We introduce partially-parity-time (pPT)-symmetric azimuthal potentials composed from individual PT-symmetric cells located on a ring, where two azimuthal directions are nonequivalent in a sense that in such potential excitations carrying topological dislocations exhibit different dynamics for different directions of energy circulation in the initial field distribution. Such nonconservative ratchetlike structures support rich families of stable vortex solitons in cubic nonlinear media, whose properties depend on the sign of the topological charge due to the nonequivalence of azimuthal directions. In contrast, oppositely charged vortex solitons remain equivalent in similar fully-PT-symmetric potentials. The vortex solitons in the pPT- and PT-symmetric potentials are shown to feature qualitatively different internal current distributions, which are described by different discrete rotation symmetries of the intensity profiles.
We address the interplay between two fundamentally different wave-packet localization mechanisms, namely, resonant dynamic localization due to collapse of quasienergy bands in periodic media and disorder-induced Anderson localization. Specifically, we consider light propagation in periodically curved waveguide arrays on-resonance and off-resonance, and show that inclusion of disorder leads to a gradual transition from dynamic localization to Anderson localization, which eventually is found to strongly dominate. While in the absence of disorder the degree of localization depends critically on the bending amplitude of the waveguide array, when the Anderson regime takes over the impact of resonant effects becomes negligible.
We show that chirped metal-dielectric waveguide arrays with focusing cubic nonlinearity can support plasmonic lattice solitons that undergo self-deflection in the transverse plane. Such lattice solitons are deeply subwavelength self-sustained excitations, although they cover several periods of the array. Upon propagation, the excitations accelerate in the transverse plane and follow trajectories curved in the direction in which the separation between neighboring metallic layers decreases, a phenomenon that yields considerable deflection angles. The deflection angle can be controlled by varying the array chirp. We also reveal the existence of surface modes at the boundary of truncated plasmonic chirped array that form even in the absence of nonlinearity.
Toroidal modes in the form of so-called Hopfions, with two independent winding numbers, a hidden one (twist s), which characterizes a circular vortex thread embedded into a three-dimensional soliton, and the vorticity around the vertical axis (m), appear in many fields, including field theory, ferromagnetics, and semi- and superconductors. Such topological states are normally generated in multicomponent systems, or as trapped quasilinear modes in toroidal potentials. We uncover that stable solitons with this structure can be created, without any linear potential, in the single-component setting with the strength of repulsive nonlinearity growing fast enough from the center to the periphery, for both steep and smooth modulation profiles. Toroidal modes with s=1 and vorticity m=0, 1, 2 are produced. They are stable for m=1, and do not exist for s>1. An approximate analytical solution is obtained for the twisted ring with s=1, m=0. Under the application of an external torque, it rotates like a solid ring. The setting can be implemented in a Bose-Einstein condensate (BEC) by means of the Feshbach resonance controlled by inhomogeneous magnetic fields.
We consider bright solitons supported by a symmetric inhomogeneous defocusing nonlinearity growing rapidly enough toward the periphery of the medium, combined with an antisymmetric gain-loss profile. Despite the absence of any symmetric modulation of the linear refractive index, which is usually required to establish a parity-time (PT ) symmetry in the form of a purely real spectrum of modes, we show that the PT symmetry is never broken in the present system, and that the system always supports stable bright solitons, i.e., fundamental and multipole ones. This fact is connected to the nonlinearizability of the underlying evolution equation. The increase of the gain - loss strength results, in lieu of the PT symmetry breaking, in merger of pairs of different soliton branches, such as fundamental and dipole, or tripole and quadrupole ones. The fundamental and dipole solitons remain stable at arbitrarily large values of the gain-loss coefficient.
We introduce Bloch-wave beatings in arrays of multimode periodically bent waveguides with a transverse refractive index gradient. The new phenomenon manifests itself in the periodic drastic increase of the amplitude of the Bloch oscillations that accompanies resonant conversion of modes guided by the individual waveguides. The Bloch-wave beatings are found to be most pronounced when the length of the resonant mode conversion substantially exceeds the longitudinal period of the Bloch oscillations. The beating frequency decreases when the amplitude of waveguide bending decreases, while the beating amplitude is restricted by the amplitude of the Bloch oscillations that emerge from the second allowed band of the Floquet-Bloch lattice spectrum.
We show, by means of numerical and analytical methods, that media with a repulsive nonlinearity which grows from the center to the periphery support a remarkable variety of previously unknown complex stationary and dynamical three-dimensional (3D) solitary-wave states. Peanut-shaped modulation profiles give rise to vertically symmetric and antisymmetric vortex states, and novel stationary hybrid states, built of top and bottom vortices with opposite topological charges, as well as robust dynamical hybrids, which feature stable precession of a vortex on top of a zero-vorticity soliton. The analysis reveals stability regions for symmetric, antisymmetric, and hybrid states. In addition, bead-shaped modulation profiles give rise to the first example of exact analytical solutions for stable 3D vortex solitons. The predicted states may be realized in media with a controllable cubic nonlinearity, such as Bose-Einstein condensates.
We study light propagation in waveguide arrays made in Kerr nonlinear media with a transverse refractive index gradient, and we find that the presence of the refractive index gradient leads to the appearance of a number of new soliton families. The effective coupling between the solitons and the localized linear eigenmodes of the lattice induces a drastic asymmetry in the soliton shapes and the appearance of long tails at the soliton wings. Such unusual solitons are found to be completely stable under propagation, and we report their experimental observation in fs-laser written waveguide arrays with focusing Kerr nonlinearity.
Guiding light at the nanoscale is usually accomplished using surface plasmons(1-12). However, plasmons propagating at the surface of a metal sustain propagation losses. A different type of surface excitation is the Dyakonov surface wave. These waves, which exist in lossless media, were predicted more than two decades ago(13) but observed only recently(14). Dyakonov surface waves exist when at least one of the two media forming the surface exhibits a suitable anisotropy of refractive indexes. Although propagating only within a narrow range of directions(15), these waves can be used to create modes supported by ultrathin films that confine light efficiently within film thicknesses well below the cutoff thickness required in standard waveguides. Here, we show that 10 nm and 20 nm dielectric nanosheets of aluminium oxide clad between an anisotropic crystal (lithium triborate) and different liquids support Dyakonov-like modes. The direction of light propagation can be controlled by modulating the refractive index of the cladding. The possibility of guiding light in nanometre-thick films with no losses and high directionality makes Dyakonov wave modes attractive for planar photonic devices in schemes similar to those currently employing long-range plasmons.
We show that the rate at which light tunnels between neighboring multimode waveguides can be drastically increased or reduced by the presence of small longitudinal periodic modulations of the waveguide properties that stimulate resonant conversion between the eigenmodes of each waveguide. Such a conversion, available only in multimode guiding structures, leads to periodic power transfer into higher-order modes, whose tails may considerably overlap with neighboring waveguides. As a result, the effective coupling constant for neighboring waveguides may change by several orders of magnitude upon small variations in the longitudinal modulation parameters. (C) 2014 Optical Society of America
We find that the recently introduced model of self-trapping supported by a spatially growing strength of a repulsive nonlinearity gives rise to robust vortex-soliton tori, i.e., three-dimensional vortex solitons, with topological charges S >= 1. The family with S = 1 is completely stable, while the one with S = 2 has alternating regions of stability and instability. The families are nearly exactly reproduced in an analytical form by the Thomas-Fermi approximation. Unstable states with S = 2 and 3 split into persistently rotating pairs or triangles of unitary vortices. Application of a moderate torque to the vortex torus initiates a persistent precession mode, with the torus' axle moving along a conical surface. A strong torque heavily deforms the vortex solitons, but, nonetheless, they restore themselves with the axle oriented according to the vectorial addition of angular momenta.
We show that an inhomogeneous defocusing nonlinearity that grows toward the periphery in the positive and negative transverse directions at different rates can support strongly asymmetric fundamental and multipole bright solitons, which are stable in wide parameter regions. In the limiting case, when nonlinearity is uniform in one direction, solitons transform into stable domain walls (fronts), with constant or oscillating intensity in the homogeneous region, attached to a tail rapidly decaying in the direction of growing nonlinearity.
We predict Anderson localization of light with nested screw topological dislocations propagating in disordered two-dimensional arrays of hollow waveguides illuminated by vortex beams. The phenomenon manifests itself in the statistical presence of topological dislocations in ensemble-averaged output distributions accompanying standard disorder-induced localization of light spots. Remarkably, screw dislocations are captured by the light spots despite the fast and irregular transverse displacements and topological charge flipping undertaken by the dislocations due to the disorder. The statistical averaged modulus of the output local topological charge depends on the initial vorticity carried by the beam.
We study the properties of bright and vortex solitons in thermal media with nonuniform thermal conductivity and
homogeneous refractive index, whereby the local modulation of the thermal conductivity strongly affects the entire
refractive index distribution. While regions where the thermal conductivity is increased effectively expel light, selftrapping
may occur in the regions with reduced thermal conductivity, even if such regions are located close to the
material boundary. As a result, strongly asymmetric self-trapped beams may form inside a ring with reduced thermal
conductivity and perform persistent rotary motion. Also, such rings are shown to support stable vortex solitons,
which may feature strongly noncanonical shapes.
Los solitones ópticos proporcionan oportunidades únicas para el control de la luz mediante luz. Actualmente, el campo de la formación de solitones en materiales naturales está bien desarrollado, ya que las principales propiedades de los posibles solitonesestán bien comprendidas y se han observado experimentalmente en una variedad de materiales y configuraciones físicas. Esto incluye materiales con non-linealidadcúbica, cuadrática,fotorrefractiva, saturable, no local y térmica.Nuevas oportunidades para la generación y control de solitones pueden ser posibles en materiales artificiales, cuyas propiedades pueden ser modificados a voluntad. Esto incluye, por ejemplo, las modulaciones de los parámetros principales del material así como las ganancias o pérdidas o materiales con propiedades lineales y no lineales con modulaciones transversales superficiales del índice de refracción lineal y el coeficiente de no linealidad. La exploración de la existencia, estabilidad y propiedades dinámicas de solitones en materiales conservativos y disipativos es el objeto de esta tesis doctoral.Estudiamos solitones fundamentales y de orden superior en materiales artificiales con índice de refracción lineal no homogéneo y no lineal.Demostramos que pueden existir solitones de dos dimensiones estables en lattices no linealescon dominios transversalmente alternos con no linealidades cúbicos y saturables. Consideramos solitones-multi-componentesen materiales donde uno de los componentes está afectado por la modulación del índice de refracción o no linealidad mientras que el otro componente se propaga como si estuviera en un medio no lineal uniforme. Estudiamos si la modulación de fase cruzada entre los componentes permite la estabilización del estado solitón.Estudiamos también medios con no linealidad auto-desfocalizante que decrece rápidamente desde el centro hacia la periferia. En contraste con la creencia común, hemos encontrado que tales materiales soportan solitones brillantes estables cuando la no linealidad crece hacia la periferia con suficiente rapidez. Consideramos diferentes formas de la no-linealidad y analizamos los tipos de solución solitón que existen en cada caso.Los materiales no lineales con distribución espacial complejas de ganancias y pérdidas también ofrecen importantes oportunidades para lageneración de solitones., incluyendo multipolos y vórtices. Estudiamos solitonesunidimensionales en materiales auto-focalizantes y auto-desfocalizantes en geometrías simples y complejas de ganancia y absorción. Abordamosestructuras con vórtices estacionarios y también vórtices con distribuciones de intensidad modulada azimutalmente. Por último, se estudia la posibilidad de formar light bullets en medios no lineales con propiedades no homogéneas.
Optical solitons provide unique opportunities for the control of light‐bylight. Today, the field of soliton formation in natural materials is mature, as the
main properties of the possible soliton states are well understood. In particular, optical solitons have been observed experimentally in a variety of materials and physical settings, including media with cubic, quadratic, photorefractive, saturable, nonlocal and thermal nonlinearities.
New opportunities for soliton generation, stability and control may become accessible in complex engineered, artificial materials, whose properties
can be modified at will by, e.g., modulations of the material parameters or the application gain and absorption landscapes. In this way one may construct
different types of linear and nonlinear optical lattices by transverse shallow modulations of the linear refractive index and the nonlinearity coefficient or
complex amplifying structures in dissipative nonlinear media. The exploration of the existence, stability and dynamical properties of conservative and dissipative solitons in settings with spatially inhomogeneous linear refractive index, nonlinearity, gain or absorption, is the subject of this PhD Thesis.
We address stable conservative fundamental and multipole solitons in complex engineered materials with an inhomogeneous linear refractive index and
nonlinearity. We show that stable two‐dimensional solitons may exist in nonlinear lattices with transversally alternating domains with cubic and saturable
nonlinearities. We consider multicomponent solitons in engineered materials, where one field component feels the modulation of the refractive index or
nonlinearity while the other component propagates as in a uniform nonlinear medium. We study whether the cross‐phase‐modulation between two
components allows the stabilization of the whole soliton state.
Media with defocusing nonlinearity growing rapidly from the center to the periphery is another example of a complex engineered material. We study such
systems and, in contrast to the common belief, we have found that stable bright solitons do exist when defocusing nonlinearity grows towards the periphery rapidly enough. We consider different nonlinearity landscapes and analyze the types of soliton solution available in each case.
Nonlinear materials with complex spatial distributions of gain and losses also provide important opportunities for the generation of stable one‐ and
multidimensional fundamental, multipole, and vortex solitons. We study onedimensional solitons in focusing and defocusing nonlinear dissipative materials
with single‐ and double‐well absorption landscapes. In two‐dimensional geometries, stable vortex solitons and complexes of vortices could be observed.
We not only address stationary vortex structures, but also steadily rotating vortex solitons with azimuthally modulated intensity distributions in radially symmetric gain landscapes.
Finally, we study the possibility of forming stable topological light bullets in focusing nonlinear media with inhomogeneous gain landscapes and uniform twophoton absorption.
Due to their unique ability to maintain an intensity distribution upon propagation, non-diffracting light fields are used extensively in various areas of science, including optical tweezers, nonlinear optics and quantum optics, in applications where complex transverse field distributions are required. However, the number and type of rigorously non-diffracting beams is severely limited because their symmetry is dictated by one of the coordinate system where the Helmholtz equation governing beam propagation is separable. Here, we demonstrate a powerful technique that allows the generation of a rich variety of quasi-non-diffracting optical beams featuring nearly arbitrary intensity distributions in the transverse plane. These can be readily engineered via modifications of the angular spectrum of the beam in order to meet the requirements of particular applications. Such beams are not rigorously non-diffracting but they maintain their shape over large distances, which may be tuned by varying the width of the angular spectrum. We report the generation of unique spiral patterns and patterns involving arbitrary combinations of truncated harmonic, Bessel, Mathieu, or parabolic beams occupying different spatial domains. Optical trapping experiments illustrate the opto-mechanical properties of such beams.
We investigate the interplay of Bloch oscillations and Anderson localization in optics. Gradual washing out of Bloch oscillations and the formation of nearly stationary averaged intensity distributions, which are symmetric for narrow and strongly asymmetric for broad input excitations, are observed experimentally in laser-written waveguide arrays. At large disorder levels Bloch oscillations are completely destroyed and both narrow and wide excitations lead to symmetric stationary averaged intensity distributions with exponentially decaying tails.
The rectification of light beams in an optical ratchet is reported. This directed transport is implemented using a photonic mesh lattice consisting of sequences of directional couplers. In such a structure, we observe light propagation at an angle that is independent of the input direction of the injected beam
We address light propagation in Vogel optical lattices and show that such lattices support a variety of stable soliton solutions in both self-focusing and self-defocusing media, whose propagation constants belong to domains resembling gaps in the spectrum of a truly periodic lattice. The azimuthally rich structure of Vogel lattices allows generation of spiraling soliton motion.
We address stationary patterns in exciton-polariton condensates supported by a narrow external pump beam, and we discover that even in the absence of trapping potentials, such condensates may support stable localized stationary dissipative solutions (quasicompactons), whose field decays faster than exponentially or even vanishes everywhere outside the pump spot. More general conditions lead to dissipative solitons which may display bistability. The bistability in exciton-polariton condensates, which manifests itself in the simultaneous existence of two stable and one unstable localized solitons with different amplitudes, widths, and exciton-photon fractions under the same physical conditions, strongly depends on the width of the pump beam and is found to disappear for sufficiently narrow pump beams.
We address stationary patterns in exciton-polariton condensates supported by a narrow external pump beam,
and we discover that even in the absence of trapping potentials, such condensates may support stable localized
stationary dissipative solutions (quasicompactons), whose field decays faster than exponentially or even vanishes
everywhere outside the pump spot. More general conditions lead to dissipative solitons which may display
bistability. The bistability in exciton-polariton condensates, which manifests itself in the simultaneous existence
of two stable and one unstable localized solitons with different amplitudes, widths, and exciton-photon fractions
under the same physical conditions, strongly depends on the width of the pump beam and is found to disappear
for sufficiently narrow pump beams
We introduce a new concept for stable spatial soliton formation, mediated by the competition between self-bending induced by a strongly asymmetric nonlocal nonlinearity and spatially localized gain superimposed on a wide pedestal with linear losses. When acting separately both effects seriously prevent stable localization of light, but under suitable conditions they counteract each other, forming robust soliton states that are attractors for a wide range of material and input light conditions.
We introduce a concept for stable spatial soliton formation, mediated by the competition between self-bending induced by a strongly asymmetric nonlocal nonlinearity and spatially localized gain superimposed on a wide pedestal with linear losses. When acting separately both effects seriously prevent stable localization of light, but under suitable conditions they counteract each other, forming robust soliton states that are attractors for a wide range of material and input light conditions.
We show that engineered photonic metamaterials composed of alternating layers of suitable dielectrics and metals can support different kinds of surface waves (SWs) under robust and readily achievable experimental conditions. The supported SWs include Dyakonov SWs, hybrid plasmons, and Dyakonov plasmons. In particular, in contrast to conventional physical settings, we show that the high form birefringence exhibited by the metamaterials allows Dyakonov SWs, or dyakonons, to exist within large angular existence domains and levels of localization similar to plasmons, thus making dyakonons available for practical applications
We show that Anderson localization is possible in waveguide arrays with periodically-spaced defect waveguides having lower refractive index. Such localization is mediated by Bragg reflection, and it takes place even if diagonal or off-diagonal disorder affects only defect waveguides. For off-diagonal disorder the localization degree of the intensity distributions monotonically grows with increasing disorder. In contrast, under appropriate conditions, increasing diagonal disorder may result in weaker localization.
We show that Anderson localization is possible in waveguide arrays with periodically spaced defect waveguides
having a lower refractive index. Such localization is mediated by Bragg reflection, and it takes place even if diagonal
or off-diagonal disorder affects only defect waveguides. For off-diagonal disorder the localization degree of the
intensity distributions monotonically grows with increasing disorder. In contrast, under appropriate conditions,
increasing diagonal disorder may result in weaker localization.
We study the gradual transition from one-dimensional to two-dimensional Anderson localization upon transformation of the di-mensionality of disordered waveguide arrays. An effective transition from one- to two-dimensional system is achieved by increas-ing the number of rows forming the arrays. We observe that, for a given disorder level, Anderson localization becomes weaker with increasing number of rows, hence the effective dimension.
We reveal that the competition among diffraction, cubic nonlinearity, two-photon absorption, and gain localized in both space and time results in arrest of collapse, suppression of azimuthal modulation instabilities for spatiotemporal wave packets, and formation of stable three-dimensional light bullets. We show that Gaussian spatiotemporal gain landscapes support bright, fundamental light bullets, while gain landscapes featuring a ringlike spatial and a Gaussian temporal shapes may support stable vortex bullets carrying topological phase dislocations.
This paper was published in Optics Letters and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://dx.doi.org/10.1364/OL.37.000869. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.
We study the spiral spectra scattered off transparent dielectric spheres when probed by different Laguerre–Gaussian
light beams, carrying nested topological wavefront dislocations. We show that such scattering data may be employed
to determine geometrical properties of the spheres, such as their position. The technique is a generalization
of standard Mie scattering, and it can be extended to study and to characterize nanospheres.
We address Anderson localization of light in disordered optical lattices where the disorder strength varies across the transverse direction. Such variation changes the preferred domains where formation of localized eigenmodes is most probable, hence drastically impacting light localization properties. Thus, step-like disorder results in formation of modes with different decay rates at both sides of the interface, while a smoothly varying disorder yields appearance of modes that are extended within weakly disordered domains and rapidly fade away in strongly disordered domains.
We uncover that, in contrast to the common belief, stable dissipative solitons exist in media with uniform gain in the presence of nonuniform cubic losses, whose local strength grows with coordinate η (in one dimension) faster than |η| . The spatially-inhomogeneous absorption also supports new types of solitons, that do not exist in uniform dissipative media. In particular, single-well absorption profiles give rise to spontaneous symmetry breaking of fundamental solitons in the presence of uniform focusing nonlinearity, while stable dipoles are supported by double-well absorption landscapes. Dipole solitons also feature symmetry breaking, but under defocusing nonlinearity.
We show the existence of stable two- and three-dimensional vortex solitons carrying multiple, spatially separated, single-charge topological dislocations nested in a common vortex-ring core. Such nonlinear states are supported by elliptical gain landscapes in focusing nonlinear media with two-photon absorption. The separation between the phase dislocations is dictated mostly by the geometry of the gain landscape, and it only slightly changes upon variation of the gain or absorption strength.
We show that bimodal systems with a spatially nonuniform defocusing cubic nonlinearity, whose strength grows toward the periphery, can support stable two-component solitons. For a sufficiently strong cross-phase-modulation interaction, vector solitons with overlapping components become unstable, while stable families of solitons with spatially separated components emerge. Stable complexes with separated components may be built not only of fundamental solitons, but of multipoles, too.
We report that defocusing cubic media with spatially inhomogeneous nonlinearity, whose strength increases rapidly enough toward the periphery, can support stable bright localized modes. Such nonlinearity landscapes give rise to a variety of stable solitons in all three dimensions, including one-dimensional fundamental and multihump states, two-dimensional vortex solitons with arbitrarily high topological charges, and fundamental solitons in three dimensions. Solitons maintain their coherence in the state of motion, oscillating in the nonlinear potential as robust quasiparticles and colliding elastically. In addition to numerically found soliton families, particular solutions are found in an exact analytical form, and accurate approximations are developed for the entire families, including moving solitons.
We show that the geometrically induced potential existing in undulated slab waveguides dramatically affects the properties of solitons. In particular, whereas solitons residing in the potential maxima do not feature power thresholds and are stable, their counterparts residing in the potential minima are unstable and may exhibit a power threshold for their existence. Additionally, the geometric potential is shown to support stable multipole solitons that cannot be supported by straight waveguides. Finally, the geometric potential results in the appearance of the effective barriers that prevent transverse soliton motion.
We introduce a general approach for generation of sets of three-dimensional quasi-nonspreading wave packets
propagating in linear media, also referred to as linear light bullets. The spectrum of rigorously nonspreading wave
packets in media with anomalous group velocity dispersion is localized on the surface of a sphere, thus drastically
restricting the possible wave packet shapes. However, broadening slightly the spectrum affords the generation of a
large variety of quasi-nonspreading distributions featuring complex topologies and shapes in space and time that are
of interest in different areas, such as biophysics or nanosurgery. Here we discuss the method and show several
illustrative examples of its potential.
We show that ringlike localized gain landscapes imprinted in focusing cubic (Kerr) nonlinear media with strong two-photon absorption support new types of stable higher-order vortex solitons containing multiple phase singularities nested inside a single core. The phase singularities are found to rotate around the center of the gain landscape, with the rotation period being determined by the strength of the gain and the nonlinear absorption.
We introduce stripe-like quasi-nondiffracting lattices that can be generated via spatial spectrum engineering. The complexity of the spatial shapes of such lattices and the distance of their almost diffractionless propagation depend on the width of their ring-like spatial spectrum. Stripe-like lattices are extended in one direction and are localized in the orthogonal one, thereby creating either straight or curved in any desired fashion optically-induced channels that may be used for optical trapping, optical manipulation, or optical lattices for quantum and nonlinear optics applications. As an illustrative example, here we show their potential for spatial soliton control. Complex networks consisting of several intersecting or joining stripe-like lattices suited to a particular application may also be constructed.
The role of Dyakonov surface waves in the transmission through structures composed of birefringent media is theoretically explored. In the case of structures using prisms, unexpected high transmission above the critical angle due to resonant excitation of Dyakonov surface waves is predicted. This transmission is produced only when TE polarized incident wave reaches the interface supporting the surface waves within a narrow interval of angles, for both the angle of incidence and the angle with respect to the optic axis of the birefringent media. As a result, over 90% transmission for a single and isolated peak confined in the two transversal directions, with hybrid TE and TM polarization, can be obtained.