CA1262580A - Wavelength division multiplexer and demultiplexer - Google Patents

Abstract

ABSTRACT
A thin-film waveguide lens and a planar wavelength division multiplexer/demultiplexer compatible with integrated optics technology and embodying such lens are disclosed. The thin film lens comprises a thin-film waveguide with completely homogeneous materials and no physically produced graded index regions, in rectangular shape. A plano-convex overlay layer integral with the waveguide extends the entire length and width of the waveguide along the axis, its thickness profile being selected so as to simulate a graded effective refractive index in the waveguide in order to collimate focussed rays and focus collimated rays entering at one end plane for substantially collimated or focussed arrival at the other end plane. The multiplexer/
demultiplexer comprises the above thin-film waveguide lens, with one of the end planes constituting an entrance/exit plane for receiving optical fibers in an abutting connection, and the other end plane bearing a diffraction/reflection grating In accordance with the preferred embodiments, the shape of the plano-convex overlay is such as to provide an effective index of refraction profile in the waveguide according to the formula n = nosech(gx), where n is the effective refractive index at the distance x from the axis of the waveguide, no is the effective refractive index at the waveguide axis, and g is a constant equal to .pi./2f, where f is the focal length of the lens, being essentially the distance between end planes.

Description

~2~

This inven-tion rela-tes -to fiber optic communica-tions systems, and particularly to a wavelength divi~ion multiplexer/demultiplexer comprising a novsl thin-film waveguide lens.
In a f`iber optic communication system, each fiber typically carries light of a particular wavelength. Existing multiplexing technology allows hundreds and thousands of sign~ls to be sent on this wavelength through a single fiber through time multiplexing. However, when the upper limi-t is reached a separate fiber has -to be installed in order to satisPy heavier traf-fic demands. Instead o~ increasing -the number of fibers, an additional wavelength may be used in one ~iber. Wavelength clivision multiplexers and demull::ipLexers are the devices which allow this to be achieved. At the transmission end, signals are fed into one single fiber by means of multiplexsrs. At the receiver end, the signals are descrambled into groups, each belonging to a separate wavelength, by a demultiplexer.
The wa~elengths currently used commonly in optical communications are in the 800 nm and 1300 nm regions. A
third potential region is the 1500 nm one. Usually commercial demultiplexers separate only two very different wavelengths. However, as the technology of light sourcas and laser ligh-t sources in particular improves, it will become possible to use many more different wavelengths to incrsase the signal-carrying capacity of each fiber. Multiplexers and demultiplexers are needed to effectively handle such increassd numbers of wavelengths.

51~
An additiona:L technology related to this invention is integrated optics, i.e. the guicling and manipula-ting of ligh-t through thin-film optical waveguides. Among -the grow-ing number of devices being developed for in-tegrated optics, the op-tical waveguide lens remains the most basic. AD opti-cal waveguide lens performs a variety of important functions, including focussing and collimating, Fourier transforma-tion, imaging, spatial filtering, and the integra-tion of guided-optical beams. For some types of mul-tiplexers/demulti-plexers, a lens is a crucial element of the device.
There are many important design criteria -for an optical waveguide lens. The position of the focal plane, focal spot size and its in-tensity profile, angular f:ield of view, the energy in the sidelobes relative to the energy of -the centraL lobe, and the -throughput losses must all be considered. ~qua]ly important, the Eabrication techniques should be simple, inexpensive and compatible with present technology. Each of these factors mus-t be considered in the application of the optical waveguide lens in a wavelength division multiplexing/demultiplexing device.
Presently-marketecl wavelength division multi-plexers and demultiplexers nearly always use thin-film filters. When more than two wavelengths, say n, are used, -the use of thin-film filters is clumsy, because n-1 filters must be used in a cascaded structure. Such devices are expensive -to fabricate, and suffer from signal attenuation.
Grating devices can be used as an alterna-tive tv thin-film filters. The use of a grating to separa-te wave-, :~

: '' ~
~.

2$~

leng-ths is of course well known in spec-troscopy, a~d is also known in optical signal ~ultiplexing and demul-tiplexing from Canadian patent no. 1,089,932, described 'below. Another method of separating wavelengths is in the use of prisms, which ilowever are not commercially practical.
One specific type of multiplexer/demultiplexer employing a -thin-film lens is that described in Canadian patent no. 1,089,932 granted to Northern Telecom Li~ited on November 18, 1980, which employs a collimating lens to collimate the light from an optical fiber, a diffraction /reflection gra-ting, a focussing lens, and an array of ligh-t de-tectors and sources.
As this device shows, all grating devices require means to collimate the light rays for arrival at the grating, and means -to focus the rays subsequently. Thus i-t is advantageous to simplify the design of the lens element, and optimize i-ts collimatingr and focussing properties.
A two-dimensional thin-fil~ optical waveguide lens can be used as the means for effectin this collimating and focussing. Three types of -thin-fil~ optical waveguide 1ens have been proposed and demonstrated for use with guided optical beams: mode index lenses such as thin-film Luneburg lenses, geodesic waveguide lenses, and Fresrlel diffraction lenses. D.B. Anderson et al, "Comparison of Optical Waveguide I,ens Technologies", J. Quant. Electron., Q.E.-13 275 (1977~. Each of these lenses utiliæes a localized change in the optical wave guide: an index gradient, a spherical depression, and a grating respectively, -to alter the . . : :

, .

~625~3~

wave~ ront curvature and ob-tain the desired focussjlng effec-t.
This loca]iæed change in the waveguide structure contributes to losses by introducing scat-tering and mode conversion at the ]ens edge.
Amongst these various lens types, the thin film Luneburg lens is the most practical for multiplexing/
demultiplexing applications with its potential -for reduced losses and scattering. This lens is fabricated by depositing the lens material through a circular mask. This deposition process is the critical step, requiring the achievement of a par-ticular thickness pro-~ile with radial symme-try. If the exact thickness profile is achieved, the circular lens wi:Ll exac-tly focus a collimated beam to a point on the focal curve, or conversely col]imate a poin-t source located on thi~ ocal curve. Such a lens can be constructed with materials having a uniform refrac-tive index. S.K. Yao;
et al., "Guided-wave optical thin-film Luneburg lenses:
Fabrication technique and properties", App. Optics, 18, 4067 ~1~79). It can also be constructed with a layer having a graded index profile, as described by United States pate~t no. 4,4~,063.
One disadvantage of the Luneburg lens :is it~l circular shape and the resulting curved image and object planes. It is di~ficult to match the curved focussing plane to the endplane of a rectangular planar optical waveguide without introducing aberrations. The circular shape necessitates a region surrounding the lens with a constant thickness, and thus constant re~rac-tive index, to carry -the , ~ :
.

"~

~L26:~5~

guided ligh-t from the circular lens edge -to the planar waveguide edge. The transition area from lens to surrounding planar waveguide is a source of loss, due to scat-tering a-t the index discontinuity n This disadvantage is observed in other devices which utili~e a circular thin-film lens in a rectangular waveguide. In the multiplexer~demultiplexer previously described in Canadian patent no. 1,089,932, we observe a thin-film lens wi-th curved edges focussing onto the straight endplane of a rectangular waveguide. Similarily, in United States patent nos. 4,253,060 and 4~348,074, a circular thin-film lens is utilized to focus a beam to a poin-t on a straigh-t edge. Thus both the~e devices suf:fer from the disadvantages mentioned above.
It i8 an object o-~ -the present invention -to provide a thin-film waveguide lens *or use in a wavelength division multiplexer/demultiplexer -to overcome some o-f the problems and o-ffer advantages over the wavelength division multiplexers an~ demultiplexers and thin-film lenses in the prior art.
Thus in accordance with one aspect of the present invention there is provided a rectangular thin-film waveuide lens for use in a multiplexer/demultiplexer, comprising a thin-film waveguide having a longitudinal axis and having` end planes essentially normal to the axis a-t each end o* the waveguide. A plano-convex overlay layer integral with the wave~uide extends the length of the waveguide along the axis, and is symmetric about this axis, the profile of " , .

: . :.,.

~;26;~

the overlay layer being selected so as -to produce a,graded effective refractive index in the lens in order to collimate focussed rays and focus collimated rays en-tering at one end plane for substan-tially collimated or focussed arrival at -the other end pla-ne.
In accordance with another aspect of the invention, there is provided a multiplexer/demultiplexer comprising the above thin-film waveguide lens, in which one of the end planes constitutes an en-trance/exit plane for ld receiving optical f'ibers in an abutting connection, and the other end plane 'bears a diffraction~reflection grating.
Focussed rays entering the lens at the entrance/exi-t plane are substantially collimated for arrival at the diffractioll/reflection grating, and rays rerlec-tillg from -the diffraction/reflection grating are substantially focussed on returning to the en-trance/exit plane, the locations -for abuttment oP the optical -fibers being matched to the spot locations of the focussed rays along the entrance/exit plane.
In accordance wi-th the prePerred embodi~ents of the above aspec-ts of the invention, the shape of the plano-csnvex overlay layer is such as to provide an ef-fective index of re-fraction profile in the waveguide according to the formula n = nosech(gx), where n is the efPective refractive index at the dis-tance x from the axis of the waveguide, no is the eff~ctive refractive index at the lens axis, sech is the trigonometrical hyperbolic secant function, and g is a constant (equal to ~/2f, where f is the focal length of the lens).

.

~12~6J2d5~30 Further *eatures nf -the invention will be described or will become apparent in the course of the following detailed description.
In order that the invention may be more clearly understood, the preferred embodiment thereof will now be described in detail by way of example, with raference to the accompanying drawings, in which:
Fig.l is a top view of the preferred embodiment of the multiplexer/demultiplexer, depicting in representational -form the case of two wavelengths;
Fig.2 is a view of the en-trance plane A--A' of the lens of the multiplexer/demultiplexer;
Fig.~ is a cross-sec-tion at the end plane B-~' of the lens;
Fig.4 is an oblique view of the entrancs pl1}le of the multiplexer/demultiplexer;
Fig.5 is a graph which shows the thickness profile of the overlay layer required for a lens with a focal length of 5.0 mm; and Fig.6 is a graph which sho~s the expected focal spot si~e as a function of wavelength for light signals passing through a multiplexer/demultiplexer with the profile of Fig.5.
Referring to Figs.l-4, there is illustrated the multiplexer/demultiplexer I of the preferred embodimen-t.
For convenience, the device will be treated as a demulti-plexer in this description, although as will hecome apparent, the device can operate in either roleg depending .

..

'` ', ",., ~ ,, 25~3~
simply on the direction of the signals.
An input fiber Fl feeds signals carried by different wavelengths in-to the device at an entrance/exit plane A-A'. Fibers F2 and F3 are representative ou-tput fibers, each carrying demultiplexed or wavelength divided signals of a dif-ferent waveleng-th. In practice, there of course may be more than two wavelengths, and there will thus be more -than the two representative output fibers F2 aDd F3.
Due to the current limitations of light sources, it is not likely that more than about ten output -Pibers would be involved at present, but as laser sources in particular improve, there could be many more output fibers corresponding to differen-t transmission wavelengths.
The device consists o~ a uniform flat substrnte 21 of cons-tant refrac-tive index on which is a uniform thin-~ilm waveguide 3 of constant refractive index, on which is an overlay layer 4 of constant refractive index and ~ith a carefully selected relief profile. The relief pro-Pile is determined, in accordance with the properties of the thin-film ~aveguide, so as to produce the ef~ective refractive index distribution n = nosech(gx).
The overlay layer 4 may be of the same material (a three-layer case, including the overlying medium, usually air), or of a different material (a four-layer case -Prom -the material of the wave~uide portion. The optical fibers Fl, F2 and F3 are butt-joined to the uniform thin-film wavegllide portion 3 o~ the entrance~exit plane A-A' of the device.
In the case of an overlay layer 4 o e the same '' ` ~ ~
,.
:: ' material as the thin-film waveguide portion 3, then instead of vj.ewing the overlay layer and the Naveguide ~ as beill~
separate, they may be properly viewed as being and the same.
For convenience of description, however, they will be referred to as separate elements -throughout -this specification. A dotted line is used in Figs.2 and 3 to indicate the plane of division between -these elements, whether real or notional.
At the end plane B-B' of the device remote from the entrance/exit plane A-A' is an integral dif-fraetion/
ref:Lection gra-ting ~. The grating 5 is scribed or embossed or otherwise prepared on the end plane B-B', and a suitable thin film is deposited on the grating ~ to turn i-t into a reflecti.on grating with high efficiency and low losses.
The profi:Le of the overlay :Layer 4 is se:lected so as to produce a graded effective re:fractive index in -the lens in order to collima-te -the rays -for arrival at the dif-fraction /reflection grating 5 and so as to focus the wavelength-divided light rays leaving the grating at spots along the entrance/exit plane A-A', corresponding to the locations of the output fibers F2, F3, etc.. This results in a plano-convex shape for the overlay layer 4, aligned alon~ the longitudinal axis of the thin-fi:Lm waveguide 3, as illustrated in the accompanying drawings, and extending from the entrance/exit plane A-A' all the way back to the end plane B-B' with the grating 5. The grating 5 is of course slightly angled :~rom the longitudinal axis so that the diffracted/re-flected signals are directed slightly away -from lQ

2~

the inpu-t fiber Fl, towards represen-tative output -~ibers F2 and F3.
The position of each ou-tpu-t fiber is of cours.e dependent on the wavelength to be picked up by that fiber.
The ceparation be-tween fibers thus dependci on the separation between wavelengths. A certain physical separation, as for exa~ple at least one fiber diame-ter be-tween cen-ters, is desirable to avoid undesirable cross-talk in the eveDt of inaccuracies in signal focussing, which are to a certain extent unavoidable. Ideally, each spot is focussed essentially entirely within the circum-ference o-f the corresponding ~iber connection point, so -that there is essentially no crosstalk, and -the focussing propertles of the present invention are such tha-t this should be essentially achievable.
To improve signal focussing, the entrance/exi-t plane A-A' is ang].ed slightly -from the plane normal to the axis of the th:in--fi.lm waveguide 3, as can be seen in exaggerated fashion from Fig.l, based on calculations from ray-trace data, to provide a flat-plane approxima-tion of the optimum ~ocal plane for the different -focal points o-f the ~arious wavelengths. Thus ~oth -the end plane B-B' with the grating 5 and the entrance/exit plane A--A' are preferably sligh-tly angled. The location of the optimum -focal plane is naturally dependent on the angle selected for the diffraction grating 5.
The details of the lens element of the inven-tion are now discussed. It is known that a GRIN rod lens with an ~. ", .
~ , index gradient according to -the fvrmula n = Dosech(gx) focusses all meridional rays e~actly. In -the preferred embodimen-t o* the present invention -this :index distribll-tion is adapted to the two dimensions of -the thin-film waveguide

3, wi-th the plane of the -thin-film wave guide 3 coincident with a meridional plane of the GRIN rod lens, the resul-t being a thin-fi]m lens which focusses all incident rays parallel to its axis. While other suitable profiles may be developed, it has been de-termined by the inventors that this particular profile is quite suitable as a profile for the overlay layer 4 in the present invention.
To fabricate the index distribution given by the above-men-tioned equation in a planar waveguide, -the concept of an effective index is used. Although the bulk refractive index in a waveguide is cons-tantl changing the thin-film thickness alters the v010ci ty of a guided optical wave. The result is that the propagating phase -~ront has a velocity that is equivalent to bulk propagation in a material having -the appropriate effective index. In a waveguide constructed with the appropriate bulk materials and with the properly shaped pro-~ile or appropriate overlay layer 4, the required effective distribution can be obtained. This effective index gradient is equivalen-t to a physically produced gradient fabricated by varying the bulk index of refraction by such means as dif-fusion or doping.
To determine ~hether there exist prac-tical materials for the realiza-tion o~ this lens, the dispersion curves for propagation in thin-film waveguides were , '~ ' ` ~ ''; ' :, ; ;
.
'; ::: , ~

,~2 ~Z58 examined. For both the three-layer and four-layer cases there exist Q ~ini~um and m~ximum waveguide thickness allowed for optimum lens performançe. It was found that for the three-layer ca6e with an SiO2 substrate a variety of focal lengths could be constructed with most waveguide materials. The only restriction occur~ with short focal lengths ~hen the bulk waveguide refractive index is close to that of the substrate. Similar construction flexibility is found for the four-layer cas~; aDd with different substr~te material~.
The exact waveguide profile re~uired to obtain the required effective index distribution iB obtained in accordance with the properties oE the thin-film waveguide.
Sun, M.J. and Muller M.W., "Measuremen-ts of four-layer i~otropic waveguides", Appl. Optics, 16, 814 ~1977). The profile is such a~ to produce the effective refractive index distribution n = nosech(gx). Accordingly, the relief profile has a thickness profile given by:

T(x) ~ r t -1~ n21x~ - nl2 ~1~2 ta~ (n32-~2~ /2 2~ ~n4 -n21xJJ~ ~ ~ n2-n21xJ J L~ ''A J

~ (n32 _ n2~A3 ~ tan ~ ( n2~r~ - n2~

where:
n(x) = n~sech(gx);
x is the distance from the lens axi~;

, , .

. , ni is -the re-frac-ti-ve index of -the ith layer;
d is the thickness of layer 3i and g~is the design wavelength This equa-tion yields the local thickness of the overlay layer 4, given the distance -from the axis of -the lens, the wavelength to be focussed and the bulk refractive indices of -the materials to be used~ In Fig.~ the -thickness profile T(x) is plotted ~or a typical lens design, whose parameters are described in Table 1.

_____ __________________________________________________ ____ Focal Length f ~.Omm Lens Wid-th 2.Omm F-number 7.8 nl (air~ 1.00 nz (S iO2 ) 1 . 47 n~ (GeO2) 1.61 n4 (~eO2) 1.61 Wavelength 1.3um ____ _ ________ ____ __~________________________ ________ By ray-tracing through the refractive index distribution produced by such a profile, it can be shown that 1~

cliffract:ion-limited focussing can be obta:ined for many focussing and collimating requirements. Ray trace results for -the lens described above and wavelengths in -the 1.3 mi.cron range are presented in Fig.6, which show that di.-ffrac-tion limited focussing can be obtained over a suitable range about the design wavelength. From the ray trace da-ta, the siæes and positions of the focal spots are determined. This permits a proper choice of the output -fiber diameters and the position of the focal plane.
The complete design of the present device can be accomplished with the abo~e procedures. When -the device parameters have been determined, as in Table ]., Eor example, the required profile of the over:Lay :Layer is determined with the above equation. This completes the lens component. Ray tracing is -then used to determine such things as optical fiber placement on the endplanes, angle of the diffraction/
re-.Flection grating and angle of the entrance/exit plane.
Such steps are considered to be routine and within the ordinary knowledge of those knowledgeable in the field.
This design enables all the wavelengths, which may be close -together, to be separated with only one lens element and grating of integral construction. The integral construction of -the device is particularly advantageou~ since there are no discontinuities in the path of -the signal, the same element being used for both collimating and focussing.
The lens structure uses the entire thin-film waveguide to obtain the collimating and focussing effect~, by virtue of the effecti~e index gradient which extends from the ,,,,, :L~

en!;rance~exi-t p]ane h--A' to -the diffrac-tion/reflec-tion grating ~. This struc-ture, in contrast to existing devices such as those in the above-men-tioned Canadian patent no.
1,089,93?, eliminates two dielec-tric boundaries, be-tween planar waveguide and lens, and between lens and planar waveguide. Such prior art devices used isola-ted thickness varying regions. This resulted in a localized ]ens element bounded by a planar ~aveguide. Eliminating these boundaries provides the potential -to minimize mode-conversion and scattering losses.
It will be readily appreciated from -the above description that the device can operate either as a mul-ti-plexer or as a demu:ltiplexer with equal -fac-ility, depending mere~ly on the direction of the signals. When operating as a rnul-tiplexer, the signals fro!n the fibers E'2, F3, e-tc. are combined at the diffraction/reflection grating 5, producing an output signal which is focussed at the fiber F1.
The lens can be manufactured using existing techniques used for Luneburg lens construction, using a wide variety of materials from low loss optical glasses to active materials such as LiNbO3.
Two methods could conceivably be used to -fabrica-te such a lens. First, sputtering or evaporation could be used with appropriate masks. Alternatively, a lens with a sui-table pro-file could be emboæsed on a dip-coated de-formable gel film on a subs-tra-te. Subsequen-t heat treatment would transform the molded gel into an inorganic hard oxide ma-terial. Each o-f -these techniques requires computer aided ~ `
, ,.. ;. :: .

design aDcl a high precision mask or mold. However, once the mask or mold is made, it can be used to fabricate many lenses.
It will be extremely important -to keep the ~abricati~n process wi-thin tolerances, since any error in th~ wavegu:ide thickness will increase aberrations and result in a corresponding error in the lens focal length. ~owever, small shif-ts in the focal plane position may be tolera-ted since the optimum focal plane can be reached through grinding and polishing of the waveguide edge.
The principal technologies involved are listed below with a brief` description of their applica-tio~s to the fabrication of -the different parts oP the device.
1. Fiber cut-ting - I:he eods of each fiber m-lst be creaved with flat surfaces perpendicular to the f'iber axis.
2. Polishing - -the two ends oP the integrated thin-film lens (surfaces AA' and BB' in Figs.2 and 3) and also the fiber ends mus-t be polished.
ZO 3. Butt-joining - -Pibers have to be very accurately butt-joined to -the end of -the in-tegrated thin-film waveguide 3 by a suitable method, possibly by using a ~uitable epoxy or by fusion.

4. Thin-film deposition - the film with an appropriate thickness must be deposited on -the flat surface of a substrate. Also, the end sur~ace BB' (Fig.3) must be provided with an appropriate ~ilm which is used to fabricate a grating ~. h dipping and baking technique is proposed.

~,~

, : ' . . .
. '~

258~1

5. ~mbossing - this technique will be used -to shape the curved surface with a pre-determined rellef profile on top of the integrated thin-film waveguide 3. I-t will also be used to produce the grating ~.

6. Conven-tional thin-film deposition - a -thick film has to be deposited on the grating ~ to turn it into a re~lection grating with very high e*ficieDcy and very low losses. A sputtering or a high-vacuum evaporation system may be used.
This new optical waveguide lens has been analyzed and its fabrication and focussing properties have been theoretically examined. It has been found to have reasonable fabrication tolerances compatible with existing shadow loasking sputtering techniques and a new emboss:ing technique. Considerable design flexibility allows lts construction with a varie-ty of materials. Ray tracing reveals that di-ffraction limited focussing should be possible. Also, low f-numbers can be obtained, making the de~slgn promising for use in miniaturized integrated optic circuits. The embossing technique will likely be suitable for mass production, resulting in lower costs when compared with other existing methods.
I-t should be noted that althou~h in Figs. 2 and 3 the plane of divi~ion between the thin-film waveguide portion 3 and the o~erlay layer 4 lS shown as be1ng such tha-t the overlay layer has zero thickness at -the edges of the waveguide, the plane could be positioned so tha-t the overlay layer do~es have a de~finite thickness at the wavé

: `, ,.; : : :

, '' ::

~, . : ,;
. ~.' . - . : `

, ~262~

guide edge. This of course is only relevent irl thelfour-layer case, since in the three-layer case the division is notional rather -than real.
It will be appreciated that the above description relates to the preferred embodiment by way of example only.
Many variations on the invention will be obvious to those knowledgeable in the field, and such obvious varia-tions are ~ithin the scope of the inven-tion as described and claimed, whether or not expressly described.

:

' ,`li 1 9 : ''' ' :' ~ , '':
, .

Claims (8)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. A multiplexer/demultiplexer comprising a thin-film waveguide lens, said lens comprising: a rectangular thin-film waveguide having a longitudinal axis; end planes essentially normal to said axis at each end of said waveguide; a plano-convex overlay layer integral with said waveguide and extending the entire length and width of said waveguide along said axis, the profile of said overlay layer being selected for the length of said lens so as to produce an effective refractive index which varies in the lens in order to collimate focussed rays and focus collimated rays entering at one said end plane for substantially collimated or focussed arrival at the other said end plane;
one said end plane constituting an entrance/exit plane for receiving optical fibers in an abutting connection;
the other said end plane bearing a diffraction/
reflection grating;
whereby focussed rays entering the lens at said entrance/
exit plane are substantially collimated for arrival at said diffraction/reflection grating, and whereby rays reflecting from said diffraction/reflection grating are substantially focussed on returning to said entrance/exit plane, the locations for abuttment of said optical fibers being matched to the spot locations of the focussed rays along the entrance/exit plane.

2. A multiplexer/demultiplexer as recited in claim 1, in which the shape of said plano-convex overlay layer is such as to provide an effective index of refraction profile in said waveguide according to the formula n = nosech(gx), where sech is the trigonometric hyperbolic secant function, n is the effective refractive index at the distance x from said axis, no is the effective refractive index at said axis, and g is a constant equal to .pi./2f, where f is the focal length of said lens, being the distance between the said entrance/exit plane and said diffraction/reflection grating.

3. A multiplexer/demultiplexer as recited in claim 1, in which the material of said overlay layer is the same as the material of said waveguide.

4. A multiplexer/demultiplexer as recited in claim 2, in which the material of said overlay layer is the same as the material of said waveguide.

5. A multiplexer/demultiplexer as recited in claim 1, in which said end plane constituting an entrance/exit plane is angled slightly from the plane normal to said waveguide axis, so as to provide a flat-plane approximation of the optimum focal plane for the spot locations of the focussed rays along the entrance/exit plane.

6. A multiplexer/demultiplexer as recited in claim 2, in which said end plane constituting an entrance/exit plane is angled slightly from the plane normal to said waveguide axis, so as to provide a flat-plane approximation of the optimum focal plane for the spot locations of the focussed rays along the entrance/exit plane.

7. A multiplexer/demultiplexer as recited in claim 3, in which said end plane constituting an entrance/exit plane is angled slightly from the plane normal to said waveguide axis, so as to provide a flat-plane approximation of the optimum focal plane for the spot locations of the focussed rays along the entrance/exit plane.

8. A multiplexer/demultiplexer as recited in claim 4, in which said end plane constituting an entrance/exit plane is angled slightly from the plane normal to said waveguide axis, so as to provide a flat-plane approximation of the optimum focal plane for the spot locations of the focussed rays along the entrance/exit plane.