Abstract

We develop a homogenization expansion approach to photonic waveguides whose transverse structures are N-fold rotationally symmetric. Examples include microstructured or holey optical fibers with air holes arranged in one or more concentric rings. We carry out a homogenization expansion for large N about the N= limit. Our multiple scale analysis applies to the scalar approximation of structures in which the microfeatures have arbitrary geometry and large index contrasts and lead to a natural efficient computational algorithm for the waveguide modes and spectral characteristics. In this paper we focus on structures that possess leaky modes. The leading order (N=) equations describe the modes of an averaged structure. We derive an expansion in powers of 1/N of corrections to the leading order behavior and show that the leading order nontrivial contribution arises at order 1/N2. We numerically calculate this leading order correction to the complex effective indices (scattering resonances) for the leaky modes of various microstructured photonic waveguides whose imaginary parts give the leakage rates. We observe that in many instances a two-term truncation of the homogenization expansion gives good agreement with full simulations, even for fairly small values of N, whereas the leading order (averaged) theory yields a substantial underestimate of the leakage rates.

© 2003 Optical Society of America

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References

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  1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
    [CrossRef] [PubMed]
  2. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25-27 (2000).
    [CrossRef]
  3. J. Jasapara, R. Bise, and R. Windeler, “Chromatic dispersion measurements in a photonic bandgap fiber,” in Optical Fiber Communication, Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2002).
  4. T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. 26, 1660-1662 (2001).
    [CrossRef]
  5. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50-56 (2000).
    [CrossRef]
  6. L. Poladian, N. A. Issa, and T. M. Monro, “Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry,” Opt. Express 10, 449-454 (2002).
    [CrossRef] [PubMed]
  7. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding-mode-resonances in air-silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084-1100 (2000).
    [CrossRef]
  8. M. J. Steel and R. M. Osgood, Jr., “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229-231 (2001).
    [CrossRef]
  9. A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, M. C. J. Large, J. Zagari, N. A. P. Nicorovici, R. C. McPhedran, and C. M. de Sterke, “Ring structures in microstructured polymer optical fibers,” Opt. Express 9, 813-820 (2001).
    [CrossRef] [PubMed]
  10. A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, Vol. 5 of Studies in Mathematics and Its Applications (North-Holland, Amsterdam, 1978).
  11. G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics (Cambridge University, Cambridge, England, 2002).
  12. F. Santosa and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 53, 1636-1668 (1993).
    [CrossRef]
  13. S. Moskow and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium: a convergence proof,” Proc. R. Soc. Edinburgh Sect. A Math. 127, 1263-1299 (1997).
    [CrossRef]
  14. S. E. Golowich and M. I. Weinstein, “Scattering resonances and homogenization theory,” Bell Labs preprint (Bell Laboratories, Murray Hill, N.J., 2002).
  15. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  16. P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory (Springer-Verlag, New York, 1996).
  17. M. Abramowitz and I. E. Stegun, eds., Handbook of Mathematical Functions (National Institute of Standards and Technology, Gaithersburg, MD., 1972).
  18. The values for the LP01 mode calculated by the Fourier expansion were taken from Fig. 3 in Ref. 6.
  19. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, Boston, 1991).
  20. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).
  21. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
    [CrossRef]

2002 (1)

2001 (3)

2000 (3)

1999 (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

1997 (1)

S. Moskow and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium: a convergence proof,” Proc. R. Soc. Edinburgh Sect. A Math. 127, 1263-1299 (1997).
[CrossRef]

1993 (1)

F. Santosa and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 53, 1636-1668 (1993).
[CrossRef]

1978 (1)

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Argyros, A.

Bassett, I. M.

Bennett, P. J.

Birks, T. A.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Botten, L. C.

Broderick, N. G. R.

Burdge, G. L.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

de Sterke, C. M.

Eggleton, B. J.

Issa, N. A.

Kerbage, C.

Knight, J. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Large, M. C. J.

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Marom, E.

McPhedran, R. C.

Monro, T. M.

Moskow, S.

S. Moskow and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium: a convergence proof,” Proc. R. Soc. Edinburgh Sect. A Math. 127, 1263-1299 (1997).
[CrossRef]

Nicorovici, N. A. P.

Osgood Jr., R. M.

Poladian, L.

Ranka, J. K.

Richardson, D. J.

Roberts, P. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Russell, P. St. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Santosa, F.

F. Santosa and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 53, 1636-1668 (1993).
[CrossRef]

Steel, M. J.

Stentz, A. J.

van Eijkelenborg, M. A.

Vogelius, M.

S. Moskow and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium: a convergence proof,” Proc. R. Soc. Edinburgh Sect. A Math. 127, 1263-1299 (1997).
[CrossRef]

F. Santosa and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 53, 1636-1668 (1993).
[CrossRef]

Westbrook, P. S.

White, C. A.

White, T. P.

Windeler, R. S.

Yariv, A.

Yeh, P.

Zagari, J.

J. Lightwave Technol. (2)

J. Opt. Soc. Am. (1)

Opt. Express (2)

Opt. Lett. (3)

Proc. R. Soc. Edinburgh Sect. A Math. (1)

S. Moskow and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium: a convergence proof,” Proc. R. Soc. Edinburgh Sect. A Math. 127, 1263-1299 (1997).
[CrossRef]

Science (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

F. Santosa and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 53, 1636-1668 (1993).
[CrossRef]

Other (10)

A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, Vol. 5 of Studies in Mathematics and Its Applications (North-Holland, Amsterdam, 1978).

G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics (Cambridge University, Cambridge, England, 2002).

J. Jasapara, R. Bise, and R. Windeler, “Chromatic dispersion measurements in a photonic bandgap fiber,” in Optical Fiber Communication, Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2002).

S. E. Golowich and M. I. Weinstein, “Scattering resonances and homogenization theory,” Bell Labs preprint (Bell Laboratories, Murray Hill, N.J., 2002).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory (Springer-Verlag, New York, 1996).

M. Abramowitz and I. E. Stegun, eds., Handbook of Mathematical Functions (National Institute of Standards and Technology, Gaithersburg, MD., 1972).

The values for the LP01 mode calculated by the Fourier expansion were taken from Fig. 3 in Ref. 6.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, Boston, 1991).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

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Figures (9)

Fig. 1
Fig. 1

Cross section of a microstructured waveguide.

Fig. 2
Fig. 2

(a) Ring of N=30 air holes approximated by a simple layered structure with 11 layers. The ratio Rhole/Rring of the hole radius to the ring radius is 0.7π/N. (b) The averaged potential Vav with Rring=12 μm.

Fig. 3
Fig. 3

(a) Ring of air wedges supported by N=20 webs of glass. The ratio Rin/Rout of the outer radii of the annulus is 10/11, whereas the air-fill fraction within the annulus is f=0.8. (b) The averaged potential Vav with Rin=10 μm.

Fig. 4
Fig. 4

(a) Eighteen-hole subset of a hexagonal lattice (N=6) with interhole spacing Λ=2.3 μm and hole radius Rhole=0.46 μm, approximated by a simple layered structure with L=51 layers. (b) The averaged potential Vav in units of μm-2 versus radius in units of micrometers.

Fig. 5
Fig. 5

Effective indices and attenuation coefficients of a set of least-lossy resonances with l{0, 1, 2, 3} of the structure depicted in Fig. 2, scaled to three different ring radii. The plotting symbol encodes angular index l: ■, ○, △, + correspond to l=0, 1, 2, 3. The results of the averaged structure are labeled N=.

Fig. 6
Fig. 6

Effective indices and attenuation coefficients of a set of least-lossy resonances with l{0, 1, 2, 3} of the structure depicted in Fig. 3 for two choices of fill fraction x and three different radii. The plotting symbol encodes the angular index l: ■, ○, △, + correspond to l=0, 1, 2, 3; the shade of gray encodes periodicity N. The results of the averaged structure are labeled N=.

Fig. 7
Fig. 7

Attenuation of the lowest-order (fundamental or LP01) resonance for a structure of the type shown in Fig. 3 with Rin=1 μm, Rout=2 μm for fill fractions f=0.8, 0.9, and 1 with N=3 and N=6 holes. The calculations were performed for free-space wavelengths λfs ranging from 1 to 2 μm. The solid curves represent the attenuations computed according to the methods outlined in the text, including the O(N-2) corrections, the dashed curves represent the attenuations of the averaged structure, and the x corresponds to the results presented in Fig. 3(a) of Ref. 6.

Fig. 8
Fig. 8

Results of weak residual calculation: (a) Fres(weak) (θ)-Fres(weak,1)(θ) (solid line) and Fres(weak)(θ) (dashed curve); (b) N2R[Fres(weak)(θ)-Fres(weak,1)(θ)] for N=20 (solid curve) and N=54 (dashed curve).

Fig. 9
Fig. 9

Example of a simple layered approximation of a structure that contains six circular air holes [see Eq. (7.19)].

Tables (1)

Tables Icon

Table 1 Comparison of Results for the First Two Leaky Modes of a Ring of Air Wedges a

Equations (127)

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[-Δ+V(r, θ, Nθ)]ψ=Eψ,
ψoutgoingasr=|x| ;
β=(k2ng2-E)1/2,
neffβ/k,
γ10 log10powerinputpoweroutputIβ
[Δ+k2n2(x)-β2]e=V1 e,
V1 e-(eln n2).
(Δ+k2n2)φ=β2φ,
V=k2(ng2-n2),E=k2ng2-β2,
Lφ(-Δ+V)φ=Eφ.
V(x)0,forall x,
V(x)0,for|x|r*.
Lφ(-Δ+V)φ=Eφ
-(Δ+E)φ=0,rr*.
φ=l=-+[cl(1)exp(ilθ)Hl(1)(Er)+cl(2)exp(ilθ)Hl(2)(Er)],
outwardgoingradiationcl(2)=0foralll.
V=V(r, θ, Nθ)=V(r, θ, Θ),
φ(r, θ; N)=Φ(N)(r, θ, Θ),Θ=Nθ
Φ(N)=Φ0+1N2 Φ2+O1N3,
E(N)=E0+1N2 E2+O1N3.
LavΦ0=E0Φ0,
Lav12π02πLdΘ-Δ+Vav(r, θ).
1r2 Θ2Φ2(p)=[V(r, θ, Θ)-Vav(r, θ)]Φ0(r, θ)
(Lav-E0)Φ2(h)=E2+r22π02π|Θ-1[V(r, θ, Θ)-Vav(r, θ)]|2dΘ×Φ0(r, θ).
E,q(x; β)exp[i(βx3-ωt)]Φ0(|x|, θ; ω)+1N2 Φ2(|x|, θ, Nθ; ω)+O1N3.
Θ=Nθ
φ(r, θ)=Φ(r, θ, Θ).
Δ=Δr+1r2 θ2,
Δrr2+1r r.
-Δr-1r2 (θ+NΘ)2+V(r, θ, Θ)Φ=EΦ,
L-2Nr2 θΘ-N2r2 Θ2Φ=EΦ.
Φ(N)=Φ0+1N Φ1+1N2 Φ2+1N3 Φ3+1N4 Φ4+Φ5(N),
E(N)=E0+1N E1+1N2 E2+E3(N).
O(N2):1r2 Θ2Φ0=0,
O(N):1r2 Θ2Φ1=-2r2 θΘΦ0,
O(1):1r2 Θ2Φ2=-2r2 θΘΦ1+(L-E0)Φ0,
O(N-1):1r2 Θ2Φ3=-2r2 θΘΦ2+(L-E0)Φ1-E1Φ0,
O(N-2):1r2 Θ2Φ4=-2r2 θΘΦ3+(L-E0)Φ2-E1Φ1-E2Φ0.
1r2 Θ2F=G(r, θ, Θ).
02πG(r, θ, p)dp=0.
1r2 Θ2Φ0=0.
Φ0(r, θ, Θ)=Φ0(r, θ).
-2r2 θΘΦ0-1r2 Θ2Φ1=0.
1r2 Θ2Φ1=0.
Φ1(r, θ, Θ)=Φ1(r, θ).
1r2 Θ2Φ2=(L-E0)Φ0.
(Lav-E0)Φ0=0.
Lav=-Δ+Vav=-Δr-1r2 θ2+Vav(r, θ),
Vav(r, θ)=12π02πV(r, θ, p)dp.
1r2 Θ2Φ2=[V(r, θ, Θ)-Vav(r, θ)]Φ0(r, θ).
Φ2=Φ2(p)(r, θ, Θ)+Φ2(h)(r, θ),
V(r, θ, Θ)-Vav(r, θ)=|j|1ηj(r, θ)exp(ijΘ).
Φ2(p)(r, θ, Θ)=Θ-2[V(r, θ, Θ)-Vav(r, θ)]r2Φ0(r, θ)=|j|1(ij)-2ηj(r, θ)exp(ijΘ)r2Φ0(r, θ).
1r2 Θ2Φ3=-2r2 θΘΦ2+(L-E0)Φ1-E1Φ0=-2r2 θΘΦ2(p)+(L-E0)Φ1-E1Φ0.
(Lav-E0)Φ1=E1Φ0,
ΘΦ3(r, θ, Θ)=-2θΦ2(p)(r, θ, Θ).
1r2 Θ2Φ4=4r2 θ2Φ2(p)+(L-E0)Φ2-E2Φ0=L-E0+4r2 θ2Φ2(p)+(L-E0)Φ2(h)-E2Φ0=Lav-E0+4r2 θ2Φ2(p)+(L-Lav)Φ2(h)+(Lav-E0)Φ2(h)-E2Φ0+(L-Lav)Φ2(p).
(Lav-E0)Φ2(h)=E2Φ0-12π02π(L-Lav)Φ2(p)dp=E2Φ0-12π02π[V(r, θ, p)-Vav(r, θ)]Φ2(p)(r, θ, p)dp=E2+r22π02π|p-1[V(r, θ, p)-Vav(r, θ)]|2dpΦ0(r, θ),
β=(k2ng2-E)1/2
neff=k-1β.
γ=2×105ln(10)Iβ.
Φ=Φ0+1N2 Φ2+Φ3(N),
E=E0+1N2 E2+E3(N).
Fres-Δ+V-E0+1N2 E2Φ0+1N2 Φ2.
Fres=E3(N)Φ0+1N2 Φ2+Φ3(N)-Δ+V-E0+1N2 E2Φ3(N).
Fres(1)=1Nr2 Θ2Φ3.
Fres(1)=-2N (θΦ0)[Θ-1(V-Vav)].
FresFres(1)O1N,Fres-Fres(1)O1N2.
Fres(weak)(θ)=0r dr-ϕR-1r ϕR-ϕR1r2 θ2+ϕRV-E0-1N2 E2×Φ0+1N2 Φ2,
Fres(weak,1)(θ)=-2N0r drϕR(θΦ0)[Θ-1(V-Vav)].
-Δr+l2r2+Vav(r)-E0f(r)=0.
f(r)=k*Hl(1)(E0r),rr*
(Lav-E0)Φ2(h)=[E2+Q(r)]Φ0,
-Δr+l2r2+Vav(r)-E0U(r)=[E2+Q(r)]f(r).
-Δr+l2r2-E0U(r)=E2f(r),rr*.
-Δr+l2r2+Vav(r)-E0Gin[F]=F(r),
0r<r*(rRin),
-Δr+l2r2-E0Gout[F]=F(r),r*r(rRout).
U(r)=E2Gin[f](r)+Gin[Qf](r)
U(r)=ξ*Hl(1)(E0r)+η*Hl(2)(E0r)+E2Gout[f](r)
ξ*(E2)η*(E2)=E2Y-1(r*)×Gin[f](r*)-Gout[f](r*)E0-1/2{Gin[f](r*)-Gout[f](r*)}+Y-1(r*)Gin[Qf](r*)E0-1/2Gin[Qf](r*),
Y(r)=Hl(1)(E0r)Hl(2)(E0r)sHl(1)(E0r)sHl(2)(E0r).
z=E0r,z*=E0r*,G(z)=U[r(z)].
z2G+zG+(z2-l2)G=E2E0 k*z2Hl(1)(z),
r(z)r*.
G(z)=U[r(z)]=ξ*(E2)Hl(1)(z)+η*(E2)Hl(2)(z)+E2E0 k*Gout[Hl(1)](z).
ξ*(p)(z)Hl(1)(z)+η*(p)(z)Hl(2)(z),
ξ*(p)[h](z)=-z*zHl(2)(ζ)h(ζ)ζ2W{Hl(1), Hl(2)}dζ,
η*(p)[h](z)=+z*zHl(1)(ζ)h(ζ)ζ2W{Hl(1), Hl(2)}dζ.
Gout[F](z)={ξ*(p)[ζ2F](z)Hl(1)(z)+η*(p)[ζ2F](z)Hl(2)(z)},
G(z)=U[r(z)]=ξ*(E2)Hl(1)(z)+η*(E2)Hl(2)(z)-iπ4E2E0 k*z*zHl(2)(ζ)Hl(1)(ζ)ζ dζHl(1)(z)+iπ4E2E0 k*z*z[Hl(1)(ζ)]2ζ dζHl(2)(z).
ζ[Hl(j)(ζ)]2dζ=ζ22 {[Hl(j)(ζ)]2-Hl-1(j)(ζ)×Hl+1(j)(ζ)}.
η*(E2)Hl(2)(z)+ik*π4E2E0ζ22 {[Hl(1)(ζ)]2
-Hl-1(1)(ζ)Hl+1(1)(ζ)}|z*zHl(2)(z)
=η*(E2)-ik*π4E2E0z*22 {[Hl(1)(z*)]2-Hl-1(1)(z*)Hl+1(1)(z*)}Hl(2)(z)+ik*π4E2E0z22 {[Hl(1)(z)]2-Hl-1(1)(z)Hl+1(1)(z)}Hl(2)(z).
η*(E2)-ik*π4E2E0z*22 {[Hl(1)(z*)]2
-Hl-1(1)(z*)Hl+1(1)(z*)}=0.
η*(E2)=η*0+E2η*1,
V(r, Θ)=i=1Lj=1Mi1[ri,ri+1](r)1[Θj, Θj+1](Θ)Vi,j,
Vav(r)=Vav,i,rRi.
Vav(r)=k2fi[ng2-nh2],rRi.
f(r)=σiy1(z(r))+τiy2(z(r)),
σ1τ1=10
σLτL=T(E0)σ1τ1,
T(E0)=TL(E0)T3(E0)T2(E0).
(01)T(E0)10=0
T21(E0)=0,
(Lav-E0)Φ2(h)=[E2+Q(r)]Φ0,
-Δr+l2r2+Vav(r)-E0U(r)=[E2+Q(r)]f(r).
Uq(r)=Uq(p)[z(r)]+ξqy1[z(r)]+ηqy2[z(r)],
ξη=Fq(E2)+Tqξq-1ηq-1.
ξL(E2)ηL(E2)=FL+TLFL-1+TLTL-1FL-2++TLTL-1T3F2.
Φ0(r, θ)=σLHl(1)(E0r).
ηL(E2)-iσLπ4E2E0zL22 {[Hl(1)(zL)]2
-Hl-1(1)(zL)Hl+1(1)(zL)}=0,
ηL(E2)ηL0+E2ηL1
Φ2(p)(r, θ, Θ)=r2Φ0(r, θ)Θ-2[V(r, Θ)-Vav(r)],
Q(r)=r22π02π|p-1[V(r, p)-Vav(r)]|2dp.
Θ-1[V(r, Θ)-Vav(r)]
=C1-Vav,iΘ+j=1Mi 1[Θj, Θj+1](θ)[Sj+Vi,j(Θ-Θj)],
Sj=k=1j-1Vi,k(Θk+1-Θk)
Θ-2[V(r, Θ)-Vav(r)]=C2+C1Θ-12 Vav,iΘ2+j=1Mi 1[Θj, Θj+1](Θ)Tj+Sj(Θ-Θj)+12 Vi,j(Θ-Θj)2,
Tj=k=1j-1[Sk(Θk+1-Θk)+12 Vi,k(Θk+1-Θk)2],
C1=πVav,i-TMi+1,
C2=-πC1+(2π)26 Vav,i-12πj=1Mi(Tj(Θj+1-Θj)+12 Sj(Θj+1-Θj)2+16 Vi,j(Θj+1-Θj)3).
12π02π|p-1[V(r, p)-Vav(r)]|2dp
=C12+1π C1TMi+1+12πj=1Mi(Sj2(Θj+1-Θj)+Sj(Vi,j-Vav,i)×(Θj+1-Θj)2+13 (Vi,j-Vav,i)2(Θj+1-Θj)3).

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