Numerical modeling of second harmonic generation in opticalwaveguides using the finite element method

A numerical study of second harmonic generation (SHG) in optical waveguides is presented using the finite element method (FEM) and the Crank-Nicholson split-step procedure. Results are given for a Cherenkov radiation scheme in both planar and channel waveguides. Also presented are results obtained on frequency doubling for guided modes in both planar and channel waveguides, using the quasi-phase-matching scheme     

Broadening of the phase-matching bandwidth in quasi-phase-matchedsecond-harmonic generation

We report on theoretical analysis and experiments for bandwidth broadening in quasi-phase-matched (QPM) second-harmonic generation (SHG) in LiTaO₃. QPM waveguides consisting of segments, where each segment has a phase-matching condition different from the others, are proposed to obtain a broad bandwidth and simultaneously, a high SHG efficiency. The waveguides were fabricated by adjusting the phase-matching condition either by changing the grating period of the SHG coefficient or by controlling the width of the waveguide. Consequently, the bandwidth of QPM-SHG was broadened to 0.35 nm with an SHG efficiency of 57 percent/W by modulation of the waveguide width and to 1.12 nm with an SHG efficiency of 29 percent/W by modulation of the grating period. The SHG efficiency and phase-matching characteristics in both types of waveguides showed good agreement with theoretical results     

Optimizing KTP and LiTaO3 channel waveguides forquasi-phase-matched second harmonic generation with high conversionefficiency

The conversion efficiency of quasi-phase-matched second harmonic generation (QPM-SHG) in waveguides can be enhanced significantly by optimizing the linear properties of the guiding structure. We describe here a method for fast and accurate computation of the modal properties of continuous, periodically poled, ferroelectric channel waveguides and a multiparameter optimization algorithm which we have used to maximize normalized internal SHG conversion efficiency. We present the phase-matching characteristics of these designs and discuss the possibility of high SHG conversion efficiency between lowest order transverse modes at the fundamental and second harmonic wavelengths near noncritically phase-matched regimes of operation     

Finite-element analysis of second-harmonic generation in AlGaAswaveguides

A rigorous and efficient finite-element method based beam propagation method is used to model second-harmonic generation (SHG) in semiconductor waveguides. The effect of loss on the efficiency of SHG is analyzed and it is shown that, under certain conditions, GaAlAs-based devices with a lower nonlinear susceptibility tensor could be more efficient than GaAs-based devices with a higher nonlinear susceptibility tensor. Numerical results are also presented, to show the effect of domain fabrication error, in the case of quasi-phase-matched devices, on the efficiency of SHG     

Three-dimensional vector beam-propagation method for secondharmonic generation analysis

A full vectorial three-dimensional beam-propagation method (BPM) based on the finite-element method is described for the analysis of second harmonic generation (SHG). Quasi-phase-matched SHG devices with periodically domain-inverted GaAlAs- and LiTaO₃-based waveguides are analyzed. The influences of the shape of domain-inverted regions and of the inversion width on the conversion efficiencies are investigated in detail. The results of full-wave analysis are compared to those of approximate scalar analysis     

Integrated optical device with second-harmonic generator,electrooptic lens, and electrooptic scanner in LiTaO3

This paper reports the first demonstration of an integrated optical device in z-cut LiTaO₃ that contains the following three functional parts: a quasiphase-matched second-harmonic generation (SHG) grating, an electrooptic (EO) lens, and an electrooptic scanner. The SHG device consists of channel waveguides passing through periodic domain-inverted gratings. The frequency of the input infrared (IR) light at 864 nm was doubled into blue light at 432 nm. A stack of EO lenses was used to collimate the light from the channel waveguide. The measured beam size at the output facet for various applied voltages to the lenses agreed with simulation. After collimation, light passes through an EO scanner that controls the angle of the output beam. A scanning sensitivity of 17 mrad/kV was measured for the scanner, compared to the calculated value of 15 mrad/kV     

Characterization of annealed proton exchanged LiNbO3waveguides for nonlinear frequency conversion

A thorough and detailed characterization of annealed proton-exchanged (APE) waveguides in Z-cut LiNbO₃ is described. The mode index measurements in planar waveguides as a function of wavelength and annealing time are reported, including useful analytical relations for the refractive index change, its dispersion, and the depth profile as a function of annealing parameters. Analytical expressions for the mode propagation characteristics are presented and experimentally verified with reasonable accuracy. It is shown that the planar waveguide characterization results can be used to model the channel waveguide characteristics accurately. The model provides closed-form expressions for the mode index and the mode field profile, and the theoretical results are in excellent agreement with the measured data. The technique is used to accurately predict the phase mismatch between the fundamental and second harmonic modes in frequency-doubling experiments using APE channel waveguides. An optimum waveguide geometry for which the phase mismatch is relatively insensitive to the waveguide nonuniformity was predicted and verified experimentally     

All-optical switching due to cascaded second-harmonic generation indirectional couplers with laterally varying phase mismatches

The local normal mode approach is employed to analyze cascaded second-harmonic generation (SHG) in directional couplers with different phase mismatches in both waveguides. The couplers are assumed to be fabricated in ferroelectric crystals, which are periodically domain inverted with different periods in both waveguide regions. These components are proposed to imitate the switching in nonlinearly asymmetric third-order directional couplers by using the cascaded second-order effect, which can mimic third-order interactions with effective third-order nonlinearities proportional to the reciprocal phase mismatch. The results for the second-order directional couplers are compared to those for the respective ideal third-order ones     

Theoretical analysis of resonant waveguide optical second harmonicgeneration devices

Theoretical analysis of resonant waveguide SHG devices is presented. Mathematical expressions for SHG interaction in waveguides for arbitrary initial conditions were derived and applied to the analysis. Dependences of SHG efficiency on pump amplitude of singly and doubly resonant devices with various cavity mirror reflectivities, phase-mismatch, and propagation losses are compared with that of traveling-wave devices. It is shown that in resonant SHG devices much higher efficiency is obtained for low pumping power than in traveling-wave device. It is found, for the first time, that complete conversion can be attained for a finite input power in the doubly resonant devices. However, the efficiency is very sensitive to the propagation losses     

Operation mechanism of electrooptic multimode X-switches

The operation mechanism of electrooptic multimode switches using crossing channel waveguides (X-switch) is investigated theoretically and experimentally. The light propagation through the X-structure as well as the switching behavior of the device is explained in terms of local normal modes. The theoretical model is confirmed by numerical calculations (BPM) and by experiments using Ti:LiNbO3multimode channel waveguides. The results indicate that the generally applied model of total internal reflection (TIR) is not suited to explain the low voltage operation of multimode X-switches.     

Measurement of the birefringence of UV-written channel silicawaveguides by magnetooptic polarization-mode coupling

We report the measurement of the birefringence in integrated-optical waveguides using magnetooptical coupling between the two principal polarizations of the fundamental mode. We demonstrate this measurement technique for directly ultraviolet (UV)-written channel waveguides in silica on-silicon and silica-on-silica. The dependence of the waveguide birefringence on the UV-writing power and UV polarization is investigated. The results are compared with the birefringence of etched waveguides in comparable material systems. An analytical formula for the form birefringence in buried channel waveguides is developed, and measured data are compared with theoretical results     

A comprehensive analysis of multilayer channel waveguides

Using a simple approach based on the scalar finite element method, propagation characteristics of multilayer channel waveguides are calculated. The effective index, modal field, confinement factor, far-field intensity pattern, and radiation angle of the far-field pattern (full width at half maximum intensity) for multilayer channel waveguides formed with multiple quantum well (MQW) materials and with the MQW materials replaced by a single homogeneous material with the root mean square value of the refractive indexes are compared. Numerical results confirm that the root-mean-square-value approximation, which has been widely used for planar MQW (two-dimensional) waveguides, is useful also for MQW channel (three-dimensional) waveguides with a large number of layers     

Rigorously modeling short bent, graded-index dielectric slabwaveguides

Circularly short bent, graded-index dielectric slab waveguides are mathematically modeled using both a direct integration method and a source-type integral equation method. The former method is easy to implement, requiring only small amounts of computer time. The latter method has the potential to be extended to a rigorous model for bent channel waveguides. Both homogeneous and inhomogeneous slab waveguide layers are considered. The mathematical models are validated through comparison of the numerical results with results obtained by the analytical resonance conditions. Numerous numerical examples are given     

Coupled-mode analysis of second harmonic generation in the form ofCerenkov radiation from a channel optical waveguide

Optical second-harmonic generation (SHG) in the form of Cerenkov radiation from a channel waveguide is analyzed by a coupled-mode theory. A simple formula is given which expresses the SHG efficiency in terms of waveguide and optical nonlinear parameters and by which one can simulate the radiation pattern of SHG. Computational examples are plotted for LiNbO₃ crystals and reasonably agree with reported results. The analysis shows that the efficiency could be increased significantly if the sign of a d-constant in a substrate were modulated with a proper period     

Computational modeling of diffused channel waveguides using adomain integral equation

A domain integral equation is proposed for the computational modeling of diffused channel waveguides. In the modeling the propagation properties and the field distributions of the lower order guided modes are computed. The method is used to design the channel waveguides that are realized by an ion-exchange process in glass substrates. In particular, the method is applied to the design of ion-exchanged waveguides with low fiber/chip coupling losses. The aim is to realize modal distributions in the channel waveguides that closely match the rotationally symmetric field distributions of the HE₁₁ fiber mode. Some technological aspects of the realization of such ion-exchange waveguides are indicated, and various numerical results relevant to the design process are presented     

Photoelastic channel optical waveguides in epitaxial GaAs layers

Photoelastic channel optical waveguides have been fabricated in GaAs epitaxial layers. These waveguides exhibit strong lateral confinement in a single region of their cross-section. They compare favourably with previously demonstrated integrated optical waveguide structures.     

光波导结构对其极化电光效应的影响

本文采用热极化技术在掺锗玻璃条形光波导中诱导出非线性光学效应,并通过变化光波导结构改变极化内电场的大小,研究热极化条件对诱导出的电光效应的影响。研究发现在最佳极化时间内,加入薄SiON层的光波导结构经极化后电光效应积累快,相同极化条件下诱导出的电光系数比原有的条形光波导增大约22%,同时极化光波导还存在一个较低的极化阈值电压。实验结果表明加入光波导结构中的薄SiON层可在一定程度上改变极化光波导内的电荷分布,实现强化其内部电场增大光波导内的非线性光学效应或电光效应的目的。     

Theory of second-harmonic generation in optical waveguides

A theory for the initial rate of second-harmonic generation (SHG) in planar optical waveguides for phase-matched and nonphase-matched cases is derived. The derivation is carried out both by a ray treatment and by a wave treatment and the results of the two are compared. Although formally and from the point of view of physical interpretation the two treatments look very different, the final results are the same if allowance is made for the Goos-Haenchen shift. Dependences of the amplitude of the second harmonic on the amplitude of the fundamental Aωand on distance are quite similar to those for bulk SHG even though the phase-matching condition is quite different. Determining the characteristics of some of the phase-matched modes, we find that, for the same Aω, it is possible to obtain a generation rate in waveguide comparable to that in the bulk. The highest generation rate is achieved, as expected, with both fundamental and harmonic having mode number 0. Phase matching can be achieved in this case without birefringence and with all three media having normal dispersion. The decrease in generation rate in going to mode pairs other than 0,0 is less than a factor 2 in some cases.     

Cerenkov-type second-harmonic generation in KNbO3channel waveguides

Cerenkov-type second-harmonic generation using KNbO₃ channel waveguides produced by MeV He⁺-ion implantation is presented from the viewpoint of device design. We derive the Cerenkov phase-matching condition for multimode waveguides and utilize Cerenkov-angle analysis as a tool for contact-free measurement of the effective indexes of guided modes of ion-implanted KNbO₃ channel waveguides at a wavelength of 860 nm. The measured mode indexes are in full agreement with calculations based on the effective-index method and the refractive index-depth profiles of ion-implanted KNbO₃ waveguides. The efficiency of Cerenkov-type second-harmonic generation is modeled using analytical approximations of the field distributions of the fundamental and the Cerenkov-radiation modes in embedded-channel waveguides. The acceptance width for Cerenkov-type frequency doubling in these ion-implanted waveguides is about one order of magnitude wider than for noncritical phase-matched second-harmonic generation in bulk KNbO₃ crystals. Based on the theoretical simulations, guidelines for optimum device design are given, and the possibility to increase the ultimate conversion efficiency to about 30% W^-1 cm^-1 through lateral-resonance enhancement of the second-harmonic field in KNbO₃ channel waveguides is demonstrated     

A variational finite-difference method for analyzing channel waveguides with arbitrary index profiles

A variational, finite-difference method for computing the normalized propagation constants and the normalized field profiles of channel waveguides with arbitrary index profiles as well as aspect ratios is presented. Mode dispersion curves and the field profiles of the fundamental mode of channel waveguides having profiles of practical interest are included.