Deducing critical coupling condition to achieve perfect absorption
for thin-film absorbers and identifying key characteristics of
absorbing materials needed for perfect absorption
Yongbing Long1a, Runmei Su1, Qiwen Wang1, Liang Shen 2b, Bowen Li1, Wenhao Zheng1
1 School of Applied Physics and Material, WuYi University, Jiangmen, 529020, China
2 State Key Laboratory on Integrated Optoelectronics, College of Electronic Science and
Engineering, Jilin University, 2699 Qianjin Street, Changchun 130012, China
Supplemental materials
The Supplemental materials include: (1) the calculating process concerning the
iterative method to solve Eq.(4) and (5); (2) key characteristics of the critical coupling
line L0 for perfect absorption; (3) key characteristics of trilayer perfect absorbers with
different substrates.
1. Calculating process concerning the iterative method to solve Eq.(4) and (5)
To solve Eq. (4) and (5), d1 is set as a given value and the complex refractive
index of the Ag substrate is set as the known values presented in Fig. 1(b)). Iterative
method is then employed to numerically solve Eq. (4) and (5) with the aim of
obtaining the refractive index (n1) and the extinction coefficient (k1) of the absorbing
material required for perfect absorption. In the calculating process, n1 and k1 are first
set as initial approximate values. With these initial values as input parameters, we
employ Fresnel equations to calculate the values of R01 , R123 , 01 and  123 . These
values are then used as input parameters for Eq.(4) and (5) to calculate the better
approximate values of n1 and k1. Using these approximate values of n1 and k1 as inputs
and repeating the calculation above by finite times, we can obtain the solutions of Eq.
(4) and (5), i.e., a pair of (n1, k1) simultaneously matching Eq. (4) and (5) when d1 is
set as the given value.
a) Corresponding author. E-mail address: yongbinglong@gmail.com;
b) Corresponding author. E-mail address: shenliang@jlu.edu.cn
2. Key characteristics of the critical coupling line L0 for perfect absorption
Tab. S1 presents key characteristics of the critical coupling line L0 for bilayer
perfect absorber with a structure of absorbing layer/Ag when the wavelength of the
incident light is set as 400nm, 500nm, 600nm, 700nm and 800nm. The critical
coupling line L0 denotes refractive index (n1) and extinction coefficient (k1) of the
absorbing materials which can be used as the absorbing layer to achieve perfect
absorption. This line can be described by the linear equation k1  an1  b with a and
b being values corresponding to the wavelength (see Tab. S1).
In other words,
materials with n1 and k1 matching this equation can be used to achieve perfect
absorption.
Table S1 Key characteristics of the critical coupling line L0 for perfect absorption
wavelength(nm)
400
500
600
700
800
n1
1.5-5.5
2.0-5.5
2.0-5.5
2.0-5.5
2.0-5.5
a
0.169
0.107
0.074
0.054
0.047
b
0.272
0.365
0.423
0.461
0.462
3. Key characteristics of trilayer perfect absorbers with different substrates
Tab. S2 presents key characteristics of trilayer perfect absorbers with different
substrates such as Au, Al, Ag or DBR. The trilayer perfect absorbers are assumed to
have a structure of InSb/TiO2/substrate. d1 and d2 in the table respectively denote the
thicknesses of InSb and TiO2. The DBR is assumed to have a structure of (TiO2/SiO2)8
with the thicknesses of TiO2 and SiO2 respectively set as 50nm and 83nm. The
reflection of the DBR is shown in Fig. S1, which demonstrates that the DBR has a
high reflection of 100% at the wavelength of 500nm.
Table S2
Key characteristics of trilayer perfect absorbers with different substrates
Substrate
d1(nm)
d2(nm)
Absorption of InSb
Absorption of
(%)
Substrate (%)
Au
2.4
19.5
48.3
51.7
Al
4.4
32.4
89.0
11.0
Ag
4.7
21.7
94.6
5.4
DBR
4.9
44.0
100
0
Fig. S1 Reflection spectrum for the DBR with a structure of (TiO2/SiO2)8