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Experimental Stability Diagram of a Diode Laser Subject to Weak Phase-Conjugate Feedback from a Rubidium Vapor Cell
Ole K. Andersen, Alexis P. A. Fischer, Ian C. Lane, Eric Louvergneaux, Steven Stolte, and Daan Lenstra

Abstract—We report on a systematic experimental investigation of the dynamical behavior of a diode laser subject to delayed feedback from a phase-conjugating mirror based on nearlydegenerate four-wave mixing in rubidium vapor. We present the first experimental stability diagram for this system. In the weak feedback regime, with feedback levels up to 030 dB, we demonstrate locking to the pump frequency with a reduced linewidth smaller than 8 MHz. In contrast to a laser with external injection, and contrary to predictions made for phaseconjugate feedback, the extent of this locking region is found to be limited. For increased feedback rates, the laser leaves the locking region and enters an unstable region through a series of multiple sidebands spectra. This is associated with self-excitation of the relaxation oscillation, and the observed scenario suggests a series of period-doublings after an initial Hopf bifurcation. Index Terms— Diode laser, four-wave mixing, Hopf bifurcation, phase conjugate feedback, relaxation oscillation, rubidium, stability diagram.

I. INTRODUCTION EMICONDUCTOR lasers are known to be extremely sensitive to any external perturbation. Different kinds of optical perturbations exist, such as optical injection or optical feedback from a conventional mirror or from a phaseconjugate mirror. They all have been intensively studied both theoretically and experimentally in order to understand why performance is sometimes improved and sometimes degraded by these perturbations. Theoretically, the problem has been approached by using rate equations [1]. It appears that the stability of the diode laser is mainly governed by two parameters: the amount of feedback and the phase or frequency difference between the emitted wave and the reflected or injected waves. During the last decade, the emphasis of research efforts has been directed toward an understanding of the conditions for which diode lasers subject to phase-conjugate feedback (PCF) are stable. To our knowledge, the first theoretical study was
Manuscript received November 16, 1998. This work was supported in part by the “Stichting voor Fundamenteel Onderzoek der Materie” (FOM) which is financially supported by the “Nederlandse Organizatie voor Wetenschappelijk Onderzoek” (NWO). O. K. Andersen, A. P. A. Fischer, S. Stolte, and D. Lenstra are with the Faculty of Sciences, Vrije Universiteit, 1081 HV Amsterdam, The Netherlands. I. C. Lane is with the School of Chemistry, University of Bristol, Bristol BS8 1TS, U.K. E. Louvergneaux is with the Universite de Lille, 59655 Villeneuve D’Ascq Cedex, France. Publisher Item Identifier S 0018-9197(99)02539-7.


done by Agrawal and Klaus [2]. They showed that the laser becomes unstable if the relaxation oscillation is undamped. This occurs if the amount of PCF exceeds a critical value. Strong noise reduction due to PCF has also been predicted [3]. However, these results are restricted to the case of identical frequencies for the solitary diode laser and for the phaseconjugate signal. For even higher feedback rates, the output of the diode laser is predicted to show rich chaotic dynamics. This has been studied using bifurcation diagrams [4], [5]. The first theoretical stability diagram was presented by Van Tartwijk et al. [6]. This diagram shows the behavior of the diode laser for every pair of parameters of the plane defined by the amount of feedback and the frequency detuning between the diode laser and the pump frequency of the phase-conjugate mirror (PCM). This stability diagram is relevant since both the amount of feedback and detuning are experimentally accessible quantities. Most of the previous treatments of semiconductor lasers subject to optical feedback from a PCM have assumed that the PCM responds instantaneously. Calculations assuming that the Kerr-type medium PCM does not respond instantaneously have also been performed [7]. In these analyses, it was found that the role of the finite response time of the mirror is to drastically enhance the stability of the diode laser and suppress chaotic output. Using the assumptions of a finite response time, a theoretical stability diagram including the regime of moderate feedback has been presented [8]. In this case, only the stable solution corresponding to locking has been addressed. Experimentally, three kinds of PCM’s are known: 1) photorefractive crystals such as BaTiO ; 2) semiconductor lasers; and 3) atomic vapors. The BaTiO crystals show high reflectivities but have very slow time response (on a scale of minutes to hours). Semiconductor lasers offer high reflectivities and fast time response on a picosecond time scale. Using atomic vapors in a four-wave mixing (FWM) configuration, one can achieve high reflectivities with response times on a nanosecond scale. The first experimental results on PCF in a diode laser were presented by Vahala et al. using a BaTiO crystal as the PCM [9]. They showed that it is possible to stabilize a diode laser with PCF and observed a linewidth reduction of more than three orders of magnitude. However, as reported by Cronin–Colomb and Yariv [10], this configuration leads to self-induced frequency scanning.

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Ohtsu et al. used an AlGaAs laser to demonstrate a stable PCM with a bandwidth of 1.9 GHz and a reflectivity of 1000 [11]. In that case, the linewidth reduced by a factor of 5. Similarly, K¨ rz and Mukai [12] reported a linewidth u reduction from 5 MHz to 25 kHz for a diode laser with external PCF from a broad-area diode laser. Their evidence for phase conjugation was the observation of the spectrum of the diode laser showing side modes spaced from the main peak by a instead of expected for conventional frequency of feedback, where is the distance from the diode laser to the PCM. Grischkowsky et al. demonstrated that time-reversed optical wavefronts can be created using FWM in atomic vapors [13]. at the atomic The PCM uses the strongly enhanced resonances. A PCM based on cesium vapor was used to stabilize the output of a diode laser [14]. However, in this paper, the stability diagram for this system was not considered. The aim of this paper is to present an experimental stability diagram of a diode laser with weak PCF from a PCM based on nearly-degenerate FWM in rubidium vapor in order to compare with the earlier mentioned calculations [8]. We also present the optical spectra that correspond to the different types of behavior of the diode laser. In this respect, not only singlefrequency locking but also different dynamical behaviors, such as period doubling, are of interest. No experimental results showing the stability diagram or parts of it, nor spectra indicating the different dynamical behaviors, were ever reported. Stable steady states have been considered, but neither the different kinds of instabilities nor routes to chaos have been distinguished. Reported experimental results that are closest to ours correspond to dynamical properties of a diode laser exposed to injection from another laser. For this case, Simpson et al. presented spectra and an experimental stability diagram [15], [16]. Indeed, the predicted stability diagram for a diode laser with weak external optical injection is close to that of a laser with PCF [8], [17]. II. EXPERIMENTAL SETUP A schematic of the experimental setup is shown in Fig. 1. It consists of a diode laser (DL) and a PCM separated by the cm and a diagnostics branch (wavemeter, distance FPI, PD2). The diode laser used was a commercial SHARP LT0277 mW Fabry–Perot type semiconductor laser with a central wavelength at 780 nm and a linewidth of 50 MHz. The temperature of the diode laser was actively stabilized to within 0.01 C. A jitter of about 20 MHz has been measured. The threshold current was 42 mA and the typical driving current was 55 mA. The diode laser was thus driven 1.3 times above threshold. The frequency of the diode laser could be tuned continuously over 0.02 nm around 780 nm by changing the driving current between 54 and 56 mA. The PCM was based on FWM in a cell filled with Rb (D2)-transition. The rubidvapor, using the 5s S -5p P ium vapor was pumped by two counterpropagating beams. These two pump beams were obtained by using a beamsplitter BS2 to divide the output from a narrow-linewidth single-mode

Fig. 1. The experimental setup. DL: diode laser; BS: beam splitter; PD: photodiode; I: optical isolator; P1: waveplate; M: mirrors; PCM: phase-conjugate mirror.

Ti : sapphire ring laser into a forward and a backward pump. The ring laser was pumped on all lines by a large-frame Spectra Physics Ar laser. Typical output power of the ring laser was 600 mW at 780 nm with 7.5 W of pump power. Additionally, the frequency of the ring laser was locked to a temperature-stabilized etalon resulting in a longterm frequency drift of about 10 MHz/h. The linewidth of the ring laser was about 500 kHz. A Faraday isolator I2 ( 40 dB isolation) was used to suppress the parasitic feedback into the ring laser. The Rb cell was made out of quartz containing the rubidium of rubidium metal in a side arm. A density of 10 cm vapor was obtained by heating the side arm to a temperature range of 130 C–150 C. The vapor was confined between two quartz windows with a diameter of 15 mm separated by 1 mm. This part of the cell was always kept at a higher temperature than the side arm, typically 180 C, to avoid condensation of the Rb vapor on the windows. The cell was placed at the point at which the pump beams cross the diode laser beam. The angle between the forward pump and the beam from the diode laser was estimated to be 1.2 . The optimum position was found by translating the cell around the intersection point while monitoring the signal from the photodiode (PD1) on an oscilloscope. The three beams all had the same linear polarization. The signal on PD1 was found to depend properly on both pump intensities and on the diode laser intensity. Part of the light emitted by the diode laser was split off by the 50 : 50 beamsplitter (BS1) and was used for diagnostics. The diagnostic branch consisted of a collection of Fabry–Perot interferometers (FPI’s) with free spectral ranges of 250 and 750 MHz. The FPI’s had a finesse of 30, resulting in accuracies of 8 and 40 MHz, respectively. A saturated absorption setup incorporating another Rb vapor cell was used to monitor the frequency of the diode laser with respect to the hyperfine lines of the D2 transition. A 30-GHz free-spectral range Burleigh wavemeter was used to monitor the absolute frequency of the diode laser. In order to avoid any feedback from the diagnostic branch into the laser diode, a Faraday isolator (I1) was added. The beamsplitter (BS1) was also used to split off and measure half of the phase-conjugate signal fed back by the mirror to the diode laser. It was detected by a 100-kHz bandwidth . photodiode



Fig. 2. (a) The saturated absorption signal when scanning the diode laser frequency across the Rb resonances. (b) The simultaneously recorded response curve of the PCM. The pump wavelength was fixed at the position marked pump .

III. RESULTS The first step was to obtain the frequency-dependent response curve of the PCM. To do this, the frequency of the pump laser was fixed close to the transition Rb 3)-5p P ] at cm [5s S ( ( nm). The frequency of the diode laser was then scanned across the rubidium resonances by increasing the driving current. For this current range (2 mA), the output power of the diode laser could be regarded as constant, 4.7 mW. During the scan, the response curve of P the PCM and the saturated absorption spectrum were recorded simultaneously (see Fig. 2). The saturated absorption spectrum shows the four resonances corresponding to the transitions Rb [5s S (F 1, 2)-5p P and Rb [5s S (F 2, 3)-5p P [18]. This spectrum is used to monitor the absolute frequency of the diode laser. The response curve shows the generated PCF power as a function of the diode laser frequency for a fixed pump frequency. Maximum response occurs when the frequency of the diode laser is equal to the frequency of the pump laser, i.e., for degenerate FWM. This main peak has a full-width at half-maximum (FWHM) of 650 MHz. In addition to this main peak, two weak side peaks displaced 1.8 GHz from the central peak are observed. These side peaks are the socalled Rabi sidebands [19]. They are generally displaced by from the central peak. Here, is the zero-detuning Rabi frequency and MHz is the detuning of the pump with respect to the resonance. In our case, the diameter of the pump beam was 2 mm, the power of the pump was 200 mW, and, assuming that the C m [19], the transition dipole-moment is given by GHz. Note that, in obtaining Rabi frequency is the response curve of the PCM, a Faraday isolator was placed in front of the diode laser so as to avoid PCF. The obtained response curve is thus centered at the pump frequency and shows strong frequency filtering. The filtering effect observed here is even more pronounced than in [8]

where 100–300-ps response times corresponding to 3–10-GHz bandwidths were considered. To investigate systematically the stability of the diode laser in the reflectivity versus plane, we proceed as follows. The pump frequency is fixed at cm ( nm). The diode laser current is fixed so that the detuning between the is kept pump frequency and the diode laser frequency constant while the reflectivity is changed using a grey filter in the pump beam. Note that the observed response curve exhibits the multiresonant behavior observed by Steel and Lind [20]. As mentioned above, the weak side peaks are attributed to the Rabi sidebands whereas the central peak consists of structures and [20]. By lowering the pump power, the at weak Rabi side peaks disappear while the relative strength of and stays the same. In our the components at case, the frequency response of the PCM, dominated by the central part of the response curve, is therefore not altered as a consequence of the changing pump power. The behavior of the diode laser is then characterized by the optical spectra obtained with the FPI and checked with the wavemeter. The spectra are measured for different amounts of the reflected PCF. Note that, due to the FSR of the Fabry–Perot, the positions of the peaks are known, modulo 250 MHz. This means that frequency components displaced more than 250 MHz from the central peak are convoluted back to appear within the 250MHz range. Therefore, the linewidths of the different spectra can be compared directly (the resolution is 8 MHz), in contrast to the positions of the peaks. However, the real positions of the peaks were determined using the 30-GHz FSR wavemeter. The various different spectra observed for increasing feedback strength are shown in Fig. 3, where the pump frequency is indicated by the dotted line. Fig. 3(a) shows the spectrum of the free-running diode laser, i.e., the laser without feedback. MHz The frequency of the diode laser is detuned from the pump frequency. Note that the spectrum in Fig. 3(a) is shown modulo 250 MHz due to the FSR of the FPI used. The FWHM of the spectrum is 50 MHz. The free-running behavior is observed for feedback levels until 50 dB (0.002%). MHz, the next regime encountered For detunings is shown in Fig. 3(b). This corresponds to an FWM process. This spectrum shows two peaks symmetrically placed around the pump frequency. A third peak, symmetrically placed with respect to the strongest peak, should be expected but was not observed, probably because it is below the detection limit. The strongest peak shows the free-running diode laser whose frequency has been pulled in the direction of the pump frequency. In addition, the linewidth of the diode laser is substantially narrowed. The small peak is the response of the PCM to the shifted frequency of the diode laser. Simulations for a diode laser with optical injection have predicted that both FWM and multiwave-mixing will occur [21]. In our case, only the behavior as in Fig. 3(b) has been observed. We believe that the strong filtering effect excludes higher order wave mixing. Further increasing the amount of feedback results in singlefrequency locking [see Fig. 3(c)]. In this case, the frequency of the diode laser is pulled to the frequency of the pump and locked on to it. The observed spectrum is characterized



Fig. 4. Stability diagram of a diode laser exposed to delayed PCF. The feedback level (%) is the ratio of the feedback power to the output power of the solitary diode laser.

Fig. 3. Optical spectra observed with a 250-MHz FSR FPI. (a) Free-running laser, linewidth 50 MHz, and  = 300 MHz. (b) FWM. (c) Locking of the diode laser to the pump frequency, linewidth 8 MHz. (d) Locking with two sidebands. Peak A is located 3.9 GHz from the central peak, (e) locking with four sidebands. Peak B is located 7.8 GHz from the central peak, (f) locking with four sidebands. Peaks C and D are located 12 and 16 GHz from the central peak. (g) Broad spectra. (h) Coherence collapse.


by a single high and narrow peak, similar to what has been observed by Ohtsu et al. [11]. In our case, the linewidth of the diode laser was reduced and was measured to 8 MHz, which was the limit of the experimental resolution. A higher feedback level causes the diode laser to develop undamped relaxation oscillations (RO’s) [see Fig. 3(d)]. The diode laser is still locked to the pump frequency but, in addition to the central peak at the pump frequency, the spectrum shows two symmetric sidebands displaced by 3.9 GHz. The height of the central peak is smaller than in (c) since a part of the radiation energy is emitted at the frequency of the sidebands. These sidebands are due to the undamped RO. For even higher feedback rates, higher harmonics of the RO frequencies appear [see Fig. 3(e) and (f)]. In the first case, a total of four sidebands is observed, and, in the latter case, eight sidebands are observed. Fig. 3(g) and (h) show two different spectra corresponding to chaotic dynamics. In Fig. 3(g), the spectrum is broad but the central peak is still at the pump frequency, in contrast to the situation shown in Fig. 3(h), where there is no central peak but only a very broad spectrum indicating coherence collapse [22]. Let us now consider the stability diagram presented in Fig. 4. The pump frequency is kept at cm ( nm) while the diode laser frequency is successively changed in steps of 125 MHz. For each of the frequency steps, the pump intensity is changed continuously

using the grey filter. For a given detuning , the amount of feedback at which the spectrum of the diode laser abruptly changes is recorded. In this way, the stability diagram is decomposed into different zones corresponding to the above listed types of behavior [see Fig. 3]. First consider the single-frequency locking range indicated by black boxes. It is seen that the lower and upper limits of this locking range are increasing functions of the detuning. Further, apart from the small feature close to zero detuning, the locking range is mainly observed for positive detuning. This can be explained as being due to the well-known fact that the self-phase-modulation parameter has a large positive value ). The frequency shift and the change (typically are connected through . in threshold gain Feedback causes lower losses, thus changing the threshold condition, so that a lower gain is required for lasing to occur. which, in the Therefore, feedback implies a negative , case of a positive -parameter, results in a negative i.e., a redshift of the diode laser frequency. This explains why locked single-frequency operation is observed for positive detunings. This trend is in accordance with what has been theoretically predicted [8]. Note, however, that in contrast to theoretical predictions, the width of this locking range dramatically decreases for detunings larger than 1 GHz. The same behavior was also observed for other pump frequencies. Below the single-frequency locking range, a region is found where FWM behavior occurs [see Fig. 3(b)]. Note that for GHz a part of this region is not detunings larger than accessible, since the diode laser suddenly jumps to the upper limit of the locking range. Above the single-frequency locking range, three different zones are found corresponding to the behaviors described in Fig. 3(d)–(f). The dark grey zone of Fig. 4 with the behavior of Fig. 3(d) is clearly dominant. The behavior corresponding to Fig. 3(f) is only observed for detunings smaller than 500 MHz (light grey). The upper part of the stability diagram corresponds to the broad spectra [see Fig. 3(g) and (h)]. The behavior in Fig. 3(h) is only observed for a limited detuning range between 200 and 600 MHz. As can be seen from Fig. 4, the maximum PCF



obtainable increases with the detuning. An explanation for this observation can be given by taking into account two facts: 1) the limited frequency response of the PCM and 2) the fact that, due to the -parameter, the route to the broad spectra shown in Fig. 3 occurs for gradually higher feedback levels as the detuning becomes larger. 1) Consider the response of the PCM to a spectrum broader than the bandwidth of the mirror. Only that part of the spectrum that falls within the PCM bandwidth is actually being conjugated. In Fig. 3(c)–(h), therefore, only the central peak is being conjugated. Since the optical power of the diode laser can be regarded as constant, an increased feedback level causes energy to be taken from the central peak of the spectrum to the sidebands [see Fig. 3]. Therefore, the reflectivity of the PCM decreases as more optical energy is displaced to frequencies outside the bandwidth of the PCM. Ultimately, the spectrum is so broad [see Fig. 3(g) and (h)] that almost no energy remains within the PCM bandwidth, thereby terminating the phase-conjugating process. This process therefore puts an upper limit on the phase-conjugate efficiency obtainable with a feedback laser. At the moment where the diode laser shows broad spectrum behavior, pumping the mirror harder will not result in an increased reflectivity. 2) For small detunings, the broad spectrum is reached for lower feedback levels as compared to the case of larger detunings [8]. This explains the gross features of the stability diagram. MHz, locking For small negative detunings, i.e., is observed as predicted by calculations [8]. For more negative MHz, only two types of behavior are detunings observed, namely FWM and the broad spectrum [see Fig. 4]. In this case, as the feedback rate is increased, the diode laser leaves the FWM regime and enters directly into the broad spectra. IV. CONCLUSION In this paper, we present for the first time an experimental stability diagram of a diode laser subject to delayed PCF. The PCM was based on nearly-degenerate FWM in rubidium vapor where the D2-transition at 780 nm was used to resonantly enhance the FWM yield. Feedback levels up to 0.2% in power were investigated ( 26 dB) (the weak feedback regime). In the stability diagram, we distinguished between the following types of behavior: solitary laser, FWM process, singlefrequency locking, locking with two sidebands, locking with four sidebands, locking with eight sidebands, broad spectra, and coherence collapse. Single-frequency locking is a stable behavior with an observed linewidth reduction of at least a factor of five, limited by the experimental resolution. In the stability diagram, the range of single-frequency locking is observed to vanish for detunings larger than 1 GHz. This result was not predicted by theory. We believe that it is a consequence of the strong filtering of the PCM, a point that we intend to clarify in a forthcoming paper. The unstable behaviors [shown in Fig. 3(d)–(f)] correspond to an undamping of the relaxation

oscillation (Hopf bifurcation) at 3.9 GHz. This behavior is a period-doubling route to chaos. The achievable amount of PCF has an upper limit, which increases with detuning. This unexpected result can be explained by: 1) the combination of the broad spectra and the strong filtering PCM and 2) the difference between the effective feedback strength used in our study and the feedback rate used in theoretical studies. REFERENCES
[1] G. P. Agrawal and N. K. Dutta, Long Wavelength Semiconductor Lasers. New York: Van Nostrand Reinhold, 1986. [2] G. P. Agrawal and J. Klaus, “Effect of phase conjugate feedback on semiconductor laser dynamics,” Opt. Lett., vol. 16, pp. 1325–1327, 1991. [3] G. P. Agrawal and G. Gray, “Effect of phase conjugate feedback on the noise characteristics of semiconductor lasers,” Phys. Rev. A., vol. 26, pp. 5890–5898, 1992. [4] G. Gray, D. Huang, and G. P. Agrawal, “Chaotic dynamics of a semiconductor laser with phase conjugate feedback,” Phys. Rev. A., vol. 49, pp. 2096–2105, 1994. [5] B. Krauskopf, G. R. Gray, and D. Lenstra, “Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations,” Phys. Rev. E, vol. 58, pp. 7190–7197, 1999. [6] G. H. M. van Tartwijk, H. J. C. van der Linden, and D. Lenstra, “Theory of a diode laser with phase conjugate feedback,” Opt. Lett., vol. 17, pp. 1590–1592, 1992. [7] D. H. DeTienne, G. R. Gray, G. P. Agrawal, and D. Lenstra, “Semiconductor laser dynamics for feedback from a finite-penetration-depth phase-conjugate mirror,” IEEE J. Quantum Electron., vol. 33, pp. 838–844, 1997. [8] W. van der Graaf, L. Pesquera, and D. Lenstra, “Stability of a diode laser with phase conjugate feedback,” Opt. Lett., vol. 23, pp. 256–259, 1998. [9] K. Vahala, K. Kyuma, A. Yariv, S. Kwong, M. Cronin-Colomb, and K. Y. Lau, “Narrow linewidth single frequency semiconductor laser with phase conjugate external cavity mirror,” Appl. Phys. Lett., vol. 49, pp. 1563–1565, 1986. [10] M. Cronin-Colomb and A. Yariv, “Self induced frequency scanning and distributed Bragg reflection in semiconductor lasers with phase conjugate feedback,” Opt. Lett., vol. 11, pp. 455–457, 1986. [11] M. Ohtsu, I. Koshiishi, and Y. Teramachi, “A semiconductor laser as a stable phase conjugate mirror for linewidth reduction of another semiconductor laser,” Jpn. J. Appl. Phys., vol. 29, pp. 2060–2062, 1990. [12] P. Kurz and T. Mukai, “Frequency stabilization of a semiconductor laser by external phase conjugate feedback,” Opt. Lett., vol. 21, pp. 1369–1371, 1996. [13] D. Grischkowsky, N. S. Shiren, and R. J. Bennett, “Generation of time-reversed wave fronts using a resonantly enhanced electronic nonlinearity,” Appl. Phys. Lett., vol. 33, pp. 805–807, 1978. [14] A. E. Korolev and V. N. Nazarov, “Optical feedback in the external cavity of a semiconductor laser using resonance four-wave mixing in cesium vapor,” Opt. Spectrosc., vol. 74, pp. 119–123, 1993. [15] T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, “Periode doubling cascades and chaos in a semiconductor laser with optical injection,” Phys. Rev. A., vol. 51, pp. 4181–4185, 1995. [16] T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclass. Opt., vol. 9, pp. 765–784, 1997. [17] A. Gavrielides, V. Kovanis, and T. Erneux, “Analytical stability boundaries for a semiconductor laser subject to optical injection,” Opt. Commun., vol. 136, pp. 253–256, 1997. [18] A. A. Radzig and B. M. Smirnov, Reference Data on Atoms, Molecules and Ions. Berlin, Germany: Springer-Verlag, 1980. [19] J. Lin, A. I. Rubiera, and Y. Zhu, “Nearly degenerate four-wave mixing with bichromatic laser fields in a Rb atomic system,” Phys. Rev. A., vol. 52, pp. 4882–4885, 1995. [20] D. G. Steel and R. C. Lind, “Multiresonant behavior in nearly degenerate four-wave mixing,” Opt. Lett., vol. 6, pp. 587–589, 1981. [21] D. Lenstra, G. H. M. van Tartwijk, W. A. van der Graaf, and P. C. De Jagher, “Multi-wave mixing dynamics in a diode laser,” in Chaos in Optics, R. Roy, Ed., Proc. SPIE, vol. 2039, pp. 11–22, 1993. [22] D. Lenstra, B. H. Verbeek, and A. J. den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron., vol. QE–21, pp. 674–679, 1985.



Ole K. Andersen was born in 1970 in Aarhus, Denmark. He received the M.Sc. degree in 1995 from the Oersted Laboratory, University of Copenhagen, Denmark. He is currently working toward the Ph.D. degree on phase-conjugate feedback effects on diode laser dynamics in the groups of Prof. D. Lenstra and Prof. S. Stolte at the Vrije Universiteit, Amsterdam, The Netherlands.

Eric Louvergneaux, photograph and biography not available at the time of publication.

Alexis P. A. Fischer was born in 1969 in Lyon, France. He received the M.Sc. degree in engineering in 1993, the DEA degree in 1994 from the University of Franche-Comte, Besancon, France, and the Ph.D. degree from the Optics Laboratory P. M. Duffieux, Besancon, France, in 1998, performing research on electronic and optical feedback in semiconductor lasers for telecommunication applications. He is currently working on semiconductor lasers with phase-conjugate feedback in the group of D. Lenstra, Faculty of Sciences, Vrije Universiteit Amsterdam, The Netherlands. His interests are optical telecommunications, signal routing, tunable semiconductor lasers, wavelength switching, laser frequency stabilization, electronic and optical feedback, phase-conjugate feedback, laser dynamics and chaos.

Steven Stolte was born in Utrecht, The Netherlands, in 1943. He received the M.Sc. and Ph.D. degrees doing his research in the Molecular and Laser Physics Group at The Catholic University of Nijmegen, The Netherlands, in 1967 and 1972, respectively. His postdoctoral work on reaction-dynamics of long-lived complexes was carried out in the group of Prof. R. B. Bernstein (1972–1974) at the University of Wisconsin, Madison, and at the University of Texas at Austin. As a Senior Scientist, he returned to the molecular and laser physics group in Nijmegen, where he broadened his research expertise in a study of the laser frequency and power dependence for the infrared multiphoton excitation of small molecules and clusters and CWovertone Raman spectroscopy of high-lying states. In 1989, he was appointed a Full Professor in experimental laser spectroscopy in the Department of Physical and Theoretical Chemistry at the Vrije Universiteit, Amsterdam, The Netherlands, where, in 1990, he started a new group aimed at laser spectroscopy studies of matter. Prof. Stolte is a member of the Dutch and German Physical Societies. He served as Chairman of the Molecular Physics Section of the Atomic and Molecular Physics Section of the European Physical Society (1990–1992).

Ian C. Lane was born in 1967 in Stoke On Trent, England. He received the B.Sc. degree in chemistry and materials science and the Ph.D. degree in physical chemistry from the University of Nottingham, England, U.K., in 1989 and 1993, respectively. His doctoral work, under the supervision of Dr. Ivan Powis, concerned the photofragmentation dynamics of molecules under ultraviolet irradiation. He was an EC Fellow (1993–1995) at the Vrije Univeristeit, Amsterdam, The Netherlands, working on four-wave mixing experiments in rubidium. Between 1995 and 1997, he was a Post-Doctoral Researcher at Oxford University’s Clarendon Laboratory, measuring the two-photon 1S–2S transition in muonium (collaboration between the Universities of Heidelberg, Oxford, Yale, and Novosibirsk) at the Rutherford Appleton Laboratory, England. In 1998, he began a Levershulme Trust Fellowship at the University of Bristol, England, working on cavity ring-down spectroscopy of small molecules. His research interests include molecular reaction dynamics, photodissociation, high-resolution spectroscopy, ab initio quantum calculations, and atom–laser interactions.

Daan Lenstra was born in Amsterdam, The Netherlands, in 1947. He received the M.Sc. degree in theoretical physics from the University of Groningen, The Netherlands, and the Ph.D. degree from Delft University of Technology, The Netherlands. His dissertation work was on polarization effects in gas lasers. He was with Delft University of Technology during 1975–1984 and with Eindhoven University of Technology during 1984–1991. He was a Research Associate at the University of Rochester, NY, from 1981–1982 and a Guest Scientist in 1984 at Philips Research Laboratories, Eindhoven, The Netherlands. From 1989 to 1991, he was appointed a parttime Professor at the University of Leiden, The Netherlands, and since 1991, he has held a chair in theoretical quantum electronics at the Vrije Universiteit, Amsterdam. Since 1979, he has researched topics in quantum electronics and solid-state physics, such as photon statistics in resonant fluorescence, coherent electron transport, resonant tunneling, semiconductor lasers, and analogies between optics and microelectronics. His present interests are fundamental nonlinear dynamics in semiconductor lasers and quantum optics in small semiconductor structures.

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