Robert A Indik

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• Feng, Q., Indik, R. A., Lega, J. C., Lega, J., Lega, J. C., Moloney, J. V., Newell, A. C., Newell, A. C., Staley, M., & Moloney, J. V. (1998). Transverse Traveling-Wave Patterns and Instabilities in Lasers. In Self-Organization in Optical Systems and Applications in Information Technology(pp 133-145). Springer Berlin Heidelberg. doi:10.1007/978-3-642-60315-0_6
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  Complex pattern formation or destruction is commonly observed in spatially extended, continuous, dissipative systems, such as convection in ordinary and binary mixture fluids and in liquid crystals, Taylor-Coullet flow, chemical reaction, and directional solidification, etc. [NPL]. Rayleigh-Benard convection has been a canonical system in developing and testing ideas for understanding patterns and transitions [NPL]. Lasers provide another physically and mathematically convenient system for studying space-time complexity, such as pattern selection and spatio-temporal chaos, widely observed in hydrodynamic systems. In this article, we will illustrate, using the two-level and Raman single, longitudinal-mode lasers as prototype systems, the rich variety of pattern-forming instability mechanisms that may appear in wide-aperture lasers.
• Abraham, E., Adachihara, H., Hess, O., Indik, R. A., Jacobsen, P., Moloney, J. V., & Ru, P. (1993). Optical Turbulence in Semiconductor Lasers. In Nonlinear Processes in Physics(pp 213-4). Springer, Berlin, Heidelberg. doi:10.1007/978-3-642-77769-1_39
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  In this paper we study the dynamics of broad-area semiconductor lasers and arrays. In the case of the former, our numerical simulations show symmetry breaking as transverse modulation instabilities develop leading to optical turbulence: temporal chaos and random filamentation of the output beam. The spatio-temporal evolution of linear arrays of otherwise individually stable lasers also becomes turbulent for currents well above threshold and separation smaller than some critical value. Stability can be restored by injecting an external field of suitable amplitude and frequency.

Journals/Publications

• Gabitov, I., Indik, R., Mollenauer, L., Shkarayev, M., Stepanov, M., & Lushnikov, P. M. (2007). Twin families of bisolitons in dispersion-managed systems. Optics Letters, 32(6), 605-607.
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  PMID: 17308575;Abstract: We calculate bisoliton solutions by using a slowly varying stroboscopic equation. The system is characterized in terms of a single dimensionless parameter. We find two branches of solutions and describe the structure of the tails for the lower-branch solutions. © 2007 Optical Society of America.
• Indik, R., Binder, R., Mlejnek, M., Moloney, J., Hughes, S., Knorr, A., & Koch, S. (2007). Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers. PHYSICAL REVIEW A, 53(5), 3614-3620.
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  Based on a microscopic theory of a two-band semiconductor light amplifier, we show that plasma heating, cooling, and ultrafast memory effects all act in concert to produce strong distortion of subpicosecond pulses propagating in semiconductor amplifiers. Plasma heating, spectral hole burning, and carrier density depletion are responsible for saturation of the gain seen by a propagating intense femtosecond pulse in the amplifier. Plasma cooling replenishes the carrier population on the trailing edge of the pulse, leading to pulse broadening as a consequence of gain regeneration. The inclusion of memory effects in the description of dephasing processes goes beyond the usual Markov assumption of constant dephasing rates; it significantly affects the dynamical pulse reshaping processes.
• BROOMHEAD, D., INDIK, R., NEWELL, A., & RAND, D. (2006). LOCAL ADAPTIVE GALERKIN BASES FOR LARGE-DIMENSIONAL DYNAMIC-SYSTEMS. NONLINEARITY, 4(2), 159-197.
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  We suggest and develop a method for following the dynamics of systems whose long-time behaviour is confined to an attractor or invariant manifold A of potentially large dimension. The idea is to embed A in a set of local coverings. The dynamics of the phase point P on A in each local ball is then approximated by the dynamics of its projections into the local tangent space. Optimal coordinates in each local patch are chosen by a local version of a singular value decomposition (SVD) analysis which picks out the principal axes of inertia of a data set. Because the basis is continually updated, it is natural to call the procedure an adaptive basis method. The advantages of the method are the following. (i) The choice of the local coordinate system in the local tangent space of A is dictated by the dynamics of the system being investigated and can therefore reflect the importance of natural nonlinear structures which occur locally but which could not be used as part of a global basis. (ii) The number of important or active local degrees of freedom is clearly defined by the algorithm and will usually be much lower than the number of coordinates in the local embedding space and certainly considerably fewer than the number which would be required to provide a global embedding of A. (iii) While the local coordinates indicate which nonlinear structures are important there, the transition matrices which glue the coordinate patches together carry information about the global geometry of A. (iv) The method also suggests a useful algorithm for the numerical integration of complicated spatially extended equation systems, by first using crude integration schemes to generate data from which optimal local and sometimes global Galerkin bases are chosen.
• Gabitov, I. R., Indik, R. A., Litchinitser, N. M., Maimistov, A. I., Shalaev, V. M., & Soneson, J. E. (2006). Double-resonant optical materials with embedded metal nanostructures. Journal of the Optical Society of America B: Optical Physics, 23(3), 535-542.
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  Abstract: We derive equations modeling the resonant interaction of electric and magnetic components of light fields with metal nanostructures. This paired resonance was recently shown to produce negative refractive index. The model equations are a generalization of the well-known Maxwell-Lorentz model. We demonstrate that in the case of nonlinear polarization and linear magnetization, these equations are equivalent to a system of equations describing the resonant interaction of light with plasmonic oscillations in metal nanospheres. A family of solitary wave solutions is found that is similar to pulses associated with self-induced transparency in the framework of the Maxwell-Bloch model. The evolution of incident optical pulses is studied numerically, as are the collision dynamics of the solitary waves. These simulations reveal that the collision dynamics vary from near perfectly elastic to highly radiative, depending on the relative phase of the initial pulses. © 2006 Optical Society of America.
• Gabitov, I. R., Wabnitz, S., Tonello, A., Indik, R. A., & Gabitov, I. R. (2006). Analyse statistique de la phase des solitons optiques. Journal De Physique Iv, 135(1), 287-288. doi:10.1051/jp4:2006135092
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  On analyse numeriquement la statistique de la phase des solitons optiques et on arrive a determiner la fonction de densite de probabilite. Nos resultats sont en accord avec les theories existantes, qui sont basees sur les proprietes des solitons.
• Indik, R. A., & Newell, A. C. (2006). Conical refraction and nonlinearity. Optics Express, 14(22), 10614-10620.
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  PMID: 19529464;Abstract: We study conical refraction in crystals where both diffraction and nonlinearity are present. We develop a new set of evolution equations. We find that nonlinearity induces a modulational instability when it is defocussing as well as focussing. We also examine the evolution of incident beams which contain analytic singularities, and in particular optical vortices, which do not feel the effect of conical refraction. © 2006 Optical Society of America.
• Tonello, A., Wabnitz, S., Gabitov, I., & Indik, R. (2006). Importance sampling of gordon-mollenauer soliton phase noise in optical fibers. IEEE Photonics Technology Letters, 18(7), 886-888.
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  Abstract: We develop an importance sampling method to perform the direct numerical computation of the probability density function of the random optical soliton phase under the influence of both amplifier spontaneous emission noise and nonlinear conversion of amplitude to phase fluctuations owing to the Kerr effect, or nonlinear phase noise. © 2006 IEEE.
• Tonello, A., Wabnitz, S., Gabitov, I., & Indik, R. (2005). Importance sampling of optical soliton phase. Conference on Lasers and Electro-Optics Europe - Technical Digest.
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  Abstract: We obtain by the importance sampling method a direct numerical estimation of the probability density function of the optical soliton phase in a periodically amplified fiber optics link. © 2005 IEEE.
• Ercolani, N., Indik, R., Newell, A. C., & Passot, T. (2003). Global description of patterns far from onset: A case study. Physica D: Nonlinear Phenomena, 184(1-4), 127-140.
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  Abstract: The Cross-Newell phase diffusion equation τ(k)ΘT = - ∇ · k→B(k), k→ = ∇Θ, |k→| = k, and its regularization describe patterns and defects far from onset in large aspect ratio systems with translational and rotational symmetry. In this paper we show how director field solutions of this equation can be used to describe features of global patterns. The ideas are illustrated in the context of a non-trivial case study of high Prandtl number convection in a large aspect ratio, shallow, elliptical container with heated sidewalls, for which we also have the results of simulation and experiment. © 2003 Elsevier B.V. All rights reserved.
• Gabitov, I. R., Podivilov, E., Lebedev, V., Kolokolov, I., Indik, R. A., Gabitov, I. R., & Chertkov, M. (2003). Phase Randomization Technique in Optical Fibers with Strong Dispersion Map: Reduction of Intra-channel Interactions. Frontiers in Optics. doi:10.1364/fio.2003.mkk3
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  The statistics of ghost pulses, generated by pseudo-random sequences of optical pulses in optical fiber links with strong dispersion map is analytically and numerically calculated. To reduce bit error due to ghost pulse generation, phase randomization of bit-pulses is proposed and simulated and analyzed.
• Gabitov, I. R., Soneson, J., Khanov, I., Indik, R. A., & Gabitov, I. R. (2003). Non-Gaussianity of amplitude jitter in DM fiber links with DPSK coding. Frontiers in Optics. doi:10.1364/fio.2003.tuh4
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  Amplitude jitter due to nonlinear intra-channel pulse interaction is investigated in dispersion managed optical fiber links with Differential Phase Shift Keying. It is shown that as in the case of amplitude modulation, nonlinear effects lead to errors. The probability distribution function is characterized for the energy of the output pulses.
• Ercolani, N. M., Indik, R., Newell, A. C., & Passot, T. (2000). The geometry of the phase diffusion equation. Journal of Nonlinear Science, 10(2), 223-274.
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  Abstract: The Cross-Newell phase diffusion equation, τ(|k→|)ΘT = -∇ · (B(|k→|) · k→), k→ = ∇ Θ, and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order "self-dual" equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit.
• McInerney, J. G., O'Brien, P., Skovgaard, P., Mullane, M., Houlihan, J., O'Neill, E., Moloney, J. V., & Indik, R. A. (2000). Towards filament-free semiconductor lasers. Proceedings of SPIE - The International Society for Optical Engineering, 3944, I/-.
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  Abstract: We outline physical models and simulations for suppression of self-focusing and filamentation in large aperture semiconductor lasers. The principal technical objective is to generate multi-watt CW or quasi-CW outputs with nearly diffraction limited beams, suitable for long distance free space transmission, focusing to small spots or coupling to single-mode optical fibers. The principal strategies are (a) optimization of facet damage thresholds, (b) reduction of the linewidth enhancement factor which acts as the principal nonlinear optical coefficient, and (c) design of laterally profiled propagation structures in lasers and amplifiers which suppress lateral reflections.
• Ning, C., Moloney, J., Egan, A., & Indik, R. (2000). A first-principles fully space-time resolved model of a semiconductor laser. QUANTUM AND SEMICLASSICAL OPTICS, 9(5), 681-691.
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  We present a semiconductor laser model which incorporates the gain bandwidth and nonlinear gain by using the multi-band microscopic theory of an electron-hole plasma in a semiconductor quantum-well medium. The approach is extremely robust, allowing us to take into account important many-body effects, as well as material and structural parameters of a given laser device. As a specific illustrative example, we resolve for the first time, the full multi-longitudinal mode and transverse filamentation instabilities of a master-oscillator power amplifier (MOPA) device.
• GEDDES, J., LEGA, J., MOLONEY, J., INDIK, R., WRIGHT, E., & FIRTH, W. (1999). PATTERN SELECTION IN PASSIVE AND ACTIVE NONLINEAR-OPTICAL SYSTEMS. CHAOS SOLITONS & FRACTALS, 4(8-9), 1261-1274.
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  The spontaneous emergence of patterns beyond some critical value of an external stress parameter, is illustrated with three examples from nonlinear optics. Counterpropagating beam patterns in an extended optically transparent system pose a challenging nonlinear boundary value problem for which self-focusing hexagons and self-defocusing squares are the preferred planforms. Polarization-induced patterns in a passive ring cavity represents a more manageable problem and rolls turn out to be the preferred planforms. Finally, a single longitudinal mode, wide-aperture Raman laser is shown to select a transverse traveling wave mode, which appears in the far-field as an off-axis emission. Complex order parameter equations are derived for each situation, and are used to illustrate how intrinsic symmetries play a role in pattern selection.
• Moloney, J. V., Indik, R. A., Hader, J., & Koch, S. W. (1999). Modeling semiconductor amplifiers and lasers: From microscopic physics to device simulation. Journal of the Optical Society of America B: Optical Physics, 16(11), 2023-2029.
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  Abstract: We combine the results of full many-body band-structure calculations of the semiconductor optical response and a full space-time-resolved laser propagation model. Two quantum-well structures are chosen, one showing a sharp increase of the line width enhancement factor with density; the other, a clamping of this factor with increasing density. The average far-field broadening of two weakly turbulent broad-area high-power semiconductor lasers is shown to be quite different for the two structures. © 1999 Optical Society of America.
• Bowman, C., Ercolani, N., Indik, R., Newell, A. C., & Passot, T. (1998). Patterns, defects and integrability. Physica D: Nonlinear Phenomena, 123(1-4), 474-492.
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  Abstract: In this paper, recent results on the behavior of roll patterns in a class of problems typified by high Prandtl number convection are presented. A key finding is that the Gaussian curvature of the "crumpled" phase surface, which consists of patches with an almost constant wave number, line defects on which most of the free energy is stored and point defects with nontrivial topologies; condenses onto line and point defects. This property allows considerable mathematical simplification in that the fourth order nonlinear diffusion equation governing stationary states can be effectively reduced to the linear Helmholtz equation. The observed patterns have much is common with the deformation of thin elastic sheets. Copyright © 1998 Published by Elsevier Science B.V.
• Egan, A., Ning, C. Z., Moloney, J. V., Indik, R. A., Wright, M. W., Bessert, D. J., & McInerney, J. G. (1998). Dynamic instabilities in master oscillator power amplifier semiconductor lasers. IEEE Journal of Quantum Electronics, 34(1), 166-170.
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  Abstract: We investigate theoretically the master oscillator power amplifier using a semiconductor laser model that is fully time and space (laterally and longitudinally) resolved. We numerically examine the stability of the device and identify the nature of the different instabilities. These can arise from undamped relaxation oscillations, beating between the longitudinal modes of any of the cavities that comprise the device, or lateral filamentation.
• Kent, A., Skovgaard, P. M., McInerney, J. G., Moloney, J. V., Indik, R. A., & Ning, C. Z. (1998). Optimisation of flared semiconductor lasers and MOPAs. Conference on Lasers and Electro-Optics Europe - Technical Digest, 88-.
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  Abstract: A detailed and realistic theory of the fast dynamics and stability of flared laser oscillators and master oscillator-power amplifier (MOPA) is described. The theory approximates a realistic gain spectrum into a complex Lorentzian susceptibility which depends non-linearly on carrier density and includes saturation effects. As an illustration, the model is used to simulate the dynamics of flared oscillators with high spatial and temporal resolution.
• Moloney, J. V., Egan, A., Ning, C. Z., & Indik, R. A. (1998). Spontaneous spatiotemporal instabilities in current modulated master oscillator power amplifier lasers. IEEE Photonics Technology Letters, 10(9), 1229-1231.
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  Abstract: We provide the first comprehensive theoretical evidence for nonlinear space-time dynamical behavior which seriously degrades the performance of a monolithically integrated master oscillator power amplifier laser under direct current modulation of the master oscillator. A host of instabilities, reflecting dynamical behavior of different sections of the device, are possible. These instabilities are due to the combination of weak feedback from the power amplifier section of the device causing leakage into the master oscillator section, from the buildup of unsaturated carriers along the narrow edges of the expanding flare and differential saturation of the passive regions under the distributed Bragg reflector grating sections.
• Ning, C., Indik, R., & Moloney, J. (1998). Effective Bloch equations for semiconductor lasers and amplifiers. IEEE JOURNAL OF QUANTUM ELECTRONICS, 33(9), 1543-1550.
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  A set of effective Bloch equations is established for semiconductor bulk or quantum-well media, The model includes the nonlinear carrier-density dependence of the gain and refractive index and their respective dispersions (frequency dependences), A comparative study is performed between the full microscopic semiconductor Bloch equations and this effective model for pulse propagation to show the range of validity of the present model. The results show that this model agrees well with the microscopic model provided carrier depletion is the dominant saturation mechanism relative to the plasma heating, The effective Bloch equations provide an accurate and practical model for modeling amplifiers with pulses of duration, greater than a few picoseconds, By capturing the large bandwidth and the carrier density dependence of the gain, it also provides a reliable model for studying the complex spatiotemporal multilongitudinal and transverse mode dynamics of a variety of wide-aperture high-power semiconductor lasers, The model goes beyond the traditional rate equations and is computationally much more efficient to simulate than the full model.
• RU, P., JAKOBSEN, P., MOLONEY, J., & INDIK, R. (1998). GENERALIZED COUPLED-MODE MODEL FOR THE MULTISTRIPE INDEX-GUIDED LASER ARRAYS. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS, 10(3), 507-515.
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  We develop a generalized coupled-mode model for multistripe index-guided laser arrays that includes explicitly the influence of carrier-induced antiguiding, gain guiding, and carrier diffusion in the gain stripe. As an illustration of an application of the model, stability criteria for two-element laser arrays are derived that show that the phase-locked solution is intrinsically unstable. We find that the phase-locked solution can be stabilized at a low external injection-locking power at the suitably chosen injection-locking frequency. We have tested our model in the large carrier-diffusion case and still find good qualitative agreement between the coupled-mode model and a full coupled partial differential equation model.
• Rössler, T., Indik, R. A., Harkness, G. K., Moloney, J. V., & Ning, C. Z. (1998). Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surface-emitting lasers. Physical Review A - Atomic, Molecular, and Optical Physics, 58(4), 3279-3292.
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  Abstract: We present a microscopically based vertical-cavity surface-emitting laser (VCSEL) model that treats plasma and lattice heating self-consistently and includes gain dispersion in a fashion facilitating the incorporation of many-body effects. This model is used to investigate the interplay of thermal effects and transverse mode behavior observed in recent experiments with large-aperture selectively oxidized VCSELs. We confirm that the highly divergent single-mode emission seen experimentally at low ambient temperatures may be caused by a redshift of the cavity resonance frequency relative to the quantum-well gain peak. Moreover, due to the dependence of the gain spectrum on temperature our model qualitatively reproduces the measured increase of the dominant spatial scale of the low-temperature steady-state field patterns with pumping. Finally, we demonstrate that spatial hole burning plays a significant role at larger ambient temperatures and explains the decrease of the spatial wavelength with pumping, in agreement with the experiments.
• Skovgaard, P. M., McInerney, J. G., Moloney, J. V., Indik, R. A., Ning, C. Z., & Egan, A. (1998). Suppression of transverse and longitudinal instabilities in high-power MFA-MOPA semiconductor devices. Conference on Lasers and Electro-Optics Europe - Technical Digest, 244-.
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  Abstract: Detailed theoretical analyses of monolithically integrated flared amplifier master oscillator power amplifiers (MFA-MOPAs) are presented. The field amplitudes and the carrier density are calculated using effective Block equations. The obtained features of the MFA-MOPAs are presented and discussed.
• JAKOBSEN, P., INDIK, R., MOLONEY, J., NEWELL, A., WINFUL, H., & RAMAN, L. (1997). DIODE-LASER ARRAY MODES - DISCRETE AND CONTINUOUS MODELS AND THEIR STABILITY. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS, 8(8), 1674-1680.
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  A new class of mode profiles appropriate to large one-dimensional evanescently coupled laser diode arrays can be conveniently approximated by solutions to a Riccatti equation derived from a continuous model of the discrete array. Solutions of the latter model are shown to agree with the continuous solutions in the large-N limit. The continuous model has the advantage that new types of discrete array modes can readily be identified. Stability analyses of the discrete array modes as a function of increasing N confirm that these laser systems are intrinsically unstable in the free-running mode.
• MOLONEY, J., ADACHIHARA, H., INDIK, R., LIZARRAGA, C., NORTHCUTT, R., MCLAUGHLIN, D., & NEWELL, A. (1997). MODULATIONAL-INDUCED OPTICAL-PATTERN FORMATION IN A PASSIVE OPTICAL-FEEDBACK SYSTEM. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS, 7(6), 1039-1044.
• Moloney, J. V., Indik, R. A., & Ning, C. Z. (1997). Full space-time simulation for high-brightness semiconductor lasers. IEEE Photonics Technology Letters, 9(6), 731-733.
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  Abstract: A semiconductor laser model is presented, which resolves the full time, longitudinal and lateral space dependences. The model is applied to an investigation of the dynamical stability of an integrated master-oscillator power-amplifier (MOPA) device. The model captures the full gain and refractive index bandwidth as a function of total carrier density. Our simulation confirms, for the first time, some recent experimental observations of high frequency whole beam oscillations and experimental reports that complex transverse filamentation occurs at high power amplifier currents.
• Moloney, J. V., Indik, R. A., Ning, C. Z., & Egan, A. (1997). Space-time simulation of high-brightness semiconductor lasers. Proceedings of SPIE - The International Society for Optical Engineering, 2994, 562-571.
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  Abstract: A full scale simulation model, that resolves the spatio- temporal behavior of competing longitudinal mode and transverse filamentation instabilities in a wide variety of high brightness edge emitter geometries, is presented. The model is highly modular and is built on a first principles microscopic physics basis. The nonlinear optical response function of the semiconductor, computed for specific QW structures, covers the low-density absorption to high density gain saturation regimes. As an illustration of its robustness as a laser design tool, the model is applied to a monolithically integrated flared amplifier master oscillator power amplifier semiconductor laser.
• Moloney, J., Indik, R., & Ning, C. (1997). Full space-time simulation for high-brightness semiconductor lasers. IEEE PHOTONICS TECHNOLOGY LETTERS, 9(6), 731-733.
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  A semiconductor laser model is presented, which resolves the full time, longitudinal and lateral space dependences, The model is applied to an investigation of the dynamical stability of an integrated master-oscillator power-amplifier (MOPA) device. The model captures the full gain and refractive index bandwidth as a function of total carrier density, Our simulation confirms, for the first time, some recent experimental observations of high frequency whole beam oscillations and experimental reports that complex transverse filamentation occurs at high power amplifier currents.
• NING, C., INDIK, R., & MOLONEY, J. (1997). SELF-CONSISTENT APPROACH TO THERMAL EFFECTS IN VERTICAL-CAVITY SURFACE-EMITTING LASERS. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS, 12(10), 1993-2004.
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  A self-consistent theory for semiconductor lasers, in which plasma and lattice temperatures are treated as two independent variables, is presented. This theory consists of a set of coupled equations for the total carrier density, field amplitude, and plasma and lattice temperatures with the coupling that is due to phonon-carrier scattering and to the band gap's dependence on lattice temperature. The self-consistent theory is then employed to study thermal effects in vertical-cavity surface-emitting lasers. We first investigate the plasma heating by solving the stationary (cw) solution of the set of equations with a fixed lattice temperature. The solution is studied systematically with respect to different parameters for both hulk and quantum-well media. Significant plasma-heating effects are found. These include the carrier-density dependence on pumping, decrease of input-output efficiency, dependence of the cw frequency shift on pumping, and a pronounced Pauli-blocking effect that is due to plasma heating. Furthermore, we solve the whole set of equations, including that for lattice temperature. We show that the output power is strongly saturated or switched off with an increase of pumping. Details of the saturation depend on the position of the cavity frequency in the gain spectrum and on the heat transfer rate from the lattice to the ambient. (C) 1995 Optical Society of America
• Ning, C. -., Chow, W. W., Bossert, D. J., Indik, R. A., & Moloney, J. V. (1997). Influences of unconfined states on the optical properties of quantum-well structures. IEEE Journal on Selected Topics in Quantum Electronics, 3(2), 129-135.
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  Abstract: The population of the unconfined states, with energies above the band edge of the barrier layers, can be significant in some regions of the active volume in high power lasers and amplifiers. This paper analyzes the influences of these states on optical properties, such as gain, refractive index, differential gain, and linewidth enhancement factor, for different quantum-well (QW) structures. Our results show that at high excitation levels, the unconfined band contributions to the real part of the optical susceptibility can be significant, especially in structures with weak quantum confinement potentials. This is in agreement with recent measurements of peak gain and carrier-induced refractive index change versus carrier density, for InGaAs-GaAs QW laser structures.
• Ning, C. Z., Indik, R. A., & Moloney, J. V. (1997). Effective bloch equations for semiconductor lasers and amplifiers. IEEE Journal of Quantum Electronics, 33(9), 1543-1550.
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  Abstract: A set of effective Bloch equations is established for semiconductor bulk or quantum-well media. The model includes the nonlinear carrier-density dependence of the gain and refractive index and their respective dispersions (frequency dependences). A comparative study is performed between the full microscopic semiconductor Bloch equations and this effective model for pulse propagation to show the range of validity of the present model. The results show that this model agrees well with the microscopic model provided carrier depletion is the dominant saturation mechanism relative to the plasma heating. The effective Bloch equations provide an accurate and practical model for modeling amplifiers with pulses of duration greater than a few picoseconds. By capturing the large bandwidth and the carrier density dependence of the gain, it also provides a reliable model for studying the complex spatiotemporal multilongitudinal and transverse mode dynamics of a variety of wide-aperture high-power semiconductor lasers. The model goes beyond the traditional rate equations and is computationally much more efficient to simulate than the full model.
• Ning, C. Z., Indik, R. A., Moloney, J. V., Chow, W. W., Girndt, A., Koch, S. W., & Binder, R. H. (1997). Incorporating many-body effects into modeling of semiconductor lasers and amplifiers. Proceedings of SPIE - The International Society for Optical Engineering, 2994, 666-677.
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  Abstract: Major many-body effects that are important for semiconductor laser modeling are summarized. We adopt a bottom-up approach to incorporate these many-body effects into a model for semiconductor lasers and amplifiers. The optical susceptibility function computed from the semiconductor Bloch equations (SBEs) is approximated by a single Lorentzian, or a superposition of a few Lorentzians in the frequency domain. Our approach leads to a set of effective Bloch equations. We compare this approach with the full microscopic SBEs for the case of pulse propagation. Good agreement between the two is obtained for pulse widths longer than tens of picoseconds.
• Ning, C. Z., Moloney, J. V., Egan, A., & Indik, R. A. (1997). A first-principles fully space-time resolved model of a semiconductor laser. Journal of Optics B: Quantum and Semiclassical Optics, 9(5), 681-691.
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  Abstract: We present a semiconductor laser model which incorporates the gain bandwidth and nonlinear gain by using the multi-band microscopic theory of an electron-hole plasma in a semiconductor quantum-well medium. The approach is extremely robust, allowing us to take into account important many-body effects, as well as material and structural parameters of a given laser device. As a specific illustrative example, we resolve for the first time, the full multi-longitudinal mode and transverse filamentation instabilities of a master-oscillator power amplifier (MOPA) device.
• Rossler, T., Indik, R., Harkness, G., Moloney, J., & Ning, C. (1997). Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surface-emitting lasers. PHYSICAL REVIEW A, 58(4), 3279-3292.
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  We present a microscopically based vertical-cavity surface-emitting laser (VCSEL) model that treats plasma and lattice heating self-consistently and includes gain dispersion in a fashion facilitating the incorporation of many-body effects. This model is used to investigate the interplay of thermal effects and transverse mode behavior observed in recent experiments with large-aperture selectively oxidized VCSELs. We confirm that the highly divergent single-mode emission seen experimentally at low ambient temperatures may be caused by a redshift of the cavity resonance frequency relative to the quantum-well gain peak. Moreover, due to the dependence of the gain spectrum on temperature our model qualitatively reproduces the measured increase of the dominant spatial scale of the low-temperature steady-state field patterns with pumping. Finally, we demonstrate that spatial hole burning plays a significant role at larger ambient temperatures and explains the decrease of the spatial wavelength with pumping, in agreement with the experiments. [S1050-2947(98)02410-X].
• Skovgaard, P. M., McInerney, J. G., Moloney, J. V., Indik, R. A., & Ning, C. Z. (1997). Enhanced stability of MFA-MOPA semiconductor lasers using a nonlinear, trumpet-shaped flare. IEEE Photonics Technology Letters, 9(9), 1220-1222.
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  Abstract: Monolithically integrated flared amplifier master oscillator power amplifier (MFA-MOPA) lasers are studied using a high-resolution computational model that resolves time as well as longitudinal and transverse space dependences and includes Lorentzian gain and dispersion dynamics. By altering the linear flare of the power amplifier into a nonlinear, trumpet-shaped flare to overlap the gain region to the expanding field, the instability threshold of the MOPA is increased by ∼2 for single-longitudinal, single-transverse mode operation and ∼3 for single-transverse mode operation. This enables the MOPA to maintain a stable, near-diffraction limited output beam for higher currents before the onset of transverse instabilities. Thus the trumpet-flared MOPA emits an output beam of significantly higher power and brightness. This increased stability is due to a large reduction in feedback from the output facet of the trumpet shaped MFA-MOPA.
• Skovgaard, P. M., McInerney, J. G., Moloney, J. V., Indik, R. A., & Ning, C. Z. (1997). Modeling of master oscillator-power amplifier (MOPA) semiconductor lasers. Proceedings of SPIE - The International Society for Optical Engineering, 2994, 801-809.
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  Abstract: Monolithically integrated flared amplifier master oscillator power amplifier (MFA-MOPA) semiconductor lasers are studied theoretically using a high resolution computational model which resolved times and longitudinal and transverse space dependencies and includes Lorentzian gain and dispersion spectra. The simulations show that, by altering the linear flare of the power amplifier into a nonlinear, trumpet- shaped flare, the dynamic stability range of the MOPA is increased by a factor of 3. This enables the MOPA to maintain a stable, nearly diffraction limited output beam for higher currents before the onset of transverse instabilities, large beam divergence and facet damage due to filamentation. Thus the MOPA will be able to emit an output beam of significantly higher power and brightness.
• Hughes, S., Knorr, A., Koch, S. W., Binder, R., Indik, R., & Moloney, J. V. (1996). The influence of electron-hole-scattering on the gain spectra of highly excited semiconductors. Solid State Communications, 100(8), 555-559.
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  Abstract: A microscopic treatment of the influence of electron-hole-scattering on the optical dephasing and the lineshape in semiconductor gain media is presented. The calculations incorporate non-diagonal- and diagonal-scattering contributions to the optical polarisation. The strong compensation between both contributions leads to gain spectra, which are significantly modified in comparison to those obtained using a pure dephasing approximation. Copyright © 1996 Published by Elsevier Science Ltd.
• Hughes, S., Knorr, A., Koch, S., Binder, R., Indik, R., & Moloney, J. (1996). The influence of electron-hole-scattering on the gain spectra of highly excited semiconductors. SOLID STATE COMMUNICATIONS, 100(8), 555-559.
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  A microscopic treatment of the influence of electron-hole-scattering on the optical dephasing and the lineshape in semiconductor gain media is presented. The calculations incorporate non-diagonal- and diagonal-scattering contributions to the optical polarisation. The strong compensation between both contributions leads to gain spectra, which are significantly modified in comparison to those obtained using a pure dephasing approximation. Copyright (C) 1996 Published by Elsevier Science Ltd
• Indik, R. A., Binder, R., Mlejnek, M., Moloney, J. V., Hughes, S., Knorr, A., & Koch, S. W. (1996). Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers. Physical Review A - Atomic, Molecular, and Optical Physics, 53(5), 3614-3620.
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  Abstract: Based on a microscopic theory of a two-band semiconductor light amplifier, we show that plasma heating, cooling, and ultrafast memory effects all act in concert to produce strong distortion of subpicosecond pulses propagating in semiconductor amplifiers. Plasma heating, spectral hole burning, and carrier density depletion are responsible for saturation of the gain seen by a propagating intense femtosecond pulse in the amplifier. Plasma cooling replenishes the carrier population on the trailing edge of the pulse, leading to pulse broadening as a consequence of gain regeneration. The inclusion of memory effects in the description of dephasing processes goes beyond the usual Markov assumption of constant dephasing rates; it significantly affects the dynamical pulse reshaping processes.
• Newell, A. C., Passot, T., Bowman, C., Ercolani, N., & Indik, R. (1996). Defects are weak and self-dual solutions of the Cross-Newell phase diffusion equation for natural patterns. Physica D: Nonlinear Phenomena, 97(1-3), 185-205.
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  Abstract: We show that defects are weak solutions of the phase diffusion equation for the macroscopic order parameter for natural patterns. Further, by exploring a new class of nontrivial solutions for which the graph of the phase function has vanishing Gaussian curvature (in 3D, all sectional curvatures) except at points, we are able to derive explicit expressions which capture the anatomies of point and line (and surface) defects in two and three dimensional patterns, together with their topological characters and energetic constraints. Copyright © 1996 Elsevier Science B.V. All rights reserved.
• Newell, A., Passot, T., Bowman, C., Ercolani, N., & Indik, R. (1996). Defects are weak and self-dual solutions of the Cross-Newell phase diffusion equation for natural patterns. PHYSICA D, 97(1-3), 185-205.
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  We show that defects are weak solutions of the phase diffusion equation for the macroscopic order parameter for natural patterns. Further, by exploring a new class of nontrivial solutions for which the graph of the phase function has vanishing Gaussian curvature (in 3D, all sectional curvatures) except at points, we are able to derive explicit expressions which capture the anatomies of point and line (and surface) defects in two and three dimensional patterns, together with their topological characters and energetic constraints.
• Skovgaard, P., McInerney, J., Moloney, J., Indik, R., & Ning, C. (1996). Enhanced stability of MFA-MOPA semiconductor lasers using a nonlinear, trumpet-shaped flare. IEEE PHOTONICS TECHNOLOGY LETTERS, 9(9), 1220-1222.
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  Monolithically integrated flared amplifier master oscillator power amplifier (MFA-MOPA) lasers are studied using a high-resolution computational model that resolves time as well as longitudinal and transverse space dependences and includes Lorentzian gain and dispersion dynamics. By altering the linear hare of the power amplifier into a nonlinear, trumpet-shaped flare to overlap the gain region to the expanding field, the instability threshold of the MOPA is increased by similar to 2 for single-longitudinal, single-transverse mode operation and similar to 3 for single-transverse mode operation. This enables the MOPA to maintain a stable, near-diffraction limited output beam for higher currents before the onset of transverse instabilities. Thus the trumpet-flared MOPA emits an output beam of significantly higher power and brightness. This increased stability is due to a large reduction in feedback from the output facet of the trumpet shaped MFA-MOPA.
• Binder, R., Moloney, J. V., Koch, S. W., Knorr, A., Indik, R. A., & Binder, R. (1995). Self-induced channeling of subpicosecond optical pulses in broad-area bulk semiconductor amplifiers.. Optics letters, 20(22), 2315. doi:10.1364/ol.20.002315
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  Propagation of a 150-fs optical pulse in a broad-area semiconductor amplifier with many-body effects, plasma heating, cooling, and memory effects included is predicted to produce a self-induced transient waveguide and cause strong transverse pulse distortion.
• Chow, W. W., Indik, R., Knorr, A., Koch, S. W., & Moloney, J. V. (1995). Time-resolved nondegenerate four-wave mixing in a semiconductor amplifier. Physical Review A, 52(3), 2479-2482.
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  Abstract: The dynamics of copropagating femtosecond pulses with different carrier frequencies is modeled for a semiconductor amplifier. The propagation induced four-wave mixing signal is studied as a function of the pulse intensity and the spectral detuning between the pulses. © 1995 The American Physical Society.
• Gabitov, I., Indik, R., Litchinitser, N., Maimistov, A., Shalaev, V., & Soneson, J. (1995). Double-resonant optical materials with embedded metal nanostructures. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS, 23(3), 535-542.
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  We derive equations modeling the resonant interaction of electric and magnetic components of light fields with metal nanostructures. This paired resonance was recently shown to produce negative refractive index. The model equations are a generalization of the well-known Maxwell-Lorentz model. We demonstrate that in the case of nonlinear polarization and linear magnetization, these equations are equivalent to a system of equations describing the resonant interaction of light with plasmonic oscillations in metal nanospheres. A family of solitary wave solutions is found that is similar to pulses associated with self-induced transparency in the framework of the Maxwell-Bloch model. The evolution of incident optical pulses is studied numerically, as are the collision dynamics of the solitary waves. These simulations reveal that the collision dynamics vary from near perfectly elastic to highly radiative, depending on the relative phase of the initial pulses. (c) 2006 Optical Society of America.
• Indik, R. A., Moloney, J. V., Binder, R. H., Chow, W. W., Knorr, A., & Koch, S. W. (1995). Many-body effects in the propagation of short pulses in a semiconductor amplifier. Proceedings of SPIE - The International Society for Optical Engineering, 2399, 650-659.
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  Abstract: The semiconductor Maxwell-Bloch equations provide a model that is grounded in the fundamental physics of semiconductors which include a variety of many body effects. Many of these effects are particularly noticeable when the semiconductor is probed with ultrashort pulses. We present computational results describing the computed behavior of model equations which describe the propagation of femto-second pulses in bulk GaAs. It is shown how the inclusion of additional physics modifies the predictions of the model. Among the effects that are discussed are plasma heating, plasma cooling, self-focusing, and memory effects.
• Newell, A. C., Ercolani, N. M., Passot, T., Newell, A. C., Indik, R. A., & Ercolani, N. M. (1995). Elementary and Composite Defects of Striped Patterns. Journal De Physique Ii, 5(12), 1863-1882. doi:10.1051/jp2:1995217
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  Labyrinthic patterns are observed both in systems where the uniform states are metastable, as a result of a front instability, and in systems displaying a cellular instability, when the band of excited Fourier modes is wide enough to support resonant interactions between modes lying on different shells. We show that the phase formalism is a suitable description for low-density labyrinthic patterns with a relatively long range correlation and is capable of describing both its smooth and singular structures. The point defects of roll patterns, the concave and convex disclinations, and the line singularities or phase grain boundaries across which the wavevector makes an order one transition, are found to be singular and weak solutions of the Cross-Newell phase diffusion equation, which take account of their energetics as well as their topologies.
• Ning, C. Z., Indik, R. A., Moloney, J. V., & Koch, S. W. (1995). Effects of plasma and lattice heating in VCSELs. Proceedings of SPIE - The International Society for Optical Engineering, 2399, 617-628.
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  Abstract: We report thermal effects revealed by a self-consistent treatment of plasma and lattice heating in vertical cavity surface-emitting lasers (VCSELs). The basic idea of our treatment is to couple the equations for carrier density and field amplitude in the conventional laser theory with those for two additional variables, the plasma and lattice temperatures. The CW operation of the VCSELs is investigated both for a fixed and for a self-consistently determined lattice temperature. In the first case plasma heating results in an increase of carrier density with pumping and thus in a pumping dependent frequency shift. In the latter case, both plasma and lattice heating induce a thermal switch-off of the laser as the pumping is increased. Furthermore, depending on the initial alignment of the cavity frequency and the ambient temperature of the device, heating can introduce a discontinuous threshold, exhibiting a bistability between lasing and nonlasing states. While some of our theoretical predictions are in qualitative agreement with known experiments, others await experimental verification.
• Ning, C. Z., Moloney, J. V., & Indik, R. A. (1995). Self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers. Journal of The Optical Society of America B-optical Physics, 12(10), 1993-2004. doi:10.1364/josab.12.001993
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  A self-consistent theory for semiconductor lasers, in which plasma and lattice temperatures are treated as two independent variables, is presented. This theory consists of a set of coupled equations for the total carrier density, field amplitude, and plasma and lattice temperatures with the coupling that is due to phonon-carrier scattering and to the band gap’s dependence on lattice temperature. The self-consistent theory is then employed to study thermal effects in vertical-cavity surface-emitting lasers. We first investigate the plasma heating by solving the stationary (cw) solution of the set of equations with a fixed lattice temperature. The solution is studied systematically with respect to different parameters for both bulk and quantum-well media. Significant plasma-heating effects are found. These include the carrier-density dependence on pumping, decrease of input–output efficiency, dependence of the cw frequency shift on pumping, and a pronounced Pauli-blocking effect that is due to plasma heating. Furthermore, we solve the whole set of equations, including that for lattice temperature. We show that the output power is strongly saturated or switched off with an increase of pumping. Details of the saturation depend on the position of the cavity frequency in the gain spectrum and on the heat transfer rate from the lattice to the ambient.
• Ercolani, N., Indik, R., Newell, A., & Passot, T. (1994). The geometry of the phase diffusion equation. JOURNAL OF NONLINEAR SCIENCE, 10(2), 223-274.
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  The Cross-Newell phase diffusion equation, t(\(k) over right arrow\)Theta(T) = -del . (B(\(k) over right arrow\) . (k) over right arrow), (k) over right arrow = del Theta, and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order "self-dual" equation.
• Geddes, J. B., Indik, R. A., Moloney, J. V., & Firth, W. J. (1994). Hexagons and squares in a passive nonlinear optical system. Physical Review A - Atomic, Molecular, and Optical Physics, 50(4), 3471-3485.
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  Abstract: In this work, substantial extensions of the studies on pattern formation were reported, particularly in the regime of self-defocusing Kerr media. It was shown that by nonlinear perturbation analysis, two very distinct pattern-forming modes coexisted in this system. One was a fairly typical hexagon-forming mode which was dominant in self-focusing media. The other was a roll-pattern mode with the rolls found to be unstable. Instead square patterns emerged and appeared to be dominant for self-defocusing media. By varying the grating parameter G one mode can be favored over the other. However, the less-favored more manifested its effect quite close to the instability threshold, mediating defect formation and a Hopf bifurcation.
• Geddes, J. B., Lega, J., Moloney, J. V., Indik, R. A., Wright, E. M., & Firth, W. J. (1994). Pattern selection in passive and active nonlinear optical systems. Chaos, Solitons and Fractals, 4(8-9), 1261-1274.
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  Abstract: The spontaneous emergence of patterns beyond some critical value of an external stress parameter, is illustrated with three examples from nonlinear optics. Counterpropagating beam patterns in an extended optically transparent system pose a challenging nonlinear boundary value problem for which self-focusing hexagons and self-defocusing squares are the preferred planforms. Polarization-induced patterns in a passive ring cavity represents a more manageable problem and rolls turn out to be the preferred planforms. Finally, a single longitudinal mode, wide-aperture Raman laser is shown to select a transverse traveling wave mode, which appears in the far-field as an off-axis emission. Complex order parameter equations are derived for each situation, and are used to illustrate how intrinsic symmetries play a role in pattern selection. © 1994.
• INDIK, R., KNORR, A., BINDER, R., MOLONEY, J., & KOCH, S. (1994). PROPAGATION-INDUCED ADIABATIC FOLLOWING IN A SEMICONDUCTOR AMPLIFIER. OPTICS LETTERS, 19(13), 966-968.
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  A propagation-induced transition from linear amplification into an adiabatic following regime is predicted for femtosecond pulses in an inverted semiconductor medium. This process is accompanied by considerable pulse shortening.
• Indik, J. H., Indik, R. A., & Cetas, T. C. (1994). Fast and efficient computer modeling of ferromagnetic seed arrays of arbitrary orientation for hyperthermia treatment planning. International Journal of Radiation Oncology, Biology, Physics, 30(3), 653-662.
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  PMID: 7928497;Abstract: Purpose: Effective hyperthermia treatment planning requires an ability to predict temperatures quickly and accurately from an arbitrary distribution of power. Our purpose was to design such a fast executing computer code, MGARRAY, to compute steady-state temperatures from ferromagnetic seed heating, allowing seeds to have arbitrary orientations and to be curved to permit more realistic modeling of clinical situations. We further required flexibility for the tissue domain, allowing inhomogeneity with respect to thermal conductivity and blood perfusion, as well as an arbitrary shaped boundary. Methods and Materials: MGARRAY uses multigrid methods and a finite volume discretization to solve the Pennes bioheat transfer equation in three dimensions. We used MGARRAY to compare temperature distributions that result from an array of straight, parallel seeds and from an array of seeds that were curved and tilted randomly by 13°. Results: On a personal workstation the Central Processing Unit (CPU) time of MGARRAY was under 4 min. We found that the median temperature in a predetermined target volume was ∼0.8°C higher in the straight array than in the curved array. At specific locations within the target volume temperature differed by ∼0.5-0.9°C, but could differ by up to several degrees, depending on proximity to a seed and the level of blood perfusion. Conclusion: These differences can impact on retrospective analyses whereby temperatures at a few locations are used to infer the overall temperature field and blood perfusion levels. The flexibility and computational speed of MGARRAY could potentially lead to a substantial improvement in both retrospective and prospective hyperthermia treatment planning. © 1994.
• Indik, R. A., & Indik, J. H. (1994). A new computer method to quickly and accurately compute steady-state temperatures from ferromagnetic seed heating. Medical Physics, 21(7), 1135-1144.
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  PMID: 7968846;Abstract: A new, very fast, yet accurate program, MGSEED, has been developed that computes steady-state temperatures from the three-dimensional bioheat transfer equation due to heating by a ferromagnetic seed. Seeds have a self- regulating power absorption characteristic such that their temperatures remain within a few degrees of their Curie transition point. The code is also very flexible, being able to model a seed of any orientation embedded in a tissue domain that can be inhomogeneous with respect to blood perfusion or thermal conductivity. MGSEED uses multigrid (or multilevel) programming techniques as well as finite volume discretization that exploits knowledge of the approximate shape of the temperature solution very near to a seed. These techniques allow the code to sample the seed very coarsely, requiring only one or two nodes to cross the seed. With these coarse samplings MGSEED calculated very accurate temperatures in under 3 min of CPU time on a Sun Sparcstation 2. The accuracy of MGSEED is demonstrated at different levels of perfusion by comparing its solution in a perpendicular plane that bisects the seed with the known analytical solution. The speed of MGSEED is compared to other methods of solution and it is found that MGSEED performs 14 times faster than successive over relaxation and conjugate gradient methods, and 2.5 times faster than a preconditioned (modified block incomplete Cholesky) conjugate gradient method. It is concluded that the techniques for discretization and solution incorporated into MGSEED can greatly improve the flexibility and speed of hyperthermia treatment planning, which could ultimately lead to an increased level of control over treatment outcome.
• Indik, R. A., & Newell, A. C. (1994). Computing for pattern forming systems in nonlinear optics. Mathematics and Computers in Simulation, 37(4-5), 385-403.
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  Abstract: Methods used to compute solutions of nonlinear PDEs arising in nonlinear optics are described. In addition a spectral method is presented for integrating dispersion dominated PDEs that works particularly well in integrating systems where a modulational ansatz is appropriate. © 1994.
• Indik, R. A., Indik, J. H., Indik, R. A., Indik, J. H., Indik, R. A., & Indik, J. H. (1994). A New Computer Method to Quickly and Accurately Compute Steady State Temperatures from Ferromagnetic Seed Heating. Medical Physics, 1135-1144.
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  Indik, R.A. and Indik, J.H. A New Computer Method to Quickly and Accurately Compute Steady State Temperatures from Ferromagnetic Seed Heating. Medical Physics, 1994;21, 1135-1144
• Indik, R. A., Moloney, J. V., & Binder, R. (1994). Femtosecond pulse compression and adiabatic following in semiconductor amplifiers. IEEE Nonlinear Optics: Materials, Fundamentals and Applications - Conference Proceedings, 163-165.
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  Abstract: Reported here we studies of a plane wave femtosecond pulse propagating in a semiconductor amplifier over gain lengths. Preliminary results on the extension of the study include transverse spatial effects necessary to model realistic waveguiding and broad area amplifier structures will also be discussed. For a plane wave pulse, it is concluded that after initial linear amplification and saturation, the pulse undergoes strong intensity and spectral deformation before settling into a strongly compressed intense superluminal pulse undergoing adiabatic following (AF) with off-resonant noninverted states, well above the chemical potential.
• Indik, R. A., Moloney, J. V., Binder, R., Knorr, A., & Koch, S. W. (1994). Femtosecond pulse compression and adiabatic following in semiconductor amplifiers. European Quantum Electronics Conference - Technical Digest.
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  Abstract: We report on our studies of a plane-wave femtosecond pulse propagating in a semiconductor amplifier over long gain lengths. The overall conclusion for a plane-wave pulse is that, after initial linear amplification and saturation, the pulse undergoes strong intensity and spectral deformation before settling into a strongly compressed intense superluminal pulse undergoing adiabatic following (AF) with off-resonant noninverted states, well above the chemical potential.
• Indik, R., Knorr, A., Binder, R., Moloney, J. V., & Koch, S. W. (1994). Propagation-induced adiabatic following in a semiconductor amplifier. Optics Letters, 19(13), 966-968.
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  Abstract: This paper presents an analytical investigation of the nonlinear interactions between the pulse and different carrier states in the amplifier medium and their effect on the pulse propagation process. For an inverted semiconductor medium, it is proposed that for femtosecond pulses, the linear amplification is converted into an adiabatic following regime due to the propagation. Substantial pulse shortening results due to this process.
• Moloney, J., Indik, R., Hader, J., & Koch, S. (1994). Modeling semiconductor amplifiers and lasers: from microscopic physics to device simulation. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS, 16(11), 2023-2029.
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  We combine the results of full many-body band-structure calculations of the semiconductor optical response and a full space-time-resolved laser propagation model. Two quantum-well structures are chosen, one showing a sharp increase of the linewidth enhancement factor with density; the other, a clamping of this factor with increasing density. The average far-field broadening of two weakly turbulent broad-area high-power semiconductor lasers is shown to be quite different for the two structures. (C) 1999 Optical Society of America [S0740-3224(99)00211-8].
• Ru, P., Moloney, J. V., & Indik, R. (1994). Mean-field approximation in semiconductor-laser modeling. Physical Review A - Atomic, Molecular, and Optical Physics, 50(1), 831-838.
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  Abstract: A mean-field model from the full counterpropagating wave equations describing a general Fabry-Perot laser structure was derived. The model was validated through straightforward comparisons with more rigorous counterpropagating field code including both transverse and axial variations of the significant physical quantities within the laser structure. The model was confined to a single-longitudinal-mode laser which should furnish an adequated description of the semiconductor laser dynamics and steady-state behavior over an extensive operating range. However, both spatial and spectral hole-burning effects were predicted to induce multi-longitudinal-mode oscillation at higher drive currents. It was also noted that the introduction of transverse effects in rate-equation approximations of laser dynamics can present spurious high-transverse-wave-number instabilities.
• ADACHIHARA, H., HESS, O., INDIK, R., & MOLONEY, J. (1993). SEMICONDUCTOR-LASER ARRAY DYNAMICS - NUMERICAL SIMULATIONS ON MULTISTRIPE INDEX-GUIDED LASERS. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS, 10(3), 496-506.
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  The space-time dynamical behavior of multistripe index-guided semiconductor laser arrays is studied by using an extension of the usual phenomenological laser model to include transverse diffraction of the counterpropagating optical fields and transverse diffusion of the excited carriers. Our results confirm that evanescently coupled multistripe lasers are a fascinating manifestation of spatiotemporal complexity in spatially extended nonlinear systems. Stabilization of the laser output can be achieved by injection locking the array with a weak external injected signal, and we show that the stability of the externally driven array depends on the transverse profile of the injected signal. A numerical algorithm is presented that takes advantage of high-performance parallel computing architectures to solve the coupled partial differential equations describing the light-matter interaction in the laser structure. Both the model and the numerical algorithm are sufficiently flexible and modular to support arbitrary laser geometries and to allow for inclusion of important many-body semiconductor effects in future studies.
• Adachihara, H., Hess, O., Indik, R., & Moloney, J. V. (1993). Semiconductor laser array dynamics: numerical simulations on multistripe index-guided lasers. Journal of the Optical Society of America B: Optical Physics, 10(3), 496-506.
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  Abstract: Time dependent, fully transverse-dimensional partial differential equations are numerically integrated to study the space-time dynamics of the laser array. The computational model provides a basis for reliable coupled mode or mean field theories for semiconductor arrays and various broad area lasers.
• GEDDES, J., INDIK, R., MOLONEY, J., & FIRTH, W. (1993). HEXAGONS AND SQUARES IN A PASSIVE NONLINEAR-OPTICAL SYSTEM. PHYSICAL REVIEW A, 50(4), 3471-3485.
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  Pattern formation is analyzed and simulated in a nonlinear optical system involving all three space dimensions as well as time in an essential way. This system, counterpropagation in a Kerr medium, is shown to lose stability, for sufficient pump intensity, to a nonuniform spatial pattern. We observe hexagonal patterns in a self-focusing medium, and squares in a self-defocusing one, in good agreement with analysis based on symmetry and asymptotic expansions.
• Newell, A. C., Chow, K. W., & Indik, R. A. (1993). Do resonantly forced internal solitary waves protect the fuel of hurricanes?. Physical Review Letters, 71(12), 1951-1954.
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  Abstract: Assuming that air-sea interaction is the primary energy source of the hurricane, we suggest that a new class of resonantly forced internal solitary waves protects the fuel of the storm by preventing or reducing entrainment.
• Ru, P., Jakobsen, P. K., Moloney, J. V., & Indik, R. A. (1993). Generalized coupled-mode model for the multistripe index-guided laser arrays. Journal of the Optical Society of America B: Optical Physics, 10(3), 507-515.
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  Abstract: A generalized coupled mode model for weakly and strongly index guided laser arrays has been derived. Exact numerical solution from a partial differential equation model to construct the coupled mode basis functions is used for the present model. Large carrier diffusion case is used to test the model. Good qualitative agreement is found.
• Ru, P., Moloney, J. V., Indik, R., Koch, S. W., & Chow, W. W. (1993). Microscopic modelling of bulk and quantum-well GaAs-based semiconductor lasers. Optical and Quantum Electronics, 25(10), 675-693.
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  Abstract: The dynamical behaviour of Fabry-Perot type semiconductor lasers is modelled, including the relevant many-body Coulomb effects of the excited carriers. Conditions are given under which a parametrization of the full model is possible, allowing simple analytic relations for local gain, refractive index and linewidth enhancement factor. The parameters of the simplified model are uniquely determined by the microscopic theory and have to be optimized for the respective operating conditions. The theory is evaluated for bulk and quantum-well GaAs active material and a variety of laser structures, including strongly and weakly index-guided structures, as well as purely guided single-and twin-stripe lasers. © 1993 Chapman & Hall.
• CHANG, R., FIRTH, W., INDIK, R., MOLONEY, J., & WRIGHT, E. (1992). 3-DIMENSIONAL SIMULATIONS OF DEGENERATE COUNTERPROPAGATING BEAM INSTABILITIES IN A NONLINEAR MEDIUM. OPTICS COMMUNICATIONS, 88(2-3), 167-172.
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  We report our findings from a detailed series of three-dimensional simulations of degenerate counterpropagating beam instabilities in a nonlinear Kerr medium. These simulations show the formation of far field intensity patterns with both radial and hexagonal symmetries in agreement with recent experiments. Moreover, we suggest that the ring patterns which are characteristic of degenerate conical emission are transient phenomena which are always replaced by hexagons for long enough times, at least close to the plane-wave threshold. For misaligned input beams we observe the formation of a stable point defect, a penta-hepta pair, in the hexagonal pattern.
• Chang, R., Firth, W. J., Indik, R., Moloney, J. V., & Wright, E. M. (1992). Three-dimensional simulations of degenerate counterpropagating beam instabilities in a nonlinear medium. Optics Communications, 88(2-3), 167-172.
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  Abstract: We report our findings from a detailed series of three-dimensional simulations of degenerate counterpropagating beam instabilities in a nonlinear Kerr medium. These simulations show the formation of far field intensity patterns with both radial and hexagonal symmetries in agreement with recent experiments. Moreover, we suggest that the ring patterns which are characteristic of degenerate conical emission are transient phenomena which are always replaced by hexagons for long enough times, at least close to the plane-wave threshold. For misaligned input beams we observe the formation of a stabel point defect, a penta-hepta pair, in the hexagonal pattern. © 1992.
• Geddes, J. B., Moloney, J. V., & Indik, R. (1992). Spontaneous transverse spatial pattern formation due to stimulated Brillouin scattering of counterpropagating optical beams. Optics Communications, 90(1-3), 117-122.
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  Abstract: Counterpropagating laser beams can couple to acoustic phonons and cause efficient simultaneous generation of both Stokes and anti-Stokes fields. This results in a combination of dynamic oscillation and spontaneous transverse spatial symmetry breaking. © 1992.
• JAKOBSEN, P., MOLONEY, J., NEWELL, A., & INDIK, R. (1992). SPACE-TIME DYNAMICS OF WIDE-GAIN-SECTION LASERS. PHYSICAL REVIEW A, 45(11), 8129-8137.
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  The space-time behavior of a wide-gain-section, single-longitudinal-mode laser is investigated. Analysis of the full Maxwell-Bloch model in one transverse spatial dimension identifies a traveling-wave solution beyond the first laser threshold, for positive detuning from resonance, which appears to be a globally attracting state of the laser. On the negative-detuning side we observe an initial bifurcation to a homogeneous plane-wave state. A second instability threshold can be identified by linearizing about the respective solutions for both positive and negative detuning. We also find that the standard approach to adiabatic elimination, whereby the polarization equation is adiabatically eliminated, can lead to a spurious high-transverse-wave-number instability in the reduced equations. The derivation of an adiabatically reduced set of equations to remove the instability on the positive-detuning side is a nontrivial matter.
• Jakobsen, P. K., Moloney, J. V., Newell, A. C., & Indik, R. (1992). Space-time dynamics of wide-gain-section lasers. Physical Review A, 45(11), 8129-8137.
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  Abstract: The space-time behavior of a wide-gain-section, single-longitudinal-mode laser is investigated. Analysis of the full Maxwell-Bloch model in one transverse spatial dimension identifies a traveling-wave solution beyond the first laser threshold, for positive detuning from resonance, which appears to be a globally attracting state of the laser. On the negative-detuning side we observe an initial bifurcation to a homogeneous plane-wave state. A second instability threshold can be identified by linearizing about the respective solutions for both positive and negative detuning. We also find that the standard approach to adiabatic elimination, whereby the polarization equation is adiabatically eliminated, can lead to a spurious high-transverse-wave-number instability in the reduced equations. The derivation of an adiabatically reduced set of equations to remove the instability on the positive-detuning side is a nontrivial matter. © 1992 The American Physical Society.
• Broomhead, D. S., Indik, R., Newell, A. C., & Rand, D. A. (1991). Local adaptive Galerkin bases for large-dimensional dynamical systems. Nonlinearity, 4(2), 159-197.
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  Abstract: The authors suggest and develop a method for following the dynamics of systems whose long-time behaviour is confined to an attractor or invariant manifold A of potentially large dimension. The idea is to embed A in a set of local coverings. The dynamics of the phase point P on A in each local ball is then approximated by the dynamics of its projections into the local tangent space. Optimal coordinates in each local patch are chosen by a local version of a singular value decomposition (SVD) analysis which picks out the principal axes of inertia of a data set. Because the basis is continually updated, it is natural to call the procedure an adaptive basis method. The advantages of the method are the following. (i) The choice of the local coordinate system in the local tangent space of A is dictated by the dynamics of the system being investigated and can therefore reflect the importance of natural nonlinear structures which occur locally but which could not be used as part of a global basis. (ii) The number of important or active local degrees of freedom is clearly defined by the algorithm and will usually be much lower than the number of coordinates in the local embedding space and certainly considerably fewer than the number which would be required to provide a global embedding of A. (iii) While the local coordinates indicate which nonlinear structures are important there, the transition matrices which glue the coordinate patches together carry information about the global geometry of A. (iv). The method also suggests a useful algorithm for the numerical integration of complicated spatially extended equation systems, by first using crude integration schemes to generate data from which optimal local and sometimes global Galerkin bases are chosen.
• Egan, A., Ning, C., Moloney, J., Indik, R., Wright, M., Bossert, D., & McInerney, J. (1991). Dynamic instabilities in master oscillator power amplifier semiconductor lasers. IEEE JOURNAL OF QUANTUM ELECTRONICS, 34(1), 166-170.
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  We investigate theoretically the master oscillator power amplifier using a semiconductor laser model that is fully time and space (laterally and longitudinally) resolved, We numerically examine the stability of the device and identify the nature of the different instabilities. These can arise from undamped relaxation oscillations, beating between the longitudinal modes of any of the cavities that comprise the device, or lateral filamentation.
• Newell, A. C., Winful, H. G., Raman, L., Newell, A. C., Moloney, J. V., Jakobsen, P. K., & Indik, R. A. (1991). Diode-laser array modes : discrete and continuous models and their stability. Journal of The Optical Society of America B-optical Physics, 8(8), 1674-1680. doi:10.1364/josab.8.001674
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  A new class of mode profiles appropriate to large one-dimensional evanescently coupled laser diode arrays can be conveniently approximated by solutions to a Riccatti equation derived from a continuous model of the discrete array. Solutions of the latter model are shown to agree with the continuous solutions in the large-N limit. The continuous model has the advantage that new types of discrete array modes can readily be identified. Stability analyses of the discrete array modes as a function of increasing N confirm that these laser systems are intrinsically unstable in the free-running mode.
• Gabitov, I., Indik, R., Mollenauer, L., Shkarayev, M., Stepanov, M., & Lushnikov, P. M. (1990). Twin families of bisolitons in dispersion-managed systems. OPTICS LETTERS, 32(6), 605-607.
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  We calculate bisoliton solutions by using a slowly varying stroboscopic equation. The system is characterized in terms of a single dimensionless parameter. We find two branches of solutions and describe the structure of the tails for the lower-branch solutions. (c) 2007 Optical Society of America
• Newell, A. C., Northcutt, R., Newell, A. C., Moloney, J. V., Mclaughlin, D. W., Lizarraga, C., Indik, R. A., & Adachihara, H. (1990). Modulational-induced optical pattern formation in a passive optical-feedback system. Journal of The Optical Society of America B-optical Physics, 7(6), 1039-1044. doi:10.1364/josab.7.001039
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  Two-dimensional metastable patterns can appear randomly across a broad laser beam profile in a passive optical ring cavity. The instability responsible for pattern formation is primarily modulational in nature, leading to a competitive interaction between strongly saturated solitary-wave ringlike and filamentary spatial structures. The complexity of pattern formation depends sensitively on saturated filament density, which can be conveniently controlled in an optical-feedback arrangement. Our preliminary computational results suggest that the instability domain can be partitioned into two general categories: (1) rapid modulational growth to a single stationary filament or a long-lived metastable pattern involving five filaments growing directly from a shelflike ridge about the central filament or (2) a significantly slower formation of localized filaments forming primarily from a slow modulational growth on fully developed saturated solitary-wave rings. One-dimensional computations with a retarded material response indicate that the general trend of unstable formation should be relatively insensitive to finite retardation effects.
• Indik, R. A. (1979). A PI-algebra which is not PI when an inverse is adjoined. Proceedings of the American Mathematical Society, 76(1), 11-13. doi:10.1090/s0002-9939-1979-0534379-6
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  An example is produced of an algebra, embedded in 2 x 2 matrices over a field, which is not PI when an element's inverse is formally adjoined. This example is used to show that the generic 2 x 2 matrices, a domain, has the same property. In commutative algebra the procedure of adjoining the inverse of an element in a ring has proven useful. Recently work has been done towards generalizing the concepts of commutative algebra and algebraic geometry to PI-algebras [1], [2], [6]. In this paper it is shown that the algebra which is obtained from a subring of a ring of 2 x 2 matrices over an extension of the base field, by formal adjunction of the inverse of an element is not PI. We do this by describing an irreducible infinite dimensional representation and applying Kaplansky's theorem [4]. For A and B algebras over a commutative ring C, we shall use the notation of the free product A *c B as found in [5]. This C-algebra is the coproduct in the category of C-algebras. For R an algebra over a field k, and x an element of R, let x' denote the residue class of z in the algebra R *k k[z]/(zx 1, xz 1). Define R{x'} to be the above algebra. Let k{X, Y} be the algebra obtained by adjoining two generic 2 x 2 matrices to a field k. THEOREM 1. k{X, Y} {X'} is not a PI-algebra. To prove this theorem we shall work with another algebra. We define A to be the quotient of the free algebra k{x, y} by the relations y2 = 0 and yx'y = 0, for i > 0. LEMMA 2. The set S = {xiyxJ, xk; ij,k > 0) spans A as a vector space over k. PROOF. The set S u {0} is closed under multiplication, since x vxJxkyxt = 0, for ij,k,n > 0. Thus since A is generated as an algebra by x andy, the ring A is spanned by S over k. Note. S is actually a basis for A. Received by the editors July 25, 1978 and, in revised form, October 18, 1978. AMS (MOS) subject classifications (1970). Primary 16A08; Secondary 16A38.

Proceedings Publications

• Skovgaard, P. M., O'neill, E., O'brien, P., Mullane, M. P., Moloney, J. V., Mcinerney, J. G., Indik, R. A., & Houlihan, J. (2001). High brightness semiconductor lasers with reduced filamentation. In AIP Conference Proceedings, 548, 173-190.
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  We outline physical models and simulations for suppression of self-focusing and filamentation in large aperture semiconductor lasers. The principal technical objective is to generate multi-watt CW or quasi-CW outputs with nearly diffraction limited beams, suitable for long distance free space transmission, focusing to small spots or coupling to single mode optical fibres. The principal strategies are (a) optimization of facet damage thresholds, (b) reduction of the linewidth enhancement factor which acts as the principal nonlinear optical coefficient, and (c) design of laterally profiled propagation structures in lasers or amplifiers which suppress lateral reflections.
• Skovgaard, P. M., O'neill, E. P., O'brien, P., Mullane, M. P., Moloney, J. V., Mcinerney, J. G., Indik, R. A., & Houlihan, J. A. (2000). Toward filament-free semiconductor lasers. In Physics and Simulation of Optoelectronic Devices VIII, 3944, 376-386.
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  We outline physical models and simulations for suppression of self-focusing and filamentation in large aperture semiconductor lasers. The principal technical objective is to generate multi-watt CW or quasi-CW outputs with nearly diffraction limited beams, suitable for long distance free space transmission, focusing to small spots or coupling to single-mode optical fibers. The principal strategies are (1) optimization of facet damage thresholds, (2) reduction of the linewidth enhancement factor which acts as the principal nonlinear optical coefficient, and (3) design of laterally profiled propagation structures in lasers and amplifiers which suppress lateral reflections.
• Moloney, J. V., Koch, S. W., Indik, R. A., & Hader, J. (1999). Modeling spatiotemporal dynamics of high power semiconductor lasers: microscopically computed gain and device simulation. In Advanced Semiconductor Lasers and Their Applications.
• Moloney, J. V., Vawter, G. A., Moloney, J. V., Mar, A., Koch, S. W., Indik, R. A., & Chow, W. W. (1999). High peak power gain-switched flared waveguide lasers. In Technical Digest. Summaries of papers presented at the Conference on Lasers and Electro-Optics. Postconference Edition. CLEO '99. Conference on Lasers and Electro-Optics (IEEE Cat. No.99CH37013), 487-488.
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  Summary form only given. Record (14.5 W) peak powers and pulse energies of 490 pJ from a gain-switched flared waveguide quantum-well laser are presented in this paper. Because of their compactness, reliability, efficiency, and relatively low cost, semiconductor lasers are attractive sources of picosecond optical pulses of high peak power. The flared waveguide expands the optical mode from a narrow region which gives a single lateral optical mode, to a wider multimode region for higher power. Flared laser amplifiers have previously been used for mode-locking, Q- and gain-switching of diode lasers. Flared structures have also been utilized for post-amplification to achieve high peak power. Gain-switching offers a considerable advantage in terms of simplicity over mode-locking, in that no resonant cavity is required to be synchronous with the electrical drive, while the external amplifier approach requires additional alignments. High single-mode peak powers have also been obtained from bulk single-heterostructure lasers with ion implanted saturable absorbers. In this work, we gain-switch a flared waveguide laser with a reverse-biased saturable absorber section to obtain high peak powers and pulse energies from a single laser diode without post-amplification. Numerical simulations based on a microscopic laser model verify the effectiveness of the saturable absorber in the generation of high peak power, single-pulse output. Insight obtained from the model suggests that the quantum-confined Stark effect is primarily responsible for saturable absorber operation.
• Skovgaard, P. M., Ning, C. Z., Moloney, J. V., Mcinerney, J. G., Kent, A., & Indik, R. A. (1998). Optimisation and Flared Semiconductor Lasers with MOPAs. In CLEO/Europe Conference on Lasers and Electro-Optics, 88-88.
• White, J. K., Ru, P., Moloney, J. V., Koch, S. W., Indik, R. A., & Chow, W. W. (1994). Modeling the nonlinear dynamics of wide aperture semiconductor lasers and amplifiers. In Physics and Simulation of Optoelectronic Devices II, 2146, 410-417.
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  Issues relating to modeling the full spatiotemporal dynamics of wide aperture semiconductor lasers and amplifiers are discussed. Included in the discussion are the limitations of the usual beam propagation approach, characteristics of the many-body light- semiconductor material interaction, spurious nonphysical instabilities which mimic numerical grid oscillations and novel subpicosecond pulse reshaping and compression effects. An explicit simulation is presented for a flared amplifier structure and the results are compared with those using a linear gain model.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
• Ning, C., Indik, R., Moloney, J., Chow, W., Girndt, A., Koch, S., Binder, R., Osinski, M., & Chow, W. (1991). Incorporating many-body effects into modeling of semiconductor lasers and amplifiers. In PHYSICS AND SIMULATION OF OPTOELECTRONIC DEVICES V, 2994, 666-677.
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  Major many-body effects that are important for semiconductor laser modeling are summarized. We adopt a bottom-up approach to incorporate these many-body effects into a model for semiconductor lasers and amplifiers. The optical susceptibility function (chi) computed from the semiconductor Bloch equations (SBEs) is approximated by a single Lorentzian, or a superposition of a, fem Lorentzians in the frequency domain. Our approach leads to a set of effective Bloch equations (EBEs). We compare this approach with the full microscopic SBEs for the case of pulse propagation. Good agreement between the two is obtained for pulse widths longer than tens of picoseconds.

Presentations

• Indik, R. A., Eichhorn, S., Landers, M., & Peterson, S. (2016, February). Supporting collegiate mathematics teaching: Current realities. Critical Issues in Mathematics Education. MSRI, Berkeley, CA: MSRI.
• Indik, R. A. (2015, October). Strategies for dealing with issues of student readiness.. MAA conference: Progress through Calculus Workshop. Washington DC, Georgetown Embassy Suites Hotel: MAA.
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  Panel: Strategies for dealing with issues of student readiness.I presented the results of placement and course redesigns for the University of Arizona's Calculus and Precalculus courses.