Abstract

Leaky mode attenuation coefficients for graded index optical fibers are derived using the WKB approximation. For the case of a parabolic index profile, the attenuation coefficients for explicitly evaluated and used to calculate the leaky mode contribution to near field and attenuation measurements. Failure to observe the expected leaky mode contributions in several graded index fibers is interpreted as evidence for the presence of an additional loss mechanism.

© 1976 Optical Society of America

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References

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  1. A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
    [CrossRef]
  2. A. W. Snyder, D. J. Mitchell, C. Pask, J. Opt. Soc. Am. 64, 608 (1974).
    [CrossRef]
  3. C. Pask, A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 65, 356 (1975).
    [CrossRef]
  4. W. J. Stewart, in Digest of OSA Topical Meeting on Optical Fiber Transmission, Williamsburg, Va. (Optical Society of America, Washington, D.C., 1975).
  5. W. J. Stewart, in Digest of IEE European Conference on Optical Fibre Communication, London (Institute of Electrical Engineers, London WC2, 1975).
  6. M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron Lett. 11, 238 (1975).
    [CrossRef]
  7. W. J. Stewart, Electron Lett., 11, 321 (1975).
    [CrossRef]
  8. M. J. Adams, F. M. E. Sladen, D. N. Payne, post deadline paper given at IEEE European Conference in Optical Fibre Communication, Londonx (1975).
  9. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).
  10. R. Olshansky, Appl. Opt. 15, 782 (1976).
    [CrossRef] [PubMed]
  11. D. N. Payne, F. M. E. Sladen, M. J. Adams, in Digest of IEE European Conference in Optical Fibre Communication, London (Institute of Electrical Engineers, London WC2, 1975).
  12. D. B. Keck, Appl. Opt. 13, 1882 (1974).
    [CrossRef] [PubMed]
  13. R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
    [CrossRef] [PubMed]
  14. R. Olshansky, D. Nolan, Appl. Opt. 15, 1045 (1976).
    [CrossRef] [PubMed]
  15. Note added in proof: After this manuscript was accepted for publication, it was brought to my attention that A. W. Snyder, J. D. Love, Elect. Letts. 12, 325 (1976) have obtained this same result using a ray analysis.

1976 (4)

Note added in proof: After this manuscript was accepted for publication, it was brought to my attention that A. W. Snyder, J. D. Love, Elect. Letts. 12, 325 (1976) have obtained this same result using a ray analysis.

R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[CrossRef] [PubMed]

R. Olshansky, Appl. Opt. 15, 782 (1976).
[CrossRef] [PubMed]

R. Olshansky, D. Nolan, Appl. Opt. 15, 1045 (1976).
[CrossRef] [PubMed]

1975 (3)

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron Lett. 11, 238 (1975).
[CrossRef]

W. J. Stewart, Electron Lett., 11, 321 (1975).
[CrossRef]

C. Pask, A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 65, 356 (1975).
[CrossRef]

1974 (3)

1973 (1)

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

Adams, M. J.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron Lett. 11, 238 (1975).
[CrossRef]

M. J. Adams, F. M. E. Sladen, D. N. Payne, post deadline paper given at IEEE European Conference in Optical Fibre Communication, Londonx (1975).

D. N. Payne, F. M. E. Sladen, M. J. Adams, in Digest of IEE European Conference in Optical Fibre Communication, London (Institute of Electrical Engineers, London WC2, 1975).

Gloge, D.

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

Keck, D. B.

Love, J. D.

Note added in proof: After this manuscript was accepted for publication, it was brought to my attention that A. W. Snyder, J. D. Love, Elect. Letts. 12, 325 (1976) have obtained this same result using a ray analysis.

Marcatili, E. A. J.

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

Mitchell, D. J.

Nolan, D.

Olshansky, R.

Pask, C.

Payne, D. N.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron Lett. 11, 238 (1975).
[CrossRef]

D. N. Payne, F. M. E. Sladen, M. J. Adams, in Digest of IEE European Conference in Optical Fibre Communication, London (Institute of Electrical Engineers, London WC2, 1975).

M. J. Adams, F. M. E. Sladen, D. N. Payne, post deadline paper given at IEEE European Conference in Optical Fibre Communication, Londonx (1975).

Sladen, F. M. E.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron Lett. 11, 238 (1975).
[CrossRef]

M. J. Adams, F. M. E. Sladen, D. N. Payne, post deadline paper given at IEEE European Conference in Optical Fibre Communication, Londonx (1975).

D. N. Payne, F. M. E. Sladen, M. J. Adams, in Digest of IEE European Conference in Optical Fibre Communication, London (Institute of Electrical Engineers, London WC2, 1975).

Snyder, A. W.

Note added in proof: After this manuscript was accepted for publication, it was brought to my attention that A. W. Snyder, J. D. Love, Elect. Letts. 12, 325 (1976) have obtained this same result using a ray analysis.

C. Pask, A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 65, 356 (1975).
[CrossRef]

A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, C. Pask, J. Opt. Soc. Am. 64, 608 (1974).
[CrossRef]

Stewart, W. J.

W. J. Stewart, Electron Lett., 11, 321 (1975).
[CrossRef]

W. J. Stewart, in Digest of OSA Topical Meeting on Optical Fiber Transmission, Williamsburg, Va. (Optical Society of America, Washington, D.C., 1975).

W. J. Stewart, in Digest of IEE European Conference on Optical Fibre Communication, London (Institute of Electrical Engineers, London WC2, 1975).

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

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

Elect. Letts. (1)

Note added in proof: After this manuscript was accepted for publication, it was brought to my attention that A. W. Snyder, J. D. Love, Elect. Letts. 12, 325 (1976) have obtained this same result using a ray analysis.

Electron Lett. (2)

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron Lett. 11, 238 (1975).
[CrossRef]

W. J. Stewart, Electron Lett., 11, 321 (1975).
[CrossRef]

J. Opt. Soc. Am. (3)

Other (4)

W. J. Stewart, in Digest of OSA Topical Meeting on Optical Fiber Transmission, Williamsburg, Va. (Optical Society of America, Washington, D.C., 1975).

W. J. Stewart, in Digest of IEE European Conference on Optical Fibre Communication, London (Institute of Electrical Engineers, London WC2, 1975).

M. J. Adams, F. M. E. Sladen, D. N. Payne, post deadline paper given at IEEE European Conference in Optical Fibre Communication, Londonx (1975).

D. N. Payne, F. M. E. Sladen, M. J. Adams, in Digest of IEE European Conference in Optical Fibre Communication, London (Institute of Electrical Engineers, London WC2, 1975).

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Figures (4)

Fig. 1
Fig. 1

The effective index profile (solid curve) is the sum of the index profile and angular momentum terms (dashed curves). Guided modes with propagation constants in the range βMAX(ν) > β > kn2 are confined to the shaded portion of the core. Modes with propagation constants in the range kn2 > β > βMIN(ν) are only partially confined to the indicated region of the core. These leaky modes eventually tunnel through the angular momentum barrier (dotted region) and are lost from the waveguide.

Fig. 2
Fig. 2

The leaky mode attenuation coefficients of a parabolic fiber are shown in dB/m as a function of the azimuthal mode number ν. As a visual aid, modes having the same principal mode number m are joined by a solid curve. Only modes with even principal mode numbers are shown.

Fig. 3
Fig. 3

Total transmitted power, divided by the power in the guided modes, is plotted as a function of the fiber length for parabolic and step index fibers. It is assumed all modes are equally excited at input.

Fig. 4
Fig. 4

The near-field pattern of leaky modes in a parabolic fiber is shown for several different lengths. For comparison, the near-field pattern of the guided modes is also shown.

Equations (39)

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n 2 ( x ) = n 1 2 [ 1 - 2 Δ f ( x ) ]             x 1 , n 2 ( x ) = n 2 2             x 1 ,
E ( x , θ , z ) = R ( x ) exp ( i ν θ ) exp ( i β z ) ,
[ 1 x d d x × d d x - V e ( x ) - β 2 a 2 ] R ( x ) = 0 ,
V e ( x ) = n 1 2 k 2 a 2 - V 2 f ( x ) - ν 2 / x 2 x 1 n 2 2 k 2 a 2 - ν 2 / x 2 x 1 ,
V = n 1 k a ( 2 Δ ) 1 / 2 .
n 1 k β n 2 k .
ν 2 V 2 f ( 1 ) / 2.
k n 2 > β > ( k 2 n 2 2 - ν 2 / a 2 ) 1 / 2 ,
( k 2 n 2 2 - ν 2 / a 2 ) 1 / 2 > β ,
W = a ( n 1 2 k 2 - β 2 ) 1 / 2 , Q = a ( n 2 2 k 2 - β 2 ) 1 / 2 .
V 2 = W 2 - Q 2 .
ν Q 0.
R ( x ) = 1 ( x u ¯ ) 1 / 2 exp ( - x x 1 d x u ¯ )             0 x < x 1 ,
R ( x ) = 2 ( x u ) 1 / 2 cos ( x 1 x d x u - π / 4 )             x 1 < x < x 2 ,
R ( x ) = 1 ( x u ¯ ) 1 / 2 [ sin ϕ 1 exp ( - x 2 x d x u ¯ ) + 2 cos ϕ 1 exp ( x 2 x d x u ¯ ) ]             x 2 < x 1 ,
R ( x ) = 1 ( x u ¯ ) 1 / 2 [ sin ϕ 1 exp ( - ϕ 2 ) exp ( x ν / Q d x v ¯ ) + 2 cos ϕ 1 exp ( ϕ 2 ) exp ( - x ν / Q d x v ¯ ) ]             1 x < ν / Q ,
R ( x ) = 1 ( x v ) 1 / 2 [ 4 cos ϕ 1 exp ( ϕ 2 ) cos ( ν / Q x d x v - π / 4 ) - sin ϕ 1 exp ( - ϕ 2 ) sin ( ν / Q x d x v - π / 4 ) ]             ν / Q < x ,
u 2 = - u ¯ 2 = W 2 - V 2 f ( x ) - ν 2 / x 2 , v 2 = - v ¯ 2 = Q 2 - ν 2 / x 2 ,
ϕ 1 = x 1 x 2 d x u ,
ϕ 2 = x 2 1 d x u ¯ + 1 ν / Q v ¯ d x .
u ( x 1 ) = u ( x 2 ) = 0.
cot ϕ 1 exp ( 2 ϕ 2 ) = + i / 4.
β μ ν = β μ ν r + i β μ ν i ,
cot ϕ 1 = 0             or            ϕ 1 ( β μ ν r ) = ( μ + 1 / 2 ) π .
β μ ν i = - 1 4 exp [ - 2 ϕ 2 ( β ) ] / d ϕ 1 d β | β = β μ ν .
γ μ ν = 2 β μ ν i .
f ( x ) = x 2 ,
β μ ν = n 1 k [ 1 - 2 Δ ( 2 μ + ν + 1 M ) ] 1 / 2 ,
M = V / 2.
M 2 μ + ν + 1 M + ν 2 / 4 M ,
1 ( 2 μ + ν + 1 ) M .
ν 2 M
N l m PARA = V 2 / 12 ,
N g m PARA = V 2 / 4.
N l m STEP = N g m STEP = V 2 / 2.
γ μ ν = 1 π ( 2 Δ ) 1 / 2 a exp ( - 2 ϕ 2 ) ,
ϕ 2 = ν log [ ν + ( ν 2 - Q 2 ) 1 / 2 Q ] + ν 2 log | ( W 4 - 4 ν 2 V 2 ) 1 / 2 2 ν ( ν 2 - Q 2 ) 1 / 2 + 2 ν 2 - W 2 | - 1 2 ( ν 2 - Q 2 ) 1 / 2 - W 4 4 V log | ν 2 - Q 2 + 2 V ( ν 2 - Q 2 ) 1 / 2 ( W 4 - 4 ν 2 V 2 ) 1 / 2 | .
N g ( r ) = E μ ν ( r ) 2 ,
N l ( r , z ) = E μ ν ( r ) 2 exp ( - γ μ ν z ) ,

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