IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 11, NOVEMBER 1987 1043

Coupling between an Abruptly TerminatedOptical Fiber and a Dielectric

Planar Waveguide

CHRISTOS N. CAPSALIS AND NIKOLAOS K. UZUNOGLU, MEMBER, IEEE

,4b.vtract —The coupling between an optical fiber and a dielectric planar

waveguide is analyzed when both guides are terminated abruptly and are

facing each other. Mixed spectrum eigenwave representations of fields are

employed inside the wavegnides while Fourier integrats are utilized to

describe the field in the space between the two guides. A coupled system of

integral equations is derived by satisfying the boundary conditions on the

terminal planes of both wavegnides. A weak guidance approximation is

assumed to facilitate the analysis. Numerical results are presented for

several coupling geometries. Misalignment losses and coupling optimiza-

tion phenomena are investigated.

I. INTRODUCTION

A PROBLEM OF great importance in optical com-

munication systems is to transfer light energy from a

planar or a linear waveguide to a single-mode optical fiber,

or vice versa. The planar waveguide could be an injection

laser or a semiconductor detector. Recently, composite

packages utilizing cylindrical microlenses have been tried

as a means of improving the coupling efficiency of laser

emission into optical fibers [1], [2], [3]. However these

techniques are still under investigation and as yet no

standard coupling method has emerged. The alignment of

microlenses is expected to be a problem. The laser active

layer thickness is around 0.2 pm while the single-mode

fiber core diameter is approximately 8 pm. Therefore, in

principle, the emitted radiation from a semiconductor laser

can be coupled into an optical fiber if the distance between

them is sufficiently small. Then it is crucial to examine the

achievable coupling efficiencies and the misalignment

tolerances of this simple coupling scheme. The positioning

of the fiber in front of the planar waveguide can be done

by using mechanical means, epoxy resin fixing, or even

monolithic integration techniques.

Theoretical analyses of laser diode to multimode fiber

coupling have been undertaken by several researchers

[3]-[6] when cylindrical or spherical lenses are employed.

Usually ray-tracing techniques are applied to estimate the

coupling efficiencies.

Manuscript received April 20, 1987; revised JLdy 15, 1987.

The authors are with the Department of Electrical Engineering, NationalTechnical University of Athens, Athens, 10682 Greece.

IEEE Log Number 8716918.

TAxL

n2

Fiber guide

Y

‘t----I

Fig. 1. Single-mode fiber to planar waveguide coupling geometry

The present paper deals with the coupling of an abruptly

terminated dielectric slab waveguide with a single-mode

fiber. In Fig. 1 the coupling geometry is defined. A single-

mode fiber of radius b is facing a planar waveguide of 2d

guiding layer thickness. The core and cladding region

refractive indices of the fiber guide are designated n ~ and

n ~, respectively. The corresponding guiding layer and

cover-substrate refractive indices of the planar waveguide

are denoted by n; and n ~, respectively. The space between

the two guides is assumed to be homogeneous and its

refractive index is denoted by n ~. The distance between

the abruptly terminated guide parallel terminal planes is

w. The planar waveguide is taken to be of infinite width

along the y axis. The displacement of the propagation axes

of the two guides along the x axis is denoted by h.

In the following analysis, an exp ( + jcot) time variation

is assumed for the field quantities and is suppressed

throughout the analysis. The free-space wavenumber is

kO = o/c, where c is the velocity of light in vacuum.

II. ANALYSIS OF THE COUPLING PROBLEM

In treating the coupling between a planar and a single-

mode fiber waveguide, an analytical technique bearing

similarities with a method developed in analyzing abruptly

terminated dielectric slab waveguides [7] will be utilized.

The method to be presented relies on the use of mixed

spectral field representations inside the guiding regions,

while a Fourier integral representation is employed in the

intermediate space between the two waveguides.

0018 -9480/87/1100-1043 $01.00 01987 IEEE

1044 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. M~-35, NO. 11, NOVEMBER 1987

In the following analysis, an incident guided wave prop-

agating inside the fiber along the positive z axis (see Fig.

1) is assumed. Then the dominant mode amplitudes are

computed inside the planar waveguide. Although in prac-

tice usually the laser diodes are used to feed power into

fibers, because of the reciprocity principle, the analysis of

the opposite problem presented in this paper provides the

anwer to the practical problem of laser diode–fiber cou-

pling.

The single-mode fiber can support two degenerate HEII

modes. In optical guides the weak guidance condition

always is satisfied [8]. This means that the validity of

n ~ > n ~ and n ~ > n{ could be used to simplify consider-

ably the field expressions and description of guided waves

in either fiber or planar waveguides. Then the guided

waves in fiber waveguide are linearly polarized. In the case

of a single-mode fiber, there are two orthogonal modes

polarized linearly along the x and y axes, respectively.

Then a single transversal electric field component, EX or

EY, could be used to describe the propagation of guided

waves. The gain-guided laser diodes operate exclusively

with the dominant TE (transverse electric) mode. There-

fore, inside the planar guide the dominant electric field

component will be along the y axis (see Fig. 1). It is

therefore of interest to examine mainly the case when the

incident wave from the fiber waveguide is polarized paral-

lel to the y axis. In addition to guided modes, the presence

of longitudinal discontinuities along the propagation axes

on both waveguides implies the excitation of radiation

modes. The mixed eigenwave spectrum used in the follow-

ing consists of the guided modes (discrete spectrum) plus

the radiation modes (continuous spectrum). The nonde-

polarizing nature of the encountered discontinuities allows

the use of a single field component in describing the

radiation modes [9]. Then it is possible to express the

radiation modes in terms of the electric field y component.

Therefore a scalar wave V(r) could be used to describe the

field either inside the fiber or planar waveguide.

A. Field Expansion inside the Fiber

In describing the field distribution inside the fiber wave-

guide (z <0, see Fig. 1) a mixed spectrum of eigenwaves is

constructed by determining the solution of the wave equa-

tion

(V2+k&,)2(p, q3,z) =0 fori=l,2 (la)

in cylindrical coordinates p, T, z, defined in Fig. 1, subject

to the boundary conditions

T(p, cp, z)lp=h_ =qp, rp, z)lp=~+ (lb)

dT(p, cp, z) dv(p, (p, z)—

ap ap(lC)

~=& p=b+

on the core–cladding p = b interface. Equations (la)–(lc)

are valid only for weakly guiding fibers, and their ap-

proximate nature should be stressed. In the Appendix the

mixed spectrum of a weakly guiding fiber is given. Then

the *f electric field (polarized along the y axis) inside the

single-mode fiber (z <0, see Fig. 1) can be expanded as

follows :

V~(p, q,z) = $O(p, q)(e-J80’+ROeJpOz)

+/

‘@qdq •~m R~(q)~~(p,qlq)e@ (2)o ,fi=-~

where $.( p, q) is the guided-mode field distribution on a

z = constant transversal plane and +~( p, p Iq ) is the corre-

sponding radiation-mode field. Furthermore /30 and ~ =

(k~n~ – q’)1/2 are the propagation constants of the guided

and radiation modes, respectively. The unknown terms R o

and R ~ ( q ) in (2) are the reflection coefficients of the

guided and radiation modes, respectively.

B. Field Expansion inside the Planar Guide

The three-dimensional nature of the fiber guide radia-

tion pattern excludes the possibility of having y-axis-inde-

pendent-type waves inside the planar guide, the type most

frequently analyzed in the literature [10]. Then it is obvi-

ous that only hybrid modes (i.e., combination of transverse

electric and magnetic waves) should be excited inside the

planar waveguide. Indeed, if a rigorous solution is desired,

hybrid-mode field expressions should be adopted in ex-

panding the field. However because of the weak guidance

condition, in a similar way to the fiber guide, an ap-

proximate linearly polarized field distribution could be

taken to simplify the analysis. This assumption ensures

tractability, and its validity can be checked by observing

the variation of the obtained solution along the y axis,

which should be slow [11]. To this end, the ~p-dominant

y-polarized electric field inside the planar guide can be

written as follows:

Wp(x’, y’, z’) =f+mda ~

{3 A$n(a)e-’o’”’’~ n()’)

—CC s=e, o n=l

+ J 1‘~dpll, (a,p)e-’p’’’rp$, p),p) e-’”” (3)o

where the subscripts s = e and o ‘are to show that the even

and odd modes; (x’ , y’, z’) are the local Cartesian coordi-

nates of the planar guide (see Fig. 1) and are connected to

the fiber coordinates with the relations

X’=y–h

y’=y

z’=~—~. (4)

CAPSALIS AND UZUNOGLU: COUPLING BETWEEN OPTICAL FIBER AND DIELECTRIC WAVEGUIDE 1045

In (3) the set of functions { U~l(x ‘), U..2(X’),. . . . U~N$x’)}

is the discrete part of the spectrum. N. and NO are the

number of even. and odd symmetry guided modes, respec-

tively. The radiation modes, constituting the continuous

spectrum, are described with the ~~(x’, p] functions. Notice

that only waves propagating along the positive z axis are

taken into account in (3). The mode functions U~n(x) and

CP$(X,p) and their basic properties are given in the “Ap-pendix. The dependence on the y’= y variable is descri-

bed with the exp ( – jay) Fourier term. The values of v,.

and the guided- and radiation-mode propagation cons-

tants, depending on the value of a, are determined by

solving the transcendental equations given in the Appen-

dix. Finally, A,.(a) and II,( a, p) are unknown coef-

ficients to be determined.

C. Fie[d Description in the Space between the Two Guides

Assuming in this region also the validity of the single-

component

drical wave

q?a(p, Q),z)

electric field approximation and

functions, the following expansion

using cylin-

is obtained:

.(C~(~)e-J7z+C(A)e~~z) (5)m

where ~ = (k~n 2 – A2)l/2, J%(x) is the m th-order Bessel

function, and”C~, C; are unknown expansion coefficients

to be determined.

D. Boundary Conditions

In order to determine the unknown coefficients appear-

ing in (2), (3), and (5), the boundary conditions on the

z = O and z = w interface planes should be satisfied. An

integral equation approach will be employed in terms

of the unknown electric field distributions #1( p, v)

and da (x, y) on the z = O and z = w planes, respec-

tively. By employing the orthogonality relations of the

{+m(p$~)~+m(p> ~1~)>~= o+l!”””~o<~<+~} fibermixed spectrum eigenwaves, given in the Appendix, the

unknown coefficients R o, R ~( q) of (2) can be written as

follows:

(6)

(7)

Applying similar considerations by using the planar wave-

guide mixed spectrum eigenwaves (see the Appendix), the

unknown coefficients in (3) are found to be

A$n(a) = & ~~~dy’ f~~dx’d,(x’, y’)eJ”y’wn(x’)

(8)

B,(a, p) = ~~~~dy’{~~dx’~,(x’, y’)e’a”i,(x’,p).

(9)

In order to express the unknown coefficients C~(A) and

C’;(A) of (5) in terms of the #l(p, q) and &z(x, y) field

distributions, use is made of the orthogonality relations,

J‘mpdp$n(A, p) JW,(A’, p) =;8(M’) (10)

o

& ~2”dTeJi’”-””)p = ~mm’. (11)

Then it can easily be shown that

c,,,(~)+%z(~)

Cm(A)e-~”w+C,:(A)eJTw

where the continuity of the electric field on the z = O and

z = w planes is also incorporated.

In addition to the electric field, the continuity of the

tangential magnetic field components on the z = O and

z = w planes should be satisfied. Then it is required to

have

a~f d+a

61Z = azatz=O (14)

and

d~u a+p

dz = 82atz=w. (15)

On substituting (2), (3), and (5) into the continuity condi-

tions (14), and (15) and then introducing the expressions

(6)-(9), (12), and (13), the following system of integral

1046 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 11, NOVEMBER 1987

equations is obtained:

[.— d-,’,d)’”dq’~+mp’dp’JW, (A, p’)e-J~p’d’l(p’, rp’) – e-J’W&2(x’, y’)

1

+,-:J2Tw~2”dq’~

‘~p’dp’Jm(Ap’)e-Jw’W’( &l(p’, p’) – eJTW&2(x’, y’) 1

(fl(d,g’)-e -“w($,(x’, y’))+,:;:,.i2”d~’~+mp’dp’Jm( ~p’)’”Jm”

1(16)

.6Y2(x’, y’)e’a-’’rp, (,p)~lel~alp,(x,x, p). (17)

The system of equations in (16) and (17) constitutes a

Fredholm integral equation of the first kind. In general it

is preferable to work with Fredholm integral equations of

the second kind, which are amenable to iterative or ap-

proximate solutions. Furthermore examining the physical

aspects of the coupling geometry, it is observed that when~z~= ~z~ and ~ { = n ~, a simple boundary value problem is

encountered. Therefore, since n ~- n‘ and n{ - n ~ is al-

ways satisfied, it is desirable to transform the system (16)

and (17) into a Fredholm system of equations of the

second kind. The homogeneous term of this system should

be identical with the field distributions &l(p, rp) and

C7z(x’, y’) obtained when the reflection from a dielectric

slab between two different media is solved by simple

analytical techniques [12]. Furthermore, when n ~= n ~ and

n ~ = n j, the contributions from the integrations should

vanish. Then following similar analytical techniques de-

scribed in [7] in treating the diffraction from an abruptly

terminated dielectric slab waveguide, after lengthy algebra

the following system is obtained:

+J2”@’L‘mp’dp’ (K1l(p. rplp’>qf)c9@,P’)

+K,, (P> TIP’, fp’)~2(P’, P’)) (18)

+L’”’q’mp’dp’(~zl(p,pip’, ~’)dl(p’>q’)

+L2(P, q4P’, def ’’(P’>@)) (19)

CAPSALIS AND UZUNOGLU: COUPLING BETWEEN OPTICAL FIBER AND DIELECTRIC WAVEGUIDE 1047

~,l(P,dP’j9’) = – :( Bo–kon2)+o(P#P)wJ’>@)+

and solving for #1 and d12.Then, substituting this solution

into the right-hand sides of (18) and (19), the first-order

solution is derived. The use of orthogonality relations

given in the Appendix and the integration formulas for the

products of two Bessel functions resulted in analytic ex-

pressions except for the cross products involving fiber and

planar guide eigenwaves. The latter integrals are computed

numerically. The exceptionally lengthy formulas and the

little practical interest in knowing directly the %1 and cf’z

field distributions restrained us from giving their expres-

sions. Instead of this, the reflection coefficient R o and the

coupling coefficient computed form the A,l( a) term are

quoted in the next section.

+m

x J+mqdq(B -kon2)+;(P,9)4m( P’jP’)L ,, ~=_ *-u

+

H‘%d~

T(1+ e2J’w) (1+ e’,%%w) +~—

~2JTw — ~ ~J 2kOw_l1

~ Jm(Ap)Jm(Ap’)eJm9e-JmT’ (22)o ~=.~

[

~eJ7w

&2( P>wP’ w’) = :J+mAdA – l_e2JTW+

1‘onoejkon”w Y J~(Ap)J~(Ap’)ejMWe-jnW’1 – ~2Jokofiow

(23)~.—~

1 +CO

[

~eJTw

I

konoe.ib%w + mK,l(p, q[p’, q’) = ;~ AdA ~_ e2JTW – 1 – ~2Jkonow ~ ~m(~p)Jm(xp’)eJm9e-Jmq’ (24)

~..~

1 +mK22(P@oP’)=-;J ~d~

[

7(1+ e~2’W) ko(l+ e2Jk0”0w) ‘~

1 – ezjrw – l_ e2jk.now1

~ J~(~p)J~(Ap’)eJm~e-J~~’=— w

+: (+” da ~ ‘Ja(Y-Y’)z : (usn-ko~[)un(~)qn(~’)+ [+mdP9s(xj P)9s(x’>P)(Pl- ko~{). (w

L77—~ s=e, on=l

Notice that in (25) the transformations x = p cos rp, x’ =

p’ cos rp’ and y = p sin rp, y’= p’ sin q’ should be incorpo-

rated.

It can easily be shown that if nl ~ n z and n~ ~ n;, thenK,J ~ () (i= 1, 2; j =1,2) and the solution of (19), (20)

reduces to the elementary problem of reflection from a

dielectric layer placed between two different media. There-

fore, following exactly the same raison d’6tre as in [7], an

iterative procedure could be used to solve approximately

the system (18) and (19). In this paper up to first-order

iterative solutions have been obtained. The zeroth-order

solution is obtained by setting

&l(p, q)=&lo(p, rp)

‘$2( P>9)=~20(P,9) (26)

“o

III. COMPUTATION OF COUPLING AND REFLECTION

COEFFICIENTS

Assuming the &l(p, q) and &’2(x’, y’) fields are known

on the z = O and z = w interface planes, it is possible to

compute the electric fields in either z <0 or z > w semi-

infinite spaces. To this end, the first-order iterative solu-

tions #1, d’2 are substituted into (6) and (8) to derive the

guided wave amplitudes inside the fiber and planar wave-

guides, respectively. The use of the first-order approxima-

tion is justified from the conclusions of the analysis of the

abruptly terminated planar dielectric waveguide [7], where

a satisfactory convergence has been observed. The use of

orthogonality relations given in the Appendix and integrals

for the product of Bessel functions resulted in the follow-

ing expression from the reflection coefficient on the fiber

guide:

[( 4noeJk0”0w

)

2p.RO=–l+B1(W) 2~o l–A2(w)1_e2JkOnOW –(130–kon2)~Al(w)

00 1

,-#B1(w)~‘mAdA [( C1(A, W)- C2(W)~1(W) -4~2(W)(CS(& W)- C4(W))] (27)

00

1048 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO, 11, NOVEMBER 1987

where

(no- nj)+eJ’O”o~(no+n~)

“(w) = (nO-n~)(n, -nO)+e’J’O”Ow(nO +n{)(nO+n2)

(28)

elhm~

“(w) = (no–nf)(n2 –no)+eJ2k0n0w(n0 +ni)(~0+n2)

Bl(w) =

C1(A, W)=

C3(A, W)=

(29)

e]’%%w-l

~On2(e2A~o~ – 1)+ kOnO(l + e2~k0”UW)

(30)

T(1 + e2J””)C,(w) =cl(o, w) (31)

~~J7w — 1

~eJTW

2TWC4(W)=C3(0, W). (32)

~1 —1

In a similar way, the amplitude coefficient of the domi-

nant-systemtric even TE mode propagating inside the

planar waveguide is obtained as

2p. + m‘4e, (a)=—

[J Jdy ‘mdx’eJ”yU,l(x’) VO(p, q)

77 —m —cc 1

[

e]%% A,(w)‘2(”’ )A1(W) ~_e~,kOw + ~(”el–kOnl)

00 I4poc’o + ‘x

–BZ(W)G/

AdAIAl(w)(C, (A, w)oo~

–Ca(w))- A2(w)(C1(A, w)– C2(w)]

- xKo(yb)J1(Ab)) (33)

where CO, a o, y, and IU( A ) are given in the Appendix. In

computing the numerical values of the A ,1(a) coefficients,

a two-dimensional numerical integration is performed in-

volving the product of the U,l(x ) and TO( p, q) mode

functions. Owing to the highly spatial concentration of

these functions near the guiding axes, no convergence

problems are encountered. Furthermore, it is necessary to

compute in both the R o and A ,1(a) coefficients improper

integrals with respect to the A integration variable. To this

end, a 12-point Gaussian quarature numerical integration

procedure is employed after dividing the integration do-

mains into an adequate number of subintervals. The trun-

cation of the infinite bounds is taken sufficiently high to

ensure convergence.

IV. NUMERICAL RESULTS

Numerical computations have been performed by using

the analytical results obtained in the previous sections. In

all the computations, the free-space radiation wavelength

is taken to be

Ao=l.55pm ( =2 T/ko).

The single-mode fiber characteristics are

b=5pm nl=l.450 n2=l.454

and the corresponding planar guide dimensions modeling a

GaAs laser diode are taken to be

d= O.5pm n; = 3.40 n$ =3.61.

The space between the two guides is taken to be air, i.e.,

n o =1. In order to compute the coupling efficiency, the

following mode power ratio is defined:

Coupling Efficiency = CE

~planar guide dominant TEO mode ~

_\ total power (– co < a < +CO) 1(34)

incident fiber mode power “

On substituting the planar guide dominant-mode function

into the Poynting theorem written on the z = w plane, the

total power coupled into the planar guide is found to be

1Pp=—

J‘~dalA.l(a)12U,l( a).

120ko .~(35)

In a similar way, the incident fiber mode total power is

P=~f 120

(36)

when the { I ~p dp IVo( p, rp) 12= 1 normalization condition

is satisfied. Then, according to (34),

Notice that in integrating the Poynting vector on the z = w

plane the total power coupled into the TE planar guide is

obtained as a superposition of IA ,1(a) 12mode amplitudes.

A Simpson rule numerical integration procedure is em-

ployed to compute the integral in (37). For a given specific

value of a, the guided wave inside the planar guide travels

parallel to the unit vector.

fiU=2’cos8+j’sin0 (38)

where

(39)

‘in’=~+ (40)

as also illustrated in Fig. 1. Numerical computation reveals

that the IA ,1(a)\ 2 has significant values only for a -0(i.e., @- O). A specific IA,(a) 12distribution is presented in

Fig. 2 using polar coordinates. This shows that the excited

TE mode inside the planar guide is almost linearly polarized

and that the assumptions in writing (3) are valid in the

framework of the present analysis.

In Fig. 3 the variation of CE with lateral displacement h

(see Fig. 1) is presented for a w = 121.8 pm interguide

CAPSALIS AND UZUNOGLU: COUPLING BETwEEN OPTICAL FIBER AND DIELECTRIC WAVEGUIDE 1049

lAe1(u)12

--0.6

,’

,’/’

(

‘9 ;;a

,’/’

CE Io 2;

I[E

I

L-A’ ,-120 121 122 60 612

w(m) w(w)

Fig. 4. Dependence of coupling efficiency (CE)oninterguide distance~, forh=Oand thesame setofparameters as in Fig. 3.

Fig. 2. Dependence of dominant TE planar guide mode amplitude

IA,1(u)12 on propagation direction 9 (see Fig. 1 and (39), (40)) forw=120pm and h=O.

CE

OJ2-

0.08-

o.04–

+

w

‘-–- - :1’Ftber -

planar guide

II I I5 10

h(ww15

Fig. 3. Variation of coupling efficiency (CE) with the lateral displace-

ment between the two guide axes h (see Fig. 1) forasingle-mode fiberwith b = 5 pm, nl = 1.450, and n ~ = 1.454 and a dielectric planar guide2d =1 pm, n( = 3.40, and n~ = 3.61 at & =1.55 pm operation wave-length. The interguide distance is w = 121.8 pm and no =1.

distance. It is observed that an h = 5 pm displacement

could reduce the coupling efficiency 6 dB below the peak

CE when h = O. The dependence of CE on the interguide

distance w is investigated for small displacements around

w = 60 pm and w = 120 pm distances. The expected stand-

ing wave patterns are observed in Fig. 4 when the two

guide axes coincide (i.e., h = O). The average CE values

are approximately equal to 0.15, and it is slightly higher

for w -60 pm distances. Furthermore, the CE fluctuates

1‘“” lR~

0.9

0.8 –

0.7 –

0.6 –

0.5

I

0.4 1i I I

60 61 62w(vm)

Fig. 5. Variation of 1RO 1reflection coefficient of the fiber guide for the

same set of parameter values as in Fig. 3 and A = O.

from 0.12 to 0.18 periodically. The period is equal to the

free-space wavelength AO = 2n/k0 = 1.55 pm.

The variation of the reflection coefficient R ~ on the

fiber guide has also been computed. In Fig. 5, the variation

of IR ~I with interguide distance w is presented. A signifi-

cant fluctuation (almost 100 percent) is observed in the

reflected power inside the fiber guide when the interguide

distance w changes by half a wavelength. Comparing Figs.

4 and 5, it is shown that when CE is high R. is low, and

vice versa.

It is necessary to point out that in practice planar

dielectric slab waveguides (i.e., laser diodes) always have

finite width (along with y’ axis, see Fig. 1). Therefore, the

infinite-width planar guide assumed in this paper is an

approximation for the fiber–laser diode coupling. Consid-

ering the rather large width of laser diodes ( -20 pm) and

the fact that the guided waves inside the planar guide are

taken with an arbitrary propagation direction on the x’y’

plane, the spillover radiation entering the planar guide for

y >> X ~ is expected to be insignificant.

1050 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35> NO. 11, NOVEMBER 1987

V. CONCLUSIONS

The coupling between an abruptly terminated single-

mode optical fiber and a dielectric planar waveguide has

been analyzed. Weak guidance approximations in both

guides are employed to simplify the analysis and to achieve

tractability. Furthermore, the same weak guidance condi-

tions are employed to obtain approximate analytical re-

sults where only a few numerical integrations are required

to obtain numerical results. Computations shows that a

coupling efficiency of around 20 percent could be achieved

between the two guides under proper alignment. The most

critical tolerance seems to be lateral (i.e., h parameter of

Fig. 1) displacement of the two guide axes. The fluctuation

of the coupling efficiency with the interguide-distance mi-

cromovements signifies the critical dependence of laser

diode operation on strong externally reflected power levels

in optical transmitter systems.

APPENDIX

A. Single-Mode Fiber Guide Spectrum

Guided Modes:

/

KO(aOb)JO(aOp)

+O(P>9) ‘co Jo(~ob) ‘or p< b (Al)

[KO(YP) forp>b

propagation constant PO is obtained by solving the tran-

scendental equation

aoJ~(aOb) yK((yb)

Jo(aob) = Ko(yb)(A2)

and the normalization constant Co of the lowest order

mode is

2J:(uob)

Co= b2(J~(aOb)K~(yb) +K~(yb)J~(aob))(A3)

so that

is satisfied.

Radiation Modes:

4’fi,(P,914)

{

JW,( Up)

= Aw,eJm19 cm(q)forp<b

(Jan+%%) forp>b

where m=O, +1, ~2, +3,..., ;O<q<+co; and

uJJ(ub)Y~(qb) –J~(ub)Y~(qb)q(AS)

cm(q) = qJJ(qb)Yn(qb) –J~(~b)Yi(qb)4

qJH(ub)JJ(qb) –Ji(ob)Jti(qb)(A6)

‘m(q) = J~(ub)Y~(qb) –qJ~(ub)yjj(4b)

~=(pn:-y)l’2 p = (/@; - 4’)’/’

Am= (l+ D:)-1’2.

Orthogonality Relations:

=;tl(q-q’)a:m(A7)

B. Planar Guide Spectrum

Guided Modes: For even modes,

forx<–d

Icos (K,x )

u..(x) ‘G.. for–d<x<dCOS ( Ked )

(e-w(x-d) forx>d

Gen = (Y.-l + Y./K$ + d/cos2 (Ked))-’”.

For odd modes,

( – eh(x+d) forx<–d

Isin(Kox)UO,,(X) = Go. for–d<xgd

sin(Kod)

(e-n(+) forx>d

Go. = (y;’+ yo/K~ + d/sin2 (Kod))-l”. (A9)

Transcendental equations:

y,= K, tan(K~d) Ko=–yotan(Kod)

( )Ye= a2+u2–k~n~ 112

Ke=(k~n~– a2–u~)l’2.

Orthogonality relations:

J+muen(x)uem(x) dx=dnm (A1O)—m o 0

/‘mdxu,. (x)u.. (x) =0. (All)

—m

Radiation Modes: For even modes,

1cos(p(x+d)– XP(p)) forx<–d

%(x, p)=l 1

Cos ( ox )

h C,(p) cos(ud)for–d<x<d

(A4)[cos(~(x-d)+ ~,(p)) forx>d.

CAPSALIS AND UZUNOGLU: COUPLING BETWEEN OPTICAL FIBER AND DIELECTWC WAVEGUIDB 1051

For odd modes, [4]

1sin[p(x+d)– XO(p)) forx–d

1[5]

sin ( ux )

fPo(x>P) =: -& Co(p) sin(tid) ,

for–d<x<d[6]

MP(X-4+XO(P))forx>d[7]

(A12)

Etan(ud))[8]

Xe(p)=tafi-l [9]

Etan(od’)

[10]

XO(p)= tan-l

[11]

FC,(p)= l+~tan (ud)

Gco(p)= 1+7

Orthogonality relations:

J+m9e(x>P)ie(x’> P)d~=~(P-P’) (A13)o

j+m:e(x,p):(x,jY) dx=O. (A14)o

C. IU(A) Integral of (33)

IU(A)=G=I

[ [

cos(~~d) 1 sin(Ke+{~) d)

WY K.+ ~~

[1]

[2]

[3]

[12]

D. Kate, “Light coupling from a stripe-geometry GaAs diode laserinto an opticRI fiber with sphericaf end,” J. Appl. Phys., vol. 44,

pp. 2756-2758, 1973.

C. A. Brackett, “On the efficiency of coupling light from stripe-

geometry GaAs lasers into multimode opticaf fibers,” J. Appl.

Phys., vol. 45, pp. 2636-2637, 1974.

K. Kawano, H. Miyazawa, and O. Mitomi, “New calculations for

cbupling laser diode to multimode fiber,” J. Lightwaue Technol.,

vol. LT-4, pp. 368–374, 1986.

C. N. Capsalis, J. G. Fikions, and N. K. Uzunoglu, “Scattering

from an abruptly terminated dielectric-slab waveguide~’ J. Light-waoe Technol., vol. LT-3, pp. 408–415, 1985.J. Senior, Optical Fiber Communications: Principles and Practises.London: Prentice-HaJl, 1985.A. Snyder and J. D. Love, Optical Waoeguide Theory. New York.Chapman and Hall, 1983, chs. 25 and 32.H. Kogelnik, “Theory of dielectric waveguides;’ in IntegratedOptics (Topics in Applied Physics, vol. 7), T. Tamir, Ed. NewYork: Springer-Verlag, 1982, pp. 13-81.

H. G. Unger, Planar Optical Waueguides and Fibres. Oxford:

Clarendon Press, 1977, ch. 4.D. S. Jones, Theory of Electromagnetism. Oxford: Pergamon Press,

1964.

Cti.stos N. Capsalis was born in Nafplion,Greece, on September 25, 1956. He received the

Diploma of E.E. and M.E. from the NationalTechnical University of Athens (NTUA) in 1979

and the bachelor’s degree in economics from theUniversity of Athens in ‘1983. He also receivedthe doctor’s degree in electrical engineering fromNTUA in 1985.

Since Janu@ 1982, he has been a ResearchAssociate in the Department of Electrical En-

xineerinx at NTUA. In November 1986 he was,“

elected Lecturer at NTUA. His &in research interests are in the electro-magnetic field area, with emphasis on scattering and propagation at

millimeter and oDtical waveleruzths.

(( Hsin Ke -i= d+

K= – i= I

,

K

=sin(~d)- yncos(/=d)—y$+A2–a2

1Nikolaos K. Uzunoglu (M82) received the B. SC.

(A15) degree in electronics engineering from theIstanbul Technical University, Turkey, in 1973.He obtained the MSC. and Ph.D. degreesfromthe University of Essex,England, in 1974 and

REFERENCES1976, respectively.

He worked for the Hellenic Navy ResearchI. Garett and J. E. Midwinter, “Optical communication systems,” and Technology Development Office from 1977in Opticci/ ,Fibre Communication~, M. J. Howes and D. V. Morgan, to 1984. During this period, he also workedEds. New York: Wiley, 1980, pp. 251-299. on a part-time basis at the National TechnicalP. A. Kirby, “Semiconductor laser sources for optical communica- University on electromagnetic theory. In 1984,tion,” Radio Eiectron Eng., vol. 51, pp. 363-376, 1981. he was electged Associate Professor at the National Technical UniversityR. Itoh, “Hybrid laser-to-fiber coupler with a cylindrical lens,” of Athens, the position that he holds presently. His research interested areApp[. opt., vol. 16, pp. 1966-1970, July 1977. microwave applications, fiber optics, and electromagnetic theory.