Anisotropy minimization via least squares method for transformation optics

Mateus A. F. C. Junqueira; Lucas H. Gabrielli; Danilo H. Spadoti

doi:10.1364/OE.22.018490

1. Introduction

In recent years, the transformation optics (TO) technique has attracted the attention of researchers due to the great freedom that it allows in designing electromagnetic devices with unconventional functionalities. To make use of TO, coordinate transformations describing the desired functionality is necessary. TO then maps that transformation to the actual non-medium that will make the final electromagnetic device. However, the media obtained via TO are, in general, both inhomogeneous and anisotropic, which, renders them difficult to obtain in practice. The former is a fundamental part of TO, but to overcome the latter, researchers have proposed the use of quasi-conformal mappings [1], which minimizes the anisotropy [2] in the original coordinate transformation.

This paper introduces a new technique of quasi-conformal mapping for transformation optics with the application of the least squares method (LSM) to reduce anisotropy. To this end, perturbation functions in the form of power series are added to the coordinate transformation. The power series coefficients can be optimized via LSM to algebraically design electromagnetic media with extremely low anisotropy while meeting all boundary conditions specified in the initial design. Finally, we present our technique particularized for the treatment of waveguide-based photonic devices, and demonstrate it by designing a 30° bend with simultaneous width scaling.

2. Theoretical development

According to Fermat's physical principle, a ray of light travels through the shortest optical path between two points. If on the one hand this principle allows one to calculate the path traveled by light from the knowledge of the refractive index distribution of the environment, on the other hand it does not permit the derivation of a suitable media based on prescribed optical paths. This inverse problem is what TO aims to solve, and not only in the ray optics domain, but also in the full vectorial electromagnetic field model.

2.1 Transformation optics

The TO theory is based on the fact that Maxwell's equations are invariant to coordinate transformations. It has been shown that the space deformation imposed by a coordinate transformation is maps the path traveled by light in a corresponding medium [3] with permittivity and permeability tensors defined by the transformation itself.

Considering the transformation from a cartesian system, the electric permittivity, $ε^{'}$ , magnetic permeability, $μ^{'}$ , of transformed medium are obtained from [3]:

{ε^{'}}^{i j} = ε \sqrt{g} g^{i j}

{μ^{'}}^{i j} = μ \sqrt{g} g^{i j}

where,

g^{i j}

is the metric tensor of the coordinate transformation and

g

is its determinant.

Despite the mathematical robustness of the TO technique, its experimental demonstration is not trivial, because, in general, coordinate transformations result in anisotropic media, as indicated by the previous expressions.

2.2 Coordinate transformation

For a two-dimensional coordinate system, one can define a coordinate transformation (CT) through the use of a complex function $w : ℂ \to ℂ$ (real and imaginary parts represent the coordinates in each dimension). If a complex function $w (z)$ is differentiable in $D \subset ℂ$ , then $w (z)$ is said to be analytic in D. The function is analytic if, and only if, the Cauchy-Riemann conditions are satisfied. In turn, the coordinate transformation represented by $w (z)$ is said to be conformal [4].

In TO, a conformal transformation results in an isotropic medium [3], a desirable characteristic to simplify the fabrication of the final device.

Special attention, however, must be paid at the edges of the transformed region. In order to avoid reflections, the CT must be continuous between the transformed and untransformed media, which translates mathematically in the boundary conditions x' = x and y' = y [5].

These conditions cannot be met by a conformal transformation (except the identity), as discussed in [6]. This follows from the uniqueness theorem [4, Theorem 10.39] from complex analysis, which states that if a conformal transformation coincides with the original system in a certain region, then the transformation and the original systems must coincide at all points. Therefore, the use of a conformal transformation for the benefit of an isotropic medium will unavoidably result in reflections at the interfaces as well.

Alternatively, one can use quasi-conforming mappings [1] (mappings presenting reduced anisotropy [2]) that may or not meet the boundary conditions to avoid reflections. These mappings have small deviations from the Cauchy-Riemann conditions [1]. For example, the well-known quasi-conformal technique proposed by Li and Pendry [2] effectively minimizes the average anisotropy, but due to its sliding boundary conditions, the technique end up generating reflections at the edges of the transformed medium.

Chang et al. [7], in a different way from the one in [2], also proposed a strategy to reduce the anisotropy, but still with the same disadvantage of using sliding boundary conditions. Other variations for reducing anisotropy via slipping boundary conditions can be found in [8] and [9]. These works demonstrated that quasi-conformal mapping can be used to perform practical devices such as lens, bends and expanders [7–13].

Another quasi-conformal mapping technique was proposed by Liu et al. in [13], called Transformation Inverse Design, where perturbation functions in the form of Chebyshev polynomials and sine and cosine series are added to the coordinate transformation. Later, numerical optimization is used to calculate the coefficients of those series. The format of the transformation media is not predefined, allowing for a larger search space in the optimization problem. Among the constraints employed are the boundary conditions at the interfaces and the limiting values for the refractive index distribution.

Although the mentioned techniques of quasi-conformal are 2D mapping, it can be easily extend to 3D by extruding or rotating the obtained refractive index profile. In [11] is presented a three-dimensional broadband and broad-angle lens by rotating an isotropic refractive index profile.

Once a quasi-conformal mapping is obtained, we can assume the medium is isotropic by ignoring the residual anisotropy. As presented in [14], the refractive index distribution may be implemented, for example, with a pattern of holes (air, n = 1.0) in a silicon (n = 3.5) substrate for integration with conventional silicon photonics devices.

In this work, we present a new method to obtain quasi-conformal mappings in 2D through the use of simpler perturbation functions that can be optimized via LSM. Our technique allows us to choose which boundaries will be free in the optimization (ensuring a large search space) and which will be constrained, such that reflections are avoided and the original device requirements are preserved.

To quantify the residual anisotropy in the transformed medium we use the measure K defined below [15]:

K = \frac{t r (J^{T} \cdot J)}{2 \cdot \det (J)}

where, J is Jacobian matrix of the CT. Assuming a quasi-conformal transformation, the isotropic index of refraction for the resulting medium can be approximated by:

{n^{'}}^{2} = n^{2} \cdot t r (\frac{\sqrt{g}}{2} g^{i j})

3. Mathematical development

To perform the minimization by LSM, we write the CT functions in the form:

x^{'} = f_{x} (x, y) + b (x, y) Σ_{i = 0}^{p} Σ_{j = 0}^{q} A_{i}_{j} x^{i} y^{j}

y^{'} = f_{y} (x, y) + b (x, y) Σ_{i = 0}^{p} Σ_{j = 0}^{q} B_{i j} x^{i} y^{j}

where,

f_{x}

and

f_{y}

are coordinate transformation functions which present desired boundary conditions; p and q are the orders of the polynomials in x- and y-directions, respectively.

A_{i j}

and

B_{i j}

are coefficients to be determined;

b (x, y)

is a helper function which ensures that the power series do not interfere with the boundary conditions of the original transformation, i.e.,

b (x, y)

is designed to be zero at the required boundaries. It is possible to demonstrate that this function does not limit the space of search if it is chosen to vanish only on the boundaries.

The coordinate transformation defined by (5) and (6) has the advantage of being general, allowing its application for any transformation (with adequate choice for $b (x, y)$ ) and, automatically, absorbing the boundary conditions (facilitating the optimization) without losing of simplicity.

In TO, the boundary conditions usually determine the functionality of the desired optical devices. Therefore, it is of paramount importance that the anisotropy minimization technique does not change the boundary conditions for the engineered transformation.

3.1 Transformation optics for planar waveguides

Although the proposed technique is general, we will give special attention to the case where TO is used to design waveguide-based devices. As previously mentioned, the term $b (x, y)$ must vanish at the device boundaries to avoid reflections. This condition alone is, however, not sufficient. To ensure that the waveguide and/or the electromagnetic wave fronts are planar at the input and output boundaries we must also satisfy:

{\frac{\partial b (x, y)}{\partial x} |}_{b n d} = 0 o r {\frac{\partial b (x, y)}{\partial y} |}_{b n d} = 0

Assuming these conditions are met for the case of a waveguide with the propagation along the y-axis in the original medium (see Fig. 1), the wave propagation vector at the output of the transformed medium will have angle θ relative to the x'-axis such that:

Fig. 1 General transformation applied to a y-directed waveguide. A LSM optimizes the transformation with sample points chosen iteratively.

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t g (θ) = \frac{\frac{\partial y^{'}}{\partial y}}{\frac{\partial x^{'}}{\partial y}} = \frac{\frac{\partial f_{y}}{\partial y}}{\frac{\partial f_{x}}{\partial y}}

We also define φ, the device output boundary angle:

t g (φ) = \frac{\frac{\partial y^{'}}{\partial x}}{\frac{\partial x^{'}}{\partial x}} = \frac{\frac{\partial f_{y}}{\partial x}}{\frac{\partial f_{x}}{\partial x}}

Figure 1 illustrates the physical meaning of angles and in the transformed structure.

If the propagating angle θ is not constant over the guide output facet, the electromagnetic wave fronts exiting the device are not planar. Similarly, if the angle φ is not constant, the waveguide output itself is not flat.

3.2 Application of the least squares method

Once the coordinate transformation is defined according to expressions (5) and (6), we use the LSM to determine the coefficients $A_{i j}$ and $B_{i j}$ that minimize anisotropy in the transformed medium. To do so, we create a set of ‘k’ sample points in the non-transformed medium as shown in Fig. 1.

Ideally, the number of points must be greater than the total number of degrees of freedom in the power series, i.e., $k > (p + 1) (q + 1)$ , to avoid oscillations in the polynomial interpolation.

Considering the Cauchy-Riemann condition and the coordinate transformation functions (5) and (6), we calculate the error, E, summed over each sampling point:

E (A, B) = \sum_{i = 0}^{k} {[{(\frac{\partial x^{'}}{\partial x} - \frac{\partial y^{'}}{\partial y})}^{2} + {(\frac{\partial x^{'}}{\partial y} + \frac{\partial y^{'}}{\partial x})}^{2}]}_{i}

Taking the gradient of E with respect to the coefficients $A_{i j}$ and $B_{i j}$ , and equating it to zero, one obtains a linear system with $2 (p + 1) (q + 1)$ equations whose solutions are the coefficients $A_{i j}$ and $B_{i j}$ minimizing E.

In order to reduce the ratio between peak and mean anisotropy in the transformed medium we selected the sampling points iteratively according to the algorithm presented in Fig. 2.

Fig. 2 Algorithm used to choose the sampling points for the LSM.

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This algorithm allows a better use of the power series because it reduces the ratio between peak and mean anisotropy to a value less than 'maxR', which must be specified in the project. To this end, the algorithm sample the point with highest anisotropy and excluding (or not) the sampled point with lowest anisotropy. In this procedure, the number of sampled points tends to increase. There are evidences [13] that the peak anisotropy has greater importance than the mean in the device performance.

The ratio reduction between peak and mean anisotropy intensify the reduction of the peak anisotropy for the media and the LSM guarantees that the error function ‘E’, with the sampled points, is the global minima, helping to avoid poor local minima for peak anisotropy that may result of the non-linear optimization. Besides, there are some evidences that the ratio reduction between peak and mean anisotropy, with simultaneous application of the LSM, may be a direction for the global minima of the peak anisotropy.

It is not possible to restrict, algebraically, the refractive index limits, however we can consider different power series orders and chose one of them with lowest anisotropy, satisfying the refractive index limits, as demonstrated in Table 1.

Table 1. Conditions. Peak and Mean Anisotropy. Maximum and Minimum Refractive Index

View Table

The Cauchy-Riemann conditions applied to expressions for the angles θ and φ result in the following expression when anisotropy reduction occurs:

t g (θ) \cdot t g (φ) \approx - 1

This expression implies that the wave propagation vector in the output tends to be perpendicular to the waveguide output edge when anisotropy reduction occurs, as one would expect. If the medium becomes completely isotropic (which does not occur due to the uniqueness theorem), then we would have equality in expression (11).

It is important to notice that on the boundaries, where both conditions in (7) are simultaneously met, the anisotropy cannot be reduced on the boundary, since it will be solely defined by functions $f_{x}$ and $f_{y}$ , or equivalently, both angles θ and φ can be defined.

In principle, we could extend the application of the LSM for a 3D transformation, achieving conformal map with low anisotropy. However, the 3D transformation is limited, as is demonstrated by the Liouville's theorem for conformal mapping [16, 17], by a composition of translations, similarities, orthogonal transformations and inversions, i.e., only the Möbius transformations [4] are conformal in 3D and it could be a startup to application of the LSM for 3D transformations.

3.3 Transformation example

We present a general transformation to create a bent waveguide of arbitrary angle ( $φ$ ), output position ( ${x^{'}}_{o}$ , ${y^{'}}_{0}$ ), and expansion or compression factor $(e)$ . These parameters are pictured in Fig. 1. We use the general transformation:

x^{'} = ({x^{'}}_{o} + e \cdot \cos φ \cdot x) \cdot (\frac{y}{y f}) + x \cdot (1 - \frac{y}{y f})

y^{'} = ({y^{'}}_{0} + e \cdot \sin φ \cdot x) \cdot (\frac{y}{y f})

Where ‘y = 0’ and ‘yf’ are, respectively, the input and output boundaries, and the original medium is symmetric across the y-axis. This coordinate transformation satisfies the boundary conditions and may become highly anisotropic depending on the chosen parameters.

4. Results

In this section, we present the results of using the LSM for anisotropy reduction in a transformation of the form (12) and (13) with $φ = - 30 °$ and expansion factor $e = 1.5$ :

x^{'} = (4 + 1.5 \cdot \frac{\sqrt{3}}{2} \cdot x) \cdot (\frac{y}{10}) + x \cdot (1 - \frac{y}{10}) + \sin (π y / 10) \sum_{i = 0}^{q} \sum_{j = 0}^{p} A_{i j} x^{i} y^{j}

y^{'} = (9 - 1.5 \cdot \frac{1}{2} \cdot x) \cdot (\frac{y}{10}) + \sin (π y / 10) \sum_{i = 0}^{q} \sum_{j = 0}^{p} B_{i j} x^{i} y^{j}

The original region is defined in $- 1.2 μ m \leq x \leq 1.2 μ m$ and $0 \leq y \leq 10 μ m$ . The input signal is a Gaussian beam with wavelength $1.55 μ m$ applied in the $y = 0$ boundary. The output is measured at $y = 10 μ m$ . In the transformed device we expect a planar Gaussian wave front at the output with 1.5 times wider waist and a 30° rotation in the propagation direction. The first condition in (7) is satisfied allowing to define the angle $φ$ . We considered different polynomial orders for the power series and an evaluation grid of 240 × 1000 points. The parameter maxR used was 3.

After determination of the coefficients with the LSM for each case, the resulting transformation was assessed by their peak and mean anisotropy. Each case and their corresponding results are shown in Table 1. In particular, case 1 represents the original transform with no anisotropy reduction.

The results in Table 1 show the effectiveness of the technique for reducing the peak and mean anisotropy. These results suggest that the residual anisotropy can be made as small as needed (but never exactly zero) by increasing the perturbation polynomial orders. This reduction comes at a price, nonetheless: the increased refractive index contrast, necessary to implement the transformed medium. The waveguide bends designed in [8] and [9], after the reduction of the anisotropy, also resulted in increased refractive index contrast.

We also assessed the conditions in Table 1 using another $b (x, y)$ function. Our results appoint an influence reduction of this function for anisotropy, and, the resulting transformations are convergent when power series orders increase, independently of the choice of $b (x, y)$ , through modification of resulting coefficients A_ij and B_ij. This is possible since the $b (x, y)$ function has no effect in the space of search in the optimization problem.

Figure 3 shows the distortion by the transformation for non-transformed medium, case 1 and case 7. We can see the effect of anisotropy reduction in preserving of the angle between oriented curves.

Fig. 3 Simple example of coordinate transformation. (a) Non-transformed media. (b) Transformed media with no anisotropy reduction (case 1). (c) Transformed media with anisotropy reduction (case 7).

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The optimized devices were simulated by finite element method, using the software COMSOL, to evaluate the behavior of the Gaussian beam in the multimode waveguide. The simulated devices were made of isotropic materials, so that the effects of any residual anisotropy would reflect in deformations in the output beam. Figure 4 presents the results for cases 1 and 7 in Table 1.

Fig. 4 Numerical simulations results. (a) Original medium. Only the core and cladding regions are transformed, not the entire simulation domain. (b) Electric filed distribution over the original transformed medium, without anisotropy reduction (case 1). The simulated medium is isotropic, therefore any anisotropy in the transformation results in propagation distortions, as demonstrated by the cross-section plot (line plot) of the device output. (c) Same as (b) but with anisotropy reduction (case 7). The beam propagates through the structure without distortions. (d) Refractive index profile for case 1. (e) Refractive index profile for case 7.

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The results show a significant improvement of the waveguide performance after the reduction of anisotropy. In case 1, the presence of beam deformation (due to mode conversion) is notable, in contrast to case 7, where the beam shape is preserved along the propagation direction (except, of course, for the intended expansion factor of 1.5).

5. Conclusion

This paper presented a new technique for reducing anisotropy in transformation optics with the application of the least squares method. In particular, we developed a mathematical framework to design and to analyze waveguide-based transformations, which was used to exemplify the proposed technique. Our method is able to effectively reduce the anisotropy in the transformed medium close to zero with the benefit of being straightforward to implement.

Acknowledgments

This work is supported by CNPq, CAPES and FAPEMIG

References and links

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Condition	p	q	$m a x (K - 1)$	$a v g (K - 1)$	n_max	n_min
1	-	-	0.1395650	0.0509228	2.046	1.196
2	0	0	0.0947237	0.0324140	2.483	1.124
3	1	1	0.0110468	0.0036061	2.566	1.083
4	2	2	0.0020726	0.0007597	2.765	1.052
5	3	3	0.0002804	0.0000946	3.062	1.038
6	4	4	0.0000880	0.0000309	3.624	1.017
7	5	5	0.0000131	0.0000044	4.332	1.015

Anisotropy minimization via least squares method for transformation optics

Abstract

1. Introduction

2. Theoretical development

2.1 Transformation optics

2.2 Coordinate transformation

3. Mathematical development

3.1 Transformation optics for planar waveguides

3.2 Application of the least squares method

3.3 Transformation example

4. Results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (15)

Optics Express