Transcript Fan

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
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, SiO2, GaAs, Ge, ….)
If the optical properties are entirely described by
e ( r ) = e¢ ( r ) + ie¢¢ ( r )
Reciprocal system has a symmetric scattering matrix
a1
a2
a3
b2
b3
Device
e ( r ) = e¢ ( r ) + ie¢¢ ( r )
b1
Input-output is defined by the scattering matrix (S-matrix)
æ b1 ö æ S11 S12
ç
÷ ç
b
ç 2 ÷ = ç S21 S22
ç b ÷ ç S
è 3 ø è 31 S32
Reciprocity theorem implies that
S13 ö æ a1 ö
֍
÷
S23 ÷ ç a2 ÷
S33 ÷ø çè a3 ÷ø
ST = S
e.g.
S13 = S31
Reciprocity relates two pathways that are related by time-reversal.
Reciprocity therefore is closely related to time-reversal symmetry.
Conventional optical isolators
5cm
Images from www.ofr.com
Use magneto-optical materials
Magneto-optical effect is non-reciprocal
e. g. YIG
M
z
Dielectric tensor
æ e
ö
e
+
i
e
0
1
2
3
ç
÷
e = ç e 2 - ie3
e1
0 ÷
çç
÷÷
0
0
e1 ø
è
• Asymmetric
• Non-reciprocal
• Hermitian
• Energy conserving
Faraday Rotation
E
k
M
M
Faraday Rotation Has An Asymetric S-matrix
Mode 1
Mode 2
E
S21 =1
S12 = 0
k
M
M
Isolator Based on Faraday Rotation
Polarizer at 0o
Polarizer at 45o
E
SMF
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
a1
a2
a3
b2
b3
Device
e ( r ) = e¢ ( r ) + ie¢¢ ( r )
b1
Input-output relation is defined by the scattering matrix
æ b1 ö æ S11 S12
ç
÷ ç
b
ç 2 ÷ = ç S21 S22
ç b ÷ ç S
è 3 ø è 31 S32
Reciprocity theorem implies that
S13 ö æ a1 ö
֍
÷
S23 ÷ ç a2 ÷
S33 ÷ø çè a3 ÷ø
ST = S
e.g.
S13 = S31
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
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
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 nW
L. Fan, A. Weiner and M. Qi, et al, Science 335, 447 (2012).
Input power 85 mW
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
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
Single-mode
waveguide
Kerr
nonlinearity
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:
e ( r, t ) ¹ e ( r, -t )
Dynamic optical isolators
Output
C
Index Change (a.u.)
A
0
-1
0
V0cos(wmt)
V0cos(wmt-p/2)
Z. Yu and S. Fan, Nature Photonics,
vol. 3, pp. 91-94 (2009);
1
InputZ. Yu, S. Fan and M. Lipson, Physical
H. Lira,
Review Letters 109, 033901 (2012).
lm/2
B
See Also: G. Shvets, Physics 5, 78 (2012).
V0cos(wmt)
+
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
e ( r, t )
gauge potential for photons
K. Fang, Z. Yu and S. Fan, Physical Review Letters 108, 153901 (2012).
Magnetic field for electrons in quantum mechanics
B = Ñ´ A
• Electron couples to the vector gauge potential
Gauge potential results in a direction-dependent phase
Propagation phase
Propagation phase
A=0
A¹0
1
2
fs
fs + q ò1 ds × A
1
2
fs
fs + q ò 2 ds × A
2
1
Direct transition
Uniform modulation along z-direction De = d cos ( Wt )
Air
Silicon
2
1
z
Oscillation between two states
1.0
2
De = d cos (Wt )
1
Probability
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
Time
0.8
1.0
Direct transition independent of the modulation phase
1.0
2
De = d cos (Wt + f )
1
Probability
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
Time
0.8
1.0
Modulation phase provides a gauge transformation of
the photon wavefunction
De = d cos (Wt + f )
Gauge potential that couples to the photon
Downward and upper-ward transition acquires a phase
difference
2
Ve
1
-if
Ve
if
De = d cos (Wt + f )
A Photonic Aharonov-Bohm Interferometer
e
jj 2
2
2
Ve
-if1
Ve
if1
Ve
-if2
Ve
if2
1
1
e
jj1
Clockwise roundtrip has a phase:
-f1 + j2 + f2 + j1
Counter-clockwise roundtrip has a phase:
-f2 + j2 + f1 + j1
Phase difference between two time-reversal related trajectories due to
a gauge degree of freedom
A Photonic Aharonov-Bohm Interferometer as an Optical
Isolator
d n ~ cos ( Wt )
d n ~ cos ( Wt + p / 2 )
2
1
silicon
air
2
K. Fang, Z. Yu and S. Fan, Physical Review Letters 108, 153901 (2012).
2
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, 060301 (2013).
The Scheme
Filter
Mixer
w
w + 2W
w
w ±W w +W
w
(
cos Wt + f1
)
(
cos Wt + f2
)
w
Df = Dfreciprocal + f1 - f2
Phase shifter
w + 2W
w +W w ±W
w
w
(
cos Wt + f1
)
(
w
cos Wt + f2
)
w
Df = Dfreciprocal + f2 - f1
Non-reciprocal oscillation as a function of modulation phase
cos (Wt + f1 )
Filter
Mixer
cos (Wt + f2 )
Filter
Mixer
Filter
Phase shifter
Dfreciprocal = p / 2
f1 - f2
AB Interferometer from Photon-Phonon Interaction
He-Ne Laser
(633nm)
AOM (AcousticOptic Modulator)
f1
f2
Local oscillator
(50MHz)
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
B
B
f = ò dl × A =
òò ds × B
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.
wA
wB
wA
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).
Vcos(Wt + f )
W = wB -wA
wB
Constructing effective magnetic field for photons
B¹0
B=0
f
f
0
f
f
f
2f
3f
4f
5f
f
2f
3f
4f
5f
f
2f
3f
4f
5f
f
2f
3f
4f
5f
• 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
n1
n1
The effect of a constant gauge potential
For electrons
1
H=
(-iÑ)2 + V
2m
1
H=
(-iÑ - A)2 + V
2m
In general, a constant gauge potential shifts the wavevector
-iÑ ® -iÑ - A
w ( k ) ® w ( k - A)
A constant gauge potential shifts the constant
frequency contour
A
n1
n1
A
G
G
Gauge field induced negative refraction
A
n1
n1
A
G
K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013).
G
Gauge field induced total internal reflection
A
n1
n1
A
G
K. Fang, S. Fan, Physical Review Letters 111, 203901 (2013).
G
A single-interface four-port circulator
A
n1
n1
• 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
n1
n1
n1
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.