Abstract

Very large mode area, active optical fibers with a low high order mode content in the actively doped core region were designed by removing the inner cladding symmetry. The relevance of the numerical approach is demonstrated here by the investigation of a standard air-silica Large Pitch Fiber, used as a reference. A detailed study of all-solid structures is also performed. Finally, we propose new kinds of geometry for 50 μm core, all-solid microstructured fibers enabling a robust singlemode laser emission from 400 nm to 2200 nm.

© 2013 OSA

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References

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  1. D. Gapontsev, “6kW CW single mode ytterbium fiber laser in all-fiber format,” Proc. Solid State and Diode Laser Technology Review (2008).
  2. F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26mJ, 130W Q-switched fiber laser system with near-diffraction-limited beam quality,” Opt. Lett.37(6), 1073–1075 (2012).
    [CrossRef] [PubMed]
  3. P. F. Moulton, T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, and G. Rines, “1-kW, all-glass Tm : fiber laser,” Proc. of SPIE 7580, paper 7580112 (2010).
  4. S. Jetschke, S. Unger, A. Schwuchow, M. Leich, and J. Kirchhof, “Efficient Yb laser fibers with low photodark-ening by optimization of the core composition,” Opt. Express16(20), 15540–15545 (2008).
    [CrossRef] [PubMed]
  5. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011).
    [CrossRef] [PubMed]
  6. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 3997–4008 (2013).
    [CrossRef]
  7. F. Stutzki, F. Jansen, C. Jauregui, J. Limpert, and A. Tünnermann, “Non-hexagonal large-pitch fibers for enhanced mode discrimination,” Opt. Express19(13), 12081–12086 (2011).
    [CrossRef] [PubMed]
  8. M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express20(7), 7263–7273 (2012).
    [CrossRef] [PubMed]
  9. F. Jansen, F. Stutzki, H. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “High-power thermally guiding index-antiguiding-core fibers,” Opt. Lett.38(4), 510–512 (2013).
    [CrossRef] [PubMed]
  10. J. Limpert, F. Stutzki, F. Jansen, H. J. Otto, T. Eidam, C. Jauregui, and A. Tünnermann, “Yb-doped large-pitch fibres: effective single-mode operation based on higher-order mode delocalisation,” Light: Science & Applications1, e8, (2012).
    [CrossRef]
  11. F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Avoided crossings in photonic crystal fibers,” Opt. Express19(14), 13578–13589 (2011).
    [CrossRef] [PubMed]
  12. R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
    [CrossRef]
  13. J. M. Fini, “Improved symmetry analysis of many-moded microstructure optical fibers,” J. Opt. Soc. Am. B21(8), 1431–1436 (2004).
    [CrossRef]
  14. P. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides,” IEEE Trans. Microwave Theory Tech.23(5), 421–433 (1975).
    [CrossRef]
  15. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express11(11), 1310–1321 (2003).
    [CrossRef] [PubMed]
  16. M. Steel, “Reflection symmetry and mode transversality in microstructured fibers,” Opt. Express12(8), 1497–509 (2004).
    [CrossRef] [PubMed]
  17. P. M. Agruzov, K. V Dukelskii, and V. S. Shevandin, “Three types of microstructured large core fibers : development and investigation,” Conference on Lasers and Electro-Optics, paper CE.P.28 (2009).
  18. V. V Demidov, K. V Dukelskii, and V. S. Shevandin, “Novel bend-resistant design of single-mode microstructured fibers,” Conference on Lasers and Electro-Optics, paper CE.P35 (2011).
  19. M.-Y. Chen, Y. Li, J. Zhou, and Y.-K. Zhang, “Design of asymmetric large-mode area optical fiber with low bending loss,” J. of Lightwave Technol.31(3), 476–481 (2013).
    [CrossRef]
  20. J. Limpert, “Large-pitch fibers: pushing very large mode areas to highest powers,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper T2A.1 (2012).
    [CrossRef]

2013 (3)

F. Jansen, F. Stutzki, H. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “High-power thermally guiding index-antiguiding-core fibers,” Opt. Lett.38(4), 510–512 (2013).
[CrossRef] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 3997–4008 (2013).
[CrossRef]

M.-Y. Chen, Y. Li, J. Zhou, and Y.-K. Zhang, “Design of asymmetric large-mode area optical fiber with low bending loss,” J. of Lightwave Technol.31(3), 476–481 (2013).
[CrossRef]

2012 (3)

2011 (3)

2010 (1)

P. F. Moulton, T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, and G. Rines, “1-kW, all-glass Tm : fiber laser,” Proc. of SPIE 7580, paper 7580112 (2010).

2008 (2)

2004 (2)

2003 (1)

1975 (1)

P. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides,” IEEE Trans. Microwave Theory Tech.23(5), 421–433 (1975).
[CrossRef]

Agruzov, P. M.

P. M. Agruzov, K. V Dukelskii, and V. S. Shevandin, “Three types of microstructured large core fibers : development and investigation,” Conference on Lasers and Electro-Optics, paper CE.P.28 (2009).

Alkeskjold, T. T.

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 3997–4008 (2013).
[CrossRef]

M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express20(7), 7263–7273 (2012).
[CrossRef] [PubMed]

Benoit, A.

R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
[CrossRef]

Broeng, J.

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 3997–4008 (2013).
[CrossRef]

Chen, M.-Y.

M.-Y. Chen, Y. Li, J. Zhou, and Y.-K. Zhang, “Design of asymmetric large-mode area optical fiber with low bending loss,” J. of Lightwave Technol.31(3), 476–481 (2013).
[CrossRef]

Dauliat, R.

R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
[CrossRef]

Demidov, V. V

V. V Demidov, K. V Dukelskii, and V. S. Shevandin, “Novel bend-resistant design of single-mode microstructured fibers,” Conference on Lasers and Electro-Optics, paper CE.P35 (2011).

Dukelskii, K. V

V. V Demidov, K. V Dukelskii, and V. S. Shevandin, “Novel bend-resistant design of single-mode microstructured fibers,” Conference on Lasers and Electro-Optics, paper CE.P35 (2011).

P. M. Agruzov, K. V Dukelskii, and V. S. Shevandin, “Three types of microstructured large core fibers : development and investigation,” Conference on Lasers and Electro-Optics, paper CE.P.28 (2009).

Ehrenreich, T.

P. F. Moulton, T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, and G. Rines, “1-kW, all-glass Tm : fiber laser,” Proc. of SPIE 7580, paper 7580112 (2010).

Eidam, T.

J. Limpert, F. Stutzki, F. Jansen, H. J. Otto, T. Eidam, C. Jauregui, and A. Tünnermann, “Yb-doped large-pitch fibres: effective single-mode operation based on higher-order mode delocalisation,” Light: Science & Applications1, e8, (2012).
[CrossRef]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011).
[CrossRef] [PubMed]

Fini, J. M.

Gaponov, D. A.

R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
[CrossRef]

Gapontsev, D.

D. Gapontsev, “6kW CW single mode ytterbium fiber laser in all-fiber format,” Proc. Solid State and Diode Laser Technology Review (2008).

Grimm, S.

R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
[CrossRef]

Guobin, R.

Hansen, K. R.

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 3997–4008 (2013).
[CrossRef]

Jansen, F.

Jauregui, C.

Jetschke, S.

S. Jetschke, S. Unger, A. Schwuchow, M. Leich, and J. Kirchhof, “Efficient Yb laser fibers with low photodark-ening by optimization of the core composition,” Opt. Express16(20), 15540–15545 (2008).
[CrossRef] [PubMed]

R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
[CrossRef]

Jørgensen, M. M.

Kirchhof, J.

Lægsgaard, J.

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 3997–4008 (2013).
[CrossRef]

M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express20(7), 7263–7273 (2012).
[CrossRef] [PubMed]

Laurila, M.

Leich, M.

Leveille, R.

P. F. Moulton, T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, and G. Rines, “1-kW, all-glass Tm : fiber laser,” Proc. of SPIE 7580, paper 7580112 (2010).

Li, Y.

M.-Y. Chen, Y. Li, J. Zhou, and Y.-K. Zhang, “Design of asymmetric large-mode area optical fiber with low bending loss,” J. of Lightwave Technol.31(3), 476–481 (2013).
[CrossRef]

Liem, A.

Limpert, J.

F. Jansen, F. Stutzki, H. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “High-power thermally guiding index-antiguiding-core fibers,” Opt. Lett.38(4), 510–512 (2013).
[CrossRef] [PubMed]

J. Limpert, F. Stutzki, F. Jansen, H. J. Otto, T. Eidam, C. Jauregui, and A. Tünnermann, “Yb-doped large-pitch fibres: effective single-mode operation based on higher-order mode delocalisation,” Light: Science & Applications1, e8, (2012).
[CrossRef]

F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26mJ, 130W Q-switched fiber laser system with near-diffraction-limited beam quality,” Opt. Lett.37(6), 1073–1075 (2012).
[CrossRef] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011).
[CrossRef] [PubMed]

F. Stutzki, F. Jansen, C. Jauregui, J. Limpert, and A. Tünnermann, “Non-hexagonal large-pitch fibers for enhanced mode discrimination,” Opt. Express19(13), 12081–12086 (2011).
[CrossRef] [PubMed]

F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Avoided crossings in photonic crystal fibers,” Opt. Express19(14), 13578–13589 (2011).
[CrossRef] [PubMed]

J. Limpert, “Large-pitch fibers: pushing very large mode areas to highest powers,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper T2A.1 (2012).
[CrossRef]

Majid, I.

P. F. Moulton, T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, and G. Rines, “1-kW, all-glass Tm : fiber laser,” Proc. of SPIE 7580, paper 7580112 (2010).

McIsaac, P.

P. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides,” IEEE Trans. Microwave Theory Tech.23(5), 421–433 (1975).
[CrossRef]

Moulton, P. F.

P. F. Moulton, T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, and G. Rines, “1-kW, all-glass Tm : fiber laser,” Proc. of SPIE 7580, paper 7580112 (2010).

Otto, H.

Otto, H. J.

J. Limpert, F. Stutzki, F. Jansen, H. J. Otto, T. Eidam, C. Jauregui, and A. Tünnermann, “Yb-doped large-pitch fibres: effective single-mode operation based on higher-order mode delocalisation,” Light: Science & Applications1, e8, (2012).
[CrossRef]

Otto, H.-J.

Petersen, S. R.

Rines, G.

P. F. Moulton, T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, and G. Rines, “1-kW, all-glass Tm : fiber laser,” Proc. of SPIE 7580, paper 7580112 (2010).

Roy, P.

R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
[CrossRef]

Salin, F.

R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
[CrossRef]

Schmidt, O.

Schreiber, T.

Schuster, K.

R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
[CrossRef]

Schwuchow, A.

Shevandin, V. S.

P. M. Agruzov, K. V Dukelskii, and V. S. Shevandin, “Three types of microstructured large core fibers : development and investigation,” Conference on Lasers and Electro-Optics, paper CE.P.28 (2009).

V. V Demidov, K. V Dukelskii, and V. S. Shevandin, “Novel bend-resistant design of single-mode microstructured fibers,” Conference on Lasers and Electro-Optics, paper CE.P35 (2011).

Shuisheng, J.

Shuqin, L.

Steel, M.

Stutzki, F.

Tankala, K.

P. F. Moulton, T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, and G. Rines, “1-kW, all-glass Tm : fiber laser,” Proc. of SPIE 7580, paper 7580112 (2010).

Tünnermann, A.

Unger, S.

Wirth, C.

Zhang, Y.-K.

M.-Y. Chen, Y. Li, J. Zhou, and Y.-K. Zhang, “Design of asymmetric large-mode area optical fiber with low bending loss,” J. of Lightwave Technol.31(3), 476–481 (2013).
[CrossRef]

Zhi, W.

Zhou, J.

M.-Y. Chen, Y. Li, J. Zhou, and Y.-K. Zhang, “Design of asymmetric large-mode area optical fiber with low bending loss,” J. of Lightwave Technol.31(3), 476–481 (2013).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides,” IEEE Trans. Microwave Theory Tech.23(5), 421–433 (1975).
[CrossRef]

J. of Lightwave Technol. (1)

M.-Y. Chen, Y. Li, J. Zhou, and Y.-K. Zhang, “Design of asymmetric large-mode area optical fiber with low bending loss,” J. of Lightwave Technol.31(3), 476–481 (2013).
[CrossRef]

J. Opt. Soc. Am. B (1)

Light: Science & Applications (1)

J. Limpert, F. Stutzki, F. Jansen, H. J. Otto, T. Eidam, C. Jauregui, and A. Tünnermann, “Yb-doped large-pitch fibres: effective single-mode operation based on higher-order mode delocalisation,” Light: Science & Applications1, e8, (2012).
[CrossRef]

Opt. Express (8)

F. Jansen, F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Avoided crossings in photonic crystal fibers,” Opt. Express19(14), 13578–13589 (2011).
[CrossRef] [PubMed]

S. Jetschke, S. Unger, A. Schwuchow, M. Leich, and J. Kirchhof, “Efficient Yb laser fibers with low photodark-ening by optimization of the core composition,” Opt. Express16(20), 15540–15545 (2008).
[CrossRef] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011).
[CrossRef] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 3997–4008 (2013).
[CrossRef]

F. Stutzki, F. Jansen, C. Jauregui, J. Limpert, and A. Tünnermann, “Non-hexagonal large-pitch fibers for enhanced mode discrimination,” Opt. Express19(13), 12081–12086 (2011).
[CrossRef] [PubMed]

M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express20(7), 7263–7273 (2012).
[CrossRef] [PubMed]

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express11(11), 1310–1321 (2003).
[CrossRef] [PubMed]

M. Steel, “Reflection symmetry and mode transversality in microstructured fibers,” Opt. Express12(8), 1497–509 (2004).
[CrossRef] [PubMed]

Opt. Lett. (2)

Proc. of SPIE 7580 (1)

P. F. Moulton, T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, and G. Rines, “1-kW, all-glass Tm : fiber laser,” Proc. of SPIE 7580, paper 7580112 (2010).

Proc. Solid State and Diode Laser Technology Review (1)

D. Gapontsev, “6kW CW single mode ytterbium fiber laser in all-fiber format,” Proc. Solid State and Diode Laser Technology Review (2008).

Other (4)

J. Limpert, “Large-pitch fibers: pushing very large mode areas to highest powers,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper T2A.1 (2012).
[CrossRef]

P. M. Agruzov, K. V Dukelskii, and V. S. Shevandin, “Three types of microstructured large core fibers : development and investigation,” Conference on Lasers and Electro-Optics, paper CE.P.28 (2009).

V. V Demidov, K. V Dukelskii, and V. S. Shevandin, “Novel bend-resistant design of single-mode microstructured fibers,” Conference on Lasers and Electro-Optics, paper CE.P35 (2011).

R. Dauliat, D. A. Gaponov, A. Benoit, F. Salin, K. Schuster, S. Jetschke, S. Grimm, and P. Roy, “Ytterbium doped all solid large pitch fiber made from powder sintering and vitrification,” International Conference on Fibre Optics and Photonics - OSA Technical Digest, paper TPo.7 (2012).
[CrossRef]

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Figures (2)

Fig. 1.
Fig. 1.

(a) 2D refractive index repartition of the air/silica LPF described in [2]. The gain region is in red, the pure silica in blue and the air holes in yellow. (b) Overlap factor of the fundamental mode (solid lines) and modal discrimination (dashed lines) computed for the air/silica LPF. Calculations have been done for three air-clad diameters: 170 μm (black), 180 μm (red), and 190 μm (blue) and various ratio d/Λ. Insets: computed intensity distributions corresponding to the fundamental mode coupling (top), and the most disturbing HOM (bottom).

Fig. 2
Fig. 2

Spectral evolution of the modal discrimination for the whole 50 μm all-solid structures proposed: (a) LPF structures (air/silica LPF and structures n°2 and 4), (b) Vortex (n°3), hexagonal symmetry free (n°5) and aperiodic quasi-pentagonal (n°6) fibers. The latter two fibers feature two different refractive index contrasts.

Tables (2)

Tables Icon

Table 1 On the left side, the structures are numbered and named. The middle column depicts the distribution of the fiber refractive index, based on which the actively (passively) doped region is in red (clear blue) and pure silica in dark blue. On the right side, a representation of the modal confinement for the first 300 guided modes is shown for different core diameters: 50 μm (in black), 60 μm (in red) and 70 μm (in blue). The intensity distributions of the FM and the most disturbing HOM are depicted in the inset.

Tables Icon

Table 2 Similar to Table 1 with three structures based on symmetry reduction.

Equations (2)

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

O F = A d | E | 2 dS A p | E | 2 dS
Δ O F = O F F M O F H O M

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