“1d” Waveguides + Cavities = Devices
Lossless Bends
[ A. Mekis et al.,
Phys. Rev. Lett. 77, 3787 (1996) ]
symmetry + single-mode + “1d” = resonances of 100% transmission
Waveguides + Cavities = Devices
“Coupling-of-Modes-in-Time”
(a form of coupled-mode theory)
[H. Haus, Waves and Fields in Optoelectronics]
input
a
s2–
resonant cavity
frequency !0, lifetime "
“tunneling”
Ugh, must we simulate this to get the basic behavior?
s1+
s1–
da
2
2
= !i" 0 a ! a +
s1+
dt
#
#
s1! = !s1+ +
2
2
a, s2! =
a
#
#
output
|s|2 = flux
|a|2 = energy
assumes only:
• exponential decay
(strong confinement)
• conservation of energy
• time-reversal symmetry
“Coupling-of-Modes-in-Time”
Wide-angle Splitters
(a form of coupled-mode theory)
[H. Haus, Waves and Fields in Optoelectronics]
input
s1+
s1–
a
output
s2–
resonant cavity
frequency !0, lifetime "
transmission T
= |! s2– |2 / |! s1+ |2
1
|s|2 = flux
|a|2 = energy
T = Lorentzian filter
1
2
=
Q ! 0"
FWHM
!0
!
=
4
!2
(" # " 0 )2 + 42
!
…quality factor Q
[ S. Fan et al., J. Opt. Soc. Am. B 18, 162 (2001) ]
Waveguide Crossings
Waveguide Crossings
1
1x1
throughput
0.8
0.6
3x3
0.4
empty
0.2
empty
1x10-2
1x1
3x3
1x10-4
3x3
1x10-6
5x5
crosstalk
empty
5x5
5x5
0
1x100
1x1
1x10-8
1x10-10
0.32
[ S. G. Johnson et al., Opt. Lett. 23, 1855 (1998) ]
0.33
0.34
0.35
0.36
frequency (c/a)
0.37
0.38
Channel-Drop Filters
waveguide 1
Perfect channel-dropping if:
Two resonant modes with:
• even and odd symmetry
• equal frequency (degenerate)
• equal decay rates
Coupler
waveguide 2
(mirror plane)
Enough passive, linear devices…
Photonic crystal cavities:
tight confinement (~ #/2 diameter)
+ long lifetime (high Q independent of size)
= enhanced nonlinear effects
e.g. Kerr nonlinearity, "n ~ intensity
[ S. Fan et al., Phys. Rev. Lett. 80, 960 (1998) ]
A Linear Nonlinear Filter
A Linear Nonlinear “Transistor”
[ Soljacic et al., PRE Rapid. Comm. 66, 055601 (2002). ]
lytical
semi-ana
in
out
Logic gates, switching,
rectifiers, amplifiers,
isolators, …
numerical
+ feedback
Linear response:
Lorenzian Transmisson
shifted peak
Linear response:
Lorenzian Transmisson
Bistable (hysteresis) response
+ nonlinear
index shift
Power threshold ~ Q2/V is near optimal
(~mW for Si and telecom bandwidth)
shifted peak
Experimental Bistable Switch
[ Notomi et al., Opt. Express 13 (7), 2678 (2005). ]
Silicon-on-insulator
420 nm
Q ~ 30,000
Power threshold ~ 40 µW
Switching energy ~ 4 pJ