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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 7 — Apr. 4, 2005
  • pp: 2707–2715
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Correlation properties and drift phenomena in the dynamics of vertical-cavity surface-emitting lasers with optical feedback

Markus Sondermann and Thorsten Ackemann  »View Author Affiliations


Optics Express, Vol. 13, Issue 7, pp. 2707-2715 (2005)
http://dx.doi.org/10.1364/OPEX.13.002707


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Abstract

We investigate experimentally the polarization dynamics of vertical-cavity surface-emitting lasers with isotropic optical feedback operating in the long-cavity regime. By means of an analysis of the correlation properties in the time domain and in the frequency domain a connection between a drift phenomenon and frequency components that deviate from the harmonics of the external cavity round-trip frequency is revealed. The latter frequency components are shown to result from an interaction of external cavity dynamics and relaxation oscillations. An analogy to the carrier-envelope effect in mode-locked lasers is drawn. Similar drift phenomena are observed also for other laser systems with delay.

© 2005 Optical Society of America

1. Introduction

In the past two decades, the dynamics of semiconductor lasers with optical feedback has attracted enormous interest (for an introduction and review see, e.g., [1

1. K. Petermann, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Selec. Top. Quantum Electron. 1, 480–489 (1995). [CrossRef]

, 2

2. G. H. M. v. Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” J. Opt. B: Quantum Semiclass. Opt. 7, 87–143 (1995). [CrossRef]

] and the contributions in [3

3. B. Krauskopf and D. Lenstra, eds., Fundamental issues of nonlinear laser dynamics, vol. 548 of AIP Conference Proceedings (American Institute of Physics, Melville, 2000).

]). This is on the one hand due to the technological importance of semiconductor lasers. On the other hand, a semiconductor laser with optical feedback is also a prominent example of a non-linear dynamical system with delay.

Whereas the majority of investigations was performed on edge-emitting semiconductor lasers, the dynamics of vertical-cavity surface-emitting lasers (VCSELs) with optical feedback is investigated only since a few years (see, e.g., [4

4. F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, “Polarization modulation dynamics of vertical-cavity surface-emitting lasers with an extended cavity,” IEEE J. Quantum Electron. 33, 2231–2238 (1997). [CrossRef]

, 5

5. M. Giudici, S. Balle, T. Ackemann, S. Barland, and J. R. Tredicce, “Effects of Optical Feedback on Vertical-Cavity Surface-Emitting Lasers: Experiment and Model,” J. Opt. Soc. Am. B 16, 2114–2123 (1999). [CrossRef]

, 6

6. M. Sciamanna, K. Panajotov, H. Thienpont, I. Veretennicoff, P. Megret, and M. Blondel, “Optical Feedback induces polarization mode hopping in vertical-cavity surface-emitting lasers,” Opt. Lett. 28(17), 1543–1545 (2003). [CrossRef]

, 7

7. M. Sondermann, H. Bohnet, and T. Ackemann, “Low Frequency Fluctuations and Polarization Dynamics in Vertical-cavity Surface-emitting Lasers with Isotropic Feedback,” Phys. Rev. A 67, 021,802 (2003). [CrossRef]

, 8

8. G. Giacomelli, F. Marin, and M. Romanelli, “Multi-time-scale dynamics of a laser with polarized optical feedback,” Phys. Rev. A 67, 053,809 (2003). [CrossRef]

, 9

9. A. V. Naumenko, N. A. Loiko, M. Sondermann, and T. Ackemann, “Description and analysis of low frequency fluctuations in vertical-cavity surface-emitting lasers with isotropic optical feedback by a distant reflector,” Phys. Rev. A 68, 033,805 (2003). [CrossRef]

]). In contrast to edge-emitting lasers, VCSELs operate in only one longitudinal mode, but can exhibit two orthogonally polarized modes, assuming operation in only the fundamental transverse mode (see, e.g., [10

10. M. San Miguel, “Polarization properties of vertical cavity surface emitting lasers,” in Semiconductor quantum optoelectronics: From quantum physics to smart devices, A. Miller, M. Ebrahimzadeh, and D. M. Finlayson, eds., pp. 339–366 (SUSSP and Institute of Physics Publishing, Bristol, 1999).

]). In the presence of feedback, these two polarization modes can exhibit rich dynamics [4

4. F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, “Polarization modulation dynamics of vertical-cavity surface-emitting lasers with an extended cavity,” IEEE J. Quantum Electron. 33, 2231–2238 (1997). [CrossRef]

, 5

5. M. Giudici, S. Balle, T. Ackemann, S. Barland, and J. R. Tredicce, “Effects of Optical Feedback on Vertical-Cavity Surface-Emitting Lasers: Experiment and Model,” J. Opt. Soc. Am. B 16, 2114–2123 (1999). [CrossRef]

, 6

6. M. Sciamanna, K. Panajotov, H. Thienpont, I. Veretennicoff, P. Megret, and M. Blondel, “Optical Feedback induces polarization mode hopping in vertical-cavity surface-emitting lasers,” Opt. Lett. 28(17), 1543–1545 (2003). [CrossRef]

, 7

7. M. Sondermann, H. Bohnet, and T. Ackemann, “Low Frequency Fluctuations and Polarization Dynamics in Vertical-cavity Surface-emitting Lasers with Isotropic Feedback,” Phys. Rev. A 67, 021,802 (2003). [CrossRef]

, 8

8. G. Giacomelli, F. Marin, and M. Romanelli, “Multi-time-scale dynamics of a laser with polarized optical feedback,” Phys. Rev. A 67, 053,809 (2003). [CrossRef]

, 9

9. A. V. Naumenko, N. A. Loiko, M. Sondermann, and T. Ackemann, “Description and analysis of low frequency fluctuations in vertical-cavity surface-emitting lasers with isotropic optical feedback by a distant reflector,” Phys. Rev. A 68, 033,805 (2003). [CrossRef]

], where the specific dynamics that is observed depends on parameters like, e.g., feedback strength, operation current and the type of the feedback to that the VCSEL is subjected.

In one of the contributions mentioned above, namely in Ref. [8

8. G. Giacomelli, F. Marin, and M. Romanelli, “Multi-time-scale dynamics of a laser with polarized optical feedback,” Phys. Rev. A 67, 053,809 (2003). [CrossRef]

], a drift phenomenon in the dynamics of a VCSEL with polarized feedback has been found in experiments. The drift has been observed in the auto correlation function of the temporal dynamics of the dominant polarization mode. Drift phenomena are also reported for other systems with feedback for which the time scale of the delay is considerably larger than the intrinsic time scale of the system without feedback [8

8. G. Giacomelli, F. Marin, and M. Romanelli, “Multi-time-scale dynamics of a laser with polarized optical feedback,” Phys. Rev. A 67, 053,809 (2003). [CrossRef]

, 11

11. F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two-dimensional representation of a delayed dynamical system,” Phys. Rev. A 45, 4225–4228 (1992). [CrossRef] [PubMed]

, 12

12. G. Giacomelli, M. Giudici, S. Balle, and J. R. Tredicce, “Experimental evidence of coherence resonance in an optical system,” Phys. Rev. Lett. 84, 3298–3301 (2000). [CrossRef] [PubMed]

].

In addition, we will demonstrate an analogy between the drift phenomena in lasers with feedback and the carrier-envelope slippage effect in mode-locked lasers.

2. Experimental results

The experiments have been performed on commercial gain-guided VCSELs from EMCORE Corp. (model 8085-2010) that operate in the fundamental transverse mode up to two times the threshold current. The lasers are subjected to isotropic feedback. To this purpose, a fraction of the light output is directed by use of a non-polarizing beam splitter towards a highly reflecting mirror. The light is focussed onto the mirror surface with an anti-reflection coated collimation lens. The longitudinal position of the lens and the tilt adjustments of the mirror are adjusted in an iterative procedure such that the threshold of the laser under study is minimized. The strength of the feedback is adjusted with neutral density filters. The external cavity frequency is νext =1/τext ≈300 MHz. The VCSEL output is split into its linear polarization components by use of a half-wave-plate and a Wollaston prism. The temporal dynamics of each polarization component is recorded simultaneously with avalanche photo diodes (1.8 GHz analogue bandwidth) and a digital oscilloscope (sampling interval 125 ps, 1 GHz analogue bandwidth). In some of the experiments an oscilloscope with 6 GHz bandwidth and a sampling interval of 50 ps was available. This oscilloscope was also used in combination with fast photo diodes of 10 GHz bandwidth but poor sensitivity. Therefore, the photo diode signals were amplified by 20 dB by the cost of cutting off frequency components below 10 MHz. Unintended back reflections into the laser are prevented by optical isolators that are located in each of the two beam paths that emerge from the Wollaston prism.

The first device that is investigated operates in the regime of the well known low-frequency fluctuations (see, e.g., [3

3. B. Krauskopf and D. Lenstra, eds., Fundamental issues of nonlinear laser dynamics, vol. 548 of AIP Conference Proceedings (American Institute of Physics, Melville, 2000).

]) for currents close to the threshold value of the solitary laser [cf. Fig. 1(a) and Refs. [7

7. M. Sondermann, H. Bohnet, and T. Ackemann, “Low Frequency Fluctuations and Polarization Dynamics in Vertical-cavity Surface-emitting Lasers with Isotropic Feedback,” Phys. Rev. A 67, 021,802 (2003). [CrossRef]

, 9

9. A. V. Naumenko, N. A. Loiko, M. Sondermann, and T. Ackemann, “Description and analysis of low frequency fluctuations in vertical-cavity surface-emitting lasers with isotropic optical feedback by a distant reflector,” Phys. Rev. A 68, 033,805 (2003). [CrossRef]

, 13

13. T. Ackemann, M. Sondermann, A. V. Naumenko, and N. A. Loiko, “Polarization dynamics and low-frequency fluctuations in vertical-cavity surface-emitting lasers subjected to optical feedback,” Appl. Phys. B 77, 739–746 (2003). [CrossRef]

]]. Due to the weak intrinsic polarization anisotropies, both polarization modes participate in the dynamics with almost equal power. The temporal correlation functions are obtained numerically from the recorded time series by use of the Wiener-Chintchine theorem [18

18. W. H. Press, B. Flannery, S. Teukolsky, and W. Vettering, Numerical recipes: the art of scientific computing (Cambridge University Press, Cambridge, 1992).

]. The spectral cross correlation or cross spectral density (CSD) is obtained from the two time traces I x,y (t) by Fourier transformation (denoted in the formula below by a tilde) and use of the relationship CSD(|f|)=Ĩx (fĨy (f)+Ĩx (-fĨy (-f). This results in a real number for the CSD at a certain frequency of modulus f with a positive (negative) number indicating correlation (anticorrelation).

Fig. 1. Characterization of the dynamics of a VCSEL with isotropic optical feedback: (a) measured time traces, (b) auto correlation functions, (c) cross correlation function, (d) power spectra of the polarization modes, (e) cross spectral density (CSD). In panels displaying data for both polarization modes, the red and blue lines denote the data corresponding to the two orthogonal polarization modes. The injection current is set to the threshold of the solitary laser. The threshold reduction with feedback is 6%. The detection bandwidth is 1 GHz.

The auto correlation functions (ACFs) of both modes [Fig. 1(b)] exhibit a slow background modulation, which can be attributed to the presence of the low-frequency fluctuations [9

9. A. V. Naumenko, N. A. Loiko, M. Sondermann, and T. Ackemann, “Description and analysis of low frequency fluctuations in vertical-cavity surface-emitting lasers with isotropic optical feedback by a distant reflector,” Phys. Rev. A 68, 033,805 (2003). [CrossRef]

]. A similar conclusion was made in Ref. [8

8. G. Giacomelli, F. Marin, and M. Romanelli, “Multi-time-scale dynamics of a laser with polarized optical feedback,” Phys. Rev. A 67, 053,809 (2003). [CrossRef]

] for the the case of polarized feedback. Superimposed on the slow modulation are peak structures that are roughly located at multiples of τext . The shape of these structures changes continuously, if the time delay in the ACFs is increased continuously: For small values of the delay, the structure is predominantly pointing upwards. With increasing delay time, the structure is first transformed such that it resembles the form of a dispersion curve [e.g., about 10 ns in Fig. 1(b)], until at approximately 20 ns the structure is predominantly pointing downwards. If the delay time is increased even further, the process is continued.

Fig. 2. Cross correlation function for the same parameters as in Fig. 1(c) but after elimination of the correlated (green line) and anticorrelated (red line) frequency components from the cross spectral density (see text for further explanations). The red line is raised by 0.2 units for better visibility.

A slow transformation process of the peak structures at multiples of τext was also observed recently for the dominant polarization mode in the case of polarized feedback [8

8. G. Giacomelli, F. Marin, and M. Romanelli, “Multi-time-scale dynamics of a laser with polarized optical feedback,” Phys. Rev. A 67, 053,809 (2003). [CrossRef]

]. There, this observation was associated with a drift phenomenon of systems with delayed feedback. It has also been visualized by applying a two-dimensional rearrangement of the ACF, following a procedure also used for other systems (e.g., [11

11. F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two-dimensional representation of a delayed dynamical system,” Phys. Rev. A 45, 4225–4228 (1992). [CrossRef] [PubMed]

]). An application of this procedure leads to comparable results in the case of the data presented here.

A drift is also observed in the cross correlation function (CCF) [Fig. 1(c)]. Here, seemingly two downward pointing peaks that are superimposed on the small envelope are observed close to zero delay. If the relative delay between the two time series is increased, these two peaks are transformed first into a single peak and then into a correlated peak, as it is observed also for the ACFs.

The observations made so far are best understood by analyzing the spectral properties. In the power spectra of the modes, two pronounced frequency components are observed at every harmonic of νext and at slightly larger frequencies [cf. Fig. 1(d)]. These components are also present in the CSD [Fig. 1(e)]. The striking observation is that the dynamics at the exact harmonics of νext is anticorrelated whereas it is correlated at the shifted frequency. This observation suggests the following interpretation of the CCF: At zero delay, a superposition of an upward pointing and a downward pointing peak is observed. Since the correlated part of the dynamics corresponds to the larger frequencies (the shorter time scale), the upward peak (correlated dynamics) and the downward peak (anticorrelated dynamics) run apart for increasing delay. Thus, the downward pointing peak is observed alone for a sufficient increase of the delay [about 17 ns in Fig. 1(c)]. If the delay is increased further, the positive peak moves continuously back into the negative peak from the direction of larger delay and a superposition of a correlated and an anticorrelated peak is observed again. For the ACFs of the modes, a similar argumentation is applicable.

This conjecture has been tested in a measurement with a detection bandwidth of 6 GHz. The experiment was performed with another VCSEL that has an intermediate intrinsic polarization anisotropy and that exhibits strong polarization mode competition in the presence of feedback. Also in this case the observed spectra consist of a series of peaks that are separated by νext (see Fig. 3). At several harmonics of νext , again, the peaks have a doublet structure and coexistence of correlated and anticorrelated dynamics is observed. Each of the two kinds of dynamics is represented in the CSD by a frequency comb, where the frequency comb corresponding to the correlated dynamics is shifted to frequencies slightly larger than the exact harmonics of νext .

Thanks to the large bandwidth also an envelope with a pronounced maximum is recognized [e.g., at 1.8 GHz in Fig. 3(a),(b)]. The (local) maximum of the envelope moves to higher frequencies [5.4 GHz in Fig. 3(c),(d)], if the current is increased. Furthermore, a pronounced ‘low’ frequency tail survives (frequencies less than 2 GHz).

The frequency of the (local) maximum of the envelope is associated with the RO frequency (similar findings about the shape of the power spectrum can be found for the case of an edge-emitter with feedback in [1

1. K. Petermann, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Selec. Top. Quantum Electron. 1, 480–489 (1995). [CrossRef]

]). As it is evident, the correlated components are strongest close to the RO frequency and the anticorrelated components are weakest there. The latter components are strong below the RO frequency [best seen in Fig. 3(d)] and far above this frequency [best seen in Fig. 3(b)]. Moreover, the positively correlated components are not detected at all at frequencies far from the RO frequency [high frequencies in Fig. 3(a),(b), low frequencies in Fig. 3(c),(d)]. This is a clear indication that the correlated, i.e., the shifted components near the harmonics of ν ext are due to an interaction of external cavity dynamics and the ROs.

Finally, we mention that the observation of drift phenomena in the auto correlation functions is not depending on the occurrence of dynamics in both of the two polarization modes of the VCSEL, but occurs also in devices (or for parameter settings) in which one of the polarization modes is strongly favoured or even only one mode is lasing [cf. Fig. 4(a)]. This is not unexpected, since the drift was reported for operation in a single polarization mode for VC-SELs with polarized feedback [8

8. G. Giacomelli, F. Marin, and M. Romanelli, “Multi-time-scale dynamics of a laser with polarized optical feedback,” Phys. Rev. A 67, 053,809 (2003). [CrossRef]

]. Our measurements show that also in these cases the power spectrum exhibits more than one frequency component at the harmonics of νext [cf. Fig. 4(b)].

Fig. 3. Power spectrum of the dominant polarization mode and cross spectral density (CSD) of the dynamics of the both polarization modes for another device with a threshold reduction of 18% for injection currents 4% below (a,b) and 32% above (c,d) the threshold of the solitary laser, respectively. Red (blue) colour denotes anticorrelated (correlated) components. The inset in panel (d) is a magnification of the spectral components for frequencies larger than 2 GHz. The detection bandwidth is 6 GHz. The corresponding time series are amplified by 20 dB and the DC-component is cut off. The features near 1.7 GHz and 3.4 GHz in panel (b) are artefacts that are captured by the setup.

However, the multiplet at each harmonic has a different shape than in the case of pronounced polarization dynamics: A frequency component at the exact multiples of νext is not discernable in the power spectrum. Instead, one weak component at slightly lower frequencies and a strong component at slightly larger frequencies is observed [see Fig. 4(b)]. If the injection current is changed, both peaks move further apart from each other, i.e., the deviation from νext increases [see the inset in Fig. 4(b)]. At the reduced threshold (current value 0.92 in the inset of the figure), only one peak with a centre frequency of approximately νext is discernable.

3. Discussion

The discussion in the preceding section has been restricted to cases where the multiplet at the harmonics of νext consists of two peaks. There are parameter regimes and/or devices for which even more peaks are observed at the harmonics of ν ext . Also in these cases, a coexistence of correlated and anticorrelated dynamics at nearby time scales is observed if polarization dynamics occur and the same conclusions on the relation of drift phenomena in the correlation functions and the components of the power spectra can be drawn. Therefore, the simplest situations that have been found in the experiments have been chosen for presentation for the sake of clarity.

Fig. 4. Auto correlation function (a) and power spectrum (b) of the dominant polarization mode of another device with isotropic feedback. The inset in (b) displays the frequencies (in GHz) of the spectral components at the first harmonic of the external cavity frequency as a function of the injection current normalized to the threshold of the solitary laser. The threshold reduction is 8%. The current is set to 0.95 times the threshold of the laser without feedback. The detection bandwidth is 1 GHz.

The observation of the drift phenomenon is not restricted to the dynamical regime of the low-frequency fluctuations, i.e., the drift in the correlation functions is also observed if the injection current is increased and the devices operate in the regime of fully developed coherence collapse (see also Refs. [7

7. M. Sondermann, H. Bohnet, and T. Ackemann, “Low Frequency Fluctuations and Polarization Dynamics in Vertical-cavity Surface-emitting Lasers with Isotropic Feedback,” Phys. Rev. A 67, 021,802 (2003). [CrossRef]

, 9

9. A. V. Naumenko, N. A. Loiko, M. Sondermann, and T. Ackemann, “Description and analysis of low frequency fluctuations in vertical-cavity surface-emitting lasers with isotropic optical feedback by a distant reflector,” Phys. Rev. A 68, 033,805 (2003). [CrossRef]

]). However, one has to take care of the fact that the frequency range in which the multiplets are observed in the power spectra moves to higher values if the injection current is increased [see also Fig. 3(b)]. Therefore, the drift phenomena might easily be missed in an experiment, if the detection bandwidth is too low.

It has to be noted that multiplets at the harmonics of νext have also been observed in experiments on edge-emitters with optical feedback with a short external cavity [25

25. T. Heil, I. Fischer, W. Elsäßer, and A. Gavrielides, “Dynamics of Semiconductor Lasers Subject to Delayed Optical Feedback: The Short Cavity Regime,” Phys. Rev. Lett. 87(24), 243,901 (2001).

], where τext is shorter than the RO period. There, the splitting between the multiplet components coincided with a pronounced low-frequency component of the observed dynamics, which indicates a wave mixing relation. However, such a relation can be excluded in the experiments reported here, since (i) the low frequency component does not exactly match the splitting of the multiplet components and (ii) the multiplets are also observed if no pronounced low-frequency components are present in the dynamics at all (see above and Ref. [13

13. T. Ackemann, M. Sondermann, A. V. Naumenko, and N. A. Loiko, “Polarization dynamics and low-frequency fluctuations in vertical-cavity surface-emitting lasers subjected to optical feedback,” Appl. Phys. B 77, 739–746 (2003). [CrossRef]

]). Furthermore, to the best knowledge of the authors there are no reports on drift phenomena in feedback systems with delay times shorter than the typical time scale of the system (which is the RO period in the case of a class-B laser).

The obtained results can also be interpreted within the framework of ultra-short laser pulses that are created by mode-locking (see, e.g., Ref. [26

26. J. Reichert, R. Holzwarth, T. Udem, and T. W. Hänsch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. 172, 59–68 (1999). [CrossRef]

, 27

27. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]

, 28

28. T. Udem, R. Holzwarth, and T. W. Hánsch, “Optical frequency metrology,” Nature 416, 233–247 (2002). [CrossRef] [PubMed]

]): The pulses in the pulse train emitted by a mode-locked laser are separated by a certain period τrep , which is the round-trip time of the pulse in the laser cavity. In the frequency domain, there is a frequency comb with a separation νrep =1/τrep and an envelope, which is centred at the optical carrier frequency. It turns out that the actual value of the frequency components of the comb is not given by m·ν rep but by νCEO +m·ν rep , where νCEO denotes the so-called carrier-envelope offset frequency. The name originates from the fact that there is a phase slippage between the optical carrier wave and the pulse envelope, if νCEO ≠=0.

There is a close analogy between these observations in mode-locked lasers and our findings for the VCSEL with delayed feedback: The harmonics of νext in frequency space correspond to a frequency comb at m·νext , m integer, that produces a train of pulses with a repetition period of τext in the correlation functions. The dynamics that underlies the pulse envelope consists of oscillations at the frequency of the envelope maximum of the frequency comb. This envelope maximum is given by the RO frequency in the case of the correlated dynamics, i.e., the shifted frequency components that deviate from the harmonics of νext . Also these components are separated by νext . Thus, these frequencies constitute a frequency comb with frequencies ν 0+m·νext , where ν 0 is the shift from the exact harmonic [e.g., ν 0≈25 MHz in the case of Fig. 1(e) and ν 0≈21 MHz in the case of Fig. 3(b)]. However, the fact that ν 0≠=0 implies that the fast oscillations that underlie the pulse envelope (i.e., the carrier signal at the frequency of the ROs) suffer a continuous phase slip with respect to the repetition frequency νext of the pulses [26

26. J. Reichert, R. Holzwarth, T. Udem, and T. W. Hänsch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. 172, 59–68 (1999). [CrossRef]

]. This phase slip results in the observed drift phenomenon, i.e., in the continuous change of the pulse structure in the correlation functions.

We mention that in the mode-locked laser the difference between phase velocity and group velocity is at the origin of the phase slippage. Here, the velocity of the ‘slower’ pulses (at νext ) is still given by the group velocity of the light in the external cavity, where the frequency of the ‘faster’ pulsing (the RO frequency) is determined by an interplay of carrier and field dynamics. Hence, it is probably not surprising that there is an offset frequency, in general.

We caution that in contrast to - usually well-behaved - mode-locked lasers the dynamics in semiconductor lasers with feedback has strong irregular components [see, e.g., the time series in Figs. 1(a)] and that we are discussing the robust statistical features in long-time averages. Nevertheless, the results provide a further connection between mode-locking and feedback-induced dynamics, after mode-locking of external cavity modes has been proposed some time ago in order to explain the shortness of the pulses [29

29. G. H. M. v. Tartwijk, A. M. Levine, and D. Lenstra, “Sisyphus effect in semiconductor lasers with optical feedback,” IEEE J. Selec. Top. Quantum Electron. 1, 466–472 (1995). [CrossRef]

].

An important question is the question of the generality of the obtained results. In several theoretical works, components of the power spectrum that deviate from exact integer multiples of νext have been predicted for coherent [30

30. J. S. Cohen, R. R. Drenten, and B. H. Verbeek, “The Effect of Optical Feedback on the Relaxation Oscillation in Semiconductor Lasers,” IEEE J. Quantum Electron. 24(10), 1989–1995 (1988). [CrossRef]

] as well as incoherent or opto-electronic feedback [31

31. N. A. Loiko and A. M. Samson, “Possible regimes of generation of a semiconductor laser with a delayed opto-electric feedback,” Opt. Commun. 93, 66–72 (1992). [CrossRef]

, 32

32. G. Giacomelli and A. Politi, “Multiple scale analysis of delayed dynamical systems,” Physica D 145, 26–42 (1998). [CrossRef]

, 33

33. M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, “Order parameters for class-B lasers with a long time delayed feedback,” Physica D 145, 110–129 (2000). [CrossRef]

]. Since furthermore from a theoretical point of view the dynamics of class-B lasers with feedback is to some extent independent of the kind of the feedback [34

34. E. V. Grigorieva, H. Haken, and S. A. Kaschenko, “Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback,” Opt. Commun. 165, 279–292 (1999). [CrossRef]

], the results obtained here may be applied for other class-B lasers with feedback of a long delay time. However, definitive statements are difficult to obtain, since in none of the contributions cited here both the correlation functions and power spectra are discussed.

Acknowledgments

The authors acknowledge fruitful discussion with N. Loiko and I. Fischer. They wish to thank the DFG und DAAD for financial support.

References and links

1.

K. Petermann, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Selec. Top. Quantum Electron. 1, 480–489 (1995). [CrossRef]

2.

G. H. M. v. Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” J. Opt. B: Quantum Semiclass. Opt. 7, 87–143 (1995). [CrossRef]

3.

B. Krauskopf and D. Lenstra, eds., Fundamental issues of nonlinear laser dynamics, vol. 548 of AIP Conference Proceedings (American Institute of Physics, Melville, 2000).

4.

F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, “Polarization modulation dynamics of vertical-cavity surface-emitting lasers with an extended cavity,” IEEE J. Quantum Electron. 33, 2231–2238 (1997). [CrossRef]

5.

M. Giudici, S. Balle, T. Ackemann, S. Barland, and J. R. Tredicce, “Effects of Optical Feedback on Vertical-Cavity Surface-Emitting Lasers: Experiment and Model,” J. Opt. Soc. Am. B 16, 2114–2123 (1999). [CrossRef]

6.

M. Sciamanna, K. Panajotov, H. Thienpont, I. Veretennicoff, P. Megret, and M. Blondel, “Optical Feedback induces polarization mode hopping in vertical-cavity surface-emitting lasers,” Opt. Lett. 28(17), 1543–1545 (2003). [CrossRef]

7.

M. Sondermann, H. Bohnet, and T. Ackemann, “Low Frequency Fluctuations and Polarization Dynamics in Vertical-cavity Surface-emitting Lasers with Isotropic Feedback,” Phys. Rev. A 67, 021,802 (2003). [CrossRef]

8.

G. Giacomelli, F. Marin, and M. Romanelli, “Multi-time-scale dynamics of a laser with polarized optical feedback,” Phys. Rev. A 67, 053,809 (2003). [CrossRef]

9.

A. V. Naumenko, N. A. Loiko, M. Sondermann, and T. Ackemann, “Description and analysis of low frequency fluctuations in vertical-cavity surface-emitting lasers with isotropic optical feedback by a distant reflector,” Phys. Rev. A 68, 033,805 (2003). [CrossRef]

10.

M. San Miguel, “Polarization properties of vertical cavity surface emitting lasers,” in Semiconductor quantum optoelectronics: From quantum physics to smart devices, A. Miller, M. Ebrahimzadeh, and D. M. Finlayson, eds., pp. 339–366 (SUSSP and Institute of Physics Publishing, Bristol, 1999).

11.

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two-dimensional representation of a delayed dynamical system,” Phys. Rev. A 45, 4225–4228 (1992). [CrossRef] [PubMed]

12.

G. Giacomelli, M. Giudici, S. Balle, and J. R. Tredicce, “Experimental evidence of coherence resonance in an optical system,” Phys. Rev. Lett. 84, 3298–3301 (2000). [CrossRef] [PubMed]

13.

T. Ackemann, M. Sondermann, A. V. Naumenko, and N. A. Loiko, “Polarization dynamics and low-frequency fluctuations in vertical-cavity surface-emitting lasers subjected to optical feedback,” Appl. Phys. B 77, 739–746 (2003). [CrossRef]

14.

M. P. v. Exter, R. F. M. Hendriks, J. P. Woerdman, and C. J. v. Poel, “Explanation of double-peaked intensity noise spectrum of an external-cavity semiconductor laser,” Opt. Commun. 110, 137–140 (1994). [CrossRef]

15.

M. Giudici, C. Green, G. Nespolo, U. Giacomelli, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414–6118 (1997). [CrossRef]

16.

M. Giudici, L. Giuggioli, C. Green, and J. R. Tredicce, “Dynamical behavior of semiconductor lasers with frequency selective optical feedback,” Chaos, Solitons & Fractals 10, 811–818 (1999).

17.

A. Gavrielides, T. C. Newell, V. Kovanis, R. G. Harrison, N. Swanston, D. Yu, and W. Lu, “Synchronous Sisyphus effect in diode lasers subject to optical feedback,” Phys. Rev. A 60, 1577–1580 (1999). [CrossRef]

18.

W. H. Press, B. Flannery, S. Teukolsky, and W. Vettering, Numerical recipes: the art of scientific computing (Cambridge University Press, Cambridge, 1992).

19.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).

20.

J. Mørk, B. Tromborg, and J. Mark, “Chaos in Semiconductor Lasers with Optical Feedback: Theory and Experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992). [CrossRef]

21.

M. Ahmed and M. Yamada, “Influence of Instantaneous Mode Competition on the Dynamics of Semiconductor Lasers,” IEEE J. Quantum Electron. 38(6), 682–693 (2002). [CrossRef]

22.

I. Leyva, E. Allaria, and R. Meucci, “Transient polarization dynamics in a CO2 laser,” Opt. Commun. 217, 335–342 (2003). [CrossRef]

23.

M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, “Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers,” Phys. Rev. A 68, 033,822 (2003). [CrossRef]

24.

A. Uchida, Y. Liu, I. Fischer, P. Davis, and T. Aida, “Chaotic antiphase dynamics and synchronization in multi-mode semiconductor lasers,” Phys. Rev. A 64, 023,801 (2001). [CrossRef]

25.

T. Heil, I. Fischer, W. Elsäßer, and A. Gavrielides, “Dynamics of Semiconductor Lasers Subject to Delayed Optical Feedback: The Short Cavity Regime,” Phys. Rev. Lett. 87(24), 243,901 (2001).

26.

J. Reichert, R. Holzwarth, T. Udem, and T. W. Hänsch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. 172, 59–68 (1999). [CrossRef]

27.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]

28.

T. Udem, R. Holzwarth, and T. W. Hánsch, “Optical frequency metrology,” Nature 416, 233–247 (2002). [CrossRef] [PubMed]

29.

G. H. M. v. Tartwijk, A. M. Levine, and D. Lenstra, “Sisyphus effect in semiconductor lasers with optical feedback,” IEEE J. Selec. Top. Quantum Electron. 1, 466–472 (1995). [CrossRef]

30.

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, “The Effect of Optical Feedback on the Relaxation Oscillation in Semiconductor Lasers,” IEEE J. Quantum Electron. 24(10), 1989–1995 (1988). [CrossRef]

31.

N. A. Loiko and A. M. Samson, “Possible regimes of generation of a semiconductor laser with a delayed opto-electric feedback,” Opt. Commun. 93, 66–72 (1992). [CrossRef]

32.

G. Giacomelli and A. Politi, “Multiple scale analysis of delayed dynamical systems,” Physica D 145, 26–42 (1998). [CrossRef]

33.

M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, “Order parameters for class-B lasers with a long time delayed feedback,” Physica D 145, 110–129 (2000). [CrossRef]

34.

E. V. Grigorieva, H. Haken, and S. A. Kaschenko, “Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback,” Opt. Commun. 165, 279–292 (1999). [CrossRef]

OCIS Codes
(140.2020) Lasers and laser optics : Diode lasers
(140.3460) Lasers and laser optics : Lasers
(140.4050) Lasers and laser optics : Mode-locked lasers
(140.5960) Lasers and laser optics : Semiconductor lasers
(270.3100) Quantum optics : Instabilities and chaos

ToC Category:
Research Papers

History
Original Manuscript: January 10, 2005
Revised Manuscript: March 18, 2005
Published: April 4, 2005

Citation
Markus Sondermann and Thorsten Ackemann, "Correlation properties and drift phenomena in the dynamics of vertical-cavity surface-emitting lasers with optical feedback," Opt. Express 13, 2707-2715 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2707


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References

  1. K. Petermann, �??External optical feedback phenomena in semiconductor lasers,�?? IEEE J. Selec. Top. Quantum Electron. 1, 480�??489 (1995). [CrossRef]
  2. G. H. M. v. Tartwijk and D. Lenstra, �??Semiconductor lasers with optical injection and feedback,�?? J. Opt. B: Quantum Semiclass. Opt. 7, 87�??143 (1995). [CrossRef]
  3. B. Krauskopf and D. Lenstra, eds., Fundamental issues of nonlinear laser dynamics, vol. 548 of AIP Conference Proceedings (American Institute of Physics, Melville, 2000).
  4. F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, �??Polarization modulation dynamics of vertical-cavity surface-emitting lasers with an extended cavity,�?? IEEE J. Quantum Electron. 33, 2231�??2238 (1997). [CrossRef]
  5. M. Giudici, S. Balle, T. Ackemann, S. Barland, and J. R. Tredicce, �??Effects of Optical Feedback on Vertical-Cavity Surface-Emitting Lasers: Experiment and Model,�?? J. Opt. Soc. Am. B 16, 2114�??2123 (1999). [CrossRef]
  6. M. Sciamanna, K. Panajotov, H. Thienpont, I. Veretennicoff, P. Megret, and M. Blondel, �??Optical Feedback induces polarization mode hopping in vertical-cavity surface-emitting lasers,�?? Opt. Lett. 28(17), 1543�??1545 (2003). [CrossRef]
  7. M. Sondermann, H. Bohnet, and T. Ackemann, �??Low Frequency Fluctuations and Polarization Dynamics in Vertical-cavity Surface-emitting Lasers with Isotropic Feedback,�?? Phys. Rev. A 67, 021,802 (2003). [CrossRef]
  8. G. Giacomelli, F. Marin, and M. Romanelli, �??Multi-time-scale dynamics of a laser with polarized optical feedback,�?? Phys. Rev. A 67, 053,809 (2003). [CrossRef]
  9. A. V. Naumenko, N. A. Loiko, M. Sondermann, and T. Ackemann, �??Description and analysis of low frequency fluctuations in vertical-cavity surface-emitting lasers with isotropic optical feedback by a distant reflector,�?? Phys. Rev. A 68, 033,805 (2003). [CrossRef]
  10. M. San Miguel, �??Polarization properties of vertical cavity surface emitting lasers,�?? in Semiconductor quantum optoelectronics: From quantum physics to smart devices, A. Miller, M. Ebrahimzadeh, and D. M. Finlayson, eds., pp. 339�??366 (SUSSP and Institute of Physics Publishing, Bristol, 1999).
  11. F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, �??Two-dimensional representation of a delayed dynamical system,�?? Phys. Rev. A 45, 4225�??4228 (1992). [CrossRef] [PubMed]
  12. G. Giacomelli, M. Giudici, S. Balle, and J. R. Tredicce, �??Experimental evidence of coherence resonance in an optical system,�?? Phys. Rev. Lett. 84, 3298�??3301 (2000). [CrossRef] [PubMed]
  13. T. Ackemann, M. Sondermann, A. V. Naumenko, and N. A. Loiko, �??Polarization dynamics and low-frequency fluctuations in vertical-cavity surface-emitting lasers subjected to optical feedback,�?? Appl. Phys. B 77, 739�??746(2003). [CrossRef]
  14. M. P. v. Exter, R. F. M. Hendriks, J. P. Woerdman, and C. J. v. Poel, �??Explanation of double-peaked intensity noise spectrum of an external-cavity semiconductor laser,�?? Opt. Commun. 110, 137�??140 (1994). [CrossRef]
  15. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, �??Andronov bifurcation and excitability in semiconductor lasers with optical feedback,�?? Phys. Rev. E 55, 6414�??6118 (1997). [CrossRef]
  16. M. Giudici, L. Giuggioli, C. Green, and J. R. Tredicce, �??Dynamical behavior of semiconductor lasers with frequency selective optical feedback,�?? Chaos, Solitons & Fractals 10, 811�??818 (1999).
  17. A. Gavrielides, T. C. Newell, V. Kovanis, R. G. Harrison, N. Swanston, D. Yu, andW. Lu, �??Synchronous Sisyphus effect in diode lasers subject to optical feedback,�?? Phys. Rev. A 60, 1577�??1580 (1999). [CrossRef]
  18. W. H. Press, B. Flannery, S. Teukolsky, and W. Vettering, Numerical recipes: the art of scientific computing (Cambridge University Press, Cambridge, 1992).
  19. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).
  20. J. Mørk, B. Tromborg, and J. Mark, �??Chaos in Semiconductor Lasers with Optical Feedback: Theory and Experiment,�?? IEEE J. Quantum Electron. 28(1), 93�??108 (1992). [CrossRef]
  21. M. Ahmed and M. Yamada, �??Influence of Instantaneous Mode Competition on the Dynamics of Semiconductor Lasers,�?? IEEE J. Quantum Electron. 38(6), 682�??693 (2002). [CrossRef]
  22. I. Leyva, E. Allaria, and R. Meucci, �??Transient polarization dynamics in a CO2 laser,�?? Opt. Commun. 217, 335�??342 (2003). [CrossRef]
  23. M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, �??Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers,�?? Phys. Rev. A 68, 033,822 (2003). [CrossRef]
  24. A. Uchida, Y. Liu, I. Fischer, P. Davis, and T. Aida, �??Chaotic antiphase dynamics and synchronization in multimode semiconductor lasers,�?? Phys. Rev. A 64, 023,801 (2001). [CrossRef]
  25. T. Heil, I. Fischer, W. Els¨a�?er, and A. Gavrielides, �??Dynamics of Semiconductor Lasers Subject to Delayed Optical Feedback: The Short Cavity Regime,�?? Phys. Rev. Lett. 87(24), 243,901 (2001).
  26. J. Reichert, R. Holzwarth, T. Udem, and T. W. H¨ansch, �??Measuring the frequency of light with mode-locked lasers,�?? Opt. Commun. 172, 59�??68 (1999). [CrossRef]
  27. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S.Windeler, J. L. Hall, and S. T. Cundiff, �??Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,�?? Science 288, 635�??639 (2000). [CrossRef] [PubMed]
  28. T. Udem, R. Holzwarth, and T. W. H´ansch, �??Optical frequency metrology,�?? Nature 416, 233�??247 (2002). [CrossRef] [PubMed]
  29. G. H. M. v. Tartwijk, A. M. Levine, and D. Lenstra, �??Sisyphus effect in semiconductor lasers with optical feedback,�?? IEEE J. Selec. Top. Quantum Electron. 1, 466�??472 (1995). [CrossRef]
  30. J. S. Cohen, R. R. Drenten, and B. H. Verbeek, �??The Effect of Optical Feedback on the Relaxation Oscillation in Semiconductor Lasers,�?? IEEE J. Quantum Electron. 24(10), 1989�??1995 (1988). [CrossRef]
  31. N. A. Loiko and A. M. Samson, �??Possible regimes of generation of a semiconductor laser with a delayed optoelectric feedback,�?? Opt. Commun. 93, 66�??72 (1992). [CrossRef]
  32. G. Giacomelli and A. Politi, �??Multiple scale analysis of delayed dynamical systems,�?? Physica D 145, 26�??42 (1998). [CrossRef]
  33. M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, �??Order parameters for class-B lasers with a long time delayed feedback,�?? Physica D 145, 110�??129 (2000). [CrossRef]
  34. E. V. Grigorieva, H. Haken, and S. A. Kaschenko, �??Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback,�?? Opt. Commun. 165, 279�??292 (1999). [CrossRef]

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