Efficient optimal design of mosaic-like PPDW devices for THz application using the adjoint variable method

Md. Iquebal Hossain Patwary; Akito Iguchi; Yasuhide Tsuji

doi:10.1364/OE.490232

1. Introduction

In order to confirm the enlargement of the communication system capacity and meet the modern communication requirements, the use of THz technology [1] is being actively investigated due to its potential applications in various scientific and technological fields, such as telecommunication [2,3], biotechnology [4,5], spectroscopy [6,7], astronomy [8], imaging and security system [9–12]. To realize the THz-wave circuits, different THz waveguides have been proposed over the last few years, such as non-radiative dielectric waveguide (NRD) [13,14], and parallel plate dielectric waveguide (PPDW) [15,16]. NRD guide is operated at higher order modes and these modes have cutoff behavior which greatly limits the practical application of this waveguide. On the other hand, PPDW devices have attracted a lot of attention for THz-wave circuits due to its simple structure and very low loss over a long distance in the THz region. PPDW consists of a dielectric strip placed between two parallel metal plates and supports fundamental TE$_0$ mode whose electric field is perpendicular to the metal plates as shown in Fig. 1. The structure of the PPDW is similar to the NRD guide, however unlike the NRD guide, PPDW can work with an arbitrary metal plate spacing and can eliminate the cutoff state by selecting the fundamental TE$_0$ mode as the guided mode. These characteristics allow the PPDW to confine almost all the power inside the dielectric strip without radiating or leaking outside of the parallel plates. All of these attractive features make PPDW a promising candidate for THz integrated circuit applications [15,16] and some basic PPDW components have been reported so far [17–19]. However, a well-established optimal design methodology to realize these devices has not been developed yet.

Fig. 1. (a) 3D model of the PPDW, (b) PPDW propagation mode.

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Recently, several kinds of optimal design techniques have been proposed based on size optimization, shape optimization, topology optimization, and mosaic-like optimization for the efficient design of high performance electromagnetic devices [20–29]. Among these optimization techniques, mosaic-like optimization method has attracted a lot of attention in the development of photonic and NRD guide devices. Since out-of-plane radiation loss can be suppressed in PPDW as with the NRD guide, we are thinking mosaic-like device is more suitable for PPDW platform. To design this mosaic like devices, various kinds of solution search algorithm have been applied so far, such as direct binary search (DBS), or some evolutionary approaches [30,31]. These approaches do not require sensitivity analysis, but design efficiency is therefore not high. Thus, an efficient mosaic-like optimization technique to improve the design efficiency is crucial in the design of high performance PPDW devices.

In this paper, in order to realize efficient mosaic-like optimization of the PPDW devices for THz application, we propose a novel gradient method based optimal design approach utilizing AVM [32]. The gradient method is a one dimensional solution search method utilizing gradient information. Therefore, search efficiency is high compared with evolutionary methods although an appropriate initial solution is required to avoid local optimum. In our design approach, the density method is utilized for the mosaic structure expression and to meet the appropriate initial solution requirements. In the gradient method, the sensitivity is efficiently analyzed by using AVM. To show the effectiveness of our design approach, initially two basic and important devices, T-branch and Three branch are designed as numerical examples. Our design strategy is also effective for larger scale design problems compared with that based on DBS or evolutionary approach. Thus, another two higher functional devices, mode splitting device and THz bandpass filter are designed in addition to the basic devices. The same PPDW devices except bandpass filter are also designed for the broadband operation by modifying the objective function and considering it for a range of frequencies within the targeted frequency band. Here, to evaluate the transmission characteristics of the devices, two-dimensional finite element method (2D-FEM) is employed.

2. Mosaic-like optimization with gradient method

2.1 Density method for mosaic structure expression

In our research, we consider a design model of PPDW with mosaic-like structure as shown in Fig. 2(a). The design region is discretized into number of pixels and initially the density parameters in each ($i,j$)-pixel are to be set, where the density parameters in every ($i,j$)-pixel are uniformly distributed as the initial structure and is taken, $\rho _{ij}$= 0.5. After completing the optimization process, the optimized design region is expressed as a mosaic-like structure, where air or dielectric is allocated at each pixel. Here, the fundamental TE$_0$ mode is used in PPDW as the guided mode. The permittivity of ($i,j$)-pixel in the design region is expressed as follows:

(1)$$\varepsilon_{r,ij} = \varepsilon_a+(\varepsilon_b-\varepsilon_a)U(\rho_{ij},p)$$

where $\varepsilon _a$ and $\varepsilon _b$ are the relative permitivities of air and dielectric, respectively, $\rho _{ij}$ is a density parameter allocated in each ($i,j$)-pixel with a value between 0 to 1. $U(\rho,p)$ is a quasi-step function with a penalty parameter $p$, used to binarize the permittivity distribution and is expressed as follows:

(2)$$U\left(\rho, p \right)=\left\{ \begin{array}{cc} 0.5\left( 2\rho\right)^{p} & \left[ 0\le\rho\le0.5\right] \\1-0.5\left(2-2\rho \right)^{p} & \left[0.5<\rho\le1 \right] \end{array} \right.$$

Figure 2(b) shows the examples of quasi-step function, where quasi-step function approaches the true step function as $p$ increases. Here, $p$ is initially set to be very small and is increased during the optimization process. In this way, a continuous transition from global search to local search is achieved, and almost binarized structure is obtained by $p$= 576. In our design approach, the density parameters are the design variables which are optimized during the optimization process to minimize the objective function, thus the optimal mosaic structure can be obtained with the desired transmission property. In this paper, the objective function to be minimized is given by

(3)$$\textrm{Minimize}\,C = \sum_{i=2}^{n_{p}}\left(\left\vert S_{i1}\right\vert^{2} - P_{i1}\right)^{2}$$

where $n_{p}$ is a number of output ports, $S_{i1}$ is the S-parameter from port 1 to port $i$ and $P_{i1}$ is the desired output power at output port $i$.

Fig. 2. (a) Design model of mosaic-like PPDW devices, (b) examples of the quasi-step function, $U(\rho,p)$.

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2.2 Formulation by 2D-FEM

In the optimization process, it is necessary to evaluate the transmission characteristics of the given PPDW devices. In this purpose, the propagation field in the given devices is simulated by the FEM. The wave equation from Maxwell’s equations for TE wave with uniform structure in $z$-direction is expressed as follows:

(4)$$\frac{\partial}{\partial x}\left(\frac{\partial \Phi}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial \Phi}{\partial y}\right)+k_{0}^{2}n^{2}\Phi=0$$

where $\Phi$ is the electric fields along $z$-axis and $k_{0}$ is the free space wave number. Dividing the analysis region into quadratic triangular elements, field $\Phi$ within each element can be approximated as follows:

(5)$$\Phi=\left\{N \right\}^{T}\left\{ \Phi\right\}_{e}$$

where $N$ is the shape function vector and $\left \{ \Phi \right \}_{e}$ is the nodal $\Phi$ vector for each element. Then, considering Eqs. (4) and (5), and applying the FEM described in [33], we can obtain a final matrix equation as follows:

(6)$$[P({\mathbf{\rho}})]\{\Phi\} = \{u\}$$

where $[P]$ is a FEM matrix, $\{\Phi \}$ is a discretized field, $\{u\}$ is an incidence condition and $ {\mathbf{\rho }}$ is a vector consists of $\rho _{ij}$.

2.3 Sensitivity analysis by AVM

In this paper, the design variables are efficiently optimized by utilizing gradient method. In the gradient method, to update the structure in the direction of the gradient of objective function, sensitivity analysis is required. Here, for an efficient sensitivity analysis, the AVM is employed. After solving the propagation field in the given PPDW devices by FEM, the scattering parameter from port 1 to port $n$ is calculated in the form of

(7)$$S_{n1} = \{\Phi\}^T\{g_n\}.$$

where $\{g_n\}$ is a vector related to the modal field at the output ports. First, we represent the derivative of S-parameter with respect to $\rho _{ij}$ by

(8)$$\frac{\partial S_{n1}}{\partial \rho_{ij}}=\sum_{k} \frac{\partial S_{n1}}{\partial \Phi_k} \frac{\partial \Phi_{k}}{\partial \rho_{ij}} = \left\{ \frac{\partial S_{n1}}{\partial \Phi}\right\}^{T} \frac{\partial \left\{ \Phi\right\}}{\partial \rho _{ij}}$$

Differentiating both sides of the final FEM matrix equation given in Eq. (6) with respect to $\rho _{ij}$, we can obtain an expression as follows:

(9)$$\frac{\partial\left\{\Phi \right\}}{\partial\rho_{ij}}={-}\left[ P\right]^{{-}1}\frac{\partial\left[ P\right]}{\partial\rho_{ij}}\left\{\Phi \right\}$$

Then, substituting this equation to the Eq. (8) and considering the Eq. (7), the sensitivity of $S_{n1}$ with respect to $\rho _{ij}$ can be obtained as follows:

(10)$$\frac{\partial S_{n1}}{\partial \rho_{ij}}={-}\left\{\lambda_{n} \right\}^{T}\frac{\partial \left[ P\right]}{\partial \rho_{ij}}\left\{\Phi \right\} ={-}\sum_{m\in X}^{}\left\{\lambda_{n} \right\}^{T}\frac{\partial \left[ P\right]}{\partial \rho_{m}}\left\{\Phi \right\}$$

with an adjoint equation

(11)$$[P]^T\{\lambda_n\} = \{g_n\}$$

where $\rho _{m}$ is the density at each element, $X$ is a set of element number that ($i,j$)-pixel includes, $\{\lambda _n\}$ is called as an adjoint variable, commonly used for each design variables and can be calculated by solving Eq. (11). Therefore, the sensitivity to all the densities can be efficiently calculated just by the inner product of known vectors. In the case of using usual density method, the sensitivity is calculated with respect to each element density. On the other hand, in our mosaic-like design method, the sensitivity with respect to pixel density parameter is the sum of the sensitivities calculated with respect to the element densities that each pixel includes.

3. Numerical examples

We show the validity of our mosaic-like design approach by designing several PPDW devices. In the design examples of basic devices, T-branch and three branch, the penalty parameter is set to be increased linearly during the optimization process. For the comparatively higher functional devices like mode splitting device and bandpass filter, the penalty parameter is set to be increased exponentially to avoid the local optimum.

3.1 T-branch waveguide

First, we consider a design example of PPDW T-branch device as shown in Fig. 3(a). Here, the structural parameters of the device are, $b = 40\,\mu$m, $l = 100\,\mu$m, $d_\textrm{PML} = 50\,\mu$m and the relative permittivities of the dielectric and air are considered to be $\varepsilon _b = 11.9$ and $\varepsilon _a = 1$, respectively. The size of the design region is set to be $W_x = W_y = 280\,\mu$m and is discretized into $28\times 28$ pixels with the size of $10\,\mu \textrm{m}\times 10\,\mu \textrm{m}$. The pixel size is considered so that the optimized structure can be fabricated using the recent high resolution 3D printers which can print with the minimum feature size of $2\,\mu \textrm{m}$. Half symmetric condition is applied along $y$-axis indicated by red line in the figure. The operating frequency is assumed to be 1 THz for a single frequency operation and port 1 is cosidered as the incident port. To split the incident power equally to the output ports 2 and 3, the objective function is modified as follows:

(12)$$\textrm{Minimize}\,C = \sum_{i=2}^{3}\left(\left\vert S_{i1}\right\vert^{2} - \frac{1}{2}\right)^{2}$$

Fig. 3. (a) Design model of PPDW T-branch device, (b) convergence behavior in the optimal design of T-branch.

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The convergence behavior in the optimization of T-branch at single frequency operation is shown in Fig. 3(b). We can see that rapid convergence is achieved compared with DBS or evolutionary approach [30,31] and around 60 iteration is enough to converge to the optimal solution. In the optimization of mosaic-like device design by using evolutionary approach referenced in [31], 64 individuals were used for each approach, thus required waveguide analysis were 6,400 in 100 iteration. But, with our proposed design approach, only 100 of waveguide analysis or less are required to find the optimal solution. Thus the required computational cost is reduced by a factor of 64 or less compared with that of evolutionary method. Figure 4(a) and 4(b) show the optimized structure of the T-branch and propagation field at 1 THz operation where the permittivity distribution is binarized. The frequency characteristics of the optimized T-branch for single frequency operation is shown in Fig. 4(c), where at 1 THz frequency it shows the desired output characteristics. The achieved normalized transmission power at the output ports are $\left \vert S_{21}\right \vert ^{2}$ = $\left \vert S_{31}\right \vert ^{2}$ = 0.497.

Fig. 4. Optimized results of T-branch at single frequency operation, (a) Optimized structure, (b) propagation field, and (c) frequency characteristics around 1 THz.

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To achieve the broad bandwidth characteristics of T-branch, we consider the device for broadband operation. In this purpose, the objective function is modified to analyze the device structure at three different frequencies, 0.95, 1.0 and 1.05 THz, and is given by

(13)$$\textrm{Minimize}\,C = \sum_{j=1}^{3}\sum_{i=2}^{3}\left( \left\vert S_{i1}\left(f_{j} \right) \right\vert^2 - \frac{1}{2}\right)^2$$

\left (f_{1, 2, 3} = {0.95,\ 1.0,\ 1.05\ \,\textrm{THz}}\right ).

The design parameters are same as the single frequency operation. Figure 3(b) depicts the convergence behavior of the device optimization at broadband operation, where we can see that the convergence behavior for the both single frequency operation and broadband operation are almost same. The optimized structure of the device and the propagation field at broadband operation are depicted in Fig. 5(a) and 5(b). The frequency characteristics of the optimized structure in Fig. 5(c) shows that the optimized device achieved broad bandwidth of more than 100 GHz around the targeted frequency band. In this study, conductor plates are assumed to be perfect conductor. Although conductor loss may reduce transmission power in practice, the attenuation is estimated to be at most a few percent because of the compactness of this device.

Fig. 5. Optimized results of T-branch at broadband operation, (a) Optimized structure, (b) propagation field, and (c) frequency characteristics around targeted frequency band.

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3.2 Three branch waveguide

Next, we consider a design example of PPDW three branch device as shown in Fig. 6(a). The structural parameters of the device are considered as same as the previous example for a single frequency operation. The objective function is adjusted in order to transfer the incident power equally to the output ports 2, 3, and 4, and is given by

(14)$$\textrm{Minimize}\,C = \sum_{i=2}^{4}\left(\left\vert S_{i1}\right\vert^{2} - \frac{1}{3}\right)^{2}$$

Fig. 6. (a) Design model of PPDW three branch device, (b) convergence behavior in the optimal design of three branch.

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In this device optimization, rapid convergence is attained as with the T-branch, and depiction of the convergence behavior in Fig. 6(b) illustrates this characteristic. Although it shows a pulsating behavior, we can reduce this behavior by adjusting the step size in the optimization process. The optimized structure of the three branch and the propagation field in the optimized structure at single frequency operation are shown in Fig. 7(a) and 7(b), where almost ideal transmission property is obtained. The frequency characteristics of the optimized three branch shows the desired output characteristics at 1 THz as depicted in Fig. 7(c). The optimized structure of the three branch achieved normalized transmission power of $\left \vert S_{21}\right \vert ^{2}$ = $\left \vert S_{41}\right \vert ^{2}$ = 0.328 and $\left \vert S_{31}\right \vert ^{2}$ = 0.329. However, the optimized device has a strong frequency dependence.

Fig. 7. Optimized results of three branch at single frequency operation, (a) Optimized structure, (b) propagation field, and (c) frequency characteristics around 1 THz.

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To realize the broad bandwidth characteristics of three branch, we increase the design region size to be $W_x = W_y = 320\,\mu$m with $32\times 32$ pixels. The other structural parameters are set to be same as the single frequency operation. The objective function is modified to analyze the device structure at three different frequencies, 0.975, 1.0 and 1.025 THz, and is given by

(15)$$\textrm{Minimize}\,C = \sum_{j=1}^{3}\sum_{i=2}^{4}\left( \left\vert S_{i1}\left(f_{j} \right) \right\vert^2 - \frac{1}{3}\right)^2$$

\left (f_{1, 2, 3} = 0.975,\ 1.0,\ 1.025\ \,\textrm{THz}\right ).

Figure 6(b) shows the convergence behavior of the objective function at broadband operation and the behavior is almost same as the single frequency operation. Figure 8(a) and 8(b) show the optimized structure and propagation field in the optimized three branch for broadband operation. The frequency characteristics of the device designed for broadband operation is depicted in Fig. 8(c), where we can see that the optimized structure achieved broad bandwidth of almost 100 GHz around the targeted frequency range.

Fig. 8. Optimized results of three branch at broadband operation, (a) Optimized structure, (b) propagation field, and (c) frequency characteristics around targeted frequency band.

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3.3 Mode splitting device

Next, a PPDW mode splitting device is considered as the third design example and the design model of the device is shown in Fig. 9(a). The waveguide width is increased to $b = 80\,\mu$m to support the higher order mode TE$_1$. The design region size of the device is set to be $W_x = W_y = 480\,\mu$m and is discretized into $48\times 48$ pixels. The other structural parameters are set to be same as the previous examples. The following is the objective function which is modified so that the incident TE$_0$ mode is propagated to the output port 2 and TE$_1$ mode is propagated to the output port 3.

(16)$$\textrm{Minimize}\,C = C_{1}+C_{2}$$

C_{1}= \left(\left\vert S_{21\left( \textrm{TE}_{0}\right)}\right\vert^2 - 1\right)^2,\,C_{2}= \left(\left\vert S_{31\left( \textrm{TE}_{1}\right)}\right\vert ^2 - 1\right)^2

Fig. 9. (a) Design model of PPDW mode splitting device, (b) convergence behavior in the optimal design of mode splitting device.

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The convergence behavior in the optimization of mode splitting device at single frequency operation is shown in Fig. 9(b), where rapid convergence is attained as with the previous design examples. Figure 10 shows the optimized structure and propagation fields in the optimized mode splitting device. In the figure it is shown that TE$_0$ and TE$_1$ modes are almost ideally splitted to the output ports 2 and 3, respectively. Figure 11 shows the frequency characteristics of the optimized mode splitting device for single frequency operation. It is shown that the optimized device achieved the intended transmission properties for the both modes at 1 THz frequency. However, there is a sudden decrease in the output power for both of the modes after crossing the aimed 1 THz frequency caused by the resonance occurred in the optimized structure. The achieved normalized transmission power for the TE$_0$ mode is 0.992, and 0.991 is for the higher order mode TE$_1$.

In order to achieve broad bandwidth characteristics of the mode splitting device, we increase the design region size of the device to be $W_x = W_y = 520\,\mu$m with $52\times 52$ pixels. The other design parameters are set to be same as the single frequency operation. The objective function is adjusted as follows:

(17)$$\textrm{Minimize}\,C =\sum_{i=1}^{5} \left(C_{1}\left( f_{i}\right)+C_{2}\left( f_{i}\right)\right)$$

C_{1}\left( f_{i}\right)= \left(\left\vert S_{21}^{\textrm{TE}_{0}}\left( f_{i}\right)\right\vert ^2 - 1\right)^2,\,C_{2}\left( f_{i}\right)= \left(\left\vert S_{31}^{\textrm{TE}_{1}}\left( f_{i}\right)\right\vert ^2 - 1\right)^2

\left (f_{1, 2, 3, 4, 5} = 0.95,\ 0.975,\ 1.0,\ 1.025,\ 1.05\ \,\textrm{THz}\right ).

Fig. 10. (a) Optimized structure, and (b, c) propagation fields of TE$_0$ and TE$_1$ modes in the optimized mode splitting device at 1 THz operation.

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Fig. 11. Frequency characteristics of the optimized mode splitting device at single frequency operation.

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The above objective function is modified so that the device structure is analyzed at five different frequencies between 0.95 to 1.05 THz to realize the broadband operation of the mode splitting device. The convergence behavior of the objective function in the optimization process of the device design for broadband operation is illustrated in Fig. 9(b). The optimized structure and the propagation fields in the optimized mode splitting device are shown in Fig. 12. We can see the device designed for broadband operation accomplishes the mode splitting function with almost ideal transmission properties for the both TE$_0$ and TE$_1$ modes. The frequency characteristics of the optimized device achieved broad bandwidth of almost 100 GHz within the aimed frequency band as depicted in Fig. 13.

Fig. 12. (a) Optimized structure, and (b, c) propagation fields of TE$_0$ and TE$_1$ modes in the optimized mode splitting device for broadband operation.

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Fig. 13. Frequency characteristics of the optimized mode splitting device at broadband operation.

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3.4 THz bandpass filter

Finally, we consider a design example of PPDW THz bandpass filter as shown in Fig. 14(a). The design region of the device has to be long shaped in order to realize the bandpass filtering function properly. Here, the design region size is set to be $W_x \times W_y = 280 \times 900\,\mu$m with $28 \times 90$ pixels, and the waveguide width is again set to $b = 40\,\mu$m as the incident mode is only the fundamental TE$_0$ mode. The other design settings are set to be same as the previous examples. The passband is assumed to be 50 GHz from 0.975 THz to 1.025 THz to get a flat top transmission band around the central frequency 1 THz. The objective function is modified as follows:

(18)$$\textrm{Minimize}\,C = \sum_{i=0}^{24}\left(\left\vert S_{21}\left(f _{i}\right)\right\vert^{2} - P\left(f_{i} \right)\right)^{2}$$

P\left(f_{i} \right) = \left\{\begin{array}{cc} {1.0\ } & {\left(\textrm{within the passband}\right)\ } \\ {0.0\ } & {\left(\textrm{otherwise}\right)\ } \end{array} \right.

where 25 frequencies are considered in between 0.94 THz and 1.06 THz with an interval of 0.005 THz to considerably suppress the unwanted resonance in the passband region as well as in the band rejection region. The desired output power $P\left (f_{i}\right )$ is set to be 1.0 within the passband and 0.0 in the rejection band. Figure 14(b) shows the convergence behavior of the objective function in the device optimization. Figure 15 shows the optimized structure of the bandpass filter and the propagation fields in the optimized structure for different frequencies within the both passband and rejection band region. We can see, the frequency within the passband is transmitted through the device and the frequencies in the rejection band are greatly rejected by the optimized structure. The frequency characteristics curve depicted in Fig. 16 illustrates that the optimized bandpass filter achieved the desired flat top transmission of almost 50 GHz at the targeted frequency band.

Fig. 14. (a) Design model of PPDW THz bandpass filter, (b) convergence behavior in the optimal design of bandpass filter.

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Fig. 15. (a) Optimized structure, and (b-d) propagation fields in the optimized bandpass filter at different frequencies.

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Fig. 16. Frequency characteristics of the optimized bandpass filter.

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Several THz bandpass filters based on the different device types, metal waveguide, plasmonic waveguide, photonic crystals, frequency selective surfaces (FSS), metamaterials, and metasurfaces have been proposed so far for the different THz circuit applications [34–42]. In comparison with these conventional filters, our designed bandpass filter has the possible application in THz communication, imaging and sensing systems. Furthermore, the proposed bandpass filter can also be designed for any desired passband in the THz region by simply modifying the objective function.

4. Conclusion

In this paper, we proposed mosaic-like PPDW devices for THz application and an efficient design strategy based on AVM and density method. The usefulness of our design approach with high computational efficiency over evolutionary approach is confirrmed by considering two basic design examples, T-branch and three branch. As the proposed design strategy is also effiective for larger scale design problems compared with that based on DBS or evolutionary approach, we confirmed this statement by considering two more comparatively higher functional design examples of mode splitting device and THz bandpass filter. The proposed PPDW bandpass filter achieved the desired flat top transmission band within the aimed frequency range and the other PPDW devices achieved high transmission efficiencies at the both single frequency operation and broadband operation. In our future work, we would like to discuss the design of PPDW device considering conductor loss and practical fabrication. In addition, we will design more higher functional PPDW devices and improve our design strategy.

Funding

Japan Society for the Promotion of Science (21K04169).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Efficient optimal design of mosaic-like PPDW devices for THz application using the adjoint variable method

Abstract

1. Introduction

2. Mosaic-like optimization with gradient method

2.1 Density method for mosaic structure expression

2.2 Formulation by 2D-FEM

2.3 Sensitivity analysis by AVM

3. Numerical examples

3.1 T-branch waveguide

3.2 Three branch waveguide

3.3 Mode splitting device

3.4 THz bandpass filter

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (16)

Equations (24)

Optics Express