Diseño del Lenguaje.

The Zeeman effect introduces a net circularity to the emission which can be understood by considering the |n, k, mliC states in the limit that /γ → 0. The pure Zeeman effect

transitions in this case are listed in Table 2.13.

Comparing Tables 2.4 and 2.13 we see the Starkπ4 transition (z-axis oscillation) cor-

responds with a σ− Zeeman transition (yz-plane circular oscillation). In the intermediate Stark-Zeeman case the dipole vector then has a large real z component and a small imagi- nary y component, overall resulting in an elliptical dipole vector in the plane perpendicular to magnetic field. Meanwhile the Starkπ−4transition corresponds with aσ+Zeeman tran-

sition resulting in an elliptical dipole vector of opposite handedness. Similarly theπ2 and π3 Stark transitions correspond with σ− Zeeman transitions while π−2 and π−3 Stark

|n, k, mliC 2,1,˜0_C 2,−1,˜0_C 2,0,˜1_C 2,0,−˜₁ C 3,2,˜0 C (4608)σ − 1 - - - 3,−2,˜0 C - (4608)σ + −1 - - 3,0,˜0 C (882)σ + −1 (882)σ − 1 (729)π0 (729)π0 3,0,˜2 C (18)σ + −1 (18)σ − 1 (1681)π0 (1)π0 3,0,−˜₂ C (18)σ + −1 (18)σ − 1 (1)π0 (1681)π0 3,1,˜1_C (1152)π0 - (1936)σ1− (16)σ − 1 3,1,−˜1_C (1152)π0 - (16)σ₁− (1936)σ−₁ 3,−1,˜1 C - (1152)π0 (1936)σ + −1 (16)σ + −1 3,−1,−˜1 C - (1152)π0 (16)σ + −1 (1936)σ + −1

Table 2.13: Transitions betweenn=3 andn=2 states for the pure Zeeman effect with the |n, k, mli_C states. The π dipole vector is oriented along the magnetic field (x-axis) and σ± dipole vectors circle the magnetic field in the yz-plane. The energy splitting is in units of γ.

transitions correspond with σ+ Zeeman transitions. These elliptical Stark-Zeeman dipole vectors are illustrated in the left of Fig. 2.9.

π2,3,4,-8 π-2,-3,-4,8 σ-1,5,6 σ1,-5,-6 σ0 ξr ξr z z y y σ+ σ− y x ξc π

Figure 2.9: Plot of the Stark-Zeeman π (Left) and σ (Middle) dipole vectors in the plane perpendicular to B. The π±2,π±4 dipole vectors lie purely in the yz plane while the π±3

has a very small component in the x direction (third order). Note that ξr is related to ξ

in Fig. 2.2, but is generally different due to projection effects. (Right) Dipole vectors in the plane perpendicular to E. For theσ transitionsξc is less than 45◦ whenγ/ >0.

Theσ0 and σ±1 Stark transitions (xy-plane circular oscillation) correspond toπ Zee-

man transitions (x-axis oscillation). In the γ regime this leads to the Starkσ transi- tions that are slightly elliptised (stretched) in the direction of the magnetic field, shown in the right of Fig. 2.9. Additionally in this intermediate regime the σ±1 dipole vectors

acquire a small component in the direction of the electric field as in the middle plot of Fig. 2.9. Both σ1 transitions acquire the same handedness about B, opposite to that of both σ−1. The interpretation for the σ emission is different for the |n, k, mliC and |n, k, mliL

states, but the net circularity is the same for the two cases when upper-states are equally populated. The ellipticity angles of the dipole vectors, as defined in Fig. 2.9, are given in Table 2.14. The concept of separate Stark and Zeeman effect linear and circular emissions introduced in Ref. [51] is non-physical as the Stark and Zeeman effects are coupled and

gives rise to elliptical dipole transitions. σ₀|3,0,0i_C σ0|3,0,±2i_C σ±1 π±2 π±3 π±4 σ±5 σ±6 π±8 ξr 0 0 ±₃γ ±17₈₁γ ±₂γ ±₁₂₃97γ ∓5₃γ ∓4₃γ ∓₃γ ξc π₄ − 31γ 2 2522 π₄ − γ2 122 π₄ − 5γ2 662 π₂ π₂ − 5γ2 1082 π₂ π₄ − γ2 22 π₄ − 25γ2 362 π₂

Table 2.14: Angles for the elliptical dipole vectors for each transition, as defined in Fig. 2.9. The angles are correct to second order inγ/

The dipole vector and dimensionless Stokes vector for an elliptical Stark-Zeeman π

transition in the yz plane are,

ˆrπ =(0,−isinξr,cosξr), (2.67) ¯sπ = sin2ψcos2ξr+ (cos2ϕ+ cos2ψsin2ϕ) sin2ξr,−sin2ψcos2ξr

+ (cos2ϕ−cos2ψsin2ϕ) sin2ξr,−cosψsin 2ϕsin2ξr,−sinψcosϕsin 2ξr

. (2.68) Expressed relative to the orientation of the magnetic field, the ¯s3 component is

(¯s3)π =−Bˆ ·ˆisin 2ξr. (2.69)

Either Eq. 2.66 can be used with the second order circularity approximations in Table 2.11 or the geometric interpretation in Eq. 2.69 can be used with the angles ξr in Table

2.14 to get the same result for the π transitions.

The angleξc is also needed for the second order approximation of theσ Stokes vector,

leading to a more complicated geometrical interpretation. The dipole vector for the σ

transitions is

r_σ± = (±cosξ_ccosξ_r,−isinξ_ccosξ_r,sinξ_csinξ_r) (2.70)

and the resulting Stokes vector is unwieldy. Nevertheless the key results are summarised in Eqs. 2.63-2.66 and Table 2.11.

Possible Applications of the Circular Polarisation

An example of the s3/s0 circular polarisation fraction expected for the summed π2−4

emission is presented in Fig. 2.10 for the KSTAR IMSE viewing geometry and an 80keV deuterium beam. It is evident that the circular polarisation fraction for standard MSE conditions is significant. The majority of the circular polarisation is from the σ±1, π±3

and π±4 lines. A number of potentially useful applications of the circular polarisation are

outlined here.

Assuming that the neutral beam velocity is known precisely then the motional electric field is restricted to lie in a 2D plane perpendicular to v, since E = v× B = v×

B⊥v. Therefore measurements relating to the motional electric field are only sensitive to B⊥v. The orientation ofB⊥v is typically determined from the polarisation orientation of

the MSE emission or alternatively from the intensity ratio of some π and σ transitions (assuming equally populated upper-state populations). The magnitude of B⊥v can also

be determined spectroscopically from a measurement of the line-splitting. However when assuming the Zeeman effect is negligible there is no information available aboutBkv. That

said, for a horizontally injected beam at the midplane it is typically valid to assumeBr = 0

1.7 1.8 1.9 2.0 2.1 2.2 2.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 R(m) Z ( m ) s_3/s_0(%) 20.0 22.5 25.0 27.5 30.0

Figure 2.10: Relative fraction of circular to linear polarisation from the π2−4 transitions

for an 80keV deuterium beam assuming equally populated states.

for an inclined beam injection or measurement away from the midplane Bkv may be of

interest.

If an individual π line can be isolated then its circular polarisation can be used to complete the measurement of the magnetic field orientation. From Eq. 2.66 the circular polarisation is proportional to

ˆ

B·ˆi=Bˆ ·ˆi⊥v+Bˆ ·ˆikv

=Bˆ⊥v·ˆi⊥v+Bˆkv·ˆikv (2.71)

The remaining unknown Bˆkvis therefore available from the circular polarisation measure-

ment. Eq. 2.66 is weakly dependent on sinρ (ρ is the angle between v and B) via the

γ/ term so the calculation can be iterated to overcome this dependence. The circular polarisation can be normalised to the known linear polarisation fraction to remove any dependence on upper-state populations and unpolarised light. In case thatπ lines are not individually resolvable an accurate measurement becomes more challenging as it requires knowledge of the relative throughputs and upper-state populations of each of the π com- ponents, as each has a different ellipticity, as given in table 2.11. It is likely that the effect of the fine-structure on the circular polarisation may also need to be considered for such a measurement.

A precise measurement of Bkv from a single channel from Eq. 2.71 requires the ra-

dial electric field Er to be negligible. In the case where Er is not negligible assumptions

could instead be made thatBr= 0 near the midplane. The circular polarisation fraction

measurement could then be used to separate out the motional and radial electric field contributions. In either case (Br 6= 0 or Er 6= 0) the circular polarisation carries addi-

tional information that is useful as a further constraint, or at the very least can be used as a crosscheck for the polarisation preservation properties of the mirrors in the optical labyrinth.

The circular polarisation fraction also has applications for a polarimetric measurement of the central σ emission. The s3 spectrum has odd symmetry about σ0, as seen in table

2.12,which could be used to verify how well the narrowband filter has been tuned. Conven- tional MSE polarimeters encodes3at different carrier frequencies to thes1ands2, making

it possible to use the digitised MSE signal to compare the net linear and circular polarisa- tion lying in the filter passband. An example of such a measurement is shown in Fig. 2.11 for a shot with variable beam voltage that causes the MSE multiplet to ‘move’ under the filter passband. As expected, the beam voltage that gives the maximum linear polarisation roughly coincides with the zero crossing of the circular polarisation. Notably the narrow- band filters for the 1.78m and 1.93m channels appear to be insufficiently blueshifted given they have circular polarisation intensities crossing zero near 77keV and linear polarisation intensities peaking near 79keV. The basic analysis here does not include asymmetries in the filter passband, temporal changes in the emission intensity and temporal changes in the Stark splitting. These effects may lead to the small differences between the beam energies of the circular polarisation zero crossings and linear polarisation maxima. The filter for the 1.63m channel appears to be too blueshifted given the linear polarisation crosses zero much earlier than the other channels and the circular polarisation fraction appears as if it would cross zero at a beam energy of≈85keV. Notably there appears to be a temporally varying contamination in the circular polarisation fraction for the 1.63m channel, possibly due to a Zeeman split impurity line in the filter passband.

Figure 2.11: Linear (left) and circular (right) polarisation intensity measured from 8 of the tangential conventional MSE channels for shot 166400 from 0.3−0.9s. The beam voltage is initially 82keV and decreases to 73keV in this period. The polarisation intensities have been scaled with the appropriate Bessel functions but the total intensity of the light is not available from the signals so it is not possible to give ‘normalised’ polarisation fractions. The system is expected to deliver the greatest linear polarisation at 81keV. More details on this shot are given in Section 4.3.6.

In Fig. 2.11 the relative magnitudes of the circular and linear polarisation appear to be consistent with the Stark-Zeeman calculation. For example the 1.83m channel has a maximum linear polarisation of 4 arbitrary units while the circular polarisation is 0.5 arbitrary units near the crossover of the π and σ emissions and it seems plausible that the circular polarisation maximum would peak would peak at approximately 1 arbitrary unit when centred over theπ emission, in agreement with the ∼25% circular polarisation fraction example in Fig. 2.10.

Detrimental Effects of the Circular Polarisation Fraction

The circular polarisation fraction can also have some detrimental effects on MSE mea- surements. The circular s3 Stokes component of the emission can couple to the linear s1

and s2 Stokes components at an imperfect mirror[20] or port window with stress induced

birefringence. Mirrors are commonly used on MSE diagnostics to reflect light towards the polarimeter and it only requires a small difference in the s and p reflectivity or dephase for the circular polarisation to alter the linear polarisation orientation. It should be noted that with a standard dual PEM polarimeter the s3 component is modulated at different

frequencies to the linear components, independently of any misalignments and only in the presence of a mirror can the measurement be corrupted. Of particular interest to polarisation coherence imaging is that for some polarimeter designs the linear and circu- lar polarisation are carried at the same spatial carrier frequency. Minimising, calibrating for and decoupling the effects of the circular polarisation requires knowledge of the s3

spectrum.

The coupling of circular to linear polarisation is of particular relevance to any ITER MSE polarimeter where the mirrors will be significantly degraded by plasma exposure such that the sand p reflectivity ratio and dephasing are expected to be far from ideal. While the heating neutral beams on ITER have energies of 1000keV the γ/ ratio only decreases by a factor of 3.5 relative to the example here of 80keV. The π3 emission will

be 5% circularly polarised, assuming aρ= 45◦ injection angle to the magnetic field and a

ϕ= 0◦ view parallel to the field. The calibration procedure therefore needs to account for this circular polarisation fraction to avoid similar effects to those observed on DIII-D[20].

2.6.3 Linear Polarisation Orientation of Elliptical Transitions - ‘Angle Defect’

The linear polarisation orientation of the Stark-Zeeman emission is slightly different to the pure Stark effect due to the elliptical nature of the dipole vectors. The ‘angle defect’ is usually negligible compared to the desired 0.1◦ accuracy of the measurement and has been partially covered in Ref. [53]. Here a simple geometric description of the effect is presented.

A three dimensional elliptical dipole vector projects onto the polarimeter as a two dimensional ellipse. However, the major axis of the projected ellipse is in general different to the direct projection of the 3D elliptical dipole vector’s major axis. In other words taking the major axis before or after projecting the dipole vector produces a different result. From Eq. 2.68 it follows that the linear polarisation angle of an elliptical π

transition is θπ = π 2 + sin 2ϕcosψ 2 sin2ψ ξ 2 r+O(ξr4), (2.72)

consistent with Eq. 2.65. This geometric effect does not occur when viewing along one of the ψ = π/2 or ϕ= nπ/2 planes but increases when looking at an angle to all three of these planes. An example of the average angle defect for the KSTAR viewing geometry is illustrated in Fig. 2.12 where the defect remains under 0.01◦ across most of the view. In the lower left corner the effect reaches 0.1◦, an effect pronounced by the 28.5◦ elevation of the view above the midplane. Similar results are expected for tangential views on other devices as the sightlines are significantly aligned with the magnetic field (ψ, ϕ) = (π/2,0) such that the effect is usually negligible.

1.7 1.8 1.9 2.0 2.1 2.2 2.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 R(m) Z ( m )

Angle Defect(deg)

0 0.02 0.04 0.06 0.08

Figure 2.12: Linear polarisation angle defect for KSTAR IMSE view (weighted average across the 15 different lines) with an 80keV beam. The defect vanishes at the contours where ψ=π/2 (green) and ϕ= 0 (purple).