Light polarization nano@nanogune.eu I www.nanogune.eu - - PowerPoint PPT Presentation

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Light polarization

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Since light is composed of oscillating electric and magnetic fields, Jones reasoned that the most natural way to represent light is in terms of the electric field vector. When written as a column vector, 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 is

• present. 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 and vertical linear polarization states (reflection plane xz).

TM or p TE or s

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Linearly polarized light at 45° Right-circular polarized light Left-circular polarized light Normalized representation

• t

Ey x y Ex x y E E

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Elliptically polarized light Normalized representation

      =         =       =

   i i

• x
• y

i

• y
• x
• be

e E E e E E E 1 1 ~

Eox Eoy x y

qK

2 2 2

1 cos 2 cos 2 2 tan b b E E E E

• y
• x
• y
• x

K

− = − =   q

2 2 2

1 sin 2 sin 2 2 sin b b E E E E

• y
• x
• y
• x

K

+ = + =   

b = e2/e1 e2 e1

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To model the effect of a medium on light's polarization 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

• utput polarization, E1.

This yields: For example, an x-polarizer can be written: So:

1

E E = A

1 11 12 1 21 22 x x y y x y

E a E a E E a E a E = + = +

1

x

  =     A

1

1

x x x y

E E E E E       = = =             A

~ ~ ~

      =

22 21 12 11

A a a a a

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Other Jones matrices

A y-polarizer:

1

y

  =     A 1 1

HWP

  =   −   A

A half-wave plate:

1 1 1 1 1 1       =       − −       1 1 1 1 1 1       =       − −      

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

1

QWP

i   =      A 1 1 1 1 i i       =              

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A wave plate is not a wave plate if it’s

• riented wrong.

Remember that a wave plate wants ±45° (or circular) polarization. If it sees, say, x polarization, nothing happens.

1 1 1 1       =       −      

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

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Summary 1 −1 −1 1 1 𝑓𝑗𝜒

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

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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:

( ) ( )

1 1

' and ' E R E E R E q q = =

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

' ' ' ' 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 R q q

−

= A A

( )

cos( ) sin( ) sin( ) cos( ) R q q q q q −   =    

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

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Rotated Jones matrix for a polarizer

Applying this result to an x-polarizer:

( )

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

x

A q q q q q q q q q −       =       −      

( ) ( )

1

' R R q q

−

= A A

( )

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

x

A q q q q q q q −     =        

( )

2 2

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

x

A q q q q q q q   =    

( )

1/ 2 1/ 2 45 1/ 2 1/ 2

x

A   =    

( )

1

x

A          

for small angles, 

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Jones Matrices for standard components

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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

E 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.

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Multiplying Jones Matrices

Crossed polarizers:

1 1       = =            

y x

A A

x y z

1 y x

E E = A A

E

1

E

x-pol y-pol

so no light leaks through. Uncrossed polarizers (slightly):

( )

1 1           = =            

y x

A A

E

1

E

rotated x-pol y-pol

( )

x x y y x

E E E E E            = =                

y x

A A So Iout ≈ 2 Iin,x

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The MagnetoOptical Effect

       

ss sp ps pp

r r r r

     

ss pp

r r

iTM rTM pp

E E r =

iTE rTE ss

E E r =

iTE rTM ps

E E r =

iTM rTE sp

E E r =

M

Fresnell reflection coefficients

          − − − = ˆ          

x y x z y z

i i i i i i

             =  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 p s p s θ 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

=

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sin sin 1 sin 1 cos n n n n n q q q q − = − = − =

rpp = r0

pp+ rpp M  my

rps  - mx - mz rsp  mx-mz

xy = i1Q mz; xz = -i1Q my; yz = i1 Q mx; xy = -yx; zx = -xz; zy = -yz;

       

ss sp ps pp

r r r r

The MagnetoOptical Effect general case: Oblique incidence and arbitrary direction of M

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MOKE configurations

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( )

          + =           =       =  



sin cos 1 1 ~ i r r e r r r r E

pp sp i pp sp sp pp r

qK K

       

pp sp

r r Re

     

ss ps

r r Re

       

pp sp

r r Im      

ss ps

r r Im

rsp << rpp

pp sp K K

r r  q q cos 2 2 2 tan  

pp sp K K

r r    sin 2 2 2 sin  

 cos Re

pp sp pp sp

r r r r =          sin Im

pp sp pp sp

r r r r =        

2 2 2

1 cos 2 cos 2 2 tan b b r r r r

sp pp sp pp K

− = − =   q

2 2 2

1 sin 2 sin 2 2 sin b b r r r r

sp pp sp pp K

+ = + =   

Elliptically polarized light Normalized representation

a b Eox Eoy x y

K qK

E(z,t)

rpp rsp

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Polar & Longitudinal MOKE qK K

qK K

       

pp sp

r r Re

     

ss ps

r r Re

       

pp sp

r r Im

     

ss ps

r r Im 18

Birifringence Dichroism

P-MOKE: eigenmodes are LCP and RCP polarized EMs

qK, K H H M

qK, K (H)  M(H)

x y E0 E0

          − =

xx xx xy xy xx

      ~           − =

xx yz yz xx xx

      ~

xy = i1Q mz; yz = i1 Q mx;

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Io

Transverse Kerr effect

Laser Polarizer Detector p-polarized light (TM)

M

Er = rpp Eo rpp = ro

pp + 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 = -i1Q my

ro

pp

rm

ppmy

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x y x’ y’ l/4plate h E0 E0sinh E0cosh Optic axis x y h E’ E

Measurement of ellipticity

      − = bi Eo 1 ~       = i AQWP 1 ~       =         =       =       −       = h h h h sin cos cos sin 1 1 1 1 ~' b bi i E

   E E i i j = +

0(cos

sin ) h h

      =             = cos sin cos 1 ~ '

'

h h h E

h

2 2 ' '

cos ) ~ ( = = E I

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

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Measurement of ellipticity & rotation: high sensitivity

l/n (n = 2 or 4)

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l/4 l/2 45° 22.5° p s p s p s p s

wollastone

45° p s

wollastone

Measurement of ellipticity & rotation: high sensitivity

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24

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 ELLIPTICTY PREAMPLIFIED PHOTODIODE POLARIZER HeNe LASER p-polarized beam s-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

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PEM

* 2 ss ss ss DC

r r r I = =

Now:

 

* * *

Re 2

ss ps ps ss ss ps

r r r r r r = +

 

* * *

Im 2

ss ps ps ss ss ps

r r i r r r r = −

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

        − =

) ( ) ( 1 ) (

Im tan 2 4

ss pp ps sp s p

r r J i 

w

        =

) ( ) ( 2 ) ( 2

Re tan 2 4

ss pp ps sp s p

r r J i 

w

m

K qK

Measuring qK and K polar and longitudinal

S-pol (I considered here the general case

• f an analyzer at b respect to

extinction with the initial polarizer)

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PEM rotated 45o

45o 45o 45o

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