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NONLINEAR NANOPTICS
M. BERTOLOTTI
Università di Roma “La Sapienza”
Dipartimento SBAI
Erice 2012
1
Index
Introduction
Linear and Nonlinear Systems
Nonlinear Susceptibility
Nonlinear Optical Interactions
Nonlinearities in Metals
Composite Materials
Erice 2012
2
Introduction (1)
It is now possible to fabricate artificial structures whose
chemical compositions and shapes are controlled with a
nanometer accuracy. These man-made systems are too small
to behave like the bulk parent compounds and too big to
behave like atoms or molecules.
They may present new properties and because we can tailor
some of these for specific purposes, nanostructures have the
potential for revolutionary applications in electronics and
optics.
Furthermore, we know from the basic principle of quantum
mechanics that when the size of a system becomes
comparable to the characteristic length scale that determines
the coherence of the wave functions, quantum size effects
occur. The properties then become size and shape dependent.
3
Introduction (2)
Optical properties in particular are very sensitive to quantum
confinement.
In semiconductors, for example, when a photon is absorbed it
creates an electron-hole pair, and the two particles interact strongly
by electrostatics. Because the effective masses are small and the
dielectric constant large, the exciton (bound state of the electronhole pair) is large on the atomic scale. Depending on the material,
the corresponding Bohr radius a0, which sets the natural length scale
for all optical processes, is about 50 Å to 500 Å.
One can observe massive changes in optical properties when the
dimension of a semiconductor sample, in one or several directions,
approaches a0.
4
Introduction (3)
The optical response of a material is the average of the material
properties in a volume larger than 3.
In ordinary materials the ultimate blocks are atoms or molecules and
the optical response is the refractive index n.
Np  P  0 E
r  1    1 
P
0 E
n  r  1 
P
0 E
p dipole moment of the atom, N number of atoms per unit volume.
If the material is made by nanostructures with some optical
properties, the optical response of the material is an average index
nav which can be tailored according to the properties of the
nanostructures.
5
Introduction (4)
If the typical dimension of the structure is of the same order of ,
confinement effects may occur and the structure may behave as a
resonant cavity. Some consequences are:
- a change of the density of electromagnetic modes,
- Electromagnetic field confinement and enhancement,
- light velocity may be slowed down or become superluminal,
- nonlinear processes may be modified,
- suppression or enhancement of spontaneous emission may occur,
- thermal radiation distribution may be modified.
6
Nonlinear Optics
Linear and Nonlinear Systems
A linear system is defined as one which has a
response proportional to an external influence and
has a well-known property, i.e. if influences, F1, F2
…., Fn are applied simultaneously, the response
produced is the sum of the responses that would be
produced if the influences were applied separately.
A nonlinear system is one in which the response is
not strictly proportional to the influence and the
transfer of energy from one influence to another can
occur.
7
If the influences are periodic in time, the response of
a nonlinear system can contain frequencies different
from those present in the influences. However, the
point to emphasize here is that, as well as the
generation of new frequencies, nonlinear optics
provides the ability to control light with light and so to
transfer information directly from one beam to
another without the need to resort to electronics.
Traditionally, nonlinear optics has been described
phenomenologically in terms of the effect of an
electric field on the polarization within a material.
8
MAXWELL EQUATIONS
Electromagnetic processes are described by
Maxwell’s equations which constitute a set of linear
equations. In SI units:
  D 
D
B B
  E E  
t t
  B 0B  0
D D
H

 H  J  J
t t
div Ddiv

D
B B
rot Erot
 E  
t t
div Bdiv
 0B  0
D D
rot Hrot
 H  J  J
t t
where E and B are the electric and magnetic fields.
The displacement fields D and H arise from the
external charge and conduction current densities 
and J. In most cases of interest in nonlinear optics,
 = 0 and J = 0.
9
Constitutive relations connect the charge and current
distributions within the medium and the displacement
fields to the electric and magnetic fields.
D  P  0E  E
B  H
where P is the induced polarization in the medium
resulting from the field E,  is defined as a dielectric
constant and 0 is the permittivity of free space
(8.85 × 10-12 F m-1 in MKS units). Optical materials
are mostly non magnetic  = r 0 = 0.
10
LINEAR THEORY
Usually one assumes a linear response of a
dielectric material to an external field
P  0E
where P is the vector representing the electric dipole
moment per unit volume induced by the external
electric field E, 0 is vacuum permittivity and  is a
quantity characteristics of the considered material
with no dimensions, called electric susceptibility.
11
In general  is a tensor
 xx  xy
Px
Py  0  yx  yy
Pz
 zx
 zy
 xz E x
 yz E y
 zz E z
The symmetry properties of the material indicate
which ones of the ij coefficients are zero.
Alternatively
Dx  xx  xy
D y   yx  yy
 xz E x
 yz E y
 zx
 zz E z
Dz
 zy
12
HOMOGENEOUS MATERIALS
 not depending on space

div E 

div E  
rot E  
If
B
t

rot E  
B
t
div B  0
div B  0
E 
B
rot     j 



t

  
E 
rot B    j 


t


0 0 j0 j0
div E div
0 E0
div B div
0 B0
B
E
B
E
rot E rot
 E
rot B rot
 B  
t
t
t
t
13
WAVE EQUATION
div E  0

B
rot
E



t

Starting from Maxwell’s Eqs
div B  0div B  0

D
rot
B



0
t

 2E 1  2E
 E  0 2  2 2
t
v t
c
c
v

r n
2
 2E 1  2E
 E  0 2  2 2
t
v t
2
v
c
c

r n
c
1
c
00
1
00
refractive index
n  r
14
DISPERSION
Experimentally the refractive index is a function of
wavelength (frequency)
n( )   r ( )
 r (  )  1  (  )
This phenomenon is called DISPERSION.
The polarization in a material medium can be
explained considering the electrons tied to the atoms
as harmonic oscillators.
Nucleus: ~2000 electron mass, i.e., infinite mass
15
DISPERSION
mx   x  kx  eE0e
Damping Restoring
Force
i t
(one-dimensional model)
Driving
Force
From the solution:
eE0ei t
x
m(02   2  i )


m
02 
k
m
E
the induced moment is calculated:
p  ex 

e2
m 02  2  i
e-

x
 E0eit
16
For N oscillators per volume unit, the polarization is:
N  e2
it
PNp
E
e
0
m 02   2  i

Calling



e2
m(02
   i )
2
 atomic polarizability
E  E 0eit
P  N E   0  E

N
0
where  is the electric susceptibility.
 N 
 0 r   0 (1   )   0  1 


0 

N
2
n  1   1
n  r
0
17
2
Ne
n2  1 
2
2
 0 m(0    i )
If the second term is lower than 1 (as it happens in
gases):
Ne2
n  1
2 0 m(02   2  i )
In the expression n comes out to be a complex
number.
18
ABSORPTION
The term i is responsible for absorption. The
complex index can be written as
Ne2 (02   2 )
Ne2
n  n  ik  1 
i
2


2
2
2 2
2
2 2
2 0 m 0  
 
2 0 m 0  
  2 2 








If we consider a plane wave
1.0
E  Aexp i(t  kz)
k    

c
n
2

-1
( a. u. )
where
0.5
0
n
-0.5
-3.0
-1.5
0
1.5
3.0
0
19
we see that, substituting the complex refractive
index, one has
2
k
(n  ik)

which gives
 2 k 
2 n  
 
E  A exp i   t 
z   exp  
z
 
 
  
The last exponential represents a term of attenuation.
The attenuation coefficient may be defined from:
1 dI
2
 
I(z)  E  I (0)e  z
I dz
By comparison with the previous equation
 2 k 
2 n  
 
E  A exp i   t 
z   exp  
z
 
 
  

4

k
20
It can be noticed that a small value of k leads to an
elevated attenuation.
k  0.0001 and   0.5m gives   25 cm 1.
1.0
( a. u. )
k
n -1
0.5
0
-0.5
-3.0
-1.5
0
1.5
3.0
0
Both n and k are functions of the frequency.
21
METALS
In a metal the electrons are free and they do not
oscillate around the atoms. Therefore k = 0 and
0 = 0.
In the equation for n2 it is sufficient to put 0 = 0.
2
Ne
n2  1 
 0m( 2  i )
N  density of electrons
If  << 
n  1
2
p 2

2
p 2
For Al, Cu, Au, Ag
Ne2

0 m
Plasma frequency
N ~ 1023 cm-3 and P~ 2.1016 s-1.
22
For  > P
n is real and the waves propagate freely.
For  P n is pure imaginary and the field is
exponentially attenuated with the distance from the
surface. Therefore the radiation is reflected from the
surface.
Therefore, for visible and infrared radiation  < P
and n is imaginary. In general, n is complex because
of :
  i 
Ne2

Ne2
Ne2
2
1 n 

 i

2
2
2
2
0m    i      i   0m     
0m     
p
i
p
2   2  2   2 


23
NONLINEAR SUSCEPTIBILITY
Dipole moment per unit volume or polarization in
the linear case
Pi  Pi   ij E j
0
The general form of polarization in a nonlinear
medium is
Pi  Pi  χ E j  χ
0
(1)
ij
(2)
ijk
E j Ek  χ E j Ek El  
(3)
ijkl
24
JUSTIFICATION OF THE PRESENCE OF A
NONLINEAR RESPONSE
If the force exercised by the electric field
of the wave becomes comparable with
the Coulomb’s force between the electron
and the nucleus, the oscillator is
perturbed (anharmonic oscillator) and,
at the lower level of the perturbation, we
can write:
x(t)  x(t)  02 x(t)  Dx 2 (t)  (e / m)E(t)
E
(4)
x
25
The solution of eq.(4) can be expressed as the sum of
two terms
(1)
(2)
x(t)  x (t)  x
(t)
(5)
in which x(1)(t) is obtained solving eq.(4) without the
anharmonic term, whereas x(2)(t) is considered a
small correction of the solution at the first order x(1)(t)
and is obtained utilizing x(1)(t) in the anharmonic term
x
(2)
(t)  x
(2)
(t)  02 x (2) (t)
2
eE(t)
(1)

 D  x (t)  .
m
(6)
In this way, considering the case in which the forcing
electric field is formed by the sum of two fields at
different frequencies
1
E(t)  E1 cos 1t  E 2 cos 2 t   E1e  j1t  E 2e  j2 t  c.c.
2
(7)
26
We have the solution at the first order
1
x (1) (t)   x (1) ( 1 )e  j1t  x (1) ( 2 )e  j2 t  c.c.
2
(8)
and subsequently the solution at the second order,
solving eq.(6) with the use of (8) is
1
x (2) (t)  [x (2)  1  2  e  j 1 2  t  x (2)  1  2  e  j 1 2  t 
2
 x (2)  21  e  j2 1t  x (2)  22  e  j2 2 t  c.c] (9)
in which
x
(2)
x
(2)
1
D(e / m) 2
E1E 2






 1 2
2 02  12  j1 02  22  j2  02   1  2 2  j  1  2 





1
D(e / m) 2  E 2k
(2k )  
; k  1,2.
2
2
2
2
2
2     j
0  4k  jk
0
k
k



(10)
27
Therefore the solution of the second order brings to
the generation of oscillations at a frequency
different from the ones of the forcing field. In
particular, it is possible to have frequencies equal to
the sum or to the difference of the field frequencies
or to the double (second harmonic). Moreover, we
emphasize that the previous formulas remain valid
also if just a single forcing field  is present. In this
case x(2)(t) will be the sum of a second harmonic
term (2) and a null pulsation term (term of optical
rectification).
28
Remembering the expression for the medium
polarization, we can write
P(t)   Ne  x (1) (t)  x (2) (t) 
(11)
where N is the number of dipoles for volume unit
P(t)  PL (t)  PNL (t)
(12)
which, compared with (10)
x
(2)
x
(2)
1
D(e / m) 2
E1E 2

 1  2   
2 02  12  j1 02  22  j2  02   1  2 2  j  1  2 





1
D(e / m) 2  E 2k
(2k )  
; k  1,2.
2
2 2  2  j
02  42k  jk
0
k
k

permits to write


(10)
PL  0(1) E
PNL  (2) E  E.
(13)
29
SECOND HARMONIC PRODUCTION
The nonlinear properties in the optical region have
been demonstrated for the first time in 1961 by
Franken et al. during an experiment of second
harmonic generation. Sending red light of a ruby
laser ( = 6.943 Å) onto a crystal of quartz, they
observed ultraviolet light.
30
The story is that when Franken published his paper on the
second harmonic production, the proof reader corrected the
photo, suppressing the tiny spot of the second harmonic
believing it was just a smash. You may see this on the photo
below in which under the arrow you see….nothing!
31
In many crystals the nonlinear polarization depends
on the direction of propagation, on the polarization
of the electric field and on the orientation of the
optical axis of the crystal. Since in such materials the
vectors P and E are not necessarily parallel, the
coefficient  is a tensor. The second order
polarization can be written as
Pi(2)   d ijk E jE k
(14)
j,k
where i, j, k represent the coordinates x, y, z. The
main part of the coefficients dijk, however, are usually
zero and so only a few of them must be considered.
32
Only the non-centrosymmetric crystals can have a
non null tensor dijk. In facts, let us consider an
isotropic crystal. In this case dijk is independent from
the direction and therefore it is constant. If now we
invert the direction of the electric field, also the
polarization must change sign, that is
 Pi(2)   d ijk (  E j )(  E k )  d ijk E jE k   Pi(2) .
It is clear that, not being able to be  Pi(2)   Pi(2) , dijk
must be null.
Moreover, in materials for which d ≠ 0, since no
physical meaning can be assigned to an exchange of
Ej with Ek, it must be dijk = dikj.
33
Now if we consider the Maxwell equations writing
we have
D  0 E  P
(15)
D
E
P
rot B  j  
 j  0

t
t
t
B
rot E   .
t
(16)
The polarization can be written as the sum of a linear
term plus a nonlinear one
P  0L E  PNL
(17)
In case of materials with second order nonlinearity PNL
may be written as
 PNL i   dijk E jE k .
(18)
34
So eq.(16) can be written, assuming j = 0
PNL
E
rot B  

t
t
(19)
from which
2
2

PNL

E
2
 E   2  
.
2
t
t
(20)
If we consider the unidimensional case of propagation
along a direction z, we have
2

(PNL )i
 Ei
 Ei
  2  
.
2
2
z
t
t
2
2
(21)
35
Let us consider now three monochromatic fields with
frequencies 1, 2, 3 using the complex notation
1
 E1i (z)e j 1t  k1z   c.c.

2
1
( 2 )
E k (z, t)  E 2k (z)e j 2 t  k 2 z   c.c.

2
1
j 3 t  k 3z 
( 3 )
E j (z, t)  E 3 j (z)e
 c.c.

2
E (i 1 ) (z, t)
(22)
where the indices i, j, k represent the components x
or y.
The polarization at frequency 1 = 3 - 2, for
example, from (18) and from (22) results
Pi( 1 )
1
j 3 2  t   k 3  k 2 z 

  d ijk E3 j (z)E 2k (z)e
 c.c.
2 j,k
(23)
36
Substituting eqs.(22) into (21) for the component E1i,
it is necessary to calculate
 2 E( 1 ) 1  2

E (z)e  1t  k1z   c.c..
2
2  1i
2 z
z
If we assume
dE1i
k1
dz
d 2 E1i
dz 2
(24)
(25)
we have
 2 E(i 1 )
1 2
dE1i (z)  j 1t  k1z 
   k1 E1i (z)  2 jk1
e
 c.c.
2

2
dz 
z
(26)
with similar expressions for
 2 E(j2 )
z
2
and
 2 E(k3 )
.
2
z
37
Finally, substituting (26) and (23) into (21) we have
dE1i (z)
 
  j 1 0  d ijk E3 jE2ke  j k3  k2  k1 z  c.c.
dz
2 1
(27)
and in analogous way
dE2k j2 0
  j k1  k 3  k 2 z

d
E
E
 c.c.

ijk 1i 3 je
dz
2 2
3 0
 j k1  k 2  k 3 z
 j
d
E
E
e
 c.c..

ijk 1i 2k
dz
2 3
dE3 j
(28)
The second harmonic field is obtained immediately
from (27) and (28) for the case of 1 = 2 and 3 =
21. Therefore it is enough to consider only, f.e., (27)
38
and the last one of (28).
To further simplify the analysis we can assume that
the power lost by the frequency 1 (fundamental) is
negligible, and therefore
dE1i
dz
0.
(29)
So it is sufficient to consider just the last one of (28)
dE2k j2 0
  j k1  k 3  k 2 z

d
E
E
 c.c.

ijk 1i 3 je
dz
2 2
dE3 j
dz
 j
3 0
 j k1  k 2  k 3 z
d
E
E
e
 c.c..

ijk 1i 2k
2 3
0
d jik E1i E1k e jkz

dz

3
  1 
2
dE3 j
where
and
  j
k  k (3j)  k1(i)  k1(k) .
(31)
(28)
(30)
39
In eq.(31) k1(i) is the constant of propagation of the
beam at 1 polarized in the direction i. The solution of
(30) for E3j(0) = 0 for a crystal of length L is
or
0
e jkL  1
E3 j (L)   j
d jik E1i E1k


jk
0 2
I(L)  E3 j (L)  

2
 d jik E1i E1k
2
2
sen
k  L / 2 

2
L
.
2
 k  L / 2 
(32)
According to (32) a requirement for an efficient
second harmonic generation is that k = 0, that is
from (31) with 3 = 2, 1 = 2 = 
k  k (3j)  k1(i)  k1(k) .
k 2  2k( ) .
(31)
(33)
40
If k ≠ 0, the second harmonic wave generated at a
generic plane z1 which propagates until another
plane z2 is not in phase with that generated in z2. This
produces an interference described by the factor
sen 2  k  L / 2 
 k  L / 2 2

I(L)  E3 j (L)  0 2

2
 d jik E1i E1k
2
2
sen
k  L / 2 

2
L
.
2
 k  L / 2 
(32)
in (32).
k 2  2k( ) .
(33)
The condition
is never
practically satisfied because, due to dispersion, the
refractive index depends on .
41
In general we have
k  k
(2 )
 2k
( )
2 (2)
( )

n
n
c


(34)
being
k
( )
( )
n

c
(35)
and therefore
k  0.
42
Coherence length
We have no more second harmonic production when
sin(ΔkL/2)/ΔkL/2 = 0
This is achieved when ΔkL/2 =π
Which means L = 2π/Δk
L is named coherence lenght.
However, it is possible to make k = 0 (phasematching condition) using various skills; the most
used of which takes advantage from the natural
birefringence of the anisotropic crystals. From (34) we
can see that k = 0 implies
n
(2 )
n
( )
(36) k  k
(2 )
 2k
( )
2 (2)

n
 n( )
c


(34)
so that the refractive indices of second harmonic and
of fundamental frequency have to be equal.
Using birefringent materials it is possible to fulfill the
condition of phase matching.
44
As another example a PBG may be used to obtain phase
matching
45
Finally we point out that very well known electro-optic
effects discovered in the XIX century are particular
cases of nonlinear optics.
Δεr = qEo
Pockels effect
P = εoχE
εr = 1 + χ
P = εoχE + χ(2) Eo E = εo(χ + χ(2) /εo Eo )E = εo χ’E
χ’ = χ+ χ(2) /εo Eo
1+ εr + Δεr = 1 + εr +χ(2) /εo Eo
Δεr = χ(2) / εo Eo
46
Kerr effect
Δεr = qEo2
P = εoχE
εr = 1 + χ
P = εoχE + χ(3) Eo2 E = εo(χ + χ(3) /εo Eo2 )E = εo χ’E
χ’ = χ+ χ(3) /εo Eo2
1+ εr + Δεr = 1 + εr +χ(3) /εo Eo2
Δεr = χ(3) / εo Eo2
47
Nonlinear Optical Interactions
• The E-field of a laser beam
~
 it
E (t )  Ee  C.C.
• 2nd order nonlinear polarization
~ ( 2)
P (t )  2  ( 2) EE *  (  ( 2) E 2 e 2it  C.C.)

2

( 2)

48
2nd Order Nonlinearities
• The incident optical field
~
 i1t
 i 2 t
E (t )  E1e
 E2 e
 C.C.
• Nonlinear polarization contains the following terms
P(21 )   (2) E12
(SHG)
P(22 )   (2) E22
(SHG)
P(1  2 )  2  (2) E1 E2
(SFG)
summ frequency generation
P(1  2 )  2  (2) E1 E2*
(DFG)
difference frequency generation
P(0)  2  (2) ( E1 E1*  E2 E2* ) (OR)
optical rectification
49
Sum Frequency Generation
2
2

( 2)
1
Application:
Tunable radiation in the
UV Spectral region.
1
 3  1   2
2
1
3
50
Difference Frequency Generation
2
1
2

( 2)
1
Application:
The low frequency
photon, 2 amplifies in
the presence of high
frequency beam  . This
1
is known as parametric
amplification.
 3  1   2
1
2
3
51
Phase Matching


( 2)
2
•Since the optical (NLO) media are dispersive,
The fundamental and the harmonic signals have
different propagation speeds inside the media.
•The harmonic signals generated at different points
interfere destructively with each other.
52
Third Order Nonlinearities
When the general form of the incident electric field is in the
following form,
~
i3t
 i1t
i 2t
E (t )  E1e
 E2e
 E3e
The third order polarization will have 22 components
i ,3i , (i   j   k ), (i   j   k )
(2i   j ), (2i   j ), i, j, k  1,2,3
The Intensity Dependent
Refractive Index
• The incident optical field
~
 it
E (t )  E ( )e  C.C.
• Third order nonlinear polarization
P ( )  3 (       ) | E ( ) | E ( )
( 3)
( 3)
2
54
The total polarization can be written as
P
TOT
( )   E ( )  3 (       ) | E ( ) | E ( )
(1)
( 3)
2
One can define an effective susceptibility
 eff   (1)  4 | E ( ) |2  ( 3)
The refractive index can be defined as usual
n  1  4eff
2
55
By definition
n  n0  n2 I
where
n0 c
2
I
| E ( ) |
2
12 2 ( 3)
n2  2 
n0 c
56
Typical values of nonlinear refractive index
Mechanism
n2 (cm2/W)
( 3)
(esu)
1111
Response time (sec)
Electronic Polarization
10-16
10-14
10-15
Molecular Orientation
10-14
10-12
10-12
Electrostriction
10-14
10-12
10-9
Saturated Atomic
Absorption
10-10
10-8
10-8
Thermal effects
10-6
10-4
10-3
Photorefractive Effect
large
large
Intensity dependent
57
Third order nonlinear susceptibility of some material
Material
3
Response
time
Air
1.2×10-17
CO2
1.9×10-12
2 Ps
GaAs (bulk room
temperature)
6.5×10-4
20 ns
CdSxSe1-x doped
glass
10-8
30 ps
GaAs/GaAlAs
(MQW)
0.04
20 ns
Optical glass
(1-100)×10-14
Very fast
58
Nonlinearities in metals
• In metals bulk production of second harmonic
radiation is usually inhibited due to symmetry
reasons and only an electric quadrupole and a
magnetic dipole terms may be active
• These reasons do not exhist at surface due to
the lack of symmetry in a sheet region of the
order of a few Å’s thickness between vacuum
and the metal. In this case the discontinuity of
the electric field is important
• Therefore, although the production of radiation
has a very low efficiency (~10-10) it is interesting
to examine what happens
59
Nonlinearities in metals
• A reasonable elementary model for a metal is an
isotropic free electron gas. A consequence of
this is that metals are opaque to visible radiation.
• For frequencies lower than the plasma
frequency, the electromagnetic radiation can
penetrate only for a small depth (skin depth) of
the order of the wavelength and due to their
symmetry, bulk metals lack of dipole sources for
second harmonic production.
60
Nonlinearities in metals
• Within the skin depth of the incident
electromagnetic field, there are, however,
Lorentz forces on free (and to some
degree also bound) electrons, as well as
interband transitions, both of which lead to
nonlinear bulk magnetic dipole polarization
sources.
61
Nonlinearities in metals
• Further, the breaking of inversion symmetry and
the large change in dielectric constant at the
metal-vacuum interface introduces strong
surface currents within a few Fermi wavelengths
of the surface which produce electric quadrupole
terms. These mechanisms are however very
weak. This notwithstanding, second harmonic
generation from the surface of metals was
studied since the beginning
62
Nonlinearities in metals
Free electron gas theory
The equation of motion for a free electron is
d 2r
e
1

   E  v  B
2
m
c
dt

(1)
Damping is neglected for simplicity.
Clearly, the only nonlinear term in this equation is the Lorentz force
term. Since v << c in a plasma, the Lorentz force is much weaker
than the Coulomb force, and then (e/mc)v  B in (1) can be treated
as a perturbation in the successive approximation of the solution.
63
Taking
E  Ε1e
ik1 r i1t
 Ε2 e
ik2 r i2 t
+ c.c. we have
e
ik1 r  0 i1t
r (i )=
Εe
 c.c.
2 i
mi
(1)
r
2
1  2  
ie2
m 12 1  2 
2
2
  Ε1   k2  Ε2   Ε2   k1  Ε1 
e
i  k1  k2 r  0 i 1 2 t
 c.c.
(2)
and so on.
64
For a uniform plasma with an electronic charge density , the
current density is given by
JJ
1
2
J 
 1  2 

r r 
t


(3)
with, for example,
J
2
  2
1  2    r 1  2 
t
65
In a more rigorous treatment of an electron gas, we must also take
into account the spatial variations of the electron density  and
velocity v. Two equations, the equation of motion and the continuity
equation, are now necessary to describe the electron plasma:
v
p e 
1

  v   v =  E  v  B
t
m m 
c

and

  v   0
t
(4)
where p is the pressure and m is the electron mass.
The pressure gradient term in the equation of motion is responsible
for the dispersion of plasma resonance, but in the following
calculation we assume p = 0 for simplicity.
66
Then, coupled with (4), is the set of Maxwell’s equations
1 B
E  
,
c t
1 E 4J 4v
B 


,
c t
c
c

(5)

0
  E  4      ,
B  0
We assume here that there is a fixed positive charge background in
the plasma to assure charge neutrality in the absence of external
perturbation. Successive approximation can be used to find J as a
function of E from (4) and (5).
67
We assume here that there is a fixed positive charge background in
the plasma to assure charge neutrality in the absence of external
perturbation. Successive approximation can be used to find J as a
function of E from (4) and (5). Let

(0)

(1)

v = v (1)  v (2) 
(2)

,
,
and
j  j(1)  j(2) 
with
(1)
j
(6)
 v
(0) (1)
and
(2)
j
 v
(0) (2)
 v
(1) (1)
(7)
68
We shall find the expression for
E  Ε  exp  ik  r  it  .
j(2) (2)
as an example assuming
Substitution of (6) into (4) and (5) yields
v (1) ( )
e
 iv (1)   E,
t
m
(1) ()
 i(1)    (0) v (1)
t
  E  4(1) (),


v (2) (2)
e (1)
 i 2v (2)   v (1)   v (1) 
v B
t
mc


(8)
The second-order current density is then given by
J
(2)
 e2

ie2
 2 2  E1    E1  2 E1  B1 
m c
m 

e

   E1  E1
i 4m
i(0)
(2) 
2
(9)
69
Neglecting for the moment the (·E)E term, we may derive for the
nonlinear polarization
PNL    E1  B1   E1   E1 
(10)
with
ie3(0)

4m2 3c

e
8m2
The term with  is sometimes referred as the “magnetic dipole” and
the one with  as “electric quadrupole” source.
The term E1 ()E1 strictly vanishes in reflection SHG, but it can be
non-zero in more complicated geometries, including it.
Both terms are always polarized in the incidence plane (TM mode).
The first term is always present and is directed along
k
The second term arises only if the fundamental wave is polarized in
the incidence plane.
70
Adler for the first time speaks, for the centrosymmetric media, of two
source terms for SH, one due to the magnetic dipole, due to the
Lorentz force, and one due to the electric quadrupole, through the
Coulomb force.
71
Including the (·E)E term of (9), which can be transformed as:
 e2

ie2
E


E

E

B
 2 2 1  1

1
1
m2 c
m 

e

   E1  E1
i 4m

i(0)
J (2) (2) 
2
where


e
e
   E E   2   (0) v(1)  E
i 4m
m


ie2 
 2 3   (0) E  E

m  
ie2  (0)  E 
 2 3
E
m  1  2p / 2 
(9)
(11)

1/ 2
is the plasma resonance frequency. With (11),
B   c / i   E, and the vector relation E    E   E  E  12   E  E ,
the
current density in (9) can be written as
 p  4(0) e / m
i(0) e2
ie2 

E

E


 2 3   (0) E  E
2 3
4m 
m 
i(0) e2
ie2  (0)  E 


E

E


 2 3
 E (12)
4m 2 3
m  1  2p / 2 
J (2)  2 


Eq.(9) shows explicitly that aside from the Lorentz term, there are
also terms related to the spatial variation of E. They arise from the
nonuniformity of the plasma. In a uniform plasma, (0) = 0 and  · E
= 0 from (11). This means that k is perpendicular to E and therefore
(E · )E also vanishes. The Lorentz term is the only term in J(2)(2)
72
Coming back to (9), we may now write
PNL    E1  E1   E1   E1 
instead of (10) where
(13)
.
e3n0
 2 4
8m 
In the bulk the electric field is divergenceless and only the first term
in (13) contributes: near the surface both terms are important.
73
Jha used the model of the free electron gas in order to explain the SH in
the metals and for first he wrote the second order polarization with the two
terms, that represent the contribution of bulk and surface.
74
We look first at the component of PNL parallel to the surface, PNL ;
since E1 is continuous across the surface, using translational
symmetry in the plane of the surface we find
PNL


E1 1z / z ,
(14)
which gives a Dirac delta function at the surface in the limit of a step
function equilibrium density profile. Thus there is an effective current
PNL
sheet radiating at 2 at the surface, and following back
the
E1z ,
derivation of eq.(14) it is clear that
results from the accumulated
charge at the surface, signaled by the rapid variation in
being
PNL
driven by the component of the electric field parallel to the surface.
The magnitude of the current sheet due to
may be calculated in
the free electron theory, since eq.(13) may be integrated across the
surface.
75
For the normal component of PNL the situation is different. From
eq.(13) we find
z
2


E
z
z 
z

PNL  
E1  E1 1 ,

 z 
z
 
(15)
which is ambiguous, since both  and E1z change discontinuously
across the surface. Obviously, PNLz , which appears in part because
of the rapid variation of the normal field and in part because the
response of electrons near the surface need not display inversion
symmetry (1), is more subtle in nature than PNL . We note that
even if the integral of eq.(15) could be performed, it is not clear
whether the current sheet should be placed just inside or just
outside the surface. This point is an ambiguity in the free
electron model.
(1) J.Rudnick and E.A.Stern, PR B4 (1971) 4274
(2) J.E.Sipe et al. Solid State Comm. 34 (1980) 523; PR B21 (1980) 4389
76
We conclude that the free electron theory only leads to physically
meaningful results for the first order fields if a step function profile is
adopted for the equilibrium electron density; in that limit, however,
well-defined predictions for the second harmonic generation cannot
be made. Of course, one would certainly expect the free electron
theory to be less than accurate in the surface region where the
actual fields vary over distances on the order of a few Fermi
wavelengths; the point we wish to stress here is that the theory is in
a sense somewhat worse: it is inherently ambiguous.
If people use the free electron theory to describe SHG, a number of
essentially arbitrary “prescriptions” are given for determining the
magnitude and placement of the second harmonic current sheet; in
z
some instances, PNL is simply neglected.
77
In conclusion, the term E x B originating from the Lorentz force acts
on the current produced at the first order by B (it is nonlinear in B). It
is a bulk current in the metal and does not irradiate except if there is
a boundary.
The second term is a surface electric quadrupole term. It is present
in nonlinear reflection from a metal and originates from the
discontinuity of the normal component of E at the interface which
generates nonlinear currents. The surface currents are a tangential
component and a bulk one directed along z.
The SH polarization from the perpendicular and parallel surface
currents can be written as
and
Pzsurface (2)  a()Ez ()Ez ()
surface
Px
(2)  b()E x ()E z ( )
78
Rudnick and Stern (1971) argued from physical grounds that the
second harmonic current sheet should be placed outside the bulk
metal, and that its magnitude should be specified in terms of
parameters a and b of order unity.
These conclusions are borne out to some extent by the application of
the hydrodynamic theory of the electron gas to the calculation of
SHG. The sheet is called selvedge (Sipe, Surf. Sci. 84, p. 75, 1979)
and is smaller than . It is to be placed outside the bulk metal.
The free electron theory does not place unambiguously the current
sheet.
J.Rudnick and E.A.Stern, PR B4 (1971) 4274
79
Dependence on fundamental beam polarization
direction
• A typical apparatus is shown in the
figure
Sample
L2
Prism

m
0n
80
HP-F
/2
BS
m
0n
L1
40
• Since the surface nonlinear sources
are in the simplest theory proportional
to (·E)E, then incident fields
polarized only in the plane of the
surface (normal to the plane of
incidence) do not contribute to the
harmonic signal.
A
Dichroic filters
VA
Laser
System
PM
• However the bulk terms proportional
to E×H lead to harmonic generation
for all incidence polarizations
80
Experimental Setup
s (TE)
f90
f
p (TM)
Sample
f0
L2
Prism

nm
0
80
High Pass band filter
HP-F
/2
Neutral density filters
800 nm
150 fs
1 kHz
m
0n
40
L1
A
BS
Dichroic filters
VA
Laser
System
PM
81
Dependence on fundamental beam
polarization direction
• Neglecting bulk terms, a [ cosf ]4 dependence on the
polarization is predicted, where f is the electric field angle
relative to the plane of incidence.
• Deviations from this relation are indicative of contributions
from the bulk term, and in particular the ratio
•
M = ISH (f90) / ISH (f0)
• yields information about the relative strengths of the surface
versus bulk terms
s (TE)
•
•
•
F.Brown et al., PRL 14 (1965) 1029
F.Brown and R.E.Parks, PRL 16 (1966) 507
N. Bloembergen et al., PR 174 (1968) 813
f90
f
p (TM)f0
82
First experimental results
83
Composite materials
•
It is possible to combine two or more
optical materials in such a manner that the
effective nonlinear optical susceptibility of
the composite material exceeds those of
its constituents.
•
Some structure is shown in the figure. In
each case, the characteristic distance
scale over which the constituent materials
are mixed is much smaller than an optical
wavelength.
•
Consequently, the propagation of a beam
of ligth through such material can be
described by spatially averaged values of
the linear and nonlinear refractive indices
obtained by performing suitable volume
averages.
•
J.E. Sipe and R.W.Boyd, PR A46 (1992) 1614; JOSA
B11 (1994) 297
G.L.Fischer et al. PRL 74 (1995) 1871
•
84
Structures
• If
the
two
constituent
materials of the composite
posses
different
linear
refractive
indices,
the
electric field of the incident
laser beam will become
spatially nonuniform within
the material. Under proper
conditions, the electric field
amplitude
will
become
concentrated in the more
nonlinear constituent of the
composite, resulting in an
enhanced overall nonlinear
optical response.
85
• Much of the early work on composite nonlinear
optical materials concerned metallic colloids.
• For such materials, an enhancement of the
electric field occurs in the vicinity of each
metallic particle.
• The enhancement is particularly large when the
laser frequency is near that of the surface
plasmon resonance of the metallic particle
•
•
D.Ricard et al. Opt. Lett. 10 (1985) 511
F.Hache et al. Appl. Phys. A47 (1988) 347
86
Spherical particles a <<  dispersed with a very low volume fraction f << 1.
A single spherical inclusion experiences a local field Eloc
Eloc  E 
and the field inside it is
Ei 
4 P
3 h
3 h
Eloc
 i  2 n
(28)
(29)
If the inclusion is made of a metal with
Re( i )  0
one may choose the host and the inclusion dielectric constant such that
Re( i  2 h )  0
corresponding to the plasmon resonances, thus obtaining resonant enhancement
effects. Thus the inclusion can experience an enhanced local field that, in case of
several inclusions randomly dispersed in the host, leads to an effective dielectric
constant eff that satisfies
 eff   h
 
 f
with   i h
(30)
 eff  2 h
 i  2 h
where f is the volume fraction of the inclusion material.
87
Similar effects can be obtained with dielectric
particles. In this case the most spectacular effects
are obtained when the size of the particle is
smaller than the exciton orbits in semiconductor.
Quantum effects appear
88
Nonlinear optical properties of CdSe quantum-dot doped
glasses
As an example Fig.1 shows typical room-temperature, linear low intensity, absorption
spectra of CdSe QDs doped glasses.
Distinct discrete absorption lines as well as blue shift of the lowest energy electrohole transition (absorption edge) with decreasing microcrystallite size are clearly
seen.
S.H.Park, M.P.Casey, J.Falk, JAP 73 (1993) 8041
89
The
results
of
a
measurement of the
567 nm transmission of
the 26 A sample as a
function
of
laser
intensity is shown in the
figure
90