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Supplementary Information
Sub-cycle control of terahertz waveform polarization
using all-optically induced transient metamaterials
N. Kamaraju, Andrea Rubano, Linke Jian, Surajit Saha, T. Venkatesan,
Jan Nötzold, R. Kramer Campen, Martin Wolf, Tobias Kampfrath
S1. Characterization of homogeneously illuminated Si. The refractive index of the Si slab
(thickness d=10µm) following homogeneous illumination with a femtosecond pump pulse
(wavelength 800nm) is determined by a THz probe pulse (see Figure 1a) as a function of the
pump-probe delay. Since the THz dielectric properties change in a virtually step-like manner, it
is sufficient to measure transmitted THz waveforms 60ps before [EUE(t)] and 60ps after [E(t)]
photoexcitation (see Figure 2). In the frequency domain, these fields are related to the incident
waveform Einc by1
𝐸(𝜔)
4𝑛exp(i𝜔𝑛𝑑/𝑐)
=
2
(𝑛 + 1) − (𝑛 − 1)2 exp(2i𝜔𝑛𝑑/𝑐)
𝐸inc (𝜔)
(S1)
where  denotes frequency, and n is the instantaneous refractive index of the slab, excited or
unexcited. Equation (S1) takes all multiple reflections at the air-Si and Si-air interfaces into
account. Einc is determined by Equation (S1) using the measured EUE and n=n where n=3.42 is
the refractive index of unexcited Si (ref. 2). Numerically solving Equation (S1) yields the
permittivity n2(), which is shown in Fig. S1a and S1b for two different pump fluences. It can be
described well by the Drude model3
𝑛2 (𝑥, 𝑦, 𝜔) = (1 −
𝜔p2
) 𝑛2
𝜔 2 + i𝜔/𝜏 ∞
(S2)
2 1/2
where  is the mean collision time of the charge carriers, p=(𝑁𝑒 2 𝑐𝑍0 ⁄𝑚∗ 𝑛∞
)
is the
screened plasma frequency determined by carrier density N, elementary charge e, light velocity
c, free-space impedance Z0=377, the effective mass m* of the Si conduction electrons
(approximately 0.26 times the free-electron mass4). Note that for plasma frequencies p larger
than the frequency  of the probing THz radiation, n has a large imaginary part, implying strong
attenuation and metallic behavior.
The carrier density extracted from the Drude fits (see Fig. S1a and S1b) is plotted in Fig. S1c as
a function of the incident pump fluence. In agreement with ref. 5, we find a virtually linear
dependence, indicating that the electron effective mass is constant over the full range of carrier
densities induced and that the charge-carrier generation is dominated by one-photon pump
absorption.
S2. Si reflectivity. The refractive index obtained from transmission-type measurements (Fig. S1)
allows us to calculate the reflectance of the Si slab before and after photoexcitation. The results
are shown in Fig. S2.
Supplementary Figure S1 THz permittivity n2 of photoexcited Si. (a) At an incident pump fluence of
0.06mJ cm-2, the negative Re n2 indicates metallic behavior. A Drude fit [Equation (S2)] yields a screened
plasma frequency of p/22.1THz and a carrier scattering time of 115fs, which allows us to estimate a
conduction-electron density of N0.151018cm-3. (b) Same as b, but for a pump fluence of 0.45mJ cm-2,
resulting in p/25.8THz and the carrier scattering time 95fs, implying a larger conduction-electron
density of N1.231018 cm-3. (c) Carrier density N as extracted from the Drude fits (see a, b) as a function
of incident pump fluence along with a linear fit.
Supplementary Figure S2 THz reflectance of the unexcited and homogeneously excited Si slab. Data
(symbols and dashed line) are inferred from transmission-type measurements (Fig. S1b), and the solid
line is obtained for Drude parameterization of the Si refractive index (solid line in Fig. S1b). The large
reflectance found is a hallmark of metallic behavior.
S3. Pump-pulse absorption pattern. The shadow mask (at z4f) has a periodic spatial
structure and so diffracts the pump beam into several discrete directions. Two best-form lenses
(at z3f and zf) collect and refocus the diffracted waves onto the Si slab surface (at z). To
calculate the density N(x,y,z) of the pump-induced electron-hole in the Si, we sum up all
diffraction orders of the (scalar) pump electric field6,
𝐸(𝑢, 𝑧, 𝜔) = ∑ 𝐴(𝑘𝑢 )exp(i𝑘𝑢 𝑢) [ei𝑘𝑧 𝑧 + 𝑟ei𝑘𝑧 (2𝑑−𝑧) ].
(S3)
𝑘𝑢
Here, u is the in-plane coordinate perpendicular () to the orientation of the wires, and z is the
coordinate perpendicular to the Si slab (see Fig. 1a). (The coordinate axis perpendicular to z and
u is ignored as the system is translation-invariant along this direction.) Furthermore, 2 is a
frequency from the pump pulse spectrum, and A(ku) is the amplitude of each Fourier component
with wave vector ku arising from diffraction by the shadow mask, as specified below by Equation
(S4). Finally, the two terms in the square bracket describe the forward- and backwardpropagating wave inside the Si slab, respectively, where r is the Fresnel reflection coefficient of
the Si-air interface6, and kz is (2n2/c2kx2)1/2. The amplitude of each diffracted wave is given by
𝐴(𝑘𝑢 ) = 𝐸inc (𝜔) ∙ ℱ[𝑇(𝑢)](𝑘𝑢 ) ∙ 𝐹(𝑘𝑢 ) ∙ 𝑡(𝑘𝑢 )
(S4)
where Einc() is the amplitude spectrum of the plane pump wave normally incident onto the
shadow mask, and T(u) is the spatial transmission function of our mask (square-wave-type,
assuming values of 0 or 1 for uncoated and Cr-coated regions), and  denotes a Fourier
transformation. Since T is periodic with period length a, only wave vectors kum2a come into
play, where the integer m is the order of each diffracted wave. For the square-wave-type T of our
mask, we have ℱ[𝐸inc (𝜔)𝑇(𝑢)](𝑘𝑢 ) ∝ sinc(𝜋𝑚𝑏/𝑎) where sincxsinxx, and ba is the width
of each shadowing stripe of the mask; The filter function F quantifies the finite aperture size of
the two lenses, and in our setup, F is 1 for |m|3 and 0 otherwise. Finally, t(ku) is the Fresnel
transmission coefficient of the air-Si interface6.
We are primarily interested in the pump-induced electron-hole density N(u,z) which is
proportional to the locally absorbed power W(u,z) of the pump pulse, that is,
𝑊(𝑢, 𝑧) ∝ ∫ d𝑡 𝐸 2 (𝑢, 𝑧, 𝑡) ∝ ∫ d𝜔 |𝐸(𝑢, 𝑧, 𝜔)|2
(S5).
We calculate W using Equation (S5) for the parameters of our experiment (Einc centered at
800nm, bandwidth 25nm, a=30µm, b=a/2), and the result is shown in Fig. S3a. One recognizes
the square-type incident beam profile that is seen to persist over the whole depth of the Si slab,
without diffraction along the transverse direction. We observe some z-dependence that arises
from pump absorption (Lambert-Beer-type decrease of overall intensity along z) and interference
of the forward-propagating and reflected pump pulse at the end of the slab (z~d=10µm) The
longitudinal extent of this pattern is determined by the coherence time of the pump pulse (~35fs)
times the phase velocity of Si (~c/3.7 at 800nm).
For comparison, Fig. S3b displays W for the case of homogeneous excitation (T=1), and we
observe the same longitudinal structure as in Fig. S3a. Note that the z-dependence of the pump
excitation pattern W(u,z) and thus refractive index n(u,z) does not prohibit us to treat the pumped
Si slab as a homogeneous medium along z: since the slab thickness (10µm) is much smaller than
the THz wavelength within the slab (~100µm), the probing THz pulse perceives the slab as an
effective medium with a z-independent effective refractive index. This homogenization along z is
independent of the pump pattern used because W(u,z)/W(u,0) is independent of u [see Fig. S3 and
Equation (S3)] such that the z-homogenized refractive index n(u) is fully determined by the
locally incident pump intensity I(u). Therefore, the relationship of n vs I can be measured
conveniently using homogeneous pump illumination (see Section S1).
a
b
Supplementary Figure S3 Color plot of the calculated absorbed pump energy density W(u,z) for (a)
excitation with a pump beam shaped by a shadow mask with a periodic array of intransparent Cr stripes
(period length a30µm, stripe width ba/2). (b) W(u,z) in case of a homogeneous pump beam profile.
Note the identical modulation along z in a and b. In both panels, W is normalized to its respective
maximum value.
S4. Calculated WGP performance under different conditions. Using our WGP model (see
main text), we calculate the WGP transmittance assuming a perfectly square-type excitation
profile (stripes of unexcited and homogeneously illuminated Si) at temperatures of 300K (see Fig.
S1a and S1b) and 30K (ref. 5). Results are shown in Fig. S4.
Supplementary Figure S4 Calculated WGP transmittance assuming a perfectly square-type excitation
profile. (a) Power transmittance and (b) spectral phase shift at 300K, that is, with homogeneously excited
stripes having the permittivity of Fig. S1b and the other areas being completely unexcited. (c), (d), same
as a, b but at a temperature of 30K. The plasma frequency of the excited rectangular stripes is
p/25.8THz as in a, b, but the Drude relaxation time of 2.1ps (ref. 5) is more than one order
magnitude longer than the 95fs in a, b. The transmittance peak at 2.5THz arises from a Fabry Perot
resonance in the Si slab. Normalized transmittance values above 100% indicate that the transmittance
(referenced to vacuum) of the excited slab is higher than that of the unexcited slab.
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