Abstract

Changes in transmission characteristics caused by an outer layer have been investigated for graded-index fibers. Equations have been derived using the WKB method for calculating baseband frequency response and excess loss in a general type of the graded-index fiber. Numerical examples have been given for a square-law fiber, mainly with cladding thickness and index difference between outer layer and cladding as parameters. Measurement of the excess loss has been carried out for fibers with various cladding thickness and outer layer-cladding index differences. The measured loss is in agreement with the theoretical value.

© 1978 Optical Society of America

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References

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  1. A. H. Cherin, E. J. Murphy, Bell Syst. Tech. J. 54, 1531 (1975).
  2. N. Kashima, N. Uchida, Y. Ishida, Appl. Opt. 16, 2732 (1977).
    [CrossRef] [PubMed]
  3. N. Kashima, N. Uchida, Appl. Opt. 16, 1038 (1977).
    [CrossRef] [PubMed]
  4. N. Kashima, N. Uchida, Appl. Opt. 16, 1320 (1977).
    [CrossRef] [PubMed]
  5. K. Petermann, A.E.Ü. 29, 345 (1975).
  6. A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
    [CrossRef]
  7. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  8. L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).
  9. M. Aramowitz, K. Stegun, Eds. Handbook of Mathematical Functions (Dover, New York, 1965).
  10. D. Marcuse, Theory of Dielectric Optical Waveguide (Academic, New York, 1974).

1977 (3)

1975 (2)

K. Petermann, A.E.Ü. 29, 345 (1975).

A. H. Cherin, E. J. Murphy, Bell Syst. Tech. J. 54, 1531 (1975).

1974 (1)

1971 (1)

A.E.Ü. (1)

K. Petermann, A.E.Ü. 29, 345 (1975).

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

A. H. Cherin, E. J. Murphy, Bell Syst. Tech. J. 54, 1531 (1975).

J. Opt. Soc. Am. (1)

Other (3)

L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).

M. Aramowitz, K. Stegun, Eds. Handbook of Mathematical Functions (Dover, New York, 1965).

D. Marcuse, Theory of Dielectric Optical Waveguide (Academic, New York, 1974).

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Figures (15)

Fig. 1
Fig. 1

Cross section and refractive-index profile of an optical fiber with a three-layer structure.

Fig. 2
Fig. 2

Calculated excess loss for each mode as a function of mode number v with compound mode number m as a parameter.

Fig. 3
Fig. 3

Calculated excess loss for each mode group as a function of normalized propagation angle θ/θ c with wavelength λ and index difference Δ as parameters.

Fig. 4
Fig. 4

Calculated excess loss as a function of propagation length with Gaussian parameter G as a parameter.

Fig. 5
Fig. 5

Calculated normalized output power as a function of propagation length Z with jacket-cladding index difference Δ′ as a parameter.

Fig. 6
Fig. 6

Calculated normalized output power as a function of Z with cladding thickness t as a parameter.

Fig. 7
Fig. 7

Calculated baseband response as a function of frequency with Δ' as a parameter: (a) amplitude response A(Z,f) and (b) phase response ϕ(Z,f).

Fig. 8
Fig. 8

Calculated baseband amplitude response A(Z,f) as a function of frequency with Δ′ as a parameter.

Fig. 9
Fig. 9

Calculated baseband amplitude response A(Z,f) as a function of frequency with t as a parameter.

Fig. 10
Fig. 10

Calculated 3-dB bandwidth as a function of Z with G and Δ′ as parameters.

Fig. 11
Fig. 11

Calculated excess loss Δα and group delay difference δτ for higher modes as a function of mode number ν with compound mode number m as a parameter.

Fig. 12
Fig. 12

Calculated 3-dB bandwidth and excess loss as a function of Δ′.

Fig. 13
Fig. 13

Relation between 3-dB bandwidth B and excess loss α for fibers with various values of Δ′ and t.

Fig. 14
Fig. 14

Measured and calculated excess loss as a function of cladding thickness t.

Fig. 15
Fig. 15

Measured and calculated excess losses as a function of Δ′.

Tables (2)

Tables Icon

Table I Comparison of Excess Loss Between Graded- and Step-Index Fibers

Tables Icon

Table II Fiber Parameters used for the Calculation in Fig. 13

Equations (39)

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n = { n ( r ) ( 0 r a ) n 2 ( a r b ) n ̂ 3 ( r b ) .
E x = f ( r ) ( sin ν ϕ cos ν ϕ ) exp ( j ω t j β z ) ,
d 2 f ( r ) d r 2 + 1 r d f ( r ) d r + p ( β , r ) f ( r ) = 0 ,
p ( β , r ) = k 2 n 2 β 2 ν 2 / r 2 , with k = 2 π / λ .
p ( β , r ) = 0 .
f 1 ( r ) = c 1 ( Q ) 1 / 4 exp [ ω ω 1 ( Q ) 1 / 2 d ω ] ( 0 r < r 1 ) f 2 ( r ) = 2 ( π ) 1 / 2 c 1 ( ξ / Q ) 1 / 4 A i ( ξ ) ( r r 1 ) f 3 ( r ) = 2 c 1 Q 1 / 4 sin [ ω 1 ω ( Q ) 1 / 2 d ω + π / 4 ] ( r 1 < r < r 2 ) f 4 ( r ) = ( η / Q ) 1 / 4 [ c 2 A i ( η ) + c 3 B i ( η ) ] ( r r 2 ) f 5 ( r ) = ( 4 π ) 1 / 2 ( Q ) 1 / 4 { c 2 exp [ ω 2 ω ( Q ) 1 / 2 d ω ] + 2 c 3 exp [ ω 2 ω ( Q ) 1 / 2 d ω ] } ( r 2 < r a ) f 6 ( r ) = c 4 H ν ( 1 ) ( j γ r ) + c 5 H ν ( 2 ) ( j γ r ) ( a r b ) f 7 ( r ) = c 6 H ν ( 1 ) ( j ρ r ) ( r b ) , }
ξ = [ 3 2 ω 1 ω ( Q ) 1 / 2 d ω ] 2 / 3 = [ 3 2 ω ω 1 ( Q ) 1 / 2 d ω ] 2 / 3 , η = [ 3 2 ω ω 2 ( Q ) 1 / 2 d ω ] 2 / 3 = [ 3 2 ω 2 ω ( Q ) 1 / 2 d ω ] 2 / 3 , }
Q = r 2 p ( β , r ) , γ 2 = β 2 n 2 2 k 2 , p 2 = β 2 n ̂ 3 2 k 2 .
H ν ( 1 ) , H ν ( 2 )
r 1 r 2 [ p ( β 0 , r ) ] 1 / 2 d r = ( μ 1 2 ) π ( μ = 1 , 2 , 3 , . . . ) ,
Δ β j Δ α β β 0 = ( δ / β 0 ) { r 1 r 2 d r [ p ( β 0 , r ) ] 1 / 2 } 1 ,
δ = R + 1 2 ( R 1 ) exp [ 2 ω 2 0 ( Q ) 1 / 2 d ω ] , R = j γ a ( Q ) 1 / 2 H ν ( 1 ) ( j γ a ) + H ν ( 2 ) ( j γ a ) S H ν ( 1 ) ( j γ a ) + H ν ( 2 ) ( j γ a ) S ,
S = ρ H ν ( 1 ) ( j γ b ) H ν ( 1 ) ( j ρ b ) γ H ν ( 1 ) ( j ρ b ) H ν ( 1 ) ( j γ b ) γ H ν ( 1 ) ( j ρ b ) H ν ( 2 ) ( j γ b ) ρ H ν ( 2 ) ( j γ b ) H ν ( 1 ) ( j ρ b ) .
n ( r ) = n 0 [ 1 2 Δ ( r / a ) 2 ] 1 / 2 ,
Δ = ( n 0 n 2 ) / n 0 .
Δ ( n 0 n 2 ) / n 2 1 .
β 0 = [ ( n 0 k ) 2 2 h m ] 1 / 2
Δ β j Δ α = H π β 0 ( 1 + R 1 R ) exp ( 2 T ) ,
m = 2 μ + ν 1 , T = q 2 e 2 4 h log | 2 h 2 a 2 e 2 + 2 h q u | + ν 2 log | ( h a 2 q ) + ν ( h a 2 q ) ν · ( e 2 + u ) 2 ν h ( e 2 + u ) + 2 ν h | , q 2 = h 2 a 4 e 2 a 2 + ν 2 , u 2 = e 4 4 h 2 ν 2 , e 2 = k 2 n 0 2 β 0 2 , h 2 = 2 Δ k 2 n 0 2 / a 2 . }
τ = 1 c d β d k = τ 0 + Δ τ ,
τ 0 = 1 c d β 0 d k , and Δ τ = 1 c d ( Δ β ) d k .
P i ( Z , f ) = P i ( 0 , 0 ) exp [ 2 ( Δ α i + j π τ i f ) Z ] ,
H ( Z , f ) = i P i ( Z , f ) i P i ( 0 , 0 ) .
α ( Z ) = H ( Z , 0 ) .
H ( Z , f ) = A ( Z , f ) exp [ j ϕ ( Z , f ) ] .
τ 0 = n 0 c ( 1 2 m ) 1 / 2 ( 1 m ) ,
= h / ( n 0 k ) 2 ,
Δ τ τ 0 .
β = n 0 k cos θ n 0 k ( 1 θ 2 ) 1 / 2 .
Δ β β 0
θ 2 = 2 m .
P m ( 0 , 0 ) = θ 2 exp [ ( θ / G θ c ) 2 ] ,
θ c = ( 2 Δ ) 1 / 2 .
Δ = ( n 3 n 2 ) / n 2 .
β c = n 3 k
B · Z = constant .
τ = τ 1 + δ τ ,
τ 1 = 4.19 × 10 9 ( sec / m ) .
B 0.8 α + 2.5 ,

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