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

χχχχ(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

χω = 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 Eχ

χ χ ω χ ωχ

( )(3)

2

0

0

3

2≈ +n n E

n

χω

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 (of

order 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

Focusing 200 millijoules in 100 femtoseconds gives n2I = 1 in air.

At a pulse energy of 2 mJ, the refractive index is changed by ~1%.

In solid and liquid materials, the threshold is lower. This is a lot of

intensity, but it is achievable using intense short pulses.

This is one process which results from the

intensity-dependent refractive index.

intensity profile

of optical beam

(e.g. Gaussian)

Innn 20 +=

Self-lensing (or self-focusing)

optical thickness

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 leads 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

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: ωωωω 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.”

Anomalous dispersion + Kerr effect = soliton

Anomalous dispersion

Kerr

Solitons are a solution to the nonlinear

Schroedinger equation

22

2

∂ ∂= −

∂ ∂E E

jD j E Ez t

ξ

2

2

1

2

∂≡

∂k

D Lω 2

2≡ n L

πξ

λwhere and

proportional to

group velocity

dispersion, GVD

proportional to

Kerr nonlinearity

( ) ( )0, sech= tE z t Eτ

The 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.

Counter-example:

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

neon

helium

Sometimes, the χχχχ(n) approach fails

How to explain high harmonic generation?

We cannot use a perturbative

approach, i.e., the χ(n) picture.

We must resort to an atomic

picture of the dynamics:

1. ionization of an atom

2. acceleration of a free electron

3. impact with the parent ion

Since all of the x-ray

harmonics are in phase, it can

also be used to generate the

world’s shortest light pulses. “attosecond physics”

current world record (Sept. 2012): 67 as = 67 × 10-18 secK. Zhao et al., Optics Letters, 37, 3891 (2012)