In this paper, we propose a theory for wideband adjoint sensitivity analysis of problems with nonlinear media. We show that the sensitivities of the desired response with respect to all shape and material parameters are obtained through one extra adjoint simulation. Unlike linear problems, the system matrices of this adjoint simulation are time varying. Their values are determined during the original simulation. The proposed theory exploits the time-domain transmission line modeling (TLM) and provides an efficient AVM approach for sensitivity analysis of general time domain objective functions. The theory has been illustrated through a number of examples.
© 2014 Optical Society of America
Nonlinear electromagnetic analysis finds wide applications over the entire electromagnetic spectrum; ranging from the microwave to the visible light [1–11]. In the microwave range, nonlinear effects in integrated circuit technology are unavoidable. Non-linear microwave effects have been widely used to realize several RF devices, e.g., frequency mixers and multipliers, power amplifiers, and oscillators .
Nonlinear electromagnetics effects have enabled the development of optical devices with interesting functionalities . The high peak power of pulsed laser induces strong non-linear effects even with the weakest material nonlinearity . Non-linear interactions in optical devices are known to broaden the output spectrum [12, 13]. Nonlinearities have been exploited to reveal material characteristics using accurate Raman spectroscopy approaches [14, 15]. Ultrafast nonlinear optics has been demonstrated [16, 17] for all optical switching utilizing the frequency mixing capabilities [18–21].
Nonlinear electromagnetic modeling has been demonstrated in both time and frequency domain towards accurate design optimization of nonlinear devices [22–24]. The simulations of large computational domains with nonlinear materials in the frequency domain usually exploit assumptions, which limit the domain of application. The time domain techniques provide a complete integrated platform for the modeling of both linear and nonlinear materials. During a time domain simulation, the material properties change at each time step according to the utilized non-linear model. In addition to the global modeling approaches, several approximate but fast approaches are proposed using the harmonic balance technique .
The design of electromagnetic structures with nonlinear materials involves determining the optimal shape parameters and material properties that achieve the target response. The design optimization of these structures has been reported using artificial neural networks (ANN) , topology optimization , genetic algorithms  and space mapping . Utilizing sensitivity information reduces the necessary number of simulations required to reach an optimal design. It provides robust information about the required changes in the design parameters that would satisfy the design specifications. Gradient calculations are an integral part of the powerful topology optimization approach towards efficient shape optimization . The classical finite difference approaches for sensitivity estimation can be time intensive [31–33].
Recently, the adjoint variable method (AVM) was utilized for simulation-based sensitivity calculations for linear structures of multiple design parameters [34–47]. Accurate mathematical approaches, and approximate mapping techniques are developed to limit the number of required simulations necessary for sensitivity calculations [38, 39]. Using AVM, at most one extra simulation is required for estimating all response sensitivities regardless of the number of design parameters. This can be contrasted with the expensive finite difference approaches whose computational overhead scales linearly with the number of optimization parameters.
Adjoint variable method has been successfully applied in the time and frequency domains solvers. Elegant self adjoint schemes were developed for the sensitivity of network parameters, where the extra adjoint simulation is avoided [40, 41]. The extension of the AVM theory to dispersive linear materials was recently reported in . The AVM has found early implementation in commercial solvers [43, 44]. Variations of the AVM have been successfully applied in the sensitivity analysis of microwave circuits and high speed interconnects [45, 46]. The reported time domain AVM approaches for electromagnetic structures are applicable only to problems with linear material properties.
In this paper, we present the first wideband adjoint sensitivity analysis approach of electromagnetic problems with nonlinear media. We show that using only one adjoint simulation with time varying material properties, the sensitivities of any objective function with respect to all parameters can be estimated. The nonlinear behavior of the adjoint problem is determined using the original problem. We show also how this technique can be implemented in an efficient way to reduce the needed storage.
The paper is organized as follows. In Section 2, we review the TLM-based AVM approach for problems with linear materials. In Section 3, we extend the linear AVM theory to the general nonlinear material case. A detailed derivation of the presented theory is given. Section 4 discusses an efficient implementation of this approach with the minimum possible memory storage. Section 5 illustrates the presented theory through a number of examples including analytical circuit examples and nonlinear electromagnetic problems. Finally, the conclusions are given in Section 6.
2. TLM-based AVM for linear media
Time domain AVM aims at estimating the sensitivities of an objective function of the form:
The Transmission Line Method (TLM) models the propagation of electromagnetic fields using a network of transmission lines. We assume here that the computational domain is discretized into N nodes. Each node has L transmission lines. The number of transmission lines per cell vary depending on the type of the node and whether the modeled problem is 1D, 2D, or 3D. The system state at the kth time step is represented by the vector of incident impulses at transmission lines associated with all nodes Vk ∈ℜNL. These incident impulses scatter at the centers of the nodes and connect to neighboring nodes. One step of the TLM method with non-dispersive boundaries is given by:
The adjoint variable method (AVM) aims at efficiently estimating the gradient ∇F = [∂F/∂x1 ∂F/∂x2 …∂F/∂xn]T using at most one extra adjoint simulation. For TLM-based AVM of problems with linear media, it was shown that this extra adjoint simulation is given by :
Once the original and adjoint system states are determined, the gradient is estimated through the formulas:
3. TLM-based AVM for nonlinear media
For the nonlinear case, the local material properties change with the field. The scattering matrix S is a function of the vector of incident voltages V, which represents the electric and magnetic fields. We assume, without loss of generality, that S = S(p,V), where p ∈ ℜK is the vector of coefficients of the assumed nonlinearity for all nodes. For example, if the relative permittivity is assumed to be a third order polynomial function of an electric field component, we can write:
The derivation of the TLM-based AVM approach for electromagnetic problems with nonlinear media follows the following steps; assuming a nonlinear original simulation of the form (3), where the scattering matrix changes with time, and assuming broadband excitation with a sufficiently small simulation time step Δt, we have:Equation (9) is a second order differential equation describing how the incident impulses depend on each parameter and time.
To derive the corresponding adjoint system, we move all quantities to one side in (9) and multiply both sides by the yet-to-be determined temporal adjoint variable λT to get:38], the left hand side of (11) is compared to the implicit integral in (2) to get:
Once the original response is evaluated using (3) at every time step, and the adjoint response is evaluated using (13), the adjoint sensitivity of (1), is given by:
4. Efficient implementation
Implementation of the algorithm illustrated by the steps (3), (13) and (14) involves a number of steps that may require extensive memory storage. First, the original simulation is run and the adjoint excitation -Δt(∂G/∂V) is stored at the observation domain for all time steps. Usually the observation domain is small relative to the computational domain and this part requires little storage. We assume that the total number of needed storages in the observation domain is M per time step. The adjoint simulation (13) also requires the summation of the two matrices:47, 48]. It follows that we need only to store the first row of this matrix at each time step. The total storage needed for constructing the scattering matrix of each nonlinear node in (16) is thus reduced form L2 + 1 to only L + 1.
Further saving in the storage can be achieved in the actual sensitivity calculation (14). First, the nodal matrix:47, 48] to have a rank of only 1. All its rows are identical. It is thus sufficient to store the first row of this matrix. The number of stored components of the original problem needed to evaluate (14) is thus reduced from LK to only K per time step. Also, there is no need to store the adjoint impulses. The expression (14) is evaluated on the fly during the adjoint simulation by calculating the connected adjoint vector
The algorithm steps for evaluating the adjoint sensitivities for all parameters are as follows:
Step 1: Run the original simulation (3). At every time step, store the adjoint excitation at the observation domain. For every nonlinear node, store the controlling field value and the L components to regenerate the first matrix in (16). Also, store the first row of (17). Total storage per time step K + L + M + 1
Step 2: Run the adjoint simulation (13) using the adjoint source and time varying matrices determined in step 1. At every time step, determine the vector given by (18) and evaluate (14) for all parameters, i = 1, 2,…, n.
The theory presented in this work can be possibly extended to the Finite Difference Time Domain (FDTD) case. The approach has to be adapted to work with electric and magnetic field components rather than with incident voltage impulses. We expect that similar results can be obtained using the appropriate methodology using any time-domain modeling techniques.
In this section, we illustrate the theory through a number of examples. We first present a simple circuit example to illustrate the application of the theorem. Two electromagnetic examples are also presented. Our In-house TLM solver is used for the modeling and adjoint sensitivity analysis of these examples.
5.1 An RC circuit
To illustrate the basic concepts of our approach, we consider the simple RC circuit shown in Fig. 1. The first order differential equation governing this circuit is given by:Figure 2 shows the original and adjoint responses obtained for a set of the parameter values. The sensitivities of the response with respect to all parameters are shown in Figs. 3 and 4 for a sweep of the parameter po.
Good match is achieved with the Central Finite Difference (CFD) approximation. The CFD approach requires 8 extra simulations while the AVM approach requires only one extra adjoint simulation.
5.2 A microwave example
We consider the EM structure shown in Fig. 5. It shows a parallel plate waveguide with a nonlinear region in its center. The waveguide has a width of 2.5 cm and a length of 10.0 cm. A TLM cell of size 0.25 cm is utilized in modeling this 2D problem. The width of the nonlinear region is 0.5 cm. The dielectric constant within this region follows the nonlinear profile:Fig. 6. Good agreement is observed. The AVM approach requires only one extra simulation while the CFD approach requires 6 extra simulations.
5.3 A photonic example
We also consider the optical waveguide structure shown in Fig. 7. The waveguide has a length of L = 3.5 μm. The width of the air regions is W1 = 1.0 μm while the width of the silicon region is W2 = 0.8 μm. The cell size is Δh = 0.05 μm which gives a domain size of 70 Δh × 56 Δh. One nonlinear region exists in the middle of this waveguide with a length of 4 Δh. This region is lossy with the dielectric and conductivity profiles:
The structure is excited with a Gaussian-modulated sinusoid TM wave with a center wavelength of 1.6 μm. The excitation has the spatial modal profile shown in Fig. 8. We utilize the objective function (24) as measure of the energy delivered to the output port. We allowed for 5000 time steps in this example. The sensitivities of the objective function are estimated with respect to all 4 coefficients in (25) and (26). Using our AVM approach, only one adjoint simulation is required. We compare our result in this case with the Central Finite Difference (CFD) approximation as shown in Fig. 9 for a sweep of the parameter . The CFD approach requires 8 extra simulations for these 4 coefficients. In general, the adjoint sensitivities match reasonably the more expensive CFD estimates. Some difference is observed for some of the parameters. We believe that this difference is due to the fact that the energy in the domain does not vanish at the end of the original simulation as required by zero terminal values condition. Using a larger number of time step will likely help improve the accuracy for all parameters. It should be noted that the objective function value is small because of the short simulation time and that the shown sensitivities are significant given the small objective function value. Only few of the sensitivity profiles are shown due to space constraints but reasonable match is achieved for all of them.
We propose a novel algorithm for wideband adjoint sensitivity analysis of nonlinear electromagnetic problems. We show that using only one extra adjoint simulation with dynamic parameters, the sensitivities of the desired objective function with respect to all parameters are estimated regardless of their numbers. We also presented efficient approaches for reducing the memory storage of the technique. Our approach was illustrated through a circuit example, a microwave example, and a photonic example. Our results reasonably match the accurate but very expensive central finite difference approach.
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