Leaky modes in mul ilaye uniaxial op ical wa eguides
Lluis To ne , Fe nando Canal, and J. He nandez-Ma co
The p opaga ion cha ac e is ics o he leaky modes in plana aniso opic wa eguides wi h a mul ilaye
s uc u e ha e been in es iga ed by means o a compac igo ous o malism. The leakage losses and leaky
ansi ion angle ha e been s udied o he undamen al and i s hyb id modes. An inhomogeneous
wa e-
guide and bu e ed s ep index ype s uc u e ha e been discussed. Pa icula a en ion has been de o ed o
he a ia ion o he loss coe icien o he leaky
modes as a unc ion o bu e hickness and bu e e ac i e in-
dex. A no ably di e en beha io has been ob ained o a ious con igu a ions. Keywo ds: Leaky modes,
wa eguide heo y, in eg a ed op ics.
1. In oduc ion
A g ea numbe o bo h ac i e and passi e in eg a ed
op ical de ices a e ab ica ed by using uniaxial aniso-
opic ma e ials, such as LiNbO
3o LiTaO
3. Elec-
oop ic modula o s, swi ches, mode con e e s, cou-
ple s, and pola iza ion con olle s ha e been made on
i anium-di used li hium nioba e.12The de ailed
knowledge o he p ope ies o he aniso opic op ical
wa eguiding s uc u es appea s highly use ul o im-
p o e he op imiza ion le el o such de ices, and, also,
i may sugges a new class o de ice based speci ically
on wa eguide aniso opy. 3
Fo homogeneous h ee-laye s uc u es exac ana-
ly ical solu ions a e a ailable in he li e a u e. This
p oblem, in addi ion o he inhomogeneous case, has
been analyzed wi h g ea de ail by se e al au ho s 2
who ha e poin ed ou he in e es ing p ope ies o such
wa eguides. In pa icula , leakage losses we e heo-
e ically desc ibed and expe imen ally measu ed.13"1
4
The bidimensional case has also been s udied by
means o he coupled-mode heo y1
5and he e ec i e
index me hod.'6Howe e , because o he ising pe -
o mances equi ed by he inc easingly sophis ica ed
op ical ci cui s, he e has been in ecen yea s g owing
in e es in mul ilaye wa eguides. These s uc u es
a e widely used since hey p o ide mo e choices o he
designe in addi ion o i s pa icula p ope ies. Con-
ce ning he leaky modes in such s uc u es, in e es in
hem comes om hei connec ion wi h he mode-
The au ho s a e wi h Poly echnic Uni e si y o Ca alonia, Depa -
men o Signal Theo y & Communica ions, Apdo. 30 002, 08080
Ba celona, Spain.
Recei ed 13 Oc obe 1988.
0003-6935/90/182805-10$02.00/0.
© 1990 Op ical Socie y o Ame ica.
dependen loss wa eguides and aniso opy based cu -
o de ices. Ne e heless, al hough he heo y o
modal dispe sion in mul ilaye iso opic wa eguides
has been e y well desc ibed, i s aniso opic coun e -
pa is no ye ully de eloped.
In his pape we use he ans e -ma ix me hod o
es ablish he wa eguiding condi ion o ligh p opaga-
ion in plana uniaxial dielec ic wa eguides wi h a
mul ilaye s uc u e wi h no es ic ions on he op ical
axes o ien a ions. This me hod p o ides a e y com-
pac o malism o analyze such s uc u es, and i has
been ex ensi ely used in he iso opic case.1
7I is
based on he well known 4 X 4 o malism, which has
been de eloped by Be eman'8and Vassell'9and in a
di e en way by Yeh.20 A new o mula ion was e-
po ed ecen ly by Knoesen e al. 2' and o special
cases by Walpi a.
22 The app oach can be summa ized
as ollows. In he homogeneous subs a e and supe -
s a e he o al ields a e a supe posi ion o he o di-
na y and ex ao dina y wa es p opaga ing in a uniaxi-
al unbounded medium. These solu ions 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 op and
bo om wa eguide in e aces a e exp essed by a de e -
minan al condi ion which yields o he wa eguiding
condi ion. The p ocedu e can be applied o he exac
analy ical s udy o uniaxial mul ilaye s uc u es and
he nume ical analysis o inhomogeneous uniaxial
wa eguides.
De ails o he analysis a e gi en in Sec. II. As an
applica ion o he o malism, in Sec. III we in es iga -
ed he leaky-mode p opaga ion in bo h an inhomoge-
neous wa eguide and a s ep index s uc u e wi h a
dielec ic bu e laye . The leakage losses o he wa e-
guide ha e been s udied o he undamen al and i s
hyb id modes o di e en op ical axis o ien a ions.
In he i s case, we mainly de o ed ou a en ion o he
20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2805
e ec s in oduced by he inhomogenei y. In he sec-
ond case, he s udy was ocused o he a enua ion
coe icien o he leaky modes and he guided- o-leaky
mode ansi ion angle as a unc ion o he bu e hick-
ness and bu e e ac i e index.
II. T ans e -Ma ix App oach
An asymme ic con igu a ion o uniaxial c ys als
o ming a h ee-laye s uc u e wi h a bi a ily o ien -
ed op ical axes is assumed (Fig. 1). In he p incipal
axis coo dina e sys em o he c ys als, he dielec ic
enso akes he o m
(EO ) (1)
Supe s a e
DT
Subs a e
[E],
[Ei (Z)
[E] S
z
Fig. 1. Schema ic wa eguiding mul ilaye s uc u e. P opaga ion
is along x.
eo and e, being he o dina y and ex ao dina y pe mi -
i i ies, espec i ely. Fo a gi en o ien a ion o he
op ical axis () de e mined by he pola and azimu hal
angles (0,sp), he componen s o he dielec ic enso
a e ob ained om Eq. (1) by means o he applica ion
o he associa ed o a ion ans o ma ion (70,(p).
Thus one ob ains
exx = e0(sin
2(p + COS2p COs
2
O) + e cos2Sp sin20,
= O(cos
2(p + sin2V cos20) + ee sin2 sin20,
EZZ = e sin20 + E Cos20,
(2)
xy = (e -e) sin p cos sin20,
exz = (e e,) cosp sinO cosO,
eyz = (Ee -) sinip sinO cosD.
In a wa eguide such as he one desc ibed abo e only
hyb id modes can p opaga e excep o specially sym-
me ic &-axis o ien a ions. I he p opaga ion di ec-
ion is aken o be along he x-axis and assuming ime
ha monic dependence, he ields a any poin ha e he
o m exp[i(x -w )], being he p opaga ion cons an .
Ou side he guiding laye he ields mus be e anescen
so ha
E,(z) = E, exp(Ycz), z < 0, (3)
E8(z) = E, exp[y9(D-z)], z > D, (4)
whe e he subsc ip s c and s s and o supe s a e and
subs a e egions, espec i ely. By subs i u ion o
Eqs. (3) and (4) in o he wa e equa ion,
V2E + pW
2[e]E = V(V E(5)
and aking in o accoun Eq. (2), an homogeneous equa-
ion sys em is ob ained, he de e minan o which mus
anish. This condi ion leads o wo possible solu ions
o he decay cons an s y,, ha co espond o he
o dina y and ex ao dina y wa es
Y
0=:I: /32-_ w2e
0(6)
'Ye = -2( 2Z) -AC0
2eoeeEzz()i -, . (7)
He e he uppe sign in he pa en heses holds o he
subs a e and he lowe o he supe s a e. Acco d-
ing o he ans e -ma ix me hod, a any pai o ans-
e sal planes Z
1,Z2 he angen ial componen s o he
ields can be ela ed by using a ma ix T, which may be
de ined as
= T[E . (8)
The exis ence o his ans e ma ix is gua an eed
by he linea i y o he Maxwell equa ions. In ac , he
ma ix T con ains he ield solu ions in he egion
be ween he planes Z1,Z2, so ha in Eq. (8) i con inues
he solu ion om z o Z2ac oss he in e media e e-
gion. Le us ake z1 =0 and Z2= D. Thus he equi ed
bounda y condi ions o he angen ial ield compo-
nen s a e necessa ily e i ied i he ield solu ion is (8)
join wi h he known solu ions (3)-(4) in he subs a e
and co e . This condi ion can be w i en in a ma ix
o m using he ollowing s anda d p ocedu e. The
exis ence o solu ions (6) and (7) equi es ha all he
componen s o he ields be exp essed in e ms o wo
o hem. Then we ha e chosen as independen a i-
ables he 9 and x componen s o he elec ic ield
associa ed wi h he o dina y and ex ao dina y wa es,
espec i ely. In his way, he elec ic ield in he su-
pe s a e (z < 0) can be w i en as
E,(z) = 1Ey exp(-yz) + Ay E exp(-yez), (9)
20ZO -Aze-
and in a mo e compac no a ion
Ax, I exP(z)}
E,(z) [; A:] {Exp PYez) (10)
Fo he magne ic ield one has
H,(Z) [yoAxo aZO ye ze Exe exp(-yz j (11)
LU 43 i 3Aye Jxe exp(yez) J
The coe icien s A appea ing in hese exp essions a e
ob ained om Eq. (5). Again in z < 0 one has
2806 APPLIED OPTICS / Vol. 29, No. 18 / 20 June 1990
-
...l
A2
1
0=_ -Y -iyy'
{E
297 +,3( -E2) + 2ie,y,68
A
20
E2(MwE 2 2 -iy i) _ ey
2(,Y2 + gco
2Ej
2
x=(Co -E )-Y2 + 2(e +2
AZO(e ),2+ /20 _ e ) + 2i z ~ '
(12) -AN34 - AeN24 -eA>eN
2j - [N34
--AN14 -'yAN1
y]
+ N24 -<N14 -4yeAN1 2] + Y[N23-N1,3 + AyN
12l = 0, (24)
(13) whe e
2 py
22) + E(zA2L'z -i3)(
Aye }'X(,Y - 2 + .2 )(h2 z _ 02) _ (W2,Ey)2 (14)
(2 _ j2 + PW2E 2 )( 22) + A 2)2
Az= -+ ,' -Y (15)
(y2 _ 2 + pU2E )(pue2-2) -(2e , 2
On he o he hand, acco ding o Eq. (4), he co e-
sponding exp essions o he subs a e w i e in he
same way as Eqs. (10)-(15) by making he ans o ma-
ion -y ---y o bo h he o dina y and ex ao dina y
wa es. Thus one has
i * 1 E; 0exp[%(D-z)]
Ez) = 1 Aye J (16)
[Az*, A,*, JEexp [*(D- z)I
H,(z) = + iA 0) 0(,y + i#A;e)
ExO exp[((D -z)]
IE*, exp[y(D - z)] (17)
Th oughou he supe sc ip * indica es a subs a e
pa ame e . Subs i u ing Eqs. (10), (11) and (16), (17)
in o Eq. (8) an homogeneous equa ion sys em is ob-
ained which can be w i en as
Axo I p1o 7,le
O Te '2o D2e R e |
1 A, >'3o V3e Eyo
_ -, -,ye '4o V4J E,,
(18)
Nj jol!e V joV (25)
Equa ion (24) has been ob ained o a bi a y wa e-
guide pa ame e s wi h no es ic ion on he op ical axis
o ien a ion o he c ys als. Then i is no iceable ha
o he o ien a ions which allow sepa able TE-TM
ield solu ions Eq. (24) is no ably simpli ied. This
si ua ion happens when one (o bo h) subs a e and
supe s a e is (o a e) iso opic (i.e., e = ee) and also
when he op ical axis lies in he same plane as he
p opaga ion di ec ion and is con ained in he plane
pe pendicula o ha o he wa eguide (in ou no a ion
s = 00). As well, when 5s = 0 = 900, he ield solu ions
allow a TE-TM decomposi ion. In he second case,
when he c-axis lies in he (P = 00 plane in bo h he
subs a e and supe s a e, he exp essions o he E
and H ields a e iden ical o Eqs. (10), (11), (16), and
(17), bu now Axo = 0, Aye = 0, Az, = 0, and
$LW 22 -i
'ze= )2 _- W z (26)
The wa eguiding condi ion is ob ained as o he gen-
e al case. Thus i can be w i en in a o mallyiden ical
way o he iso opic s uc u e 9in e ms o he new
a iables:
EoEe
1 EZZ
8 _ #X2 ¢ 2 =Y+i zE
(27)
(28)
(29)
whe e
O
2 Ayo + iAL, (19)
e; + i/Aze, (20)
j {il -1,3, (21)
11W=,2,4,
wi h
12j2 TjhAx, + Tj3 7- ±j2 (, -'y7j4), (22)
Q2- Tjl + T12AYe -(Tj2Te -eAyeTj4)- (23)
As is explici in hese exp essions, he j unc ions
mus be e alua ed in he supe s a e. The condi ion
o ha ing a non i ial solu ion in Eq. (18) is ha he
de e minan o he coe icien s anishes. This condi-
ion leads o a es ic ion o he possible alues o he
p opaga ion cons an ,, which mus now o m a dis-
c e e se , and, he e o e, i is he wa eguiding condi-
ion. A e s aigh o wa d manipula ion i can be
exp essed as
and simila exp essions o e*, an and 2e (Appendix A).
I is well known ha 5 6"9'21 i he op ical axis o he
uniaxial media o ming he guiding laye lies also in
he (,o = 0° plane, he wa eguide suppo s pu e TE and
TM modes. As usual, in his case Eq. (24) spli s in o
wo (one TE and one TM) e y simple exp essions.
Conce ning he po = = 90° case, i can be conside ed as
well by aking he co esponding exp essions o he
iso opic case bu wi h e = e0in he TM e ms and e = e
in he TE ones. Likewise in a g ea numbe o p ac i-
cal applica ions he supe s a e is iso opic and he
subs a e is a uniaxial c ys al. In his case one also has
Ax = 0, Aye = 0, Az, = 0, and Aze = -iI3/yc, whe e
_y = V#2 kgn 2, (30)
nc and ko being he e ac i e index o he iso opic
medium and he ee-space wa enumbe , espec i ely.
Then he wa eguiding condi ion is gi en by Eq. (24),
bu now Eqs. (22) and (23) simpli y o
Lj. = Tj3 + Ž Tj4,(31)
20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2807
Qje = T+ n Tj2, (32)
The ans e ma ix o an homogeneous uniaxial
dielec ic slab can be analy ically calcula ed by using
he 4 X 4 o malism. In he gene al case, i con ains a
supe posi ion o he wo o dina y and wo ex ao di-
na y wa es p opaga ing in a bi e ingen medium. In
he case o mul ilaye s ep index wa eguides, he
ans e ma ix o he s uc u e is ob ained by he
p oduc o all he ma ices associa ed wi h each sub-
laye , in acco dance wi h he o de ing c i e ion p e-
sc ibed in Eq. (8). When di used wa eguides a e
conside ed, in which he inhomogenei y akes place in
addi ion o he aniso opy, he p oblem becomes mo e
complica ed. In such a case a closed- o m solu ion o
Maxwell equa ions is no gene ally a ailable, and he
associa ed ans e ma ix mus be e alua ed wi h he
help o app oxima e me hods o nume ical echniques.
The la e can be made h ough he di ec in eg a ion
o Maxwell equa ions in he di usion egion by means
o such s anda d nume ical p ocedu es as he Runge-
Ku a me hod" o he Gea p edic o echnique,
16
which a e a ailable in mos ma hema ical lib a ies.
Also, he well known mul ilaye s ai case echnique
can be use ul in analyzing especially complex s uc-
u es con aining homogeneous addi ional laye s. In
his case he inhomogeneous egion is conside ed as a
ini e se o hin homogeneous ilms, o each he ans-
e ma ix is analy ically known.
Finally, since we ha e imposed no es ic ions on he
exis ence o complex alues o he p opaga ion con-
s an 3, he abo e de i ed exp essions a e applicable o
bo h guided and leaky modes. Thus i is wo h no ic-
ing ha he squa e oo s appea ing in Eqs. (6) and (7)
o he ans e se p opaga ion cons an a he sub-
s a e and supe s a e (oye) mus be e alua ed wi h
he igh sign consis en wi h he beha io o mode
ields a om he wa eguide. Special a en ion is
equi ed when conside ing leaky modes due o hei
imp ope na u e. 02 23
Ill. Discussion
As an applica ion o he o malism de eloped in he
p eceding sec ion, we ha e analyzed he e he p opaga-
ion cha ac e is ics o he leaky modes o a ious mul-
ilaye wa eguides ab ica ed in dielec ic uniaxial ma-
e ials. Such modes come om he us a ion on
o al in e nal e lec ion a he in e aces be ween he
guiding laye and su ounding dielec ic media and a e
ob ained as complex solu ions o he eigen alue equa-
ion. Conce ning he aniso opic case, wi h a sui able
o ien a ion o he c ys al op ical axes, one o he pola -
iza ions (o dina y and ex ao dina y) su e s leakage
losses, whe eas he o he emains guided in he ilm.
These kinds o mode. a e leaky guided modes in con-
as o he leaky unguided modes which occu in he
iso opic case.24
We ha e conside ed he e he case in which he op i-
cal axes o he c ys als lie in he wa eguide plane (0 =
900), making an angle ep wi h he posi i e i-axis wi h
he same alue in all uniaxial media. The eigen alue
equa ion associa ed wi h his case is ob ained om Eq.
(24) by aking in o accoun he sui able cha ac e is ic
ma ix o he s uc u e. The explici exp ession o
his cha ac e is ic ma ix is gi en in Appendix B. A
wa eguide such as he one desc ibed abo e can suppo
only hyb id modes wi h he six ield componen s.
When he op ical axis makes a. small angle wi h he
wa eguide axis (o -00) he ield componen s associa -
ed wi h he o dina y and ex ao dina y wa es a e
weakly coupled and he modes co espond o he TE-
TM pola iza ions. When he angle inc eases, he cou-
pling be ween he abo e componen s g ows also and
se e al modes become leaky. The na u e o hese
leaky modes is desc ibed in Re s. 7-10. He e we dis-
cuss some open ques ions om which new esul s
come.
A. Inhomogeneous Wa eguide
Fi s , we ha e analyzed an asymme ic inhomogene-
ous wa eguide wi h a Gaussian p o ile in bo h he
o dina y and ex ao dina y e ac i e indices, and in
which he co e is in ai . The a ious wa eguide pa-
ame e s a e
Inq = 2.2946 ne = 2.21081
G1:. nos = 2.2866 nes = 2.2028
I n= D=21im
He e D is he cha ac e is ic dep h o he Gaussian
p o ile, and he subsc ip s and o ilm pa ame e s.
Also we assumed X = 633 nm. When s = 0, G,
suppo s he TEO and TMO modes. The mode which is
he TEO mode a so = 0 emains guided o all alues o
so and becomes he TMo mode a p = 90° ia a p edomi-
nan ly o dina y hyb id mode. On he o he hand, he
mode ha a s = 0° is TMo becomes leaky beyond n
110, and i con e s in o he TEO a so = 900 by means o
a p edominan ly ex ao dina y hyb id mode. These
modes a e e e ed o as he [TEo,TMo] [g] and
[TMoTEo] [1], espec i ely. He e [g] and [] indica e a
guided and leaky mode. We deal now wi h he beha -
io o he loss coe icien o he [TMoTEo] [] mode.
In Fig. 2 we ha e plo ed he loss coe icien o he
abo e men ioned leaky mode as a unc ion o he angle
so. To pe o m he calcula ions use has been made o a
nume ical zoom oo - inding algo i hm o sol e di ec -
ly he eigen alue equa ion (24) o complex oo s. Al-
hough his p ocedu e equi es a conside able numbe
o i e a ions in he complex -plane, i p o ides e y
accu a e esul s. The g aded index p o ile has been
in oduced in he o malism h ough he mul ilaye
s ai case echnique. Fi s , he accu a e nume ical e-
sul s ob ained o a ew alues o he angle (o a e gi en
in Table I oge he wi h he esul s epo ed by Ko-
shiba e al.'2 om he ini e elemen echnique. As
can be seen, he ag eemen be ween bo h se s o alues
is qui e good, he di e ences being unsigni ican in
p ac ice.
On he o he hand, ou main aim in his case is o
examine he e ec s o inhomogenei y on he loss coe -
icien . The eupon, we included also in Fig. 2 he
2808 APPLIED OPTICS / Vol. 29, No. 18 / 20 June 1990
T
Sol 1P (deg ees)
Fig. 2. Loss coe icien o he [TMo,TEo][lJ mode suppo ed by Gi
as a unc ion o he op ical axis o ien a ion. ol is he guided- o-
leaky mode ansi ion angle.
Table I. Accu a e Nume ical Values Ob ained o he Wa egulde wi h a
Gaussian P o ile (G1) and Compa ison wi h he Fini e Elemen Resul s
Repo ed In Re . 12
'p T ans e Ma ix Fini e Elemen
Re(Q/ko) Loss (dB/cm) Re(//ko) Loss (dB/cm)
120 2.28576 170.1 2.28574 181.6
300 2.26757 54.0 2.26756 53.8
600 2.22573 8.9 2.22572 8.9
esul s co esponding o a wa eguide iden ical o GI
bu wi h a s ep index p o ile. This plo dese es some
commen s. Fi s , in he case o he homogeneous p o-
ile he loss coe icien shows a seconda y maximum
which is no obse ed o he Gaussian p o ile (see also
Fig. 3). Also, he s ep index wa eguide shows a p o-
nounced loss peak which does no appea in he inho-
mogeneous case. These ea u es ag ee wi h he p e i-
sions o Bu ns e a.8in he sense ha he sha p
s uc u e showed by he loss coe icien in a s ep index
wa eguide is due o in e e en ial phenomena which
o igina e in he ab up discon inui ies a he ilm-
co e and ilm-subs a e in e aces. The same con-
clusions ollow om Figs. 4 and 5, whe e he loss coe i-
cien o he [TMOTEo] [1] mode has been plo ed as a
unc ion o he co e e ac i e index o wo di e en
alues o p. In all cases, o emphasize he e ec s due
o he Gaussian p o ile he cha s ha e been e e ed o
he alue o he loss coe icien o NC = 1. Fi s , i can
40-
o gaussian
20 _ s ep-index
30 60 90
P° (deg ees)
Fig. 3. De ail o Fig. 2 showing he seconda y maximum o he loss
coe icien as a unc ion o so o he homogeneous wa eguide and he
mono onous dec ease which occu s o he Gaussian p o ile.
be seen in bo h igu es ha he loss coe icien depends
on n, in a s onge way o he Gaussian p o ile han o
he s ep index one. Also, in he case o he homoge-
neous wa eguide, he beha io o he loss coe icien on
nc is comple ely di e en o qp = 300 han o p = 40°.
This di e ence, which does no occu o he Gaussian
p o ile, comes again om in e e en ial phenomena.
Finally, i is in e es ing o no e ha he p esence o
he g aded index p o ile modi ies he alue o he
c i ical angle a which he [TMoTEO] [1) mode becomes
leaky. As men ioned abo e (ij 11° o he Gaussian
p o ile, whe eas spo 140 o he homogeneous wa e-
guide.
B. Mul ilaye S ep Index S uc u es
We deal now wi h mul ilaye uniaxial s uc u es.
The in e es in such wa eguides comes om hei anal-
ogy wi h hei iso opic coun e pa s, whose no iceable
p ope ies ha e been poin ed ou in se e al wo ks.
Ou main mo i a ion on his subjec lays in he ac
ha he p esence o addi ional laye s modi ies no only
he eal pa o he e ec i e index [ieG(/ko)] bu also
he loss coe icien o he leaky modes and he guided-
o-leaky mode ansi ion angle.
In highly asymme ic s uc u es he p opaga ion
cha ac e is ics o he leaky modes a e only sligh ly
modi ied by he p esence o addi ional laye s a he op
o he wa eguide. This is because in hese wa eguides
he leaky modes mainly accoun o adia ion o he
subs a e, so ha he co e ma e ial only a ec s he
leaky modes when i s e ac i e index amoun s o a
alue close o ha o he subs a e. The si ua ion
changes comple ely when s uc u es wi h a high deg ee
o symme y a e conside ed. Now he leaky modes
20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2809

207 2.18.12.
=400
-o ~ ~ - n
15o gaussian
h os o iie c=1
0
15
_10- /
0 s e-ine
1.7 1.9 2.1 2.3
nc
Fig. 4. Loss coe icien o he leaky mode in Fig. 2 as a unc ion o
he co e e ac i e index o ex = 40' To emphasize he e ec s due
o he Gaussian p o ile he cu es ha e been e e ed o he alue o
he loss coe icien o n oe 1.
=300
10
'I
M25-
0
'~gaussian
0
~~~~4 ~~~~s ep-index
1.7 1.9 2.1 2.3
n,
Fig. 5. Same as in Fig. 4 bu o (p = 300.
co espond o adia ion owa d bo h subs a e and
supe s a e, so ha hey a e s ongly dependen on
co e pa ame e s. 12 Fo example, his is he beha io
shown in Figs. 4 and 5, which we ha e discussed p e i-
ously.
The o malism de eloped in Sec. II allows us o
analyze in an exac and simple way he mul ilaye
uniaxial wa eguides. As al eady men ioned, he cha -
X 2.187 [TMITEo[g|
Q) 2.1[5 TEoTMo] g]
'TM:/ 1
- TEo TMO
2.183- I I I I I
0 30 60 90
P (deg ees)
Fig. 6. E ec i e indices o he hyb id modes suppo ed by G2as a
unc ion o *,. A s = 0,900 he modes a e pu e TE and TM. The
[TMo,TEo] [g] mode is a p edominan ly ex ao dina y guided hyb id
mode, whe eas he [TEO,TMo] [1] mode is a p edominan ly o dina y
leaky hyb id mode. Solid line: pu e guided mode. Dashed line:
leaky mode.
ac e is ic ma ix o he whole s uc u e is ob ained by
he o de ed p oduc o he ma ices o each laye . As
an example, we ha e analyzed a s ep index symme ic
wa eguide in which co e , subs a e, and ilm a e as-
sumed o be uniaxial media. In addi ion, he e is an
iso opic bu e laye o hickness Db and e ac i e
index nb placed be ween he co e and ilm. Opposi e
he o me case, we conside now a LiTaO3based wa e-
guide wi h he ollowing pa ame e s:
n( = 2.1856 ne/ = 2.190
G2: n = 2.1834 nes = 2.1878
D = 2 jim
In he limi ing cases so = 0 and so = 900 his wa eguide
suppo s he TEo and TMo modes. Fo any o he
alue o he angle '1, G2suppo s wo hyb id modes,
which will be deno ed as [TMoTEo][g] and
[TEo,TMoJ[1] acco ding o he no a ion in oduced
abo e. The guided- o-leaky mode ansi ion angle o
he leaky mode amoun s o p 28°. The olls o he
TE and TM modes ha e been in e changed in ela ion
o he o me case (GI) because now we a e dealing wi h
a posi i e bi e ingen ma e ial (ne > no). Figu e 6
shows he p opaga ion cha ac e is ics o he abo e
modes as a unc ion o .
The in luence o he bu e laye on he
[TEo,TMoI[1] mode has been analyzed in Figs. 7-11.
In Fig. 7 we plo ed he loss coe icien o his mode as a
unc ion o he bu e hickness, and he a ia ion o
he loss coe icien wi h he bu e e ac i e index is
shown in Fig. 8. Two di e en o ien a ions o he
c ys al op ical axis ha e been conside ed: = 300 and
2810 APPLIED OPTICS / Vol. 29, No. 18 20 June 1990
20 2.189
0
ep = 30°
0
2.10 2.15 2.20
Fib
Fig. 7. Loss coe icien as a unc ion o he bu e e ac i e index.
Mode [TEo,TMoJ [1]. Bu e hickness: Db = 0.5 ,pm.
0
u0
T4
Log[Db/A]
Fig. 8. Loss coe icien as a unc ion o he decimal loga i hm o he
A-scaled bu e hickness. Mode [TEoTMoJ[1j. Bu e e ac i e
index: nb = 2.
s = 400. The cha s in hese igu es show ha he
e ec s o he addi ional laye on he loss coe icien o
he leaky mode depends s ongly on he bu e hick-
ness and is p ac ically insensi i e o he bu e e ac-
i e index. In his case, he ob ained beha io o he
loss coe icien is simila o bo h c ys al o ien a ions.
Howe e , his esul canno be gene alized. Fo exam-
ple, in Figs. 9 and 10 he p e ious dependences ha e
15-
o=300
n ~ ~ ~~~n
W~~~
10
2.14 2.16 2.18 2.20
Fig. 9. Same as inFig. 7 o he [TE,,TMZJ mode suppo ed by
he mul imode e sion (D 6 pm) o 02.
10 5- p=30
0 5~~~~~~~~~
~~/
Log[Db/A]
Fig. 10. Same as in Fig. 8 o he [TE,TM [ mode suppo ed by
he mul imode e sion (D = 6pum)
o G2-
been plo ed o he [TETM 1J [1J mode suppo ed by a
mul imode e sion (D = 6 im) o G2. In ac , his
mul imode wa eguide also suppo s he [TM1,TE][U]
mode, bu i is a e y low loss leaky mode.7As can be
seen in Figs. 9 and 10, now he e ec s due o he bu e
laye a e comple ely di e en o he wo conside ed
c ys al o ien a ions.
Finally, i is o be emphasized ha he p esence o
20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2811
u'
C;'
W
.
Log[Db/A]
Fig. 11. Guided- o-leaky mode ansi ion angle o he
[TEo,TMoI (1] mode suppo ed by G2as a unc ion o he decimal
loga i hm o he ;-scaled bu e hickness. Dashed line: ansi ion
angle in he absence o he bu e laye .
he dielec ic low index laye also modi ies he alue o
he guided- o-leaky mode ansi ion angle. Fo exam-
ple, in Fig. 11 we plo ed he a ia ion o he leaky
ansi ion angle wi h he bu e hickness o a ious
alues o he bu e e ac i e index o he
[TEoTM 0] [1] mode suppo ed by G2. Simila beha -
io has been ob ained o o he leaky modes. As in
Figs. 7 and 9, he leaky ansi ion angle becomes insen-
si i e o he bu e hickness when Db amoun s o a
alue o he o de o X. The modi ica ion o he guid-
ed- o-leaky mode ansi ion angle w ih he wa eguide
pa ame e s should be pa icula ly in e es ing o he
aniso opy based cu o de ices in which his angle
plays a undamen al ole. We e u n o his ques ion
in a o hcoming pape .25
IV. Concluding Rema ks
The wa eguiding condi ion o ligh p opaga ion in
plana dielec ic uniaxial wa eguides wi h a mul ilaye
s uc u e has been ob ained wi h no es ic ions on he
op ical axes o ien a ion. The p ocedu e is based on
he ex ension o he ans e -ma ix o malism o in-
clude an aniso opic subs a e and supe s a e. As an
applica ion o he o malism we ha e in es iga ed he
p opaga ion cha ac e is ics o leaky modes in an inho-
mogeneous wa eguide and a mul ilaye s ep index
s uc u e in which he c ys al op ical axes lie in he
wa eguide plane. The explici exp ession o he e-
qui ed cha ac e is ic ma ix is also gi en. Bo h he
loss coe icien and he guided- o-leaky mode ansi-
ion angle ha e been analyzed as a unc ion o he
inhomogenei y and bu e laye pa ame e s. We ha e
shown ha he p esence o addi ional low index laye s
o ming mul ilaye s uc u es wi h a high deg ee o
symme y s ongly a ec s he p opaga ion cha ac e -
is ics o he leaky modes suppo ed by such wa e-
guides. These e ec s depend also on he op ical axis
o ien a ion due o in e e en ial phenomena which
come om he p esence o ab up discon inui ies on
he in e aces be ween he a ious dielec ic media.
The au ho s a e g a e ul o Ji i C y oky o he
Czechoslo ak Academy o Sciences, P ague, o alu-
able sugges ions. This wo k was pa ially suppo ed
by a g an o he CYCyT o he Spanish Go e nmen
(PB-87-0798-C0302).
Appendix A
The wa eguiding condi ion o he case when he -
axis lies in he plane so = 0° in bo h he subs a e and
supe s a e is
{21 + i T22 + T11 -2 *T12 X
{T
4+jŽAwT,
+ i2:wT
33 -2
T
34
1
{ ( * X
{ 2 T1 -e E*ye T,
{T
41+ i--T4 2 +Z--g T32 = -.
'ye A'. &' J(Al)
When he wa eguide suppo s pu e TE and TM
modes, he i s b acke in he abo e exp ession co e-
sponds o he TM pola iza ion and he second o he
TE one. In such a case, also, he o he wo b acke s
anish iden ically.
Appendix B
The ans e ma ix o he bu e ed uniaxial wa e-
guide in Sec. III is gi en by T = UaUb, whe e Ua and Ub
a e he ma ices associa ed wi h he aniso opic ilm
and bu e laye , espec i ely. Bo h ma ices can be
analy ically calcula ed by means o he 4 X 4 o mal-
ism. The de i a ion ollows a gene al p ocedu e de-
eloped by Vassell,
19 who has also epo ed he exp es-
sion o he ans e ma ix o he simple case o a
uniaxial dielec ic ilm whose op ical axis lies in he P
-00 plane. The ans e ma ix o he gene al case
can be easily calcula ed also by means o he same
p ocedu e.26 In ou case, 0 = 900, i can be w i en as
Um = (X2a2 + A o) jm (Bi)
U being an auxilia y ma ix whose elemen s a e gi en
by
2812 APPLIED OPTICS I Vol. 29, No. 18 / 20 June 1990
l = p 0cos(wDX0) + X2aC cos(wDXe),
12 = e in(wDX,) + ai (e sin(wDXe)i
U
3=- 01X[CEosmDX) +'~
U13 = a1A[cos(wDX)- cos(wDXe)],
U14 = -ialki[sin(DX) -, sin(wDXe)
Tg21 = i d [Aeo sin(wDX
0) + a2a Xe sin(wDXe)],
X"
23 = oal sin(oDX,) - x sin(wDXe)I
U24 = -pcaj[coswDX) -cos(wDXe)],
U33= a1 cos(coDX
0) + Ac6cos(wDX,),
U34 = -i E [ 1o~ea2 sinQwDX
0) + peo sin(wDXe)],
X e
U43 = -i [Xoa sin(wDX0)
U22 = ll 044 = p
33,
U
31 = U24 32 = l4,
U41 = U23 42 = _U13,
whe e
Xe = Pee a
e£ -exx
cxy
Xe .1
He e a = ,1/co and D is he hickness o he aniso opic
ilm. The ans e ma ix o he iso opic bu e laye
comes di ec ly om he abo e exp essions by making
he subs i u ion E, = = enb, e being he ee space
pe mi i i y and nb he bu e e ac i e index. One
a i es a he well known exp ession
Ub= (UTM
0U ) (B7)
whe e
[ cos(koDbb) i(i/a) sin(kODb) (B8)
=i(azn) sin(keDb b) cos(koDb b) J
He e X =_1i7; and Db s ands o he bu e hickness.
Also, use has been made o he de ini ion
ib" -N (B9)
N= - /k0being he e ec i e
index and aTE = -0b,aTM =
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20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2813