Abstract

We analyze the phase shift induced in an amplitude-modulated laser beam propagating through a water dispersion of graphene oxide sheets in a fiber-to-fiber U-bench. This phase shift arises from the thermally induced nonlinear refraction in the sample. The system exhibits strong optical limiting performance for weak continuous-wave signals. A theoretical model including beam propagation and thermal lens focal length oscillation reproduces the experimental findings.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  15. X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
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  36. C. Liu, Z. Wang, H. Jia, and Z. Li, “Efficient fluorescence resonance energy transfer between upconversion nanophosphors and graphene oxide: a highly sensitive biosensing platform,” Chem. Commun. 47, 4661–4663 (2011).

2013 (4)

N. Liaros, P. Aloukos, A. Kolokithas-Ntoukas, A. Bakandritsos, T. Szabo, R. Zboril, and S. Couris, “Nonlinear optical properties and broadband optical power limiting action of graphene oxide colloids,” J. Phys. Chem. C 117, 6842–6850 (2013).
[CrossRef]

X.-L. Zhang, Z.-B. Liu, X.-C. Li, Q. Ma, X.-D. Chen, J.-G. Tian, Y.-F. Xu, and Y.-S. Chen, “Transient thermal effect, nonlinear refraction and nonlinear absorption properties of graphene oxide sheets in dispersion,” Opt. Express 21, 7511–7520 (2013).
[CrossRef]

J. Li, Y. Zhang, H. Li, C. Yao, and P. Yuan, “Observation of tunable superluminal propagation in the single-layer graphene oxide solution,” Opt. Commun. 295, 226–229 (2013).
[CrossRef]

C. Estupiñán-López, C. T. Dominguez, and R. de Araujo, “Eclipsing thermal lens spectroscopy for fluorescence quantum yield measurement,” Opt. Express 21, 18592–18601 (2013).
[CrossRef]

2012 (1)

2011 (3)

C. Liu, Z. Wang, H. Jia, and Z. Li, “Efficient fluorescence resonance energy transfer between upconversion nanophosphors and graphene oxide: a highly sensitive biosensing platform,” Chem. Commun. 47, 4661–4663 (2011).

X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
[CrossRef]

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

2010 (2)

M. Feng, H. Zhan, and Y. Chen, “Nonlinear optical and optical limiting properties of graphene families,” Appl. Phys. Lett. 96, 033107 (2010).
[CrossRef]

K. P. Loh, Q. Bao, G. Eda, and M. Chhowalla, “Graphene oxide as a chemically tunable platform for optical applications,” Nat. Chem. 2, 1015–1024 (2010).

2009 (6)

D. R. Dreyer, S. Park, C. W. Bielawski, and R. S. Ruoff, “The chemistry of graphene oxide,” Chem. Soc. Rev. 39, 228–240 (2009).
[CrossRef]

S. Park and R. S. Ruoff, “Chemical methods for the production of graphenes,” Nat. Nanotechnol. 4, 217–224 (2009).
[CrossRef]

S. Kumar, M. Anija, N. Kamaraju, K. S. Vasu, K. S. Subrahmanyam, A. K. Sood, and C. N. R. Rao, “Femtosecond carrier dynamics and saturable absorption in graphene suspensions,” Appl. Phys. Lett. 95, 191911 (2009).
[CrossRef]

Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. 94, 021902 (2009).
[CrossRef]

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

2008 (1)

J. I. Paredes, S. Villar-Rodil, A. Martínez-Alonso, and J. M. D. Tascón, “Graphene oxide dispersions in organic solvents,” Langmuir 24, 10560–10564 (2008).
[CrossRef]

2007 (1)

H. Wang, Y. Zhang, N. Wang, W. Yan, H. Tian, W. Qiu, and P. Yuan, “Observation of superluminal propagation at negative group velocity in C60 solution,” Appl. Phys. Lett. 90, 121107 (2007).
[CrossRef]

2004 (2)

G. S. Agarwal and T. N. Dey, “Sub- and superluminal propagation of intense pulses in media with saturated and reverse absorption,” Phys. Rev. Lett. 92, 203901 (2004).
[CrossRef]

M. Hirata, T. Gotou, S. Horiuchi, M. Fujiwara, and M. Ohba, “Thin-film particles of graphite oxide 1: high-yield synthesis and flexibility of the particles,” Carbon 42, 2929–2937 (2004).

2003 (2)

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]

B.-X. Wang, L.-P. Zhou, and X.-F. Peng, “A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles,” Int. J. Heat Mass Transfer 46, 2665–2672 (2003).

2000 (1)

A. Sennaroglu, “Effect of thermal lensing on the mode matching between pump and laser beams in Cr4+: forsterite lasers: a numerical study,” J. Phys. D 33, 1478–1483 (2000).

1997 (1)

1991 (1)

D. Rojas, R. J. Silva, J. D. Spear, and R. E. Russo, “Dual-beam optical fiber thermal lens spectroscopy,” Anal. Chem. 63, 1927–1932 (1991).
[CrossRef]

1982 (1)

1974 (1)

J. R. Whinney, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7, 225–231 (1974).
[CrossRef]

1973 (1)

1965 (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and T. N. Dey, “Sub- and superluminal propagation of intense pulses in media with saturated and reverse absorption,” Phys. Rev. Lett. 92, 203901 (2004).
[CrossRef]

Aloukos, P.

N. Liaros, P. Aloukos, A. Kolokithas-Ntoukas, A. Bakandritsos, T. Szabo, R. Zboril, and S. Couris, “Nonlinear optical properties and broadband optical power limiting action of graphene oxide colloids,” J. Phys. Chem. C 117, 6842–6850 (2013).
[CrossRef]

Anija, M.

S. Kumar, M. Anija, N. Kamaraju, K. S. Vasu, K. S. Subrahmanyam, A. K. Sood, and C. N. R. Rao, “Femtosecond carrier dynamics and saturable absorption in graphene suspensions,” Appl. Phys. Lett. 95, 191911 (2009).
[CrossRef]

Arrieta-Yáñez, F.

F. Arrieta-Yáñez, O. G. Calderón, and S. Melle, “Fast light based on excited-state absorption in erbium doped fibers,” in IONS 9 International OSA Network of Students (Optical Society of America, 2011).

Astrath, N. G. C.

Bai, X.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

Bakandritsos, A.

N. Liaros, P. Aloukos, A. Kolokithas-Ntoukas, A. Bakandritsos, T. Szabo, R. Zboril, and S. Couris, “Nonlinear optical properties and broadband optical power limiting action of graphene oxide colloids,” J. Phys. Chem. C 117, 6842–6850 (2013).
[CrossRef]

Bao, Q.

K. P. Loh, Q. Bao, G. Eda, and M. Chhowalla, “Graphene oxide as a chemically tunable platform for optical applications,” Nat. Chem. 2, 1015–1024 (2010).

Bialkowski, S. E.

Bian, F.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

Bielawski, C. W.

D. R. Dreyer, S. Park, C. W. Bielawski, and R. S. Ruoff, “The chemistry of graphene oxide,” Chem. Soc. Rev. 39, 228–240 (2009).
[CrossRef]

Bigelow, M. S.

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]

Blau, W. J.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Boyd, R. W.

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]

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), Chap. 4.5.

Calderón, O. G.

F. Arrieta-Yáñez, O. G. Calderón, and S. Melle, “Fast light based on excited-state absorption in erbium doped fibers,” in IONS 9 International OSA Network of Students (Optical Society of America, 2011).

Chartier, A.

Chen, X.-D.

Chen, Y.

X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
[CrossRef]

M. Feng, H. Zhan, and Y. Chen, “Nonlinear optical and optical limiting properties of graphene families,” Appl. Phys. Lett. 96, 033107 (2010).
[CrossRef]

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. 94, 021902 (2009).
[CrossRef]

Chen, Y.-S.

Chhowalla, M.

K. P. Loh, Q. Bao, G. Eda, and M. Chhowalla, “Graphene oxide as a chemically tunable platform for optical applications,” Nat. Chem. 2, 1015–1024 (2010).

Coleman, J. N.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Couris, S.

N. Liaros, P. Aloukos, A. Kolokithas-Ntoukas, A. Bakandritsos, T. Szabo, R. Zboril, and S. Couris, “Nonlinear optical properties and broadband optical power limiting action of graphene oxide colloids,” J. Phys. Chem. C 117, 6842–6850 (2013).
[CrossRef]

de Araujo, R.

Dey, T. N.

G. S. Agarwal and T. N. Dey, “Sub- and superluminal propagation of intense pulses in media with saturated and reverse absorption,” Phys. Rev. Lett. 92, 203901 (2004).
[CrossRef]

Dominguez, C. T.

Dreyer, D. R.

D. R. Dreyer, S. Park, C. W. Bielawski, and R. S. Ruoff, “The chemistry of graphene oxide,” Chem. Soc. Rev. 39, 228–240 (2009).
[CrossRef]

Eda, G.

K. P. Loh, Q. Bao, G. Eda, and M. Chhowalla, “Graphene oxide as a chemically tunable platform for optical applications,” Nat. Chem. 2, 1015–1024 (2010).

Estupiñán-López, C.

Feng, M.

M. Feng, H. Zhan, and Y. Chen, “Nonlinear optical and optical limiting properties of graphene families,” Appl. Phys. Lett. 96, 033107 (2010).
[CrossRef]

Franko, M.

M. Franko and C. D. Tran, Encyclopedia of Analytical Chemistry: Thermal Lens Spectroscopy (Wiley, 2010).

Fujiwara, M.

M. Hirata, T. Gotou, S. Horiuchi, M. Fujiwara, and M. Ohba, “Thin-film particles of graphite oxide 1: high-yield synthesis and flexibility of the particles,” Carbon 42, 2929–2937 (2004).

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Gotou, T.

M. Hirata, T. Gotou, S. Horiuchi, M. Fujiwara, and M. Ohba, “Thin-film particles of graphite oxide 1: high-yield synthesis and flexibility of the particles,” Carbon 42, 2929–2937 (2004).

Herculano, L. S.

Hernandez, Y.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Hirata, M.

M. Hirata, T. Gotou, S. Horiuchi, M. Fujiwara, and M. Ohba, “Thin-film particles of graphite oxide 1: high-yield synthesis and flexibility of the particles,” Carbon 42, 2929–2937 (2004).

Horiuchi, S.

M. Hirata, T. Gotou, S. Horiuchi, M. Fujiwara, and M. Ohba, “Thin-film particles of graphite oxide 1: high-yield synthesis and flexibility of the particles,” Carbon 42, 2929–2937 (2004).

Hu, C.

Huang, Y.

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Jia, H.

C. Liu, Z. Wang, H. Jia, and Z. Li, “Efficient fluorescence resonance energy transfer between upconversion nanophosphors and graphene oxide: a highly sensitive biosensing platform,” Chem. Commun. 47, 4661–4663 (2011).

Kamaraju, N.

S. Kumar, M. Anija, N. Kamaraju, K. S. Vasu, K. S. Subrahmanyam, A. K. Sood, and C. N. R. Rao, “Femtosecond carrier dynamics and saturable absorption in graphene suspensions,” Appl. Phys. Lett. 95, 191911 (2009).
[CrossRef]

Knight, L. V.

Kolokithas-Ntoukas, A.

N. Liaros, P. Aloukos, A. Kolokithas-Ntoukas, A. Bakandritsos, T. Szabo, R. Zboril, and S. Couris, “Nonlinear optical properties and broadband optical power limiting action of graphene oxide colloids,” J. Phys. Chem. C 117, 6842–6850 (2013).
[CrossRef]

Kumar, S.

S. Kumar, M. Anija, N. Kamaraju, K. S. Vasu, K. S. Subrahmanyam, A. K. Sood, and C. N. R. Rao, “Femtosecond carrier dynamics and saturable absorption in graphene suspensions,” Appl. Phys. Lett. 95, 191911 (2009).
[CrossRef]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Leonard, G.

G. Leonard, “Free carrier absorption in graphene oxide thin film,” B.S. thesis (National University of Singapore, 2012), http://www.physics.nus.edu.sg/student/Honours Projects Repository/leonard Goh fyp-final.pdf .

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]

Li, H.

J. Li, Y. Zhang, H. Li, C. Yao, and P. Yuan, “Observation of tunable superluminal propagation in the single-layer graphene oxide solution,” Opt. Commun. 295, 226–229 (2013).
[CrossRef]

Li, J.

J. Li, Y. Zhang, H. Li, C. Yao, and P. Yuan, “Observation of tunable superluminal propagation in the single-layer graphene oxide solution,” Opt. Commun. 295, 226–229 (2013).
[CrossRef]

Li, X.-C.

Li, Z.

C. Liu, Z. Wang, H. Jia, and Z. Li, “Efficient fluorescence resonance energy transfer between upconversion nanophosphors and graphene oxide: a highly sensitive biosensing platform,” Chem. Commun. 47, 4661–4663 (2011).

Liaros, N.

N. Liaros, P. Aloukos, A. Kolokithas-Ntoukas, A. Bakandritsos, T. Szabo, R. Zboril, and S. Couris, “Nonlinear optical properties and broadband optical power limiting action of graphene oxide colloids,” J. Phys. Chem. C 117, 6842–6850 (2013).
[CrossRef]

Liu, C.

C. Liu, Z. Wang, H. Jia, and Z. Li, “Efficient fluorescence resonance energy transfer between upconversion nanophosphors and graphene oxide: a highly sensitive biosensing platform,” Chem. Commun. 47, 4661–4663 (2011).

Liu, Z.

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. 94, 021902 (2009).
[CrossRef]

Liu, Z.-B.

X.-L. Zhang, Z.-B. Liu, X.-C. Li, Q. Ma, X.-D. Chen, J.-G. Tian, Y.-F. Xu, and Y.-S. Chen, “Transient thermal effect, nonlinear refraction and nonlinear absorption properties of graphene oxide sheets in dispersion,” Opt. Express 21, 7511–7520 (2013).
[CrossRef]

X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
[CrossRef]

Loh, K. P.

K. P. Loh, Q. Bao, G. Eda, and M. Chhowalla, “Graphene oxide as a chemically tunable platform for optical applications,” Nat. Chem. 2, 1015–1024 (2010).

Lotya, M.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Lu, X.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

Ma, Q.

Ma, Y.

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Malacarne, L. C.

Martínez-Alonso, A.

J. I. Paredes, S. Villar-Rodil, A. Martínez-Alonso, and J. M. D. Tascón, “Graphene oxide dispersions in organic solvents,” Langmuir 24, 10560–10564 (2008).
[CrossRef]

Melle, S.

F. Arrieta-Yáñez, O. G. Calderón, and S. Melle, “Fast light based on excited-state absorption in erbium doped fibers,” in IONS 9 International OSA Network of Students (Optical Society of America, 2011).

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Ohba, M.

M. Hirata, T. Gotou, S. Horiuchi, M. Fujiwara, and M. Ohba, “Thin-film particles of graphite oxide 1: high-yield synthesis and flexibility of the particles,” Carbon 42, 2929–2937 (2004).

Paredes, J. I.

J. I. Paredes, S. Villar-Rodil, A. Martínez-Alonso, and J. M. D. Tascón, “Graphene oxide dispersions in organic solvents,” Langmuir 24, 10560–10564 (2008).
[CrossRef]

Park, S.

D. R. Dreyer, S. Park, C. W. Bielawski, and R. S. Ruoff, “The chemistry of graphene oxide,” Chem. Soc. Rev. 39, 228–240 (2009).
[CrossRef]

S. Park and R. S. Ruoff, “Chemical methods for the production of graphenes,” Nat. Nanotechnol. 4, 217–224 (2009).
[CrossRef]

Peng, X.-F.

B.-X. Wang, L.-P. Zhou, and X.-F. Peng, “A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles,” Int. J. Heat Mass Transfer 46, 2665–2672 (2003).

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Qiu, W.

H. Wang, Y. Zhang, N. Wang, W. Yan, H. Tian, W. Qiu, and P. Yuan, “Observation of superluminal propagation at negative group velocity in C60 solution,” Appl. Phys. Lett. 90, 121107 (2007).
[CrossRef]

Rao, C. N. R.

S. Kumar, M. Anija, N. Kamaraju, K. S. Vasu, K. S. Subrahmanyam, A. K. Sood, and C. N. R. Rao, “Femtosecond carrier dynamics and saturable absorption in graphene suspensions,” Appl. Phys. Lett. 95, 191911 (2009).
[CrossRef]

Rojas, D.

D. Rojas, R. J. Silva, J. D. Spear, and R. E. Russo, “Dual-beam optical fiber thermal lens spectroscopy,” Anal. Chem. 63, 1927–1932 (1991).
[CrossRef]

Ruoff, R. S.

S. Park and R. S. Ruoff, “Chemical methods for the production of graphenes,” Nat. Nanotechnol. 4, 217–224 (2009).
[CrossRef]

D. R. Dreyer, S. Park, C. W. Bielawski, and R. S. Ruoff, “The chemistry of graphene oxide,” Chem. Soc. Rev. 39, 228–240 (2009).
[CrossRef]

Russo, R. E.

D. Rojas, R. J. Silva, J. D. Spear, and R. E. Russo, “Dual-beam optical fiber thermal lens spectroscopy,” Anal. Chem. 63, 1927–1932 (1991).
[CrossRef]

Sennaroglu, A.

A. Sennaroglu, “Effect of thermal lensing on the mode matching between pump and laser beams in Cr4+: forsterite lasers: a numerical study,” J. Phys. D 33, 1478–1483 (2000).

Sheldon, S. J.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Silva, R. J.

D. Rojas, R. J. Silva, J. D. Spear, and R. E. Russo, “Dual-beam optical fiber thermal lens spectroscopy,” Anal. Chem. 63, 1927–1932 (1991).
[CrossRef]

Sood, A. K.

S. Kumar, M. Anija, N. Kamaraju, K. S. Vasu, K. S. Subrahmanyam, A. K. Sood, and C. N. R. Rao, “Femtosecond carrier dynamics and saturable absorption in graphene suspensions,” Appl. Phys. Lett. 95, 191911 (2009).
[CrossRef]

Spear, J. D.

D. Rojas, R. J. Silva, J. D. Spear, and R. E. Russo, “Dual-beam optical fiber thermal lens spectroscopy,” Anal. Chem. 63, 1927–1932 (1991).
[CrossRef]

Subrahmanyam, K. S.

S. Kumar, M. Anija, N. Kamaraju, K. S. Vasu, K. S. Subrahmanyam, A. K. Sood, and C. N. R. Rao, “Femtosecond carrier dynamics and saturable absorption in graphene suspensions,” Appl. Phys. Lett. 95, 191911 (2009).
[CrossRef]

Szabo, T.

N. Liaros, P. Aloukos, A. Kolokithas-Ntoukas, A. Bakandritsos, T. Szabo, R. Zboril, and S. Couris, “Nonlinear optical properties and broadband optical power limiting action of graphene oxide colloids,” J. Phys. Chem. C 117, 6842–6850 (2013).
[CrossRef]

Tascón, J. M. D.

J. I. Paredes, S. Villar-Rodil, A. Martínez-Alonso, and J. M. D. Tascón, “Graphene oxide dispersions in organic solvents,” Langmuir 24, 10560–10564 (2008).
[CrossRef]

Thorne, J. M.

Tian, H.

H. Wang, Y. Zhang, N. Wang, W. Yan, H. Tian, W. Qiu, and P. Yuan, “Observation of superluminal propagation at negative group velocity in C60 solution,” Appl. Phys. Lett. 90, 121107 (2007).
[CrossRef]

Tian, J.

Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. 94, 021902 (2009).
[CrossRef]

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Tian, J.-G.

X.-L. Zhang, Z.-B. Liu, X.-C. Li, Q. Ma, X.-D. Chen, J.-G. Tian, Y.-F. Xu, and Y.-S. Chen, “Transient thermal effect, nonlinear refraction and nonlinear absorption properties of graphene oxide sheets in dispersion,” Opt. Express 21, 7511–7520 (2013).
[CrossRef]

X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
[CrossRef]

Tran, C. D.

M. Franko and C. D. Tran, Encyclopedia of Analytical Chemistry: Thermal Lens Spectroscopy (Wiley, 2010).

Vasu, K. S.

S. Kumar, M. Anija, N. Kamaraju, K. S. Vasu, K. S. Subrahmanyam, A. K. Sood, and C. N. R. Rao, “Femtosecond carrier dynamics and saturable absorption in graphene suspensions,” Appl. Phys. Lett. 95, 191911 (2009).
[CrossRef]

Villar-Rodil, S.

J. I. Paredes, S. Villar-Rodil, A. Martínez-Alonso, and J. M. D. Tascón, “Graphene oxide dispersions in organic solvents,” Langmuir 24, 10560–10564 (2008).
[CrossRef]

Wang, B.-X.

B.-X. Wang, L.-P. Zhou, and X.-F. Peng, “A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles,” Int. J. Heat Mass Transfer 46, 2665–2672 (2003).

Wang, E.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

Wang, H.

H. Wang, Y. Zhang, N. Wang, W. Yan, H. Tian, W. Qiu, and P. Yuan, “Observation of superluminal propagation at negative group velocity in C60 solution,” Appl. Phys. Lett. 90, 121107 (2007).
[CrossRef]

Wang, J.

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Wang, N.

H. Wang, Y. Zhang, N. Wang, W. Yan, H. Tian, W. Qiu, and P. Yuan, “Observation of superluminal propagation at negative group velocity in C60 solution,” Appl. Phys. Lett. 90, 121107 (2007).
[CrossRef]

Wang, W.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

Wang, Y.

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. 94, 021902 (2009).
[CrossRef]

Wang, Z.

C. Liu, Z. Wang, H. Jia, and Z. Li, “Efficient fluorescence resonance energy transfer between upconversion nanophosphors and graphene oxide: a highly sensitive biosensing platform,” Chem. Commun. 47, 4661–4663 (2011).

Whinnery, J. R.

C. Hu and J. R. Whinnery, “New thermooptical measurement method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973).
[CrossRef]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Whinney, J. R.

J. R. Whinney, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7, 225–231 (1974).
[CrossRef]

Wu, R.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

Wu, Y.

X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
[CrossRef]

Xu, Y.

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. 94, 021902 (2009).
[CrossRef]

Xu, Y.-F.

Yan, S.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

Yan, W.

H. Wang, Y. Zhang, N. Wang, W. Yan, H. Tian, W. Qiu, and P. Yuan, “Observation of superluminal propagation at negative group velocity in C60 solution,” Appl. Phys. Lett. 90, 121107 (2007).
[CrossRef]

Yan, W.-B.

X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
[CrossRef]

Yao, C.

J. Li, Y. Zhang, H. Li, C. Yao, and P. Yuan, “Observation of tunable superluminal propagation in the single-layer graphene oxide solution,” Opt. Commun. 295, 226–229 (2013).
[CrossRef]

Yuan, P.

J. Li, Y. Zhang, H. Li, C. Yao, and P. Yuan, “Observation of tunable superluminal propagation in the single-layer graphene oxide solution,” Opt. Commun. 295, 226–229 (2013).
[CrossRef]

H. Wang, Y. Zhang, N. Wang, W. Yan, H. Tian, W. Qiu, and P. Yuan, “Observation of superluminal propagation at negative group velocity in C60 solution,” Appl. Phys. Lett. 90, 121107 (2007).
[CrossRef]

Zboril, R.

N. Liaros, P. Aloukos, A. Kolokithas-Ntoukas, A. Bakandritsos, T. Szabo, R. Zboril, and S. Couris, “Nonlinear optical properties and broadband optical power limiting action of graphene oxide colloids,” J. Phys. Chem. C 117, 6842–6850 (2013).
[CrossRef]

Zhan, H.

M. Feng, H. Zhan, and Y. Chen, “Nonlinear optical and optical limiting properties of graphene families,” Appl. Phys. Lett. 96, 033107 (2010).
[CrossRef]

Zhang, X.

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. 94, 021902 (2009).
[CrossRef]

Zhang, X.-L.

X.-L. Zhang, Z.-B. Liu, X.-C. Li, Q. Ma, X.-D. Chen, J.-G. Tian, Y.-F. Xu, and Y.-S. Chen, “Transient thermal effect, nonlinear refraction and nonlinear absorption properties of graphene oxide sheets in dispersion,” Opt. Express 21, 7511–7520 (2013).
[CrossRef]

X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
[CrossRef]

Zhang, Y.

J. Li, Y. Zhang, H. Li, C. Yao, and P. Yuan, “Observation of tunable superluminal propagation in the single-layer graphene oxide solution,” Opt. Commun. 295, 226–229 (2013).
[CrossRef]

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

H. Wang, Y. Zhang, N. Wang, W. Yan, H. Tian, W. Qiu, and P. Yuan, “Observation of superluminal propagation at negative group velocity in C60 solution,” Appl. Phys. Lett. 90, 121107 (2007).
[CrossRef]

Zhao, J.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

Zhao, X.

X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
[CrossRef]

Zhou, L.-P.

B.-X. Wang, L.-P. Zhou, and X.-F. Peng, “A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles,” Int. J. Heat Mass Transfer 46, 2665–2672 (2003).

Acc. Chem. Res. (1)

J. R. Whinney, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7, 225–231 (1974).
[CrossRef]

Adv. Mater. (2)

J. Wang, Y. Hernandez, M. Lotya, J. N. Coleman, and W. J. Blau, “Broadband nonlinear optical response of graphene dispersions,” Adv. Mater. 21, 2430–2435 (2009).
[CrossRef]

Y. Xu, Z. Liu, X. Zhang, Y. Wang, J. Tian, Y. Huang, Y. Ma, X. Zhang, and Y. Chen, “A graphene hybrid material covalently functionalized with porphyrin: synthesis and optical limiting property,” Adv. Mater. 21, 1275–1279 (2009).
[CrossRef]

Anal. Chem. (1)

D. Rojas, R. J. Silva, J. D. Spear, and R. E. Russo, “Dual-beam optical fiber thermal lens spectroscopy,” Anal. Chem. 63, 1927–1932 (1991).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (5)

X. Zhao, Z.-B. Liu, W.-B. Yan, Y. Wu, X.-L. Zhang, Y. Chen, and J.-G. Tian, “Ultrafast carrier dynamics and saturable absorption of solution-processable few-layered graphene oxide,” Appl. Phys. Lett. 98, 121905 (2011).
[CrossRef]

H. Wang, Y. Zhang, N. Wang, W. Yan, H. Tian, W. Qiu, and P. Yuan, “Observation of superluminal propagation at negative group velocity in C60 solution,” Appl. Phys. Lett. 90, 121107 (2007).
[CrossRef]

M. Feng, H. Zhan, and Y. Chen, “Nonlinear optical and optical limiting properties of graphene families,” Appl. Phys. Lett. 96, 033107 (2010).
[CrossRef]

S. Kumar, M. Anija, N. Kamaraju, K. S. Vasu, K. S. Subrahmanyam, A. K. Sood, and C. N. R. Rao, “Femtosecond carrier dynamics and saturable absorption in graphene suspensions,” Appl. Phys. Lett. 95, 191911 (2009).
[CrossRef]

Z. Liu, Y. Wang, X. Zhang, Y. Xu, Y. Chen, and J. Tian, “Optical properties of graphene oxide in nanosecond and picosecond regimes,” Appl. Phys. Lett. 94, 021902 (2009).
[CrossRef]

Carbon (1)

M. Hirata, T. Gotou, S. Horiuchi, M. Fujiwara, and M. Ohba, “Thin-film particles of graphite oxide 1: high-yield synthesis and flexibility of the particles,” Carbon 42, 2929–2937 (2004).

Chem. Commun. (1)

C. Liu, Z. Wang, H. Jia, and Z. Li, “Efficient fluorescence resonance energy transfer between upconversion nanophosphors and graphene oxide: a highly sensitive biosensing platform,” Chem. Commun. 47, 4661–4663 (2011).

Chem. Soc. Rev. (1)

D. R. Dreyer, S. Park, C. W. Bielawski, and R. S. Ruoff, “The chemistry of graphene oxide,” Chem. Soc. Rev. 39, 228–240 (2009).
[CrossRef]

Int. J. Heat Mass Transfer (1)

B.-X. Wang, L.-P. Zhou, and X.-F. Peng, “A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles,” Int. J. Heat Mass Transfer 46, 2665–2672 (2003).

J. Appl. Phys. (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

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

J. Phys. Chem. C (1)

N. Liaros, P. Aloukos, A. Kolokithas-Ntoukas, A. Bakandritsos, T. Szabo, R. Zboril, and S. Couris, “Nonlinear optical properties and broadband optical power limiting action of graphene oxide colloids,” J. Phys. Chem. C 117, 6842–6850 (2013).
[CrossRef]

J. Phys. D (1)

A. Sennaroglu, “Effect of thermal lensing on the mode matching between pump and laser beams in Cr4+: forsterite lasers: a numerical study,” J. Phys. D 33, 1478–1483 (2000).

Langmuir (1)

J. I. Paredes, S. Villar-Rodil, A. Martínez-Alonso, and J. M. D. Tascón, “Graphene oxide dispersions in organic solvents,” Langmuir 24, 10560–10564 (2008).
[CrossRef]

Nano Lett. (1)

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).

Nat. Chem. (1)

K. P. Loh, Q. Bao, G. Eda, and M. Chhowalla, “Graphene oxide as a chemically tunable platform for optical applications,” Nat. Chem. 2, 1015–1024 (2010).

Nat. Nanotechnol. (1)

S. Park and R. S. Ruoff, “Chemical methods for the production of graphenes,” Nat. Nanotechnol. 4, 217–224 (2009).
[CrossRef]

Opt. Commun. (1)

J. Li, Y. Zhang, H. Li, C. Yao, and P. Yuan, “Observation of tunable superluminal propagation in the single-layer graphene oxide solution,” Opt. Commun. 295, 226–229 (2013).
[CrossRef]

Opt. Express (2)

Phys. Rev. Lett. (1)

G. S. Agarwal and T. N. Dey, “Sub- and superluminal propagation of intense pulses in media with saturated and reverse absorption,” Phys. Rev. Lett. 92, 203901 (2004).
[CrossRef]

Science (1)

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 (7)

F. Arrieta-Yáñez, O. G. Calderón, and S. Melle, “Fast light based on excited-state absorption in erbium doped fibers,” in IONS 9 International OSA Network of Students (Optical Society of America, 2011).

G. Leonard, “Free carrier absorption in graphene oxide thin film,” B.S. thesis (National University of Singapore, 2012), http://www.physics.nus.edu.sg/student/Honours Projects Repository/leonard Goh fyp-final.pdf .

S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis (Wiley, 1996).

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), Chap. 4.5.

M. Franko and C. D. Tran, Encyclopedia of Analytical Chemistry: Thermal Lens Spectroscopy (Wiley, 2010).

American Institute of Physics Handbook (McGraw-Hill, 1957), Sections 4g and 6b.

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Experimental setup. C, cuvette containing the GO dispersion; DFB LD, distributed feedback laser diode at 977 nm; LD TEC, laser diode temperature and current controller; FG, function generator; OC, digital oscilloscope; PD1, reference photodetector; PD2, photodetector for the GO signal; L1, collimating lens; L2, focusing lens; Green lines, single-mode optical fibers. (b) Cuvette containing an aquoeus dispersion of GO sheets with a concentration of 0.25mg/mL. (c) Time evolution of the experimental reference signal (red line) and the signal propagated through the GO dispersion and the U-bench (black line) for a 0.5mg/mL GO water dispersion, a laser input power of P0=28.5mW, and a modulation frequency of fm=1.71Hz.

Fig. 2.
Fig. 2.

Experimental (symbols) and theoretical (lines) phase delay versus modulation frequency for a 0.5mg/mL GO aqueous dispersion and different input powers: P0=2.5mW (squares), P0=18.4mW (circles), P0=28.5mW (triangles). Inset: phase delay versus input power for a modulation frequency of fm=3.64Hz.

Fig. 3.
Fig. 3.

Experimental (symbols) and theoretical (lines) phase delay versus modulation frequency for a laser input power of 18.4 mW and different GO concentrations: 0.25mg/mL (squares), 0.5mg/mL (circles), and 1mg/mL (triangles). Simulated curves with a fixed value of the thermal conductivity κ (solid lines), and changing the value of κ according to the volume fraction of GO dispersions (dashed lines).

Fig. 4.
Fig. 4.

Experimental (symbols) output-input power curves for different GO concentrations: 0.25mg/mL (squares), 0.5mg/mL (circles), and 1mg/mL (triangles). (a) Output power measured at the output optical fiber of the U-bench. Theoretical curves (solid lines). (b) Output power measured at the output of the cuvette (before the beam enters L2). The dotted lines are linear fits to the experimental data and the corresponding slopes are also indicated in panel (b).

Fig. 5.
Fig. 5.

(a) Thermal focal length [from Eq. (4)] versus input power. (b) Spot size at the entrance plane of the single-mode fiber [from Eq. (5)] versus input power. The dotted line indicates the radius of the fiber mode wf. We used the following parameters: κ=0.6W/(mK), dn/dT=0.45×104K1, w0=1.55mm, α=1.77cm1 (0.5mg/mL GO water dispersion), L=1cm, dL=11mm, and λ=977nm.

Fig. 6.
Fig. 6.

Experimental (symbols) and theoretical (lines) distortion factor versus modulation frequency for a 0.5mg/mL GO water dispersion and different input powers: P0=18.4mW (circles), P0=28.5mW (triangles). In the insets, time evolution of both the reference signal (red line) and the signal propagated through the GO dispersion and the U-bench (black line) for the modulation frequencies that are marked with the arrows.

Equations (26)

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

ϕα0LI0Isat2πfmτc1+(2πfmτc)2.
n=n0dndTΔT,
CρΔTt=ΔIL+κ2ΔT,
f1=(dn/dT)P(1eαL)κπw02.
wL=w0dL(1f)2+(λπw02)2.
TMM=(2wfwLwL2+wf2)2,
ΔTtd=2P(1eαL)πκLe2rd2+2ΔT,
0=2P0(1eαL)πκLe2rd2+2ΔT0,
ωmΔTc=2ΔTs,
ωmΔTs=2Pm(1eαL)πκLe2rd2+2ΔTc.
D0=(dn/dT)P0(1eαL)κπw02.
ωmΔT^c=(2πq)2ΔT^s,
ωmΔT^s=2Pm(1eαL)πκLπ2e12π2q2(2πq)2ΔT^c,
ΔT^c=(2πq)2ωm2+(2πq)42Pm(1eαL)πκLπ2e12π2q2,
ΔT^s=ωmωm2+(2πq)42Pm(1eαL)πκLπ2e12π2q2.
ΔTl=2π0dqΔT^lJ0(2πqrd)q,(l=c,s),
Dl=2L(dn/dT)w02ΔTlrd2,(l=c,s).
T0MM=(2wfwL0wL02+wf2)2,
wL0=w0dLD02+(λπw02)2,
TlMM=4wf2w02dL2(wf2wL02)2D0Dl(wL02+wf2)3,(l=c,s).
ϕ=2D0Dsw02dL2(wf2wL02)wL02(wL02+wf2)1PmP0+2D0Dcw02dL2(wf2wL02)wL02(wL02+wf2).
Dtd=D+Dst,
Dc=11+ωm2D0PmP0,
Ds=ωm1+ωm2D0PmP0.
ϕ=2D02w02dL2(wf2wL02)wL02(wL02+wf2)2πfmτ1+2D02w02dL2(wf2wL02)wL02(wL02+wf2)+(2πfmτ)2.
fmax=12πτ1+2D02w02dL2(wf2wL02)wL02(wL02+wf2).

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