Abstract

Conventional couplers including the 2×2 directional coupler and the self-imaging multimode interference coupler always produce a 90° coupling phase to yield complementary power transfer functions when the relative phase of the two input fields changes. For applications in coupled-cavity lasers, a 2×2 half-wave optical coupler with a 180° coupling phase is required in order to produce synchronous output power for achieving high single-mode selectivity. We present the designs and optimization of 2×2 half-wave couplers, including a three-waveguide directional coupler and a nonimaging multimode interference coupler.

© 2009 Optical Society of America

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References

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  1. V. Jayaraman, A. Mathur, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824-1834 (1993).
    [CrossRef]
  2. Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817-1823 (1993).
    [CrossRef]
  3. R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57 nm tuning range,” Appl. Phys. Lett. 60, 3209-3211 (1992).
    [CrossRef]
  4. J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE2.
  5. D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.
  6. L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
    [CrossRef]
  7. W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” Semicond. Semimet. 22, 257 (1985).
    [CrossRef]
  8. O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
    [CrossRef]
  9. M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-branch tunable semiconductor laser with 1.0 THz tuning range,” IEEE Photonics Technol. Lett. 4, 1093-1095(1992).
    [CrossRef]
  10. L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659-682 (1984).
    [CrossRef]
  11. T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740 (1985).
    [CrossRef]
  12. D. Marcuse, “Coupling coefficients of coupled laser cavities,” IEEE J. Quantum Electron. 22, 223-226 (1986).
    [CrossRef]
  13. J.-J. He and D. Liu, “Wavelength switchable semiconductor laser using half-wave V-coupled cavities,” Opt. Express 16, 3896-3911(2008).
    [CrossRef] [PubMed]
  14. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963-983 (1994).
    [CrossRef]
  15. M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N×N multimode interference couplers including phase relations,” Appl. Opt. 33, 3905-3911(1994).
    [CrossRef] [PubMed]

2008 (1)

1994 (2)

1993 (3)

V. Jayaraman, A. Mathur, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824-1834 (1993).
[CrossRef]

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817-1823 (1993).
[CrossRef]

O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
[CrossRef]

1992 (2)

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-branch tunable semiconductor laser with 1.0 THz tuning range,” IEEE Photonics Technol. Lett. 4, 1093-1095(1992).
[CrossRef]

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57 nm tuning range,” Appl. Phys. Lett. 60, 3209-3211 (1992).
[CrossRef]

1986 (1)

D. Marcuse, “Coupling coefficients of coupled laser cavities,” IEEE J. Quantum Electron. 22, 223-226 (1986).
[CrossRef]

1985 (2)

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” Semicond. Semimet. 22, 257 (1985).
[CrossRef]

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740 (1985).
[CrossRef]

1984 (1)

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659-682 (1984).
[CrossRef]

1981 (1)

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Alferness, R. C.

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57 nm tuning range,” Appl. Phys. Lett. 60, 3209-3211 (1992).
[CrossRef]

Bachmann, M.

Barton, E.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Baums, D.

O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
[CrossRef]

Besse, P. A.

Buhl, L. L.

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57 nm tuning range,” Appl. Phys. Lett. 60, 3209-3211 (1992).
[CrossRef]

Busico, G.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Coldren, L. A.

V. Jayaraman, A. Mathur, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824-1834 (1993).
[CrossRef]

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740 (1985).
[CrossRef]

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659-682 (1984).
[CrossRef]

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Dentai, A. G.

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-branch tunable semiconductor laser with 1.0 THz tuning range,” IEEE Photonics Technol. Lett. 4, 1093-1095(1992).
[CrossRef]

Duck, J. P.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Dutting, K.

O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
[CrossRef]

Griffin, R. A.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Hammerfeldt, S.

J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE2.

He, J.-J.

Hildebrand, O.

O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
[CrossRef]

Huang, W.-P.

Idler, W.

O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
[CrossRef]

Iga, K.

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Ishii, H.

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817-1823 (1993).
[CrossRef]

Jayaraman, V.

V. Jayaraman, A. Mathur, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824-1834 (1993).
[CrossRef]

Koch, T. L.

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740 (1985).
[CrossRef]

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659-682 (1984).
[CrossRef]

Koren, U.

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57 nm tuning range,” Appl. Phys. Lett. 60, 3209-3211 (1992).
[CrossRef]

Kuznetsov, M.

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-branch tunable semiconductor laser with 1.0 THz tuning range,” IEEE Photonics Technol. Lett. 4, 1093-1095(1992).
[CrossRef]

Laube, G.

O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
[CrossRef]

Liu, D.

Marcuse, D.

D. Marcuse, “Coupling coefficients of coupled laser cavities,” IEEE J. Quantum Electron. 22, 223-226 (1986).
[CrossRef]

Mathur, A.

V. Jayaraman, A. Mathur, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824-1834 (1993).
[CrossRef]

Melchior, H.

Miller, B. I.

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Ponnampalam, L.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Reid, D. C. J.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Rentschler, J. A.

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Robbins, D. J.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Sarlet, G.

J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE2.

Schilling, M.

O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
[CrossRef]

Tohmori, Y.

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817-1823 (1993).
[CrossRef]

Tsang, W. T.

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” Semicond. Semimet. 22, 257 (1985).
[CrossRef]

Verlangieri, P.

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-branch tunable semiconductor laser with 1.0 THz tuning range,” IEEE Photonics Technol. Lett. 4, 1093-1095(1992).
[CrossRef]

Ward, A. J.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Wesström, J.-O.

J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE2.

Whitbread, N. D.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Williams, P. J.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

Wunstel, K.

O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
[CrossRef]

Yoshikuni, Y.

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817-1823 (1993).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57 nm tuning range,” Appl. Phys. Lett. 60, 3209-3211 (1992).
[CrossRef]

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

IEEE J. Quantum Electron. (4)

V. Jayaraman, A. Mathur, and L. A. Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824-1834 (1993).
[CrossRef]

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817-1823 (1993).
[CrossRef]

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659-682 (1984).
[CrossRef]

D. Marcuse, “Coupling coefficients of coupled laser cavities,” IEEE J. Quantum Electron. 22, 223-226 (1986).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-branch tunable semiconductor laser with 1.0 THz tuning range,” IEEE Photonics Technol. Lett. 4, 1093-1095(1992).
[CrossRef]

J. Appl. Phys. (1)

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740 (1985).
[CrossRef]

J. Lightwave Technol. (1)

O. Hildebrand, M. Schilling, D. Baums, W. Idler, K. Dutting, G. Laube, and K. Wunstel, “The Y-laser: a multifunctional device for optical communication systems and switching networks,” J. Lightwave Technol. 11, 2066-2074 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Semicond. Semimet. (1)

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” Semicond. Semimet. 22, 257 (1985).
[CrossRef]

Other (2)

J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE2.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tunable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference (Optical Society of America, 2004), paper TuE3.

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of (a) a 2 × 2 directional coupler and (b) a self-imaging 2 × 2 MMI coupler .

Fig. 2
Fig. 2

Power transfer functions of a conventional quarter-wave coupler.

Fig. 3
Fig. 3

Schematic diagram of a three-waveguide half-wave coupler.

Fig. 4
Fig. 4

Power transfer functions of the three-waveguide half-wave coupler.

Fig. 5
Fig. 5

Coupling phase and coupling ratio as a function of the width of the middle waveguide.

Fig. 6
Fig. 6

Coupling phase and coupling ratio as a function of the gap width between the middle waveguide and the adjacent waveguides.

Fig. 7
Fig. 7

Schematic drawing of a nonimaging multimode waveguide coupler.

Fig. 8
Fig. 8

Relation between the coupler length and the gap width for 180 ° coupling phase (solid curve) and the corresponding coupling ratio (dashed curve).

Fig. 9
Fig. 9

Power transfer function of a multimode interference half-wave coupler with a 10% coupling ratio.

Fig. 10
Fig. 10

Relationship between the coupling ratio and the coupling loss with (dashed curve) and without (solid curve) the waveguide taper.

Fig. 11
Fig. 11

Cross-coupling ratio and phase as a function of (a) the coupling length and (b) the gap width.

Equations (8)

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

d A 1 ( z ) / d z = i K A 1 ( z ) , d A 2 ( z ) / d z = i K A 2 ( z ) ,
A 1 ( z ) = C 11 A 1 ( 0 ) + C 21 A 2 ( 0 ) = A 1 ( 0 ) cos K z i A 2 ( 0 ) sin K z , A 2 ( z ) = C 12 A 1 ( 0 ) + C 22 A 2 ( 0 ) = i A 1 ( 0 ) sin K z + A 2 ( 0 ) cos K z .
i + j even : φ i j = φ 0 + π + π 4 N × ( j i ) ( 2 N j + i ) , i + j odd : φ i j = φ 0 + π 4 N × ( j + i 1 ) ( 2 N j i + 1 ) ,
P 1 = 1 2 [ | C 11 | 2 + | C 21 | 2 + 2 | C 11 C 21 | cos ( φ + ϕ ) ] ,
P 2 = 1 2 [ | C 12 | 2 + | C 22 | 2 + 2 | C 12 C 22 | cos ( φ ϕ ) ] ,
d A 1 ( z ) / d z = i K A 3 ( z ) , d A 2 ( z ) / d z = i K A 3 ( z ) , d A 3 ( z ) / d z = i K A 1 ( z ) i K A 2 ( z ) .
A 1 ( z ) = 1 2 A 1 ( 0 ) ( 1 + cos 2 K z ) + 1 2 A 2 ( 0 ) ( 1 + cos 2 K z ) , A 2 ( z ) = 1 2 A 1 ( 0 ) ( 1 + cos 2 K z ) + 1 2 A 2 ( 0 ) ( 1 + cos 2 K z ) , A 3 ( z ) = i 2 2 ( A 1 ( 0 ) + A 2 ( 0 ) ) sin 2 K z .
χ = | C 21 | 2 | C 11 | 2 + | C 21 | 2 = | C 12 | 2 | C 12 | 2 + | C 22 | 2 .

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