Invessel Calibration of the Alcator C-Mod MSE Diagnostic

Transcription

1 Invessel Calibration of the Alcator C-Mod MSE Diagnostic Steve Scott (PPPL) Howard Yuh, Bob Granetz, Jinseok Ko, Ian Hutchinson (MIT/PSFC) APS/DPP Meeting Savannah, GA November 2004

2 Introduction Previously, we calibrated the 10-channel MSE system using standard beam-into-gas shots. But there were anomalies: Measured angle not a linear function of actual pitch-angle Calibration didn t agree with EFIT q edge in plasma. We have performed a comprehensive in-vessel MSE calibration We have identified ~1 o spurious rotations of the polarization angle due to imperfect mirrors mostly phase shift. We can easily correct for this rotation in the analysis. The corrected calibration reproduces the in-vessel measurements to ~0.02 o. Conclusion: the MSE polarimeter functions well at least in the absence of vacuum, magnetic field, and atomic physics. The revised calibration still fails to reproduce beam-into-gas calibration data. We have measured Faraday rotation of the polarization angle by the MSE optics. It is insufficient to explain the anomaly.

3 Introduction, cont d We have computed the expected polarization emission pattern from the combined Motional Stark Effect + Zeeman effect New physics: keep track of phase of electric field vectors important for correctly computing polarization angle of elliptically polarized light. Surprises: 15 of 36 transitions emit light polarized parallel to magnetic field. Small (~ o ) differences in polarization angle among pure σ and π lines. Polarization angle of `composite σ lines ( i.e. pure σ + emission parallel to B at same energy) varies significantly with assumed population of n=3 states. This may explain differences between beam-into-gas and beaminto-plasma for MSE systems based on the σ line. The new atomic physics does not significantly change the expected polarization angle of π lines. C-Mod MSE narrow passband filters are tuned to π 2 and π 3 lines. ~10% leakage of σ line possible. Does not reconcile beam-into-gas anomalies.

4 We Illuminated the MSE Diagnostic with Linearly Polarized Light from a Rotating Stage Alignment pointers LED array (enclosed) Rotating stage Linear polarizer Tilt/rotating table Translating stage along DNB trajectory MSE lens

5 We Absolutely Calibrated our Polarized Rotating Stage with Respect to Gravity to ~0.01 o polarizing beamsplitter cube detector linear polarizer on precision rotating stage Kinematic prism mount beam expander Polarization Angle Azimuthal knob turns 6

6 Raw Angle Measured by MSE has Small Errors o ~1 o deviations from exact linearity have clear cos(2θ) and cos(4θ) components. Simulated MSE spectra confirm that such errors would be caused by mirror imperfections phase shifts and non-equal S/P reflectances. The deviations are largest at the outermost and innermost optical channels. These small errors can be completely accounted for in the analysis.

7 Error has cos(2θ) and cos(4θ) Components Phase shifts introduces cos(4θ) error and S/P reflectance ratio introduces cos(2θ) error. The dominant error is cos(4θ) due to phase shift. Phase shift is largest for MSE channels at the edge of the field-of-view, smallest near the optical axis.

8 MSE Invessel Calibration Data is Understood Fits include corrections of order 1 o caused by phase shifts introduced by the three MSE mirrors. Final system response is linear to within ~ 0.02 o across all channels. System behavior during the invessel calibration is fully understood (no toroidal field, no vacuum, no atomic physics).

9 Anomalous Features of Beam-into-Gas Calibration Apply in-vessel calibration data to beam-into-gas calibration shots. Edge channels show significant curvature response. Slope of measured response is greater than unity for all channels. Offset exists for all channels.

10 Possible Causes of Observed Anomalies Atomic physics: geometrical effects on I σ /I π ratio Unlike situation during in-vessel calibration, the light emitted during beam-into-gas shots has both σ and π components which are orthogonal. The unusual viewing geometry of C-Mod MSE at the plasma edge may affect the I σ /I π ratio and the measured angle. Zeeman splitting of Stark Spectrum Magnetic field causes a small Zeeman splitting of the Stark emission spectrum with circular polarization. The phase shift induced by the MSE mirrors will convert any circularly polarized light into partially linearly polarized light. Faraday effect in MSE optical components MSE designed with low-verdet glass. Faraday effect expected to be small. Will measure Verdet constant of MSE optical components in the lab. Stress-induced bi-refrigence of vacuum window Hard to reconcile with channel dependence of MSE response.

11 Faraday Rotation by MSE Optical Components Large number of optical components in high-field region: 9 lenses, 1 vacuum window, 2 photo-elastic modulators (PEMs) All components except PEMs are comprised of SFL6 low Verdet glass. PEMs are standard fused silica Lab measurements with ~1 KG field confirm that lenses are low-verdet glass Verdet constant ~ o /cm/kg Corresponds to θ FL ~ 0.6 o in typical plasmas. In-situ measurements with B radial ~50 Gauss show θ = o / Tesla PEM Verdet constant = 0.11 o /cm/kg (about half published value) Large uncertainty due to small applied field rotation only ~0.01 o Corresponds to θ FL < 1.4 o in plasma (<4.0 o beam-into-gas) In-vessel shutter has been upgraded with a linear polarizer to allow Verdet measurement during normal C-Mod operation at full field. + -

12 Quantum Calculations of Stark + Zeeman Effect Several authors have calculated the polarization due to the combined Stark + Zeeman effect. Isler (1976), Breton (1980), Souw (1983). We have done it again. What s new? Previous analysis focused on calculating the emission intensity in two directions. Unfortunately, this does not uniquely determine the polarization direction if the emission is elliptically polarized. Our approach follows the phase of both electric field vectors for each coherent transition. We add intensities only when combining incoherent emission from different transitions with degenerate energies.

13 Electric Field Intensity in Two Directions does not Uniquely Determine Polarization Direction e 2 e 2 E 2 E(t) E 2 E 1 e 1 E 1 e 1 Elliptical polarization Linear polarization Same E 1 and E 2 but different polarization direction

14 Relative Strength of MSE versus Zeeman Effect ɛ = 3q ea o v beam B Stark c γ = Bq eh Zeeman 4πm e c h γ/ɛ = 12πm e a o v beam = 0.12 On C-Mod λ Stark 4.3Å λ Zeeman 0.08Å q 0 q 1 = ɛ 1 + γ2 ɛ = 3ɛ 1 + 4γ2 9ɛ Lower States: E = q 0, 0, q 0 Upper States: E = q 1, q 1 2, 0, q 1 2, q 1

15 Quantum Calculations of Combined Stark + Zeeman Effect When electric and magnetic fields are applied to a hydrogen atom, the orbitals become hybridized. So in the presence of electric and magnetic fields, we can represent the eigenfunctions of the upper state Ψ h3 and lower state Ψ h2 as linear combinations of the unperturbed eigenfunctions: Ψ h3 = Ψ h2 = Ψ h3 l 3,m 3 l 3,m 3 3l 3 m 3 Ψ h2 l l 2,m 2,m 2 2l 2 m 2 2 There are nine hybridized upper states and four hybridized lower states. The coefficients Ψ 3j l 3,m 3 and Ψ 3j l 3,m 3 have been given by Isler (Phys. Rev. A, Vol. 14, Number 3, September 1976, p. 1015). To simplify the notation we use the labels k2 and k3 to denote the full l 2, m 2 or L 3, m 3 state: Ψ h3 = k3 Ψ h3 k3 k3

16 Calculation of Electric Fields of Emitted Radiation The dipoles are computed through matrix multiplication of the hybrid representations of the lower and upper states with the matrix containing the radial overlap integrals: D, D +, D z = Ψ Lower hybrid representation (4 4) D, D +, D z Radial Overlap Integrals (4 9) Ψ Upper hybrid representation (9 9) The complex electric field E 1 and E 2 along direction vectors ê 1, ê 2 are then given by E 1 = ê 1 ˆxD x + ê 1 ŷd y + ê 1 ẑd z E 2 = ê 2 ˆxD x + ê 2 ŷd y + ê 2 ẑd z

17 Lower Upper Line E/q 0 Ellipticity Intensity Polarization State State angle 2d 3a π b 3a σ c 3a B d 3b σ d 3c B a 3a π b 3b π c 3c π b 3c B c 3b B d 3d π d 3e π d 3f B a 3b σ a 3c B b 3d σ b 3e σ c 3f σ b 3f B c 3d B c 3e B Table 1: Polarization state for individual transitions in the mse coordinate frame for the innermost channel (channel 9), B = 5 Tesla, γ F L = 8 o. Statistical state populations are assumed.

18 Line Composite de No. of Intensity Polarization Polarization Type Transitions angle fraction 0 π σ σ π π π σ σ σ π π π σ σ π Table 2: Polarization state for composite lines summed over all transitions with the specified energy. Calculations are in the mse coordinate frame for the innermost channel (channel 9), B = 5 Tesla, γ F L = 8 o. Statistical state populations are assumed. Note that the angle difference between the composite σ lines and the composite π lines is very nearly 90 o.

19 Emission Parallel to B is Significant for Composite σ Energies for Inner MSE Channels Intensities summed over Transitions MSE Channel 9 γ FL = 8 o σ 0 emission parallel to B (black) Intensity π 3 σ 1 σ 1 π 3 π 2 π 2 π 4 π 4 π 8 σ 6 σ σ 5 σ 6 5 π 8 E / q 0

20 Non-Statistical State Populations Would Significantly Affect "Calibration" of Composite σ 1 Line MSE Channel 0 γ FL = 8 o Polarization Angle of σ 1 Line degrees Multiplier on population of upper state 3c Field line pitch angle (degrees)

21 Electric Field Vectors in Souw and MSE Coordinate Frames MSE Channel 9, γ FL = 8 o E 2 E 2 E 1 E 1 E 1

22 Electric Field Vectors in Souw Coordinate Frame -- All Transitions with Same Energy MSE Channel 9, γ FL = 8 o Zeeman lines (3) Zeeman line (1) E 2 pure σ lines (3) pure σ line (1) E 1 E 1

23 Non-Statistical State Populations Have Less Effect on "Calibration" of Composite σ0 Line MSE Channel 0 γ FL = 8 o Polarization Angle of σ 0 Line degrees Multiplier on population of upper state 3e Field line pitch angle (degrees)

24 Non-Statistical State Populations Have Minimal Effect on "Calibration" of Composite π 3 Line 96 Polarization Angle of π 3 Line degrees Multiplier on population of upper state 3b Field line pitch angle (degrees)

25 Canonical Picture of Polarized Emission from the MSE Effect π line polarized parallel to E MSE σ line polarized perpendicular to π line z T z T No emission parallel to B R φ E π E MSE = v B MSE viewing Line-of-sight s e V e H E σ E π φ

26 Polarized Emission Pattern from Combined MSE + Zeeman Effect One or more pure π lines close to E MSE One or more pure σ lines nominally perpendicular to E MSE not quite to E MSE, even in limit of large v beam Zeeman line parallel to B z T E MSE = v B E Zeeman MSE viewing Line-of-sight s e V e H E π1 E π2 E π3 E σ1 E σ3 E σ2 φ

27 Emission Pattern from Composite MSE+Zeeman Effect for Statistical Upper State Populations Composite π line is very close to E MSE Composite σ line is very close to perpendicular to E MSE Differences from nominal directions < 0.01 o Basically, the canonical picture of MSE holds if the upper states are populated statistically. z T E MSE = v B E π MSE viewing Line-of-sight s e V e H E σ φ

28 Emission Pattern from Composite MSE+Zeeman Effect for NonStatistical Upper State Populations Composite π line remains very close to E MSE Composite σ line may deviate several degrees from the perpendicular to E MSE z T E MSE = v B E π MSE viewing Line-of-sight s e V e H E σ φ

29 Results of Quantum Calculations Including effect of B in addition to E MSE = v B generates new transitions Energies are degenerate with existing pure σ or π lines. Polarization direction of these transitions is parallel to B. Intensity scales with sin(ψ) so is more important for core channels. Including the effect of B changes the direction of the pure σ and π lines. The change in angle scales with γ/ε Effect is negligible for usual tokamak parameters ( < 0.01 o in C-Mod). The directions of the pure σ and π lines are not exactly perpendicular in either the natural or MSE coordinate frames. Polarization direction of composite σ lines (i.e. pure σ + polarization parallel to B at the same energy) can change several degrees for factor ~2 changes in upper-state population levels. Polarization of the composite σ lines are exactly perpendicular to the polarization of the composite π lines if and only if the upper states are populated statistically.

30 Effect of Change in Composite σ Polarization Direction on MSE Analysis MSE diagnostics on several tokamaks (DIII-D, JET, TFTR) use the σ emission as the basis for q-profile measurements. Spectrum measurements by Levinton (1996) were consistent with statistical upper-state populations in plasmas but not during beaminto-gas: Transition Plasma Statistical Gas σ σ π π π Levinton reported ~0.5 o differences between beam-into-gas and beam-into-plasma calibrations. He attributed these differences to leakage of some π radiation through the MSE optical filters combined with non-ideal mirrors. An alternate explanation is that the actual polarization direction of the σ emission during beam-into-gas was altered by non-statistical populations of the upper states.

31 Implications for MSE on C-Mod The Zeeman effect has only a tiny effect on the polarization direction of the π line for typical tokamak conditions, including C-Mod s. The C-Mod MSE system is based on the π line. Temperature-tuned optical filters designed to pass mostly π 2 and π 3 light. Some σ 1 light might leak through ~10-20% based on the observed polarization fraction. Preliminary calculations indicate that the C-Mod MSE beam-into-gas calibration anomalies cannot be explained by the effect of B on the polarization direction of the σ emission.

32 We can Apply In-Vessel Calibration Directly to Beam-into-Plasma Data Response to Linearly Polarized Light Response to Circularly Polarized Light 20 khz Amplitude 20 khz Amplitude 40 khz Amplitude 40 khz Amplitude 44 khz Amplitude 44 khz Amplitude Polarizer angle (degrees) Polarizer angle (degrees)

33 Analysis Technique A 20 R 20 R 20 R 20 I π A 40 = R 40 R 40 R 40 I σ A 40 R 44 R 44 R 44 I circ FFT amplitudes measured in plasma Brute-force solution: Response measured during in-vessel calibration (function of pitch angle) Solve for relative amplitude of π, σ, and circular components Guess γ lookup response array from in-vessel calibration Guess ratio of I π, I σ, I circ. Compute expected ratios A 20, A 40, A 44. Compute difference between measured and expected A 20, A 40, A 44. Choose γ, I π, I σ, I circ that minimizes error. Result is consistent with all mirror imperfections & actual polarization fraction.

34 The In-Vessel Calibration Agrees with Plasma Data Better than Beam-into-Gas Channel 0 (R=86.7) Channel 1 (R=85.2) Channel 2 (R=83.5) 30 gas Measured Angle (degrees) plasma EFIT Angle (degrees) EFIT Angle (degrees) EFIT Angle (degrees)