Abstract

When a tunneling leaky ray propagates along the core of a nonabsorbing, multilayered, dielectric waveguide of slab or cylindrical geometry, the tunneling coefficient describing the power loss at a reflection or turning point is a simple product of fundamental quantities: (i) a WKB integration over the continuous part of the refractive index profile in the evanescent region between the reflection or turning point and the position at which the ray reappears; and (ii) a factor |TF| at each jump in the refractive index profile between adjacent layers in the evanescent region, where TF is the analytic continuation of the Fresnel power transmission coefficient as defined between two half-spaces of constant refractive indices corresponding to the values on either side of the jump. The ray power attenuation coefficient is equal to the tunneling coefficient divided by the distance along the waveguide axis between successive reflections or turning points at which power is lost.

© 1977 Optical Society of America

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References

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  1. A. W. Snyder and J. D. Love, “Tunneling leaky modes on optical waveguides,” Opt. Commun. 12, 325–328 (1974).
    [CrossRef]
  2. M. J. Adams, D. N. Payne, and F. M. E. Sladen, “Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles,” Electron. Lett. 11, 238–240 (1975).
    [CrossRef]
  3. J. D. Love and A. W. Snyder, “Fresnel’s and Snell’s laws for the multimode optical waveguide of circular cross section,” J. Opt. Soc. Am. 65, 1241–1247 (1975).
    [CrossRef]
  4. A. W. Snyder and J. D. Love, “Reflection at a curved dilectric interface-electromagnetic tunneling,” IEEE Trans. MTT-13, 134–141 (1975).
  5. A. W. Snyder and D. J. Mitchell, “Generalized Fresnel’s laws for determining radiation loss from optical waveguides and curved dielectric structures,” Optik 40, 438–459 (1975).
  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. W. J. Stewart, “A new technique for determining the V values and refractive index profiles of optical fibers,” paper presented at OSA/IEEE meeting on Optical Fiber Transmission, Williamsburg, Virginia (1975).
  8. K. Petermann, “The mode attenuation in general graded core multimode fibers,” Arch. Elektron. Ubertragungsteck 29, 345–348 (1975).
  9. J. D. Love and A. W. Snyder, “Generalized Fresnel’s laws for a curved absorbing interface,” J. Opt. Soc. Am. 65, 1072–1074 (1975).
    [CrossRef]
  10. C. Pask and A. W. Snyder, “Multimode optical fibers: interplay of absorption and radiation losses,” Appl. Opt. 15, 1295–1298 (1976).
    [CrossRef] [PubMed]
  11. A. W. Snyder and J. D. Love, “Attenuation coefficient for rays in graded fibers with absorbing cladding,” Electron. Lett. 12, 255–257 (1976).
    [CrossRef]
  12. J. D. Love and C. Winkler, “The effects of material absorption on ray power attenuation in multilayered optical waveguides” (unpublished).
  13. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 40.
  14. A. W. Snyder and D. J. Mitchell, “Leaky rays on circular optical fibers,” J. Opt. Soc. Am. 64, 599–607 (1974).
    [CrossRef]
  15. H. Bremmer, “The W.K. B. approximation as the first term of a geometric-optical series,” in The Theory of Electromagnetic Waves, edited by M. Kline (Dover, New York, 1951).
  16. A. W. Snyder and J. D. Love, “Goos-Hänchen shift,” Appl. Opt. 15, 236–238 (1976).
    [CrossRef] [PubMed]
  17. D. Gloge and E. A. J. Marcatili, “Multimode theory of graded fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
    [CrossRef]
  18. M. J. Adams, D. H. Payne, and F. M. E. Sladen, “Length-dependent effects due to leaky modes on multimode graded-index optical fibers,” Opt. Commun. 17, 204–209 (1976).
    [CrossRef]
  19. L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1963), p. 286.
  20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

1976 (5)

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

M. J. Adams, D. H. Payne, and F. M. E. Sladen, “Length-dependent effects due to leaky modes on multimode graded-index optical fibers,” Opt. Commun. 17, 204–209 (1976).
[CrossRef]

A. W. Snyder and J. D. Love, “Goos-Hänchen shift,” Appl. Opt. 15, 236–238 (1976).
[CrossRef] [PubMed]

C. Pask and A. W. Snyder, “Multimode optical fibers: interplay of absorption and radiation losses,” Appl. Opt. 15, 1295–1298 (1976).
[CrossRef] [PubMed]

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

1975 (6)

J. D. Love and A. W. Snyder, “Generalized Fresnel’s laws for a curved absorbing interface,” J. Opt. Soc. Am. 65, 1072–1074 (1975).
[CrossRef]

J. D. Love and A. W. Snyder, “Fresnel’s and Snell’s laws for the multimode optical waveguide of circular cross section,” J. Opt. Soc. Am. 65, 1241–1247 (1975).
[CrossRef]

M. J. Adams, D. N. Payne, and F. M. E. Sladen, “Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles,” Electron. Lett. 11, 238–240 (1975).
[CrossRef]

K. Petermann, “The mode attenuation in general graded core multimode fibers,” Arch. Elektron. Ubertragungsteck 29, 345–348 (1975).

A. W. Snyder and J. D. Love, “Reflection at a curved dilectric interface-electromagnetic tunneling,” IEEE Trans. MTT-13, 134–141 (1975).

A. W. Snyder and D. J. Mitchell, “Generalized Fresnel’s laws for determining radiation loss from optical waveguides and curved dielectric structures,” Optik 40, 438–459 (1975).

1974 (2)

A. W. Snyder and J. D. Love, “Tunneling leaky modes on optical waveguides,” Opt. Commun. 12, 325–328 (1974).
[CrossRef]

A. W. Snyder and D. J. Mitchell, “Leaky rays on circular optical fibers,” J. Opt. Soc. Am. 64, 599–607 (1974).
[CrossRef]

1973 (1)

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

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

Adams, M. J.

M. J. Adams, D. H. Payne, and F. M. E. Sladen, “Length-dependent effects due to leaky modes on multimode graded-index optical fibers,” Opt. Commun. 17, 204–209 (1976).
[CrossRef]

M. J. Adams, D. N. Payne, and F. M. E. Sladen, “Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles,” Electron. Lett. 11, 238–240 (1975).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 40.

Bremmer, H.

H. Bremmer, “The W.K. B. approximation as the first term of a geometric-optical series,” in The Theory of Electromagnetic Waves, edited by M. Kline (Dover, New York, 1951).

Gloge, D.

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

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1963), p. 286.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1963), p. 286.

Love, J. D.

A. W. Snyder and J. D. Love, “Attenuation coefficient for rays in graded fibers 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 and J. D. Love, “Goos-Hänchen shift,” Appl. Opt. 15, 236–238 (1976).
[CrossRef] [PubMed]

J. D. Love and A. W. Snyder, “Fresnel’s and Snell’s laws for the multimode optical waveguide of circular cross section,” J. Opt. Soc. Am. 65, 1241–1247 (1975).
[CrossRef]

A. W. Snyder and J. D. Love, “Reflection at a curved dilectric interface-electromagnetic tunneling,” IEEE Trans. MTT-13, 134–141 (1975).

J. D. Love and A. W. Snyder, “Generalized Fresnel’s laws for a curved absorbing interface,” J. Opt. Soc. Am. 65, 1072–1074 (1975).
[CrossRef]

A. W. Snyder and J. D. Love, “Tunneling leaky modes on optical waveguides,” Opt. Commun. 12, 325–328 (1974).
[CrossRef]

J. D. Love and C. Winkler, “The effects of material absorption on ray power attenuation in multilayered optical waveguides” (unpublished).

Marcatili, E. A. J.

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

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Generalized Fresnel’s laws for determining radiation loss from optical waveguides and curved dielectric structures,” Optik 40, 438–459 (1975).

A. W. Snyder and D. J. Mitchell, “Leaky rays on circular optical fibers,” J. Opt. Soc. Am. 64, 599–607 (1974).
[CrossRef]

Pask, C.

Payne, D. H.

M. J. Adams, D. H. Payne, and F. M. E. Sladen, “Length-dependent effects due to leaky modes on multimode graded-index optical fibers,” Opt. Commun. 17, 204–209 (1976).
[CrossRef]

Payne, D. N.

M. J. Adams, D. N. Payne, and F. M. E. Sladen, “Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles,” Electron. Lett. 11, 238–240 (1975).
[CrossRef]

Petermann, K.

K. Petermann, “The mode attenuation in general graded core multimode fibers,” Arch. Elektron. Ubertragungsteck 29, 345–348 (1975).

Sladen, F. M. E.

M. J. Adams, D. H. Payne, and F. M. E. Sladen, “Length-dependent effects due to leaky modes on multimode graded-index optical fibers,” Opt. Commun. 17, 204–209 (1976).
[CrossRef]

M. J. Adams, D. N. Payne, and F. M. E. Sladen, “Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles,” Electron. Lett. 11, 238–240 (1975).
[CrossRef]

Snyder, A. W.

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, “Goos-Hänchen shift,” Appl. Opt. 15, 236–238 (1976).
[CrossRef] [PubMed]

C. Pask and A. W. Snyder, “Multimode optical fibers: interplay of absorption and radiation losses,” Appl. Opt. 15, 1295–1298 (1976).
[CrossRef] [PubMed]

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

J. D. Love and A. W. Snyder, “Generalized Fresnel’s laws for a curved absorbing interface,” J. Opt. Soc. Am. 65, 1072–1074 (1975).
[CrossRef]

A. W. Snyder and D. J. Mitchell, “Generalized Fresnel’s laws for determining radiation loss from optical waveguides and curved dielectric structures,” Optik 40, 438–459 (1975).

J. D. Love and A. W. Snyder, “Fresnel’s and Snell’s laws for the multimode optical waveguide of circular cross section,” J. Opt. Soc. Am. 65, 1241–1247 (1975).
[CrossRef]

A. W. Snyder and J. D. Love, “Reflection at a curved dilectric interface-electromagnetic tunneling,” IEEE Trans. MTT-13, 134–141 (1975).

A. W. Snyder and J. D. Love, “Tunneling leaky modes on optical waveguides,” Opt. Commun. 12, 325–328 (1974).
[CrossRef]

A. W. Snyder and D. J. Mitchell, “Leaky rays on circular optical fibers,” J. Opt. Soc. Am. 64, 599–607 (1974).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

Stewart, W. J.

W. J. Stewart, “A new technique for determining the V values and refractive index profiles of optical fibers,” paper presented at OSA/IEEE meeting on Optical Fiber Transmission, Williamsburg, Virginia (1975).

Winkler, C.

J. D. Love and C. Winkler, “The effects of material absorption on ray power attenuation in multilayered optical waveguides” (unpublished).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 40.

Appl. Opt. (2)

Arch. Elektron. Ubertragungsteck (1)

K. Petermann, “The mode attenuation in general graded core multimode fibers,” Arch. Elektron. Ubertragungsteck 29, 345–348 (1975).

Bell Syst. Tech. J. (1)

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

Electron. Lett. (3)

A. W. Snyder and J. D. Love, “Attenuation coefficient for rays in graded fibers 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]

M. J. Adams, D. N. Payne, and F. M. E. Sladen, “Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles,” Electron. Lett. 11, 238–240 (1975).
[CrossRef]

IEEE Trans. (1)

A. W. Snyder and J. D. Love, “Reflection at a curved dilectric interface-electromagnetic tunneling,” IEEE Trans. MTT-13, 134–141 (1975).

J. Opt. Soc. Am. (3)

Opt. Commun. (2)

M. J. Adams, D. H. Payne, and F. M. E. Sladen, “Length-dependent effects due to leaky modes on multimode graded-index optical fibers,” Opt. Commun. 17, 204–209 (1976).
[CrossRef]

A. W. Snyder and J. D. Love, “Tunneling leaky modes on optical waveguides,” Opt. Commun. 12, 325–328 (1974).
[CrossRef]

Optik (1)

A. W. Snyder and D. J. Mitchell, “Generalized Fresnel’s laws for determining radiation loss from optical waveguides and curved dielectric structures,” Optik 40, 438–459 (1975).

Other (6)

W. J. Stewart, “A new technique for determining the V values and refractive index profiles of optical fibers,” paper presented at OSA/IEEE meeting on Optical Fiber Transmission, Williamsburg, Virginia (1975).

J. D. Love and C. Winkler, “The effects of material absorption on ray power attenuation in multilayered optical waveguides” (unpublished).

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 40.

L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1963), p. 286.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

H. Bremmer, “The W.K. B. approximation as the first term of a geometric-optical series,” in The Theory of Electromagnetic Waves, edited by M. Kline (Dover, New York, 1951).

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Figures (9)

FIG. 1
FIG. 1

The refractive index profile n(y) showing arbitrary variation in the continuous sections between discontinuities or jumps. Note that the continuous sections may contain kinks, i. e., points at which there is an abrupt change in the slope or derivative of n(y). The positions of the caustics at ytp and yrad depend upon both the geometry of the structure and the profile. For a slab geometry n(ytp) = n(yrad), while for cylindrical geometry, n ( y tp ) n ( y rad ) because of the azimuthal ray invariant l, Eq. (6). The evanescent region association with the tunneling ray is shown shaded.

FIG. 2
FIG. 2

Tunneling ray paths within an optical waveguide core showing distance zp between reflection points at which power is lost for (a) a graded index profile on a cylindrical waveguide and (b) a step index profile on a slab waveguide.

FIG. 3
FIG. 3

Profile n(y) for step index fiber. The evanescent region is shown shaded. Reflection occurs at ytp while reradiation is observed at yrad.

FIG. 4
FIG. 4

Direction angles (θx, α, θz) of the incident wave vector k1 relative to the Cartesian axes at the point P on the core-cladding interface of the step fiber defined in Fig. 3.

FIG. 5
FIG. 5

Visual representation of the three regions of incident ray angle space for a ray that arrives at P from the core. Trapped ray angles lie in the half-cone θ z π / 2 α c, and refracting rays lie in the half cone α i α c. The two half-cones touch along a common generator in the meridional plane. Tunneling rays are incident in the two symmetric regions on either side of the meridional plane through P and exterior to the half-cones.

FIG. 6
FIG. 6

Symmetrical slab waveguide showing a tunneling ray. The layer of refractive index n2 has thickness d, and n1 > n2, n3 > n2. The evanescent region is shown shaded.

FIG. 7
FIG. 7

The plane wave decomposition of a ray propagating in a graded index cylindrical waveguide, showing the relationship between the ray invariants l and β, and the angles θ and ϕ. θ is the angle between the wave vector and the waveguide z axis, and ϕ is the angle between the projection of the wave vector onto the waveguide cross section and the azimuthal direction.

FIG. 8
FIG. 8

Local plane wave description of tunneling in a fiber with discontinuity at y = ρ showing flux tubes (shaded areas). The fields between ytp and yrad are evanescent. P(y) is the power flow along the flux tube at position y, and α(y) is the angle the flux tube makes with the y direction. ΔZ is the flux path width at the two caustics.

FIG. 9
FIG. 9

Profile of the square of refractive index showing the linearizations about the jump at y = ρ (dotted lines) and the four regions used in the Appendix, corresponding to (1) y y tp, (2) y tp y ρ, (3) ρ y y rad, and (4) y y rad.

Equations (40)

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T E = exp ( 2 y tp y rad | k y ( y ) | d y ) i = 1 N | T F E i | ,
T H = exp ( 2 y tp y rad | k y ( y ) | d y ) i = 1 N | T F H i | ,
| T F E i | = | 4 k y 1 k y 2 / ( k y 1 + k y 2 ) 2 |
| T F H i | = | 4 n 1 2 n 2 2 k y 1 k y 2 / ( n 2 2 k y 1 + n 1 2 k y 2 ) 2 |
P ( z ) = P ( 0 ) e γ z ,
γ = T / z p ,
k y ( y ) 0 { as y y tp , y y tp , or y y rad , y y rad .
k y ( y ) 2 + ( l 2 / y 2 ) + β 2 = { n 1 2 k 2 , y < ρ n 2 2 k 2 , y > ρ
U = ρ ( n 1 2 k 2 β 2 ) 1 / 2 , V = k ρ ( n 1 2 n 2 2 ) 1 / 2 , Q = ρ ( n 2 2 k 2 β 2 ) 1 / 2 .
y rad = ρ l / Q .
T = 4 ( U 2 l 2 ) 1 / 2 ( l 2 Q 2 ) 1 / 2 V 2 × exp ( 2 ( l 2 Q 2 ) 1 / 2 2 l cosh 1 l Q ) .
z p = [ 2 ρ ( n 1 2 k 2 U 2 ) 1 / 2 ( U 2 l 2 ) 1 / 2 ] / U 2 ,
γ = 2 ρ ( U V ) 2 ( l 2 Q 2 ) 1 / 2 ( n 1 2 k 2 U 2 ) 1 / 2 × exp ( 2 ( l 2 Q 2 ) 1 / 2 2 l cosh 1 l Q ) .
β = n 1 k cos θ z , l = n 1 k ρ cos θ x , k y ( ρ ) = n 1 k cos α i , α c = sin 1 ( n 2 / n 1 ) ,
T 4 θ t θ c ( 1 θ t θ c ) 1 / 2 exp ( 2 n 1 k ρ 3 cos 2 θ x ( θ c 2 θ t 2 ) 3 / 2 ) ,
T E = 16 k y 1 | k y 2 | 2 k y 3 exp ( 2 | k y 2 | d ) ( k y 1 2 + | k y 2 | 2 ) ( k y 3 2 + | k y 2 | 2 ) ,
k y 1 = n 1 k cos α i , k y 3 = k [ n 1 2 cos 2 α i ( n 1 2 n 3 2 ) ] 1 / 2 , k y 2 = i k [ ( n 1 2 n 2 2 ) n 1 2 cos 2 α i ] 1 / 2 ,
T E = 4 k y 1 | k y 2 | 2 k y 3 [ 1 tanh 2 ( | k y 2 | d ) ] [ | k y 2 | 2 ( k y 1 + k y 3 ) 2 + tanh 2 ( | k y 2 | d ) ( k y 1 k y 3 | k y 2 | 2 ) ] .
γ = T E / 2 d tan α i ,
n 2 ( y ) = n 2 ( 0 ) [ 1 2 Δ ( y / ρ ) 2 ] , 0 y ρ = n c 1 2 , y ρ
ln T = ln ( 4 | k y 1 | | k y 2 | ) 2 ln | k y 1 + k y 2 | + ( l 2 Q 2 ) 1 / 2 ( U 2 4 V l 2 ) × ln ( U 4 4 l 2 V 2 ) + U 2 2 V ln [ 2 V 2 U 2 + 2 V ( l 2 Q 2 ) 1 / 2 ] + 2 l ln Q 2 ln [ l + ( l 2 Q 2 ) 1 / 2 ] + l ln [ U 2 2 l 2 2 l ( l 2 Q 2 ) 1 / 2 ] ,
γ = V T / π β ρ 2 ,
| k y 1 | = [ β 2 + l 2 / ρ 2 n 2 ( 0 ) ( 1 2 Δ ) ] 1 / 2 , | k y 2 | = [ β 2 + l 2 / ρ 2 n c 1 2 ] 1 / 2 .
T = [ P ( y ) Δ z cos α ( y ) ] y y rad / [ P ( y ) Δ z cos α ( y ) ] y y tp ,
| E | 2 = | n ( y ) Δ x cos α ( y ) | 1 , y < y tp = | T F | | n ( y ) Δ x cos α ( y ) | 1 × exp { 2 y tp y rad | k y ( t ) | d t } , y < y rad
T = | T F | exp ( 2 y tp y rad | k y ( t ) | d t ) .
d 2 Φ ( y ) d y 2 + [ k 2 n 2 ( y ) β 2 ] Φ ( y ) = 0 ,
n 2 ( y ) = n 1 2 + ( ρ y ) α , y < ρ
= n 2 2 + ( ρ y ) α ¯ , y > ρ
Φ ( y ) = [ Ai ( s ) i Bi ( s ) ] + R [ Ai ( s ) + i Bi ( s ) ] , y > ρ
= S [ Ai ( t ) + i Bi ( t ) ] , y < ρ
u = s | y = ρ , υ = t | y = ρ , u = s | y = ρ , υ = t | y = ρ ,
T = 1 | R | 2 = 4 δ π 2 / ( [ Ai ( u ) 2 + Bi ( u ) 2 ] [ Ai ( υ ) 2 + Bi ( υ ) 2 ] δ 2 + 2 δ { [ Ai ( υ ) Ai ( υ ) + Bi ( υ ) Bi ( υ ) ] [ Ai ( u ) Ai ( u ) + Bi ( u ) Bi ( u ) ] + 1 / π 2 } + [ Ai ( υ ) 2 + Bi ( υ ) 2 ] [ Ai ( u ) 2 + Bi ( u ) 2 ] ) ,
T = ( 4 δ / π ) υ 1 / 2 exp ( 2 ρ y rad | k y ( y ) | d y ) δ 2 υ [ Ai ( u ) 2 + Bi ( u ) 2 ] + 2 δ υ 1 / 2 [ Ai ( u ) Ai ( u ) + Bi ( u ) Bi ( u ) ] + ( 2 δ / π ) υ 1 / 2 + [ Ai ( u ) 2 + Bi ( u ) 2 ] ,
T = ( 4 δ / π ) u 1 / 2 exp ( 2 y tp ρ | k y ( y ) | d y ) δ 2 [ Ai ( υ ) 2 + Bi ( υ ) 2 ] + 2 δ u 1 / 2 [ Ai ( υ ) Ai ( υ ) + Bi ( υ ) Bi ( υ ) ] + ( 2 δ / π ) u 1 / 2 + u [ Ai ( u ) 2 + Bi ( u ) 2 ] ,
T = 4 k y 1 k y 2 ( k y 1 + k y 2 ) 2 exp ( 2 y tp y rad | k y ( y ) | d y ) ,
T = exp ( 2 y tp y rad | k y ( y ) | d y ) i = 1 N | T F i | ,
d 2 Φ ( r ) d r 2 + 1 r d Φ ( r ) d r + ( n 2 ( r ) k 2 β 2 l 2 r 2 ) Φ ( r ) = 0 .
d 2 Ψ ( y ) d y 2 + ( k 2 n 2 ( y ) β 2 ( l 2 1 4 ) y 2 ) Ψ ( y ) = 0 .
k y 2 ( y ) = n 2 ( y ) k 2 β 2 ( l 2 / y 2 ) ,