1592
JOURNAL
OF
LIGHTWAVE TECHNOLOGY
,
VOL.
11,
NO.
10,
OCTOBER
1993
Guided- o-Leaky Mode T ansi ion in
Uniaxial Op ical Slab Wa eguides
Lluis To ne ,
Membe ,
ZEEE,
Jaume
Recolons, and Juan
P.
To es,
S uden Membe ,
ZEEE
Abs ac -The guided- o-leaky hyb id mode ansi ion in slab
op ical wa eguides made
on
uniaxial c ys als such as
LiNbOs
o
LiTaOs
is analyzed.
Two
di e en guided- o-leaky ansi ions
ha e been iden i ied, namely he
o dina y cu o
and he
ex ao di-
na y cu 08
which occu when conside ing nega i e and posi i e
bi e ingen ma e ials, espec i ely. Analy ical bu anscenden al
exp essions ha e been ob ained, yielding he c i ical op ical axis
o ien a ion, ela i e o he wa eguide axis, abo e which he
o ally guided hyb id modes become leaky. The esul s indica e
ha he alue o he c i ical o ien a ion s ongly depends
on
he wa eguide pa ame e s. The possibili y
o
uning his c i ical
o ien a ion o a desi ed alue h ough he wa eguide pa ame e s
is discussed.
I.
INTRODUCTION
NTEGRATED op ical echnology ypically employs uniax-
I
ial dielec ic c ys alline ma e ials, such as li hium nioba e
(LiNb03) and li hium an ala e (LiTaOs),
o
ob ain low loss
op ical wa eguides. Se e al wa eguide ab ica ion echniques
ha e been ex ensi ely de eloped in pas yea s and, as a esul ,
he in ol ed echnology has become ma u e [l]. Modula-
o s, swi ches, pola iza ion con olle s, il e s, and co ela o s,
among o he high-pe o mance de ices o signal p ocessing,
signal ou ing and sensing, ha e been demons a ed in in e-
g a ed o m
[2],
[3].
Mos o hese de ices a e based
on
he
excellen elec oop ic and acous oop ical p ope ies o LiNb03
and LiTa03. C ys al aniso opy plays also a e y impo an
ole in many de ices, o ins ance
o
ob ain TE-TM con e sion
o
o
op imize phase-ma ching condi ions o second-ha monic
gene a ion and o he nonlinea pa ame e ic in e ac ions, bu i
is he cen al poin o he aniso opy-based cu o de ices [4].
The p inciple o ope a ion o he aniso opy-based cu o
de ices is he passi e as well as elec oop ically induced
guided- o- adia ed mode con e sion. Ligh -in ensi y (cu o
o mode exci a ion) modula o s
[SI
and passi e TE-pass
[6]
o TM-pass
[7]
pola ize s belong
o
his ca ego y o op ical
de ices. Wi h a ew excep ions
[8],
[9], in mos cases he
guided- o- adia ed mode con e sion be ween he
TE
and he
TM modes, which p opaga e in bi e ingen wa eguides when
he p opaga ion di ec ion is pa allel o pe pendicula
o
he
c ys al op ical axis, is conside ed and exploi ed [lo]-[14].
Howe e , wi h he excep ion o he o me symme ic con ig-
u a ions, all he modes suppo ed by a slab wa eguide made
Manusc ip ecei ed May
5,
1992; e ised No embe 2, 1992. This
wo k
was suppo ed by he Comision In e minis e ial de Ciencia
y
Tecnologia
o
he Spanish Go e nmen unde g an s PT89-0178 and TIC92-0094-C02-02.
The au ho s a e wi h he Poly echnic Uni e si y
o
Ca alonia, Depa men
o
Signal Theo y and Communica ions,
08080
Ba celona, Spain.
IEEE
Log
Numbe 9209997.
on bi e ingen ma e ials a e o he hyb id ype, wi h he
six ield componen s. In addi ion
o
he well-known o ally
guided modes, such wa eguides supppo he so-called leaky
modes. These a e leaky guided modes, in con as o hei
leaky unguided coun e pa s which occu in he iso opic case
[15]. They a e always lossy and ene gy leakage ou side he
guiding egion.
To ally guided modes as well as leaky modes guided by
uniaxial hin dielec ic ilms ha e been ex ensi ely s udied
since he ea ly days o in eg a ed op ics,
so
now he main
ea u es conce ning guiding p ope ies a e well es ablished
[
161-[20].
In
pa icula , leakage losses ha e been expe imen-
ally measu ed [21] and heo e ically calcula ed
[22]-[24]
o
a Ti: LiNb03 wa eguide, and ex ensi e calcula ions on he
dispe sion p ope ies o o ally guided modes as
a
unc ion
o he op ical axis o ien a ion, ela i e o he wa eguide axis,
ha e been epo ed
[25].
On he con a y,
o
he bes o ou
knowledge, he c i ical o ien a ion a which he o ally guided
modes become leaky has ne e been analyzed. Ou aim he e is
o add ess his poin . Ou goal is o iden i y he condi ions o
he guided- o-leaky mode ansi ion (GLMT)
o
occu , o he
common LiNb03 and LiTa03-based wa eguiding s uc u es,
and
o
in es iga e he dependence on he wa eguide go e ning
pa ame e s o he c i ical op ical axis o ien a ion a which
he GLMT akes place. Ul ima ely, we show ha he GLMT
exhibi s a po en ial in e es o swi ching applica ions.
This pape is o ganized as ollows. Sec ion
I1
con ains he
backg ound o ou app oach. I is de o ed o he in oduc ion
o he wa eguide con igu a ion we will deal wi h and o a
b ie e ision
o
he di e en ype o possible guided modes.
In
Sec ion I11 he eigen alue equa ion o hyb id guided
modes is ob ained using he cha ac e is ic-ma ix app oach,
and in Sec ion
IV
we discuss he p ope ies o hese modes
ha a e ele an o he GLMT. In Sec ion
V
we ocus
on
he guided- o-leaky hyb id mode ansi ion. Two di e en
si ua ions a e iden i ied, namely he
o dina y cu o
and he
ex ao dina y
cu 08
and he exp essions gi ing he alue
o he c i ical op ical axis o ien a ion a which he GLMT
akes place a e ob ained. Sec ion
VI
conce ns he possibili y
o uning he alue o his c i ical o ien a ion h ough he
wa eguide pa ame e s. Finally, in Sec ion
VI1
we emphasize
ou main esul s.
11.
GENERAL
MODAL PROPERTIES
The op ical wa eguide we conside he e is shown in Fig.
1.
The co e is an iso opic ma e ial wi h e ac i e index
nc.
whe eas he guiding laye and he subs a e a e uniaxial
0733-8724/93$03.00
0
1993 IEEE
~ __
TORNER
e
al.: GUIDE-TO-LEAKY
MODE
TRANSMISSION
1593
I
iso opic
nC
I
aniso opic
nos,
ne,
Fig.
1.
Schema ic
o
he
wa eguide
s uc u e.
ma e ials. The subs a e is homogeneous, whe eas he guiding
ilm can be ei he homogeneous o ha e an inhomogeneous
e ac i e index p o ile. In he i s case
D
is he ilm hickness
and in he second one i s ands o he cha ac e is ic dep h o
he e ac i e index p o ile. We conside a geome y in which
he c ys al op ical axes
(2)
a e placed in he wa eguide plane
o ming an angle
0
wi h he wa eguide axis, and an iden ical
o ien a ion in bo h subs a e and ilm has been aken. The
p opaga ion di ec ion
is
aken along he x-axis and a ime-
ha monic dependence has been assumed,
so
ha he ields a
any poin ha e he o m
exp
b(Px
-
u )],
wi h
,d
being he
p opaga ion cons an .
In he coo dina e sys em ha coincides wi h he p incipal
axes
o
he uniaxial c ys als
(X,
Y,
2))
he ela i e dielec ic
enso
is
diagonal and akes he
o m
whe e
no
and
ne
a e he o dina y and he ex ao dina y
e ac i e indices o he ma e ial, espec i ely.
Fo
an
a bi a y
o ien a ion o he op ical axes, he dielec ic enso is ob ained
by applica ion
o
he app op ia e o a ion ope a o o
(1).
Fo
he o ien a ion conside ed he e one ge s a symme ic enso
whose non anishing elemen s w i e
E,,
=
n,"
sin2
e
+
n,"
cos2
8
eyy
=
n:
cos2
e
+
n,"
sin2
e
E,,
=n:
ezy
=
(nz
-
nz)
sin
e
cos
8
(2)
In gene al, a wa eguide such as he o me one can only
suppo hyb id modes, wi h he
six
ield componen s. In
he limi ing cases
0
=
0'
(X, Y,-cu ; 2-p opaga ing) and
8
=
90'
(X,
Y-cu ;
Y,
X-p opaga ing), he allowed guided
modes b eak down in he usual
TE
and
TM
se s.
The
ody
non anishing componen o he elec ic ield
o
modes
is
Ey;
he e o e he beha io o such modes
is
go e ned by
eyy.
Thus,
a
0
=
0')
TE
modes a e a ec ed only by he o dina y
e ac i e index o he di e en media
o ming
he
wa e,
hence such wa es a e
pu e
o dinmy
modes.
Likewise,
a
8
=
90')
one
has
eyy
=
n:,
and hus he
TE
modes
a epu e
ex ao dina y
modes.
These
wo
ypes o modes a e e e ed
o as 0-like modes and e-like modes, espec i ely.
Conce ning
TM
modes, hey ha e he o he
wo
componen s
o he elec ic ield. In he coo dina e sys em used he e he
p edominan componen
is
E,;
hence, he beha io
o
hese
wa es
is
basidly go e ned by
eZz.
A
0
=
90°,
one has
E=,
=
E,,
=
n:;
hus
a
his
o ien a ion he
TM
modes a e
0-like
modes.
On he o he hand, a
8
=
Oo,
E,,
=
n:?
whe eas
E*,
=
nz,
so
ha
Th4
modes a e a ec ed by bo h e ac i e
indices. Ne e heless, as
ez2
=
n:,
he
TM
modes beha e as
p edominan ly
o dina y
modes.
Such modes a e e e ed o
as
(0)
-like modes.
Fo an in e media e alue o
8,
he hyb id guided modes a e
a supe posi ion
o
he o dina y and he ex ao dina y wa es
p opaga ing in an unbounded uniaxial dielec ic medium. In
he
case
o an homogeneous wa eguide, he ields in he ilm
a e a supe posi ion
o
wo
o dina y and
wo
ex ao dina y
plane wa es
[Z].
A
pa icula combina ion o hese wa es
co esponds o a pe ec ly guided mode when bo h o dina y
and ex ao dina y wa es a e o ally e lec ed a he in e aces
be ween he guiding laye and he su ounding media. Then,
ou side he guiding egion in ields a e e anescen and he
guided
powe
p opaga es pa allel o he wa eguide
axis.
This
is a
o diy
guided
hyb id
mode.
Because he o dina y and he ex ao dina y wa es p opaga e
wi h di e en phase eloci ies, he condi ion
o
o al e lec ion
a he bounda ies canno always be sa is ied
o
bo h wa es.
This means ha only one o hem will achie e o al in e nal
e lec ion, whe eas he o he will be only pa ially e lec ed.
This esul s in adia ion losses. The ac ual nons a iona y
guided wa e
can
be
app oxima ely desc ibed by a hyb id
guided mode p opaga ing wi h a enua ion, hus ha ing a
complex p opaga ion cons an . This is a
leaky
guided
hyb id
mode.
The ela i e weigh o he o dina y and he ex ao dina y
wa es in he hyb id guided mode
is
dic a ed by he eigen alue
equa ion. The hyb id modes which main con ibu ion co e-
sponds o he o dina y o ex ao dina y wa es a e e e ed o
as
[o]-like modes o [e]-like modes, espec i ely. The p ecise
na u e o he [o]-like and [e]-like modes in e ms o o ally and
pa iall
e lec ed wa es has a e y s ong in luence
on
he
loss
coe icien o leaky modes
[18]-[24], [26].
This
na u e
will also play a c ucial ole on he
GLMT.
Acco ding o
he
o me discussion, he c i ical op ical axis
o ien a ion we a e looking o ollows om he eigen alue
equa ion a he cu o poin o he o dina y o ex ao dina y
wa es. The e o e, nex we ob ain he eigen alue equa ion
o guided wa es, wi h he help o he cha ac e is ic-ma ix
app oach.
111.
CHARA~~ERISTIC-M~RIX
FORMALISM
The key o he o malism uns as ollows.
Since
we a e
looking o he eigen alue equa ion o guided solu ions, he
ields in he co e and in he subs a e mus be e anescen .
So7
hey would ha e he o m
1594
JOURNAL
OF
LIGHTWAVE
TECHNOLOGY
,
VOL.
11,
NO.
10,
OCTOBER
1993
These ields mus obey he wa e equa ion. In a lossless
uniaxial dielec ic medium i w i es
2E'
+
k&.]E
=
V(V
.
E)
(5)
wi h
ko
being he ee-space wa e numbe . In he subs a e,
aking in o accoun (2), he subs i u ion o (4) in o he wa e
equa ion leads o
wo
possible solu ions o he decay cons an
ys
ha co espond o he o dina y and ex ao dina y wa es.
One ge s, espec i ely
(7)
whe e
N
E
p/ko
is he e ec i e index and
(8)
nosnes
nes
(6)
=
Jnas
sin2
6
+
nzs
cos2
6
In he iso opic co e hese solu ions degene a e and one
ob ains
(9)
-
[N2
-
n2
1/2
c-
cl
Acco ding o
(6)
and (7), in he subs a e he e anescen ields
a e a supe posi ion
o
he o dina y and ex ao dina y wa es,
so
ha (4) becomes
2.,(z)
=
2os
exp
[koyos(~
-
.)I
+
Ees
exp
[koYes(D
-
.)I
(10)
The exis ence
o
he solu iys
(6)
an!
(7)
implies ha all he
componen s o he ields
E,,
and
E,,
can be exp essed in
e ms o wo o hem. We ha e chosen as independen a iables
he
y
and
2
componen so he elec ic ield associa ed o
he o dina y and ex ao dina y wa es, espec i ely. Then, he
elec ic ield in he subs a e can be w i en as
whe e, in o de o ende he exp essions in a compac o m, a
ma ix no a ion has been adop ed. The coe icien s
A
appea -
ing in he abo e exp essions ollow om he wa e equa ion.
One ge s
(12)
(13)
Aox
=
-
an8,
Aey
=
?Aox
4s
Yo
S
N
Yes
Yos
Yo
s
Aoz
=j-Aox,
A,,
=
jyN
In he co e , he double solu ion (9) gi es an exp ession
analogous o
(11),
bu o he TE and he TM ield componen s.
One has
Acco ding o he cha ac e is ic-ma ix o malism, he ields
(1 1)
and (14) a e con inued ac oss he in e media e dielec ic
medium by means o a cha ac e is ic ma ix con aining he
ield solu ions. Thus, he equi ed bounda y condi ions on
he wa eguide dielec ic in e aces a e e i ied i he ield
solu ions appea ing in he cha ac e is ic ma ix join wi h
he e anescen ields gi en by
(11)
and (14). Following he
Be eman app oach [27], he cha ac e is ic ma ix
T
o a
dielec ic ilm
o
hickness
D
is de ined as
Q(2
=
D)
=
T*(z
=
0)
(15)
wi h
e(
2)
being a ou -dimensional a ay con aining he
angen ial ield componen s. As is cus oma y, we ha e chosen
an o de ing c i e ion in such a way ha he anspose o
9
w i es
9'
=
(E,, Hy
,
E,,
H,).
The angen ial componen s o
he magne ic ield in he subs a e and in he co e appea ing
in
Q
ollowing immedia ely om
(11)
and (14) by means
o
Maxwell equa ions. One a i es a
Yes
Aey
gs(z)
=
L
[T::
Tes
]
jN jNAey
Eo,
exp
[koYos(D
-
.)I
Ees
~XP
[ko es(D
-
.)I
He e
q
is he cha ac e is ic impedance o ee-space and use
has been made o he de ini ions
Subs i u ing
(1
l),
(14),
(16),
and (17) in o
(15),
a homogeneous
equa ion sys em o
E,,,
E,,, E,,,
and
E,,
is ob ained, and
he condi ion o ha ing a non i ial solu ion is ha he
de e minan
o
he coe icien ma ix anishes. This condi ion
is he eigen alue equa ion. A e s aigh o wa d manipula ion
i can be exp essed as
Tos[N34
-
Aey(N14
+
YesN13)l
-
Aox[TesN34
+
Aey(N24
+
YesN23)]
+
[N24
+
TesN14
-
YesAeyNlP]
+
Yos[N23
+
~esN13
+
AeyNl~]
=
0
(19)
whe e
wi h
..
He e
Til,
wi h
i,
1
=
1
-
4,
a e he elemen s o he ma ix
T,
and
i
has been de ined by he o mal exp ession
i
-1
+
(1
+j~)(&z
+
&4)
(22)
wi h
Sil
being he K onecke symbol.
The cha ac e is ic ma ix o a homogeneous uniaxial dielec-
ic ilm can be analy ically calcula ed by using he
4
x
4
TORNER
e
al.:
GUIDE-TO-LEAKY
MODE
TRANSMISSION
o malism. The de i a ion ollows a gene al p ocedu e de i ed
by Vassell
[28].
An
al e na i e app oach based
om
he well-
known
Cayley-Hamil on heo em has been epo ed mo e
ecen ly by Wohle
e
al.
[29]. When he op ical
axis
lies
on he wa eguide plane he cha ac e is ic ma ix
is
gi en by
a simple exp ession, which can
be
ound
in
[26]
andz[2%].
In he case
o
inhomogeneous s uc u es he cha ac e is ic
ma ix mus be ob ained wi h he help o app oxima e me hods
o nume ical echniques. He e we ha e
used
he
mul ilaye
s ai case echnique
in
which he inhomogeneous
is
conside ed a iini e se o hin homogeneous
ilms.
Then, he
cha ac e is ic ma ix o he whole s uc u e ollows om
he
p oduc o
all
he ma ices associa ed o each laye , acco ding
o he o de ing c i e ion p esc ibed in
(15).
IV.
HYBRID
GUIDED
MODES
In his sec ion dicusss he p ope ies o he hyb id guided
modes ha a e ele an o he
GW.
We ha e conside ed
media wi h posi i e
and
nega i e bi e iqence,
using
in
bo h
cases he ypical pa ame e s o he LiNbO3 and LiTaOpbased
wa eguides ope a ing a
X
=
632.8
nm.
Fi s , we a k going o
deal
Wi h a ma e ial
wi h
Wga i e
bi e ingence, such as LiNb03. In Fig. 2(a) we ha e plo ed
he e ec i e indices o he hyb id modes guided by an
homogeneous wa eguide wi h
he
ollowkg pa ame e s
1
2.2946
n,
=
2.2108
n,
=
1.0
G1:
nos
=
=
2.2866
nee
=
2.2028
D
=
3.0pm
This
is a mul imode wa eguide, wi h
wo
modes
o
each
pola iza ion. When
8
=
Oo
(131 2),
G1 suppo s he
"€30
and
he TE1 modes, which a e o-like modes, uge he
$&h
Be
TMo and
TM1
ones, which a e (0)-like modes. Due
o
he
small bi e ingence o GI, modes o equal o de -ga e
wi h e y
simila
e ec i e indices. When
8
inc eases
om
0"
,
hese modes become hyb id and some in e als o alues o
6'
appea whe e he guided modes exhibi e y close e ec i e
index alues, o such an ex en ha hey a e almos degene a e.
These in e als
will
be e e ed o as nea -degene acy
ng das
(NDRs).
On he o he hand, he i s -o de
malm
a
9
=
O'
become leaky o e a c i ical 8- alue.
This
is
be
@ded- o-
leaky mode ansi ion angle
(GLMTA)
81
o each
m&.
ie
lowes o de modes,
TEo
and
TMo
a
6
=
Oo,
ema
guided
o all alues o
8.
A
8
=
WO,
he ou
modes
becolhe
again
TE-TM.
In o de o iden i y he hyb id modes,
we
a e
using
he no a ion by Knoesen, Gaylo d, and
Moha am
[SI.
Fo
ins ance, he mode which a
8
=
Oo
is he
TEo
and a
8
=
90'
is he TMo,
is
deno ed as:
[TEo,
TMo]. The beha io
o
he
e ec i e indeg
o
he allowed hyb id modes
as
a
unc ion
o
8
is summa ized in he diag am in Fig.
2@).
§ymbols
used in his igu e, which is e e ed o
as
an [o]-[e] diag am,
co espond o he abo e-in oduced no a ion and il a ows
indica e nea -degene acy egions.
Le us now ocus on he
[TEI,
TEo]
and he [TMI,
TEl]
modes. In he neighbo hood
o
8
=
0'
hew
modes
e ol e
wi h
8
as [e]-like
and
[o]-like
modes,
esped culy.
When
.8
inc eases, a e he common
NDR
he oles become in e ed,
hence he mode [TMl,
TEl]
is an
[e]-like
mode when i
I
I I
1595
2.30
1
1
2.28
5
:
c
2-24
3
3
.$
2.26
?z
0
2.22
2.20
I
I I
0
30
60
90
op ical
&
o ien a ion
(~~TWUU)
(4
L
[el
-
I4
(b)
Fig.
2. (a)
E ec i e
ipdex
o
he
hyb id
modes
guided
by
G1
as
a
unc ion
o
he
op ical
p im a ion.
Coo inusus
liges
ipdico e o aUy
guid@
modes
and
dashed
lines
co espond
o
leaky modes.
@)
Io]-[e]
diag am
o
GI.
unde gws
he
GLMI:
Simila ly, he
[TEl,
TEo]
mode, which
e d es
as
an
[e]
-&e
de in he neighbo hood
o
8
=
0'
and
as
an
[o]-&
mode
beyond
he i s
NDR,
exbibi s
a
second
NDR,
now wi h
he
[TMo,
TMl]
mode.
Bdh
modes
also in e hei na u e
in
he common
NDR,
so
ha he
i s
me
becomes again an [el-like mode.
To
sum
up,
bo h
modes
ha unde go he
GLMT
a e [e]-lie modes.
Nex , we a e going o deal wi h a wa eguide made on
a po4 iYe bi e iqen ma e ial. In
Fig.
3(a) we ha e plo -
ed he e ec i e indices
o
he hyb id modes guided by a
homogeneous wa eguide wi h a ious pa ame e s
no
=
2.1856
n,
=
2.190
n,
=
1.0
which co espond o
a
LiTaOs-based sample. The [o]-[e]
---
'
Wi h
G2
is
shows
in
Fig.
3(bz
The
poin
is
ha
in
bis
case
, he
GLMT
akes
ploce
h ough
he
[.I-
like
madas.
F&
3
~~~
o
a
yaic;+
LiTaQ3-bssed
wa eguide, bu
i
he alues
o
he wa eguide
piqm s
a e
}
G2:
{nos
=
2.1834
ne,
=
2.1878
D
=
6.0pm
1596
2.190
,
2.188
R
F:
$
2.186
3
'c'
'c'
U
0
2.184
1
2.182
!
I I
0
30
60
op ical
axis
o ien a ion
(deg ees)
(a)
Fig.
3.
(a) E ec i e index
o
he hyb id modes guided by
GZ
as a unc ion
o
he op ical axis o ien a ion.
(b)
[.]-[e]
diag am
o
Gz.
sligh ly modi ied, he o de ing
o
he e ec i e indices
o
he
TE-TM modes wi h equal o de ha occu s a
6'
=
0" may
e e se in ela ion
o
he one exhibi ed by
Gz.
In
his case,
he [.]-[e] diag am becomes iden ical o Fig. 2(b), bu wi h
in e changed TE-TM and [.]-[e] oles.
Finally, Fig.
4
co esponds o a ypical p o on-exchanged
LiNb03
wa eguide, wi h a ious pa ame e s
I
n,
=
2.2466
n,
=
2.329
nc
=
1.0
G3:
{n,,
=
2.2866
ne,
=
2.2028
D
=
1.5pm
Due o he nega i e inc emen o he o dina y e ac i e index
(no
<
nos)
ha is ob ained by he p o on exchange echnique,
his wa eguide does no allow guided solu ions a
0
=
0"
[30].
Likewise, a
6
=
90°,
only TE guided wa es (e-like modes)
a e allowed.
G3
suppo s ou modes. Bo h TE2 and TE3
modes p opaga e wi h
N
<
nos;
hus, any angula de ia ion
ende s hem leaky modes. The TEo and he
TE1
e ol e as
[e]-like modes when
6'
dec eases om
90°,
and bo h o hem
become leaky.
JOURNAL
OF
LIGHTWAVE TECHNOLOGY
,
VOL.
11,
NO.
10,
OCTOBER
1993
2.34
2.30
2
%
c
'c'
$
2.26
-4
.c1
0
K
3
2.22
2.18
/
'
0
0
0
..
0
..
TE3
------
-
-
---_.
--
I
I
30
60
s
3
op ical
axis
o ien a ion
(deg ees)
Fig.
4.
E ec i e index
o
he hyb id modes guided
by
GY
as
a unc ion
o
he op ical axis o ien a ion. The do ed line s ands
o
he cu o
locus
.
=
k9(0).
The [o]-[e] diag ams mus be in e p e ed wi h cau ion, since
he no a ions
o
[o]-like and
[e]
-like mode and nea -degene acy
egion come om somewha in ui i e insigh s. Nea cu o , he
iden i ica ion o hese cha ac e is ics
on
he
6'
-
N
plo
is
no
easy; hence, he co esponding diag ams mus be in e p e ed
acco dingly. The main use ulness o he [.]-[e] diag ams comes
om he ac ha hey summa ize he p ope ies
o
a gi en
wa eguide conce ning o -axis p opaga ion condi ions,
so
ha
a g ea deal o he in o ma ion can be immedia ely ead
o
om hem.
V.
GUIDED-TO-LEAKY MODE TRANSITION
The GLMT akes place when one o he o dina y
o
ex-
ao dina y wa es o ming he hyb id mode eaches
i s
cu o
o o al in e nal e lec ion, whe eas he o he emains o ally
e lec ed. Thus, wo di e en cases mus be dis inguished: he
cu o
o
he o dina y wa e and he cu o
o
he ex ao dina y
wa e. The i s case will be e e ed
o
as he
o dina y
cu o
(0-cu o ) and he second as he
ex ao dina y cu o
(e-cu oq.
Nex we ob ain he exp ession gi ing he exac alue
o
he
GLMTA in bo h cases.
An
al e na i e app oxima e p ocedu e
based
on
coupled-mode insigh s is gi en
in
he Appendix.
When he o dina y wa e eaches i s cu o o o al in e nal
e lec ion a he ilm-subs a e dielec ic bounda y, i s ans e -
sal p opaga ion cons an a he subs a e anishes; hence, one
has
yos
=
0. Thus, acco ding
o
(6)
we can w i e
o-cu o
+
N
=
no,
(23)
Consequen ly, he ans e sal p opaga ion cons an o he
ex ao dina y wa e in he subs a e egion

II
I
TORNER
e
aL:
GUIDE-TO-LEAKY
MODE
TRANSMISSION
1597
Con e sely, since he e
i.
=
0,
a he ex ao dina y cu o
one has
T~~
=
0.
Acco ding o
(7),
his
condi ion yields
e-cu o
*
N
=
nes(Ol,e)
(25)
Hence
22
nes
-
nos
nzs
sin2
el,e
+
n:,
cos2
el,e
(26)
70s
le-cu o
=
nos
.
sin
el,e
Exp essions
(24)
and
(26)
lead o he ollowing gene al con-
clusion: The o dina y cu o , aken as pa ial cu o ,
can
only
occu when conside ing nega i e bi e ingen c ys als. The
equi alen s a emen applies o he ex ao dina y cu o and
posi i e bi e ignen c ys als.
A.
O dina y
Cu o
A he o dina y cu o , he subs i u ion o
(23)
in o he
exp essions o he coe icien s A de ined in
(12), (13)
yields
1
70s
1
Yos
A,,
lo-cu o
-
+oo
(27)
(28)
Aey Aez
lo-cu o
-
2
+
These
di e gences allow
us
o no ably simpli y he exp ession
o he eigen alue equa ion
(19),
since
only
he p edominan
e ms mus
be
e ained on i . One a i es simply a
(N14
-k
7esN13)o-cu o
=
0
(29)
When all media o ming he wa eguide a e iso opic,
=
0
implies
^leS
=
0.
In his case, aking in o accoun
also
he
p ope ies o he cha ac e is ic ma ix
o
an iso opic
ilm,
(29)
becomes
TM
TE
which
is
he cu o condi ion o
TE
and
TM
modes
[31].
B.
Ex ao dina y
Cu o
ing ha
A he ex ao dina y cu o he coe icien s A w i e assum-
C. Allowed Values
o
81
Gene ally speaking,
he
GLMTA
can
ake all possible alues
in he
ange
[Oo
,MO].
Ne e heless, o ypical wa eguides,
81
is bounded by a milximum o minimum alue, cha ac e is ic
o each wa eguide.
The
ob ainmen o his bounding alue
uns
as
oilows. Since we a e looking o ei he o ally and
leaky
guided
solu ions, powe low mus be pa allel o he
ilm-subs a e in e ace a leas o one o he
wo
kinds o
wa es, o dina y and ex ao dina y, o ming he hyb id modes.
Fo he o dina y wa es his condi ion implies
whe eas o he ex ao dina y wa es i w i es
wi h
ne (e)
being de ined as
in
(8).
A
6
=
Oo,
(36)
educes
o
(39,
whe eas a
e
=
90'
one has
ne ,es(9Oo)
=
ne ,es,
and
(36)
becomes iden ical o
(35)
bu o he ex ao dina y
e ac i e indices. The e upon, since o he wa eguides such
as
GI,
one has
no
>
nos
>
ne,,
and he 0-cu o occu s
as
N
-3
no,,
he
maximum
alue ha
can
be eached by
comes
mn
he
equali y
n, (e ,,o)
=
no#.
This
condi ion
yields
The same exp ession holds o G3, al hough now s ands
o he minimum allowed alue o On he o he hand,
when conside ing wa eguides made on posi i e bi e ingen
c ys als such as
G2,
one has
ne
>
ne,
>
no
>
nos.
The e o e, since he e-cu o occu s when
N
=
nes(Ol,e),
he maximum alue which
can
be eached by e i ies
ne,(Olm,e)
=
n, .
Hence, one ge s
In iew o
(37)
and
(38),
we no ice ha
81,
depends on he
ilm and subs a e e ac i e indexes, bu
i
is
independen
o he co e e ac i e index, as well
as
on he a io
D/X.
Also,
el,,,,
(ei,,,)
does no depend on
ne, (ne ),
whe eas
&n,e
(&n,o)
does.
VI.
DISCUSSION
The beha io o he GLMTA as a unc ion o he a ious
wa eguide pa ame e s has been in es iga ed by sol ing
(29)
and
(34).
The ou come o ypical s uc u es
is
shown
in
Figs.
In Fig.
5
we ha e plo ed
d
as
a unc ion o he a io
D/X,
o
a
GI-like wa eguide. The cu es in he plo co espond o
he di e en hyb id modes guided by he s uc u e o each
alue o
D/X.
As
his
a io g ows, no only he numbe
o TE-TM-allowed
guided
modes a
8
=
0,90°
inc eases,
bu
he
na u e
o
he
hyb id modes
modi ies
as well, since
leaky
modes
con e in o o ally guided ones. Bo h e ec s
occu
wi h
e y
smaU
a ia ions o
D/X;
hence, e y
sha p
5-10.
JOURNAL
OF
LIGHTWAVE
TECHNOLOGY
,
VOL.
11,
NO.
10,
OCTOBER
1993
I
I
D/h
Fig.
5.
Guided- o-leaky mode ansi ion angle o he hyb id modes guided
by
GI,
as a unc ion o he A-scaled wa eguide hickness. Shaded egion
co esponds o he guided- o- adia ed mode ansi ion. He e and in Fig.
6,
he
unlabelled cu e co esponds o a mode which is cu o a
0
=
0'.
90'.
50
3
i?
3
40-
U
L
'c,
U
Y
cl
30-
E
'%
-
20-
e
'c'
10-
Od
2
Fig.
6.
Same as in Fig.
5,
bu o
Gz.
changes in
81
ake place. Highe o de TE-TM modes become
allowed a gi en c i ical hicknesses, he cu o hicknesses,
whe eas he leaky- o-guided mode ans o ma ion is a g adual
bu as p ocess as a unc ion o
DIX.
The cu o hicknesses
o he TE-TM modes aking place a
0
=
0,90°
a e gi en by
simple exp essions
[4].
In he neighbo hood o hese cu o
hicknesses, some oo s o
(29)
yielding
191,~
>
appea .
These oo s co espond
o
he a o al cu o and e ol e e y as
om
O1m,o
o
90'
as a unc ion o
DIX.
The same commen s
hold as well o he cu es in Fig.
6,
which co esponds
o
a
90
2
P
2
70
'c'
Y
Y
e
E
'Z
-
6
.:
50
30 2
4
D/
Fig.
7.
Same as in Fig.
5,
bu o
GB.
He e shaded egion indica es o bidden
alues. F om le o igh :
TEo-TE4.
a
6'
=
90O.
01
II
I
I
2.290 2.292 2.294 2.296
2.:
ilm o dina i e ac i e index
Guided- o-leaky mode ansi ion angle o he hyb id modes guided
by
GI.
as a unc ion o he ilm o dina y e ac i e index.
Fig.
8.
G2-like wa eguide. Finally, Fig.
7
co esponds
o
a p o on-
exchanged sample such as
GJ.
In his plo ,
191,~
s a s a
90",
and dec eases assymp o ically owa d
Olm,o
when
DIX
g ows.
In o de o display he beha io o
01
as a unc ion
o
he a ious e ac i e indices in ol ed, we ha e chosen a
ep esen a i e example
o
each case. The ob ained esul s
ha e been plo ed in Figs.
8-10.
As
abo e, he global pa e n
o he plo s, including as a ia ions o
01
nea
191
=
0'
and
191
=
01,,
and mul iple alues o
81
wi h a ixed alue o
n,
o
n, ,
comes om bo h he g ea e numbe o allowed modes
TORNER
e
aL:
GUIDETO-LEAKY MODE
TRANSMISSION
1599
0'
V
I
2.lb
2.185
2.186
ilm
o dina i e ac i e index
Fig.
9.
Same
as
in
Fig.
8,
bu
o
G2.
and he leaky- o-guided mode ans o ma ion, which ake place
as he
ilm
eeac i e indexes inc ease.
The
den
plo s ha ma
be
emphasized
is
he
la ge,s i#s,exhibi ed
by
61
wi h e y small inc emen s in he ilm
e ac i b
@xes.
Typically,
an
ihc emen o abou
An
N
0.001
yields
A91
N
lo,
and he la ge he alue o
6im,
he la ge
A +.
Away
nnn
he
cu o poin
o
he
TE-TM
modes,
61
depends almos linea ly
wi h
no ,
in he
case
o
GI
and
Gz,
and
wi h
n, ,
in
he
case o G3. The alues o he
ilm
e ac i e
indices
can
be
easonably con olled
in
he ab ica ion p ocess,
so
ha
81
can
be app oxima@ly uned o a desi ed alue wi hin a
wide
ange.
Modi ica ion o he ilm e ac i e index by modi ying
he empe a u e o he guiding ilm, o ins ance by op ical
abso p ion p ocesses, seems o be an in a es ing
possibili y.
Typically, he he mally induced bulk e ac i e index a ia-
ions a oom empe a u e amoun o
dn/dT
w
lo-*
-
10-50C
o LiNb03, LiTaOs, LiI03, o BaTi03
[32],
[33],
and as a
ule
dn,/dT
is la ge han
dn,/dT.
In
his line, i mus be
no ed ha he a ia ions o
61
caused by
an
inc emen
o
a
ilm e ac i e index a e p ac ically counkManced by he
same inc ease o he co esponding subs a e e ac i e index.
The e o e,
in
o de o ob ain a no able modiica ion on he
alue o
61,
an asymme ical esponse
on
he
ilm
and subs a e
e ac i e indices o a empe a u e change should be desi able.
Pho oabso p ion hea ing caused by o ally guided modes would
e i y his equi emen ,
since
mos o he guided wa e powe
a els in o he ilm egion.
VII.
CONCLUDING
REUARKS
We ha e in es iga ed he guided- o-leaky hyb id
mode
ansi ion in plana op ical wa eguides made on uniaxial
c ys als like LiNbO3 o LiTaO3.
'bo
di e en guided- o-
leaky ansi ions ha e been iden i ied, namely he
o dina y
cu o
and he
ex ao dina y
cu o i
which espec i ely occu
when conside ing nega i e and posi i e bi e ingen ma e ials,
and analy ical, ye anscenden al, exp essions yielding he
ilm
ex ao dina i e ac i e index
Pig.
10.
Guided- de Ly
mode
ansi ion
angle
o
hyb id
modw
guided
by
G3,
as
a
B lc iop
o
he
ex Po diBa y
e so i e
index.
F om
le
o
igh :
TE&,
TE1,
a
0
=
WO.
CaiW
q id
axis
o ien a ion abo e which he
o ally
guided
hyb id
modas
became
leaky,
81,
ha e
been
& ab&.
The
dependem;e
o
81
on
be
wa eguide
hickness
and
on
be
ilm
and subs a e
e ac i e
indices
has
been
in es iga ed, and as
a ia ions ha e been ound. Conce ning he co e e ac i e
index,
61
is
nea ly insensi i e o
he
pa icula alue o his
pa ame e , p o ided ha a low-index medium
is
deposi ed
o e he
guiding
ilm,
bu la ge a ia ions
a ise
again when
conside ing nea ly synme id
s uc u es.
Ac ually,
in
a
esul
opposi e ha o he p o on exchange
echnique,
i anium
indi usion yields inhomogeneous wa eg-
uides.
In
o de
o
ake in o BccouB
his
ac , g aded-index
e sions o
G1
and
Gz,
wi h a gaussian p o ile in bo h
he o dina y and ex ao dina y e ac i e indexes, ha e been
analyzed. Nume cs
indica e ha
he
g aded-index na u e o
he guiding
ilm
ende s
$1
o di e en alues han
i s
s ep-
index caun e piu .
Fo
ins ance,
conce ning he
[TEl,
TEo]
hyb id mode guided by
GI,
w
15.1'
o a s ep-index
ilm,
whe eas
12.8"
in
he g aded-index case. We ha e made
no
a emp
o
do
ex ensi e calcula ions o
he
g aded-index
wa eguides, bu in ep esen a i e
cases,
no
signi ican di e -
ences
ha e been ound in he beha io o
61
wi h he in ol ed
wa eguide pa ame e s, in ela ion
o
he
s ep-index case.
Po en ial new in e es ing applica ions o he guided- o-
leaky mode ansi ion come om he swi ching p ope ies
associa ed wi h he la ge a ia ions o
6,
caused by changes
in he e ac i e indices in ol ed.
In
sui able condi ims, leaky
modes exhibi la ge
losses
[18]-124],
[%I,
[30];
hus,
sha p
swi ching cha ac e is ics will a ise. Success depends on he
possibili y o
uning
he alue o his c i ical angIe h ough
he wa eguide pa ame e s, ei he by con olling he ahica ion
p ocedu e o by means o dynamical p ocesses. The mally
induced e ac i e index changes caused by pho oabso p ion
seem o
be
an
exci ing possibili y o be in es iga ed
[34].
1600
JOURNAL
OF
LIGHTWAVE
TECHNOLOGY,
VOL.
11,
NO.
10,
OCTOBER
1993
APPENDIX
In his appendix we b ie ly ou line an app oxima e p o-
cedu e, based on he coupled-mode app oach, o ob ain he
GLMTA. Key poin s un as ollows (see, e.g.,
[SI).
Since
eZy
<<
eyy,
E,=,
he hyb id modes guided by G1-G3 can be
app oxima ely ega ded as weakly pe u bed pu e TE and TM
modes.
TE
wa es would ende o [e]-like modes, whe eas
TM wa es would co espond o he [o]-like ones. Acco dingly,
he eigen alue equa ion
(19)
can be app oxima ed by he
well-known exp ession
whe e he in ege
m
s ands o he mode ope a o ,
61,2;~~
=
I,&;TM
=
n;/nz,
and
62;~~
=
n;/n;.
Acco ding o
(2),
he app op ia e subs a e and ilm e ac i e indexes w i e
n2
=
n:
cos2
0
+
n;
sin2
8
o
TE
modes, and
n2
=
n:
in he
TM case. Finally, he o dina y cu o akes place a
NTE
=
no,,
whe eas he ex ao dina y one comes om he condi ion
NTM
=
nes(8).
The ou come exp essions ende once again
anscenden al, bu easie , equa ions han
(29)
and
(34),
wi h
hei nume ical oo s exhibi ing an excellen ag eemen wi h
he exac solu ions. The bes
i
occu s o G3-like samples
and also o single-mode GI, G2-like wa eguides.
REFERENCES
[l]
R.
R. A. Syms, “Ad ances
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