Uniqueness Theorem and Uniqueness of Inverse Problems for Lossy Anisotropic Inhomogeneous Structures with Diagonal Material Tensors Reza Dehbashi, Konstanty S

Uniqueness theorem and uniqueness of inverse problems for lossy anisotropic inhomogeneous structures with diagonal material tensors Reza Dehbashi, Konstanty S. Bialkowski, and Amin M. Abbosh

Citation: Journal of Applied Physics 121, 203103 (2017); doi: 10.1063/1.4983768 View online: https://doi.org/10.1063/1.4983768 View Table of Contents: http://aip.scitation.org/toc/jap/121/20 Published by the American Institute of Physics

Articles you may be interested in Photoconductivity induced by nanoparticle segregated grain-boundary in spark plasma sintered BiFeO3 Journal of Applied Physics 121, 203102 (2017); 10.1063/1.4983764

W/Cu thin film infrared reflector for TiNxOy based selective solar absorber with high thermal stability Journal of Applied Physics 121, 203101 (2017); 10.1063/1.4983763

Dynamic susceptibility of concentric permalloy rings with opposite chirality vortices Journal of Applied Physics 121, 203901 (2017); 10.1063/1.4983759

Practical considerations in the modeling of field emitter arrays with line charge distributions Journal of Applied Physics 121, 203303 (2017); 10.1063/1.4983680

Vertical transport in isotype InAlN/GaN dipole induced diodes grown by molecular beam epitaxy Journal of Applied Physics 121, 205702 (2017); 10.1063/1.4983767

Fluid simulation of species concentrations in capacitively coupled N2/Ar plasmas: Effect of gas proportion Journal of Applied Physics 121, 203302 (2017); 10.1063/1.4983675 JOURNAL OF APPLIED PHYSICS 121, 203103 (2017)

Uniqueness theorem and uniqueness of inverse problems for lossy anisotropic inhomogeneous structures with diagonal material tensors Reza Dehbashi,a) Konstanty S. Bialkowski, and Amin M. Abbosh School of Information Technology and Electrical Engineering, University of Queensland, St. Lucia, Brisbane, QLD 4067, Australia (Received 3 February 2017; accepted 7 May 2017; published online 22 May 2017) The uniqueness theorem for lossy anisotropic inhomogeneous structures with diagonal material tensors is proven. For these materials, we prove that all the elements of the constitutive tensors must be lossy. Materials like cloaks and lenses designed based on transformation-optics (TO) could be examples of such materials. The uniqueness theorem is about the uniqueness of Maxwell’s equations solutions for particular sets of boundary conditions. We prove the uniqueness theorem for three cases: Single medium, media composed of two lossy anisotropic inhomogeneous materials with diagonal constitutive parameters, and media composed of two materials, where a material with diagonal material tensors is surrounded by an isotropic material. The latter case can be considered for the TO-based materials like invisibility cloaks or hyper-lenses that usually have diagonal anisotropic inhomogeneous constitutive parameters and also because cloaks or hyper-lenses are usually surrounded by free space and the sources are usually outside. For the sake of our argument in the uniqueness theorem that loss is the main condition for the validity of this theory, for cloaks as an example case of our analysis, it is analytically and numerically proven that the ideal invisibility phenomenon is possible for a simple lossy structure. We also examine the uniqueness of the inverse problem for such structures. We prove that all these materials have the same surface ﬁeld distribution on a surface enclosing the area of interest, while solutions to Maxwell’s equations inside them are different. The uniqueness of the inverse problem suggests that within the surface, the same medium should exactly be present. However, for lossy anisotropic inhomogeneous structures with diagonal constitutive parameters, this paper illustrates that this might not be true, despite the result of a previous study that shows that uniqueness could be true for some anisotropic inhomogeneous structures. For the analysis, the transverse electric Z-polarization is used. The simulation results are obtained by using a commercial Finite-Element based simulator. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4983768]

I. INTRODUCTION The uniqueness theorem is an important theory in electromagnetics and is the basis of some other important electro- Transformation optics (TO) was introduced by Pendry magnetics theories like the induction theorem, equivalence et al.1 and later used to design different shapes of invisibility principle, image theory, and conformal mapping theory. The cloaks.1–6 According to that method, an enclosed area can be investigated uniqueness theorem is about the uniqueness of transformed into a shell7–9 that can be used to conceal any the solutions of Maxwell’s equations in a medium with a par- object with any shape and material.1 The method has also been used to design other novel devices like light concentra- ticular boundary condition. Most electromagnetics textbooks 10 11 12,13 have introduced this theorem for lossy isotropic homogeneous tors, lenses, gradient-index bends and waveguides, 28 black holes,14 and illusion devices.15 Both simulation16 and mediums. However, this theorem has not yet been directly measurement results17 show the effectiveness of transforma- proven for anisotropic inhomogeneous with diagonal tensor tion optics to design invisibility cloaks. The method intro- constitutive parameters. Most TO based materials have such duced in Ref. 1 is not the only way to design invisibility structures, and invisibility cloaking is one of them. Even the cloaks. Cloaking can also be designed using the TO of a newer version of TO, called Quasi-Conformal Transformation curved, non-Euclidean space,18 the transmission-line-method Optics (QCTO), usually creates materials with diagonal con- cloaking,19,20 the external cloaking based on the comple- stitutive tensors in various applications including the antenna 29–31 mentary media concept,21 and the cloaking method to con- applications. The QCTO uses a smoother transformation ceal sub-wavelength objects.22,23 Metamaterials are the key function to limit anisotropy of the structures. There is a con- to implement such methods in reality not only to manipu- cern regarding the validity of the uniqueness theorem for late the electromagnetic waves17,22–26 but also to manipu- structures like invisibility cloaks, as these anisotropic inhomo- late electron and matter waves in the area of quantum geneous media may support sets of distinct solutions having mechanics.27 identical boundary conditions; hence, it might be concluded that the uniqueness theorem may not be valid for such anisotropic inhomogeneous media with diagonal constitutive a)E-mail: [email protected]. Telephone: þ61(7)33658316. tensors. However, this notion is a misunderstanding of

0021-8979/2017/121(20)/203103/8/$30.00 121, 203103-1 Published by AIP Publishing. 203103-2 Dehbashi, Bialkowski, and Abbosh J. Appl. Phys. 121, 203103 (2017) uniqueness in an inverse problem.32–34 It is noted that con- with an anisotropic inhomogeneous medium with diagonal cepts of the uniqueness of solutions of Maxwell equations and tensors. The permittivity and permeability tensors for this the uniqueness for inverse scattering problems are different. medium at the position r are eðrÞ and lðrÞ, respectively. The focus of the ﬁrst part of this paper is on the former one. The current densities J i and Mi are the electric and magnetic Using the procedures used to prove the uniqueness of the solu- sources within this volume, respectively. tions of the Maxwell equations in a lossy homogeneous We assume two distinct solutions of Maxwell’s equa- a a medium,28 we prove the uniqueness theorem for an aniso- tions inside the volume V. These are denoted by ðE ; H Þ b b tropic inhomogeneous medium with diagonal constitutive and ðE ; H Þ. Thus, they satisfy the following: tensors. a a The focus of the second part of this paper is on the r E ¼ Mi þ jxlðrÞH uniqueness of the inverse problem for TO based materials, a a (1) rH ¼ J þ jxeðrÞE ; like invisibility cloaks which usually have anisotropic inho- i mogeneous constitutive parameters. We show that despite and the validity of the uniqueness theorem for the structure, the uniqueness of the inverse problem for the structure is not b b 30 r E ¼ Mi þ jxlðrÞH valid, even though it is valid for a particular anisotropic (2) b b inhomogeneous structure. rH ¼ J i þ jxeðrÞE :

II. UNIQUENESS THEOREM FOR LOSSY For instance, the electric field Ea can be given by ANISOTROPIC INHOMOGENEOUS STRUCTURES WITH DIAGONAL MATERIAL TENSOR STRUCTURES a a a a E ¼ E1a1 þ E2a2 þ E3a3: (3) For the lossy homogeneous medium, the uniqueness of the solutions of Maxwell’s equations was proven.28 In con- Note that the unit vectors ai appearing in (3) form a trast, this paper focuses on TO based structures with lossy right-handed orthonormal basis at every position specified anisotropic inhomogeneous structures with diagonal material by r. Similarly, other vector fields can be cast in the above tensors. Therefore, the uniqueness is investigated for aniso- form. In our analysis, the eðrÞ and lðrÞ values are assumed tropic inhomogeneous structures with diagonal constitutive to be diagonal, i.e., tensors. It is proven for three cases; one of them (case c) can 2 3 0 00 be considered a cloaking structure. e1 je1 00 6 0 00 7 eðrÞ¼4 0 e2 je2 0 5; A. Single medium 00e0 je00 2 3 3 3 The uniqueness theorem is applied to invisibility cloaks 0 00 l1 jl1 00 1 6 7 designed through the transformation-optics method. In other lðrÞ¼4 0 l0 jl00 0 5; words, the uniqueness theorem is presented for anisotropic 2 2 00l0 jl00 inhomogeneous structures with diagonal constitutive tensors. 3 3 00 00 Figure 1 depicts a volume V enclosed by surface S and filled ei 6¼ 0; li 6¼ 0; (4)

0 00 0 00 where ðei jei Þ and ðli jli Þ are the principal permittivity and permeability along the i–th principal axis, respectively. Subtracting (1) from (2), we arrive at

r dE ¼ jxlðrÞdH (5) rdH ¼ jxeðrÞdE;

where

a b a b dE ¼ E E ; dH ¼ H H : (6)

Therefore, like the case of isotropic homogeneous media, the difference ﬁelds dE and dH satisfy the source- free Maxwell’s equations within V. Now, the conservation of energy is applied to obtain ððð ͜ FIG. 1. Volume V enclosed by the surface S and ﬁlled by an anisotropic dE dH ds þ dE ðjxeðrÞÞ dE S inhomogeneous medium with diagonal tensors of eðrÞ and lðrÞ at location V r. Sources within V are the electric and magnetic current densities J and i d xl d 0: (7) Mi, respectively. þ H j ðrÞ H dv ¼ 203103-3 Dehbashi, Bialkowski, and Abbosh J. Appl. Phys. 121, 203103 (2017)

Integrating dE dH over the outer surface of V,we obtain

͜ dE dH ands ¼ 0 (8) S or equivalently

͜ðan dEÞdH ds ¼ ͜ðdH anÞdEds¼ 0: (9) S S

Since the tangential components of Ea, Eb, Ha, and Hb must satisfy certain boundary conditions over S, the difference fields dE and dH must vanish over S. From (8), the volume integral in (7) is zero. Because the constitutive tensors are diagonal, we can put (7) into a summa- tion of three components. The following relation is obtained FIG. 2. Volume V1 enclosed by the surfaces S1 and S3, filled with an anisotropic inhomogeneous medium of diagonal tensors of eI ðrÞ and lI ðrÞ at by equating the real part of the volume integral to zero: location rI, and volume V2 enclosed by the surfaces S2 and S3, filled with an ððð hianisotropic inhomogeneous medium of diagonal tensors of eII ðrÞ and lII ðrÞ X3 at location rII. Sources within VI are the electric and magnetic current densi- 00 2 00 2 xei jdEij þ xli jdHij dv ¼ 0: (10) ties J i and Mi, respectively. i¼1 V ͜d d ͜d d In (10), the imaginary parts of eðrÞ and lðrÞ are EI HI ds þ EII HII ds S S assumed to be positive (non-zero) values, i.e., 1 ðð 2 ðð þ dEI dHI asds dEII dHII asds xe00 ; xl00 > 0; i ¼ 1; 2; 3: (11) i i S3 S3 ððð 2 Since the coefficients of positive values of jdEij and þ dEI ðjxeI ðrÞÞ dEI þdH jxlI ðrÞdHI dv dH 2 are also positive (as e00 and l00 are losses), the only j ij ðððV1 2 way for the above equation to be valid is that jdEij and 2 þ dEII ðjxeII ðrÞÞ dE þdH jxl ðrÞdHII dv ¼ 0; jdHij should be zero or II II II V2 dEi ¼ dHi ¼ 0; i ¼ 1; 2; 3: (12) (13) From (10), it is obvious that, if each of the tensor ele- then like Sec. II A, we obtain ments in (4) is not lossy, (12) would not be true as the coeffi- ððð X3 hi cient for the corresponding dE or dH would not be zero. i i xe00 jdE j2 þ xl00 jdH j2 dv Thus, like the homogeneous media, the uniqueness theorem Ii Ii Ii Ii i¼1 is valid for the transformation-optics based lossy invisibility V1 ððð hi cloaks, which are anisotropic inhomogeneous media with X3 xe00 dE 2 xl00 dH 2 dv 0: (14) diagonal constitutive tensors on a condition that all elements þ IIi j IIij þ IIij IIij ¼ i¼1 of the constitutive tensors must be lossy. V2 In (14), the imaginary parts of all the constitutive tensors B. Media composed of two materials are assumed to be positive (non-zero) values, i.e.,:

Figure 2 illustrates two volumes V1 and V2 enclosed by xe00 ; xl00 ; xe00 ; xl00 > 0; i ¼ 1; 2; 3: (15) surfaces (S1, S3) and (S2, S3), respectively, and both volumes Ii Ii IIi IIi are filled with anisotropic inhomogeneous media of diagonal Therefore, if all elements of the constitutive tensors are materials. The permittivity and permeability tensors for these lossy, then media are eI ðrÞ and lI ðrÞ, respectively, at the position rI, dEIi ¼ dHIi ¼ dEIIi ¼ dHIIi ¼ 0; i ¼ 1; 2; 3: (16) and eI ðrÞ and lII ðrÞ, respectively, at the position rII. The current densities J i and Mi are the electric and magnetic Hence, the uniqueness theorem for the medium shown sources within volume V1, respectively. in Fig. 2 is proven. Consider that in (14), all the components Assuming two distinct solutions of Maxwell’s equations a a of the constitutive tensors of both the volumes V1 and V2 for each of the volumes V1 and V2 denoted by ðEI ; HI Þ and must be lossy; otherwise, the corresponding component in b b a a b b ðEI ; HI Þ for volume V1 and ðEII; HIIÞ and ðEII; HIIÞ for vol- (16) for the lossless element of (15) could be non-zero, and ume V2 and following the same procedure as in Sec. II A,we consequently, the uniqueness theorem will not be valid. The obtain the following conservation of energy for this same procedure can be applied to structures composed of structure: more than two media, such as layered structures. To that 203103-4 Dehbashi, Bialkowski, and Abbosh J. Appl. Phys. 121, 203103 (2017) end, the case presented in this case can be considered an because example of a layered structure composed of different media. 00 00 00 00 xe ; xl ; xeIIi ; xlIIi > 0; i ¼ 1; 2; 3: (20) C. One material surrounded by the other material Therefore, as explained in the previous case, if all the Figure 3 illustrates two volumes V1 and V2 enclosed by elements of the constitutive tensors are lossy, then surfaces S1 and S2, respectively, and V2 is filled by anisotropic inhomogeneous media with diagonal material tensors. The per- dEI ¼ dHI ¼ dEIIi ¼ dHIIi ¼ 0; i ¼ 1; 2; 3: (21) mittivity and permeability for these media are eI ðrÞ and lI ðrÞ, respectively, at the position r ,ande and l, respectively, at the I III. PERFECT INVISIBILITY FOR A SIMPLE LOSSY position r . The current densities J and M are the electric and II i i STRUCTURE magnetic sources within volume V1, respectively. Like the other two cases, from the conservation of Section II shows that loss is the key for the validity of energy, we obtain the uniqueness theorem for anisotropic inhomogeneous ðððstructures with diagonal material tensors. Most invisibility ͜ cloaks could be considered examples of such structures, but dE dH ds þ dEI jxedEI þdHI jxldHI dv S previous results report that loss degrades the invisibility per- V1 2,35–37 ðððformance of the cloak, and so it might be inferred that invisibility cannot be possible with loss. Therefore, perfect þ dEII ðjxeII ðrÞÞ dEII þdHII jxlII ðrÞdHII dv ¼ 0; invisibility cloaks cannot be considered as an example of the V2 structure of our interest. In this section, we analytically prove (17) the possibility of perfect invisibility for a simple lossy cloak where structure, although conceptually such structures can exist if TO is applied on a lossy medium. ͜ dE dH ds ¼ 0: (18) To obtain such perfect lossy cloaks, we apply the S transformation-optics method to a lossy space with constitutive parameters e ¼ e0 je00 and l ¼ l0 jl00. The real parts Then, like before, for the real part, we have of the obtained constitutive tensors for the cloak would be ððð equal to that of a lossless cloak,2 and the imaginary parts 00 2 00 2 ðxe jdEIj þ xl jdHIj Þ dv would depend on the loss of the surrounding medium. In V other words, the loss tangents of the material parameters of 1 ððð X3 hithe obtained lossy cloak along any of the principal axis a1, 00 2 00 2 a , and a would be equal to that of the free space. We þ xeIIi jdEIIij þ xlIIijdHIIij dv ¼ 0(19) 2 3 i¼1 choose a cylindrical shape as an example shape for the pro- V2 posed lossy cloak, and the perfect concealment for that shape is proven. We consider a cylindrical cloak with the inner radius a and the outer radius b, located in the z-direction. It is created by the linear transformation function presented in Ref. 17 for a lossless cloak. The transformation function maps ðr; u; zÞ to ðr0; u0; z0Þ. To obtain the wave equation for z-polarized fields, we start from Maxwell’s equations, and after doing some mathematical manipulations, we arrive at the following Helmholtz equation:

2 0 0 2 0 0 r Ez0 ðr ; u ÞþccEz0 ðr ; u Þ¼0 r0 ¼ðr aÞ c ¼ b=ðb aÞc c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ x ðe0 je00Þðl0 jl00Þ; (22)

where cc and c are the propagation constants of the cloak and its surrounding medium, respectively. From (22), the ﬁelds in the cloak region are Bessel functions with the argument 0 ccr .

FIG. 3. Volume V1 enclosed by the surface S1 (free space), filled by an iso- Writing the field equations in each region of the cloak, tropic homogeneous medium with constitutive parameters of e and l at loca- the surrounding medium and the cloaked region, equating tion rI, and volume V2 enclosed by the surfaces S2, filled by an anisotropic the tangential components of the electric and magnetic fields biaxial inhomogeneous medium with diagonal tensors of e ðrÞ and l ðrÞ at I I in the inner and the outer boundaries of the cloak, and fol- location r II. Sources within VI are the electric and magnetic current densities J i and Mi, respectively. lowing some mathematical procedures, we obtain 203103-5 Dehbashi, Bialkowski, and Abbosh J. Appl. Phys. 121, 203103 (2017)

ð2Þ cloaks. Such materials, usually, have anisotropic inhomoge- aqHq ðcbÞbqJqðcbÞ¼JqðcbÞ (23) neous constitutive parameters.25,26 We prove that all of them ð2Þ0 c 0 c 0 c ; aqHq ð bÞbqJqð bÞ¼Jqð bÞ can have the same field distribution on their surface while the inside fields are different. Then, it might be concerning where J is the Bessel function of the first kind of order q, q that two different solutions to Maxwell’s equations can be Hð2Þ is the Bessel function of the second kind of order q, a q q achieved, while both have the same field distribution on a is the coefficient of the scattered electric field outside the surface enclosing the area of interest having different materi- cloak, and b is the coefficient of the electric field in the q als. Uniqueness suggests that these two solutions should cloak region. Like the analysis in Ref. 2 for a lossless cloak, have the exact same medium within the surface.17 Therefore, to avoid singularities at the inner boundary of the cloak, we for anisotropic inhomogeneous media of our interest, unique- apply a small perturbation to the ideal cloak by using r ¼ ness for the inverse problem is not valid. a þ d instead of r ¼ a, and then the limit d ! 0 is applied. A previous work for a particular anisotropic inhomoge- From (23), we obtain neous medium has shown that the uniqueness is valid for a 33 2 particular case of such media. However, we can show that J ðÞcb J ðÞcb H ðÞcb JqðÞcb q q q it is not valid for lossy and lossless anisotropic inhomoge- 0 0 20 0 JqðÞcb JqðÞcb Hq ðÞcb JqðÞcb neous structures of invisibility cloaks. aq ¼ bq ¼ D D We use different designs for cloaks within a certain 2 HqðÞcb JqðÞcb boundary while we prove that all have the same boundary D ¼ : (24) H20ðÞcb J0 ðÞcb conditions yet different materials and the same scattering q q response. 26 There is no singularity for aq and bq because A. Same boundary conditions for cloaks with different ðÞ2 0 0 ðÞ2 2 materials D ¼ JqðÞcb Hq ðÞcb JqðÞcb Hq ðÞcb ¼ 6¼ 0: (25) jpcb We will show why the boundary conditions for two invisibility materials are identical: From (15) and (16), we arrive at As it can be seen in Fig. 4, for an invisible volume V, a b ¼ 1(26)the tangential components of the fields over S, an E and q a an H , are equal to the tangential components of the inci- i i and dent fields an E and an H because there is no scattering from the invisible medium. If the volume V was free space, aq ¼ 0: (27) we would have exactly the same tangential components an i i The obtained result of (27) shows that there is no scat- E and an H over S. Therefore, as shown in Fig. 5, we can tering around the cloak and it is invisible. Therefore, we have enclosed regions filled with different materials but all have analytically proven the perfect concealment for an with identical boundary conditions or simply the same field example shape by a lossy cloak. distribution on their surfaces. In Fig. 5, the field distributions over S for all three cases are the same, while the fields inside IV. ON THE UNIQUENESS OF INVERSE their regions are different. The three cases are as follows: ELECTROMAGNETIC PROBLEMS FOR INVISIBILITY (a) The enclosed region S filled with an invisible medium CLOAKS and material parameters of e1ðrÞ, l1ðrÞ. It is noted that the concepts of the uniqueness of solu- (b) The enclosed region S filled with an invisible medium tions of Maxwell equations and the uniqueness for inverse and material parameters of e2ðrÞ, l2ðrÞ. scattering problems are different. For the uniqueness theo- (c) The enclosed region S as free space. rem, using the procedures used to prove the uniqueness of the solutions of Maxwell equations in a lossy homogeneous medium,28 we proved the uniqueness theorem for anisotropic inhomogeneous media with diagonal constitutive tensors in Sec. II. Then, we used a lossy perfect cloak as the example of such structures. However, there is a concern regarding the validity of the uniqueness theorem for structures like invisibility cloaks,17 as these anisotropic media may support sets of distinct solutions having identical boundary conditions; hence, it might be concluded that the uniqueness theorem may not be valid for such anisotropic inhomogeneous media with diagonal constitutive tensors. However, this notion is a misunderstanding with uniqueness in an inverse i i 29–31 FIG. 4. The fields E , H incident onto an invisible medium enclosed within problem. i i volume V. The tangential components of the fields an E and an H The focus of this section is on the uniqueness of the i over S are equal to the tangential components of the incident fields an E i inverse problem for TO based materials, like invisibility and an H . 203103-6 Dehbashi, Bialkowski, and Abbosh J. Appl. Phys. 121, 203103 (2017)

n r0m am 1 e 0 l 0 ; e 0 l 0 ; r ¼ r ¼ 0m u ¼ u ¼ m r er0

2 0m2 2 mb r 0m m n1 0 ðÞ ez ¼ lz0 ¼ 2 r a nbðÞm am n m 0; n > 0; (28)

where ðec; lcÞ are constitutive tensors of the cloak and ðe; lÞ are the constitutive parameters of its surrounding medium. Using TO again, we can determine the constitutive tensors for the regions a < r0 < b and r0 < a,forthe external cylindrical invisibility cloak, as well.25 However, the simulation results for the internal cloak are enough to prove our argument. Equation (28) shows that within a speciﬁed boundary of S (Fig. 6), one can have numerous cloak structures by changing the parameters (m, n), while all have the same scattering response and the same boundary conditions on S (Figs. 5–7), as proven in Sec. IV A.

C. Invisibility cloak and free space For the worst case scenario, we have considered lossy FIG. 5. The plane wave Ei, Hi incidence from left onto (a) the enclosed cloaks in this section. They can also be zero scattered if their region S filled with an invisible medium and material parameters of e1ðrÞ, surrounding medium is lossy, as well.26 As can be seen in l ðrÞ and (b) the enclosed region S filled with an invisible medium and 1 Figs. 7(a) and 7(c), the same scattering results are obtained material parameters of e2ðrÞ, l2ðrÞ. (c) The enclosed region S as free space. from two different materials (cloak and free space), while It is interesting to note that surfaces with the same field dis- both the boundaries at r ¼ a in Figs. 7(a) and 7(b) have the tributions have different solutions to Maxwell’s equations inside. same boundary conditions (as proved in Section IV A). Therefore, this is an example that shows multiple solutions B. Different cloaks within the same boundary of to an inverse scattering problem. Therefore, uniqueness for interest: S the inverse scattering problem is not true for the anisotropic inhomogeneous structure of invisibility cloaks, as it is not Using the TO method, we determine the constitutive possible to uniquely determine the material from the scat- 0 tensors for the region a < r < b (Fig. 6), i.e., the cylindrical tered signal even though they have identical boundary internal invisibility cloak as: conditions.

V. CONCLUSION For the anisotropic inhomogeneous structures with diagonal constitutive tensors, we analytically proved the uniqueness theorem for solutions to Maxwell equations if all the tensor elements of constitutive parameters are lossy. The uniqueness theorem is proven for three cases: single medium, media composed of two lossy anisotropic inhomogeneous materials with diagonal constitutive parameters, and media composed of two materials, where a material with diagonal material tensors is surrounded by an isotropic material. The third case could be considered for the TO-based materials like invisibility cloaks or hyper-lenses that have diagonal anisotropic inhomogeneous constitutive parameters. We chose cloaks as the example case for structures of our interest. Because loss is a key element for the validity of the uniqueness theorem, for our example case of cloak, we analytically proved that with loss, we could still have invisibility structures. We also examined the uniqueness of the inverse FIG. 6. The simulation results for the Ez0 ﬁeld component due to a z-polarized plane wave incidence from left onto an internal cloak with a ¼ 0:5k0 problem for such structures. We analytically proved that all and b ¼ k0 (k0 denotes the wavelength in free space) with different transfor- these materials have the same surface ﬁeld distribution on a mation orders. The concealed object is a dielectric of the arbitrary shape surface enclosing the area of interest, while solutions to with er¼9 whose surrounding medium is free space: (a) Dielectric without the cloak, (b) cloak with orders of (m, n) ¼ (0.2, 0.2), (c) cloak with orders Maxwell’s equations inside them are different. The unique- of (m, n) ¼ (1, 1), and (d) cloak with orders of (m, n) ¼ (0.2, 0.2). ness theory in the inverse problem says that within the 203103-7 Dehbashi, Bialkowski, and Abbosh J. Appl. Phys. 121, 203103 (2017)

FIG. 7. The simulation results for the Ez0 ﬁeld component due to a z-polarized plane wave incidence from left onto (a) a lossy free space with a loss tangent of 0.1, (b) a perfect cylindrical electric conductor and a lossy free space with a loss tangent of 0.1, and (c) a lossy cylindrical cloak with a ¼ 0:5k0 and b ¼ k0 in a lossy surrounding medium. The loss tangent for the cloak and the surrounding medium is 0.1. k0 is the wavelength of the illumi- nated wave in free space.

enclosed region, the same medium should be exactly present. 15Y. Lai, J. Ng, H. Y. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. T. Chan, However, this paper shows that this might not be true, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009). despite the result of a previous study that shows that unique- 16S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simu- ness could be true for some anisotropic inhomogeneous lations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036,621 structures. (2006). 17D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave ACKNOWLEDGMENTS frequency,” Science 314, 977–980 (2006). 18U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean We acknowledge the support from the Australian cloaking,” Science 323, 110–112 (2009). Government for this research, through the “Australian 19P. Alitalo, O. Luukkonen, L. Jylha, J. Venermo, and S. A. Tretyakov, Research Training Program Scholarship.” “Transmission-line networks cloaking objects from electromagnetic fields,” IEEE Trans. Antenna. Propag. 56(2), 416–424 (2008). 20P. Alitalo, F. Bongard, J. F. Zurcher, J. Mosig, and S. Tretyakov, 1J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic “Experimental verification of broadband cloaking using a volumetric cloak fields,” Science 312, 1780–1782 (2006). composed of periodically stacked cylindrical transmission-line networks,” 2 Appl. Phys. Lett. 94, 014103 (2009). Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: 21 Perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99, 113903 Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media (2007). invisibility cloak that cloaks objects at a distance outside the cloaking 3H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parame- shell,” Phys. Rev. Lett. 102, 093901 (2009). 22A. Alu and N. Engheta, “Achieving transparency with plasmonic and ter equation for elliptical cylindrical cloaks,” Phys Rev. A 77, 013825 metamaterial coatings,” Phys. Rev. E 72, 016623 (2005). (2008). 23M. G. Silveirinha, A. Alu, and N. Engheta, “Parallel-plate metamaterials 4Y. Luo, J. Zhang, B. I. Wu, and H. Chen, “Interaction of an electromag- for cloaking structures,” Phys. Rev. E 75, 036603 (2007). netic wave with a cone-shaped invisibility cloak and polarization rotator,” 24W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Phys Rev. B 78, 125108 (2008). “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 5Y. You, G. W. Kattawar, and P. Yang, “Invisibility cloaks for toroids,” (2007). Opt. Express 17(8), 6591–6599 (2009). 25R. Dehbashi and M. Shahabadi, “External cylindrical invisibility cloaks 6A. Nicolet, F. Zolla, and S. Guenneau, “Finite-element analysis of cylin- with small material dynamic range,” IEEE Trans. Antennas Propag. 62(4), drical invisibility cloaks of elliptical cross section,” IEEE Trans. Magn. 2187 (2014). 44(6), 1150–1153 (2008). 26 7 R. Dehbashi and M. Shahabadi, “Possibility of perfect concealment by R. Weder, “The boundary conditions for point transformed electro- lossy conventional and lossy metamaterial cylindrical invisibility cloaks,” magnetic invisibility cloaks,” J. Phys. A: Math. Theor. 41, 415401 J. Appl. Phys. 114(24), 244501 (2013). (2008). 27 8 R. Dehbashi, M. Fathi, S. Mohajerzadeh, and B. Forouzandeh, “Equivalent A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Improvement of left-handed/right-handed metamaterial’s circuit model for the massless cylindrical cloaking with the SHS lining,” Opt. Express 15, 12717–12734 Dirac fermions with negative refraction,” IEEE J. Select. Top. Quantum (2007). 9 Electron. 16(2), 394–400 (2010). Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. I. Kong, “A rigorous 28R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas 1961). Propag. 57, 3926–3933 (2009). 29G. Oliveri, E. T. Bekele, D. H. Werner, J. P. Turpin, and A. Massa, 10 M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. “Generalized QCTO for metamaterial-lens-coated conformal arrays,” B. Pendry, “Design of electromagnetic cloaks and concentrators using IEEE Trans. Antennas Propag. 62(8), 4089–4095 (2014). form-invariant transformation optics of Maxwell’s equations,” Photonics 30G. Oliveri, E. T. Bekele, M. Salucci, and A. Massa, “Array miniaturization Nanostruct. Fundam. Appl. 6, 87–95 (2008). through QCTO-SI metamaterial radomes,” IEEE Trans. Antennas Propag. 11 A. V. Kildishev and V. M. Shalaev, “Engineering space for light via trans- 63(8), 3465–3476 (2015). formation optics,” Opt. Lett. 33, 43–45 (2008). 31G. Oliveri, E. T. Bekele, M. Salucci, and A. Massa, 12 W. X. Jiang, T. J. Cui, X. Y. Zhou, X. M. Yang, and Q. Cheng, “Arbitrary “Transformation electromagnetics miniaturization of sectoral and bending of electromagnetic waves using realizable inhomogeneous and conical horn antennas,” IEEE Trans. Antennas Propag. 64(4), anisotropic materials,” Phys. Rev. E 78, 066607 (2008). 1508–1513 (2016). 13M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation- 32F. Cakoni and D. Colton, The Inverse Scattering Problem for Anisotropic optical design of adaptive beam bends and beam expanders,” Opt. Express Media, Advanced Topics in Scattering and Biomedical Engineering 16, 11555–11567 (2008). (World Scientic Publishing, 2008), pp. 321–331. 14E. E. Narimanov and A. V. Kildishev, “Optical black hole: 33F. Cakoni, D. Colton, P. Monk, and J. Sun, “The inverse electromagnetic Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, scattering problem for anisotropic media,” Inverse Probl. 26, 074004 041106 (2009). (2010). 14pp). 203103-8 Dehbashi, Bialkowski, and Abbosh J. Appl. Phys. 121, 203103 (2017)

34F. Cakoni and D. Colton, “A uniqueness theorem for an inverse electro- 36Q. Wu, K. Zhang, F.-Y. Meng, L.-W. Li, and G.-H. Yang, “Effects of dif- magnetic scattering problem in inhomogeneous anisotropic media,” Proc. ferent kinds of losses on the performance of regular polygonal cloak,” Edinburgh Math. Soc. II 46(2), 293–314 (2003). IEEE Trans. Magn. 45(10), 4211–4214 (2009). 35B. Zhang, H. Chen, B. I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response 37H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 76, 121101–12110R (2007). (2007).