可視赤外・サブミリ偏光観測と較正法に関する考察

A Discussion on the Calibration of PolarimetricObservations in the regime of Optical/IR andSubmillimeter wavelengths～可視近赤外偏光観測チュートリアル～

Koji S. Kawabata
Hiroshima Astrophysical Science Center, Hiroshima Univ.
広島大学 宇宙科学センター 川端 弘治

Polarization of light in mathematics•Amplitude of electrical field in x- and y-coordinate： Ex, Ey– Ex = Ax exp { i ( εx + ωt – 2πz/λ ) } = AxeiΦx– Ey = Ay exp { i ( εy + ωt – 2πz/λ ) } = AyeiΦy– Φx, Φy are the phase at time t and position z– Light amplitude is𝐴𝐴𝑥𝑥 2 + 𝐴𝐴𝑦𝑦 2– Phase difference γ≡ Φy- Φx• Ax = Ay and γ = +90°: right-handed circ..
• Ax = Ay で γ = -90°: left-handed circ.
••••γ = 0°: lineary pol.
Generally, elliptical pol. （180°>γ>0°: right-haned, -180°<γ<0°left-handed）

Combination of two monochromatic polarization is calculated by simple sum of
two monochromatic polarization vectors.
However, the combination of actual lights (rays) is too complicated to produce
meaningful result.
2Coherence of light （可干渉性）Coherent light: A group of rays of which the phases are (nearly) identical•

They can be interfered when they join with a small angle•

When they are not coherent totally, the combination is simple: The sum ofintensity.

Combination of rays: More actual ways••

For coherent rays: Stokes vector; Mueller matrix
For incoherent rays: Jones vector; Jones matrix

3Jones vector （1941-）••

Applicable for fully polarized light
Applicable for coherent lights of which the phases work predominantly•

Two parameters： Orthogonal components of electric field–••

𝐸𝐸𝑥𝑥
𝐸𝐸𝑦𝑦or𝐴𝐴𝑥𝑥 𝑒𝑒 𝑖𝑖(𝜀𝜀𝑥𝑥 +𝜔𝜔𝜔𝜔)𝐴𝐴𝑦𝑦 𝑒𝑒𝑖𝑖(𝜀𝜀𝑦𝑦 +𝜔𝜔𝜔𝜔)or 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖

𝐴𝐴𝑥𝑥 𝑒𝑒 𝑖𝑖𝜀𝜀𝑥𝑥

𝐴𝐴𝑦𝑦 𝑒𝑒𝑖𝑖𝜀𝜀𝑦𝑦orIntensity is root of sum of squares of 𝐸𝐸𝑥𝑥 and 𝐸𝐸𝑦𝑦

𝐴𝐴𝑥𝑥 𝑒𝑒 𝑖𝑖𝜀𝜀𝑥𝑥

𝐴𝐴𝑦𝑦 𝑒𝑒 𝑖𝑖𝜀𝜀𝑦𝑦

Combination of rays is calculated by sum of each parameters as
𝐸𝐸3𝑥𝑥
𝐸𝐸1𝑥𝑥 + 𝐸𝐸2𝑥𝑥=
𝐸𝐸3𝑦𝑦
𝐸𝐸1𝑦𝑦 + 𝐸𝐸2𝑦𝑦4

Stokes vector (1852-)••

Applicable for any polarized light; fully/partly polarized, unpolarized
Applicable for both monochromatic and colored light ⇒ widely usable•

Four Stokes parameters: Each parameter has dimension of intensity (or flux)– I ： 強度
– Q ： 水平直線優越分（0°-90°成分）– U ： 45°直線優越分（直線45°-135°成分）– V ： 右向き円優越分（右円－左円成分）•Stokes parameters are additive= <𝐴𝐴𝑥𝑥 2+𝐴𝐴𝑦𝑦 2 >= <𝐴𝐴𝑥𝑥 2-𝐴𝐴𝑦𝑦 2 >= < 2𝐴𝐴𝑥𝑥 𝐴𝐴𝑦𝑦 cosγ >= < 2𝐴𝐴𝑥𝑥 𝐴𝐴𝑦𝑦 sinγ ><> denotes time average– Sum of incoherent rays 1 and 2 follows as
I3 = I1 + I2, Q3 = Q1 + Q2, U3 = U1 + U2, V3 = V1 + V2,•

Usable with Mueller matrices
5Mueller matrix (1940’s-)• 4×4 square matrix： M• Usable with Stokes vector• Each optical component can be represented by a Muellermatrix
M1

M2

M3
(I1, Q1, U1, V1)(I0, Q0, U0, V0)

I1
Q1
U1
V1
Outgoing ray=

M3 M2 M1

I0
Q0
U0
V0
Incident ray

6

Description of astronomical polarization••

Generally, incoherent, and partly-polarized
Stokes vectors are useful in many cases（Especially, observational astronomy, because it is represented only by intensity/fluxdimensions）•

Intensity of unpolarized component I0、 intensity of linearly polarized
one Ilin、 intensity of circularly polarized one Icirc– I = I0 + Ilin2 + Icirc2– Ilin =•Degree of linear polarization plin、 circular pcirc– plin =••𝑄𝑄2 + 𝑈𝑈 2 、 Icirc = V𝐼𝐼𝑙𝑙𝑙𝑙𝑙𝑙
𝐼𝐼、 pcirc =𝐼𝐼𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝐼𝐼Position angle of linear polarization (polarization angle) θlin
12– θlin = arctan

𝑈𝑈
𝑄𝑄

Relational expression– Q = Ilin cos 2θ、 U = Ilin cos 2θ、7𝑄𝑄𝑈𝑈= plin cos 2θ、、 = plin sin 2θ𝐼𝐼𝐼𝐼

QU plane (QU diagram）• Distance from the origin
denotes (linear) polarization
degree plinU/I

𝑄𝑄 𝑈𝑈,𝐼𝐼 𝐼𝐼Q/IO• Half position angle denotes
polarization angle θlin
Q/I = plin cos( 2θlin )U/I = plin sin( 2θlin )• Vectorial summation isvisually (easily) recognizable(Q/I, U/I) plane is also similar to (Q,U) plane for small Q/I, U/I （<0.1）.
88Principle polarimetry in Opt/NIR regime

9Principal mechanism of polarization 1
双極子散乱
入射光により電気振動子が励起されて放射– 電子散乱– レイリー散乱– ミー散乱– 反射・屈折（入射光との合成・干渉； マ
クスウェル方程式と界面での接続条件でも解ける）

電気双極子

10

散乱面に垂直な方向に偏光

Principal mechanism of polarization ２
シンクロトロン放射

星間偏光（ミー散乱）

磁場中をらせん状に動く電子により放射（サイクロトロン放射の相対論的電子版; 高周波による連続光）（銀河磁場等により）整列した非球状粒子による選択吸収（dichroism）

B

磁場に垂直な方向に偏光© The Open University: Learning Space11透過光は磁場に平行な方向に偏光Optical component for polarimetry• Polarizer (polarizing filter)– Only a linearly polarization component transmits (polyvinyl, wiregrid,liquid crystal, etc）
– Another (orthogonally polarized) component is absorbed orreflected.

• Polarizing beam splitter– Two orthogonally polarized lights are transmitted (or reflected) todifferent direction.

• Wave plate （retarder, modulator）– Producing a difference of the phases (= retardance) for two
orthogonally polarized lights after transmission of birefringent
material（複屈折製結晶）1212

Example of beam splitter••

Wollaston/Rochon prism: Separation angle of o- and e-rays are small
Glan prism: Separation angle is nearly ~90 degs (completely separated)

13

Half-wave plate (λ/2 波長板)
A half-wave plate at position angle θ makes rotation of position angle by 2θ
without any change of degree of polarization.
Rotation of a half-wave plate at the entrance of and astronomical instrument
corresponds to the rotation of the instrument.

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A most simple (linearly) polarimetry
From telescope
Incident light(I0, Q0, U0, V0)

𝑄𝑄0
𝐼𝐼0 =

Polarizer（PA= ψ）
ψ

Outgoing light(I1, Q1, U1, V1)

𝑈𝑈0
𝐼𝐼0 =

Detector

𝐼𝐼1(0°) － 𝐼𝐼1(90°)
𝐼𝐼1(0°) ＋ 𝐼𝐼1(90°)

𝐼𝐼1(45°) － 𝐼𝐼1(135°)
𝐼𝐼1(45°) ＋ 𝐼𝐼1(135°)

Measures onlyflux I1(ψ)••

Since it measures flux ratio, some incompleteness of the system (crosstalk of polarizer, etc.) are canceled
But, since it requires a few different exposures, the result is subject tothe time variation of atmospheric condition and error of telescope guiding.
15

A better method for linear polarimetry
M1

M2

Ordinary ray(Io, Qo, Uo, Vo)(I, Q, U, V)

Extraordinary ray(Ie, Qe, Ue, Ve)

Rotatable
Half wave
plate (PA ψ)

Different efficiency for o- and erays within instrument

Beamsplitter

Variability due toatmospheric condition

𝐼𝐼𝑜𝑜 (𝜓𝜓) = 𝐼𝐼 / 2 ・ { 1 + 𝑄𝑄 / 𝐼𝐼 cos( 4𝜓𝜓 ) + 𝑈𝑈 / 𝐼𝐼 sin( 4𝜓𝜓 ) } ・ 𝑘𝑘𝑜𝑜 ・ 𝜅𝜅( 𝑡𝑡 )𝐼𝐼𝑒𝑒 (𝜓𝜓) = 𝐼𝐼 / 2 ・ { 1 − 𝑄𝑄 / 𝐼𝐼 cos( 4𝜓𝜓 ) − 𝑈𝑈 / 𝐼𝐼 sin( 4𝜓𝜓 ) } ・ 𝑘𝑘𝑒𝑒 ・ 𝜅𝜅( 𝑡𝑡 )𝑎𝑎1 =𝐼𝐼𝑒𝑒 (0°) 𝐼𝐼𝑒𝑒 (45°)
�, 𝑎𝑎2𝐼𝐼𝑜𝑜 (0°) 𝐼𝐼𝑜𝑜 (45°)

𝑄𝑄
1 − 𝑎𝑎1=
1 + 𝑎𝑎1𝐼𝐼=𝑈𝑈
1 − 𝑎𝑎2=
1 + 𝑎𝑎2
𝐼𝐼𝐼𝐼𝑒𝑒 (22.5°) 𝐼𝐼𝑒𝑒 (67.5°)
�𝐼𝐼𝑜𝑜 (22.5°) 𝐼𝐼𝑜𝑜 (67.5°)

𝑘𝑘𝑜𝑜 and 𝑘𝑘𝑒𝑒 are canceled𝜅𝜅( 𝑡𝑡 ) is also canceled•偏光に依存する透過効率ko, keや大気透過率κ(t)が打ち消せる
ため、高精度が得やすい
16Calibration for Opt/NIR Polarimetry•

Instrumental polarization（軸対象な望遠鏡焦点では0.01-1%）– 望遠鏡や観測装置内部の軸対称性のずれなどで生じる偏光– 無偏光標準星（ゼロ偏光）の観測で器械偏光のストークスベクトルを求め、差し引く•

Instrumental depolarization (~polarization efficiency factor)– 光学系の不完全性（位相のずれ、直線－円偏光変換、散乱光等）による「消」偏光– 偏光板、偏光プリズムを用いて（ノミナル）100%偏光を入射し、測定、および補正•Origin and direction of PA of polarization– 強偏光標準星（星間偏光）を観測し、方位角原点を補正– 波長板を回す観測の場合、天球面に対する回転方向の自由度が出るので、（少なくとも最初は）２つの異なる標準星の観測をするのが良い•

Calibrator– Polarizer (producing nominal 100% polarization)– Standard stars (interstellar, highly-polarized; nearby normal star, unpolarized)

17

Typically 𝑝𝑝/Δ𝑝𝑝 > ～ 1017

Subaru/FOCAS Polarimetric performance 1/2
器械偏光の安定性

0.1
0-0.1
0.2

Elevation (deg)L600 (-0.0064±0.0039%)
Y47 (-0.0023±0.0029%)

7060
50
4030

∆Qins (%)

∆Qins (%)∆λ = 2000Å(L600), 3400Å(Y47)

HD212311 (unpol)

0.1∆λ = 2000Å(L600) , 3400Å(Y47)0-0.1

L600 4000-6000Å
(ave +0.0059%)
Y47 4800-8200Å
(ave -0.0034%)

0L600 (+0.1087±0.0038%)
Y47 (+0.0914±0.0029%)

0.1

L600Y47-0.1∆Uins (%)

ADC = Atmospheric Dispersion
Corrector

0.2∆Uins (%)

Dpinstr < 0.05 % at any position
within FOV independently of– telescope position,– PA of image rotator,– with or w/o ADC,
etc.

HD212311 (unpol)

0.1

L600 4000-6000Å
(ave +0.1180%)
Y47 4800-8200Å
(ave +0.1099%)

0-0.12434 2435 2436
182533 2534

JD+2450000

500-50Instr. Rot. PA (deg)Subaru/FOCAS Polarimetric performance 2/3


視野内の器械偏光のラジアル・パターン
2003-06-20 (Grism B300)
2003-06-22 (Grism B300)

2002-09-15 (Grism 150)
2003-06-08 (Grism 150)
200

N5300-5700Å mean

1% polarization

150

500y

center slit

offset slit

DEC offset (arcsec)

100

pinstr prop.to r -b , b~4
PAinstr ~yE

W-50r

CenterCCD2CCD1

FOV: 6 arcmin in diameter-100-150-200

S

FOV: 6 arcmin diameter050

pinstr ~ 0.05 % at center of FOV
pinstr ~ 0.9 % at r = 2.5 arcmin

19-200-150-100-50

RA offset (arcsec)

100

150

200

Any position, Dpinstr < 0.05 %Application for JCMT/SCUBA2-Pol
Scan pattern of sky

Holland+ 2013
Leclerc+ 2015

非軸対象の反射系→望遠鏡の向きに応じた器械偏光半波長板を回転させながら望遠鏡を動かし続けて一定領域をスキャン
𝐼𝐼𝑜𝑜 (𝜓𝜓) = 𝐼𝐼 / 2 ・ { 1 + 𝑄𝑄 / 𝐼𝐼 cos( 4𝜓𝜓 ) + 𝑈𝑈 / 𝐼𝐼 sin( 4𝜓𝜓 ) } ・ 𝑘𝑘𝑜𝑜 ・ 𝜅𝜅( 𝑡𝑡 )𝜓𝜓→ 𝜓𝜓, 𝑡𝑡 器械偏光（＋ほかの偏光）の依存性が複雑に
20

Using a dedicated 1m-class ground-based telescope andan optical 2-band (g’i’) wide-field polarimeter
Several million stars (< 14 mag, 𝛿𝛿 > −20°) will be surveyed in
optical 2 bands simultaneouslyFirst homogeneous polarimetric survey (in northern hemisphere)
The dense optical polarimetric data may
contribute to estimation/masking of foreground interstellar polarization in CMB Bmode study. And also other fruitful
外部寄稿者
collaborative studies (interstellar matter,赤堀 卓也さん、西山 正吾さん、松村 雅文さん、etc) are expected.
新沼 浩太郎さん、郷田 直輝さん、羽澄 昌史さん
Observation period
21~4.5 years

Summary• Stokes vectors can treat incoherent light.
• Each Stokes parameter is additive.
• Most Opt/NIR polarimetry is performed for`relative photometry’ for artificially polarizationmodulated light.
• Typical accuracy of Opt/NIR polarimetry is <~0.1%
in case of using beam-splitter and axi-symmetric
optics22