Abstract

We designed a new kind of mode-locked fiber laser based on fiber arrays, where the central core is doped. A theoretical model is given for an all-fiber self-starting mode-locked laser based on this kind of doped fiber array. Two different kinds of fiber lasers with negative dispersion and positive dispersion are simulated and discussed. The stable mode-locked pulses are generated from initial noise conditions by the realistic parameters. The process of self-starting mode-locking multipulse transition and the relationship between the energy of the central core and the propagation distance of the pulses are discussed. Finally, we analyze the difference between the averaged mode-locked laser and the discrete mode-locked laser.

© 2014 Optical Society of America

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  1. J. L. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005).
    [CrossRef]
  2. J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).
    [CrossRef]
  3. J. Proctor and J. N. Kutz, “Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933–8950 (2005).
    [CrossRef]
  4. M. O. Williams, E. Shlizerman, and J. N. Kutz, “The multi-pulsing transition in mode-locked lasers: a low-dimensional approach using waveguide arrays,” J. Opt. Soc. Am. B 27, 2471–2481 (2010).
    [CrossRef]
  5. J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).
    [CrossRef]
  6. D. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
    [CrossRef]
  7. D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
    [CrossRef]
  8. Q. Chao, D. D. Hudson, J. N. Kutz, and S. T. Cundiff, “Waveguide array fiber laser,” IEEE Photon. J. 4, 1438–1442 (2012).
    [CrossRef]
  9. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
    [CrossRef]
  10. B. G. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008).
    [CrossRef]
  11. J. Proctor and J. N. Kutz, “Averaged models for passive mode-locking using nonlinear mode-coupling,” Math. Comput. Simul. 74, 333–342 (2007).
    [CrossRef]
  12. A. Winter, F. O. Ilday, and B. Steffen, “Femtosecond Yb-doped fiber laser system at 1  μm of wavelength with 100-nm bandwidth and variable pulse structure for accelerator diagnostics,” in Proceedings of the 8th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators (DIPAC 2007), Venice, Italy (2007), paper WEPB03.
  13. J. Hamazaki, N. Sekine, and I. Hosako, “Mode-locked Yb-doped fiber ring-laser for use as a pump pulse source of THz-TDS,” in Proceedings of the Progress in Electromagnetics Research Symposium (PIERS 2012) (2012), pp. 482–485.
  14. D. Ma, Y. Cai, C. Zhou, W. J. Zong, L. L. Chen, and Z. G. Zhang, “37.4  fs pulse generation in an Er:fiber laser at a 225  MHz repetition rate,” Opt. Lett. 35, 2858–2860 (2010).
    [CrossRef]
  15. Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66  fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
    [CrossRef]

2014

Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66  fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
[CrossRef]

2012

D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
[CrossRef]

Q. Chao, D. D. Hudson, J. N. Kutz, and S. T. Cundiff, “Waveguide array fiber laser,” IEEE Photon. J. 4, 1438–1442 (2012).
[CrossRef]

2010

2008

2007

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).
[CrossRef]

J. Proctor and J. N. Kutz, “Averaged models for passive mode-locking using nonlinear mode-coupling,” Math. Comput. Simul. 74, 333–342 (2007).
[CrossRef]

2005

2004

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Bale, B. G.

B. G. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008).
[CrossRef]

Bartelt, H.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Cai, Y.

Chao, Q.

Q. Chao, D. D. Hudson, J. N. Kutz, and S. T. Cundiff, “Waveguide array fiber laser,” IEEE Photon. J. 4, 1438–1442 (2012).
[CrossRef]

D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
[CrossRef]

Chen, L. L.

Christodoulides, D. N.

D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
[CrossRef]

D. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
[CrossRef]

Conti, C.

Cundiff, S. T.

Q. Chao, D. D. Hudson, J. N. Kutz, and S. T. Cundiff, “Waveguide array fiber laser,” IEEE Photon. J. 4, 1438–1442 (2012).
[CrossRef]

D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
[CrossRef]

D. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
[CrossRef]

Dong, X. Z.

Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66  fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
[CrossRef]

Farnum, E.

B. G. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008).
[CrossRef]

Hamazaki, J.

J. Hamazaki, N. Sekine, and I. Hosako, “Mode-locked Yb-doped fiber ring-laser for use as a pump pulse source of THz-TDS,” in Proceedings of the Progress in Electromagnetics Research Symposium (PIERS 2012) (2012), pp. 482–485.

Hosako, I.

J. Hamazaki, N. Sekine, and I. Hosako, “Mode-locked Yb-doped fiber ring-laser for use as a pump pulse source of THz-TDS,” in Proceedings of the Progress in Electromagnetics Research Symposium (PIERS 2012) (2012), pp. 482–485.

Hudson, D. D.

D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
[CrossRef]

Q. Chao, D. D. Hudson, J. N. Kutz, and S. T. Cundiff, “Waveguide array fiber laser,” IEEE Photon. J. 4, 1438–1442 (2012).
[CrossRef]

D. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
[CrossRef]

Ilday, F. O.

A. Winter, F. O. Ilday, and B. Steffen, “Femtosecond Yb-doped fiber laser system at 1  μm of wavelength with 100-nm bandwidth and variable pulse structure for accelerator diagnostics,” in Proceedings of the 8th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators (DIPAC 2007), Venice, Italy (2007), paper WEPB03.

Kobelke, J.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Kutz, J. N.

Q. Chao, D. D. Hudson, J. N. Kutz, and S. T. Cundiff, “Waveguide array fiber laser,” IEEE Photon. J. 4, 1438–1442 (2012).
[CrossRef]

D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
[CrossRef]

M. O. Williams, E. Shlizerman, and J. N. Kutz, “The multi-pulsing transition in mode-locked lasers: a low-dimensional approach using waveguide arrays,” J. Opt. Soc. Am. B 27, 2471–2481 (2010).
[CrossRef]

D. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
[CrossRef]

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).
[CrossRef]

B. G. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008).
[CrossRef]

J. Proctor and J. N. Kutz, “Averaged models for passive mode-locking using nonlinear mode-coupling,” Math. Comput. Simul. 74, 333–342 (2007).
[CrossRef]

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).
[CrossRef]

J. L. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005).
[CrossRef]

J. Proctor and J. N. Kutz, “Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933–8950 (2005).
[CrossRef]

Lederer, F.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Ma, D.

Morandotti, R.

D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
[CrossRef]

D. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
[CrossRef]

Nolte, S.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Pertsch, T.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Peschel, U.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Proctor, J.

J. Proctor and J. N. Kutz, “Averaged models for passive mode-locking using nonlinear mode-coupling,” Math. Comput. Simul. 74, 333–342 (2007).
[CrossRef]

J. Proctor and J. N. Kutz, “Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933–8950 (2005).
[CrossRef]

Proctor, J. L.

Sandstede, B.

Schibli, T. R.

D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
[CrossRef]

D. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
[CrossRef]

Schuster, K.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Sekine, N.

J. Hamazaki, N. Sekine, and I. Hosako, “Mode-locked Yb-doped fiber ring-laser for use as a pump pulse source of THz-TDS,” in Proceedings of the Progress in Electromagnetics Research Symposium (PIERS 2012) (2012), pp. 482–485.

Shish, K.

Shlizerman, E.

Song, Y. R.

Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66  fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
[CrossRef]

Steffen, B.

A. Winter, F. O. Ilday, and B. Steffen, “Femtosecond Yb-doped fiber laser system at 1  μm of wavelength with 100-nm bandwidth and variable pulse structure for accelerator diagnostics,” in Proceedings of the 8th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators (DIPAC 2007), Venice, Italy (2007), paper WEPB03.

Tian, J. R.

Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66  fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
[CrossRef]

Trillo, S.

Tunnermann, A.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Wang, Y. G.

Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66  fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
[CrossRef]

Williams, M. O.

Winter, A.

A. Winter, F. O. Ilday, and B. Steffen, “Femtosecond Yb-doped fiber laser system at 1  μm of wavelength with 100-nm bandwidth and variable pulse structure for accelerator diagnostics,” in Proceedings of the 8th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators (DIPAC 2007), Venice, Italy (2007), paper WEPB03.

Yu, Z. H.

Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66  fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
[CrossRef]

Zhang, X.

Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66  fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
[CrossRef]

Zhang, Z. G.

Zhou, C.

Zong, W. J.

IEEE J. Quantum Electron.

B. G. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008).
[CrossRef]

IEEE Photon. J.

Q. Chao, D. D. Hudson, J. N. Kutz, and S. T. Cundiff, “Waveguide array fiber laser,” IEEE Photon. J. 4, 1438–1442 (2012).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys.

Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66  fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
[CrossRef]

Math. Comput. Simul.

J. Proctor and J. N. Kutz, “Averaged models for passive mode-locking using nonlinear mode-coupling,” Math. Comput. Simul. 74, 333–342 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixed-point attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012).
[CrossRef]

Phys. Rev. Lett.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Other

A. Winter, F. O. Ilday, and B. Steffen, “Femtosecond Yb-doped fiber laser system at 1  μm of wavelength with 100-nm bandwidth and variable pulse structure for accelerator diagnostics,” in Proceedings of the 8th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators (DIPAC 2007), Venice, Italy (2007), paper WEPB03.

J. Hamazaki, N. Sekine, and I. Hosako, “Mode-locked Yb-doped fiber ring-laser for use as a pump pulse source of THz-TDS,” in Proceedings of the Progress in Electromagnetics Research Symposium (PIERS 2012) (2012), pp. 482–485.

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Figures (12)

Fig. 1.
Fig. 1.

Cross section of a hexagonal fiber array.

Fig. 2.
Fig. 2.

Configuration of an all-fiber DFAL.

Fig. 3.
Fig. 3.

Self-starting mode-locking evolution dynamics in the Er-DFAL. (a)–(d) Intensity versus time and distance of the central core |A1|2, the noncentral core |A2|2, |A4|2, and |A7|2, respectively. g0=1.15.

Fig. 4.
Fig. 4.

Mode-locking with single, double, and triple pulses in the Er-DFAL. (a) and (b) Mode-locking evolution dynamics and the energy of the central core as a function of distance corresponding to g0 as 1.15. (c) and (d) Mode-locking evolution dynamics as a function of distance corresponding to g0 as 1.22 and 1.27.

Fig. 5.
Fig. 5.

Distribution of peak amplitude on cross section of the Er-DFAL from the initiation of self-starting mode-locking. The figures correspond to the fiber positions at 0, 0.4, 0.8, 1.2, 1.4, 1.6, 2, and 10 m, respectively.

Fig. 6.
Fig. 6.

Self-starting mode-locking evolution dynamics in the Yb-DFAL. (a), (b), (c), and (d) Intensity versus time and distance of the central core |A1|2, the noncentral core |A2|2, |A4|2, and |A7|2, respectively. g0=1.43.

Fig. 7.
Fig. 7.

Mode-locking evolution dynamics with single, double, and triple pulses in the Yb-DFAL. (a) and (b) Mode-locking evolution dynamics and the energy of the central core as a function of distance corresponding to g0 as 1.43. (c) and (d) Mode-locking evolution dynamics when g0 are 1.75 and 2.1.

Fig. 8.
Fig. 8.

Distribution of the peak amplitude on cross section of the Yb-DFAL from the initiation of self-starting mode-locking. The figures correspond to the fiber positions at 0, 0.4, 0.8, 1.2, 2, 3, 4, and 10, respectively.

Fig. 9.
Fig. 9.

(a) Simulated pulse shapes and the phases of the DFALs with negative dispersion and positive dispersion systems and the fitted results. (b) The corresponding spectra.

Fig. 10.
Fig. 10.

Discrete mode-locked fiber laser based on a gain fiber and a fiber array.

Fig. 11.
Fig. 11.

(a) Mode-locked pulses of the averaged mode-locked fiber array laser. (b) and (c) Pulses at the end of the fiber array and the end of the gain fiber in the discrete mode-locked laser, respectively. (d) The whole evolution dynamics of pulses in different positions of the discrete mode-locked laser.

Fig. 12.
Fig. 12.

Pulse shape and corresponding spectrum in the middle of the gain fiber for the discrete model.

Equations (11)

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

A1Z=sgn(β)i22A1T2+i|A1|2A1+3iC¯(A2+A3)+g(Z)(1+τ2T2)A1r1A1,
g(Z)=2g01+|A1|2/e0,
A2Z=iC¯(A1+A4+2A3+2A5)r2A2,
A3Z=iC¯(A1+A6+2A2+2A5)r2A3,
A4Z=iC¯(A2+A7+2A5+2A8)r2A4,
A5Z=iC¯(A4+A6+A3+A2+A8+A9)r2A5,
A6Z=iC¯(A3+A10+2A5+2A9)r2A6,
A7Z=iC¯(A4+2A8)r2A7,
A8Z=iC¯(A7+A9+A5+A4)r2A8,
A9Z=iC¯(A8+A10+A6+A5)r2A9,
A10Z=iC¯(A6+2A9)r2A10.

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