Nonreciprocity without magnetooptics a tutorial Shanhui Fan Ginzton

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

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

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

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

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.

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.

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

Lorentz Reciprocity Theorem 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. H. Lorentz (1896); H. A. Haus, Waves and Fields in Optoelectronics (1984) It applies independent of structural complexity, e. g. Metal (Al, Cu, …) Dielectric (Si, Si. O 2, Ga. As, Ge, …. ) If the optical properties are entirely described by

Reciprocal system has a symmetric scattering matrix a 1 a 2 a 3 b 2 b 3 Device b 1 Input-output is defined by the scattering matrix (S-matrix) 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.

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

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

Faraday Rotation E k M M

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

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

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

Reciprocal system has a symmetric scattering matrix a 1 a 2 a 3 b 2 b 3 Device b 1 Input-output relation is defined by the scattering matrix Reciprocity theorem implies that e. g.

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

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

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

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

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

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.

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. )

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

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).

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. )

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.

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

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

Asymmetric distribution of the field 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

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

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

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. )

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.

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

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).

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

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

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

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

Direct transition Uniform modulation along z-direction Air Silicon z

Oscillation between two states

Direct transition independent of the modulation phase

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

Downward and upper-ward transition acquires a phase difference

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

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

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

The Scheme Filter Mixer Phase shifter

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

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

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

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.

Photons on a dynamic lattice • Two sub-lattices of resonators • Coupling constant between nearest neighbor resonators dynamically modulated. 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).

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).

Simple but unusual gauge potential configurations A n 1

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

A constant gauge potential shifts the constant frequency contour A n 1 A

Gauge field induced negative refraction A n 1 A K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013).

Gauge field induced total internal reflection A n 1 A K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013).

A single-interface four-port circulator A n 1 • Both regions have zero effect B-field. • A B-field sheet at the interface. K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013).

A novel one-way waveguide Light cone of the cladding Light cone of the core A n 1 n 1 Waveguide mode exists only in the positive ky region Q. Lin and S. Fan, Physical Review X 4, 031031 (2014).

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.