Complete optical isolation created by indirect interband photonic transitions

Transcription

1 Corretion notie Complete optial isolation reated by indiret interband photoni transitions Zongfu Yu and Shanhui Fan Nature Photonis 4, 9 94 (009). In the version of this Supplementary Information originally posted online, there were several mistakes that needed orreting, as desribed below. The orretions have now been made in this file on 5 April Page, last paragraph: T π sin( kl/ ) = os ( ) = 0.99 has been replaed by l k / π k k / π k T = L + + i L + = π k ( ) + ( ) os( ( ) ( ) ) sin( ( ) ( ) ) 0.95 T. Page, last paragraph: The ontrast ratio r = is above 40 d has been replaed by The T T ontrast ratio r = is above 0 d. T F 3. Page 6, equation (6) has been replaed by T T exp( iθ ( ω) γ L) ) 0 os( Cz) i sin( Cz) = T T 0 exp( i ( ) L θ ω γ ) isin( Cz) os( Cz) 4. Figures S and S3 have been replaed with updated ones. F 009 Mamillan Publishers Limited. All rights reserved.

2 doi: 0.038/nphoton Complete optial isolation reated by indiret interband photoni transitions Zongfu Yu and Shanhui Fan Department of Applied Physis and Department of Eletrial Engineering, Stanford University, Stanford, California, U.S.A. Optial Isolation Created by GHz Modulation Here we provide a detailed oupled-mode theory analysis for the optial isolation effet when a dieletri waveguide is modulated at the GHz range. We show that broad-band omplete optial isolation an be ahieved. We onsider the same slab silion waveguide in the main text and apply a dieletri modulation with profile ε '( x, z, t) = δ ( x) os( Ω t + qz). The frequeny and wavevetor of this modulation an be represented by arrows in Fig. S, whih also plots the dispersion relation of the waveguide. The wavevetor q of this modulation is hosen to phase-math a mode at ω in the first band to a mode at ω = ω+ω in the seond band (Fig. S). We onsider all possible transitions indued by this modulation. In general, due to energy onservation, from a frequeny ω transition an only our to frequenies ω ±Ω. For light at a frequeny ω in the first band, propagating along the forward diretion, exept for the phasemathed transition to ω = ω+ω, as mentioned above by design, all other transitions have strong phase mismath and an be ignored (Fig. S). On the other hand, for the same mode at ω propagating in the bakward nature photonis Mamillan Publishers Limited. All rights reserved.

3 doi: 0.038/nphoton diretion, we note that in the limit of low modulation frequeny (i.e. Ω ω ), the transition to ω3 = ω Ω in the seond band beomes nearly phase-mathed (Fig. S). For modulation frequeny in the range of 0 s of GHz, one will need to expliitly take into aount this transition. ased upon the onsiderations above, we onsider the effet of a 0GHz modulation, using oupled mode theory as in Eq. (3). We onsider a silion ( ε =.5) waveguide of 0.7 µ m wide, hosen suh that the first and seond bands of the waveguide have the same group veloity around wavelength.55 µ m (or a frequeny of 93THz). The modulation has a strength δ max Ω =, a frequeny = 0GHz ε π and a spatial period π q = 0.886µm. In the forward diretion, there is omplete transition from 93THz to a frequeny that is 0GHz higher, over a propagation distane of l =.9mm (Fig. Sa). In the bakward diretion, the near phase-mathed transition to a frequeny that is 0GHz lower, has a phase mismath k = q. Suh a phase mismath indues an inomplete photon transition for light propagating in the bakward diretion (Fig. Sb). To reate an isolator, we hose the length of the modulated region L= l. From Eq. 4, the bakward transmission π k k / π k for mode ω is, T = os( L ( ) + ( ) ) + i sin( L ( ) + ( ) ) = 0.95 π k ( ) + ( ) while the forward transmission T F = 0. Therefore near-omplete optial isolation an be ahieved with GHz modulation frequeny. Fig. S3 shows the spetra of forward and bakward transmission for the fundamental mode around T wavelength.55µ m. The ontrast ratio r = is above 0d over a bandwidth 0nm (or a frequeny bandwidth T F of.thz). Importantly, the bandwidth of the isolation is not dependent upon the modulation frequeny. Instead, broad-band isolation an be ahieved by designing the two bands involved in the transition to be parallel. nature photonis Mamillan Publishers Limited. All rights reserved.

4 doi: 0.038/nphoton Theory of non-reiproal frequeny onversion in a ring resonator To desribe the ring resonator system (Fig. S4), we onsider the transition between two anti-lokwise rotating resonanes in the ring. These resonanes have frequenies ω and ω, and wavevetors in the ring-waveguide k and k, respetively. For these two modes, the oupler is desribed by: b r jt 0 0 a jt r 0 0 A = b 0 0 r jt a 0 0 jt r A. (S) Here the subsripts label the two frequenies. A and a ( and b ) are the photon flux amplitudes in the,,,, external and ring waveguides before (after) the oupler. The transmit and transfer oeffiients r, t are real [] and r + t = In the ring, the two resonanes are oupled by applying a dieletri onstant modulation along the.,, ring with a profile δ( x) os[( ω ω ) t ( k k ) z], where z measures the propagation distane on the irumferene of the ring in ounterlokwise diretion. Thus, upon ompleting one round trip, the amplitudes a and b of the two modes are related generally by:,, a T T b = a T T b, (S) where the matrix elements are related to the transition amplitudes for a single round trip. With inident light only in mode (i.e. A =, A =0), ombined equation (S) and (S), we have r T r r T + r Det[ T ] =, (S3) + rt r T r r Det[ T ] where Det stands for determinant. Thus, the ondition for omplete frequeny onversion (i.e. = 0 ) is nature photonis Mamillan Publishers Limited. All rights reserved.

5 doi: 0.038/nphoton r T r r T + r Det[ T ] = 0. (S4) π L In the ase that ring is lossless, Det[T]= and T = T = os( ), where l is the oherene length and L is l irumferene of the ring. Complete onversion between the two modes an be ahieved when the length of the ring is hosen to be π L r + r os( ) l rr =, (S5) + with r,, L/ l 0. The devie therefore an provide omplete frequeny onversion even when its length is far smaller than the oherene length. We now onsider the frequeny response of the devie, in the presene of loss. We onsider a pair of modes ω + ω and ω + ω, so that the frequeny differene between them mathes the modulation frequeny ω ω. Using equation (3) in the paper, the oupling matrix in equation (S) beomes T T exp( ( ) ) 0 iθ ω γ L) os( Cz) i sin( Cz) = T T 0 exp( iθ ( ω) γ L ) i sin( Cz) os( Cz) (S6) where γ and γ haraterize the radiation loss. θ = ( k( ω ) k( ω + ω)) L is round-trip phase delay. In the i i i viinity of ω and ω, we assume a parallel band onfiguration, thus k = k( ω + ω) k( ω + ω) q 0 for all frequenies. C is assumed to be frequeny-independent in a small range of frequeny. The ombination of equation (S6) and (S) allows us to determine the response the devie in general. In order to ompare to the FDTD simulations of the ring resonator, we alulate, by several independent simulations, the parameters used in above derivation. For the same ring-waveguide system without modulation, the external quality fators due to waveguide-avity oupling are Q =346 and Q =887 for mode and 4 nature photonis Mamillan Publishers Limited. All rights reserved.

6 doi: 0.038/nphoton respetively, orresponding to r = 0.96 and r = 0.7 in equation (S). The two modes also have a radiation quality fator of 4 Q =.9 0 and r 4 Q =.3 0, orresponding to r γ L 3 = and γ L 3 = in equation (S6). To ompute oherent length l, one an either do diret field integral, or derive from mode onversion rate in a numerial simulation. Here, we simulate a semiirle struture with the same modulation profile. The mode onversion rate from one end of the semiirle to the other is used to derive l aording to equation (4) in the paper. Sine l π =, and C is real beause the modal profile of the waveguide an be taken C to be real, this fixes the oupling onstant C in equation (S6). [] Haus, H. A. Waves and fields in optoeletronis. (Prentie-Hall, In., Englewood Cliffs, New Jersey 984). nature photonis Mamillan Publishers Limited. All rights reserved.

7 doi: 0.038/nphoton Figure S. Shemati for all possible transitions. Arrows represent the spatial and temporal frequenies speified by modulation profile. undesirable transitions. ku, kb, and k f measure the phase mismath for the Figure S. Spatial evolution of photon flux for forward (a) and bakward (b) propagation diretion. Red, blue and green urves represent modes at ω, ω and ω3 respetively. 6 nature photonis Mamillan Publishers Limited. All rights reserved.

8 doi: 0.038/nphoton Figure S3. The transmission spetra for fundamental even modes in the forward and bakward diretion. Figure S4. Shemati of a ring resonator side-oupled to a waveguide. nature photonis Mamillan Publishers Limited. All rights reserved.