Abstract

A white-light source used in combination with a two-beam interferometer provides the same information about the dispersive nature of optical fibers as a femtosecond pulse from a laser source. The shape of the coherence curve in relation to the square of the degree of coherence, | γ |2, reflects the dispersion-induced pulse shape. We induce a third-order dispersion (TOD) effect on the | γ |2 in a two-beam interferometer composed of nondispersive and dispersive arms. A theoretical treatment predicts an oscillatory structure near the trailing edge of the | γ |2–curve due to TOD. Experiments are performed using low coherence sources, a few-mode fiber and a dispersion-shifted fiber near the zero chromatic-dispersion wavelengths of 1300 and 1550 nm, respectively. As a result, the experimentally obtained | γ |2-response well reflects that obtained theoretically for an unchirped wave with a Gaussian spectrum. Therefore, the | γ |2-response interferometric technique has the potential to simulate an ultra-short pulse transmission in the group velocity approximation.

© 2017 Optical Society of America

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References

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  1. N. Shibata, M. Tateda, S. Seikai, and N. Uchida, “Spatial technique for measuring modal delay differences in a dual-mode optical fiber,” Appl. Opt. 19(9), 1489–1492 (1980).
    [PubMed]
  2. M. Tateda, N. Shibata, and S. Seikai, “Interferometric method for chromatic dispersion measurement in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-17(3), 404–407 (1981).
  3. H. T. Shang, “Chromatic dispersion measurement by white-light interferometry on meter-length single-mode optical fibers,” Electron. Lett. 17(17), 603–605 (1981).
  4. J. Stone and L. G. Cohen, “Minimum dispersion spectra of single-mode fibers measured with 0.1-ps resolution by white light cross-correlation,” Electron. Lett. 18(16), 716–718 (1982).
  5. F. M. Sears, L. G. Cohen, and J. Stone, “Interferometric measurements of dispersion-spectra variations in a single-mode fiber,” J. Lightwave Technol. LT-2(2), 181–184 (1984).
  6. N. Shibata, M. Tsubokawa, and S. Seikai, “Measurements of polarization mode dispersion by optical heterodyne detection,” Electron. Lett. 20(25/26), 1055–1057 (1984).
  7. P. L. Francois, F. Alard, and M. Monerie, “Chromatic dispersion measurement from Fourier transform of white-light interference patterns,” Electron. Lett. 23(7), 357–358 (1987).
  8. S. Diddams and J.-C. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. A 13(6), 1120–1129 (1996).
  9. N. Shibata, K. Watanabe, M. Ohashi, and K. Aikawa, “Square of the degree of coherence and dispersion-induced pulse broadening in a fiber-optic transmission,” IEICE Commun. Express 5(12), 454–460 (2016).
  10. W. A. Hamilton, A. G. Klein, and G. I. Opat, “Longitudinal coherence and interferometry in dispersive media,” Phys. Rev. A 28(5), 3149–3152 (1983).
  11. N. Shibata, M. Tsubokawa, T. Nakashima, and S. Seikai, “Temporal coherence properties of a dispersively propagating beam in a fiber-optic interferometer,” J. Opt. Soc. Am. A 4(3), 494–497 (1987).
  12. A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9(12), 610–615 (2001).
    [PubMed]
  13. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, A. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle., “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightwave Technol. 30(4), 521–531 (2012).
  14. N. Shibata, A. Nakazono, and Y. Inoue, “Interference between two orthogonally polarized modes traversing a highly birefringent air-silica microstructure fiber,” J. Lightwave Technol. 23(3), 1244–1252 (2005).
  15. G. P. Agrawal, Nonlinear Fiber Optics, 4th Ed. (Elsevier/Academic, 2007).
  16. S. Katsumoto, Craftsmen of Quantum Mechanics (Maruzen Publisher, 2016).
  17. K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2:Temporal Pulse Dynamics in Dispersive, Attenuative Media, (Springer, 2009).
  18. N. Shibata, K. Watanabe, and M. Ohashi, “Chromatic dispersion diagnosis for the two-modes of few-mode photonic crystal fiber,” IEEE Photonics Technol. Lett. 28(4), 437–440 (2016).
  19. C.-C. Chang, H. P. Sardesai, and A. M. Weiner, “Dispersion-free fiber transmission for femtosecond pulses by use of a dispersion-compensating fiber and a programmable pulse shaper,” Opt. Lett. 23(4), 283–285 (1998).
    [PubMed]
  20. T. Yamamoto, E. Yoshida, K. R. Tamura, K. Yonenaga, and M. Nakazawa, “640-Gbit/s optical TDM transmission over 92 km through a dispersion-managed fiber consisting of single-mode fiber and reverse dispersion fiber,” IEEE Photonics Technol. Lett. 12(3), 353–355 (2000).
  21. D. Marcuse, “Pulse distortion in single-mode fibers. 3: Chirped pulses,” Appl. Opt. 20(20), 3573–3579 (1981).
    [PubMed]

2016 (2)

N. Shibata, K. Watanabe, M. Ohashi, and K. Aikawa, “Square of the degree of coherence and dispersion-induced pulse broadening in a fiber-optic transmission,” IEICE Commun. Express 5(12), 454–460 (2016).

N. Shibata, K. Watanabe, and M. Ohashi, “Chromatic dispersion diagnosis for the two-modes of few-mode photonic crystal fiber,” IEEE Photonics Technol. Lett. 28(4), 437–440 (2016).

2012 (1)

2005 (1)

2001 (1)

2000 (1)

T. Yamamoto, E. Yoshida, K. R. Tamura, K. Yonenaga, and M. Nakazawa, “640-Gbit/s optical TDM transmission over 92 km through a dispersion-managed fiber consisting of single-mode fiber and reverse dispersion fiber,” IEEE Photonics Technol. Lett. 12(3), 353–355 (2000).

1998 (1)

1996 (1)

S. Diddams and J.-C. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. A 13(6), 1120–1129 (1996).

1987 (2)

P. L. Francois, F. Alard, and M. Monerie, “Chromatic dispersion measurement from Fourier transform of white-light interference patterns,” Electron. Lett. 23(7), 357–358 (1987).

N. Shibata, M. Tsubokawa, T. Nakashima, and S. Seikai, “Temporal coherence properties of a dispersively propagating beam in a fiber-optic interferometer,” J. Opt. Soc. Am. A 4(3), 494–497 (1987).

1984 (2)

F. M. Sears, L. G. Cohen, and J. Stone, “Interferometric measurements of dispersion-spectra variations in a single-mode fiber,” J. Lightwave Technol. LT-2(2), 181–184 (1984).

N. Shibata, M. Tsubokawa, and S. Seikai, “Measurements of polarization mode dispersion by optical heterodyne detection,” Electron. Lett. 20(25/26), 1055–1057 (1984).

1983 (1)

W. A. Hamilton, A. G. Klein, and G. I. Opat, “Longitudinal coherence and interferometry in dispersive media,” Phys. Rev. A 28(5), 3149–3152 (1983).

1982 (1)

J. Stone and L. G. Cohen, “Minimum dispersion spectra of single-mode fibers measured with 0.1-ps resolution by white light cross-correlation,” Electron. Lett. 18(16), 716–718 (1982).

1981 (3)

M. Tateda, N. Shibata, and S. Seikai, “Interferometric method for chromatic dispersion measurement in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-17(3), 404–407 (1981).

H. T. Shang, “Chromatic dispersion measurement by white-light interferometry on meter-length single-mode optical fibers,” Electron. Lett. 17(17), 603–605 (1981).

D. Marcuse, “Pulse distortion in single-mode fibers. 3: Chirped pulses,” Appl. Opt. 20(20), 3573–3579 (1981).
[PubMed]

1980 (1)

Aikawa, K.

N. Shibata, K. Watanabe, M. Ohashi, and K. Aikawa, “Square of the degree of coherence and dispersion-induced pulse broadening in a fiber-optic transmission,” IEICE Commun. Express 5(12), 454–460 (2016).

Alard, F.

P. L. Francois, F. Alard, and M. Monerie, “Chromatic dispersion measurement from Fourier transform of white-light interference patterns,” Electron. Lett. 23(7), 357–358 (1987).

Bolle, C.

Burrows, E. C.

Chang, C.-C.

Cohen, L. G.

F. M. Sears, L. G. Cohen, and J. Stone, “Interferometric measurements of dispersion-spectra variations in a single-mode fiber,” J. Lightwave Technol. LT-2(2), 181–184 (1984).

J. Stone and L. G. Cohen, “Minimum dispersion spectra of single-mode fibers measured with 0.1-ps resolution by white light cross-correlation,” Electron. Lett. 18(16), 716–718 (1982).

Diddams, S.

S. Diddams and J.-C. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. A 13(6), 1120–1129 (1996).

Diels, J.-C.

S. Diddams and J.-C. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. A 13(6), 1120–1129 (1996).

Esmaeelpour, M.

Essiambre, R.-J.

Fercher, A.

Francois, P. L.

P. L. Francois, F. Alard, and M. Monerie, “Chromatic dispersion measurement from Fourier transform of white-light interference patterns,” Electron. Lett. 23(7), 357–358 (1987).

Gnauck, A. H.

Hamilton, W. A.

W. A. Hamilton, A. G. Klein, and G. I. Opat, “Longitudinal coherence and interferometry in dispersive media,” Phys. Rev. A 28(5), 3149–3152 (1983).

Hitzenberger, C.

Inoue, Y.

Karamata, B.

Klein, A. G.

W. A. Hamilton, A. G. Klein, and G. I. Opat, “Longitudinal coherence and interferometry in dispersive media,” Phys. Rev. A 28(5), 3149–3152 (1983).

Lasser, T.

Lingle, R.

Marcuse, D.

McCurdy, A. H.

Monerie, M.

P. L. Francois, F. Alard, and M. Monerie, “Chromatic dispersion measurement from Fourier transform of white-light interference patterns,” Electron. Lett. 23(7), 357–358 (1987).

Mumtaz, A.

Nakashima, T.

Nakazawa, M.

T. Yamamoto, E. Yoshida, K. R. Tamura, K. Yonenaga, and M. Nakazawa, “640-Gbit/s optical TDM transmission over 92 km through a dispersion-managed fiber consisting of single-mode fiber and reverse dispersion fiber,” IEEE Photonics Technol. Lett. 12(3), 353–355 (2000).

Nakazono, A.

Ohashi, M.

N. Shibata, K. Watanabe, M. Ohashi, and K. Aikawa, “Square of the degree of coherence and dispersion-induced pulse broadening in a fiber-optic transmission,” IEICE Commun. Express 5(12), 454–460 (2016).

N. Shibata, K. Watanabe, and M. Ohashi, “Chromatic dispersion diagnosis for the two-modes of few-mode photonic crystal fiber,” IEEE Photonics Technol. Lett. 28(4), 437–440 (2016).

Opat, G. I.

W. A. Hamilton, A. G. Klein, and G. I. Opat, “Longitudinal coherence and interferometry in dispersive media,” Phys. Rev. A 28(5), 3149–3152 (1983).

Peckham, D. W.

Randel, S.

Ryf, R.

Sardesai, H. P.

Sears, F. M.

F. M. Sears, L. G. Cohen, and J. Stone, “Interferometric measurements of dispersion-spectra variations in a single-mode fiber,” J. Lightwave Technol. LT-2(2), 181–184 (1984).

Seikai, S.

N. Shibata, M. Tsubokawa, T. Nakashima, and S. Seikai, “Temporal coherence properties of a dispersively propagating beam in a fiber-optic interferometer,” J. Opt. Soc. Am. A 4(3), 494–497 (1987).

N. Shibata, M. Tsubokawa, and S. Seikai, “Measurements of polarization mode dispersion by optical heterodyne detection,” Electron. Lett. 20(25/26), 1055–1057 (1984).

M. Tateda, N. Shibata, and S. Seikai, “Interferometric method for chromatic dispersion measurement in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-17(3), 404–407 (1981).

N. Shibata, M. Tateda, S. Seikai, and N. Uchida, “Spatial technique for measuring modal delay differences in a dual-mode optical fiber,” Appl. Opt. 19(9), 1489–1492 (1980).
[PubMed]

Shang, H. T.

H. T. Shang, “Chromatic dispersion measurement by white-light interferometry on meter-length single-mode optical fibers,” Electron. Lett. 17(17), 603–605 (1981).

Shibata, N.

N. Shibata, K. Watanabe, M. Ohashi, and K. Aikawa, “Square of the degree of coherence and dispersion-induced pulse broadening in a fiber-optic transmission,” IEICE Commun. Express 5(12), 454–460 (2016).

N. Shibata, K. Watanabe, and M. Ohashi, “Chromatic dispersion diagnosis for the two-modes of few-mode photonic crystal fiber,” IEEE Photonics Technol. Lett. 28(4), 437–440 (2016).

N. Shibata, A. Nakazono, and Y. Inoue, “Interference between two orthogonally polarized modes traversing a highly birefringent air-silica microstructure fiber,” J. Lightwave Technol. 23(3), 1244–1252 (2005).

N. Shibata, M. Tsubokawa, T. Nakashima, and S. Seikai, “Temporal coherence properties of a dispersively propagating beam in a fiber-optic interferometer,” J. Opt. Soc. Am. A 4(3), 494–497 (1987).

N. Shibata, M. Tsubokawa, and S. Seikai, “Measurements of polarization mode dispersion by optical heterodyne detection,” Electron. Lett. 20(25/26), 1055–1057 (1984).

M. Tateda, N. Shibata, and S. Seikai, “Interferometric method for chromatic dispersion measurement in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-17(3), 404–407 (1981).

N. Shibata, M. Tateda, S. Seikai, and N. Uchida, “Spatial technique for measuring modal delay differences in a dual-mode optical fiber,” Appl. Opt. 19(9), 1489–1492 (1980).
[PubMed]

Sierra, A.

Sticker, M.

Stone, J.

F. M. Sears, L. G. Cohen, and J. Stone, “Interferometric measurements of dispersion-spectra variations in a single-mode fiber,” J. Lightwave Technol. LT-2(2), 181–184 (1984).

J. Stone and L. G. Cohen, “Minimum dispersion spectra of single-mode fibers measured with 0.1-ps resolution by white light cross-correlation,” Electron. Lett. 18(16), 716–718 (1982).

Tamura, K. R.

T. Yamamoto, E. Yoshida, K. R. Tamura, K. Yonenaga, and M. Nakazawa, “640-Gbit/s optical TDM transmission over 92 km through a dispersion-managed fiber consisting of single-mode fiber and reverse dispersion fiber,” IEEE Photonics Technol. Lett. 12(3), 353–355 (2000).

Tateda, M.

M. Tateda, N. Shibata, and S. Seikai, “Interferometric method for chromatic dispersion measurement in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-17(3), 404–407 (1981).

N. Shibata, M. Tateda, S. Seikai, and N. Uchida, “Spatial technique for measuring modal delay differences in a dual-mode optical fiber,” Appl. Opt. 19(9), 1489–1492 (1980).
[PubMed]

Tsubokawa, M.

N. Shibata, M. Tsubokawa, T. Nakashima, and S. Seikai, “Temporal coherence properties of a dispersively propagating beam in a fiber-optic interferometer,” J. Opt. Soc. Am. A 4(3), 494–497 (1987).

N. Shibata, M. Tsubokawa, and S. Seikai, “Measurements of polarization mode dispersion by optical heterodyne detection,” Electron. Lett. 20(25/26), 1055–1057 (1984).

Uchida, N.

Watanabe, K.

N. Shibata, K. Watanabe, M. Ohashi, and K. Aikawa, “Square of the degree of coherence and dispersion-induced pulse broadening in a fiber-optic transmission,” IEICE Commun. Express 5(12), 454–460 (2016).

N. Shibata, K. Watanabe, and M. Ohashi, “Chromatic dispersion diagnosis for the two-modes of few-mode photonic crystal fiber,” IEEE Photonics Technol. Lett. 28(4), 437–440 (2016).

Weiner, A. M.

Winzer, P. J.

Yamamoto, T.

T. Yamamoto, E. Yoshida, K. R. Tamura, K. Yonenaga, and M. Nakazawa, “640-Gbit/s optical TDM transmission over 92 km through a dispersion-managed fiber consisting of single-mode fiber and reverse dispersion fiber,” IEEE Photonics Technol. Lett. 12(3), 353–355 (2000).

Yonenaga, K.

T. Yamamoto, E. Yoshida, K. R. Tamura, K. Yonenaga, and M. Nakazawa, “640-Gbit/s optical TDM transmission over 92 km through a dispersion-managed fiber consisting of single-mode fiber and reverse dispersion fiber,” IEEE Photonics Technol. Lett. 12(3), 353–355 (2000).

Yoshida, E.

T. Yamamoto, E. Yoshida, K. R. Tamura, K. Yonenaga, and M. Nakazawa, “640-Gbit/s optical TDM transmission over 92 km through a dispersion-managed fiber consisting of single-mode fiber and reverse dispersion fiber,” IEEE Photonics Technol. Lett. 12(3), 353–355 (2000).

Zawadzki, R.

Appl. Opt. (2)

Electron. Lett. (4)

H. T. Shang, “Chromatic dispersion measurement by white-light interferometry on meter-length single-mode optical fibers,” Electron. Lett. 17(17), 603–605 (1981).

J. Stone and L. G. Cohen, “Minimum dispersion spectra of single-mode fibers measured with 0.1-ps resolution by white light cross-correlation,” Electron. Lett. 18(16), 716–718 (1982).

N. Shibata, M. Tsubokawa, and S. Seikai, “Measurements of polarization mode dispersion by optical heterodyne detection,” Electron. Lett. 20(25/26), 1055–1057 (1984).

P. L. Francois, F. Alard, and M. Monerie, “Chromatic dispersion measurement from Fourier transform of white-light interference patterns,” Electron. Lett. 23(7), 357–358 (1987).

IEEE J. Quantum Electron. (1)

M. Tateda, N. Shibata, and S. Seikai, “Interferometric method for chromatic dispersion measurement in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-17(3), 404–407 (1981).

IEEE Photonics Technol. Lett. (2)

N. Shibata, K. Watanabe, and M. Ohashi, “Chromatic dispersion diagnosis for the two-modes of few-mode photonic crystal fiber,” IEEE Photonics Technol. Lett. 28(4), 437–440 (2016).

T. Yamamoto, E. Yoshida, K. R. Tamura, K. Yonenaga, and M. Nakazawa, “640-Gbit/s optical TDM transmission over 92 km through a dispersion-managed fiber consisting of single-mode fiber and reverse dispersion fiber,” IEEE Photonics Technol. Lett. 12(3), 353–355 (2000).

IEICE Commun. Express (1)

N. Shibata, K. Watanabe, M. Ohashi, and K. Aikawa, “Square of the degree of coherence and dispersion-induced pulse broadening in a fiber-optic transmission,” IEICE Commun. Express 5(12), 454–460 (2016).

J. Lightwave Technol. (3)

J. Opt. Soc. Am. A (2)

N. Shibata, M. Tsubokawa, T. Nakashima, and S. Seikai, “Temporal coherence properties of a dispersively propagating beam in a fiber-optic interferometer,” J. Opt. Soc. Am. A 4(3), 494–497 (1987).

S. Diddams and J.-C. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. A 13(6), 1120–1129 (1996).

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

W. A. Hamilton, A. G. Klein, and G. I. Opat, “Longitudinal coherence and interferometry in dispersive media,” Phys. Rev. A 28(5), 3149–3152 (1983).

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics, 4th Ed. (Elsevier/Academic, 2007).

S. Katsumoto, Craftsmen of Quantum Mechanics (Maruzen Publisher, 2016).

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2:Temporal Pulse Dynamics in Dispersive, Attenuative Media, (Springer, 2009).

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Figures (7)

Fig. 1
Fig. 1 Interferometer model for measuring the dispersion-induced pulse shape by evaluating the | γ |2.
Fig. 2
Fig. 2 Spectrum distributions of SLDs.
Fig. 3
Fig. 3 Wavelength dependences of D and dD/ for test fibers.
Fig. 4
Fig. 4 Variation of β2 and β3 with wavelength for test fibers.
Fig. 5
Fig. 5 Intensity I and | γ |2 as a function of time T for the interference of waves traversing the two air paths in a Michelson interferometer.
Fig. 6
Fig. 6 Intensity distributions and | γ |2 as a function of time T for FMFs with (A) L = 1.185 m and (B) L = 1.519 m, and for DSFs with (C) L = 1.175 m and (D) L = 1.990 m.
Fig. 7
Fig. 7 Normalized distance dependence of the broadening factors for test fibers.

Tables (1)

Tables Icon

Table 1 Values of τc, β2 and β3 used for calculations

Equations (12)

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

I= I 0 {1+Re(γ)}.
γ(L,2d/c)= S(ω)exp[i{[β(ω)k(ω)]Lω(2d/c)}]dω S(ω)dω ,
S(ω)=exp[ (ω ω c ) 2 /2Δ ω 2 ]/ (2π) 1/2 Δω,
β(ω)= β 0 ( ω c )+(ω ω c ) β 1 ( ω c )+{ β 2 ( ω c )/2} (ω ω c ) 2 +{ β 3 ( ω c )/6} (ω ω c ) 3 = ω c / v p +(ω ω c )/ v g +{ β 2 ( ω c )/2} (ω ω c ) 2 +{ β 3 ( ω c )/6} (ω ω c ) 3 ,
β(ω)k(ω) =(1/ ν P 1/c) ω c +(1/ ν g 1/c)(ω ω c )+{ β 2 ( ω c )/2} (ω ω c ) 2 + { β 3 ( ω c )/6} (ω ω c ) 3 .
γ(L,T)= A π 1/2 exp( iTΩ p Ω 2 + iB Ω 3 3 )dΩ
γ(L,T)= A π 1/2 B 1/3 exp( 2p3BT 3p B 2 ) exp[i{( pBT p B 4/3 )u+ u 3 3 }]du = 2 π 1/2 A B 1/3 exp( 2p3BT 3p B 2 )Ai( pBT p B 4/3 )
Ai(s){ 1 2 π 1/2 s 1/4 exp[ 2 3 s 3/2 ]                    (s) 1 π 1/2 (s) 1/4 cos[ 2 3 (s) 3/2 π 4 ]        (s)
| γ(L,T) | 2 { 1 | B | 2/3 | s | 1/2 exp[ ( τ c T 2 | p | 2 ) 2 {1+( τ c 4 3 β 2 2 L 2 48 | p | 8 ) β 3 LT}]              (T)   2 | B | 2/3 | s | 1/2 exp[2 τ c 2 ( T β 3 L τ c 4 3 β 2 2 L 2 3 β 2 2 L 2 )][cosh{ 4 3 Im (s) 3/2 }+ cos{ 4 3 Re (s) 3/2 π 4 }]                                                               (T)
| γ(L,T) | 2 1 (1 2 β 3 LT τ c 4 ) 1/2 exp[ ( T τ c ) 2 (1+ β 3 LT 3 τ c 4 )]
| γ(L,T) | 2 exp[ ( T τ c ) 2 ]
σ/ σ 0 = { 1+ ( L/ L D2 ) 2 + ( L/2 L D3 ) 2 } 1/2 .

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