Abstract

The unintentional birefringence induced by the irregular structure in photonic crystal fibers is analyzed numerically using the plane wave expansion method. The statistical correlations between the birefringence and the various irregularities are obtained. The birefringence is found to be largely dependent on the fiber design parameters as well as the degree of the irregularity. And the large pitch and the small air hole make the fiber less sensitive to the structural irregularity, which is successfully explained by the simple perturbation theory. The accuracy of our analyses is confirmed by the detailed investigation of computational errors. This study provides the essential information for the characterization and the design of low birefringence photonic crystal fibers.

© 2003 Optical Society of America

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References

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  1. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
    [CrossRef] [PubMed]
  2. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
    [CrossRef]
  3. J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. 34, 1347–1348 (1998).
    [CrossRef]
  4. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 251325–1327 (2000).
    [CrossRef]
  5. Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
    [CrossRef]
  6. T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.
  7. A. Peyrilloux, T. Chartier, A. Hideur, L. Berthelot, G. Mélin, S. Lempereur, D. Pagnoux, and P. Roy, “Theoretical and experimental study of the birefringence of a photonic crystal fiber,” J. Lightwave Technol. 21, 536–539 (2003).
    [CrossRef]
  8. M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001).
    [CrossRef]
  9. Se-Heon Kim and Yong-Hee Lee, “Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes,” IEEE J. Quantum Electron. 39, 1081–1086 (2003).
    [CrossRef]
  10. S. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. R. Simonsen, “High-birefringent photonic crystal fiber,” in Proc. Optical Fiber Communication Conference 2001, Vol. 54 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1900), pp. TuM2-1 –TuM2-3.
  11. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
    [CrossRef] [PubMed]
  12. Yong-Jae Lee, Dae-Sung Song, Se-Heon Kim, Jun Huh, and Yong-Hee Lee, “Modal characteristics of photonic crystal fibers,” J. Opt. Soc. Korea 7, 47–52 (2003).
    [CrossRef]
  13. Masanori Koshiba and Kunimasa Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315 (2001).
    [CrossRef]
  14. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 18.

2003 (3)

2001 (4)

Masanori Koshiba and Kunimasa Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315 (2001).
[CrossRef]

Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
[CrossRef] [PubMed]

M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001).
[CrossRef]

2000 (2)

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 251325–1327 (2000).
[CrossRef]

1998 (1)

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. 34, 1347–1348 (1998).
[CrossRef]

1997 (1)

Arriaga, J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 251325–1327 (2000).
[CrossRef]

Berthelot, L.

Birks, T. A.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 251325–1327 (2000).
[CrossRef]

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. 34, 1347–1348 (1998).
[CrossRef]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

Bjarklev, A.

S. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. R. Simonsen, “High-birefringent photonic crystal fiber,” in Proc. Optical Fiber Communication Conference 2001, Vol. 54 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1900), pp. TuM2-1 –TuM2-3.

Bjarklev, Anders

Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
[CrossRef]

Botten, L C.

Broeng, J.

S. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. R. Simonsen, “High-birefringent photonic crystal fiber,” in Proc. Optical Fiber Communication Conference 2001, Vol. 54 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1900), pp. TuM2-1 –TuM2-3.

Broeng, Jes

Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
[CrossRef]

Chartier, T.

Cregan, R. F.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. 34, 1347–1348 (1998).
[CrossRef]

de Sterke, C. Martijn

deSandro, J. -P.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. 34, 1347–1348 (1998).
[CrossRef]

Gisin, N.

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

Hansen, Theis P.

Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
[CrossRef]

Hideur, A.

Huh, Jun

Jensen, J. R.

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

Jensen, Jacob Riis

Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kim, Se-Heon

Yong-Jae Lee, Dae-Sung Song, Se-Heon Kim, Jun Huh, and Yong-Hee Lee, “Modal characteristics of photonic crystal fibers,” J. Opt. Soc. Korea 7, 47–52 (2003).
[CrossRef]

Se-Heon Kim and Yong-Hee Lee, “Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes,” IEEE J. Quantum Electron. 39, 1081–1086 (2003).
[CrossRef]

Knight, J. C.

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 251325–1327 (2000).
[CrossRef]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. 34, 1347–1348 (1998).
[CrossRef]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

Knudsen, E.

S. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. R. Simonsen, “High-birefringent photonic crystal fiber,” in Proc. Optical Fiber Communication Conference 2001, Vol. 54 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1900), pp. TuM2-1 –TuM2-3.

Knudsen, Erik

Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
[CrossRef]

Koshiba, Masanori

Masanori Koshiba and Kunimasa Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315 (2001).
[CrossRef]

Lee, Yong-Hee

Yong-Jae Lee, Dae-Sung Song, Se-Heon Kim, Jun Huh, and Yong-Hee Lee, “Modal characteristics of photonic crystal fibers,” J. Opt. Soc. Korea 7, 47–52 (2003).
[CrossRef]

Se-Heon Kim and Yong-Hee Lee, “Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes,” IEEE J. Quantum Electron. 39, 1081–1086 (2003).
[CrossRef]

Lee, Yong-Jae

Legré, M.

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

Lempereur, S.

Libori, S. B.

S. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. R. Simonsen, “High-birefringent photonic crystal fiber,” in Proc. Optical Fiber Communication Conference 2001, Vol. 54 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1900), pp. TuM2-1 –TuM2-3.

Libori, Stig E. B.

Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 18.

Ludvigsen, H.

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

Mangan, B. J.

McPhedran, R. C.

Mélin, G.

Niemi, T.

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

Ortigosa-Blanch, A.

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 251325–1327 (2000).
[CrossRef]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

Pagnoux, D.

Petersson, A.

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

Peyrilloux, A.

Roy, P.

Russel, P. St. J.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. 34, 1347–1348 (1998).
[CrossRef]

Russell, P. St. J.

Saitoh, Kunimasa

Masanori Koshiba and Kunimasa Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315 (2001).
[CrossRef]

Scholder, F.

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

Simonsen, H. R.

S. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. R. Simonsen, “High-birefringent photonic crystal fiber,” in Proc. Optical Fiber Communication Conference 2001, Vol. 54 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1900), pp. TuM2-1 –TuM2-3.

Simonsen, Harald

Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
[CrossRef]

Skovgaard, P. M. W.

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 18.

Song, Dae-Sung

Steel, M. J.

Wadsworth, W. J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 251325–1327 (2000).
[CrossRef]

Wegmuller, M.

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

White, T. P.

Electron. Lett. (1)

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. 34, 1347–1348 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

Se-Heon Kim and Yong-Hee Lee, “Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes,” IEEE J. Quantum Electron. 39, 1081–1086 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

Masanori Koshiba and Kunimasa Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315 (2001).
[CrossRef]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Korea (1)

Opt. Express (1)

Opt. Lett. (3)

Other (3)

T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in Proc. European Conference on Optical Communication2002, paper M.S1.09.

S. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. R. Simonsen, “High-birefringent photonic crystal fiber,” in Proc. Optical Fiber Communication Conference 2001, Vol. 54 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1900), pp. TuM2-1 –TuM2-3.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 18.

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Figures (6)

Fig. 1.
Fig. 1.

Photonic crystal fiber structure.

Fig. 2.
Fig. 2.

(a) The mode indices for two polarizations versus grid resolution. (b) Numerical birefringence error versus grid resolution at various Λ/λ.

Fig. 3.
Fig. 3.

Numerical birefringence error versus grid resolution at various Λ/λ, calculated with the rectangular supercell.

Fig. 4.
Fig. 4.

Birefringence due to variation of hole diameters. (a) Probability distribution of each hole diameter, d: d 0, original hole diameter; δd, standard deviation of d. (b) Birefringence of 20 PCF samples with δd/d 0=0.2. Two insets are the structures of two PCFs with the largest and the smallest birefringence. (c) Birefringence of PCFs at various degrees of hole diameter variation. The marker and error bar denote the mean and the standard deviation, respectively, of birefringence distribution. The dotted lines are obtained by linear fitting of mean values.

Fig. 5.
Fig. 5.

Birefringence due to variation of hole positions. (a) Probability distribution of offset q in each hole: δq, standard deviation of q. (b) Birefringence of PCFs at various degrees of hole position variation. The marker and error bar denote the mean and the standard deviation, respectively, of birefringence distribution. The dotted lines were obtained by linear fitting of mean values.

Fig. 6.
Fig. 6.

Optical intensity profile as a function of x at y=0, for (a) d/Λ=0.46 and (b) d/Λ=0.36. The dotted vertical lines denote the boundary of air holes. The solid and broken curves are intensity profiles for Λ/λ=2.09 and 10.35, respectively.

Tables (2)

Tables Icon

Table 1. The fitting coefficients A and B of Eq. (1), obtained from the data in Fig. 4 (c)

Tables Icon

Table 2. The fitting coefficients A and B of Eq. (1), obtained from the data in Fig. 5 (b)

Equations (5)

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

log 10 ( Δ n ) = A · log 10 ( δ d d · 100 ) + B
Δ n = ( δ d d · 100 ) A · 10 B
S = π ( d 2 ) 2
δ S d π 2 d · δ d = π 2 d 2 · ( δ d d )
δ S q 2 d · δ q = 2 d Λ · ( δ q Λ )

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