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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 20518–20530
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Cloaking a sensor for three-dimensional Maxwell’s equations: transformation optics approach

Xudong Chen and Gunther Uhlmann  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 20518-20530 (2011)
http://dx.doi.org/10.1364/OE.19.020518


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Abstract

The ideal transformation optics cloaking is accompanied by shielding: external observations do not provide any indication of the presence of a cloaked object, nor is any information about the fields outside detectable inside the cloaked region. In this paper, a transformation is proposed to cloak three-dimensional objects for electromagnetic waves in sensor mode, i.e., cloaking accompanied by degraded shielding. The proposed transformation tackles the difficulty caused by the fact that the lowest multipole in three-dimensional electromagnetic radiation is dipole rather than monopole. The loss of the surface impedance of the sensor plays an important role in determining the cloaking modes: ideal cloaking, sensor cloaking and resonance.

© 2011 OSA

1. Introduction

In the past few years, there have been attempts to design sensors that are able to perceive electromagnetic radiation and at the same time have negligible disturbance to the surrounding environment, which is referred to as ‘cloaking sensors’ or ‘cloak in sensor mode’. The first paper in this direction is [10

10. A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009). [CrossRef] [PubMed]

], where a plasmonic coating allows electromagnetic wave to reach the sensor and at the same time the scattering due to the sensor is canceled out by that of the cloak. The idea has been applied to cloak a NSOM tip for the purpose of near field imaging [11

11. A. Alù and N. Engheta, “Cloaked near-field scanning optical microscope tip for noninvasive near-field imaging,” Phys. Rev. Lett. 105, 263906 (2010). [CrossRef]

], which is of great industrial and scientific significance. However, there are also some limitations of the method. For example, it can only cloak sensors smaller or comparable to wavelength and the shape and material of the cloak depend on the geometrical and physical properties of the sensor. Another approach to cloak a sensor is transformation optics. In [12

12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011). [CrossRef]

], a sensor that is cladded with a sphere with surface impedance is placed inside the cloak. This method works for sensors of arbitrary size and shape as long as the sensors are able to measure the field at the boundary of the cladding sphere. As mentioned in [12

12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011). [CrossRef]

], this transformation optics approach is able to achieve in cloaking a sensor for acoustic waves and two-dimensional (2D) electromagnetic waves. However, a further calculation shows that the transformation optics approach proposed in [12

12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011). [CrossRef]

] cannot cloak a sensor for three-dimensional (3D) electromagnetic waves. This is mainly due to the fact that the lowest multipole in 3D electromagnetic radiation is dipole rather than monopole. The asymptotic behavior, for small arguments, of Riccati-Bessel functions of order one and above leads to vanishing electromagnetic fields inside the cloak, i.e., the shielding effect is still hand in hand with the cloaking effect. Another transformation optics approach is presented in [13

13. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17, 106343 (2009).

, 14

14. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011). [CrossRef]

], where both a cloak layer and an anti-cloak layer play the role to achieve cloaking sensor effect. Whereas this approach can cloak sensors of arbitrary size and works for both 2D and 3D electromagnetic waves, the sensor is modeled as a dielectric sphere and the papers do not discuss whether the method works for a generic sensor, i.e., whether both the cloak layer and anti-cloak layer depend on the shape and material of the sensor. In addition, the cloaking device consists of five layers, eight unknowns for each order of multipole, and two small parameters to be adjusted, which makes it tedious to derive analytical solutions.

In this paper, we propose a novel transformation optics method to cloak a sensor, which can be of arbitrary size, shape, and material, for 3D electromagnetic waves. The cloaking device consists in three layers, five unknowns for each order of multipole, and one small parameter to be adjusted. Analytical results can be obtained for each of five unknowns even without resource to small parameter approximation. Compared with [12

12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011). [CrossRef]

], the proposed transformation, from the physical space to the virtual space, circumvents the difficulty caused by the fact that the lowest multipole in 3D electromagnetic radiation is dipole rather than monopole. The loss of the surface impedance of the sensor plays an important role in determining the cloaking modes: ideal cloaking, sensor cloaking and resonance. Numerical simulations validate the proposed transformation model.

2. Configuration of the cloak and the sensor

As shown in Fig. 1. The sensor is placed inside a sphere of radius R0, with surface impedance boundary condition −Eθ/Hϕ = Eϕ/Hθ = α0 (see [15

15. Y.-L. Geng, C.-W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propagat.57, 572–576 (2009). [CrossRef]

] and references therein). The sensor is able to measure the tangential electric fields at the surface of the sphere.The cloak layer is within an annulus with inner and outer radii R1 and R2, respectively. All three spheres mentioned above are concentric, with the center being the origin of the coordinate system in the physical space. The space between spheres of radii R0 and R1, as well as the space outside of the sphere of radius R2, are free space. The permittivity and permeability of the cloak layer are obtained using the invariance of Maxwell’s equations under transformation of the spatial coordinate systems. Consider a coordinate transformation between the virtual space (curved free space) with the physical space. The former has spatial coordinates r′,θ′,ϕ′, permittivity ɛ0, and permeability μ0, whereas the latter has spatial coordinates r,θ,ϕ, and parameters ɛ̿,μ̿. The transformation is only in the radial direction, i.e.,
r=f(r),θ=θ,ϕ=ϕ.
(1)
The associated permittivity and permeability tensors are given by
ɛ¯¯=ɛr(r)r^r^+ɛt(r)θ^θ^+ɛt(r)ϕ^ϕ^
(2)
μ¯¯=μr(r)r^r^+μt(r)θ^θ^+μt(r)ϕ^ϕ^,
(3)
where
ɛr=ɛ0f2(r)r2f(r),ɛt=ɛ0f(r)
(4)
μr=μ0f2(r)r2f(r),μt=μ0f(r).
(5)

It is well known that the transform that satisfies both f(R1) = 0 and f(R2) = R2 yields perfect invisibility. To avoid singularities, we let the boundary of the inner cloaking material be at R1 + δ, where δ is a small positive number. The incident wave is generated by a time-harmonic [exp( − iωt)] source that is located outside of the sphere of radius R2. To achieve cloaking a sensor, we aim at obtaining a negligible scattered field outside of the sphere of radius R2 and at the same time perceivable electromagnetic field at the surface of the cladding sphere.

Fig. 1 The geometry of the sensor and the transformation function. (a) A sensor of arbitrary shape is placed inside a sphere with radius R0 and surface impedance α0, and it is able to measure the tangential electric field on the spherical surface. The cloak layer is in between spheres of radii R1 + δ and R2, where δ is an small positive number and is not shown in the figure due to its small magnitude. Other regions are free space. (b) The transformation r′ = f(r) from the physical space to the virtual space.

3. Analytical results for electromagnetic fields

Electromagnetic fields can be decomposed into two independent modes, TE and TM modes (transverse to radial direction in spherical coordinate system), which are dual to each other. For the TM mode, the B field can be expressed as
B=×(r^fΦM),
(6)
where the scalar potential ΦM satisfies
1f2sinθ[θ(sinθΦMθ)+1sinθ2ΦMϕ2]+2ΦMf2+k02ΦM=0,
(7)
where k02=ω2ɛ0μ0. Separation of variables yields
ΦM=n=1Nm=nnAnmB^n(k0f)Ynm(θ,ϕ),
(8)
where N is the highest order multipole used in the numerical simulations, n(z) = zbn(z) is the Riccati-Bessel function, and Ynm are spherical harmonics. Now the magnetic field H and the electric field E can be expressed as
H=μ011rsinθΦMϕθ^μ011rΦMθϕ^
(9)
E=μ01ɛ01iω[f(2ΦMf2+k02ΦM)r^+1r2ΦMfθθ^+1rsinθ2ΦMfϕϕ^].
(10)

The potential ΦM in the three regions is written as
ΦMext(r,θ,ϕ)=nm[KnmJ^n(k0r)+AnmH^n(1)(k0r)]Ynm(θ,ϕ),forr>R2
(11)
ΦMclo(r,θ,ϕ)=nm[BnmJ^n(k0f)+CnmH^n(1)(k0f)]Ynm(θ,ϕ),forR1+δ<r<R2
(12)
ΦMint(r,θ,ϕ)=nm[DnmJ^n(k0r)+EnmH^n(1)(k0r)]Ynm(θ,ϕ),forR0<r<R1+δ.
(13)
Since the source is given, Knm is uniquely determined. For each multipole of order n, we will solve for five unknows Anm, Bnm, Cnm, Dnm, and Enm. Eqs. (9) and (10) indicate that the continuities of tangential components of H and E across the boundaries r = R1 +δ and r = R2 amount to the continuities of ΦM and ΦM(k0f), respectively. The impedance boundary condition reduces to ΦM(k0f)αΦM=0, where α is proportional to α0. Thus, we have the following boundary conditions
BnmJ^n(k0R2)+CnmH^n(1)(k0R2)=KnmJ^n(k0R2)+AnmH^n(1)(k0R2)
(14)
BnmJ^n(k0R2)+CnmH^n(1)(k0R2)=KnmJ^n(k0R2)+AnmH^n(1)(k0R2)
(15)
BnmJ^n(k0f(R1+δ))+CnmH^n(1)(k0f(R1+δ))=DnmJ^n(k0(R1+δ))+EnmH^n(1)(k0(R1+δ))
(16)
BnmJ^n(k0f(R1+δ))+CnmH^n(1)(k0f(R1+δ))=DnmJ^n(k0(R1+δ))+EnmH^n(1)(k0(R1+δ))
(17)
DnmJ^n(k0R0)+EnmH^n(1)(k0R0)=α[DnmJ^n(k0R0)+EnmH^n(1)(k0R0)].
(18)

It is easy to see that Eqs. (14) and (15) yield Bnm = Knm and Anm = Cnm. There are two cases for Eq. (18).

3.1. Case 1

When Ĵn(k0R0) – αĴn(k0R0) ≠ 0, we have
Dnm=H^n(1)(k0R0)αH^n(1)(k0R0)J^n(k0R0)αJ^n(k0R0)Enmg(α)Enm
(19)

Now we solve for two unknowns Anm and Enm from two linear equations,
KnmJ^n(k0f(R1+δ))+AnmH^n(1)(k0f(R1+δ))=Enm[g(α)J^n(k0(R1+δ))+H^n(1)(k0(R1+δ))]
(20)
KnmJ^n(k0f(R1+δ))+AnmH^n(1)(k0f(R1+δ))=Enm[g(α)J^n(k0(R1+δ))+H^n(1)(k0(R1+δ))]
(21)

We obtain:
Enm/Knm=i/Fn,
(22)
Anm/Knm=Gn/Fn,
(23)
where
Fn=[gJ^n(k0(R1+δ))+H^n(1)(k0(R1+δ))]H^n(1)(k0f(R1+δ))[gJ^n(k0(R1+δ))+H^n(1)(k0(R1+δ))]H^n(1)(k0f(R1+δ))
(24)
Gn=[gJ^n(k0(R1+δ))+H^n(1)(k0(R1+δ))]J^n(1)(k0f(R1+δ))+[gJ^n(k0(R1+δ))+H^n(1)(k0(R1+δ))]J^n(k0f(R1+δ))
(25)
and the Wronskian J^nH^n(1)J^nH^n(1)=i is used.

It is worth highlighting that we have obtained analytical result for Enm without using any approximation. From here onwards, we will use the fact that δ is a small parameter to simplify the obtained analytical result. For an infinitesimal parameter z, we have the asymptotics Ĵn(z) ≈ pnzn+1 and H^n(1)(z)qnzn [16

16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

]. Using Taylor’s expansion and the fact that f(R1) = 0, we obtain
Fn{gJ^n(k0R1)+H^n(1)(k0R1)+k0δ[gJ^n(k0R1)+H^n(1)(k0R1)]+(k0δ)22[gJ^n(k0R1)+H^n(1)(k0R1)]}×(nqn)[k0f(R1+δ)](n+1){gJ^n(k0R1)+H^n(1)(k0R1)+k0δ[gJ^n(k0R1)+H^n(1)(k0R1)]+O(δ2)}qn[k0f(R1+δ)]n=qn[k0f(R1+δ)](n+1)(A1+A2+A3+A4+A5+A6),
(26)
where
A1=n[gJ^n(k0R1)+H^n(1)(k0R1)],
(27)
A2=nk0δ[gJ^n(k0R1)+H^n(1)(k0R1)],
(28)
A3=k0f(R1+δ)[gJ^n(k0R1)+H^n(1)(k0R1)],
(29)
A4=k0f(R1+δ)k0δ[gJ^n(k0R1)+H^n(1)(k0R1)],
(30)
A5=n(k0δ)22[gJ^n(k0R1)+H^n(1)(k0R1)],
(31)
A6=O(δ2)f(R1+δ),
(32)
where the Landau big-o notation O(·) denotes terms of the same order, i.e., neglecting constant multipliers and higher-order terms. Depending on whether or not the leading term in Fn is zero, we discuss the two following two cases.

3.1.1. Case 1.1

When gJ^n(k0R1)+H^n1(k0R1)0,
EnmKnm=iqn1[k0f(R1+δ)]n+1A11.
(33)
To further quantify the asymptotic behavior of Enm, we can express the transform function f(R1 + δ) as, considering the fact that f(R1) = 0 and δ is small,
f(R1+δ)=βδs+o(δs),
(34)
for some β and s, where s > 0. Note that the Landau little-o notation o(·) denotes higher order terms, i.e., those approaching to zero at faster rates. Thus,
EnmKnm=O(δ(n+1)s).
(35)
From Eq. (23), a straightforward calculation gives that Anm = (Oδ(2n+1)s). Since n ≥ 1, we see both Enm and Anm approach zero as δ approaches zero, indicating that cloaking effect is coupled with shielding effect, i.e., there is no sensor effect.

3.1.2. Case 1.2

When gJ^n(k0R1)+H^n(1)(k0R1)=0, the Wronskian J^nH^n(1)J^nH^n(1)=i implies that gJ^n(k0R1)+H^n(1)(k0R1)0. It is also important to stress that the differentiation of the Wronskian with respect to k0R1 yields gJ^n(k0R1)+H^n(1)(k0R1)=0. Subsequently, we see from Eq. (28) to Eq. (31) that, A2 ≠ 0, A3 ≠ 0, and A4 = A5 = 0. Due to the presence of a nonzero A3, Fn contains a term of order O(δns), which is infinite and eventually leads to zero value of Enm as δ goes to zero. In this case, there is no sensing effect. To achieve a non-vanishing Enm, we have to eliminate A3 and the only way is to use A2 to cancel it.

The condition A2 + A3 = o(δs) can be satisfied when −nδβδs = 0, i.e.,
s=1,β=n.
(36)
The condition that gJ^n(k0R1)+H^n(1)(k0R1)=0 and Eq. (36) can be satisfied by different order multipoles. It is important to note that once the aforementioned two conditions are satisfied for a particular order, say n0, they cannot be simultaneously satisfied by others orders. For n = n0, we easily obtain that Enm = O(δn0–2) and Anm = O(δ2n0–1); For nn0, we obtain from Case 1.1 that Enm = O(δn+1) and Anm = O(δ2n+1). The behaviors of cloaking and shielding effect are different for various order of multipoles.

  • Case of n0 = 1: E1m = O(δ−1) approaches infinity and A1m = O(δ1) approaches zero, which means the resonance mode. It is worth mentioning that the interior resonance does not destroy the cloaking effect, which is different from the conclusion of [12

    12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011). [CrossRef]

    ]. This difference is possibly due to the fact that [12

    12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011). [CrossRef]

    ] uses the Cauchy data, i.e., the mapping from the total field to its normal derivative, whereas this paper studies the mapping from the incidence field to the scattered one.
  • Case of n0 = 2: E2m = O(δ0) is in the same order as the incidence wave and A2m = O(δ3) approaches zero, which means sensor mode.
  • Case of n0 ≥ 3: Both Enm and Anm approach zero, which means ideal cloaking mode, i.e., cloaking effect and shielding effect are hand in hand.

From the condition of gJ^n(k0R1)+H^n(1)(k0R1)=0 and the definition of g(α) in Eq. (19), we easily obtain the value of α that leads to the resonance mode (for n0 = 1) and the sensor mode (for n0 = 2),
α=J^n0(k0R0)Y^n0(k0R0)J^n0(k0R1)/Y^n0(k0R1)J^n0(k0R0)Y^n0(k0R0)J^n0(k0R1)/Y^n0(k0R1).
(37)

3.2. Case 2

When Ĵn(k0R0) – αĴn(k0R0) = 0, we find that Enm = 0 and α = Ĵn(k0R0)/Ĵn(k0R0). We carry out an analysis similar to that of Section 3.1, simply replacing [gJ^n(k0R1)+H^n(1)(k0R1)]Enm by Ĵn(k0R1)Dnm. Thus, the necessary conditions for an electromagnetic wave to penetrate the inner boundary of the cloak are Ĵn(k0R1) = 0 and f(R1 + δ) = –nδ + o(δ) for a particular n0. The resonance mode (for n0 = 1) and the sensor mode (for n0 = 2) are the same as those discussed in Section 3.1.

3.3. Removal of singularity

As shown in Section 3.1 and Section 3.2, one of the conditions for achieving the sensor mode is f(R1 + δ) = −nδ + o(δ). We note that the condition of zero scattering outside of the cloak requires that f(R2) = R2. Thus, if r′ = f(r) is a continuous function, inevitably there is a particular value of r, say Rb, for which r′ = 0. Consequently, from Eq. (12) we conclude that a singularity appears in the cloak layer since the argument for Hn(1)() is zero. To remove such a singularity, we let the function f to be discontinuous at r = Rb, but at the same time, we keep the continuity of the permittivity and permeability. With this purpose, we see from Eqs. (4) and (5) that a function f(r) that simultaneously satisfies f(Rb+)=f(Rb)>0 and f(Rb+)=f(Rb) can achieve both the continuity of permittivity and permeability and the removal of singularity. One example of such f is depicted in Fig. 1.

It is interesting to discuss the meaning of the function r′ = f(r) for the case r′ < 0. In deriving the permittivity and permeability in the physical space (Eqs. (4) and (5)), the key step is to use x′/x = y/y = z′/z = r′/r [3

3. Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008). [CrossRef]

, 17

17. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006). [CrossRef] [PubMed]

]. When r′ < 0, a point in the physical space and the corresponding point in the virtual space are on the opposite side of the origin. The invisibility conditions f(R1) = 0 and f(R2) = R2 indicate that as we move from the inner boundary of the cloak to the outer one in the physical space, the corresponding point in the virtual space moves from the original to r′ = R2. However, since the virtual space is free space, there is no scattering at all. Consequently, it does not matter in which way we move from the original to r′ = R2 in the virtual space. The linear transform that is presented in [1

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

, 17

17. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006). [CrossRef] [PubMed]

] indicates a uniform speed displacement. The concentrator transform that is presented in [3

3. Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008). [CrossRef]

] indicates a displacement all the way outside of the outer boundary of the cloak, followed by a backward displacement to come back to the outer boundary. Here, in this paper, our transform function shows a displacement toward the opposite direction, followed a directional change and then a all-the-way displacement towards the outer boundary of the cloak.

4. Numerical simulations

Following the criteria that were presented in Section 3, we proposed a particular transformation, as depicted in Fig. 1(b), to test the performance of the cloaking device. We choose the following parameters: R0 = 0.5λ, R1 = 1.5λ, R2 = 3.0λ, Ra = 2.0λ, Rb = 2.33λ, Rc = 2.66λ, and h = 0.15λ. The transform function in the range of R1 + δrR2 is given by
f(r)={n0r+n0R1forR1+δr<Rahn0(Ra+R1)RbRa(rRa)+n0(Ra+R1)forRar<Rbhn0(Ra+R1)RbRa(rRb)+hforRb<rRcR2n0(RaR1)R2Rc(rRc)+n0(RaR1)forRc<rR2.
(38)
We observe that the values of ɛ̿ and μ̿ are both negative and finite in the region R1 + δr < Ra, and the radial components approach zero at the boundary r = R1 + δ. In other regions, both ɛ̿ and μ̿ are positive and finite. It is important to stress that although four transformations are used in the region of R1 + δrR2, the potential, and consequently the electromagnetic field, is expressed in a single formula, Eq. (12). The surface impedance α is calculated through Eq. (37).

We consider an x-polarized plane wave with a unit amplitude Einc = x̂eik0z is incident upon the cloaking device along the z direction. Since the given incidence wave contains only the |m| = 1 term, from here onwards, we drop off the m in the subscript for simplicity. For example, the coefficient Knm is written as Kn. It is well known that Kn is proportional to in(2n + 1)/[n(n +1)] for both TE and TM components of the incident plane wave [3

3. Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008). [CrossRef]

]. We aim at examining the cloaking effect and penetrating effect of the cloaking device as the parameter δ approaches zero. The cloaking effect is quantified by |An/Kn|. The lower the value of |An/Kn|, the better the cloaking effect. Since Eq. (19) shows that En/Dn is independent of δ, we can use only |Dn/Kn| to quantify the penetrating effect. The lower the value of |Dn/Kn|, the poorer the penetrating ability. The case of |Dn/Kn| ≫ 1 corresponds to the resonance effect.

First, we let n0 = 1. The quantities of |An/Kn| and |Dn/Kn| for different values of the small parameter δ and order number n are shown in Fig. 2. We see from Fig. 2(a) that the cloaking effect applies to all orders, including n = n0 = 1. In addition, the slopes of the curves, which are in the logarithmic scale, agree with the theory presented in Section 3. That is to say, whenever δ is decreased by a factor 10, the values of |An/Kn| decreases by a factor of 10, 105, and 107 for n = 1, n = 2, and n = 3, respectively. We see from Fig. 2(b) that the resonance effect applies to the order n = n0 = 1 and the shielding effect applies to n = 2 and n = 3. In addition, the slopes of the curves agree with the theory, i.e., whenever δ is decreased by a factor 10, the values of |Dn/Kn| increase by a factor of 10 for n = n0 = 1 and decreases by a factor of 103 and 104 for n = 2 and n = 3, respectively. To summarize, for n0 = 1, the wave component corresponding to n = n0 yields resonance effect without external scattering, whereas the wave component corresponding to nn0 simultaneously yields cloaking and shielding effect.

Fig. 2 For n0 = 1, the cloaking and penetrating effects for different orders of multiples in the limit of δ approaching zero. (a) The cloaking effect. (b) Penetrating effect.

Since the sensor surface will result in some energy loss, we are interested in analyzing how this loss affects the cloaking and penetrating. To this purpose, we calculate |An/Kn| and |Dn/Kn| in the case when the real surface impedance α is replaced by a complex one α(1 + iLt), where Lt denotes loss tangent. Fig. 3 depicts the values of |An/Kn| and |Dn/Kn| for δ = 10−3 for the range of loss tangent from 10−10 to 10−1. Fig. 3(a) and (b) show that the loss enhances the cloaking effect and at the same time decreases the penetrating effect for the order n = n0 = 1, but it barely affects the cloaking or penetrating effect for other orders. We see from Fig. 3(b) that for n = n0 = 1, as the loss decreases, the penetrating ability changes from the shielding mode (i.e., ideal cloaking mode) to the sensor mode and then to the resonance mode. It is interesting to see that a moderate loss, such as loss tangent of 10−4, is able to enhance the cloaking effect and at the same time to shift the resonance mode to the sensor mode. In Fig. 4, we plot the x component of the electric field in the xz plane for the loss tangent of 10−2, 10−4, and 10−7, corresponding to the shielding (i.e., ideal cloaking), sensor, and resonance mode, respectively.In comparison,when only the sensor exists, without the presence of the outer cloaking layer, the distribution of the electric field is plotted in Fig. 5, where we see that the distorted field pattern indicates the presence of a scatterer.

Fig. 3 For n0 = 1 and δ = 10−3, the effect of surface impedance loss on the cloaking and penetrating effects. (a) The cloaking effect. (b) Penetrating effect, where the horizontal dotted line denotes the sensor mode.
Fig. 4 For n0 = 1, the x component of the electric field in the xz plane for the loss tangent of (a) 10−2, (b) 10−4, and (c) 10−7, corresponding to the ideal cloaking, sensor, and resonance mode, respectively.
Fig. 5 The x component of the electric field in the xz plane for the case when only the sensor exists, without the presence of the outer cloaking layer.

Next, for n0 = 2, we also analyze the cloaking and penetrating effects. We see from Fig. 6(a) that the cloaking effect applies to all orders, including n = n0 = 2 and the slopes of the curves agree with the theories, i.e., whenever δ is decreased by a factor 10, the values of |An/Kn| decreases by a factor of 103, 103, and 107 for n = 1, n = 2, and n = 3, respectively. We see from Fig. 6(b) that the sensor mode applies to the order n = n0 = 2 and the shielding effect applies to n = 1 and n = 3. The slopes of the curves agree with the theory, i.e., whenever δ is decreased by a factor 10, the values of |Dn/Kn| keeps the same order for n = n0 = 2 and decreases by a factor of 102 and 104 for n = 1 and n = 3, respectively. To summarize, for n0 = 2, the wave component corresponding to n = n0 yields sensor effect without external scattering, whereas the wave component corresponding to nn0 simultaneously yields cloaking and shielding effects. Fig. 7 depicts the values of |An/Kn| and |Dn/Kn| for δ = 10−3 for presence of loss in the surface impedance. We observe that the loss enhances the cloaking effect and in the meanwhile decrease the penetrating effect for the order n = n0 = 2, but it barely affects the cloaking or penetrating effect for other orders. We see from Fig. 7(b) that for n = n0 = 2, as the loss decreases, the penetrating ability changes from the shielding mode to the sensor mode. In Fig. 8, we plot the x component of the electric field in the xz plane for the loss tangent of 10−4 and 10−7, corresponding to the shielding and the sensor mode, respectively. In practice, for n = n0 = 2, the sensor mode can easily be destroyed by the loss. For example, in Fig. 8 a loss tangent of 10−4 is able to shift the sensor mode to the shielding mode.

Fig. 6 For n0 = 2, the cloaking and penetrating effects for different orders of multiples in the limit of δ approaching zero. (a) The cloaking effect. (b) Penetrating effect.
Fig. 7 For n0 = 2 and δ = 10−3, the effect of surface impedance loss on the cloaking and penetrating effects. (a) The cloaking effect. (b) Penetrating effect, where the horizontal dotted line denotes the sensor mode.
Fig. 8 For n0 = 2, the x component of the electric field in the xz plane for the loss tangent of (a) 10−4 and (b) 10−7, corresponding to the ideal cloaking and sensor mode, respectively.

5. Conclusion

The ideal transformation optics cloaking is accompanied by shielding: external observations do not provide any indication of the presence of a cloaked object, nor is any information about the fields outside detectable inside the cloaked region. In this paper, we proposed a transformation that cloaks three-dimensional objects for electromagnetic waves in sensor mode, i.e., cloaking accompanied by degraded shielding. The proposed transformation tackles the difficulty caused by the fact that the lowest multipole in three-dimensional electromagnetic radiation is dipole rather than monopole. It is worth emphasizing that we have obtained the analytical solution to the electromagnetic fields in each region of the whole space. We find that for the dipole term (n=1), we are able to achieve resonance mode without external scattering. For the quadrupole term (n=2), we are able to achieve sensor mode without external scattering. However, the loss that is presented on the surface of the sensor degrades the penetrating effect so that a moderate loss is able to shift the resonance mode to the sensor mode for n = 1 and to shift the sensor mode to the ideal cloaking mode for n = 2. Thus, in real world applications, where loss is present on the surface of sensor, it is more desirable to achieve the sensor mode using the dipole term.

Acknowledgments

The author Chen acknowledges the financial support from the Singapore Temasek Defence Systems Institute under grant TDSI/09001/1A.

References and links

1.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

2.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure. 24, 413–419 (2003). [CrossRef]

3.

Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008). [CrossRef]

4.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275, 749–789 (2007). [CrossRef]

5.

B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. 100, 063904 (2008). [CrossRef] [PubMed]

6.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010). [CrossRef] [PubMed]

7.

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16, 14603–14608 (2008). [CrossRef] [PubMed]

8.

B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011). [CrossRef] [PubMed]

9.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008). [CrossRef]

10.

A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009). [CrossRef] [PubMed]

11.

A. Alù and N. Engheta, “Cloaked near-field scanning optical microscope tip for noninvasive near-field imaging,” Phys. Rev. Lett. 105, 263906 (2010). [CrossRef]

12.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011). [CrossRef]

13.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17, 106343 (2009).

14.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011). [CrossRef]

15.

Y.-L. Geng, C.-W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propagat.57, 572–576 (2009). [CrossRef]

16.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

17.

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006). [CrossRef] [PubMed]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(260.2110) Physical optics : Electromagnetic optics
(290.3200) Scattering : Inverse scattering
(260.2710) Physical optics : Inhomogeneous optical media

ToC Category:
Physical Optics

History
Original Manuscript: July 27, 2011
Revised Manuscript: September 7, 2011
Manuscript Accepted: September 9, 2011
Published: October 3, 2011

Citation
Xudong Chen and Gunther Uhlmann, "Cloaking a sensor for three-dimensional Maxwell’s equations: transformation optics approach," Opt. Express 19, 20518-20530 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20518


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References

  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
  2. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure.24, 413–419 (2003). [CrossRef]
  3. Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B77, 125127 (2008). [CrossRef]
  4. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys.275, 749–789 (2007). [CrossRef]
  5. B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett.100, 063904 (2008). [CrossRef] [PubMed]
  6. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9, 387–396 (2010). [CrossRef] [PubMed]
  7. H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express16, 14603–14608 (2008). [CrossRef] [PubMed]
  8. B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106, 033901 (2011). [CrossRef] [PubMed]
  9. W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008). [CrossRef]
  10. A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett.102, 233901 (2009). [CrossRef] [PubMed]
  11. A. Alù and N. Engheta, “Cloaked near-field scanning optical microscope tip for noninvasive near-field imaging,” Phys. Rev. Lett.105, 263906 (2010). [CrossRef]
  12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E83, 016603 (2011). [CrossRef]
  13. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express17, 106343 (2009).
  14. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48, 455–467 (2011). [CrossRef]
  15. Y.-L. Geng, C.-W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propagat.57, 572–576 (2009). [CrossRef]
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).
  17. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14, 9794–9804 (2006). [CrossRef] [PubMed]

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