Data transmission using photonic-crystal waveguides

Abstract

Methods and apparatus for optical data transmission using photonic-crystal waveguides. An example apparatus includes a waveguide core having a one-dimensional array of unit cells arranged along a longitudinal direction thereof. Each of the unit cells is characterized by an absence of a mirror-symmetry plane orthogonal to the longitudinal direction. The array of unit cells is configured to impose periodic modulation of an effective refractive index that causes the waveguide core to support a hybrid guided mode including an LE component and a TE component.

Description

Docket No.: 093386-0067-W001(VU25082) DATA TRANSMISSION USING PHOTONIC-CRYSTAL WAVEGUIDESCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U. S. Provisional Patent Application Nos.63 / 749,102 (filed on 24-Jan-2025) and 63 / 798,751 (filed on 2 -May-2025), the contents of which are incorporated herein by reference.FEDERALLY SPONSORED RESEARCH

[0002] This invention was made with government support under grant numbers 1407777 and 1809937 awarded by the National Science Foundation. The government has certain rights in this invention.FIELD OF THE DISCLOSURE

[0003] Various example embodiments relate to optical communication equipment and, more specifically but not exclusively, to transmission of modulated optical signals through optical waveguides.BACKGROUND

[0004] Optical communication signals can be generated, e.g., by encoding information (data) onto light pulses. The resulting modulated optical signals can then be coupled into optical waveguides and guided therethrough by total internal reflection. The modulation process typically involves modulating some of the light’s properties (e.g., amplitude, frequency, and / or phase) to represent the data. Example types of optical modulation include amplitude modulation (AM), frequency modulation (FM), phase modulation (PM), polarization modulation, and spatial modulation. Multiplexed optical signals are beneficially capable of carrying multiple data streams through the same optical waveguide or fiber.BRIEF SUMMARY OF SOME SPECIFIC EMBODIMENTS

[0005] Various examples disclosed herein provide subwavelength waveguide designs that enable a longitudinal electric (LE) field that is in phase with both the transverse electric (TE) and transverse magnetic (TM) fields in a one-dimensional (ID) photonic-crystal waveguide. ByDocket No.: 093386-0067-W001(VU25082) introducing controlled symmetry breakage in the unit cell(s) used to construct the ID photonic¬ crystal waveguide, we obtain a guided mode in which the LE and IE fields jointly contribute to the energy flow along the propagation direction, a property not possessed by conventional homogeneous optical waveguides. Evidence for hybrid modes is shown through band-structure calculations, simulated field profiles, transmission spectra, multipole-expansion scattering simulations, and experimental measurements. The hybrid modes can beneficially be used in various applications ranging from new dimensions of polarization multiplexing to higher coupling efficiency of quantum emitters embedded in waveguides, to minimizing back reflections from lasers. More broadly, the disclosed embodiments highlight the potential of LE field engineering as a new degree of freedom in integrated photonics, paving the way for novel device concepts in compact planar (e.g., on chip) waveguide platforms.

[0006] According to an example embodiment, provided is an apparatus comprising a waveguide core including a one-dimensional array of unit cells arranged along a longitudinal direction thereof, each of the unit cells being characterized by an absence of a mirror-symmetry plane orthogonal to the longitudinal direction, wherein the array of unit cells is configured to impose periodic modulation of an effective refractive index that causes the waveguide core to support a hybrid guided mode including an LE component and a TE component.BRIEF DESCRIPTION OF THE DRAWINGS

[0007] Other aspects, features, and benefits of various disclosed embodiments will become more fully apparent, by way of example, from the following detailed description and the accompanying drawings, in which:

[0008] FIG. 1 is a schematic diagram illustrating a unit cell used to form photonic-crystal waveguides according to some examples.

[0009] FIGS. 2A-2J pictorially, graphically, and schematically illustrate certain characteristics of a photonic-crystal waveguide constructed using an array of unit cells of FIG. 1 according to some examples.

[0010] FIGS. 3A-3C are heat maps illustrating simulated and experimentally measured transmission spectra of photonic-crystal waveguides according to some examples.Docket No.: 093386-0067-W001(VU25082)

[0011] FIG. 4 graphically shows simulated and experimentally measured transmission spectra of photonic-crystal waveguides according to some examples.

[0012] FIG. 5 graphically illustrates simulated and experimentally measured far-field projections for photonic-crystal waveguides according to some examples.

[0013] FIG. 6 is a block diagram illustrating an optical system using which at least some embodiments can be practiced.

[0014] FIGS. 7A-7B are schematic diagrams illustrating unit cells used to form photonic-crystal waveguides according to some additional examples.

[0015] FIGS. 8A-8C show photonic band diagrams corresponding to photonic-crystal waveguides constructed using the unit cells illustrated m FIGS. 7A-7B according to some examples.

[0016] FIG. 9 is a block diagram illustrating an optical transmitter employing a plurality of photonic-crystal waveguides according to some examples.

[0017] FIG. 10 is a block diagram illustrating an optical receiver employing a plurality of photonic-crystal waveguides according to some examples.

[0018] FIGS. 11A-1 IB are block diagrams illustrating anisotropic properties of a photoniccrystal waveguide according to some examples.

[0019] FIG. 12 is a block diagram illustrating an optical system configured to use a back-reflection filter according to some examples,

[0020] FIG. 13 is a schematic diagram illustrating a photonic-crystal waveguide including a rotation-angle taper section according to some examples.DETAILED DESCRIPTION

[0021] Photonic crystals are engineered materials that possess a periodic refractive index m space and can control how light propagates (e.g., amplitude and direction). Photonic crystals are characterized by photonic bandgaps, or spectral regions (range of wavelengths) in which light cannot propagate. In some examples, the refractive index periodicity' of a photonic crystal isDocket No,: 093386-0067-W001(VU25082) achieved using air holes in a dielectric material. In some examples, the shape of an individual air hole can be circular or rectangular. Additionally, the inventors developed approaches for subwavelength engineering of the photonic-crystal unit cell, for example, by adding a dielectric bowtie or a dielectric bar within the air hole. This subwavelength engineering enables control over the electric field distribution and peak energy density, which may enable lower power and smaller footprint photonic devices. The inventors also realized that, in some examples, rotating the orientation of the rectangular air hole or of the dielectric bar inside the circular air hole can give rise to a new type of photonic bandgap and to a hybrid transverse-longitud inally polarized guided mode. These developments may have important implications for optical communications technologies, as described in more detail below.

[0022] In some applications, it may be possible to achieve an optical diode (also called an optical isolator and an optical equivalent of an electrical diode) based on a new mechanism that does not involve a magnetic field and is compatible with photonic integrated circuits. An optical diode allows light to travel in one direction but not in the opposite direction, which is counter to the principle of reversibility of light propagation but would be highly beneficial in photonic integrated circuits. Some embodiments may help to achieve higher-order data encoding on chip. For example, in addition to carrying different information streams on mutually orthogonal transverse electric (TE) and transverse magnetic (TM) waves, information may be separately encoded on the hybrid transverse-longitudmally polarized wave.

[0023] Some examples demonstrate guided light waves having a hybrid transverse-longitudinal polarization. This new phenomenon may be used to design many new applications, thereby- enabling a new way to consider and use electromagnetic waves, which are foundational to the optics and photonics technologies.

[0024] In some examples, simulation and experimental results indicate the existence of the longitudinal guided light waves.

[0025] A purely transverse description of electromagnetic waves is incomplete when light is strongly confined, such as in tightly focused beams or guided-mode waveguides. In such confinement, the longitudinal electric (EE) field also needs to be considered. The contribution of l. F. field components was first investigated after it was found that scalar approximations of lightDocket No,: 093386-0067-W001(VU25082) under high numerical-aperture-focusing were inadequate. Instead, a full-vectorial model that considers the rotation of polarization caused by high-numerical aperture was used to accurately represent the behavior of light within that regime. After this development, single molecule probes with radially polarized beams were utilized, theoretically and experimentally, to demonstrate that the longitudinal electric field may dominate the on-axis intensity distribution at a focal point. In some examples, this longitudinal field component can be engineered and maximized, e.g., using annular apertures and radial polarization.

[0026] For guided-wave systems, a semi-vectorial approximation similarly fails to capture the intricate relationships between LE and TE field components under substantial mode confinement. To address this problem, mode solvers based on a full-vectorial finite- difference method are developed, enabling accurate calculation of field distributions and modal polarization mixing in dielectric waveguides. Numerical studies of silicon nanowires reveal that, in some examples, the LE field component can reach a relative amplitude of up to ~97% of the dominant TE component’s amplitude. In other examples, a full-vectorial framework is applied to optical microfibers and nanopillars, wherein the strong confinement produces substantial LE fields that influence coupling, propagation loss, second-harmonic generation, and spin-orbit interactions.

[0027] In some conventional guided-wave systems, the LE field may typically exist along the sidewall and is out-of-phase with respect to the magnetic field. Thus, the LI', component within these systems represents an energy reservoir that does not contribute to the net energy transport. However, Gauss’s law implies that an in-phase LI. component may exist if the refractive index varies along the propagation direction. This situation arises, eg., in one-dimensional photoniccrystal waveguides, where the refractive index is modulated along the propagation direction, thereby breaking longitudinal homogeneity and enabling a relatively prominent LE field component in the propagation mode.

[0028] Some embodiments disclosed herein demonstrate that a subwavelength-engineered photonic crystal (PhC) waveguide with appropriately designed refractive index modulation can give rise to spatial overlap of the TE and LE field components, thereby enabling the LE field to hybridize with the TE field, contribute to energy transport, and serve as an independently controllable polarization channel for light manipulation. Some of such embodiments rely on intentionally breaking the mirror symmetry of the unit cells of the PhC’ waveguide, which both lifts theDocket No,: 093386-0067-W001(VU25082) degeneracy of two orthogonal modes and induces hybrid LE-TE modes. We provide a comprehensive validation of this phenomenon through theory, simulation, and experiment. For example, we show from Gauss’s law that the LE field component can be partially in phase with the TE and TM fields for PhC waveguides, and we further show from Ampere’s law that the LE field J. W e use n umeri calsimulation to show that some PhC waveguides with unit cells that do not have mirror symmetry enable hybridization between LE and TE field components and open a new geometry'- induced photonic bandgap. We also experimentally measure this new bandgap in some fabricated devices and use far-field imaging of the guided-mode to reveal an independently polarized LE field component.

[0029] The above- indicated developments beneficially enable new practical applications. For example, the hybrid TE -LE mode’s unique in-plane polarization insensitivity provides a mechanism for on-chip angle-invariant coupling of emitters in 2D materials. In some examples, the additional degree of freedom obtained by tuning the degree of hybridization of the LE-TE mode can be directly exploited for high-density data transmission. More specifically, information can be encoded and decoded by modulating the relative phase between the TE and LE components, thereby enabling a novel form of polarization division multiplexing that can be leveraged to significantly increase the bandwidth of on-chip optical interconnects.

[0030] FIG. 1 is a schematic diagram illustrating a unit cell 100 used to form a photonic-crystal waveguide according to some examples. A photonic-crystal waveguide constructed using an array of unit cells 100 arranged along a light propagation direction 102 may be referred to as an antislot waveguide. A plurality of design parameters for the unit cell 100 include a pitch 110, a waveguide width 112, a waveguide thickness 114, a diameter 116 of a circular opening 118, a width 120 of a solid (e.g., silicon) beam 122 located in the circular opening 118, a rotation angle 124 of the solid beam 122 with respect to the light propagation direction 102, In some examples, the solid material used to make the unit cell 100 comprises silicon, silicon nitride, IILV semiconductors, lithium niobate, polymers, and / or hybrid material stacks. In some examples, the circular opening 118 may¬ be filled by a lower index dielectric material, such as air or silicon dioxide. In some examples, the circular opening 118 contains vacuum. In some examples, the array of unit cells 100 is used to form a waveguide core, e.g., on a substrate. The waveguide core may then optionally be encased by aDocket No,: 093386-0067-W001(VU25082) waveguide cladding, e.g., made of a suitable dielectric material, such as silicon oxide, a polymer, or the like.

[0031] The unit cell 100 can be designed for a selected operating wavelength by adjusting some or all of the above-indicated parameters. In some examples, the design causes the operating wavelength to be in the wavelength range between 1200 nm and 1600 nm. In additional examples, other operating wavelength ranges can also be accessed by appropriately changing the materials and / or unit-cell parameter values. In some examples, the following parameter ranges can be used: (i) 0.05 to 3 pm for the pitch 110; (ii) 0.01 to 3 pm for the beam width 120; (iii) 0.05 to 3 pm for the waveguide width 122; (iv) 0.05 to 3 pm for the diameter 116 of circular opening 118; (v) 0.05 to 3 pm for the waveguide thickness 114; and (vi) -90 to +90 degrees for the rotation angle 124.

[0032] In some examples, the waveguide core formed using an array of unit cells 100 has a substantially rectangular cross-section in a transverse plane that is orthogonal to the light propagation direction 102. Each of the unit cells 100 includes a pair of substantially semicylindrical openings adjacent to the solid separator beam 122 oriented at a non-orthogonal angle with respect to the longitudinal direction of the waveguide core or the light propagation direction 102. The array of unit cells 100 is configured to impose a periodic modulation of an effective refractive index along the waveguide core resulting in a photonic bandgap in the waveguide core at the selected operating wavelength. The photonic bandgap is configured to cause the waveguide core to couple a light wave having a nonzero longitudinal electric-field component into a corresponding (e.g., hybrid or LE-TE) waveguide mode of the waveguide core at the selected operating wavelength.

[0033] In some examples, the non-orthogonal rotation angle 124 belongs to an angular range between 1 degree and 89 degrees or between -1 degree and -89 degrees with respect to the light propagation direction 102. In some additional examples, the non-orthogonal rotation angle 124 may belong to an angular range between 35 degrees and 55 degrees or between -35 degrees and -55 degrees.

[0034] In some examples, the operating wavelength is selected by selecting one or more parameters from the group consisting of: a material of the waveguide core; a material of the waveguide cladding; a material of the substrate; a pitch of the array of unit cells 100; a width of the separator beam 122; a width of the waveguide core; a thickness of the waveguide core; a diameter ofDocket No,: 093386-0067-W001(VU25082) the pair of substantially semicylindrical openings corresponding to the circular opening 118; and the non-orthogonal rotation angle of the separator beam 122. In some examples, the operating wavelength is in a wavelength range between 1.45 um and 1.65 m or between 1.26 pm to 1.36 pm. In some examples, the separator beam has a planar rectangular bar shape or a bowtie shape. In some examples, the waveguide cladding includes a portion of a substantially planar dielectric substrate configured to structurally support the waveguide core. In some examples, the waveguide core comprises silicon, and the waveguide cladding comprises silicon oxide.

[0035] FIGS. 2A-2E pictorially, graphically, and schematically illustrate certain characteristics of a photonic-crystal waveguide including a waveguide core 200 constructed using an array of unit cells 100 according to some examples. In total, five different photonic-crystal waveguides including the corresponding embodiments of the waveguide core 200 are illustrated. The illustrated embodiments differ m the values of the rotation angle 124 and represent the angle values of 0, 30, 45, 60, and 90 degrees. The corresponding photonic-crystal waveguides are denoted as R0, R30, R45, R60, and R90 waveguides, respectively.

[0036] FIG. 2A schematically shows a three-dimensional (3D) perspective view of the waveguide core 200 of an R45 waveguide.

[0037] FIG. 2B schematically shows a plan view of a portion 202 (also see FIG. 2A) of the waveguide core 200, which includes three unit cells 100 labeled IOO1-IOO3, respectively.

[0038] FIGS. 2C-2E show electric-field maps corresponding to a single unit cell 100 m the R0, R45, and R90 waveguide embodiments, respectively. Each of the FIGS. 2C-2E has three respective panels. The left panel visualizes a two-dimensional (2D) slice of the electric field component Ey. The middle panel visualizes a 2D slice of the electric field component Ex. The right panel visualizes the spatial overlap between the Exand Eyfield components based on their product. Comparison of the field map portions inside the dashed ovals clearly indicates that only the R45 waveguide exhibits a significant Ex- Eyhybridization inside the antislot.

[0039] FIGS. 2F-2J show band diagrams corresponding to the R0, R30, R45, R60, and R90 waveguides, respectively. The band diagrams indicate, in accordance with the provided legend, the respective pure transverse electric bands, pure longitudinal electric bands, hybrid transverse / longitudinal electric bands, fundamental photonic band gaps (Band Gap 1), and firstDocket No,: 093386-0067-W001(VU25082) upper photonic band gaps (Band Gap 2). A fundamental photonic band gap typically arises from the waveguide structure breaking the translational symmetry of the waveguide. An upper photonic band gap may arise from the broken mirror symmetry in the unit cell.(0040] Herein, the term “photonic band gap” (or PBG) refers to a range of wavelengths (or frequencies) that are not allowed to propagate through a photonic crystal structure, resulting in relatively high reflection and a corresponding trough in transmission. The formation of a PBG may, in some respects, be analogous to the formation of an electronic band gap in a semiconductor crystal. In a photonic crystal, the periodic unit-cell arrangement causes incoming light waves at certain frequencies to undergo multiple Bragg reflections and destructively interfere, thereby preventing such frequencies from propagating through the material. Instead of being absorbed, light within the PBG frequency range is strongly reflected. A higher contrast between the refractive indices of the materials in the unit cell structure generally leads to a larger band gap. The arrangement (e.g., square, hexagonal, or diamond lattice) and dimensions of the periodic elements influence the band structure and the existence of a complete photonic band gap. Incomplete photonic band gaps may only forbid propagation for specific wave vectors or polarization states of light, while a complete PBG forbids propagation for all directions and polarizations.

[0041] For a waveguide medium that is homogenous along the propagation direction (e.g., the x-direction), such as that of a conventional ridge waveguide, the transverse electric and magnetic fields are in-phase with respect to one another and give rise to energy transport along the x-direction. The longitudinal electric (LB) field (E, however, is out-of-phase with the transverse electric field (Et) and does not propagate, acting as an energy reservoir. The magnitude of the LE field scales with the spatial gradient of the transverse electric field Etand is generally not considered except for special cases, such as nanowire waveguides that support strong transverse field gradients.

[0042] Under the assumption that the permittivity (<?) of a waveguide medium can vary along the propagation direction, a general relationship can be determined between ExandE), as shown in Eq. (1) derived based on the Gauss’s law:Exa |f (V£. (£Et)) + i£ / ? Vt-£Et(1)where ft = the effective index of the waveguide, and A is the free-space wavelength oflight. For a homogeneous waveguide, ™ = 0, and Exis purely out-of-phase withEfas indicated byDocket No,: 093386-0067-W001(VU25082) the imaginary factor z. In contrast, for a non-homogenous waveguide medium, such as a photoniccrystal waveguide, = 0, and there exists a complex relationship between Exand E:. Because the LE field is not completely out-of-phase with the TE field components, the in-phase LE field components can contribute to the propagating mode.

[0043] To qualitatively understand the phase relationship between the field components in a waveguide, we consider the spatial overlap of LE and TE in the propagation plane (x,y) where the refractive index is modulated. When the two fields are in-phase, the product of the two respective spatial distributions yields relatively high intensity, whereas when the fields are out-of-phase, there is a relatively small spatial overlap intensity in the middle of the waveguide. We consider the cases when the LE and TE field components coexist spatially to represent hybrid LE-TE modes. We note that, because the transverse component of the electric field (ft is in phase with the transverse component of the magnetic field (ft), ft is also in phase with Hzin cases when it is in phase with ft.

[0044] The magnitude and phase of the LE field can be determined directly from the Ampere’s law. For example, for a given ft distribution, the transverse and longitudinal components of the electric field can be expressed as:ft = 'j f— -'j (2)y \ ia>sj \ 8x J ’£> = ®) ®) <3>where co is the angular frequency. Eq. (2) describes the well-known relationship for TE modes. Eq. (3) shows that an LE field component can exist when —■ = 0. Therefore, a criterion for awaveguide structure to support a hybrid LE-TE mode is associated with 0.

[0045] To confirm the emergence of the LI i-TE hybrid mode via numerical simulations involving a finite-difference time-domain (FDTD) analysis, we considered a silicon antislot photonic-crystal waveguide constructed using various embodiments of the unit cell 100, Representative results of those simulations are illustrated in FIGS. 2C-2J, As indicated by FIGS. 2C-2J, the value of the rotation angle 124 determines the proportion of in-phase LE components and the magnitude of the LE field. When the rotation angle 124 is changed, the magnetic field distribution may be altered in a manner that enhances the phase matching between the LE and TEDocket No,: 093386-0067-W001(VU25082) components (i.e., increasing ) and gives rise to a hybridization of those field components. The oyresults presented in FIGS. 2C-2E visualize this hybridization using the spatial overlap of the Exand Eyfield components for three different angle values, i.e., for the waveguides RO, R45, and R90. For the waveguide R45, the overlap region in the antislot is both significant and continuous in phase, indicating relatively strong LE-TE coupling and further indicating the presence of a propagating hybridized mode with significant LE contributions. In contrast, no meaningful LE-TE coupling is observed for either one of the RO and R90 waveguides.

[0046] To further study the hybrid LE-TE mode, we carried out band-structure simulations (MEEP) for various antislot angles. As shown in FIGS. 2F, 2J, for the RO and R90 waveguides, excitation with a purely Eysource (Exsource) yields distinct, pure transverse (longitudinal) modes, corresponding to the TE (LE) bands. In contrast, for intermediate rotation angle values corresponding to the R30, R45, and R60 waveguides (see FIGS. 2G-2I) where the mirror symmetry in the unit cell is broken, hybrid LE-TE bands emerge irrespective of whether the excitation originates from the Eyor Exsource. Notably, the band structure results clearly demonstrate that these hybridized modes open a new geometry-induced photonic bandgap (Band Gap 2) centered near 1.55 pm, which does not exist in the RO and R90 waveguides having mirror-symmetric unit cells 100. The corresponding symmetric structures exhibit only the conventional photonic bandgap (Band Gap 1) at the Brillouin zone boundary, which arises due to the periodic dielectric function in the waveguide. Notably, the new bandgap (Band Gap 2) appears at different k-vectors for different values of the rotation angle 124. Moreover, the degree of LE-TE mode hybridization can be systematically tuned by varying the degree of asymmetry, which provides a flexible means to controlling light-matter interactions within the nanostructure. This tunability is evident in both the band dispersion and the corresponding field profiles, highlighting the potential of symmetry-broken photonic structures for advanced functionalities, such as selective electric- field-component confinement and polarization control,

[0047] FIGS. 3A-3C are heat maps 302-306 illustrating simulated and experimentally measured transmission spectra of photonic-crystal waveguides according to some examples. In each of the heat maps 302-306, the horizontal axis represents different values of the antislot rotation angle 124 in the range from R0 to R90, and the vertical axis represents wavelength. More specifically, FIG.3A illustrates FDTD-simulated transmission spectra. At 0° and 90°, the geometry-induced photonicDocket No,: 093386-0067-W001(VU25082) bandgap is closed. As the value of the rotation angle 124 increases from 0°, the photonic bandgap progressively opens and reaches its maximum at 45°. Thereafter, the photonic bandgap progressively narrows as the angle continues to increase from 45° to 90°. This trend is consistent with the band diagrams and spatial overlap profiles of the LE and TE modes illustrated in FIGS. 2F-2J. The experimentally measured heat map 304 (FIG. 3B) is in good agreement with the FDTD simulated heat map 302.

[0048] The heat map 306 (FIG. 3C) is obtained using FEM (Finite Element Method) simulations based on multipole expansion scattering that indicate the ratio of forward to backward scattered light. While the heat maps 302, 304 are obtained using conventional photonic-crystal analysis techniques, it is also instructive to consider an antislot photonic-crystal waveguide as comprising an array of meta-atoms where each meta-atom is a single unit cell 100. Then, FEM simulations using a multipole expansion scattering method can be carried out to probe the directionality of the hybrid LE-TE modes. As indicated by FIG. 3C, the ratio of forward-to-backward scattered light follows the same trends observed in the heat maps 302, 304. The heat map 306 provides additional evidence of the LE-TE hybridization and its influence on wave propagation.

[0049] FIG. 4 graphically shows simulated and experimentally measured transmission spectra of antislot photonic-crystal waveguides according to some examples. More specifically, the transmission spectra corresponding to the R45, R60, R75, and R90 waveguides are shown. The geometry-induced photonic bandgap (Band Gap 2) is centered near 1.55 pm. The shown spectra illustrate the evolution of the geometry-induced bandgap and the close agreement between the simulation and experimental results. Together, the results illustrated in FIGS. 3-4 suggest that the antislot rotation angle 124 can be tuned to directly control the degree of coupling between the LE and TE modes. The rotation angles of 0° and 90° represent the uncoupled limiting cases, whereas the rotation angle of 45° represents the condition of maximum hybridization.

[0050] FIG. 5 graphically illustrates simulated and experimentally measured far-field projections for antislot photonic-crystal waveguides according to some examples. More specifically, the far-field projections corresponding to the R0, R45, R65, and R90 waveguides are shown. Note that the simulated far-field scattering projections of the R45 waveguide are compared with the measured far-field scattering projections of an R44 waveguide device. This rotation angle difference is indicated by the symbol in the unit-cell-graphic column of FIG. 5. MeasurementsDocket No,: 093386-0067-W001(VU25082) are taken both on-resonance and off-resonance, as determined from the transmission spectrum characterization. Each on-resonance image is captured at two linear polarizations, e.g., 0 degrees and 90 degrees relative to the light propagation direction. These on-resonance images are then subtracted from the corresponding off-resonance images to isolate the resonant scattering signal.

[0051] The far-field projections of FIG. 5 can be used to experimentally quantify the LE field component relative to the TE field component. For the RO and R90 waveguides, the far-field scattering profiles are dominated by the TE component, which is consistent with the foregoing analysis. As the rotation angle 124 approaches 45°, the LE component becomes increasingly more significant. For example, for the R45* waveguides, the ratio of Ex2 / Ey2reaches approximately 50% in both simulation and experiment. In some cases, this ratio can be further increased, e.g., by choosing a unit cell geometry that supports even stronger electric and magnetic field gradients, such as a bowtie shape of the silicon divider in the circular opening 118.

[0052] FIG. 6 is a block diagram illustrating an optical system 600 using which at least some embodiments can be practiced. A first subsystem 601 of the optical system 600 is configured for in-plane end-coupled transmission spectrum measurements. A second subsystem 602 of the optical system 600 is configured for out-of-plane imaging of far-field scattering profiles. A waveguide device under test (DUT) 603 is shared by the subsystems 601, 602 as indicated in FIG. 6,

[0053] In some examples, transmission measurements are performed using the first subsystem 601, which implements an edge-coupling configuration and uses TE-polarized light generated by a supercontinuum laser 612 (e.g., NKT Photonics) capable of covering the wavelength range from 800 to 2500 nm, The laser light is filtered using an absorptive neutral density filter 614 (e.g., NE10A-B, Thorlabs) and optically coupled from free space into a polarization-maintaining single-mode fiber (e.g., Pl-1550PM-FC-2, Thorlabs) using a plano-convex lens 616 (e.g., LA1951-AB-ML).Customized tapered optical fibers 620, 624 (e.g., TPMJ-3U-1550-8 / 125-0.25-10-2, 5-14-1, OZ Optics) are used for coupling light into / out of the DUT 603, with polarization control achieved by connecting the optical fibers through fiber-to-fiber U-benches (e.g., FBP-A-FC). The light outputted by the fiber 624 is detected using an optical detector 630. In some examples, the optical detector 630 comprises a fiber-coupled spectrometer (e.g., NIR Quest, Ocean Optics).Docket No,: 093386-0067-W001(VU25082)

[0054] In some examples, far-field scattering imaging is performed using the second subsystem 602, which integrates an out-of-plane NIR camera 640 (e.g., SN160608, NIT) using the edge¬ coupling optical setup. A linear polarizer 636 (e.g., LPNIR100, Thorlabs) mounted on a continuous rotation mount 634 (e.g., CLR1, Thorlabs) is used for polarization analysis of the far-field scattering. Far-field scattering imaging is conducted under consistent conditions, including a camera exposure time of 20 ms, laser power of -4 dBm, and strictly controlled ambient lighting.

[0055] In some examples, a plurality of DUTs 603 with rotational angles varying from 90° to 0°, in 5° increments, are fabricated on a silicon-on-insulator wafer featuring a 270 nm silicon device layer and a 3 pm buried oxide layer. Conventional electron beam lithography and reactive ion etching are used to create these waveguides on a single chip within a single batch. An example photonic-crystal waveguide fabricated in this manner for a DUT 603 is illustrated in FIG. 2A.

[0056] FIGS. 7A-7B are schematic diagrams illustrating plan views of unit cells 700, 701 used to form photonic-crystal waveguides according to some additional examples. The unit cells 700, 701 are modifications of the unit cell 100 (FIG. 1) in which the circular opening 118 with the divider beam 122 are replaced by a subwavelength rectangular opening 718. In the unit cell 700, the rectangular opening 718 is oriented at 90 degrees with respect to the light propagation direction 102, meaning that one (e.g., a shorter one) of the sidewalls of the rectangular opening 718 is parallel to the sidewall of the corresponding waveguide core or to the light propagati on direction 102. In the unit cell 701, the rectangular opening 718 is oriented at an angle ^(labeled using the reference numeral 724) with respect to the light propagation direction 102, where 0° < 6< 90°. In some additional examples, the sides of the rectangular opening 718 may have different lengths denoted as h₁ and h₂, respectively, where h₁ ≠ h₂. In some additional examples, the opening 718 may have a square shape, i.e., h₁ = h₂. In some further examples, the rectangular opening 718 can be modified to have rounded corners. In some further examples, the rectangular opening 718 can be replaced by an elliptical opening.

[0057] Similar to the unit cell 100, the unit cell 701 can support hybridized TE-LE behavior. Due to the broken mirror symmetry in the unit cell 701, the degeneracy between the two orthogonal modes is lifted, resulting in mode hybridization between the TE and LE field components. This mode coupling can be visualized directly in the photonic band structure, where it manifests itself as band anti-crossing and the formation of a new, asymmetry-induced photonic bandgap.Docket No,: 093386-0067-W001(VU25082)

[0058] In various additional examples, other unit cell designs characterized by a broken mirror symmetry can be used to enable mode hybridization between the TE and LE field components in the corresponding photonic-crystal waveguide. Examples of such broken mirror symmetry may be realized using the rotation angle of a pertinent subwavelength component within the unit cell, such as those of the above-described divider beam 122 and the rectangular opening 718.

[0059] FIGS. 8A-8C show band diagrams 810, 820, 830 corresponding to photonic-crystal waveguides constructed using the unit cells 700, 701 according to some examples. More specifically, the band diagram 810 (FIG. 8A) corresponds to a photonic-crystal waveguide including an array of unit cells 700. The band diagram 810 indicates the presence of a fundamental photonic band gap but no upper photonic band gap. The band diagrams 820, 830 (FIGS. 8B-8C) correspond to photonic-crystal waveguides, wherein the waveguide includes an array of unit cells 701, with the rotation angle 724 of 75° and 45°, respectively. The band diagram 820 (FIG. 8B) indicates the presence of a relatively narrow upper photonic band gap 822 in addition to the fundamental photonic band gap. The band diagram 830 (FIG. 8C) similarly indicates the presence of an upper photonic band gap 832 in addition to the fundamental photonic band gap, with the band gap 832 being wider than the band gap 822,

[0060] FIG, 9 is a block diagram illustrating an optical transmitter 900 employing a plurality of photonic-crystal waveguides 930i-930naccording to some examples. In the example shown, each of the wa veguides 930; is constructed using a respective embodiment of the unit cell 100 characterized by a different respective rotation angle 124, In other examples, other suitable unit cells, such as the unit cells 701, can similarly be used to construct some or all of the waveguides 930₁-930n. In various examples, the number n may be n=2 or any other practical integer, e.g., in the range from two to one hundred.

[0061] The optical transmitter 900 includes a laser 910. In some examples, the output wavelength (λ) of the laser 910 may be tunable. The wavelength λ is the operating wavelength for the waveguides 930i-930n. The laser 910 is configured to cause an output optical beam 912 to have a polarization corresponding to a transverse electric (TE) mode of the connected waveguides.

[0062] The optical transmitter 900 further includes an optical splitter 920 configured to split the optical beam 912 into n corresponding sub-beams which are then coupled into the waveguides 930i-Docket No,: 093386-0067-W001(VU25082) 930n, respectively. The optical splitting implemented in the optical splitter 920 preserves the TE polarization. As each of the sub-beams propagates through the respective one of the waveguides 930i-930n, the TE mode is converted into a respective hybrid mode containing both longitudinal and transverse field components. Different waveguides 930i-930noperate to populate different respective hybrid modes due to the different respective rotation angles 124. After the mode conversion, a hybrid mode can propagate efficiently through a conventional uniform dielectric waveguide and / or a conventional optical fiber, either of which may be used to optically connect the waveguides 930i-930nto optical modulators 940i-940n, respectively.

[0063] In various embodiments, the optical modulators 940i-940nmay be placed downstream from (e.g., immediately after) the waveguides 930i-930nor embedded within the respective waveguides 930i-930n. Each of the optical modulators 940i-940nmay be configured to encode different respective data streams onto the light transmitted therethrough based on a corresponding electrical data signal. An nxl optical multiplexer (MUX) 950 then operates to combine the respective modulated optical beams generated by the optical modulators 940i-940nto generate a multiplexed optical signal 952. The multiplexed optical signal 952 may then be directed to a corresponding optical receiver (e.g., the optical receiver illustrated in FIG. 10).

[0064] FIG. 10 is a block diagram illustrating an optical receiver 1000 employing a plurality of photonic-crystal waveguides 1030i-1030naccording to some examples. In some examples, the optical receiver 1000 may be used in conjunction with the optical transmitter 900. In such examples, an optical input signal 1002 may be an attenuated and / or partially distorted copy of the multiplexed optical signal 952. In some examples, the plurality of photonic-crystal waveguides 1030i-1030nmay be a nominal copy of the plurality of photonic-crystal waveguides 930₁-930n. In some examples, the plurality of photonic-crystal waveguides 1030i-1030nmay have a different number of photonic-crystal waveguides than that used in the corresponding optical transmitter 900.

[0065] In the example shown, the optical receiver 1000 includes a Ixn optical demultiplexer (DMUX) 1010 configured to perform a demultiplexing operation that is inverse to the multiplexing operation performed by the optical MUX 950. The plurality of photonic-crystal waveguides 1030i-1030nis arranged to form an optical filter 1020 that serves to better isolate (decrease crosstalk between) the corresponding components of the optical input signal 1002 prior to those components being detected by an array of photodetectors 1040i-1040n. Each of the photodetectors 1040i-1040nDocket No,: 093386-0067-W001(VU25082) converts modulated light into a corresponding modulated electrical signal that may be processed by electrical circuitry located downstream from the photodetectors to recover the corresponding data streams encoded in the optical input signal 1002.

[0066] FIGS. 11A-11B are block diagrams illustrating anisotropic properties of a photonic crystal waveguide according to some examples. For illustration purposes and without any implied limitations, the R45 unit cell 100 is used as an example. FIG. 11A schematically shows a “forward” propagating mode 1102 and a “backward” propagating mode 1104. Since the rotation angle 124 is measured with respect to the light propagation direction 102, the forward-propagating mode 1102 “sees” the rotation angle of +45 degrees, whereas the backward-propagating mode 1104 “sees” the rotation angle of −45 degrees, as illustrated in FIG. 11B. As such, the forward-propagating mode 1102 and the backward-propagating mode 1104 represent different respective modes of the corresponding photonic-crystal waveguide, thereby giving rise to directional anisotropy.

[0067] FIG. 12 is a block diagram illustrating an optical system 1200 configured to use an optical back-reflection filter 1220 according to some examples. In addition to the filter 1220, the optical system 1200 includes a laser 1210. The output wavelength (λ) of the laser 1210 is the operating wavelength of the photonic-crystal waveguide used to implement the back-reflection filter 1220. The laser 1210 is configured to cause an optical output beam 1212 thereof to have a polarization corresponding to a transverse electric (TE) mode of the connected optical waveguide(s). As the optical beam 1212 propagates through the photonic-crystal waveguide of the filter 1220, the TE mode is converted into a hybrid mode containing both longitudinal and transverse field components. After the mode conversion, the hybrid mode is coupled into an optical output beam 1222 of the filter 1220, which is then directed downstream where it may impinge on one or more reflective elements (such as reflective components, facets, and / or interfaces) 1230. Partial reflection of the optical beam 1222 from the one or more reflective elements 1230 generates a reflected optical beam 1224 that is coupled back into the photonic-crystal waveguide of the back-reflection filter 1220. However, due to the directional anisotropy of the photonic-crystal waveguide explained above in reference to FIGS. 11A-11B, the photonic-crystal waveguide operates to significantly attenuate or fully reject the reflected optical beam 1224 by coupling the light thereof into radiative (non-guided) optical modes 1226 of the photonic-crystal waveguide, thereby substantially stoppingDocket No,: 093386-0067-W001(VU25082) the reflected light reaching the laser 1210 and any optical components of the system 1200 that might be located between the filter 1220 and the laser 1210.

[0068] FIG. 13 is a schematic diagram illustrating a photonic-crystal waveguide 1300 including a rotation-angle taper section 1310 according to some examples. The photonic-crystal waveguide 1300 also includes a second section 1320 end-connected to the taper section 1310 as indicated in FIG. 13. In the example shown, each of the waveguide sections 1310, 1320 includes a respective sequence of unit cells 100. In the waveguide section 1310, the value of the rotation angle 124 in the sequence of unit cells 100 gradually changes from a first value to a different second value. In contrast, all unit cells 100 in the second section 1320 have a fixed value of the rotation angle 124, which may be equal to the second value. In one example, the taper section 1310 is designed as a sequence of unit cells 100 having a gradually changing value of the rotation angle 124, e.g., starting from 0 and gradually increasing in relatively small increments (e.g., smaller than 1 degree) to the fixed rotation angle (e.g., 45 degrees) of the second section 1320. In other examples, other (than cell 100) unit cell designs may similarly be used to implement the photonic-crystal waveguide 1300.

[0069] In the example shown, an optical input beam 1302 applied to the waveguide 1300 has a TE polarization. A corresponding optical output beam 1322 generated by the photonic-crystal waveguide 1300 carries a hybrid TE and l. F. mode, as explained previously. The rotation-angle taper section 1310 beneficially helps the coupled light of the optical input beam 1302 to go from the TE mode to the hybrid mode supported by the second section 1320 with relatively low insertion losses and / or back reflections by implementing a gradual (e.g., adiabatic) mode conversion therein.

[0070] In conclusion, we note that various subwavelength waveguide designs disclosed herein enable a novel approach to achieving a longitudinal electric (LE) field that is in phase with both the transverse electric (TE) and magnetic (TM) fields in a one-dimensional (1D) photonic crystal waveguide platform. By introducing controlled symmetry breakage in the unit cell, we obtain a guided mode in which the longitudinal and transverse electric fields jointly contribute to the energy flow along the propagation direction, a property not possessed by conventional homogeneous optical waveguides. Evidence for hybrid modes is shown through band structure calculations, simulated field profiles, transmission spectra, multipole expansion scattering simulations, and transmission and far-field scattering measurements. The hybrid modes can beneficially be used in various applications ranging from new dimensions of polarization multiplexing to higher coupling efficiencyDocket No,: 093386-0067-W001(VU25082) of quantum emitters embedded in waveguides to minimizing back reflections from lasers. More broadly, the disclosed embodiments highlight the potential of longitudinal field engineering as a new degree of freedom in integrated photonics, paving the way for novel device concepts in compact planar (e.g., on chip) platforms.

[0071] According to an example embodiment disclosed above, e.g., in the summary section and / or in reference to any one or any combination of some or all of FIGS. 1-13, provided is an apparatus comprising a first waveguide core including an array of first unit cells arranged along a longitudinal direction thereof, each of the first unit cells being characterized by an absence of a mirror-symmetry plane orthogonal to the longitudinal direction, wherein the array of first unit cells is configured to impose periodic modulation of an effective refractive index that causes the first waveguide core to support a first hybrid guided mode including a longitudinal electric (LE)-mode component and a transverse electric (TE)-mode component.

[0072] In some embodiments of the above apparatus, the periodic modulation further causes the first waveguide core to exhibit a first photonic bandgap and a distinct second photonic bandgap centered at a higher frequency than the first photonic bandgap.

[0073] In some embodiments of any of the above apparatus, the first waveguide core has a substantially rectangular cross-section in a transverse plane orthogonal to the longitudinal direction.

[0074] In some embodiments of any of the above apparatus, each of the first unit cells has a pair of substantially semicylindrical openings adjacent to a separator beam oriented at a nonorthogonal angle with respect to the longitudinal direction.

[0075] In some embodiments of any of the above apparatus, the nonorthogonal angle belongs to an angular range between 1 degree and 89 degrees or between -1 degree and -89 degrees.

[0076] In some embodiments of any of the above apparatus, an operating wavelength for the first waveguide core is selected by selecting one or more parameters from the group consisting of a material of the first waveguide core; a pitch of the unit cells; a width of the separator beam; a width of the first waveguide core; a thickness of the first waveguide core; a diameter of the pair of substantially semicylindrical openings; and a value of the first nonorthogonal angle.Docket No,: 093386-0067-W001(VU25082)

[0077] In some embodiments of any of the above apparatus, the operating wavelength is in a wavelength range between 1.45 um and 1.65 pm or between 1.26 pm to 1,36 pm.

[0078] In some embodiments of any of the above apparatus, the apparatus further comprises a waveguide cladding adjacent to the first waveguide core, wherein the waveguide cladding includes a portion of a substantially planar dielectric substrate configured to structurally support the first waveguide core.

[0079] In some embodiments of any of the above apparatus, the first waveguide core comprises silicon; and wherein the waveguide cladding comprises silicon oxide.

[0080] In some embodiments of any of the above apparatus, each of the first unit cells has a substantially rectangular opening having a wall oriented at a non-orthogonal angle with respect to the longitudinal direction.

[0081] In some embodiments of any of the above apparatus, different ones of the first unit cells are characterized by different respective values of the non-orthogonal angle.

[0082] In some embodiments of any of the above apparatus, the apparatus further comprises a second waveguide core including an array of second unit cells arranged along a longitudinal direction thereof, each of the second unit cells being characterized by an absence of a mirror symmetry plane orthogonal to the longitudinal direction of the second waveguide core, wherein the array of second unit cells is configured to impose periodic modulation of an effective refractive index that causes the second waveguide core to support a second hybrid guided mode including a corresponding LE-mode component and a corresponding TE-mode component, the second hybrid guided mode being different from the first hybrid guided mode.

[0083] In some examples, once the hybrid LE-TE mode is generated, it may be possible to use that mode as a source configured to excite a “pure” LE mode. For example, light propagating through the R45 unit cells will have both LE and TE components. When that light is then incident on an R0 or R90 unit cell, the corresponding LE component may launch as a pure LE mode that is illustrated in the band diagrams of FIGS. 2F-2J. One reason why the pure LE mode cannot be typically launched initially in an R0 or R90 waveguide is that an LE light source is not practically available (e.g., lasers do not emit LE light). One challenge associated with such examples is that aDocket No,: 093386-0067-W001(VU25082) pure LE mode that is oscillating along the waveguide propagation direction may actually travel in an orthogonal direction because light, by standard electromagnetics principles, does not travel in the same direction it is oscillating. Thus, some embodiments may be constructed to include an additional waveguide branch orthogonal to the original waveguide propagation direction such that the LE mode can propagate in a direction orthogonal to the original waveguide, and the TE mode continues to propagate in the original waveguide direction.

[0084] In some embodiments of any of the above apparatus, the apparatus further comprises an optical coupler including a first optical port at a first side thereof and a plurality of second optical ports at a second side thereof, each of the second optical ports being optically coupled to the first optical port, wherein the first waveguide core is connected to one of the second optical ports; and wherein the second waveguide core is connected to another one of the second optical ports.

[0085] In some embodiments of any of the above apparatus, the apparatus further comprises: a laser configured to apply light having a selected operating wavelength to the first optical port; a first optical modulator configured to modulate, based on a first data stream, a first portion of the light received from the laser through the first waveguide core; and a second optical modulator configured to modulate, based on a different second data stream, a second portion of the light received from the laser through the second waveguide core.

[0086] In some embodiments of any of the above apparatus, the apparatus further comprises an optical multiplexer configured to multiplex the modulated first and second portions.

[0087] In some embodiments of any of the above apparatus, the apparatus further comprises: a first photodetector configured to receive light from the first optical port through the first waveguide core; and a second photodetector configured to receive light from the first optical port through the second waveguide core.

[0088] In some embodiments of any of the above apparatus, the first waveguide core further includes an array of second unit cells arranged along the longitudinal direction of the first waveguide core; wherein the array of second unit cells is end-connected to the array of first unit cells; wherein the first unit cells have respective mirror-symmetry-breaking elements oriented at a same nonorthogonal angle with respect to the longitudinal direction; and wherein different ones ofDocket No,: 093386-0067-W001(VU25082) the second unit cells have respective mirror-symmetry-breaking elements oriented at different respective angles with respect to the longitudinal direction.

[0089] In some embodiments of any of the above apparatus, the different respective angles have values in a range between 0 degrees and a value of the same nonorthogonal angle.

[0090] In some embodiments of any of the above apparatus, the apparatus further comprises: a laser configured to apply light having a selected operating wavelength to a first end of the first waveguide core; and one or more optical components optically coupled to a second end of the first waveguide core to receive the light from the laser through the first waveguide core, wherein the first waveguide core is configured to couple into radiative non-guided modes light that is back reflected to the second end of the first waveguide core by the one or more optical components.

[0091] In some embodiments of any of the above apparatus, the first waveguide core has a transverse cross-section configured to cause a corresponding homogeneous waveguide core to support a single guided mode.

[0092] With regard to the processes, systems, methods, heuristics, etc. described herein, it should be understood that, although the steps of such processes, etc. have been described as occurring according to a certain ordered sequence, such processes could be practiced with the described steps performed in an order other than the order described herein. It further should be understood that certain steps could be performed simultaneously, that other steps could be added, or that certain steps described herein could be omitted. In other words, the descriptions of processes herein are provided for the purpose of illustrating certain embodiments and should in no way be construed so as to limit the claims.

[0093] Accordingly, it is to be understood that the above description is intended to be illustrative and not restrictive. Many embodiments and applications other than the examples provided would be apparent upon reading the above description. The scope should be determined, not with reference to the above description, but should instead be determined with reference to the appended claims, along with the full scope of equivalents to which such claims are entitled. It is anticipated and intended that future developments will occur in the technologies discussed herein, and that the disclosed systems and methods will be incorporated into such future embodiments. In sum, it should be understood that the application is capable of modification and variation.Docket No,: 093386-0067-W001(VU25082)

[0094] All terms used in the claims are intended to be given their broadest reasonable constructions and their ordinary’ meanings as understood by those knowledgeable in the technologies described herein unless an explicit indication to the contrary is made herein. In particular, use of the singular articles such as “a,” “the,” “said,” etc. should be read to recite one or more of the indicated elements unless a claim recites an explicit limitation to the contrary.

[0095] The Abstract of the Disclosure is provided to allow the reader to quickly ascertain the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims. In addition, in the foregoing Detailed Description, it can be seen that various features are grouped together in various embodiments for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the claimed embodiments incorporate more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in fewer than all features of a single disclosed embodiment. Thus, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separately claimed subject matter.

[0096] While this disclosure includes references to illustrative embodiments, this specification is not intended to be construed in a limiting sense. Various modifications of the described embodiments, as well as other embodiments within the scope of the disclosure, which are apparent to persons skilled in the art to which the disclosure pertains are deemed to lie within the principle and scope of the disclosure, eg., as expressed in the following claims.

[0097] Some embodiments may be implemented as circuit-based processes, including possible implementation on a single integrated circuit.

[0098] Unless explicitly stated otherwise, each numerical value and range should be interpreted as being approximate as if the word “about” or “approximately” preceded the value or range.

[0099] Although the elements in the following method claims, if any, are recited in a particular sequence with corresponding labeling, unless the claim recitations otherwise imply a particular sequence for implementing some or all of those elements, those elements are not necessarily intended to be limited to being implemented in that particular sequence.Docket No,: 093386-0067-W001(VU25082)

[0100] Reference herein to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the disclosure. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments necessarily mutually exclusive of other embodiments. The same applies to the term “implementation.”

[0101] Unless otherwise specified herein, the use of the ordinal adjectives “first,” “second,” “third,” etc., to refer to an object of a plurality of like objects merely indicates that different instances of such like objects are being referred to, and is not intended to imply that the like objects so referred-to have to be in a corresponding order or sequence, either temporally, spatially, in ranking, or in any other manner.

[0102] Unless otherwise specified herein, in addition to its plain meaning, the conjunction “if’ may also or alternatively be construed to mean “when” or “upon” or “in response to determining” or “in response to detecting,” which construal may depend on the corresponding specific context. For example, the phrase “if it is determined” or “if [a stated condition] is detected” may be construed to mean “upon determining” or “in response to determining” or “upon detecting [the stated condition or event]” or “in response to detecting [the stated condition or event].”

[0103] Also, for purposes of this description, the terms “couple,” “coupling,” “coupled,” “connect,” “connecting,” or “connected” refer to any manner known in the art or later developed in which energy is allowed to be transferred between two or more elements, and the interposition of one or more additional elements is contemplated, although not required. Conversely, the terms “directly coupled,” “directly connected,” etc., imply the absence of such additional elements.

[0104] As used herein in reference to an element and a standard, the term compatible means that the element communicates with other elements in a manner wholly or partially specified by the standard and would be recognized by other elements as sufficiently capable of communicating with the other elements in the manner specified by the standard. The compatible element does not need to operate internally in a manner specified by the standard.

[0105] The functions of the various elements shown in the figures, including any functional blocks labeled as “processors” and / or “controllers,” may be provided through the use of dedicatedDocket No,: 093386-0067-W001(VU25082) hardware as well as hardware capable of executing software in association with appropriate software. When provided by a processor, the functions may be provided by a single dedicated processor, by a single shared processor, or by a plurality of individual processors, some of which may be shared. Moreover, explicit use of the term “processor” or “controller” should not be construed to refer exclusively to hardware capable of executing software, and may implicitly include, without limitation, digital signal processor (DSP) hardware, network processor, application specific integrated circuit (ASIC), field programmable gate array (FPGA), read only memory (ROM) for storing software, random access memory (RAM), and nonvolatile storage. Other hardware, conventional and / or custom, may also be included. Similarly, any switches shown in the figures are conceptual only. Their function may be carried out through the operation of program logic, through dedicated logic, through the interaction of program control and dedicated logic, or even manually, the particular technique being selectable by the implementer as more specifically understood from the context.

[0106] As used in this application, the terms “circuit,” “circuitry” may refer to one or more or all of the following: (a) hardware-only circuit implementations (such as implementations in only analog and / or digital circuitry); (b) combinations of hardware circuits and software, such as (as applicable): (i) a combination of analog and / or digital hardware circuit(s) with software / firmware and (ii) any portions of hardware processor(s) with software (including digital signal processor(s)), software, and memory(ies) that work together to cause an apparatus, such as a mobile phone or server, to perform various functions); and (c) hardware circuit(s) and or processor(s), such as a microprocessor(s) or a portion of a microprocessor(s), that requires software (e.g., firmware) for operation, but the software may not be present when it is not needed for operation.” This definition of circuitry applies to all uses of this term m this application, including m any claims. As a further example, as used in this application, the term circuitry also covers an implementation of merely a hardware circuit or processor (or multiple processors) or portion of a hardware circuit or processor and its (or their) accompanying software and / or firmware. The term circuitry also covers, for example and if applicable to the particular claim element, a baseband integrated circuit or processor integrated circuit for a mobile device or a similar integrated circuit in server, a cellular network device, or other computing or network device.Docket No,: 093386-0067-W001(VU25082)

[0107] It should be appreciated by those of ordinary skill in the art that any block diagrams herein represent conceptual views of illustrative circuitry embodying the principles of the disclosure. Similarly, it will be appreciated that any flow charts, flow diagrams, state transition diagrams, pseudo code, and the like represent various processes which may be substantially represented in computer readable medium and so executed by a computer or processor, whether or not such computer or processor is explicitly shown.

[0108] “BRIEF SUMMARY OF SOME SPECIFIC EMBODIMENTS” m this specification is intended to introduce some example embodiments, with additional embodiments being described in “DETAILED DESCRIPTION” and / or in reference to one or more drawings. “BRIEF SUMMARY OF SOME SPECIFIC EMBODIMENTS” is not intended to identify essential elements or features of the claimed subject matter, nor is it intended to limit the scope of the claimed subject matter.

Claims

Docket No,: 093386-0067-W001(VU25082) CLAIMSWhat is claimed is:

1. An apparatus, comprising:a first waveguide core including an array of first unit cells arranged along a longitudinal direction thereof, each of the first unit cells being characterized by an absence of a mirror- symmetry plane orthogonal to the longitudinal direction,w’herein the array of first unit cells is configured to impose periodic modulation of an effective refractive index that causes the first waveguide core to support a first hybrid guided mode including a longitudinal electric (LE) component and a transverse electric (TE) component.

2. The apparatus of claim 1, wherein the periodic modulation further causes the first waveguide core to exhibit a first photonic bandgap and a distinct second photonic bandgap centered at a higher frequency than the first photonic bandgap.

3. The apparatus of claim 1, wherein the first waveguide core has a substantially rectangular cross-section in a transverse plane orthogonal to the longitudinal direction.

4. The apparatus of claim 3, wherein each of the first unit cells has a pair of substantially semicylindrical openings adjacent to a separator beam oriented at a nonorthogonal angle with respect to the longitudinal direction.

5. The apparatus of claim 4, wherein the nonorthogonal angle belongs to an angular range between 1 degree and 89 degrees or between -1 degree and -89 degrees.

6. The apparatus of claim 4, wherein an operating wavelength for the first waveguide core is selected by selecting one or more parameters from the group consisting of:a material of the first waveguide core;a pitch of the unit cells;a width of the separator beam;a width of the first waveguide core;Docket No,: 093386-0067-W001(VU25082) a thickness of the first waveguide core;a diameter of the pair of substantially semicylindrical openings; anda value of the first nonorthogonal angle.

7. The apparatus of claim 6, wherein the operating wavelength is in a wavelength range between 1.45 pm and 1.65 pm or between 1.26 pm to 1.36 pm.

8. The apparatus of claim 1, further comprising a waveguide cladding adjacent to the first waveguide core,wherein the waveguide cladding includes a portion of a substantially planar dielectric substrate configured to structurally support the first waveguide core.

9. The apparatus of claim 8,wherein the first waveguide core comprises silicon; andwherein the waveguide cladding comprises silicon oxide.

10. The apparatus of claim 1, wherein each of the first unit cells has a substantially rectangular opening having a wall oriented at a non-orthogonal angle with respect to the longitudinal direction.

11. The apparatus of claim 10, wherein different ones of the first unit cells are characterized by different respective values of the non-orthogonal angle.

12. The apparatus of claim 1, further comprising a second waveguide core including an array of second unit cells arranged along a longitudinal direction thereof each of the second unit cells being characterized by an absence of a mirror symmetry plane orthogonal to the longitudinal direction of the second waveguide core,wherein the array of second unit cells is configured to impose periodic modulation of an effective refractive index that causes the second waveguide core to support a second hybrid guided mode including a corresponding LE component and a corresponding TE component, the second hybrid guided mode being different from the first hybrid guided mode.Docket No,: 093386-0067-W001(VU25082) 13. The apparatus of claim 12, further comprising an optical coupler including a first optical port at a first side thereof and a plurality of second optical ports at a second side thereof, each of the second optical ports being optically coupled to the first optical port,wherein the first waveguide core is connected to one of the second optical ports; and wherein the second waveguide core is connected to another one of the second optical ports.

14. The apparatus of claim 13, further comprising:a laser configured to apply light having a selected operating wavelength to the first optical port;a first optical modulator configured to modulate, based on a first data stream, a first portion of the light received from the laser through the first waveguide core; anda second optical modulator configured to modulate, based on a different second data stream, a second portion of the light received from the laser through the second waveguide core.

15. The apparatus of claim 14, further comprising an optical multiplexer configured to multiplex the modulated first and second portions.

16. The apparatus of claim 13, further comprising:a first photodetector configured to receive light from the first optical port through the first waveguide core; anda second photodetector configured to receive light from the first optical port through the second waveguide core.

17. The apparatus of claim 1,wherein the first waveguide core further includes an array of second unit cells arranged along the longitudinal direction of the first waveguide core;wherein the array of second unit cells is end-connected to the array of first unit cells; wherein the first unit cells have respective mirror-symmetry-breaking elements oriented at a same nonorthogonal angle with respect to the longitudinal direction; andwherein different ones of the second unit cells have respective mirror-symmetry-breaking elements oriented at different respective angles with respect to the longitudinal direction.Docket No,: 093386-0067-W001(VU25082) 18. The apparatus of claim 17, wherein the different respective angles have values in a range between 0 degrees and a value of the same nonorthogonal angle.

19. The apparatus of claim 1, further comprising:a laser configured to apply light having a selected operating wavelength to a first end of the first waveguide core; andone or more optical components optically coupled to a second end of the first waveguide core to receive the light from the laser through the first waveguide core,wherein the first waveguide core is configured to attenuate back-reflected light by substantially preventing the back-reflected light from remaining in a guided mode and by causing the back-reflected light to radiate into one or more non-guided modes.

20. The apparatus of claim 1, wherein the first waveguide core has a transverse cross-section configured to cause a corresponding homogeneous waveguide core to support a single guided mode.

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