Wall reflection modeling for charge exchange recombination spectroscopy (CXRS)

measurements on Textor and ITER

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2010 Plasma Phys. Control. Fusion 52 125006

(http://iopscience.iop.org/0741-3335/52/12/125006)

Download details:

IP Address: 133.5.177.210

The article was downloaded on 04/11/2010 at 02:34

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING PLASMA PHYSICS AND CONTROLLED FUSION

Plasma Phys. Control. Fusion 52 (2010) 125006 (14pp) doi:10.1088/0741-3335/52/12/125006

Wall reflection modeling for charge exchangerecombination spectroscopy (CXRS) measurementson Textor and ITER

Santanu Banerjee1, P Vasu1, M von Hellermann2 and R J E Jaspers3

1 Institute for Plasma Research, Bhat, Gandhinagar 382 428, Gujarat, India2 FOM Institute for Plasma Physics, Rijnhuizen, The Netherlands3 Applied Physics Department, Eindhoven University of Technology, Eindhoven, The Netherlands

E-mail: sbanerje@ipr.res.in

Received 8 July 2010, in final form 24 September 2010Published 3 November 2010Online at stacks.iop.org/PPCF/52/125006

AbstractContamination of optical signals by reflections from the tokamak vessel wall is amatter of great concern. For machines such as ITER and future reactors, wherethe vessel wall will be predominantly metallic, this is potentially a risk factorfor quantitative optical emission spectroscopy. This is, in particular, the casewhen bremsstrahlung continuum radiation from the bulk plasma is used as acommon reference light source for the cross-calibration of visible spectroscopy.In this paper the reflected contribution to the continuum level in Textor andITER has been estimated for the detection channels meant for charge exchangerecombination spectroscopy (CXRS). A model assuming diffuse reflection hasbeen developed for the bremsstrahlung which is a much extended source. Basedon this model, it is shown that in the case of ITER upper port 3, a wall with amoderate reflectivity of 20% leads to the wall reflected fraction being as highas 55–60% of the weak signals in the edge channels. In contrast, a completebidirectional reflectance distribution function (BRDF) based model has beendeveloped in order to estimate the reflections from more localized sources likethe charge exchange (CX) emission from a neutral beam in tokamaks. Thelargest signal contamination of ∼15% is seen in the core CX channels, wherethe true CX signal level is much lower than that in the edge channels. Similarvalues are obtained for Textor also. These results indicate that the contributionsfrom wall reflections may be large enough to significantly distort the overallspectral features of CX data, warranting an analysis at different wavelengths.

(Some figures in this article are in colour only in the electronic version)

0741-3335/10/125006+14$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

1. Introduction

For the ITER tokamak, a charge exchange recombination spectroscopy (CXRS) diagnosticbased on a 100 keV/amu diagnostic neutral beam (DNB) of hydrogen has been conceptualizedas a crucial and viable path for measuring key fusion plasma parameters, such as the helium ashconcentration, ion temperature, plasma rotation and the low-Z ion density [1–3]. Achievementof the objectives of the CXRS relies upon the determination of local absolute intensities of lightfrom various impurities and the hydrogen beam. In addition, chord integrated measurementsof continuum light serves as a diagnostic of a line-averaged effective ion charge and as a cross-reference for absolute calibration. It has been observed that reflection of light from the vesselwall often contributes to a significant part of such chord integrated signals [4, 5]. Hence toobtain prior estimates modeling of the reflected light level in the CXRS diagnostic is clearly anecessary task.

In a tokamak the bremsstrahlung (BL) radiation emitted from the entire plasma volume,as well as the CX emission from regions along the beam, can get reflected into various linesof sight . In this work the ITER and Textor tokamaks, representing two different situations,have been considered and reflected contribution both in the case of BL continuum and CXsignal has been estimated. The BL continuum emission in ITER is a few orders of magnitudehigher than the weaker CX signal riding on top of it. Also, the electron density profile and thecontinuum emissivity profile have been estimated [3] to be quite flat from the core to the edge(however, it has been shown in recent studies that peaked density profile can also be obtainedin the ITER standard scenario with low collisionalities [6]). In TEXTOR the profiles are morerounded. The DNB used for CXRS in ITER is a radial beam while the beam in TEXTOR istangential.

The reflections of BL and CX emissions have been modeled and analyzed differently. Adiffuse Lambertian reflection is used for the BL emission which is a broad source. Lambertianreflection should be a reasonable approximation because the net effect of overlap among allthe specular lobes will not differ much from a diffusely reflected light pattern [4, 5, 7]. On theother hand, the CX emission is highly localized, originating from regions only along the beam.Due to a significant attenuation of the beam density along the beam path the CX emissionstrength from the edge is a few orders of magnitude higher than that from the core. Hence, toestimate the reflected light from the CX, a full bidirectional reflectance distribution function(BRDF) model [8–11] is necessary. Bidirectional reflectance is a measure of the reflectedintensity in a particular direction given the energy incident from another specific direction.

The emissivities and chord integrated signal strengths for BL and CX signals are obtainedusing the ‘Simulation of Spectra’ package [12]. Section 2 describes the theoretical backgroundof the BL and CX emissions and section 3 contains the modeling methods for estimating thereflected contribution for both BL and CX signals. Sections 4 and 5 describe the results anddiscussion, respectively.

2. Theoretical background

2.1. Continuum emission

The BL photon flux [11] at wavelength λ may be expressed as

�brems = B

λ· gff · 〈Zeff〉

[∫dr · n2

e(r)

T1/2

e (r)

], (1)

2

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

where

B =[

e2

4πε0

]316

3√

2hc3m3/2e

√π

3

and gff is the free–free Gaunt factor. ε0, c and h are the permittivity of free space, velocity oflight and Planck’s constant, respectively. 〈Zeff〉 is the line of sight (LOS) averaged effectiveplasma ion charge. ne, Te, me and e are the usual notation of electron density, temperature,mass and charge, respectively.

2.2. CX emission

Due to the charge exchange reaction between the neutral hydrogen beam and the impurityions of density nz photons of wavelength λ0 are emitted. The active charge exchange photonflux [11] centered at λ0 and of width δλdi is given by

�CX(λ) = 1

4πnz · 1√

πδλdiexp{−(λ − λ0)

2/δλ2di}

∑E

〈σv(E)〉CX

∫nb(E, s) ds. (2)

Here the integration over ds = ds(x, y, z) covers the intersection of a LOS with the neutralbeam profile. nb(E, s) is the local density of neutral beam species with energy E.

∑E denotes

the summation over various energy components that may be present (as in the case of positiveneutral beams) in the neutral beam taking into account the fact that the effective emissionrate 〈σv〉CX is dependent on E. The width of CX line is assumed to be a Gaussian. δλdi

is determined by the instrumental width and the ion temperature prevailing over the zone ofintersection. The neutral beam profile perpendicular to the beam direction may be describedby a Gaussian profile:

nb(x, y, z) = P

eEvb

1

πw2beam(z)

exp

{− x2 + y2

w2beam(z)

}. (3)

Here the z-axis is assumed to be along the neutral beam and x- and y-axes are perpendicular tothe beam. Circular beam cross section is assumed and hence, wx = wy = wbeam. In general itis given by the product of divergence angle and the distance of the virtual neutral beam pointsource from the magnetic axis. Here, however, a constant beam size is assumed. P , E and vb

denote the local beam power, energy and velocity, respectively.

3. Modeling methods

3.1. Viewing geometry for TEXTOR and ITER CXRS diagnostics

The CX diagnostic setup on Textor and ITER is shown in figures 1(a) and (b). For Textor, theLOS for the Textor pilot geometry on the co-direction neutral beam 1 has been considered.For ITER, the LOS for the upper port number 3 has been analyzed. For both the machines20 lines of sight, numbered in ascending order from core to edge, have been considered. Thevacuum vessel of ITER has been approximated as a torus with elliptical cross section and inclose proximity to the plasma volume. In figures 1(c) and (d), the Textor equatorial (standard)and the ITER equatorial port 3 lines of sight are shown.

3.2. Diffuse reflection model for continuum emission

Intersection of each LOS with the vessel wall (i.e. the coordinate of the wall spot) is determinedby solving a ray–torus intersection problem. To calculate the total light arriving at the wall

3

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

Figure 1. (a) Viewing geometry of Textor pilot CXRS diagnostic on the co-direction neutral beam1 showing the lines of sight. (b) Viewing geometry of Textor standard CXRS diagnostic on the co-direction neutral beam 1 showing the lines of sight. (c) Viewing geometry of the CXRS diagnosticon ITER upper port 3 showing the lines of sight on the DNB. (d) Viewing geometry of the CXRSdiagnostic on ITER equatorial port 3 showing the lines of sight on the DNB.

spot, we need to launch a number of view lines into the plasma volume at elemental intervals.It is convenient to start these view lines in a polar coordinate system with the surface normal asthe z-axis. If we use a finite angular grid of small angle increments �ϑk and �ϕk (∼0.025π

radians) the elemental plasma volume dV at a distance s from the wall point, along the ϑk , ϕk

direction of the kth view line is given by

dV = s2 ds sin ϑk dϑk dϕk. (4)

An area dAwall on the wall subtends a solid angle d = dAwall cos ϑk/s2 at the emitting

volume dV . Hence, the total amount of light from the elemental volume dV incident on perunit area of the wall at the intersection point is given by

dL = εk dV d/dAwall

or

dL = εk ds sin ϑk cos ϑk dϑk dϕk (photons m−2 s−1 Å−1). (5)

Here, εk (photons m−3 s−1 Sr−1 A−1) is the volume emissivity of dV . BL volume emissivitiesof Textor and ITER are shown in figure 2. It is assumed that the fan of LOS terminates on awall region that is uniform in the sense that a single dimensionless reflectivity parameter R

defines the fraction of the incident light that is backscattered into a hemisphere in a Lambertiandistribution [12–14]. The geometry is illustrated in figure 3.

4

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

7

8

9

10x 10

19

Ele

ctro

n d

ensi

ty (m

-3)

0

10

20

30

Ele

ctro

n te

mp

erat

ure

(keV

)

0 0.2 0.4 0.6 0.8 11

2

3x 10

19

Ele

ctro

n d

ensi

ty (m

-3)

0

1

2

Flux index ρρρρ

Ele

ctro

n te

mp

erat

ure

(keV

)

a(2)

a(1)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5x 10

14

Tex

tor

Bre

mss

trah

lun

g e

mis

sivi

ty (P

h/m

3 /Sr/

s/A

)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10x 10

14

Flux index ρρρρ

ITE

R B

rem

sstr

ahlu

ng

em

issi

vity

(Ph

/m3 /S

r/s/

A)

Textor

ITER

b

Figure 2. (a) Electron density and electron temperature profiles of ITER (a(1)) and Textor (a(2)),respectively. The dotted lines represent electron density and the solid lines represent electrontemperature. (b) BL emissivity profiles on the poloidal cross section are shown for Textor and ITER.

Hence, the reflected BL light from a unit area around the wall spot towards a LOS makingan angle ϑ0 with the wall surface normal at the wall spot is given by

I sumcont,refl = R

πcos ϑ0

∑s

π/2∑ϑk=0

2π∑ϕk=0

εk ds sin ϑk cos ϑk dϑk dϕk

(photons m−2 Sr−1 s−1 Å−1). (6)

Here∑

s is the integration along a view line of length s. At each point along the integrationpath s the local torus coordinate is transformed into the corresponding magnetic flux surface

5

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

Spectrometer slit

N

0

Projection of the slit on the neutral beam in plasma

dAwall

Optical axis

Spectrometer slit

N

θ0

Projection of the slit on the neutral beam in plasma

dAwall

Optical axis

N

dϑϑϑϑ

dϕϕϕϕ

ϑϑϑϑ

ϕϕϕϕ

dr

r sinϑϑϑϑ dϕϕϕϕ

dAwall

dVN

dϑϑϑϑ

dϕϕϕϕ

ϑϑϑϑ

ϕϕϕϕ

dr

r sinϑϑϑϑ dϕϕϕϕ

dAwall

dVN

dϑϑϑϑ

dϕϕϕϕ

ϑϑϑϑ

ϕϕϕϕ

dr

r sinϑϑϑϑ dϕϕϕϕ

dAwall

dV

Figure 3. Projection of the spectrometer slit in the plasma at the CX emission zone (path of theDNB) and thereby its projection on the vessel wall is shown. θ0 denotes the angle between thenormal on the surface at the point of intersection of the LOS and vessel wall and the LOS itself.The orthogonal coordinate system used to quantify the elemental volume dV is shown in the insetat the bottom left corner.

index and thus the local BL emissivity ε (ρ) is established. At the same time it can be checkedif the opposite wall is reached. The wall surface is at an angle ϑ0 to the optic axis. Hence, thecontribution to the detector along the LOS is over an etendue Aspecspec/cos ϑ0, where Aspec

is the area and spec is the solid angle of acceptance of the detector (or spectrometer entranceslit). Hence, the amount of reflected light entering the detector is given by

Scont,refl = R

πAspecspec

∑s

π/2∑ϑk=0

2π∑ϕ=

k 0

εk ds sin ϑk cos ϑk dϑk dϕk (photons s−1 Å−1),

or equivalently, in terms of spectral radiance at the detector:

IBLR = R

π

∑s

π/2∑ϑk=0

2π∑ϕ=

k 0

εk ds sin ϑk cos ϑk dϑk dϕk (photons m−2 Sr−1 s−1 Å−1).

(7)

3.3. BRDF model for CX emission

The neutral beam in the plasma (from edge to core) has been approximated as atrain of small light sources each of length �lsource and of CX line emissivity εCX,k

6

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

NL

H

V

PW

Neutral Beam

wbeam2

∆∆∆∆ source

LOS direction

Elemental CX Volume

NL

H

V

PW

θ

θ

θ Neutral Beam

wbeam2

∆∆∆∆ source

LOS direction

Elemental CX Volume

l

Figure 4. Viewing geometry with respect to a point on the wall and any arbitrary point light source.V and L are the viewing direction and the direction of the light source, respectively. H is the unitangular bisector of V and L. N is the surface normal at that wall point.

(photons m−3 Sr−1 s−1 Å−1). A representative elemental volume emitting CX light is shownin figure 4. The charge exchange emission reflected from the wall into a particular LOS iswritten as

SCX,ref = �lsourcew2beamAspecspec

∑k

εCX,k

RkBRDF

d2k

(N · L) (photons s−1 Å−1),

or equivalently, in terms of spectral radiance at the detector:

ICXR = �lsourcew

2beam

∑k

εCX,k

RkBRDF

d2k

(N · L) (photons m−2 Sr−1 s−1 Å−1). (8)

The vectors representing different directions are shown in figure 4. Here, dk is the distancefrom the wall spot to the kth elemental CX source and εCX,k is its volume emissivity. Rk

BRDFis the BRDF for the kth elemental CX source with respect to the LOS. BRDF, as formulatedby Cook and Torrance [11], is well described in the literature and only the salient features aregiven here. It is represented as the sum of a specular part and a diffuse part:

RBRDF = sRs + dRd, (9)

where s + d = 1 and

Rs = DGF

π(N · L)(N · V )Rd = F0

π. (10)

Here D is the Beckmann distribution function characterizing the micro-facets on the reflectingwall surface and is calculated as D = 1

m2 cos4 αexp −[(tan α)/m]2, α being the angle between

the surface normal and the bisector of the angle between the incidence and the viewing vectors(see figure 4). The spread of the specular component depends on the root mean square slope m.Small values of m signify gentle facet slopes and give a distribution that is highly directionalaround the specular direction, while large values of m imply steep facet slopes and give adistribution that is spread out. G is a geometrical factor responsible for masking of light,

G = min

{1,

2(N · H)(N · V )

(V · H),

2(N · H)(N · L)

(V · H)

}.

7

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

The Fresnel term F (and F0) denotes the reflection of light from each smooth micro-facet,which also determines Rd.

F can be theoretically obtained from the Fresnel equation [15] if refractive index (n) andthe extinction coefficient (k) of the surface are available. Practically, F (and hence Rd) iscalculated from F0, the reflectivity of the material for a mirror-like finish with illumination atnormal incidence. This is discussed in detail by Cook and Torrance [11] and is summarized inthe appendix. Experimentally measured F0 values for several materials are readily availablein the literature (see [11] and references therein).

The F0 value for 832 MV austenitic steel has been used [16] in our calculations. The valuevaries from ∼55% to 65% in the wavelength range 4000–6000 Å (64% for the wavelength of5290 Å, specific to the CVI (8-7) transition considered here) and is not very different forsimilar varieties of steel. It may be noted that the reflectance of polished beryllium, whichwill predominantly constitute the plasma-facing surfaces in ITER, is ∼50% [17] for the samewavelength range.

4. Results

4.1. Reflection of BL continuum radiation

Electron density and temperature profiles assumed in the case of ITER and Textor are shownin figure 2(a). Constant Zeff profiles have been assumed along the flux index ρ both forITER (Zeff = 1.6) and Textor (Zeff = 1.4). In ITER, the BL continuum emissivityprofile is rather flat at ∼9 × 1014 photons m−3 s−1 Sr−1 Å−1 from core to edge dropping offsharply beyond ρ (r/a) � 0.8 as shown in figure 2(b). The signal strength IS (intensity inphotons m−2 s−1 Sr−1 Å−1) seen by the 20 channels of upper port 3 are shown in figure 5(a).The effect of Lambertian walls with reflectivity of 10–50% has been investigated. Thelower value can be expected to be characteristic for materials like graphite, while the highestvalue could be typical of mirror-finished metallic surfaces. The amount of reflected lightIR (intensity in photons m−2 s−1 Sr−1 Å−1) in different channels of upper port 3 for a wallreflectivity of 20% (R = 0.2) is also shown in figure 5(a). In figure 5(b), the reflectedsignals are shown as a percentage of IS. The reflected contribution being comparable to orexceeding the signal in the edge channels is, of course, due to the low signal levels at thesechannels.

Similar results have been shown for the Textor pilot geometry (see figures 6(a) and (b)).In the case of Textor, the continuum profile falls off monotonically from a peak value of∼2 × 1014 photons m−3 s−1 Sr−1 Å−1 as shown in figure 2(b). It may be noted that in Textorpilot, the channels cover the region from ρ = 0 to 0.75. Most of the channels terminate on thecarbon limiter with a low reflectivity, but a few edge lines of sight end on the metallic wall.This needs to be taken into account for any comparison with recorded experimental data onTextor machine. Of course, for a quantitative comparison, wall reflectance parameters (ideallymeasured concurrently with the recorded CX spectra) would also be needed.

A comparison of the IR/IS ratio (i.e. the extent of contamination of signal by reflection)in Textor pilot and ITER upper port 3 for R = 0.2 is shown in figure 7(a). The high ratioin Textor pilot geometry for the edge channels is due to the fact that the signal IS for thesechannels is inherently weaker which in turn is due to the BL emissivity profile. On the otherhand, the reflected signal IR remains reasonably flat from the core to the edge. Interestingly, acomparison of results for the Textor equatorial (standard) view with the ITER equatorial port3 view for R = 0.2 shows a different pattern (figure 7(b)). While for the central channels, upto ρ = 0.7, the reflected contribution in both the cases amounts to 20–30%, it is in the case of

8

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7x 10

15

Flux index ρρρρ

Co

ntin

uu

m s

ign

als

I S a

nd

I R (P

ho

ton

s/m

2 /Sr/

s/A

)

Actual continuumReflected continuum

(a)

0 0.2 0.4 0.6 0.8 10

25

50

75

100

125

150

Flux index ρρρρ

I R/I S

(per

cen

tag

e)

R = 10%R = 20%R = 30%R = 40%R = 50%

(b)

Figure 5. (a) Actual and reflected continuum signals for ITER upper port 3 are shown. Reflectedsignals are calculated for a wall reflectivity of 20% (R = 0.2). (b) Percentage continuum reflection(IR/IS) for varying reflectivity of the vessel wall for the CX channels of ITER upper port 3.

ITER’s edge channels that the reflection overwhelms the true signal. This may be significantif the very edge channels (ρ > 0.8) are accessible and of interest.

4.2. Reflection of charge exchange radiation

The charge exchange signal has been considered for CVI (8-7) transition, centered at about5290.59 Å, for a 3% abundance of carbon ions in the plasma. The neutral beam power forTextor is assumed to be 1.5 MW [18] and 2.7 MW for ITER. The ITER value is based on theoriginal target values minus 25% losses in the beam-line components due to beam divergence[19–22].

A moderately smooth metal surface (m = 0.7), with 30% of the reflected light beingdiffusively scattered (d = 0.3) and the rest emerging in a specular lobe (s = 0.7), is considered

9

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4x 10

14

Co

ntin

uu

m s

ign

als

I S a

nd

I R (P

ho

ton

s/m

2 /Sr/

s/A

)

Flux index ρρρρ

Actual ContinuumReflected Continuum

(a)

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

350

400

Flux index ρρρρ

I R/I S

(per

cen

tag

e)

R = 10%R = 20%R = 30%R = 40%R = 50%

(b)

Figure 6. (a) Actual and reflected continuum signals for Textor pilot. (b) Percentage continuumreflection (IR/IS) for varying reflectivity of the vessel wall for the CX channels of Textor pilot.

(in the following we refer to the m = 0.7, d = 0.3 and s = 0.7 case as mds = 737 for brevity intext and figures). Its reflection pattern, shown in figure 8, is quite different from a Lambertian.The effect of reflection of CX emission from this surface on the CX signals for ITER andTextor is shown in figure 9 (mds = 737 curves). In the case of ITER, for the edge channelsthe reflected light (IR) is virtually negligible in comparison with the true CX signal (IS). Itdoes not exceed 15% even for the core channels which have small signals. For the Textor pilotgeometry IR varies from 4% to 12% of IS, the increase being towards the edge chords.

Figure 9 also shows the variations in the extent of contamination by reflection with thechanges in the values of the parameters m, d and s—both in the case of Textor and ITER. Alower value of m, which tends to confine the specularly reflected part into a narrower lobe, hasa reducing effect on the IR/IS ratio (mds = 537 curve) in ITER. But changing the relativeratios of the diffuse and specular parts (compare mds = 737 and mds = 755 curves) does

10

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

Flux index ρρρρ

I R/I S

(per

cen

tag

e)

Textor PilotITER Upper Port 3

a

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

Flux index ρρρρ

I R/I S

(per

cen

tag

e)

Textor Equatorial PortITER EquatorialPort 3

b

Figure 7. Comparison of the reflected continuum in the case of ITER and Textor CX views forR = 0.2. Textor pilot geometry has been compared with ITER upper port 3 in (a), while comparisonof Textor equatorial (standard) view with the ITER equatorial port 3 view is shown in (b).

not show any such readily identifiable trend from the core to the edge in either machine. Thevariation is due to a combination of geometry, emissivity profile effects and reflectance lobepattern, all playing their roles.

5. Discussion

Reflected BL continuum levels in the CX channels of ITER may be quite high (∼55% of thetrue signal), in particular for the edge channels, even for a wall with a moderate reflectivityof 20%. It appears that the reflected fraction would be similar in the case of Textor and ITERat least up to ρ = 0.75. As the ion temperature and hence the line broadening in the CXsignal will increase as we move from the edge to the core of the tokamak, it is possible that thereflected CX light from one region may distort the line shape recorded at another flux surface

11

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

(a) (b)

(c) (d)

Figure 8. BRDFs with different parametric values are shown in 3D. False color denotes the valueof the function itself. The incident direction and the mirror reflection direction are denoted bybroken line and solid line, respectively. (a) A pure Lambertian distribution, (b) a sharp specularBRDF with a predominant diffuse contribution; the parametric values are set as m = 0.2, d = 0.9,s = 0.1. (c) A similar BRDF with a somewhat reduced diffuse contribution; the parametric valuesare set as m = 0.2, d = 0.8, s = 0.2. (d) BRDF with a rather broad specular lobe for an equalcontribution from the diffuse and specular components; the parametric values are set as m = 0.7,d = 0.5, s = 0.5. Note that, in this case, there is a fair amount of backscattering as well.

and hence affect the inference of ion temperature. On the other hand, it may be the reflectedBL light level (rather than the true BL signal) that would determine the S/N (signal-to-noise)considerations of CX signal at the edge. So, the effect of reflection of both BL and CXlight on the true CX spectrum needs to be investigated as a function of wavelength. Further,the reflected CX contribution due to other simultaneous neutral beams (the counter directionneutral beam 2, in TEXTOR and the heating neutral beams (HNB) in ITER) would need to beincluded.

For modeling the wall reflection of light in a tokamak, assumption of Lambertian reflectionmay suffice to obtain a good estimate of the severity of the problem of signal contamination.It has the advantage of low computational expense. But the use of BRDF scheme of modelingthe reflection certainly allows more detailed predictive calculations. It may allow the recoveryof correct spectrum from reflection-contaminated spectra from experiments, since only a smallnumber of parameters are involved in characterizing the reflecting surface.

Acknowledgments

The authors would like to thank all the members of the FOM team working at Textor underthe Trilateral Euregio Cluster for their generous help. One of the authors (SB) wouldlike to thank Dr E Delabie, Dr Joydeep Ghosh and Dr Vinay Kumar for many usefuldiscussions.

12

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

0 0.2 0.4 0.6 0.8 110

0

101

102

Flux index ρρρρ

Per

cen

tag

e w

all r

efle

cted

CX

sig

nal

Ref. mds = 737Ref. mds = 537Ref. mds = 755

(a)

0 0.2 0.4 0.6 0.8 110

-1

100

101

102

Per

cen

tag

e w

all r

efle

cted

CX

sig

nal

Flux index ρρρρ

Ref. mds = 737Ref. mds = 537Ref. mds = 755

(b)

Figure 9. Percentages of reflected CX signal on the CX channels of Textot pilot and ITER upperport 3 for different parametric values of m, d and s are shown in (a) and (b), respectively. Thevalues of m, d and s are denoted as ‘mds’ (e.g. mds = 737 denotes m = 0.7, d = 0.3 and s = 0.7).

Appendix

The Fresnel equation expresses the reflectance F of a perfectly smooth mirror-like surfacein terms of refractive index (n), extinction coefficient (k) of the surface and the angle ofillumination (ϑ). In general both n and k vary with wavelength, but often their values are notknown. However, the measured value of normal reflectance (F0) is more frequently available.A procedure given by Cook and Torrance [11] to obtain the spectral and angular variation ofF from F0 has been used in this work and is summarized below.

If n and k are known, we can use the Fresnel equation in its original form. On the otherhand, if these values are not known, but the normal reflectance (F0) is known, we fit the Fresnelequation to the measured normal reflectance for a polished surface. For non-metals, for which

13

Plasma Phys. Control. Fusion 52 (2010) 125006 S Banerjee et al

k = 0, a straightforward estimation of the refractive index n can be made. In the case ofmetals, for which k has a finite value, we still set k equal to 0 and get an effective value for n

from the normal reflectance. The angular dependence of F is then calculated using the Fresnelequation. To illustrate this procedure, the Fresnel equation for unpolarized incident light andk = 0 is written as

F = 1

2

(g − c)2

(g + c)2

{1 +

[c(g + c) − 1]2

[c(g − c) − 1]2

}, (11)

where

c − cos ϑ = V · H,

g2 = n2 + c2 − 1.

Refer to figure 4 for vectors and angles.At normal incidence, ϑ = 0; so c = 1, g = n. Hence,

F0 ={

n − 1

n + 1

}2

.

Solving for n, we get

n = 1 +√

F0

1 − √F0

. (12)

The values of n determined in this way are then substituted back into the original Fresnelequation to obtain the reflectance F at other angles of incidence. The procedure may berepeated at other wavelengths to obtain the spectral and directional dependence of F .

References

[1] Isler R 1994 Plasma Phys. Control. Fusion 36 171[2] von Hellermann M et al 1993 Plasma Phys. Control. Fusion 35 799[3] Tugarinov S, Krasilnikov A, Dokouka V, Khayrutdinov R, Beigman I, Tolstikhina I, Vainshtein L,

von Hellermann M and Malaquias A 2003 Rev. Sci. Instrum. 74 2075[4] Harhausen J, Fuchs J C, Kallenbach A, Schustereder W, Wischmeier M and ASDEX Upgrade Team 2007 Proc.

34th EPS Conf. on Plasma Physics (Warsaw, Poland) vol 31F (ECA) P-2.037[5] Schunke B, Huysmans G and Thomas P 2004 Proc. 31st EPS Conf. on Plasma Physics (London, UK) vol 28G

(ECA) P-4.111[6] Angioni C et al 2007 Nucl. Fusion 47 1326–35[7] Zastrow K D, Keatings S R, Marot L, O’Mullane M G, de Temmerman G and JET-EFDA Contributors 2008

Rev. Sci. Instrum. 79 10F527[8] Torrance K E and Sparrow E M 1967 J. Opt. Soc. Am. 57 1105[9] Phong B T 1975 Commun. ACM 18 311

[10] Blinn J F 1977 Comput. Graph. 11 192[11] Cook R L and Torrance K E 1982 ACM Trans. Graph. 1 7[12] von Hellermann M G , Delabie E, Jaspers R, Lotte P and Summers H P 2008 AIP Conf. Proc. 1058 187 (for

details of the simulation package contact mgvh@rijnh.nl)[13] Banerjee S, Chattopadhyay A K and Vasu P 2007 Plasma Fusion Res. 2 S1120[14] Banerjee S and Vasu P 2009 Nucl. Fusion 49 075032[15] Healey G 1989 J. Opt. Soc. Am. A 6 920–37[16] Karlsson B and Ribbing C G 1982 J. Appl. Phys. 53 6340[17] Bloxsom J T and Schroeder J B 1970 Appl. Opt. 9 539–43[18] Uhlemann R and Ongena J 1999 Fusion Technol. 35 42[19] Donne A J H and Costley A E 2008 Proc. 22nd IAEA Fusion Energy Conference (Geneva, Switzerland) (Vienna:

IAEA) IT/P6-20[20] Di Pietro E et al 2001 Fusion Eng. Des. 56–57 929–34[21] Chakraborty A et al 2010 IEEE Trans. Plasma Sci. 38 248[22] Bandyopadhyay M et al 2010 IEEE Trans. Plasma Sci. 38 242

14