beams and applications to femtosecond laser micromachining F Courvoisier A Mathis L Froehly M Jacquot R Giust L Furfaro J M Dudley FEMTOST Institute University ID: 611896
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Slide1
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-ComtéBesançon, FranceSlide2
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 2013
Siviloglou et al, Phys. Rev. Lett. 99, 213901 (2007)
PropagationTransverse dimensionIntensitySlide3
High-power accelerating beams3
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 onlySlide4
Motivations4
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.Slide5
OutlineWe have developed a caustic-based approach to synthesize arbitrary accelerating beams in the nonparaxial regime.
I- Direct space shapingII-Fourier-space shapingIII-Application to femtosecond laser micromachining
5
F. Courvoisier, ICAM 2013Slide6
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).Slide7
Sommerfeld integral for the field at M :
Condition for M to beon the caustic:
Accelerating beams are caustics
7
F. Courvoisier, ICAM 2013I0(y)MInput Beam
y
z
y
M
Phase 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).Slide8
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 caustics
8F. Courvoisier, ICAM 2013I0(y)MInput Beam
y
z
y
M
Greenfield
et al.
Phys. Rev. Lett.
106
213902 (2011)
L. Froehly
et al
, Opt. Express
19
16455 (2011)
Phase mask
F
y
=c(z)Slide9
Shaping in the direct space. Experimental setup
Polarization direction
4-f telescope
Ti:Sa, 100 fs
800 nmNA 0.8F. Courvoisier, ICAM 20139Courvoisier et al, Opt. Lett. 37, 1736 (2012)Slide10
ResultsExperimental 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)Slide11
ResultsMultiple 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) Slide12
Nonparaxial regimeArbitrary nonparaxial accelerating beams
12
F. Courvoisier, ICAM 2013
Circle
R = 35 µmParabolaQuartic
Numeric
Experiment
Courvoisier
et al, Opt. Lett
. 37, 1736
(2012)Slide13
A
Sommerfeld integral
for the
field:
An optical ray corresponds to a stationary point
Mapping & geometrical rays13F. Courvoisier, ICAM 2013I0(y)
Input
Beam
y
z
Greenfield
et al.
Phys. Rev. Lett.
106
213902 (2011)
Courvoisier
et al
,
Opt
.
Lett
.
37
,
1736
(2012)
Phase mask
F
y
=c(z)
B
C
A
f
(y)
y
C
f
(y)
y
B
f
(y)
y
Fold catastrophe associated to an Airy function
B
points realize a mapping from the SLM to the causticSlide14
Sommerfeld integral for the field at any point from distance u of M :
Non vanishing d3f/dy3
yields an Airy profile:
Transverse profile
14F. Courvoisier, ICAM 2013I0(y)MInput Beam
u
y
z
Input
intensity
profile
Local radius of
curvature
y
M
M
u
Courvoisier
et al
,
Opt
.
Lett
.
37
,
1736
(
2012)
Kaminer
et al
, Phys. Rev. Lett.
108
, 163901 (2012)Slide15
The parabolic Airy beam is not diffraction free in the nonparaxial regime
Circular accelerating beams are nondiffracting.
Transverse profile
15F. Courvoisier, ICAM 2013
Input intensity profileLocal radius of curvature
Mu
Courvoisier
et al,
Opt. Lett. 37, 1736 (2012)
Kaminer et al, Phys. Rev. Lett. 108
, 163901 (2012)Slide16
More rigourous theory also supports our resultsSlide17
The temporal profile is preserved on the caustic17
F. Courvoisier, ICAM 2013
15 fs pulse propagating along a circle
The pulse is preserved in the diffraction-free domain.Slide18
Beams are generated from the Fourier space
Fourier space shaping
18
F. Courvoisier, ICAM 2013
A/ cw, 632 nmB/ 100 fs, 800 nmD. Chremmos et al, Phys. Rev. A 85, 023828 (2012)Mathis et al, Opt. Lett., 38, 2218 (2013) Slide19
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 2013Leutenegger et al Opt. Express 14, 011277 (2006)Mathis et al, Opt. Lett., 38, 2218 (2013)
A/ cw, 632 nmB/ 100 fs, 800 nmSlide20
Arbitrary accelerating beams-nonparaxial regime20
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
ExperimentMathis 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).
Slide21
Arbitrary accelerating beams-nonparaxial regimeAn excellent agreement is then found with the target trajectories
21
F. Courvoisier, ICAM 2013
Mathis et al,
Opt. Lett.,
38
, 2218 (2013) Slide22
Periodically modulated accelerating beamsEach 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) Slide23
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.
23
F. Courvoisier, ICAM 2013
MphaseSlide24
Periodically modulated accelerating beamsAdditional amplitude modulation allows us to generate periodic beams from arbitrary trajectories.
24
F. Courvoisier, ICAM 2013
Periodic Circular beam
Periodic Weber (parabolic) beamMathis et al, Opt. Lett., 38, 2218 (2013) Slide25
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) Slide26
Spherical light
26
F. Courvoisier, ICAM 2013
Mathis et al,
Opt. Lett., 38, 2218 (2013) Slide27
Application-laser machiningBeam profile
27
F. Courvoisier, ICAM 2013
Propagation
Beam cross section3D View@ 5%@ 50%
Transverse distance (µm)Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)Slide28
Edge profiling – 3D processing concept
28F. Courvoisier, ICAM 2013Slide29
Edge profiling – 3D processing concept
29F. Courvoisier, ICAM 2013Slide30
Results on silicon100 µm thick silicon slide initially cut squared
30
F. Courvoisier, ICAM 2013
Mathis
et al, Appl. Phys. Lett. 101, 071110 (2012)R=120 µm100 µmSlide31
Results on silicon – quartic profile31
F. Courvoisier, ICAM 2013
Mathis
et al, Appl. Phys. Lett. 101
, 071110 (2012)R=120 µm100 µmSlide32
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 µm100 µmSlide33
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)Slide34
Analysis in terms of light propagation direction
Surface trench opening determines the depth of the trench
34
F. Courvoisier, ICAM 2013
Intensity on
top surfaceSlide35
Nonparaxial Debye–Wolf wave diffraction theory allows the design and experimental generation of arbitrary nonparaxial beams over arc angles exceeding 90°.E
xcellent 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 2013We have developed a novel application of accelerating beams, ie curved edge profiling.