jones matrix lecture
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Very good lecture about Jones matrix.TRANSCRIPT
7/17/2019 Jones Matrix Lecture
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PolarizationJones vector & matrices
Phys 375
7/17/2019 Jones Matrix Lecture
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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
+−
+−
=
=
7/17/2019 Jones Matrix Lecture
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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
7/17/2019 Jones Matrix Lecture
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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
~
~
~
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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
7/17/2019 Jones Matrix Lecture
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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
7/17/2019 Jones Matrix Lecture
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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
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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
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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
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lliptically polarized light
/f ox ≠ oy ' e:g: if ox) and oy
) ;
#he Jones vector can "e written
−
iB
A
iB A co$ntercloc%!ise
cloc%!ise
&ere A'(
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Jones vector and polarization
/n general' the Jones vector for the ar"itrarycase is an ellipse δ≠ mπ< δ≠m01(π(
( )
+=
=
δ δ
δ sincos
~
i B
A
e E
E E i
oy
ox
o
a
)
Eox
Eoy
x
y
22
cos22tan
oyox E E
E E oyox
−
=
δ α
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=ptical elements* .inear polarizer
4electively removes all or most of the !
vi"rations except in a given direction
TA
x
y
*inear polarizer
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Jones matrix for a linear polarizer
=
1
0
1
0
d c
ba
=
10
00 M
onsider a linear polarizer !ith transmission axis alon# theertical -y. *et a matrix represent the polarizer
operatin# on ertically polarized li#ht
The transmitted li#ht m$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,
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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
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=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
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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
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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
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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
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=ptical elements*
@uarter+alf wave plate -hen the net phase difference
?ϕ ) π1 * @uarter!wave plate
?ϕ ) π * +alf!wave plate /
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=ptical elements* Dotator Dotates the direction of linearly polarized
light "y a particular angle θ
x
y
Rotator
3A
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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