Abstract

A method is presented for calculating offset and tilt losses for fiber splices with axially symmetric arbitrary-index profiles by approximating the profile with a staircase function. This method is applied to a large-core dual-mode fiber with zero intermodal dispersion as well as to single-mode fibers with step- and parabolic-index profiles. When a splice loss of 0.2 dB is permitted, the normalized offset misalignment is found to be DN = 0.635 for the dual-mode fiber at normalized frequency υ = 4.605 and a power-law exponent α = 4.5. The DN value compares favorably with the values 0.560 and 0.614 for conventional step- and parabolic-index single-mode fibers, respectively. The dual-mode fiber is superior to the step- and parabolic-index fibers with respect to permissible splice offset tolerances.

© 1978 Optical Society of America

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References

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  1. J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).
  2. D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).
  3. E. G. Neumann, W. Weidhaas, Arch. Elektron. Übertragungstech. 30, 448 (1976).
  4. P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
    [CrossRef]
  5. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  6. J. Sakai, T. Kimura, Appl. Opt. 17, 1499 (1978).
    [CrossRef] [PubMed]
  7. J. Sakai, T. Kimura, Opt. Lett. 1, 169 (1977).
    [CrossRef] [PubMed]
  8. G. Goubau, Proc. IRE 40, 865 (1952).
    [CrossRef]
  9. M. Abramowitz, I. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 485.
  10. H. Tsuchiya, IECE Jpn. OQE 73-61, 1 (1973).
  11. W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
    [CrossRef]

1978 (1)

1977 (3)

J. Sakai, T. Kimura, Opt. Lett. 1, 169 (1977).
[CrossRef] [PubMed]

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[CrossRef]

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

1976 (1)

E. G. Neumann, W. Weidhaas, Arch. Elektron. Übertragungstech. 30, 448 (1976).

1973 (2)

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

H. Tsuchiya, IECE Jpn. OQE 73-61, 1 (1973).

1971 (1)

1970 (1)

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

1952 (1)

G. Goubau, Proc. IRE 40, 865 (1952).
[CrossRef]

Chan, K. B.

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

Clarricoats, P. J. B.

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

Cook, J. S.

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

Gambling, W. A.

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[CrossRef]

Gloge, D.

Goubau, G.

G. Goubau, Proc. IRE 40, 865 (1952).
[CrossRef]

Grow, R. J.

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

Kimura, T.

Mammel, W. L.

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

Matsumura, H.

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[CrossRef]

Neumann, E. G.

E. G. Neumann, W. Weidhaas, Arch. Elektron. Übertragungstech. 30, 448 (1976).

Payne, D. N.

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[CrossRef]

Sakai, J.

Tsuchiya, H.

H. Tsuchiya, IECE Jpn. OQE 73-61, 1 (1973).

Weidhaas, W.

E. G. Neumann, W. Weidhaas, Arch. Elektron. Übertragungstech. 30, 448 (1976).

Appl. Opt. (2)

Arch. Elektron. Übertragungstech. (1)

E. G. Neumann, W. Weidhaas, Arch. Elektron. Übertragungstech. 30, 448 (1976).

Bell Syst. Tech. J. (2)

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

Electron. Lett. (2)

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[CrossRef]

IECE Jpn. (1)

H. Tsuchiya, IECE Jpn. OQE 73-61, 1 (1973).

Opt. Lett. (1)

Proc. IRE (1)

G. Goubau, Proc. IRE 40, 865 (1952).
[CrossRef]

Other (1)

M. Abramowitz, I. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 485.

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Figures (8)

Fig. 1
Fig. 1

Approximation of refractive-index profile with a staircase function. n o and n e denote the indices at the core center and in the cladding, respectively. Arrows at Q i and G i indicate regions where propagation and boundary matrices are defined.

Fig. 2
Fig. 2

Cross section of butt-joined fibers with an offset misalignment. O s and O r are the fiber axes of the sending and receiving fibers, respectively. The hatched region indicates the small area where fields are regarded to be constant.

Fig. 3
Fig. 3

Cross-section of butt-joined fibers with an angular misalignment. φ is the angle between the axes of the two joining fibers.

Fig. 4
Fig. 4

Offset and tilt losses for the LP01 mode in a step-index single-mode fiber with υ = 0.6υ c 1, 0.8υ c 1, and υ c 1, where υ c 1 = 2.405 is the LP11 mode cutoff.

Fig. 5
Fig. 5

Offset and tilt losses for the LP01 mode in a parabolic index-profile fiber. υ c 1 = 3.518.

Fig. 6
Fig. 6

Coupling efficiencies between two modes as a function of normalized offset misalignment in a dual-mode fiber. υ = υ c 2 = 4.605 is chosen for a power-law exponent α = 4.5, where υ c 2 denotes the LP21 mode cutoff.

Fig. 7
Fig. 7

Coupling efficiencies between two modes as a function of normalized angular misalignment in a dual-mode fiber. υ = υ c 2 = 4.605 and α = 4.5.

Fig. 8
Fig. 8

Offset and tilt losses for a dual-mode fiber with α = 4.5 and υ = υ c 2 = 4.605.

Equations (26)

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s E s = r C r e r .
η T r = P r P T = | s [ S ( E s × h r * ) · e ˆ z d S ] | 2 [ s S ( E s × H s * ) · e ˆ z d S ] s ( e r × h r * + ) · e ˆ z d S .
η s r = P r P s = | S [ ( s E s ) × h r * ] · e ˆ z d S | 2 S ( E s × H s * ) · e ˆ z d S · s ( e r × h r * ) · e ˆ z d S .
η 0,1 0,1 = η T 0 + η T 1 .
[ j E z E y ] r = a i = [ d 11 , d 12 d 21 , d 22 ] [ A i B i ] .
d 11 = ( κ i / 2 β g ) [ J ν + 1 sin ( ν + 1 ) θ + J ν 1 sin ( ν 1 ) θ ] , d 12 = ( κ i / 2 β g ) [ N ν + 1 sin ( ν + 1 ) θ + N ν 1 sin ( ν 1 ) θ ] , d 21 = J ν cos ν θ , d 22 = N ν cos ν θ , }
κ i = ( n i 2 k 2 β g 2 ) 1 / 2 ,
k = 2 π / λ ,
d 11 = ( γ i / 2 β g ) [ K ν + 1 sin ( ν + 1 ) θ K ν 1 sin ( ν 1 ) θ ] , d 12 = ( γ i / 2 β g ) [ I ν + 1 sin ( ν + 1 ) θ I ν 1 sin ( ν 1 ) θ ] , d 21 = K ν cos ν θ , d 22 = I ν cos ν θ , }
γ i = ( β g 2 n i 2 k 2 ) 1 / 2 ,
H x = β g k ( 0 μ 0 ) 1 / 2 E y ,
I s = S ( E s × H s * ) · e ˆ z d S .
I s = 0 2 π 0 ( E x H y * E y H x * ) r d r d θ = π S ν β g k ( 0 μ 0 ) 1 / 2 W
S ν = { 2 ; ν = 0 1 ; otherwise
W = i W i .
I s r = s ( E s × h r * ) · e ˆ z d S .
I s r = I s r N + I s r A ,
I s r N = 0 2 π 0 ρ a E y h x * R d R d Ө ,
I s r A = 0 2 π ρ a E y h x * R d R d Ө .
I s r A = π A e s A e r C s r K s r
C s r = { 2 ; ν s = ν r = 0 , 1 ; ν s = ν r 0 , 0 ; otherwise ,
K s r = { ( ρ a ) 2 / 2 [ K ν 2 ( γ e s ρ a ) , K ν 1 ( γ e s ρ a ) K ν + 1 ( γ e s ρ a ) ] ; γ e s = γ e r ρ a / ( γ e s 2 γ e r 2 ) [ γ e s K ν + 1 ( γ e s ρ a ) K ν ( γ e r ρ a ) , γ e r K ν ( γ e s ρ a ) K ν + 1 ( γ e r ρ a ) ] ; γ e s γ e r .
I T = π a 2 0 f ( R r / a ) g ( R r / a ) [ ( j ) ν s + ν r J ν s + ν r ( β g φ R r ) + ( j ) ν s ν r J ν s ν r ( β g φ R r ) ] R r / a · d R r / a
0 π ( cos n θ ) exp ( j z cos θ ) d θ = π ( j ) n J n ( z ) .
η 0,1 0,1 = [ ( P 0 η 0 0 ) 1 / 2 + ( P 1 η 1 0 ) 1 / 2 ] 2 + [ ( P 0 η 0 1 ) 1 / 2 + ( P 1 η 1 1 ) 1 / 2 ] 2 P 0 + P 1 ,
η 0,1 0,1 = P 0 ( η 0 0 + η 0 1 ) + P 1 ( η 1 0 + η 1 1 ) P 0 + P 1 .

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