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)

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).

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

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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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