Skirting Terahertz Waves in a Photo-Excited
Nanoslit Structure
Mostafa Shalaby1,4*, Justyna Fabiańska2, Marco Peccianti1,5, Yavuz Ozturk1, Francois Vidal1,
Hans Sigg3, Roberto Morandotti1, and Thomas Feurer2*
1
INRS-EMT, Varennes, Quebec, J3X 1S2, Canada
2
Institute of Applied Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
3
LMN, Paul Scherrer Institut, Villigen 5232, Switzerland
4
SwissFEL, Paul Scherrer Institut, Villigen 5232, Switzerland
5
Dept. of Physics and Astronomy Pevensey Building II, 3A8 University of Sussex
*mostafa.shalaby@psi.ch, thomas.feurer@iap.unibe.ch
Details on the simulations
The sample is a one-dimensional array of nanoslits in a thin gold layer which is deposited on a
high resistivity silicon substrate. Details on the geometry and coordinate system are given in the
main text. The nanoslit array is pumped by a femtosecond laser pulse centered at 800 nm and the
transmission of a time-delayed, single-cycle terahertz (THz) pulse is studied. The simulations are
in two spatial dimensions, i.e. in a plane (x-z) perpendicular to the slits, and are carried out using
a finite difference time domain (FDTD) [S1] and a frequency-dependent finite element method
(FEM) [S2]. The simulations shown in Fig. 2 and 3 are performed in the time domain using a
FDTD technique and those in Fig. 4 in the frequency domain using the FEM technique,
respectively. Specifically we perform the following steps:
(i)
First, we simulate the pump field distribution (at a given fluence) in the silicon
substrate.
(ii)
From this distribution, we calculate the substrate complex THz dielectric function by
assuming that the pump-induced carrier density can be described by a Drude model.
(iii)
Finally, we simulate the THz pulse propagation through the perturbed but stationary
dielectric environment and compare the simulated THz far-field spectra with the
experimental results for different pump fluences.
To simulate the spatial pump field distribution, we use the frequency-dependent properties of
gold as described by an extended Drude model [S3] and a frequency-independent refractive
index of n = (3.712+i 0.0085) for the silicon substrate [S4]. The pump field is launched from the
boundary facing the gold layer and has an amplitude of 𝐸0 . The resulting pump field distribution
in the silicon substrate is 𝐸1 (x, z), and thus, the ratio 𝐸1 (x, z)/𝐸0 accounts for Fresnel losses at
interfaces, absorption in the silicon substrate, as well as field enhancement in the nanoslits.
From the pump field distribution 𝐸1 (x, z)/𝐸0 we calculate the local carrier density in the silicon
substrate
𝑁(x, z) =
𝐸 (x,z)
𝛼0 ∙𝐹𝑜 ∙| 1
|
𝐸0
ℏ∙𝜔0
2
,
(1)
assuming that each absorbed pump photon creates one electron hole pair [S5]. Here, 𝛼0 is the
absorption coefficient of the pump in silicon, 𝐹0 the incident pump fluence, and 𝜔0 the frequency
of a pump photon. The carrier lifetime is assumed to be much larger than the time delay between
the optical pump pulse and the THz pulse. The pump-induced carrier density modifies the
dielectric properties of the silicon substrate, which we describe by way of the homogeneous THz
dielectric constant of the unpumped silicon, i.e. 𝜀∞ , and a pump-induced spatially dependent
Drude conductivity
𝜀(x, z, 𝜔) = 𝜀∞ − 𝜀
𝜎(x, z, 𝜔) =
𝑖
0 ∙𝜔
2 (x,z)
𝜀0 ∙𝜔p
𝑖𝜔+𝛾
.
𝜎(x, z, 𝜔),
(1)
(2)
The plasma frequency 𝜔p (x, z) is given by
𝜔p2 (x, z) =
𝑁(x,z)∙𝑒 2
𝜀0 ∙𝑚∗
(3)
where 𝑒 is the electron charge, 𝜀0 the vacuum permittivity, and 𝑚∗ the effective electron mass
(𝑚∗ = 0.26 𝑚𝑒 for silicon). The damping constant is 𝛾 = 2𝜋 × 0.838 THz [S6]. The frequencydependent properties of gold at THz frequencies are modeled by a Drude conductivity [S7].
Finally, we launch a THz field from the boundary facing the nanoslits and simulate its spatial
distribution in the entire structure for different pump fluences. The field enhancement factor at
THz frequencies is calculated via 𝐹𝐸(𝜔) = 𝐸𝑔𝑎𝑝 (𝜔) / 𝐸𝑖𝑛𝑐 (𝜔) , where 𝐸𝑔𝑎𝑝 (𝜔) is the
amplitude of the THz electric field at the center of a gap and 𝐸𝑖𝑛𝑐 (𝜔) is the amplitude of the
incident THz field. The far field transmission coefficient is obtained from the relation 𝑡(𝜔) =
𝐸𝑜𝑢𝑡 (𝜔) / 𝐸𝑖𝑛𝑐 (𝜔) where 𝐸𝑜𝑢𝑡 (𝜔) is the amplitude of the transmitted THz field. In Fig. 4 of the
manuscript the simulated results for the THz transmission are compared to the experimental
results.
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