Poles of the scattering matrix: an inverse method for designing photonic resonators

Brian Slovick; Erik Matlin

doi:10.1364/OE.378116

1. Introduction

Inverse methods have found widespread application in optics and photonics [1–3]. In contrast with the traditional design approach, which relies heavily on intuition, the goal of inverse modeling is to develop and implement efficient algorithms to derive the material properties directly from the desired optical functionality. Inverse methods have been applied to optimize bandgaps of photonic crystals [4–6], enhance mode propagation and coupling in waveguides [7–12], and to improve light trapping for solar energy harvesting [13–15].

More recently, inverse methods are being applied to design resonant structures for nanophotonics applications, including metamaterials [16,17], metasurfaces [18,19], and topological photonics [20,21]. In particular, the use of resonance to enhance nonlinear interactions holds great promise [22–27]. Compared to bulk materials, resonant structures provide larger field enhancement and longer interaction times, resulting in nonlinear processes at lower pump powers and within smaller volumes [22,24,28]. In addition, the field enhancement in resonant structures relaxes the phase-matching requirement for harmonic generation [22,24].

For certain applications, it is beneficial to employ structures that support multiple resonances. For instance, the efficiency of second harmonic generation is improved in structures that support resonances at both the fundamental and second harmonic frequencies [27–37], and surface-enhanced Raman spectroscopy is enhanced in structures possessing resonances at the excitation and Stokes frequencies [38–40].

The challenge of designing structures with resonances at multiple prescribed frequencies is ideally suited for an inverse design approach [26,27,41]. For example, gradient-based topology optimization, which treats each pixel of the domain as a degree of freedom, has been applied to design doubly-resonant structures for efficient harmonic generation [27,41]. However, the existing optimization methods employ an objective which is a functional of the electric field. As a result, Maxwell’s equations must be solved for each topology, which is computationally expensive, time consuming, and limits the variety of structures that can be considered.

In this Letter, we develop a new inverse mathematical and computational framework for designing resonant scatterers that does not require iteratively solving Maxwell’s equations. Instead, our algorithm is based on optimizing a single constraint condition that defines the poles of the scattering matrix. Since the poles of the scattering matrix are associated with the eigenmodes of the system [42,43], our approach can be regarded as solving the inverse eigenvalue problem, i.e., determining a matrix from its eigenvalues [44]. This distinguishes our work from prior efforts, which focus primarily on solving the forward eigenvalue problem, e.g., to obtain photonic band structures [45]. Computationally, our algorithm operates by minimizing the determinant of the matrix representing the integral operator of the electric field with respect to the permittivity profile of the resonator.

Here we develop the approach and implement it to design several resonant structures. First, we show that a subwavelength structure of a given form factor can be designed to resonate at different prescribed wavelengths, and that the spectral cross sections are consistent with Fano lineshapes with $Q$s as high as 800. Then we show that the a structure with the same form factor can be designed to resonate at two prescribed wavelengths simultaneously. Such a doubly-resonant structure can be useful for enhancing second harmonic generation. The compact form, computational efficiency, and generality of our approach suggest it can be a useful inverse design tool for engineering resonant structures for enhanced light-matter interactions.

2. Methods

In this section we derive the condition for scattering resonances from the integral form of Maxwell’s equations. The advantage of working with the integral equation is that the boundary conditions inherent to the scattering problem are implicit. To derive the condition for scattering resonances, we discretize the integral equation, recast it into matrix form, and then solve for poles of the electric field.

We consider compact dielectric scatterers with relative permittivity $\epsilon _r(x)$, where $\epsilon _r(x)=1$ outside the compact region $\Omega$. We also assume isotropic materials with scalar $\epsilon _r$. The time-harmonic Maxwell’s equation for the electric field in the inhomogeneous medium is [46–48]

(1)$$\nabla \times(\nabla \times E(x))-k^2 (1-m(x))E(x)=0,$$

where $k=2\pi /\lambda$ and $m(x)$ is given by

m(x)\equiv 1-\epsilon_r(x),

where $m(x)=0$ outside $\Omega$. The electric field can be written as the sum of the incident field $E_i(x)$ and the scattered field $E_s(x)$, where $E_i(x)$ is a solution to the homogeneous Maxwell’s equation

(2)$$\nabla \times(\nabla \times E_i(x))-k^2 E_i(x)=0,$$

and $E_s(x)$ satisfies the Silver-Müller radiation condition [46–48]

(3)$$\lim_{|x|\rightarrow \infty} (\nabla \times E_s(x))\times x -ik|x|E_s(x)=0.$$

The equivalent integral equation form of Eq. (1), consistent with Eqs. (2) and (3), is [47]

(4)$$E(x)=E_i(x)-k^2\int_\Omega G(x-y)m(y)E(y)dy-\int_\Omega \frac{1}{1-m(y)}\nabla m(y) \cdot E(y)\nabla G(x-y)dy,$$

where $G(x)$ is the Helmholtz Green’s function. In three dimensions, $G(x)=\frac {e^{ik|x|}}{4\pi |x|}$ or in two dimensions, $G(x)=\frac {i}{4}H_0^{(1)}(k|x|)$, where $H_0^{(1)}(x)$ is the Hankel function of the first kind. We also make the following definitions

p^d(x)\equiv \frac{1}{1-m(x)}\frac{\partial m}{\partial x_d} (x), \quad G^d(x)\equiv \frac{\partial G}{\partial x_d}(x), \quad d=1,2,3.

To write Eq. (4) in matrix form, we discretize $\Omega$ on a uniform Cartesian grid with $n$ points per dimension and step size $h$, and write the convolution matrices as

G_{i,j}=h^3G(ih-jh), \quad G^d_{i,j}=h^3G^d(ih-jh),

where the subscripts denote discrete indices. With these notations, Eq. (4) can be expressed in the following matrix form [46–48]

(5)$$\begin{bmatrix} E^1 \\ E^2 \\E^3 \end{bmatrix}=\begin{bmatrix} E_i^1 \\ E_i^2 \\E_i^3 \end{bmatrix}-\begin{bmatrix} k^2Gm+G^1p^1 & G^1p^2 & G^1p^3 \\ G^2p^1 & k^2Gm+G^2p^2 & G^2p^3 \\ G^3p^1 & G^3p^2 & k^2Gm+G^3p^3 \end{bmatrix} \begin{bmatrix} E^1 \\ E^2 \\E^3 \end{bmatrix}$$

where $E^d$ and $E_i^d$ are discrete vectors and $m$ and $p^d$ are diagonal matrices. Defining the $T$-matrix

T \equiv{-}\begin{bmatrix} k^2Gm+G^1p^1 & G^1p^2 & G^1p^3 \\ G^2p^1 & k^2Gm+G^2p^2 & G^2p^3 \\ G^3p^1 & G^3p^2 & k^2Gm+G^3p^3 \end{bmatrix},

and $E$ and $E_i$ as the vectors containing all $d$ components, Eq. (5) can be written as

(6) $$E=E_i+TE.$$

Thus, the solution for the total field is

(7)$$E=(I-T)^{{-}1} E_i,$$

where $I$ is the identity matrix. Since resonances correspond to singularities of the electric field, the condition for scattering resonances leads to the requirement [49]

(8)$$|\textrm{det}(I-T)|=0,$$

where the absolute value is necessary because the $T$-matrix is complex. Equation (8) forms the basis of our inverse design approach, and our goal is to solve it for the permittivity $m(x)$. We do so numerically using the fsolve function in MATLAB, which is based on the Levenberg-Marquardt (LM) algorithm [50]. The LM algorithm is a non-linear least-squares solver that interpolates between the Gauss-Newton and gradient descent directions. To implement the constraints, we multiply $m$ by two window functions, one to encode the desired form factor and another to limit the range of permittivity values. No specific information about the mode profile is required. For the given constraints and initial guess (taken as $\epsilon _r=1$), the LM algorithm then solves for the local minimum of the determinant by varying $m$ at each point in the object domain. We emphasize that our approach only requires taking the determinant of a matrix, while solving Maxwell’s equations involves taking a matrix inverse [Eq. (7)]. Although the computational complexity of both the matrix inverse and its determinant varies as $\mathcal {O}(n^3)$, in practice the determinant is faster, about 2.5 times with our algorithm.

We also note that the poles of the scattering matrix are associated with the eigenmodes of the structure [42,43]. Specifically, they correspond to the eigenmodes of the $T$-matrix for the undriven system. This can be seen by setting $E_i=0$ in Eq. (6) to obtain $E=TE$, an eigenvalue problem for the matrix $T$. It is also important to point out that since the determinant of a matrix is equal to the product of its eigenvalues, the determinant is zero when any of one of the eigenvalues is zero. Thus, there can be as many solutions as there are eigenvalues, i.e. there is no unique solution for $m(x)$. For example, for a matrix with $n$ points per dimension, the size of the $T$-matrix is $3n^3 \times 3n^3$, so there will be $3n^3$ eigenvalues and as many as $3n^3$ possible solutions for resonant structures. In the next section, we apply this approach to design several resonant structures.

3. Results

In this section, we apply Eq. (8) to design several photonic resonators. For demonstration, we consider a two-dimensional circular structure (i.e., infinite cylinder) with transversely-polarized electric field. In this case, the second integral in Eq. (4) is zero and the condition for scattering resonances in Eq. (8) simplifies to

(9)$$|\textrm{det} (I+k^2Gm)|=0.$$

First, we design a resonator with a single high-$Q$ resonance for a wavelength of $\lambda _0$. We choose a diameter of $2a/\lambda _0=0.4$, which would allow for the structure to be incorporated into a non-diffracting array. To ensure the retrieved permittivity can be fabricated from realistic materials, we constrain $\epsilon _r$ to be purely real and between 1 and 16, the latter value corresponding to Ge.

Figure 1(a) shows a cross sectional plot of the retrieved permittivity profile. It is approximately four-point symmetric with values ranging from 4 to 12. There are also four low-permittivity regions along the polar direction. Figure 1(b) shows the scattering cross section for light incident along the $y$ direction. As expected, there is a high-$Q$ resonance at $\lambda _0/\lambda =1$. There are additional unintended higher-order resonances at $\lambda _0/\lambda =1.5$ and 2.2. The cross section also exhibits a null near the resonance peaks, indicating Fano resonance behavior [51]. To confirm this, we fit the resonance at $\lambda _0/\lambda =1$ to a Fano lineshape and find excellent agreement for $Q=800$. Figure 1(c) shows the electric field amplitude at $\lambda _0/\lambda =1$. The field resembles a quadrupole mode with four maxima along the polar direction and values exceeding $6 \times 10^9$ relative to the incident field. The maxima appear to coincide with the regions of low permittivity. For reference, Fig. 1(d) shows the field away from resonance at $\lambda _0/\lambda =2$. In this case, there is no significant field enhancement.

Fig. 1. Inverse-designed cylinder with $2a/\lambda _0=0.4$ and a resonance at $\lambda _0/\lambda =1$. (a) Permittivity, (b) cross section, and electric field at (c) $\lambda _0/\lambda =1$ and (d) $\lambda _0/\lambda =2$.

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Next, we apply the formalism to design a structure with the same form factor and constraints, but with a resonance at $\lambda _0/\lambda =2$. Figure 2(a) shows the retrieved permittivity profile. It ranges from 2 to 8 with eight distinct minima along the polar direction. The scattering cross section for light incident along the $y$ direction is shown in Fig. 2(b). As desired, there is a high $Q$ resonance at $\lambda _0/\lambda =2$. The resonance peak fits a Fano lineshape with $Q=280$ (not shown). Figures 2(c) and (d) show the field amplitudes at $\lambda _0/\lambda =1$ and 2, respectively. Away from resonance at $\lambda _0/\lambda =1$, there is no significant field enhancement, while at resonance, the field is enhanced by more than $2 \times 10^4$. The fields are concentrated around with perimeter with eight maxima along the polar direction, resembling an octupole mode. The slight asymmetry of the mode may be an artifact of the coarse discretization, since at resonance the solution is sensitive to the number of points in the domain.

Fig. 2. Inverse-designed cylinder with $2a/\lambda _0=0.4$ and a resonance at $\lambda _0/\lambda =2$. (a) Permittivity, (b) cross section, and electric field at (c) $\lambda _0/\lambda =1$ and (d) $\lambda _0/\lambda =2$.

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Lastly, we apply the formalism to design a structure with the same form factor and constraints, with high-$Q$ scattering resonances at both wavelengths, $\lambda _0/\lambda =1$ and $\lambda _0/\lambda =2$. In this case, we perform a multiobjective optimization to minimize both determinants simultaneously,

(10)$$|\textrm{det} (I+k_1^2G(k_1)m)|=0, \quad |\textrm{det} (I+k_2^2G(k_2)m)|=0,$$

where the subscripts denote the two different wavelengths. Figure 3(a) shows the permittivity profile. It is approximately two-point symmetric with values ranging from 4 to 13. Figure 3(b) shows the scattering cross section for light incident along the $y$ direction. As desired, there are resonances at both wavelengths, with an additional unintended resonance at $\lambda _0/\lambda =1.6$. Fitting the resonances at $\lambda _0/\lambda =1$ and 2 to Fano lineshapes, we obtain $Q$s of 680 and 290, respectively. Figures 3(c) and (d) show the electric field amplitudes at $\lambda _0/\lambda =1$ and 2, respectively. At $\lambda _0/\lambda =1$, the field is clearly a quadrupole mode, while at $\lambda _0/\lambda =2$ the field is multimodal, with approximately four maxima inside and outside the structure. The field enhancements are greater than $4 \times 10^6$ and $6 \times 10^4$ at $\lambda _0/\lambda =1$ and 2, respectively. Such a doubly-resonant structure could be useful for enhancing second harmonic generation, though the modal overlap would also have to be considered [26,27,41].

Fig. 3. Inverse-designed cylinder with $2a/\lambda _0=0.4$ and resonances at $\lambda _0/\lambda =1$ and 2. (a) Permittivity, (b) cross section, and electric field at (c) $\lambda _0/\lambda =1$ and (d) $\lambda _0/\lambda =2$.

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These results show the effectiveness of our algorithm in designing resonators, and we are considering several avenues for future development and exploration. First, the complex variations in permittivity shown above would be difficult to realize in practice, so we might consider imposing additional constraints to make future designs more realizable. For example, a design with variations only along one direction could be realized with epitaxial growth of a material such as Al$_x$Ga$_{1-x}$As, whose permittivity can be varied between 13 ($x=0$) and 9 ($x=1$). Second, in principle there is no limitation on the spectral separation of the resonances, which allows for the possibility of exploring higher-harmonic generation. However, the numerical implementation becomes more intensive as the spectral separation increases because a finer mesh is required to capture the higher frequency resonance. Third, our formalism is sufficiently general to allow for complex frequencies. For example, we could consider solving for the imaginary poles of the scattering matrix in order to design structures that support bound states. Lastly, we plan to apply the method to design 3D resonators. However, the computational expense of the determinant will have to be addressed. A potential solution is to sparsify the matrix using a preconditioner, e.g., by minimizing the non-local interactions in the integral equation [48].

4. Summary

We developed a new inverse computational framework for designing photonic structures with multiple scattering resonances. In contrast with conventional inverse methods, our algorithm does not require solving Maxwell’s equations for each material topology, and thus promises more efficient inverse computation. Our algorithm is based on optimizing a single constraint condition that defines the poles of the scattering matrix, which mathematically amounts to minimizing the determinant of the matrix representing the integral operator of the electric field with respect to the permittivity profile of the resonator. In this Letter, we developed the approach and implemented it to design several subwavelength resonant structures. First, we applied the method to show that a structure with a fixed form factor can be designed to resonate at different prescribed wavelengths, and the spectral variation of the cross sections are consistent with Fano lineshapes with $Q$s as high as 800. Then we showed that the same structure can be designed to resonate at two prescribed wavelengths simultaneously. Such a structure may be used to enhanced second-harmonic generation or surface enhanced Raman spectroscopy. The compact form, computational efficiency, and generality of our method suggest it can be a useful inverse design tool for engineering resonators for enhanced light-matter interactions.

Funding

Defense Advanced Research Projects Agency (HR001118C0015).

Disclosures

The authors declare no conflicts of interest.

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Poles of the scattering matrix: an inverse method for designing photonic resonators

Abstract

1. Introduction

2. Methods

3. Results

4. Summary

Funding

Disclosures

References

Cited By

Figures (3)

Equations (14)

Optics Express