1629373397-425d4de58b7aea127ffc7c337418ea8d (846389), страница 23
Текст из файла (страница 23)
(4.112)corresponding to two different waves that can propagate along B0. The dispersionrelations areen2 ¼en2 ¼ω2p =ω2c2 k 2¼1ω21 ðωc =ωÞðR waveÞð4:116Þω2p =ω2c2 k2¼11 þ ðωc =ωÞω2ðL waveÞð4:117ÞThe R and L waves turn out to be circularly polarized, the designations R andL meaning, respectively, right-hand circular polarization and left-hand circularpolarization (Problem 4.17). The geometry is shown in Fig.
4.38. The electric fieldvector for the R wave rotates clockwise in time as viewed along the direction of B0,and vice versa for the L wave. Since Eqs. (4.116) and (4.117) depend only on k2, thedirection of rotation of the E vector is independent of the sign of k; the polarizationis the same for waves propagating in the opposite direction. To summarize: Theprincipal electromagnetic waves propagating along B0 are a right-hand (R) and aleft-hand (L ) circularly polarized wave; the principal waves propagating across B0are a plane-polarized wave (O-wave) and an elliptically polarized wave (X-wave).We next consider the cutoffs and resonances of the R and L waves. For the Rwave, k becomes infinite at ω ¼ ωc; the wave is then in resonance with the cyclotronmotion of the electrons.
The direction of rotation of the plane of polarization is thesame as the direction of gyration of electrons; the wave loses its energy incontinuously accelerating the electrons, and it cannot propagate. The L wave, onthe other hand, does not have a cyclotron resonance with the electrons because itrotates in the opposite sense. As is easily seen from Eq. (4.117), the L wave does nothave a resonance for positive ω. If we had included ion motions in our treatment,Fig. 4.38 Geometry ofright- and left-handedcircularly polarized wavespropagating along B01224 Waves in PlasmasFig.
4.39 The v2ϕ /c2 vs. ω diagrams for the L and R waves. The regions of nonpropagationv2ϕ =c2 < 0 have not been shaded, since they are different for the two wavesthe L wave would have been found to have a resonance at ω ¼ Ωc, since it wouldthen rotate with the ion gyration.The cutoffs are obtained by setting k ¼ 0 in Eqs. (4.116) and (4.117).
We thenobtain the same equations as we had for the cutoffs of the X wave (Eq. (4.107)).Thus the cutoff frequencies are the same as before. The R wave, with the minus signin Eqs. (4.116) and (4.107), has the higher cutoff frequency ωR given byEq. (4.108); the L wave, with the plus sign, has the lower cutoff frequency ωL.This is the reason for the notation ωR, ωL chosen previously. The dispersiondiagram for the R and L waves is shown in Fig. 4.39. The L wave has a stop bandat low frequencies; it behaves like the O wave except that the cutoff occurs at ωLinstead of ωp. The R wave has a stop band between ωR and ωc, but there is a secondband of propagation, with vϕ < c, below ωc. The wave in this low-frequency regionis called the whistler mode and is of extreme importance in the study of ionosphericphenomena.4.17Experimental Consequences4.17.1 The Whistler ModeEarly investigators of radio emissions from the ionosphere were rewarded byvarious whistling sounds in the audio frequency range.
Figure 4.40 shows aspectrogram of the frequency received as a function of time. There is typicallya series of descending glide tones, which can be heard over a loudspeaker.This phenomenon is easily explained in terms of the dispersion characteristics4.17Experimental Consequences123Fig. 4.40 Actual spectrograms of whistler signals, showing the curvature caused by thelow-frequency branch of the R-wave dispersion relation (Fig. 4.39). At each time t, the receiverrapidly scans the frequency range between 0 and 20 kHz, tracing a vertical line.
The recordermakes a spot whose darkness is proportional to the intensity of the signal at each frequency. Thedownward motion of the dark spot with time then indicates a descending glide tone. [Courtesy ofD. L. Carpenter, J. Geophys. Res. 71, 693 (1966).]Fig. 4.41 Diagramshowing how whistlers arecreated. The channels A, B,and C refer to the signalsso marked in Fig. 4.40of the R wave. When a lightning flash occurs in the Southern Hemisphere, radionoise of all frequencies is generated. Among the waves generated in the plasmaof the ionosphere and magnetosphere are R waves traveling along the earth’smagnetic field.
These waves are guided by the field lines and are detected byobservers in Canada. However, different frequencies arrive at different times.From Fig. 4.39, it can be seen that for ω < ωc/2, the phase velocity increases withfrequency (Problem 4.19). It can also be shown (Problem 4.20) that the groupvelocity increases with frequency.
Thus the low frequencies arrive later, givingrise to the descending tone. several whistles can be produced by a single lightningflash because of propagation along different tubes of force A, B, C (Fig. 4.41).Since the waves have ω < ωc, they must have frequencies lower than the lowestgyrofrequency along the tube of force, about 100 kHz. Either the whistles liedirectly in the audio range or they can easily be converted into audio signals byheterodyning.1244 Waves in Plasmas4.17.2 Faraday RotationA plane-polarized wave sent along a magnetic field in a plasma will suffer a rotationof its plane of polarization (Fig. 4.42). This can be understood in terms of thedifference in phase velocity of the R and L waves. From Fig. 4.39, it is clear that forlarge ω the R wave travels faster than the L wave.
Consider the plane-polarizedwave to be the sum of an R wave and an L wave (Fig. 4.43). Both waves are, ofcourse, at the same frequency. After N cycles, the EL and ER vectors will return totheir initial positions. After traversing a given distance d, however, the R andL waves will have undergone a different number of cycles, since they require adifferent amount of time to cover the distance. Since the L wave travels moreslowly, it will have undergone N + E cycles at the position where the R wave hasundergone N cycles. The vectors then have the positions shown in Fig.
4.44. Theplane of polarization is seen to have rotated. A measurement of this rotation bymeans of a microwave horn can be used to give a value of ω2p and, hence, of thedensity (Problem 4.22). The effect of Faraday rotation has been verified experimentally, but it is not as useful a method of density measurement as microwaveinterferometry, because access at the ends of a plasma column is usually difficult,Fig. 4.42 Faraday rotationof the plane of polarizationof an electromagnetic wavetraveling along B0Fig. 4.43 A plane-polarized wave as the sum of left and righthanded circularly polarized waves4.17Experimental Consequences125Fig.
4.44 After traversing the plasma, the L wave is advanced in phase relative to the R wave, andthe plane of polarization is rotatedand because the effect is small unless the density is so high that refractionbecomes a problem.When powerful pulsed lasers are used to produce a dense plasma by vaporizing asolid target, magnetic fields of megagauss intensities are sometimes spontaneouslygenerated. These have been detected by Faraday rotation using laser light of higherfrequency than the main beam. In interstellar space, the path lengths are so long thatFaraday rotation is important even at very low densities. This effect has been usedto explain the polarization of microwave radiation generated by maser action inclouds of OH or H2O molecules during the formation of new stars.Problems4.14. Prove that the extraordinary wave is purely electrostatic at resonance.
Hint:Express the ratio Ey/Ex as a function of ω and set ω equal to ωh.4.15. Prove that the critical points on Fig. 4.36 are correctly ordered; namely, thatω L < ω p < ω h < ωR .4.16. Show that the X-wave group velocity vanishes at cutoffs and resonances. Youmay neglect ion motions.4.17. Prove that the R and L waves are right- and left-circularly polarized asfollows:(a) Show that the simultaneous equations for Ex and Ey can be written in theformFðωÞ Ex iE y ¼ 0,GðωÞ Ex þ iE y ¼ 0where F(ω) ¼ 0 for the R wave and G(ω) ¼ 0 for the L wave.(b) For the R wave, G(ω) 6¼ 0; and therefore Ex ¼ iEy. Recalling theexponential time dependence of E, show that E then rotates in theelectron gyration direction. Confirm that E rotates in the oppositedirection for the L wave.(c) For the R wave, draw the helices traced by the tip of the E vector inspace at a given time for (i) kz > 0 and (ii) kz < 0.
Note that the rotationof E is in the same direction in both instances if one stays at a fixedposition and watches the helix pass by.1264 Waves in Plasmas4.18. Left-hand circularly polarized waves are propagated along a uniform magnetic field B ¼ B0^z into a plasma with density increasing with z. At whatdensity is cutoff reached if f ¼ 2.8 GHz and B0 ¼ 0.3 T?4.19.
Show that the whistler mode has maximum phase velocity at ω ¼ ωc/2 andthat this maximum is less than the velocity of light.4.20. Show that the group velocity of the whistler mode is proportional to ω1/2 ifω ωc and E 1.4.21. Show that there is no Faraday rotation in a positronium plasma (equalnumbers of positrons and electrons).4.22. Faraday rotation of an 8-mm-wavelength microwave beam in a uniformplasma in a 0.1-T magnetic field is measured. The plane of polarization isfound to be rotated 90 after traversing 1 m of plasma.
What is the density?4.23. Show that the Faraday rotation angle, in degrees, of a linearly polarizedtransverse wave propagating along B0 is given byðLθ ¼ 1:5 1011 λ20 BðzÞne ðzÞdz0where λ0 is the free-space wavelength and L the path length in the plasma.Assume ω2 ω2p , ω2c :4.24. In some laser-fusion experiments in which a plasma is created by a pulse of1.06-μm light impinging on a solid target, very large magnetic fields aregenerated by thermoelectric currents.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
