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The design, growth, and steady-state and small-signal modulation characteristics of high-speed tunnel injection In0.4Ga0.6As/GaAs quantum dot lasers are described and discussed. The measured small-signal modulation bandwidth for I/Ith ~ 3.2 is f-3dB = 22GHz and the gain compression factor for this frequency response is ε = 7.2s10-16 cm3. The differential gain obtained from the modulation data is dg/dn ≈ 8.85x10-14 cm2 at room temperature. The value of the K-factor is 0.171ns and the maximum intrinsic modulation bandwidth is 55GHz. The measured high speed data are comparable to, or better than, equivalent quantum well lasers for the first time.
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1.3 μm GaAs-based quantum dot (QD) lasers demonstrate parameters improved over InP-based devices. They exhibit lower threshold current densities and losses, higher differential efficiencies and improved temerature stability. Highspeed operation is demonstrated. Reduced linewidth enhancement factor advantageous for low-chirp operation makes it possible to suppress dramatically filamentation effects destroying lateral far-field pattern. GaAs-based QD 1.3 μm VCSEL with 8 μm oxide aperture wavelength emits up to 1.2 mW CW multimode.
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Photoluminescence spectra are investigated of InAs/InGaAs QD structures prepared be MBE on GaAs substrates in a range of pumping power density up to 0.6 kW/cm2. Multiple spectra band are observed corresponding to electron shells in atom-like dots. Identification of shells is proposed on the basis of spherical oscillator model. Energy diagram of dots is proposed taking into account identical temperature dependence of PL intensity in three lowest spectral bands.
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We investigate the influence of the coupling between localized and
continuum states on the optical gain and refractive index in
self-organized quantum-dot structures under high-excitation conditions. For wide-bandgap nitride-based quantum-dot structures we show that the presence of strong many-body Coulomb interactions and the quantum-confined Stark effect result in absorption/gain features that depend on the quantum-dot dimensions in a nontrivial way. For
InAs/GaAs based quantum dots, we investigate the refractive index properties and show that negative α or linewidth enhancement factors may occur in these systems, which makes the beam quality (filamentation) properties of quantum-dot lasers very different from quantum-well lasers. This is consistent with measurements which show a reduction in quantum-dot laser filamentation as the injection level is increased.
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Due to the large energy splitting of the single-electron levels in a small quantum dot, only one single electron level and one single hole level can be made resonant with the levels in the conduction band
and valence band. This results in a closed system with nine distinct levels, which are split by the Coulomb interactions. We show that flat and tall cylindrically symmetric dots have level schemes
with different selection rules. In both cases entangled photon pairs can be efficiently produced.
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A novel multi-longitudinal-mode rate-equations description of the Fabry-Perot type semiconductor laser is presented. The model includes gain dynamics among the longitudinal modes due to e.g. spatial hole burning.
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A multimode model is necessary to describe the behavior observed in
a twin stripe diode laser. We will use a single-stripe version of the
device to calibrate the parameters enabling the model to be used in
the description and analysis of the twin-stripe lasers.
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In the present study we rely in different experimental measurements to show that the mode structure changes with the current are reflected onto the output power vs. current characteristics of the device. We address the evolution of the modal structure of a twin
ridge as we change the level of current injection among the two ridges. We show that the complex behavior during ridge coupling and the appearance of the lateral modes of the arrays are qualitatively represented in these curves. This information is important in two
related aspects: a) Determine when the two ridges start to interact, giving rise to a high frequency locking phenomena and b) Study appropriate models of the device that account for the observed phenomena.
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A model for the analysis the dynamical behaviour of a twin stripe laser oscillator is developed. It consists of a laser with two active stripes with slightly different effective indices of refraction accounting for different active indices, and for the existence of two modes (symmetric and antisymmetric). The analysis is based on two laterally coupled one-dimensional transmission lines. Oscillating optical signals at the laser output are observed, if both stripes are independently biased with direct currents, without any modulation. The
main characteristics of these oscillations, as well as their dependence on several parameters of the devices, are shown. The main features of the oscillations are, that the frequency is above the relaxation oscillation frequency, and that the phase noise of the oscillations is much lower than in the beat signal of two solitary
lasers. Finally, the modulation depth of the signal is optimized, and the linewidth is evaluated.
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Generally, the high-speed modulation dynamics ofsemiconductor lasers is determined by a complex interplay of ultrafast light-field and carrier dynamics with characteristic times-scales of inter- or intraband relaxation and scattering. Those determine the relaxation oscillations and set an upper limit to the modulation ofa single-mode semiconductor laser (cut-off frequency). In spatially extended semiconductor lasers, however, the longitudinal and transverse
dimensions enable the coexistence ofnumerous longitudinal and transverse modes. With suitable resonator design allowing segmented contact carrier injection and modulation it should thus be possible to directly influence the lateral coupling and transverse mode dynamics ofa given laser structure and modulate the laser with a beat frequency associated with these modes. In this paper, we present results ofsimulations of high-frequency modulation characteristics oftwin-stripe semiconductor lasers. We show that the lateral segmentation of the contact may with proper asymmetric application of the injection current, indeed, lead to a more than five-fold increase of the modulation band-width. Our theory on the basis of multi-mode Maxwell Bloch equations includes propagation effects and spatiotemporally varying mode competition. Numerical simulations show that the increased high-speed modulation is closely associated with the coupled lateral and longitudinal multi-mode dynamics of the laser.
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The objective of this work is to analyse the dependence of the frequency response of Laterally Coupled Diode Lasers (LCDL) focusing on the separation between the laser ridges. A detailed study of the integrated optical spectra and frequency response is presented for LCDLs with 300 μm cavity length and separation between the ridges of 2, 4, 6, 8 and 10 μm. This study is of major importance as it defines the range of frequency locking for each ridge spacing and
also its dependence on the bias conditions applied.
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We considered a model in which two equal sources are symmetrically and vertically separated with respect to a central longitudinal axis, in which a Y-waveguide is placed to couple the two output signals to one fiber. Our aim was to establish the optimum considerations of coherently adding the Gaussian fundamental mode of this twin laser/s system transmitted in the junction waveguide. The coupling efficiency between each laser/s Gaussian output and each waveguide branch, and between the output waveguide and the fiber was calculated, avoiding waveguide rotations or vertical translations. For simplicity, equal minimum Gaussian waist width for each laser/s output and each waveguide branch was assumed. In addition, the fractional power transmission coupling (normalization) was numerically simulated.
To estimate the intensity transmitted by the entire twin laser - Y waveguide - fiber compound system, we used the coupling coefficient of an output laser beam and one branch of the waveguide (assumed equal for both signals in the twin laser/s array) as a weight factor for the field transmitted by the waveguide branch.
An ideal Y-waveguide for coupling to a fiber was designed to diminish coupling losses between the two output signals for a laterally coupled semiconductor twin laser.
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Physics and Simulation of Optoelectronic Devices with Periodic Structures
We demonstrate a comprehensive multi-dimensional DBR laser simulation. The DBR laser under investigation consists of three longitudinally integrated waveguide sections: an active section providing the optical gain for the laser operation, a passive phase shift section which contains neither gratings nor active material and a DBR mirror section. This structure is representative for longitudinally integrated devices such as widely tunable sampled-grating laser diodes. In our physics-based approach, we solve the fully coupled semiconductor drift-diffusion equations for electrons and holes and the temperature diffusion equation, taking into account longitudinal current and heat flux. Gain calculation and the photon rate equation are included self-consistently. A general and comprehensive solution of the transverse optical field is combined with the longitudinal field distribution including general DBR sections. The simulator is applied for the design and optimization of state-of-the-art tunable lasers. It proofs to be an effective tool for bandgap engineering, for the optimization of the transverse confinement of the optical mode as well as the current, and for thermal management.
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In this work, we demonstrate properties of a novel dual-pump four-wave mixing (DPFWM) scheme using a distributed-feedback (DFB) laser. The DFB laser acts as the mixing medium and one of the pump sources, while the second pump and the probe are injected. New experimental results are shown, including successful demonstration of the power laws of this scheme. We also show the effects of the wavelength detunings on the conversion efficiency and the signal-to-noise rations of the output signals.
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We present a device model for a lateral p-n junction quantum-well edge-emitting laser-transistor with an extra gate contact. Such a contact provides an opportunity to control the confinement conditions of the electrons injected into the active region and, as a consequence, the threshold current and output optical power by the
gate voltage. Using the proposed model, we calculate the laser dc characteristics and estimate its modulation performance. We show that the application of negative gate voltages can lead to a substantial decrease in the threshold current. The estimated cutoff modulation frequency associated with the gate recharging can be much higher than those associated with the photon and electron lifetimes.
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We propose a compact, passive optical limiter incorporating nonlinear materials in a multi-layer photonic bandgap (PBG) structure. A coupled-mode theory that includes interaction between counter-propagating optical pulses and diffraction describes the complex transmission and reflection coefficients. Two materials with different nonlinear refraction and absorption coefficients make up the individual layers. Transverse modulational instabilities appear as the intensity increases; this effect is exploited to improve the optical limiter performance by inserting an aperture in the system. Investigating the role that material combinations play in determining the transmission characteristics optimizes the device design. Our results indicate that PBG devices can improve the dynamic range by more than a factor of two over homogeneous material designs.
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We report our proof-of-principle experiment and modeling of hexagonal micro-pillar cavities. A commercial hexagonal silica fiber (125μm plane-to-plane) was side-coupled perpendicularly with a Gaussian beam, thus the fiber acted as a μ-pillar cavity. We observed multimode resonances with typical Q ≈ 2,500 in the tangential directions that are ≈ 120° to the input-coupling cavity sidewall. The observed free spectral range (FSR) ≈ 4.5 nm is consistent with a six-bounce cavity round-trip length. By using wavefront-matching concept, the observed multimode resonances can be attributed to open-loop trajectories. The multiple wavefront-matched open-loop trajectories of the same ray incident angle can result in resonance linewidth broadening. We employed a k-space representation to calculate the hexagonal cavity normal mode locations.
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The characterization and optimization of optical microring resonator-based optical filters on deeply etched GaInAsP-Inp waveguides, using the finite element-based beam propagation approach is presented here. Design issues, such as coupling , field evolution, power and phase considerations have also been investigated.
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We propose a novel multidimensional dynamical model for description of the coherent interactions of ultrashort high-intensity optical pulses with the resonant nonlinearities in planar optical waveguides and semiconductor microresonators. The model is based on the self-consistent solution of the full-wave vectorial Maxwell’s equations
coupled via polarization source terms to the evolution equations of a discrete multilevel quantum system. The latter are derived employing a group-theoretical approach exploiting symmetric properties of the system Hamiltonian. In particular, the resonant nonlinearity is modelled by a degenerate three-level system of saturable absorbers in order to account for the two-dimensional medium polarization. The resulting Maxwell-pseudospin equations are solved in the time domain using the finite-difference time-domain (FDTD) method. The model
is applied for studying conditions of onset of self-induced transparency (SIT) lossless regime of propagation. Numerical evidence of multidimensional solitons localized both in space and in time is given for the planar optical waveguides. Pattern formation and cavity SIT-soliton formation are demonstrated for the special case of a
passive semiconductor microcavity filled with saturable absorbers.
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For the faster switching speed and the lower power consumption, we optimized the structure of a fully depleted optical thyristor (DOT) by the depletion of charge at the lower negative voltage. The fabricated optical thyristor shows sufficient nonlinear s-shape I-V characteristics with the switching voltage of 2.85 V and the complete depletion voltage of -8.73 V. In this paper, using a finite difference method (FDM), we calculate the effects of parameters such as doping concentration and thickness of each layer to determine the optimized structure in the view of the fast and low-power-consuming operation.
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This paper deals with a physical analysis of the factors determining the operation of the quantum dot and quantum wire infrared photodetectors and their features focusing on semi-qualitative approach and comparison with quantum well infrared photodetectors. We address also the problems of computer modeling of these photodetectors.
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We report on temporal response measurements of InGaAs metal-semiconductor-metal photodetectors (MSM-PDs) under high-illumination conditions. The peak current efficiency of the MSM-PDs decreases as the optical pulse energy increases due to space-charge-screening effects. The screening effects begin to occur at an optical pulse energy as low as 1.0 pJ/pulse, as predicted by a recent two-dimensional model. The fall time and full width at half maximum of the impulse response increase as the optical pulse energy increases and decrease as the bias voltage increases. For optical pulse energies between 1.0 pJ and 100 pJ, the rise time displays a U-shaped behavior as the bias voltage increases. This may be associated with the shape of the electron velocity-field characteristic in conjunction with screening of the dark field by optically generated carriers.
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The promising concept of waveguide photodetection with integrated amplification is evaluated by self-consistent device simulation. Such integrated amplification detectors have the potential to achieve simultaneously high saturation power, high speed, high responsivity, and quantum efficiencies well above one. Our example design vertically combines a bulk photodetector ridge region with laterally confined quantum wells for amplification. The current flow in the three-terminal device exhibits ground current reversal with increasing light power. The net optical gain is evaluated for different waveguide modes. For the dominating mode, the detector responsivity is shown to scale with the device length, reaching quantum efficiencies larger than 100%.
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Directly modulated lasers (DMLs) have two high performance applications: 1310 nm 10 Gb/s uncooled and 1550 2.5 Gs/s extended reach. Two key elements are gain coupled gratings and buried heterostructures. Gain coupled gratings simultaneously increase the DML's intrinsic relaxation oscillation frequency and damping, while the buried heterostructure reduces thermal chirp and parasitic capacitance. Large relaxation oscillation frequencies and reduced parasitic capacitance allow 85°C operation; large damping and reduced thermal chirp enable extended reach.
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Semiconductor lasers with nonidentical InGaAsP/InP multiple quantum wells for optical communication are experimented to show the improved temperature characteristics. We explore the dependence of carrier distribution on temperature and discover the novel temperature characteristics of semiconductor lasers with nonidentical multiple
quantum wells, which are different from conventional ones with identical multiple quantum wells. The origin is due to the strongly temperature-dependent Fermi-Dirac distribution, which favors carriers in high-energy states at high temperature. As a result, carriers redistribute among those quantum wells as temperature varies. It causes the lasing wavelength much less dependent on temperature, compared to the bandgap shrinkage. The carrier redistribution
favoring high-energy states also significantly improves the characteristic temperature of short-wavelength mode.
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The intersubband electron transition rates assisted by interface-longitudinal-optical-phonon emission in a quantum cascade laser with a four quantum-well active region are evaluated using Femi’s golden rule. The electron energy and wavefunction of nonparabolic conduction band are calculated using an eight-band k•p method with the axial approximation. The energy dispersions of interface-longitudinal-optical-phonon modes and associated electrostatic potentials are studied within the framework of macroscopic dielectric continuum model. The numerical results show that AlAs-like interface phonon modes give larger contribution to intersubband transition rates between lower energy states than other interface phonon modes.
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Intersubband absorption spectra are analyzed using the density matrix theory under the second Born approximation. The intersubband semiconductor Bloch equations are derived from the first principles including electron-electron and electron-longitudinal optical phonon interactions, whereas electron-interface roughness scattering is considered using Ando's theory. A spurious-states-free 8-band
k•p Hamiltonian is used, in conjunction with the envelope function approximation to calculate the electronic band structure self-consistently for type II InAs/AlSb multiple quantum well structures. We demonstrate the interplay of various physical processes in the absorption spectra in the mid-infrared frequency range.
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We developed a method to eliminate the spurious solutions of the
k•p Hamiltonian in the envelope function approximation applied to the quantized states of heterostructures by introducing an off-diagonal k2 term. This results in a modification in the fourth and higher order terms in k of the band dispersion, which keeps the dispersion at the Γ point but modifies it at large k so that converts spurious states to the harmless evanescent ones. We show that the modification to the Hamiltonian leads to the monotonic behavior of the conduction band as a function of k and thus removes the spurious solutions in the calculations of confined states for all popular III-V compounds and their alloys.
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We investigate the polarization dynamics of vertical-cavity surface-emitting lasers (VCSELs) when they are subject to a large current modulation, but still operate within the fundamental transverse regime. New, interesting, nonlinear dynamics are unveiled in the two fundamental transverse, orthogonal, linearly polarized (LP) modes of the VCSEL. The intensities of the two LP modes may exhibit in phase time-periodic pulsating dynamics at one of the multiples of the modulation period (subharmonic cascade), but chaotic regimes are also obtained, in whic the LP mode intensities exhibit a combination of inphase pulsing at the modulation time-scale and antiphase dynamics on a slower time-scale. Our results suggest furthermore that, in contrast to conventional edge-emitting lasers, chaotic dynamics in directly modulated VCSELs would appear even for small modulation amplitudes (a few tens of percents of DC current) and small modulation frequencies (hundreds of MHz).
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In recent years, high-density optical data storage has been attracting much interest toward future Tera byte memories. An optical near-field technology is considered to make a breakthrough for ultrahigh capacity optical storages beyond the diffraction-limit. We proposed and demonstrated a micro/nano metal aperture VCSEL for producing optical near-field. Recently, we successfully demonstrated the optical probing based on a micro/nano aperture VCSEL using reflection-induced voltage change of a VCSEL.
We fabricated a nano-aperture GaAs VCSEL with various aperture sizes ranging from 100 to 840 nm. The lasing wavelength is 850 nm. We formed a nano-aperture by using focus ion beam etch on a 130 nm thick Au film deposited on the VCSEL surface. The threshold is as low as 0.3 mA. An AFM probe is scanned just above the nano-aperture of the VCSEL. The distance between the fiber probe and the VCSEL was controlled in the contact mode of an AFM (atomic force microscope) system. We successfully demonstrated the 2D imaging of the voltage change of a nano-aperture VCSEL. The maximum voltage change is 0.47 mV.
In this paper, we carried out the modeling and the experiment of optical nano-probing with a metal-aperture VCSEL. We will discuss the effect of surface plasmon excitation in our proposed nano optical probing.
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We report and use our micro-electro-mechanically tunable vertical cavity surface emitting laser (MEM-TVCSEL) computer-aided design methodology to investigate the resonant frequency design space for monolithic and hybrid MEM-TVCSELs. For various initial optical air gap thickness, we examine the sensitivity of monolithic or hybrid
MEM-TVCSEL resonant frequency by simulating zero, two, and four percent variations in III-V material growth thickness. As expected, as initial optical airgap increases, tuning range decreases due to less coupling between the active region and the tuning mirror. However, each design has different resonant frequency sensitivity to variations in III-V growth parameters. In particular, since the monolithic design is comprised of III-V material, the shift in all growth thicknesses significantly shifts the resonant frequency response. However, for hybrid MEMTVCSELs, less shift results, since the lower reflector is an Au mirror with reflectivity independent of III-V growth variations. Finally, since the hybrid design is comprised of a MUMPS polysilicon mechanical actuator, pull-in voltage remains independent of the initial optical airgap between the tuning reflector and the III-V material. Conversely, as the initial airgap increases in the monolithic design, the pull-in voltage significantly increases.
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A novel optical signal processing using an optically pumped vertical-cavity surface-emitting laser (VCSEL) with an external optical input is proposed. The mode transition between a fundamental and a high-order transverse mode is induced by an external optical injection. When we select a fundamental transverse mode through a single-mode fiber as an output signal, we are able to realize non-linear transfer functions, which may be useful for re-amplification and re-shaping in high speed photonic networks. The switching characteristic of a 1.55 μm optically pumped two-mode VCSEL has been simulated by using a two-mode rate equation, including the effects of spatial hole burning and spectral hole burning as gain saturation coefficients. Also, the detuning effect in the injection locking is investigated. When the wavelength of an input light with a fundamental mode is slightly longer than that of a VCSEL operating in a 1st-order transverse mode, the transverse mode of the VCSEL is switched to a fundamental mode at a critical input power level. This gives us an ideal 2R function with amplification and nonlinear transfer functions.
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This work deals with the TCAD (technology computer aided design) based design of VCSEL (vertical-cavity surface emitting laser) devices. A comprehensive 2D electro-thermo-optical device model is presented. Furthermore, as examples, a micromechanical, electrostatically actuated vertical optical resonator is
investigated, a procedure for optimising the higher order mode suppression in a VCSEL is presented, and a coupled electro-thermo-optical simulation of a VCSEL is performed.
The laser device model employs the photon rate equation approach. It is based on the assumption that the shapes of the optical modes depend on the instantaneous value of the time-dependent dielectric function.
The optical fields in the VCSEL cavity are expanded into modes obtained from the complex frequency representation of the homogeneous vectorial Helmholtz equation for an arbitrary complex dielectric function. The 3D problem is transformed into a set of 2D finite element (FE) problems by using a Fourier series expansion of the optical field in azimuthal direction. For the bulk electro-thermal transport a 2D thermodynamic model is employed in a rotationally symmetric body. Heterojunctions are modeled using a thermionic emission model. Quantum wells are treated as scattering centres for
carriers. The optical gain and absorption model in the quantum well active region is based on Fermi's Golden Rule. The sub-bands in the quantum well are determined by solving the stationary effective mass Schroedinger equation with parabolic band approximation for the electrons, light and heavy holes.
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Finite difference analysis was used to determine the thermal characteristics of continuous wave (CW) 850 nm AlGaAs/GaAs implant-apertured vertical-cavity surface-emitting lasers. A novel flip-chip design was used to enhance the heat dissipation. The temperature rise in the active region can be maintained below 40 °C at 4 mW output power with 10 mA current bias. In contrast, the temperature rise reaches above 60 °C without flip-chip bonding. The transient-temperature during turn-on of a VCSEL was also investigated. The time needed for the device to reach the steady-state temperature was in the range of a few tenths of a milli-second, which is orders of magnitude larger than the electrical or optical switch time. Flip-chip bonding will reduce the shift of the wavelength, peak power, threshold current and slope efficiency during VCSEL operations.
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A three-dimensional electrical-thermal-optical numerical solver is applied to model top-emitting oxide-confined vertical-cavity surface-emitting lasers (VCSELs) with GaAs/AlGaAs multiple-quantum-well active region. CW mode of operation is simulated over a range of voltages, covering sub-threshold spontaneous emission and lasing emission. Effect of self-distribution of electrical current is demonstrated for the first time in a self-consistent electrical-thermal-optical simulation of VCSELs.
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The design of the next generation of vertical-cavity surface-emitting lasers (VCSELs) will greatly depend on the availability of accurate modeling tools. Comprehensive models of semiconductor lasers are needed to predict realistic behavior of various laser devices, such as the spatially nonuniform gain that results from current crowding. Advanced physics models for VCSELs require benchmark quality experimental data for model validation. This paper presents preliminary results of a collaborative effort at ARL to fabricate and experimentally characterize test optoelectronic structures and VCSEL devices, and at CFDRC to develop comprehensive multiphysics modeling, design and optimization tools for semiconductor lasers and photodetectors. Experimental characterization procedure and measurements of optical and electrical data for oxide-confined intracavity VCSELs are presented. A comprehensive multiphysics modeling tools CFD-ACE+ O’SEMI has been developed. The modeling tool integrates electronic, optical, thermal, and material gain data models for the design of VCSELs and edge emitting lasers (EELs). This paper presents multidimensional simulation analysis of current crowding in oxide-confined intracavity VCSELs. Computational results helped design the test structures and devices and are used as a guide for experimental measurements performed at ARL.
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An advanced non-traditional method of single expression (MSE) is applied to simulate optical functionality of 1D VCSEL type structures. In the MSE Helmholtz’s equation solution is presented in the generalized form of a single expression, contrary to the widely accepted counter-propagating waves presentation. Main principles of the MSE are briefly described. VCSEL type structure consisting of two stacks of distributed Bragg reflectors (DBRs) and a single dielectric layer between them is considered. DBRs are presented as quarter-wavelength dielectric bilayers of alternating high and low indices. Spectral dependences of VCSEL type structure when amplification is absent and included in a single dielectric layer are obtained versus the normalized thickness of a single dielectric layer. Spatial distributions of electric field amplitude, real part of permittivity and power flow density along the structure are presented for minimal and maximal transmission of the structure. By attaining gain some specific value the lasing state of VCSEL type structure is observed.
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Many integrated optical-based subsystems incorporate optical guided-wave devices and connecting optical waveguides with two-dimensional confinement and a high index contrast. The modes present in such waveguides and devices are not purely of the TE or TM type but hybrid in nature, with all the six components of the electric and magnetic fields being present. Over the last three decades, many semi-analytical and numerical approaches have been developed to study the modes in optical waveguide structures: however, to characterize polarization issues in such systems, a fully vectorial approach is necessary. In that respect the fully vectorial H-field based finite element method (FEM) [1] is one of the most rigorous and versatile of the approaches. The main advantage of the FEM over many other methods is its more accurate representation of the waveguide cross-section. In an optoelectronic system, when modes are hybrid in nature, polarization conversion can take place in the optical system, either unintentionally or deliberately at different waveguide junctions. The least squares boundary residual method [2], which is a fully vectorial and rigorously convergent method, is also used here to characterize optoelectronic systems. The beam propagation method (BPM) is field evolutionary in its approach and a versatile method for the characterization of a z-dependent guided-wave structure. A numerically efficient full-vectorial FEM-based BPM [3] has been developed to characterize z-dependent guided-wave devices. Results for the polarization cross-talk in such systems will be presented, along with their various minimization approaches. Results will also be presented for the design optimization of various compact polarization rotators using cascaded sections with or without a slanted side wall and also with curved waveguide sections.
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We present structures obtained with numerical optimization techniques capable of efficiently channeling light at a fraction of the length of a conventional taper. These results could open the way to novel designs in ultra-short light injection devices. We also consider the rather different problem of how to optimise the transmission through a photonic crystal bend. We show how, using a deterministic global optimisation algorithm, novel optimal geometries can be obtained leading to considerable performance improvements.
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The application of Liquid Crystals (LC) to integrated optics is actively investigated because the mechanical and optical properties of these materials can be exploited to realize new schemes of opto-electronic devices. In particular the molecular alignment induced in nematic LCs by external electric fields can be very useful for mode splitting, tunable filtering and optical switching.
The aim of this work is to experimentally demonstrate the possibility of dynamically reshaping the refractive index profile of a planar waveguide, and hence the propagation characteristics of the modes, by applying a small external electric field. Graded index planar waveguides have been realized by using a nematic liquid crystal in a distorted hybrid configuration as the guiding layer. The application of external electric fields enables to modify the orientation of the local molecular axis and hence to reshape in an easy and controlled way the refractive index profile. Extensive m-lines spectroscopy measurements were performed in order to determine the anisotropic refractive index profiles for various electric field strengths. The experimental results are in good agreement with what expected from a simple model based on the dielectric, optical and elastic properties of the liquid crystal guiding layer.
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Semiconductor Optical Amplifiers and High Power Lasers
A numerical simulation of Semiconductor Optical amplifiers is proposed, which self-consistently solves the vertical transport and the bidirectional propagation for both coherent signals and amplified spontaneous emission. A realistic account of taper sections is obtained in deriving the confinement factor in all component sections. Microscopic equations are used, as far as possible, for both gain and spontaneous emission rates. With few adjustable parameters we nevertheless obtain a good fit to experimental data which remains stable with geometry and active material changes.
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Novel bi-directional propagation is observed in a shallow-etched bending ridge waveguide. The lasing light propagates in two different paths, straight way and bending way. The far-field pattern is quite different before and after the lasing condition is reached. Emission spectra of light emitted from two facets are also different. This is because the bidirectional guided effect of lasing mode occurs.
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We demonstrate the integration of microscopic gain calculation into the laser design tool LaserMOD, which is derived from the Minilase II simulator. A microscopic many body theory of the semiconductor allows for the accurate modeling of the spectral characteristics of the material gain. With such a model, the energetic position of the gain peak, the collision broadening, and therefore, the absolute magnitude of the gain can be predicted based solely on material parameters [2]. In contrast, many simpler approaches rely on careful calibration of model parameters requiring additional effort due to fabrication of samples and experimental studies. In our full scale laser simulation multi dimensional carrier transport, interaction with the optical field via stimulated and spontaneous emission, as well as the optical field itself is computed self consistently. We demonstrate our approach on an example of a Fabry-Perot laser structure with GaInAsP multiple quantum wells for 1.55 μm emission wavelength.
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In this work we use a numerical model to analyze the beam filamentation in tapered lasers with the goal of optimizing the epitaxial structure. The model self-consistently solves the steady state electrical and optical equations for the flared unstable resonator. This model has been applied to simulate the performance of InGaAs/InGaAsP tapered lasers emitting at 975 nm. We investigate the role of the active material properties (carrier induced index change, differential gain) and of the epitaxial design (optical confinement factor) on the filamentation process and on the maximum achievable power.
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Nonlinear Dynamics in Solitary and Coupled Semiconductor Lasers
We show that a single-mode semiconductor laser subject to
optical injection, and described by rate equations, can produce
excitable multipulses, where the laser emits a certain number of
pulses after being triggered from its steady state by a single
perturbation.
This phenomenon occurs in experimentally accessible regions in
parameter space that are bounded by curves of n-homoclinic
bifurcations, connecting a saddle to itself only at the n-th
return to a neighborhood of the saddle. These regions are organised in what we call 'homoclinic teeth' that grow in size and shape with the linewidth enhancement factor.
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We have studied the dynamical behaviour of two semiconductor lasers subject to an optoelectronic bidirectional coupling and optionally to feedback, considering non-zero delay times in the propagation of the signals between both lasers and through each feedback loop in the case the latter exists. Starting from delayed rate equations
for the photon and carrier densities, we have investigated the stability of the fixed points and limit cycles of the system as function of the coupling and feedback strengths, as well as the delay times. From this analysis, quasiperiodic route to chaos and several interesting phenomena like the recently discovered "death by delay"
are predicted for this system.
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Faint laser pulses are practical signal sources in quantum key distribution. The photon statistics has been measured by recording every photo-counts moment and leading directly to the Mandel parameter of the photon distribution. Two different types operation of a single mode laser diode have been considered: either direct pulsing of the driving current or CW emission followed by pulse-shaping using an acousto-optic modulator. In both cases, it has been found that for practical attenuations, the photon distribution in each light pulse follows a poissonian statistics.
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Nonlinear Dynamics in Diode Lasers with Optical Feedback
We analyze the dynamics of a laser diode subject to optical feedback from a short external cavity (EC), i.e. with an EC round-trip time much smaller than the period of the laser relaxation oscillations (RO). Our numerical simulations are based on the Lang-Kobayashi (LK) equations for single mode edge-emitting lasers subject to weak/moderate optical feedback. A new, detailed, Hopf bifurcation analysis shows that LK equations admist both supercritical and subcritical Hopf bifurcation points. Subcritical Hopf points lead to time-periodic pulsating intensity solutions with a frequency close to half the RO frequency. In contrast, from supercritical Hopf bifurcatiosn emerge harmonic intensity oscillations with a frequency either close to the RO frequency or to the EC frequency. Microwave oscillations are obtained, as a result of a beating between two EC modes. In general, these high frequency dynamics are stable only for a small range of feedback parameters. However, we find that decreasing the α factor largely improves the stability of the microwave oscillations and makes it possible to observe pulsating intensity solutions for a larger range of EC length. The high frequency intensity solutions of laser diodes with short EC are thought to be of great interest for new applications in all optical signal handling. Our results motivate new theoretical studies of LK equations with short EC.
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We have applied a comprehensive simulation tool for multisection DFB lasers to systematically investigate bifurcations caused by combining a single mode DFB laser with a feedback section. Strength and phase of the optical feedback are considered as main bifurcation parameters. The recording of output power, optical and power spectra and carrier density yield evidence of fold bifurcations of stationary states as well as subcritical and supercritical Hopf bifurcations
towards self-pulsations. Furthermore, a homoclinic bifurcation is detected and indications for a fold of limit cycles are observed in qualitative agreement with a bifurcation diagram very recently calculated with path following techniques 7 under simplifying assumptions. The present simulations show that these bifurcations are stable with respect to these approximations. They offer a method how the bifurcations can be determined experimentally.
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We report on a new model of analysis in semiconductor laser dynamics under strong optical feedback (OFB) in fiber communication systems. The model treats the optical feedback due to reflection of the laser light on an external fiber grating as a time-delayed optical field injected to the laser cavity. The model is versatile applicable to an arbitrary strength of optical feedback ranging from weak to strong. The model is applied to stimulate output characteristics and intensity noise in InGaAs lasers pumping fiber amplifiers in a wavelength of 980 nm. Influence of intrinsic fluctuations in the intensity and optical phase of the lasing field on the laser dynamics is taken into account. The time-varying trajectories of the output power are presented over a wide range of the injection current. The simulation results indicated that the laser mainly operates in pulsation under strong optical feedback. Counting the intrinsic fluctuations was found to reduce the OFB-induced instabilities so as to bring the laser faster to its steady state operation. The optical feedback noise is found to be as low as the quantum noise level when the laser is injected well above its threshold level.
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Vertical-cavity surface-emitting lasers have shown similar sensitivity to optical feedback as conventional edge-emitting
lasers, but new interesting phenomena can be observed due to the coexistence of two linearly polarized (LP) fundamental modes. We report on new dynamic effects in VCSELs induced by polarization insensitive optical feedback from a distant mirror, namely the appearance of low frequency random hops between the two LP modes in a nominally stable LP solitary laser. This behavior resembles that of the mode hopping in a solitary VCSEL close to its polarization switching point. However, a careful observation shows that superimposed on the low frequency polarization mode-hopping, fast oscillatory behavior at a frequency close to the external-cavity frequency appears. A complementary study of the polarization
resolved optical spectra reveals jumps between several peaks identified as external cavity modes. We analyze the dynamics using a two-mode rate equation model with delay and noise. We numerically observe polarization mode-hopping in good qualitative agreement with our experimental findings. In particular, the low-frequency hops are complemented with fast oscillations at a frequency close to the external-cavity one and the calculated optical spectra reveal the presence of a limited number of ECMs in each LP-comb. This indicates that the dynamics is created by the interplay of noise, bistability
and optical feedback. We will further discuss the effect of noise on delayed bistable laser systems in the context of new dynamical concepts, like coherence resonance and stochastic resonance.
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A semiconductor laser subject to phase-conjugate optical feedback can be described by rate equations, which are mathematically delay differential equations (DDEs) with an infinite-dimensional phase space. We employ new numerical continuation techniques for DDEs to study the exact nature of the locking region in the parameter plane given by the feedback strength and the pump current. This reveals interesting dynamics, including heteroclinic bifurcations, near the locking region, leading to different scenarios of possible transitions into and out of locking. We show how several special points act as organizing centers for the dynamics.
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We explain how a semiconductor optical amplifier in a Sagnac-interferometric arrangement can be used for switching of 200 fs optical pulses. The switching principles are based on gain and index saturation dynamics on a sub-picosecond timescale. We present a model that accounts for bi-directional propagation of ultrashort optical pulses through the amplifier as well as free-carrier absorption and two-photon absorption. We have also carried out pump and probe experiments to measure the ultrafast refractive index dynamics of a multi-quantum well InGaAsP-InGaAs semiconductor optical amplifier that is operated in the gain regime. The pump and probe pulses are cross-linearly polarized. We observe a phase shift of 200 degrees if the amplifier is pumped with 120 mA of current, but find that the phase shift vanishes if the injection current is increased to 160 mA. Our results indicate a contribution of two-photon absorption to the nonlinear phase shift that opposes the phase shift introduced by the gain. Finally, we observe that the phase shift comes up and disappears within a picosecond.
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Active mode locking is reported for a 1.55 μm semiconductor laser with a curved waveguide and passive mode expander, placed in a wavelength tuneable external cavity. The waveguide is curved to reduce the reflectivity at only one facet, and the active region tapers down towards this facet to allow the guided mode to expand into an underlying passive waveguide. This further reduces the reflectivity due to the smaller divergence of the expanded mode, whilst also enhancing the coupling efficiency and relaxing alignment tolerances out into the external cavity. The resulting inner facet reflectivity is 8 x 10-6, while the outer facet maintains a conventional reflectivity of approximately 0.3. These features make the device ideal for use in an external cavity setup, and to the authors’ knowledge this is the first time such a design of device has been mode locked. With non-optimised pulse compression, 2.5 GHz pulses of 2.9 ps duration have been generated with a linewidth of 1.09 nm. In harmonic operation, 10 GHz pulses of 3.1 ps duration have been obtained, with a linewidth of 0.81 nm. Together with its wavelength tuning capability, these features make the setup a candidate source for a multi-wavelength optical time division multiplexed communications system.
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Terabit pulse generation can be realized with a mode locked semiconductor laser resonator.The combination of high gain values and large gain bandwidth make semiconductor materials an ideal ultrahigh speed medium. Unfortunately, standard frequency domain techniques ingore transient solutions and become numerically intensive with large bandwidth calculations. Therefore, it is necessary to directly model the response of an electromagnetic field within such media in the time domain. Futhermore, the understanding of resonant structures is critical not only for modeling semiconductor lasers but for periodic resonant structures and photonic bandgap devices as well.
As a first step, we examine electromagnetic modes in a semiconductor medium inside a standing wave resonator. Because of the large bandwidth of the field and gain medium, a direct time-domain solution to the wave equation is desireable. Also, since the laser output power is proportional to pump current, conduction properties of the gain medium are emphasized. For this problem an exact solution to the time domain wave equation is obtained for a medium containing bound charges and free-carriers. Specific examples, which illustrate the transient and steady state nature of the fields, are given for doped GaAs.
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The effect of velocity matching, impedance matching, conductor loss and dielectric loss on the optical bandwidth of an ultra-high-speed lithium niobate (LN) modulator is reported by using the finite element method. It is shown that for an etched LN modulator the product VπL could be reduced by 30% and it is also relatively easier to match both Nm and Zc simultaneously. The work indicates that both the dielectric loss and impedance matching play a key role for velocity matched high-speed modulators along with the low conductor loss. The effects of etch depth, buffer thickness, electrode width and the gap between the electrodes on device performance are also illustrated.
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We consider a new model for the active modulation component of a modelocked laser cavity which allows for the construction of exact pulse train solutions. The model begins with the nonlinear Schrodinger equation for propagation in the laser cavity which is influenced by chromatic dispersion and Kerr induced self-phase modulation. Additionally, a bandwidth limited gain term is included to capture the amplification process in the cavity. The active modelocking element allows for periodically spaced regions of preferential gain. Thus a modelocked pulse train will align itself under the peaks of the gain region while radiation energy outside this region is attenuated. We consider a novel form of the periodic, active modelocking element by making use of the Jacobi elliptic functions. Two families of pulse train solutions are generated: one in which neighboring pulses are in-phase, and a second in which neighboring pulses are out-of-phase. The model predicts that only out-of-phase pulse train solutions can be stabilized. Under large perturbation, the pulse train is often stabilized to a two-pulse per round trip configuration. All in-phase solutions are unstable and are destroyed. Further, for the out-of-phase solutions, if the pulse spacing is not sufficiently far, then the nearest neighbor interactions can dominate and lead to Q-switching behavior. For short cavities, this Q-switching can result in quasi-periodic behavior of the pulse train. For long cavities, the resulting Q-switching is chaotic in nature.
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Scanning near field optical microscopy (SNOM) has been developed to assess the recombination mechanism in low-dimensional nitride semiconductors by employing spatial and temporal photoluminescence (PL) mapping under illumination-collection at cryogenic temperatures. The near-field PL images taken at an InxGa1-xN single-quantum-well (SQW) structure revealed the variation of both intensity and peak energy according to the probing location with the scale less than a few tens of a nanometer. The PL, the linewidth of which was about 60meV in macroscopic measurements, was separated into several peaks with the linewidth of about 12 meV if the SNOM-PL was taken with the aperture size of 30 nm. Clear spatial correlation was observed between PL intensity and PL peak-photon-energy, where the regions of strong PL intensity correspond to those of low PL peak-photon-energy. Time-resolved SNOM-PL study showed the important role of exciton/carrier localization in the recombination mechanism in InxGa1-xN-based quantum structures.
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In this paper some relevant aspects of the computational electronics of wide band gap semiconductors are discussed. The most important modelling differences and challenges that make wide band gap semiconductors more complex to study than conventional cubic materials are considered. The model used to compute the bulk transport properties of wurtzite phase GaN is presented as an example. In particular, the negative differential mobility effect and the velocity overshoot are discussed. It is found that the negative differential mobility results from a two-step process. The non-parabolicity of the Γ valley is responsible for the onset of the negative differential mobility phenomena and the transfer to higher energy region of the Brillouin zone makes a contributions only after the process has already initiated.
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Properties of InGaN/AlInGaN/AlGaN single- and multiple-quantum-well (MQW) light-emitting diodes grown by MOCVD on sapphire substrates are investigated over a wide temperature range from 12 to 298 K. The room-temperature (RT) UV emission band, observed in both single-quantum-well (SQW) and MQW samples, is at 3.307 eV (375 nm) and its full width at half maximum is ~82 meV. In addition to the UV band, a blue emission band at 2.96 eV (419 nm) is observed in SQW samples. The relative intensities of these UV and blue emission bands depend on the injection current. We attribute the blue emission to the carrier overflow over the quantum well (QW) and subsequent radiative recombination involving a Mg-related-level in p-GaN. In MQW LEDs, we observe an anomalous temperature-induced "blue jump" between 170-190 K, with the main emission peak switching from blue to UV. The blue band emission dominates below 170 K, and is practically absent at RT. Thus, we demonstrate a significant advantage in utilizing MQW structures that provide a more effective capture of injected carriers into the QWs.
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A new candidate for the lattice-matched metallic substrate, i.e. ZrB2, for the growth of group-III nitrides is proposed. A low-temperature-deposited-buffer layer is found to be essential for the growth of GaN on ZrB2. Highly luminescent violet-light-emitting diodes fabricated on ZrB2 perform as well as or even superior to those fabricated on sapphire. ZrB2 is easily etched by the solution of HF and HNO3. Fabrication of a nitride-based flexible display is expected using a thin
free-standing GaN film.
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Nonlinear Dynamics in Solitary and Coupled Semiconductor Lasers
We characterize the chaotic dynamics of semiconductor lasers subject to either optical or electro-optical feedback modeled by Lang-Kobayashi and Ikeda equations, respectively. This characterization is relevant for secure optical communications based on chaos encryption. In particular, for each system we compute as function of tunable parameters the Lyapunov spectrum, Kaplan-Yorke dimension and Kolmogorov-Sinai entropy.
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A rigorous and numerically efficient method for 2D and 3D analysis of general periodic structures is described and applied to a number of passive and active optical devices.
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In this research, properties of bulk and microcavity hydrogenated amorphous silicon nitride are studied. Microcavities were realized by embedding the active hydrogenated amorphous silicon layer between two dielectric mirrors. The dielectric mirrors were realized with two distributed Bragg reflectors (DBR’s). The DBR’s are one dimensional
photonic bandgap (PBG) materials, i.e., photonic crystals, composed of alternating layers of silicon oxide and silicon nitride. All of the layers are grown by plasma enhanced chemical vapor deposition (PECVD) on silicon substrates. The temperature dependence of the amorphous silicon photoluminescence is performed to fully characterize and optimize the material in the pursuit of obtaining novel photonic microdevices. Photonics device characterization was done by means of atomic force microscopy (AFM), scanning electron microscopy (SEM), photoluminescence, and reflectance measurements. The reflectance spectra calculations were performed using the transfer matrix method (TMM).
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This report presents modeling and simulation work for analyzing three designs of Micro Electro Mechanical Systems (MEMS) Compound Pivot Mirrors (CPM). These mirrors were made using the polysilicon SUMMiTTM process. At 75 volts and above, initial experimental analysis of fabricated mirrors showed tilt angles of up to 75 degrees for one design, and 5 degrees for the other two. Nevertheless, geometric design models predicted higher tilt angles. Therefore, a detailed finite element modeling study was conducted to explain why lower tilt angles occurred and if design modifications could bemade to produce higher tilt angles at lower voltages. This study showed that the spring stiffness of the mirrors was too great to allow for desired levels of rotation at lower levels of voltage. To create lower spring stiffness, a redesign is needed.
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In this paper, the authors have studied the design parameters of a tunable multiple quantum well interference filter. Analogous to the optical thin film filter, an electron wave interference filter is analyzed using the transfer matrix method. The numerical values of the design parameters such as 3-dB bandwidth and quality factor are calculated for various filter configurations with variable number of layers and different thickness values of the film layer for a pass wavelength of 100Å. The tunability of the filter is also investigated.
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Spherical and cylindrical lenses with intensity dependent focal length are investigated experimentally. The lenses were photosensitive enough for incident laser power about 0.5 mW/cm2. Two adaptive systems are implemented to demonstrate
feasibility of a) power-in-diaphragm stabilization and b) auto-correction of defocus, in all-optical setups. With very little efforts on optimization, the first system reduced intensity variation of 150% down to less than 5%, and the second one practically removed the optical power variation in the output beam when of the input beam optical power increased up to 0.46 dptr.
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In this paper, the author presents a computer algorithm for the generation of a high-quality continuous random networks using a genetic algorithm (GA). Genetic algorithms are a part of evolutionary computing, which is a rapidly growing area of artificial intelligence. As a one can guess, genetic algorithms are inspired by Darwin's theory about evolution. Simply said, solution to a problem solved by genetic algorithms is evolved. This paper formulates the amorphous silicon atomistic model problem such that a genetic algorithm can be designed to solve it. A population of models are generated randomally at the start. A sequence of genetic processes such as individuals regeneration, feature cross-over and mutation are performed to produce new generations of the models. After many generations the optimal solution is reached. A series of computer simulations are used to predict many of the structural and electronic properties of the amorphous silicon. The results are compared with the experimental values for these physical parameters mentioned in the literature for testing the model accuracy. Also, a comparison between the suggested model and the other famous computer-based algorithms is presented. The results are discussed.
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Two InGaP/AlGaInP lasers (emitting at 660 nm and at 690 nm) and one GaAs/AlGaAs laser (emitting at 780 nm) have been studied under hydrostatic pressure up to 20 kbar and at temperatures from 240 K to 300 K. The power-current characteristics and the spectra have been measured in the specially designed pressure cell. The emission spectra shifted in agreement with the pressure/temperature variation of the bandgaps in active layers of the lasers. Since at high pressure the Γ-X separation in the conduction band is strongly reduced (both in AlGaInP and in AlGaAs) the dominant loss mechanism of the lasers is the carrier leakage to X minima in the claddings. This, in turn, leads to high sensitivity of threshold currents to temperature. The dependence of threshold currents on pressure and on temperature is in good agreement with the simple phenomenological analysis taking into account the carrier leakage and the radiative and nonradiative recombination. Good description of the pressure and temperature variation of the threshold currents is obtained using three adjustable parameters. Our fits indicate that the dominant contribution to electronic leakage is drift rather than diffusion. These results are important for the application of pressure/temperature tuning of laser diodes in the 600-800 nm range. In particular, we were able to turn red laser diodes into yellow (emitting below 600 nm) and infrared 780 nm lasers into bright red. By simultanous control of pressure and temperature it is possible to obtain constant emission power of the lasers in the full tuning range (at a fixed operating current).
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Solid state electrochromic devices (ECD) are of considerable technological and commercial interest because of their controllable transmission, absorption and/or reflectance. For instance, a major application of these devices is in smart windows that can regulate the solar gains of buildings and also in glare attenuation in automobile rear view mirrors. Other applications include solar cells, small and large area flat panel displays, satellite temperature control, food monitoring, and document authentication. A typical electrochromic device has a five-layer structure: GS/TC/EC/IC/IS/TC/GS, where GS is a glass substrate, TC is a transparent conductor, generally ITO (indium tin oxide) or FTO (fluorine tin oxide), EC is an electrochromic coating, IC is an ion conductor (solid or liquid electrolyte) and IS is an ion storage coating. Generally, the EC and IS layers are deposited separately on the TC coatings and then jointed with the IC and sealed. The EC and IS are thin films that can be deposited by sputtering, CVD, sol-gel precursors, etc.
There are different kinds of organic, inorganic and organic-inorganic films that can be used to make electrochromic devices. Thin electrochromic films can be: WO3, Nb2O5, Nb2O5:Li+ or Nb2O5-TiO2 coatings, ions storage films: CeO2-TiO2, CeO2-ZrO2 or CeO2-TiO2-ZrO2 and electrolytes like Organically Modified Electrolytes (Ormolytes) or polymeric films also based on natural polymers like starch or cellulose. These last are very interesting due to their high ionic conductivity, high transparency and good mechanical properties.
This paper describes construction and properties of different thin oxide and polymeric films and also shows the optical response of an all sol-gel electrochromic device with WO3/Ormolyte/CeO2-TiO2 configuration.
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An electrochromic material (EC) reversibly changes its optical characteristics response, coloring and bleaching states when a small voltage or current is passed through it. This phenomenon is used to develop electrochromic devices like smart windows, which control the amount of heat and light entering in a building and optimize energy consumption. The change of the transparency of these devices involves the injection and extraction of small cations and electrons into the EC material and study of the kinetics of ions injection implies on operation understanding of these devices.
Pure and doped niobium oxides (Nb2O5) are promising cathodic electrochromic materials and their electrooptical
performance depends strongly of its structural morphology. The sol-gel process allows for facile fabrication of large area coatings at a low cost and offers advantages of controlling the composition and microstructure of the films. In order to study the solid sate diffusion of lithium into Nb2O5, Nb2O5:Li+ and Nb2O5:WO3, two electroanalytical
techniques have been used i.e. galvanostatic intermittent titration technique (GITT) and electrochemical impedance spectroscopy (EIS). GITT have been applied in order to obtain the chemical diffusion coefficient of Lix in Nb2O5 doped and undoped films, where the values approaching were of the 2.5x10-11 cm2s-1 at x=0,83, 7.4x10-13 cm2s-1 at x=1.65 and 1.6x10-10 cm2s-1 at x=0.33 for Nb2O5, Nb2O5:Li+ and Nb2O5-WO3 respectively. From these measurements it was also observed that within each film, D increases as x increases.
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