Abstract

We analyzed one-dimensional photonic lattices that incorporate mirror-modulated vertical cavity surface-emitting laser arrays utilizing the Bloch formalism. First, infinitely long arrays are considered. The in-phase mode (with a main central lobe at the far field) and antiphase mode (with two main symmetrically-located lobes at the far-field) are examined. A comparison of the modal losses of the in-phase and the antiphase modes, resulted in the discovery of regimes in which the in-phase mode is dominant. Considering lattices of finite length, we compared the results of the Bloch model to the exact solutions. It is shown that the boundary conditions in these lattices select a specific mode from the continuous spectrum in the infinite case. Consequently, the lattice’s length affects the eigenmodes and the corresponding eigenvalues in a periodic manner.

© 2001 Optical Society of America

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References

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  1. E. Yablonovitch, “Inhibiting spontaneous emission in solid-state and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  2. P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Localization of light in disordered and periodic dielectrics,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1993).
  3. E. Yablonovitch, T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
    [CrossRef] [PubMed]
  4. M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
    [CrossRef]
  5. H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546–548 (1997).
    [CrossRef] [PubMed]
  6. T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999).
    [CrossRef]
  7. T. Fishman, A. Hardy, E. Kapon, “Mode switching in shear-strained and modulated photonic lattices of VCSEL by means of injection locking,” Appl. Phys. Lett. 76, 816–818 (2000).
    [CrossRef]
  8. T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” presented at Advances in Semiconductor Lasers and Applications, Santa Barbara, Calif., 1999.
  9. J. P. McKelvey, Solid State and Semiconductor Physics (Harper & Row, New York, 1966), Chap. 8.
  10. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  11. D. Botez, T. Holcomb, “Bloch-function analysis of resonant arrays of antiguided diode lasers,” Appl. Phys. Lett. 60, 539–541 (1991).
    [CrossRef]
  12. R. F. Nabiev, A. I. Onishchenko, “Laterally coupled periodic semiconductor laser structures: Bloch function analysis,” IEEE J. Quantum Electron. 28, 2024–2032 (1992).
    [CrossRef]
  13. T. Fishman, M. Orenstein, “Coupling mechanism of two dimensional reflectivity modulated vertical cavity laser arrays,” in Proceedings of the Thirteenth IEEE International Semiconductor Laser Conference, Takamatsu, Japan, 1992.
  14. G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
    [CrossRef]
  15. R. J. Ram, D. I. Babic, R. A. York, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995).
    [CrossRef]
  16. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  18. T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
    [CrossRef]
  19. A. Papoulis, Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).
  20. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
    [CrossRef]

2000 (1)

T. Fishman, A. Hardy, E. Kapon, “Mode switching in shear-strained and modulated photonic lattices of VCSEL by means of injection locking,” Appl. Phys. Lett. 76, 816–818 (2000).
[CrossRef]

1999 (1)

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999).
[CrossRef]

1997 (2)

H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546–548 (1997).
[CrossRef] [PubMed]

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

1995 (1)

R. J. Ram, D. I. Babic, R. A. York, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995).
[CrossRef]

1992 (2)

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

R. F. Nabiev, A. I. Onishchenko, “Laterally coupled periodic semiconductor laser structures: Bloch function analysis,” IEEE J. Quantum Electron. 28, 2024–2032 (1992).
[CrossRef]

1991 (2)

D. Botez, T. Holcomb, “Bloch-function analysis of resonant arrays of antiguided diode lasers,” Appl. Phys. Lett. 60, 539–541 (1991).
[CrossRef]

E. Yablonovitch, T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

1987 (1)

E. Yablonovitch, “Inhibiting spontaneous emission in solid-state and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[CrossRef]

Babic, D. I.

R. J. Ram, D. I. Babic, R. A. York, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995).
[CrossRef]

Birks, T. A.

P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Localization of light in disordered and periodic dielectrics,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1993).

Botez, D.

D. Botez, T. Holcomb, “Bloch-function analysis of resonant arrays of antiguided diode lasers,” Appl. Phys. Lett. 60, 539–541 (1991).
[CrossRef]

Bowers, J. E.

R. J. Ram, D. I. Babic, R. A. York, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995).
[CrossRef]

Dutta, N. K.

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[CrossRef]

Fishman, T.

T. Fishman, A. Hardy, E. Kapon, “Mode switching in shear-strained and modulated photonic lattices of VCSEL by means of injection locking,” Appl. Phys. Lett. 76, 816–818 (2000).
[CrossRef]

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999).
[CrossRef]

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

T. Fishman, M. Orenstein, “Coupling mechanism of two dimensional reflectivity modulated vertical cavity laser arrays,” in Proceedings of the Thirteenth IEEE International Semiconductor Laser Conference, Takamatsu, Japan, 1992.

T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” presented at Advances in Semiconductor Lasers and Applications, Santa Barbara, Calif., 1999.

Florez, L. T.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Harbison, J. P.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

Hardy, A.

T. Fishman, A. Hardy, E. Kapon, “Mode switching in shear-strained and modulated photonic lattices of VCSEL by means of injection locking,” Appl. Phys. Lett. 76, 816–818 (2000).
[CrossRef]

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999).
[CrossRef]

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” presented at Advances in Semiconductor Lasers and Applications, Santa Barbara, Calif., 1999.

Holcomb, T.

D. Botez, T. Holcomb, “Bloch-function analysis of resonant arrays of antiguided diode lasers,” Appl. Phys. Lett. 60, 539–541 (1991).
[CrossRef]

Kapon, E.

T. Fishman, A. Hardy, E. Kapon, “Mode switching in shear-strained and modulated photonic lattices of VCSEL by means of injection locking,” Appl. Phys. Lett. 76, 816–818 (2000).
[CrossRef]

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999).
[CrossRef]

H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546–548 (1997).
[CrossRef] [PubMed]

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” presented at Advances in Semiconductor Lasers and Applications, Santa Barbara, Calif., 1999.

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Lloyd-Lucas, F. D.

P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Localization of light in disordered and periodic dielectrics,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1993).

McKelvey, J. P.

J. P. McKelvey, Solid State and Semiconductor Physics (Harper & Row, New York, 1966), Chap. 8.

Nabiev, R. F.

R. F. Nabiev, A. I. Onishchenko, “Laterally coupled periodic semiconductor laser structures: Bloch function analysis,” IEEE J. Quantum Electron. 28, 2024–2032 (1992).
[CrossRef]

Onishchenko, A. I.

R. F. Nabiev, A. I. Onishchenko, “Laterally coupled periodic semiconductor laser structures: Bloch function analysis,” IEEE J. Quantum Electron. 28, 2024–2032 (1992).
[CrossRef]

Orenstein, M.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

T. Fishman, M. Orenstein, “Coupling mechanism of two dimensional reflectivity modulated vertical cavity laser arrays,” in Proceedings of the Thirteenth IEEE International Semiconductor Laser Conference, Takamatsu, Japan, 1992.

Papoulis, A.

A. Papoulis, Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).

Pier, H.

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999).
[CrossRef]

H. Pier, E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546–548 (1997).
[CrossRef] [PubMed]

T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” presented at Advances in Semiconductor Lasers and Applications, Santa Barbara, Calif., 1999.

Ram, R. J.

R. J. Ram, D. I. Babic, R. A. York, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Localization of light in disordered and periodic dielectrics,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1993).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Stoffel, N. G.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibiting spontaneous emission in solid-state and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

York, R. A.

R. J. Ram, D. I. Babic, R. A. York, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995).
[CrossRef]

Appl. Phys. Lett. (4)

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, N. G. Stoffel, “Large two dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535–1537 (1992).
[CrossRef]

T. Fishman, E. Kapon, H. Pier, A. Hardy, “Modal expansion analysis of strained photonic lattices based on vertical cavity surface emitting laser arrays,” Appl. Phys. Lett. 74, 3595–3597 (1999).
[CrossRef]

T. Fishman, A. Hardy, E. Kapon, “Mode switching in shear-strained and modulated photonic lattices of VCSEL by means of injection locking,” Appl. Phys. Lett. 76, 816–818 (2000).
[CrossRef]

D. Botez, T. Holcomb, “Bloch-function analysis of resonant arrays of antiguided diode lasers,” Appl. Phys. Lett. 60, 539–541 (1991).
[CrossRef]

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

IEEE J. Quantum Electron. (3)

R. F. Nabiev, A. I. Onishchenko, “Laterally coupled periodic semiconductor laser structures: Bloch function analysis,” IEEE J. Quantum Electron. 28, 2024–2032 (1992).
[CrossRef]

R. J. Ram, D. I. Babic, R. A. York, J. E. Bowers, “Spontaneous emission in microcavities with distributed mirrors,” IEEE J. Quantum Electron. 31, 399–410 (1995).
[CrossRef]

T. Fishman, A. Hardy, E. Kapon, “Formulations for calculating the eigenmodes of vertical cavity laser arrays,” IEEE J. Quantum Electron. 33, 1756–1762 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (2)

E. Yablonovitch, “Inhibiting spontaneous emission in solid-state and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Other (9)

P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Localization of light in disordered and periodic dielectrics,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1993).

T. Fishman, A. Hardy, E. Kapon, H. Pier, “Injection locking of shear-strain photonic lattices based on VCSEL arrays,” presented at Advances in Semiconductor Lasers and Applications, Santa Barbara, Calif., 1999.

J. P. McKelvey, Solid State and Semiconductor Physics (Harper & Row, New York, 1966), Chap. 8.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

A. Papoulis, Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

T. Fishman, M. Orenstein, “Coupling mechanism of two dimensional reflectivity modulated vertical cavity laser arrays,” in Proceedings of the Thirteenth IEEE International Semiconductor Laser Conference, Takamatsu, Japan, 1992.

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic of a VCSEL array. We created the individual elements of the lattice by metal patterning the top mirror of a uniform broad area VCSEL. (b) The effective mirror ρ(x) and the relevant lattice parameters.

Fig. 2
Fig. 2

Lowest-order eigenvalue |μˆK1| of an infinite 1-D lattice as a function of K/ K BR as obtained from Eq. (8) with N = 5 for three element sizes d. The lattice parameters are Λ = 3.4 µm, ρ0 = 0.98, Δr = 0.04. The results are given for d = 2.55 µm (d/Λ = 0.75), d = 1.7 µm (d/Λ = 0.5), and d = 0.85 µm (d/Λ = 0.25).

Fig. 3
Fig. 3

Lowest-order eigenvalue |μˆK1| of a lattice with ρ0 = 0.95 and Δr = 0.1 for three values of Λ as a function of the transverse normalized wave vector K/ K BR. The aspect ratio is d/Λ = 0.5. The solid curves represent the numerical solution of Eq. (8) with N = 5. The approximate antiphase and in-phase solutions in approximations (9) and (13) are represented by dashed curves.

Fig. 4
Fig. 4

Exact lowest-order eigenvalue |μ̂1| of a lattice with finite size as a function of the normalized lattice size D/Λ. The top axis shows the corresponding values of K as taken from Fig. 2. The lattice parameters are Λ = 3.4 µm, d = 1.7 µm, ρ0 = 0.98, and Δr = 0.04.

Fig. 5
Fig. 5

Exact lowest-order eigenmode V (1)(y) of the finite size lattice of Fig. 4 for (a) D/Λ = 20 and (b) D/Λ = 21. The field envelopes of the exact (solid curves) and the corresponding Bloch analysis (dashed curves) with the appropriate value of K (as taken from Fig. 4) are given as a reference.

Equations (35)

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ρx=ρ0+Δρx,
Δρx=n=- bn expj2π/Λnxn=-NN bn expj2π/Λnx,
bn=Δrπnsinnπd/Λexp-jnπd/Λ,n0Δr2Λ2d-Λ,n=0,
μˆVy=- ρxKω0y, xVxdx=-ρ0+ΔρxKω0y, xVxdx,
Vy=- ckexpjkydk.
μkuky=ρ0-Kω0y, xukxdx.
μk=ρ0 expjk02L1-k2/k021/2,
-uk*yuqydy=δk-q,
μˆck=μkck+μkρ0|n|<N bnck-2nπ/Λ,
CKi=ciK-2Nπ/Λ,, ciK-2π/Λ, ciK, ciK+2π/Λ, ciK+2Nπ/Λ
μˆK1μBR1+b0+|b1|1-ΔK2Bw21/2,
ΔK=K-KBR μBRρ0 expjk02L1-KBR2/k021/2, Bw|b1|k02-KBR21/2/LKBR1+b0.
μˆK1max=μBR1+b0+|b1|.
αK1=1-|μˆK1|2
VK1y=E0 expiΔKyexpjKBRy+R1ΔKexp-jKBRy+c.c.,
R1ΔKexp-jϕb11-ΔK2Bw21/2+j ΔKBw,
μˆK1μ01+b0+2|b1|×1-1+b022|b1|2L2KBR4k021/2,
VK1y=E0μˆK1-μ01-b0×cosϕb1+2KBRy+|b1|μ0+c.c.
K-Q=2KBR,
μˆKi-μK1+b0μK ciK=b1ciQ,
μˆKi-μQ1+b0μQ ciQ=b-1ciK.
μˆK1,2=1+b0μK+μQ2±1+b02μK-μQ24+μKμQ|b1|21/2.
μˆK1,2=μBR1+b0±|b1|,
μBRρ0 expjk02L1-KBR2/k021/2.
μKμBR exp-jΔK/keffμBR1-jΔK/keff,
μQμBR expjΔK/keffμBR1+jΔK/keff,
μˆK1,2=μBR1+b0±|b1|1-ΔK2Bw21/2,
VK1,2y=E0 expiΔKyexpjKBRy+R1,2ΔKexp-jKBRy+c.c.,
R1,2ΔKexp-jϕb1±1-ΔK2Bw21/2+j ΔKBw=exp-jϕb1±ϕR,
μˆKiciQciKciJ=μQ1+b0μQb-10μKb1μK1+b0μKb-10μJb1μJ1+b0 ciQciKciJ,
μˆK1,3=1+b0μ0+μQ2±1+b02μ0-μQ24+2μ0μQ|b1|21/2.
μˆK1,3μ01+b0±2|b1|×1-1+b022|b1|2L2KBR4k021/2,
VK1,3y=E0μˆK1,3-μ01-b0×cosϕb1+2KBRy+|b1|μ0+c.c.
cK=μKμˆ-μK1+b0b1ciK-2π/Λ+b-1ciK+2π/Λ+,
cK-2π/Λ=μK-2π/Λμˆ-μK-2π/Λ1+b0×b1ciK-4π/Λ+b-1ciK+.

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