IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1
Cha ac e iza ion and Supp ession o T ansmission
Dips in Glide-Symme ic Holey Gap Wa eguides
Mingzheng Chen , G adua e S uden Membe , IEEE, Johan Bellb an , Oska Ze e s om , Membe , IEEE,
F ancisco Mesa , Fellow, IEEE, and Osca Que edo-Te uel , Fellow, IEEE
Abs ac —The spu ious ansmission dips ha occu in glide-
symme ic holey gap wa eguides (GSHGWs) a e sys ema ically
cha ac e ized in his wo k, and he ob ained in o ma ion is used
o supp ess hem in he in ended ope a ing band o he gap
wa eguide. The analysis elies on he dispe sion cha ac e is ics
o he wa eguide segmen wi h elec omagne ic bandgap (EBG)
holes. These cha ac e is ics a e explo ed h ough he mul imodal
ans e ma ix app oach, pa icula ly ocusing on iden i ying
ele an edge and wa eguide modes. We ind ou ypes o
unwan ed dips in he ansmission coe icien wi hin he in ended
ope a ion equency band o he gap wa eguide unde s udy. The
i s h ee ypes a e all associa ed wi h he edge mode mos ly
concen a ed in he small ai -gap egion be ween he wa eguide
and he EBG holes, whe eas he ou h ype is caused by a na ow
s opband in he wa eguide mode. Based on a ho ough unde -
s anding o all dips, we p opose h ee iable solu ions: placing
EBG holes away om he wa eguide channel, in e sec ing EBG
holes wi h he wa eguide channel, and in e sec ing addi ional
small holes wi h he wa eguide channel and he EBG holes. A e
compa ison, he las solu ion wi h wo small holes pe EBG hole
along he wa eguide channel was demons a ed o be he mos
ad an ageous in e ms o ansmission p ope ies, compac ness,
and lexibili y. This solu ion was also expe imen ally alida ed
using a WR-19 GSHGW ope a ing om 35 o 63 GHz.
Index Te ms—Elec omagne ic bandgap (EBG), glide symme-
y, holey gap wa eguide, mul imodal ans e ma ix me hod
(MMTMM), pe iodic s uc u es, spu ious ansmission dips.
I. INTRODUCTION
METALLIC wa eguides a e widely used guiding s uc-
u es in elec omagne ic (EM) enginee ing due o hei
simplici y, low losses, and high powe -handling capabili ies
[1]. A high equencies ( ypically abo e 30 GHz), me allic
wa eguides a e e en mo e ad an ageous in e ms o ansmis-
sion losses and powe handling compa ed o dielec ic-based
Recei ed 9 Feb ua y 2025; e ised 16 Ap il 2025 and 15 May 2025;
accep ed 17 May 2025. The wo k o Osca Que edo-Te uel was sup-
po ed by he Ve enskaps ˚
ade (VR) P ojec h ough Call “Resea ch
P ojec G an Wi hin Na u al and Enginee ing Sciences” unde G an
2022-03865. The wo k o F ancisco Mesa was suppo ed in pa by
MICIU/AEI/10.13039/501100011033 unde G an PID2023-148281NB-I00
and in pa by ERDF/EU. (Co esponding au ho : Mingzheng Chen.)
Mingzheng Chen, Johan Bellb an , Oska Ze e s om, and
Osca Que edo-Te uel a e wi h he Di ision o Elec omagne ic
Enginee ing and Fusion Science, KTH Royal Ins i u e o Technology,
100 44 S ockholm, Sweden (e-mail: [email p o ec ed]; [email p o ec ed];
[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 ´
ıa In o m´
a ica, Uni e sidad de Se ille, 41012 Se ille, Spain (e-mail:
[email p o ec ed]).
Digi al Objec Iden i ie 10.1109/TMTT.2025.3572361
solu ions, such as mic os ip lines and subs a e-in eg a ed
wa eguides. Howe e , due o he small dimension o he com-
ponen s a high equencies, he manu ac u ing o high-quali y
me allic wa eguides and wa eguide-based eed ne wo ks
becomes di icul and expensi e [2],[3],[4]. In pa icula , i is
impo an o ensu e good elec ical con ac be ween wa eguide
pla es because e en a small gap o 50 µm can cause high
ansmission losses due o se e e powe leakage, as discussed
in [5],[6], and [7].
In o de o o e come he d awbacks o me allic wa eg-
uides a high equencies, gap wa eguide echnology was
p oposed [8],[9]. In his echnology, wa e p opaga ion is
es ic ed o a eas con ined by elec omagne ic bandgap (EBG)
s uc u es. As a esul , elec ical con ac be ween he op
and bo om pla es is no equi ed, which p esen s a g ea
ad an age o he manu ac u ing and assembly p ocess. Du ing
he las decade, gap wa eguide echnology has ound wide
applica ions in packaging [10], wa eguide a ay an ennas
[11],[12], wa eguide il e s [13],[14], and many o he EM
componen s [15],[16]. Pin- ype EBG s uc u es ha e been
commonly used in gap wa eguide echnology o s op wa e
p opaga ion in undesi ed di ec ions and a eas. Howe e , a
high equencies, such s uc u es equi e he manu ac u ing
o hin and all me allic pins, inc easing he de ice cos and
incu ing ab ica ion di icul ies wi h con en ional p oduc ion
echniques, such as compu e ized nume ical con ol (CNC)
machining.
Glide-symme ic holey EBG solu ions we e p oposed
as a cos -e ec i e al e na i e o pin- ype EBG s uc u es
in millime e -wa e (mmWa e) bands [17],[18]. A glide-
symme ic holey EBG s uc u e consis s o pe iodic holes
in he uppe and lowe pla es o a pa allel-pla e wa eguide
(PPW) sepa a ed by a small ai gap. The EBG holes a e shi ed
hal he pe iod in one o wo o hogonal in-plane di ec ions,
causing a subs an ial inc ease in he s op bandwid h compa ed
o i s nonglide coun e pa [19]. This EBG s uc u e is made
up o holes ha a e gene ally la ge han con en ional pins
a he same ope a ing equency, wi h he dep h o he holes
less han he co esponding pin heigh [18]. The e o e, glide-
symme ic holey EBG s uc u es can esul in highe accu acy
and lowe manu ac u ing cos s a high equencies and ha e
ound wide applica ions in wa eguides [20], phase shi e s
[21],[22],[23], il e s [23],[24],[25], langes [26], and a ay
an ennas [5],[27]. Howe e , spu ious ansmission dips ha e
been epo ed o occu wi hin he ope a ing equencies o gap
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2 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
wa eguides implemen ed using such holey EBG s uc u es,
educing hei e ec i e bandwid hs [20],[22],[23],[28]. To
a oid his p oblem, one p oposed solu ion is o use small
co uga ions o connec he EBG holes and he wa eguide
[20],[22],[23]. Al hough e ec i e in supp essing some o he
obse ed ansmission dips, he li e a u e does no p o ide a
sa is ac o y explana ion o he cause o hese dips. I is wo h
no ing ha Sipus e al. [28] ela ed he de ec ed ansmission
dip wi h he B agg e ec [29], explaining ha e lec ions
om pe iodic EBG holes cons uc i ely in e e e in phase,
which is only pa ially co ec . Mo eo e , he addi ion o small
co uga ions inc eases he manu ac u ing cos when using
CNC machining.
In his wo k, we sys ema ically cha ac e ize all ansmission
dips associa ed wi h gap wa eguides implemen ed wi h glide-
symme ic holey EBG s uc u es using he mul imodal ans e
ma ix me hod (MMTMM) [30],[31],[32]. This me hod
allows o an accu a e dispe sion analysis o he gap wa eguide
s uc u e implemen ed wi h EBG holes by ep esen ing i s
uni cell wi h a gene alized mul imodal ans e ma ix. The
inclusion o highe o de modes in his me hod ensu es ha
he ele an highe o de mode coupling be ween adjacen uni
cells can be p ope ly accoun ed o . Mo eo e , he MMTMM
also enables simul aneous compu a ion o he phase and a en-
ua ion cons an s o he uni cell unde s udy, o e ing aluable
insigh in o he complex and e anescen modes p esen in he
s uc u e. De ails abou hese complex/e anescen modes a e
no eadily p o ided by commonly used comme cial so wa e.
Wi h he MMTMM ool, we iden i y ou di e en ypes o
ansmission dips and gi e a de ailed explana ion o hem.
Mo eo e , we p opose h ee iable solu ions, aiming a gi ing
guidelines o elimina ing undesi ed ansmission dips o gap
wa eguides implemen ed wi h glide-symme ic holey EBG
s uc u es. Among he p oposed solu ions, he in oduc ion o
addi ional small holes in e sec ing he wa eguide channel and
he EBG holes ( wo small holes pe EBG hole) is p ima ily
ecommended and is also expe imen ally alida ed. Al hough
his s udy p ima ily ocuses on glide-symme ic holey gap
wa eguides (GSHGWs), simila ansmission dips may also
occu in o he pe iodic guiding s uc u es. Ou me hodology
can be applied o unde s and and add ess hese p oblems as
well.
This a icle is o ganized as ollows. In Sec ion II, a GSHGW
and i s co esponding uni cell a e s udied o in es iga e
spu ious ansmission dips. Nex , he sou ces o all he dips
a e analyzed in Sec ion III. Then, in Sec ion IV, h ee possible
solu ions a e p oposed and ho oughly discussed. In Sec ion V,
he p oposed solu ion o in e sec ing addi ional small holes
wi h he wa eguide channel and he EBG holes is expe i-
men ally alida ed. Finally, he main conclusions a e gi en
in Sec ion VI.
II. SPURIOUS TRANSMISSION DIPS
In his sec ion, we use a Bloch analysis o in es iga e
spu ious ansmission dips using a s aigh wo-pla e WR-19
gap wa eguide implemen ed wi h glide-symme ic EBG holes
in he side walls, as shown in Fig. 1. This s uc u e is he ein
Fig. 1. Sec ion o a GSHGW wi h a leng h o L=9p+2 . Cylind ical holes
along he wa eguide a e added in he pla es o o m a glide-symme ic EBG.
called a GSHGW, and he wa eguide ma e ial, i no speci ied,
is aluminum.
A. S udy o he Holey Gap-Wa eguide S uc u e
We i s conside a ini e sec ion o he GSHGW shown
in Fig. 1, whe e a pa o he op pla e is emo ed o
clea illus a ion. The gap wa eguide is sepa a ed in o wo
aluminum pla es by a small ai gap (g=50 µm). This gap is
selec ed o model he expec ed wo s -case gap in he p ac ical
ealiza ion o a gap wa eguide eeding sys em [21],[22].
The ec angula guiding channel has he same dimensions
as he s anda d WR-19 wa eguide wi h W=4.78 mm and
H=2.39 mm. The WR-19 wa eguide has a cu o equency
o a ound 31.5 GHz o he undamen al TE10 mode, and he
cu o equency o he nex highe o de mode is app oxi-
ma ely 63 GHz. In o de o s udy all spu ious ansmission
dips occu ing in he single-mode ope a ing band o he
wa eguide, he equency band o in e es in his s udy is se o
35–63 GHz. Two ows o glide-symme ic EBG holes a e
placed along he la e al sides o he wa eguide channel a
a dis ance d(edge wid h). In o de o ob ain a s opband in
he ope a ional band o he wa eguide, he dep h ho he
holes needs o be la ge enough, and he adius o he holes
is op imized o di e en p, whe e p ep esen s he pe iodic
spacing o he holes along he z-axis. The o al leng h o he
GSHGW sec ion is L=9p+2 , which is long enough o
ensu e a clea obse a ion o ansmission dips. No e ha
a sho segmen o leng h is added on bo h sides o he
wa eguide o a oid unca ing he EBG holes, an issue ha
will be discussed in de ail in Sec ions III and IV.
Be o e s udying he ansmission p ope ies o he GSHGW,
we need o cha ac e ize he s opband pe o mance o he
2-D holey pe iodic EBG s uc u e i sel . Fo his pu pose, we
show in Fig. 2 he dispe sion diag am along he edges o he
i educible B illouin zone (pa h ΓXMΓ) o pe iodic s uc u es
wi h p=5.6 and 6 mm and a small ai gap o g=50 µm.
He e, pis he diagonal leng h o he squa e uni cell, which
is also shown in Fig. 2; he holes ha e a dep h o h=1.5 mm
and an op imized adius o =0.274 p. We can obse e ha
he s opband in all di ec ions o he EBG co e s he wa eguide
ope a ional band om 35 o 63 GHz when p a ies be ween
5.6 and 6 mm. No e ha when placed along he wa eguide,
he holes a e epea ed along he Γ→Mdi ec ion.
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CHEN e al.: CHARACTERIZATION AND SUPPRESSION OF TRANSMISSION DIPS IN GSHGWs 3
Fig. 2. Dispe sion diag am along he pa h ΓXMΓo he i s B illouin zone
o he squa e glide-symme ic EBG uni cell o p=5.6 and 6 mm. The
wa eguide ope a ional band is shaded yellow.
Fig. 3. (a) T ansmission and (b) e lec ion coe icien s o he WR-19 GSHGW
wi h di e en pe iods pand wi hou (w.o.) EBG.
Nex , we s udy he ansmission and e lec ion coe icien s
o he GSHGW o leng h Lwi h di e en pe iods pusing he
CST ime-domain sol e . The edge wid h is se o d=0.6 mm.
The pe iod pis a ied om 5.6 o 6 mm in s eps o 0.1 mm.
The esul s o he GSHGW a e epo ed in Fig. 3, oge he
wi h a e e ence wa eguide o leng h 55.38 mm wi hou (w.o.)
he EBG s uc u e ( he same leng h as he GSHGW wi h
p=5.8 mm). We i s look a he case o he GSHGW
wi h p=6 mm, he esul s o which a e he ligh pu ple
cu es in Fig. 3. The EBG holes, designed o minimize powe
leakage, a e expec ed o allow e icien ansmission ac oss
he wo king EBG. Howe e , as demons a ed in Fig. 3(a),
we no e se e al ansmission dips h oughou he ope a ional
equency band. In pa icula , e ec i e ansmission is only
e iden in he 41–46- and 49–54-GHz equency in e als.
Fig. 4. (a) Uni cell o he GSHGW. (b) Equi alen ne wo k o he uni cell
cha ac e ized by a 2N-po mul imodal ans e ma ix.
Simila obse a ions can also be made o o he pe iods. In
his wo k, we ca ego ize he dips o any o he conside ed
alues o pin o ou di e en ypes, namely, Types I-IV, om
low o high equencies. The eason o such classi ica ion
is ha he dips a e ound o ha e di e en p ope ies and
sou ces, which is discussed in mo e de ail in la e sec ions.
Type I dips occu a he lowe equencies (below 45 GHz)
whe e we can obse e a ound ou - i e na ow dips. A he
equencies co esponding o he Type I dips, he ansmission
coe icien s in Fig. 3(a) ha e alues be ween −3 and −1 dB,
and we obse e ha hese dips a e co ela ed wi h he peaks
appea ing in he e lec ion coe icien s shown in Fig. 3(b). The
Type II dips appea o di e en pbe ween 46 and 52 GHz
wi h alues o |S21|a ound −3 dB. When mo ing highe
in equency, mo e se e e Type III dips wi h alues o he
ansmission coe icien a ound −10 dB and highe han −3 dB
in he e lec ion coe icien a e obse ed. Finally, Type IV
dips a e obse ed be ween 58 and 65 GHz wi h le els o
app oxima ely −5 dB o bo h |S21|and |S11|. An in e es ing
obse a ion in Fig. 3is ha inc easing he pe iod po he
EBG holes om 5.6 o 6 mm esul s in dips o Types II-
IV occu ing a p og essi ely lowe equencies, while he
appea ance o Type I dips is less p edic able. This obse a-
ion sugges s a s ong connec ion be ween he ansmission
dips and he pe iodici y o he EBG holes o he dips o
Types II-IV.
B. Analysis o he Uni Cell
To unde s and he causes o hese ansmission dips, we
no e ha he GSHGW analyzed in he p e ious sec ion can
be seen as a ini e sec ion o a 1-D pe iodic s uc u e along
he p opaga ion di ec ion z. To pe o m a Bloch analysis o
his 1-D pe iodic s uc u e, we apply he MMTMM o he
uni cell o he GSHGW shown in Fig. 4(a). Po s a e added
a he bounda ies o he uni cell along he z-axis, which is
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4 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
he pe iodici y di ec ion. To accoun o he coupling be ween
adjacen cells h ough high-o de mode in e ac ions, he uni
cell is s udied using he MMTMM [32] and is modeled as
a 2N-po equi alen ne wo k wi h inpu and ou pu po s
ep esen ing he i s Nsigni ican modes, as depic ed in
Fig. 4(b). The mul imodal sca e ing pa ame e s o he uni
cell a e i s simula ed using, o example, he CST equency-
domain sol e wi h e ahed al meshing and open bounda ies
se on he wo physical po s, each exci ed by he Nmodes.
F om he mul imodal sca e ing ma ix, we ob ain he 2N×2N
mul imodal ans e (ABCD) ma ix [T(ω)] ha cha ac e izes
he uni cell in he pe iodic en i onmen . Finally, apply-
ing Bloch’s heo em, wa e p opaga ion along he pe iodic
di ec ion zcan be cha ac e ized as he ollowing eigen alue
p oblem [32]:
V2
I2=[T(ω)] V1
I1=e−jkp V1
I1(1)
whe e V1,V2and I1,I2a e he ol age and cu en ec o s
ela ed o he N-po modes in he inpu and ou pu po s, and
k=β−jαis he Bloch wa enumbe along he z-di ec ion,
wi h βand αbeing he phase and a enua ion cons an s.
As an example, he eigenp oblem (1) is now sol ed o
he uni cell wi h p=5.8 mm and o he dimensions as
de ined in Sec ion II-A. As shown in Fig. 5(a), a good
ag eemen o he phase shi (βp/π) is ob ained be ween he
esul s o he MMTMM (solid lines) and he CST eigenmode
sol e (CST ES, ci cles). Fig. 5(b) shows he no malized-
o-k0a enua ion cons an s (k0is he wa enumbe o he
ee space) calcula ed wi h he MMTMM wi h aluminum o
pe ec elec ic conduc o (PEC) as he building ma e ial. Only
MMTMM esul s a e plo ed since he CST ES canno p o ide
he a enua ion cons an in he s opband [32]. We can iden i y
h ee signi ican modes in Fig. 5(a), which also shows he
inse s o hei co esponding modal Eyp o iles gi en by CST
ES ( he op pla e is omi ed o be e illus a ion). The i s
mode (blue cu e) is he desi ed p opaga ing wa eguide mode.
The second mode ( ed cu e) is an edge mode ha p opaga es
in he small ai -gap egions be ween he wa eguide channel
and he EBG holes. The hi d mode (o ange cu e) is ano he
edge mode in he ai -gap egions bu now bounded o he
ex e nal sides o hem. The i s ele an obse a ion is ha
since Mode 3 is physically isola ed om he wa eguide mode
by he EBG s uc u es, i will no in luence he p opaga-
ion o he wa e wi hin he wa eguide channel. In con as ,
Mode 2 a els h ough he ai gap ha is di ec ly connec ed
o he wa eguide channel. This mode has a ield dis ibu ion
ha esembles ha o he ans e se elec omagne ic (TEM)
mode o a PPW. Hence, his quasi-TEM mode is expec ed
o be coupled wi h he quasi-TE10 Mode 1 ha a els along
he wa eguide channel. Fo p=5.8 mm, Fig. 5(a) shows
ha he edge Mode 2 has wo passbands a 35–44 and
57–65 GHz, espec i ely, and a s opband om 44 o 57 GHz.
This in o ma ion is key o unde s anding he o ma ion o
spu ious ansmission dips. As plo ed in Fig. 5(c), assuming
ha he me allic ma e ial is PEC, Mode 2 couples wi h Mode 1
s a ing a abou 57 GHz o gi e ise o a pai o complex
modes wi h a high a enua ion coe icien in he ange o
Fig. 5. Bloch analysis o he GSHGW uni cell wi h p=5.8 mm. (a) Phase
shi s (βp/π) o signi ican modes wi h he inse s depic ing he dis ibu ion o
Eycomponen s ob ained wi h CST ES. (b) No malized a enua ion cons an
(α/k0) o Modes 1 and 2 wi h aluminum o PEC as he building ma e ial.
(c) Roo loci o he kzsolu ions o Modes 1 and 2 om 56.5 o 60 GHz,
whe e hese modes couple o gi e ise o a complex mode. PEC is selec ed
as he building ma e ial he e o elimina e he in luence o me allic losses in
he analysis.
57–59 GHz. The appea ance o hese complex modes is wha
ac ually causes Type III ansmission dips, as p esen ed in
Fig. 3. In ac , he sou ces o all he ansmission dips can
be e ealed om he Bloch analysis o he GSHGW uni cell,
which will be de ailed in Sec ion III.
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CHEN e al.: CHARACTERIZATION AND SUPPRESSION OF TRANSMISSION DIPS IN GSHGWs 5
Fig. 6. (a) T ansmission coe icien s o he GSHGW wi h p=5.6 and 5.8 mm
a 35–45 GHz, whe e Type I dips a e indica ed. (b) Phase shi (βL/π) o
Mode 2 conside ing he o al wa eguide leng h L. (c) E- ield dis ibu ion o
he gap wa eguide wi h p=5.8 mm a 38.3 GHz ( he op pla e is hidden o
illus a ion).
III. SOURCES OF SPURIOUS TRANSMISSION DIPS
A. Type I T ansmission Dips
The ansmission coe icien s o he GSHGW segmen
wi h pe iods p=5.6 and 5.8 mm wi hin he equency
ange o 35–45 GHz a e shown in Fig. 6(a), highligh ing he
equencies whe e Type I dips appea . In Fig. 6(b), we show
he phase shi s (βL/π) o Mode 2 along he z-di ec ion in
he same equency ange. I can be obse ed ha Type I dips
occu a equencies ha sa is y app oxima ely
βL/π =n,n=1,...,6.(2)
To illus a e his phenomenon, he E- ield dis ibu ion o he
GSHGW wi h p=5.8 mm a he equency o 38.3 GHz (one
o he Type I esonances) is shown in Fig. 6(c). The wa eguide
is exci ed om Po 1 and he EM ield p opaga es o Po 2
inside he guiding channel as a quasi-TE10 mode. Since his
equency alls wi hin he passband o he edge Mode 2, pa
o he ield a els along he hin ai channel be ween he EBG
holes and he wa eguide channel. A he end o he s uc u e
Fig. 7. (a) T ansmission coe icien s o he GSHGW wi h di e en pe iods p
a 46–52 GHz, whe e Type II dips occu . (b) No malized a enua ion cons an s
α/k0o Mode 2 a 42–58 GHz.
on he Po 2 side, he edge EM wa es a e e lec ed back wi h
an ampli ude gi en by
E =η0−η1
η0+η1
Ei(3)
whe e Eiand E ep esen he ampli udes o he inciden and
e lec ed edge wa es, espec i ely, while η0and η1deno e he
wa e impedances in ee space and wi hin he na ow ai -gap
egion, espec i ely. Gi en ha his egion can be conside ed
as a PPW, i s impedance is es ima ed o be η1=(g/d)η0, and
because gd, we conclude ha η1η0. This implies ha
E ≈Ei o he edge EM wa es a he end o he wa eguide.
I we only conside one e lec ion, when he phase shi o he
edge mode mee s he condi ion βL/π =n, he elec ic ield o
he edge wa es a any z-posi ion can be ob ained as
E=Ei+E ≈Eie−jβzˆ
y+Eiej(βz+nπ)ˆ
y
≈Eie−jβz+(−1)nejβzˆ
y.(4)
Consequen ly, s ong s anding wa es o m along he hin edge
nex o he wa eguide channel, leading o Type I dips, as
illus a ed in Fig. 6(c). I is also no ed ha he in ended TE10
ansmission mode wi hin he wa eguide channel is dis o ed
by he in ense s anding edge wa es.
B. Type II T ansmission Dips
The GSHGW ansmission coe icien s co esponding o he
equency ange whe e Type II ansmission dips occu a e
shown in Fig. 7(a) o di e en pe iods pa 46–52 GHz. We
obse e ha o each p, he e exis s only one dip o Type II,
which shi s o lowe equencies as pinc eases. Wi hin he
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6 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
Fig. 8. E- ield dis ibu ion o he gap wa eguide wi h p=5.8 mm a 49 GHz
when (a) L=9p+2 =55.38 mm and (b) L=9p=52.2 mm ( unca ed).
(c) E- ield magni ude along he middle o he wa eguide channel and i s edge
wi hou and wi h unca ion (whi e dashed lines).
equency ange whe e hese dips occu , Fig. 5(a) indica es he
p esence o wo ele an modes: a p opaga ing Mode 1 and an
e anescen Mode 2. The a enua ion cons an s α/k0o Mode 2
o di e en pa 42–58 GHz a e shown in Fig. 7(b), wi h he
equency ange o Type II dips indica ed. The i s ele an
obse a ion is ha he Type II dips occu app oxima ely in
he cen e o he s opbands o Mode 2, whe e he a enua ion
cons an s ha e a maximum.
Taking p=5.8 mm as a e e ence, in Fig. 8(a), we show
he E- ield dis ibu ion o he GSHGW a 49 GHz, whe e
he Type II dip occu s. I can be seen ha he ele an edge
Mode 2 is exci ed a he edges nea he inpu and ou pu
po s, as highligh ed in he ed ec angles in Fig. 8(a). Due
o he e anescen na u e o his mode, a e a ew pe iods
om he inpu po , only he p opaga ing mode pe sis s. When
his p opaga ing mode eaches he ou pu po , he impedance
misma ch a he end o he wa eguide causes e-exci a ion o
Mode 2, esul ing in he high ield concen a ion obse ed in
he edge egion a his end. Consequen ly, powe ans e o
Mode 2 esul s in he o ma ion o he Type II dip. To u he
alida e his hypo hesis, we educe he wa eguide leng h by
on bo h po ends, e ec i ely hal ing he edge egions
whe e Mode 2 is mos ly concen a ed, as shown in Fig. 8(a).
As clea ly seen in Fig. 8(b), Mode 2 is ha dly exci ed in
hese smalle edge egions, allowing us o achie e e ec i e
ansmission o he quasi-TE10 mode h ough he wa eguide
channel. A compa ison o he magni ude o he E- ield along
wo s aigh lines in he middle o he wa eguide channel and
a i s edge, wi hou and wi h unca ion, is shown in Fig. 8(c).
Wi hou unca ion (L=9p+2 ), s ong ields a e exci ed
along he edge nea he po s, which decay exponen ially
along he ±z-di ec ion (see he blue do ed cu e). The yellow
cu e in Fig. 8(c) shows ha he a enua ion i s well wi h
he ac o E0e−αz, whe e α=0.148 mm−1is he a enua ion
coe icien o Mode 2 ob ained wi h MMTMM and shown in
Fig. 7(b). Due o he exci a ion o hese e anescen ields, he
ampli ude o he E- ield along he cen e line o he wa eguide
signi ican ly dec eases nea he po s, whe eas he ampli ude
in he unca ed wa eguide emains nea ly unchanged. In
Fig. 7(a), we obse ed ha he e was no Type II dip o he
case p=5.8 mm a e unca ion (blue do ed cu e). In ac ,
i is ecommended o always apply such a unca ion o a oid
he Type II dip.
C. Type III T ansmission Dips
The ansmission coe icien s o he GSHGW wi h di e en
pe iods pa 53–65 GHz a e p esen ed in Fig. 9(a). The Type III
dips a e hose ha occu a 55–61 GHz wi h a le el lowe
han −6 dB. The equency ange aligns p ecisely wi h he
loca ion o he coupling o Mode 1 and Mode 2, esul ing
in he eme gence o a pai o complex modes cha ac e ized
app oxima ely by he wa enumbe s β±jα. Only in he
lossless case a e he βand αo he wo complex modes
equal. In p ac ice, due o di e en me allic losses o he wo
modes, βand αo he complex modes will di e sligh ly.
Due o he symme ic na u e o he s uc u e, an addi ional
pai o complex modes wi h −β∓jαalso eme ges. De ailed
discussions on complex modes a e gi en in [33],[34], and
[35]. Impo an ly, F ei e e al. [35] highligh ha eeding Po 1
in he equency ange ha suppo s only complex modes will
exci e a decaying ield gene a ed by he combina ion o wo
complex modes wi h kz=β−jαand −k∗
z=−β−jα. Assuming,
o ins ance, ha he ield in he pe iodic s uc u e is domina ed
by he ze o h ha monic, his combina ion would gi e ise o
a so-called “modula ed e anescen mode” wi h he ollowing
gene al o m [35, eq. (10)]:
E(x,y,z)≈A0(x,y) cos(βz)+B0(x,y) sin(βz)e−αz
≈C0(x,y) cos(βz+φ)e−αz.(5)
In his case, he ield p o ile co esponds o a s anding ield
ype wi h an ampli ude unc ion dependen on β ha expo-
nen ially decays wi h an a enua ion cons an α. The alues
o he phase and a enua ion cons an s o Mode 1 wi hin he
ange 55–61 GHz a e plo ed in Fig. 9(b) and (c). Due o
he high a enua ion cons an αo he modula ed e anescen
mode, he magni ude o he E- ield a enua es signi ican ly
along he p opaga ing z-di ec ion in he wa eguide channel, as
p esen ed in Fig. 10(a) and (b). The a enua ion o he E- ield
magni ude in Fig. 10(b) i s app oxima ely wi h E0e−αz, whe e
α=0.039 mm−1is he a e age o he a enua ion coe icien s
o he wo complex modes ob ained wi h MMTMM when he
me allic ma e ial is aluminum [see Fig. 5(b)]. Mo eo e , he
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CHEN e al.: CHARACTERIZATION AND SUPPRESSION OF TRANSMISSION DIPS IN GSHGWs 7
Fig. 9. (a) T ansmission coe icien s o he GSHGW wi h di e en pe iods p
a 53–65 GHz, whe e Types III and IV dips occu . (b) Phase shi βp/π and
(c) a enua ion cons an α/k0o Mode 1.
TABLE I
FREQUENCIES OF THE TYPE IV TRANSMISSION DIPS
equency ange o he occu ence o complex modes shi s
wi h he change o pe iod p, which is he eason o he shi
o he Type III ansmission dips.
D. Type IV T ansmission Dips
To s udy he Type IV ansmission dips, in Fig. 9(a)–(c), we
can obse e ha he dips occu ing in he equency ange o
58–64 GHz a e ela ed o s opbands o he GSHGW wi h β=
0. In hese small s opbands, Fig. 9(c) shows ha he a enua ion
cons an αis much smalle han ha causing he Type III dip,
a ac ha is consis en wi h he lowe alues obse ed o
he ansmission coe icien s in Fig. 9(a). In Fig. 11, we can
Fig. 10. (a) E- ield dis ibu ion o he GSHGW p=5.8 mm a 57.7 GHz.
(b) Ampli ude o he E- ield a 57.7 GHz along he cen al line o he guiding
channel. Me allic pa s a e aluminum.
Fig. 11. E- ield dis ibu ion o he GSHGW wi h p=5.8 mm a 60.9 GHz.
see ha o p=5.8 mm, he a enua ion o he E- ield in he
wa eguide channel is much less se e e a 60.9 GHz (Type
IV dip) han ha in Fig. 10(a) a 57.7 GHz (Type III dip).
The Type IV ansmission dips can be di ec ly a ibu ed o
he B agg e ec [29],[36],[37], whe e consecu i e e lec ions
om he EBG holes cons uc i ely in e e e in phase. As a
esul , he condi ion o he Type IV ansmission dip is
nλg=2p,n=1,2, . . . (6)
whe e λgis he guiding wa eleng h. App oxima ing he
GSHGW mode as he pu e TE10 mode in no mal ec angula
wa eguide, we ob ain ha he equencies B,nsa is ying (6)
a e gi en by
B,n=s2c
np2
+ c2(7)
whe e cis he speed o ligh in ee space and cis he cu o
equency o TE10 mode. In ou case, only B,2 alls wi hin
he ope a ional equency band. When pchanges om 5.6 o
6 mm, Table Ishows a good co ela ion be ween he alues
p edic ed by (7) and he equencies o he a enua ion peaks
shown in Fig. 9(c). I should be no ed ha he o igin o hese
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8 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
TABLE II
SOURCES OF ALL SPURIOUS TRANSMISSION DIPS
Fig. 12. (a) Top iew o he GSHGW uni cell ( op pla e is hidden
he e). Ligh /da k blue holes a e in he op/bo om pla e, and he iole
ec angle ep esen s he wa eguide channel. (b) T ansmission coe icien s o
he GSHGW wi h di e en edge wid hs dand pe iods p.
ansmission dips is e y simila o he open s opband issue
in leaky-wa e an ennas [38],[39],[40].
In summa y, we ha e in es iga ed he sou ces o ou
di e en spu ious ansmission dips in his sec ion. As sum-
ma ized in Table II, Type I dips a e caused by s ong s anding
edge wa es, while Type II dips o igina e om a enua ing
edge wa es nea wa eguide po s. Mo eo e , complex modes
gene a ed by he coupling o he p opaga ing wa eguide mode
and he ele an edge mode gi e ise o Type III dips. Finally,
unlike he Types I-III dips ha a e all ela ed o he ele an
edge mode, he Type IV dips a e caused by he appea ance o
quasi-s anding wa es in he channel.
IV. SOLUTIONS TO SPURIOUS TRANSMISSION DIPS
In his sec ion, we p opose h ee possible solu ions o
elimina ing spu ious ansmission dips in GSHGW.
A. Place EBG Holes Away F om Wa eguide
The i s p oposed solu ion migh be be e iewed as a
mi iga ion s a egy. As discussed in Sec ions II and III, he
dips come om ei he he in e ac ion be ween he wa eguide
mode and he undesi ed edge mode o in-phase e lec ions
om he EBG holes. The e o e, he dips can be mi iga ed by
educing he coupling be ween he wa eguide mode and he
edge mode, and by physically inc easing he isola ion be ween
he wa eguide and he EBG s uc u e. One way o simul ane-
ously achie ing hese goals is placing he EBG holes u he
away om he wa eguide channel ( ha is, inc easing he
Fig. 13. (a) Phase shi (βp/π) o signi ican modes o he GSHGW uni cell
wi h p=5 mm and d=−0.1 mm. Nega i e alues o dmean ha he holes
in e sec he guiding channel. (b) No malized a enua ion cons an (α/k0) o
Mode 1 wi h he inse depic ing he uni cell. He e, nega i e dmeans he EBG
holes in e sec wi h he wa eguide channel. (c) T ansmission coe icien s o
he GSHGW wi h di e en in e sec ion dis ances dand pe iods p.
wid h d), as he EM wa es a e in his case less con ined
o he edges. The ansmission coe icien s o he GSHGW
(p=5.8 mm) wi h di e en wid hs da e p esen ed in
Fig. 12(b) (blue cu es). We obse e ha when dinc eases
om 0.6 o 2 mm, all ansmission dips a e educed. An
addi ional obse a ion is ha he Types II-IV dips mo e
o lowe equencies as dinc eases, he eby dec easing he
e ec i e bandwid h o he GSHGW. Fu he mo e, he Type III
dip emains signi ican , eaching mo e han −5 dB, e en when
d=2 mm. We can u he educe he pe iod po he EBG
holes om 5.8 o 5 mm o shi he dips o Types II-IV
o highe equencies and educe hei magni udes, as demon-
s a ed in Sec ion II-A. Finally, wi h a combina ion o
p=5 mm and d=2 mm, we achie e good ansmission in
he equency ange om 40 o 60 GHz wi h a ansmission
coe icien highe han −0.5 dB, as shown by he ed do ed
cu e in Fig. 12(b).
When a use needs a speci ic ope a ing bandwid h o he
GSHGW, a po en ial app oach is o enla ge he edge wid h
dand dec ease he pe iod p, which will eloca e unwan ed
dips beyond he desi ed band. Howe e , mmWa e sys ems
equi e compac ness, which means ha dis gene ally kep
o a minimum. Mo eo e , signi ican ly educing he pe iod p
could cause he s opband o he EBG s uc u e o shi ou side
he wa eguide ope a ing band. The e o e, al e na i e me hods
o supp essing ansmission dips a e discussed Sec ions IV-B
and IV-C.
B. In e sec ing EBG Holes In o he Wa eguide Channel
The second solu ion is o in e sec he EBG holes di ec ly
in he wa eguide channel, esul ing in he GSHGW uni
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CHEN e al.: CHARACTERIZATION AND SUPPRESSION OF TRANSMISSION DIPS IN GSHGWs 9
cell shown in he inse o Fig. 13(b). The bene i o his
in e sec ion is ha he ele an edge mode (Mode 2) is elimi-
na ed, as illus a ed in Fig. 13(a). The ansmission coe icien s
o he GSHGW wi h di e en in e sec ion dis ances da e
p esen ed in Fig. 13(c) (blue cu es). In his igu e, we can
obse e ha using in e sec ing holes (d=−0.1 mm), he
Types I-III dips a e emo ed in he ange o 35–55 GHz. No e
ha he disappea ance o hese h ee ypes o dips is due o
he elimina ion o he ele an edge Mode 2. Ne e heless, his
solu ion makes he Type IV dip e en mo e p ominen because
o he inc eased e lec ions om he pe iodic EBG holes. I
he in e sec ion dis ance is sligh ly inc eased o −0.2 mm, he
co esponding Type IV dip becomes much wide and deepe .
To mo e his dip beyond ou band o in e es , we can educe
he pe iod p o 5 mm, as illus a ed by he ed-do ed cu e in
Fig. 13(c). Wi h a combina ion o p=5 mm and d=−0.1 mm,
we achie e good ansmission in he ange o 40–63 GHz.
Howe e , as shown in he zoomed-in iew inse o Fig. 13(c),
we ind signi ican dips in he ange o 35–40 GHz. This is
a ibu ed o he ac ha his equency ange is e y close o
he cu o equency o Mode 1, leading o po en ial leakage, as
also e ealed in he nonze o alues o he a enua ion cons an
in his equency ange obse ed in Fig. 13(b).
We ind ha di ec ly in e sec ing EBG holes in he wa eg-
uide channel and con enien ly selec ing he pe iod pis a
comp omise solu ion in e ms o ope a ing bandwid h. How-
e e , i is s ill a be e al e na i e compa ed o he i s solu ion
in a leas wo aspec s. Fi s , he p oblems o Types I-III dips
a e sol ed comple ely ins ead o jus being mi iga ed. Second,
he spaces needed o he EBG s uc u es a e signi ican ly
educed, leading o mo e compac designs.
C. Inse ing Addi ional Small Holes
To mi iga e he Type IV dip obse ed in he second solu ion,
we p opose inse ing a se ies o small holes a he bounda y
be ween he wa eguide channel and he EBG holes. Wi h
his solu ion, he EBG holes do no need o in e sec he
wa eguide channel (d>0). This s a egy is expec ed o
ully supp ess he associa ed edge mode while a oiding he
appea ance o p onounced in-phase e lec ions. Ou i s es
is o add an addi ional small hole o each EBG hole (deno ed
in he ollowing as “1 small hole”), as illus a ed in he
uni cell shown in he inse o Fig. 14(e). In his case, he
co esponding pa ame e s a e he adius o he small hole
s=0.6 mm, he heigh hs=h/2=1.195 mm, and he
dis ance be ween he cen e o he hole and he wa eguide
channel ws=0.48 mm. The ansmission coe icien o such
GSHGW is plo ed in Fig. 14(a) as a solid yellow cu e. As
expec ed, we success ully emo ed he Types I and III dips.
Howe e , he Type II dip emains p esen since Mode 2 can
s ill be exci ed nea he po egions in his s uc u e. Howe e ,
his dip is comple ely emo ed by implemen ing he unca ion
ope a ion discussed in Sec ion III-B, as con i med by he
yellow do ed cu e in Fig. 14(a). Finally, a less p onounced
Type IV dip is no iceable a 60 GHz compa ed o he case
wi hou small holes. This weake dip is caused by e lec ions
om he small holes. This e ec is u he co obo a ed by
he no malized a enua ion cons an o he wa eguide mode
Fig. 14. (a) T ansmission coe icien s o he GSHGW wi hou (w.o.) small
holes, wi h (w.) one small hole, wi h wo small holes, wi h one small hole
and unca ed, and wi h wo small holes and unca ed. p=5.8 mm and
d=0.6 mm o all cases. (b) Implemen a ion o he gap wa eguide wi h
wo small holes ( he op pla e is hidden o illus a ion). (c) Illus a ion o a
segmen o he EBG s uc u e wi h wo addi ional small holes pe EBG hole in
he edge a ea. (d) Phase shi s (βp/π) o signi ican modes o he wa eguide
uni cell wi h one small hole. (e) No malized a enua ion cons an (α/k0) o
Mode 1 wi h he inse depic ing he wa eguide uni cell wi h one small hole.
( ) Phase shi (βp/π) o signi ican modes o he wa eguide uni cell wi h
wo small holes. (g) No malized a enua ion cons an (α/k0) o Mode 1 wi h
he inse depic ing he wa eguide uni cell wi h wo small holes.
in Fig. 14(e), whe e a dis inc spike can be obse ed a ound
60 GHz.
Once we iden i ied he p oblem o inse ing one small hole
pe EBG hole, we now inc ease he numbe o addi ional
small holes o wo pe EBG hole in he edge a ea (deno ed
as “2 small holes”), as shown in Fig. 14(b). The de ailed
a angemen o he small holes is illus a ed in Fig. 14(c),
whe e he spacing o he adjacen small holes is ps=p/4.
The o he pa ame e s a e s=0.5 mm, hs=h/2=1.195 mm,
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