Non- paraxiality and femtosecond optics

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Non- paraxiality andfemtosecond optics
Institute of Electronics, Bulgarian Academy of
Sciences Laboratory of Nonlinear and Fiber
Optics

• Lubomir M. Kovachev

Nonlinear physics. Theory and Experiment. V 2008
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Paraxial optics of a laser beam
Solution in (x, y, z) space
Initial conditions - Gaussian beam
Analytical solution for initial Gaussian beam
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zzdiff
z0
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Numerical solution using FFT technique. Paraxial
optics. Laser beam on 800 nm (zdiffk0r02 7.85
cm r0 100µm)
Initial condition
z0
z1/3 z2/3
z1zdiff7.85 cm
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Phase modulated (by lens) Gaussian beam
a-radius of the lens, f- focus distance d0-
thickness in the centrum Seff- effective area of
the laser spot
a1,27 cm Seff0.2
f200 cm
z0
z1/3 z2/3
z1zdiff
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Paraxial optics of a laser pulse. From ns to
200-300 ps time duration
Dimensionless analyze
In air, gases and metal vapors
t0gt100-200 fs
ßltlt1 - Negligible dispersion.
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Nonlinear paraxial optics
Nonlinear paraxial equation
Initial conditions
1) nonlinear regime near to critical ? 1.2
2) nonlinear regime ?1.7
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• 1) nonlinear regime near to critical ? 1.2

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2) Nonlinear regime ?1.7
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References
Non-collapsed regime of propagation of fsec pulses
1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier,
and G. Mourou, "Self-channeling of
high-peak-power femtosecond laser pulses in
Air, Opt. Lett. 20, 73-75, 1995. 2. E. T.
Nibbering, P. F. Curley, G. Grillon, B. S. Prade,
M. A. Franco, F. Salin, and A. Mysyrowich,
"Conical emission from self-guided femtosecond
pulses", Opt. Lett, 21, 62, 1996. 3. A. Brodeur,
C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G.
Kosareva, and V. P. Kandidov, "Moving focus in
the propagation of ultrashort laser pulses in
air", Opt. Lett., 22, 304-306, 1997. 4. L. Wöste,
C. Wedekind, H. Wille, P. Rairroux, B. Stein, S.
Nikolov, C. Werner, S. Niedermeier, F.
Ronnenberger, H. Schillinger, and R. Sauerbry,
"Femtosecond Atmospheric Lamp", Laser und
Optoelektronik 29, 51 , 1997. 5. H. R. Lange, G.
Grillon, J.F. Ripoche, M. A. Franco, B.
Lamouroux, B. S. Prade, A. Mysyrowicz, E. T.
Nibbering, and A. Chiron, "Anomalous long-range
propagation of femtosecond laser pulses through
air moving focus or pulse self-guiding?", Opt.
lett. 23, 120-122, 1998.
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Nonlinear pulse propagation of fsec optical
pulses Three basic new experimental effects
1. Spectral, time and spatial modulation
2. Arrest of the collapse
3. Self-channeling
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Extension of the paraxial model for ultra short
pulses and single-cycle pulses ?
Expectations Self-focusing to be compensated by
plasma induced defocusing or high order
nonlinear terms - Periodical fluctuation of the
profile.

• Experiment
• No fluctuations - Stable profile
• 2) Self- guiding without ionization

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Arrest of the collapse and self-channeling in
absence of ionization
G. Méchian, C. D'Amico, Y. -B. André, S.
Tzortzakis, M. Franco, B. Prade, A. Mysyrowicz,
A. Couarion, E. Salmon, R. Sauerbrey, "Range of
plasma filaments created in air by a
multi-terawatt femtosecond laser", Opt. Comm.
247, 171, 2005. G. Méchian, A. Couarion, Y. -B.
André, C. D'Amico, M. Franco, B. Prade, S.
Tzortzakis, A. Mysyrowicz, A. Couarion, R.
Sauerbrey, "Long range self-channeling of
infrared laser pulse in air a new propagation
regime without ionization", Appl. Phys. B 79,
379, 2004.
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Self-Channeling of Light in Linear Regime
?? (Femtosecond pulses)
C. Ruiz, J. San Roman, C. Mendez, V.Diaz,
L.Plaja, I.Arias, and L.Roso, Observation of
Spontaneous Self-Channeling of Light in Air below
the Collapse Threshold, Phys. Rev. Lett. 95,
053905, 2005.

• Saving the Spatio -Temporal Paraxial Model
• linear and nonlinear X waves??
• X-waves - J0 Bessel functions infinite energy
• 2) X-waves - Delta functions in (kx, ky) space.

• Experiment
• Self-Channeling is observed for spectrally -
  limited (regular) pulses
• 2. Wave type diffraction for single- cycle
  pulses.

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Something happens in FS region?? Wanted for
new model to explain
1. Relative Self -Guiding in Linear Regime.
2. Wave type diffraction for single - cycle
pulses. Optical cycle 2 fs pulses
with 4-8 fs duration
Three basic new nonlinear experimentally
confirmed effects
3. Spectral, time and spatial modulation
4. Arrest of the collapse
5. Self-channeling
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Non-paraxial model
1. L. M. Kovachev, "Optical Vortices in
dispersive nonlinear Kerr-type media", Int.
J. of Math. and Math. Sc. (IJMMS) 18, 949
(2004). 2. L. M. Kovachev and L. M. Ivanov,
"Vortex solitons in dispersive nonlinear Kerr
type media", Nonlinear Optics Applications,
Editors M. A. Karpiez, A. D. Boardman, G. I.
Stegeman, Proc. of SPIE. 5949, 594907, 2005. 3.
L. M. Kovachev, L. I. Pavlov, L. M. Ivanov and D.
Y. Dakova, Optical filaments and optical
bullets in dispersive nonlinear media, Journal
of Russian Laser Research 27, 185- 203, 2006 4.
L.M.Kovachev, Collapse arrest and wave-guiding
of femtosecond pulses, Optics Express, Vol. 15,
Issue 16, pp. 10318-10323 (August 2007). 5. L.
M. Kovachev, Beyond spatio - temporal model in
the femtosecond optics, Journal of Mod. Optics
(2008), in press.
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Introducing the amplitude function of the
electrical field
and the amplitude function of the Fourier
presentation of the electrical field
The next nonlinear equation of the amplitudes is
obtained
Convergence of the series I. Number of cycles
II. Media density
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SVEA in laboratory coordinate frame
or
V. Karpman, M.Jain and N. Tzoar, D.
Christodoulides and R.Joseph, N. Akhmediev and A.
Ankewich, Boyd
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SVEA in Galilean coordinate frames
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Constants
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Dimensionless parameters
Determine number of cycles under envelope with
precise 2p
1.
Determine relation between transverse and
longitudinal initial profile of the pulse
2.
Determine the relation between diffraction and
dispersion length
3.
4.
Nonlinear constant
Constant connected with nonlinear addition to
group velocity
5.
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SVEA in dimensionless coordinates
Laboratory
Galilean
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Linear Amplitude equation in media with
dispersion (SVEA)
Laboratory
Galilean
Linear Amplitude Equation in Vacuum (VLAE)
In air
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Laboratory frame
Galilean frame
Solutions in kx ky kz space
where
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Fundamental solutions of the linear SWEA
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Fundamental linear solutions of SVEA for media
with dispersion
Fundamental solutions of VLAE for media without
dispersion
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Evolution of long pulses in air (linear
regime, 260 ps and 43 ps)
Light source form Tisapphire laser, waist on
level e-1
1) 260 ps ad21 ß12.1X10-5
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43 ps (long pulse) ad26 ß12.1X10-5
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Light Bullet (330 fs) a785 d21 ß12.1X10-5
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Light Disk (33 fs) a78,5 d2100 ß12.1X10-5
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Analytical solution of SVEA (when ß1ltlt1)and VLAE
for initial Gaussian LB (d1) (Lab coordinate)
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Analytical solution of SVEA (when ß1ltlt1)and VLAE
for initial Gaussian LB (d1)
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Shaping of LB on one zdifpulsek02r4/z0 length
Gaussian shape of the solution when t0. The
surface A(x,y0,z t0) is plotted.
Deformation of the Gaussian bullet with 330 fs
time duration obtained from exact solution of
VLAE. The surface A(x,y0,z t785) is
plotted. The waist grows by factor sqrt(2) over
normalized time-distance tz785, while the
amplitude decreases with A1/sqrt(2).
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Analytical solution of SVEA (when ß1ltlt1)and VLAE
for initial Gaussian LB (d1) (Galilean
coordinate)
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Analytical solution of SVEA (when ß1ltlt1)and VLAE
for initial Gaussian LB (d1) (Galilean
coordinate)
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Analytical solution of SVEA (when ß1ltlt1)and VLAE
for initial Gaussian LB (d1) (Galilean
coordinate)
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Fig. 5. Shaping of Gaussian pulse obtained from
exact solution of VLAE in Galilean coordinates.
The surface A(x y 0 z0 t 785) is
plotted. The spot grows by factor sqrt(2) over
the same normalized time t 785 while the pulse
remains initial position z 0, as it can be
expected from Galilean invariance.
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Linear Amplitude equation in media with
dispersion (SVEA).
Laboratory
Galilean
Linear Amplitude Equation in Vacuum (VLAE).
Analytical (Galilean invariant ) solution of
3D1 Wave equation.
In air
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2. Comparison between the solutions of Wave
Equation and SVEA in single-cycle regime
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Evolution of Gaussian amplitudude envelope of
the electrical field in dynamics of wave
equation. Single cycle regime
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t3Pi
T0
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Analytical solution of SVEA (when ß1ltlt1) and VLAE
for initial Gaussian LB in single-cycle regime
(d1 and a2).
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Conclusion (linear regime)

• Fundamental solutions k space of SVEA and
• VLAE are obtained

2. Analytical non-paraxial solution for initial
Gaussian LB.
3. Relative Self Guiding for LB and LD (agtgt1) in
linear regime.
4. Wave type diffraction for single - cycle
pulses (a1-3) .
5. New formula for diffraction length of optical
pulses is confirmed from analytical solution
zdifpulsek02W4/z0
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Nonlinear paraxial optics
Nonlinear paraxial equation
Initial conditions
1) nonlinear regime near to critical ? 1.2
2) nonlinear regime ?1.7
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• 1) nonlinear regime near to critical ? 1.2

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2) Nonlinear regime ?1.7
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Nonlinear non-parxial regime.
Laboratory frames
Galilean
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Dynamics of long optical pulses governed by the
non - paraxial equation
Nonlinear regime ?2
(x,y plane) of long Gaussian pulse. Regime
similar to laser beam.
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Dynamics of long optical pulses governed by the
non - paraxial equation
Nonlinear regime ?2
Longitudinal x, z plane of the same long Gaussian
pulse. Large longitudinal spatial and spectral
modulation of the pulse is observed.
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1/ Optical bullet in nonlinear regime ?1.4.
Arrest of the collapse.
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2/ OPTICAL DISK in nonlinear regime ?2.25
NONLINEAR WAVEGUIDING.
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Conclusion - Nonlinear regime
1/ Long optical pulse The self-focusing regime
is similar to the regime of laser beam and the
collapse distance is equal to that of a cw wave.
The new result here is that in this regime it is
possible to obtain longitudinal spatial
modulation and spectral enlargement of long
pulse. 2/ Light bullet We observe significant
enlargement of the collapse distance (collapse
arrest) and weak self-focusing near the critical
power without pedestal. 3/ Optical pulse with
small longitudinal and large transverse size
(light disk) nonlinear wave-guiding.
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Something happens in FS region?? Wanted for new
model to explain
v 1. Relative Self Guiding in Linear Regime of
light disk.
v 2. Wave type diffraction for single - cycle
pulses.
Three basic new nonlinear effects
v 3. Spectral, time and spatial modulation of
long pulse
v 4. Arrest of the collapse of light bullets
v 5. Self-channeling of light disk

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??????????? - 800 nm Ti-Sapphire laser 30 fs
100 µm ???? ???????- 1.109 W ?????? ???????
?? ??????? 1X1013 W/cm2 2-3 Pkr
H. Hasegawa, L.I. Pavlov, ....
z0
z12 zdiff