Slowing and stopping light using an optomechanical crystal array

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21 E.1.3. Optomechanical coupling rates. The optomechanical coupling arises from a shift in the optical frequency caused by a mechanical deformation. Our Hamiltonian for the single-cavity system can then be written as Ĥ = ¯hω( ˆx)â † â + ¯hω m ˆb † ˆb, (E.1) where ˆx = x ZPF (ˆb † + ˆb) is the quantized displacement of the mechanical mode, and x ZPF is the characteristic per-phonon displacement amplitude. The deformation-dependent frequency ω( ˆx) may be calculated to first order in ˆx using a variant of the Feynman–Hellman perturbation theory, the Johnson perturbation theory [38], which has been used successfully in the past to model OMC cavities [9, 39]. The Hamiltonian is then given to first order by Ĥ = ¯hω o â † â + ¯hω m ˆb † ˆb + ¯hg(ˆb † + ˆb)â † â, where ω 0 is the optical mode frequency in the absence of deformation and √ g = ω o 2 (E.2) ∫ ( ¯h dl (Q · n) ɛ|E ‖ | 2 − (ɛ −1 )|D ⊥ | 2) √ 2ω m d ∫ ∫ . (E.3) s dA ρ|Q| 2 dA ɛ|E| 2 Here, E, D and Q are the optical mode electric field, optical mode displacement field and mechanical mode displacement field, respectively, d s is the thickness of the slab and ɛ(r) is the dielectric constant. These concepts can be extended to optically multi-mode systems, represented by the Hamiltonian Ĥ = ¯h ∑ i ω o,i â † i â i + ¯hω m ˆb † ˆb + ¯h 2 ∑ g i, j (ˆb † + ˆb)â † i â j , i, j (E.4) where now the cross-coupling rates can be calculated by the following expression, √ ∫ ( ) g i, j = ω i, j ¯h dl(Q · n) ɛE ‖∗ i · E ‖ j − (ɛ−1 )Di ⊥∗ · D ⊥ j √ 2 2ω m d ∫ ∫ s dA ρ|Q| 2 dA ɛ|E i | ∫ . (E.5) 2 dAɛ|E j | 2 We denote this expression for convenience as g i, j ≡ 〈E i | Q|E j 〉. For the modes of the L2 cavity shown in figures 2(b) and (c), the optomechanical coupling was calculated to be 〈E|Q|E〉/2π = 489 kHz for silicon. When two cavities are brought in the vicinity of each other, supermodes form as is shown in figure 2(a). We denote the symmetric (+) and antisymmetric (−) combinations by E ± and Q ± . These modes can be written in terms of the modes localized at cavity 1 and 2, E ± = E 1 ± E 2 √ 2 and Q ± = Q 1 ± Q 2 √ 2 . (E.6) By symmetry, the only non-vanishing coupling term involving the Q − (antisymmetric mechanical) mode is 〈E + |Q − |E − 〉. Assuming that the two cavities are sufficiently separated, we can approximate 〈E + Q − 〉 ≈ 〈E|Q|E〉/ √ 2. For the supermodes of interest, 〈E1 + |Q − |E − 〉/2π = h/2π = 346 kHz. New Journal of Physics 13 (2011) 023003 (http://www.njp.org/)
22 (a) 14 250 (b) (c) 13 240 W W a r ν (GHz) 12 11 10 ν m ν (THz) 230 220 210 200 ν o (d) (e) w 9 8 190 180 y 7 170 x 0 0.5 1 k (π/d) 0 0.5 1 k (π/d) Figure E.1. (a) The snowflake waveguide consists of the usual snowflake lattice, with a removed row and adjusted waveguide width. The guiding region is shaded red for clarity. (b) The mechanical band structure exhibits a full phononic band gap at the resonance frequency of the mechanical mode (ν m ). (c) The optical band structure exhibits a single band that passes through the resonance frequency ν 0 of the optical mode. The waveguide acts as a single-mode optical waveguide with field pattern H z shown in (d). The H z component of the guided optical mode of opposite symmetry is shown in (e) for completeness. E.2. Properties of snowflake crystal waveguides A line defect on an OMC acts as a waveguide for light [40, 41]. Here, the line defects used consist of a removed row of holes, with the rows above and below shifted toward one another by a distance W , such that the distance between the centers of the snowflakes across the line defect is √ 3a − 2W (see figure E.1(a)). The waveguide was designed such that mechanically, it would have no bands resonant with the cavity frequency (see figure E.1(b)) and would therefore have no effect on the mechanical Q factors. Optically, it was designed to have a single band crossing the cavity frequency (see figure E.1(c)) and would therefore serve as the single-mode optical waveguide required by the proposal. The band structure of the mechanical waveguide was calculated using COMSOL [42], while for the optical simulations, MPB [43] was used. E.3. Cavity–waveguide coupling By bringing the optical waveguide near our cavity, the guided modes of the line defect are evanescently coupled to the cavity mode, and a coupling between the two may be induced, as shown in figure E.2(a). Control over this coupling rate is achieved at a coarse level by changing the distance between the cavity and the waveguide, i.e. the number of unit cells between them. New Journal of Physics 13 (2011) 023003 (http://www.njp.org/)
Page 1 and 2: Home Search Collections Journals Ab
Page 3 and 4: 2 Contents 1. Introduction 2 2. Des
Page 5 and 6: 4 (a) (b) a R (z) optical waveguide
Page 7 and 8: 6 where k 0 = ω 1 /c. We solve the
Page 9 and 10: 8 respectively. In the relevant reg
Page 11 and 12: 10 heating is removed through the o
Page 13 and 14: 12 Required Input Power (mW) 10 9 8
Page 15 and 16: 14 ∫ the wavevector annihilation
Page 17 and 18: 16 transfer matrix for a single ‘
Page 19 and 20: 18 with the parameter β(δ k ) giv
Page 21: 20 where β 1 (δ k ) = −iκ ex/2
Page 25 and 26: 24 conductivity and heat capacity o
Page 27: 26 [37] Meystre P and Sargent M III

Slowing and stopping light using an optomechanical crystal array

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