1. 30.3.2006 FunFACS meeting, Toulouse 1 USTRAT WP 1 Theory WP 2

2. 2 WP 1Theory 1. Cavity solitons in a VEGSEL with frequency-selective feedback P. V. Paulau 1,2, W. J. Firth 1, T. Ackemann 1, Andrew Scroggie 1, A. V. Naumenko 2, N. A. Loiko 2 1 Department of Physics, SUPA, University of Strathclyde, Glasgow, Scotland, UK 2 Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus 3 INFM, Como, Italy; 4 currently enjoying Italy 2. Master equation for describing VECSELs with intra-cavity elements L. Columbo 1,3, A. Yao 1, W. J. Firth 1 3. A new method for adiabatic elimination G.-L. Oppo 1,4

3. 3 CS in a VEGSEL P. V. Paulau 1,2, W. J. Firth 1, T. Ackemann 1, Andrew Scroggie 1, A. V. Naumenko 2, N. A. Loiko 2 1 Department of Physics, SUPA, University of Strathclyde, Glasgow, Scotland, UK 2 Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus

4. 4 VEGSEL To keep things simple: only one complex equation for linearly polarized field loss diffractionscaling phase-amplitude coupling feedback gain saturation finite gain bandwidth feedback: self-imaging, diffraction grating (envelope of filter: sinc-function) one round-trip in external cavity note: there are no temperature effects in this model we assume resonance in external cavity:  0  = m 2 

5. 5 Stationary states  =0, only k  = 0  supercritical bifurcation  bistability between off-state and lasing states  „old“ model with temperature: subcritical Naumenko et al., Opt. Commun. 259, 823 (2006)  =0, with k  > 0  bistability disappears  no CS  need for filtering

6. 6 Structures with spatial filtering I assume spatial filter in some Fourier plane in feedback loop filter due to gain curve spatial filter in feedback loop  =0  =0.06  f =0.06  f =0  f = 0.06 pattern forming instability potential for CS

7. 7 CS real part intensity far field this is a localized traveling wave !  is exponentially localized  exists on grids 64, 128, 256 energy=azimutally integrated intensity r (pixels) log (energy)  f = 0.06

8. 8 CS II  can exist at different locations in the plane  several LS can coexist  present or absent under the same conditions seems to be a self-localized solution, a true cavity soliton

9. 9 Symmetric structure real part intensityfar field initial condition: homogeneous high-amplitude state on zero background not a Bessel beam: energy decays exponentially is this related to experiment ?

10. 10 CS for detuning zero !? direct excitation of CS apparently not possible, but taking CS from  f = 0.06 as initial condition a localized solutions is obtained  f = 0 real part NF far field cut through FF azimutally averaged FF

11. 11 Summary: VEGSEL theory  very exciting result: localized traveling wave symmetric and asymmetric  relationship with experiment unclear: temperature, spatial filter....  need to translate to physical parameters

12. 12 Master equation 1 Department of Physics, SUPA, University of Strathclyde, Glasgow, Scotland, UK 3 INFM, Como, Italy L. Columbo 1,3, A. Yao 1, W. J. Firth 1 idea: derive a closed equation for dynamics of nonlinear non-plano-planar resonators by using ABCD matrix to decribe intra-cavity elements benefits:  ability to model complex real-world cavities (e.g. VECSELs)  address effects of small deviations from self-imaging condition in external cavity  describe properly action of grating in VEGSEL  significance for WP1 and WP2 Dunlop et al., Opt. Lett. 21, 770 (1996)

13. 13 Thin lens:: focal length=f Nonlinear medium: Vapour of two levels atoms Reference symmetric plane Perfect Mirror Perfect Mirror Master equation for an unidirectional square nonlinear resonator driven by a coherent injected field Gaussian aperture: FWHM=w Injected field: wave vector=k Mirror L L/2 Exact linear propagation taken into account by means of the ABCD matrix at the reference plane: Instantaneous nonlinear medium located at the reference plane Small variation per cavity round trip T R of the adimensional field envelope E 0 (T,r  ) at the reference plane. Intracavity field carrier wave vector=k NOTE: For f,w→∞ we get the Mean Field Limit equation considered for example in ref. M.Brambilla et al.,EPL 34, 1996 and in ref. W. J. Firth and A. J. Scroggie PRL 76, 1996 Diffusion Diffraction Space dependent gain|loss Linear absorption and dispersion Saturable absorption  <0  Injected field

14. 14 Spontaneous emerging of intensity spots in the near field Parametric regime=-1,  =-10.8, Y inj =6.52 Observation: In case (a) and (b) we managed to switch on and off a single intensity spot by superimposing a suitable addressing pulse to the holding beam !! (a). Without Gaussian aperture (initial condition: null intracavity field+Gaussian distributed white noise) Time=9T R Time=23T R Time=541T R Time=280T R (b). With Gaussian aperture (initial condition: null intracavity field+Gaussian distributed white noise) Time=11T R Time=22T R Time=37T R Time=280T R

15. 15 WP 2 1 Department of Physics, SUPA, University of Strathclyde, Glasgow, Scotland, UK 2 Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus Y. Tanguy 1, A. Smith, T. Ackemann, F. Papoff, A. Scroggie, A. Yao 1, W. J. Firth 1 P. V. Paulau 1,2, A. V. Naumenko 2, N. A. Loiko 2  VEGSEL with long cavity (task 2, overlap with WP1)  Planning and test setup for VCSEL + SA (task 1)  Modelling VEGSEL (task 2, overlap with WP1)  Modelling fast spatio-temporal dynamics with extended master equation (task 1, possibly 2)  Modelling coupled cavities (task 1, possibly 2)

16. 16 VEGSEL with long external cavity  actually setup from WP1  Yann showed (see also next slide) spot can be stationary (no peaks in RF- spectrum and FP) spot can have weak sidemodes in FP spectrum shows clearly strong excitation of sidemodes in FP and peaks in RF spectrum (possibly due to background) round-trip frequency 250 MHz  need for further analysis, but potentially the spots would also qualify as CLB  God knows how (ir)regular these might be  (simplified) model developed and coded from WP1  need to reintroduce carrier dynamics for reasonable results

17. 17 Spectra of spots

18. 18 Transfer to cavity light bullets HR BS R  2-8% self-imaging forward biased laser a)reverse biased laser (reduced reflectivity) b)QW SESAM c)QD SESAM (reduced saturation fluence, no demagnification necessary) R  0.8-0.985 f 1  8 mm SA gain possibilities: f 2  100 mm here: cut-off feedback to boundaries f 3  200 mm f 4  5 mm demagnification by factor of 3: adapt saturation fluences set-up not yet tested, but could be done in week before Eastern, if considered to be necessary for annual report R=0.8 about factor of eight in gain; R=0.9 about factor of four, R=0.985 ok

19. 19 Spectrum of SESAM very high absorption losses, possibly not useful

20. 20 Extended Master equation Dunlop et al., Opt. Lett. 21, 770 (1996) ABCDnonlinearity changes on time scale longer than round-trip time changes on time scale of pulse

21. 21 Coupled cavities coupling mirror between two Fabry-Perot cavities transfer matrix coupled master equations only valid, if no variations on time-scale of round trip in both cavities (quasi single-longitudinal mode) How to couple „extended“ master equations? How to include inertia of medium?