Abstract

Boundary conditions at a dielectric interface are expressed in terms of electrodynamic vector and scalar potentials that are expressed in terms of two-dimensional scalar solutions to the wave equation. Possible field configurations are expressed in terms of linear combinations of the scalar solutions. Their behavior at a dielectric interface and within a region of constant permittivity is investigated, and a parameter that is used to define propagation modes in cylindrical dielectric waveguides is shown to have the same value in regions with different permittivities. Scalar expressions that include this parameter are used to calculate exact expressions for fields and propagation constants in a cylindrical dielectric waveguide with three coaxial regions. The characteristics of propagation modes in these waveguides are discussed.

© 1998 Optical Society of America

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References

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  1. D. Hondros, P. Debye, “Elektromagnetische Wellen an dielektrischen Drähten,” Ann. Phys. (Leipzig) Ser. 4 32, 465–476 (1910).
    [CrossRef]
  2. D. Hondros, “Über elektromagnetische Drahtwellen,” Ann. Phys. (Leipzig) Ser. 4 30, 905–950 (1909).
    [CrossRef]
  3. E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Am. 51, 491–498 (1961).
    [CrossRef]
  4. S. Kawakami, S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
    [CrossRef]
  5. C. J. Chung, A. Safaai-Jazi, “Narrow-band spectral filter made of W-index and step-index fibers,” J. Lightwave Technol. LT-10, 42–45 (1992).
    [CrossRef]
  6. K. Petermann, “Fundamental mode microbending loss in graded-index and W fibres,” Opt. Quantum Electron. 9, 167–175 (1977).
    [CrossRef]
  7. M. H. Kuhn, “The influence of the refractive index step due to the finite cladding of homogeneous fibres on the hybrid properties of modes,” Archiv Elektron. Übertragungstech. 28, 393–401 (1974).
  8. J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1992), pp. 174–182.
  9. M. M. Astrahan, “Guided waves on hollow dielectric tubes,” Ph.D dissertation (Northwestern University, Evanston, Ill., 1949).
  10. D. G. Kiely, Dielectric Aerials (Methuen, London, 1953), pp. 82–96.
  11. H.-G. Unger, “Dielektrische Rohre als Wellenleiter,” Archiv Elektron. Übertragungstech. 8, 241–252 (1954).
  12. H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), pp. 308–321.
  13. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [CrossRef] [PubMed]
  14. M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE J. Quantum Electron. QE-18, 535–542 (1982).
    [CrossRef]
  15. G. Cancellieri, Single-Mode Optical Fibres (Pergamon, Oxford, 1991), pp. 129–145.
  16. A. W. Snyder, W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
    [CrossRef]
  17. P. K. Mishra, S. I. Hosain, I. C. Goyal, A. Sharma, “Scalar variational analysis of single mode, graded-core, W-type fibres,” Opt. Quantum Electron. 16, 287–296 (1984).
    [CrossRef]
  18. J. E. Sader, “Analysis of arbitrarily perturbed circular profiles by implementation of integral-equation theory,” J. Opt. Soc. Am. A 7, 2094–2099 (1990).
    [CrossRef]
  19. M. S. Wu, M. H. Lee, W. H. Tsai, “Variational analysis of single-mode graded core W-fibers,” J. Lightwave Technol. LT-14, 121–125 (1996).
  20. G. W. Wen, “Steepest-descent approximation theory for guided modes of weakly guiding optical waveguides and fibers,” J. Opt. Soc. Am. A 8, 295–302 (1991).
    [CrossRef]
  21. C. Vasallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991), pp. 104–107.
  22. C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
    [CrossRef]
  23. C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part II: asymptotic solution for general weak inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 250–253 (1969).
    [CrossRef]
  24. F. Irrera, C. G. Someda, “Monomode fibers with smooth core-cladding transitions and index dips: an analytical investigation,” J. Lightwave Technol. LT-6, 1177–1184 (1988).
    [CrossRef]
  25. K. S. Chiang, “Analysis of optical fibers by the effective-index method,” Appl. Opt. 25, 348–354 (1986).
    [CrossRef] [PubMed]
  26. M. R. Shenoy, K. Thyagarajan, A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. LT-6, 1285–1291 (1988).
    [CrossRef]
  27. A. W. Snyder, R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. 69, 1663–1671 (1979).
    [CrossRef]
  28. H. Matsumura, T. Suganuma, “Normalization of single-mode fibers having an arbitrary index profile,” Appl. Opt. 19, 3151–3158 (1980).
    [CrossRef] [PubMed]

1996 (1)

M. S. Wu, M. H. Lee, W. H. Tsai, “Variational analysis of single-mode graded core W-fibers,” J. Lightwave Technol. LT-14, 121–125 (1996).

1992 (1)

C. J. Chung, A. Safaai-Jazi, “Narrow-band spectral filter made of W-index and step-index fibers,” J. Lightwave Technol. LT-10, 42–45 (1992).
[CrossRef]

1991 (1)

1990 (1)

1988 (2)

F. Irrera, C. G. Someda, “Monomode fibers with smooth core-cladding transitions and index dips: an analytical investigation,” J. Lightwave Technol. LT-6, 1177–1184 (1988).
[CrossRef]

M. R. Shenoy, K. Thyagarajan, A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. LT-6, 1285–1291 (1988).
[CrossRef]

1986 (1)

1984 (1)

P. K. Mishra, S. I. Hosain, I. C. Goyal, A. Sharma, “Scalar variational analysis of single mode, graded-core, W-type fibres,” Opt. Quantum Electron. 16, 287–296 (1984).
[CrossRef]

1982 (1)

M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE J. Quantum Electron. QE-18, 535–542 (1982).
[CrossRef]

1980 (1)

1979 (1)

1978 (1)

1977 (1)

K. Petermann, “Fundamental mode microbending loss in graded-index and W fibres,” Opt. Quantum Electron. 9, 167–175 (1977).
[CrossRef]

1974 (2)

M. H. Kuhn, “The influence of the refractive index step due to the finite cladding of homogeneous fibres on the hybrid properties of modes,” Archiv Elektron. Übertragungstech. 28, 393–401 (1974).

S. Kawakami, S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
[CrossRef]

1971 (1)

1969 (2)

C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
[CrossRef]

C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part II: asymptotic solution for general weak inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 250–253 (1969).
[CrossRef]

1961 (1)

1954 (1)

H.-G. Unger, “Dielektrische Rohre als Wellenleiter,” Archiv Elektron. Übertragungstech. 8, 241–252 (1954).

1910 (1)

D. Hondros, P. Debye, “Elektromagnetische Wellen an dielektrischen Drähten,” Ann. Phys. (Leipzig) Ser. 4 32, 465–476 (1910).
[CrossRef]

1909 (1)

D. Hondros, “Über elektromagnetische Drahtwellen,” Ann. Phys. (Leipzig) Ser. 4 30, 905–950 (1909).
[CrossRef]

Astrahan, M. M.

M. M. Astrahan, “Guided waves on hollow dielectric tubes,” Ph.D dissertation (Northwestern University, Evanston, Ill., 1949).

Buck, J. A.

J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1992), pp. 174–182.

Cancellieri, G.

G. Cancellieri, Single-Mode Optical Fibres (Pergamon, Oxford, 1991), pp. 129–145.

Chiang, K. S.

Chung, C. J.

C. J. Chung, A. Safaai-Jazi, “Narrow-band spectral filter made of W-index and step-index fibers,” J. Lightwave Technol. LT-10, 42–45 (1992).
[CrossRef]

Debye, P.

D. Hondros, P. Debye, “Elektromagnetische Wellen an dielektrischen Drähten,” Ann. Phys. (Leipzig) Ser. 4 32, 465–476 (1910).
[CrossRef]

Ghatak, A. K.

M. R. Shenoy, K. Thyagarajan, A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. LT-6, 1285–1291 (1988).
[CrossRef]

Gloge, D.

Goyal, I. C.

P. K. Mishra, S. I. Hosain, I. C. Goyal, A. Sharma, “Scalar variational analysis of single mode, graded-core, W-type fibres,” Opt. Quantum Electron. 16, 287–296 (1984).
[CrossRef]

Hondros, D.

D. Hondros, P. Debye, “Elektromagnetische Wellen an dielektrischen Drähten,” Ann. Phys. (Leipzig) Ser. 4 32, 465–476 (1910).
[CrossRef]

D. Hondros, “Über elektromagnetische Drahtwellen,” Ann. Phys. (Leipzig) Ser. 4 30, 905–950 (1909).
[CrossRef]

Hosain, S. I.

P. K. Mishra, S. I. Hosain, I. C. Goyal, A. Sharma, “Scalar variational analysis of single mode, graded-core, W-type fibres,” Opt. Quantum Electron. 16, 287–296 (1984).
[CrossRef]

Irrera, F.

F. Irrera, C. G. Someda, “Monomode fibers with smooth core-cladding transitions and index dips: an analytical investigation,” J. Lightwave Technol. LT-6, 1177–1184 (1988).
[CrossRef]

Kawakami, S.

S. Kawakami, S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
[CrossRef]

Kiely, D. G.

D. G. Kiely, Dielectric Aerials (Methuen, London, 1953), pp. 82–96.

Kuhn, M. H.

M. H. Kuhn, “The influence of the refractive index step due to the finite cladding of homogeneous fibres on the hybrid properties of modes,” Archiv Elektron. Übertragungstech. 28, 393–401 (1974).

Kurtz, C. N.

C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
[CrossRef]

C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part II: asymptotic solution for general weak inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 250–253 (1969).
[CrossRef]

Lee, M. H.

M. S. Wu, M. H. Lee, W. H. Tsai, “Variational analysis of single-mode graded core W-fibers,” J. Lightwave Technol. LT-14, 121–125 (1996).

Matsumura, H.

Mishra, P. K.

P. K. Mishra, S. I. Hosain, I. C. Goyal, A. Sharma, “Scalar variational analysis of single mode, graded-core, W-type fibres,” Opt. Quantum Electron. 16, 287–296 (1984).
[CrossRef]

Monerie, M.

M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE J. Quantum Electron. QE-18, 535–542 (1982).
[CrossRef]

Nishida, S.

S. Kawakami, S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
[CrossRef]

Petermann, K.

K. Petermann, “Fundamental mode microbending loss in graded-index and W fibres,” Opt. Quantum Electron. 9, 167–175 (1977).
[CrossRef]

Sader, J. E.

Safaai-Jazi, A.

C. J. Chung, A. Safaai-Jazi, “Narrow-band spectral filter made of W-index and step-index fibers,” J. Lightwave Technol. LT-10, 42–45 (1992).
[CrossRef]

Sammut, R. A.

Sharma, A.

P. K. Mishra, S. I. Hosain, I. C. Goyal, A. Sharma, “Scalar variational analysis of single mode, graded-core, W-type fibres,” Opt. Quantum Electron. 16, 287–296 (1984).
[CrossRef]

Shenoy, M. R.

M. R. Shenoy, K. Thyagarajan, A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. LT-6, 1285–1291 (1988).
[CrossRef]

Snitzer, E.

Snyder, A. W.

Someda, C. G.

F. Irrera, C. G. Someda, “Monomode fibers with smooth core-cladding transitions and index dips: an analytical investigation,” J. Lightwave Technol. LT-6, 1177–1184 (1988).
[CrossRef]

Streifer, W.

C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part II: asymptotic solution for general weak inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 250–253 (1969).
[CrossRef]

C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
[CrossRef]

Suganuma, T.

Thyagarajan, K.

M. R. Shenoy, K. Thyagarajan, A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. LT-6, 1285–1291 (1988).
[CrossRef]

Tsai, W. H.

M. S. Wu, M. H. Lee, W. H. Tsai, “Variational analysis of single-mode graded core W-fibers,” J. Lightwave Technol. LT-14, 121–125 (1996).

Unger, H.-G.

H.-G. Unger, “Dielektrische Rohre als Wellenleiter,” Archiv Elektron. Übertragungstech. 8, 241–252 (1954).

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), pp. 308–321.

Vasallo, C.

C. Vasallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991), pp. 104–107.

Wen, G. W.

Wu, M. S.

M. S. Wu, M. H. Lee, W. H. Tsai, “Variational analysis of single-mode graded core W-fibers,” J. Lightwave Technol. LT-14, 121–125 (1996).

Young, W. R.

Ann. Phys. (Leipzig) Ser. 4 (2)

D. Hondros, P. Debye, “Elektromagnetische Wellen an dielektrischen Drähten,” Ann. Phys. (Leipzig) Ser. 4 32, 465–476 (1910).
[CrossRef]

D. Hondros, “Über elektromagnetische Drahtwellen,” Ann. Phys. (Leipzig) Ser. 4 30, 905–950 (1909).
[CrossRef]

Appl. Opt. (3)

Archiv Elektron. Übertragungstech. (2)

M. H. Kuhn, “The influence of the refractive index step due to the finite cladding of homogeneous fibres on the hybrid properties of modes,” Archiv Elektron. Übertragungstech. 28, 393–401 (1974).

H.-G. Unger, “Dielektrische Rohre als Wellenleiter,” Archiv Elektron. Übertragungstech. 8, 241–252 (1954).

IEEE J. Quantum Electron. (2)

S. Kawakami, S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974).
[CrossRef]

M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE J. Quantum Electron. QE-18, 535–542 (1982).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part I: formulation, solution for quadratic inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 11–15 (1969).
[CrossRef]

C. N. Kurtz, W. Streifer, “Guided waves in inhomogeneous focusing media. Part II: asymptotic solution for general weak inhomogeneity,” IEEE Trans. Microwave Theory Tech. MTT-17, 250–253 (1969).
[CrossRef]

J. Lightwave Technol. (4)

F. Irrera, C. G. Someda, “Monomode fibers with smooth core-cladding transitions and index dips: an analytical investigation,” J. Lightwave Technol. LT-6, 1177–1184 (1988).
[CrossRef]

M. R. Shenoy, K. Thyagarajan, A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. LT-6, 1285–1291 (1988).
[CrossRef]

C. J. Chung, A. Safaai-Jazi, “Narrow-band spectral filter made of W-index and step-index fibers,” J. Lightwave Technol. LT-10, 42–45 (1992).
[CrossRef]

M. S. Wu, M. H. Lee, W. H. Tsai, “Variational analysis of single-mode graded core W-fibers,” J. Lightwave Technol. LT-14, 121–125 (1996).

J. Opt. Soc. Am. (3)

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

Opt. Quantum Electron. (2)

K. Petermann, “Fundamental mode microbending loss in graded-index and W fibres,” Opt. Quantum Electron. 9, 167–175 (1977).
[CrossRef]

P. K. Mishra, S. I. Hosain, I. C. Goyal, A. Sharma, “Scalar variational analysis of single mode, graded-core, W-type fibres,” Opt. Quantum Electron. 16, 287–296 (1984).
[CrossRef]

Other (6)

C. Vasallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991), pp. 104–107.

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), pp. 308–321.

J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1992), pp. 174–182.

M. M. Astrahan, “Guided waves on hollow dielectric tubes,” Ph.D dissertation (Northwestern University, Evanston, Ill., 1949).

D. G. Kiely, Dielectric Aerials (Methuen, London, 1953), pp. 82–96.

G. Cancellieri, Single-Mode Optical Fibres (Pergamon, Oxford, 1991), pp. 129–145.

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Figures (6)

Fig. 1
Fig. 1

Propagation curves of a two-region dielectric waveguide and behavior of P for modes for which |n| is 1. The dashed and dashed–dotted curves describe P. Values of P in the lower group of curves, labeled he, are negative, and values in the upper group, labeled eh, are positive.

Fig. 2
Fig. 2

Geometry of a cylindrical waveguide with three dielectric regions, denoted κ1, κ2, and κ3.

Fig. 3
Fig. 3

Values of r1/λ and r2/λ for which propagation occurs when k1=1.91 and |n|=1.

Fig. 4
Fig. 4

Possible values of k1r1, r1/λ and r2/λ for which boundary conditions can be satisfied for the labeled solutions. The solid curves describe k1r1 and the dotted curves describe r1/λ.

Fig. 5
Fig. 5

Positive values of P as a function of r2/λ.

Fig. 6
Fig. 6

Negative values of P as a function of r2/λ.

Equations (85)

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

1Φ1x+Ax1t=2Φ2x+Ax2t,
Az1x-Ax1z=Az2x-Ax2z,
Ay1x-Ax1y=Ay2x-Ax2y
Φ1y+Ay1t=Φ2y+Ay2t,
Φ1z+Az1t=Φ2z+Az2t,
Az1y-Ay1z=Az2y-Ay2z.
u(x, y)exp i(ωt-ωzz),
v(x, y)exp i(ωt-ωzz),
Φ1=ϕ11u1,
Az1=az11u1,
Ay1=id11 u1y+ia11 v1x,
Ax1=id11 u1x-ia11 v1y,
Axx+Ayy+Azz+μ Φt=0
u1x-κ1ωz2k12κ1ωzωk12ωκ1q1-κ1ωzωk12c2κ1ω2k12c2ωzq100k12q1ϕ11 az11 a11=V1 V2 V3
κi=i/0,
ki2=κiω2c2-ωz2,
qi=vi/yui/x.
u1x=u2x
v1y=v2y.
ϕi1azi1ai1=ω/cωz/c0,(1/κi)ωzω/c20,(1/κi)ωzq0(ωωz)/(cki)2,
Φ1=ϕ12v1,
Az1=az12v1,
Ay1=id12 v1y+ia12 u1x,
Ax1=id12 v1x-ia12 u1y.
u1x-ωz2k12qωzωk12q-ω-ωzqωq0κ1ωzωk12c2qκ1ω2k12c2q-ωzϕ12az12a12=W1W2W3.
ϕi2azi2ai2=ω/cωz/c0,0ω/c2ωωzc21+ωz2qki2,00ωωzc2.
ϕi1azi1ai1=t1(1/κi)ωzω/c20+t2(1/κi)ωzq0(ωωz)/(cki)2,
ϕi2azi2ai2=t30ω/c2ωωzc21+ωz2qki2+t400ωωzc2,
B=×A
E=-Φ-A/t.
bi=-(ai1vi+ai2ui)
ei=(1/ki2)[(ωzϕi1-ωazi1)ui+(ωzϕi2-ωazi2)vi],
Bxi=ωz bix-κiωc2eiyExi=ω biy+ωz eix
Byi=ωz biy+κiωc2eixEyi=-ω bix+ωz eiy
Bzi=iki2biEzi=iki2ei.
bi=-ωc21ωzt31+ωz2qki2+t4ui-ωc2ωzki2t2vi,
ei=-1κit1-t2 ωz2qki2ui-1ki2ω2c2t3vi.
ui=αiofi+αigi,
vi=βiofi+βigi.
ωc21ωzt31+ωz2qki2+t4αi+ωc2ωzki2t2βi=0,
1κit1-t2 ωz2qki2αi+1ki2ω2c2t3βi=0.
t1=ωz2qki2t2-κiω2ki2c2βiαit3,
t4=-ωz2ki2βiαit2-1+ωz2qki2t3.
bi=ωc2ωzki2t2βiαiui-vi,
ei=ω2c21ki2t3βiαiui-vi.
ωz2qk12t2-κ1ω2k12c2β1α1t3=ωz2qk22t2-κ2ω2k22c2β2α2t3,
ωz2k12β1α1t2+1+ωz2qk12t3=ωz2k22β2α2t2+1+ωz2qk22t3.
1qβ1α1=i 1k221tκ2ω2c2-tωz2,
1qβ2α2=i 1k121tκ1ω2c2-tωz2,
t=-i t2t3.
k1r0 Jn(k1r0)Jn(k1r0)=-nK221tκ2ω2c2-tωz2,
K2r0 Kn(K2r0)Kn(K2r0)=nk121tκ1ω2c2-tωz2,
1t=r0n(1/K22+1/k12)Jn(k1r0)k1Jn(k1r0)+Kn(K2r0)K2Kn(K2r0),
t=ω2/c2ωz2r0n(1/K22+1/k12)κ1Jn(k1r0)k1Jn(k1r0)+κ2Kn(K2r0)K2Kn(K2r0).
n2ωz21K22+1k122
=r02ω2c2Jn(k1r0)k1Jn(k1r0)+Kn(K2r0)K2Kn(K2r0)×κ1Jn(k1r0)k1Jn(k1r0)+κ2Kn(K2r0)K2Kn(K2r0).
1P=r0|n|(1/K22+1/k12)Jn(k1r0)k1Jn(k1r0)+Kn(K2r0)K2Kn(K2r0).
u1(r)=s1u2(r)+(1-s2)v2(r),
v1(r)=(1-s1)u2(r)+s2v2(r),
b2=-ωc21ωzt31+ωz2qk22+t4×[s1u2(r1)+(1-s2)v2(r1)]-ωc2ωzk22t2[(1-s1)u2(r1)+s2v2(r1)],
e2=-1κ2t1-t2 ωz2qk22[s1u2(r1)+(1-s2)v2(r1)]-1k22ω2c2t3[(1-s1)u2(r1)+s2v2(r1)].
b2=-ωc21ωzt31+ωz2qk22+t4u2(r1)-ωc2ωzk22t2v2(r1),
e2=-1κ2t1-t2 ωz2qki2u2(r1)-1k22ω2c2t3v2(r1),
t1=s1t1+ωz2k22(s2q-s1q)t2+κ2ω2k22c2(1-s1)+1+ωz2qk22q(1-s2)t3+q(1-s2)t4,
t2=s2t2+k22ωz2(1-s2)1+ωz2qk22t3+t4,
t3=k22c2κ2ω2(1-s2)t1-ωz2qc2κ2ω2(1-s2)t2+s2t3,
t4=-k22c2κ2ω2(1-s2)1+ωz2qk22t1+ωz2qc2κ2ω2(1-s2)1+ωz2qk22+ωz2k22(1-s1)t2-1+ωz2qk22s2-1+ωz2qk22s1t3+s1t4.
t2=s2-(1-s2) β2α2t2,
t3=s2-(1-s2) β2α2t3.
k1r1 Jn(k1r1)Jn(k1r1)=nk221tκ2ω2c2-tωz2,
K3r2 Kn(K3r2)Kn(K3r2)=nk221tκ2ω2c2-tωz2,
k1r1 Jn(k1r1)Jn(k1r1)=K3r2 Kn(K3r2)Kn(K3r2).
Sn(k2r)=pJn(k2r)+Yn(k2r),
k2r1 Sn(k2r1)Sn(k2r1)=nk121tκ1ω2c2-tωz2
k2r2 Sn(k2r2)Sn(k2r2)=-nK321tκ3ω2c2-tωz2.
u=αof+αg,
v=βof+βg,
q=βof/y+βg/yαof/x+αg/x.
q=-βαff/xfxgy-fygxfxg-f gx.
f(r, θ)=Jn(k1r)exp(inθ)
g(r, θ)=Yn(k1r)exp(inθ),
f(r, θ)=Kn(K2r)exp(inθ),
g(r, θ)=In(K2r)exp(inθ).
1qβα=i k1r0nJn(k1r0)Jn(k1r0)inthecore,
=i K2r0nKn(K2r0)Kn(K2r0)inthecladding.

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