## Abstract

Modeling a nanoparticle array (NPA) inside a thin glass slab as a lumped optical resonator, we propose a systematic approach to design for an efficient optical filter with bandwidth tunability. The quality factor and bandwidth of the resonator are related to the physical, geometrical, and electrical parameters of an NPA and its surrounding medium, whose permittivity is varied to change the bandwidth. We propose a structure amenable to our design approach consisting of an NPA slab surrounded by liquid crystal whose permittivity can be altered. We validate the design procedure with examples of tunable-bandwidth filters at different frequency regimes from NIR to blue.

© 2014 Optical Society of America

## 1. Introduction

In recent years, nanophotonic devices have attracted attention as optical filters for their advantages over conventional filters such as high reflection/transmission efficiency and small size. The most successful is the guided mode resonance (GMR) based structure due to its efficient and controllable frequency response. GMR structures consist of grating and waveguide layers, and transmission occurs at certain wavelengths due to the resonant coupling between the incident wave diffracted by the grating and the guided mode of the waveguide. For a one-dimensional grating, only one polarization is coupled, while for a two-dimensional structure, polarization-independent filtering can be achieved. Due to different mode-matching conditions for TE and TM modes, a secondary sub-resonance appears in the frequency response of the device [1]. This secondary response can reach non-negligible levels relative to the primary resonance, so it has to be taken into account for design purposes. One drawback of GMR devices is their susceptibility to the angle of incidence. For oblique incidence, the primary resonance will be either shifted or split into double peaks, since the phase matching condition is highly angle- and polarization-dependent. To overcome angle sensitivity, reflective filters with good angular tolerance using metal-insulator-metal (MIM) and metal-semiconductor-metal (MSM) structures supporting plasmon polariton modes (PPM) are proposed [2, 3]. However, they come with a cost: the level of rejection is highly angle-dependent, and it can go from 90% to less than 10% in few degrees. To achieve tunability in resonance, several approaches are proposed, such as using the change in the angle of incidence or the grating period, and electrically tunable polarization rotators [4–6]. In these methods the change of center frequency involves change in bandwidth and degradation of efficiency.

To achieve bandwidth tunability, conventional approaches use spatially distributed variable
retarders and polarizers. They require precise alignment of the components and have
low efficiency. In [5], a
bandwidth-tunable plasmonic device with heterograting is proposed, where the
bandwidth depends on the incident angle of light; the center frequency changes as
well. In [7] a fixed center
frequency tunable-bandwidth is suggested, but it involves geometrical alteration of
the structure that cannot be done dynamically after the device is fabricated. In
addition to these inherent drawbacks, none of the the proposals for
bandwidth-tunable optical filters suggests a true *ab-initio* design
procedure for a given bandwidth tunability. In order to achieve acceptable coupling
in GMR and PPM based approaches, trial and error plus optimization over several
parameters should be performed using a rigorous coupled-wave analysis [8]. Here we propose a true design method
for an efficient polarization-independent optical filter with desired bandwidth
whose behavior is fairly insensitive to the angle of incidence. The key for
achieving the bandwidth tunability lies in the relationship between the bandwith of
the NPA slab and the surrounding medium. In contrast to other methods, bandwidth can
be tuned dynamically without affecting other characteristics of the filter.

## 2. Modeling NPA as a lumped optical resonator: quality factor and bandwidth

Using NPA-based metasurfaces as spatial filters has recently drawn significant attention
[9–11]. NPAs of plasmonic nanoparticles (NPs)
exhibit plasmon resonance in the near-infrared (NIR) and visible frequency regime
whose location is independent of angle of incidence. Our design is based on the
recently-derived equivalent circuit of NPAs, whose element values are related to the
physical, geometrical, and electrical parameters of the array [12, 13]. The simplified version of the equivalent circuit for an NPA
consists of a series *LC* circuit in shunt with a transmission line
representing the environment (Fig. 1(a) and
1(b)). Element values, details of the modeling that includes contribution
from both individual and collective plasmon modes, and the simplification procedure
can be found in Ref. [12].
The location of the collective plasmon resonance of an NPA with pitch size
*a* consisting NPs of volume *V* and
depolarization factor *L′* can be found as [14]:

*ε*and

_{a}*ε*are high-frequency and static permittivities of NP material,

_{b}*ω*is the plasmon resonance frequency of NP, and

_{p}*ε*is the permittivity of the slab. For modeling, the NPA is assumed to be completely surrounded by a uniform dielectric material of infinite extent and with permittivity of

_{h}*ε*. However, in [14] we showed that even for slab thicknesses as small as

_{h}*λ*/20, the host permittivity can still be replaced by the slab permittivity. Assuming

*a*≪

*λ*the distance after which the power of evanescent modes decreased by more than 20 dB can be calculated as

*d*= −

_{c}*a*ln(0.1)/(2

*π*) (coupling distance at each side of the slab), and for slabs thicker than 2

*d*,

_{c}*ω*is stabilized and the problem can be considered as if the NPA inside the dielectric slab is a fixed optical resonator component (fixed L and C, and consequently

_{CPR}*ω*) whose behavior is independent of the medium in which the NPA slab is located. Furthermore, since the slab thickness required to achieve the constant resonator is much smaller than wavelength, the whole circuit can be considered as a frequency- and environment-independent optical lumped element.

_{CPR}Resonators are important circuit elements that are typically characterized by their resonant
frequency and quality factor Q. The equivalent circuit as viewed by the
*LC* tank is shown in Fig.
1(c). The *LC* tank acts as an oscillator when it is
excited at its resonant frequency and in the absence of external source (Vg), the
finite energy stored in the tank will be dissipated by external resistors that load
the circuit (the port impedance values of *η* which is the
wave impedance of the medium surrounding the NPA slab). The bandwidth is attributed
to the loaded quality factor *Q _{L}* of the resonator, given
by:

*ω*

_{0}=

*ω*. Bandwidth is related to the quality factor as

_{CPR}*BW*= 1/

*Q*. This equation can be written in terms of the medium permittivity

*ε*as where

*η*

_{0}is the wave impedance of free space. Extracting this relation is the foundation for bandwidth-tunable filter design based on NP arrays.

## 3. Design procedure

According to Eq. (3) the key for achieving
bandwidth tunability is the relation between BW and *η*.
Changes in *η* will not affect the center frequency since the
element values are independent of the medium. Based on this idea, we see that if the
NPA slab is surrounded by a medium whose refractive index can be altered, bandwidth
controllability will be achieved.

The proposed configuration to dynamically change the permittivity of the surrounding medium around the optical resonator is shown in Fig. 2. It consists of an NPA slab (much smaller than *λ*) that is surrounded by two layers of liquid crystal (LC) whose thickness is comparable to or larger than *λ*. To fabricate this configuration, sacrificial layers (to be replaced by LC), slab and NPA are fabricated, then a microfluidic channel is etched using reactive ion etching (RIE), away from the active site. The sacrificial layers inside the stack can be etched by access through this channel. Then, the vacuum medium around the NPA slab can be filled by injecting LC into the microchannels. Using nematic liquid crystal, an electrostatic field tunes its refractive index changes from *n* ≈ 1.4 to *n* ≈ 2. This is equivalent to tuning from vacuum *n* = 1 to *n* = 2, providing almost 100% bandwidth tunability.

To design for a specific bandwidth and center frequency, one has to solve a system of nonlinear equations in Eqs. (1) and (3). There are many approaches available; we use the Levenberg-Marquardt method available in Matlab. There is a unique solution for each specific pitch size, so once the pitch size is specified based on fabrication limitations, and considering that it has to be small compared to the wavelength, a unique solution can be calculated by solving the nonlinear system. To get the bandwidth tunability range of interest, one can design initially for the widest bandwidth using the lowest refractive index of the surrounding medium (in case of vacuum *n* = 1). With higher refractive index, the bandwidth will change dynamically via Eq. (3), e.g., one can reach half of the initial bandwidth going from *n* = 1 to *n* = 2.

## 4. Design examples

We illustrate the design procedure outlined above by designing three bandwidth-tunable filters for different center frequencies, from NIR to the visible.

#### 4.1. Bandwidth-tunable filter for HeNe laser

For a 260 THz HeNe laser (1.15 *μm*), we design a filter to tune from
19% to 9.5% bandwidth. The slab is silica with
*n* = 1.5 and the pitch size is *a*
= 60 nm. The required slab thickness can be calculated as 2 ×
−*a*ln(0.1)/(2*π*) ≈
44*nm*. The geometrical parameters calculated from the
nonlinear solver for the nanodisks are 48 nm for diameter and 2 nm for
thickness. Figure 3(a) shows the
frequency response as obtained from full-wave EM simulations in CST MWS. Due to
plasmonic losses not modeled in the equivalent circuit, changing the impedance
of the surrounding medium affects the level of rejection. We extracted the
equivalent circuit of the NPA based on the lossless modified Drude model that
describes the complex dielectric function of a dispersive noble metal
[12]. At resonance,
*LC* is short circuited, therefore the non-ideal transmission
line model is a two port circuit with a shunt resistance. The magnitude of the
transmission coefficient for this circuit can be found as $\frac{2R}{2R+\eta}$, where *R* is the resistance due
to the loss of metallic nanoparticles and $\eta =\sqrt{\mu /\epsilon}$ is the wave impedance in the surrounding
medium. If *R* = 0, the transmission is zero at resonance
(ideal case). However, if *R* ≠ 0 and
*ε* goes higher, the transmission becomes greater
than zero which is seen clearly from the numerical results, in which the exact
dispersion data for the materials such as silver is used. Despite the decrease
in level of rejection, the rejection efficiency is still ≥ 80%.
The bandwidth before injecting LC is from 1070 nm to 1300 nm, ∼
19%, which decreases to ∼ 14% (from 1080 nm to 1250 nm)
for *ε* = 2 and decreases to ∼
11% (1090 nm to 1220 nm) for *ε* = 3, and
finally for *ε* = 4 to ∼ 9.5%
(1100 nm to 1210 nm).

#### 4.2. Bandwidth-tunable filter for optical communication

For 1.55 *μm*, a filter is designed to go from 20% to 10%
bandwidth. The slab is optical glass with *n* = 2.2 and
the pitch *a* = 80 nm. The required slab thickness is
calculated as 58 nm. The geometrical parameters calculated from the nonlinear
solver for the nanodisks are 52 nm for diameter and thickness 2 nm. Figure 3(b) shows that the bandwidth before
injecting LC is from 1450 nm to 1780 nm, ∼ 20%, which decreases
to ∼ 15% from 1460 nm to 1705 nm for *ε*
= 2, and then decreases to ∼ 12.5% (1470 nm to 1650 nm)
for *ε* = 3, and finally for
*ε* = 4 to ∼ 10% (1480 nm to
1625 nm). Since in this part of the spectrum, frequency dependant losses of
silver are higher than the previous design regime, the change in the level of
rejection is more obvious.

#### 4.3. Bandwidth-tunable filter for the visible

Finally we design a filter that before LC injection filters both blue and green, and after
injection and for *n* = 2 filters only blue. This
requires a bandwidth change from ∼ 27% to ∼ 14%
with center wavelength at 480 nm. The slab is cryolite with *n*
= 1.35 and the pitch size is *a* = 80 nm. The
required slab thickness is the same as the previous filter. The geometrical
parameters for the nanodisks are 48 nm for diameter and 24 nm for thickness.
Figure 4 shows that the bandwidth
before injecting LC goes from ∼ 27% (442 nm to 585 nm), to
around 15% (439 nm to 520 nm) for *ε* =
4. Due to the very small loss of silver in the visible range, almost no change
in the level of rejection is seen.

## 5. Angle sensitivity study

Transmission and reflection coefficients for the HeNe filter for an obliquely incident TE plane waves are shown in Fig. 5 for various angles of incidence from *θ* = 0° to 50° in 10 degree steps. The frequency response is not affected as the angle of incidence increases from 0° to 30°. Beyond 30°, however, the bandwidth increases for TE and decreases for TM [12]. The bandwidth variations is attributed to the change in wave impedance as *η*/cos(*θ*) for TE and *η* cos(*θ*) for TM.

## 6. Conclusion

We propose a nanostructure as a spatial optical filter whose bandwidth can be tuned dynamically without affecting other characteristics of the filter. Since the problem is formulated in terms of physical, geometrical, and material properties of the structure, one can design for a desired bandwidth tunability. The structure is polarization-independent and fairly insensitive to the angle of incidence.

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