Download Parametric solitons in isotropic media.

Fifth Conference AMITaNS
June 24 - 29, 2013, Albena, Bulgaria
Parametric Solitons in isotropic media
D. A. Georgieva, L. M. Kovachev
The effects of confinement between picosecond optical pulses due to Cross Phase
Modulation (CPM) in multimode and birefringent optical fibres have been discussed in many
papers
1) Menyuk (1989, 2004); 2) Afanas’ev, Kovachev and Serkin (1990); 3) Kovachev (1991);
4) Radhakrishnan, Lakshmanan and Hietarinta (1997); 5) Todorov and Christov (2009).
The basic effect in the collision dynamics between two or three pulses is a detachment and
confinement of part of one pulse in the other. The obtained mixed states are stable at long
distances.
The numerical experiment shows that during the propagation and interaction the localized
waves change their shape, but they preserve the quantities normally associated with the
coupled Nonlinear Schrëdinger Equations (NLSs). In a stable mixed state each of the pulses
conserves his mass (local energy).
2
The typical evolution of optical pulses governed by the coupled Nonlinear Schrëdinger
Equations.
-4
0
4
For the numerical experiments we use the Split-Step Fourier Method.
3
Is it possible a mechanism of an exchange of energy
between the pieces into the mixed states?
We will discuss this possibility on the base of two types of
parametric processes in optical fibers.
4
NONLINEAR POLARIZATION OF TWO COMPONENTS
AT ONE CARRYING FREQUENCY
The electrical field associated with linearly or elliptically polarized optical wave is


1 
E x, y, z , t   xAx  yAy exp i 0 t   c.c.,
2i
where Ax, Ay are the complex amplitudes of the polarization components of a wave with a
carrying frequency ω0.
5

LINEARLY POLARIZED COMPONENTS. MANAKOV SYSTEM
We investigate the polarization dynamics of two initially linearly polarized components of
the electrical field 
1) there is no initial phase difference between the components;
2) the complex amplitudes can be expressed as a product of two amplitude
functions with equal phases: Ax = Cxexp(i), Ay = Cyexp(i).
The nonlinear polarization in the case of isotropic medium is well known

  *  B     *
Pnl  A  E  E E 
EE E ,
2




where ω is the optical frequency, A(ω), B(ω) are nonlinear coefficients. The components of the
nonlinear polarization are

B  

  A  
 A
2 

B  
2

Pnlx   A  
 Ax  Ay
2 

Pnly
x
2
 Ay
2
A ,
A .
x
2
y
6
Manakov system
From Maxwell equations in the negative dispersion region of an optical fibre 

2

2
Ax 1  2 Ax
2
i



A
 Ay
x
z 2 t 2
Ay
2
1  Ay
2
i



A
 Ay
x
2
z 2 t
A  0,
x
A
y
 0.
where  is a dimensionless nonlinear parameter.
(The system is written in normalized by dispersion lengths z = z/zdisp and moving with the
group velocity frame (Local Time frame): z = z, t = t - z/v.)
The Manakov system admits the following fundamental stable soliton solution when  = 1
Ax 
1
 iz 
sec ht  exp  ,
2
2
Ay 
1
 iz 
sec ht  exp  .
2
2
7
The exact solution is invariant with respect to an arbitrary initial phase difference, i.e.
Ax 
1
 iz

sec ht  exp   i1 ,
2
2

Ay 
1
 iz

sec ht  exp   i 2 
2
2

also is a fundamental soliton solution of the Manakov system.

ELLIPTICALLY POLARIZED COMPONENTS.
In the case of elliptically polarized light pulses the complex amplitudes Ax and Ay as a product
of two amplitude functions with different initial phases: Ax = Cxexp(ix), Ay = Cyexp(iy).
In the case of nonresonant electronic nonlinearities  A(ω) = B(ω) = const and
2


2
1
2
P  const  Ax  Ay  Ax  Ax* Ay Ay ,
3
3



x
nl
2


2
1
2
P  const  Ay  Ax  Ay  A*y Ax Ax .
3
3



y
nl
Last terms present degenerate four wave mixing process (ω1 = ω2 = ω3).
8
Parametric system
From Maxwell equations in an optical fibre 
2
Ax 1  2 Ax


2
1 *
2


i



A

A
A

A
A
A
 x
 x
y
x y
y   0,

2
z 2 t
3
3



2
Ay 1  Ay
2


2
1 *
2
i



A

A
A

A
A
A


y
x
y
y x
x   0.

z 2 t 2
3
3





In this case x y  0 (phase difference)  periodical exchange of energy between the
elliptically polarized components.
When the phase difference vanishes (x y = 0 – linearly polarized components) 
1) the amplitude functions admit again equal phases;
2) the nonlinear polarization with elliptically polarized components can be transformed to the nonlinear orthogonal polarization (Manakov case);
3) the parametric terms appear as usual cross-phase terms and the Parametric
system is equal to the Manakov system.
9
Exact soliton solution
The Parametric system has the same fundamental soliton solution as the Manakov system
when  = 1 and if there is not an initial phase difference ( = 1 - 2 = 0)
Ax 
1
 iz 
sec ht  exp  ,
2
2
Ay 
1
 iz 
sec ht  exp  ,
2
2
but it is not invariant with respect to an arbitrary initial phase difference, i.e.
Ax 
1
 iz

sec ht  exp   i1 ,
2
2

Ay 
1
 iz

sec ht  exp   i 2 
2
2

is not a fundamental solution of the Parametric system.
Overlapped components
When we investigate initial components with phase difference  = 1 - 2  0 and the
initial amplitudes are slightly above the theoretical for soliton regime we observe a stable
propagation with a periodical exchange of energy between the components.
10
 = π / 2
11
Initially separated components. Collision dinamics
When the components of the pulse 1) are initially spatially separated; 2) have different
phases; and 3) the degenerated condition of the wave synchronism is satisfied
(2ω1 - 2ω2 = 0) the components collide. After the collision we observe a detachment and
confinement of part of one pulse in the other. We observe a generation of mixed states of
the wave packets and an intensive energy exchange during their propagation.
t
-4
0
4
12
HAMILTONIAN STRUCTURE OF
THE MANAKOV TYPE SYSTEMS AND THE PARAMETRIC SYSTEM
These systems 1) are conservative;
2) have Hamiltonian structure;
3) satisfy at least three conservation laws of a) Hamiltonian H, b) momentum P and c) total energy N.

MANAKOV TYPE SYSTEMS
We consider

Ax 1  2 Ax
2
i



A
 a Ay
x
z 2 t 2
Ay

2
2
1  Ay
i



A
 a Ax
y
z 2 t 2
2
A  0,
x
2
A
y
 0,
where a > 0.
13
The Hamiltonian of the Manakov type systems:

*
Ax Ax* Ay Ay 1
4
H 


Ax  Ay
t t
t t
2
4
 a A
2
x
2
Ay .
Let us consider the following remaining integrals
a) the total energy of the system

N  N1  N 2 


Ax dt 
2


2
Ay dt  const ,

b) the energy of each component – at every moment of the interaction, even in the case
when the components could be separated on different pieces, the localized energy is
constant

N1 

Ax dt  const ,
2


N2 

2
Ay dt  const.

14

PARAMETRIC SYSTEM
The Hamiltonian of the Parametric system has the form

*
Ax Ax* Ay Ay 1
4
H 


Ax  Ay
t t
t t
2
4

2
 Ax
3
2
2
Ay 


1 *2 2
Ax Ay  Ax2 A*y2 .
3
a) the total energy of the system is still constant

N  N1  N 2 



Ax dt 
2

2
Ay dt  const ,

b) the energy of each component is not a constant
15
The change of energy of each pulse is described by the first derivative of local energy

N 1 1  *2 2
2
2 *2
  Ax Ay  Ax Ay dt   sin 2 x  2 y  Ax
z
3i 
3 


2

N 2 1  2 *2
2
  Ax Ay  Ax*2 Ay2 dt    sin 2 x  2 y  Ax
z
3i 
3 


the complex amplitudes Ax and Ay are presented in exponential form – Ax  Ax exp i x ,
2
Ay dt ,
2
2
Ay dt.
Ay  Ay exp i y .
• When  = 2x  2y
= 0 (Manakov case) the energy of each component is a constant and
there is not an exchange of energy.
• When  = 2x  2y
 0 we observe an exchange of energy between the waves which
reaches its maximum when  = 2x  2y = π / 2.
16
DIFFERENT CARRYING FREQUENCIES, CROSS-PHASE MODULATION
AND PARAMETRIC AMPLIFICATION
Using optical fibres as a nonlinear medium, a wide variety of nonlinear effects have been
observed. One of them, the four photon parametric mixing between three laser pulses
(2ω3 = ω1 +ω2) can be used to convert the input light pulse at ω3 frequency on two different
frequencies ω1 and ω2.
The electrical field associated with three linear polarized optical waves at different frequencies
can be written in the form

1
E x, y, z, t   x  A1 exp i1t   A2 exp i 2 t   A3 exp i 3t   c.c 
2i
where A1, A2 and A3 are the complex amplitudes of the components of three waves on different
frequencies and the phase-matching condition 2ω3 =ω1+ω2 is satisfied.
17
Parametric system
The corresponding Parametric system of the amplitude equations, governing the nonlinear
propagation in a fibre, written in approximation up to second order of dispersion is


2
2
2
2
 A1 1 A1  1  A1
2 *
i



A

2
A

2
A
A


A

1
2
3
1
3 A2 exp  ik z   0,
2

z
v

t
2

t


2
2
2
2
 A2 1 A2   2  A2
2 *
i



A

2
A

2
A
A


A

2
2
3
2
3 A1 exp  ik z   0,
2

z
v

t
2

t



A3  3  2 A3
i


2
z
2 t
A
3
2

 2 A1  2 A2
2
2
A  2A A A exp  ik   0,
3
1
2
*
3
z
where v is the group velocity, βi are the dispersion parameters and  is the nonlinear
coefficient.
18
Exact soliton solution
The Parametric Nonlinear System of Equations (PNSE) admits an exact soliton solution of the
kind
A1 
2
 iz 
sec ht  exp  ,
15
2
A2 
2
1
 iz 
 iz 
sec ht  exp  , A3 
sec ht  exp  ,
15
5
2
2
when the waves have equal group velocities, kz = 0 and β1 = β2 = β3 = 1. These parametric
solitons are with equal initial phases and are relatively stable on large distances.
Overlapped puses
Let us investigate PNSE for initial pulses with phase difference – when there is an initial
phase difference Δϕ = π / 2 and the initial amplitudes are slightly above the theoretical for
soliton regime, for example
A1  A2 
1
 i 
sec ht  exp  ,
5
 4
A3 
1
sec ht ,
5
we observe an intensive exchange of the energy between the waves.
19
a) the exact soliton solution
A1  A2 
A3 
2
sec ht ;
15
1
sec ht 
5
b) overlapped pulses
A1  A2 
A3 
1
 
sec ht exp  i ;
15
 4
1
sec ht 
5
20
Initially separated pulses. Collision dinamics
When the pulses 1) are initially separated; 2) have different phases; 3) have different
group velocities and 4) the condition of the wave synchronism is satisfied (2ω3 = ω1 + ω2)
the pulses collide. We observe a generation of mixed states of the wave packets and an
intensive energy exchange between them.
t
-6
A1  sec h(t  6)e
0
i / 4
6
A3  sec h(t ) A2  sec h(t  6)ei / 4
21
1) We investigate the propagation of two and three optical pulses in an isotropic media when
phase-matched conditions of the kind (2ω1 −2ω2 = 0) or (2ω3 = ω1 +ω2) are satisfied.
2) In this case terms connected with four-photon parametric processes in the corresponding
nonlinear evolution equations appear. These terms generate a periodical exchange of the
energy between the optical waves.
3) The existence of new kind of parametrically connected solitons is discussed. We observe a
confinement of the waves and a generation of mixed states of wave packets with different
polarization or frequencies.
4) It is shown that the Parametric system of equations has Hamiltonian with structure quite
different than the Hamiltonians of the well known Manakov type systems.
22