Abstract

The possibility of using Bragg reflection in a cylindrical fiber to obtain lossless confined propagation in a core with a lower refractive index than that of the cladding medium is proposed and analyzed.

© 1978 Optical Society of America

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References

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  1. P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
    [CrossRef]
  2. A. Y. Cho and J. R. Arthur, Progress in Solid State Chemistry (Pergamon, New York, 1975), Vol. 10, Part 3, pp. 157–191.
    [CrossRef]
  3. A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
    [CrossRef]
  4. P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. 67, 423 (1977).
    [CrossRef]
  5. P. Yeh, Ph.D. thesis (Caltech, 1977) (unpublished).
  6. D. Gloge, Optical Fiber Technology (IEEE, New York, 1975).
  7. E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
    [CrossRef]
  8. E. Snitzer and H. Osterberg, J. Opt. Soc. Am. 51, 499 (1961).
    [CrossRef]
  9. A. Yariv, P. Yeh, and A. Y. Cho, “Guiding of Light in Bragg Configuration,” presented at Device Research Conference, Ithaca, New York, June (1977).
  10. E. Garmire, T. McMahon, and M. Bass, Appl. Phys. Lett. 31, 92 (1977).
    [CrossRef]
  11. A. Yariv, Appl. Phys. Lett. 28, 88 (1976).
    [CrossRef]
  12. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  13. See, for example, D. Marcuse, Theory of dielectric optical waveguides (Academic, New York, 1974), p. 21.
  14. They are called virtual modes because their energy is more or less confined.

1977 (3)

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

E. Garmire, T. McMahon, and M. Bass, Appl. Phys. Lett. 31, 92 (1977).
[CrossRef]

P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. 67, 423 (1977).
[CrossRef]

1976 (2)

A. Yariv, Appl. Phys. Lett. 28, 88 (1976).
[CrossRef]

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

1961 (2)

Arthur, J. R.

A. Y. Cho and J. R. Arthur, Progress in Solid State Chemistry (Pergamon, New York, 1975), Vol. 10, Part 3, pp. 157–191.
[CrossRef]

Bass, M.

E. Garmire, T. McMahon, and M. Bass, Appl. Phys. Lett. 31, 92 (1977).
[CrossRef]

Cho, A. Y.

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

A. Yariv, P. Yeh, and A. Y. Cho, “Guiding of Light in Bragg Configuration,” presented at Device Research Conference, Ithaca, New York, June (1977).

A. Y. Cho and J. R. Arthur, Progress in Solid State Chemistry (Pergamon, New York, 1975), Vol. 10, Part 3, pp. 157–191.
[CrossRef]

Garmire, E.

E. Garmire, T. McMahon, and M. Bass, Appl. Phys. Lett. 31, 92 (1977).
[CrossRef]

Gloge, D.

D. Gloge, Optical Fiber Technology (IEEE, New York, 1975).

Hong, C. S.

Marcuse, D.

See, for example, D. Marcuse, Theory of dielectric optical waveguides (Academic, New York, 1974), p. 21.

McMahon, T.

E. Garmire, T. McMahon, and M. Bass, Appl. Phys. Lett. 31, 92 (1977).
[CrossRef]

Osterberg, H.

Snitzer, E.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Yariv, A.

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. 67, 423 (1977).
[CrossRef]

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

A. Yariv, Appl. Phys. Lett. 28, 88 (1976).
[CrossRef]

A. Yariv, P. Yeh, and A. Y. Cho, “Guiding of Light in Bragg Configuration,” presented at Device Research Conference, Ithaca, New York, June (1977).

Yeh, P.

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. 67, 423 (1977).
[CrossRef]

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

A. Yariv, P. Yeh, and A. Y. Cho, “Guiding of Light in Bragg Configuration,” presented at Device Research Conference, Ithaca, New York, June (1977).

P. Yeh, Ph.D. thesis (Caltech, 1977) (unpublished).

Appl. Phys. Lett. (3)

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

E. Garmire, T. McMahon, and M. Bass, Appl. Phys. Lett. 31, 92 (1977).
[CrossRef]

A. Yariv, Appl. Phys. Lett. 28, 88 (1976).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

Other (7)

A. Y. Cho and J. R. Arthur, Progress in Solid State Chemistry (Pergamon, New York, 1975), Vol. 10, Part 3, pp. 157–191.
[CrossRef]

A. Yariv, P. Yeh, and A. Y. Cho, “Guiding of Light in Bragg Configuration,” presented at Device Research Conference, Ithaca, New York, June (1977).

P. Yeh, Ph.D. thesis (Caltech, 1977) (unpublished).

D. Gloge, Optical Fiber Technology (IEEE, New York, 1975).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

See, for example, D. Marcuse, Theory of dielectric optical waveguides (Academic, New York, 1974), p. 21.

They are called virtual modes because their energy is more or less confined.

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Figures (4)

FIG. 1
FIG. 1

Bragg fiber.

FIG. 2
FIG. 2

General cladding interface at r = ρ.

FIG. 3
FIG. 3

Field distribution and guided flux of a typical Bragg fiber.

FIG. 4
FIG. 4

Amplitude reduction factor η vs ray angle for N = 0,2,4,6,8,10.

Equations (62)

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n ( r ) = { n g ,             0 < r < r 1 n ν ,             r ν < r < r ν + 1 ν = 1 , 2 , 3 , , .
n ( r ) = { n g , 0 r < r 1 n 2 , r 1 r < r 2 n 1 , r 2 r < r 3 n 2 , r 3 r < r 4 n 1 , r 4 r < r 5 etc .
ψ ( r , θ , z , t ) = ψ ( r , θ ) e i ( β z - ω t ) ,
E r = i β ω 2 μ - β 2 ( r E z + ω μ β r θ H z ) ;
E θ = i β ω 2 μ - β 2 ( r θ E z - ω μ β r H z ) ;
H r = i β ω 2 μ - β 2 ( r H z - ω β r θ E z ) ;
H θ = i β ω 2 μ - β 2 ( r θ H z + ω β r E z ) .
[ t 2 + ( ω 2 μ - β 2 ) ] { E z H z } = 0 ,
E z = [ A J l ( k r ) + B Y l ( k r ) ] cos ( l θ + ϕ ) ,
H z = [ C J l ( k r ) + D Y l ( k r ) ] cos ( l θ + ψ ) ,
k = ( ω 2 μ - β 2 ) 1 / 2 .
E z = [ A 1 J l ( k 1 r ) + B 1 Y l ( k 1 r ) ] cos ( l θ + ϕ 1 ) ,             r < ρ E z = [ A 2 J l ( k 2 r ) + B 2 Y l ( k 2 r ) ] cos ( l θ + ϕ 2 ) ,             r > ρ
H z = [ C 1 J l ( k 1 r ) + D 1 Y l ( k 1 r ) ] cos ( l θ + ψ 1 ) ,             r < ρ H z = [ C 2 J l ( k 2 r ) + D 2 Y l ( k 2 r ) ] cos ( l θ + ψ 2 ) ,             r > ρ
k i = [ ( ω / c ) 2 i μ i - β 2 ] 1 / 2 ,             i = 1 , 2.
( A 2 B 2 C 2 D 2 ) = M ( A 1 B 1 C 1 D 1 ) .
[ A 1 J l ( k 1 ρ ) + B 1 Y l ( k 1 ρ ) ] cos ( l θ + ϕ 1 ) = [ A 2 J l ( k 2 ρ ) + B 2 Y l ( k 2 ρ ) ] cos ( l θ + ϕ 2 ) .
ϕ 1 = ϕ 2
ψ 1 = ψ 2 .
A 1 J l ( k 1 ρ ) + B 1 Y l ( k 1 ρ ) = A 2 J l ( k 2 ρ ) + B 2 Y l ( k 2 ρ ) ,
C 1 J l ( k 1 ρ ) + D 1 Y l ( k 1 ρ ) = C 2 J l ( k 2 ρ ) + D 2 Y l ( k 2 ρ ) .
1 k 1 2 ( - l ρ [ A 1 J l ( k 1 ρ ) + B 1 Y l ( k 1 ρ ) ] sin ( l θ + ϕ ) - ω μ 1 β k 1 [ C 1 J l ( k 1 ρ ) + D 1 Y l ( k 1 ρ ) ] cos ( l θ + ψ ) ) = 1 k 2 2 ( - l ρ [ A 2 J l ( k 2 ρ ) + B 2 Y l ( k 2 ρ ) ] sin ( l θ + ϕ ) - ω μ 2 β k 2 [ C 2 J l ( k 2 ρ ) + D 2 Y l ( k 2 ρ ) ] cos ( l θ + ψ ) ) ,
( 1 / k 1 2 ) { A 1 J l ( k 1 ρ ) + B 1 Y l ( k 1 ρ ) } ( 1 / k 2 2 ) [ A 2 J l ( k 2 ρ ) + B 2 Y l ( k 2 ρ ) ] ,
( μ 1 / k 1 ) [ C 1 J l ( k 1 ρ ) + D 1 Y l ( k 1 ρ ) ] ( μ 2 / k 2 ) [ C 2 J l ( k 2 ρ ) + D 2 Y l ( k 2 ρ ) ] ,
sin ( l θ + ϕ ) = ± cos ( l θ + ψ )
ϕ = ψ ± π / 2.
1 k 1 2 ( - l ρ [ C 1 J l ( k 1 ρ ) + D 1 Y l ( k 1 ρ ) ] sin ( l θ + ψ ) + ω 1 β k 1 [ A 1 J l ( k 1 ρ ) + B 1 Y l ( k 1 ρ ) ] cos ( l θ + ϕ ) ) = 1 k 2 2 ( - l ρ [ C 2 J l ( k 2 ρ ) + D 2 Y l ( k 2 ρ ) ] sin ( l θ + ψ ) + ω 2 β k 2 [ A 2 J l ( k 2 ρ ) + B 2 Y l ( k 2 ρ ) ] cos ( l θ + ϕ ) ) .
E z = [ A J l ( k r ) + B Y l ( k r ) ] cos l θ , H z = [ C J l ( k r ) + D Y l ( k r ) ] sin l θ ;
E z = [ A J l ( k r ) + B Y l ( k r ) ] sin l θ , H z = [ C J l ( k r ) + D Y l ( k r ) ] cos l θ .
A 1 J l ( k 1 ρ ) + B 1 Y l ( k 1 ρ ) + 0 + 0 = ( 1 2 ) ,
ω 1 k 1 β A 1 J l ( k 1 ρ ) + ω 1 k 1 β B 1 Y l ( k 1 ρ ) + l k 1 2 ρ C 1 J l ( k 1 ρ ) + l k 1 2 ρ D 1 Y l ( k 1 ρ ) = ( 1 2 ) ,
0 + 0 + C 1 J l ( k 1 ρ ) + D 1 Y l ( k 1 ρ ) = ( 1 2 )
l k 1 2 ρ A 1 J l ( k 1 ρ ) + l k 1 2 ρ B 1 Y l ( k 1 ρ ) + ω μ 1 k 1 β C 1 J l ( k 1 ρ ) + ω μ 1 k 1 β D 1 Y l ( k 1 ρ ) = ( 1 2 ) ,
M ( 1 , ρ ) ( A 1 B 1 C 1 D 1 ) = M ( 2 , ρ ) ( A 1 B 2 C 2 D 2 )
M ( i , ρ ) = [ J l ( k i ρ ) Y l ( k i ρ ) 0 0 ω i β k i J l ( k i ρ ) ω i β k i Y l ( k i ρ ) l k i 2 ρ J l ( k i ρ ) l k i 2 ρ Y l ( k i ρ ) 0 0 J l ( k i ρ ) Y l ( k i ρ ) l k i 2 ρ J l ( k i ρ ) l k i 2 ρ Y l ( k i ρ ) ω μ i β k i J l ( k i ρ ) ω μ i β k i Y l ( k i ρ ) ] ,
M = M - 1 ( 2 , ρ ) M ( 1 , ρ ) .
M = π y 2 ( m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ) .
m 11 = J l ( x ) Y l ( y ) - ( k 2 1 / k 1 2 ) J l ( x ) Y l ( y ) , m 12 = Y l ( x ) Y l ( y ) - ( k 2 1 / k 1 2 ) Y l ( x ) Y l ( y ) , m 13 = ( β l / ω 2 ) ( 1 / y - 1 / x ) J l ( x ) Y l ( y ) , m 14 = ( β l / ω 2 ) ( 1 / y - 1 / x ) Y l ( x ) Y l ( y ) , m 21 = ( k 2 1 / k 1 2 ) J l ( x ) J l ( y ) - J l ( x ) J l ( y ) , m 22 = ( k 2 1 / k 1 2 ) Y l ( x ) J l ( y ) - Y l ( x ) J l ( y ) , m 23 = ( β l / ω 2 ) ( 1 / x - 1 / y ) J l ( x ) J l ( y ) , m 24 = ( β l / ω 2 ) ( 1 / x - 1 / y ) Y l ( x ) J l ( y ) , m 31 = ( β l / ω μ 2 ) ( 1 / y - 1 / x ) J l ( x ) Y l ( y ) , m 32 = ( β l / ω μ 2 ) ( 1 / y - 1 / x ) Y l ( x ) Y l ( y ) , m 33 = J l ( x ) Y l ( y ) - ( k 2 μ 1 / k 1 μ 2 ) J l ( x ) Y l ( y ) , m 34 = Y l ( x ) Y l ( y ) - ( k 2 μ 1 / k 1 μ 2 ) Y l ( x ) Y l ( y ) , m 41 = ( β l / ω μ 2 ) ( 1 / x - 1 / y ) J l ( x ) J l ( y ) , m 42 = ( β l / ω μ 2 ) ( 1 / x - 1 / y ) Y l ( x ) J l ( y ) , m 43 = ( k 2 μ 1 / k 1 μ 2 ) J l ( x ) J l ( y ) - J l ( x ) J l ( y ) , m 44 = ( k 2 μ 1 / k 1 μ 2 ) Y l ( x ) J l ( y ) - Y l ( x ) J l ( y ) .
( A 2 B 2 ) = M TM ( A 1 B 1 ) ,
( C 2 D 2 ) = M TE ( C 1 D 1 ) .
H z = [ C J 0 ( k r ) + D Y 0 ( k r ) ] e i ( β z - ω t ) ,
E θ = - i ω μ k 2 r H z ,
H r = i β k 2 r H z ,
S r = ( ½ ) Re [ E θ H z * ]
= ( ½ ) Re ( - i ω μ / k ) [ C J 0 ( k r ) + D Y 0 ( k r ) ] × [ C * J 0 ( k r ) + D * Y 0 ( k r ) ] = 0             for all r .
outflowing flux = inflowing flux .
outflowing flux = inflowing flux ( C 2 + D 2 ) ω μ / k 2 .
H z = { H z ( k 1 r ) ,             r < ρ C J 0 ( k 2 ρ ) + D Y 0 ( k 2 ρ ) ,             r > ρ
[ 1 r r ( r r ) - k 1 2 ] H z ( k 1 r ) = 0
C J 0 ( k 2 ρ ) + D Y 0 ( k 2 ρ ) = H z ( k 1 ρ ) H ( k 1 ρ ) ,
( μ 2 / k 2 2 ) [ C k 2 J 0 ( k 2 ρ ) + D k 2 Y 0 ( k 2 ρ ) ] = ( μ 1 / k 1 2 ) k 1 H z ( k 1 ρ ) ( μ 1 / k 1 2 ) k 1 H ( k 1 ρ ) ,
C J 0 + D Y 0 = H ,
C J 0 + D Y 0 = ( k 2 μ 1 / μ 2 k 1 ) H .
C = ( π k 2 ρ / 2 ) [ H Y 0 - ( k 2 μ 1 / k 1 μ 2 ) Y 0 H ] ,
D = ( π k 2 ρ / 2 ) [ ( k 2 μ 1 / k 1 μ 2 ) J 0 H - J 0 H ] ,
J 0 ( x ) Y 0 ( x ) - J 0 ( x ) Y 0 ( x ) = 2 / π x .
ρ ( C 2 + D 2 ) = 0.
ρ ( C 2 + D 2 ) = 1 2 ( π k 2 ρ 2 ) 2 { [ k 2 ( μ 1 μ 2 - 1 ) H 2 - k 1 k 2 μ 1 k 1 μ 2 ( 1 - k 2 2 μ 1 k 1 2 μ 2 ) H 2 ] ( J 0 J 0 + Y 0 Y 0 ) + H H [ k 1 ( 1 - k 2 2 μ 1 k 1 2 μ 2 ) ( J 0 2 + Y 0 2 ) - k 2 ( μ 1 μ 2 - 1 ) k 2 μ 1 k 1 μ 2 ( J 0 2 + Y 0 2 ) ] } .
ρ [ C 2 + D 2 ] = 1 2 ( π k 2 ρ 2 ) 2 ( 1 - k 2 2 k 1 2 ) × H [ k 1 H ( J 0 2 + Y 0 2 ) - k 2 H ( J 0 J 0 + Y 0 Y 0 ) ] .
J 0 ( x ) J 0 ( x ) + Y 0 ( x ) Y 0 ( x ) J 0 2 ( x ) + Y 0 2 ( x ) ~ O ( 1 / x ) .
ρ [ C 2 + D 2 ] = 1 2 ( π k 2 ρ 2 ) 2 ( 1 - k 2 2 k 1 2 ) × k 1 [ J 0 2 ( k 2 ρ ) + Y 0 2 ( k 2 ρ ) ] H z ( k 1 ρ ) H z ( k 1 ρ ) .
η ( C 2 + D 2 ) r = ( C 2 + D 2 ) r = 0 .
η = η ( θ , N ) .