system calibration and characterisation

CHAPTER 5

System Calibration andCharacterisation

Those are my principles, and if you don’t like them... well, I haveothers.

– Groucho Marx

C orrect calibration and characterisation of the ToMOSS system is essential ifreliable tomographic reconstructions are to be performed. Characterisation

includes the determination of the position of the magnetic field structure relativeto the wheel, the verification of the wheel rotation properties and the measure-ment of the spatial response of the viewing chords.

Aspects of the spectroscopic system that require calibration are the intensity,phase and contrast of the detected interferogram. The latter two are solely in thedomain of the MOSS spectrometer—with an argon-ion laser providing a refer-ence for the demodulation of the Fourier-transformed signal—and are covered in§2.5. The relative intensity calibration is affected by all optical hardware compo-nents and is the most difficult to determine correctly. Since the detector recordsonly the light intensity—recall that the derived parameters (Ti and vD) are ob-tained from the interferogram—an accurate knowledge of the intensity responseof all parts of the system is essential. The intensity response can be modelled as asystem of linear equations with an input vector of the light levels in the viewingregion mapping to an output vector of the measured signals, via a single calibra-tion matrix. The relative sensitivities of channels and cross-talk in the detectorarray are some aspects of the system which are considered and incorporated intothe measured calibration matrix.

5. System Calibration and Characterisation 82

5.1 Characterisation

5.1.1 Wheel Location Relative to Magnetic Field

The position of the light-collection system is surveyed in situ and its location isderived relative to the H-1NF coil set. The position of the magnetic field structureis determined in relation to the coil set and relies on computation by the HELIACcode. The code itself is based on the given coil-set specifications.

For a fixed toroidal plane, the mapping from H-1NF co-ordinates to wheelco-ordinates (where the wheel radius is unity) is given by

(xwhywh

)=

1rwh

[(xH1

yH1

)−

(xoffyoff

)](5.1)

where rwh is the inner radius of the wheel and xoff, yoff are the offsets of the wheelcentre—in the horizontal and vertical directions respectively—from the centre ofthe heliac. The in situ survey gives the toroidal plane of the wheel and xoff, yoffare derived from the poloidal position. The inner radius of the wheel is givenfrom the design as 37.0 cm.

Toroidal Position

The in situ survey measurements place the viewing plane of the wheel at thetoroidal angle of 204.2◦. Figure 5.1 shows the measurements made, in centime-tres, with an error of ±0.2 cm. This is an important measurement because of thethree-fold symmetry of the magnetic field of the heliac†. A 1◦ inaccuracy in thetoroidal position equates to a 3◦ error in the poloidal direction. It is also worthnoting the change in the field position due to the finite width of the viewingchord. For a nominal width of 20 mm at 1 m from the heliac centre, the toroidalwidth is 1.15◦. This equates to a rotation of ≈ 3.5◦ in the poloidal direction. Fig-ure 5.2 shows the computed flux surfaces for the two extreme positions eitherside of 204.2◦.

Poloidal Position

The designed position of the wheel centre is coincident with the PFC centre.Figure 5.3 is a schematic of the measurements taken in the poloidal cross-section.Due to the difficulty in making the measurements, the error in distances to thewheel circumference is ±0.3 cm. The error in measurements of the PFC dimen-sions is ±0.2 cm. The survey measurements give the wheel centre a horizontaloffset of 1.25 cm and a vertical offset of 0.0 cm.

† The magnetic axis completes three poloidal rotations for every toroidal orbit.


TFC #21

TFC #20

PFC

coil support

magnetic planeof TFC @ 199.05o

viewing modules

carrier ring

HCW

4.5

8.6C−frame

driveshaft

structure

Figure 5.1: The position of the wheel relative to the H-1NF coil set. Measurements are incentimetres. The offset of the magnetic plane of TFC #20 is given by the H-1NF assemblysurvey measurements.

5.1.2 Wheel Rotation

The consistency of the rotation of the wheel is verified by checking the rangeof rotation as well as the reproducibility of the intensity response as the viewingchords are rotated past the fixed in situ light sources. The measured intensityresponse for a full rotational scan, with all channels, is known as a calibrationsinogram.

Rotation Range

Any slipping or variability in the rotation speed can be checked by runningthe stepper motor at a lower speed, for which the motor develops a higher torque†.The rotation range can be verified at the same time by counting the number ofsteps required for a full rotation of the wheel at various motor speeds. Table 5.1presents the results of these experiments. Note that the fixed number of steps perdegree of rotation is 666. The designation of ’full’ in the Step Size column meansall the steps were performed in a single sequence. Hence for a ‘full’ sequence,132534 steps (199◦) were sent to the stepper motor with additional 100-step se-

† The stepper motor specifications were chosen to be a factor of 3–4, even without the 120:1gearing ratio, over the requirement to rotate the wheel. This ensures more than sufficient torqueis available.


Figure 5.2: Computed flux surfaces which are ± 0.58◦ toroidally either side of the meantoroidal position of the viewing modules. The total toroidal separation of 1.15◦ corre-sponds to a nominal viewing chord width of 20 mm at the radial position of the PFC.


2*

26.0

HCW

PFC

11.510*

29.8

12.0

31.0

32.5

wheel outline

Figure 5.3: The position of the wheel relative to the poloidal field coil (PFC) and helicalcontrol winding (HCW). Measurements are in centimetres. Asterisked measurementswere taken from the H-1NF design specifications.


quences used to bring the wheel fully to the limit. The tabulated data also showsthe number of steps required to bring the wheel completely to the limit for mul-tiple sequences of length 0.5◦ and 1◦.

Step Frequency (Hz) Step Size (◦) Number of Steps Degrees484 Full 132568±50 199.1±0.1484 Full 132668±50 199.2±0.1484 1 132534±333 199.0±0.5484 0.5 132534±166 199.0±0.2390 Full 132668±50 199.2±0.1390 1 132534±333 199.0±0.5390 0.5 132534±166 199.0±0.2222 Full 132568±50 199.1±0.1

Table 5.1: Wheel Rotation Consistency.

The results show that the rotation range is consistent, for all motor speedsand step sizes, with a mean range of 199.08◦ and a standard deviation of 0.09◦.This indicates there is no slippage or missed steps in the rotation of the wheeleven with the motor at its highest speed (484 Hz). The in situ calibration scansare performed with the stepper motor set at its highest speed.

Rotation Consistency

To ensure that, during a rotation, the angular position of the wheel as a func-tion of time is linear and reproducible over repeated rotations, the positions ofthe intensity peaks in a sinogram are recorded and compared. Any variance inthe peak positions over repeated sinograms would indicate a rotational incon-sistency. Figure 5.4 shows the intensity traces for two channels, taken from fourdifferent sinograms of the same light source. The two intensity peaks are, firstly,the far-field view of the light source and then, as the rotation progresses, thenear-field view.

The standard deviation in the positions of the intensity peaks is <0.05% of therotation range. The reproducibility of the position of the intensity peaks, togetherwith the stepper motor speed trials, give confidence that the rotation is linear andreproducible.

Angular Start Position

Though the lines-of-sight are fixed relative to the wheel, the knowledge oftheir angular orientation—now that the rotation linearity is verified—is depen-dent on the start position of the rotation. The measure of the start position θ0 is


Figure 5.4: Intensity peaks comparison from several sinograms at a single light sourceposition to verify rotational reproducibility. Note that the range of angles are differentfor each plot and the peaks are consistent across both ranges.


the angle of the viewing axis of the wheel to the vertical, as shown in Figure 3.2).The start position is the CCW limit (see §4.1.3) and corresponds to Viewing Sta-tion 0.

The position is initially estimated by noting, during the installation of the in-vacuum systems, the position of the viewing axis to a known reference positionon the C-frame. This in situ estimate is given as −16.6 ± 1◦ and is refined duringthe measurement of the spatial response (see §5.2).

5.1.3 Viewing Chord Positions

Figure 5.5 shows the geometry for the position of the viewing chord—specifiedby an impact parameter p and angle φ with respect to the viewing axis of thewheel. A point (xwh, ywh) in the viewing plane is mapped to the reference frameof the viewing chord—with chord co-ordinates (s, t) as shown in Figure 5.5—by

(st

)=

(cos φ sin φ

− sin φ cos φ

) (xwhywh

)+

( −p√r2

wh − p2

)(5.2)

��

��

t

s

viewingchord

y

lens position

x

(x,y)

wheel outline

p

φ

Figure 5.5: The geometry for the viewing chord position (p, φ) and co-ordinates alongthe chord (s, t). Here s is positive to the right of the chord length as viewed from the lensposition.


The viewing chord configuration for the system, as designed, is shown in Fig-ure 5.6, having being mapped via (5.1) to H-1NF co-ordinates. The chord posi-tions are determined more precisely in conjunction with the measurement of thespatial response for each viewing chord, as outlined in §5.2.

Figure 5.6: A schematic of the designed configuration for the viewing chords, with chan-nel numbers and the cross-section of the poloidal field coil (PFC) shown. The wheel isshown in the CCW limit (or main viewing) position and the (*) represent the positions ofthe in situ light sources. The (+) represent the location of the coupling lens in the viewingmodule. The co-ordinate system is relative to H-1NF, with the centre of the PFC at 100 cmradially from the centre of the stellarator.

During installation of the light-collection system into the heliac, several opti-cal fibres were damaged. This resulted in some lines-of-sight being unavailableand others suffering a reduced light throughput. Figure 5.7 shows a schematic ofthe final configuration of viewing chords.


Figure 5.7: A schematic of the corrected viewing chords, including channel numbers andunavailable channels due to broken fibres. The (*) represent the positions of the in situlight sources. Note that the co-ordinate system is relative to H-1NF (See Figure 5.6).


5.1.4 Magnetic Axis

An independent measure of the location of the magnetic field structure rela-tive to the light-collection wheel can be made by injecting an electron beam alongthe magnetic axis for a known field configuration. This technique is independentof computed field lines and any alignment assumptions that might be made us-ing emissivity of a plasma. A hot filament generates the electron beam which, inthe presence of a high-pressure argon fill in the vacuum-tank, causes the argonto radiate. Depending on the radial position of the filament and the fill pressure,one or more beams may be visible as the electrons complete multiple toroidal ro-tations. The wheel is rotated slowly and, observing the emission from the beams,generates a sinogram. The analysis of this sinogram reveals the position of thebeams relative to the wheel.

Electron Gun Experiment

The electron gun [Shats et al., 1995] consists of a U-shaped thoria-coated tung-sten filament (0.3mm diameter) which is clamped to copper leads. The leads sitinside a ceramic insulator, itself housed in a stainless steel tube (6.4mm diameter).The tube ends with a cap which contains an aperture (0.7mm) that is designedto collimate the beam. The housing is set at ground potential, with the filamentbiased negatively in the range 100-400V.

The gun provides up to 7mA of electron current, for a filament current of 2A.The extracted electron current is estimated to be a few hundred microamperes.The gun housing is fitted to a long extension tube mounted on a bellows assem-bly, which provides the vacuum seal. The gun can be arbitrarily positioned ina poloidal cross-section and is located at the 5◦ toroidal position, with the beamlaunched towards the ToMOSS wheel.

To minimise light attenuation, the MOSS spectrometer has all optical com-ponents between the collimating and imaging lenses removed. To improve thelight detection sensitivity, the optical equipment is covered and the laboratorylights turned off. The positions of the optical fibres in the input array of the spec-trometer are chosen with viewing chords which are expected to collect the mostlight towards the centre, to reduce vignetting losses. The light-collection wheelis placed at Viewing Station 0 (CCW limit) and rotated through to the CW limit,pausing with a sufficient dwell time at selected viewing stations. A backgroundscan, with the electron gun not operational, is performed to create baseline lightlevels for each channel. Analysis of the background levels show them to be in-sensitive to the angular position of the wheel.

With the magnetic field in standard configuration, the electron gun is posi-tioned until a single beam is visible and is as thin as possible. This is taken tobe coincident with the magnetic axis. Other radial positions of the electron gun


produce three and four beams. Table 5.2 shows the parameters for each configu-ration observed.

Visible Beams Fill Pressure (Torr) Rotation Step Size (◦) Viewing Time (s)1 1.1×10−4 1 73 4.9×10−5 1 104 1.0×10−4 1 10

Table 5.2: Configurations for Electron Gun Experiments.

Interpretation of Electron Gun Results

The sinograms are analysed using both the tomographic ART method andthe sine-fitting technique. Using the refined value for the start position of thewheel (as given in §5.2, page 95), the position of the magnetic axis in wheel co-ordinates, using the sine-fitting technique, is (−0.197,−0.551). This technique ispreferred in these circumstances to locating the peak (or centre-of-mass) of theART-reconstructed image, since an intensity calibration was unavailable for thesystem at that stage.

The HELIAC code gives the magnetic axis, at 204.2◦ toroidally, in H-1NF co-ordinates as (96.10, -21.08) in centimetres. This is mapped, via (5.1), to wheelco-ordinates of (-0.139, -0.570). Figure 5.8 shows the ART reconstruction of theelectron beam-induced emission, the emission point given by the sine-fit tech-nique and the computed magnetic axis.

The strong ‘tail’ moving to the right of the peak of the reconstruction is anartefact due to the uncalibrated intensity values. The discrepancy between theposition of the reconstruction and the computed magnetic axis may be due toan error in (any or all of) xoff, yoff or θ0. Assuming θ0 is correct then matchingthe positions of the sine-fit result and the computed axis requires changing theoffsets to xoff = 102.89, yoff = −0.52. This is a shift of 1.64 cm and -0.52 cm respec-tively. Assuming that xoff, yoff are accurate, then making the positions coincidentrequires θ0 = −10.65◦, a change of 4.55◦. Note that the sine-fit result and thecomputed axis lay on circles whose radii differ by only 1.3 × 10−3, which whenscaled by rwh is 0.05 cm. Thus no change in xoff, yoff is required for this value ofθ0.

However, using the comparison of two points to resolve a discrepancy involv-ing three parameters is inconclusive. Further information is available to constrainthis problem by fitting the plasma emissivity reconstructions to the computedsurfaces as described in Chapter 7.


(a) (b)

Figure 5.8: (a) The tomographic reconstruction of the single electron beam, shown inwheel co-ordinates. (b) A close-up view of the reconstruction showing the peak of thereconstruction (+), the result of the sine-fitting technique (*) and the magnetic axis ac-cording to the HELIAC code (�), which has been mapped to wheel co-ordinates via (5.1).

5.2 Spatial Intensity Response

Knowledge of the spatial intensity response for each channel is required toobtain more reliable tomographic inversions. That is, the use of a measured spa-tial response should give better tomographic results than the assumption of line-integrals for the viewing chord.

When using viewing chords with a known spatial response, it is important todefine the chord positions (p, φ) more accurately as the centre-line of the viewingdirection. References hereafter to the chord position refer to the chord centre-line. The positions for all chords are given in vector form as (p, φ) with elements(pi, φi).

The spatial intensity response for each channel has been carefully measuredin situ. By placing a long, thin fluorescent light source—identical to those usedas permanent in situ calibration lights (see §4.3.1)—perpendicular to the viewingplane and rotating the viewing modules past it, a slice of the spatial intensityresponse for each channel is recorded. Repeating this procedure, with the fluo-rescent light in many positions in the viewing plane, allows a complete responsefor all channels to be constructed.

A series of measurements was performed during a vacuum-break opening ofthe heliac for a total of 92 light-source positions. The fluorescent tube was held in


a support arm which was attached to a vertical translation stage which itself wasmounted on a horizontal translation stage. This rig was mounted between theinner vertical field coil and the vacuum vessel wall. Both translation stages weredriven by stepper motors—independently controlled by twin SMC24B StepperController units—giving full two-dimensional control over the light source posi-tion. Figure 5.9(a) displays the nominal positions of the fluorescent tube as givenby the translation stage settings.

Using the position of the light source in the viewing plane (xwh, ywh) andthe position of the viewing chord, the intensity response can be mapped to chordco-ordinates using (5.2). The light source positions are determined using the sine-fitting technique (see §3.6). This is a first approximation using the design chordpositions which are denoted as (p0, φ0) and the wheel start angle of θ0 = -16.6◦.Figure 5.9(b) shows the positions of the light source determined using the sine-fitting procedure. Note that the inner-radial positions are absent from (b) sincethey reside at radii less than the lowest-valued p0 viewing chords. Data fromthese positions are excluded from further consideration in the response measure-ment. The irregularity in some positions are mostly likely due to a shifting of thefluorescent tube in its single-end support arm during prolonged use. Note theslight overall rotation of the light source positions in Figure 5.9(b).

(a) (b)

Figure 5.9: (a) Positions of the light source as given by the distances moved by the trans-lation stages. (b) Positions of the light source as determined by the sine curve fittingtechnique. Note that some positions closer to the centre of the wheel are not representedas a sufficient number of data points could not be obtained to resolve this location. Thesinograms produced at these positions hold little information on the spatial response.


The angle of the start position of the wheel rotation is adjusted by a linear-regression fit to the derived positions of the light sources such that they are, onaverage, aligned vertically. The start position is adjusted to -15.2◦ from the origi-nal estimate of -16.6◦, a change of +1.4◦.

To confirm the positions, arithmetic reconstruction tomography (ART) (see§3.2.1) was also used to reconstruct the light source emissivity from the sino-grams. Using the revised wheel start angle of -15.2◦, a total of 10 iterations wereused to converge to the solution image. Fixed-width line-integrals were usedin the computation, since the spatial response is not yet measured. Figure 5.10shows that the reconstructions compare favourably with the positions calculatedby the sine-fitting method.

Figure 5.10: Tomographic reconstructions of the light source positions using ART. Notethe smearing evident at smaller radial positions due to insufficient data contained in thesinogram to resolve the light source location.

With the positions of the light source determined, the intensity responses fromall sinograms are now mapped to chord co-ordinates—via (5.2) and using thedesign chord positions (p0, φ0). Figure 5.11 shows the raw responses from all


included sinograms in chord space for one of the channels. The (s, t) axes havebeen rescaled by rwh to give an indication of the dimensions.

Figure 5.11: Raw spatial intensity response of Channel 10 compiled from all sinograms.

A more accurate determination of the viewing chord positions can now bemade and this is done before normalising the different intensity levels in a re-sponse. A small correction is made to the design positions (p0, φ0). The centreof mass for each peak is plotted in chord co-ordinates and shown in Figure 5.12.Linear regression is applied to this data with the slope Δφi and the s-interceptΔpi used to derive the new chord positions (p′, φ′) with elements (the subscripti is excluded for clarity) given by

φ′ = φ0 − Δφ

p′ = Δp + p0 cos(Δφ) − sin(Δφ)√

1 − p20

(5.3)

The revised chord positions are used to evaluate corrected positions for thelight source using the sine-fitting method. Figure 5.13 shows the new light sourcepositions. Note that the differences are sufficiently minor that there is no need tofurther iterate the procedure.

The mapping of the intensity responses to chord co-ordinates is repeated us-ing the corrected positions of the light source and the viewing chords. Figure 5.14is a re-plot of Figure 5.12 with the corrections applied.


Figure 5.12: The centre of mass of intensity peaks for a single channel using initial es-timates for the viewing chord and light source positions. The linear regression fit lineshows the minor correction to be made to the position of the chord. Most angular correc-tions are generally < 0.10◦.


Figure 5.13: The corrected estimates for the light source positions (◦), with the first esti-mates also plotted (�).


Figure 5.14: The centre of mass of peaks from a channel’s intensity response using a cor-rected chord position (p′, φ′) and the revised light source positions. The linear regressionfit line shows an even and low-spread scatter of data about s = 0. See also Figure 5.12.

Since the response measurements were taken over a timescale of days, the in-tensity levels between sinograms vary and a normalisation is required. Due tothe conservation of the total light flux along the viewing chord, the area underthe curve of the plot of intensity versus s is used as the metric. This normalisationis a first approximation—though the intensity peak also varies in the t-direction,it is assumed to change slowly compared to the change in the s-direction. Fig-ure 5.15 shows the trajectory of the light source (for an arbitrary light sourceposition) in chord co-ordinates. Only when the trajectory of the light source (inchord co-ordinates) passes completely from one side of the chord to the othercan the responses be normalised in this way. Thus a discarding of the centralportions of a response is required (see Figure 5.16). All responses discussed here-after are normalised as described above and Figure 5.16 shows the normaliseddata displayed in Figure 5.11 with the corrections discussed above.

A complete response for each channel is obtained by interpolation betweenintensity peaks and smoothing. The data points of the intensity peaks for a givenchannel are allocated to equi-spaced bins in the s-direction. This allocation cre-ates a profile in the t-direction for a given s-bin. This profile is interpolated onto auniform grid in the t-direction and then smoothed. This fills in the removed cen-tral portions and reduces the noise in the response. The complete set of t-profiles


Figure 5.15: Trajectory of the light source in chord co-ordinates (s, t). The final spatialresponse for Channel 10 is shown for comparison.

Figure 5.16: Normalised spatial intensity response for Channel 10.


then form the response for the channel. Figure 5.17 shows a typical t-profile withthe interpolated and smoothed data also displayed.

Figure 5.17: The t-profile at s = 0 for Channel 5 showing the raw data(*), the interpolateddata (dashed line) and the final smoothed profile (solid line).

Once a response has been interpolated and smoothed it is re-normalised bysetting the area under the curve for the response in the s-direction to unity. Fig-ure 5.18 shows the completed—smoothed, interpolated and re-normalised—responsefor a channel.

The final intensity responses for Modules 1 through 5 are presented in Figures5.19 through 5.23, respectively.


Figure 5.18: Smoothed and interpolated spatial intensity response for Channel 10.


Figure 5.19: Spatial intensity response for Module 1 comprising, from left to right, Chan-nels 0 to 10.


5.3 Intensity Calibration

Since the reliability of the tomographic reconstructions depends on a trust-worthy measurement set, great care has been taken to accurately calibrate the rel-ative intensity response of the ToMOSS system. To efficiently calibrate all aspectsof the system—from light collection through to light detection—which affect themeasured light intensity, it is useful to perceive the system as a mapping of vec-tors. The input vector consists of the originally emitted light levels in the viewingregion. This can then be mapped by using a single matrix—which is determinedby system components that affect the intensity level—to a vector representingthe measured signals. The determination of this intensity calibration matrix isthe subject of this section.

In formulating an understanding of the system’s response to light emitted inthe viewing region, it is instructive to follow the path through to the detectedsignal. This can be represented diagrammatically as shown in Figure 5.24.

DetectorSource ResponseSpatial

Transmission Spectrometer

Figure 5.24: Components of the path from the light collection area to the recorded signal.

5.3.1 Description and Consideration of Components

Light Source

Consider an isotropically-radiating light source which is located in the view-ing region of the ToMOSS system. The light may be generated by plasma or byartificial means (as in the case of the in situ calibration lights) with its intensitytaken to be isotropic. In the case of the plasma, the normal to the viewing planeis approximately parallel to the magnetic fields lines so that the radiation can beassumed to be isotropic. In the case of an artificial source, an isotropic radiator ischosen. It is assumed that there is no prior knowledge of the location or extent ofthe light source, which is denoted P(r).

Spatial Response

This component describes the spatial dependency of the light collection sys-tem. Consider a set of generalised pixels {bj} covering the viewing region, eachwith a weighting (or value) of xj. The light source can be represented as a linear


combination of the pixels, that is,

P(r) = ∑j

xjbj (5.4)

The spatially-integrated light flux yi can be expressed as the sum of the pixelvalues weighted by a response matrix, as outlined in §3.1. That is,

yi � ∑j

Rijxj (5.5)

The form of R is determined by the viewing geometry and the pixel type used.It is useful to keep the pixels in a general form so that different types—such asorthonormal basis functions or spatial (traditional ‘picture’) pixels—maybe usedas best fits the scenario.

Transmission

This component includes the lenses and optical fibres up to the MOSS spec-trometer. Light intensity is reduced after passing through the lenses, the opticalfibres and the fibre joins (patching). Over the spectral range selected by the linefilter in the MOSS spectrometer (≈1 nm), it is assumed that the transmission isindependent of the wavelength λ.

Spectrometer

This section accounts for the optical components of the MOSS spectrometer.These include the imaging lenses, polarisers, beam-splitter, LiNbO3 crystals, half-wave plate and spectral line filter. Light levels are reduced, to varying degrees,by all of these items. Vignetting losses depend on the position of the optical fibrein the input array and thus also vary between channels. The transmission of thespectral line filter for off-axis rays depends on the wavelength of the light and theangle of incidence θ. Since the range of angles Δθ is fixed for any given channelthe flux loss due to angle-of-incidence effects can be included in other attenua-tions which also vary between the individual channels. Thus the transmission ofthe line filter for each channel is described only as a function of wavelength. Forthe ith channel it is given as

Λi = Λi(λ) (5.6)

The overall transmission—which is associated with the described compo-nents, but which is not due to the line filter—is denoted as Ai, where Ai ≤ 1.It is convenient to retain a separation between Λi and Ai for dealing with thedifferences between quasi-monochromatic and white light sources, as will be ex-plained later.


Detector

This component includes the 8x8 multi-anode photomultiplier tube—whichproduces a current signal for each channel—and the associated transimpedanceamplifiers—which convert those signals to voltages. Light flux incident on thesemi-transparent photo-emissive cathode (photocathode) causes the release ofelectrons which are focused onto a metal-channel dynode structure. The dynodesmultiply the electron number by a secondary emission process. The multipliedelectrons are collected by an anode which produces the current. The light fluxincident on the photo-cathode is denoted as Φi, where

Φi = ΛiAiyi (5.7)

The spectral response of the photocathode is assumed to be flat over the in-cident wavelength range. The detector output is a function of the quantum effi-ciency of the photo-cathode and the gain of the dynode-chain electron cascade.The output of the detector is also affected by the angle of incidence of the lightonto the photocathode, being reduced as the angle from the normal is increased[Shimizu et al., 2001]. Recalling that Δθ is fixed for a given channel, the quan-tum efficiency, the electron multiplication and the angular dependence of thephoto-cathode sensitivity can be combined into a single effective gain. This gainis incorporated into the equations as explained shortly.

Matters with the detector are complicated by the issue of cross-talk betweenchannels. The manufacturer’s specifications [Ham, 2000] indicate that approxi-mately 2% inter-channel cross-talk will be present. Cross-talk can also be pro-duced by optical reflections in the MOSS spectrometer, most notably by the linefilter. Misalignment of the detector with the image of the optical fibres could alsoproduce cross-talk.

Whatever the mechanism, for a set of incident fluxes {Φj} the current deliv-ered at the ith detector output ςi is given by

ςi = ∑jBijΦj (5.8)

where Bij is a diagonally-dominant conversion-efficiency matrix and it is as-sumed that the cross-talk is linear in the operating regime of the detector.

The transimpedance amplifier converts the anode current to a voltage signalwith the transformation/gain denoted by mi. Though there is electrical cross-talk between the amplifiers (it has been documented as < 0.02% [Ele, 2001]†), itis neglected hereafter. The final voltage level is denoted as Si where

Si = miςi (5.9)† Actual measurement gives the worst case cross-talk co-efficient as ∼ 10−6 − 10−7 for a

500kHz bandwidth.


Combining (5.7) with (5.8) and (5.9) gives

Si = mi ∑jBijΛjAjyj (5.10)

This can be conveniently written in matrix form as

S = mBΛAy (5.11)

and if there are n channels in the system then m, Λ and A are n × n diagonalmatrices and B is a diagonally-dominant n × n matrix. Here S and y are themeasurement and input vectors, respectively.

Since Λ and A are diagonal they can commute, thus enabling the wavelength-independent components to be collected into a single matrix G = mBA so that

S = GΛy (5.12)

Here G is known as the calibration matrix and Λ the line filter response. Thelight input vector y can thus be recovered by

y = Λ−1G−1S (5.13)

where G−1 = A−1B−1m−1.When considering the attenuation due to the line filter, the transmitted inten-

sity is given by

I ′ =∫

I(λ)Λ(λ)dλ (5.14)

where I is the incident intensity.In practice the calibration light source is white giving I = constant for wave-

lengths in the range of the filter. Though the centre of the transmission curve forthe line filter Λ(λ) still shifts as a function of θ, the value of the integral

∫Λ(λ)dλ

is approximately independent of the line centre and is regarded as constant in thissituation. Since the intensity calibration is relative†, we can set Λi = 1 and thewavelength dependence of I ′ is removed. Hence, Λ becomes the unit matrix forthe case of white light input. For quasi-monochromatic light input, in general,Λi = 1 and so Λ is measured, as discussed in §5.3.4.

5.3.2 Measurement of the Calibration Matrix

The columns of G can be revealed through the use of point input vectors. Forwhite light input (Λi = 1, ∀i), a point input vector—where the jth element is unityand all others are zero—reduces (5.12) to

Si = Gijyj (5.15)

† That is, the filter still attenuates the incident light but does so for all channels equally.


Thus the measurement vector S can be used to give the jth column of G and, sinceG is an n × n matrix, n linearly-independent point input vectors are required tocompletely describe it.

A straightforward way to produce the required point input vectors is to il-luminate each channel individually by rotating the viewing chords past the insitu light sources. Since there are a high number of intersections between view-ing chords, the careful placement of multiple in situ light sources and a block-ing baffle are employed to prevent simultaneous illumination of more than onechannel. Three light sources (see §4.3.1) are located so that, over the full rota-tion range of the wheel, every viewing chord (channel) will intersect at least onelight source. Most channels will view a light source in the far-field and thenagain in the near-field as the rotation of the wheel progresses. A baffle—a widestrip of diffuse, blackened stainless steel placed between the lenses and the lightsources—is used to block the near-field view. This reduces the number of chordsintersecting a light source simultaneously, though, in the case of several chan-nels, some intersections persist. To illustrate this, Figures 5.25 and 5.26 show thesinograms recorded as the viewing chords are swept past the lower light source,without and with the blocking-baffle in place, respectively. Note that the inten-sity scales of the two figures are different and are logarithmic so that the inter-channel cross-talk is visible. The large amount of cross-talk is due to the detectoralignment having not being optimised at that stage.

The conflict due to persistent intersections is resolved, in part, by only record-ing data in elements of S which are regarded as susceptible to cross-talk. Sus-ceptible elements include those which are immediate neighbours in the detec-tor array to the channel being illuminated. An additional element—which isdiagonally-symmetric in the detector array to the illuminated channel—is alsoincluded. This diagonally-symmetric element contains cross-talk due to an opti-cal reflection† in the spectrometer. The mapping of the input fibres to the detec-tor elements is chosen to reduce the conflict between simultaneously illuminatedchannels which may also be cross-talk elements of each other. Figure 5.27 showsa layout of the detector array and illustrates the restriction scheme. For instance,if element 45 is the channel being illuminated (shown in dark green) then its as-sociated ‘cross-talk’ channels are shown in light green. Element 20 is included asthe ‘optical’ cross-talk channel.

In practice, the point input vector for a given channel is obtained from thearea under the curve of the intensity response in the s-direction only. This area isdenoted as AIs and is determined by mapping the calibration sinogram to chordco-ordinates and integrating I over s with a suitable threshold at the edges. Thisthresholding minimises the contribution from noise.

† An inverted image of the original light incident on the photocathode.


Figure 5.25: A sinogram recorded without the blocking-baffle with most channels havinga second view of the light source. The intensity scale is logarithmic to show the inter-channel cross-talk.

A point input vector is created for as many of the light sources as each channelviews and the averaged results recorded in the final matrix, reducing the effect ofnoise on the measurement. Variations in the intensity output between differentlight sources are also accounted for. For the purposes of analysis G is sometimesdecomposed into a cross-talk matrix mB, with diagonal elements normalised tounity, and a relative sensitivities matrix A.

5.3.3 Calibration Matrix Reliability

This section addresses issues which may affect the measurement and reliabil-ity of the calibration matrix.

Light Source Output Drift

Figure 5.28 shows a plot of the intensity output of an in situ light source overtime. Maximum variations in the level are on the order of 3%. The upper trace(blue) is the same source approximately 30 minutes after the lower trace wasrecorded. For comparison, the red trace shows the response of this channel dur-ing a calibration scan (rescaled here to match the overlap the traces). The outputtraces are smoothed with a width equal to the typical number of points used in


Figure 5.26: A sinogram recorded with the blocking-baffle installed with channels hav-ing the second view of the light source blocked. The intensity scale is logarithmic anddifferent to that of Figure 5.25. The blue dashed line shows a channel with no conflictin other channels for which cross-talk is present. The red dashed line indicates channelswith a persistent conflict.


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Figure 5.27: The layout for the 8x8 detector elements. For an illuminated channel, whichis shown in dark green, it has associated ‘cross-talk’ channels–those elements which aresusceptible to cross-talk—which are shown in light green. Immediate neighbours areincluded as is element 20, due to an optical reflection in the system.


the integration which derives AIs. This drift in the light source intensity can-not be compensated for without the addition of a separate monitor for the lightsources. The flow-on consequences mean that AIs for channel viewing the samelight source may be inaccurate by up to 3%. For channels which view more thanone light source, the effect of this uncertainty is reduced.

Figure 5.28: The intensity output of an in situ light source over time. The upper trace(blue) is the same source approximately 30 minutes after the lower trace was recorded.The red trace shows the response of this channel during a calibration scan (rescaled tooverlap both traces). The maximum variation in the output level is on the order of 3%.

AIs as a Point Input Vector

The conservation of the light flux in a plane perpendicular to the length of theviewing chord means that the area under the intensity response in the s-directionis the same for all t-values. However, the trajectory in chord co-ordinates alsovaries in the t-direction as shown in Figure 5.15. It is assumed that the change inthe intensity response in the t-direction is slow compared to the change in the s-direction. Figure 5.29 shows the intensity response of a channel in the t-directionand an overplot of the measured spatial response of that channel at s = 0. Notethat no near-field views, for t ≤ 200 mm where the change in response is greatest,occur due to the blocking baffle.


Figure 5.29: The intensity response of a channel, as it is rotated past an in situ light source,in the t-direction. The measured spatial response at s = 0 is also shown (dashed line).

Reproducibility of Matrices

Successive measurements of the relative sensitivities and cross-talk matricesindicate the level of noise in the measurement process. Figure 5.30 shows thevariation in rescaling values†and Figure 5.31 shows the difference between twocross-talk matrices. The root-mean-square of uncertainties in the rescaling valuesis 2%.

Magnetic Field Effect on Relative Sensitivities

The effect of the H-1NF magnetic field on the response of the photo-multipliertube has also been checked. The mean signal level for a 30 ms period of thecalibration laser light is recorded with different magnetic field strengths. Forfield strengths of 0.043 T (600 A in the main coil set) and 0.13 T (1800 A) there wasno discernible systematic change in the signal levels across all channels in thedetector array.

† The inverse of the diagonal elements of the relative sensitivities matrix


Figure 5.30: The values for two successive measurements of the relative sensitivities ma-trix.

Figure 5.31: The difference between two sequential measurements of the cross-talk ma-trix. Recall that the diagonal elements of the cross-talk matrix are normalised to unity.


Result of Calibration Matrix Application

Removal of cross-talk contamination from the measurements is importantdue to the cumulative nature of this problem. That is, though the inter-channelcross-talk is specified as ∼ 2%, this means that a given channel may have 10–18% of the signal contributed by its associated cross-talk channels. Figure 5.32shows the result of applying the calibration matrix G to the intensity responsesfor two of the channels in a calibration sinogram. The cross-talk, which appearsas a bumpiness on the baseline of the original responses, is successfully removedfollowing the application of the calibration matrix.

Figure 5.32: Sample data chosen from the response of two different channels during thein situ calibration scan. The cross-talk is manifest as bumps on the baseline of the traceson the left. The traces on the right show the clean removal of the cross-talk.

5.3.4 Determination of the Line Filter Response

The filter used to select the spectral line is a Fabry-Perot etalon type and thusthe transmission curve, as a function of wavelength, has a Lorentzian lineshape.


Figure 5.33 shows the measured transmission curve of the line filter along withthe best fit Lorentzian. The asymmetry in the line-fitting at the wings of the curvemay be due to a small misalignment or focussing error in the detector array ofthe recording spectrometer.

Figure 5.33: The transmission curve for the 488 nm centre wavelength spectral line filteras measured with LARRY is shown. A best fit Lorentzian lineshape is also shown alongwith nearby spectral lines emitted for an argon discharge in the H-1 heliac. The spectralline near 486 nm is the Hβ emission.

The passband for rays with non-normal angles of incidence is altered suchthat there is a blue-shift of the centre transmitted wavelength, a decrease inthe maximum transmission and a broadening of the bandwidth [Lissberger andWilcock, 1959; Ingesson et al., 1995].

For a Fabry-Perot filter the centre wavelength of the transmission curve, as afunction of the incident angle, is

λc(θ) = λ0

√1 −

[n0

neff

]2

sin2 θ (5.16)

where λ0 is the centre wavelength for normal incidence, n0 is the refractive indexof the external medium, neff is the effective refractive index of the filter and θ isthe angle of incidence.

This formula, however, fails to take the other effects into account, nor thespread of rays for a given channel, and the line filter transmission is measured


Figure 5.34: The line filter response as recorded by the 8x8 multi-anode photomultipliertube, with detector element numbering starting at zero.

in situ. An in situ measurement also accounts for any slight misalignment in theorientation of the filter. The measurement is made by taking the ratio of the trans-mission of white light to the selected wavelength (typically 488 nm). The second(perpendicular) input port of the spectrometer is used so that the componentsremain undisturbed for this measurement. Figure 5.34 shows the line filter re-sponse shown as recorded by the 8x8 detector array. The values for each channelare then written to the appropriate diagonal elements of Λ.

system calibration and characterisation

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