Abstract

This paper deals with the investigation of the ray delay in gradient waveguides with any transverse refractive profile. A perturbation method is used for the calculations. It becomes apparent that the delay differences in the pure gradient medium are very small for any decreasing profile and that they are of an order of magnitude of those found in a monomode waveguide (material dispersion). Larger delay differences do, however, occur when the ray paths are to some extent determined by an outer cladding. These influences must be eliminated to allow the excellent delay time properties of a gradient medium to really be utilized.

© 1974 Optical Society of America

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References

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  1. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1965).
  3. K.-H. Steiner, Nachrichtentechn. Zeitschrift, NTZ. 26, 468 (1973).
  4. R. Bouillie, K.-H. Steiner, M. Tréheux, Optoelectron. 5, 457 (1973).
  5. S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
    [CrossRef]
  6. N. S. Kapany, Fiber Optics (Academic Press, New York, 1967).
  7. R. B. Dyott, J. R. Stern, Electron. Lett. 7, 82 (1971).
    [CrossRef]

1973

K.-H. Steiner, Nachrichtentechn. Zeitschrift, NTZ. 26, 468 (1973).

R. Bouillie, K.-H. Steiner, M. Tréheux, Optoelectron. 5, 457 (1973).

1971

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 82 (1971).
[CrossRef]

1970

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

1968

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1965).

Bouillie, R.

R. Bouillie, K.-H. Steiner, M. Tréheux, Optoelectron. 5, 457 (1973).

Dyott, R. B.

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 82 (1971).
[CrossRef]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Kapany, N. S.

N. S. Kapany, Fiber Optics (Academic Press, New York, 1967).

Kawakami, S.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Nishizawa, J.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Steiner, K.-H.

K.-H. Steiner, Nachrichtentechn. Zeitschrift, NTZ. 26, 468 (1973).

R. Bouillie, K.-H. Steiner, M. Tréheux, Optoelectron. 5, 457 (1973).

Stern, J. R.

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 82 (1971).
[CrossRef]

Tréheux, M.

R. Bouillie, K.-H. Steiner, M. Tréheux, Optoelectron. 5, 457 (1973).

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1965).

Electron. Lett.

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 82 (1971).
[CrossRef]

IEEE J. Quantum Electron.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Nachrichtentechn. Zeitschrift, NTZ.

K.-H. Steiner, Nachrichtentechn. Zeitschrift, NTZ. 26, 468 (1973).

Optoelectron.

R. Bouillie, K.-H. Steiner, M. Tréheux, Optoelectron. 5, 457 (1973).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1965).

N. S. Kapany, Fiber Optics (Academic Press, New York, 1967).

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Figures (5)

Fig. 1
Fig. 1

Ray paths in the gradient medium without cladding.

Fig. 2
Fig. 2

Refractive index profiles.

Fig. 3
Fig. 3

Examples of pulse broadenings in the gradient medium.

Fig. 4
Fig. 4

Ray paths in the gradient medium with cladding: (a) with axial-symmetric injection, (b) with axial-asymmetric injection.

Fig. 5
Fig. 5

Pulse distortions caused by cladding influences: (a) with axial-symmetric injection, (b) with edge injection (ρ0 = θ0/√α).

Equations (33)

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d 2 ρ / d z 2 = [ 1 + ( d ρ / d z ) 2 ] [ 1 / n ( ρ ) ] [ n ( ρ ) / ρ ] .
n ( ρ ) = n 1 ( 1 - ½ α ρ 2 + ¼ β ρ 4 ) .
( d 2 ρ / d z 2 ) + [ 1 + ( d ρ / d z ) 2 ] ( α ρ - β * ρ 3 ± ) = 0 ,
β * = β - ½ α 2
ρ ( z ) = θ * f ( 0 ) ( z ) + θ * 3 f ( 2 ) ( z ) +
( d 2 f ( 0 ) / d z 2 ) + α f ( 0 ) = 0 ,
( d 2 f ( 2 ) / d z 2 ) + α f ( 2 ) = β * f ( 0 ) 3 - α f ( 0 ) ( d f ( 0 ) / d z ) 2 ,
ρ ( z ) = θ / α ( { 1 + ( θ 2 / 32 ) [ 9 ( β / α 2 ) - ( 5 / 6 ) ] } sin α z + ( θ 2 / 32 ) [ ( β / α 2 ) + ( 1 / 2 ) ] sin 3 α z - ( θ 2 / 8 ) [ 3 ( β / α 2 ) - ( 5 / 2 ) z α cos α z + 0 ( θ 4 ) ) .
τ = 1 c 0 L n [ 1 + ( d ρ / d z ) 2 ] 1 / 2 d z ,
τ = τ 0 [ 1 + g 1 ( α ) θ 2 + g 2 ( α , β ) θ 4 + g 3 ( α , β , γ ) θ 6 + ] ,
τ 0 = ( L / c ) n 1
g 1 ( α ) = 0 ,
g 2 ( α , β ) = 3 / 32 [ ( 5 / 6 ) - ( β / α 2 ) ] ,
τ = τ 0 { 1 + ( 3 / 32 ) [ ( 5 / 6 ) - ( β / α 2 ) ] θ 4 }
n / n 1 = 1 / ( 1 + α ρ 2 ) 1 / 2 = 1 - ½ α ρ 2 + α 2 ρ 4 , β = / 2 3 α 2 Δ τ = - / 16 1 τ 0 θ 4 .
n / n 1 = 1 / ( 1 + ½ α ρ 2 ) = 1 - ½ α ρ 2 + ¼ α 2 ρ , β = α 2 Δ τ = - / 64 1 τ 0 θ 4 .
n / n 1 = 1 / cosh α ρ = 1 - ½ α ρ 2 + / 24 5 α 2 ρ 4 , β = α 2 Δ τ = 0.
n / n 1 = 1 - ½ α ρ 2 , β = 0 Δ τ = + / 64 5 τ 0 θ 4 .
n / n 1 = ( 1 - α ρ 2 ) 1 / 2 = 1 - ½ α ρ 2 - α 2 ρ 4 - , β = - ( α 2 / 2 ) Δ τ = + τ 0 θ 4 .
F ( t ) = 1 1 - cos θ 0 0 θ 0 f ( t - τ ) sin θ d θ .
τ = τ 0 ( 1 + a θ 4 ) .
F ( t ) = ( 1 / τ 0 ) ( 1 / 2 a θ 0 2 ) rect [ τ 0 , τ 0 ( 1 + a θ 0 4 ) ] { 1 / ( t / τ 0 ) - 1 ] 1 / 2 } ,
F ( t ) = ( 1 / τ 0 ) ( 1 / 2 a θ 0 2 ) rect [ τ 0 ( 1 - a θ 0 4 ) , τ 0 ] { 1 / [ 1 - ( t / τ 0 ) ] 1 / 2 } .
τ = τ 0 [ 1 + ( θ 2 / 2 ) ( sin2 α z * / 2 α z * ) ] .
z * = ( 1 / α ) [ ( π / 2 ) - { 2 [ 1 - θ 0 / θ ) ] } 1 / 2 ]
τ = τ 0 ( 1 + ( θ 2 / 2 ) × ( 2 / π ) { 2 [ 1 - θ 0 / θ ) ] } 1 / 2 ) .
θ ( α ρ 0 2 + θ 2 ) 1 / 2 .
( α ρ 0 2 + θ 2 ) 1 / 2 = θ 0 + φ .
F ( t ) = 1 1 - cos θ 0 0 θ 0 δ ( t - τ ) sin θ d θ 2 θ 0 2 [ ½ ( θ 0 2 - α ρ 0 2 ) δ ( t - τ 0 ) + Γ ]
Γ = 0 φ 0 δ ( t - τ ) ( θ 0 + φ ) d φ .
θ 0 = ( θ 0 2 - α ρ 0 2 + 2 θ 0 φ 0 + φ 0 2 ) 1 / 2 .
Γ = 1 τ 0 π 2 θ 0 2 τ 0 τ 0 [ 1 + θ 0 2 π ( 2 φ 0 θ 0 ) 1 / 2 ] δ ( t - τ ) ( τ τ 0 - 1 ) d τ = 1 τ 0 π 2 θ 0 2 rect { τ 0 , τ 0 [ 1 + θ 0 2 π ( 2 φ 0 θ 0 ) 1 / 2 ] } ( t τ 0 - 1 ) ,
Γ = 1 τ 0 π 2 2 τ 0 τ 0 ( 1 + 2 π φ 0 2 ) δ ( t - τ ) d τ = 1 τ 0 π 2 2 × rect { τ 0 , t , τ 0 ( 1 + 2 π φ 0 2 ) } .

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