# Theory of the fundamental linewidth of a two-mode laser

01 Jul 2004-Journal of Optics B-quantum and Semiclassical Optics (IOP Publishing)-Vol. 6, Iss: 7, pp 276-282

TL;DR: In this article, the authors investigated the dependence of the phase-diffusion linewidth on the degree of coupling between the two modes of a laser oscillating simultaneously in two modes.

Abstract: Starting from the field master equation for a laser oscillating simultaneously in two modes, we probe the dependence of its fundamental phase-diffusion linewidth on the degree of coupling between the two modes. We find that the heterodyned intrinsic linewidth shows the usual decrease when the output power of each mode increases with an increase in the gain, while the nonlinear self-saturation and cross-coupling coefficients are held constant. The linewidth also decreases with increase in the cross-coupling between the modes for constant gain when the output power of each mode is kept constant by decreasing the self-saturation coefficient. The linewidth, however, increases with increasing cross-coupling when (a) the gain and the self-saturation coefficient are held constant and hence the output power in each mode decreases and (b) the gain is increased keeping the output power and the self-saturation of each mode constant.

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Bell Labs

^{1}TL;DR: In this article, a theory of the spectral width of a single-mode semiconductor laser is presented and used to explain the recent measurements of Fleming and Mooradian on AlGaAs lasers.

Abstract: A theory of the spectral width of a single-mode semiconductor laser is presented and used to explain the recent measurements of Fleming and Mooradian on AlGaAs lasers. They found the linewidth to be inversely proportional to power and to have a value of 114 MHz at 1 mW per facet. This value is 30 times greater than can be explained by existing theories. The enhanced linewidth is attributed to the variation of the real refractive index n' with carrier density. Spontaneous emission induces phase and intensity changes in the laser field. The restoration of the laser to its steady-state intensity results in changes in the imaginary part of the refractive index \Delta n" . These changes are accompanied by changes in the real part of the refractive index \Delta n' , which cause additional phase fluctuations and line broadening. The linewidth enhancement is shown to be 1 + \alpha^{2} , where \alpha = \Delta n'/\Delta n" . A value of \alpha \approx 5.4 , needed to explain the observed linewidth, is close to the experimental values of a of 4.6 and 6.2.

2,293 citations

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TL;DR: In this paper, a detailed analysis of the three-dimensional harmonic oscillator excited in coherent states is given, with special attention to the uncertainty relations and the transition to the classical limit.

Abstract: The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state vectors under consideration are clarified in the context of the Heisenberg uncertainty relations. The familiar case of the azimuthal angle variable $\ensuremath{\phi}$ and its "conjugate" angular momentum ${L}_{z}$ is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables ($sin\ensuremath{\phi}$ and $cos\ensuremath{\phi}$ to describe the phase variable. Well-defined uncertainty relations are derived and discussed. A detailed analysis of the three-dimensional harmonic oscillator excited in coherent states is given. A detailed analysis of the simple harmonic oscillator is given. The usual assumption that a (Hermitian) phase operator $\ensuremath{\varphi}$ (conjugate to the number operator $N$) exists is shown to be erroneous. However, cosine and sine operators $C$ and $S$ exist and are the appr\'opriate phase variables. A Poisson bracket argument using action-angle (rather $J$, $cos\ensuremath{\varphi}$, $sin\ensuremath{\varphi}$) variables is used to deduce $C$ and $S$. The spectra and eigenfunctions of these operators are investigated, along with the important "phase-difference" periodic variables. The properties of the oscillator variables in the various types of states are analyzed with special attention to the uncertainty relations and the transition to the classical limit. The utility of coherent states as a basis for the description of the evolution of the density matrix is emphasized. In this basis it is easy to identify the classical Liouville equation in action-angle variables along with quantum-mechanical "corrections." Mention is made of possible physical applications to superfluid systems.

945 citations

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TL;DR: In this paper, the spontaneous emission factor of planar stripe laser with narrow stripe is calculated and it is shown that spontaneous emission is approximately proportional to the solid angle of laser radiation and nearly independent of transverse active layer dimensions.

Abstract: The fraction of spontaneous emission going into an oscillating laser mode has been calculated. It is shown that this fraction strongly depends on the strength of astigmatism in the laser output beam. Therefore the spontaneous emission factor in planar stripe lasers with narrow stripe is in the order of 10-4and by one order of magnitude larger than in injection lasers with a comparable active layer volume and with a built-in index waveguide. It is shown that the spontaneous emission factor is approximately proportional to the solid angle of laser radiation and nearly independent of the transverse active layer dimensions. Owing to the large spontaneous emission factor, the spectral width of narrow planar stripe lasers is significantly broader compared to narrow stripe lasers with a built-in index waveguide. In addition the large spontaneous emission coefficient also yields a much stronger damping of relaxation oscillations.

474 citations

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TL;DR: In this article, the authors investigated the time-dependent spectrum of light from an observational point of view and defined a timedependent "physical spectrum" of light based on the counting rate of a photodetector.

Abstract: We investigate the time-dependent spectrum of light from an observational point of view and define a time-dependent “physical spectrum” of light based on the counting rate of a photodetector. The tunable element, the filter, that allows observation of different spectral components of the light is shown to play an essential role in the time-dependent spectrum. Its bandwidth cannot be taken arbitrarily narrow. We establish the connection between our physical spectrum and other time-dependent spectra associated with Page, Lampard, Silverman, and Kolmogorov, as well as with the Wiener-Khintchine power spectrum. Also, we show the conditions under which these earlier definitions can be used as the first approximations to the complete physical spectrum, and give an expression for the correction terms.

454 citations

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Bell Labs

^{1}TL;DR: The subject of phase noise in semiconductor lasers is reviewed in this paper, where Langevin forces, laser linewidth above threshold and below threshold, line structure due to relaxation oscillations, phase fluctuations, line narrowing by a passive cavity section and by external feedback, coherence collapse due to optical feedback, and the shot noise limits of several schemes of coherent optical communication are discussed.

Abstract: The subject of phase noise in semiconductor lasers is reviewed. The description of noise in lasers and those aspects of phase noise that are relevant to optical communications are emphasized. The topics covered include: Langevin forces; laser linewidth above threshold and below threshold; line structure due to relaxation oscillations; phase fluctuations; line narrowing by a passive cavity section and by external feedback; coherence collapse due to optical feedback; the shot noise limits of several schemes of coherent optical communication, and the linewidth required to approach these ideal limits.

330 citations