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Institute of Electronics, Bulgarian Academy of Sciences Laboratory of Nonlinear and Fiber Optics. Non- paraxiality and femtosecond optics. Lubomir M. Kovachev. Nonlinear physics. Theory and Experiment. V 2008. Paraxial optics of a laser beam. Solution in (x, y, z) space. - PowerPoint PPT Presentation
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Non- paraxiality and
femtosecond optics
Lubomir M. Kovachev
Institute of Electronics, Bulgarian Academy of Sciences Laboratory of Nonlinear and Fiber Optics
Nonlinear physics. Theory and Experiment. V 2008
......2
1
0
Akz
Ai
Paraxial optics of a laser beam
Solution in (x, y, z) space
yxyxyx
yx
yx dkdkykxikkkkikkAzyxA )exp()exp()2/)(exp()0,,(),,( 022
,
Analytical solution for initial Gaussian beam
Initial conditions - Gaussian beam
200
20
220 );2/)(exp()0,,( rkzryxAyxA diff
diffdiff zizr
yx
zizAzyxA
/12
)(exp
/1
1
4),,(
20
22
0
2
220
22
220
42
/12
)(exp
/1
1
16 diffdiff zzr
yx
zzAA
z=0z=zdiff
Numerical solution using FFT technique. Paraxial optics. Laser beam on 800 nm (zdiff=k0r0
2= 7.85 cm; r0= 100µm)
Initial condition
z=0 z=1/3 z=2/3 z=1;zdiff=7.85 cm
)2/)(exp()0,,(),,( 0221 kkkikkAFzyxA yxyx
);2/)(exp()0,,( 220 yxAyxA
z=0 z=1/3 z=2/3 z=1=z diff
Phase modulated (by lens) Gaussian beam
)),(2exp(),()0,,( 00 yxidiyxAyxA )()(/(),( 222 yxafSyx eff
a-radius of the lens, f- focus distanced0- thickness in the centrum
Seff- effective area of the laser spot
nm800
f=200 cm
a=1,27 cmSeff=0.2
Paraxial optics of a laser pulse. From ns to 200-300 ps time duration
Dimensionless analyze:
;'xrx 'yry '0zzz
......'' 2
2
t
AA
z
Ai
dissp
diff
z
z
kt
rk
"/20
20
02.0
1085.7,/103";1~;330~ 140
2310
cmkcmsekkmmrfst
In air, gases and metal vapors t0>100-200 fs ; β<<1 - Negligible dispersion.
....2
"
2
12
2
0
termsnlt
AkA
kz
Ai
1/ 02
0 zrk2
00 rkz
Nonlinear paraxial optics
Nonlinear paraxial equation:
;xAA x
Initial conditions:
1) nonlinear regime near to critical γ~ 1.2
2) nonlinear regime γ=1.7
AAAz
Ai
22
)2/2/exp()0,,( 22 yxzyxAx
• 1) nonlinear regime near to critical γ~ 1.2
2) Nonlinear regime γ=1.7
References
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.
Non-collapsed regime of propagation of fsec pulses
Nonlinear pulse propagation of fsec optical pulsesThree basic new experimental effects
1. Spectral, time and spatial modulation
2. Arrest of the collapse
3. Self-channeling
Extension of the paraxial model for ultra short pulses and single-cycle pulses ?
...
...2
"
2
12
2
0
ionization
termsnlt
AkA
kz
Ai
Expectations:Self-focusing to be compensated by plasma induced defocusing or high order nonlinear terms - Periodical fluctuation of the profile.
Experiment: 1) No fluctuations - Stable profile2) Self- guiding without
ionization
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.
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.
Self-Channeling of Light in Linear Regime ??(Femtosecond pulses)
Saving the Spatio -Temporal Paraxial Model – linear and nonlinear X waves??1) X-waves - J0 Bessel functions – infinite energy2) X-waves - Delta functions in (kx, ky) space.
Experiment: 1. Self-Channeling is observed for spectrally - limited (regular) pulses 2. “Wave type” diffraction for single- cycle pulses.
)2/)(exp()0,,(),,( 0221 kkkikkAFzyxA yxyx
Something happens in FS region??
Wanted for new model to explain:
3. Spectral, time and spatial modulation
4. Arrest of the collapse
5. Self-channeling
Three basic new nonlinear experimentally confirmed effects:
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
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.
Non-paraxial model
),( trE
),(ˆ rE
Introducing the amplitude function of the electrical field
and the amplitude function of the Fourier presentation of the electrical field
tzkitrAtrE 00exp),(),(
tirAtzkitirE 0000 exp),(ˆexp)exp(),(ˆ
The next nonlinear equation of the amplitudes is obtained:
dtirAAkktrAkz
trAiktrA nl ))(exp(),(ˆˆ)()(),(
),(2),( 00
2222
00
Convergence of the series: I. Number of cycles; II. Media density:
..)()(2
1)()()()( 2
0"2
0'2
022 kkkk
"''2)( "2 kkkk vkkkk /1';......'2)'( 2
SVEA in laboratory coordinate frame
AAvnk
t
A
vkk
vA
k
v
t
AAnvkn
z
Av
t
Ai
220
2
2
20
"0
0
2
202 2
1
222
or
V. Karpman, M.Jain and N. Tzoar, D. Christodoulides and R.Joseph,N. Akhmediev and A. Ankewich, Boyd……
AAvnk
t
Avk
t
A
vz
AA
k
v
t
AAnvkn
z
Av
t
Ai
220
2
2"0
2
2
22
2
0
2
202 22
1
22
SVEA in Galilean coordinate frames
AAvnk
zt
Av
t
A
vkk
v
z
AkvA
k
v
t
Ai
220
2
2
2
20
"02
23
0 2''2
'
1
2'2
"
2'
AAvnk
t
Avk
t
A
vz
AA
k
v
z
Av
t
Ai
220
2
2"0
2
2
22
2
0 22
1
2
Constants
"'":;"';";";";" 00000 ttttttzzzzzzyryxrxAAA
20 rkzdiffr "
2
0
k
tzdis
;00 zk ;2
0
22
z
r ;dis
dif
z
z
;2
1 2
02
22
0 Anrk
2
0220
2
2
01 2
1
22
1An
nvknA
00 vtz
Dimensionless parameters
0 0k z 1.
Determine number of cycles under envelope with precise 2π 1
22
20
r
z 2. Determine relation between transverse and
longitudinal initial profile of the pulse
dispdiff zz /3.Determine the relation between diffraction and dispersion length
0 0z vt2
1 "2
01 kvk
22
0 2 0
1
2k r n A
��������������4.
5.2
2 21 2 0
1
2n A
��������������
Nonlinear constant
Constant connected with nonlinear addition to group velocity
SVEA in dimensionless coordinates
AAt
A
t
A
z
AA
t
AA
z
A
t
Ai
2
2
2
2
2
2
22
2
1
22
AAz
A
zt
A
t
AA
z
AA
t
AA
t
Ai
2
2
22
2
22
22
1
2
'''2
''''2
Laboratory
Galilean
vtzztt ';' domainpsandns ......300200......;..12 domainfs....300200;...12
domainfs....15020;...12
Linear Amplitude equation in media with dispersion (SVEA)
2
2
12
2
2
2
2222
t
A
t
A
z
AA
z
A
t
Ai
2
22
1
2
2
2
122
'''2
'1
'2
z
A
zt
A
t
AA
t
Ai
Laboratory:
Galilean:
01
2
2
2
t
E
cE
tzkitrAtrE 00exp),(),(
2
2
2
2222
t
A
z
AA
z
A
t
Ai
Linear Amplitude Equation in Vacuum (VLAE)
2
2
200 2
1
2
11
t
A
ckA
kz
Ac
t
A
ci
In air5"2
01 10 kvk042
02 / zrkzz beam
diffpulsediff
Laboratory frame
0ˆ
1ˆ2ˆ
22
2
1222222
t
AAkkkk
t
Ai L
LzzyxL
Galilean frame
0ˆ
1ˆˆ
22
2
122222
t
AAkkk
t
Aki G
GzyxG
z
tk
ikkkAtkkkA zyxLzyxL1
2
22
11 1
ˆ
11exp)0,,,(ˆ),,,(ˆ
exp)0,,,(ˆ),,,(ˆzyxGzyxG kkkAtkkkA
tkkkkk
i zyxzz
12
21
2222
11 111
Solutions in kx ky kz space :
where zzyx kkkkk 2ˆ 22222
)',',,()',,,(ˆ tzyxAFtkkkA zyxG
),,,(),,,(ˆ tzyxAFtkkkA zyxL
Fundamental solutions of the linear SWEA
t
kikkkAFtzyxA zyxLL
12
2
21
2
1
1
1
ˆ
)1(1exp)0,,,(ˆ),,,(
t
kkkkkikkkAFtzyxA zyxzz
zyxGG1
2
21
2222
11
1
111exp)0,,,(ˆ),,,(
zzyx kkkkk 2ˆ 22222
t
kikkkAFtzyxA zyxLL
12
2
21
2
1
1
1
ˆ
)1(1exp)0,,,(ˆ),,,(
Fundamental linear solutions of SVEA for media with dispersion:
t
kkkkkikkkAFtzyxA zyxzz
zyxGG1
2
21
2222
11
1
111exp)0,,,(ˆ),,,(
Fundamental solutions of VLAE for media without dispersion:
1 1 2 2 2ˆ ˆ0, , , exp /L L x y zA F A k k k F i k t
1 1 2 2 2ˆ ˆ0, , , exp /G L x y z zA F A k k k F i k k t
5"201 10 kvk zzyx kkkkk 2ˆ 22222
Evolution of long pulses in air (linear regime, 260 ps and 43 ps)
Light source form Ti:sapphire laser, waist on level e-1 : mr 100
cmkcmk air /sec103;10.85.7 231"140
1) 260 ps: αδ2=1; β1=2.1X10-5
zzttv '';.....1
1'~1' zztt
43 ps (long pulse) αδ2=6; β1=2.1X10-5
Light Bullet (330 fs) α=785; δ2=1; β1=2.1X10-5
Light Disk (33 fs) α=78,5; δ2=100; β1=2.1X10-5
Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Lab coordinate)
.)exp()exp()exp(
exp
2/exp2
1),,,(
222
2223
zyxzyx
zyx
zyx
dkdkdkzikyikxik
tkkki
kkktzyxAx
zz kk̂
.ˆ)(ˆexp()exp()exp(
ˆexp
2/ˆexp
2exp
2
1),,,(
222
222
2
3
zyxzyx
zyx
zyx
kddkdkizkiyikxik
tkkki
kkk
ztitzyxAx
22222 2)(ˆ zirizyxr
.ˆsinexp
)2/exp(ˆ1
2exp
2
1),,,(
0
22
3
rrr
rr
dkkrtki
kkr
ztitzyxAx
Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1)
rti
erfcriitrti
rti
erfcriitrti
ztir
itzyxA
ˆ2
2ˆ
2
1expˆ
ˆ2
2ˆ
2
1expˆ
2exp
ˆ2),,,(
2
2
2
22 2ˆ zirr
Gaussian shape of the solution when t=0.The surface |A(x,y=0,z; t=0) | is plotted.
Deformation of the Gaussian bullet with 330 fs time duration obtained from exact solution of VLAE. The surface |A(x,y=0,z; t=785) | is plotted. The waist grows by factor sqrt(2) over normalized time-distance t=z=785, while the amplitude decreases with A=1/sqrt(2).
Shaping of LB on one zdifpulse=k0
2r4/z0 length
785
785
zt
.)exp()exp()exp(
)(exp
2/exp2
1),,,(
2222
2223
zyxzyx
zyxz
zyx
dkdkdkzikyikxik
tkkkki
kkktzyxAx
Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)
zz kk̂
.ˆ)(ˆexp()exp()exp(
ˆexp
2/ˆexp
2exp
2
1),,,(
222
222
2
3
zyxzyx
zyx
zyx
kddkdkitzkiyikxik
tkkki
kkk
ztitzyxAx
222 )(~ itzyxr
Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)
rti
erfcriitrti
rti
erfcriitrti
ztir
itzyxA
~2
2~2
1exp~
~2
2~2
1exp~
2exp~2
),,,(
2
2
2
Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)
222 )(~ itzyxr
Fig. 5. Shaping of Gaussian pulse obtained from exact solution of VLAEin Galilean coordinates. The surface A(x; y = 0; z=0; 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.
042
02 / zrkzz beam
diffpulsediff
Linear Amplitude equation in media with dispersion (SVEA).
2
2
12
2
2
2
2222
t
A
t
A
z
AA
z
A
t
Ai
2
22
1
2
2
2
122
'''2
'1
'2
z
A
zt
A
t
AA
t
Ai
Laboratory:
Galilean:
01
2
2
2
t
E
cE
tzkitrAtrE 00exp),(),(
2
2
2
2222
t
A
z
AA
z
A
t
Ai
Linear Amplitude Equation in Vacuum (VLAE). Analytical (Galilean invariant ) solution of 3D+1 Wave equation.
In air5"2
01 10 kvk042
02 / zrkzz beam
diffpulsediff
2. Comparison between the solutions of Wave Equation and SVEA in single-cycle regime
222zyxr kkkk
tkikAF
tkkki
kkkAFtzyxA
rrL
zyx
zyxL
L
exp)0,(ˆ
)(exp
)0,,,(ˆ
),,,(
1
222
1
Evolution of Gaussian amplitudude envelope of the electrical field in dynamics of wave equation. Single – cycle regime
01
2
2
2
t
E
cE
),,,( tzyxAx
))2/(exp(),,(
)2exp(),,()0,,,(2220
00
zyxzyxA
izzyxAtzyxE
x
xx
tkikEF
tkkki
kkkEFtzyxEx
rrx
zyx
zyx
x
exp)0,(ˆ
exp
)0,,,(ˆ),,,(
1
222
1
t=3PiT=0
),,,( tzyxAx
Analytical solution of SVEA (when β1<<1) and VLAE for initial Gaussian LB in single-cycle regime (δ=1 and α=2).
rti
erfcriitrti
rti
erfcriitrti
ztir
itzyxA
ˆ2
2ˆ
2
1expˆ
ˆ2
2ˆ
2
1expˆ
2exp
ˆ2),,,(
2
2
2
Conclusion(linear regime)
1. 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 (α>>1) in linear regime.
4. “Wave type” diffraction for single - cycle pulses (α~1-3) .
5. New formula for diffraction length of optical pulses is confirmed from analytical solution zdif
pulse=k02W4/z0
Nonlinear paraxial optics
Nonlinear paraxial equation:
;xAA x
Initial conditions:
1) nonlinear regime near to critical γ~ 1.2
2) nonlinear regime γ=1.7
AAAz
Ai
22
)2/2/exp()0,,( 22 yxzyxAx
1) nonlinear regime near to critical γ~ 1.2
2) Nonlinear regime γ=1.7
Nonlinear non-parxial regime.
AAt
A
z
AA
z
A
t
Ai
2
2
2
2
222
2
AAzt
A
t
AA
t
Ai
22
2
222
''2
''2
Laboratory frames
Galilean
Dynamics of long optical pulses governed by the non - paraxial equationNonlinear regime γ=2
(x,y plane) of long Gaussian pulse. Regime similar to laser beam.
;81
120
22
z
r
Dynamics of long optical pulses governed by the non - paraxial equation
Nonlinear regime γ=2
Longitudinal x, z plane of the same long Gaussian pulse. Large longitudinal spatial and spectral modulation of the pulse is observed.
;81
120
22
z
r
1/ Optical bullet in nonlinear regime γ=1.4. Arrest of the collapse. ;12
2/ OPTICAL DISK in nonlinear regime γ=2.25 NONLINEAR WAVEGUIDING.
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.
Conclusion - Nonlinear regime
Something happens in FS region??Wanted for new model to explain:
√ 3. Spectral, time and spatial modulation of long pulse
√ 4. Arrest of the collapse of light bullets
√ 5. Self-channeling of light disk
Three basic new nonlinear effects:
√ 1. Relative Self Guiding in Linear Regime of light disk.
√ 2. “Wave type” diffraction for single - cycle pulses.
Експеримент - 800 nm: Ti-Sapphire laser30 fs; 100 μm – леща: Мощност- 1.109 Wпикова мощност на импулса 1X1013 W/cm2 ~2-3 Pkr
H. Hasegawa, L.I. Pavlov, ....
z=0 z=12 zdiff