Analysis of difference frequency generation and cascaded second-harmonic generation and difference frequency generation in lossy quasi-phase-matched semiconductor waveguides

The analytical expressions of converted wave power for difference frequency generation (DFG), cascaded second-harmonic generation and difference frequency generation (cSHG/DFG) processes have been obtained under the non-depletion approximation in lossy waveguides. It is shown that the analytical results and the numerical simulation with depletion agree very well for lossy waveguides. Employing the analytical solutions, the formulas of optimized waveguide lengths in lossy waveguides are obtained for DFG and cSHG/DFG processes. After designing an AlGaAs quasi-phase-matched ridge waveguide, we investigate and compare the characteristics of the second-order nonlinear effects with and without waveguide loss, such as conversion efficiency, conversion bandwidth, pump wavelength tolerance and temperature stability in detail.     

Novel waveguide components based on complementary electrically small resonators

In this paper, we present a new class of waveguide components based on the use of complementary electrically small resonators etched on metallic sheets. We first show a rectangular-to-circular waveguide transition that allows to effectively match waveguides with different cross-sections and to transform the linearly polarized TE₁₀-mode of a rectangular waveguide into a circularly polarized TE₁₁-mode of a circular waveguide. Then, using similar structures, we present the design and the numerical results of two power dividers and of an orthomode transducer. All these components, employing only thin metallic sheets with properly shaped slits, can be integrated in waveguide transmission systems without increasing their space occupancy. A proper set of full-wave numerical simulations proves the validity of the proposed approach.     

Limitations in the perturbation analysis of bent finite-clad fibres and waveguides

The transverse shift in the field distribution and the correction to the propagation constant of the fundamental and symmetric cladding modes on bent finite-clad single-mode fibres and slab waveguides are evaluated from perturbation theory for effective index values extending below the cladding index. Analytical results are derived in both geometries for the step-profile that are valid within the overall limitations of the theory. However, it is found that, for the fibre geometry only, the method breaks down at certain discrete wavelengths because of degeneracies that occur between the HE₁₂ (LP₀₂) and TE₀₁ (LP₁₁) mode propagation constants.     

Optical propagation of the HE11 mode and Gaussian beams in hollow circular waveguides

The propagation of the HE₁₁ mode and Gaussian beams in hollow oversized circular waveguides is analyzed using optical theories. Different types of waveguides are considered : hollow dielectric or conducting waveguides, dielectric-lined waveguides, corrugated waveguides. General formulas are derived which give the power transmission through these different guides. The best wall materials and structures are determined from a comparison of the waveguide transmissions, at the infrared and millimeter wavelengths. The question of the coupling between the HE₁₁ mode and Gaussian beams is discussed and from a review of coupling coefficients derived before, an optimum value is pointed out. The problem of matching a Gaussian beam into circular waveguides in order to achieve the maximum power transmission is analyzed. These results are of interest for infrared lasers or waveguide applications and for Electron Cyclotron Wave (ECW) systems at the millimeter wavelength.     

Prediction of TE-, TM-, and hybridmode transmission losses in gaps of oversized waveguides using a scattering matrix code

The transmission losses of different modes in gaps in oversized waveguides are calculated using a modified scattering matrix formalism. Measurements performed for the modes TE₀₁, TE₁₁, TM₀₁, TM₁₁ and HE₁₁ are in good agreement with the calculated numerical results. Both numerical and measured values are compared with existing analytical formulas.     

A simple variational analysis of diffused channel waveguides

The variational analysis of Ti:LiNbO₃ strip waveguides is resolved into two related one-dimension problems, which provides a new iterative approach for the calculation of the mode parameters and the propagation constant. The analytical expressions of the local effective index profiles can be obtained from the variational analysis. The parameters of four trial solutions for the fundamental mode in a diffused-channel waveguide have been determined by using the present method. Comparison with results using the effective method shows that these four approximate expressions are accurate enough. It is also shown that this method converges very quickly and is suitable for both diffused channel waveguides and optical waveguides with rectangular cross-section.     

Rigorous Analysis of the Disk-Loaded Waveguide Slow-Wave Structures

Recently disk-loaded waveguide has been widely used in high-power traveling wave tubes (HPTWT). Although TM ₀₁ mode is it's main mode, the asymmetry mode may be excited by slight asymmetry of the structure, misalignment of the beam, or the asymmetry associated with the input or output structure. So research about asymmetry mode of disk-loaded waveguide is necessary. The general dispersion equation and interaction impedance expression of disk-loaded waveguide is obtained with accurate field theory for the first time. Based on these results, a broadband and a narrowband disk-loaded waveguides are designed which can be used in the Ka band HPTWT. Moreover, the theoretical calculated results are compared with the simulated results from the HFSS (High Frequency Structure Simulator) code using finite element method. It is found that frequency domain overlapped by these two modes of the narrowband structure is very narrow, so the HEM ₁₁ mode may be ignored when calculating Beam-wave interaction. In the other hand, the interaction impedance of HEM ₁₁ mode is very low generally.     

Minimization of losses and calculation of the optical properties of Bragg fiber waveguides with a hollow core

A numerical method for solving the inverse problem of determining the geometry of the multilayer shell of a Bragg waveguide that has the lowest waveguide losses for a given mode has been developed with the use of the genetic algorithm. For the calculated designs of waveguides, the distribution of the coordinates of the boundaries of shell layers has been found to be aperiodic under the condition r ₁ ≤ λ due to the axial symmetry of the problem. Waveguide losses for the TE ₀₁, TM ₀₁, and HE ₁₁ modes satisfy the conditions \(\alpha _{TM_{01} } > \alpha _{HE_{11} } > \alpha _{TE_{01} } \). It follows from the dependence of n _eff obtained for these modes on the core radius and number of shell layers that any change in the structure of the waveguide leads to the violation of the optimal propagation regime for these modes. A Bragg fiber waveguide with a hollow core that is designed for the TE ₀₁ mode and directs light in the single-mode regime is presented. The main fraction of losses in this waveguide is attributed to material absorption.     

Asymptotic Theory of High-Efficiency Converters of Higher-Order Waveguide Modes into Eigenwaves of Open Mirror Lines

Based on the Debye asymptotic for Bessel functions, we develop an analytical theory of high-efficiency converters of higher-order waveguide modes into eigenwaves of open mirror transmission lines. Simple analytical formulas for the parameters of the basic types of mirror lines are derived. Direct relationship between the nonequidistance of mode wavenumbers, the diffraction lengths of the Brillouin wave beams, and the lengths of converters for oversized waveguides is shown. The results of numerical calculations for some converters are presented for comparison.     

Modeling of Optical Waveguides and Devices by Combination of the Method of Lines and Finite Differences of Second Order Accuracy

Based on generalized transmission line equations a novel impedance/admittance transformation algorithm with finite differences is proposed and substantiated. The algorithm is of second order accuracy. It is combined with the Method of Lines. The algorithm is suitable for eigenmode calculations and for the analysis of optical waveguides and devices including anisotropic materials. The proposed algorithm is verified by numerical results for rib waveguide and azimuthally inhomogeneous fibres.     

An Analytical Model for the Coupling Between the Array Waveguides in AWGs and Star Couplers

The purpose of this paper is to present an accurate analytical solution for the coupling between the array waveguides in arrayed waveguide grating (AWG) devices and star couplers. The results of this analysis will be useful for developing numerical models of AWGs and star couplers, as well as other optical waveguide components containing arrays of coupled waveguides.     

Twist Sensitivity of Cladding-Mode Resonances and Its Cross-Sensitivity to Strain and Temperature in a Mechanically Induced Long-Period Fiber Grating

Abstract

Twist sensitivity of cladding-mode resonances in a mechanically induced long-period fiber grating formed over a single-mode fiber is experimentally demonstrated and theoretically analyzed. Of the two usual cladding-mode resonances corresponding to LP₁₁ and LP₁₂, higher-order mode LP₁₂ is more sensitive to twist in comparison with the lower-order mode LP₁₁. The extent of down-shifting of resonant wavelengths depends on twist-induced circular birefringence and the modal field distribution of the cladding-modes inside the fiber. When the fiber is severely twisted to 3.5 rad/cm, a shift sensitivity is observed of 1 nm/(rad/cm) for the LP₁₁ mode and 4.23 nm/(rad/cm) for the LP₁₂ mode. The fiber breaks when the twist rate exceeds 3.5 rad/cm. In comparison with LP₁₂, the LP₁₁ resonance is almost independent of the axial strain variation with an ultra-low sensitivity of 0.18 pm/με, and it is also almost insensitive to the temperature variation with a coefficient of 35 pm/°C. Forming the LP₁₁ resonance far away from its cut-off wavelength, a widely tunable band-pass filter is also demonstrated with a very high twist sensitivity of 8.75 nm/(rad/cm) and negligible cross-sensitivity to strain and temperature. The experimental and theoretical results are very useful in selecting sensitive and stable cladding-mode resonances in the design of new mechanically induced long-period fiber gratings based torsion sensors and tunable band-pass filters.     

Photon recycling in Fabry–Perot micro-cavities based on Si3N4 waveguides

We present a numerical analysis and preliminary experimental results on one-dimensional Fabry–Perot micro-cavities in Si₃N₄waveguides. The Fabry–Perot micro-cavities are formed by two distributed Bragg reflectors separated by a straight portion of a waveguide. The Bragg reflectors are composed of a few air slits produced within the Si₃N₄ waveguides. In order to increase the quality factor of the micro-cavities, we have minimized, with a multiparametric optimization tool, the insertion loss of the reflectors by varying the length of their first pairs (those facing the cavity). To explain the simulation results, the coupling of the fundamental waveguide mode with radiative modes in the Fabry–Perot micro-cavities is needed. This effect is described as a recycling of radiative modes in the waveguide. To support the modelling, preliminary experimental results of micro-cavities in Si₃N₄ waveguides realized with the focused ion beam technique are reported.     

Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task

Modal reflection, transmission and loss of deeply etched Bragg waveguide gratings were modelled by six European laboratories using independently developed two-dimensional (2D) numerical codes based on four different methods, with very good mutual agreement. It was found that (rather weak) material dispersion of the SiO₂/Si₃N₄ system does not significantly affect the results. The existence of lossless Floquet–Bloch modes in deeply etched gratings was confirmed. Based on reliable numerical results, the physical origin of out-of-plane losses of 1D or 2D photonic band gap structures in slab waveguides is briefly discussed.     

Bragg waveguide grating as a 1d photonic band gap structure: COST 268 modelling task

Modal reflection, transmission and loss of deeply etched Bragg waveguide gratings were modelled by six European laboratories using independently developed two-dimensional (2D) numerical codes based on four different methods, with very good mutual agreement. It was found that (rather weak) material dispersion of the SiO₂/Si₃N₄ system does not significantly affect the results. The existence of lossless Floquet-Bloch modes in deeply etched gratings was confirmed. Based on reliable numerical results, the physical origin of out-of-plane losses of 1D or 2D photonic band gap structures in slab waveguides is briefly discussed.     

Semi-Analytical Simulation of Titanium-Indiffused Lithium Niobate–Integrated Optic Directional Couplers Consisting of Curved Waveguides

Integrated optic directional couplers consisting of curved waveguides are simulated analytically by solving the Riccati equation. The coupling coefficient between the curved waveguides with a parabolically varying gap and the condition of total power transfer between the waveguides are derived. In order to compute the overall coupling coefficient and hence the power distribution along the waveguides for Ti:LiNbO ₃ curved waveguide directional couplers, the coupling coefficient for straight waveguide couplers is computed for different gaps using the effective-index-based matrix method (EIMM). Finally, the power distribution in the curved waveguides along the length is computed. The method is mostly analytical except the effective-index method and is computationally simple.     

WKB analysis of guided and semileaky modes in graded-index anisotropic optical waveguides

WKB analysis of graded-index anisotropic optical waveguides is briefly presented. The angular dependencies of the propagation characteristics of both guided and semileaky modes are calculated numerically for the out-diffused and in-diffused LiNbO₃ waveguides with parabolic index profile. The results for the out-diffused waveguide are compared with those obtained from the rigorous theory and the applicability of the WKG method is discussed for semileaky mode loss calculations. The angular dependence of the propagation constants of a TiO₂-diffused LiNbO₃ waveguide is demonstrated experimentally, too.     

Analysis of arbitrary index profile planar optical waveguides and multilayer nonlinear structures: a simple finite difference algorithm

we present here a simple numerical method to obtain the mode effective indices as well as field distributions of modes of any arbitrary profile planar optical waveguide. The method is based on the solutions of scalar and semivectorial Helmoltz’s equation by finite difference algorithm and devised with a field convergence technique. This approach is quite general and can be applied straightforwardly to calculate the guided as well as quasi- or leaky modes of any arbitrary planar structure without the need to solve any eigenvalue equation or any complex matrix formalism. Besides providing the ease of application, the algorithm is particularly useful for waveguides with any graded index profile or with irregular multilayered structure and multilayered waveguides with a localised arbitrary nonlinear film. The performance of our method is verified against typical problems with analytical solutions or having results known otherwise, and is shown to yield results with good accuracy.     

斜接弯波导模式转换器

经理论分析得出,一段小曲率半径弯曲圆波导的传输特性和一段与其弯角相同、轴线长度相等的斜接弯波导的传输特性一致,从而提出了一种由多节直圆波导依次倾斜连接构成的TM01-TE11斜接弯波导模式转换器,并介绍了该模式转换器的设计方法,给出了设计实例。优化结果表明,所设计的模式转换器在中心频率上均具有99%以上的转换效率。     

Analytical solution of the scalar wave equation for slightly deformed optical Bragg waveguide

Using a simple analytical method, we study the electromagnetic-wave propagation in an optical Bragg waveguide which is slightly deformed at one side. We show the results of computations in the form of dispersion curves and provide a comparison with standard circular Bragg waveguides and step-index circular waveguides. We show that the correspondence between the cutoff V values for the standard step-index fiber and the new Bragg waveguide, which is slightly flattened on one side, is quite close for the lowest mode under the weak guidance condition.