Abstract

Disagreements on the Raman gain response of different tellurite-based glasses, measured at different wavelengths, have been recently reported in the literature. In order to resolve this controversy, a multi-wavelength Raman cross-section experiment was conducted on two different TeO2-based glass samples. The estimated Raman gain response of the material shows good agreement with the directly-measured Raman gain data at 1064 nm, after correction for the dispersion and wavelength-dependence of the Raman gain process.

© 2005 Optical Society of America

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References

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    [Crossref]
  26. R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–273 (1972)
    [Crossref]
  27. R. Stegeman, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, L. Jankovic, and H. Kim, “Raman gain measurements in bulk glass samples”, J. Opt. Soc. Am. B (to be published)
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2005 (5)

2004 (2)

C. Rivero, K. Richardson, R. Stegeman, G. Stegeman, T. Cardinal, E. Fargin, M. Couzi, and V. Rodriguez, “Quantifying Raman Gain Coefficients in Tellurite Glasses,” J. Non-Cryst. Sols. 345&346, 396–401 (2004)
[Crossref]

G. Dai, F. Tassone, A. Li Bassi, V. Russo, C.E. Bottani, and F. D’Amore, “TeO2-based glasses containing Nb2O5, TiO2, and WO3 for discrete Raman fiber amplification,” Photon. Technol. Lett. 16, 1011–1013, (2004)
[Crossref]

2003 (1)

2002 (2)

M.N. Islam, “Raman Amplifiers for Telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–559 (2002)
[Crossref]

J. Bromage, K. Rottwitt, and M.E. Lines, “A Method to Predict the Raman Gain Spectra of Germanosilicate Fibers With Arbitrary Index Profiles,” Photon. Technol. Lett. 14, 24–26 (2002)
[Crossref]

2001 (1)

A. Mori, H. Masuda, K. Shikano, K. Oikawa, K. Kato, and M. Shimizu, “Ultra-wideband tellurite-based Raman fibre amplifier,” Electron. Lett. 37, 1442–1443 (2001)
[Crossref]

2000 (1)

H.S. Seo and K. Oh, “Optimization of silica fiber Raman amplifier using the Raman frequency modeling for an arbitrary GeO2 concentration,” Opt. Commun. 181, 145–151 (2000)
[Crossref]

1999 (1)

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

1996 (1)

A. Berthereau, E. Fargin, A. Villezusanne, R. Olazcuaga, G. Le Flem, and L. Ducasse, “Determination of local geometries around tellurium in TeO2-Nb2O5 and TeO2-Al2O3 oxide glasses by XANES and EXAFS: investigation of electronic properties of evidenced oxygen clusters by ab initio calculations,” J. Sol. St. Chem. 126, 143–151 (1996)
[Crossref]

1992 (1)

T. Sekiya, N. Mochida, A. Ohtsuka, and M. Tonokawa, “Raman spectra of MO1/2-TeO2 (M=Li, Na, K, Rb, Cs, and Tl) glasses,” J. Non-Cryst. Sols. 144, 128 (1992)
[Crossref]

1991 (2)

M.E. Lines, “Oxide glasses for fast photonic switching: A comparative study,” J. Appl. Phys. 69, 6876–6884 (1991)
[Crossref]

M.E. Lines, “Influence of d orbitals on the nonlinear optical response of transparent transition-metal oxides,” Phys. Rev. B 43, 11978–11990 (1991)
[Crossref]

1988 (1)

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Sols. 99, 289–307 (1988)
[Crossref]

1987 (2)

M. E. Lines, “Raman gain estimates for high-gain optical fibers,” J. Appl. Phys. 62, 4363–4370 (1987)
[Crossref]

M.E. Lines, “Absolute Raman Intensities in Glasses, I. Theory”, J. Non-Cryst. Sols. 89, 143–162 (1987)
[Crossref]

1978 (2)

F.L Galeener, J. C. Mikkelsen, R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3, and P2O5,” Appl. Phys. Lett. 32, 34–36 (1978)
[Crossref]

N.L. Boling, A.J. Glass, and A. Owyoung, “Empirical Relationships for Predicting Nonlinear Refractive Index Changes in Optical Solids,” IEEE J. Quantum Electron. 14, 601–608 (1978)
[Crossref]

1972 (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–273 (1972)
[Crossref]

1971 (2)

Y. Kato and H. Takuma, “Experimental Study on the Wavelength Dependence of the Raman Scattering Cross Sections,” J. Chemical Phys. 54, 5398–5402 (1971)
[Crossref]

Y. Kato and H. Takuma, “Absolute Measurement of Raman-Scattering Cross Sections in Liquids,” J. Opt. Soc. Am. 61, 347–350 (1971)
[Crossref]

Apanasevich, P.A.

V.A. Lisinetskii, I.I. Mishkel, R.V. Chulkov, A.S. Grabtchikov, P.A. Apanasevich, H.J. Eichler, and V.A. Orlovich, “Raman gain coefficient of Barium Nitrate measured for the spectral region of Ti:Sapphire Laser,” J. of Nonlinear Opt. Phys. Mat. 14, 107–114 (2005)
[Crossref]

Berthereau, A.

A. Berthereau, E. Fargin, A. Villezusanne, R. Olazcuaga, G. Le Flem, and L. Ducasse, “Determination of local geometries around tellurium in TeO2-Nb2O5 and TeO2-Al2O3 oxide glasses by XANES and EXAFS: investigation of electronic properties of evidenced oxygen clusters by ab initio calculations,” J. Sol. St. Chem. 126, 143–151 (1996)
[Crossref]

Blanchandin, S.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Boling, N.L.

N.L. Boling, A.J. Glass, and A. Owyoung, “Empirical Relationships for Predicting Nonlinear Refractive Index Changes in Optical Solids,” IEEE J. Quantum Electron. 14, 601–608 (1978)
[Crossref]

Bosquet, B.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Bottani, C.E.

G. Dai, F. Tassone, A. Li Bassi, V. Russo, C.E. Bottani, and F. D’Amore, “TeO2-based glasses containing Nb2O5, TiO2, and WO3 for discrete Raman fiber amplification,” Photon. Technol. Lett. 16, 1011–1013, (2004)
[Crossref]

Bromage, J.

J. Bromage, K. Rottwitt, and M.E. Lines, “A Method to Predict the Raman Gain Spectra of Germanosilicate Fibers With Arbitrary Index Profiles,” Photon. Technol. Lett. 14, 24–26 (2002)
[Crossref]

Canioni, L.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Cardinal, T.

Champarnaud-Mesjard, J. C.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Champarnaud-Mesjard, J-C.

Chulkov, R.V.

V.A. Lisinetskii, I.I. Mishkel, R.V. Chulkov, A.S. Grabtchikov, P.A. Apanasevich, H.J. Eichler, and V.A. Orlovich, “Raman gain coefficient of Barium Nitrate measured for the spectral region of Ti:Sapphire Laser,” J. of Nonlinear Opt. Phys. Mat. 14, 107–114 (2005)
[Crossref]

Churbanov, M. F.

Couzi, M.

C. Rivero, K. Richardson, R. Stegeman, G. Stegeman, T. Cardinal, E. Fargin, M. Couzi, and V. Rodriguez, “Quantifying Raman Gain Coefficients in Tellurite Glasses,” J. Non-Cryst. Sols. 345&346, 396–401 (2004)
[Crossref]

D’Amore, F.

G. Dai, F. Tassone, A. Li Bassi, V. Russo, C.E. Bottani, and F. D’Amore, “TeO2-based glasses containing Nb2O5, TiO2, and WO3 for discrete Raman fiber amplification,” Photon. Technol. Lett. 16, 1011–1013, (2004)
[Crossref]

Dai, G.

G. Dai, F. Tassone, A. Li Bassi, V. Russo, C.E. Bottani, and F. D’Amore, “TeO2-based glasses containing Nb2O5, TiO2, and WO3 for discrete Raman fiber amplification,” Photon. Technol. Lett. 16, 1011–1013, (2004)
[Crossref]

Delfyett, P.

Dianov, E. M.

Ducasse, L.

A. Berthereau, E. Fargin, A. Villezusanne, R. Olazcuaga, G. Le Flem, and L. Ducasse, “Determination of local geometries around tellurium in TeO2-Nb2O5 and TeO2-Al2O3 oxide glasses by XANES and EXAFS: investigation of electronic properties of evidenced oxygen clusters by ab initio calculations,” J. Sol. St. Chem. 126, 143–151 (1996)
[Crossref]

Eichler, H.J.

V.A. Lisinetskii, I.I. Mishkel, R.V. Chulkov, A.S. Grabtchikov, P.A. Apanasevich, H.J. Eichler, and V.A. Orlovich, “Raman gain coefficient of Barium Nitrate measured for the spectral region of Ti:Sapphire Laser,” J. of Nonlinear Opt. Phys. Mat. 14, 107–114 (2005)
[Crossref]

Fargin, E.

C. Rivero, K. Richardson, R. Stegeman, G. Stegeman, T. Cardinal, E. Fargin, M. Couzi, and V. Rodriguez, “Quantifying Raman Gain Coefficients in Tellurite Glasses,” J. Non-Cryst. Sols. 345&346, 396–401 (2004)
[Crossref]

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

A. Berthereau, E. Fargin, A. Villezusanne, R. Olazcuaga, G. Le Flem, and L. Ducasse, “Determination of local geometries around tellurium in TeO2-Nb2O5 and TeO2-Al2O3 oxide glasses by XANES and EXAFS: investigation of electronic properties of evidenced oxygen clusters by ab initio calculations,” J. Sol. St. Chem. 126, 143–151 (1996)
[Crossref]

Frit, B.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Galeener, F.L

F.L Galeener, J. C. Mikkelsen, R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3, and P2O5,” Appl. Phys. Lett. 32, 34–36 (1978)
[Crossref]

Garcia, H.

Geils, R. H.

F.L Galeener, J. C. Mikkelsen, R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3, and P2O5,” Appl. Phys. Lett. 32, 34–36 (1978)
[Crossref]

Glass, A.J.

N.L. Boling, A.J. Glass, and A. Owyoung, “Empirical Relationships for Predicting Nonlinear Refractive Index Changes in Optical Solids,” IEEE J. Quantum Electron. 14, 601–608 (1978)
[Crossref]

Grabtchikov, A.S.

V.A. Lisinetskii, I.I. Mishkel, R.V. Chulkov, A.S. Grabtchikov, P.A. Apanasevich, H.J. Eichler, and V.A. Orlovich, “Raman gain coefficient of Barium Nitrate measured for the spectral region of Ti:Sapphire Laser,” J. of Nonlinear Opt. Phys. Mat. 14, 107–114 (2005)
[Crossref]

Grishin, I. A.

Guo, Y.

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–273 (1972)
[Crossref]

Islam, M.N.

M.N. Islam, “Raman Amplifiers for Telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–559 (2002)
[Crossref]

M.N. Islam, “Raman Amplifiers for Telecommunications 1,” Physical Principles (Springer2004)

Jankovic, L.

Jeansannetas, B.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Johnson, A.M.

Kato, K.

A. Mori, H. Masuda, K. Shikano, K. Oikawa, K. Kato, and M. Shimizu, “Ultra-wideband tellurite-based Raman fibre amplifier,” Electron. Lett. 37, 1442–1443 (2001)
[Crossref]

Kato, Y.

Y. Kato and H. Takuma, “Experimental Study on the Wavelength Dependence of the Raman Scattering Cross Sections,” J. Chemical Phys. 54, 5398–5402 (1971)
[Crossref]

Y. Kato and H. Takuma, “Absolute Measurement of Raman-Scattering Cross Sections in Liquids,” J. Opt. Soc. Am. 61, 347–350 (1971)
[Crossref]

Kim, H.

Koltashev, V. V.

Le Boiteux, S.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Le Flem, G.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

A. Berthereau, E. Fargin, A. Villezusanne, R. Olazcuaga, G. Le Flem, and L. Ducasse, “Determination of local geometries around tellurium in TeO2-Nb2O5 and TeO2-Al2O3 oxide glasses by XANES and EXAFS: investigation of electronic properties of evidenced oxygen clusters by ab initio calculations,” J. Sol. St. Chem. 126, 143–151 (1996)
[Crossref]

Li Bassi, A.

G. Dai, F. Tassone, A. Li Bassi, V. Russo, C.E. Bottani, and F. D’Amore, “TeO2-based glasses containing Nb2O5, TiO2, and WO3 for discrete Raman fiber amplification,” Photon. Technol. Lett. 16, 1011–1013, (2004)
[Crossref]

Lines, M. E.

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Sols. 99, 289–307 (1988)
[Crossref]

M. E. Lines, “Raman gain estimates for high-gain optical fibers,” J. Appl. Phys. 62, 4363–4370 (1987)
[Crossref]

Lines, M.E.

J. Bromage, K. Rottwitt, and M.E. Lines, “A Method to Predict the Raman Gain Spectra of Germanosilicate Fibers With Arbitrary Index Profiles,” Photon. Technol. Lett. 14, 24–26 (2002)
[Crossref]

M.E. Lines, “Influence of d orbitals on the nonlinear optical response of transparent transition-metal oxides,” Phys. Rev. B 43, 11978–11990 (1991)
[Crossref]

M.E. Lines, “Oxide glasses for fast photonic switching: A comparative study,” J. Appl. Phys. 69, 6876–6884 (1991)
[Crossref]

M.E. Lines, “Absolute Raman Intensities in Glasses, I. Theory”, J. Non-Cryst. Sols. 89, 143–162 (1987)
[Crossref]

Lisinetskii, V.A.

V.A. Lisinetskii, I.I. Mishkel, R.V. Chulkov, A.S. Grabtchikov, P.A. Apanasevich, H.J. Eichler, and V.A. Orlovich, “Raman gain coefficient of Barium Nitrate measured for the spectral region of Ti:Sapphire Laser,” J. of Nonlinear Opt. Phys. Mat. 14, 107–114 (2005)
[Crossref]

Liu, M.

F. Yoshino, S. Polyakov, G.I. Stegeman, and M. Liu, “Nonlinear Refraction and Absorption from 1300 to 2200 nm in Single Crystal Polymer poly [bis (p-toluene sulfonate)] of 2, 4-hexadiyne-1, 6-diol (PTS),” Quantum Electronics and Laser Science QELS Prostconference Digest, QTuG43 (2003)

Lyons, K. B.

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Sols. 99, 289–307 (1988)
[Crossref]

Marchet, P.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Martin, M. O.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Masuda, H.

A. Mori, H. Masuda, K. Shikano, K. Oikawa, K. Kato, and M. Shimizu, “Ultra-wideband tellurite-based Raman fibre amplifier,” Electron. Lett. 37, 1442–1443 (2001)
[Crossref]

Merle-Mejean, T.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Mikkelsen, J. C.

F.L Galeener, J. C. Mikkelsen, R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3, and P2O5,” Appl. Phys. Lett. 32, 34–36 (1978)
[Crossref]

Miller, A. E.

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Sols. 99, 289–307 (1988)
[Crossref]

Mishkel, I.I.

V.A. Lisinetskii, I.I. Mishkel, R.V. Chulkov, A.S. Grabtchikov, P.A. Apanasevich, H.J. Eichler, and V.A. Orlovich, “Raman gain coefficient of Barium Nitrate measured for the spectral region of Ti:Sapphire Laser,” J. of Nonlinear Opt. Phys. Mat. 14, 107–114 (2005)
[Crossref]

Mochida, N.

T. Sekiya, N. Mochida, A. Ohtsuka, and M. Tonokawa, “Raman spectra of MO1/2-TeO2 (M=Li, Na, K, Rb, Cs, and Tl) glasses,” J. Non-Cryst. Sols. 144, 128 (1992)
[Crossref]

Mori, A.

A. Mori, H. Masuda, K. Shikano, K. Oikawa, K. Kato, and M. Shimizu, “Ultra-wideband tellurite-based Raman fibre amplifier,” Electron. Lett. 37, 1442–1443 (2001)
[Crossref]

Mosby, W. J.

F.L Galeener, J. C. Mikkelsen, R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3, and P2O5,” Appl. Phys. Lett. 32, 34–36 (1978)
[Crossref]

Murugan, G.S.

G.S. Murugan, T. Suzuki, and Y. Ohishi, “Tellurite glasses for ultrabroadband fiber Raman amplifiers,” Appl. Phys. Lett. 86, 161109 (2005)
[Crossref]

Nassau, K.

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Sols. 99, 289–307 (1988)
[Crossref]

Nazabal, V.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Oguama, F.A.

Oh, K.

H.S. Seo and K. Oh, “Optimization of silica fiber Raman amplifier using the Raman frequency modeling for an arbitrary GeO2 concentration,” Opt. Commun. 181, 145–151 (2000)
[Crossref]

Ohishi, Y.

G.S. Murugan, T. Suzuki, and Y. Ohishi, “Tellurite glasses for ultrabroadband fiber Raman amplifiers,” Appl. Phys. Lett. 86, 161109 (2005)
[Crossref]

Ohtsuka, A.

T. Sekiya, N. Mochida, A. Ohtsuka, and M. Tonokawa, “Raman spectra of MO1/2-TeO2 (M=Li, Na, K, Rb, Cs, and Tl) glasses,” J. Non-Cryst. Sols. 144, 128 (1992)
[Crossref]

Oikawa, K.

A. Mori, H. Masuda, K. Shikano, K. Oikawa, K. Kato, and M. Shimizu, “Ultra-wideband tellurite-based Raman fibre amplifier,” Electron. Lett. 37, 1442–1443 (2001)
[Crossref]

Olazcuaga, R.

A. Berthereau, E. Fargin, A. Villezusanne, R. Olazcuaga, G. Le Flem, and L. Ducasse, “Determination of local geometries around tellurium in TeO2-Nb2O5 and TeO2-Al2O3 oxide glasses by XANES and EXAFS: investigation of electronic properties of evidenced oxygen clusters by ab initio calculations,” J. Sol. St. Chem. 126, 143–151 (1996)
[Crossref]

Orlovich, V.A.

V.A. Lisinetskii, I.I. Mishkel, R.V. Chulkov, A.S. Grabtchikov, P.A. Apanasevich, H.J. Eichler, and V.A. Orlovich, “Raman gain coefficient of Barium Nitrate measured for the spectral region of Ti:Sapphire Laser,” J. of Nonlinear Opt. Phys. Mat. 14, 107–114 (2005)
[Crossref]

Owyoung, A.

N.L. Boling, A.J. Glass, and A. Owyoung, “Empirical Relationships for Predicting Nonlinear Refractive Index Changes in Optical Solids,” IEEE J. Quantum Electron. 14, 601–608 (1978)
[Crossref]

Plotnichenko, V. G.

Polyakov, S.

F. Yoshino, S. Polyakov, G.I. Stegeman, and M. Liu, “Nonlinear Refraction and Absorption from 1300 to 2200 nm in Single Crystal Polymer poly [bis (p-toluene sulfonate)] of 2, 4-hexadiyne-1, 6-diol (PTS),” Quantum Electronics and Laser Science QELS Prostconference Digest, QTuG43 (2003)

Pope, A.

Richardson, K.

R. Stegeman, C. Rivero, K. Richardson, G. Stegeman, P. Delfyett, Y. Guo, A. Pope, A. Schulte, T. Cardinal, P. Thomas, and J-C. Champarnaud-Mesjard, “Raman gain measurements of thallium-tellurium oxide glasses,” Opt. Express 13, 1144–1149 (2005)
[Crossref] [PubMed]

C. Rivero, K. Richardson, R. Stegeman, G. Stegeman, T. Cardinal, E. Fargin, M. Couzi, and V. Rodriguez, “Quantifying Raman Gain Coefficients in Tellurite Glasses,” J. Non-Cryst. Sols. 345&346, 396–401 (2004)
[Crossref]

R. Stegeman, L. Jankovic, H. Kim, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, Y. Guo, A. Schulte, and T. Cardinal, “Tellurite glasses with peak absolute Raman gain coefficients up to 30 times that of fused silica,” Opt. Lett. 28, 1126–1128 (2003)
[Crossref] [PubMed]

R. Stegeman, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, L. Jankovic, and H. Kim, “Raman gain measurements in bulk glass samples”, J. Opt. Soc. Am. B (to be published)

Rivero, C.

R. Stegeman, C. Rivero, K. Richardson, G. Stegeman, P. Delfyett, Y. Guo, A. Pope, A. Schulte, T. Cardinal, P. Thomas, and J-C. Champarnaud-Mesjard, “Raman gain measurements of thallium-tellurium oxide glasses,” Opt. Express 13, 1144–1149 (2005)
[Crossref] [PubMed]

C. Rivero, K. Richardson, R. Stegeman, G. Stegeman, T. Cardinal, E. Fargin, M. Couzi, and V. Rodriguez, “Quantifying Raman Gain Coefficients in Tellurite Glasses,” J. Non-Cryst. Sols. 345&346, 396–401 (2004)
[Crossref]

R. Stegeman, L. Jankovic, H. Kim, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, Y. Guo, A. Schulte, and T. Cardinal, “Tellurite glasses with peak absolute Raman gain coefficients up to 30 times that of fused silica,” Opt. Lett. 28, 1126–1128 (2003)
[Crossref] [PubMed]

R. Stegeman, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, L. Jankovic, and H. Kim, “Raman gain measurements in bulk glass samples”, J. Opt. Soc. Am. B (to be published)

Rodriguez, V.

C. Rivero, K. Richardson, R. Stegeman, G. Stegeman, T. Cardinal, E. Fargin, M. Couzi, and V. Rodriguez, “Quantifying Raman Gain Coefficients in Tellurite Glasses,” J. Non-Cryst. Sols. 345&346, 396–401 (2004)
[Crossref]

Rottwitt, K.

J. Bromage, K. Rottwitt, and M.E. Lines, “A Method to Predict the Raman Gain Spectra of Germanosilicate Fibers With Arbitrary Index Profiles,” Photon. Technol. Lett. 14, 24–26 (2002)
[Crossref]

Russo, V.

G. Dai, F. Tassone, A. Li Bassi, V. Russo, C.E. Bottani, and F. D’Amore, “TeO2-based glasses containing Nb2O5, TiO2, and WO3 for discrete Raman fiber amplification,” Photon. Technol. Lett. 16, 1011–1013, (2004)
[Crossref]

Sarger, L.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Schulte, A.

Segonds, P.

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Sekiya, T.

T. Sekiya, N. Mochida, A. Ohtsuka, and M. Tonokawa, “Raman spectra of MO1/2-TeO2 (M=Li, Na, K, Rb, Cs, and Tl) glasses,” J. Non-Cryst. Sols. 144, 128 (1992)
[Crossref]

Seo, H.S.

H.S. Seo and K. Oh, “Optimization of silica fiber Raman amplifier using the Raman frequency modeling for an arbitrary GeO2 concentration,” Opt. Commun. 181, 145–151 (2000)
[Crossref]

Shikano, K.

A. Mori, H. Masuda, K. Shikano, K. Oikawa, K. Kato, and M. Shimizu, “Ultra-wideband tellurite-based Raman fibre amplifier,” Electron. Lett. 37, 1442–1443 (2001)
[Crossref]

Shimizu, M.

A. Mori, H. Masuda, K. Shikano, K. Oikawa, K. Kato, and M. Shimizu, “Ultra-wideband tellurite-based Raman fibre amplifier,” Electron. Lett. 37, 1442–1443 (2001)
[Crossref]

Sokolov, V. O.

Stegeman, G.

R. Stegeman, C. Rivero, K. Richardson, G. Stegeman, P. Delfyett, Y. Guo, A. Pope, A. Schulte, T. Cardinal, P. Thomas, and J-C. Champarnaud-Mesjard, “Raman gain measurements of thallium-tellurium oxide glasses,” Opt. Express 13, 1144–1149 (2005)
[Crossref] [PubMed]

C. Rivero, K. Richardson, R. Stegeman, G. Stegeman, T. Cardinal, E. Fargin, M. Couzi, and V. Rodriguez, “Quantifying Raman Gain Coefficients in Tellurite Glasses,” J. Non-Cryst. Sols. 345&346, 396–401 (2004)
[Crossref]

R. Stegeman, L. Jankovic, H. Kim, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, Y. Guo, A. Schulte, and T. Cardinal, “Tellurite glasses with peak absolute Raman gain coefficients up to 30 times that of fused silica,” Opt. Lett. 28, 1126–1128 (2003)
[Crossref] [PubMed]

R. Stegeman, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, L. Jankovic, and H. Kim, “Raman gain measurements in bulk glass samples”, J. Opt. Soc. Am. B (to be published)

Stegeman, G.I.

F. Yoshino, S. Polyakov, G.I. Stegeman, and M. Liu, “Nonlinear Refraction and Absorption from 1300 to 2200 nm in Single Crystal Polymer poly [bis (p-toluene sulfonate)] of 2, 4-hexadiyne-1, 6-diol (PTS),” Quantum Electronics and Laser Science QELS Prostconference Digest, QTuG43 (2003)

Stegeman, R.

R. Stegeman, C. Rivero, K. Richardson, G. Stegeman, P. Delfyett, Y. Guo, A. Pope, A. Schulte, T. Cardinal, P. Thomas, and J-C. Champarnaud-Mesjard, “Raman gain measurements of thallium-tellurium oxide glasses,” Opt. Express 13, 1144–1149 (2005)
[Crossref] [PubMed]

C. Rivero, K. Richardson, R. Stegeman, G. Stegeman, T. Cardinal, E. Fargin, M. Couzi, and V. Rodriguez, “Quantifying Raman Gain Coefficients in Tellurite Glasses,” J. Non-Cryst. Sols. 345&346, 396–401 (2004)
[Crossref]

R. Stegeman, L. Jankovic, H. Kim, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, Y. Guo, A. Schulte, and T. Cardinal, “Tellurite glasses with peak absolute Raman gain coefficients up to 30 times that of fused silica,” Opt. Lett. 28, 1126–1128 (2003)
[Crossref] [PubMed]

R. Stegeman, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, L. Jankovic, and H. Kim, “Raman gain measurements in bulk glass samples”, J. Opt. Soc. Am. B (to be published)

Stolen, R. H.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–273 (1972)
[Crossref]

R. H. Stolen, “Issues in Raman gain measurements,” in Tech. Dig. Symp. Optical Fiber Measurements, NIST Special Publication 953 (National Institute of Standards and Technology), (Gaithersburg, MD, 2000) pp. 139

Suzuki, T.

G.S. Murugan, T. Suzuki, and Y. Ohishi, “Tellurite glasses for ultrabroadband fiber Raman amplifiers,” Appl. Phys. Lett. 86, 161109 (2005)
[Crossref]

Takuma, H.

Y. Kato and H. Takuma, “Absolute Measurement of Raman-Scattering Cross Sections in Liquids,” J. Opt. Soc. Am. 61, 347–350 (1971)
[Crossref]

Y. Kato and H. Takuma, “Experimental Study on the Wavelength Dependence of the Raman Scattering Cross Sections,” J. Chemical Phys. 54, 5398–5402 (1971)
[Crossref]

Tassone, F.

G. Dai, F. Tassone, A. Li Bassi, V. Russo, C.E. Bottani, and F. D’Amore, “TeO2-based glasses containing Nb2O5, TiO2, and WO3 for discrete Raman fiber amplification,” Photon. Technol. Lett. 16, 1011–1013, (2004)
[Crossref]

Thomas, P.

R. Stegeman, C. Rivero, K. Richardson, G. Stegeman, P. Delfyett, Y. Guo, A. Pope, A. Schulte, T. Cardinal, P. Thomas, and J-C. Champarnaud-Mesjard, “Raman gain measurements of thallium-tellurium oxide glasses,” Opt. Express 13, 1144–1149 (2005)
[Crossref] [PubMed]

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

Tonokawa, M.

T. Sekiya, N. Mochida, A. Ohtsuka, and M. Tonokawa, “Raman spectra of MO1/2-TeO2 (M=Li, Na, K, Rb, Cs, and Tl) glasses,” J. Non-Cryst. Sols. 144, 128 (1992)
[Crossref]

Villezusanne, A.

A. Berthereau, E. Fargin, A. Villezusanne, R. Olazcuaga, G. Le Flem, and L. Ducasse, “Determination of local geometries around tellurium in TeO2-Nb2O5 and TeO2-Al2O3 oxide glasses by XANES and EXAFS: investigation of electronic properties of evidenced oxygen clusters by ab initio calculations,” J. Sol. St. Chem. 126, 143–151 (1996)
[Crossref]

Yoshino, F.

F. Yoshino, S. Polyakov, G.I. Stegeman, and M. Liu, “Nonlinear Refraction and Absorption from 1300 to 2200 nm in Single Crystal Polymer poly [bis (p-toluene sulfonate)] of 2, 4-hexadiyne-1, 6-diol (PTS),” Quantum Electronics and Laser Science QELS Prostconference Digest, QTuG43 (2003)

Appl. Phys. Lett. (3)

F.L Galeener, J. C. Mikkelsen, R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3, and P2O5,” Appl. Phys. Lett. 32, 34–36 (1978)
[Crossref]

G.S. Murugan, T. Suzuki, and Y. Ohishi, “Tellurite glasses for ultrabroadband fiber Raman amplifiers,” Appl. Phys. Lett. 86, 161109 (2005)
[Crossref]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–273 (1972)
[Crossref]

Electron. Lett. (1)

A. Mori, H. Masuda, K. Shikano, K. Oikawa, K. Kato, and M. Shimizu, “Ultra-wideband tellurite-based Raman fibre amplifier,” Electron. Lett. 37, 1442–1443 (2001)
[Crossref]

IEEE J. Quantum Electron. (1)

N.L. Boling, A.J. Glass, and A. Owyoung, “Empirical Relationships for Predicting Nonlinear Refractive Index Changes in Optical Solids,” IEEE J. Quantum Electron. 14, 601–608 (1978)
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

M.N. Islam, “Raman Amplifiers for Telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–559 (2002)
[Crossref]

J. Appl. Phys. (2)

M. E. Lines, “Raman gain estimates for high-gain optical fibers,” J. Appl. Phys. 62, 4363–4370 (1987)
[Crossref]

M.E. Lines, “Oxide glasses for fast photonic switching: A comparative study,” J. Appl. Phys. 69, 6876–6884 (1991)
[Crossref]

J. Chemical Phys. (1)

Y. Kato and H. Takuma, “Experimental Study on the Wavelength Dependence of the Raman Scattering Cross Sections,” J. Chemical Phys. 54, 5398–5402 (1971)
[Crossref]

J. Non-Cryst. Sols. (4)

C. Rivero, K. Richardson, R. Stegeman, G. Stegeman, T. Cardinal, E. Fargin, M. Couzi, and V. Rodriguez, “Quantifying Raman Gain Coefficients in Tellurite Glasses,” J. Non-Cryst. Sols. 345&346, 396–401 (2004)
[Crossref]

M.E. Lines, “Absolute Raman Intensities in Glasses, I. Theory”, J. Non-Cryst. Sols. 89, 143–162 (1987)
[Crossref]

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Sols. 99, 289–307 (1988)
[Crossref]

T. Sekiya, N. Mochida, A. Ohtsuka, and M. Tonokawa, “Raman spectra of MO1/2-TeO2 (M=Li, Na, K, Rb, Cs, and Tl) glasses,” J. Non-Cryst. Sols. 144, 128 (1992)
[Crossref]

J. of Nonlinear Opt. Phys. Mat. (1)

V.A. Lisinetskii, I.I. Mishkel, R.V. Chulkov, A.S. Grabtchikov, P.A. Apanasevich, H.J. Eichler, and V.A. Orlovich, “Raman gain coefficient of Barium Nitrate measured for the spectral region of Ti:Sapphire Laser,” J. of Nonlinear Opt. Phys. Mat. 14, 107–114 (2005)
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

J. Sol. St. Chem. (2)

B. Jeansannetas, S. Blanchandin, P. Thomas, P. Marchet, J. C. Champarnaud-Mesjard, T. Merle-Mejean, B. Frit, V. Nazabal, E. Fargin, G. Le Flem, M. O. Martin, B. Bosquet, L. Canioni, S. Le Boiteux, P. Segonds, and L. Sarger, “Glass structure and optical nonlinearities in thallium(I) tellurium (IV) oxide glasses,” J. Sol. St. Chem. 146, 329 (1999)
[Crossref]

A. Berthereau, E. Fargin, A. Villezusanne, R. Olazcuaga, G. Le Flem, and L. Ducasse, “Determination of local geometries around tellurium in TeO2-Nb2O5 and TeO2-Al2O3 oxide glasses by XANES and EXAFS: investigation of electronic properties of evidenced oxygen clusters by ab initio calculations,” J. Sol. St. Chem. 126, 143–151 (1996)
[Crossref]

Opt. Commun. (1)

H.S. Seo and K. Oh, “Optimization of silica fiber Raman amplifier using the Raman frequency modeling for an arbitrary GeO2 concentration,” Opt. Commun. 181, 145–151 (2000)
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Photon. Technol. Lett. (2)

J. Bromage, K. Rottwitt, and M.E. Lines, “A Method to Predict the Raman Gain Spectra of Germanosilicate Fibers With Arbitrary Index Profiles,” Photon. Technol. Lett. 14, 24–26 (2002)
[Crossref]

G. Dai, F. Tassone, A. Li Bassi, V. Russo, C.E. Bottani, and F. D’Amore, “TeO2-based glasses containing Nb2O5, TiO2, and WO3 for discrete Raman fiber amplification,” Photon. Technol. Lett. 16, 1011–1013, (2004)
[Crossref]

Phys. Rev. B (1)

M.E. Lines, “Influence of d orbitals on the nonlinear optical response of transparent transition-metal oxides,” Phys. Rev. B 43, 11978–11990 (1991)
[Crossref]

Other (4)

F. Yoshino, S. Polyakov, G.I. Stegeman, and M. Liu, “Nonlinear Refraction and Absorption from 1300 to 2200 nm in Single Crystal Polymer poly [bis (p-toluene sulfonate)] of 2, 4-hexadiyne-1, 6-diol (PTS),” Quantum Electronics and Laser Science QELS Prostconference Digest, QTuG43 (2003)

M.N. Islam, “Raman Amplifiers for Telecommunications 1,” Physical Principles (Springer2004)

R. H. Stolen, “Issues in Raman gain measurements,” in Tech. Dig. Symp. Optical Fiber Measurements, NIST Special Publication 953 (National Institute of Standards and Technology), (Gaithersburg, MD, 2000) pp. 139

R. Stegeman, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, L. Jankovic, and H. Kim, “Raman gain measurements in bulk glass samples”, J. Opt. Soc. Am. B (to be published)

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Figures (5)

Fig. 1.
Fig. 1.

UV-Vis-NIR absorption spectra of samples W, Nb, and SiO2. Notice that 195 nm is the lowest wavelength resolution of the Cary500 Spectrophotometer.

Fig. 2.
Fig. 2.

VV Polarized Experimental Spontaneous Raman Spectrum of samples W and Nb, normalized to SiO2

Fig. 3.
Fig. 3.

VV Polarized Spontaneous Raman Spectrum of samples W and Nb, normalized to SiO2. Excitation wavelength 514 nm

Fig. 4.
Fig. 4.

Estimated multi-wavelength Raman gain coefficient at the peak Raman vibration (TeO4 units at 665 cm-1 (Δυ=20 THz)), and W-O vibration (at 920 cm-1 (Δυ=27.6 THz)) respectively, normalized to SiO2. The dash line is used as a guide to the eye. The solid lines represent the (n 2(λ)-1)2 approximation to the dispersion.

Fig. 5.
Fig. 5.

Spontaneous Raman spectra of 85% TeO2 - 15% WO3 obtained at different wavelengths, normalized to the peak Raman gain value at 665 cm-1 (Δυ=20 THz), measured with 1064 nm pumping.

Tables (2)

Tables Icon

Table 1. Physical Properties

Tables Icon

Table 2. Calculated and Experimentally measured Raman Gain coefficient with 1064 nm pumping, at the peak Raman resonance at 665 cm-1 (Δυ=20 THz)

Equations (7)

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α ij k = r α ij k , r ( ω 1 ω k , r ) + r β { α ij k , r ( ω 1 ω k , r ) Q β k Q β k = 0 } Q β k
n 2 = 1 + 1 ε 0 k N k eal { r α ij k , r ( ω 1 ω k , r ) } ,
I β k , r ( ω 1 Ω β k ) I inc ( ω 1 ) Δ Ω = K SR k , r ( ω 1 Ω β k ) 4 [ 1 R ( ω 1 ) ] [ 1 R ( ω 1 Ω β k ) ] [ n ( ω 1 Ω β k ) ] 2 α ij k , r ( ω 1 ω k , r ) Q β k 2
γ β k , r ( ω 1 Ω β k ) = K RG k ( ω 1 Ω β r ) n ( ω 1 Ω β r ) n ( ω 1 ) α ij k , r ( ω 1 ω k , r ) Q β k 2 ,
γ β k , r ( ω 1 Ω β r ) = K RG k , r K SR k , r n ( ω 1 Ω β r ) ( ω 1 Ω β r ) 2 n ( ω 1 ) [ 1 R ( ω 1 ) ] [ 1 R ( ω 1 Ω β r ) ] I β k , r ( ω 1 Ω β r ) I inc ( ω 1 ) Δ Ω .
γ β r , k ( ω 2 Ω β r ) γ β r , k ( ω 1 Ω β r ) = ( ω 1 Ω β r ) 2 ( ω 2 Ω β r ) 3 n ( ω 2 Ω β r ) n ( ω 1 ) n ( ω 1 Ω β r ) n ( ω 2 ) [ 1 R ( ω 1 Ω β r ) ] [ 1 R ( ω 1 ) ] [ 1 R ( ω 2 Ω β r ) ] [ 1 R ( ω 2 ) ]
× I β k , r ( ω 2 Ω β r ) I inc ( ω 2 ) I inc ( ω 1 ) I β k , r ( ω 1 Ω β r )

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