upper edge of the bandgap for the fundamental (HE 11 ) mode. In general, we define the bandwidth Δ ω as a frequeny range entered at ω within whih the dispersion hanges at most by a fator of x, namely D( ω Δ ω/) = x D. Here, D = D( ω) is the value of the dispersion at the entral (arrier) frequeny, and x is a ertain design parameter ditated by the speifiations of the ommuniation system. In this paper, 3/ we hose x = ~.83 as this hoie results in a simple expression for the relation between the desired dispersion D and the bandwidth Δ ω, whih is given by: D 3/ ( ω Δ ω/) = D ω Δω * 3 1/ D (4) where ω* = ω for the WBG1, while ω * = ω bu for the WBG [see Fig. ()]. In our derivations, we assumed that Δ ω ω, ω, ωbu, whih is indeed the ase for the proposed waveguide Bragg gratings as the dispersion ompensation is done at frequenies ~14GHz, while the desired bandwidth is ~1GHz. Finally, from Eq. (4) it follows the estimate for the maximal dispersion of the dispersion ompensation waveguide WBG1 or WBG operating at ~14GHz and having ~1GHz bandwidth is D 1985ps / (THz m). 3. Fabriation of the waveguide Bragg gratings The proposed waveguide Bragg gratings are realized by introduing triangular steps inside of a hollow ore tube. Both Bragg gratings omprise 4 periodially arranged triangular steps. The shape of the steps is hosen for the ease of 3D printing. The 3D model of the proposed waveguide Bragg gratings is shown in Fig. 3(a). The strutural parameters, inluding the tube ore diameter D, the base size p and the height h of triangular steps, were optimized to enable effetively single mode propagation and large negative dispersion of a hosen guided mode (HE 11 or HE 1 ) in the viinity of.14thz, as detailed in the following setions. For the sake of omparison, we also fabriated hollow ore waveguides (without orrugation) with the same ore diameters and total lengths as those of the orresponding waveguide Bragg gratings. The prototypes of all the aforementioned waveguides were fabriated using a stereolithography 3D printer (Asiga Freeform PRO) with the photosensitive resin

9 Vol. 5, No May 17 OPTICS EXPRESS 1117 (PlasCLEAR). The printer has a resolution of 5µm along the x- and y-axes, and 1µm along the z-axis. Subsequently, the printed prototypes were leaned and immersed into isopropanol solution for ~1hrs, and then fully oated with a silver layer using a wet hemistry oating method as detailed in [37]. Fig. 3. (a) 3D shemati of the waveguide Bragg grating. Insert: zoom of the grating. (b) Disseted waveguide Bragg gratings 1 and. () and (d) SEM images of the area highlighted by the red irle in (b) with magnifiations of 5 and 1, respetively. The fabriated waveguide Bragg gratings are shown in Fig. 3(b) (only one half of the disseted waveguides are shown). The surfae of the periodi orrugations was imaged using a sanning eletron mirosope (SEM). As shown in the SEM images [Figs. 3() and 3(d)], the wet hemistry oating produes a uniform deposition of silver partiles on the waveguide surfae. The grain size of the silver partiles varies from hundreds of nanometers to several mirometers. Resulting silver layer has thikness whih is muh larger than the skin depth for the THz waves ( 1nm ) whih prevents them from leaking into the dieletri ladding. As roughness of the oated silver layer ( ~1μm ) is deeply subwavelength (THz wavelength is ~1mm), thus only a small sattering loss is expeted due to surfae roughness, whih we have onfirmed experimentally by omparing transmission through Ag-oated tubes and polished opper tubes. 4. Numerial modeling Light guidane in the proposed waveguide Bragg gratings was first analyzed numerially using COMSOL finite element software. The omputation ell is that of the unit ell of an infinitely periodi struture orresponding to the grating under study. In the transverse diretion, the unit ell is terminated by a perfet eletri ondutor that simulates the air/metal interfae. In the longitudinal diretion, Bloh-Floquet boundary ondition is used. In Fig. 4, we demonstrate the omputed band diagrams for the two waveguide Bragg gratings shown in Fig. 3(b). The olor ode for the modal dispersion relations orresponds to the absolute value of the modal oupling oeffiient to the 3D Gaussian beam foused into the enter of the WBG input faet. The oupling oeffiients are omputed using ontinuity of the transverse eletri field omponents aross the oupling interfae between the inident Gaussian beam and the guided modes [38]: C m * * ( Em Hin+ Ein Hm) drdz * * ( Em Hm ) ( Ein Hin ) = Re drdz Re drdz where integration is performed over the whole unit ell. Field distribution in the linearly polarized 3D Gaussian beam is given by [39]: (5)

10 Vol. 5, No May 17 OPTICS EXPRESS 1118 ( ) ( ) ( ) ( ) E rz, = y exp rw exp ikz (6) in where w is the beam waist size (radius). The Gaussian beam waist size is dependent both on the frequeny of THz light and the fousing optis so that w = 4 λ f /( πd), where f and D are the foal length and the diameter of the paraboli mirror (or a lens) used for beam fousing. In our CW-THz experimental setup disusses in setion 5, f = 1 m and D = 5 m, and the beam waist radius is estimated to be w ~λ. Magneti field of the external Gaussian beam an be alulated from Eq. (5) using one of the Maxwell equations, i Bin = E in. ω Fig. 4. Band diagrams of the two proposed waveguide Bragg gratings, (a) WBG 1 and (b) WBG shown in Fig. 3. Color of the modal dispersion relations refers to the value of the modal oupling oeffiient to the foused Gaussian beam with the beam waist of w ~λ. Gray regions orrespond to the spetral ranges where WBGs are effetively single mode. Beige regions refer to the bandgaps of the fundamental HE 11 mode. In Fig. 4, we present modal dispersion relations of the guided modes with angular momentum equal to 1 ( m = 1 ). Modes with other angular momenta would not be exited by a entrally oupled, linearly polarized Gaussian beam due to symmetry mismath. For larity of presentation, in Fig. 4 we only present modes with oupling oeffiients higher than.1. We design WBG1 to operate in the single mode regime using the HE 1 mode for the dispersion ompensation. The geometrial parameters that we use for WBG1 are D = 4.4mm, P = 1.mm, and h =.35mm. From Fig. 4(a) we observe two spetral regions of the effetively single mode guidane (shown in gray) whih are GHz and GHz. Note that stritly speaking, within these two frequeny ranges there are two modes that an propagate. However, the oupling oeffiient is virtually zero for one of the two modes as it orresponds to the bakward propagating mode. In these frequeny ranges, as shown in more details in Fig. 5(b), the dispersion of the HE 1 mode varies between 1336 and 78 ps / ( THz m) and 467 and 191 ps / ( THz m ) respetively. Using riterion (4) we onlude that the operation wavelengths, dispersion values and the maximal bandwidths of the dispersion ompensator WBG1 are ω = 131.5GHz, D = 1568ps / (THz m), Δ ω = 5.GHz, and ω = 141.6GHz, D = 68.6GHz, Δ ω = 8.5GHz respetively. In Figs. 5() and 5(d) we present longitudinal flux distributions S z of the HE 1 mode at 13GHz and 14 GHz. By observing the modal distribution, we onfirm that the mode used for dispersion ompensation is similar to the HE 1 mode of a simple tube, and it remains unaffeted by other guided modes and plasmoni exitations within the frequeny range of interest.

11 Vol. 5, No May 17 OPTICS EXPRESS 1119 Fig. 5. Dispersion ompensator WBG1. (a) Dispersion relations of the guided modes and (b) HE 1 mode group veloity dispersions within the spetral regions of the effetive single mode guidane (gray regions) in the viinity of 13GHz and 14 GHz. (), (d) Longitudinal flux distributions S z of the HE 1 mode at 13GHz and 14 GHz. Next, we design WBG to operate within bandgaps of the higher order modes, where waveguide an be onsidered effetively single mode with maximal oupling oeffiient to the HE 11 -like mode. The geometrial parameters that we use for WBG are D = 9. mm, P = 1.35mm, and h = 1.9mm. From Fig. 4(b), we observe that there are two spetral regions of the effetively single mode operation. One is in the viinity of 14 GHz, while another one is in the viinity of 16 GHz. Unlike the ase of WBG1, both positive and negative dispersions are possible for the WBG, as shown in Fig. 6(b). This is aused by avoiding rossing with the higher order modes (inluding plasmoni modes). In Fig. 6() and 6(d), we present longitudinal flux distributions S z of the HE 11 -like mode at two frequenies labelled by red irles and the letters A and B in Fig. 6(a). Obviously, the fundamental mode of WBG is a HE 11 -like mode at 147GHz, while it is deorated with the plasmoni exitation present near the metal surfae at 137GHz. In fat, these plasmoni exitations an signifiantly modify the dispersion relation of the ore guided modes (suh as the fundamental HE 11 mode) via the phenomenon of avoided rossing as evidened from Fig. 6. In this paper, we do not explore the regime of the hybridized ore guided / plasmoni modes, mainly for the reason that in this partiular example although the resultant negative dispersion an be extremely high, the bandwidth of operation is limited. That said, we believe that hybridized ore guided / plasmoni modes have strong potential in dispersion ompensation, and they will be studied in more details in our following works.

12 Vol. 5, No May 17 OPTICS EXPRESS 11 Fig. 6. Dispersion ompensator WBG. (a) Dispersion relations of the guided modes and (b) HE 11 mode group veloity dispersions within the spetral regions of the effetive single mode guidane (gray regions) in the viinity of 14 GHz and 16 GHz. () and (d) Longitudinal flux distributions S z of various guided modes labelled by red irles in (a). In the spetral ranges of single mode guidane of the fundamental HE 11 -like mode in the viinity of 14GHz and 16GHz, mode dispersions are negative and vary strongly. Thus, in the range of GHz (gray region at the bottom of Fig. 6(b)), the dispersion varies between 33 and 138 ps / ( THz m). For frequenies within this range, all the higher order modes are eliminated due to bandgap effets, and only the fundamental HE 11 -like mode remains. At the same time, in this frequeny range the modal dispersion value hanges markedly, therefore, the problem of inreasing the operational bandwidth by keeping the dispersion value relatively onstant still deserves further study. Similarly, in the viinity of 16 GHz, the single mode guidane is realized over GHz, where the dispersion hanges from 983 ps / ( THz m) to 13 ps / ( THz m ). Using riterion (4a) we onlude that the operation wavelengths, dispersion values and the maximal bandwidths of the dispersion ompensator WBG are ω = 139.GHz, D = 17.8ps / (THz m), Δ ω = 3.GHz, and ω = 16.GHz, D = 91.5ps / (THz m), Δ ω =.8GHz. 5. Optial haraterization of the waveguide Bragg gratings 5.1 Transmission measurements Optial haraterization of the fabriated waveguides was arried out using ontinuous wave Terahertz (CW-THz) spetrosopy system (Toptia Photonis) shown shematially in Fig. 7. The setup has two distributed feedbak (DFB) lasers with slightly different enter wavelengths and a uniform power (~3mW eah) operating in the teleom region. A 5:5 oupler ombines and splits the two wavelengths equally into the emitter and detetor arms respetively. The beat signal of the two lasers is split into two beams by the fiber oupler and then delivered to the emitting and deteting antennas using two single mode polarization maintaining fibers. Two fiber strethers, whih are driven by the piezo atuators and strethed in the opposite diretions, are onneted to both arms. The deteted signal is a photourrent I ph, whih depends both on the amplitude E THz of the measured terahertz eletri field and on the phase differene Δφ between the two optial beams [4]: ( ϕ ) ( π ν ϕ ) I E os Δ = E os Δ L + (7) ph THz THz

13 Vol. 5, No May 17 OPTICS EXPRESS 111 where Δ L denotes the optial path differene between the transmitter arm and the reeiver arm, while ϕ denotes the phase differene between the two arms for a neutral position of the strether. The optial path length Δ L is varied in a predefined fashion using fiber strethers, whih allows extration of the field amplitude and phase by fitting periodi modulation in the measured photourrent using Eq. (7). The symmetri arrangement of the fiber strethers provides additional path delay as well as phase noise anelation that an be aused by the variation in the environment. The path lengths between the emitter and the detetor arms are balaned (in the empty onfiguration) to have a flat frequeny independent phase. The THz waves are generated by the emitter photomixer whih emits the frequeny differene between the two lasers whih are furthermore modulated using the bias voltage for lok-in detetion. By tuning the lasing wavelengths, suh a system an generate a tunable THz radiation in the range of 5 GHz-1. THz, with bandwidth as narrow as 4MHz. The THz beams were foused into and olleted from the waveguides using the gold oated refletive optis (flat and paraboli mirrors). Two metalli apertures were plaed at the input and output faets of the waveguide to blok any stray light and avoid oupling into plasti ladding. The measurements were obtained in the frequeny range 1-18GHz with 1MHz frequeny resolution and 6ms integration time onstant. The generated photourrent in the detetor was reorded along with the phase information as a funtion of frequeny. Fig. 7. Shemati of the CW-THz setup for optial haraterization of WBGs. The measured THz eletri fields and the orresponding transmission spetra for the two fabriated waveguide Bragg gratings (red urves), metalized tube and opper tube (blue and magenta urves) are shown in Fig. 8. The referenes (blak urves) were aquired by removing the measured waveguides, while displaing one set of the paraboli mirrors to share the foal plane with that of the seond stationary paraboli mirror. Additionally, at the foal point, we used an aperture with an opening similar to the ore size of the measured waveguide. The measured referene photourrent shows the prominent standing wave with beat frequeny of 4 GHz, whih is due to the Fabry Pérot interferene within the silion lenses of the photomixers. The omplex transmission oeffiient through a waveguide is: T( ω) = Et / Er = t( ω) exp( iϕ ( ω) ) (8) where E t and E r are the omplex eletri fields measured with and without the tested waveguide, t( ω ) is the transmission amplitude, while ϕ ( ω ) is the phase. Firstly, we haraterized the performane of the metalized 3D printed tube, and ompare it with that of the ommerial opper tube of the same ore size and length. As seen in Fig. 8 (d), the metalized tube feature very similar transmission spetrum as that of the ommerial

14 Vol. 5, No May 17 OPTICS EXPRESS 11 opper tube in the frequeny range of 1-18GHz. From this, we onlude that our metallization reipe was hosen orretly. Next, we haraterize transmission through the fabriated waveguide Bragg gratings. For the WBG1 [left olumn in Fig. 8], there are two low-transmission spetral ranges GHz and GHz with enter frequenies of 13.8GHz and 143.3GHz (gray regions in Fig. 8()). Within these regions, the transmission is lower than 6.34 db ompared to db outside of these regions. The boundaries of the regions are defined as having e 1 value by field ( 4.34 db) relative to that outside of these regions ( db). It is remarkable that the spetral positions of these low transmission windows orrespond almost exatly to the position of the theoretially predited bandgaps for the fundamental HE 11 -like mode (gray regions in Fig. 5(a)), thus onfirming an exellent agreement between the theoretial preditions and experimental measurements. At the same time, we observe that we were not suessful in the exitation of the HE 1 mode whih we planned to use for dispersion ompensation. There are several possible reasons for this. First, although, WBG1 was designed to operate with the HE 1 mode within the bandgap of the HE 11 -like mode, its theoretial exitation effiieny is low ( C HE1 <5%, see Fig. 5(a)). Furthermore, low exitation effiieny of the HE 1 mode an be related to its higher (than HE 11 -mode) absorption and sattering losses, as well as higher sensitivities to the alignment errors. Fig. 8. (a) and (b) Measured eletri fields. () and (d) Corresponding transmission spetra of the fabriated WBGs, polished opper tubes and uniform metalized plasti tubes. Left olumn: WBG1. Right olumn: WBG. Gray regions in () and (d) orresponds to the spetral ranges where WBGs are effetively single mode. Beige regions refer to the bandgaps of the fundamental HE 11 mode. We now onsider the WBG. First, we observe the four low transmission windows (beige bands in Fig. 8(d)) with enter frequenies of 118GHz, 135GHz, 153GHz, and 187GHz that have transmission losses over 15dB. These regions oinide well with the theoretially predited regions of week exitation of the HE 11 -like mode shown as beige bands in Fig. 4(b). These spetral regions orrespond either to the bandgaps for the HE 11 -like mode or the regions of antirossing with the higher order or plasmoni modes. Outside of these low transmission regions, and near the target operational frequenies of 14GHz and 16GHz the transmission amplitude is in the 1dB to 5dB range, as seen from Fig. 8(d).

15 Vol. 5, No May 17 OPTICS EXPRESS 113 Fig. 9. (a) Numerially omputed band diagram of the WBG. (b) The measured phase (red dots), a orresponding polynomial fitting (blak solid lines). () The omparison between the experimentally measured dispersion (blak solid lines) and the theoretially omputed dispersion of the fundamental HE 11 -like mode (olored dots). Finally, in Fig. 9, we present omparison of the experimentally measured and theoretially predited group veloity dispersions for the seond waveguide Bragg grating. Experimentally measured GVD is omputed from the phase data as ( ϕ / L) / ω, where ϕ is the measured phase and L is the waveguide length. Before omputing seond order derivative, a fifth order polynomial fitting is used on the experimental data in order to alleviate the noise ontribution (see Fig. 9 (b)). Theoretial dispersion is omputed using the dispersion relation of the mode with the highest oupling oeffiient to the exitation Gaussian beam shown in Fig. 9(a). In the single mode regions (shown in gray in Fig. 9), both theoretial and experimental results onfirm the strong negative dispersions of the WBG dispersion ompensator. Thus, in the GHz range, the experimentally measured dispersion varies from 5 to 1 ps/ THz m, while in the GHz range, dispersion varies from to 6 ( ) ps/ ( THz m) experimentally measure ones. 5. Model imaging of the waveguide output. The numerial results show good math with the Fig. 1. Shemati of the setup used for near field imaging. Next, we investigate the single mode operation of our waveguide Bragg gratings by mapping the model distribution of the eletri field ( E y ) at the waveguide output end as a funtion of frequeny. Model images of the output field profiles were obtained using a terahertz timedomain spetrosopy (THz-TDS) fiber-oupled imaging system. The shemati of this imaging system is shown in Fig. 1. There, the emitter, omprising of an interdigitated antenna array (Batop ) and a silion lens, is used for generating a slightly divergent THz

16 Vol. 5, No May 17 OPTICS EXPRESS 114 beam whih is i then foused d onto the wav veguide using a Teflon lens. The detetor aantenna is plaed behind d a metalli pin nhole of 1mm diameter, d mouunted on a 3D sstage, and it is optially exited using a polarization maintaining fiber/lens fi ombbination. The ggap between thhe pinhole f is about.5mm, whih is muh smalller than the waavelength and the waveguide output faet r of 9 9m mm and a steep size of.55mm, the at 14GHz ( ~ mm ). Witth a saning range vides a modal pattern p with pixels att eah frequeny. sanning prov Fig (a) Modal images (upper row) and orrespondinng simulations (llower row) of thee normaalized output ( E ) field profilee of the WBG for frequenies iin the viinity off y 14GH Hz. The blak solid irle refers to o the waveguide ore edge, while the dashed irlee showss the boundary off entral areas with hout the presenee of periodi strutures. (b) Eletri field amplitude a distributtion at 14GHz along the ross-settion of the entire W WBG waveguide. Red solid line: input end d. Orange solid lin ne: output end. In Fig. 11(a) we presentt results obtain ned for the WB BG in the viiinity of 14GH Hz, where t numerial alulations (ssee Fig. 6) onlly the fundameental HE11-likee mode is aording to the exited by thee external Gau ussian beam. Hene, H we expeet a Gaussian--like modal disstribution at the waveg guide output at a these frequeenies. In the top row of F Fig. 11(a), wee present normalized elletri field profiles ( E ) ata seleted freqquenies as meeasured experimentally. y For ompariso on, in the botto om row of Fig.. 11(a), we preesent eletri fiield profiles at the same frequenies as a used in thee experiments omputed byy using the ffinite element software COMSOL. Th he omputation n ell was that of the ross-seetion of the enntire WBG w waveguide in the plane ontaining the waveguide w axiis of rotational symmetry. Rootational symm metry was used and onlly the angularr momentum m = 1 was reetained. The omputational ell was terminated in the transversee diretion by the t perfet eletri ondutinng boundary too simulate t input end (see red solid line in Fig. 11(b)), the sourre in the the metal-air interfae. At the w a beam waist w of λ waas used (see E Eq. (6)). The thheoretial form of a Gaaussian beam with eletri profilles were reon nstruted from the omputed modal distribbution at the ouutput end (see beige sollid line in Fig. 11(b)). For laarity of demonnstration, we hiighlight the enntral area without the presene p of periodi p orrug gation using ddashed irless, while indiating the position of th he metalli bou undary with a solid blak liine. We note tthat the measuured field