An Introduction to Nonlinear Optics for Accelerator Diagnostics

David Walsh

LA3NET, 1st Intl School on Laser Applications, GANIL, France,17 October 2012

What do we mean by nonlinear?

Linear response Nonlinear responses

x

f(x)

x

f(x)

Examples: Cheap stereos turned up too loud Electric guitar distortions (clipping)

Linear Optics

• Properties independent of light intensity – Linear refractive index – Snell’s Law – Linear absorption - Beer-Lambert Law

• Super position principle – Can look at inputs singularly and sum at the end

• Frequency of light is constant

– νin = νout

• Beams do not interact

– You can cross the beams with no effect – (not counting interference effects…)

Nonlinear Optics

• Refractive index and absorption can be a function of intensity – Two photon absorption – Self focussing

• Superposition no longer true

– Sum of responses to two inputs not the same as response to both inputs simultaneously

• Optical frequency can be changed

– Mixing processes – Harmonic generation

• Beams can interact

– Crossing the beams makes interesting things happen

Which Media are Nonlinear?

All of them!

• Solids

• Liquids

• Gasses

• Plasmas

• Vacuum

(via generation of vitual e- e+ pairs)

But it’s only observable if you pump them hard enough!

In general, lasers are required to observe nonlinear effects

The Birth of Nonlinear Optics

The field was kick started in 1961, by the first demonstration of second harmonic generation

This was made possible by the realisation of the ruby laser the previous year

Origin of the Nonlinear Response

The polarisation of atoms and molecules is affected by an incident EM wave

The potential gets very flat out at infinity, so the electron’s motion can easily go nonlinear!

Potential due to nucleus

Nucleus

Ener

gy

Position

Origin of the Nonlinear Response

Light wave

Incident light affects the polarisation of atoms/molecules throughout the medium, setting up a polarisation field.

This field reradiates EM waves, and in the linear regime is the origin of the refractive index.

Oscillating dipoles

Origin of the Nonlinear Response

Second Order Susceptibility Tensor

Third Order Susceptibility Tensor

E

P

E

P

E

P

Linear Susceptibility Tensor

Effects of the Nonlinearity

DC offset

Optical Rectification

Doubled frequency

Second Harmonic Generation

Effects of the Nonlinearity

Original frequency again!

Sum- and difference-frequency generation

Suppose there are two different-color beams present:

*

1 1 1 1

*

2 2 2 2exp( ) e exp( ) exp( ) x (( )) pE i t E i t E i t E i tt E

* *

1 2 1

* *

1 2 1 2 1 2 1 2

2 *

2

2 2

1 2

2 1 2

2

2

1 2

2 2

2

1 1 1

2

*

1

2

2 exp (

exp(2 ) exp(

2 exp (

exp(2 ) exp

) 2

) 2 exp

( )

( )

2

( 2

2

)

e )

)

x

2

p (

E i t E

E

i

E

t

E

E

t

E E

i t E E i

E i

i t E i

t

t E E

t

i t

E

Note also that, when i is negative inside the exp, the E in front has a *.

2nd-harmonic gen

2nd-harmonic gen

Sum-freq gen

Diff-freq gen

dc rectification

So:

Phasematching

• It seems that a there is potentially an awful lot going on

• Especially consider the cascading of processes that appears unavoidable

• How can we make practical use of these effect?

• Well, all the processes happen, but in general with low efficiencies

• We can drastically enhance efficiency via “phasematching”

Phasematching

Phasor Representation

( ) (2 )n n

2Frequency

Refr

active

index

Unfortunately, dispersion prevents this from happening!

The phase-matching condition for SHG:

Phasematching

Except in very rare cases near absorption features.

Birefringent Phasematching

z

x

y

nx

ny

nz

x

y

z

nx

ny

nz

The Index Ellipsoid

x The optic axis, is the direction in which both polarisations of light experience the same refractive index.

“Critical Phasematching”

nz

ny

k

z=optic axis

y

nx

ny

nz k

ne

y

z

Birefringent materials have different refractive indices for different polarizations. Ordinary and Extraordinary refractive indices can be different by up to 0.1 for SHG crystals.

( ) (2 )e on n 2Frequency R

efr

active index

ne depends on propagation angle, so we can tune for a given .

Some crystals have ne < no, so the opposite polarizations work.

We can now satisfy the phase-matching condition. Use the extraordinary polarization for and the ordinary for 2:

Birefringent Phasematching

Light created in real crystals

Input beam

Far from

phase-matching:

Closer to

phase-matching:

SHG crystal

Input beam

SHG crystal

Note that SH beam is brighter as phase-matching is achieved.

Output beam

Output beam

Walk-Off

The polarisation and electric fields are represented by vectors

2nd rank tensor

k

s E

D

Overall displacement

The Nonlinear Tensor

kjijkkjijki EEdEEP )2(0

)2(

0

Conventionally the susceptibility is replaced by the nonlinear tensor 2dijk

dijk is a 3x3x3 tensor, and maps all components of the vector E to vector P

Consider: d(i,j,k) describes the nonlinear coupling of fields Ea and Eb

along j and k into the polarisation along i

x,y,z=1,2,3

Note that the order of the E fields is not important d232 = d223

x

y

z

P Ea Eb

d232 describes

Simplifications of the Nonlinear Tensor Using the “piezoelectric contraction” the nonlinear response is now (order of fields not important)

Where elements j,k have been replaced with l

We can go one step further in lossless media, the Kleinman’s contraction

For given i,j,k then dijk = dikj = dkji = dkij = djik = djki

b

X

a

Y

b

Y

a

X

b

X

a

Z

b

Z

a

X

b

Y

a

Z

b

Z

a

Y

b

Z

a

Z

b

Y

a

Y

b

X

a

X

o

Z

Y

X

EEEE

EEEE

EEEE

EE

EE

EE

dddddd

dddddd

dddddd

P

P

P

363534333231

262524232221

161514131211

2

jk 11 22 33 23, 32 13, 31 12, 21

l 1 2 3 4 5 6

d11 d12 d13 d14 d15 d16

d16 d 22 d 23 d 24 d14 d12

d15 d 24 d 33 d 23 d13 d14

di,l =

Now there are only 10 independent values! The actual elements present depends on the crystal symmetry.

The Effective Nonlinear Coefficient

An example:

LBO crystal, E-fields along (1,1,0) and (1,-1,0)

)](,0,0[

2

1

11

0

0

0

11

11

...

.....

.....

2

2415

2

332415

24

15

ddP

ddd

d

d

P

P

P

o

O

Z

Y

X

normalising factor

Maxwell’s Equations

Evaluating the effect of an induced polarization on a wave requires solving Maxwells equations, with an induced polarization term. This gives rise to an extra term in the wave equation: This is the Inhomogeneous Wave Equation. The polarization is the driving term for a new solution to this equation.

2

2

02

2

0

2

tt

NL

PEE

(Normally = 0)

Coupled Wave Equations If we consider the interaction of 3 waves propagating in the z direction

Substituting the three waves to the inhomogeneous wave equation we get a set of 3 coupled wave equations

Ei

(1)(z, t) 1

2 (E1i

exp[i(1t k

1z)] + c.c.)

Ek(2 )

(z,t ) 12 (E2 k exp[i(2 t k2z)] + c.c.)

Ej(

3)(z,t) 1

2 (E3 j exp[i(3 t k3z)] + c.c.)

i,j,k are the wave polarisation, and can be x or y

dE1i

dz

i 1

cn1

dijk

' E3 jE

2k

* exp(ikz )

dE2 k

*

dz

i2

cn2

dkij'E1iE3 j

*exp(ikz )

dE3 j

dz i 3

cn3

djik

' E1iE

2 kexp(ikz )

Phase mismatch E is complex as it contains a static phase term

123

Manley-Rowe Relations dE1i

dz

i 1

cn1

dijk

' E3 jE

2k

* exp(ikz )

dE2 k

*

dz

i2

cn2

dkij'E1iE3 j

*exp(ikz )

dE3 j

dz i 3

cn3

djik

' E1iE

2 kexp(ikz )

E1

*n1 /1

E2n2 /2

E3n3 /3

n1d E1

2

1dz

n2d E2

2

2dz

n3d E3

2

3dz

idijk

'

cE1

*E2

*E3 exp(ikz)

I nco E

2

/2

1

1

dI1

dz

1

2

dI2

dz

1

3

dI3

dz

“Photon Splitting” Picture

ω3

ω2

ω1 “Parametric” – no absorption

Example Analysis of SHG Using the coupled wave equations we can now analyse second order processes.

All we need to do is apply appropriate assumptions and boundary conditions, and then solve the equations.

Quick example for SHG:

021 dz

dE

dz

dE

123 22

)exp(21

'

3

33 kziEEdcn

i

dz

dEjik

bkziaE )exp(3)exp(3 kzikai

dz

dE

kcn

EEda

jik

3

21

'

3 0@03 zEkcn

EEdab

jik

3

21

'

3

Assume negligible depletion

Trial solution

SHG Analysis Continued…

Signal grows quadratically

with length when phasematched

)exp(14

3

'

3 kzikcn

EdE

jik

*2

2

3

2

22'2

*

333 )exp(1)exp(1

2

4kzikzi

knc

IdEEI

jik

21

3 2

EEE

2

2

2

3

2

222'2

3

2

2sin

4

kz

kz

nc

zIdI

jik

Gain depends on wavelength Phasematching is important!

Quasiphasematching

sinc 2 ( kz / 2 )

converted field in crystal

Highest nonlinear coefficients don’t generally match with allowed phasematching conditions

Power oscillates between second harmonic and fundamental

What if we are phasematched say, here

Quasiphasematching

kLcoh

0 2 3dE1i

dz

i 1

cn1

dijk

' E3 jE

2k

* exp(ikz )

dE2 k

*

dz

i2

cn2

dkij'E1iE3 j

*exp(ikz )

dE3 j

dz i 3

cn3

djik

' E1iE

2 kexp(ikz )

iexp

iexp

iexp

1exp i

Change the sign of dijk?

Quasiphasematching

g 2lcoh 2

k kqpm k3 k2 k1 2

g

with domain reversal (quasi- phasematching)

no domain reversal (no phasematching)

con

ver

sion

g

z

This is possible in ferro-electric materials, e.g. lithium niobate, by applying a periodic high electric field which flips the material domains. Efficiency is improved by an order of magnitude!

Non-colinear phasematching The wave vector, k, is actually a vector property.

This allows us to have non-colinear waves and still achieve phasematching.

One application is the compensation of “walk-off”

In an appropriate material, it can also be arranged that multiple wavelengths phasematch with the same propagation angle, leading to a very large bandwidth for the nonlinear process.

3k

2k1k

If not compensation for walk-off, this has the effect of limiting the interaction length

Sum Frequency Generation 1

nonlinear medium

3

2

1

2

1

2sinc

22

0

3

321

2

21

2

3

3

Δkz

cnnn

zIIdI

ijk

In the general, low conversion efficiency case

Difference Frequency Generation

nonlinear medium 3

2

1

3

1

)(cosh)0()( 2

11 zIzI

)(sinh)0()( 2

1

212 zIzI

i.e. both waves experience gain at the expense of the “pump”

input

output

fields

z

2

21

2

3

2

21

cnn

Edijk

*Starting from low conversion conditions, it takes the quadratic form of the SFG solution

Optical Parametric Generation/Oscillation

• For DFG, both of the down-converted (lower frequency) waves experience gain

• Additionally, there exists spontaneously emitted photons (thermal, quantum noise)

• If the gain is sufficiently high, the few photons that are generated with the correct polarisation, frequency, and direction, will be amplified – i.e. you can get new wavelengths without the need to seed the process!

Parametric Gain

mirror mirror

Broadly tuneable laser-like sources

OPG

OPO

Pockel’s Effect

The second order nonlinearity can also be exploited to modulate the phase, amplitude or frequency of light by applying a suitable electric field

Consider the polarisation response:

And define the total susceptibility:

Now, the refractive index is:

Using aTaylor series approx.

This effect was observed >50 years before the invention of the laser!

P o((1)

E ( 2)

E2 ...)

E

P

o

n 1 (1

(1)

( 2 )E ...)

1 /2 (1

( 2 )

E

no2 )

1/2no

n no ( 2 )E

2no

Important Considerations

• Temperature range for phasematching

• Angular acceptance for phasematching

• Which element of deff to use

• Focussing trade-off

w, mm

x, mm

Intensity

Interaction Length

Walk-off

Ultrashort Pulses

kvg

Time

Starting at t=0 Say all beams overlap

3 pulses begin to separate Conversion still happens in overlap

Pulses are separate No more conversion Generated pulse “stretched”

Need to consider Group Velocities

Now, Some Applications

Photo Injector System

A short, intense, pulse of high energy photons is required at the photocathode in order to liberate electrons for acceleration.

Lasers can produce the synchronised, high intensity pulses, except they are at much longer wavelengths!

e.g. Ti:Sapphire 800nm, Nd 1064nm, Er 1550nm

The solution, as we know, is nonlinear optics.

What if we want UV light, as required by some photocathodes.

Try 3rd harmonic of Ti:Sapphire?

Third Harmonic Generation

Can be achieved via a 3rd order process but a very high pump field is required due to the weak susceptibility

Thankfully there is a better way – cascaded second order processes!

Femtosecond Ti:Sapphire

Second Harmonic Generation

800nm Sum Frequency Generation

400nm 266nm

Can be 50%+ efficient Also quite efficient

p

ppSHG

SFG

pp

SHG

33

111,

11111

Electron Bunch Length Diagnostics

• A number of techniques are being refined for the optical characterisation of the bunch temporal profile.

• Generally, this is achieved by shifting characteristics of the coulomb field onto an optical frequency carrier beam

• This optical signal is then characterised in order to infer properties of the electron bunch

The field of THz generation and detection by modelocked lasers is well developed, and as the coulomb field is similar in nature to a THz pulse, similar techniques are used.

Typically, THz detection is performed via the electro-optic effect.

Coulomb field of

relativistic bunch

probe laser

non-linear crystal

(electro-optic effect)

laser pulse

(linearly polarised) elliptically polarised

Refractive index modified by external (quasi)-DC electric field

intensity dependent

on ‘DC’ field strength

quasi-DC description ok if tlaser << time scale of EDC variations

(basis for Pockels cells, sampling electro-optic THz detection, ...)

N.B. Time-varying refractive index is a restricted approximation to the physics

( albeit a very useful and applicable formalism for majority of situations )

Physics of EO encoding ... standard description

Electron Bunch Length Diagnostics

Spectral Decoding

Spatial Encoding

Temporal Decoding

Spectral upconversion**

o Chirped optical input o Spectral readout o Use time-wavelength relationship

o Ultrashort optical input o Spatial readout (EO crystal) o Use time-space relationship

o Long pulse + ultrashort pulse gate o Spatial readout (cross-correlator crystal) o Use time-space relationship

o monochomatic optical input (long pulse) o Spectral readout o **Implicit time domain information only

(2)(;thz,opt)

opt + thz

convolution over all combinations of optical

and Coulomb frequencies

Electro-optic detection bandwidth

thz

opt

opt - thz

opt

description of EO detection as sum- and difference-frequency mixing

THz spectrum (complex)

propagation & nonlinear

efficiency

geometry dependent

(repeat for each principle axis)

optical probe spectrum (complex)

EO c

ryst

al

This is “Small signal” solution. High field effects c.f. Jamison Appl Phys B 91 241 (2008)

GRENOUILLE

• Grating Eliminated No-nonsense Observation of Ultrafast Laser Light E-fields…

• Spectrometer is replaced by a thick nonlinear crystal and a CCD – the phasematching condition now determines the angular spread

• Delay stage has been replaced by a Fresnel Biprism, thereby mapping the delay across the CCD and allowing single shot pulse characterisation

Thank you for listening