Nanoparticle array based optical frequency selective surfaces: theory and design

Chiya Saeidi; Daniel van der Weide

doi:10.1364/OE.21.016170

1. Introduction

Frequency selective surfaces (FSSs) have been investigated for their widespread applications as spatial filters for decades. FSS are well-known and widely used in the microwave domain [1–3]. There has also been a great interest in pushing FSS technology to higher frequencies for filtering applications. Far-IR FSSs fabricated by stacking multiple metallic grids and meshes with good performance are presented in [4] and [5]. Also in [6] single-layer far-IR FSS filters with strong stop-bands are proposed by using metal fractal geometry. The same group reported single-layer metallodielectric nanostructures with dual-band filtering properties at midinfrared wavelengths in [7]. A near-IR FSS with metallic sub-micron circular patch elements was demonstrated in [8] where, by optimizing the patch thickness, the desired filter performance was achieved. Thus, periodic metallic inclusions, apertures, or gratings on dielectric substrates are typical building blocks of FSSs. At visible frequencies, wave interaction with metal plasmonic concepts like plasmon resonance should be taken into account. Recently, FSSs were proposed in the visible range based on plasmonic metasurfaces. The FSS formed by annular apertures perforated in a metal film and arranged in a square array is suggested in [9] and [10]. Conversely, Zhang et al. [11] showed that periodic structuring of a metal film without violation of continuity (i.e. without perforation) is sufficient to achieve substantial modification of frequency response. In [12] an optical filter composed of a subwavelength grating formed by an array of protrusions extending from the substrate is proposed. Protrusions have side-wall profiles that should be configured and optimized to obtain desired spectral properties. A genetic algorithm (GA) is proposed to optimize the side-wall profile and unit cell size (periodicity) to obtain a desired multi-spectral response. One drawback of optimization-based methods is the need for a large computational effort, and the choice of GA to avoid converging to local minima makes the optimization even more computationally expensive while numerically simulating hundreds of populations over tens of generations. Recently, possibility of using NPAs as spatial angular filters [13] and also metasurfaces for rejecting and reflecting a target frequency range has been implemented [14]. NPAs consisting of plasmonic NPs with their plasmon resonance in the optical frequency regime exhibit almost (due to large dielectric loss) perfect reflection only at resonance. However, many applications need a flat response curve with fast roll off. Similar to microwave FSS-based spatial filters, optical spatial filters based on cascading two or more NPAs has been presented recently [15–17].

Except for optimization-based methods, no proposals for optical FSSs suggest a true ab-initio design procedure. Instead, they involve identifying the configuration under study, and the extracted characteristics are shown to be a consequence of certain features associated with plasmonic resonances and interaction of light with the structure. Formalizing the wave/configuration interaction, defining the input and output, and specifying its functional behavior by means of equations, one can follow a synthesis procedure to obtain configuration parameters for a required functional behavior. With the goal of formulating the problem to achieve a synthesis method, we choose NPAs whose theory has been extensively studied [18, 19]. In principle, an ideal FSS should exhibit a passband or stopband of constant width and spectral position with respect to angle of incidence of the waves impinging on it. The frequency response for an NPA shows a sharp peak always centered around the same resonance frequency, independent of the incidence angle [18]. The stability of the frequency of maximum reflection is associated with the subwavelength features of this array. For larger angles, the reflection bandwidth increases for TE polarization and decreases for TM polarization [18]. The array density affects the bandwidth and Q-factor of the resonance.

Here, we extended circuit and network concepts to handle NPA optical analysis and design problems of practical interest by introducing a general equivalent circuit for an NPA. Having the transmission line circuit model, it is easy to modify the original problem, or to combine several elements together to find the response, without having to analyze in detail the behavior of each element in combination with its neighbors. The equivalent circuit parameter values are related to the physical, geometrical, and electrical parameters of the FSS as design equations and also to complete the synthesis procedure. The proposed synthesis procedure allows for the design of FSSs with any higher-order bandstop responses of odd-order. This procedure based on the equivalent circuit model of the structure enables synthesizing the optical FSS from system level performance indicators such as center frequency, fractional bandwidth, response type (e.g., Chebyshev, Butterworth, etc.), and response order. The proposed synthesis procedure is validated with a design example of an FSS having a third-order bandstop response.

2. Modeling and parameter extraction for an NPA

Consider a 2-D periodical array of identical NPs that is small enough to include only electric dipoles induced in the particles and neglect the higher-order multipoles. The dipole approximation is valid for isolated NPs smaller than 50 nm [20]. The lattice has square cells of the size a × a and is excited by a normally-incident plane wave propagating along z direction and assuming the plane of incidence be x = 0 electric field is polarized along one of the transverse directions (E_x for TE and E_y for TM polarization). One approach to evaluate the properties of the NPA is by assuming that an NP can be described by a polarizability α that relates induced dipole moment P to the local electric field E_loc by P = ε₀αE_loc (α is in the direction of the applied field). Since the local field is superposition of applied field E₀ and interaction field E_int, and the interaction field is itself proportional to the dipole moment by means of an interaction constant β (for normally-incident plane wave β is the same for both TE and TM polarizations) as E_int = βP/ε₀, self-consistent solution gives the equation for dipole moment as $P = \frac{ε_{0} E_{0}}{α^{- 1} - β}$ . The interaction constant which is obtained by summing the contributions of the fields due to all electric dipole moments surrounding a NP may be converted into the form of a rapidly converging series that can be found in Refs. [18] and [21]. Here a simplified yet accurate closed-form solution derived in [22] and restudied in [23] is used. After obtaining the dipole moment, the electromagnetic problem is solved and it is possible to define average current density as $J_{a v} = \frac{j ω P}{Ω}$ (Ω = a²) from which the reflection and transmission coefficients can be conveniently found [18, 19]. The grid is modeled by a shunt load in a transmission line with impedance of the ratio of the local field to the average current density and is given as:

Z = - j \frac{a^{2}}{k} (α^{- 1} - β) - \frac{1}{2}

where k is the free-space wave number. Note that the impedance is normalized to the characteristic impedance of the transmission line that equals the wave impedance of the medium in which the NPA is located. For normally-incident plane waves, this is simply the wave impedance η. Z can be written in real and imaginary parts indicating reactive and resistive components of the impedance respectively:

X = - \frac{a^{2}}{k} (Re {α^{- 1}} - Re {β}), R = \frac{a^{2}}{k} (Im {α^{- 1} - β}) - \frac{1}{2} .

For a periodic array of lossless NPs, the imaginary part Im {α⁻¹ − β} can be found exactly from the energy conservation principle [19]:

Im {α^{- 1} - β} = \frac{k}{2 a^{2}}

which by substituting it in R in Eq. (2) yields zero, which is expected for the lossless case. For the reactive part of the impedance we use the closed-form of Maslovski and Tretyakov’s interaction constant for normal incidence of transverse plane waves in the limit where the distance between the particles is much smaller than the wavelength:

β = \frac{e^{- j k R_{0}}}{4 a^{2}} (\frac{1 - j k R_{0}}{R_{0}}) ≃ \frac{{(1 - j k R_{0})}^{2}}{4 a^{2} R_{0}}

where

R_{0} = \frac{a}{1.438}

for a square lattice [19]. Thus the second term in the reactive part in Eq. (2) which corresponds to the lattice contribution in the equivalent average impedance can be written as:

X^{lattice} = \frac{a^{2}}{k} (\frac{1 - {(k R_{0})}^{2}}{4 a^{2} R_{0}}) ≃ \frac{1}{4 R_{0} k} = \frac{c}{4 R_{0} ω}

that represents a capacitor similarly derived in [19] having value of 4R₀/c where c is the speed of light. The first term in the reactive part of Eq. (2) can be thought of as the NP-in-array (NIA) impedance defined as

X^{N I A} = - \frac{a^{2}}{k_{0}} Re {α^{- 1}}

for which we need to determine the polarizability α. We can find a static field solution for the dipole moments of small ellipsoid- and sphere-shaped NPs readily. According to Stratton’s approach [24], the dipole polarizability of an ellipsoid of permittivity ε with semiaxes l₁, l₂, and l₃ when a uniform field is applied along the l_i axis, is

α = \frac{V}{L_{i} + ε_{h} / (ε - ε_{h})}

where V is the volume of the NP, ε_h is the permittivity of the host medium, and L_i is is the depolarization factor in the i direction defined in [24] (i subscript indicates the same direction as α and we drop it for simplicity from here on). Here we use a modified Drude model to describe the complex dielectric function of dispersive materials given by:

ε (ω) = ε_{a} - \frac{(ε_{b} - ε_{a}) ω_{p}^{2}}{ω^{2}}

with the high-frequency limit of ε_a, ε_b the static dielectric constant, and ω_p plasmon resonance frequency. Values ε_a = 5.45, ε_b = 6.18, and ω_p = 1.7 × 10¹⁶rad/s are extracted from fitting with experimental data for silver [25] as the metal of interest in the optical frequency range [26, 27]. Here we ignore the damping frequency in the original modified Drude equation yet find good correspondence with literature values in the frequency range of interest (Fig. 1).

Fig. 1 Real part of dielectric function of silver. open circles from Johnson and Christy [25]. Solid lines are fitted results using (7).

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Now, substituting Eq. (7) into Eq. (6), and subsequently Eq. (6) into the X^NIA equation we describe the average NP reactance:

X^{N I A} = - \frac{a^{2}}{k_{0}} Re {α^{- 1}} = \frac{a^{2}}{V} \frac{(ε_{h} + L (ε_{a} - ε_{h}))}{(ε_{h} - ε_{a})} \frac{c}{ω} \frac{ω_{1}^{2} - ω^{2}}{ω_{2}^{2} - ω^{2}}

where

ω_{1} = \sqrt{\frac{L (ε_{b} - ε_{a})}{ε_{h} + L (ε_{a} - ε_{h})}} ω_{p}

and

ω_{2} = \sqrt{\frac{ε_{b} - ε_{a}}{ε_{a} - ε_{h}}} ω_{p}

. It simplifies to the exact form of impedance of a network of Fig. 2(a), which is

Z = \frac{1}{j ω C_{2}} \frac{ω_{1}^{2} - ω^{2}}{ω_{2}^{2} - ω^{2}}

where

ω_{1} = \frac{1}{\sqrt{L_{1} C_{1}}}

and

ω_{2} = ω_{1} \sqrt{1 + \frac{C_{1}}{C_{2}}}

. Therefor using the analogy between Eq. (8) and Eq. (9) we can derive C₂ as

C_{2} = \frac{V}{a^{2} c} \frac{ε_{a} - ε_{h}}{ε_{h} + L (ε_{a} - ε_{h})}

and from ω₁ and ω₂, we get L₁ and C₁ as

C_{1} = C_{2} ({(\frac{ω_{2}}{ω_{1}})}^{2} - 1), L_{1} = \frac{1}{C_{2}} \frac{1}{(ω_{2}^{2} - ω_{1}^{2})} .

Now, deriving the first and second terms in X in Eq. (2), the final equivalent circuit representing the NPA can be configured as shown in the left side of transformation in Fig. 2(b), where C₃ = −1/ (ωX^lattice) ≃ −a/0.36c. As it can be inferred, the equivalent circuit derived here stands valid for both TE and TM polarization, and the only parameter that changes switching between polarizations is the depolarization factor in Eq. (6).

Fig. 2 (a) equivalent circuit for NP-in-array impedance. (b) network transformation for the equivalent circuit of NPA.

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Since this equivalent circuit is difficult to design with, we use a network transformation as shown in Fig. 2(b). Here we have a resonator of series inductor $L^{'} = L_{1} {(1 + \frac{C_{2}}{C_{3}})}^{2}$ and capacitor $C^{'} = \frac{C_{1} C_{3}^{2}}{(C_{2} + C_{3}) (C_{1} + C_{2} + C_{3})}$ in parallel with a capacitor $C_{p} = {(\frac{1}{C_{2}} + \frac{1}{C_{3}})}^{- 1}$ . We compare the results obtained from full-wave EM simulations in CST MWS to those predicted by the presented equivalent circuit model for an NPA of square lattice with pitch 40 nm and NPs of radius 13 nm embedded in silica (n = 1.55). Figure 3 shows the frequency response for three different cases. In the optical range of frequency there is a good agreement except for less attenuation in the stop-band and broadening both due to dissipation loss. Besides the predicted peak for the reflection, it has a zero out of the optical range, which the circuit model fails to predict correctly. This is where the permittivity of the metal equals the permittivity of host medium (around 2.4). From Fig. 1 the Drude model reaches this point around 1350 THz; for the experimental curve, the closest it can get to 2.4 is around 950 THz, although beyond 800 THz is not of interest. In the transformed circuit in Fig. 2(b), the parallel capacitor C_p in conjunction with the resonator is responsible for the null in the reflection response, which is not in of interest for our bandstop filter design procedure. Moreover, as seen in Fig. 3, ignoring this capacitor has negligible effect on the bandstop behavior of the circuit.

Fig. 3 Frequency response for three different cases: equivalent circuit, realistic case with parallel capacitor, and realistic case without parallel capacitor.

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3. FSS design

3.1. Principles of operation

To understand the principles of operation of the optical FSS, consider its equivalent circuit in Fig. 2. Note that this circuit model is only valid for normal incidence. For oblique incidence, both the characteristic impedance in the transmission line model and the values of the capacitances and inductances need to be altered. The location of the collective plasmon resonance (ω_CPR) of an NPA can be found considering the series LC resonator in the equivalent circuit

ω_{C P R} = \frac{1}{\sqrt{L^{'} C^{'}}} = ω_{1} \sqrt{\frac{C_{1} + C_{2} + C_{3}}{C_{2} + C_{3}}}

which in case of spherical NPs with L = 1/3 (insensitive to polarization) it can be more simplified to

ω_{C P R} = ω_{1} \sqrt{\frac{1 - 4.524 F^{3}}{1 + 4.524 K F^{3}}}

where

K = \frac{ε_{h} - ε_{a}}{ε_{a} + 2 ε_{h}}

and F is the filling factor defined as r = Fa. knowing the host medium, the resonance frequency of the NPA is only a function of F. Figure 4 shows the corresponding collective plasmon resonance frequencies achievable using different host media of air, S-FPL, silica, sapphire, P-SF68, ZrO2 with refractive indexes of 1, 1.42, 1.55, 1.77, 2.01, 2.21 respectively. Using different materials and optical glasses that are available provides a range of refractive index over which the whole optical frequency range can be swept. The lowest limit of F is selected such that at least 9 dB attenuation in the stop-band can be achieved for a single layer of NPA. The highest limit for F is set to avoid emerging higher order plasmon modes which cause red-shift in the dipole resonance location and add significant complexity. Although for an isolated NP emerging higher multipoles is only function of NP size [20], for NPAs closeness of NPs is an important factor. This can be stated in terms of filling factor. We found that the effects of higher plasmon modes for normal angle of incidence starts to be significant for F > 0.45. This limit can be as low as 0.35 for higher angles of incidence.

Fig. 4 NPA’s collective plasmon resonance frequencies using different host media.

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3.2. Design procedure

Optical FSS presented here can be designed using a simple and systematic approach. The fractional stop-band width w, its center frequency of operation ω₀, and the response type are generally known a priori. To realize a spatial bandstop filter acting as FSS using NPAs, only shunt resonator branches are available. In order to convert them into series branches and to have ladder configurations of resonators, we need to use impedance inverters approximated by 90° line lengths, as shown in Fig. 5. The reactance slope parameters of this circuit in terms of the low-pass prototype parameters g₀, g₁, ..., g_n₊₁, and cut-off frequency ω′_c are given in [28] as:

\frac{x_{i}}{Z_{0}} = {(\frac{Z_{l}}{Z_{0}})}^{2} \frac{g_{0}}{{ω^{'}}_{c} g_{i} w} n = even, \frac{x_{i}}{Z_{0}} = \frac{1}{{ω^{'}}_{c} g_{0} g_{i} w} n = odd

where

{(\frac{Z_{l}}{Z_{0}})}^{2} = \frac{1}{g_{0} g_{n + 1}}

and since here we consider the case where the whole FSS structure is embedded in the same medium implying Z_l = Z₀, we design for n = odd.

Fig. 5 Band-stop FSS with shunt branches.

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We relate the design parameters (center frequency and bandwidth) to the geometrical parameters (F and a) of different layers of NPA representing different branches of the filter. If frequencies of the edges are ω′₁ and ω′₂ (the prime is to avoid confusion with ω₁ and ω₂ in Eq. (8)) stop bandwidth and the center frequency can be defined respectively as $w = \frac{{ω^{'}}_{2} - {ω^{'}}_{1}}{ω_{0}}$ and $ω_{0} = \sqrt{{ω^{'}}_{1} {ω^{'}}_{2}}$ . The first step is to set the center frequency of the spatial filter ω₀ to be the collective plasmon resonance frequency ω_CPR of the NPA. Doing so for NPA of nanospheres we get the first design equation for the first geometrical parameter F as

F = 0.605 \sqrt[3]{\frac{ω_{1}^{2} - {ω_{0}}^{2}}{{ω_{1}}^{2} + K {ω_{0}}^{2}}} .

The next step is to derive design equation for the spacing parameter a. For this goal we use equation $x_{i} = ω_{0} L_{i} = \frac{1}{ω_{0} C_{i}}$ . Using parameters for spherical NP in the equation for L′ we obtain

L^{'} = \frac{c}{4 π} \frac{{(ε_{a} + 2 ε_{h})}^{2}}{3 ε_{h} (ε_{b} - ε_{a}) ω_{p}^{2}} \frac{(1 + 4.524 K F^{3})}{F^{3}} \frac{1}{a} = \frac{E}{a} .

Equating L_i and L′ we derive the second design equation as

a = \frac{E ω_{0}}{x_{i}}

where E is as defined in Eq. (16). Now, extracted design equations for designing NPA-based branches of the spatial filter can be directly adapted to design an optical FSS.

3.3. Design example

The design procedure outlined above is carried out to design an FSS to reject the green and yellow from incident white light. For this purpose we need a stop-band over 505–610 THz, almost 100 nm. Based on these values ω₀ = 555 THz and w = 18%. We choose a third-order Butterworth response and P-SF68 as the host medium. The element values for the low-pass prototype circuit can be found in [28]. These parameters along with geometrical parameters calculated from (15) and (17) are listed in Table 1. Figure 6 shows a three-dimensional topology of different layers of the FSS. The structure is composed of three different NPA layers with quarter-wavelength separation. We briefly discuss the effect of dissipation on the response of the FSS, which happens in realistic full wave simulation. In Fig. 7 the solid line shows the response of a typical bandstop filter where the resonators have no dissipation loss. The dashed line illustrates the effect of dissipation loss on the response. For the dashed line,the attenuation no longer goes to infinity and it has greater 3 dB bandwidth. Since plasmonic NPs are involved, a noticeable level of absorbtion is experienced on the NPA, particularly near the array resonance. for normal angle of incidence 1 − |R|² − |T|² corresponds to the absorbed power. Thus the same behavior as in Fig. 7 is expected.

Table 1. The element values for the low-pass prototype circuit, and the geometrical parameters of the NPAs for the design example.

View Table

Fig. 6 Topology of the third-order bandstop FSS of the design example.

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Fig. 7 The effect of dissipation loss in a bandstop filter.

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Figure 8 shows the FSS frequency response as obtained from full-wave EM simulations in CST MWS as well as those predicted by the equivalent circuit model. The stop bandwidth is broadened, and the level of rejection is lowered, yet still a very good level of rejection is achieved. Non-normal angles of incidence are presented in Fig. 9, which shows the transmission and reflection coefficients of the FSS for an obliquely incident TE and TM plane waves at various angles of incidence from θ = 0° to 50° in 10 degree steps. The frequency response of the FSS is not affected as the angle of incidence increases from 0° to 20° for both polarizations. Beyond 20°, however, the response starts to deviate from normal incidence. The bandwidth variations can be attributed to the change in wave impedance η, which changes the loaded quality factor Q_L of the resonators of the coupled resonator FSS [3]. For the TE polarization, the wave impedance changes as η/cos(θ). Therefore, the loaded quality factor of the series resonators of Fig. 5 decreases for larger incidence angles, and subsequently the bandwidth of each resonator increases. On the other hand, for the TM polarization, the wave impedance changes as η cos(θ). Therefore, for larger incidence angles, Q_L increases which leads to decrease of the bandwidth of each resonator in the FSS. It is also observed that for an increasing angle of incidence, the attenuation decreases. The reason is due to the fact that in TM polarization, the tangential component of the field on the surface of the array decreases for larger angles of incidence. Thus, the polarization of the NPs and consequently the effective averaged currents decrease which means lower level of reflection and rejection [18].

Fig. 8 FSS frequency response obtained from full-wave EM simulations and those predicted by the equivalent circuit model for the design example.

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Fig. 9 Transmission and reflection coefficients of the designed optical FSS as a function of the angle of incidence for TE and TM polarizations.

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4. Conclusion

After modeling an NPA by extracting its equivalent circuit, we present a comprehensive method for synthesizing frequency selective surfaces composed of arrays of periodic nanospheres embedded in a common host medium. This procedure for synthesizing odd-order bandstop FSS is demonstrated theoretically. A simple example synthesizing a third-order bandstop FSS is demonstrated and validated using full-wave EM simulations. Our technique allows for designing optical frequency selective surfaces with frequency responses that are fairly insensitive to the angle of incidence. This can be extended to other realizable nanoparticle geometries such as oblate/prolate ellipsoids, nanorods, and nanodisks for more degrees of freedom. It can also be extended to multiple layers of different materials. The proposed synthesis procedure, for the first time, provide us with a powerful tool to design for a completely specified frequency response. The method paves the way for designing and developing several useful optical devices such as impedance matching networks, selective and accurate absorbers and mirrors, and spatial phase shifters.

Acknowledgments

The authors would like to thank the reviewers for their thoughtful comments and suggestions, which helped improve this paper.

References and links

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i	g_i	x_i	F	a (nm)	d (nm)
1	1	5.55	0.386	17	13
2	2	2.77	0.386	34	26
3	1	5.55	0.386	17	13

Nanoparticle array based optical frequency selective surfaces: theory and design

Abstract

Corrections

1. Introduction

2. Modeling and parameter extraction for an NPA

3. FSS design

3.1. Principles of operation

3.2. Design procedure

3.3. Design example

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (17)

Optics Express