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On the Modeling of Small Wire Scatterers with Core
International Journal of Electronics and Communications
© Urban & Fischer Verlag http://www.urbanfischer.de/journals/aeue
On the Modeling of Small Wire Scatterers with Core Jens Reinert and Arne F. Jacob Abstract A simple model for chiral wire scatterers with core is presented. The model reduces the (isotropic) core of the scatterer to its dipole moments. The resulting description of the electromagnetic problem allows a simple calculation of the complete dipole polarizability tensors of the scatterer. For spherical cores the theoretical results obtained from the model are compared with the exact solution. Measurements performed on single helices without and with spherical or cylindrical core show that the model allows a reliable characterization of scatterers with arbitrarily shaped core. Keywords Chiral material, Scatterers, Polarizabilities
1. Introduction Artificial chiral materials made of randomly distributed thin wire scatterers exhibit a narrowband material resonance. The latter can be broadened by using resistive [1] or dielectric [2] wires. The influence of the host material, though, dominates [3]. For helicoidal scatterers the use of a magnetic core and a dielectric host was proposed as an efficient solution for the bandwidth problem [4]. The properties of this category of materials were studied theoretically in [5, 6]. In this contribution the simple approximate model for wire scatterers with core used for the investigations in [6] is developed in more detail. This model reduces the core and, thus, the complete electromagnetic interaction between core and wire to the dipole polarizabilities of the core. Based on a standard method of moment (MoM) solution [7] for the current on the wire the full polarizability tensors of the scatterer with core are deduced using a superposition approach. Finally, the model is verified by measurements conducted on single helices without and with spherical or cylindrical core. Throughout the whole paper cartesian coordinates are used. Furthermore, the Einstein sum convention is employed, i.e. a subscript appearing more than once on one side of an equation implies a summation over the complete domain of this subscript.
Received December 19, 2000. Revised March 21, 2001. J. Reinert, Siemens AG, ICM Mobile Phones Salzgitter, Postfach 10 07 02, 38207 Salzgitter, Germany. E-mail: [email protected] A. F. Jacob, Inst. f. Hochfrequenztechnik, TU Braunschweig, Schleinitzstr. 22, 38106 Braunschweig, Germany. E-mail: [email protected] Correspondence to J. Reinert ¨ 55 (2001) No. 4, 266−272 Int. J. Electron. Commun. (AEU)
2. Theory This section summarizes the calculation method for the polarizability tensors of wire scatterers in free space proposed in [8]. The model of the complete scatterer with core is then developed on the basis of this procedure. 2.1 Polarizability tensors of a single wire scatterer in free space Consider an electrically small scatterer made of a perfectly electrically conducting (PEC) thin wire and centered at the origin of a cartesian coordinate system. The current induced on the wire surface by an incident plane wave E(r) = E0 exp{−jk · r} with amplitude E0 and wave vector k can be calculated using the electric field integral equation (EFIE) [9]. With the help of the dyadic Green function for free space G0 [10] this integral equation can easily be deduced from the boundary condition on the PEC wire surface Etan = t · Etotal = 0: t · Etotal = t · (E + Es ) = 0 ⇔ t · E = jωµ t · G0 · J dV V
= jωµ
t · G0 · t I(s ) ds ,
(1)
s
where t and t stand for two longitudinal unit tangential vectors along the surface and the center of the wire, respectively. The symbol J represents a (fictitious) surface current flowing in the center of the wire, s and s are longitudinal variables, and I(s ) is the total current on the wire. The superscripts “total”, “tan”, and “s” indicate the total, the tangential, and the scattered field, respectively. In (1) the thin wire assumption was used, i.e. circumferential currents on the wire are neglected as well as any circumferential variation of the longitudinal current and the exciting field E [9]. If a Galerkin MoM [7] is applied for the solution of (1), the resulting current reads t (s ) I(s ) = 2πaJ(s ) = t (s ) ξn Bn (s ) , 1 ≤ n ≤ N . (2) Here, the Bn represent the N basis functions used for the expansion of the current, a is the wire radius, and the unknown coefficients ξn can be calculated from Z mn ξn = Vm , 1 ≤ m ≤ N ,
(3)
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Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core 267
yield for the different dipole polarizability tensors (αγδ )
with Z mn = jωµ s
1 −1 Uim Z mn U jn jω µ −1 αijem = −αme Uim Z mn Vjn ji = 2 jωµ2 −1 αijmm = Vim Z mn Vjn , 4
A(s, s )Bm (s) Bn (s ) ds ds
αijee =
s
A(s, s ) = t(s) · G0 (s, s ) · t (s ) Vm = Bm (s) t(s) · E(s) ds .
(4)
s
with
Note that the system matrix Z is symmetric as a consequence of the Galerkin MoM and the properties of the dyadic Green function [11]. Using (2), (3), and (4) the current density J can be expressed as
1 −1 t (s )Bm (s ) Z mn Bn (s) t(s) · E(s) ds 2πa s = t Cn Bn (s) t(s) · E(s) ds .
(5)
s
A Taylor series expansion of the incident plane wave E(r) up to the second term then yields
jωµ = t Cn Bn (s) t · E0 − (r × t) · H0 2 s j 0 − (t r + r t) : k E ds . 2
1 jω
µ pm = 2
(6)
J dV
V
r × J dV .
(7)
V
Inserting (6) in (7) and comparing the result with the microscopic constitutive relations (plane wave incidence) [13] pe = αee · E0 + αem · H0 + · · · pm = αme · E0 + αmm · H0 + · · ·
s Vuv =
(r(s) × t(s))u Bv (s) ds .
Here, the superscript δ in αγδ indicates the electric (“e”) or magnetic (“m”) origin of the polarizability and γ = “e”, “m” stands for its electric or magnetic effect. It can be shown [8] that the symmetry relations (αee )T = αee ,
(αmm )T = αmm ,
(αem )T = −αme (10)
t(s) = t (s) and r(s) = r (s) .
Here, the tensor k E0 was split into its symmetric and antisymmetric part. In addition some tensor analysis, r · (k E0 − E0 k) = r × E0 × k, and k × E0 = ωµH0 were used. The double dot product “:” is defined as A:B = i j A ij B ji = A ij B ji . From (6) the electric (pe ) and magnetic (pm ) dipole moments of the scatterer can be calculated according to [12] pe =
tu (s) Bv (s) ds
that must be fulfilled for a reciprocal scatterer [13] lead to the following restrictions for the unit tangential vectors (t, t ) and the paths along the wire (r, r ):
Bn (s)t(s) · E0 + r · (k E0 ) ds
s
Uuv =
s
J=
J ≈ t C n
(9)
(8)
(11)
According to (11) the paths along the surface (r) and the center (r ) of the wire must be of equal length (t(s) = t (s)). This restriction is clearly fulfilled for any straight wire but not necessarily on a bent wire (see for example inner and outer circumference of a torus). Furthermore, (11) implies that the paths r und r must be equal. Thus, in the framework of the thin-wire MoM used in this paper the wire must be reduced to an infinitesimal thin current filament at the center of the wire in order fulfill the symmetry relations (10). The influence of (11) on the thin-wire MoM and possible workarounds for the problems arising from (11) are discussed in [8]. All calculations presented up to this point are valid for PEC wires, exclusively. But because all calculations rely on a standard MoM solution for the current on the wire the results can be generalized for resistive wires (and, with restrictions, to material wires) using the methods described in [14, 15]. Both methods simply add a correction term to the matrix Z that accounts for the finite surface conductivity of such wires. The methods do not affect any other part of the calculation procedure developed above.
2.2 Wire scatterers with isotropic core The electric fields scattered by an electrically small isotropic core centered at the coordinate origin can be calculated from its electric (pe,core ) and magnetic (pm,core )
268 Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core dipole moments according to
Alternatively, (18) can be written as Etotal = E + Ecore − jωµ (G0 + Gcore ) · J dr ,
E (r) = ω µG (r, 0) · p e
2
0
e,core
Em (r) = −jω∇ × G0 (r, 0) · pm,core .
(12)
These dipole moments are connected to the fields at the origin (r = 0) by the electric (αe ) and magnetic (αm ) dipole polarizabilities of the core pe,core = αe · E(0) pm,core = αm · H(0) .
with ν = ε (ν = µ) for δ = “e” (δ = “m”). The depolarization dyadic is known exactly or approximately for many core geometries and materials [12, 17, 18]. In the remainder of this contribution the L-dyadic for an isotropic spherical core (15)
where diag{a, b, c} denotes the 3-dim. diagonal matrix with entries a, b, c, and the L-dyadic for an isotropic cylindrical core (length: 2h, radius: a, cylinder axis parallel to z-axis) [12]
h h h L = diag √ , √ , 1− √ 2 a2 + h 2 2 a2 + h 2 a2 + h 2 (16) are needed. The fields at r = 0 caused by a wire scatterer centered at the origin and an incident plane wave are E(0) = E0 − jωµ G0 (0, r ) · J(r ) dr H(0) = H0 +
r
∇ × G0 (0, r ) · J(r ) dr .
with Gcore = ∇ × G0 (r, 0) · αm · ∇ × G0 (0, r ) + ω2 µ2 G0 (r, 0) · αe · G0 (0, r ) /µ .
(13)
According to (10) αe and αm are symmetric tensors for a reciprocal core. They can be calculated from the material properties of the core (ε2 , µ2 ) and the host (ε1 , µ1 ), the volume V of the core, and its depolarization dyadic L [16] −1 αδ = V (ν2 − ν1 ) 1 + ν1−1 L (ν2 − ν1 ) (14)
L = diag{1/3, 1/3, 1/3},
(17)
Here, (G0 + Gcore ) defines the first order (dipole) approximation of the dyadic Green function of the core, and Ecore , given by the second line of (18), is the field scattered by the core for plane wave incidence, alone. In analogy to the procedure described in the preceding section the boundary condition on the PEC wire leads to the EFIE for scatterers with core that can be solved with the help of a Galerkin MoM. In this representation the field exciting the wire is E + Ecore , the excitation vector Vn being calculated similarly to (4). As a consequence of (2) and (3) the unknown current density J can be separated into two independent parts, J0 and Jcore , caused by E and Ecore , respectively: J =J0 + Jcore
1 −1 = t (s )Bm (s ) Z mn Bn (s) t(s) · E ds 2πa s (20) + Bn (s) t(s) · Ecore ds . s
In contrast to (4), the system matrix Z now describes the whole scatterer, i.e. the wire with core. The linear relation (20) between current and exciting field is used in the following to express the polarizabilities. The current itself never needs to be calculated explicitely, though. With (19) and (20) the total scattered field now reads Es = −jωµ (G0 + Gcore ) · (J0 + Jcore ) ds + Ecore . s
(21)
r
The total field at any point r can now be derived from (1), (12), (13), and (17) by superposition Etotal = E − jω∇ × G0 (r, 0)· αm · H0 + ω2 µ G0 (r, 0)· αe· E0 − jω µ G0 (r, r )
Thus, the scattered field is separated into five linearly independent parts. Each of these defines different dipole polarizabilities that are calculated in the following. The expression E = −jωµ G0 · J0 ds (22) s
r
+ ∇ × G0 (r, 0)· αm · ∇ × G0 (0, r ) + ω2 µ2 G0 (r, 0)· αe· G0 (0, r ) · J(r ) dr .
(19)
r
(18)
describes the field scattered by the wire in free space due to plane wave excitation. The polarizabilities caused by this field are, thus, given by (9) with the system matrix Z from (20).
Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core 269
The field
E = −jωµ
G0 · Jcore ds
(23)
s
is the field scattered by the wire due to the scattering field of the core. From (7), (8), and (20) the polarizabilities
Finally, the polarizabilities defined by the field Ecore scattered by the core due to the plane wave excitation are given in (14). The polarizability tensors of the whole scatterer with core are the sum of the polarizabilities defined in (9), (14), (24), and (29). It can be proved that these polarizabilities fulfill the symmetry (reciprocity) relations (10).
−1 αijee = −jωµ Uim Z mn U jn
ω2 µ2 −1 Vim Z mn U jn 2 −1 αijem = −Uim Z mn Vjn jωµ −1 αijmm = − Vjn Vim Z mn 2
3. Numerical verification
αijme =
with
(24)
Uuv =
ti G 0ij (s, 0) αeju Bv ds s
=
e αui G 0ij (0, s) t j Bv ds ,
(25)
s Vuv =
ti (∇ × G0 (s, 0))ij αmju Bv ds s
=
m αui (∇ × G0 (0, s))ij t j Bv ds
(26)
s
can be derived. In (25) and (26) the symmetry properties of the dyadic Green function, namely [10] T G0 (r, r ) = G0 (r , r) , and the fact that the dipole polarizability tensors of the core are symmetric (see above) were employed. The equation E = −jωµ Gcore · (J0 + Jcore ) ds (27) s
gives the field scattered by the core due to the current on the wire. The polarizabilities caused by this field can be calculated from (see (13)) e e p = − jωµ α · G0 (0, s) · J ds pm = αm ·
s
∇ × G0 (0, s) · J ds .
(28)
s
They read −1 αijee = −jωµ Uim Z mn (U jn + ω2 µ U jn ) −1 αijme = Vim Z mn (U jn + ω2 µ U jn ) −1 µ αijem = −ω2 µ Uim Z mn Vjn + Vjn 2 mm −1 µ αij = −jω Vim Z mn Vjn + Vjn . 2
(29)
For numerical verification purposes a helix (radius: 2 mm, pitch: 1.3 mm, 3 turns (right handed), wire radius: 0.1 mm) with spherical core (radius 1.6 mm) is considered. The helix axis is assumed to coincide with the z-axis of the coordinate system. In order to get numerically stable solutions that account correctly for the nearfield coupling of wire and core entire domain basis functions, here a Fourier basis, were used for the MoM solution of the EFIE [19–21]. First, the resonance shift ∆ f = f res − f 0 ( f 0 : resonance frequency of the helix without core) due to a purely dielectric, a purely magnetic, and a core possessing both material properties was calculated using the above approximate dipole model of the core. These data were compared to the resonance shift gained from an exact description of the electromagnetic problem based on the Green function of the spherical core [5, 6, 10]. This exact model can be obtained from (19) if Gcore is substituted by the exact scattering term of the dyadic Green function Gs [10] and if Ecore is replaced by the Mie series solution [10, 22] for the field scattered by a sphere. The results for both models, shown in Figure 1, agree within 5% referred to f 0 (% = ( f res1 − f res2 )/ f 0 · 100). This substantiates the validity of the dipole approximation. Furthermore, the frequency shift caused by a core with both dielectric and magnetic properties calculated with the exact model is approximately the sum of the shifts induced by a purely dielectric and a purely magnetic core. This, again, justifies the separation of electric and magnetic effects that leads to the dipole model developed above. The errors between exact and approximate frequency shifts visible in Figure 1 are larger for a dielectric than for a magnetic core. According to the above discussion, the difference between the two curves associated with a core possessing both material properties is approximately the sum of the errors for purely dielectric and purely magnetic cores. The dipole approximation of the dielectric core properties is, therefore, the main error source in the model. A closer investigation reveals that this error is directly proportional to the radius of the spherical core. Thus, the error seems to be related to a higher order multipole moment, in particular the electric quadrupole moment of the core. This is a term of the field expansion of the core. It is necessary for an exact description of the nearfield coupling between wire and core, but was neglected here for simplicity. This quadrupole moment therefore determines the accuracy of the electromagnetic
270 Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core proximate model are – as above – in good agreement. As a consequence of the simple theoretical model introduced above computation times for the approximate model are reduced by a factor of two compared to the exact solution.
4. Measurements
Fig. 1. Resonance shift due to helices with different lossless spherical cores (core diameter: 1.6 mm, material properties: εr , µr ).
Fig. 2. Permeability of a chiral material consisting of randomly distributed helices with magnetic core (µr = 5 − j 0.5). Host material: air; volume fraction: 0.28.
model of the whole scatterer. On the other hand, the quadrupole moments of the scatterer itself do not contribute to chiral effects if random orientation is assumed [23, 24]. For a purely magnetic core the electric quadrupole moment has no influence on the results. Thus, as displayed in Figure 1, the errors should become smaller for such cores. This interpretation of the error and its origin is substantiated by the fact that for a completely consistent model of the helix with core, i.e. an origin independent description of the problem, the electric quadrupole moment has to be taken into account in addition to the electric and magnetic dipole moments [25]. In practical realizations of chiral materials the error introduced by neglecting the electric quadrupole is marginal compared to other errors caused for example by uncertainties in the helix dimensions [26]. For this reason the electric quadrupole moment of the core is not included in the model, here. For a further verification of the dipole model the constitutive parameters of a chiral material made of the above mentioned helices with a purely magnetic (µr = 5 − j 0.5) core were calculated from Lorentz-Lorenz mixing rules (volume fraction f = 0.28) [16]. As an example Figure 2 shows the permeability of the mixture. Exact and ap-
To experimentally confirm the above developed theory and the numerical results measurements on single helices with and without core were performed. To this end, the waveguide measurement procedure described in [27] was used. All measurements were conducted in a circular coaxial waveguide measurement system. Figure 3 shows the measured and calculated forward scattering parameter S21 for the above defined helix embedded in a 6 mm-diameter polyurethane (PU) foam sphere (εr = 1.11 − j 0.003) [28]. When taking into account the achievable measurement accuracy (magnitude: ±0.1 dB, phase: ±2◦ [27]) the results are in very good agreement. The small deviation in the resonance frequency visible in Figure 1 can be caused by uncertainties in the model, the numerical procedure used to calculated the theoretical values, and, most likely, by uncertainties in the helix dimensions [26]. In Figure 4 measured and calculated data for a helix with dielectric spherical core (diameter: 3 mm, εr = 4.7 − j 1.6) are displayed. In comparison to Figure3 the resonance is broadened and shifted to a lower frequency (∆ f ≈ 0.2 GHz). The slight deviations between calculated and measured phase are most likely caused by measurement errors. For the calculations the dipole model discussed in Section 2.2 was used. Thus, the dipole approximation seems to be sufficiently accurate. The values of the scattering parameter S21 calculated from the exact model based on the complete Green function of the helix with core (see above and [6]) exhibit the same frequency dependence as the approximate ones. Only the resonance frequency obtained from the exact model is lower by 0.08 GHz as implied by Figures 1 and 2. To further demonstrate the potential of the dipole approximation the scattering response of a helix with
Fig. 3. Scattering parameter S21 of a helix embedded in a 6 mmdiameter PU-foam sphere.
Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core 271
the complete dipole polarizability tensors of the scatterer. It was shown both theoretically and experimentally that the proposed calculation method is capable of handling scatterers with cores of arbitrary geometry.
References
Fig. 4. Scattering parameter S21 of a helix with spherical dielectric core (εr = 4.7 − j 1.6) embedded in a 6 mm-diameter PU-foam sphere.
Fig. 5. Scattering parameter S21 of a helix with cylindrical Teflon core in air (dimensions: see text).
cylindrical core (height: 6 mm, diameter: 3 mm, material: Teflon εr = 2.03 − j 0.001) was measured and calculated. The results are displayed in Figure 5. The measured resonance frequency, resonance width, and phase values fit quite well the theoretical ones. The errors apparent in the magnitude of S21 are higher than in Figures 2 and 3. In addition to the error sources mentioned above uncertainties in the core permittivity can cause these deviations. Indeed, the cylindrical core has a larger influence on the helix than the spherical one, not only because of its increased volume but also due to its larger (average) dipole polarizabilities compared to a sphere of equal volume. Most likely a combination of all uncertainties lead to the result in Figure 5. Apart from these (possibly) larger errors Figure 5 shows clearly that the dipole approximation is able to predict the scattering response of the helix with core even for non-spherical core geometries. Furthermore, the approximation remains basically valid for cores that fill the helix completely as in the present case.
5. Conclusion A simple model for electrically small wire scatteres with core was presented. This model allows the computation of
[1] Bingle, M.; Cloete, J. H.; Davidson, D. B.: Microwave absorption by chiral, racemic and nonchiral hooks. Proc. Bianisotropics ’98, Braunschweig, 1998. Jacob, A. F.; Reinert, J. (eds.). 321–324. [2] Mariotte, F.; Gu´erin, F.; Bannelier, P.; Bourgeade, A.: Numerical computation of the electromagnetic field scattered by complex chiral bodies. J. Electrom. Waves Appl. 9 (1995), 1459–1485. [3] Cloete, J. H.; Bingle, M.; Davidson, D. B.: Scattering and absorption by thin metal wires in rectangular waveguide — chiral cranks versus non-chiral staples. Proc. Bianisotropics 2000, Lisbon, Portugal, 2000. Barbosa, A. M.; Topa, A. L. (eds.). 353–358. [4] Cloete, J. H.; Kuehl, S. A.; Bingle, M.: The absorption of electromagnetic waves at microwave frequencies by synthetic chiral and racemic materials. Int. J. Appl. Electromagnetics and Mechanics 9 (1998), 103–114. [5] Reinert, J.; Jacob, A. F.: Chiral materials consisting of helices with core: A case study. Proc. URSI’98, Toronto, Canada, 1998. 187. [6] Reinert, J.; Jacob, A. F.: Exact and approximate modeling of helices with core. Proc. Bianisotropics 2000, Lisbon, Portugal, 2000. Barbosa, A. M.; Topa, A. L. (eds.). 367–370. [7] Harrington, R. F.: Field computation by moment methods. Piscataway, NJ: IEEE Press, 1993. [8] Reinert, J.; Jacob, A. F.: Polarizability tensors of thin-wire scatterers: A direct calculation approach. IEEE Trans. Antennas Propagat. AP– 49 (2001). (at press). [9] Peterson, A. F.; Ray, S. L.; Mittra, R.: Computational methods for electromagnetics. Piscataway, NJ: IEEE Press, 1997. [10] Tai, C.-T.: Dyadic green functions in electromagnetic theory. 2. ed. Piscataway, NJ: IEEE Press, 1993. [11] Chew, W. C.: Fields and waves in inhomogeneous media. Piscataway, NJ: IEEE Press, 1995. [12] van Bladel, J.: Singular electromagnetic fields and sources. Piscataway, NJ: IEEE Press, 1995. [13] Graham, E. B.; Raab, R. E.: Light propagation in cubic and other anisotropic crystals. Proc. R. Soc. London A 430 (1990), 593–614. [14] Richmond, J. H.: Radiation and scattering by thin-wire structures in the complex frequency domain. – In: Computational Electromagnetics. Miller, E. K.; Medgyesi-Mitschang, L.; Newman, E. H. (eds.). Piscataway, NJ: IEEE Press, 1992, 156–169. [15] Whites, K. W.: Resistive and conducting tube boundary conditions for material wire-shaped scatterers. IEEE Trans. Antennas Propagat. AP– 46 (1998), 1548–1554. [16] Sihvola, A. H.: Electromagnetic mixing formulas and applications. London: IEE, 1999, Electromagnetic Waves Series 47. [17] Weiglhofer, W. S.; Michel, B.: Explicit expresions for depolarization dyadics of biaxial anisotropic media. Proc. Bianisotropics ’98, Braunschweig, 1998. Jacob, A. F.; Reinert, J. (eds.). 245–248.
272 Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core [18] Weiglhofer, W. S.: Depolarization dyadics. Proc. Bianisotropics 2000, Lisbon, Portugal, 2000. Barbosa, A. M.; Topa, A. L. (eds.). 415–420. [19] Demarest, K. R.; Garbacz, R. J.: Anomalous behaviour of near fields calculated by the method of moments. IEEE Trans. Antennas Propagat. AP– 27 (1979), 609–614. [20] Klein, C. A.; Mittra, R.: The effect of different testing functions in the moment method solution of thin-wire antenna problems. IEEE Trans. Antennas Propagat. AP– 23 (1975), 258–261. [21] Pearson, L. W.; Butler, C. M.: Inadequacies of collocation solutions to Pocklington-type models of thin-wire structures. IEEE Trans. Antennas Propagat. AP– 23 (1975), 295–298. [22] Stratton, J. A.: Electromagnetic theory. New York: McGrawHill, 1941. [23] Raab, R. E.; Cloete, J. H.: An eigenvalue theory of circular birefringence and dichroism in a non-magnetic medium. J. Electrom. Waves and Appl. 8 (1994), 1073–1089. [24] Nakano, H.; Kimura, H.: Quantum statistical-mechanical theory of optical activity. – In: Selected Papers on Natural Optical Activity. Lakthakia, A. (ed.). Bellingham, WA: SPIE: The International Society for Optical Engineering, 1990, 362–378. [25] Graham, E. B.; Pierrus, J.; Raab, R. E.: Multipole moments and Maxwell’s equations. J. Physics B 25 (1992), 4673–4684. [26] Psilopoulos, J.; Reinert, J.; Jacob, A. F.: Fabrication effects on the resonance bandwidth of chiral material. Proc. Bianisotropics 2000, Lisbon, Portugal, 2000. Barbosa, A. M.; Topa, A. L. (eds.). 313–316. [27] Reinert, J.; Jacob, A. F.: Theoretical and waveguide characterization of small wire scatterers. IEEE Trans. Microwave Theory Techn. MTT– 49 (2001). (at press). [28] Busse, G.; Reinert, J.; Jacob, A. F.: Waveguide characterization of chiral material: Experiments. IEEE Trans. Microwave Theory Techn. MTT– 47 (1999), 297–301.
Jens Reinert was born in Einbeck, Germany, in 1970. He received the Dipl.Ing. and the Dr.-Ing. degree from the Technische Universität Braunschweig in 1995 and 2001, respectively. From 1997 to 2000 he has been a member of the research staff at the Institut für Hochfrequenztechnik, Technische Universität Braunschweig. He is currently with Siemens AG, Salzgitter, Germany working on 3rd generation mobile phones. His research interests include electromagnetic theory, wave propagation in complex materials, numerical methods in electromagnetics, and RF chipset design.
Arne F. Jacob was born in Braunschweig, Germany, in 1954. He received the Dipl.-Ing. and the Dr.-Ing. degree from the Technische Universität in Braunschweig in 1979 and 1986, respectively. From February 1986 to January 1988 he was a Fellow at CERN, the European Laboratory for Particle Physics in Geneva, Switzerland. From February 1988 to September 1990, he was with the Accelerator and Fusion Research Division of the Lawrence Berkeley Laboratory, University of California at Berkeley. Since 1990, he has been a Professor at the Institut für Hochfrequenztechnik, Technische Universität Braunschweig. His current research interests include the design and application of planar circuits at microwave and millimeter frequencies, and the characterization of complex materials.
© Urban & Fischer Verlag http://www.urbanfischer.de/journals/aeue
On the Modeling of Small Wire Scatterers with Core Jens Reinert and Arne F. Jacob Abstract A simple model for chiral wire scatterers with core is presented. The model reduces the (isotropic) core of the scatterer to its dipole moments. The resulting description of the electromagnetic problem allows a simple calculation of the complete dipole polarizability tensors of the scatterer. For spherical cores the theoretical results obtained from the model are compared with the exact solution. Measurements performed on single helices without and with spherical or cylindrical core show that the model allows a reliable characterization of scatterers with arbitrarily shaped core. Keywords Chiral material, Scatterers, Polarizabilities
1. Introduction Artificial chiral materials made of randomly distributed thin wire scatterers exhibit a narrowband material resonance. The latter can be broadened by using resistive [1] or dielectric [2] wires. The influence of the host material, though, dominates [3]. For helicoidal scatterers the use of a magnetic core and a dielectric host was proposed as an efficient solution for the bandwidth problem [4]. The properties of this category of materials were studied theoretically in [5, 6]. In this contribution the simple approximate model for wire scatterers with core used for the investigations in [6] is developed in more detail. This model reduces the core and, thus, the complete electromagnetic interaction between core and wire to the dipole polarizabilities of the core. Based on a standard method of moment (MoM) solution [7] for the current on the wire the full polarizability tensors of the scatterer with core are deduced using a superposition approach. Finally, the model is verified by measurements conducted on single helices without and with spherical or cylindrical core. Throughout the whole paper cartesian coordinates are used. Furthermore, the Einstein sum convention is employed, i.e. a subscript appearing more than once on one side of an equation implies a summation over the complete domain of this subscript.
Received December 19, 2000. Revised March 21, 2001. J. Reinert, Siemens AG, ICM Mobile Phones Salzgitter, Postfach 10 07 02, 38207 Salzgitter, Germany. E-mail: [email protected] A. F. Jacob, Inst. f. Hochfrequenztechnik, TU Braunschweig, Schleinitzstr. 22, 38106 Braunschweig, Germany. E-mail: [email protected] Correspondence to J. Reinert ¨ 55 (2001) No. 4, 266−272 Int. J. Electron. Commun. (AEU)
2. Theory This section summarizes the calculation method for the polarizability tensors of wire scatterers in free space proposed in [8]. The model of the complete scatterer with core is then developed on the basis of this procedure. 2.1 Polarizability tensors of a single wire scatterer in free space Consider an electrically small scatterer made of a perfectly electrically conducting (PEC) thin wire and centered at the origin of a cartesian coordinate system. The current induced on the wire surface by an incident plane wave E(r) = E0 exp{−jk · r} with amplitude E0 and wave vector k can be calculated using the electric field integral equation (EFIE) [9]. With the help of the dyadic Green function for free space G0 [10] this integral equation can easily be deduced from the boundary condition on the PEC wire surface Etan = t · Etotal = 0: t · Etotal = t · (E + Es ) = 0 ⇔ t · E = jωµ t · G0 · J dV V
= jωµ
t · G0 · t I(s ) ds ,
(1)
s
where t and t stand for two longitudinal unit tangential vectors along the surface and the center of the wire, respectively. The symbol J represents a (fictitious) surface current flowing in the center of the wire, s and s are longitudinal variables, and I(s ) is the total current on the wire. The superscripts “total”, “tan”, and “s” indicate the total, the tangential, and the scattered field, respectively. In (1) the thin wire assumption was used, i.e. circumferential currents on the wire are neglected as well as any circumferential variation of the longitudinal current and the exciting field E [9]. If a Galerkin MoM [7] is applied for the solution of (1), the resulting current reads t (s ) I(s ) = 2πaJ(s ) = t (s ) ξn Bn (s ) , 1 ≤ n ≤ N . (2) Here, the Bn represent the N basis functions used for the expansion of the current, a is the wire radius, and the unknown coefficients ξn can be calculated from Z mn ξn = Vm , 1 ≤ m ≤ N ,
(3)
1434-8411/01/55/4-266 $15.00/0
Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core 267
yield for the different dipole polarizability tensors (αγδ )
with Z mn = jωµ s
1 −1 Uim Z mn U jn jω µ −1 αijem = −αme Uim Z mn Vjn ji = 2 jωµ2 −1 αijmm = Vim Z mn Vjn , 4
A(s, s )Bm (s) Bn (s ) ds ds
αijee =
s
A(s, s ) = t(s) · G0 (s, s ) · t (s ) Vm = Bm (s) t(s) · E(s) ds .
(4)
s
with
Note that the system matrix Z is symmetric as a consequence of the Galerkin MoM and the properties of the dyadic Green function [11]. Using (2), (3), and (4) the current density J can be expressed as
1 −1 t (s )Bm (s ) Z mn Bn (s) t(s) · E(s) ds 2πa s = t Cn Bn (s) t(s) · E(s) ds .
(5)
s
A Taylor series expansion of the incident plane wave E(r) up to the second term then yields
jωµ = t Cn Bn (s) t · E0 − (r × t) · H0 2 s j 0 − (t r + r t) : k E ds . 2
1 jω
µ pm = 2
(6)
J dV
V
r × J dV .
(7)
V
Inserting (6) in (7) and comparing the result with the microscopic constitutive relations (plane wave incidence) [13] pe = αee · E0 + αem · H0 + · · · pm = αme · E0 + αmm · H0 + · · ·
s Vuv =
(r(s) × t(s))u Bv (s) ds .
Here, the superscript δ in αγδ indicates the electric (“e”) or magnetic (“m”) origin of the polarizability and γ = “e”, “m” stands for its electric or magnetic effect. It can be shown [8] that the symmetry relations (αee )T = αee ,
(αmm )T = αmm ,
(αem )T = −αme (10)
t(s) = t (s) and r(s) = r (s) .
Here, the tensor k E0 was split into its symmetric and antisymmetric part. In addition some tensor analysis, r · (k E0 − E0 k) = r × E0 × k, and k × E0 = ωµH0 were used. The double dot product “:” is defined as A:B = i j A ij B ji = A ij B ji . From (6) the electric (pe ) and magnetic (pm ) dipole moments of the scatterer can be calculated according to [12] pe =
tu (s) Bv (s) ds
that must be fulfilled for a reciprocal scatterer [13] lead to the following restrictions for the unit tangential vectors (t, t ) and the paths along the wire (r, r ):
Bn (s)t(s) · E0 + r · (k E0 ) ds
s
Uuv =
s
J=
J ≈ t C n
(9)
(8)
(11)
According to (11) the paths along the surface (r) and the center (r ) of the wire must be of equal length (t(s) = t (s)). This restriction is clearly fulfilled for any straight wire but not necessarily on a bent wire (see for example inner and outer circumference of a torus). Furthermore, (11) implies that the paths r und r must be equal. Thus, in the framework of the thin-wire MoM used in this paper the wire must be reduced to an infinitesimal thin current filament at the center of the wire in order fulfill the symmetry relations (10). The influence of (11) on the thin-wire MoM and possible workarounds for the problems arising from (11) are discussed in [8]. All calculations presented up to this point are valid for PEC wires, exclusively. But because all calculations rely on a standard MoM solution for the current on the wire the results can be generalized for resistive wires (and, with restrictions, to material wires) using the methods described in [14, 15]. Both methods simply add a correction term to the matrix Z that accounts for the finite surface conductivity of such wires. The methods do not affect any other part of the calculation procedure developed above.
2.2 Wire scatterers with isotropic core The electric fields scattered by an electrically small isotropic core centered at the coordinate origin can be calculated from its electric (pe,core ) and magnetic (pm,core )
268 Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core dipole moments according to
Alternatively, (18) can be written as Etotal = E + Ecore − jωµ (G0 + Gcore ) · J dr ,
E (r) = ω µG (r, 0) · p e
2
0
e,core
Em (r) = −jω∇ × G0 (r, 0) · pm,core .
(12)
These dipole moments are connected to the fields at the origin (r = 0) by the electric (αe ) and magnetic (αm ) dipole polarizabilities of the core pe,core = αe · E(0) pm,core = αm · H(0) .
with ν = ε (ν = µ) for δ = “e” (δ = “m”). The depolarization dyadic is known exactly or approximately for many core geometries and materials [12, 17, 18]. In the remainder of this contribution the L-dyadic for an isotropic spherical core (15)
where diag{a, b, c} denotes the 3-dim. diagonal matrix with entries a, b, c, and the L-dyadic for an isotropic cylindrical core (length: 2h, radius: a, cylinder axis parallel to z-axis) [12]
h h h L = diag √ , √ , 1− √ 2 a2 + h 2 2 a2 + h 2 a2 + h 2 (16) are needed. The fields at r = 0 caused by a wire scatterer centered at the origin and an incident plane wave are E(0) = E0 − jωµ G0 (0, r ) · J(r ) dr H(0) = H0 +
r
∇ × G0 (0, r ) · J(r ) dr .
with Gcore = ∇ × G0 (r, 0) · αm · ∇ × G0 (0, r ) + ω2 µ2 G0 (r, 0) · αe · G0 (0, r ) /µ .
(13)
According to (10) αe and αm are symmetric tensors for a reciprocal core. They can be calculated from the material properties of the core (ε2 , µ2 ) and the host (ε1 , µ1 ), the volume V of the core, and its depolarization dyadic L [16] −1 αδ = V (ν2 − ν1 ) 1 + ν1−1 L (ν2 − ν1 ) (14)
L = diag{1/3, 1/3, 1/3},
(17)
Here, (G0 + Gcore ) defines the first order (dipole) approximation of the dyadic Green function of the core, and Ecore , given by the second line of (18), is the field scattered by the core for plane wave incidence, alone. In analogy to the procedure described in the preceding section the boundary condition on the PEC wire leads to the EFIE for scatterers with core that can be solved with the help of a Galerkin MoM. In this representation the field exciting the wire is E + Ecore , the excitation vector Vn being calculated similarly to (4). As a consequence of (2) and (3) the unknown current density J can be separated into two independent parts, J0 and Jcore , caused by E and Ecore , respectively: J =J0 + Jcore
1 −1 = t (s )Bm (s ) Z mn Bn (s) t(s) · E ds 2πa s (20) + Bn (s) t(s) · Ecore ds . s
In contrast to (4), the system matrix Z now describes the whole scatterer, i.e. the wire with core. The linear relation (20) between current and exciting field is used in the following to express the polarizabilities. The current itself never needs to be calculated explicitely, though. With (19) and (20) the total scattered field now reads Es = −jωµ (G0 + Gcore ) · (J0 + Jcore ) ds + Ecore . s
(21)
r
The total field at any point r can now be derived from (1), (12), (13), and (17) by superposition Etotal = E − jω∇ × G0 (r, 0)· αm · H0 + ω2 µ G0 (r, 0)· αe· E0 − jω µ G0 (r, r )
Thus, the scattered field is separated into five linearly independent parts. Each of these defines different dipole polarizabilities that are calculated in the following. The expression E = −jωµ G0 · J0 ds (22) s
r
+ ∇ × G0 (r, 0)· αm · ∇ × G0 (0, r ) + ω2 µ2 G0 (r, 0)· αe· G0 (0, r ) · J(r ) dr .
(19)
r
(18)
describes the field scattered by the wire in free space due to plane wave excitation. The polarizabilities caused by this field are, thus, given by (9) with the system matrix Z from (20).
Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core 269
The field
E = −jωµ
G0 · Jcore ds
(23)
s
is the field scattered by the wire due to the scattering field of the core. From (7), (8), and (20) the polarizabilities
Finally, the polarizabilities defined by the field Ecore scattered by the core due to the plane wave excitation are given in (14). The polarizability tensors of the whole scatterer with core are the sum of the polarizabilities defined in (9), (14), (24), and (29). It can be proved that these polarizabilities fulfill the symmetry (reciprocity) relations (10).
−1 αijee = −jωµ Uim Z mn U jn
ω2 µ2 −1 Vim Z mn U jn 2 −1 αijem = −Uim Z mn Vjn jωµ −1 αijmm = − Vjn Vim Z mn 2
3. Numerical verification
αijme =
with
(24)
Uuv =
ti G 0ij (s, 0) αeju Bv ds s
=
e αui G 0ij (0, s) t j Bv ds ,
(25)
s Vuv =
ti (∇ × G0 (s, 0))ij αmju Bv ds s
=
m αui (∇ × G0 (0, s))ij t j Bv ds
(26)
s
can be derived. In (25) and (26) the symmetry properties of the dyadic Green function, namely [10] T G0 (r, r ) = G0 (r , r) , and the fact that the dipole polarizability tensors of the core are symmetric (see above) were employed. The equation E = −jωµ Gcore · (J0 + Jcore ) ds (27) s
gives the field scattered by the core due to the current on the wire. The polarizabilities caused by this field can be calculated from (see (13)) e e p = − jωµ α · G0 (0, s) · J ds pm = αm ·
s
∇ × G0 (0, s) · J ds .
(28)
s
They read −1 αijee = −jωµ Uim Z mn (U jn + ω2 µ U jn ) −1 αijme = Vim Z mn (U jn + ω2 µ U jn ) −1 µ αijem = −ω2 µ Uim Z mn Vjn + Vjn 2 mm −1 µ αij = −jω Vim Z mn Vjn + Vjn . 2
(29)
For numerical verification purposes a helix (radius: 2 mm, pitch: 1.3 mm, 3 turns (right handed), wire radius: 0.1 mm) with spherical core (radius 1.6 mm) is considered. The helix axis is assumed to coincide with the z-axis of the coordinate system. In order to get numerically stable solutions that account correctly for the nearfield coupling of wire and core entire domain basis functions, here a Fourier basis, were used for the MoM solution of the EFIE [19–21]. First, the resonance shift ∆ f = f res − f 0 ( f 0 : resonance frequency of the helix without core) due to a purely dielectric, a purely magnetic, and a core possessing both material properties was calculated using the above approximate dipole model of the core. These data were compared to the resonance shift gained from an exact description of the electromagnetic problem based on the Green function of the spherical core [5, 6, 10]. This exact model can be obtained from (19) if Gcore is substituted by the exact scattering term of the dyadic Green function Gs [10] and if Ecore is replaced by the Mie series solution [10, 22] for the field scattered by a sphere. The results for both models, shown in Figure 1, agree within 5% referred to f 0 (% = ( f res1 − f res2 )/ f 0 · 100). This substantiates the validity of the dipole approximation. Furthermore, the frequency shift caused by a core with both dielectric and magnetic properties calculated with the exact model is approximately the sum of the shifts induced by a purely dielectric and a purely magnetic core. This, again, justifies the separation of electric and magnetic effects that leads to the dipole model developed above. The errors between exact and approximate frequency shifts visible in Figure 1 are larger for a dielectric than for a magnetic core. According to the above discussion, the difference between the two curves associated with a core possessing both material properties is approximately the sum of the errors for purely dielectric and purely magnetic cores. The dipole approximation of the dielectric core properties is, therefore, the main error source in the model. A closer investigation reveals that this error is directly proportional to the radius of the spherical core. Thus, the error seems to be related to a higher order multipole moment, in particular the electric quadrupole moment of the core. This is a term of the field expansion of the core. It is necessary for an exact description of the nearfield coupling between wire and core, but was neglected here for simplicity. This quadrupole moment therefore determines the accuracy of the electromagnetic
270 Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core proximate model are – as above – in good agreement. As a consequence of the simple theoretical model introduced above computation times for the approximate model are reduced by a factor of two compared to the exact solution.
4. Measurements
Fig. 1. Resonance shift due to helices with different lossless spherical cores (core diameter: 1.6 mm, material properties: εr , µr ).
Fig. 2. Permeability of a chiral material consisting of randomly distributed helices with magnetic core (µr = 5 − j 0.5). Host material: air; volume fraction: 0.28.
model of the whole scatterer. On the other hand, the quadrupole moments of the scatterer itself do not contribute to chiral effects if random orientation is assumed [23, 24]. For a purely magnetic core the electric quadrupole moment has no influence on the results. Thus, as displayed in Figure 1, the errors should become smaller for such cores. This interpretation of the error and its origin is substantiated by the fact that for a completely consistent model of the helix with core, i.e. an origin independent description of the problem, the electric quadrupole moment has to be taken into account in addition to the electric and magnetic dipole moments [25]. In practical realizations of chiral materials the error introduced by neglecting the electric quadrupole is marginal compared to other errors caused for example by uncertainties in the helix dimensions [26]. For this reason the electric quadrupole moment of the core is not included in the model, here. For a further verification of the dipole model the constitutive parameters of a chiral material made of the above mentioned helices with a purely magnetic (µr = 5 − j 0.5) core were calculated from Lorentz-Lorenz mixing rules (volume fraction f = 0.28) [16]. As an example Figure 2 shows the permeability of the mixture. Exact and ap-
To experimentally confirm the above developed theory and the numerical results measurements on single helices with and without core were performed. To this end, the waveguide measurement procedure described in [27] was used. All measurements were conducted in a circular coaxial waveguide measurement system. Figure 3 shows the measured and calculated forward scattering parameter S21 for the above defined helix embedded in a 6 mm-diameter polyurethane (PU) foam sphere (εr = 1.11 − j 0.003) [28]. When taking into account the achievable measurement accuracy (magnitude: ±0.1 dB, phase: ±2◦ [27]) the results are in very good agreement. The small deviation in the resonance frequency visible in Figure 1 can be caused by uncertainties in the model, the numerical procedure used to calculated the theoretical values, and, most likely, by uncertainties in the helix dimensions [26]. In Figure 4 measured and calculated data for a helix with dielectric spherical core (diameter: 3 mm, εr = 4.7 − j 1.6) are displayed. In comparison to Figure3 the resonance is broadened and shifted to a lower frequency (∆ f ≈ 0.2 GHz). The slight deviations between calculated and measured phase are most likely caused by measurement errors. For the calculations the dipole model discussed in Section 2.2 was used. Thus, the dipole approximation seems to be sufficiently accurate. The values of the scattering parameter S21 calculated from the exact model based on the complete Green function of the helix with core (see above and [6]) exhibit the same frequency dependence as the approximate ones. Only the resonance frequency obtained from the exact model is lower by 0.08 GHz as implied by Figures 1 and 2. To further demonstrate the potential of the dipole approximation the scattering response of a helix with
Fig. 3. Scattering parameter S21 of a helix embedded in a 6 mmdiameter PU-foam sphere.
Reinert, J., Jacob, A. F.: Small Wire Scatterers with Core 271
the complete dipole polarizability tensors of the scatterer. It was shown both theoretically and experimentally that the proposed calculation method is capable of handling scatterers with cores of arbitrary geometry.
References
Fig. 4. Scattering parameter S21 of a helix with spherical dielectric core (εr = 4.7 − j 1.6) embedded in a 6 mm-diameter PU-foam sphere.
Fig. 5. Scattering parameter S21 of a helix with cylindrical Teflon core in air (dimensions: see text).
cylindrical core (height: 6 mm, diameter: 3 mm, material: Teflon εr = 2.03 − j 0.001) was measured and calculated. The results are displayed in Figure 5. The measured resonance frequency, resonance width, and phase values fit quite well the theoretical ones. The errors apparent in the magnitude of S21 are higher than in Figures 2 and 3. In addition to the error sources mentioned above uncertainties in the core permittivity can cause these deviations. Indeed, the cylindrical core has a larger influence on the helix than the spherical one, not only because of its increased volume but also due to its larger (average) dipole polarizabilities compared to a sphere of equal volume. Most likely a combination of all uncertainties lead to the result in Figure 5. Apart from these (possibly) larger errors Figure 5 shows clearly that the dipole approximation is able to predict the scattering response of the helix with core even for non-spherical core geometries. Furthermore, the approximation remains basically valid for cores that fill the helix completely as in the present case.
5. Conclusion A simple model for electrically small wire scatteres with core was presented. This model allows the computation of
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Jens Reinert was born in Einbeck, Germany, in 1970. He received the Dipl.Ing. and the Dr.-Ing. degree from the Technische Universität Braunschweig in 1995 and 2001, respectively. From 1997 to 2000 he has been a member of the research staff at the Institut für Hochfrequenztechnik, Technische Universität Braunschweig. He is currently with Siemens AG, Salzgitter, Germany working on 3rd generation mobile phones. His research interests include electromagnetic theory, wave propagation in complex materials, numerical methods in electromagnetics, and RF chipset design.
Arne F. Jacob was born in Braunschweig, Germany, in 1954. He received the Dipl.-Ing. and the Dr.-Ing. degree from the Technische Universität in Braunschweig in 1979 and 1986, respectively. From February 1986 to January 1988 he was a Fellow at CERN, the European Laboratory for Particle Physics in Geneva, Switzerland. From February 1988 to September 1990, he was with the Accelerator and Fusion Research Division of the Lawrence Berkeley Laboratory, University of California at Berkeley. Since 1990, he has been a Professor at the Institut für Hochfrequenztechnik, Technische Universität Braunschweig. His current research interests include the design and application of planar circuits at microwave and millimeter frequencies, and the characterization of complex materials.