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Showing papers on "Quantum well published in 1992"


Journal ArticleDOI
TL;DR: The spectral response of a monolithic semiconductor quantum microcavity with quantum wells as the active medium displays mode splitting when the quantum wells and the optical cavity are in resonance.
Abstract: The spectral response of a monolithic semiconductor quantum microcavity with quantum wells as the active medium displays mode splitting when the quantum wells and the optical cavity are in resonance. This effect can be seen as the Rabi vacuum-field splitting of the quantum-well excitons, or more classically as the normal-mode splitting of coupled oscillators, the excitons, and the electromagnetic field of the microcavity. An exciton oscillator strength of 4\ifmmode\times\else\texttimes\fi{}${10}^{12}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}$ is deduced for 76-\AA{} quantum wells.

2,006 citations


PatentDOI
TL;DR: In this article, a II-VI compound semiconductor laser diode is formed from overlaying layers of material including an n-type single crystal semiconductor substrate (12), adjacent N-type and p-type guiding lasers (14), a quantum well active layer (18), and a second electrode (30) is characterized by a Fermi energy, with shallow acceptors having a shallow acceptor energy, to a net acceptor concentration of at least 1 x 1017 cm 3.
Abstract: A II-VI compound semiconductor laser diode (10) is formed from overlaying layers of material including an n-type single crystal semiconductor substrate (12), adjacent n-type and p-type guiding lasers (14) and (16) of II-VI semiconductor forming a pn junction, a quantum well active layer (18) of II-VI semiconductor between the guiding layers (14) and (16), first electrode (32) opposite the substrate (12) from the n-type guiding layer (14), and a second electrode (30) opposite the p-type guiding layer (16) from the quantum well layer (18) Electrode layer (30) is characterized by a Fermi energy A p-type ohmic contact layer (26) is doped, with shallow acceptors having a shallow acceptor energy, to a net acceptor concentration of at least 1 x 1017 cm-3, and includes sufficient deep energy states between the shallow acceptor energy and the electrode layer Fermi energy to enable cascade tunneling by charge carriers

1,453 citations


Journal ArticleDOI
TL;DR: In this paper, a new microlaser design based on the highreflectivity whisperinggallery modes around the edge of a thin semiconductor microdisk is described and initial experimental results are presented.
Abstract: A new microlaser design based on the high‐reflectivity whispering‐gallery modes around the edge of a thin semiconductor microdisk is described and initial experimental results are presented. Optical confinement within the thin disk plane results in a microresonator with potential for single‐mode, ultralow threshold lasers. Initial experiments use selective etching techniques in the InP/InGaAsP system to achieve 3–10 μm diameter disks as thin as 500 A suspended in air or SiO2 on an InP pedestal. Optically pumped InGaAs quantum wells provide sufficient gain when cooled with liquid nitrogen to obtain single‐mode lasing at 1.3 and 1.5 μm wavelengths with threshold pump powers below 100 μW.

1,383 citations


Journal ArticleDOI
TL;DR: In this article, the authors developed a model and derived analytical expressions for the modulation response, resonance frequency, damping rate, and K factor to include carrier transport effects on the high-speed properties of quantum-well lasers.
Abstract: Carrier transport can significantly affect the high-speed properties of quantum-well lasers. The authors have developed a model and derived analytical expressions for the modulation response, resonance frequency, damping rate, and K factor to include these effects. They show theoretically and experimentally that carrier transport can lead to significant low-frequency parasitic-like rolloff that reduces the modulation response by as much as a factor of six in quantum-well lasers. They also show that, in addition, it leads to a reduction in the effective differential gain and thus the resonance frequency, while the nonlinear gain compression factor remains largely unaffected by it. The authors present the temperature dependence data for the K factor as further evidence for the effects of carrier transport. >

449 citations


Journal ArticleDOI
TL;DR: In this article, surface segregation of In atoms during molecular beam epitaxy (MBE) and its influence on the energy levels in InGaAs/GaAs quantum wells were systematically studied using secondary ion mass spectroscopy (SIMS) and photoluminescence (PL).
Abstract: Surface segregation of In atoms during molecular beam epitaxy (MBE) and its influence on the energy levels in InGaAs/GaAs quantum wells (QWs) were systematically studied using secondary‐ion mass spectroscopy (SIMS) and photoluminescence (PL). Strong dependence of In surface segregation on the growth conditions was found; when the growth temperature was raised from 370 to 520 °C, the segregation length was observed to increase from 0.8 up to 2.9 nm, accompanied by an appreciable peak energy shift in the PL spectra of the InGaAs/GaAs QWs. The correlation between In surface segregation and the energy levels in InGaAs/GaAs QWs was clarified for the first time.

434 citations


Journal ArticleDOI
TL;DR: The capacitance signal resulting from single electrons tunneling into discrete quantum levels is observed and the nature of the bound states is deduced from the magnetic field evolution of the spectrum.
Abstract: We observe the capacitance signal resulting from single electrons tunneling into discrete quantum levels. The electrons tunnel between a metallic layer and confined states of a single disk in a microscopic capacitor fabricated in GaAs. Charge transfer occurs only for bias voltages at which a quantum level resonates with the Fermi energy of the metallic layer. This creates a sequence of distinct capacitance peaks whose bias positions directly reflect the electronic spectrum of the confined structure. From the magnetic field evolution of the spectrum, we deduce the nature of the bound states.

378 citations


Journal ArticleDOI
TL;DR: It is found that the strong electronic confinement in these quantum dots reflects itself in the ground-state energy and in the impurity binding energies, which are higher than those found in GaAs-(Ga,Al)As quantum wells and quantum-well wires.
Abstract: The ground-state energy and the binding energy of shallow hydrogenic impurities in spherical GaAs-(Ga,Al)As quantum dots have been calculated as functions of the radius of the dot. The binding energy has been calculated following a variational procedure within the effective-mass approximation. We have used a finite confining potential well with depth determined by the discontinuity of the band gap in the quantum dot and the cladding. Calculations were also performed for an infinite confining potential. For the infinite potential well we found that the impurity binding energy increases as the dot radius decreases whereas in the finite potential-well situation, the binding energy reaches a peak value as the dot radius decreases and then diminishes to a limiting value corresponding to the radius for which there are no bound states in the well. We found that the strong electronic confinement in these quantum dots reflects itself in the ground-state energy and in the impurity binding energies, which are higher than those found in GaAs-(Ga,Al)As quantum wells and quantum-well wires.

300 citations


Journal ArticleDOI
01 Aug 1992-Nature
TL;DR: In this paper, it was shown that Bragg reflections can be used to localize a bound state within a quantum well at an energy greater than the potential barrier height, where this state is spatially localized by Bragg reflection.
Abstract: SHORTLY after the birth of quantum mechanics, von Neumann and Wigner made the remarkable proposal1 that certain spatially oscillating attractive potentials could support bound states at energies above the potential barriers (that is, spatially confined states within the continuum) by means of diffractive interference. Because of their unusual geometry, such potentials were regarded as mathematical curiosities2,3, although more recently it has been suggested that they might be found in certain atomic and molecular systems4,5. Following the observation of discrete electronic states in ultra-thin semiconductor layered structures6,7 (for example, in quantum wells), Stillinger8 and Herrick9 proposed that super-lattices might be used to construct potentials supporting these 'positive energy' bound states. Here we report direct evidence of such states in semiconductor heterostructures grown by molecular-beam epitaxy10. Infrared absorption measurements reveal a narrow, isolated transition from a bound state within a quantum well to a bound state at an energy greater than the barrier height; this state is spatially localized by Bragg reflections.

277 citations


Book ChapterDOI
01 Jan 1992
TL;DR: In this article, the potential energy profile in semiconductor heterostructures can now be controlled in a fascinating way that could barely be dreamed of twenty years ago, and many devices have been designed according to this quantum engineering and have shown unsurpassed properties.
Abstract: The potential energy profile in semiconductor heterostructures can now be controlled in a fascinating way that could barely be dreamed of twenty years ago 1. When dealing with interband optical transitions, additional features related to electron-hole interactions (see for instance exciton descriptions in this book) are coming into play and the one-electron wavefunctions and energy levels may fail to describe or predict experimental results. Moreover, the quantization energy is usually small compared to the forbidden band gap, so that typical interband transitions always occur in the same energy range for a given materials pair. On the contrary, intersubband transitions (ISBT) are very sensitive to the exact potential profile and transitions have been observed at wavelengths between lμm and 100μm. In addition, they can be quantitatively described by a simple formalism based on one-electron approaches and many-body effects usually appear as small corrections only. Since 19852, many devices have been designed according to this quantum engineering and have shown unsurpassed properties3. Various materials have been successfully used for these quantum well (QW) heterostructures: GaAs/AlGaAs, InP/InGaAs/InAlAs, Si/SiGe, D/VI compounds…We will focus here on the GaAs/AlGaAs system which has been the most widely studied. First, the calculation of the ISBT matrix element will evidence two major characteristic properties: the optical transitions take advantage of giant dipoles but must verify in the same time a rather drastic selection rule. Then, examples will be given in different fields of application: detection, modulation and emission. Some interesting aspects of coupling and propagation in these structures involve a photon mode density alteration. Finally, a detailed study of second order non linearities will exemplify the beauty of quantum engineering for improving optical properties.

203 citations


Journal ArticleDOI
Young-Kai Chen1, Ming C. Wu1
TL;DR: In this article, the authors present the principles of and recent progress in monolithic CPM quantum-well lasers and review their application in the field of quantum well-laser systems.
Abstract: Integration of the whole mode-locked laser onto a single piece of semiconductor offers a number of advantages, including total elimination of optical alignment processes, improved mechanical stability, and the generation of short optical pulses at much higher repetition frequencies. Semiconductor laser processing technologies were used to implement the colliding-pulse mode-locking (CPM) scheme, which is known to effectively shorten the pulses and increase stability, on a miniature monolithic semiconductor cavity. The principles of and recent progress in monolithic CPM quantum-well lasers are reviewed. >

196 citations


Journal ArticleDOI
TL;DR: In this paper, several ultrafast optical pulse generation techniques utilizing external cavity semiconductor lasers are described, including active mode locking, passive mode lock, hybrid mode locking and several chirp compensation techniques.
Abstract: Several ultrafast optical pulse generation techniques utilizing external cavity semiconductor lasers are described. These techniques include active mode locking, passive mode locking, hybrid mode locking, and several chirp compensation techniques. Utilizing these techniques, optical pulses of 200 fs in duration with over 160 W of peak power have been generated, making these pulses both the shortest and most intense ever generated with a semiconductor injection diode laser system. These pulses have been used to study the ultrafast amplification characteristics of semiconductor lasers. The results presented reveal the nature of the effects which dominate the pulse shaping mechanisms in external cavity hybrid mode-locked diode lasers. >

Journal ArticleDOI
TL;DR: In this article, the electron-optical-phonon scattering rates in GaAs/AlAs quantum wells are calculated on the basis of a fully microscopic description of the phonon spectra.
Abstract: The electron--optical-phonon scattering rates in GaAs/AlAs quantum wells are calculated on the basis of a fully microscopic description of the phonon spectra. The results indicate the great importance of confined as well as GaAs-like and AlAs-like interface phonons. By comparing our results with those of several macroscopic models, we resolve a long-standing controversy on their ability to describe the relevant vibrations.

Journal ArticleDOI
TL;DR: It is shown that gain in the context of partial phase-space filling can develop at pair densities below the onset to an electron-hole plasma and that an exciton gas is stable against ionization by optical phonons up to room temperature.
Abstract: We show spectroscopically that the origin of optical gain and laser emission in (Zn,Cd)Se/ZnSe quantum wells at blue-green wavelengths is of excitonic nature. This circumstance derives from the large enhancement in the exciton binding and its oscillator strength which occurs in the quasi-2D case, so that an exciton gas is stable against ionization by optical phonons up to room temperature and that gain in the context of partial phase-space filling can develop at pair densities below the onset to an electron-hole plasma.

Journal ArticleDOI
TL;DR: In this paper, the authors constructed a model to calculate the photoconductive gain of the quantum-well intersubband infrared detectors and showed that the gain is inversely proportional to the number of quantum wells.
Abstract: Taking into account the discrete nature of the quantum‐well intersubband infrared detectors, we construct a model to calculate the photoconductive gain. It is shown that the photoconductive gain is inversely proportional to the number of quantum wells and that the detector‐current responsivity is independent of the number of wells.

Journal ArticleDOI
TL;DR: In this paper, a detailed and thorough study of a wide variety of quantum well infrared photodetectors (QWIPs) is presented, which were chosen to have large differences in their optical and transport properties.
Abstract: We present a detailed and thorough study of a wide variety of quantum well infrared photodetectors (QWIPs), which were chosen to have large differences in their optical and transport properties. Both n- and p-doped QWIPs, as well as intersubband transitions based on photoexcitation from bound-to-bound, bound-to-quasi-continuum, and bound-to-continuum quantum well states were investigated. The measurements and theoretical analysis included optical absorption, responsivity, dark current, current noise, optical gain, hot carrier mean free path; net quantum efficiency, quantum well escape probability, quantum well escape time, as well as detectivity. These results allow a better understanding of the optical and transport physics and thus a better optimization of the QWIP performance.

Patent
02 Mar 1992
TL;DR: A quantum well structure useful for semiconducting devices comprises two barrier regions and a thin epitaxially grown monocrystalline semiconductor material quantum well sandwiched between said barrier regions.
Abstract: A quantum well structure useful for semiconducting devices comprises two barrier regions and a thin epitaxially grown monocrystalline semiconductor material quantum well sandwiched between said barrier regions Each barrier region consists essentially of alternate strain layers forming a superlattice, each of said layers being thinner than said quantum well The layers are so thin that no defects are generated as a result of the release of stored strain energy

Journal ArticleDOI
TL;DR: In this article, a (Zn,Cd)Se/ZnSe and Zn(S,Se) quantum well structures were used for laser diode operation.
Abstract: Laser diode operation has been obtained from (Zn,Cd)Se/ZnSe and (Zn,Cd)Se/Zn(S,Se) quantum well structures in the blue and the green. The devices, prepared on p‐ and n‐type (In,Ga)As or GaAs buffer layers for lattice matching purposes to control the defect density, have been operated at near‐room‐temperature conditions and briefly at room temperature with uncoated end facets. Quasi‐continuous wave operation has been obtained at T=77 K.

Journal ArticleDOI
TL;DR: The long duration of spin orientation is used to demonstrate deep transient oscillations, resulting from biexcitonic effects, of exciton spin relaxation in GaAs quantum wells in moderate magnetic fields.
Abstract: We report the experimental observation of exciton spin relaxation in GaAs quantum wells in moderate magnetic fields. We resolve the electron and hole contributions and discuss the large sensitivity of the spin-relaxation time to exciton localization and quantum well width. We use the long duration of spin orientation to demonstrate deep transient oscillations, resulting from biexcitonic effects

Journal ArticleDOI
TL;DR: In this paper, the modulation bandwidth of quantum well lasers can be reduced by a factor of six due to carrier transport across undoped layers of the laser as in the separate confinement heterostructure (SCH).
Abstract: It is shown experimentally that the modulation bandwidth of quantum well lasers can be reduced by a factor of six due to carrier transport across undoped layers of the laser as in the separate confinement heterostructure (SCH). Analytical expressions are given for the modulation response function, resonance frequency, damping rate and K factor to include carrier transport, and it is shown that carrier transport is responsible for a low-frequency rolloff which limits the modulation response of quantum-well lasers. It also shown that carrier transport leads to a reduction in the effective differential gain, while the gain compression factor remains largely unaffected by it. >

Journal ArticleDOI
TL;DR: A surface-emitting semiconductor laser that utilizes a concentric circle grating defined by electron-beam lithography is observed to oscillate in a circularly symmetric fashion.
Abstract: A surface‐emitting semiconductor laser that utilizes a concentric‐circle grating defined by electron‐beam lithography is observed to oscillate in a circularly symmetric fashion. The laser emits a circularly symmetric beam with a total beam divergence of less than 1°. Despite its broad‐area geometry, the laser shows no evidence of filamentation. The laser maintains a relatively narrow wavelength spectrum approximately 1 A in width.

Journal ArticleDOI
TL;DR: In this paper, the fabrication of thin GaAs quantum wires (120-200 A)×(200-300 A) by a novel metal-organic chemical-vapordeposition growth technique is reported.
Abstract: Successful fabrication of thin GaAs quantum wires (120–200 A)×(200–300 A) by a novel metal‐organic chemical‐vapor‐deposition growth technique is reported. The GaAs quantum wires were grown on a V groove formed by two GaAs triangular prisms which were selectively grown on SiO2 masked substrates. The V groove has a very sharp corner at the bottom, which results in reduction of the effective width of the quantum wire structures. The measurement of photoluminescence and photoluminescence excitation spectra with polarization dependence indicate the existence of the quantized state in the quantum wires.

Journal ArticleDOI
TL;DR: The first observation of coherent terahertz radiation is reported, tunable from 1.4 to 2.6 THz, emerging from GaAs/Al 0.3 Ga 0.7 As single quantum wells after the coherent optical excitation of both light hole and heavy hole excitons.
Abstract: We report on the first observation of coherent terahertz radiation, tunable from 1.4 to 2.6 THz, emerging from GaAs/Al 0.3 Ga 0.7 As single quantum wells after the coherent optical excitation of both light hole and heavy hole excitons. We attribute the radiation to charge oscillations following the coherent excitation of the excitons in an electric field. Terahertz radiation is also emitted when no electric field is present in the sample suggesting that valence band mixing leads to a significant far-infrared transition dipole moment between the light hole and heavy hole subbands

Journal ArticleDOI
TL;DR: In this paper, a theory of the grating-coupled quantum-well infrared detector is presented together with numerical simulation results, and a sensitivity analysis shows that a change in grating dimensions of the order of 0.1 μm results in relative changes of ηint of about 1%.
Abstract: A theory of the grating‐coupled quantum‐well infrared detector, based on the modal expansion method (MEM) is presented together with numerical simulation results. MEM is a near‐exact way of determining the electromagnetic‐field pattern in a reflection grating‐coupled quantum‐well infrared detector, and is suitable for the calculation of absorption quantum efficiencies. Both lamellar gratings and crossed (doubly periodic) gratings are dealt with. Numerical simulations show that quantum efficiencies integrated with respect to wavelength ηint equal to 1 μm may be obtained in a detector equipped with a crossed grating of square symmetry, and a waveguide. The waveguide is defined by the metal grating on one side of the infrared absorbing quantum wells, and a thick aluminum arsenide layer on the other. This implies a factor of 4 higher ηint than in a 45° polished edge detector with the corresponding quantum‐well characteristics. Finally, a sensitivity analysis shows that a change in grating dimensions of the order of 0.1 μm results in relative changes of ηint of about 1%.

Journal ArticleDOI
TL;DR: In this article, the first GaInAsSb/AlGaAsSsb diode laser with a quantum-well active region has been demonstrated, which can achieve a temperature of 113 K between 20 and 40°C.
Abstract: The first GaInAsSb/AlGaAsSb diode lasers with a quantum‐well active region have been demonstrated. These devices, which were grown by molecular beam epitaxy, emit at ∼2.1 μm. For room‐temperature pulsed operation of lasers 100 μm wide, threshold current density as low as 260 A/cm2 and differential quantum efficiency up to 70% have been obtained for cavity lengths of 2000 and 300 μm, respectively. One device 100 μm wide and 1000 μm long has operated pulsed at heatsink temperatures up to 150 °C, with a characteristic temperature of 113 K between 20 and 40 °C. For another device of the same dimensions, cw output power up to 190 mW/facet has been achieved at a heatsink temperature of 20 °C. These characteristics represent a dramatic improvement in performance over double‐heterostructure GaInAsSb/AlGaAsSb diode lasers.

Journal ArticleDOI
TL;DR: In this paper, the photorefractive properties of semi-insulating AlGaAs-GaAs multiple quantum wells are described for the transverse Franz-Keldysh geometry with the electric field in the plane of the quantum wells.
Abstract: The photorefractive properties of semi-insulating AlGaAs–GaAs multiple quantum wells are described for the transverse Franz–Keldysh geometry with the electric field in the plane of the quantum wells. Combining the strong electroabsorption of quantum-confined excitons with the high resistivity of semi-insulating quantum wells yields large nonlinear optical sensitivities. The photorefractive quantum wells have effective nonlinear optical sensitivities of n2 ≈ 103 cm2/W and α2/α0 ≈ 104 cm2/W for applied fields of 10 kV/cm. Photorefractive gains approaching 1000 cm−1 have been observed in two-wave mixing under dc electric fields and stationary fringes. The transverse Franz–Keldysh geometry retains the general transport properties and behavior of conventional bulk photorefractive materials. The resonant excitation of free electrons and holes in the quantum wells leads to novel behavior associated with electron–hole competition. We demonstrate that under resonant excitation of electrons and holes the device resolution is fundamentally limited by diffusion lengths but is insensitive to long drift lengths.

Journal ArticleDOI
TL;DR: In this article, a whispering-gallery mode microdisc laser with singlemode operation and submilliamp threshold current at room temperature was demonstrated, and it was shown that the single-mode operation can achieve sub-milliamps threshold current.
Abstract: Electrically pumped whispering-gallery mode microdisc lasers with singlemode operation and submilliamp threshold current at room temperature are demonstrated.

Journal ArticleDOI
TL;DR: In this paper, step-like features in the absorption and photo-current spectra were identified as 2D band edge transitions from their temperature dependence, revealing the large anisotropy of the electronic band structure, and showing a plateau in the currentvoltage curve, which can be explained in terms of Wannier-Stark localization in the materials.

Journal ArticleDOI
TL;DR: Quasi-two-dimensional excitons in (Zn,Cd)Se/ZnSe quantum wells were investigated and observed a confinement-induced enhancement of the exciton binding energy to a value greater than the longitudinal-optical (LO) -phonon energy.
Abstract: Quasi-two-dimensional (2D) excitons in (Zn,Cd)Se/ZnSe quantum wells were investigated. We observed a confinement-induced enhancement of the exciton binding energy to a value greater than the longitudinal-optical (LO) -phonon energy. This unusual condition results in an effective reduction of the exciton--LO-phonon coupling. As a consequence, excitonic absorption features in this quantum-well system are well preserved to room temperature and beyond.

Book
01 Jan 1992
TL;DR: In this paper, Shah et al. presented a critical review of hot-carrier relaxations in Semiconductor microstructures and their relationship with tunneling times in SIHeterostructures.
Abstract: J. Shah, Overview. B.K. Ridley, Electron-Phonon Interactions in 2D Systems. S. Das Sarma, Quantum Many-Body Aspects of Hot-Carrier Relaxation in Semiconductor Microstructures. W. P*adotz and P. Kocevar, Cooling of Highly Photoexcited Electron-Hole Plasma in Polar Semiconductors and Semiconductor Wells: A Balance-Equation Approach. A.P. Jauho, Tunneling Times in Semiconductor Heterostructures: A Critical Review. F. Rossi, R. Brunetti, and C. Jacoboni, Quantum Transport. S.M. Goodnick and P. Lugli, Hot-Carrier Relaxation in Quasi-2D Systems. I.C. Kizilyalli and K. Hess, Monte Carlo Simulation of GaAs-AlxGA1-xAS Field-Effect Transistors. J. Shah, Ultrafast Luminescence Studies of Carrier Relaxation and Tunneling in Semicondutor Nanostructures. W.H. Knox, Optical Studies of Femtosecond Carrier Thermalization in GaAs. J.F. Ryan and M.C. Tatham, Time-Resolved Raman Measurements of Electron-Phonon Interactions in Quantum Wells and Superlattices. R.A. H*adopfel, J. Shah, and S. Juen, Electron-Hole Scattering in Quantum Wells. M. Heiblum and U. Sivan, Ballistic Transport in Two-Dimensional Electron Gas. N. Yokoyama, H. Oshnishi, T. Mori, M. Takatsu, S. Muto, K. Imamura, and A. Shibatomi, Resonant-Tunneling Hot Electron Transistors. E.R. Brown, Resonant Tunneling in High-Speed Double Barrier Diodes. Chapter References. Index.

Journal ArticleDOI
TL;DR: In this article, an efficient ultrafast all-optical switching in nonlinear directional couplers made of AlGaAs and AlGaA/GaAs quantum wells near half the band gap is reported.
Abstract: Efficient ultrafast all‐optical switching in nonlinear directional couplers made of AlGaAs and AlGaAs/GaAs quantum wells near half the band gap is reported. The switching is limited by multiphoton absorption which is dominated by three‐photon absorption in this spectral range. The three‐photon absorption in the quantum well nonlinear directional coupler is stronger than that of bulk AlGaAs. Autocorrelations of the output pulses in the bar and cross states confirm pulse breakup through nonlinear coupling, and illustrate the effects of multiphoton absorption. All sets of experimental data are fitted well by a theoretical model.