Document 6517484

Course
„
„
Nonlinear Optics
„
„
12 Lectures: 10 lectures and 2 tutorials (in Trinity term)
Timetable of nonlinear optics lectures
Nonlinear Optics NLO LB (8 lectures) and TK (2 lectures: slots
to be determined)
Timetable
Dr. Louise Bradley
Room 2:21
[email protected]
Monday
4 pm
Wednesday Friday
10 am
9 am
Week 6
NLO
NLO
Week 7 NLO
NLO
NLO
Week 8 NLO
NLO
NLO
Week 9
NLO
NLO
Dr. L. Bradley
Nonlinear optics isn’t something you see
everyday.
Suggested Reading
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Optical Electronics in Modern Communications – A.
Yariv 1997
Nonlinear Optics – R.W. Royd
Principles of Nonlinear Optics – Butcher and D.
Cotter
Applied Nonlinear Optics – Zernike and Midwinter
Fundamentals of Photonics – Saleh and Teich 1991
and 2006
Handbook of Nonlinear optics – Sutherland 2003
Great web sites – used lots of figures/slides from these
http://www.ph.surrey.ac.uk/intranet/undergraduate/3mol
http://www.physics.gatech.edu/gcuo/UltrafastOptics/index.html
Dr. L. Bradley
What is NLO?
„
What are nonlinear optical effects and why do they occur?
Sending infrared light into a crystal
yielded this display of green light
(second-harmonic generation):
Nonlinear optics allows us to change
the colour of a light beam, to change
its shape in space and time, and to
create ultrashort laser pulses, the
shortest events ever made by Man.
NL is key element for optical data
processing
Why don't we see nonlinear optical
effects in our daily life?
1. Intensities of daily life are too weak.
2. Normal light sources are incoherent.
3. The occasional crystal we see has the wrong symmetry (for SHG).
4. “Phase-matching” is required, and it doesn't usually happen on itsDr.
own.
L. Bradley
Difference-frequency generation
Examples
SHG Medium
ω
2ω
ω1
SHG Medium
ω1+ω2
ω1
ω3
ω2
ω2 = ω3 − ω1
ω1
ω3
ω2
Optical Parametric
Amplification (OPA)
THG Medium
ω
ω1
3ω
And Frequency Sum
Dr. L. Bradley
ω2 = ω3 + ω1
Dr. L. Bradley
1
Self-diffraction
Non co-linear – third order
We can also allow two different input beams, whose frequencies can
be different.
So in addition to generating the third harmonic of each input beam,
the medium will generate interesting sum frequencies, spatially
separate.
… but the frequencies don’t have to be different to generate new
optical fields propagating in different directions…
Signal #1
ω2
THG
medium
Signal #1
ω
2ω1 +ω2
2ω2 +ω1
ω1
Nonlinear
medium
ω
ω
Signal #2
Signal #2
ω
Dr. L. Bradley
Self-focusing
Dr. L. Bradley
Optical computing and optical
data processing
„
¾
x
¾
¾
n(x)
„
¾
Nonlinear absorption
Two-photon absorption
detectors
Saturable Absorbers
Optical limiting
Nonlinearity key element for
optical switches and optical
bistability
Optical logic gates, flip-flops
Dr. L. Bradley
Phase conjugation
Dr. L. Bradley
Questions
Why do nonlinear-optical effects occur?
How can we use them?
Maxwell's equations in a medium Î Nonlinear-optical media
Reflection of a plane wave
from an ordinary and phase
conjugate mirror
Reflection of a spherical wave from
an ordinary and phase conjugate
mirror
Second Order Effects: Second-harmonic generation
Sum- and difference-frequency generation
Autocorrelation
Phase-matching and Conservation laws for photons
Third Order Non-linear Effects:
Mirror
Distorting medium
Phase conjugate mirror
Dr. L. Bradley
Frequency generation, Nonlinear refractive index, Phase
conjugation…
Consequences and Applications
Dr. L. Bradley
2
Nonlinear Response
Nonlinear Response
Nonlinear response is not confined to optics
R = ζ 1S + ζ 2 S 2 + ζ 3 S 3 + ζ 4 S 4 + ...
Dr. L. Bradley
Dr. L. Bradley
The Fourier components
Representation of Nonlinearity
Linear
R = ξ1S
S
R
R
R
S
s
Non-Linear
R
R = ξ1S − ξ 2 S 2
The same frequency
as the stimulus
S
R
Double the
frequency of the
stimulus
S
A DC component
Dr. L. Bradley
Interaction of light and matter
„
An applied electric (optical) field displaces the electrons
from the nucleus in a medium
„
Polarization = dipole moment per unit volume
Separation of charges gives rise to a dipole moment
P(t)=-Nex(t)
„
„
+
Dr. L. Bradley
Polarization
ƒOptical polarization of dielectric crystals – mostly due to outer
loosely bound valence electrons displaced by the optical electric
field.
ƒPolarization is alternating with the same frequency as the applied
E field.
ƒElectron oscillates about the equilibrium position – oscillating
dipole is a source of EM radiation.
-
E
Dr. L. Bradley
Dr. L. Bradley
3
Linear Optics
Linear optics
„
„
„
„
„
Recall that, in normal linear
optics, a light wave acts on a
atom or
atom, which vibrates and then
molecule
emits its own light wave that
interferes with the original light
wave.
input
emitted
Consequence of induced charge
photons
photons
polarization and re-radiation is a
decrease in the speed f light in
the medium i.e. increase in the
refractive index relative to the
vacuum
Internal field due to nucleus
~1011 V/m
Sunlight ~1 kV/m
Consequently, for small E, P (t ) = ε 0 χ (1)E (t )
the linear approximation is very
accurate
„
„
P(t)=-Nex(t)
x(t) is small, harmonic potential regime
Dr. L. Bradley
Nonlinear optics and anharmonic oscillators
For field strength > 1 kV/m (i.e. lasers), X(t) is sufficiently large that the
potential of the electron or nucleus (in an atom/molecule) is not a simple
harmonic potential.
Anharmonic motion occurs, and higher harmonics occur, both in the
motion and the light emission
Dr. L. Bradley
Nonlinear Polarization
In an anharmonic potential:
Polarization expanded as a power series in E to give:
(
r
r
r
r
P = ε 0 χ (1) E + χ ( 2 ) E 2 + χ ( 3) E 3 + ...
)
χ(2) = 2nd order susceptibility
χ(3) = 3rd order susceptibility
Example: vibrational motion:
atom or
molecule
input
photons
different
colour!
emitted
photons
In order for the series to converge: χ(3)E3<< χ(2)E2<< χ(1)E
Accessed for optical intensities I ~1013 W/m2
Dr. L. Bradley
Linear susceptibility
Lorentz Model
„Lorentz
model – analogous to a mass on a spring
χ (1) =
Electron of mass, m, and charge, e, is
attached to the ion by a spring.
„
„
„
„
External force applied by the E
field drives the oscillation
γ is the damping constant
ω0 is the resonant frequency
Anharmonic oscillator includes
higher order terms
Dr. L. Bradley
Ne2
ε 0m ω − 2iγω1 − ω12
[
2
0
r
r
P = ε 0 χ (1) E
r
r
r r
D = εE = ε 0 E + P
]
ω0
Optical Frequency, ω
„
F=-kx
k=spring constant
x=displacement
The electric polarization P is defined as the difference
between the electric fields D (induced) and E (imposed) in a
dielectric due to bound and free charges, respectively. In
MKS,
r
r r
r
r
r
D = ε 0 E + P = ε 0 (1 + χ (1) )E = εE = ε 0ε r E
n=
Dr. L. Bradley
c
=
v
μ0ε
μ0ε 0
=
ε
= εr =
ε0
(1 + χ )
(1)
Dr. L. Bradley
4
Refraction and Absorption
Second order susceptibility
The refractive index is a complex quantity
n = 1+ χ
ε 02mξ ( 2 )
χ (2) =
(1)
N 2 e3
n = n0 − iκ
(
)
κ = Im( 1 + χ )
n0 = Re 1 + χ (1)
(1)
Dispersion, frequency dependent speed of
propagation
1
1
1
„
For a pure frequency applied field χ(2) Can be expressed
in terms of the linear susceptibility χ(1) at two
frequencies ω1 and 2ω1
„
The susceptibilty determines the magnitude of P and
hence the strength of fields being reradiated
Is proportional to the material absorption
If we are far from the absorption, the imaginary parts
are negligible.
In the linear case the dipoles and the polarization
oscillate at the same frequency as the incident field.
χ ((ω1) ) χ ((ω1) ) χ ((21ω) )
Dr. L. Bradley
Dr. L. Bradley
Many interacting fields
Third order susceptibility
r
r r
P ( 2 ) = ∑∑ ε 0 χ ((ω2n) ,ωm ) E(ωn ) E(ωm ) e − i (ωn +ωm ) t
„
n
χ
„
„
„
(2)
(ω n ,ω m )
=
Continuing our analysis of the anharmonic oscillator, for 1
driving field we will find
m
ε 02mξ ( 2 )
N 2 e3
χ
(1)
(ω n )
χ
(1)
(ω m )
χ
χ ( 3) =
(1)
(ω n +ω m )
The 2nd order: interaction of two fields producing a third
Result: all frequencies (ωn+ ωm) for all possible values of n,m
For just two fields n,m = ±1,±2
Will get all the terms we had before:
DC component
Second harmonic generation
Frequency sum and frequency difference terms
„
„
ε 03mξ ( 3)
N 3e 4
χ ((ω1) ) χ ((ω1) ) χ ((ω1) ) χ ((31ω) )
1
1
1
1
Third order: 3 input fields producing a fourth
Nonlinear polarization will find (ωn+ ωm+ ωp) for all possible
values of n,m,p = ±1,±2, ±3
r
r r
r
− i (ω +ω +ω ) t
P ( 3) = ∑∑∑ ε 0 χ ((ω3)n ,ωm ,ω p ) E(ωn ) E(ωm ) E(ω p ) e n m p
n
χ ((ω3) ω
n , m ,ω p )
„
m
=
p
ε 03mξ ( 3)
N 3e 4
χ ((ω1) ) χ ((ω1) ) χ ((ω1) ) χ ((ω1) +ω
n
m
p
n
m +ω p )
Third harmonic generation and many frequency sum/frequency
difference terms.
Dr. L. Bradley
Maxwell’s equation in a medium
r r
∇⋅E = 0
r r
∇⋅B = 0
r
r r
∂B
∇× E = −
∂t
r
r r
r
∂D
∇ × B = μ0 J + μ 0
∂t
Dr. L. Bradley
Solving the wave equation in the presence of linear
induced polarization
For low irradiances, the polarization is proportional to the incident field:
P = ε 0 χE
In this simple (and most common) case, the wave equation becomes:
∂ 2E
1 ∂ 2E
−
∂z 2 c02 ∂t 2
These equations reduce to the wave equation:
r
r
r 1 ∂2E
∂2 P
∇2 E − 2 2 = μ 2
v ∂t
∂t
“Inhomogeneous
Wave Equation”
Simplifying:
waves of all frequencies are solutions to the wave equation;
it’s the polarization that tells which frequencies will occur.
where
„The
induced polarization, P, contains the effect of the medium or if you
prefer
„The polarization is the driving term for the solutions to this equation.
Dr. L. Bradley
1 ∂ 2E
χ
c02 ∂t 2
∂ 2E 1 + χ ∂ 2E
− 2
∂z 2
c0 ∂t 2
This equation has the solution:
„Sine
=
ω=ck
Using the fact that:
ε 0 μ 0 = 1/ c02
= 0
E ( z, t ) ∝ E0 cos(ωt − k z)
and c = c0 /n and n = (1+χ)1/2
The induced polarization, at the same frequency as the incident field
and only changes the refractive index. Dull.
If only the polarization contained other frequencies…
Dr. L. Bradley
5
Maxwell's Equations in a Nonlinear Medium
… and we now know this can occur
Nonlinear optics is what happens when the polarization is
the result of higher-order (nonlinear!) terms in the field:
P = ε 0 ⎡⎣ χ (1)E + χ (2)E 2 + χ (3)E 3 + ...⎤⎦
And we saw the effects of such nonlinear terms:
Light generated at many other frequencies, for example.
Since E
(t ) ∝ E exp(iωt ) + E * exp(−iωt ),
E (t )2 ∝ E 2 exp(2iωt ) + 2 E + E *2 exp(−2iωt )
2
2ω = 2nd harmonic!
Harmonic generation is one of many exotic effects that can
arise!
Dr. L. Bradley
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