37. 3rd order nonlinearities - brown university · 2017. 5. 8. · suppose our pulse duration is τ...
Post on 07-Mar-2021
0 Views
Preview:
TRANSCRIPT
Characterizing 3rd order effects
The nonlinear refractive index
Self-lensing
Self-phase modulation
Solitons
When the whole idea of χ(n) fails
Attosecond pulses!
37. 3rd order nonlinearities
χχχχ(2): New frequency components
( ) ( ) ( ) ( ) 222 2
0 0c 2os 2P t E t E j t Eω∝ = +
One hallmark of χ(2) nonlinear optics is the generation of
new colors of light.
We have seen this effect in detail in one specific case:
second harmonic generation
We have also seen that χ(2) is zero for most materials, so
these effects can only be seen for certain specific
materials or situations.
χ(3) effects are weaker, but they can occur in virtually any
medium.
again, a piece at the
input frequency
two components at shifted frequencies
one component at an unshifted frequency
As with second order phenomena, we expect
to find new frequency components at sum and
difference frequencies:
( ) ( ) ( ) ( )
( )( )( )3 31 1 2 2
3 (3)
0 1 2 3
(3) * * *
0 1 1 2 2 3 3
−− −
=
= + + +j t j tj t j t j t j t
P E t E t E t
E e E e E e E e E e E eω ωω ω ω ω
ε χ
ε χ
( ) ( ) ( ) ( )3 (3) (3) (3)
1 3 1 3 32 2P P P Pω ω ω ω ω= + + − +
The polarization field at an unshifted frequency is particularly interesting*
Third-order nonlinear effects
( ) ( ) ( )3 (3) (3)3= +P P Pω ω
χ(3)
ω1
ω2
ω3
Consider the case where two of the components are equal: ω1 = ω2
χ(3)
ω1
ω1
ω3
If all three frequencies are equal (e.g., all are from a single laser beam):
χ(3)ω
( ) ( ) ( ) ( )2
(1) (3)
0 3 = + TOTP E E Eω ε χ ω χ ω ω
Considering only this unshifted polarization component (and
assuming that χ(2) = 0), the total polarization is:
We can define an effective susceptibility χeff, such that PTOT = ε0χeff E
( )2
(1) (3)3eff Eχ χ χ ω= +
But( )1
0 1= +n χ is the usual refractive index.
Degenerate third-order nonlinear effects
( )(3)
2
0
0
3
2≈ +n n E
n
χω = n0 + a small intensity-dependent correction
Then, the effective index of the medium: effeffn χ+= 1
( ) ( ) ( )(3)
2 2(1) (3) (1)
(1)
31 3 1 1
1
= + + = + + +
n E Eχ
χ χ ω χ ωχ
( )(3)
2
0
0
3
2≈ +n n E
n
χω
The refractive index of a χ(3) medium has an intensity-
dependent term. This is usually written:
n = n0 + n2 I where(3)
2 ∝n χ
n2 has units of inverse intensity, or m2/ Watt. It is usually very small.
Refractive index depends on intensity
If the incident radiation is very intense (of
order 1/n2), then the index of the medium
changes.
This can lead to self-induced effects.
Typical numbers for n2:
air 4×10-23 m2/W
glass 2.4×10-20 m2/W
This is known as the “optical Kerr effect”.
How realistic is it to get to these intensities?
For air: n2 = 4×10-23 m2/W
So the interesting intensity range is: I = 1/n2 = 2.5×1022 W/m2.
Suppose our light is focused to a spot size of 10 microns.
Then: area = π R2 = 8×10-11 m2
necessary power = I × area = 2×1012 W
Suppose our pulse duration is τP =100 femtoseconds.
Then:
necessary pulse energy = power × τP = 0.2 joules
Focusing 200 millijoules in 100 femtoseconds gives n2I = 1 in air.
At a pulse energy of 2 mJ, the refractive index is changed by ~1%.
In solid and liquid materials, the threshold is lower. This is a lot of
intensity, but it is achievable using intense short pulses.
This is one process which results from the
intensity-dependent refractive index.
intensity profile
of optical beam
(e.g. Gaussian)
Innn 20 +=
Self-lensing (or self-focusing)
optical thickness
A flat slab can
act like a lens!
Kerr medium
phase fronts
x,y
I
x,y
I
High-intensity pulse
Self-lensing can be used to generate
femtosecond pulses
If a pulse experiences additional focusing due to high intensity and the
nonlinear refractive index, and we align the laser for this extra focusing,
then a high-intensity beam will have better overlap with a gain medium.
Low-intensity pulse
gain
crystal
Mirror
Additional focusing
optics can arrange
for perfect overlap of
the high-intensity
beam back in the
gain crystal.
But not the low-
intensity beam!
Thus, the non-linearity favors high intensities, which favors short
pulse formation. It is now routine to generate pulses of less than
100 femtosecond duration, using this self-lensing mechanism.
Self-focusing leads to an ever-
increasing intensity at the center
of the laser beam. Eventually,
the intensity is high enough to
ionize atoms.
In air, ionizing atoms produces a
plasma. This plasma then
contributes to the refractive index,
with a negative contribution:
n = n0 + n2 I − nplasma
If these two contributions offset
each other, then a stable filament is
formed. This filament can
propagate for many meters!
Self-lensing and the formation of filaments
Optical filaments - the Teramobile
A stable filament in air acts as a conductive channel,
which is essentially a lightning rod. This can be used
as a mobile lightning protection system.
a self-guided filament induced
in air by a high-power, infrared
(800 nm) laser pulse
guided and unguided lightning
Teramobile
Pulses can modify their own spectral phase
As a light beam propagates a distance z in a medium, it acquires a phase:
t kz t n zc
ωφ ω ω= − = −
( )0 2= − +t n n I zc
ωφ ω
( )2= = −inst
dI td n z
dt c dt
φ ωω ω
Instantaneous frequency is equal to the time derivative of the phase:
If intensity depends on time, then the pulse frequency changes with time!
“self-phase modulation”
Optical Kerr effect: the refractive index depends on intensity:
The nonlinear phase gives rise to an instantaneous
frequency which depends on time: )()( 0 tt δωωω −=
where: 02
( )( ) NL
zd dI tt n
dt c dt
ωδω = ϕ =
If the light is a pulse, then
the instantaneous
frequency is first smaller
than, and then larger than,
the central frequency ω0.
Self-phase modulation
Self-phase modulation: ωωωω depends on t
This can be extremely dramatic if the excursions
of ω(t) away from its original value are large.
ω = constant
throughoutω = higher
}
ω = lower
}
microstructured
optical fiber
output spectrum
Optics Letters, vol. 25, p. 25 (2000)
Self-phase modulation: spectral broadening
If I(t) changes very rapidly (e.g., femtosecond
pulse), then its derivative is large - so that the
excursions of the frequency δω could be larger
than the initial bandwidth of the pulse! The
spectrum of the light gets broadened!
broadened by
a factor of 100!
original
pulse
bandwidth
new
frequencies!
Supercontinuum generation
red light in* white light out
What if this occurs in a regime
of anomalous dispersion?
new frequency components at
ω < ω0 generated by the early
(rising) edge of the pulse
new frequency components at
ω > ω0 generated by the later
(falling) edge of the pulse
0<dn
dωlower ω has lower velocity
higher ω has higher velocity
Self-phase modulation + anomalous dispersion
The new (lower) frequencies which are generated on the leading edge
travel a bit slower, so the pulse catches up to them.
The new (higher) frequencies which are generated on the trailing edge
travel a bit faster, so they catch up to the pulse.
Result: the pulse shape is stable! A “solitary wave”, or “soliton”
Solitons
Soliton: a localized traveling wave whose intensity
profile is stablized by the interplay of (linear) anomalous
dispersion and non-linearity, so that its shape doesn’t
change as the wave propagates.
Discovered in 1834: John Scott
Russell observed “solitary
waves” of water propagating for
long distances along the Union
canal in Scotland.
a recreation of his observation, on the
John Scott Russell Acqueduct, 1995
“…a well-defined heap of water which
continued its course along the channel
apparently without change of form or
diminution of speed.”
Anomalous dispersion + Kerr effect = soliton
Anomalous dispersion
Kerr
Solitons are a solution to the nonlinear
Schroedinger equation
22
2
∂ ∂= −
∂ ∂E E
jD j E Ez t
ξ
2
2
1
2
∂≡
∂k
D Lω 2
2≡ n L
πξ
λwhere and
proportional to
group velocity
dispersion, GVD
proportional to
Kerr nonlinearity
( ) ( )0, sech= tE z t Eτ
The envelope of the
solution is a pulse:with duration:
2
0
2
1
2= −
A
D
ξτ
• solution only exists if ξ/D < 0 - anomalous dispersion
• pulse duration is independent of propagation distance!
Solitons in optics
Solitons in fiber optics
are the basis for many
telecommunications
transmission systems.
Dispersion of standard optical fiber
anomalous
Solitons can exist in standard fibers for λ > 1310 nm
When solitons collide
http://kasmana.people.cofc.edu/SOLITONPICS/
A cool and easy-to-understand
discussion about solitons:
Because of the non-linear
nature of the equation,
superposition does not hold
for solitons.
But they can still collide with
each other, and when they
do, they act a bit like
particles!
A soliton collision:
The same collision,
decomposed:
(1) (3) 3 (5) 5( ) ( ) ( ) ( ) ...P t E t E t E tχ χ χ= + + +
This treatment assumes |χ(3)| >> |χ(5)| >> |χ(7)| *
Consider propagation of an intense pulse in a gas.
Symmetry: all even orders of χ vanish
We would expect each successive high harmonic to
be exponentially weaker than the preceding one.
Counter-example:
Kapteyn and Murnane, Phys. Rev. Lett., 79, 2967 (1997)
neon
helium
Sometimes, the χχχχ(n) approach fails
How to explain high harmonic generation?
We cannot use a perturbative
approach, i.e., the χ(n) picture.
We must resort to an atomic
picture of the dynamics:
1. ionization of an atom
2. acceleration of a free electron
3. impact with the parent ion
Since all of the x-ray
harmonics are in phase, it can
also be used to generate the
world’s shortest light pulses. “attosecond physics”
current world record (Sept. 2012): 67 as = 67 × 10-18 secK. Zhao et al., Optics Letters, 37, 3891 (2012)
top related