Employing a Fox-Li approach, we derived the cold-cavity mode structure and a coupled mode theory for a phased array of N single-transverse-mode active waveguides with feedback from an external cavity. We applied the analysis to a system with arbitrary laser lengths, external cavity design and coupling strengths to the external cavity. The entire system was treated as a single resonator. The effect of the external cavity was modeled by a set of boundary conditions expressed by an N-by-N frequency-dependent matrix relation between incident and reflected fields at the interface with the external cavity. The coupled mode theory can be adapted to various types of gain media and internal and external cavity designs.
The existing model of the LOCSET technique for the active phase synchronization of fiber laser arrays (T. Shay, Opt.
Express, 2006) is extended to include relevant physical properties of the system, such as inherent optical path
differences (OPD), line-width and group velocity dispersion (GVD), and we also include phase “jitter” of the master
oscillator’s output in the model, which in experiments is implemented to induce spectral broadening for suppression of
nonlinear frequency conversion. Linearization of the phase error signal, which incorrectly predicts convergence to a
synchronous equilibrium state, is not performed. Instead, the closed-loop control dynamics are shown to be described
by differential equations of Kuramoto type when phase corrector response dynamics are negligible. Linear stability
analysis indicates that there is always one and no more than one dynamically stable state. The latter is shown to be
normally synchronous, except when strong “jitter” is applied. A Liapounov function is found as subject to the validity of
certain symmetry conditions.
We propose a systematic approach that may apply to many complex interactive networks, such as biological or
electronic neural assemblies, which was partly inspired by mathematical features of phased laser arrays. Using an
appropriate quasi-logarithmic transformation, a Fox-Li integral equation of linearly coupled phased laser arrays is
mapped to a semi-equivalent coupled oscillator description, of which the interaction term is decomposed into
orthogonal projections. Based on traditional ideas of symmetry, orthogonality, completeness, and the physical
concept of criticality, techniques are proposed for the description of the dynamics and organization of massively
nonlinearly interconnected networks, which may serve as memories, or perform computational operations in
biological neuron assemblies, or models of evolution, pathology, ecological and social networks, individual and
collective behavior, etc.
We solve the time-independent (o.d.e.) propagation equations for the c.w. operation of an array of externally coupled
fiber amplifiers with internal reflection, in which Kerr and resonant nonlinearities play a part. A transcendental equation
for each amplifier is obtained, collectively yielding multiple distinct array solutions, which are characterized in terms of
their mutual phase coherence. We find that the two types of nonlinearity (Kerr and resonant) affect the solutions in
distinct parameter regimes. The relation of Strehl ratio to output power at fixed wavelength and feedback level reveals
that phase-locking may occur due to nonlinearity as opposed to mode selection, in accordance with recent experiments
(H-S Chiang, J.R. Leger, J. Nilsson and J. Sahu, Optics Letters (2013)). The individual lasing instability (“rogue”
lasing) anticipated by A.E. Siegman in 2004 we observed only at low feedback levels in a small number of cases.
We present an experimental study on active coherent combining of five Yb (Ytterbium)-doped fiber laser amplifiers that
employs multiplexed volume Bragg gratings (MVBGs), reporting a combining efficiency of 82% and near-diffraction
limited beam quality at a combined input power of 380 W, and 70% combining efficiency with equal beam quality at
670 W input power.
We propose a simple physical mechanism to explain observed instabilities in the dynamics of passively phased fiber amplifier arrays that arises from two properties: First that a weak phase disturbance of the output field of the array is converted into a strong intensity disturbance through the mode-selective feedback mechanism. Second, that this intensity fluctuation regenerates a phase fluctuation due to the nonlinear properties of the amplifying media. At sufficiently high operating power levels this cyclic disturbance continues to grow upon each cavity round trip, creating instability. This simple picture is supported by the results of a linear stability analysis of the set of propagation and population rate equations, which are in good agreement with observed critical power levels. A third level of quantitative confirmation was obtained by comparison to the results of numerical integration of the original set of nonlinear equations. This predicted instability is entirely a property of passively phased arrays of more than one element.
We performed a linearized stability analysis and preliminary simulations of passive phasing in a CW operating ring-geometry fiber laser array coupled in an external cavity with a single-mode feedback fiber that functions as spatial filter. A two-element array with path length error is predicted to have a dynamically stable stationary operating state at the compputer operating wavelength.
Motivated by the nonlinear phase measurements in rare-earth-doped fibers reported by Digonnet, et al (J. Lightwave Technol. 15, 299 (1997)), I provide theoretical evidence of passive synchronous phasing in ytterbium-doped multi-core fiber laser arrays of up-to 7 cores due to the existence of a negative resonant nonlinear index to which transitions at UV frequencies contribute. The effect appears to become more elusive with increasing array size, and confirmation for a 19-core array, for example, is still sought.
We used coupled-mode theory in an Yb-doped multimode fiber amplifier to compute the effects of gain saturation, nonlinear index, and fiber curvature on the evolution of the field. A positive nonlinear index results in power transfer to lower-order modes, usually the fundamental LP10 mode, and for negative nonlinear index the reverse is predicted. The nonlinear interaction between modes breaks the core’s cylindrical symmetry, resulting in recombination of degenerate LP mode pairs into super-modes: consisting of an expected in- and anti-phase pair, but also a quadrature of super-mode that reflects an increase of “information” capacity associated with nonlinearity. Convergence to all three super-modes was observed in our simulations, but the last more often. We also present observed evidence of mode phasing in experiments with two fiber amplifiers.
We extend a model of the evanescent field-coupled multi-core Yb-doped fiber laser to include fluctuations of the level populations with the goa lof determining the stability properties of collective modes. The nonlinear differential equations for light intensity, the phases of the electric fields, and the occupation number of the upper laser level of each core are integrated numerically, while, for two cores the relaxation rates, oscillation frequencies, and stability criteria are determined analytically from the set of linearized equations. For example, for two identical lasers we find the in-phase super mode to be unstable in the range 0<κ/Δβnl<1, where κ is the inter-guide coupling constant and Δβnl the non-linear change in the mode propagation constant. The system is bi-stable for |κ/Δβnl|>1, and we describe how switching can be performed. Besides evanescent-field coupled lasers the analysis may be applicable to other coupling mechanisms, such as external cavities, although with some caution.
A theory is presented that explains the observed spontaneous transition to the high-brightness synchronous array-state of a multicore fiber laser array. The mechanism, based on nonlinear refraction, is shown to be robust. Results, so far of a 19-core array, indicate possible scalability toward much larger array size.
Defocusing, angular, and linear alignment tolerances, and the effects of transform lens aberrations are calculated for spectral beam combining of fiber lasers, first for a cavity in which no microlens array is used in the collimating optics. Two design criteria are derived and used to compare four lenses: a simple bi-spherical lens, a compound quadruplet, a plano-aspheric lens, and a plano-parabolic lens. The results point to superior performance of simple aspheric lenses over compound lenses with spherical surfaces. A kind of equivalence of efficiency and beam quality is demonstrated. Partial results on the inclusion of a microlens array for improved beam collimation predicts marked increase of the maximum array size and overall efficiency.
Spectrally multiplexed, high brightness, beam combining for power scaling and WDM communications applications has advantages over coherent combining, and is suitable for use with high gain bandwidth lasers. A diffraction grating forces oscillation at distinct wavelengths in a laser array in the external resonator configuration proposed by T.Y. Fan, et al [Opt. Lett.25 , 405-407 (2000)], producing a multi-wavelength output of overlapping beams. A unified diffraction integral-based treatment is used to formulate a design principle by which to jointly optimize most significant measures of performance, such as efficiency, bandwidth, and beam quality. The sensitivity to errors in alignment and positioning is also characterized. The transform lens aberrations are shown to ultimately determine the maximum array size, and the design criterion is applied to test and propose improvements in the lens. Initial comparisons show agreement within estimated experimental uncertainty margins with MIT/LL proof-of-principle measurements on an Yb-doped fiber laser array. To illustrate, we foresee a feasible rectilinear 100 - emitter array system using a multi-element aspheric lens, with <5%array-center-to-end efficiency drop, yielding beam quality M2 ? 1.2 in the transverse (array) direction, and close to unity in the lateral direction. Application to the efficient coupling of a laser array to a multimode fiber core is discussed as fmal example.
An analytical/numerical approach, based on the Van Vleck Green function, was developed in order to analyze spectrally multiplexed beam combining of a linear array of fiber lasers, in which a blazed diffraction grating located in an external cavity plays the role of a spatially dispersive element. The focus is on the laser-external cavity coupling, which determines excess cavity loss, as affected by primary aberrations of the transform lens. Other issues are also touched upon, however, such as beam quality and bandwidth dependence on element location, and the optimization of the latter. Analytical results were supported by a more general numerical implementation. Typical values for bandwidth were found optimally as low as a few GHz, which, being substantially narrower than the free-running fiber laser line-width, maximum power limits are foreseeably determined by stimulated light scattering. A rough but encouraging degree of agreement of the resonator theory with independent lens aberration calculations and preliminary experiments performed at MIT/LL using triplet and quadruplet transform lenses was encountered so far.
A simple optical rate equation for a distributed feedback laser is derived following an analytical procedure that is based on transforming a known integral equation into an equivalent differential equation. Using this equation to model fluctuations in the phase and photon number, when supplemented with a rate equation for the carrier number expressions for the relaxation oscillation characteristics and linewidth are derived. We find that in a simple distributed feedback structure the relation for the relaxation oscillation frequency is identical to that of the Fabry-Perot laser. An effective linewidth-broadening factor is predicted showing strong dependence on longitudinal hole- burning. Power re-broadening of the linewidth and a nearly vanishing power-independent component are predicted. Finally, rate equations for injection-locking are derived, and a symmetric dynamically stable locking band predicted.
A new, broadly applicable technique for obtaining optical rate equations is derived for laser cavities having a variety of possible exterior couplings, including field injection and feedback. The method, based on analytical continuation of the roundtrip condition, retains essential resonator physics over the whole spectral range and all magnitudes of coupling, in contrast to existing rate equations. Injection locking, external feedback, and phase conjugative interaction are discussed as applications.
The effects of phase-conjugate feedback on a semiconductor laser are studied, beginning with a derivation based on linear response theory of the single-mode field rate equation. For sufficiently high feedback levels and fast-responding PCM, a multimode extermal cavity mode spectrum emerges analogous to that of the conventional feedback case, except that the spacing of the modes is half that of the latter, and tuning is possible by varying the frequency of the PCMs pump laser relative to that of the free-running laser. The latter may also be used as a control parameter to achieve stable operation at high feedback levels, and to maximize the fraction of energy in the central 'spike' of the laser spectrum. The extermal cavity modes also show up as spikes in the RIN and phase noise spectra. Finally, the effects of pump linewidth on laser linewidth reduction and finite PCM response time are described.
The theory of feedback into a diode laser by an externally located reflector is developed for the case of arbitrary feedback strength. Assuming typical values for laser diode parameters, it is predicted that by employing a nearby (13.5mm) external partially transmitting mirror, line narrowing by a factor of at least 2000 is possible without increasing the threshold gain of the laser, while maintaining dynamical stability of the output and flattening the modulation transfer function up to about 8 GHz.
The interaction of a dipole radiator with a phase-conjugate mirror through a quantized radiation field leads to dramatic enhancement of the nonlinearity if the radiator is a slightly anharmonic oscillator. Quantum noise from the conjugator saturates the linear and nonlinear response of a two-level radiator when the reflectivity exceeds unity.
Acceleration of a relativistic electron beam by the longitudinal field component of a focused laser beam is optimal when the laser beam is radially polarized. A variational calculation shows that the electron energy gain in a laser beam of given power cannot greatly exceed that available from a Gaussian beam. Particle trajectories calculated in a radially polarized Gaussian beam indicate that transverse forces focus and recollimate appropriately phased off- axis electrons over the acceleration length.
It is shown that high energy gains are possible in a gas-loaded laser linac using a nondiffracting beam. The authors derive expression for optimum gas index as a function of particle injection energy and other parameters, including the case of longitudinal grading, for both Gaussian and non diffracting beams. The properties of pulsed laser beams are discussed.
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