THE UNIVERSITY OF MICHIGAN 5825-4-T The Radiation Pattern of an Electric Line Current Enclosed by an Axially Slotted Plasma Sheath-I Andrejs Olte November 1965 Technical Report 5825-4-T on Grant NsG-472 Prepared for National Aeronautics and Space Administration Langley Research Center Langley Station Hampton, Virginia 23365

THE UNIVERSITY OF MICHIGAN 5825-4-T ABSTRACT A circular uniform plasma sheath of complex conductivity is enclosing a unit electric line source at the origin. The plasma sheath has an infinite axial slot of arbitrary cross section. The problem has been reduced to a Fredholm integral equation of the second kind in which the unknown function is the electric current in the plasma. An approximate solution of the integral equation has been obtained in the case of a locally partially transparent plasma sheath. On the basis of this solution a formula for the radiation pattern is derived. As an example, the radiation patterns are computed for a 360 wedge slot in the plasma sheath of three different thicknesses. In addition, attenuation calculations have been performed on the unslotted plasma sheath...... —-____ iii

THE UNIVERSITY OF MICHIGAN 5825-4-T TABLE OF CONTENTS ABSTRACT I INTRODUCTION II REDUCING THE PROBLEM TO A FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND mII THE CASE OF A UNIFORM PLASMA SHEATH IV AN APPROXIMATE SOLUTION TO THE CURRENT INTEGRAL EQUATION AND THE RADIATION PATTERN V NUMERICAL RESULTS AND CONCLUSIONS VI REFERENCES ACKNOWLEDGEMENTS -_-^^ -- ^ -- - - - - -- ^-_ -i i i i_ _ _ _ _ _ _-V_ _ i_ -

THE UNIVERSITY OF MICHIGAN 5825-4-T I. INTRODUCTION A homogeneous plasma sheath of inside radii a and outside radii b is taken to enclose a unit electric line current at the origin of the coordinate system (r, O, z) as shown in Fig. 1. The z-axis coincides with the line source. The plasma sheath may contain an axial slot of area As. The area of the plasma sheath we have indicated by Ap. The structure is uniform in the z-direction. The free space permittivity c~ and the permeability o~ is taken throughout, including the plasma. The electric conduction properties of the plasma (Rose and Clark, 1961) are accounted for by a complex conductivity ~ t2 o p ca M= (1) v+jto where to, v and wp are the frequencies of field, collision and plasma respectively. Since the primary current source as well as the plasma sheath does not depend on the z-coordinate, it follows that the electric and the magnetic fields depend only on the r and ~ coordinates. Furthermore, the electric field is tangential to the line source, and the magnetic field is transverse to it. That is to say, we are dealing only with the electric field component E (r, () and the magnetic field components Hr(r, ) and H (r, 0). The other field components are zero. If we let the plasma fill in the slot, then Ez(r, )-. Ez(r), Hr (r, 0) -v 0, and H (r, }) -H Ho(r) Thus we have only one component each of the electric and the magnetic fields, and they depend only on the radial variable r. In this report we are concerned with the problem of finding the radiation pattern of the electric line source enclosed by a uniform plasma sheath that contains an infinite axial slot. That is to say, we wish to discover the 0 variation of Ez(r, 0). This field may be found from the line current and the electric current induced in the plasma. The latter is the unknown quantity for which we derive a Fredholm -------------------- -1.1... 1,..

THE UNIVERSITY OF MICHIGAN 5825-4-T FIG. 1: COORDINPlaTE SYSTEM AND SLOTTED PLASMA SHEATH. 2(rfr- _ _ I - 2 nit Electric I' - s Line Current A %' o1 10 FIG. 1: COORDINATE SYSTEM AND SLOTTED PLASMA SHEATH. ------------------------ 2 - --- --------— i —--— ii —

THE UNIVERSITY OF MICHIGAN - 5825-4-T integral equation of the second kind in Section II. We may solve this equation only under restrictions which make the solution of very limited practical value. Setting out in search of a better solution, we first solve the unslotted plasma sheath in Section III. Defining the sheath current with a slot as the unslotted sheath current plus a correction current, we are able, in Section IV, to derive an integral equation for the correction current. We use the unslotted sheath current as an approximate current from which we compute the radiation pattern for the slotted plasma sheath. An approximate solution for the correction current serves as a basis for checking the reliability of the radiation pattern calculations. The rationalized MKS system of units has been used with time dependence of ejwt understood. In section V we present numerical results and conclusions and indicate the direction of the future work. II REDUCING THE PROBLEM TO A FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND The unit electric line source induces conduction currents iz(r', 0') in the plasma sheath. The electric field (Harrington, 1961) of both currents is given by E (r., = - [HO (kor)+ J iz(rt, 0)H2)(ko r-rt I)dA] (2) A p where H(2) is the Hankel function of the second kind, k =co. o, and [r-r' is the distance from the field point to the source point. We assume that Ohm's law holds in the plasma, i. e. iz (r, )=ac Ez(r, 0) (3) We are neglecting thereby any space charge effects in the plasma. Eliminating ~_____ ___ ___ ___ ___ ____ ___ ___ ___ ___ 3. - -----------

THE UNIVERSITY OF MICHIGAN 5825-4-T Ez(r, ~) between (2) and (3), we obtain an integral equation of the form r,+ 1'(2) 1 (2) iz(r, P)+4 wIoa iz(r )Ho (koJr-r1)dA =-4woaHo (kr). (4) p We recognize that this is a Fredholm integral equation of the second kind with the kernel H(2)(kl r-r'l ). o We put the integral equation in a more convenient form by defining 0 u T — 2 (5) 4k and 2 koi(p, ) =iz(r,p) (6) where p = kor, and hence (2) (2) i(p,)+T i(p',')H )(I' )dA'= -rH( (p) (7) AP where now dA'=p'dpd1'. This is the fundamental equation of the problem. The integral equation may be solved by the method of successive approximation (Mikhlin, 1964) provided that 1rl< (8) where B2 J'f (p-p' )12 dAdA' (9) p A P P (2) By considering unslotted plasma sheath and a Hankel asymptotic form for H, we obtain from (9) that B - 27 ko(b-a), (10) 4

THE UNIVERSITY OF MICHIGAN 5825-4-T and hence from (8) that p o (11) Iv+jWl 2 7 (b -a) This estimate of the conditions of convergence is- on the pessimistic side. Nevertheless, Eq. (11) gives us a good indication that the successive approximation method of solving the Fredholm integral equation is of value only for either very thin plasma sheath or very low plasma density. HII THE CASE OF UNIFORM PLASMA SHEATH The situation is as in Fig. 1, with the slot filled in by the plasma. An exact solution to this problem can be obtained in an elementary manner. The electric field may be expressed in the form; 1 (2) Ez(r)= - o H (kor)+A1Jo(kr), 0 < r < a, (12) = AJ(kr)+B2N(kr), a < r <b, (13) (2) =A3Ho (k r) b < r, (14) 3 o kr0 where J and N are Bessel and Neumann functions of order zero, respectively, and k=k41-jT/(u ) (15) Then from the Maxwell's equations the corresponding magnetic field is given as 0 < r.< a, (16) H(r) = j H( )(kOr)+-~ AJl(kor)= O < r)< a ( (16) H 0o 1 0 1 Wju ^(AJ (kr)+BN (krl, a<r<b, (17) oJ(r) =j A3H(1)(kr)2 b<r, (18) -- ------------— = — 5/' — ------—,

THE UNIVERSITY OF MICHIGAN 5825-4-T the cylindrical functions being of order 1. From the continuity of E (r) and H (r) at r=a and r=b, we obtain four algebraic z z equations. By solving them we obtain the complex amplitudes of the waves: A= < k Jo(ka)H (koa)k [No(kb)H( o(kob)-kNl(kb)Ho (kob)] + 4Dk2 ( 1 koNo(ka)H,(2) (koa) kJl(kb)H(2)(kob)-koJo(kb) H( (kob)] + kNl(ka)Ho (koa) [kJ(kb)H^ ^ (kob)-kJl(kb)Ho (kb)] + kJl(ka)H 2)(koa) [kN 1(kb)Ho2)(kob)-koNo(kb)H( 2)(kobi] 3 (19) A2= koN(kb)H b)-kN1(kb)H ),'2 (20) 2 1 2 2 a kJ2(kb)H (kb-k (21)o A j2- (22)(2 3 22 T k abD hD= J1(koa)Hl (kob) [Jo(ka)No(kb)-Jo(kb)No(ka] + k lJl(koa)Ho (kob) [J(kb)No(ka) Jo(ka)N (kba + Jo(koa) H ()(ko b)(k)N (ka)-J (ka)No(kb)7 | + The (t -)i J(k a)He (kob) Jl(ka)N (kb)-Jk b)Nl(ka]) (23) The current induced in the uniform plasma sheath iz(r) is given by -----------— 6 ------------

THE UNIVERSITY OF MICHIGAN 5825-4-T iz(r) =oEz(r), a<r<b (24) = aA2Jo(kr)+B2No(kr (4) By a similar definition as in (6) we introduce i(p) = [A2Jo(kr)+B2N(kr)]. (25) k2 The electric field outside the plasma sheath as given by (14) is rewritten in the form E(r)= 1 H(2) (kr) A (26) ar)= w0H0 (k0r) o 3 The first factor gives the electric field of the unit electric line source in the free space, the second factor introduces the effect of the plasma sheath on the radiation field. We define the latter factor as the radiation pattern for the electric line source radiating through a concentric uniform plasma sheath, i. e. Po= - A3 (27) This pattern is a constant; it does not depend either on r or 0. However, it is rather difficult to calculate because of the involved denominator in A3. However, in the case ka>> 1 and Ika >> 1, Po can be reduced with a good approximation to an elementary form by using the Hankel asymptotic form for the cylindrical functions. Taking two terms in the Hankel asymptotic series, we obtain -j[k(b-a)-kob+(8kob)-l + ^ l 2ke Po.~,1 —----- 2(28) ko[do+dl(8koa) +o. ]I where = + k +(-1+ k )e k(b sin(ka- ) + k E- k + k e-j2k(b-a)] (ko ) (29) i _ i_ L_(-_ )e Jos( -,ia,..(29) ____________________ A _

THE UNIVERSITY OF MICHIGAN 5825-4-T and dko(b-a) k +( k e-j2k(b-a ( ) dl=J ( kb 1 - sin(koa- 4 + a_ k k ) -j2k(b-a)1 (30) o - (-1- + )e From (1) and (15) we find that k=k'-jk" where kt=ko [A+ JA2+B1 (31) k"=koj- 2 A+ 5 A+B21/ (32) and 2 y2 A- 1- +y B- Y (33) 2 l+y2 1+Y 1+Y with X=-p; Y=V. (34) From (28) we observe that the amplitude of Po is determined principally by the factor exp [k"(b-a)] as would also be the corresponding case of a normally incident plane wave radiation through a plasma slab. The asymptotic current distribution in the plasma associated with (28) is given by f i(r) _k k 1 1 1 ko.i -+jk(r-a) k ^J1 + k o jk1 r 1b-a)] e do+dl(8ka)t-1 L )j8 (k b +r y)(kj (1- - ) e. (35) Tko )+j (kob kr )(1 k e (35) The first term in the f } brackets represents outward propagating current wave and the second term the reflected current wave. Again the exponentials are the dominant factors in this formula. ii ii8

THE UNIVERSITY OF MICHIGAN. 5825-4-T The approximate forms of Po and i(r) for the case koa l and k b < 1 are given by j (keb -4 bi( P0 + j In (a) sin(koa- )-j cos (koa-4 0 (36) and X2(1+jk bi-n) bi.. b i(r) _ _ _ — 1+jkobinj] sin(koa-4)-j cos(k a- 4r) (37) (l-jY)2 /27rka (L c The two formulas have been obtained using the small argument approximation for the cylindrical functions of argument kr and the first term in the Hankel asymptotic series for those of argument kor. That is to say, these formulas are valid approximations for a plasma sheath of sufficiently large radius and wp. IV AN APPROXIMATE SOLUTION TO THE CURRENT INTEGRAL EQUATION AND THE RADIATION PATTERN We recognize that the normalized current of the uniform plasma sheath, (25), is a solution of the integral equation (7) with Ap-. (Ap+As), i. e. i(p)+ i(p') H ( Ip-p'l)dAt= - H(2)() b (38) We define a current i (p, 0) by i(p, () =i(p)+ic(p,0) (39) and thus from (7) obtain 9 _

THE UNIVERSITY OF MICHIGAN 5825-4-T i (p 0)+T- f ic(Ip' 0)H(2)([ P-P( I)dA'=-i(p)-H(2)(p) - Ap -T i(p)H(2)([ fp-P' I )dA' (40) 0 A P However, from (38) we see that the right-hand side of (40) reduces to TJ i(p')H(2) ([p-p')dA' As and thus ic(P,)+T- Jic(P',0')H(2)(lP-P' )dA' =T i(p')H2)( t)dA.(41) O i: i~Ar As By opening a slot As in the uniform plasma sheath, we change the sheath current by ic(p,j) and the integral equation (41) defines this current. Integral equations (7)and (41) differ only in their right-hand sides. Thus the condition (8) on the convergence of the resolvent applies also to the latter. The zeroth order approximation to ic(p,0) is given by i(c(p, )=T i(p)H(2)(lp-p')dA (42) A s By opening the slot we reduce i(p)=0 in the slot area As and assuming that we may neglect ic(p, 0) we obtain by a simple transformation from (2) and (14) (2) 1, Ez(r,0) A3Ho (kor)+ 4to i (r) Ho (kor-r')dA' r>b A (43) 10

THE UNIVERSITY OF MICHIGAN 5825-4-T Using the asymptotic forms for the Hankel functions, we obtain 1 2O -j krj) jkfrfcos(]-0') E,(r, -- -/o kor -e Po- i(r')ej dAr, X EA JJ As r >>b (44) and the radiation pattern Ps(0), consistent with the earlier definition in (27), is then (f t P jk rtcos(0-01) P(0) P- i(r)e 0 dA. (45) (0) It is clear that including the current ic (r, 0) in the approximation we obtain for the radiation pattern 1sc(PI) - id /* jk rtcos(-0') /f(o) jk rtcos(0-0') P (0) P - J i(r')e dA'+ i (r'')e dA' SC 0 C rAs Ap (46) Equation (45) shows us that opening a slot amounts to adding to the existing current its negative, i. e. to make the total current in the slot area zero. For overdense plasmas PO j << 1 and the negative current in the slot area then dominates the radiation pattern. We believe that (45) gives a good approximation to the radiation pattern whenever i(r) penetrates into the sheath. The adding of i (r, 5) to the c radiation pattern P s() in (46) is not so much to improve the approximation in (45) as to provide perhaps a means of feeling out its bounds of validity. Whenever Ps (0) - Ps() (47) one can reasonably say that (45) gives a good approximation to the radiation pattern. If the slot is a wedge slot of width 20o, then the double integral in (45) may be transformed into a series, and we obtain for the radiation pattern 11

THE UNIVERSITY OF MICHIGAN 5825-4-T n^o sin(no~) Ps(0)V PO+20o; esin(n)v R cosn n (48) n=o no n where e =1, n=0 n =2, n= 1,2,3,... and kob R * n f i(p)Jn(p) pdp. (49) koa It is clear that the pattern is an even function in 0. The ~ variable is measured from the center of the slot. V NUMERICAL RESULTS AND CONCLUSIONS In Figs. 2, 3 and 4 we present the radiation through a uniform plasma sheath for Y=0. 01, 0. 1 and 1. 0 respectively. With Y=0. 01 the plasma is only slightly lossy. With Y=1. 0 the plasma is very lossy. The inside radius of the plasma sheath has been kept at one free space wavelength Xo, and the sheath thickness has been taken up to 3X. The plasma parameter X is varied in steps from 0.25 to 2. 0. In Figs. 2 and 3 the calculations have been carried out using the exact formula (27) for all X values except X=2. For X=2 calculation we used the asymptotic form (28) with dl set to zero. The same asymptotic form was used for all calculations in Fig. 4. The asymptotic form for P0 gave excellent agreement with the exact results in Figs. 2 and 3 where it was applicable, and surprisingly little disagreement, even for cases (e. g. Y=0. 01, X=1) where the arguments of the cylinder functions were far too small for the Hankel asymptotic expansion to be valid. The processes that inhibit radiation through the sheath are reflections, absorptive attenuation, and reactive attenuation by the plasma. The attenuation of the radiation for 0 < X< (1l approx.) is due to reflections and absorptive attenuation, and for X> (ilapprox.) due mainly to both reactive and absorptive attenuation. In 12

THE UNIVERSITY OF MICHIGAN 5825-4-T 1.00.5 1.0 (b-a)/0.2 1.5 2.0 2.5 3.0 FIG. 2: RADIATION GRAPHS THROUGH UNIFORM PLASMA SHEATH VS SHEATH THICKNESS FOR a=lX, Y=0. 01 AND X AS A PARAMETER. 0 0~90~13.01 13

THE UNIVERSITY OF MICHIGAN 5825-4-T 1.0 0 0 25.~~~~~~~~~~~00~~~~~~.1 -90.01 0 0.5 1.0 1.5 2.0 2.5 3.0 (b-a)/Xo FIG. 3: RADIATION GRAPHS THROUGH UNIFORM CIRCULAR PLASMA SHEATH VS SHEATH THICKNESS FOR a=lo, Y=0.1, AND X AS A PARAMETER 14

THE UNIVERSITY OF MICHIGAN 5825-4-T 0.25 12 12 \1.0 \ l~0 )~ 0.5 1.0 1.5 2.0 2.5 3.0 0. 1 0 O 0 5 2. 0 2. 5 3. 0 (b-a) /X FIG. 4: RADIATION GRAPHS THROUGH UNIFORM CIRCULAR PLASMA SHEATH VS SHEATH THICKNESS FOR a=lXO, Y=1.0 AND X AS A PARAMETER 15

THE UNIVERSITY OF MICHIGAN 5825-4-T each case we may identify conditions where only one process is primarily causing the attenuation. Thus for X </2 and Y=0.01 we see, in Fig. 2, the radiation graphs oscillating because of the reflections. As Y increases to 0.1 in Fig. 3, the amplitude of the oscillations decreases, and for Y=1 in Fig. 4, the oscillations are practically gone; but now the graphs have assumed a downward slope. This is caused by the absorptive attenuation in the plasma taking over and eliminating the reflections. Also, we see that we get quite a bit more attenuation for very lossy plasma (Y=1) than for very low loss plasma (Y=0.01). For the case of X >J2 the opposite is true: low loss plasma causes more attenuation than a very lossy plasma. This is due to the effectiveness of the reactive attenuation. In Fig. 5, we show a 36~ wedge slot in the plasma sheath, and in Figs. 6 to 14 we present the radiation power pattern IPs(0)12 calculated from (48) for the sheath thickness of 0. 5Xo, 1. OXo and 2.0 A, respectively. The inside sheath radius is kept at 1. OX0. The graphs have been arranged in sets of three: Y=1. 0, 0. 1 and 0. 01, and X is a parameter in each set. Since the power patterns are even functions in 0, we have plotted them only from 0~ to 180~. By the constant lines we show the attenuation in the power radiated before the plasma slot is introduced, i. e. the radiation power pattern for the unslotted plasma sheath. These lines are obtained from Figs. 2 to 4. In some cases the attenuation is so large that it is off scale. These values we will give in a note. As a way ofcalibration we mention that the radiation power pattern of unit amplitude would correspond to the electric line source radiating in the free space. In Figs. 6, 7 and 8 we present the set Y=1. By opening a slot in this very lossy plasma we have, for all practical purposes, re-established radiation in the forward direction, i. e. 0=0, independent of the sheath thickness. Outside of the front lobe there is no significant recovery of the signal strength. Increased slot width, the calculations will show, narrows the front lobe with some additional gain 16

THE UNIVERSITY OF MICHIGAN 5825-4-T (r,4) -, E 0 Unit Electric' Line Source FIG. 5: PLASMA SHEATH WITH a=1X0, b=2Xk, AND A 36~ WEDGE SLOT 17

THE UNIVERSITY OF MICHIGAN 5825-4-T 1. 0 i' ^ - \\ v 0.50 0.50 5 1.0 1. 0 0.1 1 0. 05 - 0.01 4 135 1; 0 + 0 (degrees) 90135 FIG. 6: RADIATION POWER PATTERNS FOR 0.5Xo THICK PLASMA SHEATH WITH 36~ WEDGE SLOT, Y=1, AND X AS A PARAMETER. The parallel lines are the corresponding patterns for the unslotted case. 18

THE UNIVERSITY OF MICHIGAN 5825-4-T 1.0 0.25 ~~~~~~~0.5- \ ^ ^ ^-^ 0.50 0.5 t. i/0.50 0.10.05 o 1.0 00 45~ + 90~ 135~ 180~ FIG. 7: RADIATION POWER PATTERN FOR 1.OX THICK PLASMA SHEATH WITH A 360 WEDGE SLOT; Y = 1, AND AS A PARAMETER. The parallel lines are the corresponding patterns for the unslotted case. _1.i19 0.001 19

THE UNIVERSITY OF MICHIGAN 5825-4-T - \\ \\00.25 0.25 0.5 1..50 pj _________5 0 0 ___0. _707. ~~0.707.0 NOTE: |P2 = 2.3x10 0. 01- for X = 1 ~ 4L^~ 4o0~ +o 135~ 180~ FIG. 8: RADIATION POWER PATTERN FOR 2. OX THICK PLASMA SHEATH WITH A 360 WEDGE SLOT, Y = 1, AND X AS A PARAMETER. The parallel lines are the corresponding patterns for the unslotted case. 20

THE UNIVERSITY OF MICHIGAN 5825-4-T.707 -/1.0- ---- ^ -— /\_X NOTE: |P |= 2x10 3 for X= ~2 1. 0 p1,(0)12 \ I o., 1, l J^U^T I \ 0.01 ___________ FIG. 9: RADIATION POWER PATTERN FOR 0.5X THICK PLASMA SHEATH WITH A 360 WEDGE SLOT, Y=0. 1, ANDX AS A PARAMETER. The parallel lines are the corresponding patterns for the unslotted case. 21

THE UNIVERSITY OF MICHIGAN 5825-4-T 1. 0 0.25 1,-=XH I 0.0 0 I 8 NOTE: P |2=3.4x10-6 ~\ j for X= /1W 01. 0 450 900o 1350 180 FIG. 10: RADIATION POWER PATTERN FOR 1.0 X THICK PLASMA SHEATH WITH A 360 WEDGE SLOT, Y= 0.1, ANDTX AS A PARAMETER. The parallel lines are the corresponding patterns for the unslotted case. 22

THE UNIVERSITY OF MICHIGAN 5825-4-T \ 0^ 5 \> /\ /0. 707 for X = 1.0 WITH A 326 WEDGE SLOT, Y=0.1, AND X AS A PARAMETER. The 1.0_ parallel lines are the corresponding patterns for the unslotted case. 23 i~~l for X = 1.03

THE UNIVERSITY OF MICHIGAN 5825-4-T J0O.707 0.5; 0.707 NOTE: p02 = 2x103 for X= \T2 _____ ^~1.0 f __1.0 0. 145 90 1 I I:, I / 0. 01 0 450 900 135I 1800 FIG. 12: RADIATION POWER PATTERN FOR 0. 5X THICK PLASMA SHEATH WITH A 36~ WEDGE SLOT, Y=0.01, AND X AS A PARAMETER. The parallel lines are the corresponding patterns for the unslotted case. 24

THE UNIVERSITY OF MICHIGAN 5825-4-T 1. tl: ^ \/ I/ I \ 1 / \J\ I NOTE: |P | =3.4x10 l \ 0.707 — O.7for X = 7 0.01 _0. f2 0~ 45~ 90~ +0 135~ 180~ FIG. 13: RADIATION POWER PATTERN FOR 1. OX THICK PLASMA SHEATH WITH A 36 WEDGE SLOT, Y -0.01, AND X AS A PARAMETER. The parallel lines are the corresponding patterns for the unslotted case. 25

THE UNIVERSITY OF MICHIGAN 5825-4-T 0.707 0.5 0. 707 1.0 0.1~0.1 NOTE: P = 10 2 forX= 1.0 0.01 0 45~ 90~ +0 135~ 180~ FIG. 14: RADIATION POWER PATTERN FOR 2.OX THICK PLASMA SHEATH WITH A 36~ WEDGE SLOT, Y=0.01, ANB X AS A PARAMETER. The parallel lines are the corresponding patterns for the unslotted case. 26

THE UNIVERSITY OF MICHIGAN 5825-4-T in the signal strength for 0=0. If we halve the slot width, the forward lobe is considerably reduced with no benefit for the rest of the pattern. In Figs. 9, 10 and 11 we present the set Y=0. 1. By opening a slot in this sheath of "medium" lossy plasma we observe that for X < 0. 707 the formerly uniform pattern breaks into oscillations. The radiation is enhanced slightly in the forward direction. The oscillating character for the pattern persists as X is increased, i. e. a dominant forward mode is not formed in the range of the validity of approximation, in contrast to the case of the very lossy plasma sheath. Sections of graphs for which the approximations are expected to break down are shown in the dotted line. For X > 0. 707 the slot improves the radiation very substantially as compared to the sheath without a slot. In Figs. 12, 13 and 14 we present the set Y=0. 01, i. e. a very "low" loss plasma sheath. The comments for this one are substantially the same as for the previous case. The sets of data have been presented as they were calculated. A dominating forward lobe formed for the Y=1 set. It came as a surprise that for Y=0. 1 and 0.01 sets no such dominance was found. However, there is no doubt that a dominant forward lobe would develop as X is increased beyond J}. Only (48) is not then a valid approximation in the case of our thick sheath. The method we have presented for calculating the radiation pattern for the slotted plasma sheath makes use of the electric currents in the plasma before the slot is opened. It is an approximate method which is expected to give the radiation pattern with good accuracy only if the electric currents sufficiently penetrate the plasma sheath. If the attenuation of the unslotted plasma sheath is more than 20 db, i.e. Po2 < 0. 01, then we are approaching the bounds of validity of the approximation for the pattern as a whole. However, one expects that the forward lobe will be given to a good order of approximation even for somewhat larger attenuation 27 ___

THE UNIVERSITY OF MICHIGAN 5825-4-T than 20 db, as was shown by calculations performed in (46). If the attenuation of the unslotted plasma sheath is such that the radiation through it is practically insignificant, we then find that the electric currents penetrate the plasma only a small fraction of the free space wavelength. If we open a slot in the plasma under these conditions, the slot in fact becomes a waveguide which connects the source with the outside space. For suitable slot boundaries one can show that a finite number of modal fields can propagate in the slot waveguide. These modes may be used as a basis for calculating the radiation pattern. Work is in progress to exploit this waveguide approach for these cases where the integral equation method becomes too cumbersome. This work will be reported as Part II of this study. The two significant conclusions indicated by this study are: 1) Opening a slot in the plasma sheath re-establishes radiation in the forward direction to a substantial level. 2) The plasma slot should be at least one-half wavelength wide at its narrowest cross section when the electric field is polarized tangential to the slot walls. Increasing the slot width beyond one-half wavelength brings only diminishing returns as far as the radiation enhancement in the forward direction is concerned. VI REFERENCES Harrington, R. F. (1961) Time-Harmonic Electromagnetic Fields, McGraw-Hill, pp.223-232. Mikhlin, S. G. (1964), Integral Equations, The MacMillan Co., p. 13. Rose, D.R. and M. Clark, Jr. (1961), Plasmas and Controlled Fusion, MIT Press, p. 150. 28 _ —

THE UNIVERSITY OF MICHIGAN 5825-4-T ACKNOWLEDGEMENT The author has benefited from discussions with Wi. F. Croswell, F. Russo and T. E. Sims of the NASA-Langley Research Center and their colleagues. For the computations he thanks J. Klebers and E. K. Miller of the Radiation Laboratory. -------------- ~29 --

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