1 Passive Bias -Free Nonreciprocal Metasurfaces Based on Nonlinear Quasi- Bound States in the Continuum

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Passive Bias-Free Nonreciprocal Metasurfaces Based on Nonlinear Quasi-

Bound States in the Continuum

Michele Cotrufo1, Andrea Cordaro2,3, Dimitrios L. Sounas4, Albert Polman3 and Andrea Alù*1,5

1Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY

10031, USA

2Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam Science Park 904, 1098

XH Amsterdam, The Netherlands

3Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands

4Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202,

USA

5Physics Program, Graduate Center of the City University of New York, New York, NY 10016, USA

Nonreciprocal devices – in which light is transmitted with different efficiencies along opposite

directions – are key technologies for modern photonic applications, yet their compact and

miniaturized implementation remains an open challenge. Among different avenues, nonlinearity-

induced nonreciprocity has attracted significant attention due to the absence of external bias and

integrability within conventional material platforms. So far, nonlinearity-induced nonreciprocity

has been demonstrated only in guided platforms using high-Q resonators. Here, we demonstrate

ultrathin optical metasurfaces with large nonreciprocal response for free-space radiation based

on silicon third-order nonlinearities. Our metasurfaces combine an out-of-plane asymmetry –

necessary to obtain nonreciprocity – with in-plane broken symmetry, which finely tunes the

radiative linewidth of quasi-bound states in the continuum (q-BICs). Third-order nonlinearities

naturally occurring in silicon, engaged by q-BICs, are shown to enable over 10 dB of

nonreciprocal transmission while maintaining less than 3 dB in insertion loss. The demonstrated

devices merge the field of nonreciprocity with ultrathin metasurface technologies, offering an

exciting functionality for signal processing and routing, communications, and protection of high-

power laser cavities.

Nonreciprocal electromagnetic devices transmit light asymmetrically along opposite directions,

forming key components to achieve ultimate control over the flow of light. However,

nonreciprocal transmission is hard to achieve in conventional media: Lorentz reciprocity [1]

dictates that, in any system with permittivity and permeability tensors that are symmetric, time-

invariant and linear, the transmission between a source and a detector is invariant if these are

swapped. Breaking reciprocity, thus, requires lifting at least one of these conditions. Standard

approaches for light isolation involve applying a dc magnetic bias to magneto-optical materials,

which makes the permittivity tensor asymmetric. More recently, nonreciprocity has been achieved

with time-variant materials, whereby some material properties, such as the refractive index, are

modulated in time [2]–[11]. Finally, reciprocity can be broken by exploiting electromagnetic

nonlinearities [12]–[20]. This approach has recently received significant attention, due to the

absence of any external form of bias and the universal working principle, directly integrable in a

variety of conventional photonic platforms [21]. When an electromagnetic resonator couples

asymmetrically to two input/output ports, the same power injected from different ports gives rise

to different intra-cavity field intensities. In linear systems, such internal asymmetry is not sufficient

to break reciprocity, and the port-to-port transmission remains the same in both directions.

However, if the resonator is filled with a material with nonlinear response, such as an intensity-

dependent permittivity, different intracavity intensities create different permittivity profiles,

enabling large asymmetries in the power flow for opposite directions [21]. Remarkably, this

mechanism does not require any applied bias – in essence it is the signal itself to self-bias the

device – and it does not require any absorption, because the unwanted beam is reflected rather than

being absorbed as required in the case of magnetic-based isolators. While general constraints based

on passivity and time-reversal symmetry prevent these devices from working as conventional

isolators under simultaneous two-port excitation [22], they constitute an appealing technology for

applications such as nonreciprocal routing of pulsed signals [20] and protection of high-power

lasers. Indeed, due to its simplicity and general applicability, the nonlinearity-based route to

nonreciprocity has been successfully demonstrated in various frameworks, such as integrated Si

and InP micro-cavities operating in the near-infrared [16],[20], microwave circuits [19], and

atomic systems [22]–[24]. All the devices investigated so far involve integrated systems coupled

to optical waveguides or transmission lines, since in these devices wave-matter interactions can be

carefully controlled and enhanced, and the typically weak optical nonlinearities can be engaged in

a controllable fashion. A few theoretical proposals [17], [18], [23]–[25] have suggested that these

phenomena may be also translated to optical metasurfaces coupled to propagating free-space plane

waves, which may be used to realize free-space fully-passive flat nonreciprocal devices. However,

the experimental demonstration of nonlinearity-induced nonreciprocity in optical metasurfaces has

been so far elusive, mainly due to the weak nonlinearities of the involved materials, and the

corresponding stringent requirements in terms of operating intensities and low material loss.

In this work, we experimentally demonstrate the emergence of strong nonlinearity-induced

nonreciprocity in amorphous silicon metasurfaces coupled to free-space radiation in the near-

infrared regime (Fig. 1a). In order to create an asymmetric coupling between the metasurface and

plane waves propagating along the two normal directions, the out-of-plane symmetry of the device

is broken by leaving a thin unpatterned layer, whose thickness can be controlled to maximize

nonreciprocity when combined with the third-order nonlinearities naturally occurring in silicon

[18]. A major challenge to enable large nonreciprocity in metasurfaces is the typically weak

Figure 1. (a) Schematic of the nonreciprocal metasurface: by combining structural asymmetry and material

nonlinearity, a monochromatic beam impinging from either of the two sides of the device experiences markedly

different transmission levels. (b) Geometry and design parameters (additional details in text). Inset: field profiled

of the q-BIC excited by incoming plane waves. (c-d) Calculated transmission spectra for t = 20 nm (panel c) and

t = 40 nm (panel d), for fixed values of lattice constant (a = 750nm), gaps (G = 100 nm) and total thickness (H =

100 nm), and different in-plane asymmetry 𝑊

1/𝑊2

ranging from 1.2 (dark blue lines) to 3.4 (light green lines). (e)

Scatter plot showing, for devices with fixed value of 𝑊

1/𝑊2

=2.14, the maximum linear transmission and field

asymmetry ȁ𝐸1ȁ2/ȁ𝐸2ȁ2

for different values of t. (f) Calculated nonlinear transmission for the device marked by

the red circle in panel e, for excitation from port 1 (blue lines) and port 2 (red lines), and for two excitation

wavelengths, indicated by solid and dashed lines in the top-right inset.

interactions of light with ultrathin devices, which, combined with the poor nonlinearity of optical

materials, results in negligible transmission asymmetries, and explains the lack of an experimental

demonstration of these concepts to date. Here, we address this issue by introducing tailored in-

plane broken symmetries that carefully control a quasi-bound state in the continuum (q-BIC) [26].

In turn, the q-BIC linewidth tailors the metasurface resonant response, enhancing the nonlinear

interactions with the incoming waves and hence minimizing the operating intensity, while at the

same time keeping a large transmission contrast. Based on these principles, we are able to

demonstrate transmission contrasts larger than 10 dB and insertion loss smaller than 3 dB for peak

intensities smaller than 50 MW/cm2, only limited by the material nonlinearity strength. Moreover,

we experimentally demonstrate that the range of intensities over which nonreciprocity occurs can

be fully controlled by the vertical asymmetry of the metasurface. In agreement with recent

theoretical results [18], we also experimentally demonstrate a trade-off between the extent of input

intensity range over which nonreciprocity occurs and the minimum insertion loss. Tailoring the

metasurface geometry, we are able to operate close to the bounds allowed by this trade-off.

Results

Device Design and Numerical Optimization. Figure 1(b) shows the geometry of our

metasurface, consisting of a 1D amorphous silicon grating of total thickness H = 100 nm placed

on a glass-like substrate to allow mechanical handling. The grating is uniform along the y direction

and periodic along the x direction with period a. The two excitation/collection ports (denoted ‘Port

1’ and ‘Port 2’ in Fig. 1a) correspond to plane waves propagating in opposite directions normal to

the metasurface plane (z-direction) with impinging electric field polarized along the direction of

the nanowires forming the metasurface. Nonreciprocity is achieved if the excitations coming from

the opposite ports couple with different efficiencies to the same optical mode supported by the

metasurface. In other words, the same intensity, injected from opposite directions, must result in

different steady-state intracavity field intensities of the optical mode (see also inset of Fig. 1b). In

order to induce and control such electromagnetic asymmetry, a residual silicon layer of thickness

t < H is left unpatterned. A device with t = 0 is symmetric along z (apart from the small asymmetry

induced by the substrate), and thus it is expected to provide reciprocal wave transmission at any

input power, independent of the nonlinearity.

In order to enable strong nonlinearity-induced nonreciprocity, electromagnetic asymmetry

is not sufficient, and strong nonlinear interactions are vital. We maximize these phenomena by

carefully controlling the radiative linewidth of the targeted resonant mode. Indeed, on one hand it

is desirable to reduce the radiative linewidth and hence maximize the Q-factor of the metasurface

to strengthen the nonlinear interactions and minimize the required intensity to trigger these

nonlinear phenomena [21]; on the other hand, in order to observe a large transmission contrast the

resonance linewidth must be larger than the linewidth of the impinging laser, which is particularly

important when using pico- or femto-second pulsed lasers. Too narrowband responses also

typically imply very selective angular responses, which hinder the possibility of focusing light on

the sample to enhance the input intensity. In our device, we address these trade-offs by precisely

controlling the Q-factor and frequency response of the metasurface through q-BIC engineering

[26]. Q-BICs have been proven to be a very useful platform to boost the Q-factor of metasurfaces

and enhance nonlinear phenomena, such as lasing and second harmonic generation [27]–[31].

Here, we show that q-BICs, combined with nonlinear responses, can be used to dramatically

enhance nonreciprocal wave transmission, by enhancing the metasurface Q factor while at the

same time maintaining a large contrast in the Fano-like transmission spectra lineshape. We

consider a unit cell composed of two silicon wires of lateral widths

and

, separated by even

gaps of width G (Fig. 1b). When the unit cell is symmetric (

), the metasurface supports a

localized mode that does not couple to free-space radiation due to symmetry, realizing a symmetry-

protected BIC. Breaking the in-plane symmetry of the unit-cell (

≠

) turns the BIC into a q-

BIC with finite radiative decay rate. This leads to the appearance of a Fano profile in the

transmission spectrum, whose linewidth is carefully controlled by the unit cell asymmetry. Figures

1(c-d) show the numerically calculated linear transmission spectra of devices with a = 750 nm, G

= 100 nm and different values of in-plane asymmetry

/WW

ranging from 1.2 to 3.4 (see

horizontal arrow in Fig. 1d), and for t = 20 nm (Fig. 1c) and t = 40 nm (Fig. 1d). These results

confirm that the linewidth of the Fano resonance can be continuously tuned by controlling the

value of

/WW

. The inset of Fig. 1b shows the electric field intensity profile induced in a

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1PassiveBias-FreeNonreciprocalMetasurfacesBasedonNonlinearQuasi-BoundStatesintheContinuumMicheleCotrufo1,AndreaCordaro2,3,DimitriosL.Sounas4,AlbertPolman3andAndreaAlù*1,51PhotonicsInitiative,AdvancedScienceResearchCenter,CityUniversityofNewYork,NewYork,NY10031,USA2VanderWaals-ZeemanInstitute,Institu...

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1 Passive Bias -Free Nonreciprocal Metasurfaces Based on Nonlinear Quasi- Bound States in the Continuum

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