Abstract

Under certain conditions a changing speckle pattern exists at the output plane of a multimode fiber, resulting in modal noise which can degrade the error performance of a fiber data link. Fiber motion is the usual cause of such speckle change, and Daino et al. have studied the first-order statistics of such modal noise, assuming a single frequency source. But source frequency variation can also cause modal noise; and source frequency diversity has been shown effective in its reduction. In this paper we use a speckle theory approach to study the frequency dependence of modal noise. We have measured and analyzed the correlation of two speckle patterns as a function of source frequency difference, and the speckle spatial frequency distribution as a function of fiber parameters. We have also measured the speckle contrast as a function of fiber length for several sources and fiber types. Such information permits the prediction of the modal noise statistics, from which corresponding changes in error rates can be derived.

© 1980 Optical Society of America

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References

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  1. R. E. Epworth, “The phenomenon of modal noise in analogue and digital optical fibre systems,” Proceedings of the Fourth European Conference on Optical Communications (4ECOC), p. 492, Genoa, Italy, September 12–15, 1968 (unpublished).
  2. R. E. Epworth, “Phenomenon of modal noise in fibre systems,” Technical Digest of the Topical Meeting on Optical Fiber Communication, Washington, D.C., March 6–8, 1979, p. 106 (unpublished).
  3. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
    [Crossref]
  4. B. Daino, G. de Marchis, and S. Piazzola, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
    [Crossref]
  5. J. Vanderwall and J. Blackburn, “Suppression of some artifacts of modal noise in fiber-optic systems,” Opt. Lett. 4, 295–296 (1979).
    [Crossref] [PubMed]
  6. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
    [Crossref]
  7. M. Born and E. Wolf, Principles of Optics, 2nd ed. (MacMillan, New York, 1964), pp. 508–513.
  8. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 115.
  9. L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247 (1965).
    [Crossref]
  10. R. Bracewell (private communication) has proposed “chat(.)” as an abbreviation for the “chinese hat function” given in Eq. (8).
  11. R. Bracewell, Ref. 8, pp. 262–266.
  12. See, for example, D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
    [Crossref]
  13. D. R. Scifres, W. Streifer, and R. D. Burnham, “Curved stripe GaAs: GaALAs diode lasers and waveguides,” Appl. Phys. Lett. 32, 231–234 (1978).
    [Crossref]
  14. R. D. Burnham, D. R. Scifres, W. Streifer, and S. Peled, “Nonplanar large optical cavity GaAs/GaAlAs semiconductor laser,” Appl. Phys. Lett. 35, 734–736 (1979).
    [Crossref]
  15. M. Mansuripur, J. W. Goodman, E. G. Rawson, and R. E. Norton, “Fiber optics receiver error rate prediction using the Gram-Charlier series,” IEEE Trans. Commun. 28, 402–407 (1980).
    [Crossref]

1980 (1)

M. Mansuripur, J. W. Goodman, E. G. Rawson, and R. E. Norton, “Fiber optics receiver error rate prediction using the Gram-Charlier series,” IEEE Trans. Commun. 28, 402–407 (1980).
[Crossref]

1979 (3)

R. D. Burnham, D. R. Scifres, W. Streifer, and S. Peled, “Nonplanar large optical cavity GaAs/GaAlAs semiconductor laser,” Appl. Phys. Lett. 35, 734–736 (1979).
[Crossref]

B. Daino, G. de Marchis, and S. Piazzola, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
[Crossref]

J. Vanderwall and J. Blackburn, “Suppression of some artifacts of modal noise in fiber-optic systems,” Opt. Lett. 4, 295–296 (1979).
[Crossref] [PubMed]

1978 (1)

D. R. Scifres, W. Streifer, and R. D. Burnham, “Curved stripe GaAs: GaALAs diode lasers and waveguides,” Appl. Phys. Lett. 32, 231–234 (1978).
[Crossref]

1976 (1)

1973 (1)

See, for example, D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[Crossref]

1965 (1)

Blackburn, J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (MacMillan, New York, 1964), pp. 508–513.

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 115.

R. Bracewell (private communication) has proposed “chat(.)” as an abbreviation for the “chinese hat function” given in Eq. (8).

R. Bracewell, Ref. 8, pp. 262–266.

Burnham, R. D.

R. D. Burnham, D. R. Scifres, W. Streifer, and S. Peled, “Nonplanar large optical cavity GaAs/GaAlAs semiconductor laser,” Appl. Phys. Lett. 35, 734–736 (1979).
[Crossref]

D. R. Scifres, W. Streifer, and R. D. Burnham, “Curved stripe GaAs: GaALAs diode lasers and waveguides,” Appl. Phys. Lett. 32, 231–234 (1978).
[Crossref]

Daino, B.

B. Daino, G. de Marchis, and S. Piazzola, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
[Crossref]

de Marchis, G.

B. Daino, G. de Marchis, and S. Piazzola, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
[Crossref]

Epworth, R. E.

R. E. Epworth, “The phenomenon of modal noise in analogue and digital optical fibre systems,” Proceedings of the Fourth European Conference on Optical Communications (4ECOC), p. 492, Genoa, Italy, September 12–15, 1968 (unpublished).

R. E. Epworth, “Phenomenon of modal noise in fibre systems,” Technical Digest of the Topical Meeting on Optical Fiber Communication, Washington, D.C., March 6–8, 1979, p. 106 (unpublished).

Gloge, D.

See, for example, D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[Crossref]

Goldfischer, L. I.

Goodman, J. W.

M. Mansuripur, J. W. Goodman, E. G. Rawson, and R. E. Norton, “Fiber optics receiver error rate prediction using the Gram-Charlier series,” IEEE Trans. Commun. 28, 402–407 (1980).
[Crossref]

J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
[Crossref]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
[Crossref]

Mansuripur, M.

M. Mansuripur, J. W. Goodman, E. G. Rawson, and R. E. Norton, “Fiber optics receiver error rate prediction using the Gram-Charlier series,” IEEE Trans. Commun. 28, 402–407 (1980).
[Crossref]

Marcatili, E. A. J.

See, for example, D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[Crossref]

Norton, R. E.

M. Mansuripur, J. W. Goodman, E. G. Rawson, and R. E. Norton, “Fiber optics receiver error rate prediction using the Gram-Charlier series,” IEEE Trans. Commun. 28, 402–407 (1980).
[Crossref]

Peled, S.

R. D. Burnham, D. R. Scifres, W. Streifer, and S. Peled, “Nonplanar large optical cavity GaAs/GaAlAs semiconductor laser,” Appl. Phys. Lett. 35, 734–736 (1979).
[Crossref]

Piazzola, S.

B. Daino, G. de Marchis, and S. Piazzola, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
[Crossref]

Rawson, E. G.

M. Mansuripur, J. W. Goodman, E. G. Rawson, and R. E. Norton, “Fiber optics receiver error rate prediction using the Gram-Charlier series,” IEEE Trans. Commun. 28, 402–407 (1980).
[Crossref]

Scifres, D. R.

R. D. Burnham, D. R. Scifres, W. Streifer, and S. Peled, “Nonplanar large optical cavity GaAs/GaAlAs semiconductor laser,” Appl. Phys. Lett. 35, 734–736 (1979).
[Crossref]

D. R. Scifres, W. Streifer, and R. D. Burnham, “Curved stripe GaAs: GaALAs diode lasers and waveguides,” Appl. Phys. Lett. 32, 231–234 (1978).
[Crossref]

Streifer, W.

R. D. Burnham, D. R. Scifres, W. Streifer, and S. Peled, “Nonplanar large optical cavity GaAs/GaAlAs semiconductor laser,” Appl. Phys. Lett. 35, 734–736 (1979).
[Crossref]

D. R. Scifres, W. Streifer, and R. D. Burnham, “Curved stripe GaAs: GaALAs diode lasers and waveguides,” Appl. Phys. Lett. 32, 231–234 (1978).
[Crossref]

Vanderwall, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (MacMillan, New York, 1964), pp. 508–513.

Appl. Phys. Lett. (2)

D. R. Scifres, W. Streifer, and R. D. Burnham, “Curved stripe GaAs: GaALAs diode lasers and waveguides,” Appl. Phys. Lett. 32, 231–234 (1978).
[Crossref]

R. D. Burnham, D. R. Scifres, W. Streifer, and S. Peled, “Nonplanar large optical cavity GaAs/GaAlAs semiconductor laser,” Appl. Phys. Lett. 35, 734–736 (1979).
[Crossref]

Bell Syst. Tech. J. (1)

See, for example, D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[Crossref]

Electron. Lett. (1)

B. Daino, G. de Marchis, and S. Piazzola, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
[Crossref]

IEEE Trans. Commun. (1)

M. Mansuripur, J. W. Goodman, E. G. Rawson, and R. E. Norton, “Fiber optics receiver error rate prediction using the Gram-Charlier series,” IEEE Trans. Commun. 28, 402–407 (1980).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Other (7)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 2nd ed. (MacMillan, New York, 1964), pp. 508–513.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 115.

R. Bracewell (private communication) has proposed “chat(.)” as an abbreviation for the “chinese hat function” given in Eq. (8).

R. Bracewell, Ref. 8, pp. 262–266.

R. E. Epworth, “The phenomenon of modal noise in analogue and digital optical fibre systems,” Proceedings of the Fourth European Conference on Optical Communications (4ECOC), p. 492, Genoa, Italy, September 12–15, 1968 (unpublished).

R. E. Epworth, “Phenomenon of modal noise in fibre systems,” Technical Digest of the Topical Meeting on Optical Fiber Communication, Washington, D.C., March 6–8, 1979, p. 106 (unpublished).

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Figures (10)

FIG. 1
FIG. 1

Speckle pattern on the end face of a fiber core is imaged onto a 256-element linear detector array. The direction cosines of α and β define the incident ray direction.

FIG. 2
FIG. 2

Far-field radiation distribution at the fiber end face Is assumed constant for θ < θm, zero elsewhere. The power spectrum is then the so-called Chinese hat function [Eq. (8)], shown in (b).

FIG. 3
FIG. 3

Apparatus used to measure spatial power spectra. A microprocessor computer (μP) collects and analyses the linear detector array data.

FIG. 4
FIG. 4

Spatial power spectra for large and small NA fibers. The solid lines are theoretical curves obtained by numerically evaluating Eq. (14). The data points are experimental results obtained using the apparatus of Fig. 3.

FIG. 5
FIG. 5

Planar waveguide of length L and width W, illustrating the 2K + 1 regions of width λgL/w Irresolvable by an aperture of width W. The complement of θm is the critical angle for total internal reflection.

FIG. 6
FIG. 6

Apparatus used to measure the frequency correlation function for several step- and graded-index fibers.

FIG. 7
FIG. 7

Frequency correlation functions for several lengths of step-index fiber. The solid curves are theoretical, obtained by evaluating Eq. (28); the data points are experimental values.

FIG. 8
FIG. 8

Frequency correlation functions for several lengths of step-index fiber.

FIG. 9
FIG. 9

Measured ensemble average speckle contrast versus length for step-index fibers illuminated by various sources. The broken lines are approximate fits to the data.

FIG. 10
FIG. 10

Measured spectra of two injection lasers used to obtain the contrast data in Fig. 9.

Equations (46)

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Γ U ( Δ x , Δ y ) = κ R ( α , β ) e j 2 π / λ ( α Δ x + β Δ y ) d α d β ,
Γ I ( Δ x , Δ y ) = ( I ¯ ) 2 [ 1 + | γ U ( Δ x , Δ y ) | 2 ] ,
γ U ( Δ x , Δ y ) Γ U ( Δ x , Δ y ) / Γ U ( 0 , 0 ) .
Γ I ( Δ x , Δ y ) = ( I ¯ ) 2 | γ U ( Δ x , Δ y ) | 2 .
G I ( ν x , ν y ) = Γ I ( Δ x , Δ y ) e j 2 π ( ν x Δ x + ν y Δ y ) d Δ x d Δ y
= ( I ¯ ) 2 | γ U ( Δ x , Δ y ) | 2 e j 2 π ( ν x Δ x + ν υ Δ y ) d Δ x d Δ y .
G ̂ I ( ν x , ν y ) = R ( ξ , η ) R ( ξ λ ν x , η λ ν y ) d ξ d η R 2 ( ξ , η ) d ξ d η ,
G ̂ ( ν x , ν y ) = chat ( ρ / ρ 0 ) 2 π [ cos 1 ρ / ρ 0 ρ / ρ 0 1 ( ρ / ρ 0 ) 2 ] ρ ρ 0 ,
G I ( ν ) = Γ I ( Δ x , 0 ) e j 2 π ν Δ x d Δ x ,
Γ I ( Δ x , Δ y ) = G I ( ν x , ν y ) e j 2 π ( ν x Δ x + ν y Δ y ) d ν x d ν y .
G ( ν ) = G I ( ν x , ν y ) d ν x d ν y e j 2 π ( ν x ν ) Δ x ) d Δ x .
G ( ν ) = G I ( ν x , ν y ) d ν y .
G I ( ν ) = A [ G I ( ρ ) ] 2 ν ρ G I ( ρ ) ρ 2 ν 2 d ρ .
G I ( ν ) = A [ chat ( ρ / ρ 0 ) ] .
l k = L + 1 2 k 2 λ g 2 L W 2 k λ g x W + 1 2 x 2 L
l k l m = ( k 2 m 2 ) λ g 2 L 2 W 2 ( k m ) λ g x W .
U ( x , ν ) = k = K K a k 2 K + 1 exp [ j 2 π ν n c ( l k + δ k ( x ) ) ] ,
ϕ k = ( 2 π ν n / c ) δ k ( x )
I ( x , ν ) = k = K K m = K K α k α m * 2 K + 1 × exp [ j 2 π ν n c ( ( k 2 m 2 ) λ g 2 L 2 W 2 ( k m ) λ g x W ) ] × exp [ j ( ϕ k ϕ m ) ] ,
c ( Δ ν ) = I ( x , ν ) I ( x , ν + Δ ν ) ¯ I ( x , ν ) ¯ I ( x , ν ) + Δ ν ¯ ,
f ( x ) ( 1 / W d ) W d / 2 W d / 2 f ( x ) d x .
I ( x , ν ) ¯ = k = K K m = K K α k α m * ¯ 2 K + 1 exp ( j π c L n ν W 2 ( k 2 m 2 ) ) × exp ( j 2 π ( k m ) x W ) exp [ j ( ϕ k ϕ m ) ] ¯ .
I ( x , ν ) ¯ = k = K K | a k | 2 ¯ 2 K + 1 ,
I ( x , ν ) ¯ I ( x , ν ) + Δ ν ¯ = k = K K m = K K | α k | 2 ¯ | α m | 2 ¯ ( 2 K + 1 ) 2 .
I ( x , ν ) I ( x , ν + Δ ν ) ¯ = k m p q α k α m * α p α q * ¯ ( 2 K + 1 ) 2 × exp [ j π c L n ν W 2 ( k 2 m 2 + p 2 q 2 ) ] × exp [ j π c L Δ ν n ν 2 W 2 ( p 2 q 2 ) ] exp ( j 2 π ( k m + p q ) x W ) × exp [ j ( ϕ k ϕ m + ϕ p ϕ q ) ] ¯ .
c ( Δ ν ) = k k m m | α k | 2 ¯ | α m | 2 ¯ ( 2 K + 1 ) 2 exp [ j π c L Δ ν n ν 2 W 2 ( m 2 k 2 ) ] + k | α k | 4 ¯ ( | α k | 2 ¯ ) 2 ( 2 K + 1 ) 2 .
ρ ( Δ ν ) = 1 ( 2 K + 1 ) 2 ( 2 K + 1 ) k = K K m = K K × exp ( j π c L Δ ν n ν 2 W 2 ( m 2 k 2 ) ) k m ,
ρ ( Δ ν ) = 1 ( 2 K + 1 ) 2 ( 2 K + 1 ) × p = 0 2 K 1 q = Q 2 K p 2 p cos ( π c L Δ ν n ν 2 W 2 p q ) ,
ρ ( Δ ν ) | 1 / 2 + 1 / 2 exp ( j 4 π n L Δ ν tan 2 θ m c s 2 ) d s | 2 ,
ρ ( Δ ν ) = [ C ( y ) / y ] 2 ,
C ( y ) = 0 y cos ( π t 2 / 2 ) d t ,
y = ( 2 L ( NA ) 2 Δ ν c n ) 1 / 2 .
Δ ν 1.5 c n / L ( NA ) 2 .
C i = ( 1 / 256 ) j = 1 256 | I i ( j ) I ¯ ( j ) | I ¯ ( j ) .
C = ( 1 / i m ) i = 1 i m C i ,
I = k = 1 N I k .
C = 1 / N eff ,
I ¯ = k = 1 N I ¯ k
σ I 2 = I ¯ 2 ( I ¯ ) 2 = k = 1 N l = 1 N ( I k I l ¯ I ¯ k I ¯ l ) .
ρ k l = I k I l ¯ I ¯ k I ¯ l ( I k I ¯ k ) 2 ( I l I ¯ l ) 2 ¯ .
σ I 2 = k = 1 N 1 l = 1 N 1 I ¯ k I ¯ l ρ k l ,
I k 2 ¯ = 2 ( I ¯ k ) 2 .
C = σ I I ¯ = ( k = 1 N l = 1 N I ¯ k I ¯ l ρ k l ) 1 / 2 / k = 1 N I ¯ k .
N eff = k = 1 N l = 1 N I ¯ k I ¯ l / k = 1 N l = 1 N I ¯ k I ¯ l ρ k l .
ρ k l = { 1 k = l 0 k l
N eff = k = 1 N l = 1 N I ¯ k I ¯ l / k = 1 N ( I ¯ k ) 2 .