Frequency-Modulated Combs Obey a Variational Principle

Abstract

Laser dynamics encompasses universal phenomena that can be encountered in many areas of physics, such as bifurcation and chaos, mode competition, resonant nonlinearities, and synchronization-or lockingof oscillators. When a locking process occurs in a multimode laser, an optical frequency comb is produced, which is an optical spectrum consisting of equidistant modes with a fixed phase relationship. Describing the formation of self-starting frequency combs in terms of fundamental laser equations governing the field inside the cavity does not allow one, in general, to grasp how the laser synchronizes its modes. Our finding is that, in a particular class of lasers where the output is frequency modulated with small or negligible intensity modulation, a greatly simplified description of self-locking exists. We show that in quantum cascade lasers-solid-state representatives of these lasers characterized by an ultrashort carrier relaxation time-the frequency comb formation obeys a simple variational principle, which was postulated over 50 years ago and relies on the maximization of the laser output power. The conditions for the breakdown of this principle are also experimentally identified, shedding light on the behavior of many different types of lasers, such as dye, diode, and other cascade lasers. This discovery reveals that the formation of frequency-modulated combs is an elegant example of an optimization problem solved by a physical system.

Observation of Self Mode-Locking in Terahertz Quantum Cascade Lasers through Intra-Cavity Self-Detection

“…8 we also find that the optical and carrier density pulses are ≅180deg out of phase, as expected. The observation of shallow pulses sitting on a CW background is also expected since ‰ is much shorter than the roundtrip time, which has been shown to hamper the generation of strong pulses as in standard mode-locking, and rather lead to a CW dominated, FM-comb type of emission [15,16]. We note that in the actual self-detection configuration it is not possible to establish experimentally the presence of a CW background.…”

“…The simulated carrier density (i.e. current) pulses are found to be 180deg out of phase with respect to the intracavity optical pulses, and the QCLs shows a predominantly FM-type dynamics, with both (optical and carrier density) pulses appearing as a rather shallow modulation on top of a CW background [16,33]. Importantly, our model confirms the coexistence of regimes with one or two pulses per roundtrip found experimentally at specific bias points, which we interpret as a signature of the solitonic character of the observed QCL dynamics.…”

Femtosecond pulses from a mid-infrared quantum cascade laser

“…limited in resolution by residual timing jitter between signal and sampling pulses 36 . Autocorrelation measurements, on the other hand, are by definition phase coherent and are well suited for the characterization of pulses; however, they only provide limited phase information and require considerable effort to be applied to strongly chirped fields 37 .…”

General Canonical Variational Principle and Noether Theorem, Their New Classical and Quantum Physics, Solution to Crisis Deducing All Physics Laws in Phase Space

“…A lot of references, e.g., [23][24][25][26][27][28] have very well investigated various variational principles and their useful actual applications, and lots of good works, e.g., [29], [30], [31] have very well researched on Noether's Theorem and the useful pratical applications in different branches of physics.…”