The Kolakoski absorption structure we designed can be simply denoted as air/SN/air, where S is the basic absorption unit and is composed of three materials, including monolayer MoS2 (labeled as A), SiO2 (labeled as B), Si (labeled as C). The thicknesses of the materials A, B, and C are characterized by \({d_a}\), \({d_b}\) and \({d_c}\). The order of which is determined by the Kolakoski sequence, and N is the number of repetitions. Figure 1 is a case of the Kolakoski absorption structure, where the thickness of the monolayer MoS2 is set as \({d_a}\), the thickness of SiO2 is set as \({d_b}\), the thickness of Si is set as \({d_c}\). They are arranged from top to bottom in the order of the Kolakoski sequence.
A Kolakoski sequence is an infinite sequence composed of two elements. The sequence has obvious "self-describing" characteristics. The specific rule is that the nth term of the sequence is equal to the length of the nth group of the sequence, where consecutive numbers whose positions in the sequence are the same elements are defined as a group. Assuming that the Kolakoski sequence consists of {1, 2} and the initial elements are a (1) = 2, a (2) = 2, the Kolakoski sequence can be expressed as:
If the initial elements of a given sequence a (1) = 1, a (2) = 2, the Kolakoski sequence can be expressed as:
In the structure presented here, we take the first ten terms of the sequence in both cases and then replace element 1 (denoted as AB) with a combination of molybdenum disulfide and silica, and element 2 (denoted as C) with silicon, this paper mainly studies the structure S generated according to the Kolakoski sequence expression (2) (another case will be briefly discussed later), which can be expressed as:
\(S=ABCCABABCABCCAB\)
When transverse electric (TE) wave or transverse magnetic (TM) wave is incident on the structure at an angle\(\theta\), the absorptivity of the structure will vary with the thickness of its constituent materials.
The refractive index of SiO2 is related to the wavelength of the incident wave, and its calculation formula [31] is as follows
\({n_{SiO2}}\left( \lambda \right)=\sqrt {{\varepsilon _B}} ={\left( {1+\frac{{0.6961663{\lambda ^2}}}{{{\lambda ^2} - {{0.0684043}^2}}}+\frac{{0.4079426{\lambda ^2}}}{{{\lambda ^2} - {{0.1162414}^2}}}+\frac{{0.8974794{\lambda ^2}}}{{{\lambda ^2} - {{9.896161}^2}}}} \right)^{\frac{1}{2}}}\) \(\left( 1 \right)\)
where \(\lambda\) represents the incident wave wavelength. In the wavelength range of 390nm − 780nm, we take the refractive index of Si as 3.42 [12]. The dielectric constant of MoS2 at room temperature can be obtained from the following formula [30]:
\({\varepsilon _A}={\varepsilon _{Mo{S_2}}}\left( \omega \right)={\varepsilon _{1,Mo{S_2}}}+\sum\limits_{{p=1}}^{5} {\left[ {\frac{{\left( {{f_p}+i{g_p}} \right){\omega _p}}}{{{\omega _p}+i{\gamma _p}+\omega }}+\frac{{\left( {{f_p} - i{g_p}} \right){\omega _p}}}{{{\omega _p} - i{\gamma _p} - \omega }}} \right]}\) \(\left( 2 \right)\)
The relevant parameters in the calculation formula are shown in [30].
In this paper, the transfer matrix method is used to construct the simulation model of the optical absorber. The transmission matrix method deals with photonic crystals by converting the single dielectric layer characteristic matrix into periodic transmission matrix and converting Maxwell's equations into eigenvalues of the transmission matrix. Therefore, this method can be used to study the propagation characteristics of light in multilayer structures. For the k-th layer material of the optical absorber, the transmission matrix method can be expressed as:
\({M_k}=\left[ {\begin{array}{*{20}{c}} {\cos {\delta _k}}&{ - \frac{i}{{{q_k}}}\sin {\delta _k}} \\ { - i{q_k}\sin {\delta _k}}&{\cos {\delta _k}} \end{array}} \right]\) \(\left( 3 \right)\)
where \({\delta _k}=\frac{{2\pi }}{{{\lambda _0}}}{n_k}{h_k}\cos {\theta _k}\) is the refractive index, \({h_k}\) is the material thickness, and the refraction angle is k.
For different types of plane electromagnetic waves, \({q_k}\) has different expressions, the details are as follows:
\({q_k}=\left\{ {\begin{array}{*{20}{c}} {\sqrt {\frac{{{\varepsilon _0}}}{{{\mu _0}}}} {n_k}\cos {\theta _k} \leftarrow TE} \\ {\sqrt {\frac{{{\varepsilon _0}}}{{{\mu _0}}}} \frac{{{n_k}}}{{\cos {\theta _k}}} \leftarrow TM} \end{array}} \right.\) \(\left( 4 \right)\)
Where \({\varepsilon _0}\) and \({\mu _0}\) are the permittivity and permeability of vacuum.
The complete transmission matrix is obtained by multiplying the transmission matrices of each layer of material, and its expression is as follows:
\(M=\prod\limits_{k} {{M_k}=\left[ {\begin{array}{*{20}{c}} {{m_{11}}}&{{m_{12}}} \\ {{m_{21}}}&{{m_{22}}} \end{array}} \right]}\) \(\left( 5 \right)\)
The reflection coefficient (r) and transmission coefficient (t) can be expressed as:
\(r=\frac{{({m_{21}}+{m_{22}}{q_{sub}}){q_0} - ({m_{21}}+{m_{22}}{q_{sub}})}}{{({m_{21}}+{m_{22}}{q_{sub}}){q_0}+({m_{21}}+{m_{22}}{q_{sub}})}}\) \(\left( 6 \right)\)
\(t=\frac{{2{q_0}}}{{({m_{21}}+{m_{22}}{q_{sub}}){q_0}+({m_{21}}+{m_{22}}{q_{sub}})}}\) \(\left( 7 \right)\)
Among them,\({q_0}\)and\({q_{sub}}\)are the corresponding parameters of the cover layer and the substrate layer, respectively. Therefore, the reflectance , transmittance and absorptivity of the system are respectively expressed as:
\(R={\left| r \right|^2}\) , \(T=\frac{{{q_{sub}}}}{{{q_0}}}{\left| t \right|^2}\) and \(A=1 - R - T\).
Experiments have shown that the thickness of a single layer of MoS2 (\({d_a}\))is about 0.65nm [1, 29], while the thickness of SiO2 (\({d_b}\))and the thickness of Si (\({d_c}\)) will be optimized for the operating wavelength (300nm-700nm).