Abstract

The reflectivity of a multilayer dielectric coating deposited upon the end facet of a symmetrical dielectric slab waveguide is investigated for both TE and TM waves. The integral equation satisfied by the field in the junction plane is solved numerically through a Fourier representation and an iteration process. The process converges rapidly and can easily provide a solution of as many as five or six exact digits in the case of problems involving semiconductor optical waveguides. In this case of sufficiently weak guidance, the integral equation can be simplified considerably, leading to an explicit solution for the Fourier transform of the field. This approximation greatly simplifies the computation of the reflectivity but is accurate enough to meet the current needs for the design of antireflection coatings.

© 1988 Optical Society of America

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References

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  1. C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifier,” Electron Lett. 21, 333–334 (1985).
    [CrossRef]
  2. C. Vassallo, “Polarization-independent antireflection coatings for semiconductor optical amplifiers,” Electron. Lett. 24, 61–62 (1988).
    [CrossRef]
  3. M. J. O’Mahony, “Semiconductor laser amplifiers as repeaters,” in Technical Digest of IOOC-ECOC’85 (Istituto Internazionale delle Comunicazoni, Genova, Italy, 1985), Vol. 2, pp. 39–46.
  4. T. Saitoh, T. Mukai, O. Mikami, “Theoretical analysis and fabrication of antireflection coatings on laser diode facets,” IEEE J. Lightwave Technol. LT-3, 288–293 (1985).
    [CrossRef]
  5. J. C. Simon, “GaInAsP semiconductor laser amplifiers for single-mode fiber communications,” IEEE J. Lightwave Technol. LT-5, 1286–1294 (1987).
    [CrossRef]
  6. T. E. Rozzi, “Rigorous analysis of the step discontinuities in the planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
    [CrossRef]
  7. T. E. Rozzi, G. H. Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and of dielectric millimiter wave antenna,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
    [CrossRef]
  8. S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
    [CrossRef]
  9. K. Morishita, S. I. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
    [CrossRef]
  10. P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
    [CrossRef]
  11. G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
    [CrossRef]
  12. M. A. A. Pudensi, L. G. Ferreira, “Method to calculate the reflection and transmission of guided waves,” J. Opt. Soc. Am. 72, 126–130 (1982).
    [CrossRef]
  13. T. Ikegami, “Reflectivity of mode of facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
    [CrossRef]
  14. H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity in dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
    [CrossRef]
  15. D. R. Kaplan, P. P. Deimel, “Exact calculation of the reflection coefficient for coated optical waveguide devices,” Bell Syst. Tech. J. 63, 857–877 (1984).
  16. C. Vassallo, “On a direct use of edge condition in modal analysis,” IEEE Trans. Microwave Theory Tech. MTT-24, 208–212 (1976).
    [CrossRef]
  17. C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
    [CrossRef]
  18. N. I. Muskhelishvili, Singular Integral Equations (Noordhoff, Groningen, The Netherlands, 1953), Chap. 3.
  19. A. W. Snyder, W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
    [CrossRef]
  20. C. Vassallo, Théorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985), Chap. 3, Sec. 7.

1988 (2)

C. Vassallo, “Polarization-independent antireflection coatings for semiconductor optical amplifiers,” Electron. Lett. 24, 61–62 (1988).
[CrossRef]

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
[CrossRef]

1987 (1)

J. C. Simon, “GaInAsP semiconductor laser amplifiers for single-mode fiber communications,” IEEE J. Lightwave Technol. LT-5, 1286–1294 (1987).
[CrossRef]

1986 (1)

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity in dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

1985 (2)

T. Saitoh, T. Mukai, O. Mikami, “Theoretical analysis and fabrication of antireflection coatings on laser diode facets,” IEEE J. Lightwave Technol. LT-3, 288–293 (1985).
[CrossRef]

C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifier,” Electron Lett. 21, 333–334 (1985).
[CrossRef]

1984 (1)

D. R. Kaplan, P. P. Deimel, “Exact calculation of the reflection coefficient for coated optical waveguide devices,” Bell Syst. Tech. J. 63, 857–877 (1984).

1982 (2)

G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

M. A. A. Pudensi, L. G. Ferreira, “Method to calculate the reflection and transmission of guided waves,” J. Opt. Soc. Am. 72, 126–130 (1982).
[CrossRef]

1981 (1)

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

1980 (1)

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

1979 (1)

K. Morishita, S. I. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

1978 (2)

T. E. Rozzi, “Rigorous analysis of the step discontinuities in the planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

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

1976 (1)

C. Vassallo, “On a direct use of edge condition in modal analysis,” IEEE Trans. Microwave Theory Tech. MTT-24, 208–212 (1976).
[CrossRef]

1975 (1)

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

1972 (1)

T. Ikegami, “Reflectivity of mode of facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

Beal, J. C.

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

Brooke, G. H.

G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

Citerne, J.

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

Deimel, P. P.

D. R. Kaplan, P. P. Deimel, “Exact calculation of the reflection coefficient for coated optical waveguide devices,” Bell Syst. Tech. J. 63, 857–877 (1984).

Ferreira, L. G.

Gelin, P.

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

Ikegami, T.

T. Ikegami, “Reflectivity of mode of facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

Inagaki, S. I.

K. Morishita, S. I. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Kaplan, D. R.

D. R. Kaplan, P. P. Deimel, “Exact calculation of the reflection coefficient for coated optical waveguide devices,” Bell Syst. Tech. J. 63, 857–877 (1984).

Kharadly, M. M.

G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

Kumagai, N.

K. Morishita, S. I. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Mahmoud, S. F.

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

Mikami, O.

T. Saitoh, T. Mukai, O. Mikami, “Theoretical analysis and fabrication of antireflection coatings on laser diode facets,” IEEE J. Lightwave Technol. LT-3, 288–293 (1985).
[CrossRef]

Morishita, K.

K. Morishita, S. I. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Mukai, T.

T. Saitoh, T. Mukai, O. Mikami, “Theoretical analysis and fabrication of antireflection coatings on laser diode facets,” IEEE J. Lightwave Technol. LT-3, 288–293 (1985).
[CrossRef]

Muskhelishvili, N. I.

N. I. Muskhelishvili, Singular Integral Equations (Noordhoff, Groningen, The Netherlands, 1953), Chap. 3.

O’Mahony, M. J.

M. J. O’Mahony, “Semiconductor laser amplifiers as repeaters,” in Technical Digest of IOOC-ECOC’85 (Istituto Internazionale delle Comunicazoni, Genova, Italy, 1985), Vol. 2, pp. 39–46.

Petenzi, M.

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

Pudensi, M. A. A.

Rozzi, T. E.

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

T. E. Rozzi, “Rigorous analysis of the step discontinuities in the planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

Saitoh, T.

T. Saitoh, T. Mukai, O. Mikami, “Theoretical analysis and fabrication of antireflection coatings on laser diode facets,” IEEE J. Lightwave Technol. LT-3, 288–293 (1985).
[CrossRef]

Shigesawa, H.

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity in dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

Simon, J. C.

J. C. Simon, “GaInAsP semiconductor laser amplifiers for single-mode fiber communications,” IEEE J. Lightwave Technol. LT-5, 1286–1294 (1987).
[CrossRef]

Snyder, A. W.

Tsuji, M.

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity in dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

Vassallo, C.

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
[CrossRef]

C. Vassallo, “Polarization-independent antireflection coatings for semiconductor optical amplifiers,” Electron. Lett. 24, 61–62 (1988).
[CrossRef]

C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifier,” Electron Lett. 21, 333–334 (1985).
[CrossRef]

C. Vassallo, “On a direct use of edge condition in modal analysis,” IEEE Trans. Microwave Theory Tech. MTT-24, 208–212 (1976).
[CrossRef]

C. Vassallo, Théorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985), Chap. 3, Sec. 7.

Veld, G. H.

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

Young, W. R.

Bell Syst. Tech. J. (1)

D. R. Kaplan, P. P. Deimel, “Exact calculation of the reflection coefficient for coated optical waveguide devices,” Bell Syst. Tech. J. 63, 857–877 (1984).

Electron Lett. (1)

C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for travelling-wave diode laser amplifier,” Electron Lett. 21, 333–334 (1985).
[CrossRef]

Electron. Lett. (2)

C. Vassallo, “Polarization-independent antireflection coatings for semiconductor optical amplifiers,” Electron. Lett. 24, 61–62 (1988).
[CrossRef]

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
[CrossRef]

IEEE J. Lightwave Technol. (2)

T. Saitoh, T. Mukai, O. Mikami, “Theoretical analysis and fabrication of antireflection coatings on laser diode facets,” IEEE J. Lightwave Technol. LT-3, 288–293 (1985).
[CrossRef]

J. C. Simon, “GaInAsP semiconductor laser amplifiers for single-mode fiber communications,” IEEE J. Lightwave Technol. LT-5, 1286–1294 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Ikegami, “Reflectivity of mode of facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (8)

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity in dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

C. Vassallo, “On a direct use of edge condition in modal analysis,” IEEE Trans. Microwave Theory Tech. MTT-24, 208–212 (1976).
[CrossRef]

T. E. Rozzi, “Rigorous analysis of the step discontinuities in the planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

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

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

K. Morishita, S. I. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

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

G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

Other (3)

C. Vassallo, Théorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985), Chap. 3, Sec. 7.

N. I. Muskhelishvili, Singular Integral Equations (Noordhoff, Groningen, The Netherlands, 1953), Chap. 3.

M. J. O’Mahony, “Semiconductor laser amplifiers as repeaters,” in Technical Digest of IOOC-ECOC’85 (Istituto Internazionale delle Comunicazoni, Genova, Italy, 1985), Vol. 2, pp. 39–46.

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Figures (3)

Fig. 1
Fig. 1

Investigated system. Fields do not depend on the y coordinate. An N-layer coating is understood as N different layers, the air being an (N + 1)th medium.

Fig. 2
Fig. 2

Results of various computations for the optimum width of a perfect (zero-reflectivity) monolayer antireflection coating for the waveguide shown in the inset. The optimum width is shown versus the waveguide thickness. Solid curves correspond to an adaptation of the method of Rozzi,6 with, respectively 10, 16, and 18 Laguerre–Gauss functions. The dashed curve corresponds to my approximate derivation.1 The points correspond to my new rigorous method. The arrow at the bottom of the diagram gives an estimation of the refractive-index error that causes the reflectivity to increase from 0 (on the exact optimum curve) to 10−5; the reflectivity increases as δn2.

Fig. 3
Fig. 3

Reflectivity of air at the end of a symmetrical waveguide with ncore = 3.6, nclad = ncore (1 − Δ), λ = 0.86 μm, and a variable thickness from 0 to 0.9 μm. Curves are labeled according to Δ. Solid curves represent my new results; dashed curves represent the results of Ikegami.13

Tables (4)

Tables Icon

Table 1 TE Reflectivity for a Junction between Air and a Waveguide with Parameters nclad = 3.24, ncore = 3.6, λ = 0.86 μm, and 2a = 0.25 μm

Tables Icon

Table 2 TE Reflectivity for a Junction between Air and a Waveguide with Parameters nclad = 1.0, ncore = 3.6, λ = 0.86 μm, and 2a = 0.1815 μm

Tables Icon

Table 3 TM Reflectivity for a Junction between Air and a Waveguide with Parameters nclad = 3.42, ncore = 3.6, λ = 0.86 μm, and 2a = 0.20 μm

Tables Icon

Table 4 TM Reflectivity for a Junction between Air and a Waveguide with Parameters nclad = 3.24, ncore = 3.6, λ = 0.86 μm, and 2a = 0.20 μm

Equations (78)

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e 00 ( x ) + E ref ( x , 0 ) = E tr ( x , 0 ) = E ( x ) ,
β 00 e 00 ( x ) + i z E ref ( x , 0 ) = i z E tr ( x , 0 ) ,
B wg E ( x ) = [ i z E ( x , z ) ] z = 0
B ext E ( x ) = [ i z E tr ( x , z ) ] z = 0 .
B wg ( e 00 E ref ) = B ext E .
( 1 + B wg 1 B ext ) E ( x ) = 2 e 00 ( x ) .
α φ λ ( 1 + B wg 1 B ext ) φ α X α = 2 φ λ e 00 ( for any λ in N ) ,
E ( x ) = 0 E ˜ α cos ( α x ) d α , E ˜ α = 1 π E ( x ) cos ( α x ) dx .
E ˜ tr α ( z ) = S p α exp ( i γ p z ) + T p α exp ( + i γ p z ) ,
γ p 2 = k 2 p α 2 ,
i z E ˜ tr α ( z = 0 ) = β ext ( α ) E ˜ α .
e 00 2 = 1 , e 00 e ν = 0 , e ν e ν = δ ( ν ν ) ,
B wg E ( x ) = e 00 ( x ) β 00 e 00 E + 0 e ν ( x ) β ν e ν E d ν .
B wg p E ( x ) = e 00 ( x ) β 00 p e 00 E + 0 e ν ( x ) β ν p e ν E d ν .
f g = π 0 f ˜ α g ˜ α d α .
FT [ B wg 1 E ] λ = π 0 β 00 1 e ˜ 00 λ e ˜ 00 α E ˜ α d α + π 0 d ν ( β ν 1 e ˜ ν λ 0 e ˜ ν α E ˜ α d α ) ,
FT [ B wg 1 E ] λ = 0 K ( λ , α ) E ˜ α d α .
K ( λ , α ) = β α 1 δ ( α λ ) + ρ ( λ , α ) ,
( 1 + β ext ( λ ) / β λ ) E ˜ λ = 2 e ˜ 00 λ 0 ρ ( λ , α ) β ext ( α ) E ˜ α d α .
E ˜ λ ( 0 ) = 2 ( 1 + β ext ( λ ) / β λ ) 1 e ˜ 00 λ .
E ( x ) = ( 1 + R ) e 00 ( x ) + radiating modes ,
B ext E ( x ) = ( 1 R ) β 00 e 00 ( x ) + radiating modes .
1 + R = e 00 E = π 0 e ˜ 00 α E ˜ α d α ,
β 00 ( 1 R ) = e 00 B ext E = π 0 e ˜ 00 α β ext ( α ) E ˜ α d α .
β 00 ( 1 R ) / ( 1 + R ) = 0 e ˜ 00 α β ext ( α ) E ˜ α d α / 0 e ˜ 00 α E ˜ α d α .
δ R = π 0 e ˜ 00 α δ E ˜ α d α ,
β 00 δ R = π 0 e ˜ 00 α β ext ( α ) δ E ˜ α d α .
2 β 00 δ R ( 1 + R ) 0 e ˜ 00 α [ β ext ( α ) σ ] δ E ˜ α d α .
1 h 00 2 = 1 , 1 h ν h λ = δ ( ν λ ) , 1 h 00 h ν = 0 ,
B ext Φ ( x ) = [ i z Φ ( x , z ) ] z = 0 ,
B wg Ψ ( x ) = [ i z Ψ ( x , z ) ] z = 0 ,
B wg Ψ = β 00 1 h 00 Ψ h 00 ( x ) + 0 β ν 1 h ν Ψ h ν ( x ) d ν .
FT [ B ext Φ ] λ = β ext ( λ ) Φ ˜ λ .
h 00 ( x ) + H ref ( x , 0 ) = H tr ( x , 0 ) ,
( x ) 1 [ β 00 h 00 ( x ) + i z H ref ( x , 0 ) ] = 1 1 i z H tr ( x , 0 ) .
( 1 + B wg 1 ( x ) 1 1 B ext ) H ( x ) = 2 h 00 ( x ) .
H ( x ) + 1 1 [ h 00 B ext H β 00 1 h 00 ( x ) + 0 h ν B ext H β ν 1 h ν ( x ) d ν ] = 2 h 00 ( x ) .
H ˜ α + 1 1 { β 00 1 h ˜ 00 α π 0 h ˜ 00 λ β ext ( λ ) H ˜ λ d λ + 0 d ν β ν 1 h ˜ ν α [ π 0 h ˜ ν λ β ext ( λ ) H ˜ λ d λ ] } = 2 h ˜ 00 α .
( 1 + clad β ext ( α ) / 1 β α ) H ˜ α + 0 ρ ( α , λ ) β ext ( λ ) H ˜ λ d λ = 2 h ˜ 00 α ,
H ref ( x , 0 ) = R h 00 ( x ) + 0 R ν h ν ( x ) d ν ,
1 + R = 1 h 00 H = π 0 H ˜ α h ˜ 00 α d α ,
β 00 ( 1 R ) = 1 1 h 00 B ext H = π 1 1 × 0 h ˜ 00 α β ext ( α ) H ˜ α d α ,
β 00 ( 1 R ) / ( 1 + R ) = 0 h ˜ 00 α β ext ( α ) H ˜ α d α / 1 0 h ˜ 00 α H ˜ α d α ,
( 1 R * ) ( 1 + R ) β 00 0 β ν | R ν | 2 d v ,
1 1 H * · B ext H = π 1 1 0 β ext ( α ) | H ˜ α | 2 d α .
R ν = 1 h ν H = π 0 H ˜ α h ˜ ν α d α
R ν = β ν 1 h ν 1 1 B ext H = π ( β ν 1 ) 1 0 β ext ( α ) H ˜ α h ˜ ν α d α .
B wg 2 = z 2 = k 2 ( x ) + 2 1 · .
B wg 2 = k 2 n 2 [ 1 + 2 / k 2 n 2 + ( n 2 ) / n 2 ] + 0 ( 2 ) .
B wg 1 = ( B wg 2 ) 1 / 2 = ( k n ) 1 [ 1 2 / 2 k 2 n 2 ( n 2 ) / 2 n 2 ] + 0 ( 2 ) .
( 1 + n 2 1 1 ( k 2 n 2 + 2 ) 1 / 2 B ext ) H ( x ) = 2 h 00 ( x ) ( n 2 ) · B ext H / 2 k n 1 .
[ 1 + n 2 β ext ( α ) / 1 β n ( α ) ] H ˜ α = 2 h ˜ 00 α ,
n 2 = h 00 2 / 1 h 00 2
n 2 = h 00 2 / h 00 2 .
[ 1 + β ext ( α ) / β n ( α ) ] E ˜ α = 2 e ˜ 00 α ,
n 2 = e 00 2 / e 00 2 .
Z = Z η p / η p + 1 ( continuity across interface p / p + 1 ) , β p = ( k 2 p α 2 ) 1 / 2 ( any determination can be used ) , Z = [ Z + β p tan ( β p l p ) ] / [ 1 Z tan ( β p l p ) / β p ] ( l p is the width of layer p ) , p = p 1.
e 00 ( x ) = ( υ a ( 1 + υ ) ) 1 / 2 × { cos ( u x / a ) ( core ) cos u exp [ υ ( x / a 1 ) ] ( cladding , x > a ) ,
u 2 + υ 2 = V 2 ( V is the normalized frequency ) and υ = u tan u .
e ˜ 00 α = ( a υ 1 + υ ) 1 / 2 2 V u π α a sin ( α a ) υ cos ( α a ) ( α 2 a 2 u 2 ) ( α 2 a 2 + υ 2 )
e ν ( x ) = π 1 / 2 { cos [ ν ( x a ) + θ ν ] ( cladding , x > a ) B ν cos ( ν 1 x ) ( core ) ,
( ν 1 a ) 2 = ( ν a ) 2 + V 2 , B ν = cos θ ν cos ( ν 1 a ) = ν sin θ ν ν 1 sin ( ν 1 a ) , ν a tan θ ν = ν 1 a tan ( ν 1 a ) .
e ˜ ν α = 1 π [ cos ( θ α α a ) δ ( ν α ) + sin ( α a θ α ) π ( ν α ) + φ ν α ] ,
φ ν α = 2 V 2 π a 2 α sin ( α a ) cos θ ν ν sin θ ν cos ( α a ) ( ν 2 α 2 ) ( ν 1 2 α 2 ) sin ( α a θ α ) π ( ν α ) .
d ν [ S λ β ν ( ν λ ) S α ν α f ( α ) d α ] = π 2 S λ 2 β λ f ( λ ) + d α [ S α S λ f ( α ) d ν β ν ( ν λ ) ( ν α ) ]
K ( λ , α ) = β α 1 δ ( α λ ) + ρ ( α , λ ) ,
ρ ( λ , α ) = π β 00 1 e ˜ 00 α e ˜ 00 λ + β α 1 C α × ( S λ α λ + φ α λ ) + β λ 1 C λ ( S α α λ + φ λ α ) + ( S α φ α λ + S α S λ α λ ) 0 β ν 1 d ν ν α + ( S λ φ λ α S α S λ α λ ) 0 β ν 1 d ν ν λ + 0 β ν 1 d ν ( φ ν α φ ν λ + S α φ ν λ φ α λ ν λ + S λ φ ν α φ λ α ν λ ) .
β α 1 β λ 1 ( β α + β λ ) 1 = β λ 2 [ β α 1 ( β α + β λ ) 1 ]
h 00 ( x ) = A 0 a { cos ( u x / a ) ( core ) cos ( u ) exp [ υ ( x / a 1 ) ] ( cladding , x > a ) ,
A 0 2 = clad / [ 1 + clad core V 2 / υ ( core 2 υ 2 + clad 2 u 2 ) ] .
h ˜ 00 α = 2 π A 0 a cos ( u ) [ V 2 α a sin α a υ ( core / clad ) cos α a ( α 2 a 2 u 2 ) ( α 2 a 2 + υ 2 ) ( core clad 1 ) υ cos α a α 2 a 2 + υ 2 ]
h ˜ 00 α = 2 π A 0 a cos ( u ) core [ V 2 α a sin α a υ ( core / clad ) cos α a ( α 2 a 2 u 2 ) ( α 2 a 2 + υ 2 ) ( core clad 1 ) α a sin α a α 2 a 2 + υ 2 ] .
h ν ( x ) = ( clad / π ) 1 / 2 × { B ν cos ( ν 1 x ) ( core ) cos [ ν ( x a ) + θ ν ] ( cladding , x > a ) , ν 1 2 = ν 2 + ( V / a ) 2 , tan θ ν = ( clad / core ) ν 1 tan ( ν 1 a ) / ν , B ν = cos θ ν / cos ν 1 a .
h ˜ ν α = ( clad π ) 1 / 2 [ cos ( θ α α a ) δ ( ν α ) + sin ( α a θ α ) π ( ν α ) + φ ν α ] .
φ ν α = 2 π ( ν 2 α 2 ) [ α sin ( α a ) cos ( θ ν ) ν sin ( θ ν ) cos ( α a ) ] sin ( α a θ α ) π ( ν α ) + 2 π ( ν 1 2 α 2 ) [ core clad ν sin ( θ ν ) cos ( α a ) α sin ( α a ) cos ( θ ν ) ] .
1 0 α max e ˜ 00 α 2 d α .
0 α max f ( α ) d α = i = 1 N W i f ( α i ) .
FT [ B wg 1 e 00 ] λ = β 00 1 e ˜ 00 λ = K ( λ , α ) e ˜ 00 α d α = β λ 1 e ˜ 00 + ρ ( λ , α ) e ˜ 00 α d α .

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