An exact transition matrix was formulated for electromagnetic scattering by an orthorhombic dielectric–magnetic sphere whose permeability dyadic is a scalar multiple of its permittivity dyadic. Calculations were made for plane waves incident on the sphere. As the size parameter increases, the role of anisotropy evolves; multiple lobes appear in the plots of the differential scattering efficiency in any scattering plane; the total scattering, extinction, and forward-scattering efficiencies exhibit a prominent maximum each; and the absorption efficiency generally increases with weak undulations. Certain orientations of the sphere with respect to the directions of propagation and the electric field of the incident plane wave make it highly susceptible to detection in a monostatic configuration, whereas other orientations make it much less vulnerable to detection. Impedance match to the ambient free space decreases backscattering efficiency significantly, although anisotropy prevents null backscattering.
© 2013 Optical Society of America
CorrectionsA. D. Ulfat Jafri and Akhlesh Lakhtakia, "Scattering of an electromagnetic plane wave by a homogeneous sphere made of an orthorhombic dielectric-magnetic material: erratum," J. Opt. Soc. Am. A 31, 2630-2630 (2014)
The homogeneous sphere is ubiquitous in electromagnetic-scattering literature, whether the sphere is electrically small [1–3] or electrically huge [4,5] or anywhere in between in size [6,7]. Exact analytical solutions are known for scattering by isotropic dielectric [8–10], isotropic dielectric–magnetic [11,12], and bi-isotropic spheres , because vector spherical wavefunctions are available in closed form for these materials. However, similarly analytical solutions for scattering by anisotropic spheres remain unknown, because the relevant vector spherical wavefunctions in closed form have not been found.
Recently, however, closed-form vector spherical wavefunctions were found for orthorhombic dielectric–magnetic material with gyrotropic-like magnetoelectric properties , thereby extending our options in solving scattering problems analytically. For the simplest application of the newly found wavefunctions, we decided to calculate scattering by a homogeneous sphere of an orthorhombic dielectric–magnetic material described by the frequency-domain constitutive relations
Belonging to the magnetic point group or [15, p. 68], the chosen material is anisotropic but pathologically unirefringent because its permeability dyadic is a scalar multiple of its permittivity dyadic . Suppose that a plane wave propagates in an arbitrary direction in this material with wavenumber . Then, is bounded by either (i) and or (ii) and , where . In naturally occurring materials, both and are expected be very small (), based on available data for solid dielectric crystals . However, both of these quantities could be larger for manufactured composite materials. An example is a composite material made by dispersing a mixture of electrically small springs that are aligned along any one of the three Cartesian axes, provided that every spring has a coaligned enantiomorph present in close proximity. Although simple techniques for making composite materials containing aligned springs  have been around for at least a century, nano/microscale additive manufacturing techniques  should play a major role in the very near future toward engineering complex materials. Furthermore, suspensions of ferromagnetic grains in liquid crystals  are also candidate materials.
Although the closed-form vector spherical wavefunctions applicable for the chosen material form a complete set [14,21], these functions are orthogonal not on a unit sphere described solely by the angular variables and in space, but on a unit sphere in a transformed space. Therefore, scattering by this sphere cannot be described in terms of analogs of the “Mie coefficients” found in modern treatments of scattering by homogeneous isotropic spheres [11,12]. Instead, a transition matrix (commonly called the “T matrix”) is needed to describe scattering by a homogeneous sphere made of the chosen material. The derivation of the T matrix for general nonspherical scatterers being available , we provide essential expressions and final results in Section 2. Section 3 presents numerical results to explicate the effects of anisotropy on the scattering of an incident plane wave. Special attention is paid to the scattering cross section, extinction cross section, and differential cross section in relation to (i) the size parameter and the anisotropy of the sphere as well as (ii) the polarization state and the direction of propagation of the incident plane wave. The time dependency is present but suppressed, is the free-space wavenumber, and is the intrinsic impedance of free space. The asterisk denotes the complex conjugate.
Let a homogeneous sphere of radius be located with its center at the origin of the spherical coordinate system . The sphere is made of a material that obeys the constitutive relations stated via Eqs. (1) and (2). The region is vacuous.
A. Incident Electromagnetic Field
With the assumption that the source of the incident field lies in the region , the incident electric field can be expressed as11,12]
The coefficients and are presumed to be known. Suppose that the incident electromagnetic field is that of a plane wave with a propagation vector22, p. 1866] 22, p. 1899]
B. Scattered Electromagnetic Field
With the assumption that the field scattered by the sphere does not interact with (and modify) the source of the incident field, we can state the scattered electric and magnetic fields as follows:11,12], 14].
In the far zone, the relation
The foregoing expressions can be used to verify the satisfaction of the Sommerfeld radiation conditions 24] 13) and (14).
C. Internal Electromagnetic Field
The electric and magnetic fields excited inside the sphere are represented by 
The vector spherical wavefunctions appearing in these expressions are given as25,26].
D. Solution of Boundary–Value Problem
Application of the standard boundary conditions,14]:
E. Uniaxial Dielectric–Magnetic Sphere47).
F. Isotropic Dielectric–Magnetic Sphere12] for an isotropic dielectric–magnetic sphere embedded in free space is recovered.
G. Scattering Cross Sections and Efficiencies
The plane wave scattering response is usually quantified through the differential scattering cross section [27, Section 8.8]
H. Extinction and Absorption Cross Sections and Efficiencies
In order to quantify the removal of energy from the incident plane wave through both absorption and scattering, the extinction cross section
The excess of over is the absorption cross section , which is null valued if the sphere is made of a nondissipative material (i.e., both and are both positive and real). When is normalized by , the result is the absorption efficiency .
3. NUMERICAL RESULTS AND DISCUSSION
In order to compute the T matrix as well as the differential, total scattering, and extinction cross sections, we set up a Mathematica program. For that purpose, we made the replacement
For partial validation of the Mathematica program, was set therein so that the scattering sphere was taken to be made of an isotropic dielectric–magnetic material. First, the forward-scattering efficiency,12,28]. Second, a Mathematica program was written to implement the Lorenz–Mie solution for an isotropic dielectric–magnetic sphere (, ) and was used to check the results yielded by the program written for the anisotropic dielectric–magnetic sphere. The latter program satisfied both tests.
Additional tests for validation involved symmetry. For example, for a uniaxial dielectric–magnetic sphere (), we found that the total scattering cross section does not change if (i) is replaced by , (ii) is replaced by , or (iii) is replaced by . Similarly, we verified that for remains the same as for , provided that the values of and are interchanged.
Finally, we also verified that the absorption cross section is null valued if the sphere is made of a nondissipative material.
A. Rayleigh Scattering
Scattering by an electrically small object [6, Section 6.4] is called Rayleigh scattering. Not only is it studied to explain the electromagnetic response of particulate materials [29,30], but it is also used to design composite materials [31,32].
A long-wavelength approximation is very useful to obtain closed-form analytical results for scattering by homogeneous and electrically small objects . Accordingly, scattering by an electrically small sphere described by the constitutive relations (1) is equivalent to radiation jointly by an electric dipole moment66) in Eqs. (57), (58), and (60), respectively. These expressions can be expected to hold for [6, Section 10.6.1]
We computed the differential scattering, total scattering, and extinction efficiencies, exactly using Eq. (20) and approximately using Eq. (66), of an electrically small uniaxial dielectric–magnetic sphere (, , ) illuminated by a plane wave with . No distinction can be seen in the curves of and versus presented in Fig. 1(a). The same curves also hold for . Although the long-wavelength approximation is expected to be valid for according to the inequalities (67), the Rayleigh expressions hold well in Fig. 1(a) for as high as 0.5.
For Fig. 1(a), the sphere is transversely isotropic with respect to the direction of the incident plane wave, because and . By setting and unequal without any other changes, the condition of transverse isotropy can be changed. Figure 1(b) presents and versus for an electrically small biaxial dielectric–magnetic sphere (, , , ) illuminated by a plane wave with . Clearly now, the long-wavelength approximation is highly satisfactory for , as predicted by the inequalities (67), but the discrepancies between the exact and the approximate results begin to grow as increases further.
The biaxiality of the electrically small dielectric–magnetic sphere in Fig. 1(b) also enhances the roles of the directions of both and . Comparing the plots of versus presented in67).
Although we presented data in Subsection 3.A on only the total scattering efficiencies of electrically small spheres here, the conclusions drawn on the adequacy of the Rayleigh expression (66) therefrom are also applicable to the differential scattering and extinction efficiencies.
B. Differential Scattering Efficiency
In order to present representative results for differential scattering efficiency, we fixed , , , , and . Plots of and are presented in Figs. 3 and 4 for and either or , where . A very notable feature of these plots is the appearance of multiple lobes as the size parameter increases. This happens in any scattering plane, as exemplified by the plots of versus in the and planes. However, when the electrical size is small, lobes may not appear in certain scattering planes—e.g., in the plane in Fig. 3(b) and the plane in Fig. 4(a). Even though the sphere is made of an anisotropic material lacking transverse isotropy with respect to the direction of propagation of the incident plane wave, the sphere appears to be isotropic just because it is electrically small. As the size parameter increases, anisotropy asserts its presence.
With the size parameter fixed at , we also compared the effects of uniaxiality and biaxiality. Figures 5 and 6 show plots of and , when , either or , , and . The sphere is transversely isotropic with respect to the direction of propagation of the incident plane wave if the material is uniaxial (), but not if it is biaxial (, ). The curve of versus for in Fig. 5(a) is identical to that of versus for in Fig. 6(b), and the curve of versus for in Fig. 5(b) is identical to that of versus for in Fig. 6(a), for the uniaxial sphere. This symmetry is not evident for the biaxial sphere.
C. Total Scattering and Extinction Efficiencies
The total scattering and extinction efficiencies calculated using Eqs. (58) and (60), respectively, are shown in Figs. 7–9 as functions of for six combinations of the directions of and . The sphere was taken to be dissipative and biaxial: , , , and .
Because of dissipation, for all in these figures. Furthermore, tracks so well that the absorption efficiency rises steadily with and becomes a weak function of , regardless of the directions of and .
All six panels in Figs. 7–9 clearly indicate that and are maximum at some . The plots suggest that both and would settle down to show a weak dependence on as that parameter increases further—just like an isotropic sphere [12, Section 4.4]—but calculations could not be carried out on a laptop computer for in a reasonable period of time. This limitation arises from the functions and not displaying orthogonality properties on the surface of any sphere centered at the origin when . We conjectured from Figs. 7–9 that the highest peaks of and versus are obtained when the direction of propagation of the incident plane wave is such that the scalar has a minimum value. This conjecture was verified by several calculations for different values of and .
In contrast to the clear maximums of both and versus in Figs. 7–9 the absorption efficiency —despite some undulations—shows a generally increasing trend with for six combinations of the directions of and . In all six panels in these figures, first increases slower than and then decreases faster than , as increases.
D. Backscattering and Forward-Scattering Efficiencies
The backscattering and forward-scattering efficiencies computed using Eqs. (62) and (63), respectively, are also shown in Figs. 7–9 as functions of for six combinations of the directions of and , when , , , and .
We observe that keeps on generally increasing as the size parameter increases up to a limit and then exhibits undulations with further increases of . The undulations are most pronounced for the smallest value of .
In contrast to , the plots of versus are oscillatory regardless of the directions of and . Monostatic detection of the sphere would be easier at frequencies where peaks, but it will be difficult at frequencies for which reduces to a minimum, which can be substantially smaller than the adjacent peak values. In this characteristic, a sphere made of the chosen material responds similarly to a sphere made of an isotropic achiral material [28, p. 246]. But the anisotropy of the chosen material shows its effect too. Higher peaks are favored by the highest value of , as can be concluded from Fig. 10. Even more importantly, the backscattering efficiency of a sphere of a specific size parameter can be halved from its maximum value simply by reorienting the sphere with respect to the directions of the wave vector and the electric field of the incident plane wave. Thus, certain orientations of the sphere make it highly susceptible to detection in a monostatic configuration, whereas other orientations make it much less vulnerable to detection.
E. Impedance Match
When , the chosen material can be said to be impedance-matched to the ambient medium (i.e., free space). If additionally, the sphere would be made of an isotropic dielectric–magnetic material, and then by virtue of a theorem of Weston . Figure 11 shows plots of versus for six different combinations of and , when , , and . Whereas is independent of in Fig. 11, it does depend on . Comparing with Fig. 10, we conclude that the independence of from is due to the impedance match with the ambient free space.
Even though the values of are quite similar in Figs. 10 and 11, a significant consequence of impedance match to the ambient free space is a tenfold decrease in the backscattering efficiency. That did not actually drop to zero is because anisotropy makes the chosen sphere noninvariant under a rotation by about an axis parallel to the direction of incidence, a condition required by Weston’s theorem  for null backscattering.
4. CONCLUDING REMARKS
Electromagnetic scattering by an orthorhombic dielectric–magnetic sphere whose permeability dyadic is a scalar multiple of its permittivity dyadic was formulated exactly in terms of a transition matrix, simplifying a recent development wherein closed-form vector spherical wavefunctions were formulated to capture anisotropy. Calculating the total scattering, extinction, differential scattering, backscattering, and forward-scattering efficiencies, we examined the evolving role of anisotropy as the size parameter increases.
Multiple lobes appear in the plots of the differential scattering efficiency in any scattering plane, as the size parameter increases. However, lobes may not appear in certain scattering planes when the sphere is electrically small in size. As the size parameter increases, the total scattering, extinction, and forward-scattering efficiencies exhibit a prominent maximum each, but the absorption efficiency generally increases with weak undulations. Certain orientations of the sphere with respect to the directions of propagation and the electric field of the incident plane wave make it highly susceptible to detection in a monostatic configuration, whereas other orientations make it much less vulnerable to detection. Finally, the backscattering efficiency can be reduced significantly, if the sphere is impedance-matched to the ambient free space.
A. D. U. J. acknowledges the support of the Higher Education Commission of Pakistan for sponsoring a six-month visit to the Pennsylvania State University, where this work was done. A. L. is grateful to the Charles Godfrey Binder Endowment at the Pennsylvania State University for ongoing support of his research.
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