IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 37, NO. IO, OCTOBER 1989 I235

Axial Slot Antenna on Dielectric-Coated Elliptic Cylinder

Abstract-The radiation properties are investigated for an axial slot antenna on a conducting elliptic cylinder with a homogeneous dielectric coating. In the dielectric coating and in the exterior free-space region, the field is expanded in elliptic waves via the Mathieu functions. The Mathieu angular functions are employed as basis and testing functions to enforce the boundary conditions via Galerkin’s method at the interface between the dielectric and the free-space regions. Numerical results are presented in graphical form for the transverse electric (TE) and transverse magnetic (TM) polarizations to illustrate the far-field radiation patterns, the gain versus coating thickness, and the aperture conductance versus coating thickness.

I. INTRODUCTION

HE AXIAL SLOT antenna on a dielectric-coated circular T cylinder has been investigated by Hurd [l], Wait [2], b o p [3], Shafai [4] and Richmond [5]. Wong [6], [7] and Wait [2] have investigated slot antennas on an uncoated elliptic cylinder, and Richmond [8] has considered the scattering properties of a dielectric-coated elliptic cylinder, but the slot antenna on a coated elliptic cylinder appears to have been neglected.

In certain cases the elliptic cylinder provides a useful model for the fuselage of an aircraft. Of practical concern is the influence of the heat-shielding tiles on the performance of a slot antenna on the space shuttle. In addition, the plasma surrounding a reentry vehicle has often been represented by an equivalent dielectric coating, as by Croswell et al. [9].

In this paper we present the theoretical analysis of radiation by an axial slot antenna on a dielectric coated elliptic cylinder for the transverse electric (TE) and transverse magnetic (TM) polarizations. Numerical results are presented in graphical form for the far-field radiation patterns, the gain versus coating thickness, and the aperture conductance versus coating thickness.

II. THEORY FOR TM AXIAL SLOT Fig. 1 illustrates a perfectly conducting elliptic cylinder

with a dielectric coating. The structure is infinitely long, and the cylinder axis coincides with the z-axis. The semimajor and semiminor axes are denoted by al and bl for the conducting cylinder, and by a2 and b2 for the outer surface of the coating.

Manuscript received June 14, 1988; revised November 22, 1988. This work was supported in part by the Joint Services Electronics Program under Contract N00014-78-C-0049 with The Ohio State University Research Foundation.

The author is with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212.

IEEE Log Number 8929259.

Fig. 1. cylinder with dielectric coating.

Cross-sectional view of axial slot antenna (dotted line) on elliptic

In the elliptic system, U and U denote the radial and angular coordinates, respectively. Thus U = u1 on the conducting ellipse and U = u2 on the dielectric interface. We require both of these ellipses to have the same foci, located at x = k d on the x axis, and this results in a nonuniform coating thickness as shown.

Let ( p , E ) and (pl, c l ) denote the permeability and permittivity of the ambient medium and the dielectric coating, respectively. We require these media to be homogeneous and, until more general subroutines become available for the Mathieu eigenvalues, lossless. Excellent discussions of the Mathieu functions and elliptic cylinders have been presented by Stratton [ 101 and Morse and Feshbach [ 111.

The aperture of the slot antenna is indicated by the dotted line in Fig. 1, and u l and u2 denote the angular coordinate values at the edges of the slot. The slot width, as well as the angular location of the slot, may be selected arbitrarily.

We consider time-harmonic fields with the time dependence ejot suppressed. The field is considered to have no z- dependence. In this two-dimensional problem, the TE and TM polarizations may be analyzed independently. The TM field in a source-free homogeneous region can be constructed with the aid of a vector potential A’ = 2 as follows:

(1)

(2)

H,, = ( -joE/h)dA/ih (3)

H, = (-jwE/h)dA/du (4)

V2A = - k2A ( 5 )

(6)

E,, = E, = H, = 0

E, = k2A = w 2 p d

h = dJcosh2 U -cos2 U.

OO18-926X/89/1OO0-1235$01 .OO 0 1989 IEEE

1236 I E E E TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. IO. OCTOBER 1989

The Mathieu functions are the eigenfunctions for the elliptic cylinder. In the exterior free-space region, let us express the potential as a series of elliptical waves (in terms of Mathieu functions) as follows:

A“= (l/k2) m

[CEHE(S, cosh u)SE(S, COS u ) / M ~ ( s ) n=O

+ C:H:(s, cosh u)SO,(s, cos u)/MO,(s)l (7)

where s = kd. The Mathieu angular functions are denoted by S , , and we denote the even and odd versions with the superscripts “e” and “0,” respectively. It should be noted that S : vanishes if n = 0. The normalization constants M,, are defined by Morse and Feshbach [ l l ] on page 1409, the angular functions on page 1408, and the Mathieu radial functions H, on page 1573 with i replaced by ( - j ).

We can save a great deal of space without undue confusion by representing (7) in simplified notation as follows:

C,H,(s, cosh u)S,(s, COS u)/M,(s). (8)

In this same simplified notation, the potential in Region I (the dielectric coating) is given by A’=(l/kT) [AmJm(sl, cosh U )

A”=(l/k2) n

m

+BmNm(sl, cosh u)lSm(s1, COS u ) / M m ( s l ) (9)

where s1 = k ld and kl = U&. The Mathieu radial functions Jm and Nm are defined in [l 1 , pp. 1569-15721. From (4), one component of the magnetic field intensity is

Ht= - j ( l / (~p lh) ) C [A,JA(sl, cosh U) m

+BmNA(sl, cosh U)] * Sm(s1, COS u)/Mm(sl) (IO)

Hi*= - - j ( l / (~ph) ) C CnH:(s, cosh U ) n

S,(s, cos u)/M,(s). (11)

For the functions Ji , N i , and H i , the prime denotes the derivative with respect to U.

We require E, to be continuous across the interface at U =

Equations (12) and (13) are valid for all values of U. To solve these equations, let us apply Galerkin’s method. We multiply (12) and (13) by Sm(sl, cos U) and integrate over the region 0 < U < 2n to obtain

Am J m ( S l , cosh ~ 2 ) +BmNm(sl, cosh ~ 2 )

= C n TmnHn(s, Cosh U Z ) / M ~ ( S ) (14) n

Am Jh (SI 3 Cosh ~ 2 ) + B m NA (SI, cosh uZ)

= ~ r Cn TmnHi (S, cosh U Z ) / M ~ ( S ) (15) n

2%.

Tmn= S Sm(sI, COS U ) S n ( S , COS U ) du. (16)

Equations (14)-(16) actually represent two sets of equations. In the first set, each capital-letter quantity has the superscript e, and in the second set the superscript is o. T:n and Tkn are evaluated from (16) with the aid of the Fourier series expressions for Sm and S,, as given by Morse and Feshbach [ 1 11. It is easy to show that T,, vanishes unless the integers m and n are both even or both odd. Tmn also vanishes unless Sm and S, both have the same superscript in (1 6).

The tangential electric field intensity must vanish over the perfectly conducting portion of the boundary at U = u1 , and it must match the prescribed aperture field distribution F(u) in the slot. Thus

[ A m J m ( s l , cosh U l ) + B m N m ( S I , Cosh UI)] m

* COS u ) / M m ( s d

(17)

We multiply (17) by S , (sl , cos U) and integrate over 0 < U < 2?r to obtain

AmJm(sl, cosh ~l )+B,N, ( s l , cosh U])

”2

“1

=Fm= s F(u)Sm(sl, cos U) du. (18)

Equation (18) actually represents two equations. The first equation is obtained by inserting a superscript e on each capital-letter quantity except F(u) in (18), and the second is obtained in a similar manner by inserting the superscript 0. From (18),

Bm= [Fm-Am Jm(S1, cosh ~ l ) ] / N m ( ~ l , cosh ~ 1 ) . (19)

Equation (19) can be employed to eliminate B, in (14) and (15), yielding

A m p m = -FmNm(sl, cosh ~ 2 )

+ Cn T m n H n ( S , cosh u Z ) N ~ ( S ~ , cosh u ~ ) / M , ( s ) (20) n

Amp;= -FmNA(sl, cosh U Z )

+ Pr Cn TmnHi (s, cosh ~dNm(s13 cosh U l ) / M n ( s ) n

where pr = p l / p .

RICHMOND: AXIAL SLOT ANTENNA

S ( P , ~ ) = ( I / ( T ~ P ) ) jnCnSn(S, COS 4 ) / M n ( S ) I n

1237

(39)

P,=J, (s~ , cosh u ~ ) N , ( s ~ , cosh U,)

- J , ( s ~ , cosh U~)N,(SI, cosh ~ 2 ) (22)

PA= J ; ( s ~ , cosh u ~ ) N , ( s ~ , cosh U I )

-J , (SI , cosh u ~ ) N A ( s ~ , cosh ~ 2 ) . (23)

From (20) and (21), we obtain two expressions for A,. By equating these expressions, we obtain the following system of simultaneous linear equations

C C , ~ , , = V , , m=O, 1, 2, (24) n

V m = F m [ N m ( ~ l , cosh UZ) /P , -N; (S~ , cosh u~)/PA] (25)

Z,, = [ Hn (s, cosh UZ)/P, - prH; (s, cosh u ~ ) / P ; ]

TmnN,(s1, cosh U ~ ) / M , ( S ) . (26)

Equations (24)-(26) represent two cases. In one case each capital-letter quantity has the superscript e, and in the other case the superscript is 0. Since Z,, vanishes unless m and n are both even or both odd. (24) actually represents four independent sets of simultaneous linear equations as follows:

C ; z ; , = V ; , m = O , 2 , 4 (27)

C ; Z ; , = v ; , m = l , 3, 5 (28)

C ; Z ; , , = V ; , m = 2 , 4 , 6 (29)

n = 0,2,4

n = 1,3,5

n = 2,4,6

C ; Z ; , = V ; , m = l , 3, 5 e . . . (30)

The summations in (27)-(30) represent infinite series, but they are truncated for numerical calculations.

With matrix inversion techniques, (27)-(30) can be solved to determine the coefficients C, . Then the coefficients A , can be calculated from (20) or (21), and finally the B, are obtained from (19). This procedure is based on the assumption that the aperture field distribution is known or prescribed. Following the lead of Harrington [12], we assume a cosine field distribution as follows:

n = 1,3,5

E : ( u ~ , u ) = F ( u ) = E ~ COS [ ~ ( U - U O ) / ( ~ $ ) ] (31)

(32)

(33)

U0 = (U1 + u2)/2

$ = ( U 2 - u1)/2.

Thus E, is maximum at the center of the aperture where U = u 0 , and it vanishes at the edges where U = uI or U = u 2 .

The Mathieu angular functions are expanded in Fourier series as follows [ l l ] :

(34)

(35)

s ; ( s ~ , cos U) = C' B;(s, , m) cos iu

s ; ( s ~ , cos u)=C' B P ( S ~ , m) sin iu

I

i

m is even, and over odd values of i if m is odd. From (18), (31), (34) and (351,

F ; = E ~ 2' s;(sl, m) jU2 cos (iu) cos [r(u-u0)/(21c/)1 du "1

i

(36)

F : = E ~ E' ~ p ( s ~ , m) j u 2 sin (iu) cos [r(u-uo)/(2$)1 du. i U 1

(37)

From (8) and the asymptotic form for the radial function H,,, the far-zone field of the slot antenna is

E z ( p , 4 ) = Y j / o e - J k p jnCnSn(s, cos 4) /Mn(s ) (38) n

and the antenna gain is given by

G ( + ) = S ( p , 4YSavb). (41)

Let W denote the time-average power radiated through an imaginary cylindrical surface with length L and radius p . Then

W=2rpLSaV(p)= 1 VI2GuL (42)

where G, denotes the aperture conductance per unit length and Vdenotes the aperture voltage. Following Harrington [ 121, we take Vequal to Eo (see (31)) to obtain

Gu=2rpSav(p)/(Eo12* (43)

The theory for the TE axial slot is presented in the Appendix.

ILI. NUMEFUCAL RESULTS Only a few numerical results can be presented here. The

ambient medium is taken to be free space, and denotes the wavelength in free space. Table I lists the parameters for Figs. 2-7. The slot width w is measured along the elliptic arc from U I to u2 . It is convenient to let t denote the coating thickness at the slot location (uo = 90") as follows:

where the primed summations extend over even values of i if t = b2 - bl .

1238 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. 10, OCTOBER 1989

TABLE 1 PARAMETERS FOR FIGS. 2-7

Coating Parameters

E , = 4, p, = 1

Elliptic Cylinder Parameters

= &, bl = 0.5&

Axial Slot Parameters

vo = W O , pb = 2.8657"

w = O.l&

8 1 ' ~ ~ ~ " " ~ ' ' " ~ ' " ' ~ ' ' ' ~ ~ ' ~ ~ ' ' 30 0.05 0.10 0.15 0.20 0.25 0

COATING THICKNESS t / A o Fig. 2. Aperture conductance versus coating thickness for TM axial slot.

ELLIPT IC CY LlNDER I - - . - - - - - CIRCULAR CYLINDER . T M

0 COATING THICKNESS t / A , Gain versus coating thickness for TM axial slot. Fig. 3 .

D.

Fig. 4. Radiation patterns for TM axial slot on coated elliptic cylinder.

Jd.

z Jt w .

0 - 0

W '

E L L I P T I C CYLINDER U . _ _ - - _ - _ CIRCULAR CYLINDER

$ b o ' ' . . 0.05 ' . . . . 0.10 ' . . . . 0.15 ' . . ' . 0.20 COATING THICKNESS t /A,

Fig. 5. Aperture conductance versus coating thickness for TE axial slot.

We consider first the numerical results for the TM polarization. Fig. 2 illustrates the aperture conductance versus the coating thickness. For comparison, Fig. 2 also shows the results for an axial slot on a coated circular cylinder whose radius ( a = 2 A,,) matches the radius of curvature of the elliptic cylinder at the slot location. When the thickness t tends toward zero, our calculated conductance approaches the value given by Harrington [12] for a slotted ground plane.

For a conducting plane with a uniform dielectric coating ( E ,

= 4 and p, = l), the lowest order surface wave with perpendicular polarization begins to propagate when the thickness is t = 0.1443A,,. It may be noted in Fig. 2 that the

1239 RICHMOND: AXIAL SLOT ANTENNA

m

ELLIPTIC CYLINDER TE CIRCULAR CYLINDER

:1 1 _ _ _ _ _ _ -

$

$

" ,'.-: i d - bl/al = 0.5 0 ) I , I I I

, I , I 0 8 , 8

n m .

A surface wave cannot propagate all the way around the perimeter of the elliptic cylinder with reasonably small attenuation unless to exceeds 0.1443 X,, and in the present example this requires that t exceed 0.25X,. In Fig. 2 the elliptic cylinder displays a small narrow ripple centered at t =

b/al = 0.5 TE

t = 0.05A0

- - - - - - - t =0.15X0 i-

polarization. Figs. 5 and 6 illustrate the aperture conductance and the gain versus coating thickness, while Fig. 7 presents the radiation patterns with coating thicknesses of 0.05 X, and 0.15 X, . Once again, our calculated conductance approaches the value given by Harrington [12] when the thickness t

1240 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. 10, OCTOBER 1989

the TM and TE polarizations. With some difficulty, this work could be extended to lossy dielectric coatings, axial slots with finite length, circumferential slots, coupled slots, and partially coated cylinders.

Our knowledge will advance when it becomes possible to duplicate the numerical results with moment methods, finite elements and asymptotic formulations. With this in mind, we have presented results for an elliptic cylinder which is small enough for the low-frequency techniques and large enough for the high-frequency techniques to apply.

N

APPENDIX

THEORY FOR TE AXIAL SLOT

The TE field in a source-free homogeneous region can be constructed with the aid of a vector potential A' = U as follows:

(44)

H, = k2A (45)

Fig. 8. Aperture conductance versus coating thickness for TM axial slot. E,,= (jwp/h)dA/du (46)

E, = ( j w p / h ) d A / d ~ (47)

E, = H,, = H, = 0 .OO 0.05 0.10 0.15 0.20 0.25 0.30

COATING THICKNESS t / A 0

where h is given by (6). For the TE axial slot, the potentials A I

and A" are given once again by (8) and (9). The electric field intensities E t and E t' are given by ( 10) and (1 1) with pl , p and ( - j ) replaced by e l , E and j , respectively. We require H, and E, to be continuous across the interface at U = u2, and this leads to (12) and (13) with p, replaced by E, = /E. Applying Galerkin's method, we obtain once again (14) and (15) with p, replaced by E , .

At the surface U = u1 we require E, to vanish on the perfect conductor and to match the prescribed aperture field distribu- tion as follows:

( j / (a~lhd) [ A , J A ( s l , cosh u1) m

+B,NA(s~, cosh u~) ]S , (SI , COS u)/M,(sJ

COATING THICKNESS t /Ao where Fig. 9. Aperture conductance versus coating thickness for TE axial slot. hl = d.\/cosh2 ~ 1 - cos2 U (49)

unknowns) in the simultaneous linear equations represented by (27), for example. The value of N must increase as one increases the quantities such as k l a l , klaz or ka2 which represent the characteristic dimensions of the elliptic cylinder,

for example, Nvaries from a value of 8 at the left-hand side to 9 at the right side of the graph. In Fig. 8, N increases from 11 up to 12 as one progresses from left to right.

V. SUMMARY AND CONCLUSION

We multiply (48) by S,(sl, COS U ) and integrate Over 0 < U

< 2?r to obtain

(j/(UEi))[A, JA(%, cosh UI) + BmNh(s17 cosh ui)] but N i s independent of the polarization (TE or TM). In Fig. 2,

U2

"1 = F, = F(u)S,(s,, cos U ) du. (50)

Equation (50) can be employed to eliminate B, in (14) and (15) (with p, replaced by E,) to obtain the following:

We have presented the theoretical formulation and a few A ~ Q ~ = ~ ~ ~ ~ ~ ~ ~ ~ ( ~ ~ , cosh u2) numerical results for the aperture conductance and the gain of an axial slot antenna on a dielectric-coated elliptic cylinder for + Cn T,,H,(s, cosh u ~ ) N A ( s ~ , cosh u~) /M, ( s ) (51)

n

RICHMOND: AXIAL SLOT ANTENNA

Two expressions can be obtained for A, from (51) and (52). By equating these expressions, one obtains the following system of simultaneous linear equations:

C,,Z,,= V,, m=O, 1 , 2, . - . ( 5 5 ) n

V, = jwcl F,[N;(sl, cosh u2)/Q; - N,(sl, cosh u2)/Qm]

(56)

Z,, = [H,,(s, cosh u2)/Qm - (s, cosh uz)/Q;]

T,,NA(s~, cosh ul) /Mn(~) . (57)

Again, (55) represents four independent sets of simultaneous linear equations as in (27)-(30).

In (48) the aperture field distribution is represented as follows:

E , ( u ~ , ~ ) = F ( v ) / h l . (58)

For numerical calculations it is convenient to choose F(u) to be a constant as follows:

F(u) =Eo. (59)

If the angular extent of the aperture is not too large, this choice for F(u) makes E, nearly uniform in the aperture. It may be noted that Harrington [12] selected a uniform aperture distribution for this polarization. From (34), (39 , (50) and (59), the F, are given by

I

For the TE polarization, the far-zone field H&, 4) of the slot antenna is given by the right-hand side of (38). The time- average power density is given by (39) with moved to the numerator. The antenna gain G(d) is given once again bv

1241

(41), where S,,(p) is given by (40) with q moved to the numerator. The aperture voltage is determined from (58) and (59) as follows:

V = J U 2 E,(ul, u)hl du=2$Eo (62) “1

where $ is defined in (33). From (42) and (62), the aperture conductance per unit length is

G, = 2VS,”( P 12 W O I 2 . (63)

REFERENCES R. A. Hurd, “Radiation patterns of a dielectric coated axially-slotted cylinder,” Canadian J. Phys., vol. 34, pp. 638-642, July 1956. J. R. Wait, Electromagnetic Radiation from Cylindrical Struc- tures. Elmsford, NY: Pergamon, 1959. C. M. Knop, “External admittance of an axial slot on a dielectric coated metal cylinder,” Radio Sci., vol. 3, pp. 803-818, Aug. 1968. L. Shafai, “Radiation from an axial slot antenna coated with a homogeneous material,” Canadian J. Phys., vol. 50, no. 23, 1972. J. H. Richmond, “Flush-mounted dielectric-loaded axial slot on circular cylinder,” ZEEE Trans. Antennas Propagat., vol. AP-23, pp. 348-351, May 1975. J. Y. Wong, “Radiation conductance of axial and transverse slots in cylinders of elliptical cross section,” Proc. IRE, vol. 41, pp. 1172- 1177, Sept. 1953. -, “Radiation patterns of slotted-elliptic cylinder antennas,” ZEEE Trans. Antennas Propagat., vol. AP-3, pp. 200-203, Oct. 1955. J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci., vol. 23, pp. 106-1066, Nov.-Dec. 1988. W. F. Croswell, G. C. Westrick, and C. M. Knop, “Computations of the Aperture Admittance of an axial slot on a dielectric coated cylinder,” ZEEE Trans. Antennas Propagat., vol. AP-20, pp. 89-92, Jan. 1972. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, vols. I and II. R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961, pp. 180-185. N. Wang, “Electromagnetic scattering from a dielectric-coated circular cylinder,” ZEEE Trans. Antennas Propagat., vol. AP-33, pp. 960- 963, Sept. 1985. D. B. Hodge, “The calculation of the eigenvalues and eigenfunctions of Mathieu’s equation,” ElectroSci. Lab., Ohio State Univ., Columbus, Tech. Rep. 2902-4, prepared under Grant NGL 36008-138, for NASA, Washington, DC, 1971. R. J. Lytle and D. L. Lager, “Solutions of the scalar Helmholtz equation in the elliptic cylinder coordinate system,” Lawrence Livermore Lab., Tech. Rep. UCRL-51420 Rev. 1, prepared under Contract W-7405-Eng-48 for U.S. Atomic Energy Commission, 1973. N. Toyama and K. Shogen, “Computation of the value of the even and odd Mathieu functions of order N for a given parameter S and an argument X , ” (Computer Program Description), ZEEE Trans. Anten- nas Propagat., vol. AP-32, pp. 537-539, May 1984.

New York: McGraw-Hill, 1953.

H. Richmond (S’49-M’56-SM’59-F’80), for a photograph and biography please see page 79 of the January 1987 issue oithis ?R~NSAC- TIONS.