9' Thirdorder Nonlinearities: Fourwave mixing

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9. Third-order Nonlinearities Four-wave mixing
Third-harmonic generation Induced gratings Phase
conjugation Nonlinear refractive
index Self-focusing Self-phase modulation Continuu
m generation
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Third-harmonic generation
We must now cube the input field
Third-harmonic generation is weaker than
second-harmonic and sum-frequency generation, so
the third harmonic is usually generated using SHG
followed by SFG, rather than by direct THG.
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Noncollinear Third-Harmonic Generation
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.
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Third-order difference-frequency generation
Self-diffraction
Consider some of the difference-frequency terms
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The excite-probe geometry
One field can contribute two factors, one E and
the other E. This will involve both adding and
subtracting the frequency and its k-vector.
This effect is automatically phase-matched!
The excite-probe beam geometry has many
applications, especially to ultrafast
spectroscopy. The signal beam can be difficult to
separate from the input beam, E1, however.
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Polarization gating
The input beam is the signal beam direction is
rejected by polarizer!
Here field 2 contributes two factors, one E and
the other E. But one is vertically polarized,
while the other is horizontally polarized. This
yields a signal beam thats orthogonally
polarized to the input beam E1.
If E1 is horizontally polarized, the signal will
be vertically polarized
This arrangement is also automatically
phase-matched. Its also referred to as
polarization spectroscopy due to its many uses in
both ultrafast and frequency-domain spectroscopy.
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Many nonlinear-optical effects can beconsidered
as induced gratings.
The irradiance of two crossed beams is
sinusoidal, inducing a sinusoidal absorption or
refractive index in the mediuma diffraction
grating!
An induced grating results from the cross term in
the irradiance
Time-independent fringes
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Diffraction off an induced grating
A third beam will then diffract into a different
direction. This results in a beam thats the
product of E1, E2, and E3
This is just a generic four-wave-mixing effect.
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Induced gratings
Assume
but
Phase-matching condition
The diffracted beam has the same frequency and
k-vector magnitude as the probe beam, but its
direction will be different.
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Phase-matching induced gratings
Phase-matching condition
The minus sign is just the excite-probe effect.
z-component
x-component
The Bragg Condition
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Induced gratingswith different frequencies
This effect is called non-degenerate four-wave
mixing. In this case, the intensity fringes
sweep through the medium a moving grating.
Phase-matching condition
The set of possible beam geometries is complex.
See my thesis!
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Induced gratings with plane waves and more
complex beams (of the same frequency)
A plane wave and a very distorted wave
A plane wave and a slightly distorted wave
Two plane waves
A plane wave and a slightly distorted wave
All such induced gratings will diffract a plane
wave, reproducing the distorted wave
E2 and E3 are plane waves.
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Holography is an induced-grating process.
One of the write beams has a complex spatial
pattern that is the image. Different incidence
angles correspond to different fringe spacings,
so different views of the object are stored as
different fringe spacings. A third beam (a plane
wave) diffracts off the grating, acquiring the
image information. In addition, different fringe
spacings yield different diffraction
angleshence 3D! The light phase stores the
angular info.
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Phase conjugation
When a nonlinear-optical effect produces a light
wave proportional to E, the process is called a
phase-conjugation process. Phase conjugators can
cancel out aberrations.
A normal mirror leaves the sign of the phase
unchanged
A phase-conjugate mirror reverses the sign of
the phase
The second traversal through the medium cancels
out the phase distortion caused by the first pass!
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Phase Conjugation Time Reversal
A light wave is given by
If we can phase-conjugate the spatial part, we
have
Thus phase conjugation produces a time-reversed
beam!
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Degenerate Four-Wave Mixing
Consider only processes with three input
frequencies and an output frequency that are
identical.
Because the k-vectors can have different
directions, well distinguish between them (as
well as the fields)
Degenerate four-wave mixing gives rise to an
amazing variety of interesting effects. Some are
desirable. Some are not. Some are desirable some
of the time and not the rest of the time.
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Single-Field Degenerate Four-Wave Mixing
If just one beam is involved, all the k-vectors
will be the same, as will the fields
So the polarization becomes
Single-field degenerate four-wave mixing gives
rise to self effects. These include Self-pha
se modulation Self-focusing (whole-beam and
small-scale) Both of these effects participate
in the generation of ultrashort pulses!
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Degenerate 4WM means a nonlinearrefractive index.
Recall the inhomogeneous wave equation
and the polarization envelope (the linear and
nonlinear terms)
Substituting the polarization into the wave
equation (assuming slow variation in the envelope
of E compared to 1/w)
since
So the refractive index is
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Nonlinear refractive index (contd)
The refractive index in the presence of linear
and nonlinear polarizations
Now, the usual refractive index (which well call
n0) is
So
Assume that the nonlinear term ltlt n0
So
Usually we define a nonlinear refractive index,
n2
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The nonlinear refractive index magnitude and
response time
A variety of effects give rise to a nonlinear
refractive index. Those that yield a large n2
typically have a slow response.
Thermal effects yield a huge nonlinear refractive
index through thermal expansion due to energy
deposition, but they are very very slow. As a
result, most media, including even Chinese tea,
have nonlinear refractive indices!
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Whole-Beam Self-Focusing
The nonlinear refractive index,
, causes beams to self-focus.
If the beam has a spatial Gaussian intensity
profile, then any nonlinear medium will have a
spatial refractive index profile that is also
Gaussian
Near beam center
The phase delay vs. radial co-ordinate will be
This is precisely the behavior of a lens! But
one whose focal power scales with the intensity.
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Small-Scale Self-Focusing
If the beam has variations in intensity across
its profile, it undergoes small-scale
self-focusing.
Each tiny bump in the beam undergoes its own
separate self-focusing, yielding a tightly
focused spot inside the beam, called a
filament.
Such filaments grow exponentially with distance.
And they grow from quantum noise in the beam,
which is always there. As a result, an intense
beam cannot propagate through any medium without
degenerating into a mass of tiny highly intense
filaments, which, even worse, badly damage the
medium.
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Self-Phase Modulation Continuum Generation
The self-phase-modulated pulse develops a phase
vs. time proportional to the input pulse
intensity vs. time.
Pulse intensity vs. time
The further the pulse travels, the
more modulation occurs.
That is
A flat phase vs. time yields the narrowest
spectrum. If we assume the pulse starts with a
flat phase, then SPM broadens the spectrum. This
is not a small effect! A total phase variation of
hundreds can occur! A broad spectrum generated
in this manner is called a Continuum.
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The instantaneous frequency vs. time in SPM
A 10-fs, 800-nm pulse thats experienced
self-phase modulation with a peak magnitude of 1
radian.
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Self-phase-modulated pulse in the frequency domain
The same 10-fs, 800-nm pulse thats experienced
self-phase modulation with a peak magnitude of 1
radian.

Its easy to achieve many radians for phase
delay, however.
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A highly self-phase-modulated pulse
A 10-fs, 800-nm pulse thats experienced
self-phase modulation with a peak magnitude of 10
radians
Note that the spectrum has broadened
significantly. When SPM is very strong, it
broadens the spectrum a lot. We call this effect
continuum generation.
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Experimental Continuum spectrum in a fiber
Continua created by propagating 500-fs 625nm
pulses through 30 cm of single-mode fiber.
Low Energy Medium Energy High Energy
The Supercontinuum Laser Source, Alfano, ed.
Broadest spectrum occurs for highest energy.
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Continuum Generation Simulations
Instantaneously responding n2 maximum SPM phase
72p radians
Input Intensity vs. time (and hence output phase
vs. time)
The Supercontinuum Laser Source, Alfano, ed.
Original spectrum is negligible in width compared
to the output spectrum.
Output spectrum
w
Oscillations occur in spectrum because all
frequencies occur twice and interfere, except for
inflection points, which yield maximum and
minimum frequencies.
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Continuum Generation Simulation
Noninstantaneously responding n2 maximum SPM
phase 72p radians
Output phase vs. time (? input intensity vs.
time, due to slow response)
Output spectrum
Asymmetry in phase vs. time yields asymmetry in
spectrum.
The Supercontinuum Laser Source, Alfano, ed.
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Experimental Continuum Spectra
625-nm (70 fs and 2 ps) pulses in Xe gas
p 15 40 atm
L 90 cm
The Supercontinuum Laser Source, Alfano, ed.
Data taken by Corkum, et al.
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Ultraviolet continuum
4-mJ 160-fs 308-nm pulses in 40 atm of Ar 60-cm
long cell.
Lens focal length 50 cm.
Good quality output mode.
The Supercontinuum Laser Source, Alfano, ed.
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UV Continuum in Air!
308 nm input pulse weak focusing with a 1-m
lens.
The Supercontinuum Laser Source, Alfano, ed.
Continuum is limited when GVD causes the pulse to
spread, reducing the intensity.
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Continuum GenerationGood news and bad news
Good news It broadens the spectrum, offering a
useful ultrafast white-light source and
possible pulse shortening. Bad news Pulse
shapes are uncontrollable. Theory is struggling
to keep up with experiments. In a bulk medium,
spectral broadening is not really that
great you need a log scale to see it In a
bulk medium, continuum can be high-energy, but
its a mess spatially. In a fiber, continuum
is clean, but its low-energy. In hollow fibers,
things get somewhat better. Main problem
dispersion spreads the pulse, limiting the
spectral broadening.
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Microstructure optical fiber
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Microstructure optical fibers modify dispersion.
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The continuum from microstructure optical fiber
is ultrabroadband.
Cross section of the microstructure fiber.
The spectrum extends from 400 to 1500 nm and is
relatively flat (when averaged over time).
This continuum was created using unamplified
TiSapphire pulses. J.K. Ranka, R.S. Windeler,
and A.J. Stentz, Opt. Lett. Vol. 25, pp. 25-27,
2000
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Continuum is quite beautiful!
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Other third-order nonlinear-optical effects
Two-photon absorption
Raman scattering