22P121-P
Highly Accurate Approximate Solutions of Stokes Equation forPolarimetry of ITER-like Plasmas
ITERをはじめとする核融合プラズマにおける偏光計測のためのストークス方程式の高精度近似解
Ryota Imazawa, Yasunori Kawano and Yoshinori Kusama
今澤良太, 河野康則，草間義紀
Japan Atomic Energy Agency
801-1 Mukoyama, Naka, Ibaraki 311-0193, Japan
日本原子力研究開発機構 〒311-0193 那珂市向山801-1

Polarimetry has been applied in many magnetic confinement fusion devices in order to measure
the magnetic field and the electron density. The Faraday effect rotates the polarization ellipse,
and the Cotton-Mouton effect changes the ellipticity of the polarization ellipse. In the dense
plasmas like ITER plasma, these two effects interfere with each other, and the formulas for thetwo effects are no longer valid. In order to comprehend the plasma state from the polarimetricdata, the Stokes equation should be solved. We have found the new equations equivalent to theStokes equation and the highly accurate approximate solutions. Our solutions hold even in highdensity region (~1021 m-3) and exhibit the highest accuracy among approximate solutions.
1. Introduction
Plasma polarimetric measurements have beeninstalled in many fusion devices. Physicalbackgrounds of polarimetry are usually explainedby the Faraday and the Cotton-Mouton effects. Inthe former effect the polarization ellipse (and, forlinear polarization, the plane) rotates, while in thelatter effect the ellipticity of the polarization
ellipse changes. Although these two effects areusually treated as independent events, theyinterfere with each other in dense plasmas like
ITER plasma. In such a case, formulas of the
Faraday and Cotton-Mouton effects are not valid,
and the Stokes equation expresses the change ofthe polarization state. In this study, we transformthe Stokes equation to the new equations to obtain
the highly accurate approximate solutions.
2. Faraday and Cotton-Mouton Effects
Polarization state can be defined by thepolarization ellipse parameters; the orientation
angle, ψ (with 0≤ψ≤π), the ellipticity angle, χ(with -π/4<χ≤π/4), the auxiliary angle, α (with0≤α≤π/2), and the phase shift angle, δ (with0≤δ≤2π). Figure 1 shows the relation between thepolarization ellipse and these parameters. Whenone of two effects is small enough, the Faraday
and Cotton-Mouton effects are expressed as:zΔψ = C1 ∫z 1 ne B∥ dz ,(1)0zΔδ = C2 ∫z 1 ne B⊥2 dz,(2)0respectively. Here, z denotes the coordinate axis

along propagation of electromagnetic radiation;ne denotes the electron density; B∥ denotes thecomponent of the magnetic field parallel to z; B⊥
denotes the component of the magnetic fieldorthogonal to z; and C1 and C2 denote theconstant values.
3. Stokes Equation
The Stokes equation expresses the change ofthe polarization state and is written asds�⃗��⃗ × s⃗,
=Ω(3)
dz

where the vector of s⃗ is the reduced Stokes
��⃗ is the vector
vector and the vector of Ω
associated with the Mueller matrix representing
the optical properties of the plasma[1]. The
reduced Stokes vector, s⃗, is expressed asFig. 1. The polarization ellipse andpolarization parameters (ψ, χ, α, and δ).

thecos2χcos2ψ
cos2αs⃗ = � cos2χsin2ψ � = �sin2αcosδ�.
(4)
sin2χ
sin2αsinδ�⃗ is expressed as [1]:The vector of �Ω3
CCM λ ne B⊥2 cos2β�Ω�⃗ = �−CCM λ3 ne B⊥2 sin2β�,(5)2−2CFR λ ne B∥
where CFR and CCM denote the constant values, λis the laser wavelength, β is the angle between
the y direction and B⊥ .

4. New Expressions and Approximated Solutionsof Stokes Equation
Although Stokes polarization parameters(components of s⃗) are observables, they are notas intuitive as the polarization ellipse parameters.
We transformed the Stokes equation to more usefulequations as follow:dχC= CM
λ3 ne B⊥2 sin(2ψ + 2β),
(6)
2dz
dψ
= −CFR λ2 ne B∥dz
C− CM λ3 ne B⊥2 tan2χsin(2ψ + 2β), (7)2dα= −CFR λ2 ne B∥ cosδdz
C+ CM λ3 ne B⊥2 sin2βsin2δ,(8)2sinδdδ= 2CFR λ2 ne B∥
tan2αdzcosδ
23�. (9)+CCM λ ne B⊥ �cos2β + sin2βtan2αWhen measuring objects are magnetic
confinement fusion plasmas and the probing laser
wavelength is in far-infrared range, the second termof RHS of eq. (7) is smaller than the first term.
Assuming that the second term is negligible, weobtain the approximated solution related to theFaraday effect:zΔψ = −CFR ∫z 1 λ2 ne B∥ dz.
(10)
0Substituting eq. (10) into eq. (6) leads to theapproximatedsolution
relatedtotheCotton—Mouton effect:
Δχ =CCM2z1∫z λ3 ne (p)B⊥ (p)2 sin{2ψ0 + 2β(p)0q−2 ∫z CFR λ2 ne (q)B∥ (q)dq� dp.
0(11)5. Comparison among Approximated Solutions
Several approximated solutions to the Stokes
equation have been proposed [2-5]. We havecompared the error of our approximated solutions
and the error of the approximated solution called as
Type II [3]. The approximated solutions of Type II
are expressed as:1z−1

ψ1 = − arctan ��tan �CFR ∫z 1 λ2 ne B∥ dz�� �, (12)2Fδ1 = arctan � �,G0(13)Fig.2. Errors of approximated solutions related to the (a)Faraday and (b) Cotton-Mouton effects as a function ofdensity.
wherezF = CCM λ3 ∫z 1 ne (p)�By (p)2 − Bx (p)2 �0pcos �∫z 2CCM λ3 ne (q)B∥ (q)dq� dp, (14)z0(15)
Eqs. (12) and (13) are related to the Faraday andCotton-Mouton effects, respectively. Conditions forcomparison of the error of the approximatedsolutions are λ = 10−4 [m] B∥ = 1 [T], B⊥ = 5[T], z1 − z0 = 4 [m], and β = 0. Figure 2 shows
the differences between the true values calculatedby Stokes equation and approximated solutions
related to the Faraday and Cotton-Mouton effects asa function of electron density. Our new solutions
are accurate even in dense plasmas and are moreaccurate than Type II. Comparing with othersolutions [2-5], our solutions are the most accurate
in the above condition and need the fewest
assumptions.
G = cos �2CFR ∫z 1 ne (p)Bz (p)dp�.
06. Conclusions
We have transformed the Stokes equation tomore useful equations using the polarization ellipse
parameters. We have obtained the new
approximated solutions of the Stokes equation for
the Faraday and Cotton-Mouton effect. Our
solutions hold even in high density region (~1021m-3) and exhibit the highest accuracy among
approximated solutions.
References[1] S. E. Segre: Plasma Physics and Controlled Fusion,
41, R57, (1999).
[2] S. E. Segre: Physics of Plasmas, 2, 2908 (1995).
[3] C. Mazzotta et al.: Report EFDA JET, CP(06)03/02(2006).
[4] K Guenther et al.: Plasma Physics and Controlled
Fusion, 46, 1423 (2004).
[5] S. E. Segre et al.: Plasma Physics and Controlled
Fusion, 48, 339 (2006).