Abstract

Three-dimensional (3D) femtosecond laser direct structuring in transparent materials is widely used for photonic applications. However, the structure size is limited by the optical diffraction. Here we report on a direct laser writing technique that produces subwavelength nanostructures independently of the experimental limiting factors. We demonstrate 3D nanostructures of arbitrary patterns with feature sizes down to 80 nm, less than one tenth of the laser processing wavelength. Its ease of implementation for novel nanostructuring, with its accompanying high precision will open new opportunities for the fabrication of nanostructures for plasmonic and photonic devices and for applications in metamaterials.

© 2009 OSA

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  1. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
    [CrossRef]
  2. K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71(23), 3329–3331 (1997).
    [CrossRef]
  3. A. Zoubir, M. Richardson, L. Canioni, A. Brocas, and L. Sarger, “Optical properties of infrared femtosecond laser-modified fused silica and application to waveguide fabrication,” J. Opt. Soc. Am. B 22(10), 2138–2143 (2005).
    [CrossRef]
  4. E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, “Three-dimensional optical storage inside transparent materials,” Opt. Lett. 21(24), 2023–2025 (1996).
    [CrossRef] [PubMed]
  5. L. Canioni, M. Bellec, A. Royon, B. Bousquet, and T. Cardinal, “Three-dimensional optical data storage using third-harmonic generation in silver zinc phosphate glass,” Opt. Lett. 33(4), 360–362 (2008).
    [CrossRef] [PubMed]
  6. S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412(6848), 697–698 (2001).
    [CrossRef] [PubMed]
  7. F. Huo, Z. Zheng, G. Zheng, L. R. Giam, H. Zhang, and C. A. Mirkin, “Polymer pen lithography,” Science 321(5896), 1658–1660 (2008).
    [CrossRef] [PubMed]
  8. A. Chimmalgi, C. P. Grigoropoulos, and K. Komvopoulos, “Surface nanostructuring by nano-/femtosecond laser-assisted scanning force microscopy,” J. Appl. Phys. 97(10), 104319 (2005).
    [CrossRef]
  9. S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77(22), 3490–3492 (2000).
    [CrossRef]
  10. M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
    [CrossRef] [PubMed]
  11. H. Schneckenburger, D. F. Regulla, and E. Unsöld, “Time-resolved investigations of radiophotoluminescence in metaphosphate glass dosimeters,” Appl. Phys., A Mater. Sci. Process. 26, 23–26 (1981).
    [CrossRef]
  12. A. V. Dmitryuk, S. E. Paramzina, A. S. Perminov, N. D. Solov’eva, and N. T. Timofeev, “The influence of glass composition on the properties of silver-doped radiophotoluminescent phosphate glasses,” J. Non-Cryst. Solids 202(1-2), 173–177 (1996).
    [CrossRef]
  13. I. Belharouak, C. Parent, B. Tanguy, G. Le Flem, and M. Couzy, “Silver aggregates in photoluminescent phosphate glasses of the ‘Ag2O-ZnO-P2O5’ system,” J. Non-Cryst. Solids 244(2-3), 238–249 (1999).
    [CrossRef]
  14. H. D. Jones and H. R. Reiss, “Intense-field effects in solids,” Phys. Rev. B 16(6), 2466–2473 (1977).
    [CrossRef]
  15. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53(4), 1749–1761 (1996).
    [CrossRef]
  16. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).
  17. C. B. Shaffer, J. F. Garcia, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76, 351–354 (2003).
    [CrossRef]
  18. S. M. Eaton, H. Zhang, P. R. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. I. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rates,” Opt. Express 13(12), 4708–4716 (2005).
    [CrossRef] [PubMed]
  19. L. A. Pipes, and L. R. Harvill, “Applied mathematics for engineers and physicists” (McGraw-Hill International Edition, 1987).
  20. Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
    [CrossRef]
  21. A. Podlipensky, V. Grebenev, G. Seifert, and H. Graener, “Ionization and photomodification of Ag nanoparticles in soda-lime glass by 150 fs laser irradiation: a luminescence study,” J. Lumin. 109, 135–142 (2004).
  22. L. A. Peyser, A. E. Vinson, A. P. Bartko, and R. M. Dickson, “Photoactivated fluorescence from individual silver nanoclusters,” Science 291(5501), 103–106 (2001).
    [CrossRef] [PubMed]
  23. M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
    [CrossRef]
  24. T. Tani, “The theory of the photographic process” (Oxford University Press, New York, 1995).
  25. S. Stookey, “Photosensitive Glass : A New Photographic Medium,” Ind. Eng. Chem. 41, 856–861 (1949).
    [CrossRef]
  26. G. Mie, “Contribution to the optics of turbid media, particularly of colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  27. H. C. Van De Hulst, “Light scattering by small particles” (J. Wiley & Sons, 3rd edition, New York, 1964).
  28. L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
    [CrossRef]
  29. O. M. Efimov, K. Gabel, S. V. Garnov, L. B. Glebov, S. Grantham, M. Richardson, and M. J. Soileau, “Color-center generation in silicate glasses exposed to infrared femtosecond pulses,” J. Opt. Soc. Am. B 15(1), 193–199 (1998).
    [CrossRef]
  30. J. Crank, “The mathematics of diffusion” (Clarendon Press, 1956).
  31. W. T. Doyle, “Absorption of Light by Colloids in Alkali Halide Crystals,” Phys. Rev. 111(4), 1067–1072 (1958).
    [CrossRef]

2008

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[CrossRef]

L. Canioni, M. Bellec, A. Royon, B. Bousquet, and T. Cardinal, “Three-dimensional optical data storage using third-harmonic generation in silver zinc phosphate glass,” Opt. Lett. 33(4), 360–362 (2008).
[CrossRef] [PubMed]

F. Huo, Z. Zheng, G. Zheng, L. R. Giam, H. Zhang, and C. A. Mirkin, “Polymer pen lithography,” Science 321(5896), 1658–1660 (2008).
[CrossRef] [PubMed]

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

2007

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

2005

M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
[CrossRef]

S. M. Eaton, H. Zhang, P. R. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. I. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rates,” Opt. Express 13(12), 4708–4716 (2005).
[CrossRef] [PubMed]

A. Chimmalgi, C. P. Grigoropoulos, and K. Komvopoulos, “Surface nanostructuring by nano-/femtosecond laser-assisted scanning force microscopy,” J. Appl. Phys. 97(10), 104319 (2005).
[CrossRef]

A. Zoubir, M. Richardson, L. Canioni, A. Brocas, and L. Sarger, “Optical properties of infrared femtosecond laser-modified fused silica and application to waveguide fabrication,” J. Opt. Soc. Am. B 22(10), 2138–2143 (2005).
[CrossRef]

2004

L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
[CrossRef]

A. Podlipensky, V. Grebenev, G. Seifert, and H. Graener, “Ionization and photomodification of Ag nanoparticles in soda-lime glass by 150 fs laser irradiation: a luminescence study,” J. Lumin. 109, 135–142 (2004).

2003

C. B. Shaffer, J. F. Garcia, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76, 351–354 (2003).
[CrossRef]

2001

S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412(6848), 697–698 (2001).
[CrossRef] [PubMed]

L. A. Peyser, A. E. Vinson, A. P. Bartko, and R. M. Dickson, “Photoactivated fluorescence from individual silver nanoclusters,” Science 291(5501), 103–106 (2001).
[CrossRef] [PubMed]

2000

S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77(22), 3490–3492 (2000).
[CrossRef]

1999

I. Belharouak, C. Parent, B. Tanguy, G. Le Flem, and M. Couzy, “Silver aggregates in photoluminescent phosphate glasses of the ‘Ag2O-ZnO-P2O5’ system,” J. Non-Cryst. Solids 244(2-3), 238–249 (1999).
[CrossRef]

1998

1997

K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71(23), 3329–3331 (1997).
[CrossRef]

1996

E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, “Three-dimensional optical storage inside transparent materials,” Opt. Lett. 21(24), 2023–2025 (1996).
[CrossRef] [PubMed]

A. V. Dmitryuk, S. E. Paramzina, A. S. Perminov, N. D. Solov’eva, and N. T. Timofeev, “The influence of glass composition on the properties of silver-doped radiophotoluminescent phosphate glasses,” J. Non-Cryst. Solids 202(1-2), 173–177 (1996).
[CrossRef]

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53(4), 1749–1761 (1996).
[CrossRef]

1981

H. Schneckenburger, D. F. Regulla, and E. Unsöld, “Time-resolved investigations of radiophotoluminescence in metaphosphate glass dosimeters,” Appl. Phys., A Mater. Sci. Process. 26, 23–26 (1981).
[CrossRef]

1977

H. D. Jones and H. R. Reiss, “Intense-field effects in solids,” Phys. Rev. B 16(6), 2466–2473 (1977).
[CrossRef]

1965

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

1958

W. T. Doyle, “Absorption of Light by Colloids in Alkali Halide Crystals,” Phys. Rev. 111(4), 1067–1072 (1958).
[CrossRef]

1949

S. Stookey, “Photosensitive Glass : A New Photographic Medium,” Ind. Eng. Chem. 41, 856–861 (1949).
[CrossRef]

1908

G. Mie, “Contribution to the optics of turbid media, particularly of colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Arai, A. I.

Bartko, A. P.

L. A. Peyser, A. E. Vinson, A. P. Bartko, and R. M. Dickson, “Photoactivated fluorescence from individual silver nanoclusters,” Science 291(5501), 103–106 (2001).
[CrossRef] [PubMed]

Belharouak, I.

I. Belharouak, C. Parent, B. Tanguy, G. Le Flem, and M. Couzy, “Silver aggregates in photoluminescent phosphate glasses of the ‘Ag2O-ZnO-P2O5’ system,” J. Non-Cryst. Solids 244(2-3), 238–249 (1999).
[CrossRef]

Bellec, M.

Bousquet, B.

Bovatsek, J.

Brocas, A.

Callan, J. P.

Canioni, L.

Cardinal, T.

L. Canioni, M. Bellec, A. Royon, B. Bousquet, and T. Cardinal, “Three-dimensional optical data storage using third-harmonic generation in silver zinc phosphate glass,” Opt. Lett. 33(4), 360–362 (2008).
[CrossRef] [PubMed]

M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
[CrossRef]

Chen, D.

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

Chimmalgi, A.

A. Chimmalgi, C. P. Grigoropoulos, and K. Komvopoulos, “Surface nanostructuring by nano-/femtosecond laser-assisted scanning force microscopy,” J. Appl. Phys. 97(10), 104319 (2005).
[CrossRef]

Corkum, P. B.

L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
[CrossRef]

Couzy, M.

I. Belharouak, C. Parent, B. Tanguy, G. Le Flem, and M. Couzy, “Silver aggregates in photoluminescent phosphate glasses of the ‘Ag2O-ZnO-P2O5’ system,” J. Non-Cryst. Solids 244(2-3), 238–249 (1999).
[CrossRef]

Dai, Y.

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

Dickson, R. M.

L. A. Peyser, A. E. Vinson, A. P. Bartko, and R. M. Dickson, “Photoactivated fluorescence from individual silver nanoclusters,” Science 291(5501), 103–106 (2001).
[CrossRef] [PubMed]

Dmitryuk, A. V.

A. V. Dmitryuk, S. E. Paramzina, A. S. Perminov, N. D. Solov’eva, and N. T. Timofeev, “The influence of glass composition on the properties of silver-doped radiophotoluminescent phosphate glasses,” J. Non-Cryst. Solids 202(1-2), 173–177 (1996).
[CrossRef]

Doyle, W. T.

W. T. Doyle, “Absorption of Light by Colloids in Alkali Halide Crystals,” Phys. Rev. 111(4), 1067–1072 (1958).
[CrossRef]

Eaton, S. M.

Efimov, O. M.

Feit, M. D.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53(4), 1749–1761 (1996).
[CrossRef]

Finlay, R. J.

Gabel, K.

Gaier, L. N.

L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
[CrossRef]

Garcia, J. F.

C. B. Shaffer, J. F. Garcia, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76, 351–354 (2003).
[CrossRef]

Garnov, S. V.

Gattass, R. R.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[CrossRef]

Giam, L. R.

F. Huo, Z. Zheng, G. Zheng, L. R. Giam, H. Zhang, and C. A. Mirkin, “Polymer pen lithography,” Science 321(5896), 1658–1660 (2008).
[CrossRef] [PubMed]

Glebov, L. B.

Glezer, E. N.

Graener, H.

A. Podlipensky, V. Grebenev, G. Seifert, and H. Graener, “Ionization and photomodification of Ag nanoparticles in soda-lime glass by 150 fs laser irradiation: a luminescence study,” J. Lumin. 109, 135–142 (2004).

Grantham, S.

Grebenev, V.

A. Podlipensky, V. Grebenev, G. Seifert, and H. Graener, “Ionization and photomodification of Ag nanoparticles in soda-lime glass by 150 fs laser irradiation: a luminescence study,” J. Lumin. 109, 135–142 (2004).

Grigoropoulos, C. P.

A. Chimmalgi, C. P. Grigoropoulos, and K. Komvopoulos, “Surface nanostructuring by nano-/femtosecond laser-assisted scanning force microscopy,” J. Appl. Phys. 97(10), 104319 (2005).
[CrossRef]

Her, T.-H.

Herman, P. R.

Herman, S.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53(4), 1749–1761 (1996).
[CrossRef]

Hirao, K.

K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71(23), 3329–3331 (1997).
[CrossRef]

Hu, X.

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

Huang, L.

Huo, F.

F. Huo, Z. Zheng, G. Zheng, L. R. Giam, H. Zhang, and C. A. Mirkin, “Polymer pen lithography,” Science 321(5896), 1658–1660 (2008).
[CrossRef] [PubMed]

Inouye, H.

K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71(23), 3329–3331 (1997).
[CrossRef]

Jiang, X.

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77(22), 3490–3492 (2000).
[CrossRef]

Johnson, S. G.

S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77(22), 3490–3492 (2000).
[CrossRef]

Jones, H. D.

H. D. Jones and H. R. Reiss, “Intense-field effects in solids,” Phys. Rev. B 16(6), 2466–2473 (1977).
[CrossRef]

Kawata, S.

S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412(6848), 697–698 (2001).
[CrossRef] [PubMed]

Keldysh, L. V.

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

Knight, P. L.

L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
[CrossRef]

Komvopoulos, K.

A. Chimmalgi, C. P. Grigoropoulos, and K. Komvopoulos, “Surface nanostructuring by nano-/femtosecond laser-assisted scanning force microscopy,” J. Appl. Phys. 97(10), 104319 (2005).
[CrossRef]

Le Flem, G.

I. Belharouak, C. Parent, B. Tanguy, G. Le Flem, and M. Couzy, “Silver aggregates in photoluminescent phosphate glasses of the ‘Ag2O-ZnO-P2O5’ system,” J. Non-Cryst. Solids 244(2-3), 238–249 (1999).
[CrossRef]

Le Nestour, A.

M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
[CrossRef]

Lein, M.

L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
[CrossRef]

Lelong, G.

M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
[CrossRef]

Linden, S.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Lounis, B.

M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
[CrossRef]

Maali, A.

M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
[CrossRef]

Mazur, E.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[CrossRef]

C. B. Shaffer, J. F. Garcia, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76, 351–354 (2003).
[CrossRef]

E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, “Three-dimensional optical storage inside transparent materials,” Opt. Lett. 21(24), 2023–2025 (1996).
[CrossRef] [PubMed]

Mie, G.

G. Mie, “Contribution to the optics of turbid media, particularly of colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Milosavljevic, M.

Mirkin, C. A.

F. Huo, Z. Zheng, G. Zheng, L. R. Giam, H. Zhang, and C. A. Mirkin, “Polymer pen lithography,” Science 321(5896), 1658–1660 (2008).
[CrossRef] [PubMed]

Mitsuyu, T.

K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71(23), 3329–3331 (1997).
[CrossRef]

Miura, K.

K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71(23), 3329–3331 (1997).
[CrossRef]

Paramzina, S. E.

A. V. Dmitryuk, S. E. Paramzina, A. S. Perminov, N. D. Solov’eva, and N. T. Timofeev, “The influence of glass composition on the properties of silver-doped radiophotoluminescent phosphate glasses,” J. Non-Cryst. Solids 202(1-2), 173–177 (1996).
[CrossRef]

Parent, C.

I. Belharouak, C. Parent, B. Tanguy, G. Le Flem, and M. Couzy, “Silver aggregates in photoluminescent phosphate glasses of the ‘Ag2O-ZnO-P2O5’ system,” J. Non-Cryst. Solids 244(2-3), 238–249 (1999).
[CrossRef]

Perminov, A. S.

A. V. Dmitryuk, S. E. Paramzina, A. S. Perminov, N. D. Solov’eva, and N. T. Timofeev, “The influence of glass composition on the properties of silver-doped radiophotoluminescent phosphate glasses,” J. Non-Cryst. Solids 202(1-2), 173–177 (1996).
[CrossRef]

Perry, M. D.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53(4), 1749–1761 (1996).
[CrossRef]

Peyser, L. A.

L. A. Peyser, A. E. Vinson, A. P. Bartko, and R. M. Dickson, “Photoactivated fluorescence from individual silver nanoclusters,” Science 291(5501), 103–106 (2001).
[CrossRef] [PubMed]

Plet, C.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Podlipensky, A.

A. Podlipensky, V. Grebenev, G. Seifert, and H. Graener, “Ionization and photomodification of Ag nanoparticles in soda-lime glass by 150 fs laser irradiation: a luminescence study,” J. Lumin. 109, 135–142 (2004).

Qiu, J.

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71(23), 3329–3331 (1997).
[CrossRef]

Regulla, D. F.

H. Schneckenburger, D. F. Regulla, and E. Unsöld, “Time-resolved investigations of radiophotoluminescence in metaphosphate glass dosimeters,” Appl. Phys., A Mater. Sci. Process. 26, 23–26 (1981).
[CrossRef]

Reiss, H. R.

H. D. Jones and H. R. Reiss, “Intense-field effects in solids,” Phys. Rev. B 16(6), 2466–2473 (1977).
[CrossRef]

Richardson, M.

Rill, M. S.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Rocco, F.

M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
[CrossRef]

Royon, A.

Rubenchik, A. M.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53(4), 1749–1761 (1996).
[CrossRef]

Sarger, L.

Schneckenburger, H.

H. Schneckenburger, D. F. Regulla, and E. Unsöld, “Time-resolved investigations of radiophotoluminescence in metaphosphate glass dosimeters,” Appl. Phys., A Mater. Sci. Process. 26, 23–26 (1981).
[CrossRef]

Seifert, G.

A. Podlipensky, V. Grebenev, G. Seifert, and H. Graener, “Ionization and photomodification of Ag nanoparticles in soda-lime glass by 150 fs laser irradiation: a luminescence study,” J. Lumin. 109, 135–142 (2004).

Shaffer, C. B.

C. B. Shaffer, J. F. Garcia, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76, 351–354 (2003).
[CrossRef]

Shah, L.

Shore, B. W.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53(4), 1749–1761 (1996).
[CrossRef]

Soileau, M. J.

Solov’eva, N. D.

A. V. Dmitryuk, S. E. Paramzina, A. S. Perminov, N. D. Solov’eva, and N. T. Timofeev, “The influence of glass composition on the properties of silver-doped radiophotoluminescent phosphate glasses,” J. Non-Cryst. Solids 202(1-2), 173–177 (1996).
[CrossRef]

Staude, I.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Stockman, M. I.

L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
[CrossRef]

Stookey, S.

S. Stookey, “Photosensitive Glass : A New Photographic Medium,” Ind. Eng. Chem. 41, 856–861 (1949).
[CrossRef]

Stuart, B. C.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53(4), 1749–1761 (1996).
[CrossRef]

Sun, H. B.

S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412(6848), 697–698 (2001).
[CrossRef] [PubMed]

Takada, K.

S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412(6848), 697–698 (2001).
[CrossRef] [PubMed]

Tanaka, T.

S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412(6848), 697–698 (2001).
[CrossRef] [PubMed]

Tanguy, B.

I. Belharouak, C. Parent, B. Tanguy, G. Le Flem, and M. Couzy, “Silver aggregates in photoluminescent phosphate glasses of the ‘Ag2O-ZnO-P2O5’ system,” J. Non-Cryst. Solids 244(2-3), 238–249 (1999).
[CrossRef]

Thiel, M.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Timofeev, N. T.

A. V. Dmitryuk, S. E. Paramzina, A. S. Perminov, N. D. Solov’eva, and N. T. Timofeev, “The influence of glass composition on the properties of silver-doped radiophotoluminescent phosphate glasses,” J. Non-Cryst. Solids 202(1-2), 173–177 (1996).
[CrossRef]

Treguer, M.

M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
[CrossRef]

Unsöld, E.

H. Schneckenburger, D. F. Regulla, and E. Unsöld, “Time-resolved investigations of radiophotoluminescence in metaphosphate glass dosimeters,” Appl. Phys., A Mater. Sci. Process. 26, 23–26 (1981).
[CrossRef]

Vinson, A. E.

L. A. Peyser, A. E. Vinson, A. P. Bartko, and R. M. Dickson, “Photoactivated fluorescence from individual silver nanoclusters,” Science 291(5501), 103–106 (2001).
[CrossRef] [PubMed]

von Freymann, G.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Wang, C.

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

Wegener, M.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Yoshino, F.

Yu, B.

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

Yu Ivanov, M.

L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
[CrossRef]

Yudin, G. L.

L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
[CrossRef]

Zhang, H.

Zheng, G.

F. Huo, Z. Zheng, G. Zheng, L. R. Giam, H. Zhang, and C. A. Mirkin, “Polymer pen lithography,” Science 321(5896), 1658–1660 (2008).
[CrossRef] [PubMed]

Zheng, Z.

F. Huo, Z. Zheng, G. Zheng, L. R. Giam, H. Zhang, and C. A. Mirkin, “Polymer pen lithography,” Science 321(5896), 1658–1660 (2008).
[CrossRef] [PubMed]

Zhu, C.

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

Zoubir, A.

Ann. Phys.

G. Mie, “Contribution to the optics of turbid media, particularly of colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Appl. Phys. Lett.

K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71(23), 3329–3331 (1997).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77(22), 3490–3492 (2000).
[CrossRef]

Appl. Phys., A Mater. Sci. Process.

C. B. Shaffer, J. F. Garcia, and E. Mazur, “Bulk heating of transparent materials using a high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76, 351–354 (2003).
[CrossRef]

H. Schneckenburger, D. F. Regulla, and E. Unsöld, “Time-resolved investigations of radiophotoluminescence in metaphosphate glass dosimeters,” Appl. Phys., A Mater. Sci. Process. 26, 23–26 (1981).
[CrossRef]

Chem. Phys. Lett.

Y. Dai, X. Hu, C. Wang, D. Chen, X. Jiang, C. Zhu, B. Yu, and J. Qiu, “Fluorescent Ag nanoclusters in glass induced by an infrared femtosecond laser,” Chem. Phys. Lett. 439(1-3), 81–84 (2007).
[CrossRef]

Ind. Eng. Chem.

S. Stookey, “Photosensitive Glass : A New Photographic Medium,” Ind. Eng. Chem. 41, 856–861 (1949).
[CrossRef]

J. Appl. Phys.

A. Chimmalgi, C. P. Grigoropoulos, and K. Komvopoulos, “Surface nanostructuring by nano-/femtosecond laser-assisted scanning force microscopy,” J. Appl. Phys. 97(10), 104319 (2005).
[CrossRef]

J. Lumin.

A. Podlipensky, V. Grebenev, G. Seifert, and H. Graener, “Ionization and photomodification of Ag nanoparticles in soda-lime glass by 150 fs laser irradiation: a luminescence study,” J. Lumin. 109, 135–142 (2004).

J. Non-Cryst. Solids

A. V. Dmitryuk, S. E. Paramzina, A. S. Perminov, N. D. Solov’eva, and N. T. Timofeev, “The influence of glass composition on the properties of silver-doped radiophotoluminescent phosphate glasses,” J. Non-Cryst. Solids 202(1-2), 173–177 (1996).
[CrossRef]

I. Belharouak, C. Parent, B. Tanguy, G. Le Flem, and M. Couzy, “Silver aggregates in photoluminescent phosphate glasses of the ‘Ag2O-ZnO-P2O5’ system,” J. Non-Cryst. Solids 244(2-3), 238–249 (1999).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. B

L. N. Gaier, M. Lein, M. I. Stockman, P. L. Knight, P. B. Corkum, M. Yu Ivanov, and G. L. Yudin, “Ultrafast multiphoton forest fires and fractals in clusters and dielectrics,” J. Phys. B 37(3), L57–L67 (2004).
[CrossRef]

Nat. Mater.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Nat. Photonics

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[CrossRef]

Nature

S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412(6848), 697–698 (2001).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev.

W. T. Doyle, “Absorption of Light by Colloids in Alkali Halide Crystals,” Phys. Rev. 111(4), 1067–1072 (1958).
[CrossRef]

Phys. Rev. B

H. D. Jones and H. R. Reiss, “Intense-field effects in solids,” Phys. Rev. B 16(6), 2466–2473 (1977).
[CrossRef]

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53(4), 1749–1761 (1996).
[CrossRef]

Science

F. Huo, Z. Zheng, G. Zheng, L. R. Giam, H. Zhang, and C. A. Mirkin, “Polymer pen lithography,” Science 321(5896), 1658–1660 (2008).
[CrossRef] [PubMed]

L. A. Peyser, A. E. Vinson, A. P. Bartko, and R. M. Dickson, “Photoactivated fluorescence from individual silver nanoclusters,” Science 291(5501), 103–106 (2001).
[CrossRef] [PubMed]

Solid State Sci.

M. Treguer, F. Rocco, G. Lelong, A. Le Nestour, T. Cardinal, A. Maali, and B. Lounis, “Fluorescent silver oligomeric clusters and colloidal particles,” Solid State Sci. 7(7), 812–818 (2005).
[CrossRef]

Sov. Phys. JETP

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

Other

L. A. Pipes, and L. R. Harvill, “Applied mathematics for engineers and physicists” (McGraw-Hill International Edition, 1987).

T. Tani, “The theory of the photographic process” (Oxford University Press, New York, 1995).

H. C. Van De Hulst, “Light scattering by small particles” (J. Wiley & Sons, 3rd edition, New York, 1964).

J. Crank, “The mathematics of diffusion” (Clarendon Press, 1956).

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Figures (8)

Fig. 1
Fig. 1

Schematic view of the nanostructures formation. (A) Following laser irradiation, released photoelectrons are trapped by Ag+ ions to form silver atoms Ag0, represented by the blue spots. The Ag0 distribution follows the interaction area delimited by the blue dashed circle. The area is smaller than the laser beam size because of the nature of the nonlinear interaction. (B) After 1000 pulses, the local temperature increases and the diffusion occurs, as illustrated by the red arrows. Ag0 and Ag+ interact to give rise to silver clusters Agm x+ with m<10, illustrated by the red spots. (C) Subsequent laser pulses photo-dissociate the newly formed Agm x+ except on the edges of the interaction area, leaving a cylindrical nanostructure composed of silver clusters. The silver cluster photo-dissociation threshold is represented by the green dotted circle.

Fig. 2
Fig. 2

Modeling of the silver species density. (A) The calculated Ag0 density is represented in the center of the laser beam for different numbers of pulses (from 103 to 105). The Ag0 atoms density decreases with the number of pulses because diffusion process brings them to the edges of the laser beam. In this simulation, it is an effective diffusion process which takes into account the photo-dissociation effect, see Appendix B.B. The boundary condition on the edges keeps the Ag0 density to zero because they are consumed to form silver clusters. (B) The calculation of the corresponding created Agm x+ is represented and compared to the fluorescence measurements.

Fig. 3
Fig. 3

HRSEM and AFM characterization of the nanostructures. The irradiated sample was acid-etched to reveal the nanostructures. HRSEM allows topological and chemical characterization. AFM allows topological characterization. (A) HRSEM image of a spot irradiated by a focused femtosecond laser at 1 J.cm−2. As expected, a circular shape is observed. (B) The transversal profile (small white segment) of the HRSEM image has a measured width of ~80 nm (FWHM). (C) and (D) Comparison between the transversal profiles (following long white segment) of the HRSEM image (C) and the corresponding AFM image (D) shows that the annular shape is due to chemical modifications (accumulation of Ag species). The central peak indicates a topological modification following acid-etching.

Fig. 4
Fig. 4

Optical properties of silver nanostructures. The silver clusters exhibit strong fluorescence when excited in the UV-blue range. The silver nanoparticles reveal a surface plasmon resonance. (A) 3D reconstitution of confocal fluorescence microscopy images of a 1 J.cm−2 laser irradiated spot. (B) Fluorescence spectrum of the photo-induced silver clusters with an excitation wavelength of 405 nm. A characteristic band centered at 580 nm is observed. (C) Fluorescence microscopy and corresponding HRSEM images of a cross pattern created by moving the sample first in the x direction and second in the y direction. During the second pass, the previously-created silver clusters are photo-dissociated. (D) Following laser irradiation, the sample is thermally treated at 400°C during 20 minutes to transform the silver clusters into silver nanoparticles. A differential absorbance spectrum is performed in the region where the nanostructures are formed (red dots). The band centered at 460 nm is characteristic of the surface plasmon resonance of the silver nanoparticles arrangement. The experimental measurement fitted with the Mie’s theory gives a mean diameter of 10 nm (blue curve).

Fig. 5
Fig. 5

Evolution of the plasma-induced-absorption versus the pump-probe delay for a pump irradiance of 7.3 TW.cm−2. The curve is an average of 30 acquisitions. The standard deviation of these acquisitions gives an uncertainty of the measurement of 40%.

Fig. 6
Fig. 6

Evolution of the plasma-induced-absorption versus the pump irradiance at the sample.

Fig. 7
Fig. 7

Pulse to pulse transmission measurements (blue dot). The red line represents the mean value of the transmission which is less than 1%.

Fig. 8
Fig. 8

Temperature versus the number of pulses. (A) The comparison between 10 kHz and 10 MHz repetition rate illustrates the cumulative effects of a high repetition rate laser. (B) Temperature elevation in our experimental condition. The thermal diffusion temperature (blue line) is achieved for 3000 pulses. The glass transition temperature is represented by the grey dotted line.

Tables (1)

Tables Icon

Table 1 Experimental parameters for the temperature evolution calculation

Equations (16)

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

Ne=cn0ε0mee2τcL(1+4π2c2τc2λ02)ln(1T)
T(x,t)tDth2T(x,t)x2=Q(x,t)ρCp
Q(x,t)=E0exp(Nx2w02)i=0Mδ(tiΔt)H(tiΔt)
T(x,t)T0=ΔT=αEN2π3/2w0wzρCpi=0Mexp{x24[Dth×(tiΔt)+w024N]}[Dth×(tiΔt)+w024N]1/2H(tiΔt)
N(x,t)tDAg2N(x,t)x2=0
N(x,t=0)=N0exp(kx2w02)
N(x=0,t>0)=N(x=a,t>0)=0
α(λ0)=24π2n03VNλ0εr2(εr1+2n02)2+εr22(m−1)
εrD(ω)=εr1+iεr2=1ωp2ω2+iΓω(unitless)
ωp=Nee2ε0me(rad.s−1)
ΓD(R) ~VfR(rad.s−1)
D(λ0)=0.43lγNVα(λ0)=Aλ02(λ02λm2)2+ADmλ02(unitless)
A=0.43(48π3ε0n03clγmeVf)Nee2R(m2)
Dm=0.43(24π2n03lγ)(1+2n02)1Δλ(unitless)
λm2=4π2ε0c2meNee2(1+2n02) (m2 )
Δλ=λm2Vf2πcR(m)

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