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J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011
L. Sun and K. W. Yu
Broadband electromagnetic transparency by graded metamaterials: scattering cancellation scheme L. Sun* and K. W. Yu Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong *Corresponding author:
[email protected] Received November 10, 2010; revised January 19, 2011; accepted February 12, 2011; posted February 17, 2011 (Doc. ID 137943); published April 5, 2011 The electromagnetic scattering by a radially inhomogeneous isotropic metamaterial sphere whose electric permittivity is described by the lossless graded Drude model is studied according to the generalized Mie theory in full-wave condition. The distribution of electromagnetic field is calculated by solving Maxwell’s equations, and the exact analytic solutions are obtained in terms of confluent Heun and hypergeometry functions. This allows us to achieve the full-wave scattering cross section (SCS) analytically. The corresponding numerical analysis indicates that the full-wave SCS can be extremely small over a broad frequency band, representing a broadband electromagnetic transparency. Moreover, the analytic expression of the full-wave SCS also reveals the conditions for achieving the broadband electromagnetic transparency and makes tunable electromagnetic transparency feasible. © 2011 Optical Society of America OCIS codes: 160.3918, 260.2110, 290.4020, 290.5839.
1. INTRODUCTION Achieving electromagnetic transparency and cloaking by using metamaterials has been extensively studied in physics and engineering communities since it has significant impact in various fields, including optics, medicine, biology, and nanotechnology. In order to attain this aim, Kerker and Chew [1,2] promoted the basic idea of the scattering cancellation technique, and Alù and Engheta developed this scheme [3], in which a properly designed metamaterial cover near its plasma resonance induces a dramatic drop in the scattering cross section (SCS), making an object nearly “invisible” or “transparent” to the probing electromagnetic radiation. Later, this technique was extended to achieve the electromagnetic transparency for a collection of particles [4] and multifrequency optical invisibility [5]. Meanwhile, the mechanism, robustness, and physical insights as well as the experiment demonstration of this technique are discussed [6,7] and carried out [8]. On the other hand, Leonhardt proposes the conformal mapping technique [9] to achieve the perfect invisibility in the geometric optics limit. However, the conformal mapping only represents a special case; most spatial transformations are nonconformal [10]. In contrast, Pendry and coworkers introduce the transformation optics technique [11] that can exclude the electromagnetic field from an object without disturbing its propagation in a general case. In principle, this technique is valid in the whole electromagnetic spectrum. The corresponding experiment [12] and computer simulations [13,14] prove the possibility to fabricate the invisible devices based on this technique. In addition, the interactions of the electromagnetic wave with a general spherical cloak based on a full-wave Mie scattering model are established analytically. The results show that, for an ideal cloak, the full-wave SCS is absolutely zero, but for a cloak with a specific type of loss, only the backscattering is exactly zero [15]. Later, the transformation optics technique is modified to make cylindrical cloak structures, which work in multilayered and gradually changing media. 0740-3224/11/050994-08$15.00/0
This modified technique can also be applied to the design of cloaks in an arbitrary isotropic background [16]. Nevertheless, due to the singularity in the transformation, invisible models based on this method only work in a single frequency, and the required anisotropic structure poses a difficult practical engineering design. To overcome this disadvantage, Leonhardt and Tyc introduced a non-Euclidean cloaking technique [17] that avoids the transformation singularity and obtained the broad frequency band invisibility theoretically. Moreover, Tyc and coworkers performed a two-dimensional computer simulation for the non-Euclidean cloaking, showing that the cloaking is nearly perfect for a spectrum of frequencies corresponding to the spherical harmonics and becomes perfect in the limit of geometric optics [18]. Nonetheless, this technique needs a even more complicated media structure that is more difficult in design. In this work, the electromagnetic scattering by a graded isotropic sphere with a radially inhomogeneous electric permittivity described by a lossless graded Drude model, i.e., ϵs ðrÞ ¼ 1 − ω2p ðrÞ=ω2 , is studied. The plasma frequency ωp ðrÞ depends on the position r as ω2p ðrÞ ¼ ω2p ð0Þðc0 − c1 ðr= r 0 Þn Þ, where c0 , c1 , and n are positive constants (gradient parameters). The radius of the sphere and the initial plasma frequency at the origin are denoted by r 0 and ωp ð0Þ, individually. The distribution of the electromagnetic field is calculated by applying the generalized Mie theory in full-wave condition. Exact analytic expressions of the electromagnetic field are obtained in terms of confluent Heun and hypergeometric functions, when the gradient parameter n equals 2. In addition, the exact analytic expression of the full-wave SCS is also attained from the amplitudes of the scattering electromagnetic field and is found to be dependent on both the graded profile and the frequency of the incident electromagnetic wave. The corresponding numerical results show that the graded isotropic sphere slightly disturbs the propagation of the electromagnetic field in a broad band spectrum, which indicates © 2011 Optical Society of America
L. Sun and K. W. Yu
Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B
the possibility of broadband electromagnetic transparency. Moreover, the exact analytic expression of the full-wave SCS enables us to assess the conditions for achieving broadband transparency and makes tunable electromagnetic transparency feasible.
2. MIE THEORY FOR GRADED ISOTROPIC SPHERE
↔
ϵ ðrÞ ¼ ϵϵs ðrÞ I ;
↔
↔
μ ðrÞ ¼ μμs ðrÞ I ;
d 1 dðrf u ðrÞÞ lðl þ 1Þ rf u ðrÞ ¼ 0; þ ω2 ϵðrÞμðrÞ − dr ϵðrÞ dr r2
ð7Þ d 1 dðrf v ðrÞÞ lðl þ 1Þ rf v ðrÞ ¼ 0; þ ω2 ϵðrÞμðrÞ − μðrÞ dr μðrÞ dr r2 ð8Þ
respectively. Meanwhile, the boundary conditions expressed in terms of the scalar potentials are
Mie theory is an analytical solution of Maxwell’s equations for the scattering of an electromagnetic wave by homogeneous isotropic spherical particle in full-wave condition [19–23]. To analyze the scattering of an electromagnetic wave by graded isotropic spheres, modifications on the original Mie theory are needed. Therefore, we first introduce the Mie theory modification in this section. Without loss of generality, we consider the scattering of an electromagnetic wave by a graded isotropic sphere embedded in a homogenous isotropic host with electric permittivity ϵ and magnetic permeability μ. In contrast, the electric permittivity and magnetic permeability of the sphere read ↔
ϵðrÞ
995
ð1Þ
ð2Þ
uð1Þ ðr 0 Þ ¼ uð2Þ ðr 0 Þ;
ð9Þ
vð1Þ ðr 0 Þ ¼ vð2Þ ðr 0 Þ;
ð10Þ
1 ∂ðruð1Þ Þ 1 ∂ðruð2Þ Þ ¼ ; ϵð1Þ ∂r r¼r0 ϵð2Þ ∂r r¼r 0
ð11Þ
1 ∂ðrvð1Þ Þ 1 ∂ðrvð2Þ Þ ¼ ; μð1Þ ∂r r¼r0 μð2Þ ∂r r¼r0
ð12Þ
where the superscripts denote different media beside the boundary. According to Eqs. (5) and (6) as well as the boundary conditions Eqs. (9)–(12), we can finally determine the electromagnetic field in the whole space from Eqs. (3) and (4).
↔
where I ¼ er er þ eθ eθ þ eϕ eϕ is the unit tensor in the spherical coordinate system and r 0 is the radius of the sphere. The total electromagnetic field can be expressed by a pair of scalar potentials uðrÞ and vðrÞ, similar to the Debye potentials [24,25], as E¼
1 ∇ × ∇ × ðruÞ þ ∇ × ðrvÞ; iωϵðrÞ
ð3Þ
3. SCATTERING OF PLANE ELECTROMAGNETIC WAVE Here, we explore the scattering of a monochrome plane electromagnetic wave by the graded isotropic sphere mentioned previously. The electromagnetic wave propagates along the positive z axis in the host, characterized by Ein ¼ ex eikr cos θ ;
H¼
1 ∇ × ∇ × ðrvÞ − ∇ × ðruÞ: iωμðrÞ
ð4Þ
By substituting Eqs. (3) and (4) into Maxwell’s equations under the free space ðρ ¼ 0; j ¼ 0Þ and harmonic ðE; H ∼ e−iωt Þ conditions, we obtain following equations for the scalar potentials: ∂ 1 ∂ðruÞ 1 ∂ ∂ðruÞ 1 ∂2 ðruÞ þ 2 sin θ þ 2 2 ϵðrÞ ∂r ϵðrÞ ∂r ∂θ r sin θ ∂θ r sin θ ∂ϕ2 þ ω2 ϵðrÞμðrÞru ¼ 0;
μðrÞ
ð5Þ
∂ 1 ∂ðrvÞ 1 ∂ ∂ðrvÞ 1 ∂2 ðrvÞ þ 2 sin θ þ 2 2 ∂r μðrÞ ∂r ∂θ r sin θ ∂θ r sin θ ∂ϕ2 þ ω2 ϵðrÞμðrÞrv ¼ 0;
ð6Þ
where ω is the angular frequency of electromagnetic wave in the host. Equations (5) and (6) can be solved by applying the ðmÞ variable separation method, i.e., uðr; θ; ϕÞ ¼ f u ðrÞY l ðθ; ϕÞ ðmÞ ðmÞ ðmÞ and vðr; θ; ϕÞ ¼ f v ðrÞY l ðθ; ϕÞ, where Y l ðθ; ϕÞ ¼ P l ðmÞ imϕ ðcos θÞe and P l ðcos θÞ is the associated Legendre polynomials. The radial components f u ðrÞ and f v ðrÞ satisfy
ð13Þ
H in ¼ ðey =ZÞeikr cos θ ; ð14Þ pffiffiffiffiffiffiffiffi pffiffiffiffiffi where k ¼ ω ϵμ is the wavenumber and Z ¼ μ=ϵ is the impedance. The graded isotropic sphere possesses a relative electric permittivity ϵs ðrÞ described by the lossless graded Drude model ϵs ðrÞ ¼ 1 −
ω2p ð0Þ ðc0 − c1 ðr=r 0 Þn Þ; ω2
ð15Þ
with positive gradient parameters c0 , c1 , and n, as well as a relative magnetic permeability μs ðrÞ ¼ 1. For convenience, we set the initial plasma frequency ωp ð0Þ ¼ 1, i.e., the frequency unity, and rewrite the relative electric permittivity as ϵs ðrÞ ¼ a þ br n , where a ¼ 1 − c0 =ω2 and b ¼ c1 =ðω2 r n0 Þ. A. General Formalism To determine the electromagnetic field in the whole space, we regard the total field as two parts: a superposition of the incident field and the scattering field in the host, individually noted by superscripts “in” and “sc” as well as the electromagnetic field in the sphere, noted by superscript “sp.” Because the electric permittivity and magnetic permeability of the host are both constants, the scalar potentials determining
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J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011
L. Sun and K. W. Yu
the incident field and the scattering field are simply uin ðr; θ; ϕÞ ¼ −
∞ ωϵ X
k
il
l¼1
2l þ 1 ð1Þ j ðkrÞP l ðcos θÞ cos ϕ; ð16Þ lðl þ 1Þ l
∞ ωμ X 2l þ 1 ð1Þ v ðr; θ; ϕÞ ¼ − j ðkrÞP l ðcos θÞ sin ϕ; il kZ l¼1 lðl þ 1Þ l in
ð17Þ
∞ ωϵ X 2l þ 1 ðlÞ ð1Þ ð1Þ Au hl ðkrÞP l ðcos θÞ cos ϕ; il usc ðr; θ; ϕÞ ¼ − k l¼1 lðl þ 1Þ
ð18Þ ωμ v ðr; θ; ϕÞ ¼ − kZ sc
∞ X l¼1
∞ X
ðaÞ
cj ðx=x0 Þj ; ð30Þ
where
ðlÞ and Av
lðl þ 1Þ v ðrÞÞ 2 rf v ðrÞ ¼ 0: þ k ϵs ðrÞ − dr 2 r2
ð29Þ
j¼0
ð19Þ
d2 ðrf
B3 ¼ −ð1 þ lÞ=2:
gðxÞ ¼ eiwx HcðaÞ ðp; α; γ; δ; σ; x=x0 Þ ¼ eiwx
pffiffiffi p ¼ iak= 4 b ;
ð31Þ
h i pffiffiffi pffiffiffi α ¼ iak þ bð1 þ 2lÞ = 4 b ;
ð32Þ
γ ¼ ð3 þ 2lÞ=2;
ð33Þ
δ ¼ −1;
ð34Þ
2l þ 1 ðlÞ ð1Þ ð1Þ Av hl ðkrÞP l ðcos θÞ sin ϕ; i lðl þ 1Þ
are called scattering coefficients, determined by the boundary conditions, i.e., Eqs. (9)–(12). Moreover, j l ðkrÞ is the spherical Bessel function ð1Þ and hl ðkrÞ is the first kind spherical Hankel function in Eqs. (16)–(19). According to the electromagnetic constitutive properties of the graded isotropic sphere, the electromagnetic field in the sphere is determined by the solutions of the simplified Eqs. (7) and (8): d 1 dðrf u ðrÞÞ lðl þ 1Þ 2 ϵs ðrÞ rf u ðrÞ ¼ 0; þ k ϵs ðrÞ − dr ϵs ðrÞ dr r2 ð20Þ
ð28Þ
The solution to Eq. (22), which converges at the origin, can be expressed in terms of angular confluent Heun function HcðaÞ [27] as
l
ðlÞ according to Eqs. (5) and (6). Here, Au
B2 ¼ ð1 þ 2lÞ=2;
σ¼
hpffiffiffi pffiffiffi i b þ iak 2 bð1 þ lÞ þ iak =ð4bÞ:
The coefficients
ðaÞ cj
satisfy a three-term recurrence relation ðaÞ
ð36Þ
ðaÞ
ð37Þ
c−1 ¼ 0;
ð21Þ c0 ¼ 1;
With regard to Eq. (20), we find that, only when the gradient parameter n equals 2, it has an exact analytic solution that can be expressed in terms of a well-known special function. This offers an opportunity to explore the distribution of the electromagnetic field inside the graded isotropic sphere. When the gradient parameter n equals 2, by introducing a pair of variable substitutions f u ðrÞ ¼ r l g1 ðr n Þ and f u ðrÞ ¼ r l g2 ðr 2 Þ, Eq. (20) transforms into the generalized spheroidal wave equation in Leaver version [26]
ðaÞ ðaÞ
ðaÞ ðaÞ
ðaÞ ðaÞ
f j cjþ1 þ gj cj þ hj cj−1 ¼ 0;
ð38Þ
where ðaÞ
gj
¼ jðj − 4p þ γ þ δ − 1Þ − σ;
ðaÞ
fj
xðx − x0 Þg00 ðxÞ þ ðB1 þ B2 xÞg0 ðxÞ þ ½w2 xðx − x0 Þ − 2wηðx − x0 Þ þ B3 gðxÞ ¼ 0;
ð35Þ
¼ −ðj þ 1Þðj þ γÞ;
ð39Þ
ð40Þ
ð22Þ ðaÞ
hj
where x ¼ r2;
ð23Þ
x0 ¼ −a=b;
ð24Þ
pffiffiffi w ¼ −k b=2;
ð25Þ
¼ 4pðj þ α − 1Þ:
Thus, the final expression of f u ðrÞ for Eq. (20) is pffiffi 2 1 f u ðrÞ ¼ r l e−2ik br HcðaÞ ðp; α; γ; δ; σ; −br 2 =aÞ:
ð26Þ
B1 ¼ að3 þ 2lÞ=ð2bÞ;
ð27Þ
ð42Þ
The solution to Eq. (21) can be achieved in a similar way. In the condition of n ¼ 2, Eq. (21) transforms into a form as follows: 4xz00 ðxÞ þ ð6 þ 4lÞz0 ðxÞ þ k2 ða þ bxÞzðxÞ ¼ 0;
pffiffiffi η ¼ ka= 4 b ;
ð41Þ
ð43Þ
One of its solutions, converging at the origin, can with x ¼ be expressed in terms of confluent hypergeometric function F 11 [28] as r2 .
L. Sun and K. W. Yu
Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B pffiffi bx
pffiffiffi F 11 ðκ; ν; ik bxÞ;
ð44Þ
h i pffiffiffi pffiffiffi κ ¼ ð3 þ 2lÞ b þ ika = 4 b ;
ð45Þ
ν ¼ 3=2 þ l:
ð46Þ
zðxÞ ¼ e−2ik 1
where
Thus, the final expression of f v ðrÞ for Eq. (21) is pffiffi 2 br F
f v ðrÞ ¼ r l e−2ik 1
pffiffiffi br 2 Þ:
11 ðκ; ν; ik
ð47Þ
From the foregoing analysis, the scalar potentials determining the electromagnetic field in the graded isotropic sphere can be written as usp ðr; θ; ϕÞ ¼ −
∞ ωϵ X 2l þ 1 ðlÞ ð1Þ Bu f u ðrÞP l ðcos θÞ cos ϕ; il k l¼1 lðl þ 1Þ
ð48Þ
vsp ðr; θ; ϕÞ ¼ −
∞ ωμ X 2l þ 1 ðlÞ ð1Þ Bv f v ðrÞP l ðcos θÞ sin ϕ; il kZ l¼1 lðl þ 1Þ
ð49Þ ðlÞ Bu
ðlÞ Bv
and to be determined. with two coefficients According to Eqs. (16)–(19), (48), and (49) and the boundary conditions Eqs. (9)–(12), we can determine the scattering ðlÞ ðlÞ coefficients Au and Av as ðlÞ
Au ¼
ðlÞ
Av ¼
ψðkr 0 ÞF 0u ðr 0 Þ − ϵs ðr 0 Þψ 0 ðkr 0 ÞF u ðr 0 Þ ; ϵs ðr 0 Þξ0 ðkr 0 ÞF u ðr 0 Þ − ξðkr 0 ÞF 0u ðr 0 Þ
ð50Þ
ψðkr 0 ÞF 0v ðr 0 Þ − μs ðr 0 Þψ 0 ðkr 0 ÞF v ðr 0 Þ ; μs ðr 0 Þξ0 ðkr 0 ÞF v ðr 0 Þ − ξðkr 0 ÞF 0v ðr 0 Þ
ð51Þ
ð1Þ
where ψðkrÞ ¼ krj l ðkrÞ and ξðkrÞ ¼ krhl ðkrÞ are the Riccati– Bessel functions [29,30], while F u ðrÞ ¼ krf u ðrÞ and F v ðrÞ ¼ krf v ðrÞ. The primes on ψðkr 0 Þ, ξðkr 0 Þ, F u ðr 0 Þ, and F v ðr 0 Þ denote differentiations with respect to the arguments. Therefore, the formula of the full-wave SCS for the graded isotropic sphere, written in terms of scattering coefficients, is ðMÞ
C sca ¼
∞ 2π X ðlÞ ðlÞ ð2l þ 1ÞðjAu j2 þ jAv j2 Þ: k2 l¼1
ð52Þ
Equation (52) implies that the full-wave SCS is determined by the scattering coefficients up to some order l ¼ N max because the successive scattering coefficients beyond N max are negligible. The value of N max usually increases with the physical or electrical size of the sphere, and there is a rule of thumb that N max ≈ kr 0 . Actually, when the size of the sphere is sufficiently small, i.e., kr 0 < 1, only the dipolar term l ¼ 1 is retained in Eq. (52). In such a case, we can make a quasi-static approximation on the full-wave SCS by using the Rayleigh scattering theory, thus ðMÞ ðRÞ C sca ≈ C sca
128π 5 r 60 ϵeff − ϵ 2 μeff − μ 2 128π 5 r 60 2 ¼ ¼ þ jbj ; 4 ϵeff þ 2ϵ μeff þ 2μ 3λ 3λ4 ð53Þ
997
where λ is the wavelength of the incident electromagnetic wave in the host and b is the dipole factor [31] of the graded isotropic sphere. Here, ϵeff and μeff denote the effective electric permittivity and magnetic permeability [32] of the graded isotropic sphere, respectively. B. Numerical Analysis In this section, we perform numerical analysis on the SCS (normalized by the square of wavelength λ2 ) in the full-wave ðMÞ ðRÞ condition C sca =λ2 and the quasi-static approximation C sca =λ2 , according to Eqs. (52) and (53), respectively. The SCS of the graded isotropic sphere is calculated with different radii to reveal the broadband electromagnetic transparency. In the numerical analysis, the gradient parameter c0 is set to be 1=2, while the length unity is set to be λp ð0Þ=ð2πÞ, where λp ð0Þ is the wavelength of the electromagnetic wave corresponding to the frequency ωp ð0Þ in the host. In the calculation of the full-wave SCS, the condition kr 0 ≤ 1 is considered. In that case, the expression of the full-wave SCS [Eq. (52)] only counts in the dipole term. That is to say, the scattering field of the graded isotropic sphere is mainly determined by its induced dipole moment given by an integral of the electric polarization P ¼ ϵðϵs ðrÞ − 1ÞE over the whole sphere. Therefore, it is clear that while the electric permittivity ϵs ðrÞ is around unity over a wide frequency range, the integral of the electric polarization will be close to zero and leads to a near-zero induced dipole moment. Hence, the scattering field of the graded isotropic sphere will be extremely small over a broad frequency band. In other words, the graded isotropic sphere will be transparent to the probing electromagnetic wave due to the dipole cancellation. This dipole cancellation mechanism is revealed more clearly in Fig. 1, which displays the electric permittivity and magnetic permeability distributions inside the graded isotropic sphere when the gradient parameters are c0 ¼ 1=2 and c1 ¼ 0:9 and the frequency is ω=ωp ð0Þ ¼ 0:65. The two vertical dashed lines indicate the position where the electric permittivity equals 0 and 1, individually. Meanwhile, they divide this figure into three parts. In the first part, the electric permittivity is negative, while the magnetic permeability is unity. This region is forbidden to electromagnetic wave, i.e., electromagnetic radiation will decay to zero exponential at the boundary of this region and cannot propagate in it. Therefore, this region can be regarded as a cloaking zone. We can replace this region with other objects, but retain the zero-permittivity boundary, thus the objects cannot be detected by probing electromagnetic wave. Outside this region, there are two zones in which the electric permittivities are less and greater than unity, respectively. Since electric permittivity and magnetic permeability in these two regions are both positive, the electromagnetic wave can propagate in them. However, the polarizations in these regions have opposite directions, according to the definition of polarization. Hence, the scattering wave of these two regions can cancel each other and make the probing electromagnetic wave be nearly undisturbed. In addition, the position- and frequency-dependent electric permittivity can make this scattering cancellation work well over a wide frequency range, thus leading to the broadband electromagnetic transparency. It should be noted that the dipole cancellation becomes poor while kr 0 ≥ 1 because the higher-order multipole
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J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011
Fig. 1. (Color online) Electric permittivity and magnetic permeability of the graded isotropic sphere.
L. Sun and K. W. Yu
moments may dominate the scattering in that condition. However, the expression of the full-wave SCS [Eq. (52)] still offers an opportunity to find a proper design to achieve a broadband transparency by cancelling the leading term in the fullwave SCS. Figure 2 displays the results for the graded isotropic sphere with r 0 ¼ 0:1. The full-wave results are shown in Figs. 2(a) and 2(b), while the quasi-static results are in Figs. 2(c) and 2(d). In the full-wave results, the contour plot in Fig. 2(a) shows the variation of the full-wave SCS versus both frequency and gradient parameter c1 . The darkest region in the plot corresponds ðMÞ to the smallest C sca =λ2 , implying a broadband electromagnetic transparency. In addition, Fig. 2(b) shows the variation of the full-wave SCS versus the frequency when the parameter c1 is fixed on some special values located in the darkest region of Fig. 2(a). This plot clearly indicates that, for values of the ðMÞ parameter c1 located in the darkest region, C sca =λ2 can be quite small over a frequency range, which indicates a broadband electromagnetic transparency of the graded isotropic sphere. Similar results are also found in the quasi-static state [Figs. 2(c) and 2(d)]. Figs. 2(c) and 2(d) also indicate that when r 0 is much less than the wavelength of the incident electromagnetic wave, the quasi-static approximation gives quite accurate results with respect to the full-wave results.
Fig. 2. (Color online) Contour plots and the cross section diagrams of the SCS for (a) r 0 ¼ 0:1, (b) the full-wave condition, and (c), (d) quasi-static approximation versus the frequency ω of the incident electromagnetic wave and the gradient parameter c1 . Darker regions in the plots correspond to lower SCS.
L. Sun and K. W. Yu
Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B
999
Fig. 3. (Color online) Similar to Fig. 2, but for (a) r 0 ¼ 1:0, (b) the full-wave condition, and (c), (d) quasi-static approximation.
A further increase of r 0 is studied and the results are shown in Fig. 3 for r 0 ¼ 1:0. At this point, the size of the graded isotropic sphere is comparable to the wavelength of the incident electromagnetic wave, but the broadband electromagnetic transparency is still visible in the full-wave [Figs. 3(a) and 3(b)] and the quasi-static results [Figs. 3(c) and 3(d)]. In addition, it is clear that the quasi-static approximation becomes poor compared with the full-wave results while the wavelength is comparable with the size of the graded sphere. To emphasize the importance of the graded electric permittivity profile, Fig. 4 displays the full-wave SCS of a homogenous isotropic sphere with radius r 0 ¼ 1:0, directly calculated by Mie theory. In addition, the electric permittivity of this sphere is described by lossless Drude model as εg ðωÞ ¼ 1 − c0 · ω2p ð0Þ=ω2 ;
ð54Þ
where the parameter c0 still equals 1=2. Compared with the results in Fig. 3(b), the homogenous isotropic sphere without the graded electric permittivity profile obviously gets a higher SCS, which is around the order of 10−2 . Furthermore, there is one more thing that should be pointed out in Fig. 4. The vertical dashed p line ffiffiffi in Fig. 4 denotes the frequency position ω=ωp ð0Þ ¼ 1= 2, where εg ðωÞ ¼ 0. This is a critical point be-
Fig. 4. (Color online) Full-wave SCS of a homogenous isotropic sphere.
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J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011
cause the electromagnetic wave is totally forbidden in an object with zero electric permittivity, thus, the SCS gets its pffiffiffimaximum here. Moreover, while ω=ωp ð0Þ is less than 1= 2, the electric permittivity εg ðωÞ is negative, thus, the homogenous isotropic sphere is metalliclike, while the homogenous isotropic sphere is dielectriclike.
4. DISCUSSION AND CONCLUSIONS To conclude, we have extended the original Mie theory to study the electromagnetic scattering by a graded isotropic spherical structure. By using the generalized Mie theory, we have analytically and numerically investigated the scattering electromagnetic wave of the graded isotropic sphere whose electric permittivity is characterized by the graded Drude model. The results indicate that while the full-wave SCS depends on both the frequency ratio and the graded structure profile, the full-wave SCS can achieve extremely small values over a broad frequency band. That is to say, it is possible to achieve a small full-wave SCS over a broadband by using graded spherical structures. Regarding further work on electromagnetic transparency, it is valuable to extend the broadband transparency research to anisotropic graded media. According to the results of former research about the perfect electromagnetic transparency and cloaking, we believe bringing anisotropic structure into the graded media may lead to a better transparency over a broad frequency range. In addition, based on our research on composite materials [33] and recent research on the optical negative refraction in ferrofluids [34], we may find a proper way to achieve the broadband transparency practically. In addition, loss is an essential physical parameter in characterizing the properties of materials and plays an important role in the electromagnetic behavior of materials, especially in that of metamaterials, because loss is a key parameter for controlling the electromagnetic resonance, and the resonance is the origin of the extraordinary properties of metamaterials. According to [6], in an extremely high loss, the structure does not possess transparency characteristics due to large material absorption. However, the main propose of our article is to demonstrate broadband electromagnetic transparency by the scattering cancellation scheme. In order to emphasize the physical mechanism and simplify the analytical calculation, we ignored the loss of the graded spherical structure. However, for a moderate loss, e.g., γ ¼ 0:1ωp , the transparency is still present. In summary, it is worth mentioning that the scattering cancellation mechanism belongs to the passive transparency and cloaking technique in which the local-responding material is used. As the local-responding material can never provide a perfect invisibility in a wide frequency range, an active exterior cloaking mechanism is proposed and studied by other groups [35–37]. In this mechanism, three or more active devices can create a region, in which the amplitude of the electromagnetic wave is extremely small, but does not disturb the propagation of the probing electromagnetic wave. Therefore, objects placed in this region are virtually invisible. Furthermore, this technique is demonstrated by computer simulations with a broadband incident pulse. Regarding the advantage of this mechanism, we may introduce it into our research in order to achieve a perfect broadband electromagnetic transparency and cloaking.
L. Sun and K. W. Yu
ACKNOWLEDGMENTS This work is supported by the General Research Fund of the Hong Kong SAR Government. Meanwhile, we give our acknowledgment to J. J. Xiao and M. J. Zheng for their useful suggestions on our manuscript.
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