Conclusiones

Interferometers are devices which cause two or more waves of light to interfere. Interfer- ometers used in plasma studies include, but are not limited to, mechanical interferometers of the Michelson-Morley, Mach-Zehnder and Fabry-Periot [49] type which require movable components and rely on a spatial variation between branches of the device to introduce the delay between the orthogonally polarised components of the light.

Figure 2.5: Standard components of a polarising Michelson-Morley interferometer, showing how light is channeled through the instrument and combined onto the screen to form interference fringes.

By following the light emission of a single pixel as the delay τ is increased will give the contrast

and phase associated with the coherence.

A Michelson-Morley interferometer is shown in figure 2.5. Quasi-monochromatic light is selected from a light source using a filter and is then passed through a beam splitter so that each of the components are transmitted along one arm of the instrument. The components are reflected by a mirror and recombined at the beam splitter. The light is focused by a lens onto a screen. When the lengths of the arms are equal the light recombines in phase. When the length of one of the arms is extended, there is a time delay,τ, introduced between the two orthogonal components. The components now interfere on recombination causing the intensity image on the screen to split into a series of light and dark circular fringes. The circular fringes are caused by slight difference in τ due to angular effects from the dispersion of the light through the lens.

Features such as shifts and broadening of the spectral line are related to changes in the phase and contrast (envelope) of the coherence and can be determined using a Michelson- Morley type interferometer using the plasma emission as the source. A standard set-delay interferometer is able to sample the coherence at the fixed delay. Interferometers which scan the coherence by varying the delay between the emission components are called Fourier transform interferometers.

The following discussion on the interferometer for measuring spectral shifts and broadening is based on established theory presented in [48] and [42]. The interferometric signal for a

16 Spectroscopic and probe diagnostics on MAGPIE

given pixel on the screen at delayφ= 2πντ, is given by

S±(φ) =

I0

2 1± <[˜γ(φ)]

(2.4)

whereI0 is the spectrally integrated irradiance of the incident beam (the amount of light collected by the interferometer over all wavelengths) and ˜γ(φ) is the phase dependent complex-coherence. The ± indicates the transmitted and reflected signal components, respectively.

For an incident field u(t), the spectrally integrated irradiance and the complex coherence are defined asI0 =huu∗i and ˜γ(φ) =hu(t)u∗(t+τ)i/I0 respectively. The brackets denote the time average.

The Wiener-Khinchine theorem states ifu(t) has a Fourier transform given byU(ν), then its autocorrelation functionhu(t)u∗(t+τ)iwill have the Fourier transform,|U(ν)|2_{, which} is equal to the power spectrum of the radiation i.e.

˜ γ(τ) = 1 I0 Z ∞ −∞ |U(ν)|2exp(i2πντ) dν (2.5)

The power spectrum I(ν) = |U(ν)|2 _{of the emission field u(t) describes the intensity} contributed from each of the different frequencies in the frequency spread ∆ν present in the emission.

Considering a quasi-monochromatic light source, where the spectral line has a central frequency ν0 and has a small spread, ∆ν << ν0, the complex scalar electric field may be written as an analytic signal

u(t) =A(t) exp(i2πν0t) (2.6)

Where the amplitude, A(t), varies slowly with respect to the complex phasor.

For the snapshot polarisation interferometers used in this work, the emission must pass through birefringent optics which can result in optical dispersion effects. The time delay is therefore a function of frequencyτ(ν) and the phaseφ= 2πντ(ν) can be approximated as a first order Taylor series expansion

φ= 2π ν0τ0+ τ0+ν0 dτ dν (ν−ν0) =φ0+κφ0ξ (2.7)

where τ0 = τ(ν0). The phase delay at the central frequency is φ0 = 2πν0τ0 and ξ = (ν−ν0)/ν is a normalised frequency difference coordinate. The constantκ describes the

§2.1 Spectroscopic diagnostics in plasmas 17

optical dispersion in the delay

κ=1 +ν0 τ0 dτ dν (2.8)

Allowing for dispersion and changing the variable of integration the complex coherence may be rewritten as ˜ γ(φ0) = exp(iφ0) I0 Z ∞ −∞ I(ξ) exp(iφˆ0ξ) dξ (2.9) = γ(φ0) exp(iφ0) (2.10)

where ˆφ0 =κφ0 is the group delay and

γ(φ0) = 1 I0 Z ∞ −∞ I(ξ) exp(iφˆ0ξ) dξ (2.11)

The term I(ξ) is the spectral distribution of the irradiance. It is convenient to separate the complex coherence into an amplitude and a complex phasor as each quantity can independently be related to the contrast and phase of the interferogram. This is shown by substituting equation 2.10 into equation 2.4 so that the interferometric signal for a quasi-monochromatic light source is given as

S±( ˆφ0) =

I0

2 1± <[γ(φ0) exp(iφ0)]

(2.12)

This derivation shows the connection between the interferogram generated by an inter- ferometer and the coherence in the temporal domain. For the case of the Doppler effect, spectral shifts and broadening of the spectral line in the spectral domain will be observed as shifts in the phase and contrast of the interferometric fringe pattern measured by an interferometer.

An interferometer coupled to a photo detector array or camera can provide even greater spatial resolution (1D or 2D) compared to spectrometers and they are not limited by light throughput which in theory, makes them a powerful diagnostic tool for spectral measurements of plasmas. In practice however, mechanical interferometers such as the Michelson-Morley interferometer are rarely used for measuring spectral broadening or shifts in plasmas as these effects are usually very small and therefore such systems require very precise alignment and stability of the optical components in order to detect the resulting changes in the coherence. For measurements using short wavelengths, such as the visible region, vibrational noise becomes the major limitation, scaling as 1/λ2 [49]. For this reason many interferometers operate in the 100 µm−2000µm wavelengths [49].

Typically, plasma features which are difficult to obtain using spectroscopic or interfero- metric methods, are instead measured using electronic probes or laser techniques, such as

18 Spectroscopic and probe diagnostics on MAGPIE

laser induced fluorescence (LIF) or laser absorption.