Abstract

Mode conversion at splices of multimode optical fibers with power-law graded-index is investigated. With the assumption that the multitude of modes may be regarded to be a continuum, the boundary value problem is approximated by a geometric analysis, which eases the interpretation of mode splitting at splices. Although the approximation is not rigorous it provides a new approach for the determination of mode splitting at splices, generating results supported by published data. These results are used to verify the realizability of the optical equalization technique.

© 1983 Optical Society of America

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References

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  1. C. M. Miller, Bell Syst. Tech. J. 55, 917 (1976).
  2. D. Gloge, Bell Syst. Tech. J. 55, 905 (1976).
  3. C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).
  4. S. C. Mettler, Bell Syst. Tech. J. 58, 2163 (1979).
  5. T. Matsumoto, K. Nakagawa, Appl. Opt. 18, 1449 (1979).
    [CrossRef] [PubMed]
  6. N. Kashima, Appl. Opt. 19, 2597 (1980).
    [CrossRef] [PubMed]
  7. N. Kashima, Appl. Opt. 20, 3859 (1981).
    [CrossRef] [PubMed]
  8. M. Eve, Opt. Quantum Electron. 10, 41 (1978).
    [CrossRef]
  9. D. Gloge, Bell Syst. Tech.. 51, 1767 (1972).
  10. R. Olshansky, Appl. Opt. 14, 935 (1975).
    [PubMed]
  11. Y. Daido, Trans. IECE Jpn. E64, 13 (1981).
  12. D. Marcuse, Bell Syst. Tech. J. 52, 1423 (1973).

1981 (2)

N. Kashima, Appl. Opt. 20, 3859 (1981).
[CrossRef] [PubMed]

Y. Daido, Trans. IECE Jpn. E64, 13 (1981).

1980 (1)

1979 (2)

S. C. Mettler, Bell Syst. Tech. J. 58, 2163 (1979).

T. Matsumoto, K. Nakagawa, Appl. Opt. 18, 1449 (1979).
[CrossRef] [PubMed]

1978 (2)

C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).

M. Eve, Opt. Quantum Electron. 10, 41 (1978).
[CrossRef]

1976 (2)

C. M. Miller, Bell Syst. Tech. J. 55, 917 (1976).

D. Gloge, Bell Syst. Tech. J. 55, 905 (1976).

1975 (1)

1973 (1)

D. Marcuse, Bell Syst. Tech. J. 52, 1423 (1973).

1972 (1)

D. Gloge, Bell Syst. Tech.. 51, 1767 (1972).

Daido, Y.

Y. Daido, Trans. IECE Jpn. E64, 13 (1981).

Eve, M.

M. Eve, Opt. Quantum Electron. 10, 41 (1978).
[CrossRef]

Gloge, D.

D. Gloge, Bell Syst. Tech. J. 55, 905 (1976).

D. Gloge, Bell Syst. Tech.. 51, 1767 (1972).

Kashima, N.

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 52, 1423 (1973).

Matsumoto, T.

Mettler, S. C.

S. C. Mettler, Bell Syst. Tech. J. 58, 2163 (1979).

C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).

Miller, C. M.

C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).

C. M. Miller, Bell Syst. Tech. J. 55, 917 (1976).

Nakagawa, K.

Olshansky, R.

Appl. Opt. (4)

Bell Syst. Tech. J. (5)

D. Marcuse, Bell Syst. Tech. J. 52, 1423 (1973).

C. M. Miller, Bell Syst. Tech. J. 55, 917 (1976).

D. Gloge, Bell Syst. Tech. J. 55, 905 (1976).

C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).

S. C. Mettler, Bell Syst. Tech. J. 58, 2163 (1979).

Bell Syst. Tech.. (1)

D. Gloge, Bell Syst. Tech.. 51, 1767 (1972).

Opt. Quantum Electron. (1)

M. Eve, Opt. Quantum Electron. 10, 41 (1978).
[CrossRef]

Trans. IECE Jpn. (1)

Y. Daido, Trans. IECE Jpn. E64, 13 (1981).

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Figures (11)

Fig. 1
Fig. 1

Transfer of energy between mode x1 in the left fiber to mode x2 in the right fiber can occur only in the overlapping area along a given curve. Points C, A1, and A 1 are on such a curve. The two fibers with centers at O1 and O2 are separated by a distance S. The modes x1 and x2 are distributed over the area of the circle with radii ρ1(x1) and ρ2(x2).

Fig. 2
Fig. 2

Definition of energy transfer efficiency levels (zero, low, moderate, and high) from mode number x1 to all modes in the second fiber smaller than x2. Horizontal scale shown as either x2 or Dρ = ρ2(x2) − ρ1(x1). Vertical axis is fiber offset S.

Fig. 3
Fig. 3

Transfer of energy between mode x1 = 1 in the left fiber to mode x2 = 1 in the right fiber takes place along curve A 1 G A 1. The hatched area around O1 represents the region where mode x1 = 1 cannot be coupled to any mode x2 < 1 in the second fiber.

Fig. 4
Fig. 4

Circles are loci of equal transverse wave vectors in the two fibers. Their value increases toward the center. Same circles could represent different transverse wave vectors for different modes, i.e., circle 2 is k T 2 = 0 for mode 2, but at the same time k T 2 = k T max 2 / ( m 1 ) for mode 1. m is an arbitrary number of circles drawn in the figure, in our case m = 5. The intersection is marked by two values, indicating the two modes that provide equal kT values. The lines marked P1, Z, N1, etc. are connection curves: P1 for mode 1 in the left fiber to mode 2 in the right fiber, Z for mode 1 in the left fiber to mode 1 in the right fiber, N1 for mode 2 in the left fiber to mode 1 in the right fiber, etc. At the same time P1 acts as the connection curve from mode 2 to mode 3, from mode 3 to mode 4, etc. as well as for all other curves as marked in the figure.

Fig. 5
Fig. 5

Energy transfer from mode x1 in the left fiber to the mode band (x2dx2) to x2 in the right fiber takes place in the hatched area.

Fig. 6
Fig. 6

Definition of energy transfer efficiency for coupling of two identical parabolic index fibers. Horizontal axis is either x2 or D ¯ = [ ρ ( x 2 ) ρ ( x 2 ) ] / a; vertical axis is the normalized fiber offset S ¯ = S / a.

Fig. 7
Fig. 7

Computation of the energy transfer efficiency from mode x1 in the left fiber to all modes smaller than x2 in the right fiber for three particular cases. The parabolic index fibers are identical but displaced one with respect to the other by normalized distance S ¯ (vertical axis). Horizontal axis is shown either in terms of mode x2 or D ¯ = ( 1 / a ) [ ρ ( x 2 ) ρ ( x 1 ) ]: (i) [ρ(x1)]/a = 1.0; (ii) [ρ(x1]/a = 0.3; (iii) [ρ(x1)]/a = 0.1.

Fig. 8
Fig. 8

Same as Fig. 7 but plotted for a fixed x2 mode in the right fiber. Energy transfer efficiency is shown as a function of mode number x1 (horizontal axis) and fiber displacement S ¯ (vertical axis): (i) [ρ(x2)]/a = 1.0; (ii) [ρ(x2)]/a = 0.6; (iii) [ρ(x2)]/a = 0.1.

Fig. 9
Fig. 9

Experimental and theoretical comparisons of losses [Fig. 9(i)] and transmission [Fig. 9(ii)] at splices of two identical parabolic index fibers in the presence of lateral displacement: dashed line, calculations based on this paper; solid line, Gloge2 steady-state calculations (available only for small fiber displacements); circles, experimental data from Miller.1

Fig. 10
Fig. 10

Energy transfer from mode x1 in the left fiber to a narrow range of modes in the right fiber. This range of modes is bound by the two existence circles of x2max and x2min which are tangential from outside and inside, respectively, to the circle defining area of existence of mode x1.

Fig. 11
Fig. 11

Energy transfer from mode x1 in the left fiber to mode x2 in the right fiber takes place along chord A 1 A 1. The two fibers are identical parabolic index fibers. Hatched area Ac represents the area of energy transfer from mode x1 in the left fiber to all modes smaller than x2 in the right fiber.

Equations (54)

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β x = n 0 k ( 1 2 Δ x 2 α / α + 2 ) 1 / 2 ,
n 2 ( r ) = { n 0 2 [ 1 2 Δ ( r / a ) α ] r < a , n 0 2 ( 1 2 Δ ) r a ,
n 0 k { 1 2 Δ [ ρ ( x ) / a ] α } 1 / 2 = n 0 k ( 1 2 Δ x 2 α / α + 2 ) 1 / 2 .
ρ ( x ) = a x 2 / α + 2 .
k T 1 = k T 2 ,
k T 2 = k 2 n 2 ( r ) β x 2 .
k T 2 = 2 k 2 n 0 2 Δ [ x 2 α / α + 2 ( r / a ) α ] .
n 01 2 Δ 1 [ x 1 2 α 1 / α 1 + 2 ( r 1 / a 1 ) α 1 ] = n 02 2 Δ 2 [ x 2 2 α 2 / α 2 + 2 ( r 2 / a 2 ) α 2 ] ,
F = ( n 0 1 2 Δ 1 / n 0 2 2 Δ 2 ) ,
( r 2 / a 2 ) α 2 F ( r 1 / a 1 ) α 1 = x 2 2 α 2 / α 2 + 2 F x 1 2 α 1 / α 1 + 2 .
S > ρ 1 ( x 1 ) + ρ 2 ( x 2 ) .
S < ρ 1 ( x 1 ) ρ 2 ( x 2 ) .
S < ρ 2 ( x 2 ) ρ 1 ( x 1 ) .
S < ρ 1 ( x 1 ) + ρ 2 ( x 2 ) , S > | ρ 1 ( x 1 ) ρ 2 ( x 2 ) | .
[ ( S u 1 ) / a 2 ] α 2 F ( u 1 / a 1 ) α 1 = 1 F .
k T max 2 = k 2 n 01 2 2 Δ 1 [ 1 ( u 1 / a 1 ) α 1 ] .
k T 1 2 ( i ) = i 1 m 1 k T max 2 i = 1 , 2 , , m .
r 1 ( i ) = a 1 [ 1 k T 1 2 ( i ) / ( 2 Δ 1 k 2 n 0 1 2 ) ] 1 / α 1 .
r 2 ( i ) = a 2 [ 1 k T 1 2 ( i ) / ( 2 Δ 2 k 2 n 0 2 2 ) ] 1 / α 2 .
k T 1 2 | x 1 = p 1 ( i ) r 1 = r 1 ( i ) = 0.
p 1 ( i ) = [ r 1 ( i ) / a 1 ] α 1 + 2 / 2 i = 1 , 2 , , m .
p 2 ( i ) = [ r 2 ( i ) / a 2 ] α 2 + 2 / 2 i = 1 , 2 , , m .
d P 2 ( x 1 , x 2 ) = d A c ( x 1 , x 2 ) A ( x 1 ) P 1 ( x 1 ) ,
P 2 | ( x < x 2 ) = A c ( x 1 , x 2 ) A ( x 1 ) P ( x 1 ) .
η ( x 1 , x < x 2 ) = P 2 ( x < x 2 ) P 1 ( x 1 ) 100 % .
D ¯ = ρ 2 ( x 2 ) ρ 1 ( x 1 ) a 1 ,
S ¯ = S / a 1 .
A ¯ = ρ 2 ( x 2 ) + ρ 1 ( x 1 ) a 1 .
S ¯ < D ¯ ,
S ¯ > A ¯ ,
S ¯ < D ¯ .
D ¯ = ρ ( x 2 ) ρ ( x 1 ) a = ρ ¯ 2 ρ ¯ 1 .
P ( x < x 2 ) = i P ( x 1 i ) η ( x 1 i , x 2 ) .
τ ( x < x 2 ) = P ( x < x 2 ) i P ( x 1 i ) .
x 2 min = ( ρ 2 min a 2 ) ( 2 + α 2 ) / 2 ,
x 2 max = ( ρ 2 max a 2 ) ( 2 + α 2 ) / 2 ,
Δ x 2 = min ( x 2 max , 1 ) x 2 min .
S < a 2 ρ 1 ( x 1 ) ,
min ( x 2 max , 1 ) = x 2 max .
x 2 = x 2 max x 2 min ,
Δ x 2 = [ ρ 1 ( x 1 ) a 2 ] ( 2 + α 2 ) / 2 ( 2 + α 2 ) S ρ 1 ( x 1 ) = ( 2 + α 2 ) [ ρ 1 ( x 1 ) a 2 ] α 2 / 2 S a 2 .
Δ x 2 = 4 S a 2 ρ 1 ( x 1 ) a 2 = 4 S a 1 a 2 2 x 1 .
x 2 = ( a 1 a 2 x 1 2 / ( 2 + α 1 ) ) ( 2 + α 2 ) / 2 | α 1 = α 2 = 2 = x 1 ( a 1 a 2 ) 2 .
x 1 ( r 1 / a ) 2 = x 2 ( r 2 / a ) 2 .
r 1 2 = u 2 + υ 2 ,
r 2 2 = ( u S ) 2 + υ 2 ,
u = ( x 1 x 2 ) a 2 + S 2 2 S .
S ¯ > | x 2 x 1 | = | D ¯ | ,
S ¯ < x 2 + x 1 = A ¯ .
A c = ( 2 γ / 2 π ) A ( x 1 ) Δ O 1 A 1 A 1 ,
cos γ = a 2 ( x 1 x 2 ) + S 2 2 S a x 1 = u 1 ρ ( x 1 ) .
A ( x 1 ) = π ρ 2 ( x 1 ) = a 2 x 1 ,
Δ O 1 A 1 A 1 = u 1 2 tan γ = [ ρ ( x 1 ) cos γ ] 2 tan γ = a 2 x 1 cos 2 γ tan γ = 1 2 a 2 x 1 sin 2 γ .
η % = { 0 S ¯ > A ¯ , 0 S ¯ < D ¯ , A c π a 2 x 1 100 = [ γ π sin ( 2 γ ) 2 π ] 100 A ¯ > S ¯ > | D ¯ | , 100 S ¯ < D ¯ .

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