CHAPTER 2

Doppler Spectroscopy

And God said, Let there be light: and there was light.– Genesis 1:3

T his chapter provides an understanding for the basis of—and the informa-tion that can be derived from—spectroscopy where the dominant spectral

lineshape broadening mechanism is the Doppler effect. In particular, a treat-ment is given for the measurement and interpretation of line-integrated measure-ments. It outlines the theory and reasoning for the use of a Fourier-transformspectrometer in making these measurements and gives a conceptual descrip-tion of the Modulated Optical Solid State spectrometer. Consideration is givento the properties and characteristics of this Fourier-transform instrument. Thischapter also establishes the framework—including the concepts, definitions andnomenclature—for the experimental work described later in this thesis. Through-out this and subsequent chapters, the terms ion and atom are often used inter-changeably in the description of electronic transition processes.

2.1 Introduction

The transition of an electron, bound to a nucleus, from an upper fixed energystate k (with energy Ek) to a lower level i emits radiation with energy given by

Eki = Ek − Ei = hν =hc

λvac(2.1)

where ν is the frequency of the radiation and λvac its wavelength in vacuum. hand c are the Planck and speed of light constants, respectively. However becausethe radiation process is affected by other influences the observed emission doesnot have the single frequency ν.

The spectral lineshape—the energy distribution as a function of frequency—has a natural width Δν = ΔE/h (also called the linewidth) due to the Heisenberguncertainty principle which is given by ΔνΔt ∼ 1/2π. That is, the energy spread

2. Doppler Spectroscopy 14

is inversely proportional to the radiative lifetime† of the electron in that upperenergy state. The measured spectral lineshape may not be narrower than thisnatural width but can be broadened by different effects such as the Doppler, Zee-man and Stark (also called collisional or pressure broadening). Natural linewidthbroadening can become significant when observing the high-energy lines fromhighly charged ions and yields information on the lifetime of the energy states.

The Doppler effect is the apparent shift in the emitted wavelength due tomotion of the radiating source away (increase in wavelength or ‘red-shift’) ortowards (wavelength decrease or ‘blue-shift’) the observer. Doppler broadeningof the observed line arises from the distribution in ionic velocities which createsa corresponding distribution in the observed wavelengths.

Doppler broadening is dominant in many laboratory plasmas [Hutchinson,2002] and the linewidth provides a quite direct measurement of the ionic temper-ature [Hilliard and Shepherd, 1966]. Additionally, the observation of a shift in themean (or centre) wavelength of a spectral lineshape gives a measurement of thebulk flow velocity of the emitting species [Rees and Greenaway, 1983]. Localisedturbulence can distort the interpretation of temperature information and regimesin which this may occur require closer scrutiny. Doppler spectroscopy can also beused to infer the flow velocities of very hot, fully-ionised majority species plas-mas by observation of impurity ions [Benjamin et al., 1990] or through charge-exchange with injected particles.

The Stark effect is the perturbation to the energy levels of a bound electronin the presence of an external electric field. This perturbation alters the emittedwavelength of the radiation. In a plasma, nearby electrons generally provide theperturbing electric field. Thus for high pressures (or densities) or collisionality,the linewidth can be broadened by this effect. Stark broadening becomes an im-portant consideration for lower-temperature, higher density plasmas and in suchcircumstances allows the determination of the electron density.

The Zeeman effect results in the splitting of a spectral line into three (or more)components in the presence of a magnetic field. The unshifted π componentis polarised parallel to the magnetic field direction while the equally shifted σ

components are orthogonally polarised. The change in the potential energy ofthe electron is qehB/2me so that the spectral line splitting caused by the Zeemaneffect becomes non-negligible in high magnetic field strength plasmas. Techni-cally, this effect allows the magnetic field strength and direction to be determined,though the splitting is often obscured by Doppler broadening.

For plasma conditions typical of low-field argon discharges in the H-1NF stel-larator, the spectral lineshapes are dominated by Doppler broadening, as will bediscussed in detail below.

†Which is the inverse of the sum of the probabilities of all transitions from that excited energystate.

2. Doppler Spectroscopy 15

2.2 Doppler Broadening

While Doppler spectroscopy is a powerful technique for sensing gross fea-tures of the emitting medium, the interpretation of the spectrum is complicatedby the fact that the measurement is intrinsically a line-integrated one. This meansthat the inferred velocity can only be known in the direction of view and that thespectral lineshape is the result of intensity-weighted contributions from all pointsalong the line-of-sight for an extended and inhomogeneous medium.

For a medium where the velocity distribution function is in local thermalequilibrium, the three plasma properties are conveyed by the spectral lineshapewhich is dominated by Doppler-broadening. They are the emitted radiance (de-rived from the area under the lineshape), the species temperature (linewidth)and the flow velocity (line centre shift). The physical mechanisms underlyingeach property and how the quantities are inferred are discussed below.

2.2.1 Emission Intensity

The local species emission for a given spectral line depends on the electronenergy-state populations and the transition (emission and absorption) proba-bilities. The population of the energy states is governed by the particle den-sity, the thermal conditions and dynamic processes. These processes includetransitions which are both upward—such as collisional excitation and radiativeabsorption—and downward—such as radiative decay (both spontaneous andstimulated) and collisional de-excitation. The probabilities of the radiative tran-sitions are intrinsic properties of the species and can, in theory, be calculated ifthe quantum wave functions are known.

Consider the spectral line of central wavelength λki (and frequency νki), re-lated to the transition between energy levels Ek (upper) and Ei (lower). In thepresence of radiation, a stimulated (also called induced) radiative decay may oc-cur with probability per unit time of Bkiρ(νki) where ρ(νki) is the energy densityper unit frequency of the radiation at the ion. The ion may also absorb an inci-dent photon causing a transition from state i to k with probability per unit timeof Bikρ(νki).

The total emission co-efficient εki, which describes the power radiated per unitvolume and per solid angle by the ions in the upper state k due to spontaneousradiative decay, is given by

εki =hνki

4πAkink (2.2)

where Aki is the spontaneous transition probability per unit time (the Einsteinco-efficient) and nk is the population density of the energy state k.

If the mean free path of an emitted photon is greater than the dimensions of

2. Doppler Spectroscopy 16

the emitting medium, that medium is said to be optically thin. That is, radiationfrom any part of the medium will escape without being reabsorbed. For opticallythin media, induced radiative decay can be neglected. Except for possibly strongtransition lines in very dense (or large) plasmas [Hutchinson, 2002, p. 252], mostlaboratory plasmas are considered to be optically thin.

The population density nk is determined by the balance between collisionalexcitation and de-excitation and radiative decay. A model is sought, whethertime-dependent or static, which describes this balance in plasmas such as arestudied in this thesis.

For a system in complete radiative thermal equilibrium (CTE), the radiationenergy density for all possible transitions equates to the blackbody spectrum forthe system temperature. Since CTE really only applies to stellar interiors, a lesscomprehensive regime is offered in local thermal equilibrium (LTE). In LTE theenergy state populations follow the form of the Boltzmann distribution, as is alsothe case for CTE, but differ in that the radiation is not necessarily thermal. Addi-tionally, collisional transitions dominate the radiative transitions and usually thisrequires quite a high density. The density is considered high enough if it satisfiesthe following rule-of-thumb inequality [McWhirter, 1965].

ne � 1019(

Te

qe

) 12(

ΔEqe

)3

(2.3)

where ΔE/qe and Te/qe are the difference in energy levels and temperature, re-spectively, in electron volts. qe is the electron charge and the unit of the electrondensity ne is m−3.

For typical H-1NF argon plasmas (Te ∼10 eV, ΔE = 2.54 eV for the 488 nmArII transition and ne ∼ 1018m−3), the right-hand side evaluates to ∼ 5 × 1020

and, since this condition is not satisfied, LTE as a model regime must be rejected.Since thermal equilibrium models are unable to determine the energy state

populations, a rate-equation model is used. For a low density plasma, the coro-nal equilibrium model—which assumes that all downward transitions are spon-taneous radiation and all upward transitions are the result of collisions—is anappropriate choice. If the upper energy state k is populated by collisions fromthe ground state†, then the large ground state population will make this tran-sition the dominant process. In steady-state conditions, the rate of collisionalupward transitions can be equated with the photon emission rate due to spon-taneous decay (see (2.2)). This balance leads to an expression for the populationdensity of state k. That is,

nk =n1ne〈σ1kve〉

∑i

Aki(2.4)

†Denoted as energy state 1.

2. Doppler Spectroscopy 17

where 〈σ1kve〉 is the collisional excitation rate coefficient for the 1 to k transitionand the angled brackets denote an averaging over the velocity distribution func-tion. Here ve denotes the electron velocity. For a Maxwellian electron distribu-tion at temperature Te, it can be shown that (as an approximation for optically-allowed dipole transitions)

〈σ1kve〉 =f1kKA

E1k√

Teexp

(−E1k

kBTe

)(2.5)

where f1k is the 1 − k transition oscillator strength, KA is a constant and kB is theBoltzmann constant.

Noting (2.2), and substituting (2.4) and (2.5) gives

εki =hνki

4πAki

n1ne

∑i

Aki

f1kKA

E1k√

Teexp

(−E1k

kBTe

)(2.6)

In terms of plasma parameters, this may be written more simply as

εki ∝ n1neχ1k(Te) (2.7)

Assuming quasi-neutrality (ne ≈ Zeffni, where Zeff is effectively the ‘charge’ onthe ions), this leads to

εki ∝ n2eχ1k(Te) (2.8)

Hence a measurement of the local emission co-efficient, via tomographic means,can yield information on the electron density or temperature if the other param-eter is known.

2.2.2 Ion Temperature and Flow Velocities

As the following discussion deals only with a single electronic transition, the‘rest frame’ emitted frequency is now denoted as ν0. Recall that the Doppler effectdescribes the shift in radiation frequency of a particle radiating in its own framewith frequency ν0 and moving with velocity v, relative to the observer. For smallshifts, the observed frequency is given by

ν = ν0(1 +|v |c

). (2.9)

For observation of an extended radiating medium, along a line in the directionl, only the component of v in the direction of the line-of-sight contributes to theshift in frequency. Then |v |= v.l and (2.9) gives

ν − ν0

ν0=

v.lc

= ξ (2.10)

2. Doppler Spectroscopy 18

where ξ is a normalised frequency co-ordinate. Figure 2.1 shows the viewinggeometry where the observation line is denoted L. The distance p from the originto L is known as the impact parameter and is in the direction p = (cos φ, sin φ),which is orthogonal to l as shown.

y

x

L

p

p

l

^

emitting medium

φ

Figure 2.1: The viewing geometry for an ideal line-integral through an extendedmedium.

Consider an emitting plasma with an inhomogeneous, isotropic distributionof ion velocities v specified by the function f (r, V) where r is a position vectorin the plasma and V(r) = v(r)/c is the normalised velocity. Note that the dis-tribution may be drifting with velocity vD (normalised as VD) which is the firstmoment of the velocity distribution function. The non-drifting function is de-noted as f0 = f (r, V − VD). The normalised local spectral lineshape at positionr, as viewed in the direction l, is given by an integral in velocity-space over thevelocity distribution [Shaw, 1987],

g(r, l; ξ) =∫ ∞

−∞f (r, V)δ(ξ − V .l)dV (2.11)

where dV = dVx dVy dVz. When observing in the direction l, through an arbi-trary velocity distribution, the contribution at frequency ν to the observed inten-sity spectrum g(ν) is given by particles with velocities which satisfy (2.10). Thus,if the velocity distribution function f (r, V) is specified in three dimensions, then

2. Doppler Spectroscopy 19

the emitting particles which contribute to the local emission spectrum g(ξ) havevelocities which are found in the plane a distance ξ from the origin and which isorthogonal to l. Figure 2.2 illustrates this concept.

vy

vz

vx

vχ

ψv⊥

l^

Plane ξ = v l^.f(v)

v

ξ

Figure 2.2: Only velocities which lie on the plane a distance ξ from the origin of thethree-dimensional velocity distribution function f contribute to the spectral lineshape gobserved in the direction l. Figure courtesy of J. Howard.

The delta function in (2.11) selects the velocities in the distribution that con-tribute, in the l direction, to the emission. It is apparent from (2.11) that the line-shape is directly and simply related to the ion velocity distribution. However, themeasured emission spectrum is line-integrated and is the sum of many varying lo-cal lineshapes weighted by the local intensity (or brightness) function I0(r) (see(2.6)). If the line-of-sight is regarded as an ideal line-integral, the measurement yof the emission for the viewing line L(p, φ) is given by

y(p, φ; ξ) =∫ ∞

−∞I0(r)g(r, l; ξ)δ(p − r.p)dr ≡

∫L

I0(r)g(r, l; ξ)dl (2.12)

where dr = dxdy. Here, I0(r) is the radiated power into a fixed collection solidangle from an infinitessimal volume element of projected area δA along the line-of-sight and in the direction of the detector element. This quantity is colloquiallyreferred to in this thesis as the ‘radiance’ or ‘local emission intensity’.

If the velocity distribution is an isotropic Maxwellian, as is the case for localthermal equilibrium of velocities, and is shifted by some bulk flow vD(r), then

2. Doppler Spectroscopy 20

the local lineshape is described by a Gaussian profile. That is,

g(r, l; ξ) =1√

πV2th

exp

(−(ξ − VD.l)2

V2th

)(2.13)

where VD = vD(r)/c and is related to the Doppler-shift of the central frequency.The local linewidth is described by the normalised thermal velocity of the particleVth(r)(= vth/c) which is given by

Vth =

√2kBTi

mic2 (2.14)

where Ti and mi are the temperature and mass of the ion respectively. However,since the measured emission spectrum is line-integrated, it is not itself a Gaussianbut is the sum of many local Gaussians of various linecentre shifts and widths.

For a velocity distribution as just described, it can be shown that, with a suf-ficient number of line-integrals covering the region of interest, the spatial distri-bution of the parameters I0, Ti and vD can be recovered. Scalar tomography tech-niques, as discussed in Chapter 3, are used for the first two parameters while thesolenoidal component only of the intensity-weighted drift velocity vector fieldcan be recovered using vector tomography [Howard, 1996]. §2.3 below outlinesthe optical methods used in this thesis to measure the parameters of Doppler-broadened emission lines.

2.3 Fourier Transform Spectroscopy

Fourier transform spectroscopy (FTS), where the spectrometer measures theFourier transform of the spectral lineshape, has many advantages [Bell, 1972].These include high light throughput and the ability to encode the spectral infor-mation in the time domain. This allows a single detector to record the spectrumfor a single sight-line as opposed to a grating spectrometer, which requires a de-tector array to record the dispersed lineshape. A single detector per spatial chan-nel gives scope to perform two-dimensional imaging of an extended light sourceusing a detector array. This is not possible using traditional frequency-domaininstruments.

The following discussion is based on [Born and Wolf, 1959; Howard et al.,2003] and deals with Doppler-broadened emission lines. Consider the analyticrepresentation of the field of a quasi-monochromatic light wave as given by

u(t) = A(t) exp(−i2πν0t) (2.15)

where the complex amplitude A(t) varies slowly compared with 2πν0t. If awave is delayed by a time τ with respect to a second wave a phase difference

2. Doppler Spectroscopy 21

φ (=2πτν0) is created between the waves. The intensity pattern formed whenthe waves are recombined is known as an interferogram. If the waves are nom-inally of the same amplitude, the intensity of the interferogram as recorded at asquare-law detector can be written as

S(φ) =Iy2

(1 + [γ exp(iφ)]) (2.16)

where Iy is the line integral of the spectrally-integrated local emission intensity(zeroth moment of the measured spectrum) and is given by

Iy =∫

LI0(r)dl (2.17)

The symbol in (2.16) denotes that the real part of the complex number is beingtaken. The quantity γ is termed the complex degree of optical coherence† (or justthe optical coherence) and is given by γ = 〈A(t)A∗(t + τ)〉/Iy where the anglebrackets denote a time average. The optical coherence is related to the measuredspectrum via the Wiener-Khintchine theorem [Wiener, 1930; Khintchine, 1934] as

γ(φ; l) =1Iy

∫ ∞

−∞y(ξ; l) exp(iφξ)dξ (2.18)

The local optical coherence is obtained by taking the Fourier transform of thelocal lineshape g(r, l; ξ) (defined in (2.11)):

G(r, l; φ) ≡ F [g(r, l; ξ)] = exp(iφVD.l)G0(r, φ) (2.19)

where the contributions from the drift velocity VD and the body of the distribu-tion are separated using the shift theorem of the Fourier transform. G0(r, φ) isa central slice of the Fourier transform of the spherically symmetric distributionfunction f0 (that is, the distribution function in the local (non-drifting) frame ofreference.)

As the time delay τ increases beyond the coherence time of the original wave(see [Hecht, 1987, p. 264]), there is a decrease in the visibility (or contrast) of theinterference fringes. This loss of coherence is due to the finite spectral bandwidthof the light and is related to the linewidth. The visibility of the fringes is equiv-alent to the modulus of the optical coherence. Utilising (2.11), (2.12), (2.19) and(2.18) it can be shown that the fringe visibility can be written as

ζ ≡|γ(φ; l) |= 1Iy

∫L

G0(r, φ)dl (2.20)

†γ is equivalent to the cross-correlation function in the theory of stationary random processes.

2. Doppler Spectroscopy 22

Note that, due to the Fourier transform given in (2.19), the fringe visibility isindependent of the spatially varying drift VD. For a Maxwellian velocity distri-bution (LTE), the Fourier transform of the lineshape in the local drifting framebecomes

G0(r, φ) = exp

[−φ2V2

th4

]= exp

[−Ti(r)Tc

](2.21)

where Ti(r) is the local species temperature. Tc is an instrumental ‘characteristictemperature’ and is given by

kBTc =12

miv2c (2.22)

with the ‘characteristic velocity’ vc set by the total interferometer phase delay φ:

vc =2cφ

. (2.23)

The fringe visibility (2.20) can now be expressed in terms of the temperature as

ζ =1Iy

∫L

I0(r) exp[−Ti(r)

Tc

]dl (2.24)

Information regarding the drift velocity is found in the Doppler shift of thecentral frequency and is manifested in the phase of the interferogram. Providedthat φVD < 1—which is true if the drift velocity is less than the thermal velocityand a suitable phase delay is chosen—then the change in the interferogram phaseis given by

φD =φ

ζ Iy

∫L

I0(r) exp[−Ti(r)

Tc

]VD.l dl (2.25)

2.3.1 Measurement Principle

The phase delay φ in (2.16) can be obtained by splitting an input light beamand having the resulting two beams then traverse separate paths, one of whichmay have a different length, before recombining the beams at a detector. The ba-sic layout of this scheme (a Michelson interferometer) is shown in Figure 2.3, withthe displacement of the mirror in the measurement arm creating the time delayτ. The interferometer is desirable as the instrument used to create the phase de-lay due to its high etendue (optical throughput) [Jacquinot, 1954, 1960; Van Heel,1967].

To measure the contrast ζ and phase shift φD of the interferogram the phasedelay φ is modulated sinusoidally about some initial delay φ0–which is set by thepath length delay τ—with a frequency Ω and amplitude φ1. This gives the totaldelay as

φ = φ0 + φ1 sin Ωt (2.26)

2. Doppler Spectroscopy 23

measurementarm

referencearm

beamsplitter

source

light

detector

xcombinedbeam

Figure 2.3: Schematic of a Michelson interferometer.

Using some simple trigonometric identities, (2.16) then becomes

S(φ) = Iy[1 + γc cos(φ1 sin Ωt) − γq sin(φ1 sin Ωt)] (2.27)

where γ = γc + iγq =|γ | exp(iφ0 + iφD). The Bessel expansion shows that γc andγq are proportional to the even and odd harmonics of the modulation frequency,respectively. This approach allows line-integrated versions of the velocity distri-bution function parameters I0, Ti and vD to be directly related to Iy, γc and γq,which are carried by the DC level and the even and odd harmonics of Ω.

Figure 2.4 illustrates the measurement concept, where the visibility of the in-terferogram fringes is recorded by tracing out the fringe amplitude via the mod-ulation and the shift in the interferogram phase is apparent by a shift in the oper-ating point of the modulation. The signals generated and the harmonics of thosesignals are also shown.

2.4 Modulated Optical Solid State Spectrometer

This section describes a robust, compact implementation of the concepts dis-cussed in §2.3 and which can easily be adapted for multi-channel operation.

Michelson interferometers operating in the visible spectrum can be very sus-ceptible to vibrational and acoustic noise interference due to the small wave-length and the separation of the light beams before recombination. The Modu-

2. Doppler Spectroscopy 24

Interferogram

0 2 4 6 8 10 12 14Path delay in half waves

0.5

1.0

1.5

2.0

Frin

ge c

ontr

ast

0.5

1.0

1.5

2.0

Frin

ge c

ontr

ast

Wavelength w1=1.0

Temperature=T_C

Wavelength w2=0.98

Temperature=1.4*T_C

FTS signal

0 1 2 3Time (periods)

Power spectrum

0 1 2 3 4Harmonic No.

Figure 2.4: Simulated interferograms showing the effect on the interferogram phase of ashift in line-centre frequency (exaggerated for illustrative purposes). The dashed verticalline is the operating point and corresponds to the interferometer delay while the boldsection is the portion of the interferogram traced out by the modulating path length.Information regarding vD is carried in the odd harmonics. The fringe contrast (visibility)varies with changes in the temperature of the emitting species with the even harmonicsconveying this information.

2. Doppler Spectroscopy 25

lated Optical Solid-State (MOSS) spectrometer is essentially a Michelson interfer-ometer that uses a solid-state birefringent optical element to produce the phasedelay φ0.

The extraordinary (‘E ’) and ordinary (‘O ’) axes of the birefringent materialare at right angles to each other, have different refractive indices and are usuallyoriented normal to the light propagation direction. The phase speed for lightpropagating through a material of refractive index n is vφ = c/n. The E axis isknown as the ‘fast’ (or Z) axis due to its lower refractive index. For monochro-matic light propagating perpendicular to the fast axis of a birefringent plate ofthickness L there is a phase difference between the E and O waves given by

φ0 =2πν0

c(LnE − LnO) (2.28)

where nE and nO are the extraordinary and ordinary refractive indices respec-tively. In terms of the crystal birefringence,

B = nE − nO (2.29)

(2.28) can be written more succinctly as

φ0 =2πν0

cLB = 2πν0τ = 2πN (2.30)

where τ is the time delay introduced between the waves and N is the order ofinterference or the total difference in the number of waves.

By polarising the incident light at 45◦ to the E axis the electric field is re-solved into equal amplitude E and O waves by the birefringent element and arephase-shifted according to (2.30). A second polariser, placed after the birefrin-gent material, transmits the components of the orthogonal E and O waves thatare in the same polarisation state. These components then interfere to producethe intensity variations at the detector. The polarisation sequence is depicted inFigure 2.5.

It is a property of some birefringent materials that they are also electro-optic,meaning that the refractive index changes in the presence of an electric field.Thus, the effective path length difference between the E and O waves can be var-ied by modulating the birefringence using an oscillating voltage applied acrossthe crystal (parallel to the E or O axis) and transverse to the light propagationdirection. This gives a robust method, not reliant on critical alignment, for pro-ducing the required optical path length modulation of (2.26).

The electro-optic, birefringent element is a Y-cut lithium niobate (LiNbO3)crystal whose fast axis is at 45◦ to the plane of polarisation. LiNbO3 is an artificialcrystal which—in addition to the two mentioned properties—also exhibits pyro-, piezo- and ferroelectric effects as well as photo-elasticity [Weiss and Gaylord,

2. Doppler Spectroscopy 26

Plane polarised light

Polarisation resolvedinto components at 45°

Slow wave delayed withrespect to fast wave in birefringent wave plate

Time delay

t

Components interfere at final polarising cube and onecomponent is selected

Polarisation States

Z axis

L

d

rejected light

(used for other

instruments)

possibly to

second

detector

calibration light

plasma light

to detection

system

Figure 2.5: The polarisation sequence to produce an interferogram using an electro-opticand birefringent element with suitable polarisers. Figure courtesy of C. Michael.

2. Doppler Spectroscopy 27

1985]. It has nE = 2.25 and nO = 2.35 for a birefringence of B ≈ −0.1. The typicalcrystal aperture is 40 mm, while the length L is in the range 5-100 mm dependingon the expected coherence of the spectral line. Narrow lines require a larger delayin order to produce a significant change in the interferogram fringe contrast.

The additional small delay modulation φ1 sin(Ωt) of amplitude φ1 = π/2(see (2.26)) is obtained by electro-optically modulating the crystal at a frequencywell above or below its natural mechanical resonant frequency (which is typicallyaround 100 kHz). This is achieved by placing electrodes across the Z faces ofthe crystal, transverse to the light direction. The phase difference φ1 is given by[Kaminow, 1974]

φ1 =πLVν0

d c(n3

Er33 − n3Or13) (2.31)

Here d is the width of the crystal (see Figure 2.5). r33 = 28.8 × 10−12 m/V andr13 = 7.7 × 10−12 m/V are electro-optic tensor co-efficients. Voltages in the range1.5–2.5 kV are required to produce a change in the refractive indices. It can beshown that a phase modulation of ∼140◦ gives optimal signal-to-noise ratio forthis particular encoding method. It is also possible to apply a DC field to finetune φ0.

In practice, the optics are enclosed in a solid light-tight enclosure which al-lows a number of ports for input and output of various light beams. The inputlight can be coupled directly from the source through appropriate apertures orby using optical fibres and a collimating lens to produce parallel light. An in-terference filter isolates the desired emission line from the remainder of the vis-ible spectrum before the interferogram is detected using a photomultiplier tube.A polarising beamsplitter cube is used as the initial polariser. This allows thetransmission of the horizontally polarised component of the input light and thereflected component can be absorbed or relayed to another device (e.g. anotherMOSS or detector). Through the opposite port of the polarising cube anotherlight source, such as a calibration laser, can be introduced. The second polariseris a high efficiency, thin film type. A typical layout of the optical elements of aMOSS spectrometer is shown in Figure 2.6.

The MOSS spectrometer can be adapted easily for imaging applications by theaddition of a multi-channel detector and imaging lens after the final polariser.§4.2.1 describes in more detail the implementation of the spectrometer as usedfor experiments covered in this thesis.

2.5 MOSS Spectrometer Properties

It is important to know the spectral response of the spectrometer as well asmore detailed information including the instrument dispersion, the instrumentcontrast (analogous to a grating spectrometer’s slit function) and the absolute

2. Doppler Spectroscopy 28

Figure 2.6: The optical layout for the Modulated Optical Solid-State (MOSS) spectrome-ter.

delay (wavelength calibration). The light throughput, or etendue, as well as thesensitivity of the linewidth and lineshift estimates to photon noise, are also con-sidered. The following discussion is based largely on [Howard, 2002].

2.5.1 Spectral response

The spectral response shown in Figure 2.7 was measured with white light in-put using a high resolution grating spectrometer. The ArII 488 nm plasma emis-sion line is also shown. The spacing between fringe peaks is a rough measure ofthe spectral resolution ν0/Δν = N and is proportional to the phase delay φ0. Re-call that the delay is chosen to be comparable to the linewidth. The fringe depth(visibility or contrast) is governed by the instrument contrast and the spectralwidth of the radiation (in this case, the grating spectrometer resolution). Theapodizing envelope is a result of the interference filter that is used to reject lightoutside the passband of interest. Modulating the phase delay shifts the inter-ference pattern and allows the intensity variations, due to changes in the centrefrequency and/or spectral width, to be determined.

2. Doppler Spectroscopy 29

Figure 2.7: Overlay of plasma spectrum in the vicinity of the 488 nm ArII line and themoss spectral response function. The interferogram is apodized by the narrow bandpassinterference filter that is used to isolate the spectral line from the plasma background.The period of the interferogram is inversely proportional to the MOSS interferometerdelay (in this case, the crystal thickness is 24 mm) and is chosen to be comparable withthe Doppler-broadened width of the emission line.

2. Doppler Spectroscopy 30

2.5.2 Phase delay dispersion

In general, the crystal birefringence is a function of frequency B(ν), and, be-cause interferometers integrate over all frequency components (see (2.18)), it isnecessary to take account of the dispersive properties when evaluating the effec-tive (or group) phase delay. The delay at frequency ν = ν0 + δν, where δν is asmall shift, is given approximately by

φ(ξ) ≈ φ0 + κφ0ξ (2.32)

whereκ = 1 +

ν0

B0

∂B∂ν

(2.33)

accounts for the frequency dependence of the birefringence. The Sellmeier equa-tions give nE and nO as functions of wavelength and for lithium niobate are[Weiss and Gaylord, 1985]

n2E(λ) = 4.5820 + 0.099169/(λ2 − 0.04443) − 0.021950λ2

n2O(λ) = 4.9048 + 0.11768/(λ2 − 0.04750) − 0.027169λ2 (2.34)

where the wavelength λ is in μm. Thus the phase delay due to the crystal mustbe scaled by the factor κ and at 488 nm the correction term is κ = 1.54. This has aflow-on effect for measured contrast and phase shifts such that

vc =⇒ vc/κ

φD =⇒ κφD(2.35)

Figure 2.8 shows the wavelength dependence of the birefringence and thegroup birefringence κB. Experimental estimates of the group birefringence de-termined by fitting the spectral response† at a number of points in the visiblespectrum are also shown.

2.5.3 Instrument contrast

Including the delay dispersion, the signals at the output of the interferometer(2.27) become

S± = I0 ± I0ζ cos [φ0(1 + κVD.l) + φ1 sin (Ωt)] (2.36)

where φ0 + δφ = φ0(1 + κVD.l) and VD.l is the component of the normalised flowvelocity in the direction of view as specified in (2.13). The total fringe visibilityis given by | γ |≡ ζ = ζIζs. The term ζs ≡ exp (−Ti/Tc), where Tc is given by(2.22) and uses the modified vc in (2.35), is the change in fringe contrast related

†As for Figure 2.7 but with the interference filter removed.

2. Doppler Spectroscopy 31

Figure 2.8: Plot of the wavelength dependence of the birefringence for lithium niobateobtained using the Sellmeier equations. The dashed curve is the monochromatic birefrin-gence while the solid curve shows the calculated effective birefringence brought aboutby inclusion of the refractive index dispersion. The square symbols are measurementsobtained as discussed in [Howard, 2002].

to the source temperature. The instrumental fringe contrast ζI, which is analo-gous to the familiar slit function for grating spectrometers, is determined by thesolid angle of light collected and optical imperfections in the spectrometer. It isa constant factor and can be represented with an equivalent ‘instrument temper-ature’ TI where ζI = exp (−TI/Tc). Because the interferogram is monitored at asingle fixed delay, the source temperature Ti can be obtained from the measuredfringe contrast through subtraction of exponents which are proportional to themeasured and instrumental temperatures. This compares favourably to gratingspectrometers which require noise-prone deconvolution of the instrument pro-file.

For Ti = Tc the contrast reduces to 1/e of its maximum. This condition is usedas a measure of the instrument resolving power RC [Thorne, 1988] and (2.22)gives

RC ≡ ν0

Δν= πNeff (2.37)

where Neff = φeff/2π = κφ/2π.The variation of fringe contrast ζs with phase delay φ0 for argon ion temper-

atures in the range 10 to 100 eV is shown in Figure 2.9. The vertical line corre-sponds to the delay introduced by the LiNbO3 crystal at 488 nm and representsa characteristic temperature Tc ≈ 14 eV. Observe that a 90% variation in fringe

2. Doppler Spectroscopy 32

40 eV 5 eV

Tc = 36 eV

Tc = 14 eV

Figure 2.9: Plot of fringe contrast versus FTS phase delay for a uniform plasma with iontemperatures 5, 10, 15, . . . 40 eV. The dashed lines correspond to the delays introduced by25 mm and 40 mm lithium niobate cells at 488 nm. Note particularly the wide dynamicrange for sensitivity to temperature changes.

contrast represents roughly an order of magnitude variation in temperature.The instrument fringe contrast (or temperature) is limited fundamentally by

the range of angles subtended by the source, or the ‘field-of-view’. For smallangles of incidence θ, the delay difference between the characteristic E and Owaves after propagation through the birefringent crystal is given by [Steel, 1967]

φ ≈ φ0

[1 − θ2

2nO

(cos2 ϑ

nO− sin2 ϑ

nE

)](2.38)

where ϑ is the azimuthal angle from the optic axis. The mutual coherence ofthe recombining waves at the detector is obtained by integrating over the sourceirradiance distribution (van Cittert-Zernike theorem). For a uniform, monochro-matic circular source with solid angle Ω (= πθ2

max, for θ small), the resultinginstrument coherence is given by

γI =(

1 − Ω2N2

12n4 + . . .)

exp[−iφ0

(1 − BΩ

8πn3 + . . .)]

(2.39)

meaning that the instrument temperature evaluates to

TI ≈ Tc

(Ω2N2

12n4

)(2.40)

2. Doppler Spectroscopy 33

where n is the mean refractive index. The reduction in contrast due to the inte-gration over the solid angle is negligible provided the parameter ε = ΩN/n2 issmall. For the stated assumptions this parameter can be re-written as

ε =πNθ2

n2 (2.41)

Hence there is the usual trade-off between resolving power RC(∼ N) and col-lection solid angle, but it is mitigated by a gain factor of n2/

√2 compared with

a free-space Michelson interferometer. Widening the field-of-view also gives afurther gain [Steel, 1967; Michael et al., 2001]. Note the strong variation in thephase delay as a function of incident angles which is shown in the top left of Fig-ure 2.10. To improve the contrast a second crystal is placed after the first and withits fast axis orthogonal to that of the initial crystal. A half-wave plate is insertedbetween the crystals, as shown in Figure 2.11, to rotate the polarisation statesemerging from the first crystal to match the orientation of the second. Crossingthe patterns in this way results in a very flat phase variation.

θxθ

y

phas

e sh

ift

="Field widened"

system

θxθ

y

phas

e sh

ift

+

1st crystal [at 0°] 2nd crystal [at 90°]

θxθ

y

phas

e sh

ift

Figure 2.10: The addition to the phase spatial variation of its complement (90◦ rotation)creates a flat phase shift for a large field of view. Figure courtesy of C. Michael.

For a typical implementation of the spectrometer as discussed in §4.2.1, theuse of crystals of total thickness 50 mm produces an effective delay Neff ∼ 16000

2. Doppler Spectroscopy 34

Figure 2.11: Polarisation sequence of a ‘field widened’ configuration. The half-waveplate rotates the polarisation states emerging from the first birefringent crystal to matchthe fast axis of the second.

waves (from (2.30)) and a nominal wavelength resolution of 0.01 nm at 488 nmfrom (2.37). Light input, via optical fibres of diameter 1 mm and numerical aper-ture NA 0.3, is collimated by an f /# = 1.8 lens with focal length of 85 mm.Noting (2.42) gives the resulting collimated beam a half angle of θmax ∼ 0.006giving an evaluation of (2.41) as ε ≈ 0.34. Thus (2.40) yields TI as ≈ 1% of Tc

and the decrement in fringe contrast due to the field-of-view is small. How-ever, imperfections in the crystals degrade the contrast further and may includesuch things as unequal surface reflection losses for different polarisation com-ponents, crystal birefringence inhomogeneities and non-parallelism of surfaces.Other factors such as non-unity extinction ratio for the polarising cubes and mis-alignment of the optical components may also contribute to a measured instru-ment contrast which is lower than (2.40) indicates. It is generally in the rangeζI = exp (−TI/Tc) ∼ 70 − 90% at 488 nm corresponding to TI/Tc ∼ 0.35 − 0.1.

The instrument contrast can be compromised further by the combined actionof piezo-electric and acousto-optic effects at high electric field strengths appliedto the crystal. These distortions, which become significant for modulation fre-quencies near the crystal acoustic resonances, result in a variation of the instru-ment contrast at the modulation frequency with the effective decrease being afunction of φ0. However, for the typical drive voltages used in experiments de-scribed in this thesis, measurements of the instrument contrast variation with φ0

have shown this to be a small effect (standard deviation of 1% of the mean ζI)[Howard, 2002].

2. Doppler Spectroscopy 35

2.5.4 Light throughput

To better understand the Jacquinot (or light throughput) advantage the im-plementation of the MOSS spectrometer, discussed in §4.2.1, is compared with agrating spectrometer of equivalent resolving power. The etendue of an opticalcomponent is given by

E = AΩ (2.42)

where A is the area of the element aperture and Ω is the collection solid anglesubtended by the source at the optic axis.

A standard F/5 grating spectrometer of focal length 0.5 m equipped with a2400 lines/mm grating operating in first order and using a 25μm slit width willgive RC ∼ 5000. For a slit height equal to the 40 mm clear diameter of the lithiumniobate crystal† the grating spectrometer etendue is EG = AGΩG = 0.016 Sr mm2,where AG = 25μm × 40 mm is the slit area and ΩG is the collection solid angleset by the spectrometer F-number. The limiting field-of-view for the MOSS spec-trometer is obtained by setting Ti = Tc in (2.40) to obtain for the MOSS etendueEM = 0.9 Sr mm2 ∼ 120EG. Note that with field widening and larger apertures,this etendue disparity is greater than previously reported [Howard, 2002]. Ulti-mately though, the amount of light collected from the source may be limited byspatial resolution considerations rather than the etendue of the spectrometer.

2.5.5 Instrument delay

For absolute flow velocity measurements, the calibration of the interferogramphase requires comparison with light from a standard source. This requirementis because of uncertainties in the determination of the absolute crystal delay φ0

due to, for example, thermal drifts of the refractive indices. Preferably the sourceshould generate an identical emission, with lasers or spectral calibration lampsparticularly suitable. This ‘instrument phase’ φI also requires that light from bothplasma and calibration sources have a common geometry through the spectrom-eter so that this phase offset is compensated. This can be arranged, for example,by using lens-coupled optical fibres to relay light from the standard source to thespectrometer. In the absence of a suitable calibration source, or careful tempera-ture stabilisation, only relatively fast temporal changes in the mean normalisedflow speed 〈VD〉 can be measured accurately. The absolute value of these changesis easily calculated from the measured phase shift and the estimated initial phaseoffset φ0.

†The primary limiting aperture for this implementation of the MOSS spectrometer.

2. Doppler Spectroscopy 36

2.5.6 Noise sensitivity

The sensitivity of the MOSS signal to changes in the spectral linewidth relieson the instrument temperature. Without regard to the details of the demodula-tion scheme, a rough estimate of the sensitivity of the signal described in (2.36)to temperature changes is given by

αT ≡ maxφ

(∂S∂Ti

)= − I0

Tcζ (2.43)

The sensitivity deteriorates when the instrument fringe contrast is poor and at-tains a maximum with respect to Tc when Ti + TI = Tc (provided TI < Tc).

The ability to distinguish changes in light intensity arising from temperaturevariations is determined, at best, by the Poisson noise of the intensity. The signalsensitivity to this noise, averaged over a modulation period, is given by αI ≡(∂S/∂I0) = 1. The sensitivity of the derived temperature to photon shot noise isestimated by setting dS = αIdI0 + αTdTi = 0 to obtain

dI0

I0= ζ

dTi

Tc. (2.44)

Thus, for a given light noise level, the absolute uncertainty in the derived sourcetemperature increases as fringe visibility decreases. (2.44) can be re-written as

dTi

Ti= kT(r)

dI0

I0(2.45)

where r = Ti/Tc and kT(r) = exp r/r is the factor relating the fractional lightnoise level to the relative uncertainty in the inferred species temperature. Thebehaviour of kT(r) as a function of normalised species temperature r is shown inFigure 2.12.

The MOSS spectrometer is most sensitive to relative changes in Ti when Ti =Tc. For the H-1 system, and at Ti = Tc = 36 eV, intensity noise on the order of5% corresponds to temperature uncertainties of about 4 eV. The divergence whenr is small is due to the exponential relationship between measured contrast andsource temperature. The dynamic range can be increased effectively by usingmultiple birefringent plates mutually aligned at 45◦, as is the case for a field-widened system.

To maintain an optimum signal-to-noise ratio (SNR), higher source tempera-tures require a higher value for Tc. Combining (2.22), (2.23) and (2.30) indicatesthat Tc is inversely proportional to L2 so that the voltage required to obtain amodulation depth of at least π/2 is proportional to T1/2

c . This difficulty can beovercome through a reduction of the crystal dimensions perpendicular to the op-tic axis, but at the cost of light throughput. An alternative approach is to combine

2. Doppler Spectroscopy 37

kTkv

Figure 2.12: Dependence of the scaling functions kT(r) and kv(r) relating the fractionallight noise and the relative uncertainty in the inferred temperature Ti and drift velocityvD as a function of the normalised temperature r = Ti/Tc.

crystals so that their individual delays subtract while the modulation voltagesare of opposite polarity. In this case, however, the instrument temperature is alsoincreased unless a half wave matching plate (field widening) is used.

The sensitivity to flow speed changes is estimated by

αv ≡ maxφ

(∂S

∂vD

)= I0ζκφ (2.46)

With respect to the characteristic temperature Tc, the maximum sensitivity toflow speed variations occurs when Ti + TI = Tc/2. This optimum value is ob-tained because the increase in sensitivity to vD with delay φ is opposed by adecrease in the contrast ζ.

When the temperature is constant, the ability to resolve small flow velocitychanges depends on the light signal to noise ratio. Setting dS = αIdI0 + αvdvD =0 gives

dvD

vth= kv(r)

dI0

I0(2.47)

where the scaling factor kv = exp (r)/2√

r, which is analogous to kT, is alsoshown in Figure 2.12. For argon at Ti = Tc/2 = 18 eV, vth ≈ 9 × 103 m/swhich implies a velocity resolution δvD ∼ 600 m/s for δI0/I0 = .05. Simulations[Howard, 2002] confirming the form of the noise sensitivity factors displayed in

2. Doppler Spectroscopy 38

Figure 2.12 also indicate a slow dependence on the modulation depth φ1>∼ π/2

which is optimum for φ1 ∼ 130◦ [Sasaki and Okazaki, 1986].In summary, the MOSS is an ideal instrument for measuring the ion emis-

sivity, ion temperature and bulk flow velocities in the radiation-dominated low-temperature plasmas of the H-1NF heliac.