Abstract

A birefringent launching fiber (BLF) is proposed as a means of reducing backscattered power fluctuations in single-mode fibers that occur during polarization-sensitive measurements. The BLF consists of birefringent single-mode fibers spliced in series. The arrangement of the birefringent single-mode fibers is discussed theoretically and experimentally. Results show the usefulness of the BLF in an optical-time-domain reflectometer.

© 1985 Optical Society of America

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References

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  1. M. K. Barnoski, S. M. Jensen, “Fiber waveguides: a novel technique for investigating attenuation characteristics,” Appl. Opt. 15, 2112–2115 (1976).
    [CrossRef] [PubMed]
  2. S. D. Personick, “Photon probe—an optical time domain reflectometer,” Bell. Syst. Tech. J. 56, 355–356 (1977).
    [CrossRef]
  3. A. J. Rogers, “Polarization optical time domain reflectometery,” Electron. Lett. 16, 489–490 (1980).
    [CrossRef]
  4. A. H. Hartog, D. N. Payne, A. J. Conduit, “Polarization optical time-domain reflectometry: experimental results and application to loss and birefringence measurements in single-mode fibers,” in Proceedings of the Sixth European Conference on Optical Communications (Institution of Electrical Engineers, York, England, 1980), p. 5–8 (postdeadline papers supplement).
  5. B. Y. Kim, S. S. Choi, “Backscattering measurement of bending-induced birefringence in single-mode fibers,” Electron. Lett. 17, 193–194 (1981).
    [CrossRef]
  6. M. Nakazawa, T. Horiguchi, M. Tokuda, N. Uchida, “Polarization beat length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–515 (1981).
    [CrossRef]
  7. K. Aoyama, K. Nakagawa, T. Itoh, “Optical time domain reflectometry in a single-mode fiber,” IEEE J. Quantum Electron. QE-17, 862–868 (1981).
    [CrossRef]
  8. B. Lyot, Annates de I’Observatoire d’Astronomie de Paris (Mendon, Paris, 1929), Vol. VIII, p. 102.
  9. B. H. Billings, “A monochromatic depolarizer,” J. Opt. Soc. Am 41, 966–975 (1951).
    [CrossRef]
  10. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am 31, 488–493 (1941).
    [CrossRef]
  11. E. Brinkmeyer, “Forward-backward transmission in birefringent single-mode fibers: interpretation of polarization-sensitive measurements,” Opt. Lett. 6, 575–577 (1981).
    [CrossRef] [PubMed]
  12. S. Ezekiel, H. J. Arditty, Fiber-Optic Rotation Sensors (Springer-Verlag, Berlin, 1982), pp. 14–15.
  13. T. Horiguchi, K. Suzuki, N. Shibata, M. Nakazawa, S. Seikai, “A novel technique for reducing polarization noise in optical time domain reflectometers for single-mode fibers,” IEEE J. Lightwave Technol. (to be published).
  14. T. Asano, T. Okumura, M. Aizawa, “1.3-μ m band high power BH laser on p– InP substrate,” in Proceedings of the Ninth IEEE International Semiconductor Laser Conference (Institute of Electrical and Electronics Engineers, New York, 1984).
  15. A. R. Resinger, C. D. David, K. L. Lawley, A. Yariv, “Coherence of a room-temperature cw GaAs/GaAlAs injection laser,” IEEE J. Quantum Electron. QE-15, 1382–1387 (1979).
    [CrossRef]
  16. N. Shibata, M. Tsubokawa, S. Seikai, “Measurements of polarization mode dispersion by an optical heterodyne detection,” Electron. Lett. 20, 1055–1057 (1984).
    [CrossRef]
  17. T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low-loss single-polarization fibers with asymmetrical strain birefringence,” Electron. Lett. 17, 530–531 (1981).
    [CrossRef]
  18. K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot depolarizers with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
    [CrossRef]
  19. W. K. Burns, “Degree of polarization in the Lyot depolarizer,” IEEE J. Lightwave Technol. LT-1, 475–479 (1983).
    [CrossRef]

1984 (1)

N. Shibata, M. Tsubokawa, S. Seikai, “Measurements of polarization mode dispersion by an optical heterodyne detection,” Electron. Lett. 20, 1055–1057 (1984).
[CrossRef]

1983 (2)

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot depolarizers with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

W. K. Burns, “Degree of polarization in the Lyot depolarizer,” IEEE J. Lightwave Technol. LT-1, 475–479 (1983).
[CrossRef]

1981 (5)

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low-loss single-polarization fibers with asymmetrical strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

B. Y. Kim, S. S. Choi, “Backscattering measurement of bending-induced birefringence in single-mode fibers,” Electron. Lett. 17, 193–194 (1981).
[CrossRef]

M. Nakazawa, T. Horiguchi, M. Tokuda, N. Uchida, “Polarization beat length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–515 (1981).
[CrossRef]

K. Aoyama, K. Nakagawa, T. Itoh, “Optical time domain reflectometry in a single-mode fiber,” IEEE J. Quantum Electron. QE-17, 862–868 (1981).
[CrossRef]

E. Brinkmeyer, “Forward-backward transmission in birefringent single-mode fibers: interpretation of polarization-sensitive measurements,” Opt. Lett. 6, 575–577 (1981).
[CrossRef] [PubMed]

1980 (1)

A. J. Rogers, “Polarization optical time domain reflectometery,” Electron. Lett. 16, 489–490 (1980).
[CrossRef]

1979 (1)

A. R. Resinger, C. D. David, K. L. Lawley, A. Yariv, “Coherence of a room-temperature cw GaAs/GaAlAs injection laser,” IEEE J. Quantum Electron. QE-15, 1382–1387 (1979).
[CrossRef]

1977 (1)

S. D. Personick, “Photon probe—an optical time domain reflectometer,” Bell. Syst. Tech. J. 56, 355–356 (1977).
[CrossRef]

1976 (1)

1951 (1)

B. H. Billings, “A monochromatic depolarizer,” J. Opt. Soc. Am 41, 966–975 (1951).
[CrossRef]

1941 (1)

R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am 31, 488–493 (1941).
[CrossRef]

Aizawa, M.

T. Asano, T. Okumura, M. Aizawa, “1.3-μ m band high power BH laser on p– InP substrate,” in Proceedings of the Ninth IEEE International Semiconductor Laser Conference (Institute of Electrical and Electronics Engineers, New York, 1984).

Aoyama, K.

K. Aoyama, K. Nakagawa, T. Itoh, “Optical time domain reflectometry in a single-mode fiber,” IEEE J. Quantum Electron. QE-17, 862–868 (1981).
[CrossRef]

Arditty, H. J.

S. Ezekiel, H. J. Arditty, Fiber-Optic Rotation Sensors (Springer-Verlag, Berlin, 1982), pp. 14–15.

Asano, T.

T. Asano, T. Okumura, M. Aizawa, “1.3-μ m band high power BH laser on p– InP substrate,” in Proceedings of the Ninth IEEE International Semiconductor Laser Conference (Institute of Electrical and Electronics Engineers, New York, 1984).

Barnoski, M. K.

Billings, B. H.

B. H. Billings, “A monochromatic depolarizer,” J. Opt. Soc. Am 41, 966–975 (1951).
[CrossRef]

Böhm, K.

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot depolarizers with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

Brinkmeyer, E.

Burns, W. K.

W. K. Burns, “Degree of polarization in the Lyot depolarizer,” IEEE J. Lightwave Technol. LT-1, 475–479 (1983).
[CrossRef]

Choi, S. S.

B. Y. Kim, S. S. Choi, “Backscattering measurement of bending-induced birefringence in single-mode fibers,” Electron. Lett. 17, 193–194 (1981).
[CrossRef]

Conduit, A. J.

A. H. Hartog, D. N. Payne, A. J. Conduit, “Polarization optical time-domain reflectometry: experimental results and application to loss and birefringence measurements in single-mode fibers,” in Proceedings of the Sixth European Conference on Optical Communications (Institution of Electrical Engineers, York, England, 1980), p. 5–8 (postdeadline papers supplement).

David, C. D.

A. R. Resinger, C. D. David, K. L. Lawley, A. Yariv, “Coherence of a room-temperature cw GaAs/GaAlAs injection laser,” IEEE J. Quantum Electron. QE-15, 1382–1387 (1979).
[CrossRef]

Edahiro, T.

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low-loss single-polarization fibers with asymmetrical strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Ezekiel, S.

S. Ezekiel, H. J. Arditty, Fiber-Optic Rotation Sensors (Springer-Verlag, Berlin, 1982), pp. 14–15.

Hartog, A. H.

A. H. Hartog, D. N. Payne, A. J. Conduit, “Polarization optical time-domain reflectometry: experimental results and application to loss and birefringence measurements in single-mode fibers,” in Proceedings of the Sixth European Conference on Optical Communications (Institution of Electrical Engineers, York, England, 1980), p. 5–8 (postdeadline papers supplement).

Horiguchi, T.

M. Nakazawa, T. Horiguchi, M. Tokuda, N. Uchida, “Polarization beat length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–515 (1981).
[CrossRef]

T. Horiguchi, K. Suzuki, N. Shibata, M. Nakazawa, S. Seikai, “A novel technique for reducing polarization noise in optical time domain reflectometers for single-mode fibers,” IEEE J. Lightwave Technol. (to be published).

Hosaka, T.

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low-loss single-polarization fibers with asymmetrical strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Itoh, T.

K. Aoyama, K. Nakagawa, T. Itoh, “Optical time domain reflectometry in a single-mode fiber,” IEEE J. Quantum Electron. QE-17, 862–868 (1981).
[CrossRef]

Jensen, S. M.

Jones, R. C.

R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am 31, 488–493 (1941).
[CrossRef]

Kim, B. Y.

B. Y. Kim, S. S. Choi, “Backscattering measurement of bending-induced birefringence in single-mode fibers,” Electron. Lett. 17, 193–194 (1981).
[CrossRef]

Lawley, K. L.

A. R. Resinger, C. D. David, K. L. Lawley, A. Yariv, “Coherence of a room-temperature cw GaAs/GaAlAs injection laser,” IEEE J. Quantum Electron. QE-15, 1382–1387 (1979).
[CrossRef]

Lyot, B.

B. Lyot, Annates de I’Observatoire d’Astronomie de Paris (Mendon, Paris, 1929), Vol. VIII, p. 102.

Miya, T.

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low-loss single-polarization fibers with asymmetrical strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Nakagawa, K.

K. Aoyama, K. Nakagawa, T. Itoh, “Optical time domain reflectometry in a single-mode fiber,” IEEE J. Quantum Electron. QE-17, 862–868 (1981).
[CrossRef]

Nakazawa, M.

M. Nakazawa, T. Horiguchi, M. Tokuda, N. Uchida, “Polarization beat length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–515 (1981).
[CrossRef]

T. Horiguchi, K. Suzuki, N. Shibata, M. Nakazawa, S. Seikai, “A novel technique for reducing polarization noise in optical time domain reflectometers for single-mode fibers,” IEEE J. Lightwave Technol. (to be published).

Okamoto, K.

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low-loss single-polarization fibers with asymmetrical strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Okumura, T.

T. Asano, T. Okumura, M. Aizawa, “1.3-μ m band high power BH laser on p– InP substrate,” in Proceedings of the Ninth IEEE International Semiconductor Laser Conference (Institute of Electrical and Electronics Engineers, New York, 1984).

Payne, D. N.

A. H. Hartog, D. N. Payne, A. J. Conduit, “Polarization optical time-domain reflectometry: experimental results and application to loss and birefringence measurements in single-mode fibers,” in Proceedings of the Sixth European Conference on Optical Communications (Institution of Electrical Engineers, York, England, 1980), p. 5–8 (postdeadline papers supplement).

Personick, S. D.

S. D. Personick, “Photon probe—an optical time domain reflectometer,” Bell. Syst. Tech. J. 56, 355–356 (1977).
[CrossRef]

Petermann, K.

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot depolarizers with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

Resinger, A. R.

A. R. Resinger, C. D. David, K. L. Lawley, A. Yariv, “Coherence of a room-temperature cw GaAs/GaAlAs injection laser,” IEEE J. Quantum Electron. QE-15, 1382–1387 (1979).
[CrossRef]

Rogers, A. J.

A. J. Rogers, “Polarization optical time domain reflectometery,” Electron. Lett. 16, 489–490 (1980).
[CrossRef]

Sasaki, Y.

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low-loss single-polarization fibers with asymmetrical strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

Seikai, S.

N. Shibata, M. Tsubokawa, S. Seikai, “Measurements of polarization mode dispersion by an optical heterodyne detection,” Electron. Lett. 20, 1055–1057 (1984).
[CrossRef]

T. Horiguchi, K. Suzuki, N. Shibata, M. Nakazawa, S. Seikai, “A novel technique for reducing polarization noise in optical time domain reflectometers for single-mode fibers,” IEEE J. Lightwave Technol. (to be published).

Shibata, N.

N. Shibata, M. Tsubokawa, S. Seikai, “Measurements of polarization mode dispersion by an optical heterodyne detection,” Electron. Lett. 20, 1055–1057 (1984).
[CrossRef]

T. Horiguchi, K. Suzuki, N. Shibata, M. Nakazawa, S. Seikai, “A novel technique for reducing polarization noise in optical time domain reflectometers for single-mode fibers,” IEEE J. Lightwave Technol. (to be published).

Suzuki, K.

T. Horiguchi, K. Suzuki, N. Shibata, M. Nakazawa, S. Seikai, “A novel technique for reducing polarization noise in optical time domain reflectometers for single-mode fibers,” IEEE J. Lightwave Technol. (to be published).

Tokuda, M.

M. Nakazawa, T. Horiguchi, M. Tokuda, N. Uchida, “Polarization beat length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–515 (1981).
[CrossRef]

Tsubokawa, M.

N. Shibata, M. Tsubokawa, S. Seikai, “Measurements of polarization mode dispersion by an optical heterodyne detection,” Electron. Lett. 20, 1055–1057 (1984).
[CrossRef]

Uchida, N.

M. Nakazawa, T. Horiguchi, M. Tokuda, N. Uchida, “Polarization beat length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–515 (1981).
[CrossRef]

Weidel, E.

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot depolarizers with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

Yariv, A.

A. R. Resinger, C. D. David, K. L. Lawley, A. Yariv, “Coherence of a room-temperature cw GaAs/GaAlAs injection laser,” IEEE J. Quantum Electron. QE-15, 1382–1387 (1979).
[CrossRef]

Appl. Opt. (1)

Bell. Syst. Tech. J. (1)

S. D. Personick, “Photon probe—an optical time domain reflectometer,” Bell. Syst. Tech. J. 56, 355–356 (1977).
[CrossRef]

Electron. Lett. (5)

A. J. Rogers, “Polarization optical time domain reflectometery,” Electron. Lett. 16, 489–490 (1980).
[CrossRef]

B. Y. Kim, S. S. Choi, “Backscattering measurement of bending-induced birefringence in single-mode fibers,” Electron. Lett. 17, 193–194 (1981).
[CrossRef]

M. Nakazawa, T. Horiguchi, M. Tokuda, N. Uchida, “Polarization beat length measurement in a single-mode optical fiber by backward Rayleigh scattering,” Electron. Lett. 17, 513–515 (1981).
[CrossRef]

N. Shibata, M. Tsubokawa, S. Seikai, “Measurements of polarization mode dispersion by an optical heterodyne detection,” Electron. Lett. 20, 1055–1057 (1984).
[CrossRef]

T. Hosaka, K. Okamoto, T. Miya, Y. Sasaki, T. Edahiro, “Low-loss single-polarization fibers with asymmetrical strain birefringence,” Electron. Lett. 17, 530–531 (1981).
[CrossRef]

IEEE J. Lightwave Technol. (2)

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot depolarizers with birefringent single-mode fibers,” IEEE J. Lightwave Technol. LT-1, 71–74 (1983).
[CrossRef]

W. K. Burns, “Degree of polarization in the Lyot depolarizer,” IEEE J. Lightwave Technol. LT-1, 475–479 (1983).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. Aoyama, K. Nakagawa, T. Itoh, “Optical time domain reflectometry in a single-mode fiber,” IEEE J. Quantum Electron. QE-17, 862–868 (1981).
[CrossRef]

A. R. Resinger, C. D. David, K. L. Lawley, A. Yariv, “Coherence of a room-temperature cw GaAs/GaAlAs injection laser,” IEEE J. Quantum Electron. QE-15, 1382–1387 (1979).
[CrossRef]

J. Opt. Soc. Am (2)

B. H. Billings, “A monochromatic depolarizer,” J. Opt. Soc. Am 41, 966–975 (1951).
[CrossRef]

R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am 31, 488–493 (1941).
[CrossRef]

Opt. Lett. (1)

Other (5)

S. Ezekiel, H. J. Arditty, Fiber-Optic Rotation Sensors (Springer-Verlag, Berlin, 1982), pp. 14–15.

T. Horiguchi, K. Suzuki, N. Shibata, M. Nakazawa, S. Seikai, “A novel technique for reducing polarization noise in optical time domain reflectometers for single-mode fibers,” IEEE J. Lightwave Technol. (to be published).

T. Asano, T. Okumura, M. Aizawa, “1.3-μ m band high power BH laser on p– InP substrate,” in Proceedings of the Ninth IEEE International Semiconductor Laser Conference (Institute of Electrical and Electronics Engineers, New York, 1984).

B. Lyot, Annates de I’Observatoire d’Astronomie de Paris (Mendon, Paris, 1929), Vol. VIII, p. 102.

A. H. Hartog, D. N. Payne, A. J. Conduit, “Polarization optical time-domain reflectometry: experimental results and application to loss and birefringence measurements in single-mode fibers,” in Proceedings of the Sixth European Conference on Optical Communications (Institution of Electrical Engineers, York, England, 1980), p. 5–8 (postdeadline papers supplement).

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Figures (7)

Fig. 1
Fig. 1

Configuration of an OTDR and a BLF assumed in the calculation.

Fig. 2
Fig. 2

Relation between the group-delay differences ϕ ˙ 1 and ϕ ˙ 2 of the birefringent single-mode fibers of which the BLF is made. Hatched areas satisfy expressions (23).

Fig. 3
Fig. 3

Experimental setup for measuring backward Rayleigh scattering of a single-mode fiber. (APD, avalanche photodiode.)

Fig. 4
Fig. 4

Degree of coherence of the laser diode used in the experiments.

Fig. 5
Fig. 5

(a) Backscattered power measured without using the BLF and (b) backscattered power fluctuation normalized by an optical dc component.

Fig. 6
Fig. 6

Normalized backscattering fluctuation for various values of the polarization direction of the incident light when using the BLF’s with (a) θ1 = 27° and (b) θ1 = 45°.

Fig. 7
Fig. 7

Normalized backscattering fluctuation at 2.1 km in the single-mode fiber as a function of the polarization direction of the incident light, θp, when using the BLF’s with θ1 = 27° and 45°.

Equations (37)

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M i = R ( θ i ) F i F 1 R ( θ 0 ) F 0 R ( θ 0 ) F 1 F i R ( θ i ) = ( a i b i c i d i ) ( i = 0 , 1 , , N 1 ) ,
R ( θ ) = ( cos θ sin θ sin θ cos θ ) .
F i = ( exp ( j ϕ i / 2 ) 0 0 exp ( j ϕ i / 2 ) ) ( i = 1 , 2 , , N ) ,
b i = c i .
| a i | 2 = | d i | 2 = r | b i | 2 .
( E s x E s y ) = F N M N 1 F N ( cos θ p sin θ p ) .
| E s | 2 = | E s x cos θ a + E s y sin θ a | 2 = r sin 2 ( θ p + θ a ) + g N + h N | a N 1 | 2 ,
g N = [ a N 1 d N 1 * exp ( j 2 ϕ N ) sin ( 2 θ p ) sin ( 2 θ a ) / 4 + a n 1 b N 1 * exp ( j ϕ N ) sin ( θ p + θ a ) cos θ p cos θ a + b N 1 d N 1 * exp ( j ϕ N ) sin ( θ p + θ a ) sin θ p sin θ a ] + c . c .
h N = cos ( 2 θ p ) cos ( 2 θ a ) sin ( 2 θ p ) sin ( 2 θ a ) / 2 ,
M i = R ( θ i ) F i M i 1 F i R ( θ i ) ,
| a i | 2 = r sin 2 ( 2 θ i ) + g i + h i | a i 1 | 2 ( i = 1 , 2 , , N 1 ) ,
g i = [ a i 1 d i 1 * exp ( j 2 ϕ i ) sin 2 θ i cos 2 θ i a i 1 b i 1 * exp ( j ϕ i ) sin ( 2 θ i ) cos 2 θ i b i 1 d i 1 * exp ( j ϕ i ) sin ( 2 θ i ) sin 2 θ i ] + c . c .
h i = h ( θ i ) = cos 2 ( 2 θ i ) sin 2 ( 2 θ i ) / 2 .
| E s | 2 = I 1 + I 2 + I 3 ,
I 1 = r { i = 1 N 1 [ sin 2 ( 2 θ i ) j = i + 1 N h j ] + sin 2 ( θ a + θ p ) } ,
I 2 = i = 1 N 1 ( g i j = i + 1 N h j ) + g N ,
I 3 = | a 0 | 2 i = 1 N h i .
| E s | 2 ¯ = Q ( ω ) | E s | 2 d ω = I ¯ 1 + I ¯ 2 + I ¯ 3 .
Φ m = ϕ 0 m + n = 1 N u n ϕ n ,
I ¯ 2 = m C m exp ( j Φ m ) ¯ = 2 m C m ( ω 0 ) γ ( Φ ˙ m ) cos Φ m ,
γ ( τ ) = Q ( ω ) cos { ( ω ω 0 ) τ } d ω .
γ ( τ ) = exp [ ln 2 ( τ / t c ) 2 ]
γ ( τ ) = exp [ ln 2 ( τ / t c ) ] ,
| ϕ ˙ 0 m + n = 1 N u n ϕ ˙ n | A t c ,
ϕ ˙ 1 A t c + ( ϕ ˙ 0 ) max = T c
ϕ ˙ 1 , ϕ ˙ 2 T c ,
| ϕ ˙ 1 ϕ ˙ 2 | T c ,
| 2 ϕ ˙ 1 ϕ ˙ 2 | T c ,
| ϕ ˙ 1 2 ϕ ˙ 2 | T c ,
ϕ ˙ 1 T c .
ϕ ˙ 2 2 ϕ ˙ 1 + T c 3 T c .
ϕ ˙ N 3 N 1 T c .
h i = 0 ( i = 1 , 2 , , N ) .
I ¯ 3 = | a 0 | 2 ¯ ( h i θ i θ i = θ opt × δ θ ) k = | a 0 | 2 ¯ ( 2 δ θ ) k ,
I 1 = r { i = 1 k [ sin 2 ( 2 θ i ) j = i + 1 N h j ] + sin 2 ( θ p + θ a ) } .
sin 2 ( 2 θ opt ) = 2 h i / 3 + sin 2 ( 2 θ i ) = 2 / 3 ,
I 1 = r [ 1 + cos 2 ( θ p θ a ) ] / 3 .

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