Abstract

The homogeneous linewidth of the transition I15/24I13/24 in highly doped erbium fibers and its dependence with temperature in the range from 10 to 50 K are experimentally characterized using spectral hole burning. The homogeneous linewidth dependence with temperature is quadratic above 20 K where homogeneous broadening is dominated by two-phonon Raman processes, and linear at lower temperatures where direct phonon processes occur. This characteristic power-law dependence was also derived from transmittance measurements. The solution of nonlinear field equations using the results obtained from our experiments predicts that Gaussian probe pulses propagate at subluminal speed through the narrow spectral holes burned in erbium-doped fibers. For gigahertz pulses in the telecommunication window, a fractional delay as high as 0.6 is predicted.

© 2012 Optical Society of America

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  6. K.-Y. Song, M. González-Herráez, and L. Thévenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005).
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    [CrossRef]
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    [CrossRef]
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  38. The Cauchy principal value integral was evaluated by a double exponential quadrature. Coding in Matlab is due to Mohankumar and Natarajan http://www.mathworks.com/matlabcentral/fileexchange/13871-hilbertf .

2011

2009

R. Lauro, T. Chanelière, and J. L. Le Gouët, “Slow light using spectral hole burning in a Tm3+-doped yttrium-aluminum-garnet crystal,” Phys. Rev. A 79, 063844 (2009).
[CrossRef]

2008

2007

S. Melle, O. G. Calderón, F. Carreño, E. Cabrera, M. A. Antón, and S. Jarabo, “Effect of ion concentration on slow light propagation in highly doped erbium fibers,” Opt. Commun. 279, 53–63 (2007).
[CrossRef]

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef]

2006

R. M. Camacho, M. V. Pack, and J. C. Howell, “Slow light with large fractional delays by spectral hole-burning in rubidium vapor,” Phys. Rev. A 74, 033801 (2006).
[CrossRef]

R. M. MacFarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602 (2006).
[CrossRef]

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silica fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
[CrossRef]

2005

R. N. Shakhmuratov, A. Rebane, P. Mégret, and J. Odeurs, “Slow light with persistent hole burning,” Phys. Rev. A 71, 053811 (2005).
[CrossRef]

K.-Y. Song, M. González-Herráez, and L. Thévenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

2004

2003

G. S. Agarwal and T. N. Dey, “Slow light in Doppler-broadened two-level systems,” Phys. Rev. A 68, 063816 (2003).
[CrossRef]

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

2002

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, “Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium,” Phys. Rev. A 66, 013805 (2002).
[CrossRef]

L. Bigot, A. M. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glasses for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

2001

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

1999

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

1998

T. Suemoto, T. Okuno, and D. Nakano, “Defect-induced persistent hole burning in MgO-doped Pr3+:YAG systems,” Opt. Commun. 145, 113–118 (1998).
[CrossRef]

1993

P. F. Wysocki, J. L. Wagener, M. J. F. Digonnet, and H. J. Shaw, “Evidence and modelling of paired ions and other loss mechanisms in erbium-doped silica fibers,” Proc. SPIE 1789, 66–79 (1993).
[CrossRef]

1992

R. Yano, M. Mitsunaga, and N. Uesugi, “Stimulated-photon-echo spectroscopy. I. Spectral diffusion in Eu3+:YAIO3,” Phys. Rev. B 45, 12752–12759 (1992).
[CrossRef]

Y. Silberberg, V. L. da Silva, J. P. Heritage, E. W. Chase, and M. J. Andrejco, “Accumulated photon echoes in doped fibers,” IEEE J. Quantum Electron. 28, 2369–2381 (1992).
[CrossRef]

1990

E. Desurvire, J. L. Zyskind, and J. R. Simpson, “Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 2, 246–248 (1990).
[CrossRef]

J. L. Zyskind, E. Desurvire, J. W. Sulhoff, and D. J. Di Giovanni, “Determination of homogeneous linewidth by spectral gain hole-burning in an erbium-doped fiber amplifier with GeO2:SiO2 core,” IEEE Photon. Technol. Lett. 2, 869–871 (1990).
[CrossRef]

1987

R. M. MacFarlane and R. M. Shelby, “Homogeneous line broadening of optical transitions of ions and molecules in glasses,” J. Lumin. 36, 179–207 (1987).
[CrossRef]

M. J. Weber, ed., Special feature on “Optical linewidths in glasses,” J. Lumin. 36, 179–329 (1987).

1986

M. M. Broer, B. Golding, W. H. Haemmerle, and J. R. Simpson, “Low-temperature optical dephasing of rare-earth ions in inorganic glasses,” Phys. Rev. B 33, 4160–4165 (1986).
[CrossRef]

1984

D. L. Huber, M. M. Broer, and B. Golding, “Low temperature optical dephasing of rare-earth ions in glass,” Phys. Rev. Lett. 52, 2281–2284 (1984).
[CrossRef]

1972

S. Haroche and F. Hartmann, “Theory of saturated-absorption line shapes,” Phys. Rev. A 6, 1280–1300 (1972).
[CrossRef]

Afzelius, M.

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silica fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and T. N. Dey, “Slow light in Doppler-broadened two-level systems,” Phys. Rev. A 68, 063816 (2003).
[CrossRef]

Andrejco, M. J.

Y. Silberberg, V. L. da Silva, J. P. Heritage, E. W. Chase, and M. J. Andrejco, “Accumulated photon echoes in doped fibers,” IEEE J. Quantum Electron. 28, 2369–2381 (1992).
[CrossRef]

Anton, M. A.

Antón, M. A.

S. Melle, O. G. Calderón, F. Carreño, E. Cabrera, M. A. Antón, and S. Jarabo, “Effect of ion concentration on slow light propagation in highly doped erbium fibers,” Opt. Commun. 279, 53–63 (2007).
[CrossRef]

Arrieta-Yañez, F.

Arrieta-Yáñez, F.

Balic, V.

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

Bayart, D.

L. Bigot, A. M. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glasses for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

Bigot, L.

Boivin, D.

Böttger, T.

C. W. Thiel, T. Böttger, and R. L. Cone, “Rare-earth-doped materials for applications in quantum information storage and signal processing,” J. Lumin. 131, 353–361 (2011).
[CrossRef]

Boyd, R. W.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

Braje, D. A.

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

Broer, M. M.

M. M. Broer, B. Golding, W. H. Haemmerle, and J. R. Simpson, “Low-temperature optical dephasing of rare-earth ions in inorganic glasses,” Phys. Rev. B 33, 4160–4165 (1986).
[CrossRef]

D. L. Huber, M. M. Broer, and B. Golding, “Low temperature optical dephasing of rare-earth ions in glass,” Phys. Rev. Lett. 52, 2281–2284 (1984).
[CrossRef]

Burov, E.

R. Peretti, B. Jacquier, D. Boivin, E. Burov, and A. M. Jurdyc, “Inhomogeneous gain saturation in EDF: experiment and modeling,” J. Lightwave Technol. 29, 1445–1452 (2011).
[CrossRef]

R. Peretti, A. M. Jurdyc, B. Jacquier, E. Burov, and L. Gasca, “Resonant fluorescence line narrowing and gain spectral hole burning in erbium-doped fiber amplifier,” J. Lumin. 128, 1010–1012 (2008).
[CrossRef]

Cabrera, E.

S. Melle, O. G. Calderón, F. Carreño, E. Cabrera, M. A. Antón, and S. Jarabo, “Effect of ion concentration on slow light propagation in highly doped erbium fibers,” Opt. Commun. 279, 53–63 (2007).
[CrossRef]

Cabrera-Granado, E.

Calderón, O.

Calderón, O. G.

O. G. Calderón, S. Melle, F. Arrieta-Yañez, M. A. Anton, and F. Carreño, “Effect of ion pairs in fast-light bandwidth in high-concentration erbium-doped fibers,” J. Opt. Soc. Am. B 25, C55–C60 (2008).
[CrossRef]

S. Melle, O. G. Calderón, F. Carreño, E. Cabrera, M. A. Antón, and S. Jarabo, “Effect of ion concentration on slow light propagation in highly doped erbium fibers,” Opt. Commun. 279, 53–63 (2007).
[CrossRef]

Camacho, R. M.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Slow light with large fractional delays by spectral hole-burning in rubidium vapor,” Phys. Rev. A 74, 033801 (2006).
[CrossRef]

Carreño, F.

O. G. Calderón, S. Melle, F. Arrieta-Yañez, M. A. Anton, and F. Carreño, “Effect of ion pairs in fast-light bandwidth in high-concentration erbium-doped fibers,” J. Opt. Soc. Am. B 25, C55–C60 (2008).
[CrossRef]

S. Melle, O. G. Calderón, F. Carreño, E. Cabrera, M. A. Antón, and S. Jarabo, “Effect of ion concentration on slow light propagation in highly doped erbium fibers,” Opt. Commun. 279, 53–63 (2007).
[CrossRef]

Chanelière, T.

R. Lauro, T. Chanelière, and J. L. Le Gouët, “Slow light using spectral hole burning in a Tm3+-doped yttrium-aluminum-garnet crystal,” Phys. Rev. A 79, 063844 (2009).
[CrossRef]

Chase, E. W.

Y. Silberberg, V. L. da Silva, J. P. Heritage, E. W. Chase, and M. J. Andrejco, “Accumulated photon echoes in doped fibers,” IEEE J. Quantum Electron. 28, 2369–2381 (1992).
[CrossRef]

Choblet, S.

Cone, R. L.

C. W. Thiel, T. Böttger, and R. L. Cone, “Rare-earth-doped materials for applications in quantum information storage and signal processing,” J. Lumin. 131, 353–361 (2011).
[CrossRef]

Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
[CrossRef]

R. M. MacFarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602 (2006).
[CrossRef]

da Silva, V. L.

Y. Silberberg, V. L. da Silva, J. P. Heritage, E. W. Chase, and M. J. Andrejco, “Accumulated photon echoes in doped fibers,” IEEE J. Quantum Electron. 28, 2369–2381 (1992).
[CrossRef]

Desurvire, E.

E. Desurvire, J. L. Zyskind, and J. R. Simpson, “Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 2, 246–248 (1990).
[CrossRef]

J. L. Zyskind, E. Desurvire, J. W. Sulhoff, and D. J. Di Giovanni, “Determination of homogeneous linewidth by spectral gain hole-burning in an erbium-doped fiber amplifier with GeO2:SiO2 core,” IEEE Photon. Technol. Lett. 2, 869–871 (1990).
[CrossRef]

Dey, T. N.

G. S. Agarwal and T. N. Dey, “Slow light in Doppler-broadened two-level systems,” Phys. Rev. A 68, 063816 (2003).
[CrossRef]

Di Giovanni, D. J.

J. L. Zyskind, E. Desurvire, J. W. Sulhoff, and D. J. Di Giovanni, “Determination of homogeneous linewidth by spectral gain hole-burning in an erbium-doped fiber amplifier with GeO2:SiO2 core,” IEEE Photon. Technol. Lett. 2, 869–871 (1990).
[CrossRef]

Digonnet, M. J. F.

P. F. Wysocki, J. L. Wagener, M. J. F. Digonnet, and H. J. Shaw, “Evidence and modelling of paired ions and other loss mechanisms in erbium-doped silica fibers,” Proc. SPIE 1789, 66–79 (1993).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Ezquerro, J.

Gaeta, A. L.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

Gasca, L.

R. Peretti, A. M. Jurdyc, B. Jacquier, E. Burov, and L. Gasca, “Resonant fluorescence line narrowing and gain spectral hole burning in erbium-doped fiber amplifier,” J. Lumin. 128, 1010–1012 (2008).
[CrossRef]

L. Bigot, A. M. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glasses for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

Gauthier, D. J.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

Gisin, N.

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silica fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

Golding, B.

M. M. Broer, B. Golding, W. H. Haemmerle, and J. R. Simpson, “Low-temperature optical dephasing of rare-earth ions in inorganic glasses,” Phys. Rev. B 33, 4160–4165 (1986).
[CrossRef]

D. L. Huber, M. M. Broer, and B. Golding, “Low temperature optical dephasing of rare-earth ions in glass,” Phys. Rev. Lett. 52, 2281–2284 (1984).
[CrossRef]

González-Herráez, M.

Haemmerle, W. H.

M. M. Broer, B. Golding, W. H. Haemmerle, and J. R. Simpson, “Low-temperature optical dephasing of rare-earth ions in inorganic glasses,” Phys. Rev. B 33, 4160–4165 (1986).
[CrossRef]

Hahn, J.

Ham, B. S.

J. Hahn and B. S. Ham, “Observations of self-induced ultraslow light in a persistent spectral hole burning medium,” Opt. Express 16, 16723–16728 (2008).
[CrossRef]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Haroche, S.

S. Haroche and F. Hartmann, “Theory of saturated-absorption line shapes,” Phys. Rev. A 6, 1280–1300 (1972).
[CrossRef]

Harris, S. E.

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Hartmann, F.

S. Haroche and F. Hartmann, “Theory of saturated-absorption line shapes,” Phys. Rev. A 6, 1280–1300 (1972).
[CrossRef]

Hastings-Simon, S. R.

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silica fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Hemmer, P. R.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Heritage, J. P.

Y. Silberberg, V. L. da Silva, J. P. Heritage, E. W. Chase, and M. J. Andrejco, “Accumulated photon echoes in doped fibers,” IEEE J. Quantum Electron. 28, 2369–2381 (1992).
[CrossRef]

Howell, J. C.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Slow light with large fractional delays by spectral hole-burning in rubidium vapor,” Phys. Rev. A 74, 033801 (2006).
[CrossRef]

Huber, D. L.

D. L. Huber, M. M. Broer, and B. Golding, “Low temperature optical dephasing of rare-earth ions in glass,” Phys. Rev. Lett. 52, 2281–2284 (1984).
[CrossRef]

Jaccard, D.

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silica fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

Jacquier, B.

R. Peretti, B. Jacquier, D. Boivin, E. Burov, and A. M. Jurdyc, “Inhomogeneous gain saturation in EDF: experiment and modeling,” J. Lightwave Technol. 29, 1445–1452 (2011).
[CrossRef]

R. Peretti, A. M. Jurdyc, B. Jacquier, E. Burov, and L. Gasca, “Resonant fluorescence line narrowing and gain spectral hole burning in erbium-doped fiber amplifier,” J. Lumin. 128, 1010–1012 (2008).
[CrossRef]

Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
[CrossRef]

L. Bigot, S. Choblet, A. M. Jurdyc, and B. Jacquier, “Transient spectral hole burning in erbium-doped fluoride glasses,” J. Opt. Soc. Am. B 21, 307–312 (2004).
[CrossRef]

L. Bigot, A. M. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glasses for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

Jarabo, S.

S. Melle, O. G. Calderón, F. Carreño, E. Cabrera, M. A. Antón, and S. Jarabo, “Effect of ion concentration on slow light propagation in highly doped erbium fibers,” Opt. Commun. 279, 53–63 (2007).
[CrossRef]

Javan, A.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, “Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium,” Phys. Rev. A 66, 013805 (2002).
[CrossRef]

Jurdyc, A. M.

R. Peretti, B. Jacquier, D. Boivin, E. Burov, and A. M. Jurdyc, “Inhomogeneous gain saturation in EDF: experiment and modeling,” J. Lightwave Technol. 29, 1445–1452 (2011).
[CrossRef]

R. Peretti, A. M. Jurdyc, B. Jacquier, E. Burov, and L. Gasca, “Resonant fluorescence line narrowing and gain spectral hole burning in erbium-doped fiber amplifier,” J. Lumin. 128, 1010–1012 (2008).
[CrossRef]

L. Bigot, S. Choblet, A. M. Jurdyc, and B. Jacquier, “Transient spectral hole burning in erbium-doped fluoride glasses,” J. Opt. Soc. Am. B 21, 307–312 (2004).
[CrossRef]

L. Bigot, A. M. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glasses for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

Kocharovskaya, O.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, “Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium,” Phys. Rev. A 66, 013805 (2002).
[CrossRef]

Lauro, R.

R. Lauro, T. Chanelière, and J. L. Le Gouët, “Slow light using spectral hole burning in a Tm3+-doped yttrium-aluminum-garnet crystal,” Phys. Rev. A 79, 063844 (2009).
[CrossRef]

Le Gouët, J. L.

R. Lauro, T. Chanelière, and J. L. Le Gouët, “Slow light using spectral hole burning in a Tm3+-doped yttrium-aluminum-garnet crystal,” Phys. Rev. A 79, 063844 (2009).
[CrossRef]

Lee, H.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, “Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium,” Phys. Rev. A 66, 013805 (2002).
[CrossRef]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

MacFarlane, R. M.

R. M. MacFarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602 (2006).
[CrossRef]

R. M. MacFarlane and R. M. Shelby, “Homogeneous line broadening of optical transitions of ions and molecules in glasses,” J. Lumin. 36, 179–207 (1987).
[CrossRef]

Mégret, P.

R. N. Shakhmuratov, A. Rebane, P. Mégret, and J. Odeurs, “Slow light with persistent hole burning,” Phys. Rev. A 71, 053811 (2005).
[CrossRef]

Melle, S.

Mitsunaga, M.

R. Yano, M. Mitsunaga, and N. Uesugi, “Stimulated-photon-echo spectroscopy. I. Spectral diffusion in Eu3+:YAIO3,” Phys. Rev. B 45, 12752–12759 (1992).
[CrossRef]

Musser, J. A.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Nakano, D.

T. Suemoto, T. Okuno, and D. Nakano, “Defect-induced persistent hole burning in MgO-doped Pr3+:YAG systems,” Opt. Commun. 145, 113–118 (1998).
[CrossRef]

Odeurs, J.

R. N. Shakhmuratov, A. Rebane, P. Mégret, and J. Odeurs, “Slow light with persistent hole burning,” Phys. Rev. A 71, 053811 (2005).
[CrossRef]

Okawachi, Y.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

Okuno, T.

T. Suemoto, T. Okuno, and D. Nakano, “Defect-induced persistent hole burning in MgO-doped Pr3+:YAG systems,” Opt. Commun. 145, 113–118 (1998).
[CrossRef]

Pack, M. V.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Slow light with large fractional delays by spectral hole-burning in rubidium vapor,” Phys. Rev. A 74, 033801 (2006).
[CrossRef]

Peretti, R.

R. Peretti, B. Jacquier, D. Boivin, E. Burov, and A. M. Jurdyc, “Inhomogeneous gain saturation in EDF: experiment and modeling,” J. Lightwave Technol. 29, 1445–1452 (2011).
[CrossRef]

R. Peretti, A. M. Jurdyc, B. Jacquier, E. Burov, and L. Gasca, “Resonant fluorescence line narrowing and gain spectral hole burning in erbium-doped fiber amplifier,” J. Lumin. 128, 1010–1012 (2008).
[CrossRef]

Rebane, A.

R. N. Shakhmuratov, A. Rebane, P. Mégret, and J. Odeurs, “Slow light with persistent hole burning,” Phys. Rev. A 71, 053811 (2005).
[CrossRef]

Schweinsberg, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

Scully, M. O.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, “Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium,” Phys. Rev. A 66, 013805 (2002).
[CrossRef]

Sellin, P. B.

R. M. MacFarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602 (2006).
[CrossRef]

Shahriar, M. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Shakhmuratov, R. N.

R. N. Shakhmuratov, A. Rebane, P. Mégret, and J. Odeurs, “Slow light with persistent hole burning,” Phys. Rev. A 71, 053811 (2005).
[CrossRef]

Sharping, J. E.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

Shaw, H. J.

P. F. Wysocki, J. L. Wagener, M. J. F. Digonnet, and H. J. Shaw, “Evidence and modelling of paired ions and other loss mechanisms in erbium-doped silica fibers,” Proc. SPIE 1789, 66–79 (1993).
[CrossRef]

Shelby, R. M.

R. M. MacFarlane and R. M. Shelby, “Homogeneous line broadening of optical transitions of ions and molecules in glasses,” J. Lumin. 36, 179–207 (1987).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986), Chap. 30.

Silberberg, Y.

Y. Silberberg, V. L. da Silva, J. P. Heritage, E. W. Chase, and M. J. Andrejco, “Accumulated photon echoes in doped fibers,” IEEE J. Quantum Electron. 28, 2369–2381 (1992).
[CrossRef]

Simpson, J. R.

E. Desurvire, J. L. Zyskind, and J. R. Simpson, “Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 2, 246–248 (1990).
[CrossRef]

M. M. Broer, B. Golding, W. H. Haemmerle, and J. R. Simpson, “Low-temperature optical dephasing of rare-earth ions in inorganic glasses,” Phys. Rev. B 33, 4160–4165 (1986).
[CrossRef]

Song, K.-Y.

Staudt, M. U.

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silica fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

Sudarshanam, V. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Suemoto, T.

T. Suemoto, T. Okuno, and D. Nakano, “Defect-induced persistent hole burning in MgO-doped Pr3+:YAG systems,” Opt. Commun. 145, 113–118 (1998).
[CrossRef]

Sulhoff, J. W.

J. L. Zyskind, E. Desurvire, J. W. Sulhoff, and D. J. Di Giovanni, “Determination of homogeneous linewidth by spectral gain hole-burning in an erbium-doped fiber amplifier with GeO2:SiO2 core,” IEEE Photon. Technol. Lett. 2, 869–871 (1990).
[CrossRef]

Sun, Y.

Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
[CrossRef]

R. M. MacFarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602 (2006).
[CrossRef]

Thévenaz, L.

Thiel, C. W.

C. W. Thiel, T. Böttger, and R. L. Cone, “Rare-earth-doped materials for applications in quantum information storage and signal processing,” J. Lumin. 131, 353–361 (2011).
[CrossRef]

Tittel, W.

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silica fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

Turukhin, A. V.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Uesugi, N.

R. Yano, M. Mitsunaga, and N. Uesugi, “Stimulated-photon-echo spectroscopy. I. Spectral diffusion in Eu3+:YAIO3,” Phys. Rev. B 45, 12752–12759 (1992).
[CrossRef]

Wagener, J. L.

P. F. Wysocki, J. L. Wagener, M. J. F. Digonnet, and H. J. Shaw, “Evidence and modelling of paired ions and other loss mechanisms in erbium-doped silica fibers,” Proc. SPIE 1789, 66–79 (1993).
[CrossRef]

Wysocki, P. F.

P. F. Wysocki, J. L. Wagener, M. J. F. Digonnet, and H. J. Shaw, “Evidence and modelling of paired ions and other loss mechanisms in erbium-doped silica fibers,” Proc. SPIE 1789, 66–79 (1993).
[CrossRef]

Yano, R.

R. Yano, M. Mitsunaga, and N. Uesugi, “Stimulated-photon-echo spectroscopy. I. Spectral diffusion in Eu3+:YAIO3,” Phys. Rev. B 45, 12752–12759 (1992).
[CrossRef]

Yin, G. Y.

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

Zhu, Z.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

Zyskind, J. L.

E. Desurvire, J. L. Zyskind, and J. R. Simpson, “Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 2, 246–248 (1990).
[CrossRef]

J. L. Zyskind, E. Desurvire, J. W. Sulhoff, and D. J. Di Giovanni, “Determination of homogeneous linewidth by spectral gain hole-burning in an erbium-doped fiber amplifier with GeO2:SiO2 core,” IEEE Photon. Technol. Lett. 2, 869–871 (1990).
[CrossRef]

IEEE J. Quantum Electron.

Y. Silberberg, V. L. da Silva, J. P. Heritage, E. W. Chase, and M. J. Andrejco, “Accumulated photon echoes in doped fibers,” IEEE J. Quantum Electron. 28, 2369–2381 (1992).
[CrossRef]

IEEE Photon. Technol. Lett.

E. Desurvire, J. L. Zyskind, and J. R. Simpson, “Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 2, 246–248 (1990).
[CrossRef]

J. L. Zyskind, E. Desurvire, J. W. Sulhoff, and D. J. Di Giovanni, “Determination of homogeneous linewidth by spectral gain hole-burning in an erbium-doped fiber amplifier with GeO2:SiO2 core,” IEEE Photon. Technol. Lett. 2, 869–871 (1990).
[CrossRef]

J. Lightwave Technol.

J. Lumin.

R. Peretti, A. M. Jurdyc, B. Jacquier, E. Burov, and L. Gasca, “Resonant fluorescence line narrowing and gain spectral hole burning in erbium-doped fiber amplifier,” J. Lumin. 128, 1010–1012 (2008).
[CrossRef]

C. W. Thiel, T. Böttger, and R. L. Cone, “Rare-earth-doped materials for applications in quantum information storage and signal processing,” J. Lumin. 131, 353–361 (2011).
[CrossRef]

M. J. Weber, ed., Special feature on “Optical linewidths in glasses,” J. Lumin. 36, 179–329 (1987).

R. M. MacFarlane and R. M. Shelby, “Homogeneous line broadening of optical transitions of ions and molecules in glasses,” J. Lumin. 36, 179–207 (1987).
[CrossRef]

J. Opt. Soc. Am. B

Nature

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Opt. Commun.

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silica fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

S. Melle, O. G. Calderón, F. Carreño, E. Cabrera, M. A. Antón, and S. Jarabo, “Effect of ion concentration on slow light propagation in highly doped erbium fibers,” Opt. Commun. 279, 53–63 (2007).
[CrossRef]

T. Suemoto, T. Okuno, and D. Nakano, “Defect-induced persistent hole burning in MgO-doped Pr3+:YAG systems,” Opt. Commun. 145, 113–118 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

G. S. Agarwal and T. N. Dey, “Slow light in Doppler-broadened two-level systems,” Phys. Rev. A 68, 063816 (2003).
[CrossRef]

R. N. Shakhmuratov, A. Rebane, P. Mégret, and J. Odeurs, “Slow light with persistent hole burning,” Phys. Rev. A 71, 053811 (2005).
[CrossRef]

R. M. Camacho, M. V. Pack, and J. C. Howell, “Slow light with large fractional delays by spectral hole-burning in rubidium vapor,” Phys. Rev. A 74, 033801 (2006).
[CrossRef]

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

R. Lauro, T. Chanelière, and J. L. Le Gouët, “Slow light using spectral hole burning in a Tm3+-doped yttrium-aluminum-garnet crystal,” Phys. Rev. A 79, 063844 (2009).
[CrossRef]

S. Haroche and F. Hartmann, “Theory of saturated-absorption line shapes,” Phys. Rev. A 6, 1280–1300 (1972).
[CrossRef]

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, “Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium,” Phys. Rev. A 66, 013805 (2002).
[CrossRef]

Phys. Rev. B

L. Bigot, A. M. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glasses for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

M. M. Broer, B. Golding, W. H. Haemmerle, and J. R. Simpson, “Low-temperature optical dephasing of rare-earth ions in inorganic glasses,” Phys. Rev. B 33, 4160–4165 (1986).
[CrossRef]

R. Yano, M. Mitsunaga, and N. Uesugi, “Stimulated-photon-echo spectroscopy. I. Spectral diffusion in Eu3+:YAIO3,” Phys. Rev. B 45, 12752–12759 (1992).
[CrossRef]

Phys. Rev. Lett.

R. M. MacFarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602 (2006).
[CrossRef]

D. L. Huber, M. M. Broer, and B. Golding, “Low temperature optical dephasing of rare-earth ions in glass,” Phys. Rev. Lett. 52, 2281–2284 (1984).
[CrossRef]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef]

Proc. SPIE

P. F. Wysocki, J. L. Wagener, M. J. F. Digonnet, and H. J. Shaw, “Evidence and modelling of paired ions and other loss mechanisms in erbium-doped silica fibers,” Proc. SPIE 1789, 66–79 (1993).
[CrossRef]

Science

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef]

Other

A. E. Siegman, Lasers (University Science, 1986), Chap. 30.

L. F. Shampine, M. W. Reichelt, and J. Kierzenka, “Solving boundary value problems for ordinary differential equations in MATLAB with BVP4C,” http://www.mathworks.com/bvp_tutorial .

The Cauchy principal value integral was evaluated by a double exponential quadrature. Coding in Matlab is due to Mohankumar and Natarajan http://www.mathworks.com/matlabcentral/fileexchange/13871-hilbertf .

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Figures (11)

Fig. 1.
Fig. 1.

Experimental setup for SHB in EDFs. DFB LD, distributed feedback laser diode; LD TEC, laser diode and temperature controller; TDL, tunable diode laser; FG, function generator; VOA, variable optical attenuator; CIR, circulator; EDF, erbium-doped fiber; PD, photodetector; OS, oscilloscope.

Fig. 2.
Fig. 2.

Absorption spectrum of the Er80 fiber measured at 10.8 K. Probe input power, 1 μW; control power at 1536.3 nm, 10 μW; hole width (FWHM) Δ λ , 0.031 nm. (Inset) Zoom of the spectral hole where experimental data (thick gray line) are overlapped to the Lorentzian fit (solid red line). Inset axes units are the same as in main figure.

Fig. 3.
Fig. 3.

Hole-width dependence with control power for the Er80 fiber at 22.5 K. (a) Experimental data at low control powers (squares) and linear fit (solid line) to obtain the hole-width value at zero control power: Δ λ = 0.047 ± 0.006 nm . From Eq. (19), Γ h = 3.0 ± 0.4 GHz . (b) Experimental data for a wide control power range (squares), simulated curve (solid line) from Eqs. (15) and (16), and analytical result (dotted line) using Eq. (19).

Fig. 4.
Fig. 4.

Simulation results showing the dependence of the hole width normalized to Γ h with control power for different optical depths: α 0 RT L = 1.5 (solid green line), α 0 RT L = 5 (dotted line), α 0 RT L = 15 (dashed line), and α 0 RT L = 30 (dotted–dashed line). The thick gray line corresponds to the analytical result given by Eq. (19), which neglects the attenuation of the control field.

Fig. 5.
Fig. 5.

(a) Experimental homogeneous linewidth dependence with temperature for Er30 and Er80 fibers (symbols). Solid lines are power-law fits to the data. The dashed line is the theoretical homogeneous linewidth dependence with temperature given by Eq. (20). (b) Comparison of our experimental data (solid symbols) with other experimental results (open symbols) performed on similar long time scales by using different techniques: accumulated photon echoes [17], SHB [18], spectral gain hole burning [19,20,22], and resonant fluorescence line narrowing [21,22].

Fig. 6.
Fig. 6.

Transmittance (percent) dependence with input power at different temperatures for (a) Er80 fiber and (b) Er30 fiber. Symbols, experimental data; lines, simulation results from Eq. (15).

Fig. 7.
Fig. 7.

(a) Dependence of the saturation power obtained from fitting the transmittance curves using Eq. (15) with temperature; (b) homogeneous linewidth dependence with temperature obtained from SHB experiments (squares) and transmittance measurements (open circles).

Fig. 8.
Fig. 8.

Experimental (symbols) and theoretical (solid lines) relative change of the complex susceptibility χ . The experimental curves were obtained through the relative change of the absorption spectrum of Fig. 2 (imaginary part of χ ) and the Kramers–Kronig relation (real part of χ ). The theoretical curves were obtained through Eqs. (13) and (14).

Fig. 9.
Fig. 9.

Simulated output pulse after propagation through the fiber at 20 K for several values of the normalized control power P ^ c . The input pulse is plotted for comparison purpose (thick gray line). Simulation parameters: α 0 RT L = 5 , P sat RT = 0.4 mW , Δ in = 0.0027 Γ h RT , and Γ h RT / Γ h = 244.4 .

Fig. 10.
Fig. 10.

Simulations of the fractional delay F d versus the normalized control power P c / P sat RT at different temperatures. Simulation parameters: α 0 RT L = 5 ; P sat RT = 0.4 mW ; Δ in = 0.0027 Γ h RT ; and Γ h RT / Γ h = 478 , 244.4, 83.3, and 23.4 for temperatures T = 11 , 20, 40, and 77 K, respectively.

Fig. 11.
Fig. 11.

Comparison of the output pulse predicted by the analytical solution in Eq. (22) (dashed line) and the full numerical integration of Eq. (22) (solid line) at a given temperature T = 11 K . The input pulse is also plotted for comparison purposes (dotted line). Simulation parameters: α 0 RT L = 5 , P sa t RT = 0.4 mW , Δ in = 0.0054 Γ h RT , and Γ h RT / Γ h = 478 .

Tables (1)

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Table 1. Fiber Properties a

Equations (25)

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E c ( z , t ) = 1 2 E c ( z , t ) e i ω c t + c.c. , E p ( z , t ) = 1 2 E p ( z , t ) e i ω p t + c.c. ,
D ( n ) t = γ ( D ( n ) + 1 ) + 2 i ( Ω c + Ω p e i δ t ) ρ 12 ( n ) + c.c , ρ 21 ( n ) t = ( γ 21 i Δ n ) ρ 21 ( n ) i ( Ω c + Ω p e i δ t ) D ( n ) ,
D ( n ) = D 0 + D 1 e i δ t + D 2 e i δ t , ρ 21 ( n ) = p 0 + p 1 e i δ t + p 2 e i δ t ,
D 0 t = γ ( D 0 + 1 ) + 2 i ( Ω c p 0 * Ω c * p 0 + Ω p p 1 * Ω p * p 1 ) , D 1 t = ( γ i δ ) D 1 + 2 i ( Ω c p 2 * Ω c * p 1 + Ω p p 0 * ) , p 0 t = ( γ 21 i Δ n ) p 0 i ( Ω c D 0 + Ω p D 2 ) , p 1 t = ( γ 21 i Δ n i δ ) p 1 i ( Ω c D 1 + Ω p D 0 ) , p 2 t = ( γ 21 i Δ n + i δ ) p 2 i Ω c D 2 .
D 0 t = γ ( D 0 + 1 ) + 2 i ( Ω c p 0 * Ω c * p 0 ) , p 0 t = ( γ 21 i Δ n ) p 0 i Ω c D 0 , p 1 t = ( γ 21 i Δ n i δ ) p 1 i Ω p D 0 .
D 0 = 1 + I ^ c 1 + I ^ c + Δ n 2 / γ 21 2 , p 0 = i Ω c D 0 γ 21 i Δ n , p 1 = i Ω p D 0 γ 21 i ( Δ n + δ ) ,
I sat = c ϵ 0 2 γ γ 21 2 μ 21 2 .
E ˜ c ( z , ω ) z + i ω c E ˜ c ( z , ω ) = i ω c 2 ϵ 0 P ˜ c ( z , ω ) , E ˜ p ( z , ω ) z + i ω c E ˜ p ( z , ω ) = i ω p 2 ϵ 0 P ˜ p ( z , ω ) ,
P ˜ c ( z , ω ) = 2 N μ 21 p 0 ( ω n ) G ( ω n ) d ω n , P ˜ p ( z , ω ) = 2 N μ 21 p 1 ( ω n ) G ( ω n ) d ω n ,
G ( ω n ) = Γ inh π 1 Δ n 2 + Γ inh 2 .
E ˜ c ( z , ω ) z + i ω c E ˜ c ( z , ω ) = N μ 21 2 ω c 2 c ϵ 0 E ˜ c ( z , ω ) ( Γ inh + γ 21 1 + I ^ c ) 1 + I ^ c ,
E ˜ p ( z , ω ) z + i ω c E ˜ p ( z , ω ) = ω p 2 c [ χ I ( ω ) i χ R ( ω ) ] E ˜ p ( z , ω ) ,
χ R ( ω ) = N μ 21 2 ϵ 0 ω ω 2 + Γ inh 2 [ 1 γ 21 Γ inh I ^ c 1 + I ^ c ( 1 + Γ inh ( Γ inh + Δ h ) ω 2 + Δ h 2 ) ] ,
χ I ( ω ) = N μ 21 2 ϵ 0 Γ inh ω 2 + Γ inh 2 [ 1 γ 21 2 Γ inh 2 I ^ c 1 + I ^ c ( 1 + Γ inh ( Γ inh + Δ h ) Δ h / γ 21 ω 2 + Δ h 2 ) ] ,
P ^ c ( z ) z = α 0 γ 21 Γ inh P ^ c 1 + P ^ c ,
P ^ p ( z ) z = α 0 γ 21 Γ inh [ 1 P ^ c 1 + P ^ c 1 + 1 + P ^ c ( δ / γ 21 ) 2 + ( 1 + 1 + P ^ c ) 2 ] P ^ p ,
P p ( L ) = P p ( 0 ) e α L ,
α = α 0 γ 21 Γ inh [ 1 P ^ c 1 + P ^ c 1 + 1 + P ^ c ( δ / γ 21 ) 2 + ( 1 + 1 + P ^ c ) 2 ] .
Δ ν = Γ h [ 1 + 1 + P ^ c ] ,
Γ h ( T ) = β 1 1 e Δ E 1 / ( k T ) 1 + β 2 1 e Δ E 2 / ( k T ) 1 + η ( T T D ) 7 0 T D / T x 6 e x ( e x 1 ) 2 d x ,
E p ( z = 0 , t ) = E 0 e π 2 Δ in 2 t 2 / ( 4 log 2 ) ,
E p ( z = L , t ) = E 0 e α 0 L b I / 2 e π 2 Δ in 2 ( t τ d ) 2 / ( 4 log 2 ) ,
b R = γ 21 Γ inh 2 [ 1 γ 21 Γ inh P ^ c 1 + P ^ c ( 1 + Γ inh ( Γ inh + Δ h ) Δ h 2 ) ] , b I = γ 21 Γ inh [ 1 γ 21 2 Γ inh 2 P ^ c 1 + P ^ c ( 1 + Γ inh ( Γ inh + Δ h ) / γ 21 Δ h ) ] .
τ d α 0 L Γ inh P ^ c 1 + P ^ c ( 1 + 1 + P ^ c ) 2 ,
τ d MAX = α 0 RT L ( 1 + 2 ) 2 1 γ 21 .

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