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2. physics
3. optics

First demonstration of an NLO effect
„
„
First demonstration of SHG: The Data
Laser demonstrated in 1960 (Maiman)
The actual published results…
P.A. Franken, et al, Physical Review Letters 7, p. 118 (1961)
The second harmonic
Focused 3kW of pulsed red light (694.3 nm) from a Ruby laser onto a
quartz crystal and observed some UV light (347.15 nm) coming out
(SHG).
Input beam
Note that the very weak spot due to the second harmonic is missing.
It was removed by an overzealous Physical Review Letters editor,
who thought it was a speck of dirt.
„At
that time the conversion efficiency was 1 in 108.
Dr. L. Bradley
Why is the conversion efficiency
so low?
Generation of light at the new frequencies is
not just a question of having a high intensity beam or
the size of the susceptibility
Dr. L. Bradley
SHG
To see the implication of this lets look at the specific
example of SHG
Conditions:
„
„
„
Do we have the whole story?
„
„
Under what conditions do we achieve the maximum
conversion?
Coupled Wave Equations
„
The issue of phase-matching?
There must be a relationship between the frequencies ω1,
ω2 and ω3 in order to drive an oscillation at ω3
ω1=ω2 = ω
E(ω1)=E(ω2 )=E(ω)
ω3=ω1 + ω2=2ω
E(ω3)=E(2ω)
„
„
The amplitude of each field depends on the amplitude of the
other fields as they propagate.
They are coupled through the nonlinear susceptibility.
And there is a term dependent on their relative propagation
speeds, i.e their phase relationship.
Dr. L. Bradley
Dr. L. Bradley
Maximum I2ω
SHG
⎛ Δkl ⎞
Sin 2 ⎜
⎟
1 ε 2ω ⎡ 2 2 2 μ 4 ⎤
⎝ 2 ⎠
I 2ω =
E0ω ⎥
ω d l
⎢
2
ε 2ω
2 μ ⎣
⎦ ⎛ Δkl ⎞
⎟
⎜
„ I2ω ∝ Iω2 ∝ Eω4
⎝ 2 ⎠
⎛ Δkl ⎞
Sin 2 ⎜
⎟
⎝ 2 ⎠ =1
2
⎛ Δkl ⎞
⎜
⎟
⎝ 2 ⎠
Δkl
=0
2
Δk = 0
Intensity of the second harmonic signal generated is
proportional to the square of the input intensity at ω
„The
Dr. L. Bradley
Maximum 2nd harmonic intensity will be
achieved when Δk=0
Dr. L. Bradley
1
But isn’t this conservation of
momentum?
Conservation Laws for Photons in NLO
Energy must be conserved:
r
r
hkout = hkin
r
r
r
hk2ω = hkω + hkω
r
r
k2ω = 2kω
And light of this frequency will
have the corresponding
wavevector:
or
r
r
k2ω − 2kω = Δk = 0
Dr. L. Bradley
ω1 + ω 2 + ω3 − ω 4 + ω5 = ω0
is the k-vector of the induced light
c
λ
=
ωo nω
o
c
Momentum must be conserved:
r
r r
p p = p1 + p2
kp is the wavevector of the
induced polarization, and it
may not correspond to the kvector of the induced light
E h
p = = = hk
c λ
r
r
r
hk p = hk1 + hk 2
r
r r
k p = k1 + k2
Satisfying these two relations simultaneously
is called "phase-matching."
„
Dr. L. Bradley
Phase matching condition for SHG
Δk = 0 = k2ω − 2kω
k2ω = 2kω
Momentum :
r
r r r r r
k p = k1 + k 2 + k 3 − k 4 + k 5
2πnω
r
k p = ko ?
Satisfying these two relations
simultaneously
is called "phase-matching."
Thanks to Rick Trebino http://www.physics.gatech.edu/gcuo/UltrafastOptics/index.html
2πnωo
ω2= ω1
energy
Energy :
nωo ωo
kω o =
Phase Matching Condition
More general picture - multiphoton
kω o =
hωout = hω1 + hω 2
ωout = ω1 + ω 2
2ωn2ω
=
kω =
, k 2ω =
c
c
λ
2ω (n2ω − nω )
Δk = k2ω − 2kω =
=0
c
n2ω = nω
ω0=2ω1
ω1
ωnω
Dr. L. Bradley
y
2k1
k1
k1
Dr. L. Bradley
What is this telling us?
Coherence length
Applied E field induces polarization field which travels at the same
speed in the medium as the incident field ->The resulting
oscillating dipoles emit radiation.
Only when the radiation from each dipole adds constructively (are
in phase) will a propagating EM wave be generated.
„
The distance it takes for the charge polarization wave
and the radiated light to get 1800 out of phase is the
coherence length – if crystal length = odd multiple of lc
no light 2nd harmonic light will be emitted
lc
Unless the light generated is travelling at the same speed as the
charge polarization wave, light radiated from one part of the
crystal will not be in phase with light radiated from another
part, and they will destructively interfere at some point
Phase-matching must be considered in
all nonlinear-optical problems.
Dr. L. Bradley
Dr. L. Bradley
2
Coherence length
Coherence length
When z=lc
lc =
lc =
π
Δk
=
„
lc is a measure of the maximum crystal length that is useful
in producing 2nd harmonic power
„
If the crystal is longer the 2nd harmonic waves will interfere
and the efficiency will be very low.
„
This distance is typically <10-2 cm.
Z/lc
„
πc
2ω (n2ω − nω )
1
πc
λ0
=
ω 2(n2ω − nω ) 4(n2ω − nω )
Dr. L. Bradley
Achieving phase matching
„
Have interference described by the factor
„
Phase matching condition
⎛ Δkl ⎞
Sin 2 ⎜
⎟
⎝ 2 ⎠
2
⎛ Δkl ⎞
⎜
⎟
⎝ 2 ⎠
Δk = 0 = k2ω − 2kω
k2ω = 2kω
this from ever happening!
Dr. L. Bradley
Phase-matching Second-Harmonic
Generation using Birefringence
Birefringent materials have different refractive indices for different
polarizations. “Ordinary” and “extraordinary” refractive indices
can be different by up to ~0.1 for SHG crystals.
We can now satisfy the
phase-matching condition.
Use the extraordinary polarization
for ω and the ordinary for 2ω.
Refractive
index
n2ω = nω
Unfortunately, dispersion
prevents
I2ω ∝ l2, so phase matching is crucial to be able to use
longer crystals
no (2ω ) = ne (ω )
ω
Frequency
2ω
ω
Frequency
2ω
Dr. L. Bradley
SHG crystal
Input beam
Input beam
no
Birefringence
Light created in real crystals
Closer to
phase-matching:
ne
ne depends on propagation angle, so we can tune for a given ω.
Some crystals have ne < no, so the opposite polarizations work.
Dr. L. Bradley
Far from
phase-matching:
Refractive index
⎛ Δkz ⎞
Sin 2 ⎜
⎟
2
⎝ 2 ⎠
I 2ω ( z ) ∝ I ω
2
⎛ Δk ⎞
⎜ ⎟
⎝ 2 ⎠
Δkz
π
Max when
=
2
2
Output beam
The molecular "spring
constant" can be
different for different
directions.
SHG crystal
Output beam
Note that SH beam is brighter as phase-matching is achieved.
Dr. L. Bradley
Dr. L. Bradley
3
Birefringence
Birefringence
Angle tuning
The x- and y-polarizations
can see different refractive
index curves.
Or can also use
temperature tuning – not shown
Dr. L. Bradley
Tuning
With birefringence can
even tune the refractive
index as a function of
angle
Dr. L. Bradley
Crystals
1
Cos 2θ Sin 2θ
=
+ 2
ne2 (θ )
n02
ne
1
(n )
ω 2
0
=
Cos 2θ m
(n )
2ω 2
0
+
„
Sin 2θ m
(n )
2ω 2
e
matched direction at an angle
of approx 350 to the optic axis.
Usually divided into 2 groups – depending on whether
the crystal structure remains unchanged under inversion
„ Centrosymmetric e.g. NaCl
Inversion of any ion about the
central Na+ ion leaves the structure
unchanged.
„Index
„ Non-centrosymmetric
e.g. ZnS zinc blende class,
GaAs, CdTe
Dr. L. Bradley
Crystal Symmetry
Dr. L. Bradley
d
Due to crystal symmetry there are no 2nd order polarization
effects in centrosymmetric media.
Dr. L. Bradley
NB lack of uniformity in textbooks
Dr. L. Bradley
4
A couple of examples from the
laser called FRED, want to generate very low
lab UV
wavelength (high energy photons) laser to investigate
novel wideband gap materials e.g. ZnO for optoelectronic
(polariton lasers, efficient white lights) and medical
applications.
No easy source
available at these
wavelengths
< 300 nm
So we use SHG.
Light at 488 nm,
easily produced by a
standard Ar+ laser is
converted to 244 nm
See only 6-7%
conversion efficiency
but it is the only
way!
Intra cavity conversion
Using a BBO crystal – transparent from 200 nm to 2000
nm
To get appreciable conversion requires large power
densities (W/m2) at the fundamental
2
Usually not available from CW lasers I 2ω ∝ I ω
„
„ BUT can place the NL crystal in the laser resonantor
From your laser physics: the intensity inside the cavity
exceeds its value outside by (1-R)-1, R is the mirror
reflectivity.
For R~1 the enhancement is large, so more efficient
conversion is achieved.
Dr. L. Bradley
Intracavity SHG Conversion
Dr. L. Bradley
Photon energy level representation – energy
conservation
„
Ar+ laser
„
„
„
Under the proper conditions with the proper mirrors, can extract
the total available power at 2ω
„ IR laser ω (λ= 1064 nm), 2ω (λ = 532 nm)
Dr. L. Bradley
Some experimental results
„
„
„
„
NL crystal
ω3
ω1
„
ω1
ω3=ω1+ ω2
ω2
ω2
BBO
UV laser ω (λ= 488 nm), 2ω (λ = 244 nm)
The intracavity intensity is >100 kW/m2
The phase matching is achieved via the birefringence
ω light propagates as the ordinary ray, 2ω as the extraordinary
ray
„
Frequency Sum
P = ε 0 χ (2) E1 E2
Second Harmonic Generation
ω1= ω2
ω3=2ω1
P = ε 0 χ (2) E12
ω1
ω1
ω3
Dr. L. Bradley
Expt: Frequency Sum
2nd
A new material for
nonlinear effects: PPLN
Can achieve phasematching over long lengths, e.g. 8 cm
Highly efficient, but very temperature dependent
Working at optical telecommunications wavelengths
1521 nm
1527 nm
1532 nm
763.2 nm
763.5 nm
Dr. L. Bradley
Dr. L. Bradley
5
Induced polarization for nonlinear optical
effects
Frequency Difference
ω1
NL crystal
ω2
ω3=ω1- ω2
ω1
ω2
ω3
P = ε 0 χ (2) E1 E2*
For every photon created at the difference frequency ω3 a
photon at frequency ω1 must be destroyed and a photon at ω2
created.
„ Incoming photon at ω1 excites the atom to the highest virtual
level, incoming photon at ω2 stimulates the emission of a photon
at ω2 and for energy conservation a photon at ω3 is also emitted.
⇒ Amplification of the input light at ω2
Optical Parametric Amplification
„
Dr. L. Bradley
„Arrows
pointing upward
correspond to absorbed
photons and contribute a factor
of their field, Ei; arrows pointing
downward correspond to
emitted photons and contribute
a factor the complex conjugate
of their field:
P = ε 0 χ (5) E1 E2 E3 E4* E5
r
r r
r
P ( n ) (Ω = ω1 + ω 2 + ... + ω n ) = ε 0 χ ( n ) (− Ω, ω1 , ω 2 ,..., ω n )E(ω1 ) E(ω2 ) ...E(ωn )
„
Ω is the resulting frequency
Dr. L. Bradley
Difference-Frequency Generation: Optical
Parametric Generation, Amplification, Oscillation
Difference-frequency generation takes many useful forms.
ω1
ω3
ω2 = ω3 − ω1
ω1
ω3
Parametric Down-Conversion
(Difference-frequency generation)
ω1
ω1
ω3
ω2
ω2
Optical Parametric
Generation (OPG)
"idler"
By convention:
ωsignal > ωidler
ω1
ω3
ω2
mirror
Optical Parametric
Amplification (OPA)
"signal"
mirror
Optical Parametric
Oscillation (OPO)
Dr. L. Bradley
6