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IEEE Journal of Quantum Electronics

Quantum Noise of Actively Mode-Locked Lasers With Dispersion and Amplitude/Phase Modulation

Farhan Rana, Rajeev J. Ram, H. A. Haus

Published Jan 1, 2004• 39 citations
Publisher

Abstract

A quantum theory for the noise of optical pulses in actively mode-locked lasers is presented. In the presence of phase modulation and/or group velocity dispersion, the linear operator that governs the time evolution of the pulse fluctuations inside the laser cavity is not Hermitian (or normal) and the eigenmodes of this operator are not orthogonal. As a result, the eigenmodes have excess noise and the noise in different eigenmodes is highly correlated. We construct quantum operators for the pulse photon number, phase, timing, and frequency fluctuations. The nonorthogonality of the eigenmodes results in excess noise in the pulse photon number, phase, timing, and frequency. The excess noise depends on the frequency chirp of the pulse and is present only at low frequencies in the spectral densities of the pulse noise operators.

DOI
10.1109/jqe.2003.820831

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A quantum theory for the noise of optical pulses in actively mode-locked lasers is presented. In the presence of phase modulation and/or group velocity dispersion, the linear operator that governs the time evolution of the pulse fluctuations inside the laser cavity is not Hermitian (or normal) and the eigenmodes of this operator are not orthogonal. As a result, the eigenmodes have excess noise and the noise in different eigenmodes is highly correlated. We construct quantum operators for the pulse photon number, phase, timing, and frequency fluctuations. The nonorthogonality of the eigenmodes results in excess noise in the pulse photon number, phase, timing, and frequency. The excess noise depends on the frequency chirp of the pulse and is present only at low frequencies in the spectral densities of the pulse noise operators.
Classical optical solitons propagate in optical fibers without temporal or spectral distortion as a result of a balance between phase shifts due to group velocity dispersion and those due to nonlinear self-phase modulation. This balance is achieved by the proper choice of pulse shape and duration for a given pulse energy. The quantum fluctuations associated with a coherent soliton do not undergo stationary propagation, however, and a number of effects are predicted, including the evolution of vacuum noise associated with the input coherent pulse into squeezed fluctuations and the temporal spreading of the soliton pulse because of dispersion acting on quantum fluctuations of the pulse frequency.
We theoretically investigate the quantum dynamics of an unstable resonator laser. Compared to a stable cavity laser of equal gain and loss it exhibits a K-fold enhanced linewidth. This excess noise factor K is a measure of the nonorthogonality of the resonator eigenmodes and amounts to an enlargement of the quantum vacuum fluctuations. Using a quantum treatment starting from first principles based on the nonorthogonal eigenmodes, we put previous theoretical predictions onto a more firm ground. While we find a position-dependent enhancement of the spontaneous emission rate into an empty mode of only K,\sqrt{K},K​, the constructive quantum interference of the spontaneous emission with a single oscillating mode lets the Petermann excess noise factor K reappear in the phase diffusion (laser linewidth). Hence locally enhanced spontaneous emission as well as noise enhanced by interference (amplified spontaneous emission) play an equal role in the origin of excess noise.
We describe the quantum-noise behavior of a laser that has nonorthogonal polarization modes. The nonorthogonality of the modes leads to excess quantum noise. Theoretically, we derive the excess noise dynamics of the laser, including the saturation of the gain medium. Experimentally, we have measured the noise dynamics in the polarization, intensity and phase degrees of freedom of a 3.51-$\ensuremath{\mu}\mathrm{m}\mathrm{}$ HeXe laser and we obtain good agreement with theory. When the modes are made nonorthogonal we observe: (i) excess intensity and phase noise, (ii) spectral coloring of the excess intensity noise and (iii) correlations between the polarization-angle noise and the intensity noise. The excess phase noise is found to be partly suppressed by the polarization-anisotropic part of the saturation of the gain medium.
The basic equations governing noise phenomena are derived from first principles and applied to examples in optical communications. Quantum noise arises from two sources, the momentum fluctuations of electrons at optical frequencies and the uncertainty-related fluctuations of the electromagnetic field. Shot noise results from the beating of the noise sources with the signal field. In high-gain amplifiers, the spontaneous-emission noise dominates shot noise and results in a noise figure of at least 3 dB. It is shown explicitly how, at high power, both the laser field and the laser noise source become classical. The increase in noise in lasers with open cavities is not due to enhanced spontaneous emission as once thought, but to single-pass amplification. The noise fields and spontaneous currents have Gaussian distributions, while nonlasing modes have exponential photon-number distributions. Low-frequency intensity fluctuations arise from the electric current driving the laser and can be sub-Poissonian, in contrast to shot noise, which has a Poissonian distribution. The calculational tools are a wave equation for the field operator and a rate equation for the carrier-number operator, each containing spontaneous current noise sources. The correlation functions of these sources are determined by the fluctuation-dissipation theorem. [S0034-6861(96)00503-X]
Optical solitons are pulses that propagate in optical fibers without temporal or spectral distortion, owing to balancing of the second-order dispersion of the refractive index by nonlinear self-phase modulation induced by the intensity-dependent refractive index. Classically, this stationary distortion-free propagation only requires the proper choice of pulse width and shape for a given pulse energy. However, quantum fluctuations do not undergo stationary propagation but rather evolve due to dispersion and self-phase modulation. As a soliton pulse propagates, zero-point or vacuum noise associated with the pulse evolves into squeezed noise. Vacuum fluctuations are responsible for the fundamental detector noise floor known as the shot noise limit. Reduction of the noise in the detection of squeezed soliton pulses has achieved a noise level 34% below this limit. In optical fibers, thermal fluctuation of the refractive index limit the noise reduction that can be achieved by adding phase fluctuations in excess of the vacuum noise. At low frequencies thermal noise originates from local structural dynamics of fused silica which produces a 1/f power spectrum and from guided acoustic wage Brillouin scattering which modulates the refractive index at frequencies characteristic of the fiber structure. The bandwidth of these fluctuations is ~2 GHz; and for soliton pulses with spectral bandwidth much >2 GHz, we have shown that this noise can be made insignificant. For very short pulses, however, noise from Raman scattering must be considered. Recent calculations incorporating this noise mechanism are compared with experiment.