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Polarization-independent broad-band absorbers in the visible regime are theoretically investigated. The absorbers are three-layered structures consisting of a lossy dielectric grating on top of a low-loss dielectric layer and a substrate of the same lossy dielectric placed at the bottom. Enhanced absorption in the underlying structure is attained over a broad range of frequency for both TE and TM polarizations. In particular, a nearly perfect absorbance (over 99.6%) is achieved at λ ≈ 600 nm, around which the absorption spectra show a substantial overlap between two polarizations. The enhanced absorption is attributed to cavity resonance and its hybridization with a weakly bound surface wave. This feature is illustrated with the electric field patterns and time-averaged power loss density associated with the resonances.
© 2011 Optical Society of America
1. Introduction
Extraordinary optical transmission through subwavelength holes or slits has been the subject of intensive research over the last decade [1–3]. The enhanced transmission arises from the excitation of coupled surface plasmons on metal films and/or cavity resonances associated with the holes or slits [4, 5]. This mechanism may also lead to enhanced absorption due to the resonance nature in the underlying structure [6–8]. On the one hand, the extreme light concentration can profoundly increase the optical absorption rate in the nanostructure [9]. Extraordinary optical absorption or blackbody phenomenon becomes an intriguing topic in most recent years [10, 11]. On the other hand, the enhanced absorption may find important applications in solar cells [12, 13], thermo-photovoltaics [14, 15], photodetectors [16, 17], and thermal emitters [18].
Various approaches have been proposed to greatly enhance the absorption in embedded nanostructures [19,20], metallic gratings with guiding layer [12,18], multilayer structures [21], and metamaterials [22–24]. Under suitable conditions, a nearly perfect absorption can even be attained in a certain frequency range [10, 18, 22, 24–26]. This is a special feature that all the incident power is absorbed in the structure without reflection from or transmission through the structure. The absorption efficiency is therefore far beyond the theoretical limit of absorption (50%) in thin planar structures [2]. Most designs for enhanced absorption rely on the excitation of surface waves (that is, surface plasmons) at the metal/dielectric interface. The enhancement of field near the surface gives rise to a large absorption. Some structures may also exploit the feature of cavity resonance to increase the absorption through field confinement. In fact, the two mechanisms can be supported in lossy dielectric structures with careful arrangement. A suitable material is tungsten, whose real part of the dielectric constant is positive in the optical regime [3, 27].
In the present study, we investigate the feature of enhanced absorption for polarization-independent broad-band absorbers in the visible regime. The absorbers are three-layered structures consisting of a lossy dielectric grating on top of a low-loss dielectric layer and a substrate of the same lossy dielectric placed at the bottom. Enhanced absorption in the underlying structure is attained over a broad frequency range for both TE and TM polarizations. In particular, a nearly perfect absorbance (over 99.6%) is achieved around λ ≈ 600 nm, the absorption spectra showing a substantial overlap between the two polarizations. The enhanced absorption is attributed to the cavity resonance and its hybridization with a weakly bound surface wave. This feature is illustrated with the electric field patterns and time-averaged power loss density associated with the resonances.
2. Results and discussion
Consider a three-layered structure consisting of a top layer of tungsten (W) grating, a middle layer of polysilicon (p-Si) slab, and a bottom layer of tungsten substrate. The schematic diagram and geometric parameters are shown in Fig. 1. In the present study, the underlying structure serves as a nearly perfect absorber in the visible regime for TE and TM polarizations. Here, TE (TM) refers to the electric (magnetic) field parallel to the slits. In order to characterize the absorption, the frequency domain finite element solver is used to calculate the reflectance (R), transmittance (T), and the corresponding field distributions [28]. The absorbance (A) is then obtained by the conservation of energy: 1 − R − T. In another aspect, the absorption of light is related to the power loss in the material. The time-averaged power loss density (per unit volume): , where ɛ″ is the imaginary part of the dielectric constant, is used to illustrate the distribution of absorption in the structure. The power loss Ploss, obtained by integrating dPloss/dV over the region of nonzero ɛ″, is equal to the absorbance A times the incident power Pinc, that is, A = Ploss/Pinc. In the present study, the optical constants for W and p-Si are taken from the solid handbook [29].
Fig. 1 Schematic diagram of the light absorber consisting of a grating layer and a substrate made of tungsten (W), spaced by a polysilicon (p-Si) slab, where p is the grating period, b is the grating depth, a is the slit width, w = p − a, h is the p-Si slab thickness, and t is the W substrate thickness.
2.1. Nearly perfect absorption
Figure 2 shows the absorption spectra for the grating structure (cf. Fig. 1) for normal incidence (θ = 0°), where p = 500 nm, w = 170 nm, a = 330 nm, b = 420 nm, h = 497 nm and t = 200 nm. Strong absorption is observed over the whole visible regime for both TE and TM polarizations. The absorbance A exceeds 0.6 in the wavelength range from 400 nm to 800 nm. Between 500 nm and 700 nm, the absorption efficiency is even greater than 80%. In particular, a nearly perfect absorption (A ≈ 1) is achieved around 600 nm for either polarization; A ≈ 0.999 at λ ≈ 600 nm for TE polarization and A ≈ 0.996 at λ ≈ 609 nm for TM polarization. In this situation, all the incident power is absorbed in the system, without reflection from or transmission through the structure. Due to the tungsten slab placed at the bottom, which acts as a reflector as well as an absorber, the transmittance T is in fact negligible. The absorbance is equal to unity minus the reflectance: A = 1 − R.
Fig. 2 Absorbance of the light absorber as sketched in Fig. 1 for TE and TM polarizations, where p = 500 nm, a = 330 nm, b = 420 nm, h = 497 nm and t = 200 nm.
Note that there is a substantial overlap (roughly from λ ≈ 520 nm to 670 nm) between the absorption curves for TE and TM polarizations. The present structure is therefore eligible to be a polarization-independent absorber. This property, however, is gradually changed as the angle of incidence increases from zero [cf. Fig. 3]. Note also that a small absorption peak is observed around λ ≈ 500 nm, which is more evident for TM polarization. This feature corresponds to the occurrence of Wood’s anomaly [2, 30], where the reflection experiences a rapid variation within a small frequency interval. The anomaly comes from the onset of a new diffraction order tangential to the surface [30]. According to the grating equation
the angle θm of the mth-order diffracted beam (m = 0, ±1, ±2,...) is related to the angle of incidence θ through the wavelength λ and the grating period p. For normal incidence (θ = 0°), the first nonzero order of diffraction (m = ±1) emerges as the wavelength approaches the grating period (λ ≈ p). The respective diffracted wave goes into surface modes (θm = ±90°) and a reflection dip is associated with this wavelength (also known as the Rayleigh wavelength). As the transmission is negligible in the present configuration, the anomaly appears as an absorption peak.Fig. 3 Absorbance as a function of wavelength and angle of incidence for the same absorber in Fig. 2 for (a) TE polarization and (b) TM polarization. White dashed lines indicate the onset of grating lobes with nonzero diffraction order m. Black solid triangles denote the absorption peaks (A > 0.9) at different angles of incidence.
The absorbance as a function of wavelength and angle of incidence is plotted in Fig. 3. It is shown that the absorption experiences a marked change when the angle of incidence exceeds the onset of grating lobe with diffraction order m = −1: θ = sin−1 (λ/p − 1) (denoted by the white dashed line). For TE polarization [Fig. 3(a)], the absorption simply attenuates as the angle of incidence increases. The absorption peaks (denoted by the black solid triangles) are located around λ ≈ 600 nm until they are distracted along the onset of grating lobe to larger wavelengths, where the nonzero-order of diffracted wave emerges and adds to the zero-order reflection. These waves are diffracted away from the structure and will not be absorbed in the system. The maximum absorption efficiency, however, is still over 80% for θ ≈ 20° and 60% for θ ≈ 50°.
For TM polarization [Fig. 3(b)], the absorption pattern is somewhat complicated. The absorption peaks basically move toward longer wavelengths as θ increases from zero. Beyond the onset of grating lobe (m = −1), the major absorption band is separated into three branches and spreads over a wider wavelength range. An array of absorption peaks is located around λ ≈ 475 nm (θ ≈ 30° to 60°), while another array is situated near λ ≈ 800 nm (θ ≈ 60° to 80°). The absorption for TM polarization is apparently stronger than for TE polarization, especially at large angles of incidence. The maximum absorbance is over 60% even at θ ≈ 80°.
2.2. Mechanism of enhanced absorption
In order to identify the mechanism of enhanced absorption, the pattern of electric field (Ez) associated with the nearly perfect absorption (A ≈ 0.999) at λ ≈ 600 nm for TE polarization is plotted in Fig. 4(a). It is shown that the field is strongly confined within the slits of the grating and depicts a typical feature of cavity resonance. For an ideal cavity, the resonant wavelength of the TEmn mode is given by
where a and b are the width and length, respectively, of the cavity, and m and n are integers. The field pattern in Fig. 4(a) is analogous to the cavity mode with m = n = 1. This feature has also been identified as the origin of enhanced transmission through broad slits for TE polarization [31, 32]. Note that the field of the resonant mode extends somehow through the open end to the outside and partially penetrates into the layer of p-Si, leading to a weaker confinement [33,34]. The field also penetrates into the side walls and effectively increases the cavity width due to the skin depth of tungsten. As a result, the effective resonant wavelength (λ ≈ 600 nm) of the TE11-like mode is slightly longer than λ11 ≈ 519 nm [cf. Eq. (2)]. In fact, if the skin depth of tungsten (about 33.5 nm at λ ≈ 600 nm) is taken into account for the cavity width a in Eq. (2), the estimation of λ11 ≈ 577 nm is closer to the simulation result.Fig. 4 Contours of the electric field Ez at (a) λ ≈ 600 nm (TE11-like mode) associated with the absorption peak and (b) λ ≈ 407 nm (TE12-like mode) for the same absorber in Fig. 2 for TE polarization. In (b), the black arrows denote the directions of diffraction order m = ±1.
Meanwhile, the feature of Fabry-Perot like resonance with somewhat weaker strength is observed in the middle layer of p-Si. This layer acts as a parallel-plate waveguide with a higher-order oscillation of field than in the slits. Note that very little field penetrates into in the bottom layer of tungsten, where it is either reflected or absorbed.
The origin of enhanced absorption has a strong correspondence with the quality of field confinement. For TE polarization, the fundamental TE11-like cavity mode depicts a higher quality of confinement than the higher-order modes. For comparison, the TE12-like mode at λ ≈ 407 nm is plotted in Fig. 4(b). The respective absorbance is significantly reduced (A ≈ 0.7). On the one hand, the first-order diffraction emerges at a reflection angle around ±54.5° at λ ≈ 407 nm [cf. Eq. (1) with m = ±1], which can be noticed by the interference pattern in Fig. 4(b) (the reflected beams are denoted by the black arrows). On the other hand, both the real and imaginary parts of the complex refraction index n (= n′ + in″) of p-Si increase significantly from λ ≈ 600 nm (n ≈ 3.89 + 0.05i) to λ ≈ 400 nm (n ≈ 5.22 + 0.44i). As a result, less field is allowed to enter into the p-Si layer and the W substrate. The absorption is therefore weaker.
The electric field associated with the nearly perfect absorption (A ≈ 0.996) at λ ≈ 609 nm for TM polarization is plotted in Fig. 5. The pattern of horizonal electric field Ex in Fig. 5(a) shows a typical feature of TM02-like mode in the cavity. As in the case of TE polarization, the resonant mode is not completely confined within the cavity due to the open end at the top. The Fabry-Perot like resonance is also observed in the middle layer of p-Si. A special feature for TM polarization is the appearance of a surface wave, which is manifest on the pattern of vertical electric field Ey in Fig. 5(b). The enhanced absorption is therefore characterized by the coupling of cavity modes with surface waves. This hybridization feature has also been identified in the study of enhanced transmission in nanostructured materials [6]. In the present study, the tungsten behaves as a lossy dielectric since its dielectric constant has a positive real part in the optical regime [27]. The corresponding mode is weakly bound to the surface and known as the Zenneck wave [3], rather than the surface plasmon that usually occurs on the metal surface [35]. This wave is also termed as structured surface plasmon or surface charge density wave [36].
Fig. 5 (a) Contours of horizontal electric field Ex and (b) vertical electric field Ey associated with the absorption peak at λ ≈ 609 nm for the same absorber in Fig. 2 for TM polarization.
The feature of enhanced absorption is further illustrated with the time-averaged power loss density dPloss/dV for the absorption peaks. For TE polarization in Fig. 6(a), the incident power is mainly absorbed in the tungsten grating walls. A small portion of power is absorbed by the tungsten substrate and the polysilicon layer. This feature is consistent with the TE11-like mode [cf. Fig. 4(a)] associated with the absorption peak. For TM polarization in Fig. 6(b), the absorption occurs on the grating top and bottom as well as on the walls. In addition, the absorption in the middle layer and substrate is more intensified. This is due to the concurrence of cavity modes and surface waves. The alignment of surface charges is overlaid in the figure to show the feature of surface waves.
Fig. 6 Contours of the time-averaged power loss density dPloss/dV associated with the absorption peaks for the same absorber in Fig. 2 at (a) λ ≈ 600 nm for TE polarization and (b) λ ≈ 609 nm for TM polarization. In (b), the alignment of surface charges are denoted by symbols “+” and “−”.
Figure 7 shows the distribution of time-averaged power loss Ploss by integrating dPloss/dV over separate layers. It is shown that for TE polarization [Fig. 7(a)] the incident power is largely absorbed in the grating, especially at longer wavelengths. Around the absorption peak, 80% of the power loss is due to the grating. For TM polarization [Fig. 7(b)], the absorption in the substrate becomes more significant. This is more evident at λ ≈ 700 nm, where 38% of the incident power is absorbed in the substrate.
Fig. 7 Ratios of the time-averaged power loss Ploss in different layers of the same absorber in Fig. 2 for (a) TE polarization and (b) TM polarization.
2.3. Effect of geometrical parameters
Figure 8(a) shows the dependence of absorbance on the grating period p, all other parameters being unchanged. The absorption band is basically centered around λ ≈ 600 nm for both polarizations as the grating period changes. This feature depicts that the absorption in the underlying absorber is more of the site resonance that depends on the individual structure in the unit cell. The absorption is, however, blocked by the line λ = p (denoted by the white dashed line), above which the nonzero order of diffractions emerge. A substantial portion of incident light is reflected away from the absorber and the absorption is therefore significantly reduced. Note that for TM polarization (right plot), the dependence of absorption on the grating period is a bit more complicated, due to the hybridization of cavity modes and surface waves. The main absorption band, however, is still around λ ≈ 600 nm and substantially blocked by the line λ = p.
Fig. 8 Dependence of absorbance on (a) the grating period p and (b) the slit width a for TE polarization (left plots) and TM polarization (right plots). All geometry parameters are the same as in Fig. 2 except the grating period p for (a) and the slit width a for (b). In each plot, the white circle denotes the absorption peak for the optimized structure in Fig 2.
The dependence of absorbance on the slit width a is shown in Fig. 8(b). For TE polarization (left plot), the absorption peak moves to longer wavelengths as the slit width is increased. This feature is consistent with the character of TE11-like mode associated with the enhanced absorption [cf. Fig. 4(a)]. The absorption is rather weak when the slit width is less than 200 nm. Note that the strong absorption (A ≈ 96%) still occurs over a large wavelength range (λ ≈ 610 nm to 800 nm) for a = 490 nm. The corresponding grating width w is 10 nm, which is smaller than the skin depth of tungsten in the visible regime (20 nm to 40 nm). For TM polarization (right plot), the redshift of the absorption peak with the slit width is less obvious. This feature is consistent with the character of TM02-like mode associated with the enhanced absorption. The change of slit width basically does not alter the resonant frequency. The enhanced absorption, however, is significant when the slit width is larger than 150 nm.
The effect of grating depth b on the absorption is shown in Fig. 9(a). For TE polarization (left plot), the lower and higher absorption bands correspond to TE11-like and TE12-like modes, respectively, associated with the enhanced absorptions. The respective resonant modes are red-shifted with the grating depth, as they are with the slit width. For TM polarization (right plot), the three major absorption bands are associated with the TM01-like, TM02-like, and TM03-like modes (from the lower to the higher bands). As in the case of TE polarization, more resonant modes are excited as the grating depth is increased. Meanwhile, the absorption bands move to longer wavelengths for larger grating depth. Finally, the dependence of absorption on the p-Si layer thickness h is shown in Fig. 9(b). For TE polarization (left plot), the absorption band is nearly unchanged with the p-Si layer thickness, except that there are small variations near the band edges. For TM polarization (right plot), on the other hand, significant variations occur on the absorption band edges, which is more evident on the long wavelength side. The periodic pattern along the layer thickness indicates that the Fabry-Perot like resonance in the p-Si layer increases its importance in the enhanced absorption. This feature is consistent with the increased portion of time-average power loss in the corresponding layer for TM polarization, as depicted in Fig. 7. As the absorption band width for TM polarization is strongly dependent on h, a suitable p-Si layer thickness is therefore required to give a maximum overlap of absorption curves between TE and TM polarizations. In the present study, an optimal design for this purpose is attained with h = 497 nm (λ ≈ 520 nm to 670 nm).
Fig. 9 Dependence of absorbance on (a) the grating depth b and (b) the p-Si layer thickness h for TE polarization (left plots) and TM polarization (right plots). All geometry parameters are the same as in Fig. 2 except the grating depth b for (a) and p-Si layer thickness h. In each plot, the white circle denotes the absorption peak for the optimized structure in Fig 2.
3. Concluding remarks
In conclusion, we have investigated the feature of polarization-independent broad-band absorbers in the visible regime. Enhanced absorption (A > 80%) occurs over a wide range of wavelength (200 nm) for both polarizations. In particular, a nearly perfect absorption efficiency is achieved around λ ≈ 600 nm (A ≈ 99.9% at λ ≈ 600 nm for TE polarization and A ≈ 99.6% at λ ≈ 609 nm for TM polarization). The extraordinary optical absorption in the underlying structure comes from the occurrence of cavity-like resonance (for both polarizations) as well as the weakly bound surface wave (for TM polarization). The electromagnetic field is trapped inside the grating structure, giving rise to a strong absorption. With careful arrangement of the geometric parameters of the absorber, the absorption spectra show a substantial overlap in the visible regime between two polarizations. The underlying grating structure is therefore eligible to be a polarization-independent absorber.
Acknowledgments
This work was supported in part by National Science Council of the Republic of China under Contracts No. NSC 99-2221-E-002-121-MY3 and NSC 99-2221-E-002-140.
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