Light and MatterTim Freegarde

School of Physics & Astronomy

University of Southampton

The tensor nature of susceptibility

2

Birefringence

• asymmetry in crystal structure causes polarization dependent refractive index

• ray splits into orthogonally polarized components, which follow different paths through crystal• note that polarization axes are not related to plane of incidence

3

Anisotropic media

• difference in refractive index (birefringence) or absorption coefficient (dichroism) depending upon polarization

• recall that EEP 100 2

2

21

ciwhere

• susceptibility cannot be a simple scalar

• difference in or implies difference

in

1

2

EP

4

Linear dichroism

• fields normal to the conducting wires are transmitted

• current flow is not parallel to field

WIRE GRID POLARIZER

• susceptibility not a simple scalar

• fields parallel to the conducting wires are attenuated

I

5

Birefringence – mechanical model

x

y

z• springs attach electron cloud to fixed ion• different spring constants for x, y, z axes• polarization easiest along axis of weakest spring

• polarization therefore not parallel to field• susceptibility not a simple scalar

6

Susceptibility tensor

• the Jones matrix can map

• similarly, a tensor can describe the susceptibility of anisotropic media

y

x

y

x

a

a

aa

aa

a

a

2221

1211

JONES MATRIX

ya

xa

ya

xa ,,

EP

333231

232221

131211

0

SUSCEPTIBILITY TENSOR

• e.g. zyxx EEEP 1312110

7

Diagonizing the susceptibility tensor

• if the polarization axes are aligned with the principal axes of birefringent crystals, rays propagate as single beams• the susceptibility tensor is then diagonal

EP

33

22

11

0

00

00

00

DIAGONAL SUSCEPTIBILITY

TENSOR

• a matrix may be diagonalized if symmetrical:

jiij • the optical activity tensor is not

symmetrical; it cannot be diagonalized to reveal principal axes

8

The Fresnel ellipsoid

• surface mapped out by electric field vector for a given energy density

FRESNEL ELLIPSOID

• symmetry axes x’,y’,z’ are principal axes

UDE

zzyyxx

1,

1,

1• semi-axes are

xE

yE

zE

9

The Fresnel ellipsoid

• allows fast and slow axes to be determined:

FRESNEL ELLIPSOID

xE

yE

zE

• establish ray direction through Poynting vector

• electric field must lie in normal plane

• fast and slow axes are axes of elliptical cross-section

• axis lengths are

slowfast

1,

1

S

• allows fast and slow axes to be determined:

10

The optic axis

• if the cross-section is circular, the refractive index is independent of polarization

FRESNEL ELLIPSOID

xE

yE

zE

S• the Poynting vector then defines an optic axis• if , the single optic axis lies

along (uniaxial crystals)

yyxx z

• if , there are two, inclined, optic axes (biaxial crystals)

yyxx

11

Uniaxial crystals

FRESNEL ELLIPSOID

xE

yE

zE

S

• the single optic axis lies alongz optic axis

• one polarization is inevitably perpendicular to the optic axis (ordinary polarization)• the second polarization will be orthogonal to both the ordinary polarization and the Poynting vector

(extraordinary polarization)• positive uniaxial: oe • negative uniaxial: eo

12

Poynting vector walk-off

• Poynting vector obeys 0SE• wavevector vector obeys 0kD• in anisotropic media, and are not

necessarily parallelE D

• the Poynting vector and wavevector may therefore diverge

S

E

D

kwavevector

Poynting vector Fresnel

ellipsoid

13

How light interacts with matter

• atoms are polarized by applied fields

• Lorentz model: harmonically bound classical particles

220

2x

mEP 0 EPED

10

0

xmx

Vxm 2

0d

d

xV

x

E

14

Quantum description of atomic polarization

1

tie 02

tibat 0exp21, r

energy

0

0

1 2

• harmonic oscillator• two-level atom

2exp1,0

tit

r • electron density depends upon relative phase of superposition components

• weak electric field

15

Quantum description of atomic polarization

x/a0 x/a0

2exp1,0

tit

r

1 2

• electron density depends upon relative phase of superposition components

1

tie 02

energy

0

0• harmonic

oscillator• two-level atom

• weak electric field

16

Optical nonlinearity

• potential is anharmonic for large displacements

43220

2cxbxx

mxV

xV

x

220

2x

m

332210 EEEP

• polarization consequently varies nonlinearly with field

• in quantum description, • uneven level spacing• distortion of

eigenfunctions• higher terms in perturbation

17

Optical nonlinearity

• potential is anharmonic for large displacements

43220

2cxbxx

mxV

xV

x

332210 EEEP

• polarization consequently varies nonlinearly with field

• in quantum description, • uneven level spacing• distortion of

eigenfunctions• higher terms in perturbation

x

V

d

d

18

Electro-optic effect

• nonlinearity mixes static and oscillatory fields

332210 EEEP

x

x

V

d

d

EEEP 320

20

10 32

• susceptibility at hence controlled by 0E

• exploit the nonlinear susceptibility

• Pockels effect; Kerr effect

0E

19

Second harmonic generation

x

x

V

d

d

332210 EEEP

• again exploit the nonlinear susceptibility

• distortion introduces overtones (harmonics)

2210 EEP

where 22cos1cos 22 tEtE

20

Nonlinear tensor susceptibilities

• nonlinear contributions to the polarization depend upon products of electric field components

• each product corresponds to a different susceptibility coefficient

zyxkj

kjijkzyxj

jiji EEEP,,,

2

,,

10

e.g. zyxjzyxiEEEE ji ,,;,,,2,121

• terms in the susceptibility expansion are therefore tensors of increasing rank

• the induced polarization has three components (i =x,y,z):

21

Nonlinear tensor susceptibilities

• the susceptibility depends upon the frequencies of the field and polarization components

213 23

113

1

0

1 :: EEP

e.g. if , 21 EEE

2211 223

2113

2 ,:,: EEEE

1221 1232

2132 ,:,: EEEE

• any susceptibility unless 0,: 2132 213

22

Symmetry in susceptibility

• the susceptibility tensor may be invariant under certain symmetry operations

e.g. • rotation• reflection

• the symmetries of the susceptibility must include – but are not limited to – those of the crystal point group

• optically active materials fall outside the point group description (nonlocality)

• materials showing inversion symmetry have identically zero terms of even rank

• inversion zyxzyx ,,,,

23

Properties of susceptibility

• depends upon frequency

• dispersion and absorption in material response

• depends upon field orientation

• anisotropy in crystal and molecular structure

• depends upon field strength

• anharmonicity of binding potential hence nonlinearity

• tensor nature of susceptibility

• series expansion of susceptibility

24

Pockels (linear electro-optic) effect

x

x

V

d

d

• nonlinearity mixes static and oscillatory fields

0E

EEχEχP 021

0 0,: 02 ,0: EEχ

• applying intrinsic permutation symmetry,

021 0,:21 Eχχε 2

03 0,0,:3 Eχ

• in non-centrosymmetric materials, dominates

2χ

25

Kerr (quadratic electro-optic) effect

x

x

V

d

d

• nonlinearity mixes static and oscillatory fields

0E

EEχEχP 021

0 0,: 02 ,0: EEχ

• applying intrinsic permutation symmetry,

021 0,:21 Eχχε 2

03 0,0,:3 Eχ

• in centrosymmetric materials, 02 χ

26

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