1. FunFACS 2005 FunFACS report of the CNQO group of the USTRAT partner USTRAT personnel of the CNQO group: Andrew Scroggie, William Firth, Damia Gomila, Francesco Papoff, Alison Yao, and Gian-Luca Oppo Contributions from: J. Jeffers, T. Ackemann and G. McCartney Department of Physics, University of Strathclyde Glasgow, Scotland, U.K.

2. FunFACS 2005 Content WORK ALREADY COMPLETED Cavity Solitons on modulated backgrounds Conversion from Phase to Amplitude modulations FUTURE WORK Semiconductor, VCSEL and VECSEL simulations Soliton speed on gradients for Soliton Force Microscopy (WP3) Theoretical and numerical investigation of VCSEL with external cavity (Thorsten’s experiment, WP1) and LB (WP2)

3. FunFACS 2005 Introduction: Utilisation of Cavity Solitons Nonlinear Material External Wide Pump Output Address Laser Beam Cavity Soliton

4. FunFACS 2005 Relevance of modulated backgrounds Arrays of Cavity Solitons (CS) can be used to store and retrieve information in an all-optical way but one needs a good control of the CS position in space. Optical Register In general modulations of the pump or the refractive index are used. It is generally believed that cavity solitons get stuck at the maxima of the background modulation.

5. FunFACS 2005 First Effect of Modulated Backgrounds: Inhibition of Modulational Instabilities D. Gomila et al., PRL 92, 253904 (2004) D. Gomila and GLO, Phys. Rev. E 72, 016614 (2005) FunFACS f(x) =  cos(Kx)

6. FunFACS 2005 Aim: Motion of Cavity Solitons Induced by Background Modulations Nonlinear Material Phase and/or Amplitude Modulator Cavity Soliton PUMP General belief: CS get stuck at maxima

7. FunFACS 2005 General Theory with a stationary spatial soliton solution E SOL (x) when  =0. Then, The velocity vanishes at the extrema of the modulation (sin(Kx)=0) If A(K) changes sign at K=K 0, the soliton reverses its direction of motion For K<K 0 it climbs modulation gradients (NORMAL behaviour) For K>K 0 it descends modulation gradients (ANOMALOUS behaviour) VERY For K=K 0 the soliton does not move in any point in space in spite of the presence of a background modulation ( VERY ANOMALOUS !!) velocity v(K) of the soliton induced by  cos(Kx)

8. FunFACS 2005  (2) E0E0 A0A0 A1A1 Phase Modulated cw Degenerate OPO Phase Modulator ReA 1 |A 1 | |A 0 |

9. FunFACS 2005 Reversible Motion of Cavity Solitons K Reversed motion REVERSED MOTION v(K) K E We have predicted and measured the CS-velocity v(K) induced by the phase modulation when changing its wave-vector K for fixed  In these points CS are stationary EVERYWHERE in spite of background modulations A. Scroggie et al., Phys. Rev. E 71, 046602 (2005) FunFACS

10. FunFACS 2005 Kerr Media and Saturable Absorbers Medium E0E0 E Phase Modulator General but not Universal v(K) K REVERSED MOTION Kerr Saturable

11. FunFACS 2005 Swift-Hohenberg Equation. Amplitude Modulation v(K) K REVERSED MOTION Screwed-up Optical Register

12. FunFACS 2005 Reversible Motion of Patterns The theory is so general that can be applied to other spatial structures such as patterns (A. Scroggie, D. Gomila, W. Firth and GLO, to appear in App. Phys. (2005) FunFACS). For the saturable absorber equations we have patterns moving to maxima and patterns moving to minima depending on the K of the input modulation.

13. FunFACS 2005 Transverse Phase (Amplitude) Modulations Nonlinear Material Phase or Amplitude Modulator Output Address Laser Beam Optical Cavity

14. FunFACS 2005 The Optical Parametric Oscillator Case  (2) Nonlinear Material Phase or Amplitude Modulator Output Address Laser Beam Optical Cavity Pump Signal

15. FunFACS 2005 The Empty Cavity Case Phase Modulated Pump Cavity Field after n round-trips ReflectivityDetuningDiffraction Conversion from Phase to Amplitude modulation happens when: A. Scroggie et al., Physical Review A 72, 023824 (2005) FunFACS

16. FunFACS 2005 Mean Field Conditions In order to understand the conversion formula F=0, we move to the MF Limit: Conversion from Phase to Amplitude modulation happens at: This condition can be satisfied ONLY for negative detunings 

17. FunFACS 2005 Why Negative Detunings Off-Axis On-Axis Off-Axis On-Axis Only for negative detunings the on-axis and off-axis components of the field lie in adjacent quadrants after a large number of round trips. For appropriate values of K=K 0, the two amplitudes are exactly perpendicular to each other. This means that an input amplitude (phase) modulation has been fully converted into a phase (amplitude) modulation.

18. FunFACS 2005 Generic Input Modulation For an empty cavity, K 0 is independent of  and given by: K02K02 For a Kerr cavity K 0 is a function of  First effect of nonlinearity: K 0 is a function of the kind of input modulation

19. FunFACS 2005 The Optical Parametric Oscillator Case  (2) Nonlinear Material Phase or Amplitude Modulator Output Address Laser Beam Optical Cavity Pump Signal

20. FunFACS 2005 The OPO case at Resonance ! Full conversion from amplitude (phase) to phase (amplitude) modulation of the pump to the signal is possible at resonance Conversion from Phase to Amplitude modulation Conversion from Amplitude to Phase modulation

21. FunFACS 2005 Remarkable Cavity Effects Apart from the spatial response of cavity systems to input modulations the CNQO group has also worked on giant excess (8 orders of magnitude) noise amplification of misaligned cavities with apertures. Details of the analytical calculations can be found in W.J. Firth and A. Yao, Phys. Rev. Lett. 95, 073903 (2005) This work may have connections on the external cavity configuration of Thorsten’s experiment WP1

22. FunFACS 2005 FunFACS Future Work I A) Models of VCSEL and VECSEL a) a)We do not like the model of J. Yao et al., Opt. Comm. 119, 246 (1995) also known as the ‘Agrawal model’. b) b)We prefer models based on a careful description of the susceptibility such as those of S. Balle (see Phys. Rev. A 57, 1304 (1998)) c) c)Numerical implementation is not easy but feasible. Need of several tests and checks. d) d)We hope to report about this numerical implementation soon.

23. FunFACS 2005 FunFACS Future Work II B) Motion of CS in VCSEL and VECSEL with noisy backgrounds. WP3 A. Scroggie, very private communication (2005)

24. FunFACS 2005 FunFACS Future Work III a) a)Extension of soliton motion analysis on models of CS in VCSEL and VECSEL b) b)Relevance of Temporal Scales (see Thorsten’s talk) D) New configurations a) a)Modeling of VCSEL with external cavity (Thorsten experiment at USTRAT). WP1 b) b)Modeling of VECSEL with saturable absorbers for Light Bullets. WP2 C) CS Motion in VCSEL for Soliton Force Microscope. WP3.

25. FunFACS 2005 Conclusions Cavity solitons an patterns can either climb or descend (anomalous motion) modulation gradients and get stuck at either maxima or minima of the background modulations At K=K 0 cavity solitons are stationary everywhere in spite of the breaking of the spatial symmetry by the modulations The phenomenon survives in two transverse dimensions The phenomenon is so general that it should be observable in a variety of other systems. More information in A. Scroggie et al., Phys. Rev. E 71, 046602 (2005)

26. FunFACS 2005 Final Message! For some values of the modulation wave-vector K, a CS moves up phase gradients and for others it moves down SEE-SAWLITON

27. FunFACS 2005 Conclusions 1. 1.Cavity Talbot-like effect can convert phase (amplitude) transverse modulations into amplitude (phase) modulations at precise values of the wave-vector K of the input modulation. 2. 2.In the empty cavity (under paraxial and MF approximations) this can happen only for negative detunings. 3. 3.Nonlinearity introduces a dependence of the critical wave- vectors from the kind of modulation 4. 4.Nonlinearity shifts the detuning values of the cavity Talbot effect making is possible to observe it at resonance ! A.J. Scroggie et al., to appear in Physical Review A (2005)

28. FunFACS 2005 Further results for the OPO case Input Phase modulationInput Amplitude modulation When pump-phase is converted into signal-amplitude modulation, the output pump reproduces the input The output pump reproduces the input at a K-value different from that where pump-amplitude is converted into signal-phase modulation