General design method of ultra-broadband perfect absorbers based on magnetic polaritons

Yuanbin Liu; Jun Qiu; Junming Zhao; Linhua Liu

doi:10.1364/OE.25.00A980

1. Introduction

The utilization of solar energy or radiative cooling can serve as an important and effective way to reduce the excessive consumption of traditional energy resources [1, 2]. These applications are involved with many fields such as thermophotovoltaic (TPV) systems, solar cells and thermal emitter. As a matter of fact, the broadband perfect absorption is of critical significance in these fields. With remarkable progress of nanoscience, metamaterials have been demonstrated to achieve the exotic optical properties unattainable with naturally-existing materials by creating independent tailored magnetic and electric responses to incident electromagnetic radiation [3–5]. In particular, magnetic and electric responses can excite magnetic polaritons (MPs) and surface plasmon polaritons (SPPs) through the coupling of external electromagnetic fields to induced electric currents and oscillations of the conductor’s electron plasma, respectively. Both resonance modes can make the electric and magnetic fields associated with light confined to metamaterials [6, 7]. Such as split-ring resonators [4], arrays of pairs of parallel nanorods [8] and rectangular gratings [9] are typical cases of applications of MPs and SPPs to induce a resonance absorption. Nowadays, numerous researchers have been attracted to fabricate an ideal absorber by metamaterials. Dramatic progress has been made, yet plenty of them concentrate mainly on wide-angle and polarization independent perfect absorbers with one or several narrow peaks [10–13]. It is because the design of plasmonic nanostructures is still challenging to effectively couple broadband and close resonance wavelengths together and keep high performance [14–21]. Even so, there is still great essentiality and interest in achieving broadband perfect absorption by the use of metamaterials supporting plasmonic modes.

The one-dimension (1D) rectangular grating in Fig. 1(a) is well known as its simple structure and remarkable property of enhancing the absorption of metals in specific wavelengths. The resonances for transverse magnetic (TM) waves in 1D gratings have been successfully predicted and elucidated by the theory of MPs [22]. The MP resonance can be described by an inductor-capacitor (LC) circuit. The metal ridge and substrate act as inductive elements L and the dielectric in the trench acts as capacitive elements C. The fundamental resonance wavelength can be obtained from $λ_{R} = 2 π c_{0} \sqrt{L C}$ . Compared with SPPs, the unique property of MPs is its robustness to the incident angle of light [10, 22, 23]. However, the 1D rectangular grating also suffers from the common disadvantage of narrow bandwidth. Figure 1(b) shows five cases of 1D Ag gratings with different trench widths b, where period $Λ = 500 nm$ and height $h = 300 nm$ are fixed. The trench widths change from 6 nm to 10 nm and the step size is 1 nm. The optical constants of Ag are derived from a Drude model [24]. The rigorous coupled-wave analysis (RCWA) is applied to compute the spectral absorptance of gratings at normal incidence of TM waves. Five narrow peaks corresponding different trenches are shown in Fig. 1(c) and peak positions make a red shift with the trench width decreasing. In addition, peak positions are close to each other due to the near trench widths. If the difference of trench widths reduces further, there are more narrow peaks with closer positions and the family of their absorptance curves can be encompassed by an envelope curve as shown in Fig. 1(c). The envelope curve shows that if there are plenty of gratings with close resonance wavelengths or continuous trench widths blended together, the absorption band will be broadened.

Fig. 1 (a) One-dimension rectangular grating with plane of incidence and polarization. (b) Five cases of 1D grating with different trench widths, other parameters are fixed: Λ = 500 nm and h = 300 nm. (c) Spectral absorptance of five cases of 1D gratings corresponding Fig. 1(b). (d) Schematic of 1D metal-dielectric-metal sandwich structure proposed in this work.

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Though directly packing so many gratings together is not feasible in practice, it provides a useful enlightenment for the design of the ultra-broadband perfect absorber. Figure 1(d) shows that the five cases of different trench widths in Fig. 1(b) are marked with black dashed rectangles and packed along the direction of Z axis where trench widths are tapering. When the number of gratings with different trench widths is enough large and trench widths change linearly and continuously, in limited cases of geometry, we can take the metal-dielectric-metal structure in Fig. 1(d) as an approximate substitute of plentiful gratings with continuous variation in trench widths. Yellow regions in Fig. 1(d) represent the dielectric material sandwiched between metals. As a result, the 1D sandwich structure in Fig. 1(d) can excite MPs in a wide wavelength band for TM waves at various parts of the structure. Besides, the MP resonance can simultaneously happen in both the metal-air-metal and metal-dielectric-metal regions. Resonances from the two places make joint efforts to enhance absorption. The coupling of resonances at different wavelengths brings about the broadband absorption.

Because the transverse electric (TE) waves cannot excite MPs in 1D grating, to make the absorber insensitive to both TM and TE waves, the metal-dielectric-metal pyramid nanostructure with four-fold rotational symmetry is proposed in this work and shown in Fig. 2(a) as the 2D counterpart of the 1D sandwich structure. The structure period is represented by Λ_x and Λ_y (Λ_x = Λ_y); l_x and l_y are the edge lengths of inner dielectric pyramid (l_x = l_y); β shows the dihedral angle; δ represents the thickness of the metal membrane. As shown in Fig. 2(a), the left half part represents the external view of the proposed nanostructure and the right half part is the perspective neglecting the uppermost metal membrane. For the practical devices, the electron beam lithography is a potential and effective fabrication method to obtain the pyramid geometry in the dielectric layer, which has been utilized to manufacture similar nanocone structures [25, 26]. Then using physical vapor deposition (PVD) can grow a metallic thin film over the dielectric pyramid. To facilitate the application of the pyramid nanostructured absorber, this work gives the design process of the structure dimension in the target wavelength band firstly. Based on the proposed design method, the spectral absorptance of the obtained nanostructure is investigated from 0.2 μm to 4 μm. Furthermore, the feature of wide-angle and polarization independent absorption of the pyramid grating has been demonstrated.

Fig. 2 (a) Schematic of the proposed pyramid nanostructure, external view in left half and perspective drawing in right half. (b) Model of calculating equivalent impedance which is intercepted from the whole pyramid. (c) Equivalent LC circuit of the calculation model.

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2. Theoretical model and analysis

The MP resonance can be described by an inductor-capacitor (LC) circuit. The resonance in the pyramid is taken for study due to the advantage of four-fold rotational symmetry of pyramid which can simplify the formulas of inductance and capacitance. It is noting that the MP resonance generated in the space between ridges of two pyramids has been similarly studied [27]. In the pyramid, the metal membrane acts as inductive elements L and the dielectric part acts as capacitive elements C. Inductive elements include both kinetic inductance L_k and mutual inductance L_m. Because the incident radiation with different wavelengths can be accumulated and absorbed in various parts of the pyramid nanostructure, we have intercepted a part of the pyramid shown in Fig. 2(b) to obtain specific expressions of L and C shown in Fig. 2(c). L_k is contributed by the drifting electrons in metal and its formula is [22]:

L_{k} = \frac{- ε^{'}}{ε_{0} ω^{2} (ε^{'}^{2} + {ε^{″}}^{2})} \int_{0}^{s} \frac{d s}{δ (l - 2 s / \tan γ)} = \frac{- ε^{'} \tan γ}{2 ε_{0} ω^{2} (ε^{'}^{2} + {ε^{″}}^{2})} \ln \frac{l}{l - 2 s / \tan γ}

where s is the total length of current path in the metal;

ε^{'}

and

ε^{″}

are the real and imaginary part of dielectric function of metals, respectively. Via the mean value theorem of integrals and simplification treatment, Eq. (1) can be changed as another form:

L_{k} \approx \frac{- ε^{'}}{ε_{0} ω^{2} (ε^{'}^{2} + {ε^{″}}^{2})} \frac{s}{δ (l - s / \tan γ)}

In the later analysis, it can be found that the form of Eq. (2) is helpful. Through integration, the mutual inductance is

L_{m} = 0.5 μ_{0} s

. And the capacitance of two metallic plates is

C = c^{'} ε_{0} ε_{d} \int_{0}^{s} \frac{l_{y} d s}{l_{x}} = c^{'} ε_{0} ε_{d} s

where

ε_{d}

represents the dielectric function of the dielectric material; the numerical factor

c^{'}

between 0 and 1 accounts for the non-uniform charge distribution. The resonance wavelength can be predicted by

λ_{R} = 2 π c_{0} \sqrt{(2 L_{k} + 2 L_{m}) C}

. After some manipulations, it can be shown that:

λ_{R} = 2 π s \sqrt{\frac{c^{'} ε_{d}}{1 + A}}

where the parameter A has been defined as

A = \frac{2 ε^{'} s^{2} c^{'} ε_{d}}{(ε^{'}^{2} + {ε^{″}}^{2}) δ (l - s / \tan γ)}

Equation (4) is an implicit function about the resonance wavelength due to the fact that materials are usually dispersive for different wavelengths. The optical constants of solid materials can be described by the Lorentz-Drude (LD) oscillator model [28]. The LD model shows that a complex dielectric function is expressed in the form: $ε = ε_{D} + ε_{L}$ , which consists of the intraband part (Drude model, referred to free-electron effects) and interband part (Lorentz model, referred to bound-electron effects). Because the interband part is the semiquantum model, it is complicated and infeasible to obtain the explicit expression about $λ_{R}$ of Eq. (4). However, in the low frequency limit ( $| A | < < 1$ ), an explicit asymptotic value can be yielded from Eq. (4) for the resonance wavelength:

λ_{R} = 2 π s \sqrt{c^{'} ε_{d}}

The insight of MP resonance from above equations gives some useful implications in the design process of the nanostructure:

(a) Based on the geometric feature of the pyramid, when l is fixed as l_x or l_y, the maximum length $s_{\max}$ of current path is proportional to l_x. If the proportionality coefficient is set as ρ ( $s_{\max} = ρ l$ ), the form of A in Eq. (5) is changed as $A = \frac{2 ε^{'} ρ^{2} l_{x} c^{'} ε_{d}}{(ε^{'}^{2} + {ε^{″}}^{2}) δ (1 - ρ / \tan γ)}; 0 < ρ \leq \tan γ / 2$
Hence, when l_x rises, others being held constant, the term (1 + A) will go down (noting $ε^{'}$ is minus for general metals) and the lower limits of a frequency will reduce, which means that the bandwidth can be broadened;
(b) In Eq. (7), if we assume s reaches the mathematical maximum $s_{\max} = l \tan γ / 2$ . The form of A has the form: $A = \frac{ε^{'} l c^{'} ε_{d} \tan^{2} γ}{(ε^{'}^{2} + {ε^{″}}^{2}) δ};$
Therefore, with γ increasing, others being held constant, the term (1 + A) will reduce and likewise the bandwidth is broadened;
(c) With in a certain range, when δ decreases, others being held constant, the bandwidth can be broadened. Specially, δ cannot continue to increase to narrow the bandwidth because of the skin effect of metals. In another word, the light cannot penetrate the metal membrane and reach at the dielectric pyramid when δ increases up to a certain value.

In the design of the nanostructure, the relationship between resonance wavelengths and dimension parameters is provided from Eq. (4). As can be seen from Eq. (4), there are three dimension parameters l_x (or l_y), δ and γ but one equation and at this time the equation is underdetermined. In addition, s is a function of resonance wavelengths and s cannot be obtained accurately at beginning excepting the situation in the low frequency limit. Hence, it is hard to determine all parameter dimensions quantitatively by utilizing above equations directly. Equation (4) is better to act as an expression to verify resonance wavelengths. Yet we can design general dimensions semi-quantitatively on the basis of Eq. (4) and original idea for proposal of the nanostructure. The nanostructure proposed in this work stems from the packing of 1D gratings and MPs. Increasing the height of the pyramid is an effective way to provide more diverse inductances and capacitances to support more resonance wavelengths. The insight is consistent with the conclusion (b) obtained from MPs that the bandwidth is broadened with γ going up. Accordingly, γ should be chosen as a large angle. Based on Eq. (4), at the maximum resonance wavelength ( $λ_{\max}$ ), s is also maximized. As mentioned before, the mathematical maximum that s can reach is $l_{x} \tan γ / 2$ . From Eq. (4), for general metals ( $ε^{'} \leq 0$ ), it can be found:

s \leq \frac{λ}{2 π \sqrt{c^{'} ε_{d}}}

To make the geometry support the resonance at the maximum wavelength, the relationship should be satisfied

\frac{λ}{2 π \sqrt{c^{'} ε_{d}}} \leq \frac{l_{x} \tan γ}{2}

When γ is chosen as 63.44°, the maximum of s is exactly l_x mathematically. Practically, because γ needs to be selected as a large angle, the value of γ more than 63.44° can be set. As a simplified process and design, we can firstly gain l_x ( = l_y) by assuming

l_{x} = \frac{λ_{\max}}{2 π \sqrt{c^{'} ε_{d}}}

Then, δ is designed from conclusion (c). At last, the final dimensions are determined by numerical simulations and optimization. Besides, we should emphasize that the detection of the MP resonance still requires the electromagnetic field distribution and whether Eq. (4) is satisfied at the wavelength detected.

3. Results and discussion

As a practical application of the proposed design method to harvest the solar energy, a case is provided as follows. The considered solar spectrum is from 0.2 to 4 μm which covers from ultraviolet, visible to near infrared regime. The schematic of the nanostructure is shown in Fig. 2(a). The material of the metal membrane and substrate is chosen as tungsten and the material of dielectric part is selected as SiO₂ due to fine corrosion resistance and high melting point of both materials. According to the design method mentioned before, edge length of inner pyramid l_x or l_y is gained as 0.566 μm from Eq. (11), where $c^{'} = 0.22$ is set [9]. Other geometric parameters are set as: dihedral angle β = 80°, film thickness δ = 15 nm, Λ_x = Λ_y = 0.6 μm. The optical constants of tungsten and SiO₂ come from tabulated data with interpolation [29]. The spectral reflectance R of the structure is calculated by the finite difference time domain (FDTD) method and the absorptance can be obtained from $α = 1 - R$ , for the tungsten substrate has been assumed to be thick enough so that the transmittance T is equal to zero.

Figure 3 shows the spectral absorptance of the proposed pyramid nanostructure at normal incidence of TM waves. For TE waves, they have the same results as TM waves due to the symmetry of the structure. As can be seen from Fig. 3, a broadband perfect absorption has been achieved in the spectral range of 0.2 μm < λ < 2.21 μm. The average of spectral absorptance is over 99% for 0.2 μm < λ < 2.21 μm. The result also shows that the maximum resonance wavelength is located at 2.21 μm which is more than the target maximum wavelength of 2 μm. The error is mainly attributed to the simplified process of Eq. (11). To verify whether the pyramid nanostructure or the nature optical properties of the material tungsten is the important factor to achieve the broadband high absorption, the spectral absorption of the smooth tungsten plate at normal incidence is also calculated in Fig. 3. Through comparing two results from the nanostructure and plate, it is obvious that the broadband perfect absorption is attributed to the function of the proposed structure.

Fig. 3 Absorptance spectra of the metal-dielectric-metal pyramid nanostructure at normal incidence of TM waves (θ = 0°). The graph is plotted in logarithmic scale of λ.

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Figure 4 shows the effect of the edge length on the band width of spectral absorptance of the proposed pyramid nanostructure at normal incidence of TM waves. Two sets of parameters are used to simulate the spectral absorptance: (1) l_x = l_y = 0.672 μm; Λ_x = Λ_y = 0.7 μm; δ = 14 nm; β = 80°; (2) l_x = l_y = 1 μm; Λ_x = Λ_y = 1.03 μm; δ = 14 nm; β = 80°, respectively. As demonstrated in Fig. 4, the maximum resonance peak moves toward longer wavelengths and the absorption band can be broadened further when l_x increases, which is consistent with the conclusion (a) mentioned before. When l_x increases up to 1μm, the absorptance is over 95% at the resonance wavelength of 3 μm where a semi-infinite slab of tungsten only has an absorptance that is less than 2.6%. Hence, it is flexible to extend the spectral band of perfect absorption from ultraviolet, visible to near infrared regime by adjusting the structure dimension.

Fig. 4 Spectral absorptance of the proposed pyramid nanostructure with different edge lengths l_x at normal incidence of TM waves. Two sets of parameters used for simulation are: (1) l_x = l_y = 0.672 μm; δ = 14 nm; Λ_x = Λ_y = 0.7 μm; β = 80°; (2) l_x = l_y = 1 μm; δ = 14 nm; Λ_x = Λ_y = 1.03 μm; β = 80°, respectively.

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In practice, the direction of solar radiation is random, which requires the absorber to be stable for a wide range of both polarization and incident angle. Figure 5(a) shows the spectral absorptance contour at normal incidence as a function of polarization angles and wavelengths. The geometric parameters applied are: grating period Λ_x = Λ_y = 400 nm, edge length of inner pyramid l_x = l_y = 335 nm, dihedral angle β = 80°, film thickness δ = 14 nm. The yellow bands indicate the region of high absorptance. As demonstrated in Fig. 5(a), the pyramid nanostructure is insensitive to the polarization of incident light with wavelength range from 0.2 to 2 μm. It can be attributed to the four-fold rotational symmetry of the proposed structure. Figure 5(b) shows the spectral absorptance in terms of incident angles and wavelengths for both TM and TE waves. It can be found that the structure is also robust at a wide range of incident angles within the wavelength range of 0.2 μm < λ < 2 μm. To obtain more clear quantification results, the average of spectral absorptance within the wavelength range of 0.2 μm < λ < 2 μm is calculated as the function of incident angles in Fig. 6. The average absorptance keeps over 99% within 40° incidence and remains 80% even at 70° incidence. The angular independence of the proposed structure is ascribed to the natural property of MP resonance [10, 23], which is discussed in detail later.

Fig. 5 (a) Spectral absorptance contour as a function of polarization angles and wavelengths at normal incidence. (b) Spectral absorptance contour at various incident angles for both TM and TE waves. Both graphs are plotted in logarithmic scale of λ.

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Fig. 6 Polar plots of average spectral absorptance within the wavelength range of 0.2 μm < λ < 2 μm as a function of incident angles for both TM and TE waves.

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To figure out the mechanism of high broadband absorption of the pyramid nanostructure and the effect of the pyramid shape on broadening the absorption spectrum, we investigate the magnetic field intensity distributions |H| of a symmetrical section of the structure at normal incidence of TM waves in Fig. 7 with λ₀ = 0.3, 1.2, 1.4 and 1.6 μm. The amplitude of the instantaneous magnetic field is showed by the color contour and the short black arrows represent the direction of current density vectors at λ₀ = 0.3 μm. It can be obviously seen from the subfigure at λ₀ = 0.3 μm that multiple closed loops of the induced electric current are formed in both the air gap between ridges and the inside of pyramids. These circulating currents can generate the new magnetic field whose vibrating direction is parallel to y axis. Simultaneously, the excited magnetic field interacts strongly with the incident electromagnetic waves, which results in the formation of the magnetic polaritons (MPs) and the enhancement of the spectral absorption. The MP resonances from the interspace between ridges and the inside of pyramids make joint efforts to enhance absorption. As shown in the subfigure at λ₀ = 0.3 μm, the resonance areas locate exactly at the places where the magnetic field concentrates. Besides, it is demonstrated in Fig. 7 that the incident radiation with different wavelengths has been accumulated and absorbed in various parts of the pyramid nanostructure. For the light with a shorter incident wavelength at λ₀ = 0.3 μm, the energy is harvested in the separated space with relative small areas. With the incident wavelength increasing, the number of separated areas reduces and larger resonance areas are developed to support the absorption for the light with longer wavelengths. When the incident wavelength increases from 1.2 to 1.6 μm, part of magnetic field concentration areas enlarges at the bottom of the air gap between the ridges (indicated by the black dashed arrow in corresponding subfigures) and another part moves towards the top-side from the middle waist air region (indicated by the white dashed arrow in corresponding subfigures). It is exactly the geometric feature of the proposed structure, which increases in width of air gap and reduces in width of pyramids from the bottom to top, that provides diverse dimensions to support the resonance in a broadband wavelength range. The coupling of the different resonance wavelengths brings about the broadband absorption. By LC circuit and equations mentioned before, the resonance wavelength can be predicted by $λ_{R} = 2 π c_{0} \sqrt{(2 L_{k} + 2 L_{m}) C}$ and $λ_{R} = 300 .57nm$ is gotten when $c^{'} = 0.23$ is set [30]. The result can agree well with the wavelength λ = 300nm from FDTD. Hence, on the basis of all these properties, the mechanism of high absorptance can be elucidated by MP resonance.

Fig. 7 Contour of the electromagnetic field at normal incidence of TM waves with different incident wavelengths. The arrow represents the direction of electric current and the unit of H is 10⁻³ A/m.

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4. Conclusions

In summary, we show a general design method of a broadband perfect absorber. Starting from 1D rectangular gratings and the theory of MPs, the metal-dielectric-metal pyramid nanostructure was proposed and its semi-quantitative dimension design method was demonstrated. The remarkable property of the structure to harvest broadband light energy has been shown at normal incidence for both TM and TE waves. Besides, it is found that the perfect absorption spectrum can be extended flexibly to near infrared regime by adjusting the structure dimension. Furthermore, the feature of wide-angle and polarization independent absorption of the proposed structure has been demonstrated. The mechanism of broadband high absorption is attributed to the MP resonance, which was supported by the analysis of electromagnetic field and the prediction of LC circuits. The findings in this work will facilitate the design of perfect absorbers or emitters in many fields including solar energy harvesting, radiative cooling and thermophotovoltaic (TPV) systems.

Funding

The National Natural Science Foundation of China (51336002, 51421063); the China Postdoctoral Science Foundation (2014T70331, 2014M560258); and the International Postdoctoral Exchange Fellowship Program 2015 by the Office of China Postdoctoral Council (No.20150039).

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General design method of ultra-broadband perfect absorbers based on magnetic polaritons

Abstract

1. Introduction

2. Theoretical model and analysis

3. Results and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (7)

Equations (11)

Optics Express