Abstract

Gain-guided eigenmodes in open vertical-cavity surface-emitting laser cavities are constructed by superposition of paraxial (i.e., Gauss–Laguerre) modes employing the unfolded-cavity hard-mirror equivalent to distributed Bragg reflectors. The round-trip matrix is obtained analytically for simple gain profiles, including finite-mirror-size losses, diffraction spreading, and gain-confinement effects. Diagonalization yields the full range of stable, unstable, and steady-state complex eigenmodes and gain eigenvalues, in terms of the cavity parameters. More importantly, it is demonstrated that in cases of interest the lower-order cavity eigenmodes can be approximated by pure Gauss–Laguerre modes with optimum waist size prescribed through a simple variational principle. The Gaussian nature of the cavity modes is confirmed by comparison with experiments. Finally, the new eigenmode properties self-consistently account for wavelength blueshifting and reduction in the mode waist with increasing bias current, without invoking index guiding.

© 2001 Optical Society of America

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References

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  1. R. Michalzik and K. J. Ebeling, “Generalized BV diagrams for higher order transverse modes in VCSEL diodes,” IEEE J. Quantum Electron. 31, 1371–1379 (1995).
    [CrossRef]
  2. D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
    [CrossRef]
  3. M. J. Noble, J. P. Loehr, and J. A. Lott, “Analysis of microcavity VCSEL lasing modes using a full-vector weighted index method,” IEEE J. Quantum Electron. 34, 1890–1903 (1998).
    [CrossRef]
  4. G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607–616 (1996).
    [CrossRef]
  5. J. D. Jackson, ed., Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 364–368.
  6. D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
    [CrossRef]
  7. D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in VCSELs,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
    [CrossRef]
  8. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 57–70.
  9. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 108–118.
  10. D. I. Babic, R. J. Ram, J. E. Bowers, M. Tan, and L. Yang, “Scaling laws for gain guided VCSELs with distributed Bragg reflectors,” Appl. Phys. Lett. 64, 1762–1765 (1994).
    [CrossRef]
  11. See, for example, J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, Mass., 1995), pp. 313–316.
  12. K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616–2618 (1994).
    [CrossRef]
  13. See, for example, C. C. Davis, Lasers and Electro-optics (Cambridge University, Cambridge, UK, 1996), pp. 416–423.
  14. Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950–1958 (1996).
    [CrossRef]
  15. G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
    [CrossRef]

1998 (2)

M. J. Noble, J. P. Loehr, and J. A. Lott, “Analysis of microcavity VCSEL lasing modes using a full-vector weighted index method,” IEEE J. Quantum Electron. 34, 1890–1903 (1998).
[CrossRef]

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

1997 (1)

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
[CrossRef]

1996 (2)

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950–1958 (1996).
[CrossRef]

1995 (1)

R. Michalzik and K. J. Ebeling, “Generalized BV diagrams for higher order transverse modes in VCSEL diodes,” IEEE J. Quantum Electron. 31, 1371–1379 (1995).
[CrossRef]

1994 (2)

D. I. Babic, R. J. Ram, J. E. Bowers, M. Tan, and L. Yang, “Scaling laws for gain guided VCSELs with distributed Bragg reflectors,” Appl. Phys. Lett. 64, 1762–1765 (1994).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616–2618 (1994).
[CrossRef]

1993 (1)

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in VCSELs,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

1992 (1)

D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[CrossRef]

Babic, D. I.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

D. I. Babic, R. J. Ram, J. E. Bowers, M. Tan, and L. Yang, “Scaling laws for gain guided VCSELs with distributed Bragg reflectors,” Appl. Phys. Lett. 64, 1762–1765 (1994).
[CrossRef]

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in VCSELs,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[CrossRef]

Barnes, D. C.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Binder, R.

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
[CrossRef]

Bowers, J. E.

D. I. Babic, R. J. Ram, J. E. Bowers, M. Tan, and L. Yang, “Scaling laws for gain guided VCSELs with distributed Bragg reflectors,” Appl. Phys. Lett. 64, 1762–1765 (1994).
[CrossRef]

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in VCSELs,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

Burak, D.

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
[CrossRef]

Choquette, K. D.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616–2618 (1994).
[CrossRef]

Chuang, S. L.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Chung, Y.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in VCSELs,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

Corzine, S.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

Corzine, S. W.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[CrossRef]

Dagli, N.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in VCSELs,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

Ebeling, K. J.

R. Michalzik and K. J. Ebeling, “Generalized BV diagrams for higher order transverse modes in VCSEL diodes,” IEEE J. Quantum Electron. 31, 1371–1379 (1995).
[CrossRef]

Hadley, G. R.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

Kilcoyne, S. P.

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616–2618 (1994).
[CrossRef]

Lear, K. L.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616–2618 (1994).
[CrossRef]

Liu, G.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Loehr, J. P.

M. J. Noble, J. P. Loehr, and J. A. Lott, “Analysis of microcavity VCSEL lasing modes using a full-vector weighted index method,” IEEE J. Quantum Electron. 34, 1890–1903 (1998).
[CrossRef]

Lott, J. A.

M. J. Noble, J. P. Loehr, and J. A. Lott, “Analysis of microcavity VCSEL lasing modes using a full-vector weighted index method,” IEEE J. Quantum Electron. 34, 1890–1903 (1998).
[CrossRef]

McInerney, J.

Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950–1958 (1996).
[CrossRef]

Michalzik, R.

R. Michalzik and K. J. Ebeling, “Generalized BV diagrams for higher order transverse modes in VCSEL diodes,” IEEE J. Quantum Electron. 31, 1371–1379 (1995).
[CrossRef]

Noble, M. J.

M. J. Noble, J. P. Loehr, and J. A. Lott, “Analysis of microcavity VCSEL lasing modes using a full-vector weighted index method,” IEEE J. Quantum Electron. 34, 1890–1903 (1998).
[CrossRef]

Ram, R. J.

D. I. Babic, R. J. Ram, J. E. Bowers, M. Tan, and L. Yang, “Scaling laws for gain guided VCSELs with distributed Bragg reflectors,” Appl. Phys. Lett. 64, 1762–1765 (1994).
[CrossRef]

Schneider, R. P.

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616–2618 (1994).
[CrossRef]

Scott, J. W.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

Seurin, J.-F.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Tan, M.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

D. I. Babic, R. J. Ram, J. E. Bowers, M. Tan, and L. Yang, “Scaling laws for gain guided VCSELs with distributed Bragg reflectors,” Appl. Phys. Lett. 64, 1762–1765 (1994).
[CrossRef]

Tiouririne, T. N.

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

Warren, M. E.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

Yang, L.

D. I. Babic, R. J. Ram, J. E. Bowers, M. Tan, and L. Yang, “Scaling laws for gain guided VCSELs with distributed Bragg reflectors,” Appl. Phys. Lett. 64, 1762–1765 (1994).
[CrossRef]

Zhao, Y. G.

Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950–1958 (1996).
[CrossRef]

Appl. Phys. Lett. (3)

D. I. Babic, R. J. Ram, J. E. Bowers, M. Tan, and L. Yang, “Scaling laws for gain guided VCSELs with distributed Bragg reflectors,” Appl. Phys. Lett. 64, 1762–1765 (1994).
[CrossRef]

K. L. Lear, K. D. Choquette, R. P. Schneider, and S. P. Kilcoyne, Appl. Phys. Lett. 66, 2616–2618 (1994).
[CrossRef]

G. Liu, J.-F. Seurin, S. L. Chuang, D. I. Babic, S. W. Corzine, M. Tan, D. C. Barnes, and T. N. Tiouririne, “Mode-selectivity study of VCSELs,” Appl. Phys. Lett. 73, 726–728 (1998).
[CrossRef]

IEEE J. Quantum Electron. (7)

Y. G. Zhao and J. McInerney, “Transverse-mode control of VCSELs,” IEEE J. Quantum Electron. 32, 1950–1958 (1996).
[CrossRef]

R. Michalzik and K. J. Ebeling, “Generalized BV diagrams for higher order transverse modes in VCSEL diodes,” IEEE J. Quantum Electron. 31, 1371–1379 (1995).
[CrossRef]

D. Burak and R. Binder, “Cold-cavity vectorial eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. 33, 1205–1215 (1997).
[CrossRef]

M. J. Noble, J. P. Loehr, and J. A. Lott, “Analysis of microcavity VCSEL lasing modes using a full-vector weighted index method,” IEEE J. Quantum Electron. 34, 1890–1903 (1998).
[CrossRef]

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. Corzine, “Comprehensive numerical modeling of VCSELs,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

D. I. Babic and S. W. Corzine, “Analytic expressions for reflection delay, penetration depth and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[CrossRef]

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, “Modal reflection of quarter-wave mirrors in VCSELs,” IEEE J. Quantum Electron. 29, 1950–1962 (1993).
[CrossRef]

Other (5)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 57–70.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 108–118.

J. D. Jackson, ed., Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 364–368.

See, for example, C. C. Davis, Lasers and Electro-optics (Cambridge University, Cambridge, UK, 1996), pp. 416–423.

See, for example, J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, Mass., 1995), pp. 313–316.

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Figures (13)

Fig. 1
Fig. 1

Illustration of a typical mesa-structure VCSEL cavity and its equivalent unfolded cavity.

Fig. 2
Fig. 2

Resonant cavity wavelength (free space) λr versus cavity length niLo. The DBR midband wavelength λo=780 nm.

Fig. 3
Fig. 3

Plots of (a) eigenvalue magnitude |ζ(0)| for the first cavity eigenmode and (b) expansion coefficients (magnitude) of that eigenmode into GL modes, versus the elected GL mode waist w. At w values far from optimum (here ∼1.2 µm) the number of GL modes necessary to cover the cavity eigenmodes exceeds the number employed (15), causing an artificial reduction in ζ.

Fig. 4
Fig. 4

Expansion coefficients (magnitude) of cavity eigenmodes into GL modes, given by the bar heights for an elected GL mode waist (a) about optimum and (b) twice optimum.

Fig. 5
Fig. 5

Effective cavity model for (i) proton-implant and (ii) etched-mesa VCSELs.

Fig. 6
Fig. 6

Diffraction, overlap (gain/mirror), and total cavity losses versus waist size.

Fig. 7
Fig. 7

Mode-waist to active-region-radius ratio plotted against active-region radius in wavelengths for (a) a proton-implant cavity, and (b) an etched-mesa cavity.

Fig. 8
Fig. 8

Round-trip losses 1-Rmp2 for the first 16 (m, p) modes with optimum waist wo for the cavity parameters of Section 10: (a) proton-implant and (b) etched-mesa VCSELs.

Fig. 9
Fig. 9

Ratio of waist to active-area radius versus normalized gain g/go for (a) proton-implant and (b) etched-mesa VCSELs.

Fig. 10
Fig. 10

Ratio of cavity-threshold gain to nominal (infinitely wide cavity) threshold g/go versus cavity Fresnel number N for (a) proton-implant and (b) etched-mesa VCSELs.

Fig. 11
Fig. 11

Frequency correction for the fundamental eigenmode versus Fresnel number.

Fig. 12
Fig. 12

Comparison of experimental fundamental intensity profile (points) versus best-fitting Gaussian at bias currents 2 mA and 4 mA. The best-fitting waist of 1.30 µm is in good agreement with the predicted wo=1.26 µm.

Fig. 13
Fig. 13

Comparison of the optimum-waist GL mode profile versus the fundamental waveguide LP mode profile corresponding to an index-step radius at 3 µm.

Equations (95)

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

Ψ(x, y, z=2Lc)=S Ψ(x, y, 0),
STclRlTlcGcTcrRrTrcGc,
S Ψ(x, y, 0)=ζ Ψ(x, y, 0).
Ψ(x, y, z)=T (x-xo, y-yo, z-zo)Ψ(xo, yo, zo)=-ik2π(z-zo) dxodyo×expik (x-xo)2+(y-yo)22(z-zo)×Ψ(xo, yo, zo).
r*=reiχ.
r=1-vqpp2M1+vqpp2M,
χτ=π2 ωωo-1q 1-p2M1-p v2p2M-1-11-v2q2p4M-2,
χD=π4 k2ko2 ωωoq 1-p2M1-p 1n12+1n22 v2p2M-1-11-v2q2p4M-2-21n12-1n22 v2p2M-1+11-v2q2p4M-2.
Lτ=ωωo-1h=ωωo-1×λo8niq 1-p2M1-p v2p2M-1-11-v2q2p4M-2,
LD=λo4ni ωωoq 1-p2M1-p 12 ni2n12+ni2n22 1-v2p2M-1 1-v2q2p4M-2+ni2n12-ni2n22 1+v2p2M-11-v2q2p4M-2.
Lˆ=Lo+LD1+LD2,
Lˆϕ=Lo+Lτ1+Lτ2,
Le=-λo2ni Mln(1-r),
LE=Lo+Le1+Le2
1λr1λo+1niLo-1λo1+h1+h2niLo,
Ψ=m,pCmpψmp(ρ, θ, z),
ψmp(ρ, θ, z)=Ump2ρ2W2exp(ipθ)expik ρ2R×exp[-i(2m+p+1)φ],
Ump(Υ)=2πW2 m!(m+p)! Υp/2Lmp(Υ)exp(-Υ/2).
W(z)=w1+z2/b2,
R(z)=(z2+b2)/z,
φ=tan-1(z/b);
b=12kw2.
T (z2)ψ[W(z1),R(z1), φ(z1)]
ψ[W(z1+z2),R(z1+z2),φ(z1+z2)].
T=τ100001000010,
C00C10C20:C01C11C21::out=R0000R1000R2000:R0010R1010R2010:OR0020R1020R2020:........R0101R1101R2101:OR0111R1111R2111:R0121R1121R2121:........
×C00C10C20:C01C11C21::in.
Qmpnq=02πdθ0adρρψmpψnq*.
Qmpnq=exp{-i[2(m-n)+(p-q)]φ}×02πdθ0adρρUmp2ρ2W2Ump2ρ2W2×exp[i(p-q)θ],
|Qmpnq|=δpq[m!n!(m+p)!(n+q)!]1/2×k=0ml=0n×(-1)k+l+p(k+l+p)!k!l!(m-k)!(n-l)!(p+k)!(p+l)!×1-exp(-2µ)i=0k+l+p (2µ)k+l+p-i(k+l+p-i)!,
R0000=r[1-exp(-2µ)],R1000=R0010=r2µ exp(-2µ).
Gmpnq=δmpnp+σΓ02πdθ0adρρψmp[N(ρ)-Ntr]ψnq*.
G=I+G0000G1000G2000G0010G1010G2010OG0020G1020G2020G0101G1101G2101OG0111G1111G2111G0121G1121G2121.
G=I+g(1+iα)×Q0000Q1000Q2000:Q0010Q1010Q2010:OQ0020Q1020Q2020:........Q0101Q1101Q2101:OQ0111Q1111Q2111:Q0121Q1121Q2121:........,
Pmpnq=02πdθ0dρρψpm(2ρ2/W2)ψnq(2ρ2/w2)*=exp[-i(2m+p+1)φ]×02πdθ0dX Unq*(ξX)Ump(X)×coskρ22R+i sinkρ22R,
Pmpnq=exp[-i(2m+p+1)ϕ](Pcmpnq+iPsmpnq)
Pcmpnq=j=0 (-1)2j(2j)! 12 b/2Lˆ1+(b/2Lˆ)22j[Qmpnq](2j)(ξ, ),
Psmpnq=j=0 (-1)2j+1(2j+1)! 12 b/2Lˆ1+(b/2Lˆ)22j+1×[Qmpnq](2j+1)(ξ, ).
Pmpnq=|Pmpnq|exp(iΘmpnq)exp[-i(2m+p+1)φ],
|Pmpnq|[(Pcmpnq)2+(Psmpnq)2]1/2,
Θmpnqtan-1PsmpnqPcmpnq.
|P0p0q|δpqQ0p0q(ξ, )1-(p+2)!p!-(p+1)!p!2×21+ξ2212 b/2Lˆ1+(b/2Lˆ)221/2,
Θ0p0q12 b/2Lˆ1+(b/2Lˆ)2 (p+1)!p! 21+ξ2×12 b/2Lˆ1+(b/2Lˆ)2Q0p0q(, ξ).
S=PTclR1TlcGcTcrRrTrcGc=rrrlPTclQlTlc(I+g*Qc)TcrQrTrc(I+g*Qc).
SC(k)=ζ(k)C(k)det|S-ζ(k)I|=0,
w|ψ00|S|ψ00|=0.
ψ00|S |ψ00=C*·S·C=S0000(w)=P0000 Tcl0000 Rl0000 Tlc0000 Gc0000 Tcr0000 Rr0000 Tcr0000 Gc0000+O(ε2).
|S(w)|=r1r2|PC|,To2Q2T2o(1+gQo)To1Q1T1o×(1+gQo),
Arg[S(w)]=(IS0000/RS0000)Θ0000+gα(Qo+Qo).
R1=r11+ΔR1,
R2=r21+ΔR2,
G1=1+g[1-exp(-2a2/w2)]1+ΔG1,
G2=1+g1-exp-2a2/w21+Lˆ2/b21+ΔG2,
|P|=2[1+(2Lˆ)2/b2]1/22+(2Lˆ)2/b2×1-2! 12 (2Lˆ)2/b2[1+(2Lˆ)2/b2]2×1{1+[1+(2Lˆ)2/b2]2}21+ΔP,
Nb(a)πLˆ=ka22πLˆ,
ΔR1=r1-1,
ΔR2=r2-1,
ΔG1=g[1-exp(-2µ)],
ΔG2=g1-exp-2µ1+(1/πN)2μ2,
Δ|P|-2πN2μ218-38-182πN2μ2.
|S(μ)|1+ΔG1+ΔR1+ΔG2+ΔR2+ΔP.
|S|μ2g exp(-2µ)+2g 1-(1/πN)2μ2[1+(1/πN)2μ2]2 exp-2µ1+(1/πN)2μ2-μ2πN214-32-122πN2μ2=0,
R1=1+r11-exp-2a2/w21+L2/4b21+ΔR1,
ΔR1r11-exp-2µ1+1/4(1/πN)2μ2-1,
2g exp(-2µ)
+2r1 1-(1/πN)2μ2/4[1+(1/πN)2μ2/4]2 exp-2µ1+(1/πN)2μ2/4
+2g 1-(1/πN)2μ2[1+(1/πN)2μ2]2 exp-2µ1+(1/πN)2μ2
-μ2πN214-32-122πN2μ2=0.
(1+go)2r1r2=1go(1-r1r2)/2.
2g0=1-RR
2g0=1R1R2P-1Q1+Q22.
R=1+1R1R2P-1Q1+Q22-11-1R1R2P-1Q1+Q22.
gˆogo=2+21-r1 2πN2μ218-28 2πN2μ22-exp(-2µ)-exp-2µ1+(1/πN)2μ2,
gˆogo=2+21-r1 2πN2μ218-28 2πN2μ2+r1 exp-2µ1+(1/πN)2μ4/42-exp(-2µ)-exp-2µ1+(1/πN)2μ2.
kiLϕ-(2m+p+1)φ(Lˆ)=lπ,
ki=ni 2πλr2πLo,
ki(p, m)=2πLo+(2m+p+1) 1b0,
Δω=cniΔk=cni 1bo=cni 2kiwo2.
Δω(0, p)c2niLˆ 12 2Lˆb0(2Lˆ)2+bo2 2ξ1/2ξ+1p+1 (p+1)!p!+gα(Q0p0p+Q0p0p).
Δω(0, p)ω=-Δλλ12 1kibo (p+1)1+(2/πN)2(a/wo)4.
ω(0, p)-ω(0, 0)ω(0, 0)=-λ(0, p)-λ(0, 0)λ(0, 0)=2ki2ωo2 p+12 (p+1)-11+(2/πN)2(a/wo)4.
1λr=1λo 1+-1+h1+h2niLo+1+h1+h2niLo2-4 h1+h2niLo 1-λoLo1/22 h1+h2niLo 1λ0+1niLo-1λo1+h1+h2niLo,
hj=λo8niq 1-p2M1-p v2p2M-1-11-v2q2p4M-2.
k2mp0d2kk2Umpˆ(k)=1+2m+pw2,
Pmpnq=2πw2αmp2δpq exp{-i[(m-n)+2(p-q)]φ}×14 0dXUpm(ξ2X)exp(ikρ2/2R)Uqn*(X).
14 0dX 1-12 b2R2X2+i b2R X
×Upm(ξ2X)Uqn*(X).
[Qmpnq(ξ, μ)](j)14 02µdXX1+jUpm(ξ2X)Uqn*(X),
[Qmpnq(ξ, μ)](j)
=δpq[m!n!(m+p)!(n+q)]1/2 (2ξ)p+1(1+ξ2)p+1×k=0ml=0nξ2k21+ξ2k+l+j×(-1)k+l+p+j(k+l+p)!k!l!(m-k)!(n-l)!(p+k)!(p+l)!×1-exp(-2µ)i=0k+l+p (2µ)k+l+p-i(k+l+p-i)!.
[Qmpnq(ξ, μ)](0)=|Qmpnq|.
Pmpnq=[Qmpnq(ξ, )](0)-12 b2R2[Qmpnq(ξ, )](2)+ib2R2[Qmpnq(ξ, )](1),
(Qmpnq)(j)(ξ, )
=δpq[m!n!(m+p)!(n+q)!]1/2×(2ξ)p+1(1+ξ2)p+1 k=0ml=0nξ2k21+ξ2k+l+j×(-1)k+l+p(k+l+p+j)!k!l!(m-k)!(n-l)!(p+k)!(p+l)!.
P0p0p=Q0p0p(ξ, )1-12 L/2b1+(2L/b)22×(p+2)!p! 21+ξ22,iL/2b1+(2L/b)2×(p+1)!p! 21+ξ2,

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