Download Introduction to Nonlinear Optics

Introduction to Nonlinear Optics
H. R. Khalesifard
Institute for Advanced Studies in Basic
Sciences
Email: [email protected]
Contents
1.
2.
3.
4.
5.
6.
Introduction
The essence of nonlinear optics
Second order nonlinear phenomena
Third order nonlinear phenomena
Nonlinear optical materials
Applications of nonlinear optics
Introduction
input
Answer:
Not without a laser light
NLO sample
Question:
Is it possible to change the
color of a monochromatic
light?
output
Stimulated emission, The MASER
and The LASER



(1916) The concept of stimulated emission Albert
Einstein
(1928) Observation of negative absorption or stimulated
emission near to resonant wavelengths, Rudolf
Walther Ladenburg
(1930) There is no need for a physical system to always
be in thermal equilibrium, Artur L. Schawlow
E2
h
Absorption
h
h
E1
E2
E1
Spontaneous
Emission
h
E2
E1
Stimulated
Emission
h
Light (Microwave) Amplification
by
Stimulated
Emission of Radiation
LASER
(MASER)
The Maser
Two groups were working on Maser in 50s
 Alexander M. Prokhorov and Nikolai
G. Bassov (Lebedev institute of
Moscow)
 Charles H. Townes, James P. Gordon
and Herbert J. Zeiger (Colombia
University)
Left to right: Prokhorov, Townes and Basov at the Lebede
institute (1964 Nobel prize in Physics for developing the
“Maser-Laser principle”)
Townes (left) and
Gordon (right) and
the ammonia maser
they had built at
Colombia University
The LASER






(1951) V. A. Fabrikant “A method for the application of
electromagnetic radiation (ultraviolet, visible, infrared, and radio
waves)” patented in Soviet Union.
(1958) Townes and Arthur L. Schawlow, “Infrared and Optical
Masers,” Physical Review
(1958) Gordon Gould definition of “Laser” as “Light Amplification
by Stimulated Emission of Radiation”
(1960) Schawlow and Townes
U. S. Patent No. 2,929,922
(1960) Theodore Maiman Invention of the first Ruby Laser
(1960) Ali Javan The first He-Ne Laser
Maiman
and the
first ruby
laser
Ali Javan and
the first He-Ne
Laser
Properties of Laser Beam
A laser beam
 Is intense
 Is Coherent
 Has a very low divergence
 Can be compressed in time up to few
femto second
Applications of Laser


(1960s) “A solution looking for a
problem”
(Present time) Medicine, Research,
Supermarkets, Entertainment,
Industry, Military, Communication,
Art, Information technology, …
Start of Nonlinear Optics
Nonlinear optics started
by the discovery of
Second Harmonic
generation shortly
after demonstration
of the first laser.
(Peter Franken et al
1961)
2. The Essence of Nonlinear Optics
Output
When the intensity of
the incident light to
a material system
increases the
response of
medium is no
longer linear
Input intensity
Response of an optical Medium
The response of an
optical medium to
the incident
electro magnetic
field is the
induced dipole
moments inside
the medium
h
h


h
h



Nonlinear Susceptibility
Dipole moment per unit volume or polarization
Pi  Pi   ij E j
0
The general form of polarization
Pi  Pi  χ E j  χ
0
(1)
ij
(2)
ijk
E j Ek  χ E j Ek El  
(3)
ijkl
Nonlinear Polarization




Permanent
Polarization
First order
polarization:
Second order
Polarization
Third Order
Polarization
P   Ej
1
i
(1)
ij
Pi   E j Ek
2
( 2)
ijk
Pi   E j Ek El
3
( 3)
ijkl
How does optical nonlinearity
appear
The strength of the
electric field of the light
wave should be in the
range of atomic fields
h
a0
N
Eat  e / a
2
0
a0   / me
2
e
2
7
Eat  2 10 esu
Nonlinear Optical Interactions

The E-field of a laser beam
~
E (t )  Eeit  C.C.

2nd order nonlinear polarization
~ ( 2)
P (t )  2  ( 2) EE*  (  ( 2) E 2e 2it  C.C.)

2

( 2)

2nd Order Nonlinearities

The incident optical field
~
 i1t
 i 2t
E (t )  E1e
 E2e
 C.C.

Nonlinear polarization contains the following terms
2
1
P(21 )   E
(SHG)
P(2 2 )   ( 2 ) E22
(SHG)
( 2)
P(1   2 )  2  E1 E2
(SFG)
P(1   2 )  2  ( 2 ) E1 E2*
(DFG)
( 2)
P(0)  2  ( 2) ( E1 E1*  E2 E2* ) (OR)
Sum Frequency Generation
2
2

( 2)
1
Application:
Tunable radiation in the
UV Spectral region.
1
3  1   2
2
1
3
Difference Frequency
Generation
2
2

1
( 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
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.
SHG Experiments

We can use a
resonator to increase
the efficiency of SHG.
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
which their frequency dependent are
 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
The total polarization can be written as
P
TOT
( )   E ( )  3 (       ) | E ( ) | E ( )
(1)
( 3)
2
One can define an effective susceptibility
 eff    4 | E ( ) | 
(1)
2
( 3)
The refractive index can be defined as usual
n  1  4eff
2
By definition
n  n0  n2 I
where
n0c
2
I
| E ( ) |
2
12 2 ( 3)
n2  2 
n0 c
Typical values of nonlinear refractive index
Mechanism
n2 (cm2/W)
( 3)
1111
(esu)
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
Third order nonlinear susceptibility of some material
Material

1111
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
(1-100)×10-14
Very fast
Optical glass
Processes due to intensity
dependent refractive index
1. Self focusing and self defocusing
2. Wave mixing
3. Degenerate four wave mixing
and optical phase conjugation
Self focusing and self defocusing

The laser beam has Gaussian intensity
profile. It can induce a Gaussian refractive
index profile inside the NLO sample.

( 3)
Wave mixing


2n0Sin( /2)
Optical Phase Conjugation

Phase conjugation mirror
PCM
M
s
M
PCM
Aberration correction by PCM
Aberrating
medium
s
Aberrating
medium
PCM
PCM
What is the phase conjugation
The signal wave
~
it
Es (r, t )  Es e  C.C.
Es  ε̂ s As e
The phase conjugated wave
~
*  it
Ec (r , t )  rEs e  C.C.
iks .r
Degenerate Four Wave Mixing
A1
A2

( 3)
A3
A4
•All of the three incoming beams A1, A2 and A3 should be originated
from a coherent source.
•The fourth beam A4, will have the same Phase, Polarization, and
Path as A3.
•It is possible that the intensity of A4 be more than that of A3
Mathematical Basis
The four interacting waves
~
i ( ki .r t )
Ei (r.t )  Ai (r )e
 C.C.
The nonlinear polarization
P
NL
* i (( k1  k 2  k3 ).r t )
3
 6  E1E2 E  6  A1 A2 A e
( 3)
*
3
( 3)
The same form as the phase
conjugate of A3
Holographic interpretation of
DFWM
A1
A2

( 3)
A3
A4
Bragg diffraction
from
induced dynamic
gratings