Abstract

We report the observation of thermally-induced self-diffraction in carbon nanotube (CNT) solutions under the influence of the gravity. We present a theoretical model in which CNTs are assumed to obey the Boltzmman distribution law. Under the approximations of small temperature rise and a very narrow distribution of CNT masses, the model simulation is consistent with the data measured at low laser powers. An immediate application of such a gravitation-dependent characteristic is the optical measurement for molecular weights of CNTs.

© 2006 Optical Society of America

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References

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  1. R. F. Service, "Superstrong nanotubes show they are smart, too," Science 281, 940-942 (1998).
    [CrossRef]
  2. M. S. Dresslhaus, G. Dresslhuas, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubes, (Academic, New York, 1996).
  3. M. Endo, S. Iijima, and M. S. Dresselhaus, Carbon Nanotubes, (Pergamon, Oxford, 1996).
  4. T. W. Ebbesen, Carbon Nanotubes: Preparation and Properties, (CRC, Boca Raton, FL, 1997).
  5. X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optical limiting with multiwalled carbon nanotubes," Appl. Phys. Lett. 73, 3632-3634 (1998).
    [CrossRef]
  6. P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999).
    [CrossRef]
  7. L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
    [CrossRef]
  8. S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
    [CrossRef]
  9. J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, "Optical limiting properties of suspended and solubilized carbon nanotubes," J. Phys. Chem. B 104, 7071-7076 (2000).
    [CrossRef]
  10. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd Edition, (Oxford Univ Press, Oxford, 1995).
  11. I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals, (World Scientific, Singapore, 1993).
  12. S. Brugioni and R. Meucci, "Thermally induced nonlinear optical effects in an isotropic liquid crystal at 10.6 µm," Appl. Opt. 41,7627-7630 (2002).
    [CrossRef]
  13. Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
    [CrossRef]
  14. 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]
  15. H. El-Kashef, "Thermo-optical and dielectric constants of laser dye solvents," Rev. Sci. Instrum. 69, 1243-1245 (1998).
    [CrossRef]
  16. M. S. Dresselhaus, G. Dresselhaus, and Ph. Avouris, Eds, Carbon Nanotubes: Synthesis, Structure, Properties and Applications, Topics in Applied Physics, vol 80 (Springer, New York, 2000).
  17. R. A. Serway, Physics for Scientists and Engineers with Modern Physics, 4th Edition, (Saunders College Publishing, Philadelphia, 1996).

2003 (1)

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

2002 (1)

2000 (2)

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, "Optical limiting properties of suspended and solubilized carbon nanotubes," J. Phys. Chem. B 104, 7071-7076 (2000).
[CrossRef]

1999 (2)

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999).
[CrossRef]

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

1998 (3)

R. F. Service, "Superstrong nanotubes show they are smart, too," Science 281, 940-942 (1998).
[CrossRef]

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optical limiting with multiwalled carbon nanotubes," Appl. Phys. Lett. 73, 3632-3634 (1998).
[CrossRef]

H. El-Kashef, "Thermo-optical and dielectric constants of laser dye solvents," Rev. Sci. Instrum. 69, 1243-1245 (1998).
[CrossRef]

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]

Andrieux, M.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Anglaret, E.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Bacou, F.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Bandyopadhyay, R.

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

Bernier, P.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Brugioni, S.

Brunet, M.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Carroll, D. L.

J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, "Optical limiting properties of suspended and solubilized carbon nanotubes," J. Phys. Chem. B 104, 7071-7076 (2000).
[CrossRef]

Chen, P.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999).
[CrossRef]

El-Kashef, H.

H. El-Kashef, "Thermo-optical and dielectric constants of laser dye solvents," Rev. Sci. Instrum. 69, 1243-1245 (1998).
[CrossRef]

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]

Govindaraj, A.

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

Goze, C.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Guo, Z. X.

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

Hache, F.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Hor, T. S. A.

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optical limiting with multiwalled carbon nanotubes," Appl. Phys. Lett. 73, 3632-3634 (1998).
[CrossRef]

Ji, W.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999).
[CrossRef]

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optical limiting with multiwalled carbon nanotubes," Appl. Phys. Lett. 73, 3632-3634 (1998).
[CrossRef]

Journet, C.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Lafonta, F.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[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]

Li, Y. F.

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

Lin, J.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999).
[CrossRef]

Liu, L. Q.

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

Mehendale, S. C.

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

Meucci, R.

Mishra, S. R.

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

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]

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]

Qin, Y. J.

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

Rao, C. N. R.

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

Rawat, H. S.

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

Riehl, D.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Riggs, J. E.

J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, "Optical limiting properties of suspended and solubilized carbon nanotubes," J. Phys. Chem. B 104, 7071-7076 (2000).
[CrossRef]

Rustagi, K. C.

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

Service, R. F.

R. F. Service, "Superstrong nanotubes show they are smart, too," Science 281, 940-942 (1998).
[CrossRef]

Shi, J. H.

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

Sood, A. K.

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

Sun, X.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999).
[CrossRef]

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optical limiting with multiwalled carbon nanotubes," Appl. Phys. Lett. 73, 3632-3634 (1998).
[CrossRef]

Sun, Y.-P

J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, "Optical limiting properties of suspended and solubilized carbon nanotubes," J. Phys. Chem. B 104, 7071-7076 (2000).
[CrossRef]

Tan, K. L.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999).
[CrossRef]

Vivien, L.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

Walker, D. B.

J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, "Optical limiting properties of suspended and solubilized carbon nanotubes," J. Phys. Chem. B 104, 7071-7076 (2000).
[CrossRef]

Whinnery, J. R.

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]

Wu, W.

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

Wu, X.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999).
[CrossRef]

Xu, G. Q.

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optical limiting with multiwalled carbon nanotubes," Appl. Phys. Lett. 73, 3632-3634 (1998).
[CrossRef]

Yu, R. Q.

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optical limiting with multiwalled carbon nanotubes," Appl. Phys. Lett. 73, 3632-3634 (1998).
[CrossRef]

Zhang, J.

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

Zhu, D. B.

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optical limiting with multiwalled carbon nanotubes," Appl. Phys. Lett. 73, 3632-3634 (1998).
[CrossRef]

Chem. Mater. (1)

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003).
[CrossRef]

Chem. Phys. Lett. (2)

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999).
[CrossRef]

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000).
[CrossRef]

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. Phys. Chem. B (1)

J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, "Optical limiting properties of suspended and solubilized carbon nanotubes," J. Phys. Chem. B 104, 7071-7076 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999).
[CrossRef]

Rev. Sci. Instrum. (1)

H. El-Kashef, "Thermo-optical and dielectric constants of laser dye solvents," Rev. Sci. Instrum. 69, 1243-1245 (1998).
[CrossRef]

Science (1)

R. F. Service, "Superstrong nanotubes show they are smart, too," Science 281, 940-942 (1998).
[CrossRef]

Other (7)

M. S. Dresslhaus, G. Dresslhuas, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubes, (Academic, New York, 1996).

M. Endo, S. Iijima, and M. S. Dresselhaus, Carbon Nanotubes, (Pergamon, Oxford, 1996).

T. W. Ebbesen, Carbon Nanotubes: Preparation and Properties, (CRC, Boca Raton, FL, 1997).

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd Edition, (Oxford Univ Press, Oxford, 1995).

I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals, (World Scientific, Singapore, 1993).

M. S. Dresselhaus, G. Dresselhaus, and Ph. Avouris, Eds, Carbon Nanotubes: Synthesis, Structure, Properties and Applications, Topics in Applied Physics, vol 80 (Springer, New York, 2000).

R. A. Serway, Physics for Scientists and Engineers with Modern Physics, 4th Edition, (Saunders College Publishing, Philadelphia, 1996).

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Figures (4)

Fig. 1.
Fig. 1.

Gravitation-dependant, thermally-induced self-diffraction in carbon nanotube solutions. (a) and (b) Schematic diagrams showing two experimental set-ups. (c) and (d) Diffraction patterns recorded at 532 nm with the set-ups shown in (a) and (b), respectively. (e) and (f) Diffraction patterns observed at 780 nm with the set-ups shown in (a) and (b), respectively. The input laser powers used are ~100 mW.

Fig. 2.
Fig. 2.

Far-field distribution of the transmitted irradiance measured at 780 nm with various laser powers. The squares, triangles, circles, and stars are the experimental data obtained from the set-up in Fig. 1(a). The linear transmittance of the carbon nanotube solution is 85.2%. The half angle is defined as the ratio of the radial distance, ρ , on the observation screen to the distance of z. The solid lines are the numerical simulations using Eqs. (2) and (3) described in the text. The inset in A shows a Z-scan measured at 780 nm with an aperture in front of the detector.

Fig. 3.
Fig. 3.

Far-field distribution of the transmitted irradiance recorded at 780 nm and an incident power of 9 mW with various carbon nanotube concentrations. The squares, triangles, circles, and stars are the experimental data obtained from the set-up in Fig. 1(a). The half angle is defined as the ratio of the radial distance, ρ , on the observation screen to the distance of z. The solid lines are the numerical simulations using Eqs. (2) and (3) described in the text.

Fig. 4.
Fig. 4.

Far-field distribution of the transmitted irradiance measured at 780 nm with various laser powers. The squares, triangles, circles, and stars are the experimental data obtained from the set-up shown in Fig. 1(b). The transmittance of the carbon nanotube solution is 85.2%. The half angle is defined as the ratio of the x’-coordinate on the observation screen to the distance of z. The solid lines are the numerical simulations using Eqs. (7) and (8) with an approximation of very narrow weight distribution described in the text.

Equations (8)

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k 2 T = A exp ( 2 ρ 2 ω 2 )
Δ T ( ρ ) = 0.25 α 0 P k [ E n ( 2 ρ 2 ω 2 ) E n ( 2 a 2 ω 2 ) 2 ln ( ρ a ) ]
E ( ρ ' ) = E ( 0 ) 0 exp [ i 2 π λ Δ n ( ρ ) L ρ 2 w 2 ] J 0 ( 2 π ρ ρ ' λ z ) ρ d ρ ,
α = i σ i N i
α = i σ i N i ( 0 ) exp ( M i g x k B T )
k 2 T = exp [ 2 ( x 2 + y 2 ) ω 2 ] i A i exp ( M i g x k B T )
Δ T = i 0.25 q i σ i N i ( 0 ) P k [ E n ( 2 ρ i 2 ω 2 ) E n ( 2 a 2 ω 2 ) 2 ln ( ρ i 2 a ) ]
E ( x ' , y ' ) = E ( 0 , 0 ) exp ( i 2 π λ d n d T Δ T L x 2 + y 2 ω 2 ) exp [ i 2 π λ z ( x x ' + y y ' ) ] d x d y ,

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