We demonstrate analytically and numerically that the anomalous extraordinary optical absorption through thin metal grating with air slit arrays can be excited simultaneously by long and short range surface plasmonic modes supported on the surfaces of thin metal film in weak coupling while this absorption can only be excited by short range surface plasmonic mode in strong coupling. In particular, we predict localized plasmonic mode inside air slits holding double-effect to generate the anomalous extraordinary optical absorption when only this mode is present and to suppress this absorption when the pathways excited by localized and surface plasmonic modes are present simultaneously. Furthermore, we present that double-layer metal grating consisting of two identical single-layer metal gratings can be exploited to enhance absorption efficiency more than 99% and 90% for excited localized plasmonic modes and surface plasmonic modes respectively.
©2009 Optical Society of America
Discovery of the extraordinary optical transmission (EOT)  through subwavelength nano-apertures perforated in metals has attracted a mass of experimental and theoretical studies devoting to interpreting the essential physics of the process in both hole [2–9] and slit arrays [10–15]. Generally speaking, the classical mechanism  has interpreted the EOT as excitation of the surface plasmon modes supported at the metal film interfaces, and the phase-matching mechanism obeying momentum and energy conservation has been exploited to predict the locations of EOT peaks with resonant excitation of these modes. Although this mechanism didn’t take account into the effects of the air slits or holes to the surface plasmonic modes propagation, the predicted locations for the EOT peaks were usually accurate [2,4,8,16]. However, the existence of other mechanisms leads to a heated debate about this EOT effect. For example, for metal with slit arrays ref.13 held the opposing view to the role of surface plasmonic modes and emphasized that surface plasmonic modes play a negative role in EOT in which the transmission of predicted positions by the phase-matching mechanism is always nearly zero; for metal with hole arrays the localized surface plasmonic modes inside subwavelength holes were regarded as a new pathway to generate the EOT [17-21]. Especially, when the pathway by exciting localized surface plasmonic modes in holes and the pathway by exciting surface plasmonic modes are present simultaneously the interference between these pathways may lead to the rich and complex transmission behavior as discussed in references [22,23, and 24]. Thus, in order to mediate the controversy of the mechanisms of EOT, a self-consistent theory  through arrays of subwavelength slits and a microscopic theory  of EOT through subwavelength holes arrays have been derived recently, where the self-consistent theory incorporating the propagating modes、evanescent modes and anomalous modes clarified a number of fundamental issues of EOT through metal with slit arrays, and the microscopic theory turned out that surface plasmonic mode through metallic films perforated by hole arrays holds different contributions to EOT at different wavelength range and the excitation efficiency of quasi-cylindrical wave is not affected in whole wavelength range.
As we all know, most of past researches mainly focused on adopting a thick metal film to study the EOT and there is only one surface plasmonic mode supported in single metal-dielectric interface. It is until quite recently thin metal films with shallow corrugations on both surfaces  and with slit arrays  were exploited to achieve the anomalous extraordinary optical absorption (EOA), where the EOA holding the property of concentrating light into subwavelength structure is an opposite effect to the EOT, and its efficiency is acquired from the transmission (T) and reflection (R) values according to energy conservation (A=1-R-T). In these studies, this anomalous EOA was confirmed by exciting the short range surface plasmon polariton supported in thin metal film. However, as we all know that thin metal, due to the coupling between the surface plasmonic modes in top and bottom surfaces, could shape two coupled surface plasmonic modes : the long range (LR) and short range (SR) surface plasmonic modes, and the decreasing of thickness of metal will lead to the coupling between the modes in both metal surfaces from weak to strong. For weak and strong coupling the LR and SR surface plasmonic modes illustrate different properties in which the LR surface plasmonic modes with long optical interaction have been exploited to achieve high sensitivity chemical sensors . Before, in ref.28 the EOA was achieved mainly in strong coupling corresponding to ultrathin metal film (10nm). In fact as we will discuss in this letter this EOA also can be obtained in weak coupling in which this anomalous effect can be excited by the SR and LR plasmonic surface modes simultaneously. Meanwhile, The anomalous EOA  and the terahertz field enhancement beyond the skin-depth limit  also have been studied for the isolated metallic nano slit, where the air slit can behave as a truncated metal-dielectric-metal plasmonic waveguide holding the localized plasmonic mode along the propagating direction of incidence pulse and can shape a resonant cavity resulting in strong reflections from the slit terminations.
In this letter, we will study the EOA excited the surface plasmonic modes (LR and SR) and localized plasmonic modes simultaneously supported on thin metallic grating with periodic air slits. The property of surface plasmonic modes (LR and SR) supported in thin metal film from the state of weak to strong coupling is demonstrated firstly, and the surface plasmonic modes supported in thin metal grating structure for weak coupling are exploited to predict an approximate frequency for the EOT/EOA peaks by using the phase-matching mechanism. Numerical results indicate that the EOT/EOA can be excited by SR and LR plasmonic surface modes simultaneously in weak coupling while this EOA can only be excited by SR surface plasmonic mode in strong coupling. In particular, we predict a dual effect of localized plasmonic modes inside the air slits that it can be exploited to generate the anomalous EOA when only this mode is present and it will cause destructive interference to suppress the absorption when the pathway by exciting high-order localized plasmonic mode and the pathway by exciting surface plasmonic mode are present simultaneously. Further studies for the metal grating demonstrate that the EOA efficiency excited by LR surface plasmonic mode reduces as the thickness of metal film decreases corresponding to the coupling changes from weak to strong. However, the EOA efficiency excited by surface plasmonic modes (LR and SR) and localized plasmonic mode is not more than 0.5. We next adopt double-layer structure containing two identical single-layer thin metal gratings to enhance the EOA efficiency. Before this double-layer structure  with thick metal film was studied for the EOT. Here our studies indicate that this double-layer structure also can be exploited to enhance the anomalous EOA efficiency more than 99% in the case of excited localized plasmonic modes and 90% in the case of excited surface plasmonic modes by modulating the distance and the materials filled in separating space between two single-layer metal gratings.
2. Dispersion analysis
It is well-established that the excitation of surface plasmonic modes supported on top and bottom interfaces of metal film regarded as the most-studies pathway for extraordinary optical transmission (EOT) through subwavelength slit . When the thickness of metal film is very large, the EOT peaks could be predicted by combing with the dispersion relation for surface modes on a single flat metal-dielectric interface and the phase-matching mechanism  due to the presence of a periodic array. However, when the metal is thin with the thickness less than 100nm, the interaction of surface modes supported on the top and bottom interfaces of metal film could form two coupled surface plasmonic modes : the long range (LR) surface plasmonic mode and the short range (LR) surface plasmonic mode. Thus, we need exploit the dispersion band provided by the dielectric-metal-dielectric (DMD) three-layer waveguide as shown in the inset of Fig. 1 to understand the properties of EOT. Here we assume air (ε=1) for the dielectric environment and describe the metal using a Drude model with the dielectric function:34].
Surface plasmonic modes of DMD waveguide have wave vector β related to the frequency by the following equation :Figure 1 shows the dispersion relations calculated by this equation supported in DMD waveguide with three different thicknesses. We identify different thickness of metal film with different properties, at large thickness L=1.0λp for example in Figure, the LR and SR plasmonic band is gathered for a band holding the same property with single flat metal-dielectric interface. However, as the thickness of metal decreases, the coupling of surface plasmonic modes leads to the formation of two bands corresponding to the symmetric LR and anti-symmetric SR plasmonic mode. The LR plasmonic mode increasingly closes to the air light line which means that the excited fields for the LR plasmonic mode extend far away from the metal film and intersects with the air light line to shape a low cutoff frequency at the small wave vector. The SR plasmonic mode substantially deviates from the air light line at small frequencies, and the excited fields for the SR plasmonic mode are highly concentrated near the metal interface. At large wave vectors the LR and SR plasmonic modes lying far to the right of air light line approach the surface plasmonic frequency ωsp=ωp/21/2.
In general, in a metal film with a periodic air slit arrays, the phase-matching mechanism provided by the property of period allows that the interaction between incident light and surface waves obeys momentum and energy conservation as following :Eq. (1), ksinθ is the component of the incident wave vector in x-direction and is zero in the case of normal incidence; here we assume the period of the slit arrays in metal film is along the x-direction. Gx is the reciprocal lattice vectors with the values of 2π/d, d is the period for the slit arrays and n is integer. Thus, we can combine Eq. (1) and (2) to analytically predict the approximate excitation frequencies corresponding to the wave vectors determined by period d and n at which the EOT happens.
Before a comprehensive analysis of the structure proposed next section, we begin by exploiting the method discussed above to study the past research results observed on a thin silver grating [27,28], where it turned out that the EOT/EOA is caused mainly by exciting the SR plasmonic mode, and the excitation of LR plasmonic mode has an almost zero contribution with ultrathin metal film. In fact, this phenomenon can be understood by using the property of LR plasmonic mode with a low cutoff frequency. When the excitation frequencies predicted by using Eq. (2) are less than this cutoff frequency, it very easy to recognize that the excitation of SR plasmonic mode is the dominating pathway to generate the EOT because of only SR plasmonic mode supported in thin metal. This phenomenon will show more prominent because the low cutoff frequency for the LR plasmonic mode moves to the high frequency as the thickness of metal film decreases corresponding to strong coupling. In order to give a fully insight into the properties of EOT caused by exciting both plasmonic modes, in next section we mainly study the metal film with the suitable thickness in which the LR and SR plasmonic modes don’t deviate from each other greatly corresponding to weak coupling, for example, in the case of thickness of 0.4λp in Fig. 1.
3. Numerical Results and Discussion
3.1. Numerical method
In what follows, we perform numerical simulations by using a finite-difference time domain method  to investigate the transmission of the subwavelength slit arrays in metal film as shown in Fig. 2 Left. The parameters in the figure are defined as: d is the periodicity, a is the slit width, and L is the thickness of metal film. The computational domain consists of a single period of the structure as shown in Fig. 2 Right, where d is the periodicity of the structure. We use the periodic boundary condition at the left and right boundaries in the x direction. Perfectly matched layer (PML) absorbing boundary conditions  are used at the top and bottom boundaries in the z direction. Specifically, a periodic line-dipole Gaussian source with the same size of grating’s period d in the x direction for TM ( Hy ) polarization is excited from the top of the structure. Two detectors are arranged at top and bottom of structure to obtain the reflection and transmission of energy respectively, Here we only concern the case of normal incidence and also assume air ( ε1=1 ) for the ambient dielectric environment and the slit regions. Meanwhile, we need to normalize our flux by running the simulation without the slit arrays structure to get the quantitative transmission and reflection spectrums.
3.2. Mechanisms discussion
The calculated transmission and reflection spectrums at normal incidence for the aforementioned metal grating with metal film thickness of 0.5λp and air slit width of 0.1λp are shown in Fig. 3 (a) for period of 0.25λp and (b) for period of 0.1λp respectively. Meanwhile, we show the dispersion relation given by Eq. (1) for the metal thickness of 0.5λp and the predicted exciting frequencies for the EOT calculated by the phase-matching Eq. (2) for different values of n. In Fig. 3 (a) for the periodic of 2.5λp four surface wave phase-matching locations (n=1, 2, 3, 4) are shown with the longitudinal dot lines. The intersecting points between the longitudinal dot lines and the dispersion relation highlighted by horizontal blue lines represent the predicted frequency points to shape the EOT. We can observe that there are two frequency points for every phase-matching location due to the LR and SR plasmonic modes supported in thin metal as discussed above, so two EOT frequencies can be predicted for every phase-matching location. This predication for the EOT can be verified from the transmission and reflection spectrums calculated by the numerical simulation for the metal grating as shown in the right of Fig. 3 (a). We can observe that two EOT peaks are generated at the phase-matching locations of n=1. Meanwhile, the Hy-field distributions for these two peaks B and C are shown in Fig. 4 . It can be seen that the anti-symmetric field distributed on the top and bottom interface of metal at the transmission peak B coincides with the anti-symmetric SR plasmonic mode, and the symmetric field at the transmission peak C illustrates that the EOT is excited by the symmetric LR plasmonic mode. The transmission peaks at the locations n≥2 become very weak because of increasing reflection caused by structure. However, there are three interesting phenomena which can’t be explained by the phase-matching mechanism. The first one is the predicted frequencies are slightly smaller than those achieved in simulations differing from the previous report in which it predicts the predicted peaks slightly bigger than those observed numerically and experimentally. Before this redshift phenomenon has also been observed in ref.9. The second one is the lowest high transmission peak at small frequency, and this may be caused by exciting the localized plasmonic mode based on the Hy-field distribution as shown in Fig. 4 (A). The last one happens at the location n=3 at which there is only one transmission/reflection peak, and this phenomenon doesn’t agree with the predicted results with two peaks at n=3 by phase-matching mechanism. Next, we carry the simulation for the metal grating with the periodic of d=1.0λp as shown in Fig. 3 (b), where we perform the same analysis method with Fig. 3 (a) to insight the property of EOT. The same phenomenon with only one peak can be observed at the location n=1 as shown in Fig. 3 (b).
Based on the discussion above we can conclude that the phase-matching mechanism can account for the EOT due to the exciting surface plasmonic modes, but some special phenomena listed above can’t be predicted. Thus, we need study another mechanisms or pathways to provide a complete physical picture. For the metal grating discussed here we need analyze every surface not just in top and bottom surfaces of metal film but also in the inner surfaces of air slits, because the air slit sandwiched between metal behaves as a truncated metal-dielectric-metal (MDM) plasmonic waveguide holding the localized plasmonic mode along the propagating direction of incidence pulse and can shape a resonant cavity resulting in strong reflections from the slit terminations. This resonant cavity behavior also can be excited to generate the EOT. Meanwhile, when this resonant cavity pathway and surface plasmonic mode pathway are present simultaneously, the interference between these pathways may lead to the rich and complex transmission behavior. The former studies  indicated that these pathways can be separated by varying the structures filling another material in the air slit. Here, we explore ultrathin layer of perfect electric conductor (PEC) coated on the surface of metal to demonstrate the role of every metal surface and Fig. 5 show the results for the metal grating with the PEC coated on the inner surfaces of air slit and coated on top and bottom surfaces of metal film respectively. Other parameters of metal grating are same with the structure in Fig. 3 (b). It can be seen that the inhibition of localized plasmonic mode in air slit results the appearance of two peaks in the transmission spectrum at locations of large frequency and the disappearance of the transmission at location of small frequency. This property illustrates that localized surface plasmonic modes inside air slits has a dual effect to generate the extraordinary transmission and to suppress the transmission. Meanwhile, there is an asymmetric-lineshape transmission peak at frequency of 0.9 (2πc/λp) which may correlate with the wood anomaly . But the frequency location of this peak deviates from the positions of the wood anomaly (carmine arrow) determined from the equation ωwood=m (2πc/d) (m=1, 2 …) in which d=1.0λp is the period of grating. This phenomenon can be explained according as the studying in Ref. . When surface plasmonic modes on top and bottom surfaces of metal film are suppressed as shown in Fig. 5, there are several transmission peaks caused by localized plasmonic modes which can be observed clearly from the Hy-field distributions as shown in the inset of Fig. 5.
3.3. EOA on single/double-layer grating
More importantly, the results in Fig. 3 (a) illustrate that the moderate transmission values are accompanied by small reflection values at the locations A, B, and C corresponding to the excited localized plasmonic mode, excited SR and LR surface plasmonic modes for the phase-matching of n=1 respectively. This property means anomalous absorption effect, defined as A=1-T-R, can be achieved at this location, and it is interesting to observe that this absorption effect disappears at the phase-matching locations n≥2 and can hold for grating with different period as shown in Fig. 3 (b). Thus, we next focus on studying this anomalous absorption effect as shown in Fig. 6 , where we depict transmission, reflection and absorption spectrums for the metal grating with different thicknesses of metal film. Results demonstrate that the anomalous absorption efficiency achieved by these plasmonic mode discussed above can reach to 50% as shown in Fig. 6 (d) for the metal grating with thickness of 0.5λp. Meanwhile, the absorption efficiency excited by LR surface plasmonic mode becomes weak along with the blueshift of absorption frequency as the thickness of metal film decreases. This blueshift of absorption frequency also can be observed for the excited localized plasmonic mode but with unaffected absorption efficiency. When the thickness of metal film reduces to 0.2λp, there is only one absorption peak corresponding to the excited SR surface plasmonic mode (the field distribution doesn’t show here) with the disappearance of absorption peaks excited by LR surface and localized plasmonic mode, and this property is analogous to that of the studying in ref.28. So we can conclude that the anomalous absorption effect can be excited by the SR/LR surface plasmonic mode and localized plasmonic mode simultaneously for the metal grating corresponding to the weak coupling while this effect is excited only by the SR surface plasmonic mode for the metal grating with ultrathin metal film corresponding to the strong coupling.
However, the absorption efficiency excited by this mode is not more than 0.5. Thus, we next propose double-layer metal grating containing two identical single-layer gratings to enhance the absorption efficiency in which the parameters for every single-layer grating are same with the structure in Fig. 3(a) and the separating distance between this single-layer grating is G. We begin by studying the structure with air filled in the separating space as shown in Fig. 7(a) and absorption spectrums simulated by numerical method with different separating distance G are illustrated in the left of Fig. 7(c). We can observe that two peaks with 50% efficiency excited by SR and LR surface plasmonic modes in single-layer grating become one peak with enhanced efficiency more than 90% and most of field energy concentrated on the top surface of metal as illustrated in C of Fig. 7(d) for double-layer grating with small separating space, and when the separating space G increases to 1.1λp two peaks appear again along with the reduction of absorption efficiency. Meanwhile, the absorption excited by localized plasmonic mode illustrates the opposite property where interference effect between two gratings separates one peak for single-layer grating into two peaks with unaffected absorption values for double-layer grating with small separating space. The field distributions as shown in Fig. 7(d) indicate that most of field energy at this peak distributes in the separating space and in the air slit respectively. When the separating space increases to 1.1λp only one peak is formed holding enhanced absorption efficiency more than 90%. However, this double-layer grating is more difficult to fabricate in practice. Figure 7(b) describes another easy-to-fabricate design that the separating space between two single-layer grating is filled with SiO2. Through the modulation of separating distance we can obtain a series of absorption spectrums in the right of Fig. 7(c) excited by surface and localized surface plasmonic modes holding the analogous absorption features with the design in Fig. 7(a), but when the separating space G equals to 0.4λp we can obtain a novel property for second double-layer grating that the enhanced absorption efficiencies can be excited simultaneously by the localized plasmonic mode (more than 99%) and the surface plasmonic mode (more than 80%), where the first double-layer grating doesn’t hold this feature.
Next, we recur to the coupled cavities model [38,39] to give a qualitative explanation for this high absorption efficiency. The single-layer metal grating can be treated as a dissipative cavity in which this dissipation is characterized by the imaginary part of dielectric constant of metal. Thus, two-layer metal grating can be described by two coupled dissipative cavities. Based on the model established in ref.38 we can understand that critical coupling with near-perfect absorption efficiency can be achieved by adjusting the relation between the dissipative factor of cavity、the coupling coefficient between two cavities and the leakage factor of cavity. For the system with this critical coupling, the second single-layer grating is opaque playing a role of suppressing the transmission and the separating space between two gratings can be modulated to achieve sharp dip for the reflection spectrum. Thus, optical energy will be concentrated mainly in the first metal grating and in the separating space, and this property also can be observed clearly from Hy-field distributions as shown in Fig. 7(d) corresponding to the high absorption peaks (D, E).
In conclusion, we demonstrated analytically and numerically that the anomalous extraordinary optical absorption through thin metal grating with air slit arrays could be excited simultaneously by long and short range surface plasmonic modes supported on the surfaces of thin metal film in the case of weak coupling and this effect is excited only by the SR surface plasmonic mode for the metal grating with ultrathin metal film corresponding to the strong coupling. We predicted that localized surface plasmonic modes inside air slits holds double-effect to generate the anomalous extraordinary optical absorption and to cause destructive interference to suppress the transmission. We proposed double-layer metal grating containing two identical single-layer gratings to enhance the absorption efficiency, and results indicated that the design grating could hold the enhanced absorption efficiency more than 99% for the excited localized plasmonic mode and more than 90% for the excited surface plasmonic mode.
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