Symmetry and Finite-Size Effects in Quasi-Optical Extraordinarily THz Transmitting Arrays of Tilted Slots
Miguel Camacho, Ajla Nekovic, Suzanna Freer, Pavel Penchev, Rafael R. Boix, Stefan Dimov, Miguel Navarro-Cía
aa r X i v : . [ phy s i c s . op ti c s ] A ug Symmetry and Finite-size Effects in Quasi-opticalExtraordinarily THz Transmitting Arrays of TiltedSlots
Miguel Camacho,
Member, IEEE , Ajla Nekovic, Suzanna Freer, Pavel Penchev, Rafael R. Boix,
Member, IEEE ,Stefan Dimov and Miguel Navarro-C´ıa,
Senior Member, IEEE
Abstract —Extraordinarily transmitting arrays are promisingcandidates for quasi-optical (QO) components due to their highfrequency selectivity and beam scanning capabilities owing tothe leaky-wave mechanism involved. We show here how bybreaking certain unit cell and lattice symmetries, one canachieve a rich family of transmission resonances associated withthe leaky-wave dispersion along the surface of the array. Bycombining two dimensional and one dimensional periodic Methodof Moments (MoM) calculations with QO Terahertz (THz) time-domain measurements, we provide physical insight, numericaland experimental demonstration of the different mechanismsinvolved in the resonances associated with the extraordinarytransmission peaks and how these evolve with the number ofslots. Thanks to the THz instrument used, we are also able toexplore the time-dependent emission of the different frequencycomponents involved.
Index Terms —Extraordinary transmission, frequency selectivesurface, method of moments, quasi-optics, terahertz, time-domainspectrometer.
I. I
NTRODUCTION I N the early 1990s, the extraordinary optical tranmission(EOT) phenomenon through subwavelength apertures wasdiscovered [1]–[3] opening the door to new and excitingphysics. The transmission peak frequency and magnitude wereinitially thought to invalidate Bethe’s prediction for subwave-length apertures [4], although this comparison lacked a verymeaningful feature of the experimental sample: the periodicity.
The work of M. Camacho was supported by the Engineering and PhysicalSciences Research Council (EPSRC) of the United Kingdom, via the EPSRCCentre for Doctoral Training in Metamaterials [Grant No. EP/L015331/1]. Thework of S. Freer was supported by the EPSRC [Studentship No. 2137478]. Thework of R.R. Boix was supported by the Ministerio de Ciencia, Innovaci´ony Universidades [Grant TEC2017-84724-P]. The work of M. Navarro-C´ıawas supported by the EPSRC [Grant No. EP/S018395/1], the Royal Society[Grant No. RSG/R1/180040], and the University of Birmingham [BirminghamFellowship]. (
Corresponding author: Miguel Navarro-C´ıa )M. Camacho is with the Department of Electrical and Systems Engineering,University of Pennsylvania, Philadelphia, PA 19104-6390, USA (e-mail:[email protected]).Ajla Nekovic is with the Faculty of Electrical Engineering, University ofSarajevo, Sarajevo 71000, Bosnia and HerzegovinaR. R. Boix is with Microwaves Group, Department of Electronics andElectromagnetism, College of Physics, University of Seville, Avda. ReinaMercedes s/n, 41012, Seville, Spain (e-mail: [email protected])S. Freer and M. Navarro-C´ıa are with the School of Physics and Astronomy,University of Birmingham, Birmingham B15 2TT, United Kingdom (e-mail:[email protected])P. Penchev and S. Dimov are with the Department of Mechanical Engi-neering, School of Engineering, University of Birmingham, Birmingham, B152TT, UK xy z w l a b α Fig. 1. Schematic diagram of the freestanding subwavelength tilted slot N x by N y array along with the collimated incident Gaussian beam. Lattice periods a and b , and slot dimensions l × w rotated by an angle α . This characteristic held the key for the explanation of the, atthat point, unexplained mystery.The relation between the high transmissivity and the pe-riodicity became apparent when the transmission spectrumwas mapped for different planes of incidence [5], finding thatthese peaks correspond to the excitation of optical-frequencysurface waves known as surface plasmons [6], [7]. Thesesurface waves, usually tightly bound to the surface of a plas-monic material, can become leaky when supported by periodicstructures, following the same principle used nowadays byperiodically-modulated leaky wave antennas [8] and that canbe traced back to [9]. This intuitive explanation was laterconfirmed by means of plane wave expansion [10], showingthat these surface waves are capable of providing large trans-mission enhancements even for screens with non-negligiblethicknesses thanks to the evanescent coupling between surfaceplasmons supported by opposite faces.Experiments similar to those published in [3] were con-ducted at microwave regime, in the absence of plasmonicmaterials, shockingly finding similar results [11]–[14]. Also,EOT was reported at such regimes for s-polarized illumination that cannot excite surface plasmons even if plasmonic mate-rials were used [15], [16]. Without plasmons, the explanationof microwave extraordinary transmission was challenging.However, not one but two complementary explanations wereprovided. The first one followed the same logic as that oneprovided at optical frequencies: although metals do not supportplasmons at microwave frequencies, when properly geomet-rically engineered, they can support analogous surface wavemodes [17], [18], which produce extraordinary transmissioneffects analogous to those of surface plasmons [19]. It isworth noting that although these surface waves received a largeattention subsequently to their role in EOT, they were knownby the engineering community since the mid twentieth century[20], [21]. An alternative explanation, based on impedancematching formulation well known for transmission lines wasprovided over a decade ago [22], [23].The development of extraordinary transmission was subse-quent to similar periodic structures, widely-used by the mi-crowave community, known as Frequency Selective Surfaces(FSS) [24], [25]. However, the latter make use of the natural resonance of the unit element in each until cell, defined this asthe one dictated by their electrical size, with little or no effectsarising from their arrangement [26]. This main difference is theroot of interesting phenomena, as imposing or breaking certainsymmetries has drastic effects on the existence of EOT peaksassociated with the periodicities of the system. For instance,when the unit cell presents a symmetry plane aligned with theperiodicity (electric wall, for instance), this inhibits the EOTpeak associated with such lattice vector [27]. As it was shownthere, this means that when finite arrays are considered, onecan ignore the finite size effects along one of the periodicitiesif the unit cell presents symmetries that inhibit that resonance,as the surface wave along the short direction will not coupleto the radiation and therefore cannot sense the truncations ofthe system [28], [29].EOT arrays, with their narrow linewidth and strong suppres-sion out of band, have been shown to be excellent candidatesfor quasi-optical (QO) components such as filters and waveplates [15], [30], [31]. However, the potential added by theexploitation of the (lack of) symmetries is still to be explored.In an effort to bridge this gap, in this paper we study theextraordinary transmission phenomenon through arrays of non-symmetric slot arrays. This lack of symmetry is two-fold: wemake use of rectangular arrays (with periodicities given by a and b , with a different from b ), whose unit cell containsslots that present a rotation with respect to the lattice vectorsand also insert a slot-shaped aperture which is not colinearwith any of the lattice vectors. In the first part of the paperwe show how these two conditions are required for us toexplore the whole set of resonances associated with the twodifferent periodicities, and in the second part we experimen-tally demonstrate the presence of these resonances, and showhow finite effects also play an important role when realisticcollimated illuminations are used. Finally we explore boththe time-response of the system through spectrography andillumination effects such as polarization and angle of incidencedependencies. II. T HEORY
Let us consider the array shown in Fig.1, which containsan array of slots of dimensions l × w , rotated with respectto the two periodicities, with values a and b . These slots arecut into an infinite perfectly-conducting zero-thickness screen,which is a valid approximation for terahertz (THz) frequencies,such as those studied here. This rectangular geometry for theapertures is very convenient for the semi-analytical solution ofthe scattering problem in terms of analytical basis functions,yet the phenomena presented in the following can be foundfor any other geometry that lacks mirror symmetry. Let usconsider the case in which this array of slots is illuminatedby a plane wave propagating along the ˆz direction, whoseelectric field is given by E i . The problem of the scattering bythe array can be studied by means of an integral equation forthe tangential electric field in the slots, E sc t , given by J as + Z ∞−∞ Z ∞−∞ G M ( x − x ′ , y − y ′ ) · E sc t ( x ′ , y ′ , z = 0) dx ′ dy ′ = ( x, y ) ∈ slots (1)where G M ( x, y ) is a dyadic Green’s function, relating thein-plane components of the electric field on the surface of theslots and the surface current density on the metallic screen, and J as is the surface current density on the perfectly conductingplane in the absence of the slots (dictated by E i ). It iswell-known that for the case of a doubly-infinitely periodicarray, the integral equation can be reduced to a single unitcell defining a periodic dyadic Green’s function, G per M whichgroups the contributions from all the arrays through an infinitesummation, in which the fields on the array are consideredFloquet-periodic as imposed by the impinging wave [32]. Theperiodic expression of (1) is then given by J as + Z a Z b G per M ( x − x ′ , y − y ′ ) · E sc t ( x ′ , y ′ , z = 0) dx ′ dy ′ = ( x, y ) ∈ δ (2)where the integral is limited to the single unit cell δ bydefining a two-dimensionally periodic Green’s function givenby G per M ( x, y ) = + ∞ X m,n = −∞ G M ( x − ma, y − nb ) e j( k x ma + k y nb ) (3)where k x and k y are the phasing wavevectors defined as in[27].Both equations (1) and (2) can be solved by means ofthe Method of Moments (MoM) [33], which was used in[28] to tackle the problem of periodic non-rotated slots, andtherefore will be just summarized here. In this method, theunknown tangential electric field E sc t is expanded as a linearcombination of proposed basis functions. When this linearexpansion is introduced into either (1) or (2), one can builda system of linear equations for the weights of those basisfunctions, that can be readily solved by numerical means. Thekey of this method is the use of basis functions that can renderthe features of the fields in the slot aperture, reproducingthe singular behaviour of the fields near the edges for both k a/(2 ) (cid:1) (cid:2) m (cid:0) m (cid:3) (cid:4) m (cid:5) m (cid:6) (cid:7) m (cid:8) m | (cid:9) p e r | ( d B ) (1,0) (0,1) (1,1) | (cid:10) p e r | ( d B ) (1,0) (0,1)(1,1) | (cid:11) p e r | ( d B ) (0,1)(1,1) (1,0) (a) (c) (b) k x0 a/ k x0 a/ k x0 a/ k a/(2 ) k a/(2 ) k a / ( ) k a / ( ) k a / ( ) T ( d B ) T ( d B ) T ( d B )
45 deg45 deg o o o (cid:12) = (cid:13) = (cid:14) = o o o (cid:15) = (cid:16) = (cid:17) = o o o (cid:18) = (cid:19) = (cid:20) = o (cid:21) = o (cid:22) = o (cid:23) = Fig. 2. Top figures show the transmission spectra at normal incidence for the lowest Floquet mode for vertical, diagonal ( α = 45 ◦ ) and horizontal slots, withthe electric field polarization perpendicular to the long side of the slots, for (a) a = 540 µ m and b = 480 µ m , (b) a = b = 540 µ m and (c) a = 540 µ m and b = 600 µ m . Bottom figures show the real part of the associated dispersion diagrams calculated for the diagonal slot ( α = 45 ◦ ), over a color map representinga cut of the three dimensional complex space value of the MoM matrix determinant. In the dispersion diagrams, dashed lines represent the labelled lightlinespatial harmonics. All the data in this figure was obtained for l = 220 µ m and w = 175 µ m . polarization. Previous studies show that only a very reducedset of them is needed (5 at most) when these are chosen to beChebyshev polynomials weighted by the singular square-rootsingular behaviour of the field for each polarization [34]–[36].Once the system of equations has been inverted, the totalelectric field on the surface of the array can be directlycalculated together with its Fourier transform, which can beused to calculate the far-field and the scattering parameters[32].As discussed in the introduction, the presence of EOT phe-nomenon is linked to the existence of surface waves supportedby the array. Mathematically this link is clear, as the existenceof surface waves can be studied by looking for the zeroes ofthe determinant of the matrix of the system of linear equationsresulting from the application of the MoM, | ∆ per | , which isa function of the geometrical parameters, the frequency and the phasing imposed between adjacent unit cells. This meansthat when the excitation is close in frequency, phasing andpolarization to that of the surface wave, resonant behaviouris found. When the position of these zeroes is tracked for arange of frequencies and phasings, one can build the dispersiondiagram of the surface modes [37].Using the analysis techniques and the concepts presentedhere, we have studied the transmission through a two dimen-sional periodic array of slots for three different periodicityconfigurations, i.e. a > b , a = b and a < b , as shown in Fig.2(a-c), which contain both the transmission spectra at normalincidence (top figures) and the dispersion diagram of the leakysurface waves supported by the array along the x direction forthe case of 45 ◦ slot rotation. Fig. 2 has been obtained for a doubly infinite periodic array of slots (i.e., through the solutionof Eqn. (2) with a periodic dyadic Green’s function).In Fig. 2(b), the well-studied EOT through a square slotarray is presented for benchmarking purposes, for which wepresent both transmission spectra for three different rotationangles at normal incidence and the dispersion diagram for the45 ◦ slot rotation. Given the equality between the lattice vec-tors associated with both periodicities, their EOT resonancesappear at the same frequency, and the system has a singleEOT peak as the screen zero-thickness inhibit the so-calledodd mode that appears very close to the frequency of the firstWood-Rayleigh’s anomaly [22]. In terms of the surface wavedispersion shown in the bottom figure, one can see only onemode crosses the broadside direction.When the periodicity along the y direction is smaller thanthat along the x , the degeneracy of the EOT peak is broken,and we would expect to find two peaks in the transmissionspectrum in Fig. 2(a). However, when the slot is aligned witheither of the periodicities and the impinging electric field isperpendicular to that, one see that the peak associated with thelattice vector parallel to the slot disappears. In contrast, bothEOT peaks are present for any intermediate value. The fact thatthe existence of one of the periodicities can be completelyhidden through the use of symmetries is remarkable, andits explanation can be found both in the mathematics andin the physics of the problem. Mathematically, the EOT islinked with the pole-type divergence of the Green’s functionat the onset of a diffraction lobe. However, the symmetryof the problem can lead to a pole-zero cancellation whenthe scattered electric field points perpendicularly to the in- plane wavevector of the diffraction mode that is generatingsuch pole divergence, via a zero in the diagonal term of thedyadic Green’s function [27]. From the point of view of thesurface wave modes responsible for the EOT phenomenon,this absence of EOT peaks under certain illuminations canbe explained in terms of polarization mismatch between theimpinging radiation and the modes supported by the array.Note that freestanding connected conducting structures, suchas arrays of apertures, support transverse magnetic (TM)modes, while disconnected conducting surfaces such as arraysof patches, support transverse electric (TE) modes. Therefore,if one wants to excite the surface mode propagating along the x direction, one would need to be able to excite a magneticcomponent along the y direction, and vice versa . In thepresence of symmetry (i.e., slot parallel to one of the axes),the impinging polarization across or along the slot axis willbe preserved, and only one of the two modes will be excited.Therefore, the EOT peak associated with the not-excited modewill be suppressed. However, if one considers a slot that is notaligned with the axes, the slot field distribution will couple toboth surface waves, therefore exciting both EOT peaks.In Fig. 2(c) we present the case in which the EOT associatedwith the periodicity along y appears at a frequency lower thanthat associated with the periodicity along x . As seen earlier forFig. 2(a), when the slot is aligned with either of the axes, theperiodicity along the direction parallel to the long sides of theslot will not be present in the transmission spectrum, althoughits presence can be noticed due to the loss in transmissionof the zero-th order Floquet (see the solid blue line of thetop Fig. 2(c)), due to the interesting fact that although theEOT peak does not appear, its associated Floquet mode cancouple to the energy of the scattered fields once it is above itscut-off frequency, while avoiding the singularity of its onset.In terms of the dispersion diagram, in the bottom Fig. 2(c),one can see that the two modes associated with the surfacewaves propagating along x and y are pushed together bytheir corresponding lightlines, giving place to a mode couplingthrough a Morse critical point [38].The fact that the position of the two resonant peaks canbe dictated by the periodicities chosen along the x and y direction allows for a large freedom in the design of quasi-optical filters. The electrical size of the aperture can be thenused to engineer the width of the resonances, as done instandard single-resonance EOT filters [32]. In the particularcase of tilted slots, the angle α can be used for the designof the transmission levels of the two peaks, as it tunes thecoupling between the scattered field and the two surface wavefamilies, each associated to one periodicity.As one can see, the EOT through tilted slot arrays opensa wide variety of situations through the coupling of theimpinging radiation into a wider variety of Floquet modes,and therefore of EOT resonances. As it was shown in [39], forthe cases of highly symmetrically positioned slots, the finitesize along the non-excited EOT resonance can be disregarded,however, this is not the case when one considers tilted slots. Inthe following let us focus on the truncation effects that have tobe considered for any realistic implementation of quasi-opticalfilters based on this multi-EOT phenomenon. Experiment (a) (cid:24) m (cid:25) m E i,t Frequency (THz) (cid:26) m (cid:27) m E i,t Theory (b)
Frequency (THz)
Fig. 3. On-axis transmission through freestanding arrays of diagonal ( α =45 ◦ ) slots and varying number of rows N y with lattice periods a ≈ µ m and b ≈ µ m , and slot dimensions l ≈ µ m and w ≈ µ m :measurements (a) and MoM results (b). III. M
EASUREMENTS AND D ISCUSSION
In order to verify the theoretical results and physical insightprovided in the previous section, we performed a series ofexperiments at THz frequencies using QO experimentationtechniques as discussed in the Appendix. The samples un-der test were laser micromachined (further details about thefabrication can also be found in the Appendix) subwavelengthhole arrays on 10 µ m-thick aluminium foils. The geometricaldimensions of the fabricated samples were inspected with amicroscope; the in-plane lattice periods were a = 540 ± µ m and b = 600 ± µ m , and the slot dimensions were l = 226 ± µ m , w = 176 ± µ m .The transmission spectrum at normal incidence with theelectric field polarization perpendicular to the long side ofthe slots, l , increases with the number of rows N y (Fig. 3) Frequency (THz) T i m e ( p s )
11 rows
Frequency (THz) T i m e ( p s )
21 rows
Frequency (THz) T i m e ( p s )
61 rows
Frequency (THz) T i m e ( p s ) -155-150-145-140-135-130-125 P o w e r / fr e qu e n c y ( d B / H z ) Fig. 4. From left to right, spectrogram (i.e. time-frequency-maps) of the detected waveform (co-polar measurement) for samples with N y = 5, 11, 21 and 61when the incident electric field is polarized perpendicularly to the long side of the slots. for diagonal slots. Notice that N x > across the wholewafer in all fabricated samples, and thus, the truncationalong x direction was assumed to have no impact in theelectromagnetic response. The MoM approach described in[35] can be then used for the very efficient analysis of thestructures involved in the experiments by reducing the problemto a single strip of slots which is repeated periodically along asingle direction. Under this approach, integral equation in (1)becomes J as ( x, y ) + N y − X j =0 Z Z η j G per1D ( x − x ′ , y − y ′ ) · E sc t ( x ′ , y ′ , z = 0) dx ′ dy ′ = ( x, y ) ∈ η j (4) ( j = 0 , . . . , N y − , where η j represents the surface of the j-th slot within theunit strip and G per1D ( x, y ) is the 1-D periodic dyadic Green’sfunction given in terms of the free space Green’s function G M ( x, y ) G per1D ( x, y ) = + ∞ X i = −∞ G M ( x − ia, y )e j ik x a . (5)To introduce the illumination effects, a 1-D Gaussian beamprofile is used as the impinging wave for the calculationof J as ( x, y ) , as in [28]. In the experiment, almost totaltransmission is obtained when N y = 61 , whereas the MoMpredicts total transmission. The slight quantitative disagree-ment between the results for this N y is arguably, to a largeextent, due to the absence of loss (i.e. ohmic and scatteringlosses due to surface imperfections) in the MoM, and, to alower extent, due to the two-dimensional MoM calculation inwhich the incident Gaussian beam has a beam-waist of 4 mmalong y and infinite along x . For such array with N y = 61 ,both the direct transmission and the contribution of the surfacemode propagating along the y direction are saturated. Indeed,saturation of the direct transmission actually happens alreadyfor N y = 21 given the beam-waist of the Gaussian beam.The existence of two transmission channels involved in theEOT (i.e. direct transmission and leaky-wave-mediated trans-mission) becomes evident in the time-frequency analysis of the scattering phenomenon, which allows for the time-dependentanalysis of the radiated waves [40]. To this end, Fig. 4 showsthe spectrograms for the different truncated arrays of diagonalslots with varying number of slots along the y direction, N y .Regardless of N y , we find that a significant amount of theenergy arriving at the detector does it around 28 ps across thewhole spectral window. This is associated with the contributionof the direct transmission, as the whole-array standing wavehas not had enough time to form [32]. Meanwhile, there isa prolonged presence of energy arriving beyond 28 ps onlyat the frequency of the EOT peak. The larger N y is, thelonger this ringing of energy lasts. This is due to the factthat more elements are involved in the resonance, increasingits quality factor (and therefore the frequency selectivity asshown in Fig. 3). This resonance is formed as the superpositionof counter-propagating leaky waves, which are excited at aprecise frequency, the EOT frequency, dictated by the leaky-wave complex dispersion relation. Such a slow build-up ofthe resonance is possible due fact that the crossing of theleaky-wave dispersion with the broadside direction occurs ata zero slope (and therefore zero group velocity). As it wasshown in [28], [41], when Gaussian beams are considered,the transmission peak is slightly red-shifted, meaning that theresonance is mediated by slowly-propagating waves, but notcompletely static ones, contrarily to that found when usingplane waves in infinite periodic arrays.We have also studied the complete angular dependence ofthe system by varying both angle of incidence and polarization,as these allow to control the matching of the impinging waveto the two leaky waves responsible for the two different EOTresonances of the tilted slot array.Fig. 5 shows the angle of incidence dependence of themeasured transmission spectrum under both TE and TM waveillumination, whose planes of incidence are defined by the z direction and the vectors shown in the insets of the figure. Incontrast to what we find when exploring the dispersion alongthe x axis in Fig. 2, at both planes of incidence associatedwith TE and TM incidence, the two lightlines associated withthe (1,0) and (0,1) harmonics show very similar dispersionrelation, enforcing a similar angle-dependence on the EOT S (dB) (cid:28) m (cid:29) m E i,t H i,t (cid:30) m (cid:31) m E i,t H i,t TMTE
Fig. 5. Transmission spectrum as a function of the angle of incidence for TE(rotation of the sample along the E -axis) and TM polarization (rotation of thesample along the H -axis), with black circles marking the largest transmissionfor each angle of incidence. The electric field is perpendicular to the longside of the diagonal ( α = 45 ◦ ) slots. peaks. Additionally, one can see the excitation of higherorder spatial harmonics, such as the (-1,0) and (0,-1), whoseassociated resonant frequencies increase with the angle ofincidence. Although for the sake of brevity we only showhere the position of the peaks, we have found that the relativetransmissivity of the two EOT peaks can be controlled bychoosing the wave to be either TE or TM, therefore promotingthe excitation of one of the two leaky waves while suppressingthe other. In particular, by using TE oblique incidence thelower frequency resonance is favored, while the TM obliqueincidence enhances the higher EOT resonance. This effect canbe explained by the increasing wavevector matching betweenthe impinging and the leaky wave, which can ultimately leadto huge resonant amplitudes as theoretically shown for boundsurface waves supported by semi-infinite arrays [29].We have found experimentally, however, that the excitationof the two leaky waves can also be controlled by playing ξ = 0º ξ = 45º ξ = 90º μ m μ m ξ E i,t Fig. 6. Measured on-axis transmission through freestanding arrays of diagonalslots for different rotation of the N y = 61 sample along the z -axis as indicatedby the inset in the figure. with the field configuration of the scattered fields on theslot. In Fig. 6, we show the normal incidence transmissioncoefficient for varying angles of electric field polarization.We find that the higher frequency EOT, associated with theperiodicity along the x direction can be almost completelysuppressed when the impinging excitation is polarized alongthe y direction. This is due to the modification of the fielddistribution, as in contrast with all the cases studied before,now there is a non-negligible electric field component excitedalong the long side of the slot, which allow for the couplingwith a larger number of modes.IV. C ONCLUSION
In this paper we have provided an extensive study of theEOT phenomenon through periodic and finite arrays of tiltedslots, which allow for a very distinguishable simultaneouscoupling to two different EOT resonances, in contrast to thehighly symmetric arrays traditionally studied in the literature.We have shown that the two resonances can be linked tothe existence of two different leaky waves, propagating alongperpendicularly to each other, as demonstrated by the calcu-lation of their dispersion diagrams and also by showing theirmatching (or absence of matching) to the scattered fields undercertain symmetries. By performing experiments at terahertzfrequencies we have both demonstrated the validity of ourtheory, but have also shown the role that finite effects play inthe quality factor of the theoretically highly selective EOT res-onance. Thanks to the time-domain spectroscopy, we are ableto show the consequences that the dispersion characteristicsof the leaky-waves involved have in the time response asso-ciated with each frequency component of the scattered fields,namely its slow resonance build-up. By exploring the angulardependence under two differently polarized illuminations, wehave shown the off-axis dispersion of the leaky waves whichis dominated by the linear lightline harmonic dispersion and
TABLE IL
ASER PARAMETERS FOR THE FABRICATION OF THE FINITE SIZE TILTEDSLOT ARRAYS . Laser Parameter Unit Value
Power W 3.9Frequency kHz 500Pulse energy µ J 7.8Scanning speed m/s 2Pulse duration fs 310Beam spot diameter µ m 30Laser beam polarization - CircularHatch style - RandomHatch pitch µ m 4Ablation rate per layer µ m 1 also the ability to suppress or enhance the EOT resonances viaoff-axial excitation of the slots. The insight provided here setsthe design guidelines for advanced quasi-optical componentsbased on extraordinary transmission phenomenon.A PPENDIX
Fabrication . The laser fabrication trials were performed ona state-of-art laser micromachining system that integrates afemtosecond (fs) laser source (Amplitude Systems Satsumamodel) with a maximum average power of 5W, central emis-sion wavelength of 1030 nm, pulse duration of 310 fs andmaximum pulse repetition rate of 500 KHz. A stack of fivemechanical axes (three linear and two rotary stages) were usedto move the samples with very high accuracy and precision,i.e. the positioning resolutions of the linear and rotary stagesas stated by the manufacturer were 0.25 µ m and 45 µ rad,respectively [42]. The laser system was also equipped witha high dynamics 3D optical scan head for structuring withscanning speeds up to 2 m/s and thus to obtain relatively highmaterial ablation rates. A 100 mm focal length telecentric lensis mounted at the exit of the beam path to achieve a focusedbeam spot diameter of 30 µ m. The achievable processingaccuracy and repeatability with the 3D scan head is betterthan ± µm for all scanning speeds used in the experiments,while its processing volume is 30 mm (X) ×
30 mm (Y) × Measurements . The samples were characterized with theall fibre-coupled THz time-domain spectrometer TERA K15from Menlo Systems in a QO configuration without purging.A couple of TPX planoconvex lenses (effective focal length ≈
54 mm) were used to work with a collimated beam whosefrequency-dependent beam-waist at the sample position wasestimated to be 4.1 mm and 4.0 mm along the E - and H -planeat 0.5 THz, respectively. For the angle-resolved measurementsof Fig. 5 (with an angular step of 2.5 ◦ until 10 ◦ and 5 ◦ until30 ◦ ) and Fig. 6, the N y = 61 sample was installed on amanual rotation stage and on a high-precision rotation mount.The lock-in constant was set to 300 ms for the angle-resolvedmeasurements of Fig. 5 and 100 ms for the rest, whereas thetotal temporal length of the recorded waveforms was at least260 ps to have a spectral resolution of 3.8 GHz in the worstcase. Calibration was done by comparing the measurementswith the line-of-sight configuration (i.e. emitter and detector on-axis) without the sample on the sample holder. To generatethe spectrograms (i.e. short-time Fourier transform) in Fig. 4,each waveform is divided into segments of length 8.9 ps andan overlap of 98%, which are windowed with a Hann window.A CKNOWLEDGEMENTS
The authors would like to thank Dr. J. Churm (Universityof Birmingham) for measuring the unit cell dimensions of thefabricated truncated arrays.R
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