1. Introduction

Perfect optical absorption (POA) is an efficient way to transfer optical energy to other forms of energy. Since the total absorption of polarized light was firstly demonstrated in 1976 [1, 2] by metallic gratings, it has attracted academic attention for the mechanism, device design and the wide applications such as photovoltaics, sensing, modulator, surface-enhanced Raman spectroscopy and optical coupling [3]. The perfect optical absorbers can be classified as resonant absorption and non-resonant absorption. For the resonant absorption, the light is absorbed by interference of resonant modes. In the case of non-resonant absorption, lossy materials and gradient refractive index materials are utilized. Compared to the non-resonant absorption, the resonant absorption has advantages of high sensitivity, good tunability and ultra-thin structure design, and is promising for sensing [4, 5], filtering, modulating [6, 7], and surface-enhanced Raman spectroscopy [8]. The mechanism behind resonant perfect optical absorption is the phase matching of different modes, such as surface plasmon modes [9], lattice resonance modes [10], Fabry-Perot modes in slits [11, 12], cavity modes of trenches on grating surfaces [13] and quasi waveguide modes in dielectric films [14]. To physically understanding the mode interactions and their effects on optical transmission/absorption of periodical metallic structures, a thorough mode analysis for a three region system (e.g. a free-standing grating structure) by the coupled mode method was reported [11]. In addition, the coupling strength and decay rate of different modes were analyzed by transmission Fano lineshape analysis [15, 16]. Though there are many studies on resonant perfect optical absorption of grating structures, the sufficient and necessary conditions to realize the perfect optical absorption have not been discussed. The rules for the design of a perfect optical absorption structure for a specific wavelength are still not determined. This limits the practical applications of the resonant perfect optical absorber.

In this paper, we proposed a full mode analysis by the coupled mode method for the air/grating/film/air four region system, one of the promising resonant perfect optical absorber structures. The sufficient and necessary conditions for the perfect optical absorption are derived from the interface scattering coefficients analyses in the infrared wavelength region, which offer the prompt and accurate design rules for the perfect optical absorbers. The coupling of the Fabry-Perot modes in the grating slits and the quasi waveguide modes in dielectric film play a key role for the perfect optical absorption. The features of the electromagnetic field distribution and energy loss under the perfect optical absorption are studied by numerical simulations. Furthermore, we investigated the fabrication window for geometry parameters for specific POA wavelength.

2. Metallic grating structure and methods

The air/grating/film/air four region structures can be realized by attaching a freestanding metallic grating to a dielectric film. Here we mainly consider the classical model and p-polarized incident light from the grating side, as seen in Fig. 1. In this case the surface modes can be coupled to the incident light by grating momentum compensation. The four region system is labeled from input to output as layer 1 to layer 4 as shown in Fig. 1. The permittivity of gold is taken for the grating and is modeled by the Drude model [15]:${\epsilon }_m={\epsilon }_{\infty }-{\omega }_p^2/({\omega }^2+i\omega \gamma )$ with ${\epsilon }_{\infty }=1$, ${\omega }_p=1.15\times {10}^{16}{s}^{-1}$ and $\gamma =0.9\times {10}^{14}{s}^{-1}$. The refractive index of the dielectric film (layer 3) is taken as $n_3=1.4$ for example, while other regions are all air with the refractive index as $n_1=n_2=n_4=1$. The grating geometry parameters are grating period (D), slit width (s), grating thickness (T_g) and dielectric film thickness (T_d). Commercial finite differential time domain software (FDTD Solutions, Lumerical Solutions Inc.) was used to calculate the transient electric field distribution and the time averaged ohmic loss [17]. The scattering coefficients for each interface are derived by the coupled mode method [18–20] with two approximations. Firstly, the surface impedance boundary condition (SIBC) is imposed on the metallic surface, $Z_s=1/\sqrt{{\epsilon }_m}$, which relates the field in the dielectric with the field in the metal at the interface and it fits well as long as the interface geometry size is much larger than the skin depth. Secondly, only the fundamental Fabry-Perot mode in the slits is considered since the wavelength of light is five times larger than the slit width, which is valid as$s<\raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$5n_2$}\right.$. These two approximations make the scattering coefficients calculated analytically and apply well for the structures in the infrared incident case. The coupled mode method is a mode expansion method. To the air and film layer, the fundamental modes are plane wave modes. While, to the grating layer in the infrared region, the fundamental modes are the Fabry-Perot modes in the slits. The interface scattering coefficients of the modes are defined in Fig. 1. For example, ${\rho }_{24}$ is the effective reflection coefficient at the interface of layer 2 and layer 4, which contains the contributions of both Fabry-Perot modes in the grating slits and the waveguide modes in the film. The propagation of the Fabry-Perot modes in the slits is expressed as $u=\exp (i{n}_{eff}{k}_0{T}_g)$. The propagation of waveguide modes in the film is expressed as $w_p=\exp (i{k}_{3z,p}{T}_d)$ with a diffraction order of p. Up to 50 orders waveguide modes are calculated in the coupled mode analysis with the effective refractive index $n_{eff}=\sqrt{1+\frac{2i{Z}_s}{k_{0}s}}$ of the Fabry-Perot mode in the slit [21]. The reflection and transmission coefficients from region 1 to region 4 are expressed by the interface scattering coefficients as:

 

Fig. 1 Schematic for the air/grating/film/air four region system. The yellow part and white part of layer 2 represents the metal part and the air in slit of the grating respectively. Layer 3 is the dielectric film. Layer 1 and layer 4 are both air. Some interface reflection coefficients and the refractive index for each layer are shown. The p-polarized light is incident from layer 1.

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3. Results and discussions

3.1 Fundamental mode analysis

First, the fundamental mode analyses are performed from the interface scattering coefficients of the modes. As seen in Eqs. (1) and (2), the pole of transmission coefficient ${\tau }_{14,p}$ and reflection coefficient ${\rho }_{14,p}$ are directly related to the denominator $1-{\rho }_{21}{\rho }_{24}{u}^2$. ${\rho }_{21}$, ${\rho }_{24}$ and $u^2$ can be treated as the influence of air/grating lattice mode, quasi waveguide modes in the film and the Fabry-Perot modes in the grating slits. To prove this, we performed the phase dispersion of ${\rho }_{21}$, ${\rho }_{24}$ and $u^2$ in Fig. 2 with D = 10 µm, s = 1 µm, T_g = 16 µm, T_d = 12.5 µm. The incident light energy is scaled to grating period as 1/D and the phase is presented by the unit of $2\pi$. It is clearly shown that the line with $\mathrm{phase}({\rho }_{21})=0$is the air/grating lattice modes, the lines with $\mathrm{phase}(u^2)=2n\pi$ are the Fabry-Perot modes in the grating slits. The lines with$\mathrm{phase}({\rho }_{24})=2n\pi$, are quite close with to the condition $\mathrm{phase}({\rho }_{34,p}{w}_p^2)=2n\pi$ with integer n and p = + 1, −1 and −2 as shown in Fig. 3(b) thus are named as quasi waveguide modes in the film. Only three quasi waveguide lines for p = −1 are due to the film thickness T_d = 12.5 µm and n₃ = 1.4. More waveguide modes for p = −1 will appear with larger film thickness.

 

Fig. 2 The phase dispersion of reflection coefficients (a) ${\rho }_{21}$, (b) ${\rho }_{24}$ and (c) coefficient$u^2$^{in the grating slits normalized by $2\pi$) with D = 10 µm, s = 1 µm, T_g =16 µm, T_d = 12.5µm.}

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Fig. 3 (a) Absorption dispersion calculated by FDTD and (b) modes analysis by the coupled mode method for air/grating/film/air system with D = 10 µm, s = 1 µm, T_g = 16 µm, T_d = 12.5 µm under p-polarized incidence. The yellow circles in (a) indicate the POA with abs>95%. In (b), the black lines mark $\mathrm{phase}({\rho }_{21}{\rho }_{24}{u}^2)=2n\pi$, the magenta lines are input surface (air/grating) lattice modes, the cyan lines are output surface (grating/film) lattice modes, and the quasi waveguide modes in the dielectric film are plotted in green(−1), blue( + 1) and red lines(−2) by the equation $\mathrm{phase}({\rho }_{34,p}{w}_p^2)=2n\pi$.

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Then, the mode interactions and the corresponding effects on the absorption peaks are studied. The absorption dispersion is calculated by FDTD for the same structure in Fig. 3(a) and compared with the coupled mode dispersion analysis in Fig. 3(b). In the coupled mode dispersion plot, the black lines mark the condition $\mathrm{phase}({\rho }_{21}{\rho }_{24}{u}^2)=2n\pi$ with a deviation of ± 2%. The condition $\mathrm{phase}({\rho }_{34,p}{w}_p^2)=2n\pi$ with a deviation of ± 2% is shown by green, blue and red lines with p = −1, + 1 and −2, respectively. The magenta lines are the input surface (grating/air) lattice modes and the cyan lines are the output surface (grating/film) lattice modes. First of all, all the absorption resonances correspond to black lines, which mean that the resonant absorption is indeed caused by the coupling of the different modes mentioned above [11], with the features of energy shifts and energy anti-crossings. Secondly, the grating/air lattice modes on the input surface reduce the absorption peaks. This is for the reason that the surface lattice modes short out the input surface and lead to high reflectance. Contrarily, the grating/film lattice modes on the output surface enhance the absorption because they can couple with the quasi waveguide modes in the film. Thirdly, the quasi waveguide modes in the dielectric film introduce new absorption resonance peaks and new energy band distortions. All the enhanced absorption lines (A > 0.5) are related with one quasi waveguide mode. Therefore, the quasi waveguide modes seem to play a dominant role for the enhanced absorption. The enhanced absorption peak wavelength are within the range from 10 µm ($n_{1}D$) to 14 µm ($n_{3}D$) because of the contributions of the waveguide modes.

3.2 Perfect optical absorption conditions

As marked by yellow circles for the POA (Abs > 95%) in Fig. 3(a), the perfect optical absorption is realized in the structure with D = 10 µm, s = 1 µm, T_g = 16 µm, T_d = 12.5 µm under the normal incident wavelength 13.796 µm. However, the yellow circles only cover parts of the mode lines. Therefore, the sufficient and necessary conditions for perfect optical absorption for the four region system need to be studied. As shown in Fig. 4(a), the transmittance/reflectance spectra, which are calculated by the coupled mode analysis, match well with the spectra calculated by FDTD with the same structure used in Fig. 3. It means that the coupled mode analysis is accurate enough for the study of the perfect optical absorption conditions.

 

Fig. 4 (a) Absorption/transmittance/reflectance spectra calculated by FDTD (dots) and by the coupled mode method (solid line). (b) The phase spectra of${\rho }_{21}{\rho }_{24}{u}^2$, ${\rho }_{24}$, and ${\rho }_{24,p}{w}_p^2$ (p =± 1) calculated by the coupled mode method with D = 10 µm, s = 1 µm, T_g = 16 µm, T_d = 12.5µm under p-polarized normal incidence.

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As shown in Fig. 4(a), the perfect optical absorption at 13.796 µm is companied with zero transmittance and zero reflectance. Consequently, we analyzed the components for both ${\tau }_{14,p}$ and ${\rho }_{14,p}$ for the reasons of the zeros. The related key coefficients are${\rho }_{21}{\rho }_{24}{u}^2$, ${\rho }_{24}$ and${\rho }_{14,0}$. We plotted the phase spectra of ${\rho }_{21}{\rho }_{24}{u}^2$ and ${\rho }_{24}$as shown in Fig. 4(b). Considering ${\rho }_{24}$ is mainly contributed by the quasi waveguide modes in the film, we also added the phase spectra of ${\rho }_{34,p}{w}_p^2$ (p = ± 1) in Fig. 4(b). First, we can see the perfect matching between $\mathrm{phase}({\rho }_{21}{\rho }_{24}{u}^2)=2n\pi$ and the reflectance valleys marked as the pink triangles in Fig. 4(a) and Fig. 4(b). Hence, $\mathrm{phase}({\rho }_{21}{\rho }_{24}{u}^2)=2n\pi$ is the first necessary condition for the perfect optical absorption and it means that the perfect phase matching in the grating slits and in the film. Next, the two enhanced absorption peaks with roughly A > 0.5 are located at the wavelength with the condition $\mathrm{phase}({\rho }_{24})=2n\pi$marked as the blue squares in Figs. 4(a) and 4(b). This means that the film perfectly blocks the transmitted energy and causes both the minimum of ${\tau }_{14,p}$ and the enhanced absorption peak. Besides, a close relationship is found between $\mathrm{phase}({\rho }_{24})=2n\pi$ and $\mathrm{phase}({\rho }_{34,p}{w}_p^2)=2n\pi$ as shown in Fig. 4(b). Thus, the quasi waveguide modes (p =± 1) in the film is the main reason to block the transmitted energy because of the total internal reflection at the film/air interface for p =± 1 order waveguide modes. The small deviation between $\mathrm{phase}({\rho }_{34,p}{w}_p^2)=2n\pi$ and $\mathrm{phase}({\rho }_{24})=2n\pi$ is due to the small contributions of the grating/film lattice modes. The quasi waveguide modes is dominant to the enhanced absorption peaks, however, the precise necessary condition for the perfect optical absorption is $\mathrm{phase}({\rho }_{24})=2n\pi$. The last necessary condition for the perfect optical absorption is the destructive interference of the reflectance, i.e. ${\rho }_{14,0}=0$ because only p = 0 order is the propagation diffraction order in our case ($D<\lambda$), which determines the perfect optical absorption peak at 13.796 µm.

The sufficiency of the three perfect optical absorption conditions for the four region system is verified by the predictions of the perfect optical absorption structures and the related wavelength. For this purpose, a geometry parameters scan for the perfect optical absorption under normal incidence is performed by FDTD and by using the three perfect optical absorption conditions separately with D = 10 µm and s = 1 µm. The search range is 0~20 µm for both T_g and T_d with step size 0.1 µm, 10~20 µm for incident wavelength with step size 0.01 µm. The cases with Abs > 99% are plotted as blue lines and dots in Fig. 5 calculated by FDTD with the corresponding absorption wavelength, grating thickness and film thickness. For prediction of the three conditions, the deviation standard 2% is chosen for all three conditions$\mathrm{phase}({\rho }_{21}{\rho }_{24}{u}^2)=2n\pi$, $\mathrm{phase}({\rho }_{24})=2n\pi$ and ${\rho }_{14,0}=0$. The results of the prediction of the three conditions give the same plots in Fig. 5. So the sufficiency of the three perfect optical absorption conditions is verified for the four region system under normal incidence. For the oblique incident case, the coupled mode method is still accurate as long as the grazing incidence is excluded.

 

Fig. 5 Geometry and wavelength scan for the POA with abs>99% under p-polarized normal incidence for the air/grating/film/air four region system by FDTD. The fixed parameters are D = 10 µm, s = 1 µm and the POA wavelength range from 10µm to 14µm with different grating thickness and film thickness.

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The sufficiency and necessary conditions of the perfect optical absorption for the four region system provide an efficient tool to understand and design the perfect optical absorber structures. For the incident light near the air/grating lattice mode, the contribution of the grating/film lattice modes can be neglected, so we can simplify the perfect optical absorption conditions to $\mathrm{phase}({\rho }_{21}{u}^2)=2n\pi$ and$\mathrm{phase}({\rho }_{34,\pm 1}{w}_{\pm 1}^2)=2n\pi$, therefore the analytical conditions are:

As a complementary part, we also studied the four region case with light incident from the film side. The absorption and mode dispersions for these structures are plotted in Fig. 6 with the same structure as in Fig. 3. It is interesting to see that the absorption spectra are so different but the fundamental modes are the same. This difference is due to the different role of grating surface lattice modes at the input or output side. The input surface lattice modes short out the surface and reduce the incident coupling thus reduce the absorption which is similar with the study that corrugation at input surface determine the transmission spectra [22]. The output surface lattice modes enhance both the transmission and the absorption [23, 24]. When the film is at the input side shown as in Fig. 6, it is like the asymmetric dielectric/grating/air system, which only has the perfect optical absorption at oblique incidence. The perfect optical absorption conditions for the four region system with light incident from the film side are $\mathrm{phase}({\rho }_{21}{\rho }_{24}{u}^2)=2n\pi$, ${\tau }_{14,0}=0$, and ${\rho }_{14,0}=0$, and are different from the conditions with light incident from the grating side [25,26]. The perfect optical absorption cases indicated by the yellow open circles in Fig. 6(a) are fewer than in Fig. 3(a). Therefore, for the prominent enhanced optical absorption it is better to illuminate the air/grating/film/air four region system from the grating side.

 

Fig. 6 (a) Absorption dispersion by FDTD, (b) modes analysis and (c) the schematic of the four region system with the incident light from the film side.

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3.3 Perfect optical absorption field and loss analyses

The electric field amplitude distributions and energy loss distributions are calculated by FDTD for the perfect optical absorption of the four region system. Two geometries are studied and are named as the thin case and the thick case which compared grating/film thickness with the incident wavelength. The geometry parameters of the thin case are D = 10 µm, s = 1 µm, T_g = 1 µm, and T_d = 0.5 µm, so the total thickness is less than a fifth of the POA wavelength 10.145 µm. Even though the film thickness is only 1/20 of the incident wavelength, the perfect optical absorption peak is dominated by the quasi waveguide modes from the mode dispersion analysis. The instantaneous electric field distribution for the thin case is shown in Fig. 7(a), where the color represents intensity and the arrows represent the direction. Strong surface electric field enhancement is observed, with the asymmetric charge distributions on the top and bottom surface. The interesting point is that the strongest electric field enhancement as large as 16 is observed at the film surface rather than at the grating surface. This is promising for surface enhanced fluorescence substrates. The distribution of the ohmic loss as shown in Fig. 7(b) is purely on the grating surface, where most of the energy is absorbed within a 100 nm depth. This is smaller than the penetration depth of 200 nm of the incident light (n_m = 13.4 + 54.8i @10.156 µm). The shallow absorption depth along with the high Q factors ~200 estimated from the absorption spectrum indicate a long lifetime resonance in this system. The ultra thin resonant perfect optical absorber is quite promising for enhanced light-matter interaction and hot electron genaration.

 

Fig. 7 (a) Instantaneous electric field distribution and (b) ohmic loss distribution for an air/grating/film/air system with D = 10 µm, s = 1 µm, T_g = 1 µm, and T_d = 0.5 µm under normal (θ=0) incident wavelength 10.156 µm. Black lines mark the outlines of the structure.

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The geometry parameters for the thick case are D = 10 µm, s = 1 µm, T_g = 16 µm, and T_d = 12.5 µm. The instantaneous electric field distribution for the thick case is given in Fig. 8 for two enhanced optical absorption wavelengths 10.47 µm and 13.796 µm. Clear quasi waveguide mode patterns are seen for the two wavelengths with different node numbers. The charge distribution is symmetric on the top and bottom grating surfaces for the 10.47 µm wavelength and is asymmetric for the 13.796 µm wavelength. The electric fields are distributed mainly in the slits and at the grating/film interface for the POA wavelength 13.796 µm, which is consistent with the energy loss distrubutions.

 

Fig. 8 Instantaneous electric field distribution and ohmic loss distribution for (a) wavelength 10.47 µm and (b) wavelength 13.796 µm (POA) under normal (θ=0) incidence. The structure is an air/grating/film/air system with D = 10 µm, s = 1 µm, T_g = 16 µm, and T_d = 12.5 µm. Black lines mark the outlines of the structures.

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3.4 Fabrication window for perfect optical absorber

The coupled mode method provides a prompt and accurate tool for perfect optical absorber design at specific wavelength. By the coupled mode method we offered the fabrication windows for ultra thin perfect optical absorbers at the 9 µm wavelength under the normal incidence. The basic geometry parameters are D = 8.85 µm, s = 0.761 µm, T_g = 1 µm and T_d = 0.5 µm. As shown in Fig. 9 the fabrication windows for the perfect optical absorption at the 9 µm wavelength are within D = 8.8~8.85 µm, s = 0.5~0.9 µm, T_g = 0.7~1.1 µm and T_d = 0.3~0.5 µm. The period is the most critical parameter with relatively low fabrication tolerance for the specific POA wavelength. Once the period is chosen, the other parameters should be fabricated with ~50 nm tolerance. The surface roughness should also be optimized otherwise it would jeopardize the perfect optical absorption by additional light scattering. The influence of the refractive index of the film can be fully adjusted by the film thickness owing to the quasi waveguide behavior. The grating structures can be fabricated by the core-shell structure process method [15] and then attached with a thin film such as silicon nitride. Therefore, the dielectric film supporting metallic grating structures are practical for the perfect optical absorber in the infrared wavelength.

 

Fig. 9 Absorption dependence on geometry parameters for the four region system under p-polarized normal incident wavelength 9 µm. The basic geometry parameters are D = 8.85 µm, s = 0.761 µm, T_g = 1 µm and T_d = 0.5 µm.

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4. Conclusion

The mechanism of resonant perfect optical absorption in the air/grating/film/air four region system is analyzed by the coupled mode method in the infrared wavelength incidence and sub-wavelength slits conditions. The contributions of fundamental modes (surface lattice modes, Fabry-Perot modes in slits and quasi waveguide modes in films) to the absorption are revealed by the phase analyses of the interface scattering coefficients and propagation coefficients. The perfect phase matching between the Fabry-Perot modes in the slits and quasi waveguide modes in the dielectric film have been found to play a key role for the perfect optical absorption for the four region system. The sufficient and necessary conditions for the perfect optical absorption in the four region system under normal incidence from the grating side are provided as three equations: $\mathrm{phase}({\rho }_{21}{\rho }_{24}{u}^2)=2n\pi$, $\mathrm{phase}({\rho }_{24})=2n\pi$, and ${\rho }_{14,0}=0$. These equations are prompt and accurate tools for the perfect optical absorber design for specific wavelength. The features of enormous electric field enhancement on the surface of dielectric film, total energy loss within the grating surface and high Q factors (long lifetime) for the perfect optical absorption were demonstrated. Benefitting from the prompt and accurate coupled mode calculations, the fabrication windows for the geometry parameters in the four region system for a specific POA wavelength are provided. The realizability of the dielectric film supporting metallic sub-wavelength grating structures and the advantages of wide-range wavelength tunability and high Q factors under the perfect optical absorption make these structures promising for wide applications in sensing, modulation, surface enhanced spectroscopy and hot electron based applications.