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Localization of an electromagnetic field can be achieved by transformation optics using metamaterials. A coordinate transformation structure different from traditional resonator is proposed. Wherein, arbitrary frequency of the whole band of electromagnetic wave can be localized without energy loss, i.e., the modes in this structure are continuous. Theoretical analysis and numerical simulation show that the material parameter variations at the outer boundary of the structure have little influence on the localization property. When realizable physical structure is considered, multi-layer approximation should be applied. The calculated results show that the estimated localization time is about 100 ns for an 8-layer inhomogeneous approximation, and it could reach several seconds for a 30-layer homogeneous approximation. The present work may present a new application of transformation optics.
©2010 Optical Society of America
1. Introduction
In recent research, invisibility cloaks [1–13] have become especially attractive. A cloak is implemented based on the transformation optics proposed by J. Pendry [1,2] and U. Leonhardt [3]. However, for transformation optics, attention is now mainly concentrated on the invisibility property of the cloak, while the localization property, which is also interesting, has not yet been of major concern. A cloak has no internal modes, it only responds to an external field. Based on transformation optics, this research proposes an electromagnetic localization structure (ELS) which possesses the property of electromagnetic localization, i.e., it only responds to the internal field. It can achieve localization of the electromagnetic wave with arbitrary frequency. From the view of mechanical space [14], the size of the structure is considered as finite, but in the electromagnetic space [14] it is infinite. It is known that a cavity with infinite length will induce a spectrum with continuous eigenfrequency, and the distribution of its material parameters is different from that of an invisibility cloak. The material parameter at its outer boundary is infinite, but the electromagnetic field localized inside this structure is extremely insensitive to changes in the outer boundary parameters. The localization property of this structure could still be maintained when the method of multilayer (homogeneous) approximation [6] is used in its analyses, so it is relatively easy to implement the proposed structure. The present work may open up new applications in transformation optics.
2. Theoretical considerations
Figure 1 is the diagram of the coordinate transformations of a cloak and the electromagnetic localization structure. Figures 1(a)-(b) show the transformation process of the cloak in which the central point expands (Fig. 1(a)) to form the inner boundary of the cloak structure (Fig. 1(b)). When an electromagnetic wave passes through this structure from the outer region of R2, there is no field distribution within the region of R1, and the wave front and power flow of the electromagnetic wave are not disturbed after passing through the structure. This is just an invisibility cloak. Figure 1(c) is the complementary structure of Fig. 1(a) geometrically in which R4 approaches infinity. The combination of Fig. 1(a) and Fig. 1(c) forms the whole plane. Figure 1(d) shows the coordinate transformation formed by compressing the boundary R4 from infinity to R3, i.e. the structure shown in Fig. 1(d) is obtained by compressing a semi-infinity space r>R2 (Fig. 1(c)) into an annular domain R2<r<R3. For this transformation, if the wave source is in the inner region of R3, the wave front reached at the outer boundary of R3 (Fig. 1(d)) is equivalent to that reached at infinity before the transformation (Fig. 1(c)). So, there is no electromagnetic field distribution in the outer region of R3, and electromagnetic localization is achieved in the mechanical space.
Fig. 1 Coordinate transformation of cloak and the electromagnetic localization structure (ELS). (a)→(b): coordinate transformation of cloak. (c)→(d): coordination transformation of ELS. R1< R2< R3< R4, R4→ + ∞.
In cylindrical coordinates, a general transformation function can be written as [15]:
When the transformation order p is larger than zero, Eq. (1) expresses the transformation of the invisibility cloak, including low order transformations (0<p<1), linear transformations [2,5,15] and high order transformations (p>1) in Refs. 16-19.When the transformation order p is less than zero, Eq. (1) expresses the transformation of the electromagnetic localization proposed in Fig. 1(c)-1(d). The material parameters of the ELS in Fig. 1(d) can be expressed as:
The distribution of permittivity and permeability in the coordinate transformation region of this structure are shown in Fig. 2 , where the smoothed curves show their ideal distributions (Eqs. (2)-(4)).Fig. 2 Distribution of permittivity and permeability in the coordinate transformation region of the proposed structure (R2<r’<R3, R2 = 0.1 m, R3 = 0.4 m in Fig. 1(d)).
3. Numerical simulation and analysis
Full-wave simulations by finite element method were made to demonstrate the localization property of the structure using software COMSOL Multiphysics. The computational domain and details of the calculations are shown in Fig. 3 . The simulations are in two parts, i.e., the solution of the eigenfrequency of the structure and the simulation of the response characteristics of a single frequency in the time domain.
Fig. 3 Calculated region and simulation parameters. (a) Parameters for eigenfrequency simulation; (b) Parameters for harmonic propagation simulation.
3.1 Calculations of eigenmodes
The calculation was done for solving the eigenmodes (using RF Module/In-Plane Waves/Hybrid-Mode Waves/Eigenfrequency analysis, and the solver was chosen as “Eigenfrequency”). The calculating conditions were shown in Fig. 3(a). The simulated fields were computed with approximately 170,000 elements and 681,000 unknowns (the maximum element size is 0.00301 m). The calculated results showed that the electromagnetic wave with arbitrary frequency can be localized in the ELS, i.e., the eigenmodes are continuous in the structure. For example, when solving 20 eigenmodes (it should be addressed that the number of eigenmodes can be specified to be any value) around the frequency of 1.9 GHz, the obtained eigenmodes are: 1.899686 GHz (double degeneracy), 1.899687 GHz (double degeneracy), 1.899742 GHz (double degeneracy), 1.899744 GHz (double degeneracy), 1.900256 GHz (double degeneracy), 1.900257 GHz (double degeneracy), 1.90059 GHz (fourfold degeneracy), 1.900769 GHz (double degeneracy), and 1.900775 GHz (double degeneracy). The difference between the wavelengths (the magnitude of micrometer) of two adjacent modes is much smaller than the maximum grid size (0.00301 m). Figure 4 shows the field distribution of the mode at 1.899744 GHz. It should be noticed that the characteristic length of the mode shown in (b) is in same order of magnitude with the grid size shown in (c). The ideal parameter distributions (the smoothed curves in Fig. 2) at the outer boundary correspond to infinite value of ELS, and cannot be achieved rigorously in numerical simulations. Therefore, in actual simulations the grid size close to the outer boundary of the ELS should be as small as possible. Our simulations showed that the smaller the grid size the smaller difference between the two adjacent eigenmodes. It reveals actually that the eigenvalue of the structure will be continuous when the grid size approaches zero. In this case the electromagnetic field with arbitrary frequency can persist inside the ELS.
Fig. 4 Field distributions of the calculated modes (). (a) at 1.899744 GHz. (b) the enlargement of the red box in (a). (c) the meshed grids of simulation region in (b). The colored vertical bar represents field amplitude (arb. units).
3.2 Calculations of response characteristics of a single frequency wave in the time domain
As an example, a 2 GHz transverse electric (TE) polarized time harmonic cylindrical wave was used for calculation (using RF Module/In-Plane Waves/Hybrid-Mode Waves/Harmonic Propagation, and the solver was chosen as “Stationary”), which originates from a circle with a radius of 0.02 m in the center of the ELS (shown in Fig. 3(b)). The white area shows the free space, the grey area is the perfect matched layer (PML), while the blue area is the coordinate transformed area with an inner radius of 0.1 m (0.67 wavelength) and an outer radius of 0.4 m (2.67 wavelength).
The internal electromagnetic field distribution of the structure under ideal conditions (Eqs. (2)-(4) or the smoothed curves in Fig. 2) was simulated. The results are shown in Fig. 5 . (p = −1 in (a), p = −0.55 in (b)). It can be seen that the localization effect is better when the electromagnetic field is in the internal area by low order transformation, and there is no field distribution near the outer surface (r’ = R3).
Fig. 5 Influence of transformation order n on internal power localization. Red lines denote the inner and outer boundaries of the proposed structure. (a) p = −1; (b) p = −0.55. The colored vertical bar represents field amplitude (arb. units).
Figure 6 shows the internal electromagnetic field (Ez) distribution under various transformation orders of the structure. The frequency of Ez is still 2 GHz. It can be seen that the amplitude of Ez attenuated rapidly with the increase of transmission distance in the transformed region for different transformation orders, the wave front compressed rapidly and there was no field distribution near the outer boundary. It can also be seen that the lower the transformation order, the faster the attenuation of the field. For p = −0.4, the field approaches zero at 2/3 thickness of the transformation structure (0.3 m). It implies that the lower the transformation order, the higher the insensitivity of the localization effect to the variations of the material parameters at its outer boundary. It means that the material parameters within a certain thickness near the outer boundary would have no effect on the electromagnetic field localized in the structure. To demonstrate this property, we introduce a perturbation to the structure, i.e. the outer boundary is shifted inward for a distance (L in the insert of Fig. 7 ) toward the center as shown in Fig. 7, so that the new outer radius is located at . However, the permittivity and the permeability are still calculated according to Eqs. (2)-(4) as if the outer boundary is unchanged. It can be seen that the high order transformation of p = −1 is sensitive to the inward shift of the outer boundary, while for p = −1, p = −0.7 or p = −0.4, the influence will begin to appear only when the shift is larger than 3%, 13%, 23% (indicated by the colored arrows in Fig. 7) of the thickness of the total transformation area (R3-R2).
Fig. 6 Electric field distribution (Ez) in the proposed structure calculated using different transformation orders.
Fig. 7 Influence of shifted length of the outer boundary of the proposed structure on its localization function. Iouter is the energy at the new outer boundary (r’ = R3-L). Iinner is the energy at the field source (r’ = 0.02 m).
Figures 5 and 6 show that a perfect ELS has no energy leakage, so the localization time will be infinite. However, considering that a theoretically perfect structure (shown in Fig. 5) cannot be achieved practically, the simulations were made using multilayer (homogeneous) approximation method used by Pendry et al. [6], where the thickness of the transformed region was uniformly divided into eight layers according to the abscissa in Fig. 1. However, the material parameters were non-uniformly divided into 8 layers according to the slope of the curves, i.e., the layer thickness is thin at the steeper part of the slope and thick at the shallower part of the slope (shown by the stepped curves in Fig. 2). This corresponds to that the transformed thickness of 0.3 m was divided into eight layers. Then the localization effect of the proposed structure after the multilayer (homogeneous) approximation was simulated and shown in Fig. 8 .
Fig. 8 (a) Localization of electromagnetic field in the proposed structure using an 8-layer approximation. The colored vertical bar represents field amplitude (arb. units). (b) Localized field distribution (Ez) of the proposed structure under an 8-layer approximation.
Figure 8 shows the electric field distribution (Ez) of the proposed structure. It can be seen that the wave energy inside the ELS leaked out when an eight layer (homogeneous) approximation was used. It means that for a practical ELS with multilayer approximation the localization time will be finite.
A rough estimate was made of the localization time of the internal electric field of the ELS under an eight non-uniform layer approximation, basing on the simple consideration that the electromagnetic field transmits from the center of the structure towards the outer boundary, and then transmits along the outer boundary with certain losses. Supposing that an electromagnetic pulse originates within the structure, then the total energy within the structure and the total energy flow leakage rate through the outer boundary of the structure can be found. The localization time of the electromagnetic field in the structure can be estimated by the ratio between the total energy within the structure and the leakage rate of the energy flow through the outer boundary, and this time can be considered as the life time of a mode existing in the ELS.
Figure 9 shows the calculated values of the ratio for different intensities of the pulse. It can be seen that there is a linear relationship between the total energy and the total energy flow leakage rate through the outer boundary (shown in Fig. 9). The slope of the line in Fig. 9 can be roughly taken as the localization time, which is about the magnitude of 100 ns. Several sources with different frequencies were also calculated, the obtained life time was in the same order of magnitude. It should be addressed that when a 30-layer homogeneous approximation is chosen, the calculated localization time could reach several seconds. This is attractive enough.
Fig. 9 Localization time of the proposed structure. In the left upper insert, R2 = 0.1 m, R3 = 0.4 m, the radius of the outer boundary is 0.402 m.
Since the delay effects and the reflection by every layer within the structure are not considered, this estimated localization time is only a rough estimation.
This structure can be treated as a kind of equivalent cavity physically with infinite length, so, some concepts of the classical cavity theory are still valid. For an ideal ELS, fields with arbitrary frequency can exist in the cavity, and the quality factor Q will be infinite. It means that electromagnetic field can be localized inside the equivalent cavity without energy loss and with infinite time. However, for an ELS with multilayer approximation, the Q factor will be a finite value. It means that electromagnetic field can still be localized inside the equivalent cavity, but with energy loss and finite time. The corresponding Q factor can be estimated by the formula , where E is the total energy inside the cavity, P is the rate of energy loss, ν is the frequency of the electromagnetic field. This Q factor can be used to estimate the localization time, and the estimation is equivalent to the above simulation.
3. Conclusion
A structure based on coordinate transformation was proposed in which the electromagnetic field could be localized. Changes of the material parameters near the outer boundary of the structure have little influence on its localization property. The lower limit of the electromagnetic field localization time of the structure is of the magnitude of 100 ns.
Acknowledgments
The authors thank the National Natural Science Foundation of China (grant No. 60277014 and No. 60677006) for financial support. The authors thank Dr. Jun Zheng and Prof. Zhengming Sheng in Shanghai Jiaotong University for their help in numerical simulations. Also, the authors would like to thank the unknown reviewer for his/her valuable comments and suggestion.
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