Slide 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 InstituteUniversity of Franche-ComtBesanon, France

Accelerating beamsAiry beams are invariant solutions of the paraxial wave equation.

Airy beams follow a parabolic trajectory: they are one example of accelerating beam.

2F. Courvoisier, ICAM 2013Siviloglou et al, Phys. Rev. Lett. 99, 213901 (2007)

Propagation

Transverse dimensionIntensityHigh-power accelerating beams3F. 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 onlyMotivations4F. Courvoisier, ICAM 2013Aside 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.

OutlineWe 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 micromachining5F. Courvoisier, ICAM 2013Accelerating 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.

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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).Sommerfeld integral for the field at M :

Condition for M to beon the caustic:

Accelerating beams are caustics7F. Courvoisier, ICAM 2013I0(y)MInput Beam

yzyMPhase mask F

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).Sommerfeld integral for the field at any point from distance u of M :

Condition for M to beon the caustic:

This provides the equation for the phase mask:

Accelerating beams are caustics8F. Courvoisier, ICAM 2013I0(y)MInput Beam

yzyMGreenfield et al. Phys. Rev. Lett. 106 213902 (2011)L. Froehly et al, Opt. Express 19 16455 (2011) Phase mask F

y=c(z)Shaping in the direct space. Experimental setup

Polarization direction4-f telescopeTi:Sa, 100 fs800 nmNA 0.8F. Courvoisier, ICAM 20139Courvoisier et al, Opt. Lett. 37, 1736 (2012)ResultsExperimental results are in excellent agreement with predictions from wave equation propagation using the calculated phase profile.

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L. Froehly et al., Opt. Express 19 16455 (2011) Propagation dimension z (mm)Transverse dimension z (mm)ResultsMultiple caustics can be used to generate Autofocusing waves11F. 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)

Nonparaxial regimeArbitrary nonparaxial accelerating beams12F. Courvoisier, ICAM 2013

Circle R = 35 mParabolaQuarticNumeric

ExperimentCourvoisier et al, Opt. Lett. 37, 1736 (2012)

A

Sommerfeld integral for the field:

An optical ray corresponds to a stationary point

Mapping & geometrical rays13F. Courvoisier, ICAM 2013I0(y)Input Beam

yzGreenfield et al. Phys. Rev. Lett. 106 213902 (2011)Courvoisier et al, Opt. Lett. 37, 1736 (2012)Phase mask F

y=c(z)BCAf(y)y

Cf(y)y

Bf(y)yFold catastrophe associated to an Airy functionB points realize a mapping from the SLM to the caustic

Sommerfeld integral for the field at any point from distance u of M :

Non vanishing d3f/dy3yields an Airy profile:

Transverse profile14F. Courvoisier, ICAM 2013I0(y)MInput Beam

uyzInput intensity profileLocal radius of curvature

yM

MuCourvoisier et al, Opt. Lett. 37, 1736 (2012)Kaminer et al, Phys. Rev. Lett. 108, 163901 (2012)The parabolic Airy beam is not diffraction free in the nonparaxial regime

Circular accelerating beams are nondiffracting.

Transverse profile15F. Courvoisier, ICAM 2013Input intensity profileLocal radius of curvature

MuCourvoisier et al, Opt. Lett. 37, 1736 (2012)Kaminer et al, Phys. Rev. Lett. 108, 163901 (2012)More rigourous theory also supports our results

The temporal profile is preserved on the caustic17F. Courvoisier, ICAM 2013

15 fs pulse propagating along a circle

The pulse is preserved in the diffraction-free domain.

Beams are generated from the Fourier space

Fourier space shaping

18F. Courvoisier, ICAM 2013A/ cw, 632 nmB/ 100 fs, 800 nmD. Chremmos et al, Phys. Rev. A 85, 023828 (2012)Mathis et al, Opt. Lett., 38, 2218 (2013)

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 shaping19F. Courvoisier, ICAM 2013Leutenegger et al Opt. Express 14, 011277 (2006)Mathis et al, Opt. Lett., 38, 2218 (2013)

A/ cw, 632 nmB/ 100 fs, 800 nmArbitrary accelerating beams-nonparaxial regime20F. Courvoisier, ICAM 2013Bending 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 Maxwells equations.

NumericExperimentMathis 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). Arbitrary accelerating beams-nonparaxial regimeAn excellent agreement is then found with the target trajectories21F. Courvoisier, ICAM 2013

Mathis et al, Opt. Lett., 38, 2218 (2013) Periodically modulated accelerating beamsEach Fourier frequency corresponds to a single point on the caustic trajectory.

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MMathis et al, Opt. Lett., 38, 2218 (2013) Periodically modulated accelerating beamsEach 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.23F. Courvoisier, ICAM 2013

M

phasePeriodically modulated accelerating beamsAdditional amplitude modulation allows us to generate periodic beams from arbitrary trajectories.

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Periodic Circular beamPeriodic Weber (parabolic) beamMathis et al, Opt. Lett., 38, 2218 (2013) Spherical light

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

26F. Courvoisier, ICAM 2013Mathis et al, Opt. Lett., 38, 2218 (2013) Application-laser machiningBeam profile 27F. Courvoisier, ICAM 2013Propagation

Beam cross section

3D View@ 5%@ 50%Transverse distance (m)Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)Edge profiling 3D processing concept28F. Courvoisier, ICAM 2013

Edge profiling 3D processing concept29F. Courvoisier, ICAM 2013

Results on silicon100 m thick silicon slide initially cut squared30F. Courvoisier, ICAM 2013

Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)R=120 m100 mResults on silicon quartic profile31F. Courvoisier, ICAM 2013Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)

R=120 m100 mIt also works for transparent materials diamond32F. Courvoisier, ICAM 2013

Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)50 mR=120 mR=70 m100 mDirect trench machining in siliconDebris distribution is highly asymmetric.

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Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)Mathis et al, JEOS:RP , 13019 (2013)

Analysis in terms of light propagation directionSurface trench opening determines the depth of the trench34F. Courvoisier, ICAM 2013

Intensity on top surfaceNonparaxial DebyeWolf 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.

Conclusions35F. Courvoisier, ICAM 2013We have developed a novel application of accelerating beams, ie curved edge profiling.