Abstract

A coupled-local-mode theory is presented for a nonuniform fiber grating with slowly varying background induced-index change. On this basis the transmission spectrum of a Gaussian fiber grating is calculated, and the underlying physical mechanism is studied in detail.

© 1999 Optical Society of America

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References

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  1. K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides,” Appl. Phys. Lett. 32, 647–649 (1978).
    [CrossRef]
  2. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  3. R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fibre,” Int. J. Optoelectron. 9, 33–57 (1994).
  4. R. Kashyap, “Photosensitive optical fibers: devices and applications,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 17–34 (1994).
    [CrossRef]
  5. I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum Electron. 28, 93–135 (1996).
    [CrossRef]
  6. V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
    [CrossRef]
  7. T. Erdogan, J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
    [CrossRef]
  8. A. M. Vengstarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
    [CrossRef]
  9. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
    [CrossRef]
  10. Y. Zhao, J. C. Palais, “Fiber Bragg grating coherence spectrum modeling, simulation, and characteristics,” J. Lightwave Technol. 15, 154–160 (1997).
    [CrossRef]
  11. P. Fonjallaz, H. G. Limberger, R. P. Salathé, “Bragg gratings with efficient and wavelength-selective fiber out-coupling,” J. Lightwave Technol. 15, 371–376 (1997).
    [CrossRef]
  12. J. E. Sipe, L. Poladian, C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
    [CrossRef]
  13. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
    [CrossRef]
  14. L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, UK, 1980).
  15. H. C. Huang, Microwave Approach to Highly Irregular Fiber Optics (Wiley, New York, 1998).
  16. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  17. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, San Diego, Calif., 1991).
  18. D. K. W. Lam, B. K. Garside, “Characterization of single-mode optical fiber filters,” Appl. Opt. 20, 440–445 (1983).
    [CrossRef]
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).
  20. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  21. R. Kashyap, “Fibre laser and beat frequency sources based on fibre gratings for microwave and ultrafast processing,” Int. J. Optoelectron. 11, 87–92 (1997).

1997 (5)

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

Y. Zhao, J. C. Palais, “Fiber Bragg grating coherence spectrum modeling, simulation, and characteristics,” J. Lightwave Technol. 15, 154–160 (1997).
[CrossRef]

P. Fonjallaz, H. G. Limberger, R. P. Salathé, “Bragg gratings with efficient and wavelength-selective fiber out-coupling,” J. Lightwave Technol. 15, 371–376 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

R. Kashyap, “Fibre laser and beat frequency sources based on fibre gratings for microwave and ultrafast processing,” Int. J. Optoelectron. 11, 87–92 (1997).

1996 (3)

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum Electron. 28, 93–135 (1996).
[CrossRef]

T. Erdogan, J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
[CrossRef]

A. M. Vengstarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

1994 (3)

R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fibre,” Int. J. Optoelectron. 9, 33–57 (1994).

R. Kashyap, “Photosensitive optical fibers: devices and applications,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 17–34 (1994).
[CrossRef]

J. E. Sipe, L. Poladian, C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

1993 (2)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

1989 (1)

1983 (1)

1978 (1)

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Bennion, I.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum Electron. 28, 93–135 (1996).
[CrossRef]

Bhatia, V.

A. M. Vengstarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Campbell, R. J.

R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fibre,” Int. J. Optoelectron. 9, 33–57 (1994).

Doran, N. J.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum Electron. 28, 93–135 (1996).
[CrossRef]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

T. Erdogan, J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
[CrossRef]

A. M. Vengstarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Fonjallaz, P.

P. Fonjallaz, H. G. Limberger, R. P. Salathé, “Bragg gratings with efficient and wavelength-selective fiber out-coupling,” J. Lightwave Technol. 15, 371–376 (1997).
[CrossRef]

Fujii, Y.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Garside, B. K.

Glenn, W. H.

Hill, K. O.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Huang, H. C.

H. C. Huang, Microwave Approach to Highly Irregular Fiber Optics (Wiley, New York, 1998).

Johnson, D. C.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Judkins, J. B.

A. M. Vengstarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Kashyap, R.

R. Kashyap, “Fibre laser and beat frequency sources based on fibre gratings for microwave and ultrafast processing,” Int. J. Optoelectron. 11, 87–92 (1997).

R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fibre,” Int. J. Optoelectron. 9, 33–57 (1994).

R. Kashyap, “Photosensitive optical fibers: devices and applications,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 17–34 (1994).
[CrossRef]

Kawasaki, B. S.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Lam, D. K. W.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, UK, 1980).

Lemaire, P. J.

A. M. Vengstarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, UK, 1980).

Limberger, H. G.

P. Fonjallaz, H. G. Limberger, R. P. Salathé, “Bragg gratings with efficient and wavelength-selective fiber out-coupling,” J. Lightwave Technol. 15, 371–376 (1997).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, San Diego, Calif., 1991).

Martijn de Sterke, C.

Meltz, G.

Mizrahi, V.

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

Morey, W. W.

Palais, J. C.

Y. Zhao, J. C. Palais, “Fiber Bragg grating coherence spectrum modeling, simulation, and characteristics,” J. Lightwave Technol. 15, 154–160 (1997).
[CrossRef]

Poladian, L.

J. E. Sipe, L. Poladian, C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

Salathé, R. P.

P. Fonjallaz, H. G. Limberger, R. P. Salathé, “Bragg gratings with efficient and wavelength-selective fiber out-coupling,” J. Lightwave Technol. 15, 371–376 (1997).
[CrossRef]

Sipe, J. E.

A. M. Vengstarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

T. Erdogan, J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
[CrossRef]

J. E. Sipe, L. Poladian, C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Sugden, K.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum Electron. 28, 93–135 (1996).
[CrossRef]

Vengstarkar, A. M.

A. M. Vengstarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Williams, J. A. R.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum Electron. 28, 93–135 (1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Zhang, L.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum Electron. 28, 93–135 (1996).
[CrossRef]

Zhao, Y.

Y. Zhao, J. C. Palais, “Fiber Bragg grating coherence spectrum modeling, simulation, and characteristics,” J. Lightwave Technol. 15, 154–160 (1997).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Int. J. Optoelectron. (2)

R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fibre,” Int. J. Optoelectron. 9, 33–57 (1994).

R. Kashyap, “Fibre laser and beat frequency sources based on fibre gratings for microwave and ultrafast processing,” Int. J. Optoelectron. 11, 87–92 (1997).

J. Lightwave Technol. (5)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

A. M. Vengstarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Y. Zhao, J. C. Palais, “Fiber Bragg grating coherence spectrum modeling, simulation, and characteristics,” J. Lightwave Technol. 15, 154–160 (1997).
[CrossRef]

P. Fonjallaz, H. G. Limberger, R. P. Salathé, “Bragg gratings with efficient and wavelength-selective fiber out-coupling,” J. Lightwave Technol. 15, 371–376 (1997).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Fiber Technol.: Mater., Devices Syst. (1)

R. Kashyap, “Photosensitive optical fibers: devices and applications,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 17–34 (1994).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum Electron. 28, 93–135 (1996).
[CrossRef]

Phys. Rev. E (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

Other (5)

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, UK, 1980).

H. C. Huang, Microwave Approach to Highly Irregular Fiber Optics (Wiley, New York, 1998).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, San Diego, Calif., 1991).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

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Figures (9)

Fig. 1
Fig. 1

Plot of transmission spectrum of a Gaussian fiber grating by a straightforward numerical calculation of the coupled-local-mode equations (17). The parameters and the boundary conditions used in the calculation are given in the text.

Fig. 2
Fig. 2

(a) Plot of the characteristic quantity P(z, λ) against position z for wavelength λ=1.54006 µm satisfying relations (20), where two symmetrical P(z, λ)>0 regions are separated by a P(z, λ)<0 region. (b) Plot of |a+(z)| (solid curve) and |a-(z)| (dashed curve) against position z for λ=1.54006 µm. Here |a+(z)| and |a-(z)| are trapped in the P(z, λ)<0 region as two standing waves. The reflectivity is 97%.

Fig. 3
Fig. 3

(a) Plot of the characteristic quantity P(z, λ) against position z for wavelength λ=1.5410 µm satisfying relations (21), where two symmetrical P(z, λ)>0 regions meet at the center of the Gaussian grating. (b) Plot of |a+(z)| (solid curve) and |a-(z)| (dashed curve) against position z for λ=1.5410 µm. Here the two curves overlap each other with reflectivity of nearly 100%.

Fig. 4
Fig. 4

Schematic diagram of the two-effective-grating model. Here region 1 between z1 and z2 and region 3 between z3 and z4 represent the two effective gratings, respectively, which are separated by region 2 between z2 and z3.

Fig. 5
Fig. 5

Plot of transmission spectrum of a Gaussian fiber grating with the use of Eqs. (23). The parameters and the boundary conditions used in the calculation are identical with those used in Fig. 1.

Fig. 6
Fig. 6

Plots of the spatial reflection distribution function Rs(z) against position z for (a) λ=1.53925 µm, (b) λ=1.53958 µm, and (c) λ=1.54006 µm.

Fig. 7
Fig. 7

Schematic diagram for the consideration of the two (external and internal) small regions of space immediately outside the two effective gratings. The external pair of small regions is denoted by z1zz1 and z4zz4. The internal pair of small regions is denoted by z2zz2 and z3zz3. The two effective gratings are still denoted by z1zz2 and z3zz4, but the center region is shortened to be represented by z2zz3.

Fig. 8
Fig. 8

Plot of transmission spectrum of a Gaussian fiber grating with the use of Eqs. (33). The parameters and the boundary conditions used in the calculation are identical with those used in Fig. 1.

Fig. 9
Fig. 9

Plot of the spatial reflection distribution function Rs(z) against position z for λ=1.53914 µm.

Equations (165)

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n(r, z)=n1(z)=nco1+σ(z)1+m cos2πΛzran2r>a,
σ(z)=σ0 exp[-2(ln 2)(z-z0)2/w2],
n0(r, z)=n1(z)=nco[1+σ(z)],ran2,r>a.
Ej=ej(x, y, βj(z))expi0zβj(z)dz,
Hj=hj(x, y, βj(z))expi0zβj(z)dz,
etj=-1kn02μ001/2βjzˆ×htj-1kt×(t+etj),
ezj=ikn02μ001/2zˆ·(t×htj),
htj=1k0μ01/2βjzˆ×etj+1kt×t×htjn02,
hzj=-ik0μ01/2zˆ·(t×etj),
Et=1kn2iμ001/2zˆ×Htz+1kt×(t×Et),
Ez=ikn2μ001/2zˆ·(t×Ht),
Ht=-1ki0μ01/2zˆ×Etz-1kt×t×Htn2,
Hz=-ik0μ01/2zˆ·(t×Et)
Et=j[bj(z)+b-j(z)]etj(x, y, βj(z)),
Ht=j[bj(z)-b-j(z)]htj(x, y, βj(z)),
b±j(z)=a±j(z)exp±i0zβj(z)dz.
12Azˆ·(etj×htk)dA=δjk,
dajdz=kCjk(+,+)ak expi0z(βk-βj)dz+Cjk(+,-)a-k exp-i0z(βk+βj)dz,
da-jdz=kCjk(-,+)ak expi0z(βk+βj)dz+Cjk(-,-)a-k exp-i0z(βk-βj)dz,
Cjk(p, q)=14βj(p)|βj(p)|Azˆ·htj(p)×etk(q)z-etj(p)×htk(q)zdA+i4k0μ01/2βj(p)|βj(p)|A(n2-n02)etj(p)·etk(q)+n02n2hzj(p)·hzk(q)dA
βj(+)=-βj(-1)=βj,
etj(+)=etj(-)=etj, ezj(+)=-ezj(-)=ezj,
htj(+)=-htj(-)=htj,hzj(+)=hzj(-)=hzj.
da+dz=iM(z)a- exp-2i0zβ(z)dz+i2πΛz,
da-dz=-iM(z)a+ exp2i0zβ(z)dz-i2πΛz.
M(z)=k4mσ(z)nco2b(z)neff(z)×1+J02[V(z)1-b(z)]J12[V(z)1-b(z)],
1-b(z)J1[V(z)1-b(z)]J0[V(z)1-b(z)]=b(z)K1[V(z)b(z)]K0[V(z)b(z)],
a¯+(z)=a+(z)expi0zβ(z)dz-iπΛz,
a¯-(z)=a-(z)exp-i0zβ(z)dz+iπΛz,
da¯+dz=iδβ(z)a¯+(z)+iM(z)a¯-(z),
da¯-dz=-iM(z)a¯+(z)-iδβ(z)a¯-(z),
δβ(z)=β(z)-πΛ=2πλneff-πΛ.
P(z, λ)=M2(z, λ)-δβ2(z, λ).
δβ(0, λ)δβ(z0, λ)<0,P(z0, λ)<0,
δβ(0, λ)<0,δβ(z0, λ)>0,P(z0, λ)>0
δβ(z0, λ)0,P(z0, λ)>0,
δβ(0, λ)>0,P(0, λ)0,
δβ(z0, λ)<0,P(z0, λ)0,
T=1+M1S1sinh(2S¯1lg)cos(Q¯2lc)+M2Q2-2M1(δβ1δβ2-M1M2)S12Q2sinh2(S¯1lg)×sin(Q¯2lc)2-1
R1-T.
δβ1=1z2-z1z1z2δβ(z)dz,
M1=1z2-z1z1z2M(z)dz,
S1=(M12-δβ12)1/2,
S¯1=1z2-z1z1z2[M2(z)-δβ2(z)]1/2 dz,
δβ2=1z0-z2z2z0δβ(z)dz,
M2=1z0-z2z2z0M(z)dz,
Q2=(δβ22-M22)1/2,
Q¯2=1z0-z2z2z0[δβ2(z)-M2(z)]1/2 dz.
T=11+ηF sin2(Q¯2lc+θ),
Rg=sinh2(S¯1lg)1+sinh2(S¯1lg)
F=4Rg(1-Rg)2,
η=1+M2Q21+Rg2Rg2,
θ=tan-1Q2M22Rg1+Rg.
2z2z0[δβ2(z)-M2(z)]1/2 dz=m+12π-θ.
T=11+(M12/S1)2 sinh2(2S¯1lg),R1-T.
Rs(z)
=M2(z)sinh2[S(z)z]S2(z)+M2(z)sinh2[S(z)z]zz0M2(z)sinh2[S(z)(L-z)]S2(z)+M2(z)sinh2[S(z)(L-z)]z>z0,
S(z)=[M2(z)-δβ2(z)]1/2,
T=11+|(|a1|2+|a2|2)r2+(a1a2r1-a1*a2*r1*)|2,
R1-T.
r1=[d1 cos(Q¯2lc)+is1 sin(Q¯2lc)]cosh(2S¯1lg)+id1δβ1S1-d2M1S1cos(Q¯2lc)+is1δβ1S1-s2M1S1sin(Q¯2lc)sinh(2S¯1lg)+M1S12{[2d2δβ1-(d1-d1*)M1]cos(Q¯2lc)+i[2s2δβ1-(s1+s1*)M1]sin(Q¯2lc)}sinh2(S¯1lg),
r2=[d2 cos(Q¯2lc)+is2 sin(Q¯2lc)]cosh(2S¯1lg)+iM1S1[(d1+d1*)cos(Q¯2lc)+i(s1-s1*)sin(Q¯2lc)]sinh(2S¯1lg)+δβ1S12{[2d2δβ1-(d1-d1*)M1]cos(Q¯2lc)+i[2s2δβ1-(s1+s1*)M1]sin(Q¯2lc)}sinh2(S¯1lg),
d1=b12-b22,d2=b1*b2-b1b2*,
s1=(b12+b22)δβ2Q2-2b1b2M2Q2,
s2=-(b1b2*+b1b2*)δβ2Q2+(|b1|2+|b2|2)M2Q2,
δβ2=1z0-z2z2z0δβ(z)dz,
M2=1z0-z2z2z0M(z)dz,
Q2=(δβ22-M22)1/2,
Q¯2=1z0-z2z2z0[δβ2(z)-M2(z)]1/2 dz,
lc=z3-z2.
δβ2(z)-M2(z)δβ2(0)-M2(0)=0.45forz1,
[δβ2(z)-M2(z)]1/2z=3π/4forz2.
dAdz=K(z)A,
K(z)=iδβ(z)iM(z)-iM(z)-iδβ(z).
A=OW,
O(z)=O-1(z)=1{M2(z)-[Q(z)+δβ(z)]2}1/2×i[Q(z)+δβ(z)]iM(z)-iM(z)-i[Q(z)+δβ(z)],
Q(z)=[δβ2(z)-M2(z)]1/2,
dWdz=ΛW-OdOdz,
Λ=iQ(z)00-iQ(z).
1δβdδβdz-1MdMdz|δβ|M-M|δβ|.
δβ(z0, λ)<0,|δβ(z0)|1.7M(z0).
W+(z)=W+(0)expi0zQ(z)dz,
W-(z)=W-(0)exp-i0zQ(z)dz.
A(L)=O(L)Λ˜O(0)A(0),
Λ˜=expi0LQ(z)dz00exp-i0LQ(z)dz.
O(0)=O(L)=0i-i0.
A(L)=TA(0),
T=exp-i0LQ(z)dz00expi0LQ(z)dz.
|a¯-1(0)|=exp-i0LQ(z)dza¯-(L)=0,
|a¯+(L)|=exp-i0LQ(z)dza¯+(0)=1.
a¯+(z)=a+1(z)expiz2zQ(z)dz+a+2(z)exp-iz2zQ(z)dz,
a¯-(z)=a-1(z)expiz2zQ(z)dz+a-2(z)exp-iz2zQ(z)dz,
Q(z)=[δβ2(z)-M2(z)]1/2.
da+1dz=i(δβ-Q)a+1+iMa-1,
da+2dz=i(δβ+Q)a+2+iMa-2,
da-1dz=-i(δβ+Q)a-1-iMa+1,
da-2dz=-i(δβ-Q)a-2-iMa+2.
a+1=-Q2+δβ2M2a-1=M2Q2-δβ2a-1,
a+2=Q2-δβ2M2a-2=-M2Q2+δβ2a-2,
δβ2=1z0-z2z2z0δβ(z)dz,
M2=1z0-z2z2z0M(z)dz,
Q2=(δβ22-M22)1/2.
a+1(z)=a+1(z2)1-iz2z(Q-Q2)dz,
a+2(z)=a+2(z2)1+iz2z(Q-Q2)dz,
a-1(z)=a-1(z2)1-iz2z(Q-Q2)dz,
a-2(z)=a-2(z2)1+iz2z(Q-Q2)dz.
a¯+(z2)=a+1(z2)+a+2(z2),
a¯-(z2)=a-1(z2)+a-2(z2),
A(z3)=KA(z2)=[K(0)+K(1)]A(z2),
k11(0)=cos(Q¯2lc)+iδβ2Q2sin(Q¯2lc),
k12(0)=iM2Q2sin(Q¯2lc),
k21(0)=-iM2Q2sin(Q¯2lc),
k22(0)=cos(Q¯2lc)-iδβ2Q2sin(Q¯2lc),
Q¯2=1lcz2z3Q dz,lc=z3-z2,
k11(1)=η2-sin(Q¯2lc)+iδβ2Q2cos(Q¯2lc),
k12(1)=η2iM2Q2cos(Q¯2lc),
k21(1)=η2-iM2Q2cos(Q¯2lc),
k11(1)=η2-sin(Q¯2lc)-iδβ2Q2cos(Q¯2lc),
η2=(Q2-Q¯2)lc.
A(z2)=KA(z1)=[K(0)+K(1)]A(z1).
k11(0)=cosh(S¯1lc)+iδβ1S1sinh(S¯1lg),
k12(0)=iM1S1sin(S¯1lg),
k21(0)=-iM1S1sinh(S¯1lg),
k22(0)=cosh(S¯1lg)-iδβ1S1sinh(S¯1lg),
δβ1=1lgz1z2δβ(z)dz,M1=1lgz1z2M(z)dz,
lg=z2-z1,S1=(M12-δβ12)1/2,
S¯1=1lgz1z2S dz,S(z)=[M2(z)-δβ2(z)]1/2.
k11(1)=η1sinh(S¯1lg)+iδβ1S1cosh(S¯1lg),
k12(1)=η1iM1S1cosh(S¯1lg),
k21(1)=η1-iM1S1cosh(S¯1lg),
k11(1)=η1sinh(S¯1lg)-iδβ1S1cosh(S¯1lg),
η1=(S1-S¯1)lg.
A(z4)=KA(z3)=[K(0)+K(1)]A(z3),
ψ(z)=a¯+(z)+a¯-(z),ϕ(z)=a¯+(z)-a¯-(z),
dψdz=-iκ1(z)ϕ,dϕdz=-iκ2(z)ψ,
κ1(z)=|δβ(z)|+M(z),κ2(z)=|δβ(z)|-M(z).
d2ψdz2=-idκ1dzϕ-iκ1dϕdz.
1κ1dκ1dz1ϕdϕdz,
d2ψdz2+U(z)ψ=0,
U(z)=-P(z)=κ1(z)κ2(z).
U(z)|(dU/dz)z=z1|(z1-z),
U(z1)=0,(dU/dz)z=z1<0.
ξ(z)=|(dU/dz)z=z1|1/3(z1-z).
d2ψdz2+ξψ=0.
ψ(z)=c1q1/3(z)J1/3(q)+c2q1/3(z)J-1/3(q),
q(z)=23ξ3/2(z).
q(z)=zz1Q(z)dz,
ϕ(z)=-iQ(z)κ1(z)[c1q1/3(z)J-2/3(q)-c2q1/3(z)J2/3(q)].
a¯+(z1)=a1a¯+(z1)+a2a¯-(z1),
a¯-(z1)=a2*a¯+(z1)+a1*a¯-(z1),
a1=AJ-2/3[q(z1)]+Bκ1(z1)Q(z1)J-1/3[q(z1)]-iAκ1(z1)Q(z1)J1/3[q(z1)]+BJ2/3[q(z1)],
a2=AJ-2/3[q(z1)]-Bκ1(z1)Q(z1)J-1/3[q(z1)]+iAκ1(z1)Q(z1)J1/3[q(z1)]-BJ2/3[q(z1)],
A=21/3π3Γ(2/3)q2/3(z1),
B=22/3π3Γ(1/3)Q(z1)κ1(z1)q1/3(z1).
a¯+(z4)=a1a¯+(z4)-a2*a¯-(z4),
a¯-(z4)=-a2a¯+(z4)+a1*a¯-(z4),
a¯+(z3)=b1a¯+(z3)+b2a¯-(z3),
a¯-(z3)=b2*a¯+(z3)+b1*a¯-(z3),
b1=AJ-2/3[q(z3)]+Bκ1(z3)Q(z3)J-1/3[q(z3)]-iAκ1(z3)Q(z3)J1/3[q(z3)]+BJ2/3[q(z3)],
b2=AJ-2/3[q(z3)]-Bκ1(z3)Q(z3)J-1/3[q(z3)]+iAκ1(z3)Q(z3)J1/3[q(z3)]-BJ2/3[q(z3)],
A=21/3π3Γ(2/3)q2/3(z3),
B=22/3π3Γ(1/3)Q(z3)κ1(z3)q1/3(z3),
q(z3)=z3z3Q(z)dz.
a¯+(z2)=b1a¯+(z2)-b2*a¯-(z2),
a¯-(z2)=-b2a+(z2)+b1*a¯-(z2),

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