Plasma interferometry at high pressuresKamran Akhtar, John E. Scharer, Shane M. Tysk, and Enny Kho Citation: Review of Scientific Instruments 74, 996 (2003); doi: 10.1063/1.1533104 View online: http://dx.doi.org/10.1063/1.1533104 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/74/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Diagnosing collisions of magnetized, high energy density plasma flows using a combination of collectiveThomson scattering, Faraday rotation, and interferometry (invited)a) Rev. Sci. Instrum. 85, 11E502 (2014); 10.1063/1.4890564 Core density gradient fluctuation measurement by differential interferometry in the helically symmetric experimentstellaratora) Rev. Sci. Instrum. 83, 10E308 (2012); 10.1063/1.4730999 A Stark-tuned laser application for interferometry and polarimetry on the National Spherical Torus Experiment Rev. Sci. Instrum. 74, 1621 (2003); 10.1063/1.1530352 Microwave interferometer for steady-state plasmas Rev. Sci. Instrum. 72, 1672 (2001); 10.1063/1.1347971 Multichannel interferometer system for the helically symmetric experiment Rev. Sci. Instrum. 72, 1081 (2001); 10.1063/1.1319366

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Plasma interferometry at high pressuresKamran Akhtar,a) John E. Scharer, Shane M. Tysk, and Enny KhoDepartment of Electrical and Computer Engineering, University of Wisconsin, Madison, Wisconsin 53706

~Received 17 September 2002; accepted 1 November 2002!

A general formulation for the measurement of plasma density and effective collision frequency forlowly as well as highly, collisional plasmas using millimeter wave interferometry is presented. In thepresence of high density and collisionality at high gas pressures where the collision frequency~n!is of the order of both the plasma (vp) and the wave frequency of the millimeter wave~v! (n;v,vp), the measured line-average plasma density has a complex dependence on phase shift aswell as the amplitude change of the millimeter wave signal. The measurement scheme and analysispresented in this article show that for collisional plasmas, simultaneous measurement of the phasechange and the amplitude change data is required to uniquely determine the plasma density andcollision frequency. The treatment allows the application of millimeter wave interferometry to awide range of relative collision frequency, wave frequency and plasma frequency since it uniquelydetermines the line-average plasma density and effective collision frequency using the phase andamplitude change data. This diagnostic is particularly valuable at plasma densities (ne

,1014/cm3) and high pressures (P.10 Torr) where Langmuir probe and optical emissiondiagnostics including the Stark effect are not accurate for density measurements. ©2003American Institute of Physics.@DOI: 10.1063/1.1533104#

I. INTRODUCTION

Millimeter ~mm! wave interferometry1,2 techniquesalong with polarimetry3,4 and optical emission diagnostics5,6

have been widely used as nonperturbing methods for plasmadiagnostics. Faraday rotation-based polarimetry originallydeveloped for magnetic field measurement can be used as anaccurate densitometer for magnetized plasmas. However, theapplication of such polarimetry is limited to high-densitymagnetized plasmas at low gas pressure where electron–neutral and electron–ion collision frequencies are small. Athigh gas pressure and collisionality, optical diagnostics in-cluding the Stark effect5 are used for plasma density andtemperature characterizations. But the minimum plasma den-sity requirement for these optical diagnostics to be accurateis very high (ne>1014– 15/cm3).5,6 Interferometry provides aviable diagnostic tool to measure the average plasma densityaccurately over wide plasma density and gas pressure ranges.

Interferometry is primarily a technique to measure therefractive properties of the plasma which in turn depends onthe plasma properties including the plasma density and col-lision frequency. The interferometer works on the Mach–Zehnder principle1 in which the plasma is in one arm of thetwo-beam interferometer. Let us consider the mm wavepropagation in an infinite, uniform Lorentz plasma. At lowcollisionalities, the phase difference between the two armswith the plasma present to that without the plasma is a mea-sure of the plasma density.1 In low pressure discharges wherethe wave frequency~v! is always much greater than theplasma (vp) and collision frequency (neff) (v@vp@neff),the mm wave suffers almost little or no attenuation as ittravels through the plasma. Therefore, only phase change

data are needed for a density measurement. In this case, alinear relationship exists between the line-average plasmadensity and the phase shift for a radially uniform plasmacolumn.2 Also if vp*v in low collisionality plasmas, theordinary wave mode~o-mode! is in cutoff and interferometrydata can not be obtained. On the other hand for high pressuredischarges, where the collision frequency is higher than boththe plasma and the wave frequency (neff>v;vp), we experi-mentally observe o-mode transmission forvp*v as pre-dicted by the theory. The average plasma density for suchhighly collisional plasmas is proportional to the square rootof the phase shift and is linearly related to the collisionfrequency.7 In this limiting case, an approximate refractiveindex is obtained for Lorentz plasma and the effect of colli-sion frequency is incorporated in the formulation onlythrough the phase function. This stronger dependence of theplasma density on collision frequency and reduced depen-dence on the phase shift introduce considerable error in theplasma density measurement at an intermediate pressurerange (100 Torr,P,300 Torr) where the collision fre-quency is of the order of the wave frequency.

In a previous experiment carried out by our group, a 105GHz interferometer was used to measure the average plasmadensity of a laser-initiated and rf sustained plasma.8 Usingthe collisionless approximation, an average plasma densityof 331012 cm23 was measured for an argon plasma at 150Torr with 2.5 kW of coupled rf power. The correspondingplasma phase change was 70° over a 5 cmdiameter chamber.In such high-pressure discharges, the dominant collisionmechanism is the electron–neutral collision for which colli-sion rates are compiled by Vahedi.9 This can be used to cal-culate the electron–neutral collision frequency as10a!Electronic mail: kamran@cae.wisc.edu

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 74, NUMBER 2 FEBRUARY 2003

9960034-6748/2003/74(2)/996/6/$20.00 © 2003 American Institute of Physics

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n5ngK'331093P ~Torr!. ~1!

Here, ng is the neutral particle density~m23!, K is thecollision rate constant~m3/s!, andP is the neutral gas pres-sure in Torr. The corresponding electron–neutral collisionfrequency at 150 Torr is 1.531011 s21. If the effect of col-lision frequency in the highly collisional limit is consideredas in Ref. 7 and discussed in more detail in Sec. III B, thepredicted average plasma density increases by an order ofmagnitude to 2.831013 cm23. In this pressure and densityregime whereneff;v>vp , neither the collisionless nor thehigh-collisional limits are valid and the predicted plasmadensity is not accurate.

At high gas pressures, the mm wave propagating throughthe plasma arm undergoes a phase change as well as strongattenuation. The wave attenuation is caused by the presenceof high collisionality. In this situation, the plasma density hasa complex dependence on phase change as well as on ampli-tude change and, therefore, the plasma density estimate usingonly phase change data is inaccurate. Therefore, a correctevaluation of plasma density can only be obtained if bothphase change and amplitude change data are used.

In this article, we present a measurement and analysistechnique where both amplitude change and phase changedata are used simultaneously to uniquely determine bothplasma density and effective collision frequency. The treat-ment does not limit the application of interferometry to therelative values of collision frequency and, hence, can be usedfor measurements at both low gas pressure (v@vp@neff)and high gas pressure (neff>v,vp). The analysis does notassume,ab initio, a particular value of the collision fre-quency, rather than it calculates the collision frequency alongwith density using the phase and amplitude change data. Inaddition, insight in to the dominant loss mechanism can alsobe obtained from the temporal plasma decay plot. The ex-perimental arrangement is presented in Sec. II, with the the-oretical analysis presented in Sec. III, and results are dis-cussed in Sec. IV.

II. EXPERIMENTAL SETUP

A 105 GHz quadrature mm wave interferometry system~QBY-1A10UW, Quinstar Technology! is used to measurethe plasma density and the effective collision frequency of arf produced plasma. The rf source is a 10 kW solid-state unit~CX10K-S, Comdel, Inc.! with variable duty cycle~90%–10%!, variable pulse repetition frequency~100 Hz–1 kHz!and very fast~ms! turn-on/off time. This fast turn-on/off timeis needed to determine the dominant loss mechanisms as theplasma decays when the rf power is turned off. The rf poweris coupled through a helical antenna that excites them50transverse electric mode very efficiently using a capacitivematching network. The helical antenna is a five-turn coil ofquarter-inch copper tube wound tightly over the 5 cm diam-eter pyrex plasma chamber. The coil is 10.0 cm long axiallyand has a 6 cminternal diameter. Figure 1 shows the sche-matic of the experimental system.

The interferometer works by using an in-phase andquadrature phase mixer to determine the phase and ampli-tude change of the 105 GHz mm wave signal going through

the plasma. The data acquisition set up is shown schemati-cally in Fig. 2. The two outputs are transferred to the com-puter through an oscilloscope with a general purpose inter-face bus and stored using aLABVIEW program. As the Gunnsource and detector diodes are extremely sensitive to the rfradiation, the entire interferometer assembly is enclosed in aconducting box with slots cut in the box for the waveguides.In addition, cables with very high shielding~>90 dB, TimesMicrowave Systems! have been used to reduce the noiselevel on the interferometer signal. The interferometric traceshown in the Fig. 3 is a function of time as the argon plasmadecays when the rf power is turned off. A numerical programutilizing MATLAB solves for both phase and attenuation datato obtain the plasma density as well as the effective collisionfrequency. The density plot versus time also reveals the lossmechanisms including effective recombination coefficients athigher gas pressures.

III. THEORY

The propagation of electromagnetic waves in a plasma isdetermined by the value of the dielectric function. We as-sume a cold plasma approximation model since for most of

FIG. 1. Schematic of the laser-initiated and rf sustained plasma experiment.

FIG. 2. Interferometer and data acquisition setup.

997Rev. Sci. Instrum., Vol. 74, No. 2, February 2003 Plasma interferometry

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the applications, the wave phase velocity is much greaterthan the electron thermal velocities and we can ignore ther-mal particle motion. Electron interaction is only through col-lective space-charge forces and the ions are assumed to forma stationary background.

The equation of motion for plasma electrons in the ab-sence of a magnetic field is written as

mr52eE2neffmr , ~2!

where r is the electron displacement vector andE is theelectromagnetic field. We also defineneff , the effective col-lision frequency for momentum transfer. If the electric fieldvaries as exp(jvt), the displacement vectorr is given as

r5eE

mv~v2 j neff!; r5

2eE

m

~neff2 j v!

~v21neff2 !

. ~3!

The complex conductivitys is given as

s[s r1 j s i52neer

E5

nee2

m

~neff2 j v!

~v21neff2 !

, ~4!

and the complex relative dielectric constant for a linear me-dium is given by

«

«05k5k r1 j k i512 j

s

«0v512

vp2

v21neff2 S 11 j

neff

v D ,

~5!

while vp is the plasma frequency and«0 is the free spacepermittivity. The complex refractive index and the complexpropagation constants are

n5c

v5m r2 j x5k1/2; g5a1 j b5

v

c~ j m!5

v

cAk.

~6!

Here a5xv/c is the attenuation constant in Np/m andb5m rv/c is the phase constant in rad/m. The solution for theplane wave phase and attenuation constants yields

bp5v

c H 1

2 S 12vp

2

v21neff2 D 1

1

2 F S 12vp

2

v21neff2 D 2

1S vp2

v21neff2

neff

v D 2G1/2J 1/2

, ~7!

ap5v

c H 21

2 S 12vp

2

v21neff2 D 1

1

2 F S 12vp

2

v21neff2 D 2

1S vp2

v21neff2

neff

v D 2G1/2J 1/2

. ~8!

Assuming a plasma slab of uniform average density profile,the total change in phase and amplitude for interferometricsignal are given as

Dw5E0

d

~b02bp!dr; DA5E0

d

~a02ap!dr. ~9!

Hereb0 anda0 are the free space values andbp andap arethe plasma values. A numerical program inMATLAB is usedto solve for plasma density andneff simultaneously from ex-perimentally measuredDf andDA values.

A. Low collision frequency limit

In the approximation of low collision frequencies (v@vp@neff) where most interferometry operates, the phaseconstant and attenuation constant are given as

bp5v

c S 12vp

2

v2D 1/2

'v

c S 12vp

2

2v2D ;

ap5neffvp

2

2v2c S 12vp

2

v2D 21/2

. ~10!

In the limit of v.vp , ap is small and, therefore, the wavesuffers almost no attenuation.2 Hence, the plasma density canbe expressed in this limit for uniform density profile usingEq. ~7! as

ne5S 4pc«0me

e2 D f Df

d51.1873106

f Df

d

52.073f Df

dcm23. ~11!

Here, the phase change is in degrees, the diameter in centi-meters, and wave frequency is in s21.

B. High collision frequency limit

For highly collisional plasmas at high gas pressures(neff@v>vp), the phase constant is written as

bp5v

c H 1

21

1

2 F11vp

4

v2neff2 G1/2J 1/2

'v

c F11vp

4

8v2neff2 G .

~12!

For a radially uniform plasma profile,

FIG. 3. Interferometer trace showing a nearly cutoff density of 931013 cm23 in 10 Torr argon plasma at 1.0 kW using a five-turn helicalantenna. The vacuum circle is represented by the dotted line. Measurementsare made 10 cm away from the antenna.

998 Rev. Sci. Instrum., Vol. 74, No. 2, February 2003 Akhtar et al.

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ne538.6neffS f Df

d D 1/2

55.0931025neffS f Df

d D 1/2

cm23.

~13!

Here again, frequencies are in s21, the diameter is in centi-meter, and phase changes are in degrees.7

C. Determination of recombination coefficients

A measurement of the dominant loss mechanism in aplasma can be obtained from the temporal plot of plasmadensity decay. Each point on the trace in Fig. 3 correspondsto the instantaneous phase and amplitude change of the mmwave signal traveling through the plasma. The instantaneousplasma density and corresponding effective collision fre-quency are obtained by simultaneously solving for the in-stantaneous phase and amplitude change data and a plot ofthe plasma density and effective collision frequency as afunction of time is obtained. From the density decay plot, anestimate of the effective recombination rate for the plasma athigher gas pressures is obtained using the continuity equa-tion.

The continuity equation, including the ambipolar diffu-sion term, the ionization frequency term,n iz , and two- andthree-body recombination terms, is given by

]ne /]t5Dn¹2ne1n izne2a rne22b rngne

2. ~14!

Here,a r , b r , andng are the two-body recombination rate,three-body recombination rate, and the neutral gas pressure,respectively. For a temporally decaying plasma without anyionizing source, the continuity equation takes the form

]ne /]t5Dn¹2ne2a rne22b rngne

2. ~15!

In low to moderate pressure discharges, plasma diffusionto the chamber is the principal loss mechanism for plasmadecay. However, at higher gas pressures (P.100 mTorr),two-body recombination starts to dominate the plasma decaycharacteristics and limits the peak plasma density obtainableat high pressures. The number of particles lost due to two-body recombination in argon above 100 mTorr is muchhigher than the number lost due to diffusion.8 With a furtherincrease in pressure, three-body recombination processes in-volving an ion, an electron, and a neutral, contributes to theplasma decay and becomes the dominant loss mechanism atvery high pressures. An estimate of the gas pressure, wherethe three-body effect in argon becomes comparable to the

two-body effect, can be obtained. For argon gas, the two-body recombination coefficient11 a r

Ar56.531028 cm3 s21

and three-body recombination coefficient~Ref. 12! b rAr

510226 cm6 s21. Using a rAr5b r

Arng , the argon pressure atwhich these two effects become comparable is 202 Torr.However, the pressure range where the contribution toplasma decay from three-body recombination is important inthe calculation differs widely for different gases. In somecases, it is more appropriate to define an effective recombi-nation coefficient,aeff , which includes two- and three-bodyrecombination effects, and can be easily determined from theplasma density decay plot. For the pressure range of interest,we have

]ne /]t52~a r1b rng!ne252aeffne

2. ~16!

Here aeff is the effective recombination term that includestwo- and three-body recombination effects.

IV. RESULTS AND DISCUSSION

The results for the plasma density computation usingEqs. ~9!, ~11!, and ~13! are presented in Table I, for typicalphase change and collision frequency data in a rf producedplasma source at high gas pressures. From the experimen-tally determined phase and attenuation data, the plasma den-sity and effective collision frequency are determined usingthe analysis presented herein. The collision frequency andphase change data are then used to calculate the plasma den-sity using Eqs.~11! and ~13!. Plasma density and effectivecollision frequency have a complex dependence on phasechange and attenuation data. At low gas pressure~;1 Torrargon!, where the collision frequency is low (1.03109 s21), the plasma density estimate using the collision-less approximation is the same as that obtained by thepresent method. However, the plasma density estimated us-ing the highly collisional plasma approximation is two ordersof magnitudes lower. When the pressure is raised to 300 Torrand the rf power is also raised to obtain a similar phase shiftof 100° as obtained for gas pressure for 1 Torr, the effectivecollision frequency is 1.031012 s21. The plasma densitycomputed using the collisionless approximation remains thesame, since it is assumed to be independent of the collisionfrequency. However, the predicted density increases from7.3831010 cm23 at low neff to 7.3831013 cm23 when com-puted using the highly collisional plasma approximation in

TABLE I. Comparison of plasma density computation using 105 GHz interferometer for 5 cm diameter tube.

DF~degrees!

Attenuation~dB!

neff

~s21!

ne(cm23)Using both phase

and amplitude data@Eq. ~9!#

ne(cm23)Collisionless limit

@Eq. ~11!#a

ne(cm23)Highly collisionalplasma@Eq. ~13!#b

100.0 0.023 1.03109 4.3031012 4.3231012 7.3831010

479.6 6.51 5.731010 2.0131013 2.0731013 9.2231012

175.0 4.07 1.031011 7.6531012 7.5731012 9.7731012

70.0 8.60 5.331011 4.9931012 3.031012 3.2731013

30.0 6.93 1.031012 4.3031012 1.2931012 4.0431013

100.0 23.65 1.031012 1.4531013 4.3231012 7.3831013

150.0 56.1 1.531012 4.2531013 6.4931012 1.3531014

aSee Ref. 2.bSee Ref. 7.

999Rev. Sci. Instrum., Vol. 74, No. 2, February 2003 Plasma interferometry

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Eq. ~13!. The plasma density when computed using the moreaccurate phase change and attenuation data changes only byan order of magnitude from 4.3031012 to 4.7731013 cm23. It is clear from this analysis that density mea-surements in the presence of high collisionality at intermedi-ate pressures where the collision frequency is less than thewave frequency must include the attenuation data, otherwise,it will lead to significant error. Since both the phase changeand attenuation data are used in the present method, it isindependent of the collision frequency range.

The characteristics of plasma decay and the nature of theloss mechanism can only be investigated if the rf generatorturn-on/off time is faster than the plasma decay time. A solid-state rf generator with the fast turn-on/off time is suitable forthese measurements. From the interferometer trace shown inFig. 3, the average plasma density and effective collisionfrequency are calculated for a temporally decaying plasma.The density decay plot is obtained by numerical differentia-tion of the density plot. Using Eq.~16!, the recombinationcoefficient is obtained. An interactive graphical program inMATLAB has been developed to analyze the interferometerdata and a typical plot is shown in Fig. 4. In an experiment toinvestigate the effect of gas flow on rf produced plasmas, thegas flow rates were adjusted using a mass flow meter. Twotest cases of 50 Torr argon plasmas are presented here toexplain the determination of recombination coefficients. Inthe static case, or zero-flow rate case, the evacuated chamberis raised to 50 Torr by opening the gas inlet valve and closingthe pump valve. In the flow case, a mechanical pump is keptrunning with the flow rates adjusted to 20 standard liters perminute~slm! using a mass flow meter at 50 Torr. A delay of5–20 ms can be introduced between the oscilloscope triggerand rf power turn off. As shown in Figs. 5 and 6, the plasmadensity remains constant initially for 5–7 ms~delay! andthen it decays sharply. A higher density decay rate and re-combination coefficient (1.031028 cm3/s) are obtained forthe static case. The recombination rate is much lower (2.0310210 cm3/s) when gas flow is maintained at 20 slm.

This diagnostic is particularly valuable for collisionalplasmas of moderate densities (ne,1014/cm3) at higher gaspressures, where probe and optical emission diagnostics arenot suitable for density measurements. We are currently uti-lizing this diagnostic for characterization of mixed flow air–argon, laser-initiated, rf-sustained collisional plasmas.

ACKNOWLEDGMENTS

This research was supported by Air Force Office of Sci-entific Research Grant No. F49620-00-1-0181 and in part byNSF Grant No. ECS-9905948.

FIG. 4. For a temporally decaying plasma,MATLAB plots of ~a! plasmadensity,~b! density decay, and~c! effective collision frequency as a functionof time obtained from mm waves amplitude and phase shift data.

FIG. 5. Density decay plot as a function of time for a temporally decayingargon plasma. For a 50 Torr argon discharge at 3.0 kW coupled rf power andflow rate of 20 slm, the effective recombination coefficient isaeff52.0310210 cm3/s.

FIG. 6. The plasma density decay plot for the static 50 Torr argon dischargeat 3.0 kW. The corresponding recombination coefficient for this case isaeff51.031028 cm3/s.

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1001Rev. Sci. Instrum., Vol. 74, No. 2, February 2003 Plasma interferometry

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