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Nonlinear optics!• Nonlinear optical media"• Second-order nonlinear optics"• Three-wave mixing phenomena"• Phase matching"• Third-order nonlinear optics"

"

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What is nonlinear optics?!It is the branch of optics that studies the behavior of light in nonlinear media""

where P and E have a nonlinear relation!

It is responsible for a number of interesting and exotic phenomena such as:""•  Creating new frequencies"•  Change the frequency of a light beam"•  Changing its shape in space and time"•  Mixing different laser pulses"•  Create ultrashort laser pulses"…""In nonlinear optics the superposition principle is not valid anymore."

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Nonlinear optics is not easy to observe!

Nonlinear optical phenomena require very high intensities and / or electric fields, and several other conditions. 1.  The light intensities of “normal” optics are too weak

2. Normal light sources are incoherent 3. Nonlinear media must possess certain symmetry conditions 4. Phase-matching is required

"

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Linear vs. nonlinear optics!Linear optics!

Optical properties of materials are independent of intensity!"The principle of superposition applies!"The frequency of light going through a medium does not change!"Two beams of light in the same region of a medium have no effect on each other: light can not control light!

Nonlinear optics!Optical properties of materials are dependent of intensity""The principle of superposition doses not apply""The frequency of light going through a medium may change""Two beams of light in the same region of a medium affect each other: light can control light"

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Where does nonlinear optics come from?!

We can understand the basics of nonlinear optics in terms of waves or in terms of photons."Using a wave description, nonlinear behavior is associated with distortion."

new!

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Where does nonlinear optics come from?!

Linear optics Nonlinear optics

new!

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Describing nonlinear optics!We are going to use a description based on the nonlinear wave equation, that can be derived from Maxwell’s equations:"

�

∇ ×H = ∂D∂t

∇ ⋅B = 0

∇ × E = − ∂B∂t

∇ ⋅D = 0

Electric field E Magnetic field H Electric flux density D Magnetic flux density B Polarization density "P"Magnetization density "M"El. permittivity (free sp.) "ε0"El. susceptibility "χ"

�

D = ε0E + P

B = µ0H + µ0M

The specific shape of the wave equation depends on the relation between P and E for each type of medium (constitutive relation)."

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A linear medium is characterized by a linear constitutive relation!

The relation at every (r,t ) is simply"(χ is the electric susceptibility)"

�

P = ε0χE

E P χ

�

F = −kxYou can think of this as similar to the restoring force of a spring"

�

∇ ×H = ε ∂E∂t

∇ ⋅H = 0

∇ × E = −µ ∂H∂t

∇ ⋅E = 0

�

∇2E − 1c 2

∂2E∂t 2

= 0

�

c = 1/ εµ

n = c0c

= εµε0µ0

= 1+ χ

Maxwell’s equations and wave equation for a linear medium:"

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A nonlinear medium is characterized by a nonlinear constitutive relation!

The previous Maxwell’s equation in a linear medium and the wave equation are not valid.!

�

P ∝a1E +a2E2 +a3E

3 +

Starting from the general Maxwell’s eqns. in a medium, and considering a homogeneous and isotropic medium:"

�

∇ × E = −µ ∂H∂t

⇔

∇× ∇ × E( ) = −∇ × µ ∂H∂t

⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

∇2E − 1c02∂2E∂t 2

= µ0∂2P∂t 2

general wave equation for a homogeneous, isotropic medium"

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Linear vs. nonlineardielectric media!

Most dielectric media are approximately linear, except when using very high electric fields – e.g. powerful, focused laser beams. In that case, a Taylor expansion should be used."The nonlinearity becomes relevant for E ~ 105 – 108 V/m.""

�

P = ε0χE

�

P ∝a1E +a2E2 +a3E

3 +

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The nonlinear polarization!It is more practical to write the nonlinear relation in the form"

This classifies the corresponding related phenomena in second-order and third-order nonlinear optics.

�

P ∝ ε0χE + 2dE 2 + 4χ(3)E 3 +

the usual linear susceptibility!

quadratic nonlinearity!d ≈ 10-24 – 10-21 C/V2"

cubic nonlinearity!χ(3) ≈ 10-34 – 10-29 Cm/V3"

�

= PNL

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The nonlinear wave equation!Let’s look again at the general wave equation in a medium:""Now we separate P into its linear and nonlinear components and replace:""By using the well-known relations between c, c0, n, ε0, μ0 and χ:"

�

P ∝ ε0χE + PNL

�

∇2E − 1c02∂2E∂t 2

= µ0∂2P∂t 2

�

∇2E − 1c 2

∂2E∂t 2

= −S

S = −µ0∂2PNL∂t 2

Wave equation in a nonlinear medium

“Source” term

This equation represents a wave propagating in a medium with a “source” term linked to E 2 – it is a nonlinear PDE in E."

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How to solve the nonlinear wave equation!

A first approach is called the Born Approximation:"

Given E0, calculate S(E0)"Solve the equation to obtain E1"

Given E1, calculate S(E1)"Solve the equation to obtain E2"

1st Born approximation"

2nd Born approximation"

For a small nonlinearity, the first approximation is enough – this is what we will use in the next derivations. "

. . . "

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Harmonic field in a 2nd order nonlinear medium!

Let’s calculate the response of this nonlinear medium to a harmonic field:""Following the 1st Born Approximation, we now calculate S(E0):""Replacing the expression for E0, we can write the nonlinear polarization as: ""The polarization has the form of a sum of"•  a DC component PNL(0)"•  a component PNL(2ω) at twice the

frequency"

�

E0(t ) = Re E ω( )eiωt{ }= 1

2 E ω( )eiωt +E * ω( )e−iωt[ ]

This is an example of a second-order process:

�

PNL = 2dE 2

�

PNL(t ) = 2dE02

S E0(t )[ ] = −µ0∂2PNL / ∂t

2

�

PNL(t ) = PNL(0) +Re PNL(2ω)ei2ωt{ }

�

dE(ω)E * ω( )

�

dE 2(ω)

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Harmonic field in a 2nd order nonlinear medium!

Harmonic input field E0(t)"

Medium response"

�

PNL = 2dE 2

�

PNL(t ) = PNL(0) +Re PNL(2ω)ei2ωt{ }

DC component" Second harmonic"

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Second harmonic generation (SHG)!

As we have seen, the nonlinear polarization PNL(t) has a component at a frequency 2ω."So the source term S(t) will also “radiate” at 2ω, with a complex amplitude S(2ω):"

PNL(t ) = PNL(0)+Re dE 2(ω)ei 2ωt{ }S t( ) = −µ0∂

2PNL / ∂t2

S 2ω( ) = 4µ0ω2dE 2

The field E1 that is a solution of the wave equation will also have components at 2ω. This is a mechanism for generating light at twice the frequency."

nonlinear medium

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SHG explained by photons!If the intensity is high enough, two photons may be absorbed at once in a transition to a virtual level, resulting in the emission of a single photon at twice the frequency."

Pantazis et al, PNAS 2010"

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The first demonstration of SHG!P.A. Franken, et al, Physical Review Letters 7, p. 118 (1961)

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Efficiency of SHG!The amplitude of the emitted 2ω light is proportional to S(2ω), so for the intensity we have:"""It is also prop. to the L2 (length) of the medium:""So we can write for the efficiency of SHG:"( I = Power / Area )"

�

I 2ω( ) ∝ S 2ω( ) 2 ∝d 2E ω( ) 4 ∝d 2I 2 ω( )

I 2ω( )∝L2d 2I 2 ω( )

�

I 2ω( )I ω( ) ∝d 2 L2

AP

For a given crystal length L the power P should be concentrated in the smallest possible area A

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The electro-optic effect!

"The polarization has the form of a sum of"•  a DC component PNL(0)"•  a comp. PNL(ω) at the original frequency"•  a comp. PNL(2ω) at twice the frequency"

�

E0(t ) = E 0( ) +Re E ω( )eiωt{ }In this case we consider an electric field E(t) consisting of a steady state component and a harmonic component:

�

PNL(t ) = PNL 0( ) +Re PNL ω( )eiωt{ } +Re PNL 2ω( )ei2ωt{ }

�

d 2E 2 0( ) + E ω( )2[ ]

�

4dE 0( )E ω( )

steady-state (ω=0)

harmonic (ω)

�

4E 2 ω( )

But if E(ω) << E(0) then PNL(2ω) is much smaller than the other two, so the total polarization is essentially linear with E(ω)!

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The electro-optic effect!In the presence of a strong steady field E(0), the nonlinear polarization becomes linear with the optical field E(ω):"

�

PNL ω( ) = ε0ΔχE ω( )Δχ = 4d / ε0( )E 0( ) the constant of

proportionality is governed by E(0)"

�

Δn = 2dnε0

E 0( )

The medium is linear with a refractive index that can be controlled by the applied electric field = Pockels effect."

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Pockels cells!= electrooptic devices used to rotate the polarization of light"

One of the main applications of Pockels cells is as an electrooptic switch (together wih a polarizer) in q-switched lasers (E ~ 3 kV)."

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Three wave mixing!

A 2nd order nonlinear medium can be used to mix two different frequencies and generate a third wave:"•  at the sum frequency (frequency up-conversion)"•  at the difference frequency (frequency down-conversion)"

�

E0(t ) = Re E ω1( )eiω1t +E ω2( )eiω2t{ }Now let’s consider two harmonic waves at different frequencies:

PNL 0( ) = d E ω1( ) + E ω2( )( )PNL 2ω1( ) = dE ω1( )E ω1( )PNL 2ω2( ) = dE ω2( )E ω2( )PNL ω1 +ω2( ) = 2dE ω1( )E ω2( )PNL ω1 − ω2( ) = 2dE ω1( )E ∗ ω2( )

it can be demonstrated that the polarization will have components at:

0, 2ω1, 2ω2, ω1+ω2 and ω1-ω2,

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Sum frequency generation!

�

ω1

�

ω2

�

ω3

�

ω3 = ω1 + ω2

1λo3

= 1λo1

+ 1λo2

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Sum frequency generation vibrational spectroscopy!

This technique allows the study of surfaces and interfaces by mixing an IR and a visible laser beams."

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Frequency and phase matching!In a three-wave mixing process not necessarily all frequencies are generated: certain additional conditions are required"

�

E ω1( ) = A1exp −i k1 ⋅ r( )

�

E ω2( ) = A2 exp −i k2 ⋅ r( )

�

E ω3( ) = A3 exp −i k3 ⋅ r( )

�

ω1 + ω2 = ω3

k1 + k2 = k3

frequency matching condition!

phase matching condition!

These two conditions ensure that the phases of the three waves are matched in space and time, which is necessary for their mutual interaction"

PNL ω1 +ω2( ) = 2dA1A2 exp −i k1 + k2( ) ⋅r⎡⎣ ⎤⎦

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Modalities of three-wave mixing!Optical

frequency conversion

Optical parametric amplifier

Optical parametric oscillator

Spontaneous parametric downcon-

version

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Three wave mixing in terms of photons!

•  two photons of frequency ω1 and ω2 are annihilated and a new photon of higher frequency ω3 is created, or"

•  one photon of higher frequency ω3 is annihilated and two new photons of frequency ω1 and ω2 are created"

�

ω1 + ω2 = ω3

k1 + k2 = k3

conservation of energy!

conservation of momentum!

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How to achieve phase matching?!Let’s look at the equations again (assuming that all beams are collinear):""•  the first one is automatically met"•  the second one requires the following:"

�

ω1 + ω2 = ω3

k1 + k2 = k3

�

ω1n1 + ω2n2 = ω3n3k = ωn /c0( )

It would be easy to match the 2nd one if n1 = n2 = n3, but dispersion does not allow. The higher the frequency, the higher the value of n. What can we do?

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Phase matching is made possible by birefringence!

Anisotropic materials exhibit birefringence i.e. the refractive index “seen” by light depends on its polarization and the angle it makes with special axes of the medium."We can use these degrees of freedom to control the refractive index at the desired frequencies."

""

�

1n2 θ,ω( ) = cos2 θ

no2 ω( ) + sin2 θne2 ω( )

Example: phase matching for SHG in a uniaxial crystal

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Phase matching in SHG!

weak phase matching:intensity is low and “spatial pulsation” is short"

strong phase matching: intensity is high and “spatial pulsation” is long"

SHG crystal

SHG crystal

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Third order nonlinear optics!In some types of optical media the second-order term vanishes (d = 0) and we have a dominant cubic nonlinearity:"

�

PNL(ω) = 3χ(3)E ω( ) 2E ω( )PNL(3ω) = χ(3)E 3 ω( )

�

PNL = 4χ(3)E 3

This is called a Kerr medium."The nonlinear polarization has terms of the type"

third harmonic generation

Kerr effect

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Optical Kerr effect!PNL(ω) corresponds to an increase in the refractive index proportional to the intensity, where n2 = optical Kerr coefficient.""This corresponds to a modulation in the phase of a laser pulse:""Applications:"•  generation of new frequencies in Kerr media (e.g. optical fibers)"

•  self-focusing (e.g. Kerr-lens mode locking)"

�

n I( ) = n +n2I

E z,t( ) =E0 exp ik0n I( )z⎡⎣ ⎤⎦=E0e

iknz exp ik0n2 P / A( )z⎡⎣ ⎤⎦

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Gaussian pulse in a dispersive and nonlinear medium !

https://youtu.be/c5oud_h8R6I

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Soliton in a positive dispersion and nonlinear medium !

https://youtu.be/ua0GQHmlZaQ

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Kerr lens mode locking!Remember that mode locking needs a mechanism to discriminate between low intensity and high intensity pulses. In this case, self focusing is that mechanism."

no pulses:"low intensity, no self focusing""short pulses:"high intensity, self focusing""

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Extreme nonlinear optics!High harmonic generation allows the creation of laser pulses down to only a few nm from a source at e.g. 800 nm"

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High harmonic generation!Three step model for HHG"

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Extreme nonlinear optics!Photonic crystal fibers can exhibit strong nonlinearity leading to intense self-phase modulation and generation of a very broad frequency range."

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Extreme nonlinear optics!Supercontinuum generation in solid state media can also lead to a very broadband spectrum in only a few mm."

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Optical parametric amplification!

nonlinear medium"pump"

signal"

amplification"

idler (necessary because of phase

matching!)"

This is an alternative laser amplification technique, with no population inversion – energy is transferred from the pump to the signal pulse through three wave mixing.

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OPCPA = Optical Parametric Chirped Pulse Amplification!

nonlinear medium"pump"

short pulse"

stretching"

amplification"

signal"

idler"

compressed pulse"

(Optical Parametric Amplification + Chirped Pulse Amplification)

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Research topics at the Laboratory for Intense Lasers!

Laser development and laser applications"High intensity lasers"CPA and OPCPA amplifiers"Supercontinuum generation"Ultrashort pulse diagnostics"High harmonic generation"Coherent x-ray sources"X-ray imaging"Laser particle acceleration"Optical physics""http://xgolp.ist.utl.pt/"