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Dual-carrier Floquet circulator with time-modulated optical resonators Ian A.D. Williamson, S. Hossein Mousavi, and Zheng Wang ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00576 • Publication Date (Web): 20 Aug 2018 Downloaded from http://pubs.acs.org on August 22, 2018
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Dual-carrier Floquet circulator with time-modulated optical resonators Ian A. D. Williamson, 1 S. Hossein Mousavi, 2 Zheng Wang * Microelectronics Research Center, The University of Texas at Austin, Austin, TX 78758 USA 1 Present
address: Ginzton Laboratory, Stanford University, Stanford, California 94305 USA address: Infinera Corporation, 140 Caspian Ct., Sunnyvale, CA 94089 USA *Corresponding author:
[email protected] 2 Present
Spatio-temporal modulation has shown great promise as a strong time-reversal symmetry breaking mechanism that enables integrated nonreciprocal devices and topological materials at optical frequencies. However, ideal circulator and isolator performance has relied on spatial symmetry or momentum matching between modulation and optical modes. The resulting systems have been challenging to experimentally realize, due to the prohibitively complex and lossy biasing networks and tight fabrication tolerances that maintain the desired rotational and mirror symmetries. In this work, we propose a micro-resonator Floquet circulator that leverages the previously untapped degrees of freedom of the modulation, through waveforms with strong harmonic components. The Floquet circulator response exhibits ideal on-resonance isolation and supports broadband forward transmission with no tradeoff in insertion loss. We present a numerical demonstration in an onchip photonic crystal platform with just two modulated resonators requiring no rotational symmetry. Moreover, this approach is general and can leverage a variety of modulation mechanisms while not being limited by pump depletion and signal distortion associated with parametric nonlinear systems. Keywords:
modulator, circulator, nonreciprocity, resonator, photonic crystal
Nonreciprocal devices, such as circulators, play essential roles in modern optical systems to prevent feedback-induced instability in lasers1 and amplifiers2 and to protect interferometers3, full-duplex transceivers4, and read-out circuits in quantum computers from interference5. When combined into periodic lattices and arrays, nonreciprocal devices also provide the broken time-reversal symmetry needed to create topologically protected photonic edge states that are immune to disorder-induced backscattering6,7. The realization of chip-scale circulators and isolators for integrated photonics remains a grand challenge due to weak magnetic effects at optical frequencies and issues with the compatibility of magneto-optical materials and silicon photonics. Spatio-temporally modulated systems have recently emerged as a promising alternative to realize nonreciprocal responses without magnets, either via photonic transitions between spatial modes in waveguides and resonators8, mode splitting in traveling-wave resonators9,10, topological edge states in resonator arrays11, or parametric modulators12,13. Many of these schemes have been successfully demonstrated at microwave frequencies using lumped elements14, and the use of resonators has led to wavelength-scale nonreciprocal devices, which significantly reduce device footprint and propagation-induced insertion
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loss from traveling-wave spatial-temporal modulation. However, at optical frequencies, further miniaturization and broad bandwidth remain challenging for spatio-temporally modulated circulators15 within the realistic limitations of modulation index and available bandwidth of the mechanisms such as stimulated Brillouin scattering16,17, cavity optomechanics18–20, Kerr nonlinearities21, and carrier-induced nonlinearity22. These challenges are fundamentally linked to the conventional structures of three-port Y-junction circulators or four-port circulators, originally conceived to harness the large frequency splitting from magnetized ferrites23 and still retained by most spatio-temporally modulated circulators. These circulators are narrowband devices: off resonance, both forward transmission and isolation degrade rapidly and the operational bandwidth is proportional to the mode splitting. At optical frequencies, relative to the operating frequency, mode splitting from either magneto-optical effects or spatiotemporal modulation is orders of magnitude lower than that at microwave frequencies, resulting in significant bandwidth reduction, large device footprints24, and large networks of biasing elements7. In addition, high structural symmetry is required to simultaneously realize complete transmission in the forward directions and large isolation23. This operating condition is crucial to the improvement of a system’s signal-to-interference ratio – the difference between the forward transmission for the desired signals and the backward isolation for the interference – and remains challenging for fabricated optical devices. Floquet states that arise in periodically time-modulated systems have been leveraged to break timereversal symmetry to achieve non-reciprocity25 and topological orders11,26 in photonics and acoustics7 using materials that are naturally reciprocal. In general, a time-invariant system is transformed into a Floquet system by an externally applied periodic modulation, with every static eigenstate spawning a set of Floquet modes equally spaced in frequency, with relative amplitudes and phases determined by the modulating waveform. Purely sinusoidal modulation produces an frequency distribution given by Bessel functions of the first kind, which is identical to the sideband amplitudes produced in phase modulation27 (Fig. 1a). However, these amplitudes can be individually tailored with more general forms of modulation beyond pure single-frequency sinusoidal waveforms28,29 (Fig. 1b). This opens new opportunities for creating unconventional nonreciprocal responses that are not limited by the spatial symmetry or narrowband responses discussed earlier. These sideband amplitudes are well understood in the framework of parametric resonances in dynamical systems30 and quantum field theory31, but the deliberate control over them remains to be exploited as a new degree of freedom in nonreciprocal photonics or acoustics. In this article, we present a compact three-port circulator based on individually tailored Floquet modes that provides broadband nonreciprocal transmission distinct from that of conventional circulators. We first discuss the nonreciprocal phase shift and general scattering properties of a single parametrically modulated resonator side-coupled to two waveguides. We then apply temporal coupled mode theory to study the necessary conditions that produce unique circulator responses from a low-symmetry cascade of two such Floquet resonators. Finally, a photonic crystal realization of such a Floquet circulator is presented from the results of first-principle numerical simulations.
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Fig. 1. Schematics of the Floquet mode frequency distributions from (a) single-tone sinusoidal modulation and (b) multi-tone modulation waveforms. Single-frequency modulation results in a amplitude distribution defined by Bessel functions of the first kind at intervals of the fundamental modulation frequency. (c) Schematic of a dual resonator Floquet circulator with resonators supporting identical modes at frequency ωa . The resonators are modulated by single-frequency sinusoidal waveforms with fundamental frequency Ω and phases φa and φb . The top waveguide (red) targets the n = 1 Floquet mode with an evanescent coupling rate of γ 1 , and the bottom waveguide (blue) targets the n = 0 Floquet mode with an evanescent coupling rate of
γ 0 . The phase delay between the resonators can be different in the top and bottom waveguides, which is defined by θ0 and θ1 . (d) Schematic of compound even and odd modes in the two waveguides, corresponding to a rotation of the scattering matrix into a basis defined by the vectors in Eqn. 5. The ports of T the circulator are defined in compound modes on the left reference plane as α E = (1 1) and
α O = (1 −1) and on the right reference plane as β E = (1 1) and βO = (1 −1) , where the first (second) vector element indicates the amplitude of the wave in the bottom (top) waveguide. T
T
T
Results Floquet Resonator Nonreciprocal Phase Shifter The building block of the proposed circulator is a single time-modulated resonator supporting a set of Floquet modes, which exhibits a highly nonreciprocal phase response when coupled to two narrowband waveguides. As shown in Fig. 1a, an infinite number of sidebands, i.e. Floquet modes, reside in the resonator. The sidebands are spectrally distributed on both sides of the intrinsic structural resonance frequency ωa at intervals given by the fundamental modulation frequency Ω. We denote the complex instantaneous amplitude of the nth Floquet mode by a( ) . In a high quality factor resonator, where the modulation driven energy exchange between the sidebands exceeds the external coupling, the amplitudes of the sidebands are coherently correlated. Thus, the amplitude n
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a( ) can be expressed as the product a ( n ) = u ( ) a , between the total Floquet mode amplitude a and n
n
the relative Floquet mode amplitude u ( ) . a is proportional to the total energy aggregated over all n
2
Floquet modes and u ( ) is the solution of Hill’s differential equation for the case of a general periodic modulation waveform29,32. n
Although the amplitudes of the Floquet modes are correlated, each couples very differently to the external environment, such that the non-trivial phase between the modulating waveform and the optical carrier waves produces a nonreciprocal response. Similarly to the well-known channel adddrop filter configuration33, we side-couple two parallel narrowband waveguides to either side of the Floquet resonator (Fig. 1c), with each waveguide having a narrow enough bandwidth that allows it to couple only to one of the Floquet modes. In the system considered here, the bottom waveguide targets the zero-order Floquet mode and the top waveguide targets the first-order Floquet mode. The evanescent coupling from the resonator to the bottom (top) waveguide results in a decay rate γ 0 ( γ 1 ) . In a high quality factor resonator, temporal coupled mode theory (CMT) accurately describes the time evolution of the Floquet mode amplitudes. For the system considered in Fig. 1c, but with only a single modulated resonator, two coupled mode equations are used to capture the time evolution of each mode, s d (0) 0 u a = ( jωa − γ ) u ( ) a + (κ 0 κ 0 ) 1+ dt s2 + s d (1) 1 u a = ( jωa + jΩ − γ ) u ( ) a + (κ1 κ1 ) 3+ dt s4 +
(1)
where sm + is the instantaneous amplitude of the incident wave from the mth port and
γ = γ 0 + γ 1 + γ L for the loss rate due to absorption and radiation given by γ L . Note that only two Floquet modes need to be explicitly considered, and all others are uncoupled from the external environment and maintained at relative amplitudes dictated by the modulation waveform. Unlike conventional nonreciprocal systems involving Floquet states7, the system considered here is unique in that the signal can be transferred between multiple Floquet modes. To clearly differentiate the signal wave from the carriers, we decompose the instantaneous amplitudes of the incoming and j ω + nΩ t outgoing waves at the mth port, sm ± (ω ) = s%m ± ( ∆ ) e ( a ) to a slowly varying envelope s%m ± (∆ ) , i.e. the signal wave, and the carrier e ( a ) . The instantaneous frequency ω is related to the frequency detuning parameter by ∆ = ω − ( nΩ + ωa ) where the integer n is the Floquet mode targeted by the j ω + nΩ t
particular waveguide or port. The scattering parameters throughout the remainder of this paper are defined in terms of the signal wave, i.e. Smp = s%m− ( ∆ ) / s% p + ( ∆ ) , where the optical carrier frequencies at ports m and p are generally different. The slowly varying envelope of the resonator a% is defined similarly.
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The coefficients κ i are determined from time-reversal symmetry and energy conservation34 (see supporting information). Eqn. 1 can be converted to a single expression for the transfer function between the waves and the resonator as
a% =
(
1 0 d 0 u ( )* j∆ + γ
d 0u (
0 )*
d1u ( )* 1
d1u ( )* 1
)
s%1+ % s ⋅ 2+ , s%3+ s%4 +
(2)
where d0 ( d1 ) represent the structural coupling between the bottom (top) waveguide. The associated decay rates are γ 0 = d 0 ⋅ u ( 0)
2
2
and γ 1 = d1 ⋅ u (1) , which is the product of a structural
coupling factor and the relative mode amplitude (see supporting information). When absorption or radiation loss is negligible ( γ L
