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InTech eBooks

Quantum Noise in Optical Amplifiers

Kyo Inoue

Published Jul 25, 2018• 6 citations
Publisher

Abstract

Noise is one of the basic characteristics of optical amplifiers. Whereas there are various noise sources, the intrinsic one is quantum noise that originates from Heisenberg’s uncertainty principle. This chapter describes quantum noise in optical amplifiers, including population-inversion–based amplifiers such as an Erbium-doped fiber amplifier and a semiconductor optical amplifier, and optical parametric amplifiers. A full quantum mechanical treatment is developed based on Heisenberg equation of motion for quantum mechanical operators. The results provide the quantum mechanical basis for a classical picture of amplifier noise widely used in the optical communication field.

DOI
10.5772/intechopen.72992

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Noise is one of the basic characteristics of optical amplifiers. Whereas there are various noise sources, the intrinsic one is quantum noise that originates from Heisenberg’s uncertainty principle. This chapter describes quantum noise in optical amplifiers, including population-inversion–based amplifiers such as an Erbium-doped fiber amplifier and a semiconductor optical amplifier, and optical parametric amplifiers. A full quantum mechanical treatment is developed based on Heisenberg equation of motion for quantum mechanical operators. The results provide the quantum mechanical basis for a classical picture of amplifier noise widely used in the optical communication field.
The concept of noise figure F and noise measure M applicable to radio frequency and microwave amplifiers is reviewed and extended to cover optical amplifiers. Two noise figures are defined in the optical regime: one gives the ASE output, while the other gives the ratio of signal-to-noise ratios at the input and output. In the quantum regime, the lower limit on these optimum values is set by a generalization of Heisenberg's uncertainty relation. A Raman back-pumped fiber link with post-amplification with an erbium-doped fiber amplifier is used to illustrate the use of noise measure.
Introduction In previous chapters we studied wave propagation in fiber OPAs from a classical standpoint, i.e. starting from Maxwell's equations. That approach provides only a partial view of the rich nature of the electromagnetic field, namely its wave nature. However, a complete description of the electromagnetic field requires quantum mechanics. This quantum description of the electromagnetic field (called quantum electrodynamics, or QED) predicts fluctuations in measurable quantities due to the Heisenberg uncertainty principle and sets ultimate limits on the performance of devices such as optical amplifiers. A remarkable result of this theory is that the statistics and thus the noise performance of a system are affected by the quantum statistics of the source and by the physical measurement that is made. Thus a different analysis is needed depending on whether homodyne, heterodyne, or direct detection (where the field simply impinges on a photodiode) is used. Direct detection reveals the particle nature of the electromagnetic field. More specifically, energy arrives in discrete units, called photons, each of which may create an electron–hole pair in a semiconductor photodetector. In particular, the number of photons in an optical pulse can be thought of as a random variable having a mean and a variance. By deriving the mean and variance of the photocurrent that would be present if one performed direct detection of ideal laser light at the input of an optical amplifier and then a similar measurement at its exit, one can quantify the degradation of the signal-to-noise ratio (SNR) caused by the amplifier.
Optical amplification in a population-inverted medium is quantum-mechanically described, using the Heisenberg picture. Unlike conventional approaches that use a photon rate equation or a phenomenological field equation, the time evolution of the field operator, the photon number operator, and their squares are directly derived from first principles. The results give a logical theoretical base for the traditional phenomenological expressions describing quantum-mechanical characteristics of optical amplifiers.
A field noise spectral density matrix for the noise from a linear optical device, modeled as an optical multiport, is derived semi-classically. The noise is formulated in the scattering parameters, population inversion factor, and internal efficiency of the device. From this noise expression, a new equation for the amplified spontaneous emission spectral density from an optical amplifier is derived. This equation is more general than those previously published. The derivations are accomplished with the aid of methods from ordinary (classical) electrical network theory and only one quantum mechanical result regarding the noise from a conductance. As a result, the first-mentioned noise expression also includes the so-called vacuum field fluctuations. Further, the current noise spectrum of a square-law detector is derived with classical methods. The optical input to the detector includes the vacuum field fluctuations, which are shown to be the cause of the detector shot noise. Here, a second quantum mechanical result has to be employed, viz., that the vacuum field fluctuations cannot be detected alone. The results in this paper agree with the special cases found in the literature. The noise expressions are well suited for examination of the optical noise performance of arbitrary linear optical networks including, e.g., amplifiers, attenuators, isolators, reflections, filters, and couplers.>