1. Arbitrary nonparaxial accelerating beams and applications to femtosecond laser micromachining F. Courvoisier, A. Mathis, L. Froehly, M. Jacquot, R. Giust, L. Furfaro, J. M. Dudley FEMTO-ST Institute University of Franche-Comté Besançon, France

2. Accelerating beams Airy beams are invariant solutions of the paraxial wave equation. Airy beams follow a parabolic trajectory: they are one example of accelerating beam. 2 F. Courvoisier, ICAM 2013 Siviloglou et al, Phys. Rev. Lett. 99, 213901 (2007) Propagation Transverse dimension Intensity

3. High-power accelerating beams 3 F. Courvoisier, ICAM 2013 Polynkin et al, Science 324, 229 (2009) Airy beams can generate curved filaments. Lotti et al, Phys. Rev. A 84, 021807 (2011) BUT: paraxial trajectories, parabolic only

4. Motivations 4 F. Courvoisier, ICAM 2013 Aside from the fundamental interest for novel types of light waves, accelerating beams provide a novel tool for laser material processing. Nonparaxial and arbitrary trajectories are needed.

5. Outline We have developed a caustic-based approach to synthesize arbitrary accelerating beams in the nonparaxial regime. I- Direct space shaping II-Fourier-space shaping III-Application to femtosecond laser micromachining 5 F. Courvoisier, ICAM 2013

6. Accelerating beams are caustics Accelerating beams can be viewed as caustics – an envelope of rays that forms a curve of concentrated light. The amplitude distribution is accurately described diffraction theory and allows us to calculate the phase mask. 6 F. Courvoisier, ICAM 2013 S. Vo et al, J.Opt.Soc. Am. A 27 2574 (2010) M. V. Berry & C. Upstill, Progress in Optics XVIII (1980) "Catastrophe optics" J. F. Nye, “Natural focusing and fine structure of light”,IOP Publishing (1999).

7. Sommerfeld integral for the field at M : Condition for M to be on the caustic: Accelerating beams are caustics 7 F. Courvoisier, ICAM 2013 I 0 (y) M Input Beam y z yMyM Phase mask  y=c(z) M. V. Berry & C. Upstill, Progress in Optics XVIII (1980) "Catastrophe optics" J. F. Nye, “Natural focusing and fine structure of light”,IOP Publishing (1999).

8. Sommerfeld integral for the field at any point from distance u of M : Condition for M to be on the caustic: This provides the equation for the phase mask: Accelerating beams are caustics 8 F. Courvoisier, ICAM 2013 I 0 (y) M Input Beam y z yMyM Greenfield et al. Phys. Rev. Lett. 106 213902 (2011) L. Froehly et al, Opt. Express 19 16455 (2011) Phase mask  y=c(z)

9. Shaping in the direct space. Experimental setup Polarization direction 4-f telescope Ti:Sa, 100 fs 800 nm NA 0.8 F. Courvoisier, ICAM 2013 9 Courvoisier et al, Opt. Lett. 37, 1736 (2012)

10. Results Experimental results are in excellent agreement with predictions from wave equation propagation using the calculated phase profile. 10 F. Courvoisier, ICAM 2013 L. Froehly et al., Opt. Express 19 16455 (2011) Propagation dimension z (mm) Transverse dimension z (mm)

11. Results Multiple caustics can be used to generate Autofocusing waves 11 F. Courvoisier, ICAM 2013 N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010). I. Chremmos et al, Opt. Lett. 36, 1890 (2011). L. Froehly et al, Opt. Express 19 16455 (2011)

12. Nonparaxial regime Arbitrary nonparaxial accelerating beams 12 F. Courvoisier, ICAM 2013 Circle R = 35 µmParabolaQuartic Numeric Experiment Courvoisier et al, Opt. Lett. 37, 1736 (2012)

13. A Sommerfeld integral for the field: An optical ray corresponds to a stationary point Mapping & geometrical rays 13 F. Courvoisier, ICAM 2013 I 0 (y) Input Beam y z Greenfield et al. Phys. Rev. Lett. 106 213902 (2011) Courvoisier et al, Opt. Lett. 37, 1736 (2012) Phase mask  y=c(z) B C A f(y) y C y B y Fold catastrophe associated to an Airy function B points realize a mapping from the SLM to the caustic

14. Sommerfeld integral for the field at any point from distance u of M : Non vanishing d 3 f/dy 3 yields an Airy profile: Transverse profile 14 F. Courvoisier, ICAM 2013 I 0 (y) M Input Beam u y z Input intensity profile Local radius of curvature yMyM M u Courvoisier et al, Opt. Lett. 37, 1736 (2012) Kaminer et al, Phys. Rev. Lett. 108, 163901 (2012)

15. The parabolic Airy beam is not diffraction free in the nonparaxial regime Circular accelerating beams are nondiffracting. Transverse profile 15 F. Courvoisier, ICAM 2013 Input intensity profile Local radius of curvature M u Courvoisier et al, Opt. Lett. 37, 1736 (2012) Kaminer et al, Phys. Rev. Lett. 108, 163901 (2012)

16. More rigourous theory also supports our results

17. The temporal profile is preserved on the caustic 17 F. Courvoisier, ICAM 2013 15 fs pulse propagating along a circle The pulse is preserved in the diffraction-free domain.

18. Beams are generated from the Fourier space Fourier space shaping 18 F. Courvoisier, ICAM 2013 A/ cw, 632 nm B/ 100 fs, 800 nm D. Chremmos et al, Phys. Rev. A 85, 023828 (2012) Mathis et al, Opt. Lett., 38, 2218 (2013)

19. Beams are generated from the Fourier space Debye-Wolf integral is used to accurately describe the microscope objective and the precise mapping of the Fourier frequencies. Fourier space shaping 19 F. Courvoisier, ICAM 2013 Leutenegger et al Opt. Express 14, 011277 (2006) Mathis et al, Opt. Lett., 38, 2218 (2013) A/ cw, 632 nm B/ 100 fs, 800 nm

20. Arbitrary accelerating beams-nonparaxial regime 20 F. Courvoisier, ICAM 2013 Bending over more than 95 degrees. Numerical results are obtained from Debye integral and plane wave spectrum method. The phase masks that we can calculate analytically (circular and Weber beams) are the same as those obtained from Maxwell’s equations. Numeric Experiment Mathis et al, Opt. Lett., 38, 2218 (2013) Aleahmad et al Phys. Rev. Lett. 109, 203902 (2012). P. Zhang et al Phys. Rev. Lett. 109, 193901 (2012).

21. Arbitrary accelerating beams-nonparaxial regime An excellent agreement is then found with the target trajectories 21 F. Courvoisier, ICAM 2013 Mathis et al, Opt. Lett., 38, 2218 (2013)

22. Periodically modulated accelerating beams Each Fourier frequency corresponds to a single point on the caustic trajectory. 22 F. Courvoisier, ICAM 2013 M Mathis et al, Opt. Lett., 38, 2218 (2013)

23. Periodically modulated accelerating beams Each Fourier frequency corresponds to a single point on the caustic trajectory. An additional amplitude modulation is performed by multiplying the phase mask by a binary function and Fourier filtering of zeroth order. 23 F. Courvoisier, ICAM 2013 M phase

24. Periodically modulated accelerating beams Additional amplitude modulation allows us to generate periodic beams from arbitrary trajectories. 24 F. Courvoisier, ICAM 2013 Periodic Circular beam Periodic Weber (parabolic) beam Mathis et al, Opt. Lett., 38, 2218 (2013)

25. Spherical light 25 F. Courvoisier, ICAM 2013 Half-sphere with 50 µm radius Alonso and Bandres, Opt. Lett. 37, 5175 (2012) Mathis et al, Opt. Lett., 38, 2218 (2013)

26. Spherical light 26 F. Courvoisier, ICAM 2013 Mathis et al, Opt. Lett., 38, 2218 (2013)

27. Application-laser machining Beam profile 27 F. Courvoisier, ICAM 2013 Propagation Beam cross section 3D View @ 5% @ 50% Transverse distance (µm) Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)

28. Edge profiling – 3D processing concept 28 F. Courvoisier, ICAM 2013

29. Edge profiling – 3D processing concept 29 F. Courvoisier, ICAM 2013

30. Results on silicon 100 µm thick silicon slide initially cut squared 30 F. Courvoisier, ICAM 2013 Mathis et al, Appl. Phys. Lett. 101, 071110 (2012) R=120 µm 100 µm

31. Results on silicon – quartic profile 31 F. Courvoisier, ICAM 2013 Mathis et al, Appl. Phys. Lett. 101, 071110 (2012) R=120 µm 100 µm

32. It also works for transparent materials – diamond 32 F. Courvoisier, ICAM 2013 Mathis et al, Appl. Phys. Lett. 101, 071110 (2012) 50 µm R=120 µmR=70 µm 100 µm

33. Direct trench machining in silicon Debris distribution is highly asymmetric. 33 F. Courvoisier, ICAM 2013 Mathis et al, Appl. Phys. Lett. 101, 071110 (2012) Mathis et al, JEOS:RP, 13019 (2013)

34. Analysis in terms of light propagation direction Surface trench opening determines the depth of the trench 34 F. Courvoisier, ICAM 2013 Intensity on top surface

35. Nonparaxial Debye–Wolf wave diffraction theory allows the design and experimental generation of arbitrary nonparaxial beams over arc angles exceeding 90°. Excellent agreement is found between experimental results and target trajectories. Additional amplitude modulation yields high contrast periodic accelerating beams. 3D half-spherical fields have been reported. Conclusions 35 F. Courvoisier, ICAM 2013 We have developed a novel application of accelerating beams, ie curved edge profiling.