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Europhysics Letters (EPL)

Circuit Theory of Amplitude Noise for a Laser Diode with Electronic Feedback

Jacques Arnaud

Published Feb 15, 1989• 6 citations
Publisher

Abstract

The amplitude noise of a laser diode submitted to electronic feedback is evaluated using a new circuit theory. It is postulated that the electron-hole injection rate equals the photon generation rate at any time, and Nyquist-like noise sources are introduced. Previous results based on quantum mechanics are recovered. It is found that the amplitude fluctuations of an optical beam in the coherent state can be squeezed below shot noise by feeding back the driving current of a laser amplifier.

DOI
10.1209/0295-5075/8/4/008

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Reviews of Modern Physics•Jul 1, 1996•220 citations
The amplitude noise of a laser diode submitted to electronic feedback is evaluated using a new circuit theory. It is postulated that the electron-hole injection rate equals the photon generation rate at any time, and Nyquist-like noise sources are introduced. Previous results based on quantum mechanics are recovered. It is found that the amplitude fluctuations of an optical beam in the coherent state can be squeezed below shot noise by feeding back the driving current of a laser amplifier.
It is generally believed that the amplitude fluctuations of a light beam initially in the coherent state cannot be squeezed below those of shot noise by the simple arrangement of a beamsplitter and a detector, the current from the detector being fed back to the light source or to a modulator. A simple semiclassical theory shows that arbitrary amounts of squeezing can in fact be obtained if a negative opticalconductance device such as a constant-voltage-driven (nonself-oscillating) laser diode is used in place of a conventional detector.
A new scheme for generating an amplitude-squeezed state is proposed. The photon-flux fluctuation of a semiconductor-laser output wave is measured with a quantum nondemolition (QND) detector, and is negatively fed back to the laser pumping current. The operator Langevin equations are derived by combining the quantum-mechanical analyses on the laser internal-external field fluctuations, the quantum nondemolition detector based on an optical Kerr effect, and a negative-feedback circuit. The output wave features a reduced photon number noise below the standard quantum limit, 〈(\ensuremath{\Delta}n^)2{)}^{2})2〉^〉, and an enhanced phase noise above that, 〈(\ensuremath{\Delta}\ensuremath{\psi}^)2{)}^{2})2〉>(1/4)〈n^${〉}^{\mathrm{\ensuremath{-}}1}$, while the minimum uncertainty product is still preserved. The observed photoelectron statistics in a negative-feedback GaAs laser diode using a conventional p-i-n photodiode (not a QND detector) are shown to exhibit sub-Poissonian statistics with the variance 〈(\ensuremath{\Delta}n^)2{)}^{2})2〉apeq20.26〈n^〉. The measured photocurrent fluctuation spectral density is also indicated to be below the standard quantum limit by a factor of 0.2 (=-7 dB). The experimental results also confirm that such amplitude squeezing is only observed inside the feedback loop and cannot be extracted from the loop unless a quantum nondemolition detector is used.
We have produced amplitude-squeezed light whose noise is reduced by ~10 dB below the quantum noise limit in four different feedback systems. In our experimental method we have directly compared the noise power spectrum of the laser diode with the quantum noise limit in the low-frequency region, where the classical noise is stronger than the quantum noise. In addition to the well-known methods for extracting the out-loop beams from the feedback system by use of a 50:50 beam splitter, a polarizing beam splitter, or the leakage light from the rear side of a laser diode, we also have tried to use the spatial coherence of the laser light. We also have examined the vacuum fluctuations that prohibited the out-loop beam from being amplitude squeezed. In all these systems we have verified that the noise power spectrum for the in-loop beam is reduced below the quantum noise limit but not that for the out-loop beam; the difference arises mainly because vacuum fluctuations enter each system differently.
We detected a portion of the output from a semi-conductor diode laser and used the detected signal as a source of negative feedback to the laser drive current. At frequencies below the characteristic frequency of the feedback loop, the power spectral density of the detected photocurrent was as much as 15 dB below the shot noise limit. This reduction can be described by a semiclassical theory of the effects of feedback on an injected shot noise current. Physically, this corresponds to a process whereby the laser output is changed in response to changes in the noise current.
Amplitude fluctuations in the light from a laser and the partition noise attributable to a beam splitter are analyzed using an equivalent-circuit model that incorporates amplitude squeezing. When the energy is not carried by photons but by electrons and also when electrons can be treated as classical particles, the circuit model need not be changed. One simply applies the general circuit model to low frequencies h\ensuremath{\nu}\ensuremath{\ll}kB{\mathit{k}}_{\mathit{B}}kB​T. Conversion of an electric current to photon flux, and vice versa, converts classical sub-shot-noise spectra of electric currents into quantum squeezing of photon fluxes, and vice versa. The sub-shot-noise character of electric currents, unlike that of quantum squeezing, is not necessarily destroyed by current loss.
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]