Light and MatterTim Freegarde

School of Physics & Astronomy

University of Southampton

Controlling light with light

2

Optical nonlinearity• potential is anharmonic for large displacements

43220

2cxbxxmxV

x 332210 EEEP

• polarization consequently varies nonlinearly with field

xVdd

• restoring force is nonlinear function of displacement

xV

• is a tensor of rank n 1n

3

Electro-optic effect

• nonlinearity mixes static and oscillatory fields

332210 EEEP

x

xVdd

EEEP 320

20

10 32

• exploit the nonlinear susceptibility

• Pockels (linear) and Kerr (quadratic) effects

0E

• consider withEEE 0 0EE

4

Pockels (linear electro-optic) effect

x

xVdd

• nonlinearity mixes static and oscillatory fields

0E

EEχEχP 021

0 0,: 02 ,0: EEχ

• applying intrinsic permutation symmetry, 0

21 0,:21 Eχχε 2

03 0,0,:3 Eχ

• in non-centrosymmetric materials, dominates

2χ

5

Kerr (quadratic electro-optic) effect

x

xVdd

• nonlinearity mixes static and oscillatory fields

0E

EEχEχP 021

0 0,: 02 ,0: EEχ

• applying intrinsic permutation symmetry,

021 0,:21 Eχχε

20

3 0,0,:3 Eχ • in centrosymmetric materials, 02 χ

6

Second harmonic generation

x

xVdd

332210 EEEP

• again exploit the nonlinear susceptibility

• distortion introduces overtones (harmonics) 221

0 EEP

where 22cos1cos 22 tEtE

• consider strong field EE

7

Second harmonic generation

x

xVdd

• generated intensities depend upon square of fundamental intensity

• incident field: fundamental• constant

component:optical

rectification• new frequency: second

harmonic

• focussed and pulsed beams give higher conversion efficiencies

• non-centrosymmetric materials required

8

Second harmonic generation

x

xVdd

i

iEE

• if the fundamental field contains differently polarized components

then the harmonic field contains their products

ijjiji EEχP ,:3

203

• the harmonic polarization need not be parallel to ,iE jE

9

Sum and difference frequency generation

x

xVdd

i

iEE

• if the fundamental field contains different frequency components

then the harmonic field contains their products

ijji ji EEχP ,:3

203

where ji 3

10

Pockels cell

polarizer

polarizermodulation voltage

• voltage applied to crystal controls birefringence and hence retardance• mounted between crossed linear polarizers

• longitudinal and transverse geometries for modulation field• allows fast intensity modulation and beam switching

11

Sideband generation2V• bias Pockels cell to

VVT2

sin 2

tVV 10 cos

Pockels cell

V0 V

V1

trans

miss

ion

• add r.f. field to modulate transmitted intensity

• transmitted field contains sum and difference frequency sidebands

0 0 10 10

12

Harmonic generation• fields may be at optical, radio or quasistatic

frequencies 2• combines in pairs, to produce sums and

differences• higher terms in susceptibility may combine more

frequencies• frequency tripling, quadrupling• high harmonic generation

• total photon energy conserved:

ener

gy

1 2 3 43214 e.g.

13

Phase matching• transit time through crystal x

l

cxlxt 2• harmonic beam is superposition of

contributions from all positions in crystal• for contributions to emerge in phase, 2

• use birefringence to offset dispersion 2• choose opposite polarizations for and 2

• conservation of photon momentum

14

Faraday (magneto-optic) effect

• consider effect of longitudinal field upon bound electrons

2220

2

20

3

cmNeV

• optical properties may also be influenced by magnetic fields

B

magneto-optical glass

• induced circular birefringence, characterized by Verdet constant

• non-reciprocal

15

Quantum description of atomic polarization

x/a0 x/a0

1,2exp1,0

tit

r

1 2

• electron density depends upon relative phase of superposition components

16

Faraday optical isolator

• optical ‘diode’ passes incident light but rejects reflection

polarizer

polarizermagneto-

optical glass

B

• 45º rotation in permanent magnetic field

• http://physics.nadn.navy.mil/physics/faculty/mungan/scholarship/FaradayIsolators.pdf