Electromagnetic bandgap based on a compact three-hole double-layer periodic structure

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 1045 Communica ion Elec omagne ic Bandgap Based on a Compac Th ee-Hole Double-Laye Pe iodic S uc u e Luis Fe nando He an , Qiao Chen , F ancisco Mesa , and Osca Que edo-Te uel Abs ac — We p opose and s udy a new ype o double-laye holey s uc u e wi h a wide bandgap. The s uc u e can ha e glide symme y in wo o hogonal di ec ions bu no 2-D glide symme y. We epo esul s in e ms o dispe sion diag ams calcula ed wi h he eigensol e o a comme cial sol e , as well as wi h a mul imode ans e ma ix app oach ha pe mi s an accu a e calcula ion o he a enua ion cons an . The esul s demons a e ha he bandgap o he p oposed s uc u e can p o ide a wide ac ional bandwid h and a la ge a enua ion cons an han hose o a 2-D glide-symme ic holey con igu a ion. The e o e, his new ype o pe iodic s uc u e can be ad an ageous in p e en ing leakage in gap wa eguide echnology o , in gene al, pa allel pla e con igu a ions and il e s. The ope a ion o his new uni cell is expe imen ally demons a ed wi h a double- lange con igu a ion be ween 40–60 GHz. Index Te ms— Elec omagne ic bandgap (EBG), lange ansi ion, glide symme y, holey pe iodic s uc u e, mul imodal analysis. I. INTRODUCTION Fully me allic wa eguides a e o en used in millime e -wa e appli- ca ions due o hei low inse ion losses. When used o design de ices ha in eg a e in o high- equency sys ems, wa eguides a e usually spli in o wo pieces, so hei in e nal de ails can be ab ica ed using he machining echnique. In p ac ice, his spli ing in oduces a small ai gap be ween me al pieces due o manu ac u ing/assembly impe ec ion and/o me al de o ma ion, which can cause signi ican ene gy leakage a millime e -wa e equencies [1]. Leakage may also occu in he in e connec ion be ween he wa eguides, whe e i is di - icul o ensu e pe ec elec ical con ac [2]. Al hough con en ional 1-D pe iodic co uga ions/g oo es only p o ide a high supp ession o leakage in he di ec ion o pe iodici y [3], he gap wa eguide echnology [4],[5] based on bed-o -nails s uc u es can p ohibi wa e p opaga ion in unwan ed di ec ions inside a pa allel-pla e wa eguide (PPW). Howe e , he manu ac u ing p ocess o pe iodic pins can be agile and cos ly when hei physical dimensions a e scaled down o a magni ude o submillime e s in millime e -wa e equencies Manusc ip ecei ed 16 July 2023; e ised 19 Oc obe 2023; accep ed 31 Oc obe 2023. Da e o publica ion 15 No embe 2023; da e o cu - en e sion 9 Feb ua y 2024. This wo k was suppo ed in pa by COST Ac ion SyMa unde G an CA18223 and in pa by COST (Eu opean Coope a ion in Science and Technology). The wo k o Luis Fe nando He an was suppo ed in pa by unde G an PID2020-114172RB- C21/AEI/10.13039/501100011033 and in pa by he Gobie no del P incipado de As u ias unde G an IDI/2021/000097. The wo k o F ancisco Mesa was suppo ed by MCIN/AEI/10.13039/501100011033 unde G an PID2020- 116739GB-I00. (Co esponding au ho : Osca Que edo-Te uel.) Luis Fe nando He an is wi h he Depa men o Elec ical Enginee ing, Uni e si y o O iedo, 33003 O iedo, Spain (e-mail: [email p o ec ed]). Qiao Chen and Osca Que edo-Te uel a e wi h he Di ision o Elec omag- ne ic Enginee ing and Fusion Science, School o Elec ical Enginee ing and Compu e Science, KTH Royal Ins i u e o Technology, SE 100 44 S ockholm, Sweden (e-mail: [email p o ec ed]; [email p o ec ed]). F ancisco Mesa is wi h he Depa men o Applied Physics 1, ETS Ingenie ía In o má ica, Uni e sidad de Se illa, 41012 Se ille, Spain (e-mail: [email p o ec ed]). Colo e sions o one o mo e igu es in his communica ion a e a ailable a h ps://doi.o g/10.1109/TAP.2023.3331502. Digi al Objec Iden i ie 10.1109/TAP.2023.3331502 [6]. As an al e na i e, holey elec omagne ic bandgap (EBG) s uc- u es ha e ecei ed inc easing in e es due o hei obus ness and cos -e ec i eness [7]. I has ecen ly been demons a ed ha glide symme y [8],[9] enables a numbe o ad an ages when applied o pe iodic s uc u es, and in pa icula o holey EBG s uc u es [10]. Two-dimensional glide-symme ic holey s uc u es we e i s s udied in [11], e ealing ha glide symme y can be used o inc ease he ope a ing bandwid h o a Lunebu g lens. Fu he mo e, glide symme y has been shown o imp o e EBG bandwid h [12],[13] and i s a enua ion [14], which a e bene icial p ope ies o gap wa eguide echnology [15], il e s [16], leaky wa e an ennas [17],[18], and educ ion o leakage in langes [19],[20]. The cha ac e is ics o pe iodic EBG s uc u es a e s udied p ima ily by means o dispe sion analysis. The commonly used eigenmode sol e (ES) o comme cial ull-wa e simula o s only compu es he p opaga ion cons an s o pu ely p opaga i e modes, and hence only p o ides he equency ange o he s opband. Howe e , knowledge o he na u e o he s opband and i s a enua ion is c ucial o ully cha ac e ize he beha io o he bandgap. This in o ma ion can be ob ained om he comple e modal solu ions o e ed by ad hoc analy ical/nume ical me hods such as he equi alen ci cui app oach [21],[22] and he ans e se esonance me hod [23],[24] widely used in single-sided me asu aces. When a pai o pe iodic su aces possess glide symme y, highe o de in e ac ions be ween hem canno always be easily modeled wi h an equi alen ci cui o homogenized using he ans e se esonance me hod [25]. Al hough a emp s o use he mode-ma ching echnique [26],[27] ha e been epo ed o glide-symme ic co uga ions [26] and holey su aces [27], his echnique is es ic ed o he analysis o some canonical geome ies. The mul imode ans e ma ix app oach [28],[29],[30],[31],[32], [33] combines he abili y o ull-wa e simula o s [34],[35],[36] o deal wi h complex geome y s uc u es [14],[37] and inhomogeneous ma e ials [38],[39],[40] wi h he ad an age o ad hoc app oaches o ob ain comple e modal solu ions (bo h phase and a enua ion cons an s) and p o ide physical insigh [41].In[14], his app oach p o ed o be capable o accu a ely cha ac e izing he glide-symme y holey EBG s uc u e, including he ejec ion bandwid h, he le el o a enua ion, he na u e o he mode (complex/e anescen /bound), pa i y o he mode, and he di ec ional p ope y o he s opband. In his wo k, we p opose and s udy he EBG p ope ies o a new ype o glide-symme ic holey s uc u e ha has “b oken” glide sym- me y in wo o hogonal di ec ions a he han 2-D glide symme y. The ope a ion o his new uni cell is expe imen ally alida ed wi h a double- lange expe imen al se -up. II. DESCRIPTION OF THE PROPOSED PERIODIC UNIT CELL The common opology o holey pe iodic me asu aces consis s o wo me allic laye s sepa a ed by a na ow gap ha o ms a PPW. In bo h op and bo om laye s, he e is a pe iodic dis ibu ion o holes, and depending on i s speci ic pe iodic con igu a ion (o symme y), © 2023 The Au ho s. This wo k is licensed unde a C ea i e Commons A ibu ion-NonComme cial-NoDe i a i es 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ 1046 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 Fig. 1. Holey uni cells. (a) Mi o . (b) One edge displaced mi o . (c) Glide symme ic. (d) P oposed pe iodic uni cell. Top and bo om heigh s o he holes (h) a e he same. he s uc u e beha es di e en ly. The mos common con igu a ions o a uni cell wi h holes in he uppe and lowe laye s a e: mi o ing, mi o ing wi h a displacemen o one edge, and symme ic glide [12]. A schema ic o hese uni cells is illus a ed in Fig. 1(a)–(c). In all cases, pis he pe iodici y o he uni cell, his he heigh o he op and bo om holes, gis he ai gap be ween he bo om and op laye s, and diis he diame e o he hole. Fo s uc u es wi h di e en diame e s in he op and bo om holes, d1will e e o he bo om holes and d2 o he op holes. The no el uni cell p oposed in his wo k is illus a ed in Fig. 1(d). This uni cell has one hole o diame e d1in he bo om me allic laye and wo holes wi h diame e d2in he op me allic laye . The op-laye holes a e shi ed in bo h plane di ec ions o hal he uni -cell pe iod wi h espec o he mi o o he bo om hole. Unlike he o he uni cells shown in Fig. 1, which ha e wo holes in o al, he p oposed con igu a ion has h ee holes pe uni cell (p.u.c.). The diame e d2o he holes in he uppe laye mus be selec ed so ha he adjacen holes do no o e lap; ha is, i s maximum alue is gi en by d2,max =p/√2. In he ollowing, uni cells will also be deno ed in e ms o he numbe o holes p.u.c.; namely, he ones in Fig. 1(a)–(c) as “ wo-hole” uni cells, while he p oposed no el uni cell will be deno ed as a “ h ee- hole” uni cell. III. PARAMETRIC STUDY In his sec ion, we s udy he beha io o he p oposed h ee-hole uni cell, Fig. 1(d), and compa e i wi h he o he h ee wo-hole uni cells illus a ed in Fig. 1(a)–(c) h ough a de ailed pa ame ic s udy. A. Cells Wi h he Same Pe iod Fi s , we show he dispe sion diag ams o he ou holey s uc u es in Fig. 2when he size o hei uni cells is he same. The emaining pa ame e s o each uni cell a e op imized o ob ain he maximum s opband. As p e iously epo ed in [10] and [15], a glide-symme ic uni cell [Fig. 2(c)] exhibi s a wide s opband bandwid h han i s co esponding nonglide e sions [Fig. 2(a) and (b)]. Howe e , he p oposed h ee-hole uni cell has a much lowe equency bound o he s opband wi h he same pe iodici y and simila bandwid h [Fig. 2(d)]. We can obse e ha he beha io o he undamen al mode o he mi o ed wo-hole uni cell and he one o he h ee-hole uni cell a e quali a i ely simila , al hough he h ee-hole uni cell limi s he p opaga ion in he 45◦di ec ion (M0) a a signi ican ly lowe equency, hus educing he lowe equency bound o he s opband and inc easing i s bandwid h. This is a ele an ea u e, since i means ha his newly p oposed uni cell is mo e compac han o he p e iously epo ed holey s uc u es. An explana ion o he abo e ea u e could be expec ed o be ha he p oposed h ee-hole uni cell has a highe illing ac o han i s wo-hole coun e pa s. Howe e , a u he analysis o he s opband beha io e eals ha he speci ic geome y o he h ee-hole uni cell also has a signi ican impac . Fo example, Fig. 3compa es he beha io o wo h ee-hole uni cells wi h he same illing ac o : he Fig. 2. Dispe sion diag ams o he uni s cells in Fig. 1wi h he same pe iodici y (p=3 mm). In all cases, h=2 mm and g=50 µm. (a) Mi o (d=2.8 mm). (b) One edge displaced mi o (d1=2.6 mm, d2=1.6 mm). (c) Glide symme ic (d=2.4 mm). (d) P oposed h ee-hole uni cell (d1= 2.6 mm, d2=1.6 mm). Fig. 3. Dispe sion diag ams o wo h ee-hole uni cells. (a) P oposed h ee-hole cell. (b) Th ee-hole uni cell wi h he op-laye holes shi ed by p/2. In he inse o (b), he holes in he bo om laye a e colo ed blue, while hose in he op laye a e colo ed ed. one p oposed in Fig. 1(d) and a a ia ion o his cell wi h i s op laye shi ed by an addi ional leng h o p/2. The compa ison o bo h plo s in Fig. 3clea ly shows ha he alue o he illing ac o appea s o be less ele an han he ela i e posi ion o he bo om and op holes. I is somewha unexpec ed ha changing he ela i e posi ion o he holes a ec s he s opband loca ion and bandwid h so d as ically. I can be in e ed ha he beha io o he shi ed h ee-hole opology in Fig. 3(b) is a so o hyb idiza ion be ween he wo-hole mi o and glide wo-hole cases shown in Fig. 2(a) and (c), in he sense ha he beha io o he i s mode in he M0 egion is simila o he wo-hole mi o case in Fig. 2(a), while i s s opband is con ined be ween he second and hi d o de o p opaga ion as in he wo-hole glide case in Fig. 2(c). A pa ame ic s udy has also e ealed ha among he pa ame e s ha ha e mo e in luence on he bandwid h pe o mance o he p oposed h ee-hole uni cell, bo h he op and bo om hole diame e s ha e he g ea es impac on i . The e ec o he in luence o hese wo pa ame e s is plo ed in Fig. 4, which shows he equencies in he i s and second modes, and hus he absolu e bandwid h, o di e en combina ions o d1and d2. This analysis was ca ied ou using he same pe iod as in he p e ious igu es, p=3 mm. Bo h diame e alues a e aken o hei p ac ical size limi s, he diame e o he bo om hole is less han he pe iod o he uni cell, and he diame e o he op hole is chosen o a oid o e lap [ hese limi s a e (d1/p)lim =1 and (d2/p)lim =0.7]. An impo an obse a ion is IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 1047 Fig. 4. Pa ame ic analysis o he p oposed h ee-hole cell o (a) equency bounds o he i s and second modes and (b) ac ional bandwid hs. All da a ha e been ob ained using a no malized heigh o h/p=0.66. Fig. 5. Holey pe iodic me asu aces wi h he same lowe equency bound o he s opband. (a) Mi o . (b) Mi o wi h one displaced edge. (c) Glide symme ic. (d) P oposed pe iodic uni cell. ha he s opband bandwid h inc eases as he diame e o he holes g ows, sugges ing ha he la ge he a ea o he su ace occupied by he holes, he wide he bandwid h ob ained. Howe e , his inc ease shows a limi o an op imum combina ion o d1and d2. This ac is be e obse ed in Fig. 4(b), whe e he ac ional bandwid h is plo ed e sus he no malized hole diame e wi h espec o he uni - cell pe iod. The op imal ac ional bandwid h is ob ained o d1/p= 0.86 and d2/p=0.54. I has been obse ed ha he a ia ion in hole dep h does no ha e a signi ican in luence on ac ional bandwid h o alues g ea e han h/p=0.5. B. Cells Wi h Di e en Pe iods Nex , we s udy he beha io o he s opband o all holey s uc u es when hei uni -cell dimensions a e adjus ed o ha e he same lowe equency bound so he ac ional bandwid h can be p ope ly compa ed in all he s uc u es. The co esponding dispe sion diag ams a e shown in Fig. 5, wi h he inse s ep esen ing each holey uni cell wi h i s ac ual aspec a io o help compa e he sizes o he di e en cells. All pa ame e s used o his compa ison ha e been selec ed o ob ain he op imum s opband bandwid h [14]. These pa ame e s a e shown in Table I. The esul s o Fig. 5demons a e ha he h ee-hole uni cell p oposed in his wo k exhibi s a wide ac ional s opband bandwid h and a smalle pe iod size compa ed o he o he h ee wo-hole s uc u es. Compa ed o he glide case, he h ee-hole uni cell has TABLE I HOLEY UNIT CELL PARAMETERS FOR FIG.5(DIMENSIONS IN mm) Fig. 6. Bloch’s analysis o he p oposed uni cell [ ep oduced in (d)] wi h dimensions in Table Iusing MMTMM o i s p opaga ion and a enua ion cons an s along (a) 0X and (b) 0M, in compa ison wi h CST ES. (c) Roo loci o he complex mode in (b) om 40.8 o 81 GHz. (d) Ske ch o he p oposed h ee-hole uni cell. 36% less size and 20% mo e ac ional bandwid h. This esul shows ha he p oposed h ee-hole uni cell has a e y high po en ial o de ices whe e a high s opband is equi ed, wi h he addi ional bene i o a educed size compa ed o o he wo-hole uni cells while main aining i s obus ness and ease o manu ac u ing. Fu he mo e, he educed size o he h ee-hole uni cell makes i s use con enien a lowe equencies, whe e he la ge size o he o he wo-hole uni cells could be imp ac ical. Howe e , i should be no ed ha size educ ion can lead o a lowe a enua ion ac o p.u.c. (e−2αp). In ha case, he minimum numbe o uni cells equi ed o achie e he a ge a enua ion should be de e mined by in es iga ing he cha ac e is ics o he s opband. IV. BLOCH ANALYSIS USING MMTMM In addi ion o he p e ious s udy, he mul imode ans e ma ix me hod (MMTMM) [41] is applied in his sec ion o analyze an op i- mal h ee-hole p.u.c. holey me asu ace (see dimensions in he i h column o Table I). MMTMM allows us o ob ain no only he phase cons an bu also he a enua ion cons an o any pe iodic s uc u e. Al hough accu a e alues o he a enua ion cons an canno be easily ob ained using comme cial elec omagne ic so wa e, MMTMM has al eady demons a ed being a eliable ool o his ask [14],[41]. Fig. 6(a) shows bo h he phase shi and he no malized ( o he wa enumbe o he ee space) a enua ion cons an along he edge 0X in he i s B illouin zone o he p oposed h ee-hole uni cell wi h op imal dimensions using MMTMM. The igu e also included he phase shi ob ained wi h he ES o he CST S udio Sui e using pe iodic bounda ies in he p opaga ion di ec ions wi hin he B illouin zones and PEC bounda ies on he es . Bo h esul s a e 1048 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 Fig. 7. F equency beha io o he a enua ion o a ious no malized alues o d2 o he h ee-hole uni cell wi h dimensions gi en in Table I. (a) Fi s B illouin edge 0X. (b) Thi d B illouin edge M0. pe ec ly co ela ed, showing bo h he alidi y o he MMTMM and he pe o mance o he p oposed cell. The esul s also demons a e he high le el o a enua ion/ ejec ion p o ided by he h ee-hole s uc u e in he o bidden band, simila o he glide-symme ic holey EBG epo ed in [14]. As demons a ed in Fig. 6(b), when applying he MMTMM o he hi d edge 0M o he B illouin zone, a good ag eemen in he phase cons an wi h he CST-ES esul s is ound again excep o he appea ance o a complex mode in he o bidden band. This mode canno be de ec ed wi h he CST-ES because i only compu es p opaga i e modes. A de ailed discussion on he beha io o his kind o complex modes can be ound in [14]. Simila o he complex mode in [14], he complex mode in Fig. 6(b) spli s a abou 77 GHz in o wo eal p opaga i e o wa d and backwa d modes. The lowe b anch o he a enua ion cons an co esponds o he e anescen modes in he s opband, while he uppe b anch co esponds o he complex mode. To ob ain a be e pic u e o he beha io o he complex mode, i s oo loci a e p esen ed in Fig. 6(c) om i s onse equency a 40.8 GHz un il 81 GHz. I can be clea ly obse ed ha he complex mode spli s a 77.4 GHz in o wo eal p opaga i e o wa d and backwa d modes. The e ec o a ying he op no malized diame e , d2(keeping d1 ixed), on he a enua ion cons an is shown in Fig. 7. The alue d1/pis aken as he op imal one ha p oduces he highes limi o he s opband, as can be seen in Fig. 4(b). Fig. 7(a) shows ha he e ec o no malized d2on he i s B illouin edge is no e y signi ican , excep o i s minimum alue. The maximum le el o a enua ion is ob ained a he op imal no malized alue o d2/p=0.54, which is he same alue ha achie es he maximum ac ional s opband bandwid h. The in luence o d2/pis mo e signi ican a he hi d B illouin edge, as obse ed in Fig. 7(a). In his egion, i can be seen ha he change om he minimum no malized alue o i s maximum signi ican ly a ec s he a enua ion ac o . Again, he op imal alue o d2/p o a enua ion ma ches he op imal alue o he s opband. The esul s shown in Fig. 7(b) show ha mos o he in luence on he p opaga ion cha ac e is ics comes om he hi d B illouin edge (kx=ky). This means ha p opaga ion in he 0M di ec ion is he main esponsible o he good pe o mance o he p oposed uni cell in e ms o s opband. V. EXPERIMENTAL RESULTS In o de o expe imen ally alida e he p e ious analysis, wo pai s o WR-19 wa eguide space s we e manu ac u ed and measu ed in a back- o-back con igu a ion. All space s a e compa ible wi h he UG-383/U lange, wi h a 2.5 mm leng h. One o he pai s is a egula space wi h no holes on i , as shown in Fig. 8(a), while he o he uses he h ee-hole double-laye pe iodic s uc u e p oposed in his wo k, as shown in Fig. 8(b). The dimensions o he holes a e gi en in Table Ico esponding o he dispe sion diag am in Fig. 5(d). The measu emen se up can be seen in Fig. 8(c), showing he back- o-back Fig. 8. Manu ac u ed WR-19 wa eguide space s. (a) No holey egula space . (b) Th ee-hole double-laye space . (c) Space s measu emen se up in back- o-back con igu a ion. Fig. 9. Measu emen s o he ansmission coe icien . (a) Compa ison o he esponse wi h and wi hou holey s uc u e o ai gaps o 0 and 100 µm. (b) Response o he holey s uc u e o di e en alues o he ai gap. case. To measu e di e en ai gaps, a sepa a o is added be ween bo h space s o achie e he desi ed ai gap. Two di e en measu emen s ha e been pe o med o bo h pai s: one wi h ze o ai gap be ween bo h space s and he o he wi h an ai gap o 100 µm. The ansmission coe icien s (S21) a e illus a ed in Fig. 9. Bo h s uc u es pe o m as expec ed when no gap exis s be ween each space ; ha is, minimal losses a e ob ained. Howe e , when he e is a 100 µm ai gap, he s anda d s uc u e IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 1049 exhibi s conside able losses, wi h a dip appea ing a ound 47.5 GHz. In con as , he holey s uc u e has small losses wi h no peaks in he ope a ing band, 40–60 GHz. Fig. 9(b) shows a compa ison o ansmission losses o he holey s uc u e wi h a ying ai gaps. I is e iden ha he smalle he ai gap, he lowe he losses, al hough hey emain almos insigni ican in compa ison o he nonholey case. No e ha , in all cases, he measu ed e lec ion coe icien s a e below −20 dB. The e o e, he losses in he nonholey s uc u e can only be explained by he leakage due o he ai gap in he in e ace be ween bo h space s. In conclusion, he h ee-hole s uc u e d as ically educes leakage due o he space . These expe imen al esul s clea ly demons a e he po en ial o he p oposed h ee-hole s uc u e. This con igu a ion is a compac and e sa ile candida e o di e en p ac ical scena ios whe e leakage mus be a oided. VI. CONCLUSION In his communica ion, a no el holey EBG is p esen ed, which is based on a h ee-hole opology wi h di e en alues o he op and bo om diame e s. The pe o mance o his uni cell has been compa ed wi h o he simila holey EBGs, such as mi o , one- edge displaced mi o , and glide-symme ic opologies. In pa icula , we ha e s udied he s opband beha io , main aining he uni cell pe iod equal in all s uc u es and wi h he same lowe equency bound o he s opband. The esul s show ha he p oposed uni cell exhibi s a educ ion in size and an inc ease in he ac ional bandwid h wi h espec o he glide-symme ic s uc u e. These esul s sugges ha he p oposed uni cell could ha e po en ial uses in p ac ical applica ions wi h limi ed space. Wi h a pa ame ic analysis, he op imum pa ame e s o his uni cell ha e been ound o achie e he maximum ac ional bandwid h o he s opband. A MMTMM analysis was pe o med o de e mine he a enua ion o he p oposed uni cell, which shows a desi ed high le el o a enua ion in he s op- band, simila o o he glide-symme ic s uc u es p e iously s udied. To illus a e he po en ial o he p oposed EBG s uc u e in a p ac ical scena io, i has been used o educe leakage a he in e ace be ween wo space s. Ou expe imen al esul s show a d as ic imp o emen wi h espec o i s nonholey coun e pa . 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