The ABCD matrix in graded index tapers used for beamexpansion and compression

James N. McMullin

A new form is proposed for the ABCD matrix in a graded index taper with a large variation in cross sectionsuch as might be used for single-mode beam expansion. Expressions are given for the loss of power from thefundamental mode and the coupling efficiency between fibers when two tapers are used in an expanded beamconnector. Exact solutions are found for linear tapers and for a class of tapers with zero slope ends. Thedistinction between adiabatic and nonadiabatic tapers is made clear from the functional form of the matrix inthe linear case. Comparisons are made with previously published results and the effect of taper shape on thecoupling efficiency is discussed.

1. Introduction

In recent years, tapered waveguides have been pro-posed and tested1-4 as the basic optical elements inexpanded beam coupling between single-mode fibers.Losses below 0.1 dB were reported with tapers severalcentimeters long. The expanded beam method showsless sensitivity to lateral and axial displacements andincreased but tolerable sensitivity to angular misalign-ments. A numerical study by Marcuse5 showed thatcomparable coupling losses (<0.1 dB) could occur intapers as short as 1 cm if the derivative of the taperradius, da/dz, vanishes at both ends. The basic idea isthat the narrow end of the taper supports only onebound mode which should be exactly matched to themode in the fiber. As the beam propagates into thewidening taper, which becomes multimode, its energywill remain predominantly in the fundamental modeas long as the taper radius varies slowly enough. Theexpanded beam is then coupled into the wide end of asimilar taper which guides the beam into the otherfiber. The da/dz = 0 condition attempts to minimizethe phase mismatch between the fundamental modesin adjoining waveguides.

The expanded beam couplers based on this concepthave so far been constructed from the leftovers ofsingle-mode fiber preforms which are naturally ta-pered. Another concept being developed6' 7is to let the

The author is with University of Alberta, Department of ElectricalEngineering, Edmonton, Alberta T6G 2G7, Canada.

Received 6 June 1988.0003-6935/89/071298-07$02.00/0.© 1989 Optical Society of America.

core near the fiber end expand by thermal diffusion sothat the fundamental mode in that region will be en-larged. The expected index profile in that case will bemore nearly parabolic than square.7 Marcuse5 consid-ered graded index tapers in his numerical study andfound very similar properties for these as for the stepindex variety.

Graded index tapers have been the subject of manystudies over many years and a powerful method for thedescription of light propagation in this type of wave-guide is in terms of the ABCD ray matrix. 8 All thebeam guiding properties of a taper are contained in itsABCD matrix. For example, the coupling of non-Gaussian or misaligned Gaussian beams between fi-bers can be calculated by expressing the fields in termsof the higher-order Hermite-Gaussian modes of thequadratic index taper whose propagation is also de-scribed by the matrix.9

In this paper, I investigate the use of the ABCDmatrix for calculating the coupling efficiencies or para-bolic index tapes used for fiber-to-fiber coupling. InSec. II, results for single linear and raised cosine tapersare reported which are essentially identical to thoseobtained by Marcuse.5 The theory is then extended toback-to-back tapers used for expanded beam fibercoupling. Three interesting by-products of this inves-tigation are (1) a new description of the ABCD matrixwhen the physical size of the taper varies greatly; (2) alocal condition for the rate of change of the taperradius which distinguishes between adiabatic and non-adiabatic tapers; and (3) the first explicit expressionfor the matrix in a linear parabolic index taper.

In Sec. III, the form for the matrix in adiabatic lineartapers is used as a guide for developing a new form forarbitrary but adiabatic tapers. The exact solution in a

1298 APPLIED OPTICS / Vol. 28, No. 7 / 1 April 1989

special case is described and is used to calculate ananalytic expression for the coupling coefficients of awhole family of adiabatic tapers with zero slope ends.Using this solution, some comparisons of the effective-ness in coupling by tapers of different shapes are made.Conclusions and a discussion are given in Sec. IV.

II. The ABCD Matrix in Tapers with Large Variation in

Radius

Some of the tapers studied numerically by Marcuse5

vary in cross section by over a factor of 90 and fabricat-ed tapers have had factors ranging from 12 to 20.1-4

The principle in all is the same: that a beam predomi-nantly in the fundamental mode should stay in thatmode (as defined by the local taper radius) despite thelarge changes in taper size. For a square law taper thebeam behavior is contained in the elements of theABCD matrix. It makes sense then in analyzingtapers from the ABCD approach to explicitly accountfor the expansion (or contraction) of the beam by writ-ing each element of the matrix as the product of ascaling factor and another factor which contains thedetails of the difference of the beam propagation fromthe locally defined waveguide modes.

A. Modified Matrix and Equations

By studying linear tapers, I have found that a conve-nient form for the matrix at any axial position is

A B] = [[Q()/Q(Z)]11/2a(z) [Q(O)Q(z)]-1/2(z) 1

IC D L-[Q(O)Q(ZW]-/`Y(Z) [Q(O)/Q(ZW]-/%6Z)1

where Q(z) is the inverse scale length parameter in thequadratic index profile

n2(r,z) = n[1 - Q2 (z)r2]. (2)

If the taper has the structure of a fiber with a core andcladding (which it would have if constructed from afiber preform), Q(z) is related to the core radius a(z)and relative index change from core to cladding A by

Q(z) = 2) -a(z)

(3)

da/dt = [-Q(t)a -]; (6a)

df3/dt = [-Q(t)f3 + 6];

d'y/dt = [Q(t)-y + a];

db/dt = [Q(t)6 - ].

where

Q = - nd Q(z)] =1 lna(z)] = (8Ar1 2 da2 dt J2 dt Idz

(6b)

(6c)

(6d)

(7)

Q(t) is a dimensionless parameter proportional to thetaper slope. It will be seen below that the behavior of abeam is fundamentally different depending on wheth-er Q > 1 or Q < 1. Note that a linear taper has aconstant slope parameter Q. The initial conditions atthe input end (call itz = 0 and t = 0) are A = D = 1, andB = C = 0, and therefore a(O) = 6(0) = 1, 0(0) = -y(O) =0. The ABCD determinant property 1" becomes

AD - BC = ab + fly = 1. (8)

Before examining the special case of a taper withconstant slope, I derive expressions (in terms of a, 3, -y,and 5) for (1) the loss of the fundamental mode propa-gation through one taper, and (2) the coupling efficien-cy of two back-to-back tapers used for expanded beamcoupling.

B. Fundamental Mode Loss Coefficients

The coupling efficiency of beam a to beam b withnormalized electric fields, E0 (r,z) and Eb(r,z), respec-tively, is

7(Z)= 2w j E.(r,z)E(r,z)rdrj - (9)

A normalized Gaussian beam has a field (in the nota-tion of Yariv 8 by

E(r,z) = - -tk Im[q(z)]1 exp[ 2qkzi21

Hence we find4[Imq(a)(z)] [Imq(b)(z)]

-q(z) [q(b)(z)] *1

(10)

(11)

Equations for the new a, , y, and can be deter-mined from the following first-order ordinary differen-tial equations for A, B, C, and D which have beendescribed previously 1 01 1 (here we exclude the factor njfrom the elements of ABCD which is the usual conven-tion):

dA/dz = C(z), (4a)

dB/dz = D(z), (4b)

dCldz = _Q2 (z)A(z), (4c)

dD/dz =-Q 2 (z)B(z). (4d)

If we define a dimensionless axial distance variable t(z)by

t(z) =JQ(z)dz = a 4) ' (5)

we find

Let us assume that beam a when launched is in thefundamental mode of the taper at z = 0 and hence has

qo) = ja(0)/(2A) 2 = j/Q(O) = j/Qo. (12)

The complex beam parameter q(a)(z) is transformedfrom qla) to q(a) by the ABCD matrix between z = 0 andz = L according to

- Aqoa) + BCqa) + D

(13)

If at z = L in the taper, beam b is taken to be the locallydefined fundamental mode with

q(b) = j/Q(L) = I/QL' (14)

then 77(L) = f7L will be the fraction of the power of thelaunched beam which remains in the fundamentalmode at L. Substituting Eqs. (12), (13), and (14) intoEq. (11) and using Eq. (1) for the ABCD matrix and

1 April 1989 / Vol. 28, No. 7 / APPLIED OPTICS 1299

Eq. (8) for the determinant property, the followingsimple expression for qL is obtained:

4=[2 + a' + #L2 + y' + 6] (15)

This is the quantity (when expressed in decibels) re-ferred to by Marcuse 5 as "loss of mode ." From nowon, assume that L is the length of one taper.

For fiber-to-fiber coupling efficiency, assume twoidentical tapers will be used back-to-back. We makeuse of the property that the ray matrix of an opticalsystem operated in reverse is

D B _[Q./QLI-1 /26L [QoQL]-1 2OL (16)

[C A L-[QOQL1/2YL [Q0/QL 1 /2 I

in this case. The total ray matrix is the product of thesuccessive matrices

A BJ= B B] _ (LbL-L 3L) (20L)

IC D C A C D - QO(2aLYL) (aQO - )

(17)

This has the form of Eq. 1 where the diagonal scalingfactors are unity since Q(2L) = Q(O) = Q for two back-to-back identical tapers. So atot = tot = (aLhL - LYL),Otot = 2LaL, and ytot = 2 aLhL. Therefore we may makeuse of Eq. (15) which is true for any aofy3 to find theefficiency of fiber-to-fiber coupling through two tapersof length L:

7ltot 2 + 2 (aLL-ILYL) 2 + 4iLfL + 4aj7L (18)

Again, Eq. (8) is useful and Eq. (18) reduces to (keep-ing in mind that subscript L means these parametersare for one taper of the coupling pair)

1(aot:.- + o2&y + 62) (19)

An important point to note is that the loss, ntot, throughtwo back-to-back tapers is not L. In fact, as theresults below show, even though flL may be very smallcompared to 1, 77tot may be very close to unity. Let usnow apply this theory to the case of coupling by lineartapers.

C. Linear Taper

The propagation of a Gaussian beam in a linear,graded index waveguide has been described before,1 2 1 3

but not in terms of the ABCD matrix. It demonstratesthe difference between fast and slow tapers. A lineartaper may be described by the core radius function

a(z) = a, + (a2 - a,) L I (20)

where a1 and a2 are the radii of the small and largeends, respectively, of the taper. We then have fromEq. (7)

Q(t) = (8A)-"2 da = (8A)- /(a2-al)/L = constant. (21)

Equations (6a)-(6d) are then easy to solve yielding

a(z) = 1' cos[Pt(z) + cosIti],

A(z) = y(z) = NY` sin[it(z)],

b(z) = I-' cos['It(z) - os-1 f],

where

'1 = [ - 02]1/2,

and from Eq. (5)

(2A)11 2L [ a(z)l(a2 - a,) I a, I

(22a)

(22bc)

(22d)

(23)

(24)

Despite the complexity of the solution, some fea-tures of the propagation in a linear taper can be seenfrom the ABCD matrix. For a GRIN rod lens withconstant radius ao, Q = 0, T = 1, and t = Qoz. Then theABCD matrix for a length z of the lens reduces to thefamiliar form

(25)LA BCD] cos(Qz) 1 sin(Qoz)IC DJ =-Qo sin(Qoz) cQOz

When Q < 1, each element of the matrix consists of afactor which scales with the size of the taper and afunction which is periodic in t. This corresponds to aGaussian beam oscillating about the local fundamentalmode which scales in size as the square root of the taperradius. As a result, the coupling coefficient showsrapid variation with the same periodicity. The singletaper loss of mode 0 is found to be

(1 0)?L (1 - U2) + 112 sin2('It)

and the back-to-back taper coupling coefficient is(1 - 2)2

7tt = (1 -2)2 + 4Q2 sin 4(*t)

(26)

(27)

These formulas predict small losses by linear taperswith Q << 1 and perfect coupling for periodic values of t,a result demonstrated in the numerical results of Mar-cuse5 and described below. Tapers with Q < 1 may bethought of as slow or adiabatic.

When Q > 1, T becomes imaginary and the trigono-metric functions must be replaced by their hyperboliccounterparts. In this case, the expansion of a Gauss-ian beam by diffraction cannot keep up with the ex-pansion of the local fundamental mode of the taper.This we may call a fast or nonadiabatic taper. Thecoefficients are

7 L ( 21)- 1) + Q2 sinh 2('It)

(02 - 1)21 tot = (2 - 1)2 + 4Q2 sinh4(blJt)

(28)

(29)

Another adiabatic condition was given by Stewartand Love14 which may be written in the form

da << aAfl Qadz 2wr 2r (30)

since the increment in propagation constants, 4/, be-

1300 APPLIED OPTICS / Vol. 28, No. 7 / 1 April 1989

0 1 RAISED COSINE TAPERz

W 0.1-00La0 0.010CO)

o0.001:

10 10

0.001 o1 o 1 11 100

TAPER LENGTH IN MM

Fig. 1. Loss of mode 0 in decibels for single linear and raised cosine

graded index tapers. Parameters: a = 4 jum, a 2 = 361 8m, A0.0056.

100,

10- AAYLNEAR TAPER

o RAISED (r).ThsreutarnoinOSINE TAPERzco

0.

.I 0.01

0C.)

0.001 0.01 01110 10TAPER LENGTH IN MM

Fig. 2. Coupling loss in decibels for linear and raised cosine gradedindex expanded beam couplers. Parameters: a 4 tim, a2 = 48

ptm, A = 0.0056. Irregularities at large L are due to having an

insufficient number of points.

tween modes is just Q.9 This condition is equivalent toQ2 « (4r)- 1 These results are not inconsistent sincethe definition is necessarily qualitative. The condi-tion I have derived defines the value of Q at which thefunctional form changes between fast and slow tapers,not the point at which loss of mode 0 is small.

In Fig. 1, I present results from the ABCD analysisfor a case studied by Marcuse, 5 namely, the loss ofmode 0 in single linear tapers and raised cosine tapeswith a1 = 4 mn, a 2 = 361 Aim, A = 0.0056, and L varyingfrom 1 to 100 mm. The curves in Fig. 1 appear to beindistinguishable from his (Ref. 5, Fig. 7) althoughcloser inspection shows that the raised cosine curve inFig. 1 has a slight oscillation which is not noticeable inhis results. Equations (26) and (28) were used tocalculate the loss in linear tapers. The crossover fromfast to slow linear tapers is at L = 1.687 mm. Thisexplains the high loss as a function of length for lengths

<2 mm. The calculation of 1501 linear taper couplingcoefficients on a standard IBM PCXT with an 8087math coprocessor using Turbo Pascal takes only 20 s.The aOf3yb equations [Eqs. (6a)-(6d)] were integratednumerically for the raised cosine tapers, a considerablyslower process.

In Fig. 2, I present the coupling coefficients for thesame tapers when they are used in pairs for expandedbeam coupling [Eqs. (27) and (29) for the lineartapers]. I have also included more fast tapers with Lbeing as small as 1 ,m. This shows that two very shortback-to-back tapers actually give quite good couplingalthough the loss of mode 0 in each taper is large. Thisis easily explained since in reality we are only removingthe fiber core for a short distance so the fundamentalmode expands slightly before entering the second fi-ber. An interesting result for the raised cosine cou-plers is that for long tapers (approximately L > 8 mm)the loss is actually smaller than that predicted by thesingle taper loss of mode 0. A possible explanation isthat the loss of mode 0 in the first taper is mainly due toa radial size mismatch rather than a phase mismatchand that during the compression of the beam in thesecond taper the size error is in the opposite direction,e.g., the beam is first underexpanded, then undercom-pressed.

Ill. Analytic Study of Tapers with Zero Slope Ends

Exact solutions in quadratic index tapers have beenstudied by many authors.11-1315 They are useful notonly for aiding our understanding of light propagationin these structures but also for providing easily calcu-lable checks for numerical solutions. In some cases,they may themselves be reasonable approximations toreal situations. Exact solutions for the ABCD matrixare especially useful since they provide informationabout the general optical properties of the taper andnot just the solutions for particular fields. In thissection, a class of adiabatic tapers with zero slope endsand the associated ABCD matrix are analytically de-scribed.

A. Improved Form for Long, Slow Tapers

We have seen that the difference between the con-stant grin rod and the linear taper is that an amplitudefactor Tip and a phase factor cos-1'I were introducedinto the matrix components, and that the combinationQoz was replaced by xI' So Q(z')dz' = Tt(z). Since aslowly varying taper of arbitrary shape should behaveover a short distance as a linear taper, let us assumeforms for a, fi, -y, and 6 which are similar to the lineartaper forms but contain two new amplitudes, A and B,and two new phases, A and v, which vary along the taperlength. We also replace the combination At by a newvariable

u(t) = Jt 4(t')dt' = J [1- (0) dt' (31)

For simplicity, all functions will be expressed in termsof the variable t. A form which leads to relativelysimple equations for the amplitudes and phases is

1 April 1989 / Vol. 28, No. 7 / APPLIED OPTICS 1301

cx(t) = A(t)[*(t)(O)]-112

X Cos u(t) + cs" I(0) + cos"V(t) + t])<cs Ut 2 + Mt)

/3(t) = B(t)[*(t)y(O)]-1/2

X sin [u(t) - cosI'F(0) - cos"P(t) + v(t)]

(32a)

(32b)

y(t) = (t)[p(t)y(0)]-l/2

X sin [u + cos1 (0) -cos"I'(t) + 1M

(32c)

6(t) = 3t[~~~)-/

X cos [u(t) - cosI(O) + cos1 'I(t) + v(t)

(32d)

It is clear than when T(t) = T(0) = I, a constant, Eqs.(32a)-(32d) reduce to the linear solution if A = = 1and A = v = 0. These are also the initial conditions forA, .B , and v in the general case. The Ajfu3v equa-tions are found by substituting these forms into Eqs.(6a)-(6d) (dots) mean d/dt):

- = + 2 sin[2u(t) + 2(t)], (33a).A 2[1 - 2]

2[=+ - 2 cos[2u(t) + 2(t)], (33b)

!3 1 -n n2] sin[2u(t) + 2v(t)], (34a)

11 ]

=[1 - 2] cos[2u(t) + 2(t)]. (34b)

For tapers of length L [where t(L) = I] with zero slopeends, AF(0) = (T) = 1. Therefore

cient of a class of tapers with zero slope ends. Consid-er Eq. (33b) and let u be the independent variable. Aone-to-one relationship between u and t is given by Eq.(31). If we define M(u) = u + tt, the A-A equationsbecome

d In.u = (u) sin(2M), (37a)

dM 1 + d = + ddt= + (u) cos(2M), (37b)du dui du/dt

where [after using Eq. (31) for du/dt]

2[1 - u213/2 (38)

If we now assume e(u) to be no worse than a piecewiseconstant, solutions of Eq. (37) are obtainable in termsof simple functions as described below. First I shallshow the form of the tapers implied by this assump-tion.

Let us assume that the taper has its narrow end at z= 0 where da/dz (and therefore Q) is zero. Using thedefinitions of t, Q, and Q from Sec. II, it is easily shownthat Q is proportional to a(d2a/dz2) and therefore emust be positive at z = 0 where a = a1 . Similarly, emust become negative for Q to return to zero at z = Lwhere a = a2. Therefore, let us define a taper com-posed of two sections in which e = El for 0 < t < t and e= -2 for t < t < T = t + t2, where t and t2 are to bedetermined. In the first section of the taper, Eq. (38)integrates to give

2 elt[1 + (2et) 2]1/2

(39)

and the radius as a function of t follows from Eqs. (5)and (7):

a(t) = a1 exp [1 + (2elt)2112 - * (40)

In the second section, these two functions are found tobe

aL = AL cos[u(T) + AL],

/3L = BL sin[u(T) + vL],

YL = AL sin[u(T) + ML],

6L L cos[u(T) + VL] .

(35a)

(35b)

(35c),

(35d)

Again, note that for a linear taper, ~2 = 0, and thereforeA = B = 1, A = v = 0 and the linear solution is obtained.

The one-taper loss of mode 0 coefficient followssimply from Eq. (15):

4[2 + LA + L] (36)

The two-taper expanded beam coupling loss does nothave such a simple formula but is easy to calculatefrom Eq. (19) once a /3L, YL, and L are known.

B. Analytic Solution

An exact solution is obtainable for Eq. (33) and (34)leading to a closed expression for the coupling coeffi-

QM=-2E2 (T -t) Q2(t) = $1 + [2E2(T -t)]211/2

a(t) = a 2 exp {---[1 + [2E2(T-t)2] /2 -{ C2 E2 J

(41)

(42)

We now demand that the radius a(t) and the slope, orQ(t), be continuous at a = a, the radius at which theinflection in the taper occurs. This leads to two condi-tions:

el n a,) = 2 2n ,

eltl = e2t2.

(42a)

(43b)

Given a1, a2, and aj, the selection el will determine E2, t 2,and t2. The taper shape a(z) may be determined bynumerically integrating Eq. (5) in the form

z(t) = I dt' (44)

1302 APPLIED OPTICS / Vol. 28, No. 7 / 1 April 1989

co

00z

IC

300

200

100M_ ,-' // ~~~~EIE2 TAPER, Al - 1825

.- / ElE2 TAPER, Al - 330.0

, 1_ RAISED COSINE TAPER

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NORMALIZED TAPER LENGTH

Fig. 3. Shapes of elE2 tapers with three different radii compared

with raised cosine tapers with aj = 182.5 Am, El = 0.01.

and compiling a set of (a,z) points. As an example,Fig. 3 compares the shape Ele2 tapers with El = 0.01 andthree different inflection radii with a raised cosinetaper. All have a, = 4.0, a2 = 361.0, and A = 0.0056.In Fig. 4, the shapes of ElE2 tapers with a, = 182.5 Mmand three different el are compared with the raisedcosine taper, again with the same a1 , a2, and A. Forlarge el, the taper is very linear with d2 a/dz2 being largeat each end to meet the zero slope conditions. As l -- the taper becomes perfectly linear with a maximumslope corresponding to Q = 1 and length L given by

(a2- a 1)= ~ = 1.687 nm.

[8A] 1 /2

(45)

In general since El and E2 turn out to be not too differentin magnitude, the tapers of this type have a larger d2a/dz2 near z = 0 and hence reach the inflection radiusfaster than the raised cosine taper.

To find the coupling coefficient of a pair back-to-back elE2 tapes between two fibers, we need expressionsfor a2, i2, -Y2, and AL of one of the tapers. These aregiven in the Appendix. We may then use the analyticexpressions obtained to quickly investigate the cou-pling efficiencies of Ele2 tapers for various lengths andinflection radii a,. Some results are presented in Fig.5, again with the raised cosine taper for comparison.Since the tapers must everywhere be slow (Q < 1) forthe trigonometric functions forms of the solution toapply, the shortest taper allowable is 1687 mm long[Eq. (45)].

The results of Fig. 5 show that the position of theinflection point has some impact on the coupling coef-ficient, especially at longer taper lengths where thetapers with large inflection radius suffer the most loss.Except for a few small ranges of L, the tapers withinflection radii at (a, + a2)/2 = 182.5 gm have thesmallest loss and there is a window between 6 and 8mm where the loss is very small. In any event, for alltapers with zero slope ends longer than 2 cm, the cou-pling loss is <0.03 dB, a number which is probablyunobtainable in practice due to other effects such asaberrations and misalignments, but that much shorterlow loss expanded beam connectors can be built ifattention is paid to taper shape.

CO20 C 300-

CS200 /

Ir ~ ~ ~~/ElE2 TAPER, El - 3.0CC 100 7'ElE2 TAPER E 0.030L / ElE2 TAPER, El - 0.0003

7 / / RAISED COSINE TAPER

0 0.1 0.2 0.3 0.4 0.5 o.6 0.7 0.8 0.9

NORMALIZED TAPER LENGTH

Fig. 4. Shapes of flE2 tapers with three different l values and al182.5 Am compared with raised cosine tapers.

10m0tCS

0CO

z0. 0.01

0.00011

E1E2 TAPERSE1E2 TAPER'

ElE2 TAPER'RAISED COS

10TAPER LENGTH IN MM

;:Al- 35.0: Al - 182.5,A = 0.0INE TAPER

N1A . .

100

Fig. 5. Coupling loss in decibels for ele2 and raised cosine expandedbeam couplers. Parameters: a = 4 Am, a2 = 361 ,um, A = 0.0056.

IV. Conclusion

I have developed the general theory for the ABCDmatrix in graded index tapers which show a large varia-tion in radius of the type which could be used forexpanded fiber-to-fiber coupling. The exact solutionfor linear tapers was given and a condition for specify-ing when tapers are adiabatic is derived. Results ob-tained for the propagation of the fundamental modethrough a single taper were compared with previouslypublished results5 obtained by a different method.The extension to calculating the coupling coefficient inan expanded beam connector is straightforward usingthis method.

The form of the ABCD matrix for linear tapers sug-gests a further refinement of the theory for slowlyvarying nonlinear tapers. An analytic solution for aclass of tapers with zero slope ends was derived. Theadjustable parameters are the initial and final radii,the radius at the point of inflection and taper length.The performance of some of these taper when used inexpanded beam connectors is calculated. It wasshown that the taper shape can affect the couplingefficiency but that sufficiently long tapers of any shapewill have low loss. The analytic solution also may beused for tapers with nonzero slopes at the ends. Awide range of taper shapes can be approximated by theE1e2 tapers without the need to use special func-

tions. 11 13

1 April 1989 / Vol. 28, No. 7 / APPLIED OPTICS 1303

nn

tog i

n

Long tapers have already been shownl-4 to be usefulfor fiber-to-fiber coupling and recently have been usedin laser-to-fiber coupling.16 In this application, thetapers need not have zero slope at the laser end. Thena knowledge of the ABCD matrix should be useful forthe analysis and design of such couplers.

This research was supported by the Natural Sci-ences & Engineering Research Council of Canada andthe University of Alberta.

Appendix: Equations for E 1 E 2 Tapers

Where e(u) is constant, we can show from Eqs. (37a)and (37b) that

d [In ( dM)] =-d ln, jl , (Al)

from which it follows that_42 = constant - constant (A2)

dM/du 1 + cos(2M)

Similarly, we find

2= constant constant (A3)dN/du 1 - cos(2N)

where N = u + v. Hence at the wide end of the taper(subscript I denotes the taper inflection point)

2= 2 - 2 cos(2M,) 1 + El 1 - 62 cos(2M,)

L I 1 - 62 cos(2 ML) 1 + el cos(2M) 1 - E2 cos(2 ML)

(A4)

since the initial condition at the narrow end is A 0 = 1.Solutions of Eq. (37b) for the two taper sections can beexpressed in terms of simple functionsl7:

7+ tan(MI) = tan(vl), (A5a)

+ tan(ML) = tan V2 + tan-1 1 tan(MI)I,v1 -62 12 1 + 2

where

V [1 - El, 2]"21 1,2 , (A6)

u1,2 = [2l2]-112 ln{[1 + (2el,2 tl,2 )2

]1/2 + 2(1,2t1,21 (A7)

are the amounts of u(t) defined by Eq. (31) within therange of the first and second taper sections, respective-ly. Using Eqs. (A4)-(A7) and considerable algebraiceffort, the following expressions for aL = A2 cos2 MLand -y2 = A2 sin2ML in terms of el, E2, ti, and t2 may befound:

a= [COS(V1 ) COS(V2) - R1R2 sin(vl) sin(v2 )J2 , (A8)

= [R cos(vl) sin(v 2) - R, sin(vl) cos(v2 )]] . (A9)

Here

R 2 - 1 + E,2 1/2R1,2 -l (A10)

and v1,2 were defined earlier. The solutions for L =

.L cos2NL and ,BL = .L sin2NL are found by replacing elwith -El and 2 with -E2.

References1. N. Amitay, H. M. Presby, F. V. DiMarcello, and X. T. Nelson,

"Optical Fiber Tapers-A Novel Approach to Self-AlignedBeam Expansion and Single-Mode Hardware," IEEE/OSA J.Lightwave Technol. LT-5, 70 (1987).

2. H. M. Presby, N. Amitay, F. V. DiMarcello, and K. T. Nelson,"Optical Fiber Tapers at 1.3 ,um for Self-Aligned Beam Expan-sion and Single-Mode Hardware," IEEE/OSA J. LightwaveTechnol. LT-5, 1123 (1987).

3. F. Martinez, G. Wylangowski, C. D. Hussey, and F. P. Payne,"Practical Single-Mode Fibre-Horn Beam Expander," Elec-tron. Lett. 24, 14 (1988).

4. H. M. Presby, N. Amitay, and A. Benner, "Straight-Tip OpticalFibre Up-Tapers for Single-Mode Hardware Applications,"Electron. Lett. 24, 34 (1988).

5. D. Marcuse, "Mode Conversion in Optical Fibers with Monoton-ically Increasing Core Radius," IEEE/OSA J. Lightwave Tech-nol. LT-5, 125 (1987).

6. C. P. Botham, "Theory of Tapering Single-Mode Optical Fibresby Controlled Core Diffusion," Electron. Lett. 24, 243 (1988).

7. J. S. Harper, C. P. Botham, and S. Hornung, "Tapers in Single-Mode Optical Fibre by Controlled Core Diffusion," Electron.Lett. 24, 244 (1988).

8. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, NewYork, 1985).

9. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984), pp. 128-129.

10. S. Yamamoto and T. Makimoto, "Equivalence Relations in aClass of Distributed Optical Systems-Lenslike Media," Proc.IEEE 34, 1254 (1971).

11. J. N. McMullin, "The ABCD Matrix in Arbitrarily TaperedQuadratic-Index Waveguides," Appl. Opt. 25, 2184 (1986).

12. M. S. Sodha and A. K. Ghatak, Inhomogeneous Optical Wave-guides (Plenum, New York, 1977).

13. L. W. Casperson, "Beam Propagation in Tapered Quadratic-Index Waveguides: Analytical Solutions," IEEE/OSA J.Lightwave Technol. LT-3, 264 (1985).

14. W. J. Stewart and J. D. Love, "Design Limitation on Tapers andCouplers in Single-Mode Fibres," in Technical Digest, FifthInternational Conference on Integrated Optics and OpticalFiber Communication-Eleventh European Conference on Op-tical Communication, Venice (1985), pp. 559-562.

15. D. Bertilone, A. Ankiewicz, and C. Pask, "Wave Propagation in aGraded-Index Taper," Appl. Opt. 26, 2213 (1987).

16. H. M. Presby, N. Amitay, R. Scotti, A. Benner, "SimplifiedLaser to Fibre Coupling via Optical Fibre Up-tapers," Electron.Lett. 24, 323 (1988)

17. H. B. Dwight, Tables of Integrals and Other MathematicalData (Macmillan, New York, 1965), p. 105.

1304 APPLIED OPTICS / Vol. 28, No. 7 / 1 April 1989