WO2002052315A1

WO2002052315A1 - Coupled waveguide systems

Info

Publication number: WO2002052315A1
Application number: PCT/GB2001/005789
Authority: WO
Inventors: Ivan Evans
Original assignee: Bookham Technology Plc
Priority date: 2000-12-22
Filing date: 2001-12-21
Publication date: 2002-07-04
Also published as: GB2370372A; GB0031623D0

Abstract

A coupled waveguide system, such as a codirectional coupler 1', comprises a first waveguide 2 formed on a substrate, a second waveguide 3 formed on the substrate closely adjacent to the first waveguide 2, the first and second waveguides being non-symmetrical with respect to one another, and a grating 4 provided within the second waveguide 3 to effect phase matching between the waveguides 2 and 3. Furthermore a stress-controlling layer provides stressing of the first waveguides 2 and 3 to increase the waveguide birefringence such that light of an optical mode, for example a TE mode, within an appropriate wavelength range in the waveguide 2 is transferred to the waveguide 3 by interaction between the waveguides and is thereby separated from light of another optical mode, for example a TM mode, in the waveguide 2. Such a system can be totally integrated utilising SOI technology, and can be applied to a wide range of applications.

Description

"Coupled Waveguide Systems"

The present invention relates to coupled waveguide systems, and is concerned more particularly, but not exclusively, with such systems for use as wavelength- selective codirectional polarisers in optical communication systems.

In integrated optical circuits light is transmitted down waveguides formed in materials such as silicon, other semiconductors or silica. A silicon waveguide structure typically comprises a rib formed in the upper silicon layer of a SOI (silicon-on- insulator) chip, the rib having a top surface and side walls, and serving to confine an optical transmission along the waveguide structure.

It is often desirable to modify the basic waveguide structure to perform a number of different functions, for example to spatially separate orthogonal polarisation components, such as TE and TM components, or to spatially combine such components. Integrated polarisation beam splitters and combiners are known. "An integrated optic adiabatic TE/TM mode splitter in Silicon", R.D. de Ridder, A.F.M. Sander, A. Driessen, J.H. Fluitman, IEEE J. of Lightwave Technology, vol. 11, no. 11, p. 1806, 1993 discloses a polarisation beam splitter in a silicon-oxynitride-on-silicon material system having an asymmetric layered waveguide structure which relies on shape birefringence to introduce a different sign of birefringence in each branch of the Y junction.

Furthermore US 5333216 and EP 0431527 A2 disclose a wavelength-selective coupler comprising two waveguide layers formed one on top of the other with a cladding layer therebetween, the two waveguide layers being coupled together by a grating formed by corrugations in an upper surface of one of the waveguides. Depending on the period, mark/space ratio and index contrast between the grating elements, light of an appropriate wavelength propagating within one of the waveguides in a first guided mode is coupled to the other waveguide by the grating so that optical power is transferred from one waveguide to the other with the result that light is then propagated in the other waveguide in a second guided mode. Further references relating to directional or grating-assisted couplers are as follows: "Directional couplers made of nonidentical asymmetric slabs- part II grating assisted couplers" D. Marcuse, Journal of Lightwave Technology, vol. LT-5. no 2, p. 268, Feb 1987; and "Coupled- mode theory for guided-wave optics", Amnon Yariv, IEEE Journal of Quantum Electronics, vol QE 9, no. 9, p.919, September 1973.

It is an object of the invention to provide a coupled waveguide system which can be produced in a particularly straightforward manner, for example during fabrication of an integrated device utilising SOI technology, and which can be produced for a range of applications.

According to one aspect of the present invention there is provided a coupled waveguide system comprising a substrate, a first waveguide formed on the substrate, a second waveguide formed on the substrate closely adjacent to the first waveguide, the first and second waveguides being non-symmetrical with respect to one another, perturbation means provided within the second waveguide to effect phase matching between the first and second waveguides, and stress-controlling means providing stressing of the first and second waveguides to increase the waveguide birefringence such that an optical mode within an appropriate wavelength range in one of the first and second waveguides is transferred to the other one of the first and second waveguides by interaction between the first and second waveguides and is thereby separated from any other optical mode in said one waveguide.

Such a coupled waveguide system differs from prior systems, such as the systems of US 5333216, in that it incorporates a stress-controlling means which may in the form of an added layer of a material having a sufficiently different coefficient of expansion to the material of the underlying layer such that the resulting stress causes waveguide birefringence of a magnitude sufficient to separate the optical modes within the waveguides. Although US 5333216 discloses systems in which added layers are grown at elevated temperatures, such added layers are not such as to produce stress of the required magnitude to effect optical separation in this manner, and indeed there is no implication that such layers are provided for this purpose. Such a system can be totally integrated utilising SOI technology for example with its inherent advantage of simplicity of fabrication, and can be applied to a wide range of functions, including polarising functions for which a high level of waveguide birefringence is may be required. Examples of possible applications of such a system are as a codirectional or contradirectional polariser, a polarisation selector for an integrated extended cavity laser, an optical isolator, a polarisation extinction enhancer for an integrated laser, an adjustable polariser and an adjustable polarisation router or mixer. Furthermore the system can be constructed without metallisation (although metallisation may be provided in certain embodiments) and without a thin dielectric layer, and this renders fabrication easier, as well as possibly making the system less susceptible to manufacturing tolerances.

According to another aspect of the present invention there is provided a method of splitting apart or combining together two optical modes comprising passing the optical modes through closely spaced first and second waveguides which are non- symmetrical with respect to one another, one of the first and second waveguides being provided with a grating structure to effect phase matching between the waveguides, and stressing the first and second waveguides to increase the waveguide birefringence, whereby the optical modes are split apart or combined together by interaction between the first and second waveguides.

For a better understanding of the present invention and to show how the same may be carried into effect, reference will now be made, by way of example, to the accompanying drawings, in which:

Figures 1 and 2 are highly schematic explanatory diagrams illustrating the principle behind the invention;

Figure 3 is an explanatory diagram showing a plan view of an embodiment in accordance with the invention; Figure 4 is a schematic diagram of an embodiment in accordance with the invention;

Figure 5 is a block diagram showing a possible application of an embodiment of the invention; and.

Figures 6(a) to 6(d) are explanatory diagrams illustrating a method of fabrication of an embodiment of the invention.

Codirectional coupling or grating-assisted directional coupling allows the efficient transfer of power between two optical modes which are propagating in the same direction. The modes may be quasi-linear polarisation states, either Transverse Electric (TE) or Transverse Magnetic (TM). For the TE mode the electric field lies predominantly in the transverse plane, whereas for the TM mode the magnetic field lies predominantly in the transverse plane, so the electric field lies in the vertical plane.

Referring to Figure 1 a known codirectional coupler 1 comprises two closely spaced waveguides 2 and 3, which are non-symmetrical and which conduct light in the same direction. For example, the two waveguides 2 and 3 may differ in size, by having different widths and/or heights, and/or may be differently shaped and/or may be formed on different slab regions. As a further alternative the waveguides may be rendered non- symmetrical by being formed of different materials or materials having different refractive indices as a result of doping, carrier injection, heating, acousto-optical effects, electro-optical effects, or the use of metallic or semiconductor cladding layers providing periodic variation in stress.

The two waveguides 2 and 3, which interact with one another due to their close proximity, have effective refractive indices of n_a and n_b respectively. Where these indices n_a and nb are sufficiently dissimilar, very little power transfer occurs between two waveguides 2 and 3, because the optical modes in each waveguide are not phase matched. However, if the waveguide 3 is periodically perturbed by a linear grating 4 (or non-linear grating as described more fully below) with a suitable period, as shown in the coupler 1 ' of Figure 2, quasi-phase matching is achieved and a periodic transfer of power between the waveguides 2 and 3 occurs. An alternative view is that power transfer occurs at each perturbance by the grating 4, and the grating 4 is selected so that the coupled light adds up in phase. For a particular wavelength the period Λ necessary for phase matching is given by,

2 -π β_a ^~β_b were __ and are the propagation constants of the two system supermodes, where

_ 2 - π - n ^{P ~} λ and λ specifies the vacuum wavelength and the effective index of the appropriate waveguide mode is denoted by n.

In order for the device to provide highly polarisation dependent behaviour, a stress-inducing top layer is arranged to stress the waveguides, thereby increasing the waveguide birefringence to such an extent as to ensure that a mode within an appropriate wavelength range is transferred from the waveguide 2 to the waveguide 3 only for a given polarisation. In this context the birefringence is the difference between the effective refractive indices for the two optical modes, and can be considered as having two constituents, namely (i) waveguide (or shape) birefringence which is caused by the asymmetry of the waveguide geometry, and (ii) material birefringence which is related to the symmetry of the crystal structure. For example, in a symmetric structure like silicon, the material birefringence should be zero. The application of stress leads to asymmetry in the crystal lattice which in turn leads to birefringence. The stress results from the differing thermal expansion coefficients of the stress-inducing layer and the underlying material of the waveguides, the stress-inducing layer generally being deposited at high temperature when the underlying structure is unstressed so that, on cooling, stressing occurs as a result of the differing expansion coefficients. The stress-inducing layer may take a variety of forms, and may induce either compressive or tensile stress. Examples of suitable materials for such a layer include silicon oxides, silicon nitrides and silicon oxynitrides. Such materials are particularly suitable because of their ease of deposition onto silicon and compatibility with standard semiconductor processing techniques, but films of many other materials may also be used including dielectric materials such as CND diamond / carbon, polysilicon, TEOS (tetraethylorthosilicate Si(OCH₂CH₃)₄),), AlO₃, TiO₂ and other oxides. Other techniques, such as flame hydrolysis or spin coating, may be used to deposit glass coatings. Polymer coatings may also prove suitable. Metal films on silicon can also be used to produce the necessary stress effects. In addition crystalline semiconductor materials with a slight lattice mismatch may be grown on top of the silicon to produce similar stress effects.

Other embodiments of the invention may utilise one or more of the following features: (i) stress releasing grooves, (ii) an undercut waveguide portion to enhance stress, (iii) films on both sides or film removal on one side of the substrate, (iv) annealing to relieve stress, (v) physical application of stress or pressure by a pressure plate pressing downwardly on the waveguide portion, and (vi) piezoelectric material (or a similar class of material which exhibits change in length when a voltage is suitably applied).

Thus the coupler V of Figure 2 has a wavelength dependant transfer characteristic, as phase matching only occurs around a given wavelength, as well as exhibiting high birefringence so that one of the modes is transferred from one waveguide to the other whilst the other mode remains within the original waveguide. The grating periods required are typically long, that is of the order of 100 μm, which can be easily realised by conventional lithography. Furthermore, by appropriate design of the grating 4 and the length of the coupler 1', it is be possible to construct an efficient wavelength filter with the desired bandwidth. Whilst the grating 4 of Figure 2 is shown as a linear grating, that is a grating having a constant period and index variation, it may be preferable for a non-linear grating to be used for which the period or mark-space ratio vary along the device. For example, by chirping the grating 4, that is by increasing or decreasing the period of the grating as a function of distance along the grating, a required filter response may be obtained. One disadvantage of the coupler 1 ' is that the grating 4 introduces losses, but such losses would generally only significantly affect the routed signal, that is the signal which is transferred from one waveguide to the other.

In a practical implementation of such a codirectional coupler 5 shown in Figure 3 utilising one relatively thick waveguide 2 and one relatively thin waveguide 3, it is advantageous for the separating bends 6 and 7 to be provided only in the relatively thick waveguide 2 and not in the relatively thin waveguide 3 as the thin waveguide 3 would experience considerable bend losses. Such bends 6 and 7 serve to bring the waveguides 2 and 3 closely adjacent one another in the region 8 over which coupling is to occur whilst separating the waveguides 2 and 3 outside this region. The coupler 5 shown in Figure 3 may be designed to provide 1310/1550 run routing, although the technique could also be used for other wavelength bands. In this case light comprising a first optical mode of 1310 nm wavelength and a second optical mode of 1550 run wavelength is input to the waveguide 2. In the coupling region 8 the first optical mode is transferred from the waveguide 2 to the waveguide 3 whereas the second optical mode is not transferred, with the result that light of 1550 nm wavelength is outputted by the waveguide 3 and light of 1310 nm wavelength is outputted by the waveguide 2.

There is considerable potential for device optimisation by appropriate selection of the grating period, the index step, and the length for example. The filter bandwidth can be controlled by appropriately varying the grating index step(determined by the amount change in step between steps the periodic changes in the waveguide dimension). In some implementations the grating may have an index step whereas in other implementations the grating may have arm index step in dependence on the distance along the waveguide. In general codirectional couplers formed by perturbing one waveguide in a coupled waveguide system will be lossy. However, by suitable design, loss can be minimised at the wavelength which does not couple. The grating pitch affects the losses of the device, and it is believed that such losses will be less in devices with a wide bandwidth. As with other grating devices a flatter passband can be achieved if an appropriately chirped grating is used. The use of nonlinear or chirped gratings or gratings with weighted coupling to manipulate the filter characteristics will be better understood by referring to the following references: "Design of grating-assisted waveguide couplers with weighted coupling", Kim A. Winick, Journal of Lightwave Technology, vol 9, no. 11, p. 1481, 1991; and "Fabrication of sidelobe-supressed InP- InGaAsP vertical coupler optical filter using grating pair structure", D-B Kim et al, IEEE Photonics Technology Letters, Nol 10, no. 11, p. 1593, November 1998.

Other methods of achieving phase matching which do not use such a grating are possible, such as periodically meandering waveguide systems in which the waveguides lie closely adjacent to one another, but are more widely spaced apart at intermediate positions. In this case the meander distance between the positions at which the waveguides lie closely adjacent to one another may be varied, and may constitute a pattern which is repeated along the length of the device, so as to effect the required phase matching. As in the case described above in which phase matching is achieved by use of a grating, optimisation of the phase matching can be achieved by appropriate selection of the meander distance and the period of pattern repeats, as well as the device length and bandwidth. The meander distance will be small, typically between 3μm and lOμm.

In general any periodic perturbation of one or both of the waveguides may be used to effect the necessary phase matching, and it is possible to conceive of other arrangements which could be used to achieve the required effect in a device in accordance with the invention. For example the width or height of one or both of the waveguides may be varied along the length of the device, as well as the positions of the waveguides with respect to one another and the width of the slab region on which the waveguides are fabricated. Alternatively the perturbations may be achieved by variation of the refractive index of the material of the waveguides, for example by means of doping, carrier injection, heating, acousto-optical effects, electro-optical effects, or the use of metallic or semiconductor cladding layers providing periodic variation in stress.

Figure 4 shows the coupling region 10 of a codirectional coupler in accordance with the invention comprising two closely coupled waveguides 2 and 3 formed by etching in an epitaxial layer 11 on a silicon substrate 12 having a buried silicon dioxide layer 13. The grating 4 within the waveguide 3 is shown in broken lines and is constituted by a series of discontinuities in one of the walls of the waveguide 3, such as indents in the wall which are produced during the etching process. A stress-inducing silicon dioxide layer 14 formed on top of the waveguides 2 and 3 causes stressing of the waveguides 2 and 3 so as to increase the waveguide birefringence to an extent that ensures that a mode within an appropriate wavelength range is transferred from the waveguide 2 to the waveguide 3.

Figure 5 is a block diagram of an integrated transmitter/receiver module 20 comprising a transmitter laser 21 for transmitting light of 1310 nm wavelength and an optical receiver 24 for receiving light of 1550 nm wavelength. In order to prevent back reflections from disturbing the operation of the laser 21, the 1550 nm signal transmitted by a transmitter laser 21 is conducted to an optical fibre 22 by way of an optical isolator comprising two codirectional polarisers 25 and 26 and an intervening rotator assembly 27 comprising a Faraday rotator and an integrated optic rotator (which are each designed to rotate the polarisation of a 1310 nm signal by 45°). The polariser 25 is arranged to permit light of a particular optical mode, e.g. TE mode, to pass whereas the polariser 26 is arranged to permit light of a different optical mode, e.g. TM mode, to pass. Thus, if the 1310 nm transmitted light is appropriately polarised and is rotated through 90° on passing through the rotator assembly 27, the transmitted light can be arranged to pass through the isolator substantially unattenuated. The light transmitted back along the optical fibre 22 will include both the light to be received of 1550 nm wavelength, as indicated by the arrow 28, and unwanted backreflected light of 1310 nm wavelength, as indicated by the arrow 29. The light of 1550 nm wavelength passes through the isolator to the receiver 24 substantially unattenuated by virtue of the fact that such light is in an appropriate wavelength range such that it is not decoupled by the polarisers 25 and 26. By contrast, the backreflected light of 1310 nm wavelength is strongly attenuated by the isolator by virtue of the fact that such light is in the wavelength range such that it is decoupled by both of the polarisers 25 and 26, the 1310 nm light being transferred to the other waveguides of the polarisers so that it is prevented from reaching the receiver 24 Thus the wavelength-dependent polarisation of the optical isolator 23 provides effective optical isolation of the receiver 24 from the backreflected light of the transmitter wavelength, 1550 nm.

A brief description will now be given, with reference to Figure 6, of the fabrication steps which may be used in the fabrication of a codirectional coupler in accordance with the invention in a SOI structure. Initially an epitaxial layer 11 is grown on top of a buried silicon dioxide layer 13 on a silicon substrate 12, and a layer 30 of silicon dioxide is optionally formed in the epitaxial layer 11 by thermal oxidation, as shown in Figure 6(a). The substrate 12 has a thickness in the range 100-1000 μm, typically about 500 μm, whereas the buried layer 13 has a thickness in the range 0.05-2 μm, typically about 0.4 μm, and the epitaxial layer 11 has a thickness in the range 0.5- 20 μm, typically about 5 μm.

Photoresist is then applied on top of the layer 30 (where present) and is exposed through a mask and developed prior to being removed in selected areas in order to define windows through which the layer 30 may be etched. Wet or dry etching (preferably dry etching using a chlorine or fluorine based chemistry) is then effected through the windows in order to remove the silicon dioxide layer 30 in those regions below the windows to produce the structure having two ribs 31 and 32 of silicon dioxide of different widths, typically in the range of 0.5-10 μm separated by a gap in the range of 0.2-5 μm, as shown in Figure 6(b), after removal of the remaining photoresist. As indicated by broken lines 36 the rib 32 has regularly spaced indentations along its side wall facing the rib 31.

Wet or dry etching of the epitaxial layer 11 is then effected using the ribs 31 and 32 as a mask (or simply using the photoresist as a mask where no layer 30 is provided) so as to produce the structure having ribs 33 and 34 of silicon beneath the ribs 31 and 32 as shown in Figure 6(c). The total depth etched may be in the range of 0.5-15 μm, and is typically about 1.7 μm. As indicated by broken lines 37 the rib 34 has regularly spaced indentations along its side wall facing the rib 33, corresponding to the indentations along the side wall of the rib 32. Finally a layer 35 of silicon dioxide is applied by thermal oxidation over both the ribs 33 and 34 and the remainder of the substrate to produce the stress-inducing layer and the structure defining the closely spaced waveguides 2 and 3 as shown in Figure 6(d). The thermal oxide layer 35 has a thickness of 0.05-3 μm, typically about 0.4 μm, and is grown by reaction of the silicon in a wet atmosphere (steam) at a temperature of between 700 and 1200°C for 0.2 to 10 hours. The precise growth conditions of this stress-inducing layer are chosen to optimise the birefringence to the desired device response. It will be appreciated that the waveguide 3 incorporates a grating formed by the indentations in the side wall of the rib 34.

Various modifications of the above described arrangements are possible within the scope of the invention. For example, instead of the coupling between the waveguides being chosen to transfer one of the optical modes fully from one waveguide to the other, the coupling between the waveguides may be chosen to obtain the desired ratio of the optical modes in one or both of the output waveguides. Additionally the device may be used in the reverse manner to that described with reference to Figure 3, that is to combine the two optical modes rather than to split the optical modes apart. A similar device may be used as a wavelength selective polariser in which only light of a wavelength within the selected passband is polarised by the device, whereas light of a wavelength outside the passband is not polarised by the device.

Furthermore a device in accordance with the invention the invention may be used as an integrated polarisation dependent loss (PDL) compensator and controller which serves to compensate for PDL by separating the input light signal into two polarisation components, for example a TE polarisation component and a TM polarisation component, and by inducing more loss in one polarisation component than the other polarisation component before recombining the polarisation components to obtain the output light signal. Generally such a device could be used to provide a variable ratio of TE/TM polarisation components in a waveguide by varying the relative powers of the two polarisation components before recombining them.

Alternatively the device could serve to totally extinguish one of the polarisation components in which case the device would serve as a polariser producing light having only one of the polarisation components. Such a device can be produced to have a high extinction ratio with a low loss for the remaining polarisation component.

Furthermore the device may be incorporated in a polarisation switch or active wavelength router or in a polarisation diversity receiver, or may be applied to demultiplex/multiplex cross-polarised channels in an optical communication system.

To reduce sensitivity, for example, in order to achieve polarisation commonality, a device in accordance with the invention may be designed to split an input signal into two polarisation components. One component may then be rotated into the same state as the other component using a polarisation rotator. The two identical polarisation components may then be supplied together to the polarisation sensitive system. A similar device in accordance with the invention may be used in polarisation diversity receivers for coherent communications, whereby the two components are split, and mixed separately with orthogonal local oscillators. Furthermore a device in accordance with the invention may be used in a digital communication system in which 0 is denoted by one polarisation state and 1 by the other, in order to reduce sensitivity to variation in polarisation due to propagation which would otherwise lead to cross-talk.

Additionally devices in accordance with the invention may be used as switching devices. By using a splitter, the routing of the signal will be dependent on the polarisation. This may be extended to active wavelength routers, for example electro- optic tunable filters and acousto-optic tunable filters using polarisation splitters and converters which are wavelength sensitive. In the context of wavelength division multiplexed systems, it may be advantageous to launch adjacent channels with cross polarisations to reduce non-linear beating effects. Polarisation splitters could be used in such systems for demultiplexing.

Claims

CLAIMS:

1. A coupled waveguide system comprising a substrate, a first waveguide (2) formed on the substrate (12), a second waveguide (3) formed on the substrate (12) closely adjacent to the first waveguide (2), the first and second waveguides (2, 3) being non-symmetrical with respect to one another, perturbation means (4) provided within the second waveguide (3) to effect phase matching between the first and second waveguides (2, 3), and stress-controlling means (14) providing stressing of the first and second waveguides (2, 3) to increase the waveguide birefringence such that an optical mode within an appropriate wavelength range in one of the first and second waveguides (2, 3) is transferred to the other one of the first and second waveguides (2, 3) by interaction between the first and second waveguides (2, 3) and is thereby separated from any other optical mode in said one waveguide .

2. A coupled waveguide system according to claim 1, wherein the stress- controlling means comprises a stress-inducing layer (14) on the waveguides (2, 3).

3. A coupled waveguide system according to claim 2, wherein the stress-inducing layer (14) is a thermal oxide layer grown on the waveguides (2, 3).

4. A coupled waveguide system according to claim 1, 2 or 3, wherein the perturbation means comprises a grating structure (4) within the second waveguide (3).

5. A coupled waveguide system according to claim 4, wherein the grating structure (4) comprises a series of discontinuities in one of the walls of the second waveguide (3).

6. A coupled waveguide system according to any one of claims 1 to 5, wherein the perturbation means (4) has a constant period related to said wavelength range.

7. A coupled waveguide system according to any preceding claim, wherein the first and second waveguides (2, 3) have different widths.

8. A coupled waveguide system according to any preceding claim, wherein one of the first and second waveguides (2, 3) is bent so as to run closely adjacent to the other of the first and second waveguides (2, 3) for only a limited proportion of the overall length of each waveguide.

9. A coupled waveguide system according to any preceding claim, wherein the interaction between the first and second waveguides (2, 3) is codirectional.

10. A coupled waveguide system according to any preceding claim, which is integrated with an optical circuit.

11. A coupled waveguide system according to any one of claims 1 to 10, which constitutes a wavelength selective polariser.

12. A coupled waveguide system according to any one of claims 1 to 10, which constitutes an optical isolator.

13. A coupled waveguide system according to any one of claims 1 to 10, which is incorporated in a polarisation mode dispersion (PMD) compensation device .

14. A coupled waveguide system according to any one of claims 1 to 10, which is incorporated in a device which compensates for polarisation dependent effects, such as a polarisation dependent loss (PDL) reduction device .

15. A coupled waveguide system according to any preceding claim, wherein the substrate (12) is a silicon-on-insulator (SOI) substrate.

16. A method of splitting apart or combining together two optical modes comprising passing the optical modes through closely spaced first and second waveguides (2, 3) which are non-symmetrical with respect to one another, one of the first and second waveguides (2, 3) being provided with a grating structure (4) to effect phase matching between the waveguides (2, 3), and stressing the first and second waveguides (2, 3) to increase the waveguide birefringence, whereby the optical modes are split apart or combined together by interaction between the first and second waveguides (2, 3).