Abstract

A local plane wave analysis is employed to derive the ray power transmission and attenuation coefficients for refracting leaky rays on graded-index fibers. These coefficients also reduce to established forms for tunneling rays and are compared with modal attenuation coefficients in the limit of strongly refracting rays.

© 1978 Optical Society of America

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References

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  1. A. W. Snyder, J. D. Love, Electron. Lett. 12, 324 (1976).
    [CrossRef]
  2. A. W. Snyder, D. J. Mitchell, Optik 40, 438 (1975).
  3. R. Olshansky, Appl. Opt. 16, 1639 (1977).
    [CrossRef] [PubMed]
  4. A. Ankiewicz, C. Pask, Opt. Quant. Electron. 9, 87 (1977).
    [CrossRef]
  5. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.
  6. A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
    [CrossRef]
  7. J. D. Love, C. Winkler, J. Opt. Soc. Am. 67, 1627 (1977).
    [CrossRef]

1977 (3)

1976 (1)

A. W. Snyder, J. D. Love, Electron. Lett. 12, 324 (1976).
[CrossRef]

1975 (1)

A. W. Snyder, D. J. Mitchell, Optik 40, 438 (1975).

1974 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

Ankiewicz, A.

A. Ankiewicz, C. Pask, Opt. Quant. Electron. 9, 87 (1977).
[CrossRef]

Love, J. D.

J. D. Love, C. Winkler, J. Opt. Soc. Am. 67, 1627 (1977).
[CrossRef]

A. W. Snyder, J. D. Love, Electron. Lett. 12, 324 (1976).
[CrossRef]

Mitchell, D. J.

A. W. Snyder, D. J. Mitchell, Optik 40, 438 (1975).

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

Olshansky, R.

Pask, C.

A. Ankiewicz, C. Pask, Opt. Quant. Electron. 9, 87 (1977).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Electron. Lett. 12, 324 (1976).
[CrossRef]

A. W. Snyder, D. J. Mitchell, Optik 40, 438 (1975).

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

Winkler, C.

Appl. Opt. (1)

Electron. Lett. (1)

A. W. Snyder, J. D. Love, Electron. Lett. 12, 324 (1976).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Quant. Electron. (1)

A. Ankiewicz, C. Pask, Opt. Quant. Electron. 9, 87 (1977).
[CrossRef]

Optik (1)

A. W. Snyder, D. J. Mitchell, Optik 40, 438 (1975).

Other (1)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

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Figures (5)

Fig. 1
Fig. 1

The continuous core–cladding profile n(r)2, where n ( ρ ) 2 = n c 1 2 and r = ρ is the interface. If χ is the angle the tangent to the core profile at r = ρ makes with the r axis, then δ = −tanχ in Eq. (6).

Fig. 2
Fig. 2

The incident, reflected, and transmitted rays lie in a plane and make an angle αi = π/2 − θt with the normal at P, where θt is the angle between each ray and the tangent to the interface in the plane of the rays. For meridional rays θt = θz(ρ) of Eq. (8).

Fig. 3
Fig. 3

Qualitative description of tunneling rays. The core ray path has a turning point or caustic at rtp and reappears at rrad in the cladding. Between rtp and rrad the fields of the rays are evanescent.

Fig. 4
Fig. 4

( l ˜ 0 )  . Plots of the full transmission coefficient Eq. (4) and the two WKB forms Eqs. (11) and (12) against θz for skew rays. The dashed line is the division between tunneling rays to the left and refracting rays to the right, with θt ≥ 0 for the latter. (θz is in radians; θt is in degrees.)

Fig. 5
Fig. 5

( l ˜ = 0 )  . Plots of the full transmission coefficient Eq. (4) and the WKB form Eq. (12) against θz for meridional rays. The dashed line is the division between bound rays to the left and refracting rays to the right, with θt ≥ 0 for the latter. (θz is in radians; θt is in degrees.)

Equations (24)

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T = power transmitted across the interface power of the incident ray ,
P ( z ) = P ( o ) exp ( γ z ) ,
γ = T / z p  ,
T = 4 π 2 κ ( C 0 + 2 C 1 κ + C 2 κ 2 ) ,
C 0 = [ A i ( υ ) 2 + B i ( υ ) 2 ] [ A i ( u ) 2 + B i ( u ) 2 ]  , C 1 = 1 π 2 + [ A i ( u ) A i ( u ) + B i ( u ) B i ( u ) ] × [ A i ( υ ) A i ( υ ) + B i ( υ ) B i ( υ ) ]  , C 2 = [ A i ( u ) 2 + B i ( u ) 2 ] [ A i ( υ ) 2 + B i ( υ ) 2 ]  , κ = + [ ( δ ρ / 2 l ˜ 2 ) 1 ] 1 / 3  ,
δ = d n ( r ) 2 d r | r = ρ ;
u = ( k ρ ) 2 / 3 ( n c l 2 β ˜ 2 l ˜ 2 ) / ( ρ δ 2 l ˜ 2 ) 2 / 3 , υ = u [ ( ρ δ / 2 l ˜ 2 ) 1 ] 2 / 3 ,
β ˜ = n c o ( r ) cos θ z ( r ) , l ˜ = ( r / ρ ) n c o ( r ) sin θ z ( r ) cos ϕ ( r ) ,
sin θ t = ( n c l 2 β ˜ 2 l ˜ 2 ) 1 / 2 / n c l  ,
u = [ k 1 / 3 n c l sin θ t / ( δ 2 l ˜ 2 / ρ ) 1 / 3 ] 2 , θ t 0.
T = exp  { 2 k r t p r rad [ β ˜ 2 + ( ρ l ˜ / r ) 2 n ( r ) 2 ] 1 / 2 d r }  .
T = 1 ( δ 8 n c l 3 k ) 2 1 sin 6 θ t , θ t > 0.
n ( r ) 2 = n ( o ) 2 [ 1 2 Δ ( r / ρ ) 2 ]  , 0 r ρ n c l 2 = n ( o ) 2 [ 1 2 Δ ]  , r ρ
β = k β ˜ , ν = k ρ l ˜ .
n ( r ) 2 = n c l 2 + δ ( ρ r )  , r ρ , = n c l 2 r ρ  .
d 2 Ψ d r 2 + 1 r d Ψ d r + k 2 [ n ( r ) 2 β ˜ 2 ( ρ l ˜ r ) 2 ] Ψ = 0.
d 2 Φ d r 2 + k 2 [ n ( r ) 2 β ˜ 2 ( ρ l ˜ r ) 2 ] Φ = 0 ,
d 2 Φ d r 2 + k 2 [ n c l 2 β ˜ 2 l ˜ 2 + ( 1 r / ρ ) ( δ ρ 2 l ˜ 2 ) ] Φ = 0 , r ρ , d 2 Φ d r 2 + k 2 [ n c l 2 β ˜ 2 l ˜ 2 ( 1 r / ρ ) 2 l ˜ 2 ] Φ = 0 , r ρ .
ξ 1 ( r ) = ( k ρ ) 2 / 3 [ ( r ρ 1 ) ( δ ρ 2 l ˜ 2 ) 1 / 3 ( n c l 2 β ˜ 2 l ˜ 2 ) ( δ ρ 2 l ˜ 2 ) 2 / 3 ]  , ξ 2 ( r ) = ( k ρ ) 2 / 3 [ ( 1 r / ρ ) ( 2 l ˜ 2 ) 1 / 3 ( n c l 2 β ˜ 2 l ˜ 2 ) / ( 2 l ˜ 2 ) 2 / 3 ]
d 2 Φ d ξ 1 2 ξ 1 Φ = 0 , d 2 Φ d ξ 2 2 ξ 2 Φ = 0 ,
Φ i = A i ( ξ 1 ) i B i ( ξ i )  , Φ r = R [ A i ( ξ 1 ) + i B i ( ξ 1 ) ]  , Φ s = S [ A i ( ξ 2 ) + i B i ( ξ 2 ) ]  ,
Φ i + Φ r = Φ s , d d r ( Φ i + Φ s ) = d Φ s d r , r = ρ ,
T = 1 | R | 2 .
u = ξ 1 ( ρ ) , υ = ξ 2 ( ρ ) , κ = d ξ 2 d r / d ξ 1 d r | r = ρ .

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