Detailed design of a waveguide structure based on conformal transformation

introduction

Pendry et al. [1] proposed an interesting idea. The idea is to use the idea of ​​coordinate transformation (ie, transform optics) to achieve the ability to regulate electromagnetic waves. Unfortunately, most of the relative permittivity and magnetic permeability of electromagnetic wave-wave device based on coordinate transformation are often heterogeneous anisotropic materials [2][3][4], which brings about great realization. difficult. Conformal transformation is a special case of optical transformation. It can make isotropic materials in two dimensions. After the conformal transformation, the Laplacian equation itself becomes a coefficient. After the conformal transformation of the two-dimensional Helmholtz equation, the coefficient is reflected in the refractive index change, that is, after the conformal transformation, if the refractive index The rate changes with the coefficient, then satisfies the same wave equation, so as to achieve the ability to modulate the electromagnetic wave. In this paper, we design an electromagnetic waveguide wave structure based on the conformal transformation theory. Its material is non-uniformly isotropic, which reduces the difficulty in implementation.

1. Conformal shape optical transformation theory

The space before transformation coordinate conversion is w, the refractive index of this space is 1; the space after coordinate transformation is z, the refractive index of this space is n, then the refractive index of z space: [5][6] n=|dw /dz|, optical theory has been proposed for several years, although its idea is very new, but because the electrical parameters of most of the electromagnetic devices designed according to this idea are inhomogeneous anisotropy, many scholars and experts continue to simplify the parameters, the paper reference [7] According to the logarithmic transformation, a waveguide adapter that can be turned is designed, but we do not think it can achieve a perfect switching effect because the truncation of the spatial refractive index will inevitably bring about scattering. In this paper we have Based on the principle of conformal transformation, an isotropic and non-uniform guided waveguide device is designed. Although the optical transformation theory has been popularized in guided wave structures for many years, electromagnetic wave-wave devices with uniform electrical anisotropy have been implemented, and the structure of guided wave turning based on the logarithmic transformation of the logarithm has also been realized. It is believed that it cannot achieve a true waveguide transit structure, including a very complex function that is difficult to apply in a guided wave structure, because a spatial refractive index cutoff occurs at the boundary, thereby failing to achieve a perfect transition effect. In this dissertation, we seek out some feasible design functions to make this guided wave structure based on the conformal transformation become feasible. However, due to the problem of solving the function of the guided wave structure, the waveguide transfer can only occur in the case of going forward. This is also the limitation of our paper. We can mathematically express this by bending a certain area in the middle and leaving the guided wave structure in this area and returning to the original situation:

z=f(w)

(1)

Or for

w=g(z)

(2)

Condition (1) and condition (2) are intercepted where the index of refraction is close to one, and a bend occurs after the conformal transformation in the middle, so that obstacles encountered in the traveling direction of the waveguide can be bypassed.

2、Specific design of guided wave structure (unit m)

According to the above requirements we design a function whose expression is:

Z2=w2+k (3)

(4)

In the above expression we have k=0.03.

Figure 1. Transformation from w-space to z-space (coordinates are marked according to the above coordinate system)

Fig. 2. Ez distribution of H10 mode before conformal transformation (f = 4.6 GHz)

Fig. 3 Ez distribution of H10 mode after conformal transformation (f = 4.6 GHz)

Fig.4 Ez distribution of H10 mode before and after conformal transformation in three-dimensional case (f=4.6GHz)

Fig. 5 Refractive index profile after conformal transformation in three-dimensional case

In the upper three-dimensional case, the middle is a conformal transformation region. In the region where the refractive index is close to one away from the optical transformation region, the height is one-half the width, and the cross-section is the same as the two-dimensional case. Both sides are truncated at a refractive index close to one, and it can be seen that the conformal transformation can achieve a very good waveguide transition structure because the conformal transformation theory is strictly derived from Maxwell's equation, and we are only at the refractive index Truncation occurs close to one place, so this waveguide structure can be transmitted as long as it can be transmitted in a rectangular waveguide in the original space. From the above implementation of the conformal transformation and the non-performing conformal transformation, it can be more clearly seen that the conformal transformation can make the waveguide pass around a certain area and the electromagnetic wave can still be transmitted well. Of course, it is only limited to the rectangular waveguide H10. Modulus, but in the guided wave communication rectangular waveguide has always been used in the mode of the main mode, so this can not be considered as a limitation of this method, as long as the two conditions (1), (2) are satisfied, then use today's computer technology. It is possible to achieve guided wave structures that bypass various shapes. We also believe that with the development of computers, after the solvability of equations is improved, we can also achieve arbitrary guided wave structures because we do not have a good computer. We hope that People are exploring this issue. We believe that this method can be generalized to dielectric waveguides. The middle field concentration area of ​​the following image is the same as the above two-dimensional waveguide, and the refractive index in the middle is changed from 2 in the original space to the refractive index in the current space:

The refractive index of the region on both sides is one, and the frequency is 3 GHz.

Figure 6. Ez distribution of two-dimensional dielectric waveguides

From the above figure we also see that our design method based on conformal transformation is not limited to rectangular waveguides. For dielectric waveguides, if the electromagnetic field is concentrated in a certain area, only the conformal transformation of the region is also applicable, of course, The plasmon waveguide can be used. The metal above is the medium. In the plasmon waveguide, the dielectric constant of the metal is negative. To make the theory of the optical transformation based on the conformal transformation more effective, the absolute value of the dielectric constant of the metal is required. It is much larger than the dielectric constant of the medium, ensuring that the electromagnetic wave decays rapidly in the metal and conformal transformation in the medium.

(5)

(6)

Simulation parameters are as follows:

1). Co = 0.03, the relative dielectric constant of the metal is -50, the original medium has a spatial refractive index of 1 and a frequency of 4 GHz.

2). The original media space geometry is as follows:

3). The above w-space region passes the following conformal transformation: As shown in the middle area of ​​FIG. 6 , the refractive index in the middle is already the distribution of the electric field in the z direction after the conversion, and the refractive index of both sides is truncated to be 1, so that an obstacle can be placed under the plasmon waveguide.

The simulation results are as follows:

Fig.7 Ez distribution after conformal transformation of 2D plasmon waveguide

Fig. 8 Ez distribution of the parameters of the two-dimensional dielectric wave dielectric still maintaining the original w-space

Obviously, the above plasmon waveguide is an impractical model, because its frequency is very low, gold and silver can only show negative relative permittivity at a frequency much higher than this simulation frequency, but Maxwell's equation itself has a similarity principle. We can easily transform the frequency to the corresponding frequency and adjust the corresponding geometric parameters and the size of Co. From the above simulations, we can see that the value of the conformal transformation theory is not only reflected in the waveguide, but also in the plasmon waveguide. There are dielectric waveguides.

3, the conclusion

Based on the theory of conformal transformation, this paper designs the transfer problem of the electromagnetic wave in the guided wave structure when it bypasses an obstacle in the original direction of advancement. It can be seen that this design method can be used to achieve almost the same field with two-dimensional analysis. Perfect transfer. The electrical parameters of this design method (the magnetic permeability and relative permittivity are not uniform, but they are isotropic. Although the refractive index is less than one, as long as the electrical parameter is increased by a certain value, it will be All designs with an index of refraction greater than one are easier to implement, although non-uniform, but as long as discrete sampling is used, this method is limited in that it must leave the obstacle and must be restored to its original space. The main limitations of this paper.

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