## Abstract

In conjunction with our previous work on spectral transmission dependence on geometric parameters for subwavelength slits, we have investigated the optical behavior with the inclusion of an absorber on the extraordinary optical transmission of sliver slits resonant in the mid-wave infrared (MWIR). The placement of an absorbing layer causes a dramatic change to the dielectric environment of the subwavelength slit causing the cavity to become energetically leaky. We have found this broadens the spectral response of the cavity by increasing the imaginary component of the effective cavity index, reducing the cavity quality by a factor of more than seven. To mitigate this undesired effect, we have found that partially decoupling the absorber with a thin isolating layer helps restore the slit’s narrow spectral response and we explore the dependence of optical properties on the isolating layer’s depth. The optimum thickness of a silicon dioxide isolation layer for best quantum efficiency (Q.E.) was found to be 100 nm with a maximum Q.E. of 37.5%. This is more than double the Q.E. of the directly coupled absorber and the cavity Q increase is of the same order, with a corresponding narrowing of the resonance bandwidth. In addition, we explore the effect on the cavity of changing the dielectric environment at the input of the slit to improve the resonance properties in the propagation direction improving the Q by 20% and improving the Q.E. to 40.2%.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

There have been countless efforts to make optical systems smaller. The discovery of extraordinary optical transmission (EOT) took a leap in this direction with the discovery of light’s ability to transmit through deep subwavelength hole arrays in metals [1]. The same team found that a major mechanism that enabled this phenomena was plasmonics [2,3]. When a photon is incident on a metallic surface it causes the surface electrons to oscillate and in turn send a wave across the surface [4]. This gives a pathway for the light to effectively flow through. However, this is not the only mechanism, it was also found that certain Fabry-Perot resonances exist [5]. These resonances arise from both the individual geometry of the deep subwavelength cavities [6–8] and also the arrangement and periodicity of cavities [9].

While much work was done with cavities in a vacuum environment, work was also done by exploring the effects on changing the dielectric environments [10]. These investigations have included dependency of substrate material and cavity structures [11] and is closely related to metal-insulator-metal (MIM) interfaces [12] and environmental monitoring in biotechnology [13]. Theoretical development was made by determining the Green’s function with an arbitrary dielectric environment [14] while modeling the metal as a perfect electric conductor (PEC). Many other studies of resonant nano-cavities make the work within the PEC approximation [7,14–16]. While valid in the THz regimes, this approximation breaks down for other wavelength bands important for sensing and imaging technologies. The atmospheric transmission window from 3-5 microns, commonly designated mid-wave infrared (MWIR), is important for defense, firefighting, semiconductor wafer inspection and other applications. Previous studies have suggested that resonant nano-cavities could be useful for the demonstration of more efficient multi-spectral detectors [17]. However, we know of no published studies that examine resonant nano-cavities coupled to an absorbing material. Coupling to an absorbing material may be essential for nano-cavities to be used as high efficiency spectrally selective optical devices. Previous investigations of nanoscale devices have explored control of third-harmonic generation via plasmon-plasmon coupling through dielectric spacer layers [18]. Here we investigate plasmonic cavity coupling to an absorber, using a dielectric spacer layer to control the coupling strength and maintain desirable cavity properties. Investigation of a dielectric environment and the inclusion of the imaginary component of all indices is necessary. Direct detection of the cavity field via placement of an absorber in the vicinity of the aperture necessarily introduces loss, which broadens the spectral response.

We have previously examined the spectral properties dependence on geometric parameters of subwavelength slits [19] and found how changing extent of a slit in the x, y, and z dimensions change the corresponding spectral response in the MWIR. In order to account for real metal response we solved these systems using a finite difference frequency domain (FDFD) solver [20]. Exact numerical solvers such as FDFD and FDTD are common techniques that have been used for investigating many types of geometries and shapes in addition to many different wavelength regimes [8,17,21–23]. We continue our investigation of these deep subwavelength systems by exploration of a varied dielectric environment in terms of an absorber. For applications that depend on conversion of optical energy to an electrical signal, as in solar cells or photodetectors, an absorber is necessary. Understanding the potential for near-field coupling of resonant nano-cavities may allow for greater efficiency in the optical to electrical conversion and improve device performance. In particular, multi-spectral imaging detectors may be able to achieve greater efficiency via sub-wavelength localized absorption rather than pixel-scale optical filtering. As a foundation for the integration of resonant nano-cavities into sensors or photovoltaics, we calculate optical performance of nano-cavities in the presence of an absorber (detector) and with varying dielectric environment. Our analysis of the numerical data includes determination of the Quality Factor, Q, of a resonant cavity, as well as the theoretical quantum efficiency for different amounts of coupling to the absorber.

## 2. Simulation

We performed 3-dimensional simulations utilizing a finite-difference, frequency domain solver (FD3D) [24], which is described in our previous work [19]. The solver enforces Maxwell’s equations at discrete points in space for a given frequency in an iterative manner. The framework uses perfectly matched layers (PML) [25] to isolate the simulation area, which mitigates boundary reflections in the finite volume. These simulations were performed on a Linux server equipped with 48 processors having 12 cores each at 2.50 GHz and 256 GB of RAM.

The basic simulation volume for this work is shown in Fig. 1. The total simulation area goes from -250 to 250 nm in the x-direction, -1000 to 1000 nm in the y-direction, and -400 to 700 nm in the z-direction. The illumination source is a normally incident plane wave (green) polarized in the x-direction and located 100 nm prior to the first non-vacuum layer. Due to typical fabrication constraints, patterning design of the devices was in the x and y dimensions, while structure in the z-dimension is designed for deposition of layers of controlled thickness. We examine 200 wavelengths, evenly distributed from 2.5 to 4.5 microns. A 300 nm thick (z-extent) opaque silver layer prevents transmission except through a small aperture where the silver is absent. This basic volume is modified with different material layers in order to change the dielectric environment for the aperture as shown in Fig. 1. Following the aperture is 550 nm of either vacuum or absorbing material, GaSb (blue). Dielectric material layers of SiO2 (red) with variable thickness are added on either side of the silver layer. The aperture dimensions are held at a constant 50 nm in the x-direction while varying in the y-direction to maintain a constant resonant wavelength around 3.15 microns. The extent of the apertures in the y-direction can be seen in Table 1 for all cases within this study. In general, the grid spacing in the x, y, and z directions were 10, 50, and 20 nm, respectively. However, in order to capture near-field effects, the z-spacing was reduced with a maximum of 10 nm for the volume of the aperture and within 100 nm of the aperture with smaller z-spacing in the case of thin layers to establish at least 5 steps through the layer. The PML layers (gray) extend 100, 500, and 200 nm from each surface of the basic simulation volume in the x, y, and z directions. This is to provide 10 steps in the PML layer so the field can completely dampen. Standard values were used for the complex index of refraction values of Ag [26], $SiO_2$ [27], and GaSb [28].

## 3. Calculations and theory

We determined the spectral response using the same procedure as our previous work [19] by summing the magnitude of Poynting vector within the aperture at each specified frequency:

A commonly calculated property, or figure of merit, of resonant cavities is the Quality Factor, Q, representing the ratio of how much energy is stored in the cavity vs the dissipation rate of the energy. This number can be calculated from the spectral response of a cavity for any mode by taking the resonant frequency divided by the full width at half maximum, $\Delta \omega$, of the resonance,

We use the resonant frequency and bandwidth determined using a skewed Cauchy distribution [19],

Another important quantity is the near-field power enhancement, $T_E$, similar to the normalized transmission [23], describes the amount of power in a plane of the simulation volume normalized by the power of the illumination source, $P_0$:

## 4. Results and discussion

#### 4.1 Directly coupled absorber

A description of apertures directly coupled only to vacuum (DCV), has been published previously [19], and here we will only briefly describe the attributes of these systems for comparison to systems with absorbers. The aperture in the silver forms a resonant optical cavity. For the DCV cavity calculated here, $Q=14.71$ and $\tau _p = 24.61 fs$. Large values of $T_E$ on resonance for z values in and near the cavity are observed due to the energy storage of the cavity.

When the absorber is placed directly at the output plane of the aperture (DCA), this strongly couples the absorber material to the cavity and changes the effective index within the aperture. This has the undesirable effect of lowering the photon lifetime and broadening the resonance. For the DCA cavity, $Q=1.97$ and $\tau _p =3.3 fs$. The power within the cavity is much smaller than in the DCV case due to the loss within the cavity. The lineshapes of DCA and DCV cavities are plotted in Fig. 2. The spectral selection capability of the DCV cavity is nearly completely undone by direct coupling to an absorber.

In addition to the quality factor, one can examine the field distribution or mode of the cavity. In Fig. 3, we plot transparent scatter points of the magnitude of the Poynting vector and observe a change in the distribution from DCV (a) to DCA (b). The DCA mode has been skewed toward the absorber and has lost nearly all of its y-dimension structure. This z-gradient is not at all observed in the vacuum case.

#### 4.2 Weakly coupled absorber

One method of coupling to an absorber while maintaining a measure of isolation for its resonance is to add a dielectric isolation layer in between the cavity and the absorber. The isolation layer can make the cavity less leaky by effectively closing a valve to the absorber. This works by decreasing the spatial overlap with the field resonating in the slit and the absorber, reducing the influence of the imaginary part of the complex refractive index of the absorber on the effective complex refractive index within the slit. We examine the effect of this valve by changing the thickness of our isolation layer $(SiO_2)$ between 10 and 800 nm.

In Fig. 3(c) the Poynting vector magnitude for the isolation layer of 100 nm is plotted. The WCA mode with an isolation layer returns to similar spatial characteristics as the DCV case. The spectral response for the DCA and WCA of two depths (25 and 100 nm) can be seen in Fig. 4(a). In Fig. 4(b) we plot the quality factor for WCA cavities and find an increase in the quality factor of the cavity as the isolation layer thickness increases. Though the quality factor increases almost linearly with isolation layer thickness, the effect of increasing Q on device linewidth is partially offset by efficiency loss as the fields sampled by the absorbing material are correspondingly weaker at greater distances. The resonant frequency was maintained at less than a 1% difference. However, the skew parameter which dictates the tilt within the spectral response is an order of magnitude higher for the DCA case than the WCA cases.

As an example, for the WCA system with a 100 nm isolation layer thickness, we find $Q_{100}=3.72$ which gives a measure of the quantitative decoupling from the absorber. Comparing this case to the directly coupled absorber we found a Q enhancement of 46% over the DCA case and the photon lifetime for the same $WCA_{100}$ system increases to 6.3 fs. It is not generally possible to detect a strongly enhanced field due to the perturbation of the enhancement by the detection event. However, we find here that weakly coupling an absorber through use of an isolation layer provides a method to partially restore the quality of a resonant cavity, while still detecting the spectrally filtered and enhanced field within the slit. The results presented here show that isolation layer thickness is one way to control a trade-off between cavity quality and coupling into the detector. Figure 5 illustrates this tradeoff for the Ag, $SiO_2$, and GaSb system.

Examining $T_E$ in Fig. 5(a) & (b), we observe an exponential decay of the resonant power through the isolation layer followed by a drop $(\gamma _0-\gamma _1)$ across the isolation layer-absorber interface. The ratio $(\frac {\gamma _1}{\gamma _0})$ quantifies the coupling of both the propagating and evanescent fields emitted from the cavity in which the evanescent field is stronger in the near field. Across any given interface the reflection power coefficient (R) can be found from, $R=1-\frac {|E_T |^2}{|E_i |^2}$ [29], in which $E_T$ is the transmitted field and $E_i$ is the incident field. We propose a simple analytical model to describe the reflection at the isolation layer-absorber boundary $(R_\gamma )$ (as seen in Fig. 5(b)) that assumes the propagating and evanescent fields have orthogonal polarization and therefore:

As a theoretical upper limit for the potential quantum efficiency (Q.E.) of a detection system, the power of the field entering the absorber can be used. For this study, we assume a thick absorber layer, and therefore, the Q.E. of the cavity system is approximated as the transmission efficiency into the first layer of the absorber. This represents power flow into the absorber and is plotted in Fig. 5(d). We found a maximum to occur at an isolation layer depth of 100 nm with a $Q.E.=0.375$. The Q.E. for these coupled absorber systems can be found in Fig. 5(d), the Q from Fig. 4(b) indicates the photon lifetime for each system because all systems have equivalent resonant frequencies. The Q.E. decrease at large isolation layer thickness is due to the the limited z-extent of the resonance-enhanced field. The spatial extent of this field may be inferred from a comparison of Fig. 4(b) and Fig. 5(d). These results show the difficulty of near-field detection of resonant effects. A large photon lifetime indicates a strong field enhancement, but due to the localized nature of these fields, their detection cannot be efficiently accomplished with high cavity quality. We are reporting only on the Q.E. of coupling the optical fields into the absorbing material. We do not attempt to report a device Q.E., which would have to take into account the semiconductor doping and structure, and other operational variables, which may further limit the achievable Q.E.. Therefore, the values reported here are theoretical upper limits for coupling resonant nano-cavities with the described materials, layering and patterning.

#### 4.3 Weakly coupled absorber with front layer enhancement

Another method to enhance the resonant cavity in the presence of an absorber involves the dielectric environment on the front side of the aperture. This approximately has an effect on cavity quality without deteriorating the coupling to the detector. We investigated the use of a front layer at the input of the slit. The WCAF system simulated has a front layer of 100 nm $SiO_2$ with an isolation layer of 100 nm. The y-dimension of the WCAF slit is 1010 nm. This system is compared to the $WCA_{100}$ simulation, where coupling between cavity and detector should be roughly the same. The spectral response of $WCA_{100}$ and WCAF are shown in Fig. 6. As done previously, we calculate the Quality Factor and find $Q_{WCAF}=4.22$. This Q is an enhancement of 20% over $WCA_{100}$ case. In addition to an increase of Q, the WCAF system has an increased quantum efficiency of 0.402. We find a photon lifetime of 7.13 fs.

In the case that a Fabry-Perot mechanism is responsible for the cavity resonance, the change in photon lifetime upon addition of the front layer can be understood through a change in the Fresnel reflection at the surfaces of the cavity. In general, the number of photons in a cavity can be represented by [29],

in which $N_p$ is the number of photons in a cavity at time (t), $N_{p0}$ is the initial number of photons, and $\tau _p$ is the photon lifetime. One can associate the time it takes to make a single round trip $(\tau _{RT})$ to the resonant frequency with $\tau _{RT}=\frac {2\pi }{\omega _r}$. Neglecting contributions from absorption, we can find the number of photons after this round trip by the reflection coefficients of the surfaces with in which $R_n$ represents the reflection from the nth surface of the cavity. Considering two different systems having the same resonance center and same reflection coefficients except at the i-th surface, Eq. (9) shows that the ratio of $\frac {N_{p1}}{N_{p2}}$ reduces to a simple ratio of $\frac {R_{i1}}{R_{i2}}$. Substituting Eq. (8) for the number of photons and suppressing the subscript for the i-th surface gives, where $R_1$ and $R_2$ are the reflection coefficients for system 1 and 2, respectively. In our study, these two systems correspond to the WCA and WCAF cases in which only the front surface reflection coefficient is changed. From Eq. (10), we can use the calculated photon lifetimes for the $WCA_{100}$ and WCAF cases to determine ratio of the power reflection coefficients which corresponds to $\frac {R_{WCA_{100}}}{R_{WCAF}} =0.821$.We use waveguide theory to calculate an expected ratio of power reflection coefficients. Utilizing the complex effective index method of metallic waveguides [30] allows for an approximation of the effective index of the cavity. This waveguide approximation is in some cases valid for vacuum filled cavities in silver films, provided the films are thick enough to approach the waveguide limit. For cavities uncoupled to detectors, we previously found for a silver film thickness of 300 nm the observed resonance centers in numerical calculation approach the predicted waveguide resonance centers to within 2% [19]. For waveguides consisting of silver and vacuum with extent of 50 nm in the x direction while 1010 and 1050 nm in the y direction, the waveguide theory predicts $n_{eff}=0.140$ with resonance centers of $\lambda _0=2.87 \mu m$ and $2.98 \mu m$. The resonance centers are different than that observed in the WCAF case within 10% and this is due to the detector coupling being not accounted for in the waveguide theory. However, the effective index on-resonance should not have a strong dependence on the cavity center wavelength as the index for silver is relatively flat over this region. Within this approximation, we can use the Fresnel equation to calculate the power coefficients for this effective index transitioning to vacuum and $SiO_2$ and find a ratio of $\frac {R_{vac}}{R_{SiO_2}}=0.845$, in very good agreement with the value observed from numerical calculations.

## 5. Conclusions

We examined the spectral behavior of deep subwavelength cavities when they are coupled to an absorber through FDFD calculations. We found that directly coupling an absorber to the output of a cavity greatly diminishes the quality factor by nearly 85%. This was caused by a greater rate of dissipation of energy from the slit. With the inclusion of an isolation layer, we were able to find a way to weakly couple the absorber from the cavity and improve the quality factor. We explored the effect of isolation layer depth and found a relationship between the coupling of propagating and evanescent fields produced by the cavity. From this relationship an optimal depth corresponding to the largest quantum efficiency was found at 100 nm. In addition to being weakly coupled, we examined a way to enhance the weakly coupled slit by an additional front layer which is found to enhance through changing the reflection coefficient at the input of the cavity increasing the quantum efficiency by 2.5%. We believe this work sets the foundation of coupling nanocavities to detectors which is directly applicable to light sensing applications such as photovoltaics and imaging.

## Funding

U.S. Naval Research Laboratory.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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