Microwave interferometer

By launching waves across a plasma volume, this diagnostic infers the electron density by measuring the refractive index of the plasma. To penetrate the plasma without any significant interaction electromagnetic waves with frequencies much higher than the plasma frequency are required; generally, in the microwave range. The refractive index is measured from the phase delay of the wave as it traverses the volume of plasma. By combining the “probing” beam with a coherent “reference” beam the phase delay is inferred from an interference pattern. For microwave frequencies much greater than the plasma frequency the phase delay and the line

integrated electron density are related by Eq. 1 [60]. The terms∆𝜙𝜙,𝑛𝑛_𝑒𝑒and𝑛𝑛_𝑐𝑐represent the

phase delay, electron density and cut-off density for the given microwave

frequency𝜔𝜔respectively. The variable “s” describes the path length of the beam. The

terms𝑐𝑐,𝜖𝜖₀,𝑚𝑚_𝑒𝑒and𝑒𝑒represent the speed of light, permittivity of free space, electron mass and electron charge respectively. For more details on this diagnostic refer to the following books [60], [61] or articles [62], [63], [64]. Due to their non-intrusive nature and simpler theoretical framework based on the cold plasma approximation MWI measurements are considered to be more reliable than electrostatic probes. The main contribution of the MWI in this work is to provide absolute electron density measurements and allow calibration of the DLPs.

∆𝜙𝜙=_{2𝑐𝑐𝑛𝑛}𝜔𝜔

𝑐𝑐� 𝑛𝑛𝑒𝑒(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑛𝑛𝑐𝑐=

𝜖𝜖0𝑚𝑚𝑒𝑒𝜔𝜔2

𝑒𝑒2 Eq. 1

3.2.1

Description

The type of MWI implemented in MAGPIE is referred to as a “double pass 140 GHz heterodyne polarization interferometer”. The term “heterodyne” refers to the use of two distinct frequencies; namely, (1) a microwave and (2) a modulating (intermediate) frequency. The topology of the MWI in MAGPIE corresponds to a Michelson interferometer and its setup is illustrated in Figure 26. A Voltage Controlled Oscillator (VCO) provides plane polarized microwave radiation with a Gaussian radial profile at a central frequency of 140 GHz and a

bandwidth∆𝑓𝑓of 2 GHz. A 100 kHz sawtooth waveform is used to linearly modulate the VCO’s

radiation from 139 to 141 GHz. At this frequency (140 GHz), quasi-optical theory has to be used to describe the propagation of the microwave beam [64]. The microwave radiation from the VCO is separated into two coherent beams using a 45 degree wire beam splitter. We refer to these as the “reference” and “probe” beams. Using an elliptical mirror, the “probe” beam is focused down to 3 cm in diameter and injected into the plasma through a TPX window in Port 2 (Figure 25) at z = 14 cm.

Figure 26, Schematic of the microwave interferometer in MAGPIE

The TPX window is tilted a few degrees to reduce reflections back into the interferometric setup. The “probe” beam traverses the plasma volume twice as it is reflected from a mirror across the plasma. The reflected outgoing “probe” beam is separated from the incoming “probe” beam using a quarter wave plate by rotating the beam’s polarization 45 degrees per pass for a total rotation of 90 degrees. The incoming “probe” and “reference” beam are combined at a microwave diode detector and the interference pattern is measured as a function of time. As the plasma density increases at the start of every RF pulse, the “probe” beam experiences a phase delay with respect to the “reference” beam due to the changing refractive index of the plasma and the resulting interference pattern changes accordingly.

3.2.2

Principle of operation

The equations describing the operation of the heterodyne polarization interferometer are described next. The “reference” and “probe” beams are described by two complex electric field plane waves as in Eq. 2 and denoted with subscripts 1 and 2 respectively. We represent the length traversed by the beams with the symbol “s”. In addition, both beams are coherent and thus share the same frequency and wavenumber. However, due to path length difference and the

effect of plasma phase shift they will in general have different phases𝜙𝜙_𝑛𝑛. If we assume both

beams have the same magnitude, the resultant electric field at the diode detector is given by the

𝐵𝐵 ⨀ Plasma Mirror TPX 𝜆𝜆/4 wave plate 𝜆𝜆/2 reflector 45∘wire beam splitter Diode detector 140 GHz 𝜇𝜇wave source VCO 100 kHz sawtooth

superposition of the “reference” and “probe” beams (Eq. 3). The terms∆𝐿𝐿and∆𝜙𝜙represent the

difference in path length and phase between the beams. The termsΣ𝐿𝐿andΣ𝜙𝜙represent the sum

of the path lengths and phases of the beams. By linearly varying the radiation’s wavenumber using the 100 kHz sawtooth modulation on the VCO, the total electric field (Eq. 3) can be viewed as the product of a high frequency GHz carrier modulated by a low frequency kHz envelope. In general, diode detectors measure the square of the electric field (power) and only the slow frequency modulation component. Therefore, after neglecting the GHz component and using trigonometric identities, the kHz signal measured by the diode detector is given by Eq. 4.

The time varying component of Eq. 4 is labelled as 𝛿𝛿𝐸𝐸_𝑡𝑡2 and is given in Eq. 5. We notice that

the plasma induced phase shift∆𝜙𝜙can be extracted from this expression by eliminating the

known VCO induced phase shift (𝑘𝑘∆𝐿𝐿) by means of Fourier or Hilbert transforms. Using Eq. 1

and the extracted plasma induced phase shift∆𝜙𝜙one can estimate the line integrated electron

density. These results are equally valid when the “probe” and “reference” beams are not of equal amplitude. In Figure 32 to Figure 34, examples of MWI measurements are shown and compared with DLP measurements under various operating conditions.

𝑬𝑬𝑛𝑛(𝑠𝑠,𝑡𝑡) =𝑬𝑬����𝒏𝒏exp[𝑖𝑖(𝑘𝑘𝑠𝑠 − 𝜔𝜔𝑡𝑡+𝜙𝜙𝑛𝑛)] 𝑛𝑛= 1,2 Eq. 2

𝑬𝑬𝑡𝑡= 2𝑬𝑬����𝟏𝟏cos�𝑘𝑘∆𝐿𝐿₂+∆𝜙𝜙�exp�𝑖𝑖 �𝑘𝑘𝑘𝑘𝐿𝐿₂+𝑘𝑘𝜙𝜙− 𝜔𝜔𝑡𝑡�� Eq. 3

𝐸𝐸𝑡𝑡2= 2𝐸𝐸���12+ 2���𝐸𝐸12cos(𝑘𝑘∆𝐿𝐿+∆𝜙𝜙) Eq. 4