Volume 62A, number 7 PHYSICS LETTERS 3 October 1977

PROPAGATION AND REFLECTION OF L O W E R H Y B R I D WAVES L A U N C H E D BY SLOW B ' A V E G U I D I N G C I R C U I T S *

Shigetoshi TANAKA, Yasushi TERUMICHI, Mitsuaki FUKUSHIMA and Shigeki NISHITANI Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan

Received 14 July 1977

Using finite length, modified slow waveguiding circuits, cold lower hybrid waves can be made to propagate in one or two ways along the axial direction, corresponding to the 7r/2- or ~r-modes characteristic for these periodic struc- tures. On reflection at a metal wall, phase reversal is demonstrated.

The lower hybrid resonance heating has received considerable attention in connection with further heating of toroidal plasmas to thermonuclear temper- atures [1 -6 ] . It is predicted that in an inhomogeneous plasma an incident cold lower hybrid wave (LHW),

satisfying the accessibility condition [2] can propagate toward the lower hybrid resonance layer, where the wave is converted into a hot plasma one. Recently, Bellan et al. [4] have shown that LHWs can be excited by a multiple-ring, slow waveguiding circuit, whose

rings are fed with appropriate phases of rf power. In this letter we report the propagation and reflec-

tion of LHWs launched by slow waveguiding circuits (SWC), similar to that used by Bellan et al. [4]. It is emphasized that these circuits, though the length is finite, can excite the two modes which are character- istic for periodic structures: the ~/2-mode which propagates in one axial direction only and the 7r-mode which can propagate in two directions [7]. One of the SWCs used in our experiments is a nmltiple-ring structure, where the N rings (N = 2 - 1 3 ) are connected in series by coaxial cables of suitable length, as shown in fig. l(a). The other is a multiple-plate structure (a modified Millman circuit) where rectangular plates are placed with a period L and connected in series by a delay line, replacing the corrugated line. It is noted that this SWC is analogous, in principle, to an array of phased waveguide (so-called Grill) [8].

In the experiments the plasma was produced by the electron cyclotron resonance (2.4 GHz, 70 W cw) near an end of a mirror field and diffused into a vacu-

* Reported in part in Kakuyugo-Kenkyu (in Japanese) 34 (1975) 438.

/I gloss tube Ii r probe ___3_ .~t.-~ 1~ L

"(" fr'~Tg__ Z ",jr-: z-~robe "~ " " ~ ~ load

m ~ " ~ ( b ) n -

rf-So rc

\ / Axial (c)n;/z-rnode_15 ~ 3 g i s t a n ~ ; Z(cm)

Fig. 1. (a) SWC consisting of multiple rings (right half, 2a = 9.4, 1 = 2.8 and L = 5.8 cm). Its wave field is similar to that of TMm-mode in diskloaded circular waveguide (left half; hi2 mode being shown), lnterferometer patterns for (b) 7r mode and (c) 7r/2 mode. The dotted curve shows the pattern in vacuum, while the solid curve that of lower hybrid wave when the plasma is present; w/2 = 31.5 MHz, co/S2 i = 39.4 and COpe(r = 0)/fZ e = 1.3.

um vessel (10 cm in dia.) located in a uniform mag- netic field region (B ~ 530 G, 150 cm in length). A plasma with a density n O (r = 0) = 1 ~ 10 X 1010 cm -3 and an electron temperature T e = 1 ~ 2 eV could be obtained in hydrogen gas. The SWC was placed near the end of the uniform field region and the excited waves were picked up by the T-shape antenna probes, axially and radially movable, and wave patterns were measured by a radio interferometer.

In figs. l(b) and (c), the dotted curves are the interferometer patterns as function of axial position z

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Volume 62A, number 7 PHYSICS LETTERS 3 October 1977

in SWC in vacuum, which represent the slow waves of the n-mode (Xz = 2L) and ~r/2-mode (Xz = 4L), respec- tively, when each ring is connected in series by the co- axial cables with effective length of XO/2 and ~0/4, where X0 is a wavelength in free space. The different character of two modes manifests itself clearly, when LHW is excited in the presence of plasma, as follows. In the case o f the 7r/2-mode, (c), the excited LHW (solid curve) propagates only in that direction, which corresponds to that of the rf power flow in the delay line, and the direction of the wave pattern is reversed if the latter flow is reversed. On the contrary, in the case of the 7r-mode, (b), the excited two LHWs are propagating in both directions and their patterns are apparently not changed when the direction of rf power flow in the delay line is reversed. Their ampli- tudes are smaller than that of 7r/2-mode, which is to be expected if the same amount o f rf power is trans- ferred to the two LHWs propagating in opposite di- rections. These results show that the waves launched by SWC are the 7r/2- and n-modes, while only the latter mode was tested by Bellan et al. [4]. The cou- pling efficiency of SWC is examined, by measuring the transmitted rf power at each delay line between the rings. In the case of 7r/2-mode, rf power is strongly damped and the apparent at tenuat ion amounts to 4.6 dB/Xz (~'z = 23 cm) when LHWs are excited in the plasma, while the at tenuation due to the r f loss in the glass tube etc. is 0.46 dB/)t z without plasma. On the contrary, in the case of the n-mode, the rf power at- tenuation is rather weak and amounts to 2.7 dB/X z at most.

In fig. 2(a) the LHW are seen to penetrate deeper and deeper into the plasma with increasing axial dis- tance z. This behavior is shown clearly in (b), where the position of peaks and troughs of wave patterns are plot ted in r - z plane and the contours of constant phase of the wave are drawn. The experimental deter- mination of the dispersion relation is made by using the local value of kz /k measured from the inclination o f k to the z-axis. Over the experimental range o f co2/co2 n = 4 - 1 0 , we find good agreement with the theoretical dispersion relation, w2 /W2LH = 1 + (M/m)(k2/k2) , valid in the electrostatic, cold ~lasma approximation and where CO2H = w2i{1 + (¢.O~e/~'22)) -1. In a plasma slab with varying density along the r- direction in a uniform magnetic field, the contours of constant phase of the LHWs are given by z =

-2

~ 0

4

L

¢ -

~ 2 "13

-6

r r

Axial distance Z(cm)

Fig. 2. (a) Interferometer patterns as functions ofr in succes- sive planes downstream from SWC (the most upstream-side ring is set at z = 0), where the ~r-mode is excited by a SWC of four rings, no(r = 0) = 4 X 10 l° cm -3, T e = 2 eV, p = 0.8 m Torr, co/2 = 63 MHz, ¢o/s2 i = 78.8 ckz/w = 41.4. (b) Experi- mental points defining contours of constant phase (o, peaks; ~, trough) and theoretical curves.

+ f~)(- KII/K±) 1/2 dr in a region where the WKB ap-

proximation is valid [3 ,4] . Here KII, K l are the com- ponents of dielectric tensor, parallel and perpendicular to the field. The same equation can be obtained when LHWs with axial symmetry are excited in a cylindrical plasma [9]. In fig. 2(b) we show the contours calcu- lated from the above equation, where the curves are shifted with respect to each other by the separation L of the SWC. The experimental plots are in good agree- ment with the theoretical curves. The best fit (solid lines) is obtained after multiplying the experimental density by 1.5, while the dot ted curve corresponds to the uncorrected value of n 0. Substi tution of a delay line in the reference path of interferometer shows that the phase of the propagating LHW is delayed for increasing z as well as for increasing r. On the other hand, the trajectory of energy flow can be measured from the envelope of wave pattern, which is the super-

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Volume 62A, number 7 PHYSICS LETTERS 3 October 1977

~ I - E

~ 2 -

~3-

"5 ~5/4- O

I I I I ~ / " 1 ~ , I1/v I

, i / 'o

, , o

i i I i 20 ~0

Axial distance Z(cm)

Fig. 3. Experimental plots are shown for the contours of con- stant phase, where e, • are plotted for peaks of the wave and o, ~ for troughs and e, o are obtained from the interferometer patterns as functions of z, while =, ~ as functions of r. LHW is launched by the modified Millman circuit of eight plates (0.5 X 5 cm2), where the ~r/2 mode is excited, n o = 4 X 10 l° cm -3, T e = 1 eV, p = 3 m Torr, w/2~r = 63 MHz, kzc /w = 43.3.

position of many patterns excited by the SWC (two

rings of 7r-mode) for the various values of phase in the reference path. The observed trajectory of the group

velocity Vg is shown to be coincident with the contour of constant phase (fig. 2(b)). It is known that for the electrostatic wave (ckz/CO = 40 in this experiments) the direction of phase velocity Vp is nearly orthogonal to that of group velocity Vg [4, 8]. Further, LHWs ex-

cited by a finite length SWC have the following charac-

teristics; (i) The axial wavelength X z is the same as that given by SWC (X z = 2L or 4L for 7r or 7r/2 mode) and

independent of plasma parameters such as he(r), B and co. (ii) The energy flow trajectory is independent of Xz" (iii) The number of radial wavelengths is the same as that of axial ones which is determined by that of SWC. (iv) As LHW is propagated toward a high density layer its perpendicular wavelength Xi becomes short and its amplitude small (fig. 2(a)).

Using the second SWC, we can excite and trace rather localized LHWs, which can be approximated to be propagating in a two-dimensional, inhomogeneous plasma slab, and then we can show that LHWs are re- flected near a metal wall. The wave patterns are meas- ured as a function of z as well as r and their peaks and troughs are plotted in the r - z plane, thus the contours of constant phase of the LHW are obtained as shown in fig. 3. It is clear from fig. 3 that the excited, incident LHW is propagating in ( - z ) direction and arrives near the metal wall (z = 0), where the LHW is reflected and

propagating in (+z) direction. Using delay-line tech- niques we have verified that both incident and reflect- ed LHW are forward in the axial direction, while they are backward in the radial direction, It is seen that the phase of incident wave is changed by 7r when it is re- flected. The result can be interpreted that the incident LHW is reflected at the metal wall, where the resultant

field Ere s = Ein c + Ere f = --LEresle z and the boundary condition is fulfilled, since Ein c II kinc and Ere f II ( -k re f Here the suffixes, inc and ref, refer to the incident and reflected waves, respectively, and e z the unit vector in the z-direction. This interpretation may be reasonable, since the wavelength X z (= 11 cm) is much longer than

the sheath width of plasma X s (~10 ~tDe ~ 0.07 cm) near the metal wall.

In summary, we have experimentally showed that

the finite length SWC, such as the multiple-ring struc- ture and the modified Millman circuit, can efficiently launch LHWs in one or two directions corresponding to excitation of the ~r/2- or 7r-mode, which are charac-

teristic for those periodic structures. Tracing the localized LHW, the reflection of LHW is observed at

the metal wall, where the phase of the wave is inverted.

References

[1 ] T.H. Stix, in: Symposium on plasma heating in toroidal device, Varenna, Italy, 1974 (Editice Compositori, Bologna, 1974).

[2] T.H. Stix, Phys. Rev. Lett. 15 (1965) 878; V.E. Golant, Zh. Tekh. Fiz. 41 (1971) 2492 [Soy. Phys- Tech. Phys. 16 (1972) 1980]; S. Puri and M. Tutter, Nuclear Fusion 14 (1974) 93; M.D. Simonutti, Phys. Fluids 18 (1975) 1524.

[3] R.J. Briggs and R.R. Parker, Phys. Rev. Lett. 29 (1972) 852.

[4] P. Bellan and M. Porkolab, Phys. Rev. Lett. 34 (1975) 124; Phys. Fluids 17 (1974) 1592.

[5] R.L. Stenzel and W. Gekelman, Phys. Rev. Lett. 35 (1975) 1708; Phys. Rev. Al l (1975) 2057.

[6] P. Javel, G. Muller, U. Weber and R.R. Weynants, Plasma Phys. 18 (1976) 51.

[7] J.C. Slater, Microwave electronics (D. Van Nostrand, New York, 1950). chap. 8; Rev. Mod. Phys. 20 (1948)473; C. Johnson, Fields and wave electrodynamics (McGraw Hill, New York, 1965) chap. 7.

[8] S. Bernabei, M.A. Heald, W.M. Hooke and F.J. Paoloni, Phys. Rev. Lett. 34 (1975) 866; M. Brambilla, P. Lallia and K.N. Trong, Report EUR-CEA- FC-792 (1975) (unpublished).

[9] S. Tanaka et al., Phys. Lett. 59A (1976) 290.

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