Abstract

We present the z-scan studies of low-excited laser plasma showing a self-interaction of a femtosecond beam during the high-order harmonic generation in the laser plumes produced on the surfaces of different targets. This process was attributed to the ionization of neutrals and singly charged ions at the leading part of the laser pulse with further self-defocusing caused by the free electrons. The spatial distribution of the radiation propagating through the plasma showed an appearance of a ringlike shape at the intensities exceeding the barrier suppression intensity for singly charged ions. We show that this process leads to the restriction of the cutoff energy of high harmonics and the conversion efficiency from laser plasma.

© 2006 Optical Society of America

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  1. Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
    [CrossRef] [PubMed]
  2. S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
    [CrossRef] [PubMed]
  3. C.-G. Wahlström, S. Borgström, J. Larsson, and S.-G. Petterson, "High-order harmonic generation in laser-produced ions using a near-infrared laser," Phys. Rev. A 51, 585-591 (1995).
    [CrossRef] [PubMed]
  4. W. Theobald, C. Wülker, F. R. Schäfer, and B. N. Chichkov, "High-order harmonic generation in carbon vapor and low charged plasma," Opt. Commun. 120, 177-183 (1995).
    [CrossRef]
  5. R. Ganeev, M. Suzuki, M. Baba, H. Kuroda, and T. Ozaki, "High-order harmonic generation from boron plasma in the extreme-ultraviolet range," Opt. Lett. 30, 768-770 (2005).
    [CrossRef] [PubMed]
  6. M. Bellini, C. Corsi, and M. C. Gambino, "Neutral depletion and beam defocusing in harmonic generation from strongly ionized media," Phys. Rev. A 64, 023411 (2001).
    [CrossRef]
  7. J. H. Kim and C. H. Nam, "Plasma-induced frequency chirp of intense femtosecond lasers and its role in shaping high-order harmonic spectral lines," Phys. Rev. A 65, 033801 (2002).
    [CrossRef]
  8. V. Tosa, H. T. Kim, I. J. Kim, and C. H. Nam, "High-order harmonic generation by chirped and self-guided femtosecond laser pulses. I. Spatial and spectral analysis," Phys. Rev. A 71, 063807 (2005).
    [CrossRef]
  9. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, andE. W. van Stryland, "Sensitive measurements of anisotropy of nonlinear refraction and absorption in crystals," IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  10. R. A. Ganeev, M. Suzuki, M. Baba, and H. Kuroda, "Generation of strong coherent extreme ultraviolet radiation from the laser plasma produced on the surface of solid targets," Appl. Phys. B 81, 1081-1089 (2005).
    [CrossRef]
  11. X. Lui, S. Guo, H. Wang, and L. Hou, "Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption," Opt. Commun. 197, 431-437 (2001).
    [CrossRef]
  12. S. I. Anisimov, Y. A. Imas, G. S. Romanov, and Y. V. Khodyko, High Power Radiation Effect in Metals (Nauka, 1970).

2005 (3)

V. Tosa, H. T. Kim, I. J. Kim, and C. H. Nam, "High-order harmonic generation by chirped and self-guided femtosecond laser pulses. I. Spatial and spectral analysis," Phys. Rev. A 71, 063807 (2005).
[CrossRef]

R. A. Ganeev, M. Suzuki, M. Baba, and H. Kuroda, "Generation of strong coherent extreme ultraviolet radiation from the laser plasma produced on the surface of solid targets," Appl. Phys. B 81, 1081-1089 (2005).
[CrossRef]

R. Ganeev, M. Suzuki, M. Baba, H. Kuroda, and T. Ozaki, "High-order harmonic generation from boron plasma in the extreme-ultraviolet range," Opt. Lett. 30, 768-770 (2005).
[CrossRef] [PubMed]

2002 (1)

J. H. Kim and C. H. Nam, "Plasma-induced frequency chirp of intense femtosecond lasers and its role in shaping high-order harmonic spectral lines," Phys. Rev. A 65, 033801 (2002).
[CrossRef]

2001 (2)

M. Bellini, C. Corsi, and M. C. Gambino, "Neutral depletion and beam defocusing in harmonic generation from strongly ionized media," Phys. Rev. A 64, 023411 (2001).
[CrossRef]

X. Lui, S. Guo, H. Wang, and L. Hou, "Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption," Opt. Commun. 197, 431-437 (2001).
[CrossRef]

1995 (2)

C.-G. Wahlström, S. Borgström, J. Larsson, and S.-G. Petterson, "High-order harmonic generation in laser-produced ions using a near-infrared laser," Phys. Rev. A 51, 585-591 (1995).
[CrossRef] [PubMed]

W. Theobald, C. Wülker, F. R. Schäfer, and B. N. Chichkov, "High-order harmonic generation in carbon vapor and low charged plasma," Opt. Commun. 120, 177-183 (1995).
[CrossRef]

1993 (1)

S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
[CrossRef] [PubMed]

1992 (1)

Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
[CrossRef] [PubMed]

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, andE. W. van Stryland, "Sensitive measurements of anisotropy of nonlinear refraction and absorption in crystals," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Akiyama, Y.

S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
[CrossRef] [PubMed]

Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
[CrossRef] [PubMed]

Anisimov, S. I.

S. I. Anisimov, Y. A. Imas, G. S. Romanov, and Y. V. Khodyko, High Power Radiation Effect in Metals (Nauka, 1970).

Baba, M.

R. A. Ganeev, M. Suzuki, M. Baba, and H. Kuroda, "Generation of strong coherent extreme ultraviolet radiation from the laser plasma produced on the surface of solid targets," Appl. Phys. B 81, 1081-1089 (2005).
[CrossRef]

R. Ganeev, M. Suzuki, M. Baba, H. Kuroda, and T. Ozaki, "High-order harmonic generation from boron plasma in the extreme-ultraviolet range," Opt. Lett. 30, 768-770 (2005).
[CrossRef] [PubMed]

Bellini, M.

M. Bellini, C. Corsi, and M. C. Gambino, "Neutral depletion and beam defocusing in harmonic generation from strongly ionized media," Phys. Rev. A 64, 023411 (2001).
[CrossRef]

Borgström, S.

C.-G. Wahlström, S. Borgström, J. Larsson, and S.-G. Petterson, "High-order harmonic generation in laser-produced ions using a near-infrared laser," Phys. Rev. A 51, 585-591 (1995).
[CrossRef] [PubMed]

Chichkov, B. N.

W. Theobald, C. Wülker, F. R. Schäfer, and B. N. Chichkov, "High-order harmonic generation in carbon vapor and low charged plasma," Opt. Commun. 120, 177-183 (1995).
[CrossRef]

Corsi, C.

M. Bellini, C. Corsi, and M. C. Gambino, "Neutral depletion and beam defocusing in harmonic generation from strongly ionized media," Phys. Rev. A 64, 023411 (2001).
[CrossRef]

Gambino, M. C.

M. Bellini, C. Corsi, and M. C. Gambino, "Neutral depletion and beam defocusing in harmonic generation from strongly ionized media," Phys. Rev. A 64, 023411 (2001).
[CrossRef]

Ganeev, R.

Ganeev, R. A.

R. A. Ganeev, M. Suzuki, M. Baba, and H. Kuroda, "Generation of strong coherent extreme ultraviolet radiation from the laser plasma produced on the surface of solid targets," Appl. Phys. B 81, 1081-1089 (2005).
[CrossRef]

Guo, S.

X. Lui, S. Guo, H. Wang, and L. Hou, "Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption," Opt. Commun. 197, 431-437 (2001).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, andE. W. van Stryland, "Sensitive measurements of anisotropy of nonlinear refraction and absorption in crystals," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Hou, L.

X. Lui, S. Guo, H. Wang, and L. Hou, "Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption," Opt. Commun. 197, 431-437 (2001).
[CrossRef]

Imas, Y. A.

S. I. Anisimov, Y. A. Imas, G. S. Romanov, and Y. V. Khodyko, High Power Radiation Effect in Metals (Nauka, 1970).

Khodyko, Y. V.

S. I. Anisimov, Y. A. Imas, G. S. Romanov, and Y. V. Khodyko, High Power Radiation Effect in Metals (Nauka, 1970).

Kim, H. T.

V. Tosa, H. T. Kim, I. J. Kim, and C. H. Nam, "High-order harmonic generation by chirped and self-guided femtosecond laser pulses. I. Spatial and spectral analysis," Phys. Rev. A 71, 063807 (2005).
[CrossRef]

Kim, I. J.

V. Tosa, H. T. Kim, I. J. Kim, and C. H. Nam, "High-order harmonic generation by chirped and self-guided femtosecond laser pulses. I. Spatial and spectral analysis," Phys. Rev. A 71, 063807 (2005).
[CrossRef]

Kim, J. H.

J. H. Kim and C. H. Nam, "Plasma-induced frequency chirp of intense femtosecond lasers and its role in shaping high-order harmonic spectral lines," Phys. Rev. A 65, 033801 (2002).
[CrossRef]

Kubodera, S.

S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
[CrossRef] [PubMed]

Kuroda, H.

R. Ganeev, M. Suzuki, M. Baba, H. Kuroda, and T. Ozaki, "High-order harmonic generation from boron plasma in the extreme-ultraviolet range," Opt. Lett. 30, 768-770 (2005).
[CrossRef] [PubMed]

R. A. Ganeev, M. Suzuki, M. Baba, and H. Kuroda, "Generation of strong coherent extreme ultraviolet radiation from the laser plasma produced on the surface of solid targets," Appl. Phys. B 81, 1081-1089 (2005).
[CrossRef]

Larsson, J.

C.-G. Wahlström, S. Borgström, J. Larsson, and S.-G. Petterson, "High-order harmonic generation in laser-produced ions using a near-infrared laser," Phys. Rev. A 51, 585-591 (1995).
[CrossRef] [PubMed]

Lui, X.

X. Lui, S. Guo, H. Wang, and L. Hou, "Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption," Opt. Commun. 197, 431-437 (2001).
[CrossRef]

Matsunawa, Y.

Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
[CrossRef] [PubMed]

Midorikawa, K.

S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
[CrossRef] [PubMed]

Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
[CrossRef] [PubMed]

Nagata, Y.

S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
[CrossRef] [PubMed]

Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
[CrossRef] [PubMed]

Nam, C. H.

V. Tosa, H. T. Kim, I. J. Kim, and C. H. Nam, "High-order harmonic generation by chirped and self-guided femtosecond laser pulses. I. Spatial and spectral analysis," Phys. Rev. A 71, 063807 (2005).
[CrossRef]

J. H. Kim and C. H. Nam, "Plasma-induced frequency chirp of intense femtosecond lasers and its role in shaping high-order harmonic spectral lines," Phys. Rev. A 65, 033801 (2002).
[CrossRef]

Obara, M.

S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
[CrossRef] [PubMed]

Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
[CrossRef] [PubMed]

Ozaki, T.

Petterson, S.-G.

C.-G. Wahlström, S. Borgström, J. Larsson, and S.-G. Petterson, "High-order harmonic generation in laser-produced ions using a near-infrared laser," Phys. Rev. A 51, 585-591 (1995).
[CrossRef] [PubMed]

Romanov, G. S.

S. I. Anisimov, Y. A. Imas, G. S. Romanov, and Y. V. Khodyko, High Power Radiation Effect in Metals (Nauka, 1970).

Said, A. A.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, andE. W. van Stryland, "Sensitive measurements of anisotropy of nonlinear refraction and absorption in crystals," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Schäfer, F. R.

W. Theobald, C. Wülker, F. R. Schäfer, and B. N. Chichkov, "High-order harmonic generation in carbon vapor and low charged plasma," Opt. Commun. 120, 177-183 (1995).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, andE. W. van Stryland, "Sensitive measurements of anisotropy of nonlinear refraction and absorption in crystals," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Suzuki, M.

R. Ganeev, M. Suzuki, M. Baba, H. Kuroda, and T. Ozaki, "High-order harmonic generation from boron plasma in the extreme-ultraviolet range," Opt. Lett. 30, 768-770 (2005).
[CrossRef] [PubMed]

R. A. Ganeev, M. Suzuki, M. Baba, and H. Kuroda, "Generation of strong coherent extreme ultraviolet radiation from the laser plasma produced on the surface of solid targets," Appl. Phys. B 81, 1081-1089 (2005).
[CrossRef]

Tashiro, H.

S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
[CrossRef] [PubMed]

Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
[CrossRef] [PubMed]

Theobald, W.

W. Theobald, C. Wülker, F. R. Schäfer, and B. N. Chichkov, "High-order harmonic generation in carbon vapor and low charged plasma," Opt. Commun. 120, 177-183 (1995).
[CrossRef]

Tosa, V.

V. Tosa, H. T. Kim, I. J. Kim, and C. H. Nam, "High-order harmonic generation by chirped and self-guided femtosecond laser pulses. I. Spatial and spectral analysis," Phys. Rev. A 71, 063807 (2005).
[CrossRef]

Toyoda, K.

S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
[CrossRef] [PubMed]

Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
[CrossRef] [PubMed]

van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, andE. W. van Stryland, "Sensitive measurements of anisotropy of nonlinear refraction and absorption in crystals," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Wahlström, C.-G.

C.-G. Wahlström, S. Borgström, J. Larsson, and S.-G. Petterson, "High-order harmonic generation in laser-produced ions using a near-infrared laser," Phys. Rev. A 51, 585-591 (1995).
[CrossRef] [PubMed]

Wang, H.

X. Lui, S. Guo, H. Wang, and L. Hou, "Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption," Opt. Commun. 197, 431-437 (2001).
[CrossRef]

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, andE. W. van Stryland, "Sensitive measurements of anisotropy of nonlinear refraction and absorption in crystals," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Wülker, C.

W. Theobald, C. Wülker, F. R. Schäfer, and B. N. Chichkov, "High-order harmonic generation in carbon vapor and low charged plasma," Opt. Commun. 120, 177-183 (1995).
[CrossRef]

Appl. Phys. B (1)

R. A. Ganeev, M. Suzuki, M. Baba, and H. Kuroda, "Generation of strong coherent extreme ultraviolet radiation from the laser plasma produced on the surface of solid targets," Appl. Phys. B 81, 1081-1089 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, andE. W. van Stryland, "Sensitive measurements of anisotropy of nonlinear refraction and absorption in crystals," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Opt. Commun. (2)

W. Theobald, C. Wülker, F. R. Schäfer, and B. N. Chichkov, "High-order harmonic generation in carbon vapor and low charged plasma," Opt. Commun. 120, 177-183 (1995).
[CrossRef]

X. Lui, S. Guo, H. Wang, and L. Hou, "Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption," Opt. Commun. 197, 431-437 (2001).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (5)

M. Bellini, C. Corsi, and M. C. Gambino, "Neutral depletion and beam defocusing in harmonic generation from strongly ionized media," Phys. Rev. A 64, 023411 (2001).
[CrossRef]

J. H. Kim and C. H. Nam, "Plasma-induced frequency chirp of intense femtosecond lasers and its role in shaping high-order harmonic spectral lines," Phys. Rev. A 65, 033801 (2002).
[CrossRef]

V. Tosa, H. T. Kim, I. J. Kim, and C. H. Nam, "High-order harmonic generation by chirped and self-guided femtosecond laser pulses. I. Spatial and spectral analysis," Phys. Rev. A 71, 063807 (2005).
[CrossRef]

S. Kubodera, Y. Nagata, Y. Akiyama, K. Midorikawa, M. Obara, H. Tashiro, and K. Toyoda, "High-order harmonic generation in laser-produced ions," Phys. Rev. A 48, 4576-4581 (1993).
[CrossRef] [PubMed]

C.-G. Wahlström, S. Borgström, J. Larsson, and S.-G. Petterson, "High-order harmonic generation in laser-produced ions using a near-infrared laser," Phys. Rev. A 51, 585-591 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

Y. Akiyama, K. Midorikawa, Y. Matsunawa, Y. Nagata, M. Obara, H. Tashiro, and K. Toyoda, "Generation of high-order harmonics using laser-produced rare-gas-like ions," Phys. Rev. Lett. 69, 2176-2178 (1992).
[CrossRef] [PubMed]

Other (1)

S. I. Anisimov, Y. A. Imas, G. S. Romanov, and Y. V. Khodyko, High Power Radiation Effect in Metals (Nauka, 1970).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup: T, target; VC, vacuum chamber; L1, L2, L3, lenses; FP, femtosecond pulse; PP, picosecond pulse; CCD, charge-coupled device.

Fig. 2
Fig. 2

Harmonic spectra in the cases of (1) boron, (2) chromium, and (3) indium plasmas.

Fig. 3
Fig. 3

Spatial shapes of femtosecond radiation after the propagation through the Mo plume at the different intensities at the focal plane. (a) 4 × 10 13 W cm 2 , (b) 5 × 10 14 W cm 2 .

Fig. 4
Fig. 4

Spatial shapes of the main beam propagating through the In plume at (a) I fp = 1 × 10 13 , (b) 7 × 10 13 , (c) 1 × 10 14 , and (d) 6 × 10 14 W cm 2 .

Fig. 5
Fig. 5

Closed-aperture z scans of indium plasma at low (circles) and high (triangles) intensities of femtosecond radiation. Fitting solid curve is a theoretical calculation.

Fig. 6
Fig. 6

Open-aperture z scan of indium plasma. Fitting thin and thick solid curves are the theoretical calculations for the two- and four-photon absorption mechanisms, respectively.

Equations (9)

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

T = 1 4 x ( x 2 + 9 ) ( x 2 + 1 ) Δ Φ 0 2 ( x 2 + 3 ) ( x 2 + 9 ) ( x 2 + 1 ) Δ Ψ 0 ,
T = 1 2 ( ρ x 2 + 2 x + 3 ρ ) ( x 2 + 9 ) ( x 2 + 1 ) Δ Φ 0 .
T ( z ) = m = 0 [ q 0 ( z ) ] m ( m + 1 ) 3 2 for q 0 < 1 ,
α ( I ) = α 0 + β ( 2 ω ) I ,
α ( I ) = α 0 + β ( 2 ω ) I + β ( 3 ω ) I 2 + β ( 4 ω ) I 3 = α 0 + β eff I ,
β eff = β ( 2 ω ) + β ( 3 ω ) I + β ( 4 ω ) I 2
N 0 = A ( M ν D 2 k T 0 ) 3 2 exp ( λ 1 k T 0 ) ,
I s = B M ν D 3 ( k T 0 + C λ 1 ) k T 0 exp ( λ 1 k T 0 ) .
Δ n = ω p 2 2 ω 2 ,

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