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arXiv:1008.0628 · Physical Review A

Steady-stateab initiolaser theory: Generalizations and analytic results

Li Ge, Y. D. Chong, A. Douglas Stone

Published Dec 21, 2010• 141 citations
arXivHTMLPDF

Abstract

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-$Q$ cavities. Within this range of validity, it provides an extraordinarily efficient technique for modeling realistic and complex lasers.

DOI
10.1103/physreva.82.063824

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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 derive and test a generalization of the steady-state ab initio laser theory (SALT) to treat complex gain media. The generalized theory (C-SALT) is able to treat atomic and molecular gain media with diffusion and multiple lasing transitions, and semiconductor gain media in the free carrier approximation including fully the effect of Pauli blocking. The key assumption of the theory is stationarity of the level populations, which leads to coupled self-consistent equations for the populations and the lasing modes that fully include the effects of openness and non-linear spatial hole-burning. These equations can be solved efficiently for the steady-state lasing properties by a similar iteration procedure as in SALT, where a static gain medium with a single transition is assumed. The theory is tested by comparison to much less efficient finite difference time domain (FDTD) methods and excellent agreement is found. Using C-SALT to analyze the effects of varying gain diffusion constant we demonstrate a cross-over between the regime of strong spatial hole burning with multimode lasing to a regime of negligible spatial hole burning, leading to gain-clamping, and single mode lasing. The effect of spatially inhomogeneous pumping combined with diffusion is also studied and a relevant length scale for spatial inhomogeneity to persist under these conditions is determined. For the semiconductor gain model, we demonstrate the frequency shift due to Pauli blocking as the pumping strength changes.
Abstract We review and interpret a modern approach to laser theory, steady-state ab initio laser theory (SALT), which treats lasing and amplification in a unified manner as a non-unitary scattering problem described by a nonlinear scattering matrix. Within the semiclassical version of the theory the laser line has zero width as the lasing mode corresponds to the existence of an eigenvector of the S -matrix with diverging eigenvalue due to the occurrence of a pole of the scattering matrix on the real axis. In this approach the system is infinite from the outset and no distinction is made between cavity modes and modes of the Universe; lasing modes exist both in the cavity and in the external region as solutions satisfying Sommerfeld radiation boundary conditions. We discuss how such solutions can be obtained by a limiting procedure in a finite box with damping according to the limiting absorption principle. When the electromagnetic and matter fields are treated as operators, quantum fluctuations enter the relevant correlation functions and a finite linewidth is obtained, via a generalization of SALT to include noise (N-SALT). N-SALT leads to an analytic formula for the linewidth that is more general than all previous corrected versions of the Schawlow–Townes formula, and can be evaluated simply from knowledge of the semiclassical SALT modes. We derive a simpler version of this formula which emphasizes that the noise is dominated by the fluctuations in the polarization of the gain medium and is controlled by the rate of spontaneous emission.
We generalize and test the recent "ab initio" self-consistent (AISC) time-independent semiclassical laser theory. This self-consistent formalism generates all the stationary lasing properties in the multimode regime (frequencies, thresholds, internal and external fields, output power and emission pattern) from simple inputs: the dielectric function of the passive cavity, the atomic transition frequency, and the transverse relaxation time of the lasing transition.We find that the theory gives excellent quantitative agreement with full time-dependent simulations of the Maxwell-Bloch equations after it has been generalized to drop the slowly-varying envelope approximation. The theory is infinite order in the non-linear hole-burning interaction; the widely used third order approximation is shown to fail badly.
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.