Avoided resonance crossing and non-reciprocal nearly perfect absorption in plasmonic nanodisks with near-field and far-field couplings

Shih-Hui Gilbert Chang; Chia-Yi Sun

doi:10.1364/OE.24.016822

1. Introduction

Avoided resonance crossing [1] is a general phenomenon occurring in almost all physical interactions. It describes the splitting behavior in a coupled system [2]. For example, in dielectric hexagonal dielectric resonators [3], the degenerated triangular resonant modes exhibit energy level and linewidth anti-crossing by varying the height of one hexagonal edge. One of the modes leads to longer life time with higher quality factor [3–5]. In microdisk resonators, the avoided crossing can lead to directional emission [6,7]. Whether the energy level or linewidth exhibits either crossing or anti-crossing depends on the mechanism of interaction [1,3]. In this paper, we show that similar anti-crossing behavior can be observed in plasmonic nanostructures due to either near field or far field coupling. Near field coupling in metal disk dimmer can lead to both energy and linewidth anti-crossing. This is well understood under the hybridization model [8]. For heterodimer where the two disks are of different size, other resonant modes appear due to near field coupling [9]. This anticrossing phenomena can be explained by simple Hamiltonian model [1,3] and in the first part of the paper we show the corresponding phenomena for both vertically and horizontally aligned two metal disks. By varying the size of one disk as the heterodimer approaching the homodimer, the anticrossing in both energy and linewidth appears. The Hamiltonian models also predict the energy crossing and linewidth anticrossing for far field coupling [3]. However, there is little literature discussion on the avoided crossing by far field coupling in plasmonic structure. In the second part of this paper we found that far field coupling in double layered disk array with gap size close to Fabry-Perot resonant condition leads to linewidth anti-crossing but energy crossing by varying either the gap size or the diameter of one disk. Asymmetric reflection and absorption spectra from different side of the double layered disk arrays with mismatched disk size (disk arrays without mirror system) show the disappearing of Fabry-Perot resonant mode and non-reciprocal nearly perfect absorption properties. This nearly perfect absorption is fundamentally connected to the anticrossing phenomena in asymmetric disk arrays. We use a simple frequency-selective surface model [10,11] to represent the individual disk array and use the FP model to connect the two arrays [12–15]. This simple FSS-FP model matches well with the full wave finite-different time-domain modeling. This model can also explain the perfect absorption properties for ultrathin metamaterial surface observed in literatures [16–26]. The other nonreciprocal phenomena observed in other optical system [27–32] might also be explained by this model. We believe that the observed avoided resonance crossings and nonreciprocal absorption in plasmonic nanostructure would lead to many photonics applications such as high Q resonators for future sensing applications.

This paper is organizing as follows: Section 2 introduces the Hamiltonian approach to explain different avoided resonance crossing phenomena with different interaction Hamiltonian. Section 3 and 4 show the finite-different time-domain simulations for avoided resonance crossing phenomenon in plasmonic nanodisk structures by near field and far field coupling. Section 5 illustrates the frequency-selective-surface coupled with Fabry-Perot model to explain the result in section 3 and 4. Section 6 is the conclusion.

2. Avoided resonance crossing and the Hamiltonian approach

The near field and far field coupling in a two nano disks system can be described by an effective Hamiltonian. The 2x2 matrix of the Hamiltonian for two coupled resonant modes E₁ and E₂ can be expressed in a general form

H = (\begin{matrix} E_{1} & V \\ V & E_{2} \end{matrix})

where V is the coupling constants, and E₁, E₂, V can be complex numbers. The eigenvalues are

E_{\pm} = \frac{E_{1} + E_{2}}{2} \pm Δ, Δ = \sqrt{{(\frac{E_{1} - E_{2}}{2})}^{2} + V^{2}} .

The eigenvectors are

ψ_{\pm} = (\begin{matrix} V \\ \frac{E_{2} - E_{1}}{2} \pm Δ \end{matrix}) .

In the open or dissipative system, the Hamiltonian matrix is non-Hermitian. E_i can be complex with the imaginary part related to its linewidth of a quasi-bound state with decay rate γ_i. For the un-coupled resonant modes, E₁ = ε₁ -jγ₁, and E₂ = ε₂ -jγ₂. The $e^{- j ω t}$ convention is adopted here. The real part of coupling constant V = k’-jk” can be either positive for the vertical aligned disks or negative for the horizontal aligned disks, or zero for external coupling such as far field. For far field coupling, there is no mode overlapping but the scattering waves affect its oscillating phase. Thus for far field coupling the energy level is not shifted and V is purely imaginary. The near field coupling between vertically aligned disks induces charge oscillating in phase which increases the total energy as seen in the inset of Fig. 2(b), and thus k’ is positive. While the horizontal alignment induces charge oscillating horizontally out of phase, it reduces the total energy. The effective Hamiltonian can be expressed as

H_{e f f} = (\begin{matrix} ε_{1} & \pm k' \\ \pm k' & ε_{2} \end{matrix}) - j (\begin{matrix} γ_{1} & k " \\ k " & γ_{2} \end{matrix})

To understand the avoided crossing behavior, we vary

ε_{1}

=

ε_{0}

+

δ

by

δ

from below and across

ε_{0}

, while keeping

E_{2}

at

ε_{0} - j γ_{0}

. At crossing

δ = 0

and

E_{1} = E_{2}

,

Δ = \pm k' - j k "

is complex.

For vertically aligned disks with positive coupling constant, the Eigen states $E_{-} = ε_{0} - k' - j (γ_{0} - k ")$ and $E_{+} = ε_{0} + k' - j (γ_{0} + k ")$ exhibit both energy and linewidth anti-crossing as shown in Fig. 1(a). For the lower (higher) energy Eigen state has smaller (larger) linewidth. The Eigen states Ψ₋ = (1,−1) and Ψ₊ = (1,1) indicate that at lower (higher) energy Eigen state, vertical disks are anti-parallel (parallel) oscillating with respect to each other. The lower energy state has zero dipole moment and become a dark mode with higher Q factor. The higher energy state contains nonzero dipole moment and become the bright mode with lower Q factor.

Fig. 1 Eigen energy real ε (upper row) and imaginary γ part (lower row) of Hamiltonian of E₁ = 1 + δ - 0.1j and E₂ = 1 - 0.1j with (a) V = 0.05 - 0.05j (b) V = - 0.05 - 0.05j (c) V = - 0.05j. The color of the line indicates the same mode.

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When the disks are parallel aligned, the coupling is negative $V = - k' - j k "$ . As shown in Fig. 1(b), the lower Eigen states $E_{-} = ε_{0} - k' - j (γ_{0} + k ")$ has larger linewidth and it eigenvector $ψ_{-} = (1, 1)$ shows the parallel bright mode oscillation. While the higher energy state $E_{+} = ε_{0} + k' - j (γ_{0} - k ")$ has higher Q factor and shows the anti-parallel oscillation $ψ_{+} = (1, - 1)$ exhibiting a dark mode property.

For far field coupling, the energy level is not affected and thus is purely imaginary $V = - j k "$

H_{e f f} = (\begin{matrix} E_{1} & 0 \\ 0 & E_{2} \end{matrix}) - j (\begin{matrix} γ_{1} & k " \\ k " & γ_{2} \end{matrix})

At crossing E₁ = E₂_, the energy eigenvalue is $E_{+} = ε_{0} - j (γ_{0} - k "), E_{-} = ε_{0} - j (γ_{0} + k ")$ . Now E₊ and E_- have the same real part. And one mode has longer life time, while the other is the short lived state. The energy level is attracting while the linewidth is repulsive. The linewidth anticrossing but energy crossing is a signature of purely imaginary coupling Hamiltonian. The increase of life time is due to the reduction of the decay channel by destructive interference of the imaginary interaction Hamiltonian. And the decrease of life time is due to the constructive interference.

3. Near field coupling

To illustrate the avoided crossing phenomena revealed by the Hamiltonian model, we use numerical modeling to calculate the resonant spectrum of two coupled nanodisks. Figure 2 illustrates the case of vertically aligned plasmonic disks. The scattering, absorption and extinction cross sections are calculated by finite-difference time-domain method. The dielectric constant of the silver disk is modeled by Drude Lorentz model with parameters fitting to the bulk Ag. Figure 2(b) indicates two resonant modes splitting when top disk and bottom disk are identical with radius 80nm, thickness 16nm and gap distance 50nm. This is traditionally explained by hybridization model [7]. In the hybridization theory, as the two disks approach each other, the gap distance decreases, and the overlapping of their field distribution leads to interaction. The increase of interaction will split the two degenerate modes into one bright and one dark mode. This splitting can be treated with avoided resonance crossing described in Section 2. At a fixed gap distance supporting strong interaction between the top and bottom disk, by varying one of the disk size, the avoided crossing phenomena can be observed. Here we fixed r₁ at 80nm and vary the radius of disk 2 by δ as r₂ = r₁ + δ. As δ changes from negative to positive, the resonant peak starts to deviate from the individual localized surface plasmon resonance (LSPR) and split into two lines as shown in Fig. 2(c), the quality factor of I increase, while the quality factor of II decreases. Figure 2(d) depicts the charge distribution. Group I is the anti-parallel mode with lower energy and higher Q factor (dark mode). Group II is parallel oscillating mode with higher energy and lower quality factor (bright mode) as shown in Fig. 2(e).

Fig. 2 Near field coupling of vertically aligned disks. (a) Geometry of heterodimer disks. (b) Extinction spectrum of r₂ = r₁ = 80 nm with schematic charge distribution for dark (I) and bright (II) modes. (c) Extinction spectrum vs. r₂. The black(red) line indicates the dark(bright) mode resonant peak. (d) Surface charge distribution of disk dimmers. (e) Q factor for dark (black), bright (red) modes and single disk (green) vs. r₂.

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The avoided resonant crossing can also be observed when the thickness of the disk varies. Decreasing thickness redshifts the LSPR and leads to similar avoided crossing in both energy level and quality factor.

When the disks are placed side by side, the induced charge coupling will lead to lower energy state as shown in Fig. 3. Thus the coupling coefficient is negative. To excite both the bright mode and dark mode in horizontally aligned disks, the incident wave must be lunched with oblique angle. By varying one the disk radius, both the energy level and quality factor reveal anti-crossing as shown in Fig. 3(b) and 3(c). Due to the coupling coefficient is negative, the eigenstate of the lower energy mode is parallel, and higher energy state is anti-parallel with higher quality factor. We have demonstrated that the Hamiltonian model of the avoid resonance crossing can well explain the mode splitting in plasmonic system with near field couplings.

Fig. 3 Near field coupling of horizontally aligned disks. (a) Geometry of heterodimer disks. (b) Q factor for dark (black), bright (red) modes and single disk (green) vs. r2 (c) surface charge distribution of disk dimers.

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4. Far field coupling

The far field coupling could also lead to the avoided resonance crossing. By increasing the gap distance between two disks, a Fabry-Perot (FP) cavity is formed due to the multiple scatterings between the two disks. However, the wide angle distribution of scattering from individual disk will lead to very weak cavity confinement between one disk pair. In order to enhance the Fabry-Perot effect, disk arrays are chosen as shown in Fig. 4(a). The period is chosen to be much smaller than the LSPR of the disk such that grating effect from the disk array will not interfere with the gap FP resonance. The small period eliminates the linewidth narrowing effect in array structure as discussed in other LSPR sharpening effect [33]. In our case, the disk array behaves as a frequency selective surface (FSS) which functions as a RLC broadband band-stop filter with band center related to the LSPR of individual constitute disk. As shown in Fig. 4(b), the transmission and reflection spectra of each disk array with radius of 70 nm or 80nm, period of 200nm, and disk thickness of 16nm are well matched between the FDTD simulation and the FSS model described in section 5. The extra sharp resonant peak in shorter wavelength regime from FDTD calculation is due to the higher quadrupoles of the individual disk and grating effect which we are not interested in present study. We thus fit the spectrum from 400nm to 1000nm as our main spectral domain for FSS model.

Fig. 4 Far field coupling of two disk arrays. (a) Geometry (b) Reflection/transmission spectrum of individual disk array with r₁ = 70 nm or r₂ = 80 nm. (c) Fabry Perot model of two disk arrays. (d) Phase of two disk arrays around its frequency selective surface stopband center (e) 2 port network schemes for ABCD and S matrix. Single admittance Y diagram represents an individual disk array. The pi-network of admittances Y₁, Y₂, and Y₃ represent the two disk array system.

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To adopt similar scheme used to observe avoided resonance crossing in near field coupling, we fixed the r₂ radius at 80nm and the FP gap distance at 495 nm with FP peak around the stopband center of the r₂ FSS array at wavelength 540nm. We then vary the radius of r₁ to across r₂ and observe the spectrum by top and bottom illumination which we identified as the two Eigen modes. As we swipe r₁ from 60 nm to 90 nm, the transmission spectrum is almost zero around the broad FSS stopband center. We observe identical transmission spectrum for top and bottom illumination as shown in Fig. 5(g) and 5(h). And the Fabry-Perot resonant peak is observed as a dip in the reflection spectrum or equivalently a peak in the absorption spectrum as shown in Fig. 5(c)–5(f). However, as r₁ approach 70nm, the reflection spectrum by top illumination shows no FP dip and the absorption spectrum is completely flat as indicated by circles in the upper row of Fig. 5. Meanwhile, the bottom illumination shows a big dip in reflection spectrum or equivalently a peak with nearly-perfect absorption in the absorption spectrum as shown in the lower row of Fig. 5. The intensity distributions at such condition are shown in the inset of Fig. 5(a) and 5(b). The top illumination shows complete flat reflection spectrum and the Fabry-Perot gap cavity does not support any resonant mode even though the excitation wavelength is close to FP resonant condition. The bottom illumination shows no reflection and no transmission. The electric field is completely trapped inside the FP gap and lead to nearly perfect absorption by the ohmic loss of metal disks.

Fig. 5 Disk arrays with top (upper row) and bottom (lower row) illumination for different r₁ size at fixed gap = 495nm. (a)(b) Absorption spectrum. The insect is the intensity distribution at λ₁. (c)(d) Color maps of absorption spectrum. (e)(f) Reflection spectrum. (g)(h) Transmission spectrum. The circles indicate mode at λ₁

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Since the range of r₁ is limited by the chosen short period of 200 nm as we reasoned earlier, we adopt a different approach to further understand this asymmetric reflection/absorption phenomenon. To further investigate the non-reciprocal nearly-perfect absorption phenomena, we fix r₁ at 70 nm and r₂ at 80 nm but now vary the gap distance. When the gap size is approaching the r₁ array’s FSS stopband center which is around 500 nm, the FP peak disappears for top illumination, while the bottom illumination shows no reflection and almost perfect absorption at λ₁ as shown in Fig. 6. This occurs at the same condition as in Fig. 5 with gap distance equals 495nm. When the gap size further increases as the FP condition approaches the r₂ (larger radius) array’s FSS stopband center (longer wavelength), the top illumination shows strong absorption at λ₂. The absorption spectrum shows a peak with value close to 1 and a dip in reflection spectrum without any reflection. The intensity patterns depict that the electric field is completely trapped within the FP cavity. While the bottom illumination shows strong reflection with no FP resonance and no absorption. λ₂ case is simply the reverse scenario as oppose to λ₁ case. The non-reciprocal nearly-perfect absorption is observed again in the gap varying approach.

Fig. 6 Disk arrays with top (upper row) and bottom (lower row) illumination for different gap size with r₁ = 70 nm and r₂ = 80 nm (a)(b) Absorption spectrum. The insect is the intensity distribution at λ₁ and λ₂. (c)(d) Color maps of absorption spectrum. (e)(f) Reflection spectrum (g)(h) Transmission spectrum. The circles indicate mode at λ₁ or λ₂

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To further illustrate the transition from λ₁ to λ₂, the transmission, reflection and absorption spectrum by top and bottom illumination are plotted in Fig. 7. Figure 7(a) and 7(b) correspond to the λ₁ case. Figure 7(e) and 7(f) correspond to the λ₂ case. Figure 7(c) and 7(d) corresponds to the regime in-between. It is clearly seen that the transmission spectrum (blue line in Fig. 7) is kept the same for both the top and bottom illumination. And the linewidth of the reflection (green line) is closely related to the absorption (red line) linewidth when the FP is around the FSS band center. We can use the linewidth of the absorption peak to illustrate the quality factor of the Eigen modes. And the peak wavelength of the absorption peak is the Eigen energy. As shown in Fig. 8(a) and 8(c), the energy level is crossing (attracting), the FP cavity couples the top and bottom FSS resonator through far field without affecting their resonant peak, thus the coupling term V in the 2x2 Hamiltonian of the system, is pour imaginary. And the Eigen states should show the level crossing as illustrated in section 2. There is only slightly energy level splitting around the λ₁ and λ₂ regimes which indicates there is only very small (almost negligible) real part in coupling constant V. The Q factor is evidently anticrossing as shown in Fig. 8(b) and 8(d). We further show the charge distributions for the λ₁ and λ₂ regime. For total absorption, the top and bottom surface currents are oscillating antiparallel. For the high reflection or the disappearing of FP resonance, disk on the reflection side shows strongly charged pattern, and disk on the transmission side shows only slightly charged pattern parallel to the reflection side. The signs of these charge patterns reveal the Eigen vector properties of the two modes. It also indicates that the asymmetric absorption/reflection is simply due to the interference effect. Constructive interference leads to almost total reflection. While the destructive interference leads to the trapping of light and thus nearly-perfect absorption. We now can predict the condition for total reflection or total absorption. When the gap FP resonance matches the incident side FSS stopband center, it leads to constructively interfered reflection and thus the disappearing of FP resonance. When the gap FP resonance matches the FSS stopband center of other side, the destructive interference cancels the reflection wave. While the transmission is extremely low on the exit side due to the low transmission at the FSS stopband center, the light is trapped within the FP cavity and eventually dissipated through ohmic loss by multiple reflections on loss metals. This phenomenological explanation can be illustrated with the FSS-FP mode discussed in next section.

Fig. 7 Reflection, transmission and absorption spectrum for top (upper row) and bottom (lower row) illumination at gap (a)(b) 495nm (c)(d) 525nm (e)(f) 555nm.

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Fig. 8 Energy (peak wavelength) crossing and linewidth (Q) anticrossing for hetero disk arrays with top and bottom illuminations by varying (a)(b) disk array radius r₁ and (c)(d) gap distance. (e)(f) The charge and intensity distributions at λ₁ and (g) (h) at λ₂

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5. FSS-Fabry-Perot model

First we quickly go through the circuit analysis for frequency selective surface to lay out the FSS model to explain the avoided crossing and the non-reciprocal nearly-perfect absorption in far field coupling of asymmetric disk arrays. Each disk array is a frequency selective surface constituted with capacitance c from the spacing between adjacent metal disk, inductance L from the current loop around the disk, and the intrinsic Ohm loss R in the metal. The array can be expressed as a series RLC resonant circuit with an admittance of

Y = \frac{1}{Z} = 1 / (\frac{1}{j ω c} + j ω L + R) .

placed in a transmission line as illustrated in Fig. 4(e). In the 2 port network, the ABCD matrix

(\begin{matrix} V_{1} \\ I_{1} \end{matrix}) = (\begin{matrix} A & B \\ C & D \end{matrix}) (\begin{matrix} V_{2} \\ - I_{2} \end{matrix})

can represent the disk array as an admittance Y with A = 1, B = 0, C = Y, D = 1, where C is defined as in open port 2 circuit I₁ = CV₂ with I₂ = 0 and is equal to Y in Eq. (6). For incoming wave and outgoing wave in transmission line,

V_{1, i n} + V_{1, o u t} = V_{1}, \frac{V_{1, i n}}{η_{0}} - \frac{V_{1, o u t}}{η_{0}} = I_{1}, V_{2, i n} + V_{2, o u t} = V_{2}, \frac{V_{2, o u t}}{η_{0}} - \frac{V_{2, i n}}{η_{0}} = (- I_{2})

V_1,in and V_1,out correspond to the incident and reflected E fields respectively in port 1. V_2,in V_2,out correspond to the incident and transmitted E field from port 2. Similar to the electromagnetic field treatment of a dielectric slab, the V₁ and I₁ correspond to the total E field and H field respectively on the incident side (port 1) of the transmission line. The V₂ and I₂ correspond to the total E field and H field on the port 2 side. The direction of current I₂ is defined as the incoming current to port 2 with the negative sign as in ABCD matrix. The scattering matrix is defined as

(\begin{matrix} V_{1, o u t} \\ V_{2, o u t} \end{matrix}) = (\begin{matrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{matrix}) (\begin{matrix} V_{1, i n} \\ V_{2, i n} \end{matrix})

To obtain the S parameters from ABCD parameters, we first express (V_1,in, V_2,in) and (V_1,out, V_2,out) in forms of matrix multiplications to (V₁, I₁) and (V₂, -I₂), and then replace (V₁, I₁) as (V₂, -I₂) with Eq. (8). By matrix inversion to express (V₂, -I₂) in terms of (V_1,in, V_2,in), we can get the scattering matrix with elements as

S_{11} = \frac{A + B / η_{0} - C η_{0} - D}{A + B / η_{0} + C η_{0} + D} = \frac{- Y}{2 + Y}, S_{12} = \frac{2 (A D - B C)}{A + B / η_{0} + C η_{0} + D} = S_{21}

S_{21} = \frac{2}{A + B / η_{0} + C η_{0} + D} = \frac{2}{2 + Y}, S_{22} = \frac{- A + B / η_{0} - C η_{0} + D}{A + B / η_{0} + C η_{0} + D} = S_{11}

The reflection and transmission coefficients of one disk layer are simply r = S₁₁ and t = S₂₁ respectively. Since the array is surrounded by air. The reflection and transmission from the other side are the same and thus S₁₂ = S₂₁ and S₁₁ = S₂₂.

Each disk array behaves as a stopband frequency selective surface with high reflectivity approximately around the LSPR of a single disk. The RLC parameter can be readily fit with the FDTD transmission spectrum as shown in Fig. 4(b) with R₁ = 2.052e-2, L₁ = 4.256e-1, C₁ = 3.865e-1 for r₁ = 70nm, and R₂ = 1.596e-2, L₂ = 3.162e-1, C₂ = 6.131e-1 for r₂ = 80 nm in units of combinations with eV and η₀ .

By stacking two layers of disk arrays, gap in between acts as transmission line in 2 port network. The system is a π-network. The ABCD matrix for a transmission line is

(\begin{matrix} V_{1} \\ I_{1} \end{matrix}) = (\begin{matrix} \cos δ & j η_{0} \sin δ \\ \frac{j}{η_{0}} \sin δ & \cos δ \end{matrix}) (\begin{matrix} V_{2} \\ - I_{2} \end{matrix})

where δ = 2kd is the round trip optical phase change. By cascade three ABCD matrix of first FSS, transmission line and second FSS, as the new ABCD matrix, the transmission and reflection coefficient can be calculated using the S₁₁ and S₁₂ parameter. For different side illumination, the matrix multiplication is performed reversely.

To better understand the physical meaning of the gap effect, the gap can also be treated as a Fabry-Perot cavity sandwiched between two FSS mirrors as shown in Fig. 4(c). The total transmission and reflection spectra for illumination from r₁ disk array side are

τ = \frac{t t' e^{j δ / 2}}{1 - r r' e^{j δ}}, Γ = r + \frac{r' t^{2} e^{j δ}}{1 - r r' e^{j δ}}

respectively. And the total transmission and reflection spectra for illumination from r₂ disk array side are

τ' = \frac{t' t e^{j δ / 2}}{1 - r' r e^{j δ}}, Γ' = r' + \frac{r t'^{2} e^{- j δ}}{1 - r r' e^{j δ}}

respectively. And the reflection and transmission coefficients in each individual disk array (prime indicates disk 2 array) are

r = - Y_{1} / (2 + Y_{1}), t = 2 / (2 + Y_{1}), r' = - Y_{2} / (2 + Y_{2}), t' = 2 / (2 + Y_{2})

Y_{1} = 1 / (\frac{1}{j ω c_{1}} + j ω L_{1} + R_{1}), Y_{2} = 1 / (\frac{1}{j ω c_{2}} + j ω L_{2} + R_{2})

The total transmission spectrum τ’ for illumination form disk 2 array is identical to the total transmission spectrum τ for illumination from disk 1 array. However, the reflection spectrum Γ ’ is different from that with side 1 incidence. With the three parameters L, R, C from fitting in Fig. 4(b), one can model the top illumination and bottom illumination with Eqs. (13) and (14) respectively.

At FSS resonance, the transmission from one disk array is approximately 2/(2 + 1/R) which is a very small number for typical metal loss R. While the reflection is approximately –1/(1 + 2R) which is almost totally reflected due to the band stop filter characteristics of FSS. Thus when the gap PF resonance matches the FSS peak of the incident side, the transmission t term is minimum and the t² term in Γ is almost negligibly zero. And thus Γ is approaching r which is large and functioned as the original FSS stop band filter, and the FP resonance dip in Γ disappears. In this case, the incident layer is excited and the other side is weakly excited with both oscillating in phase. However, when illumination from the other side with hetero disk arrays, r’ is not approaching its reflection peak and is less than r (at its max), and the t’² term is not approaching zero. The transmitted t’ encounters the highly reflected r and results in strong reflecting multiple interference and since the charge oscillation of two disk arrays is out of phase, it cancels out the reflection. In the meantime, the transmission is also very small. This leads to no reflection, no transmission, and the EM wave is dissipated due to the ohmic loss of the metal disks. The charge distribution also reveals the anti-parallel oscillation in Fig. 8(f) and 8(g). As the gap FP resonance shifts toward the FSS resonance of the other disk array, the whole process reversed. This nonreciprocity occurs most significantly between the top and bottom FSS band center where reflection is still high to support multiple reflection interference. The FSS-Fabry-Perot model is illustrated in Visualization 1.

For very thin layer, surprisingly the nonreciprocal absorption also appears. This is originated from the fact that this nonreciprocal effect occurs around FP condition. Thus it is the extra phase effect in addition to the FP around trip 2π contribution. By skipping 2π contribution, slightly phase change would also lead to the same effect. Note that the strong absorption always occurs on the higher resonant side (smaller disk size array). This ultra thin case modeled by FSS-FP model is illustrated in Visualization 2.

Furthermore, the FSS-FP model can work on different type of perfect absorbers. For substrate with thick metals, the bottom r’ = −1 and t’ = 0. Only the top illumination can be operated. By varying the gap size, strong absorption peak appears around both sides of the r₁ FSS band center in the spectrum as the gap FP resonance swipes through the FSS band center right before and after. This case is illustrated in Visualization 3.

For symmetry top and bottom disk array, only around 50% of absorption can be observed and no non-reciprocal effect exists. The metal loss plays an important role in the non-reciprocal nearly perfect absorption. For large loss, for example the damping loss in metal increased by 3 times, the asymmetric disk arrays still exhibit nonreciprocal absorption except that the absorption peak is broaden and less than perfect. This is illustrated by FSS-FP model in Visualization 4. For less loss, for example the damping loss in metal reduced by 10 times, it still exhibits narrow nonreciprocal absorption but less than perfect as illustrated by FSS-PF model in Visualization 5. For lossless case, such as dielectric or metal with zero damping loss, the nonreciprocal effect disappears as illustrated by the FSS-FP model in Visualization 6. Since there is no loss, the energy conservation implies the time reversal symmetry and the system should exhibit reciprocity. To break the parity time symmetry, loss is one of the few methods (such as magneto-optic effect, nonlinear effect, etc.) to achieve nonreciprocity.

This FSS-FP model provides an efficient design tool and could explain all type of thin and thick metamaterial absorbers. This model also reveals the mechanism of non-reciprocal asymmetric absorption properties of the nearly perfect absorbers which is not discussed by earlier literature. This nonreciprocal nearly-perfect absorption eigenstate might be viewed as the bound states in the continuum. To summarize, under the avoided resonance crossing scheme discussed in this paper, we have shown the nonreciprocal absorption in asymmetric plasmonic nanodisk arrays and provide a physical model to intuitively interpret the observed phenomena.

6. Conclusion

We adopt the avoided resonant crossing Hamiltonian model to explain the heterodimer energy and linewidth anticrossing for near field coupling. The energy eigenvalue and eigenvectors explains the energy shift and charge distribution of bright and dark modes for both vertically and horizontally aligned disk dimmer. Energy crossing and linewidth anticrossing for far field coupling by two disk arrays are demonstrated by FDTD numerical simulations with varying either disk array size or the gap size between arrays. The Far-field avoid resonance crossing studies unveil a novel non-reciprocal nearly-perfect absorption in the asymmetric disk arrays. The asymmetric reflection and absorption by top and bottom illumination is rooted in the avoided resonance crossing with pure imaginary coupling constant in the interaction Hamiltonian. These non-reciprocal phenomena can be explained in a simple frequency-selective surface Fabry-Perot model. This model also explains the perfect absorption metasurface with ultra-thin or thick layers. The linewidth anticrossing could lead to ultra-high Q resonators and provide a new route to design highly sensitive bio-chemical sensors or efficient energy harvesting applications.

Funding

Ministry of Science and Technology, Taiwan (MOST) (104-2112-M-006-017, 103-2112-M-006-006, 102-2112-M-006-011, 101-2112-M-006-015)

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Name	Description
Visualization 1: MP4 (1316 KB)	FSS-FP model for metal
Visualization 2: MP4 (1762 KB)	FSS-FP model for thin gap
Visualization 3: MP4 (1036 KB)	FSS-FP model for PEC substrate
Visualization 4: MP4 (1247 KB)	FSS-FP model for 3 times loss
Visualization 5: MP4 (1141 KB)	FSS-FP model for small loss
Visualization 6: MP4 (936 KB)	FSS-FP model for no loss

Avoided resonance crossing and non-reciprocal nearly perfect absorption in plasmonic nanodisks with near-field and far-field couplings

Abstract

1. Introduction

2. Avoided resonance crossing and the Hamiltonian approach

3. Near field coupling

4. Far field coupling

5. FSS-Fabry-Perot model

6. Conclusion

Funding

References and Links

Supplementary Material (6)

Cited By

Figures (8)

Equations (16)

Optics Express