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NON LINEAR OPTICSNON LINEAR OPTICSDR. N. VENKATANATHAN
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Non Linear OpticsNon Linear Opticsy Light of one wavelength is transformed to
light of another wavelength.y The red light was already present in the
white light before it hit the piece of red
glass.
y The glass only filters out the other
wavelengths; it does not generate a newwavelength.
y But in nonlinear optics, new wavelengths
are generated.
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1. The 2nd harmonic, green light at a
wavelength of 532 nm is generated from the
1.06-mm beam of infrared light from an
Nd:YAG laser.
2. It's important to note that only part of the1.06-mm light is converted to the second
harmonic, remaining part is unchanged.
3. In many cases, it is very important to
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Electrons Oscillation inElectrons Oscillation in
CrystalsCrystalsy Electrons are bound in "potential wells,"
which act very much like tiny springsholding the electrons to lattice points
y External force pulls an electron away from
its equilibrium position.
y The spring pulls it back with a force
proportional to displacement.y The spring's restoring force increases
linearly with the electron's displacement
from its equilibrium position.
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Electrons in a nonlinear crystal are bound in
potential wells, which act something likesprings, holding the electrons to lattice
points in the crystal.
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Linear OscillationsLinear Oscillationsy The electric field in a light wave passing
through the crystal exerts a force on theelectrons that pulls them away from theirequilibrium positions.
y In an ordinary (i.e., linear) optical material,the electrons oscillate about theirequilibrium positions at the frequency ofthis electronic field.
y Fundamental law of physics says that anoscillating charge will radiate at itsfrequency of oscillation, so theseelectrons in the crystal "generate" light atthe frequency of the original light wave.
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Non Linear OscillationsNon Linear Oscillations
y Nonlinear material as one whose
electrons are bound by very shortsprings.
y If the light passing through the material
is intense enough, its electric field canpull the electrons.
y The restoring force is no longer
proportional to displacement and it
becomes nonlinear.
y
The electrons are jerked back roughlyrather than ulled back smoothl .
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y They oscillate at frequencies other than
the driving frequency of the light wave.y These electrons radiate at the new
frequencies, generating the new
wavelengths of light.
y The exact values of the new
wavelengths are determined byconservation of energy.
y The energy of the new photons
generated by the nonlinear interaction
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yThe infrared and second-harmonic
photons involved in the secondharmonic generation process.
y
The nonlinear process as weldingtwo infrared photons together to
produce a single photon of green
light.
yThe energy of the two 1.06-mm
photons is equal to the energy of the
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Calculating New WavelengthCalculating New Wavelength
yE = hc/ y
hc/ 3 = hc/ 1 + hc/2y3 = 1 2 / (1 + 2)
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SHG as a welding process: two photons are welded
together to produce a single photon with the energy of both
original photons.
Optical mixing is similar to SHG, except that theoriginal photons have different energies.
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REQUIREMENTS FOR NONREQUIREMENTS FOR NON
LINEAR OPTICSLINEAR OPTICS
yIntense light
y
Conservation of energyyConservation of momentum
- fulfilled by phase matching
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Second Harmonic GenerationSecond Harmonic Generationy It is otherwise called as Frequency
Doubling.y It has relatively higher conversion
efficiency.y It depends on several factors.
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y PSH is the second-harmonic power,
y l is the length of the nonlinear crystal,y Pf is the fundamental power,
y A is the cross-sectional area of the beam
in the nonlinear crystal, and
y the quantity inside the brackets is a
phase-match factor that can vary betweenzero and one.
y Obviously, it is important to ensure that
this factor be as close to unity as possible,
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If the crystal length in the upperexperiment is doubled, the second
harmonic is generated four times
earlier one.
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The fundamental power in the upper
experiment is doubled, the second
harmonic generated increases by 4
times.
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The pulse energy in the upperexperiment is compacted into a pulse
half as long and it is quadrupled in the
second case.
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If the beam in the upper experiment is focused to
one-half its original diameter and second-
harmonic power is inversely proportional to
beam area, which is proportional to the square of
the beam radius (A = r2). So by reducing the
beam radius by a factor of two, you reduce the
beam area by a factor of four and increase the
second-harmonic power by a factor of four.
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Phase MatchingPhase Matchingy If the second-harmonic power generated at
point B is out of phase with the second-harmonic power generated at point A.
y They will interfere destructively and result ina total of zero second-harmonic from the twopoints.
y If the crystal isn't phase matched, thesecond harmonic generated at nearly everypoint in the crystal will be canceled by a
second harmonic from another point.y Practically no second-harmonic will be
produced, no matter how tightly you focus or
how long the crystal is.
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A nonlinear crystal is not phase
matched and harmonic lightgenerated at one point will interfere
destructively with that generated at
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Effect of DispersionEffect of Dispersion
y The refractive index of the nonlinear
crystal is slightly different for the two
wavelengths.y Although the second-harmonic
wavelength is exactly half as long as
the fundamental wavelength in a
vacuum, that is not true inside the
crystal.y The frequency of the second-harmonic
is still exactly twice that of the
fundamental, but the wavelength
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The (horizontally polarized) second-
harmonic wave generated at point A isexactly out of phase with the wave
generated at point B (dotted). The
fundamental wave that creates the second-
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y
The second-harmonic is being generated ina polarization orthogonal to the fundamental.
y Dispersion causes the phase between thefundamental and the second-harmonic toshift slightly as the two travel along togetherinside the nonlinear crystal.
y Eventually, the phase shift becomes large
enough so that new second-harmonic light isgenerated exactly 180 out of phase with theoriginal second-harmonic.
y It only takes several wavelengths for this tohappen; in reality, dispersion is a small effectand the full 180 phase mismatch requires
hundreds or thousands of wavelengths.
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y The solution to this problem hinges on
the fact that the second-harmonic is
generated in a polarization orthogonal to
the fundamental.y In a birefringent crystal, the two
orthogonal polarizations experience
different refractive indices.y First, they are different wavelengths, so
dispersion will cause them to experience
different refractive indices.
y Second, they are orthogonally polarized,
so birefringence will cause them to
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Solution to the ProblemSolution to the Problem
y Making the index difference due todispersion be exactly opposite to the
index difference due to birefringence.y As a result, they both experience the
same refractive index.
y But it's not quite that easy becausenature doesn't supply us with crystals thathave all the requirements of nonlinear
materials.y In practice, it is find a crystal with the right
nonlinear properties, and then fine-tune
its birefringence.
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Techniques to tunethe Birefringence
Temperaturedependent Crystals
TemperatureTuning
TemperatureIndependent
Crystals
ngle Tuning
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y Temperature tuning, takes advantage of the
temperature dependence of some crystals'
refractive indices.
y The nonlinear crystal is placed in an oven (or a
cryostat) and heated (cooled) to a temperature
where its birefringence exactly compensates for
dispersion.
y Angle tuning, can be used with crystals whose
indices aren't temperature dependent.
y The amount of birefringence depends on the
angle of propagation through the crystal, so the
crystal can be rotated with respect to the
incoming beam until the proper birefringence is
obtained.
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y Type I phase matching, in which the
fundamental light is in the ordinarypolarization of the nonlinear crystal.
y The second-harmonic light is generated
in the extraordinary polarization.y Type II phase matching, the fundamental
is evenly divided between the ordinary
and extraordinary polarizations.
y The second-harmonic is generated in the
extraordinary polarization.y SHG with very high-power, solid-state
lasers, Type II phase matching is more
efficient than Type I.
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INTRACAVITY HARMONICINTRACAVITY HARMONIC
GENERATIONGENERATIONy Normally, a nonlinear crystal is placed in
the output beam of a laser.y For intracavity doubling, the crystal is
placed between the mirrors, inside the
resonator.y The efficiency of frequency doubling can
be enhanced by placing the non linear
crystal inside the resonator.
y The amount of second-harmonic power
generated is proportional to the square of
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y There are problems with putting a
nonlinear crystal inside a laser.y If the crystal introduces even a small
lossdue to imperfect surfaces, for
exampleit can drastically decrease thecirculating power.
y One percent additional loss can cut the
circulating power of some lasers by half,
and the advantage of placing the crystal
inside the resonator is immediately lost.
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y It is difficult to fabricate the output
mirror of an internally frequencydoubled laser.
yThe mirror must have maximum
reflectivity at the fundamental to keepthe circulating power inside theresonator.
yAt the same time, it must transmit allthe second harmonic that falls on it.
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HIGHER HARMONICSHIGHER HARMONICSy Third-harmonic light can be generated
with an arrangement quite similar toSHG.
y But phase-matching requirements make
it impossible to generate the third-harmonic in a single step in a crystal.
y So a two-step process is common.
y The second-harmonic is generated in
the first crystal and is then "mixed" with
the fundamental in the second crystal to
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(a) Single-step, third-harmonicgeneration (b) generation of third-
harmonic light by SHG and mixing.
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