Abstract

The scattering of propagating waves from an abruptly terminated single-mode-fiber waveguide is investigated analytically by means of an integral equation. Weak-guidance conditions on the fiber guide are assumed to formulate the corresponding boundary value problem, and then a Neumann-series iterative solution is developed to solve the basic integral equation. The reflection coefficients of the propagating waves and the radiation patterns of the diffracted waves are also computed. Numerical results are presented for several abruptly terminated fiber guides.

© 1987 Optical Society of America

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References

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  1. K. F. Barrell, C. Pask, “Ray launching and observation in graded-index optical fibers,”J. Opt. Soc. Am. 69, 294–300 (1979).
    [Crossref]
  2. G. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,”IRE Trans. Antennas Propag. AP-5, 100–109 (1957).
    [Crossref]
  3. P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
    [Crossref]
  4. T. E. Rozzi, G. H. Int Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
    [Crossref]
  5. C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
    [Crossref]
  6. A. D. Yaghjian, E. T. Kornhauser, “A model analysis of the dielectric rod antenna excited by the HE11mode,”IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
    [Crossref]
  7. A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Invest. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).
  8. A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change of parameters. I. Solution by the factoring method,” Izvest. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1329–1338 (1982).
  9. A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change in parameters. II. Solution by a variational method,” Izvest. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1484–1490 (1982).
  10. A. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
    [Crossref]
  11. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983), Chaps. 25 and 32.
  12. S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), Secs. 5–14.
  13. A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), Vol. 6, Chap. 6.
  14. D. S. Jones, Theory of Electromagnetism (Pergamon, Oxford, 1964), Chap. 8.

1985 (2)

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Invest. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).

1982 (2)

A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change of parameters. I. Solution by the factoring method,” Izvest. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1329–1338 (1982).

A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change in parameters. II. Solution by a variational method,” Izvest. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1484–1490 (1982).

1981 (1)

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[Crossref]

1980 (1)

T. E. Rozzi, G. H. Int Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[Crossref]

1979 (1)

1972 (1)

A. D. Yaghjian, E. T. Kornhauser, “A model analysis of the dielectric rod antenna excited by the HE11mode,”IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
[Crossref]

1971 (1)

A. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
[Crossref]

1957 (1)

G. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,”IRE Trans. Antennas Propag. AP-5, 100–109 (1957).
[Crossref]

Angulo, G. M.

G. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,”IRE Trans. Antennas Propag. AP-5, 100–109 (1957).
[Crossref]

Barrell, K. F.

Capsalis, C. N.

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

Citerne, J.

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[Crossref]

Fikioris, J. G.

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

Gelin, P.

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[Crossref]

Int Veld, G. H.

T. E. Rozzi, G. H. Int Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[Crossref]

Jones, D. S.

D. S. Jones, Theory of Electromagnetism (Pergamon, Oxford, 1964), Chap. 8.

Kornhauser, E. T.

A. D. Yaghjian, E. T. Kornhauser, “A model analysis of the dielectric rod antenna excited by the HE11mode,”IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
[Crossref]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983), Chaps. 25 and 32.

Manenkov, A. B.

A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Invest. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).

A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change of parameters. I. Solution by the factoring method,” Izvest. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1329–1338 (1982).

A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change in parameters. II. Solution by a variational method,” Izvest. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1484–1490 (1982).

Pask, C.

Petenzi, M.

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[Crossref]

Rozzi, T. E.

T. E. Rozzi, G. H. Int Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[Crossref]

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), Secs. 5–14.

Snyder, A.

A. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
[Crossref]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983), Chaps. 25 and 32.

Sommerfeld, A. R.

A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), Vol. 6, Chap. 6.

Uzunoglu, N. K.

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

Yaghjian, A. D.

A. D. Yaghjian, E. T. Kornhauser, “A model analysis of the dielectric rod antenna excited by the HE11mode,”IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
[Crossref]

IEEE J. Lightwave Technol. (1)

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. LT-3, 408–415 (1985).
[Crossref]

IEEE Trans. Antennas Propag. (1)

A. D. Yaghjian, E. T. Kornhauser, “A model analysis of the dielectric rod antenna excited by the HE11mode,”IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
[Crossref]

IEEE Trans. Microwave Theory Tech. (3)

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[Crossref]

T. E. Rozzi, G. H. Int Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[Crossref]

A. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
[Crossref]

Invest. Vyssh. Uchebn. Zaved. Radiofiz. (1)

A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Invest. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).

IRE Trans. Antennas Propag. (1)

G. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,”IRE Trans. Antennas Propag. AP-5, 100–109 (1957).
[Crossref]

Izvest. Vyssh. Uchebn. Zaved. Radiofiz. (2)

A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change of parameters. I. Solution by the factoring method,” Izvest. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1329–1338 (1982).

A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change in parameters. II. Solution by a variational method,” Izvest. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1484–1490 (1982).

J. Opt. Soc. Am. (1)

Other (4)

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983), Chaps. 25 and 32.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), Secs. 5–14.

A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), Vol. 6, Chap. 6.

D. S. Jones, Theory of Electromagnetism (Pergamon, Oxford, 1964), Chap. 8.

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Figures (4)

Fig. 1
Fig. 1

Geometry of an abruptly terminated single-mode fiber.

Fig. 2
Fig. 2

Real part of the reflection coefficient versus the fiber core diameter α (micrometers) for n1 = (1 − Δ)n2, n2 = 1.46, and Δ = 0.002, 0.004, 0.006, 0.01, and 0.03. The free-space radiation wavelength is λ0 = 1.3 μm, and n0 = 1.

Fig. 3
Fig. 3

Real part of the reflection coefficient versus the refractive index n0 of the semi-infinite space z > 0 for Δ = 0.005, n2 = 1.46, n1 = (1 − Δ)n2, and α = 2 and 3 μm.

Fig. 4
Fig. 4

Radiation patterns of three abruptly terminated fiber guides for Δ = 0.005, n2 = 1.46, n1 = (1 − Δ)n2, free-space wavelength λ0 = 1.3 μm, and α = 2 and 3 μm.

Tables (1)

Tables Icon

Table 1 Convergence Pattern for an Abruptly Terminated Fiber Waveguidea

Equations (47)

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ψ g x = U m ( ρ , φ , α 0 ) exp ( - j β 0 z ) ,
U m ( ρ , φ , α 0 ) = C m e j m φ { K m ( γ a ) J m ( α 0 a ) J m ( α 0 ρ ) for ρ < a K m ( γ ρ ) for ρ > a ,
α 0 2 = k 0 2 n 1 2 - β 0 2 ,             γ 2 = β 0 2 - k 0 2 , m = 0 , ± 1 , ± 2 , ,
( α 0 a ) J m + 1 ( α 0 a ) J m ( α 0 a ) = ( γ a ) K m + 1 ( γ a ) K m ( γ a ) .
1 2 π φ = 0 2 π d φ 0 + ρ d ρ U m ( ρ , φ , α 0 ) U m * ( ρ , φ , α 0 ) = 1 ;
C m = J m ( α 0 a ) a [ 2 K m 2 ( γ a ) J m + 1 2 ( α 0 a ) + K m + 1 2 ( γ a ) J m 2 ( α 0 a ) ] 1 / 2
Ψ r x = Ψ m ( ρ , φ , q ) e j β z ,
β = ( k 0 2 n 1 2 - q 2 ) 1 / 2 ,             0 < q < + ,
Ψ m ( ρ , φ , q ) = A m ( q ) e j m φ × { 1 C m ( q ) J ( σ ρ ) for ρ < a [ J m ( q ρ ) + D m ( q ) Y m ( q ρ ) ] for ρ > a ,
σ = ( k 0 2 n 2 2 - β 2 ) 1 / 2 , D m ( q ) = σ J m + 1 ( σ a ) J m ( q a ) - q J m ( σ a ) J m + 1 ( q a ) q J m ( σ a ) Y m + 1 ( q a ) - σ J m + 1 ( σ a ) Y m ( q a ) ,
C m ( q ) = q J m ( σ a ) Y m + 1 ( q a ) - σ J m + 1 ( σ a ) Y m ( q a ) q J m ( q a ) Y m + 1 ( q a ) - q J m + 1 ( q a ) Y m ( q a ) , Re ( β ) > 0 ,             Im ( β ) < 0 ,
A m ( q ) = 1 [ 1 + D m 2 ( q ) ] 1 / 2
1 2 π φ = 0 2 π d φ ρ = 0 + ρ d ρ Ψ m ( ρ , φ , q ) Ψ m * ( ρ , φ , q ) = 1 q δ ( q - q ) δ m m
φ = 0 2 π d φ ρ = 0 + ρ d ρ Ψ m ( ρ , φ , q ) U 0 ( ρ , φ , α 0 ) = 0.
U 0 ( ρ , φ , α 0 ) U 0 * ( ρ , φ , α 0 ) + m = - + q = 0 + q d q Ψ m ( ρ , φ , q ) Ψ m * ( ρ , φ , q ) = 2 π ρ δ ( ρ - ρ ) δ ( φ - φ ) .
Ψ I ( ρ , φ , z ) = U 0 ( ρ , φ , α 0 ) e - j β 0 z + U 0 ( ρ , φ , α 0 ) e j β 0 z R ( α 0 ) + 0 + q d q m = - + R m ( q ) Ψ m ( ρ , φ , q ) e j β z .
Ψ I I ( ρ , φ , z ) = 0 + λ d λ m = - + Φ m ( ρ , φ , λ ) e - j W z C m ( λ ) ,
Φ m ( ρ , φ , λ ) = J m ( λ ρ ) e j m φ             ( 0 < λ < + )
1 2 π 0 2 π d φ 0 + ρ d ρ Φ m ( ρ , φ , λ ) Φ m * ( ρ , φ , λ ) = δ ( λ - λ ) λ ,
m = - + 0 + λ d λ Φ m ( ρ , φ , λ ) Φ m ( ρ , φ , λ ) = 2 π ρ δ ( ρ - ρ ) δ ( φ - φ ) .
Ψ I ( ρ , φ , z ) z = 0 = Ψ I I ( ρ , φ , z ) z = 0 = E ( ρ , φ ) ,
Ψ i ( ρ , φ , z ) z | z = 0 = Ψ I I ( ρ , φ , z ) z | z = 0 .
2 β 0 U 0 ( ρ , φ , α 0 ) = φ = 0 2 π d φ ρ = 0 + ρ d ρ Ξ ( ρ , φ / ρ , φ ) E ( ρ , φ ) ,
Ξ ( ρ , φ / ρ , φ ) = 1 2 π ( β 0 U 0 ( ρ , φ , α 0 ) U 0 * ( ρ , φ , α 0 ) + q = 0 + q d q m = - + { β Ψ m ( ρ , φ , q ) Ψ m ( ρ , φ , q ) + J m q ρ ) J m ( q ρ ) exp [ j m ( φ - φ ) ] W ( q ) } ) .
E ( ρ , φ ) = E 0 ( ρ , φ ) + φ = 0 2 π d φ ρ = 0 + ρ d ρ K ( ρ , φ / ρ , φ ) E ( ρ , φ ) ,
E 0 ( ρ , φ ) = 2 β 0 U 0 ( ρ , φ , α 0 ) k 0 ( n 0 + n 1 ) ,
K ( ρ , φ / ρ , φ ) = - 1 2 π k 0 ( n 1 + n 0 ) ( ( β 0 - k 0 n 1 ) U 0 ( ρ , φ , α 0 ) × U 0 * ( ρ , φ , α 0 ) + q = 0 + q d q m = - + { ( β - k 0 n 1 ) × Ψ m ( ρ , φ , q ) Ψ m * ( ρ , φ , q ) + [ w ( q ) - k 0 n 0 ] × J m ( q ρ ) J m ( q ρ ) exp [ j m ( φ - φ ) ] } ) .
E ( ρ , φ ) = E 0 ( ρ , φ ) = 2 n 1 n 0 + n 1 U 0 ( ρ , φ , 0 ) ,
E N ( ρ , φ ) = E 0 ( ρ , φ ) + n = 1 N C n ( ρ , φ ) ,             N = 1 , 2 , 3 , ,
C n ( ρ , φ ) = 0 + ρ 1 d ρ 1 0 2 π d φ 1 0 + ρ 2 d ρ 2 0 2 π d φ 2 0 + ρ n d ρ n × 0 2 π d φ n K ( ρ , φ / ρ 1 , φ 1 ) K ( ρ 1 , φ 1 / ρ 2 , φ 2 ) K ( ρ n - 1 , φ n - 1 / ρ n , φ n ) E 0 ( ρ n , φ n ) .
E 1 ( ρ , φ ) = 2 β 0 ( k 0 n 0 - β 0 + 2 k 0 n 1 ) k 0 2 ( n 0 + n 1 ) 2 U 0 ( ρ , φ , α 0 ) - 2 β 0 a C 0 k 0 2 ( n 0 + n 1 ) 2 λ = 0 + λ d λ J 0 ( λ ρ ) S ( λ ) [ W ( λ ) - k 0 n 0 ] ,
S ( λ ) = k 0 2 ( n 2 2 - n 1 2 ) γ K 1 ( γ a ) J 0 ( λ a ) - λ K 0 ( γ a ) J 1 ( λ a ) ( α 0 2 - λ 2 ) ( γ 2 + λ 2 ) .
R ( α 0 ) = ρ = 0 + ρ d ρ φ = 0 2 π d φ E ( ρ φ ) U 0 * ( ρ , φ , α 0 ) - 1 .
R ( 0 ) ( α 0 ) = - 1 + 2 β 0 / [ k 0 ( n 0 + n 1 ) ] ,
R ( 1 ) ( α 0 ) = - 1 + 2 β 0 ( k 0 n 0 - β 0 + 2 k 0 n 1 ) [ k 0 ( n 0 + n 1 ) ] 2 - 2 β 0 a 2 C 0 2 [ k 0 ( n 0 + n 1 ) ] 2 × λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S 2 ( λ ) .
Ψ II ( ρ , φ , z ) = 2 β 0 a C 0 ( k 0 n 0 - β 0 + 2 k 0 n 1 ) [ k 0 ( n 0 + n 1 ) ] 2 × λ = 0 + λ d λ J 0 ( λ ρ ) S ( λ ) exp [ - j W ( λ ) z ] - 2 β 0 a C 0 [ k 0 ( n 0 + n 1 ) ] 2 λ = 0 + λ d λ J 0 ( λ ρ ) S ( λ ) × exp [ - j W ( λ ) z ] [ W ( λ ) - k 0 n 0 ] .
E ϑ = cos ϑ cos φ Ψ II , E φ = - sin φ Ψ II
Ψ II ~ 2 β 0 a C 0 ( n 0 + n 1 ) 2 exp ( - j k 0 n 0 r ) r S ( k 0 n 0 sin ϑ ) n 0 cos ϑ × [ n 0 - β 0 k 0 + 2 n 1 - n 0 ( cos ϑ - 1 ) ] .
n 2 = 1.46 ,             n 1 = n 2 ( 1 - Δ ) ,
I α = ρ = 0 a ρ d ρ A m ( q ) A m ( q ) C m ( q ) C m ( q ) J m [ σ ( q ) ρ ) J m [ σ ( q ) ρ ] = A m ( q ) A m ( q ) C m ( q ) C m ( q ) a σ 2 - σ 2 [ σ J m + 1 ( σ a ) J m ( σ a ) - σ J m ( σ a ) J m + 1 ( σ a ) ] ,
δ ( x ) = lim k + sin ( k x ) π x .
1 2 π 0 2 π d φ 0 + ρ d ρ Ψ m ( ρ , φ , q ) Ψ m ( ρ , φ , q ) = δ m m δ ( q - q ) q [ A m ( q ) ] 2 { 1 + [ D m ( q ) ] 2 } ,
E 2 ( ρ , φ ) = 2 β 0 [ ( k 0 n 0 ) 2 + β 0 2 + 3 ( k 0 n 1 ) 2 + 3 k 0 2 n 1 n 0 - k 0 n 0 β 0 - 3 k 0 β 0 n 1 ] [ k 0 ( n 1 + ( n 0 ) ] 3 U 0 ( ρ , φ , α 0 ) - 2 β 0 a C 0 ( k 0 n 0 - β 0 + 2 k 0 n 1 ) [ k 0 ( n 0 + n 1 ) ] 3 × λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] J 0 ( λ ρ ) S ( λ ) + 2 β 0 a 2 C 0 2 ( β 0 - k 0 n 1 ) U 0 ( ρ , φ , α 0 ) [ k 0 ( n 0 + n 1 ) ] 3 λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S 2 ( λ ) + 2 β 0 a 2 C 0 [ k 0 ( n 0 + n 1 ) ] 3 × q = 0 + q d q A 0 ( q ) ( β - k 0 n 1 ) Ψ 0 ( ρ , φ , q ) λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S ( λ ) Q ( q , λ , σ ) + 2 β 0 a C 0 [ k 0 ( n 0 + n 1 ) ] 3 λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] 2 × J 0 ( λ ρ ) S ( λ ) ,
Q ( q , λ , σ ) = - k 0 ( n 2 2 - n 1 2 ) a ( q J 0 ( σ a ) Y 1 ( q a ) - σ J 1 ( σ a ) Y 0 ( q a ) λ J 1 ( λ a ) J 0 ( σ a ) - σ J 0 ( λ a ) J 1 ( σ a ) ( λ 2 - σ 2 ) ( λ 2 - q 2 ) .
E 3 ( ρ , φ ) = 2 β 0 k 0 ( n 1 + n 0 ) U 0 ( ρ , φ , α 0 ) - 2 β 0 [ k 0 ( n 0 + n 1 ) ] 4 ( ( β 0 - k 0 n 1 ) U 0 ( ρ , φ , α 0 ) { ( k 0 2 n 0 2 + β 0 2 + 3 k 0 2 n 1 2 + 3 k 0 2 n 1 n 0 - k 0 n 0 β 0 - 3 k 0 β 0 n 1 ) + a 2 C 0 2 ( 2 β 0 - k 0 n 0 - 3 k 0 n 1 ) λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S 2 ( λ ) + a 2 C 0 2 λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] 2 S 2 ( λ ) } + a 2 C 0 q = 0 + q d q ( β - k 0 n 1 ) A 0 ( q ) Ψ 0 ( ρ , φ , q ) λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S ( λ ) Q ( q , λ , σ ) [ β 0 - 2 k 0 n 0 + β - 3 k 0 n 1 + W ( λ ) ] + a C 0 λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] J 0 ( λ ρ ) { ( k 0 2 n 0 2 + β 0 2 + 3 k 0 2 ( n 1 2 + n 1 n 0 ) - k 0 n 0 β 0 - 3 k 0 β 0 n 1 ) S ( λ ) + [ W ( λ ) - k 0 n 0 ] × S ( λ ) · [ W ( λ ) - 2 k 0 n 0 + β 0 - 2 k 0 n 1 ] + a 2 C 0 2 S ( λ ) ( β 0 - k 0 n 1 ) λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S 2 ( λ ) + a 2 λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S ( λ ) q = 0 + q d q A 0 2 ( q ) ( β - k 0 n 1 ) Q ( q , λ , σ ) Q ( q , λ , σ ) } ) .
R 2 ( α 0 ) = - 1 + 2 β 0 ( k 0 2 n 0 2 + β 0 2 + 3 k 0 2 n 1 2 + 3 k 0 2 n 0 n 1 - k 0 n 0 β 0 - 3 k 0 β 0 n 1 ) [ k 0 ( n 0 + n 1 ) ] 3 - 2 β 0 a 2 C 0 2 ( k 0 n 0 - 2 β 0 + 3 k 0 n 1 ) [ k 0 ( n 0 + n 1 ) ] 3 λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S 2 ( λ ) + 2 β 0 a 2 C 0 2 [ k 0 ( n 0 + n 1 ) ] 3 λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] 2 S 2 ( λ ) .
R ( 3 ) ( α 0 ) = - 1 + { 2 β 0 [ ( k 0 n 0 ) 3 - β 0 3 + 4 k 0 3 n 1 3 + 4 k 0 3 n 1 n 0 2 + 6 k 0 3 n 1 2 n 0 - k 0 2 n 0 2 β 0 - 6 k 0 2 β 0 n 1 2 - 4 k 0 2 β 0 n 0 n 1 + k 0 n 0 β 0 2 + 4 k 0 β 0 2 n 1 ] } / [ k 0 ( n 0 + n 1 ) ] 4 - 2 β 0 a 2 C 0 2 [ 3 β 0 2 + ( k 0 n 0 ) 2 + 4 k 0 2 n 0 n 1 + 6 k 0 2 n 1 2 - 2 k 0 n 0 β 0 - 8 k 0 β 0 n 1 ] [ k 0 ( n 0 + n 1 ) ] 4 λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S 2 ( λ ) - 2 β 0 α 2 C 0 2 [ k 0 ( n 0 + n 1 ) ] 4 λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] 2 [ W ( λ ) - 2 k 0 n 0 + 2 β 0 - 3 k 0 n 1 ] S 2 ( λ ) - 2 β 0 a 4 C 0 4 ( β 0 - k 0 n 1 ) [ k 0 ( n 0 + n 1 ) ] 4 × { λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S 2 ( λ ) } 2 - 2 β 0 a 4 C 0 2 [ k 0 ( n 0 + n 1 ) ] 4 q = 0 + q d q A 0 2 ( q ) ( β - k 0 n 1 ) { λ = 0 + λ d λ [ W ( λ ) - k 0 n 0 ] S ( λ ) Q ( q , λ , σ ) } 2 .

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