The shape of fiber tapers

432 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 10. NO. 4, APRIL 1992

The Shape of Fiber Tapers Timothy A. Birks and Youwei W. Li

Abstract-A model for the shape of optical fiber tapers, formed by stretching a fiber in a heat source of varying length, is presented. Simple assumptions avoid any need for the techniques of fluid mechanics. It is found that any decreasing shape of taper can be produced. The procedure for calculating the hot- zone length variation required to produce a given shape of taper is described, and is used to indicate how an optimal adiabatic taper can be made. A traveling burner tapering system is capable of realizing the model’s predictions, and a complete practical procedure for the formation of fiber tapers with any reasonable shape is thus presented.

UNSTRETCHED TAPER TAPER TAPER UNSTRETCHED FIBRE TRANSITION WAIST TRANSITION FIBRE

1. INTRODUCTION

ANY important optical fiber components, such as the M directional coupler [ l ] and the beam expander [2], are based on the single-mode fiber taper. A taper is made by stretching a heated fiber, forming a structure comprising a narrow stretched filament (the “taper waist”), each end of which is linked to unstretched fiber by a conical tapered section (the “taper transition”), as depicted in Fig. 1. Optically, the taper transitions transform the local fundamental mode from a core mode in the untapered fiber to a cladding mode in the taper waist, and this is the basis of many of its applications. However, if this transformation is to be accompanied by little loss of light from the fundamental mode, the shape of the taper transitions must be sufficiently gradual to satisfy a criterion for adiabaticity at every point [3], [4]. On the other hand, it is desirable for the transition to be as short as possible, allowing the resulting component to be compact and insensitive to environmental degradation. The theoretical shape of the shortest taper satisfying the adiabatic criterion for a particular fiber has been described by Love and Henry, and this was shown to be very much shorter than an equivalent sinusoidal taper [4]. There is, though, no indication of how such an optimal taper may be fabricated in practice.

The shape of a taper is also important in situations where the taper is to be controllably deformed; for example, by bending to produce miniature devices [SI or sensors [6], or by twisting to tune a coupler [7]. For low loss, these deformations need to be taken up as completely as possible by the narrowest parts of the tapered fiber, where the light is guided most strongly. Thus a taper with a long narrow waist but with short transitions is preferred. Different shape considerations apply if the taper is to be used as a sensor. Taper shape is also relevant to the behavior of taper beam expanders [2]. The particular shape of the taper is therefore of great importance.

Manuscript received May 15. 1991; revised Augwt 20, 1991. This work

The authors are with the Lightwave Technology Research Centre. Univcr-

IEEE Log Number Y 105HYO.

wa5 supported by Sumicem Optoelectronic\ (Ireland) Limited.

sity of Limerick, Plassey Technological Park. I.inierick, Ireland.

Fig. 1 . The structure of a fiber taper, indicating the terminology used in the text.

However, to the authors’ knowledge, no complete description of how the shape of a taper can be controlled has yet been published. The optical performance of taper components has been modeled by assuming parabolic, sinusoidal, polynomial or other taper profiles [3], [4], [SI, [9]. These taper shapes have been deduced by approximating the measured shapes of real tapers, and not by analysis of the processes by which the taper was formed. The problem of finding what particular taper profile results from stretching a fiber in a particular heat source, with its own temperature distribution, is a more or less complicated problem in fluid mechanics. It has been tackled by Dewynne et al., who were more interested in the effect of fiber perturbations on the resultant taper [lo]. However, they do describe a simple tapering model in which a fixed- length cylindrical section of fiber is heated to a uniform temperature and stretched. A similar model is also described by Eisenmann and Weidel [ l l ] and in both cases an exponential taper profile is predicted, the dimensions of which depend only on the hot-zone length and the taper extension. Significantly, no knowledge of fluid mechanics, beyond elementary mass conservation, is required to obtain this result.

The simple analysis is extended here to include the case where the length of the heated region is allowed to vary as tapering proceeds. This leads, for the first time, to a detailed procedure by which any shape of taper with any length of waist can be formed. The idea that the fiber section being heated is always cylindrical and is always heated to a uniform temperature (and hence is given a uniform viscosity) is retained, so that the techniques of fluid mechanics are still not required. Expressions for the shape of the taper resulting from any given variation of hot-zone length are derived, as is the solution for the inverse problem in which the taper shape is given and the hot-zone variation which will produce this shape is to be found. This yields a general recipe for the

0733-X724/92$03.00 0 1992 IEEE

BIRKS AND LI: SHAPE OF FIBER TAPERS 433

fabrication of any reasonably shaped taper, and the conditions required for the fabrication of an optimal adiabatic taper in a standard single-mode fiber are described by way of example. Finally, the applicability of the model for predicting the shape of tapers made in real fabrication systems is considered, and a practical fabrication procedure is described which matches the assumptions of the model. The production of tapers with any desired shape is thus made possible.

This paper is mainly theoretical in nature and is concerned only with the simplified analysis of the mechanical process of taper formation, a process which has up to now been poorly understood. An experimental study of the optical properties of real tapers produced with the aid of this analysis is of interest, but can only proceed if the procedure for controlling taper shape has first been demonstrated. Thus the optical properties of tapers are not considered here.

11. DESCRIPTION O F THE MODEL

A. Terminology

Fig. 1 illustrates the quantities used to describe the shape of a complete fiber taper. It is assumed that the taper is formed symmetrically (i.e., the ends of the taper are pulled apart at equal and opposite speeds relative to the center of the heat source) so that the two taper transitions are identical, although the analysis can readily be extended to cases where the pulling speeds are different. The radius of the untapered fiber is r0, and the uniform taper waist has length 1 , (which may be zero) and radius T,. Each identical taper transition has a length z , and a shape described by a decreasing local radius function ~ ( z ) , where z is a longitudinal coordinate. The origin of z at the beginning of each taper transition (point P for the representative left-hand transition in Fig. 1); hence ~ ( 0 ) = T , and ~ ( 2 , ) = T,. The taper extension II: is the net distance through which the taper has been stretched-it is equal to the present distance PQ minus the distance PQ before tapering commenced. Typically, the variation of .7: with time t is determined directly by the relative velocity of the two translation stages which pull the ends of the taper apart during its fabrication. It is assumed that .7: is an increasing function of t; taper compression is not considered. All of these quantities can apply to the finished taper, and also to the instantaneous states of the taper as it is being elongated. The final extension when the taper is finished and elongation ceases is denoted by xo.

B. The Model

Referring to Fig. 2, at any instant t during taper elongation, a symmetrically placed length L of the taper waist (the “hot- zone” AB) is uniformly heated and is a deformable cylinder of low viscosity glass. The particular temperature and viscosity values are unimportant, though the hot glass is assumed always to be soft enough to be stretched, while not being so soft that the taper sags under its own weight. Outside the hot-zone the glass is cold and solid. The ends of the taper are steadily pulled apart, so that at time t + St the hot glass cylinder stretches to form a narrower cylinder AB of length L + Sx, where Sx is

f

Fig. 2. Schematic diagrams of (a) a cylindrical taper waist at time t , a part AB of which is uniformly heated, and (b ) the same glass elements at a slightly later time t + h t . AB has been stretched through a distance h.r to form a narrower cylindrical taper waist, a part A’B’ of which is still heated.

the increase in extension during the interval S t . The hot-zone length is changed to L + SL in the same time, where SL may be negative. As the taper is elongated, the extremities AA’ and B’B of the stretched heated cylinder leave the hot-zone and solidify, forming new elements of the taper transitions. The portion A’B’ of the taper waist that is still within the hot-zone remains deformable and will be further stretched and narrowed.

L is changed by the appropriate control of a heat source and may vary arbitrarily. However, for the above description of the model to be valid, the variation of L is subject to two constraints:

dL - < 1. dx -

The first constraint is obvious. The second ensures that the heated section of glass is always cylindrical, i.e., that the hot-zone does not overtake the retreating taper transitions.

It is clear that the instantaneous length 1, of the taper waist at time t is equal to the hot-zone length at that time

l,(t) = L( t ) . ( 2 )

The final waist length therefore equals the final hot-zone length (i.e., at the time when elongation stops). Besides this rather simple equation and the conditions on the variation of L given in inequalities (l), the analyses which follow from the model are based on two fundamental equations. The first arises from the conservation of mass (and hence of glass volume) in the stretched taper waist. Thus the volume of the stretched glass

434 JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. 10. NO. 4. APRIL 1992

P Q --r r-

Fig. 3. (a) A fiber at time t = 0 at the start of tapering. A section PQ of length L,, is heated. (b) The fiber at time t during tapering. P and Q are further separated by a distance .r. The distance PQ is also equal to 23,, + L .

The validity of the volume law does not depend on whether the taper is formed symmetrically. On the other hand, the above statements of the distance law are specific to the case where the taper is formed symmetrically. Other modes of taper formation require their own versions of the distance law, with left and right z coordinates distinguished. Note that the time t does not appear explicitly in either law. Dependent quantities such as rtL and L can thus be expressed as functions of T ,

regardless of the rate of elongation or of any variations thereof, and the model predicts that the taper shape will not depend on the rate of elongation.

Mathematically, the relationship between the shape of the tapered fiber (expressed by r,, , l , , . z,, and r ( z ) ) and the elongation conditions which produce it (expressed by r , and L(.r)) is entirely determined by (2), (4), and (6), subject to the constraints (1) on the variation of L. The application of these equations depends on the problem at hand. Two particular situations are now considered. These are the “forward problem,” in which the elongation conditions are specified and the resultant taper shape is to be found, and the “reverse problem,” in which a particular taper shape is specified and the conditions to produce it are to be found.

111. THE FORWARD PROBLEM

L(.r) and .ro are given, and I , . r,,, z,. and r ( z ) are to be found. L ( s ) must satisfy the conditions (1). The length of the taper waist is given straight away from (2)

cylinder AB at time t + h t must equal the volume of the heated glass cylinder AB at time t

7r(rw + SruJ2(L + 61.) = 7rri,L (3)

where br,. is the change in the cylinder’s radius and is negative. In the limit S t + 0, this can be arranged to give a differential equation-the “volume law”-governing the variation of waist radius r,, with extension .I’

(4)

L may vary and may in general be regarded as being a function of z, since 2 is an increasing function of time t.

The second fundamental equation relates the instantaneous taper transition length z, to the taper extension 2:. From Fig. 3, comparing the total length PQ of the tapered fiber with the initial distance PQ at t = 0, we obtain the “distance law”

The variation of the waist radius ru; with .r is obtained by integrating the volume law (4), with the initial condition T,,(O) = T o .

r” 0

giving the general expression

(9) 22, + L = T + L,

where L can be a function of .r and Lo is its initial value, at x = 0. Now, according to the model, the local radius r ( z ) at a general point z along the taper transition is equal to the waist radius r,(z) as that point was pulled out of the hot- zone. The extension ~ ( z ) corresponding to this event is given by the distance law with z, = z

22 = x + Lo - L

where r in this expression is specifically the extension at which the point z was pulled out of the hot-zone. This is the generalized distance law. The solution x ( z ) of this equation depends on how L varies with 2. Thus the taper profile r ( z ) can be determined by substituting this ~ ( z ) into the ryI(.r) found from the volume law.

(5 1

(6)

Since L ( s ) is known, rw(.r) can be found, and the final waist radius is simply r = ( x , ) . The distance law (6) gives the taper transition length z as a function of x

(10) 1 2

z ( r ) = -[.r + L, - L ( x ) ] .

The final transition length z, is thus z ( . r , ) . To obtain ~ ( z ) , it is necessary to invert (10) to find . r (z) . This may be achieved analytically or numerically, depending on the particular functional form of L(.r). The taper profile function is then found by substituting this r ( r ) into (9) for r,(z)

.(.) = T u (.(.I). (1 1)

The complete taper shape has thus been found.


1 ) Example: Constant Hot-Zone: The simplest example 1 A illustrating the above procedure is for a constant value of

L ( x )

L ( x ) = Lo. (12)

Hence 1, = Lo, from (7). Equation (9) then gives a = 0.5

with the final waist radius being r,(x,). a=O U The distance law

(10) gives v A a = -0.5 z ( x ) = x / 2 (14)

so that z , = x,/2, and x as a function of z is simply x = 22. Substitution into (13) gives the taper profile function

v A

a = -1

r (z> = r,e-z/Lo (15)

a decaying-exponential profile. This is the result found else- where [lo], [11].

2) Example: Linear Hot-Zone Variation: As a further example, consider a hot-zone with a length which changes linearly with taper extension

L ( x ) = Lo + ax (16)

where a is a constant which determines the relative rates of hot-zone change and taper elongation. The conditions (1) require that a 5 1, and also that x , 5 Lo/(al if a is negative. With these requirements satisfied, (7) gives

1, = Lo + cyx,. (17)

Equation (9) becomes

dx' Lo + ax'

-1/2a

= r, [I+ 3 again with the final waist radius being r,(a,). The distance law (10) gives

z(x) = L(1- 2 cy)a (19)

so that z , = (1 - a)x,/2, and x can easily be expressed as a function of z. Hence the taper profile function is

Some specific cases with the linearly varying hot-zone are worth further consideration. If a = -0.5, with the hot-zone shrinking at half the rate of taper elongation, the resulting taper profile is linear

r ( z ) = r, 1 - - [ :?,I

Fig. 4. The calculated taper shapes resulting from a linearly varying hot-zone f.!.r) = Lo +os, for the same fiber radius r,,, waist radius rl,. = r , / i and initial hot-zone length L o , but for a range of values of cy. The c\ = 1 waist has length lGL, while the n = 0 waist has length Lo.

becomes the exponential profile (15) as required. For cy = +1, the limiting case satisfying (lb), the taper transitions have zero length (19) and there is an abrupt junction between untapered fiber and taper waist. Complete taper shapes are drawn in Fig. 4 for a range of values of cy, but corresponding to the same values of r, and Lo, and the same final waist radius of r, = r,/4. Given values of a and r,, a particular waist length is specified by (17). Note that the greater values of cy

give tapers with a very long waist region. To obtain the same taper profile with a different waist length, a more complicated L ( x ) is necessary, the determination of which is an example of the reverse problem.

Iv. THE REVERSE PROBLEM

In this case, the desired taper shape (defined by l,, r,, z,, and r ( z ) , where r(0) = r, and r ( z o ) = ruj) is given, and L ( a ) and xo are required. The solution of this problem is less compact than that of the forward problem. The volume law can be arranged to give

2L dx r dr - - - _ _

dxldr is unknown but, by differentiating the distance law (6), can be written as

dx dz dL - = 2 - + - dr dr dr

substitution into (22) gives a first-order linear differential equation for L as a function of r , with dzldr a known function since r ( z ) is given

dz dL 2 dr r - + - L + 2 - dr = 0.

This is a shape which, because of its simplicity, has received much theoretical attention [6]. If cy = +0.5 the taper transition has the shape of a reciprocal curve. In the limit as a -+ 0, (20)

With initial conditions r = r, and L = Lo for z = 0, this (21) equation integrates to

436 JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. IO, NO. 4, APRIL 1YY2

i.e.,

where L ( z ) is the hot-zone length as point z was pulled out of the hot-zone. Lo is as yet unknown, but can be determined by evaluating (25) at z = z , since L ( z , ) = ltL, is given. Hence (25) becomes

L ( z ) is now completely known, and so the distance law (6) gives x as a known function of z

x ( z ) = 22 + L ( z ) - Lo. (27)

This function is inverted to give ~ ( 3 ; ) . Again, this may be done analytically or numerically, depending on the functional form of r ( z ) . L ( x ) then follows from the distance law again

L ( x ) = x + L, - 24.). (28)

Finally, x, is found from the distance law with z = z ,

xo = 22, + I , - Lo.

Equations (28) and (29), solved using . ( :E) found from (27) and L ( z ) from (26), constitute the complete solution of the reverse problem.

(29)

A. Constraints on L

The constraints on L should be considered. Using (26), condition (la) requires that

Since the right-hand side is an increasing function of z and has a maximum value of zero at z = z,, the condition becomes

I , 2 0. (31)

Hence (la) is always satisfied, since it would be nonsensical to seek to fabricate a taper with a negative waist length.

Condition (lb) must also be considered. The distance law can be differentiated to give

d z = 1 - 2 - dL dx dx -

d z dr = 1 . - 2 - - dr d z

using the chain rule for derivatives. The volume law provides dr ldx, hence

r 1 - dL - - l + L - dX (33)

Since r and L are always positive quantities, ( lb) becomes

that is, that the local taper radius in the transition decreases with z . This is a condition which any “reasonable” required taper transition will satisfy. Hence both conditions (1) are automatically satisfied if the pre-specified taper shape is reasonable, and any such taper shape can be produced by suitably varying the hot-zone length with extension.

B. Optimal Adiabatic Tapers

As an example of the reverse problem, the fabrication of the shortest taper satisfying the adiabatic criterion is considered. Before the procedures outlined above can be carried out, the shape r ( z ) of the optimal taper is first calculated from the adiabatic criterion [3], [4]

(35)

where /-/I and 02 are respectively the local propagation constants, in the transition, of the fundamental (LPo,) mode and the mode to which power is most likely to be lost (the LP02 mode). Inequality (3.5) is converted into a differential equation by introducing a factor f , which specifies how much less the left-hand side of (35) should be than the right-hand side [4], and by writing Idrldzl = - d r / d z , since r ( z ) is a decreasing function. This yields

where /I1 and /j2 are calculated (numerically) as functions of r , since the local waveguide at z is determined by the local cladding radius r . With the initial condition r = ro at z = 0, the solution to this equation is

dr z ( r ) = -- (37)

T”

from which the required inverse function r ( z ) can be calculated numerically. For a standard fiber (core diameter 10 pm, cladding diameter 125 pm, cut-off wavelength 1200 nm, oper- ating wavelength 1300 nm), this optimal profile is drawn for f = 1 in Fig. 5. For different values of f , the shape is the same but with the z axis suitably rescaled. For example, if f = 0.5 then the z axis labels in Fig. 5 should be doubled. The value of f to choose depends on the amount of light loss which can be tolerated. As a guide, Love and Henry calculated the low loss of 0.03 dB for an optimal taper with f = 0.5, though this result applies strictly only to their particular fiber and wavelength [4]. The optimal taper profile will depend on the optical wavelength being considered via the propagation constants and 0 2 in (37).

Having found the optimal r ( z ) , the procedure for the solution of the reverse problem can be followed to find the hot-zone variation L(:r) which will give this taper profile. This solution will depend on what taper waist length l,,, is required. It will also depend on how much of the optimal profile is needed, i.e., on the value of r?,). For the f = 1 taper profile drawn in Fig. 5 with r,, = ro/4, the required hot-zone

dr - < o d z

iariations L ( L ) are drawn in Fig. 6 as functions of the taper extension x, for a range of final waist lengths lu,. Since I , (34)


80 I

I I I I I I 0.0 0.5 1.0 1.5 2.0 2.5

z (“1 Fig. 5. The optimal f = 1 adiabatic taper profile for 1300-nm operation in a standard single-mode fiber (cladding diameter 2r, 125 pm, core diameter 10 pm, cut-off wavelength 1200 nm). The profile is drawn from r = r , to r = r 0 f 4

~

OO 2 4 6 x (“1

Fig. 6. A selection of hot-zone variations L(.r) , all of which will form the optimal adiabatic taper profile drawn in Fig. 5 but with differing taper waist lengths.

equals the final hot-zone length L(zo) , in each case 1, is the value of L ( x ) at the rightmost end of the curve. For other values of f, the curves of Fig. 6 are retained but with both the z and L axes rescaled. For example, if f = 0.5 the labels on both axes should be doubled, a process which of course also doubles the I , values.

V. PRACTICAL ASPECTS

This paper has described a simple model for the dependence of taper shape on the tapering conditions, particularly the variation in hot-zone length as taper elongation proceeds. As a result, a procedure for the manufacture of any reasonable taper shape has been outlined. However, in order to follow this procedure in practice, a taper fabrication technique should be devised which matches the assumptions of the theory. Specifically, a heat source is needed which can heat the glass equally throughout a length L of fiber, without heating the fiber outside L at all. This is to ensure that the heated fiber section remains cylindrical and uniformly susceptible to stretching. It is also necessary that the heated length L be known, controllable, and variable in a prescribed way as tapering proceeds.

Fig. 7. A schematic diagram of a traveling-burner tapering system which imitates the assumptions of the model. While translation stages TS slowly elongate the fiber, a small burner B oscillates with constant speed between limit switches LS, a distance L apart. L can be controllably changed during taper elongation simply by moving the limit switches.

Conventional tapering heat sources, such as large fixed gas burners and electric furnaces, are not suitable. Not only are these difficult to change in length controllably, but they also heat different parts of the fiber to different temperatures. Consequently, the theory cannot be expected to match the shapes of tapers made with such heat sources. Tapers we have made using a large fixed-length gas burner have profiles which do not match the theoretical exponential shape, although Eisenmann and Weidel did find that their tapers matched the predicted exponential shape quite closely [ 113.

However, there is a practical tapering process which does mimic the assumptions of the model; this is based on the “flame brush” technique [12] and is depicted in Fig. 7. The point-like heat source is a gas burner which heats only a very small section of fiber at any particular time. This burner is made to travel with constant speed in an oscillatory manner along a distance L, so that in each cycle of oscillation every element in the length L of fiber is heated identically. If the burner’s speed of travel is large compared to the ‘speed of taper elongation dx/dt, then a time-averaged hot-zone is set up in the fiber which effectively satisfies the assumptions of the model. The effective hot-zone length will clearly equal the burner’s distance of travel, so that L is known, controllable, and insensitive to the type of burner used, and can easily be varied in real time in any desired manner. Complete control of the L ( z ) of the model is thus possible simply by changing the travel of the burner as elongation proceeds.

Such a fabrication system has recently been constructed and, for the simple variations L ( z ) so far attempted, indeed produces tapers with shapes corresponding to the predictions of the theory [13]. As an example, Fig. 8 is a graph of the longitudinal variation of the local radius of a real taper produced with such a heat source. L was varied linearly as described by (16) with (I! = -0.5, for which the model predicts the linear taper profile of (21). This L(z ) was chosen because it is particularly simple to implement, and it is also easy to check whether the real profile corresponds to the predicted straight line. The profile in Fig. 8 is indeed linear, and the dimensions of the profile closely matched those predicted. Hence, for the first time, a model is presented which provides a procedure for the formation of any reasonable taper shape, together with the practical means for the fabrication of the taper in accordance with this theoretical procedure.

438

1 1 -

1

0 9

0 8

local 0 7

radius 0 6 -

0 5

0 4

0 3

0 2

rlr,

-..... ....

...........

- .... -

-

- -

~

-

IOURNAL OF 1.lGHTWAVE TFCHNOLOGY. VOL. IO. NO 4. APRIL 1402

0 1

0 0 4 8 12 16 20 24 28 32 36

distance along taper (mm)

Fig. 8. The profile of a real taper fabricated using the variable flame brush method. L was varied according to (16) with o = - ( I .> , from an initial value of 15 mm to a final value of 5 mm. The taper elongation and flame traversal rates were 6.4 mm/min and 2 mm/s, respectively, and the flame was approximately 2 mm wide. The initial fiber diameter ? I , , , was 125 p m .

VI. CONCLUSIONS

A model for the shape of fiber tapers is presented, in which it is assumed that the fiber section being heated is always cylindrical and is always heated uniformly. This side-steps any detailed fluid mechanical analysis. By generalizing this previously considered model by allowing the hot-zone length to change as taper elongation proceeds, any shape of fiber taper consisting of two identical decreasing taper transitions linked by a uniform taper waist (of any length, including zero) can be produced. The recipe for the calculation of the hot-zone variation required to produce any such taper shape is described. Although the assumptions of the model are inapplicable to most conventional tapering systems, except in a limited way, a traveling burner tapering system is described which does closely follow the model’s assumptions. Early experimental results have confirmed this model’s correspondence with the shapes of tapers produced with this system. A complete procedure for the formation of fiber tapers of any reasonable shape is thus presented.

ACKNOWLEDGMENT

The authors would like to thank Prof. C.D. Hussey, R.P . Kenny, and K.P. Oakley for useful discussions and for their traveling burner results.

REFERENCES

B. S. Kawasaki. K. 0. Hill, and R. G. Lamont, “Biconical-taper single- mode fiber coupler,” O p t . Le(/.. vol. 6. pp. 327-328. 1981. K. P. Jedrzcjewski. F. Martinez, J. D. Minclly, C. D. Hussey. and F. P. Paync, “Tapered-beam cxpander for single-mode optical-fibre gap dc- vices,” Electrori. Lcztt. vol. 22, pp. 105 - 106. 1986. W. J. Stewart and J. D. Lovc. “Design limitation on tapers and couplers in \ingle mode fibres.” in Proc. ECOC‘ ‘85 (Venice), 1985, pp. 55Y-562. J . D. Love and W. M . Ilenry, “Quantifying loss minimisation in single- mode fibre tapers,“ Elecrrori. Lctr., vol. 22. 1986, pp. 912-914. C. Caspar and E. J . Bachus, “Fibre-optic micro-ring-resonator with 2 m m diameter.” Elcctrori. Lett.. vol. 25, pp. 1506-1508, 1989. L. C. Bohb. P. M . Shankar, and H. D. Krumholtz, “Bending effects in hiconically tapered single-mode fibers.“ J . Lrg/zrwcrw Teclmol., vol. 8, 1090, pp. 1084- 1090. T. A. Birks. “Twist-induced tuning in tapered fiber couplers,” Appl . Opt.,

J . Bures. S. Licroix, and J. Lapierre, “Analyse d’un coupleur hidirec- tionnel a fibres optiques monomodes fusionnees,” Appl. Opt.. vol. 22,

W. K. Burns. M. Abebe. and C. A. Villarruel. “Parabolic model for shape of fiber taper,” Appl. Opt., vol. 24, pp. 2753-2755. 1985. J. Dewynne, J . R. Ockendon. and P. Wilmott. “On a mathematical model for fiber tapering,“ SIAM J . Appl. Murhernutics, vol. 4Y, pp. 983-990, IYXY. M . Eisenmann and E. Weidel. “Single-mode fused hiconical couplers for wavelength division multiplexing with channel spacing between 100 and 300 nm,”J . Lightwove Zdzrrd., vol. 6. pp. 113- 119, 1988. F. Bilodeau. K. 0. Hill, S. Faucher, and D. C. Johnson, “Low-loss highly overcoupled fused couplers: Fabrication and sensitivity to external pressure,” J . Lighrwavc Techno/.. vol. 6, pp. 1476- 1482, 1988. R. P. Kenny, T. A. Birks, and K. P. Oakley, “Control of optical fibre taper shape.“ vol. 27. pp. 1654- 16.56, Electrori. Lett., 1991.

vol. 28. pp. 4226-4233, 1989.

pp. 1018- 1Y22, 1983.

TimothyA. Birks joined the Optical Fibre Group at the Universlty of Southampton as a research student in 1986, after receiving the B.A. degree in physics at Brasenose College, Oxford. He became Senior Reseach Fellow at the Lightwave Technology Research Centre, University of Limerick in 1Y89, where in I Y O O he was awarded the Ph.D. degree for work carried out at both universities. His main research interests are passive fiber components, particularly fused tapered couplers. and waveguide theory.

Dr. Birks is a mcmber of the Optical Society of America.

Youwei W. Li was born in Kaifeng City in Henan Province, People‘s Republic of China. He received the BSc . degree in physics from Henan Shifan University in 1982 and a postgraduate diploma from Beijing University in 1986. From 1986 lo 198Y he was a lecturer at Zhengzhou Institute of Technology. He joined the Lightwave Technlogy Research Center, University of Limerick, in 1989, and received the M.Eng. degree in 1990. He is now pursuing the Ph.D. degree in fiber optics.

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The shape of fiber tapers