- 1 -

NOVEL RADOMES USING CHIRAL MATERIALS

Field of the Invention

Generally this invention relates to methods of constructing radome structures and antenna radomes. More _ specifically, this invention relates to antenna radomes comprising chiral materials.

Background of the Invention It has been shown that, for rime-harmo ic electromagnetic fields with exp(-iωt excitation, a nooo- 10 qeneous, low loss, isotropic chiral (optically active) Micd-un can be described electromagneticaily by the following constitutive relations:

D = eE + i ? (I;

f H =i _E - (l/μ)B (2)

where E, B, D and H are electromagnetic field vectors and e, μ, 4 ^' _c represent the dielectric constant, permeability and chirality admittance of the chiral medium, respectively. A "chiral medium" comprises chiral objects of the same handedness, randomly oriented and uniformly distributed. A chiral object is a three-dimensional body that cannot be brought into congruence with its mirror image by translation and rotation. Therefore, all chiral objects can be classified in terms of their "handedness." The term "handedness,' ^• as known by those with skill in the art, refers to whether a chiral object is "right-handed" or "left-handed." That is, if a chiral object is right-handed (left-handed) , its mirror image is left-handed (right-handed) . Therefore, the mirror image of a chiral object is its enantiomorph. Chiral media exhibit electromagnetic chirality which embraces optical activity and circular dichroism. Optical activity refers to the rotation of the plane of polarization of optical waves by a medium while circular dichroism indicates a change in the polarization ellipticity of optical waves by a medium. There exists a variety of materials that exhibit optical activity. For example, for 0.63-μm wavelength, Te0 ₂ exhibits optical activity with chiiality admittance magnitude of 3.83 x 10 ^"7 mho. This results in a rota-cion of the plane of polarization of 87° per mm. These phenomena, known since the mid-nineteenth century, are due to the presence of the two unequal characteristic v/avεnu bεrs corresponding to two circularly polarized eigenmodes with opposite handedness.

The fundamentals of electromagnetic chirality are known. See, e.g., J. A. Kong, Theory of Electromagnetic

W&ves , 1975; E.J. Post, Formal Structure cf Electromagnetics , 1962. More recent work includes the macroscopic treatment of electromagnetic waves with chiral structures, D.L. Jaggard et al., Applied Physics 18 , 211, 1979; the analysis of dyadic Green's functions and dipole radiation in chiral media, 5. Bassiri et al., Alta Frequenza 2 , S3, 1986 and N. Engheta et ai. IEEE Trans , on Ant . & Propag . 37 . 4, 1989; and the

reflection and refraction of waves at a dielectric-chiral interface, S. Bassiri et al., J . Opt . Soc . Am . A5 , 1450, 1988; and guided-wave structures comprising chiral materials, N. Engheta and P. Pelet, Opt . Lett . , 14, 593, 1989. The following table compares the electromagnetic properties of chiral and non-chiral materials:

Prior radomes have been known and used to protecr antenna elements from the adverse effects of a harsh environment. Radomes have also been used to control the radiation cross-section ("RCS") parameters of antennas and antenna arrays that receive and transmit electromagnetic energy. Typically, prior radomes have been made of low loss dielectric materials which have been designed to have minimal interaction with the antennas found therein. However, prior radomes have been unable to efficiently and effectively provide control of radiation and scattering properties for antennas and antenna arrays both in low RCS applications and high efficiency antenna design. There therefore exists a long-felt need in the art for sophisticated and effective

antenna radomes to provide sensitive radiation control and RCS management for low profile antennas.

Summary of the Invention

Antenna structures and chiral radomes claimed and described herein solve the aforementioned long-felt needs and provide efficient and ultipolarized antenna structures and radomes. In a preferred embodiment, a radome for covering an antenna comprises a chiral medium. Methods of manufacturing antenna structures provided in accordance with the present invention also solve tne aforementioned long-felt needs. In a preferred embodiment, a method of manufacturing an antenna structure comprising the step of embedding an antenna element in a radome which further comprises a chiral medium is provided.

Brief Description of the Drawings

Figure 1 depicts a helix which is a preferred embodiment of a chiral structure used in the practice of the present invention.

Figure 2 is an illustration of two two-element point arrays.

Figure 3 is the radiation pattern of the turnstyle antenna of Figure 2.

Figure 4 is a linear array of N-element dipoi.es spaced a distance d apart along the x axis. Figures 5a through 5c are the radiation patterns of the linear array of Figure 4 which illustrates beam splitting and mode suppression.

Figure 6 is a spherical chiral radome covering an antenna element. Figure 7 is a plot of the normalized radiation resistance of the antenna structure of Figure 6.

Figure 8 is a plot of the ellipticity of the polarization ellipse of the radiated field of antenna structures in Figure 6.

Figure 9 is a preferred embodiment of an antenna structure provide in accordance with the present invention comprising a chiral radome and a plurality of antenna elements. Figures 10a through 10c are preferred embodiments of antenna structures comprising chiral radomes provided in accordance with the present invention.

Detailed Description of Preferred Embodiments

An exceedingly wide variety of chiral structures are amendable to the practice of this invention, so long as such structures exhibit an effective capacity to conduct electric current and have the same handedness. Chiral structures employable in the practice of this invention can be naturally- occurring or man-made. A preferred chiral structure is the single-turn wire helix given in Figure 1, having total stem length 2/, loop radius a, and thickness t as noted therein.

Preferred materials for constructing helices include copper, gold, silver, iron, and aluminum. As will be appreciated by those skilled in the art, chiral structures can be produced, for example, by molding, extruding or otherwise shaping a suitable metal, alloy, polymer or other conducting structure. These chiral structures are embedded in a suitable host material which is generally constituted so as to contain the chiral moieties and to cause them to adhere to or form articles or coatings upon articles. Polymerizable materials such as acrylics, epoxies and the like are exemplary host materials. Other solidifiable materials may be used as well. Suitable host media comprise liquids, polymeric, polymerizabi.e or otherwise solidifiable materials, and certain solids witn varying degrees of loss. The chiral material may be homogeneous or may comprise chiral structures of varying size, shape, and constitution to provide broadband characteristics. Materials which can be either naturally occurring or man-made may be employed. Chiral molecular species are also suitable in accordance with certain embodiments of the invention.

Thus, natural or synthetic molecules or molecules having chirality introduced by electromagnetic forces may be used.

Chiral materials used in connection with the invention described and claimed herein can be described in accordance with the following examples:

EXAMPLES

Example 1 - Construction of Helices

Elemental copper having conductivity of about 5.0 x 10 ⁷ mhos is drawn into a cylindrically shaped wire having diameter (t) of about 0.1 millimeters. The wire is then shaped into single-turn helices having stem half-length (/) and loop radius (a) of about 3.0 millimeters, as shown in Figure 1.

Example 2 - Incorporation of Helices into Lossy Host Medium The components of an ECCOSORB (Emerson and Cuming) lossy material preparation are mixed in an open-top, cardboard box having known internal volume. Before the preparation solidifies, an appropriate number of the copper helices constructed in Example 1 is uniformly added to reach the desired concentration (N) helices per square centimeter. The matrix is stirred well and allowed to fully solidify.

Derivation of the Radiation Characteristics of Antenna Arrays in Chiral Materials

Using the time-harmonic Maxwell equations for both electric sources J and r and magnetic sources J _m and r _m yields

Vx E= iωB -J (3)

VxH =J-iωD ₍ )

V • B = p m (5)

V • D = p. (6)

From these relations, the following inhomogeneous differential equations for the field quantities can be found with the aid of (1) and (2) .

D _c E = ∑ ^' ωμ[J-iξ _c Jm] - x J _m (7)

D H = /ωμ[ιξ _c J + J _m /η _c J + V x J (8)

(9)

D_ B = μ{V x [J + zξ _c J _m ] + zωεj _m }

(10)

0 = _{μ(iωεJ +} Vx _KcJ - _Jm /ώ

where the chiral differential operator is defined by the relation

O { }≡ VxVx { }-2ωμξ _c Vx { } -j_2 { } (11)

and where

s a generalized chiral impedance with η ₀ (= γμ/ε)as the background intrinsic wave impedance. The introduction of both the chiral impedance by relation (12) and the chiral admittance through expressions (1) - (2) leads naturally to

the definition of a dimensionless chirality factor k given by their product. Explicitly,

K ^≡ .e-c ( ¹³ )

where the absolute value of K is bounded by zero and unity. It is this parameter that is a quantitative measure of the degree of the chirality of the medium and it is a measure of the chirality of a medium.

Since the fields E, B, D and H are linearly dependent on the current sources J and J _m , these fields can be written in terms of integrals over the sources and an appropriately defined dyadic Green's function. Furthermore, these expressions can be simplified so that each field eigen ode, denoted by a "+" subscript is written in the form below:

E( ^x x)' _± -*- <-__' ₍₁₄₎

H(x) _± = -^j r±( _XlX ') ^• (± ι)[J(x') ± iJm x /T ] dx' ( 15)

B (x)+ = -f μk± J E±(x,x') • (± [ (χ') ± i J _m (χ')/τι,3 dx' ^{{ 16 )}

iμk± r ( 17

D(x) _± = — = J r±(x.x') ■ [J( ') ± zJmCx'Vri _. ] dx',

^c

where the dyadic Green's function £(x,x ^' ) is given below. Here boldface quantities denote vectors while underbars indicate dyads. It is noted that the total field quantities- are the sum of the "+" and the "-" eigenmodes given in Equations (14) - (17) . Each eigenmode represents a circularly polarized wave of a given handedness. The above-referenced dyadic Green's function r(x,x') can be rewritten in the compact form

Q —

E(x.x') +E"(χ,χ')=β *-(κ ₊ ) G+(χ,χ') +[i -β]:r( _) G.Cx.x ¹ ) (is)

where the "+" and "-" superscripts refer to the first and second terms, respectively, on the right-hand side of (18) and the dyadic operators for the two eigenmodes are given in terms of the unit dyad 1 by

Q-±) = {1 ± k^IxV +k± VV) (19)

and where exp[k+lx-x'!J

G+(x,x') =

(20)

4πlx-x'l

k ² - k ² R 1 _rι , , (22) β=^-^- = [1 + ]

The wavenumbers k _± are the propagation constants for the two eigenmodes (' ^■ +" and "-") supported by the medium. The factors β and 1 - β are denoted "handedness factors". These. quantities will play a role in the far-field radiation patterns of antennas and arrays and represent the relative amplitude of waves of each handedness. Here k ₀ (=ω ^" \με) is the host or background wavenumber of the achirai media with identical permittivity and permeability.

From a far-field expansion of the Green's dyad (1_) the electric field eigenmodes corresponding to (14) can be written in the form

^E(X) ± kr >l ^{6rX 6rX ±} * ^{X ]}

J _e -/k _± _ _r -X' _[J(χ . _{) ±} ,-J _m(χ .y j _dχ .

^ (24)

for general current sources where r = jxj, e _r is a unit vector along the position vector x. It is understood here and in the following equations that in the triple cross product involving e _r , the cross products are carried out right to left. Like¬ wise, using (15) it can be shown that the magnetic field in this limit is given by the relation

Je-zk+_ χ- _[J(X .) _{± Z} -J _m(x ') _{/7 ]} x'

(25)

Of particular note from (24) - (25) is that either eigenmode can be excited while the other is suppressed through the appropriate choice of electric and magnetic sources. Referring now to the drawings wherein like reference numerals refer to like elements. Figure 2 represents point arrays comprising electric and/or magnetic dipoles. In the far zone, the expression for the electric field eigenmodes due to a point electric dipole p and point electric magnetic dipole m located at the origin is immediately found from (24; with the relations J(x')= -iωp<.(x') and J_(x')=-ic_μ_._ (x') as:

( 26 ;

with Vc ≡ being the generalized chiral

1.1 -

velocity . This expression teaches ways in which one or both of the eigenmodes of the medium can be excited or sensed. Th two-element point arrays 50, in preferred embodiments, are formed by coincident parallel electric and magnetic dipoles 20 and 30. The turnstyle antenna 60 is formed, in preferred embodiments, by two coincident orthogonal electric dipoles 20. These configurations are displayed in Figure 2.

Consider the case of the parallel electric dipole 20 and the magnetic dipole 30 in antenna array 50 located at the origin 40, preferably in a spheroid coordinate system (r , θ , φ) . Two special cases are especially illuminating. Assume as the first special case the relation where the currents in the two dipoles are in phase and give rise to fields cf equal magnitude. If p = im/v. = pe_, only the positive eigenmode is excited and the total electric field is found to be

E(x)=E(x) ₊ => -2V2θ)2μβpsinθ— έ ₊ ^κr>>1 4πr ( 2 "7 )

while if p = -im/v_ = pe. , only the negative eigenmode is excited and the result is

E ⁽ x) = E(x)_ => -2V2 ω2μ (i - β _{) p sinθ} 5_!£ _ό

^Kr>>1 4πr ( 28 )

for the total electric field where the circular polarization basis vectors are e _± ≡ (e„±ie _φ ) /V2 and the angles . and φ are the polar and azi uthal angles measured from the z and x axes, respectively. The sole excitation of a single eigenmode of the chiral medium is particular to the case where both electric and magnetic sources are present since this cannot be accomplished in chiral medium with only electric sources. Further, the far field is directly circularly polarized, regardless of direction. As in the achiral case, however, the radiation pattern has the sin. dependance characteristic of all electrically small sources.

As the second special case, consider the case where the currents in the two dipoles are fed out of phase so that the moments are in phase and are given by p = m/v _ς . = pe ₇ . The far-zone electric field calculation using (26) yields an expression for both eigenmodes as

in a manner similar to that of the electric dipole alone. Here the total electric field is not circularly polarized but instead is elliptically polarized. As the third case, consider the turnstyle antenna 60 where the electric current distribution is given by J(x')= -iωp(e. + ie _x ) 5(x'). Using (26) the far-zone electric field exhibits the two circularly polarized eigenmodes as given by

^E(X)± S»l ^{~ fIω2μ{} 1Λ-βiP 4π^p [l±sinθsinφ] ^« _i ₍₃ o)

where e' _± denotes the right- and left-handed circular polarization vectors with rotated axes. These two eigenmodes possess considerably different angular dependences. Referring tc Figure 3, the two eigenmodes 70 and 80 of (30) access two different half spaces divided by the plane of the turnstyle antennas 60. Therefore, ih this case, each half space has essentially a circularly polarized wave of opposite handedness.

Referring to Figure 4, a distributed source antenna array 90 which is comprised of a linear array of dipoles N or other radiating elements 100 embedded in chiral media spaced a distance d apart along the x axis is shown. Since there is an inherent geometrical spacing which defines the array 90, the two eigenmodes of the medium will "see" an array of differing effective geometry, i.e., spacing and total length. The case

of a linear array of dipoles is displayed in Figure 3 where N elements are spaced a distance d apart: along the x axis. The phase shift per element (in free space) is taken to be α. Equation (24) produces the eigenmode expressions for the far- zone electric field as:

E(x)± j^ _j = ω2 _μ r±(χ,ϋ)'[p±z /v _c ] AF+ ₍₃₁₎

where the angular dependence of the array factor AF _± is given by

and where Ω is the angle between the array axis (x axis in this exemplary case) and the position vector 120 of the observer 130. Here both eigenmodes play an important role except for the special case of ±p = i _m /v _c when only one of the eigenmodes is excited as noted above. Consider the case when dipoles 100 are strictly electric dipoles given by p •- pe _z . The total electric field is found to be:

^E(X) kS>l -^^PsinθUβ] ^~ _4π ^ ^. A.~F ₊₊ ^. _ ^, ₊₊ + _τ [_l_--β _H ] _J ^ _4m __-A*Fr ₊ +<*_.. }. ₍₃₃ )

which is an elliptically polarized wave (combined from the two circular eigenmodes) at broadside but for nonzero phase shifts, can also exhibit two beams of opposite handedness. Referring to Figures 5a through 5c, the far zone radiation patterns of electric field (33) for an array of preferably fifteen elements (N=15) , spaced a half-wavelength apart (k ₀ d=7r) are shown. The phase shift _ is varied from

broadside (α = 0) to increasing values to illustrate beam splitting. Beam splitting occurs when the main lobes of the two eigenmodes have the same first null and are not overlapping. The criterion for beam splitting is:

N|<_| = 2π/κ. (34)

This indicates that for values of NI | larger than those of (34) , the array exhibits two distinct main beams, each circularly polarized with opposite handedness.

Degrees of beam separation for six exemplary values of phase shift with positive chiral admittance ξ _c are shown. The broadside case, α = 0 shown generally at 140 illustrates complete beam overlap when the elements are fed in phase. Then the phase shift is used to satisfy (34) , phase shift until condition (34) is met in part (b) . As the phase shift is increased, α = 90°, α = 115°, a. = 130°, = 180°, respectively 160, 170, 180 and 190 grating lobes 200 appear. In the limit as → 180°, generally at 190, almost all of the beam energy in the negative eigenmode vanishes and is converted to the positive eigenmode. This is generally noted "beam suppression" and is of interest when it is desired to use the antenna array of Figure 4 as a source of circular polarization.

The canonical cases discussed above are of practical interest in a variety of problems. In preferred embodiments, chiral antennas provided in accordance with this invention may comprise chiral radomes. In ^" further preferred embodiments, chiral antennas provided in accordance with this invention may comprise lenses of chiral material which focus electromagnetic waves on conventional antennas, antenna arrays, waveguide antennas, horn antennas, or dielectric antennas. The results given here for unbounded chiral media in preferred embodiments provide an upper bound or first-order approximation for the effect of finite non-resonant chiral slabs.

Derivation of the Dyadic Green's Function for a Source Interior to a Chiral Sphere: Radome Design

The geometry of interest is shown in Figure 6. It consists of a chiral sphere 210 of radius a located at the origin and embedded in a non-chiral dielectric 220 of infinite extent. As depicted in Figure 6, the exterior 220 and interior of the chiral sphere are denoted regions 1 and 2, respectively. Since the dyadic Green's function for sources in region 1 is different from that for sources in region 2, the two cases are considered separetly. Here, the case where the source 230 is at the interior of the sphere is examined.

The boundary conditions require that the total tangential components of E and H be. continuous across the interface 240:

e _r x E, = e _r x E ₂ (35)

e _r x H] = e _r x H ₂ (36)

where Ε* and H[ are the fields in region 1 and E ₂ and H ₂ are those in region 2. The electric fields can be written as:

and

E ₂ (r)=iωμ _c J rSr,r J ₂ (r') dV (38)

The first superscript of the dyadic indicates the location of the observation point, while the second gives that of the source. With the electric field representation in (37) and (38) , boundary condition (35) at __ = a becomes

,02) xOr.r ^' ) = e _r xrSr,r') (3s>

- 16 -

In order to express boundary condition (36) in terms of these same functions, replace H by E with the help of the appropriate constitutive relation and Maxwell's equation V x E = iωB, which yields:

J[ , - V x r2(r.r ^• )] =-ωξ _c [e _r x rjr )] ₊ - [e _r x V x r< _t ,r')].

(40)

From scattering superposition, the total dyadic Green's functions in (37) and (33) may be written as:

r ,(t1o2.)cr. = r V, r>a (41)

,(22) r«fr. -r _c (r,r') ₊ r ( _rfr . ₎ r<a (42)

(12) (22) where good choices for the forms of F. (_,_') and T (r,r') are:

and

∑* ^(r ' ^{r) =} 2π( _{+ 4} +- klc_; _{, !} ^,=! _{. m} =,o ^ ^m o) n(n+l) (n- ^Hii )!

d!Ve 43b)

:mn (k ₊ J + (

Here,

and

_T ^p; ₍ co,θ -gJ(»φ (3 _ll C _K r)e _β -i 2

⁺ [rj _n ( r)]e _θ -j κr)e _ώ j ₍₄₅₎ n(n+l) P^(cosθ) ^(raφ) ^J - -^-e_ ι_X

where 5y is the Kronecker delta, j _n (κr) is a spherical Bessei function with order n and P"(cos.) is an associated Legendre function of the first kind with order (.._-_?) . Only integer values of n and m will be used herein. It should also be noted that the subscripts e and o do not refer to the nature of ^ _m -(κ) or Ne _mn (κ) _ }-, _u t _ra ther to the even or odd character of the generating function. The coordinates (r,_ , φ) , with their corresponding unit vectors, are standard spherical coordinates readily understood by those with skill in the art. When unprimed (primed) , these coordinates represent the location of the observation (source) point. τ ⁽¹² V Furthermore, as is required to match ^s ⁽ - ^r ' ^r - ⁾ and s - ^r ' ^r > ⁾ with =c ' ' at the boundary, the arguments of the primed spherical vector wave functions agree with those for £ _c (r,r') . Also, here, four unknowns are present in each of

(43a) and (43b) . In general it is not possible to satisfy these conditions separately with each eigenmode, which explains why different coefficients are needed for _emπ and We _mπ .

Now, all that remains to be done is to solve for the unknown coefficients a , a*, b , b", c , c„, d and d". The following simplifying notation is introduced for this task:

³ =i| ^[ * ^)] _

Substituting (43a) and 43b) into (39) and (40) , yields the following sets of linear relations:

(46)

and

with the matrix T given by

and where /, the impedance ratio between the sphere's interior and its exterior, is

The inverse of T can be found by Gaussian-Jordan elimination. However, because of the tediousnesε of the task, the

Mathematica™ computer mathematics system developed by Stephen Wolfar , Inc., Urbana, Illinois, yields the following result:

with hi = V ^~ l)j_a. ₊ h - (/ + l)j ₊ 9j_h ÷ 2/j _3h t ₁₂ = - (/ + l)j_3j ₊ 3h + ( - l)j ₊ 3j_3h + 2/3j ₊ 3j_h t ₁₃ = - (/ - l)j_3j ₊ h - (/ + i)j ₊ 3j_h + 2j _3h _M = - ⁽ 7 + l)j.-j ₊ 3h - (/ - l)j ₊ 3j_9h + 23j ₊ 3j_h *2i = V + lj_θj ₊ h - (/ - l)j ₊ 3j_h - 2/j _3h i = (■ ^~ ϊ)j_3j ₊ 3h - (/ ^" + l)j ₊ 3j_3h + 2/3j ₊ 3j_h t^ = - (/ + i)j_aj ₊ _ / _ ₁₎ j ₊₃ j_ _{h + 2j} . j_ _9h hi = V ^~ l) oU ₊ 9h + (7 + -)j ₊ 3j_3h - 23j ₊ 3j_h t ₃₁ =-2/3j_h ² + 2/ ² j_h3h t3 ₂ =-2/j_ah ² + 2/ ² 9j _ah ^t 33=- ² ^j_h ² + 2j_h3h t ₃₄ =-2/j_3h ² + 23j_h3h

t ₄₁ = 2/3j ₊ h ² -2/ ² j ₊ h3h t ₄₂ =-2/j ₊ 3h ² + 2/ ² 3j ₊ h3h t ₄₃ =-2/3j ₊ h ² + 2j ₊ h3h t ₄₄ = 2/j ₊ 3h ² -23j ₊ h3h and

D = 2/ [li 3j ₊ 3j_ + 3h 2 ^z j.j •_] - h3h(. + l)G ₊ 3j_ + j_3j ₊ ) (51)

Hence, the coefficients of the dyadic Green's functions for sources at the interior of the chiral sphere are

=k ² (/+ l)G_3h-3j_h)G ₊ 3h ₊ -3j ₊ h ₊ )/D ₍ 52)

_ζ = __ kj ₍ / _ i) _3 + 3j_h)G ₊ 3h ₊ - 3j ₊ h ₊ ) /D (53)

bj = -£(l - l)G ₊ 3h + 3j ₊ h)(j_3h_ - 3 j _. _h_) /D

(54)

b^ = k (/ + l)G ₊ 3h - 3j ₊ h)G_dh_ - 3j_hJ /D

(55)

<ζ = k ² [( ² + l)h3hG 3h ₊ + 3j_h ₊ ) - 2/ G_h ₊ 3h ² + 3j 3h h ² )] /D

(56)

_c ζ = _ k ² ( - l)b3hG ₊ 3h ₊ - 3j ₊ h ₊ ) /D _{( 57 ,}

(58)

£ = k:[(/ ² + 1 )h3hG ₊ 3h_ ÷ 3j ₊ h - 2/ Q ₊ h 3h ² + 3j ₊ 3h_h ² )] /D

(59)

Substituting the coefficients (52) through (59) into (43a) and (43b), the complete expression of the dyadic Green's function for electromagnetic sources at the interior of the chiral sphere is obtained.

Application of the Interior Dyadic Green's

Functions to a Radiating Dipole

In a preferred embodiment, the radiation pattern from an electric dipole source at the center of a chiral sphere may be derived. The current distribution J ₂ (r') is again given by:

Inserting this distribution and (43a) into (37), yields:

E ₁ (r)=-

For the far field, i . e . kr>>l , ( 61) reduces to

Limiting this expression to large spheres, for which:

. -__> ^C ° ^S( ± ^a) f 63 )

^J ± _± ^" _>>ι ^" ~ i^a

Eqn. (62) thus becomes:

iωμ _c I _o sinθ _e * ^(r -*> [λ ₊ (e _θ + ie _φ ) + λ_(e _θ - ie _φ )]

4π(k ₊ + k_) ^r 2/cos[(k _j +k_)a] - i(/ ² + 1 )sin[(k ₊ +k_)a]

( 67 )

where

— ik a λ ₊ = k ₊ (/+ l)e *- ^a + _(/- l)e ^ik + ⁱ ( 68 )

λ_ = k_<7 + De ^" ^ + k ₊ (/ - De ik- ¹ a

( 69 )

It is worth noting that, due to the geometry of the problem, the angular dependence of the dipole's radiated fields are similar to those of a dipole in an unbounded chiral or non- chiral medium.

The total radiated power P is given by:

P Re n • (E^H.JdS. (70)

Since, outside the sphere, the ratio of the electric field to the magnetic field in the far zone is , this may be written in terms of the . and φ components of the electric field as:

Substituting (67) into this relation and performing the

O

- 23 -

integration yields:

ω ² μ ² I ² (k ² + k ² )(/ ² + 1) + 2k ₊ k_(/ ² - l)cos[(k ₊ +k_)a]

P =

3πVμ/ε ^" (k ₊ + k_) ² 4/ ² cos ² [(k ₊ +k_)a] + (I ² + l) ² sin ² [(k ₊ +k_)a] ( 72 )

Therefore, the radiation resistance is

(X + k_)(r+ 1) + 2k ₊ k_(r-l)cos[(k ₊ +k_)aJ 4/ ⁱ cos ⁱ [(k ₊ +k_)a] + (l ¹ + 1) r(k ₊ +k_)a] (73)

since P = I ² R/2. The graph of (73) as a function of (k ₊ ⁺ k_)a is shown in Figure _{7 f} or ξ/ = 0.16. Two notable ef ^f ects are seen in the graph: the first is the strong resonance that occurs when ₍ k ₊ +k_ ₎ a is an odd multiple of π and the secon ^d is the increase in radiation resistance due to increased chirality. The former effect is simply a result of constructive interference occurring within the sphere. The secon _d one enters into the problem by changing both the impedance of the sphere and the radiation characteristics of the dipole, i.e., _b y exciting the RCP mode more strongly than the LCP mode.

Using the standard representation for the Poincare sphere as known by those with skill in the art, the polarization of a point on the Poincare sphere with latitude _2χ may be expressed as:

(74 )

since

For right hand elliptically polarized waves -1 < sin2χ < 0, whereas for left hand elliptically polarized waves 0 < sin2χ < 1. At the extremes, sin2χ = -1 and sin2χ = 1, the waves are RCP and LCP, respectively. Furthermore, when sin2χ = 0 the polarization is linear. Therefore, it follows from (74) , that for positive (negative) ξ _c the radiated field is always of right (left) handed polarization. A plot of (74) is found in Figure 8 for several values cf positive _c . It is evident from Figure 8, that when (k ₊ +k__)a is an even multiple cf π, one may achieve complete right circular polarization. The physical conditions which permit this phenomenon are of particular interest to radome design. To examine these conditions, we must seek the roots of (75) or, in a more intuitive form, cf:

k_R ² - 2k ₊ k_cos[(k ₊ +kJa]R + k ² = 0 ^{( 76 }}

where the reflection coefficient R =- ( -l) (l÷l ) • The solution cf (76) :

which requires that (k ₊ +k_)a = nπ in order to have real values for R. Therefore, for |λ ₊ | ² = 0

and for λ r =- 0

and for | λ_| ² = 0

k_/k ₊ n even

^R -^-k_/k ₊ nodd- ₍₇₉₎

However, since ξ. is fixed at a particular value, it is only possible to satisfy one of the two equations in a given medium. We can suppress the LCP wave only when λ _c is positive and the RCP wave only when it is negative. Physically, the elimination of one of the modes, let us say the LCP mode, may be explained as follows. When condition (79) is satisfied, a fraction k_/k ₊ of the RCP wave is reflected at the sphere's boundary and becomes an LCP wave. This latter wave now has the same magnitude as the original LCP wave since, for an unbounded chiral medium, the ratio of the amplitude of the RCP mode to that of the LCP mode is k ₊ /k_. If the latter wave is 180° out of phase with the original LCP wave radiated by the source, the LCP mode is completely cancelled. Furthermore, as the chirality progressively increases, a smaller and smaller portion of the RCP mode is needed for the cancellation, which results in the greater radiation efficiency seen in (37) .

The effects of matching the sphere's impedance to that of the surrounding medium, that is the case where /=1 is next considered in a preferred embodiment. In this case (67) then simplifies to

^(80>

It must be noted that due to the impedance matching offered by /=1, the above radiation resistance is independent of the sphere's radius.

2.6 -

Referring again to Figure 6, a preferred embodiment of an antenna structure is shown. The antenna structure comprises a radome further comprising a chiral medium shown generally at 210. The radome 210 preferably has a finite volume and an antenna element 230, generally as described above, is substantially embedded within the infinite volume of the radome 210. In further preferred embodiments the chiral radome 210 is any three-dimensional structure having a finite volume, for example, a sphere, a cube, a hemisphere, an ellipsoid, a pyramid, or any other finite size three- dimensional body. All such structures and equivalents thereof are intended to be within the scope of the present invention.

Referring to Figure 9, yet a further preferred embodiment of antenna structures provided in accordance with the present invention is shown. In this embodiment, a plurality of antenna elements 230 are embedded in chiral radome 210. The antenna elements 230 may be as described previously, for example, turnstile antenna elements, magnetic or electric dipoles, horn antennas, or other receiving and transmitting antenna elements. As the spherical radome 210 approaches an infinitely large volume as compared to the wavelength of electromagnetic energy interacting with antenna elements 230, the radiation patterns approach the patterns as substantially illustrated in Figures 5a through 5c in preferred embodiments.

In still further preferred embodiments, the antenna structures as shown in Figure 10a are described. A dielectric substrate 250 having material parameters e, μ provides a non- chiral substrate to the structure. An antenna element 230 is interfaced with the non-chiral dielectric substrate 250 such that the substrate 250 holds the antenna element 230 in a substantially fixed position. Preferably, a radome 210 comprising a chiral medium is further interfaced with antenna element 230 and provides a cover for the antenna element. In still further preferred embodiments, the chiral radome 210 is a substantially hemispherical finite volume of chiral material.

- n -

Referring to Figure 10b, a dielectric non-chiral layer 250 also provides a substrate to the antenna structure. In a preferred embodiment, a plurality of antenna elements 23 may be interfaced with the dielectric substrate and a radome may further comprise a plurality of chiral layers 260 and 270 having different chiral parameters ( e* , μ. , ξ _c . ) and (e ₂ ,μ ₂ ,ξ _c2 ). ground plane 280 is interfaced with the non-chiral dielectric layer 250 in preferred embodiments and is provided to both ground the antenna structure and to hold the antenna structur in a preferred orientation. The ground plane 280 may be, for example, the body of an aircraft when the antenna structure i fixed to an aircraft or, more generally, is any conducting ground structure. It may also be a conducting metal ground plate provided specifically for an antenna structure as shown in Figure 10b.

In still further preferred embodiments of antenna structures provided in accordance with the present invention, an antenna element 230 may be embedded in a radome shell 290 as shown in Figure 10c. Radome shell 290 preferably has a finite volume and further comprises a chiral medium such that the antenna element is embedded in radome chiral shell 290 in a substantially non-chiral portion 300 of the radome shell. The non-chiral portion 300 could be a dielectric material, or alternately, is simply an airspace which also exhibits certai dielectric properties. In still further embodiments, a plurality of non-chiral layers such as dielectrics could be interfaced with a plurality of chiral layers to provide radom structures in antenna structure configurations such as those shown in Figures 10b or 10c. Again, all such configurations and equivalents thereof are intended to be within the scope o the present invention.

With all antenna structures described and claimed herein, it is expected that the radiation resistance characteristics as illustrated in Figure 7 for the chiral sphere radome will be achieved. It should be noted as shown in Figure 7 that by introducing a chiral radome to a chiral structure in accordance with the present invention, the

radiation resistance can be greatly increased as compared to a non-chiral radome structure depending on particular design parameters that are desired. While this is true for the single antenna element described in Figure 6, having the particular shape of the curve shown in Figure 7, by introducing a plurality of antenna elements the shape of the curve will change but increased radiation resistance is expected.

Another interesting and important aspect of antenna structures comprising chiral radomes provided in accordance with the present invention is the change of the state of polarization of the radiated field as compared with a non- chiral radome as illustrated in Figure 3. As can be seen in this Figure, by introducing chirality in radome structures for the preferred embodiment in Figure 6, the radiated field is elliptically polarized as opposed with the non-chiral case. More specifically, as was described earlier, the circularly polarized radiated field can be achieved for the antenna structure in Figure 6 depending on particular design parameters. Such features evince s artlingly unexpected results not heretofore achieved in the antenna and radome art and solve long-felt needs for efficient and multipolarized antenna structures.

There have thus been described certain preferred embodiments of methods of constructing radomes of chiral marerials. While preferred embodiments have been described and disclosed, it will be recognized by those «f/ith skill in the art that modifications are. ithin the true spirit and scope of the invention. The appended claims are intended to cover ail such modifications.