Giant non-reciprocity at the subwavelength scale

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Received 8 May 2013 | Accepted 6 Aug 2013 | Published 2 Sep 2013

Breaking time-reversal symmetry enables the realization of non-reciprocal devices, such as isolators and circulators, of fundamental importance in microwave and photonic communication systems. This effect is almost exclusively achieved today through magneto-optical phenomena, which are incompatible with integrated technology because of the required large magnetic bias. However, this is not the only option to break reciprocity. The OnsagerCasimir principle states that any odd vector under time reversal, such as electric current and linear momentum, can also produce a non-reciprocal response. These recently analysed alternatives typically work over a limited portion of the electromagnetic spectrum and/or are often characterized by weak effects, requiring large volumes of operation. Here we show that these limitations may be overcome by angular momentum-biased metamaterials, in which a properly tailored spatiotemporal modulation is azimuthally applied to subwavelength Fano-resonant inclusions, producing largely enhanced non-reciprocal response at the sub-wavelength scale, in principle applicable from radio to optical frequencies.

DOI: 10.1038/ncomms3407

Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials

Dimitrios L. Sounas1, Christophe Caloz2 & Andrea Al1

1 Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712, USA. 2 Poly-grames Research Center,cole Polytechnique de Montral, Montral, Quebec, Canada H3T1J4. Correspondence and requests for materials should be addressed to A.A.(email: mailto:[email protected]

Web End [email protected] ).

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Non-reciprocity in ferromagnetic materials originates from the unidirectional precession of electron spins1 or the Zeeman splitting of atomic orbitals2 in the presence of a

static magnetic bias. There is a large interest in mimicking this effect in articial materials without requiring applied magnetic elds, but instead relying on bias with other odd vectors under time reversal in compliance with the general principle of microscopic reversibility3. Such possibility would relax the current requirements of large magnetic biasing devices in non-reciprocal components, with the potential of revolutionizing the microwave and photonic component industry. Kodera et al.4, for instance, present a non-reciprocal metamaterial composed of transistor-loaded ring resonators, where suitably designed active loads, biased with direct electric current, can suppress one of the two azimuthally propagating eigenstates, producing a unidirectionally rotating magnetic moment functionally equivalent to ferromagnetic effects. Although this approach can be used to realize some non-reciprocal components5,6, it is limited to microwave and millimetre-wave frequencies, where transistors are available, and it involves signicant power consumption in the transistors biasing network. Another transistor-based, current-biased non-reciprocal metamaterial was proposed in ref. 7, consisting of cross-polarized dipoles interconnected to each other via transistors to achieve symmetry breaking and microwave Faraday rotation, with similar limitations.

Refs 813 present a different class of non-reciprocal components. In this case, biasing is provided by the linear momentum vector imparted by longitudinal spatiotemporal modulation suitably tailored to break time-reversal symmetry. This concept can provide optical isolation by means of direct or indirect interband photonic transitions in spatiotemporally modulated waveguides. Direct transitions10 can be described as the photonic equivalent of the AharonovBohm effect, whereas indirect transitions8 occur in the presence of a traveling wave modulation breaking space-inversion and time-reversal symmetries. The proposed devices can be broadband but their size is determined by the coherence length of the transitions, which, for reasonable modulation amplitudes, is several wavelengths, making these devices rather bulky. Nonlinearities have also been proposed to induce non-reciprocity1418 but the operation of the resulting devices is limited to specic ranges of input power. All these solutions rely on inherently weak effects, requiring large volumes, and are therefore not necessarily preferable to conventional solutions based on magneto-optical effects. The eld of optical metamaterials has also offered opportunities for boosting the non-reciprocal response of magneto-optical materials by means of large eld enhancement and localization. Two recent examples are plasmonic-enhanced Faraday rotation19 and isolation enabled by parity-time symmetry20.

In the following, we suitably combine the strong wavematter interaction of resonant metamaterial inclusions with the non-reciprocal properties of a new form of spatiotemporal modulation aimed at imparting a bias based on the angular momentum vector. We show that this combination can induce giant non-reciprocal response at the subwavelength scale, leading to the introduction of a new type of linear, non-reciprocal phenomenon based on angular momentum-biased metamaterials, which can overcome the limitations of the aforementioned approaches with potentially groundbreaking applications in microwave and nanophotonic integrated systems.

ResultsPrinciple of angular momentum-biased metamaterials. The simplest constituent inclusion suited to form the proposed meta-material is a ring resonator with an azimuthal spatiotemporal

permittivity modulation De(j,t) Dem cos (omt Lmj), as

depicted in Fig. 1a, where j is the azimuthal coordinate in a cylindrical reference system co-centred with the inclusion and Lm is

the modulation orbital angular momentum. In absence of mod

ulation, Dem 0, the ring supports degenerate counter-propagating

resonant states

j ie iot with azimuthal dependence e ij,

resonating when the ring circumference is times the guided wavelength, which implies that, for the fundamental |1S states used herein, the ring dimensions are smaller than the wavelength. The resonant size can be further reduced by adding capacitances along the loop, as in split-ring resonator designs21. As it will be shown shortly, introducing the aforementioned spatiotemporal azimuthal modulation lifts the degeneracy between the |1S states and produces non-reciprocity, an effect that can be interpreted as the metamaterial analogue of a static magnetic bias removing the degeneracy between atomic states of opposite orbital angular momenta in magnetic materials.

The proposed permittivity modulation is a type of amplitude modulation and, as such, it results in the generation of two intermodulation products k Lm

j ie iok omt and

k Lm

j ie iok omt for each state |kSe iokt. If any of them

overlaps in frequency with another state

j ie iot, resonant

coupling between the |kS and

j i state occurs, signicantly

affecting both resonances. As our goal is to lift the degeneracy

between |1S states, Lm 1 and om o2 o1 might appear

the most reasonable choice, so that the | 1S state gets

resonantly coupled to the | 2S state, whereas no coupling

occurs for the | 1S state, as illustrated in Fig. 1b, left panel.

However, o2 is usually close to 2o1 and, as a result, om should be close to o1, which may be challenging to achieve especially at terahertz and optical frequencies. For Lm 2, in contrast, the

states |1S resonantly couple to each other for om 0, as

illustrated in Fig. 1b, right panel. For om identically zero, the structure is obviously reciprocal but any small departure from zero can break reciprocity and, no matter how large o1 is, strong non-reciprocal response may be obtained by properly choosing

Dem and the resonator Q-factor, as will be discussed in more detail in the following.

This concept may be analysed using coupled-mode theory as in ref. 22: the amplitudes a 1 of the |1S states satisfy the

equations (see Methods for details)

a 1 io1a

1 i 12 o1kme iomta 1;

a 1 io1a

1 i 12 o1kmeiomta 1;

where

Dem Et1j j2rdrdz 2

is the coupling coefcient between the |1S states, with St being the resonator cross-section and Et1 the corresponding normalized electric eld distribution. For homogeneous resonators (Dem and e uniform, where e is the background permittivity), km Dem/2e.

The solution of equation (1) provides the eigenstates of the modulated ring (see Supplementary Note 1 for details):

j i 1

j ie ioat

o1km 1

2p Z

j ie ioa omt;

j i 1

where oa o1 Do/2, ob o1 Do/2 and Do

o2m o21k2m

om. This solution may be extended to take into account the presence of loss and coupling with the excitation signals, as discussed in the Methods section.

j ie iobt

o1km 1

j ie iob omt;

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[afii9830] = [afii9830]m cos([afii9853]mt [afii9850]) [afii9830] = [afii9830]m cos([afii9853]mt 2[afii9850])

0.5

[afii9830]s

[afii9853]3

y [afii9850]

[afii9853] [afii9853] [afii9853]3

[afii9853]2

[afii9853]1

[afii9853]2

[afii9853]1

[afii9853]m

3 1

1 0 +2

+1 +3 0 +2

+1 +3

[afii9830] = [afii9830]m cos([afii9853]mt Lm[afii9850])

[afii9853]

[afii9853] + [afii9853]m

[afii9853]

[afii9853] [afii9853]m

[afii9853]1

[afii9853]1[afii9837]m

[afii9853]1

[afii9853]m [afii9853]

[afii9853]

[afii9853]m

Energy

[afii9853]m

Figure 1 | Principle of non-reciprocal angular momentum-biased metamaterials. (a) Azimuthally symmetric ring resonator with a spatiotemporal modulation of permittivity. The modulation follows the form of an azimuthally propagating wave in the j direction. In the absence of modulation, the

resonator supports two degenerate counter-propagating states |1S. The applied modulation lifts the degeneracy and produces non-reciprocity.(b) Transformation of the |1S states in the frequency and angular momentum plane for Lm 1 (left) and Lm 2 (right). Red crosses: states of the

unmodulated ring. Blue crosses: intermodulation products. (c) Frequency diagram of the ring eigenstates without (left and right columns) and with (middle column) modulation for Lm 2. The states of the modulated ring are hybridizations of the |1S states. Each hybrid state consists of a dominant

(thick line) and a secondary (thin line) sub-state. (d) Sub-state eigenfrequencies versus the modulation frequency for Lm 2. (e) Sub-state energies

versus the modulation frequency for Lm 2.

both sides of a thin dielectric layer, as in Fig. 2a. Permittivity modulation is effectively obtained by loading the rings with time-variable capacitors DCn DCm cos (omt 2jn) at the azimuthal

positions jn np/4, where n 0y7, which is equivalent to

applying a continuous capacitance modulation DC (4DCm/

p)cos(omt 2j), as shown in the Methods section. It is note

worthy that the modulation amplitude 4DCm/p is the average of the localized capacitance DCm over the discretization period p/4, as it may be intuitively expected. A possible practical implementation of this capacitance modulation is illustrated in Fig. 2b: it consists of a varactor, as its core element, a direct current-biasing source, an alternating current modulation source with frequency om and appropriate lters that minimize the interference between the ring and the biasing network (see Supplementary Note 2 for details). Such a circuit may be easily integrated into the ring substrate within conventional printed circuit technology. Further, as varactors and lters are low-loss components, the overall power consumption is expected to be very low (in principle zero for ideally lossless circuit elements).

In the absence of modulation, the metasurface exhibits two resonances, shown in the inset of Fig. 2c, a low-Q bright mode at23 GHz with parallel currents induced in the two rings and a coupled high-Q dark mode at 9 GHz with anti-parallel currents. Suitable coupling between these two modes results in a peculiar Fano-resonant response24,25 at 8.9 GHz, with a sharp transition from full to no transmission (Fig. 2c). This response is ideal for our purpose, as its sharp frequency response relaxes the requirements on the modulation capacitance and, at the same time, leads to strong non-reciprocal effects because of the associated anti-parallel currents in the rings, maximizing the excitation of the modulation capacitors.

Figure 2d shows the transmission of circularly polarized (CP) waves through the metasurface (along z) for DCm 0.02 pF

The states |aS and |bS are hybridizations of the non-modulated ring states | 1S and | 1S, which are generally

characterized by different frequencies and amplitudes, as illustrated in Fig. 1c. In the absence of modulation (om 0),

the sub-states |1S of each hybrid state share the same frequency and energy, as shown in Fig. 1d,e, respectively, and the system is reciprocal as expected. However, when modulation is introduced (oma0), the sub-states split (Fig. 1d) and non-reciprocity arises. This splitting follows from the simultaneous spatial and temporal nature of the proposed modulation, generating the states k 2

j ie ioomt and k 2

j ie io omt

from |kSe iot (Fig. 1b). Therefore, if the sub-state | 1S exists

at frequency o, the sub-state | 1S can only exist at frequency

o om. For oma0, the energy is unevenly distributed between

sub-states (Fig. 1e) with the unbalance increasing with om. The

dominant sub-states for |aS and |bS are | 1S and | 1S,

respectively, as indicated in Fig. 1ce with thick lines.

The amount of non-reciprocity is determined by the minimum distance between sub-states of opposite handedness Domin

min{om,ob oa Do} and by the resonance width o1/Q, where

Q is the resonance quality factor, corresponding to the inverse of the fractional bandwidth. In practical devices such as polarization rotators and circulators, which are based on interference between states, Domin and o1/Q must be of the same order23. In the

Supplementary Note 1, we prove that DominrDo1km/

p , with

the maximum value holding for om Do o1km/ 3

p . Therefore,

Qkm 3

p , consistently with the expectation that a lower Q resonator requires a higher km and subsequently a higher Dem.

Microwave Fano-resonant metasurface. Consider now a spatiotemporally modulated metasurface consisting of periodically arranged pairs of broadside-parallel metallic rings patterned on

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B X

[afii9853]1 [afii9853] [afii9853]1 [afii9853]

[afii9830]s

C[afii9853]1 (t) = C0 + Cm cos([afii9853]mt + [afii9850]m)

[afii9853]m

~ BSF

f fm f f f + fm

BPF

c d e

Transmission (dB)

0 10 20 30

8.6 8.8 9 9.2

Frequency (GHz)

Figure 2 | Angular momentum-biased metasurface. (a) Constituent inclusion of the metasurface: a ring-pair resonator periodically loaded with time-variable capacitors. (b) Practical implementation of the time-variable capacitors: a band-stop lter (BSF) and a band-pass lter (BPF) are used to minimize the leakage of ring and modulation signals to the biasing network and to the ring, respectively. The two lters are designed to have zero susceptance and zero reactance at the ring resonance o1, respectively. (c) Transmission through the unmodulated metasurface. A Fano-type resonance is supported at 8.9 GHz, resulting from the interference of a high-Q magnetic (anti-parallel currents) resonance and a low-Q electric (parallel currents)

one. (d) Transmission across the metasurface in the z direction with DCm 0.02 pF and fm 0.1 GHz. Blue line: RHCP incidence. Red line: LHCP

incidence. (e) Same as in d but for fm 0.5 GHz. If the propagation direction is reversed (from z to z), the RHCP and LHCP curves ip. At 8.91 GHz

(dashed line), the metamaterial operates as an electromagnetic diode for CP waves; RHCP and LHCP waves can only propagate along the z and z

directions, respectively. The geometrical parameters of the structure are R 2 mm, w 1 mm, t 0.4 mm, es 9 and p 3.4R.

and fm 100 MHz. The chosen value of DCm corresponds to an

effective capacitance modulation of 0.026 pF rad 1, which, considering the ring-pair capacitance 0.48 pF rad 1, leads to km

0.027. This yields o1kmD0.24 GHz, which is enough for a clear separation between |aS and |bS states, consistent with the discussion in the previous subsection. Small variations of DCm that may occur in practice can be easily compensated by adjusting om as long as the |aS and |bS states are distinguishable.

As expected, the response is different for right-handed CP (RHCP) and left-handed CP (LHCP) excitations, and in each case two resonant dips are observed, with the stronger one resulting from coupling with the state whose dominant sub-state is of the same handedness as the incident wave. RHCP incident waves strongly couple to |aS at frequency fa, whereas LHCP waves strongly couple to |bS at fb. The weaker resonant dips correspond to the secondary sub-states at fb fm (RHCP

excitation, substate | 1S of |bS) and fa fm (LHCP waves,

| 1S of |aS). If the structure is excited from z, the

transmission curves of Fig. 2d switch handedness as the incident wave feels opposite modulation spin, a clear demonstration of non-reciprocity. This polarization transmission asymmetry can be exploited to realize, for instance, a CP isolator by placing the transmission null of one polarization at the same frequency as the transmission peak of the other polarization. This condition is fullled for fm 0.5 GHz, as shown in Fig. 2e; at 8.91 GHz RHCP

and LHCP waves can penetrate the metasurface only from z

and z, respectively. This operation may be the basis of different

types of polarization-dependent microwave isolators.

Another important non-reciprocal effect, common in ferro-magnetic materials, is Faraday rotation, that is, the non-reciprocal

rotation of the polarization plane of a wave, as it propagates through the material. The rotation is opposite for opposite propagation directions, as it is determined by the (xed) bias direction. We can achieve the same effect in the proposed spatiotemporally modulated metasurface, as shown in Fig. 3a, which plots the polarization rotation angle y for different fm and

DCm 0.02 pF. As the modulation frequency increases, y

increases and the bandwidth decreases. The bandwidth reduction is clearly because of the decrease of Do as fm increases, but the monotonic increase of y may seem contradictory with the fact that the separation between states, which determines the amount of non-reciprocity, is actually reduced as fm increases. As shown in Fig. 3b, this peculiar monotonic increase of y results from the fact that at resonance, for fm40.5 GHz, the transmission coefcient for x-polarized waves Txx decreases faster than the transmission coefcient of x- to y-polarized waves Tyx, so that

Tyx/Txx, which is proportional to y, actually increases. For fm 0.5 GHz, Tyx is maximum and y 60, corresponding to a

giant rotation of 6,000 per free-space wavelength, without the need of any magnetic bias. In Fig. 3c, we show the ellipticity angle w at the output for linearly polarized inputs. It is easy to see that this angle is zero at the frequency of maximum y, implying that the transmitted eld is linearly polarized, making the proposed metasurface particularly exciting for applications requiring non-reciprocal linear polarization rotation.

Optical isolator. As pointed out above, the proposed scheme of spatiotemporal modulation with Lm 2 poses no restriction on

the modulation frequency, opening the possibility to apply the proposed concept to optical frequencies. As a proof of concept, an

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b c

Rotation angle [afii9835](degree)

Ellipticity angle [afii9851](degree)

fm=1 GHz

fm=0.5 GHz

fm=0.1 GHz

fm=0.05 GHz

fm=1 GHz

[afii9851] [afii9835]

fm=0.5 GHz

fm=0.1 GHz

fm=0.05 GHz

Txx

Tyx

T xx, T yx(dB)

8.6 8.8 9 9.2

Frequency (GHz)

Figure 3 | Faraday rotation for the angular momentum-biased metasurface. (a) Rotation angle. (b) Transmission coefcient from x- to x-polarized waves, Txx, and from x- to y-polarized waves, Tyx. (c) Ellipticity angle: at the frequency of maximum rotation the ellipticity angle is zero, corresponding to ideal Faraday rotation. All the results were derived for the same ring parameters as in Fig. 2.

S21 S12

drop waveguide

S21,S12

S-parameters (dB)

12 4 channel waveguide

0.6686 0.6687 0.6688

Frequency (c/a)

Figure 4 | Angular momentum-biased optical isolator based on a channel-drop lter. (a) Reciprocal transmission versus frequency of a conventional channel-drop lter. Waves entering from ports 1 and 2 couple to the right- and left-handed resonances of the ring, as illustrated in the left and right insets, respectively, creating a transmission dip at the ring resonance. (b) Non-reciprocal transmission versus frequency when spatiotemporal modulation is applied to the ring. The insets show the electric eld amplitude distribution at the frequency indicated by the dashed line for excitation from port 1 (left) or 2 (right). The geometrical parameters of the structure are R 0.88a, w 0.2a and g 0.3a, where a is an arbitrary reference length. The modulation

frequency for the case of panel b is fm 2 10 4 (c/a). For operation at 1.55 mm, the corresponding absolute values are a 1.04 mm, R 0.92 mm,

w 0.21 mm and fm 60 GHz.

optical isolator based on a spatiotemporally modulated channel-drop lter is presented in Fig. 4, with the design details provided in the caption. Without modulation, the power entering the structure through the channel waveguide (right-hand-side waveguide) from either port 1 or 2 couples to the right- or left-handed ring resonance, respectively, creating a transmission dip at resonance (Fig. 4a). Splitting the ring resonances with proper azimuthal spatiotemporal modulation moves the transmission dips to different frequencies for opposite propagation directions, thus creating non-reciprocity. The isolator can be realized on silicon (Si), which exhibits the strongest electro-optic effect observed to date, with typical values around Dem 5 10 4es,

where es is its permittivity, leading to kmD2.5 10 4 (refs

26,27). Such permittivity modulation can be obtained using pin diodes, as analytically described in refs 9,26,27. According to our bandwidth criterion QCm 3

p , a Q-factor B7,000 would be sufcient for adequate separation of the |aS and |bS states. Such level of Q-factor is common in Si-photonics integrated systems, and in the design of Fig. 4a it is achieved using the 11th azimuthal resonance of the ring. Although the theory above was derived for the |1S states, it can be easily extended to any pair of states

j i by substituting Lm 2 with Lm 2. Therefore, Lm 22

was used in the design of Fig. 4, which may be achieved by uniformly integrating 88 pin diodes along the ring perimeter, leading to a separation of 65 nm between consecutive diodes. It should be noted that this design has not been optimized, and our

theory indicates that a signicantly lower number of pin diodes may still be sufcient to achieve a similar effect in optimized geometries.

The simulated scattering parameters of the structure without and with modulation are presented in Fig. 4a,b, respectively. In the absence of modulation, S21 S12 (Sij being the transmission

coefcient from port j to port i), indicating that the system is reciprocal. When the modulation is applied, the right- and left-handed resonances of the ring split and non-reciprocity occurs, as shown in Fig. 4b. For instance, at the right-handed resonance, indicated in Fig. 4b with the dashed line, transmission from port 1 to 2 is signicantly lower than transmission from port 2 to 1, an effect that can also be seen in the corresponding eld plots in the inset. This operation is obtained within a ring structure that is comparable in size to the operation wavelength l 1.55 mm, and

without the need of magnetic bias.

DiscussionA new approach to achieve magnetic-free non-reciprocity via angular momentum biasing was presented here on the basis of resonant rings with specically tailored spatiotemporal azimuthal modulation. The proposed form of modulation removes the degeneracy between opposite resonant states, which, combined with suitably induced high-Q response, realizes giant non-reciprocity in subwavelength components with moderate

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modulation frequencies and amplitudes. A few applications based on this approach have been presented, including an ultrathin radio frequency isolator, giant Faraday rotation and an optical isolator, all realized without requiring bulky magnetic biasing elements. The proposed approach opens pathways towards many non-reciprocal integrated microwave and nanophotonic components, without requiring magnetic bias, for a variety of applications.

Methods

Coupled-mode equations neglecting loss and excitation. The starting point of our analysis is the coupled-mode theory from Winn et al.22 for a resonator with a small permittivity perturbation De(r,t). In particular, if ak, ok and Ek are the amplitude, the eigenfrequency and the electric eld of the kth mode,

ak iokak iok X

normalization coefcient. For the discretely modulated ring,

DC DCm X

7 cos omt 2j

d j

np 4

: 13

Then, the coupling coefcients between the 1 and 1 states read k 1; 1 4DCm V10

j j2e io t: 14

If the modulation was continuous, DC DCm cos (omt 2j) and k 1; 1 pDCm V01

j j2e io t: 15

Comparing equation (14) with equation (15), we conclude that the discrete modulation is equivalent to a continuous one with amplitude 4DCm/p.

Full-wave simulations. All the numerical simulations of the varactor-loaded ring were performed in the frequency domain by combining a full-wave nite-element electromagnetic simulation (CST Microwave Studio) with a harmonic balance circuit simulation (Agilent Advanced Design System). The numerical domain was terminated with unit cell boundary conditions along the x and y directions and with wave ports along the z direction. The wave ports were used to excite the incident wave and absorb the reected and transmitted waves. Further, lumped ports were inserted where the variable capacitors were supposed to be connected. The S-parameters obtained through the electromagnetic simulations were imported into the circuit simulator. Ideal variable capacitors, implemented in Agilent Advanced Design System via equation-based non-linear components as shown in the Supplementary Fig. S1, were connected to the lumped ports of the electromagnetic model and the structure was solved for the S-parameters at the wave ports.

The optical ring of Fig. 4 was simulated with a home-made nite-difference time-domain code, where the time-domain permittivity modulation was realized by continuously changing the permittivity of the structure at each time step. The computational domain was terminated with perfectly matched absorbing layers.

kka; 4

where

Z dr3Der; tE k E 5

is the coupling coefcient between the kth and th modes. The modes are normalized as

Z dr3 e Ek

1; 6

where Hk is the magnetic eld of the kth mode. Noticing that the integral in equation (6) is essentially the energy of the kth mode and that the electric and magnetic energies of a resonator are equal,

Z dr3e Ek

j j2

j j2 m0 Hk

j j2

2 7

The eigenspectrum of the ring of Fig. 1a consists of pairs of counter-propagating modes |kS with electric elds

E k Etke ikj; 8 where Etk is the electric eld of the kth pair in the transverse r z plane. Sub

stituting equation (8) into equation (7), we arrive at the following normalization condition for Etk

Z S e Etk

References

1. Lax, B. & Button, K. J. Microwave ferrites and ferrimagnetics (McGraw-Hill, 1962).

2. Dionne, G. F. Simple derivation of four-level permittivity relations for magneto-optical applications. J. Appl. Phys. 97, 10F103 (2005).

3. Casimir, H. B. G. On Onsagers principle of microscopic reversibility. Rev. Modern Phys. 17, 343350 (1945).

4. Kodera, T., Sounas, D. L. & Caloz, C. Articial Faraday rotation using a ring metamaterial structure without static magnetic eld. Appl. Phys. Lett. 99, 03114 (2011).

5. Kodera, T., Sounas, D. L. & Caloz, C. Magnet-less non-reciprocal mematerial technology and application to microwave components. IEEE Trans. Microw. Theory Tech. 61, 10301042 (2013).

6. Kodera, T., Sounas, D. L. & Caloz, C. Non-reciprocal magnet-less CRLH leaky-wave antenna based on a ring metamaterial structure. IEEE Antennas Wireless Propag. Lett. 10, 15511554 (2012).

7. Wang, Z. et al. Gyrotropic response in the absence of a bias eld. Proc. Natl Acad. Sci. USA 109, 1319413197 (2012).

8. Yu, Z. & Fan, S. Complete optical isolation created by indirect interband photonic transitions. Nat. Photonics 3, 9194 (2009).

9. Lira, H., Yu, Z., Fan, S. & Lipson, M. Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip. Phys. Rev. Lett. 109, 033901 (2012).

10. Fang, K., Yu, Z. & Fan, S. Photonic Aharonov-Bohm effect based on dynamic modulation. Phys. Rev. Lett. 108, 153901 (2012).

11. Kang, M. S., Butsch, A. & Russell, P. S. t. J. Recongurable light-driven opto-acoustic isolators in photonic crystal bre. Nat. Photonics 5, 549553 (2011).

12. Kamal, A., Clarke, J. & Devoret, M. H. Noiseless non-reciprocity in a parametric active device. Nat. Phys. 7, 311315 (2011).

13. Wang, D.-W. et al. Optical diode made from a moving photonic crystal. Phys. Rev. Lett. 110, 093901 (2013).

14. Shadrivov, I. V., Fedotov, V. A., Powell, D. A., Kivshar, Y. S. & Zheludev, N. I. Electromagnetic wave analogue of an electronic diode. New J. Phys. 13, 033025 (2011).

15. Soljai, M., Luo, C., Joannopoulos, J. D. & Fan, S. Nonlinear photonic crystal microdevices for optical integration. Opt. Lett. 28, 637639 (2003).

16. Fan, L. et al. An all-silicon passive optical diode. Science 335, 447450 (2012).17. Ramezani, H., Kottos, T., El-Ganainy, R. & Christodoulides, D. N. Unidirectional nonlinear PT-symmetric optical structures. Phys. Rev. A 82, 043803 (2010).

18. Gallo, K., Assanto, G., Parameswaran, K. R. & Fejer, M. M. All-optical diode in a periodically poled lithium niobate waveguide. Appl. Phys. Lett. 79, 314316 (2001).

2 ; 9

where St is the ring cross section. For a permittivity modulation De(r,t) Dem cos

(omt 2j) with omooo1, the coupling between the |1S modes and the higher-

order modes can be neglected and the coupled-mode equations (4) and (5) reduce to equations (1) and (2). Dividing equations (2) and (9) yields

j j2rdrdz

S Dem Et1

j j2rdrdz

: 10

If the modulation is applied to a region, which concentrates most of the eld and where e varies very little, km Dem/(2e).

Coupled-mode equations including loss and excitation. In the presence of loss, described via a relaxation time t, the io1 term multiplying a 1 in equation (1)

should be substituted by io1 t 1 (ref. 28). In addition, when excitation is

taken into account the terms im1s1 and im 1s 1 should be added to the

right-hand side of equation (1), where s1 and s 1 are the components of the

incident wave that couple to the | 1S and | 1S states, respectively, and m1

and m 1 are the corresponding coupling coefcients, which are generally functions

of the geometry. Then, the coupled-mode equations read

a1 io1

1 t1

S e Et1

j j2rdrdz

a1 i12o1kme io ta 1 im1s1;

a 1 io1

These equations support a family of solutions similar to those of equation (3).

Discrete capacitance modulation. The ring of Fig. 2a is best analysed in terms of voltage V and current I instead of the electric and magnetic elds E and H. Then, the coupled-mode equations (4) and (5) are expected to be valid under the substitutions E-V, H-I, e-C and m0-L, so that the coupling coefcient is given by

a 1 i

2 o1kmeio ta1 im 1s 1:

1 t1

0 DCV kVdj; 12

where Vk Vk0eikj is the voltage of the kth mode, with Vk0 an appropriate

6 NATURE COMMUNICATIONS | 4:2407 | DOI: 10.1038/ncomms3407 | http://www.nature.com/naturecommunications

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3407 ARTICLE

19. Chin, J. Y. et al. Nonreciprocal plasmonics enables giant enhancement of thin-lm Faraday rotation. Nat. Commun. 4, 1559 (2013).

20. Ramezani, H. et al. Taming the ow of light via active magneto-optical impurities. Opt. Express 20, 2620026207 (2012).

21. Pendry, J. B., Holden, A. J., Robbins, D. J. & Stewart, W. J. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans. Microw. Theory Tech. 47, 20752084 (1999).

22. Winn, J. N., Fan, S., Joannopoulos, J. D. & Ippen, E. P. Interband transitions in photonic crystals. Phys. Rev. B 59, 15511554 (1999).

23. Wang, Z. & Fan, S. Optical circulators in two-dimensional magneto-optical photonic crystals. Opt. Lett. 30, 19891991 (2005).

24. Lukyanchuk, B. et al. The Fano resonance in plasmonic nanostructures and metamaterials. Nat. Mater. 9, 707715 (2010).

25. Shaei, F. et al. A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance. Nat. Nanotech. 8, 9599 (2013).

26. Xu, Q., Schmidt, B., Pradhan, S. & Lipson, M. Micrometer-scale silicon electro-optic modulator. Nature 435, 325327 (2005).

27. Xu, Q., Manipatruni, S., Schmidt, B., Shakya, J. & Lispon, M. 12.5 Gbits/s carrier-injection-based silicon micro-ring silicon modulators. Opt. Express 15, 430436 (2007).

28. Little, B. E., Chu, S. T., Haus, H. A., Foresi, J. & Laine, J.-P. Microring resonator channel dropping lter. J. Lightwave Technol. 15, 9981005 (1997).

Acknowledgements

This work was partially supported by the DTRA YIP Award No. HDTRA1-12-1-0022 and the AFOSR YIP Award No. FA9550-11-1-0009.

Author contributions

D.L.S. and A.A. developed the concept. D.L.S. carried out the analytical and numerical modelling. A.A., as the principal investigator, planned, coordinated and supervised the project. C.C. co-supervised the project. All authors discussed the results and commented on the article.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

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Competing nancial interests: The authors declare no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

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Web End =reprintsandpermissions/

How to cite this article: Sounas, D.L. et al. Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials. Nat. Commun. 4:2407doi: 10.1038/ncomms3407 (2013).

NATURE COMMUNICATIONS | 4:2407 | DOI: 10.1038/ncomms3407 | http://www.nature.com/naturecommunications

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Breaking time-reversal symmetry enables the realization of non-reciprocal devices, such as isolators and circulators, of fundamental importance in microwave and photonic communication systems. This effect is almost exclusively achieved today through magneto-optical phenomena, which are incompatible with integrated technology because of the required large magnetic bias. However, this is not the only option to break reciprocity. The Onsager-Casimir principle states that any odd vector under time reversal, such as electric current and linear momentum, can also produce a non-reciprocal response. These recently analysed alternatives typically work over a limited portion of the electromagnetic spectrum and/or are often characterized by weak effects, requiring large volumes of operation. Here we show that these limitations may be overcome by angular momentum-biased metamaterials, in which a properly tailored spatiotemporal modulation is azimuthally applied to subwavelength Fano-resonant inclusions, producing largely enhanced non-reciprocal response at the subwavelength scale, in principle applicable from radio to optical frequencies.

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Title

Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials

Author

Sounas, Dimitrios L; Caloz, Christophe; Alù, Andrea

Pages

2407

Publication year

2013

Publication date

Sep 2013

Publisher

Nature Publishing Group

e-ISSN

20411723

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms3407

ProQuest document ID

1429233018

Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials

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