Abstract

In this paper we analyze the transmission and time delay properties of light propagating through a microring resonator (MRR) consisting of a solid core waveguide surrounded by an atomic vapor cladding. Using the atomic effective susceptibility of Rubidium we derive the complex transmission spectrum of the integrated system. We show, that when the system is under-coupled, the transmission can exceed the standalone MRR’s background transmission and is accompanied by enhanced positive time delay. It is shown that in this case the contrast of the atomic lines is greatly enhanced. This allows achieving high optical densities at short propagation length. Furthermore, owing to its features such as small footprint, high tunability, and high delay-transmission product, this system may become an attractive choice for chip scale manipulations of light.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2011 (3)

A. Sargsyan, C. Leroy, Y. Pashayan-Leroy, R. Mirzoyan, A. Papoyan, and D. Sarkisyan, “High contrast D1 line electromagnetically induced transparency in nanometric-thin rubidium vapor cell,” Appl. Phys. B105(4), 767–774 (2011).
[CrossRef]

K. Saha, V. Venkataraman, P. Londero, and A. L. Gaeta, “Enhanced two-photon absorption in a hollow-core photonic-band-gap fiber,” Phys. Rev. A83(3), 033833 (2011).
[CrossRef]

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on the rubidium D1 line including resonant dipole–dipole interactions,” J. Phys. At. Mol. Opt. Phys.44(19), 195006 (2011).
[CrossRef]

2010 (2)

T. Baluktsian, C. Urban, T. Bublat, H. Giessen, R. Löw, and T. Pfau, “Fabrication method for microscopic vapor cells for alkali atoms,” Opt. Lett.35(12), 1950–1952 (2010).
[CrossRef] [PubMed]

S. M. Hendrickson, M. M. Lai, T. B. Pittman, and J. D. Franson, “Observation of Two-photon Absorption at Low Power Levels Using Tapered Optical Fibers in Rubidium Vapor,” Phys. Rev. Lett.105(17), 173602 (2010).
[CrossRef] [PubMed]

2009 (1)

A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable delay of Einstein-Podolsky-Rosen entanglement,” Nature457(7231), 859–862 (2009).
[CrossRef] [PubMed]

2008 (4)

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett.100(23), 233602 (2008).
[CrossRef] [PubMed]

J. Zhang, G. Hernandez, and Y. Zhu, “Slow light with cavity electromagnetically induced transparency,” Opt. Lett.33(1), 46–48 (2008).
[CrossRef] [PubMed]

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. At. Mol. Opt. Phys.41(15), 155004 (2008).
[CrossRef]

D. Goldring, U. Levy, I. E. Dotan, A. Tsukernik, M. Oksman, I. Rubin, Y. David, and D. Mendlovic, “Experimental measurement of quality factor enhancement using slow light modes in one dimensional photonic crystal,” Opt. Express16(8), 5585–5595 (2008).
[CrossRef] [PubMed]

2007 (3)

D. Goldring, U. Levy, and D. Mendlovic, “Highly dispersive micro-ring resonator based on one dimensional photonic crystal waveguide design and analysis,” Opt. Express15(6), 3156–3168 (2007).
[CrossRef] [PubMed]

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, “Atomic spectroscopy on a chip,” Nat. Photonics1(6), 331–335 (2007).
[CrossRef]

J. Vanier and C. Mandache, “The passive optically pumped Rb frequency standard: the laser approach,” Appl. Phys. B87(4), 565–593 (2007).
[CrossRef]

2006 (3)

R. Kondo, S. Tojo, T. Fujimoto, and M. Hasuo, “Shift and broadening in attenuated total reflection spectra of the hyperfine-structure-resolved D_{2} line of dense rubidium vapor,” Phys. Rev. A73(6), 062504 (2006).
[CrossRef]

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A73(6), 063812 (2006).
[CrossRef]

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett.88(11), 111107 (2006).
[CrossRef]

2005 (4)

M. Soljacić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(2), 026602 (2005).
[CrossRef] [PubMed]

D. K. Sparacin, C. Y. Hong, L. C. Kimerling, J. Michel, J. P. Lock, and K. K. Gleason, “Trimming of microring resonators by photooxidation of a plasma-polymerized organosilane cladding material,” Opt. Lett.30(17), 2251–2253 (2005).
[CrossRef] [PubMed]

K. Zhao and Z. Wu, “Regionally specific hyperfine polarization of Rb atoms in the vicinity (10^{−5}cm) of surfaces,” Phys. Rev. A71(1), 012902 (2005).
[CrossRef]

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, “Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres,” Nature434(7032), 488–491 (2005).
[CrossRef] [PubMed]

2004 (2)

S. Knappe, L. Liew, V. Shah, P. Schwindt, J. Moreland, L. Hollberg, and J. Kitching, “A microfabricated atomic clock,” Appl. Phys. Lett.85(9), 1460 (2004).
[CrossRef]

J. Heebner, A. Vincent Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron.40(6), 726–730 (2004).
[CrossRef]

2003 (1)

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature422(6932), 596–599 (2003).
[CrossRef] [PubMed]

2002 (1)

L. Wang, “Causal “all-pass” filters and Kramers–Kronig relations,” Opt. Commun.213(1-3), 27–32 (2002).
[CrossRef]

2000 (1)

1999 (1)

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

1997 (1)

G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, “Optical resonator with steep internal dispersion,” Phys. Rev. A56(3), 2385–2389 (1997).
[CrossRef]

1996 (1)

J. Guo, J. Cooper, and A. Gallagher, “Selective reflection from a dense atomic vapor,” Phys. Rev. A53(2), 1130–1138 (1996).
[CrossRef] [PubMed]

1994 (1)

J. Guo, J. Cooper, A. Gallagher, and M. Lewenstein, “Theory of selective reflection spectroscopy,” Opt. Commun.110(1-2), 197–208 (1994).
[CrossRef]

1988 (1)

G. Nienhuis, F. Schuller, and M. Ducloy, “Nonlinear selective reflection from an atomic vapor at arbitrary incidence angle,” Phys. Rev. A38(10), 5197–5205 (1988).
[CrossRef] [PubMed]

Adams, C. S.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on the rubidium D1 line including resonant dipole–dipole interactions,” J. Phys. At. Mol. Opt. Phys.44(19), 195006 (2011).
[CrossRef]

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. At. Mol. Opt. Phys.41(15), 155004 (2008).
[CrossRef]

Allred, J. C.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature422(6932), 596–599 (2003).
[CrossRef] [PubMed]

Baluktsian, T.

Beausoleil, R. G.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett.100(23), 233602 (2008).
[CrossRef] [PubMed]

Benabid, F.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, “Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres,” Nature434(7032), 488–491 (2005).
[CrossRef] [PubMed]

Bettles, R. J.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on the rubidium D1 line including resonant dipole–dipole interactions,” J. Phys. At. Mol. Opt. Phys.44(19), 195006 (2011).
[CrossRef]

Birks, T. A.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, “Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres,” Nature434(7032), 488–491 (2005).
[CrossRef] [PubMed]

Boyd, R. W.

J. Heebner, A. Vincent Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron.40(6), 726–730 (2004).
[CrossRef]

Boyer, V.

A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable delay of Einstein-Podolsky-Rosen entanglement,” Nature457(7231), 859–862 (2009).
[CrossRef] [PubMed]

Bublat, T.

Burkett, W. H.

Camacho, R. M.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A73(6), 063812 (2006).
[CrossRef]

Campbell, K.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett.88(11), 111107 (2006).
[CrossRef]

Conkey, D. B.

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, “Atomic spectroscopy on a chip,” Nat. Photonics1(6), 331–335 (2007).
[CrossRef]

Cooper, J.

J. Guo, J. Cooper, and A. Gallagher, “Selective reflection from a dense atomic vapor,” Phys. Rev. A53(2), 1130–1138 (1996).
[CrossRef] [PubMed]

J. Guo, J. Cooper, A. Gallagher, and M. Lewenstein, “Theory of selective reflection spectroscopy,” Opt. Commun.110(1-2), 197–208 (1994).
[CrossRef]

Couny, F.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, “Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres,” Nature434(7032), 488–491 (2005).
[CrossRef] [PubMed]

Danzmann, K.

G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, “Optical resonator with steep internal dispersion,” Phys. Rev. A56(3), 2385–2389 (1997).
[CrossRef]

David, Y.

Dotan, I. E.

Ducloy, M.

G. Nienhuis, F. Schuller, and M. Ducloy, “Nonlinear selective reflection from an atomic vapor at arbitrary incidence angle,” Phys. Rev. A38(10), 5197–5205 (1988).
[CrossRef] [PubMed]

Fainman, Y.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett.88(11), 111107 (2006).
[CrossRef]

Franson, J. D.

S. M. Hendrickson, M. M. Lai, T. B. Pittman, and J. D. Franson, “Observation of Two-photon Absorption at Low Power Levels Using Tapered Optical Fibers in Rubidium Vapor,” Phys. Rev. Lett.105(17), 173602 (2010).
[CrossRef] [PubMed]

Fry, E. S.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Fujimoto, T.

R. Kondo, S. Tojo, T. Fujimoto, and M. Hasuo, “Shift and broadening in attenuated total reflection spectra of the hyperfine-structure-resolved D_{2} line of dense rubidium vapor,” Phys. Rev. A73(6), 062504 (2006).
[CrossRef]

Gaeta, A. L.

K. Saha, V. Venkataraman, P. Londero, and A. L. Gaeta, “Enhanced two-photon absorption in a hollow-core photonic-band-gap fiber,” Phys. Rev. A83(3), 033833 (2011).
[CrossRef]

Gallagher, A.

J. Guo, J. Cooper, and A. Gallagher, “Selective reflection from a dense atomic vapor,” Phys. Rev. A53(2), 1130–1138 (1996).
[CrossRef] [PubMed]

J. Guo, J. Cooper, A. Gallagher, and M. Lewenstein, “Theory of selective reflection spectroscopy,” Opt. Commun.110(1-2), 197–208 (1994).
[CrossRef]

Ge, C.

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. At. Mol. Opt. Phys.41(15), 155004 (2008).
[CrossRef]

Giessen, H.

Gleason, K. K.

Goldring, D.

Goorskey, D. J.

Groisman, A.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett.88(11), 111107 (2006).
[CrossRef]

Guo, J.

J. Guo, J. Cooper, and A. Gallagher, “Selective reflection from a dense atomic vapor,” Phys. Rev. A53(2), 1130–1138 (1996).
[CrossRef] [PubMed]

J. Guo, J. Cooper, A. Gallagher, and M. Lewenstein, “Theory of selective reflection spectroscopy,” Opt. Commun.110(1-2), 197–208 (1994).
[CrossRef]

Hall, M.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett.100(23), 233602 (2008).
[CrossRef] [PubMed]

Hasuo, M.

R. Kondo, S. Tojo, T. Fujimoto, and M. Hasuo, “Shift and broadening in attenuated total reflection spectra of the hyperfine-structure-resolved D_{2} line of dense rubidium vapor,” Phys. Rev. A73(6), 062504 (2006).
[CrossRef]

Hau, L. V.

M. Soljacić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(2), 026602 (2005).
[CrossRef] [PubMed]

Hawkins, A. R.

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, “Atomic spectroscopy on a chip,” Nat. Photonics1(6), 331–335 (2007).
[CrossRef]

Heebner, J.

J. Heebner, A. Vincent Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron.40(6), 726–730 (2004).
[CrossRef]

Hendrickson, S. M.

S. M. Hendrickson, M. M. Lai, T. B. Pittman, and J. D. Franson, “Observation of Two-photon Absorption at Low Power Levels Using Tapered Optical Fibers in Rubidium Vapor,” Phys. Rev. Lett.105(17), 173602 (2010).
[CrossRef] [PubMed]

Hernandez, G.

Hollberg, L.

S. Knappe, L. Liew, V. Shah, P. Schwindt, J. Moreland, L. Hollberg, and J. Kitching, “A microfabricated atomic clock,” Appl. Phys. Lett.85(9), 1460 (2004).
[CrossRef]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Hong, C. Y.

Howell, J. C.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A73(6), 063812 (2006).
[CrossRef]

Hughes, I. G.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on the rubidium D1 line including resonant dipole–dipole interactions,” J. Phys. At. Mol. Opt. Phys.44(19), 195006 (2011).
[CrossRef]

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. At. Mol. Opt. Phys.41(15), 155004 (2008).
[CrossRef]

Jackson, D. J.

J. Heebner, A. Vincent Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron.40(6), 726–730 (2004).
[CrossRef]

Joannopoulos, J. D.

M. Soljacić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(2), 026602 (2005).
[CrossRef] [PubMed]

Kash, M. M.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Kimerling, L. C.

Kitching, J.

S. Knappe, L. Liew, V. Shah, P. Schwindt, J. Moreland, L. Hollberg, and J. Kitching, “A microfabricated atomic clock,” Appl. Phys. Lett.85(9), 1460 (2004).
[CrossRef]

Knappe, S.

S. Knappe, L. Liew, V. Shah, P. Schwindt, J. Moreland, L. Hollberg, and J. Kitching, “A microfabricated atomic clock,” Appl. Phys. Lett.85(9), 1460 (2004).
[CrossRef]

Knight, J. C.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, “Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres,” Nature434(7032), 488–491 (2005).
[CrossRef] [PubMed]

Kominis, I. K.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature422(6932), 596–599 (2003).
[CrossRef] [PubMed]

Kondo, R.

R. Kondo, S. Tojo, T. Fujimoto, and M. Hasuo, “Shift and broadening in attenuated total reflection spectra of the hyperfine-structure-resolved D_{2} line of dense rubidium vapor,” Phys. Rev. A73(6), 062504 (2006).
[CrossRef]

Kornack, T. W.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature422(6932), 596–599 (2003).
[CrossRef] [PubMed]

Kumar, P.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett.100(23), 233602 (2008).
[CrossRef] [PubMed]

Lai, M. M.

S. M. Hendrickson, M. M. Lai, T. B. Pittman, and J. D. Franson, “Observation of Two-photon Absorption at Low Power Levels Using Tapered Optical Fibers in Rubidium Vapor,” Phys. Rev. Lett.105(17), 173602 (2010).
[CrossRef] [PubMed]

Leroy, C.

A. Sargsyan, C. Leroy, Y. Pashayan-Leroy, R. Mirzoyan, A. Papoyan, and D. Sarkisyan, “High contrast D1 line electromagnetically induced transparency in nanometric-thin rubidium vapor cell,” Appl. Phys. B105(4), 767–774 (2011).
[CrossRef]

Lett, P. D.

A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable delay of Einstein-Podolsky-Rosen entanglement,” Nature457(7231), 859–862 (2009).
[CrossRef] [PubMed]

Levy, U.

Lewenstein, M.

J. Guo, J. Cooper, A. Gallagher, and M. Lewenstein, “Theory of selective reflection spectroscopy,” Opt. Commun.110(1-2), 197–208 (1994).
[CrossRef]

Lidorikis, E.

M. Soljacić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(2), 026602 (2005).
[CrossRef] [PubMed]

Liew, L.

S. Knappe, L. Liew, V. Shah, P. Schwindt, J. Moreland, L. Hollberg, and J. Kitching, “A microfabricated atomic clock,” Appl. Phys. Lett.85(9), 1460 (2004).
[CrossRef]

Lock, J. P.

Londero, P.

K. Saha, V. Venkataraman, P. Londero, and A. L. Gaeta, “Enhanced two-photon absorption in a hollow-core photonic-band-gap fiber,” Phys. Rev. A83(3), 033833 (2011).
[CrossRef]

Löw, R.

Lukin, M. D.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Mandache, C.

J. Vanier and C. Mandache, “The passive optically pumped Rb frequency standard: the laser approach,” Appl. Phys. B87(4), 565–593 (2007).
[CrossRef]

Marino, A. M.

A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable delay of Einstein-Podolsky-Rosen entanglement,” Nature457(7231), 859–862 (2009).
[CrossRef] [PubMed]

Mendlovic, D.

Michel, J.

Mirzoyan, R.

A. Sargsyan, C. Leroy, Y. Pashayan-Leroy, R. Mirzoyan, A. Papoyan, and D. Sarkisyan, “High contrast D1 line electromagnetically induced transparency in nanometric-thin rubidium vapor cell,” Appl. Phys. B105(4), 767–774 (2011).
[CrossRef]

Mookherjea, S.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett.88(11), 111107 (2006).
[CrossRef]

Moreland, J.

S. Knappe, L. Liew, V. Shah, P. Schwindt, J. Moreland, L. Hollberg, and J. Kitching, “A microfabricated atomic clock,” Appl. Phys. Lett.85(9), 1460 (2004).
[CrossRef]

Müller, G.

G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, “Optical resonator with steep internal dispersion,” Phys. Rev. A56(3), 2385–2389 (1997).
[CrossRef]

Müller, M.

G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, “Optical resonator with steep internal dispersion,” Phys. Rev. A56(3), 2385–2389 (1997).
[CrossRef]

Nienhuis, G.

G. Nienhuis, F. Schuller, and M. Ducloy, “Nonlinear selective reflection from an atomic vapor at arbitrary incidence angle,” Phys. Rev. A38(10), 5197–5205 (1988).
[CrossRef] [PubMed]

Oksman, M.

Pack, M. V.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A73(6), 063812 (2006).
[CrossRef]

Papoyan, A.

A. Sargsyan, C. Leroy, Y. Pashayan-Leroy, R. Mirzoyan, A. Papoyan, and D. Sarkisyan, “High contrast D1 line electromagnetically induced transparency in nanometric-thin rubidium vapor cell,” Appl. Phys. B105(4), 767–774 (2011).
[CrossRef]

Pashayan-Leroy, Y.

A. Sargsyan, C. Leroy, Y. Pashayan-Leroy, R. Mirzoyan, A. Papoyan, and D. Sarkisyan, “High contrast D1 line electromagnetically induced transparency in nanometric-thin rubidium vapor cell,” Appl. Phys. B105(4), 767–774 (2011).
[CrossRef]

Pati, G. S.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett.100(23), 233602 (2008).
[CrossRef] [PubMed]

Pfau, T.

Pittman, T. B.

S. M. Hendrickson, M. M. Lai, T. B. Pittman, and J. D. Franson, “Observation of Two-photon Absorption at Low Power Levels Using Tapered Optical Fibers in Rubidium Vapor,” Phys. Rev. Lett.105(17), 173602 (2010).
[CrossRef] [PubMed]

Pooser, R. C.

A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable delay of Einstein-Podolsky-Rosen entanglement,” Nature457(7231), 859–862 (2009).
[CrossRef] [PubMed]

Rinkleff, R.-H.

G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, “Optical resonator with steep internal dispersion,” Phys. Rev. A56(3), 2385–2389 (1997).
[CrossRef]

Romalis, M. V.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature422(6932), 596–599 (2003).
[CrossRef] [PubMed]

Rostovtsev, Y.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Rubin, I.

Russell, P. S. J.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, “Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres,” Nature434(7032), 488–491 (2005).
[CrossRef] [PubMed]

Saha, K.

K. Saha, V. Venkataraman, P. Londero, and A. L. Gaeta, “Enhanced two-photon absorption in a hollow-core photonic-band-gap fiber,” Phys. Rev. A83(3), 033833 (2011).
[CrossRef]

Salit, K.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett.100(23), 233602 (2008).
[CrossRef] [PubMed]

Sargsyan, A.

A. Sargsyan, C. Leroy, Y. Pashayan-Leroy, R. Mirzoyan, A. Papoyan, and D. Sarkisyan, “High contrast D1 line electromagnetically induced transparency in nanometric-thin rubidium vapor cell,” Appl. Phys. B105(4), 767–774 (2011).
[CrossRef]

Sarkisyan, D.

A. Sargsyan, C. Leroy, Y. Pashayan-Leroy, R. Mirzoyan, A. Papoyan, and D. Sarkisyan, “High contrast D1 line electromagnetically induced transparency in nanometric-thin rubidium vapor cell,” Appl. Phys. B105(4), 767–774 (2011).
[CrossRef]

Sautenkov, V. A.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Schmidt, H.

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, “Atomic spectroscopy on a chip,” Nat. Photonics1(6), 331–335 (2007).
[CrossRef]

Schuller, F.

G. Nienhuis, F. Schuller, and M. Ducloy, “Nonlinear selective reflection from an atomic vapor at arbitrary incidence angle,” Phys. Rev. A38(10), 5197–5205 (1988).
[CrossRef] [PubMed]

Schweinsberg, A.

J. Heebner, A. Vincent Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron.40(6), 726–730 (2004).
[CrossRef]

Schwindt, P.

S. Knappe, L. Liew, V. Shah, P. Schwindt, J. Moreland, L. Hollberg, and J. Kitching, “A microfabricated atomic clock,” Appl. Phys. Lett.85(9), 1460 (2004).
[CrossRef]

Scully, M. O.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Shah, V.

S. Knappe, L. Liew, V. Shah, P. Schwindt, J. Moreland, L. Hollberg, and J. Kitching, “A microfabricated atomic clock,” Appl. Phys. Lett.85(9), 1460 (2004).
[CrossRef]

Shahriar, M. S.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett.100(23), 233602 (2008).
[CrossRef] [PubMed]

Siddons, P.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on the rubidium D1 line including resonant dipole–dipole interactions,” J. Phys. At. Mol. Opt. Phys.44(19), 195006 (2011).
[CrossRef]

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. At. Mol. Opt. Phys.41(15), 155004 (2008).
[CrossRef]

Soljacic, M.

M. Soljacić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(2), 026602 (2005).
[CrossRef] [PubMed]

Sparacin, D. K.

Spillane, S. M.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett.100(23), 233602 (2008).
[CrossRef] [PubMed]

Tojo, S.

R. Kondo, S. Tojo, T. Fujimoto, and M. Hasuo, “Shift and broadening in attenuated total reflection spectra of the hyperfine-structure-resolved D_{2} line of dense rubidium vapor,” Phys. Rev. A73(6), 062504 (2006).
[CrossRef]

Tsukernik, A.

Urban, C.

Vanier, J.

J. Vanier and C. Mandache, “The passive optically pumped Rb frequency standard: the laser approach,” Appl. Phys. B87(4), 565–593 (2007).
[CrossRef]

Venkataraman, V.

K. Saha, V. Venkataraman, P. Londero, and A. L. Gaeta, “Enhanced two-photon absorption in a hollow-core photonic-band-gap fiber,” Phys. Rev. A83(3), 033833 (2011).
[CrossRef]

Vincent Wong, A.

J. Heebner, A. Vincent Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron.40(6), 726–730 (2004).
[CrossRef]

Wang, H.

Wang, L.

L. Wang, “Causal “all-pass” filters and Kramers–Kronig relations,” Opt. Commun.213(1-3), 27–32 (2002).
[CrossRef]

Welch, G. R.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Weller, L.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on the rubidium D1 line including resonant dipole–dipole interactions,” J. Phys. At. Mol. Opt. Phys.44(19), 195006 (2011).
[CrossRef]

Wicht, A.

G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, “Optical resonator with steep internal dispersion,” Phys. Rev. A56(3), 2385–2389 (1997).
[CrossRef]

Wu, B.

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, “Atomic spectroscopy on a chip,” Nat. Photonics1(6), 331–335 (2007).
[CrossRef]

Wu, Z.

K. Zhao and Z. Wu, “Regionally specific hyperfine polarization of Rb atoms in the vicinity (10^{−5}cm) of surfaces,” Phys. Rev. A71(1), 012902 (2005).
[CrossRef]

Xiao, M.

Yang, W.

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, “Atomic spectroscopy on a chip,” Nat. Photonics1(6), 331–335 (2007).
[CrossRef]

Yin, D.

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, “Atomic spectroscopy on a chip,” Nat. Photonics1(6), 331–335 (2007).
[CrossRef]

Zhang, J.

Zhao, K.

K. Zhao and Z. Wu, “Regionally specific hyperfine polarization of Rb atoms in the vicinity (10^{−5}cm) of surfaces,” Phys. Rev. A71(1), 012902 (2005).
[CrossRef]

Zhu, Y.

Zibrov, A. S.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Appl. Phys. B (2)

J. Vanier and C. Mandache, “The passive optically pumped Rb frequency standard: the laser approach,” Appl. Phys. B87(4), 565–593 (2007).
[CrossRef]

A. Sargsyan, C. Leroy, Y. Pashayan-Leroy, R. Mirzoyan, A. Papoyan, and D. Sarkisyan, “High contrast D1 line electromagnetically induced transparency in nanometric-thin rubidium vapor cell,” Appl. Phys. B105(4), 767–774 (2011).
[CrossRef]

Appl. Phys. Lett. (2)

S. Knappe, L. Liew, V. Shah, P. Schwindt, J. Moreland, L. Hollberg, and J. Kitching, “A microfabricated atomic clock,” Appl. Phys. Lett.85(9), 1460 (2004).
[CrossRef]

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microring resonator,” Appl. Phys. Lett.88(11), 111107 (2006).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Heebner, A. Vincent Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron.40(6), 726–730 (2004).
[CrossRef]

J. Phys. At. Mol. Opt. Phys. (2)

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on the rubidium D1 line including resonant dipole–dipole interactions,” J. Phys. At. Mol. Opt. Phys.44(19), 195006 (2011).
[CrossRef]

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. At. Mol. Opt. Phys.41(15), 155004 (2008).
[CrossRef]

Nat. Photonics (1)

W. Yang, D. B. Conkey, B. Wu, D. Yin, A. R. Hawkins, and H. Schmidt, “Atomic spectroscopy on a chip,” Nat. Photonics1(6), 331–335 (2007).
[CrossRef]

Nature (3)

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, “Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres,” Nature434(7032), 488–491 (2005).
[CrossRef] [PubMed]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature422(6932), 596–599 (2003).
[CrossRef] [PubMed]

A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable delay of Einstein-Podolsky-Rosen entanglement,” Nature457(7231), 859–862 (2009).
[CrossRef] [PubMed]

Opt. Commun. (2)

L. Wang, “Causal “all-pass” filters and Kramers–Kronig relations,” Opt. Commun.213(1-3), 27–32 (2002).
[CrossRef]

J. Guo, J. Cooper, A. Gallagher, and M. Lewenstein, “Theory of selective reflection spectroscopy,” Opt. Commun.110(1-2), 197–208 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. A (7)

G. Nienhuis, F. Schuller, and M. Ducloy, “Nonlinear selective reflection from an atomic vapor at arbitrary incidence angle,” Phys. Rev. A38(10), 5197–5205 (1988).
[CrossRef] [PubMed]

R. Kondo, S. Tojo, T. Fujimoto, and M. Hasuo, “Shift and broadening in attenuated total reflection spectra of the hyperfine-structure-resolved D_{2} line of dense rubidium vapor,” Phys. Rev. A73(6), 062504 (2006).
[CrossRef]

J. Guo, J. Cooper, and A. Gallagher, “Selective reflection from a dense atomic vapor,” Phys. Rev. A53(2), 1130–1138 (1996).
[CrossRef] [PubMed]

K. Saha, V. Venkataraman, P. Londero, and A. L. Gaeta, “Enhanced two-photon absorption in a hollow-core photonic-band-gap fiber,” Phys. Rev. A83(3), 033833 (2011).
[CrossRef]

K. Zhao and Z. Wu, “Regionally specific hyperfine polarization of Rb atoms in the vicinity (10^{−5}cm) of surfaces,” Phys. Rev. A71(1), 012902 (2005).
[CrossRef]

G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, “Optical resonator with steep internal dispersion,” Phys. Rev. A56(3), 2385–2389 (1997).
[CrossRef]

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A73(6), 063812 (2006).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

M. Soljacić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(2), 026602 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett.100(23), 233602 (2008).
[CrossRef] [PubMed]

S. M. Hendrickson, M. M. Lai, T. B. Pittman, and J. D. Franson, “Observation of Two-photon Absorption at Low Power Levels Using Tapered Optical Fibers in Rubidium Vapor,” Phys. Rev. Lett.105(17), 173602 (2010).
[CrossRef] [PubMed]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett.82(26), 5229–5232 (1999).
[CrossRef]

Other (2)

L. Stern, B. Desiatov, I. Goykhman, and U. Levy, “Evanescent light-matter Interactions in Atomic Cladding Wave Guides,” arXiv:1204.0393 (2012).

D. A. Steck, “Rubidium 87 D Line Data,” http://steck.us/alkalidata , (unpublished)

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic diagram of an atomic vapor cell bonded to a chip consisting of SiN MRR (b) Calculated TE-like electric field distribution (in absolute values) for the D1 wavelength of Rb (795 nm) superimposed on the geometry of the proposed MRR’s wave guide. Rb atoms are illustrated as balisitcaly striking the surface, entering the evanescent portion of the optical mode. Plotted is the norm of the electric field. (c) Absorption coefficient of the fundamental waveguide mode as a function of frequency detuning (d) Effective index of the atomic cladding fundamental waveguide mode (e) Group index of the atomic cladding fundamental waveguide mode, calculated using the effective index from (d)

Fig. 2
Fig. 2

(a) MRR time delay as a function of the coupling coefficient. The MRR time delay inverts its sign at critical coupling. (b): transmission-group delay product as a function of the coupling coefficient, illustrating that high values of time delay are accompanied with vanishing transmission. Optimal positive (negative) values of transmission- group delay product are obtained for over (under) coupled system. A loss parameter of a = 0.99 was assumed.

Fig. 3
Fig. 3

(a) Round trip transmission (normalized linear scale) of the atomic cladding wave guide which constitutes the MRR (b) Combined (blue) and standalone (green) MRR transmission as a function of frequency detuning (c) Combined (blue) and standalone (green) MRR phase delay as a function of frequency detuning (d) Combined (blue) and standalone (green) MRR’s group index as a function of frequency detuning

Fig. 4
Fig. 4

(a) Combined (blue) and standalone (green) MRR transmission as a function of coupling coefficient. Illustrated are the different turning points: the negative (dip) and positive (peak) turning point and the turning point between slow and fast light. (b) MRR transmission in the over-coupled regime (c) MRR transmission in the under-coupled regime (d) MRR group index in the over-coupled regime (e) MRR group index in the under-coupled regime. In all figures, the dashed green figures correspond to the standalone MRR, and the blue figures to the combined system.

Fig. 5
Fig. 5

(a) Round trip optical density (b) Atomic Cladding wave guide effective index (c) MRR transmission in the case we have an atomic cladding wave guide MRR (blue), in the case we have an fictitious atomic cladding wave dispersion only (red), the case we have a fictitious atomic cladding with absorption only (cyan).

Equations (12)

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

χ(ν)= e 2 π ε 0 m e i=1,2 C i F,F' N f FF' 2(2I+1) ν F i ', F i d v z 0 d v t W( v z , v t ) 2πΔν k z v z i(γi k t v t )
f FF' =f W 2 (J'F'JF;I1)(2F'+1)(2F+1)/(2I+1)
h(ω)= ta e i ϕ rt 1ta e i ϕ rt
T(ω)= | h(ω) | 2 = a 2 + t 2 2atcos ϕ rt 1+ a 2 t 2 2atcos ϕ rt
Φ Total =angle(h)=π+ ϕ rt +atan( tsin ϕ rt atcos ϕ rt )+atan( tasin ϕ rt 1tacos ϕ rt )
τ( ω 0 )= L c [ n eff +( n eff t at + n eff ta 1ta ) ]= L c n g L v g
(n g T) | ω= ω 0 = a(at) (1at) 2 + ta (at) 2 (1at) 3
a e i ϕ rt =( a r e ω/c k eff L ) e iω/c n eff L
T(ω)= ( at 1at ) 2
ϕ rt = ωL c n eff (ω) ωL c ( n eff ( ω 0 )+ d n eff dω Δω)
τ(ω)= d Φ total (a, ϕ rt ) dω = d Φ total d ϕ rt d ϕ rt dω + d Φ total da da dω d Φ total d ϕ rt L c ( n eff ( ω 0 )+ d n eff dω Δω+ω d n eff dω ) + d Φ total da da dω
τ( ω 0 )= τ ring (a( ω 0 ))( n eff ( ω 0 )+ ω 0 d n eff ( ω 0 ) dω )= τ ring n group,eff

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