Light polarization - Magnetismmagnetism.eu/esm/2018/slides/vavassori-practical.pdf · polarization state, we use Jones matrices. Since we can write a polarization state as a (Jones)

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Light polarization


Since light is composed of oscillating electric and magnetic fields,Jones reasoned that the most natural way to represent light is interms of the electric field vector. When written as a columnvector, this vector is known as a Jones vector and has the form:

Jones vectors and matrices

These values can be complex numbers, so both amplitude and phase information ispresent. Oftentimes, however, it is not necessary to know the exact amplitudes and phases of the vector components. Therefore Jones vectors can be normalized and common phase factors can be neglected.

Horizontal andvertical linearpolarization states(reflection plane xz).

TM or p TE or s


Linearly polarized light at 45°

Right-circular polarized light

Left-circular polarized light

Normalized representation

-

t

Ey

x

yEx

x

y

E

E


Elliptically polarized light


=

=

=

ii

ox

oyi

oy

ox

obee

E

EeE

EE

11

~

Eox

Eoy

x

y

qK

222 1

cos2cos22tan

b

b

EE

EE

oyox

oyox

K−

=−

=

q

222 1

sin2sin22sin

b

b

EE

EE

oyox

oyox

K+

=+

=

b = e2/e1

e2

e1


To model the effect of a medium on light'spolarization state, we use Jones matrices.

Since we can write a polarization state as a (Jones) vector, we use

matrices, A, to transform them from the input polarization, E0, to the

output polarization, E1.

This yields:

For example, an x-polarizer can be written:

So:

1 0E E= A

1 11 0 12 0

1 21 0 22 0

x x y

y x y

E a E a E

E a E a E

= +

= +

1 0

0 0x

=

A

0 0

1 0

0

1 0

0 0 0

x x

x

y

E EE E

E

= = =

A

~

~

~

=

2221

1211A

aa

aa


Other Jones matrices

A y-polarizer:0 0

0 1y

=

A

1 0

0 1HWP

=

− A

A half-wave plate: 1 0 1 1

0 1 1 1

=

− −

1 0 1 1

0 1 1 1

=

− −

A half-wave plate rotates 45-degree-polarization to -45-degree, and vice versa.

A quarter-wave plate:1 0

0QWP

i

=

A

1 0 1 1

0 1i i

=


A wave plate is not a wave plate if it’s oriented wrong.

Remember that a wave plate wants ±45° (or circular) polarization.

If it sees, say, x polarization, nothing happens.

1 0 1 1

0 1 0 0

=

−

So use Jones matrices until you’re really on top of this!!!

AHWP

Wave plate w/ axes at 0° or 90°

0° or 90° Polarizer


Summary

1 00 −1

−1 00 1

1 00 𝑒𝑖𝜑

Half-wave plate,fast axis horizontal

Half-wave plate,fast axis vertical

General retarder,fast axis horizontal

In terms of waves (wavelength l),

this is a retarder l*j/2p

Retardation l/2 Retardation l/2

Retardation l/4

Retardation l/4


Rotated Jones matrices

Okay, so E1 = A E0. What about when the polarizer or wave plate responsible for A is rotated by some angle, q ?

Rotation of a vector by an angle q means multiplication by a rotation matrix:

( ) ( )0 0 1 1' and 'E R E E R Eq q= =

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1

1 1 0 0

1 1

0 0 0

'

' ' '

E R E R E R R R E

R R R E R R E E

q q q q q

q q q q q

−

− −

= = =

= = =

A A

A A A

( ) ( )1

' R Rq q−

=A A

( )cos( ) sin( )

sin( ) cos( )R

q qq

q q

− =

Thus:

Rotating E1 by q and inserting the identity matrix R(q)-1 R(q), we have:

where:


Rotated Jones matrix for a polarizer

Applying this result to an x-polarizer:

( )cos( ) sin( ) 1 0 cos( ) sin( )

sin( ) cos( ) 0 0 sin( ) cos( )xA

q q q qq

q q q q

− =

−

( ) ( )1

' R Rq q−

=A A

( )cos( ) sin( ) cos( ) sin( )

sin( ) cos( ) 0 0xA

q q q qq

q q

− =

( )2

2

cos ( ) cos( )sin( )

cos( )sin( ) sin ( )xA

q q qq

q q q

=

( )1/ 2 1/ 2

451/ 2 1/ 2

xA

=

( )1

0xA

for small angles,


Jones Matrices for standard components


To model the effect of many media on light's polarization state, we use many Jones matrices.

To model the effects of more than one component on the polarization state, just multiply the input polarization Jones vector by all of the Jones matrices:

1 3 2 1 0E E= A A A

Remember to use the correct order!

A single Jones matrix (the product of the individual Jones matrices) can describe the combination of several components.


Multiplying Jones Matrices

Crossed polarizers:

0 0 1 0 0 0

0 1 0 0 0 0

= =

y xA A

x

y z

1 0y xE E= A A0E

1E

x-pol

y-pol

so no light leaks through.

Uncrossed polarizers

(slightly):

( )0 0 1 0 0

0 1 0 0

= =

y xA A

0E1E

rotatedx-pol

y-pol

( )00 0

0

x x

y y x

E E

E E E

= =

y xA A So Iout ≈ 2 Iin,x


The MagnetoOptical Effect

sssp

pspp

rr

rr

ss

pp

r

r

0

0

iTM

rTMpp

E

Er =

iTE

rTEss

E

Er =

iTE

rTMps

E

Er =

iTM

rTEsp

E

Er =

M

Fresnell reflection coefficients

−

−

−

=

0

0

0

ˆ

xy

xz

yz

ii

ii

ii

=

0

0

0

00

00

00

ˆ

x = 0 Q mx

y = 0 Q my

z = 0 Q mz;

M

Dielectric tensor

is

ip

E

E

rs

rp

E

E=

x

z

y

ps

psθ θ

s

p

s

p

s

p

Reflected Light

z

θ

x

Transmitted Light

Polarization Plane

Sample

rs

rp

E

E

is

ip

E

E=


1

220

212

21

20

12

1

sinsin1sin1cos

n

nn

n

n qqqq

−=−=−=

rpp = r0pp+ rpp

M my

rps - mx - mz

rsp mx-mz

xy = i1Q mz; xz = -i1Q my; yz = i1 Q mx; xy = -yx; zx = -xz; zy = -yz;

sssp

pspp

rr

rr

The MagnetoOptical Effect general case:

Oblique incidence and arbitrary direction of M


MOKE configurations


( )

+=

=

= sincos

11~

ir

re

r

rr

rE

pp

spi

pp

sp

sp

pp

r

qK K

pp

sp

r

rRe

ss

ps

r

rRe

pp

sp

r

rIm

ss

ps

r

rIm

rsp << rpp pp

sp

KKr

r qq

cos222tan

pp

sp

KKr

r

sin222sin

cosRepp

sp

pp

sp

r

r

r

r=

sinIm

pp

sp

pp

sp

r

r

r

r=

222 1

cos2cos22tan

b

b

rr

rr

sppp

sppp

K−

=−

=

q

222 1

sin2sin22sin

b

b

rr

rr

sppp

sppp

K+

=+

=

Elliptically polarized light


a

b

Eox

Eoy

x

y

K

qK

E(z,t)

rpp

rsp


Polar & Longitudinal MOKE

qK

K

qK

K

pp

sp

r

rRe

ss

ps

r

rRe

pp

sp

r

rIm

ss

ps

r

rIm

18

Birifringence

Dichroism

P-MOKE: eigenmodes are LCP and RCP polarized EMs

qK, K

HH

M

qK, K (H) M(H)

x

y

E0

E0

−=

xx

xxxy

xyxx

00

0

0~

−

=

xxyz

yzxx

xx

0

0

00~

xy = i1Q mz; yz = i1 Q mx;



Io

Transverse Kerr effect

Laser

Polarizer

Detector

p-polarized light (TM)M

Er = rpp Eo

rpp = ropp + rm

ppmy

Ir = Er (Er)*

Ir = Io+ DIm DIm/Io a my

The reflected beam is p-polarized.Variation of intensity and phase.

y

Y. Souche et al.JMMM 226-230, 1686 (2001);JMMM 242-245, 964 (2002).

M fm

Dfm a my

E

E

Polarizer

l/4

xz = -i1Q my

ropp

rmppmy


x

y

x’

y’

l/4plate

h

E0

E0sinh

E0cosh

Opticaxis

x

y

h

E’

E

Measurement of ellipticity

−=

biEo

1~

=

iAQWP

0

0

1~

=

=

=

−

=

h

h

h

hsin

cos

cos

sin111

0

01~'0

bbiiE

E E i i j= +0(cos sin )h h

=

=

0

cos

sin

cos

00

01~''

0

h

h

hE h22''

0 cos)~

( == EIAdding an analyzed (i.e. A polarazier in front of the detector I get h (Malus law)


Measurement of ellipticity & rotation: high sensitivity

l/n (n = 2 or 4)


l/4

l/2

45°

22.5°

p

sp

s p

s

p

s

wollastone

45°p

s

wollastone

Measurement of ellipticity & rotation: high sensitivity

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Measuring qK and K

Modulation polarization technique for recording the longitudinal and polar Kerr effects, which are proportional to the magnetization components mx mz..

POLARIZER Glan-Thompson

PHOTOELASTIC MODULATOR (50kHz)

ORIGINAL ELLIPTICTY

MODULATED ELLIPTICTYPREAMPLIFIED PHOTODIODE

POLARIZER

HeNe LASER

p-polarized beams-p polarized reflected

beam (elliptical polarization)

y

x

Lock-in

Electromagnet

E

More details in: P. Vavassori, Appl. Phys. Lett. 77, 1605 (2000)

qK K

spol - mx – mz

ppol mx- mz


PEM

*2ssssssDC rrrI ==

Now:

*** Re2 sspspsssssps rrrrrr =+ *** Im2 sspspsssssps rrirrrr =−

If I can measure the normalized photodiode intensity at w and 2w

−=

)(

)(1)(Imtan

2

4

sspp

psspsp

r

rJi w

=

)(

)(2)(2 Retan

2

4

sspp

psspsp

r

rJi w

m

K

qK

Measuring qK and K polar and longitudinal

S-pol

(I considered here the general case of an analyzer at b respect to extinction with the initial polarizer)


PEM rotated 45o

45o

45o

45o

Measuring qK and K transverse (sensitive to my along y-direction)

Light polarization - Magnetismmagnetism.eu/esm/2018/slides/vavassori-practical.pdf · polarization state, we use Jones matrices. Since we can write a polarization state as a (Jones)

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