Abstract

An accurate analysis of the perturbation methods of propagation constants and attenuation coefficients of TE and TM modes in an asymmetric metal/dielectric/dielectric layer structure and four-layer metal-clad waveguides is presented. Simple formulas are deduced, and numerical results and illustrations are given for several guide configurations. The results are in good agreement with exact solutions in the method of solving the eigenvalue equation in the complex plane. Approximate methods and formulas for analyzing metal-clad waveguides with buffer layers are also given.

© 1990 Optical Society of America

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References

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  1. A. Reisinger, “Characteristics of optical guided modes in lossy waveguides,” Appl. Opt. 12, 1015–1023 (1973).
    [CrossRef] [PubMed]
  2. J. N. Polky, G. L. Mitchell, “Metal-clad planar dielectric waveguide for integrated optics,” J. Opt. Soc. Am. 64, 274–279 (1974).
    [CrossRef]
  3. I. P. Kaminow, W. L. Mammel, H. P. Weber, “Metal-clad optical waveguides: analytical and experimental study,” Appl. Opt. 13, 396–405 (1974).
    [CrossRef] [PubMed]
  4. Y. Yamamoto, T. Kamiya, H. Yamai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with the low index dielectric buffer,” IEEE J. Quantum Electron. QE-11, 729–737 (1975).
    [CrossRef]
  5. S. J. Al Bader, “Ohmic loss in metal-clad graded index optical waveguides,” IEEE J. Quantum Electron. QE-22, 8–11 (1986).
    [CrossRef]
  6. S. J. Al Bader, H. A. Jamid, “Comparison of absorption loss in metal-clad optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-34, 300–314 (1986).
  7. C. Ma, S. Liu, “Optical characteristics analysis and TE mode selection for asymmetric metal-clad waveguide,” Opt. Quantum Electron. 20, 323–328 (1988).
    [CrossRef]
  8. S. J. Al Bader, H. A. Jamid, “Guided wave characteristics of metal-clad graded index planar optical waveguides: analytical approach,” IEEE J. Quantum Electron. QE-23, 539–544 (1987).
    [CrossRef]
  9. M. Masuda, J. Koyama, “Effect of a buffer layer on TM modes in a metal-clad optical waveguide using Ti diffused LiNbNO3C plate,” Appl. Opt. 16, 2994–3000 (1977).
    [CrossRef] [PubMed]
  10. M. C. Amann, “Calculation of metal-clad ridge waveguide (MCRW) laser modes by mode coupling technique,” IEEE J. Lightwave Technol. LT-4, 689–693 (1986).
    [CrossRef]
  11. S. J. Al Bader, “Metal-clad ridge waveguide laser modes,” Proc. Inst. Electr. Eng. Part J 135, 79–84 (1988).
  12. A. Hosaka, K. Okamoto, J. Noda, “Single mode fiber type polarizer,” IEEE J. Quantum Electron. QE-18, 1569–1572 (1982).
    [CrossRef]
  13. T. Yu, Y. Wu, “Theoretical study of metal-clad optical waveguide polarizers,” IEEE J. Quantum Electron. 25, 1209–1213 (1989).
    [CrossRef]

1989 (1)

T. Yu, Y. Wu, “Theoretical study of metal-clad optical waveguide polarizers,” IEEE J. Quantum Electron. 25, 1209–1213 (1989).
[CrossRef]

1988 (2)

S. J. Al Bader, “Metal-clad ridge waveguide laser modes,” Proc. Inst. Electr. Eng. Part J 135, 79–84 (1988).

C. Ma, S. Liu, “Optical characteristics analysis and TE mode selection for asymmetric metal-clad waveguide,” Opt. Quantum Electron. 20, 323–328 (1988).
[CrossRef]

1987 (1)

S. J. Al Bader, H. A. Jamid, “Guided wave characteristics of metal-clad graded index planar optical waveguides: analytical approach,” IEEE J. Quantum Electron. QE-23, 539–544 (1987).
[CrossRef]

1986 (3)

S. J. Al Bader, “Ohmic loss in metal-clad graded index optical waveguides,” IEEE J. Quantum Electron. QE-22, 8–11 (1986).
[CrossRef]

S. J. Al Bader, H. A. Jamid, “Comparison of absorption loss in metal-clad optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-34, 300–314 (1986).

M. C. Amann, “Calculation of metal-clad ridge waveguide (MCRW) laser modes by mode coupling technique,” IEEE J. Lightwave Technol. LT-4, 689–693 (1986).
[CrossRef]

1982 (1)

A. Hosaka, K. Okamoto, J. Noda, “Single mode fiber type polarizer,” IEEE J. Quantum Electron. QE-18, 1569–1572 (1982).
[CrossRef]

1977 (1)

1975 (1)

Y. Yamamoto, T. Kamiya, H. Yamai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with the low index dielectric buffer,” IEEE J. Quantum Electron. QE-11, 729–737 (1975).
[CrossRef]

1974 (2)

1973 (1)

Al Bader, S. J.

S. J. Al Bader, “Metal-clad ridge waveguide laser modes,” Proc. Inst. Electr. Eng. Part J 135, 79–84 (1988).

S. J. Al Bader, H. A. Jamid, “Guided wave characteristics of metal-clad graded index planar optical waveguides: analytical approach,” IEEE J. Quantum Electron. QE-23, 539–544 (1987).
[CrossRef]

S. J. Al Bader, “Ohmic loss in metal-clad graded index optical waveguides,” IEEE J. Quantum Electron. QE-22, 8–11 (1986).
[CrossRef]

S. J. Al Bader, H. A. Jamid, “Comparison of absorption loss in metal-clad optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-34, 300–314 (1986).

Amann, M. C.

M. C. Amann, “Calculation of metal-clad ridge waveguide (MCRW) laser modes by mode coupling technique,” IEEE J. Lightwave Technol. LT-4, 689–693 (1986).
[CrossRef]

Hosaka, A.

A. Hosaka, K. Okamoto, J. Noda, “Single mode fiber type polarizer,” IEEE J. Quantum Electron. QE-18, 1569–1572 (1982).
[CrossRef]

Jamid, H. A.

S. J. Al Bader, H. A. Jamid, “Guided wave characteristics of metal-clad graded index planar optical waveguides: analytical approach,” IEEE J. Quantum Electron. QE-23, 539–544 (1987).
[CrossRef]

S. J. Al Bader, H. A. Jamid, “Comparison of absorption loss in metal-clad optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-34, 300–314 (1986).

Kaminow, I. P.

Kamiya, T.

Y. Yamamoto, T. Kamiya, H. Yamai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with the low index dielectric buffer,” IEEE J. Quantum Electron. QE-11, 729–737 (1975).
[CrossRef]

Koyama, J.

Liu, S.

C. Ma, S. Liu, “Optical characteristics analysis and TE mode selection for asymmetric metal-clad waveguide,” Opt. Quantum Electron. 20, 323–328 (1988).
[CrossRef]

Ma, C.

C. Ma, S. Liu, “Optical characteristics analysis and TE mode selection for asymmetric metal-clad waveguide,” Opt. Quantum Electron. 20, 323–328 (1988).
[CrossRef]

Mammel, W. L.

Masuda, M.

Mitchell, G. L.

Noda, J.

A. Hosaka, K. Okamoto, J. Noda, “Single mode fiber type polarizer,” IEEE J. Quantum Electron. QE-18, 1569–1572 (1982).
[CrossRef]

Okamoto, K.

A. Hosaka, K. Okamoto, J. Noda, “Single mode fiber type polarizer,” IEEE J. Quantum Electron. QE-18, 1569–1572 (1982).
[CrossRef]

Polky, J. N.

Reisinger, A.

Weber, H. P.

Wu, Y.

T. Yu, Y. Wu, “Theoretical study of metal-clad optical waveguide polarizers,” IEEE J. Quantum Electron. 25, 1209–1213 (1989).
[CrossRef]

Yamai, H.

Y. Yamamoto, T. Kamiya, H. Yamai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with the low index dielectric buffer,” IEEE J. Quantum Electron. QE-11, 729–737 (1975).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto, T. Kamiya, H. Yamai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with the low index dielectric buffer,” IEEE J. Quantum Electron. QE-11, 729–737 (1975).
[CrossRef]

Yu, T.

T. Yu, Y. Wu, “Theoretical study of metal-clad optical waveguide polarizers,” IEEE J. Quantum Electron. 25, 1209–1213 (1989).
[CrossRef]

Appl. Opt. (3)

IEEE J. Lightwave Technol. (1)

M. C. Amann, “Calculation of metal-clad ridge waveguide (MCRW) laser modes by mode coupling technique,” IEEE J. Lightwave Technol. LT-4, 689–693 (1986).
[CrossRef]

IEEE J. Quantum Electron. (5)

Y. Yamamoto, T. Kamiya, H. Yamai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with the low index dielectric buffer,” IEEE J. Quantum Electron. QE-11, 729–737 (1975).
[CrossRef]

S. J. Al Bader, “Ohmic loss in metal-clad graded index optical waveguides,” IEEE J. Quantum Electron. QE-22, 8–11 (1986).
[CrossRef]

A. Hosaka, K. Okamoto, J. Noda, “Single mode fiber type polarizer,” IEEE J. Quantum Electron. QE-18, 1569–1572 (1982).
[CrossRef]

T. Yu, Y. Wu, “Theoretical study of metal-clad optical waveguide polarizers,” IEEE J. Quantum Electron. 25, 1209–1213 (1989).
[CrossRef]

S. J. Al Bader, H. A. Jamid, “Guided wave characteristics of metal-clad graded index planar optical waveguides: analytical approach,” IEEE J. Quantum Electron. QE-23, 539–544 (1987).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. J. Al Bader, H. A. Jamid, “Comparison of absorption loss in metal-clad optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-34, 300–314 (1986).

J. Opt. Soc. Am. (1)

Opt. Quantum Electron. (1)

C. Ma, S. Liu, “Optical characteristics analysis and TE mode selection for asymmetric metal-clad waveguide,” Opt. Quantum Electron. 20, 323–328 (1988).
[CrossRef]

Proc. Inst. Electr. Eng. Part J (1)

S. J. Al Bader, “Metal-clad ridge waveguide laser modes,” Proc. Inst. Electr. Eng. Part J 135, 79–84 (1988).

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Figures (13)

Fig. 1
Fig. 1

Asymmetric metal-clad waveguide.

Fig. 2
Fig. 2

Propagation constant β/k0 versus k0w for APAg guide.

Fig. 3
Fig. 3

β/k0 versus k0w for an APAg guide.

Fig. 4
Fig. 4

Effective index N versus guide thickness w for an Al/GaAs/AlGaAs guide: Solid curves, the present method and the exact method; dashed curves from Ref. 7.

Fig. 5
Fig. 5

Power-attenuation coefficient α versus guide thickness w for an Al/GaAs/AlGaAs guide: Solid curves, the present method and exact method; dashed curves from Ref. 7.

Fig. 6
Fig. 6

Four-layer metal-clad dielectric waveguides.

Fig. 7
Fig. 7

Attenuation coefficient of several modes versus the core layer thickness. n3 = 1.2-i7.0 (for Al); n2 = 1.54 (for Corning 7059 glass); n1 = 1.758 (for Al2O3); n0 = 1.457 (for SiO2); λ = 0.6328 μm. Buffer thickness s = 0.1 μm.

Fig. 8
Fig. 8

The attenuation coefficients of several modes versus the buffer thickness. Core thickness w = 4λ.

Fig. 9
Fig. 9

Attenuation coefficients of the TE mode and the second low-order TM mode (TM0 → TM1) versus the buffer-layer thickness for Al cladding. n0 = 1.457 (for SiO2); n1 = 1.758 (for Al2O3); (n1n2)/n1 = 0.2, 0.1, 0.05, 0.25 (denoted A, B, C, D, respectively).

Fig. 10
Fig. 10

Variation of the real part of modal index N of the TM0 and TM1 modes with s. Au cladding is used, and w = 0.2 μm.

Fig. 11
Fig. 11

Variation of the imaginary part of the modal index N of the TM0 and TM−1 modes with s. Au cladding is used, and w = 0.2 μm.

Fig. 12
Fig. 12

Attenuation coefficients versus core thickness w for TE0 and TM0 modes with buffer thickness s = 0.1 μm. n0 = 1 (air); n1 = 1.95 (dielectric film); n2 = 1.45 (SiO2 buffer); n3 = 1.2 − 7.0 (Al2O3); λ = 0.6328 μm. Solid curves, exact method and Eqs. (29) and (33); dashed curves, Eq. (57).

Fig. 13
Fig. 13

Attenuation coefficients versus core thickness w for TE0 and TM0 modes with buffer thickness s = 0.4 μm. n0 = 1; n = 1.95; n2 = 1.45; n3 = 1.2 − 7.0; λ = 0.6328 μm. Solid curves, exact method and Eqs. (29) and (33); dashed curves, Eq. (57).

Equations (68)

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n 3 = n 3 i n 3 ,
1 = n 1 2 , 2 = n 2 2 ,
3 = 3 i 3 ,
3 = n 3 2 n 3 2 , 3 = 2 n 3 n 3 .
κ w = m π + ϕ 12 + ϕ 13 , m = 0,1,2 , ,
ϕ 12 = tan 1 ( C 1 p / κ ) , ϕ 13 = tan 1 ( C 2 q / κ ) ,
p = k 0 ( N 2 2 ) 1 / 2 , q = k 0 ( N 2 3 ) 1 / 2 , κ = k 0 ( 1 N 2 ) 1 / 2 ,
C 1 = C 2 = 1 for TE modes , C 1 = 1 / 2 , C 2 = 1 / 3 for TM modes .
tan 1 = 1 2 i ln 1 + i z 1 i z
ln ( a + b i ) = ln ρ + i θ ,
ρ = a 2 + b 2 , θ = ta n 1 b a + { 0 a > 0 π a < 0 b 0 π a < 0 b < 0 ,
q q 0 ( 1 + i δ ) ,
δ = 3 2 ( N 2 3 ) .
tan 1 q κ tan 1 q 0 κ + i q 0 κ k 0 2 ( 1 3 ) δ ,
i κ q 0 k 0 2 ( 1 3 ) δ = m π + tan 1 ( p / κ ) + tan 1 ( q 0 / κ ) κ w .
Δ β = i δ κ q 0 k 0 2 ( 1 3 ) ( d F / d β ) ,
d F / d β = ( β / κ ) w e .
w e = w + 1 p + 1 q 0 ;
Δ β = i 1 2 [ ( κ k 0 ) 2 1 N w e 1 ( N 2 3 ) 3 ( 1 3 ) ] .
α = 2 Im ( Δ β ) = 1 N w e ( 1 N 2 1 3 ) 3 ( N 2 3 ) 1 / 2 .
κ w = m π + tan 1 ( 1 2 p κ ) + tan 1 ( 1 3 q κ ) ,
Δ β = i q κ 2 β w e 1 3 κ 2 3 2 + q 2 1 2 [ 1 3 + 1 2 ( N 2 3 ) ] 3 ,
w e = w + D 1 p + D 2 q ,
D 1 = 1 2 1 2 2 2 ( 1 N 2 ) + 1 2 ( N 2 2 ) , D 2 = 1 3 1 3 3 2 ( 1 N 2 ) + 1 2 ( N 2 3 ) .
α = 2 Im ( Δ β ) = 1 N 2 N w e 1 3 ( N 2 3 ) 1 / 2 3 2 ( 1 N 2 ) + 1 2 ( N 2 3 ) × ( 2 3 + 1 N 2 3 ) 3 .
i k 0 ( N 2 1 ) 1 / 2 w = π tan 1 [ i 2 1 ( N 2 1 N 2 2 ) 1 / 2 ] tan 1 [ i 3 1 ( N 2 1 N 2 3 ) 1 / 2 ] .
tan 1 ( i z ) = 1 2 ln ( 1 + z 1 z ) ,
k 0 w ( N 2 1 ) 1 / 2 = 1 2 ln ( 1 + x 2 1 x 2 ) 1 2 ln ( 1 + x 1 1 x 1 ) i π ,
x 1 = 2 1 ( N 2 1 N 2 2 ) 1 / 2 , x 2 = 3 1 ( N 2 1 N 2 3 ) 1 / 2 .
Ψ = A 0 exp [ p 0 ( x + s + w ) ] ( , s w ) , Ψ = { A 1 cos ( κ 1 x θ 1 ) ( cases 1 and 2 ) ( s w , s ) , A 1 exp ( p 1 x ) B 1 exp ( p 1 x ) ( case 3 ) Ψ = { A 2 cos ( κ 2 x θ 2 ) ( cases 2 ) ( s , 0 ) , A 2 exp ( p 2 x ) B 2 exp ( p 2 x ) ( case 1 and 3 ) Ψ = A 3 exp ( p 3 x ) ( 0 , ) ,
κ i = k 0 ( n i 2 N 2 ) 1 / 2 , i = 1,2 , p i = k 0 ( N 2 n i 2 ) 1 / 2 , i = 0,1,2,3.
C 1 = C 2 = C 3 = 1 C 1 = ( n 1 n 0 ) 2 , for TE modes ; C 2 = ( n 1 n 2 ) 2 , C 3 = ( n 2 n 3 ) 2 for TM modes .
κ 1 w = m π + tan 1 ( C 1 p 0 κ 1 ) + tan 1 ( C 2 p 2 κ 1 Z ) ,
Z = C 3 p 3 / p 2 + tanh p 2 s 1 + C 3 p 3 / p 2 tanh p 2 s .
Z 0 = p 3 / p 2 + tanh p 2 s 1 + p 3 / p 2 tanh p 2 s .
i p 2 / κ 1 Z 0 1 + ( p 2 / κ 1 Z 0 ) 2 δ = κ 1 w m π tan 1 ( p 0 / κ 1 ) tan 1 ( p 2 / κ 1 Z 0 ) ,
δ = p 3 p 2 3 2 ( N 2 3 ) × ( 1 p 3 / p 2 + tanh p 2 s tanh p 2 s 1 + p 3 / p 2 tanh p 2 s ) .
Δ β = i p 2 / κ 1 Z 0 1 + ( p 2 / κ 1 Z 0 ) 2 δ d F / d β ,
Z 0 = C 3 p 3 / p 2 + tanh p 2 s 1 + C 3 p 3 / p 2 tanh p 2 s , C 3 = 2 3 ( 3 2 + 3 2 ) .
i C 2 p 2 / κ 1 Z 0 1 + ( C 2 p 2 / κ 1 Z 0 ) 2 δ = κ 1 w m π tan 1 ( C 1 p 0 / κ 1 ) tan 1 ( C 2 p 2 / κ 1 Z 0 ) ,
δ = C 3 p 3 p 2 [ 3 2 ( N 2 3 ) + 3 3 ] × ( 1 C 3 p 3 / p 2 + tanh p 2 s tanh p 2 s 1 + C 3 p 3 / p 2 tanh p 2 s ) .
Δ β = i C 2 p 2 / κ 1 Z 0 1 + ( C 2 p 2 / κ 1 Z 0 ) 2 δ d F / d β ,
κ 1 w = m π tan 1 ( C 1 p 0 / κ 1 ) tan 1 ( C 2 κ 2 / κ 1 Z 0 ) ,
Z = tan κ 2 s C 3 p 3 / κ 2 1 + C 3 p 3 / κ 2 tan κ 2 s .
Z 0 = tan κ 2 s p 3 / κ 2 1 + p 3 / κ 2 tan κ 2 s .
i κ 2 / κ 1 Z 0 1 + ( κ 2 / κ 1 Z 0 ) 2 δ = κ 1 w m π tan 1 ( p 0 / κ 1 ) + tan 1 ( κ 2 / κ 1 Z 0 ) ,
δ = p 3 κ 2 3 2 ( N 2 3 ) × ( 1 p 3 / κ 2 tan κ 2 s tan κ 2 s 1 + p 3 / κ 2 tan κ 2 s ) .
Δ β = i κ 2 / κ 1 Z 0 1 + ( κ 2 / κ 1 Z 0 ) 2 δ d F / d β ,
Z 0 = tan κ 2 s C 3 p 3 / κ 2 1 + C 3 p 3 / κ 2 tan κ 2 s .
i C 2 κ 2 / κ 1 Z 0 1 + ( C 2 κ 2 / κ 1 Z 0 ) 2 δ = κ 1 w m π tan 1 ( C 1 p 0 / κ 1 ) tan 1 ( C 2 κ 2 / κ 1 Z 0 ) ,
δ = C 3 p 3 p 2 [ 3 2 ( N 2 3 ) + 3 3 ] × ( 1 p 3 / κ 2 tan κ 2 s tan κ 2 s 1 + p 3 / κ 2 tan κ 2 s ) .
Δ β = i C 2 κ 2 / κ 1 Z 0 1 + ( C 2 κ 2 / κ 1 Z 0 ) 2 δ d F / d β ,
p 1 w = i π + 1 2 ln ( x 2 + 1 x 2 1 ) 1 2 ln ( x 1 + 1 x 1 1 ) ,
x 1 = C 1 p 0 p 1 Z , x 2 = C 2 p 2 p 1 Z , Z 0 = C 3 p 3 / p 2 + tanh p 2 s 1 + C 3 p 3 / p 2 tanh p 2 s .
Z 0 = C 3 p 3 / p 2 + tanh p 2 s 1 + C 3 p 3 / p 2 tanh p 2 s .
i C 2 p 2 / p 1 Z 0 1 + ( p 2 / p 1 Z 0 ) 2 δ = 1 2 ln ( x 2 + 1 x 2 1 ) 1 2 ln ( x 1 + 1 x 1 1 ) p 1 w i π ,
δ = C 3 p 3 p 2 [ 3 2 ( N 2 3 ) + 3 3 ] × ( 1 C 3 p 3 / p 2 + tanh p 2 s tanh p 2 s 1 + C 3 p 3 / p 2 tanh p 2 s ) .
Δ β = i C 2 p 2 / p 1 Z 0 1 ( C 2 p 2 / p 1 Z 0 ) 2 δ d F / d β ,
exp [ i 2 ( κ 1 w ϕ 10 ϕ 12 ) ] 1 = { exp [ i 2 ( κ 1 w ϕ 10 ) ] exp ( i 2 ϕ 12 ) } C 3 p 3 p 2 C 3 p 3 + p 2 ,
ϕ 10 = tan 1 ( C 1 p 0 / κ 1 ) , ϕ 12 = tan 1 ( C 2 p 2 / κ 1 ) .
exp [ i 2 ( κ 1 w ϕ 10 ϕ 12 ) ] = 1.
exp [ i 2 ( κ 1 w ϕ 10 ) ] exp ( i 2 ϕ 12 ) ,
exp ( i 2 ( κ 1 w ϕ 10 ϕ 12 ) ] 1 = i 2 sin 2 ϕ 12 × exp ( 2 p 2 s ) C 3 p 3 p 2 C 3 p 3 + p 2
Δ β = 1 d F / d β i 2 sin 2 ϕ 12 exp ( 2 p 2 s ) C 3 p 3 p 2 C 3 p 3 + p 2 .
d F / d β = i 2 w e β / κ 1 ,
w e = w + G 0 / p 0 + G 2 / p 2 , G 0 = C 1 1 + ( p 0 / κ 1 ) 2 1 + ( C 1 p 0 / κ 1 ) 2 , G 2 = C 2 1 + ( p 2 / κ 1 ) 2 1 + ( C 2 p 2 / κ 1 ) 2 .
Δ β = κ 1 β w e 2 C 2 p 2 / κ 1 1 + ( C 2 p 2 / κ 1 ) exp ( 2 p 2 s ) C 3 p 3 p 2 C 3 p 3 + p 2 ,
β = β 0 + Re ( Δ β ) , α = 2 Im ( Δ β ) ,

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