Abstract

The layer-peeling method for reconstruction of fiber and waveguide gratings is extended to the case of birefringent reflective gratings with polarization-dependent background index and polarization-dependent effective index contrast. Using a discrete grating model, we characterize the set of possible reflection and transmission Jones matrices and show that for a given wavelength, the total structure can be represented by a discrete reflector sandwiched between two retardation sections. In reflection the discrete reflector acts as a partial polarizer. A method for designing birefringent gratings is developed and tested numerically.

© 2004 Optical Society of America

Full Article  |  PDF Article

Errata

Ole Henrik Waagaard and Johannes Skaar, "Synthesis of birefringent reflective gratings: errata," J. Opt. Soc. Am. A 23, 1796-1796 (2006)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-23-7-1796

References

  • View by:
  • |
  • |
  • |

  1. M. Yamada, K. Saduka, “Analysis of almost periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
    [CrossRef] [PubMed]
  2. J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
    [CrossRef]
  3. R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  4. J. Skaar, L. Wang, T. Erdogan, “On the Synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
    [CrossRef]
  5. G. Meltz, W. W. Morey, “Bragg grating formation and germanosilicate fiber photosensitivity,” in International Workshop on Photoinduced Self-Organization Effects in Optical Fiber, F. Ouellette, ed., Proc. SPIE1516, 185–199 (1991).
    [CrossRef]
  6. K. O. Hill, F. Bilodeau, B. Malo, D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: application to external writing of rocking filters,” J. Opt. Soc. Am. B 27, 1548–1550 (1991).
  7. T. Erdogan, V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B 11, 2100–2105 (1994).
    [CrossRef]
  8. S. Pereira, J. E. Sipe, R. E. Slusher, S. Spälter, “Enhanced and suppressed birefringence in fiber Bragg gratings,” J. Opt. Soc. Am. B 19, 1509–1515 (2002).
    [CrossRef]
  9. P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
    [CrossRef]
  10. L. Bjerkan, K. Johannessen, X. Guo, “Measurements of Bragg grating birefringence due to transverse compressive forces,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 60–63.
  11. C. M. Lawrence, D. V. Nelson, E. Udd, T. Bennett, “A fiber optic sensor for transverse strain measurement,” Exp. Mech. 39, 202–209 (1999).
    [CrossRef]
  12. R. Gafsi, M. A. El-Sherif, “Analysis of induced-birefringence effects on fiber Bragg gratings,” Opt. Fiber Technol. 6, 299–323 (2000).
    [CrossRef]
  13. A.-P. Zhang, B.-O. Guan, X.-M. Tao, H.-Y. Tam, “Experimental and theoretical analysis of fiber Bragg gratings under lateral compression,” Opt. Commun. 206, 81–87 (2002).
    [CrossRef]
  14. M.-J. Li, S. I. Najafi, “Polarization dependence of grating-assisted waveguide Bragg reflectors,” Appl. Opt. 32, 4517–4521 (1993).
    [CrossRef] [PubMed]
  15. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–503 (1941).
    [CrossRef]
  16. E. Rønnekleiv, M. N. Zervas, J. T. Kringlebotn, “Modeling of polarization-mode competition in fiber DFB lasers,” IEEE J. Quantum Electron. 34, 1559–1569 (1998).
    [CrossRef]
  17. D. Sandel, R. Noé, G. Heise, B. Borchert, “Optical network analysis and longitudinal structure characterization of fiber Bragg grating,” J. Lightwave Technol. 16, 2435–2442 (1998).
    [CrossRef]
  18. A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1983), Chap. 6.4.
  19. R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999), Chap. 5.
  20. A. Buryak, “Iterative schema for ‘mixed’ scattering problems,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, postconference digest, Vol. 94 of OSA Topic in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003).
  21. L. V. Ahlfors, Complex Analysis (McGraw-Hill International Editions, 1979).

2003

J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

2002

S. Pereira, J. E. Sipe, R. E. Slusher, S. Spälter, “Enhanced and suppressed birefringence in fiber Bragg gratings,” J. Opt. Soc. Am. B 19, 1509–1515 (2002).
[CrossRef]

A.-P. Zhang, B.-O. Guan, X.-M. Tao, H.-Y. Tam, “Experimental and theoretical analysis of fiber Bragg gratings under lateral compression,” Opt. Commun. 206, 81–87 (2002).
[CrossRef]

2001

J. Skaar, L. Wang, T. Erdogan, “On the Synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

2000

R. Gafsi, M. A. El-Sherif, “Analysis of induced-birefringence effects on fiber Bragg gratings,” Opt. Fiber Technol. 6, 299–323 (2000).
[CrossRef]

1999

C. M. Lawrence, D. V. Nelson, E. Udd, T. Bennett, “A fiber optic sensor for transverse strain measurement,” Exp. Mech. 39, 202–209 (1999).
[CrossRef]

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

1998

E. Rønnekleiv, M. N. Zervas, J. T. Kringlebotn, “Modeling of polarization-mode competition in fiber DFB lasers,” IEEE J. Quantum Electron. 34, 1559–1569 (1998).
[CrossRef]

D. Sandel, R. Noé, G. Heise, B. Borchert, “Optical network analysis and longitudinal structure characterization of fiber Bragg grating,” J. Lightwave Technol. 16, 2435–2442 (1998).
[CrossRef]

1995

P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
[CrossRef]

1994

1993

1991

K. O. Hill, F. Bilodeau, B. Malo, D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: application to external writing of rocking filters,” J. Opt. Soc. Am. B 27, 1548–1550 (1991).

1987

1941

Ahlfors, L. V.

L. V. Ahlfors, Complex Analysis (McGraw-Hill International Editions, 1979).

Bennett, T.

C. M. Lawrence, D. V. Nelson, E. Udd, T. Bennett, “A fiber optic sensor for transverse strain measurement,” Exp. Mech. 39, 202–209 (1999).
[CrossRef]

Bernage, P.

P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
[CrossRef]

Bilodeau, F.

K. O. Hill, F. Bilodeau, B. Malo, D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: application to external writing of rocking filters,” J. Opt. Soc. Am. B 27, 1548–1550 (1991).

Bjerkan, L.

L. Bjerkan, K. Johannessen, X. Guo, “Measurements of Bragg grating birefringence due to transverse compressive forces,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 60–63.

Boj, S.

P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
[CrossRef]

Borchert, B.

Buryak, A.

A. Buryak, “Iterative schema for ‘mixed’ scattering problems,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, postconference digest, Vol. 94 of OSA Topic in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003).

Delevaque, E.

P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
[CrossRef]

Douay, M.

P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
[CrossRef]

El-Sherif, M. A.

R. Gafsi, M. A. El-Sherif, “Analysis of induced-birefringence effects on fiber Bragg gratings,” Opt. Fiber Technol. 6, 299–323 (2000).
[CrossRef]

Erdogan, T.

J. Skaar, L. Wang, T. Erdogan, “On the Synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

T. Erdogan, V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B 11, 2100–2105 (1994).
[CrossRef]

Feced, R.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Gafsi, R.

R. Gafsi, M. A. El-Sherif, “Analysis of induced-birefringence effects on fiber Bragg gratings,” Opt. Fiber Technol. 6, 299–323 (2000).
[CrossRef]

Guan, B.-O.

A.-P. Zhang, B.-O. Guan, X.-M. Tao, H.-Y. Tam, “Experimental and theoretical analysis of fiber Bragg gratings under lateral compression,” Opt. Commun. 206, 81–87 (2002).
[CrossRef]

Guo, X.

L. Bjerkan, K. Johannessen, X. Guo, “Measurements of Bragg grating birefringence due to transverse compressive forces,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 60–63.

Heise, G.

Hill, K. O.

K. O. Hill, F. Bilodeau, B. Malo, D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: application to external writing of rocking filters,” J. Opt. Soc. Am. B 27, 1548–1550 (1991).

Johannessen, K.

L. Bjerkan, K. Johannessen, X. Guo, “Measurements of Bragg grating birefringence due to transverse compressive forces,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 60–63.

Johnson, D. C.

K. O. Hill, F. Bilodeau, B. Malo, D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: application to external writing of rocking filters,” J. Opt. Soc. Am. B 27, 1548–1550 (1991).

Jones, R. C.

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999), Chap. 5.

Kringlebotn, J. T.

E. Rønnekleiv, M. N. Zervas, J. T. Kringlebotn, “Modeling of polarization-mode competition in fiber DFB lasers,” IEEE J. Quantum Electron. 34, 1559–1569 (1998).
[CrossRef]

Lawrence, C. M.

C. M. Lawrence, D. V. Nelson, E. Udd, T. Bennett, “A fiber optic sensor for transverse strain measurement,” Exp. Mech. 39, 202–209 (1999).
[CrossRef]

Li, M.-J.

Malo, B.

K. O. Hill, F. Bilodeau, B. Malo, D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: application to external writing of rocking filters,” J. Opt. Soc. Am. B 27, 1548–1550 (1991).

Meltz, G.

G. Meltz, W. W. Morey, “Bragg grating formation and germanosilicate fiber photosensitivity,” in International Workshop on Photoinduced Self-Organization Effects in Optical Fiber, F. Ouellette, ed., Proc. SPIE1516, 185–199 (1991).
[CrossRef]

Mizrahi, V.

Morey, W. W.

G. Meltz, W. W. Morey, “Bragg grating formation and germanosilicate fiber photosensitivity,” in International Workshop on Photoinduced Self-Organization Effects in Optical Fiber, F. Ouellette, ed., Proc. SPIE1516, 185–199 (1991).
[CrossRef]

Muriel, M. A.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Najafi, S. I.

Nelson, D. V.

C. M. Lawrence, D. V. Nelson, E. Udd, T. Bennett, “A fiber optic sensor for transverse strain measurement,” Exp. Mech. 39, 202–209 (1999).
[CrossRef]

Niay, P.

P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
[CrossRef]

Noé, R.

Pereira, S.

Poumellec, B.

P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
[CrossRef]

Rønnekleiv, E.

E. Rønnekleiv, M. N. Zervas, J. T. Kringlebotn, “Modeling of polarization-mode competition in fiber DFB lasers,” IEEE J. Quantum Electron. 34, 1559–1569 (1998).
[CrossRef]

Saduka, K.

Sandel, D.

Sipe, J. E.

Skaar, J.

J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

J. Skaar, L. Wang, T. Erdogan, “On the Synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Slusher, R. E.

Spälter, S.

Tam, H.-Y.

A.-P. Zhang, B.-O. Guan, X.-M. Tao, H.-Y. Tam, “Experimental and theoretical analysis of fiber Bragg gratings under lateral compression,” Opt. Commun. 206, 81–87 (2002).
[CrossRef]

Tao, X.-M.

A.-P. Zhang, B.-O. Guan, X.-M. Tao, H.-Y. Tam, “Experimental and theoretical analysis of fiber Bragg gratings under lateral compression,” Opt. Commun. 206, 81–87 (2002).
[CrossRef]

Taunay, T.

P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
[CrossRef]

Udd, E.

C. M. Lawrence, D. V. Nelson, E. Udd, T. Bennett, “A fiber optic sensor for transverse strain measurement,” Exp. Mech. 39, 202–209 (1999).
[CrossRef]

Waagaard, O. H.

J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

Wang, L.

J. Skaar, L. Wang, T. Erdogan, “On the Synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Yamada, M.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1983), Chap. 6.4.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1983), Chap. 6.4.

Zervas, M. N.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

E. Rønnekleiv, M. N. Zervas, J. T. Kringlebotn, “Modeling of polarization-mode competition in fiber DFB lasers,” IEEE J. Quantum Electron. 34, 1559–1569 (1998).
[CrossRef]

Zhang, A.-P.

A.-P. Zhang, B.-O. Guan, X.-M. Tao, H.-Y. Tam, “Experimental and theoretical analysis of fiber Bragg gratings under lateral compression,” Opt. Commun. 206, 81–87 (2002).
[CrossRef]

Appl. Opt.

Exp. Mech.

C. M. Lawrence, D. V. Nelson, E. Udd, T. Bennett, “A fiber optic sensor for transverse strain measurement,” Exp. Mech. 39, 202–209 (1999).
[CrossRef]

IEEE J. Quantum Electron.

J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

J. Skaar, L. Wang, T. Erdogan, “On the Synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

E. Rønnekleiv, M. N. Zervas, J. T. Kringlebotn, “Modeling of polarization-mode competition in fiber DFB lasers,” IEEE J. Quantum Electron. 34, 1559–1569 (1998).
[CrossRef]

IEEE Photon. Technol. Lett.

P. Niay, P. Bernage, T. Taunay, M. Douay, E. Delevaque, S. Boj, B. Poumellec, “Polarization selectivity of gratings written in Hi-Bi fibers by the external method,” IEEE Photon. Technol. Lett. 7, 391–393 (1995).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Commun.

A.-P. Zhang, B.-O. Guan, X.-M. Tao, H.-Y. Tam, “Experimental and theoretical analysis of fiber Bragg gratings under lateral compression,” Opt. Commun. 206, 81–87 (2002).
[CrossRef]

Opt. Fiber Technol.

R. Gafsi, M. A. El-Sherif, “Analysis of induced-birefringence effects on fiber Bragg gratings,” Opt. Fiber Technol. 6, 299–323 (2000).
[CrossRef]

Other

L. Bjerkan, K. Johannessen, X. Guo, “Measurements of Bragg grating birefringence due to transverse compressive forces,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 60–63.

G. Meltz, W. W. Morey, “Bragg grating formation and germanosilicate fiber photosensitivity,” in International Workshop on Photoinduced Self-Organization Effects in Optical Fiber, F. Ouellette, ed., Proc. SPIE1516, 185–199 (1991).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1983), Chap. 6.4.

R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999), Chap. 5.

A. Buryak, “Iterative schema for ‘mixed’ scattering problems,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, postconference digest, Vol. 94 of OSA Topic in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003).

L. V. Ahlfors, Complex Analysis (McGraw-Hill International Editions, 1979).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Discrete model of the birefringent grating. The cylinders represent the retardation or phase-delay sections Φj, and the arrows represent the eigenvalues of the discrete reflectors (ρj(1) and ρj(2)).

Fig. 2
Fig. 2

Top, reflection-response amplitude |R11| (solid curve), |R12|=|R21| (dashed curve), and |R22| (dotted curve) for grating example 1. The inner product between the eigenvectors of R is shown by the dashed–dotted curve. Bottom, singular values squared of R: ρ(1)2 (solid curve), ρ(2)2 (dashed curve), and (ρ(1)2+ρ(2)2)/2 (dotted curve).

Fig. 3
Fig. 3

Reconstructed spatial profile of grating example 1. Top, birefringence (from the difference in eigenvalue phases of Φ). Middle, index-modulation amplitudes (from the eigenvalues of q) (solid and dashed curves) and the index-modulation amplitude calculated with scalar layer peeling (dashed–dotted curve). Bottom, rotation angle θΦ of Φ eigenvectors (solid curve) and θq of q eigenvectors (dashed curve).

Fig. 4
Fig. 4

Reflection response R11=R22 (solid curve) and R12=R21 (dashed curve) for grating example 2. Top, reflection response. Middle, in-band details of the reflection response. Bottom, group-delay response.

Fig. 5
Fig. 5

Spatial profile of grating example 2. Top, phase delays (phase of eigenvalues of Φ). Middle, coupling-coefficient amplitudes (eigenvalues of q). Bottom, rotation angle of Φ eigenvectors (solid curve) and q eigenvectors (dashed curve).

Fig. 6
Fig. 6

Alternative realization of a decoupled grating.

Fig. 7
Fig. 7

Scattering matrix formulation for a linear optical device.

Equations (110)

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

n(x)=Udc(x)n¯+Δndc(s)(x)00n¯+Δndc(f)(x)Udc(x)+Uac(x)nac(1)(x)cos[2πx/Λ+ϕ(1)(x)]00nac(2)(x)cos[2πx/Λ+ϕ(2)(x)]Uac(x).
dE/dx=iCE,
C=D+Cψ+Cq.
D=δI00-I,I=1001,
Cq=0-qq0,
q=Uacπnac(1)2n¯Λexp(iϕ(1))00πnac(2)2n¯Λexp(iϕ(2))Uac,
Cψ=ψ00-ψT,ψ=UdcβΔndc(s)00βΔndc(f)Udc.
ψ=PdcβΔndc(lin,s)00βΔndc(lin,f)PdcT+0iα-iα0.
E(xj+Δx)=exp(iCΔx)E(xj),xj=jΔx.
Tjexp(iCΔx)=TzTρjTΦj,
Tzexp(iDΔx)=z-1/2I00z1/2I,z-1=exp(i2δΔx),
Tρjexp(iCqΔx)=tj-1-tj-1ρj-tj-1ρjtj-1,ρj=iq(qq)-1/2tanh[(qq)1/2Δx],tj=cosh[(qq)1/2Δx]-1=(I-ρjρj)1/2,
TΦjexp(iCψΔx)=Φj00Φj*,Φj=Udcexp(iβΔndc(s)Δx)00exp(iβΔndc(f)Δx)Udc.
ΔβΔndc(s,f)Δx=πΔndc(s,f)n¯.
Φj=Udcexp(iφ(s))00exp(iφ(f))Udc,
Sρj=ρjtjtj-ρj.
Tj=TzTρjTΦj=z-1/2tj-1Φj-z-1/2tj-1ρjΦj*-z1/2tj-1ρjΦjz1/2tj-1Φj*.
Sj=ΦjTρjΦjz-1/2ΦjTtjz-1/2tjΦj-z-1ρj.
MN=TNTN-1T1T0.
R(δ)=k=0h(k)exp(i2kδΔx)=k=0h(k)z-k.
Tj=z-1/2tj-1Φj00ztj-1Φj* I-ΦjρjΦj*-ΦjTρjΦjI.
uj+1(k)vj+1(k-1)=tj-1Φj00tj-1Φj* I-ΦjρjΦj*-ΦjTρjΦjI uj(k)vj(k).
vj(k)uj+1(k)=ΦjTρjΦjΦjTtjtjΦj-ρj uj(k)vj+1(k-1).
vj(0)=ΦjTρjΦjuj(0).
Υ=ΦjTρjΦj=QΦPΦTPρΛPρTPΦQΦ=QΦPΛPTQΦ,
TN=Φ00Φ* tN-1-tN-1ρN-tN-1ρNtN-1 ΦN00ΦN*=ΦtN-1ΦN-ΦtN-1ρNΦN*-Φ*tN-1ρNΦNΦ*tN-1ΦN*.
MN=A*(z)B*(z)B(z)A(z).
A(z)=zN/2k=0Na(k)z-k,
B(z)=zN/2k=0Nb(k)z-k,
AA*T-BB*T=I,
ABT-BAT=0,
ATB*-B*TA=0.
|αj| <1,j=1,,2N.
A=Φr*t-1Φl*,
B=-Φr*t-1ρΦl,
R=-A-1B=ΦlTρΦl,
T=A-1T=ΦrtΦl,
A=ΦlTt-1DΦl*.
arg(detD)=H{ln(det t-1)}.
B=-AR.
B=-ΦlTt-1ρΦl.
A=ΦlTt-1DΦl*.
v1u2=Su1v2=S11S12S21S22 u1v2,
u2v2=Tu1v1=T11T12T21T22 u1v1.
v1=S12v2,
u2=S22v2,
u2=T12v1,
v2=T22v1,
0=S11u1+S12v2,
u2=S21u1+S22v2,
u2=T11u1,
v2=T21u1,
T=S21-S22S12-1S11S22S12-1-S12-1S11S12-1.
S=-T22-1T21T22-1T11-T12T22-1T21T12T22-1.
S21=S12T,
S11=S11T,
S22=S22T.
T11-T12T22-1T21=T22T-1,
T22-1T21=T21TT22T-1,
T12T22-1=T22T-1T12T.
T11T22T-T12T21T=T22T11T-T21T12T=I,
T22TT11-T12TT21=T11TT22-T21TT12=I.
T22T21T-T21T22T=0,
T22TT12-T12TT22=0.
S11S11+S21S21=I,
S12S12+S22S22=I,
S12S11+S22S21=0.
T22T22-T21T21=I,
T22T22-T12T12=I,
T22T21T-T12*T22T=0.
T=A*B*BA,
AA-BB=I,
ABT-BAT=0,
ATB*-BA=0.
A=UAΣAVA,
B=UBΣBVB.
VAΣA2VA-VBTΣB2VB*=I.
ΣA=VAVBT(I+ΣB2)1/2VB*VA,
A=UAVAVBT(I+ΣB2)1/2VB*.
D2ΣB2=ΣB2D2,
A=Φr*t-1Φl*,
B=-Φr*t-1ρΦl,
MN=T0T1TN-1TN,
Tj=z-1/2tj-1Φj-z-1/2tj-1ρjΦj*-z1/2tj-1ρjΦjz1/2tj-1Φj*
T0=Φt0-1Φ0-Φt0-1ρ0Φ0*-Φ*t0-1ρ0Φ0Φ*t0-1Φ0*.
MN+1=z-1/2A*t-1Φ-z1/2B*t-1ρΦ-z-1/2A*t-1ρΦ*+z1/2B*t-1Φ*-z1/2At-1ρΦ+z-1/2Bt-1Φz1/2At-1Φ*-z-1/2Bt-1ρΦ*.
z1/2At-1Φ*-z-1/2Bt-1ρΦ*
=z1/2A(I-z-1A-1Bρ)t-1Φ*.
R(z)v2=|R11(z)v1+R12(z)v2|2+|R21(z)v1+R22(z)v2|2.
TN-1=z1/2ΦNtN-1z-1/2ΦNtN-1ρNz1/2ΦNTtN-1ρNz-1/2ΦNTtN-1
MN-1=[ z1/2(A*+B*Φ NTρNΦN)Φ Nt N-1z-1/2(B*+A*Φ NρNΦ N*)Φ NTt N-1z1/2(B+AΦ NTρNΦN)Φ Nt N-1z-1/2(A+BΦ NρNΦ N*)Φ NTt N-1 ].
b(0)+a(0)ΦNTρNΦN=a(N)+b(N)ΦNρNΦN*=0.
ΦNTρNΦN=-a(0)-1b(0)
a(N)-b(N)a*(0)-1b*(0)=0.
R(z)-k=0Na(k)z-k-1k=0Nb(k)z-k=k=0h(k)z-k.
ρN=a(0)-1b(0)=h(0)=12π-ππR[exp(iθ)]dθ12π-ππR[exp(iθ)]dθ.
h(0)=-a(0)-1b(0)=QPTΛPQ.
N=UΛU,
UΛU=U*ΛUT,
UTUΛ=ΛUTU.
N=PΛPT,
Υ=V1ΣV2,
WΣ2=Σ2W,
WΣ=ΣW.
Υ=UTΣU,
Υ/s=UTU,
U=u1u2-u2*u1*exp(iϕ),
P=p1p2-p2p1,p12+p22=1.
QP-1U=PTU=p1u1+p2u2*p1u2-p2u1*p2u1-p1u2*p2u2+p1u1*exp(iϕ).
p1(u2+u2*)=p2(u1+u1*).

Metrics