Optical waveguide

Abstract

An example optical waveguide includes: a chain of dielectric particles configured to propagate an optical wave in a propagation direction along the chain; wherein each dielectric particle comprises a shape defining a radiation profile having a radiative region and at least one node,; wherein adjacent dielectric particles are spaced to support a resonant mode to transmit the optical wave between the adjacent dielectric particles in the propagation direction, and said particles are spatially arranged into an array with a pitch such that radiation phase matching is achieved at a diffraction angle substantially aligned with the direction covered by at least one node in the particle radiation profile.

Description

P12545PC00OPTICAL WAVEGUIDE FIELD

[0001] The specification relates generally to optical waveguides, and more particularly to an on-chip optical waveguide with low mode index and low loss.BACKGROUND

[0002] Optical waveguides are employed in building integrated photonic circuits and function to guide electromagnetic waves in the optical spectrum, for example in a given dimension from a source location to a destination. In the field of functional waveguides composed of periodic structures, the subwavelength grating waveguides have minimal diffraction losses, however operate in the lowest band of the dispersion diagram, below the fundamental photonic bandgap and the light line. Mie resonant particles can also be used in waveguide structures, however feature a large wavevector and mode index, operating below the light line.SUMMARY

[0003] According to one aspect of the present specification, an example optical waveguide includes: a chain of dielectric particles configured to propagate an optical wave in a propagation direction along the chain; wherein each dielectric particle comprises a shape defining a radiation profile having a radiative region and at least one node; wherein adjacent dielectric particles are spaced to support a resonant mode to transmit the optical wave between the adjacent dielectric particles in the propagationP12545PC00direction, and wherein said dielectric particles are spatially arranged along the chain such that radiation phase matching is achieved at a diffraction angle substantially covered by the at least one node in the particle radiation profile.BRIEF DESCRIPTION OF DRAWINGS

[0004] The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

[0005] Implementations are described with reference to the following figures, in which:

[0006] FIG. 1 depicts an example optical waveguide with low light loss and small mode index, in accordance with the present disclosure.

[0007] FIGS. 2A and 2B depict cross sections and a magnetic quadrupole excited in an example particle as described herein.

[0008] FIG. 2C depicts the electric field distribution of the particle of FIGS. 2A and 2B in an optical waveguide.

[0009] FIG. 3 depicts a schematic diagram of the operation of light propagation and radiation suppression in the optical waveguide of FIG. 1.

[0010] FIG.4A depicts a schematic diagram of light propagation and radiation in a surface grating optical waveguide.

[0011] FIG. 4B depicts a band diagram of the optical modes of the surface grating optical waveguide of FIG. 4A.

[0012] FIGS. 5A and 5B depict band diagrams of the optical modes of the optical waveguide of FIG. 1.P12545PC00

[0013] FIG. 5C depicts the electric field and magnetic field profiles of the optical waveguide of FIG. 1.

[0014] FIG.6 depicts a band diagram of the optical modes for use of the optical waveguide of FIG. 1 in a phase shifter.DETAILED DESCRIPTION

[0015] Guided modes of a waveguide are generally characterized by a wavevector that is larger than that of the radiation modes for a given operating wavelength. In particular, for periodic waveguides, the wavelength range is limited by coupling to radiation modes, which is particularly strong in the region above the light line (i.e., with a propagative wave vector smaller than the wavevector of cladding modes). Thus, such optical waveguides typically operate below the light line to reduce light loss via radiation, resulting in a larger operating wavevector and therefore an effective mode index larger than the material index of the claddings. Confining and manipulating light in in-plane integrated optical waveguides with small wavevector, and particularly an effective mode index smaller than the material index of cladding, remains a challenge.

[0016] For example, some optical waveguides may employ periodic structures which confine light for propagation via total internal refraction. Such waveguides generally operate in the lowest band of the dispersion diagram, below the fundamental photonic bandgap and the light line. Other waveguides comprised of chains of Mie resonant particles may also manipulate and guide the flow of light. Such waveguides feature a large wavevector and mode index, operating below the light line to achieve intrinsically low optical losses. One way to confine an optical mode above the light line is to suppressP12545PC00radiation with a symmetry-protected mode based on bound-state-in-continuum in two-dimensional slab photonic crystals. However, this condition can only be fulfilled for extended structures along the transverse direction, and not in one-dimensional waveguides with wavelength-scale cross-sections. Such waveguides do not support inplane modes and do not easily conduce to practical integrated photonic circuits.

[0017] FIG. 1 depicts an optical waveguide 100 including a plurality of dielectric particles 104 arranged in a chain 108 to propagate light (i.e., in the form of an optical wave) in a propagation direction 112 along the chain 108. That is, the dielectric particles 104 (also referred to herein as simply the particles 104) are aligned one-dimensionally to form the chain 108. The direction in which the chain 108 extends, which in the present example coordinate system 116 is in the x-direction, defines the propagation direction 112 of light guided by the optical waveguide 100.

[0018] The optical waveguide 100 is configured to operate above the light line (i.e., having a propagative wavevector smaller than the wavevector of the cladding modes) while maintaining a small wavevector and therefore small effective mode index with minimal loss via radiation. In particular, the particles 104 are substantially equivalent and form a periodic array and are formed of a dielectric material, such as silicon, germanium, lithium niobates, or lll-V semiconductor particles. To confine the optical mode above the light line, the geometries of the individual particles 104 are controlled to support multipole resonances which are used to engineer the radiation pattern of the particles.

[0019] That is, the particles 104 have a shape selected to define a particular radiation profile to enable the propagation of light with small wavevector and low radiation loss. Specifically, each particle 104 defines a radiation profile having a radiative region and atP12545PC00least one node (i.e., at least one nonradiative region, or region where radiation is substantially suppressed). The particles 104 are arranged along the chain 108 and spaced to generate resonant modes. That is, the radiative regions of adjacent particles 104 may be aligned and spaced to optically couple with one another to transmit optical wave between the particles 104 in the propagation direction 112 along the chain 108. Thus, when the optical mode of a first particle 104 in the chain 108 is excited, the resonant mode allows a corresponding optical mode of the second adjacent particle 104 in the chain 108 to be similarly excited, allowing the optical wave (i.e., the light) to be propagated along the chain 108.

[0020] Further, the particle 104 is shaped to such that the at least one node in the radiation profile covers an in-phase radiation angle of the optical wave of the particle chain (e.g., as defined by the momentum matching condition). That is, the optical mode of the optical waveguide 100 operates at a small wavevector situated above the light line in the upper band, which propagates via the resonant mode according to the radiation profile and spacing of the particles 104 along the chain 108. Thus, the optical mode couples to a radiation mode defined by the radiation angle according to the momentum matching condition, is covered by the at least one node of the radiation profile of the particles 104, thereby suppressing optical loss via radiation.

[0021] For example, referring to FIG. 2A, a first cross section 200-1 of one of the particles 104 is depicted. In particular, the particle 104 has a substantially square first cross section 200-1 in the x-y plane. Accordingly, the substantially square first cross section 200-1 has a radiation pattern 204-1 of a magnetic quadrupole with four radiative poles 208-1 corresponding to each corner of the square and four nodes 212-1 at the edges of theP12545PC00square. Similarly, referring to FIG. 2B, a second cross section 200-2 of the particle 204 is depicted. The particle 104 also has a substantially square second cross section 200-2 in the x-z plane. Accordingly, the substantially square second cross section 200-2 has a similar radiation pattern 204-2 of a magnetic quadrupole with four radiative poles 208-2 corresponding to each corner of the square and four nodes 212-2 at the edges of the square.

[0022] Together, the substantially square cross sections 200-1 and 200-2 (referred to herein collectively as the cross sections 200 and generically as a cross section 200; this nomenclature is also used elsewhere herein) define a substantially cuboid shape for the particle 104. In some examples, the particle 104 may be a cube, such that the length, width and thickness (i.e., x-, y- and z- dimensions of the particle 104) are equal. For example, the dimensions of the cubic particle 104 may each be 500 nm for C Band optical wavelengths. The dimensions are chosen based on the material indexes and the optical wavelength considered through optical band simulation. Numerically, a longer wavelength and a lower waveguide material index may result in selection of larger dimensions for the cubic particle 104. For example, each dielectric particle may have a dimension of approximately 1 / n of a vacuum wavelength of the light to be transmitted by the waveguide 100, where n is the material refractive index of the particle.

[0023] Based on the substantially square cross sections 200, the magnetic moments of the particle 104 orientate along the normal directions of the cube surfaces. When the resonant quadrupole modes are optically excited, the particle 104 acts effectively as a directional nanoantenna. The excitation of the two quadrupole modes results in an overall radiation profile of the particle 104 having radiative fields or regions along the diagonalP12545PC00directions, pointing from the center to the eight corners of the cube. Further, there are nodes in the radiation profile, substantially forming a cone about the magnetic moments of the particle and having an apex at the center of the cube and base at the surfaces of the cube.

[0024] The combination of the quadrupole modes in the unit cell of the said waveguide forms an electric field distribution as shown in FIG. 2C.

[0025] Leveraging the resonant and radiative features, the optical waveguide 100 is able to transmit light in accordance with the principles as illustrated in FIG. 3. In particular, light is transmitted in the propagation direction 112 (i.e., along the chain 108 between adjacent particles 104). In particular, the particles 104 are spaced apart by a grating period A (or pitch) to support a resonant coupling between the radiative fields or regions 300 of adjacent particles 104 of the chain 108. Accordingly, the resonant coupling allows light 304 to be guided in-plane, along the optical waveguide 100 in the propagation direction 112, in a Floquet-Bloch mode.

[0026] In typical surface diffraction grating couplers, and as depicted schematically in FIG.4, a waveguide 400 may have a grating structure 402 with unit cells 404 spaced apart by a grating period of A. The waveguide 400 may operate above the light line and have an effective mode index smaller than nciad(i.e., the material index of the cladding). The optical mode may therefore couple to the radiation modes in the cladding and cause optical losses of the waveguide transmitted mode. In particular, the input waveguide mode of the wavevector = koneff(where k0is the vacuum optical wavevector and neff is the effective mode index of the waveguide) will be partially diffracted to the cladding with a wide radiation cone 408, with diffracted light from adjacent unit cellsP12545PC00having a relative phase difference of cp = ^A. Due to the phase difference of the diffracted light, a constructive wavefront is formed at an angle 6 satisfying the phasematching condition given by equation (1 ):< P ~ kciad^ sin(0) = m2n (1)

[0027] where m is an integer and kctadis the wavevector of the cladding material.

[0028] In the simplest case, where m = 1, then sin(0) =where?2= ft - is the in-plane wavevector component of the radiation mode in the cladding. That is, the diffraction grating behaves in accordance with the band diagram as depicted in FIG. 4B. Dictated by the Floquet-Bloch boundary condition, the optical mode of the frequency a>0operates at a small wavevector= ft, situated above the light line in the upper band. This mode simultaneously propagates and couples to the radiation mode which has an in-plane wavevector component equal to f$Band therefore a radiation angle 0 = sin-1(TT2-). That'kclad' is, a radiative beam 412, as depicted in FIG. 4A, is emitted as a result of the coupling to the radiation mode in the cladding, resulting in leakage of light propagating in-plane in the optical mode of the waveguide. The effective mode index of this leaky Bloch mode can therefore be computed as.

[0029] Returning to FIG. 3, and applying the principles of the diffraction grating couplers that descripted in the context of FIG. 4 above, the particles 104 in the chain 108 forming the optical waveguide 100 may be spaced in a periodic array having a grating period of A, and accordingly, may form a radiated beam 308 at an angle 6 = sin-1f-^2-). However,'■kclad' according to the radiation profile of each of the particles 104, a nonradiative, cone-shapedP12545PC00node 312 extends from the center of the particle 104 to the surfaces of the particles 104. In particular, the dimensions of the particles 104 and the grating period A may be selected such that the node 312 covers the angle 0, thereby intrinsically suppressing the radiative beam 308 due to the multipole resonances.

[0030] Thus, in operation and referring to FIG. 5A, a new band curve 500 is obtained with respect to the optical waveguide 100. In particular, radiative loss is substantially suppressed, and hence the curve no longer vanishes at the chosen operating point < D0and a small ftB.

[0031] For example, referring to FIG. 5B, a simulated band structure 510 is depicted. The band structure 510 is based on the optical waveguide 100 being formed of silicon dielectric particles 104, a silicon dioxide cladding material, a cube length of 500 nm, and a period A of 600 nm. More generally, the cube length may be approximately 1 / n of the vacuum wavelength of the optical wave (i.e., the light input to be transmitted along the optical waveguide 100), where n is the material refractive index of the particle. Color difference between bands results from different optical excitation and do not represent absolute loss differences from one mode to another. Variation of the color shade on the same band does relate to the difference in optical loss for different parts of the curve.

[0032] In particular, while the upper band of the fundamental mode 514 vanishes above the light line, the proposed resonant mode 518 of the operating near« 0 is as strong as when working below the light line. The highest loss of the band occurs at a wavevector (indicated by the dashed circle) near [3B« 2^, and therefore a diffraction angle of 9 -P12545PC00sin1» 46°. That is, such a loss profile corroborates the principle of using the'■kclad'particle radiation profile to control waveguide losses.

[0033] Furthermore, FIG. 5C depicts the electric field (E) and the magnetic field (H) profiles. The optical mode is excited in the left most cuboid 520, and the resonant optical modes transfer from one cube to another, without losing energy to radiation modes.

[0034] Accordingly, as described herein, the presently described optical waveguide 100 including particles having a radiation profile capable of substantially suppressing loss in the resonant waveguide operating above the light line with small Bloch wavevector. In the present example, the particles are substantially cuboid to generate magnetic quadrupole modes; in other examples, other multipoles, such as electric multipole and octupole modes are also contemplated and may similarly define a radiation profile in which a node covers the angle of the in-phase radiative beam.

[0035] The presently described optical waveguide may have many applications, one of which may be an optical phase shifter. Conventional optical phase shifters rely on the material refractive index change An, which can be introduce using a thermo-optic effect, an electro-optic effect, a plasma dispersion effect, or the like. The achievable material index changes generated by these effects are usually at the level of 10-3to 10-2, which employs a long physical length for the phase shifter to yield a sufficient phase change (e.g., up to 2n).

[0036] To improve the efficiency of the phase shifter and reduce the footprint for a given material refractive index change, one option is using the slow light effect, i.e., operating within the flat optical band. The phase change can be calculated using equation (2):P12545PC00(dneff\,rTI An (2)dn /

[0037] Where k0and L are the vacuum optical wavevector and the length of the phase shifter, respectively. The factor FI=describes the rate of change of mode effective index neffwith material refractive index n. In conventional wire or rib waveguides, the factor FIis approximately 1.1, but this factor can be increased by operating in a flat region of an optical band.

[0038] In particular, the presently described optical waveguide allows a wider selection of the working point above the light line, and in particular, the working point may be selected to be in a flat region of an optical band to better optimize the phase shifter.

[0039] Referring to FIG. 6, a band diagram 600 is depicted, in which an original working point may be selected as (W0>^BI). located on a band 604. By inducing a material index change (e.g., through a similar electro-optic, thermo-optic or other effects), the band is downshifted to the band 608. The working point is then relocated to (ci>0,pB2). The overall phase change may be expressed as A<p = (J3B2- @BI)L> resulting in a factor FI=

[0040] In an example simulation, assuming a material index variation of An = 0.018 for approximately 100 degrees of temperature change, the variation of Bloch wavevectorβB2− βB1is increased to 0.056 ^)> yielding a mode index change Δneff= 0.133 and FI= 7.4. As will be appreciated, the factor FImay be further tuned by tailoring the band curve.

[0041] As described herein an optical waveguide may be composed of dielectric particles whose shapes are engineered to generate multipole modes. Specifically, the multipoleP12545PC00modes interact to generate a radiation profile with radiative regions or fields and at least one node, in which radiation is substantially suppressed. The dielectric particles are aligned in a chain and are spaced to support resonant modes to propagate along the chain via the resonance when optically excited. Further, the shape and grating period of the dielectric particle and the chain are selected such that an in-phase radiative beam emitted from each particle is substantially covered by the at least one node of the radiation profile, thereby suppressing radiation loss during propagation of the optical wave.

[0042] The scope of the claims should not be limited by the embodiments set forth in the above examples but should be given the broadest interpretation consistent with the description as a whole.

Claims

P12545PC00CLAIMS1. An optical waveguide comprising:a chain of dielectric particles configured to propagate an optical wave in a propagation direction along the chain;wherein each dielectric particle comprises a shape defining a radiation profile having a radiative region and at least one node;wherein adjacent dielectric particles are disposed to support a resonant mode to transmit the optical wave between the adjacent dielectric particles in the propagation direction; andwherein said dielectric particles are spatially arranged along the chain such that radiation phase matching is achieved at a diffraction angle substantially covered by the at least one node in the particle radiation profile.

2. The optical waveguide of claim 1, wherein the dielectric particles are substantially equivalent.

3. The optical waveguide of claim 1, wherein the dielectric particles form a periodic array along the chain having a pitch such that a diffraction loss of the optical waveguide is suppressed by the diffraction angle being covered by the at least one node in the particle radiation profile.

4. The optical waveguide of claim 1, wherein the shape of each dielectric particle comprises:P12545PC00a first cross section parallel to the propagation direction, the first cross section being substantially square; anda second cross section parallel to the propagation direction and perpendicular to the first cross section, the second cross section being substantially square, such that the dielectric particle forms a cube.

5. The optical waveguide of claim 4, wherein each of the first and second cross sections supports a magnetic quadrupole mode having a magnetic moment along a respective normal direction of the respective cross section.

6. The optical waveguide of claim 5, wherein the two magnetic quadrupole modes contribute to the radiation profile of the dielectric particle to define radiative fields along diagonal directions towards corners of the cube.

7. The optical waveguide of claim 4, wherein each dielectric particle has a dimension of approximately 1 / n of a vacuum wavelength of the optical wave, where n is the material refractive index of the particle8. The optical waveguide of claim 1, wherein the dielectric particles comprise silicon, germanium, lithium niobates or lll-V semiconductor particles.

9. The optical waveguide of claim 1, wherein the shape of each dielectric particle supports a plurality of electric or magnetic multipole modes, wherein the multipoleP12545PC00modes are configured to contribute to the radiation profile to define the radiative region and the at least one node covering the in-phase radiation angle of the optical wave.

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