Reciprocal Vectors Research Articles

Generation of conical and spherical second harmonics in three-dimensional nonlinear photonic crystals with radial symmetry

Two types of radially symmetric three-dimensional nonlinear photonic crystals with the cylindrical structure and the egglike structure are proposed, from which the conical and the spherical quadratic harmonic waves can be produced, respectively, by three-dimensional quasi-phase matching. First, the cylindrical structures with periodic and aperiodic modulations of the nonlinear coefficient are both studied, showing their significant advantages compared to the corresponding two-dimensional structures. The dependencies of the transverse and the longitudinal phase-matching periods on harmonic propagating directions are also calculated and analyzed. Then, the egglike structure is designed by programming and the distribution of reciprocal vectors is presented, indicating its ability to generate the spherical harmonic as a point light source. The investigation of the intensity distribution on the spherical wavefront is also performed, showing its strong dependence on the harmonic polarization and the quadratic nonlinear coefficients.

Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure

We report simultaneous generation of multiple conical second-harmonic waves in a two-dimensional nonlinear photonic structure when illuminated by two overlapping noncollinear fundamental beams. We show that this phenomenon is caused by the nonlinear Čerenkov radiation emitted due to the interaction of photons from each constituent fundamental beam as well as the virtual one propagating along the bisector of the two beams. In addition, by studying the asymmetric geometry of the interaction, we uniquely verify the effects of reciprocal vectors on the Čerenkov-type second-harmonic generation in nonlinear photonic structures.

Nonlinear plasmonic frequency conversion through quasiphase matching

Received 9 September 2010; published 6 October 2010An efficient frequency conversion scheme of surface-plasmon polariton SPP is proposed for nonlinearplasmonic applications. Lossy coupling wave equations are deduced to describe the interaction among involvedSPP frequency components. The reciprocal vector of a microstructure could partially compensate the SPPwave-vector mismatch through quasiphase matching QPM . As an example, the frequency doubling of SPPover a periodically poled lithium niobate is studied. The high-field concentration of SPP greatly enhanced thesecond-harmonic generation SHG over a short traveling distance. Although the high attenuation of SPPaffects the SHG severely, a long-range SPP SHG is further proposed and investigated, which shows greatefficiency improvement. The applications of QPM SPP frequency conversion are also discussed showing greatpotentials in miniature plasmonic sources and processors.DOI: 10.1103/PhysRevB.82.155107 PACS number s : 73.20.Mf, 42.65.Ky, 78.20.Bh

Lattice Green Functions on a Two-Dimensional Rectangular Lattice with Next-Nearest Neighbor Interaction

Lattice Green functions appear in many branches of physics, such as: problems of bound states on a lattice, theory of diffusion, random walks, band structure calculations as well as in calculations of resistivity of infinite resistor lattices. They were a subject of examination in sixties, in connection with the investigations of spin waves interactions in the ferromagnetic Heisenberg model (see, e.g. [1]). After a break, an interest in this field slowly grows again, in connection with high-temperature superconductivity theories resorting to Bose-condensation of bound pairs of charges. For two-electron bound pairs in an empty lattice not only we can give the exact eigenenergies and eigenvalues of such pairs but also express them in analytic form — a rare case in the solid state theory. The formulae for lattices with nearest neighbor interactions using elliptic integrals can be found in literature. An interesting problem is considering next-nearest neighbor interaction, especially in the context of relatively large values of such interactions found in some high-temperature superconductors (on quasi-two-dimensional lattices). These interactions have a large influence on, e.g., phase diagrams of the Hubbard type models or on the evolution of the superconductivity between the limits of BCS and Bose-condensation in these models. Numerically such Green functions on a simple cubic lattice in the direction [111] were calculated by Krompiewski [2], the analytic form for the reciprocal space vector (π, π, π) in the same lattice was given by Bahurmuz [3]; for two-dimensional rectangular lattice the recursion formula was published by Morita [4]. Let us note that the imaginary part of the on-site lattice Green function is proportional to the density of states for a given lattice. In case of the rectangular lattice it

Efficient orthogonalization in gas sensor arrays using reciprocal kernel support vector regression

In this paper support vector regression is presented, and it is used to model the responses of metal oxide gas sensors to combustion byproducts. A new version of the reciprocal kernel is presented for use in the regression, and it is tested in multiple dimensions. The orthogonality of the sensors is also calculated to determine if the sensors are suitable for use in arrays. A fast numerical approximation of the sensor orthogonality, which takes advantage of the reciprocal kernel, is presented as a way of quickly optimizing the effective response of large arrays. Comparison reveals advantages over standard approaches like principal component analysis.

Vector method for studying the second-harmonic-generation light derived from complex periodic ferroelectric domains

In this Letter, in order to overcome the disadvantages of controlling the second-harmonic-generation (SHG) light derived from the traditional one-dimensional (1D) periodic ferroelectric domains we propose a kind of so-called complex periodic ferroelectric structure (CPFS), which unit cell is composed of even layers of positive and negative domains arranged alternatively following aperiodic sequence. It is found that comparing with the traditional periodic structure, CPFS cannot offer more reciprocal vector compensations for the mismatching phase, but CPFS may provide larger effective nonlinear coefficients (ENCs) in high-order quasi-phase-matching (QPM) and possesses advantages of the amplitude modulation for SHG peaks. In this Letter we study CPFS by use of vector method (VM), where the contribution to ENC for each domain or each unit cell will be treated as a vector and the QPM condition for CPFS and the modulation effect of aperiodic unit cells have been obtained. Without any Fourier transformation VM treats the grating function in real space and will be very convenient and intuitive. Both VM and CPFS would possess potential applications in the field of SHG investigations.

Absorption Enhancement of Solar Concentrators via New Surface Photonic Designs

Enhancing absorption of solar cells over the solar spectrum is one of the most important ways to improve such devices’ performances. In this research, two-dimensional surface photonic designs were suggested to introduce rotationally distributed reciprocal vectors, which will match the guiding modes of the thin planar layer of an active material, and thus to further couple the incident light laterally into the layer. This allows the use of a fewer amount of active materials, increases the devices’ angle acceptance, and reduces costs for both fabrication and system installation.

X-ray dynamical diffraction from partly relaxed epitaxial structures

An approach to calculation of reciprocal space maps of x-ray diffraction from partly relaxed multilayered epitaxial structures is reported. The theory takes into account the additional harmonics of wave field caused by the difference in the lateral projections of reciprocal vectors in the sample layers. The reciprocal space maps shown can be simulated on the basis of the dynamical diffraction theory and matrix method for boundary conditions, which are applicable to arbitrary experimental geometry. The developed theory explains the experimental results from several typical epitaxial structures in partly relaxed state.

Space Vector Modulation for Multiphase Inverters Based on a Space Partitioning Algorithm

Since the late 1990s, multiphase drives have become a serious alternative to three-phase drives in some particular applications such as electric ship propulsion, locomotive traction, electric vehicles, and high-power industrial applications. Nowadays, the research activity is focused on the development of control strategies that can exploit the degrees of freedom that exist in multiphase machines. As known, a multiphase motor cannot be analyzed using the space vector representation in a single <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> plane, but it is necessary to introduce multiple <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> planes. In this paper, the problem of the space vector modulation (SVM) of multiphase inverters is solved extending the theory of SVM used for traditional three-phase voltage source inverters and introducing the concept of reciprocal vector. The proposed approach, confirmed by experimental tests, allows the full exploitation of the dc input voltage and the simultaneous modulation of voltage space vectors in different <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> planes.

Poling Quality Evaluation of Optical Superlattice Using 2D Fourier Transform Method

In this article we develop a method to evaluate the poling quality of optical superlattice (OSL) based on two-dimensional (2D) Fourier transform. To demonstrate this method, -Z or +Z face etched OSL samples with desired patterns are fabricated by standard electric field poling technique. By analyzing the processed micrograph of the etched surfaces, the magnitude of the reciprocal vectors of the OSL are calculated directly and rapidly. Second harmonic generation (SHG) experiment is performed to validate the evaluation result.

Analysis of three-spacecraft data using planar reciprocal vectors: methodological framework and spatial gradient estimation

Abstract. In the context of ESA's Cluster mission, four-point array techniques are widely used to analyze space plasma phenomena such as shocks and discontinuities, waves and turbulence, and spatial gradients. Due to failures of single instruments on the Cluster spacecraft fleet, there is also need for array processing of three-point measurements. In this paper we identify planar reciprocal vectors as a generic tool for this purpose. The class of three-point techniques introduced here includes methods for discontinuity analysis, wave identification, and spatial gradient determination. Parameter vectors can be resolved fully in the spacecraft plane but further assumptions or physical constraints have to be specified to estimate the normal components. We focus on the gradient estimation problem where we check and illustrate our approach using Cluster measurements.

Spatial aliasing and distortion of energy distribution in the wave vector domain under multi-spacecraft measurements

Abstract. Aliasing is a general problem in the analysis of any measurements that make sampling at discrete points. Sampling in the spatial domain results in a periodic pattern of spectra in the wave vector domain. This effect is called spatial aliasing, and it is of particular importance for multi-spacecraft measurements in space. We first present the theoretical background of aliasing problems in the frequency domain and generalize it to the wave vector domain, and then present model calculations of spatial aliasing. The model calculations are performed for various configurations of the reciprocal vectors and energy spectra or distribution that are placed at different positions in the wave vector domain, and exhibit two effects on aliasing. One is weak aliasing, in which the true spectrum is distorted because of non-uniform aliasing contributions in the Brillouin zone. It is demonstrated that the energy distribution becomes elongated in the shortest reciprocal lattice vector direction in the wave vector domain. The other effect is strong aliasing, in which aliases have a significant contribution in the Brillouin zone and the energy distribution shows a false peak. These results give a caveat in multi-spacecraft data analysis in that spectral anisotropy obtained by a measurement has in general two origins: (1) natural and physical origins like anisotropy imposed by a mean magnetic field or a flow direction; and (2) aliasing effects that are imposed by the configuration of the measurement array (or the set of reciprocal vectors). This manuscript also discusses a possible method to estimate aliasing contributions in the Brillouin zone based on the measured spectrum and to correct the spectra for aliasing.

Acousto-optic tunable second harmonic generation in periodically poled LiNbO_3

Acousto-optic (AO) tunable second harmonic generation (SHG) was proposed in periodically poled lithium niobate (PPLN). The acoustic wave could either be induced from an external transducer or self-generated in PPLN driving with a cross-field radio frequency field. The reciprocal vector of PPLN compensates the SHG wave-vector mismatch when quasi-phase- matching (QPM) condition is satisfied, while phonons with suitable frequencies may affect it by scattering photons to different polarization state. The QPM SHG and AO polarization rotation are coupled together. Second harmonic waves' intensities, polarization states and even phases thus could be manipulated instantly through AO interaction.

Collinear second harmonic generation of 20 wavelengths in a single two-dimensional decagonal nonlinear photonic quasi-crystal

We report on the observation of quasi-phase matched collinear second harmonic generation (SHG) at 20 wavelengths in a two-dimensional nonlinear photonic quasi-crystal with decagonal lattice. We show that at some wavelengths the second harmonics are generated via standard quasi-phase matching, namely a reciprocal vector exists that equals the phase-mismatch vector, while at others the SHG is as a result of the projection-based quasi-phase matching in which the momentum conservation is satisfied up to a projection of a reciprocal vector onto the direction of propagation. In spite of different generation mechanisms, the reciprocal vectors (or their projections) involved in the collinear QPM SHG can be described by a generalized equation.

Cascaded third-harmonic generation in a single short-range-ordered nonlinear photonic crystal

Collinear third-harmonic generation at 526.7 nm was realized by the simultaneous phase matching of two second-order processes in a single quadratic crystal: second-harmonic generation (SHG) and sum-frequency mixing (SFM). The measured conversion efficiency was 12%. As a nonlinear medium a LiNbO(3) nonlinear photonic crystal with short-range order was used that allowed simultaneous phase matching by use of discrete reciprocal vector (for the SHG process) and continuous reciprocal vectors (for the SFM process). It was demonstrated that the third harmonic could be generated efficiently in such a crystal even if the intermediate process of SHG was not perfectly phase matched.

Comparative structural study of thin films of a columnar liquid crystal aligned by mechanical shearing and zone casting

Zone casting is a promising method for fabrication of open highly oriented crystalline and liquid crystalline (LC) films for various applications in (opto)electronics. We have performed a comparative structural analysis of mechanically sheared and zone-cast films of a model columnar LC. Grazing Incidence X-ray diffraction and UV–vis spectroscopy show that, contrary to the mechanically sheared films, the columns in the zone-cast films are aligned perpendicular to the casting direction. In the films, two LC domains with [20] or [11] reciprocal space vectors perpendicular to the substrate plane are observed. This can be explained by a small lattice mismatch allowing epitaxial growth of the LC domains on each other.

Substantial gain enhancement for optical parametric amplification and oscillation in two-dimensional χ^(2) nonlinear photonic crystals

We have analyzed optical parametric interaction in a 2D NPC. While in general the nonlinear coefficient is small compared to a 1D NPC, we show that at numerous orientations a multitude of reciprocal vectors contribute additively to enhance the gain in optical parametric amplification and oscillation in a 2D patterned crystal. In particular, we have derived the effective nonlinear coefficients for common-signal amplification and common-idler amplification for a tetragonal inverted domain pattern. We show that in the specific case of signal amplification with QPM by both G(10) and G(11), symmetry of the crystal results in coupled interaction with the corresponding signal amplification by G(10) and G(1,-1). As a consequence, this coupled utilization of all three reciprocal vectors leads to a substantial increase in parametric gain. Using PPLN we demonstrate numerically that a gain that comes close to that of a 1D QPM crystal could be realized in a 2D NPC with an inverted tetragonal domain pattern. This special mechanism produces two pairs of identical signal and idler beams propagating in mirror-imaged forward directions. In conjunction with this gain enhancement and multiple beams output we predict that there is a large pulling effect on the output wavelength due to dynamic signal build-up in the intrinsic noncollinear geometry of a 2D NPC OPO.

Engineering the dielectric function of plasmonic lattices

We have systematically measured epsilon(omega) of subwavelength aperture arrays fabricated in metal films as a function of aperture size and incidence angle using terahertz time-domain spectroscopy. This approach simultaneously yields both the real and imaginary epsilon(omega) components, enabling deeper insight into the underlying mechanism of the 'enhanced optical transmission' (EOT) phenomenon. For random aperture arrays we find that epsilon(omega) has a plasma response, with an effective plasma frequency that is determined by the waveguide mode cutoff frequency of the individual apertures. However epsilon(omega) in plasmonic lattices is strongly modulated at discrete resonant frequencies that correspond to the reciprocal vectors in the structure factor that are superposed on the plasma envelope response and appear as dips in the EOT spectrum. The existence of a sum rule for the discrete resonance oscillator strengths when the aperture size or incidence angle are changed validates our approach and allows for engineering of the individual resonances in the EOT spectrum.

Light transmission through Fibonacci and periodic sub-wavelength slit arrays

In this paper, the light transmission spectra of quasi-periodic and periodic sub-wavelength slit arrays are studied. The results show that the spectral shape and transmission efficiency depend on the reciprocal vectors as well as the Fourier coefficients. The analysis suggests that the transmission minimum can be attributed to the surface plasmon polaritons (SPP) whereas the transmission maximum is associated with the Fabry–Perot-like resonance.

Effect of flexibility on surface tension and coexisting densities of water

Molecular dynamics simulations of pure water at the liquid-vapor interface are performed using direct simulation of interfaces in a liquid slab geometry. The effect of intramolecular flexibility on coexisting densities and surface tension is analyzed. The dipole moment profile across the liquid-vapor interface shows different values for the liquid and vapor phases. The flexible model is a polarizable model. This effect is minor for liquid densities and is large for surface tension. The liquid densities increase from 2% at 300 K to 9% at 550 K when the force field is changed from a fully rigid simple point charge extended (SPCE) model to that of a fully flexible model with the same intermolecular interaction parameters. The increases in surface tension at both temperatures are around 11% and 36%, respectively. The calculated properties of the flexible models are closer to the experimental data than those of the rigid SPCE. The effect of the maximum number of reciprocal vectors (h(z) (max)) and the surface area on the calculated properties at 300 K is also analyzed. The coexiting densities are not sensitive to those variables. The surface tension fluctuates with h(z) (max) with an amplitude larger than 10 mN m(-1). The effect of using small interfacial areas is slightly larger than the error in the simulations.