“1d” Waveguides + Cavities = DevicesLossless Bendssymmetry + single-mode + “1d” = resonances of 100% transmission[ A. Mekis et al.,Phys. Rev. Lett. 77, 3787 (1996) ]Waveguides + Cavities = Devices“tunneling”Ugh, must we simulate this to get the basic behavior?“Coupling-of-Modes-in-Time”(a form of coupled-mode theory)[H. Haus, Waves and Fields in Optoelectronics]ainput outputs1+s1–s2–resonant cavityfrequency !0, lifetime "|s|2 = flux|a|2 = energydadt= !i"0a !2#a +2#s1+s1!= !s1++2#a, s2!=2#aassumes only:• exponential decay (strong confinement)• conservation of energy• time-reversal symmetry“Coupling-of-Modes-in-Time”(a form of coupled-mode theory)[H. Haus, Waves and Fields in Optoelectronics]ainput outputs1+s1–s2–resonant cavityfrequency !0, lifetime "|s|2 = flux|a|2 = energytransmission T= |! s2– |2 / |! s1+ |2 1!0T = Lorentzian filter=4!2"#"0( )2+4!2!FWHM1Q=2!0"…quality factor QWide-angle Splitters[ S. Fan et al., J. Opt. Soc. Am. B 18, 162 (2001) ]Waveguide Crossings[ S. G. Johnson et al., Opt. Lett. 23, 1855 (1998) ]Waveguide Crossings00.20.40.60.81throughput1x10-101x10-81x10-61x10-41x10-21x1000.32 0.33 0.34 0.35 0.36 0.37 0.38crosstalkfrequency (c/a)empty1x13x35x55x53x3empty1x1empty5x53x31x1Channel-Drop Filters[ S. Fan et al., Phys. Rev. Lett. 80, 960 (1998) ]Perfect channel-dropping if:Two resonant modes with:• even and odd symmetry• equal frequency (degenerate)• equal decay ratesCouplerwaveguide 1waveguide 2(mirror plane)Enough passive, linear devices…Photonic crystal cavities:tight confinement (~ #/2 diameter)+ long lifetime (high Q independent of size)= enhanced nonlinear effectse.g. Kerr nonlinearity, "n ~ intensityA Linear Nonlinear Filterin outLinear response:Lorenzian Transmissonshifted peak+ nonlinearindex shiftA Linear Nonlinear “Transistor”Linear response:Lorenzian Transmissonshifted peak+ feedbackBistable (hysteresis) responsePower threshold ~ Q2/V is near optimal(~mW for Si and telecom bandwidth)semi-analyticalnumericalLogic gates, switching,rectifiers, amplifiers,isolators, …[ Soljacic et al., PRE Rapid. Comm. 66, 055601 (2002). ]Experimental Bistable SwitchSilicon-on-insulator420 nmQ ~ 30,000Power threshold ~ 40 µWSwitching energy ~ 4 pJ[ Notomi et al., Opt. Express 13 (7), 2678 (2005).