Antimatched Electromagnetic Metasurfaces for Broadband Arbitrary

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Antimatched Electromagnetic Metasurfaces for Broadband Arbitrary Phase Manipulation in Reflection Odysseas Tsilipakos, Thomas Koschny, and Costas M. Soukoulis ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b01415 • Publication Date (Web): 03 Jan 2018 Downloaded from http://pubs.acs.org on January 9, 2018

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Antimat hed Ele tromagneti Metasurfa es for Broadband Arbitrary Phase Manipulation in Ree tion ∗,†

Odysseas Tsilipakos,

Thomas Kos hny,

‡

and Costas M. Soukoulis

† ,‡

Institute of Ele troni Stru ture and Laser, FORTH, GR-71110 Heraklion, Crete, Gree e ‡Ames LaboratoryU.S. DOE and Department of Physi s and Astronomy, Iowa State University, Ames, Iowa 50011, USA

†

E-mail: otsilipakosiesl.forth.gr

Abstra t Metasurfa es impart phase dis ontinuities on impinging ele tromagneti waves that are typi ally limited to 0 − 2π . Here, we demonstrate that multi-resonant metasurfa es

an break free from this limitation and supply arbitrarily-large, tunable time delays over ultra-wide bandwidths. As su h, ultra-thin metasurfa es an a t as the equivalent of thi k bulk stru tures by emulating the multiple geometri resonan es of threedimensional systems whi h originate from phase a

umulation with ee tive material resonan es implemented on the surfa e itself via suitable subwavelength meta-atoms. We des ribe a onstru tive pro edure for dening the required sheet admittivities of su h metasurfa es. Importantly, the proposed approa h provides an exa tly linear phase response so that broadband pulses an experien e the desired group delay without any distortion of the pulse shape. We fo us on operation in ree tion by exploiting an antimat hing ondition, satised by interleaved ele tri and magneti Lorentzian resonan es in the surfa e admittivities, whi h ompletely zeroes out transmission through 1

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the metasurfa e. As a result, the proposed metasurfa es an perfe tly ree t a broadband pulse imparting a pres ribed group delay. The group delay an be tuned by modifying the implemented resonan es, thus opening up diverse possibilities in the temporal appli ations of metasurfa es.

Keywords metasurfa es, metamaterials, multiple resonan es, tunable, broadband, ree tion, phase delay, group delay Metasurfa es are the two-dimensional versions of metamaterials, typi ally formed by arranging subwavelength resonant meta-atoms on a single plane. They have been investigated intensely in re ent years for an abundan e of fun tionalities spanning perfe t absorption, 1,2 dispersion ompensation, 3 ele tromagneti ally indu ed transparen y, 4 wavefront transformations, 521 polarization ontrol, 22,23 and nonre ipro al response. 24,25 Operation in both transmission and ree tion has been examined and the main a hievements an be found in re ent review papers. 2629 An important realization along the way has been that metasurfa es with both ele tri and magneti response an extend the π phase span oered by purely ele tri resonant sheets and provide a phase modulation approa hing 2π . This, by Huygens' prin iple, allows for full ontrol over the wavefront; at the same time, the ex itation of both ele tri and magneti surfa e

urrents allows for unidire tional s attering. Thus, impedan e-mat hed gradient metasurfa es that provide maximum e ien y in transmission and perform pres ribed wavefront transformations be ame possible. 7 Still, even ele tromagneti sheets exhibit a limited delaybandwidth produ t, restri ted by the maximum 2π shift obtained over the narrow bandwidth of the mat hed resonan e pair. In this work, we demonstrate that metasurfa es an over ome this longstanding limitation and provide arbitrarily-large time delays over broad bandwidths. This is a hieved by implementing

multiple, properly-arranged resonan es in the ee tive surfa e admittivities. 2


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As su h, ultra-thin metasurfa es an a t as the equivalent of thi k bulk stru tures, by emulating the multiple geometri (e.g. Fabry-Pérot) resonan es of three-dimensional systems whi h originate from phase a

umulation with ee tive material resonan es on the surfa e itself via suitable subwavelength meta-atoms. Both ele tri and magneti resonan es are ne essary for obtaining ontrol over unidire tional radiation. Importantly, we require at amplitude response and an exa tly linear phase prole orresponding to zero group delay dispersion. As a result, broadband pulses an intera t with the metasurfa e and experien e the desired group delay with zero pulse distortion. The group delay an be readily tuned (or swit hed o) by modifying (quen hing) the implemented resonan es, thus opening up diverse possibilities in the temporal appli ations of metasurfa es. We fo us on operation in ree tion mode, drawing on an idea introdu ed in ref 30 for operation in transmission. Operating in ree tion is highly desirable for the tunable delay appli ations sin e any ontrol ir uitry for tuning the implemented resonan es an be a

ommodated behind the metasurfa e without interfering with the ele tromagneti wave. Notably, e ient operation in ree tion is a hieved by exploiting an admittivity antimat hing ondition whi h zeroes out transmission. As a result, the proposed metasurfa es are highly e ient with power lost only through absorption.

Creating a monotoni a

umulative ree tion phase In ultra-thin metasurfa es where geometri resonan es are absent, phase delay an only be provided by ee tive material resonan es, i.e., those of the onstituent meta-atoms. For arbitrarily-large delays one needs to break free from the typi al singly-resonant metasurfa es and enter the multi-resonant regime. Our rst on ern is to determine the proper way of arranging multiple resonan es so that the respe tive phase shifts ombine onstru tively produ ing a monotoni aggregate phase shift. Let us onsider a metasurfa e des ribed by ele tri and magneti surfa e admittivities

3


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( omplex ondu tivities) σse and σsm measured in S and Ω, respe tively. 31 We assume that the ma ros opi admittivities are the result of homogenized mi ros opi meta-atom responses and an feature multiple Lorentzian resonan es. The equations relating the surfa e admittivities with ree tion and transmission oe ients are 32,33

−˜ σse + σ˜sm , 1+σ ˜se σ ˜sm + σ ˜se + σ ˜sm 1−σ ˜se σ ˜sm , t(ω, θ) = 1+σ ˜se σ ˜sm + σ ˜se + σ ˜sm

(1a)

r(ω, θ) =

(1b)

1−r−t ζσse = , 2 1+r+t σsm 1+r−t σ ˜sm (ω, θ) = = , 2ζ 1−r+t

(2a)

σ ˜se (ω, θ) =

where we have dened dimensionless

ee tive admittivities σ˜

se (ω, θ)

(2b)

= ζσse /2 and σ ˜sm (ω, θ) =

σsm /(2ζ). Note that ζ TE (θ) = ωµ/k⊥ = η sec(θ) and ζ TM (θ) = k⊥ /(ωε) = η cos(θ) for the p TE and TM polarization, respe tively, where θ is the in iden e angle and η = µ/ε is the ˜se , σ ˜sm

hara teristi impedan e of the homogeneous host medium. Both r, t oe ients and σ depend on the polarization, with the notation suppressed for brevity. A single resonan e in the ele tri or magneti surfa e admittivity results in a spe tral region of high ree tion and an underlying π phase variation. The bandwidth of high ree tion is limited, being dire tly asso iated with the resonan e linewidth. Trying to in rease the bandwidth and the available phase span, one ould think of pla ing two resonan es of the same kind (ele tri or magneti ) side by side (Figure 1a). However, the resonan es do not ombine in a single high-ree tion region. The respe tive sus eptivity (imaginary part of admittivity) ontributions ompensate ea h other approximately halfway between the two resonant frequen ies. At the zero rossing we get a ree tion zero sin e no urrents are indu ed on the metasurfa e. Importantly, while traversing the zero rossing the

4


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Refl./Trans. Phase Power Coefficients Eff. Admittivities

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(a) Two of same kind e

(b) Two of different kind m e

(c) Alternating e m

5 0 -5

Im Re

1 T

T

T .5 R A

0 3π

R

R A

A

2π π r

r

r

0 t

t

t

70

90 110 130 70 90 110 130 70 90 110 130 Frequency (GHz) Frequency (GHz) Frequency (GHz)

Figure 1: Metasurfa e with resonant surfa e admittivities. (a) Adja ent resonan es of the same kind (ele tri ). The sus eptivity zero rossing leads to a ree tion minimum. The ree tion phase is not monotoni . (b) Adja ent ele tri and magneti resonan es. Ele tri and magneti sus eptivities annot ompensate ea h other and the ree tion phase is monotoni . ( ) Alternating ele tri and magneti resonan es. The magneti resonan e masks the ele tri sus eptivity zero rossing. The ree tion zero is avoided and the ree tion phase is monotoni . For the parameters of the Lorentzian resonan es see the Supporting Information.

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polarization shifts from anti-phase to in-phase and the ree tion phase from 3π/2 to π/2 (Eref ∝ Jse ∝ P˙ se ). Thus, the ree tion phase is not monotoni and respe tive π shifts do not add up to 2π . Ree tion phase monotoni ity an be a hieved by utilizing adja ent resonan es of different kind (Figure 1b). Sin e ele tri and magneti sus eptivities annot ompensate ea h other, the ree tion phase be omes monotoni and respe tive π shifts add up to 2π . Moreover, by avoiding the zero rossing and the asso iated ree tion zero we end up with a single high-ree tion region. This on ept an be generalized to multiple resonan es despite the fa t that sus eptivity zero rossings will be inevitably present. Consider for example alternating ele tri , magneti , ele tri resonan es (Figure 1 ). Although there is a zero rossing in the ele tri sus eptivity, it is masked by the magneti resonan e and the asso iated ree tion zero is avoided. As a result, the ree tion phase is monotoni and respe tive π shifts add up to 3π . Note that out of the available 3π span more than 2π is obtained under high ree tion. Thus, Figure 1 illustrates a way of a hieving a 2π span in ree tion using passive isotropi metasheets. This an be exploited in designing gradient metasurfa es in ree tion, presenting an alternative to magnetoele tri oupling 13 or stru tures with a ground plane. 10,11 Three resonan es are required in ontrast to the transmission ase where two mat hed ele tri and magneti resonan es provide the entire 2π shift under high transmission.

Broadband uniform ree tion with zero group delay dispersion We now seek to build on the on ept of alternating resonan es. Our goal is to perfe tly ree t broadband pulses imparting on them a tunable time delay. Observing Figure 1 we re ognize that the ree tion bandwidth is nite, the amplitude is not at, and the phase is not linear. To a

ommodate arbitrarily-broadband pulses without any pulse distortion we need wideband at ree tion with zero group delay dispersion. Mathemati ally, we 6


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require r(ω) = −A exp(iτ0 ω) and t(ω) = 0 for the s attering amplitudes of the metasurfa e, where τ0 is the desired group delay and the prefa tor A allows for some absorption in the metasurfa e. 34 Substituting this mathemati al pres ription in eq 2 we nd

σ ˜se =

1 + Aeiτ0 ω 1 = , σ ˜sm 1 − Aeiτ0 ω

(3)

stating that the surfa e admittivities should exhibit a spe i frequen y dependen e while

˜se oin ide with the zeros of σ ˜sm and vi ebeing omplex inverses. Noti e that the poles of σ versa in agreement with the on ept of interleaved resonan es guaranteeing a monotoni a

umulative ree tion phase. Importantly, in analogy with the well-known admittivity mat hing ondition σ ˜se = σ ˜sm 7 whi h zeroes-out ree tion, eq 1a, there is an

antimat hing

˜se = 1/˜ σsm that zeroes out transmission, eq 1b. This ondition has been identied

ondition σ independently in refs 35,36, albeit exploited for a single frequen y point. Equation 3 des ribes the target spe trum. However, only ertain types of resonant behavior are available in nature. Thus, we seek a good approximation of the target spe trum using Lorentzian resonan es whi h an be provided by subwavelength meta-atoms. Using a partial fra tion de omposition at the poles of eq 3, as detailed in the Supporting Information, we end up with a

physi al re ipe that mat hes the target spe trum almost perfe tly.

It requires trains of interleaved ele tri and magneti Lorentzian resonan es with proper frequen y spa ing, strength (κ) and damping (Γ): +∞

X iκe /2 iκe ω σ ˜se = , + 2 cor 2 + iΓe ω ω + iΓe /2 ω − ωe,k

(4a)

k=1

σ ˜sm =

+∞ X k=0

ω2

iκm ω , 2 − ωm,k + iΓm ω

(4b)

where ωe,k = [(2kπ)2 + ln2 (A)]1/2 /τ0 with k = 1, 2, . . . , ωm,k = {[(2k + 1)π]2 + ln2 (A)}1/2 /τ0 with k = 0, 1, 2, . . ., κe = κm = 4/τ0 , Γe = Γm = 2| ln(A)|/τ0, and Γcor = [2(1 − e

A)/(A| ln(A)|)]Γe . Physi ally, the interleaved resonan es guarantee the monotoni ity of 7


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the ree tion phase, as illustrated in Figure 1 . The spe i frequen y spa ing, strength and damping is required for providing uniform ree tion and a linear phase response. The only input parameters of the re ipe are the desired group delay τ0 and the allowed attenuation in the metasurfa e A. For large (of pra ti al interest) values of A the normalized

(˜ ω , ωτ0 /π) ele tri and magneti resonant frequen ies are given by ω ˜ e,k ≈ 2k (even inte˜ m,k ≈ 2k + 1 (odd integers), respe tively. This an be seen in Figure 2a where we gers) and ω ˜ = 0 . . . 8, involving the rst four ele tri plot the re ipe for A = 0.9 in the spe tral region ω and magneti resonan es. Noti e that the k = 0 (Drude) ele tri term is hara terized by a dierent damping frequen y Γcor as eviden ed by the peaking of Im(˜ σse ) at ∼ 5 instead of e

∼ 10. Im(σ~se)

Eff. Sheet Admittivities

10

Im(σ~sm)

Re(σ~sm)

(a)

5 +1

+1

0 -1

-1

−5 −10 1

Power Coefficients

Re(σ~se)

(b)

0.8

+π/2

R

w/o corr.

0.6

9π 8π 7π 6π 5π 4π

A = 0.9

-π/2

A

w/o corr.

T

0.4

3π 2π π

0.2 0

0

1

2 3 4 5 6 7 Normalized Frequency ωτ0/π

8

Reflection Phase (rad)

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0

Figure 2: (a) Lorentzian sum re ipe of eq 4 for A = 0.9. First four ele tri (ω ˜ e,k ≈ 2k ) and magneti (ω ˜ m,k ≈ 2k + 1) resonan es. Note the orre ted ele tri Drude (k = 0) term for supplying proper loss at DC. (b) Metasurfa e response: The ree tion amplitude is at and the phase linear. The latter relies on hara teristi points of unity negative-equal ˜se ≈ ±i + ǫ, σ ˜sm ≈ ∓i + ǫ) whi h lead to a ree tion phase Arg(r) = ∓π/2 sus eptivities (σ o

urring at the midpoints between resonan es (ω ˜ ≈ k + 1/2). The response of the re ipe is plotted in Figure 2b by substituting eq 4 into eq 1. Ree tion

8


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R = |r|2 is at and equal to the pres ribed A2 = 0.81. This behavior extends to arbitrarilyhigh frequen ies for the untrun ated sums of eq 4. Transmission is zero sin e we have satised the antimat hing ondition. Noti e the ee t of the Drude term orre tion meant to provide proper loss at DC (see Supporting Information). Importantly, the phase is exa tly linear. Physi ally, this relies on hara teristi points with Arg(r) = ∓π/2 showing up at the

˜ ≈ k + 1/2). Then, starting from an ele tri resonant midpoints between resonan es (ω frequen y the ree tion phase

equidistantly traverses {−π, −π/2, 0, +π/2, +π, . . .} leading

σs | = 1 and the to a linear phase response. Note that Arg(r) = ∓π/2 is attained when |˜ admittivities have negative-equal imaginary parts: σ ˜se = ±ia + b and σ ˜sm = 1/˜ σse = ∓ia + b. Substituting in eq 1a one nds r = ∓ia/(1 + b), i.e., Arg(r) = ∓π/2. For A values of pra ti al interest the admittivity real parts are small at the midpoints leading to a ≈ 1, i.e., to

unity negative-equal sus eptivities σ˜

se

≈ ±i + ǫ and σ ˜sm ≈ ∓i + ǫ with ǫ2 ≪ 1.

Toleran e on in iden e angle angular bandwidth Let us now investigate the ee t of in iden e angle on the re ipe des ribed by eq 4 and the orresponding ele tromagneti response. This is important for assessing the angular spe trum of wave pa kets with a nite spatial extent that an be a

ommodated by the metasurfa e. We start by noti ing that eq 4 is written for the

ee tive admittivities. The

a tual physi al admittivities σse and σsm an be spe ied on e the desired in iden e angle and polarization are determined by utilizing σ ˜se = ζσse /2 and σ ˜sm = σsm /(2ζ). Given that

ζ TE/TM depends on the in iden e angle, the re ipe holds exa tly only for the pres ribed angle of in iden e unless the physi al ondu tivities are spatially dispersive (nonlo al) with a very spe i dispersion that exa tly ountera ts the angle dependen e of ζ TE/TM . Limiting ourselves to lo al admittivities, some performan e degradation is inevitable when the a tual in iden e angle θact is dierent from the pres ribed θpre . Spe i ally, it is easy to show that for TE polarization the ele tri ee tive admittivity s ales with γ = sec(θact )/ sec(θpre ) and

9


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the magneti ee tive admittivity with 1/γ . The opposite holds for TM polarization. This interferes with the re ipe of equally strong ele tri and magneti resonan es ( f. Figure 2a). The ree tion amplitude a quires a periodi ripple about the pres ribed value A, as demon√ strated in Figure 3a for γ = 2 (TE polarization). Resistive damping is not masked evenly leading to a periodi ally varying absorption whi h shows up in the ree tion (see Supporting Information). The ripple period is equal to the spa ing between two adja ent ele tri (or magneti ) resonan es, i.e., equal to 2π/τ0 in ω or 1/τ0 in f . In Figure 3a we have hosen

τ0 = 0.5 ns leading to a period of 2GHz. 90

1 A

0.8 (a)

e-1

0.2 0 90

1

12GHz

0.4

θact (deg)

Uin ~ Uin

0.6

100

(b)

30 0

0.1 0.2

0.5

-30 -60

95 100 105 110 Frequency (GHz) (c)

50

Rp-p(%)

60

4GHz

Reflection Coefficient |r|

0

Output Pulsetrain |u~out(t)|

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-90

0 15 30 45 60 75 90 θpre (deg)

Actual (γ=√2) Ideal (γ=0)

A = 0.9

0.5

0 0

0.5

1

1.5 Time (ns)

2

2.5

3

√ Figure 3: Impa t of θact 6= θpre on pulsed input. (a) Ree tion amplitude when γ = 2 and pulse/pulsetrain input spe tra. The periodi r(ω) leads to pulse repli as in the temporal domain. (b) Color oded fra tional peak-to-peak amplitude of R ripple as θact deviates from θpre . As Rp−p in reases more power is transferred to the pulse repli as. Contours indi ate √ the relative strength of the strongest repli a ompared to the main pulse. For γ = 2, Rp−p = 15% and the strongest repli a is at 0.2 of the main pulse (dashed ontour). ( ) Output pulsetrain. Tpt is ommensurate with τ0 (250 and 500 ps, respe tively) and pulse repli as fall on the bit positions modifying logi al 1's and 0's. Next, we study the impa t on a broadband input pulse. Consider a single Gaussian pulse

uin(t) entered at 100 GHz with a bandwidth of 12 GHz measured at the e−1 amplitude points. Multipli ation of the pulse spe trum Uin (ω) with the transfer fun tion r(ω) translates into 10


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onvolution in the temporal domain. Sin e the transfer fun tion is periodi , its temporal response is dis rete with a period of 0.5 ns, equal to τ0 . As a result, besides the main output pulse at t = τ0 we get ausal pulse repli as at multiples of τ0 (t = mτ0 , m = 0, 2, 3, . . .), with the strongest one at t = 0 (see Supporting Information). They an be ome detrimental in a pulsetrain s enario, espe ially in real-world onditions with additive noise and jitter

˜in(t) with (see Supporting Information). Consider for example a pseudorandom pulsetrain u ˜in (ω), is depi ted in Figure 3a and is a period of Tpt = 250 ps. The input spe trum, U

hara terized by prominent peaks every 4 GHz, i.e., the pulse repetition frequen y. In this worst- ase s enario where Tpt is ommensurate with τ0 , pulse repli as fall on the bit positions, modifying both logi al 1's and 0's. This an be learly seen in Figure 3 where we plot the √ output pulsetrain. For the γ = 2 ase we have onsidered, the strongest (t = 0) repli a is at 0.2 (only 4% intensity) of the main pulse. If the in iden e angle deviates even more from the pres ribed, repli as be ome stronger. The fra tional peak-to-peak amplitude of the ripple (Rp−p ) in reases ( olor oded in Figure 3b) and more energy is transferred to the repli as. In addition, the ripple be omes less sinusoidal with higher-order repli as oming into play (see Supporting Information). Depending on the appli ation, only a ertain relative strength of the repli as an be tolerated. In turn, this sets upper and lower bounds on the a tual in iden e angle for a given pres ribed angle and denes the minimum spatial extent for an in ident wave pa ket. These bounds are plotted in Figure 3b for hara teristi ases of the √ t = 0 repli a relative strength: 0.1, 0.2, 0.5 orresponding to γ = 1.2, 2, 2.25, respe tively. Obviously, the proposed metasurfa es are hara terized by ample angular bandwidth. Only for very steep θpre angles does the toleran e on θact deteriorate noti eably sin e it depends on the se ant ratio γ .

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Realisti implementation with few Lorentzian resonan es The re ipe des ribed by the innite sums in eq 4 guarantees perfe tly at ree tion amplitude and exa tly linear ree tion phase for arbitrarily-wide bandwidths. In pra ti e, however, one

an implement only a limited number of resonan es on the metasurfa e. Figure 4 examines the ee t of trun ating the innite sum to a pra ti al number of terms keeping 3 ele tri

˜ = 100. Their positions are and 4 magneti resonan es around the normalized frequen y ω marked in Figure 4 with ir les and rosses, respe tively. Obviously, trun ation results in a nite ree tion bandwidth, as shown in Figure 4 . The full width half maximum (FWHM) of the ree tion band is ∼ 6. In addition, it interferes with the antimat hing ondition due to the absen e of lower- and higher-order resonan es (see Supporting Information). Consequently, the ree tion amplitude and group delay slightly deviate from the pres ribed values, predominantly near the band edges (Figure 4 ,d). Let us onsider now an in ident pulse with a spe tral bandwidth of 4 measured at the e−2 intensity points (Figure 4 ) impinging on the metasurfa e. The orresponding output pulse is depi ted in Figure 4b. It is somewhat shifted and attenuated ompared to the ideal ase of no trun ation sin e, on average, R and τg are slightly lower than pres ribed (Figure 4 ,d). However, there is negligible broadening or distortion as an be seen by properly shifting and s aling the ideal output pulse. Importantly, it is not the non-ideal response in the ree tion band, but rather the nite bandwidth that onstitutes the bottlene k of the trun ated re ipe performan e; in ident pulses with higher bandwidths would simply experien e windowing (see Supporting Information). Finally, we study the ombined ee t of sum trun ation and θact 6= θpre . We onsider a pulsetrain impinging on the metasurfa e at an in iden e angle dierent than the pres ribed. The pulsetrain period is set to Tpt = 700 ps and τ0 = 500 ps; they are in ommensurate meaning that pulse repli as will not fall on the bit positions ( f. Figure 3 ). The metasurfa e √ response for γ = 2/ 3 is depi ted in Figure 4e,f along with the pulsetrain input spe trum. Both ree tion amplitude and group delay deviate from their pres ribed values. One period 12


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(a)

Actual Ideal

(b)

1

Output Pulse |uout(t)|2

shifted Δt=τ0

.5 θact

bit clock

−0.5

0.6 0.4 0.2 0

A

R

95

100 ωτ0/π ~ |Uin|2

(e)

0.8

105

A2

0.6 0.4

0

R

95

100 105 Frequency (GHz)

1 0.8 (g) 0.6

A

0 2

|Uin|2

3π π

1

-π

-3π

95

100

ωτ0/π

105

5π

T

0.2

(d)

0.5

0 2

(f)

~ |Uin|2

3π π

1

-π

-3π

95

100

105

Group Delay τg /τ0

Power Coefficients

1

Amplitude

5π A2

Reflection Phase

0.8

|Uin|2

(c) T

Reflection Phase

Power Coefficients

1

0 (t-τ0)/τ0

Group Delay τg /τ0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

0

Frequency (GHz)

2 bits (1.4ns)

Jitter + AWGN γ=2/√3

0.4 0.2 0

Figure 4: Broadband pulsed input signal impinging on a metasurfa e with 3 ele tri and 4 magneti resonan es. First ase: Single pulse impinging with θact = θpre . (b) Output pulse with negligible broadening or distortion. ( ) R, T, A oe ients and input pulse spe trum. Positions of ele tri and magneti resonan es are marked with solid ir les and rosses, respe tively. (d) Ree tion phase and √ group delay. Se ond ase: Pulsetrain impinging at an in iden e angle θact 6= θpre (γ = 2/ 3). The pulsetrain period is Tpt = 700 ps and τ0 is set to 500 ps. (e) R, T, A oe ients and input pulsetrain spe trum. (f) Ree tion phase and group delay. Noti e the added ee t of θact 6= θpre . (g) Output eye diagram with jitter and additive white Gaussian noise.

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of the 1/τ0 = 2 GHz ripple due to θact 6= θpre is learly visible in the enter of the ree tion band, before the ee t of trun ation also starts ontributing near the band edges. The impa t on the pulsetrain an be assessed by onstru ting the output eye diagram (Figure 4g). To emulate as realisti a s enario as possible, we have in luded jitter (normal distribution with a standard deviation of Tpt /50) and additive white Gaussian noise (signal to noise ratio of 20 dB). Even in su h real-world onditions, the input pulsetrain an be readily re overed as the levels of logi al 1 and 0 are learly distinguishable. In Figure 4 we have demonstrated how a multi-resonant metasurfa e an be used to ree t a broadband pulsetrain imparting a pres ribed group delay on the in ident pulses. It is worth noting that the number of resonan es needed to over a given bandwidth depends on the desired group delay. For example, the seven resonan es in Figure 4a an over a 7 GHz bandwidth if the desired group delay is 0.5 ns, or 35 GHz if it is 0.1 ns. Finally, we stress that the proposed approa h is general and not limited to any parti ular region of the ele tromagneti spe trum. For example, broadband tunable delay omponents are parti ularly useful in wavelength-division-multiplexed tele ommuni ation systems operating at infrared wavelengths. In su h ases, providing the tunable time delay over broad bandwidths while preserving signal integrity, as demonstrated in Figure 4g for Gbit/s transmission rates, is of utmost importan e. The examples in Figures 3,4 refer to GHz arrier frequen ies, sin e implementing multiple ele tri and magneti resonan es on the metasurfa e unit ell should be easier ompared to the infrared or opti al regime, allowing for a rst demonstration. For example, one ould employ printed elements on multi-sided ir uit boards as exer ised in refs 7,27. For the meta-atoms, one potential hoi e is the ut-wire pair whi h supports losely spa ed ele tri and magneti resonan es. By tuning the geometry, the resonant frequen ies

an be ontinuously varied so that the resonan es progressively approa h and ultimately

ross, 37 providing a valuable degree of freedom. Then, arranging multiple ut-wire pairs inside the unit ell in an appropriate topology and tuning the system parameters to a 14


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ACS Photonics

urately spa e the resonan es and balan e their strength ould oer a route to a physi al implementation of the proposed on ept. Apart from ut-wire pairs, pra ti ally any resonant meta-atom studied in metamaterials and metasurfa es resear h ould be investigated for this purpose, sin e they have been shown to result in Lorentzian spe tral features in the ele tri and/or magneti admittivities, as required by the derived re ipe (eq 4). These in lude plasmoni resonan es in variants of ut-wires, split ring resonators and shnet stru tures, as well as ele tri and magneti resonant Mie modes in diele tri parti les. Finally, one an use one meta-atom for ea h resonan e or resonan e pair, or instead rely on multi-resonant meta-atoms for either the ele tri or magneti response.

Con lusion In on lusion, we have shown that multi-resonant metasurfa es an greatly ex eed the typi al limitation of a 0 − 2π imparted phase modulation. For operation in ree tion, the proper way of arranging the resonan es is interleaving ele tri and magneti resonan es, leading to an a

umulative monotoni ree tion phase a ross their aggregate bandwidth. Combining this prin iple with an admittivity antimat hing ondition that zeroes out transmission and requiring an exa tly linear ree tion phase, we have su

essfully demonstrated metasurfa es that an perfe tly ree t a broadband pulsetrain imparting a spe ied group delay on the in ident pulses. To assess the performan e of the proposed metasurfa es, we have thoroughly studied their toleran e on in iden e angle and found that they are hara terized by ample angular bandwidth. In addition, we have shown that few resonan es an a

ommodate a broadband input pulsetrain without pulse distortion, indi ating the potential of our approa h for pra ti al appli ations. Finally, we have established their performan e for realisti Gbit/s transmission rates in real-world onditions with timing and amplitude noise. In this paper, we have solved the physi al problem of establishing the theoreti al prin-

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iples and foundations for obtaining an arbitrarily-broadband linear ree tion phase from a metasurfa e. A next step would be to engineer an a tual implementation of a metasurfa e that approximates the derived surfa e admittivities by appropriately ombining dis rete meta-atoms. This is a omplex engineering task that will be the subje t of future work. Our results highlight that ubiquitous phase modulation and dispersion engineering operations (su h as tunable group delay, dispersion ompensation, pulse ompression, and slow light ee ts) that ustomarily rely on bulky resonators, an be instead performed a ross the deeply subwavelength thi kness of a metasurfa e. Thus, they an push metasurfa es into un harted territories of broadband temporal appli ations, previously onsidered a privilege of three-dimensional systems relying on phase a

umulation.

A knowledgement This work was supported by the European Resear h Coun il under ERC Advan ed Grant No. 320081 (PHOTOMETA) and the European Union's Horizon 2020 resear h and innovation programme Future Emerging Te hnologies (FETOPEN) under grant agreement No. 736876 (VISORSURF). Work at Ames Laboratory was partially supported by the Department of Energy (Basi Energy S ien es, Division of Materials S ien es and Engineering) under Contra t No. DE-AC02-07CH11358.

Supporting Information Available We provide a detailed a

ount of the physi al re ipe derivation, we thoroughly assess the ee t of ex eeding the available spe tral and angular bandwidth, and we examine real-world pulsetrain s enarios with timing and amplitude noise. This material is available free of harge via the Internet at http://pubs.a s.org.

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Graphi al TOC Entry σse Δt=τ0

4π

σsm

φ

ω bit clock

-3π

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Antimatched Electromagnetic Metasurfaces for Broadband Arbitrary

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