Transcript Slide 1

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
A
1
i

  A  ......
z
2k 0
Solution in (x, y, z) space
A( x, y, z )   A(k x , k y ,0) exp( i (k x2  k y2 ) / 2k0 ) exp( ik x x) exp( k y y )dk x dk y
x, y
Initial conditions - Gaussian beam
A( x, y,0)  A0 exp(( x2  y 2 ) / 2r02 );zdiff  k0r02
Analytical solution for initial Gaussian beam
2
2

1
(
x

y
)

A( x, y, z ) 
A0
exp 
1  iz / zdiff   2r02 1  iz / zdiff
4
2
A
2

4
16
2
0
A
1
1  z / z diff


2
2
2

(
x

y
)
exp 
 2r 2 1  z / z
0
diff




 

2




z=0
z=zdiff
Numerical solution using FFT technique. Paraxial optics.
Laser beam on 800 nm (zdiff=k0r02= 7.85 cm; r0= 100µm)

A( x, y, z)  F 1 A(kx , k y ,0) exp(i(kx2  k y2 ) / 2k0 )

Initial condition
A( x, y,0)  A0 exp(( x 2  y 2 ) / 2);
z=0
z=1/3
z=2/3
z=1;zdiff=7.85 cm
Phase modulated (by lens) Gaussian beam
A( x, y,0)  A0 ( x, y) exp(i 2d0  i ( x, y))
 ( x, y)  Seff ( / f )(a2  ( x2  y 2 )
a-radius of the lens, f- focus distance
d0- thickness in the centrum
Seff- effective area of the laser spot
a=1,27 cm
Seff=0.2
  800 nm
f=200 cm
z=0
z=1/3
z=2/3
z=1=z diff
Paraxial optics of a laser pulse. From ns to 200-300 ps time duration
A
1
k"  2 A
i

 A 
 .nl..term s  .
2
z
2k0
2 t
Dimensionless analyze:
x  r x' ;
y  r y'
2
0 
zdiff
k r
  2

t0 / k "
zdissp
z  z0 z'
2
A
 A

 k0r / z0  1
i
  A  

......
z '
t '2
2
z0  k0r
2
t0 ~ 330 fs;r ~ 1mm;k"  3 1031 sek 2 / cm,k0  7.85104 cm1
  0.02
In air, gases and metal vapors t0>100-200 fs ;
β<<1 - Negligible dispersion.
Nonlinear paraxial optics
Nonlinear paraxial equation:


2 
A
 2i
  A   A A
z
Initial conditions:


A  Ax x;
Ax ( x, y, z  0)  exp(x2 / 2  y 2 / 2)
1) nonlinear regime near to critical
γ~ 1.2
2) nonlinear regime
γ=1.7
• 1) nonlinear regime near to critical γ~ 1.2
2) Nonlinear regime γ=1.7
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.
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
Extension of the paraxial model for
ultra short pulses and single-cycle pulses ?
A
1
k"  A
i

 A 
 .nl..term s
2
z
2k 0
2 t
2
 ionization ...
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 profile
2) 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.
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??
1) X-waves - J0 Bessel functions – infinite energy
2) X-waves - Delta functions in (kx, ky) space.

A( x, y, z)  F 1 A(kx , k y ,0) exp(i(kx2  k y2 ) / 2k0 )

Experiment:
1. Self-Channeling is observed for spectrally - limited (regular) pulses
2. “Wave type” diffraction for single- cycle pulses.
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
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.
Introducing the amplitude function of the electrical field

E (r , t )

ˆ
and the amplitude function of the Fourier presentation of the electrical field E
( r , )


E(r, t )  A(r, t ) expik0 z  0t 
ˆ
ˆ
E (r ,  ) exp( it )  exp ik0 z  0t A(r ,   0 ) exp  i  0 t 
The next nonlinear equation of the amplitudes is obtained:




ˆ 2  ˆ
 2
2
A(r , t ) 2
A(r , t )  2ik0
 k 0 A(r , t )    k ()  k nl () A  A(r ,   0 ) exp(i(  0 )t )d
z




Convergence of the series: I. Number of cycles; II. Media density:
1 2 "
k ( )  k (0 )  (k ) (  0 )  (k ) (  0 ) 2  ..
2
2
2
2 '
(k 2 )'  2kk ' ;......k '  1 / v
(k 2 )"  2k ' k 'k"
SVEA in laboratory coordinate frame
 2 


 
2

A
A
k 0 v n2 
A
A 
v  v "
1   A k 0 vn2  2 


 2 
i
 v   n2 

A   k 0 
A A

2
 t
z 
2   t  2k 0
2
2
k 0 v  t


or
 2 



 
  2
" 2
2 

A
A
k0v n2 
A A 
v 
 A 1  A  vk0  A k0vn2  2 


i
 v   n2 

 A  2  2 2  

A A

2

 t
z 
2   t  2k0 
z v t  2 t
2




V. Karpman, M.Jain and N. Tzoar, D. Christodoulides and R.Joseph,
N. Akhmediev and A. Ankewich, Boyd……






"
2
2
2
 A A  v    A 1  A  vk0  A k0vn2  2 
 i  v  
 A  2  2 2  

A A
2

z  2k0 
z v t  2 t
2
 t


SVEA in Galilean coordinate frames




3
2
2
2


 A 
v
v k"  A v  "
1   A
 A  k0vn2  2 

 i  
A 
  k0 
 2v

A A
2
2 
2

2 z ' 2 
k0v  t '
t ' z ' 
2
 t '  2k0
Constants

A  A0 A"; x  r x"; y  r y"; z  z0 z"; z'  z0 z"; t  t0t":t '  t0t"
2
z dis 
zdiffr  k0r2
t0
k
z0  vt0
"
2
  k0 z0 ;

2

r
2
z0
 2
1
  k 0 r n2 A0 ;
2
2 2
;

z dif
z dis
;
k0 v n2  1  2
1  2
1  A0  n2 
  n2 A0
2
2   2

Dimensionless parameters
1.
  k0 z0
r 2
 2
z 0
2.
2
3.
  zdiff / zdisp
  1
4.
Determine number of cycles under
envelope with precise 2π
Determine relation between transverse and
longitudinal initial profile of the pulse
Determine the relation between diffraction
and dispersion length
2
1
  k0 r 2  n2 A0
2
  1
2
1  k 0 v k
"
z0  vt0
2
Nonlinear constant
2
1 2
5.   1   n2 A0 Constant connected with nonlinear addition
2
to group velocity
2
SVEA in dimensionless coordinates
Laboratory
 2 
 


2 
2 
2 
 A A  


2 
2 A
2  A
A
 A 
 A



   A  
 2i 

 1
 2  2   A A
2


t
 t z


z

t
t






Galilean
 
  A 2 A   A 2 A   
2 
2 
2 











2 
2 A
2
 A
 A  A







   A    
 2i
 1 

2
 2   A A
2


z '  
t ' z ' 
 t '
 t '

t
'
z '









t '  t; z'  z  vt

  1;..ns..and....200 300.. ps....domain
2
  1;...200 300 fs....domain
2
 2  1;...20 150 fs....domain
Linear Amplitude equation in media with dispersion (SVEA)





2
2
2
Laboratory:



A 
 A 
 A
2  A
2  A
2
 2i

  A  
 2   1 2
 t z 
 z 2


t
t








2
2
2

 A
 A 
2 A
2
2  A





2
i



A


1



2



Galilean:

1 
1
2
2

t '

t
'

z
'

t
'

z
'


Linear Amplitude Equation in Vacuum (VLAE)



2
 1  E
E(r, t )  A(r, t ) expik0 z  0t 
E  2 2  0



2
c t

1  A
A 
1
1  A
i
c
A 


c  t
z  2k0
2k0 c 2 t 2




2
2



A 
 A
2 A
2  A
   A    2  2 
 2i 


 z


t

z

t




pulse
beam
zdiff
  2 zdiff
 k02r4 / z0
In air
1  k0v 2k "  105
Laboratory frame

ˆ

AL (k x , k y , k z , t )  F A( x, y, z, t )


ˆ

AG (k x , k y , k z , t ' )  F A( x, y, z ' , t ' )

ˆ

2 ˆ

A
 A
ˆ
 2i 2 L   k x2  k y2   2 k z2  2k z AL   2 1  1  2L  0
t
t



Galilean frame
ˆ

2 ˆ

AG
 AG
2
2
2
2 ˆ
2
 2i   k z 
  k x  k y  k z AG   1  1  2  0
t
t


Solutions in kx ky kz space :
 
ˆ
ˆ
 

AL (k x , k y , k z , t )  AL (k x , k y , k z ,0) exp i 

  1  1 
 
2
 
  
kˆ 2
 

  2
t

 1  1   
 1  1 
 
2

2
2
2
2
   k z  k x  k y   1 k z
    kz
ˆ
ˆ
AG (k x , k y , k z , t )  AG (k x , k y , k z ,0) exp  i  1     1      2 1   

1
1 

1

where

2
2
2
2
2
ˆ
k  k x  k y   k z  2k z



t 


Fundamental solutions of the linear SWEA


2
2


ˆ



k
ˆ
t  
AL ( x, y, z, t )  F 1  AL (k x , k y , k z ,0) exp i 


2
2
  1 

(
1


)

1  1    

1
1
 






   kz
    kz
ˆ
AG ( x, y, z , t )  F 1  AG (k x , k y , k z ,0) exp i 
 
1  1
  1  1





2
2
2
2
2
ˆ
k  k x  k y   k z  2k z




2

k x2  k y2   2 1k z2   

t
2

 1  1    
 
Fundamental linear solutions of SVEA for media with dispersion:


2
2


ˆ



k
ˆ
t  
AL ( x, y, z, t )  F 1  AL (k x , k y , k z ,0) exp i 


2
2
  1 

(
1


)

1  1    

1
1
 






   kz
    kz
1 ˆ


AG ( x, y, z , t )  F  AG (k x , k y , k z ,0) exp i 
 
  1  1
 1  1




2
2
2
2
 k x  k y   1 k z
 
2
1  1 


2
 
 
t 

 
Fundamental solutions of VLAE for media without dispersion:
AL  F
AG  F
1
1
 
ˆ 0, k , k , k
A
L
x
y
z
 







2
2
2
ˆ
 F  exp  i     k /  t  



1



ˆA 0, k , k , k  F 1  exp  i    k    2  kˆ 2 /  2 t  


L
x
y
z
z





2
2
2
2
2
ˆ
k  k x  k y   k z  2k z

1  k0v 2k "  105
Evolution of long pulses in air (linear regime, 260 ps and 43 ps)
Light source form Ti:sapphire laser, waist on level e-1 : r  100m
"
v  1;.....t  t '  z '  z
k0  7.85.104 cm1; kair
 31031 sec2 / cm
1) 260 ps: αδ2=1; β1=2.1X10-5
t  t '  1 ~ z'  z  1
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)

xA( x, y, z, t ) 
exp k

2 
1
3
2
x
 
k k /2 
2
y
2
z
2  
2
2


exp i   k x  k y  k z    t  
 

exp(ik x x) exp(ik y y ) exp(ik z z )dkx dky dkz .
kˆz  k z  

x A( x, y, z , t ) 

 

exp 
 i t  z 
 2

2
1
2 
3
 
2
2
ˆ2 / 2 
exp

k

k

k
x
y
z

exp i k x2  k y2  kˆz2 t  


exp(ik x x) exp(ik y y ) exp(ikˆz ( z  i ) dkx dky dkˆz .
rˆ  x  y  ( z  i )  r  2iz  
2
2
2
2
2

xA( x, y, z , t ) 

 
1
2


exp 
 i t  z   k r exp(k r / 2) 
3
2   2
 rˆ 0
1
exp i  k r t sin rˆk r dkr .
2
Analytical solution of SVEA (when β1<<1)
and VLAE for initial Gaussian LB (δ=1)
2


i

A( x, y, z , t ) 
exp 
 i t  z 
2rˆ
2



i 2
 
1

2
 i t  rˆ  exp it  irˆ  erfc
ˆ   

t

r



2
2



 


i 2


1
2 
t  rˆ  
  i t  rˆ  exp it  irˆ  erfc
2
2





rˆ  r  2iz  
2
2
Shaping of LB on one zdifpulse=k02r4/z0 length
Gaussian shape of the
solution when t=0.
The surface
|A(x,y=0,z; t=0) | is plotted.
  785
t  z  785
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).
Analytical solution of SVEA (when β1<<1)
and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)

xA( x, y, z , t ) 
exp k

2 
1
3

2
x
k k
 
2
2
2
exp i (  k z )  k x  k y  k z  
 
2
y

2
2
z
/ 2
t  
 
 
exp(ik x x) exp(ik y y ) exp(ik z z )dkx dky dkz .
kˆz  k z  
Analytical solution of SVEA (when β1<<1)
and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)
 2

exp 
 i t  z 
3
2 
 2

2
2
2
ˆ
 exp  k x  k y  k z / 2 

xA( x, y, z , t ) 

1
 
exp i k x2  k y2  kˆz2 t  
 
 
exp(ik x x) exp(ik y y ) exp(ikˆz ( z  t  i )dkx dky dkˆz .
2
2
2
~
r  x  y  ( z  t  i )
Analytical solution of SVEA (when β1<<1)
and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)
2


i

A( x, y, z , t )  ~ exp 
 i t  z 
2r
 2





1
i
2


2
 i t  ~
t  ~
r  exp it  i~
r  erfc
r   

2

 2
 


i 2


1
2
~
~
~
t  r  
  i t  r  exp it  ir  erfc
2
2





2
2
2
~
r  x  y  ( z  t  i )
Fig. 5. Shaping of Gaussian pulse obtained from exact solution of VLAE
in 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.
pulse
beam
zdiff
  2 zdiff
 k02r4 / z0
Linear Amplitude equation in media with dispersion (SVEA).





2
2
2
Laboratory:



A 
 A 
 A
2  A
2  A
2
 2i

  A  
 2   1 2
 t z 
 z 2


t
t








2
2
2

 A
 A 
2 A
2
2  A





2
i



A


1



2



Galilean:

1 
1
2
2

t '

t
'

z
'

t
'

z
'


Linear Amplitude Equation in Vacuum (VLAE).
Analytical (Galilean invariant ) solution of 3D+1 Wave equation.

2
 1  E


E  2 2  0
E(r, t )  A(r, t ) expik0 z  0t 
c t




2
2



A 
 A
2 A
2  A
   A    2  2 
 2i 


 z


t

z

t




pulse
beam
zdiff
  2 zdiff
 k02r4 / z0
In air
1  k0v 2k "  105
2. Comparison between the solutions of Wave Equation
and SVEA in single-cycle regime
ˆ



A
(
k
,
k
,
k
,
0
)


L
x
y
z
1 
AL ( x, y, z , t )  F 


 exp i    k x2  k y2  (k z   ) 2 t 



1  ˆ
F  AL ( k r ,0) expi  k r t 



kr  k x2  k y2  k z2

Evolution of Gaussian amplitudude envelope of the electrical field in
dynamics of wave equation. Single – cycle regime
1 2E
E  2
0
2
c t
E x0 ( x, y, z , t  0)  Ax0 ( x, y, z ) exp(2iz )
Ax0 ( x, y, z )  exp(( x 2  y 2  z 2 / 2))
Ax ( x, y, z, t )
ˆ (k , k , k ,0) 


E
x
y
z


xEx ( x, y, z, t )  F 1 
 exp i k x2  k y2  k z2 t 


F 1 Eˆ (k ,0) expi  k t 


x
r
r

T=0

t=3Pi
Ax ( x, y, z, t )
Analytical solution of SVEA (when β1<<1) and VLAE for initial
Gaussian LB in single-cycle regime (δ=1 and α=2).
 2

i
A( x, y , z , t ) 
ex p

 i t  z 


2rˆ
2



i 2
 
1

2 
 i t  rˆ  ex p it  irˆ  erfc
ˆ   

t

r
 2
 

2





i 2


1
2 
ˆ  

t

r
  i t  rˆ  ex p it  irˆ  erfc
 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 zdifpulse=k02W4/z0
Nonlinear paraxial optics
Nonlinear paraxial equation:


2 
A
 2i
  A   A A
z
Initial conditions:


A  Ax x;
Ax ( x, y, z  0)  exp(x2 / 2  y 2 / 2)
1) nonlinear regime near to critical
γ~ 1.2
2) nonlinear regime
γ=1.7
1) nonlinear regime near to critical γ~ 1.2
2) Nonlinear regime γ=1.7
Nonlinear non-parxial regime.
Laboratory frames


2 
2 


2 
2  A
2  A
A 

A
   A    2  2    A A
 2i 





t

z

z

t




Galilean

2 
2 


2 
2 A
2  A

A
 A A
 2i
  A    2  2
 t '

t '

t
'

z
'


Dynamics of long optical pulses governed
2
by the non - paraxial equation
r
 2  2
Nonlinear regime γ=2
z0

(x,y plane) of long Gaussian pulse. Regime similar to laser
beam.
1
;
81
Dynamics of long optical pulses governed
by the non - paraxial equation
Nonlinear regime γ=2

2
r2
1


;
2
z0
81
Longitudinal x, z plane of the same long Gaussian pulse.
Large longitudinal spatial and spectral modulation of the pulse is observed.
1/ Optical bullet in nonlinear regime γ=1.4.
Arrest of the collapse.
 2  1;
2/ OPTICAL DISK in nonlinear regime γ=2.25
NONLINEAR WAVEGUIDING.
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.
Something happens in FS region??
Wanted for new model to explain:
√ 1. Relative Self Guiding in Linear Regime of light disk.
√ 2. “Wave type” diffraction for single - cycle pulses.
Three basic new nonlinear effects:
√ 3. Spectral, time and spatial modulation of long pulse
√ 4. Arrest of the collapse of light bullets
√ 5. Self-channeling of light disk
Експеримент - 800 nm: Ti-Sapphire laser
30 fs; 100 μm – леща: Мощност- 1.109 W
пикова мощност на импулса 1X1013 W/cm2 ~2-3 Pkr
H. Hasegawa, L.I. Pavlov, ....
z=0
z=12 zdiff