Abstract

Waveguide propagation of a pulse-modulated carrier wave is formulated to include distortion due to dispersion in both attentuation and phase velocity. An optimum input gaussian pulse width exists for maximum information carrying capacity. Results are applied to a numerical study of several single-mode glass optical waveguides in which mode and dielectric dispersion may total zero at some wavelength. For our low-loss (20 dB/km) guides in kilometer lengths, information rates of at least 3 × 1010 bits/sec should be attainable.

© 1971 Optical Society of America

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References

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  1. Conference on Trunk Telecommunications by Guided Waves, Electronics Division of the Institution of Electrical Engineers, 29 Sept.–2 Oct. 1970, London.
  2. 72nd Annual Meeting of the Amer. Ceram. Soc., May 1970, Philadelphia.
  3. Proc. IEEE, Special Issue on Optical Communication 58, No. 10 (1970).
  4. F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
    [Crossref]
  5. D. Williams, K. C. Kao, Proc. IEEE 56, 197 (1968).
    [Crossref]
  6. W. A. Gambling, P. J. R. Laybourn, M. D. Lee, Electronics Lett. 6, 364 (1970).
    [Crossref]
  7. P. J. R. Laybourn, Electronics Lett. 4, 508 (1968).
    [Crossref]
  8. R. B. Dyott, J. R. Stern, presented at the Conference on Trunk Telecommunications by Guided Waves, Electronics Division of the Institution of Electrical Engineers, 29 Sept.–2 Oct. 1970, London.
  9. D. Gloge, Bell Telephone Laboratories, private communication, has pointed out that a more general discussion of this topic is given by C. G. B. Garrett, D. E. McCumber, Phys. Rev. 1, 305 (1970).
    [Crossref]
  10. N. S. Kapany, Fiber Optics—Principles and Applications (Academic, New York, 1967).
  11. W. A. Gambling, P. J. R. Laybourn, Electronics Lett. 6, 661 (1970) have used an approximate pulse rate equation and erroneously consider dispersion in phase velocity rather than group velocity and have apparently missed some of the above points.
    [Crossref]
  12. M. A. Duguay, J. W. Hansen, IEEE Conference on Laser Engineering and Applications, paper 5.5, Washington, D.C., May 1969.
  13. T. L. Paoli, J. E. Ripper, Appl. Phys. Lett. 15, 105 (1969).
    [Crossref]
  14. M. A. Duguay, J. W. Hansen, Appl. Phys. Lett. 15, 192 (1969).
    [Crossref]

1970 (4)

Proc. IEEE, Special Issue on Optical Communication 58, No. 10 (1970).

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[Crossref]

W. A. Gambling, P. J. R. Laybourn, M. D. Lee, Electronics Lett. 6, 364 (1970).
[Crossref]

W. A. Gambling, P. J. R. Laybourn, Electronics Lett. 6, 661 (1970) have used an approximate pulse rate equation and erroneously consider dispersion in phase velocity rather than group velocity and have apparently missed some of the above points.
[Crossref]

1969 (2)

T. L. Paoli, J. E. Ripper, Appl. Phys. Lett. 15, 105 (1969).
[Crossref]

M. A. Duguay, J. W. Hansen, Appl. Phys. Lett. 15, 192 (1969).
[Crossref]

1968 (2)

P. J. R. Laybourn, Electronics Lett. 4, 508 (1968).
[Crossref]

D. Williams, K. C. Kao, Proc. IEEE 56, 197 (1968).
[Crossref]

Duguay, M. A.

M. A. Duguay, J. W. Hansen, Appl. Phys. Lett. 15, 192 (1969).
[Crossref]

M. A. Duguay, J. W. Hansen, IEEE Conference on Laser Engineering and Applications, paper 5.5, Washington, D.C., May 1969.

Dyott, R. B.

R. B. Dyott, J. R. Stern, presented at the Conference on Trunk Telecommunications by Guided Waves, Electronics Division of the Institution of Electrical Engineers, 29 Sept.–2 Oct. 1970, London.

Gambling, W. A.

W. A. Gambling, P. J. R. Laybourn, M. D. Lee, Electronics Lett. 6, 364 (1970).
[Crossref]

W. A. Gambling, P. J. R. Laybourn, Electronics Lett. 6, 661 (1970) have used an approximate pulse rate equation and erroneously consider dispersion in phase velocity rather than group velocity and have apparently missed some of the above points.
[Crossref]

Gloge, D.

D. Gloge, Bell Telephone Laboratories, private communication, has pointed out that a more general discussion of this topic is given by C. G. B. Garrett, D. E. McCumber, Phys. Rev. 1, 305 (1970).
[Crossref]

Hansen, J. W.

M. A. Duguay, J. W. Hansen, Appl. Phys. Lett. 15, 192 (1969).
[Crossref]

M. A. Duguay, J. W. Hansen, IEEE Conference on Laser Engineering and Applications, paper 5.5, Washington, D.C., May 1969.

Kao, K. C.

D. Williams, K. C. Kao, Proc. IEEE 56, 197 (1968).
[Crossref]

Kapany, N. S.

N. S. Kapany, Fiber Optics—Principles and Applications (Academic, New York, 1967).

Kapron, F. P.

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[Crossref]

Keck, D. B.

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[Crossref]

Laybourn, P. J. R.

W. A. Gambling, P. J. R. Laybourn, M. D. Lee, Electronics Lett. 6, 364 (1970).
[Crossref]

W. A. Gambling, P. J. R. Laybourn, Electronics Lett. 6, 661 (1970) have used an approximate pulse rate equation and erroneously consider dispersion in phase velocity rather than group velocity and have apparently missed some of the above points.
[Crossref]

P. J. R. Laybourn, Electronics Lett. 4, 508 (1968).
[Crossref]

Lee, M. D.

W. A. Gambling, P. J. R. Laybourn, M. D. Lee, Electronics Lett. 6, 364 (1970).
[Crossref]

Maurer, R. D.

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[Crossref]

Paoli, T. L.

T. L. Paoli, J. E. Ripper, Appl. Phys. Lett. 15, 105 (1969).
[Crossref]

Ripper, J. E.

T. L. Paoli, J. E. Ripper, Appl. Phys. Lett. 15, 105 (1969).
[Crossref]

Stern, J. R.

R. B. Dyott, J. R. Stern, presented at the Conference on Trunk Telecommunications by Guided Waves, Electronics Division of the Institution of Electrical Engineers, 29 Sept.–2 Oct. 1970, London.

Williams, D.

D. Williams, K. C. Kao, Proc. IEEE 56, 197 (1968).
[Crossref]

Appl. Phys. Lett. (3)

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[Crossref]

T. L. Paoli, J. E. Ripper, Appl. Phys. Lett. 15, 105 (1969).
[Crossref]

M. A. Duguay, J. W. Hansen, Appl. Phys. Lett. 15, 192 (1969).
[Crossref]

Electronics Lett. (3)

W. A. Gambling, P. J. R. Laybourn, Electronics Lett. 6, 661 (1970) have used an approximate pulse rate equation and erroneously consider dispersion in phase velocity rather than group velocity and have apparently missed some of the above points.
[Crossref]

W. A. Gambling, P. J. R. Laybourn, M. D. Lee, Electronics Lett. 6, 364 (1970).
[Crossref]

P. J. R. Laybourn, Electronics Lett. 4, 508 (1968).
[Crossref]

Proc. IEEE (2)

D. Williams, K. C. Kao, Proc. IEEE 56, 197 (1968).
[Crossref]

Proc. IEEE, Special Issue on Optical Communication 58, No. 10 (1970).

Other (6)

M. A. Duguay, J. W. Hansen, IEEE Conference on Laser Engineering and Applications, paper 5.5, Washington, D.C., May 1969.

Conference on Trunk Telecommunications by Guided Waves, Electronics Division of the Institution of Electrical Engineers, 29 Sept.–2 Oct. 1970, London.

72nd Annual Meeting of the Amer. Ceram. Soc., May 1970, Philadelphia.

R. B. Dyott, J. R. Stern, presented at the Conference on Trunk Telecommunications by Guided Waves, Electronics Division of the Institution of Electrical Engineers, 29 Sept.–2 Oct. 1970, London.

D. Gloge, Bell Telephone Laboratories, private communication, has pointed out that a more general discussion of this topic is given by C. G. B. Garrett, D. E. McCumber, Phys. Rev. 1, 305 (1970).
[Crossref]

N. S. Kapany, Fiber Optics—Principles and Applications (Academic, New York, 1967).

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of input and output pulse train intensities vs time. Note the time lag and intensity decrease of the output. The full width at half maximum is 2√log e 2 times the 1/e half-width. If δ is the maximum allowed intensity fraction at pulse intersection, then 1/N is the corresponding minimum temporal separation.

Fig. 2
Fig. 2

Schematic representation of the effective dielectric guide index n e vs wavelength λ. (a) For two core radii a1, a2, dispersionless indices n1, n ¯ 1 , and dispersionless cladding index n2. (b) For glasslike core and cladding indices n1, n2.

Fig. 3
Fig. 3

Calculated difference of guide and cladding indices, n e n2 vs wavelength λ for several optical waveguides. The first number is the bulk core-cladding index difference at 632.8 nm followed by the core radius in μm: guide A—0.0034, 2; B—0.0034, 3; C—0.0094, 1; D—0.0094, 2; E—0.0287, 1.

Fig. 4
Fig. 4

Calculated second derivative h″ of guide wavenumber with respect to circular frequency plotted vs wavelength λ for the same quides as in Fig. 3 and for the bulk cladding (CL).

Fig. 5
Fig. 5

Calculated maximum information rate of waveguide A for two guide lengths L in kilometers and two gaussian pulse separation parameters Δ = −10 log10 δ in dB vs wavelength λ.

Tables (1)

Tables Icon

Table 1 Calculated Pulse Transmission Characteristics for Waveguide A with Several Mode-Locked Laser Sources a

Equations (26)

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A ( x , y , z ) = B ( x , y ) exp ( - 1 2 α z ) exp [ i ( h z - ω t ) ] .
A ( x , y , L ) = S ( ω ) A ( x , y , 0 ) ,
S ( ω ) = exp [ ( - 1 2 α + i h ) L ]
F ( ω ) = - f ( t ) exp ( i ω t ) d t ,
q ( t ) = ( 1 / 2 π ) - F ( ω ) S ( ω ) exp ( - i ω t ) d ω .
q ( t ) = k f ( t - t 0 ) ,             k , t 0 const .
α = - ( 2 / L ) log e k ,             ω / h v p = L / t 0 ,
S ( ω ) = m = 0 S m ( ω - ω 0 , ω 0 ) ,
S m ( ω , ω 0 ) = exp { ω m m ! [ d m d ω m ( - 1 2 α + i h ) ] ω 0 L } .
f ( t ) = g ( t ) exp ( - i ω 0 t )
q ( t ) = exp ( - 1 2 α 0 L ) exp [ - i ω 0 ( t - L / v p 0 ) ] r ( t - L / v g 0 ) ,
v g = 1 / h ,
R ( ω ) = G ( ω ) exp ( - 1 2 ω α 0 L ) m = 2 S m ( ω , ω 0 ) .
g ( t ) = ( a π ) - 1 2 exp ( - t 2 / 2 a 2 ) ,             a ω 0 - 1 3 × 10 - 16 sec ,
r ( t ) = ( a π ) - 1 2 ( 1 + Δ ) - 1 2 exp [ ( 1 2 α 0 L + i t ) 2 / 2 a 2 ( 1 + Δ ) ] ,
r ( t ) 2 = ( b π ) - 1 ( 1 + α 0 L / 2 a 2 ) - 1 2 exp [ 1 2 ( α 0 L ) 2 / ( 2 a 2 + α 0 L ) ] · exp { - b - 2 [ t + α 0 h 0 L 2 / ( 2 a 2 + α 0 L ) ] 2 } .
b = [ a 2 + 1 2 α 0 L + ( h 0 L ) 2 / ( a 2 + 1 2 α 0 L ) ] 1 2
N = 1 2 a [ a 4 + ( h 0 L ) 2 ] - 1 2 log e δ - 1 2 bits / sec
b min = a m 2 , where a m = ( h 0 L ) 1 2
N max = 8 h 0 L log e δ - 1 2 bits / sec .
ξ ( u ) - 1 2 ( κ 2 + 1 ) η ( w ) + [ u - 2 + 1 2 ( κ 2 + 1 ) w - 2 ] + { ( κ 2 + 1 ) 2 ( 1 4 η 2 + 1 2 η w - 2 ) + [ u - 2 + 1 2 ( κ 2 + 1 ) w - 2 ] } 1 2 = 0 ,
( u / a ) 2 = ( 2 π n 1 / λ ) 2 - h 2 ,             ( w / a ) 2 = h 2 - ( 2 π n 2 / λ ) 2 ,             ξ ( u ) = J 0 ( u ) u J 1 ( u ) ,             η ( w ) = K 0 ( w ) w K 1 ( w ) ,             κ = n 2 n 1 .
V = ( u 2 + w 2 ) 1 2 = ( 2 π a / λ ) ( n 1 2 - n 2 2 ) 1 2 < 2.405.
n e ( λ ) λ h ( λ ) / 2 π ,
n e 2 ( λ ) = 1 + j A j λ 2 λ 2 - λ j 2 ,             j = 1 , 2 ,
h ( λ ) λ 3 2 π c 2 d 2 n e d λ 2 ,

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