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The feature of enhanced absorption in two-layered grating structures is theoretically investigated. The underlying structures make the most use of resonance mechanism to achieve a nearly perfect absorption in an intrinsically low-loss medium. For standalone gratings, the maximum absorption efficiency is shown to be 50%, which is attributed to the coupling of short range (bonding) or long range (antibonding) surface plasmons with cavity resonances. By attaching a dielectric slab on top or bottom to the metallic grating, the maximum absorption efficiency can be raised to nearly 100%. The presence of guided waves in the dielectric slab causes the strong concentration of fields and reinforces the absorption to its extreme value. The efficient absorption mechanism is illustrated with the pattern of resonance fields and the distribution of power loss density. A phenomenological theory is also used to characterize the absorption anomaly in terms of complex pole and zero.
©2011 Optical Society of America
1. Introduction
Extraordinary optical absorption in nanostructures has been the subject of intensive research in recent years [1–10] due to the potential applications in photovoltaics [11–13] and photodectors [14]. The ultimate feature of enhanced absorption is the total absorption of incident power in the structure, without either reflection or transmission. This special character can be realized from the aspect of scattered waves, which are expected to be drastically different between two sides of the structure. On the incident side, the scattered field is evanescent and therefore no reflection exists in the far field. This is the typical feature of surface waves. On the transmission side, on the other hand, the scattered field has to be propagative so that the transmission through the structure is possibly canceled out by a destructive interference with the incident field. Once the two distinct conditions are fulfilled in the same structure, the total absorption may appear.
The phenomenon of total absorption by metallic gratings has been discovered in the mid-1970s from a phenomenological theory and confirmed by numerical and experimental results [15–17]. The resonance generated by surface plasmons is attributed to the origin of absorption anomaly. In most recent years, various designs of subwavelength structures also give rise to nearly perfect absorption, including porous metallic films [1], metallic resonators [6,9], crossed gratings [3], lamellar gratings [4, 5], and metallic gratings [7, 8, 10], among other structures. In contrast to the anomalous absorption in the medium having a large imaginary part of the permittivity [18], a small absorption coefficient is usually sufficient for the enhanced absorption in the subwavelength structures. The extreme light concentration can profoundly increase the optical absorption rate [19].
A typical way to concentrate the field is to bring about the resonance. The metallic grating serves as a simple structure with strong absorption. Due to the excitation of surface plasmons, the absorption efficiency can be as high as 50% [20]. For thin metal films, there are two basic types of surface plasmons [21], characterized by the alignment of surface charges on the two sides of the film. The symmetric charge alignment is referred to as the bonding mode, while the antisymmetric alignment is called the antibonding mode [22, 23]. In particular, the bonding mode is strongly damped (with the electric field essentially parallel to the surface inside the metal) and also named the short range surface plasmon, while the antibonding mode is weakly damped (with the electric field predominantly normal to the surface) and is called the long range surface plasmon [24, 25].
For metallic gratings with narrow slits, surface plasmon modes are coupled with cavity resonances associated with the slits. The cavity-coupled surface plasmons have been identified as the origin of enhanced transmission [26–28]. The same mechanism is also accompanied with the enhanced absorption [29, 30]. For ideally thin planar structures, where the scattered fields are symmetric on the two sides, the theoretical limit of absorption efficiency is 50% [31]. The other half incident power is either reflected from or transmitted through the structure. In order to increase the absorption based on the grating design, a certain degree of complexity is to be added in the system so that the scattered fields on the two sides could be as much different as possible. The attachment of a dielectric slab to the metallic grating serves as a simple way to introduce such a complexity. In a previous study, the anomalous optical absorption was achieved in aluminum compound gratings in the near-infrared frequency [32]. Later on, three-layered tungsten grating structures were designed to enhance the broad-band absorption in the visible regime [33].
In the present study, we investigate the feature of nearly perfect absorption in more simplified grating structures. They are two-layered structures consisting of a low-loss metallic grating (made of silver) and a lossless dielectric slab. The underlying structures make the most use of resonance mechanism to yield strong absorption in an intrinsically low-loss medium. For standalone metallic gratings with narrow slits, the maximum absorption efficiency is close to 50%, which is attributed to the coupling of short range (bonding) or long range (antibonding) surface plasmons with cavity resonances. By attaching a lossless dielectric slab on either top or bottom to the metallic grating, the maximum absorption efficiency can be raised to nearly 100%. The presence of guided waves in the dielectric slab causes the strong concentration of fields and reinforces the absorption in the underlying structure. In particular, the top slab enhances the absorption associated with the short range bonding mode, while the bottom slab enhances the absorption with the long range antibonding mode. These features are illustrated with the resonant field patterns associated with the absorption peaks. The distribution of time-averaged power loss density is used to highlight the enhanced absorption in the underlying structure. Finally, the phenomenological theory [34] is used to characterize the absorption anomaly in terms of complex pole and zero.
2. Problem description
Consider a metallic grating made of silver (Ag) with narrow slits as the baseline structure. A dielectric slab is attached to either the top or bottom of the grating to introduce a certain complexity into the system. The schematics of the underlying structures are depicted in Fig. 1. A plane wave with the magnetic field oriented perpendicular to the propagation direction, that is, the transverse magnetic (TM) polarization, is incident from above. A frequency domain finite element solver [35] is employed to solve the time-harmonic (with time dependance e − iωt) wave equation in terms of Hz:
where the incident plane coincides with the xy plane, k 0 = ω/c is the wave number in the surrounding medium, which is assumed to be free space for simplicity, and ε is the dielectric constant of the constituent materials. In the present study, ε for Ag is taken from the solid handbook [36].
Fig. 1 Schematics of the metallic grating with narrow slits (left), the grating structure with a dielectric slab attached to the top surface (middle), and the grating structure with a dielectric slab attached to the bottom surface (right). The shaded region in yellow color stands for the incident plane.
For periodic structures, it is sufficient to solve the problem in a unit cell, along with the Bloch condition, Hz (x + d, y) = eik||dHz (x, y), imposed on the cell boundary, where d is the grating period (along the x direction), and k || = k 0 sinθ is the wave number parallel to the surface, with θ being the angle of incidence measured from the surface normal. Once the electromagnetic fields are solved, the reflectance R and transmittance T, the ratios of reflected and transmitted powers, respectively, to the incident one are determined by the fields at the far boundary. According to the conservation of energy, the absorbance A in the system is given as A = 1 – R – T.
The absorption of energy comes from the power loss in the system. For nonmagnetic materials, the power loss is associated with the nonzero imaginary part of the permittivity. According to the Poynting theorem [37],
the power loss in a system is due to the work done by the electromagnetic forces on electric charges, where u is the electromagnetic energy density, S is the Poynting vector, and J is the total current density. The time-averaged power loss density (per unit volume) is given by where ε″ is the imaginary part of ε. This relation shows that the enhanced absorption can be attributed to high ε″ and/or large |E|2. For low-loss materials such as silver, a feasible way to enhance the absorption is to bring about the resonance so that the electric fields are intensified in the medium. In the present problem, the grating provides a mechanism to excite resonant modes. The absorption mainly comes from silver when the resonance occurs. Note that the time-averaged power loss P loss, obtained by integrating dP loss /dV over the region of nonzero ε″, is equal to the absorbance A times the incident power P inc, that is, A = P loss /P inc. The relation (3) can also be used to identify the distribution of absorbed energy in the system.3. Results and discussion
3.1. Single layer grating with maximum 50% absorbance
Figure 2(a) shows the absorbance A, transmittance T, and reflectance R for a thin metallic grating with narrow slits, where a = 600 nm, w = 30 nm, and h = 30 nm. Note that the grating thickness is on the order of skin depth for silver (Ag) around λ ≈ 600 nm, where the refractive index n = n′ +in″ ≈ 0.124+3.72i [36] and the skin depth δ = λ/(2πn″) ≈ 26 nm. An absorption peak A ≈ 0.497 occurs at λ ≈ 612 nm, which is close to the transmission peak at λ ≈ 615 nm. The maximum absorbance approaches the theoretical limit of absorption, A = 0.5, for thin planar structures. This feature can be realized by noting that the total field is considered as the sum of the incident field and the scattered field by the grating. As the wavelength is larger than the grating period, there exist a single reflected and a single transmitted wave of non-evanescent order. For a very thin grating, the scattered fields on two sides tend to be symmetric with respect to the grating plane. The optimal absorbance is then achieved when the reflected power is equal to the transmitted power.
Fig. 2 (a) Absorbance, transmittance, and reflectance for the metallic (Ag) grating with narrow slits, where d = 600 nm, w = 30 nm, and h = 30 nm. (b) Contours of vertical electric field Ey (overlaid with electric field vectors) and time-averaged power loss density dP loss /dV [cf. Eq. (3)] for the absorption peak A ≈ 0.497 at λ ≈ 612 nm. A schematic profile of horizonal electric field Ex in the slit is plotted in the inset. The symbols “+” and “-” denote the signs of surface charges.
In Fig. 2(a), the transmission profile shows a peak and a dip located at nearby frequencies, which is the standard feature of Fano resonance [38]. This feature has been interpreted in the transmission characteristic of hole arrays [39, 40]. In particular, the transmission peak for slit arrays can be characterized by the resonant condition: ϕ = 2nπ, where ϕ is the sum of scattering phase due to the lattice and geometric phase determined by the optical path of the slit [41]. For thin metal films, the total phase is dominated by the scattering phase and the transmission resonance appears at a wavelength slightly larger than the grating period. Meanwhile, the transmission dip is very close to the grating period, indicating that it is lattice resonance in nature. The above phenomenon is also known as Wood’s anomaly [42], a feature referring to the drastic change of transmission (or reflection) within a small frequency interval.
For the present configuration, the maximum absorption is accompanied with the Fano resonance for transmission, which has also been identified in thin metal films [20, 43]. In this circumstance, both the enhanced transmission and enhanced absorption are attributable to the occurrence of surface plasmons. For grating structures, the excitation of surface plasmons is fulfilled by the momentum matching condition [44]:
where k sp is the wave number of surface plasmons. Compared to the parallel component of incident wave number k 0 sinθ, k sp acquires an additional amount due to the multiple scattering of fields over the grating structure. This amount covers the momentum deficit needed for coupling the incident light with surface plasmons. The momentum of surface plasmon on a flat metallic surface is given as [45] where εm and εd are the dielectric constants of the metal and surrounding dielectric, respectively. For a thin grating, Eq. (5) serves as an approximation to the actual surface plasmon momentum.The feature of surface plasmons is illustrated with the distribution of electric fields. In Fig. 2(b), the pattern of vertical electric field Ey associated with the absorption peak at λ ≈ 612 nm is antisymmetric with respect to the top and bottom surfaces. The surface charges on the two surfaces (denoted by the symbols “+” and “-”) are inphase. This alignment of charges is identified as the bonding mode [22, 23], one of the two basic types of surface plasmons in metal films [21]. For thin metal films, the bonding mode is strongly damped and referred to as short range surface plasmon [24, 25]. The corresponding electric fields are essentially parallel to the surface inside the metal. Note that the horizontal electric field Ex inside the slit depicts a typical feature of even mode (sketched in the inset), which is therefore able to couple the short range surface plasmon. In particular, the short range surface plasmon is responsible for the enhanced absorption in thin metal films [20, 43]. The excitation of surface plasmons gives rise to strong enhancement of fields in the metal, and the extreme concentration of fields leads to large absorption rate [19]. In terms of the time-averaged power loss density dP loss /dV [cf. Eq. (3)], the absorption is shown to be distributed over a large portion of the grating, with higher strength near the slits as well as in between.
If the grating thickness is increased to h = 100 nm, two absorption peaks appear, as shown in Fig. 3(a). The maximum absorbance A ≈ 0.496 moves to a longer wavelength (λ ≈ 640 nm), while another absorption peak with slightly weaker absorbance A ≈ 0.467 occurs around the lattice period (λ ≈ 606 nm). Note that the two absorption peaks are very close to the two transmission peaks (λ ≈ 641 nm and 605 nm). As mentioned earlier, the transmission through slit arrays is characterized by the phase resonance, the total phase (sum of scattering phase and geometric phase) being an integer times 2π. As the thickness increases, the geometric phase (determined by the optical path of the slit) becomes more dominated and the transmission peak moves to a longer wavelength than the grating period [41]. The role of cavity or Fabry-Perot resonance associated with the slit increases its importance and the coupling of surface plasmon with cavity resonance is more evident. Meanwhile, the Fano resonance feature of the transmission becomes less significant.
Fig. 3 (a) Absorbance, transmittance, and reflectance for the metallic (Ag) grating with narrow slits, where d = 600 nm, w = 30 nm, and h = 100 nm. (b) Contours of vertical electric field Ey (overlaid with electric field vectors) and time-averaged power loss density dP loss /dV for the absorption peak A ≈ 0.467 at λ ≈ 606 nm. (c) Same as (b) for the absorption peak A ≈ 0.496 at λ ≈ 640 nm.
The field pattern and surface charge alignment associated with the absorption peak at λ ≈ 606 nm in Fig. 3(b) is analogous to the short range bonding mode as in the case of h = 30 nm [cf. Fig. 2(b)]. The time-averaged power loss density dP loss /dV [cf. Eq. (3)] is more concentrated toward the top and bottom surfaces. On the other hand, the pattern of Ey at λ ≈ 640 nm in Fig. 3(c), where the maximum absorption (A ≈ 0.496) occurs, is symmetric with respect to the top and bottom surfaces. The surface charges on the two surfaces are out of phase. This alignment of charges is identified as the antibonding mode [22, 23], the other one of the two types of surface plasmons in metal films [21]. For thin metal films, the antibonding mode is weakly damped and referred to as long range surface plasmon [24, 25]. The corresponding electric fields are predominantly normal to the surface inside the metal. In the present configuration, however, the long range surface plasmon is also associated with an enhanced absorption. This is because the antibonding mode is now coupled to a cavity resonance, the damping feature being different from that in thin films without slits. Note that the horizontal electric field Ex inside the slit depicts a typical feature of odd mode (sketched in the inset), which is able to couple the long range surface plasmon. In addition, the time-averaged power loss density dP loss /dV tends to be distributed more around the slit.
For an even larger thickness h = 325 nm as shown in Fig. 4(a), the absorption peak A ≈ 0.498 at λ ≈ 665 nm becomes broader, while the other absorption peak A ≈ 0.365 at λ ≈ 609 nm is narrower. Unlike the case of h = 100 nm, the maximum absorption for the present case is associated with the short range surface plasmon [cf. Fig. 4(c)], which is coupled with the even mode of cavity resonance in the slit (sketched in the inset). Note that the cavity mode has a higher oscillation order than that in the case of h = 30 nm. Another absorption peak is associated with the long range surface plasmon, coupled with the odd mode of cavity resonance [cf. Fig. 4(b)]. Note also that the bonding of charges occurs as well on the slit walls. This feature is consistent with the distribution of time-averaged power loss density dP loss /dV [cf. Eq. (3)], which is concentrated near the slit walls and less significant on the top and bottom surfaces.
Fig. 4 (a) Absorbance, transmittance, and reflectance for the metallic (Ag) grating with narrow slits, where d = 600 nm, w = 30 nm, and h = 325 nm. (b) Contours of vertical electric field Ey (overlaid with electric field vectors) and time-averaged power loss density dP loss /dV for the absorption peak A ≈ 0.365 at λ ≈ 609 nm. (c) Same as (b) for the absorption peak A ≈ 0.498 at λ ≈ 665 nm.
3.2. Attachment of dielectric slab with nearly 100% absorbance
If a dielectric slab is attached to the metallic grating, the absorption can be greatly enhanced. Figure 5(a) shows the absorbance, transmittance, and reflectance for the same metallic grating as in Fig. 3, with a dielectric slab of thickness 70 nm and dielectric constant ε = 2 attached to the top surface. The maximum absorbance becomes nearly perfect: A ≈ 0.997 at λ ≈ 623 nm. In this configuration, the transmittance is small, especially in the range λ > a. The reflectance exhibits an inverse Lorentzian line shape and reaches a rather small value at resonance, where both the transmittance and reflectance approach zero. As a result, a nearly perfect absorption occurs.
Fig. 5 (a) Absorbance, transmittance, and reflectance for the same grating structure as in Fig. 3, with a dielectric slab of thickness 70 nm and ε = 2 attached to the top. (b) Contours of vertical electric field Ey (overlaid with electric field vectors) and time-averaged power loss density dP loss /dV for the absorption peak A ≈ 0.997 at λ ≈ 623 nm.
The pattern of Ey in Fig. 5(b) associated with the absorption peak shows a typical feature of bonding mode or short range surface plasmon, as in the case of standalone grating [cf. Fig. 3(b)]. The attachment of a dielectric slab to the top of the grating causes the fields to be strongly confined inside the slab, showing a typical feature of guided wave [46]. The hybridization of a cavity-coupled surface plasmon with the guided wave significantly increases the absorption from the theoretical limit of 50% for thin films to nearly 100%. Note that the fields in the dielectric slab change signs around the slit and the profile of Ex inside the slit (sketched in the inset) depicts an odd mode with substantial asymmetry (due to the different environments between the top and bottom surfaces of the grating). In this situation, the short range surface plasmon is still able to couple with the odd mode of cavity resonance. The enhanced absorption is manifest on the time-averaged power loss density dP loss /dV [cf. Eq. (3)], which is much more intense than for the standalone grating [cf. Fig. 3(b)].
If the same dielectric slab is attached to the bottom of the grating, a nearly perfect absorption A ≈ 0.992 is achieved at λ ≈ 707 nm, as shown in Fig. 6(a). As in the case with the top slab, the transmittance is small. The reflection dip moves to a longer wavelength, substantially away from the lattice period. In Fig. 6(b), the pattern of Ey associated with the absorption peak shows a typical feature of antibonding mode or long range surface plasmon, which is similar to that for the standalone grating [cf. Fig. 3(c)]. Note that the profile of Ex inside the slit (sketched in the inset) also depicts an odd mode with asymmetry. Compared to the case with the top slab, the fields in the bottom slab show a more regular pattern and do not change signs around the slit. The long range surface plasmon is able to couple with the odd mode of cavity resonance. In the bottom slab, the fields are concentrated and exhibit the character of a guided wave [46]. Likewise, the hybridization of a cavity-coupled surface plasmon with the guided wave leads to nearly perfect absorption, despite the fact that the long range surface plasmon is weakly damped in thin films. For this configuration, the corresponding time-averaged power loss density dP loss /dV [cf. Eq. (3)] tends to be more concentrated toward the bottom surface.
Fig. 6 (a) Absorbance, transmittance, and reflectance for the same grating structure as in Fig. 3, with a dielectric slab of thickness 70 nm and ε = 2 attached to the bottom. (b) Contours of vertical electric field Ey (overlaid with electric field vectors) and time-averaged power loss density dP loss /dV for the absorption peak A ≈ 0.992 at λ ≈ 707 nm.
Figure 7 is a plot showing the effect of the angle of incidence θ on the absorbance for the grating structure with the attachment of a top slab (cf. Fig. 5) or a bottom slab (cf. Fig. 6). As the angle of incidence is increased from zero (normal incidence), the enhanced absorption splits into two bands; one goes to longer wavelengths and the other to shorter wavelengths. The two branches basically follow the onset of grating lobes with nonzero diffraction orders [47]. For the case with the top slab [Fig. 7(a)], the right absorption branch forms an anticrossing scheme with the left branch of another pair of much weaker absorption bands located to the right. The appearance of an additional pair of branches is due to the asymmetric configuration by attaching a dielectric slab to the top of the grating. The latter branches are weakly damped as they are not coupled with cavity resonances. For the case with the bottom slab [Fig. 7(b)], the enhanced absorption moves to the right pair of branches, which is complementary to the pair of absorption bands for the case with the top slab. Meanwhile, the left absorption branch forms an anticrossing scheme with the right branch of another pair of much weaker absorption bands located to the left. For either the case with the top slab or the bottom slab, the anticrossing phenomenon comes from the existence of like symmetry of modes between the two interacting branches, the crossing being avoided due to the repulsion against each other [48]. In the present configuration, the fields inside the slits present such a symmetry. This is manifest on the pattern of Ex in the slit for the two cases [cf. the insets in Fig. 5(b) and Fig. 6(b)].
Fig. 7 Effect of the angle of incidence θ on the absorbance for the grating structure with a dielectric slab attached to the top as in Fig. 5 and (b) attached to the bottom as in Fig. 6.
3.3. The phenomenological theory
The feature of nearly perfect absorption is further interpreted from the phenomenological theory [16, 34], which characterizes the reflection (or transmission) coefficient in terms of a pole and a zero. Let r be the reflection coefficient, defined as the complex amplitude of reflected field normalized by the incident field strength. The phenomenological formula for r is given as
where λp and λz are the pole and zero, respectively, in the complex plane of wavelength, and r 0 is the reflection coefficient for a basis structure, which is taken here to be the structure without slits; that is, w = 0. The values of pole and zero are determined from the reflection coefficient in the vicinity of resonance, where a pair of local maximum and minimum appear. A similar formula can be used to characterize the transmission coefficient.Figure 8(a) shows the phases of reflection coefficients for the same grating structures as in Figs. 5 and 6. Around the nearly perfect absorption, the reflection phase experiences a drastic variation for both structures. This is a standard feature associated with the appearance of a pole and a zero. For the grating structure with a top slab [cf. Fig. 5], λz ≈ (1.0372 + 0.0002i)d and λp ≈ (1.0329 – 0.0053i)d. For the grating structure with a bottom slab [cf. Fig. 6], λz ≈ (1.1773 – 0.0009i)d and λp ≈ (1.1809 – 0.0073i)d. Note that the zeros are very close to the real axis, indicating the occurrence of nearly perfect absorption at real wavelengths. The poles, on the other hand, are deviated from the real axis. In addition, the real parts of the zero and the pole for each structure are close to each other. In Fig. 8(b), the locations of complex poles and zeros are plotted for different slit width w. As the slit width varies from its optimal value (around w = 30 nm), the zeros begin to move away from the real axis and the absorption anomaly gradually disappears. The above features occur as well for the transmission coefficient. The transmission level in the present configuration, however, is rather low over a substantial range near the resonance [cf. Figs. 5(a) and 6(a)]. The corresponding features for the transmission poles and zeros are slightly inconspicuous than for the reflection.
Fig. 8 (a) Reflection phase of the grating structure with a dielectric slab attached to the top as in Fig. 5 and to the bottom as in Fig. 6. (b) Locations of the poles and zeros in the complex plane of wavelength for different slit width w.
4. Concluding remarks
In conclusion, we have investigated the feature of strong absorption in intrinsically low-loss grating structures. A nearly perfect absorption was attained in two-layered structures made of a low-loss metallic grating and a lossless dielectric slab. The enhanced absorption is attributed to the hybridization of cavity-coupled surface plasmons and guided waves. The former gives rise to enhanced absorption in standalone gratings, whereas the latter reinforces the absorption in the composite grating structures to its extreme value. Due to the highly resonance nature, the absorption spectrum depicts a very sharp profile and sensitive to the angle of incidence. These features can be exploited for application in high quality detectors and sensors.
Acknowledgments
The authors thank Prof. C. T. Chan for his helpful comments and suggestions. This work was supported in part by National Science Council of the Republic of China under Contract No. NSC 99-2221-E-002-121-MY3.
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