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Arbitrary nonparaxial accelerating beams and applications to femtosecond laser micromachining

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. Accelerating beams.

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Arbitrary nonparaxial accelerating beams and applications to femtosecond laser micromachining

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  1. Arbitrary nonparaxial accelerating beamsand applications to femtosecond laser micromachining F. Courvoisier, A. Mathis, L. Froehly, M. Jacquot, R. Giust, L. Furfaro, J. M. Dudley FEMTO-ST Institute Universityof 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. Propagation Intensity Transverse dimension Siviloglouet al, Phys. Rev. Lett. 99, 213901 (2007) F. Courvoisier, ICAM 2013

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

  4. Motivations • 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. F. Courvoisier, ICAM 2013

  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 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. 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). F. Courvoisier, ICAM 2013

  7. Accelerating beams are caustics • Sommerfeld integral for the field at M: • Condition for M to be • on the caustic: y y=c(z) Phase mask F M Input Beam I0(y) yM 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). F. Courvoisier, ICAM 2013

  8. Accelerating beams are caustics • 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: y y=c(z) Phase mask F M Input Beam I0(y) yM z Greenfieldet al. Phys. Rev. Lett. 106 213902 (2011) L. Froehly et al, Opt. Express 19 16455 (2011) F. Courvoisier, ICAM 2013

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

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

  11. Results • Multiple caustics can be used to generate Autofocusing waves 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) F. Courvoisier, ICAM 2013

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

  13. Mapping & geometrical rays • Sommerfeld integral for the field: • An optical ray • corresponds to a • stationary point y y=c(z) Phase mask F C B Input Beam A I0(y) z Fold catastrophe associated to an Airy function B points realize a mapping from the SLM to the caustic f(y) f(y) f(y) A B C y y y Greenfieldet al. Phys. Rev. Lett. 106 213902 (2011) Courvoisier et al, Opt. Lett. 37,1736(2012) F. Courvoisier, ICAM 2013

  14. Transverse profile • Sommerfeld integral for the field at any point from distance u of M: • Non vanishing d3f/dy3 • yields an Airy profile: y M Input Beam u I0(y) u M yM z Input intensity profile Local radius of curvature Courvoisier et al, Opt. Lett. 37,1736(2012) Kaminer et al, Phys. Rev. Lett. 108, 163901 (2012) F. Courvoisier, ICAM 2013

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

  16. More rigourous theory also supports our results

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

  18. Fourier space shaping • Beams are generated from the Fourier space 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) F. Courvoisier, ICAM 2013

  19. Fourier space shaping • 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. A/ cw, 632 nm B/ 100 fs, 800 nm • Leutenegger et al Opt. Express 14, 011277 (2006) • Mathis et al, Opt. Lett., 38, 2218 (2013) F. Courvoisier, ICAM 2013

  20. Arbitrary accelerating beams-nonparaxial regime • Bending over more than95 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. Experiment Numeric • Mathis et al, Opt. Lett., 38, 2218 (2013) • Aleahmad et al Phys. Rev. Lett. 109, 203902 (2012). • P. Zhang et alPhys. Rev. Lett. 109, 193901 (2012). F. Courvoisier, ICAM 2013

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

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

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

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

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

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

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

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

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

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

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

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

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

  35. Conclusions • 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. We have developed a novel application of accelerating beams, ie curved edge profiling. F. Courvoisier, ICAM 2013

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