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. The terms and interpreted as operators. The propagation of an optical field in describe the deviation of these parameters from their values an SOA is described by the following equation [4]: without noise. The gain coefficient can be described in a similar manner as . Substituting (6) in (4) and (5), (1) we get the following relations for the amplitude and phase of the optical field in the frequency domain: where is the slowly varying envelope of the optical field. The parameter is the longitudinal space coordinate and (7) is a shifted time coordinate with and being the real time coordinate and the group velocity in the (8) amplifier. The input is at and the output at . 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

(19) (9) 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 (10) .

The terms and are the Fourier transforms of III. SEMICLASSICAL TREATMENT OF THE NOISE SPECTRA and respectively. Equation (7) has the simple solution Equations (11)–(18) enable us to calculate the noise spectra at the amplifier output. We use the notation for the (11) cross correlation power spectrum with given by (20)

(12) of the mutually stationary processes and . The brack- ets “ ” denote ensemble averaging. For simplicity we write Using the result (11) for (8) can be easily integrated for the autocorrelation power spectrum of to give . 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 (13) RIN (21) The expression for the electric field is then given by and from (11) the output spectrum at becomes

RIN RIN (22)

(14) We have here defined by the relation where is where is given by the Dirac delta function and we make use of the fact that (15) noise components at different locations in the amplifier are uncorrelated. The function is seen to transform the The parameters and which appear in (11)–(15) are noise contribution at position to the output end at . derived by setting the time derivative in (5) to zero and solving Thus, the first term on the right-hand side of (22) is the RIN of (4) and (5) without the noise sources and . Thus we get the input signal transformed to the output end, and the integral in (22) gives the contribution from the noise sources in the (16) amplifier. In order to calculate the latter, we need to know the correlation relations of the Langevin forces and . with In both the semiclassical and the quantum description of the (17) 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], An efficient method for solving (16) for numerically is described in [12]. [13], [14] or from the observation that the diffusion constant is the sum of all rates which change the photon or carrier Finally, as we show in the appendix, can be expressed number [14]. At this stage we use the commonly adopted analytically as with given by expressions for the semiclassical correlation relations for and as derived by Henry [15] (18) (23) where is the ratio between the scattering losses and gain coefficients . The parameter must be less than (24) 872 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998 where is the inversion factor given approximately by emission noise, carrier noise, and the cross correlation between [16]. The Langevin force carrier noise and spontaneous emission, respectively. The is a real process describing the gain fluctuations caused by index indicates classical treatment. These terms can be carrier noise. It results from two kinds of processes and can expressed explicitly by substituting the correlation relations be written in the following way [9]: (23), (24), (29), and the modified relation (28) into (22) and by using the definition of in (9) (25) The first term on the right-hand side of (25) describes (31) the noise of the injection current and the noise caused by (32) nonradiative recombination of carriers. This noise term is not correlated with the process of spontaneous emission and its diffusion coefficient is given by [9]

(33)

(26) (34) where is a parameter describing the noise of the current Fig. 1 shows an example of the RIN and its various com- source. It is equal to 1 when the current source exhibits ponents for three different values of input power assuming shot noise behavior and it satisfies for a sub- a coherent (noiseless) input signal. The amplifier parameters Poissonian current source. The second term on the right- used in the calculation are specified in Table I. The dominant hand side of (25) results from radiative recombination and is noise contribution is that of spontaneous emission, which therefore correlated to the spontaneous emission noise .In is often referred to as the signal spontaneous beat noise. At fact, since every emitted photon implies the recombination high frequencies this contribution has a constant of one electron hole pair, the carrier noise resulting from value, consistent with the wide-band nature of the spontaneous photon emission is proportional to the fluctuation of the optical emission process. At low frequencies it is characterized by intensity. Therefore [9] a dip indicating a significant narrow-band noise suppression. (27) This effect results from gain saturation and it was dicussed in [4]. When the optical intensity is increased due to noise, the From (23)–(27), the autocorrelation for and the cross gain is reduced so that the increase in intensity is suppressed correlation between and become 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, (28) are relatively small and their contribution is limited to low and frequencies due to the limited bandwidth in which carriers can exhibit fluctuations. They become significant only when the (29) input power is large, as can be seen in Fig. 1(c), where and for which the amplifier gain is reduced from Unfortunately, (28) deviates from the result obtained by adding 30 to 5 dB due to gain saturation. Such high input powers are, the rates that change the carrier number [14]. It can easily be however, impractical in most applications, so we conclude that shown that the latter method will replace by . carrier noise contributions to RIN can safely be neglected. As we shall see, the same modification is obtained from a In order to gain some intuition for the frequency dependence quantum description. The procedure which is often applied in a of the various RIN components, it is useful to assume the semiclassical treatment (and which we will adopt) is therefore limiting case of where is given by (19). In this to use the relations (23), (24), and (29) as they are, but replace case (32)–(34) can easily be manipulated in a way that the in (28) by . We will show that this leads to the frequency dependence is pulled out of the integrals. This quantum mechanical expression for the RIN except for a shot yields that the dip in as well as the shapes of and noise term. are Lorenzian and proportional to The RIN given by (22) can be written as . Qualitatively, this frequency dependence is a good RIN (30) approximation for the case when scattering losses are not neglected. This is illustrated in Fig. 2 which shows the RIN where the four terms on the right-hand side of (30) describe the spectrum for and in dashed, solid, and contributions of the noise of the injected signal, spontaneous dotted lines, respectively. The contribution of scattering losses SHTAIF et al.: NOISE SPECTRA OF SEMICONDUCTOR OPTICAL AMPLIFIERS 873

Fig. 2. The total RIN spectrum normalized to its wide-band value and r aH r aHXP r aHXR (a) calculated for (dashed), (solid), and (dotted). The input power is € @HA a HXHI€s—t.

TABLE I AMPLIFIER PARAMETERS

IW "h3H 1.310 J qH 30 dB $ 1  5 €s—t 9 mW ( 400 ps xtv 10 r a s™agH 0.2

written explicitly as (b)

RIN

(36)

(37)

(38)

(39) (c) Fig. 1. The RIN spectrum (dashed curve) and its three components. (a) Input The phase noise spectrum as well as its various components power € @HA a HXHI€s—tY corresponding to saturated gain q a 21 dB. (b) € @HA a HXI€s—tYq a 14 dB. (c) € @HA a €s—tYq a 5 dB. 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 is evident only around since their effect is equivalent the shape of the RIN spectrum shown in Fig. 1. Qualitatively, to the reduction of the average gain value. the two spectra have a similar shape, but while the RIN is In a way which is similar to the one used for the derivation decreased at low frequencies, the phase noise increases. This is of the RIN spectrum, we derive the optical phase noise so because the same carrier density fluctuations that suppress spectrum the fluctuations of the optical intensity cause fluctuations of (35) the refractive index and increase phase noise. For all input signal powers, the dominant noise contribution is still that of where the meaning of the terms on the right-hand side of (35) spontaneous emission. However, the contribution of carrier is analogous to that of (30). Using (13) these terms can be noise is in this case much more significant than in the case 874 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998

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 (a) only give the approximate spectrum neglecting carrier noise and assuming a coherent input signal

(40)

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 spec- trum (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. (b) 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 (c) frequency and may therefore be neglected. In the linear amplification regime, when and for Fig. 3. Phase noise power density spectrum (dashed curve) and its three components. Same parameters as in Fig. 1. so that , the spectra obey the familiar relations

RIN (41) of the RIN spectrum. This is mainly because here and have the same sign while the corresponding terms for where is the output power . 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 IV. QUANTUM MECHANICAL can be shown to be a very good approximation for the terms TREATMENT OF THE NOISE SPECTRA and . In the case of , the inclusion of scattering In a quantum description of optical amplification, the elec- losses reduces the noise at zero frequency, which becomes tric field is represented by an operator. However, since the significant at high input powers [see Fig. 3(c)] and causes the noise phenomena are manifested only as small perturbations creation of a double-peak structure. As we mentioned earlier, relative to the average field values, it is quite sufficient to apply 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)

where “ ” indicates Hermitian conjugation. All other combina- tions 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 (b) 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 me- chanical noise spectra. We start again by analyzing the RIN. As in (30), it is the sum of four contributions

RIN (50) (c) Fig. 4. The normalized electric field power spectrum. The dashed curve where stands for “quantum mechanical” treatment. The corresponds to the complete spectrum including the effect of carrier noise, interpretation of the various terms is similar to that of (30), and the solid curve corresponds to the approximated spectrum where carrier noise is neglected. Same parameters as in Fig. 1. and in this case we find

RIN (51) 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, (52) where following the approach of [9], consists of two contributions (42) The term is related to the spontaneous emission process whereas is a new Langevin force which accounts for (53) 876 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998

the amplifier for quantum-limited detection [18]. The SNR of the output signal is (54) where, as before, denotes the power of the input signal. SNR RIN (61) 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 in- where is the detector bandwidth. The SNR of the input troduction of . Second, (52) contains the quantum noise signal is given by the same expression with RIN replaced by contribution of the loss term which is added to the numer- the input shot noise The noise figure is therefore ator inside the integral. Third, the term which appeared given by 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 RIN (62) semiclassical results, we write (52) for in the following form: and it depends on the detector bandwidth . For broad-band detection with the noise figure reduces to (55) RIN (63) The integral on the right-hand side of (55) can be expressed as which by (60) and (41) leads to the familiar expression [18]

(64)

(56) in the case of weak saturation. For narrow-band detection with the noise figure becomes We have here used the fact that RIN (65) (57) For the three input power levels 0.01, 0.1, which results from (16). From (12) we get 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 (58) substantial improvement in SNR for narrow-band detection and intermediate levels of saturation. Note that in all examples Substituting these relations into (55) we find 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- (59) Poissonian current injection and negligible internal loss . Finally, substituting (59) into (50) yields 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 RIN RIN (60) (66) 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 i.e., the spectra coincide except for the shot noise term. for the shot noise term. The quantum optics expression for which represents It is interesting to see how the reduction of RIN at low the measured spectrum is given by [19] frequencies influences the noise figure of the amplifier. The noise figure is defined as the ratio between the signal- (67) to-noise ratio (SNR) of the signal at the input and output of SHTAIF et al.: NOISE SPECTRA OF SEMICONDUCTOR OPTICAL AMPLIFIERS 877

Here the ordering of and is essential since only REFERENCES (67) ensures that for and hence that no [1] C. H. Henry and R. Kazarinov, “Quantum noise in photonics,” Rev. signal is detected in vacuum. In this case, the semiclassical Mod. Phys., vol. 68, pp. 801–853, July 1996. and quantum mechanical calculations give identical spectra [2] J. M. Wiesenfeld, “Gain dynamics and associated nonlinearities in semiconductor amplifiers,” Int. J. High Speed Electron., vol. 7, pp. (68) 179–222, Mar. 1996. [3] A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, and there is no additional shot noise term. The absence of J. Eckner, and G. Guekos, “Efficiency and noise performance of wave- the term can be ascribed to the fact that the field power length converters based on FWM in semiconductor optical amplifiers,” spectrum is obtained by a narrow-band measurement. It may IEEE Photon. Technol. Lett., vol. 7, pp. 357–359, 1995. [4] M. Shtaif and G. Eisenstein, “Noise characteristics of nonlinear semi- be performed with a Fabry–Perot interferometer followed by a conductor optical amplifiers in the Gaussian limit,” IEEE J. Quantum slow photodetector. The shot noise contribution to the variance Electron., vol. 32, 1801–1809, Oct. 1996. [5] K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, of the detector current is the familiar where is the and H. G. Weber, “Noise analysis of frequency converters utilizing electron charge and is the single-sided detector bandwidth. semiconductor-laser amplifiers,” IEEE J. Quantum Electron., vol. 33, The shot noise can therefore be ignored if the bandwidth is pp. 81–88, Jan. 1997. [6] Y. Yamamoto and N. Imoto, “Internal and external field fluctuations sufficiently small. of a laser oscillator—Part I: Quantum mechanical Langevin treatment,” IEEE J. Quantum. Electron., vol. QE-22, pp. 2032–2042, Oct. 1986. [7] Y. Yamamoto, S. Machida, and O. Nilsson, “Squeezed-state generation V. CONCLUSIONS by semiconductor lasers,” in Coherence, Amplification and Quantum We have analyzed the noise spectra of an SOA and com- Effects in Semiconductor Lasers, Y. Yamamoto, Ed. New York: Wiley, 1991. pared quantum mechanical and semiclassical derivations of [8] S. Prasad, “Theory of a homogeneously broadened laser with arbitrary the spectra with a focus on the regime of strong saturation. mirror outcoupling: Intrinsic linewidth and phase diffusion,” Phys. Rev. A, vol. 46, pp. 1540–1559, Aug. 1992. The two formalisms are conceptually rather different and lead [9] B. Tromborg, H. E. Lassen, and H. Olesen, “Traveling wave analysis to different expressions for the various noise contributions. of semiconductor lasers: Modulation responses, mode stability, and However, with appropriate choise of noise correlation relations quantum mechanical treatment of noise spectra,” IEEE J. Quantum Electron., vol. 30, pp. 939–956, Apr. 1994. the differences add up to the effect of transforming the shot [10] M. Shtaif and G. Eisenstein, “Noise properties of nonlinear semicon- noise of the input signal into the shot noise of the output signal. ductor optical amplifiers,” Opt. Lett., vol. 21, pp. 1851–1853, Nov. Hence, when all the noise terms are included, the quantum 1996. [11] A. Uskov, J. Mørk, and J. Mark, “Wave mixing in semiconductor laser and semiclassical spectra only deviate by shot noise terms. amplifiers due to carrier heating and spectral holeburning,” IEEE J. The analysis shows how to make the semiclassical approach Quantum Electron., vol. 30, pp. 1769–1776, 1994. consistent with a quantum description. This is important [12] A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, “Four- wave mixing in traveling-wave semiconductor amplifiers,” IEEE J. when it comes to studying noise properties of more complex Quantum Electron., vol. 31, pp. 689–699, Apr. 1995. photonic systems where semiclassical calculations are much [13] L. D. Landau and E. M. Lifschitz, Statistical Physics, 2nd ed. New York: Pergaman, 1994. simpler than quantum calculations. [14] D. Marcuse, “Computer simulation of laser photon fluctuations: Theory of single-cavity laser,” IEEE J. Quantum Electron., vol. QE-20, pp. 1139–1148, Oct. 1984. APPENDIX [15] C. H. Henry, “Theory of spontaneous emission noise in open resonators DERIVATION OF and its applications to lasers and optical amplifiers,” J. Lightwave Technol., vol. LT-4, pp. 288–297, Mar. 1986. To solve the integral in the exponent of (12) we make the [16] T. Saitoh and T. Mukai, “Traveling-wave semiconductor optical ampli- variable substitution . From (16) and (17), one fiers” in Coherence, Amplification and Quantum Effects in Semiconduc- tor Lasers, Y. Yamamoto, Ed. New York: Wiley, 1991. can easily find that [17] A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous inter- action of spectral modes in a semiconductor laser,” IEEE J. Quantum (A1) Electron., vol. QE-11, pp. 510–515, 1975. [18] G. Agrawal, Fiber-Optic Communication Systems. New York: Wiley, 1992, pp. 334–335. Substituting in the exponent of (12) we get [19] R. Loudon, The Quantum Theory of Light, 2nd ed. Oxford, U.K.: Oxford University, 1991, ch. 6.

Mark Shtaif was born on April 21, 1966. He received the B.Sc., M.Sc., and Ph.D. degrees from the Technion, Israel Institute of Technology, Haifa, (A2) in 1990, 1993, and 1997, respectively. His Ph.D. work concentrated on the gain and noise properties Substitution of (A2) in (12) leads to (18). of nonlinear semiconductor optical amplifiers. In 1997, he joined the lightwave technology research group at AT&T Laboratories Research, Red ACKNOWLEDGMENT Bank, NJ, as a Senior Member of the Technical Staff. His current research interests concentrate on One of the authors, M. Shtaif, wishes to thank Dr. M. the theoretical analysis of noise and optical nonlin- Margalit from MIT, Cambridge, for interesting discussions. earities in the context of optical communication systems. 878 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998

Bjarne Tromborg was born in 1940 in Denmark. Gadi Eisenstein was born on June 20, 1949, in He received the M.Sc. degree in physics and math- Haifa, Israel. He received the B.Sc. degree from ematics from the Niels Bohr Institute, University of the University of Santa Clara, Santa Clara, CA, in Cophenhagen, Denmark, in 1968. 1975 and the M.Sc. and Ph.D. degrees from the From 1968 to 1977, he was a Research Associate University of Minnesota, Minneapolis, in 1978 and at NORDITA and the Niels Bohr Institute. His 1980, respectively. research fields were theoretical elementary particle In 1980, he joined AT&T Bell Laboratories where physics, in particular analytic S-matrix theory and he was a Member of the Technical Staff in the Pho- electromagnetic corrections to hadron scattering. tonic Circuits Research Department. His research From 1977 to 1979, he taught at a high school. at AT&T Bell Laboratories was in the fields of In 1979, he joined Tele Danmark Research (for- diode lasers dynamics, high-speed optoelectronic merly TFL), Horsholm, where he was Head of the Optical Communications devices, optical amplification, optical communication systems, and thin-film Department from 1987 until the end of 1995. As Head of the department, technology. In 1989, he joined the faculty of the Technion, Israel Institute he established semiconductor and optics laboratories for fabrication and of Technology, Haifa, where he is Professor of Electrical Engineering and characterization of laser diodes and for communication systems experiments. a member of the Technion Advanced Optoelectronics Center. His current Since 1991, he has been Adjunct Professor in physics at the Niels Bohr activities are in the fields of quantum-well lasers, nonlinear semiconductor Institute, University of Copenhagen. In 1996, he was Project Manager in Tele optical amplifiers, and compact short-pulse generators. 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 1997.