Abstract

The behavior of a system of fiber-optic white-light interferometric sensors is analyzed. Analytical expressions for contrasts of all the interference patterns that may occur at the setup’s output are derived as a function of coupling coefficients between successive sensors. Two cases of exemplary systems consisting of highly birefringent optical fibers, one with nondichroic the other with some polarizing fibers, are analyzed.

© 1997 Optical Society of America

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References

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  1. Y.-J. Rao, D. A. Jackson, “Recent progress in fibre optic low-coherence interferometry,” Meas. Sci. Technol. 7, 981–999 (1996).
    [CrossRef]
  2. E. Udd, Fiber Optic Sensors: An Introduction for Engineers and Scientists (Wiley, New York, 1991).
  3. S. A. Al-Chalabi, B. Culshaw, D. E. N. David, “Partially coherent sources in interferometric sensors,” in Proceedings of IEE First International Conference on Optical Fibre Sensors, London, 1983 (Institute of Electrical Engineers, Stevenage, UK, 1989), pp. 132–135.
  4. T. Bosselmann, R. Ulrich, “High-accuracy position-sensing with fibre-coupled white-light interferometers,” in Second International Conference on Optical Fiber Sensors, R. T. Kersten, R. Kist, eds., Proc. SPIE514, 361–364 (1984).
    [CrossRef]
  5. V. Gusmeroli, P. Vavassori, M. Martinelli, “A coherence-multiplexed quasi-distributed polarimetric sensor suitable for structural monitoring,” in Sixth International Conference on Optical Fibre Sensors, Paris, 1989, Springer Proceedings in Physics, H. J. Arditty, J. P. Dakin, R. T. Kersten, eds. (Springer-Verlag, Berlin, 1989), Vol. 44, pp. 513–518.
  6. F. Farahi, T. P. Newson, J. D. C. Jones, D. A. Jackson, “Coherence multiplexing of remote fibre Fabry–Pérot sensing system,” Opt. Commun. 65, 319–321 (1988).
    [CrossRef]
  7. M. T. Velluet, P. Graindorge, H. J. Arditty, “Fibre optic pressure sensors using white-light interferometry,” in Fiber Optic and Laser Sensors, R. P. De Paula, P. Ramon, E. Udd, eds., Proc. SPIE838, 78–83 (1987).
  8. W. J. Bock, W. Urbańczyk, “Electronically scanned white-light interferometric sensor for high hydrostatic pressure measurements,” in Proceedings of the Ninth International Conference on Optical Fibre Sensors, Florence, 1993 (The Instituto of Florence, Florence, 1993), pp. 135–138.
  9. K. Takada, K. Chida, J. Noda, “Precise method for angular alignment of birefringent fibers based on an interferometric technique with a broadband source,” Appl. Opt. 26, 2979–2987 (1987).
    [CrossRef]
  10. P. Cielo, C. Delisle, “Multiplexing in optical communications by interferometry with a large path-length difference in white light,” Can. J. Phys. 54, 2322–2331 (1976).
    [CrossRef]
  11. J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence multiplexing of fiber-optic interferometric sensors,” J. Lightwave Technol. 3, 1062–1071 (1985).
    [CrossRef]
  12. K. Blotekjaer, R. H. Wentworth, H. J. Shaw, “Choosing relative optical path delays in series topology interferometric sensor arrays,” J. Lightwave Technol. 5, 229–235 (1987).
    [CrossRef]
  13. R. H. Wentworth, “Theoretical noise performance of coherence-multiplexed interferometric sensors,” J. Lightwave Technol. 7, 941–956 (1989).
    [CrossRef]
  14. W. Urbańczyk, W. J. Bock, “Visibility of white-light interference patterns for a chain of coherence multiplexed sensors based on highly birefringent fibers,” Opt. Eng. 32, 2100–2106 (1993).
    [CrossRef]
  15. W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis of a tandem of white-light fiber-optic interferometers,” Opt. Commun. 135, 1–6 (1997).
    [CrossRef]
  16. W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis for temperature-compensated white-light interferometric fiber sensors,” Optik 104, 153–158 (1997).
  17. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1968).

1997 (2)

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis of a tandem of white-light fiber-optic interferometers,” Opt. Commun. 135, 1–6 (1997).
[CrossRef]

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis for temperature-compensated white-light interferometric fiber sensors,” Optik 104, 153–158 (1997).

1996 (1)

Y.-J. Rao, D. A. Jackson, “Recent progress in fibre optic low-coherence interferometry,” Meas. Sci. Technol. 7, 981–999 (1996).
[CrossRef]

1993 (1)

W. Urbańczyk, W. J. Bock, “Visibility of white-light interference patterns for a chain of coherence multiplexed sensors based on highly birefringent fibers,” Opt. Eng. 32, 2100–2106 (1993).
[CrossRef]

1989 (1)

R. H. Wentworth, “Theoretical noise performance of coherence-multiplexed interferometric sensors,” J. Lightwave Technol. 7, 941–956 (1989).
[CrossRef]

1988 (1)

F. Farahi, T. P. Newson, J. D. C. Jones, D. A. Jackson, “Coherence multiplexing of remote fibre Fabry–Pérot sensing system,” Opt. Commun. 65, 319–321 (1988).
[CrossRef]

1987 (2)

K. Takada, K. Chida, J. Noda, “Precise method for angular alignment of birefringent fibers based on an interferometric technique with a broadband source,” Appl. Opt. 26, 2979–2987 (1987).
[CrossRef]

K. Blotekjaer, R. H. Wentworth, H. J. Shaw, “Choosing relative optical path delays in series topology interferometric sensor arrays,” J. Lightwave Technol. 5, 229–235 (1987).
[CrossRef]

1985 (1)

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence multiplexing of fiber-optic interferometric sensors,” J. Lightwave Technol. 3, 1062–1071 (1985).
[CrossRef]

1976 (1)

P. Cielo, C. Delisle, “Multiplexing in optical communications by interferometry with a large path-length difference in white light,” Can. J. Phys. 54, 2322–2331 (1976).
[CrossRef]

Al-Chalabi, S. A.

S. A. Al-Chalabi, B. Culshaw, D. E. N. David, “Partially coherent sources in interferometric sensors,” in Proceedings of IEE First International Conference on Optical Fibre Sensors, London, 1983 (Institute of Electrical Engineers, Stevenage, UK, 1989), pp. 132–135.

Arditty, H. J.

M. T. Velluet, P. Graindorge, H. J. Arditty, “Fibre optic pressure sensors using white-light interferometry,” in Fiber Optic and Laser Sensors, R. P. De Paula, P. Ramon, E. Udd, eds., Proc. SPIE838, 78–83 (1987).

Blotekjaer, K.

K. Blotekjaer, R. H. Wentworth, H. J. Shaw, “Choosing relative optical path delays in series topology interferometric sensor arrays,” J. Lightwave Technol. 5, 229–235 (1987).
[CrossRef]

Bock, W. J.

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis of a tandem of white-light fiber-optic interferometers,” Opt. Commun. 135, 1–6 (1997).
[CrossRef]

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis for temperature-compensated white-light interferometric fiber sensors,” Optik 104, 153–158 (1997).

W. Urbańczyk, W. J. Bock, “Visibility of white-light interference patterns for a chain of coherence multiplexed sensors based on highly birefringent fibers,” Opt. Eng. 32, 2100–2106 (1993).
[CrossRef]

W. J. Bock, W. Urbańczyk, “Electronically scanned white-light interferometric sensor for high hydrostatic pressure measurements,” in Proceedings of the Ninth International Conference on Optical Fibre Sensors, Florence, 1993 (The Instituto of Florence, Florence, 1993), pp. 135–138.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1968).

Bosselmann, T.

T. Bosselmann, R. Ulrich, “High-accuracy position-sensing with fibre-coupled white-light interferometers,” in Second International Conference on Optical Fiber Sensors, R. T. Kersten, R. Kist, eds., Proc. SPIE514, 361–364 (1984).
[CrossRef]

Brooks, J. L.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence multiplexing of fiber-optic interferometric sensors,” J. Lightwave Technol. 3, 1062–1071 (1985).
[CrossRef]

Chida, K.

Cielo, P.

P. Cielo, C. Delisle, “Multiplexing in optical communications by interferometry with a large path-length difference in white light,” Can. J. Phys. 54, 2322–2331 (1976).
[CrossRef]

Culshaw, B.

S. A. Al-Chalabi, B. Culshaw, D. E. N. David, “Partially coherent sources in interferometric sensors,” in Proceedings of IEE First International Conference on Optical Fibre Sensors, London, 1983 (Institute of Electrical Engineers, Stevenage, UK, 1989), pp. 132–135.

David, D. E. N.

S. A. Al-Chalabi, B. Culshaw, D. E. N. David, “Partially coherent sources in interferometric sensors,” in Proceedings of IEE First International Conference on Optical Fibre Sensors, London, 1983 (Institute of Electrical Engineers, Stevenage, UK, 1989), pp. 132–135.

Delisle, C.

P. Cielo, C. Delisle, “Multiplexing in optical communications by interferometry with a large path-length difference in white light,” Can. J. Phys. 54, 2322–2331 (1976).
[CrossRef]

Farahi, F.

F. Farahi, T. P. Newson, J. D. C. Jones, D. A. Jackson, “Coherence multiplexing of remote fibre Fabry–Pérot sensing system,” Opt. Commun. 65, 319–321 (1988).
[CrossRef]

Graindorge, P.

M. T. Velluet, P. Graindorge, H. J. Arditty, “Fibre optic pressure sensors using white-light interferometry,” in Fiber Optic and Laser Sensors, R. P. De Paula, P. Ramon, E. Udd, eds., Proc. SPIE838, 78–83 (1987).

Gusmeroli, V.

V. Gusmeroli, P. Vavassori, M. Martinelli, “A coherence-multiplexed quasi-distributed polarimetric sensor suitable for structural monitoring,” in Sixth International Conference on Optical Fibre Sensors, Paris, 1989, Springer Proceedings in Physics, H. J. Arditty, J. P. Dakin, R. T. Kersten, eds. (Springer-Verlag, Berlin, 1989), Vol. 44, pp. 513–518.

Jackson, D. A.

Y.-J. Rao, D. A. Jackson, “Recent progress in fibre optic low-coherence interferometry,” Meas. Sci. Technol. 7, 981–999 (1996).
[CrossRef]

F. Farahi, T. P. Newson, J. D. C. Jones, D. A. Jackson, “Coherence multiplexing of remote fibre Fabry–Pérot sensing system,” Opt. Commun. 65, 319–321 (1988).
[CrossRef]

Jones, J. D. C.

F. Farahi, T. P. Newson, J. D. C. Jones, D. A. Jackson, “Coherence multiplexing of remote fibre Fabry–Pérot sensing system,” Opt. Commun. 65, 319–321 (1988).
[CrossRef]

Kim, B. Y.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence multiplexing of fiber-optic interferometric sensors,” J. Lightwave Technol. 3, 1062–1071 (1985).
[CrossRef]

Kurzynowski, P.

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis of a tandem of white-light fiber-optic interferometers,” Opt. Commun. 135, 1–6 (1997).
[CrossRef]

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis for temperature-compensated white-light interferometric fiber sensors,” Optik 104, 153–158 (1997).

Martinelli, M.

V. Gusmeroli, P. Vavassori, M. Martinelli, “A coherence-multiplexed quasi-distributed polarimetric sensor suitable for structural monitoring,” in Sixth International Conference on Optical Fibre Sensors, Paris, 1989, Springer Proceedings in Physics, H. J. Arditty, J. P. Dakin, R. T. Kersten, eds. (Springer-Verlag, Berlin, 1989), Vol. 44, pp. 513–518.

Newson, T. P.

F. Farahi, T. P. Newson, J. D. C. Jones, D. A. Jackson, “Coherence multiplexing of remote fibre Fabry–Pérot sensing system,” Opt. Commun. 65, 319–321 (1988).
[CrossRef]

Noda, J.

Rao, Y.-J.

Y.-J. Rao, D. A. Jackson, “Recent progress in fibre optic low-coherence interferometry,” Meas. Sci. Technol. 7, 981–999 (1996).
[CrossRef]

Shaw, H. J.

K. Blotekjaer, R. H. Wentworth, H. J. Shaw, “Choosing relative optical path delays in series topology interferometric sensor arrays,” J. Lightwave Technol. 5, 229–235 (1987).
[CrossRef]

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence multiplexing of fiber-optic interferometric sensors,” J. Lightwave Technol. 3, 1062–1071 (1985).
[CrossRef]

Takada, K.

Tur, M.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence multiplexing of fiber-optic interferometric sensors,” J. Lightwave Technol. 3, 1062–1071 (1985).
[CrossRef]

Udd, E.

E. Udd, Fiber Optic Sensors: An Introduction for Engineers and Scientists (Wiley, New York, 1991).

Ulrich, R.

T. Bosselmann, R. Ulrich, “High-accuracy position-sensing with fibre-coupled white-light interferometers,” in Second International Conference on Optical Fiber Sensors, R. T. Kersten, R. Kist, eds., Proc. SPIE514, 361–364 (1984).
[CrossRef]

Urbanczyk, W.

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis of a tandem of white-light fiber-optic interferometers,” Opt. Commun. 135, 1–6 (1997).
[CrossRef]

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis for temperature-compensated white-light interferometric fiber sensors,” Optik 104, 153–158 (1997).

W. Urbańczyk, W. J. Bock, “Visibility of white-light interference patterns for a chain of coherence multiplexed sensors based on highly birefringent fibers,” Opt. Eng. 32, 2100–2106 (1993).
[CrossRef]

W. J. Bock, W. Urbańczyk, “Electronically scanned white-light interferometric sensor for high hydrostatic pressure measurements,” in Proceedings of the Ninth International Conference on Optical Fibre Sensors, Florence, 1993 (The Instituto of Florence, Florence, 1993), pp. 135–138.

Vavassori, P.

V. Gusmeroli, P. Vavassori, M. Martinelli, “A coherence-multiplexed quasi-distributed polarimetric sensor suitable for structural monitoring,” in Sixth International Conference on Optical Fibre Sensors, Paris, 1989, Springer Proceedings in Physics, H. J. Arditty, J. P. Dakin, R. T. Kersten, eds. (Springer-Verlag, Berlin, 1989), Vol. 44, pp. 513–518.

Velluet, M. T.

M. T. Velluet, P. Graindorge, H. J. Arditty, “Fibre optic pressure sensors using white-light interferometry,” in Fiber Optic and Laser Sensors, R. P. De Paula, P. Ramon, E. Udd, eds., Proc. SPIE838, 78–83 (1987).

Wentworth, R. H.

R. H. Wentworth, “Theoretical noise performance of coherence-multiplexed interferometric sensors,” J. Lightwave Technol. 7, 941–956 (1989).
[CrossRef]

K. Blotekjaer, R. H. Wentworth, H. J. Shaw, “Choosing relative optical path delays in series topology interferometric sensor arrays,” J. Lightwave Technol. 5, 229–235 (1987).
[CrossRef]

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence multiplexing of fiber-optic interferometric sensors,” J. Lightwave Technol. 3, 1062–1071 (1985).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1968).

Wozniak, W. A.

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis of a tandem of white-light fiber-optic interferometers,” Opt. Commun. 135, 1–6 (1997).
[CrossRef]

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis for temperature-compensated white-light interferometric fiber sensors,” Optik 104, 153–158 (1997).

Youngquist, R. C.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence multiplexing of fiber-optic interferometric sensors,” J. Lightwave Technol. 3, 1062–1071 (1985).
[CrossRef]

Appl. Opt. (1)

Can. J. Phys. (1)

P. Cielo, C. Delisle, “Multiplexing in optical communications by interferometry with a large path-length difference in white light,” Can. J. Phys. 54, 2322–2331 (1976).
[CrossRef]

J. Lightwave Technol. (3)

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence multiplexing of fiber-optic interferometric sensors,” J. Lightwave Technol. 3, 1062–1071 (1985).
[CrossRef]

K. Blotekjaer, R. H. Wentworth, H. J. Shaw, “Choosing relative optical path delays in series topology interferometric sensor arrays,” J. Lightwave Technol. 5, 229–235 (1987).
[CrossRef]

R. H. Wentworth, “Theoretical noise performance of coherence-multiplexed interferometric sensors,” J. Lightwave Technol. 7, 941–956 (1989).
[CrossRef]

Meas. Sci. Technol. (1)

Y.-J. Rao, D. A. Jackson, “Recent progress in fibre optic low-coherence interferometry,” Meas. Sci. Technol. 7, 981–999 (1996).
[CrossRef]

Opt. Commun. (2)

F. Farahi, T. P. Newson, J. D. C. Jones, D. A. Jackson, “Coherence multiplexing of remote fibre Fabry–Pérot sensing system,” Opt. Commun. 65, 319–321 (1988).
[CrossRef]

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis of a tandem of white-light fiber-optic interferometers,” Opt. Commun. 135, 1–6 (1997).
[CrossRef]

Opt. Eng. (1)

W. Urbańczyk, W. J. Bock, “Visibility of white-light interference patterns for a chain of coherence multiplexed sensors based on highly birefringent fibers,” Opt. Eng. 32, 2100–2106 (1993).
[CrossRef]

Optik (1)

W. Urbańczyk, P. Kurzynowski, W. A. Woźniak, W. J. Bock, “Performance analysis for temperature-compensated white-light interferometric fiber sensors,” Optik 104, 153–158 (1997).

Other (7)

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1968).

M. T. Velluet, P. Graindorge, H. J. Arditty, “Fibre optic pressure sensors using white-light interferometry,” in Fiber Optic and Laser Sensors, R. P. De Paula, P. Ramon, E. Udd, eds., Proc. SPIE838, 78–83 (1987).

W. J. Bock, W. Urbańczyk, “Electronically scanned white-light interferometric sensor for high hydrostatic pressure measurements,” in Proceedings of the Ninth International Conference on Optical Fibre Sensors, Florence, 1993 (The Instituto of Florence, Florence, 1993), pp. 135–138.

E. Udd, Fiber Optic Sensors: An Introduction for Engineers and Scientists (Wiley, New York, 1991).

S. A. Al-Chalabi, B. Culshaw, D. E. N. David, “Partially coherent sources in interferometric sensors,” in Proceedings of IEE First International Conference on Optical Fibre Sensors, London, 1983 (Institute of Electrical Engineers, Stevenage, UK, 1989), pp. 132–135.

T. Bosselmann, R. Ulrich, “High-accuracy position-sensing with fibre-coupled white-light interferometers,” in Second International Conference on Optical Fiber Sensors, R. T. Kersten, R. Kist, eds., Proc. SPIE514, 361–364 (1984).
[CrossRef]

V. Gusmeroli, P. Vavassori, M. Martinelli, “A coherence-multiplexed quasi-distributed polarimetric sensor suitable for structural monitoring,” in Sixth International Conference on Optical Fibre Sensors, Paris, 1989, Springer Proceedings in Physics, H. J. Arditty, J. P. Dakin, R. T. Kersten, eds. (Springer-Verlag, Berlin, 1989), Vol. 44, pp. 513–518.

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Figures (3)

Fig. 1
Fig. 1

Typical scheme of a white-light fiber-optic interferometric system.

Fig. 2
Fig. 2

Scheme of fiber-optic interferometric system made of (a) HB fiber, (b) single-mode fiber with fiber couplers (FC). Note that the orientation of the main axes of each fiber is rotated relative to the preceding element.

Fig. 3
Fig. 3

Calculation method. Positions of elements that produce visible contrasts (black) among elements with group delays τn ≫ τs (white) between the low-coherent source S and the analyzer A: (a) average intensity; (b) detection of pattern produced by single kth sensor; (c) detection of cross pattern produced by adjacent kth and (k + 1)th sensors; (d) detection of cross patterns produced by separated kth and mth sensors.

Equations (41)

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

Knω=JnωKn-1ωJnω,
Iout=0trKoutωdω,
Iout=I¯out+pI¯outVp cosδ¯pΓτp,
Ĩout=pI¯outVp cosδ¯pΓτp=pi˜p cosδ¯pΓτp,
Vp=i˜p/I¯out.
Jnω=cos αnsin αn-sin αn exp-iδnωcos αn exp-iδnω.
Knω=KxxnωKxynωKyxnωKyynω,
KxxnKyyn=0.51+cos2αn1-cos2αn2-cos2αn1+cos2αnKxxn-1Kyyn-1+0.5 sin2αn11-1-1Kxyn-1Kyxn-1,
KxynKyxn=expiδn00exp-iδn×0.5cos2αn+1cos2αn-1cos2αn-1cos2αn+1Kxyn-1Kyxn-1-0.5 sin2αn1-11-1Kxxn-1Kyyn-1.
τnτs,
Kinω=Einω1000,
KxxNKyyN=n=1N0.51+cos2αn1-cos2αn1-cos2αn1+cos2αnKxxinKyyin=0.51+P1,N1-P1,N1-P1,N1+P1,NKxxinKyyin=0.51+P1,N1-P1,NEin,
Pk,1=m=k1cos2αm.
Kout=JAKNJA=0.51+P1,N+1Ein000,
JA=cos αN+1sin αN+100.
Einωdω=1,
I¯out=0.51+P1,N+1.
Kxxk-1Kyyk-1=0.51+P1,k-11-P1,k-1Ein,
KxxkKyyk=0.51+P1,k1-P1,kEin,
KxykKyxk=-0.5P1,k-1 sin2αkexpiδkexp-iδkEin.
Kxxk+1Kyyk+1=0.51+P1,k+11-P1,k+1Ein-0.5P1,k-1×sin2αksin2αk+1cosδk1-1Ein.
KxxNKyyN=n=k+2N0.51+cos2αn1-cos2αn1-cos2αn1+cos2αn×Kxxk+1Kyyk+1,
Iout=0.5I01+P1,N+1-0.5I0P1,k-1×sin2αksin2αk+1Pk+2,N+1 cosδ¯kΓτk,
Vk=P1,k-1 sin2αksin2αk+1Pk+2,N+11+P1,N+1=P1,N+11+P1,N+1 tan2αktan2αk+1.
Kxyk+1Kyxk+1=-0.5P1,k-1 sin2αkcos2 αk+1 expiδk+1+δk-sin2 αk+1 expiδkn+1-δkcos2 αk+1 exp-iδk+1+δk-sin2 αk+1 exp-iδk+1-δkEin,
Kxxk+2Kyyk+2=-0.5P1,k-1 sin2αkcos2αk+1sin2αk+2×cosδk+1+δk1-1Ein+0.5P1,k-1 sin2αksin2αk+1sin2αk+2×cosδk+1-δk1-1Ein.
Vk,+k+1=P1,k-1 sin2αkcos2 αk+1 sin2αk+2Pk+3,N+11+P1,N+1,
Vk,-k+1=P1,k-1 sin2αksin2 αk+1 sin2αk+2Pk+3,N+11+P1,N+1,
Vk,±k+1=P1,N+11+P1,N+1 tan2αk×0.5sec2αk+1±1]tan2αk+2,
D=sec2αk+1±1.
V=P1,N+11+P1,N+1 tan 2αk0.5K-1×i=1K-1sec 2αk+i±1tan 2αk+K,
Vk,±m=0.5P1,N+11+P1,N+1 tan2αk×tan2αk+1tan2αmtan2αm+1,
V=0.5K-1P1,N+11+P1,N+1i=1Ktan2αkitan2αki+1,
V=0.5K-11+P1,N+1P1,N+1K-1i=1KVki,
V=0.5K-1P1,N+11+P1,N+10.5i=1Kmi-1i=1K×tan2αkij=1mi-1sec2αki+j±1tan2αki+m,
Īout=i=1K0.51+Pki+1,ki+1,
Vki<j<ki+1=Pki+1,ki+1 tan2αjtan2αj+11+Pki+1,ki+1,
Vki<j,±j+1<ki+1=Pki+1,ki+11+Pki+1,ki+1 tan2αj×0.5sec2αj+1±1tan2αj+2,
Vki<j,±m<ki+1=0.5Pki+1,ki+11+Pki+1,ki+1 tan2αjtan2αj+1×tan2αmtan2αm+1.
Vki<j<ki+1,k1<m<kl+1=0.5Pki+1,ki+1 tan2αjtan2αj+11+Pki+1,ki+1×Pkl+1,kl+1 tan2αmtan2αm+11+Pkl+1,kl+1,
Vki<j<ki+1,k1<m<kl+1=0.5Vki<j<ki+1Vkl<m<kl+1.

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