Polarization dynamics in a transverse multimode class B laser: role of the optical feedback

Transcription

1 Polarization dynamics in a transverse multimode class B laser: role of the optical feedback R. Meucci, K. Al Naimee Department of Physics, College of Science University of Baghdad Baghdad, Iraq M. Ciszak Firenze,Italy S. De Nicola, Pozzuoli,Italy INFN Sez.di Napoli Napoli,Italy S. F. Abdalah F. T. Arecchi Dipartimento di Fisica, Università di Firenze, Firenze, Italy Abstract-We investigate the polarization dynamics in a quasiisotropic CO 2 laser emitting on the annular mode subjected to an optical feedback. We observe a complex dynamics in which spatial mode and polarization competition are involved. The observed dynamics is well reproduced by a model that discriminates between the intrinsic asymmetry due to the kinetic coupling of molecules with different angular momenta and the anisotropy induced by the polarization feedback.we observe various dynamical regimes including chaotic dynamics and show that feedback changes these states from regular to chaotic and vice versa. Such a dynamics is also accompanied by spontaneous between the patterns that indicate the existence of bistability in the system. Finally the possible applications to polarization coding are discussed. Keywords-Laser;chaoticdynamics;polarization;optical feedback;control. I. INTRODUCTION Laser dynamics in quasi-isotropic gas lasers, where there exists a competition between intrinsic anisotropies, due to the kinetic coupling of molecules with different angular momenta and residual cavity and detuning anisotropies (extrinsic anisotropies) is a topic of particular interest [1,2,3,4]. These phenomena were studied on lasers emitting on the fundamental transverse mode(tem 00 mode); however, allowing the laser emission on higher order transverse modes makes it possible to investigate the interplay between spatial and polarization effects [5,6]. Particular importance is played by the annular mode, more precisely by the TEM 01 * mode. This mode can be considered as the superposition of two Laguerre Gauss modes or the superposition of two Hermite Gauss modes TEM 01 and TEM 10.In a previous study[7], we observed that, if the polarization state is analyzed along the H- V axis, two configurations can exist. In particular, depending on the cavity detuning, the annular pattern consists of a TEM 01 mode polarized along a given eigendirection (H or V) and a TEM01 mode polarized along the orthogonal eigendirection. We named this configuration splitted. When the cavity is set far from the atomic resonance the pattern changes to a polarized ring pattern along the vertical or the horizontal eigendirection ( homogenous configuration). However, in these experiments no spontaneous jumps between these configurations where observed, in other terms, no pattern alternation was observed in two eigendirections. Improvements in the cavity symmetry has led the actual configuration to observe interesting spatio- temporal dynamics with evidence of complex dynamics including chaotic behaviours in the polarized output intensity. Such a dynamics is also accompanied by spontaneous jumps between the patterns that indicates the existence of bistability in the system. We also demonstrate that such a dynamics can be controlled or enhanced by a suitable optical feedback. By inserting a polarized beam splitter, a fraction of one of the polarized radiations is rejected in the optical cavity, allowing a control of the polarization state of emission. This offers the possibility to use coding schemes where both spatial patterns and their polarizations can be exploited, offering a better performance with respect to the laser emission on the fundamental mode where the only degree of freedom for security is offered by the direction of polarization. Finally, we present a mathematical model that reproduces most of the phenomena observed in the experiments. II. EXPERIMENTAL RESULTS The polarization dynamics of a quasi-isotropic CO 2 laser controlled by optical feedback has been investigated by using the experimental setup shown in Ref.[8]. It consists of a quasi Complexity in Engineering (COMPENG) /14/$31.00 c 2014 IEEE

2 isotropic CO 2 laser, with a Fabry-Perot optical resonator. The laser is allowed to operate on the first order transverse mode (TEM 01 *mode) by means of an intracavity iris diaphragm. The set-up is supported by infrared camera (Pyrocam III- pixel size 85 x 85 m) to observe the spatial laser output. The beam transmitted by LP is vertically polarized(vb) and the reflected part is horizontally polarized(hb) and it is directed towards the laser by the feedback mirror. The Feedback Mirror (FM) is a ZnSe non-polarizing beam splitter (50/50). The reflected beam by FM is re-injected into the laser cavity as a horizontally polarized optical feedback beam, whereas the beam transmitted by FM is reflected towards the infrared camera T. Both HB and VB beams are focused onto different pixel areas of the infrared camera detector surface by a convex lens L.Together with the polarized spatial patterns on the infrared camera, we recorded also the temporal dynamics by means of a fast HgCdTe detector at room temperature inserted in one of the two polarized beams. Setting the discharge current at r p =1.8 times the threshold value and adjusting the cavity detuning, we see the occurrence of a complex dynamics showing competition between polarization and spatial modes. Figure1 shows polarized spatial patterns recorded by the infrared camera in absence of optical feedback (g = 0) for increasing values of the detuning, normalized to the free spectral range of the optical cavity. Polarized spatial patters recorded close to the laser threshold are shown in Fig.2. Now, the spatial patterns tend to align and are quite insensitive to detuning variations. This means that the optical feedback (g = 0.005) induces a symmetry breaking of the unperturbed annular transverse mode (TEM 01 *mode). We identified various dynamical regimes occuring in the absence and presence of optical feedback. In Fig.3(a-b) we report the maxima of oscillations x m for varying δ parameter, (a) (b) (e) Fig.1. Polarized spatial patters recorded by the infrared camera for increasing values of the detuning in the absence of optical feedback. In each recorded frame, the upper right patterns are the spatial profile of the vertically polarized beam; the lower left patterns are the spatial profile of the horizontally polarized beam: (a) ; (b) ;(c) ; (d) (e) ; (f). (f) (a) (b) (c) Fig.2. Polarized spatial patters recorded near threshold at for different values of the detuning: (a) ;(b) ; (c). at zero and non-zero optical feedback g, respectively. Application of feedback changes the dynamics of the system, mostly consisting of the transition from periodic oscillations to steady state, and from periodic oscillations to chaotic. We have used the embedding technique to reconstruct the attractor from the single time series. In order to decide if the time series in the figures are periodic, quasiperiodic or chaotic we have used the reconstructions of embedded phase spaces to estimate corresponding correlation dimensions [according to Ref.(9)]. As an example, we show the times series for a free running laser with periodic, chaotic and quasiperiodic oscillations in the intensity (Fig. 3(c-e) respectively). (c) (d)

3 Fig.3. Bifurcation diagrams from the experiment (a) g = 0 and (b) g = for. Reconstructed attractors from the time series running in the absence of optical feedback for (c), (d) and (e) with x 1 = power (t) and x 2 = power (t-τ) for τ = 0.8μs. Moreover, we have found the bistability in the system. A bifurcation diagram demonstrating the existence of hysteresis obtained experimentally for g = 0 is shown in Fig.4(a), where the maxima of oscillations x m have been reported. In Fig.4(b) the time series in bistable region (for ) are shown. In the following section we introduce theoretical model of the experiment and demonstrate numerically that the phenomenology observed in the experiment may be well reproduced. Fig.4. Bifurcation diagram demonstrating the existence of hysteresis obtained from experiment for g = 0.(a) Bifurcation diagram showing the maxima of oscillations. (b) The time series in the bistable region are shown for. III. THERORETICAL AND NUMERICAL RESULTS The theoretical analysis is based on the theory of the isotropic laser [2, 3] where the single-longitudinal-mode Maxwell-Bloch equations for a polarized two-level laser in the rotating and slowly varying amplitude approximations and first order coherences between upper levels are considered. The theory was developed for the simplest case of a transition but it has also been applied successfully to explain the polarization dynamics on the fundamental transverse mode [10] and on the TEM 01 * mode [7] in a higher order transition such as a CO 2 laser emitting on the P (20) line. The evolution equation for the electric, matter polarization and population inversion fields can decomposed in a circularly polarized basis and written for each component of polarization as: (1) In Eqs. (1), are the slowly varying electric fields for right R and left L polarization, respectively. They are related

4 to the fields Ex and E in the H-V orthogonal basis by the y relations &. The matter polarization fields are, and, the corresponding population inversions, is the first order coherence between the sublevels in the case of the transition from a state J =19 to J =20. The coherence accounts for the induced anisotropies in the laser medium responsible for the competition between the two polarized modes, is the pump normalized to its threshold value, the parameter represents the cavity losses whose value, according to our resonator specifications, is. The optical feedback caused by the presence in the experimental set up [see details on Ref.(8)] of the feedback mirror FM is modeled in the equations for the electric fields by including terms proportional to the field and the feedback strength g, with g <<1. In the orthogonal basis H-V the feedback term which couples to the horizontal x polarization state takes the simple form The transverse coordinate x-y are rescaled with respect to the minimum beam waist, is the transverse Laplacian, and is the square of the radial transverse coordinate to the mirror center. The parameter is the diffraction coefficient with γ =10.6 m the laser wavelength, δ represents the cavity detuning of the field modes from the matter transition frequency, i.e. the frequency of the P (20) line which is the only active transition. In our low-pressure, homogeneously broadening CO 2 laser, the polarization decay is and the population inversion decay rate is.the parameter is the coherence decay rate whose value is chosen between and [10]. From the experimental results we know that just the TEM 01 and TEM 10 modes take part in the dynamics, and therefore the most general expression for the slowly varying electric field operating on these modes can be written as time dependent superimpositions of mode function E r,t t A( r) t A r (2) R R 1 R 2 E r,t t A( r) t A r (2) L L 1 L 2 In Eq.(2), the mode functions and ) with ) are the standard orthonormal Gauss-Hermite modes TEM 01 &TEM 10 and the time dependent functions, describe the slowly varying time dependence of the fields. The dynamics of the system can be studied according to the above described modal expansion. The matter variables, &, and the coherence C operating on these modes can be similarly expanded and we obtain a set of coupled equations for the electric fields amplitudes and the amplitudes of the matter variables and coherence. The complete set of equations is reported in Ref. [1]. The modal expansion method Ref.[8] reproduces fairly well the experimentally recorded spatial pattern. The evolution of the transverse pattern intensity distributions is calculated for different values of the optical feedback strength, pump value and cavity detuning. Fig.5.displays the spatial distribution across the x - y plane of the polarized patterns calculated for increasing values of the detuning from to. The numerical results are calculated at a normalized pump value.. The upper right and lower left are the horizontal and vertical polarized spatial patterns. The numerical simulations of the polarized patterns calculated at resonance for increasing pump parameter are shown in Fig.6. It can be clearly seen that that the polarized patterns tend to stay aligned up to a (pump parameter) around.7. This fact confirms that the observed pattern alternation far from threshold depends on the energy provided by the pump mechanism. This phenomenon was also observed on the fundamental mode in a condition leading to coherence resonance between the two allowed polarization configurations [11]. Fig.5. Numerical results for the polarized spatial patterns at a pump value.(a) ; (b) ; (c) ; (d) ; (e) ; (f). The time dependence of the functions, embodies the observed rich spatio-temporal dynamics. To further analyze the time evolution of the polarized spatial pattern we have calculated the time behavior of the power of the x and y- polarized pattern. Fig.7 shows the total intensity time dependence of the horizontal x- and vertical y- polarized spatial pattern for increasing values of the pump parameter.

5 Power fluctuation can be clearly seen for values of the pump parameter greater than. These fluctuations cause the observed pattern alternation phenomenon (compare Fig.7 (b)-(c) and (d))and demonstrates the competition between polarized spatial modes. The high spikes denote the transition from below to above threshold when the cavity loss parameter k is changed. Such spikes are the typical ones in class B lasers [12]. For comparison, in Fig.7 (a) we also plot the condition below threshold where the laser intensity relaxes to the steady state condition with zero intensity. Fig.7. Time evolution of the power of the x-polarized spatial pattern (red curve) and y-polarized spatial pattern(blue curve)at zero detuning for increasing values of the pump parameter:(a) ; (b) ; (c) ;(d). We have shown in Section 2, that laser produces a rich complex dynamics. In order to understand how the optical feedback changes these dynamical states of the system, we realize a detailed numerical analysis by scanning various system parameters in the presence and in the absence of the feedback. The systematic scanning of the parameters (through bifurcation diagrams) allows us to obtain a global image of the feedback effect on the system. Analyzing the phase space of parameters r p and δ (see Fig.8), we can observe quantitatively the effect of feedback of the system. Here, we consider only one value of optical feedback (assumed to be small with respect to the laser intensity), however a similar phenomenology is observed for other (still small) values. Fig.6. Calculated polarized spatial patterns at detuning for increasing values of the pump parameter: (a) ; (b) ;(c) ; (d) (e) ; (f). Fig.8. Dynamical regimes calculated numerically as the control parameters are modulated: steady state(black colour), periodic oscillations(gray colour) and aperiodic dynamics(white colour) for (a) g = 0 and (b) g = In Fig.9(a)we show the bifurcation diagrams where parameter r p was increased and subsequently decreased. The resulting diagrams revealed the existence of hysteresis in the system. In a certain parameter regions we observe the existence of bistability, between(i)steady and periodic states,

6 (ii) two various amplitude periodic oscillations and (iii) periodic and chaotic oscillations. Thus numerical results confirm the existence of bistable regimes observed in the experiment. Curiously, numerical simulations demonstrate that optical feedback eliminates bistable states (see Fig.9 (b)). where the codification of information also utilizes the polarization properties of the laser modes. Recently, experimental evidence of polarization chaos in vertical-cavity surface emitting lasers was given by Olejniczak et al. [15] and by Virte et al. [16]. The security enhancement of optical chaotic communications based on the polarization properties of these lasers was demonstrated by Xiang et al.[17]. Other communication schemes based on polarization-rotated optical feedback and polarization-rotated optical injection were proposed by J.Liu [18]. ACKNOWLEDGMENT Work was partly supported by Ente Cassa di Risparmio di Firenze. MC and RM acknowledge Regione Toscana for financial support. Fig.9. Bifurcation diagrams demonstrating the existence of hysteresis obtained numerically for (a) g = 0 and (b) g = CONCLUSIONS We have analyzed the evolution of the spatial polarization profiles under the influence of a polarized optical feedback. Near threshold, a symmetry breaking of the annular pattern weakly dependent on the cavity detuning is observed. Far from threshold, pattern and polarization instabilities are induced leading to an alternation of different polarized patterns critically dependent on the cavity detuning. We have demonstrated the existence of rich complex dynamics in a system, including chaos, quasiperiodicity and bistability, that may be efficiently controlled by an application of a weak optical feedback. Moreover we derived a theoretical model that well reproduces all the phenomenology observed in the experiment. The exhaustive numerical simulation of a model delivered an useful information about the effect of feedback in a wide range of system parameters. We show that this modal expansion method reproduces fairly well the experimentally recorded spatial pattern with little computational cost. The evolution of the transverse pattern intensity distributions is calculated for different values of the optical feedback strength, pump value and cavity detuning. Complex polarization dynamics is analyzed in terms of the competition of among these two spatial modes and we compare the evolution of the spatial polarization profiles for small values of the cavity detuning near resonance and large detuning values where symmetry breaking slightly increases The possibility to alter the polarization direction in an annular pattern, which is the usual one in unstable resonators, by means of a weak optical feedback is of crucial importance in industrial high power applications for cutting processes. Communication systems for near earth space applications based on a CO2 laser were proposed by J. H. McElory et al. [13]. 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