## Abstract

Absorption properties in one-dimensional quasiperiodic photonic crystal composed of a thin metallic layer and dielectric Fibonacci multilayers are investigated. It is found that a large number of photonic stopbands can occur at the dielectric Fibonacci multilayers. Tamm plasmon polaritons (TPPs) with the frequencies locating at each photonic stopband are excited at the interface between the metallic layer and the dielectric layer, leading to almost perfect absorption for the energy of incident wave. By adjusting the length of dielectric layer with higher refractive-index or the Fibonacci order, the number of absorption peaks can be tuned effectively and enlarged significantly.

©2011 Optical Society of America

## 1. Introduction

High absorption efficiency is particularly desirable at present for various technologies such as solar cells [1], thermal detectors [2], and time-reversed lasers [3]. The concept of perfect absorber (PA) was first proposed by Padilla *et al.* [4] who had demonstrated that total absorption was possible in metamaterials by properly engineering electric and magnetic responses. The concept of PA initiates a new research area and henceforth many attentions have been paid to design metamaterial PAs from microwave to optical frequency [5,6]. Also, there are some other proposals of PA realized in critically coupled resonators [7], subwavelength hole-arrays [8], nanostructured metal surfaces [9], and thick metallic films [10]. As applications of the PA, recently, a refractive index sensing [11] and a subsampling infrared imaging [12] are achieved experimentally. The PAs proposed above have different structure designs and absorption phenomena, which can find applications in various fields. However, the aforementioned PAs have only single-band response, which may limit the potential applications such as spectroscopic detection, multi-parameter sensing, and dense wavelength division multiplexing (DWDM) communications which require multiple channels with narrow bandwidths. Motivated by this issue, quite recently Tao *et al.* obtained dual absorption bands at 1.6 THz and 2.8 THz by separating the split ring resonator (SRR) layer from the metal plate layer with a polyimide layer [13]. Dual absorption bands were also achieved at 0.45 THz and 0.92 THz by cascading two symmetrical single-resonant metamaterials in SRR unit cell [14]. PAs possessing more absorption peaks with narrow bandwidths, although desirable, remain unreported to our best knowledge.

In this paper, based on one-dimensional metal/Fibonacci quasiperiodic photonic crystal, a new kind of PA with multiple channels and ultranarrow bandwidths is proposed, and its optical properties are investigated analytically and numerically. At the interface between the metal and Fibonacci photonic crystal, the incident electromagnetic wave is transformed to tamm plasmon polaritions (TPPs). The TPPs, a new type of light-matter mode in crystals, were recently proposed by Kaliteevski *et al.* in a metal/dielectric Bragg mirror structure both theoretically and experimentally [15,16]. In contrast to the well known surface plasmon polaritons (SPPs) that are strictly TM-polarized with the dispersion locating outside the light cone, TPPs can be both TE- and TM-polarized with the dispersion locating inside the light cone [17–19]. Taking advantage of the TPP effect, Brand *et al.* recently studied the effects of quasibound photonic state within a substrate/metal film/Bragg reflector/air structure [20], and Du *et al.* experimentally realized optical absorber in heterostructures composed of thick metallic films and truncated all-dielectric photonic crystals [21]. In our proposed structure, the Fibonacci photonic crystal can simultaneously generate multiple photonic stopbands. The TPPs with eigenfrequencies lying at each photonic stopband will be excited, and are totally trapped due to the strong local-field enhancement. Therefore, perfect absorption with multiple channels is obtained in the proposed structure.

This paper is organized as follows. In Section 2, transmission properties of the metal/Fibonacci quasiperiodic photonic crystal are investigated, and analytical formulas for the eigenfrequency of are derived. In Section 3, by means of the Transfer matrix method (TMM) and the Finite-difference time-domain (FDTD) method, multiple response of perfect absorption with ultranarrow bandwidth is demonstrated. In Section 4, dependence of the absorption spectra on structure geometric parameters is investigated, and the underlying physics is explained by the Fourier Transform (*FT*) of the structure index profile. Finally, conclusions are made in the last section.

## 2. Analysis method and optical propagation properties

The proposed PA consists of a semi-infinite thin metallic layer *M* on the left, and two semi-infinite dielectric layers *A* and *B* on the right arranged with a Fibonacci-sequence. The Fibonacci-sequence was first introduced in optical system by Kohmoto *et al.* [22]. Then, existence of bandgaps in the spectrum of Fibonacci structures was revealed experimentally by Gellermann *et al.* [23]. Omni-directional bandgaps using Fibonacci quasi-periodic structures were also reported [24]. Some useful mathematical expressions to demonstrate transmission properties of light propagating through such structures were presented [25]. It is also found that Fibonacci multilayers can generate perfect transmission peaks that are useful for filtering applications [26,27]. The concept of multiple bands in Fibonacci structure has been discovered long time ago [28], and recently Nguyen *at al.* experimentally realized a new kind of mirrors with multiple reflection spectral windows based on the one-dimensional Fibonacci chains [29]. In our proposed structure, the Fibonacci-sequence applied to the arrangement of dielectric layers *A* and *B* can be depicted by the following recursive formula:

*n*and

*j*are positive integers known as the Fibonacci order and Fibonacci generation number, respectively. In our design,

*j*is set as 3, thus the proposed metal/Fibonacci quasiperiodic photonic crystal is

*MF*

_{3}(

*n*)

*= M*(

*B*

^{n}^{−1}

*A*)

*as schematically shown in Fig. 1 .*

^{n}BFirst we consider the optical propagation of the Fibonacci photonic crystal of *F*
_{3}(*n*) *=* (*B ^{n}*

^{−1}

*A*)

*. According to the Maxwell boundary condition that the tangential components of electric and magnetic fields must be continuous at the interface between two dielectrics, when optical wave propagates to the structure of*

^{n}B*F*

_{3}(

*n*), the characteristic matrix for dielectric layers

*A*and

*B*can be obtained as [30]:

*θ*is the incident angle,

*n*, and

_{a,b}*L*are refractive index and length for dielectric layers

_{a,b}*A*and

*B*, respectively.

*β*= 2π

_{a,b}*n*cos(

_{a,b}L_{a,b}*θ*)/λ,

*q*=

_{a,b}*n*cos(

_{a,b}*θ*) is for the

*s*-polarized wave while

*q*= cos(

_{a,b}*θ*)/

*n*is for the

_{a,b}*p*-polarized wave. The operation direction of the transfer matrix is from right to left, and the total characteristic matrix for the structure of of

*F*

_{3}(

*n*) is

*q*

_{1,2}=

*n*

_{1,2}cos(

*θ*) is for the

*s*-polarized wave, while

*q*

_{1,2}= cos(

*θ*)/

*n*

_{1,2}is for the

*p*-polarized wave. Here,

*n*

_{1}and

*n*

_{2}are the refractive index of the left and the right ambient mediums, respectively. In this paper, we assume

*n*

_{1}and

*n*

_{2}to be 1.

In the proposed structure of *F*
_{3}(*n*), TPPs can only be excited when the phase matching condition is satisfied [16]:

*r*is reflection coefficient of the structure of

_{F}*F*

_{3}(

*n*),

*r*

_{m}is reflection coefficient for the wave incident on the metallic layer

*M*and is determined by the Fresnel formula

*r*

_{m}= (

*n*

_{b}-

*n*

_{m})/(

*n*

_{b}+

*n*

_{m}).

*n*

_{m}is refractive index of the metal assuming to be gold in our design. The gold permittivity can be described by the Drude model of

*ε*= 1-

_{m}*w*

_{p}

^{2}/(

*w*

^{2}+ i

*w*γ), where

*w*is the incident optical frequency,

*w*

_{p}= 2.17 × 10

^{15}Hz is plasma frequency, and

*γ =*1.95 × 10

^{13}Hz is the damping constant [11], respectively. Since

*w*

_{p}

^{2}/(

*w*

^{2}+ i

*w*γ) is far larger than 1 and

*w*is far larger than γ, the refractive index of gold

*n*

_{m}can be approximated to

*iw*

_{p}/

*w*. Therefore, the reflection coefficient for the electromagnetic wave incident on the metallic layer

*M*can be expressed as:

In the numerical calculation, the incident wave is normally s-polarized, and our interested frequencies are in the visual range from 400 to 800 nm. The dielectrics *A* and *B* are selected to be *Si* (*n _{a}* = 1.23) and

*TiO*

_{2}(

*n*= 2.13), respectively. The lengths for the gold layer

_{b}*M*, the dielectric

*A*, and the dielectric

*B*are

*L*= 25 nm,

_{m}*L*

_{a}=

*λ*

_{c}/4/

*n*= 120 nm and

_{a}*L*=

_{b}*λ*/4/

_{c}*n*= 70 nm, respectively, where

_{b}*λ*

_{c}is the central wavelength with value of 600 nm. With above parameters, reflection coefficient

*r*for the structure of

_{F}*F*

_{3}(

*n*) are plotted in Table 1 . It shows that when the Fibonacci order

*n*equals to four there is only single photonic stopband, which is similar to the results designed in Ref [15]. When

*n*is enlarged, the number of the photonic stopbands will be increased correspondingly. As depicted in Table 1 and Fig. 2 , when

*n*is enlarged to 14, the structure of

*F*

_{3}(

*n*) will possess five photonic stopbands with central wavelengths of 416 nm, 462 nm, 520 nm, 598 nm, and 700nm, respectively. Therefore, the structure of

*F*

_{3}(

*n*) can generate multiple photonic stopbands, with each photonic stopband locating at

*r*−1 and having central frequency of

_{F}=*w*. Since TPPs is generated in the photonic stopbands, we only need to consider the frequencies in each photonic stopband. When an optical wave with frequency

_{i}*w*sufficiently close to the

*w*, the refection coefficient

_{i}*r*in the frequency range near the

_{F}*i-*th photonic stopband can be expressed as [15,31]

*i*is integer,

*η*is a factor determining the bandwidth of

_{i}*i-*th photonic stopband, and

*w*is the central frequency of the

_{i}*i-*th photonic stopband.

Substituting the Eqs. (6)-(7) into Eq. (5) and applying some algebraic manipulations, the eigenfrequencies of the TPPs in the structure of *M*(*B ^{n}*

^{−1}

*A*)

*can be derived as*

^{n}B*N*photonic stopbands (

*i*= 1, 2,…,

*N*), TPPs with

*N*eigenfrequencies will be excited simultaneously. Meanwhile, because 2

*n*

_{b}/(

*η*) is positive and far smaller than 1, each eigenfrequency will locate in the

_{i}w_{p}*i-*th photonic stopband and is a little smaller than the central frequency of

*w*.

_{i}## 3. Absorption properties and discussions

Now, we investigate the absorption properties of the metal/Fibonacci photonic crystal of *M*(*B ^{n}*

^{−1}

*A*)

*. As a design example, the Fabonacci order*

^{n}B*n*is set as 9, so the studied structure becomes

*M*(

*B*)

^{8}A*. With the parameters of*

^{9}B*n*

_{a}= 1.23,

*n*= 2.13,

_{b}*L*= 25 nm,

_{m}*L*= 120 nm, and

_{a}*L*= 70 nm, the spectral properties under normal incidence are calculated by the TMM and demonstrated in Fig. 3 , where transmission (T), reflection (R), and absorption (A) are all plotted. It can be obtained that the transmission is small in the whole range and reaches zero at wavelengths locating at the photonic stopbands of the Fibonacci structure (

_{b}*B*

^{8}

*A*)

^{9}

*B*. There are thee dips in the reflection spectra of Fig. 3(a), which thereby leads to three high absorption peaks in Fig. 3(b) because of A = 1-R-T. The three absorption peaks at the wavelengths of 447.6, 543.8, and 689.3 nm possess value as high as 0.81, 0.97, and 0.96, respectively. Meanwhile, the FWHM of the three stopbands are as narrow as 0.5, 0.9, and 2.2 nm, respectively. The three peaks are in the photonic stopbands while are a little larger than the central wavelength of the photonic stopbands (i.e., 447 nm, 537 nm, and 671 nm, as illustrated in Table 1). The phenomenon is in consistent with the Eq. (8). The absorption spectra for

*s*- and

*p*-polarization at incident angle of 0°, 20°, and 40° are plotted in Figs. 4(a) and 4(b), respectively. When the angle of the incident wave is increased, all the absorption peaks demonstrate a blueshift. It arises from that the photonic stopbands will shift to the shorter wavelength when increasing the incident angle [32]. In contrast to SPPs which can only work with

*p*-polarization as shown in Fig. 4, TPPs are able to work with both the

*s*- and the

*p*-polarization. Without loss of generality, we only consider the absorption properties of

*s*-polarization under normal incidence in the following paper.

Figure 5
plotted the intensity distributions of electric field |*E*| simulated by the FDTD method for the structure of *M*(*B*
^{8}
*A*)^{9}
*B*. The FDTD computational domain is schematically shown in Fig. 5(a). The mediums on both sides of the structure are air. The source locates at the left sides with position of *x* = −1 μm, and normally travels to the right with *s*-polarization. Perfect matching layer (PML) and periodic boundary condition (PBC) are set along *x* and *y* directions at the edge of the computational domain, respectively. The spatial sizes are Δ*x* = Δ*y* = 5 nm, and the temporal cell size is Δ*t* = Δ*x*/(2*c*), where *c* is the velocity of light in vacuum. Figures 5(b) and (c) show the intensity distributions |*E*| in the structure of *M*(*B*
^{8}
*A*)^{9}
*B* at eigenfrequency of 689.3 nm and off-eigenfrequency of 520 nm, respectively. At the eigenfrequency, the electromagnetic field is trapped at the Au/*TiO _{2}* interface and decrease exponentially inside

*TiO*

_{2}/

*Si*Fibonacci photonic crystal. Electromagnetic field is barely reflected or transmitted, and almost all the incident energy is absorbed. However, at the off-eigenfrequency of 520 nm

_{,}the electromagnetic field is strongly reflected by the entrance face and no absorption occurs.

## 4. Dependence of the absorption on the structure geometric parameters

In this part, we investigate dependence of the absorption spectra on the structure geometric parameters. Figure 6(a)
shows the influence of *n _{b}* on the absorption by varying

*n*from 2.03 to 2.23 with a step of 0.1. When

_{b}*n*is 2.03, the three absorption peaks are at λ

_{b}_{1}= 427.9 nm, λ

_{2}= 518 nm, λ

_{3}= 657.6 nm, respectively. While it is increased to 2.13 and 2.23, the peaks shift to λ

_{1}= 447.6 nm, λ

_{2}= 542.8 nm, λ

_{3}= 689.3 nm, and λ

_{1}= 467.5 nm, λ

_{2}= 567.5 nm, λ

_{3}= 721.3 nm, respectively. Increasing

*n*with the same step of 0.1 as indicated in Fig. 6(b), each absorption peak will exhibit a redshift, which is similar to the behavior of increasing

_{a}*n*. Note that the redshifts induced by increasing

_{b}*n*is significantly smaller than that by increasing

_{a}*n*. To explain this phenomenon, Fourier transform (FT) of the structure refractive index can be utilized [33]. The refractive index profile for the structure (

_{b}*B*

^{8}

*A*)

^{9}

*B*is

*i*= 1, 2,…,

*n*, and

*z*is the coordinate along the structure (

*B*)

^{8}A*.*

^{9}B*FT* of refractive index profile is expressed as

*L*is the length of the structure (

*B*)

^{8}A*, and*

^{9}B*β*is wave number in the medium. Inserting Eq. (9) to Eq. (10) we can obtain that

*FT*result gives characteristics of photonic stopband in the structure (

*B*)

^{8}A*, and each peak at a given*

^{9}B*β*domain corresponds to a photonic stopband. The

*FT*spectra of the structure

*F*

_{3}(9) under different values of

*n*are calculated by Eq. (11) and plotted in Fig. 7 . It shows that the

_{b}*FT*results agree well with the results calculated by the TMM in Fig. 6(a). In Eq. (11) which contains three polynomials, the coefficient for

*n*in the first and third polynomial is far larger than the coefficient for

_{b}*n*in the second polynomial. Therefore, although with the same changes of 0.1,

_{a}*n*have a larger influence on absorption than

_{b}*n*does as observed in Fig. 6.

_{a}Effects of *L _{a}* and

*on the absorption spectra are depicted in Fig. 8 . Seen from Eq. (11), dependence of*

_{.}L_{b}*FT*on

*L*is determined by the term of exp(

_{b}*jβ*(

*i*-1)(

*n*-1)

*L*). Since

_{b}*n*-1 is a large value and any change in

*L*multiplies it, the peak density will be increased dramatically even with a small change of

_{b}*L*in the

_{b}*FT*spectrum at a given range. As shown in Fig. 8(b), when

*L*varies from 25 nm to 50 nm, 70 nm, 95 nm, and 120 nm, the number of absorption peaks in visual frequency is tuned from 1 to 2, 3, 4, and 5, respectively. The dependence of

_{b}*FT*on

*L*is determined by the term of exp(

_{a}*jβ*(

*i*-1)

*L*) which do not involve the large

_{a}*n-*1. As a result, increasing

*L*as illustrated in Fig. 8(a) will not introduce obvious influence on the number of absorption peaks as that by increasing

_{a}*L*, and can only redshift the absorption peaks as increasing

_{b}*n*and

_{a}*n*does as shown in Fig. 5. We can also obtain from Eq. (11) that increasing

_{b}*n*will also increase the number of absorption peaks as the behavior of increasing

*L*, which can well explain the phenomena in Table 1. When

_{b}*n*= 9,

*n*

_{a}= 1.23,

*n*= 2.13,

_{b}*L*= 25 nm,

_{m}*L*= 120 nm, and

_{a}*L*= 280 nm, PA with twelve channels and 1-nm FWHM for each channel are designed as shown in Fig. 9 . This ultranarrow FWHM is significantly narrower than that of the PAs proposed in [11,12] and is more attractive for sensing and imaging applications. The number of absorption peaks can be further enlarged and the FWHM can be further reduced by increasing

_{b}*n*or

*L*.

_{b}## 5. Conclusions

In this paper, we have demonstrated and investigated multiple responses of near-perfect absorption with ultranarrow bandwidth in one-dimensional metal/Fibonacci quasiperiodic photonic crystal. The results show that a large number of photonic stopbands can be generated simultaneously in the structure of *F*
_{3}(*n*) at *r _{F}* = −1. TPPs with the frequencies locating at each photonic stopband will be excited at the interface between the metallic layer and the dielectric layer, leading to multiple peaks in the absorption spectra. The dependence of the absorption spectra on the structure geometric parameters of

*n*,

*n*,

_{a}*n*,

_{b}*L*, and

_{a}*L*are studied in detail. It is found that

_{b}*n*and

*L*play the dominate roles in the multiple absorption responses. To explain above phenomena, a

_{b}*FT*technique is utilized. As a design example, finally, a PA with channel number as high as twelve and bandwidth as narrow as 1 nm is achieved, which is very attractive in sensor/detector technology and highly integrated dense wavelength division multiplexing networks. Although the PAs in this paper are designed to work in the visual frequency, it can be further applied to other frequency ranges by scaling the structure geometric.

## Acknowledgments

This work was supported by the “Hundreds of Talents Programs” of the Chinese Academy of Sciences and by the National Natural Science Foundation of China under Grants 10874239, 10604066, and 60537060. Corresponding author (X. Liu). Tel.: + 862988881560; fax: + 862988887603; electronic mail: liuxueming72@yahoo.com and liuxm@opt.ac.cn.

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