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Sub-cycle pulse propagation in a cubic medium
Ajit Kumar
Department of Physics,
Indian Institute of Technology, Delhi, India
@ NONLINEAR PHYSICS. THEORY AND EXPERIMENT. V
JUNE 17, 2008
Applications
Telecommunications: Multi-terabit per second transfer of information
in a single fiber.
Industry : Dramatic improvement in the resolution of optical
microscopy. Provides the only means of machining
and structuring materials (Nanophotonics).
Medicine: New diagnostic and therapic tools.
Frequency and Time Metrology: Frequency ruler of unprecedented
precision.
Ultra-fast Metrology: Attosecond metrology allows to observe the motion
of electrons on atomic length scale in real time and
record the electric field of the visible light.
Allow to create coherent light sources in all the way from the infra-red
to the soft X-ray regimes.
Ultra-short pulses have also been used to study:
# Light-matter interaction leading to higher harmonics and attosecond
pulse generation
# Nonlinear optical phenomena, like, spatio-temporal self-focusing and
filamentation
# Controlled manipulation of chemical reactions and bond formation
# Coherent quantum control of microscopic dynamics
etc.
Study of propagation characteristics of such pulses: Important
EVOLUTION EQUATION ?
Problem: SVEA doesn’t hold . What to do?
Main attempts:
# Brabec and Krausz (PRL,1997): Showed that it may be possible to define
envelope for optical pulses in the single-cycle regime.
exp[ i t  i ]  cc,
0


    | E ( ) |2 d /  | E ( ) |2 d
0
0
0
E  A(t )
The phase is defined in such a way that the imaginary part of the complex
envelope A(t) is zero at t=0. Equivalently, the center frequency  remains
unchanged under a phase shift of the electric field:
0
~
~
E (t )  E (t )ei
Using, then, the SVEA, they derived the following evolution equation (NEE)
 A  
0
2i
i
1 2
2
A  iDˆ A 
[1  (i /  0 )  ] ( t A)  2 0 [1  (i /  0 ) ] A A


2
20
n0


Using this NEE they studied the propagation properties of a single-cycle
at 0.8 micron wavelength up to a propagation distance close to the
characteristic distance of self-steepening.
WHAT IS MISSING?
# The expected coupling between spatial and temporal dynamics of the
pulse (Rothenberg, 1992) in single- and sub-cycle regimes is missing.
# The SEWA: requires, besides the envelope A(t), the relative phase 
not to vary significantly as the pulse covers a distance equal to the
wavelength 0 . Leads to applicability condition  0 Lchar  1 ,
which can not be satisfied always, for instance in the case of strong
resonant coupling.
# In the sub-cycle regime, the introduced concept of envelope does not
hold good.
# Schaefer and Wayne (Physica D, 2004): derived a short pulse equation
(SPE)
2
( 3) 
2
 2u  c1u  
(
u
u ),   (t  x) /  ,   x
2


where u is the real dimensionless electric field.
# This equation was derived by approximating the susceptibility function
by
in the region from 1.6-3.0 microns .
 (1) ( )    2 ,  ,   const
 (1) ( )
# This equation is integrable via IST (Sakovich, 2004) and possesses soliton
solutions but is valid only for a particular form of susceptibility.
# The novel feature of this SPE is the coupling between spatial and temporal
dynamics which was pointed out by Rothenberg in 1992.
# Recently Tsurumi (J. Phys. Soc. Japan,2006) has investigated Gaussian
beam propagation in a linear medium on the basis of a model equation,
for the real electric field.
In almost all the problems in nonlinear optics and plasma physics, involving
ultra-short pulses, the Brabec and Krausz NEE is used.
However, in the sub-cycle regime it is difficult to accept the concept
of an envelope and hence, it is desirable to have a pulse evolution equation
for the real electric field.
In the given work a nonlinear pulse evolution equation, for the real electric
field, is derived without separating the pulse into a slowly varying envelope
and the fast carrier oscillation parts.
The wave equation:
 2   
1     
  t  2  E ( r , t )  2 2   (t ' ) E ( r , t  t ' )
z 
c t 0

2
2
dt ' 

( 3)

2
t
c
2
2
E (r,t )E (r,t ) E (r,t )
(3)
where  (t ) is the third-order nonlinear susceptibility,  (t ) is the linear
permittivity of the medium and c is the speed of light in vacuum.
Nondimensionlization:
where
l0 ,
t0 ,
E0

r  r / l0 , t  t / t 0 ,

E  E / E0
are constants having dimensions of length, time and the
electric field respectively.
The non-dimensional wave equation is
2 
2 

 3 2
'
'
'
  2   E ( x , y, z , t )  1 
E  E E ( x , y, z , t )
 (t )E ( x , y , z , t  t )dt 

2
2 t 2 
2 t 2

z
c
c
0


t c
c 0 ,
l
0
where
  t ,
0
 3  E 2  3,
0
1 
 i t d , k    i
 (t ) 
  ( )e
2 0
Assuming the spectrum to be localized near zero frequency, we expand
k ( )
 0
around
to obtain

2 

  n 
 
  2    E ( x , y, z , t )   1    k 
i

  n 
2
 n!
t

z


  0 
 n  0
where
k  l k,
0
2

 3 2

E  E E ( x , y, z , t )
 E ( x , y, z , t ) 

c 2 t 2

x  x/l , y  y/l , z  z /l
0
0
0
 2   2 / x 2   2 / y 2
As a result, we obtain
2
 1    n k     n   2 (1) 2    2

 ˆ
2
ˆ

  i      0 
   
D(T )  D (T ) 
  2i
n 


T
 T 
 n! n0    0  t   

where the dispersion operator

Dˆ   i
2 m3
m 0
Dˆ (T )
is given by
2 m 2
 2 m 3  0 2 m 3    2 m 3

 0 2 m 3   
2 m 2







(2m  3)!  T 
2(2m  2)!  T 


If now we go over to the moving frame by introducing new variables
  T   0(1) z '  T 
1
z' ,
vg
    z'
where  is a constant and v g is the group velocity which is identical
with the phase velocity at   0, we obtain.
  E ( R ,  , )  2  
2a
2
R
(1)
0
2 
2
2
2  2 
E ( R ,  , )  
E ( R ,  , )  2i 3  0(1)  Dˆ ( ) E ( x , y , z , T )
2
 

3
2
ˆ
2
2 
E ( R ,  , )  E ( R ,  , )E ( R ,  , )
  D ( ) E ( x , y , z , T )   2
c  2
4
To account for dispersion, dissipation and diffraction consistently, we
introduce
a
 a

R   x,

 y,
z  z ,
T  t 

We get
 1   nk   
 2a 2 2  
   R  
 E ( x , y , z , T )      n   i
2 

z
 n! n 0    0  T


2
Note that
i
k   ( )   ( ),
2
2
3
2

2 
E  E E ( x , y, z , T )
 E ( x , y , z , T )   2
2
c

T

  
   cos  ,
c
  c
 Im  
  tan 1 
.
Re



 ( )  Re 
  
    sin  ,
c

 c
 ( )  Im
and the electric field is real. As a consequence

( 2m)
0
for m=0,1,2,3,… and
 d 2m  
  o,
 
2m 
d


 0
 (  0)  0,

( 2 m 1)
0
 d 2 m1 
  o.
 
2 m 1 
d


 0
 (  0)  0.
The dispersion operator
Dˆ ( )

is given by
2 m 3
2 m 2
( 2 m  3)
2 m 2










ˆ
2 m3
2 m 1
2m
0
0
D  i

 
 


(
2
m

3
)!


2
(
2
m

2
)!


 
 
m 0



Consider now the case when diffraction and dispersion are comparable
 . This leads to     2
in the leading order in
. Further, for
a weakly nonlinear and weakly dissipative medium, we assume the
dissipative and nonlinear effects to be of the order of 
and  2 ,
respectively:
0
(2)
  0( 2 ) ,

As a result, in the leading order in
3
  2  3.

, we obtain
  2 E  0(3)  4 E  0( 2)  3 E   3  2
 2

E  E E 
 E  2 


4
3 
2
6 
4   c 
 
2
t
(1)
0
The above partial differential equation describes the spatio-temporal
evolution of the real electric field in a cubic medium.
If we include the delayed response of the medium, we have

 t2 

1 2    
2  

)
t
,
r
(
E
  (t ') E ( r ,t t ') dt ' 
c 2 t 2 0
z 2
 3  2
c t
2




2
 E ( r ,t )E ( r ,t ) E ( r ,t ) 

 
 
(1 )  3  2    
'
dt
)
'
t

t
,
r
(
E

)
'
t

t
,
r
(
E
)
'
t
(
g
)
t
,
r
(
E
 R

t 2 
c2
0


where  is a numerical constant representing the fraction of the fast
electronic response and
g R (t )

1
2
  22
 1 2
expt /  2  sin t /  1 
is the Raman Response function for the case when one vibrational mode with
linewidth 1/ 
and eigenfrequency 1/ 
is important.
2
1
Going through the same steps of calculations, we arrive at
2
(1 )
t E  2  0


( 3)
( 2)
 3  2
 2 E 0  4 E 0 3E


 2
c  2

6  4
4  3

(1 )  3  2


2
2 E 0 g R (  ) E (   ) E (   ) d 
c


 E E  E 
Three features of this equation
# There is a coupling between the spatial and temporal evolution.
# Dispersion comes from the third-order derivative of the wave vector
with respect to frequency.
# Dissipation emanated from the second-order derivative of the wave
vector with respect to frequency.
Results of Numerical Simulations: For half a-cycle pulse at   0.8 m.
  2 E  0(3)  4 E  0(2)  3 E   3  2

 2 0(1) 


E E E
4
3 
2
2
 
6
4




c






Parameters used in numerical calculations:
#
#
#
 3  0.05
corresponding to
fs3
3
  0.0274
,
m
l0  ld 

10 V
E0  1.496 10 m ,


2
2
 3  2.234 10 22 m  n  0.6 10 22 m
2

V2
V2
fs2
2
  0.03592
( Diddams and Diels ,1996)
m
( p / 1.67 )
3
3
 18 .58 m ,
Lnl 
z s  0.39  Lnl   0  T0  7.23 m ,  
# Input pulse shape: At
 0
1
1
2
 9.85 m , P0   0 n0 cAeff E 0
P0
2
2 n20
 0 n0 c 2 Aeff
is taken to be
2



 

E  exp     cos(  )
0
T 
0
  0 


Fig.1: Dimensionless Electric field as a function of time. The solid curve
corresponds to the pulse after 3.7 micron propagation. The dashed
curve corresponds to the input pulse shape.
Fig.2: Dimensionless intensity as a function of dimensionless frequency.
The solid curve corresponds to the pulse after 3.7 micron propagation.
The dashed curve corresponds to the input pulse shape.
Fig.3: The dimensionless Electric Field of the pulse, after7.4 microns
of propagation as a function of dimensionless time.
Fig.3: The dimensionless Electric Field of the pulse, after16.7 microns
of propagation as a function of dimensionless time.
Fig.5: The dimensionless Electric Field of the pulse, after 22.3 microns
of propagation as a function of dimensionless time.
Fig.6: Dimensionless intensity as a function of dimensionless frequency,
after 22.3 microns of propagation.
Conclusions:
# We have presented a novel nonlinear evolution equation for a fewand sub-cycle pulses, which has been derived from Maxwell’s
equations for the real electric field.
Novel Features
# In the leading order, the dispersion effects come from the fourth-order
derivative of the electric field with the coefficient equal to the third-order
derivative of the propagation constant with respect to frequency at   0.
# Dissipation arises from the third-order derivative of the electric field
with the coefficient equal to the second-order derivative of the propagation
constant with respect to frequency at   0.
# By numerical calculations, we have shown that the given evolution equation
is capable of describing pulse propagation up to and beyond the critical
distance of self-steepening without any pulse break-up taking place.