Full‐Control and Switching of Optical Fano Resonance by Continuum State Engineering

Abstract Fano resonance, known for its unique asymmetric line shape, has gained significant attention in photonics, particularly in sensing applications. However, it remains difficult to achieve controllable Fano parameters with a simple geometric structure. Here, a novel approach of using a thin‐film optical Fano resonator with a porous layer to generate entire spectral shapes from quasi‐Lorentzian to Lorentzian to Fano is proposed and experimentally demonstrated. The glancing angle deposition technique is utilized to create a polarization‐dependent Fano resonator. By altering the linear polarization between s‐ and p‐polarization, a switchable Fano device between quasi‐Lorentz state and negative Fano state is demonstrated. This change in spectral shape is advantageous for detecting materials with a low‐refractive index. A bio‐particle sensing experiment is conducted that demonstrates an enhanced signal‐to‐noise ratio and prediction accuracy. Finally, the challenge of optimizing the film‐based Fano resonator due to intricate interplay among numerous parameters, including layer thicknesses, porosity, and materials selection, is addressed. The inverse design tool is developed based on a multilayer perceptron model that allows fast computation for all ranges of Fano parameters. The method provides improved accuracy of the mean validation factor (MVF = 0.07, q‐q') compared to the conventional exhaustive enumeration method (MVF = 0.37).


Supplementary Note 1. Thin-film Fano resonator and equivalent pendulum oscillator model.
The weakly coupled resonator, which makes a Fano resonance, is manifested in the absorption spectrum, σ(E), calculated by the Fano formula: [1] (E)= D 2 (q+ Ω) 2 1+Ω 2 , where E is the energy and Ω = 2(E -E0)/Γ (Γ is the resonance width, E0 is the energy at the resonance frequency and q = cot(δ), where δ is the phase difference between the two modes).where γ1 and γ2 are the damping rates and ω1 and ω2 the resonant frequencies of resonators 1 and 2, respectively.As Lm becomes large (approaching infinity), the transmission coefficients across the spacer (tla and tld) vanish, resulting in zero coupling terms.In this situation, the system behaves as two uncoupled oscillators. [2]However, for finite Lm, all the necessary conditions for Fano resonance are met, with a strongly damped, driven oscillator (Resonator 1) weakly coupled to a less damped oscillator (Resonator 2).Therefore, we chose a suitable thickness of 25 nm that fulfills these conditions.
As shown in Figure N1, the damping rate tuning of resonator 1 affects the whole system in terms of oscillator intensities |Ak(ω)| 2 and q, where Ak is the intensity ratio of the field of the k th resonator (Ek) with regard to the input field into the k th resonator (E i k), Ak = (Ek/E i k) 2 .By putting the parameters, these relations are expressed in terms of E1 and E2 as follows: [2] ( where The total injected field is expressed as By using the relation of d  ~   ,  2 () is defined as where For resonator 1 (ultrathin resonator), there is no exact analytical expression for the resonance frequency ω1.Nevertheless, it can be approximated under the assumption that Im(nl) is typically smaller than Re(nl).Thus, the resonant frequency can be approximated by the following condition: [2] 2[∅  ( 1 )] ≈ ∅ 0 − ∅  + 2, where ∅ is oscillator phase.
In case of  1 (), the injected field is expressed as Assuming that    is generally smaller than    , the approximation gives the oscillator intensity as where The subscripts represent the layers (a indicates the absorbing layer, m the metal layer and d the dielectric layer).ϕ and ω represent the phase and frequency, respectively.As a strategical approach for universal realization, we considered the effective complex refractive index using a porous medium based on volume-averaging theory.Then, we derived the damping rate corresponding to the porous medium of the lossy layer of resonator 1 as follows: (Fabry-Perot cavity, i.e., MIM structure).Control of structure parameters ⅰ) thickness of modelling layer, ⅱ) complex refractive index, and ⅲ) resonant wavelength.
-2.000 -         To achieve the desired spectral shape, the target q parameter should be addressed, which is represented in Figures N3-N5.However, the desired configuration with matched complex refractive index is difficult to realize with limited conventional materials.Therefore, we tailored the effective refractive index through porosity change as follows: [3] where For the derivation of the relationship in calculation process, we selected a-Si as the lossy layer.Consequently, we controlled the damping factor and oscillator intensities corresponding to the porosity change (Figure N6).In the calculations, we used a complex refractive index from the literature (Ag, SiO2, Co, and TiN) [4][5][6][7] and measured data (a-Ge, a-Si, c-GST, and a-GST).
Figure N7 shows the effective complex refractive indices of representative lossy materials (i.e., a-GST, c-GST, a-Si, a-Ge, Co, and TiN) with a changing porosity.Hence, the cover range of complex refractive indices is enlarged.

Figure N1 .
Figure N1.Fano resonator and pendulum oscillator model.a) Thin-film Fano resonator comprised of resonator 1 (ultrathin resonator, i.e., lossy film on metal reflector) and resonator 2 (Fabry-Perot cavity, i.e., metal/insulator/metal (MIM) structure).The variables are Ei and Er are the total field injected into resonator 1 and the reflected field from the Fano resonator, respectively; Ll, Lm, and Ld are the thickness of absorbing layer (lossy layer), thin metal film and dielectric layer, respectively; the subscripts of reflection and transmission coefficients designate the medium (i.e., 0, l, d, and m represent air, lossy layer, dielectric, and metal layer, respectively).b) Coupled pendulum model with forced oscillation representing Fano resonance, Figure N2.Computational model and structural parameters.Fano resonator comprised of resonator 1 (ultrathin resonator, i.e., modelling layer on metal reflector) and resonator 2

Figure N6 .
Figure N6.Damping factor and intensity with porosity.a) Oscillator intensities for MIM layer and ultrathin resonator corresponding to various porosities of a-Si layer.b) Closed view of oscillator intensities of continuum state.

PorosityFigure N7 .
Figure N7.Complex refractive indices of lossy materials.Representative lossy materials and their complex refractive indices with the porosity, Pr, changing from 0% to 80% at the three representative wavelengths.In the calculations, we used a complex refractive index from the

Figure S1 .
Figure S1.Calculation process of q parameter with varying material combination, dimension and complex refractive index.a) Overall design process and q parameter extraction.b) Material, porosity and dimension tuning of resonator 1. c) Calculated q parameter corresponding to Pr and Ll at three resonance wavelengths.

Figure S2 .
Figure S2.Complex refractive index of six lossy materials n and kq map.Complex refractive index of six lossy materials and n and kq map.Each circle represents target design.

Figure S3 .
Figure S3.Procedure of designing and validating four different spectral line shapes.a)Modelling process of four spectral line shapes at the target wavelength 400 nm.b) Absorption spectra of each shape calculated by TMM with designed parameters and materials.c) Evaluation of spectral line shape with validation q parameter, q'.d) Reflectance spectra of designed parameters.e) Measured spectra of designed structures realized.

Figure S4 .
Figure S4.Fabrication of porous/anisotropic layer.Schematic of fabrication process via glancing angle deposition method (left).Cross-sectional transmission electron microscopy image of Fano filter with porous a-Ge layer (right).
Figure S5.a) Schematic of self-aligned nano-columns with lossy material.b) Conceptual schematic image of top view of nanocolumns corresponding to the polarization angle.c) Schematic of self-aligned nano-columns with deposition angle.d) Measured/simulated reflectance and measured effective index.

Figure S6 .
Figure S6.Design and operation principle of polarization-sensitive bidirectional display (PSBD).a) Schematic illustration of PSBD, which shows bi-directional property.When observer view the PSBD from an external perspective, the pattern is revealed, and conversely, the pattern is hidden when viewing it from inside of car.b) Photographs of Fano filter.For contrast with the original photographs, a customized color map was applied.c) Each section is distinguished by the function.Lossy layer of section a is dense a-Ge, which has isotropic medium (polarization insensitive), on the other hand, section b and c has anisotropic medium, enabling polarization sensitive operation.Section A and B are participated in pattern area and C is background area.Schematic illustration of self-aligned nanocolumns and variation of Pr of effective medium corresponding to polarization state.

Figure S7 .
Figure S7.Design and experimental result of PSBD.a) Spatial placement of sections A, B, and C. b) Simulation result of four states of PSBD.c) Original photographs of experimental result of four states of PSBD.

Figure
Figure S8.a) Schematic representation of transmission and reflection from the Fano resonator, illustrating color palettes with varying Ld from 0 to 400 nm.The reflected color also exhibits polarization-sensitive variations.b) Pattern mask and its corresponding pixels used in the experiment.We note that the reflected color of pixels A, C, and D was matched under the polarized light.

Figure S9 .
Figure S9.a) RCWA model with a monitor for measuring absorption.b) Absorption profiles of each structure (MIM, MIM/Ge, and MIM/Pr-Ge) corresponding to changes in wavelength.c) TMM simulation model of each structure.d) Reflectance, transmittance, and absorptance of each structure.The integrated absorptance of MIM/Pr-Ge reached the highest level at 75%, whereas MIM/Ge showed 63% and MIM exhibited 25% (wavelength range, λ = 350 -750 nm) for λres = 550 nm.

Figure S10 .Figure S11 .Figure S12 .Figure S13 .Figure S14 .
Figure S10.Fabrication/immunoassay process of bidirectional sensor.a) Change of reflection and transmission spectral shapes by attaching virus clusters on Fano filter.b) Surface functionalization by grafting SARS-CoV-2 antibody and sensing process.

Figure S16 .FigureFigure S18 .
Figure S16.Data amount of parameter space.Data fraction from the total data set, which is expressed by the number of data set with smaller parameter q divided by the number of total data sets).This range includes 90% of the data set of FigureS1.