Supplementary Material
Perfect Electromagnetic Absorption at One-Atom-Thick Scale
Sucheng Li,1 Qian Duan,1 Shuo Li,1 Qiang Yin,1
Weixin Lu,1 Liang Li,1 Bangming Gu,2 Bo Hou,1 and Weijia Wen3
1
College of Physics, Optoelectronics and Energy & Collaborative Innovation Center
of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
2
Institute of Applied Physics, Zhejiang Wanli University,
8 Qianhu South Road, Ningbo 315101, China
3
Department of Physics, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong
1. PET substrate
The PET substrate has the 0.16mm thickness, and the nonmagnetic nature. Its relative
permittivity is measured to be εr=3.4+0i (i being the imaginary unit) from the
microwave spectra, as shown in Fig. S1, using the Agilent 85071 material
measurement software. Note that the optical path of the substrate is about one percent
of wavelength at 10GHz.
1.0
R
T
A
4
3
0.6
r
R/T/A (100%)
5
0.8
Real Part
Imaginary Part
2
0.4
1
0
0.2
7
8
9
10
11
12
13
Frequency (GHz)
0.0
7
8
9
10
11
Frequency (GHz)
1
12
13
Figure S1. The measured reflectance, transmittance and absorbance of the bare PET
substrate. The microwave measurement was done with ordinary incidence and the
same operative procedure as for the results in Fig. 2(b). The absorbance was found
less than 0.2% over the X-band, and was overlapping with the reflectance. The inset
shows the relative permittivity with average value 3.4+0i.
2. Theoretical calculation
Wave 1
Wave 2
0
t1
r2
t2
r1
Figure S2. The schematic diagram for theoretical calculation. The thin (grey) layer
represents the graphene, and the thick (green) layer denotes the PET substrate.
We consider two coherent TE-polarized plane waves are incident oppositely upon the
graphene/PET structure in free space at the oblique angle θ0. The reflection and
transmission problem can be solved by the classical electromagnetic theory. For wave
1, the reflection and transmission coefficients, r1 and t1, are expressed as:
i (1 '  1 / 1 ' )sin1cos 2  i ( 2 '  1 /  2 ' )cos1sin 2  (1 '/  2 '   2 '/ 1 ' )sin1sin 2
i (1 '  1 / 1 ' )sin1cos 2  i (2 '  1 /  2 ' )cos1sin 2  (1 '/  2 '   2 '/ 1 ' )sin1sin 2  2cos1cos 2 ,
2
t1 
i (1 '  1 / 1 ' )sin1cos 2  i ( 2 '  1 /  2 ' )cos1sin 2  (1 '/ 2 '  2 '/ 1 ' )sin1sin 2  2cos1cos 2
r1 
(S1)
where  1  n1k 0 h cos 1 ,  2  n2 k 0 d cos  2 , 1 ' 
Z / cos  2
Z1 / cos 1
, and  2 '  2
.
Z 0 / cos  0
Z 0 / cos  0
Z 0 , Z1 and Z 2 are the wave impedance in free space, graphene layer and PET
substrate, respectively; 0 , 1 and  2 are the incidence angle, refractive angle in
2
graphene layer, and refractive angle in PET substrate, respectively; k0 is the vacuum
wave vector; n1 ( n2   r ) is the refractive index of the graphene (substrate); h (d) is
the thickness of the graphene (substrate).
Because the graphene is atomically thin and has the good conductivity, σ, at the
radio wave and microwave regime, Eq. S1 can be simplified within the two
approximation conditions, h<<λ and (σ/ε0)/ω>>1, as:
i ( 2 '1 /  2 ' ) sin  2  ( Z 0 / cos  0 / Rs ) cos  2  i 2 ' ( Z 0 / cos  0 / Rs ) sin  2
i ( 2 '1 /  2 ' ) sin  2  ( Z 0 / cos  0 / Rs ) cos  2  i 2 ' ( Z 0 / cos  0 / Rs ) sin  2  2 cos  2 ,
2
t1 
i ( 2 '1 /  2 ' ) sin  2  ( Z 0 / cos  0 / Rs ) cos  2  i 2 ' ( Z 0 / cos  0 / Rs ) sin  2  2 cos  2
r1 
where the expression, n1  i
(S2)
 /  0 , has been applied for the graphene, and the sheet

resistance, Rs, is defined as 1/(σh). ε0 is the permittivity of free space, and ω is the
radian frequency.
Due to the fact that the optical path, δ2, of the substrate is negligibly short within
our measuring frequencies, Eq. S2 can be further approximated as:
r1  
t1 
(0.5Z 0 / cos  0 / Rs )
(0.5Z 0 / cos  0 / Rs )  1 .
(S3)
1
(0.5Z 0 / cos  0 / Rs )  1
Equation S3 is just the formula for a freestanding graphene, which justifies that the
dielectric substrate has negligible EM influence on our investigation. The absorbance
at single beam illumination (Eq. 1 in main text) is obtained according to Eq. S3 and
the relation A  1 | r1 |2  | t1 |2 .
Likewise, we can also calculate r2 and t2 for wave 2. It should be noted that for
coherent illumination, four partial waves, r1, t1, r2 and t2, are received as reflected
signal by the network analyzer, since all waves return to port 1 in the measurement.
Finally, the coherent absorbance is calculated as:
A  1  | 0 .5 (r 1 t 2)
0
. 5r( 22 t. 1| )
(S4)
Casting Eq. S3 into Eq. S4 with r2=r1 and t2=t1 will give the expression, Eq. 2, in
main text. A full calculation including the PET substrate gives the result consistent
3
with the freestanding graphene case, as shown in Fig. S3.
Note that though the above derivation is for the plane wave incidence where θ0 is
a geometrical constant, all expressions will apply to TE10 waveguide mode if making
the change, θ0=acos(β/k0), where β is the propagation constant of TE10 mode.
Absorbance (100%)
1.00
0.95
0.90
Graphene+PET substrate
Freestanding graphene
0.85
0.80
0.2
0.0
7
8
9
10
11
Frequency (GHz)
12
13
Figure S3. The comparison of the calculated coherent absorption between the
graphene/PET structure (black solid line) and freestanding graphene (red solid line).
Note that there is almost no difference for the two curves.
3. Phase modulation
We have also tested the case of phase modulation for the coherent illumination. The
delay phase is introduced to one beam via inserting a section of waveguide, as
illustrated in Fig. S4(a). The measured and calculated results, as well as the delay
phase, are plotted in Fig. S4(b), and exhibit the desired absorbance modulation which
is in accordance with the value of delay phase.
4
(b)
section delay
8
0.8
6
0.6
4
0.4
2
0.2
Delay phase ()
sample
1.0
Absorbance (100%)
(a)
0
0.0
7
8
9
10
11
12
13
Frequency (GHz)
Figure S4. (a) The photo of the setup, where the inset shows the power divider. (b)
The measured (symbols) and calculated (solid line) coherent absorbance and the delay
phase (dash line) βL (β being the propagation constant and L=14.5mm being the delay
length). The calculation of coherent absorption can be done through adding the phasor
term, eiL , to partial waves, t1 & t2, and ei 2 L to r1 in Eq. S4.
5