Three-dimensional electromagnetic wave polarizations in a plasma

Transcription

1 ADIO SCIENCE, VO. 37, NO. 6, 1100, doi:10.109/001s00496, 00 Three-dimensional electromagnetic wave olarizations in a lasma Bao Dong Chengdu Institute of Information Technology, Chengdu, China K. C. Yeh University of Illinois at Urbana-Chamaign, Urbana, Illinois, USA eceived 18 May 001; revised 10 June 00; acceted 19 July 00; ublished November 00. [1] It is nown that electromagnetic waves roagating in a magnetized cold lasma are not necessarily urely transverse waves. The existence of a longitudinal comonent imlies that the normal to the olarization lane is not, in general, arallel to the roagation vector. The offset angle between these two directions is comuted for each characteristic wave. Our investigation shows that the offset angle can be large under certain conditions. For a wave field that is a suerosition of two characteristic waves, its olarization lane wobbles in sace as the field roagates while its normal recesses. Several examles are considered, comuted, and given to illustrate the 3D nature of wave olarizations. INDEX TEMS: 0654 Electromagnetics: Plasmas; 6984 adio Science: Waves in lasma; 6964 adio Science: adio wave roagation; 471 Ionoshere: Plasma waves and instabilities Citation: Dong, B., and K. C. Yeh, Three-dimensional electromagnetic wave olarizations in a lasma, adio Sci., 37(6), 1100, doi:10.109/001s00496, Introduction [] The discovery of Faraday rotation was made over 150 years ago [Faraday, 1846]. A theoretical exlanation of this henomenon was offered 51 years after its discovery [Bacquerel, 1897]. The issue of wave roagation in a magnetized cold lasma has been attaced for many decades by many researchers. Mature as it seems to be, attention is still attracted in recent years on Faraday rotation henomenon since the comlexity of its 3D nature does not yet close the question. While theoretical foundation has been soundly laid, the interretation of the solutions and laconic and intuitive aroach to lay out the theoretical results are still worthwhile. ecently, Yeh et al. [1999a, 1999b] introduce a D olarization diagram which maes the nature of olarization transformation clearer and simler. They tae advantage of the bilinear maing in comlex function theory to ma roagation factor in the comlex lane = E y (z)/e x (z), called olarization ratio lane, which determines geometric features of the transverse olarization. It lays an equivalent role as Smith chart does in transmission line theory. [3] Polarization of the wave field in a magnetized cold lasma, however, is 3D in nature because the electric field Coyright 00 by the American Geohysical Union /0/001S is not urely transverse in general. The wor in this aer is intended to extend the earlier D results to 3D case. Under some circumstances the longitudinal comonent lays an imortant role as discussed later in this aer. [4] esearch interest on the toic is artly stimulated by the ractical objective of NASA s IMAGE roject, which has been on board a radio lasma imager (PI) used for magnetosheric sounding. In a aer authored by einisch et al. [1999], the woring rincile of the active Doler radar sounding in IMAGE has been outlined. One of the PI exeriments determines the normal vector of the olarization lane of the returned echo. If the returned electric field is urely transverse, this normal vector gives the arrival direction of the echo. However, if the returned echo has a longitudinal comonent, as is generally true in a magnetized cold lasma, care must be exercised in its interretation. [5] In this aer we will discuss this imortant issue in detail to see under what conditions neglect of longitudinal comonent is justified and, if not, what is the angle difference between the olarization normal vector of a characteristic wave and the actual arrival angle. [6] The calculation given in section 4 shows that a general ellitically olarized field, constructed as a suerosition of ordinary and extraordinary waves, has the olarization lane sanned by two frame vectors which are turning their directions in two different lanes, maing the olarization lane either nutating or rolling along roagation ath.

2 1 - DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA [7] This aer adots a Hermitian coordinate system, which is constituted by two oositely rotating circular olarization base vectors and the unit roagation vector. In this system every comonent of the wave field is found to be real valued. This turns out to be convenient in exressing the solutions and esecially in determining the normal vector of a wave olarization. With these exlicit vectorial exressions the roblem of a suerosition of two characteristic wave modes can be discussed further in detail, esecially those ertaining to 3D geometric features.. Wave Disersion Equation in a Circular Polarization Basis [8] We start with Maxwell s two curl equations. For lane waves, all field variables have the time sace deendence of the form ex(wt r) for which the differential oerators are simlified to algebraic ones. After eliminating the magnetic field, the electric field is found to satisfy E ¼ w m 0 D ð1þ [9] In a magnetized cold lasma, the relative ermittivity K $ taes the dyadic form, h i $ K ¼ aw ð Þ $ I þ jbðwþ^b $ I þ gðwþ^b^b ðþ where 8 aw ð Þ ¼ 1 w w w ¼ 1 X 1Y b >< bw ð Þ ¼ w w b wðw w bþ ¼ XY 1Y >: gðwþ ¼ w w b w ð Þ ¼ XY w w b 1Y ð3þ In (3), the coefficients a, b, and g are exressed in two forms using the angular lasma frequency w and angular gyrofrequency w b or the normalized frequencies X and Y oularly used in the ionosheric literature. These quantities are defined as follows: w b ¼ eb 0 m ; w ¼ Ne e 0 m and X ¼ w w ; Y ¼ w b w where N, m, and e are the density, mass and electric charge of an electron resectively. [10] Substituting the relative dielectric ermittivity exression () into (1), the resulting equation becomes $ P E ¼ 0 ð4þ where, $ P ¼ n ð $ I ^^Þ $ K, n = / 0 is the refractive index and 0 = w m 0 e 0. [11] Consider a coordinate system constructed with the following three base vectors: ^ ¼ =, a unit vector in the wave number direction ^y ¼ ^b ^=sin q, where ^b ¼ B 0 =B 0 is a unit vector in the steady magnetic field direction, the angle between B 0 and is q ¼ ff ^b; ^ ; ^x ¼ ^y ^ ¼ 1 ^b ^ ^ ^b ¼ tan q ^ csc q ^b sin q ð5þ It is convenient to combine ^x, ^y, and ^ into the following Hermitian basis: 8 ^x þ j^y ^e ¼ ffiffi >< ^x j^y ^e ¼ ffiffi ð6þ >: ^ The left-hand circular olarization vector ^e and the right-hand circular olarization vector ^e have their roerties discussed by ee [1988]. With these coordinate transformations, the dyadic equation (4) exressed in the basis system (6) taes the matrix form where the coefficient matrix is P ¼ n a þ b cos q þ g sin q 6 4 g sin q ðg cos qb ffiffi Þsin q 3 E P4 E 5 ¼ 0 ð7þ E 3 g sin ðg cos qbþsin q q ffiffi n a b cos q þ g sin ðg cos qþbþsin q q ffiffi 7 5 ffiffi ða þ g cos qþ ðg cos qþbþsin q Inserting (3) into P given by (8) yields P ¼ l X ð 1Y cos q ð Y =Þsin qþ XY sin q Y 1 ðy 1Þ XY sin q ðy 1Þ l X ð 1þY cos q ð Y = 6 Y 1 4 XY ð1y cos qþsin q XY ð1þy cos qþsin q ffiffi ffiffi ð Y 1Þ ð Y 1Þ Þsin q Þ ffiffi ð8þ XYð1Y cos qþsin q ð Y 1Þ ffiffi XY ð1þy cos qþsin q ð Y 1Þ 1Y Xð1Y cos q Y 1 ð9þ where l = n 1. Notice that all the elements of P aearing in (8) and (9) are real. As we shall see later, the reality of P is an imortant roerty useful in later develoment Þ

3 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA 1-3 [1] For nontrivial solutions to (7) we require det P = 0, obtaining the following disersion equation: 1X Y þ XY cos q l þ X ð1 X ÞXY sin q l þ X ð1 X Þ ¼ 0 ð10þ The two roots of (10) give two characteristic wave numbers, nown as Aleton assen formula: ¼ 0 ð 8 l þ 1Þ 9 >< ¼ 0 X >= 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >: 1 Y sin q 1X ð Þ Y 4 sin 4 q þ Y cos q>; ð11þ 41X ð Þ It is interesting to see how simle (9) becomes for the secial case of q =0, n 1 þ X 3 1þY 0 0 P ¼ 4 0 n 1 þ X 1Y 0 5 ð1þ 0 0 X 1 In this secial q = 0 case, the disersion relation simlifies to det P 0 ¼ n 1 X n 1 X 1 þ Y 1 Y ðx 1Þ ¼ 0 ð13þ The two cases, reresented by the vanishing of the first factor or the second factor of (13), corresond to the urely left-handed or the urely right-handed circularly olarized wave modes resectively. The case reresented by the vanishing of the third factor corresonds to longitudinal lasma oscillations. 3. Characteristic Waves [13] Taing first two rows of the matrix equation (7) yields " # ð1þ P 11 P 1 U P 1 P ð Þ ¼U ð1þ P 13 ð14þ P 3 U 1 where the root n (1) is assumed as n in P 11 and P Solving (14) by taing U as nown and then multilying the three comonents of the 3D vector U with an aroriate common factor gives the following ordinary characteristic wave mode vector. U ð1þ ¼ 6 4 U ð1þ U ð1þ U ð1þ ¼ 4 ffiffi þ 1 ð 1 þy cos qþxy sin q 1Y X ð1y cos qþ 3 5 ð15þ Similarly, the extraordinary characteristic wave mode associated with the root n () can be shown to be 3 U ðþ 3 1 U ðþ 6 ¼ 4 ð Þ 7 5 ¼ cos q4 ffiffi 1 þ 5 ð16þ U U ðþ ð þy cos qþxy sin q 1Y X ð1y cos qþ In (15) and (16), the olarization ratio for the transverse comonents is, for resective modes, " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 1; ¼ 1 Y sin q cos q 1 ð X Þ Y sin 4 q 41 ð X Þ þ cos q ð17þ A cos q factor in (16) is used to assure continuity of the comonents over q. In (17), the uer sign alies to 1 and the lower sign alies to. The two characteristic wave vectors can be normalized as follows: ð Þ ¼ u ð1;þ ^e þ u ð1;þ ^e þ u ð1;þ ^ 1 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1; þ U ð1;þ ð18þ 1; U U ^u 1; U ð1;þ It is easy to see that 1 = 1, and therefore u ð1þ u ðþ ð Þ þ u 1 ð Þ u ðþ ð Þ ¼ 0 ð19þ As it has been noted before, all elements of the characteristic vectors (15) and (16) are real valued. The hysical imlication of (19) is that all comonents in the characteristic wave modes are either in hase or 90 out of hase to one another. But, since two of the three base vectors are comlex valued (see (6)), the characteristic wave vectors ^u ð1;þ are comlex valued vectors when exressed in real hysical sace. et ð a 1;Þ ¼ e ^u ð1;þ and ð1;þ b ¼ Im ^u ð1;þ, then ð a 1;Þ u ð1;þ þ u ð1;þ ^x þ u ð1;þ ^ ¼ 1 ffiffi ð1;þ b u ð1;þ þ u ð1;þ ^y ¼ 1 ffiffi ð0aþ ð0bþ These two exressions turn out to be very useful in determining 3D olarization features of the wave modes as will be discussed later. [14] The variations of the characteristic wave vectors with q are illustrated in Figures 1a and 1b. The arameters used are: X = 0.005, Y = and X = 0.005, Y = , resectively. Even though the longitudinal comonent is very small for these two examles, it is not necessarily always the case, even when X is small. This is shown next. [15] Figure shows surfaces describing absolute value of the longitudinal comonent of the normalized characteristic wave electric field vector as a function of X and Y

4 1-4 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA Figure 1. This figure dislays three comonents of the two characteristic wave modes as a function of q. (a) For a lasma with X = 0.005, Y = Note the small scale of the longitudinal comonent. (b) For a lasma with X = 0.005, Y = The longitudinal comonent is still small but now larger than that given in (a). for ordinary and extraordinary wave modes resectively for three roagation angles q =30, 74, and 89. Itis seen that the longitudinal comonent of the extraordinary mode may be large even when X 1, rovided Y is close to 1. [16] To hel discuss the wave roagation reresented by a comlex vector hasor, we consider the following general case. et A be a constant vector hasor with A = A r + ja i. For a lane wave of hase y = wt r, the instantaneous vector field becomes Uðt; rþ ¼ e Ae jy ¼ cos yar sin ya i ð1þ When y =0,U(t, r) coincides with A r ; while when, y = /, U(t, r) coincides with A i. At a fixed osition r the ti of the vector U traces out an ellise as a function of time in the lane formed by A r and A i. This lane is nown as the olarization lane. In the olarization lane, the vector U(t, r) rotates with time in the sense from A i to A r. Define a normal to this olarization lane as N ¼ A i A r ðþ For transverse waves, N must necessarily be either arallel to as for right-handed olarization or

5 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA 1-5 Figure 1. antiarallel to as for left-handed olarization. Because of the ossible resence of a longitudinal comonent in a magnetized cold lasma, there exists an angle F between N and in general. If the angle F is acute, the sense of rotation can be defined as right handed. On the other hand, if F is obtuse, the sense of rotation is left handed. Such a definition is in comlete agreement with the usual sense determination using the transverse comonents only. This is so because the acuteness of angle F (i.e., the sign of cos F) is determined by the transverse comonent A Trans only and not by the longitudinal comonent A ong as shown below N ^ ¼ jnjcos F ¼ A Trans i A Trans r ^ ð3þ The calculation of (3) follows by noting the following vector relation, (continued) N ¼ A Trans i ¼ A Trans i A Trans r : þ A ong i A Trans r ^ A Trans r þ A ong r A Trans i ^ þ A ong r ^ þ A ong i [17] We now aly the above result to the characteristic wave vectors (18). Using (0a) and (0b), the normals can be calculated to be ð Þ ¼ Im ^u ð1;þ e ^u ð1;þ ¼ 1 u ð1;þ ð1;þ u ^ þ u ð1;þ ffiffi u ð1;þ u ð1;þ ^x N 1; ^ ð4þ

6 1-6 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA Figure. The variation of the longitudinal comonent of the normalized characteristic vectors in the (X, Y) arameter sace for roagation angles q =30, 74, and 89 as mared on to of each grah.

7 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA 1-7 for each of the two characteristic modes. This means that the normal vector is in the lane sanned by and B 0 (see (5)). The angle F between N and, nown as the offset angle, can be comuted from F ð1;þ ¼ arctan ffiffiffi! ð1;þ u ð Þ þ u ð1;þ u 1; ð5þ For urose of investigating how large this offset angle F can be, let us adot some numbers alicable to the PI [einisch et al., 1999]. PI uses an active Doler sweeing in frequency from 3 Hz to 3 MHz for sensing lasma structures in the magnetoshere. As a first examle, tae f = 75 Hz, f = 5 Hz, f b = 1.5 Hz, and q =74. Theses arameters yield X = and Y = 0.0. The comuted offset angles are and for the ordinary and extraordinary waves, resectively. Since the offset angles are very small, the normal to the olarization lane can be considered as the roagation direction with negligible error. In the second examle, we dro the frequency to the middle of PI range, say 30 Hz. Keeing other arameters the same, i.e., f = 5 Hz, f b = 1.5 Hz, and q =74, roduces X = and Y = The comuted offset angle in this second examle comes out to be and for the ordinary and extraordinary waves resectively. The offset angles are now larger and begin to matter in some alications. Actually, even in the same lasma (i.e., X = and Y = 0.05) the offset angle has a strong deendence on the roagation angle q, esecially for the extraordinary wave. This is shown in Figure 3a. In this figure the magnitude of the offset angle for the extraordinary wave is seen to increase drastically as the roagation direction becomes erendicular to the steady magnetic field. If the local lasma frequency and gyrofrequency at PI can be determined from other exeriments, (5) can be used to correct the aarent echo arrival angle determined by using the wave olarization data from the roagation direction. [18] In assing we mention two secial cases of interest, both corresonding to the cutoff condition, one for the ordinary mode, the other for the extraordinary mode. For ordinary mode at X =1, 1 turns out to be Figure 3. (oosite) The variation of the offset angle as a function of q for both characteristic waves. (a) In a lasma with X = , Y = (b) In a lasma with X = , Y = Notice F being aroximately equal to / q for the ordinary mode in this case for a wide range of q. (c) In a lasma with X = 0.5, Y = Notice F being aroximately equal to q for the extraordinary mode in this case.

8 1-8 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA zero. The corresonding characteristic vector (15) simlifies to 3 1 U ð1þ ¼ 4 1 ffiffiffi 5 ð6þ cot q When this is substituted into (5), the offset angle is found simly as ð1þ ¼ q ð7þ For this secial case the normal N to the olarization lane is always erendicular to B 0. In ionosheric literature, a vertically incident ordinary ray on a horizontally stratified ionoshere is nown to bend toward the nearest geomagnetic ole [Forsgren, 1951]. At the oint of reflection, i.e., X = 1, the ray is exactly erendicular to B 0, imlying N and the ray are arallel to each other. Figure 3b shows a case in which X = , Y = This figure shows clearly that (7) is aroximately satisfied for a wide range of q. [19] A second secial case of interest occurs when X = 1 Y for the extraordinary mode. In this case in (17) simlifies to 1/cosq and its characteristic vector becomes 3 U ðþ ¼ cos q 1 4 cos ffiffiffi q þ 1 5 ð8þ sin q Using (8), the offset angle reduces to F ðþ ¼ arctan sin q ¼q cos q ð9þ That is to say, the offset angle F is equal to q. Because of the way F is defined, the normal vector of the olarization lane is always arallel to B 0.Inthis case, however, the refractive index n = 0, corresonding to the cutoff condition. But if Y tends to 1 X from below, (9) still holds aroximately. One such examle is shown in Figure 3b where X = 0.5, Y = 0.49 for which X is not exactly equal to 1 Y. Itis seen that the offset angle F is very close to q, showing aroximate erendicularity of the olarization lane to B 0 for whatever direction in. [0] As seen from Figures 3a 3c, N of the ordinary mode always deviates away from B 0 because F is ositive when q < / and is negative when q > /. This henomenon may be called the divergence effect. [1] The instantaneous electric field value of a characteristic mode, with the unimortant amlitude set to unity, is E ð1;þ ðz; tþ ¼ e ^u ð1;þ e jy ¼ ð a 1;Þ cos wt ð1; Þ z ð1;þ b sin wt ð1; Þ z ð30þ where (1,) a and (1,) b are defined in (0a) and (0b) resectively. It is obvious that (1,) a (1,) b imlying the olarization ellise described by (30) has a major axis aligned along either (1,) a or (1,) b. Note that (1,) a is the lane P b, sanned by the roagation ath and the steady magnetic field, while (1,) b is erendicular to P b. The determination of the major axis can be decided by calculating the difference in squared lengths between a and b. ð a 1;Þ ð1;þ b ¼ 1 þ u ð1;þ þ u ð1;þ ð1;þ u 1 u ð1;þ u ð1;þ ¼ u ð1;þ u ð1;þ þ u ð1;þ ð31þ Under the assumtion of negligible u as considered in Figure 1, the major axis of the ellise is determined by the relative signs between u and u.ifu and u are of the same sign as in mode 1 shown in Figure 1, a is the major axis. On the other hand, when u and u have oosite signs as for mode, b is the major axis. 4. Wobbling of the Polarization Plane and Nutation [] In this section, we wish to consider the olarization transformation of a wave field roagating in a given direction, say z, and at a given frequency. Such a wave field can be decomosed as a sum of two characteristic wave modes as follows: E ¼ ^u ð1þ e j ð wtð1þ zþ þw^u ð Þ e j ð wtðþ zþ h ¼ ^u ð1þ e j ð1þ ðþ z þ w^u ðþ e j ð1þ ðþ z ie j wt 1 ð Þ þ ðþ z ð3aþ where the comlex w taes care of the ossible existence in relative magnitude and hase shift between the two characteristic modes. [3] et q ¼ 1 ðþ 1 ðþ z þ fw and fw = Arg(w), (3a) can be converted into the following form: E ¼ e jq^u ð1þ þ jwje jq^u ðþ e j wt ð1þ þ ðþ ð Þ zf w ¼ ½F r ðqþþjf i ðqþše jy ð3bþ where ð1þ þ ðþ z fw y ¼ wt F r ðqþ ¼ cos q r þ sin q i ð33þ

9 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA 1-9 F i ðqþ ¼ sin q M r þ cos q M i ¼ cos q þ M r þ sin q þ M i ð34þ r ¼ e ^u þ jwj^u u þ u þ þ u ¼ ffiffiffi ^x þ u þ jwju i ¼ Im ^u jwj^u u u u ¼ ffiffi ^y ð36þ M r ¼ e ^u jwj^u u þ u þ u ¼ ffiffi ^x þ u jwju M i ¼ Im ^u þ jwj^u u u þ u ¼ ffiffiffi ^y ð38þ The instantaneous value of E, i.e., the electrical field olarization, is eðeþ ¼ F r ðqþcos y F i ðþsin q y ð39þ For a fixed osition, the ti of the electric field E still traces out an ellise in time. However, because of the deendence of F r and F i on q and thus also on z, the olarization ellise changes as the wave field roagates. [4] From (3b), the normal vector (unnormalized) of the olarization lane can be exressed as where N ¼ F i F r ¼ A c cos q þ A s sin q þ A 0 ð40þ A c ¼ 1 ð i M r þ r M i Þ ð41þ A s ¼ 1 r M r ð4þ A 0 ¼ 1 ð i M r r M i Þ ð43þ As discussed in connection with (3), the sense of rotation of the comosite wave field is determined by the inner roduct N ¼ N ^ ¼ A c cos q þ A 0 ð44þ where A 0 ¼ A 0 ^ u ¼ u ð1þ ð1þ jwj u ðþ ðþ u A c ¼ A c ^ ¼ jwj u ð1þ u ðþ u ð1þ u ðþ ¼ 4jwju 1 ð45þ ð Þ u ðþ ¼ C Y ð46þ 1 X In above equations, C ¼ 4jwjsin q ju ð1þ jju ðþ j > 0, U (1) and U () are given in (15) and (16) resectively. Since C is a ositive number, A c is negative when X > 1 and ositive when X < 1. Therefore, when X < 1, as in some ractical cases, N is varying between two extreme values. N max ¼ A 0 þ A c ¼ 1 h i h i u ð1þ þ jwju ðþ u ð 1 Þ þ jwju ðþ N min ¼ A 0 A c ¼ 1 h i h i u ð1þ þ jwju ðþ u ð 1 Þ þ jwju ðþ ð47þ ð48þ As mentioned above, the sense of rotation is determined by the sign of the N. Consequently, the behavior of the normal vector N(q) falls into the following three tyes: 1. Always right handed. This haens when N min > 0, i.e., h i > h i jwju ðþ u 1 jwju ðþ u ð1þ. Always left handed. This haens when N max <0, i.e., u ð1þ ð Þ h i < h i þ jwju ðþ u 1 þ jwju ðþ 3. Sense of rotation alternating between right handed and left handed. This haens when N min N max <0, [5] To connect the results obtained here with existing results given by Yeh et al. [1999a, 1999b], we calculate the electrical field comonents along x and y axes, resectively, E x ¼ 1 h i ffiffiffi e jq u ð1þ þ u ð1þ þ jwje jq u ðþ þ u ðþ e jy ð49þ E y ¼ j h i ffiffiffi e jq u ð1þ þ u ð1þ þ jwje jq u ðþ þ u ðþ e jy ð50þ ð Þ

10 1-10 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA Figure 4. Illustrating the wobbling of the olarization lane and the nutation of the normal vector for a lasma with X = 0.5, Y = 0.9, and relative amlitude jwj = 1. For each case, the normal vector s zenith angle q n and the azimuthal angle j n, both being functions of q, are shown on the left. The normalized normal vector in 3D sace is deicted in the lower right anel. The olarization transformation for the transverse comonents is reresented by a circle in the olarization ratio lane as deicted in the uer right anel. (a) The angle between the roagation vector and the geomagnetic field is 170. The wave olarization transforms as it roagates, but its sense of rotation stays right handed. (b) The angle between the roagation vector and the geomagnetic field is 8. The wave olarization transforms as it roagates, but its sense of rotation stays left handed. (c) The angle between the roagation vector and the geomagnetic field is 110. The wave olarization transforms as it roagates, but its sense of rotation alternates between being right handed and left handed. From these exressions, the olarization ratio of the transversal comonents can be comuted as ¼ E y E x ¼ j A þ Bx C þ Dx ð51þ where the several quantities aeared in (51) are defined as x ¼ e jq ; A ¼ u ð1þ u ð1þ ; B ¼ jwj u ðþ u ðþ ; C ¼ u ð1þ þ u ð1þ ; D ¼ jwj u ðþ þ u ðþ (5) The bilinear transformation (51) mas circles in comlex x-lane into circles in comlex -lane. In Figure 4, all three tyes of normal vector s behavior are illustrated by (a), (b), and (c), resectively. The arameters are mared in the figures. [6] To describe the behavior of the olarization lane, the zenith angle q n (q) and azimuth angle j n (q) of its normal vector can be exressed as N ðqþ q n ðþ¼arccos q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nx ðqþþn y ðqþ ð53þ

11 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA 1-11 Figure 4. (continued) j n ðqþ ¼ arctan N yðqþ N x ðqþ ð54þ In Figure 4, q n (q) is shown on the to left anel, j n (q) on the bottom left anel, the unit normal vector ^n ðþof q the olarization lane on the bottom right anel and the olarization ratio lane, including the circular locus indicating the olarization transformation along the roagation ath, on the to right anel. The tyes of olarization transformation reresented by (a) and (b) are lie nutation of a sin to where the normal vector recesses, while (c) can be called olarization lane rolling since the sense of rotation alternates along the roagation ath. In this case, the ti of the electric wave vector at a fixed osition still draws an ellise in 3D sace as a function of time on the olarization lane. But according to (39), this lane, sanned by two vectors F r (q) and F i (q) (called frame vectors of the ellise), wobbles as a function of q and thus also as a function of the roagation distance z. [7] When X 1 and Y 1, the longitudinal comonent of a wave field is much smaller than the transversal ones (Figure 1a describes a tyical case of this). Therefore, the angle between the lanes sanned by ( r, i ) and (M r, M i ), resectively, is very small. Even then, rovided that the angle is not exactly zero, it is clear that the two frame vectors F r and F i can never be collinear with each other for the case of nonzero vectors ( r, i ) and (M r, M i ). Because if so, F r and F i must be on the intersection line of the lanes sanned by ( r, i ) and (M r, M i ) resectively; and this line is along the direction of ^y (since i, M i, and ^y are collinear, see (36) and (38)). But from (33) and (34) this is imossible since sin(q) and cos(q) can never be zero simultaneously. Therefore, in three dimensions, the case of the linear olarization never occurs. Thus, on the wave front, the change of sense of rotation of the transverse olarization across a critical linear olarization case in the roagation ath (as seen in the wor of Yeh et al. [1999a, Figure 4] and Figure 4c in this aer when the loci are crossing real axis of comlex -lane) turns out to be a rojection on

12 1-1 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA wave front of a continuous rolling rocess. The D critical linear olarization occurs when the 3D ellitical olarization is rojected on the wave front lane. 5. Conclusions Figure 4. [8] From discussions given in revious sections, we can conclude as follows: 1. The olarization ellises of the two characteristic electromagnetic wave modes in a magnetized cold lasma are in lanes whose normals are in the lane P b sanned by the roagation wave number vector and the steady magnetic field. The normal vector to the olarization lane offsets from the roagation direction by an angle F. This offset angle can be very large, esecially for extraordinary waves near their cutoff when the roagation direction tends to be orthogonal to the magnetic field. Besides, the offset angle can be large even in the case X 1, rovided Y is not much less than 1. When the offset angle is large the normal vector (continued) to the olarization lane can no longer be taen as the echo arrival angle without a correction.. The olarization ellise of a monochromatic comosition of the two characteristic waves roagating in the same direction is sanned by two frame vectors. These two frame vectors determine a olarization lane that wobbles as the wave roagates. On the wobbling olarization lane, the ti of the electric field traces out an ellise as a function of time. This maes the normal vector of the olarization lane either nutating (sense of rotation unchanged) or rolling (sense of rotation alternating). Thus, the use of normal vector as an indicator of wave roagation direction must be done with care. 3. In the cases X 1 and Y 1, olarization lane rolling effect is not easily erceivable since the angle between the two lanes on which the two frame vectors reside resectively is very small. Even though, awareness of its existence may imrove our understanding of the abrutly changed sense of olarization rotation along the roagation ath as being caused by the rojection of

13 DONG AND YEH: 3D EECTOMAGNETIC WAVE POAIZATIONS IN A PASMA 1-13 the electric field on the wave front of a continuous rolling rocess. 4. One of the two frame vectors shrins into the origin when the roagation direction tends to the steady magnetic field direction and when the two modes have the equally amlitude (i.e., jwj = 1). Then only a single frame vector exists and olarization becomes linear with its electric field direction rotating along the roagation ath. This is then reduced to the classical Faraday rotation case. [9] Acnowledgments. This wor was artially suorted by the National Science Foundation under grants ATM and ATM eferences Bacquerel, H., Sur interrétation alicable au hénomène de Faraday et au hénomène de Zeeman, C.. Acad. Sci., 15, , Faraday, M., On the magnetization of light and the illumination of magnetic lines of force, Trans.. Soc. ondon, 136, 1 6, Forsgren, S. K. H., Some Calculations of ay Paths in the Ionoshere, Trans. Chalmers Univ. Technol., Gothenburg, 104, ee, S. W., Chater 1: Basics, in Antenna Handboo, edited by Y. T. o and S. W. ee,. 1 15, Van Nostrand einhold, New Yor, einisch, B. W., G. S. Sales, D. M. Haines, S. F. Fung, and W. W. Taylor, adio wave active Doler imaging of sace lasma structures: Arrival angle, wave olarization and Faraday rotation measurements with the radio lasma imager, adio Sci., 34, , Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New Yor, Yeh, K. C., and C. H. iu, Theory of Ionosheric Waves, Academic, New Yor, 197. Yeh, K. C., H. Y. Chao, and K. H. in, A study of the generalized Faraday effect in several media, adio Sci., 34, , 1999a. Yeh, K. C., H. Y. Chao, and K. H. in, Polarization transformation of a wave field roagating in an anisotroic medium, IEEE Antennas Proag. Mag., 41, 19 33, 1999b. B. Dong, Chengdu Institute of Information Technology, Chengdu , China. K. C. Yeh, University of Illinois at Urbana-Chamaign, Urbana, I 61801, USA. (-yeh@uiuc.edu)