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IEEE Journal of Quantum Electronics

Internal and external field fluctuations of a laser oscillator: Part I--Quantum mechanical Langevin treatment

Y. Yamamoto, N. Imoto

Published Oct 1, 1986• 67 citations
Publisher

Abstract

The amplitude and phase noise spectra of the internal and external fields of a laser oscillator are calculated using the quantum mechanical Langevin equations. The outgoing field spectra are demonstrated to be different from the internal field spectra. The outgoing photon flux\hat{N}from a laser oscillator operating at far above the threshold has a spectral density equal to the classical shot noise level,2\langle\hat{N}\rangle. On the other hand, the internal photon number spectrum is Lorentzian, and the variance is equal to the average photon number,\langle\hat{N}\rangle. The difference stems from the quantum mechanical interference between the transmitted internal field and the reflected zero-point fluctuation at the output coupling mirror. The relation between the present analysis and the previous noise theory of a laser is also discussed.

DOI
10.1109/jqe.1986.1072889

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The amplitude and phase noise spectra of the internal and external fields of a laser oscillator are calculated using the quantum mechanical Langevin equations. The outgoing field spectra are demonstrated to be different from the internal field spectra. The outgoing photon flux\hat{N}from a laser oscillator operating at far above the threshold has a spectral density equal to the classical shot noise level,2\langle\hat{N}\rangle. On the other hand, the internal photon number spectrum is Lorentzian, and the variance is equal to the average photon number,\langle\hat{N}\rangle. The difference stems from the quantum mechanical interference between the transmitted internal field and the reflected zero-point fluctuation at the output coupling mirror. The relation between the present analysis and the previous noise theory of a laser is also discussed.
The amplitude and phase noise spectra of the internal and external fields of a negative resistance oscillator are derived using electrical circuit theory. It is shown that the difference between internal and external fluctuations becomes important in the quantum domain (h\omega > kT) where external noise sources are not negligible as they often are in the classical domain (h\omega < kT). Using established methods to describe quantum fluctuations in electrical circuits, we obtain results in full agreement with the more exact but less general direct quantum mechanical analysis of a Fabry-Perot laser in Part I.
The amplitude and phase noise spectra of the internal and external field of a microcavity are calculated using the quantum mechanical Langevin equations. We assume that two-photon absorber works inside a cavity, which brings photon-number squeezing in the internal field as well as the external field. The individual frequency components of the output field have strong squeezing at cavity resonance due to the interference between the input and the cavity field, which is similar to the quadrature squeezing in a parametric oscillator and the photon number squeezing in a saturated laser oscillator. However, the degree of the squeezing is not perfect becuase the material fluctuation remains.
Using population rate equations, we have calculated the intrinsic intensity fluctuations expected in the output of 3-level and 4-level CW laser oscillators. Such fluctuations stem from the quantum shot noise implicit in a restriction of the atomic and photon population-operator eigenvalues to discrete integral values (in the manner of Shimoda, Takahasi, and Townes). The intrinsic fluctuations below threshold are typical of a Gaussian distribution ofe-mfield amplitudes (Bose-Einstein distribution of photons) and above threshold of an amplitude-stabilizede-mfield (Poisson distribution of photons). The noise properties of 3-level and 4-level lasers are qualitatively similar, although for a given output power the intensity fluctuations of 3-level oscillators are greater than (or equal to) those in 4-level oscillators, principally as a consequence of the increased spontaneous emission associated with the larger upper-state occupation necessary to achieve a net gain in 3-level systems.
This paper clarifies the origins of the standard quantum limit for the amplitude noise of a laser-oscillator outgoing field. The amplitude noise within the cavity bandwidth, \ensuremath{\Omega}\ensuremath{\le}\ensuremath{\omega}/Q, is ultimately caused by the pump amplitude fluctuation, while that above the cavity bandwidth, \ensuremath{\Omega}\ensuremath{\ge}\ensuremath{\omega}/Q, is due to the field zero-point fluctuation. The uncertainty product of the amplitude- and phase-noise spectra at an extremely high pumping level is still larger than the Heisenberg minimum-uncertainty product because of the presence of nonstationary phase-diffusion noise. In this sense, an ordinary laser oscillator is not a quantum-limited device. This paper suggests that a laser oscillator can produce an amplitude-squeezed state in itself if the pump amplitude fluctuation is suppressed below the ordinary shot-noise level. The paper discusses possible schemes for suppressing pump fluctuation, commutator bracket preservation without pump fluctuation, and resulting amplitude and phase spectra. The similarity of and difference between a pump-noise-suppressed laser and a cavity degenerate parametric amplifier are delineated.
The random nature of spontaneous emission leads to unavoidable fluctuations in a laser's output. This is often included through random Langevin forces in laser rate equations, but this approach falls short for nanolasers. In this paper, we show that the laser quantum noise can be quantitatively computed for a very broad class of lasers by starting from simple and intuitive rate equations and merely assuming that the number of photons and excited electrons only takes discrete values. While the approach has seen previous success, we here derive it rigorously from an open quantum system master equation, whereas it was previously introduced only on phenomenological grounds. We further show that in the many-photon limit, the model simplifies to Langevin equations. We perform an extensive comparison of different approaches for computing quantum noise in lasers, identifying the best approach for different system sizes, ranging from nanolasers to macroscopic lasers, and different levels of excitation, i.e., cavity photon number. In particular, we show that below the laser threshold, stochastic fluctuations in the numerical solution to the Langevin equations can drive populations to unphysical negative values, requiring the introduction of population bounds, which in turn skew the noise statistics, leading to inaccuracies. The Laser Markov Chain model, on the other hand, is accurate for all pump values and laser sizes when collective emitter effects are excluded.
Quantum fluctuations in the radiation of multitransverse-mode laser are investigated theoretically and numerically, based on quantum c-number Langevin stochastic equations for the complex amplitudes of transverse modes. It is shown that strong antiphase quantum fluctuations of transverse-mode amplitudes occur, which corresponds to spatial fluctuations of the laser field. Fluctuation spectra are calculated, showing that antiphase fluctuations dominate at low frequencies (smaller than relaxation oscillation frequency), and in-phase oscillations prevail at high frequencies.