High-temperature nonlinear phenomena in LNB crystals

Dedicated to jubilee of
Vladimir Zakharov
High-temperature nonlinear
phenomena in LNB crystals
Boris Sturman
Institute of Automation and Electrometrey,
Novosibirsk, Russia
In collaboration with the University of Bonn, Germany

Physical background: Nonlinear optical material
LiNbO 3 crystals – Silicon of optics? Why?
É Cheap and robust ferroelectric, T c ª 1200°C,
C 3v symmetry
É Transparent in the visible-to-infrared range (ª 4 eV forbidden gap)
É
High c (2) nonlinearity, large electro-optic coefficients
É Domain engineering: PPLN structuring, photonic crystals,
quasi-phase-matching, etc
É Sensitization by doping (Fe, Cu,...). Light-induced charge transport
É Generation of large electric fields and refractive index changes
(reversible optical recording)
Serious drawback: Optical damage
Unwanted generation of strong nonlinear index changes
and deterioration of light beams for I > (10 3 - 10 4 ) W/cm 2

A closer look at light-induced charge transport
j
e
=
−β
I
+
e µ n
e
e
E
+
eD
e
∂n
∂z
e
hw
m e
τ e
CB
ne
N e
α =
N
e
σ
e
Neσ
e
I
ne
=
hω
Neσ
e
β = e
hω
µ
eT
De
=
e
τ
l
e
pv
Light absorption coefficient
Free electron concentration
Photovoltaic coefficient
Einstein relation
Fe 2+ Fe 3+
n e

Why high temperature?
An ultimate goal of electron engineering:
To remove photo-excitable electrons
far below the background level, N e 200 °C , ions (H + , Li + , …) become mobile: j i
= e m i
(T) N i
E
In the electric equilibrium j S
= j e
+ j i
= 0, but j e
π 0
The electron removal persists
In other words, ions compensate the removed electrons and
maintain the charge quasi-neutrality

A trivial example
Steady state
+
+
+
+
LNB
-
-
-
-
j
j i e
1+
σ
e
/ σ
i
σ
eσ
i
| je,
i
| = E
pv
j
e
E
+
=
j
i
= −
E
e
pv
j
pv
σ + σ
+ ( σ
i
e
+ σ ) E
i
=
0
There are permanent fluxes of electrons and ions to the right

Ultra-slow shock wave of electron density
(PRL 2008)
First experiments:
U
± - 0
± +
LNB:Fe, N e0
~ 10 19 cm -3
T = 600 °C, U 0
= 1 kV
t ~ 1 hour, w < 100 mm
Nobody observed things like this earlier

Model
Assumptions and notation:
N
−
+
= N , N = N ( N +
e
i
Li
≈10
cm
20 −3
)
±
j± = eµ ±
N E, µ
±
∝ exp( −ε
±
/ T )
z = z t),
v = v(t)
0
0 (
E = U l,
ξ z / l
0 0
/ =
0
More predictions:
Time dependence
of the potential U l/2
(t)
Main predictions:
v = µ E
ξ =
a
t
t
0
Σ
=
=
=
Main physical requirement:
v = −µ
−E 2
/(1 +
( −1+
1+
2at
/ t )
( ) (
+ −
µ / µ + 1 N / N −1)
l
t
−
0
−
0
µ E
−
+
0
( 1+
a / 2)
aξ
)
0
0
a
0

Comparison with experiment
1: T = 650 °C, a = 7.4, t 0
= 15 min
2: T = 600 °C, a = 9.2, t 0
= 35 min
3: T = 550 °C, a = 9.8, t 0
= 94 min
No doubts remain

Optical cleaning: How to remove electrons optically
(NP-09)
Initial set of equations
b I(z)
∂N
∂t
e
+
∂J
∂z
e
=
0,
∂N
∂t
i
+
∂J
i
∂z
=
0
J
e
= β I
N
e
− µ n E
e
e
− D
e
∂n
e
∂z
J
i
= µ N E
i
i
− D
i
∂N
i
∂z
β I(z)
The photovoltaic drift velocity
∂E
=
∂z
n
e
4π
ε
sNeIτ
e
=
hω
0 0
( N − N − N + N )
i
e
i
e

Zero-model of optical cleaning
σ
I
0
e

Results: Static light beam, v = 0
Works, but takes a long time

Results: Moving light beams
A different situation
An exponential decrease/increase
at the divergence/convergence points
N
e
γ =
0
( ± z , t) / N = exp( ± γ t / t
1
2 (v / v
0
)
e
ln(v
0
/ v)
0
)
Two zero (fixed) points (±z 1
)
of the effective
velocity profile u(z)

Concentration profiles for moving beams
A v > v 0 scenario
Abandoning
of the cleaned area
An exponential cleaning,
No profile breacking

Beyond the zero-model:
Impact of nonlinear phenomena
Two parameters to control
the nonlinear behavior
A small parameter
to simplify theory
a
b
=
=
≈10
−3
0 d
0 0
σ
e
/ σ
i
< 1
0
td
= ε / 4π σ
Σ
0 0
Ne
/ Ni
< 1
t
/ t

Nonlinear effects
a = b = 0.01
b = 0
Breaking of
the concentration profile
Stabilization by electron
diffusion
Increase of N min
Impairment of
the cleaning performance

More nonlinear effects
b = 0
Ç Shift of the discontinuity backwards
Ç Separation of the beam from the cleaned area

More tricks
An asymmetric
flat-top I-profile
This ensures:
Ç Increasing of the exponential decay rate g
Ç Separation of the zero (fixed) points
Ç Flattening of the concentration peak
Ç Suppression of the impact of nonlinear effects

About experiment
At least four variable experimental parameters, I 0 , z 0 , v, s i (T)
Uncertainty of material parameters
At least two different cases:
Doped crystals (N e
0
> 10 16 cm -3 ). The profile N e
(z) can be monitored
Via absorption scan. The model parameters are known (measurable).
Not useful to remove electrons up to (10 11 – 10 12 ) cm -3 , but useful to
test the method
Undoped crystals (optically transparent). The initial concentration
N e
0
< 10 15 cm -3 cannot be measured. Model parameters are uncertain.
Promising for applications
Long-time treatments, t > 1 day, T = 180 °C, H + ions
The cleaning performance is yet far from optimum

N e 0 ª 6 ¥ 10 16 cm -3 , N i 0 ª 2 ¥ 10 18 cm -3 ,
z 0 ª 70 mm, l = 514 nm, I 0 = 10 W/cm 2 ,
v = 0, a ª 0.5, b

Thank you!