Localization of electromagnetic wave with continuous eigenmodes in free space cavities of cylindrical or arbitrary shapes

Xuan Liu; Xinxin Li; Jing Zhou; Zhaona Wang; Jinwei Shi; Dahe Liu

doi:10.1364/OE.21.030746

1. Introduction

The technique of transformation optics (TO) together with the metamaterials provides a powerful means to precisely control the light flow in almost arbitrary ways [1–6]. Numerous remarkable optical devices based on TO have been demonstrated, such as invisibility cloaks [7–17], beam rotators [18], omnidirectional retroreflectors [19] plasmonic open resonators [20], and the control of SPP propagation [21]. The TO method is a general method whatever the shape and function of the transformed media [2, 3, 8], however the determination of the corresponding material parameters for the device with irregular shape is a difficult task. Some techniques for designing cloaks/anticloaks with irregular shapes have been proposed including analytic method, semi-analytic and semi-numerical method [17, 22–25], and numerical method [26]. To our knowledge so far, only various cloaks/anticloaks with arbitrary/irregular shapes have been reported and attracted great attention, while the localization element with arbitrary shape, which is also interesting, has not yet been reported. Tianrui Zhai et al and V. Ginis’ team proposed cylindrical Electromagnetic localization structures (ELSs) [27, 28] based on TO, both of which can localize the electromagnetic energy inside the cylindrical cavity. However, for both of these two kinds of structures, a certain part of the electromagnetic energy distributes in the metamaterials layer which may cause some difficulty in applications. In this paper, we propose a scheme for constructing the ELS that can localize the electromagnetic wave in the central free space cavity with no energy leaked in the metamaterials layer. And the cavity can be designed to have cylindrical or regular/arbitrary shape.

2. Theoretical consideration

For the convenience of description, we describe the principle of the proposed scheme through cylindrical ELS. Figure 1(a) shows the original flat space, where $R_{2}$ approaches infinity. Figure 1(b) shows the transformed space formed by compressing the boundary $R_{2}$ from infinity to $R_{3}$ , while keeping the boundary $R_{1}$ unchanged. In the transformed space, the layer $I I$ between $R_{1}$ and $R_{3}$ constructs the transformed layer which could be made by metamaterials, while the central area $I$ enclosed by the boundary R₃ is a free space cavity. The electromagnetic wave emitted by a source inside the free space $I$ will be totally localized in this cavity because $R_{3}$ corresponds to the infinity in the original space.

Fig. 1 The scheme of the space transformation for constructing ELS

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This scheme of space transformation can be confirmed by analyzing the bounded electromagnetic modes of this system. Since the structure is cylindrical symmetric, cylindrical coordinates is used in our analysis. The coordinates in the original and transformed spaces are represented by $o - r φ z$ and $o' - r' φ' z'$ respectively. The above space transformation can be expressed as

f (r') = r, φ' = φ, z' = z

With the space boundary conditions:

f (R_{1}) = R_{1}, f (R_{3}) = R_{2}, R_{2} \to \infty

Where,

f (r')

should be a harmonic function with arbitrary form. The bounded electromagnetic modes of the transformed structure are determined by the Maxwell’s equations and the electromagnetic boundary conditions at the interfaces of the areas. It is known that the bounded electromagnetic field in such a cylindrical symmetric system can be written as:

\overset{⇀}{E} (r', φ', t) = \overset{⇀}{E} (r') e^{i (m φ' - ω t)}, \overset{⇀}{H} (r', φ', t) = \overset{⇀}{H} (r') e^{i (m φ' - ω t)}

Where

m

is the angular momentum and

ω

is the eigenfrequency. Without loss of generality, we assume transverse electric (TE) polarization. When no wave source exists in any of the areas, using the general solution of the Maxwell’s equations for the cylindrical symmetrical system together with the finite energy and Sommerfeld’s radiation conditions, we can write the electric fields in areas I and III as follows:

E_{z}^{I} (r') = A J_{m} (k_{0} r'), E_{z}^{I I I} (r') = B H_{m}^{(1)} (k_{0} r')

Because of the form-invariance of Maxwell’s equations under spatial transformation, the electric field in the transformed layer (region II) has a similar form to the general solution with the parameter

r'

replaced by

f (r')

[4,28,29]:

E_{z}^{I I} (r') = C J_{m}^{} [k_{0} f (r')] + D Y_{m}^{} [k_{0} f (r')]

Where,

J_{m}

and

Y_{m}

are the Bessel functions of the first and second kind,

H_{m}^{(1)}

is the Hankel function of the first kind, and (A, B, C, D) are arbitrary complex constants. At the interface between two areas, the tangential components of the electric and magnetic fields,

E_{z'}^{}

and

H_{φ'}^{}

, must be continuous. For the electric field, we have:

A J_{m} (k_{0} R_{3}) = C J_{m} (k_{0} R_{2}) + D Y_{m} (k_{0} R_{2})

C J_{m} (k_{0} R_{1}) + D Y_{m} (k_{0} R_{1}) = B H_{m}^{(1)} (k_{0} R_{1})

where the space boundary conditions of Eq. (2) is used. For TE polarization, the tangential component of the magnetic field is determined by:

H_{φ^{'}} = \frac{1}{i ω μ_{φ}'} \frac{\partial E_{z^{'}}}{\partial r'}

where

μ_{φ}'

is the permeability of the material in the transformed space, which is 1 in areas I and III. While in the area II,

μ_{φ}'

can be devised by TO theory. According to the TO theory, under a space transformation from a flat space

x

to a distorted space

x^{'} (x)

, the permittivity

ε^{'}

and permeability

μ^{'}

in the transformed space are given by [30].

\begin{array}{l} ε^{'} = A ε A^{T} / \det A \\ μ^{'} = A μ A^{T} / \det A \end{array}

where

ε

and

μ

are the permittivity and permeability of the original space.

A

is the Jacobian transformation tensor with the components

A_{i j} = \partial {x^{'}}_{i} / \partial x_{j}

, which characterizes the geometrical variations between the original space and the transformed space. Applying the space transformation of Eq. (1) into the Eq. (9), we find

μ_{φ}' = \frac{r'}{f (r')} \frac{\partial f (r')}{\partial r'}

. Then substituting

μ_{φ}'

into the Eq. (8), we get the tangential magnetic field in area II as:

H_{φ^{'}}^{I I} = \frac{1}{i ω} \frac{f (r')}{r' \cdot \frac{\partial f' (r')}{\partial r'}} \frac{\partial E_{z^{'}} [f (r')]}{\partial r'}

Subsequently combining with the space boundary conditions of Eq. (2), we get the boundary conditions of

H_{φ^{'}}

:

A J'_{m} (k_{0} R_{3}) = C \frac{R_{2}}{R_{3}} J'_{m} (k_{0} R_{2}) + D \frac{R_{2}}{R_{3}} Y'_{m} (k_{0} R_{2})

C J'_{m} (k_{0} R_{1}) + D Y'_{m} (k_{0} R_{1}) = B H'_{m}^{(1)} (k_{0} R_{1})

Where the prime (') denotes differentiation with respect to the parameter of the Bessel or Hankel function. Using the recursive formula of the Bessel function in Eq. (11), we have

A R_{3} [J_{m - 1} (k_{0} R_{3}) - J_{m + 1} (k_{0} R_{3})] = C R_{2} [J_{m - 1} (k_{0} R_{2}) - J_{m + 1} (k_{0} R_{2})] + D R_{2} [Y_{m - 1} (k_{0} R_{2}) - Y_{m + 1} (k_{0} R_{2})]

When

R_{2} \to \infty

, the Bessel functions

J_{m} (R_{2})

and

Y_{m} (R_{2})

can be written as the following asymptotic forms:

J_{m} (R_{2}) ~ \sqrt{\frac{2}{π R_{2}}} \cos (R_{2} - m π / 2 - π / 4), \begin{matrix}  \end{matrix} Y_{m} (R_{2}) \sim \sqrt{\frac{2}{π R_{2}}} \sin (R_{2} - m π / 2 - π / 4)

Thus, the constants C and D have to be zeros otherwise the right side of the Eq. (11) will be infinity. Then using the Eqs. (6), (7), (11) and (12), we concluded that the constants A and B should be zeros too because

J_{m} (k_{0} R_{3})

,

{J^{'}}_{m} (k_{0} R_{3})

,

H_{m}^{(1)} (k_{0} R_{1})

and

{H^{'}}_{m}^{(1)} (k_{0} R_{1})

are usually not zeros. This means that the electromagnetic field is zero everywhere when there is no wave source in any of the areas. Nevertheless, when an electromagnetic wave source exists inside the free space cavity (area I), the field energy will be totally localized inside this area no matter what the frequency of the wave is. And therefore the eigenmodes of the cavity is continuous.

This space transformation scheme is feasible for the structure of irregular shape too. Since it is difficult to give the analytic expressions of the bounded electromagnetic modes in this case, its localization character will be numerically demonstrated by full-wave simulations in section 4.

3. The cylindrical ELS

As mention above, the material parameters of the transformed layer (or the metamaterials layer) are determined by the TO theory. It can be seen from Eq. (9) that the transformation function $x^{'} (x)$ and the Jacobian transformation tensor $A$ are the crucial points. In our transformation scheme, cylindrical coordinates system is used, and $x^{'} (x)$ can be written as $r' (r)$ , which is a harmonic function of arbitrary form. Since the Laplace’s equations with proper boundary conditions always give rise to harmonic solutions, $f (r')$ can be determined using the solution of the Laplace’s equation [26]. The calculated formula of the coordinate transformation corresponding to the transformation in Fig. 1 is:

\ln r^{'} = (\frac{\ln R_{3} - \ln R_{1}}{R_{1} - R_{2}} (R_{1} - r) + \ln R_{1})

Figure 2 shows the coordinate lines of this transformation.

Fig. 2 The calculated coordinate lines of the cylindrical ELS.

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The formulas of the material parameters of transformed layer can be devised using the Eq. (9). In order to demonstrate the localization property of the cylindrical ELS structure, we made the full-wave simulations using the finite element method with software COMSOL Multiphysics. According to the scheme in Fig. 1, $R_{3}$ should be transformed from infinity, or $R_{2} = \infty$ , which is impossible in practice. It is found that, as long as $R_{2} > > R_{3}$ , the ELS will have no energy leakage. A 6GHz TE polarized time harmonic cylindrical wave was used for calculation, which originates from a circle with a radius of 0.01m in the center of the free space cavity. Figure 3(a) shows the simulated electric field pattern for $R_{1} = 0.7 m$ , $R_{3} = 0.3 m$ and $R_{2} = 6 m$ . The structure and the field distribution are circularly symmetric. Figure 3(b) shows the amplitude distributions of E_z along the radial direction for frequencies of 5, 5.5, and 6GHz.

Fig. 3 (a) The simulated electric field pattern, E_z for 6GHz. (b) The amplitude distribution of E_z along the radial direction for 5, 5.5, and 6GHz. $R_{1} = 0.7 m$ , $R_{3} = 0.3 m$ , $R_{2} = 6 m$ . The colored vertical bar in (a) represents field amplitude (arb. units)

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It can be seen from Fig. 3 that most of the electric energy is localized in the central free space cavity, and there is almost no field distribution in the metamaterials layer. This is a novel property that the reported ELSs could not realize. The localization time of the electromagnetic field can be roughly estimated by the ratio between the total energy within the free space cavity and the leakage rate of the energy flow through the inner boundary of the metamaterials layer [27]. The calculated localization time of the electromagnetic field shown in Fig. 3 is 1.95 × 10⁴ second, which corresponds to the quality factor about 10¹⁴. Figure 4 shows the material parameters of the metamaterials layer, ε_r = μ_r, ε_ϕ = μ_ϕ, ε_z = μ_z. All other components are zeros.

Fig. 4 The material parameters of the metamaterials layer of the cylindrical ELS with R₁ = 0.7m, R₃ = 0.3m, R₂ = 6m.

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It can be found that all components of the material parameters change smoothly with finite values. Such distributions are convenience for achieving multilayer structure, which is a method commonly used for approximate realization of TO design [9,27]. On the other hand, ε_z (μ_z) is relatively large, with maximum absolute value of |ε_{z_max}| = 417. In order to reduce |ε_z (μ_z)|, we can decrease R₂. For example, when R₂ = 5m, |ε_{z_max}| is about 282, nevertheless the quality factor reduces to 10⁹. The smaller the |ε_{z_max}| is, the lower the quality factor will be. Table 1 shows the calculated quality factors, values of ε_{z_max} and the estimated localization time corresponding to different values of R₂, all other parameters remain unchanged. It can be seen that the quality factor approaches infinity quickly when R₂>6m. The value of R₂ should be set by comprehensively considering the requirement of the quality factor and the practical material manufacturing technology.

Table 1. The calculated ε_{z_max}, quality factor, and estimated localization time of the cylindrical ELS corresponding to different R₂

View Table

When R₂ goes infinity, the metamaterial is mapped from the entire free space; thus the eigenmodes are continuous. The eigenmodes of the structure were also solved using the eigenfrequency analysis of the software. Figure 5 shows some typical field patterns of the calculated eigenmodes around 6GHz. The eigenmodes with close frequencies have similar field patterns that are difficult to be distinguished.

Fig. 5 Electric field patterns of some eigenmodes around 6GHz. The colored vertical bars represent field amplitudes (arb. units).

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4. ELS with arbitrary shape

For the ELS with arbitrary shape, it is difficult to give analytic solutions of the bounded electromagnetic modes. The feasibility of it will be numerically demonstrated in this section. Figure 6 shows the scheme of the space transformation for an ELS with irregular/arbitrary shape, which is formed by compressing the outer boundary $S_{2}$ from infinity to $S_{3}$ , while keeping the boundary $S_{1}$ unchanged. The boundary conditions can be expressed as

x (S_{3}) = S_{2}, x (S_{1}) = S_{1}, S_{2} \to \infty

Since the Dirichlet boundary conditions are flexible to describe whatever the shape of the boundary, the ELS with arbitrary shape can be designed using the solutions of the Laplace’s equation. The process of solving the Laplace’s equations was finished by the PDE (Partial Differential Equation) solver provided by the commercial software COMSOL Multiphysics.

Fig. 6 The scheme of constructing an ELS with irregular shape

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In the calculation we firstly found the function $x_{i} ({x^{'}}_{j})$ , which is the inverse function of ${x^{'}}_{i} (x_{j})$ . Then the Jacobian transformation tensor $A$ was deduced using the derivative of implicit function. After that we got the material parameters of the transformed layer (area II, or the metamaterials layer). The localization property of this ELS can be demonstrated by full-wave simulations. The computational domain and details are shown in Fig. 7. The white areas represent the free spaces, the grey area is the perfect matched layer (PML), while the blue area is the metamaterials layer (or transformed layer) of the ELS. The red circle inside the central free space cavity represents wave source with radius, which could be anywhere in the cavity and there could be several sources existing simultaneously.

Fig. 7 Computational domain and details for full-wave simulation of ELS with irregular shape.

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The electromagnetic waves used in the simulation were still TE polarized time harmonic cylindrical waves. Figure 8 illustrates the simulated electric field patterns for waves of 6GHz and 2GHz respectively. It can be seen that the waves are well localized in the central irregular free space cavity and no field distributions in the metamaterials layer. The calculated localization times corresponding to both cases are greater than 10³³ hours (or the quality factors are 3.77 × 10⁴⁴).

Fig. 8 The simulated electric field patterns of TE waves with frequencies of 6GHz (a) and 2GHz (b) for ELS with irregular shape. The colored vertical bars represent field amplitudes (arb. units).

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Figure 9 demonstrates the contour plots of the nonzero components of the material parameters. All of them change smoothly with finite values, nevertheless the absolute value of ε_zz(or μ_zz) is large. As discussed in section 3, the value of ε_zz (or μ_zz) could be reduced by properly decreasing the quality factor.

Fig. 9 The contour plots of the nonzero material parameters of the ELS with irregular shape.

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5. Conclusion

A scheme of space transformation is proposed to construct the electromagnetic localization structure (ELS). The ELS can localize the electromagnetic energy in the free space cavity of cylindrical or arbitrary shape. All the material parameter components of the proposed cylindrical ELS change smoothly with finite values. Theoretical analysis and full-wave simulations demonstrated the feasibility of the proposed scheme.

Acknowledgments

The authors thank the National Natural Science Foundation of China (grant No. 61275130) for financial support. The authors also thank Dr. Jun Zheng and Prof. Zhengming Sheng in Shanghai Jiaotong University for their help in numerical simulations.

References and links

1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43(4), 773–793 (1996). [CrossRef]

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006). [CrossRef]

4. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

5. J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094), 549–552 (2012). [CrossRef] [PubMed]

6. Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012). [CrossRef] [PubMed]

7. D. 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 frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

8. U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8(7), 118 (2006). [CrossRef]

9. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 74(3), 036621 (2006). [CrossRef] [PubMed]

10. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).

11. J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17(15), 12922–12928 (2009). [CrossRef] [PubMed]

12. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]

13. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]

14. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009). [CrossRef] [PubMed]

15. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

16. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]

17. L. Li, F. Huo, Y. Zhang, Y. Chen, and C. Liang, “Design of invisibility anti-cloak for two-dimensional arbitrary geometries,” Opt. Express 21(8), 9422–9427 (2013). [CrossRef] [PubMed]

18. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett. 102(18), 183903 (2009). [CrossRef] [PubMed]

19. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).

20. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010). [CrossRef] [PubMed]

21. H. Xu, X. Wang, T. Yu, H. Sun, and B. Zhang, “Radiation-suppressed plasmonic open resonators designed by nonmagnetic transformation optics,” Sci. Rep. 2, 784 (2012). [CrossRef] [PubMed]

22. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. 10, 043040 (2006).

23. W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 77(6), 066607 (2008). [CrossRef] [PubMed]

24. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. 33(14), 1584–1586 (2008). [CrossRef] [PubMed]

25. H. Ma, S. Qu, Z. Xu, and J. Wang, “Numerical method for designing approximate cloaks with arbitrary shapes,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 78(3), 036608 (2008). [CrossRef] [PubMed]

26. J. Hu, X. Zhou, and G. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009). [CrossRef] [PubMed]

27. T. Zhai, Y. Zhou, J. Shi, Z. Wang, D. Liu, and J. Zhou, “Electromagnetic localization based on transformation optics,” Opt. Express 18(11), 11891–11897 (2010). [CrossRef] [PubMed]

28. V. Ginis, P. Tassin, C. M. Soukoulis, and I. Veretennicoff, “Confining light in deep subwavelength electromagnetic cavities,” Phys. Rev. B 82(11), 113102 (2010). [CrossRef]

29. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” New J. Phys. 12(9), 093030 (2010). [CrossRef]

30. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8(10), 248 (2006). [CrossRef]

R₂(m)	4	4.5	5	5.5	6	6.5	7
ε_{z_max}	−173	−224	−282	−346	−417	−494	−578
quality factor	1.6 × 10³	3 × 10⁶	4.3 × 10⁹	3.7 × 10¹⁰	7.4 × 10¹⁴	8.5 × 10¹⁶	8.5 × 10¹⁷
estimated localization time	42ns	78μs	114ms	0.98s	5.4h	628h	6291h

Localization of electromagnetic wave with continuous eigenmodes in free space cavities of cylindrical or arbitrary shapes

Abstract

1. Introduction

2. Theoretical consideration

3. The cylindrical ELS

4. ELS with arbitrary shape

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (16)

Optics Express