Abstract

A new and efficient method for analyzing a chirped-grating distributed-feedback (DFB) laser is presented. We show that coupled-wave equations can be solved by use of the power series method. The single-mode gain margin of a linearly chirped-grating DFB laser is calculated for different chirping factors and coupling constants. We found that clearly optimum chirping exists for which the single-mode gain margin is maximum. The gain margins were also calculated for different positions of the cavity center. The effect of facet reflectivities and their phases on the gain margin was investigated. We found that the gain margin is maximum and the spatial hole burning is minimum for the cavity center at the middle of the laser cavity.

© 1999 Optical Society of America

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References

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  1. H. Kogelnik, C. V. Shank, “Coupled wave theory of distributed-feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  2. S. R. Chinn, “Effect of mirror reflectivity in a distributed-feedback laser,” IEEE J. Quantum Electron. QE-9, 570–574 (1973).
  3. W. Streifer, R. D. Burnham, D. R. Scifres, “Effect of external reflectors on longitudinal modes of distributed feedback lasers,” IEEE J. Quantum Electron. QE-11, 154–161 (1975).
    [CrossRef]
  4. H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase adjusted lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
    [CrossRef]
  5. J. E. A. Whiteway, B. Garrett, G. H. B. Thompson, A. J. Collar, C. J. Armistead, M. J. Fice, “The static and dynamic characteristics of single and multiple phase-shifted DFB laser structures,” IEEE J. Quantum Electron. 28, 1277–1293 (1992).
    [CrossRef]
  6. K. O. Hill, “Aperiodic distributed-parameter waveguides for integrated optics,” Appl. Opt. 13, 1753–1756 (1974).
    [CrossRef]
  7. A. Suzuki, K. Tada, “Theory and experiment on distributed-feedback lasers with chirped grating,” in Guided Wave Optical and Surface Acoustic Wave Devices: Systems and Applications, C. S. Tsai, ed. Proc. SPIE239, 10–18 (1980).
    [CrossRef]
  8. P. Zhou, G. S. Lee, “Phase-shifted distributed-feedback laser with linearly chirped grating for narrow linewidth and high-power operation,” Appl. Phys. Lett. 58, 331–333 (1991).
    [CrossRef]
  9. P. Vankwikelberge, G. Morthier, R. Baets, “CLADISS-A longitudinal model for the analysis of the static, dynamic, and stochastic behavior of diode lasers with distributed feedback,” IEEE J. Quantum Electron. 26, 1728–1741 (1990).
    [CrossRef]
  10. K. B. Kahen, “Analysis of distributed-feedback lasers using a recursive Green’s function approach,” IEEE J. Quantum Electron. 29, 368–373 (1993).
    [CrossRef]
  11. K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “Analysis of quarter-wave-shifted DFB laser,” Electron. Lett. 20, 326–327 (1984).
    [CrossRef]
  12. N. Chen, Y. Nakano, K. Okamoto, K. Tada, G. I. Morthier, R. G. Baets, “Analysis, Fabrication, and Characterization of Tunable DFB Lasers with Chirped Gratings,” IEEE J. Sel. Top. Quantum Electron. 3, 541–546 (1997).
    [CrossRef]
  13. P. Zhou, G. S. Lee, “Mode selection and spatial hole burning suppression of a chirped grating distributed feedback laser,” Appl. Phys. Lett. 56, 1400–1402 (1990).
    [CrossRef]
  14. H. Hillmer, A. Grabmaier, H. Burkhard, “Continuously distributed phase shifts by chirped distributed-feedback gratings for 1.55 µm distributed-feedback lasers,” IEE Proc. Optoelectron. 144, 256–260 (1997).
    [CrossRef]
  15. J. D. Freeze, M. A. Jensen, R. H. Selfridge, “A unified Green’s function analysis of complicated DFB lasers,” IEEE J. Quantum Electron. 33, 1253–1259 (1997).
    [CrossRef]

1997

N. Chen, Y. Nakano, K. Okamoto, K. Tada, G. I. Morthier, R. G. Baets, “Analysis, Fabrication, and Characterization of Tunable DFB Lasers with Chirped Gratings,” IEEE J. Sel. Top. Quantum Electron. 3, 541–546 (1997).
[CrossRef]

H. Hillmer, A. Grabmaier, H. Burkhard, “Continuously distributed phase shifts by chirped distributed-feedback gratings for 1.55 µm distributed-feedback lasers,” IEE Proc. Optoelectron. 144, 256–260 (1997).
[CrossRef]

J. D. Freeze, M. A. Jensen, R. H. Selfridge, “A unified Green’s function analysis of complicated DFB lasers,” IEEE J. Quantum Electron. 33, 1253–1259 (1997).
[CrossRef]

1993

K. B. Kahen, “Analysis of distributed-feedback lasers using a recursive Green’s function approach,” IEEE J. Quantum Electron. 29, 368–373 (1993).
[CrossRef]

1992

J. E. A. Whiteway, B. Garrett, G. H. B. Thompson, A. J. Collar, C. J. Armistead, M. J. Fice, “The static and dynamic characteristics of single and multiple phase-shifted DFB laser structures,” IEEE J. Quantum Electron. 28, 1277–1293 (1992).
[CrossRef]

1991

P. Zhou, G. S. Lee, “Phase-shifted distributed-feedback laser with linearly chirped grating for narrow linewidth and high-power operation,” Appl. Phys. Lett. 58, 331–333 (1991).
[CrossRef]

1990

P. Vankwikelberge, G. Morthier, R. Baets, “CLADISS-A longitudinal model for the analysis of the static, dynamic, and stochastic behavior of diode lasers with distributed feedback,” IEEE J. Quantum Electron. 26, 1728–1741 (1990).
[CrossRef]

P. Zhou, G. S. Lee, “Mode selection and spatial hole burning suppression of a chirped grating distributed feedback laser,” Appl. Phys. Lett. 56, 1400–1402 (1990).
[CrossRef]

1987

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase adjusted lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

1984

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “Analysis of quarter-wave-shifted DFB laser,” Electron. Lett. 20, 326–327 (1984).
[CrossRef]

1975

W. Streifer, R. D. Burnham, D. R. Scifres, “Effect of external reflectors on longitudinal modes of distributed feedback lasers,” IEEE J. Quantum Electron. QE-11, 154–161 (1975).
[CrossRef]

1974

1973

S. R. Chinn, “Effect of mirror reflectivity in a distributed-feedback laser,” IEEE J. Quantum Electron. QE-9, 570–574 (1973).

1972

H. Kogelnik, C. V. Shank, “Coupled wave theory of distributed-feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Akiba, S.

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “Analysis of quarter-wave-shifted DFB laser,” Electron. Lett. 20, 326–327 (1984).
[CrossRef]

Armistead, C. J.

J. E. A. Whiteway, B. Garrett, G. H. B. Thompson, A. J. Collar, C. J. Armistead, M. J. Fice, “The static and dynamic characteristics of single and multiple phase-shifted DFB laser structures,” IEEE J. Quantum Electron. 28, 1277–1293 (1992).
[CrossRef]

Baets, R.

P. Vankwikelberge, G. Morthier, R. Baets, “CLADISS-A longitudinal model for the analysis of the static, dynamic, and stochastic behavior of diode lasers with distributed feedback,” IEEE J. Quantum Electron. 26, 1728–1741 (1990).
[CrossRef]

Baets, R. G.

N. Chen, Y. Nakano, K. Okamoto, K. Tada, G. I. Morthier, R. G. Baets, “Analysis, Fabrication, and Characterization of Tunable DFB Lasers with Chirped Gratings,” IEEE J. Sel. Top. Quantum Electron. 3, 541–546 (1997).
[CrossRef]

Burkhard, H.

H. Hillmer, A. Grabmaier, H. Burkhard, “Continuously distributed phase shifts by chirped distributed-feedback gratings for 1.55 µm distributed-feedback lasers,” IEE Proc. Optoelectron. 144, 256–260 (1997).
[CrossRef]

Burnham, R. D.

W. Streifer, R. D. Burnham, D. R. Scifres, “Effect of external reflectors on longitudinal modes of distributed feedback lasers,” IEEE J. Quantum Electron. QE-11, 154–161 (1975).
[CrossRef]

Chen, N.

N. Chen, Y. Nakano, K. Okamoto, K. Tada, G. I. Morthier, R. G. Baets, “Analysis, Fabrication, and Characterization of Tunable DFB Lasers with Chirped Gratings,” IEEE J. Sel. Top. Quantum Electron. 3, 541–546 (1997).
[CrossRef]

Chinn, S. R.

S. R. Chinn, “Effect of mirror reflectivity in a distributed-feedback laser,” IEEE J. Quantum Electron. QE-9, 570–574 (1973).

Collar, A. J.

J. E. A. Whiteway, B. Garrett, G. H. B. Thompson, A. J. Collar, C. J. Armistead, M. J. Fice, “The static and dynamic characteristics of single and multiple phase-shifted DFB laser structures,” IEEE J. Quantum Electron. 28, 1277–1293 (1992).
[CrossRef]

Fice, M. J.

J. E. A. Whiteway, B. Garrett, G. H. B. Thompson, A. J. Collar, C. J. Armistead, M. J. Fice, “The static and dynamic characteristics of single and multiple phase-shifted DFB laser structures,” IEEE J. Quantum Electron. 28, 1277–1293 (1992).
[CrossRef]

Freeze, J. D.

J. D. Freeze, M. A. Jensen, R. H. Selfridge, “A unified Green’s function analysis of complicated DFB lasers,” IEEE J. Quantum Electron. 33, 1253–1259 (1997).
[CrossRef]

Garrett, B.

J. E. A. Whiteway, B. Garrett, G. H. B. Thompson, A. J. Collar, C. J. Armistead, M. J. Fice, “The static and dynamic characteristics of single and multiple phase-shifted DFB laser structures,” IEEE J. Quantum Electron. 28, 1277–1293 (1992).
[CrossRef]

Grabmaier, A.

H. Hillmer, A. Grabmaier, H. Burkhard, “Continuously distributed phase shifts by chirped distributed-feedback gratings for 1.55 µm distributed-feedback lasers,” IEE Proc. Optoelectron. 144, 256–260 (1997).
[CrossRef]

Hill, K. O.

Hillmer, H.

H. Hillmer, A. Grabmaier, H. Burkhard, “Continuously distributed phase shifts by chirped distributed-feedback gratings for 1.55 µm distributed-feedback lasers,” IEE Proc. Optoelectron. 144, 256–260 (1997).
[CrossRef]

Imai, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase adjusted lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

Ishikawa, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase adjusted lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

Jensen, M. A.

J. D. Freeze, M. A. Jensen, R. H. Selfridge, “A unified Green’s function analysis of complicated DFB lasers,” IEEE J. Quantum Electron. 33, 1253–1259 (1997).
[CrossRef]

Kahen, K. B.

K. B. Kahen, “Analysis of distributed-feedback lasers using a recursive Green’s function approach,” IEEE J. Quantum Electron. 29, 368–373 (1993).
[CrossRef]

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled wave theory of distributed-feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Kotaki, Y.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase adjusted lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

Lee, G. S.

P. Zhou, G. S. Lee, “Phase-shifted distributed-feedback laser with linearly chirped grating for narrow linewidth and high-power operation,” Appl. Phys. Lett. 58, 331–333 (1991).
[CrossRef]

P. Zhou, G. S. Lee, “Mode selection and spatial hole burning suppression of a chirped grating distributed feedback laser,” Appl. Phys. Lett. 56, 1400–1402 (1990).
[CrossRef]

Matsushima, Y.

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “Analysis of quarter-wave-shifted DFB laser,” Electron. Lett. 20, 326–327 (1984).
[CrossRef]

Morthier, G.

P. Vankwikelberge, G. Morthier, R. Baets, “CLADISS-A longitudinal model for the analysis of the static, dynamic, and stochastic behavior of diode lasers with distributed feedback,” IEEE J. Quantum Electron. 26, 1728–1741 (1990).
[CrossRef]

Morthier, G. I.

N. Chen, Y. Nakano, K. Okamoto, K. Tada, G. I. Morthier, R. G. Baets, “Analysis, Fabrication, and Characterization of Tunable DFB Lasers with Chirped Gratings,” IEEE J. Sel. Top. Quantum Electron. 3, 541–546 (1997).
[CrossRef]

Nakano, Y.

N. Chen, Y. Nakano, K. Okamoto, K. Tada, G. I. Morthier, R. G. Baets, “Analysis, Fabrication, and Characterization of Tunable DFB Lasers with Chirped Gratings,” IEEE J. Sel. Top. Quantum Electron. 3, 541–546 (1997).
[CrossRef]

Okamoto, K.

N. Chen, Y. Nakano, K. Okamoto, K. Tada, G. I. Morthier, R. G. Baets, “Analysis, Fabrication, and Characterization of Tunable DFB Lasers with Chirped Gratings,” IEEE J. Sel. Top. Quantum Electron. 3, 541–546 (1997).
[CrossRef]

Sakai, K.

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “Analysis of quarter-wave-shifted DFB laser,” Electron. Lett. 20, 326–327 (1984).
[CrossRef]

Scifres, D. R.

W. Streifer, R. D. Burnham, D. R. Scifres, “Effect of external reflectors on longitudinal modes of distributed feedback lasers,” IEEE J. Quantum Electron. QE-11, 154–161 (1975).
[CrossRef]

Selfridge, R. H.

J. D. Freeze, M. A. Jensen, R. H. Selfridge, “A unified Green’s function analysis of complicated DFB lasers,” IEEE J. Quantum Electron. 33, 1253–1259 (1997).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled wave theory of distributed-feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Soda, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase adjusted lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

Streifer, W.

W. Streifer, R. D. Burnham, D. R. Scifres, “Effect of external reflectors on longitudinal modes of distributed feedback lasers,” IEEE J. Quantum Electron. QE-11, 154–161 (1975).
[CrossRef]

Sudo, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase adjusted lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

Suzuki, A.

A. Suzuki, K. Tada, “Theory and experiment on distributed-feedback lasers with chirped grating,” in Guided Wave Optical and Surface Acoustic Wave Devices: Systems and Applications, C. S. Tsai, ed. Proc. SPIE239, 10–18 (1980).
[CrossRef]

Tada, K.

N. Chen, Y. Nakano, K. Okamoto, K. Tada, G. I. Morthier, R. G. Baets, “Analysis, Fabrication, and Characterization of Tunable DFB Lasers with Chirped Gratings,” IEEE J. Sel. Top. Quantum Electron. 3, 541–546 (1997).
[CrossRef]

A. Suzuki, K. Tada, “Theory and experiment on distributed-feedback lasers with chirped grating,” in Guided Wave Optical and Surface Acoustic Wave Devices: Systems and Applications, C. S. Tsai, ed. Proc. SPIE239, 10–18 (1980).
[CrossRef]

Thompson, G. H. B.

J. E. A. Whiteway, B. Garrett, G. H. B. Thompson, A. J. Collar, C. J. Armistead, M. J. Fice, “The static and dynamic characteristics of single and multiple phase-shifted DFB laser structures,” IEEE J. Quantum Electron. 28, 1277–1293 (1992).
[CrossRef]

Utaka, K.

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “Analysis of quarter-wave-shifted DFB laser,” Electron. Lett. 20, 326–327 (1984).
[CrossRef]

Vankwikelberge, P.

P. Vankwikelberge, G. Morthier, R. Baets, “CLADISS-A longitudinal model for the analysis of the static, dynamic, and stochastic behavior of diode lasers with distributed feedback,” IEEE J. Quantum Electron. 26, 1728–1741 (1990).
[CrossRef]

Whiteway, J. E. A.

J. E. A. Whiteway, B. Garrett, G. H. B. Thompson, A. J. Collar, C. J. Armistead, M. J. Fice, “The static and dynamic characteristics of single and multiple phase-shifted DFB laser structures,” IEEE J. Quantum Electron. 28, 1277–1293 (1992).
[CrossRef]

Yamakoshi, S.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase adjusted lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

Zhou, P.

P. Zhou, G. S. Lee, “Phase-shifted distributed-feedback laser with linearly chirped grating for narrow linewidth and high-power operation,” Appl. Phys. Lett. 58, 331–333 (1991).
[CrossRef]

P. Zhou, G. S. Lee, “Mode selection and spatial hole burning suppression of a chirped grating distributed feedback laser,” Appl. Phys. Lett. 56, 1400–1402 (1990).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

P. Zhou, G. S. Lee, “Phase-shifted distributed-feedback laser with linearly chirped grating for narrow linewidth and high-power operation,” Appl. Phys. Lett. 58, 331–333 (1991).
[CrossRef]

P. Zhou, G. S. Lee, “Mode selection and spatial hole burning suppression of a chirped grating distributed feedback laser,” Appl. Phys. Lett. 56, 1400–1402 (1990).
[CrossRef]

Electron. Lett.

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “Analysis of quarter-wave-shifted DFB laser,” Electron. Lett. 20, 326–327 (1984).
[CrossRef]

IEE Proc. Optoelectron.

H. Hillmer, A. Grabmaier, H. Burkhard, “Continuously distributed phase shifts by chirped distributed-feedback gratings for 1.55 µm distributed-feedback lasers,” IEE Proc. Optoelectron. 144, 256–260 (1997).
[CrossRef]

IEEE J. Quantum Electron.

J. D. Freeze, M. A. Jensen, R. H. Selfridge, “A unified Green’s function analysis of complicated DFB lasers,” IEEE J. Quantum Electron. 33, 1253–1259 (1997).
[CrossRef]

P. Vankwikelberge, G. Morthier, R. Baets, “CLADISS-A longitudinal model for the analysis of the static, dynamic, and stochastic behavior of diode lasers with distributed feedback,” IEEE J. Quantum Electron. 26, 1728–1741 (1990).
[CrossRef]

K. B. Kahen, “Analysis of distributed-feedback lasers using a recursive Green’s function approach,” IEEE J. Quantum Electron. 29, 368–373 (1993).
[CrossRef]

S. R. Chinn, “Effect of mirror reflectivity in a distributed-feedback laser,” IEEE J. Quantum Electron. QE-9, 570–574 (1973).

W. Streifer, R. D. Burnham, D. R. Scifres, “Effect of external reflectors on longitudinal modes of distributed feedback lasers,” IEEE J. Quantum Electron. QE-11, 154–161 (1975).
[CrossRef]

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase adjusted lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

J. E. A. Whiteway, B. Garrett, G. H. B. Thompson, A. J. Collar, C. J. Armistead, M. J. Fice, “The static and dynamic characteristics of single and multiple phase-shifted DFB laser structures,” IEEE J. Quantum Electron. 28, 1277–1293 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

N. Chen, Y. Nakano, K. Okamoto, K. Tada, G. I. Morthier, R. G. Baets, “Analysis, Fabrication, and Characterization of Tunable DFB Lasers with Chirped Gratings,” IEEE J. Sel. Top. Quantum Electron. 3, 541–546 (1997).
[CrossRef]

J. Appl. Phys.

H. Kogelnik, C. V. Shank, “Coupled wave theory of distributed-feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Other

A. Suzuki, K. Tada, “Theory and experiment on distributed-feedback lasers with chirped grating,” in Guided Wave Optical and Surface Acoustic Wave Devices: Systems and Applications, C. S. Tsai, ed. Proc. SPIE239, 10–18 (1980).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic of a linearly chirped-grating DFB laser.

Fig. 2
Fig. 2

Normalized threshold gain margin versus the chirping parameter (Λ10) for coupling constants |κ|l = 1.0, 2.0, and 3.0.

Fig. 3
Fig. 3

Normalized threshold gain margin as a function of the left facet reflectivity r 1 for Λ10 = 0, 3.33 × 10-5, and 1.67 × 10-4. Phase of r 1 = 0 and r 2 = 0.

Fig. 4
Fig. 4

Normalized threshold gain margin as a function of the phase of the left facet reflectivity r 1 for Λ10 = 0 and 1.67 × 10-4. |r 1| = 0.1 and r 2 = 0.

Fig. 5
Fig. 5

CF number (a measure of SHB) as a function of the chirping parameters Λ10 for |κ|l = 1.0, 2.0, and 3.0.

Fig. 6
Fig. 6

Normalized threshold gain margin as a function of the position of the cavity center l 1/l.

Fig. 7
Fig. 7

CF number (a measure of SHB) as a function of the position of the cavity center l 1/l.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

Λz=Λ0+Λ1fz,
βz=πΛzβ01-Λ1Λ0|z|,
dAzdz=iΔβ+2 Λ1Λ0 β0|z|Az+iκabBz,
dBzdz=-iΔβ+2 Λ1Λ0 β0|z|Bz+iκbaAz,
d2Alzdz2=-p1z2-p2z+p3-ip4Alz,
d2Blzdz2=-p1z2-p2z+p3+ip4Blz,
d2Arzdz2=-p1z2+p2z+p3-ip4Arz,
d2Brzdz2=-p1z2+p2z+p3+ip4Brz,
p1=4Λ1Λ0 β02, p2=4 Λ1Λ0 β0Δβ, p3=Δβ2+κabκba, p4=2 Λ1Λ0 β0.
k kk-1ak-zk-2=-p1k ak-zk+2+p2k ak-zk+1-p3-ip4k ak-zk.
ak=-p1ak-4-p2ak-3+p3-ip4ak-2kk-1k=2, 3,.
Alz=α1k ak-zk+α2k bk-zk,
ak, bk=-p1ak-4, bk-4-p2ak-3, bk-3+p3-ip4ak-2, bk-2kk-1 k=2, 3,. .
Blz=-1iκabα1k kak-zk-1+i Δβ+2 Λ1Λ0 β0zk ak-zk+α2k kbk-zk-1+iΔβ+2 Λ1Λ0 β0zk bk-zk.
Arz=γ1k ckzk+γ2k dkzk,
Brz=1iκabγ1k kckzk-1-iΔβ+2 Λ1Λ0 β0zk ckzk+γ2k kdkzk-1-iΔβ+2 Λ1Λ0 β0zk dkzk.
ck, dk=-p1ck-4, dk-4+p2ck-3, dk-3+p3-ip4ck-2, dk-2kk-1 k=2, 3,.
γ1=-Δβa0κabc0 α1-b1iκabc0 α2,
γ2=i κab2-Δβ2κaba0d1 α1-Δβκab α2.
M11M12M21M22 α1α2=0,
M11=iκabr1+Δβ+2 Λ1Λ0 β0l1k ak-l1k+k kak-l1k-1,
M12=iκabr1+Δβ+2 Λ1Λ0 β0l1k bk-l1k+k kbk-l1k-1,
M21=ia0κab2-Δβ2d1κabiκabr2+Δβ+2 Λ1Λ0 β0l2k dkl2k-k kdkl2k-1-Δβa0κabc0iκabr2+Δβ+2 Λ1Λ0 β0l2×k ckl2k-k kckl2k-1,
M22=-b1Δβd1κabiκabr2+Δβ+2 Λ1Λ0 β0l2×k dkl2k-k kdkl2k-1-b1iκabc0×iκabr2+Δβ+2 Λ1Λ0 β0l2×k ckl2k-k kckl2k-1.
iκabr1+Δβ+2 Λ1Λ0 β0l1k ak-l1k+k kak-l1k-1-b1Δβd1κabiκabr2+Δβ+2 Λ1Λ0 β0l2k dkl2k-k kdkl2k-1-b1iκabc0iκabr2+Δβ+2 Λ1Λ0 β0l2k ckl2k-k kckl2k-1-iκabr1+Δβ+2 Λ1Λ0 β0l1k bk-l1k+k kbk-l1k-1ia0κab2-Δβ2d1κab×iκabr2+Δβ+2 Λ1Λ0 β0l2k dkl2k-k kdkl2k-1-Δβa0κabc0iκabr2+Δβ+2 Λ1Λ0 β0l2k ckl2k-k kckl2k-1=0.

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