Abstract

We present a quantum-mechanical theory to describe narrow-band photon-pair generation via four-wave mixing in a Silicon-on-Insulator (SOI) micro-resonator. We also provide design principles for efficient photon-pair generation in an SOI micro-resonator through extensive numerical simulations. Microring cavities are shown to have a much wider dispersion-compensated frequency range than straight cavities. A microring with an inner radius of 8 μm can output an entangled photon comb of 21 pairwise-correlated peaks (42 comb lines) spanning from 1.3 μm to 1.8 μm. Such on-chip quantum photonic devices offer a path toward future integrated quantum photonics and quantum integrated circuits.

© 2011 OSA

Full Article  |  PDF Article
Related Articles
Ultrabroadband parametric generation and wavelength conversion in silicon waveguides

Qiang Lin, Jidong Zhang, Philippe M. Fauchet, and Govind P. Agrawal
Opt. Express 14(11) 4786-4799 (2006)

Nonlinear optical phenomena in silicon waveguides: Modeling and applications

Q. Lin, Oskar J. Painter, and Govind P. Agrawal
Opt. Express 15(25) 16604-16644 (2007)

A proposal for highly tunable optical parametric oscillation in silicon micro-resonators

Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter
Opt. Express 16(14) 10596-10610 (2008)

References

  • View by:
  • |
  • |
  • |

  1. D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, 1st Ed., (Springer2000).
    [PubMed]
  2. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
    [Crossref] [PubMed]
  3. T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, “Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Type-II Parametric Down-conversion,” Phys. Rev. Lett. 71, 3893 (1993).
    [Crossref] [PubMed]
  4. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
    [Crossref]
  5. H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A 70, 031802(R) (2004).
    [Crossref]
  6. X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber Source of Polarization-Entangled Photons in the 1550 nm Telecom Band,” Phys. Rev. Lett. 94, 053601 (2005).
    [Crossref] [PubMed]
  7. J. Fan, A. Migdall, and L. J. Wang, “Efficient generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. 30, 3368 (2005).
    [Crossref]
  8. J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534 (2005).
    [Crossref] [PubMed]
  9. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646 (2008).
    [Crossref] [PubMed]
  10. A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
    [Crossref]
  11. K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. 26, 1367 (2001).
    [Crossref]
  12. M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express 15, 7479 (2007).
    [Crossref] [PubMed]
  13. J. Chen, A. Pearlman, A. Ling, J. Fan, and A. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17, 6727 (2009).
    [Crossref] [PubMed]
  14. T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled KTiOPO4 waveguide,” Opt. Express 17, 12019 (2009).
    [Crossref] [PubMed]
  15. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388 (2006).
    [Crossref] [PubMed]
  16. K. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Generation of high-purity entangled photon pairs using silicon wire waveguide,” Opt. Express 16, 20368 (2008).
    [Crossref] [PubMed]
  17. S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, Ph. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558 (2009).
    [Crossref] [PubMed]
  18. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554 (2000).
    [Crossref]
  19. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16, 4881 (2008).
    [Crossref] [PubMed]
  20. Q. Lin and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett. 31, 3140 (2006).
    [Crossref] [PubMed]
  21. H. J. Kimble, “The quantum internet,” Nature 453, 1023 (2008).
    [Crossref] [PubMed]
  22. J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033801 (2005).
    [Crossref]
  23. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604 (2007).
    [Crossref] [PubMed]
  24. We note that the optical modes in our proposed microring resonators are not ideal whispering gallery modes, in the sense that the modes have non-zero interatction with the inner wall of the ring resonators. However, the electric field strength at the inner wall is typically no more than 1% of its peak value near the outer wall (see Fig. 2). Even though this interaction is negligibly small, we used the numerically determined modes in our calculations, rather than analytical expressions of ideal whispering gallery modes. We still refer to our modes as whispering gallery modes throughout the text, but we recognize this is an approximation.
  25. M. Scholtz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal-idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518 (2009).
    [Crossref]
  26. J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
    [Crossref]
  27. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304 (2000).
    [Crossref] [PubMed]
  28. S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
    [Crossref]
  29. L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
    [Crossref]
  30. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295 (2006).
    [Crossref] [PubMed]
  31. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguide,” Opt. Express 14, 4357 (2006).
    [Crossref] [PubMed]
  32. A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904 (2010).
  33. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron.11, 75 (1975).
    [Crossref]
  34. E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press1985), p. 548.
  35. http://www.comsol.com
  36. Certain trade names and company products are mentioned in the text or identified in an illustration in order to specify adequately the experimental procedure and equipment used. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it necessarily imply that the products are the best available for the purpose.
  37. Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express 16, 10596 (2008).
    [Crossref] [PubMed]
  38. I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A 76, 043837 (2007).
    [Crossref]
  39. M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Tran. Micro. Theory 55, 1209 (2007).
    [Crossref]
  40. Private communications with M. Oxborrow and P. Del’Haye.

2010 (2)

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
[Crossref]

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

2009 (5)

2008 (5)

2007 (4)

I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A 76, 043837 (2007).
[Crossref]

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Tran. Micro. Theory 55, 1209 (2007).
[Crossref]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604 (2007).
[Crossref] [PubMed]

M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express 15, 7479 (2007).
[Crossref] [PubMed]

2006 (4)

2005 (4)

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033801 (2005).
[Crossref]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber Source of Polarization-Entangled Photons in the 1550 nm Telecom Band,” Phys. Rev. Lett. 94, 053601 (2005).
[Crossref] [PubMed]

J. Fan, A. Migdall, and L. J. Wang, “Efficient generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. 30, 3368 (2005).
[Crossref]

J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534 (2005).
[Crossref] [PubMed]

2004 (1)

H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A 70, 031802(R) (2004).
[Crossref]

2002 (1)

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[Crossref]

2001 (1)

2000 (2)

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554 (2000).
[Crossref]

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304 (2000).
[Crossref] [PubMed]

1999 (1)

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

1993 (1)

T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, “Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Type-II Parametric Down-conversion,” Phys. Rev. Lett. 71, 3893 (1993).
[Crossref] [PubMed]

1904 (1)

A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904 (2010).

Absil, P. P.

Agha, I. H.

I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A 76, 043837 (2007).
[Crossref]

Agrawal, G. P.

Alley, C. O.

T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, “Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Type-II Parametric Down-conversion,” Phys. Rev. Lett. 71, 3893 (1993).
[Crossref] [PubMed]

Baets, R. G.

Banaszek, K.

Battle, P.

Beausoleil, R. G.

Benson, O.

M. Scholtz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal-idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518 (2009).
[Crossref]

Bogaerts, W.

Bouwmeester, D.

D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, 1st Ed., (Springer2000).
[PubMed]

Brendel, J.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Chen, J.

Cho, P. S.

Clemmen, S.

Cryan, M. J.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

Cussey, J.

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

Duligall, J.

Dwir, B.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
[Crossref]

Eberly, J. H.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304 (2000).
[Crossref] [PubMed]

Ekert, A. K.

D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, 1st Ed., (Springer2000).
[PubMed]

Emplit, P.

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

Emplit, Ph.

Faist, J.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
[Crossref]

Fan, J.

Fedrizzi, A.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Felici, M.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
[Crossref]

Fiorentino, M.

Foster, M. A.

Fukuda, H.

Fulconis, J.

Gaeta, A. L.

Gallo, P.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
[Crossref]

Gisin, N.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Harada, K.

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron.11, 75 (1975).
[Crossref]

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron.11, 75 (1975).
[Crossref]

Ho, P.-T.

Hryniewicz, J. V.

Huy, K. P.

Inoue, K.

H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A 70, 031802(R) (2004).
[Crossref]

Itabashi, S.

Johnson, T. J.

Joneckis, L. G.

Kapon, E.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
[Crossref]

Kiess, T. E.

T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, “Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Type-II Parametric Down-conversion,” Phys. Rev. Lett. 71, 3893 (1993).
[Crossref] [PubMed]

Kimble, H. J.

H. J. Kimble, “The quantum internet,” Nature 453, 1023 (2008).
[Crossref] [PubMed]

Koch, L.

M. Scholtz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal-idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518 (2009).
[Crossref]

Kumar, P.

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388 (2006).
[Crossref] [PubMed]

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033801 (2005).
[Crossref]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber Source of Polarization-Entangled Photons in the 1550 nm Telecom Band,” Phys. Rev. Lett. 94, 053601 (2005).
[Crossref] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[Crossref]

Kwiat, P. G.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

Langford, N. K.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Law, C. K.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304 (2000).
[Crossref] [PubMed]

Lee, K. F.

Li, X.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber Source of Polarization-Entangled Photons in the 1550 nm Telecom Band,” Phys. Rev. Lett. 94, 053601 (2005).
[Crossref] [PubMed]

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033801 (2005).
[Crossref]

Lin, Q.

Ling, A.

Lipson, M.

Little, B. E.

Manolatou, C.

Massar, S.

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, Ph. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558 (2009).
[Crossref] [PubMed]

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

Merolla, J.-M.

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

Michael, C. P.

Migdall, A.

Mohan, A.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
[Crossref]

Munro, M. W.

Nguyen, A. T.

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

O’Brien, J. L.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

Okawachi, Y.

I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A 76, 043837 (2007).
[Crossref]

Olislager, L.

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

Oxborrow, M.

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Tran. Micro. Theory 55, 1209 (2007).
[Crossref]

Painter, O. J.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press1985), p. 548.

Pearlman, A.

Perahia, R.

Phan Huy, K.

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

Politi, A.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

Ramelow, S.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Rarity, J. G.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534 (2005).
[Crossref] [PubMed]

Ratschbacher, L.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Roberts, T. D.

Rudra, A.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
[Crossref]

Russell, P. S.

Salem, R.

A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904 (2010).

Schmidt, B. S.

Scholtz, M.

M. Scholtz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal-idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518 (2009).
[Crossref]

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, “Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Type-II Parametric Down-conversion,” Phys. Rev. Lett. 71, 3893 (1993).
[Crossref] [PubMed]

Sharping, J. E.

I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A 76, 043837 (2007).
[Crossref]

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388 (2006).
[Crossref] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguide,” Opt. Express 14, 4357 (2006).
[Crossref] [PubMed]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber Source of Polarization-Entangled Photons in the 1550 nm Telecom Band,” Phys. Rev. Lett. 94, 053601 (2005).
[Crossref] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[Crossref]

Shih, Y.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

Shih, Y. H.

T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, “Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Type-II Parametric Down-conversion,” Phys. Rev. Lett. 71, 3893 (1993).
[Crossref] [PubMed]

Spillane, S. M.

Takesue, H.

K. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Generation of high-purity entangled photon pairs using silicon wire waveguide,” Opt. Express 16, 20368 (2008).
[Crossref] [PubMed]

H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A 70, 031802(R) (2004).
[Crossref]

Tittel, W.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Tokura, Y.

Tsuchizawa, T.

Turner, A. C.

Turner-Foster, A. C.

A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904 (2010).

U’Ren, A. B.

Voss, P. L.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber Source of Polarization-Entangled Photons in the 1550 nm Telecom Band,” Phys. Rev. Lett. 94, 053601 (2005).
[Crossref] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[Crossref]

Wadsworth, W. J.

Walmsley, I. A.

K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides,” Opt. Lett. 26, 1367 (2001).
[Crossref]

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304 (2000).
[Crossref] [PubMed]

Wang, L. J.

Watanabe, T.

Weinfurter, H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

Wilson, R. A.

Wong, F. N. C.

Yamada, K.

Yin, L.

Yu, S.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

Zbinden, H.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

Zeilinger, A.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, 1st Ed., (Springer2000).
[PubMed]

Zhong, T.

IEEE Photon. Technol. Lett. (1)

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002).
[Crossref]

IEEE Tran. Micro. Theory (1)

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Tran. Micro. Theory 55, 1209 (2007).
[Crossref]

Nature (1)

H. J. Kimble, “The quantum internet,” Nature 453, 1023 (2008).
[Crossref] [PubMed]

Nature Photon. (1)

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nature Photon. 4, 302 (2010).
[Crossref]

Opt. Commun. (1)

M. Scholtz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal-idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518 (2009).
[Crossref]

Opt. Express (12)

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604 (2007).
[Crossref] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguide,” Opt. Express 14, 4357 (2006).
[Crossref] [PubMed]

A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904 (2010).

Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express 16, 10596 (2008).
[Crossref] [PubMed]

A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16, 4881 (2008).
[Crossref] [PubMed]

M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express 15, 7479 (2007).
[Crossref] [PubMed]

J. Chen, A. Pearlman, A. Ling, J. Fan, and A. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17, 6727 (2009).
[Crossref] [PubMed]

T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled KTiOPO4 waveguide,” Opt. Express 17, 12019 (2009).
[Crossref] [PubMed]

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388 (2006).
[Crossref] [PubMed]

K. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Generation of high-purity entangled photon pairs using silicon wire waveguide,” Opt. Express 16, 20368 (2008).
[Crossref] [PubMed]

S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, Ph. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558 (2009).
[Crossref] [PubMed]

J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534 (2005).
[Crossref] [PubMed]

Opt. Lett. (5)

Phys. Rev. A (4)

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy, “Frequency-bin entangled photons,” Phys. Rev. A 82, 013804 (2010).
[Crossref]

I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres,” Phys. Rev. A 76, 043837 (2007).
[Crossref]

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72, 033801 (2005).
[Crossref]

H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A 70, 031802(R) (2004).
[Crossref]

Phys. Rev. Lett. (6)

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber Source of Polarization-Entangled Photons in the 1550 nm Telecom Band,” Phys. Rev. Lett. 94, 053601 (2005).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, “Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Type-II Parametric Down-conversion,” Phys. Rev. Lett. 71, 3893 (1993).
[Crossref] [PubMed]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
[Crossref]

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304 (2000).
[Crossref] [PubMed]

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103, 253601 (2009).
[Crossref]

Science (1)

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-Silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

Other (7)

D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, 1st Ed., (Springer2000).
[PubMed]

We note that the optical modes in our proposed microring resonators are not ideal whispering gallery modes, in the sense that the modes have non-zero interatction with the inner wall of the ring resonators. However, the electric field strength at the inner wall is typically no more than 1% of its peak value near the outer wall (see Fig. 2). Even though this interaction is negligibly small, we used the numerically determined modes in our calculations, rather than analytical expressions of ideal whispering gallery modes. We still refer to our modes as whispering gallery modes throughout the text, but we recognize this is an approximation.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron.11, 75 (1975).
[Crossref]

E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press1985), p. 548.

http://www.comsol.com

Certain trade names and company products are mentioned in the text or identified in an illustration in order to specify adequately the experimental procedure and equipment used. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it necessarily imply that the products are the best available for the purpose.

Private communications with M. Oxborrow and P. Del’Haye.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(a) Schematic of a microring resonator side-coupled to a bus waveguide, both integrated on a SOI chip. (b) Top-down view of photon pair production in the SOI device shown in (a). Pump is injected into the microring via the bus waveguide; copolarized photon pairs are generated and resonantly enhanced and evanescently coupled out of the microring. Waves propagate in the z direction in the bus waveguide. Inside the microring, there exist two polarization eigenmodes: TM (Electric field perpendicular to the plane of propagation) and TE (Electric field in the plane of propagation but perpendicular to the propagation direction). R1, inner ring radius; R2, outer ring radius. (c) Cross section of both the microring and the bus waveguide. The crystallographic axes are designated for the bus waveguide only. (d) An entangled comb of photon pairs is generated when pump frequency is tuned to mode number Mp. A signal photon occupying mode ms can always find its partner idler photon symmetrically placed around pump occupying mode mi = ms. Also shown is the simulated signal output with a relative mode number ms = 1, which has a full width at half-maximum of 20 GHz for a cavity damping rate of 31.25 GHz.

Fig. 5
Fig. 5

(a) Frequency mismatch for TM modes for a microring resonator of R1 = 8μm. Blue, λp = 1.498 μm with Mp = 114; red, λp = 1.555 μm with Mp = 109; black, λp = 1.616 μm with Mp = 104. One can see that Mp = 109 corresponds to the optimal pump mode. Frequency mismatch when pump is chosen optimally for TE (black) and TM (red) for several different bending radii: (b) R1 = 8μm, Mp = 111 for TE, Mp = 109 for TM; (c) R1 = 7μm, Mp = 97 for TE, Mp = 96 for TM; and (d) R1 = 5μm, Mp = 69 for TE, Mp = 68 for TM. Optimal pump wavelengths are labelled on the figures.

Fig. 2
Fig. 2

Conformal transformation from a bent waveguide to its equivalent straight waveguide, along with the fundamental TE mode shape (R 1 = 8μm, λ = 1.528 μm) in its corresponding coordinate.

Fig. 3
Fig. 3

Numerically simulated group index ng and group velocity dispersion k″ for (a) straight waveguide, (b) bent waveguide with R1 = 8μm, and (c) bent waveguide with R1 = 3μm. TE mode: black; TM mode: red. The curves for bulk Si are plotted in blue for reference.

Fig. 4
Fig. 4

Zero-dispersion wavelength vs. the inverse of the bending radius of an SOI waveguide for both TE (black dots) and TM (red squares) modes.

Fig. 6
Fig. 6

Frequency mismatch in a straight cavity for quasi-TE modes with λp = 1.407 μm (solid blue), 1.516 μm (solid red), and 1.649 μm (solid black), and quasi-TM modes with λp = 1.405 μm (hollow blue), 1.514 μm (hollow red), and 1.646 μm (hollow black).

Equations (23)

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

E p ( + ) ( z , t ) = E p e i [ k p ( ω p ) z ω p t ] e i Γ P z ,
E s ( ) ( z , t ) = h ¯ ω s 2 ɛ 0 n s c A eff , s γ s Δ ω s 2 π m s d Ω s a s ( ω s , m s + Ω s ) γ s / 2 i Ω s e i [ k s z ( ω s , m s + Ω s ) t ] .
| Ψ = η L m s m i d Ω s d Ω i γ s γ i δ ( m s Δ ω s + m i Δ ω i + Ω s + Ω i ) ( γ s / 2 i Ω s ) ( γ i / 2 i Ω i ) e i L [ k 4 ( m s Δ ω s Ω s ) 2 + k 4 ( m i Δ ω i + Ω i ) 2 + Γ P ] a s ( ω s , m s + Ω s ) a i ( ω i , m i + Ω i ) | 0 sinc { L [ k 4 ( m s Δ ω s Ω s ) 2 + k 4 ( m i Δ ω i + Ω i ) 2 + Γ P ] } ,
e i u t d t = 2 π δ ( u ) ,
L 0 e i β x d x = L e i β L / 2 sinc ( β L / 2 ) .
| Ψ = η L m d Ω γ s γ i e i L [ k ( m Δ ω Ω ) 2 / 2 + Γ P ] ( γ s / 2 i Ω ) ( γ i / 2 + i Ω ) sinc { L [ k ( m Δ ω Ω ) 2 / 2 + Γ P ] } a s ( ω p m Δ ω + Ω ) a i ( ω p + m Δ ω Ω ) | 0 .
S ( ω s ) = ( η L ) 2 m γ s γ i sinc 2 { L [ k ( ω p ω s ) 2 / 2 + Γ P ] } | γ s / 2 i ( ω s ω p + m Δ ω ) | 2 | γ i / 2 + i ( ω s ω p + m Δ ω ) | 2 .
S ( ω i ) = Ψ | a i ( ω i ) a i ( ω i ) | Ψ = ( η L ) 2 m γ s γ i sinc 2 { L [ k ( ω i ω p ) 2 / 2 + Γ P ] } | γ s / 2 i ( ω p ω i + m Δ ω ) | 2 | γ i / 2 + i ( ω p ω i + m Δ ω ) | 2 .
u = R 2 ln ( ρ / R 2 ) ,
v = R 2 θ .
n Air = 1 ,
n SiO 2 = 1 + 0.6961663 λ 2 λ 2 0.0684043 2 + 0.4079426 λ 2 λ 2 0.1162414 2 + 0.8974794 λ 2 λ 2 9.896161 2 ,
n Si = 3.41906 + 0.123172 λ 2 0.028 + 0.0265456 ( λ 2 0.028 ) 2 2.66511 × 10 8 λ 2 + 5.45852 × 10 14 λ 2 ,
Δ = 1 2 π ( 2 ω 0 p ω 0 s ω 0 i ) ,
Δ nl = 2 n 2 I n ( ω p ) ω p 2 π ,
R ijkl ( 3 ) ( τ ) = γ e δ ( τ ) [ σ 3 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) + ( 1 σ ) δ ij δ jk δ kl ] + γ R h R ( τ ) ( δ ik δ jl + δ il δ jk 2 δ ij δ jk δ kl ) .
[ a x a ρ a θ ] = [ M xx M xy M xz M ρ x M ρ y M ρ z M θ x M θ y M θ z ] [ a x a y a z ] ,
[ M xx M xy M xz M ρ x M ρ y M ρ z M θ x M θ y M θ z ] = [ 1 0 0 0 cos θ sin θ 0 sin θ ρ cos θ ρ ] .
R ρ ρ ρ ρ ( 3 ) = γ e δ ( τ ) ( cos 4 θ + sin 4 θ + 2 σ sin 2 θ cos 2 θ ) + 4 γ R h R ( τ ) sin 2 θ cos 2 θ ,
R xxxx ( 3 ) = γ e δ ( τ ) ,
R x ρ ρ x ( 3 ) = R ρ xx ρ ( 3 ) = σ 3 γ e δ ( τ ) + γ R h R ( τ ) .
n g L = M λ ,
ω d k d ω = 2 π M L .

Metrics