THE NOISE properties of the optical signal from photonic

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1 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 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 semiclassical and a quantum description of the output noise spectra of semiconductor optical amplifiers. The noise sources are represented by Langevin noise functions or operators and by the vacuum field operator, and the analysis takes into account the effects of increasing gain saturation along the amplifier waveguide. It is shown that the difference between the semiclassical and quantum descriptions sums up to a shot noise term and to a correction to the semiclassical autocorrelation relation for the carrier noise. A discussion is given of the shapes of the noise spectra, the relative importance of various noise contributions, and their dependence on input power. Index Terms Nonlinear optics, optical amplifiers, optical noise. I. INTRODUCTION THE NOISE properties of the optical signal from photonic devices like semiconductor lasers, optical amplifiers, and wavelength converters have been the subject of extensive studies for the last 15 years and are still a very active field of research (see [1] for an excellent review). The strong interest is due to the practical importance of the topic as well as its relation to fundamental issues in quantum optics. The practical applications have mainly been within optical communications, where a detailed understanding of the noise properties is necessary for optimizing systems performance of optical communication systems. The recent promising development of devices for wavelength conversion by the use of nonlinear effects in semiconductor optical amplifiers [2] has stimulated new studies of the relation between noise and nonlinearity [3] [5]. The main source of noise in the optical field in a semiconductor optical gain medium is the spontaneous emission of photons by recombination of electron hole pairs. The Manuscript received October 1, 1997; revised January 27, This work was supported in part by the Israeli Academy of Sciences and Humanities under Grant M. Shtaif was with the Advanced Optoelectronic Center, Electrical Engineering Department, Technion, Haifa Israel. He is now with AT&T Labs Research, Red Bank, NJ USA. B. Tromborg is with the Advanced Optoelectronic Center, Electrical Engineering Department, Technion, Haifa Israel, on leave from Tele Danmark R&D, DK-2630 Taastrup, Denmark. G. Eisenstein is with the Electrical and Computer Engineering Department, University of Minnesota, Minneapolis, MN USA, on sabbatical from the Advanced Optoelectronic Center, Electrical Engineering Department, Technion, Haifa Israel. Publisher Item Identifier S (98) emission is a genuine quantum phenomenon which can only be properly described by quantizing the electromagnetic field and interpreting it as an operator. It is, however, very convenient and rather common to deal with the noise properties of photonic devices within the framework of the semiclassical approximation. According to this approach, the various noise contributions are described in terms of white Gaussian noise processes, commonly refered to as Langevin forces, which are added to the wave equation for the electric field and to the rate equation for the carrier density. From these equations and the correlation relations between the Langevin noise functions, one can determine the noise spectra of the output field such as the intensity noise, the phase noise, and the field power spectra. In this paper, we analyze the noise properties of semiconductor optical amplifiers (SOA s). We show that with the right choice of correlation relations the semiclassical result becomes identical to that of a quantum description with one exception. One has explicitly to add a shot noise term to the semiclassical intensity and phase noise spectra to make them agree with the spectra derived from a quantum approach. The quantum description that we shall adopt was developed by Yamamoto et al. [6], [7] to deal with lumped laser diodes with high- cavities and uniform carrier density in the gain section. The basic idea in that description is that vacuum fields are injected into the laser cavity from outside. The fields are partly reflected at the laser facets and the transmitted and reflected vacuum fields interfere with the fields generated in the gain medium. With this procedure, the shot noise of the signal appears as an inherent property of the total field and does not have to be added as an independent noise term. The vacuum fields and the Langevin noise terms are interpreted as operators given by their commutation relations. The approach was generalized by Prasad [8] and Tromborg et al. [9] to describe lossy (open) extended cavities with nonuniform carrier density distribution along the cavity. For this case, one does not have to introduce Langevin noise forces to describe the field fluctuations caused by radiation losses from the cavity. The fluctuations are included through the boundary conditions for the combined vacuum fields and the internally generated fields. One of the advantages of the approach is that it avoids introducing photons as excitations of modes of the radiation field. For fields in a dispersive gain medium in an open cavity, the concept of a mode is a subtle matter [8]. This applies to semiconductor lasers, but even more /98$ IEEE

2 870 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998 so to SOA s which may be considered as a limiting case of a laser with a completely open cavity. However, the noise properties are given by the field operator algebra and one does not have to construct specific representations of the algebra to derive the noise spectra. It was shown in [9] for a general class of laser structures including distributed feedback (DFB) lasers that apart from shot noise terms the quantum description gives the same intensity noise and phase noise spectra as the semiclassical calculation when the correlation relations for the semiclassical Langevin noise functions are appropriately chosen. However, the general formalism makes it difficult to capture the main issues. In this paper, we will consider the simple case of a nonlinear SOA for which the relation between the semiclassical and the quantum descriptions is more transparent. The semiclassical calculation of the intensity noise and field power spectra were presented in [4] and compared to experiments in [10]. The analysis in [4] only included the spontaneous emission noise sources. In this paper, we also include the noise source in the carrier density rate equation and discuss the relative importance of the two sources and their cross correlation. Furthermore, we present the analogous results for the phase noise spectrum for which we find a stronger influence of the carrier noise than for the intensity noise and field power spectra. The paper is organized as follows. In Section II, we solve the equations for the electric field and carrier density in a SOA and relate the fluctuations of the output field to the noise in the injected field and the noise generated in the amplifier. Section 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 input powers. Section IV presents the quantum formalism which we then use for calculating the intensity noise spectrum. For this spectrum, we discuss in detail how the various noise contributions compare with the semiclassical calculation. For the analogous calculations of the phase noise and the field power spectra, we present and discuss the final results. Finally, Section V is devoted to conclusions. 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 and no internal reflecting interfaces or imperfections. We choose to use a semiclassical notation in the analysis presented in this section; it applies with minor modifications to the quantum mechanical description where the classical fields are interpreted as operators. The propagation of an optical field in an SOA is described by the following equation [4]: where is the slowly varying envelope of the optical field. The parameter is the longitudinal space coordinate and is a shifted time coordinate with and being the real time coordinate and the group velocity in the amplifier. The input is at and the output at. (1) The coefficient is the amplifier modal gain, is a loss coefficient introduced to account for the waveguide scattering losses, is the linewidth enhancement factor, and the term is a Langevin force (white Gaussian noise source) describing the noise contribution of the spontaneous emission process. The modal gain is approximated by where is the differential gain coefficient, the carrier density, and its value at transparency. In (2), we have neglected the effect of intraband mechanisms [11]. This neglect is enabled by the wide-band nature of spontaneous emission noise in the amplifier and it will be justified in more detail later. The rate equation for the carrier density is [4], [9] where is normalized such that is the optical power in the amplifier. The other parameters are the pumping rate the carrier lifetime the photon energy of the injected optical field, and the effective cross-section area of the active region. The term is a Langevin force accounting for carrier noise. In order to simplify the analysis, it is convenient to normalize the intensity of the optical field to the saturation power of the amplifier, which is defined as Thus, defining and making use of (2), we get where and are given by and respectively, and. A basic assumption in deriving the noise spectra is that the effect of the noise on the total output electric field is small and can be treated as a perturbation. This applies especially 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 where and are the values of its amplitude and its phase, respectively, in the absence of amplifier noise. The terms and describe the deviation of these parameters from their values without noise. The gain coefficient can be described in a similar manner as. Substituting (6) in (4) and (5), we get the following relations for the amplitude and phase of the optical field in the frequency domain: (2) (3) (4) (5) (6) (7) (8)

3 SHTAIF et al.: NOISE SPECTRA OF SEMICONDUCTOR OPTICAL AMPLIFIERS 871 where unity for the SOA to have net gain. In the limiting case the expression for reduces to (9) (10) The terms and are the Fourier transforms of and respectively. Equation (7) has the simple solution (11) (19) which describes a circle in the complex plane for varying. As we shall see, (19) is a useful approximation for visualizing the qualitative behavior of the spectra which we calculate for. III. SEMICLASSICAL TREATMENT OF THE NOISE SPECTRA Equations (11) (18) enable us to calculate the noise spectra at the amplifier output. We use the notation for the cross correlation power spectrum with given by (20) (12) Using the result (11) for (8) can be easily integrated to give (13) The expression for the electric field is then given by of the mutually stationary processes and. The brackets denote ensemble averaging. For simplicity we write for the autocorrelation power spectrum of. Let us first calculate the spectrum of the relative intensity noise (RIN) at the amplifier output. Since,the double-sided RIN spectrum at position is given by RIN (21) and from (11) the output spectrum at becomes RIN RIN (22) where is given by (14) (15) The parameters and which appear in (11) (15) are derived by setting the time derivative in (5) to zero and solving (4) and (5) without the noise sources and. Thus we get with (16) (17) An efficient method for solving (16) for numerically is described in [12]. Finally, as we show in the appendix, can be expressed analytically as with given by (18) where is the ratio between the scattering losses and gain coefficients. The parameter must be less than We have here defined by the relation where is the Dirac delta function and we make use of the fact that noise components at different locations in the amplifier are uncorrelated. The function is seen to transform the noise contribution at position to the output end at. Thus, the first term on the right-hand side of (22) is the RIN of the input signal transformed to the output end, and the integral in (22) gives the contribution from the noise sources in the amplifier. In order to calculate the latter, we need to know the correlation relations of the Langevin forces and. In both the semiclassical and the quantum description of the noise phenomena, the Langevin forces and are assumed to be Gaussian noise processes, and the corrrelation relations are determined from the fluctuation-dissipation theorem [1], [13], [14] or from the observation that the diffusion constant is the sum of all rates which change the photon or carrier number [14]. At this stage we use the commonly adopted expressions for the semiclassical correlation relations for and as derived by Henry [15] (23) (24)

4 872 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998 where is the inversion factor given approximately by [16]. The Langevin force is a real process describing the gain fluctuations caused by carrier noise. It results from two kinds of processes and can be written in the following way [9]: (25) The first term on the right-hand side of (25) describes the noise of the injection current and the noise caused by nonradiative recombination of carriers. This noise term is not correlated with the process of spontaneous emission and its diffusion coefficient is given by [9] emission noise, carrier noise, and the cross correlation between carrier noise and spontaneous emission, respectively. The index indicates classical treatment. These terms can be expressed explicitly by substituting the correlation relations (23), (24), (29), and the modified relation (28) into (22) and by using the definition of in (9) (31) (32) (33) (26) where is a parameter describing the noise of the current source. It is equal to 1 when the current source exhibits shot noise behavior and it satisfies for a sub- Poissonian current source. The second term on the righthand side of (25) results from radiative recombination and is therefore correlated to the spontaneous emission noise.in fact, since every emitted photon implies the recombination of one electron hole pair, the carrier noise resulting from photon emission is proportional to the fluctuation of the optical intensity. Therefore [9] From (23) (27), the autocorrelation for correlation between and become and (27) and the cross (28) (29) Unfortunately, (28) deviates from the result obtained by adding the rates that change the carrier number [14]. It can easily be shown that the latter method will replace by. As we shall see, the same modification is obtained from a quantum description. The procedure which is often applied in a semiclassical treatment (and which we will adopt) is therefore to use the relations (23), (24), and (29) as they are, but replace in (28) by. We will show that this leads to the quantum mechanical expression for the RIN except for a shot noise term. The RIN given by (22) can be written as RIN (30) where the four terms on the right-hand side of (30) describe the contributions of the noise of the injected signal, spontaneous (34) Fig. 1 shows an example of the RIN and its various components for three different values of input power assuming a coherent (noiseless) input signal. The amplifier parameters used in the calculation are specified in Table I. The dominant noise contribution is that of spontaneous emission, which is often referred to as the signal spontaneous beat noise. At high frequencies this contribution has a constant value, consistent with the wide-band nature of the spontaneous emission process. At low frequencies it is characterized by a dip indicating a significant narrow-band noise suppression. This effect results from gain saturation and it was dicussed in [4]. When the optical intensity is increased due to noise, the gain is reduced so that the increase in intensity is suppressed and vice versa. The bandwidth of this effect is limited to frequencies in which the gain can respond to the intensity fluctuations. Its value increases with the optical power (see Fig. 1) and it is typically of the order of several gigahertz. The terms and which involve carrier noise, are relatively small and their contribution is limited to low frequencies due to the limited bandwidth in which carriers can exhibit fluctuations. They become significant only when the input power is large, as can be seen in Fig. 1(c), where and for which the amplifier gain is reduced from 30 to 5 db due to gain saturation. Such high input powers are, however, impractical in most applications, so we conclude that carrier noise contributions to RIN can safely be neglected. In order to gain some intuition for the frequency dependence of the various RIN components, it is useful to assume the limiting case of where is given by (19). In this case (32) (34) can easily be manipulated in a way that the frequency dependence is pulled out of the integrals. This yields that the dip in as well as the shapes of and are Lorenzian and proportional to. Qualitatively, this frequency dependence is a good approximation for the case when scattering losses are not neglected. This is illustrated in Fig. 2 which shows the RIN spectrum for and in dashed, solid, and dotted lines, respectively. The contribution of scattering losses

5 SHTAIF et al.: NOISE SPECTRA OF SEMICONDUCTOR OPTICAL AMPLIFIERS 873 (a) Fig. 2. The total RIN spectrum normalized to its wide-band value and calculated for r =0(dashed), r =0:2(solid), and r =0:4(dotted). The input power is P (0) = 0:01P sat. TABLE I AMPLIFIER PARAMETERS h! J G 0 30 db 1 5 P sat 9 mw 400 ps N tl 10 r = sc=g (b) written explicitly as RIN (36) (37) (38) (c) Fig. 1. The RIN spectrum (dashed curve) and its three components. (a) Input power P (0) = 0:01P sat ; corresponding to saturated gain G = 21 db. (b) P (0) = 0:1P sat ;G = 14 db. (c) P (0) = P sat ;G = 5 db. is evident only around since their effect is equivalent to the reduction of the average gain value. In a way which is similar to the one used for the derivation of the RIN spectrum, we derive the optical phase noise spectrum (35) where the meaning of the terms on the right-hand side of (35) is analogous to that of (30). Using (13) these terms can be (39) The phase noise spectrum as well as its various components are illustrated in Fig. 3, where a coherent input signal has been assumed similar to the calculation of Fig. 1. The shape of the phase noise spectrum at low frequencies is closely related to the shape of the RIN spectrum shown in Fig. 1. Qualitatively, the two spectra have a similar shape, but while the RIN is decreased at low frequencies, the phase noise increases. This is so because the same carrier density fluctuations that suppress the fluctuations of the optical intensity cause fluctuations of the refractive index and increase phase noise. For all input signal powers, the dominant noise contribution is still that of spontaneous emission. However, the contribution of carrier noise is in this case much more significant than in the case

6 874 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998 (a) such high powers are rather impractical so that in most realistic cases the spectrum can safely be approximated as Lorentzian. Finally, note that the phase noise spectrum assumes a constant value at high frequencies, where carrier density fluctuations are no longer effective. This results from the fact that an SOA without any reflective elements has no frequency filtering effects on the signal. For comparison, the phase noise spectrum of a semiconductor laser has a high frequency tail that, apart from a shot noise term, falls off as [7]. The field power spectrum is obtained from (14). It describes the noise spectrum that is measured at the optical frequency when the amplified signal is examined with an optical spectrum analyzer. Due to the complexity of the complete expression for the noise spectrum in this case, we only give the approximate spectrum neglecting carrier noise and assuming a coherent input signal (40) (b) (c) Fig. 3. Phase noise power density spectrum (dashed curve) and its three components. Same parameters as in Fig. 1. of the RIN spectrum. This is mainly because here and have the same sign while the corresponding terms for the RIN have opposite signs. The functional dependence of the various components of the phase noise on frequency in the limiting case of is, again, Lorentzian. This shape can be shown to be a very good approximation for the terms and. In the case of, the inclusion of scattering losses reduces the noise at zero frequency, which becomes significant at high input powers [see Fig. 3(c)] and causes the creation of a double-peak structure. As we mentioned earlier, The accuracy of this approximation is illustrated in Fig. 4, in which the dashed and solid curves correspond to the complete spectrum (including carrier noise) and the approximate spectrum (40), respectively, for the same input powers used in the calculation of Figs. 1 and 3. Since the field spectrum combines both the effects of amplitude and phase, the significance of the carrier noise is larger than in Fig. 1 but smaller than in Fig. 3. However, for reasonable levels of gain saturation, the effect of carrier noise can be safely negleted, as can be seen in Fig. 4(a) and (b). The asymmetric shape of the field spectrum is strongly related to the so-called Bogatov effect [17], known in the context of nondegenerate four-wave mixing (NDFWM). The nonlinear interaction, which determines the shape of the output noise spectra, can in fact be viewed as a NDFWM interaction between the amplified signal and the spontaneous emission noise [4]. This interpretation provides an explanation for the neglect of intraband dynamics in our treatment of the gain in (2). The contribution of intraband mechanisms to the NDFWM process is significant only at high frequencies [11]. Since at such high frequencies the NDFWM efficiency is much lower than 1, the contribution of the NDFWM process is much smaller than the component of spontaneous emission at that frequency and may therefore be neglected. In the linear amplification regime, when and for so that, the spectra obey the familiar relations RIN (41) where is the output power. IV. QUANTUM MECHANICAL TREATMENT OF THE NOISE SPECTRA In a quantum description of optical amplification, the electric field is represented by an operator. However, since the noise phenomena are manifested only as small perturbations relative to the average field values, it is quite sufficient to apply

7 SHTAIF et al.: NOISE SPECTRA OF SEMICONDUCTOR OPTICAL AMPLIFIERS 875 internal absorption. The latter must be included to be consistent with the dissipation-fluctuation theorem. The field which enters the amplifier is represented by where is a classical field and is an operator which describes vacuum field fluctuations. All three operators are uncorrelated with one another and their correlation relations are given by [9] (43) (44) (45) (a) (46) (47) (48) (b) where indicates Hermitian conjugation. All other combinations of and and their conjugates are uncorrelated. Note that and are non-hermitian operators which do not commute with their conjugates. These relations were introduced in [9] and determined by the requirement that they should match the correlation relations for lumped laser structures derived by Yamamoto and Imoto [6]. They have recently been rederived from first principles by Henry and Kazarinov [1]. The carrier noise is a Hermitian operator and its components are still described by (26) and (27) with in (27) replaced by and with Hermitian instead of complex conjugation. As we have already stated, this leads to the correct correlation relation for and no phenomenological corrections are required (49) The above relations enable us to calculate the quantum mechanical noise spectra. We start again by analyzing the RIN. As in (30), it is the sum of four contributions (c) Fig. 4. The normalized electric field power spectrum. The dashed curve corresponds to the complete spectrum including the effect of carrier noise, and the solid curve corresponds to the approximated spectrum where carrier noise is neglected. Same parameters as in Fig. 1. the quantum mechanical analysis only to the fluctuations of the electric field and leave the large-signal analysis semiclassical. The Langevin forces and are now treated as operators, where following the approach of [9], consists of two contributions The term whereas (42) is related to the spontaneous emission process is a new Langevin force which accounts for RIN (50) where stands for quantum mechanical treatment. The interpretation of the various terms is similar to that of (30), and in this case we find RIN (51) (52) (53)

8 876 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998 (54) where, as before, denotes the power of the input signal. Note the three differences between (51) (54) and (31) (34). First, the injected noise signal in (51) contains the input shot noise contribution that resulted from the introduction of. Second, (52) contains the quantum noise contribution of the loss term which is added to the numerator inside the integral. Third, the term which appeared in (32) and (34) is replaced by in the quantum mechanical expressions (52) and (54). To see the relation between the quantum mechanical and the semiclassical results, we write (52) for in the following form: (55) The integral on the right-hand side of (55) can be expressed as the amplifier for quantum-limited detection [18]. The SNR of the output signal is SNR RIN (61) where is the detector bandwidth. The SNR of the input signal is given by the same expression with RIN replaced by the input shot noise The noise figure is therefore given by RIN (62) and it depends on the detector bandwidth. For broad-band detection with the noise figure reduces to RIN (63) which by (60) and (41) leads to the familiar expression [18] (64) We have here used the fact that which results from (16). From (12) we get Substituting these relations into (55) we find Finally, substituting (59) into (50) yields (56) (57) (58) (59) RIN RIN (60) The net result is that by using the relation (49) in both the semiclassical and the quantum mechanical descriptions, the two approaches lead to the same total RIN spectrum except for the shot noise term. It is interesting to see how the reduction of RIN at low frequencies influences the noise figure of the amplifier. The noise figure is defined as the ratio between the signalto-noise ratio (SNR) of the signal at the input and output of in the case of weak saturation. For narrow-band detection with the noise figure becomes RIN (65) For the three input power levels 0.01, 0.1, and 1 in Fig. 1(a) (c), we find 7.4, 8.2, and 11.2 db for broad-band detection and 2.5, 4.3, and 0.45 db for narrow-band detection. One can therefore obtain a substantial improvement in SNR for narrow-band detection and intermediate levels of saturation. Note that in all examples the calculated noise figure is well above which is the level that corresponds to a shot-noise-limited output. In principle, the RIN at can be squeezed below its shot noise value, but this would require the amplifier gain as well as the injected signal power to be unreasonably large even for the case of sub- Poissonian current injection and negligible internal loss. The calculation of the spectra and is straightforward but cumbersome. The result for the phase noise spectrum is analogous to (60) for the RIN spectrum (66) i.e., the spectra coincide except for the shot noise term. The quantum optics expression for which represents the measured spectrum is given by [19] (67)

9 SHTAIF et al.: NOISE SPECTRA OF SEMICONDUCTOR OPTICAL AMPLIFIERS 877 Here the ordering of and is essential since only (67) ensures that for and hence that no signal is detected in vacuum. In this case, the semiclassical and quantum mechanical calculations give identical spectra (68) and there is no additional shot noise term. The absence of the term can be ascribed to the fact that the field power spectrum is obtained by a narrow-band measurement. It may be performed with a Fabry Perot interferometer followed by a slow photodetector. The shot noise contribution to the variance of the detector current is the familiar where is the electron charge and is the single-sided detector bandwidth. The shot noise can therefore be ignored if the bandwidth is sufficiently small. V. CONCLUSIONS We have analyzed the noise spectra of an SOA and compared quantum mechanical and semiclassical derivations of the spectra with a focus on the regime of strong saturation. The two formalisms are conceptually rather different and lead to different expressions for the various noise contributions. However, with appropriate choise of noise correlation relations the differences add up to the effect of transforming the shot noise of the input signal into the shot noise of the output signal. Hence, when all the noise terms are included, the quantum and semiclassical spectra only deviate by shot noise terms. The analysis shows how to make the semiclassical approach consistent with a quantum description. This is important when it comes to studying noise properties of more complex photonic systems where semiclassical calculations are much simpler than quantum calculations. APPENDIX DERIVATION OF To solve the integral in the exponent of (12) we make the variable substitution. From (16) and (17), one can easily find that Substituting in the exponent of (12) we get (A1) REFERENCES [1] C. H. Henry and R. Kazarinov, Quantum noise in photonics, Rev. Mod. Phys., vol. 68, pp , July [2] J. M. Wiesenfeld, Gain dynamics and associated nonlinearities in semiconductor amplifiers, Int. J. High Speed Electron., vol. 7, pp , Mar [3] A. D Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall Ara, J. Eckner, and G. Guekos, Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers, IEEE Photon. Technol. Lett., vol. 7, pp , [4] M. Shtaif and G. Eisenstein, Noise characteristics of nonlinear semiconductor optical amplifiers in the Gaussian limit, IEEE J. Quantum Electron., vol. 32, , Oct [5] K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, Noise analysis of frequency converters utilizing semiconductor-laser amplifiers, IEEE J. Quantum Electron., vol. 33, pp , Jan [6] Y. Yamamoto and N. Imoto, Internal and external field fluctuations of a laser oscillator Part I: Quantum mechanical Langevin treatment, IEEE J. Quantum. Electron., vol. QE-22, pp , Oct [7] Y. Yamamoto, S. Machida, and O. Nilsson, Squeezed-state generation by semiconductor lasers, in Coherence, Amplification and Quantum Effects in Semiconductor Lasers, Y. Yamamoto, Ed. New York: Wiley, [8] S. Prasad, Theory of a homogeneously broadened laser with arbitrary mirror outcoupling: Intrinsic linewidth and phase diffusion, Phys. Rev. A, vol. 46, pp , Aug [9] B. Tromborg, H. E. Lassen, and H. Olesen, Traveling wave analysis of semiconductor lasers: Modulation responses, mode stability, and quantum mechanical treatment of noise spectra, IEEE J. Quantum Electron., vol. 30, pp , Apr [10] M. Shtaif and G. Eisenstein, Noise properties of nonlinear semiconductor optical amplifiers, Opt. Lett., vol. 21, pp , Nov [11] A. Uskov, J. Mørk, and J. Mark, Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral holeburning, IEEE J. Quantum Electron., vol. 30, pp , [12] A. Mecozzi, S. Scotti, A. D Ottavi, E. Iannone, and P. Spano, Fourwave mixing in traveling-wave semiconductor amplifiers, IEEE J. Quantum Electron., vol. 31, pp , Apr [13] L. D. Landau and E. M. Lifschitz, Statistical Physics, 2nd ed. New York: Pergaman, [14] D. Marcuse, Computer simulation of laser photon fluctuations: Theory of single-cavity laser, IEEE J. Quantum Electron., vol. QE-20, pp , Oct [15] C. H. Henry, Theory of spontaneous emission noise in open resonators and its applications to lasers and optical amplifiers, J. Lightwave Technol., vol. LT-4, pp , Mar [16] T. Saitoh and T. Mukai, Traveling-wave semiconductor optical amplifiers in Coherence, Amplification and Quantum Effects in Semiconductor Lasers, Y. Yamamoto, Ed. New York: Wiley, [17] A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, Anomalous interaction of spectral modes in a semiconductor laser, IEEE J. Quantum Electron., vol. QE-11, pp , [18] G. Agrawal, Fiber-Optic Communication Systems. New York: Wiley, 1992, pp [19] R. Loudon, The Quantum Theory of Light, 2nd ed. Oxford, U.K.: Oxford University, 1991, ch. 6. Substitution of (A2) in (12) leads to (18). (A2) ACKNOWLEDGMENT One of the authors, M. Shtaif, wishes to thank Dr. M. Margalit from MIT, Cambridge, for interesting discussions. Mark Shtaif was born on April 21, He received the B.Sc., M.Sc., and Ph.D. degrees from the Technion, Israel Institute of Technology, Haifa, in 1990, 1993, and 1997, respectively. His Ph.D. work concentrated on the gain and noise properties of nonlinear semiconductor optical amplifiers. In 1997, he joined the lightwave technology research group at AT&T Laboratories Research, Red Bank, NJ, as a Senior Member of the Technical Staff. His current research interests concentrate on the theoretical analysis of noise and optical nonlinearities in the context of optical communication systems.

10 878 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998 Bjarne Tromborg was born in 1940 in Denmark. He received the M.Sc. degree in physics and mathematics from the Niels Bohr Institute, University of Cophenhagen, Denmark, in From 1968 to 1977, he was a Research Associate at NORDITA and the Niels Bohr Institute. His research fields were theoretical elementary particle physics, in particular analytic S-matrix theory and electromagnetic corrections to hadron scattering. From 1977 to 1979, he taught at a high school. In 1979, he joined Tele Danmark Research (formerly TFL), Horsholm, where he was Head of the Optical Communications Department from 1987 until the end of As Head of the department, he established semiconductor and optics laboratories for fabrication and characterization of laser diodes and for communication systems experiments. Since 1991, he has been Adjunct Professor in physics at the Niels Bohr Institute, University of Copenhagen. In 1996, he was Project Manager in Tele Danmark R&D in charge of introducing computer-based tools for planning of telecommunications networks, and, since 1997, he has been with Technion, Haifa, Israel, as a Visiting Professor. He has co-authored a research monograph on dispersion theory and authored or co-authored more than 70 scientific journal or conference publications. Recently, he was Guest Editor of an issue of the IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS. His research interests include physics and technology of optoelectronic devices with focus on noise and dynamics of semiconductor lasers. Mr. Tromborg is a member of the Danish Natural Science Research Council (1995 present). In 1981, he reeived the Electro-prize from the Danish Society of Engineers, and he was awarded the Lady Davis Visiting Professorship for nine months at Technion, Haifa, Israel, in Gadi Eisenstein was born on June 20, 1949, in Haifa, Israel. He received the B.Sc. degree from the University of Santa Clara, Santa Clara, CA, in 1975 and the M.Sc. and Ph.D. degrees from the University of Minnesota, Minneapolis, in 1978 and 1980, respectively. In 1980, he joined AT&T Bell Laboratories where he was a Member of the Technical Staff in the Photonic Circuits Research Department. His research at AT&T Bell Laboratories was in the fields of diode lasers dynamics, high-speed optoelectronic devices, optical amplification, optical communication systems, and thin-film technology. In 1989, he joined the faculty of the Technion, Israel Institute of Technology, Haifa, where he is Professor of Electrical Engineering and a member of the Technion Advanced Optoelectronics Center. His current activities are in the fields of quantum-well lasers, nonlinear semiconductor optical amplifiers, and compact short-pulse generators.