Understanding the bifurcation to traveling waves in a class-Blaser using a degenerate Ginzburg-Landau equation

Abstract

We analyze the Risken-Nummendal -Graham-Haken instability occurring in a homogeneously broadened two-level unidirectional ring laser modeled by the Maxwell-Bloch equations. We investigate the class-B limit of these equations and derive partial differential equations describing the evolution of traveling waves. In particular, we obtain a degenerate Ginzburg-Landau equation with higher-order nonlinearities. We then investigate this equation and determine periodic traveling wave solutions. We discuss their stability in terms of the laser parameters and predict unusual properties for the long-time behavior of the laser. PACS number(s): 42.50.p, 42.55.f, 42.60.MiAT=m, Azz+mzA+m3~A~A where A is the order parameter, T is a slow time, Z is a long space variable and the coe%cients m; are real.An important feature of our problem is the change of the direction of the bifurcation at the minimum of the neutral stability curve. As a result, we obtain a different and more complicated equation of the form

Oscillation regimes of a solid-state ring laser with active beat-note stabilization: From a chaotic device to a ring-laser gyroscope

“…It is worth noting that the zone corresponding to the absence of beat note around Ω = 0 is not a lock-in zone as in the case of gas ring laser gyroscopes, but rather a zone of self-modulation of the first kind with an average value of the phase difference equal to zero. It is also worth noting that the condition for the occurrence of the natural beat note regime, as described in the previous section, is with our parameters weaker than the validity condition (11). The size of the zone of insensitivity to rotation is thus determined by the condition of occurrence for the natural beat regime, rather than by condition (13).…”

“…Interest in ring lasers developed almost simultaneously with the invention of laser itself [1,2,3,4,5]. Intensive work on this device has been motivated both by fundamental aspects (especially in the field of non linear dynamics, phase transitions, instabilities and chaos [6,7,8,9,10,11,12,13,14,15]) and by practical applications (amongst which are the ring laser gyroscope [16,17,18] and the single-frequency unidirectional ring laser [19,20,21]). The recent achievement of active beat note stabilization in a diode-pumped Nd-YAG ring laser [22] revived interest for homogeneously broadened (e.g.…”

Asymptotically correct error estimation for collocation methods applied to singular boundary value problems

“…In this paper, we treat the most general problem class, where the spectrum of the matrix M(0) contains both eigenvalues with positive and negative real parts. This case is of particular importance in physical applications, see for instance [5], [7], [22], [25].…”

The convergence of shooting methods for singular boundary value problems

“…The search for a method to solve problems (1.1) is strongly motivated by numerous applications from physics (see [3], [5], [6]), chemistry (cf. [30]) and mechanics (buckling of spherical shells, [4], [7]), as well as research activities in related areas ( [8], [26], [27], [28]).…”