Nodal domains of the equilateral triangle billiard

We characterize the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains. The number of nodal domains has a quadratic form in terms of the quantum numbers, with a non-trivial number-theoretic factor. The patterns of the eigenfunctions follow a group-theoretic connection in a way that makes them predictable as one goes from one state to another. Extensive numerical investigations bring out the distribution functions of the mode number and signed areas. The statistics of the boundary intersections is also treated analytically. Finally, the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula. We believe that the results belong generically to non-separable systems, thus extending the previous works which are concentrated on separable and chaotic systems.

Medium-Voltage Drive for Induction Machine With Multilevel Dodecagonal Voltage Space Vectors With Symmetric Triangles

Multilevel inverters with dodecagonal (12-sided polygon) voltage space vector structure have advantages, such as complete elimination of fifth and seventh harmonics, reduction in electromagnetic interference, reduction in device voltage ratings, reduction of switching frequency, extension of linear modulation range, etc., making it a viable option for high-power medium-voltage drives. This paper proposes two power circuit topologies capable of generating multilevel dodecagonal voltage space vector structure with symmetric triangles (for the first time) with minimum number of dc-link power supplies and floating capacitor H-bridges. The first power topology is composed of two hybrid cascaded five-level inverters connected to either side of an open-end winding induction machine. Each inverter consists of a three-level neutral-point-clamped inverter, which is cascaded with an isolated H-bridge making it a five-level inverter. The second topology is for a normal induction motor. Both of these circuit topologies have inherent capacitor balancing for floating H-bridges for all modulation indexes, including transient operations. The proposed topologies do not require any precharging circuitry for startup. A simple pulsewidth modulation timing calculation method for space vector modulation is also presented in this paper. Due to the symmetric arrangement of congruent triangles within the voltage space vector structure, the timing computation requires only the sampled reference values and does not require any offline computation, lookup tables, or angle computation. Experimental results for steady-state operation and transient operation are also presented to validate the proposed concept.

An Open-End winding IM drive with Multilevel 12-sided Polygonal Vectors with Symmetric Triangles

Multilevel inverters with hexagonal voltage space vector structures have improved performance of induction motor drives compared to that of the two level inverters. Further reduction in the torque ripple on the motor shaft is possible by using multilevel dodecagonal (12-sided polygon) voltage space vector structures. The advantages of dodecagonal voltage space vector based PWM techniques are the complete elimination of fifth and seventh harmonics in phase voltages for the full modulation range and the extension of linear modulation range. This paper proposes an inverter circuit topology capable of generating multilevel dodecagonal voltage space vectors with symmetric triangles, by cascading two asymmetric three level inverters with isolated H-Bridges. This is made possible by proper selection of DC link voltages and the selection of resultant switching states for the inverters. In this paper, a simple PWM timing calculation method is proposed. Experimental results have also been presented in this paper to validate the proposed concept.

Effect of Channel Confinement on Mixed Convective Flow Past an Equilateral Triangular Cylinder

The present work investigates the mixed convective flow and heat transfer characteristics past a triangular cylinder placed symmetrically in a vertical channel. At a representative Reynolds number, Re = 100, simulations are carried out for the blockage ratios beta = 1/3; 1/4; and 1/6. Effect of aiding and opposing buoyancy is brought about by varying the Richardson number in the range -1.0 <= Ri <= 1.0. At a blockage ratio of 1/3, suppression of vortex shedding is found at Ri = 1, whereas von Karman vortex street is seen both at beta = 1/4 and 1/6, respectively. This is the first time that such behavior of blockage ratio past a triangular cylinder in the present flow configuration is reported. Drag coefficient increases progressively with increasing Ri and a slightly higher value is noticed at beta = 1/3. For all b, heat transfer increases with increasing Ri. Flattening of Nu(avg)-Ri curve beyond Ri > 0: 75 is observed at beta = 1/3.

Conductance calculation of a long tube with equilateral triangle cross section

Using the stratified gas flow model for calculating the conductance of long tubes with constant cross section, an analytical expression for calculating the conductance of along tube with equilateral triangle cross section has been derived. The formula given is applicable to the full pressure range. A minimum in the conductance in the intermediate flow state is shown. 2002 American vacuum Society.

GaSb based midinfrared equilateral-triangle-resonator semiconductor lasers

Theoretical calculations of the mode characteristics of an equilateral-triangle resonator (ETR) with a 10 mu m cavity side length show that the fundamental mode, with longitudinal mode index of 25, has a wavelength of 2.185 mu m and a longitudinal mode separation of 100 nm. This mode has a quality factor (similar to 2x10(5)) that is much larger than the first (similar to 5x10(4)) and second (similar to 3x10(4)) order modes, indicating that single fundamental mode lasing should be accessible over a broad wavelength tuning range. An electrically injected ETR based on this design is fabricated from an InGaAsSb/AlGaAsSb/GaSb, graded-index separate-confinement heterostructure, laser diode wafer with a 2.1 mu m emission wavelength. This device achieved single mode, continuous wave operation at 77 K with a threshold current of 0.5 mA and a single mode wavelength tuning range of 3.25 nm, which is accomplished by varying the injection current from 0.5 to 6.0 mA. (C) 2008 American Vacuum Society.

Optical bistability in InP/GaInAsP equilateral-triangle-resonator microlasers

Optical bistability is reported in InP/GaInAsP equilateral-triangle-resonator (ETR) microlasers, which are fabricated by planar technology. For a 30 mu m side ETR microlaser with a 2-mu m-wide output waveguide connected to one of the vertices of the ETR, hysteresis loops are observed for the output power versus the injection current from 215 to 235 K. The laser output spectra are measured in the upper and lower states of the hysteresis loop, which show strong mode competition among transverse modes. The hysteresis loops are demonstrated by two-mode rate equations with asymmetric cross gain saturation and different output efficiencies. (C) 2009 Optical Society of America

Equilateral-Triangle and Square Resonator Semiconductor Microlasers

The characteristics of equilateral-triangle resonator (ETR) and square resonator microlasers are reported, which are potential light sources in the photonic integrations. Based on the numerical simulations, we find that high-efficiency directional emission can be achieved for the triangle and square microlasers by directly connecting an output waveguide to the resonators. The electrically injected InP/InGaAsP ETR and square resonator microlasers with a 2-mu m-wide output waveguide were fabricated by standard photolithography and inductively coupled plasma etching techniques. Room-temperature continuous-wave (CW) operations were achieved for the ETR microlasers with the side length from 10 to 30 mu m and the square resonator microlasers with the side length of 20 mu m. The output power versus CW injection current and the laser spectra are presented for an ETR microlaser up to 310 K and a square resonator microlaser to 305 K. The lasing spectra with mode wavelength intervals as that of whispering-gallery-type modes and Fabry-Perot modes are observed for two square lasers, which can lase at low temperature and room temperature, respectively.

Mode Analysis for Equilateral-Triangle-Resonator Microlasers with Metal Confinement Layers

Mode characteristics are analyzed for electrically injected equilateral-triangle-resonator (ETR) semiconductor microlasers, which are laterally confined by insulating barrier SiO2 and electrode metals Ti-Au. For the ETR without metal layers, the totally confined mode field patterns are derived based on the reflection phase shifts, and the Q-factors are calculated from the far-field emission of the analytical near field distribution, which are agreement very well with the numerical results of the finite-difference time-domain (FDTD) simulation. The polarization dependence reflections for light rays incident on semiconductor-SiO2 -Ti-Au multi-layer structures are accounted in considering the confinement of TE and TM modes in the ETR with the metal layers. The reflectivity will greatly reduce with a Ti layer between SiO2 and Au for light rays with incident angle less than 30 especially for the TE mode, even the thickness of the Ti layer is only 10 nm. If the ETR is laterally confined by SiO2-Au layers without the Ti layer, the Fabry-Perot type modes with an incident angle of zero on one side of the ETR can also have high Q-factor. The FDTD simulation for the ETR confined by metal layers verifies the above analysis based on multi-layer reflections. The output spectra with mode intervals of whispering-gallery modes and Fabry-Perot type modes are observed from different ETR lasers with side length of 10 m, respectively.

Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 mu m

Semiconductor equilateral triangle microresonators (ETRs) with side length of 5, 10, and 20 mum are fabricated by the two-step inductively coupled plasma (ICP) etching technique. The mode properties of fabricated InGaAsP ETRs are investigated experimentally by photoluminescence (PL) with the pumping source of a 980-nm semiconductor laser and distinct peaks are observed in the measured PL spectra. The wavelength spacings of the distinct peaks agree very well with the theoretical longitudinal mode intervals of the fundamental transverse modes in the ETRs, which verifies that the distinct peaks are corresponding to the enhancement of resonant modes. The mode quality factors are calculated from the width of the resonant peaks of the PL spectra, which are about 100 for the ETR with side length of 20 mum.

Symmetry analysis and numerical simulation of mode characteristics for equilateral-polygonal optical microresonators

Mode characteristics for two-dimensional equilateral-polygonal microresonators are investigated based on symmetry analysis and finite-difference time-domain numerical simulation. The symmetries of the resonators can be described by the point group C-Nv, accordingly, the confined modes in these resonators can be classified into irreducible representations of the point group C-Nv. Compared with circular resonators, the modes in equilateral-polygonal resonators have different characteristics due to the break of symmetries, such as the split of double-degenerate modes, high field intensity in the center region, and anomalous traveling-wave modes, which should be considered in the designs of the polygonal resonator microlasers or optical add-drop filters.

Room-temperature continuous-wave electrically injected InP-GaInAsP equilateral-triangle-resonator lasers

Equilateral-triangle-resonator (ETR) microlasers with an output waveguide connected to one of the vertices of the ETR are suitable to be a light source for photonic integrated circuits. InP-GaInAsP ETR lasers with side length from 10 to 30 pm and the output-waveguide width of 1 or 2 pm are fabricated using standard photolithography and inductively coupled-plasma etching techniques. Continuous-wave electrically injected 1520-nm ETR laser with 20-mu m sides is realized with the maximum output power 0.17 and 0.067 mW and the threshold current 34 and 43 mA at 290 K and 295 K, respectively.

Equilateral-triangle-resonator injection lasers with directional emission

Equilateral-triangle-resonator (ETR) lasers with an output waveguide jointed at one vertex of the resonator are fabricated on (100) GaInAsP-InP wafers using photolithography and a two-step inductively coupled plasma (ICP) etching technique. Distinct peaks with the mode spacing of longitudinal mode intervals are observed in the luminescence spectra at room temperature. Furthermore, some minor peaks appear in the middle of the main peaks, which can be attributed to the first-order transverse modes as predicted in the theoretical results. CW directional lasing emissions are achieved for ETR lasers with side lengths ranging from 15 to 30 pm up to 200 K. The temperature dependences of the threshold current and lasing wavelength are measured for an ETR laser with the side length of 20 mu m from 80 to 200 K. The observed threshold current rapidly increases as temperature increases over 170 K.

Mode characteristics for equilateral triangle optical resonators

Mode characteristics of equilateral triangle resonators (ETRs) are analyzed based on the symmetry operation of the point group C-3v. The results show that doubly degenerate eigenstates can be reduced to the A(1) and A(2) representations of C-3v, if the longitudinal mode number is a multiple of 6; otherwise, they form the E irreducible representation Of C-3v. And the one-period length for the mode light ray is half of the perimeter of the ETR. Mode Q-factors are calculated by the finite-difference time-domain (FDTD) technique and compared with those calculated from far-field emission based on the analytical near-field pattern for TE and TM modes. The results show that the far-field emission based on the analytical field distribution can be used to estimate the mode Q-factor, especially for TM modes. FDTD numerical results also show that Q-factor of TE modes reaches maximum value as the longitudinal mode number is a multiple of 7. In addition, photoluminescence spectra and measured Q-factors are presented for fabricated ETR with side lengths of 20 and 30 mu m, and the mode wavelength intervals are compared with the analytical results.

Analysis of mode quality factors for equilateral triangle semiconductor microlasers with rough sidewalls

The eigenmode characteristics for equilateral triangle resonator (ETR) semiconductor microlasers are analysed by the finite-difference time-domain technique and the Pade approximation. The random Gaussian correlation function and sinusoidal function are used to model the side roughness of the ETR. The numerical results show that the roughness can cause the split of the degenerative modes, but the confined modes can still have a high quality factor. For the ETR with a 3 mum side length and the sinusoidal fluctuation, we can have a quality factor of 800 for the fundamental mode in the wavelength of 1500 nm, as the amplitude of roughness is 75 mn.