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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 851 The Modulation Response of a Semiconductor Laser Amplifier Jesper Mørk, Antonio Mecozzi, Member, IEEE, and Gadi Eisenstein, Fellow, IEEE Abstract—We present a theoretical analysis of the modulation response of a semiconductor laser amplifier. We find a resonance behavior similar to the well-known relaxation oscillation resonance found in semiconductor lasers, but of a different physical origin. The role of the waveguide (scattering) loss is investigated in detail and is shown to influence the qualitative behavior of the response. In particular, it is found that a certain amount of waveguide loss may be beneficial in some cases. Finally, the role of the microwave propagation of the modulation signals is investigated and different feeding schemes are analyzed. The nonlinear transparent waveguide, i.e., an amplifier saturated to the point where the stimulated emission balances the internal losses, is shown to be analytically solvable and is a convenient vehicle for gaining qualitative understanding of the dynamics of modulated semiconductor optical amplifiers. Index Terms—Laser amplifiers, modeling, optical modulation, semiconductor lasers. I. INTRODUCTION S EMICONDUCTOR optical amplifiers (SOA’s) have been the subject of intense research over the past decade or so [1]. The main characteristics of an SOA compared to other optical amplifiers are perhaps its small size, made possible by the huge gain available in direct-bandgap semiconductors, and the fact that it is electrically pumped. The small size and compatibility with semiconductor laser sources and semiconductor detectors offer the possibility of photonic integration and the electrical injection offers the possibility of simple modulation and control schemes. The main application of SOA’s may be within optical signal processing. It has been demonstrated that cross-gain and crossphase saturation in an SOA (the modulation of the amplitude or the phase of an optical beam passing through the waveguide by injection of a control beam) can be used for various types of signal-processing [2]–[4]. Also, coherent techniques such as four-wave mixing (FWM) have been shown to have a remarkable potential [5]–[7]. The success of these various schemes can be traced to the large value of the differential in semiconductor media; this leads to a large gain gain for modest injection currents, but also implies that the semiconductor gain medium can be readily saturated. The carrier lifetime of the semiconductor is on the order of several Manuscript received December 2, 1998; revised May 19, 1999. J. Mørk is with the Center for Communications, Optics and Materials, Technical University of Denmark, DK-2800 Lyngby, Denmark. A. Mecozzi is with Fondazione Ugo Bordoni, 00142 Rome, Italy. G. Eisenstein is with the Department of Electrical Engineering, Technion, Haifa 32000, Israel. Publisher Item Identifier S 1077-260X(99)06945-2. hundreds of picoseconds. This would seem to imply that the characteristic frequencies of schemes relying on the saturation of the carrier density should be of the order of a few gigahertz at most. It has been recently demonstrated, however, that long waveguides offer responses with speeds in the range of tens of gigahertz [3], [8]. The large gain of long amplifiers reduces the stimulated carrier lifetimes to values of the order of tens of picoseconds and propagation effects also increase the amplifier bandwidth [9], [10]. In this paper, we shall analyze the modulation response of a semiconductor laser amplifier, i.e., the frequency dependence of the amplitude modulation imposed on an injected CW optical beam when the bias current is modulated. This mode of operation has not been dealt with extensively, although it seems important to understand whether SOA’s can be used as efficient high-speed modulators. Also, switching fabrics based on SOA gates rely on current modulation [1]. These may only need moderate switching speeds but the requirements to the turn-on and turn-off times can be severe. We show here that the modulation response of an SOA displays a number of interesting features. A resonance behavior, resembling the well-known relaxation resonance in semiconductor lasers, is predicted. Since the cavity feedback is absent, the underlying physical mechanisms are different. We analyze this behavior in detail and find that the waveguide (scattering loss) plays a very important role. This is similar to the case of cross-gain modulation (XGM) in SOA’s [11], [12], and the present analysis yields further insight into this behavior. We show that the case of a transparent waveguide, where stimulated emission exactly balances the scattering losses, is particularly simple to analyze, but it still displays many of the general features of the SOA propagation dynamics and is convenient for understanding the underlying physics. It has been demonstrated that at microwave frequencies above 10 GHz, the typical electrode geometry of semiconductor lasers and amplifiers yields a large propagation loss [13], which means that the details of the way the modulation signals are fed to the device becomes important. We analyze different modulation schemes that give qualitatively different results. The paper is organized as follows: In Section II, we give the basic model describing the large-signal modulation behavior of SOA’s and investigate the small-signal response by numerical integration. Section III presents an analytical investigation of the so-called transparent waveguide, where stimulated emission exactly balances the waveguide loss. On the basis of this simplified model, Section IV investigates the effects of the microwave propagation characteristics of 1077–260X/99$10.00 1999 IEEE 852 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 the electrode. Section V demonstrates the application of an alternative technique for studying the modulation response, where the amplifier is considered as a cascade of discrete elements [14], each of which can be simply analyzed. This yields further insight into the dynamics of SOA’s. Finally, a brief summary and conclusion is given in Section VI. II. BASIC EQUATIONS AND SMALL-SIGNAL ANALYSIS Considering small-signal modulation, form and take the (6) (7) , we get the and neglecting terms, but of first order in following equation for the average photon density along the amplifier The equations for the propagation of the photon density and the rate equation for the carrier density are (8) where is the saturated gain (1) (9) (2) is the saturation photon density Here, is the spatial coordinate along the amplifier and is the local time, measured in a coordinate system moving with the group velocity . The relationship with the coordinates and measured in a static reference coordinate system , where is are: one of the space and time dependent variables, i.e., photon density , carrier density , or current . Furthermore, is the is the carrier differential gain, is the confinement factor, is the active region volume, is density at transparency, is the internal loss, which is mainly the carrier lifetime and due to waveguide scattering and free-carrier absorption. The photon density is related to the instantaneous power via (10) and the small-signal material gain is (11) For the small-signal amplitude we find (3) (12) is the modal area of the waveguide, where related to the geometric area, , by the confinement factor. The modal gain in the equations above is Since a cw signal is applied at the input of the amplifier, the initial conditions are (4) (13) (14) These equations neglect ultrafast gain nonlinearities like carrier heating and spectral holeburning, which is a good approximation for moderate saturation and low to moderate modulation frequencies, say below 20 GHz. While the inclusion of these effects lead to obvious quantitative changes, they do not modify the qualitative conclusions of the theory. We have also neglected the influence of spontaneous emission on the saturation of the gain. For long amplifiers with large gain, the level of amplified spontaneous emission can be high enough to saturate the amplifier [11], [15] but this case is not treated here. The implicit assumption is therefore that the injected power is large enough to dominate the saturation behavior. We shall consider sinusoidal current modulation of the form (5) being the constant (dc) current, which is assumed with to be uniform along the amplifier. The form of the current depends on the way the current modulation amplitude is applied, i.e., uniformly, in the form of a traveling wave, several discrete contacts (multielectrode amplifier), etc. Equation (8) can easily be solved numerically for the longitudinal variation of the average photon density, and once this dependency is known, (12) can be solved for the small-signal amplitude. To illustrate the intrinsic propagation effects we shall first consider the case of traveling-wave modulation, i.e., the current modulation amplitude follows the propagating optical becomes infield, in which case the current amplitude dependent of . Fig. 1 shows examples of the calculated modulation response for different amplifier lengths and different values of the internal loss. Notice, that since we are performing a small-signal analysis, the curves in Fig. 1 just scale linearly with the current modulation amplitude. Fig. 1(a) shows the relative response, i.e., normalized with respect to the transmitted dc-power, and Fig. 1(b) shows the absolute amplitude. The parameters used for these calculations are: mW, ps, m , m , , and m . The most prominent feature seen in Fig. 1(a) is the appearance of a resonance for the 5-mm-long amplifier when the internal loss is increased. The resonance appears for MØRK et al.: THE MODULATION RESPONSE OF A SEMICONDUCTOR LASER AMPLIFIER 853 (a) (a) (b) (b) Fig. 1. Modulation amplitude at amplifier output versus modulation frequency for different amplifier lengths L. (a) Relative modulation response. (b) Absolute modulation amplitude. The modulation was assumed to be a traveling wave matched to the optical propagation speed. Line signatures correspond to different values of the internal loss: i 20 cm01 (solid line), i = 10 cm01 (dashed–dotted line is only shown for L = 5 mm for the normalized response), and i = 0 (dashed). The small-signal gain was fixed at g0 = 138 cm01 . = modulation frequencies on the order of 10 GHz and resembles the well-known relaxation oscillation resonance seen in (semiconductor) lasers. Due to the absence of a laser cavity, however, the physical origin must be quite different. While, in the present example, the resonance is seen only for the long amplifier, it appears also for the shorter amplifiers, i.e., when the internal loss or the small-signal gain are increased. This is illustrated in Fig. 2, which is for a 2-mm-long amplifier. Fig. 1 shows that the relative response actually increases when the internal loss is increased. The increase is larger for smaller modulation frequencies and in the vicinity of the resonance (if present) and approaches zero for large modulation frequencies. It can be seen from the figure that except for frequencies in the vicinity of the resonance, the improvement comes from the increase in the response at zero frequency. Since Fig. 1(b) shows that the absolute amplitude of the modulation always decreases when the loss is increased, it can be concluded that the improvement of the relative response for increasing loss is due to the lowering of the transmitted dc or average power component in combination with the appearance of the resonance phenomenon. From Fig. 1(b), it can be seen that the absolute amplitude of the modulation at the output never exceeds the zero-loss case. (c) Fig.. 2. Modulation response for a 2-mm-long amplifier in dependence of: (a) internal loss i , (b) small-signal gain factor g0 , and (c) injected dc power P (0). Unless otherwise mentioned, the injected dc power was 1 mW, the loss was 20 cm01 , and the unsaturated small-signal gain was g0 = 276 cm01 . The modulation was assumed to be a traveling wave matched to the optical propagation speed. The mechanisms responsible for this behavior can be analyzed by considering the much simpler case of a so-called transparent waveguide. III. THE TRANSPARENT WAVEGUIDE In the case where (15) we have , and therefore (16) independently of . 854 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 (18) In the last line we assumed to be independent of , an assumption that will be relaxed later when we analyze the influence of the microwave propagation characteristics. In the . following, we shall assume (corresponding to the “steady-state” limit of a For ) we get long waveguide, (19) (29) In this case, the equation for the small-signal amplitude becomes (17) with defined by (20) therefore becomes independent of the frequency in and this limit. Let us define the relative modulation as (21) Equation (19) shows that the parameter governs both the amplitude of the source term, i.e., the last term in (19), as well as the propagation of the generated modulation compohas a low-pass nent. The frequency dependent amplitude . characteristic with a cutoff frequency of evolves according to The absolute value of Re where (22) is the phase of the small-signal amplitude, , and (30) The high- and low-frequency limits of are (31) and for (32) has a simple physical meaning, since for The parameter we have , at which the response If we assume that the frequency is down by 3 dB from the low-frequency limit, is attained in the high-frequency limit (which seems to be a reasonable approximation from numerical examples), we get (24) (33) is therefore the effective absorption coefficient and of the waveguide for transmitting a modulation component at displays a the angular frequency . It can be seen that high-pass filter characteristic with the properties while The 3-dB bandwidth is a decreasing function of the low-frequency modulation efficiency increases with . also displays a resonance The expression (28) for behavior as a function of frequency similar to the effect seen are attained, in the numerical results. If large values of several resonances are seen. The behavior can be understood versus geometrically by plotting the surface and and considering the locus curve that follows for increasing . Equations (28) and (23) show that for vanishing imaginary , the part of the effective propagation constant , Im modulation amplitude at the output decreases monotonically with the modulation frequency , and the resonance is, therefore, associated with the phase of . The sluggish response of the carriers to the modulated current thus acts to increase the modulation amplitude at the output, since the attenuation of the waveguide is effectively decreased. From the expression (27) it can be seen that the total modulation at coordinate is the sum of local contributions along the waveguide with a phase factor from coordinates which may add in-phase or out of phase. It can be seen that if the current amplitude is allowed dependent and chosen as Re (23) (25) for (26) The high-pass transmission characteristic of the waveguide reflects that components with frequency higher than the stimulated carrier lifetime cannot excite oscillations of the carrier density and are therefore transmitted without loss for , whereas lower frequency components are attenuated due to the transfer of energy to carrier density oscillations. The exact solution of (19) for a nonzero current modulation amplitude reads (27) (28) (34) MØRK et al.: THE MODULATION RESPONSE OF A SEMICONDUCTOR LASER AMPLIFIER where is constant, the phase factor of different contributions is cancelled (they always add in-phase) and we get (35) Even in this “phase-matched” case (which is different from considered above) it is found that the the case modulation response exhibits a resonance. As already dishas a low-pass filter characteristic, while the cussed, , which reflects the transmission factor characteristic of the waveguide, has a high-pass characteristic. When multiplied, the two factors act to give a maximum for certain values of the parameters. This shows that in this case the resonant modulation behavior occurs because for certain parameters the drop of the carrier density modulation with frequency is not strong enough to suppress the increase of the waveguide transmission with frequency. It is found that in this phasematched case the maximum of the modulation response occurs for the average (dc) power will depend on the distance traversed up to the position of the modulation. The modulation also shows a maximum for a certain modulation frequency in this case. We find that maximum modulation at the output, is attained for (39) Also in this case, there is a requirement on the propagation distance to see the maximum, i.e., (40) B. Traveling-Wave Modulation The case of a traveling-wave type of modulation corresponds to (in static reference-frame coordinates , , neglectbelow), ing the constant phase of c.c. (36) meaning that for given magnitudes of the loss and the smallsignal gain there is a critical length below which no resonance will be observed. The numerical value of 0.796 above arises from the numerical solution of a simple nonlinear equation. 855 (41) (42) , where is the phase with propagation constant velocity. The corresponding current in local coordinates becomes c.c. (43) where IV. THE DISTRIBUTED NATURE CURRENT MODULATION OF THE (44) In the above, we considered the case of a current modulation amplitude which was constant in the moving coordinate system, i.e., corresponding to a traveling microwave which is matched exactly with the propagating optical signal. We now consider more realistic situations where the current modulation is either very localized or in the form of a traveling wave that propagates with a speed different from the optical mode. and are the phase index for the microwave and propagating on the electrode and the group index for light propagating in the waveguide. This case can, thus, be analyzed by making in (27) the substitution (45) obtaining A. Localized Current Modulation The results of Tauber et al. [13] show that the propagation losses for modulated currents applied to the contacts can be very large at microwave frequencies, i.e., in excess of 300 dB/cm for frequencies larger than 20 GHz. If this is the case, the modulation is to be considered rather as a “point source” located at the position of the bonding wire or transmission line. This case can be analyzed by approximating the current as a delta function located at the coordinate (37) which upon insertion in (27) yields (38) The result obviously depends only on the distance from where the modulation is applied, the first part of the amplifier is completely inactive. Notice that this only holds for the transparent case considered here, since in the general case (46) we get the result previously derived for a For current modulation considered to be constant in the moving corresponds to the phase coordinate system. Since velocity of the propagating microwave coinciding with the , group velocity of light propagating in the waveguide, i.e., the constant translation speed of the local coordinate system, this is as expected. The case of a uniform current amplitude along the amplifier , which means that the phase velocity corresponds to or the wavelength of the microwave are infinite. One expects here the modulation amplitude of light at the output to be very small when the light propagation delay through the waveguide corresponds to one period of the modulation, i.e., . According to the results of Tauber et al. [13] the phase velocity of the microwave is in the range of 7%–12% of the 856 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 A. Single SOA with Zero Internal Loss: Large-Signal Analysis Neglecting the internal loss, the equations for the photon density and carrier density can be formulated as (47) (48) where (49) Fig. 3. Effect of the propagation speed of a traveling-wave type current modulation. The group index of the optical field is fixed at ng 3:5 while the group index n for the microwave field is varied. A negative value for n indicates counter-propagation. The case of n = 0 corresponds to an infinite wavelength of the microwave field and the current modulation amplitude is thus uniform along the waveguide. = velocity of light in vacuum for frequencies in the range of in the copropagating case 5–40 GHz. This means that . since Fig. 3 shows calculations, using (46), of the effect of different propagation speeds of the propagating microwave. The figure shows that the frequency at which mismatch between the microwave and the light leads to a dip in the modulation response is smaller for the case of a copropagating ) than for the traveling-wave (in the realistic case of ). This might case of a uniform current modulation ( at first sight be somewhat surprising. It can be understood, however, by considering the other extreme case where light propagates much faster than the microwave. Then, a dip in the modulation amplitude at the output is expected when the wavelength of the microwave corresponds to the cavity length, . Since this gives a lower i.e., frequency limit than the propagation delay of light. At intermediate frequencies, below the “dip,” travelingwave effects may, however, increase the response over that of the phase-matched case. In an actual case of a single bonding wire attached to the center part of an electrode, the result will probably be a damped microwave propagating in both directions and one gets a combination of the effects discussed above. Since the propagation losses increase with frequency (from about 100–150 dB/cm at a few GHz to about 500 dB/cm at 40 GHz according to [13]), the type of the field may even change with frequency. Defining a new variable (50) we get (51) (52) Here, we have explicitly shown the transformation to the local coordinate system, by using , the current in the static coordinate system, as the variable. Notice that (52) is a large signal equation, only the absence of internal loss was assumed. Thus, even though it is an ordinary differential equation, as opposed to the set of partial differential equations we started with, it takes full account of the propagation effects. We see that for zero internal loss the effect of the walkoff is to change the modulation current according to the transformation (53) B. Single SOA with Zero Internal Loss: Small-Signal Analysis Let us expand (54) (55) (56) (57) with unknown , lowest order , , and . Equation (52) gives, to (58) V. CASCADED AMPLIFIERS We develop here an alternative technique for analyzing the dynamics of long SOA’s by studying a cascaded chain of several amplifiers. The basics of this technique were developed in [14]. The technique utilizes that, for zero internal loss, a very simple analytical solution for the dynamics of an SOA including propagation effects can be found. The effects of loss can be analyzed by cascading amplifiers and including a finite loss in-between. as the integrated gain where (59) and, to first-order (60) MØRK et al.: THE MODULATION RESPONSE OF A SEMICONDUCTOR LASER AMPLIFIER where (61) From (51), we get (62) Inserting (60) in the above, we obtain (63) where (64) (65) 857 All amplifiers are equal, and the loss between the amplifiers is constant . The input power of the line is such that the saturated gain of each amplifier exactly compensates for the . The second, is the case of zero loss between loss, the amplifiers. We will show that, as obvious for physical reasons, this case is equivalent to a single amplifier with gain equal to the product of the saturated gains of all the amplifiers. 1) Transparent Line: If the input power of the line is such that the saturated gain of each of the amplifiers exactly , we have and compensates for the loss . Assume also that the applied modulation has the same amplitude at each amplifier, but include the possibility that the going from an phase of the modulation increases by amplifier to the next. Assume also that the single amplifier is so short that the effect of the walkoff of the modulation is negligible within the amplifier. We, therefore, have (66) From (58), is the solution of (73) where (67) where (74) with (68) C. Amplifier Chain (75) independent of frequency. Solution of (69) is thereby blocks, each one being the sequence Consider a chain of of an SOA and a linear element of loss . The output of the th linear loss, is fed into the input of the th amplifier, with a possible frequency-dependent phase shift that we set . We have (76) (69) The saturated gain of the amplifiers are the solution of (70) where is the input of the th amplifier th Note that the modulation at the input of the fictitious is the modulation at the th amplifier amplifier output times . If the phase of the modulation is such that , (or if and the amplifier chain for lying in the region where is such that the response of the chain is large) we get (77) (71) and are given by (65) and (66) with , In (69), and replaced by , , and . . The solution of (69) can be obtained Assume iteratively. The result is Here, we have defined the frequency independent normalized as modulation amplitude (78) Assume now that (79) Expanding (65) to first order in , we obtain (80) (72) It is convenient at this point to separate two limit cases. The first, the easiest to analyze, is the case of a transparent line. Inserting this expansion in (77) we obtain (81) 858 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 (84) into (83) and using in the product the cancellation of the numerator of the th term with the denominator of the th, we obtain (85) where (86) N Fig. 4. Pictorial representation of the dependence of the high-frequency limit of the term [(1 + )N 1], that appears in the modulation response, (77), of a transparent line of modulated amplifiers. R 0 The amplitude of the modulation is is Let us now again assume, for simplicity, that independent of except for a phase factor that exactly cancels the phase factor in the sum of (85). Inserting (86) into (85), we finally obtain (82) The modulation shows maxima, equal to twice the normalized modulation amplitude, for , and zeros for , with and . The number of maxima and minima is obviously finite. They are the points that are given by the above equations and meet condition (79) for the validity of (82). A pictorial representation of the high-frequency modulation response of a chain of amplifiers is shown in Fig. 4. The arrows represent the result of the successive addition to a vector of unit length of a small vector in quadrature. The occurrence of maxima and minima of the modulation response is the result of the constructive and destructive interference at the amplifier output of the modulation response of each amplifier in the chain, rotated by the effect of the delayed response of all the amplifiers downstream, which at highfrequency act like pure phase shifters. ) should be studied with 2) Zero Loss: This case ( the help of the general solution (72), which becomes (83) and using (84) We have used in (84) that the input of the is the input of the th times its saturated gain th amplifier . Inserting (87) As expected from physical reasons, the modulation response in this loss-free case is equal to the response of a single amplifier with a saturated gain equal to the product of the saturated gains of all the amplifiers in the cascade, and subject to the . Unlike the case of a transparent total modulation line where maxima and minima may appear, the modulation response shows a pure low-pass characteristic. The reason for the qualitatively different behavior of the two cases is related to the way each amplifier in the chain contributes to the total modulation at the chain output. Each amplifier contributes with a term of the product in the first line of (72). , In the transparent line case they are and these terms are whereas in the zero loss case, . In the transparent line case and in the high frequency limit, each amplifier contributes with a pure rotation and this leads, as we discussed above, to interference effects in the output modulation. In the zero loss case, on the other is never a pure rotation unless . hand, , In this case, however, also the in-quadrature contribution , is zero and altogether. proportional to The interface effects of the transparent case cannot therefore take place. D. Limit : Saturated Amplifier with Internal Loss Assume that the amplifiers have length . If we set and , and let , we MØRK et al.: THE MODULATION RESPONSE OF A SEMICONDUCTOR LASER AMPLIFIER F. Inclusion of Ultrafast Dynamics obtain by expanding to first order (67) (88) If the loss between the amplifiers is implies condition 859 , the The inclusion of the ultrafast dynamics may be done by using the integral approach developed in [16]. Equation (56) of that paper can be easily extended to a modulated current, by arguments similar to those used above. We get (89) From (76), substituting into , we get given by (65) (98) where (see [16, eqs. (27), (45)]) (99) (90) Expanding to first order, using (54)–(57) together with where (100) (91) we get Assuming phase matching of the modulation, we get for the efficiency (101) where unchanged, with [see (61)]. Equation (63) remains formally (102) (103) (92) (104) E. Two Cascaded Amplifiers It is easy, using (63), to find the efficiency of the modulation in between, in the for two amplifiers in a cascade with loss general case (and for matched modulation, for simplicity). The is given by output of the second amplifier is the solution of (105) where (106) (93) where and The results obtained for a chain of amplifiers are therefore easily extended to the case in which the fast response is included. VI. CONCLUSION are given by (94) (95) and are the unsaturated gain of the two and amplifiers. In two limit cases, we obtain (96) (97) we have which is an indication For that the loss may increase the modulation efficiency (at zero detuning). We have shown, through numerical simulations and approximate analytical treatments, that the modulation response of a semiconductor optical amplifier is heavily influenced by propagation effects. In particular, it was found that a finite amount of waveguide internal loss, due to waveguide scattering or freecarrier absorption, can lead to the appearance of a resonance in the modulation response. Comparison to the case of zero internal loss shows that the loss decreases the response at all frequencies, but not uniformly so. Two different techniques were used for this theoretical study, namely a standard smallsignal analysis of the well-known propagation equations as well as description in terms of cascaded amplifiers. Experimental results for a modulated SOA were presented in [17] and clearly show the appearance of a resonance in the modulation response. It is to be noted that the mount used 860 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 in that case was carefully designed for high-frequency laser applications [18]. This is a very important requirement if one attempts to measure the intrinsic modulation characteristics without the masking effect due to parasitic effects of the package and microwave propagation effects of the electrodes, and may explain why resonance effects usually are not seen in measured amplifier modulation responses. We notice, however, that the electrically modulated SOA bear some common features with the optically modulated SOA (XGM) where an intensity modulated control beam is used to modulate the gain of the amplifier. In the latter case, resonance behavior has been observed both experimentally [8], [9] and theoretically [9], [12], but was not explained. Our results show that the resonance characteristic is not related to the special case of optical modulation but rather is a general consequence of the propagation characteristics of long and lossy amplifiers. [13] D. A. Tauber, R. Spickermann, R. Nagarajan, T. Reynolds, A. L. Holmes, and J. E. Bowers, “Inherent bandwidth limits in semiconductor lasers due to distributed microwave effects,” Appl. Phys. Lett., vol. 64, pp. 1610–1612, Mar. 1994. [14] A. Mecozzi and D. Marcenac, “Theory of optical amplifier chains,” J. Lightwave Technol., vol. 16, pp. 745–756, May 1998. [15] 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. 81–88, Jan. 1997. [16] A. Mecozzi and J. Mørk, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Select. 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Mikkelsen, “Optical amplifiers and their system applications,” Ph.D. dissertation, Dept. Electromagnetic Systems, Tech. Univ. Denmark, Rep. no. LD 112, Dec. 1994. [12] A. Mecozzi, “Small-signal theory of wavelength converters based on cross-gain modulation in semiconductor optical amplifiers,” IEEE Photon. Technol. Lett., vol. 8, pp. 1471–1473, Nov. 1996. Jesper Mørk was born in Denmark in 1962. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Technical University of Denmark, Lyngby, in 1986 and 1988, respectively. From 1988 to the shutdown in 1996, he was with Tele Danmark Research; since 1994, as a group leader. After half a year with Ericsson in Denmark as a Project Manager, he moved to his present position as an Associate Research Professor at Technical University of Denmark in November 1996. His main research interests are ultrafast carrier dynamics, photonic switching, and nonlinear dynamics in semiconductor lasers. Antonio Mecozzi (M’97) was born in Rome, Italy, in 1959. He received the laurea degree in chemical engineering from the University of Rome, “La Sapienza,” Italy, in 1983. He joined the Optics Division of the Fondazione Ugo Bordoni, Rome, Italy, in December 1985. From February 1991 to June 1992, he was with the Optics Group of the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, working with Prof. H. Haus. His main research interests are quantum optics, nonlinear optics, long-distance optical transmission, and laser theory. He is a topical editor of the OSA publication, Optics Letters for Nonlinear Optics. Dr. Mecozzi is a fellow of the Optical Society of America. Gadi Eisenstein (S’80–M’80–SM’90–F’99) was born in Haifa, Israel, on June 29, 1949. He received the B. Sc. degree in electrical engineering from the University of Santa Cara, Santa Clara, CA, and in 1975, he started his graduate studies at the University of Minnesota under Prof. K. Champlin and the late Prof. A. Van der Ziel. His Ph.D. work dealt with specialized Schottky barrier detectors for far-infrared radiation. In 1980, he joined Bell Laboratories, Holmdel, NJ, where he worked for nine years in the field of devices and concepts for optical fiber communication making significant contributions in the areas of optical amplifiers, short optical pulse generators, and very fast systems. In 1989, he joined the Technion—Israel Institute of Technology, Haifa, Israel, where he is a Professor of Electrical Engineering. He is one of the founders of the Barbara and Norman Seiden Center for Advanced Optoelectronics Research. He spent one sabbatical year at the University of Minnesota from August 1997 to June 1998.