Abstract

Novel solution algorithms are presented for obtaining the complete power dispersion curve of nonlinear-optical channel waveguides using full vectorial finite-element analysis. With the new algorithm the jump in the modal index in power dispersion curves reported in the literature is successfully explained, and the origins of a useful bistable behavior in nonlinear-optical channel waveguides is then identified. A comparison between the full E and H formulations of nonlinear-optical waveguides is given for the first time. The structure analyzed may find applications in all-optical switching and bistability devices.

© 1993 Optical Society of America

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References

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  1. G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
    [CrossRef]
  2. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. LT-6, 953–970 (1988).
    [CrossRef]
  3. G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
    [CrossRef]
  4. G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
    [CrossRef]
  5. X. H. Wang, Finite element methods for nonlinear optical waveguides, Ph.D. thesis (Monash University, Melbourne, Australia, 1992).
  6. K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988).
    [CrossRef]
  7. X. H. Wang, L. N. Binh, and G. K. Cambrell, “Vectorial finite element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1988), pp. 129–132.
  8. R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
    [CrossRef]
  9. X. H. Wang, G. K. Cambrell, and L. N. Binh, “Scalar and vector formulations of nonlinear optical waveguides: a comparison,” in Proceedings of IREECON International 1989, Melbourne, Australia (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1989), pp. 551–554.
  10. B. M. A. Rahman and J. B. Davies, “Finite element solution of nonlinear bistable optical waveguides,” Int. J. Optoelectron. 4, 153–161 (1989).
  11. X. H. Wang and G. K. Cambrell, “All-optical switching and bistability phenomena in nonlinear optical waveguides: Part I. Power dispersion relations,” in 16th Australian Conference on Optical Fibre Technology (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1991), pp. 314–317.
  12. B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
    [CrossRef]
  13. X. H. Wang, G. K. Cambrell, and L. N. Binh, “A package for nonlinear optical waveguides based on E-vector finite elements,” in Advances in Electrical Engineering Software, P. P. Silvester, ed., proceedings of the First International Conference on Electrical Engineering Analysis and Design (Computational Mechanics, Southampton, 1990), pp. 151–162.
  14. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. E. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculation of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
    [CrossRef]
  15. N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
    [CrossRef]
  16. N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
    [CrossRef]

1991 (1)

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

1990 (1)

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

1989 (3)

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[CrossRef]

B. M. A. Rahman and J. B. Davies, “Finite element solution of nonlinear bistable optical waveguides,” Int. J. Optoelectron. 4, 153–161 (1989).

G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
[CrossRef]

1988 (3)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. LT-6, 953–970 (1988).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
[CrossRef]

K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988).
[CrossRef]

1985 (2)

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. E. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculation of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

1984 (1)

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
[CrossRef]

Binh, L. N.

X. H. Wang, L. N. Binh, and G. K. Cambrell, “Vectorial finite element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1988), pp. 129–132.

X. H. Wang, G. K. Cambrell, and L. N. Binh, “Scalar and vector formulations of nonlinear optical waveguides: a comparison,” in Proceedings of IREECON International 1989, Melbourne, Australia (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1989), pp. 551–554.

X. H. Wang, G. K. Cambrell, and L. N. Binh, “A package for nonlinear optical waveguides based on E-vector finite elements,” in Advances in Electrical Engineering Software, P. P. Silvester, ed., proceedings of the First International Conference on Electrical Engineering Analysis and Design (Computational Mechanics, Southampton, 1990), pp. 151–162.

Cambrell, G. K.

X. H. Wang, G. K. Cambrell, and L. N. Binh, “Scalar and vector formulations of nonlinear optical waveguides: a comparison,” in Proceedings of IREECON International 1989, Melbourne, Australia (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1989), pp. 551–554.

X. H. Wang, G. K. Cambrell, and L. N. Binh, “A package for nonlinear optical waveguides based on E-vector finite elements,” in Advances in Electrical Engineering Software, P. P. Silvester, ed., proceedings of the First International Conference on Electrical Engineering Analysis and Design (Computational Mechanics, Southampton, 1990), pp. 151–162.

X. H. Wang and G. K. Cambrell, “All-optical switching and bistability phenomena in nonlinear optical waveguides: Part I. Power dispersion relations,” in 16th Australian Conference on Optical Fibre Technology (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1991), pp. 314–317.

X. H. Wang, L. N. Binh, and G. K. Cambrell, “Vectorial finite element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1988), pp. 129–132.

Chilwell, J. T.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. E. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculation of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Davies, J. B.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

B. M. A. Rahman and J. B. Davies, “Finite element solution of nonlinear bistable optical waveguides,” Int. J. Optoelectron. 4, 153–161 (1989).

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

Ettinger, R. D.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

Fernandez, F. A.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

Finlayson, N.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. LT-6, 953–970 (1988).
[CrossRef]

Hayata, K.

Koshiba, M.

Nabiev, R. F.

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
[CrossRef]

Popov, Yu. M.

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
[CrossRef]

Rahman, B. M. A.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

B. M. A. Rahman and J. B. Davies, “Finite element solution of nonlinear bistable optical waveguides,” Int. J. Optoelectron. 4, 153–161 (1989).

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. LT-6, 953–970 (1988).
[CrossRef]

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. E. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculation of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Shoemaker, R. L.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. E. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculation of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Smith, S. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. E. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculation of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Stegeman, G. E.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. E. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculation of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. LT-6, 953–970 (1988).
[CrossRef]

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

Stolen, R. H.

Valera, J. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. E. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculation of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

Wang, X. H.

X. H. Wang, G. K. Cambrell, and L. N. Binh, “A package for nonlinear optical waveguides based on E-vector finite elements,” in Advances in Electrical Engineering Software, P. P. Silvester, ed., proceedings of the First International Conference on Electrical Engineering Analysis and Design (Computational Mechanics, Southampton, 1990), pp. 151–162.

X. H. Wang and G. K. Cambrell, “All-optical switching and bistability phenomena in nonlinear optical waveguides: Part I. Power dispersion relations,” in 16th Australian Conference on Optical Fibre Technology (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1991), pp. 314–317.

X. H. Wang, G. K. Cambrell, and L. N. Binh, “Scalar and vector formulations of nonlinear optical waveguides: a comparison,” in Proceedings of IREECON International 1989, Melbourne, Australia (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1989), pp. 551–554.

X. H. Wang, Finite element methods for nonlinear optical waveguides, Ph.D. thesis (Monash University, Melbourne, Australia, 1992).

X. H. Wang, L. N. Binh, and G. K. Cambrell, “Vectorial finite element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1988), pp. 129–132.

Wright, E. M.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. LT-6, 953–970 (1988).
[CrossRef]

Zanoni, R.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. LT-6, 953–970 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. E. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculation of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, and J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

Int. J. Optoelectron. (1)

B. M. A. Rahman and J. B. Davies, “Finite element solution of nonlinear bistable optical waveguides,” Int. J. Optoelectron. 4, 153–161 (1989).

J. Appl. Phys. (1)

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

J. Lightwave Technol. (1)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. LT-6, 953–970 (1988).
[CrossRef]

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

Opt. Commun. (1)

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[CrossRef]

Opt. Quantum Electron. (1)

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

Solid State Commun. (1)

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
[CrossRef]

Other (5)

X. H. Wang and G. K. Cambrell, “All-optical switching and bistability phenomena in nonlinear optical waveguides: Part I. Power dispersion relations,” in 16th Australian Conference on Optical Fibre Technology (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1991), pp. 314–317.

X. H. Wang, G. K. Cambrell, and L. N. Binh, “A package for nonlinear optical waveguides based on E-vector finite elements,” in Advances in Electrical Engineering Software, P. P. Silvester, ed., proceedings of the First International Conference on Electrical Engineering Analysis and Design (Computational Mechanics, Southampton, 1990), pp. 151–162.

X. H. Wang, Finite element methods for nonlinear optical waveguides, Ph.D. thesis (Monash University, Melbourne, Australia, 1992).

X. H. Wang, L. N. Binh, and G. K. Cambrell, “Vectorial finite element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1988), pp. 129–132.

X. H. Wang, G. K. Cambrell, and L. N. Binh, “Scalar and vector formulations of nonlinear optical waveguides: a comparison,” in Proceedings of IREECON International 1989, Melbourne, Australia (Institutions of Radio and Electronics Engineers Australia, Sydney, Australia, 1989), pp. 551–554.

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Figures (7)

Fig. 1
Fig. 1

Channel/strip nonlinear-optical waveguide structure and normalized coordinate system.

Fig. 2
Fig. 2

Mesh network of the nonlinear strip-loaded optical channel waveguide structure (126 second-order elements, 281 nodes).

Fig. 3
Fig. 3

Power dispersion relation of the nonlinear strip-loaded optical channel waveguide resulting from the conventional solution technique for both the electric- (solid curve) and the magnetic-(dashed curve) field formulations: β/k0 versus P ˜.

Fig. 4
Fig. 4

Magnitude distribution of the normalized magnetic field || H ˜||2 for (a) P ˜ = 20.0 and (b) P ˜ = 80.0.

Fig. 5
Fig. 5

Comparison of power dispersion curves (β/k0 versus P ˜) resulting from the conventional and the nonconventional solution algorithms for both the electric- and the magnetic-field formulations, where Elec and Mag denote the electric- and magnetic-field formulations and the extensions c and n denote the conventional and the nonconventional solution algorithms, respectively.

Fig. 6
Fig. 6

Magnitude distribution of the normalized electric field || E ˜||2 at (a) the upper portion, (b) the intermediate portion, and (c) the lower portion of the power dispersion curve for P ˜ = 44.

Fig. 7
Fig. 7

Comparison of power dispersion curves (β/k0 versus P ˜) resulting from the modified conventional and the nonconventional solution algorithms for both the electric- and the magnetic-field formulations, where Elec and Mag denote the electric- and magnetic-field formulations and the extensions m and n denote the modified conventional and the nonconventional solution algorithms, respectively.

Tables (1)

Tables Icon

Table 1 Mapping of the Variables for the Two Different Formulations

Equations (17)

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

^ r ( E ˜ ) = { I ^ x ¯ 30 15 < y ¯ 30 8 < x ¯ 30 0 < y ¯ 15 ( 1.52 2 + E ˜ 2 2 1 + α E ˜ 2 2 ) I ^ x ¯ 8 0 < y ¯ 15 1.55 2 I ^ x ¯ 8 - 12 < y ¯ 0 1.52 2 I ^ 8 < x ¯ 30 - 12 < y ¯ 0 x ¯ 30 - 30 y ¯ - 12 ,
E ˜ ( n ) 1 / 2 E ,
n = c 0 0 r l n 2
H ˜ Z 0 ( n ) 1 / 2 H ,
H ˜ = j μ ^ r - 1 · ¯ × E ˜ ,
E ˜ = - j ^ r - 1 · ¯ × H ˜ ,
¯ × ( p ^ - 1 · ¯ × V ˜ ) - p q ^ · ¯ ¯ · ( q ^ · V ˜ ) = q ^ · V ˜
P = 1 2 Z 0 n k 0 2 ( Im { - Ω [ V ˜ * × ( p ^ - 1 · ¯ × V ˜ ) ] · u ^ z d x ¯ d y ¯ } ) ,
P ˜ = 2 Z 0 n k 0 2 P .
V ˜ A m V
Im { - Ω [ V * × ( p ^ - 1 · ( ¯ × V ) ] · u ^ z d x ¯ d y ¯ } 1 ,
P ˜ A m 2 .
^ r ^ r l + ^ r n .
¯ × { μ ^ r - 1 · [ ¯ × E ( i + 1 ) ] } - p ^ r ( i ) · ¯ ¯ · [ ^ r ( i ) · E ( i + 1 ) ] - ^ r l · E ( i + 1 ) = A m ( i + 1 ) 2 [ ^ r ( i ) n A m ( i ) 2 ] · E ( i + 1 )
¯ × { ( ^ r l ) - 1 · [ ¯ × H ( i + 1 ) ] } - p μ ^ r · ¯ ¯ · [ μ ^ r · H ( i + 1 ) ] - μ ^ r · H ( i + 1 ) = A m ( i + 1 ) 2 ¯ × { [ ( ^ r l ) - 1 - ^ r ( i ) - 1 A m ( i ) 2 ] · [ ¯ × H ( i + 1 ) ] }
^ r ( i ) ^ r l + ^ r ( i ) n ,             ^ r ( i ) n ^ r n [ A m ( i ) E ( i ) ] ,
E ( i ) = { - j ^ r ( i - 1 ) - 1 · ¯ × H ( i ) i 1 - j ^ r ( i ) - 1 · ¯ × H ( i ) i = 0 .

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