Mixed integer programming with kriging surrogate model technique for dispersion control of photonic crystal fibers

In the recent years, special features of photonic crystal fibers (PCFs) have made them an efficient candidate in many applications in different fields. These features include single mode behavior along wide band of frequency range (Knight et al. 1996), large birefringence (Schreiber et al. 2014; Mohammadd et al. 2022), high nonlinearity (Broeng and Tünnermann 2005; Su et al. 2014), and large effective mode area (Knight et al. 1998). Further, the chromatic dispersion in PCFs could be easily controlled along wide range of wavelengths (Hameed et al. 2017). Consequently, the PCFs may be exploited in many potential applications, such as supercontinuum generation devices (Lanh et al. 2019), dispersion compensating fibers (Islam et al. 2017; Ranathive et al. 2022), ultra-flattened-dispersion (Le et al. 2018), and polarization maintaining fibers (Yin and Xiong 2014), lasers (Cui et al. 2019). The PCF properties can be also adjusted by changing the shape, size, pitch, number of air-holes rings around the core, and the positions of cladding air-holes. Also, filling PCFs with other materials increases the degree of freedom (Eggleton et al. 2001; Xuan et al. 2017; Yiou et al. 2015; Sun et al. 2016). Therefore, new applications have been reported using PCFs such as plasmonic sensors (Chen et al. 2023a, b), cancer cell detector (Habib et al. 2021; Mohammed et al. 2023) and photonic analog-to-digital converters (Mohammadi et al. 2022).

To improve data capacity in optical communication through PCFs, the dispersion should be controlled. Initially, Saitoh et al. (2003) have shown that the PCFs can be controlled to obtain nearly zero flattened dispersion over studied wavelength range. Further, several optimization algorithms have been exploited to control the dispersion of PCFs such as metaheuristic algorithms (Hameed et al. 2016), genetic algorithms (Kerrinckx et al. 2004; Poletti et al. 2005; Abdelaziz et al. 2013), radial basis function artificial neural network (RBF-ANN) (El-Mosalmy et al. 2014; Hameed et al. 2008) and trust region (TR) techniques (Hameed et al. 2017). In (Hameed et al. 2016), a central force optimization algorithm (CFO) is used to optimize 6 design parameters to reach an ultra-flat zero dispersion between + 0.1 and − 0.609 Ps/km nm over a range of wavelengths from 1.25 to 1.6 μm. Also, a particle swarm optimization (PSO) algorithm could obtain a dispersion between + 0.1974 and − 0.7404 Ps/km nm over the same range of wavelengths. The previous results are obtained by a specified range for each design parameter. Further, Kerrinckx et al. (2004) have used genetic algorithm (GA) to optimize only two design parameters to achieve a dispersion between + 7.75 and + 1.41 Ps/km nm over a wavelength range from 1.25 to 1.6 μm. Furthermore, a dispersion between + 0.1 and − 0.1 Ps/km nm is obtained by Poletti et al. (2005) using another GA over a limited range of wavelengths from 1.5 to 1.6 μm. Moreover, a GA is used by Abdelaziz et al. (2013) to obtain a dispersion between \(\pm\) 0.1 Ps/km nm over a wide wavelength range from 1.335 to 1.584 μm. Besides, RBF-ANN (El-Mosalmy et al. 2014) is used to obtain a dispersion between + 0.28 and − 0.59 Ps/km nm over a range of wavelengths from 1.25 to 1.6 μm. In addition, a trust region algorithm (Hameed et al. 2017) could achieve a nearly flat zero dispersion using several designs over wavelength range from 1.45 to 1.6 μm. Also, the trust region algorithm (Hameed et al. 2017) has obtained a highly negative flat dispersion with an average value – 155 Ps/km nm through wavelength from 1.4 to 1.6 μm for the quasi TM mod.

The metaheuristic algorithms and genetic algorithms (Hameed et al. 2016; Kerrinckx et al. 2004; Poletti et al. 2005) have several drawbacks as they may stuck in local optimal point and reaching to the global optimum is not guaranteed. Further, they can be sensitive for the initial conditions and have a slow rate of convergence in the optimization process. Additionally, they are not based on any mathematical basis compared to the classical optimization techniques (Viana et al. 2005). Also, trust region algorithms suffer from the absence of a general rule to choose the trust region radius and their dependence on the solved problem (Kamandi et al. 2015). In addition, the newly ANN algorithms (Chen et al. 2023a, b; Yang et al. 2023) suffer from several disadvantages (Zhang et al. 2023; Lin et al. 2023; Abdolrasol et al. 2021) such as the unexplained way of selecting the outputs of the network. Further, the ANN structure is determined by the experience without any deterministic laws. Furthermore, the newly ANNs may have a gradual corruption during the optimization process which increases the simulation time where large amount of memory is needed. Moreover, the network training is complex and may be trapped in local optimum.

In this paper, mixed integer nonlinear programming (MINLP) optimization technique called branch and bound optimization (Dakin 1965) is exploited to optimize the dispersion characteristics of PCFs. In this class of optimization technique, some of the design parameters take real values while the others are integers. This feature will be very helpful to control the filling material in each hole of the PCFs during the optimization process. However, in the previous studies, only the geometrical parameters are optimized assuming fixed infiltration. The branch and bound technique is considered one of the most efficient optimization methods for MINLP problems which have some or all of its design parameters are restricted to take discrete values only and the performance of the objective function is affected by nonlinear dynamics. In addition, due to the computational cost of the objective function, computationally efficient surrogate-based models could be constructed to replace the objective function. Further, the suggested technique branches the original problem into sequence of subproblems. Then, some of the subproblems are solved to obtain upper and lower bounds for the objective function value. These bounds enable us to discard some other subproblems to save computational time. Therefore, the suggested technique covers all the design space with less complexity and time than the ANN algorithms.

In this study, the proposed algorithm relies on kriging surrogate models (Lophaven et al. 2002). Further, a MINLP branch and bound optimization algorithm with kriging surrogate model is used along with full vectorial finite difference method (FVFDM) (Fallahkhair et al. 2008) to achieve an ultra-flat dispersion with zero value over a wide band of wavelengths from 1.25 to 1.6 μm. Moreover, the introduced technique is employed to reach a highly negative flat dispersion of − 163 ± 0.9 and − 170 ± 1.2 ps/Km. nm over the same wavelength range for the quasi TM and the quasi TE modes, respectively. These results are compared with other algorithms (Hameed et al. 2008; Hameed et al. 2016; Hameed et al. 2017; Kerrinckx et al. 2004; Poletti et al. 2005; El-Mosalmy et al. 2014) to show the strength of the proposed algorithm.

It should be noticed that, in the proposed algorithm all design parameters can be changed at the same time which enable us to benefit from the entire design space and to obtain a global optimal solution. However, most of the preceding designs are obtained using the parametric sweep methodology where only one parameter can be changed at a time while the other parameters are kept constant leading to reduced design space.

MINLP problems are optimization problems which involve discrete design parameters and nonlinear dynamics. The general form of the MINLP problem iswhere \(\varvec{c}, \varvec{n}_1\) and \({\varvec{n}}_2\) denote the number of constraints, the number of continuous design parameters and the number of discrete design parameters respectively. The function \({\varvec{f}}\left( {{\varvec{x}}_1 , {\varvec{x}}_2 } \right)\) represents the objective function, the function \({\varvec{h}}_i \left( {{\varvec{x}}_1 ,{\varvec{x}}_2 } \right)\) used to represent ith constraint. \({\varvec{x}}_1 , {\varvec{x}}_2\) denote the continuous and discrete design parameters respectively. The reported algorithm belongs to the mixed integer nonlinear programming optimization class where the computationally expensive objective function is approximated by a computationally cheaper kriging surrogate model \({\varvec{\hat{y}}}\left( {\varvec{x}} \right)\) (Lophaven et al. 2002):

The first part is a quadratic regression polynomial with \(m = \left( {n + 1} \right){ }\left( {n + 2} \right)/2\) where \(n\) is the total number of continuous and discrete design parameters, unknown regression coefficients \({\varvec{\alpha}} = [\alpha_1 , \alpha_2 , \ldots , \alpha_m ] \in {\mathbb{R}}^m\) and \(g_j ({\varvec{x}})\) are the basis functions of the regression model where \(g_1 \left( {\varvec{x}} \right) = 1\), \(g_2 ({\varvec{x}}) = {\varvec{x}}_1\), …, \(g_{n + 1} ({\varvec{x}}) = {\varvec{x}}_n\), \(g_{n + 2} ({\varvec{x}}) = {\varvec{x}}_1^2\),…, \(g_{2n + 1} ({\varvec{x}}) = {\varvec{x}}_1 {\varvec{x}}_n\), …, \(g_{2n + 2} ({\varvec{x}}) = {\varvec{x}}_2^2\),…, \(g_{3n + 1} ({\varvec{x}}) = {\varvec{x}}_2 {\varvec{x}}_n\),…, \(g_m ({\varvec{x}}) = {\varvec{x}}_n^2\).

The second part, \(e({\varvec{x}})\), is a Gaussian random error function with zero mean and variance \(\sigma^2\). The covariance matrix \({\varvec{\varSigma}} \in {{\mathbb{R}}}^{s \times s}\) of \(e({\varvec{x}})\) is calculated as follows:where \(s\) is the number of sample data; \({\varvec{\rho}}\) \(\in {{\mathbb{R}}}^{s \times s}\) is a symmetric positive semidefinite matrix with diagonal elements of all ones that represents the Gaussian spatial correlation function. The entries of \({\varvec{\rho}}\) is computed by the kernel functionwhere \(\varphi_k \ge 0\) and \(\varvec{\varphi } = \left[ {\begin{array}{*{20}c} {\varphi _1 } & \ldots & {\varphi _n } \\ \end{array} } \right]\) represents distance weights. Let \({\varvec{x}}^{\left( 1 \right)}\), \({\varvec{x}}^{\left( 2 \right)}\), …, \({\varvec{x}}^{\left( s \right)} \in {{\mathbb{R}}}^n\) be a set of \(s\) inputs to the high-fidelity model and the resulting outputs are \({\varvec{Y}} = [y({\varvec{x}}^{\left( 1 \right)} )\), \(y({\varvec{x}}^{\left( 2 \right)} )\), …, y(\({\varvec{x}}^{\left( s \right)} )] \in {{\mathbb{R}}}^s\). The kriging model \(\hat{y}\left( {\varvec{x}} \right)\) linearly predict \(y\left( {\varvec{x}} \right)\) at an unknown point \({\varvec{x}} \in {{\mathbb{R}}}^n\) as follows

The optimal of \({\varvec{\mu}}\left( {\varvec{x}} \right)\) is then obtained through reducing the mean square error of the prediction and this leads to the following predictor equationwhere \({\varvec{g}}\left( {\varvec{x}} \right)\) is \(m \times 1\) vector contains the basis functions of the quadratic regression model, \(\varvec{G} = \left[ {\begin{array}{lll} {\varvec{g}({\varvec{x}}^{(1)} )} & \ldots & {\varvec{g}(\varvec{x}^{(s)} )} \\ \end{array} } \right]^T \in \mathbb{R}^{s \times m}\) is the regression matrix and \({\varvec{r}}\left( {\varvec{x}} \right) = \left[ {{\varvec{\rho}}\left( {{\varvec{x}}, {\varvec{x}}^{\left( 1 \right)} } \right){ } \ldots {\varvec{\rho}}\left( {{\varvec{x}},{\varvec{x}}^{\left( s \right)} } \right)} \right]^{\varvec{T}} \in {{\mathbb{R}}}^s\). The value of \({\varvec{\alpha}}\) is given by:

The MINLP branch and bound algorithm starts by solving a relaxed version of MINLP in (1) obtained by relaxing the integrality condition. If the solution to the relaxed problem satisfies the integrality conditions on the discrete design parameters, then it also solves the original MINLP. Otherwise, the algorithm branches on any discrete design parameter with fractional value and creates two MINLP subproblems with additional constraint on the value of the selected discrete design parameter. In the first subproblem, the selected discrete design parameter takes at least the ceil of the fractional value. In the second subproblem, it takes at most the floor of it. Then the algorithm solves the new subproblems after relaxing the integrality condition. This process continues till the algorithm solves all the created subproblems. The optimal solution is the solution with the minimum value for the objective function. A flowchart for the steps of the proposed algorithm is shown in Fig. 1.

In this paper, MINLP with kriging model algorithm is employed in the design and optimization of dispersion characteristics of PCFs. The reported technique is very reliable and gives an accurate prediction for the objective function values compared with other surrogate models like polynomials, radial basis functions…etc. The effectiveness of the proposed algorithm is illustrated by designing a nearly zero flat dispersion over a broad wavelength range from 1.25 to 1.6 \({\upmu }\) m using silica PCF selectively filled with Ethanol material. Further, the proposed algorithm is used for optimizing two soft glass NLC-PCFs for dispersion compensation through the same wavelength range for the quasi TM and quasi TE modes. The suggested MINLP technique with kriging model could achieve a highly negative flat dispersion up to − 163 Ps/Km nm and up to − 170 Ps/Km nm for the quasi TM mode and the quasi TE mode, respectively over the wavelength range from 1.25 to 1.6 μm. Therefore, the MINLP algorithms could prove strength and effectiveness for the dispersion control of selectively filling photonic devices. Further, the proposed MINLP algorithm can be integrated with other surrogate models other than kriging models such as neural networks models and space mapping techniques to effectively replace the computationally expensive objective function.