Transformation optics provides a promising way to guide waves in the open space. It is shown that a small waveguide coated with transformation medium will behave as a big virtual tunnel connecting two waveguide ports separated faraway. The waves are transmitted and guided smoothly in the open space through this ‘invisible tunnel’. The transformation medium is obtained by squeezing the space between the two ports into a small region. Numerical simulations are performed to illustrate this idea.
© 2009 Optical Society of America
Based on the form invariant coordinate transformation of Maxwell’s equations , Pendry et al. suggested the idea of directing electromagnetic (EM) fields with metamaterials to achieve invisibility cloak . Similar optical conformal mapping method was described by Leonhardt where two-dimensional (2D) Helmholtz equation is transformed to exhibit similar effect in geometry limit . The transformation-based cloak design which generalized a similar idea on the transformation of conductivity equation [4–6] has aroused enormous interests [7–16], since it offers great convenience to the manipulation of EM fields [17, 18]. Apart from cloak of invisibility, some other EM devices, such as twisted EM waveguides [19, 20], perfect corner reflector , hyperlens , EM field and polarization rotator [23, 24], beam expander , and antennae [26–28], can also be realized with this methodology. In this paper, we further use the coordinate transformation approach to build an ‘invisible tunnel’ between two waveguide ports and make the EM wave propagate from one port to another in the open space through this tunnel. This guidance of EM wave through open space is achieved by inserting a small waveguide into a transformation medium whose parameters are obtained by judiciously compressing the space between the two ports into a very small region. In this condition, this small waveguide covered by transformation medium will behave exactly the same as a big waveguide connecting the two ports. Therefore, an observer will see the amazing phenomenon that the EM waves transmitted from one port would not disperse in the free space, but are all collected and guided smoothly to the other port. The idea of using a small transmission medium to control the field distribution of a large region can be used in remote controlling or camouflage, and has potential applications in many other fields of engineering.
Traditional physical realizations of point-to-point communication are always based on transmission wires including electric cables and optical fibers. Compared with wireless communication, the wire-based communication can localize the signals in a small region. However, the installations of wired transmission are always affected by the environment. Many practical factors, such as the terrain and the geology, will increase the difficulty of building such wire-based communication systems. If the EM waves can be transmitted in the free space without dispersion, the construction difficulty would be greatly reduced and it will not be necessary to take the terrain factor into consideration any more.
In Fig. 1(a), ports A and B, which are far away from each other, are connected by a waveguide so that the EM wave can be transmitted from A to B. Now we remove the waveguide between the two ports and only place a block of transformation medium with a small waveguide inside at arbitrary position between A and B, as shown in Fig. 1(b) (the transformation medium is colored in blue). By properly designing the parameters of the transformation medium, an invisible tunnel, which has the equivalent functionality to that of the waveguide in Fig. 1(a), is built between the ports and the wave can be smoothly transmitted from A to B in the free space. This transformation medium can be obtained by compressing the space (denoted by the dashed sphere in Fig. 1(b), though actually the space can be of arbitrary shape) containing the original waveguide in Fig. 1(a) into a small region (denoted by the dark blue sphere in Fig. 1(b)).
For simplicity, we limit our discussion in two dimensional (2D) cases in the following analysis (three dimensional problems can be solved in similar ways). Suppose we utilize the mapping ρ’ = f (ρ), then the material parameters of the 2D transformation medium will take the following form:
where the transformation function f (ρ) is selected as
We can see from the transformation that a circular space ρ < D is squeezed into a smaller circle ρ < R 1, and the annular layer R 1 ρ < R 2 is designed to meet the impedance matching requirement. The constitutive parameters of the whole system [described by Eq. (1)] can be realized with metamaterials. It is worthwhile to notice that the transformation function f(ρ) here is not one-to-one, which is different from traditional cases. Moreover, f(ρ) is monotonically decreasing in the region R 1 < ρ < R 2, resulting in the negative-refractive-index (NRI)  of the transformation media. Thereby, the waveguide located in the interior region (ρ < R 1) is magnified by a factor of D/R 1 and its image outside the device creates a “virtual tunnel”, which connects the two waveguide ports. This is similar to the super concentration case of Ref.  where the NRI outer shell of the concentrator makes the energy flow inside the concentrator enhanced. This super concentration effect was also shown in some later proposed transformation-based devices, such as scattering shifter and superscatterer; .
3. Numerical simulation
To verify our idea, we use finite element based numerical simulations to study a typical 2D case. Fig. 2(a) depicts the electric field distribution of a waveguide whose width (In all the following discussions, the width refers to the distance between the two planes of the planar waveguide) is 0.2m. When the waveguide is cut off and the two ports are separated 0.4m away, as depicted in Fig. 2(b), it can be observed clearly that a lot of energy will disperse in the space and only a small portion of the energy can be received at the end port. Next we put a circular transformation medium device (Here R 1= 0.05m, R 2 =0.1m, and D = 0.5m) between the open waveguides, and place a 0.02m wide, 0.04m long waveguide in the core (ρ < R 1) of the device. From Fig. 2(c), we can see that the wave transmitted from the left port no longer disperse in the open space as in Fig. 2(b), but propagate harmoniously in an invisible tunnel in the air. After passing the transformation medium device, the wave goes on propagating through the ‘tunnel’ until finally most of the power is received by the other port.
Even when the two ports are far away from each other, our idea still works. The simulation result is shown in Fig. 3(a), where the two waveguide ports are separated 0.8m away and the small waveguide inside the device is 0.02m wide and 0.08m long. Although the figure is not perfect due to the precision limitation of the numerical simulation, we can still clearly see the effect of this amazing ‘invisible tunnel’ which combines the two ports and inducts the EM wave. In addition, it is worth mentioning that, the transformation medium does not have to locate in center between the two ports. Introducing a shift term to the transformation function in Eq. (2) will yield a medium which can work at the side of the waveguide ports, as illustrated by Fig. 3(b).
A more interesting phenomenon which can be observed in Fig. 2(c) and Fig. 3 is the highly oscillating behavior of the field at the interfaces of the NRI coating. This is due to the enhancement of fields of the evanescent wave, as pointed out in Ref. . The field distributions in these figures are all in time harmonic state. Before reaching this steady state, total transmission cannot be achieved, since wave will leak from the left waveguide port to free space. However, the outer NRI shell will enhance the evanescent fields, causing more energy to concentrate inside the coating . When the steady harmonic state is reached, the energy will no longer be radiated outside. Hence, complete transmission between the two remote waveguide ports is realized. Similar phenomena can also be observed in the imaging process of perfect lens .
Now we come to the question that whether we are still able to build a tunnel in more general cases, for example, where the two ports are not face to face. As an answer to this question, we study the case where the two ports are facing the orthogonal direction. The simulation result shows that, as long as we place a small bent waveguide inside the core of the transmission medium, the bend angle of which depends on the angle between the two ports, a curved tunnel would be created, which bend the path of the EM wave and guide it to the other port, as depicted by Fig. 4. Note that though the transmission is dramatically enhanced, some scattered wave can still be observed around the transformation device. This phenomenon is partly due to the precision limitation of simulation, and partly because some reflection occurs at the inner boundary of the bent waveguide. This problem can be solved by applying another transformation to the original space before our mapping process, as expounded in .
In conclusion, a transformation approach which aims to build an invisible tunnel for the EM waves to propagate through by means of space compression is proposed in this paper. Though here we limit ourselves in 2D waveguide examples, our method is free to be extended to deal with more general cases. The construction of an invisible tunnel provides great convenience in some specific practical cases, where it is difficult to build cables between two targets, and is likely to influence the concept of point-to-point transmission in engineering.
This work is sponsored by the National Natural Science Foundation of China under Grants 60801005, 60531020, and 60701007, in part by the NCET-07-0750, the Zhejiang Provincial Natural Science Foundation under grants R1080320 and R105253, the Ph.D Programs Foundation of MEC (No. 200803351025), the Office of Naval Research under contract N00014-06-1-0001, the Department of the Air Force under contract F19628-00-C-0002, and the excellent doctoral thesis foundation of Zhejiang University (08009A).
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