Experimental Physics EP3 Optics€¦ · Experimental Physics III - Crystal optics 3 Light...

Experimental Physics III - Crystal optics 2

Experimental Physics EP3 Optics

– Crystal and molecular optics –

https://bloch.physgeo.uni-leipzig.de/amr/


Light propagation in a medium

-1.0

-0.5

0.0

0.5

1.0

Ampl

itude

X

vDt

2

22

2

2

tEk

xE

¶¶

÷øö

çèæ=

¶¶

w

vTk 12

2==

plp

wphase velocity

( )xw 00

ktiepp -=!!

( )rkkti

erpkEPA

00020 --

=xw

! = !#$%('()*#+)

xD x

rA

qsin20

rcvqkE!

=^ = -./0.1

L04


r

x-x

xD

( )rkkti

erpkEPA

00020 --

=xw

A

Light propagation in a medium

FZPA EE 121

=

ò=åFZst

AAPA dVnEE1

21

,

xrpr D×= ddV 2

r

222 )( xr --= xr rdrd =rr

( ) dreepknEx

x

rikktiAPA ò

+-

-

--å D=

2/

020,

00

lx

x

xwxp

å+= ,PAP EEE

( ) ( )xktiAP epkniEE 0

000 2 -D-= wxp( )DF--@ xkti

P eEE 00

w

000 /2 EpknA xp DºDF

( ) xD-»DF 01 kn

small angles

( ) 11 <<-n

00 Ep b=

bpe Ann 412 +==

( )xktiAPA e

ikpBknE 0

00

20,

2 -å D= wxp


Maxwell laws in differential form

r=×Ñ D!

Gauss’ law for electricity ED!!

e=

Gauss’ law for magnetism 0=×Ñ B!

Faraday’s law of inductiontBE¶¶

-=´Ñ!

!

Ampere’s lawtDH¶¶

=´Ñ!

!HB!!

µ=

å= k kjkj ED e

HkiH!!!

´-=´Ñ

( )rktieHH!!!!

-= w0

( )rktieEE!!!!

-= w0

( )rktieDD!!!!

-= w0

DitD !!

w=¶¶

EkiE!!!

´-=´Ñ

BitB !!

w=¶¶

HNDv!!

´-= ˆ

( )Nvk ˆ/wº!

ENHv!!

´= ˆµ


Transverse plane wave in a crystal

( )BES!!!

´= -10µHNDv

!!´-= ˆ ENHv

!!´= ˆµ

Direction of light propagation

D!

HD!!

^

NS!H

!

E!

HE!!

^ HS!!

^

( ) ( ) ( ) cbabcacba !!!!!!!!!××-××=´´

( )ENNv

D ˆˆˆ12 ´´-=

µ

!

( )2

2 1DEDv!!

×=µ eµe

eµ

1122

2

==EE

E defines all other quantities, such as D (exactly), H, N, S and phase velocity.


Uniaxial crystals

D!

E!

optical axis

|||||| ED e=

^^^ = ED e||D

^D

Cubic symmetry: ||ee =^

N

Main cross-section M

MD ^!

optical axisD!

N

ED!!

^= e HNDv!!

´-= ˆ

ENHv!!

´= ˆµ

vHE /=^e

µvEH /= ^

=µe1v

Ordinary wave – speed is independent of the propagation direction.


MD ||!

HNDv!!

´= ˆ

ENHv!!

´= ˆµ

vHD /=

µe vDH eff/=

eff

vµe1

=

Extraordinary wave:speed is direction dependent.

optical axis

D!

N

E! DN EEE

!!!+=

DN ENENHv!!!

´+´= ˆˆµ DEN!

´= ˆ

DDEED

!!×

=D

DEDE ^^+= ||||

÷÷ø

öççè

æ+=

^

^

ee

2

||

2||1 DD

D

÷÷ø

öççè

æ+=

^

^

ee

2||

||

2 NNDED Deffe1

º DENHv!!

´-= ˆµ

Uniaxial crystals

0=^N

^

=µe1v

0|| =N

||

1µe

=v

Ordinary and extraordinary wave travel at

the same speed along the optical

axis.


Birefringence

Example

oe

^

=µe1

ov||

1µe

=ev

oo

nvcn ==^

21

2||

||

2

||

-

^

^÷÷ø

öççè

æ+==eeNN

vcne

:1;0|| == ^NN enn == |||| e


Ø When light travels through a medium, all atoms may be considered

as sources of the secondary waves.

Ø Their interference yields a complex wave amplitude at a distant

point, i.e., leads to an additional phase delay.

Ø Dielectric permittivity in crystals is, in general, tensor.

Ø In uniaxial crystals there are two light rays,

ordinary and extraordinary.

Ø They, generally, propagate in different directions

with different speeds and are linearly polarized

perpendicular to one another.

Ø Along the optical axis ordinary and extraordinary

they propagate with equal speeds.

To remember!


Polarisators

a

oe

!64

Nicol prism Glan-Thompson prism

Wollaston prism


Electro-optic Kerr effect

John Kerr 1824 – 1907

+

-oe nn -

0E

20qEnn oe =- ( ) 2

00 22 KlElnne plpj =-=

l/qK º

Kerr constant

Substance T,°C K, m/V2

Nitrobenzene 20 4.4×10-12

Water 20 9.4×10-14

Chloroform 20 -3.5×10-15

pj 3104 -´»nenitrobenze

d = 1 cmV = 1 kV

E = 100 kV/ml = 5 cm ( )( )120 ddnnk e --=Dj

1d 2d


Theory of Kerr effect

x

zy

sEp s ˆb=!

sE!

( ) ssEp ˆˆ ××=!! b

sEsp E ˆb=!

ixEE =! :xE ss =Þ 2

xx Esp b= yxy sEsp b= zxz sEsp b=2xx sEp b= 0=yp 0=zp

2xAx sEnP b= 0=yP 0=zP

22 41 xAex snn bpe +==

0E

221 xAe snn bp+»

a22 cos=xs abp 2cos21 Ae nn +»

0E! s

a

E!



jyEE =!

:yE ss =Þ

yxx sEsp b= 2yy Esp b= zyz sEsp b=

2yy sEp b= 0== zx pp

2yAy sEnP b= 0== zx PP

22 41 yAoy snn bpe +== 221 yAo snn bp+»

abp 2sin1 Ao nn +»

acos=xs ja cossin=ys ja sinsin=xs0E! s

a

E! j ja 222 cossin=ys ja 22 cossin= a22

1 sin=

x

zy


abp 2cos21 Ae nn +» abp 2sin1 Ao nn +»

E0 = 0222zyx sss == 3

1= bpAnn 3

21+=

( )312cos2 -=- abp Ae nnn

( )322sin -=- abp Ao nnn ( )abp 231 cos-= An

2-=--nnnn

o

e

( )aabp 2212 sincos2 -=- Aoe nnn ( )312cos3 -= abp An

( )123 -= nnAbp ( ) ÷

øö

çèæ --=-

31cos1

29 2annn oe



0E! s

a

a2cos

Wþýü

îíì-= dkTUCdw exp1 aadeC kTU sin/

2-=

acos00 pEEpU EFindipole -=×-=!!

b2/2pU creationdipole = ab 2202

1 cosEU -=

U << kT aaabp

dkTEC sincos2

10

220

2 ò ÷÷ø

öççè

æ+ ÷÷

ø

öççè

æ+=

kTEC

622

20

2b 1=

aaa dA

Adw sin1cos1

21

21

261

++

=


÷øö

çèæ +º AC

6112 2

aaa dA sin31cos

211

21 2 ÷

ø

öçè

æ÷øö

çèæ -+=

ò=p

aa0

22 coscos dw452

31 A+=

( )51 Annn oe

-=- ( )

kTnKl

b51-

=


Ø By electro-optic Kerr effect a phenomenon is referred in which

optically isotropic medium becomes optically anisotropic upon

applying an electric field.

Ø Similar to this is optical Kerr effects, when a medium becomes

polarized by the incident light itself.

Ø Thus, an incident light ray splits into two rays,

ordinary and extraordinary.

Ø The difference between the refractive indices

associated with these two rays are proportional

to a medium-dependent Kerr constant and

to square of the applied electric field.

Ø Kerr effects becomes weaker with increasing T.

To remember!


Dispersion

Eedtrdgrk

dtrdm

!!!

!+--=2

2

Emer

dtrd

dtrd !!

!!=++ 2

02

2

2 wg

( ) tierAE w!!!=

Ei

mer!!

wgww 2/22

0 +-= Erep

!!! b==

tierr w0!!

=

PEED!!!!

pe 4+==

EnpnP AA

!!!b==

( )wgww

peimenA2/41 22

0

2

+-+= ce in +=

2202 wwwg -<<

( )22

0

22 /41

wwpe

-+==

menn A


Pockels effect

20Enn oe µ-Kerr effect

Pockels effect0Enn oe µ-

02 20 =++ rrr wg!!!

0202 Errr m

e=++ wg!!!

02 20 =++ qqq wg !!!

20

0

wmeErq -=

022

02 Errrr me=+++ awg!!!

0200

20 Err m

e=+aw

( ) 022 020 =+++ qrqq awg !!!

0rrq -=

020

20

2 Emewaw »D 2

0

0

wwb

w mEenn

¶¶

=D


Zeeman effect

0wW-0w W+0w

25.0 25.0

5.0

S NW+0w

W-0w5.0

5.0

Brrr ´-=+ !!! me2

0w BΩ me2=

ïî

ïí

ì

=+

=+W-

=+W+

0

02

02

20

20

20

zz

yxy

xyx

w

w

w

!!

!!!

!!!

z

( )W-=´ jiΩr ˆˆ xy !!!

tieiyx w=+

220 W+±W= ww

02,1 ww ±W»tieiyx 2,1

2,12,1w=+


Polarization rotation in active medium

circular dichroism

(M)-Hexaheliece


Faraday effect

S NW+0w

W-0w5.0

5.0

RlB=b

Verdet constant

llddn

mceR 22

-=


Ø Dielectric permittivity and its dispersion can be understood within

a simple model of damped harmonic oscillator.

Ø Pockels effect originates from anharmonicity of oscillations.

Ø Zeeman effect can be rationalized by considering orbiting elctron

under action of the Lorentz force.

Ø In active optical media, polarization of a

linearly polarized light rotates with propagation.

Ø Such a media is typically composed of chiral

molecules leading to circular dichroism.

Ø Faraday effect does refer to polarization plane

rotation for light traveling along magnetic field.

To remember!

Experimental Physics EP3 Optics€¦ · Experimental Physics III - Crystal optics 3 Light...

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