Transcript Lecture 9

Miscellaneous Other Third Order Nonlinearities
There are many mechanisms through which light interacts with matter. For example, there are
local third order nonlinearities associated with the coupling of light to molecular degrees of
freedom like vibrational and rotational molecular motions which are relatively fast and occur on
100fs to the 10’s ps times scales.
New nonlinear effects occur in the transition from single molecules to condensed matter. Van der
Waals, coulomb and other interactions can lead to co-operative behavior on the scale of optical
wavelengths or larger, and/or to the weak breaking of molecular symmetries. For example, in
liquid crystals intermolecular interactions and geometry effects lead to strong inter-molecule
coupling in the relative orientation of molecules and the nonlinear response can be highly
non-local, and very slow. In photorefractive materials, charge transport due to Coulomb
interactions subsequent to the absorption of light also results in non-locality. The nonlinearities
in these two classes of materials can be very large and hence can be very accessible for simple
experiments at low powers. Unfortunately, in some cases they have erroneously been referred
to as Kerr nonlinearities.
Electrostriction, thermal effects and cascading of second order nonlinearities also lead to
nonlinear changes in the refractive index (or the phase of light beams) non-locally. They are
characterized by the propagation of mechanical effects (sound waves), thermal effects (heat)
and the coupling between optical waves at different frequencies via (2) (“cascading’) respectively.
Single Molecule Re-orientation Effects in Liquids
Anisotropic molecules in a liquid re-orient in response to a light-induced torque, hindered by
viscosity and randomized by thermal fluctuations in the positional, rotational and vibrational
degrees of freedom. The “turn-on” time depends on the strength of the applied field, the
liquid viscosity and molecular shape. The “turn-off”, ps – ns, depends on the latter two.
Consider CS2
S
C
S
  
In molecule's frame of reference, x , y , z : P   : E

P induced in a molecule
Euler angles relate the polarization

to the induced polarization P in laboratory frame of reference

 0
0
  0  0
0
0  ||
 
1
The applied field at molecule : Eloc (r , t )  eˆz E z f (1) ( )ei ( kz t )  c.c.
2
There is a torque exerted on a molecule by the strong field
which re-orients it towards z. The corresponding potential is given by
2
1 
Vint   p  Eloc  Eloc
2
Temperature fluctuations of energy  kBT tend to randomize the
molecular orientation where kB is Boltzman’s constant.
Molecular reoriention times (e.g. ps in CS2) >> period of EM field oscillation
2
→ molecular re-orientation cannot follow the field at 2. It follows the time average of Eloc
.
(1)
4
2
2
1 (||    )[ f ( )] | E z |
(1)
4 2(||    )
 Pz  NE z
(||    )  NE z [ f ( )]
I
45
k BT
45k BTnc 0
 n2||,or ( ;  )  
N
(||    )2
n02 02c
45k BT
[  (1) ]4
similarly in x  y plane  n2 ,or ( ;  )  
For circularly polarized light : n2 ( ;  ) 
N
(||    )2
n02 02c
90k BT
N
(||    )2
n02 02c 180k BT
[  (1) ]4 
1
n2||,or ( ;  )
2
[  (1) ]4
By definition, the net index change saturates when all of the molecules are lined up!
Approximate “turn-on” and “turn-off” times can be obtained from the Debye rotational diffusion
equation in terms of the “order parameter” Q given by Q  3 cos 2 ( )  1  .
2
2
When the molecules are randomly oriented, Q=0 and when they are all aligned, Q=1.
2 (1) 2  D E z2
 Q  [ ]
( ||    ){1  exp[t /  D ]}  D   / 5k BT   viscosity
3

Also when field turned off, E z2  0  Q  Qmax exp[t /  D ]
Typical  D range from a few ps for simple molecules like CS2 to ns for large molecules.
Liquid Crystals
Nonlinear optics of liquid crystals is in some ways closely related to the previous case. Strong
inter-molecular forces between liquid molecules in the liquid state can lead to a unique form of
matter in which molecular “clusters” exist, aligned along a direction in space (“director”). Note
that in contrast to the solid state where X-ray diffraction patterns reveal 3D positional
correlation, there is no such positional correlation between the molecules. The orientational
correlation only exists over a finite temperature range above the melting point.
General Properties of Liquid Crystals
There are many “families” of liquid crystals. Most of the molecules can be considered to have
ellipsoidal shapes as shown below. The most commonly used and extensively studied molecule is
the nematic (at room temperature) 5CB, shown below.
CN
Examples of R and R’ are CnH2n+1,
CnH2n+1O, nitro, cyano (e.g. 5CB) etc.
C5H11
A single molecular structure can take on different liquid crystal ordering as temperature or the
side groups are changed. For example nCB is not a liquid crystal for n4, it is nematic for n=5-7
and then smectic for larger n. Note that the ordering is not perfect and is described by the “order
parameter” Q. The average of the direction of α|| over all molecules, n̂ , called the “director”.
Liquid Crystals Nonlinear Optics - Re-orientation Effects
10-11cm2/W
r 
 ( viscosity)
a(T  T *)
Temperature
(0C)
Temperature
(0C)
n2 (arbitrary units)
Relaxation tin time r ( 100 nsec)
As the temperature is increased above the nematic-isotropic phase transition, limited orientational
order persists over sub-wavelength volumes with directors ( n̂ ) not parallel to each other.
These clusters behave like large, highly polarizable molecules and can be oriented by strong
optical fields as discussed before for the single molecule case. The larger the cluster size, the
larger the nonlinearity and the slower the response time, as shown below. As the temperature is
increased, the cluster size decreases until this re-orientational nonlinearity reaches the single
molecule value (10-13cm2/W).
Temperature (0C)
Enhanced Orientational Nonlinearity: Freedericksz Transition in Nematic Phase
For an increasing DC field, a phase transition (Freedericksz
transition) occurs at which the molecules begin to re-orient. The
 K1
required field is given by
EF 
d 
where “d” is the plate separation, ε=ε||-ε, and K1 is the “splay”
Frank elastic constant (10-11 newtons).

E
Planar molecule alignment
in zero field
ne θ  
n n||
n||2 cos 2 θ  n 2 sin 2 θ
The Freedericksz transition can also be induced by the
2 as shown previously for single
time averaged E opt
molecule re-orientation.
Application of both a bias DC field and an optical
field in transparent liquid crystals leads to nonlinear
effects with very small optical powers, of order mW.
Giant Orientational Optical Nonlinearities in Doped Nematic Liquid Crystals
A photosensitive dye or molecule dopant is used to mediate, facilitate and enhance the
reorientation process. Largest effects are obtained with molecules which undergo
trans-cis isomerization, e.g. azobenzenes.
Change in liquid crystal molecular
Conformational change on photon absorption
alignment due to isomerization.
by azobenzene molecules
n2  103 cm2 / W
Thermal Nonlinearities
The large n/T in the region is a consequence of the
rapid decrease with increasing temperature of the size of
the aligned regions as the nematic to isotropic liquid
crystal transition is approached. At the temperature at
which the difference ne-no vanishes, the domains
become sub-wavelength and the material strongly
scatters the light. These effects can be enhanced by
including other strongly absorbing molecules. The
resulting nonlinearities can attain values of 1 cm2/W ,
usually with large attenuation coefficients.
Photorefractive Nonlinearities
These non-local nonlinearities occur due to a combination of physical phenomena which
indirectly lead an index change that depends on the input intensity in electro-optic active media.
Photorefractive materials have electron donor and acceptor states (defects, dopants etc.) which
are located between the “valence” and “conduction” bands.
Conduction band
Ionized electron donor states
Electron donor states
Electron acceptor states
Valence band
The magnitude of n2,eff can be very large. However, there is a nonlinearity - response time
 constant trade-off. It is the accumulated energy absorbed from a beam that is the key
parameter. There are a number of different photorefractive mechanisms possible which give
rise to nonlinear effects.. We focus on the steady state response due to the diffusion and
screening mechanisms which in electro-optic active materials lead to index changes.
Screening Nonlinearity
No Applied DC field
1. Electrons are promoted from neutral electron donor states into the “conduction
band” by the absorption of light, resulting in ionized electron donor states.
  N
Typically N D  N A
e
Ionized electron donor states
Electron (neutral) donor states
Electron acceptor states




Ne N D
N D
1

I  I d  N D  N D 

t
h
R
recombination time
equivalent (dark) intensity due to
thermally excited electrons




1

Continuity condition :
N D  Ne    J  0
t
e
Initially density of electrons in the conduction band is highest at beam
maximum.
-- -- --
-
-
x
2. Transport of electrons in response to:
- Diffusion to regions of lower electron
density Ne in conduction band which
creates local space charge fields Esc
electron mobility
carrier diffusion constant
Diffusion current : J x  eD

N e
N
 k BT e
x
x
 



Gauss' es Law :   Esc  e N D
 Ne  N A
Conduction band
- +
+ ++
- +
x
Ionized electron donor states
Electron donor states
Electron acceptor states
Valence band
3.
Electrons trapped in acceptor sites.
Conduction band
+
+ ++
- -
-
Valence band
+
- -
Ionized electron donor states
Electron donor states
Electron acceptor states
-++ +
- -+ + +-++ +
- -+ + +-++ +
----

Since electrons trapped in new sites, charge separation produces “space charge” field ESC
N e
J x  e[ N e  N th ]Esc  k BT
x
“background” Nth in conduction band (thermal excitation etc.)
N e
k BT
Steady State : J x  0  Esc  
e[ N e  N th ] x
k BT I
Since N e  I  E sc  
e[ I  I d ] x
5. Refractive index change via electro-optic effect n  n03reff Esc
4.
The “turn-on “and “turn-off “ response times are usually very slow
The “turn-on” time, s→secs, is determined by the input integrated
absorbed flux and the carrier diffusion time. The “turn-off” time
depends on the thermal excitation rate of carriers and their diffusion time.
Screening Nonlinearity: Applied DC Bias Field
An additional possibility is to use a strong bias field so as to “overcome”
diffusion effects. In this regime the net field across the beam eclipses
small diffusion effects and hence the net space charge field varies with
the optical intensity (not with its derivative as with diffusion). The space
charge field opposes the applied field, reducing the net field in the region
of the optical beam. Furthermore, steady state “turn-on” time can also
be reduced by illuminating the whole sample uniformly, called I
which contributes an extra uniform background Ne0.
In steady-state J = constant, and in the limit that the diffusion terms can be neglected and
relatively broad beams, i.e, 1>> Esc / x , the space charge field is given by
Esc  E0
I  Id
I  Id
I I
1
1
n   n03reff Esc   n03reff E0  d
2
2
I  Id


E0

E0
Etotal
n  0
Index depressed less in this region
Material
Dopant

(m)
n3reff
(pm/V)
Sr0.75 Ba0.25 Nb2O6
Ce
0.4-0.6
Sr0.6 Ba0.4 Nb2O6
Ce
BaTiO3
InP
*
diel(sec)
nmax
EDC
(KV/cm)
17390
0.1-1.0
0.005
3
0.4-0.6
3000
0.1-1.0
0.0014
3
Fe
0.4-0.9
21,500
0.1-1.0
0.005
2.5
Fe
0.9-1.3
52
10-6-10-4
**5x10-5
8
*at an intensity of 1 Watt/cm2
**
with enhancement can go to 5x10-4
Novel /2 Phase Shift Intensity Between Index Change Maxima
- Gratings induced in photorefractive
media have some unusual properties.
E1
Spatial Intensity Distribution
E2
I(x)
 

1
I (r , t )  c 0 n{| E1 |2  | E 2 |2
2
 E1E 2*e 2ik sin( ) x  c.c.}

1
For real fields : I (r , t )  c 0 n{| E1 |2
2
(x)
Esc(x)
 | E 2 |2 2E1E 2 cos[ 2k cos( ) x]}
Note the /2 phase shift between I(x) and n(x).
n(x)
Nuclear (Vibrational) Contributions to n(I)
e.g. CO2 molecule
No vibration
(at rest)
Light couples via electric dipole effects to the vibrational
(non-electronic) normal modes in matter. This leads to
significant contributions to n2 (10-20% in glasses). The
formulation below is for the cw case normally valid for
pulse widths > 10ps.
The vibrations modulate the molecular polarizability


 mn
L


 m  polarizabi lity tenso r  αm   qn

q

0

n


qn



Summation over all
Vibration amplitude
vibrational modes “”
Raman
(optically driven)
hyperpolarizability.
When an optical field of frequency  is applied, this
gives rise to a nonlinear polarizability in the molecule
pNL
Vibrating
molecule
  qn


  n
qn

qn
(1)
[

]E
0
From classical mechanics, there is an all-optical force which induces the vibration in the ’th
mode with frequency  (approximated as a simple harmonic oscillator) in the molecule given by
1  n
(1) 2

1  
2 


[

]
E
E

m
[
q

2

q


q
]

 
 n
 n
 n
qn 0
2 qn
where m is the effective mass associated with the vibration.
Fn 
1  n
2 qn


qn
(1) 2
[

] E E
0




1 2
2i ( k r t )
E (r , t ) E (r , t )  E ( )e
 c.c.
4
Vibration driven at 2,
“virtual” state
small response

Vibration driven at 0,
“virtual” state
net displacement of atoms

1
qn (r , t )  Qn  c.c. 
-
2
vibrational states



1  2i ( k r t )
qn (r , t )  Qn e
 c.c.
2
vibrational states
 
1
| E ( ) |2 c.c.
4

| E ( ) |2   n
Keeping only the zero frequency term : Qn (r , t ) 
4m D(0) qn


qn
(1) 2
[

]
0
Solving for the optically driven displacement and substituting into the nonlinear polarization gives
2

  


1
 n   [  (1) ]4 | E ( ) |2 E ( )ei (k r t )  c.c.}
p NL (r , t )  
2 
 qn  0 
8
m


q

  
n


Proceeding as before for p NL ( r , t )  n NL
SVEA
 n2||,nuc
   n
1

2
2
2 

 4n ( )m  0 c   qn
2
 (1) 4
[ ] .
qn  0 

Fractional contribution of n2,nuc to the total n2
Glass
Wavelength
Nuclear
Fraction
Measurement
Method
Fused silica (SiO2)
visible
15-18%
Raman
87% GeS2-13%Ga2S3
825 nm
135%
35fs OKE
64%PbO-14%Bi2O3-7%B2O3-15%SiO2
825 nm
125%
35fs OKE
0%-50%GeO2 in GeO2-SiO2
800 nm
13-18%
18fs SRTBC
e.g. linear molecule CS2 (liquid)
Single pulse measurements
n2,ef cm2/W)
n2,ef cm2/W)
Re-orientational
SRTBC – spectrally resolved two beam coupling
Vibrational
Kerr
Electrostriction
Consider a capacitor
+++++++++++++
x

E
----------------------------
[Universal mechanism, always has the same sign (+ve)]
Due to the presence of the + and – charges there is an electric
field and a compressive force squeezing the medium
This compressive force produces a strain field S k , S11 in this
case, associated with the electric field.
 
ΔV


u



   u   k   Skk
V0


xk
generalized strain Sk 
1  uk u 

2  x xk 
Elasto-optic interaction


Pi NL (r , t )   0ni2n 2j pijk Sk E j (r , t ) pijk  elasto - optic coefficien ts
compressive forces →  increase (density change>0) → increase in local EM field energy density
Work done in compressing the medium (U) = Increase in EM energy density (W)
2
n x6 p11
n2||,est ( ;  ) 
4 Kc
Note that  0n x4 p11   e " electrostrictive constant"
For a weak (probe) y - polarized beam n2 y,est (;  )  n2,est (;  ) 
 0n3y n3x p11 p21
4 Kc
“Turn-on” and “turn-off” times are a complex issue because turning on or off an optical beam
involves compressive forces. They lead to the generation of a spectrum of acoustic waves. The
acoustic decay time s(s)  s-2 and the details of beam shape, sample boundaries etc.
influence the acoustic spectrum generated which includes both compressional and shear waves.
Material in beam path densified. When beam turned off,
sound waves generated
Sound waves generated
In an “infinite” medium, the shortest “turn-on” and “turn-off” times are given by the acoustic
transit time across the optical beam [beam diameter]/vS, with vS ~1 micron/nsec giving s-ns.
Polarization
 (m)
Elasto-optic
coefficient
K (1010
m2/newton)
n
Fused silica
(0.63)
p11= 0.12
3.69
1.46
0.4x10-16
GaAs
[110] (1.15)
= 0.14
7.6
3.37
1.6x10-13
Al2O3
[001] (0.63)
p33= 0.20
27.0
1.76
3.7x10-17
polystyrene
(0.63)
p11= 0.31
0.54
1.59
2.4x10-15
p11= 0.32
0.083
1.33
5.7x10-15
Material
Methanol
n2,eff
(cm2/W)
Thermo-Optic Effect
This is a very complex problem in general which can be simplified in some useful limits.
 n 
 n 
δn      
 T .
 T  
  T
α1 (absorption coefficient) ρ (density)
Cp (specific heat)
 (thermal diffusion constant
The local temperature is given by the thermal diffusion equation
(T )

Q


 2 (T ) 
 1 I
t
C p
C p C p
where Q is the absorbed power per unit volume per unit time.
1
Note that  2 / C p has the units of inverse time which we define as  th .

2
Gaussian Input Beam : I (r , t )  I 0 ( z ) exp[r 2 / w02  t 2 /  opt
]
Assume  th   opt so that maximum temperature distribution has the spatial distribution
2
2
  t /  opt


1

δTmax (r ) 
I (r )  e
dt    opt 1 I (r )

C p
C p
1  2 
4
r2
2
 δTmax (r )  
 [δTmax (r )]  

{1 
}δTmax (r )
2
2
2
 r r r 
w0
w0


δTmax (r , t )
4
 2
Tmax (r , t )
t
w0 C p

 δTmax (r , t )  δTmax (0, t )e

t
 th
with  th  w02 C p / 4 .
Material
GaAs
Al2O3
NaCl
ZnO
Acetone
C6H6
methanol
th (ms)
0.080
3.1
0.72
0.39
45
24
20
dn/dT x10-4(/oC )
1.6-2.7
0.13
0.25
0.1
-5.6
-6.2
-4.0
For  th   opt
t
t  1.76 opt
n (t )
Decay time th
t



n
n

n

δnmax (r )  [ ]δTmax (r )   [ ] opt 1 I (r )  n2,th   [ ] opt 1
T
T
C p
T
C p
 2
23 / 2 n 1
3
3
 pulse energy E pulse   / 2  opt I (r ) w0  δnmax 
[ ]
E pulse
2 T  C
w0
p
For high repetition rates (mode-locked lasers), the key question is the
energy accumulation over all the pulses within the time window th!
e.g. For a mode-locked laser operating with 1ps pulses at a repetition rate
of 100 MHZ accumulates energy from 103 pulses over th giving
a cumulative
12
2
n2, th  1.2 x10
cm / W (bigger th an Kerr!)
(3) Via Cascaded (2) Nonlinear Processes: Non-local
A nonlinear phase shift in
the fundamental beam
occurs when the
non-phase-matched
second harmonic
generated from the
fundamental interacts
with it on propagation.
That is, propagation
is required making the
process non-local.
Low Fundamental Depletion Approximation
By analogy to Kerr :  n2,nlcas ( z ) 
( 2) 2
4[d eff
]
c 2 0 n 2 ( )n(2 )k
sin 2 (
kz
).
2
VNB: Sign of nonlinearity depends on sign of k, i.e. can be self-focusing or self-defocusing!
But, this is not really a n2 process since there is no refractive index change. What can be
measured is a nonlinear phase shift NL.

NL
( L) 
( 2) 2
2 2 [d eff
]
c  0n ( )n( 2 ) k
2
2
Transmission : T ( )  1 
L{1  sinc[ kL]}I ( )
( 2) 2 2
2[d eff
] kvac ( )
n(2 )n 2 ( ) 0c
L2 I (0,  )sinc2 (
kL
)
2
Maximum n2,casc occurs at kL  1.6
 n2,nlcas  0.36
( 2)
( 2)
4[k vac( ) L] mj
(

2

;

,

)


imk ( ;2 , )
n2 n2 c 0
e.g. DSTMS,  coh  3.6m : maximum n2, nlcas =4x10-13cm2 /W
e.g. QPM LiNbO3, L=1cm: maximum n2, nlcas =2x10-12cm2 /W