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Arbitrary connector for waveguides of different cross sections is proposed and designed theoretically based on the embedded optical transformation theory. First, the general expressions of constitutive tensors of the metamaterials filled in the connector are derived. Second, there are some full-wave simulations that validate the constitutive tensors derived. The results show that the connector with metamaterials inclusions with designed constitutive parameters can fulfill the reflectionless transmission of electromagnetic waves between waveguides of different cross sections. Finally, connectors of several forms are investigated parametrically, and two sets of constitutive tensors that can be physically achieved by existing metamaterials are gotten. It is believed that this study provides a feasible way to fulfill the efficient transmission of electromagnetic waves between waveguides of different cross sections.
©2010 Optical Society of America
1. Introduction
In the past four years, great interests have been attracted in optical transformation, which was first proposed by Pendry. By employing the form invariance property of Maxwell’s equations in different spaces, transformation optics opens up many possibilities and methods to control the behavior of electromagnetic fields, among which the most attractive application is the invisibility cloak [1,2]. Since 2006, cloaks have been widely investigated, which results in many analytical, numerical and experimental results, including the analytical solutions of the cylindrical and spherical cloaks based on Mie scattering model [3,4], simplified cloak with suppressed 0th order scattering [5], the open cloak [6], the non-conformal cloak [7], cloak with layered structure [8–10], active exterior cloaking [11], theory and experiment on the carpet cloak [12–14], polygonal cloak [15], etc. Also there are many other interesting designs based on the embedded optical transformation, such as the super scatter [16], polarization controller [17], beam splitter [18], flat force lens [19], etc.
Recently, optical transformation theory has been applied to waveguide. Many theoretical studies, numerical simulations and parameter designs have been devoted to the wave-bending structures [20–25]. In this paper, we put forward arbitrary waveguide connector based on embedded optical transformation, which is more general than waveguide taper in symmetrical form, and the constitutive tensor gotten is easier to be physically realized by 2D metamaterials [26]. We construct the coordinate transformation from free space to an arbitrarily compressed region, from which the inhomogeneous and anisotropic material properties are obtained. This research provides a feasible way to fulfill the lossless transmission of electromagnetic waves between different waveguides, and there also exists potential application in manipulating the direction of the beam and enhancing the gain of the horn antennas.
2. Derivation of the connector’s constitutive tensors based on embedded optical transformation
In this study, we focus our concentration on arbitrary waveguide connector, which is feasible for realization. Here finite embedded optical transformation is applied to the design of waveguide connector. The sketch for the waveguide connector is shown in Fig. 1(a) . Considering a two-dimensional structure in the Cartesian coordinate system, as shown in Fig. 1(b), the optical transformation that compresses/expands (the inverse problem) the original space ACBD into the transformed space ACB’D’ can be defined as:
where the length of AC is assumed to be 2a, the curve that connects C and D’ is defined as y 1(x), and the curve that connects A and B’ is defined as y 2(x). The functions of the two curves can be selected arbitrarily, so long as they can satisfy the numerical values at the points of A, B’ and C, D’, respectively. Hence the Jacobian transformation matrix can be gotten based on Eq. (1):which represents the derivative of the transformed coordinates with respect to the original coordinates. Using the property that Maxwell’s equations are form invariant in the original space and transformed space, the permittivity and permeability tensors of the medium in the transformed space can be expressed as: whereand represent the constitutive tensors of the original space. In this study, the original space is supposed to be free space, so the constructive tensors of the original space can be expressed as: Hence we can easily get the relative permittivity and permeability tensors in the transformed region as:
Fig. 1 Sketch of the waveguide connector. (a) Sketch of the connector, (b) Connector in Cartesian coordinate.
Furthermore, the symmetrical constitutive matrix can be transform into the diagonal matrix through rotating the coordinate, which will be more useful in the construction of metamaterial. The diagonal matrix can be expressed through the eigen values of the symmetrical constitutive matrix:
Above all, Eq. (5a)–(5c) provide full design parameters for the permittivity and permeability tensors of metamaterials filled in the waveguide connector. Next the constitutive tensors gotten above will be utilized to make full-wave simulations on arbitrary waveguide connectors.
3. Full-wave simulations and parametric investigations
In order to validate the constitutive tensors gotten above, we made full-wave simulations on arbitrary waveguide connectors using the COMSOL Multiphysics Software, which is based on the finite element method (FEM). The geometrical sizes of the waveguides are properly selected to make sure that the TE10 mode of 2GHz can be transmitted. The simulation domain is just like Fig. 1(a), and the port 1 is selected to illuminate the incident wave. Here it should be noticed that the simulations are done in the transformed space, but the constitutive parameters in Eq. (5a)–(5c) are expressed as the function of x and y, which are variables of the original space. Here the variables of the original space should be replaced by the variables of transformed space, which can be gotten through Eq. (1):
First, a simple connector of symmetrical structure is simulated to verify the designed formulae. The electric field distribution of the connector filled with metamaterials of designed constitutive parameters is shown in Fig. 2(a) , and the connector just filled with air is also simulated for comparison, as shown in Fig. 2(b). It can be seen that although the connector filled only with air can fulfill the transmission of electromagnetic waves from a big waveguide into a small one, there exists reflections and part of the energy is lost. For the connector filled with metamaterial of designed constitutive tensors, it is obvious that the electromagnetic waves are properly guided from the big waveguide to the small one without any impact on the guided mode. So the results’ validation is proven. Then several general models are also simulated, including the sharper connector and the unsymmetrical connectors, and the results are shown in Fig. 2(c)-(f). From the electric field distributions it can be seen that the connectors all work very well. There are few scattering fields, and the electromagnetic waves can transmit from the big waveguide into the small one properly.
Fig. 2 Electric fields distributions of different waveguide connectors (a) symmetrical connector, (b) connector filled only with air, (c) sharper symmetrical connector, (d) symmetrical curve connector, (e) unsymmetrical connector I, (f) unsymmetrical connector II, (g) connector for sloping transmission.
Second, it does make sense to carry out the parametric study to find out a set of constitutive tensors that is easy to be achieved by artificial metamaterials. The corresponding numerical values of the constitutive tensors of the models above are calculated and shown in Fig. 3 . It can be seen that due to the symmetrical structure, the numerical value of ε xy owes a part of negative values, no matter what kinds of boundary function y 2(x’) and y 1(x’) are selected. This also can be explained through Eq. (5b). When the connector is symmetrical, the non-diagonal component of the constitutive tensor can be simplified as:
Fig. 3 Numerical Values of the constitutive tensors, (a)-(c) Symmetrical connector, (d)-(f) Sharper symmetrical connector, (g)-(i) Symmetrical curve connector, (j)-(l) Unsymmetrical connector I, (m)-(o) Unsymmetrical connector II
For that case, the function y 2(x’) must be monotone decreasing at some numerical range of x’, this results in the negative value of ε xy. For the unsymmetrical structures, the function y 2(x’) and y 1(x’) can be selected properly, and all the components of the constitutive tensor can be kept positive, as shown in Fig. 3(j)–3(o). Although one or two components maybe less than 1, these two sets of constitutive tensors are easy to be physically achieved by existing 2D metamaterials.
Finally, the connector for waveguides of the same geometrical size but with displacement in y-direction is simulated, as shown in Fig. 2(g). The constitutive tensor of the connector can be simplified on the condition that the functions of y 2(x’) and y 1(x’) are selected as lines with the same slope of k 0. The simplified constitutive tensor can be expressed as:
It is obvious that it can be physically realized even with ordinary homogeneous and anisotropic material when k 0≥1.
4. Conclusion
Based on the embedded optical transformation, an arbitrary connector for waveguides of different cross sections is proposed and designed theoretically in this paper. The general expressions of constitutive tensors of the connector are derived. Then there are full-wave simulations. From the results it could be seen that the connector can guide the incident electromagnetic waves propagating from the big waveguide to the small one efficiently. Furthermore, the parametric study is also conducted. The factors that have impacts on the numerical values of the constitutive tensors are pointed out and analyzed, and then two sets of constitutive tensors that are easy to be achieved with existing metamaterials are obtained. Finally, the connector for waveguide of the same sized but with displacement in y-direction is also studied. The results show that it even can be constructed by ordinary homogeneous and anisotropic materials. It is believed that the results are helpful to fulfill lossless transmission of electromagnetic waves between different waveguides.
Acknowledgement
Project supported by the National Natural Science Foundation of China (Grant Nos. 60801015 and 60971064), Ph.D. programs foundation of Ministry of Education of China (No. 200802131075), the Open Project Program of State Key Laboratory of Millimeter Wave (No. K201006, K201007).
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