Integrated Nanophotonics Dmitry Yu. Fedyanin Laboratory of Nanooptics and Plasmonics, MIPT Lecture 6: Photonic Resonators and Filters 

2 FABRY-PEROT RESONATOR 2k n 1 d cosθ2 −ϕ21 −ϕ23 =2π N ,N=0,1,2,3,... 

3 FABRY-PEROT RESONATOR TE wave n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) d = 20 μm λ = 1550 nm 2k n 1 d cosθ2 −ϕ21 −ϕ23 =2π N , N=0,1,2,3,... 

4 FABRY-PEROT RESONATOR n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) d = 20 μm λ = 1550 nm 

5 FABRY-PEROT RESONATOR n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) d = 20 μm λ = 1550 nm 

6 FABRY-PEROT RESONATOR n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) d = 3 μm λ = 1550 nm 

7 FABRY-PEROT RESONATOR n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) d = 3 μm λ = 1550 nm Problems: 1. Resonance peaks are not sharp. 2. There is no spectral spacing between peaks. 3. High background level. 

8 FABRY-PEROT RESONATOR TE wave n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) d = 20 μm λ = 1550 nm High reflectance at large angles due to the high reflection coefficients at the interfaces! 

9 FABRY-PEROT RESONATOR n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) d = 3 μm λ = 1550 nm Problems: 1. Resonance peaks are not sharp. 2. There is no spectral spacing between peaks. 3. High background level. 

10 FABRY-PEROT RESONATOR n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) mirror thickness = 20 nm d = 3 μm λ = 1550 nm Mirror reflectance is as high as 90%. 

11 FABRY-PEROT RESONATOR n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) mirror thickness = 20 nm d = 3 μm λ = 1550 nm Losses in the metal decrease the mirror reflectance from 90% to 80%. The absor- bance is higher than the transmittance and reflectance. Therefore Bragg (DBR) reflectors are preferred when possible. 

12 FABRY-PEROT RESONATOR What is inside the resonator? Try to evaluate the intensity of electromagnetic field inside the resonator using this plot and the values of the reflectance (80%) and transmittance (10%) of the metal mirrors. n 1 = n 3 = 1.4 (SiO 2 ) n 2 = 3 (semiconductor) mirror thickness = 20 nm d = 3 μm 

13 FABRY-PEROT RESONATOR What is inside the resonator? A 2 B 2 A 3 T M = 0.1; |r M |2 = 0.8 | ⟹ t M |≈ 0.46; |r M | ≈ 0.9 A 2 = A 3 / t M | ⟹ A 2 | = 2.16 |A 3 | B 2 = A 3 r M / t M | ⟹ B 2 | = 1.93 |A 3 | max|E cavity | max[ ∝ A 2 sin(...) + B 2 sin(...)] ≈ 4|A 3 | |A 3 |2 = 0.36 max|E cavity |2 ≈ 5.8 |E incident |2 

15 FABRY-PEROT RESONATOR Silicon waveguide with DBR mirrors M.W. Pruessner, T.H. Stievater, W.S. Rabinovich, Opt. Lett. 32, 533 (2007). Resonator based on a photonic crystal P. Velha et al., Ultra-High Q/V Fabry-Perot microcavity on SOI substrate, Opt. Express 15, 16090 (2007). 

16 FABRY-PEROT LASER exp(2g L−2α L)R front R back ≃1 g≃α + 1 2 L log( 1 R front R back ) Common laser diode VCSEL Figures: www.photonics.com 

17 FABRY-PEROT LASER -K. Takeda et al., Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers, Nat. Photon. 7, 569–575 (2013). -K. Takeda et al., Heterogeneously integrated photonic-crystal lasers on silicon for on-off chip optical interconnects, Opt. Express 22, 702 (2015). Photonic-crystal on-chip laser 

18 FABRY-PEROT LASER Electrically pumped metal coated nanolaser -K. Ding et al., Record performance of electrical injection sub-wavelength metallic-cavity semiconductor lasers at room temperature, Opt. Express 21, 4728 (2013). 

19 FABRY-PEROT LASER Electrically pumped plasmonic nanolaser -D.Yu. Fedyanin, Toward an electrically pumped spaser, Optics Letters 37, 404-406 (2012). 

20 RING RESONATOR 

21 RING RESONATOR 

22 RING RESONATOR -W. R. McKinnon et al., Extracting coupling and loss coefficients from a ring resonator, Opt. Express 17, 18971 (2009). ‖B D ‖=‖t WG κWR κRW t R ‖×‖A C ‖ 

23 RING RESONATOR ‖B D ‖=‖t WG κWR κRW t R ‖×‖A C ‖ C=D exp(iβ 2π R)=a Dexp(iθ) a=exp(−Imβ 2π R) θ=β 2π R t R =t WG * =t* κWR =−κRW * =−κ* |t |2 +| κ|2 =1 -A. Yariv, Universal relations for coupling of optical power between microresonators and dielectric waveguides, Electronic Lett. 36, 321 (2000). -W. R. McKinnon et al., Extracting coupling and loss coefficients from a ring resonator, Opt. Express 17, 18971 (2009). 

24 RING RESONATOR ‖B D ‖=‖t WG κWR κRW t R ‖×‖A C ‖ C=D exp(iβ 2π R)=a Dexp(iθ) a=exp(−Imβ 2π R) θ=β 2π R t R =t WG * =t* κWR =−κRW * =−κ* |t |2 +| κ|2 =1 B= −a+t exp(−i θ) −at* +exp(−iθ) C= −a κ* −at* +exp(−iθ) D= −κ* 1−at exp(iθ) 

25 RING RESONATOR B= −a+t exp(−i θ) −at* +exp(−iθ) C= −a κ* −at* +exp(−iθ) D= −κ* 1−at exp(iθ) T=| B|2 = a2 +|t |2 −2a|t |cos(θ−ϕt ) 1+a2 |t |2 −2a|t |cos(θ−ϕt ) | D |2 = a2 (1−|t |) 1+a2|t |2 −2a|t |cos(θ+ϕt ) | A |2 =1 

26 RING RESONATOR T=| B|2 = a2 +|t |2 −2a|t |cos(θ−ϕt ) 1+a2 |t |2 −2a|t |cos(θ−ϕt ) λ = 1550 nm; n eff = 2; R = 5 μm; a2 = 0.64; |t|2 = 0.81 

27 RING RESONATOR T=| B|2 = a2 +|t |2 −2a|t |cos(θ−ϕt ) 1+a2 |t |2 −2a|t |cos(θ−ϕt ) λ = 1550 nm; n eff = 2; R = 5 μm; a2 = |t|2 = 0.81 Transmittance goes to zero 

28 RING RESONATOR T=| B|2 = a2 +|t |2 −2a|t |cos(θ−ϕt ) 1+a2 |t |2 −2a|t |cos(θ−ϕt ) λ = 1550 nm; n eff = 2; R = 5 μm; a2 = 0.9, 0.7, 0.5; |t|2 = 0.81 a2 

29 RING RESONATOR T=| B|2 = a2 +|t |2 −2a|t |cos(θ−ϕt ) 1+a2 |t |2 −2a|t |cos(θ−ϕt ) λ = 1550 nm; n eff = 2; R = 5 μm; a2 = 0.64; |t|2 = 0.9, 0.8, 0.5 |t|2 

30 RING RESONATOR −T=| B|2 = a2 +|t |2 −2a|t |cos(θ−ϕt ) 1+a2 |t |2 −2a|t |cos(θ−ϕt ) λ = 1550 nm; n eff = 2; R = 5.01, 5.02, 5.03 μm; a2 = 0.64; |t|2 = 0.81 θ=2π Rβ 

31 RING RESONATOR MODULATOR T=|B|2 = a2 +|t |2 −2a|t |cos(θ−ϕt ) 1+a2 |t |2 −2a|t |cos(θ−ϕt ) λ = 1550 nm; n eff = 2; R = 5 μm; a2 = 0.64; |t|2 = 0.81 ΔL = 1 μm; 

32 RING RESONATOR MODULATOR -Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, Micrometre-scale silicon electro-optic modulator, Nature 435, 325–327 (2005). They fabricated the p-i-n ring resonator on a silicon- on-insulator substrate with a 3-µm-thick buried oxide layer. Both the waveguide coupling to the ring and that forming the ring had a width of 450 nm and a height of 250 nm. The diameter of the ring is 12 µm, and the spacing between the ring and the straight waveguide was 200 nm. 

33 RING RESONATOR MODULATOR -Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, Micrometre-scale silicon electro-optic modulator, Nature 435, 325–327 (2005). quality factor Q = λ/Δλ = 39 350 photon lifetime of τ cav = λ2/(2πcΔλ) = 33 ps The resonance is blue-shifted owing to the lowering of n eff caused by the increase of the electron–hole pair density in the cavity 

34 RING RESONATOR MODULATOR -Z. Peng, D. Fattal, M. Fiorentino, R. Beausoleil, CMOS-Compatible Microring Modulators for Nanophotonic Interconnect, Integrated Photonics Research, Silicon and Nanophotonics 2010, DOI: 10.1364/IPRSN.2010.IWA2 6 Gb/s, 45 fJ per bit 

35 RING RESONATOR MODULATOR -Z. Peng, D. Fattal, M. Fiorentino, R. Beausoleil, CMOS- Compatible Microring Modulators for Nanophotonic Interconnect, DOI: 10.1364/IPRSN.2010.IWA2 

36 ADD-DROP FILTER α=α1/2 2 ; θ =2θ1/2 At resonance: 

38 RING RESONATOR INTEGRATED LASER 

40 NEXT CLASS Numerical simulation of optical waveguides and resonators. Homework: Article mentioned in the presentation D. G. Rabus, Integrated Ring Resonators: The Compendium, Springer Series in Optical Sciences, Springer, 2007. Chapter 2 S. L. Chuang, Physics of Photonic Devices, Wiley, 2009. Section 8.4