Rib waveguide optimized for low loss coupling to optical fibers and method of determining same

A raised-rib waveguide provides a low-loss coupling to a conventional single-mode optical fiber propagating light of a wavelength (.lambda.). An exemplary embodiment of the waveguide is disclosed for a wavelength of 0.85 .mu.m. In order to identify the values of the structural and compositional parameters of the raised-rib waveguide, a method employing a computer is used to identify a set of parameters that optimize the efficiency of the coupling of the waveguide to an optical fiber while maintaining the waveguide's ability to propagate a selected mode.

FIELD OF THE INVENTION 
This invention relates generally to planar optical waveguide devices for 
use in optical communication systems, and more particularly to the design 
of such waveguides with low-loss coupling to standard optical fibers. 
BACKGROUND OF THE INVENTION 
Planar optical waveguides are typically used in optical communications to 
implement devices such as directional coupler switches, phase modulators 
and interferometric amplitude modulators. In such applications, planar 
waveguides are typically coupled to optical fibers at their input and 
output facets. Among the potential significant sources of power losses are 
those deriving from a mismatch between the fundamental modes of the planar 
waveguide and the optical fibers connected at these facets. Without 
special care, power losses at each facet can be very high, as much as 75 
percent or greater. 
In order to achieve low coupling losses between an optical waveguide and a 
fiber, the distribution of electromagnetic radiation at the facets of the 
waveguide should be roughly equivalent to the distribution provided by the 
optical fiber coupled at the facet. It has proven extremely difficult, 
however, to manufacture planar waveguide devices in which the distribution 
of electromagnetic radiation is roughly equivalent to that of the optical 
fibers commonly in use today. In particular, planar optical waveguides are 
made up of layers so as to have a rectangular geometry at their facets, 
whereas optical fibers are cylindrical in shape and have a circular or 
elliptical geometry at their facets. 
In an optical fiber, the usual arrangement is that guiding and confinement 
of the optical fields are produced by changes in the refractive index that 
are distributed in a circularly symmetric or elliptical manner with 
respect to the cross-section of the fiber. The majority of optical fiber 
now used in telecommunication systems, particularly in long-distance 
systems, is monomode with a core of higher refractive index of the order 
of 15 microns or less wide, and a cladding of lower refractive index whose 
outer diameter is of the order of 125 microns. These fibers are used to 
transmit radiation of a wavelength in the range of 0.8 to 1.65 microns, 
the radiation propagating along the fiber in a single transverse mode. The 
beam spot generally has dimensions in the range of 5 to 15 microns and the 
cross-section of the beam is circularly symmetric or elliptical as a 
result of the distribution of refractive index changes in the fiber. 
In contrast to the geometry of an optical fiber, a planar waveguide device 
is generally based on a slab of material in which changes in refractive 
index are more easily produced along flat interfaces rather than in curved 
distributions. For instance, a semiconductor planar waveguide device may 
be manufactured in the form of epitaxially grown layers of material on a 
substrate. Changes in refractive index can then most easily be produced in 
each of two perpendicular directions. First, changes can be produced at 
the interfaces between the layers of material by using material of 
different refractive indices. Second, changes in the perpendicular 
direction can be produced by making steps in the layers of materials, for 
instance by etching using a mask. The steps may then either be left 
exposed to air, which has a low refractive index compared to 
semi-conductor materials, or buried in suitable material of preselected 
refractive index. In general, these planar waveguides can be classified 
into a number of different basic types, including rib guides, strip-loaded 
guides, buried-channel guides and slab guides of various kinds. 
In order to provide a low-loss coupling between the optical fiber and 
planar waveguide, it is known to modify the modal shape at the output of 
the fiber in an attempt to match the shape to the modal shape of the 
waveguide. For example, the output of an optical fiber can be focused 
somewhat through tapers and spherical lenses, but control over its 
fundamental mode is generally limited. Most of the available options for 
tailoring the modal shape of the fiber involve changing the radial 
distance scale while leaving the field pattern essentially circular and 
thus still poorly matched to the elliptically shaped modes typically 
associated with waveguides. Although it may be possible to alter the modal 
shapes of the fibers by using special lenses, the small sizes involved 
create experimental and production difficulties--e.g., alignment of 
special lenses used to interface the waveguide. 
Some success has been achieved in altering the shape of the waveguide modes 
for a certain class of waveguides for the purpose of matching the modal 
shape of the waveguide to an optical fiber. In U.S. Pat. No. 4,776,655 to 
Robertson et al., a type of rib waveguide is disclosed that has values for 
its compositional and structural parameters that provide a modal shape 
which is approximately matched to the modal shape of a mating optical 
fiber. 
For a rib waveguide of the type illustrated by Robertson et al., the 
guiding zone is well defined in the lateral direction by the presence of a 
material of lower refractive index, typically air, on either side of the 
rib. Therefore, in a lateral direction, there is a large change in 
refractive index that provides strong optical confinement. According to 
the Robertson et al. patent, the modal shape in the waveguide can be 
adjusted to provide an elliptical shape that approximates the circular 
shape of the optical fiber by providing a small change in the refractive 
index between the core and cladding on the order of 0.01 to 0.0001. This 
range of differences in the refractive indices provides a measure of 
control of the modal shape in the direction perpendicular to the layers of 
the waveguide. The Robertson et al. patent also provides ranges of values 
for the structure of the rib with respect to its height and width to 
further sculpt the shape of the modal structure propagated by the 
waveguide. 
In contrast to the rib guide described in the Robertson et al. patent, 
which laterally confines a light beam by means of etching the rib into the 
guiding layer, a raised-rib waveguide is a less common guide that provides 
a rib etched into an upper cladding layer grown over the guiding layer. 
Both types of waveguides utilize the rib to laterally confine the light 
beam as it propagates along the guiding layer. Conventional rib waveguides 
are the most common and have been studied to various degrees. A raised-rib 
waveguide, however, has been examined in less detail and is less 
widespread. 
A raised-rib waveguide shares many of the functional characteristics of the 
conventional rib waveguide, but it offers some advantages that make it an 
attractive alternative to the conventional rib guide. In both the 
conventional and raised-rib guide, some light is inevitably lost by 
scattering due to roughness at the air-guide interface. Such roughness 
develops when the upper layer is partially etched away to create the rib. 
By including an upper cladding, however, the raised-rib guide provides a 
dielectric buffer between the air and the core, thereby reducing the modal 
fields at the air-waveguide interface and also reducing the scattering at 
the surface of the waveguide. At the same time, the weaker fields at the 
surface of the raised-rib guide also limit to some extent the ability of 
the rib to confine the modes laterally. 
In a raised-rib waveguide, the relationship between the values of each of 
the structural and compositional parameters and the shape of the modal 
structure is complex since they strongly affect both lateral and 
perpendicular confinement. In the rib waveguide of the Robertson et al. 
patent, the lateral confinement of the modal structure is well defined 
because the guiding region is bounded, laterally, by material of lower 
refractive index. In the raised-rib waveguide, lateral confinement is 
relatively weak because the guiding region is not bounded laterally by 
material of low refractive index and changes such as those suggested in 
the Robertson et al. patent in order to shape the modal structure provided 
by the waveguide have different effects in both the perpendicular and 
lateral dimensions of the guide. 
SUMMARY OF THE INVENTION 
It is a primary object of the present invention to provide a raised-rib 
waveguide having highly efficient coupling between an input or output 
facet of the waveguide and a standard optical fiber, which can be 
successfully implemented in a production environment. In this connection, 
it is also an object of the invention to provide such a highly efficient 
coupling without requiring substantial re-shaping of the mode at the end 
of the fiber. 
In order to achieve the foregoing objects and others, the invention 
provides a raised-rib waveguide whose modal structure approximates the 
circular symmetry and size of the modal structure provided by an optical 
fiber coupled to the waveguide at an input or output facet, thereby 
providing a low-loss coupling between the two. The relationship between 
the structural and compositional parameters of a raised-rib waveguide and 
its modal structure is complex and not susceptible to an analytical 
solution. Therefore, the invention utilizes the computing power of a large 
computer to approximate numerical solutions for the purpose of defining a 
relation between each of the structural and compositional parameters of 
the waveguide and its modal shape. With these relationships defined, a set 
of values for the parameters can be identified that optimize the coupling 
to conventional optical fibers while maintaining the waveguide's ability 
to propagate light in the selected mode. Once this set of values have been 
identified, conventional epitaxial growth techniques and lithographic 
techniques can be employed in order to create the desired waveguide. 
In order to identify the set of values for the compositional and structural 
parameters that best approximate a modal structure for the waveguide that 
matches to the modal structure of the optical fiber, an initial set of 
values are identified that are known to provide guided wave propagation at 
the selected wavelength. From this starting point, the effect of each 
parameter on the shape of the modal structure is investigated by varying 
the selected parameter over a range of values while the other parameters 
are held constant. Some parameters are found to have a strong effect on 
the shape of the modal structure, while others appear to have only a 
relatively weak effect. When varying the values of each parameter, an 
optimum value is identified that tailors the shape of the modal structure 
most closely to a circular one while at the same time keeping the 
waveguide above cutoff. 
Using established mathematical methods, the shape of the modal structure 
can be determined for each new set of values of the parameters created by 
varying the values of the parameters in turn. By simply selecting values 
for the variable parameter while the values of the other parameters are 
held constant and observing a trend in the shape of the modal structure as 
the value is changed, the optimum value can be identified for maximizing 
the coupling efficiency between an optical fiber and the waveguide. By 
repeating this procedure for each of the structural and compositional 
parameters (i.e., varying one parameter while holding the others constant) 
an optimum set of values for the parameters can be identified that will 
define a waveguide having an approximately maximized coupling efficiency 
with an optical fiber at a selected wavelength (.lambda.). 
In the preferred approach to identifying the set of values for the 
structural and compositional parameters of the waveguide, parameters are 
selected as variables in a sequence beginning with the parameter that most 
strongly affects the modal structure and continuing with the next 
parameter in an order of decreasing effect on the shape of the modal 
structure. After the optimum value for a parameter has been identified, 
the value is substituted for the original value in the set of values and 
the new set is used as the set for varying the value of the next 
parameter. For example, the set of values used to identify the optimum 
value of the second most sensitive parameter includes a value of the most 
sensitive parameter that has been previously identified as optimizing the 
coupling. At the end of the sequence, a set of values for the structural 
and compositional parameters is identified that provides an excellent 
low-loss coupling between the waveguide and the optical fiber. 
In the process of sequencing through each of the parameters, it may be that 
a slight adjustment of a previously determined optimum value for a 
parameter will allow a subsequently selected parameter to be varied over a 
wider range of values and, thereby, achieve a greater degree of coupling 
efficiency. Typically, this may occur if a parameter is at its optimum 
value very close to or at the cutoff of the desired mode.

DETAILED DESCRIPTION OF THE PRESENT INVENTION 
Turning now to the drawings and referring first to FIG. 1a, a raised-rib 
guide 21 has an upper layer 23 formed by low-index cladding just above a 
film core 25. A light beam is confined laterally within the guide 21 by 
means of a rib 27 etched into the upper layer 23 of the guide 21. For 
convenience, this structure etched from the upper layer 23 shall be 
hereinafter referred to as a "raised rib." In more conventional rib 
waveguides, there is no upper layer of cladding over the core 25. In order 
to provide a rib guide, the core itself is etched to form a rib. 
The raised-rib guide 21 of FIG. 1a is specified by five structural and 
three compositional parameters. In FIG. 1a, these parameters are the 
heights (h.sub.f), (h.sub.lc), (h.sub.uc) and (h.sub.r) of the core 25, 
lower and upper cladding 29 and 23 and the rib 27, respectively, the width 
(w) of the rib and the refractive indices (n.sub.f), (n.sub.lc) and 
(n.sub.uc) of the core 25 and lower and upper cladding 29 and 23, 
respectively. The values of these parameters can be controlled during the 
process of fabricating the waveguide 21. By appropriate choices of these 
values, one can shape the cross-sections of the guided modes in various 
ways as well as manipulate the conditions for cutoff and for single-mode 
behavior. 
For many optical applications, it is useful to introduce into the core one 
or more thin layers of material with refractive index greater than 
n.sub.f. Called quantum wells, these additional layers allow for further 
electro-optic control. By way of illustration, the dashed line 30 in FIG. 
1A shows an exemplary quantum well in the core 25. In the event that the 
core contains a number q of quantum wells, each of height (h.sub.w) and 
refractive index (n.sub.w) we use as a model a single equivalent bulk 
layer with refractive index (n.sub.f ') given approximately by 
EQU n.sub.f '.sup.2 =[q h.sub.w n.sub.w.sup.2 +(h.sub.f -q h.sub.w) 
n.sub.f.sup.2 ]/h.sub.f (1) 
for a core of total height (h.sub.f). Henceforth, we will understand by 
"refractive index of the core" either the refractive index of the 
homogeneous material constituting the core or the index (n.sub.f ') 
defined above for the case of a core containing quantum wells. We will use 
the symbol (n.sub.f) without the prime for either case. 
Designing a waveguide for optimum coupling with a fiber requires both a 
qualitative grasp of the modal characteristics of the guide 21 and a 
quantitative analysis of the coupling efficiency. In keeping with the 
invention, iterative numerical computations generate a data base that 
leads to a qualitative understanding of how the different parameters 
affect the modal shape. From such an understanding, both lateral 
confinement and coupling efficiency can be optimized in a raised-rib 
waveguide. 
In general, the best way to achieve this optimization is to relax the 
vertical confinement, generally by decreasing the height (h.sub.f) of the 
core 25 and reducing the difference in refractive indices (n.sub.f) and 
(n.sub.uc) between the core 25 and the upper cladding 23. Excessively 
strong vertical confinement reduces the penetration of the electromagnetic 
field in the upper cladding 23 and the rib 27. If the field emerging from 
the core 25 is too weak, then the structure begins to resemble a slab 
guide; since the field barely reaches the rib 27, the rib itself can have 
little effect on guiding the wave and hence the field spreads laterally 
into a long and narrow mode. By permitting somewhat more field to emerge 
from the core 25, however, one can significantly enhance the effect of the 
rib 27 and, at the same time, reshape the mode so that it becomes 
approximately circular. With simultaneous adjustment of the height 
(h.sub.r) of the rib, its width (w) and its distance (h.sub.uc) from the 
core 25, the mode can be tightened laterally to the point where it extends 
roughly equally in both vertical and horizontal dimensions. 
As will be shown, for a particular wavelength (.lambda.) there exists a 
range of values for the parameters over which the coupling between the 
waveguide and the fiber should be highly satisfactory, with losses between 
-0.3 and -1.0 dB for both -5-.mu.m and -9-.mu.m fibers. For example, at 
wavelengths near 0.85 .mu.m, best results are achieved for the core 
ranging in height (h.sub.f) from 0.3 to 0.5 .mu.m, core-cladding index 
differences (n.sub.f)-(n.sub.uc) between 0.01 and 0.02 (approximately 2% 
to 3% difference in Al composition in AlGaAs), (n.sub.uc)=(n.sub.lc), ribs 
3 to 5 .mu.m wide (w) and 1 to 3 .mu.m high (h.sub.r) with a height of the 
upper cladding of approximately 0.4 .mu.m (h.sub.uc). 
I. COMPUTATION OF MODES OF THE FIBER AND THE RAISED-RIB WAVEGUIDE 
Referring to FIG. 1b, the guided modes of a circular fiber 31 butt coupled 
to the waveguide and those of the waveguide itself may be computed by any 
of several numerical methods. For the modes of the optical fiber 31, the 
Fourier-Grid method is particularly convenient. (see "Application of the 
Fourier-Grid Method to Guided-Wave Problems," M. Munowitz and D. J. 
Vezzetti, Journal of Lightwave Technology, Vol. 8, No. 6, p. 889, Jun. 
1990.) For the raised-rib waveguide 21, however, the Fourier-series method 
outlined by Henry and Verbeek (C. H. Henry and B. H. Verbeek, "Solution of 
the Scalar Wave Equation for Arbitrarily Shaped Dielectric Waveguides by 
Two-Dimensional Fourier Analysis," J. Lightwave Tech., vol. LT-7, pp. 
308-311, 1989), provides a convenient way to compute the shape of the 
discrete modes. Both the Munowitz et al. and Henry et al. publications are 
hereby incorporated by reference. 
According to the Fourier-series method, one assumes that the modes of the 
full-vector problem (particularly the transverse electric modes, with the 
electric vector largely parallel to the major interfaces of the structure) 
are well approximated by the solutions of the scalar wave equation. The 
guided modes then are of the following form: 
EQU u(x,y,z)=E(x,y) exp(i.beta.z), (2) 
EQU where 
EQU .beta.=N.sub.m k.sub.0 (3) 
with k.sub.0 being the vacuum wavenumber and (N.sub.m) the modal refractive 
index. Inserting these expressions into the scalar wave equation provides 
the following: 
EQU [d.sup.2 /dx.sup.2 +d.sup.2 /dy.sup.2 +k.sub.0.sup.2 (n(x,y).sup.2 
-N.sub.m.sup.2)] E(x,y)=0 (4) 
We assume that the field is zero on the boundaries of a large rectangular 
domain 
EQU 0.ltoreq.x.ltoreq.L.sub.x (5) 
EQU 0.ltoreq.y.ltoreq.L.sub.y (6) 
completely surrounding the guide, where L.sub.x and L.sub.y are the lengths 
of the domain along the x and y axes, respectively. The mode is then 
expanded in a complete set of functions as follows: 
##EQU1## 
Substituting equations (7) and (8) into equation (4) and integrating over 
the rectangular domain yields the following matrix eigenvalue equation: 
##EQU2## 
where A is a matrix given by 
EQU A.sub.l'm',lm =[(l.pi./L.sub.x).sup.2 +(m.pi./L.sub.y).sup.2 
].delta..sub.ll' .delta.mm'-k.sub.0.sup.2 .phi..sub.l'm' .phi..sub.lm 
n(x,y).sup.2 dxdy. (10) 
The solution of equation (9) gives both the modal refractive index N.sub.m 
of the guided mode and the expansion coefficients, a.sub.lm, from which 
the mode amplitude may be constructed via equation (7). 
Several practical matters must be addressed before one can apply equation 
(9) to the solution of a particular waveguide. For use in a computer 
program, the equation must be simplified by appropriate approximations so 
that numerical results can be obtained. First, the infinite sums in 
equations (7) and (9) are truncated at l.sub.max and m.sub.max, giving a 
matrix A of order l.sub.max .times.m.sub.max by l.sub.max .times.m.sub.max 
that can be evaluated using a properly programmed computer. The basis must 
be chosen large enough so that the shape of the modes can be reproduced 
with sufficient accuracy, but also small enough for the matrix eigenvalue 
problem to be solved in reasonable time with available computer memory. 
Using a Cray-2 computer system, for example, it has been found that 
typical CPU times for evaluating the terms in equation (10) (as described 
below) and solving the eigenvalue problem are: 
______________________________________ 
l.sub.max 
m.sub.max size of A CPU sec 
______________________________________ 
30 40 1200 .times. 1200 
22 
40 40 1600 .times. 1600 
45 
50 50 2500 .times. 2500 
170 
______________________________________ 
A second computational issue concerns the numerical representation of the 
refractive index profile n(x,y), which is most easily approximated by a 
number of rectangles wherein the index is constant. Taking n(x,y) as 
piece-wise constant in this way allows the integrals in equation (10) to 
be computed analytically and the results inserted directly into the code 
of the computer program. Other simple shapes for which the integrals can 
be computed analytically may also be used. The alternative, numerical 
integration, generally is impractical because of the large number of terms 
involved. 
A final consideration is the presence of additional solutions arising from 
the imposition of boundary conditions over the finite domain. These 
solutions, although legitimate for the finite model, are not present in 
the real waveguide since in that case both the lateral dimensions and the 
air layer above are virtually infinite. These extraneous modes may cause 
confusion if not understood properly, and one must separate them carefully 
from the bound modes according to the procedure given in "Analysis of 
Finite Rib Waveguides by Matrix Methods", D. J. Vezzetti and M. Munowitz, 
Journal of Lightwave Technology. Vol. 8, No. 8, p. 1228, Aug. 1990. 
In the foregoing approach, the refractive index of a waveguide is specified 
on a grid of 55 by 55 points covering a space of 20 .mu.m by 20 .mu.m. For 
the optical fiber, the dimensions and indices used are 
TABLE I 
______________________________________ 
Dimensions Index 
Core Clad Core Clad Core Clad 
(.mu.m) (.mu.m) 
(0.8 .mu.m) 
(0.8 .mu.m) 
(1.3 .mu.m) 
(1.3 .mu.m) 
______________________________________ 
"9 mi- 8.3 125 1.4580 1.4528 1.4535 1.4483 
cron"* 
"5 mi- 5.0 125 1.4580 1.4528 1.4535 1.4483 
cron" 
______________________________________ 
*Corning SMF28 
These 9 and 5 micron fibers support single modes at light wavelengths 
(.lambda.) of 1.3 .mu.m and 0.85 .mu.m, respectively. Note, however, that 
because of circular symmetry a "single-mode fiber" actually has two 
degenerate modes with orthogonal polarizations. The 9-.mu.m core fiber, by 
contrast, supports more than one mode at wavelengths near 0.85 .mu.m, with 
approximate ranges of wavelengths as set forth in TABLE II below. 
TABLE II 
______________________________________ 
Number of modes 
Wavelength (.mu.m) 
______________________________________ 
1 &gt;1.3 
3 .85-1.3 
4 .84-.85 
5 .83-.84 
.gtoreq.6 &lt;.83 
______________________________________ 
The second and third scalar modes are degenerate in each instance where the 
fiber is multimode. 
Mesh plots showing the fields of the first and second modes of the 9-.mu.m 
fiber at a wavelength (.lambda.) equal to 0.85 .mu.m are illustrated in 
FIGS. 2a and 2b, respectively, while the contour maps in FIGS. 3a and 3b 
show the lines of constant field amplitude of the fundamental modes of 
both the 9-.mu.m and 5-.mu.m fibers, respectively, at 0.85 .mu.m. The 
contours are equally spaced in FIGS. 3a and 3b from minimum to maximum. 
Just the coupling of the fundamental mode of each fiber to the waveguide 
will be considered. Note that the fields of the two fibers differ mainly 
in extent but not in shape. A useful measure is the diameter at which the 
intensity of the mode falls to 1/e.sup.2 of its peak value, computed as 
7.8 .mu.m and 5.6 .mu.m for the 9 .mu.m and 5 .mu.m fiber, respectively. 
II. CALCULATING THE SHAPE OF THE MODES 
The foregoing mesh plots and contour maps of FIGS. 2 and 3 for the 5 and 
9-.mu.m fibers are derived using FORTRAN programs FIBFGH and EVEC, 
attached hereto as Appendices A and B, respectively. The program FIBFGH 
implements the Fourier-Grid method as described by Munowitz and Vezzetti 
in the previously identified publication in order to approximate the 
eigenvectors on a grid of 55 by 55 points. From these values, the modal 
amplitude for each point is determined and a commercially available 
program PC-MATLAB, published by The Math Works, Inc., of South Natick, 
Mass., generates the mesh plots and contour maps. 
As previously mentioned, for determining the modal shapes of raised-rib 
waveguides, the Fourier-series method of Henry and Verbeek is preferred. 
Attached as Appendix C is a FORTRAN program RIB2DIM that implements the 
Fourier-series method on a raised-rib waveguide in order to approximate 
the eigenvalues and expansion coefficients of the modal eigenfunctions. 
From these coefficients, the amplitude of a selected mode is found at each 
point of a 55.times.55 grid, using the FORTRAN program PAT55 attached as 
Appendix D. From the values found for the modal amplitudes at the grid 
points, the mesh plots and contour maps of FIGS. 4, 6, 8-9, 11-12 and 17 
are generated using the commercially available program PC-MATLAB. 
III. EVALUATING THE MODE OF THE WAVEGUIDE 
Guided modes of the raised-rib waveguide 21 are characterized by the number 
of oscillations of the field vertically (i.e., the growth direction, x, 
perpendicular to the major interfaces) and laterally (i.e., the lateral 
direction, y, parallel to the major interfaces). Mesh plots showing field 
amplitudes for two exemplary guided modes for the waveguide 21 are shown 
in FIGS. 4a-4b to illustrate the basic shape of the modes, and especially 
the perturbing effect of the rib 27. As always, the fundamental mode has 
just a single lobe in each of the vertical and lateral directions x and y. 
The second guided mode, if one exists, may have two lobes either in the 
vertical direction (x) or two in the lateral direction (y), depending on 
the precise structure of the guide 21. Such a mode shall be referred to as 
the second vertical or second lateral mode. 
A. Lateral Guidance 
Lateral guidance in the raised-rib waveguide 21 can be understood 
qualitatively by effective-index theory, according to which the 
two-dimensional profile n(x,y) in FIG. 1A is reduced to an approximate 
one-dimensional form as shown in FIG. 5. One considers, separately, each 
of the three regions I, II and III shown in FIG. 5, and computes the modal 
refractive indices of each region as if it were a four-layer slab guide 
made from lower cladding, core, upper cladding, and air. Knowing the 
effective indices of the three regions I, II and III, one then constructs 
a fictitious three-layer guide laterally, where the central layer (region 
II), with effective index n.sub.eff (II), is bounded by the two outer 
layers (regions I and III), having effective indices n.sub.eff (I) and 
n.sub.eff (III), respectively. If both n.sub.eff (I) and n.sub.eff (III) 
are less than n.sub.eff (II), then a guided mode can exist in region II. 
The lateral confinement becomes stronger as the difference in effective 
index increases, and especially so as the vertical mode in regions I and 
III is cut off. Even more than in a conventional rib waveguide, a change 
in the structural and compositional parameters of the core 25 
significantly affects both the lateral and vertical confinement of a 
raised-rib waveguide. Therefore, the relationship between the parameters 
of the raised-rib waveguide and the modal shape of the guide is more 
complex than the same relationship for a conventional rib waveguide. 
In FIG. 5, region II differs from the regions I and III on either side of 
it only by the extra dielectric material of the rib 27, giving region II 
in effect a thicker layer of the upper cladding 23. As a result, the 
effective index n.sub.eff (II) in region II generally exceeds the 
effective indices n.sub.eff (I) and n.sub.eff (III) in regions I and III. 
Thus the geometry of the raised rib 27 always allows for the possibility 
of a guided lateral mode; how many actually exist, and what shapes they 
take, however, is determined by the detailed interplay of the various 
structural and compositional parameters of the waveguide. 
The ability of the raised rib 27 to effect lateral confinement is 
determined in part by the vertical decay of the modal field. Two simple 
limiting cases help illustrate the important points. First, if the field 
reaching the rib 27 in FIG. 5 (after decaying in the upper cladding) is 
too weak, then the structure more closely resembles a slab guide and the 
relatively unperturbed mode thus is poorly confined laterally. This limit 
is approached when the decay length of the guided mode is small compared 
to the height (h.sub.uc) of the upper cladding 23. Hence lateral 
confinement worsens with (1) increasing difference in the refractive 
indices (n.sub.f) and (n.sub.uc) of the core 25 and cladding 23, 
respectively, (2) increasing the height (h.sub.f) of the core, and (3) 
increasing the height (h.sub.uc) of the upper cladding. 
Also, the perturbing effect of the rib 27 similarly decreases if the field 
over the rib is relatively uniform and nearly identical to that existing 
in the upper cladding 23 in regions I and III. Lateral confinement 
therefore is degraded when the rib 27 is too short relative to the decay 
length of the field. Moreover, the mode may be cut off under such 
conditions, especially when the waveguide 21 in the absence of the rib 27 
is unable to support a guided mode. Lateral confinement also is controlled 
by the width (w) of the raised rib 27 just as in conventional rib 
waveguides, increasing as the width (w) of the rib is increased to the 
point where additional lateral modes can be supported. 
B. The Effect Of Each Structural And Compositional Parameter On The Modal 
Structure 
These structural and compositional parameters of the waveguide clearly do 
not influence the modal structure independently. For example, the 
difference in refractive indices (n.sub.f) and (n.sub.uc) of the core 25 
and the cladding 23, the height (h.sub.f) of the core, and the value of 
the wavelength (.lambda.) all combine to determine the decay length of the 
mode. Additionally, the height (h.sub.r) of the rib 27 exerts different 
effects, depending on the height (h.sub.uc) of the upper cladding layer 
23. To achieve an analytical understanding of the effect on the modal 
structure of the values of the structural and compositional parameters of 
the waveguide, it is convenient to discuss each structural and 
compositional parameter separately, noting in certain instances how it may 
interact with others. 
In the following examples, values are assigned to each of the compositional 
and structural parameters. In each example, the value of one of the 
parameters is varied while the others are held constant in order to 
determine the sensitivity of the modal shape to the selected parameter. In 
all of the examples, the wavelength is taken to be 0.85 .mu.m, typical of 
that produced by GaAs/AlGaAs diode lasers. The material of the core 25 and 
upper and lower claddings 23 and 29 is assumed to be Al.sub.x Ga.sub.l-x 
As of different compositions, with the bulk refractive indices (n.sub.f), 
(n.sub.uc) and (n.sub.lc) given by the model described in D. W. Jenkins, 
"Optical Constants of Al.sub.x Ga.sub.l-x As," Journal of Applied 
Physics., Vol. 68, p. 1848, 1990. The indices (n.sub.f), (n.sub.uc) and 
(n.sub.lc) chosen for the examples run from 3.3601 (x=0.4) to 3.4165 
(x=0.3) at the wavelength of 0.85 .mu.m. 
i. The Height (h.sub.lc) of the Lower Cladding 
The height (h.sub.lc) of lower cladding must be enough to allow the modal 
amplitude to decay practically to zero at the point where the cladding 
meets the supporting substrate. If it is not, unwanted losses due to free 
carrier absorption in the substrate may result. The decay length in the 
lower cladding 29, which determines the height (h.sub.lc) necessary, 
depends on how close the guided mode is to cutoff. The penetration of the 
field into this layer of the lower cladding 29 thus depends on the 
difference in refractive indices (n.sub.f) and (n.sub.lc) between core 25 
and lower cladding 29, and on the closeness of the modal index to the 
index of the lower cladding. 
ii. The Refractive Index (n.sub.f) Of the Core 
A refractive index of the core (n.sub.f) or, more precisely, the difference 
in index (n.sub.f)-(n.sub.uc) between core 25 and upper cladding 23, 
affects both the lateral and vertical confinement of the guided modes. 
FIGS. 6a-6d show a typical sequence of fundamental modes as the index 
(n.sub.f) alone is increased, with the indices (n.sub.uc) and (n.sub.lc) 
of the upper and lower claddings 23 and 29, respectively, taken as being 
equal for simplicity. The outermost contour in each plot, and in the 
others to follow represents approximately 10% of the peak amplitude of the 
modal field (1% intensity). 
In FIGS. 6a-6d, the value of the refractive index (n.sub.f) of the core 25 
is varied from 3.3700 to 3.4165. All of the other parameters are held 
constant at the following values (in microns): (w)=3.0; (h.sub.r)=0.6; 
(h.sub.uc)=0.4; (h.sub.f)=0.5 and (n.sub.uc)=(n.sub.lc)=3.3601. The values 
of the refractive index (n.sub.f) of the core in FIGS. 6a-6d are as 
follows: FIG. 6a - 3.3700; FIG. 6b - 3.3850; FIG. 6c - 3.4000 and FIG. 6d 
- 3.4165. 
As in a conventional rib waveguide, increasing the difference between the 
indices (n.sub.f) and (n.sub.uc) increases the vertical confinement of the 
mode. The effect on the penetration of the field into the lower cladding 
29 is clear from the profiles displayed in FIGS. 6a-6d. As the mode 
becomes better confined vertically, its shortened decay in the upper 
cladding 23 eventually reduces the ability of the rib 27 to effect lateral 
confinement [FIGS. 6b, 6c and 6d]. 
In comparing FIGS. 6a and 6d, an apparently anomalous improvement is noted 
in lateral confinement with increasing difference in the indices 
(n.sub.f)-(n.sub.uc). Here, however, the mode in FIG. 6a is almost at its 
cutoff point, and the effective index (n.sub.eff) in region II is only 
3.3604 relative to the effective index (n.sub.eff) in region I and III of 
3.3601. For a slightly smaller value of (n.sub.f), this mode in fact 
extends to the computational boundaries and then ceases to be a guided 
mode altogether. Under these conditions, the field emerging from the core 
25 decays so slowly that the rib 27 is too short to support any mode. As 
the index (n.sub.f) of the core 25 increases, the decay is brought into a 
range where lateral confinement is optimized before the "slab" limit is 
approached and the rib 27 loses effectiveness. 
The non-monotonic dependence of lateral confinement on the refractive index 
(n.sub.f) of the core 25, illustrated further in FIG. 7, is an important 
feature of the structure, and shows as well that prediction of modal 
characteristics is not always straightforward in these waveguides. 
As lateral confinement improves, the guide becomes able to support a second 
(antisymmetric) lateral mode. Such a mode develops, for example, under a 
5-.mu.m wide rib 27 when the refractive index (n.sub.f) of the core 25 
exceeds the index (n.sub.uc) of the upper cladding 23 by approximately 
0.02, and its lateral confinement similarly passes through an optimum 
range as the difference in indices increases. Although the guide 21 
subsequently retains both modes for all values of the index (n.sub.f), 
each mode spreading more and more as the indices of the core and upper 
cladding diverge. It is observed, nevertheless, that the second mode 
apparently loses its local character more readily than the fundamental. As 
the index (n.sub.f) is increased still further, the core 25 eventually is 
able to support a second vertical mode as well, for which the lateral 
confinement again is determined by the same considerations. 
iii. The Height (h.sub.f) of the Core 
The height (h.sub.f) of the core 25 influences the number of allowed 
vertical modes and the vertical extent of the mode pattern. The effect on 
the modal shape caused by varying the value of the height (h.sub.f) of the 
core 25 is shown by the contours in FIGS. 8a-8b, in which all parameters 
are held fixed except (h.sub.f). In FIGS. 8a and 8b, the value of the 
height (h.sub.f) of the core 25 is 0.7 and 0.5 .mu.m, respectively, 
whereas the other parameters are valued as follows (in microns): (w)=5.0; 
(h.sub.r)=0.6; (h.sub.uc)=0.4; (n.sub.f)=3.3700 and 
(n.sub.uc)=(n.sub.lc)=3.3601. 
iv. The Height (h.sub.uc) of the Upper Cladding 
The height (h.sub.uc) of the upper cladding 23, which helps determine the 
field amplitude at both the cladding-air interface and the rib 27, 
directly affects the lateral confinement of the guided modes. As the 
height (h.sub.uc) of the upper cladding 23 increases, the rib 27 moves 
farther away from the core 25 and consequently is less able to influence 
the fields. Lateral confinement worsens as the height (h.sub.uc) of the 
upper cladding 23 is increased, and the modes become more like those of a 
slab waveguide. Conversely, decreasing the height (h.sub.uc) of the upper 
cladding 23 increases the lateral confinement and, depending on the other 
waveguide parameters, lateral modes of higher order may develop. A 
smaller, but still important, effect of increasing the height (h.sub.uc) 
is to allow the mode to spread vertically. 
FIGS. 9a-9c clearly illustrate these effects by a sequence of contours of 
the fundamental mode in which all waveguide parameters except the height 
(h.sub.uc) of the upper cladding 23 are held fixed. In each of the modal 
contours of FIGS. 9a-9c, the values of the height (h.sub.uc) of the upper 
cladding 23 is 0.2, 0.6 and 1.0 .mu.m, respectively. The values of the 
other parameters are held constant for all three contours and are as 
follows (in microns): (w)=5.0; (h.sub.r)=1.0; (h.sub.f)=0.3; 
(n.sub.f)=3.3775 and (n.sub.uc)=(n.sub.lc)=3.3601. Other parameters 
influence the field here as well, and thus interact strongly with the 
height (h.sub.uc) of the upper cladding 23 in establishing lateral 
guidance. 
The upper cladding 23 is all that distinguishes the raised-rib guide 21 
from a conventional strip-loaded guide. This additional layer 23 serves to 
remove the rib 27 to some distance from the core 25, so that the modes of 
the structure evolve smoothly from those of a strip-loaded guide when the 
layer is very thin to those of a slab guide when the layer 23 is very 
thick. The dispersion curve displayed in FIG. 10 shows exactly how the 
modal refractive index, defined by Equation (3), approaches the limiting 
value of the slab as the height (h.sub.uc) of the upper cladding 23 is 
increased for a particular example. The quantity plotted is the normalized 
modal refractive index b, 
EQU b=(N.sub.m.sup.2 -n.sub.uc.sup.2)/(n.sub.f.sup.2 -n.sub.uc.sup.2) (11) 
v. The Height (h.sub.r) of the Rib 
The height (h.sub.r) of the rib 27 protruding above the upper cladding 23 
has two important effects on the shape of the guided modes. The first 
effect concerns the extent of lateral confinement. In the limit where the 
height (h.sub.r) goes to zero, the waveguide reduces to a slab and hence 
the guided modes are unconfined laterally. With increasing height 
(h.sub.r) of the rib 27, the effective index (n.sub.eff) in region II 
under the rib increases with respect to regions I and III, and the modes 
become better confined laterally. 
The second effect of the height (h.sub.r) of the rib 27 concerns the degree 
to which the mode extends vertically, and whether the natural decay of the 
field is altered by the presence of the rib. Decay into the rib is 
determined in part by the difference in index between the core 25 and the 
upper cladding 23, with the decay length increasing as this difference 
narrows. If the rib 27 extends for less than an exponential decay length, 
the mode naturally will reach the upper edge of the rib. Then as the 
height (h.sub.r) of the rib 27 increases, the vertical extent of the field 
grows with it until the height (h.sub.r) exceeds the decay length. Once 
the height (h.sub.r) becomes equal to a few decay lengths, further 
increase has little effect. 
Lateral confinement also is influenced by the decay of the field into the 
rib 27, and so the height (h.sub.r) required is in turn influenced by the 
other parameters governing the vertical extent of the mode--i.e., 
principally the difference in index between core 25 and the upper cladding 
23, and the value of the height (h.sub.uc) of the upper cladding. The 
field emerging from the core 25 must have sufficient space to decay over 
the rib 27 if indeed the rib is to have any effect. A field that barely 
decays over the height (h.sub.r) of the rib 27 simply passes over the rib 
unperturbed. 
The primary effects of the rib's height (h.sub.r) are illustrated by the 
sequence of modal shapes shown in FIGS. 11a-11d, in which the height 
(h.sub.r) of the rib increases from the contour of FIG. 11a to the contour 
of FIG. 11d while all other parameters are held fixed. Here the lateral 
confinement of the mode increases noticeably from FIGS. 11a-11b, but 
beyond that remains approximately fixed with increasing height (h.sub.r) 
of the rib 27. The upward vertical extent of the mode increases as well, 
essentially following the rib 27 from FIG. 11a to FIG. 11b. Between FIG. 
11c and 11d, however, the mode grows very little vertically, thereby 
indicating that the rib 27 already exceeds the decay length substantially. 
There is of course a slight change in the modal refractive index (N.sub.m) 
in region II of the waveguide from FIGS. 11a-11d, and therefore a slight 
change in decay length as well. The heights (h.sub.r) of the rib 27 in 
FIGS. 11a-11d are 0.6, 1.1, 1.6 and 2.1 .mu.m, respectively. The values of 
the remaining parameters are the same for all four contours and are as 
follows (in microns): (w)=5; (h.sub.uc)=0.4; (h.sub.f)=0.5; 
(n.sub.f)=3.3700 and (n.sub.uc)=(n.sub.lc)=3.3601. 
vi. The Width (w) of the Rib 
The effects of the width (w) of the rib 27 on the modal structure are 
illustrated in FIGS. 12a-12c, where (w) alone is varied. Here with no rib 
at all (i.e., (w) equals zero), the slab guide that results does not 
support a guided mode. The mode also remains cut off for relatively narrow 
ribs. As the width (w) of the rib 27 is increased, the mode that appears 
initially is very close to cutoff, the effective index (n.sub.eff) of the 
region II differing from that of regions I and III in the fifth decimal 
place, and shows very weak lateral confinement. Lateral confinement 
steadily improves as the width (w) of the rib 27 increases, and the mode 
begins to acquire more the character of a rib mode than a slab mode. 
Further widening of the rib 27 induces little additional change in the 
general shape of the mode, but a second lateral mode can be supported at 
some larger width (w). In each of the contours of FIGS. 12a-12c, the value 
of the width (w) of the rib 27 is 2.0, 4.0, 6.0 .mu.m, respectively. The 
remaining parameters are held constant and are as follows (in microns): 
(h.sub.r)=0.6; (h.sub.uc)=0.4; (h.sub.f)=0.5; (n.sub.f)=3.3700; 
(n.sub.uc)=(n.sub.lc)=3.3601. 
vii. Other Parameters 
The overall width of the structure typically should be sufficient to 
suppress any effects due to finite size, although boundary conditions do 
influence the modes computed for a model system with finite width. It is 
possible also to consider guides in which the indices (n.sub.uc) and 
(n.sub.lc) of the two cladding layers 23 and 29, respectively, are 
different. Such cases may be handled with the arguments developed above. 
Finally, the rib 27 may be capped by a thin layer of doped material or by 
a metal contact (not shown). If any mode in such a structure penetrates to 
the top of the rib, then the additional layers can be added to the 
computational model as needed. 
The modes generally are well confined vertically for index differences of 
approximately one percent and more, and in such instances a height 
(h.sub.lc) of the lower cladding 29 of 1 .mu.m usually is adequate. 
Thicker cladding below the core 25 may be required to accommodate modes 
optimized for coupling to the fiber 31. Such quasi-circular modes 
typically are very close to cutoff, and these often are supported only 
when the difference between core and upper cladding indices (n.sub.f) and 
(n.sub.uc) is very small. For small but realistic values of the index 
difference, lower cladding thicknesses (n.sub.lc) of 4 .mu.m or more may 
be needed. As a preliminary guideline for fabrication of such waveguides, 
the lower cladding is required to be sufficiently thick so that no more 
than two (2) percent of the total integrated intensity of the mode travels 
in the substrate. 
III. OPTIMIZING THE AMETERS TO MATCH MODAL STRUCTURE OF WAVEGUIDE TO 
THAT OF THE OPTICAL FIBER 
For the waveguide 21 to propagate light at a wavelength (.lambda.) of 0.85 
.mu.m, the layers may be made of AlGaAs. For example, the upper and lower 
claddings 23 and 29 may be fabricated from Al.sub.0.4 Ga.sub.0.6 As and 
the core from Al.sub.0.3 Ga.sub.0.7 As, for which the indices (n.sub.lc) 
and (n.sub.f) are expected to be 3.3601 and 3.4165, respectively. Although 
these values provide good propagation characteristics, they promote strong 
vertical confinement, with the mode contained almost entirely within the 
core. The rib 27, typically has a width (w) of 3 or 5 .mu.m, a height 
(h.sub.r) of 0.4 to 0.6 .mu.m and is removed from the core 25 by a height 
(h.sub.uc) of the upper cladding layer 23 of 0.4 to 0.6 .mu.m. This 
removal from the core 25, combined with the weak evanescent fields in the 
cladding 23, lessens the ability of the rib 27 to confine the light 
laterally. The result often is a mode whose cross-section resembles a 
highly eccentric ellipse with minor axis of about 0.5 .mu.m and major axis 
of 6 to 10 .mu.m. The fundamental mode of a circular fiber, however, 
regardless of the fiber's diameter and refractive indices, is circularly 
symmetric with a nearly Gaussian intensity profile as shown in FIGS. 
3a-3b. 
To see how these differences in shape between the single mode of the 
waveguide and that of the fiber prevent efficient fiber-waveguide 
coupling, one begins by writing the modes of the waveguide as 
EQU u.sub.j (x,y)exp(i.beta..sub.j z), j=1, 2, . . . (12) 
and the fundamental mode of the fiber as 
EQU u.sub.fib (x,y)exp(i.beta..sub.fib z) . (13) 
With these expressions normalized to unity, 
EQU .intg..vertline.u.sub.fib.sup.2 .vertline. dx dy=.intg..vertline.u.sub.j 
.vertline..sup.2 dx dy=1, (14) 
the expansion 
EQU u.sub.fib (x,y)=.SIGMA..alpha..sub.j u.sub.j (x,y) (15) 
then is valid at the input facet of the waveguide--i.e., where power from 
the fiber enters the waveguide. Since the waveguide modes are mutually 
orthogonal, the expansion coefficients .alpha..sub.j are determined by the 
overlap integrals 
EQU .alpha..sub.j =.intg. u.sub.j.sup.* (x,y) u.sub.fib (x,y) dx dy . (16) 
The fraction of the power that couples into the fundamental waveguide mode 
(j=1), 
EQU P.sub.l =.vertline..alpha..sub.l .vertline..sup.2, (17) 
thus attains its maximum value of unity only when the fiber and waveguide 
modes are identical. The more dissimilar the modes, the larger are the 
coupling losses. The problem of optimizing the coupling between waveguide 
and fiber consequently becomes a matter of adjusting the shapes of the two 
modes to achieve a better match. 
Efficiency of coupling between the fundamental mode of the fiber 31 and any 
mode of the waveguide 21 is given by the square of the overlap integral 
(equation 16). For numerical evaluation, one calculates the modes of the 
waveguide 21 by the previously mentioned Fourier-series method, computing 
the values on the same 55 by 55 grid of points (20 .mu.m by 20 .mu.m) used 
to represent the field of the fiber 31. As with the fiber 31, the modal 
amplitude is set to zero on the boundaries of the grid. Both the 
normalization integral and the overlap integral can be evaluated simply by 
summing the products of the two fields at each grid point. 
Since the fundamental mode of the waveguide 21 necessarily is symmetric in 
the lateral direction, maximum coupling with the fiber 31 is obtained when 
the fiber's axis lies somewhere on the vertical centerline of the 
waveguide, y equals 0. One can determine the maximum coupling by 
translating the fiber's field along this line one grid point at a time and 
evaluating the overlap at each position. The maximum generally lies within 
two or three grid points (about 1 .mu.m) of the center of the core 25. 
The second mode of the waveguide 21, if one exists, is antisymmetric with 
respect to the vertical center line of the waveguide and thus will not 
couple to the fiber 31 placed with its axis anywhere along this line. 
Coupling will occur, however, as the fiber 31 is moved parallel to the y 
axis. 
A. Implementing A Numerical Solution 
Referring to the flow diagram of FIG. 13, optimization of the compositional 
and structural parameters of a raised-rib waveguide utilizes the programs 
RIB2DIM and PAT55 discussed in Section II in order to implement steps A, 
B, C and D of the diagram. The steps A and B consisting of initializing 
the values of the structural and compositional parameters and determining 
the eigenvector coefficients for each point of the 55 by 55 grid to be 
plotted are accomplished by the program RIB2DIM, attached hereto as 
Appendix C. The amplitude of the mode at each point of the grid is 
provided in step C by the program PAT55 of Appendix D. A matrix M.sub.w of 
the amplitudes is filled in step C for the purpose of comparing the modal 
shape of the raised-rib waveguide with that of the optical fiber. Plotting 
of the mesh plot and contour map from the amplitudes is done by the 
commercial program PC-MATLAB, previously mentioned in Section II. 
In order to optimize the values of the structural and compositional 
parameters of the raised-rib waveguide so as to achieve a low-loss 
coupling with a commercially available optical fiber, the modal amplitudes 
of a selected fiber are determined for the 55 by 55 grid (as discussed in 
Section II) and entered into a matrix M.sub.f as indicated by step E in 
FIG. 13. With both the matrices M.sub.w and M.sub.f filled with values, 
the coupling efficiency for the two waveguides described by the values in 
the matrices M.sub.w and M.sub.f can be determined quite simply in step F 
by summing the products of the two fields at each grid point. 
From information garnered in the experiments of Section III, a parameter of 
the waveguide 21 is selected in step G whose value has the greatest effect 
on the modal shape. The selected value is varied while the others are held 
constant and the steps B, C, D, F and G are repeated. Once the value is 
identified that provides the optimum coupling and still propagates the 
selected mode, the value of the parameter is fixed at the identified value 
and another parameter is chosen as a variable. Preferably, the parameters 
are selected as variables in an order of decreasing effect on the modal 
shape. The following is a particular example. 
B. An Example of Optimization 
The process of optimization begins by selecting as a variable the one 
parameter of the waveguide 21 for which the shapes of the fundamental mode 
is most sensitive at the selected wavelength (.lambda.) of 0.85 .mu.m. An 
initial set of values is selected for the guide 21 that provide a 
reference point from which improvements in coupling efficiency may be 
measured in accordance with the invention. In the present example, the 
initial value of the parameters are (w)=5.0, (h.sub.r)=0.6, 
(h.sub.uc)=0.4, (h.sub.f)=0.5, (.mu.)=0.85 .mu.m, (n.sub.f)=3.4165, and 
(n.sub.lc)=(n.sub.uc)=3.3601, the latter three values corresponding to a 
Al.sub.x Ga.sub.l-x As composition for the core 25 and upper and lower 
claddings 23 and 29 of x equals 0.3 and 0.4, respectively, as previously 
mentioned. Computation of the overlap integral (equation 16) gives 
coupling efficiencies of 34% (-4.66 dB insertion loss) for the 5-.mu.m 
fiber and 27% (-5.76 dB) for the 9-.mu.m fiber. 
To improve the coupling in accordance with the invention, one first 
decreases the refractive index (n.sub.f) of the core 25, thereby causing 
the mode to spread vertically. The spread is mainly into the lower 
cladding since the extent of the mode upward is essentially limited by the 
interface with air. As the difference decreases between the indices of the 
core (n.sub.f) and the upper cladding (n.sub.uc), the shape of the mode 
becomes increasingly circular, and the computed coupling losses decrease 
correspondingly as shown in FIG. 14. The efficiency reaches 80% (-0.99 dB 
insertion loss) for the 5-.mu.m fiber and 69% (-1.61 dB) for the 9-.mu.m 
fiber at an index (n.sub.f) equal to 3.370. The improvement in efficiency 
cannot be continued further without changing some other parameter of the 
waveguide 21, since the mode already has a modal refractive index 
(N.sub.m) for its cutoff wavelength of only 3.3607, very close to that of 
the upper and lower claddings. With further reduction in the index 
(n.sub.f) of the core, this mode is cut off and ceases to exist at 
approximately an index (n.sub.f) equal to=3.368. 
The second most sensitive parameter for effecting changes to the modal 
shape is the height (h.sub.f) of the core 25. Reducing the initial value 
of the height (h.sub.f) causes the mode to spread vertically and generally 
become more circular. All parameters in this sequence are fixed except the 
height (h.sub.f) of the core, which decreases. The fixed values are the 
same as the initial values, except the value of the index (n.sub.f) of the 
core 25 has been amended to the optimized value of 3.370. 
Coupling losses to 5-.mu.m and 9-.mu.m fibers are shown in the table I 
below. 
TABLE III 
______________________________________ 
core height (.mu.m) 
coupling loss (dB) 
h.sub.f 5-.mu.m 9-.mu.m 
______________________________________ 
0.7 -1.54 -2.47 
0.5 -0.99 -1.61 
0.4 -0.83 -0.89 
______________________________________ 
The value of the other parameters for the waveguides 21 of TABLE I are held 
constant and are: w=5, h.sub.c =0.4, h.sub.r =0.6, n.sub.f =3.37, n.sub.c 
=3.3601. The mode for the waveguide having a core height (h.sub.f) of 0.4 
.mu.m has a modal refractive index (N.sub.m) of 3.3602, which is almost at 
cutoff. Therefore, at the height (h.sub.f) of the core 25 equal to 0.3 and 
below, there are no guided modes in this structure. Excellent coupling may 
be obtained with values of the height (h.sub.f) equal to 0.3, however, for 
guides where the refractive index (n.sub.f) of the core 25 ranges between 
3.371 and 3.378. Just by concerted adjustment of both the height (h.sub.f) 
of the core 25 and the difference between the refractive indices (n.sub.f) 
and (n.sub.uc) of the core 25 and upper cladding 23, respectively, leaving 
the other parameters untouched, one can reduce theoretical insertion 
losses from--5.76 dB to -0.89 dB and from -4.66 dB to -0.83 dB for the 
9-.mu.m and 5-.mu.m fibers, respectively. 
The remaining parameters of the waveguide 21 also may be adjusted to 
improve coupling. Most effective among these is the height (h.sub.r) of 
the rib. FIG. 15 shows the corresponding insertion losses for the guide 21 
where all parameters but the height (h.sub.r) of the rib 27 are fixed. All 
of these guides share the parameter values of (w)=5.0, (h.sub.f)=0.3, 
(h.sub.uc)=0.4, (n.sub.f)=3.3775, (n.sub.c)=3.3601, where the values of 
(h.sub.f) and (n.sub.f) have been previously optimized. 
As the height (h.sub.r) of the rib increases, the contour lines of the 
field extend further into the rib 27 and, at the same time, the pattern 
shrinks laterally because of the increased effective index (n.sub.eff) 
below the rib with respect to the effective indices in regions I and III 
(FIG. 5). These two effects cause the mode to become generally more 
circular and therefore better able to couple to the fibers, although a 
counter-example is provided between the rib heights (h.sub.r) of 0.6 and 
1.6 .mu.m. Here the coupling to the 9-.mu.m fiber actually becomes worse, 
because the wider field pattern for the waveguide 21 with the 0.6 .mu.m 
rib more closely matches the wider fiber than does the narrower pattern 
created by the waveguide with a rib width (w) of 1.6 .mu.m. Any effect of 
the height (h.sub.r) of the rib 27 also is limited, of course, by the 
extent of the evanescent field, for the rib is able to effect little 
marginal change once its height substantially exceeds the decay length of 
the mode. These decays can be prolonged, though, since the modes under 
consideration usually are close to their cutoff points. 
The several effects exhibited above may be combined to yield still better 
coupling efficiencies, of which one example is shown in FIG. 16. Here the 
difference in refractive indices again is relatively small (n.sub.f 
=3.3715, n.sub.uc =3.3601) and the core is narrow (h.sub.f =0.3 .mu.m). 
The height (h.sub.uc) of the upper cladding 23 is 0.4 .mu.m. The first 
five points of the curves in FIG. 16 correspond to a width (w) of the rib 
27 equal to 5.0 .mu.m, with the height (h.sub.r) increasing from 1.6 .mu.m 
to 5.6 .mu.m in steps of 1 .mu.m. For the last five points, the rib's 
height (h.sub.r) has been fixed at 5.6 .mu.m, while its width is increased 
from 6.0 to 10.0 .mu.m in steps of 1 .mu.m. Coupling losses to either the 
5 or 9 .mu.m fiber are less than -1.0 dB over the entire curve and, for 
the 5 .mu.m fiber, an optimum value of -0.29 dB loss is obtained. A second 
lateral mode also exists for the last guide in this sequence (w=10), 
whereas no guided mode is supported when the width (w) of the rib 27 
equals 5 .mu.m and its height (h.sub.r) equals 0.6 .mu.m (before the first 
point on the graph). 
The coupling efficiencies cited above are all "best" values, obtained by 
moving the fiber 31 vertically along the centerline of the waveguide until 
maximum coupling is achieved. Often this maximum coupling occurs with the 
fiber axis slightly below the center of the guiding film, but practical 
difficulties in attaching fibers naturally prevent the precise positioning 
of the fiber. It is therefore desirable to estimate how much leeway exists 
and understand what penalties in lost power will be incurred by 
misalignment. 
The computational procedure is straightforward. Both the mode of the fiber 
31 and the mode of the waveguide 21 are represented on the 55 by 55 grid 
mentioned above, the points spaced at approximately 0.37 .mu.m. The 
overlap of the two fields is computed by multiplying their values at each 
grid point and summing these products over the grid points. The modal 
pattern of the fiber then is shifted by one grid point in either the x or 
y direction, and the process repeated. Both fields are taken strictly zero 
outside the original 20 .mu.m by 20 .mu.m square. 
Two illustrations of alignment are given in FIGS. 17a and 17b. In both, 90% 
of the peak coupling is realized if the fiber axis lies within the 
innermost contour; 80% is achieved within the next contour, and so forth. 
FIG. 17a shows the coupling of the 5-.mu.m fiber to the guide 21 having 
the parameters of the waveguide in TABLE I, with (h.sub.f)=0.4 .mu.m. The 
mode profile is "semicircular" and most of the power is carried in the 
lower cladding. The alignment contours confirm that optimum coupling to 
this mode is achieved when the center of the fiber is approximately 1.3 
.mu.m below the center of the guiding film, with 90% of optimum achieved 
inside a distorted circle of approximately 1.7 .mu.m diameter. 
The second illustration, FIG. 17b, shows alignment for a 5-.mu.m fiber 
coupled to the guide of FIG. 15, with the height (h.sub.r) of the rib 27 
equal to 1.6 .mu.m. Here the fundamental mode is more nearly circular with 
its intensity distributed approximately equally above and below the 
guiding film. The map of coupling efficiency correspondingly shows roughly 
circular areas of equal coupling, centered on the core 25 of the waveguide 
21. The area corresponding to 90% of maximum coupling efficiency is about 
1.5 .mu.m in diameter, typical for the range of guides examined herein. 
From the foregoing, it will be appreciated that the invention provides for 
a raised-rib waveguide 21 whose structural and compositional parameters 
have values that optimize the coupling efficiency of the waveguide to an 
optical fiber such as commercially available 5 or 9-.mu.m fiber. In the 
exemplary embodiment, a raised-rib waveguide 21 is described whose 
structural and compositional parameters are optimized for coupling to 5 
and 9-.mu.m fibers at a wavelength (.lambda.) of 0.85 .mu.m. Using the 
same methodology as disclosed herein to identify the exemplary embodiment, 
other embodiments of raised-rib waveguides may be identified for other 
wavelengths (.lambda.) such as 1.3 .mu.m.