bullet holes (section 3), stabilisation, selection, and tracking of unstable optical patterns (section. 4 (a)) and elimination of spatio-temporal disorder in nonlinear ...
Controlling Complexity in Nonlinear Optics Final Report of EPSRC Grant J / 30998 Gian-Luca Oppo and William J Firth with contributions from G K Harkness, A J Scroggie, G D'Alessandro, A J Kent, R Martin, and A Lord
Department of Physics and Applied Physics, University of Strathclyde, Glasgow
April-June 1997
1. Introduction Controlling Spatio-Temporal Dynamics in Non Linear Optics (NLO) has been a key area of research for the Computational Nonlinear Optics group at Strathclyde during the last three and half years. A detail description of all the activity is impossible within the few pages of this report, so we refer the reader to the list of publications in the rst page of this document. Here we describe control of dynamics of multi-transverse mode lasers (section 2), control of the position of optical bullet holes (section 3), stabilisation, selection, and tracking of unstable optical patterns (section 4 (a)) and elimination of spatio-temporal disorder in nonlinear optics (section 4(b)).
2. Controlling Spatio-temporal Dynamics in Multi-Transverse Mode Lasers In the rst part of the project we focused on learning and applying techniques devised for the control of chaos in the temporal domain [1-3] to the case of spatially extended nonlinear optical systems. A prototype model for the description of a ring laser with an active two level medium is the system of Maxwell-Bloch equations with the electric eld projected onto a set of GaussLaguerre modes [4] dfpm = ?f[l ? i( ? (2p + jmj))]f ? 2C Z A (; )P d d g pm AC pm pm d @P = ?P + FD (1) @ @D = ? fD ? (x; y) + 1 (FP + F P )g @ 2 where fpm is the modal amplitude of the corresponding Gauss-Laguerre mode Apm, P the polarisation and D the population inversion, fpm, Apm and P are complex quantities. Also is the transverse radial coordinate in the (x; y) plane, AC is the detuning between the atom and the fundamental cavity mode, is the mode-spacing, 2C is the pump intensity, is the eld decay rate, normalised by the polarisation decay rate ?, lpm is the cavity loss parameter (normalised by ) of the mode denoted by the indices pm, is the population inversion decay rate ( k) normalised by ?, is the time normalised by 1= ? and (x; y) the spatially dependent pump of either symmetric or asymmetric (o -axis) Gaussian shape.
The behaviour of the multimode laser was simulated by considering 10 radial and 64 angular grid points and integrating the resulting 2050 coupled ordinary di erential equations using a variable order variable step Adam's method. All of the integrations were performed with parameters aimed to simulate a CO2 laser; = 0:3, = 0:01, mode-spacing = 0:027 and the detuning AC = ?0:161. We implemented the Occasional Proportional Feedback (OPF) method in the form used by Roy [3] who successfully controlled a chaotic laser. This consists by choosing rst a reference value of the intensity Iref , sampling the intensity of the laser at some inherent frequency (the frequency of the relaxation oscillations) and calculating a forcing term according to: forcing = ?s (Isample ? Iref )
(2)
where s is a proportionality or strength factor. This yields four control parameters: Iref , the sampling frequency, the strength s and the time t which is the duration of the forcing. The forcing term is then added to the chosen control parameter. We considered and compared the pump intensity, the cavity losses l, the detuning AC and the mode spacing as possible control variables [5]. By applying the OPF technique, we were able to control the laser output (i.e. suppress chaos) with cylindrically symmetric and asymmetric pump shapes. Figure 1 presents a successful result. A table of the orbits controlled, along with their control parameter values can be found (a)
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Figure 1: Time evolution of the output power of the fteen transverse mode laser before (a) and after (b) control. The lower trace in (b) is the intensity of the control signal. in the table of Ref. [5]. Although di erent periodic orbits were obtained, the relative intensities of the individual modes before and after the control remained very similar. This means that the output shape of the controlled and uncontrolled cases, when averaged in time, are not very di erent from each other (see Figure 2 (a) and (b)). The method apparently does not allow any selective control of desired pattern, but produces whichever pattern is dictated by the underlying dynamics. Moreover, it is important to note that we only obtained control where the chaos was weak. In order to determine the degree of chaotic motion, we calculated the Fractal Dimension of the attractor using a standard method [6]. When control is achieved (2C around 2.0), we found that the dimension is between 4 and 6 (see lower plot of Figure 2 (c)). Also in Figure 2 (c), is a similar plot for the uncontrolled state with 2C = 2:5 which gives some indication of how the complexity increases with increasing pump. The diagram shows that for the values of
the embedding dimension considered, the Fractal Dimension shows no sign of convergence. This means that small control kicks can activate dormant modes (recall the high dimension of the phase space) increasing the degree of disorder instead of suppressing it. (a)
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Figure 2: Time average output power of the laser oscillating in a chaotic (a) and in a periodic (controlled) (b) state. Panel (c) shows the Fractal Dimension D2 vs the Embedding Dimension m for a controllable case (2C = 2:0) and an uncontrollable one (2C = 2:5). From our analysis it became clear that full spatio-temporal control cannot be reached by simply generalising methods devised for the elimination of chaos in the temporal domain. Such methods do not allow for the stabilisation of desired output shapes and work only in regimes of weak (low dimensional) chaos. For these reasons we moved to develop an entirely new spatio-temporal control method based on the Fourier space as described in Section 4.
3. Control of Optical Bullet Holes We have studied a system consisting of a two{level medium in an optical cavity for the case where the material variables evolve on a much faster time scale than the eld (the good cavity limit). Under such conditions our model equation for the intra{cavity eld, E , is [7] : "
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2 @t E = ?E (1 + i) + jE2Cj2 (1+ ?1 +i) + E I + air E 2
(3)
where EI is the input eld, is the cavity detuning, is the detuning of the pump from the atomic resonance frequency, and C = L=2T with the absorption co{ecient of the medium, L the length of the medium and T the transmission co{ecient of the mirrors. Nonlinear analysis and numerical integration of Equation (3) have shown that in two transverse dimensions it exhibits the formation of the two possible types of hexagonal solution, as well as both sub{ and super{critical rolls [8]. Sub{critical rolls or transcritical hexagons imply a coexistence region of the at solution with either a roll or a hexagon solution. It is precisely in these regions that localised states are found [5]. These are like one or more peaks of a roll (or hexagon in two dimensions). The rst panel of Figure 3 shows an example of a localised state for the case of = 0. Such soliton-like states are time-independent if the at background is stable and have been labelled Optical Bullet
Holes (OBH) since, in transmission, they appear as transparent disks on an absorbing background, created by a localised address pulse (bullet). In order to nd applications for these spots it is necessary to be able to control them, i.e. to generate, store and erase them at will. The rst process is straightforward: a local address pulse will create a spot at the point of address (at least for appropriate parameter ranges). The spot is stationary in principle, but because the its location is arbitrary (due to the translational symmetry of (3)), there is no restraint against its wandering away from its original location. We note, however, that the basic equation (3) has a Galilean-like \boost" symmetry. If the input beam has a phase gradient (e.g. it is slightly misaligned) then this symmetry implies that there are solutions which are transversely moving versions of any static solution with a at input phase. So, OBH will move up a phase gradient. Therefore if a pattern of phase peaks and troughs is present on the input beam, and spots are created at any point, they should move to seek out the phase peaks, and remain there. Since at each peak we may choose to have a spot or not, we have a binary optical memory, with each peak a \pixel". Figure 3 (a) and (b) show the writing of the acronym \IT" (Information Technology) from an initial bunch of localised structures. The capacity of such a memory is 2mn states for an m x n array of phase peaks. As well as this passive control, active control has been also demonstrated, the OBH following a moving phase trough.
Figure 3: Intensity pro le of an Optical Bullet Hole. (a) Initial randomly mis-addressed OBH's. (b) Control of OBH via a phase modulated pump (letters IT). Recently, generalisations to semiconductor media, described by two coupled partial di erential equations for the complex eld E and the carrier density D have shown that OBH are universal features in nonlinear optics. Stable OBH are again found over a wide range of experimentally interesting parameters. More importantly, control of OBH is still possible. In all semiconductor models we have been able to create, erase and x the OBH at any location with gaussian address pulses of suitable width, duration and phase. The semiconductor models suggest typical OBH sizes about 15 m assuming a cavity length of 5 m and 1% losses. This is comparable to etched optical memory array pixel pitches [9]. Smaller OBH can be achieved by reducing the cavity length and possibly the di usion length. The paraxial approximation inherent in our present model is not a real restriction on OBH size, and we are con dent that OBH down to wavelength scales are possible.
4. Control via Fourier Space Techniques (a) Stabilisation, Selection, and Tracking of Unstable Optical Patterns The behaviour of spatially extended nonlinear dynamical systems, including those found in optics, often has a more convenient description in Fourier space than in real space. Because of the spontaneous breaking of the translational symmetry, pattern forming systems possess a huge number (in principle an in nity) of stationary states, some stable and the largest majority unstable, for any given set of parameter values. These features suggest the possibility of using control techniques which operate in the spatial Fourier domain to stabilise unstable states or to choose between alternative stable states. This is a potential advantage in optics since the Fourier transform of an optical eld is simply its far{ eld image (in the paraxial approximation), meaning that the operations of Fourier transformation and inverse transformation can each be performed simply with a lens system. To investigate this idea we have chosen the model of a passive two{level medium in an optical cavity given by Equation (3). For simplicity we concentrate on the purely absorptive ( = 0) case. Time{independent, spatially homogeneous solutions of Equation (3) become unstable for suitable values of C [10]. Above this \modulational instability"p(MI) threshold perturbations of the form eiK:r experience growth if jKj ' Kc " where Kc = ? [10], and " is small close to threshold. The condition on jKj corresponds to an annulus in Fourier space. Competition between modes in this annulus leads eventually to a stead state consisting of either two (\rolls") or six (hexagons) peaks. Depending on the parameter values either one, or both, of the roll or hexagon solutions may be stable. The model therefore allows investigations into both the stabilisation of unstable patterns and the selection of a given stable pattern from a set of alternatives. The characteristics of the states we wish to stabilise provide the physical basis for our control technique. First, we take the Fourier transform of the output eld and lter it to discourage growth of the undesired modes and to stimulate growth of the selected pattern. We then take the inverse Fourier transform of the resulting eld, multiply it by a small strength parameter and add it from the input pump eld. Thus, the pump eld acquires a spatial modulation which is determined by the selection mechanism operated by ltering in the Fourier space. The pump eld can then be written as EI (x; y) = EI 0(1 ? s1f1 (x; y) + s2f2 (x; y)) (4) ? 1 fi(x; y) = F UiF E (5) where EI 0 is the magnitude of the plane{wave pump, si are the feedback strengths, F denotes the operation of Fourier transform of the electric eld E , Ui describe the ltering operations in Fourier space and F ?1 is the inverse Fourier transform. To stabilise the homogeneous solution, we suppress the growth of the modes on the annulus Kc by removing all the Fourier components but the annulus ( ltering U1 ) and by superposing it to the pump as a negative control f1 which discourages pattern formation. To stabilise rolls, instead, we remove two diametrically opposite modes from the Fourier circle of magnitude Kc to build up f1 while s2 remains zero. This suppresses the growth of the homogeneous solution as well as all patterns but the rolls. Note that the feedback vanishes as the rolls stabilise, ensuring that these rolls are indeed a solution of equation (3). Due to the rotational degeneracy, we are free to choose the orientation of the stabilised rolls.
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If we now try to stabilise squares (four wave-vectors) by removing the corresponding wave{ vectors from the feedback, we end with the stabilisation of rolls in one of the two possible orientations, the roll pattern being more stable than squares. We must ensure the presence of all four wave{vectors and this is done by ltering the Fourier eld to obtain the amplitudes of the four modes and passing the eld through an interferometer with a eld rotating element in one arm. We then take the inverse Fourier transform to construct f2(x; y) which is fed back as positive feedback to the pump. Such rotation guarantees the simultaneous growth of both pairs of roll wavevectors, thus stabilising the square pattern. Similar ideas can be applied to the stabilisation and tracking of hexagons [11]. Figure 4 shows a dynamical sequence of pattern selection and stabilisation. The sequence starts with the formation of rolls, the stable pattern for the given parameters. The control was applied since this allowed faster convergence to the steady state. Then H + hexagons, H ? hexagons, squares and nally, the homogeneous solution are all stabilised with appropriate feedback control. The feedback vanishes once control is achieved. Since the time is in cavity-response units, the control is established in just ten or so response times.
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Figure 4: Succession of successful stabilisation of four patterns all at the same parameter values. The shaded trace corresponds to the maximum amplitude of the feedback and the time is in cavity-response units.
(b) Elimination of Spatio-Temporal Disorder in Nonlinear Optics The coupling of spatial and temporal degrees of freedom in nonlinear dynamical systems often leads to loss of spatio-temporal coherence. A simple-minded suppression of such behaviour can be obtained by reducing either the nonlinearity or the number of spatial degrees of freedom (as in the use of apertures in optical cavities). Such approaches always result in serious limitations for practical applications. A less restrictive restoration of spatial and temporal order is a highly desirable feature in elds as diverse as laser and plasma physics as well as hydrodynamics. We have applied our Fourier space method devised originally to stabilise unstable patterns to eliminate spatio-temporal disorder. We applied it to two speci c optical systems which display spatio-temporal disorder: a liquid-crystal light valve (LCLV) [12], used in a con guration modelling a Kerr slice with feedback mirror [13] and a broad-area laser [14]. The successful elimination of spatial and temporal disorder for both systems is an indication of the generality of our method. The LCLV is a hybrid electrical/optical device which allows the phase shift of a `read' beam to be controlled by the intensity of a `write' beam. The feedback loop is provided by allowing the phase shifted input eld to fall on the `write' side of the LCLV after some distance of free space propagation. For input intensities I above some threshold, the plane wave output beam loses stability and forms a hexagonal pattern. For larger values of I , the hexagonal pattern breaks
into uctuating spots of light [13]. The consequent `turbulent' motion is shown in panel (a) of Figure 5. We can inhibit this instability by including an additional feedback loop into the LCLV setup with a fraction of the eld propagating to the `write' side of the LCLV extracted, ltered in Fourier space and then re-combined with the backward eld. A typical lter mask shape is shown in panel (b) of Figure 5. The form of the mask is chosen based on the target pattern, in this case a hexagonal pattern comprising six wavevectors on a circle in k-space together with higher harmonics associated with their nonlinear mixing. The mask is chosen so as to allow all wavevectors relevant to the pattern to pass through it una ected while undesired wavevectors will see an additional loss due to the mask, proportional to s [11]. In this way, we have been able to stabilise hexagonal patterns well beyond their normal regime of stability. Panel (c) of Figure 5 shows the stabilised hexagonal pattern for the same parameters as in panel (a). As in reference [11], by suitable modi cation of the mask we have also successfully stabilised rolls and squares, both unstable solutions in this system. It is important to note that this technique is non-invasive; the patterns it stabilises are exact solutions of the original system without control and the control signal vanishes when control is established. As a quantitative measure of this fact, we compared the output eld integrated over the transverse plane in the cases of `turbulent' and controlled dynamics. The hexagonal pattern contains around 97% of the `energy' of the `turbulent' one. (a)
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Figure 5: (a) A disordered state of the LCLV model. (b) The lter mask used to select hexagons (black areas are transparent). (c) A control-stabilised hexagonal pattern. Finally, a second system where spatio-temporal disorder has been suppressed via Fourier techniques is a broad area laser with nite size gain pro le. For certain ranges of parameter values, an erratic emission of `optical vortices', which travel across the beam, takes place. By study of spatio-temporal correlations, such patterns have been shown to be weakly `turbulent' [15]. Again, this instability can be suppressed by injecting into the laser an additional eld derived from the output eld by suitable ltering in Fourier space, as described above [16]. Needless to say that we have been able to eliminate the vortex stream and to stabilise the laser output. The power output of the controlled state is the same, within the bounds of numerical accuracy, as for the uncontrolled state. This further emphasizes the advantage of our technique over more conventionally used `pinhole' methods.
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