Supplementary Information
An all-optical modulator based on a stereo graphene-microfiber
structure
Jin-hui Chen, Bi-cai Zheng, Guang-hao Shao, Shi-jun Ge, Fei Xu,* and Yan-qing Lu*
This Supplementary Information includes:
1.
Small signal amplification model
2.
Numerical simulations and results
Supplementary References
1.
Small signal amplification model
According to the signal amplification model 1, we have:
dI ( z )

dz
0
1  I ( z) / I s
I ( z)
(1)
where I(z) is the signal intensity, γ is the amplification coefficient, and Is is the saturation intensity. By
integrating the differential equation, we have:
ln G 
I0
(G  1)   L
(2)
Is
where G is the amplification of the signal (equally regarded here as modulation depth), I0 is the original
signal intensity, and L is the effective interaction length. Because I s>>I0 in our platform, the second
term in the left-hand side of (2) can be neglected. In a graphene-waveguide-integrated structure, the
higher pump the power is, the longer the interaction length that can be expected and the higher the
modulation depth that can be achieved naturally. In this manner, on average, we can suppose that
 L  (W  Wth )
(3)
where W is the pump power, Wth is the threshold of the pump power, and α is the polynomial order. As
a result,
ln G  (W  Wth )
(4)
The fitting order we chose in Figure 3d in the main text is 1. Although higher order fitting will be much
better, they need more fitting parameters. Here we only want to give a qualitative explanation as simple
as possible and the linear fitting is sufficient to conclude and estimate the evolution roughly. It should
be noted that the linear relation between modulation depth and pump power only lasts for considerably
large pump power. When the pump power is large enough, the neglected item (second term in the
left-hand side of (2)) will play a role. As a result, the relation will become nonlinear and modulation
depth will reach a saturate value, which will be discussed in the next section.
2.
Numerical simulations and results
We also employed the finite element method (FEM) to investigate the modulation depth of our
platform numerically. The modeled graphene simulation parameters are based on Ref 2. Considering
the substrate-induced doping in graphene, we set the initial Fermi level (Ef) of graphene as 0.02 eV. In
our simulations, we changed the Fermi level of graphene to simulate the optical-pump-induced
saturation absorption in our platform. By utilizing the experimental parameters mentioned in the main
text, we obtained the results listed in Table 1 for the 3-μm-diameter microfiber. From Table 1, we can
find that the maximized modulation depth for the HAM and LAM are 1.94 dB/mm and 0.45 dB/mm,
respectively. As a result, in our experiment, the maximized modulation depth should be 23 dB and 5.4
dB (L ~ 12 mm) for the HAM and LAM, respectively. For the 2.5-μm-diameter microfiber, we found
that the modulation depth should be 42 dB and 9.6 dB (L ~ 12 mm) for the HAM and LAM,
respectively. As we can see that thanks to the ultra-long light-graphene interaction length and large
evanescent field of the microfiber, the modulation depth of our platform is very sensitive to the
structure parameters, such as the diameter of the microfiber.
Table 1. Propagation loss for eigen-modes of the 3μm-diameter microfiber
Eigen mode loss (dB/mm)
HAM
LAM
Fermi level 0.02 eV
1.96
0.53
Fermi level 0.5 eV
0.018
0.081
MDs
1.94
0.45
Supplementary References
1.Yariv A, Yeh P. Optical Electronics in Modern Communications (Sixth Edition). Oxford University
Press; 2007; 230-232.
2. Bao Q, Zhang H, Wang B, Ni Z, Lim CHYX, Wang Y, et al. Broadband graphene polarizer. Nat.
Photonics 2011; 5(7): 411-415.