Transcript Wave-mixing solitons in ferroelectric crystals

Wave-mixing solitons in
ferroelectric crystals
S. Bugaychuk1, L. Kovacs2, G. Mandula2,
K. Polgar2, R. A. Rupp3
1Institute of Physics,
Prospect Nauki, 46, Kiev-39, 03039, Ukraine,
Ph.(38-044) 265-40-69;Fax: (38-044) 265-15-89;
e-mail: [email protected]
Abstract
Experimental set-up
2Research
Institute for Solid State Physics and Optics,
H-1121 Budapest, Konkoly-Thege M. ut 29-33, Ph.: 36-1-39-22222
ext.1249, Fax: 36-1-3922223;
e-mail: [email protected]; [email protected]
• Although the sine-Gordon equation was originally
obtained for the description of four wave-mixing in
transmission geometry, it describes self-diffraction of the
wave from shifted gratings as well. The sine-Gordon
equation governs the soliton propagation.
• The photoinduced amplitude of the refractive-index grating
exhibits also a soliton shape in the crystal volume. The
origin of this effect is the change of the contrast of light
due to energy transfer between coupled waves during their
propagation, which occurs in bulk crystals with strong
photorefractive gain.
• The theoretical description shows the possibility to control
the soliton properties by changing the input intensity ratio
and/or input phase difference of the wave.
• The effect can lead to diffraction efficiency management,
auto-oscillations and bistability of the output waves due to
wave-mixing in ferroelectrics.
• Results on the first experimental observation of nonuniform distribution of the grating amplitude profile and its
changes versus input intensity ratio are presented.
3University
Vienna, Institute for Experimental Physics,
Strudlhofgasse 4, A-1090 Vienna, Austria,
Ph.: (+43-1)31367/3005; Fax: 3105239 or 3102683;
e-mail: [email protected]
The theory of the
transmission four-wave mixing with
non-local gratings
Four writing beams from Ar-ion laser are expanded and converged in the
crystal to record a transmission grating. The expanded probe beam from
He-Ne laser is entered at the Bragg angle relative to the recorded grating.
We use specially grown congruently melted LiNbO3:Fe crystal with input
faces ac=35 mm and the largest dimension as a thickness b=12 mm. The
crystallographic dimensions a, b and c are connected with axes y, z and x,
respectively. Axis x coincides with the polar crystallographic axis c.
The diffracted pattern of the probe beam displays the grating
amplitude distribution along the crystal.
The system that describes the changes of coupled wave amplitudes
inside the crystal has the following form:
 A 3*
  i   A 4*
z
 A4
 i   A3
z
 A1
  i  A2
z
 A 2*
 i   A1*
z
where  is the photoinduced changes of complex grating amplitude:
 (t , r )   0   (t , r ),
Band-transport model of recording of
non-local phase gratings
The photorefractive mechanism of recording refractive index grating
includes photo-excitation of free carriers by light interference pattern; their
movement due to diffusion and drift with systematic trapping by defects;
creation of the internal non-uniform distributed space charge field, which
modulate the refractive index.
In the case of diffusion dominant mechanism the grating is shifted relative
to light interference pattern by a quarter of the period.
  /  0  1
The space area of the
soliton formation inside
the crystal volume.
The optical scheme of four-wave
mixing with four input waves.
For the value  we can write the kinetic equation that describes the process
of grating recording due to photorefractive mechanism and the grating
relaxation with the time constant T0:

 
 Fˆ E
t
2
  T1
Optical control of localized gratings.
Diffraction efficiency management.

0
In the case of diffusion dominant mechanism of the photorefraction the
change of the grating amplitude is proportional to the light intensity:
 
2
Fˆ E  ( 1  i ) Ei (t , r ) E *j (t , r )
where  is the photorefractive gain for the non-local grating and 1 is the
maximum grating amplitude for local gratings.
In case of only non-local gratings the kinetic equation has the following
form:
 
 i  A 1 A 2*  A 4 A 3*   


Single level band
transport model

I10/I20=I4d/I2d=1; d=10
I10/I20=I4d/I2d=0.1; d=10
By changing the input intensity ratio or/and the phase difference of the input
waves, the localization degree of the grating amplitude profile changes. Thus
the wave-mixing diffraction efficiency defined as the integral under the grating
amplitude shape changes as well.
All optical space switching
Photo-excitation of free carriers.
I is the light interference pattern
Trapping of free carriers.
Creation of internal space charge
distribution.
E is the space-charge field that
modulates the refractive index.
Solution for non-local gratings: the soliton
u (t, z ) 
1
1
d 1 sin[  2 u ( z )  C 1 ]  d 1
2
2
I10/ I20>I4d/I3d or I4d=0; d=10.
A22  d1  A12
1
1
d 2 sin[  2 ( u ( z )  u d )  C 2 ]  d 2
2
2
d 1 , 2  A12, 3  A 22, 4
A42  d 2  A32
1, 2  (1, 4   2 ,3 )  (

2
  )  0; 
ud=u(z=d)
where n is the nth wave phase and  is the grating phase.
Substituting these solutions in the kinetic equation, we obtained a damped
sine-Gordon equation:
 2u
u

 R sin( 2 u   )
z
z
where =t/T0. R and  are constants defined from input intensities
I10, I20, I3d, I4d and input phase differences of coupled waves.
The steady state solution is:
 
where both
Two-wave mixing. The energy transfer between two coupled waves (I1
and I2) is started from the input surface of the crystal. The grating
amplitude maximum is located near the input boundary.
  (t, )d 
z
A12 ( z ) 
Two- and four-wave mixing.
Formation of the grating amplitude
profile along axis z.

we have found the solutions for wave intensities from the equation set of
coupled waves:
C
C
cosh( 2  Cz  p )
1
( I 2  I 4  I1  I 3 ) 2  4 
2
2
 const
p  const
and
may be found from the input conditions.
The grating amplitude has a soliton shape.
The solution describes a single stationary soliton in the coordinate
system (, z).
The wave-mixing diffraction efficiency is defined by the integral under
the grating amplitude shape: ~sin(2ud).
Optical control of the grating amplitude shape.
Diffraction efficiency management.
The stationary soliton shape is unequivocally defined by the input
intensity ratio and input wave phase difference.
Four-wave mixing. In the case of equal input intensity ratios at the crystal
boundaries (I10/I20=I4d/I3d) and a strong photorefractive response, the energy
transfer takes place only in the center of the crystal. The maximum grating
amplitude is located in the middle of the crystal.
The grating amplitude maximum is located near a crystal boundary in the
case of I10/I20I4d/I3d, or two-wave mixing, or usual four-wave mixing with
three input waves.
I10/ I20<I4d/I3d or I10=0; d=10.
Braking the equal intensity ratio I10/I20I4d/I3d on crystal boundaries leads to
a movement of the grating amplitude maximum from one boundary to the
other one.
The scheme of optical space
switching. The waves 1-4 record the
grating with different amplitude
profile that determine the angle of
space diffraction for the probe beam
5-6.
Alteration of the single wave-mixing soliton by changing input intensity ratio:
1) I4d/I3d=0.1 ; 2) I4d/I3d=0.5, I20=I3d=I4d ; 3) I10/I20=I4d/I3d=1 ; 4) I10/I20=I4d/I3d=0.01.
(Itotal=1; d=10).
The localized soliton can be moved through the medium from one side to
another.
The grating amplitude distributions for
some input intensity ratios. The numbers
on the top indicate lg((I10+I20)/(I3d+I4d)).
The dashed curves correspond to
disphased input coupled waves by  at the
crystal boundaries (10-20=0; 4d-3d=)
or (10-20=; 4d-3d=0).
Location of the grating amplitude
maximum of steady state gratings in
the case of FWM with four input
waves.
J1=J10+J20,
J2=J3d+J4d.
J10/J20=J4d/J3d=0.1, d=10.
The sine-Gordon equation admits multi-soliton solutions that describe the
interaction of several solitons as well as bond soliton states (pulsing, or so
called breather solutions). We obtained the breather solutions numerically
for the case of usual four-wave mixing with three input waves and for
certain area of input intensity ratios.
The breather solutions. I10/I20=3, I3d=0,87, I4d=0, d=15, (Itotal=1).
Conclusion
The wave self-diffraction from non-local phase gratings
in a photorefractive medium can be described by a sineGordon equation with a damped term in the case of
transmission geometry. The sine-Gordon equation reveals the
changes of the grating amplitude induced by light beam
interaction in the medium. The grating amplitude distribution
has a soliton shape in the direction of wave propagation. In
steady state the soliton is motionless and its parameters, i.e.
the amplitude, the half-with and the amplitude maximum
position, are unequivocal defined by input intensity ratio and
input wave phase difference. The crucial parameter of the
soliton is the energy transfer gain on a given distance z in the
medium. The photorefractive gain determines the change of
the light contrast during the wave propagation and this way
the soliton localization degree.
Alteration of the soliton shape of the grating amplitude
opens the ways to control the parameters of output waves. The
one of them is the all-optical management of the diffraction
efficiency, as the wave-mixing diffraction efficiency is
determined by the integral under the soliton shape.
Multisoliton and bond-soliton behaviors lead to bistability and
auto-oscillations of output intensities.
For the first time, we observed experimentally the nonuniform distribution of the grating amplitude and the control
of its profile by means of changes of input intensity ratio.
Research was supported by OTKA contract Nos. T23092, T26088, T35044,
by the Austrian-Hungarian Intergovernmental ST Program A-8/2001, and by
the Centre of Excellence Program ICA1-1999-75002.
Distribution of the grating amplitude and the intensities of coupled waves
along the crystal. d=10.
We change the input intensity ratios of writing beams.
The left pictures show the photos of the space distribution of light induced
refractive index. The right pictures show the measured intensity distribution
of the diffracted probe beam along axis z.
(a) - I4d/I3d=0.08, I20=I3d=I4d ; (b) – I4d/I3d=0.14, I20=I3d=I4d ;
(c) - I10=I20=I3d=I4d.
In case of equal input intensities of all waves the maximum of the grating
amplitude is located in the center of the crystal.
The obtained experimental results of the soliton shape changes versus input
intensity ratio are in good qualitative agreement with the theoretically
calculated curves.
Breather solutions of bond soliton states
Introducing the new real variable u:
A 32 ( z ) 
Non-local and local gratings recorded in the crystal
Experimental observation of
the grating amplitude distribution
The oscillation of the grating amplitude causes the auto-oscillations of the
diffraction efficiency  and by this way the auto-oscillations of every output
intensities.
The reason of the bond-soliton behavior is the emergence of a local
component of the grating that causes the changes of wave phases during
their propagation. In this way, the light contrast changes with time, and the
grating is repeatedly erased and rerecorded. Hence the auto-oscillations can
be observed only in optically reversible media.
Abstract
• Although the sine-Gordon equation was originally
obtained for the description of four wave-mixing in
transmission geometry, it describes self-diffraction of the
wave from shifted gratings as well. The sine-Gordon
equation governs the soliton propagation.
• The photoinduced amplitude of the refractive-index grating
exhibits also a soliton shape in the crystal volume. The
origin of this effect is the change of the contrast of light
due to energy transfer between coupled waves during their
propagation, which occurs in bulk crystals with strong
photorefractive gain.
• The theoretical description shows the possibility to control
the soliton properties by changing the input intensity ratio
and/or input phase difference of the wave.
• The effect can lead to diffraction efficiency management,
auto-oscillations and bistability of the output waves due to
wave-mixing in ferroelectrics.
• Results on the first experimental observation of nonuniform distribution of the grating amplitude profile and its
changes versus input intensity ratio are presented.
Band-transport model of recording of
non-local phase gratings
The photorefractive mechanism of recording refractive index grating
includes photo-excitation of free carriers by light interference pattern; their
movement due to diffusion and drift with systematic trapping by defects;
creation of the internal non-uniform distributed space charge field, which
modulate the refractive index.
In the case of diffusion dominant mechanism the grating is shifted relative
to light interference pattern by a quarter of the period.
Single level band
transport model
Photo-excitation of free carriers.
I is the light interference pattern
Trapping of free carriers.
Creation of internal space charge
distribution.
E is the space-charge field that
modulates the refractive index.
Non-local and local gratings recorded in the crystal
The theory of the
transmission four-wave mixing with
non-local gratings
The system that describes the changes of coupled wave amplitudes
inside the crystal has the following form:
A1
 i A2
z
A2*
 i A1*
z
A3*
 i A4*
z
A4
 i A3
z
where  is the photoinduced changes of complex grating amplitude:
 (t , r )   0   (t , r ),
 /  0  1
For the value  we can write the kinetic equation that describes the process
of grating recording due to photorefractive mechanism and the grating
relaxation with the time constant T0:
 

ˆ E
F
t
2
1


T0
In the case of diffusion dominant mechanism of the photorefraction the
change of the grating amplitude is proportional to the light intensity:
 
2
ˆ
F E  ( 1  i ) Ei (t , r ) E *j (t , r )
where  is the photorefractive gain for the non-local grating and 1 is the
maximum grating amplitude for local gratings.
In case of only non-local gratings the kinetic equation has the following
form:

 i A1 A2*  A4 A3*  



Solution for non-local gratings: the soliton
Introducing the new real variable u:
u(t , z ) 
  (t ,  )d
z
we have found the solutions for wave intensities from the equation set of
coupled waves:
A12 ( z ) 
A32 ( z ) 
1
1
d1 sin[ 2u ( z )  C1 ]  d1
2
2
A22  d1  A12
A42  d 2  A32
1
1
d 2 sin[ 2(u ( z )  u d )  C2 ]  d 2
2
2
d1, 2  A12,3  A22, 4

1, 2  (1, 4   2,3 )  (   )  0; 
2
ud=u(z=d)
where n is the nth wave phase and  is the grating phase.
Substituting these solutions in the kinetic equation, we obtained a damped
sine-Gordon equation:
 2u
u

 R sin(2u   )
z z
where =t/T0. R and  are constants defined from input intensities
I10, I20, I3d, I4d and input phase differences of coupled waves.
The steady state solution is:
C
 
cosh(2Cz  p)
where both
C
1
( I 2  I 4  I1  I 3 ) 2  4 
2
2
 const
and
p  const
may be found from the input conditions.
The grating amplitude has a soliton shape.
The solution describes a single stationary soliton in the coordinate
system (, z).
The wave-mixing diffraction efficiency is defined by the integral under
the grating amplitude shape: ~sin(2ud).
Two- and four-wave mixing.
Formation of the grating amplitude
profile along axis z.
Two-wave mixing. The energy transfer between two coupled waves (I1
and I2) is started from the input surface of the crystal. The grating
amplitude maximum is located near the input boundary.
Four-wave mixing. In the case of equal input intensity ratios at the crystal
boundaries (I10/I20=I4d/I3d) and a strong photorefractive response, the energy
transfer takes place only in the center of the crystal. The maximum grating
amplitude is located in the middle of the crystal.
Distribution of the grating amplitude and the intensities of coupled waves
along the crystal. d=10.
Optical control of the grating amplitude shape.
Diffraction efficiency management.
The stationary soliton shape is unequivocally defined by the input
intensity ratio and input wave phase difference.
The grating amplitude maximum is located near a crystal boundary in the
case of I10/I20I4d/I3d, or two-wave mixing, or usual four-wave mixing with
three input waves.
Alteration of the single wave-mixing soliton by changing input intensity ratio:
1) I4d/I3d=0.1 ; 2) I4d/I3d=0.5, I20=I3d=I4d ; 3) I10/I20=I4d/I3d=1 ; 4) I10/I20=I4d/I3d=0.01.
(Itotal=1; d=10).
The localized soliton can be moved through the medium from one side to
another.
The grating amplitude distributions for
some input intensity ratios. The numbers
on the top indicate lg((I10+I20)/(I3d+I4d)).
The dashed curves correspond to
disphased input coupled waves by  at the
crystal boundaries (10-20=0; 4d-3d=)
or (10-20=; 4d-3d=0).
Location of the grating amplitude
maximum of steady state gratings in
the case of FWM with four input
waves.
J1=J10+J20,
J2=J3d+J4d.
J10/J20=J4d/J3d=0.1, d=10.
Experimental set-up
Four writing beams from Ar-ion laser are expanded and converged in the
crystal to record a transmission grating. The expanded probe beam from
He-Ne laser is entered at the Bragg angle relative to the recorded grating.
We use specially grown congruently melted LiNbO3:Fe crystal with input
faces ac=35 mm and the largest dimension as a thickness b=12 mm. The
crystallographic dimensions a, b and c are connected with axes y, z and x,
respectively. Axis x coincides with the polar crystallographic axis c.
The diffracted pattern of the probe beam displays the grating
amplitude distribution along the crystal.
The optical scheme of four-wave
mixing with four input waves.
The space area of the
soliton formation inside
the crystal volume.
Experimental observation of
the grating amplitude distribution
We change the input intensity ratios of writing beams.
The left pictures show the photos of the space distribution of light induced
refractive index. The right pictures show the measured intensity distribution
of the diffracted probe beam along axis z.
(a) - I4d/I3d=0.08, I20=I3d=I4d ; (b) – I4d/I3d=0.14, I20=I3d=I4d ;
(c) - I10=I20=I3d=I4d.
In case of equal input intensities of all waves the maximum of the grating
amplitude is located in the center of the crystal.
The obtained experimental results of the soliton shape changes versus input
intensity ratio are in good qualitative agreement with the theoretically
calculated curves.
Optical control of localized gratings.
Diffraction efficiency management.
I10/I20=I4d/I2d=1; d=10
I10/I20=I4d/I2d=0.1; d=10
By changing the input intensity ratio or/and the phase difference of the input
waves, the localization degree of the grating amplitude profile changes. Thus
the wave-mixing diffraction efficiency defined as the integral under the grating
amplitude shape changes as well.
All optical space switching
I10/ I20>I4d/I3d or I4d=0; d=10.
I10/ I20<I4d/I3d or I10=0; d=10.
Braking the equal intensity ratio I10/I20I4d/I3d on crystal boundaries leads to
a movement of the grating amplitude maximum from one boundary to the
other one.
The scheme of optical space
switching. The waves 1-4 record the
grating with different amplitude
profile that determine the angle of
space diffraction for the probe beam
5-6.
Breather solutions of bond soliton states
The sine-Gordon equation admits multi-soliton solutions that describe the
interaction of several solitons as well as bond soliton states (pulsing, or so
called breather solutions). We obtained the breather solutions numerically
for the case of usual four-wave mixing with three input waves and for
certain area of input intensity ratios.
The breather solutions. I10/I20=3, I3d=0,87, I4d=0, d=15, (Itotal=1).
The oscillation of the grating amplitude causes the auto-oscillations of the
diffraction efficiency  and by this way the auto-oscillations of every output
intensities.
The reason of the bond-soliton behavior is the emergence of a local
component of the grating that causes the changes of wave phases during
their propagation. In this way, the light contrast changes with time, and the
grating is repeatedly erased and rerecorded. Hence the auto-oscillations can
be observed only in optically reversible media.
Conclusion
The wave self-diffraction from non-local phase gratings
in a photorefractive medium can be described by a sineGordon equation with a damped term in the case of
transmission geometry. The sine-Gordon equation reveals the
changes of the grating amplitude induced by light beam
interaction in the medium. The grating amplitude distribution
has a soliton shape in the direction of wave propagation. In
steady state the soliton is motionless and its parameters, i.e.
the amplitude, the half-with and the amplitude maximum
position, are unequivocal defined by input intensity ratio and
input wave phase difference. The crucial parameter of the
soliton is the energy transfer gain on a given distance z in the
medium. The photorefractive gain determines the change of
the light contrast during the wave propagation and this way
the soliton localization degree.
Alteration of the soliton shape of the grating amplitude
opens the ways to control the parameters of output waves. The
one of them is the all-optical management of the diffraction
efficiency, as the wave-mixing diffraction efficiency is
determined by the integral under the soliton shape.
Multisoliton and bond-soliton behaviors lead to bistability and
auto-oscillations of output intensities.
For the first time, we observed experimentally the nonuniform distribution of the grating amplitude and the control
of its profile by means of changes of input intensity ratio.
Research was supported by OTKA contract Nos. T23092, T26088, T35044,
by the Austrian-Hungarian Intergovernmental ST Program A-8/2001, and by
the Centre of Excellence Program ICA1-1999-75002.