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

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

3. Optical isolator: a one-way street for light
Single-mode signal
Any backreflection 3

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

5. 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.

6. 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.

7. 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.

8. 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, SiO2, GaAs, Ge, ….)
If the optical properties are entirely described by

9. Reciprocal system has a symmetric scattering matrixb1
Device a2 b2 a3 b3
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.

10. Conventional optical isolators
5cm
Images from
Use magneto-optical materials 10

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

12. Faraday Rotation E k M M

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

14. Isolator Based on Faraday Rotation
Polarizer at 0o
Polarizer at 45o M E k
SMF
SMF
SMF
SMF 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

15. The main question of the tutorial Silicon Photonics Platform
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.

16. Reciprocal system has a symmetric scattering matrix
Device b1 b2 b3
Input-output relation is defined by the scattering matrix
Reciprocity theorem implies that
e.g.

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

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

19. 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, (2012).
S. Fan et al, Science 335, 38 (2012) [Comment on Feng et al, Science 333, 729, 2011]

20. 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, (2012)
S. Fan et al, Science 335, 38 (2012) [Comment on Feng et al, Science 333, 729, 2011]

21. How does one really test non-reciprocity?
Device
Device
High transmission, left to right
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).

22. How does one really test non-reciprocity?
Single-mode waveguide
Single-mode waveguide
Device
Device
High transmission, left to right
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).

23. 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.

24. 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.)

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

26. Combination of Si and Magneto-Optical Material
37dB insertion coupling loss between fiber and waveguide, about 8dB additional loss on-chip at transmission peak. Maximum isolation ratio is 21dB in APL 2008.
Y. Shoji, T. Mitzumoto, R. M. Osgood et al, Applied Physics Letters 92, (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, (2011).

27. 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.)

28. 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.

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

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

31. 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
Single-mode waveguide

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

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

34. 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.)

35. 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.

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

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

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

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

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

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

42. Direct transition Uniform modulation along z-direction z Air Silicon42

43. Oscillation between two states43

44. Direct transition independent of the modulation phase44

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

46. Downward and upper-ward transition acquires a phase difference

47. 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

48. A Photonic Aharonov-Bohm Interferometer as an Optical Isolator
silicon
air
Connect with previous slides: size. Number of pads
K. Fang, Z. Yu and S. Fan, Physical Review Letters 108, (2012).

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

50. The Scheme
Filter
Mixer
Phase shifter

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

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

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

54. 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.

55. 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).

56. 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).

57. Simple but unusual gauge potential configurations

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

59. A constant gauge potential shifts the constant frequency contour59

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

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

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

63. A novel one-way waveguide
Light cone of the cladding
Light cone of the core n1 n1 n1 A
Waveguide mode exists only in the positive ky region
Q. Lin and S. Fan, Physical Review X 4, (2014).

64. 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.