# Biomimicry-Gradient-Based Algorithm as Applied to Photonic Devices Design: Inverse Design of Flat Plasmonic Metalenses

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## Abstract

**:**

## 1. Introduction

## 2. Methods: Concept and Mathematical Model

- Location phase
- Stalking phase
- Chasing phase

#### 2.1. Location Phase

#### 2.2. Stalking Phase

#### 2.3. Chasing Phase

#### 2.3.1. Team ${\tilde{G}}_{1}$ Strategy: Exploitation Mode with a Hard Besiege

#### 2.3.2. Team ${\tilde{G}}_{2}$ Strategy: Exploitation Mode with a Soft Besiege

#### 2.3.3. Team ${\tilde{G}}_{3}$ Strategy: Exploration Mode

#### 2.4. Constraints Inside the Pack

#### 2.5. Repeated Escape Attempt of the Prey

## 3. Results and Discussions

#### 3.1. Numerical Method: Polynomial Modal Method

#### 3.2. Plasmonic Lens Design

_{2}substrat. To ensure the convergence of the polynomial expansion, $n=4$ polynomials are used on each subinterval ${I}_{x}^{\left(k\right)},k=1:{N}_{p}$.

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Sketch of variables ${e}_{k}$ distribution. (

**b**) Sketch of design variables ${x}_{k}$. In the proposed optimization process, the increment of ${x}_{k}$ increasing favorably the FOM is kept as the best current optimal profile.

**Figure 2.**Flowchart of the proposed optimization method using a wolf pack biomimicry-gradient-based strategy.

**Figure 3.**Design of a one dimensional flat metalens. (

**a**) Schematic of dispersive metal film perforated with a non periodic subwavelength array of 1D nanoslits. (

**b**) the adjoint-based optimization method involved two different sources: an electric dipole and a polarized incident plane wave. (

**a**) Sketch of the plasmonic dispersive metalens; (

**b**) Assessment of the FOM variation computing.

**Figure 4.**Design of a $10.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$-wide plasmonic metalens consisting of ${N}_{p}=105$ silver-nanoridges and nanoslits focusing a normal incident TM-polarized plane wave at a wavelength $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ at a distance of $15\lambda $. Line scans of the normalized magnetic field intensity in the transverse (${z}_{f}=0$) (

**a**) and in the longitudinal $x=0$ (

**b**) focal plane respectively, for ten random realizations of initial conditions. Numerical parameters: $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, ${N}_{p}=105$, $h=400$ nm, TM polarization. (

**a**) $|{H}_{y}{(x,0)|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$; (

**b**) $|{H}_{y}({x}_{0},z){|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$.

**Figure 5.**Design of d-wide silver plasmonic metalense consisting of ${N}_{p}$ silver-nanoridges and air-gap focusing a normal incident TM-polarized plane wave at a wavelength $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ at a distance of $15\lambda $. Line scans of the normalized magnetic field intensity in the transverse (${z}_{f}=0$) (

**a**) and in the longitudinal $x={x}_{0}$ (

**b**) focal plane respectively, for four values of the couple $({N}_{p},d)$: $({N}_{p},d)\in \{(105,10.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}),(125,12.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}),(165,16.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}),(185,18.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m})\}$. Numerical parameters: $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, $h=400$ nm, TM polarization. (

**a**) $|{H}_{y}{(x,0)|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$; (

**b**) $|{H}_{y}({x}_{0},z){|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$.

**Figure 6.**Design of a plasmonic silver-flat-metalens of $d=18.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$-wide consisting of ${N}_{p}=185$ silver-nanoridges and air-gap focusing a normal incident TM-polarized plane wave at a wavelength $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ at a distance of $15\lambda $. (

**a**) shows the values of the (nanorodes, nanoslits)-widths sequence ${\left[{e}_{k}\right]}_{k}$ with respect to their locations labelled by k. (

**b**) presents the cartography of the normalized magnetic field’s intensity in the plane $(X,O,Z)$. Numerical parameters: $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, ${N}_{p}=52$, ${e}_{z}=400$ nm, TM polarization. (

**a**) $|{H}_{y}({x}_{0},z){|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$; (

**b**) $|{H}_{y}{(x,0)|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$.

**Figure 7.**Design of a one dimensional flat metalens: illustration of the two-metacells-stitched approach.

**Figure 8.**Design of a plasmonic silver-flat-metalens of $d=24.99\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$-wide consisting of ${N}_{p}=248$ silver-nanoridges and nanoslits focusing a normal incident TM-polarized plane wave at a wavelength $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ at a distance of $15\lambda $. (

**a**,

**b**) presents the cartography of the normalized magnetic field intensity in the plane $(X,O,Z)$. (

**c**,

**d**) shows the values of the (nanoridges, nanoslits)-widths sequence ${\left[{e}_{k}\right]}_{k}$ with respect to their locations labelled by k. Numerical parameters: $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, ${N}_{p}=125$, ${e}_{z}=400$ nm, TM polarization. (

**a**) $d=12.55\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$; (

**b**) $d=24.99\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$; (

**c**) ${n}_{p}=125$; (

**d**) ${N}_{p}=248=125+(125-2)$.

**Figure 9.**Design of a plasmonic silver-flat-metalens of $d=41\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$-wide consisting of ${N}_{p}=248$ silver-nanoridges and nanoslits focusing a normal incident TM-polarized plane wave at a wavelength $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ at a distance of $15\lambda $. (

**a**,

**b**) presents the cartography of the normalized magnetic field intensity in the plane $(X,O,Z)$. (

**c**,

**d**) shows the values of the (nanoridges, nanoslits)-widths sequence ${\left[{e}_{k}\right]}_{k}$ with respect to their locations labelled by k. Numerical parameters: $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, ${N}_{p}=205$, ${e}_{z}=400$ nm, TM polarization. (

**a**) $d=20.55\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$; (

**b**) $d=41\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$; (

**c**) ${n}_{p}=205$; (

**d**) ${N}_{p}=408=205+(205-2)$.

**Figure 10.**Design of d-wide gold plasmonic metalens consisting of ${N}_{p}$ gold nanoridges and nanoslits focusing a normal incident TM-polarized plane wave at a wavelength $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ at a distance of $3\lambda $. Line scans of the normalized magnetic field intensity in the transverse (${z}_{f}=0$) (

**a**) and in the longitudinal $x={x}_{0}$ (

**b**) focal plane respectively, for four values of the couple $({N}_{p},d)$: $({N}_{p},d)\in \{(65,6.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}),(85,8.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}),(105,10.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}),(125,12.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m})\}$. Numerical parameters: $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, $h=400$ nm, TM polarization. (

**a**) $|{H}_{y}{(x,0)|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$; (

**b**) $|{H}_{y}({x}_{0},z){|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$.

**Figure 11.**Design of a plasmonic gold-flat-metalens of $d=12.75\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$-wide consisting of ${N}_{p}=125$ gold-nanoridges and air-gap focusing a normal incident TM-polarized plane wave at a wavelength $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ at a distance of $3\lambda $. (

**a**) presents the cartography of the normalized magnetic field intensity in the plane $(X,O,Z)$. (

**b**) shows the values of the (ridges, air-gaps)-widths sequence ${\left[{e}_{k}\right]}_{k}$ with respect to their locations labelled by k. Numerical parameters: $\lambda =0.637\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$, ${N}_{p}=125$, ${e}_{z}=400$ nm, TM polarization. (

**a**) $|{H}_{y}{(x,0)|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$; (

**b**) $|{H}_{y}({x}_{0},z){|}^{2}/{\int}_{\mathcal{I}}{\left|{H}_{y}^{inc}(x,{z}_{s})\right|}^{2}dx$.

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**MDPI and ACS Style**

Edee, K.
Biomimicry-Gradient-Based Algorithm as Applied to Photonic Devices Design: Inverse Design of Flat Plasmonic Metalenses. *Appl. Sci.* **2021**, *11*, 5436.
https://doi.org/10.3390/app11125436

**AMA Style**

Edee K.
Biomimicry-Gradient-Based Algorithm as Applied to Photonic Devices Design: Inverse Design of Flat Plasmonic Metalenses. *Applied Sciences*. 2021; 11(12):5436.
https://doi.org/10.3390/app11125436

**Chicago/Turabian Style**

Edee, Kofi.
2021. "Biomimicry-Gradient-Based Algorithm as Applied to Photonic Devices Design: Inverse Design of Flat Plasmonic Metalenses" *Applied Sciences* 11, no. 12: 5436.
https://doi.org/10.3390/app11125436