Abstract

This paper reports on some theoretical and experimental investigations of the radial refractive index gradient that maximizes the information-carrying capacity of a multimode optical waveguide. The primary difference between this work and previous studies is that the dispersive nature of core and cladding materials is taken into consideration. This leads to a new expression for the index gradient parameter αc which characterizes the optimal profile. Using the best available refractive index data, it is found that in highsilica waveguides, the dispersive properties of the glasses significantly influence the pulse broadening of near-parabolic fibers, and that the parameter ac must be altered by 10–20% to compensate for dispersion differences between core and cladding glasses. These predictions are supported by pulse broadening measurements of two graded-index fibers. A comparison is made between the widths and shapes of measured pulses and pulses calculated using the WKB approximation and the near-field measurement of the index profiles. The good agreement found between theory and experiment not only supports the predictions made for the value of αc, but demonstrates an ability to predict pulse broadening in fibers having general index gradients.

© 1976 Optical Society of America

Full Article  |  PDF Article

Corrections

Goran Einarsson, "Pulse broadening in graded-index optical fibers: correction," Appl. Opt. 25, 1030-1030 (1986)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-25-7-1030

References

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  1. S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).
  2. S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MIT-16, 814 (1968).
    [CrossRef]
  3. T. Ochida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
    [CrossRef]
  4. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).
  5. M. Ikeda, IEEE J. Quantum Electron. QE-10, 362 (1974).
    [CrossRef]
  6. C. C. Timmermann, AEU 28, 344 (1974).
  7. R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
    [CrossRef]
  8. D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 562 (1972).
    [CrossRef]
  9. D. B. Keck, R. D. Maurer, in (to be published) Proc. Microwave Research Institute Int. Symp. (1975), Vol. 13.
  10. L. G. Cohen, P. Kaiser, J. B. MacChesney, P. B. O'Connor, H. M. Presby, in Technical Digest of OSA Topical Meeting on Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).
  11. S. D. Personick, Bell Syst. Tech. J. 52, 843 (1973).
  12. M. C. Hudson, D. B. Keck, R. Olshansky (investigation of the near-field technique is currently in progress and will be reported).
  13. R. Olshansky, D. B. Keck, in Technical Digest of OSA Topical Meeting an Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).
  14. I. M. Malitson, J. Opt. Soc. Am. 55, 1205 (1965).
    [CrossRef]
  15. D. B. Keck, Proc. IEEE 62, 649 (1974).
    [CrossRef]
  16. D. B. Keck, Appl. Opt. 13, 1882 (1974).
    [CrossRef] [PubMed]
  17. R. Olshansky, Appl. Opt. 14, 935 (1975).
    [CrossRef] [PubMed]
  18. M. C. Hudson, Corning Glass Works; private communication.

1975 (1)

1974 (4)

D. B. Keck, Proc. IEEE 62, 649 (1974).
[CrossRef]

D. B. Keck, Appl. Opt. 13, 1882 (1974).
[CrossRef] [PubMed]

M. Ikeda, IEEE J. Quantum Electron. QE-10, 362 (1974).
[CrossRef]

C. C. Timmermann, AEU 28, 344 (1974).

1973 (2)

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

S. D. Personick, Bell Syst. Tech. J. 52, 843 (1973).

1972 (2)

R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
[CrossRef]

D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 562 (1972).
[CrossRef]

1970 (1)

T. Ochida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

1968 (1)

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MIT-16, 814 (1968).
[CrossRef]

1965 (2)

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

I. M. Malitson, J. Opt. Soc. Am. 55, 1205 (1965).
[CrossRef]

Andrews, J. R.

R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
[CrossRef]

Bouille, R.

R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
[CrossRef]

Chinnock, E. L.

D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 562 (1972).
[CrossRef]

Cohen, L. G.

L. G. Cohen, P. Kaiser, J. B. MacChesney, P. B. O'Connor, H. M. Presby, in Technical Digest of OSA Topical Meeting on Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).

Furukawa, M.

T. Ochida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Gloge, D.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 562 (1972).
[CrossRef]

Hudson, M. C.

M. C. Hudson, D. B. Keck, R. Olshansky (investigation of the near-field technique is currently in progress and will be reported).

M. C. Hudson, Corning Glass Works; private communication.

Ikeda, M.

M. Ikeda, IEEE J. Quantum Electron. QE-10, 362 (1974).
[CrossRef]

Kaiser, P.

L. G. Cohen, P. Kaiser, J. B. MacChesney, P. B. O'Connor, H. M. Presby, in Technical Digest of OSA Topical Meeting on Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).

Kawakami, S.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MIT-16, 814 (1968).
[CrossRef]

Keck, D. B.

D. B. Keck, Appl. Opt. 13, 1882 (1974).
[CrossRef] [PubMed]

D. B. Keck, Proc. IEEE 62, 649 (1974).
[CrossRef]

M. C. Hudson, D. B. Keck, R. Olshansky (investigation of the near-field technique is currently in progress and will be reported).

D. B. Keck, R. D. Maurer, in (to be published) Proc. Microwave Research Institute Int. Symp. (1975), Vol. 13.

R. Olshansky, D. B. Keck, in Technical Digest of OSA Topical Meeting an Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).

Kitano, J.

T. Ochida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Koizumi, K.

D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 562 (1972).
[CrossRef]

T. Ochida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

MacChesney, J. B.

L. G. Cohen, P. Kaiser, J. B. MacChesney, P. B. O'Connor, H. M. Presby, in Technical Digest of OSA Topical Meeting on Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).

Malitson, I. M.

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Matsumura, H.

T. Ochida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Maurer, R. D.

D. B. Keck, R. D. Maurer, in (to be published) Proc. Microwave Research Institute Int. Symp. (1975), Vol. 13.

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

Nishizawa, J.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MIT-16, 814 (1968).
[CrossRef]

Ochida, T.

T. Ochida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

O'Connor, P. B.

L. G. Cohen, P. Kaiser, J. B. MacChesney, P. B. O'Connor, H. M. Presby, in Technical Digest of OSA Topical Meeting on Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).

Olshansky, R.

R. Olshansky, Appl. Opt. 14, 935 (1975).
[CrossRef] [PubMed]

R. Olshansky, D. B. Keck, in Technical Digest of OSA Topical Meeting an Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).

M. C. Hudson, D. B. Keck, R. Olshansky (investigation of the near-field technique is currently in progress and will be reported).

Personick, S. D.

S. D. Personick, Bell Syst. Tech. J. 52, 843 (1973).

Presby, H. M.

L. G. Cohen, P. Kaiser, J. B. MacChesney, P. B. O'Connor, H. M. Presby, in Technical Digest of OSA Topical Meeting on Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).

Timmermann, C. C.

C. C. Timmermann, AEU 28, 344 (1974).

AEU (1)

C. C. Timmermann, AEU 28, 344 (1974).

Appl. Opt. (2)

Bell Syst. Tech. J. (3)

S. D. Personick, Bell Syst. Tech. J. 52, 843 (1973).

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Electron. Lett. (2)

R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
[CrossRef]

D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 562 (1972).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Ikeda, IEEE J. Quantum Electron. QE-10, 362 (1974).
[CrossRef]

T. Ochida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MIT-16, 814 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

D. B. Keck, Proc. IEEE 62, 649 (1974).
[CrossRef]

Other (5)

M. C. Hudson, Corning Glass Works; private communication.

M. C. Hudson, D. B. Keck, R. Olshansky (investigation of the near-field technique is currently in progress and will be reported).

R. Olshansky, D. B. Keck, in Technical Digest of OSA Topical Meeting an Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).

D. B. Keck, R. D. Maurer, in (to be published) Proc. Microwave Research Institute Int. Symp. (1975), Vol. 13.

L. G. Cohen, P. Kaiser, J. B. MacChesney, P. B. O'Connor, H. M. Presby, in Technical Digest of OSA Topical Meeting on Optical Fiber Transmission (Optical Society of America, Wash. D.C., 1975).

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Figures (11)

Fig. 1
Fig. 1

Refractive index data for 3.4 wt % TiO2 doped silica (n1) and fused silica (n2) are shown for (A) Sellmeier fit to the refractive index, (B) λdn/dλ, (C) λ2d2n/dλ2.

Fig. 2
Fig. 2

The index difference A determined from the data of Fig. 1(A) is shown in (A), and the derivative λdΔ/dλ is shown in (B).

Fig. 3
Fig. 3

The α value that minimizes the pulse broadening is shown as a function of wavelength.

Fig. 4
Fig. 4

Assuming equal power in all modes, the rms pulse width is shown as a function of α for three different sources, all operating at 0.9 μm. The sources are taken to be an LED, a gallium arsenide injection laser, and a distributed feedback laser having rms spectral widths of 150 Å, 10 Å, and 2 Å, respectively. The dashed curve shows the pulse width that would be predicted if all material dispersion effects were neglected.

Fig. 5
Fig. 5

For a source with 2-Å spectral width, the rms pulse width is shown as a function of wavelength for several different values of α

Fig. 6
Fig. 6

A schematic representation of the pulse broadening measurement.

Fig. 7
Fig. 7

The index profile n2(r) − n2(a), determined by a near-field measurement at 900 nm, is shown for two fibers. Fiber A is best fit by α ≈1.7 and fiber B by α ≈ 2.5. The dashed curves show the optimal profile that passes through the end points determined from the best α fit.

Fig. 8
Fig. 8

Output pulses measured at five wavelengths are shown for fibers A and B.

Fig. 9
Fig. 9

Two calculated pulses for each fiber are shown in (A) and (B), and the corresponding measured pulses are shown below in (C) and (D). The larger of the two calculated pulses corresponds to equal power in all modes and the smaller pulse to the power distribution given by Eqs. (47)(49).

Fig. 10
Fig. 10

Calculated pulses for fibers A and B are shown at three wavelengths. The pulses are shown for 500 nm (solid lines), 700nm (dashed lines), and 900 nm (dotted lines). Zero relative delay corresponds to the arrival of the lowest order mode.

Fig. 11
Fig. 11

Calculated rms pulse widths (solid lines) and the rms width determined from experiment (dashed lines) are shown.

Equations (50)

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n 2 ( r ) = n 1 2 [ 1 2 Δ f ( r / a ) ] ,
f ( 0 ) = 0
f ( r / a ) = 1 for r a .
n 2 = n 1 ( 1 2 Δ ) 1 / 2 .
Δ = ( n 1 2 n 2 2 ) / 2 n 1 2 .
β μ , ν = β μ , ν ( n 1 , Δ , a , λ ) .
τ μ , ν = d β μ , ν d ω .
τ μ , ν = 1 c d β μ , ν d k .
P ( t , z , λ ) = Σ P μ , ν ( λ , z ) δ [ t z τ μ ν ( λ ) ] ,
P ( t , z ) = 0 d λ P ( t , z , λ ) .
M n ( z ) = 0 d t t n P ( t , z ) .
M n ( z ) = z n 0 d λ Σ P μ ν ( λ , z ) τ μ ν n ( λ ) .
P μ ν ( λ , z ) = S ( λ ) p μ ν ( λ , z ) ,
0 d λ S ( λ ) = 1 .
λ 0 = 0 d λ λ S ( λ ) ,
σ s = [ 0 d λ ( λ λ 0 ) 2 S ( λ ) ] 1 / 2 .
M n ( z ) = z n 0 d λ S ( λ ) p μ ν ( z ) { τ μ ν n ( λ 0 ) + n ( λ λ 0 ) τ μ ν n 1 ( λ 0 ) τ μ ν ( λ 0 ) + n ( λ λ 0 ) 2 / 2 τ μ ν n 1 ( λ 0 ) τ μ ν ( λ 0 ) + n ( n 1 ) ( λ λ 0 ) 2 / 2 τ μ ν n 2 ( λ 0 ) [ τ μ ν ( λ 0 ) ] 2 + . . . } .
M n ( z ) = z n p μ ν ( z ) ( τ μ ν n ( λ 0 ) + σ s 2 / ( 2 λ 0 2 ) × { n τ μ ν n 1 ( λ 0 ) λ 0 2 τ μ ν ( λ 0 ) + n ( n 1 ) τ μ ν n 2 ( λ 0 ) [ λ 0 τ μ ν ( λ 0 ) ] 2 } ) + 0 ( σ s 3 / λ 0 3 ) .
M 0 ( z ) = p μ ν ( z ) ;
τ ( z ) = M 1 ( z ) / M 0 ( z ) ;
σ ( z ) = [ M 2 ( z ) / M 0 ( z ) τ 2 ( z ) ] 1 / 2 .
A p μ ν ( z ) A μ ν / M 0 .
τ ( z ) = z [ τ ( λ 0 ) + σ s 2 / ( 2 λ 0 2 ) λ 0 2 τ ( λ 0 ) ] .
σ ( z ) = ( σ 2 INTERMODAL + σ 2 INTERMODAL ) 1 / 2 + 0 ( σ s 3 / λ 0 3 ) ,
σ 2 INTERMODAL = z 2 { τ 2 ( λ 0 ) τ ( λ 0 ) 2 + σ s 2 λ 0 2 [ λ 0 2 τ ( λ 0 ) τ ( λ 0 ) λ 0 2 τ ( λ 0 ) τ ( λ 0 ) ] }
σ 2 INTERMODAL = z 2 σ s 2 λ 0 2 [ λ 0 τ ( λ 0 ) ] 2
τ μ ν = N 1 / c + δ τ μ ν ,
N 1 = n 1 λ d n 1 / d λ ,
τ μ ν = λ n 1 + δ τ μ ν .
σ 2 INTERMODAL = z 2 σ s 2 λ 0 2 [ ( λ 0 2 n 1 ) 2 2 λ 0 2 n 1 λ 0 δ τ + ( λ 0 δ τ ) 2 ] .
σ INTRAMODAL σ s λ 0 ( λ 0 2 n 1 ) .
( μ + 1 / 2 ) π = R 1 R 2 d r K μ ν ( r ) ,
K μ ν ( r ) = [ k 2 n 2 ( r ) ν 2 / r 2 β μ ν 2 ] 1 / 2 ,
τ μ ν = k / β μ ν [ ( N 1 / n 1 ) R 1 R 2 d r n 2 ( r ) / K μ ν ( r ) ( λ Δ / 2 Δ ) R 1 R 2 d r [ n 2 ( r ) n 1 2 ] / K μ ν ( r ) ] / R 1 R 2 d r / K μ ν ( r ) .
f ( r / a ) = ( r / a ) α ,
β n = n 1 k { 1 2 Δ [ n / N ( α ) ] α / α + 2 } 1 / 2 ,
N ( α ) = α α + 2 a 2 k 2 n 1 2 Δ .
n 1 k β β n .
τ n = N 1 [ 1 + Δ ( α 2 α + 2 ) ( n / N ) α / α + 2 + Δ 2 2 ( 3 α 2 2 ) α + 2 ( n / N ) 2 α / α + 2 ] + 0 ( Δ 3 ) ,
= 2 n 1 N 1 λ Δ Δ .
α = 2 + .
σ INTERMODAL = L N 1 Δ 2 c α α + 1 ( α + 2 3 α + 2 ) 1 / 2 × [ C 1 2 + 4 C 1 C 2 Δ ( α + 1 ) 2 α + 1 + 4 Δ 2 C 2 2 ( 2 α + 2 ) 2 ( 5 α + 2 ) ( 3 α + 2 ) ] 1 / 2 ,
C 1 = α 2 α + 2 ,
C 2 = 3 α 2 2 2 ( α + 2 ) .
α c = 2 + Δ ( 4 + ) ( 3 + ) ( 5 + 2 ) .
λ τ n = λ 2 n 1 + N 1 Δ ( α 2 α + 2 ) ( 2 α α + 2 ) ( n N ) α / α + 2 .
σ INTERMODAL = σ λ λ [ ( λ 2 n 1 ) 2 2 λ 2 n 1 ( N 1 Δ ) ( α 2 α + 2 ) × ( 2 α 2 α + 2 ) + ( N 1 Δ ) 2 ( α 2 α + 2 ) 2 2 α 3 α + 2 ] 1 / 2 .
p μ ν = 1 ( m / M c ) m M c , p μ ν = 0 m > M c ,
m = 2 μ + ν ,
M c 0.9 M MAX ,

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