Metal-clad graded-index planar optical waveguides: accurate perturbation analysis

We present a perturbation analysis of propagation constants and attenuation coefficients of TE and TM modes in a metal-clad linear index planar optical waveguide. The imaginary part N″ of the complex modal index N=N′ + iN″ is given by N″ = (∂N′ / ∂′), where =′ + i″ is the complex dielectric constant of the metal cladding and ∂N′ / ∂′ is obtained by numerical differentiation of the solution of the real eigenvalue equation. The cumbersome solution of a complex transcendental equation is thus completely eliminated. The results are in good agreement with those obtained by solving the eigenvalue equation in the complex plane. By taking the metal-clad linear planar waveguide as a preselected waveguide, we can use our RWKB method to solve metal-clad planar waveguides with parabolic, exponential, gaussian and complementary error function index profiles.     

Elimination of phase-matching requirement for optical waveguide isolator

A novel optical waveguide isolator based on self-phase compensated TE–TM mode conversion was proposed. In this work, the troublesome phase-match condition for the TE–TM mode conversion is eliminated without degradation on the performances. An isolator based on this mechanism is designed and simulated. An insertion loss of −0.1 dB and an isolation of −25 dB was obtained. The tolerance of fabrication and bandwidth of the isolator was also discussed.     

PECVD SiC optical waveguide loss and mode characteristics

We report the fabrication of single mode SiC (silicon carbide) waveguides and the measurement of their propagation loss. By studying the effect of sidewalls scattering loss due to surface roughness and by reducing it, minimal propagation loss of 2.3 dB/cm for the TM polarization is measured in the visible at 0.633 μm. This loss can be used as a benchmark for further development of SiC microphotonic components and circuit for sensor systems in harsh environment.     

Analytical propagation equation of astigmatic Hermite–Gaussian beams through a 4×4 paraxial optical system and their symmetrizing transformation

Starting from the Collins formula, a closed-form propagation equation of astigmatic Hermite–Gaussian (H–G) beams through a 4×4 paraxial optical system is derived, which permits us to calculate the irradiance distribution at any propagation plane and to study the symmetrization of astigmatic standard and elegant H–G beams. A detailed symmetrizing transformation of astigmatic H–G beams through a three-cylindrical-lens mode converter is illustrated both analytically and numerically. It is found that in accordance with the second-order moments characterization, there are two types of beam symmetrization. The transformation of standard H–G beams through the three-cylindrical-lens mode converter belongs to the perfect symmetrization, whereas the transformation of elegant H–G beams is the imperfect one.     

Numerical modelling of TE and TM modes in lossy nonlinear optical waveguides

The intensity-dependent characteristics of nonlinear TE and TM waves in a thin film bounded by lossy nonlinear media are analysed by the finite-element method. In this approach, the power-dependent complex propagation constants and local complex electromagnetic field distributions of lossy nonlinear TE and TM waves are obtained directly from the given lossy nonlinear waveguides, without any approximations. Numerical results for these modes in a lossy nonlinear waveguiding system with different absorption coefficients are given. It is shown that the complex dispersion relations, for both TE and TM modes, are strongly power dependent.     

Variational analysis of Er:Ti:LiNbO3 waveguide lasers based on Gaussian/two-half Gaussian mode field distribution

Starting with two dimensional, scalar wave equation, a variational equation was established for the fundamental TE and TM modes guided in Ti:LiNbO₃ waveguides on the basis of assuming a symmetric Gaussian mode field function in the width direction and two-half Gaussian trial functions in the depth direction. The controllable waveguide fabrication parameters, including channel width, diffusion temperature, initial Ti-strip thickness and diffusion time, dependent of fundamental mode size, effective pump area, coupling efficiency between pump and laser modes, and the coupling loss between a Ti:LiNbO₃ waveguide and a fiber were numerically calculated for Z-cut Er:Ti:LiNbO₃ channel waveguide lasers at three possible emission wavelengths 1532,1563 and 1576 nm and two possible pump wavelengths 1480 and 980 nm. The calculated results were compared with those of Gaussian/Hermite–Gaussian mode field distribution in detail.     

Polymer-based waveguides with low propagation loss and polarization-dependent loss

Polymeric optical waveguides based on a new synthesized bisphenol A–aldehyde polymer (PA-1 resin) have been fabricated using photolithography and reactive ion etching technique. The polymer is novel with relatively high refractive index, low birefringence and absorption at the optical communication wavelengths. The single-mode channel waveguides exhibit a propagation loss of 0.41±0.05 dB/cm at a wavelength of and 0.5±0.05 dB/cm at for both the TE and TM polarizations. The polarization-dependent loss of the waveguides is 0.1±0.05 dB/cm at these wavelengths.     

Generalized Hermite–Gaussian functions and pulse propagation in optical fiber systems

Chirped Hermite–Gaussian functions shows outstanding potentialities in mathematical models for linear and non linear pulse propagation in optical fibers. We propose three generalized subclasses of Hermite–Gaussian functions as natural extension to the complex domain. We exemplify the properties of these subclasses by analyzing some relevant cases of pulse oscillatory dynamics governed by nonlinear Schrödinger equation with constant or periodic coefficfients.     

A Simple and Effective Method for Calculating the Bending Loss and Phase Enhancement of a Bent Planar Waveguide

A simple and effective method is introduced to calculate the bending loss and phase enhancement of a bent planar waveguide. The wave field is represented in terms of Airy functions and an eigenvalue equation is derived by matching the boundary conditions and the radiation condition in the outer cladding layer. The complex propagation constant is obtained by solving the eigenvalue equation with the Newton-Raphson method, and the imaginary part of the propagation constant gives directly the bending loss of the bent waveguide. The results are compared with the previous experimental and numerical results and are shown to be highly accurate and effective. The phase enhancement due to the bending is also studied.     

Effective dielectric constant of two-dimensional photonic crystals in optical bands

By using the Fourier expansion method, we have developed an approach to calculate the effective dielectric index of a two-dimensional photonic crystal. The approach is very general: it can take into account various Bravais lattice structure as well as arbitrary spatial variation of the dielectric index. It has been found that near a nondegenerate frequency ω_n(Γ) at Γ point, the transverse magnetic (TM) mode is ordinary, as it is independent of the propagation direction, whereas in general the transverse electric (TE) mode depends on the propagation direction, it is extraordinary. Therefore, a two-dimensional photonic crystal can always be described by an effective dielectric index for TM mode near the nondegenerate frequency ω_n(Γ). However, the TE mode is much more complicated unless the lattice structure is highly symmetric. Moreover, a two-dimensional square photonic crystal has been identified as an effective birefringent crystal having two negative refractive indexes from the perspective of Snell's law.     

Field Analysis of Pillbox Resonator with Radiation Loss Utilizing Finite-Element Beam Propagation Method Formulation

To deal with the radiation from an axially symmetric pillbox resonator embedded in LiNbO₃ substrate, an eigenvalue equation is derived from the expression of the finite-element beam propagation method in a cylindrical coordinate system. By solving the eigenvalue equation iteratively, the field distributions and the angular phase constants of the pillbox resonator are evaluated. The iterative scheme of solving the eigenvalue equation is an extension of our previous direct solution method which was described for a lossless case. The results obtained by this method are compared with those obtained analytically, and excellent agreement is found.     

Fabrication and characterization of β-Ga2O3 optical nanowires

Monoclinic gallium oxide (β-Ga₂O₃) nanowires with lengths of tens of micrometers and diameters ranging from 100 to 250 nm are synthesized using simple physical evaporation based on vapor–liquid–solid (VLS) mechanism. The as-synthesized straight β-Ga₂O₃ nanowires show excellent diameter uniformity and sidewall smoothness, making them suitable for optical wave-guiding. Light from a fiber taper is launched into the nanowire by means of evanescent coupling. Measured propagation loss of the nanowire at 633 nm wavelength is on the order of 10 dB/mm. Favorable mechanical strength of these nanowires for elastic bending is also observed. Our results suggest that β-Ga₂O₃ nanowires are promising building blocks for micro- and nanophotonic circuits and devices.     

Numerical Method of Determining the Cutoff Frequency and Mode Speckle Pattern of Optical Waveguides

Based on a numerical solution of the nonstationary wave equation, an algorithm for determining the eigenfunctions and eigenvalues of the LP_im modes of axially symmetric optical waveguides is suggested. The cutoff frequency of the LP₁₁ mode of different waveguides is determined using the algorithm constructed. The results obtained are compared with the well-known analytical formulas and results of numerical calculations for the equivalent step-index waveguide model. Based on the analytical results obtained, an eigenfunction and eigenvalue search algorithm is suggested for a waveguide with a two-dimensional refractive index profile (in particular, for an axially nonsymmetric waveguide).__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 11–16, January, 2005.     

Propagation of generalized Mathieu–Gauss beams through paraxial misaligned optical systems

Based on the generalized diffraction integral, we derive an analytical formula for generalized Mathieu–Gauss beams (gMGBs) passing through an apertured misaligned optical system. Furthermore, we use the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions to establish an approximate propagation equation of gMGBs through paraxial circularly apertured optical system. As an example, the propagation of ordinary and modified zeroth order MGBs through a misaligned thin lens is studied numerically.     

An improved design of an integrated optical isolator based on non-reciprocalMach–Zehnder interferometry

Non-reciprocal rib waveguide structures can be used to realize integrated optical isolators. In this paper, we propose a concrete design for a Mach–Zehnder interferometer type isolator for TM modes. Just one of the arms, which are of equal length, is a non-reciprocal magneto-optic waveguide. The rest of the interferometer is reciprocal. Required fabrication tolerances are estimated, and the entire isolator is simulated by applying a finite difference beam propagation method.     

有损耗介质中非线性光导波的数值模拟

发展了一种以有限元法为基础的数值方法，用它分析由损耗任意大小材料构成的平面非线性波导结构所传导的非线性有损耗光波。在这种方法中，从给出的非线性有损耗波导直接得到ＴＥ和ＴＭ波与功率有关的复传播常数和局域电磁场分布。     

Experimental study of laser-induced thermal ignition in O2/O3 mixtures

O₂/O₃ mixtures are ignited by absorption of laser pulses of a TEA CO₂ laser along the axis of a cylindrical cell. The dependence of the radial propagation of the O₃ decomposition, detected by uv absorption of the ozone, on laser fluence and on O₃ concentration is investigated. Oscillations of the signals are identified to be the first radial acoustic mode of the cell. For mixtures of 0.35 bar and 0.70 bar total pressure and O₃ percentages within 20–50%, the ignition limit is in the order of 0.1–0.2 J/cm³ (absorbed energy density). These values are in reasonable agreement with the results of the corresponding numerical simulations.     

Amplification oand absorption of an electromagnetic wave in a moderating structure with a paramagnetic crystal

Conclusions An analysis of the paramagnetic amplification G_e and absorption L_e of an electromagnetic wave in a moderating structure shows that the effects of saturation in some cases are significantly different. In consequence of this, the saturation threshold in the crystal in the amplification mode is lower than the saturation threshold in the absorption mode approximately by a factor of G_eM/2L_att. This conclusion must be borne in mind when investigating paramagnetic crystals in distributed structures, in particular when investigating the invesion characteristics of potentially active substances for lasers.It follows from the analysis that the investigation of the invesion properties of the crystal must be conducted on small samples of the crystal, as the possible error in measuring the inversion coefficient will be less for lower values of the paramagnetic amplification factor (all conditions being equal).A moderating structure with a radiating paramagnetic crystal is an example of an active medium, in which the wave propagation process can be described and analyzed by means of Eq. (1). As E_p. (1) postulates the solution in analytic form, and all the parameters of the equation have a clear physical meaning and can be monitored easily during the experimental investigation, then the structure considered here obviously can be a suitable model for studying the propagation of waves in other active media (see, e. g., [9]).Institute of Radiophysics and Electronics, Academy of Sciences of the Ukrainian SSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 22, No. 7, pp. 819–825, July, 1979.     

Dual-core leaky optical waveguide as an integrated-optic polarizer

We present a design of a dual-core leaky waveguide that can be used as an integrated-optic polarizer. The proposed polarizer works on the principle of mode filtering. The structure is characterized by two cores, namely core-1 and core-2 and a high index layer in the upper-most region, such a structure supports leaky modes. The leakage losses of the modes have been calculated by using the transfer matrix method (TMM). Single polarization operation is ensured by high differential leakage loss between fundamental TE and TM modes. We show TE-pass operation with 0.5 dB/mm loss in TE₀ mode and 13 dB/mm loss in TM₀ mode, and TM-pass operation with 0.36 dB/mm loss in TM₀ mode and 7.45 dB/mm loss in TE₀ mode. Besides single-polarization operation, single-mode operation of the structure is ensured by high leakage loss of all the other higher-order modes.     

Intra-cavity spectroscopy with diode lasers

A multi-mode diode laser with an external cavity is studied experimentally and theoretically for its application to intra-cavity spectroscopy. One facet of a typical Ga_0.89Al_0.11As laser diode was antireflection-coated by deposition of HfO₂ such that 10^–3 residual reflectivity was left over. This diode was placed in an external optical cavity. The emission spectrum of this diode laser is highly sensitive to any frequency-dependent loss in the cavity, and the detectivity of such a loss grows with the pump rate. Even close to threshold, the absorption at 780 nm of Rb atoms with a density of 5×10¹⁰ cm^–3 has been detected. An adequate model for diode lasers based on rate equations and including frequency-dependent gain saturation is developed and applied to the calculations of output spectra. The sensitivity of these spectra to intra-cavity absorption is determined by the overall cavity loss — which is rather high — and the fraction of spontaneous emission in the total emission, in contrast with dye lasers where it is limited by nonlinear mode coupling. Various criteria for the sensitivity are suggested. The smallest detectable absorption with a perfectly antireflection-coated laser is 10^–6 cm^–1. Improvement of the characteristics of the laser diode would increase the sensitivity.