Abstract

The computation of the reflectivity of a multilayer dielectric coating deposited on the end facet of single-mode or multimodal fiber is presented. The calculation is done through a plane-wave expansion of the incident fields, and the waveguide is replaced by a homogeneous medium of refractive index neq while the incident electric (or magnetic) field remains that of the guided mode. The modal reflectivities (coupling factors) between modes are calculated, and the method of computation is accurate enough to satisfy current needs for the design of coatings on fibers and is helpful for the design of coatings deposited on other waveguide structures.

© 1998 Optical Society of America

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References

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  1. J. Chen, D. Li, Y. Lu, “Experimental and theoretical studies on monitored signals from semiconductor diodes undergoing antireflection coatings,” Appl. Opt. 30, 4554–4559 (1991).
    [CrossRef] [PubMed]
  2. T. E. Rozzi, G. H. Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antenna,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
    [CrossRef]
  3. P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “New formula for semiconductor laser facet reflectivity,” IEEE Photon. Technol. Lett. 5, 148–151 (1993).
    [CrossRef]
  4. P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space rediation modes,” IEE Proc. J 140, 49–55 (1993).
  5. C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facet,” Electron. Lett. 29, 1352–1353 (1993).
    [CrossRef]
  6. C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifiers,” Electron. Lett. 21, 333–335 (1985).
    [CrossRef]
  7. C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
    [CrossRef]
  8. C. Vassallo, “Polarisation independent antireflection coatings for semiconductor optical amplifiers,” Electron. Lett. 24, 61–62 (1988).
    [CrossRef]
  9. C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
    [CrossRef]
  10. C. Vassallo, “Theory and practical calculation of antireflection coatings on semiconductor laser diode optical amplifiers,” IEE Proc. Optoelectron. 137, 193–202 (1990).
    [CrossRef]
  11. J. Xu, D. Yevick, M. Gallant, “Approximate methods for modal reflectivity at optical waveguide facets,” J. Opt. Soc. Am. A 12, 725–728 (1995).
    [CrossRef]
  12. F. Favre, D. Le Guen, “82 nm of continuous tunability for an external cavity semiconductor laser,” Electron. Lett. 27, 183–184 (1991).
    [CrossRef]
  13. O. Vasseur, “Calcul et réalisation de traitements optiques multicouches pour diodes laser. Etude de l’influence des défauts de structure des couches sur les performances des diodes traitées,” Ph.D. thesis (Université d’Aix-Marseille III, Marseille, France, 1990).
  14. C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991), pp. 269–308.
  15. A. Snyder, J. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), pp. 238–326.
  16. C. Vassallo, Theorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985), pp. 379–402.
  17. J. A. Savage, Infrared Optical Materials and Their Antireflection Coatings (Hilger, Bristol, 1985), pp. 150–184.
  18. H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, 1986), pp. 434–443.
  19. O. Vasseur, “Analyse modale de la réflexion sur des traitements multicouches déposés en extremité de fibres,” in 15èmes Journées Nationales d’Optique Guidée 95 (Société Française d’Optique, Orsay, France, 1995), Paper 32bis.
  20. O. Vasseur, “Multilayer coatings for waveguide components,” in Proceedings of the International Symposium on Optronics and Defence (Association Aéronautique et Astronautique de France, Paris, 1996), Paper 1.43.

1995 (1)

1993 (3)

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “New formula for semiconductor laser facet reflectivity,” IEEE Photon. Technol. Lett. 5, 148–151 (1993).
[CrossRef]

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space rediation modes,” IEE Proc. J 140, 49–55 (1993).

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facet,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

1991 (2)

J. Chen, D. Li, Y. Lu, “Experimental and theoretical studies on monitored signals from semiconductor diodes undergoing antireflection coatings,” Appl. Opt. 30, 4554–4559 (1991).
[CrossRef] [PubMed]

F. Favre, D. Le Guen, “82 nm of continuous tunability for an external cavity semiconductor laser,” Electron. Lett. 27, 183–184 (1991).
[CrossRef]

1990 (1)

C. Vassallo, “Theory and practical calculation of antireflection coatings on semiconductor laser diode optical amplifiers,” IEE Proc. Optoelectron. 137, 193–202 (1990).
[CrossRef]

1988 (3)

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
[CrossRef]

C. Vassallo, “Polarisation independent antireflection coatings for semiconductor optical amplifiers,” Electron. Lett. 24, 61–62 (1988).
[CrossRef]

C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
[CrossRef]

1985 (1)

C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifiers,” Electron. Lett. 21, 333–335 (1985).
[CrossRef]

1980 (1)

T. E. Rozzi, G. H. Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antenna,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[CrossRef]

Adams, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “New formula for semiconductor laser facet reflectivity,” IEEE Photon. Technol. Lett. 5, 148–151 (1993).
[CrossRef]

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space rediation modes,” IEE Proc. J 140, 49–55 (1993).

Benson, T. M.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facet,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

Chen, J.

Favre, F.

F. Favre, D. Le Guen, “82 nm of continuous tunability for an external cavity semiconductor laser,” Electron. Lett. 27, 183–184 (1991).
[CrossRef]

Gallant, M.

Kendall, P. C.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “New formula for semiconductor laser facet reflectivity,” IEEE Photon. Technol. Lett. 5, 148–151 (1993).
[CrossRef]

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facet,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space rediation modes,” IEE Proc. J 140, 49–55 (1993).

Le Guen, D.

F. Favre, D. Le Guen, “82 nm of continuous tunability for an external cavity semiconductor laser,” Electron. Lett. 27, 183–184 (1991).
[CrossRef]

Li, D.

Love, J.

A. Snyder, J. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), pp. 238–326.

Lu, Y.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, 1986), pp. 434–443.

Roberts, D. A.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “New formula for semiconductor laser facet reflectivity,” IEEE Photon. Technol. Lett. 5, 148–151 (1993).
[CrossRef]

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space rediation modes,” IEE Proc. J 140, 49–55 (1993).

Robertson, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space rediation modes,” IEE Proc. J 140, 49–55 (1993).

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “New formula for semiconductor laser facet reflectivity,” IEEE Photon. Technol. Lett. 5, 148–151 (1993).
[CrossRef]

Robson, P. N.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “New formula for semiconductor laser facet reflectivity,” IEEE Photon. Technol. Lett. 5, 148–151 (1993).
[CrossRef]

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space rediation modes,” IEE Proc. J 140, 49–55 (1993).

Rozzi, T. E.

T. E. Rozzi, G. H. Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antenna,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[CrossRef]

Savage, J. A.

J. A. Savage, Infrared Optical Materials and Their Antireflection Coatings (Hilger, Bristol, 1985), pp. 150–184.

Smartt, C. J.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facet,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

Snyder, A.

A. Snyder, J. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), pp. 238–326.

Vassallo, C.

C. Vassallo, “Theory and practical calculation of antireflection coatings on semiconductor laser diode optical amplifiers,” IEE Proc. Optoelectron. 137, 193–202 (1990).
[CrossRef]

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
[CrossRef]

C. Vassallo, “Polarisation independent antireflection coatings for semiconductor optical amplifiers,” Electron. Lett. 24, 61–62 (1988).
[CrossRef]

C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
[CrossRef]

C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifiers,” Electron. Lett. 21, 333–335 (1985).
[CrossRef]

C. Vassallo, Theorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985), pp. 379–402.

C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991), pp. 269–308.

Vasseur, O.

O. Vasseur, “Calcul et réalisation de traitements optiques multicouches pour diodes laser. Etude de l’influence des défauts de structure des couches sur les performances des diodes traitées,” Ph.D. thesis (Université d’Aix-Marseille III, Marseille, France, 1990).

O. Vasseur, “Analyse modale de la réflexion sur des traitements multicouches déposés en extremité de fibres,” in 15èmes Journées Nationales d’Optique Guidée 95 (Société Française d’Optique, Orsay, France, 1995), Paper 32bis.

O. Vasseur, “Multilayer coatings for waveguide components,” in Proceedings of the International Symposium on Optronics and Defence (Association Aéronautique et Astronautique de France, Paris, 1996), Paper 1.43.

Veld, G. H.

T. E. Rozzi, G. H. Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antenna,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[CrossRef]

Xu, J.

Yevick, D.

Appl. Opt. (1)

Electron. Lett. (5)

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facet,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifiers,” Electron. Lett. 21, 333–335 (1985).
[CrossRef]

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
[CrossRef]

C. Vassallo, “Polarisation independent antireflection coatings for semiconductor optical amplifiers,” Electron. Lett. 24, 61–62 (1988).
[CrossRef]

F. Favre, D. Le Guen, “82 nm of continuous tunability for an external cavity semiconductor laser,” Electron. Lett. 27, 183–184 (1991).
[CrossRef]

IEE Proc. J (1)

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space rediation modes,” IEE Proc. J 140, 49–55 (1993).

IEE Proc. Optoelectron. (1)

C. Vassallo, “Theory and practical calculation of antireflection coatings on semiconductor laser diode optical amplifiers,” IEE Proc. Optoelectron. 137, 193–202 (1990).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “New formula for semiconductor laser facet reflectivity,” IEEE Photon. Technol. Lett. 5, 148–151 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

T. E. Rozzi, G. H. Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antenna,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[CrossRef]

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

Other (8)

O. Vasseur, “Calcul et réalisation de traitements optiques multicouches pour diodes laser. Etude de l’influence des défauts de structure des couches sur les performances des diodes traitées,” Ph.D. thesis (Université d’Aix-Marseille III, Marseille, France, 1990).

C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991), pp. 269–308.

A. Snyder, J. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), pp. 238–326.

C. Vassallo, Theorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985), pp. 379–402.

J. A. Savage, Infrared Optical Materials and Their Antireflection Coatings (Hilger, Bristol, 1985), pp. 150–184.

H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, 1986), pp. 434–443.

O. Vasseur, “Analyse modale de la réflexion sur des traitements multicouches déposés en extremité de fibres,” in 15èmes Journées Nationales d’Optique Guidée 95 (Société Française d’Optique, Orsay, France, 1995), Paper 32bis.

O. Vasseur, “Multilayer coatings for waveguide components,” in Proceedings of the International Symposium on Optronics and Defence (Association Aéronautique et Astronautique de France, Paris, 1996), Paper 1.43.

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Figures (8)

Fig. 1
Fig. 1

Investigated system: infinite cladding fiber. n2 =1.48, R=1.37 µm.

Fig. 2
Fig. 2

Δ influence on modal reflectivity. Fiber: n2 =1.48, R=1.37 µm; filter: HBHB2HBHB2HBHBHBHBHB2HBHB2HBHBH at λ=1.85 µm. H means quarter wave of high index, nH =2.25 at the wavelength given. B means quarter wave of low index nB =1.48 at the wavelength given. Half-waves are denoted by 2H or 2B. (In Figs. 27, commas in axis numbers are to be read as decimals.)

Fig. 3
Fig. 3

Modal and power reflectivity. Fiber: n1=1.55, n2 =1.48, R=1.37 µm. filter: HBHB2HBHB2HBHBHBHBHB2HBHB2HBHBH at λ=1.85 µm.

Fig. 4
Fig. 4

Four-layer antireflection coatings on slab (TE, TM) and on fiber (HE1, 1). Diode: n1=3.512, n2=3.17, d=0.2 µm; fiber: n1=3.512, n2=3.17, R=0.2 µm; filter: n1=2.25, e1=0.2035 µm, n2=1.48, e2=0.2349 µm, n3=2.25, e3 =0.0694 µm, n4=1.48, e4=0.3755 µm.

Fig. 5
Fig. 5

Two-layer antireflection coatings on slab (TE, TM) and fiber (HE1, 1). Diode: n1=3.512, n2=3.17, d=0.25 µm; fiber: n1=3.512, n2=3.17, R=0.25 µm; filter: n1=2.4917, e1=0.1689 µm, n2=1.3924, e2=0.3096 µm.

Fig. 6
Fig. 6

Equivalent index influence on reflectivity RHE1,1;HE1,1. Fiber: n1=3.512, n2=3.17, R=0.9140 µm. filter; n1 =2.25, e1=0.2035 µm, n2=1.48, e2=0.2349 µm, n3 =2.25, e3=0.0694 µm, n4=1.48, e4=0.3755 µm. Curve (1) neq,HE1,1 case, curve (2) neq case, curve (3) neq-with-corrective-terms case.

Fig. 7
Fig. 7

Modal reflectivities: curve (1) RHE1,1;HE1,1, curve (2) RHE1,2;HE1,2, curve (3) RHE1,1;HE1,2. Fiber: n1=1.55, n2=1.48, R=3 µm; filter: HBHB2HBHB2HBHBHBHBHB2HBHB2HBHBH at λ=1.85 µm.

Fig. 8
Fig. 8

Modal reflectivities RHE1,1;HE1,2 versus R. Fiber: n1=1.55, n2=1.48; filter: HBHB2HBHB2HBHBHBHBHB2HBHB2HBHBH at λ=1.85 µm.

Equations (54)

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e0,T+Er,T=Etr,T=E(x, y),
h0,T+Hr,T=Htr,T=H(x, y),
1+R=E, h0e0, h0,
A, B=0+02πA(x, y)  B*(x, y)·zdxdy,
(1+Zwg0Yar)E=2e0-δZYarE+O(2),
Zwg=Zwg0+δZ,
(1+Zwg0Yar)E0=2e0.
1+R0=E0, h0e0, h0.
δR=K(n2(r)-neq2)e0, h0,
neq2=(n2(x, y)e0, h0e0, h0.
R0=Er, h0e0, h0.
u4v4K1(v)vK1(v)+J1(u)uJ1(u)n12 J1(u)uJ1(u)+n22 K1(v)vK1(v)-γ2k2V4=0,
u=αR,v=aR,α2=k2n12-γ2
a2=γ2-k2n22,k=2πλ.
Ex=12J0(αr)α(γA1+kB1)+12J2(αr)α×exp(i2θ)(-γA1+kB1),
Ey=12J0(αr)α(iγA1+ikB1)+12J2(αr)α×exp(i2θ)(iγA1-ikB1),
Hx=12J0(αr)αη(-ikn12A1-iγB1)+12J2(αr)α×η exp(i2θ)(-ikn12A1+iγB1),
Hy=12J0(αr)αη(kn12A1+γB1)+12J2(αr)α×η exp(i2θ)(-kn12A1+γB1).
Ex=12K0(ar)a(γA2+kB2)+12K2(ar)a×exp(i2θ)(γA2-kB2),
Ey=12K0(ar)a(iγA2+ikB2)+12K2(ar)a×exp(i2θ)(-iγA2+ikB2),
Hx=12K0(ar)aη(-ikn22A2-iγB2)+12K2(ar)a×η exp(i2θ)(ikn22A2-iγB2),
Hy=12K0(ar)aη(kn22A2+γB2)+12K2(ar)a×η exp(i2θ)(kn22A2-γB2),
J1(u)A1=K1(v)A2,J1(u)B1=K1(v)B2,
A1u2v2-1v2-K0(v)vK1(v)+J0(u)uJ1(u)-1u2=-B1γkV2,
η=0/μ0.
ϕ=12Re0+02πE  H*·zrdθdr=12Re0+02π(ExHy*-EyHx*)rdθdr,
Ex˜(kx, ky)=-+-+Ex(x, y)exp[ikxx+ikyy]dxdy
Ex˜(s, ψ)=0+02πEx(r, θ)exp[irs cos(θ-ψ)]rdrdθ.
Ex,y,zr˜=(kx2rTM+ky2rTE)Ex˜+kxky(rTM-rTE)Ey˜kxky(rTM-rTE)Ex˜+(ky2rTM+kx2rTE)Ey˜rTMEz˜,
neq2=n2(r)E, HE, H,
A, B=0+02πA(r, θ)  B*(r, θ)·zrdrdθ,
0+02πn2(r)E  H*·zrdrdθ=0+02πneq2E  H*·zrdrdθ.
Rm=Er, H0E0, H02,
Rm=-+-+ErxH0y*-EryH0x*dxdy-+-+E0xH0y*-E0yH0x*dxdy2=0+02πErx˜H0y*˜-Ery˜H0x*˜sdsdψ0+02πE0x˜H0y*˜-E0y˜H0x*˜sdsdψ2.
Rp=Re0+02πErx˜Hry*˜-Ery˜Hrx*˜sdsdψ0+02πE0x˜H0y*˜-E0y˜H0x*˜sdsdψ
andTp=1-Rp.
neq,i2=n2(r)Ei, HiEi, Hi.
neq2=n2(r)E, HE, H,
Ex=12Jm-1(αr)αexp[i(m-1)θ](γA1+kB1)+12Jm+1(αr)αexp[i(m+1)θ](-γA1+kB1),
Ey=12Jm-1(αr)αexp[i(m-1)θ](iγA1+ikB1)+12Jm+1(αr)αexp[i(m+1)θ](iγA1-ikB1),
Hx=12Jm-1(αr)αη exp[i(m-1)θ](-ikn12A1-iγB1)+12Jm+1(αr)αη exp[i(m+1)θ]×(-ikn12A1+iγB1),
Hy=12Jm-1(αr)αη exp[i(m-1)θ](kn12A1+γB1)+12Jm+1(αr)αη exp[i(m+1)θ]×(-kn12A1+γB1);
Ex=12Km-1(ar)aexp[i(m-1)θ](γA2+kB2)+12Km+1(ar)aexp[i(m+1)θ](γA2-kB2),
Ey=12Km-1(ar)aexp[i(m-1)θ](iγA2+ikB2)+12Km+1(ar)aexp[i(m+1)θ](-iγA2+ikB2),
Hx=12Km-1(ar)aη exp[i(m-1)θ](-ikn22A2-iγB2)+12Km+1(ar)aη exp[i(m+1)θ]×(ikn22A2-iγB2),
Hy=12Km-1(ar)aη exp[i(m-1)θ](kn22A2+γB2)+12Km+1(ar)aη exp[i(m+1)θ]×(kn22A2-γB2),
Jm(u)A1=Km(v)A2,Jm(u)B1=Km(v)B2,
A1u2v2Jm(u)uJm(u)+Km(v)vKm(v)=-B1mγkV2.
Rm,i=Er, HiEi, Hi2Rp=Er, HrE, Hand
Ri,j=Eri, HjEj, Hj2,
HBHB2HBHB2HBHBHBHBHB2HBHB2HBHBH
HBHB2HBHB2HBHBHBHBHB2HBHB2HBHBH
HBHB2HBHB2HBHBHBHBHB2HBHB2HBHBH
HBHB2HBHB2HBHBHBHBHB2HBHB2HBHBH

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