Time-domain modeling of semiconductor optical ... - IEEE Xplore

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999

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Time-Domain Modeling of Semiconductor Optical Amplifiers for OTDM Applications Gueorgui Toptchiyski, Stefan Kindt, Klaus Petermann, Senior Member, IEEE, Enno Hilliger, Stefan Diez, and Hans G. Weber

Abstract— An advanced time-domain dynamical model for the investigation of semiconductor optical amplifiers (SOA) is presented. The model accounts for the ultrafast gain dynamics, the gain saturation and the gain spectral profile. It is also suitable for analyzing the amplifier in a system environment. As an example the model is used to investigate the gain dynamics of an SOA as well as the characteristics of an interferometer switch semiconductor laser amplifier in a loop mirror (SLALOM). Good agreement between modeling and experiment is shown. The model can be applied to the investigation of other optically timedivision multiplexed (OTDM) applications, too. Index Terms—Nonlinear optics, optical gain saturation, semiconductor optical amplifiers, ultrafast carrier dynamics.

I. INTRODUCTION

S

EMICONDUCTOR optical amplifiers (SOA’s) are important components for optical networks. In the linear regime they can be used for both booster and in-line amplifiers, e.g., in the 1.3- m window [1]. On the other hand, potential use of SOAs’ nonlinearities for all-optical signal processing has led to research in various application fields [2]–[6]. One application is demultiplexing and switching with use of an SOA as a nonlinear element in a short fiber loop, a configuration also known as terahertz optical asymmetric demultiplexer (TOAD) or semiconductor laser amplifier in a loop mirror (SLALOM) [7]–[9]. We will consider this example later in this paper. The potential of SOA’s has led to the development of various theoretical models, see, e.g., [10]–[16]. A quite successful description of the SOA gain dynamics that includes the ultrafast gain dynamics and its saturation has been presented by Mecozzi and Mørk in [15] and [16]. One of the assumptions there is the spectral independence of the gain. In the present paper we will present a time-domain amplifier model based on the gain (and index) dynamics and taking into account the spectral gain profile. The model is suitable for investigating and optimizing the performance of an SOA in a system environment. The paper is organized as follows. Section II is divided into three parts. Section II-A describes some general characteristics of the model and its implementation. Section II-B describes the Manuscript received April 5, 1999; revised August 13, 1999. This work was supported by the Deutsche Forschungsgemeinschaft (DFG). G. Toptchiyski and K. Petermann are with the Fachgebiet Hochfrequenztechnik, Technische Universität Berlin, Berlin D-10587 Germany. S. Kindt is currently with Siemens AG, München D-81359 Germany. E. Hilliger, S. Diez, and H. G. Weber are with the Heinrich-Hertz-Institut für Nachrichtentechnik, Berlin D-10587 Germany. Publisher Item Identifier S 0733-8724(99)09663-2.

Fig. 1. Schematic diagram of the SOA, divided into longitudinal sections. E + und E 0 denote the forward- and backward-traveling fields, respectively. The other symbols are explained in the text.

modeling of the gain dynamics, expressed in the appropriate rate equations, whereas Section III-C deals with the modeling of the spectral profile of the gain and the numerical solution of the propagation equation. For verification we present simulation results in Section III, and conclude in Section IV. II. MODELING A. Numerical Model The model of the SOA is shown in Fig. 1. the SOA is divided into In the propagation direction ), which fact takes into account sections (of the length and the longitudinal variation of the carrier density (the subscripts rethe carrier temperatures fer to the conduction and valence band, respectively). The effect of spectral hole burning (SHB) is considered in the standard way, which leads to an instantaneous reduction of the gain (see next subsection). The propagating electrical where the sign denotes the field is designated by sign in the backward propagation in the forward, and the direction (see Fig. 1). In each section two computational steps are performed: First, the local gain (with its linear and nonlinear components) as well as the local nonlinear phase change are calculated. Then, the propagation equations for the forward- and backward-traveling fields are solved, whereby the spectral profile of the gain through the implementation of a finite-impulse response (FIR) filter is taken into account. The numerical filter implementation does not increase strongly the computational time. System simulations and optimization procedures thus become possible. Preliminary results of such a model with respect to four-wave mixing (FWM) were presented in [17]. B. Rate Equations For reasons of clarity we present briefly the rate equations that describe the carrier dynamics in an active semiconductor material. A rigorous derivation of the rate equations, based

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on the concepts of local carrier densities, total carrier density, and carrier temperatures can be found in Mecozzi and Mørk in [15] and [16]. and the The rate equations for the carrier density are carrier temperatures (1)

differential gain and the carrier density at transparency. With the above assumptions (1) may be transformed to (7) is the unsaturated value of the carrier where density (in the absence of an input signal). The saturation where is the photon density is familiar saturation power. Equation (2) may be transformed to

(2)

(8)

is the time in a retarded time frame and where is the time. The other symbols are: group velocity of light current electron charge active region volume total and lattice temperature With an applied energy density electromagnetic field with the angular optical frequency are the mean carrier energies where being the band-gap energy. The time parameters and temperature relaxation times are: carrier lifetime The calculation of the product

denoting the nonlinear gain suppression factor with where due to carrier heating and

(3)

represent the nonlinear gain suppression factors where due to SHB. Equations (7)–(9) determine the variation of the gain from (6) with time at a certain (or, actually, in one amplifier section). In order to find the intensity profile of a propagating pulse in the SOA one has to solve the propagation equation for the photon density (or power) This is done in the next along the longitudinal coordinate subsection.

was chosen in order to account for the gain saturation from In each amplifier section the local photon both fields and the local gains are calculated densities according to (4) (5) and are the optical confinement factor and where the internal scattering loss, respectively. In our calculations (in units of Watt) and the relation between optical power is expresses the electrical field relation between optical power and photon density: where is the effective area of the waveguide, and are the waveguide width and thickness, respectively. The rate equations (1) and (2) treat simultaneously the effects of carrier depletion (also referred to as carrier density pulsation, CDP) and carrier heating (CH). Equation (1) describes carrier density changes due to the processes of pumping, spontaneous and stimulated recombination whereas (2) includes the effects of stimulated recombination and relaxation to the lattice temperature. With the assumptions and algebra similar to [15], the gain at the spectral gain peak may generally (i.e., including the effects of CDP, CH, and SHB) be written as a sum of terms (6) CH and where the contributions of CDP to the gain saturation have been separated. We SHB use the linear approximation for the carrier density dependence with being the of the gain, so that

Because the carrier-carrier scattering times are much shorter than the temperature relaxation times and the carrier lifetime, the rate equation for the nonlinear gain contribution from SHB (see [15]) transforms to (9)

C. Modeling of the Gain Spectral Profile and Numerical Solution of the Propagation Equation In this subsection the derivation of the propagation equation for the electrical field, taking into account the spectral profile of the gain, will be shown. We start by parameterizing the gain in the frequency domain

(10) with the gain peak quency of the gain peak are defined as

according to (6). The optical freand the spectral gain width (11) (12)

is the band-gap optical frequency and the shift of and as a function of the carrier density variation is linearized using In Fig. 2, the gain modeled with (10) at three different carrier densities is schematically shown. For reasons of clarity are shown; in the modelonly the gain curves above ing, through appropriate choice of a reference frequency and spectral positioning of the incoming signals, only fields at

where

TOPTCHIYSKI et al.: TIME-DOMAIN MODELING OF SOA’S

Fig. 2. gpar densities.

(!; N; 1T )

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modeled according to (10) at different carrier

frequencies above will be calculated. Another point worth mentioning is that the asymmetry in the physical gain spectral profile is not taken into account by the parabolic model in (10), which means that the best approximation of the gain spectral profile is realized in the vicinity of the gain peak. The accuracy of the actual gain modeling is discussed later on. Additionally, is presumed. operation above transparency The above gain modeling is in the frequency domain. What we need, however, is a gain function in the time domain, which takes into account the gain spectral profile, as described above. For the time dependence of the electrical field at a given coordinate we write (13) is a reference optical frequency, is the comwhere plex amplitude of the field and c.c. means complex conjugate. through one amplifier Let us introduce the propagation of segment in the time domain in the following way:

Fig. 3. Schematical representation of the gain in one amplifier segment calculated according to (10) (solid line) and the intensity gain corresponding to (14) in the frequency domain (dashed line).

means a larger frequency region in which the parabolic gain can be approximated accurately through the filter. On the other hand, because of our choice of the FIR gain function in (14), additional phase information is added to the phase changes due to the processes of CDP, CH and SHB, as means accurate gain modeling described below. A smaller for the amplitudes in a larger frequency region but also larger phase distortions. We made calculations for a value range of from 6 fs to 48 fs and compared the phase distortions added by the FIR. The smallest phase distortion was observed at fs. A rough estimate with typical parameters amounts to an effective phase-amplitude coupling coefficient of the FIR of up to 1. The value of 48 fs for the sampling interval was used in all simulations. is the total phase change in one segment In (14) due to changes of the carrier density and carrier temperatures, which changes affect not only the gain but the refractive index (because of the Kramers-Kronig relations) too, and, accordingly, the phase of the pulse propagating through the waveguide. The SHB contribution to the phase change is also considered. For the total phase change we write (15)

(14) is the time interval between two samples where of the propagation field. We define the reference optical ie, the frequency correfrequency as sponding to the gain spectral peak at the unsaturated carrier density. Equation (14) corresponds to a Mach-Zehnder filter Put it another way, it represents configuration with a delay a finite-impulse response filter (FIR) with the real coefficients and In order to take from (10) into and as shown in account, we calculate the coefficients the Appendix. In Fig. 3, the gain according to (10) and the and in (14)—see also gain described by the coefficients (23) in the Appendix—are shown in the frequency domain. The asymmetry of the physical gain spectral profile notwithstanding, the FIR approximation allows the accurate calculation of the parabolic gain in a broad spectral region, e.g., full amplifier bandwidths of 25 THz for carrier densities up to several 1018 cm 3 . In the FIR approximation the choice of the sampling interval becomes important because of the cosine A smaller term, the period of which is determined by

where (16) (17) (18) , , and and coupling coefficients.

are the respective phase-amplitude

III. RESULTS In this section we present numerical examples, calculated with the SOA model described in the previous sections. We compared the theoretical with the experimental results concerning, first, the gain dynamics of a solitary SOA (pumpprobe measurements) and then the so-called switching windows of a SLALOM (semiconductor laser amplifier in a loop mirror). In the simulations we considered one equation for the carrier temperature dynamics, thus in effect taking into account nonlinear gain due to changes of both the electron and hole

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Fig. 4. Schematic experimental setup for measurements of the gain dynamics of the solitary SOA. The mean powers of the pump and probe pulses were 10 dBm and 17 dBm, respectively.

Fig. 6. Schematic experimental setup for switching window measurements on a SLALOM. The mean powers of the pump and probe pulses were 12 dBm and 4 dBm, respectively.

0

Fig. 5. SOA gain for the probe pulses co- and counterpropagating to the pump pulses.

carrier temperatures as a single effect. For both the pumpprobe and SLALOM simulations, a continuous-wave signal for the probe was used, which experiences the gain and phase dynamics caused by the propagating pump pulses. The schematic experimental set-up for pump-probemeasurements of the gain dynamics is shown in Fig. 4. The SOA is an InGaAsP-bulk SOA with a length of 980 m [18]. Two tunable external-cavity mode locked lasers (TMLL) [19] are used as sources for the pump and probe pulses. For both lasers the pulse width is about 2 ps and the repetition rate is 10 GHz. Varying the optical delay line in the probe arm, the temporal dependence of the gain can be measured with the help of an optical spectrum analyzer. Two different wavelengths are used for pump and probe in order to discriminate between them. The gain curves are measured for two cases, namely, when the pump and probe pulses are coand counterpropagating through the SOA. In Fig. 5, simulated and measured gain curves of the solitary SOA are shown. As can be seen, the gain curves depend on the propagation direction of the pump pulses through the SOA. This effect arises from the finite length of the amplifier. The difference in the gain curves has, in particular, to be taken into account for the modeling of the interferometric switching windows, as shown below. Next, measurements on a SLALOM switch were carried out with the schematic set-up shown in Fig. 6. A SLALOM consists of a Sagnac interferometer with an incorporated SOA as a nonlinear optical medium. The temporal dependence of the transmission is called the switching window, which can

Fig. 7. Switching windows of the SLALOM for different asymmetries.

be used to characterize the performance of a switch. The measurement principle is the same as for the solitary SOA described above. Additionally a second optical delay line is inserted into the fiber loop to vary the spatial asymmetry of the SOA in the loop. In the calculations, in the case of an ideal 50 : 50 coupler, we obtain for the SLALOM output (see Figs. 6 and 7)

(19) are the experienced gains and the accumulated where phases of the co- and counterpropagating data signals, after traversing the loop. The time for the round trip in the loop . In the simulations, the exact pump powers as in the is experiment were used. Furthermore, we assumed that the probe power did not contribute to the saturation of the gain as the experimental mean pump power was much higher than the mean probe power. Simulated and measured switching windows are shown In this in Fig. 7 for different settings of the asymmetry measurement the asymmetry was chosen to be negative, i.e., the counterpropagating probe pulses reach the SOA before the copropagating ones. The probe pulses, after traversing


TABLE I SOA PARAMETERS USED IN THE SIMULATIONS

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the material and the fabrication of the amplifier influence the theoretical results. In particular, high confinement factor and high differential gain contribute to better switching windows, in terms of high transmission inside and low transmission outside the window. We also found that high and are advantageous for high switching contrast. The low calculations showed that yet another critical parameter is the with lower values leading to gain suppression factor improved switching performance. IV. CONCLUSIONS We presented a numerical time domain model that accounts for the ultrafast gain dynamics, the gain saturation and the spectral dependence of the gain. Comparisons of simulated with experimental results for a SLALOM interferometric switch showed good agreement. The model is well suited to investigate the SOA performance in a system environment.

CALCULATION

APPENDIX FILTER COEFFICIENTS

OF THE

and In this appendix we derive the filter coefficients in (14), which are calculated in every section of the SOA. Applying the Fourier transform, defined as (20) to (14), we obtain in the frequency domain (21)

the SOA and the fiber loop, interfere at the 3-dB coupler, and thus the gain and phase differences between the coand counterpropagating probe signals are responsible for the “switching window” transmission function. As a result of the different dynamics for the probe pulses traveling in opposite directions in the loop (see Fig. 5), the left side of the switching windows in Fig. 7 is flattened, whereas the right side is steeper. As can be seen, good qualitative agreement between experiment and simulations was achieved. The set of amplifier parameters used for the above simulations are presented in Table I. We also compared the pump-probe and SLALOM results with the model when gain dispersion is not taken into account. The results were very similar meaning that for the used pulse widths of about 2 ps the different spectral components experience a similar gain. Although gain dispersion is not required for the particular examples presented here, which are restricted to the case of data signals at one wavelength, the model can be applied to signals with different wavelengths without modification. As mentioned above, the length of the amplifier section is calculated from the sample time, and considerations for its choice when gain dispersion is considered were already described. In the case when no gain dispersion is considered the theoretical results were identical for amplifier section lengths in the range of 0.9–3.66 m. The parameters associated with

by in the For modeling purposes we replace remaining part of this Appendix. We write for the gain function (22) In order to take calculate the filter coefficients

from (10) into account we and according to: (23)

Expanding the left side of the above equation for small and taking only the first two terms, we write (24) For the right side of (23), we obtain (25) Expanding first two terms (justified for small

and again taking only the (25) may be written as (26)

Setting equal (24) to (26) (27) (28)

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and using the definition of equations for the filter coefficients

in (10) leads to two (29) (30)

After some algebra and defining the coefficients

and

as (31) (32)

we obtain for

and

[15] A. Mecozzi and J. Mørk, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Select. Topics Quantum Electron., vol. 3, pp. 1190–1207, Oct. 1997. , “Saturation induced by picosecond pulses in semiconductor [16] optical amplifiers,” J. Opt. Soc. Amer. B, vol. 14, pp. 761–770, Apr. 1997. [17] S. Kindt, I. Koltchanov, K. Obermann, and K. Petermann, “New timedomain model of a semiconductor laser amplifier suitable for system simulations,” in Proc. Tech. Dig. OAA’96, 1996, Paper FD 13, pp. 170–173. [18] G. Toptchiyski, K. Obermann, K. Petermann, E. Hilliger, S. Diez, and H. G. Weber, “Modeling of semiconductor optical amplifiers for interferometric switching applications,” accepted for publication in Proc. Tech. Dig. CLEO’99, Paper CTuW2, 1999, pp. 215-216. [19] R. Ludwig and A. Ehrhardt, “Turn-key-ready wavelength-, repetition rate- and pulsewidth-tunable femtosecond hybrid modelocked semiconductor laser,” Electron. Lett., vol. 31, pp. 1165–1157, 1995.

(33) (34) REFERENCES [1] S. Reichel, J. Eckert, R. Leppla, R. Zengerle, A. Mattheus, and L. C. Garcia, “Simulation and experimental verification of a 10-Gb/s NRZ field trial at 1.3 m using semiconductor optical amplifiers,” IEEE Photon. Technol. Lett., vol. 10, pp. 1498–1500, Oct. 1998. [2] K. Kikuchi and K. Matsuura, “Transmission of 2-ps optical pulses at 1550 nm over 40-km standard fiber using midspan optical phase conjugation in semiconductor optical amplifiers,” IEEE Photon. Technol. Lett., vol. 10, pp. 1410–1412, Oct. 1998. [3] S. Kawanishi, “Ultrahigh-speed optical time-division-multiplexed transmission technology based on optical signal processing,” IEEE J. Quantum Electron., vol. 34, pp. 2064–2079, Nov. 1998. [4] K. S. Jepsen, B. Mikkelsen, A. T. Clausen, H. N. Poulsen, K. E. Stubkjaer, and M. Vaa, “High-speed OTDM switching,” in Proc. Tech. Dig. CLEO’98, p. 1, Paper CMA1, 1998. [5] C. Joergensen, S. L. Danielsen, K. E. Stubkjaer, M. Schilling, K. Daub, P. Doussiere, F. Pommerau, P. B. Hansen, H. N. Poulsen, A. Kloch, M. Vaa, B. Mikkelsen, E. Lach, G. Laube, W. Idler, and K. Wunstel, “All-optical wavelength conversion at bit rates above 10 Gb/s using semiconductor optical amplifiers,” IEEE J. Quantum Electron., vol. 3, pp. 1168–1180, Oct. 1997. [6] S. Diez, C. Schmidt, R. Ludwig, H. G. Weber, K. Obermann, S. Kindt, I. Koltchanov, and K. Petermann, “Four-wave mixing in semiconductor optical amplifiers for frequency conversion and fast optical switching,” IEEE J. Select. Topics Quantum Electron., vol. 3, pp. 1–15, Oct. 1997. [7] A. E. Kelly, R. J. Manning, A. J. Poustie, and K. J. Blow, “All-optical clock division at 10 and 20 GHz in a semiconductor optical amplifier based nonlinear loop mirror,” Electron. Lett., vol. 34, pp. 1337–1339, June 1998. [8] J. P. Sokoloff, P. R. Prucnal, I. Glesk, and M. Kane, “A terahertz optical asymmetric demultiplexer (TOAD),” IEEE Photon. Technol. Lett., vol. 5, pp. 787–790, July 1993. [9] M. Eiselt, W. Pieper, and H. G. Weber, “SLALOM: Semiconductor laser amplifier in a loop mirror,” J. Lightwave Technol., vol. 13, pp. 2099–2112, Oct. 1995. [10] T. Durhuus, B. Mikkelsen, and K. E. Stubkjaer, “Detailed dynamic model for semiconductor optical amplifiers and their crosstalk and intermodulation distortion,” J. Lightwave Technol., vol. 10, pp. 1056–1065, Aug. 1992. [11] J. M. Tang, P. S. Spencer, and K. A. Shore, “Influence of fast gain depletion on the dynamic response of TOAD’s,” J. Lightwave Technol., vol. 16, pp. 86–90, Jan. 1998. [12] R. Hess, M. Caraccia-Gross, W. Vogt, E. Gamper, P. A. Besse, M. Duelk, E. Gini, H. Melchior, B. Mikkelsen, M. Vaa, K. S. Jepsen, K. E. Stubkjaer, and S. Bouchoule, “All-optical demultiplexing of 80 to 10 Gb/s signals with monolithic integrated high-performance MachZehnder interferometer,” IEEE Photon. Technol. Lett., vol. 10, pp. 165–167, Jan. 1998. [13] J. M. Tang and K. A. Shore, “Strong picosecond optical pulse propagation in semiconductor optical amplifiers at transparency,” IEEE J. Quantum Electron., vol. 34, pp. 1263–1269, July 1998. [14] K. J. Blow, R. J. Manning, and A. J. Poustie, “Model of longitudinal effects in semiconductor optical amplifiers in a nonlinear loop mirror configuration,” Opt. Commun., vol. 148, pp. 31–35, 1998.

Gueorgui Toptchiyski was born in Sofia, Bulgaria, on June 22, 1971. He received the Dipl.-Ing. degree in telecommunications from the Technical University of Sofia in 1995, after accomplishing his diploma thesis at the Technische Hochschule Darmstadt, Germany. Since 1997, he has been working toward the Ph.D. degree at the Fachgebiet Hochfrequenztechnik of the Technische Universität Berlin, Germany, on the field of nonlinear applications of semiconductor optical amplifiers. From 1996 to 1997, he received a DAAD scholarship for work on optical amplifiers at the Technische Hochschule Darmstadt (now Technische Universität Darmstadt).

Stefan Kindt, photograph and biography not available at the time of publication.

Klaus Petermann (SM’85) was born in Mannheim, Germany, on October 2, 1951. He received the Dipl.-Ing. degree and the Dr.-Ing. Degree both in electrical engineering from the Technische Universität Braunschweig, Germany, in 1974 and 1976, respectively. From 1974 to 1976, he was a Research Associate at the Institut für Hochfrequenztechnik, Technische Universität Braunschweig, Germany, where he worked on optical waveguide theory. From 1977 to 1983, he was with AEG-Telefunken, Forschungsinstitut Ulm, Germany, where he was engaged in research work on semiconductor lasers, optical fibers, and optical fiber sensors. In 1983, he became a Full Professor at the Technische Universität Berlin, Germany, where his research interests are concerned with optical fiber communications and integrated optics. In 1993, he was awarded with the Leibniz-award from the “Deutsche Forschungsgemeinschaft.” Dr. Petermann is a member of the Optical Society of America (OSA). He is member of the board of the Verein der Elektrotechnik (VDE) and member of the Berlin-Brandenburg academy of science.

Enno Hilliger was born in Berlin, Germany, on June 8 1973. He received the Dipl.-Phys. degree in physics from the Technische Universität Berlin in 1998. He is currently working towards the Ph.D. degree. ˚ From 1995 to 1996 he was ERASMUS student at the Arhus Universitet, Denmark. Since 1997, he has been with the Heinrich-Hertz-Institut für Nachrichtentechnik Berlin GmbH, Berlin, where he has been engaged in research work on all-optical switches.


Stefan Diez was born in Dresden, Germany, in 1969. He stduied physics at the Friedrich-Schiller Universität Jena, Germany, in 1989 and then changed to the Technische Universität Berlin, Germany, in 1991. He received the Dipl.Phys. degree from the Technische Universität Berlin in 1996. From 1992 to 1993, he was Fulbright Fellow at the University of Washington, Seattle. Since 1994, he has been working at the Heinrich-Hertz-Institut für Nachrichtentechnik Berlin GmbH, Germnay, where he is involved in research on semiconductor-optical amplifiers and their applications for fast optical signal processing.

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Hans G. Weber received the Dr.rer.nat degree in physics from Marburg University, Germany, in 1971. From 1972 to 1976, he was Research Associate at the Physikalisches Institut der Universität Heidelberg, Germany. From 1977 to 1978, he was Max–Kade Fellow at Stanford University, Stanford, CA, and from 1979 to 1984, he was Heisenberg Fellow at Heidelberg University. He joined the Heinrich-HertzInstitut für Nachrichtentechnik Berlin GmbH, Germany, in 1985, where he is Project Leader in the Department of Optical Signal Processing. Since 1996, he is also Professor at the Optisches Institut of the Technische Universität Berlin.