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Non-reciprocity without magneto-optics: a tutorial Shanhui Fan Ginzton Laboratory and Department of Electrical Engineering Stanford University

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Large-scale on-chip network Towards large-scale on-chip information network Large-scale communication network

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Optical isolator: a one-way street for light Single-mode signal Any backreflection

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Silicon Photonics Platform The main question of the tutorial How does one achieve optical isolation on a standard optoelectronic platform?

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Outline of my talk The basics of reciprocity. Options for on-chip non-reciprocity. Nonlinear optical isolator: fundamental limitation. Dynamic modulation: effective gauge potential for photons.

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Outline of my talk The basics of reciprocity. Options for on-chip non-reciprocity. Nonlinear optical isolator: fundamental limitation. Dynamic modulation: effective gauge potential for photons.

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What do you need isolator for? Device Output signal Parasitic reflection Device Isolator Output signal Parasitic reflection Parasitic reflection is assumed to be unknown in system design. Therefore isolator needs to be non-reciprocal device.

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Lorentz Reciprocity Theorem H. Lorentz (1896); H. A. Haus, Waves and Fields in Optoelectronics (1984) The theorem applies to any electromagnetic system that is: linear, time-independent, has a symmetric permittivity and permeability tensor, including medium that has gain or loss. It applies independent of structural complexity, e.g. Dielectric (Si, SiO 2, GaAs, Ge, .) Metal (Al, Cu,) If the optical properties are entirely described by

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Reciprocal system has a symmetric scattering matrix Input-output is defined by the scattering matrix (S-matrix) a1a1 b1b1 Device a2a2 b2b2 a3a3 b3b3 Reciprocity theorem implies that e.g. Reciprocity relates two pathways that are related by time-reversal. Reciprocity therefore is closely related to time-reversal symmetry.

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5cm Conventional optical isolators Images from www.ofr.com Use magneto-optical materials

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Magneto-optical effect is non-reciprocal M e. g. YIG z Dielectric tensor Asymmetric Non-reciprocal Hermitian Energy conserving

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Faraday Rotation M M E k

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Faraday Rotation Has An Asymetric S-matrix M M E k Mode 1Mode 2

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Isolator Based on Faraday Rotation Polarizer at 0 o Polarizer at 45 o M M E k X High transmission in the forward direction. Suppress backward propagation for every mode of reflection. Suppress backward propagation independent of the existence of forward signal SMF

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Silicon Photonics Platform The main question of the tutorial How does one achieve optical isolation on a standard optoelectronic platform? As a matter of principle, one can not construct a passive, linear, silicon isolator.

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Reciprocal system has a symmetric scattering matrix Input-output relation is defined by the scattering matrix a1a1 b1b1 Device a2a2 b2b2 a3a3 b3b3 Reciprocity theorem implies that e.g.

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Isolator needs to suppress reflection from every mode High transmission, left to right Necessarily implies that one can create a input mode profile to achieve high transmission from right to left For reciprocal structure Therefore, one cannot construct an isolator out of reciprocal structure. Device

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But I see asymmetry in my experiment and simulations! High transmission, left to rightLow transmission, right to left Is this an isolator? Silicon Unidirectionality, Optical Diode, ..

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Nonreciprocal light propagation in an aperiodic silicon photonic circuits? S. Fan et al, Science 335, 38 (2012) [Comment on Feng et al, Science 333, 729, 2011] Near perfect transmission, left to right Near perfect reflection, right to left V. Liu, D. A. B. Miller and S. Fan, Optics Express 20, 28318 (2012).

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Mode-to-mode transmission coefficient always symmetric S. Fan et al, Science 335, 38 (2012) [Comment on Feng et al, Science 333, 729, 2011] Nonreciprocal light propagation in an aperiodic silicon photonic circuits? V. Liu, D. A. B. Miller and S. Fan, Optics Express 20, 28318 (2012)

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How does one really test non-reciprocity? D. Jalas et al, Nature Photonics 7, 579 (2013). Device High transmission, left to rightLow transmission, right to left Send time-reversed output back into the device Detect asymmetry in transmission between two modes.

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How does one really test non-reciprocity? D. Jalas et al, Nature Photonics 7, 579 (2013). Device High transmission, left to rightLow transmission, right to left which is how isolator in practice will be used in an on-chip setting Single-mode waveguide Test transmission asymmetry between two single-mode waveguides

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Outline of my talk The basics of reciprocity. Options for on-chip non-reciprocity. Nonlinear optical isolator: fundamental limitation. Dynamic modulation: effective gauge potential for photons.

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Only ways to achieve on-chip optical isolation Lorentz reciprocity theorem applies to any electromagnetic system that is: linear, time-independent, has a symmetric permittivity and permeability tensor. Therefore, to create optical isolation on-chip, the only options are: On-chip integration of magneto-optical materials. Exploit nonlinearity. Consider time-dependent systems. (e.g. systems where the refractive index varies as a function of time.)

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On-chip integration of magneto-optical materials Silicon Photonics Platform Yittrium Iron Garnet

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Combination of Si and Magneto-Optical Material Y. Shoji, T. Mitzumoto, R. M. Osgood et al, Applied Physics Letters 92, 071117 (2008). For related experimental developments, See L. Bi, L. C. Kimering and C. A. Ross et al, Nature Photonics 5, 758 (2011) M. Tien, T. Mizumoto, and J. E. Bowers et al, Optics Express 19, 11740 (2011).

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Only ways to achieve on-chip optical isolation Lorentz reciprocity theorem applies to any electromagnetic system that is: linear, time-independent, has a symmetric permittivity and permeability tensor. Therefore, to create optical isolation on-chip, the only options are: On-chip integration of magneto-optical materials. Exploit nonlinearity. Consider time-dependent systems. (e.g. systems where the refractive index varies as a function of time.)

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Outline of my talk The basics of reciprocity. Options for on-chip non-reciprocity. Nonlinear optical isolator: fundamental limitation. Dynamic modulation: effective gauge potential for photons.

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An optical isolator using intensity dependent index Input power 85 nW Input power 85 W L. Fan, A. Weiner and M. Qi, et al, Science 335, 447 (2012).

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The idea of a nonlinear isolator: starting point Single-mode waveguide Start with a linear, reciprocal, spatially asymmetric structure Single-mode waveguide Weak transmission in the linear regime Transmission completely reciprocal

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Asymmetric distribution of the field Single-mode waveguide While the transmission is reciprocal, the field distribution in the structure depends on incident light direction Single-mode waveguide Weak transmission in the linear regime

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Nonlinear structure breaks reciprocity Single-mode waveguide Forward and backward light now sees a different dielectric structure Single-mode waveguide High transmission in the forward direction Low transmission in the backward direction Kerr nonlinearity So there is a contrast in the forward and backward direction! Kerr nonlinearity

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Nonlinear optical isolators in fact do not isolate Forward signal When forward signal is present, there is no isolation High transmission for noise in the forward direction High transmission for noise in the backward direction Kerr nonlinearity Y. Shi, Z. Yu and S. Fan, Nature Photonics 9, 388 (2015).

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Only ways to achieve on-chip optical isolation Lorentz reciprocity theorem applies to any electromagnetic system that is: linear, time-independent, has a symmetric permittivity and permeability tensor. Therefore, to create optical isolation on-chip, the only options are: On-chip integration of magneto-optical materials. Exploit nonlinearity. Consider time-dependent systems. (e.g. systems where the refractive index varies as a function of time.)

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Outline of my talk The basics of reciprocity. Options for on-chip non-reciprocity. Nonlinear optical isolator: fundamental limitation. Dynamic modulation: effective gauge potential for photons.

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Time-reversal symmetry and reciprocity breaking in time- dependent systems Break time-reversal symmetry and reciprocity as long as:

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Dynamic optical isolators Z. Yu and S. Fan, Nature Photonics, vol. 3, pp. 91-94 (2009); H. Lira, Z. Yu, S. Fan and M. Lipson, Physical Review Letters 109, 033901 (2012). See Also: G. Shvets, Physics 5, 78 (2012).

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Static magnetic field breaks time-reversal symmetry for electrons Can we create an effective magnetic field for photons? B B

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gauge potential for photons K. Fang, Z. Yu and S. Fan, Physical Review Letters 108, 153901 (2012). Si Metal electrode: applying a time-dependent voltage

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Magnetic field for electrons in quantum mechanics Electron couples to the vector gauge potential

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1 2 Propagation phase Gauge potential results in a direction-dependent phase 1 2

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Direct transition Air Silicon z Uniform modulation along z-direction

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Oscillation between two states

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Direct transition independent of the modulation phase

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Modulation phase provides a gauge transformation of the photon wavefunction Gauge potential that couples to the photon

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Downward and upper-ward transition acquires a phase difference

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A Photonic Aharonov-Bohm Interferometer Clockwise roundtrip has a phase: Counter-clockwise roundtrip has a phase: Phase difference between two time-reversal related trajectories due to a gauge degree of freedom

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K. Fang, Z. Yu and S. Fan, Physical Review Letters 108, 153901 (2012). A Photonic Aharonov-Bohm Interferometer as an Optical Isolator silicon air

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Experimental demonstration of photonic AB effect FilterMixerFilterMixerFilter Phase shifter Mixer provides the modulation K. Fang, Z. Yu, and S. Fan, Phys. Rev. B Rapid Communications 87, 060301 (2013).

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The Scheme Filter Mixer Phase shifter

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FilterMixerFilterMixerFilter Phase shifter Non-reciprocal oscillation as a function of modulation phase

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AB Interferometer from Photon-Phonon Interaction E. Li, B. Eggleton, K. Fang and S. Fan, Nature Communications 5, 3225 (2014). Local oscillator (50MHz) AOM (Acoustic- Optic Modulator) He-Ne Laser (633nm)

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AB interferometer on a silicon platform L. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, Nature Photonics 8, 701 (2014).

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Electron on a lattice Electron hopping on a tight-binding lattice Single unit cell Magnetic field manifests in terms of a non-reciprocal round-trip phase as an electron hops along the edge of a unit cell.

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Two sub-lattices of resonators Coupling constant between nearest neighbor resonators dynamically modulated. Photons on a dynamic lattice K. Fang, Z. Yu and S. Fan, Nature Photonics 6, 782 (2012). See also M. Hafezi et al, Nature Physics 7, 907 (2011); M. C. Rechtsman et al, Nature 496, 196 (2013).

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Constructing effective magnetic field for photons Lorentz force for photons Analogue of Integer quantum hall effects for photons. K. Fang, Z. Yu and S. Fan, Nature Photonics 6, 782 (2012).

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Simple but unusual gauge potential configurations n1n1 n1n1 A

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The effect of a constant gauge potential For electrons In general, a constant gauge potential shifts the wavevector

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A constant gauge potential shifts the constant frequency contour n1n1 n1n1 A A

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Gauge field induced negative refraction n1n1 n1n1 A K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013). A

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Gauge field induced total internal reflection n1n1 n1n1 A K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013). A

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n1n1 n1n1 A A single-interface four-port circulator K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013). Both regions have zero effect B-field. A B-field sheet at the interface.

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A novel one-way waveguide n1n1 n1n1 n1n1 A Waveguide mode exists only in the positive k y region Light cone of the cladding Light cone of the core Q. Lin and S. Fan, Physical Review X 4, 031031 (2014).

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Summary To create optical isolation on a silicon platform, Isolators need to suppress all reflections. Therefore, there is no passive, linear, silicon isolator. The only options for optical isolations on silicon chip are: Integration of magneto-optical materials on chip. Significant material science challenges are being overcome. Nonlinear isolators. Innovative concepts. But does not provide complete optical isolation. Dynamic isolators from refractive index modulation. Can completely reproduce standard magneto-optical isolator functionality. Does require energy input. There is exciting fundamental physics in on-chip non-reciprocal photonics.