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

χ(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

χ ω = 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 Eχχ χ ω χ ωχ

( )(3)

2

0

0

3

2≈ +n n E

n

χ ω

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 (i.e., approaching 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

Thus: focusing a pulse with an energy of 2 millijoules and a duration of 100 femtoseconds gives n2I = 0.01 in air.

The refractive index is changed by ~1% at the peak of the pulse.

This is a big intensity, but it is achievable using high-energy short pulses. In solid and liquid materials, the threshold is lower, because n2 is bigger.

Suppose we focus a Gaussian beam into a χ(3) medium. The center of the beam is more intense than the wings…

intensity profileof Gaussian beam Innn 20 +=

One result: self-lensing (or self-focusing)

optical thickness (i.e., travel time*c0)

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 can be a positive feedback effect, leading 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

Another effect: 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: ω 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.”

Solitons are a solution to the nonlinear Schroedinger equation

22

2

∂ ∂= −∂ ∂E E

jD j E Ez t

ξ2

2

1

2

∂≡∂

kD L

ω 2

2≡ n Lπξλ

where and

proportional to group velocity dispersion, GVD

proportional to Kerr nonlinearity

( ) ( )0, sech= tE z t E τThe 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.

Therefore, we would never expect observe very high-order harmonics.

Sometimes, the χ(n) approach fails

High harmonic generation

Very high-order harmonics can be observed.

We cannot use a perturbative approach, i.e., the χ(n)

picture fails completely.

Counter-example (the first report):

Kapteyn and Murnane, Phys. Rev. Lett., 79, 2967 (1997)

neon

helium

We must resort to an atomic description of the dynamics:1. ionization of an atom2. acceleration of a free electron3. impact with the parent ion