Abstract

The HE11 fields of weakly guiding fibers with graded refractive-index profiles are nearly identical to the fields of a step-index fiber with dimensionless frequency V¯ and radius ρ¯, where V¯ and ρ¯ are found by an elementary variational method. The results are remarkably accurate, with errors of a fraction of a percent at most, so that a simple, closed-form expression for the HE11 fields of graded fibers is now available. The step-fiber approximation is more widely applicable than the Gaussian field approximation which is inadequate for small V and for describing the evanescent field. However, the variational procedure also complements the Gaussian field approximation by providing analytical expressions for the spot size and propagation constant in several profiles of interest.

© 1979 Optical Society of America

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References

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  1. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  2. A. Ankiewicz and C. Pask, “Geometric optics approach to light acceptance and propagation in graded-index fibres,” Opt. Quantum Electron. 9, 87–109 (1977).
    [CrossRef]
  3. W. Streiffer and C. N. Kurtz, “Scalar analysis of radially inhomogeneous guiding media,” J. Opt. Soc. Am. 57, 779–786 (1967).
    [CrossRef]
  4. D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
    [CrossRef]
  5. A. W. Snyder and J. D. Love, “Attenuation coefficient for rays in graded fibres with absorbing cladding,” Electron. Lett. 12, 255–257 (1976).
    [CrossRef]
  6. A. W. Snyder and J. D. Love, “Attenuation coefficient for tunneling leaky rays in graded fibers,” Electron. Lett. 12, 324–326 (1976).
    [CrossRef]
  7. A. W. Snyder and W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
    [CrossRef]
  8. R. Yamada, T. Meiri, and N. Okamoto, “Guided waves along an optical fiber with parabolic index profile,” J. Opt. Soc. Am. 67, 96–103 (1977).
    [CrossRef]
  9. W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31–40 (1978).
    [CrossRef]
  10. J. D. Love, “Power series solutions of the scalar wave equation for cladded, power-law profiles of arbitrary exponent,” Opt. Quantum Electron. 11, September, 1979.
    [CrossRef]
  11. W. A. Gambling and H. Matsumura, “Simple characterization factor for practical single-mode fibres,” Electron. Lett. 13, 691–693 (1977).
    [CrossRef]
  12. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [CrossRef]
  13. D. Marcuse, “Gaussian approximation of the fundamental modes of graded-index fibers,” J. Opt. Soc. Am. 68, 103–109 (1978).
    [CrossRef]
  14. M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics 6, 271–286 (1974); see also references cited therein.
    [CrossRef]
  15. M. Matsuhara, “Analysis of TEM modes in dielectric waveguides, by a variational method,” J. Opt. Soc. Am. 63, 1514–1517 (1973).
    [CrossRef]
  16. T. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
    [CrossRef]
  17. K. Okamoto and T. Okoshi, “Vectorial wave analysis of inhomogeneous optical fibers using finite element method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
    [CrossRef]
  18. S. E. Miller, “Light propagation in generalized lens-like media,” Bell Syst. Tech. J. 44, 2017–2064 (1965).
    [CrossRef]
  19. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1112.
  20. The HE11mode is the mode with the largest value of β, or equivalently, the smallest value of U.
  21. A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
    [CrossRef]
  22. C. Pask and R. A. Sammut, “Developments in the theory of fibre optics,” Proc. IREE (Aust.) 40, 89–101 (1979).
  23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  24. A. W. Snyder, “Mode propagation in an optical waveguide,” Electron. Lett. 6, 561–562 (1970).
    [CrossRef]

1979 (2)

J. D. Love, “Power series solutions of the scalar wave equation for cladded, power-law profiles of arbitrary exponent,” Opt. Quantum Electron. 11, September, 1979.
[CrossRef]

C. Pask and R. A. Sammut, “Developments in the theory of fibre optics,” Proc. IREE (Aust.) 40, 89–101 (1979).

1978 (4)

K. Okamoto and T. Okoshi, “Vectorial wave analysis of inhomogeneous optical fibers using finite element method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
[CrossRef]

D. Marcuse, “Gaussian approximation of the fundamental modes of graded-index fibers,” J. Opt. Soc. Am. 68, 103–109 (1978).
[CrossRef]

A. W. Snyder and W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
[CrossRef]

W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31–40 (1978).
[CrossRef]

1977 (4)

W. A. Gambling and H. Matsumura, “Simple characterization factor for practical single-mode fibres,” Electron. Lett. 13, 691–693 (1977).
[CrossRef]

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

A. Ankiewicz and C. Pask, “Geometric optics approach to light acceptance and propagation in graded-index fibres,” Opt. Quantum Electron. 9, 87–109 (1977).
[CrossRef]

R. Yamada, T. Meiri, and N. Okamoto, “Guided waves along an optical fiber with parabolic index profile,” J. Opt. Soc. Am. 67, 96–103 (1977).
[CrossRef]

1976 (2)

A. W. Snyder and J. D. Love, “Attenuation coefficient for rays in graded fibres with absorbing cladding,” Electron. Lett. 12, 255–257 (1976).
[CrossRef]

A. W. Snyder and J. D. Love, “Attenuation coefficient for tunneling leaky rays in graded fibers,” Electron. Lett. 12, 324–326 (1976).
[CrossRef]

1974 (2)

M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics 6, 271–286 (1974); see also references cited therein.
[CrossRef]

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

1973 (2)

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

M. Matsuhara, “Analysis of TEM modes in dielectric waveguides, by a variational method,” J. Opt. Soc. Am. 63, 1514–1517 (1973).
[CrossRef]

1970 (1)

A. W. Snyder, “Mode propagation in an optical waveguide,” 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. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[CrossRef]

1967 (1)

1965 (1)

S. E. Miller, “Light propagation in generalized lens-like media,” Bell Syst. Tech. J. 44, 2017–2064 (1965).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Ankiewicz, A.

A. Ankiewicz and C. Pask, “Geometric optics approach to light acceptance and propagation in graded-index fibres,” Opt. Quantum Electron. 9, 87–109 (1977).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1112.

Gambling, W. A.

W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31–40 (1978).
[CrossRef]

W. A. Gambling and H. Matsumura, “Simple characterization factor for practical single-mode fibres,” Electron. Lett. 13, 691–693 (1977).
[CrossRef]

Gloge, D.

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

Kurtz, C. N.

Love, J. D.

J. D. Love, “Power series solutions of the scalar wave equation for cladded, power-law profiles of arbitrary exponent,” Opt. Quantum Electron. 11, September, 1979.
[CrossRef]

A. W. Snyder and J. D. Love, “Attenuation coefficient for rays in graded fibres with absorbing cladding,” Electron. Lett. 12, 255–257 (1976).
[CrossRef]

A. W. Snyder and J. D. Love, “Attenuation coefficient for tunneling leaky rays in graded fibers,” Electron. Lett. 12, 324–326 (1976).
[CrossRef]

Marcatili, E. A. J.

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

Marcuse, D.

D. Marcuse, “Gaussian approximation of the fundamental modes of graded-index fibers,” J. Opt. Soc. Am. 68, 103–109 (1978).
[CrossRef]

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Matsuhara, M.

Matsumura, H.

W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31–40 (1978).
[CrossRef]

W. A. Gambling and H. Matsumura, “Simple characterization factor for practical single-mode fibres,” Electron. Lett. 13, 691–693 (1977).
[CrossRef]

Meiri, T.

Miller, S. E.

S. E. Miller, “Light propagation in generalized lens-like media,” Bell Syst. Tech. J. 44, 2017–2064 (1965).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1112.

Okamoto, K.

K. Okamoto and T. Okoshi, “Vectorial wave analysis of inhomogeneous optical fibers using finite element method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
[CrossRef]

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

Okamoto, N.

Okoshi, T.

K. Okamoto and T. Okoshi, “Vectorial wave analysis of inhomogeneous optical fibers using finite element method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
[CrossRef]

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

Pask, C.

C. Pask and R. A. Sammut, “Developments in the theory of fibre optics,” Proc. IREE (Aust.) 40, 89–101 (1979).

A. Ankiewicz and C. Pask, “Geometric optics approach to light acceptance and propagation in graded-index fibres,” Opt. Quantum Electron. 9, 87–109 (1977).
[CrossRef]

Sammut, R. A.

C. Pask and R. A. Sammut, “Developments in the theory of fibre optics,” Proc. IREE (Aust.) 40, 89–101 (1979).

Snyder, A. W.

A. W. Snyder and W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
[CrossRef]

A. W. Snyder and J. D. Love, “Attenuation coefficient for tunneling leaky rays in graded fibers,” Electron. Lett. 12, 324–326 (1976).
[CrossRef]

A. W. Snyder and J. D. Love, “Attenuation coefficient for rays in graded fibres with absorbing cladding,” Electron. Lett. 12, 255–257 (1976).
[CrossRef]

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

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

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Streiffer, W.

Vassell, M. O.

M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics 6, 271–286 (1974); see also references cited therein.
[CrossRef]

Yamada, R.

Young, W. R.

Bell Syst. Tech. J. (3)

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

S. E. Miller, “Light propagation in generalized lens-like media,” Bell Syst. Tech. J. 44, 2017–2064 (1965).
[CrossRef]

Electron. Lett. (4)

A. W. Snyder and J. D. Love, “Attenuation coefficient for rays in graded fibres with absorbing cladding,” Electron. Lett. 12, 255–257 (1976).
[CrossRef]

A. W. Snyder and J. D. Love, “Attenuation coefficient for tunneling leaky rays in graded fibers,” Electron. Lett. 12, 324–326 (1976).
[CrossRef]

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

W. A. Gambling and H. Matsumura, “Simple characterization factor for practical single-mode fibres,” Electron. Lett. 13, 691–693 (1977).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

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

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

K. Okamoto and T. Okoshi, “Vectorial wave analysis of inhomogeneous optical fibers using finite element method,” IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
[CrossRef]

J. Opt. Soc. Am. (5)

Opt. Quantum Electron. (3)

W. A. Gambling and H. Matsumura, “Propagation in radially-inhomogeneous single-mode fibre,” Opt. Quantum Electron. 10, 31–40 (1978).
[CrossRef]

J. D. Love, “Power series solutions of the scalar wave equation for cladded, power-law profiles of arbitrary exponent,” Opt. Quantum Electron. 11, September, 1979.
[CrossRef]

A. Ankiewicz and C. Pask, “Geometric optics approach to light acceptance and propagation in graded-index fibres,” Opt. Quantum Electron. 9, 87–109 (1977).
[CrossRef]

Opto-electronics (1)

M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics 6, 271–286 (1974); see also references cited therein.
[CrossRef]

Proc. IREE (Aust.) (1)

C. Pask and R. A. Sammut, “Developments in the theory of fibre optics,” Proc. IREE (Aust.) 40, 89–101 (1979).

Other (4)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1112.

The HE11mode is the mode with the largest value of β, or equivalently, the smallest value of U.

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Figures (12)

FIG. 1
FIG. 1

Refractive index profiles for waveguides with circular symmetry. (a) Power law profiles for fibers with a finite core radius; (b) Profiles with an infinite core radius.

FIG. 2
FIG. 2

Fraction η of the HE11 mode’s power that travels within the core of an arbitrary graded fiber of radius ρ. The parameters ρ ¯ and V ¯ are found from Eq. (15a) once V,ρ, and the profile shape of the graded fiber are specified. These curves are constructed from Eq. (19).

FIG. 3
FIG. 3

Step fiber, characterized by n ¯ co and ρ ¯ and shown as the dashed curve, whose HE11 fields are nearly identical to those of the given graded fiber shown by the solid curve. (a) q = 2; (b) q = 0.25. Knowing V ¯ and ρ ¯ from Eq. (15a), n ¯ co is found from the equation ( n ¯ co / n cl ) 2 = 1 + ( V ¯ ρ / V ρ ¯ ) 2 ( n co 2 / n cl 2 1 ) .

FIG. 4
FIG. 4

Field amplitude et for graded fibers (solid curve) determined numerically compared to the fields of a step fiber (dashed curve) with V = V ¯ and ρ = ρ ¯. The dotted line is the Gaussian field approximation discussed in Sec. V. All fields are normalized so that 0 e t 2 r d r = 1. Figs, (a), (b), (c), and (d) apply to the q = 2 profile of Fig. 1(a) for increasing values of V.

FIG. 5
FIG. 5

Same as Fig. 4 with q = 0.25.

FIG. 6
FIG. 6

Cutoff V for the 2nd mode, Vc, determined from Eq. ( 16) using the step fiber approximation.

FIG. 7
FIG. 7

Dimensionless waveguide parameter U defined by Eq. (14) vs V for power law profiles. The solid curve is calculated from Eq. (15a) using the step fiber approximation while the dashed curve is calculated from Eq. (15a) using the Gaussian field approximation discussed in Sec. V. The solid curves are indistinguishable from the exact numerical solution of the wave equation.13 The format for these results is taken from Ref. 13 where they have been determined by other methods.

FIG. 8
FIG. 8

Parameters ρ ¯ and V ¯ necessary for the step fiber approximation of a graded fiber as a function of grading parameter q. For each q, V is at the cutoff of the second mode given by Fig. 6 The fraction of HE11 mode’s power propagating inside the core of the graded fiber, Eq. (19), is also shown.

FIG. 9
FIG. 9

Waveguide parameter U vs V for the Gaussian profile of Fig. 1(b), where the arrow marks the cutoff of the 2nd mode at V = 2.62. The value of ρ used to define V is given in Sec. IV B. The solid curve is found by the step fiber approximation from Eq. (15a) as shown in Table I while the dashed curve is determined using the Gaussian field approximation, Eq. (32).

FIG. 10
FIG. 10

Same as Fig. 8 only for the Gaussian profile of Fig. 1(b). The arrow is cutoff for the 2nd mode at V = 2.62 where the definition of ρ used in V is discussed in Sec. IV B.

FIG. 11
FIG. 11

Spot size parameter r ¯ 0 / ρ for power law profiles as computed from Eq. (30). Dark circles are V for cutoff of 2nd mode as shown in Fig. 6. The range 2 ≤ V ≤ 5 describes single mode operation for 1 ≤ q ≤ ∞.

FIG. 12
FIG. 12

Gaussian profile (bold solid line) compared with the field amplitude et for various values of Vc. The fields are computed using the Gaussian field approximation Eq. (22) with r ¯ 0 / ρ given by Eq. (31). The 0.5 Vc curve is slightly in error due to the inherent inaccuracy of the Gaussian approximation for small V and also for large r. The fields are normalized to 0 e t 2 r d r = 0.5.

Tables (3)

Tables Icon

TABLE I Integrals used in equivalent step approximation I ¯ = ( U ¯ / V ¯ ) 2 { 1 [ K 0 ( W ¯ ) / K 1 ( W ¯ ) ] 2 } I = 2 [ U ¯ K 0 ( W ¯ ) / V ¯ K 1 ( W ¯ ) ] 2 [ I 1 / J 0 2 ( U ¯ ) + I 2 / K 0 2 ( W ¯ ) ]where I 1 = 0 1 f J 0 2 ( U ¯ R ) R d R ; I 2 = 1 f K 0 2 ( W ¯ R ) R d R ; R = r / ρ ¯ .

Tables Icon

TABLE II Results for the parabolic profile fiber of finite radius ρ and dimensionless frequency V, where the profile shape is given by Eq. (14) with q = 2 as illustrated in Fig. 1(a). The step fiber whose HE11 fields are nearly identical to this q = 2 fiber is characterized by radius ρ ¯ and dimensionless frequency V ¯ found as the pair that minimize U2 in Eq. (15a). This value of U is shown in the 4th column.

Tables Icon

TABLE III Cutoff frequency of second mode; V ¯ c = 2.405.

Equations (44)

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E t ( x , y , z ) = e t ( x , y ) e i ( β z ω t ) ,
[ t 2 + k 2 ( x , y ) ] e t = β 2 e t ,
k = 2 π n ( x , y ) / λ ,
β 2 = A ( e t t 2 e t + k 2 e t 2 ) / A e t 2 d A ,
[ t 2 + k ¯ 2 ( x , y ) ] e ¯ t = β ¯ 2 e ¯ t
β 2 β ¯ 2 + A ( k 2 k ¯ 2 ) e ¯ t 2 d A / A e ¯ t 2 d A .
e t e ¯ t = J 0 ( U ¯ r / ρ ¯ ) / J 0 ( U ¯ ) ; r ρ ¯
= K 0 ( W ¯ r / ρ ¯ ) / K 0 ( W ¯ ) ; r ρ ¯ .
U ¯ 2 + W ¯ 2 = V ¯ 2 = ( 2 π ρ ¯ / λ ) 2 ( n ¯ co 2 n cl 2 ) .
W ¯ K 1 ( W ¯ ) J 0 ( U ¯ ) = U ¯ K 0 ( W ¯ ) J 1 ( U ¯ ) .
n 2 ( r ) = n co 2 [ 1 θ c 2 f ( r ) ]
n 2 ( r ) = n ¯ co 2 [ 1 θ ¯ c 2 f ¯ ( r ) ]
θ c 2 1 ( n cl / n co ) 2 ; θ ¯ c 2 1 ( n cl / n ¯ co ) 2 .
( β ¯ ρ ¯ ) 2 = ( V ¯ / sin θ ¯ c ) 2 U ¯ 2 .
( ρ β ) 2 = ( V / sin θ c ) 2 U 2 ,
V 2 = ( 2 π ρ / λ ) 2 ( n co 2 n cl 2 ) .
U 2 ( ρ / ρ ¯ ) 2 ( U ¯ 2 V ¯ 2 I ¯ ) + V 2 I ,
I = 0 e ¯ t 2 f ( r ) r d r / 0 e ¯ t 2 r d r
I ¯ = 0 e ¯ t 2 f ¯ ( r ) r d r / 0 e ¯ t 2 r d r .
V c 2 ( 2.405 ) 2 ( ρ / ρ ¯ ) 2 ( 1 I ¯ ) / ( 1 I ¯ ) ,
e ¯ t J 1 ( 2.405 r / ρ ¯ ) / J 1 ( 2.405 ) ; r < ρ ¯
ρ ¯ / r , r > ρ ¯ .
η = 0 ρ e t 2 r d r / 0 e t 2 r d r .
η 1 ( ρ U ¯ ρ ¯ V ¯ K 1 ( W ¯ ) ) 2 [ K 1 2 ( W ¯ ρ ρ ¯ ) K 0 2 ( W ¯ ρ ρ ¯ ) ] ; ρ ¯ < ρ
η ( ρ W ¯ ρ ¯ V ¯ J 1 ( U ¯ ) ) 2 [ J 1 2 ( U ¯ ρ ρ ¯ ) + J 0 2 ( U ¯ ρ ρ ¯ ) ] ; ρ ¯ > ρ .
f ( r ) = ( r / ρ ) q r ρ
= 1 r > ρ .
f ( r ) = 1 e ( r / ρ ) 2 .
e t e ¯ t = e 1 / 2 ( r / r ¯ 0 ) 2 ,
r ¯ 0 = ρ ¯ / V ¯ 1 / 2 .
U ¯ = ( 2 V ¯ ) 1 / 2 .
η = 1 e ( ρ / r ¯ 0 ) 2 .
U 2 ( ρ / r ¯ 0 ) 2 + V 2 e ( ρ / r ¯ 0 ) 2
r ¯ 0 = ρ ( ln V 2 ) 1 / 2
U = ( 1 + ln V 2 ) 1 / 2 .
U 2 ( ρ / r ¯ 0 ) 2 + V 2 [ I q + e ( ρ / r ¯ 0 ) 2 ] ,
I q = ( r ¯ 0 / ρ ) q Γ ( q / 2 + 1 , ( ρ / r ¯ 0 ) 2 ) ,
e ( ρ / r ¯ 0 ) 2 = ( 1 / V ) 2 + ( r ¯ 0 / ρ ) 2 I q I q + 2 .
U 2 ( ρ / r ¯ 0 ) 2 + V 2 ( ρ 2 / r ¯ 0 2 + 1 ) 1 ,
r ¯ 0 = ρ ( V 1 ) 1 / 2 .
U = ( 2 V 1 ) 1 / 2 .
( n ¯ co / n cl ) 2 = 1 + ( V ¯ ρ / V ρ ¯ ) 2 ( n co 2 / n cl 2 1 ) .
I ¯ = ( U ¯ / V ¯ ) 2 { 1 [ K 0 ( W ¯ ) / K 1 ( W ¯ ) ] 2 } I = 2 [ U ¯ K 0 ( W ¯ ) / V ¯ K 1 ( W ¯ ) ] 2 [ I 1 / J 0 2 ( U ¯ ) + I 2 / K 0 2 ( W ¯ ) ]
I 1 = 0 1 f J 0 2 ( U ¯ R ) R d R ; I 2 = 1 f K 0 2 ( W ¯ R ) R d R ; R = r / ρ ¯ .