1. Introduction

Recently, scientists have paid much attention to artificial crystal structures, known as photonic crystals (PCs), which could manipulate the flow of light in the same way that semiconductors control the flow of electrons [1]. These PCs are composed of dielectric–dielectric or metallic–dielectric nanostructures that are repeated regularly in a way that may result in the appearance of what is being called a photonic band gap (PBG). The PBG is defined as a range of forbidden frequencies within which transmission of light is blocked as it is totally reflected or absorbed. This PBG exists as a result of the multiple Bragg scattering of the incident electromagnetic waves (EMW). According to the number of directional axes in which dielectric materials exhibit periodicity, PCs can be classified into three types: one, two, or three-dimensional PC structures. The one-dimensional photonic crystal (1D PC) structure consists of a periodically repeated configuration of double/triple layers. Although they are the simplest among these types due to the ease of fabrication and analyzing, they have received the most attention of researchers and engineers because of their numerous promising applications. Among these applications are high-reflecting omnidirectional mirrors, antireflection coatings, low-loss waveguides, low threshold lasers, high-quality resonator cavities, and photonic-based active devices [1,2]. Moreover, these 1D PC structures form the basis of photonic filters operating over a frequency spectrum ranging from radio waves up to optical wavelengths, passing by infrared range.

Nowadays 1D PC filters play a vital role in a wide range of applications for many areas of science. For instance, these applications include ultra-high-speed wireless communications, eye protection glasses, and antireflecting coating for solar cells. They are also utilized in biological and chemical imaging as well as security screening. In addition to this, photonic filters are commonly used in space science and laser applications as well as thermophotovoltaic (TPV) applications [3–19]. According to the application type and the desired specifications, the required photonic filter differs among wide band, narrow band, or selective pass/stop filters at selected wavelength ranges. However, the only common factor among these cases remains the problem of finding the optimal design that best fits the desired performance.

In order to improve the characteristics and performance of a given 1D photonic filter, it is necessary to change the design approach from the traditional design of quarter-wave-thick layers to a non-quarter-wave-thick layers design [10]. This leads to an increase of the designable parameters, making the filter design problem much more complicated. Also, it becomes hard to predict, by intuitive PBG analysis, the behavior of the filter response due to specific variations of parameters. Therefore, the need for employing a simulation-driven optimization approach, seeking to achieve the optimal filter design, highly increases.

In general, the structure of a 1D PC filter is characterized by a set of designable parameters $\mathit{\varphi }\in {\mathbb{R}}^n$, which might be the refractive index of the layer $n_j$, the layer thickness $d_j$, and the number of periods $N$. The filter performance is described in terms of some measurable quantities $f_i(\mathit{\varphi }),i=1,2,\dots ,m$. These performance measures may be the output transmittance, reflectance or absorbance response of the filter and are usually evaluated through numerical system simulations. According to the application, design specifications are suggested by the designer through specifying bounds on the performance measures. These design specifications define a region in the design parameter space called feasible region ${\mathbf{R}}_{\mathbf{f}}$. If a design point $\mathit{\varphi }$ lies inside ${\mathbf{R}}_{\mathbf{f}}$ then all the corresponding design specifications are satisfied. The objective is to find the optimal design point within ${\mathbf{R}}_{\mathbf{f}}$ that best fits the predefined design specifications. Generally, the problem of finding the optimal design point can be formulated as an optimization problem according to the used criterion [20–23], i.e., finding the optimal design point of the 1D PC filter necessitates the solution of an optimization problem.

The optimal design of 1D PC filters has been treated, in literature, through different strategies [10–17]. For example, Asghar et al. obtained the optimal design of a wide bandpass optical filter by employing an approach based on the genetic optimization algorithm [10]. Other design techniques based on genetic algorithms were also studied, by Jia et al. [11] and Xu [12]. Besides, Celanovic et al. [13] and Xuan et al. [14] applied genetic algorithm to achieve the optimal design of EMW spectral filters. Moreover, Baedi et al. have achieved the optimal design of a narrow bandpass filter by using an approach based on particle swarm optimization [15]. Recently, Badaoui and Abri have proposed an optimization technique based on simulated annealing to obtain the optimal design of selective filters [16]. However, these presented approaches have some drawbacks, mainly, the excessive number of required function evaluations (some practical cases need more than thousands) and the low convergence rate in reaching to an optimal design point. Also, all of these approaches are not guaranteed to converge to an optimal design point, starting from any initial point, since their optimization algorithms are based on uncertainties. Inflexibility in adapting the aforementioned techniques to other filter design problems is another pitfall.

Another strategy, seeking to find the optimal refractive index values of multilayer structures using convex optimization, was proposed by Swillam et al. [17]. However, this approach fixes the layers’ thicknesses to their quarter-wave-thick values, in order to guarantee a convex optimization problem. In addition, the complexity of the corresponding design problem increases in order to realize values of the optimized refractive indices.

In this paper, an optimization approach is proposed to find the optimal design of 1D PC structures exploiting the minimax optimization approach [24–27]. The approach is powerful and general enough to be utilized in achieving the optimal design of any 1D PC-based EMW filter. This approach has been widely used in microwave and electronic circuit design problems. However, here, and for the first time, it is being used in the field of photonic crystals with very promising results and a significant support in the design cycle. The filter design problem is treated as an optimization problem with a minimax objective function and predefined design specifications (constraints). The design problem is formulated in a rigorous mathematical form avoiding the search-based heuristic techniques’ drawbacks, e.g., lower convergence rate and high computational cost. This makes the proposed approach superior in accelerating the design cycle and, hence, time-to-market. It also has the advantages of reaching the optimal design that achieves a given ripple level and converging to the optimum even from an infeasible starting design point. The presented approach shows its ability to get the optimal practical material type and optimal layer thicknesses of a given filter at the same time as well.

Although the concept of minimax is rather traditional [24], it is still effectively used in many engineering design problems, especially filter design problems [20,28,29]. In fact, the minimax criterion is preferable for filter design problems because it tends to achieve an equal-ripple response of the obtained optimal design [26]. In addition, the minimax approach searches for a better design point even if all the desired specifications were satisfied, which allows for much-exceeded satisfactions of the desired specifications. The efficiency, reliability, and flexibility of the proposed approach are demonstrated by applying it to obtain the optimal design of two practical filters, wide bandpass optical filters (WBP-OF) and spectral filters, that are operating in two different spectrum regions and are used in two separate applications.

The paper is organized as follows: In Section 2 the general structure of 1D PC filters, as well as the governing equations of the propagated EMW, are described. The transfer matrix method used to model the transmittance response is also presented. Section 3 represents the formulation of the design problem as a minimax optimization problem. Besides, the methodology used to transform the minimax problem to a conventional nonlinear programming (NLP) problem is also described. Furthermore, Section 3 includes a description for the algorithm of the proposed optimal design approach. Sections 4 and 5 are devoted to review the optimization procedures and the obtained results when optimizing two examples, namely, ${(\mathrm{SiC}/\mathrm{Ag}/{\mathrm{SiO}}_2)}^N$ WBP-OF and ${(\mathrm{Ag}/{\mathrm{SiO}}_2)}^N$ spectral filter, respectively. Finally, conclusions of the work are drawn in Section 6.

2. 1D Photonic Crystal Filters—Structure and Governing Equations

The general structure of these photonic filters is a 1D PC comprising a unit cell repeated $N$ times. This unit cell consists of two or three dielectric/metallic layers, as shown in Fig. 1. The filter is surrounded from the front and back by incident and substrate media with refractive indices $n_0$ and $n_s$ respectively. This periodic layered structure of the filter configuration is defined by the number of periods $N$, the layer thicknesses $d_j$, and their refractive indices $n_j$.

 

Fig. 1. Basic structure of a 1D PC filter.

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Generally, the propagation of EMW onto periodic photonic crystal structures is governed by the decoupled Maxwell’s equation for nonmagnetic materials [1], which is stated as

The standard transfer matrix method (TMM) [30], which is frequently used in optics, is used to analyze the propagation of the EMW signals through the layered media of the filter. The TMM is based on applying the simple continuity conditions of the electric field that follows from Maxwell’s equations, at each interface. The tangential component of the electric field and its first derivative must be continuous across boundaries from one medium to the next. Hence, by imposing these boundary conditions, the electric field amplitudes between any two successive layers can be related by the following transfer matrix:

The forward amplitude of the incident medium $a_0$ is assigned to a fixed value $r_0$, and by introducing one more boundary condition on the backward amplitude of the substrate $b_{\alpha N+1}=b_s$ to be equal to zero, the amplitude of the transmission and reflection coefficients ($t$ and $r$) can be related by the total transfer matrix $M_T$ as

Thus, from this relation, the amplitude transmission and reflection coefficients can be derived from the elements of the total transfer matrix, yielding

The associated transmittance $T$ and reflectance $R$, which are often of more practical use, are calculated from

3. Optimal Design of 1D PC Filters

In general, the photonic filter system parameters ($d_j$ and $n_j$) are assembled in a vector $\mathit{\varphi }\in {\mathbb{R}}^{2p}$ where $\mathit{\varphi }={[\begin{array}{cccc}{d}_1& d_2\dots d_p& n_1& n_2\dots n_p\end{array}]}^T$, $N$ is the number of periods, and $p=\alpha N$ is the total number of layers.

A certain photonic filter is required to pass the incident EMW signals related to wavelength values located at passband region(s) and to stop/reject the signals within the stopband region(s). The ranges and locations of both the passband and stopband regions vary according to the concerned application and needed filter type. The desirable ideal transmittance response of any filter can be expressed as satisfying the following condition:

In practice, the condition required in Eq. (9) can never be achieved. Therefore, a set of some design specifications $U(\lambda )$ is introduced to describe an acceptable transmittance response:

Practically, the design specifications parameters ($\tau$, $\beta$, and $\mathrm{\Gamma }$) are chosen by the designer, according to the considered application and the allowed relaxation range of the design problem output response. There is a correlation between the parameters $\tau$ and $\beta$; therefore one can take larger $\tau$, but at the expense of larger $\beta$ too, in order to have a feasible solution. In other words, the designer needs to compromise between passband constraints $\tau$ and stopband constraints $\beta$ according to the considered application. In fact, in some applications, the highest passband transmittance is very important regardless to how low the stopband transmittance is.

In practice, we consider a finite number of wavelength samples in the spectrum range such that satisfying the specifications at these points implies satisfying them almost everywhere. Theoretically, the number of samples and samples spacing can be employed as optimizable parameters for given response and design variables. This optimization process can be considered as a part of the sensitivity analysis phase, which usually precedes the actual design problem solution. Let $m_p$ and $m_s$ be the number of sample points in the passband and stopband regions, respectively. In this case the continuous specification function (10) is approximated by the discrete specification function:

On the other hand, it should be emphasized that not all the design parameters $\mathit{\varphi }$ are supposed to be optimized; only a subset of $\mathit{\varphi }$, denoted by the design variables $\mathbf{x}\subset \mathit{\varphi }$, is selected. Fabrication techniques or technologies may also introduce additional restrictions on the possible values of these design variables. Thus, we may have upper bounds ${\xi }_{uj}$ or lower bounds ${\xi }_{lj}$ on some design variables, $x_j$. These constraints can be stated as follows:

Error functions $e_i$ arise from the deviation between each desired specification $U_i$ and its corresponding calculated response $T_i(\mathbf{x})$. However, to have a single type of errors, the upper and lower specifications should be formulated properly so that the constraints can be defined as

A. Minimax Problem Formulation

Now, starting from an initial point, ${\mathbf{x}}^{(0)}$, which is usually infeasible point, we seek to move not only to a feasible point but also to the optimal feasible design point that best meets the desired specifications. Thus, an optimization problem is formed from the objective function that seeks a single point in the space of the design variables ${\mathbf{x}}^{*}\in {\mathbb{R}}^n$, which best satisfies all of these defined error functions (13) under the restrictions on the optimization variables (12). This optimization problem is formulated as a minimax optimization problem:

B. Solving the Minimax Problem

One of the most efficient methods used for solving a minimax problem is to convert it to a nonlinear programming problem [24]. The transformation can be done by introducing an additional slack variable into the design variables of Eq. (15). The new independent variable is denoted by $z$, and it refers, implicitly, to the maximum calculated errors at the current optimizing iteration. By using this new slack variable $z$, we can formulate problem Eq. (15) as follows:

Now to obtain the optimum minimax solution, we solve problem (17). Actually, such conventional nonlinear programming problem can be solved using any nonlinear programming algorithm. In this work, MATLAB [31] is used.

C. Algorithm of the Proposed Optimal Design Approach

The flow diagram illustrated in Fig. 2 summarizes the algorithm of the proposed optimal design approach. The approach used is general enough to be applied on any 1D PC design problem.

 

Fig. 2. Flow diagram for the proposed optimization algorithm.

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4. Optimal Design of a Wide Bandpass Optical Filter—Example 1

A. Brief Description of the Filter

As a practical example, the proposed optimal design approach is applied for achieving the optimal design of a wide bandpass optical filter (WBP-OF). The suggested structure of the filter is a 1D photonic crystal, which comprises a unit cell repeated $N$ times. This unit cell consists of two different dielectric layers with a single metal layer between them. This structure is denoted by ${({\text{dielectric}}_1/\text{metal}/{\text{dielectric}}_2)}^N$. Particularly, the structure of ${(\mathrm{SiC}/\mathrm{Ag}/{\mathrm{SiO}}_2)}^N$ is considered, due to its good performance as a WBP-OF working in the visible range [18]. Moreover, such a materials configuration is applicable and of ease to be fabricated [4,32,33].

In this example, we focus on the EMW spectrum ranging from 300 nm to 900 nm. The objective is to achieve a WBP-OF that passes EMW in the visible range (from 450 nm to 700 nm) and rejects both infrared and ultraviolet ranges. The range under study is divided into three main regions: the lower stopband region (LSBR) below 350 nm, the passband region (PBR) from 450 nm to 700 nm, and the upper stopband region (USBR) above 800 nm.

All results of this section are obtained while the refractive indices of the incident and substrate media are assumed to be 1 and 1.52, respectively. A unity value is assigned to the forward amplitude of the incident medium $(r_0)$, and the refractive index of ${\mathrm{SiO}}_2$ is set to 1.45, whereas the refractive indices of Ag and SiC are assigned to practical measured values obtained from [34] so that the frequency dependency of the layered media can be considered.

The performance of the filter transmittance response is measured with respect to some figures of merit (FOM). The first of these is the band factor BF, which indicates the passband sharpness of the calculated transmittance response and is defined as [18]

Then, we seek to design a filter having the highest possible $T_{\max }$, BF, and $T_{\text{avg\_PBR}}$. In contrast, the filter should have the smallest possible $T_{\text{avg\_LSBR}}$ and $T_{\text{avg\_USBR}}$. Also, the filter should have ${\lambda }_{l50}$ and ${\lambda }_{u50}$, which best match the visible range limits.

B. Optimization Procedures and Results

1. Periodic Structure with Constant Thicknesses Optimization

First, we consider the case of a periodic structure with constant thicknesses (PSCT). Hence, the optimization variables are limited only to three variables, which are assembled to the design vector, $\mathbf{x}={[\begin{array}{ccc}{d}_1& d_2& d_3\end{array}]}^T$, where $d_1$, $d_2$, and $d_3$ are the thickness of SiC, Ag, and ${\mathrm{SiO}}_2$ layers, respectively. The variables are restricted to ${\xi }_{lj}=3$ and ${\xi }_{uj}=300$, for $j=1$, 2, and 3. Furthermore, the discretization parameters ($m_p$ and $m_s$) are set to 25 and 20, respectively. However, in order to guarantee a minimal transmittance of 75% for all samples located at the PBR and a maximal transmittance of 7% for those sample points located either at the LSBR or at the USBR, the design specifications’ parameters ($\tau$ and $\beta$) are set to 75% and 7%, respectively. Here, the ripples constraint is ignored, by neglecting the upper bound on bandpass transmittance, defined in Eq. (13), and by setting $\mathrm{\Gamma }$ to 0. We assign the design point suggested in [18], $\mathbf{x}={[\begin{array}{cc}2010& 70\end{array}]}^T$, to be our initial point, ${\mathbf{x}}_{\mathbf{0}}$. After applying our optimization approach, we move (only after 13 iterations with 65 TMM system simulations) to an optimal design point ${\mathbf{x}}^{*}={[\begin{array}{ccc}18.14& 8.94& 53.68\end{array}]}^T$ that achieves the desired specifications. Let’s denote solution A to this optimal design point. Figure 3 shows how the filter response is improved at solution A. The optimized $T_{\text{avg\_PB}}$ is incremented from 82.55% to 83.33% without violating any of the defined specifications, as declared in Table 1. Besides, the transmittance response is slightly shifted to the left such that it becomes more centered around the visible spectrum. Finally, the achieved enhancement in the FOM of the filter, for solution A, is summarized in Table 1.

 

Fig. 3. (a) Transmittance and (b) absorbance of ${(\mathrm{SiC}/\mathrm{Ag}/{\mathrm{SiO}}_2)}^5$, before and after optimization, starting from an initial point: $d_1=20\mathrm{nm}$, $d_2=10\mathrm{nm}$, and $d_3=70\mathrm{nm}$. The case of PSCT is considered.

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Table 1. Figures of Merit of the ${(\mathsf{SiC}/\mathsf{Ag}/{\mathsf{SiO}}_{\mathsf{2}})}^{\mathsf{5}}$ Filter^a

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2. Periodic Layered Structure with Variable Thickness Optimization

In order to enhance the performance of the filter response, the dimensionality of the problem is increased by considering a periodic layered structure with variable thickness (PLSVT) assigned to each layer. Thus, the designable variables become $\mathbf{x}={[\begin{array}{cc}{d}_1& d_2\dots d_{15}\end{array}]}^T$, where $d_j$ is the $j$-th layer thickness. The values of the optimization approach’s parameters are set as follows: $m_p=25$, $m_s=20$; $\tau =82\%$, $\beta =5\%$; ${\xi }_{lj}=3$ and ${\xi }_{uj}=300$, for $j=1,2,\dots 15$. We also ignored the ripples constraint for this design problem. Starting from the design point of solution A, we apply the proposed design approach under the new considerations by repeating the optimal design values of the three layers for the five periods to cover the whole structure as indicated in Table 2. Accordingly, we can move (only after 108 iterations with 1862 TMM system simulations) into the new optimal design point ${\mathbf{x}}^{*}\in {\mathbb{R}}^{15}$, namely, solution B (see Table 2). Figure 4 shows the optimal response at solution B, where $T_{\text{avg\_PB}}$ is incremented to 84.53% without violating any other specification. In fact we could achieve a transmittance lower than 3% over the whole stopband spectrum except for wavelength value of 800 nm at which 5% of transmittance is obtained. Moreover the sharpness (as indicated in Fig. 4) of the response’s edges is considerably improved. The characteristics of the optimized filter are illustrated in Table 1.

 

Fig. 4. (a) Transmittance and (b) absorbance of ${(\mathrm{SiC}/\mathrm{Ag}/{\mathrm{SiO}}_2)}^5$ before and after the optimization. Solution A is assigned to the initial design point. The case of PLSVT is considered.

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Table 2. Design Data of the Filter ${(\mathrm{SiC}/\mathrm{Ag}/{\mathrm{SiO}}_2)}^5$^a

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3. Testing the Convergence Ability of the Proposed Approach

In order to measure the efficiency of our proposed approach, the approach is applied when different initial points are being considered. The proposed approach shows a very good ability to converge into the optimal solution whatever the initial point is. Among those various results obtained, Figs. 5 and 6 show the obtained optimal response related to initial points: ${[\begin{array}{ccc}3& 3& 3\end{array}]}^5$ and ${[\begin{array}{ccc}70& 10& 70\end{array}]}^5$, respectively. Figure 5 shows that the approach can converge (after 81 iterations with 1395 TMM system simulations) to a solution even from a trivial guess ${[\begin{array}{ccc}3& 3& 3\end{array}]}^5$, whereas in Fig. 6 the proposed approach shows how it can converge (after 112 iterations with 1998 TMM system simulations) from an initial point ${[\begin{array}{ccc}70& 10& 70\end{array}]}^5$ almost corresponding to the complete converse of the desired response.

 

Fig. 5. (a) Transmittance and (b) absorbance of ${(\mathrm{SiC}/\mathrm{Ag}/{\mathrm{SiO}}_2)}^5$ before and after optimizing. Starting from initial point: $d_1=3\mathrm{nm}$, $d_2=3\mathrm{nm}$ and $d_3=3\mathrm{nm}$. The case of PLSVT is considered.

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Fig. 6. (a) Transmittance and (b) absorbance of ${(\mathrm{SiC}/\mathrm{Ag}/{\mathrm{SiO}}_2)}^5$, before and after optimizing starting from initial point: $d_1=70\mathrm{nm}$, $d_2=10\mathrm{nm}$, and $d_3=70\mathrm{nm}$. The case of PLSVT is considered.

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4. Optimizing for the Least Level of Ripples at the PBR

Now, we try to find the optimal flat response that corresponds to the highest possible PBR transmittance with the least possible level of ripples. Thus, this time the ripples constraint, defined in Eq. (13), is considered. We set the acceptable ripples parameter $\mathrm{\Gamma }$ to be 2% and by adjusting the other approach’s parameters as $m_p=25$, $m_s=20$; $\tau =82\%$, $\beta =5\%$; ${\xi }_{lj}=3$, ${\xi }_{uj}=300$, for $j=1,2,\dots 15$. The approach is applied, starting from solution B. We obtain an optimal design point (after 79 iterations with 1375 TMM system simulations) denoted by solution C (see Table 2). The obtained response and the FOM of solution C are illustrated in Fig. 7 and Table 1, respectively. Although the PBR’s transmittance is slightly degraded, the transmittance response became almost flat around 82%. Actually, such a flat response is much preferable for many applications, as it prevents phase noise, resulting from the passband ripples.

 

Fig. 7. (a) Transmittance and (b) absorbance of ${(\mathrm{SiC}/\mathrm{Ag}/{\mathrm{SiO}}_2)}^5$ before and after optimizing. Solution B is assigned to the initial design point. The case of PLSVT is considered, and the ripples constraint is taken into consideration.

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5. Optimal Design of a Spectral Filter—Example 2

A. Brief Description of the Filter

In this section, the proposed approach is applied to obtain the optimal design of spectral control filters, required for enhancing the efficiency of TPV systems.

The TPV system is an energy converter that converts thermal heat into electrical energy [9]. It consists of an emitter, a photovoltaic (PV) cell, and a spectral control filter. The emitter is a thermal heater. It emits EMW onto the PV cell. Transmitted photons having energies above or equal to the bandgap energy of the PV cell can be absorbed by the cell and electron-hole pairs are generated. Contrarily, photons having energies below the bandgap energy will not be absorbed by the cell and will be lost, which limits the overall efficiency of the TPV system. Hence, in order to enhance the TPV efficiency, a spectral filter is located between the emitter and the PV cell to transmit photons that are in band to the PV cell and reflect the remaining back to the emitter. In other words, an ideal spectral filter can be considered as a low bandpass photonic filter that passes all the radiations with wavelength below the PV cell bandgap wavelength (${\lambda }_g$) and reflects all the radiations corresponding to wavelength higher than ${\lambda }_g$.

The performance of the spectral filter and the whole TPV system is assessed with respect to three suggested FOM, which are (i) the passband efficiency of the filter (${\eta }_p$), which is the ratio between the above bandgap power density transmitted from the filter to the PV cell ($P_{\mathrm{abg}}$) and that transmitted from an ideal filter ($P_{\mathrm{abg}|I}$); (ii) the filter stopband efficiency (${\eta }_s$), which is the ratio between the amount of below bandgap power density ($P_{\mathrm{bbg}}$) reflected from the filter back to the emitter and that amount of power reflected in the case of an ideal filter ($P_{\mathrm{bbg}|I}$); and (iii) the spectral efficiency of the TPV system (${\eta }_{sp}$), which is the ratio of $P_{\mathrm{abg}}$ to the net power density ($P_{\text{net}}$) radiated by the filter, where $P_{\text{net}}$ is the total power density radiated from the emitter minus the amount of the power reflected from the filter and returned back to the emitter. These power density quantities are estimated as follows:

For an ideal spectral filter ${\eta }_p$, ${\eta }_s$, and ${\eta }_{sp}$ equal 100%. However, a practical filter does not perfectly transmit all power in the passband, which negatively affects the above bandgap power radiated to the PV cell, i.e, decreases ${\eta }_p$. Besides, in order to obtain a very high passband transmittance response, the filter will not be able to effectively reflect all the below bandgap power, which decreases ${\eta }_s$, as it is practically impossible to have a sharp edge transition at ${\lambda }_g$. Therefore, we should compromise between the passband efficiency and the stopband efficiency to achieve the highest possible spectral efficiency.

B. Optimization Procedures and Results

All results of this subsection are obtained, assuming the emitter radiation as an ideal blackbody radiation with 1500 K temperature. The PV material is assumed to be gallium antimonide (GaSb), which is the most common material used for the fabrication of TPV cells [9,14]. GaSb has a refractive index of 3.8 and a bandgap energy of 0.7 eV, which is equivalent to bandgap wavelength ${\lambda }_g=1.78\mathrm{\mu m}$. Thus, the spectral filter is supposed to transmit all photons below 1.78 μm to the PV cell and to reflect all photons above 1.78 μm. However, due to low energy of the blackbody radiation below 0.85 μm and above 6.5 μm, the filter is just designed to have a transmittance as high as possible in the wave band of 0.85–1.78 μm and becomes as low as possible in the band of 1.79–6.5 μm. In order to guarantee a steep transition at ${\lambda }_g$, the passband and stopband regions are defined strictly closed to each other, and as an alternative approach the design specifications ($\tau$ and $\beta$) are defined on the average transmittance of the passband and stopband regions, respectively. Moreover, the ripples constraint is ignored for this design problem due to the minor impact of the passband ripples in this application.

The filter structure is suggested as a 1D PC comprising of $N$ periodically repeated unit cells. The unit cell consists of two consecutive metallic ($M$) and dialectic ($D$) layers. Thus, the filter is denoted by $D{(MD)}^N$, where the first dielectric layer is added to improve the filter matching with the incident medium. Initially, the metal and dielectric layers are assumed to be Ag and ${\mathrm{SiO}}_2$, respectively [19]. The number of periods, $N$, is fixed to three. The refractive indices of the incident and substrate media are assumed to be one and 3.8, respectively, and a unity value is assigned to the forward amplitude of the incident medium $(r_0)$. The refractive index of ${\mathrm{SiO}}_2$ is set to 1.5. Besides, the absorption and frequency dependency of Ag layers are considered, by using the Drude model [9] to calculate the refractive index of Ag.

1. Thickness Optimization

First, the designable parameters are supposed to be the thickness of the layers. Thus, the design vector is considered as $\mathbf{x}={[\begin{array}{cc}{d}_1& d_2\dots d_7\end{array}]}^T$, where $d_j$ is the thickness of $j$-th layer. The variables are restricted to ${\xi }_{lj}=3$ and ${\xi }_{uj}=300$, for $j=1,2,\dots 7$, and the discretization parameters ($m_p$ and $m_s$) are set to 20 and 40, respectively. Initially, the Ag layers’ thickness is set to 10 nm and the thickness of ${\mathrm{SiO}}_2$ is assumed to follow the well-known quarter-wave-thick design (QWTD), at which $n_{{\mathrm{SiO}}_2}{d}_{{\mathrm{SiO}}_2}=\frac{{\lambda }_g}{4}$, where $n_{{\mathrm{SiO}}_2}$ and $d_{{\mathrm{SiO}}_2}$ are the refractive index and thickness of the ${\mathrm{SiO}}_2$ layers, respectively. ${\lambda }_g=1.78\mathrm{\mu m}$ is the bandgap wavelength of the GaSb PV cell. Figures 8 and 9 show optimal responses of two solutions, obtained using the proposed optimization approach, namely solution 1 and solution 2, respectively. In Fig. 8, we optimize for the highest possible transmittance in the passband, while we optimize for the least possible transmittance within the stopband in Fig. 9. In solution 1, the design specifications are set as $\tau =83\%$ and $\beta =3.5\%$ to have the highest possible transmittance in the passband. By contrast, to obtain the least possible stopband transmittance in solution 2, the design specifications are adjusted as $\tau =79\%$ and $\beta =2\%$. Solution 1 is obtained after 54 iterations and 486 TMM system simulations. In contrast, 62 number of iterations and 558 TMM system simulations are executed, till achieving solution 2. Table 3 shows the optimized values obtained for the two solutions. The FOM regarding the two solutions and the initial point are compared in Table 4. Starting from initial passband efficiency equal to 15.2% in the initial point, ${\eta }_p$ significantly improved to 83.7% in solution 1, as opposed to only 77.6%, achieved, in solution 2. However, solution 2 is practically better than solution 1, because it achieves much reflection of the radiation in the stopband, resulting in higher spectral efficiency than solution 1 (67% comparing to 59.8% in solution 2).

 

Fig. 8. (a) Transmittance and (b) absorbance of the ${\mathrm{SiO}}_2{(\mathrm{Ag}/{\mathrm{SiO}}_2)}^3$ spectral filter, before and after optimizing for the highest passband transmittance. The initial point is assumed as the QWTD for ${\mathrm{SiO}}_2$-layers, while the Ag layers are fixed to 10 nm. The obtained optimal solution is referred to as solution 1.

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Fig. 9. (a) Transmittance and (b) absorbance of the ${\mathrm{SiO}}_2{(\mathrm{Ag}/{\mathrm{SiO}}_2)}^3$ spectral filter, before and after optimizing for the least stopband transmittance. The initial point is assumed as the QWTD for ${\mathrm{SiO}}_2$ layers, while the Ag layers are fixed to 10 nm. The obtained optimal solution is referred to as solution 2.

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Table 3. Design Data of the $D{(\mathrm{Ag}/D)}^3$ Spectral Filter, where $D$ Refers to a Dielectric Layer^a

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Table 4. Figures of Merit of the $D{(\mathrm{Ag}/D)}^3$ Spectral Filter, where $D$ Refers to a Dielectric Layer^a

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2. Dielectric Material Optimization

In order to improve the performance and efficiency of the filter, we try to find the optimum dielectric material that achieves the best characteristics of the spectral filter. Thus, the designable parameters are, now, supposed to be the thickness of the layers ($d_j$), as well as the refractive index of the dielectric material, $n_D$, i.e., the design vector becomes $x={[\begin{array}{ccc}{d}_1& d_2\dots d_7,& n_D\end{array}]}^T$.

Although performing optimization on the refractive index of the dielectric material may seem just a theoretical study as it may result on nonrealizable materials, it is still a potential future study as we may be capable of fabricating materials with these exact refractive indices values once day. Another solution to address this problem is to replace the optimized refractive index with its nearest, in refractive index value, realizable (already exact) material. The optimization parameters are set as $m_p=20$, $m_s=40$; $\tau =82.5\%$, $\beta =1.5\%$; ${\xi }_{lj}=3$ and ${\xi }_{uj}=300$, for $j=1,2,\dots 7$. Besides, the dielectric refractive index is bounded as $1\le n_D\le 5$. Starting from solution 2, we apply the proposed design approach under these considerations. Accordingly, we obtain (after 77 iterations with 798 TMM system simulations) the new optimal design point ${\mathbf{x}}^{*}\in {\mathbb{R}}^8$, namely, solution 3 (see Table 3). Figure 10 shows the obtained optimized response, where the passband transmittance is significantly enhanced. The FOM of the optimized filter are illustrated in Table 4. The spectral efficiency is noticeably incremented to 77.5% with considerably high passband efficiency, ${\eta }_p=79.5\%$. Fortunately, the optimized refractive index value ($n_D^{*}=2.38$) is very near to the refractive index of titanium dioxide (${\mathrm{TiO}}_2$) dielectric material, which is commonly used in the fabrication of 1D PC structures [10]. That makes the achieved spectral efficiency in solution 3 applicable in reality.

 

Fig. 10. (a) Transmittance and (b) absorbance of the $D{(\mathrm{Ag}/D)}^3$ spectral filter, before and after optimizing for the least possible stopband transmittance, where $D$ refers to a dielectric layer. Solution 2 is considered as an initial point. Both the refractive index of the dielectric layers and the thickness of layers are optimized.

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6. Conclusion

An approach for optimal design of 1D photonic crystal filters is introduced. The filter design problem is formulated as a minimax optimization problem. The design problem is entirely performed within MATLAB environment. In addition to the precise control of the required filter specifications, the proposed approach allows the designer to consider problems of higher dimension than usually treated before, using intuitive PBG analysis. The validity, flexibility, and efficiency of the approach have been demonstrated by applying it to obtain the optimal design of two practical examples, namely, a wide bandpass optical filter and a spectral filter. Different designable variables, desired specifications, and initial design points were considered and studied. A comparative study between different design solutions for several FOM is also given. The obtained numerical results convincingly show that the proposed approach is powerful and effective in satisfying the predefined design specifications, starting even from a trivial guess or from an initial point almost corresponding to the complete converse of the desired response. The presented study (simulations and optimizations) is usually a vital step in the design cycle. It provides an important guiding map to save time and money before doing the fabrication step.