jones matrix lecture

7/17/2019 Jones Matrix Lecture

http://slidepdf.com/reader/full/jones-matrix-lecture 1/21

1

PolarizationJones vector & matrices

Phys 375



2

Matrix treatment of polarization

Consider a light ray with an instantaneous !

vector as shown

( ) ( ) ( )t z E jt z E it z E y x ,ˆ,ˆ, +=

x

y

Ex

Ey

( )

( ) y

x

t kz i

oy y

t kz i

ox x

e E E

e E E

ϕ ω

ϕ ω

+−

+−

=

=



3

Matrix treatment of polarization Com"ining the components

#he terms in "rac$ets represents the complexamplitude of the plane wave

( ) ( )

[ ] ( )

( )t kz i

o

t kz ii

oy

i

ox

t kz i

oy

t kz i

ox

e E E

ee E je E i E

e E je E i E

y x

y x

ω

ω ϕ ϕ

ϕ ω ϕ ω

−

−

+−+−

=

+=

+=

~

ˆˆ

ˆˆ



4

Jones %ectors #he state of polarization of light is determined "y

the relative amplitudes ox' oy( and'

the relative phases δ ) ϕy ! ϕx (

of these components

#he complex amplitude is written as a two!

element matrix' the Jones vector

=

=

= δ

ϕ

ϕ

ϕ

i

oy

oxi

yi

oy

i

ox

oy

ox

o e E E e

e E e E

E E E x

x

~

~

~



5

Jones vector* +orizontally polarized light

#he electric field oscillations areonly along the x!axis

#he Jones vector is then written'

where we have set the phase ϕx ),' for convenience

=

=

=

=

0

1

00~

~~

A Ae E

E

E E

xi

ox

oy

ox

o

ϕ x

y

#he arrows indicate

the sense ofmovement as the

"eam approaches you

The normalized form

is

0

1

E



6

x

y

Jones vector* %ertically polarized light

#he electric field oscillations

are only along the y!axis

#he Jones vector is then

written'

-here we have set thephase ϕy ) ,' for

convenience

=

=

=

=

1

000

~

~~

A

Ae E

E

E E

yi

oyoy

ox

o ϕ

The normalized form

is

1

0

E



7

Jones vector* .inearly polarized light at

an ar"itrary angle /f the phases are such that δ ) mπ for

m ) ,' ±0' ±1' ±3' 2

#hen we must have'

and the Jones vector is simply a line

inclined at an angle α ) tan!0oyox(

since we can write

( )oy

oxm

y

x

E

E

E

E 1−=

( )

−=

=

α

α

sin

cos1~

~~ m

oy

ox

o A

E

E E

x

y



8

Circular polarization

4uppose ox ) oy ) and x leads y "y6,o)

π1

t the instant x reachesits maximumdisplacement (' y iszero

fourth of a period later'x is zero and y)

x

y

t=0, Ey = 0, Ex = +A

t=T/4, Ey = +A, Ex = 0

t=T/8, Ey = +Asin 45o, Ex = Acos45o



9

Circular polarization

#he Jones vector for this case 8 where x leads y is

#he normalized form is'

#his vector represents circularly polarized light' where

rotates countercloc$wise' viewed head!on

#his mode is called left!circularly polarized light -hat is the corresponding vector for right!circularly

polarized light9

=

=

=

i A

Ae

A

e E

e E E ii

oy

i

ox

o y

x 1~

2π ϕ

ϕ

i

1

21

− i

1

2

1Replace / !ith " / to #et



10

lliptically polarized light

/f ox ≠ oy ' e:g: if ox) and oy

) ;

#he Jones vector can "e written

−

iB

A

iB A co$ntercloc%!ise

cloc%!ise

&ere A'(



11

Jones vector and polarization

/n general' the Jones vector for the ar"itrarycase is an ellipse δ≠ mπ< δ≠m01(π(

( )

+=

=

δ δ

δ sincos

~

i B

A

e E

E E i

oy

ox

o

a

)

Eox

Eoy

x

y

22

cos22tan

oyox E E

E E oyox

−

=

δ α



12

=ptical elements* .inear polarizer

4electively removes all or most of the !

vi"rations except in a given direction

TA

x

y

*inear polarizer



13

Jones matrix for a linear polarizer

=

1

0

1

0

d c

ba

=

10

00 M

onsider a linear polarizer !ith transmission axis alon# theertical -y. *et a matrix represent the polarizer

operatin# on ertically polarized li#ht

The transmitted li#ht m$st also )e ertically polarized Th$s,

Th$s,*inear polarizer !ith TA

ertical

=

0

0

0

1

d c

ba

1peratin# on horizontally polarized li#ht,



14

Jones matrix for a linear polarizer

>or a linear polarizer with a transmission

axis at θ

=

θ θ θ

θ θ θ 2

2

sincossin

cossincos M



15

=ptical elements* Phase retarder

/ntroduces a phase difference ?ϕ( "etweenorthogonal components

#he fast axis>( and slow axis 4( are shown

2A

x

y

Retardation plate

3A



16

Jones matrix of a phase retarder

-e wish to find a matrix which will transform the elements

as follows*

/t is easy to show "y inspection that'

+ere εx and εy represent the advance in phase of the

components

( )

( ) y y y

x x x

i

oy

i

oy

i

ox

i

ox

e E oe E

e E oe E

ε ϕ ϕ

ε ϕ ϕ

+

+

int

int

= y

x

i

i

e

e

M ε

ε

0

0



17

Jones matrix of a @uarter -ave Plate

Consider a Auarter wave plate for which B?εB )π1

>or εy ! εx ) π1 4low axis vertical( .et εx ) !π and εy ) π

#he matrix representing a @uarter wave plate'

with its slow axis vertical is'

=

=

−−

ie

e

e M

i

i

i

0

01

0

0 4

4

4 π

π

π



18

Jones matrices* +alf!wave Plate

>or B?εB ) π

−

=

=

−=

=

−

−−

10

01

0

0

10

01

0

0

2

2

2

2

2

2

π

π

π

π

π

π

i

i

i

i

i

i

ee

e M

ee

e M &, 3A ertical

&, 3A horizontal



19

=ptical elements*

@uarter+alf wave plate -hen the net phase difference

?ϕ ) π1 * @uarter!wave plate

?ϕ ) π * +alf!wave plate /



20

=ptical elements* Dotator Dotates the direction of linearly polarized

light "y a particular angle θ

x

y

Rotator

3A



21

Jones matrix for a rotator

n !vector oscillating linearly at θ is rotated "y

an angle β

#hus' the light must "e converted to one thatoscillates linearly at β θ (

=ne then finds

( )

( )

+

+=

θ β

θ β

θ

θ

sin

cos

sin

cos

d c

ba

−=β β

β β

cossin

sincos M

jones matrix lecture

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