iv) diagnostics based on plasma refractivityweisen/cours_diag_2014/cours_diag_12_4.… ·...

IV) Diagnostics based on plasma refractivity

Electromagnetic wave propagation in the geometri-cal optics approximation

The wave equation can be written as (eq. 4.1):

where k0=ω ⁄c band E is the electrical field, where we already dropped the exp(-iωt) time dependence. A general solution takes the form of the scatteringequation seen in the chapter on Thomson scattering. A useful approximation, thegeometrical optics approximation, corrresponding to scattering into the direction ofthe incident beam (propagation) is obtained by the Ansatz:

E(r)=exp(ρ(r)+iφ(r)) (Rytov transformation)

where ρ(r) and φ(r) are real functions describing amplitude and phase and thepolarization is ignored. It leads to :

and (eq. 4.2)

Defines rays which are parallel to ∇φ and perpendicular to the wavefrontsdefined by φ=constant. Solution:

( ) ( )EnkEk 1220202 −−=+∇

( ) ( )222202 ρρφ ∇+∇+=∇ nk

neglectedeikonal equation

ρφφ ∇∇−=∇ 22

eq. 4.3

eq. 4.4

∫=−r

r

dllnkrr0

)()()( 00φφ

)(2

1exp)( 02

0 0

rEdlnk

rEr

r ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ ∇−

= ∫φ

IV) Diagnostics based on plasma refractivity 24 March 2009 2

Local condition for geometrical optics: ⏐∇n/n⏐

Effects of refraction in geometrical optics limit:

1) Phase shift (see above)

2) Ray deflection (often called ’refraction’)

Consider a plane wave propagating at an angle α with respect to z-axis. The gradi-

ent of the wave phase transverse to z is:

(eq. 4.6)

For small deflections, treat coordinates // and ⊥ to k separately:

(eq. 4.7)

Nuisance effect for interferometry and microwave diagnostics. Diagnostics

use in high density plasmas such as z-pinches, inertial fusion, gaseous flows:

‘Schlieren’ methods from German for ‘streaks’.

French: ‘stries’, d’où ‘strioscopie’.

z

x1

k

λ2λ

3λ

φ=2π+φ0

φ=4π+φ0

φ=φ0α φ x( )⊥∇

2πλ

------ αsin=

αsin x( )φ x( )⊥∇k0

------------------ n l( )⊥∇ ldz0

z

∫≅=

αr0r=(x1,x2,z)

wavefronts

x1x2

z

l


3) Intensity modulations

Intensity modulations (shadow effects, lensing effects)

appear depending on local convergence or divergence of

rays.

At a distance L from an object having produced a phase

shift φ, to first order (eq. 4.8):

This can be obtained from eq.4.4, assuming the wavefield

characterised by φ propagates through a homogeneous medium with n=n0 after the

phase object.

This is effect is at the basis of ‘schadowgraphy’, a simple method used in high den-

sity plasmas and gaseous flows.

Everyday observations of this effect include refraction of sunlight by waves at the

bottom of a lake or river, the scintillation of stars and other objects viewed through

a turbulent medium.

L

I x y,( )ΔI

-------------------L φ x y,( )2⊥∇–

n0k0---------------------------------- L–

n0------≅ n l( )2⊥∇ ld

z∫≅


Interferometers

Many types (see optical textbooks). Most commonly used in fusion plasmas are ofMach-Zehnder type:

S1, S2: semi-transparent beamsplitters, M1,M2 mirrors. System shown is a classi-cal imaging interferometer. Lenses L1 & L2 form a telescope and image a phaseobject at Σ onto Σ’. A similar telescope arrangement is used to provide a parallelbeam entering the system at S1, using a laser beam.

Intensity distributon detected with semiconductor device (CDD, diode array...) orphotographic film (in the old times).

I(x’,t)∝⏐Eref+Eobj⏐2 =⏐Eref⏐2+⏐Eobj⏐2+2⏐ErefEobj⏐cos(Δφ(x’,t)),

where Δφ is the phase difference between thereference and object path’s.Usually the two beams are recombined with asmall angle, generating a background of straightfringes, which is deformed in the presence ofthe phase object. The background fringes out-side the phase object allow Δφ to be determinedwithout ambiguity (‘spatial heterodyning’)

L1 L2


Time resolved phase measurements are best performed with frequency heterodyn-

ing, using different frequencies for the reference and the object beam, hence

I(x’,t)∝⏐Eref+Eobj⏐2 =⏐Eref⏐2+⏐Eobj⏐2+2⏐ErefEobj⏐cos(Δωt+Δφ(x’,t))

The phase shift introduced by the plasma is determined (up to a constant) by coher-

ent detection:

Finally Δφ=atan2(sin(Δφ),cos(Δφ))+2πΝ. Phase is nulled in absence of plasma

(Δφ=0). Slow (τ>1/Δω) continuous evolution of Δφ allows Ν to be determined.

ref

plasma

π/2 shifterΑsin(Δωt)

Αcos(Δωt)

Αcos(Δωt+Δφ)

Multipliers or mixers

Low pass filters removeall harmonics of Δω

−Α2sin(Δφ)

Α2cos(Δφ)

to dataaquisition& fringecounter


Frequency shift Δω can be obtained

1) using two sources at different frequencies

- Δω feedback stabilised by phase locked loop as in future λ=1mm single channel

microwave interferometer on TCV

2) by Doppler shifting a reference wave derived from the same source as the probe

wave

- by diffracting in a ‘Bragg cell’ (acousto-optical modu-

lator) with a traveling acoustic wave with frequency Δω.

Available at visible to mid-infrared wavelength

(~40MHz). Can use two at different frequencies.

- by diffracting off a rotating

grating (≤100kHz), as for TCV

multichannel far infrared interfer-

ometer at λ=0.214mm.

ω ω+Δω

ω


Sources and detectors for refractive diagnostics in fusion plasmas

• Requirements: - Sufficiently large phase shift due to plasma (Δφ ∝ λ)- Small phase shift (

Beam transmission optical (Gaussian beams) and/or in dielectricwaveguides (quartz tubes).

• Visible lasers were (or still are) used in high density plasmas, such as plasma focii. These devices are no longer considered serious con-tenders in magnetic fusion research.

Detectors:

- pyroelectric crystals (internal dipole depends on temperature) 0

Polarimetry

Based on Faraday rotation in magnetised plasma. Measure line integral of neB//.Normally combined with interferometry.

Several methods for measuring the small (

Polarimeter currently under implementation on TCV


Data inversion• General case of arbitrary flux surfaces:- As for tomography. Because number of chords is small, assume pixels having the shape of the flux sur-faces determined from equilibrium reconstruction.Transfer matrix from lengths of chords in each pixel.- Important to have data near edge- Polarimetry data (if available) used as constraint in equilibrium reconstruction- Reconstructions using basis functions, preferably ’suggested’ by topos present in a local diagnostic

(Thomson scattering), see Furno et al, Plasma Phys. Control. Fusion 47 (2005) 49.

• Special case of circular symetry (Abel inversion)

Formal solution:

• Polarimetry: Abel inversion of α(x)/x provides c2ne(r)Bp(r)/r -> two inversions to get Bp! Soltwisch, 1986, Varenna Course on Basic & Advanced Diagnostics for Fusion Plasmas.

z

xa

r

z0(x)

-z0(x)

φ c1λ ne z 2c1λ ne r( )r

x2 a2–--------------------- rd

x

a

∫=dz0–

z0

∫=

ne r( )1c1----- xd

dφ 1

x2 r2–--------------------⋅ xd

r

a

∫–=


Examples: Interfero-Polarimetry at the RTP tokamak from J. Rommers, PhD thesis, University of Utrecht, 1996


ω p c

Reflectometry

• An O-mode beam is reflected where

ε=0, or ne=ε0meω2/e2

O-mode is absorbed at harmonics of ωc, especially first harmonic

• For an X-mode beam ε=0 for

cutoff layer

X-mode absorption occurs if the beam encounters the upper hybrid resonance

ωUH = 2 + ω2

or a ωc harmonic (especially 2ωc)

JET is currently equipped with a reflectometer allowing a spatial resolution of 1cm

4

22

,c

pcLU

Interferometric setup for reflected wave phase

In geometrical optics approximation, for a return trip:

, where the π/2 shift is due to reflection.

In O-mode

or n=1-Δn(r). Decompose phase into a part due to distance and a part due to the plasma:

from Costley 1986

φrefl2ωc------- n r( ) r( )d

rc

a

∫π2---–=

n

1

0 r rc

ne(r)

n(r)

Δn

nc

a

n r( ) 1ωp

2 r( )

ω2-------------– 1

ne r( )nc r( )------------–= =

φrefl2ωc------- a rc–( )

2ωc-------+ Δn r( ) rd

rc

a

∫π2---–=


For a linear density gradient:

Phase shift can be written as: , where s is a

shape factor for the density profile (s>0 concave, s

Flat profiles problematic: especially near center (Ln→∞): ambigueous signals, wave can tunnel to other side, φrefl erratic if fluctuations.

Very steep gradients (Ln≤λ/n) appear as discontinuities and cause partial reflection (ambigueous signals)

Absorption at optically thick ωc harmonics and upper hybrid layer.

Examples

Density profile in L-mode and evolution after an ELM in H-mode (ASDEX)


Fixed frequency reflectometer signal (bottom) near plasma edge. Particle pulse after sawtooth crash in JET (Costley 1991)causes a sudden movement of rc.


Reflectometry specialties:

Correlation reflectometry:

Analyse coherence and phase of two closely spaced (few cm) fixed fre-quency channels to get radial structure of fluctuations.

Pulsed radar reflectometry:

group velocity gives group delay

Time delay for echo from microwave pulses (1ns duration) can be meas-ured with MHz repetition rate. (Hugenholtz, Rev. Sci .Instrum 70 (1999) 1034.)

Problem: Pulse broadening through dispersion because .

But dispersive effects can be modelled to improve measurement.

Swept frequency reflectometry

Fast tunable microwave sources can scan the entire radius in microsec-onds, providing a snapshot of the profiles and revealing localised pertur-bations (MHD modes, turbulence).

The smallest scales give rise to backscattering (not ’simple’ reflection) and quantitative information on the turbulence can be obtained by model-ling of the backscattered signal.

Doppler reflectometry

When the angle of incidence to the cutoff layer is non-normal, propagat-ing modes will give rise to Doppler-shifted backscattering.

For further information see for instance:

Recent results on turbulence and MHD activity achieved by reflectometryR Sabot et al, Plasma Phys. Control. Fusion 48 (2006) B421–B432

Study of turbulence and radial electric field transitions in ASDEX Upgrade using Doppler reflectometry, Conway, G.D. et al, IAEA-CN-149 / EX / 2-1, Chengdu, China 2006. http://edoc.mpg.de/get.epl?fid=36176&did=316699&ver=0

vg k∂∂ω= τ ω( )

ω∂∂φ=

ω2

2

∂

∂ φ 0≠


The Phase Contrast Imaging (PCI) technique for plasma density fluctuations

Plasma density fluctuations are revealing of MHD activity, transport processes, suchas drift waves and the effects of waves produced by low frequency RF heating. Imagingoffers attractive alternatives to the more traditional scattering techniques (discussed later)in the case of perturbations with spatial scales such that the plasma can be considered tobe a thin phase object. If a fluctuating wavefield has a characteristic spatial structure, areal space representation – an image – is likely to be more directly interpretable than itsspectrum. Low amplitude fluctuations causing small phase shifts (Δφ

Motivation:• Plasma waves in hot plasmas

(TCA: kinetic Alfvén waves, CMOD: ion Bernstein waves)

• Drift wave turbulence (TEM, ITG) and relation with transport(TCA, DIII-D)

• MHD fluctuations (ELMs & precursors on DIII-D, Alfvén cascades on C-MOD)

• Better suited for long wavelength fluctuations and inhomogeneous situations than collective scattering

• For a more complete exposition see http://crppwww.epfl.ch/~weisen/publications/lrp_639_99.pdf


Fourier transform by a lens

Lets consider an wavefield on a plane Σo which we’ll call our object field. Thewave will contain components of the form exp(ikr)=expi(kxx+kyy+kzz). If the inci-dent wave is of the form expi(kzz), then we’ll consider all waves with kx or ky≠0 tobe scattered waves, produced by the object. These are of course the transverse (2D)Fourier components of the wavefield.

These wavefields are plane waves propagate with angles sin(kx/k0) and sin(ky/k0).A lens of focal length f will focuss them onto a position displaced by distances f⋅sin(kx/k0) and f⋅sin(ky/k0) from the optical axis.

At the focal plane of the lens, the different transferse Fourier components are sepa-rated and can be acted upon separately, to reconstruct a filtered image. For a refer-ence on wave optics (or so-called Fourier optics, have a look at G. Gaskill’s ’LinearSystems, Fourier transforms and Optics, Wiley, N.Y. 1978)


Phase Contrast Imaging on the TCA tokamak


for detailed calculations of impulse response of PCI and other methods seehttp://crppwww.epfl.ch/~weisen/publications/lrp_639_99.pdf


∝


Acoustic line integrated and local signals from microphone are verysimilar

Transfer properties from acoustic calibration, impulse response and from response to ’phase edge’ (Mylar sheet), agree. Cf. H. Weisen, “The phase contrast method as an imaging diagnostic for plasma density fluctuations“, Rev. Sci. Instrum. 59, 1544 (1988).


Detection of externally excited kinetic Alfvén waves in TCA


The two left side plots show propagating, localised wave structures at theirspecific resonance positions. The fact that they are propagating is seen fromthe fact that the phase varies continuously with position. The plot on the rightis a global oscillation, essentially a standing wave, because the phase jumps byπ where the amplitude crosses zero. In TCA relative density fluctuations of

KAW were of order 10-3 for some 40kW of coupled RF power. In TCA thefrequency was 2MHz, whereas at C-MOD it was about 80 MHz. At C-MOD aheterodyne version of PCI was used, with part of the beam shifted to near79MHz using two acousto-optical modulators. The reference signal from theRF power supply was downmixed to about 1MHz, using the driver signals forthe modulator.Y . L IN e t a l , “O b se rv a tio n an d m o d ellin g o f io n c yc lo tro n ran ge o f freq u en c ies w av es in th e m o d e co n v ers io n reg io n o f A lca to r C -M o d ” , P la sm a P h ys . C o n tro l. F u s io n 4 7 , 1 2 0 7 (2 0 0 5 ).


Diagnostic potential of Alfvén Waves

Alfvén Wave resonance condition ω/k//=vA(1-ω2/ω2ci)1/2. Since k//=(n+m/q)/R0:

where ωci > ω is ion cyclotron frequency.AW resonance can be used to measure plasma parameters such as

- mass density ρ (->D/T ratio in reactor)- safety factor q

Toroidal (n) and poloidal (m) mode numbers can be selected by antenna phasing.Profiles of ρ and q can be measured together using frequency sweeps for two (n,m)pairs, then solving for ρ and q in the resulting 2 versions of above equation.

Example: Mass density and safety factor profiles in TCA, measured using(n,m)=(2,0) and (1,1) AW resonances. (H. Weisen et al, PRL 62 (1989) 434).

ω2n mq r )( )-------------+⎝ ⎠

⎛ ⎞ 2BT2

μ0ρ r( )R02-------------------------------------- 1

ω2

ωci2-------–⎝ ⎠

⎜ ⎟⎛ ⎞

12---

=

∝ρ(0)/ρ(r)

qa=3.2


Assumed that q-profile and shape of mass density profile did not depend on ne(0) during density ramp. Solving for q(r) for four discharges with dif-ferent qa yielded:

Note AW wave resonance condition and damping are routinely exploited on JETfor global AW eigenmodes (TAE, EAE) which can be excited/detected using mag-netic coils. Provide info on q, ρ and potential for instability in presence of fast ions(see A Fasoli, Plasma Phys. Contr. Fusion 44 (2002) B159 or CRPP report LRP728/02 or crppwww.epfl.ch/conferences (Montreux, EPS 2002))

q(0)>1no sawteeth

x indicates edge safety factor from magnetics

q(0)>1no sawteeth


Low frequency drift wave turbulence measured by PCI on TCA


2-point cross correlation as function of detector separation

i∗ x ω,( )i x Δx+ ω,( )〈 〉i∗ x ω,( )i x ω,( )〈 〉 i∗ x Δx+ ω,( )i x Δx+ ω,( )〈 〉

-----------------------------------------------------------------------------------------------------------------


Relationship between local and line integrated fluctuating quantities

Define .

How is the variance of N related to the variance of n?

(we changed z’ to z+Δz)

The quantity in 〈〉 is the autocorrelation function of in the z direction. The brack-ets mean an expectance value, in practice usually a temporal average, if the systemis statistically stationary. Let’s introduce the local coherence length in the z direc-tion by

For a homogeneous slab of length L of turbulent plasma we then have:

See exercises for an alternative calculation based on a random walk argument. Theabove approach can be extended to the complete correlation functions of n and Nwith the result (H. Weisen, Plasma Phys. Contr. Fusion 30 (1988) 293, appendix):

and similarly for the tem-

poral Fourier transforms . The autocorrelation functionof N is proportional to the line integral of the autocorrelation function of n.

Ñ x t,( ) ñ x z t, ,( ) zd∫=

N2˜ x t,( )〈 〉 ñ x z t, ,( ) z ñ x z' t, ,( ) z'd∫×d∫〈 〉=

ñ x z t, ,( )ñ x z' t, ,( ) z'd zd∫∫〈 〉 ñ x z t, ,( )ñ x z zΔ+ t, ,( ) zΔ( )d zd∫∫〈 〉==

ñ x z t, ,( )ñ x z zΔ+ t, ,( )〈 〉 zΔ( ) zdd∫∫=

ñ

lz x z t, ,( )ñ x z t, ,( )ñ x z zΔ+ t, ,( )〈 〉 zΔ( )d∫

ñ x z t, ,( )ñ x z t, ,( )〈 〉-------------------------------------------------------------------------------=

N2˜〈 〉 lzL n2̃〈 〉=

Ñ x t,( )Ñ x xΔ+ t tΔ+,( )〈 〉

ñ x z t, ,( )ñ x xΔ+ z zΔ+ t tΔ+, ,( )〈 〉 zΔ( ) zdd∫∫=

Ñ x ω,( )Ñ x xΔ+ ω,( )〈 〉


Fourier transform of autocorrelation function yields k,ω spectrum:

Intergration over ω yields wavenumber spectrum:

Peak at kθ=1.3 rad/cm corresponds to kθρs~0.2


Some background info: correlations and spectra

Notes: a) The above turbulent fluctuations in TCA are consistent with the ‘Kadomt-sev mixing length’ rule Δne/ne~1/Ln with 1/~~. b) Density fluctuations on their own do not allow us to infer what the particle orheat fluxes are: for this we need and , which can be measured withmultiple Langmuir probes in the edge or a Heavy Ion Beam Probe in the core.

ṽrñe i,〈 〉 ṽrp̃〈 〉


IV) Diagnostics based on plasma refractivityElectromagnetic wave propagation in the geometrical optics approximationEffects of refraction in geometrical optics limit:InterferometersSources and detectors for refractive diagnostics in fusion plasmasPolarimetryData inversionExamples: Interfero-Polarimetry at the RTP tokamak from J. Rommers, PhD thesis, University of Utrecht, 1996ReflectometryInterferometric setup for reflected wave phaseLimitationsExamplesThe Phase Contrast Imaging (PCI) technique for plasma density fluctuationsFourier transform by a lensDiagnostic potential of Alfvén WavesAssumed that q-profile and shape of mass density profile did not depend on ne(0) during density ramp. Solving for q(r) for four discharges with different qa yielded:Relationship between local and line integrated fluctuating quantitiesSome background info: correlations and spectra

iv) diagnostics based on plasma refractivityweisen/cours_diag_2014/cours_diag_12_4.… ·...

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