Experimental control of non linear dynamics and chaos using filtered optical feedback in a diode laser A.P.A. FISCHER(1), M. YOUSEFI(2), G.VEMURI(3), D. LENSTRA(2) (1) Laboratoire de Physique des Lasers, Université Paris XIII, UMR CNRS 7538, FRANCE [email protected]
This work has been supported by the European Community under the contract HPRICT-1999-00064 (2) Theoretical physics Department - Vrije Universiteit Amsterdam, THE NETHERLANDS (3) Indiana University, Indianapolis, IN, USA ABSTRACT: We report on experimental results for the dynamical regime of a diode laser with delayed frequency selective optical feedback from a Fabry-Pérot interferometer type of filter located in the external feedback cavity of a diode laser. Three effects of the filter on the dynamical behavior of the diode laser are observed. First of all we report an Optical impedance effect resulting from the detuning between the filter center frequency and the closest external cavity mode. Second of all there is a filtering effect and it is shown how with an appropriate filter width it is possible to suppress and control some of the different dynamics usually present in conventional optical feedback. A reduction of the RO oscillations is reported for narrow bandwidth filters. Third of all, we report a non-linear shaping effect arising from the non-linear response in frequency of the filter profile which produces a new dynamics in frequency. To our best knowledge it is the first time that a frequency dynamics occurring with a period related to the external cavity round trip time is reported. Potential applications deal with all-optical controlled chaos for secure communications, and with all-optical signal processing.
I-INTRODUCTION Diode lasers are known to be extremely sensitive to external perturbations, such as optical injection , optical feedback from a conventional mirror  or from a phase-conjugate mirror . They have all been intensively studied both theoretically and experimentally in order to understand why the performances of diode lasers (DL) are sometimes improved while at other times degraded by these perturbations. Theoretically the problem has been approached by using a set of rate equations *. From this it appeared that the stability of a DL is mainly governed by two parameters: the amount of feedback and, in the case of coherent feedback, the phase difference between the emitted wave and the perturbating wave. Of the above mentioned perturbation schemes, conventional optical feedback (COF) has proven to be a very interesting way of controlling the stability of a DL. For instance it has been shown that the operation frequency of a DL can be locked to one of the external cavity modes allowing a linewidth reduction . In fact a parasitic change in the external-cavity length of the order of a wavelength may cause the frequency to jump spontaneously from one external cavity mode to another. This phenomenon called mode hopping has also been reported . Multistability, i.e. the fact that the laser has several stable operation frequencies to choose from depending on the electrode current, has been reported in . Multistability in frequency has also been reported, however direct application, for example in the field of telecommunications with wavelength division multiplexing (WDM), would require a precise mechanical and thermal control of the external cavity in order to achieve long-term 2/25
19/08/2002 11:33 stability which is very difficult. So far, only a few applications already have been reported, although it seems very promising to use fast frequency-switching all-optical devices for WDM or other all-optical signal processing. In that case, fast switching times with nanosecond time scale is required especially in the telecommunication field. Switching times are rarely reported, and moreover no solution are available yet to shorten them when necessary. Actually the main problem remains the mechanical and thermal stability of the external cavity, that may cause parasitic frequency changes. Moreover it is difficult to reproduce exactly such a configuration so as to obtain exactly the same reference frequency in different lasers. This is the reason why other structures have been reported in which a medium, such as an alkali vapor, or a filter is placed within the external cavity. In such a configuration the laser frequency is locked to the transition lines of rubidium or barium or to a resonance of a Fabry-Pérot interferometer. All these systems are equivalent to a DL with an external cavity containing a frequency-selective filter. In a previous paper  we showed that the effect of such a filter is to select one or a group of external cavity modes to which the laser locks, thus enhancing the stability of the laser, and allowing a better control on its stability or its switching time. Indeed fast switching devices could find useful application for example in the DWM systems but also in the signal-routing domain. Moreover the selection of a group of external cavity mode might become interesting for the hyper dense –WDM. The question of stability for a diode laser with COF has been widely studied, especially the dynamical behaviour. It has been shown that the laser undergoes usually unwanted chaotic dynamics . Different routes to chaos , LFF  and coherence collapse has been simulated or observed experimentally. One potential application is encrypted 3/25
19/08/2002 11:33 communication using a chaotic carrier . It has been recently reported  that this is fully feasible, although the security based on the similarity of the emitter laser and the receiver-laser, is not really reliable. A lack for chaos encrypted all optical lines is the control of the chaos complexity, in order to obtain reliable systems. In a previous paper we mainly interested in the steady state regime of the FOF systems, here we address the more specifically their dynamical behaviours. The main purpose of this paper is thus to explain the principles behind FOF that may affects the dynamical behaviour of a diode laser and its switching time. At least three effects will be distinguished. Indeed, the optical filter is both a frequency selective element and a phase shift inducers but it can also be seen as non-linear element in frequency.
II- DESCRIPTION OF THE SETUP. The experimental setup is shown in fig. 1. It consists of a diode laser (DL) with an externalcavity loop which includes a filter and a diagnostic branch. The DL is a commercial single mode 5mW Fabry-Pérot type semiconductor laser (SHARP LT027) emitting at 780nm, with a threshold current of 46 mA (free running). The external cavity loop consists of a neutral density filter NF, a beam splitter BS1, and the filter F. This filter consists of two mirrors M1 and M2 resulting in a finesse f, and spaced by the distance d, while the distance between the DL and M1 is L. The diagnostic branch consists of 5 arms A,B, C, D, and E) isolated from the rest of the setup by three optical isolators I1 I2 and I3 respectively. Here I1 and I3 prevent parasitic reflections from the arms A, B, C and E through BS1, whereas I2 protects from parasitic reflection from arm D through the filter and the mirrors M1 and M2. Part of the light emitted by the DL is split off by BS1 into the arm A, B and C. Arm A consists of a scanning 250MHz FSR Fabry-Pérot interferometer (FPI) with a finesse of 25 and a photodiode PDA, arm B is just a 1 GHz bandwidth photodiode PDB with a 30dB amplifier, arm C consists of a diagnostic filter followed by a 1 GHz bandwidth photodiode PDC without DC component. Arms D and E are both a 1 GHz bandwidth Thorlabs DT210 photodiode. FPI measures the DL optical spectrum with a 10MHz resolution, PDB the direct power, whereas the diagnostic filter measures the "instantaneous" frequency by converting it into power detected by PDC. The linewidth of this diagnostic filter is a trade off between covering the full frequency range of the laser and the maximum sensitivity. In practice diagnostic filter bandwidth is chosen 2 to 5 times larger than the width of filter F. The light 5/25
19/08/2002 11:33 transmitted through F is measured in arm D with PDD while part of the light reflected by F is detected by PDE. The path of the light in the external cavity is as follows : Light emitted by the DL propagates through ND to BS1. Both elements are misaligned slightly so as to avoid spurious feedback from their surfaces. It has been carefully checked that the total parasitic feedback is much below -55dB and has no influence on the DL dynamics. Part of the light is split off and used for diagnostic in arms A, B, C, and E, the rest enters the Fabry-Pérot filter F (M1 and M2). M1 is a 3mm thick wedged mirror so that no multiple reflection occurs inside the mirror M1. The transmitted light then encounters a similar wedge mirror M2 spaced by a distance d from M1. Both mirrors are fixed on an accurate fine tuning mechanical mount so that M2 can be aligned to reflect the light directly into the DL. When M2 is slightly misaligned multiple reflections between M1 and M2 do not coincide, and one can thus measure approximately the finesse f of the filter F by counting the number of reflected spots. Multiple pass interference is built up by aligning M2 so that all the spots overlap onto the front facet of the DL. In this manner the filtered optical feedback situation is achieved.
After realignment of M2, the behavior of the DL can be analyzed through the diagnostic part of the setup. Since F is a Fabry-Pérot interferometer, a frequency selection takes place and both the amount of light fed back and transmitted depend on the DL frequency . The transmission is monitored in arm C. When the DL makes a frequency scan by modulation of the pump current I, while the total feedback is kept below –55dB using the density filter D, one obtains the respective transmission and the reflection transfer curves of F on PDD
19/08/2002 11:33 and PDE respectively. The spectral properties of F such as the linewidth, and the free spectral range can thus be measured. The current system is characterised by several time scales : I-Like in the COF case, the relaxation oscillation (RO) frequency νRO of the DL , related to the carrier lifetime, is one of the most important time scale for a DL. For a given DL, the RO frequency is a fixed parameter in the 1GHz-5GHz range:
υRO = 1 ξΓo P 2π where ξ is the differential gain , Γ is the photon decay term, P the photon number. II-The external cavity mode (ECM) free spectral range ∆νECM =c/2L (COF case) appears in the presence of coherent feedback in a DL. It is possible to vary the ECM value by changing the external cavity length. The external cavity length is usually on the order of centimeters up to few meters corresponding to a time scale of few GHz up to few tens of MHz. III- Due to the presence of the filter one needs to add two more time scales ; the filter bandwidth (FWHM) δνf=c/2fd, and the filter free spectral range (FSR) ∆νf=c/2d. The use of a tunable filter in the current system makes from the FWHM and the FSR the easiest controllable parameters. In our experiment the FWHM δνf is few MHz up to few GHz and covers all the different time scale ranges. One can thus distinguish three cases depending on the filter bandwidth δνf compared to the RO frequency and the ECM spacing. Each case leads to different dynamical behaviours that will be investigated in the rest of the paper. 1-The filter bandwidth δνf is larger than the RO frequency so that when the filter is centered at the solitary laser frequency, the RO-side-peak falls within the filter 7/25
19/08/2002 11:33 profile. This case is very close to COF, which can be considered as FOF in the limit of an infinitely broad filter, and has been partially reported in a previous paper . It will be referred to wide filter case. This also includes special cases where the RO mechanism plays a major role, but that will not be investigated in the current paper : 2 -The filter width δνf is narrower than the RO but a multiple of ∆νf coincides with νRO, this means that a higher order of the transmission peak let light modulated by the RO being fed back into the DL.. 3 – The filter is detuned from the solitary laser frequency by νRO such that the RO frequency coincide with the filter frequency. Here also, the light modulated at the RO frequency is fed back to the DL. These cases are favourable to a dynamical behaviour of the DL where the undamping of the RO take place. 2- The filter width δνf is smaller than the RO frequency νRO but larger than the external cavity mode FSR ∆νECM : (L>df >νRO /2c ). This case will be referred to intermediate case. 3- The filter width δνf is much smaller than the RO frequency νRO and smaller or of the same order of magnitude than the ECM FSR ( df ≥L). This case will be referred to "narrow" filter.
III-SIMULATIONS AND EXPERIMENTAL RESULTS In this section we present a combined experimental and theoretical study of delayed filtered optical feedback effects on a diode laser for different filter width. The experimental parts have been obtained with the setup mentioned in section II whereas the simulations are based on the theory presented in . Through the above mentioned three cases (wide, intermediate and narrow filter case) the simulations present the new frequency ω of the FOF system shifted from the solitary DL frequency ωo as a consequence of the optical feedback. These simulations include three types of calculations : 1/ First of all the fixed points are calculated. They corresponds to solutions with monochromatic field and constant inversion. They mean steady-state operation of the system. 2/The averaged operating points are the mean value of a series of solutions obtained with the integration of Eq. 1 ,2 and 3 of  started with a fixed point. When the serie converge that solution is stable and its mean value is plotted on the corresponding figure together with the above mentioned fixed point. This indicates the stable solutions reached by the laser. 3/The instantaneous operating point is a projection on a Poincaré map of the trajectory obtained from the serie calculated with the integration of Eq. 1 ,2 and 3 of  . This indicates the dynamics followed by the laser. The three types of calculations are collected and plotted on the same figures the so called frequency shift figure or snake contour figure because of its wiggling aspect. The comparison between the three filter cases relies on a two steps study of the system which include first, the steady state behaviour of the FOF system and second, its dynamics. 9/25
1. Wide filter case. (2Λ≥νRO) For this case the parameter in this experiment are L=1.5m external cavity length, a d=2 cm spacing between M1 and M2 and a finesse f=10. This results in an ECM spacing
∆νECM=100MHz and a Λ=4GHz filter width (HWHM). For this choice of parameter values, there are about 80 ECM present within the filter profile. Fig. 2 summarises the dynamical behaviour of the system as predicted by the theory Eq. 13 of ref. . It shows the frequency ω of the system relative to threshold as a function of the solitary laser frequency ωo (also relative to threshold) for an optical feedback rate
γ=… s-1 which corresponds in the experiment to (1µW –37db). Also the steady-state structure with all the available external cavity modes and antimodes is shown. This figure is similar but for different parameter value to fig.3 in . In this paper more reference about the method can be found. If the solitary laser frequency ωo is increased by decreasing the pump current starting at well below the filter center frequency, the operating point moves from A along a mode (the DL locks on to the modes) until it ultimately reaches the top of the filter (H) where a jump occurs to the tail of the filter (I) several external cavity modes away. Here two points should be made. First of all, point H does not match necessarily with the top of the filter profile (Hmax at ωo=ωf). Apparently some fixed points within the filter between H and Hmax are not stable. In an experiment this would imply that the maximum of feedback power i.e. the net amount of light coupled back into the laser is notably smaller than the power available from the filter measured at its center. In our experimental setup
19/08/2002 11:33 we measured a maximum feedback power of 18.5µW whereas the filter was observed to deliver 20µW at its center frequency in the absence of feedback. These values depend very much on the external cavity length L and the mirror spacing of the filter d. In the rest of the paper it will be referred to as the optical impedance effect. A similar behavior was observed earlier in ref  were in Fig.2 the jump is observed to occur clearly before the feedback power reaches its maximum potential (i.e. without feedback). We will now address the issue of dynamics on the basis of the instantaneous operating points of fig.2a, and the experimental DL spectra of Fig.3. Calculated time series show in fig.2 that in most cases the dynamics occurs on the RO time scale. in that current case is characterised by a time scale related to the RO. This is due to the undamping of the relaxation oscillations in a similar as in COF. The experimental spectra in Fig.3 show as well this progressive growth of the RO with the increase of the feedback rate, ultimately resulting in the coherence collapsed state. Clearly, a FOF system where the filter width is wider than the RO frequency is in many respects very similar to the COF case. .
2. Intermediate filter case 1/τ=c/2L