Abstract

A simple method is presented for finding the modes on those optical waveguides with a cladding refractive index that differs only slightly from the refractive index of the core. The method applies to waveguides of arbitrary refractive index profile, arbitrary number of propagating modes, and arbitrary cross section. The resulting modal fields and their progagation constants display the polarization properties of the waveguide contained within the ∇ term of the vector wave equation. Examples include modes on waveguides with circular symmetry and waveguides with two preferred axes of symmetry, e.g., an elliptical core. Only a minute amount of eccentricity is necessary for the well-known LP modes to be stable on an elliptical core, while the circle modes couple power among themselves.

© 1978 Optical Society of America

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  1. A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides,” IEEE Trans. MTT 17, 1130–1138 (1969).
    [Crossref]
  2. R. Yamada and Y. Inabe, “Guided waves in an optical square-law medium,” J. Opt. Soc. Am. 64, 964–969 (1974).
    [Crossref]
  3. J. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. MTT 22, 938–945 (1974).
    [Crossref]
  4. C. N. Kurtz, “Scalar and vector mode relations in gradient-index light guides,” J. Opt. Soc. Am. 65, 1235–1240 (1975). This paper is specialized to waveguides of circular symmetry. Then standard perturbation methods can be applied to the vector wave equation, with the vector operator ∇t2 expanded in circular coordinates. Solutions to the ∇t∊ = 0 portion of the equation automatically obey the appropriate symmetry properties of the structure because they are built into ∇t2. The ∇t∊ = 0 portion of the vector wave equation can then be reduced to a cylindrical scalar wave equation.
    [Crossref]
  5. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  6. A. W. Snyder and C. Pask, “Light absorption in the bee photoreceptor,” J. Opt. Soc. Am.998–1008 (1972).
    [Crossref]
  7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 116.
  8. R. B. Adler, “Waves on inhomogeneous cylindrical structures,” Proc. IRE 40, 339–348 (1952).
    [Crossref]
  9. A. W. Snyder and J. D. Love, Optical waveguide theory [Wiley and Chapman and Hall, London (in press)].
  10. Recall that the potential of an electrostatic dipole can be expressed by two independent parameters, the dipole moment p= Qd and the separation distance d between a plus and minus charge, each of strength Q. If d is sufficiently small, then the potential ϕ at an arbitrary positions is ϕ(p,d) ≃ ϕ(p,o) where ϕ(p,o) is the potential of a point (d= o) dipole. A point dipole is unphysical because (a) charges that occupy the same position cancel and (b) p is arbitrary because Q= ∞. Nevertheless, the fields of a physical (d ≠ o) dipole are well approximated by those of the point dipole. Note that p, d, and Q of the dipole play the roles of V, θc and λ−1, respectively, of the waveguide.
  11. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1963), p. 285.
  12. It may appear that the ∇t ln∊ has a large effect for rapidly varying ∊ profiles; however, this term is bounded by θc2. Taking the extreme situation of a step profile, where ∊ = ∊co + (∊cl − ∊co)s(r − ρ) and s(r − ρ) is a step function, we see that ∇t ln∊=θc2δ(r-ρ). In other words, ∇t ln∊ is zero except at r= ρ, but the delta function δ(r − ρ) has a minute strength of θc2, where θc is given by Eq. (8). While it is true that the ∇t ln∊ term is most significant for step profiles, it is usually incorporated into the mathematics via the boundary conditions. We show that our procedure leads to highly accurate results even for this extreme case (Sec. IV).
  13. A circularly symmetric waveguide is unchanged if it is rotated through an arbitrary angle or reflected in an axis. Hence if a mode of the waveguide is rotated through an arbitrary angle it must remain a mode (not necessarily the same mode) with the same β. Now, if the pattern e˜xe of Fig. 2(a), for example, is rotated through an arbitrary angle it is then represented by a linear combination of all four patterns in Fig. 2(a). Thus if the individual patterns are modes of the waveguide, all four must have the same β. But if the fields e˜xe,e˜xo,e˜yo are substituted into Eq. (17) we find that the four corrected β’s are not all equal. Therefore nco = ncl modes are not nco ≅ ncl modes.
  14. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).
  15. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, 1972).
  16. The patterns e1 and e4 of Fig. 2(b) are unchanged by reflection in an arbitrary axis and by rotation through an arbitrary angle, consistent with their being nondegenerate modes of a circularly symmetric waveguide. However, under arbitrary rotation and reflections e2 changes into a pattern which is a linear combination of e2 and e3. Symmetry demands that this new combination is also a mode, which in turn requires that e2 and e3 have identical β’s. Indeed, by substituting e2 and e3 into Eq. (17) we verify that they have identical β’s. Thus the linear combinations given by Eq. (23) are consistent with the requirements of symmetry and the results of Eq. (17). Analogous arguments show that these consistencies remain when l ≠ 1.
  17. M. Born and E. Wolf, Principles of Optics, 3rd edition (Pergamon, New York, 1964).
  18. E. Snitzer and H. Osterberg, “Observed dielectric waveguide modes in the visual spectrum,” J. Opt. Soc. Am. 51, 499–505 (1961).
    [Crossref]
  19. This result is found by substituting the fields e˜x=ψ xˆ and e˜y=ψy^ into Eq. (17) to determine βx and βy where ψ is the solution to the scalar wave equation in elliptical geometry. It is not sufficient to approximate ψ by the solutions of the circular symmetric fiber as done for β˜e-β˜o in Appendix D. Instead, higher-order terms are necessary. Alternatively,9 one can perturb about Maxwell’s equations in circular symmetry as formulated in Ref. 21; however, because et must include terms of order θc2, the previously reported result of Marcuse (Ref. 22) is inaccurate. We have included θc2 terms using the expansion presented in Ref. 1.
  20. A. W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1268–1277 (1972). Note that the coupling coefficient is C=(β˜+-β˜-)/2, where C is given by Eq. (26a) of the 1972 paper.
    [Crossref]
  21. A. W. Snyder, “Mode propagation in optical waveguides,” Electron. Lett. 6, 561–562 (1970).
    [Crossref]
  22. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 159, 164.

1975 (1)

1974 (2)

R. Yamada and Y. Inabe, “Guided waves in an optical square-law medium,” J. Opt. Soc. Am. 64, 964–969 (1974).
[Crossref]

J. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. MTT 22, 938–945 (1974).
[Crossref]

1972 (2)

A. W. Snyder and C. Pask, “Light absorption in the bee photoreceptor,” J. Opt. Soc. Am.998–1008 (1972).
[Crossref]

A. W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1268–1277 (1972). Note that the coupling coefficient is C=(β˜+-β˜-)/2, where C is given by Eq. (26a) of the 1972 paper.
[Crossref]

1971 (1)

1970 (1)

A. W. Snyder, “Mode propagation in optical waveguides,” Electron. Lett. 6, 561–562 (1970).
[Crossref]

1969 (1)

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides,” IEEE Trans. MTT 17, 1130–1138 (1969).
[Crossref]

1961 (1)

1952 (1)

R. B. Adler, “Waves on inhomogeneous cylindrical structures,” Proc. IRE 40, 339–348 (1952).
[Crossref]

Adler, R. B.

R. B. Adler, “Waves on inhomogeneous cylindrical structures,” Proc. IRE 40, 339–348 (1952).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd edition (Pergamon, New York, 1964).

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 116.

Gloge, D.

Inabe, Y.

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Kurtz, C. N.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1963), p. 285.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1963), p. 285.

Love, J. D.

A. W. Snyder and J. D. Love, Optical waveguide theory [Wiley and Chapman and Hall, London (in press)].

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, 1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 159, 164.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 116.

Okamoto, K.

J. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. MTT 22, 938–945 (1974).
[Crossref]

Okoshi, J.

J. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. MTT 22, 938–945 (1974).
[Crossref]

Osterberg, H.

Pask, C.

A. W. Snyder and C. Pask, “Light absorption in the bee photoreceptor,” J. Opt. Soc. Am.998–1008 (1972).
[Crossref]

Snitzer, E.

Snyder, A. W.

A. W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1268–1277 (1972). Note that the coupling coefficient is C=(β˜+-β˜-)/2, where C is given by Eq. (26a) of the 1972 paper.
[Crossref]

A. W. Snyder and C. Pask, “Light absorption in the bee photoreceptor,” J. Opt. Soc. Am.998–1008 (1972).
[Crossref]

A. W. Snyder, “Mode propagation in optical waveguides,” Electron. Lett. 6, 561–562 (1970).
[Crossref]

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides,” IEEE Trans. MTT 17, 1130–1138 (1969).
[Crossref]

A. W. Snyder and J. D. Love, Optical waveguide theory [Wiley and Chapman and Hall, London (in press)].

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd edition (Pergamon, New York, 1964).

Yamada, R.

Appl. Opt. (1)

Electron. Lett. (1)

A. W. Snyder, “Mode propagation in optical waveguides,” Electron. Lett. 6, 561–562 (1970).
[Crossref]

IEEE Trans. MTT (2)

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides,” IEEE Trans. MTT 17, 1130–1138 (1969).
[Crossref]

J. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. MTT 22, 938–945 (1974).
[Crossref]

J. Opt. Soc. Am. (5)

Proc. IRE (1)

R. B. Adler, “Waves on inhomogeneous cylindrical structures,” Proc. IRE 40, 339–348 (1952).
[Crossref]

Other (12)

A. W. Snyder and J. D. Love, Optical waveguide theory [Wiley and Chapman and Hall, London (in press)].

Recall that the potential of an electrostatic dipole can be expressed by two independent parameters, the dipole moment p= Qd and the separation distance d between a plus and minus charge, each of strength Q. If d is sufficiently small, then the potential ϕ at an arbitrary positions is ϕ(p,d) ≃ ϕ(p,o) where ϕ(p,o) is the potential of a point (d= o) dipole. A point dipole is unphysical because (a) charges that occupy the same position cancel and (b) p is arbitrary because Q= ∞. Nevertheless, the fields of a physical (d ≠ o) dipole are well approximated by those of the point dipole. Note that p, d, and Q of the dipole play the roles of V, θc and λ−1, respectively, of the waveguide.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1963), p. 285.

It may appear that the ∇t ln∊ has a large effect for rapidly varying ∊ profiles; however, this term is bounded by θc2. Taking the extreme situation of a step profile, where ∊ = ∊co + (∊cl − ∊co)s(r − ρ) and s(r − ρ) is a step function, we see that ∇t ln∊=θc2δ(r-ρ). In other words, ∇t ln∊ is zero except at r= ρ, but the delta function δ(r − ρ) has a minute strength of θc2, where θc is given by Eq. (8). While it is true that the ∇t ln∊ term is most significant for step profiles, it is usually incorporated into the mathematics via the boundary conditions. We show that our procedure leads to highly accurate results even for this extreme case (Sec. IV).

A circularly symmetric waveguide is unchanged if it is rotated through an arbitrary angle or reflected in an axis. Hence if a mode of the waveguide is rotated through an arbitrary angle it must remain a mode (not necessarily the same mode) with the same β. Now, if the pattern e˜xe of Fig. 2(a), for example, is rotated through an arbitrary angle it is then represented by a linear combination of all four patterns in Fig. 2(a). Thus if the individual patterns are modes of the waveguide, all four must have the same β. But if the fields e˜xe,e˜xo,e˜yo are substituted into Eq. (17) we find that the four corrected β’s are not all equal. Therefore nco = ncl modes are not nco ≅ ncl modes.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, 1972).

The patterns e1 and e4 of Fig. 2(b) are unchanged by reflection in an arbitrary axis and by rotation through an arbitrary angle, consistent with their being nondegenerate modes of a circularly symmetric waveguide. However, under arbitrary rotation and reflections e2 changes into a pattern which is a linear combination of e2 and e3. Symmetry demands that this new combination is also a mode, which in turn requires that e2 and e3 have identical β’s. Indeed, by substituting e2 and e3 into Eq. (17) we verify that they have identical β’s. Thus the linear combinations given by Eq. (23) are consistent with the requirements of symmetry and the results of Eq. (17). Analogous arguments show that these consistencies remain when l ≠ 1.

M. Born and E. Wolf, Principles of Optics, 3rd edition (Pergamon, New York, 1964).

This result is found by substituting the fields e˜x=ψ xˆ and e˜y=ψy^ into Eq. (17) to determine βx and βy where ψ is the solution to the scalar wave equation in elliptical geometry. It is not sufficient to approximate ψ by the solutions of the circular symmetric fiber as done for β˜e-β˜o in Appendix D. Instead, higher-order terms are necessary. Alternatively,9 one can perturb about Maxwell’s equations in circular symmetry as formulated in Ref. 21; however, because et must include terms of order θc2, the previously reported result of Marcuse (Ref. 22) is inaccurate. We have included θc2 terms using the expansion presented in Ref. 1.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 116.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 159, 164.

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Figures (11)

FIG. 1
FIG. 1

(a) A waveguide with cylindrical symmetry. (b) The refractive index profile in some arbitrary cross section.

FIG. 2
FIG. 2

(a) The nco = ncl or LP modes for l = 1. Note that e ˜ x e and e ˜ y o are symmetric under reflections in the x and y axes, while e ˜ x o and e ˜ y e are antisymmetric. If any one of the above fields is rotated through an arbitrary angle it transforms into a linear combination of all four. Note also that apart from normalization, e ˜ x e = e1 + e2, e ˜ x o = e3e4, e ˜ y e = e3 + e4 and e ˜ y o = e1e2, where the e’s are shown in (b). (b) The nconcl modes for l = 1. Under an arbitrary reflection and rotation, e1 and e4 are unchanged, while either e2 or e4 transform into linear combinations of e2 and e4.

FIG. 3
FIG. 3

Waveguides with preferred axes of symmetry. (a) Composite, two-parallel-waveguide system, and (b) an elliptical core.

FIG. 4
FIG. 4

Transition from circle to ellipse modes for l = 1 modes. An electric field vector maintains its orientation to the interface, i.e., if it was initially perpendicular it remains perpendicular, as the eccentricity increases. Using this heuristic principle one can anticipate the way in which a particular circle mode changes s the eccentricity increases.

FIG. 5
FIG. 5

An example of a solution of the scalar wave equation corresponding to the l = 1 mode. The β ˜ ’s of the even and odd circle mode are identical unlike the β ˜ ’s for the even and odd modes of the elliptical core.

FIG. 6
FIG. 6

The difference in β ˜ ’s for l = 1 modes of the circularly symmetric, step profile waveguide. Each LP mode of Fig 2(a) is formed by linear combination of β1, β2 modes or β3, β4 modes.

FIG. 7
FIG. 7

The parameter Λ defined by Eq. (39) determines the ratio ai/bi of the ellipse i = 1 fields, Eq. (25). When |Λ| ≫ 1, the modes are uniformly polarized (LP modes), while when |Λ| ≪ 1 the modal fields are those of a circular core.

FIG. 8
FIG. 8

The difference in β’s of the x- and y-polarized, fundamental ellipse modes.

FIG. 9
FIG. 9

The four fundamental modes of the two-parallel-waveguide system shown in Fig. 3(a).

FIG. 10
FIG. 10

The transition of an l = 1 mode of the two-parallel-waveguide system as the separation increases. When the fibers are close, the composite mode appears like a superposition of two e ˜ x e modes of Fig. 2(a). When the fibers are well separated, the composite mode appears like a superposition of the e2 model of Fig. 2(b).

FIG. 11
FIG. 11

The parameter Λ defined by Eq. (4b) which determines the composition of l = 1 modes.

Tables (1)

Tables Icon

TABLE I Modal parameters for a step profile, circularly symmetric waveguide when nconcl.

Equations (93)

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E ( x , y , z ) = e ( x , y ) e i β z = ( e t + e z ) e i β z ,
H ( x , y , z ) = h ( x , y ) e i β z = ( h t + h z ) e i β z ,
t 2 e t + ( k 2 - β 2 ) e t = - t ( e t · t ln ) ,
t = - z ˆ ( / z ) ,
k ( x , y ) = ω [ μ ( x , y ) ] 1 / 2 = 2 π n ( x , y ) / λ ,
k cl β k co ,
sin θ c = { 1 - n cl 2 / n co 2 } 1 / 2 θ c .
V = ρ { k co 2 - k cl 2 } 1 / 2 = k co ρ sin θ c ,
e ( V , θ c ) = e 0 ( V , 0 ) + θ c e 1 ( V , 0 ) + θ c 2 e 2 ( V , 0 ) .
n co = n cl .
β = k co = k cl = k = 2 π n / λ .
h ˜ z = e ˜ z = 0 ,
h ˜ t = ( / μ ) 1 / 2 z ˆ × e ˜ t ,
e ˜ x = ψ x ˆ ,             e ˜ y = ψ ŷ ,
{ t 2 + k 2 - β ˜ 2 } ψ = 0 ,
h t ( / μ ) 1 / 2 z ˆ × e t .
h z = ( i / β ) t · h t ,
e z = ( i / β ) { t · e t + ( t ln ) · e t } ,
( i / β ) t · e t .
β - β ˜ β 2 - β ˜ 2 2 k = A e ˜ t · t ( e t · t ln ) d A 2 k A e ˜ t · e t d A ,
A e ˜ t · t ( e t · t ln ) d A = θ c 2 core ( t · e ˜ t ) ( e t · n ˆ ) d l
= i θ c 2 β ˜ core ( e ˜ z · z ˆ ) ( e t · n ˆ ) d l ,
ψ e ( r , ϕ ) = f l ( r ) cos l ϕ ,             ψ o ( r , ϕ ) = f l ( r ) sin l ϕ .
( d 2 d r 2 + 1 r d d r + k 2 ( r ) - β ˜ 2 - l 2 r 2 ) f l ( r ) = 0.
e ˜ x e = f l ( r ) cos ( l ϕ ) x ˆ ,             e ˜ x o = f l ( r ) sin ( l ϕ ) x ˆ ,
e ˜ y e = f l ( r ) cos ( l ϕ ) ŷ ,             e ˜ y o = f l ( r ) sin ( l ϕ ) ŷ .
e x = f 0 ( r ) x ˆ ,             e y = f 0 ( r ) ŷ ,
e t 1 = e ˜ x e + e ˜ y o ,             e t 2 = e ˜ x e - e ˜ y o ,
e t 3 = e ˜ x o + e ˜ y e ,             e t 4 = e ˜ x o - e ˜ y e ,
E x = e x e i β x z = ψ e i β x z x ˆ ,
E y = e y e i β y z = ψ e i β y z ŷ ,
e t 1 = a i ψ e x ˆ + b i ψ o ŷ = a i e ˜ x e + b i e ˜ y o ,
Λ = ( β ˜ e - β ˜ o ) / ( β EH - β HE ) ,
a i / b i = Λ ± ( Λ 2 + 1 ) 1 / 2
β i 2 = [ ( β ˜ e 2 + β ˜ o 2 ) / 2 ] ± { [ ( β ˜ o 2 - β ˜ e 2 ) / 2 ] 2 + C 2 } 1 / 2 ,
C = A e ˜ x e · t ( e ˜ y o · t ln ) d A / A e ˜ x e 2 d A
β 1 2 - β 2 2 2 k ( β 1 - β 2 ) ,
Λ = ( β ˜ e 2 - β ˜ o 2 ) / 2 C ( β ˜ e - β ˜ o ) / ( β 1 - β 2 ) ,
f l ( r ) = J l ( Ũ r / ρ ) / J l ( Ũ ) ,             r ρ
f l ( r ) = K l ( W ˜ r / ρ ) / K l ( W ˜ ) ,             r ρ
Ũ J l + 1 ( Ũ ) K l ( W ˜ ) = W ˜ K l + 1 ( W ˜ ) J l ( Ũ ) ,
V 2 = Ũ 2 + W ˜ 2 .
( ρ β ˜ ) 2 = ( ρ k co ) 2 - Ũ 2 = ( ρ k cl ) 2 + W ˜ 2 ,
β 1 - β 2 = θ c 3 2 ρ ( Ũ 2 V 3 ) K 1 2 ( W ˜ ) K 0 ( W ˜ ) K 2 ( W ˜ ) { 2 - W ˜ K 0 ( W ˜ ) K 1 ( W ˜ ) } .
β 1 - β 2 = l θ c 3 ρ ( Ũ 2 V 3 ) K l 2 ( W ˜ ) K l - 1 ( W ˜ ) K l + 1 ( W ˜ ) ,
Λ = ( β ˜ e - β ˜ o ) / ( β EH - β HE ) ,
β ˜ e - β ˜ o = ( θ c e 2 / 4 ρ ) ( Ũ 2 / V ) K 1 2 ( W ˜ ) K 0 ( W ˜ ) K 2 ( W ˜ ) ,
Λ = ( 1 / 2 ) ( e / θ c ) 2 V 2 { 2 ± W ˜ K 0 ( W ˜ ) / K 1 ( W ˜ ) } - 1 ,
β x - β y = ( e 2 θ c 3 8 ρ ) ( Ũ 2 W ˜ 2 V 3 ) { 1 + Ũ K 0 2 ( W ˜ ) J 2 ( Ũ ) K 1 2 ( W ˜ ) J 1 ( Ũ ) } ,
β ˜ + - β ˜ - = 2 θ c ρ Ũ 2 V 3 K 0 ( W ˜ d / ρ ) K 1 2 ( W ˜ )
β x - β y = θ c 3 ρ Ũ 2 V 3 K 0 ( W ˜ d / ρ ) K 1 2 ( W ˜ ) [ 1 - 2 I 1 ( W ˜ ) K 1 ( W ˜ ) ]
β x - β y β ˜ + - β ˜ - = 1 2 θ c 2 [ 1 - 2 I 1 ( W ˜ ) K 1 ( W ˜ ) ] ~ θ c 2 2 ( 1 - 1 V ) .
Λ = ( β ˜ e - β ˜ o ) / ( β EH - β HE ) ,
β ˜ e - β ˜ o = θ c ρ Ũ 2 V 3 K 2 ( W ˜ d / ρ ) K 0 ( W ˜ ) K 2 ( W ˜ ) ,
Λ = 2 θ c 2 K 2 ( W ˜ d / ρ ) K 1 2 ( W ˜ ) ( 2 ± W ˜ K 0 ( W ˜ ) K 1 ( W ˜ ) ) - 1 .
[ t 2 + ( k 2 - β 2 ) ] e t = - t ( e t · t ln ) ,
[ t + ( k 2 - β ˜ 2 ) ] e ˜ t = 0.
A ( e ˜ t · t 2 e t - e t · t 2 e ˜ t ) d A
( ln ) d A = [ ln ( cl / co ) ] δ ( B ) n ˆ d l = - θ c 2 δ ( B ) n ˆ d l ,
A e ˜ t · t ( e t · t ln ) d A = θ c 2 B ( t · e ˜ t ) ( e t · n ˆ ) d l .
e t i = a i e ˜ a + b i e ˜ b ,
A e ˜ a · e ˜ b d A = 0 ,
( C a a + β ˜ a 2 - β i 2 ) a i + ( C a b ) b i = 0 ,
C a j = A e ˜ a · t ( e ˜ j · t ln ) d A A e ˜ a 2 d A ,
( C b a ) a i + ( C b b + β ˜ b 2 - β i 2 ) b i = 0 ,
( C a a + β ˜ a 2 - β i 2 ) ( C b b + β ˜ b 2 - β i 2 ) - C a b C b a = 0 ,
β i 2 = C a b + β ˜ a 2 + C b b + β ˜ b 2 2 ± [ ( C a a + β ˜ a 2 - C b b - β ˜ b 2 2 ) 2 + C a b C b a ] 1 / 2 ,
a i / i = ( β i 2 - β b 2 - C b b ) / C b a .
β i 2 = [ ( β ˜ a 2 + β ˜ b 2 ) / 2 ] ± C ( Λ 2 + 1 ) ,
Λ = ( β ˜ a 2 - β ˜ b 2 ) / 2 C ,
a i / b i = Λ ± ( Λ 2 + 1 ) 1 / 2 ,
{ t 2 + k 2 } ψ = β ˜ 2 ψ ,
{ t 2 + k ¯ 2 } ψ ¯ = β ¯ 2 ψ ¯ .
{ ψ ¯ t 2 ψ - ψ t 2 ψ ¯ } + { k 2 - k ¯ 2 } ψ ψ ¯ = { β ¯ 2 - β ˜ 2 } ψ ψ ¯ .
β ˜ 2 - β ¯ 2 = A ( k 2 - k ¯ 2 ) ψ ψ ¯ d A A ψ ψ ¯ d A .
ψ i = a i ψ ¯ a + b i ψ ¯ b ,
( C a a + β ¯ a 2 - β i 2 ) a i + [ C a b + D a ( β ¯ a 2 - β i 2 ) ] b i = 0 ,
[ C b a + D b ( β ¯ b 2 - β i 2 ) ] a i + ( C b b + β ¯ b 2 - β i 2 ) b i = 0 ,
C p q = A ( k 2 - k ¯ 2 ) ψ ¯ p ψ ¯ q d A A ψ ¯ p 2 d A ,
D j = A ψ ¯ a ψ ¯ b d A A ψ ¯ j 2 d A ,
e t 1 = e ˜ x e + e ˜ y o
= ( cos l ϕ x ˆ + sin l ϕ ŷ ) f l ( r )
= ( cos ( l - 1 ) ϕ r ˆ + sin ( l - 1 ) ϕ ϕ ˆ ) f l ( r ) .
t · e ˜ x e = f l ( r ) cos l ϕ cos ϕ + [ l f l ( r ) / r ] sin l ϕ sin ϕ
β 2 = β ˜ 2 + θ c 2 ρ 0 2 π d ϕ [ f l ( ρ ) cos ( l - 1 ) ϕ cos l ϕ cos ϕ + { l f l ( ρ ) / ρ } cos ( l - 1 ) ϕ sin l ϕ sin ϕ A f l 2 ( r ) cos 2 l ϕ d A ,
= β ˜ 2 - ( θ c ρ ) 2 ( Ũ V ) 2 W ˜ K l ( W ˜ ) K l + 1 ( W ˜ ) ,             l 1
= β ˜ 2 - 2 ( θ c ρ ) 2 ( Ũ V ) 2 W ˜ K 1 ( W ˜ ) K 2 ( W ˜ ) ,             l = 1
β ˜ e 2 - β ¯ 2 = 3 4 ( e ρ ) 2 Ũ 2 K 1 2 ( W ˜ ) K 2 ( W ˜ ) K 0 ( W ˜ ) .
β ˜ 0 2 - β ¯ 2 = ( β ˜ e 2 - β ¯ 2 ) / 3.
β ˜ ± 2 = β ¯ 2 ± ( θ c k co ) 2 core 2 ψ ¯ 1 ψ ¯ 2 d A / A ψ ¯ 1 2 d A .
β x + 2 = β ˜ + 2 + ϕ c 2 ψ + ( ψ + · x ˆ ) ( n ˆ · x ˆ ) d l / A ψ + 2 d A
β x + 2 = β ˜ 2 + 2 ( n ˆ · x ˆ [ ψ ¯ 1 ( t ψ ¯ 1 · x ˆ ) + ψ ¯ 2 ( t ψ ¯ 1 · x ˆ ) + ψ 1 ( t ψ ¯ 2 · x ˆ ) ] d l ) .
K 0 W ˜ r 1 ρ = l = - ( - 1 ) l cos l ϕ 2 I l W ˜ r 2 ρ K l W ˜ d ρ ,