DOCSLIB.ORG

Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers R ESEARCH A RTICLES cavity. Active control of the relative carrier- envelope phase prepares a stable pulse-to- Carrier-Envelope Phase Control pulse phase relation, as presented below, and will dramatically impact extreme nonlinear of Femtosecond Mode-Locked optics. Although it may be natural to think about Lasers and Direct Optical the carrier-envelope phase in the time do- main, it is also apparent in a high-resolution measurement of the frequency spectrum. The Frequency Synthesis output spectrum of a mode-locked laser con- David J. Jones,1* Scott A. Diddams,1* Jinendra K. Ranka,2 sists of a comb of optical frequencies sepa- rated by the repetition rate. However, the Andrew Stentz,2 Robert S. Windeler,2 1 1 comb frequencies are not necessarily integer John L. Hall, * Steven T. Cundiff *† multiples of the repetition rate; they may also have an offset (Fig. 1B). This offset is due to We stabilized the carrier-envelope phase of the pulses emitted by a femto- the difference between the group and phase second mode-locked laser by using the powerful tools of frequency-domain velocities. Control of the carrier-envelope laser stabilization. We confirmed control of the pulse-to-pulse carrier-envelope phase is equivalent to control of the absolute phase using temporal cross correlation. This phase stabilization locks the ab- optical frequencies of the comb, and vice solute frequencies emitted by the laser, which we used to perform absolute versa. This means that the same control of the optical frequency measurements that were directly referenced to a stable carrier-envelope phase will also result in a microwave clock. revolutionary technique for optical frequency metrology that directly connects the micro- Progress in femtosecond pulse generation has velocities differ inside the laser cavity (Fig. wave cesium frequency standard to the opti- made it possible to generate optical pulses that 1A). To date, techniques of phase control of cal frequency domain with a single laser (14). are only a few cycles in duration (1–4). This femtosecond pulses have employed time do- We used a self-referencing technique to has resulted in rapidly growing interest in con- main methods (5). However, these techniques control the absolute frequencies of the optical trolling the phase of the underlying carrier wave have not used active feedback, and rapid comb generated by a mode-locked laser. with respect to the envelope (1, 5–7). The dephasing occurs because of pulse energy Through the relation between time and fre- absolute carrier phase is normally not important fluctuations and other perturbations inside the quency described below, this method also in optics; however, for such ultrashort pulses, it can have physical consequences (6, 8). Concur- rently, mode-locked lasers, which generate a train of ultrashort pulses, have become an im- portant tool in precision optical frequency mea- surement (9–14). There is a close connection between these two apparently disparate topics. We exploited this connection to develop a fre- quency domain technique that stabilizes the carrier phase with respect to the pulse envelope. Using the same technique, we performed abso- lute optical frequency measurements using a single mode-locked laser with the only input being a stable microwave clock. Mode-locked lasers generate a repetitive train of ultrashort optical pulses by fixing the relative phases of all of the lasing longitudi- nal modes (15). Current mode-locking tech- niques are effective over such a large band- width that the resulting pulses can have a duration of 6 fs or shorter, i.e., approximately two optical cycles (2–4). With such ultra- short pulses, the relative phase between peak of the pulse envelope and the underlying electric-field carrier wave becomes relevant. In general, this phase is not constant from pulse to pulse because the group and phase Fig. 1. Time-frequency correspondence and relation between ⌬␾ and ␦.(A) In the time domain, the 1JILA, University of Colorado and National Institute of relative phase between the carrier (solid) and the envelope (dotted) evolves from pulse to pulse by Standards and Technology, Boulder, CO 80309Ð0440, ⌬␾ ␾ϭ⌬␾ ␶ ϩ␾ ␾ the amount . Generally, the absolute phase is given by (t/ ) 0, where 0 is an USA. 2Bell Laboratories, Lucent Technologies, Murray unknown overall constant phase. (B) In the frequency domain, the elements of the frequency comb Hill, NJ 07733, USA. of a mode-locked pulse train are spaced by f . The entire comb (solid) is offset from integer rep␦ϭ⌬␾ ␲ ␦ *These authors contributed equally to this work. multiples (dotted) of frep by an offset frequency f rep/2 . Without active stabilization, is †To whom correspondence should be addressed. E- a dynamic quantity, which is sensitive to perturbation of the laser. Hence, ⌬␾ changes in a mail: [email protected] nondeterministic manner from pulse to pulse in an unstabilized laser. www.sciencemag.org SCIENCE VOL 288 28 APRIL 2000 635 R ESEARCH A RTICLES serves to stabilize the carrier phase with re- the frequency domain offset ␦ enabled us to use frequency doubling a single-frequency laser spect to the envelope. This phase stabilization the powerful techniques developed for stabili- and then measuring the difference between was verified with temporal cross correlation. zation of single-frequency lasers. It is possible the fundamental and second harmonic by The utility of this method for absolute optical to lock the position of the frequency comb to a subdividing it. The subdivision may be done frequency metrology was demonstrated by known optical frequency, such as that of a by optical bisection (20, 21), comb genera- measuring the frequency of a stable single- single-frequency laser (12). However, employ- tion (22), or, most typically, a combination of frequency laser directly from a microwave ing this approach to determine and lock ␦ is the two (9, 10, 17, 23). The goal of the clock. problematic because the optical frequencies are subdivision is finally to obtain a frequency Time versus frequency. The connection Ͼ106 times the repetition rate. In addition, it interval small enough that it can be directly between the pulse-to-pulse carrier-envelope introduces the complication of a highly stabi- compared to the cesium frequency. A recent phase shift and the absolute frequency spec- lized single-frequency laser, not to mention the series of landmark experiments by Ha¨nsch trum can be understood by considering how uncertainty in the single-frequency continuous and co-workers demonstrated that mode- the spectrum is built up by a temporal train of wave (CW) laser itself. locked lasers are the preferred implementa- pulses (7, 9, 14). It is easily shown with A more elegant approach is to use a self- tion of optical comb generators (9–11). We Fourier transforms that a shift in time corre- referencing technique, which is based on recently performed absolute frequency mea- sponds to a phase shift that is linear with comparing the frequency of comb lines on the surements using only the comb generated by frequency; that is, the phase at angular fre- low-frequency side of the optical spectrum to a mode-locked laser (13). The laser and mi- quency ␻ is ␻t for a time shift of t. Within a those on the high-frequency side that have crostructure fiber were similar to those used narrow spectral bandwidth, successive pulses approximately twice the frequency. Let the here, although the technique is quite different interfere, but a signal will be observed only at frequency of comb line n, which is on the red in that the laser comb was not stabilized in ϭ frequencies where they add constructively, side of the spectrum, have a frequency fn position but rather only used to divide down ␲ ϩ␦ i.e., have a phase shift of 2n . For a pulse nfrep . The comb line corresponding to 2n, an optical frequency interval. train with time ␶ between pulses, these fre- which will be the blue side of the spectrum, The self-referencing technique described ϭ ␶ϭ ϭ ϩ␦ quencies are fn n/ nfrep, where n is an will have frequency f2n 2nfrep .We here is a dramatic step beyond these previous ϭ ␶ ␦ integer and frep 1/ is the repetition fre- obtain by frequency doubling fn and then techniques because it uses only a single ϭ␦ quency of the pulse train. Thus, we recover taking the difference 2fn – f2n . To imple- mode-locked laser and does not need any the fact that the frequency spectrum consists ment the technique in a simple way, we need stabilized single-frequency lasers. We think of a comb of frequencies spaced by the rep- an optical spectrum that spans a factor of 2 in that it will make precision absolute optical etition rate of the pulse train. If we include frequency, known as an optical octave. This frequency metrology into an easily accessible the pulse-to-pulse phase shift ⌬␾, then the is obtained by spectrally broadening the laser laboratory tool. phase difference between successive pulses at pulse in air-silica microstructure fiber (16). Laser and stabilization. The heart of the angular frequency ␻ is now ␻␶Ϫ⌬␾. Again, Full experimental detail is given below. experiment is a titanium-doped sapphire for constructive interference, this phase dif- Optical frequency metrology tech- (Ti:S) laser (shown in Fig. 2) that is pumped ference is set equal to 2n␲, which shows that niques. Optical frequencies are preferred for with a single-frequency, frequency-doubled ϭ ϩ␦ now the frequencies are fn nfrep , where the precision measurements that test funda- Nd:YVO4 laser operating at 532 nm. The Ti:S ␲␦ϭ⌬␾ 2 frep. Hence, a pulse-to-pulse phase mental physical theories (17, 18). In part, this laser generates a 90-MHz pulse train with pulse shift between the carrier and envelope corre- preference is associated with the very narrow widths as short as 10 fs using Kerr lens mode- sponds to an offset of the frequency comb fractional linewidths displayed by optical res- locking (2).
Load more

Recommended publications
  • First-Principles Envelope-Function Theory for Lattice-Matched Semiconductor Heterostructures
    PHYSICAL REVIEW B 72, 165345 ͑2005͒ First-principles envelope-function theory for lattice-matched semiconductor heterostructures Bradley A. Foreman* Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China ͑Received 17 June 2005; revised manuscript received 26 August 2005; published 28 October 2005͒ In this paper a multiband envelope-function Hamiltonian for lattice-matched semiconductor heterostructures is derived from first-principles self-consistent norm-conserving pseudopotentials. The theory is applicable to isovalent or heterovalent heterostructures with macroscopically neutral interfaces and no spontaneous bulk polarization. The key assumption—proved in earlier numerical studies—is that the heterostructure can be treated as a weak perturbation with respect to some periodic reference crystal, with the nonlinear response small in comparison to the linear response. Quadratic response theory is then used in conjunction with k·p perturbation theory to develop a multiband effective-mass Hamiltonian ͑for slowly varying envelope functions͒ in which all interface band-mixing effects are determined by the linear response. To within terms of the same order as the position dependence of the effective mass, the quadratic response contributes only a bulk band offset term and an interface dipole term, both of which are diagonal in the effective-mass Hamiltonian. The interface band mixing is therefore described by a set of bulklike parameters modulated by a structure factor that determines the distribution of atoms in the heterostructure. The same linear parameters determine the interface band-mixing Hamiltonian for slowly varying and ͑sufficiently large͒ abrupt heterostructures of arbitrary shape and orientation. Long-range multipole Coulomb fields arise in quantum wires or dots, but have no qualitative effect in two-dimensional systems beyond a dipole contribution to the band offsets.
    [Show full text]
  • Laser Envelope Model In
    Laser Envelope model in Maison de la Simulation Francesco– March 2019 Massimo Francesco Massimo Outline I. Motivations II. Modeling Laser Wakefield Acceleration with a laser envelope III. Case study: Laser Wakefield Acceleration with external injection IV. Perspectives and future developments Laser Wakefield Acceleration (LWFA): Characteristic Scales 1.5 Laser Beam Duration: 28 fs Transverse 1.2 Electric Field Ponderomotive (A.U.) Force: 0.9 F= - ILaser 0.6 Electrons Immobile Ions Mi >>me 0.003 Plasma density: 3.1018 cm-3 0.002 Electron Charge Density Plasma 0.001 (A.U.) wavelength: ~20 μm 0.000 Laser Wakefield Acceleration (LWFA): Characteristic Scales E > 100 GV/m 0.04 Longitudinal Electric Field Electrons 0.00 (A.U.) injected here on propagation axis are accelerated -0.04 0.003 Laser wavelength: Electron ~1 μm 0.002 Charge Laser duration ~ Density Plasma wavelength: 0.001 (A.U.) ~20 μm 0.000 0.000 LWFA PIC simulations are cumbersome X. Davoine, Physics of Plasmas 15, 113102 (2008) 2D cartesian simulations are not accurate enough for LWFA! 3D LWFA simulations: 1 mm plasma ~ 320 kcpu-heures ~ 10.2 k€ Laser Envelopes need less sampling points Spatial variation scales: Laser Envelope Laser length ~ Plasma wavelength (10-100 μm) Laser wavelength Laser “Standard” (~1 μm) Points sampling Laser “Standard” = 10 Points sampling Laser Envelope Larger Δx and Δt can be used! Structure of a PIC code with Laser Envelope Standard PIC Envelope (/Ponderomotive) PIC ~ E, B field E,B field Envelope A (laser) (plasma fields+laser fields) (plasma fields) Lorentz Lorentz Force Ponderomotive Force Force Plasma Current Current Density J Density J Susceptibility Particles Particles The Laser Envelope evolution: wave equation D.
    [Show full text]
  • The K·P Method Lokc.Lewyanvoon· Morten Willatzen
    The k·p Method LokC.LewYanVoon· Morten Willatzen The k·p Method Electronic Properties of Semiconductors 123 Dr. Lok C. Lew Yan Voon Dr. Morten Willatzen Wright State University University of Southern Denmark Physics Dept. Mads Clausen Institute for 3640 Colonel Glenn Highway Product Innovation Dayton OH 45435 Alsion 2 USA 6400 Soenderborg [email protected] Denmark [email protected] ISBN 978-3-540-92871-3 e-ISBN 978-3-540-92872-0 DOI 10.1007/978-3-540-92872-0 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926838 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Uber¨ Halbleiter sollte man nicht arbeiten, das ist eine Schweinerei; wer weiss ob es uberhaupt¨ Halbleiter gibt.
    [Show full text]
  • Introducing the Rossby Wave Packet Tool
    Introducing the Rossby Wave Packet Tool Mike Bodner HPC/DTB 25 August 2011 Acknowledgements The work on developing a wave packet tool is a collaborative effort between HPC and Stony Brook University, and is part of C-Star funding. Special thanks to Brian Colle and Edmund Chang for leading this effort, and to Matt Souders for sharing computer code and his research materials with HPC. Gratitude extended to Bob Daniels from OPC for assistance in configuring Matlab. Training Outline • Overview of Wave Packets • How to View Wave Packets • Methods of Computing Wave Packet Envelopes • Wave Packet Climatology • Utilization of Wave Packets Why Wave Packets? • High impact weather has been found to be associated with wave packets (Archambault et al. 2010) • 500 hPa geopotential heights move with phase velocity, while wave packets propagate with group velocity which can be an indicator of downstream development • Potential indicator of uncertainty – forecast errors can develop and propagate like wave packets (Hakim 2005) What are Wave Packets ? • Baroclinic waves propogate chaotically and are driven primarily by phase speed • A wave packet encompasses several waves and will propagate with the group velocity which is much faster than the phase speed of individual troughs ▪ Red curves are two sine curves which can represent troughs and ridges ▪ Blue curve is the wave packet which encompasses the toughs/ridges The figure on the right shows 300 hPa geopotenital heights for the period 18-28 December, 1985 (Chang 1993). ▪ Wave packets are useful in forecasting because since they propagate with this much faster group velocity, the packets will often result in downstream development.
    [Show full text]
  • Wavepacket and Dispersion
    Wavepacket and Dispersion Andreas Wacker1 Mathematical Physics, Lund University September 18, 2017 1 Motivation A wave is a periodic structure in space and time with periods λ and T , respectively. Common examples are water waves, electromagnetic waves, or sound waves. The spatial structure is 2π 2π A cos(kz − !t + ') = RefA~ei(kz−!t)g with A~ = ei'A; k = and ! = (1) λ T where the complex representation often simplifies the maths significantly. The angular fre- quency ! and the angular wavenumber k (Note that the term wavenumber refers to 1/λ in spectroscopy) are related to each other by a dispersion relation !disp(k), which is a property of the material considered. For linear media considered here, the dispersion relation is indepen- dent of the amplitude A and different waves can be superimposed (i.e. added) without affecting each other. The structure of waves is described by the phase φ = kz − !t + ', where certain values, such as φ = 0 or φ = π=2, correspond to positions of maximal or vanishing amplitude, respectively. If we considers the position zp for a point with constant phase in time, we find dzp=dt = !=k = vp, which defines the phase velocity vp = !disp(k)=k. This is the velocity of the maxima in a single monochromatic wave as given by Eq. (1). In order to transmit any information between two places, the recipient cannot handle a single wave, as all peaks look the same. Thus we require some modulation, so that characteristic spatial structures are transmitted. This can be achieved by interfering waves with different wavenumbers/frequencies.
    [Show full text]
  • Laser Wakefield Simulation Using a Speed-Of-Light Frame Envelope Model
    Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Title Laser wakefield simulation using a speed-of-light frame envelope model Permalink https://escholarship.org/uc/item/41h1k8kf Author Cowan, B. Publication Date 2009-01-22 Peer reviewed eScholarship.org Powered by the California Digital Library University of California Laser wakefield simulation using a speed-of-light frame envelope model , B. Cowan∗, D. Bruhwiler∗, E. Cormier-Michely, E. Esareyy ∗∗, C. G. R. Geddesy, P. Messmer∗ and K. Paul∗ ∗Tech-X Corporation, 5621 Arapahoe Ave., Boulder, CO 80303 yLOASIS program, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 ∗∗University of Nevada, Reno, Reno, NV 89557 Abstract. Simulation of laser wakefield accelerator (LWFA) experiments is computationally inten- sive due to the disparate length scales involved. Current experiments extend hundreds of laser wave- lengths transversely and many thousands in the propagation direction, making explicit PIC simula- tions enormously expensive and requiring massively parallel execution in 3D. We can substantially improve the performance of laser wakefield simulations by modeling the envelope modulation of the laser field rather than the field itself. This allows for much coarser grids, since we need only resolve the plasma wavelength and not the laser wavelength, and therefore larger timesteps. Thus an envelope model can result in savings of several orders of magnitude in computational resources. By propagating the laser envelope in a frame moving at the speed of light, dispersive errors can be avoided and simulations over long distances become possible. Here we describe the model and its implementation, and show simulations and benchmarking of laser wakefield phenomena such as channel propagation, self-focusing, wakefield generation, and downramp injection using the model.
    [Show full text]
  • Laser Wakefield Simulation Using a Speed-Of-Light Frame Envelope Model
    Laser wakefield simulation using a speed-of-light frame envelope model , B. Cowan∗, D. Bruhwiler∗, E. Cormier-Michely, E. Esareyy ∗∗, C. G. R. Geddesy, P. Messmer∗ and K. Paul∗ ∗Tech-X Corporation, 5621 Arapahoe Ave., Boulder, CO 80303 yLOASIS program, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 ∗∗University of Nevada, Reno, Reno, NV 89557 Abstract. Simulation of laser wakefield accelerator (LWFA) experiments is computationally inten- sive due to the disparate length scales involved. Current experiments extend hundreds of laser wave- lengths transversely and many thousands in the propagation direction, making explicit PIC simula- tions enormously expensive and requiring massively parallel execution in 3D. We can substantially improve the performance of laser wakefield simulations by modeling the envelope modulation of the laser field rather than the field itself. This allows for much coarser grids, since we need only resolve the plasma wavelength and not the laser wavelength, and therefore larger timesteps. Thus an envelope model can result in savings of several orders of magnitude in computational resources. By propagating the laser envelope in a frame moving at the speed of light, dispersive errors can be avoided and simulations over long distances become possible. Here we describe the model and its implementation, and show simulations and benchmarking of laser wakefield phenomena such as channel propagation, self-focusing, wakefield generation, and downramp injection using the model. INTRODUCTION Laser-driven plasma wakefield accelerators (LWFAs) are capable of producing acceler- ating gradients several orders of magnitude higher than conventional accelerators. Sev- eral years ago, high-quality electron beams were produced by self-trapping and acceler- ated to 100MeV in a few millimeters [1, 2, 3].
    [Show full text]
  • The Influence of a Pressure Wavepacket's Characteristics on Its
    The influence of a pressure wavepacket’s characteristics on its acoustic radiation R. Serré, Jean-Christophe Robinet, F. Margnat To cite this version: R. Serré, Jean-Christophe Robinet, F. Margnat. The influence of a pressure wavepacket’s character- istics on its acoustic radiation. Journal of the Acoustical Society of America, Acoustical Society of America, 2015, 137 (6), pp.3178-3189. 10.1121/1.4921031. hal-02569482 HAL Id: hal-02569482 https://hal.archives-ouvertes.fr/hal-02569482 Submitted on 11 May 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The influence of a pressure wavepacket’s characteristics on its acoustic radiation R. Serre and J.-C. Robinet Arts et Metiers ParisTech, DynFluid, 151 Boulevard de l’Hopital, 75013 Paris, France a) F. Margnat Institut PPRIME, Department of Fluid Flow, Heat Transfer and Combustion, Universite de Poitiers - ENSMA - CNRS, Building B17 - 6 rue Marcel Dore, TSA 41105, 86073 Poitiers Cedex 9, France Noise generation by flows is modeled using a pressure wavepacket to excite the acoustic medium via a boundary condition of the homogeneous wave equation. The pressure wavepacket is a generic representation of the flow unsteadiness, and is characterized by a space envelope of pseudo- Gaussian shape and by a subsonic phase velocity.
    [Show full text]
  • 2.2 the Nonlinear Schrödinger Equation As an Envelope Equation
    20 V. Hakim Another interesting elementary example of application of the multiscale method is the computation of the size of the limit cycle of a Van der Pol oscillator: x¨ + ǫ x2 1 x˙ + x =0 for ǫ 1 (2 .13) − ≪ The zeroth order solution is x0 = A cos( t + φ) (2 .14) With an arbitrary amplitude A. Introducing a slow time scale like in (2.6), the cancellation of secular terms at first order gives the evolution of the amplitude dA A3 2 = A (2 .15) dT − 4 This shows that the amplitude of the limit cycle is equal to 2 and that (2.15) describes the approach to the limit cycle on a time scale that is long compared to the oscillation frequency. At next order, the change of the period of the limit cycle appears 1 ω = 1 ǫ2 (2 .16) − 16 This computation is left as an exercise to the reader. It is described in detail in S. Fauve’s lectures. 2.2 The nonlinear Schr¨odinger equation as an envelope equation for small amplitude gravity waves in deep water 2.2.1 A short introduction to water waves The problem of water waves has played a central role in the development of nonlinear physics, with Stokes’s discovery that dispersion relation in- volves the amplitude for periodic nonlinear wave trains, Russell’s first observation of a solitary wave, the derivation of model equations by Boussinesq, Rayleigh and Korteweg-de Vries . Moreover, the observa- tion of water waves is a common but fascinating experience. A good introduction to the subject can be found in Stoker (1957) and Whitham (1974).
    [Show full text]
  • A Spatial Envelope Soliton
    A Soliton Model of the Electron with an internal Nonlinearity cancelling the de Broglie-Bohm Quantum Potential ROALD EKHOLDT§ Abstract The paper proposes an envelope soliton model of the electron that propagates as a protuberance on a fictitious waveguide, which acts as trajectory. The model is based on de Broglie’s original electron wave-particle relativistic theory, and on the observation that the Klein-Gordon equation governs the propagation mode of a common microwave waveguide—with the Schrödinger equation as a low group velocity approximation. This analogy opened for practical physical models, including solitons. In the linear case a conceived corpuscle zigzags within the fictitious waveguide—a zigzagging that resembles Penrose’s picture of Dirac’s electron theory. The soliton envelope is defined by a special nonlinear version of the Schrödinger equation. The nonlinearity cancels the de Broglie/Bohm Quantum Potential of the envelope. The wavefunction is confined by the envelope, and thus the concept of the collapse of the wavefunction is eliminated. Since the electron model is based directly on the Special theory of relativity it also illustrates that theory. The corpuscle, incorporating spin and charge, which moves at the speed of light, needs further study relating it to the photon. A nonlinear model of the photon, based on mutual coupling of two modes of an optical fiber at the Planck scale, is suggested. One of these modes is thought to carry two orthogonal polarized electromagnetic external fields, while the other, internal, mode carries a corpuscle that moves helically and thus represents the spin. A single photon is considered to have one linear polarization only, without any possibility for a circular polarization; hence, the concept of collapse in a polarization measurement is eliminated.
    [Show full text]
  • The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time
    fluids Article The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time Amin Chabchoub 1,2,†,* and Roger H. J. Grimshaw 3,† 1 Graduate School of Frontier Sciences, The University of Tokyo, Chiba 277-8563, Japan 2 Department of Mechanical Engineering, Aalto University, Espoo 02150, Finland 3 Department of Mathematics, University College London, London, WC1E 6BT, UK; [email protected] * Correspondence: [email protected]; Tel./Fax: +81-7136-4880 † These authors contributed equally to this work. Academic Editor: Pavel Berloff Received: 11 April 2016; Accepted: 11 July 2016; Published: 19 July 2016 Abstract: The nonlinear Schrödinger equation (NLS) is a canonical evolution equation, which describes the dynamics of weakly nonlinear wave packets in time and space in a wide range of physical media, such as nonlinear optics, cold gases, plasmas and hydrodynamics. Due to its integrability, the NLS provides families of exact solutions describing the dynamics of localised structures which can be observed experimentally in applicable nonlinear and dispersive media of interest. Depending on the co-ordinate of wave propagation, it is known that the NLS can be either expressed as a space- or time-evolution equation. Here, we discuss and examine in detail the limitation of the first-order asymptotic equivalence between these forms of the water wave NLS. In particular, we show that the the equivalence fails for specific periodic solutions. We will also emphasise the impact of the studies on application in geophysics and ocean engineering. We expect the results to stimulate similar studies for higher-order weakly nonlinear evolution equations and motivate numerical as well as experimental studies in nonlinear dispersive media.
    [Show full text]
  • CHAPTER 7 Envelope Solitary Waves
    CHAPTER 7 Envelope solitary waves R.H.J. Grimshaw Department of Mathematical Sciences, Loughborough University, Loughborough, UK. Abstract In this chapter, we discuss envelope solitary waves that arise when a plane peri- odic wave is amplitude modulated. In general, the modulation leads to a rapidly oscillating carrier wave with a slowly varying envelope. In the weakly nonlinear regime, envelope waves are described by the nonlinear Schrödinger equation when attention is confined to just one spatial dimension. We shall focus our attention on capillary–gravity waves, although envelope waves can arise in many other fluid flow situations. When the nonlinear Schrödinger equation is of the focusing type, it supports solitary waves (solitons). However, these solitary waves do not lead to a steady solution of the full physical system, as the carrier’s phase speed and the envelope’s group velocity are not usually equal. But in the special circumstances when the phase and group velocities do coincide, there is a bifurcation to a steady envelope solitary wave. Our main interest is in capillary–gravity waves, and for these we give many details. Extensions to waves in more than one space dimension is based on the Benney–Roskes system, and leads to ‘lump’ solitary waves. 1 Linear waves and group velocity As noted in Chapter 1, there are two classes of solitary waves of interest, each of which can be regarded as a bifurcation from those points in the linear spec- trum where the group velocity and phase velocity are equal. When this bifurcation occurs in the long-wave limit, it leads typically to the Korteweg–de Vries equation and its solitary wave solutions.
    [Show full text]