1 Other polarimeters
• Wave plates can be used to build
polarimeters sensitive to linear or to circular polarization.
• Half-wave plate (φ=180
o)
• Quarter-wave plate (φ=90
o)
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
= −
0 1 0 0
1 0 0 0
0 0 1 0
0 0 0 1 QWP
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
= −
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1 HWP
Polarimeter with HWP
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
=
0 0 0 0
0 0 0 0
0 0 1 1
0 0 1 1
PH
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
= −
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1 HWP
θ
source Polarizer
Intensity detector HWP
( ) ( )
{ P R HWP R } S D
W = ×
H× − θ × × θ ×
( )
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
= −
1 0 0 0
0 0
0 0
0 0 0 1
2 2
2 2
c s
s Rθ c
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
= V U Q I
S D=
(
1 0 0 0) ( )
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
= −
−
1 0 0 0
0 0
0 0
0 0 0 1
2 2
2 2
c s
s R θ c
Polarimeter with HWP
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
= −
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
−
1 0 0 0
0 0 0
0 0 0
0 0 0 1
1 0 0 0
0 0
0 0
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
2 2 2 2
2 2
c c c s
s c θ
source Polarizer
Intensity detector HWP
( ) ( )
{ P R HWP R } S
D
W = ×
H× − θ × × θ ×
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
= −
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
⎟⎟ −
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
1 0 0 0
0 0 0
0 0 0
0 0 0 1
1 0 0 0
0 0 0
0 0 0
0 0 0 1
1 0 0 0
0 0
0 0
0 0 0 1
2 2 2 2 2 2 2 2
2 2
c c c c c s
s c
Polarimeter with HWP
θ
source Polarizer
Intensity detector HWP
( ) ( )
{ P R HWP R } S
D
W = ×
H× − θ × × θ ×
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
=
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
⎟⎟ −
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
0 0 0 0
0 0 0 0
0 0 1
0 0 1
1 0 0 0
0 0 0
0 0 0
0 0 0 1
0 0 0 0
0 0 0 0
0 0 1 1
0 0 1 1
2 2 2 2
2 2 2
2 c
c
c c
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛ + +
=
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
0 0 0 0 0 0
0 0 0 0
0 0 1
0 0 1
2 2 2 2 2
2 2 2
Q c I
Q c I
V U Q I c c
( )
2(
22)
1 2 2 2 2
2 1
0 0 0 0 0
1 I cQ I Qc
Q c I
W = +
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛ + +
=
( )
f
of
Q I W
4
2 cos
22 1
= +
= θ
Polarimeter with QWP
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
=
0 0 0 0
0 0 0 0
0 0 1 1
0 0 1 1
PH
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
= − 0 1 0 0
1 0 0 0
0 0 1 0
0 0 0 1 QWP
θ
source Polarizer
Intensity detector QWP
( ) ( )
{ P R QWP R } S D
W = ×
H× − θ × × θ ×
( )
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
= −
1 0 0 0
0 0
0 0
0 0 0 1
2 2
2 2
c s
s Rθ c
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
= V U Q I
S D=
(
1 0 0 0) ( )
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
= −
−
1 0 0 0
0 0
0 0
0 0 0 1
2 2
2 2
c s
s R θ c
Polarimeter with QWP
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
= −
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
0 0
1 0 0 0
0 0
0 0 0 1
1 0 0 0
0 0
0 0
0 0 0 1
0 1 0 0
1 0 0 0
0 0 1 0
0 0 0 1
2 2
2 2 2
2 2 2
c s
s c c s
s c θ
source Polarizer
Intensity detector QWP
( ) ( )
{ P R HWP R } S D
W = ×
H× − θ × × θ ×
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
−
−
= −
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
⎟⎟ −
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
0 0 0 0 0 0
0 0 0 1
0 0
1 0 0 0
0 0
0 0 0 1
1 0 0 0
0 0
0 0
0 0 0 1
2 2 2 2 2
2 2 2 2 2
2 2
2 2 2 2
2 2
c s s c
s s c c
c s
s c c s
s c
2 Polarimeter with QWP
θ
source Polarizer
Intensity detector QWP
( ) ( )
{ P R HWP R } S
D
W = ×
H× − θ × × θ ×
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
−
=
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
−
−
−
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
0 0 0 0
0 0 0 0 1 1
0 0 0 0 0 0
0 0 0 1
0 0 0 0
0 0 0 0
0 0 1 1
0 0 1 1
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
2 2 2 2
2 c cs s
s s c c
c s s c
s s c c
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
− + +
− + +
=
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
−
0 0 0
0 0 0
0 0 0 0 1 1
2 2 2 2 2
2 2 2 2 2 2
2 2 2 2
2 2 2 2 2
V s U s c Q c I
V s U s c Q c I
V U Q I s s c c
s s c c
( cos
22 θ cos 2 θ sin 2 θ sin 2 θ )
2
1 I Q U V
W
= + + +
fo
4 4
fo2
foModulation techniques for polarization experiments
All these techniques suffer for the need of long integration time:
One needs to point to the same sky pixel during many cycles of the analyzer. It is thus difficult to produce extended maps of the CMB polarization.
