Abstract

Thin glass fibers imbedded into a glass cladding of slightly lower refractive index represent a promising medium for optical communication. This article presents simple formulas and functions for the fiber parameters as a help for practical design work. It considers the propagation constant, mode delay, the cladding field depth, and the power distribution in the fiber cross section. Plots vs frequency of these parameters are given for 70 modes

© 1971 Optical Society of America

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References

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  1. F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
    [CrossRef]
  2. E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
    [CrossRef]
  3. K. C. Kao, G. A. Hockham, Proc. IEE 113, 1151 (1966).
  4. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
  5. A. W. Snyder, IEEE Trans. Microwave Theory Techniques MTT-17, 1130 (1969).
    [CrossRef]
  6. S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943), p. 94.
  7. A. W. Snyder, IEEE Trans. Microwave Theory Techniques MTT-17, 1138 (1969).
    [CrossRef]
  8. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [CrossRef] [PubMed]
  9. P. Moon, D. E. Spencer, Field Theory Handbook (Springer, Berlin, 1961), p. 192.

1970

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

1969

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

A. W. Snyder, IEEE Trans. Microwave Theory Techniques MTT-17, 1130 (1969).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Techniques MTT-17, 1138 (1969).
[CrossRef]

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
[CrossRef] [PubMed]

1966

K. C. Kao, G. A. Hockham, Proc. IEE 113, 1151 (1966).

1961

Hockham, G. A.

K. C. Kao, G. A. Hockham, Proc. IEE 113, 1151 (1966).

Kao, K. C.

K. C. Kao, G. A. Hockham, Proc. IEE 113, 1151 (1966).

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]

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Maurer, R. D.

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

Moon, P.

P. Moon, D. E. Spencer, Field Theory Handbook (Springer, Berlin, 1961), p. 192.

Schelkunoff, S. A.

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943), p. 94.

Snitzer, E.

Snyder, A. W.

A. W. Snyder, IEEE Trans. Microwave Theory Techniques MTT-17, 1138 (1969).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Techniques MTT-17, 1130 (1969).
[CrossRef]

Spencer, D. E.

P. Moon, D. E. Spencer, Field Theory Handbook (Springer, Berlin, 1961), p. 192.

Toraldo di Francia, G.

Appl. Phys. Lett.

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

Bell Syst. Tech. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

IEEE Trans. Microwave Theory Techniques

A. W. Snyder, IEEE Trans. Microwave Theory Techniques MTT-17, 1130 (1969).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Techniques MTT-17, 1138 (1969).
[CrossRef]

J. Opt. Soc. Am.

Proc. IEE

K. C. Kao, G. A. Hockham, Proc. IEE 113, 1151 (1966).

Other

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943), p. 94.

P. Moon, D. E. Spencer, Field Theory Handbook (Springer, Berlin, 1961), p. 192.

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Figures (8)

Fig. 1
Fig. 1

Sketch of the fiber cross section and the four possible distributions of LP11.

Fig. 2
Fig. 2

The regions of the parameter u for modes of order l = 0,1.

Fig. 3
Fig. 3

Normalized propagation parameter b = (β/kn)/(ncn) as a function of the normalized frequency v.

Fig. 4
Fig. 4

Normalized group delay d(vb)/dv as a function of v.

Fig. 5
Fig. 5

Portion of the mode power which propagates in the cladding plotted vs v.

Fig. 6
Fig. 6

Normalized power density at the core–cladding interface plotted vs v.

Fig. 7
Fig. 7

Cladding parameter w plotted vs v.

Fig. 8
Fig. 8

Sketch of the fiber front face and the cone of light that the fiber accepts.

Equations (56)

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u = a ( k 2 n c 2 - β 2 ) 1 2
w = a ( β 2 - k 2 n 2 ) 1 2 ,
v 2 = u 2 + w 2
v = a k ( n c 2 - n 2 ) 1 2 ,
Δ = ( n c - n ) / n 1.
E y = H x { Z 0 / n c Z 0 / n } = E l { J l ( u r / a ) / J l ( u ) K l ( w r / a ) / K l ( w ) } cos l ϕ .
E z = i Z 0 k { 1 / n c 2 1 / n 2 } H x y ,
and             H z = ( i / k Z 0 ) ( E y / x ) .
E z = - i E l 2 k a { u n c J l + 1 ( u r / a ) J l ( u ) sin ( l + 1 ) ϕ + u n c J l - 1 ( u r / a ) J l ( u ) sin ( l - 1 ) ϕ w n K l + 1 ( w r / a ) K l ( w ) sin ( l + 1 ) ϕ - w n K l - 1 ( w r / a ) K l ( w ) sin ( l - 1 ) ϕ } ,
H z = - i E l 2 k Z 0 a { u J l + 1 ( u r / a ) J l ( u ) cos ( l + 1 ) ϕ - u J l - 1 ( u r / a ) J l ( u ) cos ( l - 1 ) ϕ w K l + 1 ( w r / a ) K l ( w ) cos ( l + 1 ) ϕ + w K l - 1 ( w r / a ) K l ( w ) cos ( l - 1 ) ϕ } .
E ϕ = 1 2 E l { J l ( u r / a ) / J l ( u ) K l ( w r / a ) / K l ( w ) } [ cos ( l + 1 ) ϕ + cos ( l - 1 ) ϕ ] ,
H ϕ = - 1 2 E l Z 0 { n c J l ( u r / a ) / J l ( u ) n K l ( w r / a ) / K l ( w ) } × [ sin ( l + 1 ) ϕ - sin ( l - 1 ) ϕ ] .
u [ J l - 1 ( u ) / J l ( u ) ] = - w [ ( K l - 1 ( w ) / K l ( w ) ] .
( u / n c ) [ J l ± 1 ( u ) / J l ( u ) ] = ± ( w / n ) [ K l ± 1 ( w ) / K l ( w ) ] .
d u / d v = ( u / v ) [ 1 - κ l ( w ) ] ,
where             κ l ( w ) = K l 2 ( w ) / K l - 1 ( w ) K l + 1 ( w ) .
κ l 1 - ( w 2 + l 2 + 1 ) - 1 2 ,
w ( v 2 - u c 2 ) 1 2 ,
u ( v ) = u c exp [ arcsin ( s / u c ) - arcsin ( s / v ) ] / s ,
with             s = ( u c 2 - l 2 - 1 ) 1 2 .
u ( v ) = ( 1 + 2 ) v / [ 1 + ( 4 + v 4 ) 1 4 ]             for HE 11 .
u ( ν ) = u [ 1 - ( 1 / v ) ]
b ( v ) = 1 - ( u 2 / v 2 ) = [ ( β 2 / k 2 ) - n 2 ] / ( n c 2 - n 2 ) ,
b [ ( β / k ) - n ] / ( n c - n ) .
β = n k ( b Δ + 1 ) = n k [ 1 + Δ - Δ ( u 2 / v 2 ) ] .
τ g r = ( L / c ) ( d β / d k ) .
τ g r = L c { [ d ( n k ) / d k ] + n Δ [ d ( v b ) / d v ] } .
d ( v b ) / d v = 1 - ( u / v ) 2 ( 1 - 2 κ ) ,
P core = [ 1 + ( w 2 / u 2 ) ( 1 / κ ) ] ( π a 2 / 2 ) ( Z 0 / n c ) E l 2
and             P clad = [ ( 1 / κ ) - 1 ] ( π a 2 / 2 ) ( Z 0 / n ) E l 2
P = P core + P clad = ( v 2 / u 2 ) ( 1 / κ ) ( π a 2 / 2 ) ( Z 0 / n ) E l 2 .
P core / P = 1 - ( u 2 / v 2 ) ( 1 - κ )
and             P clad / P = ( u 2 / v 2 ) ( 1 - κ ) ,
p ( r ) = κ u 2 v 2 2 P π a 2 { J l 2 ( u r / a ) / J l 2 ( u ) K l 2 ( w r / a ) / K l 2 ( w ) } cos 2 l ϕ .
p ¯ ( r ) = κ ( u 2 / v 2 ) P π a 2 { J l 2 ( u r / a ) / J l 2 ( u ) K l 2 ( w r / a ) / K l 2 ( w ) } .
p ¯ ( a ) = κ ( u 2 / v 2 ) ( P / π a 2 ) .
p ¯ ( r ) κ ( u 2 / v 2 ) ( P / π a r ) exp [ - 2 w ( r - a ) / a ]             for r a ,
p ¯ ( r ) κ l ( P / π a 2 ) ( a / r ) l             for r > a , w = 0.
θ sin θ = ( n c 2 - n 2 ) 1 2
δ = λ / π a .
N 2 ( θ / δ ) 2
N v 2 / 2.
u c ( 2 ν ) 1 2 .
b = 1 - ( u c 2 / v 2 ) = 1 - ( ν / N ) .
d ( v b ) / d v = 1 + ( u c 2 / v 2 ) = 1 + ( ν / N ) .
p ¯ ( a ) = ( P / π a 2 ) ( u c 2 / v 2 ) = ( P / π a 2 ) ( ν / N ) .
P clad P u c 2 / v 2 ( v 2 - u c 2 ) 1 2 = P ν / N ( 2 N - 2 ν ) 1 2 .
[ p ¯ ( a ) / P ] tot = 1 π a 2 N 0 N ν N d ν = 1 2 π a 2 .
( P clad / P ) tot = 1 N 0 N ν d ν N ( 2 N - 2 ν ) 1 2 = 4 3 N - 1 2 .
( Q - D - 2 Δ { [ ( l ± 1 ) / ω 2 ] ± [ K l ( ω ) / ω K l ± 1 ( ω ) ] } ) ( Q - D ) = Q 2 [ 1 - 2 Δ ( u 2 / v 2 ) ] ,
Q = ( l ± 1 ) ( v 2 / u 2 w 2 ) ,
D = [ J l ( u ) / u J l ± 1 ( u ) ] [ K l ( w ) / w K l ± 1 ( w ) ] ,
and             2 Δ = ( n c 2 - n 2 ) / n c 2 .
D = Δ { Q ( u 2 / v 2 ) - [ ( l ± 1 ) / w 2 ] [ K l ( w ) / w K l ± 1 ( w ) ] }
D = Δ [ K l ( w ) / w K l ± 1 ( w ) ] .
( u / n c ) [ J l ± 1 ( u ) / J l ( u ) ] = ± ( w / n ) [ K l ± 1 ( w ) / K l ( w ) ] .

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