1965 Daly, et al.: Radiation Resistance of an Osmllating Dipole 583
y go•
~~~::::::=---!~~-l.~~_._Jc--_JL...J.~x 2. -- 4 6 8 10 PIRECTION INCIDENT 80\INOARY WAVE - NORMALIZED RADIATIOti ANGLE FUNCTION [R'Fl8!] 1 OIRECTlON OF REFLECTED BOUNDARY WAVE .,.__ R's RAOIATION COEl'flCIENT R • ),,_ vZ1 Fig. 4. The radiation pattern.
part of the impedance (x <0) becomes large within a boundary waves (low z1) the pattern is more directive, few wavelengths. This results in a boundary wave which with the maximum pointing closer and to the is tightly coupled to the surface and is consequently not positive x axis. easily transformed into radiation. Figure 4 shows the radiation pattern for different ACKNOWLEDGMENT values of z1. For tightly coupled boundary waves The author wishes to express his gratitude to Dr. (high z1) the pattern is broad. For loosely coupled C. Marcinkowski for his many and valuable suggestions.
Radiation Resistance of an Oscillating Dipole in a Moving Medium
P. DALY, K. s. H. LEE, AND c. H. PAPAS, MEMJ3ER, IEEE
Abstract-The radiation resistance of an oscillating electric di I. I~TRODUCTION pole immersed in a moving medium is calculated by Brillouin's "EMF method.'' Two cases are studied in detail: in case 1) the X ANTENNA THEORY the radiation resistance moving medium is sinlple, and in case 2) the moving medium is an of the dipole antenna plays an important role since ionized gas (plasma). It is found that in case 1), the motion of the I it is effectively the terminating resistance of the medium tends to increase the radiation resistance, whereas in case transmission line supplying the antenna. The radiation 2), the motion of the medium tends to decrease the radiation resis resistance of an elementary dipole radiating in vacuum tance. A comparison is made between Frank's well-known theory of the radiation from an oscillating dipole moving through a material has been known for some time, and more recent investi medium and the results obtained in this paper. gations into the subject have dealt with more compli cated situations such as dipole radiation in anisotropic media [1], [2] and in dissipative media [3]. However, Manuscript received 27, 1964; revised February 15, 1965. The research in this paper supported by the U. S. Air Force Office no attention has been given in the literature to finding of Scientific Research. P. Daly is with the Dept. of Mathematics, L niversity o[ Glasgow, the radiated power from a dipole in a medium moving Scotland. He was formerly with the Antenna Lab., California Insti uniformly with respect to the source. Recently, Lee and tute of Technology, Pasadena, Calif. Papas [4], [5] have formulated the problem of electro K. S. H. Lee and C. H. Papas are with the Antenna Lab., Cali fornia Institute of Technology, Pasadena, Calif. magnetic radiation in the presence of a moving medium. 584 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION July
In the present paper, we apply their theory to calculate Thus, we see that P may be calculated by using (ld), the radiation resistance of an elementary dipole--1) in i.e., by the Poynting vector method, or by using expres a simple (i.e., nondispersive, lossless, homogeneous, and sion (le), i.e., by Brillouin's Ei\IF method. Both meth isotropic) moving medium, and 2) in a moving ionized ods will, of course, yield the same result, but in the gas (plasma). present instance where the source is an oscillating dipole The total time-average power radiated by a source it is easier to use Brillouin's method. may be calculated by either Brillouin's "E:\IF method" [6] or Poynting's vector method. To formulate these POWER RADIATED IN A SIMPLE }lOVING MEDIUM two methods quantitatively we note that in an inertial To compute by Brillouin's method the total time frame K at rest with respect to an electromagnetic average power P radiated by an oscillating dipole im sourc:e] the :.\:J axwell equations are mersed in a simple moving medium, we need to know a a only the electric field E of the dipole. Since the vector 'ilXE=--B 11XH=J+-D potential A and the scalar potential cf> for such a dipole at at source are already known [4], E can be obtained V'·B=O 11 · D = p. (la) through the use of the relation E= -Vef>+iwA. Thus we find that the electric field E of an electric dipole of These equations are valid regardless of the state of the moment pis given by medium surrounding the source. However, they are quite meaningless until we introduce constitutive rela E = w 2µ'p·LG. (2a) tions that connect the four field vectors E, B, D, H. In the present paper we assume that in the rest frame K' Here the dyadic operator L has the form of the medium moving past the source with velocity v 1 2 1 2 2 K "( iK C "( 1 the electromagnetic properties of the medium are de L = u + - vv + --(11v + vv) + - 1111 (2b) n'2 wn'2 k2 scribed by the constitutive parameters e' (dielectric constant) and µ' (permeability). This assumption, ac 1 2 2 1 2 where K = (n' - 1)/c , n' = c(µ'e') ' , k = n'aw/c, cording to l\tlikowski's electrodynamics of moving media, 2 2 2 2 2 1 2 a = (1 - {3 )/(1 - n' {3 ), {3 = v/c, 'Y = (1 - {3 )- 1 , leads to the following constitutive relations in K: and u =unit dyadic. Also the Green's function G for the 1 case where n'fJ < 1 is known to be D+-vXH= e'(E+vXB) c2 1 B - -v XE= µ'(H- v X D) (lb) c2 • ; a2 - 2 1 2 J where c is the vacuum speed of light. exp [ ik 'V r + -----;;;--- (v · r) 1 For harmonic time-dependence (e-i"' ), the :.\faxwell (2c) equations (la) yield the following relation a 2 -1 r 2 + --(v·r) 2 V v2 ~Ref (EX H*)-ndS = - ~Ref E·]*dV 2 s 2 v where 2 2 b = a13"( [(n' - l)/n']. - .!:!__Im f (H*·B - E·D*)dV (le) 2 v Substituting (2a) into (le), we find that the total time where Re and Im denote, respectively, the real and average radiated power is given by imaginary parts, and n is the unit outward normal to w3µ' the surface S surrounding a volume V which includes P = - - Im Jp·LG(r) ·po(r)dV (2d) 2 the source]. By virtue of the constitutive relations (1 b), the second term on the right-hand side of (le) is identi where o(r) is the Dirac delta function. If one attempts cally zero, and we see that the total time-average power to evaluate this integral by taking the value of the inte Pis given by the surface integral grand at the origin, he finds that Pis infinite. It is perti nent to note that a similar infinity arises also in the P =~Ref (EX H*)·ndS (ld) evaluation of the radiation resistance of a dipole im 2 s mersed in a magneto-ionic medium f2 ]. It seems that or by the volume integral this "infinity catastrophe" is an outcome of the non isotropic nature of the ambient medium. In the present case the infinity is removed by noting that P = - Re E·]*dV. (le) ~ J: o(r) =o(x)o(y)o(z) and that o(x) = o(-x), o(y) =o(-y), 1965 Daly, et al.: Radiation Resistance of an Oscillating Dipole 585 o(z) =o(-z). Thus, all the odd functions of x, y, z in the formly through a material medium. By using the integrand vanish on integration. The justification of Poynting vector method he found in the rest frame K' removing the singularity in this way rests on the fact of the medium that the total time-average radiated that the result so obtained checks with that derived by power is given by use of the Poynting vector method (see Appendix). 1 - 132 )3 \Ve now proceed to find P by evaluating (2d) for two P~' = ( (2k) dipole orientations. In one orientation the dipole is lined 1 - n'2132 Po up with the moving medium, and in the other the dipole is normal to the direction of flow of the medium which for parallel orientation, and by is taken to be along the z axis of the coordinate system. 1 - 2 In the parallel orientation v) we have p p,e,, and, 13 )2 (pJJ = P1-' = ( Po (21) hence, (2d) for the total time-average radiated power 1 - n'2132 reduces to for perpendicular orientation. Comparing (2k) and (21) with (2f) and (2i), we see that his results exceed ours by Pii = wap,2µ' Im f [i + ~ + 2ib ~ + ~ ~] 2 2 2 a factor (1-{3 )/(1-n' /3 ). However, this discrepancy is 2 an' k oz k2 fJz 2 only an apparent one and can be explained as follows. ·G (r) o(r) d V · (2e) The po\ver radiated by the dipole, as measured in the rest frame K' of the medium, is the sum of two parts. Carrying out the integration \Ve obtain One part is the electromagnetic radiation which is equal to our expression (2f) or (2i), while the other part is due (2f) to the mechanical work done on the dipole to keep it moving uniformly through the medium. This mechani
4 2 cal work done appears in the form of electromagnetic where P 0 (=w p, µ'vfµ'e'/l27r) is the total time-average power the dipole would radiate if the medium were not radiation. In the rest frame K of the dipole, however, in motion. The corresponding expression for the radia only the first part contributes to the radiated power; tion resistance of the dipole is the second part contributes nothing because the dipole is at rest. 1 - 132 )2 The mechanical force necessary to keep the dipole R 1 = Ro (2g) ( 1 - n'2132 stationary in K is equal in magnitude but opposite in direction to the reaction force which can be calculated where Ro is the radiation resistance when the medium with the help of Minkowski's stress tensor. Then, by a is not in motion. proper Lorentz transformation, one can obtain the In the perpendicular orientation (p_.L v) \Ve choose mechanical force required to keep the dipole moving p = Pxex, and hence (2d) for the total time-average radi uniformly in reference frame K'. From a knowledge of ated power becomes this mechanical force in K' one can deduce the part of the mechanical work done on the dipole which appears w3px2µ' 1 fJ2) Pl_ = --Im f (1 + - - G(r)o(r)dV (2h) in the form of electromagnetic radiation in K'. 2 2 2 k fJx So far we have confined our attention to situations which in turn yields where n'/3<1. It is pertinent to note that when n'/3?:.1 the radiation is of the Cerenkov type, i.e., the fields are 2 1 - 13 ) confined within a conical region and are infinite on the Po. (2i) ( 1 - n'2132 conical surface [4], and the total radiated power turns out to be infinite. The infinite value arises from the as The radiation resistance in this case is given by sumption that the medium is nondispersive.
2 1 - 13 ) Rl_ = Ro. (2j) PowER RADIATED IN A l\:IovING loNrzEn GAs ( 1 - n'2132 Now we consider the case where the moving medium For both parallel and perpendicular orientations \\re is an ionized gas. In the rest frame K' of the gas the con 2 2 see that for n' > 1 the radiated power and the radiation stitutive parameters are µ'=µ 0 and e'=e0 (1-w/ /w' ) resistance exceed their stationary values. This means where eo, µo are the free-space dielectric constant and that to maintain the same dipole moment p and angular permeability, w/ is the plasma frequency, and w' is the frequency w as in the stationary case, more power has frequency of the radiation. to be fed into the dipole. \Ve recall that to compute by Brillouin's method the We recall that Frank [7] has considered the comple total time-average power radiated by the dipole we need mentary problem of an oscillating dipole traveling uni- to know the electric field E of the dipole. It has been 586 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Jul,y found that, in the rest frame K of the dipole, Eis related Similarly the third term Pu<•> on the right-hand side of to the dipole moment p through the relation [5] (3d) yields
2 2 E = µoc p · (k0 u + VV)G(O) iwµocfJ(e,p· V + e,·pV)GOl
2 2 2 - µ. 0c p · (k0 {3 e,e, - VV)Gm (3a) After performing the integrations in (3g) and (3h) we where u =unit dyadic, ko w/ c, and w =angular fre find that the total time-average power radiated by the quency of dipole. The functions c(oJ, G(ll are defined by dipole is given by
e"'' 3 [ k2 k 2 GCO) =- Pi! = Po 1 - -- - _P_ (2 + j) J (3i) (3b) 2 2 2 47!'1' 2 3k o 2ko fJ
where f is defined by (3c) 1 [ k - {Jk 0 2 2 2 f = 2ykok loa --- where kp=w ,/c, Wp plasma frequency, k k -kp , 1 0 kk;/31'3 P " k + {Jko 2 2 2 and p =x +y • To compute the total time-average power radiated by (3j) the dipole, we substitute (3a) into (le) and obtain for Pl] v the following expression For the case p..Lv, we choose p and find m a 2 ,.iµ 0p, 1 a2 ) P11 = --Im J( 1 + - - cc 0lo(r)dV similar manner that the total time-average radiated 2 ko 2 iJz2 power PJ.. is given by
3 PJ. = Po[t - ~+ kl (2 + 7 2f)] (3k) 2 3ko2 4ko2fJ 2
WJJ.oP 2 2 f a2 2 2 c Im - - {3 k0 G(llo(r)dV. (3d) where f is defined by (3j). + --· 2 2 az Expressions (3i) and (3k) can be cast into power series 2 The first term P10l on the right-hand side of (3d) is in fl. Keeping terms up to {3 we obtain easily evaluated and is given by
3 2 (31) oo µ.op. k(1_ ~). (3e) 2 8'Jf 3ko 2 2 k 2(3ko - 4k ) P P 4 2 Po 1- fJ2 0((3 ) (3m) To evaluate the second term PiC ) on the right-hand side [ 10k2ko2 J. of (3d), we note that for s> Ikl the integral Examination of (3i), (3k), and (3j) shows that Pg and !__ ccoo(r)dV PJ.. are invariant under the transformation (3--'>-{3. This f az means that Pa and PJ. have power-series expansions 1 has no real part since the Hankel function H 0( l arising which involve only even powers of {:J. For small fl we see in the expression for G
Plasma Simulation with an Artificial Dielectric in a Horn Geometry
K. E. GOLDEN, MEMBER, IEEE
Abtsract-The simulation of a plasma sheath using an artificial I. INTRODL'CTION dielectric is studied in this investigation and is applied to an antenna geometry which is similar to some configurations encountered in HE PURPOSE of this investigation was to study aerospace applications. The antenna configuration is equivalent to a plasma simulation using a rodded medium and to horn in an infinite ground plane with an unbounded plasma layer T study the radiation patterns of an X-band horn in front of the horn. The plasma layer is simulated by a rodded me antenna in the presence of a simulated plasma sheath. dium, and the radiation patterns of the antenna system are studied The plasma sheath is assumed to be under-dense or experimentally at 9, 10, and 11 Gc/s. · Two constructions of rodded media are utilized in the pattern operating above the plasma frequency. The antenna measurements. The experimentally determined radiation patterns geometry is shown in Fig. 1. Several authors [ 1 ]-[4] have a secondary ma.xi.mum associated with the properties of the have studied this geometry or very similar geometries plasma sheath. More energy is radiated in the OFF-axis region when previously, but to obtain a clean experiment using a the plasma sheath is present. laboratory plasma is very difficult. Typically, the experi mental and theoretical geometries are not the same. Manuscript received September 21, 1964; revised February 25, Usually, the theoretical models are highly idealized and 1965. The author is with the Aerospace Corp., Los Angeles, Calif. only approximate the actual experimental geometry be-