Floquet-Mode Solutions of Space-Time Modulated Huygens' Metasurfaces

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by  Shulabh Gupta

MANUSCRIPTDRAFT 1
Floquet-Mode Solutions of
Space-Time Modulated Huygens’ Metasurfaces
Shulabh Gupta, Tom. J. Smy and Scott A. Stewart
Abstract—A rigorous Floquet mode analysis is proposed for exoticeffectssuchasgenerationofnewharmoniccomponents
a zero thickness space-time modulated Huygens’ metasurface and Lorentz non-reciprocity [9][10]. Space-time modulated
to model and determine the strengths of the new harmonic
metasurfaces fall within the general framework of space-time
componentsofthescatteredfields.Theproposedmethodisbased
modulated mediums [11][12], which have found important
on Generalized Sheet Transition Conditions (GSTCs) treating
a metasurface as a spatial discontinuity. The metasurface is applications in acousto-optical systems for spectrum analysis,
described in terms of Lorentzian electric and magnetic surface parametricoscillatorsandamplifiers,forinstance[13][14][15].
7 susceptibilities, χee and χmm, respectively, and its resonant fre- Consequently, a combination of the wave-shaping capabil-
1 quencies are periodically modulated in both space and time.
ities of Huygens’ metasurfaces with space-time modulation
0 The unknown scattered fields are then expressed in terms of
principles, is an interesting avenue to explore for advanced
2 Floquet modes, which when used with the GSTCs, lead to a
system of field matrix equations. The resulting set of linear electromagnetic wave control, in both space and time. To
n
equations are then solved numerically to determine the total investigate into the properties of space-time modulated Huy-
a
J scattered fields. Using a finite-difference time domain (FDTD) gens’ metasurfaces, a finite-difference time-domain (FDTD)
9 solver, the proposed method is validated and confirmed for technique has recently been proposed to analyze a zero
several examples of modulation depths (∆ ) and frequencies
1 p thickness model of Huygens’ metasurfaces [16][17], based
(ω ). Finally, the computed steady-state scattered fields are
Fopurier propagated analytically, for visualization of refracted on Generalized Sheet Transition Conditions (GSTCs) [18].
]
s harmonics.Theproposedmethodissimpleandversatileandable In contrast to the other existing techniques for analyzing
c todeterminethesteady-stateresponseofaspace-timemodulated static metasurfaces in frequency domain [19][20], the FDTD
i Huygen’smetasurface,forarbitrarymodulationfrequenciesand
t analysis is naturally applicable to the problem of space-time
p depths.
modulated metasurfaces, considered herein.
o
Index Terms—Electromagnetic Metasurfaces, Electromagnetic While such numerical methods are useful to determine the
.
s Propagation,FloquetAnalysis,ExplicitFinite-Difference,Gener-
time-evolution of scattered waves from space-time modulated
c alizedSheetTransitionConditions(GSTCs),LorentzDispersions,
i metasurfaces for a given input wave, efficient determination
s Parametric Systems.
y of steady-state response of the metasurface is an equally
h important problem to solve. This issue is addressed in this
p I. INTRODUCTION work, whereby exploiting the periodic nature of the spatio-
[ Recently, there has been a strong growing interest in temporal perturbation on the metasurface, the scattered fields

1 Huygen’s metasurfaces due to their impedance matching ca- areexpressedintermsofFloquetmodes.TheHuygens’meta-
v pabilities with free-space and their versatile applications in surface is modelled using surface susceptibilities following a
1 wavefront shaping [1][2][3]. They are constructed using a physically motivated Lorentzian profile, whose resonant fre-
7
2-D array of electrically small Huygen’s sources, exhibiting quenciesareparametrizedtoemulateaspace-timemodulation
2
perfect cancellation of backscattered fields, due to optimal of the metasurface. Combined with GSTCs, the Floquet mode
5
0 interactionsoftheirelectricandmagneticdipolarmoments[4]. amplitudes are computed by solving a set of linear equations.
. Some efficient implementations of Huygens’ metasurfaces are The proposed method thus efficiently computes the steady-
1
0 based on all-dielectric resonators [5][1][6] and orthogonally state response of a zero-thickness space-time modulated Huy-
7 collocated small electric and magnetic dipoles [7][8]. gens’ metasurface, consistent with the FDTD field solutions.
1 While the majority of the work on metasurfaces has been Furthermore, integrating the proposed method with analytical
v: focussed on static (linear time invariant) metasurfaces, there Fourier methods [13], the propagation of scattered fields are
i is also a growing interest in dynamic metasurfaces, where the conveniently visualized in free-space.
X
constitutive parameters of the metasurface unit cells are con- The paper is structured as follows. Section II describes
ar trolled in real-time. An important class of such dynamic elec- the problem statement of this work, and develops the fields
tromagnetic structures is space-time modulated metasurfaces, equations governing the scattering fields from a space-time
wheretheirconstitutiveparametersareperiodicallymodulated modulated metasurface based on GSTCs and Lorentz surface
in both space and time, at comparable frequency scales to susceptibilities. Section III presents the proposed method
the input excitations. This leads to a complex interaction of based on Floquet mode expansions, forming the set of lin-
the incident wavefronts with the metasurfaces resulting in ear equations to be solved numerically. Several results are
then presented for both time-only and space-time modulated
S. Gupta, T. J. Smy and S. A. Stewart are with the Department of
metasurfaces.ConclusionsareprovidedinSec.IV,andabrief
Electronics, Carleton University, Ottawa, Ontario, Canada. Email: shu-
[email protected] summaryoftheFDTDmethodisprovidedinAppendixV,for
MANUSCRIPTDRAFT 2
the sake of self-consistency and completeness of the paper. x t
SPACE-TIME MODULATED
METASURFACE
II. SPACE-TIMEMODULATEDMETASURFACES
A. Problem Statement χee(x,t),χmm(x,t) ψ2(r,t)sin{(ω0−ωp)t}
Metasurfaces are zero thickness electromagnetic structures
ψ(r,t)=ψ0(r,t)sin(ω0t)
that act as discontinuities in space. The exact zero thickness
natureofelectromagneticstructureswasdevelopedbyIdemen ψ1(r,t)sin{ω0t}
t
in terms of Generalized Sheet Transition Conditions (GSTCs)
t
[21], which were later applied to metasurfaces [18]. Consider
a Huygens’ metasurface illustrated in Fig. 1, where the meta- ψ3(r,t)sin{(ω0+ωp)t}
surfaceliesinthex y planeatz =0,withawaveincidence z
on the left, normal t−o the surface. The wave interacts with the z=0
metasurface and produces a transmitted and a reflected wave,
t
along the forward and backward direction, respectively. This
interaction of the metasurface with the electromagnetic waves Fig.1. AgeneralIllustrationofaspace-timemodulatedHuygens’metasur-
faceundernormallyincidentCWplane-waveresultingingenerationofseveral
is described by the GSTCs, using the transverse components
frequency harmonics, refracted along different angles. ωp is the pumping
of electric and magnetic surface polarizabilities P and M, as frequency.
[22]
dP (x,t)
zˆ ∆H(x,t)= || (1a) where ωn = ω0 +nωp, and βn = βx0 +nβp with βx0 = 0
× dt
due to assumed normal input incidence. Each harmonic term
dM (x,t) ofthisexpansionwithatemporalfrequencyω ,representsan
∆E(x,t)×zˆ=µ0 |d|t , (1b) oblique forward propagating plane-wave in thne +z direction,
where and making an angle θn measured from the normal of the
metasurface, as illustrated in Fig. 1. These refraction angles
are given by
∆E=(E E E ), ∆H=(H H H ).
t− 0− r t− 0− r (cid:20) n β (cid:21)
The surface polarizabilities on the Huygens’ metasurface are θ(ωn)=sin−1 p , (4)
(1+nω /ω )k
p 0 0
related to the average fields around the metasurface and can
where k is the free-space wavenumber of the fundamental
be described in terms of scalar electric and magnetic surface 0
susceptibilities χ and χ , respectively, as1. frequency. Based on this simple physical argument, the angle
ee mm
of refraction of these newly generated frequency components
Q˜ (ω)=χ˜ E˜ (ω), (2a) fromaspace-timemodulatedmetasurface,canbedetermined.
ee av
|| However, the strengths of these harmonics (weights qn) are
M˜ (ω)=χ˜ H˜ (ω), (2b) still unknown and must be determined taking the exact elec-
mm av
|| tromagneticinteractionofthewaveswiththemetasurfaceinto
account using the GSTCs of (1).
(cid:34) (cid:35) (cid:34) (cid:35)
E˜ +E˜ +E˜ H˜ +H˜ +H˜
where E˜ = 0 t r , H˜ = 0 t r
av 2 av 2 B. Field Equations
The space-time modulated problem considered here is nat-
and Q=P/(cid:15) is the normalized electric polarizability2.
0 urally treated in the time domain. Consequently, the surface
Next, consider a space-time modulated metasurface, whose
susceptibilities must be defined in time domain as well. The
electric and magnetic susceptibilities are both a function of
most common, and causal, description of the susceptibilities
space and time, i.e. χ (x,t) and χ (x,t). Let us restrict
ee mm is in terms of Lorentz dispersion, which is also typical of
heretoaperiodicmodulationonly,withapumpingfrequency
the Huygens’ sources used to construct the metasurfaces
ω and the spatial frequency β . Due to this periodic spatio-
p p [5][23][24]. Consequently, the electric and magnetic suscepti-
temporal perturbation on the metasurface, assumed to be
blities can be expressed as
infiniteinsize,thescatteredtransmission(andreflection)fields
can be expressed in terms of Floquet series as ω2
χ˜ (ω)= ep (5a)
ee (ω2 ω2)+iα ω
(cid:88)∞ e0− e
Et(x,z =0+,t)= qnejωntejβnx, (3) ω2
n=0 χ˜mm(ω)= (ω2 ωm2)p+iα ω, (5b)
m0− m
1Assumingnorotationofpolarizationandonlyonecomponentofthefields where (ω , ω ), (ω , ω ), (α ,α ) are the electric and
alongtheprincipalaxis. e0 m0 ep mp e m
2The fields expressions with a tilde, ψ˜ denote the frequency domain magnetic resonant frequencies, plasma frequencies and loss
quantities. coefficients, respectively.
MANUSCRIPTDRAFT 3
The space-time modulation can now be introduced into III. PROPOSEDFLOQUETMODEEXPANSIONSOLUTIONS
the metasurface, by sinusoidally modulating, for instance, the A. Field Matrix Equations
resonant frequencies of the two Lorentzian susceptibilities, as
Let us consider a metasurface with a periodic space-time
[17]
modulation of the resonant frequencies following (6), and
expand the unknown reflected and transmitted fields using
ω (x,t)=ω 1+∆ cos(ω t+β x) (6a) Floquet expansion as
e0 e0 e p p
{ }
(cid:88) (cid:88)
ωm0(x,t)=ωm0{1+∆mcos(ωpt+βpx)}, (6b) Et(t)= qnejω0tejn(ωpt+βpx) = qnejω0tejnΩ, (12a)
n n
where ∆ and ∆ are the modulation depths and the spatial (cid:88) (cid:88)
e m E (t)= p ejω0tejn(ωpt+βpx) = p ejω0tejnΩ, (12b)
frequencies of the perturbation, for electric and magnetic r n n
n n
resonances, respectively. They can now be inserted into the
where a new variable Ω = (ω t + β x) is introduced for
GSTCs of (1), for a prescribed input fields, to determine the p p
compact notation. Similarly, the corresponding unknown po-
total scattered fields.
larizabilities, can also be expanded in Floquet series as
To illustrate the procedure, let us assume a normally in-
cident plane-wave (E = const.) where the corresponding
| 0| (cid:88) (cid:88)
transmitted and reflected fields, given by Q (t)= a ejω0tejnΩ, M (t)= d ejω0tejnΩ,
0 n 0 n
n n
(cid:88) (cid:88)
Q (t)= b ejω0tejnΩ, M (t)= e ejω0tejnΩ,
ˆz E (z,t) t n t n
E0(z,t)=E0ej(ω0t−k0z) yˆ, H0(z,t)= × 0 (7) n n
η (cid:88) (cid:88)
0 Q (t)= c ejω0tejnΩ, M (t)= f ejω0tejnΩ.
ˆz E (z,t) r n r n
Et(z,t)=Et(x,t)e−jk0z yˆ, Ht(z,t)= × t n n
η
0 Next, substituting the above expressions in (8) and (9) and
E (z,t) ˆz
E (z,t)=E (x,t)e+jk0z yˆ, H (z,t)= r × . re-arranging the terms, for incident, transmitted and reflected
r r r
η0 E- and H-fields, we get (10), where
Each of these field components are related to their respective
∆2ω2 ∆ ω2
polarizabilities on the metasurface (at z = 0) through the ∆ = e e0, ∆ = e e0,
Lorentz susceptibilities of (5) with (6), which can then be e1 4ωe2p e2 ωe2p
expressed in the time domain as3 [16] (cid:26)ω2 (ω +nω )2 ∆2ω2 (cid:27)
A = e0 0 p + e e0 .
e,n ω2 − ω2 2ω2
ep ep ep
dd2tQ2i +ωe20{1+∆ecos(ωpt+βpx)}2Qi =ωe2pEi(t) (8a) ∆m1 = ∆42mωωm2m2p0, ∆m2 = ∆ωmm2ωpm20,
(cid:26)ω2 (ω +nω )2 ∆2 ω2 (cid:27)
A = m0 0 p + m m0
d2M ω2 m,n ω2 − ω2 2ω2
i +ω2 1+∆ cos(ω t+β x) 2M = epE (t),4 mp mp mp
dt2 m0{ m p p } i η i B =j(ω +jnω ). (13)
0 n 0 p
(8b)
Theseequationsareobtainedbyexpressingthecosinefunction
where the subscript i = 0,t,r for incident, transmitted and
using Euler’s form and then grouping the terms of common
reflectedfields,respectively.Finally,allthescatteredfieldsare
complex exponentials. Each of these series equations, can be
related to their polarizabilities following the GSTCs of (1), as
truncated to 2N + 1 harmonic terms5, and be written in a
matrix form as shown in (11). In compact form, the matrix
dQ dQ dQ 2
0 + t + r = (E E E ), (9a) equation (11) can be written as
0 r t
dt dt dt η (cid:15) − −
0 0
[C ][V ]=[E ], (14)
n n n
dM dM dM 2
0 + t + r = (E E E ) (9b)
t r 0 which represents 8 (2N + 1) linear equations with same
dt dt dt µ − −
0 ×
number of unknowns. The sought field solutions can now be
Equations (8) and (9) thus represent two sets of field numerically computed using this matrix equation as
equations, that must be solved to determine the transmitted
andreflectedfields,Et(x,z,t)andEr(x,z,t),insteadystate, [Vn]=inv{[Cn]}[En], (15)
for a given input excitation E0 = E0ejω0t yˆ at the input of from which the corresponding p ’s and q ’s can be extracted
n n
the metasurface at z =0 .
to construct the reflected and transmitted fields, respectively,
−
following (12).
3forthelosslesscase,forsimplicity.
4withpropersignoftherighthandsideterm,dependingonincident(−), 5assumingthattheharmonicamplitudesrapidlyfallstozerowithincreasing
reflected(+)ortransmittedfields(−),accordingto(7). indexn,whichisthecaseinalltheforthcomingresults.
MANUSCRIPTDRAFT 4
(cid:88)
[∆ a +∆ a +A a +∆ a +∆ a ]ejnΩt =E (Incident E-Fields) (10)
e1 n 2 e2 n 1 e,n n e2 n+1 e1 n+2 0
− −
n
(cid:88)
[∆ b +∆ b +A b +∆ b +∆ b q ]ejnΩt =0 (Transmitted E-Fields)
e1 n 2 e2 n 1 e,n n e2 n+1 e1 n+2 n
− − −
n
(cid:88)
[∆ c +∆ c +A c +∆ c +∆ c p ]ejnΩt =0 (Reflected E-Fields)
e1 n 2 e2 n 1 e,n n e2 n+1 e1 n+2 n
− − −
n
(cid:88)[∆ d +∆ d +A d +∆ d +∆ d ]ejnΩt = E0 (Incident H-Fields)
m1 n 2 m2 n 1 m,n n m2 n+1 m1 n+2
− − −η0
n
(cid:88)
[∆ e +∆ e +A e +∆ e +∆ e b ]ejnΩt =0 (Transmitted H-Fields)
m1 n 2 m2 n 1 m,n n m2 n+1 m1 n+2 n
− − −
n
(cid:88)
[∆ f +∆ f +A f +∆ f +∆ f f ]ejnΩt =0 (Reflected H-Fields)
m1 n 2 m2 n 1 m,n n m2 n+1 m1 n+2 n
− − −
n
(cid:88)
[2cB a +2cB b +2cB c +p +q ]ejnΩt =E (GSTC Equation)
n n n n n n n n 0
n
(cid:20) (cid:21)
(cid:88) µ0Bnd + µ0Bne + µ0Bnf +p q ejnΩt = E (GSTC Equation)
n n n n n 0
2 2 2 − −
n
Cn Vn En
(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123)
 Se 0 0 0 0 0 0 0  an   I0   ... 
 0 Se 0 0 0 0 0 −I  bn   0   0 
 0 0 Se 0 0 0 −I 0  cn   0   0 
 00 00 00 S0m S0m 00 00 I/0η0  denn = −I00/η0 ,where I0 = 10 
 0 0 0 0 0 Sm −I/η0 0  fn   0   0 
 Sg/(2c0) Sg/(2c0) Sg/(2c0) 0 0 0 I I  pn   I0   . 
.
0 0 0 S S S I I q I .
g g g n 0
− −
   
A ∆ ∆ 0 0 A ∆ ∆ 0 0
e, N e2 e1 m, N m2 m1
.− . . . ··.· .− . . . ··.·
 . . . . .   . . . . . 
 . . . . .   . . . . . 
 ···   ··· 
 ∆e2 Ae, 1 ∆2 ∆e1 0   ∆m2 Am, 1 ∆2 ∆m1 0 
 − ···   − ··· 
Se = ∆e1 ∆e2 Ae,0 ∆e2 ∆e1 , Sm = ∆m1 ∆m2 Am,0 ∆m2 ∆m1 
 ···   ··· 
 0 ∆e1 ∆e2 Ae,+1 ∆e2   0 ∆m1 ∆m2 Am,+1 ∆m2 
 ···   ··· 
 .. .. .. .. ..   .. .. .. .. .. 
 . . . . .   . . . . . 
··· ···
0 0 ∆ ∆ A 0 0 ∆ ∆ A
e1 e2 e,+N m1 m2 m,+N
··· ···
S =diag B ,B , ,B ,B ,B , ,B ,B . (11)
g N N+1 1 0 1 N 1 N
{ − − ··· − ··· − }
B. Results FouriertransformsandforaclosercomparisonwiththeFDTD
method, given by
To validate the results of the proposed method, an FDTD
solver is used, which has been recently proposed in [16][17]
to directly simulate (8) and (9), for an arbitrary time-domain (cid:88)+N (cid:40) (cid:18) t (cid:19)2(cid:41)
input. A brief summary of the method is described in the Et(x,t)= qnexp cos(ωnt+βnx),
− T
0
appendix for self-consistency of the paper. In the results n= N
described next, the modulation depths of the electric and (cid:88)+−N (cid:40) (cid:18) t (cid:19)2(cid:41)
E (x,t)= p exp cos(ω t+β x).
magnetic resonant frequencies are assumed to be equal for r n n n
− T
0
simplicity, i.e. ∆ = ∆ = ∆ . Furthermore, the output n= N
e m p − (16)
fields from the proposed method are constructed using a
Gaussian envelopes6, for better conditioning of the numerical Figure 2 shows several examples of the transmission and
reflection spectrum for different modulation depths and fre-
6ThisprovidesafinitebandwidthinthetemporalFFTsasopposedtoideal quencies. While the modulation frequency ω only fixes the
p
deltafunctionswhicharedifficulttocaptureinnumericalcomputations.These
location of the newly generated harmonic components, the
Gaussianenvelopesshouldnotbeseenastheinputpulseshapes,butrather
aswindowingfunction. modulationdepth∆p controlstherelativestrengthsofthehar-
MANUSCRIPTDRAFT 5
monics, compared to the fundamental frequency of excitation. ∆ . For simplicity, and considering the practical point of
p
An excellent agreement is observed with FDTD solver, for view, a perfectly matched metasurface with χ˜ = χ˜ is
ee mm
all the cases considered, validating the proposed procedure. assumedexhibitingzeroreflections.Thespectralmapprovides
Minor discrepancies are observed between the two, which interesting and useful information about the interaction of the
are attributed to several factors. Firstly, the Floquet solution metasurface with the input fields. For example, at ω 2ω ,
p 0
≈
assumed zero losses here7, while the FDTD time-domain a substantial amplification of the fundamental frequency ω
0
results consider finite losses. Secondly, the Floquet solution is is observed, which is also the dominant spectral component
the steady state response of the metasurface, while the FDTD in the transmitted fields. This is reminiscent of wave ampli-
solvertakesintoaccountthedispersionbaseddistortionofthe fications in diverse class of mechanical and electromagnetic
input pulse, which is naturally not taken into account in the parametric systems, where specific parameters of the system
Floquet analysis in (16). are modulated at twice the excitation frequency leading to
wave instabilities [11][12].
∆p=0.05,fm=0.05f0
0.7 1
FDTD FDTD χee=χmm,∆p=0.1
0.6 Floquet 0.8 Floquet 0 20log|Et(z=0+,ω)| 10 -10 0 10
0.5
E(t)ttF{}00..34 E(t)trF{}00..46 ω0 000...246 --02100 0.5
0.2 0.2 yω/p 0.8 -30
0.10 0.8 0.9 1 1.1 1.2 0 0.8 0.9 1 1.1 1.2 frequenc 1 --5400 1
0.7 ∆p=0.1,fm=01.1f0 Modulation 11..24 --7600 1
FDTD FDTD 1.6 .5
0.6 Floquet Floquet -80
0.8
0.5 1.8 -90
E(t)ttF{}00..34 E(t)trF{}00..46 20 0.5 1ffrreeqquueenn1cc.yy5,,ωω//ωω00 2 2.5 3 -100 20log|Et(ω0)|
0.2
0.2 Fig. 3. Transmission spectrum of a time-only modulated metasurface
0.1 for a varying modulation frequency and fixed modulation depth, ∆p. The
0 0 metasurfaceisassumedtobematchedwithχee=χmm.
0.5 1 1.5 0.5 1 1.5
∆p=0.25,fm=0.25f0 Next,anexampleofaspace-timemodulatedmetasurfaceis
0.7 0.7
considered. Due to the spatial dependence of the susceptibili-
FDTD FDTD
0.6 0.6
Floquet Floquet ties following (6), the metasurface is spatially discretized and
0.5 0.5 the Floquet analysis is performed at each x location to de-
E(t)ttF{} 00..34 E(t)trF{}00..34 taenrdmEi˜nre(xth,eztr=an0s+m,itωte)d, raensdpercetfliveecltye.dAfineledxsa,mE˜pt(lex,iszs=ho0w+n,ωin)
Fig. 4, for an input Gaussian beam, showing the spectrum
0.2 0.2
of the transmitted fields at the output of the metasurface at
0.1 0.1
each location, with the mismatched parameters of Fig. 2. The
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Gaussianwaveformnatureisclearlyobservedforeachspectral
frequency,ω/ω0 frequency,ω/ω0 component. More interestingly, the phase profile across the
Fig.2. Spectrumoftransmissionandreflectionfieldsfromaninfinitetime- metasurface as a function of frequency is also shown in
only (βp = 0) modulated metasurface incident with a normally incident Fig. 4. A linear phase gradient with different spatial slopes,
plane-wave, for the case of a) ∆p = 0.05, ωp = 0.05ω0, b) ∆p = 0.1, d∠E (x)/dx, across the metasurface is clearly evident for
ωp =0.1ω0 andc)∆p =0.25,ωp =0.25ω0.Totalnumberofharmonics t
2N + 1 = 61 and T0 = 100 fs in (16). The metasurface parameters each harmonic component, except the fundamental frequency.
are: ωe0 = 2π(224.63 THz), ωep = 0.36 Trad/s, αe = 500×109, Therefore, each of these harmonics are expected to refract
ωm0 = 2π(224.40 THz), ωmp = 0.29 Trad/s, αm = 100×109, and at different angles. Similar observations are also made for
excitationfrequencyω0=2π(230THz).Thespectrumisnormalizedtothe
non-modulatedcasewith∆p=0. reflected fields (not shown here).
To confirm the refraction angles of the transmitted and
reflected fields from the metasurface, the output fields at a
While Fig. 2 showed results for few discrete points for
specific harmonic frequency ω , are then Fourier propagated
∆ and ∆ , Fig. 3 shows a spectral map for a continuously n
p m in free-space as [13]
variedmodulationfrequencyω ,foragivenmodulationdepth
p
7only for simplicity, and clearer compact description of the associated
matrices. E˜t(x,z,ωn)=Fx−1[Fx{Et(x,z+,ωn)}exp{−jkz,nz}]
MANUSCRIPTDRAFT 6
20log|E(ω0−ωp)| 0 20log|E(ω0)| 0 20log|E(ω0+ωp)| 0
30 30 30
15 15 15
-20 -20 -20
0 0 0
m m m
µ µ µ
x -15 x -15 x -15
-40 -40 -40
-30 -30 -30
-200 0 200 -200 0 200 -200 0 200
zµm zµm zµm
1 1 1
θ−1=−8.67◦ θ0=0◦ θ+1=+7.08◦
2,ω)0}±00..68 kxp,e−ak1 2,ω)0}±00..68 20,ω)0}± 00..68 kxp,e+ak1
0 0 =
E(x,z=xF{00..24 kx,in E(x,z=xF{00..24 E(x,zxF{ 00..24 kx,in
kx,in
0 0 0
-2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2
spatialfrequencykx×10−6 spatialfrequencykx×10−6 spatialfrequencykx×10−6
Fig.5. Steady-statescatteredfieldsinthetransmissionandreflectionregioncorrespondingtothespace-timemodulatedmetasurfaceofFig.4,obtainedusing
analyticalFourierpropagationmethod,forthefirstup-convertedanddown-convertedspectralcomponentalongwiththefundamental,inagreementwiththe
predictionsof(4).AlsoshownarethespatialFouriertransformsoftheindividualharmonicswithsolidanddashedcurvescorrespondingtotransmittedand
reflectedfields,respectively.Allfieldsarenormalizedtotheirownmaximum.
∆p=0.025,fm=0.1f0
12.5 20log|Et(x,ω)| 0 12.5 6 Et(x,ω)rad 3 θ(ωn =ω0+nωp)=sin−1(cid:34)kxpe,nak(cid:35). (18)
k
7.5 -10 7.5 2 n
1
m 2.5 -20 mm 2.5 where kxpe,nak is the peak location of the spatial spectrum of the
xµ -2.5 -30 xµxµ -2.5 0 output fields, kn is the wavenumber of the nth harmonic, and
-1
the angle θ is measured from the normal of the metasurface.
-7.5 -40 -7.5 -2 For the n = 1 harmonics, the refraction angles are found
±
-12.5 -50 -12.5 -3 to be θ = 8.67◦ and θ = 7.08◦, respectively, for both
0.5 0.75 1 1.25 1.5 0 0.5 1 1.5 2
transmission and reflection, with a very good agreement with
frequency,ω/ω0 frequency,ω/ω0
the theoretical values of 8.33 and 6.81 obtained from (4).
◦ ◦
Fig. 4. Transmission spectrum (amplitude and phase) of a space-time
modulated metasurface, when incident with a normally incident Gaussian
beam.ThebeamisgivenbyE0(x)=exp{−(x/2wx)2},withwx=5µm. IV. CONCLUSIONS
Themetasurfacesize(cid:96)=25µmandtheexcitationfrequencyf0=230THz.
Thepumpingspatialfrequencyβp=5π/(cid:96). A rigorous Floquet mode analysis has been proposed for
a zero thickness space-time modulated Huygens’ metasur-
face to model and determine the strengths of the harmonics
E˜ (x,z,ω )= 1[ E (x,z ,ω ) exp +jk z ] of the scattered fields. The proposed method is based on
r n Fx− Fx{ r + n } { z,n } GSTCs treating the metasurface as a spatial discontinuity.
(cid:112)
where k = k2 k2, is the free-space wave number in The metasurface has been modelled using space-time varying
z,n n− x
the z direction. Fig. 5 shows the resulting field in the x z resonant frequencies of the associated electric and magnetic
−
plane for the first down-converted (ω ω ) and first up- surface susceptibilities, χ and χ , respectively. The un-
0 p ee mm
−
converted harmonic (ω +ω ), in addition to the fundamental knownscatteredfieldshavebeenexpressedintermsofFloquet
0 p
frequency ω . As expected from Fig. 4 and consistent with modes, which when used with the GSTCs, led to a system of
0
(4), the fundamental frequency ω propagates without any field matrix equations. The resulting set of linear equations
0
refraction. On the other hand, the down-converted and up- were then solved to determine the transmitted and reflected
converted harmonics are refracted in the lower and upper half scattered fields. Using an FDTD solver, the proposed method
of the x z plane, respectively. Their reflection angles using isvalidatedandconfirmedforseveralexamplesofmodulation
−
transverse wavenumbers are given by depths (∆ ) and frequencies (ω ). Finally, the computed
p p
MANUSCRIPTDRAFT 7
steady-state scattered fields are Fourier propagated analyti-
cally,forconvenientvisualizationofrefractedharmonics.The
[C]
proposed method is fast, simple and versatile, and is expected (cid:122) (cid:125)(cid:124) (cid:123)
0 0 0
to be a useful tool in determining the steady-state scattered
 0 0 0 
fieldsfromaspace-timemodulatedHuygen’smetasurface,for  d[V]
arbitrary modulation frequencies and depths.  W1 0 0  dt
 0 T1 0 
0 0 T
1
V. APPENDIX [G(t)]
(cid:122) (cid:125)(cid:124) (cid:123)
 
A A A
Summary of Finite-Difference Time Domain (FDTD) Formu- 1 2 3
lation of Space-Time Modulated Metasurface [16][17]  B1 B2 B3 
 
+ W2(t) 0 0 [V]=[E(t)], (21)
Consider a space-time modulated metasurface of Fig. 1  0 T2(t) 0 
with the plane-wave input and output fields described in (7). 0 0 T (t)
2
The field equations governing the transmitted and reflected
where [V] is the solution vector containing the transmission
fieldsarethetime-domainLorentzrelations(8)andtheGSTC
and reflection field E and E , and [G(t)] contains the exact
equations (9). t r
description of the metasurface. For a space-time modulated
Each Lorentz resonator equation of (8) corresponding to
metasurface considered here, the resonant frequencies of the
incident, transmitted and reflected fields, is a second order
Lorentziansusceptibilitiesareassumedtobeafunctionofboth
differential equation. By introducing an auxiliary variable,
spaceandtime,i.e.ω (x,t)andω (x,t).Theabovematrix
each second-order differential equation can be decomposed e0 m0
equationscanbere-writteninacomplexmatrixequationform
into two first-order differential equations. For instance, the
as
electric and magnetic polarizabilities corresponding to the
incident fields are given by d[V]
[C] +[G(t)][V]=[E(t)], (22)
dt
dQ
ω Q¯ = 0 +α Q , (19a) which now can be easily solved using standard finite-
e0 0 e 0
dt
difference technique based on trapezoidal integration as
dQ¯ 1 dω ω2
dt0 +Q¯0ωe0 dte0 +ωe0Q0 = ωeep0E0 (19b) [V] =(cid:18)[C]+ ∆t[G] (cid:19)−1(cid:20)∆t[E]i+[E]i−1
i 2 i 2
(cid:18) (cid:19) (cid:21)
dM ∆t
ω M¯ = 0 +α M , (19c) + [C] [G] [V] .
m0 0 dt m 0 − 2 i i 1
−
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