Abstract

For the dispersion-optimized range of power-law refractive-index profiles in optical fibers, the phase and the eigenvalue integrals appearing in Wentzel–Kramers–Brillouin solutions are solved. These new solutions provide an explicit description of the electromagnetic-field distributions and phase constants in weakly guiding fibers with power-law refractive-index profiles in terms of usual fiber parameters.

© 1985 Optical Society of America

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References

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  1. D. Gloge, E. A. J. Marcantilli, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
    [CrossRef]
  2. M. K. Barnoski, ed., Fundamentals of Optical Fiber Communications (Academic, New York, 1981).
  3. W. Streifer, C. N. Kurtz, “Scalar analysis of radially inhomogeneous guiding media,” J. Opt. Soc. Am. 57, 779 (1967).
    [CrossRef]
  4. T. Okoshi, Optical Fibers (Academic, New York, 1982).
  5. I. S. Gradstein, I. M. Ryshik, Tables of Series, Products and Integrals (Verlag Harri Deutsch, Thun, Frankfurt/Main, 1981), Vol. 1.
  6. M. Abramovitz, J. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

1973 (1)

D. Gloge, E. A. J. Marcantilli, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

1967 (1)

Abramovitz, M.

M. Abramovitz, J. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Gloge, D.

D. Gloge, E. A. J. Marcantilli, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

Gradstein, I. S.

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products and Integrals (Verlag Harri Deutsch, Thun, Frankfurt/Main, 1981), Vol. 1.

Kurtz, C. N.

Marcantilli, E. A. J.

D. Gloge, E. A. J. Marcantilli, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

Okoshi, T.

T. Okoshi, Optical Fibers (Academic, New York, 1982).

Ryshik, I. M.

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products and Integrals (Verlag Harri Deutsch, Thun, Frankfurt/Main, 1981), Vol. 1.

Stegun, J. A.

M. Abramovitz, J. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Streifer, W.

Bell Syst. Tech. J. (1)

D. Gloge, E. A. J. Marcantilli, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (4)

M. K. Barnoski, ed., Fundamentals of Optical Fiber Communications (Academic, New York, 1981).

T. Okoshi, Optical Fibers (Academic, New York, 1982).

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products and Integrals (Verlag Harri Deutsch, Thun, Frankfurt/Main, 1981), Vol. 1.

M. Abramovitz, J. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (5)

Fig. 1
Fig. 1

Normalized function P 2 n determined by the modulus M2

Fig. 2
Fig. 2

Normalized phase integral f(yn) for parabolic-index fibers with the modulus M2 as a parameter. Normalized turning points y1n and y2n and eigenvalue integral f0 as functions of the modulus M2.

Fig. 3
Fig. 3

Transformation of the asymmetrical function P α to the symmetrical and integrable function P 2 by multiplication with the ratio function r(y).

Fig. 4
Fig. 4

Normalized function f(yn) − g(yn) with the modulus M2 as a parameter.

Fig. 5
Fig. 5

Normalized phase integral h(yn) for parabolic-index fibers with the modulus M2 as a parameter; valid for the damping regions in the core of the fiber.

Equations (44)

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r 1 r [ q 2 ( r ) ] 1 / 2 d r = r 1 r [ k 2 ( r ) β μ 2 m 2 1 / 4 r 2 ] 1 / 2 d r ,
2 υ r 1 r [ q 2 ( r ) ] 1 / 2 d r = y 1 y ( P α / y ) d y = y 1 y { [ A α + b α y y ( α + 2 ) / 2 ] 1 / 2 y 1 } d y
y = R 2 = ( r / a ) 2
b α = δ Δ = k 1 2 ( 0 ) β μ 2 k 1 2 ( 0 ) k 2 2 = ( u υ ) 2 ,
b α = { μ υ [ 2 ( α + 2 ) α ] 1 / 2 } 2 α / ( α + 2 ) 1 .
A α = m 2 1 / 4 υ 2 ( m 2 2 μ max 2 ) ( α α + 2 ) < 0 , | A α | < 1 / 2 .
[ P 2 ( y ) ] 1 / 2 = [ ( y y 1 ) ( y y 2 ) ] 1 / 2 .
A 2 = y 1 y 2 = m 2 1 / 4 υ 2 m 2 υ 2 ,
b 2 = y 1 + y 2 = μ μ max = 2 μ υ 1 .
P 2 = b 2 P 2 n = b 2 [ ( A 2 / b 2 2 ) + y n y n 2 ] 1 / 2 .
4 A 2 / b 2 2 = 4 y 1 y 2 / ( y 1 + y 2 ) 2 = m 2 / μ 2 .
M 2 = m / μ = 2 y 1 y 2 / ( y 1 + y 2 ) = 2 y 1 n y 2 n = 2 A 2 / b 2 .
P 2 = b 2 [ y n y n 2 ( M 2 2 / 4 ) ] 1 / 2 .
y 1 n , 2 n = 0.5 [ 1 ( 1 M 2 2 ) 1 / 2 ] .
ρ n ( y 2 n y 1 n ) / 2 = 0.5 ( 1 M 2 2 ) 1 / 2 ,
ψ ( y n ) = arccos 2 y n 1 ( 1 M 2 2 ) 1 / 2 .
y 1 y ( P 2 / y ) d y = ρ 2 ψ π sin 2 ψ d ψ y ϕ + ρ cos ψ = y ϕ f ( y n ) ,
f ( y n ) = 2 ρ n 2 ψ π sin 2 ψ d ψ y n ( ψ ) = π ψ + ( 1 M 2 2 ) 1 / 2 sin ψ + 2 M 2 [ arctan ( y 1 n y 2 n tan ψ 2 ) π 2 ] .
( 2 / μ ) r 1 r q 2 ( r ) d r = ( 2 / b 2 ) y 1 y ( P 2 / y ) d y = f ( y n ) .
f 0 = ( 2 / μ ) r 1 r 2 q 2 ( r ) d r = π ( 1 M 2 ) .
μ = ( m 2 1 / 4 ) 1 / 2 + 2 l 1 m + 2 l 1.
r ( y ) = P α / P 2 = r ϕ ( y ϕ / h ) Δ r + ( y / h ) Δ r ,
Δ r = r ϕ r , h = ρ cos ψ .
y 1 y ( P α / y ) d y = y ϕ [ r ϕ f ( y ϕ / h ) Δ r ( f g ) ] ,
y 1 y ( P 2 / y ) d y = y ϕ f ( y n ) , y 1 y ( P 2 d y = ( ρ 2 / 2 ) ( π ψ + sin ψ cos ψ ) = y ϕ 2 g ( y n ) .
( 2 / υ ) r 1 r q 2 ( r ) d r = y 1 y ( P α / y ) d y = P α ϕ ( 1 M 2 2 ) 1 { f ( 1 M 2 2 ) 1 / 2 2 [ 2 1 ( P α / P α ϕ ) 1 / 2 ] ( f g ) } .
f 0 = π ( 1 M 2 ) and f 0 g 0 = ( π / 2 ) ( 1 M 2 ) 2 ,
( 2 / υ ) r 1 r 2 [ q 2 ( r ) ] 1 / 2 d r = π ( 1 M 2 1 + M 2 P α ϕ ) 1 / 2 × { 1 [ 2 1 ( P α / P α ϕ ) 1 / 2 ] ( 1 M 2 1 M 2 ) 1 / 2 } .
[ P 2 ( y ) ] 1 / 2 = j Q 2 = j ( y 2 b 2 y + | A 2 | ) 1 / 2 .
y ( Q 2 / y ) d y = Q 2 | A 2 | ln { ± [ 2 ( | A 2 | Q 2 ) 1 / 2 + 2 | A 2 | b 2 y ] y 1 } ( b 2 / 2 ) ln [ ( 2 Q 2 + 2 y b 2 ) ] ,
y n ( Q 2 n / y n ) d y n = h ( y n ) ( 1 / 2 ) ( 1 + M 2 ) ln b 2 .
h ( y n ) = y i n y n ( Q 2 n / y n ) d y n = [ Q 2 n ( M 2 / 2 ) × ln | M 2 Q 2 n + ( M 2 2 / 2 ) y n y n | ( 1 / 2 ) ln | 2 Q 2 n + 2 y n 1 | ] y i n y n .
( 2 / υ ) r 2 r [ | q 2 ( r ) | ] 1 / 2 d r = y i y ( Q 2 / y ) d y = 2 y ϕ h ( y n ) .
y i y ( Q α / y ) d y = y i y ( Q α / Q 2 ) 1 / 2 ( Q 2 / y ) d y = y i y p ( y ) ( Q 2 / y ) d y
y i y ( Q α / y ) d y = ( y y i ) 1 y i y p ( y ) d y y i y ( Q 2 / y ) d y = G ( y i , y ) y i y ( Q 2 / y ) d y .
G ( y i , y ) = ( 1 / 6 ) { [ ( α + 2 2 y i α / 2 b α ) / ( 2 y i b 2 ) ] 1 / 2 + 4 p ( y 1 + y 2 ) + p ( y ) } , i = 1 , 2 .
( 2 / υ ) r i r [ | q 2 ( r ) | ] 1 / 2 d r = ( y ϕ / 3 ) { [ ( α + 2 2 y i α / 2 b α ) × ( 2 y i b 2 ) 1 ] 1 / 2 + 4 p ( y 1 + y 2 ) + p ( y ) } h ( y n ) .
1 r d d r ( r d R d r ) + [ k 2 ( r ) β 2 ( n 1 ) 2 r 2 ] R = 0 ,
R ( r ) = { C { 2 π r [ q 2 ( r ) ] 1 / 2 } 1 / 2 cos { r i r [ q 2 ( r ) ] 1 / 2 d r + π / 4 } , q 2 > 0 D { π 2 r [ q 2 ( r ) ] 1 / 2 } 1 / 2 exp { r i r | q 2 ( r ) | 1 / 2 d r } , q 2 < 0 ;
R ( r ) r = [ η q 2 ( r ) ] 1 / 4 [ c 1 B i ( η ) + c 2 A i ( η ) ] .
R ( r ) = { ( c 1 / r ) [ η q 2 ( r ) ] 1 / 4 B i ( η ) , q 2 ( r ) > 0 ( c 2 / r ) [ η | q 2 ( r ) | ] 1 / 2 A i ( η ) , q 2 ( r ) < 0 .
η ( r ) = [ ( 3 / 2 ) r [ q 2 ( r ) ] 1 / 2 d r ] 2 / 3 , η ( r ) = [ ( 3 / 2 ) r [ | q 2 ( r ) | ] 1 / 2 d r ] 2 / 3 .
B i ( η ) [ ( π η ) 1 / 2 ] 1 cos [ ( 2 / 3 ) η 3 / 2 + π / 4 ] , A i ( η ) [ ( π η ) 1 / 2 ] 1 exp ( 2 η 2 / 3 / 3 ) .
r 1 r 2 [ k 2 ( r ) β 2 ( m 2 1 / 4 ) / r 2 ] 1 / 2 d r = ( l 1 / 2 ) π .

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