14 non linear optics

8/14/2019 14 Non Linear Optics

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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.

14 non linear optics

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