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arXiv:1611.01676 · Physical Review A

Symmetry, stability, and computation of degenerate lasing modes

David Liu, Bo Zhen, Li Ge, Felipe Hernández, Adi Pick, Stephan Burkhardt, Matthias Liertzer, Stefan Rotter, Steven G. Johnson

Published Feb 23, 2017• 7 citations
arXivHTMLPDF

Abstract

We present a general method to obtain the stable lasing solutions for the steady-state ab initio lasing theory (SALT) for the case of a degenerate symmetric laser in two dimensions (2D). We find that under most regimes (with one pathological exception), the stable solutions are clockwise and counterclockwise circulating modes, generalizing previously known results of ring lasers to all 2D rotational symmetry groups. Our method uses a combination of semianalytical solutions close to lasing threshold and numerical solvers to track the lasing modes far above threshold. Near threshold, we find closed-form expressions for both circulating modes and other types of lasing solutions as well as for their linearized Maxwell-Bloch eigenvalues, providing a simple way to determine their stability without having to do a full nonlinear numerical calculation. Above threshold, we show that a key feature of the circulating mode is its ``chiral'' intensity pattern, which arises from spontaneous symmetry breaking of mirror symmetry, and whose symmetry group requires that the degeneracy persists even when nonlinear effects become important. Finally, we introduce a numerical technique to solve the degenerate SALT equations far above threshold even when spatial discretization artificially breaks the degeneracy.

DOI
10.1103/physreva.95.023835

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We present a general method to obtain the stable lasing solutions for the steady-state ab initio lasing theory (SALT) for the case of a degenerate symmetric laser in two dimensions (2D). We find that under most regimes (with one pathological exception), the stable solutions are clockwise and counterclockwise circulating modes, generalizing previously known results of ring lasers to all 2D rotational symmetry groups. Our method uses a combination of semianalytical solutions close to lasing threshold and numerical solvers to track the lasing modes far above threshold. Near threshold, we find closed-form expressions for both circulating modes and other types of lasing solutions as well as for their linearized Maxwell-Bloch eigenvalues, providing a simple way to determine their stability without having to do a full nonlinear numerical calculation. Above threshold, we show that a key feature of the circulating mode is its ``chiral'' intensity pattern, which arises from spontaneous symmetry breaking of mirror symmetry, and whose symmetry group requires that the degeneracy persists even when nonlinear effects become important. Finally, we introduce a numerical technique to solve the degenerate SALT equations far above threshold even when spatial discretization artificially breaks the degeneracy.
We improve the steady-state ab initio laser theory (SALT) of T\"ureci et al. by expressing its fundamental self-consistent equation in a basis set of threshold constant flux states that contains the exact threshold lasing mode. For cavities with nonuniform index and/or nonuniform gain, the new basis set allows the steady-state lasing properties to be computed with much greater efficiency. This formulation of the SALT can be solved in the single-pole approximation, which gives the intensities and thresholds, including the effects of nonlinear hole-burning interactions to all orders, with negligible computational effort. The approximation yields a number of analytic predictions, including a ``gain-clamping'' transition at which strong modal interactions suppress all higher modes. We show that the single-pole approximation agrees well with exact SALT calculations, particularly for high-QQQ cavities. Within this range of validity, it provides an extraordinarily efficient technique for modeling realistic and complex lasers.
In this thesis we generalize and extend the steady-state ab initio laser theory (SALT), first developed by Türeci and Stone, and apply it problems in laser design. SALT as first formulated modeled the gain medium as identical two-level atoms, leading to the well-known Maxwell Bloch laser equations. The result is a set of coupled non-linear wave equations that treats the openness of the cavity exactly and the non-linear modal interactions to infinite order. Most gain media have more than two atomic levels, and in this thesis we generalize the SALT equations to treat realistic and complex gain media, specifically N-level atomic media with a single lasing transition, N-level atomic media with multiple lasing transitions and semiconductor gain media with particle-hole band excitations. The extension to multiple transitions requires fundamentally enlarging the set of SALT equations, by adding a set of population equations that must be solved self-consistently with the non-linear wave equations for the lasing modes in standard SALT. In addition, the population equation can be generalized to treat gain diffusion, an important problem in a number of laser systems, not treated in SALT or in most earlier laser theories. The semiconductor version (Semi-SALT) includes the continuum of particle-hole transitions and the effect of Pauli blocking of transitions, but is only developed and applied in the free-carrier approximation. The resulting theory is termed complex SALT (C-SALT). We also demonstrate how to incorporate amplification and injected signals naturally within the SALT framework, yielding injection SALT (I-SALT). The generalization to I-SALT leads to a larger set of self-consistent coupled non-linear wave equations, a set for the lasing modes and a set for the injected and amplified fields, coupled through cross gain saturation. It clearly distinguishes the lasing modes, which correspond to poles of the scattering matrix, from the injected fields, which do not; in this limit the locking of a lasing mode corresponds to the injected signal forcing the lasing pole off the real axis, reducing its amplitude to zero. I-SALT is shown to reduce to a version of the standard Adler theory of injection-locked lasing in a certain limit (the single pole approximation). We apply SALT to design a highly multimode cavity for use as a spatially incoherent light source for applications to imaging and microscopy. Laser illumination typically leads to coherent artifacts that degrade optical images; this can be alleviated by having a very large number of modes 500) which are spatially independent and average out such artifacts. We used SALT to model a D-shaped laser cavity with chaotic ray dynamics and showed that a certain shape greatly increases the number of lasing modes for the same cavity size and pump strength, due to a flat distribution of Q-values and reduced mode competition. An on-chip electrically-pumped semiconductor laser was realized using this cavity design and showed negligible coherent artifacts in imaging, as well as much better efficiency and power per mode than traditional incoherent light sources such as LEDs. The thesis also goes beyond semiclassical laser theory to treat quantum noise and the laser linewidth in a SALT-based approach. We demonstrate that SALT solutions can be used in conjunction with a temporal coupled mode theory (TCMT) to derive an analytic formula for the quantum limited laser linewidth in terms of integrals over SALT solutions. This linewidth formula is a substantial generalization of the well-known Schawlow-Townes result and includes all previously known corrections: the Petermann factor, Henry alpha factor, incomplete inversion factor and the "bad cavity factor". However, unlike previous theories these corrections are not simply multiplicative and are not separable in general. The predictions of TCMT linewidth theory are tested quantitatively by means of an FDTD algorithm that includes the Langevin noise as a source term.
We provide a comprehensive analysis of the standard model of the bidirectional ring laser in which only one mode can be supported in each direction and in the limit that the polarization can be eliminated adiabatically. The interaction between the two counterpropagating modes can be derived and it is most naturally viewed as a coupling between them via scattering from a spatial grating formed in the population inversion. If the grating is a sufficiently small modulation of the spatial average of the population inversion, it can be approximated by a sinusoidal function. A systematic derivation of the model with only a sinusoidal grating for the homogeneously broadened case is presented that reveals and corrects errors in several previously published analyses. The stability of the steady-state solution is analyzed. The bidirectional steady-state solution is unstable and the unidirectional steady-state solutions may be stable or unstable depending on the parameters. The well-established result of bistability between the two modes when the cavity is tuned to resonance is recovered, a result that persists, in part, even when the losses between the two modes are different.With detuning, the unidirectional steady-state solutions can be destabilized, creating regions in the parameter space where only time-dependent solutions are found. For parameters characteristic of solid-state or molecular gas lasers, the instability can occur for very small detunings and even very close to the lasing threshold. Asymptotic limits of the instability boundaries for these parameters are presented. Additional results are derived formally for inhomogeneous broadening with a Lorentzian line shape and for arbitrary inhomogeneous linewidths. Explicit analytic results are presented in the limits of very small and very large inhomogeneous linewidths compared with the homogeneous linewidth. Time-dependent solutions in the homogeneously broadened case for relaxation rates appropriate to CO2{\mathrm{CO}}_{2}CO2​ lasers show that there are broad regions in the parameter space of gain and detuning for which the behavior is dynamically chaotic in the form of nearly ``square-wave'' alternation between the counterpropagating modes of the laser with irregular switching times. In other regions of that parameter space we find more irregular pulsations and pulse alternations. Regions of periodic pulsations also have been found.
Studying time-dependent behavior in lasers is analytically difficult due to the saturating nonlinearity inherent in the Maxwell-Bloch equations and numerically demanding because of the computational resources needed to discretize both time and space in conventional finite-difference time-domain approaches. We describe here an efficient spectral method to overcome these shortcomings in complex lasers of arbitrary shape, gain medium distribution, and pumping profile. We apply this approach to a quasidegenerate two-mode laser in different dynamical regimes and compare the results in the long-time limit to the steady-state ab initio laser theory (SALT), which is also built on a spectral method but makes a more specific ansatz about the long-time dynamical evolution of the semiclassical laser equations. Analyzing a parameter regime outside the known domain of validity of the stationary inversion approximation, we find that for only a narrow regime of pump powers the inversion is not stationary, and that this, as pump power is further increased, triggers a synchronization transition upon which the inversion becomes stationary again. We provide a detailed analysis of mode synchronization (also known as cooperative frequency locking), revealing interesting dynamical features of such a laser system in the vicinity of the synchronization threshold.
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