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Physical Review A

Laser line broadening due to classical and quantum noise and the free-electron-laser linewidth

Avraham Gover, Avner Amir, Luis R. Elias

Published Jan 1, 1987• 28 citations
Publisher

Abstract

The problem of fundamental laser line broadening due to random spontaneous emission of radiation and amplification of thermal radiation noise is analyzed in terms of a classical fluctuating field phasor model. We derive a general expression for the intrinsic linewidth, given in terms of the spectral power of the radiation noise source, which can be classical or quantum mechanical in nature. In the case of a two-level atomic laser, we recover by the use of Einstein relations, the traditional linewidth formula of the Schalow-Townes form. In the case of the free-electron laser (FEL), using the explicit expression for the spontaneous emission, we present calculation of the laser linewidth by purely classical methods. The result agrees with the one obtained in the framework of a quantum-mechanical model. By using ``extended Einstein relations'' which are applicable to classical radiators, we show that a Schalow-Townes-type formula can also be obtained for the FEL. The theory predicts extremely narrow intrinsic linewidth ${(10}^{\mathrm{\ensuremath{-}}7}$ Hz) for cw FEL's with parameters similar to those of the FEL experiment of Elias et al.

DOI
10.1103/physreva.35.164

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International Conference on Transparent Optical Networks•Jul 1, 2017•0 citations
The problem of fundamental laser line broadening due to random spontaneous emission of radiation and amplification of thermal radiation noise is analyzed in terms of a classical fluctuating field phasor model. We derive a general expression for the intrinsic linewidth, given in terms of the spectral power of the radiation noise source, which can be classical or quantum mechanical in nature. In the case of a two-level atomic laser, we recover by the use of Einstein relations, the traditional linewidth formula of the Schalow-Townes form. In the case of the free-electron laser (FEL), using the explicit expression for the spontaneous emission, we present calculation of the laser linewidth by purely classical methods. The result agrees with the one obtained in the framework of a quantum-mechanical model. By using ``extended Einstein relations'' which are applicable to classical radiators, we show that a Schalow-Townes-type formula can also be obtained for the FEL. The theory predicts extremely narrow intrinsic linewidth (10\ensuremath−7{(10}^{\mathrm{\ensuremath{-}}7}(10\ensuremath−7 Hz) for cw FEL's with parameters similar to those of the FEL experiment of Elias et al.
A simple method for calculating the linewidth of sources of stimulated coherent radiation caused by spontaneous noise and thermal fluctuations has been proposed in the framework of the two-level model of the active medium. An expression for the spectral width has been derived beyond the adiabatic approximation under an arbitrary relation between relaxation times in a self-sustained oscillation system, taking into account a finite spectral band of spontaneous radiation. The result is applicable for a wide class of coherent radiation sources.
As discussed in the last three chapters, the fundamental source of noise in a laser is spontaneous emission. A simple pictorial model for the origin of the laser linewidth envisions it as being due to the random phase diffusion process arising from the addition of spontaneously emitted photons with random phases to the laser field. In this chapter we show that the quantum noise leading to the laser linewidth can be suppressed below the standard, i.e., Schawlow–Townes limit by preparing the atomic systems in a coherent superposition of states as in the Hanle effect and quantum beat experiments discussed in Chapter 7. In such coherently prepared atoms the spontaneous emission is said to be correlated. Lasers operating via such a phase coherent atomic ensemble are known as correlated spontaneous emission lasers (CEL). An interesting aspect of the CEL is that it is possible to eliminate the spontaneous emission quantum noise in the relative linewidths by correlating the two spontaneous emission noise events. A number of schemes exist in which quantum noise quenching below the standard limit can be achieved. In two-mode schemes a correlation between the spontaneous emisson events in two different modes of the radiation field is established via atomic coherence so that the relative phase between them does not diffuse or fluctuate. In a Hanle laser and a quantum beat laser this is achieved by pumping the atoms coherently such that every spontaneously emitting atom contributes equally to the two modes of the radiation, leading to a reduction and even vanishing of the noise in the phase difference.
Abstract The Schawlow–Townes limit established spontaneous emission as the fundamental quantum mechanism defining the minimum linewidth of an ideal laser oscillator. According to this description, random phase perturbations introduced by spontaneous photons become progressively less significant as the intracavity photon population increases, resulting in a continuous narrowing of the emission linewidth with increasing optical power. Although this relation remains a cornerstone of laser physics, it represents an ideal limit in which other sources of phase instability are absent. Practical laser systems frequently deviate from this behaviour: after an initial power-dependent narrowing regime, the linewidth often approaches a residual value that cannot be removed by further increasing the optical power. Such saturation suggests that linewidth should be treated not only as a photon-statistical quantity, but also as a property of the complete laser system, including the active medium. In this work, we develop a phenomenological framework in which laser linewidth is described as the combined result of spontaneous-emission noise, pump-induced fluctuations, and medium induced phase perturbations. The proposed approach preserves the Schawlow–Townes limit as the fundamental quantum contribution while introducing an active-medium-dependent term associated with the finite spectral response and microscopic fluctuations of the gain medium. The physical origin of this residual contribution is analysed using optical line-shape theory, considering homogeneous broadening, inhomogeneous broadening and the role of local environments in solid-state gain media. Er³+-doped systems are considered as representative rare earth active media, since their shielded 4f–4f transitions remain sufficiently narrow to support coherent emission while still retaining measurable signatures of crystal-field splitting, host disorder and spectral broadening. The model provides a physical interpretation for linewidth saturation and highlights that the ultimate optimisation of laser coherence requires not only improved cavity design and pump stabilisation but also control of the microscopic stability of the active medium. This viewpoint reframes the linewidth floor as a system-level property emerging from the interaction between quantum fluctuations, excitation dynamics and the gain medium.
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.
A recent theory of the fundamental laser linewidth developed by the authors is extended to include the effects of the Henry \ensuremath{\alpha} parameter. The usual (1+${\mathrm{\ensuremath{\alpha}}}^{2}$) enhancement of the linewidth is shown to be increased for arbitrary output coupling by spatial hole burning. A total enhancement factor is derived that multiplies the Schawlow-Townes linewidth and takes into account line broadening due to amplification of quantum noise as well as that due to phase-amplitude coupling.
The processes of absorption and stimulated emission are investigated via (i) energy conservation in a Fabry-Perot resonator, (ii) the Lorentz-oscillator model, and (iii) the Kramers-Kronig relations. Vacuum fluctuations are added and the relation between stimulated and spontaneous emission is clarified. It is shown that previous models start from incorrect assumptions about the physical foundations of spontaneous emission. The non-orthogonality of longitudinal optical modes in a linear resonator is investigated and its consequences for the laser linewidth are discussed. Based on these results, a semi-classical approach to the laser linewidth below, around, and above the laser threshold which is consistent with resonator theory is presented. These results also shed new light on the usually considered linewidth-narrowing and -broadening factors.