Transcript 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 Institute
University of Franche-Comté
Besançon, France
Accelerating beams
Airy beams are invariant solutions of the paraxial wave equation.
Propagation
Intensity
Transverse dimension
Siviloglou et al, Phys. Rev. Lett. 99, 213901 (2007)
Airy beams follow a parabolic trajectory: they are one example of
accelerating beam.
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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)
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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.
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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
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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).
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Accelerating beams are caustics
Sommerfeld integral for the field at M :
y
Phase mask F
y=c(z)
M
Condition for M to be
on the caustic:
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).
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Accelerating beams are caustics
Sommerfeld integral for the field at any point from distance u of M :
y
Phase mask F
y=c(z)
M
Condition for M to be
on the caustic:
Input
Beam
I0(y)
yM
z
This provides the equation for
the phase mask:
Greenfield et al. Phys. Rev. Lett. 106 213902 (2011)
L. Froehly et al, Opt. Express 19 16455 (2011)
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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)
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Results
Transverse dimension z (mm)
Experimental results are in excellent agreement with predictions from
wave equation propagation using the calculated phase profile.
Propagation dimension z (mm)
L. Froehly et al., Opt. Express 19 16455 (2011)
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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)
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Nonparaxial regime
Arbitrary nonparaxial accelerating beams
Circle R = 35 µm
Parabola
Quartic
Numeric
Experiment
Courvoisier et al, Opt. Lett. 37, 1736 (2012)
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Mapping & geometrical rays
Sommerfeld integral for the field:
y
Phase mask F
A
Fold catastrophe associated to an Airy function
B points realize a mapping from the SLM to the caustic
f(y)
f(y)
A
y
y=c(z)
B
Input
Beam
I0(y)
An optical ray
corresponds to a
stationary point
C
z
f(y)
B
y
C
y
Greenfield et al. Phys. Rev. Lett. 106 213902 (2011)
Courvoisier et al, Opt. Lett. 37, 1736 (2012)
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Transverse profile
Sommerfeld integral for the field at any point from distance u of M :
y
M
Non vanishing d3f/dy3
yields an Airy profile:
Input
Beam
I0(y)
u
yM
z
M
Input intensity profile
u
Local radius of curvature
Courvoisier et al, Opt. Lett. 37, 1736 (2012)
Kaminer et al, Phys. Rev. Lett. 108, 163901 (2012)
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Transverse profile
The parabolic Airy beam is not diffraction free in the nonparaxial regime
Circular accelerating beams are nondiffracting.
M
Input intensity profile
u
Local radius of curvature
Courvoisier et al, Opt. Lett. 37, 1736 (2012)
Kaminer et al, Phys. Rev. Lett. 108, 163901 (2012)
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More rigourous theory also supports our results
The temporal profile is preserved on the caustic
15 fs pulse propagating along a circle
The pulse is preserved in the diffraction-free domain.
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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)
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Fourier space shaping
Beams are generated from the Fourier space
A/ cw, 632 nm
B/ 100 fs, 800 nm
Debye-Wolf integral is used to accurately
describe the microscope objective and the
precise mapping of the Fourier frequencies.
Leutenegger et al Opt. Express 14, 011277 (2006)
Mathis et al, Opt. Lett., 38, 2218 (2013)
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Arbitrary accelerating beams-nonparaxial regime
Bending over more than 95 degrees.
Numerical results are obtained from Debye integral and plane wave
spectrum method.
Experiment
Numeric
The phase masks that we can calculate analytically (circular and Weber
beams) are the same as those obtained from Maxwell’s equations.
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).
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Arbitrary accelerating beams-nonparaxial regime
An excellent agreement is then found with the target trajectories
Mathis et al, Opt. Lett., 38, 2218 (2013)
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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)
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Periodically modulated accelerating beams
Each Fourier frequency corresponds to a single point on the caustic
trajectory.
M
phase
An additional amplitude modulation is performed by multiplying the
phase mask by a binary function and Fourier filtering of zeroth order.
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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)
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Spherical light
Half-sphere with 50 µm radius
Alonso and Bandres, Opt. Lett. 37, 5175 (2012)
Mathis et al, Opt. Lett., 38, 2218 (2013)
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Spherical light
Mathis et al, Opt. Lett., 38, 2218 (2013)
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Application-laser machining
Beam profile
@ 5%
Propagation
@ 50%
3D View
Beam cross section
Transverse
distance (µm)
Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)
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Edge profiling – 3D processing concept
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Edge profiling – 3D processing concept
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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)
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Results on silicon – quartic profile
100 µm
R=120 µm
Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)
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It also works for transparent materials – diamond
100 µm
50 µm
R=120 µm
R=70 µm
Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)
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Direct trench machining in silicon
Mathis et al, Appl. Phys. Lett. 101, 071110 (2012)
Mathis et al, JEOS:RP , 13019 (2013)
Debris distribution is highly asymmetric.
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Analysis in terms of light propagation direction
Surface trench opening determines the depth of the trench
Intensity on
top surface
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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.
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