Transformation optical design of a bendingwaveguide by use of isotropic materials
Xiaojiong Wu,1,* Zhifang Lin,1 Huanyang Chen,2 and C. T. Chan3
1Surface Physics Laboratory, Department of Physics, Fudan University, Shanghai 200433, China2Department of Physics, Soochow University, Suzhou, Jiangsu 215006, China
3Department of Physics, The Hong Kong University of Science and Technology,Clear Water Bay, Hong Kong
Transformationmedia [1–5] have drawnmuch atten-tion from researchers in many fields, includingphysics, mathematics, and various branches of engi-neering. The popularity is mainly fueled by thepossibility of realizing invisible cloaking devices[1–14]. Transformation media actually extends be-yond the ability to make things invisible since theycan lead to many novel wave manipulation strate-gies, including field concentrators [15], field rotators[16,17], electromagnetic wormholes [18], impedance-matched hyperlenses [19], field shifters [20,21],anticloaks [22], cylindrical superlenses [23], super-scatterers [24,25], superabsorbers [26], and externalcloaks [27]. Recently, the transformation mediabending waveguide [28,29] was also proposed basedon the finite embedded coordinate transformation[20] from a rectangular flat space to a bending space.The related simplified version was also given withrealizable inhomogeneous and anisotropic materials[30]. As is well known, anisotropic materials with re-
quired permittivity tensors can be implemented by alayered structure of thin alternating layers of metaland dielectrics [31]. The applications of a layeredstructure include subwavelength imaging [31], nega-tive refraction [32], optical hyperlenses [33–37], in-visibility cloaking [38], and field rotators [21,39].Here we use three kinds of isotropic material in analternating layered structure to design the simplifiedtransformation media bending waveguide describedin Ref. [30].
2. Transformation Formulation
We begin with the following transformation [30] (seealso Fig. 1):
x0 ¼ x cosðαy=lÞ; y0 ¼ x sinðαy=lÞ; z0 ¼ z ð1Þto obtain an ideal transformation media bendingwaveguide with permittivity and permeability ten-sors, written as
in the circular cylindrical coordinate ðr;ϕ; zÞ, wherewe drop the primes for aesthetic reasons. By keepingthe principal refractive indices unchanged, the sim-plified material parameters can be written as [30]
ϵr;ϕ;z ¼ μr;ϕ;z ¼0@
l2=ðαrÞ2 0 00 1 00 0 1
1A: ð3Þ
If we consider only transverse magnetic (TM) polar-ization incident waves (whose magnetic field is alongthe z direction), the above simplified transformationmedia bending waveguide has no magnetic response.As the principal values of its permittivity tensor arethe functions of radial coordinate r, we first break thesimplified bending waveguide into several shellsalong the radial direction to gain the required inho-mogeneity. Each shell has a constant permittivitytensor. To implement such an anisotropy for eachshell, we can use the alternating layered structurewith its normal direction either along the radial di-rection (r direction) or the tangential direction (ϕ di-rection) [33]. The thickness of each layer should bemuch smaller than the wavelength. Here we chosethe latter, i.e., the normal is along the ϕ direction.The principal values of the effective permittivity ten-sor are functions of the permittivity and the thick-ness of each layer. To obtain the two requiredprincipal values of the permittivity tensor in eachshell, we should build a separation set that hastwo freedoms.Let us suppose that the alternatinglayered structure contains three kinds of isotropicmaterial: A, B, and C. Figure 2 shows the alternatinglayered structure for each shell of the waveguide. Weset dA þ dB þ dC ¼ d or a ¼ dA=d, b ¼ dB=d, c ¼ dC=dso that
aþ bþ c ¼ 1: ð4ÞFrom the effective medium theory [31] we have
aϵa
þ bϵb
þ cϵc
¼ 1; ð5Þ
aϵa þ bϵb þ cϵc ¼ l2=ðαrÞ2: ð6ÞFor each shell coordinate r is set as the average valueof the inner and outer radii of each shell so that wecan obtain the values of a, b, and c uniquely by sol-ving Eqs. (4)–(6). For example, we chose ϵa ¼ 0:5,ϵb ¼ 1, and ϵc ¼ 10. Note that, to ensure that0 < a < 1, 0 < b < 1, and 0 < c < 1, the inner andouter radii of the bending waveguide must have lim-itations that are determined by the values of l or α.
3. Simulation
To investigate the performance of the bendingwaveguide, we used full-wave simulations with theCOMSOL Multiphysics finite element-based electro-magnetics solver (COMSOL, Incorporated, Burling-ton Massachusetts). The bending waveguide isbounded by perfect electrical conducting (PEC)sheets at its inner and outer boundaries. We con-nected two straight air waveguides (that are alsobounded by PEC sheets) to the two ends of the bend-ing waveguide. The whole structure is then sealed bytwo perfectly matched layers at the ends as the ab-sorbing boundaries. If the structure is filled with air(including the bending part), the incoming wavesfrom one side of the waveguide will undergo reflec-tions from the bending part and the wavefront willbe significantly changed when the waves reach theother side. If we replace the bending part with theideal transformation media described by Eq. (2) butkeep other parts unchanged, the incoming waves willbe smoothly guided along the structure to the other
Fig. 1. (Color online) Coordinate transformation from a rectangu-lar flat space to a bending space to design the transformation med-ia bending waveguide.
Fig. 2. (Color online) Alternating layered structure to implementthe anisotropy of each shell of the bending waveguide. The struc-ture for each shell shares the same three isotropic materials: A, B,and C.
side without any distortion. Such a functionality isthe motivation for designing the ideal bending wave-guide. We performed full-wave simulations to showthat the functionality is also maintained for a simpli-fied bending waveguide as described by Eq. (3) and
eventually for the current implementation with analternating layered structure.
We suppose that a TM-polarized plane wave is in-cident from the bottom part of the structure with afrequency of 8GHz (or a free-space wavelength λ0 ¼37:5mm). We set α ¼ π=2 and l ¼ 8λ0. The inner andouter radii of the bending part were chosen as 0.1and 0.2 m. Figure 3(a) shows the magnetic field dis-tributions inside the waveguide with the bendingpart filled with the ideal transformation media bend-ing waveguide. We can see that the incoming planewave is smoothly guided by the transformationmedia and reaches the other side as a plane wavewithout any distortion. Figure 3(b) shows the mag-netic field distributions inside the waveguide withthe bending part filled with a simplified bendingwaveguide. The plane wave reaches the other side
Fig. 3. (Color online) Distribution of magnetic fields inside thewaveguide with the bending part filled by (a) the ideal transforma-tion media bending waveguide, (b) the simplified transformationmedia bending waveguide, (c) air.
Fig. 4. (Color online) (a) Computation domain and geometricstructure of the waveguide with the bending part implementedby the alternating layered structure and (b) the distribution ofmagnetic fields inside the waveguide with the bending part imple-mented by the alternating layered structure.
with slight distortions. Figure 3(c) shows the mag-netic field distributions inside the waveguide filledwith air; the wave reaches the other side with signif-icant distortions.We now show the functionality of the current im-
plementation with an alternating layered structure.As mentioned above, there are some limitations forthe inner and outer radii of the bending waveguide.We carefully chose the inner radius as 0:1m and theouter radius as 0:15m (but not 0:2m as used above)to ensure that 0 < a; b; c× < 1 for α ¼ π=2 andl ¼ 8λ0. The simplified bending waveguide is dividedinto five shells of equal thickness (0:01m) with eachradial coordinate r chosen to be the average radius ofeach shell. Each shell is divided into 24 equal sec-tions. The three components (A, B, and C) are alter-nately and periodically put into in each section alongthe ϕ direction as shown in Figs. 2 and 4(a). For eachshell, a, b, and c can be obtained uniquely by solvingEqs. (4)–(6). In Fig. 4(b) we plot the magnetic fielddistributions inside the waveguide implemented bythe above alternating layered structure. Althoughthere is some slight distortion, the plane wavefrontsare almost maintained after the waves reach theother side. Therefore, the functionality of the simpli-fied bending waveguide can be implemented by ouralternating layered structure with only three kindsof isotropic material. If each shell is divided into12 sections or less, the effective medium theory willfail because the thickness of each alternating layer isnow comparable to the working wavelength. In addi-tion, the functionality is also slightly compromised.
4. Conclusion
In conclusion, we have given a simple design of thesimplified bending waveguide with only three kindsof isotropic material in an alternating layered struc-ture. A similar design can be applied to optical de-vices composed of metal dielectrics, the size ofwhich should be of the order of submicrometers. Tomanufacture the optical devices, one can refer toRef. [34], which provides details of a layered hyper-lens created on a nanometer scale. Since the layeredstructures can be fabricated precisely [32,34,40] evenfor optical devices, the current design is thereforecloser to reality for future realizations.
This research was supported by the China 973 pro-gram from the Chinese National Science Foundation(NSF) , the Program for Changjiang Scholars and In-novative Research Team (PCSIRT), Ministry of Edu-cation (MOE) of China, B06011, and the ShanghaiScience and Technology Commission. H. Chen andC. T. Chan were supported by a Central AllocationGrant, Project HKUST3/06C, from the Hong KongResearch Grants Council (RGC). The computationresources were also supported by the Shun HingEducation and Charity Fund.
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