Abstract

We consider spontaneous four-wave mixing (SFWM) in a single-channel side-coupled integrated spaced sequence of resonators within a fully quantum formalism. We show that the probability of photon pair production can scale quadratically with the number of resonators, a phenomenon we call super SFWM, in analogy with super-radiant spontaneous emission. Remarkably, in this situation the spectral probability density of the generated photons is independent of the number of rings.

© 2012 Optical Society of America

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  1. D. Ljunggren, M. Tengner, P. Marsden, and M. Pelton, Phys. Rev. A 73, 032326 (2006).
    [CrossRef]
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    [CrossRef]
  3. C. Xiong, C. Monat, A. S. Clark, C. Grillet, G. D. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, J. G. Rarity, and B. J. Eggleton, Opt. Lett. 36, 3413 (2011).
    [CrossRef]
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    [CrossRef]
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  6. E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).
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  8. M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
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    [CrossRef]
  12. Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
    [CrossRef]
  13. While θr(ω) is not slowly varying in ω, it can be considered constant over an interval δω centered at a resonant frequency ω0 provided δω≪ΔΩ, where ΔΩ is the resonance width. In CW experiments the laser line width δω is several orders of magnitude smaller than the resonance width (e.g., Q=10,000 @200 THz versus δω≃1  MHz).
  14. L. G. Helt, M. Liscidini, and J. E. Sipe, J. Opt. Soc. Am. B 29, 2199 (2012).
    [CrossRef]
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    [CrossRef]

2012 (6)

S. Azzini, D. Grassani, M. Galli, L. C. Andreani, M. Sorel, M. J. Strain, L. G. Helt, J. E. Sipe, M. Liscidini, and D. Bajoni, Opt. Lett. 37, 3807 (2012).

S. Azzini, D. Grassani, M. J. Strain, M. Sorel, G. Helt, J. E. Sipe, M. Liscidini, M. Galli, and D. Bajoni, Opt. Express 20, 23100 (2012).

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[CrossRef]

M. Liscidini, L. G. Helt, and J. E. Sipe, Phys. Rev. A 85, 013833 (2012).
[CrossRef]

L. G. Helt, M. Liscidini, and J. E. Sipe, J. Opt. Soc. Am. B 29, 2199 (2012).
[CrossRef]

2011 (1)

2010 (1)

2009 (1)

2008 (1)

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[CrossRef]

2006 (2)

2004 (1)

1954 (1)

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[CrossRef]

Abolghasem, P.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[CrossRef]

Agha, I.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Andreani, L. C.

Assefa, S.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Azzini, S.

Baets, R. G.

Bajoni, D.

Bijlani, B. J.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[CrossRef]

Bogaerts, W.

Bonneau, D.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Boyd, R. W.

Chak, P.

Clark, A.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Clark, A. S.

Clemmen, S.

Davanço, M.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Dicke, R. H.

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[CrossRef]

Dorenbos, S. N.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Eggleton, B. J.

Emplit, Ph.

Engin, E.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Ezaki, M.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Foster, M. A.

Gaeta, A. L.

Galli, M.

Grassani, D.

Green, W. M. J.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Grillet, C.

Hadfield, R. H.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Heebner, J. E.

Helmy, A. S.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[CrossRef]

Helt, G.

Helt, L. G.

Horn, R.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[CrossRef]

Huy, K. Phan

Iizuka, N.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Kang, D.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[CrossRef]

Krauss, T. F.

Li, J.

Lipson, M.

Liscidini, M.

Ljunggren, D.

D. Ljunggren, M. Tengner, P. Marsden, and M. Pelton, Phys. Rev. A 73, 032326 (2006).
[CrossRef]

Manolatou, C.

Marsden, P.

D. Ljunggren, M. Tengner, P. Marsden, and M. Pelton, Phys. Rev. A 73, 032326 (2006).
[CrossRef]

Marshall, G. D.

Massar, S.

Monat, C.

Mookherjea, S.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Natarajan, C. M.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

O’Brien, J. L.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

O’Faolain, L.

Ohira, K.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Ong, J. R.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Pelton, M.

D. Ljunggren, M. Tengner, P. Marsden, and M. Pelton, Phys. Rev. A 73, 032326 (2006).
[CrossRef]

Pereira, S.

Rarity, J. G.

Schmidt, B. S.

Sharping, J. E.

Shehata, A. B.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Sipe, J. E.

Sorel, M.

Srinivasan, K.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Steel, M. J.

Strain, M. J.

Suzuki, N.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Tanner, M. G.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Tengner, M.

D. Ljunggren, M. Tengner, P. Marsden, and M. Pelton, Phys. Rev. A 73, 032326 (2006).
[CrossRef]

Thompson, M. G.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Tosi, A.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Turner, A. C.

Weihs, G.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[CrossRef]

Xia, F.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

Xiong, C.

Yang, Z.

L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, Opt. Lett. 35, 3006 (2010).
[CrossRef]

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[CrossRef]

Yoshida, H.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Zwiller, V.

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

Appl. Phys. Lett. (1)

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, Appl. Phys. Lett. 100, 261104 (2012).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. (1)

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[CrossRef]

Phys. Rev. A (3)

M. Liscidini, L. G. Helt, and J. E. Sipe, Phys. Rev. A 85, 013833 (2012).
[CrossRef]

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[CrossRef]

D. Ljunggren, M. Tengner, P. Marsden, and M. Pelton, Phys. Rev. A 73, 032326 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[CrossRef]

Other (2)

E. Engin, D. Bonneau, C. M. Natarajan, A. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon micro-ring resonator and enhancement via reverse bias,” arXiv:1204.4922v1 (2012).

While θr(ω) is not slowly varying in ω, it can be considered constant over an interval δω centered at a resonant frequency ω0 provided δω≪ΔΩ, where ΔΩ is the resonance width. In CW experiments the laser line width δω is several orders of magnitude smaller than the resonance width (e.g., Q=10,000 @200 THz versus δω≃1  MHz).

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Figures (2)

Fig. 1.
Fig. 1.

(a) Sketch of a one-channel SCISSOR and (b) transmission phase delay as a function of frequency detuning from ωP for a SCISSOR composed of N=10 rings with R=10μm, Q=104, and Λ=30μm. Possible signal and idler frequencies are indicated. Inset: Magnification near ωP.

Fig. 2.
Fig. 2.

Coherence number as a function of the frequency difference between the pump and high energy generated photons for triply resonant generation. We consider R=10μm, Q=104, and take ωP=2π·200THz, v108ms1, ξ0.01ps2/m.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

ϕRR;LL(ω1,ω2)=22πα2βiω1ω2vg(ω1)vg(ω2)328ϵ0×dω3[ϕP(ω1+ω2ω3)ϕP(ω3)ω3(ω2+ω1ω3)vg(ω3)vg(ω2ω1ω3)×JRR;LL(ω1,ω2,ω3,ω2+ω1ω3)],
JRR;LL(ω1,ω2,ω3,ω4)=drΓ3ijkl(r)[DRk(ω1)i,asy-in(r)×DRk(ω2)j,asy-in(r)DLk(ω3)k,asy-in(r)DLk(ω4)l,asy-in(r)]
JRR;LL(ω1,ω2,ω3,ω4)=J1,RR;LL(ω1,ω2,ω3,ω4)×m=1NeiΘm(ω1,ω2,ω3,ω4),
Θm(ω1,ω2,ω3,ω4)=θ¯(ω1,ω2)+θm(ω1,ω2,ω3,ω4),
θ¯(ω1,ω2)=(N1)[θr(ω1)+θr(ω2)],
θm(ω1,ω2,ω3,ω4)=(m1){θr(ω3)+θr(ω4)θr(ω1)θr(ω2)+[k(ω3)+k(ω4)k(ω1)k(ω2)]Λ},
θr(ω)=π+k(ω)L+2tan1(σsin[k(ω)L]1σcos[k(ω)L]).
ϕRR;LL(ω1,ω2)=β1βϕ1(ω1,ω2)m=1NeiΘm(ω1,ω¯,ωP,ωP),
ϕ1(ω1,ω2)=22πα2β1vg(ωP)iω1ω2vg(ω1)vg(ω2)328ϵ0×dω3[ϕP(ω1+ω2ω3)ϕP(ω3)ω3(ω1+ω2ω3)×J1,RR;LL(ω1,ω2,ω3,ω1+ω2ω3)]
|β|2=|β1|2dω1dω2|ϕ1(ω1,ω2)|2|m=1Neiθm(ω1,ω¯,ωP,ωP)|2.
ϕRR;LL(ω1,ω2)=ϕ1(ω1,ω2),
k(ω)k(ωP)+1v(ωωP)+12ξ(ωωP)2,
δk2k(ωP)k(ω1)k(ω¯)=ξ(ω1ωP)2,
θm(ω1,ω¯,ωP,ωP)(m1)δk(Λ+L+2Lσ1σ),
Ncoh=π|θm/(m1)|πωP4Qv|δk|,

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