IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998 869 Noise Spectra of Semiconductor Optical Amplifiers: Relation Between Semiclassical and Quantum Descriptions Mark Shtaif, Bjarne Tromborg, and Gadi Eisenstein Abstract—The paper presents a comparison between a semi- emission is a genuine quantum phenomenon which can only be classical and a quantum description of the output noise spectra properly described by quantizing the electromagnetic field and of semiconductor optical amplifiers. The noise sources are rep- interpreting it as an operator. It is, however, very convenient resented by Langevin noise functions or operators and by the vacuum field operator, and the analysis takes into account the and rather common to deal with the noise properties of effects of increasing gain saturation along the amplifier wave- photonic devices within the framework of the semiclassical guide. It is shown that the difference between the semiclassical approximation. According to this approach, the various noise and quantum descriptions sums up to a shot noise term and to contributions are described in terms of white Gaussian noise a correction to the semiclassical autocorrelation relation for the processes, commonly refered to as “Langevin forces,” which carrier noise. A discussion is given of the shapes of the noise spectra, the relative importance of various noise contributions, are added to the wave equation for the electric field and to the and their dependence on input power. rate equation for the carrier density. From these equations and the correlation relations between the Langevin noise functions, Index Terms— Nonlinear optics, optical amplifiers, optical noise. one can determine the noise spectra of the output field such as the intensity noise, the phase noise, and the field power spectra. I. INTRODUCTION In this paper, we analyze the noise properties of semicon- HE NOISE properties of the optical signal from photonic ductor optical amplifiers (SOA’s). We show that with the right Tdevices like semiconductor lasers, optical amplifiers, and choice of correlation relations the semiclassical result becomes wavelength converters have been the subject of extensive identical to that of a quantum description with one exception. studies for the last 15 years and are still a very active field of One has explicitly to add a shot noise term to the semiclassical research (see [1] for an excellent review). The strong interest intensity and phase noise spectra to make them agree with the is due to the practical importance of the topic as well as its spectra derived from a quantum approach. relation to fundamental issues in quantum optics. The practical The quantum description that we shall adopt was developed applications have mainly been within optical communica- by Yamamoto et al. [6], [7] to deal with lumped laser diodes tions, where a detailed understanding of the noise properties with high- cavities and uniform carrier density in the gain is necessary for optimizing systems performance of optical section. The basic idea in that description is that vacuum communication systems. The recent promising development fields are injected into the laser cavity from outside. The of devices for wavelength conversion by the use of nonlinear fields are partly reflected at the laser facets and the transmitted effects in semiconductor optical amplifiers [2] has stimulated and reflected vacuum fields interfere with the fields generated new studies of the relation between noise and nonlinearity in the gain medium. With this procedure, the shot noise of [3]–[5]. the signal appears as an inherent property of the total field The main source of noise in the optical field in a semi- and does not have to be added as an independent noise conductor optical gain medium is the spontaneous emission term. The vacuum fields and the Langevin noise terms are of photons by recombination of electron–hole pairs. The interpreted as operators given by their commutation relations. The approach was generalized by Prasad [8] and Tromborg Manuscript received October 1, 1997; revised January 27, 1998. This work et al. [9] to describe lossy (open) extended cavities with was supported in part by the Israeli Academy of Sciences and Humanities nonuniform carrier density distribution along the cavity. For under Grant 050 831. this case, one does not have to introduce Langevin noise M. Shtaif was with the Advanced Optoelectronic Center, Electrical Engi- neering Department, Technion, Haifa 32000 Israel. He is now with AT&T forces to describe the field fluctuations caused by radiation Labs Research, Red Bank, NJ 07701-7033 USA. losses from the cavity. The fluctuations are included through B. Tromborg is with the Advanced Optoelectronic Center, Electrical En- the boundary conditions for the combined vacuum fields and gineering Department, Technion, Haifa 32000 Israel, on leave from Tele Danmark R&D, DK-2630 Taastrup, Denmark. the internally generated fields. One of the advantages of the G. Eisenstein is with the Electrical and Computer Engineering Department, approach is that it avoids introducing photons as excitations University of Minnesota, Minneapolis, MN 55455 USA, on sabbatical from of modes of the radiation field. For fields in a dispersive gain the Advanced Optoelectronic Center, Electrical Engineering Department, Technion, Haifa 32000 Israel. medium in an open cavity, the concept of a mode is a subtle Publisher Item Identifier S 0018-9197(98)03051-6. matter [8]. This applies to semiconductor lasers, but even more 0018–9197/98$10.00 1998 IEEE 870 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998 so to SOA’s which may be considered as a limiting case of The coefficient is the amplifier modal gain, is a loss a laser with a completely open cavity. However, the noise coefficient introduced to account for the waveguide scattering properties are given by the field operator algebra and one does losses, is the linewidth enhancement factor, and the term not have to construct specific representations of the algebra to is a Langevin force (white Gaussian noise source) derive the noise spectra. describing the noise contribution of the spontaneous emission It was shown in [9] for a general class of laser structures in- process. The modal gain is approximated by cluding distributed feedback (DFB) lasers that apart from shot noise terms the quantum description gives the same intensity (2) noise and phase noise spectra as the semiclassical calculation where is the differential gain coefficient, the carrier when the correlation relations for the semiclassical Langevin density, and its value at transparency. In (2), we have ne- noise functions are appropriately chosen. However, the general glected the effect of intraband mechanisms [11]. This neglect formalism makes it difficult to capture the main issues. In this is enabled by the wide-band nature of spontaneous emission paper, we will consider the simple case of a nonlinear SOA for noise in the amplifier and it will be justified in more detail which the relation between the semiclassical and the quantum later. The rate equation for the carrier density is [4], [9] descriptions is more transparent. The semiclassical calculation of the intensity noise and field power spectra were presented (3) in [4] and compared to experiments in [10]. The analysis in [4] only included the spontaneous emission noise sources. In this where is normalized such that is the optical power in paper, we also include the noise source in the carrier density the amplifier. The other parameters are the pumping rate the rate equation and discuss the relative importance of the two carrier lifetime the photon energy of the injected optical field sources and their cross correlation. Furthermore, we present , and the effective cross-section area of the active region the analogous results for the phase noise spectrum for which . The term is a Langevin force accounting for we find a stronger influence of the carrier noise than for the carrier noise. In order to simplify the analysis, it is convenient intensity noise and field power spectra. to normalize the intensity of the optical field to the saturation The paper is organized as follows. In Section II, we solve power of the amplifier, which is defined as the equations for the electric field and carrier density in a SOA Thus, defining and making use of (2), we get and relate the fluctuations of the output field to the noise in the injected field and the noise generated in the amplifier. Section (4) III presents a semiclassical derivation of the intensity noise, the phase noise, and the field power spectra together with calculations of the different noise contributions for various (5) input powers. Section IV presents the quantum formalism which we then use for calculating the intensity noise spectrum. where and are given by and For this spectrum, we discuss in detail how the various noise respectively, and . contributions compare with the semiclassical calculation. For A basic assumption in deriving the noise spectra is that the the analogous calculations of the phase noise and the field effect of the noise on the total output electric field is small power spectra, we present and discuss the final results. Finally, and can be treated as a perturbation. This applies especially Section V is devoted to conclusions. in cases of nonlinear amplification when the amplified signal power is large. The assumption implies that the electric field in the amplifier can be written as II. GENERAL FORMALISM We start by introducing the general formalism to be used for the derivation of the noise spectra at the amplifier output. The amplifier is assumed to have ideally nonreflective facets (6) and no internal reflecting interfaces or imperfections. We choose to use a semiclassical notation in the analysis presented where and are the values of in this section; it applies with minor modifications to the its amplitude and its phase, respectively, in the absence of quantum mechanical description where the classical fields are amplifier noise.