Patent Description:
A low voltage, low insertion-loss electro-optical modulator (EOM) is a key element for high-speed optical switching. Quantitatively, an insertion loss of <NUM> dB or less is desirable. For a switching of the electro-optical modulator in a GHz frequency rage, a control voltage lower than <NUM> V is beneficial, since providing larger control voltages at such high frequencies becomes increasingly challenging. To achieve a control voltage of <NUM> V or less using known opto-electronic materials, an optical path in the electro-optical material of several centimeters and a distance between electrodes of the EOM and the path of light of few decades of micrometers are simultaneously required. These may be achieved using a waveguide architecture for the electro-optical modulator. A planar waveguide manufactured in a LiNbO<NUM> film is a typical main unit in a high-speed modulator. However, such a waveguide possesses a high propagation loss, which typically exceeds <NUM> dB/cm. A waveguide manufactured in an electro-optical crystal (or crystalline film) with a propagation loss lower than <NUM> dB/cm could provide a key element for optical switching operations in the GHz frequency range.

<CIT> describes a dynamically tunable electro-optic cladding. <NPL>describes waveguide beam splitters produced in an LiNbO<NUM> crystal by direct femtosecond laser writing of optical-lattice-like cladding structures.

In view of the technical problems laid out above, there is a need for a waveguide device with a propagation loss below <NUM> dB/cm and a control voltage of at most <NUM> V to promote fast (≥ <NUM>) switching. In particular, these operation parameters ought to be achieved in an electro-optical modulator, for example a phase shifter such as a π phase shifter.

This objective is achieved with a waveguide device according to claim <NUM>. Claim <NUM> provides an electro-optical modulator comprising the waveguide device. Claim <NUM> refers to a method for fabricating a waveguide device suitable to provide the key parameters laid out above. The dependent claims relate to preferred embodiments.

In a first aspect, a waveguide device comprises a substrate comprising an electro-optical material; a waveguide formed in the electro-optical material; and a plurality of electrodes formed in a vicinity of the waveguide. The electro-optical material has a first refractive index. The waveguide comprises a plurality of tracks. The tracks comprise a second refractive index smaller than the first refractive index, are parallel to each other with a common direction defining a direction of the waveguide, and form an arrangement in a plane perpendicular to the direction of the waveguide. The arrangement comprises at least <NUM> equilateral triangles of identical side lengths, wherein all three corners of each of the equilateral triangles each coincide with a different track of the plurality of tracks in the plane perpendicular to the direction of the waveguide.

In the context of the present disclosure, the equilateral triangles may serve as a reference for the arrangement of the tracks, but do not add any physical structure to the waveguide device.

A corresponding waveguide may provide a propagation loss as low as <NUM> dB/cm, or <NUM> dB for a waveguide with a length of <NUM>, respectively, corresponding to the length typically applied in an electro-optical modulator suitable to provide a π phase shifter. The low propagation loss is a consequence of the arrangement of the tracks of the waveguide and the method of track manufaturing. Together with the low refractive index of the tracks, the arrangement provides an embedded cladding for the waveguide. The waveguide geometry supports a low control voltage, which may be below <NUM> V according to embodiments. Particularly beneficial arrangements of the electrodes may further reduce the control voltage.

At least <NUM>, in particular at least <NUM> or, in particular at least <NUM> or all of the equilateral triangles may form a lattice, wherein at least one corner of each equilateral triangle of the lattice coincides with a corner of another equilateral triangle of the lattice. Any triangle of the lattice may be interconnected with any other triangle of the lattice via triangles of the lattice, in particular via sides of triangles of the lattice. The sides of the triangles of the lattice may connect corners of triangles of the lattice. The lattice may be hexagonal and/or form a section of a hexagonal lattice. In particular, at least two corners of each equilateral triangle of the lattice may each coincide with a corner of a different equilateral triangle of the lattice. In particular, all three corners of each equilateral triangle of the lattice may each coincide with a corner of a different equilateral triangle of the lattice.

Any pair of two different equilateral triangles may share at most one same track. Any pair of two different corners of equilateral triangles may share at most one same track. According to embodiments, when corners of a first equilateral triangle coincide with a first track, a second track, and a third track, no more than one corner of any other equilateral triangle coincides with the first track, the second track, or the third track.

The arrangement may comprise at least <NUM> or at least <NUM> of the equilateral triangles with the identical side lengths.

The identical side lengths may be at least <NUM> or at least <NUM>.

Alternatively, or in addition, the identical side lengths may be at most <NUM>, in particular at most <NUM> or at most <NUM>.

In an embodiment, each of the equilateral triangles share their orientation in space, in particular their orientation in the plane perpendicular to the direction of the waveguide.

According to an embodiment, each of the equilateral triangles in the arrangement may be construed as a shifted image of any other equilateral triangle in the arrangement, wherein the shift may be a translation in the plane perpendicular to the direction of the waveguide. The shifted image may be a non-rotated image that is translated linearly in the plane perpendicular to the direction of the waveguide.

The arrangement in the plane perpendicular to the direction of the waveguide may refer to centers of cross sections of the tracks in the plane perpendicular to the direction of the waveguide.

The arrangement may comprise an outer boundary of an essentially hexagonal shape.

The arrangement may comprise at least <NUM>, in particular at least <NUM> or at least <NUM>, tracks, preferably each located in a center of an equilateral hexagon formed by other tracks in the plane perpendicular to the direction of the waveguide, wherein each equilateral hexagon has the identical side length.

According to an embodiment, the waveguide device may have a translational symmetry along the direction of the waveguide.

According to an embodiment, a plurality of planes perpendicular to the direction of the waveguide, in particular all planes perpendicular to the direction of the waveguide, form an arrangement with some or all of the features described above or elaborated further below.

The tracks may form the arrangement in any plane perpendicular to the direction of the waveguide along a length of the waveguide device.

The electro-optical material may comprise or be a nonlinear optical material and/or a crystalline material without inversion symmetry, such as RbTiOPO<NUM> or KTiOPO<NUM>.

Such an electro-optical material may provide a high electro-optical coefficient, which may be beneficial for establishing an electro-optical modulator, such as a π phase shifter, with a short optical path in the electro-optical material. The short optical path in the electro-optical material may reduce propagation losses and reduce a capacitance of the waveguide device, which may promote a high switching speed.

The waveguide may be adapted to operate as a single-mode waveguide. The waveguide may have an extension along at least one direction in the plane perpendicular to the direction of the waveguide smaller than <NUM>, in particular smaller than <NUM> or smaller than <NUM>. In particular, the extension of the waveguide may be smaller than <NUM>, in particular smaller than <NUM> or smaller than <NUM> along any direction in the plane perpendicular to the direction of the waveguide.

The first refractive index may be a refractive index of the electro-optical material for light with a polarization perpendicular to the direction of the waveguide. The first refractive index may be a refractive index of the electro-optical material for an electromagnetic wave with a telecommunication wavelength, such as an electromagnetic wave with a wavelength in vacuum of <NUM> or <NUM>.

Therefore, an electro-optical modulator applying the waveguide device may be adapted to work at a telecommunication wavelength, which is beneficial for communication applications.

The second refractive index may be smaller than the first refractive index by at least <NUM>%, in particular by at least <NUM>%, in particular by at least <NUM>% or by at least <NUM>%.

The second refractive index may be a refractive index of the tracks for an electromagnetic wave with the telecommunication wavelength.

The second refractive index may be a refractive index of the tracks for light with the polarization perpendicular to the direction of the waveguide.

Each track of the plurality of tracks may have an extension smaller than <NUM>, in particular smaller than <NUM> or smaller than <NUM> along at least one direction in the plane perpendicular to the direction of the waveguide. Each track of the plurality of tracks may have an extension of at least <NUM>, in particular of at least <NUM> or of at least than <NUM> along a second direction in the plane perpendicular to the direction of the waveguide, wherein the second direction in the plane perpendicular to the direction of the waveguide is perpendicular to the at least one direction in the plane perpendicular to the direction of the waveguide. Each track of the plurality of tracks may have an extension of at most <NUM>, in particular of at most <NUM> or of at least most <NUM> along the second direction in the plane perpendicular to the direction of the waveguide.

The plurality of tracks may form at least a section of an outer boundary of the waveguide.

The waveguide may comprise a first end, and the waveguide device may comprise a first optical fiber optically coupled to the first end of the waveguide. The waveguide may comprise a second end, and the waveguide device may comprise a second optical fiber optically coupled to the second end of the waveguide.

At least one electrode of the plurality of electrodes or all electrodes of the plurality of electrodes may comprise or be composed of a noble metal, such as copper or gold.

The waveguide comprises a core. The core may be defined by a first interruption of the arrangement of the tracks comprising the equilateral triangles in the plane perpendicular to the direction of the waveguide.

A minimum distance between a center of the core and a nearest electrode of the plurality of electrodes may be at most <NUM>, in particular at most <NUM>, in particular at most <NUM> or at most <NUM>.

A shortest distance between the center of the core and the nearest electrode may reduce the control voltage required to establish an electro-optical modulator with the waveguide device, for example a phase shifter such as a π phase shifter with the waveguide device. A reduced control voltages may improve a switching speed of the electro-optical modulator.

The first interruption may comprise the electro-optical material without the tracks.

The center of the core may refer to the center of the core in the plane perpendicular to the direction of the waveguide.

The nearest electrode may be the electrode of the plurality of electrodes with the smallest distance to the center of the core in the plane perpendicular to the direction of the waveguide. The core may be enclosed by the arrangement in the plane perpendicular to the direction of the waveguide on at least two sides, in particular on at least three sides. The sides may correspond to directions, wherein angles between any two of the directions may be integer multiples of <NUM>°.

The arrangement of the track enclosing the core may be suitable to confine an electromagnetic wave to the core and reduce propagation losses related to a leakage of the electromagnetic wave to regions with an increased dissipation, such as absorptive and/or metallic regions.

The core may have an outer boundary with an essentially hexagonal shape.

A cladding thickness may be a thickness of the arrangement of the tracks in the plane perpendicular to the direction of the waveguide. In particular, the cladding thickness may refer to a distance in the plane perpendicular to the direction of the waveguide from the core of the waveguide to an outer boundary of the arrangement of the tracks. The outer boundary of the arrangement of the tracks may be defined as a polygon comprising all tracks of the arrangement in the plane perpendicular to the direction of the waveguide, in particular as a polygon of a minimum size, such as a hexagon with a minimum size.

The cladding thickness is asymmetric, such that a section of the arrangement of the tracks in the plane perpendicular to the direction of the waveguide with a minimum cladding thickness is arranged between the core and an electrode of the plurality of electrodes, in particular between the core and the nearest electrode. The minimum cladding thickness may correspond to one track. The section of the arrangement with the minimum cladding thickness may comprise or be an interruption of a hexagonal symmetry of the arrangement of the tracks. The minimum thickness of the waveguide and/or of the arrangement of tracks between the core and the electrode reduces the distance between the two. The reduced distance may promote a large electric field, and therefore an operation of the waveguide device, for example as a phase shifter such as a π phase shifter, with a moderate control voltage applied to the electrodes. The moderate control voltage may improve a switching speed of the waveguide device.

The minimum cladding thickness may be smaller than <NUM>, in particular smaller than <NUM> or smaller than <NUM>.

According to an embodiment, the minimum cladding thickness may be at least <NUM>, in particular at least <NUM> or at least <NUM>.

This thickness range may allow for a short distance between the waveguide and/or its core to the nearest electrode, while providing a sufficient confinement of an electromagnetic wave to the waveguide and/or its core to avoid losses related to a leakage of the electromagnetic wave towards the electrodes.

The minimum cladding thickness may be zero, and/or the arrangement of the tracks may be absent in the section of the arrangement with the minimum cladding thickness.

The arrangement of the tracks and the plurality of electrodes may together enclose the core in the plane perpendicular to the direction of the waveguide. In particular, the arrangement of the tracks and the nearest electrode may together enclose the core in the plane perpendicular to the direction of the waveguide. For example, any half-line in the plane perpendicular to the direction of the waveguide starting from the center of the core may intersect the arrangement of the tracks or an electrode of the plurality of electrodes, in particular the nearest electrode.

An absence of the waveguide cladding and/or the arrangement of the tracks in the section with the minimum thickness may further reduce the distance between the waveguide and/or its core to the nearest electrode, and thus a control voltage. However, propagation losses due to leakage of an electromagnetic wave from the waveguide and/or its core to the electrode(s) may be increased in such embodiments.

The at least one electrode of the plurality of electrodes may be arranged on a first surface, wherein the first surface may be a surface of the substrate and/or a surface of the electro-optical material.

Arranging the electrode on the surface of the substrate and/or of the electro-optical material may provide a device design which may be implemented readily and economically using available techniques.

Alternatively, at least one electrode of the plurality of electrodes may be at least partially embedded into the substrate and/or the electro-optical material. In such embodiments, the at least one electrode may be at least partially arranged below a first surface, wherein the first surface is a surface of the substrate and/or a surface of the electro-optical material.

A first electrode of the plurality of electrodes and a second electrode of the plurality of electrodes may be arranged on opposite sides of the waveguide.

At least a section of the first electrode, and/or at least a section of the second electrode may be concentric with an outer shape of the waveguide and/or with the core.

Embedding at least one electrode into the substrate and/or the electro-optical material may bring the electrode closer to the waveguide or its core, and may therefore reduce a control voltage further, yet possibly at the cost of a more complex and potentially more expensive fabrication of the electrodes.

A minimum distance between a center of the core and the first surface may be at most <NUM>, in particular at most <NUM>, in particular at most <NUM>, in particular at most <NUM> or at most <NUM>.

In embodiments with the section of the waveguide with the minimum thickness, the section of the waveguide with the minimum thickness may be arranged in part or completely between the core of the waveguide and the first surface.

The at least one electrode may have an extension along the direction of the waveguide of at least <NUM>, in particular at least <NUM>, at least <NUM> or at least <NUM>.

A corresponding extension may promote the application of the waveguide device as an electro-optical modulator, for example as a phase shifter such as a π phase shifter.

The at least one electrode may be in direct contact with the electro-optical material.

The direct contact may minimize the distance between the electrode and the waveguide and/or its core, and thereby the control voltage of an electro-optical modulator applying the waveguide device.

The at least one electrode may comprise the nearest electrode.

The plurality of electrodes may further comprise at least one counter electrode different from the at least one electrode, wherein the at least one counter electrode is arranged on the first surface.

Arranging also the counter electrode on the first surface may provide a device design which may be implemented readily and economically using known technologies.

The at least one counter electrode may be in direct contact with the electro-optical material.

The at least one counter electrode may have an extension along the direction of the waveguide of at least <NUM>, in particular at least <NUM>, at least <NUM> or at least <NUM>.

The at least one counter electrode may be arranged parallel to the at least one electrode.

In some embodiments, a minimum distance between the at least one counter electrode and the at least one electrode may not exceed <NUM>, in particular not exceed <NUM>, in particular not exceed <NUM> or <NUM>.

The at least one counter electrode may comprise at least two counter electrodes.

The at least two counter electrodes may be arranged on opposite sides of the at least one electrode on the first surface. Alternatively, or in addition, the at least two counter electrodes may be arranged symmetrically in a vicinity of the at least one electrode on the first surface, in particular with respect to a mirror plane intersecting the at least one electrode and/or the core of the waveguide.

The at least two counter electrodes may be arranged parallel to each other.

The at least two counter electrodes may be arranged parallel to the at least one electrode.

A minimum distance between any of the at least two counter electrodes and the at least one electrode may not exceed <NUM>, in particular not exceed <NUM>, in particular not exceed <NUM> or <NUM>.

The first surface may be planar. A surface of the at least one electrode and a surface of the at least one counter electrode may be coplanar with each other, and in particular also be coplanar with the first surface.

In a second aspect, an electro-optical modulator comprises a waveguide device as described above. In particular, the electro-optical modulator may be a phase shifter, in particular a π phase shifter.

The electro-optical modulator may be adapted to operate at a frequency of at least <NUM>.

The electro-optical modulator may be adapted to operate with a control voltage between the at least one electrode and the at least one counter electrode of at most <NUM> V.

The electro-optical modulator may comprise a first lead connected to the at least one electrode and a second lead connected to the at least one counter electrode, wherein the first lead and the second lead may be adapted to connect the at least one electrode and the at least one counter electrode to a voltage source adapted to provide the control voltage.

In a third aspect, a method for fabricating a waveguide device comprises providing a substrate comprising an electro-optical material with a first refractive index; and forming a waveguide in the electro-optical material. The forming the waveguide comprises forming a plurality of tracks of the waveguide such that the tracks are parallel to each other with a common direction defining a direction of the waveguide, and such that the tracks comprise an arrangement in a plane perpendicular to the direction of the waveguide. The arrangement comprises at least <NUM> equilateral triangles of identical side lengths. All three corners of each of the equilateral triangles each coincide with a different track of the plurality of tracks in the plane perpendicular to the direction of the waveguide. The forming of each track of the plurality of tracks comprises focusing a laser beam into the electro-optical material to permanently reduce a refractive index in a focus of the laser beam from the first refractive index to a second refractive index smaller than the first refractive index; and propagating the focus of the laser beam along the direction of the waveguide to form the track with the second refractive index in the electro-optical material.

The method of the present disclosure may provide a technique for laser-writing a waveguide with tracks of a reduced refractive index, also sometimes referred to as an embedded cladding waveguide. The embedded cladding waveguide may provide a low propagation loss. The arrangement of the tracks with the equilateral triangles may reduce the propagation loss even further.

The method may further comprise generating the laser beam using a laser. The laser may be a pulsed laser, in particular a pulsed laser providing a laser beam with a pulse duration below <NUM> ps, in particular below <NUM> ps. The laser may be an infrared laser, providing the laser beam, for example, with a wavelength of at most <NUM>, in particular at most <NUM>.

A repetition rate of the pulsed laser may be at last <NUM>. A repetition rate of the pulsed laser may be at most <NUM>.

The propagating the focus of the laser beam may comprise translating the substrate, in particular while keeping a position of the laser fixed. The translating the substrate may use a translation stage, in particular an at least partially automatized translation stage. Alternatively, or in addition, the propagating the focus of the laser beam may comprise translating the position of the laser beam.

The waveguide device, the substrate, the electro-optical material, the plurality of tracks, and the arrangement in the plane perpendicular to the direction of the waveguide may be characterized by features corresponding to the ones described above in the context of the waveguide device.

A polarization of the laser beam may be linear and perpendicular to the direction of the waveguide.

The focus may refer to a plane perpendicular to a direction of the laser beam, in particular to a focal plane.

In the focus, a width of the laser beam may be minimal along at least one direction perpendicular to the direction of the laser beam.

The focusing the laser beam into the electro-optical material comprises generating a longitudinal width of the laser beam in the focus along the direction of the waveguide and perpendicular to a direction of the laser beam, and generating a transverse width of the laser beam in the focus perpendicular to the direction of the waveguide and perpendicular to the direction of the laser beam, wherein the longitudinal width is larger than the transverse width. The longitudinal width and the transverse width may each refer to a width of the laser beam perpendicular to the direction of the laser beam.

The longitudinal width and the transverse width may each refer to a width of the laser beam in the focus, in particular in a focal plane perpendicular to the direction of the laser beam.

The longitudinal width may refer to a width of the laser beam along the direction of the waveguide. The transverse width may refer to a width of the laser beam perpendicular to the direction of the waveguide. The laser beam may have a non-circular or asymmetric cross section in the focus.

In particular, the laser beam may have an elliptical cross section in the focus. A long axis of the elliptical cross section may be parallel to the direction of the waveguide. A short axis of the elliptical cross section may be perpendicular to the direction of the waveguide.

The non-circular or asymmetric cross section of the laser beam in the focus may result in a reduced roughness of the formed tracks, and in a further reduction of the propagation loss of the waveguide.

The longitudinal width and/or the transverse width may each refer to an extension of the laser beam in the focus and perpendicular to the direction of the laser beam.

The method may further comprise, prior to the focusing, providing the laser beam as an essentially parallel beam with a first extension along the direction of the waveguide and a second extension along a direction perpendicular to both the direction of the waveguide and the direction of the laser beam, wherein the second extension exceeds the first extension.

The essentially parallel beam with the first extension and the second extension may readily be transformed into a laser beam in the focus with the longitudinal width exceeding the transverse width by focusing.

Alternatively, or in addition, the method may comprise introducing an anisotropic focusing element, such as a cylindrical lens or mirror and/or an ellipsoidal lens or mirror and/or a tilted lens, into the laser beam to generate the longitudinal width larger than the transverse width. In such embodiments, the method may comprise generating a second focus in addition to the focus.

In such embodiments, the laser beam may have a second longitudinal width in the second focus along the direction of the waveguide, and a second transverse width in the second focus along a direction perpendicular to both the direction of the waveguide and the direction of the laser beam, wherein the second transverse width is larger than the second longitudinal width.

The method may further comprise, prior to the providing the laser beam as an essentially parallel beam with the first extension and the second extension, providing the laser beam as an essentially parallel beam with an essentially circular cross-section; and shaping the essentially parallel beam with the essentially circular cross-section into the essentially parallel beam with the first extension and the second extension.

The essentially parallel beam may comprise a cross section essentially corresponding to an ellipse. The first extension may correspond to a short axis of the ellipse, and the second extension may correspond to a long axis of the ellipse.

The shaping the essentially parallel beam with the essentially circular cross-section into the essentially parallel beam with the first extension and the second extension may comprise introducing a collimator, such as a slit, to the essentially parallel beam. The collimator may be introduced to the essentially parallel beam along one direction perpendicular to a direction of the essentially parallel beam.

The collimator, such as the slit, may have a width of at least <NUM>. The collimator, such as the slit, may have a width of at most <NUM>.

The method may further comprise forming a plurality of electrodes in a vicinity of the waveguide.

The plurality of electrodes may be characterized by features corresponding to the ones described above in the context of the waveguide device.

The focusing the laser beam into the electro-optical material may comprise transmitting the laser beam through a first surface, wherein the first surface is a surface of the substrate and/or a surface of the electro-optical material.

The forming the plurality of electrodes may further comprise forming at least one electrode of the plurality of electrodes on the first surface.

The first surface may be characterized by features corresponding to the ones described above in the context of the waveguide device.

The forming the plurality of the tracks of the waveguide may comprise forming a track of the plurality of tracks located further away from the first surface prior to a track of the plurality of tracks located closer to the first surface.

In particular, for any pair of tracks of the plurality of tracks with different distances from the first surface, the forming the plurality of the tracks of the waveguide may comprise forming the track of the pair with the larger distance from the first surface prior to the track of the pair with the smaller distance from the first surface.

The forming the plurality of the tracks of the waveguide may comprise forming a track of the plurality of tracks located closer to a center of the waveguide in a plane parallel to the first surface prior to a track of the plurality of tracks located further away from the center of the waveguide in the plane parallel to the first surface.

In particular, for any pair of tracks of the plurality with different distances from the center of the waveguide in the plane parallel to the first surface, the forming the plurality of the tracks of the waveguide may comprise forming the track of the pair with the smaller distance from the center of the waveguide in the plane parallel to the first surface prior to the track of the pair with the larger distance from the center of the waveguide in the plane parallel to the first surface.

The forming the plurality of electrodes may further comprise forming at least one counter electrode of the plurality of electrodes on the first surface, wherein the at least one counter electrode is different from the at least one electrode.

The at least one counter electrode may be characterized by features corresponding to the ones described above in the context of the waveguide device.

The techniques of the present disclosure and the advantages associated therewith will be best apparent from a description of exemplary embodiments in accordance with the accompanying drawings, in which:.

<FIG> is a schematic illustration of a waveguide device <NUM>. The waveguide device <NUM> is formed on and in a substrate <NUM>. It comprises a waveguide <NUM> and a plurality <NUM> of electrodes <NUM>, <NUM>. The waveguide <NUM> is formed in an electro-optical material <NUM>, such as Rubidium titanyl phosphate (RTP, RbTiOPO<NUM>), contained in the substrate <NUM>.

The substrate <NUM> may consist entirely of RTP <NUM>, or it may comprise additional structures, for example for optical or plasmonic wave guiding, or to establish electronic functionality. For this purpose, the additional structure may comprise sections composed of a linear dielectric, a semiconductor, and/or metallic structures. Alternatively or in addition to RTP, other electro-optical materials <NUM> may be applied in the substrate <NUM>, such as potassium titanyl phosphate (KTP, KTiOPO<NUM>).

Applying a control voltage between an electrode <NUM> and counter electrodes <NUM> of the plurality <NUM> modulates the refractive index of a core <NUM> of the waveguide <NUM> and permits to operate the waveguide device <NUM> as an electro-optical modulator. The waveguide <NUM> and the electrodes <NUM>, <NUM> overlap along a distance L defining a length L of the waveguide device <NUM>. Light may be coupled into and out of the waveguide device <NUM> by coupling optical fibers (not shown) to the first end 100a and the second end 100b of the waveguide device <NUM>.

The detailed view of <FIG> illustrates the cross section of the waveguide <NUM> of the waveguide device <NUM> in a plane perpendicular to the waveguide <NUM>. The waveguide <NUM> may have a similar cross section in any plane perpendicular to the waveguide <NUM>. In particular, the waveguide <NUM> may have a similar cross section in any plane perpendicular to the waveguide along the length L of the waveguide device <NUM>. The cross section may deviate slightly along the length L of the waveguide device <NUM>, for example, if the waveguide device <NUM> curves, or if additional structures of the substrate <NUM> are formed in a vicinity of the waveguide device <NUM>.

The waveguide <NUM> is formed by an arrangement <NUM> of tracks <NUM> in the electro-optical material <NUM>. The tracks <NUM> contain modified RTP with a lower refractive index than a refractive index of the RTP <NUM>. Their arrangement <NUM> may be characterized in terms of neighboring tracks <NUM> that intersect corners of equilateral triangles with identical sizes and orientations. The arrangement <NUM> and/or the tracks <NUM> may therefore form a section of a hexagonal lattice, but in some embodiments they deviate from the perfect hexagonal lattice. In the central region <NUM> of the waveguide, the arrangement <NUM> of tracks has an interruption <NUM>, where the tracks <NUM> are absent, and the central region <NUM> consists of essentially unmodified RTP <NUM>.

As the central region <NUM> has a higher refractive index than the tracks <NUM>, the waveguide <NUM> is suitable to confine an electromagnetic wave to the central region <NUM> in the x-y-plane <NUM>. The central region thus acts as the core <NUM> of the waveguide <NUM>. The waveguide <NUM> may be referred to as an embedded cladding waveguide <NUM>. The embedded cladding waveguide <NUM> guides the electromagnetic wave along a direction z of the waveguide <NUM> perpendicular to the x-y-plane <NUM>. Along the direction z of the waveguide <NUM>, the waveguide <NUM> has translational symmetry along its length, and a constant cross sectional structure, possibly apart from minor modifications due to, for example, curving or bending of the waveguide <NUM>.

Embedded cladding waveguides have previously been implemented in materials with garnet-type structures, such as cubic YAG:Nd (propagation loss of <NUM> dB/cm at <NUM>) and orthorhombic YAP:Nd (propagation loss of <NUM> dB/cm at <NUM>), as described in <NPL>) and<NPL>). These embedded cladding waveguides provide low propagation losses. However, for a waveguide device such as an electro-optical modulator, a waveguide in an electro-optical material with a significant electro-optical coefficient, and therefore in a crystalline material without inversion symmetry, such as RTP or KTP, is desirable.

<FIG> and <FIG> illustrate the arrangement <NUM> of the tracks <NUM> in more detail. The arrangement <NUM> may have a strong influence on the propagation losses of the electromagnetic wave propagating in the core <NUM> along the direction z of the waveguide. The waveguide <NUM> according to this description has a minimized propagation loss, which is achieved using the depicted arrangement <NUM> of tracks <NUM>. The arrangement <NUM> may be illustrated with equilateral triangles <NUM> of identical sizes or side lengths, respectively. These equilateral triangles <NUM> are introduced for illustration and reference, and do not add any physical structure to the waveguide <NUM> of <FIG>.

Any of the equilateral triangles <NUM> is constructed in such a way, that each of its corners is located in a cross section of a different track <NUM> in the x-y-plane <NUM> perpendicular to the direction z of the waveguide. Any pair of two different equilateral triangles shares at most one cross section of the same track <NUM>. In other words, if a corner of a first equilateral triangle <NUM> is located in the same cross section of the same track <NUM> as a corner of a second equilateral triangle <NUM>, the other two corners of the first equilateral triangle <NUM> are located in cross sections of tracks different from the cross sections of tracks wherein the other two corners of the second equilateral triangle are located.

<FIG> and <FIG> also illustrate an alternative characterization of the arrangement <NUM> using lattices 118a, 118b, 118c, 118d of equilateral triangles <NUM>. In contrast to the individual equilateral triangles <NUM> described above, which may each be identified at arbitrary positions of the arrangement <NUM> (apart from the restrictions laid out above), the lattices 118a, 118b, 118c, 118d comprise multiple, interrelated equilateral triangles <NUM>.

According to a first definition, any equilateral triangle <NUM> of the lattice 118a, 118b, 118c, 118d shares at least one corner with a neighboring triangle <NUM> of the lattice 118a, 118b, 118c, 118d. For example, the lattices 118a, 118b, 118c, 118d comprise <NUM>, <NUM>, <NUM>, and <NUM> such equilateral triangles <NUM>.

The lattice 118a, 118b, 118c, 118d may alternatively be defined such that any equilateral triangle <NUM> of the lattice 118a, 118b, 118c, 118d shares at least two corners with neighboring triangles of the lattice 118a, 118b, 118c, 118d. According to this definition, for example, the lattices 118a, 118b, 118c, 118d comprise <NUM>, <NUM>, <NUM>, and <NUM> equilateral triangles <NUM>.

The lattice 118a, 118b, 118c, 118d may also be defined such that any equilateral triangle <NUM> of the lattice 118a, 118b, 118c, 118d shares all its corners with neighboring triangles of the lattice 118a, 118b, 118c, 118d. According to this definition, for example, the lattices 118c, 118d comprise <NUM> and <NUM> equilateral triangles <NUM>. The sections 118a, 118b of the arrangement <NUM> of <FIG> do not contain any such equilateral triangles <NUM>, and are therefore not lattices according to this definition.

The arrangement <NUM> with the equilateral triangles <NUM> corresponds to an at least locally and/or approximately hexagonal arrangement. In other words, the arrangement <NUM> exhibits a hexagonal symmetry at least locally and/or approximately. In particular, any set of equilateral triangles <NUM> that may be described as a lattice 118a, 118b, 118c, 118d forms a section of a hexagonal lattice and exhibits an exact, local hexagonal symmetry. Consequently, each of the lattices 118a, 118b, 118c, 118d of equilateral triangles <NUM> corresponds to a section of a hexagonal lattice and has a local hexagonal symmetry. However, this is not necessarily the case for the tracks <NUM> corresponding to the lattices 118a, 118b, 118c, 118d. The tracks <NUM> may have a shape that reduces the symmetry and/or the centers of the tracks <NUM> may be offset from the corners of the equilateral triangles <NUM>, as long as the corners of the equilateral triangles <NUM> are located within the cross sections of the tracks <NUM>.

The arrangement <NUM> with the equilateral triangles <NUM> optimizes the confinement of an electromagnetic wave propagating along the direction z of the waveguide, and minimizes propagation losses. This is particularly important for the embedded cladding waveguide <NUM>, as the relative difference between the refractive indices of the modified RTP of the tracks <NUM> and the unmodified RTP <NUM> is only <NUM> - <NUM>. As the arrangement <NUM> with the equilateral triangles <NUM> is at least locally and/or approximately hexagonal, it provides an optimized (approximately closest) packing density of the tracks <NUM>. The arrangement <NUM> therefore makes best possible use of the limited refractive indices difference for achieving the strongest possible confinement and the lowest possible propagation loss.

<FIG>, <FIG>, <FIG>, and <FIG> illustrate an apparatus and a method for fabricating the waveguide <NUM>. The apparatus comprises a laser system <NUM> with a laser. The laser setup <NUM> emits an essentially parallel laser beam <NUM> at wavelength of <NUM> with a pulse duration of <NUM> fs at a repetition rate of <NUM>, a pulse energy in the range from <NUM> to <NUM> nJ and with a diameter of <NUM>. The laser beam <NUM> is focused into the KTP <NUM> substrate <NUM> through the surface <NUM> using a microscope objective <NUM> with a numerical aperture of <NUM>. Under such conditions, the modified RTP for the tracks forms in the focus <NUM> of the laser beam <NUM>, with a refractive index that is reduced by <NUM> - <NUM> relative to the refractive index of the unmodified RTP. The extension of the focus <NUM> along the direction y of the laser beam <NUM> is in the range from <NUM>-<NUM>. Therefore, also the extension along this direction y of the regions of modified RTP and ultimately of the tracks <NUM> of <FIG>, <FIG> and <FIG> is in the range from <NUM>-<NUM>.

The substrate <NUM> is positioned on a translation stage <NUM> with a translation direction <NUM>, z for propagating the focus <NUM> in the electro-optical material <NUM> to write the tracks <NUM>. <FIG> and <FIG> show the apparatus viewed along the translation direction <NUM>, z, whereas <FIG> and <FIG> show the apparatus viewed along a direction perpendicular to the translation direction <NUM>, z. The translation direction <NUM>, z of the translation stage <NUM> corresponds to the direction z of the waveguide <NUM> to be produced.

A polarization of the laser beam <NUM> is perpendicular to the translation direction <NUM>, z, as this geometry has been found to give the strongest reduction of the refractive index of the modified RTP.

As depicted in <FIG> and <FIG>, along the direction x perpendicular to the direction <NUM>, z of the translation stage <NUM>, the laser beam <NUM> is focused to a transverse width w2 in the focus <NUM>.

As depicted in <FIG> and <FIG>, along the direction z parallel to the direction <NUM>, z of the translation stage <NUM>, the laser beam <NUM> is focused to a longitudinal width w1 in the focus <NUM>. The longitudinal width w1 and the transverse width w2 refer to the widths of the laser beam <NUM> in the focus <NUM> along and perpendicular to the direction <NUM> of the translation stage and of the waveguide, respectively, and are both perpendicular (transverse) to the direction of the laser beam <NUM>. The longitudinal width w1 is larger than the transverse width w2.

As is illustrated in <FIG>, a spectroscopic slit <NUM> with a width of <NUM> may be introduced into the essentially parallel laser beam <NUM> emitted by the laser system <NUM> to achieve the larger longitudinal width w1. The spectroscopic slit <NUM> results in a reduced extension of the essentially parallel laser beam <NUM> along one direction <NUM>, z perpendicular to the direction y of the laser beam <NUM>. Consequently, focusing the laser beam <NUM> using the microscope objective <NUM> results in a larger longitudinal width w1 along this direction <NUM>, z.

As an alternative to the spectroscopic slit <NUM>, an anisotropic focusing element <NUM> such as a cylindrical lens <NUM> may be introduced into the laser beam, in particular into the essentially parallel laser beam <NUM>, to generate the longitudinal width w1 along the direction <NUM>, z exceeding the transverse width w2 along the perpendicular direction y.

A corresponding embodiment is illustrated in <FIG>. In such an embodiment, the focus <NUM> refers to a plane <NUM> perpendicular to the direction of the laser beam <NUM>, at a position along the direction of the laser beam <NUM> where the width w2 of the laser beam <NUM> along the transverse direction (with respect to the translation direction <NUM>, or direction of the waveguide <NUM>, respectively) is minimal. Consequently, in the focus, the laser beam <NUM> exhibits a beam waist w2 along the transverse direction x. In addition, a second focus <NUM>' forms, corresponding to a plane <NUM> perpendicular to the direction of the laser beam <NUM>, at a position beam of a waist of the laser beam <NUM> along the translation direction <NUM>. In the second focus <NUM>', the order of the beam widths is reversed as opposed to the focus <NUM>, i. the transverse width of the laser beam <NUM> exceeds the longitudinal width. A larger longitudinal width in the focus as compared to the transverse width leads to the formation of smoother tracks and a reduced propagation loss of the waveguide formed, and is therefore desirable. Consequently, the focus <NUM> is used for forming the waveguide rather than the second focus <NUM>'. For details regarding the focusing with an anisotropic focusing element such as a cylindrical lens see, e.

A writing of tracks into YAG:Nd using a laser beam with an elliptical cross section has previously been demonstrated in <NPL>). However, the question whether such a technique may be applied to electro-optical materials and/or to produce a triangular arrangement has so far remained elusive.

<FIG> shows a cross section of the apparatus for fabricating the waveguide <NUM> along the plane 216e of <FIG>, <FIG>. In this plane 216e, the laser beam <NUM> has an essentially circular cross section <NUM>.

<FIG> shows a cross section of the apparatus for fabricating the waveguide <NUM> along the plane 216f of <FIG>, <FIG>. In this plane 216f, the laser beam <NUM> has a cross section <NUM> corresponding to an ellipse. An extension e2 along a direction x perpendicular to the translation direction <NUM>, z exceeds an extension e1 along the translation direction <NUM>, z.

<FIG> shows a cross section of the apparatus for fabricating the waveguide <NUM> along the plane <NUM> of <FIG>, <FIG>. This plane <NUM> corresponds to the focus <NUM> of the laser beam. In this plane <NUM>, <NUM>, the laser beam <NUM> has the longitudinal width w1 along the direction <NUM>, z of the translation stage <NUM> and the transverse width w2 along the direction x perpendicular to the direction <NUM>, z of the translation stage <NUM>. The longitudinal width w1 along the direction <NUM>, z of the translation stage <NUM> exceeds the transverse width w2 along the direction x perpendicular to the direction <NUM>, z of the translation stage <NUM>.

<FIG> shows a cross section of the apparatus for fabricating the waveguide <NUM> along the plane <NUM> of <FIG>. This plane <NUM> corresponds to the second focus of the laser beam. In this plane <NUM>, <NUM>', the laser beam <NUM> has a longitudinal width along the direction <NUM>, z of the translation stage <NUM> and a transverse width along the direction x perpendicular to the direction <NUM>, z of the translation stage <NUM>. The longitudinal width along the direction <NUM>, z of the translation stage <NUM> is smaller than the transverse width along the direction x perpendicular to the direction <NUM>, z of the translation stage <NUM>.

The flow diagram of <FIG> summarizes the essential process steps of the method <NUM> for forming the waveguide device. The method starts with providing <NUM> the substrate <NUM> comprising the electro-optical material <NUM> with the first refractive index, and proceeds with forming <NUM> the waveguide <NUM> in the electro-optical material <NUM>. The forming <NUM> the waveguide <NUM> comprises forming <NUM> the plurality of tracks <NUM> of the waveguide <NUM>. The forming <NUM> of any of the track <NUM> comprises focusing <NUM> the laser beam <NUM> into the electro-optical material <NUM> to reduce the refractive index in the focus <NUM>. The forming <NUM> of any of the tracks <NUM> further comprises propagating <NUM> the focus <NUM> of the laser beam <NUM> along the direction z of the waveguide <NUM> to form the track <NUM>.

<FIG> illustrates a preferred embodiment of the method for fabricating the waveguide. The numbers <NUM>' in <FIG> give an order, according to which the tracks <NUM> are written using the method described in the context of <FIG>. Tracks with lower numbers <NUM>' are written first, and tracks with higher numbers <NUM>' are written later. Tracks located further away from the surface <NUM> are written prior to tracks located close to the surface <NUM>. This way, reflections and scattering of the laser beam <NUM> by tracks <NUM> already written into the substrate <NUM> are minimized. When a set of tracks <NUM>, <NUM>' has a same distance from the surface <NUM>, tracks <NUM>, <NUM>' of the set located closer to the center <NUM> of the waveguide are written first.

<FIG> shows a waveguide device with a waveguide <NUM> in an electro-optical material <NUM> of a substrate <NUM>. The waveguide <NUM> may be similar to the one described in the context of <FIG>, <FIG>, <FIG>, <FIG> and/or <FIG>. In addition, a plurality <NUM> of electrodes <NUM>, <NUM> are formed on a surface <NUM> of the substrate <NUM>. A distance h separates the center <NUM> of the core <NUM> of the waveguide <NUM> from a nearest electrode <NUM> with a width w. Two counter electrodes <NUM> are arranged symmetrically at a distance g in the vicinity of the electrode <NUM> on the surface <NUM>. Along the direction z of the waveguide <NUM>, the waveguide <NUM> and the plurality <NUM> of electrodes <NUM>, <NUM> have translational symmetry along their respective lengths, and an identical cross section apart from minor modifications due to, for example, curving or bending of the waveguide device.

A control voltage is applied between the electrode <NUM> and the counter electrodes <NUM> to operate the waveguide device. Applying the control voltage induces an electric field at the position of the core <NUM>, which modifies the refractive index of the core <NUM>. Consequently, the waveguide device acts as an electro-optical modulator to modulate an electromagnetic wave confined to the core <NUM> and propagating along the direction z of the waveguide.

The extent of the modulation depends on the magnitude of the electric field at the location of the core <NUM> induced by applying the control voltage to the electrodes <NUM>, <NUM>. A larger extent of the modulation is desirable, since it allows achieving a preselected modification of the electromagnetic wave, such as a π phase shift, using a waveguide device with a shorter length. The shorter length of the waveguide device reduces propagation losses suffered by the electromagnetic wave as it propagates in the waveguide <NUM> while the preselected modification is carried out.

Generally, providing a waveguide device with a shorter distance h between the core <NUM> and the nearest electrode <NUM> enhances the electric field at the location of the core <NUM> and the extent of the modulation. In the embodiment of <FIG>, wherein a thickness t of the waveguide <NUM> formed by the arrangement <NUM> of tracks <NUM> is essentially the same in each direction (isotropic), the minimum distance h between the center <NUM> of the core <NUM> and the nearest electrode <NUM> is dictated by the thickness t of the waveguide <NUM>. According to the depicted embodiment, the smallest minimum distance h is given by the sum of the thickness t and a radius of the core <NUM>.

<FIG> illustrates the electric field <NUM> induced by applying the control voltage between the electrode <NUM> and the counter electrodes <NUM>. At the position <NUM> of the core, the electric field <NUM> is directed approximately perpendicular to the surface <NUM> and/or to the interface between the electrode <NUM> and the substrate <NUM> (or the electro-optical material <NUM>), respectively. The electric field <NUM> at the position <NUM> of the core may therefore be characterized by its y-component Ey. The y-component Ey is largest at the interface between the electrode <NUM> and the substrate <NUM>, and drops with increasing distance d to the electrode <NUM>.

<FIG> shows the maximum y-component max(Ey) and the minimum y-component min(Ey) of the electric field in the region <NUM> of the core for different widths w of the electrode <NUM>. A small deviation between the maximum y-component max(Ey) and the minimum y-component min(Ey) is desirable to facilitate a uniform modulation of an electromagnetic wave confined in the core <NUM>. <FIG> demonstrates that the deviation is generally small for the waveguide device and may be further minimized using a width w of the electrode <NUM> of around <NUM> to <NUM>.

<FIG> shows the resulting change Δnyy of the refractive index for light with a polarization along a direction y perpendicular to the surface <NUM> achieved by applying a control voltage of <NUM> V between the electrode <NUM> and the counter electrodes <NUM>. The change Δnyy of the refractive index has been determined for a waveguide device with a width w of the electrode <NUM> of <NUM> and a distance g between the electrode <NUM> and the counter electrodes <NUM> of <NUM>.

Using the change of the refractive index, a length L of the waveguide device is calculated which is suitable for operating the waveguide device as a π phase shifter. <FIG> gives the corresponding length L for different widths w of the electrode <NUM>, while the distance g between the electrode <NUM> and the counter electrodes <NUM> amounts to <NUM> and the distance h between the center <NUM> of the core <NUM> and the nearest electrode <NUM> amounts to <NUM>. For an optimized width w of the electrode of approximately <NUM> to <NUM>, such as <NUM>, a length L of the waveguide device of <NUM> is sufficient to establish a π phase shifter. The short length reduces the optical path length of the propagation of the electromagnetic wave in the core <NUM> as it undergoes the π phase shift, and therefore minimizes the propagation losses of the electro-optical modulator.

The length L of the waveguide device for the electro-optical modulator, such as the π phase shifter, may be reduced further by increasing the electric field <NUM> at the location at the core <NUM>, or by increasing the overlap between the electric field <NUM> and an electromagnetic wave confined in the core, respectively.

<FIG> shows an embodiment of the waveguide device with a reduced distance h between the center <NUM> of the waveguide <NUM> and the nearest electrode <NUM>. The reduced distance h is achieved by forming the arrangement <NUM> of the tracks <NUM> with an anisotropic thickness, i. with different thicknesses t, t' along different directions. The thickness t, t' is minimal in the region between the core <NUM> and the nearest electrode <NUM>. Therefore, the minimum thickness t' of the waveguide <NUM> dictates the minimum distance h between the center <NUM> of the core <NUM> and the nearest electrode <NUM>, rather than an isotropic thickness t as in the embodiment of <FIG>. The minimum thickness t' of the waveguide <NUM> is smaller than the isotropic thickness t of the embodiment of <FIG>. This enhances the electric field induced at the position of the core <NUM> by applying a control voltage between the electrode <NUM> and the counter electrodes <NUM>. Consequently, the electro-optical modulator, such as the π phase shifter, may be implemented with a reduced length L and/or a reduced control voltage of the waveguide device. Therefore, a propagation loss of the electro-optical modulator is reduced and/or a switching frequency of the electro-optical modulator is increased. For example, the waveguide device according to the embodiment of <FIG> is operable as an electro-optical modulator in the form of a π phase shifter when it is formed with a length of <NUM> and a control voltage of <NUM> V is applied. Thus, the waveguide device according to the embodiment of <FIG> provides a low-loss, high-speed electro-optical modulator.

<FIG> depicts iso-electrical-field lines <NUM> of an electromagnetic wave propagating in the waveguide <NUM>, for a waveguide device corresponding to the embodiment of <FIG>. In this embodiment, the minimum thickness t' of the arrangement <NUM> of tracks <NUM> corresponds to one single track. In this geometry, the iso-electrical-field lines <NUM> are confined to the core <NUM>, with a negligible overlap with the electrode <NUM>. Consequently, the arrangement <NUM> of tracks <NUM> according to the embodiment of <FIG>, <FIG> provides a functional waveguide with a low propagation loss, while the distance h and the length L are minimized.

<FIG> and <FIG> present an embodiment with a further reduction of the minimum thickness t' of the waveguide <NUM> and consequently of the minimum distance h between the core <NUM> and the nearest electrode <NUM>. In this embodiment, the minimum thickness t' is essentially zero, with no tracks in the region between the core <NUM> and the nearest electrode <NUM>. The distance between the center <NUM> of the core <NUM> and the electrode <NUM> is minimized, and corresponds essentially to a radius of the core <NUM>. Therefore, the electric field <NUM> at the position of the core <NUM> induced by applying a control voltage between the electrode <NUM> and the counter electrodes <NUM> is further enhanced, and a modulation of an electromagnetic wave propagating along the core <NUM> is enhanced.

However, the absence of tracks <NUM> in the region between the core <NUM> and the nearest electrode <NUM> results in a leakage of the electromagnetic wave from the core <NUM> to the electrode <NUM>. This is illustrated in <FIG>, which depicts iso-electrical-field lines <NUM> of the electromagnetic wave for such an embodiment. The overlap between the electromagnetic wave and the electrode <NUM> results in an absorption of the electromagnetic wave in the electrode <NUM> and an increased propagation loss.

In summary, the embodiment of <FIG> and <FIG> may provide an enhanced electric field <NUM> and modulation of the refractive index at the position of the core <NUM>, yet possibly at the cost of a stronger distortion of the electromagnetic field and larger propagation losses.

The graph of <FIG> illustrates the propagation losses 700a, 700b, 702a, 702b, 704a, 704b of waveguide devices with a minimum thickness t' corresponding to one track <NUM> (embodiment of <FIG>, <FIG>) and with a minimum thickness t' of essentially zero (embodiment of <FIG>, <FIG>).

In detail, the individual data sets 700a, 700b, 702a, 702b, 704a, <NUM> indicate the propagation losses of devices with the following characteristics:.

<FIG> demonstrates that the devices with the minimum thickness t' corresponding to one track <NUM> (embodiment of <FIG>, <FIG>) provide propagation losses as low as <NUM> dB/cm (copper electrode) or <NUM> dB/cm (gold electrode) for an optimized distance h between the center <NUM> of the core and the nearest electrode <NUM> of <NUM>. A corresponding waveguide device with a length L of <NUM> therefore has a total propagation loss of <NUM> dB (copper electrode) or <NUM> dB (gold electrode). In contrast, for the devices with the minimum thickness t' of essentially zero (embodiment of <FIG>, <FIG>) propagation losses are around <NUM> dB/cm.

<FIG> and <FIG> depict waveguide devices according to alternative embodiments. The waveguide <NUM> of the embodiments depicted in <FIG> and <FIG> is similar to the waveguide described in the context of the embodiments of <FIG>, <FIG>, and <FIG>, and is fabricated using the apparatus and method described in the context of <FIG> and <FIG>. Each of the waveguide devices also comprises a plurality <NUM> of electrodes <NUM>, <NUM>.

However, the waveguide devices according to the embodiments of <FIG> and <FIG> differ from each other and from the embodiments described above in the geometric arrangement of the electrodes <NUM>, <NUM>, <NUM>. In particular, the electrode <NUM> and the counter electrode <NUM> of the embodiments of <FIG> and <FIG> are arranged at least partially below the surface <NUM>. In other words, the surface <NUM> is located at a position along the direction y perpendicular to the surface <NUM>, which is above at least a part of the electrode <NUM> and the electrode <NUM>.

In the embodiment of <FIG>, the electrode <NUM> and the counter electrode <NUM> are located on opposite sides of the waveguide <NUM>. The electrodes <NUM>, <NUM>, <NUM> are essentially flat in a plane y, z parallel to the waveguide <NUM> and perpendicular to the surface <NUM>. Such electrodes are fabricated by first forming the waveguide <NUM> in the electro-optical material <NUM> of the substrate <NUM> as described in the context of <FIG> and <FIG>. Afterwards, an etching or laser cutting step is performed to selectively remove a section of the substrate <NUM> and produce a void structure with the shape of the electrodes <NUM>, <NUM>, <NUM> to be formed. The etching or laser cutting step may use a photolithographic etching step, in particular an anisotropic etching step, or a maskless laser cutting step or a combination of the two. Afterwards, the structured void is filled with a conductive material, such as titanium, tantalum, gold or copper or a combination thereof, deposited for example from the gas phase in a vacuum chamber.

The embodiment of <FIG> is similar to the embodiment of <FIG>, yet with curved electrodes <NUM>, <NUM>, <NUM>. The curved electrodes <NUM>, <NUM>, <NUM> further improve the overlap between the electric field induced by the control voltage applied between the electrode <NUM> and the counter electrode <NUM> and an electromagnetic wave propagating in the core <NUM>. The curved electrodes <NUM>, <NUM>, <NUM> are produced using at least one photolithographic etching step, in particular using a combination of anisotropic and an isotropic etching steps, and/or a maskless laser cutting step. The more complex shape of the electrodes <NUM>, <NUM>, <NUM> of the embodiment of <FIG> as compared to the electrodes <NUM>, <NUM>, <NUM> described above may make the production of the waveguide device of <FIG> more challenging and expensive.

Claim 1:
A waveguide device (<NUM>), comprising:
a substrate (<NUM>) comprising an electro-optical material (<NUM>) with a first refractive index;
a waveguide (<NUM>) formed in the electro-optical material (<NUM>), the waveguide (<NUM>) comprising a plurality of tracks (<NUM>);
a plurality (<NUM>) of electrodes (<NUM>, <NUM>) formed in a vicinity of the waveguide (<NUM>);
wherein the tracks (<NUM>) comprise a second refractive index smaller than the first refractive index, are parallel to each other with a common direction (z) defining a direction (z) of the waveguide (<NUM>), and form an arrangement (<NUM>) in a plane (<NUM>) perpendicular to the direction (z) of the waveguide (<NUM>); and
wherein the arrangement (<NUM>) comprises at least <NUM> equilateral triangles (<NUM>) of identical side lengths, wherein all three corners of each of the equilateral triangles (<NUM>) each coincide with a different track (<NUM>) of the plurality of tracks (<NUM>) in the plane (<NUM>) perpendicular to the direction (z) of the waveguide (<NUM>);
wherein the waveguide (<NUM>) comprises a core (<NUM>);
wherein the core (<NUM>) is defined by a first interruption (<NUM>) of the arrangement (<NUM>) of the tracks (<NUM>) comprising the equilateral triangles (<NUM>) in the plane (<NUM>) perpendicular to the direction (z) of the waveguide (<NUM>); and
wherein the waveguide (<NUM>) comprises an asymmetric cladding thickness (t, t'), such that a section of the waveguide (<NUM>) with a minimum thickness (t') is arranged between the core (<NUM>) and an electrode (<NUM>) of the plurality (<NUM>) of electrodes (<NUM>, <NUM>), in particular between the core (<NUM>) and the nearest electrode (<NUM>).