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Physical Review A

Intensity phase coherence in three-mode Fabry-Pérot lasers

Nguyen Ba An, Paul Mandel

Published Aug 1, 1996• 9 citations
Publisher

Abstract

We study analytically the intensity phase coherence in a three-mode Fabry-P\'erot laser. We consider in detail the case of a central mode with maximum gain and two side modes with smaller but equal gains. This laser is characterized by three relaxation oscillation frequencies ${\mathrm{\ensuremath{\Omega}}}_{\mathit{R}}^{\mathrm{\ensuremath{''}}}$\ensuremath{\gtrsim}${\mathrm{\ensuremath{\Omega}}}_{\mathit{L}1}^{\mathrm{\ensuremath{''}}}$\ensuremath{\gtrsim}${\mathrm{\ensuremath{\Omega}}}_{\mathit{L}2}^{\mathrm{\ensuremath{''}}}$. In the framework of a linearized theory, the laser dynamics is, respectively, inphased and perfectly antiphased at ${\mathrm{\ensuremath{\Omega}}}_{\mathit{R}}^{\mathrm{\ensuremath{''}}}$ and ${\mathrm{\ensuremath{\Omega}}}_{\mathit{L}2}^{\mathrm{\ensuremath{''}}}$, irrespective of the modal gains. At ${\mathrm{\ensuremath{\Omega}}}_{\mathit{L}1}^{\mathrm{\ensuremath{''}}}$ the antiphase is only partial if the side mode gains are smaller than the central mode gain. Analytic gain- and pump-dependent relations between the three frequencies and between the heights of the peaks in the power spectra at these frequencies are established. We also derive universal relations between the peaks of the power spectra of the modal and the total intensities at the same frequencies that do not involve any parameter at all. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1638

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We study analytically the intensity phase coherence in a three-mode Fabry-P'erot laser. We consider in detail the case of a central mode with maximum gain and two side modes with smaller but equal gains. This laser is characterized by three relaxation oscillation frequencies \ensuremathΩR\ensuremath′′{\mathrm{\ensuremath{\Omega}}}_{\mathit{R}}^{\mathrm{\ensuremath{''}}}\ensuremathΩR\ensuremath′′​\ensuremath{\gtrsim}\ensuremathΩL1\ensuremath′′{\mathrm{\ensuremath{\Omega}}}_{\mathit{L}1}^{\mathrm{\ensuremath{''}}}\ensuremathΩL1\ensuremath′′​\ensuremath{\gtrsim}\ensuremathΩL2\ensuremath′′{\mathrm{\ensuremath{\Omega}}}_{\mathit{L}2}^{\mathrm{\ensuremath{''}}}\ensuremathΩL2\ensuremath′′​. In the framework of a linearized theory, the laser dynamics is, respectively, inphased and perfectly antiphased at \ensuremathΩR\ensuremath′′{\mathrm{\ensuremath{\Omega}}}_{\mathit{R}}^{\mathrm{\ensuremath{''}}}\ensuremathΩR\ensuremath′′​ and \ensuremathΩL2\ensuremath′′{\mathrm{\ensuremath{\Omega}}}_{\mathit{L}2}^{\mathrm{\ensuremath{''}}}\ensuremathΩL2\ensuremath′′​, irrespective of the modal gains. At \ensuremathΩL1\ensuremath′′{\mathrm{\ensuremath{\Omega}}}_{\mathit{L}1}^{\mathrm{\ensuremath{''}}}\ensuremathΩL1\ensuremath′′​ the antiphase is only partial if the side mode gains are smaller than the central mode gain. Analytic gain- and pump-dependent relations between the three frequencies and between the heights of the peaks in the power spectra at these frequencies are established. We also derive universal relations between the peaks of the power spectra of the modal and the total intensities at the same frequencies that do not involve any parameter at all. \textcopyright{} 1996 The American Physical Society.
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We use the semiclassical strong-signal theory of the laser to predict and explain the onset of side-mode buildup in lasers with one oscillating mode. Two general categories are considered: one for which the side modes and the oscillating mode all have the same wavelength and the other for which they have different wavelengths. The treatments include an arbitrary amount of inhomogeneous broadening. Our approach unifies the treatments of the side-mode instabilities presented earlier and extends them to handle standing waves in addition to the previously treated running waves. We write the field and the population matrix elements as Fourier series in the adjacent-mode beat frequency. This approach has been used extensively in both multimode laser theory and saturation spectroscopy. This technique coincides with linear stability analyses used by others, provided that our beat frequency includes a contribution that is proportional to the complex side-mode gain. We give a solution that allows for detuned operation along with its simpler, centrally tuned special case. The connection with saturation spectroscopy clearly reveals that the side-mode instabilities require side-mode gain. For the single-wavelength case, nonlinear anomalous dispersion is also required. The side-mode gain and dispersion result from both inhomogeneous broadening and population pulsations. The lowest instability thresholds occur when both of these mechanisms play a role. The approach can also be used to treat instabilities in optical bistability by substituting the appropriate equation of state for the strong-mode intensity and by changing the sign of the absorption coefficient. In homogeneously broadened, standing-wave lasers, we show that multiwavelength instabilities depend strongly on the position of the medium in the cavity. We illustrate the theory by giving numerical results for the output pulsation frequency and for the instability threshold by using parameters that are appropriate for the He–Xe laser. These results correlate well with experimental observations.
This dissertation investigates three different optical instabilities: cw conical emission, bifurcations of optical transverse solitary waves, and asymmetric gain modification in a vertical cavity surface emitting laser (VCSEL). The first two instabilities occur in sodium vapor while the third occurs in a new type of semiconductor laser. For the conical emission instability, the first comparison and agreement between theory and experiment is given. The conical emission process is modeled as a series of nonlinear effects including self-trapping of the pump beam, Doppler-broadened Raman-gain amplification of Doppler-broadened resonance fluorescence, propagational four-wave mixing, and diffraction and pump-induced refraction of the new frequencies. Each step of the emission process is thoroughly examined and explained. Detailed comparisons are made between experimental observables and numerical calculations including near-field and far-field spatial profiles, frequency spectra, and cone angles. All comparisons show good agreement. In a different but related experiment, bifurcations of optical transverse solitary waves are studied for one-way propagation through a sodium vapor cell. Two types of phase encoding seed transverse bifurcations which result in transverse cell-exit profiles with the beauty rivaling that of a kaleidoscope. The cell-exit profiles are stationary in time, reproduce completely when the power or frequency is scanned, and agree well with one-way computations. Evidence is also presented for the first observation of a double-peaked Raman gain. In the third experiment, a temporal instability is examined in a GaAs VCSEL. Strong evidence is presented for an asymmetric gain modification induced when a strong optical signal is injected into the Fabry-Perot transmission peak of the VCSEL while it is lasing. This gain modification results in increased absorption on the high frequency side of the injected signal and increased gain on the low frequency side as manifested by the formation of an optical sideband. The frequency of the sideband depends on the intensity of the injected signal with frequencies (> 50 GHz) well beyond the predicted relaxation oscillation frequency being observed. Very efficient nondegenerate four-wave mixing is also observed.
This communication has considered the case of a Fabry-Perot cavity with light impinging on one end while the field distribution on the other end is observed. It has been shown that if one starts with a wave that can be expanded into the modes of the integral operator, and if the medium in the cavity is passive, more than one mode would be observed at the other end although the amplitude of the dominant mode can be much greater than that of the other modes. It was further shown that if there is present an active medium with sufficient gain there will be only a single mode, and the single mode is necessary and sufficient for complete spatial coherency.