Abstract

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

1. Introduction

Nonlinear electromagnetic analysis finds wide applications over the entire electromagnetic spectrum; ranging from the microwave to the visible light [111]. 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 [1].

Nonlinear electromagnetics effects have enabled the development of optical devices with interesting functionalities [10]. The high peak power of pulsed laser induces strong non-linear effects even with the weakest material nonlinearity [11]. 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 [1821].

Nonlinear electromagnetic modeling has been demonstrated in both time and frequency domain towards accurate design optimization of nonlinear devices [2224]. 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 [25].

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) [26], topology optimization [27], genetic algorithms [28] and space mapping [29]. 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 [30]. The classical finite difference approaches for sensitivity estimation can be time intensive [3133].

Recently, the adjoint variable method (AVM) was utilized for simulation-based sensitivity calculations for linear structures of multiple design parameters [3447]. 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 [42]. 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:

F=0TmΩg(x,V) dΩ dt=0TmG(x,V)  dt,
where Tm is the simulation time and Ω is the observation domain where the response is calculated. x∈ℜn is the vector of design parameters which may include geometrical and material properties. The vector V is the vector of system state. It represents electric and magnetic fields values at different computational cells. The analytical derivative of (1) with respect to the ith parameter xi, i = 1, 2, …, n, is given by:
Fxi=0TmeGxidt+ 0Tm(GV)TVxi dt,
where e/xidenotes the explicit dependence which is zero in most practical objective functions. The classical approach for evaluating (2) involves repeatedly simulating perturbed structures. A total of extra n electromagnetic simulations are required for forward or backward differences. The more accurate central differences require 2n extra simulations. This overhead can be significant even for a small number of designable parameters especially for time intensive problems.

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:

Vk+1=CSVk+Vks,
where C∈ℜNL × NL is the connection matrix of the whole domain. S∈ℜNL × NL is the block diagonal scattering matrix of the computational domain. The jth block of this matrix is the scattering matrix of the jth node S(j) ∈ℜL × L. This nodal scattering matrix is a function of the local material properties p(j) and the utilized spatial and time discretizations. For a linear problem, the nodal scattering matrices do not change with time. The electric and magnetic fields are estimated as linear functions of the incident nodal voltages.

The adjoint variable method (AVM) aims at efficiently estimating the gradient ∇F = [∂F/∂x1F/∂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 [38]:

λk1=STCTλk+Vks,λ, λNT=0,
where NT is the number of time steps with Tm = NTΔt. The adjoint simulation (4) is backward running with zero terminal conditions. The order of the connection and scattering procedures is reversed as compared to the original simulation (3). Also, the scattering and connection matrices in (4) are the transpose of those in (3). These matrices are static matrices for problems with linear media. The adjoint excitation Vks,λis determined using the original simulation and the objective function (1).

Once the original and adjoint system states are determined, the gradient is estimated through the formulas:

FxieFxi -ΔtkλkTΔAiΔxiVk, i=1,2,, n,
where A = (CS-I)/Δt is the system matrix. The change in the ith parameter xi usually changes only a subset of the computational domain. It follows that only a small subset of the vectors Vk and λk needs to be stored to evaluate (5).

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:

εr=p0+p1E+p2E2+p3E3,
where p = [p0 p1 p2 p3]T is the vector of nonlinearity coefficients and E = f(V). Notice that (6) defaults to the linear case when p1 = p2 = p3 = 0. An expression similar to (6) can be written for nonlinearities of the electric conductivity and magnetic permeability

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:

Vk+(V t)kΔtCSVk+Vks.
Reorganizing (7), we have the first order differential equation governing the incident impulses:
Vt=1Δt(CSI)V+VsΔt.
Differentiating both sides of (8) with respect to the ith parameter xi, i = 1,2,…, n, and reorganizing, we get:
2Vtxi=1Δt(C[(SV¯)pTpxi+(SV¯)VTVxi+SVxi]Vxi),
where V¯is the value of the vector of incident impulses and should be treated as a constant in (9). 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:

0TmλT[2Vtxi1ΔtC[(SV¯)pTpxi+(SV¯)VTVxi+SVxi]+1ΔtVxi] dt=0.
Integrating the mixed derivative term by parts, assuming zero terminal values (λ(Tm)=0andV/xi|Tm=0), and assembling, we get the expression:
0Tm(λTt+λT[1ΔtC(SV¯)VT+1ΔtCS1ΔtI])Vxi dt= 0TmλTΔtC(SV¯)pTpxidt.
Similar to the linear case derivation [38], the left hand side of (11) is compared to the implicit integral in (2) to get:
(λTt+λT[1ΔtC(SV¯)VT+1ΔtCS1ΔtI])=GVT.
Multiplying both sides by Δt, and utilizing the discrete time approximation λk-1 = λkt(∂λ/∂t)k, one obtains the discrete time adjoint system
λk1=((SV¯)VT+S)TCTλkΔtGV.
The adjoint system (13) is a backward running adjoint system where the order of the connection and scattering processes are reversed. Comparing (13) and (4), one could see that the connection matrix of the adjoint problem is still the transpose of the connection matrix of the original problem. Unlike the linear AVM case, the scattering matrix of the adjoint problem is not the transpose of the scattering matrix as in (4). Rather, a correction term (SV¯)/VTis added to the scattering matrix of the adjoint problem at every time step. This correction term is zero for the linear case as the scattering matrix is not a function of the field values.

Once the original response V¯ is evaluated using (3) at every time step, and the adjoint response is evaluated using (13), the adjoint sensitivity of (1), is given by:

Fxi=0TmeGxidt0TmλTC(SV¯)pTpxidt,  i=1,2,, n.
Calculating (14) requires knowing the derivative of the scattering matrix with respect to the material properties. These derivatives are stored during the original simulation.

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:

Sn=((SV¯)VT+S).
The actual implementation of (15) does not require building these matrices with large dimensions. Instead, on a node by node basis, the following matrices are built:
Sn(j)=((S(j)V¯(j))V(j)T+S(j)),j=1,2,...,N.
If the jth node is a linear node, the scattering matrix is not a function of the incident voltage and the nodal scattering matrix is reduced to static matrix Sn(j)=S(j). No extra storage is needed in this case for the scattering matrix. If the jth node is a nonlinear node, we need to store the field values at this node at all time steps to determine the time varying matrix S(j) for all time steps. Also, the first matrix in (16) has L2 time varying components. Using the unique properties of the scattering matrices in TLM, it can be shown that this matrix has only a rank of 1 with all identical rows. This interesting property was proved in earlier work [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:

J(j)=(S(j)V¯(j))p(j)T
can also be shown using a derivation similar to [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
λ˜k=CTλk
from (13) and substituting it in (14) during the adjoint simulation for all parameters. It follows from the previous discussion that the storage needed for running the adjoint simulation and evaluating (14) is L + K + 1 per nonlinear node per time step. Additional M storages are needed for the adjoint excitation as stated earlier.

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 λ˜kgiven 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.

5. Examples

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:

Vct=VSR1C(R1+R2R1R2C)Vc.
Comparing this equation with (8), one can see the analogy between the two systems. The resistor R2 is assumed to be nonlinear with a voltage dependence profile:
R2=po+p2Vc2.
The coefficients p = [p0 p2]T define different nonlinearity profiles. The parameters of this circuit are x = [R1 C p0 p1]T. The objective function of this example is taken as:
F=0Tm(Vs(t)Vc(t))2dt=0Tmψ(t)dt.
The source Vs(t) has a Gaussian profile with a unity amplitude. Using the theory developed in this paper, the adjoint system of (19) is given by:
λt=λSR1C[(R1+R2R1R2C)+1CddVc(V¯cR2)]λ,
whereλs=ψ(t)/Vc=2(VcVs)is the adjoint excitation. The original response Vc(t) and the adjoint response λ(t) are determined through a finite difference integration scheme of (19) and (22). However, (22) is integrated backwards in time. 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.

 figure: Fig. 1

Fig. 1 The RC circuit of example 1.

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 figure: Fig. 2

Fig. 2 The original and adjoint responses of the circuit in Fig. 1 for the set of values R1 = 1000.0 Ω, R2 = 1000.0 Ω, and C = 60.0 µF.

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 figure: Fig. 3

Fig. 3 The sensitivities of the objective function (21) with respect to the nonlinear coefficients of the resistor R2 (p0 and p2) for a sweep of the coefficient po with R1 = 1000.0 Ω, C = 10.0 µF, and p2 = 50.0 shown in (a) and (b) respectively; the sensitivities estimated using AVM ( + ) and sensitivities obtained using the central finite difference method (CFD) (o).

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 figure: Fig. 4

Fig. 4 The sensitivities of the objective function (21) with respect to the circuit parameters C and R1 for a sweep of the parameter po with R1 = 1000.0 Ω, C = 10.0 µF, and p2 = 50.0 shown in (a) and (b) respectively; the sensitivities estimated using AVM ( + ) and sensitivities obtained using the central finite difference method (CFD) (o).

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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:

εr=po+p1Ey+p2Ey2,
with the nonlinear profile parameter values p = [30.0 70.0 1000.0]T. The structure is excited with a plane wave exhibiting a Gaussian modulated sinusoid time dependence with a center frequency of 1.5 GHz. The objective function is the energy delivered to the output port which is given by the integral:
F=0TmΩEy2 dΩdt,
where Ω is the last column of cells. The number of allowed time steps is 3000. The sensitivities of the objective function relative to the parameters of the nonlinearity profile are estimated using both the AVM approach and the expensive CFD approach for a sweep of the parameter p0. The results are shown in Fig. 6. Good agreement is observed. The AVM approach requires only one extra simulation while the CFD approach requires 6 extra simulations.

 figure: Fig. 5

Fig. 5 A nonlinear region in a parallel plate waveguide.

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 figure: Fig. 6

Fig. 6 The adjoint sensitivities ( + ) as compared to the central finite difference (o) for a sweep of the parameter po for the microwave example.

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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:

εr=p0ε+p1εEy+p2εEy2,
σ=p2σEy2,
with p = [p0εp1εp2εp2σ]T = [3.5 0.15 1.0 0.1]T. Even though the rest of the nonlinearity coefficients for both the permittivity and conductivity are zeros, we can still evaluate the adjoint sensitivities with respect to these parameters. The Silicon waveguide is lossless and has a dielectric constant of 3.5.

 figure: Fig. 7

Fig. 7 The structure of the optical example.

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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 p0ε. 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.

 figure: Fig. 8

Fig. 8 The modal profile of the excitation used in the photonic example.

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 figure: Fig. 9

Fig. 9 The adjoint sensitivities ( + ) as compared to the central finite difference (CFD) (o) for a sweep of the parameter p0ε for the photonic example.

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6. Conclusion

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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37. J. P. Webb, “Design sensitivity of frequency response in 3-D finite-element analysis of microwave devices,” IEEE Trans. Magn. 38(2), 1109–1112 (2002). [CrossRef]  

38. M. H. Bakr and N. K. Nikolova, “An adjoint variable method for time-domain transmission line modeling with fixed structured grids,” IEEE Trans. Microw. Theory Tech. 52(2), 554–559 (2004). [CrossRef]  

39. M. H. Bakr, P. Zhao, and N. K. Nikolova, “Adjoint first order sensitivities of transient responses and their applications in the solution of inverse problems,” IEEE Trans. Antenn. Propag. 57(7), 2137–2146 (2009). [CrossRef]  

40. M. H. Bakr, N. K. Nikolova, and P. A. W. Basl, “Self-Adjoint S-Parameter Sensitivities for Lossless Homogeneous TLM Problems,” International Journal of Numerical Modeling: Electronic Networks, Devices and Fields 18(6), 441–455 (2005). [CrossRef]  

41. P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, “Theory of self-adjoint S-parameter sensitivities for lossless nonhomogeneous transmission-line modeling problems,” IET Microwave Antennas Propag. 2(3), 211–220 (2008). [CrossRef]  

42. O. S. Ahmed, M. H. Bakr, X. Li, and T. Nomura, “A Time-Domain Adjoint Variable Method for Materials With Dispersive Constitutive Parameters,” IEEE Trans. Microw. Theory Tech. 60(10), 2959–2971 (2012). [CrossRef]  

43. Optimetrics: parametrics and optimization using Ansoft HFSS,” Microwave Journal, (1999). [Online]. Available: http://www.microwavejournal.com/articles/2779-parametrics-and-optimization-using-ansoft-hfss. [Accessed 18 Nov. 2013].

44. Press Release, “Sensitivity analysis and adapted optimization strategies for CST MICROWAVE STUDIO transient solver,” May 2010. [Online]. Available: www.cst.com/Content/News/Documents/2010_5_Timedomain2011_web.pdf. [Accessed 23 Oct. 2013].

45. A. S. Saini, M. S. Nakhla, and R. Achar, “Generalized time-domain adjoint sensitivity analysis of distributed MTL networks,” IEEE Trans. Microw. Theory Tech. 60(11), 3359–3368 (2012). [CrossRef]  

46. S. Lum, M. Nakhla, and Q.-J. Zhang, “Sensitivity analysis of lossy coupled transmission lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech. 42(4), 607–615 (1994). [CrossRef]  

47. O. S. Ahmed, M. H. Bakr, and X. Li, “A memory-efficient implementation of TLM- based adjoint sensitivity analysis,” IEEE Trans. Antenn. Propag. 60(4), 2122–2125 (2012). [CrossRef]  

48. O. S. Ahmed, M. H. Bakr, and X. Li, “An impulse sampling approach for efficient 3D TLM-based adjoint sensitivity analysis,” Prog. Electromagnetics Res. 142, 485–503 (2013). [CrossRef]  

References

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  1. S. A. Maas, Nonlinear microwave circuits. (IEEE, 1997).
  2. A. E. Schmitz, R. H. Walden, L. E. Larson, S. E. Rosenbaum, R. A. Metzger, J. R. Behnke, and P. A. Macdonald, “A deep-submicrometer microwave/digital CMOS/SOS technology,” IEEE Electron Device Lett. 12(1), 16–17 (1991).
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  4. P. Russer and N. Fichtner, “Nanoelectronics in radio-frequency technology,” IEEE Microw. Mag. 11(3), 119–135 (2010).
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  5. S. A. Sørngård, S. I. Simonsen, and J. P. Hansen, “High-order harmonic generation from graphene: Strong attosecond pulses with arbitrary polarization,” Phys. Rev. A 87(5), 053803 (2013).
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  6. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16(2), 1300–1320 (2008).
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  7. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15(10), 5976–5990 (2007).
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  8. N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. 24(20), 1395–1397 (1999).
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  9. P. L. Kelley, “Self-Focusing of Optical Beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965).
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  10. J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics 3(2), 85–90 (2009).
    [Crossref]
  11. T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
    [Crossref]
  12. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Efficient visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” in Summaries of Papers Presented at the Conference on Lasers and Electro-Optics, CLEO ’99 (1999), CPD8/1–CPD8/2.
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  13. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25(1), 25–27 (2000).
    [Crossref] [PubMed]
  14. Y. Gontier and M. Trahin, “Frequency conversion involving Raman-like processes above the ionization threshold,” IEEE J. Quantum Electron. 18(7), 1137–1145 (1982).
    [Crossref]
  15. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in Silicon waveguides,” Opt. Express 12(12), 2774–2780 (2004).
    [Crossref] [PubMed]
  16. R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D Appl. Phys. 40(14), R249–R271 (2007).
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  17. T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 607–609 (2012).
    [Crossref]
  18. S. Kawanishi and O. Kamatani, “All-optical time division multiplexing using four-wave mixing,” Electron. Lett. 30(20), 1697–1698 (1994).
    [Crossref]
  19. A. M. Darwish, E. P. Ippen, H. Q. Le, J. P. Donnelly, and S. H. Groves, “Optimization of four‐wave mixing conversion efficiency in the presence of nonlinear loss,” Appl. Phys. Lett. 69(6), 737–739 (1996).
    [Crossref]
  20. Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express 12(17), 4094–4102 (2004).
    [Crossref] [PubMed]
  21. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15(10), 5976–5990 (2007).
    [Crossref] [PubMed]
  22. V. Rizzoli, A. Costanzo, D. Masotti, A. Lipparini, and F. Mastri, “Computer-aided optimization of nonlinear microwave circuits with the aid of electromagnetic simulation,” IEEE Trans. Microw. Theory Tech. 52(1), 362–377 (2004).
    [Crossref]
  23. R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antenn. Propag. 45(3), 364–374 (1997).
    [Crossref]
  24. P. Russer, P. P. M. So, and W. J. R. Hoefer, “Modeling of nonlinear active regions in TLM (distributed circuits),” IEEE Microw. Guided Wave Letts. 1(1), 10–13 (1991).
    [Crossref]
  25. M. Nakhla and J. Vlach, “A piecewise harmonic balance technique for determination of periodic response of nonlinear systems,” IEEE Trans. Circ. Syst. 23(2), 85–91 (1976).
    [Crossref]
  26. J. E. Rayas-Sanchez, “EM-based optimization of microwave circuits using artificial neural networks: the state-of-the-art,” IEEE Trans. Microw. Theory Tech. 52(1), 420–435 (2004).
    [Crossref]
  27. J. S. Jensen, “Topology optimization of nonlinear optical devices,” Struct. Multidisc. Optim. 43(6), 731–743 (2011).
    [Crossref]
  28. M. Gao, C. Jiang, W. Hu, and J. Wang, “Optimized design of two-pump fiber optical parametric amplifier with two-section nonlinear fibers using genetic algorithm,” Opt. Express 12(23), 5603–5613 (2004).
    [Crossref] [PubMed]
  29. M. H. Bakr, J. W. Bandler, K. Madsen, J. E. Rayas-Sanchez, and J. Sondergaard, “Space-mapping optimization of microwave circuits exploiting surrogate models,” IEEE Trans. Microw. Theory Tech. 48(12), 2297–2306 (2000).
    [Crossref]
  30. L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech. 53(9), 2752–2767 (2005).
    [Crossref]
  31. H. Igarashi and K. Watanabe, “Complex adjoint variable method for finite-element analysis of eddy current problems,” IEEE Trans. Magn. 46(8), 2739–2742 (2010).
    [Crossref]
  32. I.-H. Park, B.-T. Lee, and S.-Y. Hahn, “Design sensitivity analysis for nonlinear magnetostatic problems using finite element method,” IEEE Trans. Magn. 28(2), 1533–1536 (1992).
    [Crossref]
  33. B. Lojek, “Sensitivity analysis of nonlinear circuits,” Electronic Circuits and Systems, IEE Proceedings 129(3), 85–88 (1982).
    [Crossref]
  34. Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microw. Theory Tech. 48(12), 2289–2296 (2000).
    [Crossref]
  35. Y. Chung, J. Ryu, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part I: FETD case,” IEEE Trans. Magn. 37(5), 3289–3293 (2001).
    [Crossref]
  36. Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part II: FDTD case,” IEEE Trans. Magn. 37(5), 3255–3259 (2001).
    [Crossref]
  37. J. P. Webb, “Design sensitivity of frequency response in 3-D finite-element analysis of microwave devices,” IEEE Trans. Magn. 38(2), 1109–1112 (2002).
    [Crossref]
  38. M. H. Bakr and N. K. Nikolova, “An adjoint variable method for time-domain transmission line modeling with fixed structured grids,” IEEE Trans. Microw. Theory Tech. 52(2), 554–559 (2004).
    [Crossref]
  39. M. H. Bakr, P. Zhao, and N. K. Nikolova, “Adjoint first order sensitivities of transient responses and their applications in the solution of inverse problems,” IEEE Trans. Antenn. Propag. 57(7), 2137–2146 (2009).
    [Crossref]
  40. M. H. Bakr, N. K. Nikolova, and P. A. W. Basl, “Self-Adjoint S-Parameter Sensitivities for Lossless Homogeneous TLM Problems,” International Journal of Numerical Modeling: Electronic Networks, Devices and Fields 18(6), 441–455 (2005).
    [Crossref]
  41. P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, “Theory of self-adjoint S-parameter sensitivities for lossless nonhomogeneous transmission-line modeling problems,” IET Microwave Antennas Propag. 2(3), 211–220 (2008).
    [Crossref]
  42. O. S. Ahmed, M. H. Bakr, X. Li, and T. Nomura, “A Time-Domain Adjoint Variable Method for Materials With Dispersive Constitutive Parameters,” IEEE Trans. Microw. Theory Tech. 60(10), 2959–2971 (2012).
    [Crossref]
  43. Optimetrics: parametrics and optimization using Ansoft HFSS,” Microwave Journal, (1999). [Online]. Available: http://www.microwavejournal.com/articles/2779-parametrics-and-optimization-using-ansoft-hfss . [Accessed 18 Nov. 2013].
  44. Press Release, “Sensitivity analysis and adapted optimization strategies for CST MICROWAVE STUDIO transient solver,” May 2010. [Online]. Available: www.cst.com/Content/News/Documents/2010_5_Timedomain2011_web.pdf . [Accessed 23 Oct. 2013].
  45. A. S. Saini, M. S. Nakhla, and R. Achar, “Generalized time-domain adjoint sensitivity analysis of distributed MTL networks,” IEEE Trans. Microw. Theory Tech. 60(11), 3359–3368 (2012).
    [Crossref]
  46. S. Lum, M. Nakhla, and Q.-J. Zhang, “Sensitivity analysis of lossy coupled transmission lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech. 42(4), 607–615 (1994).
    [Crossref]
  47. O. S. Ahmed, M. H. Bakr, and X. Li, “A memory-efficient implementation of TLM- based adjoint sensitivity analysis,” IEEE Trans. Antenn. Propag. 60(4), 2122–2125 (2012).
    [Crossref]
  48. O. S. Ahmed, M. H. Bakr, and X. Li, “An impulse sampling approach for efficient 3D TLM-based adjoint sensitivity analysis,” Prog. Electromagnetics Res. 142, 485–503 (2013).
    [Crossref]

2013 (2)

S. A. Sørngård, S. I. Simonsen, and J. P. Hansen, “High-order harmonic generation from graphene: Strong attosecond pulses with arbitrary polarization,” Phys. Rev. A 87(5), 053803 (2013).
[Crossref]

O. S. Ahmed, M. H. Bakr, and X. Li, “An impulse sampling approach for efficient 3D TLM-based adjoint sensitivity analysis,” Prog. Electromagnetics Res. 142, 485–503 (2013).
[Crossref]

2012 (4)

O. S. Ahmed, M. H. Bakr, X. Li, and T. Nomura, “A Time-Domain Adjoint Variable Method for Materials With Dispersive Constitutive Parameters,” IEEE Trans. Microw. Theory Tech. 60(10), 2959–2971 (2012).
[Crossref]

A. S. Saini, M. S. Nakhla, and R. Achar, “Generalized time-domain adjoint sensitivity analysis of distributed MTL networks,” IEEE Trans. Microw. Theory Tech. 60(11), 3359–3368 (2012).
[Crossref]

O. S. Ahmed, M. H. Bakr, and X. Li, “A memory-efficient implementation of TLM- based adjoint sensitivity analysis,” IEEE Trans. Antenn. Propag. 60(4), 2122–2125 (2012).
[Crossref]

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 607–609 (2012).
[Crossref]

2011 (1)

J. S. Jensen, “Topology optimization of nonlinear optical devices,” Struct. Multidisc. Optim. 43(6), 731–743 (2011).
[Crossref]

2010 (3)

H. Igarashi and K. Watanabe, “Complex adjoint variable method for finite-element analysis of eddy current problems,” IEEE Trans. Magn. 46(8), 2739–2742 (2010).
[Crossref]

P. Russer and N. Fichtner, “Nanoelectronics in radio-frequency technology,” IEEE Microw. Mag. 11(3), 119–135 (2010).
[Crossref]

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

2009 (2)

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics 3(2), 85–90 (2009).
[Crossref]

M. H. Bakr, P. Zhao, and N. K. Nikolova, “Adjoint first order sensitivities of transient responses and their applications in the solution of inverse problems,” IEEE Trans. Antenn. Propag. 57(7), 2137–2146 (2009).
[Crossref]

2008 (2)

M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16(2), 1300–1320 (2008).
[Crossref] [PubMed]

P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, “Theory of self-adjoint S-parameter sensitivities for lossless nonhomogeneous transmission-line modeling problems,” IET Microwave Antennas Propag. 2(3), 211–220 (2008).
[Crossref]

2007 (3)

2005 (2)

M. H. Bakr, N. K. Nikolova, and P. A. W. Basl, “Self-Adjoint S-Parameter Sensitivities for Lossless Homogeneous TLM Problems,” International Journal of Numerical Modeling: Electronic Networks, Devices and Fields 18(6), 441–455 (2005).
[Crossref]

L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech. 53(9), 2752–2767 (2005).
[Crossref]

2004 (6)

M. H. Bakr and N. K. Nikolova, “An adjoint variable method for time-domain transmission line modeling with fixed structured grids,” IEEE Trans. Microw. Theory Tech. 52(2), 554–559 (2004).
[Crossref]

Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express 12(17), 4094–4102 (2004).
[Crossref] [PubMed]

J. E. Rayas-Sanchez, “EM-based optimization of microwave circuits using artificial neural networks: the state-of-the-art,” IEEE Trans. Microw. Theory Tech. 52(1), 420–435 (2004).
[Crossref]

M. Gao, C. Jiang, W. Hu, and J. Wang, “Optimized design of two-pump fiber optical parametric amplifier with two-section nonlinear fibers using genetic algorithm,” Opt. Express 12(23), 5603–5613 (2004).
[Crossref] [PubMed]

V. Rizzoli, A. Costanzo, D. Masotti, A. Lipparini, and F. Mastri, “Computer-aided optimization of nonlinear microwave circuits with the aid of electromagnetic simulation,” IEEE Trans. Microw. Theory Tech. 52(1), 362–377 (2004).
[Crossref]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in Silicon waveguides,” Opt. Express 12(12), 2774–2780 (2004).
[Crossref] [PubMed]

2002 (1)

J. P. Webb, “Design sensitivity of frequency response in 3-D finite-element analysis of microwave devices,” IEEE Trans. Magn. 38(2), 1109–1112 (2002).
[Crossref]

2001 (2)

Y. Chung, J. Ryu, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part I: FETD case,” IEEE Trans. Magn. 37(5), 3289–3293 (2001).
[Crossref]

Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part II: FDTD case,” IEEE Trans. Magn. 37(5), 3255–3259 (2001).
[Crossref]

2000 (3)

Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microw. Theory Tech. 48(12), 2289–2296 (2000).
[Crossref]

M. H. Bakr, J. W. Bandler, K. Madsen, J. E. Rayas-Sanchez, and J. Sondergaard, “Space-mapping optimization of microwave circuits exploiting surrogate models,” IEEE Trans. Microw. Theory Tech. 48(12), 2297–2306 (2000).
[Crossref]

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25(1), 25–27 (2000).
[Crossref] [PubMed]

1999 (1)

1997 (1)

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antenn. Propag. 45(3), 364–374 (1997).
[Crossref]

1996 (1)

A. M. Darwish, E. P. Ippen, H. Q. Le, J. P. Donnelly, and S. H. Groves, “Optimization of four‐wave mixing conversion efficiency in the presence of nonlinear loss,” Appl. Phys. Lett. 69(6), 737–739 (1996).
[Crossref]

1994 (2)

S. Kawanishi and O. Kamatani, “All-optical time division multiplexing using four-wave mixing,” Electron. Lett. 30(20), 1697–1698 (1994).
[Crossref]

S. Lum, M. Nakhla, and Q.-J. Zhang, “Sensitivity analysis of lossy coupled transmission lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech. 42(4), 607–615 (1994).
[Crossref]

1992 (1)

I.-H. Park, B.-T. Lee, and S.-Y. Hahn, “Design sensitivity analysis for nonlinear magnetostatic problems using finite element method,” IEEE Trans. Magn. 28(2), 1533–1536 (1992).
[Crossref]

1991 (2)

P. Russer, P. P. M. So, and W. J. R. Hoefer, “Modeling of nonlinear active regions in TLM (distributed circuits),” IEEE Microw. Guided Wave Letts. 1(1), 10–13 (1991).
[Crossref]

A. E. Schmitz, R. H. Walden, L. E. Larson, S. E. Rosenbaum, R. A. Metzger, J. R. Behnke, and P. A. Macdonald, “A deep-submicrometer microwave/digital CMOS/SOS technology,” IEEE Electron Device Lett. 12(1), 16–17 (1991).
[Crossref]

1982 (2)

Y. Gontier and M. Trahin, “Frequency conversion involving Raman-like processes above the ionization threshold,” IEEE J. Quantum Electron. 18(7), 1137–1145 (1982).
[Crossref]

B. Lojek, “Sensitivity analysis of nonlinear circuits,” Electronic Circuits and Systems, IEE Proceedings 129(3), 85–88 (1982).
[Crossref]

1976 (1)

M. Nakhla and J. Vlach, “A piecewise harmonic balance technique for determination of periodic response of nonlinear systems,” IEEE Trans. Circ. Syst. 23(2), 85–91 (1976).
[Crossref]

1965 (1)

P. L. Kelley, “Self-Focusing of Optical Beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965).
[Crossref]

Achar, R.

A. S. Saini, M. S. Nakhla, and R. Achar, “Generalized time-domain adjoint sensitivity analysis of distributed MTL networks,” IEEE Trans. Microw. Theory Tech. 60(11), 3359–3368 (2012).
[Crossref]

Ahmed, O. S.

O. S. Ahmed, M. H. Bakr, and X. Li, “An impulse sampling approach for efficient 3D TLM-based adjoint sensitivity analysis,” Prog. Electromagnetics Res. 142, 485–503 (2013).
[Crossref]

O. S. Ahmed, M. H. Bakr, and X. Li, “A memory-efficient implementation of TLM- based adjoint sensitivity analysis,” IEEE Trans. Antenn. Propag. 60(4), 2122–2125 (2012).
[Crossref]

O. S. Ahmed, M. H. Bakr, X. Li, and T. Nomura, “A Time-Domain Adjoint Variable Method for Materials With Dispersive Constitutive Parameters,” IEEE Trans. Microw. Theory Tech. 60(10), 2959–2971 (2012).
[Crossref]

Arpin, P.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Badolato, A.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 607–609 (2012).
[Crossref]

Bakr, M. H.

O. S. Ahmed, M. H. Bakr, and X. Li, “An impulse sampling approach for efficient 3D TLM-based adjoint sensitivity analysis,” Prog. Electromagnetics Res. 142, 485–503 (2013).
[Crossref]

O. S. Ahmed, M. H. Bakr, and X. Li, “A memory-efficient implementation of TLM- based adjoint sensitivity analysis,” IEEE Trans. Antenn. Propag. 60(4), 2122–2125 (2012).
[Crossref]

O. S. Ahmed, M. H. Bakr, X. Li, and T. Nomura, “A Time-Domain Adjoint Variable Method for Materials With Dispersive Constitutive Parameters,” IEEE Trans. Microw. Theory Tech. 60(10), 2959–2971 (2012).
[Crossref]

M. H. Bakr, P. Zhao, and N. K. Nikolova, “Adjoint first order sensitivities of transient responses and their applications in the solution of inverse problems,” IEEE Trans. Antenn. Propag. 57(7), 2137–2146 (2009).
[Crossref]

P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, “Theory of self-adjoint S-parameter sensitivities for lossless nonhomogeneous transmission-line modeling problems,” IET Microwave Antennas Propag. 2(3), 211–220 (2008).
[Crossref]

M. H. Bakr, N. K. Nikolova, and P. A. W. Basl, “Self-Adjoint S-Parameter Sensitivities for Lossless Homogeneous TLM Problems,” International Journal of Numerical Modeling: Electronic Networks, Devices and Fields 18(6), 441–455 (2005).
[Crossref]

M. H. Bakr and N. K. Nikolova, “An adjoint variable method for time-domain transmission line modeling with fixed structured grids,” IEEE Trans. Microw. Theory Tech. 52(2), 554–559 (2004).
[Crossref]

M. H. Bakr, J. W. Bandler, K. Madsen, J. E. Rayas-Sanchez, and J. Sondergaard, “Space-mapping optimization of microwave circuits exploiting surrogate models,” IEEE Trans. Microw. Theory Tech. 48(12), 2297–2306 (2000).
[Crossref]

Bandler, J. W.

M. H. Bakr, J. W. Bandler, K. Madsen, J. E. Rayas-Sanchez, and J. Sondergaard, “Space-mapping optimization of microwave circuits exploiting surrogate models,” IEEE Trans. Microw. Theory Tech. 48(12), 2297–2306 (2000).
[Crossref]

Basl, P. A. W.

P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, “Theory of self-adjoint S-parameter sensitivities for lossless nonhomogeneous transmission-line modeling problems,” IET Microwave Antennas Propag. 2(3), 211–220 (2008).
[Crossref]

M. H. Bakr, N. K. Nikolova, and P. A. W. Basl, “Self-Adjoint S-Parameter Sensitivities for Lossless Homogeneous TLM Problems,” International Journal of Numerical Modeling: Electronic Networks, Devices and Fields 18(6), 441–455 (2005).
[Crossref]

Behnke, J. R.

A. E. Schmitz, R. H. Walden, L. E. Larson, S. E. Rosenbaum, R. A. Metzger, J. R. Behnke, and P. A. Macdonald, “A deep-submicrometer microwave/digital CMOS/SOS technology,” IEEE Electron Device Lett. 12(1), 16–17 (1991).
[Crossref]

Bennett, P. J.

Boyraz, Ö.

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T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
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Y. Chung, J. Ryu, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part I: FETD case,” IEEE Trans. Magn. 37(5), 3289–3293 (2001).
[Crossref]

Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microw. Theory Tech. 48(12), 2289–2296 (2000).
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Y. Chung, J. Ryu, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part I: FETD case,” IEEE Trans. Magn. 37(5), 3289–3293 (2001).
[Crossref]

Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part II: FDTD case,” IEEE Trans. Magn. 37(5), 3255–3259 (2001).
[Crossref]

Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microw. Theory Tech. 48(12), 2289–2296 (2000).
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V. Rizzoli, A. Costanzo, D. Masotti, A. Lipparini, and F. Mastri, “Computer-aided optimization of nonlinear microwave circuits with the aid of electromagnetic simulation,” IEEE Trans. Microw. Theory Tech. 52(1), 362–377 (2004).
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A. M. Darwish, E. P. Ippen, H. Q. Le, J. P. Donnelly, and S. H. Groves, “Optimization of four‐wave mixing conversion efficiency in the presence of nonlinear loss,” Appl. Phys. Lett. 69(6), 737–739 (1996).
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R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D Appl. Phys. 40(14), R249–R271 (2007).
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Ding, R.

L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech. 53(9), 2752–2767 (2005).
[Crossref]

Donnelly, J. P.

A. M. Darwish, E. P. Ippen, H. Q. Le, J. P. Donnelly, and S. H. Groves, “Optimization of four‐wave mixing conversion efficiency in the presence of nonlinear loss,” Appl. Phys. Lett. 69(6), 737–739 (1996).
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R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D Appl. Phys. 40(14), R249–R271 (2007).
[Crossref]

Dudley, J. M.

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics 3(2), 85–90 (2009).
[Crossref]

Fichtner, N.

P. Russer and N. Fichtner, “Nanoelectronics in radio-frequency technology,” IEEE Microw. Mag. 11(3), 119–135 (2010).
[Crossref]

Först, M.

R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D Appl. Phys. 40(14), R249–R271 (2007).
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Freude, W.

Gaeta, A. L.

Gao, M.

Gontier, Y.

Y. Gontier and M. Trahin, “Frequency conversion involving Raman-like processes above the ionization threshold,” IEEE J. Quantum Electron. 18(7), 1137–1145 (1982).
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Groves, S. H.

A. M. Darwish, E. P. Ippen, H. Q. Le, J. P. Donnelly, and S. H. Groves, “Optimization of four‐wave mixing conversion efficiency in the presence of nonlinear loss,” Appl. Phys. Lett. 69(6), 737–739 (1996).
[Crossref]

Hahn, S.

Y. Chung, J. Ryu, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part I: FETD case,” IEEE Trans. Magn. 37(5), 3289–3293 (2001).
[Crossref]

Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part II: FDTD case,” IEEE Trans. Magn. 37(5), 3255–3259 (2001).
[Crossref]

Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microw. Theory Tech. 48(12), 2289–2296 (2000).
[Crossref]

Hahn, S.-Y.

I.-H. Park, B.-T. Lee, and S.-Y. Hahn, “Design sensitivity analysis for nonlinear magnetostatic problems using finite element method,” IEEE Trans. Magn. 28(2), 1533–1536 (1992).
[Crossref]

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S. A. Sørngård, S. I. Simonsen, and J. P. Hansen, “High-order harmonic generation from graphene: Strong attosecond pulses with arbitrary polarization,” Phys. Rev. A 87(5), 053803 (2013).
[Crossref]

Hennessy, K. J.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 607–609 (2012).
[Crossref]

Hoefer, W. J. R.

P. Russer, P. P. M. So, and W. J. R. Hoefer, “Modeling of nonlinear active regions in TLM (distributed circuits),” IEEE Microw. Guided Wave Letts. 1(1), 10–13 (1991).
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Hu, E. L.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 607–609 (2012).
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Igarashi, H.

H. Igarashi and K. Watanabe, “Complex adjoint variable method for finite-element analysis of eddy current problems,” IEEE Trans. Magn. 46(8), 2739–2742 (2010).
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T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 607–609 (2012).
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Ippen, E. P.

A. M. Darwish, E. P. Ippen, H. Q. Le, J. P. Donnelly, and S. H. Groves, “Optimization of four‐wave mixing conversion efficiency in the presence of nonlinear loss,” Appl. Phys. Lett. 69(6), 737–739 (1996).
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R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antenn. Propag. 45(3), 364–374 (1997).
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S. Kawanishi and O. Kamatani, “All-optical time division multiplexing using four-wave mixing,” Electron. Lett. 30(20), 1697–1698 (1994).
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T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
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S. Kawanishi and O. Kamatani, “All-optical time division multiplexing using four-wave mixing,” Electron. Lett. 30(20), 1697–1698 (1994).
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[Crossref]

Le, H. Q.

A. M. Darwish, E. P. Ippen, H. Q. Le, J. P. Donnelly, and S. H. Groves, “Optimization of four‐wave mixing conversion efficiency in the presence of nonlinear loss,” Appl. Phys. Lett. 69(6), 737–739 (1996).
[Crossref]

Lee, B.-T.

I.-H. Park, B.-T. Lee, and S.-Y. Hahn, “Design sensitivity analysis for nonlinear magnetostatic problems using finite element method,” IEEE Trans. Magn. 28(2), 1533–1536 (1992).
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Leuthold, J.

Li, X.

O. S. Ahmed, M. H. Bakr, and X. Li, “An impulse sampling approach for efficient 3D TLM-based adjoint sensitivity analysis,” Prog. Electromagnetics Res. 142, 485–503 (2013).
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O. S. Ahmed, M. H. Bakr, and X. Li, “A memory-efficient implementation of TLM- based adjoint sensitivity analysis,” IEEE Trans. Antenn. Propag. 60(4), 2122–2125 (2012).
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O. S. Ahmed, M. H. Bakr, X. Li, and T. Nomura, “A Time-Domain Adjoint Variable Method for Materials With Dispersive Constitutive Parameters,” IEEE Trans. Microw. Theory Tech. 60(10), 2959–2971 (2012).
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Lipparini, A.

V. Rizzoli, A. Costanzo, D. Masotti, A. Lipparini, and F. Mastri, “Computer-aided optimization of nonlinear microwave circuits with the aid of electromagnetic simulation,” IEEE Trans. Microw. Theory Tech. 52(1), 362–377 (2004).
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Lipson, M.

Lojek, B.

B. Lojek, “Sensitivity analysis of nonlinear circuits,” Electronic Circuits and Systems, IEE Proceedings 129(3), 85–88 (1982).
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S. Lum, M. Nakhla, and Q.-J. Zhang, “Sensitivity analysis of lossy coupled transmission lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech. 42(4), 607–615 (1994).
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Macdonald, P. A.

A. E. Schmitz, R. H. Walden, L. E. Larson, S. E. Rosenbaum, R. A. Metzger, J. R. Behnke, and P. A. Macdonald, “A deep-submicrometer microwave/digital CMOS/SOS technology,” IEEE Electron Device Lett. 12(1), 16–17 (1991).
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Madsen, K.

M. H. Bakr, J. W. Bandler, K. Madsen, J. E. Rayas-Sanchez, and J. Sondergaard, “Space-mapping optimization of microwave circuits exploiting surrogate models,” IEEE Trans. Microw. Theory Tech. 48(12), 2297–2306 (2000).
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Masotti, D.

V. Rizzoli, A. Costanzo, D. Masotti, A. Lipparini, and F. Mastri, “Computer-aided optimization of nonlinear microwave circuits with the aid of electromagnetic simulation,” IEEE Trans. Microw. Theory Tech. 52(1), 362–377 (2004).
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Mastri, F.

V. Rizzoli, A. Costanzo, D. Masotti, A. Lipparini, and F. Mastri, “Computer-aided optimization of nonlinear microwave circuits with the aid of electromagnetic simulation,” IEEE Trans. Microw. Theory Tech. 52(1), 362–377 (2004).
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Metzger, R. A.

A. E. Schmitz, R. H. Walden, L. E. Larson, S. E. Rosenbaum, R. A. Metzger, J. R. Behnke, and P. A. Macdonald, “A deep-submicrometer microwave/digital CMOS/SOS technology,” IEEE Electron Device Lett. 12(1), 16–17 (1991).
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Monro, T. M.

Murnane, M. M.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Nakhla, M.

S. Lum, M. Nakhla, and Q.-J. Zhang, “Sensitivity analysis of lossy coupled transmission lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech. 42(4), 607–615 (1994).
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M. Nakhla and J. Vlach, “A piecewise harmonic balance technique for determination of periodic response of nonlinear systems,” IEEE Trans. Circ. Syst. 23(2), 85–91 (1976).
[Crossref]

Nakhla, M. S.

A. S. Saini, M. S. Nakhla, and R. Achar, “Generalized time-domain adjoint sensitivity analysis of distributed MTL networks,” IEEE Trans. Microw. Theory Tech. 60(11), 3359–3368 (2012).
[Crossref]

Nikolova, N. K.

M. H. Bakr, P. Zhao, and N. K. Nikolova, “Adjoint first order sensitivities of transient responses and their applications in the solution of inverse problems,” IEEE Trans. Antenn. Propag. 57(7), 2137–2146 (2009).
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P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, “Theory of self-adjoint S-parameter sensitivities for lossless nonhomogeneous transmission-line modeling problems,” IET Microwave Antennas Propag. 2(3), 211–220 (2008).
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M. H. Bakr, N. K. Nikolova, and P. A. W. Basl, “Self-Adjoint S-Parameter Sensitivities for Lossless Homogeneous TLM Problems,” International Journal of Numerical Modeling: Electronic Networks, Devices and Fields 18(6), 441–455 (2005).
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M. H. Bakr and N. K. Nikolova, “An adjoint variable method for time-domain transmission line modeling with fixed structured grids,” IEEE Trans. Microw. Theory Tech. 52(2), 554–559 (2004).
[Crossref]

Nomura, T.

O. S. Ahmed, M. H. Bakr, X. Li, and T. Nomura, “A Time-Domain Adjoint Variable Method for Materials With Dispersive Constitutive Parameters,” IEEE Trans. Microw. Theory Tech. 60(10), 2959–2971 (2012).
[Crossref]

Park, I.

Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part II: FDTD case,” IEEE Trans. Magn. 37(5), 3255–3259 (2001).
[Crossref]

Y. Chung, J. Ryu, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part I: FETD case,” IEEE Trans. Magn. 37(5), 3289–3293 (2001).
[Crossref]

Y. Chung, C. Cheon, I. Park, and S. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microw. Theory Tech. 48(12), 2289–2296 (2000).
[Crossref]

Park, I.-H.

I.-H. Park, B.-T. Lee, and S.-Y. Hahn, “Design sensitivity analysis for nonlinear magnetostatic problems using finite element method,” IEEE Trans. Magn. 28(2), 1533–1536 (1992).
[Crossref]

Popmintchev, T.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Poulton, C.

Raghunathan, V.

Ranka, J. K.

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J. E. Rayas-Sanchez, “EM-based optimization of microwave circuits using artificial neural networks: the state-of-the-art,” IEEE Trans. Microw. Theory Tech. 52(1), 420–435 (2004).
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M. H. Bakr, J. W. Bandler, K. Madsen, J. E. Rayas-Sanchez, and J. Sondergaard, “Space-mapping optimization of microwave circuits exploiting surrogate models,” IEEE Trans. Microw. Theory Tech. 48(12), 2297–2306 (2000).
[Crossref]

Reinhard, A.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 607–609 (2012).
[Crossref]

Richardson, D. J.

Rizzoli, V.

V. Rizzoli, A. Costanzo, D. Masotti, A. Lipparini, and F. Mastri, “Computer-aided optimization of nonlinear microwave circuits with the aid of electromagnetic simulation,” IEEE Trans. Microw. Theory Tech. 52(1), 362–377 (2004).
[Crossref]

Rosenbaum, S. E.

A. E. Schmitz, R. H. Walden, L. E. Larson, S. E. Rosenbaum, R. A. Metzger, J. R. Behnke, and P. A. Macdonald, “A deep-submicrometer microwave/digital CMOS/SOS technology,” IEEE Electron Device Lett. 12(1), 16–17 (1991).
[Crossref]

Russer, P.

P. Russer and N. Fichtner, “Nanoelectronics in radio-frequency technology,” IEEE Microw. Mag. 11(3), 119–135 (2010).
[Crossref]

P. Russer, P. P. M. So, and W. J. R. Hoefer, “Modeling of nonlinear active regions in TLM (distributed circuits),” IEEE Microw. Guided Wave Letts. 1(1), 10–13 (1991).
[Crossref]

Ryu, J.

Y. Chung, J. Ryu, C. Cheon, I. Park, and S. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis—Part I: FETD case,” IEEE Trans. Magn. 37(5), 3289–3293 (2001).
[Crossref]

Saini, A. S.

A. S. Saini, M. S. Nakhla, and R. Achar, “Generalized time-domain adjoint sensitivity analysis of distributed MTL networks,” IEEE Trans. Microw. Theory Tech. 60(11), 3359–3368 (2012).
[Crossref]

Schmitz, A. E.

A. E. Schmitz, R. H. Walden, L. E. Larson, S. E. Rosenbaum, R. A. Metzger, J. R. Behnke, and P. A. Macdonald, “A deep-submicrometer microwave/digital CMOS/SOS technology,” IEEE Electron Device Lett. 12(1), 16–17 (1991).
[Crossref]

Simonsen, S. I.

S. A. Sørngård, S. I. Simonsen, and J. P. Hansen, “High-order harmonic generation from graphene: Strong attosecond pulses with arbitrary polarization,” Phys. Rev. A 87(5), 053803 (2013).
[Crossref]

So, P. P. M.

P. Russer, P. P. M. So, and W. J. R. Hoefer, “Modeling of nonlinear active regions in TLM (distributed circuits),” IEEE Microw. Guided Wave Letts. 1(1), 10–13 (1991).
[Crossref]

Sondergaard, J.

M. H. Bakr, J. W. Bandler, K. Madsen, J. E. Rayas-Sanchez, and J. Sondergaard, “Space-mapping optimization of microwave circuits exploiting surrogate models,” IEEE Trans. Microw. Theory Tech. 48(12), 2297–2306 (2000).
[Crossref]

Sørngård, S. A.

S. A. Sørngård, S. I. Simonsen, and J. P. Hansen, “High-order harmonic generation from graphene: Strong attosecond pulses with arbitrary polarization,” Phys. Rev. A 87(5), 053803 (2013).
[Crossref]

Stentz, A. J.

Taflove, A.

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antenn. Propag. 45(3), 364–374 (1997).
[Crossref]

Taylor, J. R.

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics 3(2), 85–90 (2009).
[Crossref]

Trahin, M.

Y. Gontier and M. Trahin, “Frequency conversion involving Raman-like processes above the ionization threshold,” IEEE J. Quantum Electron. 18(7), 1137–1145 (1982).
[Crossref]

Turner, A. C.

Usechak, N.

R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D Appl. Phys. 40(14), R249–R271 (2007).
[Crossref]

Vlach, J.

M. Nakhla and J. Vlach, “A piecewise harmonic balance technique for determination of periodic response of nonlinear systems,” IEEE Trans. Circ. Syst. 23(2), 85–91 (1976).
[Crossref]

Volz, T.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 607–609 (2012).
[Crossref]

Walden, R. H.

A. E. Schmitz, R. H. Walden, L. E. Larson, S. E. Rosenbaum, R. A. Metzger, J. R. Behnke, and P. A. Macdonald, “A deep-submicrometer microwave/digital CMOS/SOS technology,” IEEE Electron Device Lett. 12(1), 16–17 (1991).
[Crossref]

Wang, J.

Watanabe, K.

H. Igarashi and K. Watanabe, “Complex adjoint variable method for finite-element analysis of eddy current problems,” IEEE Trans. Magn. 46(8), 2739–2742 (2010).
[Crossref]

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J. P. Webb, “Design sensitivity of frequency response in 3-D finite-element analysis of microwave devices,” IEEE Trans. Magn. 38(2), 1109–1112 (2002).
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Windeler, R. S.

Winger, M.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 607–609 (2012).
[Crossref]

Xu, J.

L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech. 53(9), 2752–2767 (2005).
[Crossref]

Yagoub, M. C. E.

L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech. 53(9), 2752–2767 (2005).
[Crossref]

Zhang, L.

L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech. 53(9), 2752–2767 (2005).
[Crossref]

Zhang, Q.-J.

L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech. 53(9), 2752–2767 (2005).
[Crossref]

S. Lum, M. Nakhla, and Q.-J. Zhang, “Sensitivity analysis of lossy coupled transmission lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech. 42(4), 607–615 (1994).
[Crossref]

Zhao, P.

M. H. Bakr, P. Zhao, and N. K. Nikolova, “Adjoint first order sensitivities of transient responses and their applications in the solution of inverse problems,” IEEE Trans. Antenn. Propag. 57(7), 2137–2146 (2009).
[Crossref]

Appl. Phys. Lett. (1)

A. M. Darwish, E. P. Ippen, H. Q. Le, J. P. Donnelly, and S. H. Groves, “Optimization of four‐wave mixing conversion efficiency in the presence of nonlinear loss,” Appl. Phys. Lett. 69(6), 737–739 (1996).
[Crossref]

Electron. Lett. (1)

S. Kawanishi and O. Kamatani, “All-optical time division multiplexing using four-wave mixing,” Electron. Lett. 30(20), 1697–1698 (1994).
[Crossref]

Electronic Circuits and Systems, IEE Proceedings (1)

B. Lojek, “Sensitivity analysis of nonlinear circuits,” Electronic Circuits and Systems, IEE Proceedings 129(3), 85–88 (1982).
[Crossref]

IEEE Electron Device Lett. (1)

A. E. Schmitz, R. H. Walden, L. E. Larson, S. E. Rosenbaum, R. A. Metzger, J. R. Behnke, and P. A. Macdonald, “A deep-submicrometer microwave/digital CMOS/SOS technology,” IEEE Electron Device Lett. 12(1), 16–17 (1991).
[Crossref]

IEEE J. Quantum Electron. (1)

Y. Gontier and M. Trahin, “Frequency conversion involving Raman-like processes above the ionization threshold,” IEEE J. Quantum Electron. 18(7), 1137–1145 (1982).
[Crossref]

IEEE Microw. Guided Wave Letts. (1)

P. Russer, P. P. M. So, and W. J. R. Hoefer, “Modeling of nonlinear active regions in TLM (distributed circuits),” IEEE Microw. Guided Wave Letts. 1(1), 10–13 (1991).
[Crossref]

IEEE Microw. Mag. (1)

P. Russer and N. Fichtner, “Nanoelectronics in radio-frequency technology,” IEEE Microw. Mag. 11(3), 119–135 (2010).
[Crossref]

IEEE Trans. Antenn. Propag. (3)

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antenn. Propag. 45(3), 364–374 (1997).
[Crossref]

M. H. Bakr, P. Zhao, and N. K. Nikolova, “Adjoint first order sensitivities of transient responses and their applications in the solution of inverse problems,” IEEE Trans. Antenn. Propag. 57(7), 2137–2146 (2009).
[Crossref]

O. S. Ahmed, M. H. Bakr, and X. Li, “A memory-efficient implementation of TLM- based adjoint sensitivity analysis,” IEEE Trans. Antenn. Propag. 60(4), 2122–2125 (2012).
[Crossref]

IEEE Trans. Circ. Syst. (1)

M. Nakhla and J. Vlach, “A piecewise harmonic balance technique for determination of periodic response of nonlinear systems,” IEEE Trans. Circ. Syst. 23(2), 85–91 (1976).
[Crossref]

IEEE Trans. Magn. (5)

H. Igarashi and K. Watanabe, “Complex adjoint variable method for finite-element analysis of eddy current problems,” IEEE Trans. Magn. 46(8), 2739–2742 (2010).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 The RC circuit of example 1.
Fig. 2
Fig. 2 The original and adjoint responses of the circuit in Fig. 1 for the set of values R1 = 1000.0 Ω, R2 = 1000.0 Ω, and C = 60.0 µF.
Fig. 3
Fig. 3 The sensitivities of the objective function (21) with respect to the nonlinear coefficients of the resistor R2 (p0 and p2) for a sweep of the coefficient po with R1 = 1000.0 Ω, C = 10.0 µF, and p2 = 50.0 shown in (a) and (b) respectively; the sensitivities estimated using AVM ( + ) and sensitivities obtained using the central finite difference method (CFD) (o).
Fig. 4
Fig. 4 The sensitivities of the objective function (21) with respect to the circuit parameters C and R1 for a sweep of the parameter po with R1 = 1000.0 Ω, C = 10.0 µF, and p2 = 50.0 shown in (a) and (b) respectively; the sensitivities estimated using AVM ( + ) and sensitivities obtained using the central finite difference method (CFD) (o).
Fig. 5
Fig. 5 A nonlinear region in a parallel plate waveguide.
Fig. 6
Fig. 6 The adjoint sensitivities ( + ) as compared to the central finite difference (o) for a sweep of the parameter po for the microwave example.
Fig. 7
Fig. 7 The structure of the optical example.
Fig. 8
Fig. 8 The modal profile of the excitation used in the photonic example.
Fig. 9
Fig. 9 The adjoint sensitivities ( + ) as compared to the central finite difference (CFD) (o) for a sweep of the parameter p 0 ε for the photonic example.

Equations (26)

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F= 0 T m Ω g(x,V)  dΩ dt= 0 T m G(x,V)   dt,
F x i = 0 T m e G x i dt+   0 T m ( G V ) T V x i  dt,
V k+1 =CS V k + V k s ,
λ k1 = S T C T λ k + V k s,λ ,  λ N T =0,
F x i e F x i  -Δt k λ k T Δ A i Δ x i V k , i=1,2,, n,
ε r = p 0 + p 1 E+ p 2 E 2 + p 3 E 3 ,
V k + ( V  t ) k ΔtCS V k + V k s .
V t = 1 Δt (CSI)V+ V s Δt .
2 V t x i = 1 Δt (C[ (S V ¯ ) p T p x i + (S V ¯ ) V T V x i +S V x i ] V x i ),
0 T m λ T [ 2 V t x i 1 Δt C[ (S V ¯ ) p T p x i + (S V ¯ ) V T V x i +S V x i ]+ 1 Δt V x i ]  dt=0.
0 T m ( λ T t +λ T [ 1 Δt C (S V ¯ ) V T + 1 Δt CS 1 Δt I ] ) V x i  dt=  0 T m λ T Δt C (S V ¯ ) p T p x i dt.
( λ T t +λ T [ 1 Δt C (S V ¯ ) V T + 1 Δt CS 1 Δt I ] )= G V T .
λ k1 = ( (S V ¯ ) V T +S ) T C T λ k Δt G V .
F x i = 0 T m e G x i dt 0 T m λ T C (S V ¯ ) p T p x i dt,  i=1,2,, n.
S n = ( (S V ¯ ) V T +S ) .
S n (j) = ( ( S (j) V ¯ (j) ) V (j)T + S (j) ), j=1,2,...,N.
J (j) = ( S (j) V ¯ (j) ) p (j)T
λ ˜ k = C T λ k
V c t = V S R 1 C ( R 1 + R 2 R 1 R 2 C ) V c .
R 2 = p o + p 2 V c 2 .
F= 0 T m ( V s (t) V c (t)) 2 dt = 0 T m ψ(t)dt .
λ t = λ S R 1 C [ ( R 1 + R 2 R 1 R 2 C )+ 1 C d d V c ( V ¯ c R 2 ) ]λ,
ε r = p o + p 1 E y + p 2 E y 2 ,
F= 0 T m Ω E y 2  dΩdt,
ε r = p 0 ε + p 1 ε E y + p 2 ε E y 2 ,
σ = p 2 σ E y 2 ,

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