Realization of a spherical boundary by a layer of wave-guiding medium
aa r X i v : . [ phy s i c s . c l a ss - ph ] A p r Realization of a Spherical Boundary by a Layer ofWave-Guiding Medium
Ismo V. Lindell, Johannes Markkanen,Ari Sihvola, and Pasi Yl¨a-OijalaAalto University School of Electrical EngineeringDepartment of Radio Science and EngineeringBox 13000, FI–00076 AALTO, FinlandNovember 21, 2018
Abstract
In this paper the concept of wave-guiding medium, previously intro-duced for planar structures, is defined for the spherically symmetric case.It is shown that a quarter-wavelength layer of such a medium serves asa transformer of boundary conditions between two spherical interfaces.As an application, the D’B’-boundary condition, requiring vanishing ofnormal derivatives of the normal components of D and B field vectors, isrealized by transforming the DB-boundary conditions. To test the theory,scattering from a spherical DB object covered by a layer of wave-guidingmaterial is compared to the corresponding scattering from an ideal D’B’sphere, for varying medium parameters of the layer.
Electromagnetic problems are often mathematically defined in terms of differ-ential equations and boundary conditions which appear on surfaces limiting theregion of interest. Here we may separate two different ways to connect mathe-matical boundary conditions and a physical structure. In the analytical way onestarts from a given physical problem involving a structure of electromagneticmedia and tries to replace existing medium interfaces by boundary conditionsfor the purpose of simplifying the mathematical problem. In such cases theboundary conditions are approximative. In the synthetic way a desired struc-ture is mathematically designed in terms of boundary conditions, which raisesthe question how to realize them in terms of physical media. This latter way isof concern here.In [1] it was shown that any given impedance boundary, defined by boundaryconditions of the type u z × E = Z s · H (1)1n a planar surface z = 0 with unit normal u z , can be realized by an interfaceof a wave-guiding medium defined by medium dyadics of the form ǫ = ǫ z u z u z + ǫ t , (2) µ = µ z u z u z + µ t , (3)when the axial parameters grow without limit as ǫ z → ∞ , µ z → ∞ . It wasshown that the dyadics ǫ t and µ t transverse to the medium axis can be designedto realize the surface impedance dyadic Z s .As another example, the DB and D’B’ boundary conditions on a plane [2, 3],respectively defined in terms of normal field components as u z · D = 0 , u z · B = 0 , (4)and ∂ z ( u z · D ) = 0 , ∂ z ( u z · B ) = 0 , (5)have been shown to be realizable in terms of physical structures. In fact, alreadyin 1959, it was found that the DB boundary can be realized by the interfaceof a uniaxially anisotropic medium (2) and (3) satisfying ǫ z → µ z → u r · D = D r = 0 , (6) u r · B = B r = 0 , (7)and ∇ · ( u r u r · D ) = ∂ r ( r D r ) = 0 , (8) ∇ · ( u r u r · B ) = ∂ r ( r B r ) = 0 . (9)It was shown in [10] that objects with either DB or D’B’ boundary conditionswith a certain symmetry in their geometry have no backscattering, i.e., they areinvisible for the monostatic radar. 2 Fields in spherical wave-guiding medium
Let us consider the spherically symmetric anisotropic medium defined by ǫ = ǫ r u r u r + ǫ t ( I − u r u r ) , (10) µ = µ r u r u r + µ t ( I − u r u r ) , (11)where ǫ r , ǫ t , µ r and µ t are constant parameters. The Maxwell equations can berepresented in spherical coordinates as [11]1 r sin θ det u r r u θ r sin θ u ϕ ∂ r ∂ θ ∂ ϕ E r rE θ r sin θE ϕ = − jωµ · H , (12)1 r sin θ det u r r u θ r sin θ u ϕ ∂ r ∂ θ ∂ ϕ H r rH θ r sin θH ϕ = jωǫ · E . (13)Expressing the fields in their radial and transverse components as E = u r ǫ r D r + u θ E θ + u ϕ E ϕ , (14) H = u r µ r B r + u θ H θ + u ϕ H ϕ , (15)and assuming that the radial components of the medium parameters satisfy thewave-guiding medium condition ǫ r → ∞ , µ r → ∞ , (16)we have E r → H r →
0, whence the Maxwell equations for the transversefield components are simplified to ∂ r ( rE ϕ ) = jk t η t rH θ , (17) ∂ r ( rE θ ) = − jk t η t rH ϕ , (18) η t ∂ r ( rH ϕ ) = − jk t rE θ , (19) η t ∂ r ( rH θ ) = jk t rE ϕ , (20)where we denote k t = ω √ µ t ǫ t , η t = p µ t /ǫ t . (21)After elimination of fields, the equations become( ∂ r + k t ) rE ϕ = 0 , (22)( ∂ r + k t ) rE θ = 0 , (23)( ∂ r + k t ) rH ϕ = 0 , (24)3 ∂ r + k t ) rH ϕ = 0 , (25)It is noteworthy that there is no restriction to the θ and ϕ dependence for thefields. Thus, the four field components satisfying (22) – (25) can be expressedin the form F ( r ) = F + ( θ, ϕ ) e − jk t r + F − ( θ, ϕ ) e jk t r . (26)Let us omit the dependence on θ and ϕ in the notation.After solving the transverse field components, the radial fields B r and D r can be obtained from the Maxwell equations as B r = jωr sin θ ( ∂ θ (sin θE ϕ ) − ∂ ϕ E θ ) , (27) D r = − jωr sin θ ( ∂ θ (sin θH ϕ ) − ∂ ϕ H θ ) . (28) The solutions for the transverse field components can be expressed as E ϕ ( r ) = 1 r ( A + e − jk t r + A − e jk t r ) , (29) η t H θ ( r ) = − r ( A + e − jk t r − A − e jk t r ) , (30) E θ ( r ) = 1 r ( C + e − jk t r + C − e jk t r ) , (31) η t H ϕ ( r ) = 1 r ( C + e − jk t r − C − e jk t r ) , (32)where the coefficients A ± , C ± are functions of θ and ϕ . The radial fields can beobtained by substituting these in (27) and (28). In view of the D’B’-boundaryconditions (8) and (9), let us define modified radial field quantities as B m ( r ) = r B r ( r )= jω sin θ ( ∂ θ (sin θ ( rE ϕ )) − ∂ ϕ ( rE θ )) , (33) D m ( r ) = r D r ( r )= − jω sin θ ( ∂ θ (sin θ ( rH ϕ )) − ∂ ϕ ( rH θ )) . (34)The modified fields have the general form B m ( r ) = B + e − jk t r + B − e jk t r , (35) D m ( r ) = D + e − jk t r + D − e jk t r , (36)4ith coefficients depending on those of the transverse fields as B ± = jω sin θ ( ∂ θ (sin θA ± ) − ∂ ϕ C ± ) , (37) D ± = ∓ jωη t sin θ ( ∂ θ (sin θC ± ) + ∂ ϕ A ± ) . (38)The quantities B m , D m propagate radially for any transverse coordinate θ, ϕ just like waves in a linear transmission line. Forming two new functions involvingradial derivatives as B ′ m ( r ) = jk t ∂ r B m = B + e − jk t r − B − e jk t r , (39) D ′ m ( r ) = jk t ∂ r D m = D + e − jk t r − D − e jk t r , (40)the coefficents B ± , D ± can be solved from (35), (36), (39) and (40) as B ± = 12 ( B m ( r ) ± B ′ m ( r )) e ± jk t r , (41) D ± = 12 ( D m ( r ) ± D ′ m ( r )) e ± jk t r . (42)Let us now consider the region between two spherical surfaces, a ≤ r ≤ b .Assuming values of B m , D m , B ′ m and D ′ m known on r = a we can find theirvalues on r = b as (cid:18) B m ( b ) B ′ m ( b ) (cid:19) = K (cid:18) B m ( a ) B ′ m ( a ) (cid:19) , (43) (cid:18) D m ( b ) D ′ m ( b ) (cid:19) = K (cid:18) D m ( a ) D ′ m ( a ) (cid:19) , (44)through the matrix K = (cid:18) cos k t ( b − a ) − j sin k t ( b − a ) − j sin k t ( b − a ) cos k t ( b − a ) (cid:19) . (45)Similarly, we can find the following relations between the transverse fieldcomponents, (cid:18) bE ϕ ( b ) bη t H θ ( b ) (cid:19) = L (cid:18) aE ϕ ( a ) aη t H θ ( a ) (cid:19) , (46) (cid:18) bE θ ( b ) bη t H ϕ ( b ) (cid:19) = K (cid:18) aE θ ( a ) aη t H ϕ ( a ) (cid:19) , (47)with L = (cid:18) cos k t ( b − a ) j sin k t ( b − a ) j sin k t ( b − a ) cos k t ( b − a ) (cid:19) . (48)5 The quarter-wave transformer
Let us now assume that the distance between the two spherical surfaces b − a satisfies k t ( b − a ) = π/ , (49)whence cos k t ( b − a ) = 0 and sin k t ( b − a ) = 1. Defining the wavelength λ t by k t λ t = 2 π , we have b − a = λ t /
4. In this case the layer of wave-guiding mediumhas quarter wavelength thickness and the fields at the two surfaces obey therelations B m ( b ) = − jB ′ m ( a ) , B ′ m ( b ) = − jB m ( a ) , (50) D m ( b ) = − jD ′ m ( a ) , D ′ m ( b ) = − jD m ( a ) , (51) bE ϕ ( b ) = jaη t H θ ( a ) , bη t H θ ( b ) = jaE ϕ ( a ) , (52) bE θ ( b ) = − jaη t H ϕ ( a ) , bη t H ϕ ( b ) = − jaE θ ( a ) . (53)The quarter-wavelength layer serves as a transformer of boundary conditions on r = a to other boundary conditions on r = b . Let us list a few examples. • DB-boundary to D’B’-boundary B m ( a ) = 0 ⇒ B ′ m ( b ) = 0 , (54) D m ( a ) = 0 ⇒ D ′ m ( b ) = 0 . (55) • D’B’-boundary to DB-boundary B ′ m ( a ) = 0 ⇒ B m ( b ) = 0 , (56) D ′ m ( a ) = 0 ⇒ D m ( b ) = 0 . (57) • PEC boundary to PMC boundary E ϕ ( a ) = 0 ⇒ H θ ( b ) = 0 , (58) E θ ( a ) = 0 ⇒ H ϕ ( b ) = 0 . (59) • PMC boundary to PEC boundary H ϕ ( a ) = 0 ⇒ E θ ( b ) = 0 , (60) H θ ( a ) = 0 ⇒ E ϕ ( b ) = 0 . (61) • PEMC boundary to PEMC boundary [12] H ϕ ( a ) + M E ϕ ( a ) = 0 ⇒ H θ ( b ) − M η t E θ ( b ) = 0 , (62) H θ ( a ) + M E θ ( a ) = 0 ⇒ H ϕ ( b ) − M η t E ϕ ( b ) = 0 . (63)6he spherical quarter-wave transformer is quite similar to the previouslystudied planar quarter-wave transformer [9]. In particular, the D’B’-boundaryconditions can be realized on the spherical surface by applying the transformerlayer upon a DB boundary whose realization is previously known.Because the parameters ǫ t and µ t may be freely chosen, the distance betweenthe two spherical surfaces can be made as small as we wish by choosing largevalues for ǫ t , µ t . However, we must take care that the parameters ǫ r , µ r must belarger by an order of magnitude, because ǫ t /ǫ r and µ t /µ r must be small. Whenthese conditions are met, the transformer layer can be made a thin sheet on thesphere of radius a . As an example, let us consider plane-wave scattering from a layered spheresimulating the sphere on which a D’B’ boundary is forced. The sphere is locatedin free space defined by parameters ǫ o , µ o . In the following, we compare thescattering behavior of an ideal D’B’ sphere and its different material realizations. The scattering fields are found by first solving the equivalent polarization cur-rents in the whole sphere (region V ) from integral equations. The total time-harmonic electric E and magnetic H fields can be expressed via the volumeequivalence principle [13] as E ( r ) = E inc ( r ) + 1 jωǫ o ( ∇∇ + k o I ) · S ( J ) − ∇ × S ( M ) , (64) H ( r ) = H inc ( r ) + ∇ × S ( J ) + 1 jωµ o ( ∇∇ + k o I ) · S ( M ) , (65)where E inc and H inc are the incident electric and magnetic fields, and k o = ω √ ǫ o µ o . The volume integral operator in (64) and (65) is defined by S ( F ) = Z V G o ( r , r ′ ) F ( r ′ ) dV ′ , (66)where G o is the free-space Green function. The equivalent electric and magneticpolarization currents are defined as J ( r ) = jω ( ǫ ( r ) − ǫ o I ) · E ( r ) M ( r ) = jω ( µ ( r ) − µ o I ) · H ( r ) . (67)By applying the volume equivalence principle, definitions of the volumeequivalent currents J and M , vector identities, and the fact that the operator766) satisfies the Helmholtz equation, we obtain the following volume integralequation formulation, − jωǫ o τ E · E inc ( r ) = − ǫ · J + τ E · ( ∇ × ∇ × S ( J )) − jωǫ o τ E · ∇ × S ( M ) , (68) − jωµ o τ M · H inc ( r ) = − µ · J + jωµ o τ M · ∇ × S ( J ) + τ M · ( ∇ × ∇ × S ( M )) , (69)with the contrast dyadics defined by ǫ o τ E = ǫ − ǫ o I , µ o τ M = µ − µ o I . (70)The formulation (68), (69) can be discretized using Galerkin’s method withpiecewise constant vector basis and testing functions. The hypersingularity ofthe kernel is reduced by moving one derivative into the testing function throughintegrating by parts. The remaining derivatives are then moved to operate onthe Green function. Let us now consider scattering from a layered sphere of radius b = 1 m, for anincident plane wave with free-space wavelength λ o = 3 . z axis ( θ = 0). Fig. 1 presents two possible realizationsfor the D’B’ sphere with transverse parameters either µ t = µ o , ǫ t = ǫ o or µ t = 2 µ o , ǫ t = 2 ǫ o . In order to obtain DB conditions at the surface r = a ,the permittivity and permeability of the inner sphere are set to zero [14]. InFig. 1 (left) the thickness of the quarter-wave transformer layer is b − a = 0 . λ t = λ o = 3 . λ t = λ o / . b − a = 0 . ǫ r = 10 ǫ o , µ r = 10 µ o yield a poor approxi-mation to the quarter-wave transformer, whence the sphere markedly deviatesfrom one with the D’B’ boundary. However, for larger values of radial permit-tivity and permeability, the scattering cross sections of the two coated spheresapproach to that of the ideal D’B’ sphere. The ideal D’B’ scattering is com-puted from an exact Mie scattering solution [10]. From the numerical results itappears that, for larger values for the transverse parameters ǫ t , µ t , lower valuesof the radial parameters ǫ r , µ r can be accepted for the realization. Furthermore,one may note that, for all parameter values used in the computation, the layeredsphere has very small scattering in the backward direction ( θ = 180 ◦ ). This isin accord with the theory stating that both DB and D’B’ spheres are invisiblefor the monostatic radar [10]. 8Sfrag replacements ǫ = µ = 0 ǫ = µ = 0 ǫ r = µ r = ∞ ǫ r = µ r = ∞ ǫ t /ǫ o = µ t /µ o = 2 ǫ t /ǫ o = µ t /µ o = 10 . . . . b = 1 m canbe realized by a quarter-wave transformer layer upon a sphere of radius a withDB-boundary conditions. The thickness b − a of the layer can be controlledby the components of permittivity and permeability transverse to the radialdirection in the layer, ǫ t , µ t . The two cases are considered here as examples. An anisotropic medium with infinitely large axial permittivity and permeabil-ity has been called waveguiding medium and its properties in transformingimpedance-boundary conditions between two planar interfaces has been studiedin [1]. In [9] it was shown that a layer of waveguiding medium can be alsoused to transform between planar DB and D’B’ boundary condition which in-volve vanishing of normal components of the D and B fields or their normalderivatives. In the present paper the spherical waveguiding medium is definedand its transforming properties are studied. In particular, it is shown that aquarter-wavelength layer of waveguiding medium can be used to transform a DBboudary to a D’B’ boundary. Since it is known that a spherical DB boundarycan be realized by a medium with vanishing permittivity and permeability [14],this gives a means to realize a spherical D’B’ boundary which until now has hadno realization whatever. The theory is verified numerically through volume-integral equation approach by considering scattering from a layered sphericalobject and comparing with that from an ideal D’B’ sphere. It is seen that therealization approaches the ideal case when the radial parameters of the quarter-wavelength layer grow large. Such an object is of interest because it has zerobackscattering like the corresponding DB sphere [10]. References [1] I.V. Lindell and A. Sihvola, “Realization of impedance boundary,”
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30 60 90 120 150 180−60−50−40−30−20−10010 θ [deg] σ / λ [ d B ] D’B’ Mie100010010
Figure 2: Calculated bistatic scattering cross section corresponding to the re-alization of the D’B’ sphere. Different values of radial permittivity and per-meability ǫ r /ǫ o = µ r /µ o are used to approximate the quarter-wave transformerwhile the relative tangential components are ǫ t = ǫ o , µ t = µ o . The ideal D’B’scattering is computed using Mie series. The geometry of the realization can beseen in Fig. 1 (left).[2] I.V. Lindell and A. Sihvola: “Electromagnetic boundary condition andits realization with anisotropic metamaterial,” Phys. Rev. E , vol.79, no.2,026604 (7 pages), 2009.[3] I.V. Lindell and A. Sihvola, “Electromagnetic boundary conditions definedin terms of normal field components,”
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30 60 90 120 150 180−60−50−40−30−20−10010 θ [deg] σ / λ [ d B ] D’B’ Mie100010010
Figure 3: Calculated scattering cross section of the realization of the D’B’ spherewith b − a = 0 . ǫ r /ǫ o = µ r /µ o .[7] R. Weder, “The boundary conditions for point transformed electromagneticinvisible cloaks,” J. Phys. A , vol.41, 415401 (17 pages), 2008.[8] A.D. Yaghjian, “Extreme electromagnetic boundary conditions and theirmanifestation at the inner surfaces of spherical and cylindrical cloaks,”
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