Abstract

Theoretical and experimental investigations are described for determining the transmission characteristics of a multimode fiber with microbending for coherent and partially coherent illumination. The measured values of the average excess power loss are shown to be in close agreement with the theory. Also, an estimate of the excess transient loss due to mode coupling is found to be in good agreement with previously published data. Mode–mode interference is shown to be the cause of temporal fluctuations in the microbending loss, from which expressions for modal noise and baseband/subcarrier nonlinearity are derived on a statistical basis. For a given overall loss, the results show that many uniformly distributed small amplitude microbends cause much less modal noise and distortion than a few large amplitude microbends.

© 1985 Optical Society of America

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References

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  1. R. E. Epworth, “Modal Noise—Causes and Cures,” Laser Focus 17, 109 (1981).
  2. K. Petermann, “Nonlinear Distortions and Noise in Optical Communication Systems due to Fiber Connectors,” IEEE J. Quantum Electron. QE-16, 761 (1980).
    [CrossRef]
  3. S. Das, C. G. Englefield, P. A. Goud, “Modal Noise and Distortion Caused by a Longitudinal Gap Between Two Multimode Fibers,” Appl. Opt. 23, 1110 (1984).
    [CrossRef] [PubMed]
  4. E. G. Rawson, J. W. Goodman, R. E. Norton, “Experimental and Analytical Study of Modal Noise in Optical Fibers,” in Technical Digest, Sixth European Conference on Optical Communication, York (1980).
  5. D. R. Hjelme, A. R. Mickelson, “Microbending and Modal Noise,” Appl. Opt. 23, 3874 (1983).
    [CrossRef]
  6. S. Das, C. G. Englefield, P. A. Goud, “Polarization Modal Noise due to Microbending of Single-Mode Fibers,” in Technical Digest, Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1985), paper TuQ9.
  7. R. Olshansky, D. A. Nolan, “Mode-Dependent Attenuation of Optical Fibers: Excess Loss,” Appl. Opt. 15, 1045 (1976).
    [CrossRef] [PubMed]
  8. H. Unger, Planar Optical Waveguides and Fibers (Oxford U. P., London, 1977).
  9. S. Kawakami, H. Tanji, “Evolution of Power Distribution in Graded-Index Fibers,” Electron. Lett. 19, 100 (1983).
    [CrossRef]
  10. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  11. D. Marcuse, “Microbending Losses of Single-Mode, Step-Index and Multimode, Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 937 (1976).
  12. R. Olshansky, “Distortion Losses in Cabled Optical Fibers,” Appl. Opt. 14, 20 (1975).
    [PubMed]
  13. D. Marcuse, “Mode Mixing with Reduced Losses in Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 777 (1976).
  14. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), p. 360.
  15. M. D. Rourke, “Measurement of the Insertion Loss of a Single Microbend,” Opt. Lett. 6, 440 (1981).
    [PubMed]
  16. R. Olshansky, M. G. Blankenship, D. B. Keck, “Length-Dependent Attenuation Measurements in Graded-Index Fibers,” in Technical Digest, Second European Conference on Optical Communication, Paris (1976), p. 111.
  17. W. B. Gardner, “Microbending Loss in Optical Fibers,” Bell Syst. Tech. J. 54, 457 (1975).
  18. D. J. Eccleston, “Environmental Testing of UV-Cured Acrylate Coated Fibers,” Corning Application Notes (1982).
  19. D. Gloge, “Optical-Fiber Packaging and its Influence on Fiber Straightness and Loss,” Bell Syst. Tech. J. 54, 245 (1975).
  20. D. Marcuse, H. M. Presby, “Mode Coupling in an Optical Fiber with Core Distortions,” Bell Syst. Tech. J. 54, 3 (1975).
  21. K. J. Blow, N. J. Doran, S. Hornung, “Power Spectrum of Microbends in Monomode Optical Fibres,” Electron. Lett. 18, 448 (1982).
    [CrossRef]
  22. N. Yoshizawa, T. Yabuta, K. Noguchi, “Residual Nylon-Jacketed Fiber Shrinkage Caused by Cooling,” Electron. Lett. 19, 411 (1983).
    [CrossRef]
  23. Y. Murakami, K. Ishihara, Y. Negishi, N. Kojima, “Micro-bending Losses of P2O5-Doped Graded-Index Multimode Fiber,” Electron. Lett. 18, 774 (1982).
    [CrossRef]
  24. K. Stubkjaer, “Distortion of Light Signals Transmitted via Multimode Graded-Index Fibers,” Electron. Lett. 15, 797 (1979).
    [CrossRef]
  25. G. J. Meslener, “Dispersion-Induced Harmonic Distortion in Fiber Systems Using Coherent Light Sources,” in Technical Digest, Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1982), paper THFF4.
  26. K. Stubkjaer, IBM Watson Research Center; private communication.
  27. S. Kobayashi, Y. Yamamoto, T. Kimura, “Modulation Frequency Characteristics of Directly Optical Frequency Modulated AlGaAs Semiconductor Laser,” Electron. Lett. 17, 350 (1981).
    [CrossRef]

1984

1983

D. R. Hjelme, A. R. Mickelson, “Microbending and Modal Noise,” Appl. Opt. 23, 3874 (1983).
[CrossRef]

S. Kawakami, H. Tanji, “Evolution of Power Distribution in Graded-Index Fibers,” Electron. Lett. 19, 100 (1983).
[CrossRef]

N. Yoshizawa, T. Yabuta, K. Noguchi, “Residual Nylon-Jacketed Fiber Shrinkage Caused by Cooling,” Electron. Lett. 19, 411 (1983).
[CrossRef]

1982

Y. Murakami, K. Ishihara, Y. Negishi, N. Kojima, “Micro-bending Losses of P2O5-Doped Graded-Index Multimode Fiber,” Electron. Lett. 18, 774 (1982).
[CrossRef]

K. J. Blow, N. J. Doran, S. Hornung, “Power Spectrum of Microbends in Monomode Optical Fibres,” Electron. Lett. 18, 448 (1982).
[CrossRef]

1981

S. Kobayashi, Y. Yamamoto, T. Kimura, “Modulation Frequency Characteristics of Directly Optical Frequency Modulated AlGaAs Semiconductor Laser,” Electron. Lett. 17, 350 (1981).
[CrossRef]

M. D. Rourke, “Measurement of the Insertion Loss of a Single Microbend,” Opt. Lett. 6, 440 (1981).
[PubMed]

R. E. Epworth, “Modal Noise—Causes and Cures,” Laser Focus 17, 109 (1981).

1980

K. Petermann, “Nonlinear Distortions and Noise in Optical Communication Systems due to Fiber Connectors,” IEEE J. Quantum Electron. QE-16, 761 (1980).
[CrossRef]

1979

K. Stubkjaer, “Distortion of Light Signals Transmitted via Multimode Graded-Index Fibers,” Electron. Lett. 15, 797 (1979).
[CrossRef]

1976

D. Marcuse, “Microbending Losses of Single-Mode, Step-Index and Multimode, Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 937 (1976).

D. Marcuse, “Mode Mixing with Reduced Losses in Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 777 (1976).

R. Olshansky, D. A. Nolan, “Mode-Dependent Attenuation of Optical Fibers: Excess Loss,” Appl. Opt. 15, 1045 (1976).
[CrossRef] [PubMed]

1975

R. Olshansky, “Distortion Losses in Cabled Optical Fibers,” Appl. Opt. 14, 20 (1975).
[PubMed]

W. B. Gardner, “Microbending Loss in Optical Fibers,” Bell Syst. Tech. J. 54, 457 (1975).

D. Gloge, “Optical-Fiber Packaging and its Influence on Fiber Straightness and Loss,” Bell Syst. Tech. J. 54, 245 (1975).

D. Marcuse, H. M. Presby, “Mode Coupling in an Optical Fiber with Core Distortions,” Bell Syst. Tech. J. 54, 3 (1975).

Blankenship, M. G.

R. Olshansky, M. G. Blankenship, D. B. Keck, “Length-Dependent Attenuation Measurements in Graded-Index Fibers,” in Technical Digest, Second European Conference on Optical Communication, Paris (1976), p. 111.

Blow, K. J.

K. J. Blow, N. J. Doran, S. Hornung, “Power Spectrum of Microbends in Monomode Optical Fibres,” Electron. Lett. 18, 448 (1982).
[CrossRef]

Das, S.

S. Das, C. G. Englefield, P. A. Goud, “Modal Noise and Distortion Caused by a Longitudinal Gap Between Two Multimode Fibers,” Appl. Opt. 23, 1110 (1984).
[CrossRef] [PubMed]

S. Das, C. G. Englefield, P. A. Goud, “Polarization Modal Noise due to Microbending of Single-Mode Fibers,” in Technical Digest, Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1985), paper TuQ9.

Doran, N. J.

K. J. Blow, N. J. Doran, S. Hornung, “Power Spectrum of Microbends in Monomode Optical Fibres,” Electron. Lett. 18, 448 (1982).
[CrossRef]

Eccleston, D. J.

D. J. Eccleston, “Environmental Testing of UV-Cured Acrylate Coated Fibers,” Corning Application Notes (1982).

Englefield, C. G.

S. Das, C. G. Englefield, P. A. Goud, “Modal Noise and Distortion Caused by a Longitudinal Gap Between Two Multimode Fibers,” Appl. Opt. 23, 1110 (1984).
[CrossRef] [PubMed]

S. Das, C. G. Englefield, P. A. Goud, “Polarization Modal Noise due to Microbending of Single-Mode Fibers,” in Technical Digest, Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1985), paper TuQ9.

Epworth, R. E.

R. E. Epworth, “Modal Noise—Causes and Cures,” Laser Focus 17, 109 (1981).

Gardner, W. B.

W. B. Gardner, “Microbending Loss in Optical Fibers,” Bell Syst. Tech. J. 54, 457 (1975).

Gloge, D.

D. Gloge, “Optical-Fiber Packaging and its Influence on Fiber Straightness and Loss,” Bell Syst. Tech. J. 54, 245 (1975).

Goodman, J. W.

E. G. Rawson, J. W. Goodman, R. E. Norton, “Experimental and Analytical Study of Modal Noise in Optical Fibers,” in Technical Digest, Sixth European Conference on Optical Communication, York (1980).

Goud, P. A.

S. Das, C. G. Englefield, P. A. Goud, “Modal Noise and Distortion Caused by a Longitudinal Gap Between Two Multimode Fibers,” Appl. Opt. 23, 1110 (1984).
[CrossRef] [PubMed]

S. Das, C. G. Englefield, P. A. Goud, “Polarization Modal Noise due to Microbending of Single-Mode Fibers,” in Technical Digest, Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1985), paper TuQ9.

Hjelme, D. R.

D. R. Hjelme, A. R. Mickelson, “Microbending and Modal Noise,” Appl. Opt. 23, 3874 (1983).
[CrossRef]

Hornung, S.

K. J. Blow, N. J. Doran, S. Hornung, “Power Spectrum of Microbends in Monomode Optical Fibres,” Electron. Lett. 18, 448 (1982).
[CrossRef]

Ishihara, K.

Y. Murakami, K. Ishihara, Y. Negishi, N. Kojima, “Micro-bending Losses of P2O5-Doped Graded-Index Multimode Fiber,” Electron. Lett. 18, 774 (1982).
[CrossRef]

Kawakami, S.

S. Kawakami, H. Tanji, “Evolution of Power Distribution in Graded-Index Fibers,” Electron. Lett. 19, 100 (1983).
[CrossRef]

Keck, D. B.

R. Olshansky, M. G. Blankenship, D. B. Keck, “Length-Dependent Attenuation Measurements in Graded-Index Fibers,” in Technical Digest, Second European Conference on Optical Communication, Paris (1976), p. 111.

Kimura, T.

S. Kobayashi, Y. Yamamoto, T. Kimura, “Modulation Frequency Characteristics of Directly Optical Frequency Modulated AlGaAs Semiconductor Laser,” Electron. Lett. 17, 350 (1981).
[CrossRef]

Kobayashi, S.

S. Kobayashi, Y. Yamamoto, T. Kimura, “Modulation Frequency Characteristics of Directly Optical Frequency Modulated AlGaAs Semiconductor Laser,” Electron. Lett. 17, 350 (1981).
[CrossRef]

Kojima, N.

Y. Murakami, K. Ishihara, Y. Negishi, N. Kojima, “Micro-bending Losses of P2O5-Doped Graded-Index Multimode Fiber,” Electron. Lett. 18, 774 (1982).
[CrossRef]

Marcuse, D.

D. Marcuse, “Mode Mixing with Reduced Losses in Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 777 (1976).

D. Marcuse, “Microbending Losses of Single-Mode, Step-Index and Multimode, Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 937 (1976).

D. Marcuse, H. M. Presby, “Mode Coupling in an Optical Fiber with Core Distortions,” Bell Syst. Tech. J. 54, 3 (1975).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), p. 360.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Meslener, G. J.

G. J. Meslener, “Dispersion-Induced Harmonic Distortion in Fiber Systems Using Coherent Light Sources,” in Technical Digest, Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1982), paper THFF4.

Mickelson, A. R.

D. R. Hjelme, A. R. Mickelson, “Microbending and Modal Noise,” Appl. Opt. 23, 3874 (1983).
[CrossRef]

Murakami, Y.

Y. Murakami, K. Ishihara, Y. Negishi, N. Kojima, “Micro-bending Losses of P2O5-Doped Graded-Index Multimode Fiber,” Electron. Lett. 18, 774 (1982).
[CrossRef]

Negishi, Y.

Y. Murakami, K. Ishihara, Y. Negishi, N. Kojima, “Micro-bending Losses of P2O5-Doped Graded-Index Multimode Fiber,” Electron. Lett. 18, 774 (1982).
[CrossRef]

Noguchi, K.

N. Yoshizawa, T. Yabuta, K. Noguchi, “Residual Nylon-Jacketed Fiber Shrinkage Caused by Cooling,” Electron. Lett. 19, 411 (1983).
[CrossRef]

Nolan, D. A.

Norton, R. E.

E. G. Rawson, J. W. Goodman, R. E. Norton, “Experimental and Analytical Study of Modal Noise in Optical Fibers,” in Technical Digest, Sixth European Conference on Optical Communication, York (1980).

Olshansky, R.

R. Olshansky, D. A. Nolan, “Mode-Dependent Attenuation of Optical Fibers: Excess Loss,” Appl. Opt. 15, 1045 (1976).
[CrossRef] [PubMed]

R. Olshansky, “Distortion Losses in Cabled Optical Fibers,” Appl. Opt. 14, 20 (1975).
[PubMed]

R. Olshansky, M. G. Blankenship, D. B. Keck, “Length-Dependent Attenuation Measurements in Graded-Index Fibers,” in Technical Digest, Second European Conference on Optical Communication, Paris (1976), p. 111.

Petermann, K.

K. Petermann, “Nonlinear Distortions and Noise in Optical Communication Systems due to Fiber Connectors,” IEEE J. Quantum Electron. QE-16, 761 (1980).
[CrossRef]

Presby, H. M.

D. Marcuse, H. M. Presby, “Mode Coupling in an Optical Fiber with Core Distortions,” Bell Syst. Tech. J. 54, 3 (1975).

Rawson, E. G.

E. G. Rawson, J. W. Goodman, R. E. Norton, “Experimental and Analytical Study of Modal Noise in Optical Fibers,” in Technical Digest, Sixth European Conference on Optical Communication, York (1980).

Rourke, M. D.

Stubkjaer, K.

K. Stubkjaer, “Distortion of Light Signals Transmitted via Multimode Graded-Index Fibers,” Electron. Lett. 15, 797 (1979).
[CrossRef]

K. Stubkjaer, IBM Watson Research Center; private communication.

Tanji, H.

S. Kawakami, H. Tanji, “Evolution of Power Distribution in Graded-Index Fibers,” Electron. Lett. 19, 100 (1983).
[CrossRef]

Unger, H.

H. Unger, Planar Optical Waveguides and Fibers (Oxford U. P., London, 1977).

Yabuta, T.

N. Yoshizawa, T. Yabuta, K. Noguchi, “Residual Nylon-Jacketed Fiber Shrinkage Caused by Cooling,” Electron. Lett. 19, 411 (1983).
[CrossRef]

Yamamoto, Y.

S. Kobayashi, Y. Yamamoto, T. Kimura, “Modulation Frequency Characteristics of Directly Optical Frequency Modulated AlGaAs Semiconductor Laser,” Electron. Lett. 17, 350 (1981).
[CrossRef]

Yoshizawa, N.

N. Yoshizawa, T. Yabuta, K. Noguchi, “Residual Nylon-Jacketed Fiber Shrinkage Caused by Cooling,” Electron. Lett. 19, 411 (1983).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

D. Marcuse, “Mode Mixing with Reduced Losses in Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 777 (1976).

D. Marcuse, “Microbending Losses of Single-Mode, Step-Index and Multimode, Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 937 (1976).

W. B. Gardner, “Microbending Loss in Optical Fibers,” Bell Syst. Tech. J. 54, 457 (1975).

D. Gloge, “Optical-Fiber Packaging and its Influence on Fiber Straightness and Loss,” Bell Syst. Tech. J. 54, 245 (1975).

D. Marcuse, H. M. Presby, “Mode Coupling in an Optical Fiber with Core Distortions,” Bell Syst. Tech. J. 54, 3 (1975).

Electron. Lett.

K. J. Blow, N. J. Doran, S. Hornung, “Power Spectrum of Microbends in Monomode Optical Fibres,” Electron. Lett. 18, 448 (1982).
[CrossRef]

N. Yoshizawa, T. Yabuta, K. Noguchi, “Residual Nylon-Jacketed Fiber Shrinkage Caused by Cooling,” Electron. Lett. 19, 411 (1983).
[CrossRef]

Y. Murakami, K. Ishihara, Y. Negishi, N. Kojima, “Micro-bending Losses of P2O5-Doped Graded-Index Multimode Fiber,” Electron. Lett. 18, 774 (1982).
[CrossRef]

K. Stubkjaer, “Distortion of Light Signals Transmitted via Multimode Graded-Index Fibers,” Electron. Lett. 15, 797 (1979).
[CrossRef]

S. Kawakami, H. Tanji, “Evolution of Power Distribution in Graded-Index Fibers,” Electron. Lett. 19, 100 (1983).
[CrossRef]

S. Kobayashi, Y. Yamamoto, T. Kimura, “Modulation Frequency Characteristics of Directly Optical Frequency Modulated AlGaAs Semiconductor Laser,” Electron. Lett. 17, 350 (1981).
[CrossRef]

IEEE J. Quantum Electron.

K. Petermann, “Nonlinear Distortions and Noise in Optical Communication Systems due to Fiber Connectors,” IEEE J. Quantum Electron. QE-16, 761 (1980).
[CrossRef]

Laser Focus

R. E. Epworth, “Modal Noise—Causes and Cures,” Laser Focus 17, 109 (1981).

Opt. Lett.

Other

E. G. Rawson, J. W. Goodman, R. E. Norton, “Experimental and Analytical Study of Modal Noise in Optical Fibers,” in Technical Digest, Sixth European Conference on Optical Communication, York (1980).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), p. 360.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

S. Das, C. G. Englefield, P. A. Goud, “Polarization Modal Noise due to Microbending of Single-Mode Fibers,” in Technical Digest, Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1985), paper TuQ9.

H. Unger, Planar Optical Waveguides and Fibers (Oxford U. P., London, 1977).

D. J. Eccleston, “Environmental Testing of UV-Cured Acrylate Coated Fibers,” Corning Application Notes (1982).

R. Olshansky, M. G. Blankenship, D. B. Keck, “Length-Dependent Attenuation Measurements in Graded-Index Fibers,” in Technical Digest, Second European Conference on Optical Communication, Paris (1976), p. 111.

G. J. Meslener, “Dispersion-Induced Harmonic Distortion in Fiber Systems Using Coherent Light Sources,” in Technical Digest, Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1982), paper THFF4.

K. Stubkjaer, IBM Watson Research Center; private communication.

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Figures (11)

Fig. 1
Fig. 1

Two sections of fiber with core–cladding interface distortions connected via a piece of ideal fiber L0 long. A single bump is assumed to lie between z = Z and z = Z + d.

Fig. 2
Fig. 2

Schematic showing the relationship between mechanisms causing temporal fluctuations in the microbending loss.

Fig. 3
Fig. 3

Theoretical microbending loss vs linear pressure for parabolic-index fibers. N.A. = 0.2, α = 25 m. Same power spectrum is used for each value of V.

Fig. 4
Fig. 4

Theoretical values of the microbending loss vs linear pressure at λ = 0.82 μm for parabolic-index fibers with different N.A. values. Same power spectrum is used for each value of N.A.

Fig. 5
Fig. 5

Experimental arrangement for microbending loss studies. CMS designates the Cladding Mode Stripper.

Fig. 6
Fig. 6

Experimental microbending loss values as a function of the linear pressure f0 for load–unload cycles of 10- and 100-sec time intervals. Theoretical values are shown for a = 25 μm, N.A. = 0.2, and V = 39.

Fig. 7
Fig. 7

Microbending loss as a function of the linear pressure f0 for V = 39. Experimental results of Eccleston18 are compared with theoretical predictions of Eq. (19).

Fig. 8
Fig. 8

Theoretical values of dc-SNR vs microbending loss for coherent illumination. L = 1 km, and the number of imperfections = 1000. Steady state and uniform power distributions give almost identical curves.

Fig. 9
Fig. 9

Theoretical values of dc-SNR vs microbending loss for different lengths of fiber (coherent source). a = 25 μm, N.A. = 0.2, V = 21, and the number of imperfections = 1/m.

Fig. 10
Fig. 10

Theoretical values of modal distortion R2 f / f for a partially coherent source. Microbending loss = 1 dB/km, τrms = 1 nsec/km, Ω m = 109 rad/sec, and the number of imperfections = 1/m.

Fig. 11
Fig. 11

Theoretical modal distortion results as a function of microbending loss for the experimental parameters taken from Refs. 24 and 26 (see Table I). Nb denotes the number of fiber imperfections. The experimental value of R2 f / f obtained by Stubkjaer24 fluctuates between the limits shown.

Tables (1)

Tables Icon

Table I Values of the Fiber and Source Parameters Studied in Ref. 24 (Fig. 11)

Equations (46)

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a x = A x exp [ j ( ω t β x z ) ] ,
β x ( ω ) β x ( ω 0 ) + d β x d ω | ( ω ω 0 ) ω = ω 0 = β x 0 + τ x ( ω ω 0 ) ,
x N d Q x d z = x υ y N K x υ K υ y ( Re ( A * x A y ) { cos [ ( β x β y ) z ] H ( β υ β y ) + sin [ ( β x β y ) z ] S ( β υ β y ) } Im ( A * x A y ) { sin [ ( β x β y ) z ] H ( β υ β y ) cos [ ( β x β y ) z ] S ( β υ β y ) } ) + x υ y N K x υ K x y ( Re ( A υ A * y ) { cos [ ( β y β υ ) z ] H ( β x β y ) + sin [ ( β y β υ ) z ] S ( β x β y ) } Im ( A υ A * y ) { sin [ ( β y β υ ) z ] H ( β x β y ) cos [ ( β y β υ ) z ] S ( β x β y ) } ) ,
x d Q x d z = ν M μ M 1 K μ ν 2 Q μ H ( β ν β μ ) + x μ ν M μ M 1 K μ ν K ν x Q μ Q x × { cos [ ( β μ β x ) z ] H ( β ν β μ ) sin [ ( β μ β x ) z ] S ( β ν β μ ) } .
x d Q x d z
r ( x , y , z ) = a + b ( z ) cos ( n ϕ + ψ ¯ ) ,
n i 2 = n 0 2 ( r ) + n p 2 ( r , ϕ , z )
n p 2 ( r , ϕ , z ) = n f n ( z ) g n ( r ) cos ( n ϕ + ψ ) ,
K μ ν = k 2 2 β μ E μ t E * ν t g n ( r ) cos ( n ϕ + ψ ) r d r d ϕ ,
l μ ± l = ± n l μ = azimuthal number of mode μ .
R ( u ) = ƒ ( z ) f ( z ± u ) = σ 2 exp ( | u | / D ) ,
Φ ( Ө ) = H ( Ө ) = 2 σ 2 D [ Ө 2 + ( 1 / D ) 2 ] ,
B ( Ө ) = Φ ( Ө ) [ a 2 / ( 16 n 1 4 Δ 2 ) ] ,
F S ( Ө ) = S ( Ө ) = 2 σ 2 Ө [ Ө 2 + ( 1 / D ) 2 ] .
F S ( Ө ) = G · Φ ( Ө ) ,
Δ Q ( z ) = μ ( M 1 ) 2 ( M 1 ) F μ μ Q μ H ( Δ β 0 ) d + μ ( M 1 ) 2 ( M 1 ) x ( M 2 ) 2 ( M 2 ) F μ x Q μ Q x Δ β 0 × ( H ( Δ β 0 ) { sin [ ϕ μ x + τ μ x ( ω ω 0 ) ( Z + d ) ] sin [ Ψ μ x + τ μ x ( ω ω 0 ) Z ] } + S ( Δ β 0 ) { cos [ ϕ μ x + τ μ x ( ω ω 0 ) ( Z + d ) ] cos [ Ψ μ x + τ μ x ( ω ω 0 ) Z ] } ) ,
F μ x = ν K μ ν K ν x = interference cofficient ,
Δ β 0 = | β ν 0 β μ 0 | | β μ 0 β x 0 | 2 Δ / a since ( m ν m μ = m μ m x = 1 ) ,
Δ P ( z ) = Δ Q ( z ) = μ F μ μ P μ Φ ( Δ β 0 ) d ,
η ( z ) = 1 Δ Q ( z ) Q
η ( z ) | = 1 γ,
γ = 1 P μ P μ F μ μ Φ ( Δ β 0 ) d
P = P 0 exp ( γ N b ) ,
δ ( n ) = η 2 η 2
( η η ) 2 | d = 1 P 2 μ x ν k Q μ Q x Q ν Q k F μ x F ν k Φ 2 ( Δ β 0 ) Δ β 0 2 × [ sin ( ϕ μ x ) sin ( Ψ μ x ) + G cos ( ϕ μ x ) G cos ( Ψ μ x ) ] × [ sin ( ϕ ν k ) sin ( Ψ ν k ) + G cos ( ϕ ν k ) G cos ( Ψ ν k ) ] .
δ 2 ( η ) | d = η 2 η 2 = ( η η ) 2 = μ x μ x P μ P x F μ x 2 ξ 2 γ 2 ( 1 + G 2 ) d 2 Δ β 0 2 = μ x μ x P μ P x F μ x 2 γ 2 / ξ 2 ,
δ 2 ( η ) | L = N b δ 2 ( η ) | d = μ x μ x P μ P x F μ x 2 · γ t 2 / ( ξ 2 · N b ) ,
dc-SNR = [ η | L δ ( η ) | L ] 2 = exp ( 2 γ t ) δ 2 ( η ) | L .
d η d ω = 1 Q μ x μ x Q μ Q x F μ x Δ β 0 { H ( Δ β 0 ) [ τ μ x ( Z + d ) cos ( ϕ μ x ) τ μ x Z cos ( Ψ μ x ) ] + S ( Δ β 0 ) [ τ μ x ( Z + d ) sin ( ϕ μ x ) + τ μ x Z sin ( Ψ μ x ) ] } .
( d η d ω ) 2 = μ x μ x P μ P x F μ x 2 τ μ x 2 Z 2 γ 2 ξ 2 .
Z = L N b · x ,
( d η d ω ) 2 | L = μ x μ x P μ P x F μ x 2 τ μ x 2 γ t 2 3 ξ 2 N b ,
δ 2 ( η s ) = η s 2 η s 2 = 1 P 2 μ x μ x Γ ( τ μ x ) P μ P x F μ x 2 Φ 2 ( Δ β 0 ) Δ β 0 2 ( 1 + G 2 ) ,
δ 2 ( η s ) | L = μ x μ x P μ P x ( F μ x 2 / τ μ x ) [ 1 exp ( | τ μ x | / τ c ) ] τ c · γ t 2 ξ 2 · N b .
( d η s d ω c ) 2 | d = 1 P 2 μ x μ x Γ ( τ μ x ) τ μ x 2 Z 2 P μ P x F μ x 2 Φ 2 ( Δ β 0 ) Δ β 0 2 ( 1 + G 2 ) .
( d η s d ω c ) 2 = 1 ξ 2 N b μ x μ x P μ P x F μ x 2 τ μ x 2 [ 2 α 3 e α ( 1 α + 2 α 2 + 2 α 3 ) ] γ t 2 ,
R 2 f / f [ d b ] 10 log { γ t 2 Ω m 2 4 ξ 2 N b · M l μ x μ x P μ P x F μ x 2 τ μ x 2 × [ 2 ε 3 + e ε ( 1 ε 2 ε 2 + 2 ε 3 ) ] } ,
( d η d ω c ) 2 1 τ r m s η
n p 2 = ƒ ( z ) ( r / a ) n cos ( n ϕ ) | r | a .
K l p , ( l + 1 ) p = k 4 n 1 ε l ( p + l ) V ,
K l p , ( l 1 ) ( p + 1 ) = k 4 n 1 ε l 1 p V ,
K l p , ( l 1 ) p = k 4 n 1 ε l 1 ( p + l 1 ) V ,
K l p , ( l + 1 ) ( p 1 ) = k 4 n 1 ε l ( p l ) V ,
K l p , ( l + 2 ) p = k 4 n 1 ε l ( p + l ) ( p + l + 1 ) V 2 ,
K l p , ( l 2 ) p = k 4 n 1 ε l 2 ( p + l 2 ) ( p + l 1 ) V 2 ,
ε n = { 1 for n 0 , 2 for n = 0 ,

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