APPLIED PHYSICS LETTERS 108, 031601 (2016)
Broadband metasurfaces enabling arbitrarily large delay-bandwidth products
Vincent Ginis,1,a) Philippe Tassin,2 Thomas Koschny,3 and Costas M. Soukoulis3,4
1
Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium
Department of Applied Physics, Chalmers University, SE-412 96 G€
oteborg, Sweden
3
Ames Laboratory—U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA
4
Institute of Electronic Structure and Lasers (IESL), FORTH, GR-71110 Heraklion, Crete, Greece
2
(Received 6 November 2015; accepted 5 January 2016; published online 19 January 2016)
Metasurfaces allow for advanced manipulation of optical signals by imposing phase discontinuities
across flat interfaces. Unfortunately, these phase shifts remain restricted to values between 0 and
2p, limiting the delay-bandwidth product of such sheets. Here, we develop an analytical tool to
design metasurfaces that mimic three-dimensional materials of arbitrary thickness. In this way, we
demonstrate how large phase discontinuities can be realized by combining several subwavelength
Lorentzian resonances in the unit cell of the surface. Our methods open up the temporal response
of metasurfaces and may lead to the construction of metasurfaces with a plethora of new optical
C 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4939979]
functions. V
A sheet metamaterial or metasurface is a two-dimensional
patterned structure whose electromagnetic properties are determined by subwavelength resonant elements. These singlelayer extensions of bulky metamaterials1–5 have revolutionized
the manipulation of optical signals as they allow for controlling
light in a deep subwavelength regime. Starting from the observation that the laws of reflection and refraction can be generalized to include phase jumps at the interface,6 a new line of
research has emerged around artificial surfaces that reflect and
refract light in unconventional ways.7–9 This approach has led
to a class of surfaces that introduce nontrivial phase discontinuities and may replace many conventional bulky devices,
whose functionality relies on the specific phase accumulation
of light. Indeed, metasurfaces with spatially varying phase discontinuities allow for ultra-thin analogues of traditional lenses
and axicons,10 focusing mirrors,11 compact polarimeters,12 and
anomalous reflection and refraction.13,14 In addition, several
groups have demonstrated perfect absorption using metasurfaces that exhibit a nontrivial phase shift in reflection.15–17 It is
also possible to manipulate the polarization of light using surfaces with different spectral response along two orthogonal directions.18 In this way, anisotropic metasurfaces can implement
wave plates, vortex plates, chromatic polarizers, and surface
plasmon polariton couplers.19–23 Two generalizations have
considerably increased the efficiency of metasurfaces.
Electromagnetic metasurfaces that exhibit both an electrical and
a magnetic phase discontinuity,24 and meta-transmit-arrays that
consist of a sequence of metasurfaces25 may facilitate highly efficient, reflectionless manipulation of the polarization, phase and
amplitude of incoming pulses. So far, the overall phase shift that
is imposed by these metasurfaces remains bounded between 0
and p for electric response sheets and between 0 and 2p for
electromagnetic sheets. In addition, their bandwidth is normally
restricted due to their explicit resonant operation.
In this letter, we demonstrate that it is possible to expand
the range of phase discontinuities to much larger values,
opening up the temporal response of metasurfaces, and we
a)
Author to whom correspondence should be addressed. Email: vincent.ginis@
vub.ac.be
0003-6951/2016/108(3)/031601/4/$30.00
discuss the application of such metasurfaces for broadband
pulse delay. In addition, we demonstrate that it is possible to
achieve a metamaterial in an a priori determined frequency
band, i.e., the frequency band of operation is not restricted
by the resonant properties associated with the desired
response function. We start our analysis from the observation
that there is a one-to-one correspondence between the electric (rse ) and magnetic sheet (rsm ) conductivities of metasurfaces and their scattering amplitudes24,26
rse ðx; k? Þ ¼
2
1 Rðx; k? Þ T ðx; k? Þ
;
nðx; k? Þ 1 þ Rðx; k? Þ þ T ðx; k? Þ
rsm ðx; k? Þ ¼ 2nðx; k? Þ
1 þ Rðx; k? Þ T ðx; k? Þ
;
1 Rðx; k? Þ þ T ðx; k? Þ
in which Rðx; k? Þ and Tðx; k? Þ are the reflection and transmission amplitudes, respectively. The wave impedance nðx; k? Þ
equals xl0 =k? . For normal incident waves, this simplifies to the
free-space impedance n 377 X. To obtain the sheet conductivities that impose an arbitrarily large phase shift on a broadband incoming pulse, we can insert the scattering parameters of
a dispersionless matched slab of thickness d and refractive index
n, i.e., RðxÞ ¼ 0 and TðxÞ ¼ A exp ½ixðn 1Þd=c; where the
parameter A accounts for a possible amplitude decrease in the
transmitted field (A < 1) and where the reference planes of the
incoming (outgoing) waves coincide with the leftmost (rightmost) boundary of the slab. Writing the corresponding fraction
as a series expansion, we arrive at
1
X
nrse ðxÞ rsm ðxÞ
ð AÞm expð ixmt0 Þ;
¼
¼1þ2
2
2n
m¼1
(1)
where t0 ¼ ðn 1Þd=c is the difference between the phase
delay time through the matched slab and the phase delay
time through a vacuum region of the same thickness. These
conductivities are causal periodic functions of x with periodicity p=t0 . In Fig. 1, we plot the first two periods of this
function. The resonances of the conductivities occur at those
frequencies for which the optical path difference between
108, 031601-1
C 2016 AIP Publishing LLC
V
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Ginis et al.
Appl. Phys. Lett. 108, 031601 (2016)
FIG. 1. The first two periods of the real and the imaginary parts of the surface conductivities as a function of frequency for several values of the transmission amplitude A.
the slab and vacuum of the same thickness equals an odd
multiple of half the free-space wavelength. These are the frequencies for which a wave propagating through vacuum and
a wave propagating through the slab are in anti-phase. The
zeros of the susceptance [Im(rse )], on the other hand, correspond to the frequencies for which the aforementioned
waves are in phase with each other and for which the sheet
should only generate an amplitude decrease, in correspondence with the nonzero real part of the conductivities.
The physics behind the time delay of this metasurface
can be understood by evaluating the time-domain representation of the conductivities. Therefore, we calculate the inverse
Fourier transform of Eq. (1), which yields
1
X
nrse ðtÞ rsm ðtÞ
ð AÞm dðt mt0 Þ:
¼
¼ dðtÞ þ 2
2
2n
m¼1
(2)
The electric and magnetic currents flowing on the metasurface are given by the convolution of these conductivity kernels with the local electromagnetic field. In Fig. 2(a), we
show that the impulse response of the currents flowing on
the surface is a Dirac pulse train whose amplitude is decreasing with a factor A at each subsequent pulse. It can be shown
that for an arbitrary incident field Ein ðtÞ; Hin ðtÞ the electric
and magnetic sheet currents are given by (see the supplementary material)27
njse ðtÞ ¼ Ein ðtÞ AEin ðt t0 Þ;
The highly singular nature of the impulse response of
the conductivities is a result of the fact that this sheet mimics
a nondispersive slab that adds a propagation phase to the
incident waves of all frequencies. However, in any physical
realization of this surface, there exists an upper limit on the
frequency for which this assumption remains valid.
Furthermore, only specific types of resonant behavior are
available in nature. Therefore, we investigated whether the
spectrum defined in Eq. (1) can be approximated by a sum of
Lorentzian resonances, which can be easily obtained with
subwavelength meta-atoms. Using a partial fraction decomposition at the poles pm of Eq. (1), we can show that the
desired spectrum of conductivities can be written as the conductivities corresponding to a sum of Lorentzians in the
susceptibility27
þ1
X
m¼0 jpm j
4 ix=t0
2
m¼0 jpm j
As a result, the outgoing field is a replica of the incident
one modulated with the appropriate amplitude A and phase
shift expð ixt0 Þ
Eout ðtÞ ¼ AEin ðt t0 Þ;
(3)
Hout ðtÞ ¼ AHin ðt t0 Þ;
(4)
þ 2 iIm ðpm Þx x2
;
(5)
with an extra residual term
þ1
X
n1 jsm ðtÞ ¼ Hin ðtÞ AHin ðt t0 Þ:
as shown in Fig. 2(b).
FIG. 2. (a) The time-domain representation of the ideal surface conductivities is a Dirac pulse train whose amplitude decays with a factor A at each
consecutive pulse. The surface currents on the sheet are defined by the convolution of the local field on the surface with these conductivities. (b) For an
arbitrary incident pulse (red), the outgoing pulse is an exact replica of the
incident one with a shifted phase and amplitude (blue).
4Im ðpm Þ=t0
2
þ 2 iIm ðpm Þx x2
;
where the mth pole of the transmission function is located at
pm ¼ ð2m þ 1Þp=t0 i log ð1=aÞ=t0 . The residual term,
whose relative magnitude rapidly decreases at nonzero frequencies, accounts for the fact that the real part of Eq. (1)
has a nonzero contribution at DC. Apart from the DC discrepancy, our technique also works perfectly in the non-ideal
case of lossy oscillators. In other words, the nonzero background in the real part of the conductivities in between the
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Ginis et al.
resonances can be fitted arbitrarily good using a specific sum
of Lorentzians oscillators.
In Fig. 3, we evaluate the transmission through a metasurface approximating the ideal sheet conductivities defined in
Eq. (1) by a finite sum of Lorentzians, corresponding to the
first N terms of Eq. (5). The impulse response of these realistic
conductivities is no longer a singular Dirac train, but instead
an oscillatory function whose extrema decrease as a function
of time, as shown in Fig. 3(a). Consequently, there exists an
upper limit on the bandwidth of operation for any given time
delay. More generally, the delay-bandwidth product depends
on the number of Lorentzian resonances that is implemented
in the surface. By truncating the series of resonances, the
FIG. 3. (a) The impulse response of the sheet conductivities (A ¼ 0.9) of the
approximated sheets, incorporating 1, 2, or 3 Lorentzian resonances. (b) The
numerically calculated cross-correlation between the pulse transmitted
through the approximated sheet and the ideal amplitude-shifted (A ¼ 0.9)
and phase-shifted pulse as a function of the number of Lorentzian resonances. This incident wave is a broadband Gaussian pulse (pulse width
Dt ¼ 5=x0 ). (c) The extracted minimal number of Lorentzians versus delaybandwidth product requiring a 95% cross-correlation level between the
pulses transmitted through the ideal and the realistic sheets.
Appl. Phys. Lett. 108, 031601 (2016)
shape of the delayed pulse will not be preserved when the
spectral content of the incident pulse is too broad. To investigate this effect, we numerically simulate the transmission of a
Gaussian pulse through the metasurface and compare this
delayed pulse with the transmitted pulse of the ideal system,
given in Eqs. (3) and (4), as a function of the resulting time
delay t0. The pulse distortion is quantified by the crosscorrelation between both pulses. In Fig. 3(b), we plot this
cross-correlation for several surfaces implementing a different
number of Lorentzian resonances. As expected, the crosscorrelation decreases as the delay-bandwidth product
increases and the gradient of this curve highly depends on the
number of resonances. From this figure, we can extract the
minimal number of resonances that must be implemented for
a given delay-bandwidth product by imposing a crosscorrelation of at least 95% between the transmitted pulses
through the ideal and the approximated surface. This relation,
shown in Fig. 3(c), demonstrates that there is no fundamental
limit on the delay-bandwidth product of a matched surface.
Our approach thus allows to design a metasurface with an
arbitrarily large delay-bandwidth product. In contrast with
most other metamaterials, we can also choose the bandwidth
at the design stage. The bandwidth is not fundamentally limited by the resonant response. A surface with 4 resonances,
e.g., outperforms traditional metamaterial slow-light implementations by an order of magnitude and approaches the
delay-bandwidth products of atomic vapors.28–33
FIG. 4. (a) The numerically retrieved transmission parameters of a metasurface in which the macroscopic conductivities of one Lorentzian pair are
implemented with electric and magnetic dipoles with Lorentzian response
(setup shown in (b)). The inset in (a) shows the correlation between incident
and transmitted pulse versus the delay-bandwidth product.
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Ginis et al.
Subsequently, we have verified these results using finite
element numerical simulations.27 We simulated an implementation of the desired conductivities approximated by the first
Lorentzian in the electric and in the magnetic response. The
resonance is achieved through electric and magnetic point
dipoles with a microscopic Lorentzian response. This
response can be implemented with lumped elements up to
microwave frequencies, with standard metallic metamaterials
up to infrared frequencies1–4 and with nano circuit elements at
optical frequencies.34,35 The results in Fig. 4 show the numerically retrieved values of the magnitude of the transmittance
and group delay of this metasurface. We extracted the timedelay-bandwidth product of these parameters by calculating
the cross-correlation between the incident and transmitted
pulse as a function of the incident pulse bandwidth and found
agreement (DxDsg ¼ 3:0560:06) with the analytical findings
from Fig. 3 (DxDsg ¼ 3:04). The design procedure decouples
the losses, determined by the transmission amplitude A, from
the delay-bandwidth product, determined by the number of
Lorentzian oscillators and their periodicity p=t0 in the frequency domain.
In conclusion, we have demonstrated that thin-sheet
metamaterials allow for broadband phase manipulation. The
phase shift experienced by an incoming pulse passing
through the sheet can become arbitrarily large. Furthermore,
we have shown that the material response to achieve this
functionality can be approximated by a sum of Lorentzian
resonances that can be implemented by meta-atoms. We
studied the dependence of the delay-bandwidth product versus the number of implemented resonances and found that
our approach dramatically extends the reach of metasurfaces
for the manipulation of electromagnetic signals, introducing
electromagnetic sheets for broadband phase manipulation.
The understanding of metamaterials as a function of multiple Lorentzian resonators may also facilitate the further development
of
nonlinear,
tunable,
and
active
metamaterials.36–38 Our methods can be further used to
achieve other optical functions with subwavelength metamaterial structures based on a finite number of superimposed
Lorentzian meta-atoms.
Work at the VUB (simulations) was supported by
BelSPO (Grant No. IAP P7-35 photonics@be) and the
Research Foundation—Flanders. Work at Ames Lab (theory)
was supported by the US Department of Energy, Office of
Basic Energy Science, Division of Materials Sciences and
Engineering under Contract No. DE-AC02- 07CH11358).
Work at FORTH (modeling) was supported by the European
Research Council under ERC Advanced Grant No. 320081
(PHOTOMETA). V.G. acknowledges support as a postdoctoral Fellow of the Research Foundation—Flanders
(FWO- Vlaanderen).
Appl. Phys. Lett. 108, 031601 (2016)
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Broadband metasurfaces enabling arbitrarily large delay-bandwidth products
Vincent Ginis,1 Philippe Tassin,2 Thomas Koschny,3 and Costas M. Soukoulis3
1)
Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel,
Belgium
2)
Department of Applied Physics, Chalmers University, SE-412 96 Göteborg,
Sweden
3)
Ames Laboratory—U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA
(Dated: 23 December 2015)
Metasurfaces allow for advanced manipulation of optical signals by imposing phase discontinuities across
flat interfaces. Unfortunately, these phase shifts remain restricted to values between 0 and 2π, limiting the
delay-bandwidth product of such sheets. Here we develop an analytical tool to design metasurfaces that
mimic three-dimensional materials of arbitrary thickness. In this way, we demonstrate how large phase
discontinuities can be realized by combining several subwavelength Lorentzian resonances in the unit cell of
the surface. Our methods open up the temporal response of metasurfaces and may lead to the construction
of metasurfaces with a plethora of new optical functions.
Eq. (S3) can now be used to calculate the transmitted
magnetic field for an arbitrary incoming magnetic field:
I. BROADBAND PHASE DELAY THROUGH A
METASURFACE
We demonstrate that an incoming pulse of arbitrary
frequency is delayed in time as it propagates across a thin
sheet with the following specific kernel for the electric and
magnetic conductivities:
∞
X
σsm (τ )
ξσse (τ )
m
(−A) δ (τ − mt0 ) .
=
= δ (τ ) + 2
2
2ξ
m=1
(S1)
The currents generated on the surface of the sheet are
found by convolving the local fields with these conductivities:
Z ∞
jse (t) =
σse (τ ) Eloc (t − τ ) dτ .
(S2)
0
∞
X
δ (τ ) + 2
(−A) Eloc (t − mt0 ) .
m=1
The local field on the surface equals the average of the
fields to the left and to the right of the surface, Eloc =
(Eleft + Eright ) /2, hence
Hright (t) = Hleft (t) −
−
2 Eleft (t) + Eright (t)
ξ
2
∞
4 X
m Eleft (t − mt0 ) + Eright (t − mt0 )
(−A)
ξ m=1
2
m
(−A) δ (τ − mt0 ) Eloc (t − τ ) dτ
m=1
Hright (t) +
∞
X
(−A)
m
Z
∞
X
m
(−A) Hright (t − mt0 )
m=1
δ (τ ) Eloc (t − τ ) dτ
0
+2
!
m
This can now be rearranged into:
!
∞
=
Eloc (t) + 2
∞
X
m=1
0
Z
2
= Hleft (t) −
ξ
= Hleft (t) − Hleft (t) − Hright (t)
∞
X
m
−2
(−A) (Hleft (t − mt0 ) + Hright (t − mt0 )) .
Substitution of Eq. (S1) into Eq. (S2) yields
ξjse (t)
2
Z ∞
=
Hright (t) = Hleft (t) − jse (t)
=−
∞
δ (τ − mt0 ) Eloc (t − τ ) dτ
∞
X
m
(−A) Hleft (t − mt0 ) .
m=1
0
m=1
The electromagnetic boundary conditions relate the discontinuity of the electric and the magnetic fields across
the sheet to the surface currents that are present on the
sheet:
The solution of this equation is given by Hright =
AHleft (t − t0 ), as can be verified by inserting this solution in the previous equation. The same reasoning holds
for the electric fields using the equation for the magnetic
sheet currents. Moreover, since the sheet is impedance
matched, we know that the total field on the left side of
the sheet equals the incident field (there is no reflected
field) and, therefore, we can conclude that:
Eleft (t) − Eright (t) = jsm (t) ,
Hleft (t) − Hright (t) = jse (t) .
Eout (t) = AEin (t − t0 ) ,
Hout (t) = AHin (t − t0 ) .
= Eloc (t) + 2
∞
X
m
(−A) Eloc (t − mt0 ) .
m=1
(S3)
(S4)
(S5)
2
Poles and zeros of σse and σsm in the complex plane
II. FITTING THE TARGET SPECTRUM TO A SUM OF
LORENTZIANS
pole
ξσse (ω)
σsm (ω)
1 − Ae iωt0
.
=
=
2
2ξ
1 + Ae iωt0
lim
ω→∞
f (ω)
ω
A = 0.9
0
A = 0.5
Im(ωt0)
We now demonstrate that this function corresponds to a
sum of Lorentzian resonances. Hereto, we use the Cauchy
residue theorem. If a function f (ω) does not diverge at
infinity:
zero
1
The desired conductivities σse and σsm that implement
a specific delay t0 and amplitude change A are given by:
A = 0.2
= 0,
then this function can be written as a sum over partial
fractions
-40
The locations of the poles are shown in Fig. S1, for several values of the outgoing amplitude A. On the same
graph, we also show the locations of the zeros, which are
lying symmetrically in between the poles. When A becomes one (the incoming amplitude equals the outgoing
amplitude), the poles and zeros coincide with the poles
and zeros of a tangent function. The residues of the poles
are calculated as
rm = lim ((ω − pm ) f (ω)) .
The residue in each pole pm equals 2 i/t0 . Let us now
consider the infinite sum over all poles:
+∞
+∞
X
X
rm
1
= 2i
p
−(2m
−
1)π
+ i log (A)
m=−∞ m
m=−∞
= 2i
=
+
This limit can be evaluated using L’Hôpital’s rule, which
yields
1
(2m
+
1)π
+ i log (A)
m=+∞
−∞
X
2 i(2m + 1)π
m=+∞
((2m + 1)π) + (log (A))
−∞
X
2 log (A)
2
2
m=+∞
((2m + 1)π) + (log (A))
2
2.
The first sum (the imaginary part) converges to zero as
for every term of the sum there exists a term with opposite sign. The second sum can be rewritten as:
2 log (A)
m=−∞
=
(ω − pm ) − it0 Ae
+ 1 − Ae
it0 Ae iωt0
1 − Ae iωt0
= lim
ω→pm
it0 Ae iωt0
2i
= .
t0
−∞
X
+∞
X
ω→pm
iωt0
40
FIG. S1. Poles and zeros of the desired conductivities for
several outgoing amplitudes, A = 0.9, A = 0.5, and A = 0.2.
in which rm is the residue of the function in the pole pm .
We now proceed by calculating the poles and residues of
the function that we want to expand. The poles of the
function coincide with the zeros of the denominator:
1
1
2πm
pm =
log −
−
it0
A
t0
i
(2m − 1)π
1
= − log
−
t0
A
t0
i
(2m − 1)π
= log (A) −
.
(S6)
t0
t0
iωt0
20
Re(ωt0)
+∞
+∞
X
X
rm
rm
+
,
f (ω) = f (0) +
p
ω − pm
m=−∞
m=−∞ m
0
-20
((2m + 1)π)2 + (log (A))
+∞
X
2
log (A) m=−∞
rm = lim
2
1
π
log(A) (2m
+ 1)
2
,
+1
ω→pm
and can be evaluated using the identity
+∞
X
1
2
m=−∞
(C(2m + 1)) + 1
=
π π
tanh
.
2C
2C
3
(a)
Therefore, we find that
Target spectrum (A = 0.9)
10
+∞
X
2 π log (A)
π log (A)
rm
=
tanh
p
log (A)
2π
2π
m=−∞ m
log (A)
= tanh
.
2
Re
Im
ξσse/2
5
We can further simplify this equation, using the identity
tanh (log(x)) =
0
-5
x2 − 1
,
x2 + 1
-10
-20
-10
0
ω/t0
and finally find
10
Re
Im
5
The partial fraction decomposition of the desired conductivities is therefore given by
ξσse/2
= −f (0).
+∞
ξσsm (ω)
2i X
1
σse (ω)
=
=
,
2ξ
2
t0 m=−∞ ω − pm
-10
+∞
ξσsm (ω)
2 i X ω + p̄m + ω − pm
σse (ω)
=
=
2ξ
2
t0 m=0 (ω − pm ) (ω + p̄m )
+∞
X
2i
2ω − 2 i Im (pm )
=
t0 m=0 (ω − pm ) (ω + p̄m )
2
|pm | + 2 i Im (pm ) ω − ω 2
.
σLorentzian (ω) =
ω2
m=0 0m
− iωfm
.
− 2 iγm ω − ω 2
−4 iω/t0
2
m=0
|pm | + 2 i Im (pm ) ω − ω 2
Residual function
Re
Im
0.5
0.0
-0.5
-1.0
-20
-10
0
ω/t0
+∞
X
10
m=0
(S8)
−4 Im (pm ) /t0
2
|pm | + 2 i Im (pm ) ω − ω 2
20
.
The parameters of the Lorentzians are given by
fm =
4
,
t0
γm = − Im (pm ) = −
,
20
with an extra residual term
Comparing Eq. (S7) with Eq. (S8), indicates that the
desired spectrum can be written as a sum of Lorentzians
+∞
X
10
(S7)
A sum of Lorentzian resonances in the conductivity can
be written as
+∞
X
0
ω/t0
FIG. S2. The desired spectrum (top), together with the fitted sum of Lorentzian resonances and the additional residual
term. The relative magnitude of the residual term rapidly
decreases for higher frequencies.
+∞
2ω − 2 i Im (pm )
2i X
=
2
t0 m=0 ω + (p̄m − pm ) ω − pm p̄m
m=0
-10
1.0
ξσse/2
where p̄m is the complex conjugate of pm . This now adds
up to
-20
(c)
+∞
σse (ω)
ξσsm (ω)
2i X
1
1
=
=
+
,
2ξ
2
t0 m=0 ω − pm
ω + p̄m
+∞
X
−4 iω/t0 − 4 Im (pm ) /t0
0
-5
where the poles pm are given by Eq. (S6). We can rewrite
this expression by adding the poles, two by two, symmetrically around the imaginary axis:
=
20
Fitted sum of Lorentzian resonances
(b)
X rm
A−1
=
p
A+1
m
m
10
ω0m = |pm | .
1
log (A),
t0
4
(a)
40
20
Im(ξσse/2)
This residual term accounts for the fact that we need a
nonzero contribution at DC. The relative magnitude of
this term rapidly decreases at higher frequencies. This
result is visualized in Fig. S2, in which we plot the desired spectrum, the Lorentzian fit and the residual term
together.
0
-20
-40
FINITE ELEMENTS SIMULATIONS
The preceding results have been verified using finiteelements numerical simulations (COMSOL multiphysics)
from which we extracted the scattering parameters of a
sheet in which electric and magnetic dipoles are positioned on a square lattice (with lattice constant a = 5 ×
µm). The simulation domain consists of a unit cell, which
is terminated by periodic boundary conditions in one direction and PEC boundaries along the symmetry axis of
the setup. The electric and magnetic dipoles are simulated as deep subwavelength line currents (length =
a/5) and circles (diameter = a/10) whose polarizabilities
correspond to a Lorentzian function. We implemented
a Lorentzian resonance for the microscopic polarizability of the electric and magnetic dipoles, as applicable
for subwavelength meta-atoms. In the macroscopic response, these resonances are renormalized as a result of
the interaction between neighboring circuits. It is important to note the generality of the methodology de-
0.8
4.0
0.6
3.5
0.4
3.0
0.2
2.5
Reflection phase
Reflection magnitude
(a) 1.0
2.0
0.0
85
90
95
100 105
Frequency (THz)
110
115
-2.0
0.8
-2.5
0.6
-3.0
0.4
-3.5
0.2
-4.0
Transmission phase
Tranmission magnitude
(b) 1.0
-4.5
0.0
85
90
95
100 105
Frequency (THz)
110
115
FIG. S3. The scattering parameters of the simulated sheet of
electric and magnetic dipoles with Lorentzian polarizabilities.
80
85
90
95 100 105 110 115 120
Frequency (THz)
80
85
90
95 100 105 110 115 120
Frequency (THz)
20
(b)
Im(σsm/(2ξ))
III.
10
0
-10
-20
FIG. S4. The electric (a) and magnetic (b) macroscopic sheet
conductivities retrieved from the scattering parameters shown
in Fig. S3.
scribed in this manuscript. Indeed, our method can be
used for many other microscopic implementations. For
instance, if the dipoles are oriented along perpendicular
orientations, the metasurface will generate the same effect for the orthogonal polarization. Since the system is
purely linear, both polarizations can be treated in linear
superposition, although such a polarization-independent
metasurface would be challenging from a fabricational
point of view. A different lattice geometry, on the other
hand, would in most cases influence the magnitude of
the material response due to a different near field coupling between adjacent cells. As a result, different lattice
geometry would change the microscopic response (polarizability) that is required for the individual dipoles.
The scattering parameters obtained for this sheet are
plotted in Fig. S3 and the associated macroscopic sheet
conductivities are shown in Fig. S4. Subsequently, we
have calculated the maximal delay-bandwidth product
of this implementation by evaluating the same criterion
as used in Fig. 2, i.e., the delay bandwidth for which
the cross-correlation between the transmitted and shifted
incident pulse drops below 0.95. Hereto, we calculated
the cross-correlation between the time-delayed incident
pulse, centered around 100 THz, and its transmitted
pulse and repeated this for several bandwidths of the incident pulse. The resulting cross-correlation versus time
delay bandwidth is shown in Fig. S5.
5
corr(Eout(t), AEin(t-t0))
1.00
FUNDING INFORMATION
0.98
0.96
0.94
0.92
0.90
1.0
2.0
3.0
ΔωΔτg
4.0
FIG. S5. The cross-correlation between the time-shifted incident and transmitted pulse versus the delay-bandwidth product, calculated from the numerically retrieved scattering parameters.
Work at the Vrije Universiteit Brussel (simulations)
was supported by BelSPO (Grant IAP P7-35 photonics@be) and the Research Foundation—Flanders (Grant
No. K2.185.13N). Work at Ames Laboratory (theory)
was supported by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering (Ames Laboratory is operated for
the U.S. Department of Energy by Iowa State University
under Contract No. DE-AC02-07CH11358). Work at
FORTH (modeling) was supported by the European Research Council under ERC Advanced Grant No. 320081
(PHOTOMETA). V. G. is a post-doctoral Fellow of the
Research Foundation—Flanders (FWO-Vlaanderen).