Lower hybrid wave launchers

RADIOFREQUENCY WAVES, HEATING AND CURRENT DRIVE IN MAGNETICALLY CONFINED PLASMAS

6.2. THEORY OF RF WAVE PROPAGATION IN A MAGNETIZED PLASMA The theory of wave propagation in magnetized plasmas has been

6.3.4. Lower hybrid wave launchers

RF waves in the lower hybrid range of frequencies (ci  ce), and especially drive current, have been extensively used to heat in tokamak plasmas.

In typical tokamaks this means utilizing frequencies in the range 1–10 GHz.

Since the free space wavelength ( 2 / c ) of such waves is 0.0–0.1 m, wave launching structures can be relatively compact. As shown previously, two possible polarizations of cold plasma waves exist in this frequency range — the fast (whistler) and the slow (cold lower hybrid or Trivelpiece–Gould) branches.

In fusion applications the slow lower hybrid wave branch has been used since it is efficiently absorbed by Landau damping in present day tokamaks at temperatures from 1 to about 12 keV. More importantly, it is possible to construct a phase array of waveguides with N>1 in vacuum with the TE10 polarization in each guide (the fundamental mode of propagation in a rectangular waveguide) so that at the plasma edge ERF is parallel to the toroidal magnetic field. The individual waveguides are typically 0.750 (where 0 is the free space wavelength) high in the poloidal direction and their individual widths in the toroidal direction are typically 0 / 8 so that a stack of waveguides phased at 90° relative to their neighbours would produce a net wavelength in front of the guides of 0/ 2 or N=2 as required for good accessibility and efficient coupling to the slow wave in the edge of a tokamak plasma (called the “grill” or “grille”) [6.173]. The wave would then propagate into the plasma, where it could either damp on electrons via Landau damping or on ions after conversion to a hot ion wave. Because of the specific waveguide dimensions, each guide would remain in the fundamental mode of propagation. Higher order modes are cut off and no dielectric loading is necessary for mode propagation. On the contrary, to couple to the fast wave, the polarization of the wave at the edge is such that ERF must bein the poloidal direction, and a phase arraywith N>1 is not possible unless the individual waveguide is loaded with a dielectric so that its dimension can be reduced in the toroidal direction relative to 0 / 2 without cutting off the fundamental mode of propagation. Such a dielectric loaded waveguide array experiment was carried out with some success [6.174]. Alternatively, a slotted slow wave structure was tested for fast wave launching in Versator II [6.175]. From the practical point of view, such experiments were abandoned after some initial trials due to technical difficulties. Instead, early experiments demonstrated the effectiveness of coupling to the slow wave with a “grill” and excellent coupling efficiencies (low reflectivity) were obtained without the use of tuning devices such as those required for lower frequency (ICRF) techniques. Consequently the standing wave ratio (SWR) in the launching structure is quite low. The practical power limit of such waveguide arrays is set by high voltage breakdown in the waveguides.

Referring back to Fig. 6.4 for N<Ncrit, the fast and slow wave roots coalesce in the plasma core and the inward propagating slow wave converts to

an outward propagating fast wave before the centre of the plasma is reached.

For typical applications of LHCD in tokamaks, this mode conversion region is placed in the core of the plasma, preferably at the plasma centre. Typically this means that N is between ~1.3 (high magnetic field and low density) and 3 (low magnetic field and high density). This accessibility condition places a lower bound on the N coupled by the launching structure and the upper bound is determined by the desired location of wave absorption by electron Landau damping. As shown by Eq. (6.63), Landau absorption occurs when N~ /a Te1/2, where   5 7 depending on whether linear or quasi-linear damping is assumed [6.176]. Therefore, we can summarize the spectral requirements placed on a lower hybrid launcher as follows: for RF current drive, the antenna must launch a directed spectrum with respect to the magnetic field such that the momentum transferred to the electrons is unidirectional. The N values in the spectrum must exceed the critical value for accessibility but be below those at which absorption would take place in the plasma periphery. Note that for tokamaks with central temperatures less than ~5 keV a window in N values between the lower (accessibility) and upper (electron damping) limits of Eq. (6.133) can exist while for higher temperature tokamaks the lower hybrid wave will typically damp well before reaching the centre of the plasma. This is why LHCD can be very effective for driving current well off-axis. Because of the requirements on the Nspectrum for proper coupling and absorption, the slow wave antenna is typically a structure periodic in the toroidal direction. Adjacent elements in the array are phased such as to launch the preferred spectrum. The RF electric field polarization at the antenna–plasma interface must also be in the toroidal direction.

The N spectrum of the antenna can be estimated by Fourier transformation of the antenna fields; the representative or peak value of the spectrum is roughly

~ /

N cDφ wDz, where z is the separation between radiating elements and  is the phase difference (in radians) between elements. Of course the actual launched spectrum is a combination of the applied spectrum and the plasma response. If significant parts of the launcher spectrum fall outside of the region of significant plasma response then there will be either RF energy in the plasma periphery, potentially leading to impurity production from the walls and limiters, or a large reflection of power into the launching structure, leading to large values of the RF electric field, in turn also leading to RF breakdown and damage. Other consider- ations for the antenna include the materials used, which must be compatible with the tokamak environment, and the necessity for an insulating vacuum window at some point between the RF source and the plasma.

Lower hybrid waves that satisfy the accessibility condition are evanescent in a thin layer near the waveguide mouth where 2pe 2, which necessitates that the ends of the waveguides of the grill be close to the plasma edge. In fact, in modern tokamaks this layer may essentially be non-existent (note that for an RF frequency of 5 GHz this corresponds to a density of ~ 2.5 10 m 17 –3. For

conditions with sufficient density and proper density gradient, strong coupling to the plasma is achieved with little power reflected back along the waveguide towards the RF source. Such a waveguide array is shown in Fig. 6.37.

FIG. 6.37. Diagram of a waveguide grill launcher inserted through a port in the vacuum vessel wall [6.19].

Simple waveguide arrays grew from simple double waveguides to as many as 24 in a given row. Further growth has been by adding more rows and by more modern designs (see below). Each waveguide is oriented such that the fundamental TE10 mode carries the incident power, automatically aligning the applied electric field with the toroidal direction. The relative phase between guides is then set by phase shifters, either in the waveguide structure in the high power feed lines or in the low power RF drive preceding the (typically) klystron amplifiers (see Fig. 6.38). Thus the N spectrum and directionality can be varied, even actively if electronic phase shifters are employed, allowing much greater flexibility than that of the longitudinal slow wave structure.

FIG. 6.38. The 4.6 GHz klystron amplifiers for the Alcator C-Mod LH system [6.79]. Reprinted from Ref. [6.79]. Copyright (2011), American Nuclear Society.

The major limitation on power in the waveguide grill antenna is RF breakdown in the vacuum portion of the waveguides. The location of the ceramic window used to isolate the machine vacuum is a compromise between placing them close to the plasma so as to ensure that electron cyclotron resonance is located in the pressured region of the waveguide array. This approach clearly requires brazing of a large number of ceramic windows into individual waveguides, a difficult technology which often prevents repair once the window has failed. However, the window is typically a sturdy ceramic block made of alumina (Al2O3) or beryllium oxide (BeO) of thickness 0/2 and has been employed successfully in the Alcator C and C-Mod experiments for years. Placing the windows far away from the plasma introduces the cyclotron resonance in the waveguide in the vacuum region, and extra pumping of the waveguides must be provided to prevent RF breakdown. This makes it possible to combine guides and expand them to conventional sizes in externally accessible regions where commercial windows can be utilized. This approach has been deployed in the JET and TORE-Supra experiments, in particular with PAM launcher designs (see below).

6.3.4.1. RF coupling theory

The general problem of coupling lower hybrid waves to the edge plasma has been approached in a variety of ways [6.33, 6.173, 6.177]. In general it is one of matching the impedance of the launching structure to that of the lower hybrid wave in the plasma. Typically a plane is defined at the waveguide mouth where the tangential components of the electric and magnetic fields are required to be continuous. A simple model for understanding coupling utilizes the density profile in Fig. 6.39.

FIG. 6.39. Idealized density behaviour near the grill mouth, where à is a step in density at the grill mouth, n Lc/ is the density gradient and x is the radial distance into the plasma [6.19].

The plasma density is assumed to be finite at the mouth of the waveguides and increase linearly away from the launcher. Since the perpendicular wavelength of the slow wave is significantly shorter than the density scale length in the edge plasma, coupling can be treated as a local problem independent of absorption or evanescence in the interior of the plasma. For cold lower hybrid waves with Ey = 0, the wave equation can be written (for Kzz P1) as:

( )

2 2

2 2Ez 2 P N2 1 Ez 0

x c

∂ w

- - =

∂  (6.134)

where P is the cold plasma zz component of the dielectric tensor (Eq. (6.14)).

For the density profile in Fig. 6.39, n x( )n0n x Lc / , where nc 0me2/e2 (where nc is coupling density and L is the density scale length) and with the substitution:

2 1/3

2 0

2

( ) 1 1

c

N n

u x L x

L n

c

w -     

 

=     - + 

 (6.135)

Eq. (6.134) becomes:

2

2Ez z 0

u uE

  

 (6.136)

The solutions to Eq. (6.136) are the Airy functions. A linear combination can be chosen which yields a wave radiating in the plus x direction. For N <1 the evanescent solution is chosen and for N >1 the oscillatory solution is chosen. Since the lower hybrid wave is backward in the direction perpendicular to the magnetic field the outgoing asymptotic solution should be proportional to exp(ik xx ). For this general discussion we will consider only the propagating waves, as lower hybrid launchers are phased to avoid exciting N <1. However, the numerical coupling codes calculate the match over the entire range of N. The propagating solution to Eq. (6.136) is:

( ) Ai( ) Bi( )

E uz  i u  u (6.137)

where Ai and Bi are the Airy functions of the first and second type. The RF magnetic field is obtained from Faraday’s Law:

( )

2 1/3 2 2 2

1 Ai( ) Bi( )

( ) 1

y

c N u u

B u i i

L u u

N c

w w

 -   ∂ ∂ 

 

= -   - ∂ + ∂ 





(6.138) The procedure for finding the reflection coefficient is to match Ez and By to the corresponding antenna components at x = 0:

( )0 (0) (0)

z zi zr

E u E E (6.139)

( )0 (0) (0)

y yi yr

B u B B (6.140)

where u0 u x( 0) and (0) means at x = 0. Ezi and Byi are the incident electric and magnetic fields imposed by the antenna, and Ezr and Byr are the reflected fields of the antenna. The field reflection coefficient Ezr(0) /Ezi(0) is:

 

ZZpp//ZZ0011

   (6.141)

where Zp E u B uz( ) / y( ) is the wave impedance in the plasma and

0 zi( 0) / yi( 0)

Z E x B x is the wave impedance of the launcher. With

/ 0 i

Zp Z  Z e (with ϕ being phase) the power reflection coefficient, , is minimized for Z 1, – /2 < < /2   .

6.3.4.2. Numerical coupling codes

A variety of numerical codes have been written [6.176, 6.178, 6.179] to calculate coupling from an array of waveguides to lower hybrid waves. For illustrative purposes the initial approach used by Brambilla [6.173] for an array of N rectangular waveguides is given here. For infinitely high wave guides (an approximation of no great consequence since in a rectangular guide the field is sinusoidal in the vertical direction and the dimension is large compared to the horizontal) the waveguide fields can be written:

 

1 0

( )

( , ) N i p ( ) ik xn ik xn cos p

zwg p np np

p n

n z z

E x z e z e e

b

 

    

 

  

   

 

 

  (6.142)

1

( , ) N i p ( )

wgy p

p

B x z eφ θ z

=

=∑ ⋅

( )

0

( )

cos

n n p

ik x ik x

np np

n n

n z z

e e

ck b

w a b p

∞ -

=

 - 

- -

 

 

∑  (6.143)

( , ) wgy

xwg

c B E x z i

 z

  

 (6.144)

where wg stands for waveguide, b is the width of the guide, p is the waveguide number, zp is the z coordinate of the edge of the pth waveguide and p is the phase of the pth waveguide. The function p( )z is a “window” function: it is equal to one for zp z (zpb) and zero elsewhere. The sum over n waveguide modes includes both propagating and evanescent ones. The coefficients np

and np are the amplitudes of the incident and reflected nth mode fields in the pth waveguide respectively. The waveguide wave vector of the nth mode is

2 2 2 2 2 1/2

( / / )

kn  c n  b . The matching of the antenna fields to the plasma is performed by matching the forward and reflected waveguide fields at the array mouth to generalized vacuum fields in front of the array (written in terms of Fourier integrals):

( / )

( , ) ( ) ik xx ( ) ik xx i x N z

E x zz N e N e e dN

c

w σ ρ w

∞

- -∞

 

= ∫   +     (6.145)

( / )

2 1/2

( / )

( , ) ( ) ( )

( 1) x x

i c N z

ik x ik x

y i c

B x z N e N e e dN

N

w σ ρ w

∞

- -∞

 

= ∫ -   -    



(6.146) Here ( )σ N is the coefficient corresponding to the field incident on the plasma and ( )ρ N is the coefficient of the reflected field from the plasma. For a given N the field reflectivity of the plasma is ( )Y N =ρ( ) / ( )N σ N . The vacuum region in front of the grill is employed as a mathematical convenience and is usually assumed to have zero width, although if close fitting guard limiters are present around a relatively narrow coupler it may be required to match the experimentally observed reflection coefficient. The vacuum fields (Eqs (6.142 and 6.143)) are matched to the plasma fields (Eqs (6.144) and (6.145)). The unknown quantities ( )σ N , ( )ρ N and np are determined. The reflection in the pth waveguide is |0p| / |2 0p|2. The N spectrum of the RF power launched into the plasma is obtained from the Poynting flux in the x direction at the plasma–vacuum boundary:

( )

2 2

* 2 1/2

( )

( ) Re( ) Im( ( ))

8 4 1

x z y

c c N

S N E B Y N

N σ

p p

= - = -

-



 



(6.147) Note that the calculated power is zero for N <1 since the waves are evanescent for overdense plasma. Actual spectra have been measured and compared with numerical codes. In Fig. 6.40 we show such measurements performed on the TdeV tokamak [6.180]. Additional confirmation has been provided by CO2 laser scattering experiments in the plasma near the antenna in Alcator C, and the results were in good agreement with expectations (Ref. [6.181]

and as shown in Fig. 6.41).

FIG. 6.40. Left: Amplitude and phase of RF fields at the mouth of the grill; crosses correspond to measured data and lines to the SWAN code [6.178] calculations. ϕ is phase and E the electric field strength. Right: Reconstructions of the spectrum [6.180]. Reprinted from Ref. [6.180].

Copyright (2011), American Institute of Physics.

To provide some guidance for estimating the optimal density for coupling, note that for an overdense plasma, u0 1, asymptotic expressions for the Airy functions can be taken and

2 1 1/2 p 1

Z N

à

 - 

 

= - - 

 (6.148)

Optimal coupling is for Zp /Z0 1 or à=n n0/ c = +1

2 2

1 (cp w/ h) (N 1)

 -  -

   , which yields the value of density n0 at the waveguide mouth for optimum coupling at a given N. Non-zero reflectively in practice is due to the finite width of the N spectrum. The optimal density increases with increasing N.

FIG. 6.41. Experimental verification of the Brambilla spectrum (solid curve) from Alcator-C for 90° phasing of adjacent waveguides. The dots are measurements with CO2 laser scattering.

Reprinted from Ref. [6.181]. Copyright (2011), American Institute of Physics.

6.3.4.3. Evolution of LH launchers in tokamaks

Early launchers were arrays of simple waveguides starting with 1 × 2 (1 row of 2 adjacent waveguidwes) such as Alcator A and later reaching up to as many as 4 rows and 24 columns. It was quickly found that the maximum power in a single waveguide was limited by voltage breakdown. This limit was found to be a strong function of materials and cleanliness in the vacuum portion of the launcher.

Sources of the breakdown plasma are secondary electron emission (SEE), electron stimulated desorption (ESD) of contaminants on the waveguide walls and impact ionization of residual gas in the waveguides. If the vacuum window is located sufficiently far from the vacuum vessel the electron cyclotron resonance layer at the RF frequency due to the tokamak fields can exist within the vacuum guide. This allows resonant acceleration of free electrons to higher energies, increasing SEE and ESD. Proper choice of materials (those with low secondary emission coefficients) and cleaning of the guides (RF or glow discharge) was found to allow operation up to high power levels (up to 50 MWãm–2 short pulse).

The power per waveguide limit was found to be a function of both waveguide dimension (width) and RF frequency (see Fig. 6.42).

FIG. 6.42. Maximum power density delivered from lower hybrid phased grills (from Ref. [6.182]). Ps is the power density at the grill mouth transmitted without RF breakdown; it is shown as a linear fit to the data points, where f is frequency.

Because of the power per waveguide limit and the spectral purity requirements, the number of waveguides required for large high power experiments has grown to a level (hundreds of guides) where simple couplers

have become impractical. A fuller description of the basic couplers and their limitations is given in Ref. [6.183]. A new approach (the multi-junction grill) was proposed that would still have the plasma see TE10 wave fields but where the power is split by passive elements into waveguides in the evacuated region of the grill after the RF power has passed through the vacuum window [6.184] (see Fig. 6.43).

FIG. 6.43. (a) Top view of an eight-waveguide multi-junction grill designed to launch a travelling wave spectrum (  /2). The grill is fed by two standard waveguides. The labelled blocks represent passive phase shifters. (b) Sections of reduced height waveguide act as phase shifters to set the relative phase between adjacent waveguides. Taken from Ref. [6.184].

The advantages of such a design include:

• Reduces greatly the number of vacuum windows required (and increases their size to where commercially available ones can be used);

• Minimizes the number and complexity of transmission line components (also decreases the RF losses in the transmission system);

• Reduces the maximum power that can be reflected to the RF sources (the passive elements also act as tuning elements).

Drawbacks include a reduced flexibility of spectrum (Fig. 6.44) since the phase shifts are fixed, and a small reduction in power handling since the tuning

property of the elements (and the reduced dimensions of the guides) increases the RF electric fields for a given power.

FIG. 6.44. Spectrum of the Tore-Supra multifunction grill. Taken from Ref. [6.178].

Most modern LH experiments in large machines with ample access, namely JET [6.185] (384 waveguides), Tore-Supra [6.186] and JT-60U [6.187] (212 waveguides) (see Fig. 6.45), utilize this approach.

FIG. 6.45. Front internal section of the JT-60 divided multi-junction launcher for JT-60.

Reprinted from Ref. [6.187]. Copyright (2011), American Nuclear Society.

However, Alcator C-Mod, a modern state-of-the-art compact high field device, still uses individually brazed windows. Such an approach allows great

flexibility and total control of the spectrum (see Fig. 6.46 for spectra for different phasings from Alcator C-Mod [6.188]). Further improvements of the Alcator C-Mod launcher design are evolving even now [6.189].

FIG. 6.46. Phase control of the of the Alcator C-Mod launcher power spectrum with alternate columns fixed phased at 120º. Shown are ideal (lines) and achieved (shaded) spectra.

Reprinted from Ref. [6.188]. Copyright (2011), American Nuclear Society.

A challenge for continued use of lower hybrid waveguide launchers in future devices is the heat load to the launcher from the plasma and from RF losses in steady state. The multi-junction design has been extended to one in which every other waveguide is passive — the passive active multi-junction (PAM) approach [6.190]. In this approach the passive guides provide a path for cooling to be applied near the mouth of the grill. This is the approach foreseen for applying lower hybrid to ITER and beyond. For ITER (Fig. 6.47) a system with an N=2 capable of coupling more than 16 MW at a power density of 33 MWãm–2 has been proposed [6.190].

FIG. 6.47. Illustration of the proposed ITER PAM-LH launcher. Reprinted from Ref. [6.190].

Copyright (2011), with permission from Elsevier.