An official website of the United States government
A significant challenge in the modeling of short pulse fiber lasers is that with each successive generation there has been a dramatic increase in the amount by which the pulse varies over each round trip. Therefore, lumped rather than averaged models are required to accurately compute the periodically stationary (breather) solutions generated by these lasers. We use a spectral method to assess the linear stability of periodically stationary pulses in lumped models. This approach extends previous work by Menyuk and Wang on stationary pulses in averaged models. We first present a gradient based optimization method inspired by the work of Ambrose and Wilkening to compute periodically stationary pulses. Then, we use Floquet theory to characterize the linear stability of the pulses obtained using optimization in terms of the spectrum of the monodromy operator,M, obtained by linearization of the round trip operator about a periodically stationary pulse.
more »
« less
The essential spectrum of periodically stationary pulses in lumped models of short‐pulse fiber lasers
Abstract In modern short‐pulse fiber lasers, there is significant pulse breathing over each round trip of the laser loop. Consequently, averaged models cannot be used for quantitative modeling and design. Instead, lumped models, which are obtained by concatenating models for the various components of the laser, are required. As the pulses in lumped models are periodic rather than stationary, their linear stability is evaluated with the aid of the monodromy operator obtained by linearizing the round‐trip operator about the periodic pulse. Conditions are given on the smoothness and decay of the periodic pulse that ensure that the monodromy operator exists on an appropriate Lebesgue function space. A formula for the essential spectrum of the monodromy operator is given, which can be used to quantify the growth rate of continuous wave perturbations. This formula is established by showing that the essential spectrum of the monodromy operator equals that of an associated asymptotic operator. Since the asymptotic monodromy operator acts as a multiplication operator in the Fourier domain, it is possible to derive a formula for its spectrum. Although the main results are stated for a particular experimental stretched pulse laser, the analysis shows that they can be readily adapted to a wide range of lumped laser models.
more »
« less
- Award ID(s):
- 2106203
- PAR ID:
- 10443386
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- Volume:
- 150
- Issue:
- 1
- ISSN:
- 0022-2526
- Page Range / eLocation ID:
- p. 218-253
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract Radiofrequency (RF) injection locking and spectral broadening of a terahertz (THz) quantum‐cascade vertical‐external‐cavity surface‐emitting laser (QC‐VECSEL) is demonstrated. An intracryostat VECSEL focusing cavity design is used to enable continuous‐wave lasing with a cavity length over 30 mm, which corresponds to a round‐trip frequency near 5 GHz. Strong RF current modulation is injected to the QC‐metasurface electrical bias to pull and lock the round‐trip frequency. The injection locking range at various RF injection powers is recorded and compared with the injection locking theory. Moreover, the lasing spectrum broadens from 14 GHz in free‐running mode to a maximum spectral width around 110 GHz with 20 dBm of injected RF power. This experimental setup is suitable for further exploration of active mode‐locking and picosecond pulse generation in THz QC‐VECSELs.more » « less
-
The starting point of analysis of state space models is investigating existence, uniqueness and solution properties such as the semigroup property, and various formulas for the solutions. Several concepts such as the state transition matrix, the matrix exponential, the variations of constants formula (the Cauchy formula), the Peano-Baker series, and the Picard iteration are used to characterize solutions. In this note, a tutorial treatment is given where all of these concepts are shown to be various manifestations of a single abstract method, namely solving equations using an operator Neumann series involving the Volterra operator of forward integration. The matrix exponential, the Peano-Baker series, the Picard iteration, and the Cauchy formula can be "discovered" naturally from this Neumann series. The convergence of the series and iterations is a consequence of the key property of asymptotic nilpotence of the Volterra operator. This property is an asymptotic version of the nilpotence property of a strictly-lower-triangular matrix.more » « less
-
Abstract This work offers a new prospective on asymptotic perturbation theory for varying self‐adjoint extensions of symmetric operators. Employing symplectic formulation of self‐adjointness, we use a version of resolvent difference identity for two arbitrary self‐adjoint extensions that facilitates asymptotic analysis of resolvent operators via first‐order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati‐type differential equation and the first‐order asymptotic expansion for resolvents of self‐adjoint extensions determined by smooth one‐parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard–Rellich‐type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self‐adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig–Penney model, elliptic second‐order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.more » « less
-
Abstract The semiclassical (small dispersion) limit of the focusing nonlinear Schrödinger equation with periodic initial conditions (ICs) is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that solutions arising from a certain class of ICs, referred to as “periodic single‐lobe” potentials, share the same qualitative features, which also coincide with those of solutions arising from localized ICs. The spectrum of the associated scattering problem in each of these cases is then numerically computed, and it is shown that such spectrum is confined to the real and imaginary axes of the spectral variable in the semiclassical limit. This implies that all nonlinear excitations emerging from the input have zero velocity, and form a coherent nonlinear condensate. Finally, by employing a formal Wentzel‐Kramers‐Brillouin expansion for the scattering eigenfunctions, asymptotic expressions for the number and location of the bands and gaps in the spectrum are obtained, as well as corresponding expressions for the relative band widths and the number of “effective solitons.” These results are shown to be in excellent agreement with those from direct numerical computation of the eigenfunctions. In particular, a scaling law is obtained showing that the number of effective solitons is inversely proportional to the small dispersion parameter.more » « less
