Robert A Indik
 Associate Professor, Mathematics
 Associate Head, Undergraduate Programs
Contact
 (520) 6214599
 Mathematics, Rm. 302
 Tucson, AZ 85721
 indik@math.arizona.edu
Degrees
 Ph.D. Mathematics
 Princeton University, Princeton, New Jersey, United States
 FOURIER COEFFICIENTS OF NONHOLOMORPHIC EISENSTEIN SERIES ON A TUBE DOMAIN ASSOCIATED TO AN ORTHOGONAL GROUP
 B.S. Mathematics
 Massachusetts Institute of Technology, Cambridge, Massachusetts, United States
 A PI algebra which is not PI when an inverse is adjoined
Awards
 Thinktank Faculty award
 Think tank, Fall 2014
Interests
No activities entered.
Courses
202021 Courses

Combinatorial Math
MATH 447 (Spring 2021) 
Math of BioSystems
MATH 119A (Fall 2020)
201920 Courses

Applied Linear Algebra
MATH 310 (Summer I 2020) 
Math of BioSystems
MATH 119A (Spring 2020) 
Math of BioSystems
MATH 119A (Fall 2019)
201819 Courses

Math of BioSystems
MATH 119A (Spring 2019) 
Numerical Analysis
MATH 575B (Spring 2019) 
Numerical Analysis
MATH 575A (Fall 2018)
201718 Courses

Discrete Math Cmptr Sci
MATH 243 (Fall 2017) 
Pro Dev Teaching Math
MATH 597T (Fall 2017)
201617 Courses

Intro Ord Diff Equations
MATH 254 (Summer I 2017) 
2nd Crs Abstract Algebra
MATH 415B (Spring 2017) 
2nd Crs Abstract Algebra
MATH 515B (Spring 2017) 
Col Alg Cncpts+Aplcns
MATH 112 (Fall 2016) 
Precalculus Sup Instruction
MATH 196L (Fall 2016) 
Pro Dev Teaching Math
MATH 597T (Fall 2016)
201516 Courses

2nd Crs Abstract Algebra
MATH 415B (Spring 2016) 
Intro to Linear Algebra
MATH 313 (Spring 2016)
Scholarly Contributions
Journals/Publications
 Gabitov, I., Indik, R., Mollenauer, L., Shkarayev, M., Stepanov, M., & Lushnikov, P. M. (2007). Twin families of bisolitons in dispersionmanaged systems. Optics Letters, 32(6), 605607.More infoPMID: 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 lowerbranch 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), 36143620.More infoBased on a microscopic theory of a twoband 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 LARGEDIMENSIONAL DYNAMICSYSTEMS. NONLINEARITY, 4(2), 159197.More infoWe suggest and develop a method for following the dynamics of systems whose longtime 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). Doubleresonant optical materials with embedded metal nanostructures. Journal of the Optical Society of America B: Optical Physics, 23(3), 535542.More infoAbstract: 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 wellknown MaxwellLorentz 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 selfinduced transparency in the framework of the MaxwellBloch 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.
 Indik, R. A., & Newell, A. C. (2006). Conical refraction and nonlinearity. Optics Express, 14(22), 1061410620.More infoPMID: 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 gordonmollenauer soliton phase noise in optical fibers. IEEE Photonics Technology Letters, 18(7), 886888.More infoAbstract: 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 ElectroOptics Europe  Technical Digest.More infoAbstract: 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(14), 127140.More infoAbstract: The CrossNewell 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 nontrivial 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.
 Ercolani, N. M., Indik, R., Newell, A. C., & Passot, T. (2000). The geometry of the phase diffusion equation. Journal of Nonlinear Science, 10(2), 223274.More infoAbstract: The CrossNewell 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 fourthorder 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 wellapproximated by a secondorder "selfdual" equation. Moreover, the selfdual 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 selfdual 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 filamentfree semiconductor lasers. Proceedings of SPIE  The International Society for Optical Engineering, 3944, I/.More infoAbstract: We outline physical models and simulations for suppression of selffocusing and filamentation in large aperture semiconductor lasers. The principal technical objective is to generate multiwatt CW or quasiCW outputs with nearly diffraction limited beams, suitable for long distance free space transmission, focusing to small spots or coupling to singlemode 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 firstprinciples fully spacetime resolved model of a semiconductor laser. QUANTUM AND SEMICLASSICAL OPTICS, 9(5), 681691.More infoWe present a semiconductor laser model which incorporates the gain bandwidth and nonlinear gain by using the multiband microscopic theory of an electronhole plasma in a semiconductor quantumwell medium. The approach is extremely robust, allowing us to take into account important manybody 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 multilongitudinal mode and transverse filamentation instabilities of a masteroscillator power amplifier (MOPA) device.
 GEDDES, J., LEGA, J., MOLONEY, J., INDIK, R., WRIGHT, E., & FIRTH, W. (1999). PATTERN SELECTION IN PASSIVE AND ACTIVE NONLINEAROPTICAL SYSTEMS. CHAOS SOLITONS & FRACTALS, 4(89), 12611274.More infoThe 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 selffocusing hexagons and selfdefocusing squares are the preferred planforms. Polarizationinduced 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, wideaperture Raman laser is shown to select a transverse traveling wave mode, which appears in the farfield as an offaxis 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), 20232029.More infoAbstract: We combine the results of full manybody bandstructure calculations of the semiconductor optical response and a full spacetimeresolved laser propagation model. Two quantumwell 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 farfield broadening of two weakly turbulent broadarea highpower 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(14), 474492.More infoAbstract: 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), 166170.More infoAbstract: 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 ElectroOptics Europe  Technical Digest, 88.More infoAbstract: A detailed and realistic theory of the fast dynamics and stability of flared laser oscillators and master oscillatorpower amplifier (MOPA) is described. The theory approximates a realistic gain spectrum into a complex Lorentzian susceptibility which depends nonlinearly 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), 12291231.More infoAbstract: We provide the first comprehensive theoretical evidence for nonlinear spacetime 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), 15431550.More infoA set of effective Bloch equations is established for semiconductor bulk or quantumwell media, The model includes the nonlinear carrierdensity 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 wideaperture highpower 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 COUPLEDMODE MODEL FOR THE MULTISTRIPE INDEXGUIDED LASER ARRAYS. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA BOPTICAL PHYSICS, 10(3), 507515.More infoWe develop a generalized coupledmode model for multistripe indexguided laser arrays that includes explicitly the influence of carrierinduced antiguiding, gain guiding, and carrier diffusion in the gain stripe. As an illustration of an application of the model, stability criteria for twoelement laser arrays are derived that show that the phaselocked solution is intrinsically unstable. We find that the phaselocked solution can be stabilized at a low external injectionlocking power at the suitably chosen injectionlocking frequency. We have tested our model in the large carrierdiffusion case and still find good qualitative agreement between the coupledmode 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 nativeoxideconfined verticalcavity surfaceemitting lasers. Physical Review A  Atomic, Molecular, and Optical Physics, 58(4), 32793292.More infoAbstract: We present a microscopically based verticalcavity surfaceemitting laser (VCSEL) model that treats plasma and lattice heating selfconsistently and includes gain dispersion in a fashion facilitating the incorporation of manybody effects. This model is used to investigate the interplay of thermal effects and transverse mode behavior observed in recent experiments with largeaperture selectively oxidized VCSELs. We confirm that the highly divergent singlemode emission seen experimentally at low ambient temperatures may be caused by a redshift of the cavity resonance frequency relative to the quantumwell 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 lowtemperature steadystate 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 highpower MFAMOPA semiconductor devices. Conference on Lasers and ElectroOptics Europe  Technical Digest, 244.More infoAbstract: Detailed theoretical analyses of monolithically integrated flared amplifier master oscillator power amplifiers (MFAMOPAs) are presented. The field amplitudes and the carrier density are calculated using effective Block equations. The obtained features of the MFAMOPAs are presented and discussed.
 JAKOBSEN, P., INDIK, R., MOLONEY, J., NEWELL, A., WINFUL, H., & RAMAN, L. (1997). DIODELASER ARRAY MODES  DISCRETE AND CONTINUOUS MODELS AND THEIR STABILITY. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA BOPTICAL PHYSICS, 8(8), 16741680.More infoA new class of mode profiles appropriate to large onedimensional 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 largeN 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 freerunning mode.
 MOLONEY, J., ADACHIHARA, H., INDIK, R., LIZARRAGA, C., NORTHCUTT, R., MCLAUGHLIN, D., & NEWELL, A. (1997). MODULATIONALINDUCED OPTICALPATTERN FORMATION IN A PASSIVE OPTICALFEEDBACK SYSTEM. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA BOPTICAL PHYSICS, 7(6), 10391044.
 Moloney, J. V., Indik, R. A., & Ning, C. Z. (1997). Full spacetime simulation for highbrightness semiconductor lasers. IEEE Photonics Technology Letters, 9(6), 731733.More infoAbstract: 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 masteroscillator poweramplifier (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). Spacetime simulation of highbrightness semiconductor lasers. Proceedings of SPIE  The International Society for Optical Engineering, 2994, 562571.More infoAbstract: 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 lowdensity 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 spacetime simulation for highbrightness semiconductor lasers. IEEE PHOTONICS TECHNOLOGY LETTERS, 9(6), 731733.More infoA 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 masteroscillator poweramplifier (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). SELFCONSISTENT APPROACH TO THERMAL EFFECTS IN VERTICALCAVITY SURFACEEMITTING LASERS. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA BOPTICAL PHYSICS, 12(10), 19932004.More infoA selfconsistent 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 phononcarrier scattering and to the band gap's dependence on lattice temperature. The selfconsistent theory is then employed to study thermal effects in verticalcavity surfaceemitting 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 quantumwell media. Significant plasmaheating effects are found. These include the carrierdensity dependence on pumping, decrease of inputoutput efficiency, dependence of the cw frequency shift on pumping, and a pronounced Pauliblocking 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 quantumwell structures. IEEE Journal on Selected Topics in Quantum Electronics, 3(2), 129135.More infoAbstract: 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 quantumwell (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 carrierinduced refractive index change versus carrier density, for InGaAsGaAs 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), 15431550.More infoAbstract: A set of effective Bloch equations is established for semiconductor bulk or quantumwell media. The model includes the nonlinear carrierdensity 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 wideaperture highpower 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 manybody effects into modeling of semiconductor lasers and amplifiers. Proceedings of SPIE  The International Society for Optical Engineering, 2994, 666677.More infoAbstract: Major manybody effects that are important for semiconductor laser modeling are summarized. We adopt a bottomup approach to incorporate these manybody 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 firstprinciples fully spacetime resolved model of a semiconductor laser. Journal of Optics B: Quantum and Semiclassical Optics, 9(5), 681691.More infoAbstract: We present a semiconductor laser model which incorporates the gain bandwidth and nonlinear gain by using the multiband microscopic theory of an electronhole plasma in a semiconductor quantumwell medium. The approach is extremely robust, allowing us to take into account important manybody 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 multilongitudinal mode and transverse filamentation instabilities of a masteroscillator 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 nativeoxideconfined verticalcavity surfaceemitting lasers. PHYSICAL REVIEW A, 58(4), 32793292.More infoWe present a microscopically based verticalcavity surfaceemitting laser (VCSEL) model that treats plasma and lattice heating selfconsistently and includes gain dispersion in a fashion facilitating the incorporation of manybody effects. This model is used to investigate the interplay of thermal effects and transverse mode behavior observed in recent experiments with largeaperture selectively oxidized VCSELs. We confirm that the highly divergent singlemode emission seen experimentally at low ambient temperatures may be caused by a redshift of the cavity resonance frequency relative to the quantumwell 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 lowtemperature steadystate 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. [S10502947(98)02410X].
 Skovgaard, P. M., McInerney, J. G., Moloney, J. V., Indik, R. A., & Ning, C. Z. (1997). Enhanced stability of MFAMOPA semiconductor lasers using a nonlinear, trumpetshaped flare. IEEE Photonics Technology Letters, 9(9), 12201222.More infoAbstract: Monolithically integrated flared amplifier master oscillator power amplifier (MFAMOPA) lasers are studied using a highresolution 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, trumpetshaped flare to overlap the gain region to the expanding field, the instability threshold of the MOPA is increased by ∼2 for singlelongitudinal, singletransverse mode operation and ∼3 for singletransverse mode operation. This enables the MOPA to maintain a stable, neardiffraction limited output beam for higher currents before the onset of transverse instabilities. Thus the trumpetflared 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 MFAMOPA.
 Skovgaard, P. M., McInerney, J. G., Moloney, J. V., Indik, R. A., & Ning, C. Z. (1997). Modeling of master oscillatorpower amplifier (MOPA) semiconductor lasers. Proceedings of SPIE  The International Society for Optical Engineering, 2994, 801809.More infoAbstract: Monolithically integrated flared amplifier master oscillator power amplifier (MFAMOPA) 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 electronholescattering on the gain spectra of highly excited semiconductors. Solid State Communications, 100(8), 555559.More infoAbstract: A microscopic treatment of the influence of electronholescattering on the optical dephasing and the lineshape in semiconductor gain media is presented. The calculations incorporate nondiagonal and diagonalscattering 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 electronholescattering on the gain spectra of highly excited semiconductors. SOLID STATE COMMUNICATIONS, 100(8), 555559.More infoA microscopic treatment of the influence of electronholescattering on the optical dephasing and the lineshape in semiconductor gain media is presented. The calculations incorporate nondiagonal and diagonalscattering 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), 36143620.More infoAbstract: Based on a microscopic theory of a twoband 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 selfdual solutions of the CrossNewell phase diffusion equation for natural patterns. Physica D: Nonlinear Phenomena, 97(13), 185205.More infoAbstract: 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 selfdual solutions of the CrossNewell phase diffusion equation for natural patterns. PHYSICA D, 97(13), 185205.More infoWe 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 MFAMOPA semiconductor lasers using a nonlinear, trumpetshaped flare. IEEE PHOTONICS TECHNOLOGY LETTERS, 9(9), 12201222.More infoMonolithically integrated flared amplifier master oscillator power amplifier (MFAMOPA) lasers are studied using a highresolution 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, trumpetshaped flare to overlap the gain region to the expanding field, the instability threshold of the MOPA is increased by similar to 2 for singlelongitudinal, singletransverse mode operation and similar to 3 for singletransverse mode operation. This enables the MOPA to maintain a stable, neardiffraction limited output beam for higher currents before the onset of transverse instabilities. Thus the trumpetflared 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 MFAMOPA.
 Chow, W. W., Indik, R., Knorr, A., Koch, S. W., & Moloney, J. V. (1995). Timeresolved nondegenerate fourwave mixing in a semiconductor amplifier. Physical Review A, 52(3), 24792482.More infoAbstract: The dynamics of copropagating femtosecond pulses with different carrier frequencies is modeled for a semiconductor amplifier. The propagation induced fourwave 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). Doubleresonant optical materials with embedded metal nanostructures. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA BOPTICAL PHYSICS, 23(3), 535542.More infoWe 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 wellknown MaxwellLorentz 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 selfinduced transparency in the framework of the MaxwellBloch 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). Manybody effects in the propagation of short pulses in a semiconductor amplifier. Proceedings of SPIE  The International Society for Optical Engineering, 2399, 650659.More infoAbstract: The semiconductor MaxwellBloch 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 femtosecond 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, selffocusing, and memory effects.
 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, 617628.More infoAbstract: We report thermal effects revealed by a selfconsistent treatment of plasma and lattice heating in vertical cavity surfaceemitting 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 selfconsistently 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 switchoff 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.
 Ercolani, N., Indik, R., Newell, A., & Passot, T. (1994). The geometry of the phase diffusion equation. JOURNAL OF NONLINEAR SCIENCE, 10(2), 223274.More infoThe CrossNewell 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 fourthorder 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 wellapproximated by a secondorder "selfdual" 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), 34713485.More infoAbstract: In this work, substantial extensions of the studies on pattern formation were reported, particularly in the regime of selfdefocusing Kerr media. It was shown that by nonlinear perturbation analysis, two very distinct patternforming modes coexisted in this system. One was a fairly typical hexagonforming mode which was dominant in selffocusing media. The other was a rollpattern mode with the rolls found to be unstable. Instead square patterns emerged and appeared to be dominant for selfdefocusing media. By varying the grating parameter G one mode can be favored over the other. However, the lessfavored 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(89), 12611274.More infoAbstract: 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 selffocusing hexagons and selfdefocusing squares are the preferred planforms. Polarizationinduced 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, wideaperture Raman laser is shown to select a transverse traveling wave mode, which appears in the farfield as an offaxis 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). PROPAGATIONINDUCED ADIABATIC FOLLOWING IN A SEMICONDUCTOR AMPLIFIER. OPTICS LETTERS, 19(13), 966968.More infoA propagationinduced 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), 653662.More infoPMID: 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 steadystate 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 ∼0.50.9°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 steadystate temperatures from ferromagnetic seed heating. Medical Physics, 21(7), 11351144.More infoPMID: 7968846;Abstract: A new, very fast, yet accurate program, MGSEED, has been developed that computes steadystate temperatures from the threedimensional 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(45), 385403.More infoAbstract: 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, 11351144.More infoIndik, 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, 11351144
 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, 163165.More infoAbstract: 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 offresonant 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.More infoAbstract: We report on our studies of a planewave femtosecond pulse propagating in a semiconductor amplifier over long gain lengths. The overall conclusion for a planewave 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 offresonant noninverted states, well above the chemical potential.
 Indik, R., Knorr, A., Binder, R., Moloney, J. V., & Koch, S. W. (1994). Propagationinduced adiabatic following in a semiconductor amplifier. Optics Letters, 19(13), 966968.More infoAbstract: 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 BOPTICAL PHYSICS, 16(11), 20232029.More infoWe combine the results of full manybody bandstructure calculations of the semiconductor optical response and a full spacetimeresolved laser propagation model. Two quantumwell 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 farfield broadening of two weakly turbulent broadarea highpower semiconductor lasers is shown to be quite different for the two structures. (C) 1999 Optical Society of America [S07403224(99)002118].
 Ru, P., Moloney, J. V., & Indik, R. (1994). Meanfield approximation in semiconductorlaser modeling. Physical Review A  Atomic, Molecular, and Optical Physics, 50(1), 831838.More infoAbstract: A meanfield model from the full counterpropagating wave equations describing a general FabryPerot 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 singlelongitudinalmode laser which should furnish an adequated description of the semiconductor laser dynamics and steadystate behavior over an extensive operating range. However, both spatial and spectral holeburning effects were predicted to induce multilongitudinalmode oscillation at higher drive currents. It was also noted that the introduction of transverse effects in rateequation approximations of laser dynamics can present spurious hightransversewavenumber instabilities.
 ADACHIHARA, H., HESS, O., INDIK, R., & MOLONEY, J. (1993). SEMICONDUCTORLASER ARRAY DYNAMICS  NUMERICAL SIMULATIONS ON MULTISTRIPE INDEXGUIDED LASERS. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA BOPTICAL PHYSICS, 10(3), 496506.More infoThe spacetime dynamical behavior of multistripe indexguided 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 highperformance parallel computing architectures to solve the coupled partial differential equations describing the lightmatter 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 manybody semiconductor effects in future studies.
 Adachihara, H., Hess, O., Indik, R., & Moloney, J. V. (1993). Semiconductor laser array dynamics: numerical simulations on multistripe indexguided lasers. Journal of the Optical Society of America B: Optical Physics, 10(3), 496506.More infoAbstract: Time dependent, fully transversedimensional partial differential equations are numerically integrated to study the spacetime 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 NONLINEAROPTICAL SYSTEM. PHYSICAL REVIEW A, 50(4), 34713485.More infoPattern 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 selffocusing medium, and squares in a selfdefocusing 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), 19511954.More infoAbstract: Assuming that airsea 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 coupledmode model for the multistripe indexguided laser arrays. Journal of the Optical Society of America B: Optical Physics, 10(3), 507515.More infoAbstract: 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 quantumwell GaAsbased semiconductor lasers. Optical and Quantum Electronics, 25(10), 675693.More infoAbstract: The dynamical behaviour of FabryPerot type semiconductor lasers is modelled, including the relevant manybody 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 quantumwell GaAs active material and a variety of laser structures, including strongly and weakly indexguided structures, as well as purely guided singleand twinstripe lasers. © 1993 Chapman & Hall.
 CHANG, R., FIRTH, W., INDIK, R., MOLONEY, J., & WRIGHT, E. (1992). 3DIMENSIONAL SIMULATIONS OF DEGENERATE COUNTERPROPAGATING BEAM INSTABILITIES IN A NONLINEAR MEDIUM. OPTICS COMMUNICATIONS, 88(23), 167172.More infoWe report our findings from a detailed series of threedimensional 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 planewave threshold. For misaligned input beams we observe the formation of a stable point defect, a pentahepta pair, in the hexagonal pattern.
 Chang, R., Firth, W. J., Indik, R., Moloney, J. V., & Wright, E. M. (1992). Threedimensional simulations of degenerate counterpropagating beam instabilities in a nonlinear medium. Optics Communications, 88(23), 167172.More infoAbstract: We report our findings from a detailed series of threedimensional 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 planewave threshold. For misaligned input beams we observe the formation of a stabel point defect, a pentahepta 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(13), 117122.More infoAbstract: Counterpropagating laser beams can couple to acoustic phonons and cause efficient simultaneous generation of both Stokes and antiStokes 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). SPACETIME DYNAMICS OF WIDEGAINSECTION LASERS. PHYSICAL REVIEW A, 45(11), 81298137.More infoThe spacetime behavior of a widegainsection, singlelongitudinalmode laser is investigated. Analysis of the full MaxwellBloch model in one transverse spatial dimension identifies a travelingwave solution beyond the first laser threshold, for positive detuning from resonance, which appears to be a globally attracting state of the laser. On the negativedetuning side we observe an initial bifurcation to a homogeneous planewave 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 hightransversewavenumber instability in the reduced equations. The derivation of an adiabatically reduced set of equations to remove the instability on the positivedetuning side is a nontrivial matter.
 Jakobsen, P. K., Moloney, J. V., Newell, A. C., & Indik, R. (1992). Spacetime dynamics of widegainsection lasers. Physical Review A, 45(11), 81298137.More infoAbstract: The spacetime behavior of a widegainsection, singlelongitudinalmode laser is investigated. Analysis of the full MaxwellBloch model in one transverse spatial dimension identifies a travelingwave solution beyond the first laser threshold, for positive detuning from resonance, which appears to be a globally attracting state of the laser. On the negativedetuning side we observe an initial bifurcation to a homogeneous planewave 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 hightransversewavenumber instability in the reduced equations. The derivation of an adiabatically reduced set of equations to remove the instability on the positivedetuning 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 largedimensional dynamical systems. Nonlinearity, 4(2), 159197.More infoAbstract: The authors suggest and develop a method for following the dynamics of systems whose longtime 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), 166170.More infoWe 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.
 Gabitov, I., Indik, R., Mollenauer, L., Shkarayev, M., Stepanov, M., & Lushnikov, P. M. (1990). Twin families of bisolitons in dispersionmanaged systems. OPTICS LETTERS, 32(6), 605607.More infoWe 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 lowerbranch solutions. (c) 2007 Optical Society of America
Proceedings Publications
 Ning, C., Indik, R., Moloney, J., Chow, W., Girndt, A., Koch, S., Binder, R., Osinski, M., & Chow, W. (1991). Incorporating manybody effects into modeling of semiconductor lasers and amplifiers. In PHYSICS AND SIMULATION OF OPTOELECTRONIC DEVICES V, 2994, 666677.More infoMajor manybody effects that are important for semiconductor laser modeling are summarized. We adopt a bottomup approach to incorporate these manybody 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.More infoPanel: 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.