1. Introduction

Transformation Optics (TO) [1–3] is a powerful technique to control the propagation of electromagnetic waves using coordinate transformations, under which Maxwell’s equations are invariant. TO can be used to design devices with very interesting applications, such as: invisibility [1, 4–8], perfect lenses [9–11], among others. Moreover, theoretical and practical applications for waveguides using TO were demonstrated in [12–17]. Unfortunately, TO usually results in unconventional material properties, particularly anisotropic and inhomogeneous permeability and permitivity tensors with strong magnetic response. Quasi-conformal maps [4, 14–16] can be used to avoid anisotropy, but most techniques to obtain such maps are exclusively 2-dimensional, i.e., they can only be applied for 2D transformations, whereas, in a 3D device extrusion or rotation [11, 18, 19] could be used to obtain the resulting refractive index map. However, extrusion is not applicable for some geometries lacking the necessary symmetries, such as a cylindrical waveguide bend. Moreover, the wave propagation control is only achieved for waves traveling in the original 2D planes and do not directly extend to full 3D propagation.

A 3D quasi-conformal map does not have these limitations. However, obtaining this map could be a hard task due to Liouville’s theorem for conformal mapping [20]. According to this theorem, only Mobius transformations are conformal in the 3D and they are limited to a combination of translations, similarities, rotations, and inversions which has very limited application in TO. Moreover, a 3D version of the Cauchy-Riemann equations does not exist, which represents an additional difficulty to achieve a 3D quasi-conformal mapping for TO application. In [21] the authors show that anisotropy reduction is possible in 3D TO by adapting the method from [22] for a 30° bend and in a compressor. However, their method does not exactly match the boundary conditions at interfaces and reflections may arise there. Moreover, their method can not be applicable for domains, an essential feature for use in invisibility.

In this paper, a new technique to achieve 3D quasi-conformal TO is proposed. It is based on the parametrization and numerical optimization of the coordinate transformation via a quasi-Newton method. Our technique is demonstrated in a cylindrical waveguide 90° bend and in a cylindrical waveguide S bend. As indicated by the results, the anisotropy can be made as small as wanted in the whole transformed volume.

2. Development

In TO, the coordinate transformation boundary is the key for the device functionality, while internal points play a less important role and can be adjusted to minimize the anisotropy in the resulting medium. In our work, the coordinate transformation is embedded with parameters that allow the use of numerical optimization algorithms to perform the minimization. We take special care in parametrizing the transformation so that it does not interfere with the predefined boundaries and, consequently, with the device functionality. The general parametrized transformation used here is an extension of our previous method [14] for 3D geometries:

Therefore, if the off-diagonal terms in (2) vanish and the diagonal terms are equal, the resulting medium will be isotropic. In that case, ignoring the magnetic response, the refractive index is $n(x,y,z)=n_0{\left(h_{11}^2+h_{21}^2+h_{31}^2\right)}^{-1/2}$. Thus, we define the following cost function:

The term K_3D is dimensionless and it can be understood as a measurement of the local material anisotropy. In this work, the maximum and average values for this term over the evaluation grid are considered the maximum and mean anisotropy measurements, respectively. Thus, the minimization of F also represents the minimization of the mean anisotropy.

A quasi-Newton method with algebraic gradient is used for optimization. Any method using Hessian matrix, such as the Newton Method in the optimization, would become prohibitive due to its computing, which increases quadratically with the number of degrees of freedom (DoF) in the parametrization. If the objective were to minimize the maximum anisotropy (maxK_3D), we could use just the numeric gradient, which represents an additional computational work.

3. Results and Discussion

We use 2 cylindrical waveguides curves to demonstrate the effectiveness of the proposed method for 3D TO. Both waveguides have the same initial domain: $\sqrt{x^2+z^2}\le 600\hspace{0.17em}\text{nm}$, 0 ≤ y ≤ 10μm, and the same evaluation grid with 25 × 100 × 25 points. The non-transformed waveguides have a cylindrical core with radius of 200 nm and refractive index of 2.5, and cladding index of 1.5 extending over the whole domain outside the core. All simulations are performed for a free-space wavelength of 1.55 μm. The waveguides initial transformation is:

The waveguide input and output facets define the device functionality, therefore the boundary function must vanish at those surfaces. One choice for this function is: b(x, y, z) = sin(πy/y₁). This approach also guarantees the transformation continuity at the boundaries.

Depending on the parameters in (5), different bends can be designed. We present results for both a 90° and an S bends, where we investigate the effect of the number of DoF provided by the parametrization in the maximum and mean anisotropies, as well as the compromise between anisotropy minimization and the maximal refractive index contrast in the resulting medium. Finally, wave propagation simulations were performed using the finite element method in order to confirm modal preservation in the waveguide bends when the residual anisotropy is ignored.

Our parametrization approach allows its application in definition of domains not connected such as the case of invisibility through the convenient choice of b(x, y, z). It is important to note, however, that the anisotropy may not be reduced to arbitrarily small values in this cases. Furthermore, our method is applicable for any area of the physics which uses coordinates transformation once the J^T J term is found in these applications such as acoustics and quantum mechanics.

Although the cost function is not convex, the optimization converges when the initial transformation is bijective. However, approaches to avoid poor local minima can be used, such as the use of a multistage procedure starting with low number of parameters [23].

3.1. 90° cylindrical waveguide bend

An initial transformation for a 90° bend is obtained with x₀ = y₀ = 7μm, e = 1, and ϕ = −π/2 in (5). Table 1 shows the results for the maximum and mean anisotropies normalized by the mean anisotropy of the original transformation, as well as the maximum and minimum refractive indexes for increasing the number of DoF. The results indicate that anisotropy can be made as small as wanted with the increase of the number of DoF. However, there is a trade-off between the anisotropy minimization and the increase of the refractive index contrast, which could render the device more difficult to be fabricated and, additionally, increases its losses.

Table 1. Anisotropy and refractive index for different DoF for the 90° waveguide bend.

View Table | View all tables in this article

Although complex, fabrication of 2D TO devices has been demonstrated [24–26]. For 3D transformations, these methods can be extended using, e. g., a 3D printing approach [27, 28].

The original and final coordinate transformations are shown in Fig. 1. After the optimization, the coordinate lines are perpendicular to each other at intersections using p = r = 4 and q = 5, a distinctive sign of a quasi-conformal transformation. Figure 1 also shows the frequency domain simulations and field profiles at input and output facets of the waveguides. Here the requirement of anisotropy minimization is confirmed: the optimized bend preserves the modal profile almost perfectly along the waveguide length, in strong contrast to the original transformation.

 

Fig. 1 Results for the 90° bend. (a), (b), (c) and (d) are the top view, the perspective view, the media simulation and the refractive index map, respectively. (e), (f), (g) and (h) the same as (a), (b), (c) and (d) but for the optimized media with p = r = 4 and q = 5.

Download Full Size | PDF

We have also investigated the impact of the bend radius of the waveguide in our optimization method. By decreasing the radius without changing the diameter of the waveguide we have found that it is still possible to reach anisotropy values as small as wanted at the price of increased both refractive index contrast and DoF’s (increased values of p, q and r).

3.2. Cylindrical waveguide S bend

For the waveguide S bend, we define x₀ = 3μm, y₀ = 10μm, e = 1, and ϕ = 0. Similar results to those presented for the 90° bend are shown in Table 2 and Fig. 2. Once again we see that the anisotropy could be made arbitrarily small with the increase in the number of DoF at the expense of higher refractive index contrasts in the resulting medium. The optimized transformation preserves the angles from the initial domain and the modal profile along the device as it did in the previous case. We point out that all of the above results are similar to what is observed in the 2D case [4, 14, 15], where the use of the Cauchy-Riemann conditions make it straightforward to define a measurement for anisotropy.

Table 2. Anisotropy and refractive index for different DoF for the waveguide S bend.

View Table | View all tables in this article

 

Fig. 2 Results for the S bend. (a), (b), (c) and (d) are the top view, the perspective view, the media simulation and the refractive index map, respectively. (e), (f), (g) and (h) the same as (a), (b), (c) and (d) but for the optimized media with p = r = 4 and q = 5.

Download Full Size | PDF

We point out that for a set number of DoF, the optimization result is independent of the exact shape of the initial transformation or boundary function, i.e., the choices of f_x, f_y, f_z, and b, as long as the fixed boundaries remain the same. The convergence to the same solution independent of the initial conditions indicates that the proposed method could be achieving the global minimum of the cost function at least in the presented examples.

4. Conclusion

This paper presented a parametrization and optimization technique to achieve quasi-conformal mapping in 3D TO, giving designers much more flexibility than previously possible simply by extruding a 2D transformation. The numerical results confirm the possibility of achieving negligible residual anisotropy, whose omission does not affect mode propagation along both cylindrical waveguide examples. The method does not use boundary sliding conditions and undesired reflections are avoided. Moreover, the method can be applied for domains not simple connected enabling its application in invisibility. Finally, the presented results may find direct application in the development of multimode fibers with minimal mode coupling.