Abstract

Parabolic grading of the core index in a multimode fiber (Selfoc) diminishes mode dispersion and interface loss. This paper shows that this grading affects the mode volume and the loss in bends very little, if the index difference of the graded core (between the core axis and the cladding) is twice that of the homogeneous core. Curvature radii of several centimeters are tolerable. Mode coupling (or ray deflection) in random bends is slightly decreased by grading. Both the graded and the homogeneous multimode fiber are particularly sensitive to certain critical deviations of the guide axis from straightness. These deviations must be less than a fraction of a micrometer in order that catastrophic mode loss be avoided.

© 1972 Optical Society of America

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References

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  1. T. Uchida, M. Furukawa, I. Kitano, K. Kaizuki, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
    [Crossref]
  2. E. G. Rawson, D. R. Herriott, J. McKenna, Appl. Opt. 9, 753 (1970).
    [Crossref] [PubMed]
  3. S. Kawakami, T. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1969).
  4. D. Gloge, Proc. IEEE 58, 1513 (1970).
    [Crossref]
  5. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
  6. E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).
  7. V. V. Shevchenko, Izv. Vuz. Radiofiz. 14, 768 (1971).
  8. E. A. J. Marcatili, Bell Syst. Tech. J. 49, 1645 (1970).
  9. H. I. Heyke, H. Kirchhoff, H. G. Unger, “Analysis of Optical Wave Launching and Propagation in Monomode and Multimode Fibers with Imperfections,” Topical Meeting on Integrated Optics-Guided Waves, Materials, and Devices, 7–10 February 1972, Las Vegas, Nevada.
  10. E. A. J. Marcatili, SPIE J. 8, 101 (1970).
  11. H. G. Unger, Arch. Elektrischen Uebertragung 19, 189 (1964).
  12. H. G. Unger, Arch. Elektrischen Uebertragung 24, 101 (1970).
  13. D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).
  14. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [Crossref] [PubMed]
  15. D. Gloge, D. Weiner, Bell Syst. Tech. J. 47, 2095 (1966).
  16. E. A. J. Marcatili, BTL; private communication.

1971 (3)

V. V. Shevchenko, Izv. Vuz. Radiofiz. 14, 768 (1971).

D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).

D. Gloge, Appl. Opt. 10, 2252 (1971).
[Crossref] [PubMed]

1970 (6)

H. G. Unger, Arch. Elektrischen Uebertragung 24, 101 (1970).

E. A. J. Marcatili, Bell Syst. Tech. J. 49, 1645 (1970).

E. A. J. Marcatili, SPIE J. 8, 101 (1970).

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

E. G. Rawson, D. R. Herriott, J. McKenna, Appl. Opt. 9, 753 (1970).
[Crossref] [PubMed]

D. Gloge, Proc. IEEE 58, 1513 (1970).
[Crossref]

1969 (3)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).

S. Kawakami, T. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1969).

1966 (1)

D. Gloge, D. Weiner, Bell Syst. Tech. J. 47, 2095 (1966).

1964 (1)

H. G. Unger, Arch. Elektrischen Uebertragung 19, 189 (1964).

Furukawa, M.

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

Gloge, D.

D. Gloge, Appl. Opt. 10, 2252 (1971).
[Crossref] [PubMed]

D. Gloge, Proc. IEEE 58, 1513 (1970).
[Crossref]

D. Gloge, D. Weiner, Bell Syst. Tech. J. 47, 2095 (1966).

Herriott, D. R.

Heyke, H. I.

H. I. Heyke, H. Kirchhoff, H. G. Unger, “Analysis of Optical Wave Launching and Propagation in Monomode and Multimode Fibers with Imperfections,” Topical Meeting on Integrated Optics-Guided Waves, Materials, and Devices, 7–10 February 1972, Las Vegas, Nevada.

Kaizuki, K.

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

Kawakami, S.

S. Kawakami, T. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1969).

Kirchhoff, H.

H. I. Heyke, H. Kirchhoff, H. G. Unger, “Analysis of Optical Wave Launching and Propagation in Monomode and Multimode Fibers with Imperfections,” Topical Meeting on Integrated Optics-Guided Waves, Materials, and Devices, 7–10 February 1972, Las Vegas, Nevada.

Kitano, I.

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

Marcatili, E. A. J.

E. A. J. Marcatili, SPIE J. 8, 101 (1970).

E. A. J. Marcatili, Bell Syst. Tech. J. 49, 1645 (1970).

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).

E. A. J. Marcatili, BTL; private communication.

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).

Matsumura, H.

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

McKenna, J.

Miller, S. E.

E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).

Nishizawa, T.

S. Kawakami, T. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1969).

Rawson, E. G.

Shevchenko, V. V.

V. V. Shevchenko, Izv. Vuz. Radiofiz. 14, 768 (1971).

Uchida, T.

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

Unger, H. G.

H. G. Unger, Arch. Elektrischen Uebertragung 24, 101 (1970).

H. G. Unger, Arch. Elektrischen Uebertragung 19, 189 (1964).

H. I. Heyke, H. Kirchhoff, H. G. Unger, “Analysis of Optical Wave Launching and Propagation in Monomode and Multimode Fibers with Imperfections,” Topical Meeting on Integrated Optics-Guided Waves, Materials, and Devices, 7–10 February 1972, Las Vegas, Nevada.

Weiner, D.

D. Gloge, D. Weiner, Bell Syst. Tech. J. 47, 2095 (1966).

Appl. Opt. (2)

Arch. Elektrischen Uebertragung (2)

H. G. Unger, Arch. Elektrischen Uebertragung 19, 189 (1964).

H. G. Unger, Arch. Elektrischen Uebertragung 24, 101 (1970).

Bell Syst. Tech. J. (5)

D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).

D. Gloge, D. Weiner, Bell Syst. Tech. J. 47, 2095 (1966).

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).

E. A. J. Marcatili, Bell Syst. Tech. J. 49, 1645 (1970).

IEEE J. Quantum Electron. (1)

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

IEEE Trans. Microwave Theory Tech. (1)

S. Kawakami, T. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1969).

Izv. Vuz. Radiofiz. (1)

V. V. Shevchenko, Izv. Vuz. Radiofiz. 14, 768 (1971).

Proc. IEEE (1)

D. Gloge, Proc. IEEE 58, 1513 (1970).
[Crossref]

SPIE J. (1)

E. A. J. Marcatili, SPIE J. 8, 101 (1970).

Other (2)

H. I. Heyke, H. Kirchhoff, H. G. Unger, “Analysis of Optical Wave Launching and Propagation in Monomode and Multimode Fibers with Imperfections,” Topical Meeting on Integrated Optics-Guided Waves, Materials, and Devices, 7–10 February 1972, Las Vegas, Nevada.

E. A. J. Marcatili, BTL; private communication.

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Figures (9)

Fig. 1
Fig. 1

Sketch of two cladded multimode fibers (a) with a homogeneous core and (b) with a graded core (parabolic index profile). The broken lines show possible guided and unguided rays. The critical ray is indicated by a solid line.

Fig. 2
Fig. 2

Curved dielectric guide and its cladding field distribution at the outside of the bend (after Shevchenko7).

Fig. 3
Fig. 3

Curvature loss in dB/km vs the relative mode angle for the graded and the flat index profile. Core diameter 0.1 mm, curvature radius R = 1 cm, wavelength λ = 1 μm, relative index difference Δ = 1%.

Fig. 4
Fig. 4

Sketch of a curved dielectric guide showing the critical ray (a) for a flat and (b) for a graded core profile.

Fig. 5
Fig. 5

Ray deflection in a curvature increment c(z)dz (a) for a graded core and (b) for a homogeneous core.

Fig. 6
Fig. 6

Plot of the function rec(x) and tri(x) as used in the text.

Fig. 7
Fig. 7

Range of the double integration of Eq. (32). The rec-product in Eq. (32) is +1 in the white and −1 in the shaded areas. Also shown is the u,v-coordinate system used in Eq. (34).

Fig. 8
Fig. 8

Spectrum of possible guide curvatures and probability distribution of possible ray periods to be expected in a ray injected off-axis. The functional dependence is strictly speculative and merely meant for illustration.

Fig. 9
Fig. 9

Sinusoidal deviation from guide straightness used to interpret the curvature spectrum.

Equations (57)

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θ c = [ 1 - ( n c / n ) 2 ] 1 2 ( 2 Δ ) 1 2 .
P f = 1 2 ( a k n θ c ) 2 = ( a k n ) 2 Δ ,
n ( r ) = n [ 1 - Δ ( r 2 / a 2 ) ] ,
P g = ( a k n ) 2 ( Δ / 2 ) = P f 12
p f = ( 2 a k n / π ) θ
p g = ( a k n / 2 ) θ
α = 2 γ 2 ( 0 ) β exp [ - 2 a r 0 γ ( r ) d ] ,
γ 2 ( r ) = β 2 R 2 / ( r + R ) 2 - n c 2 k 2 ,
r 0 = R β / n c k - R ,
exp [ - a r γ ( r ) d r ] .
γ 2 ( r ) = γ 2 ( 0 ) - 2 β 2 r / R .
α = 2 γ 2 ( 0 ) n k exp [ - 2 3 n k R ( γ 2 ( 0 ) n 2 k 2 - 2 a R ) / 2 3 ] .
β f 2 = ( n k ) 2 - ( π p f / 2 a ) 2
γ f 2 ( 0 ) = ( n k ) 2 ( θ c 2 - θ 2 ) .
α f = 2 n k ( θ c 2 - θ 2 ) exp [ - 2 3 n k R ( θ c 2 - θ 2 - 2 a R ) / 2 3 ] .
β g = n k - ( 2 Δ ) 1 2 ( p g + 1 2 ) a
γ g 2 ( 0 ) = ( n k ) 2 ( θ c 2 - θ θ c + 1 4 θ c 2 θ 2 ) .
α g = 2 n k ( θ c 2 - θ c θ ) exp [ - 2 3 n k R ( θ c 2 - θ c θ - 2 a R ) / 2 3 ] .
θ f = θ c ( 1 - 2 a / R θ c 2 ) 1 2
θ g = θ c ( 1 - 2 a / R θ c 2 )
cos θ f = ( 1 + a / R ) cos θ c .
θ f = θ c ( 1 - 2 a / R θ c 2 ) 1 2 ,
r ( z ) = a 2 R θ c 2 + ( a - a 2 R θ c 2 ) sin z θ c / a .
θ f = d r d z | r = 0 = θ c ( 1 - 2 a / R θ c 2 ) 1 2 ,
y f = ( θ c 2 - θ f 2 ) / θ c 2 = a / R Δ
y g = ( θ c 2 - θ g 2 ) / θ c 2 = 2 a / R Δ .
c ( z ) = 1 / R ( z )
r ( L ) = a θ c 0 L sin [ θ c a ( L - c ) ] c ( z ) d z .
θ g ( L ) = 0 L cos [ θ c a ( L - z ) ] c ( z ) d z .
θ ( L ) 0 = θ 0 + 0 L rec [ θ ( z ) ( L - z ) / a ] c ( z ) d z ,
θ ( L ) = θ 0 ,
c ( z ) c ( z - u ) = f ( u ) ,
θ 2 ( L ) f = θ 0 2 + 0 L 0 L rec [ θ 0 ( L - z 1 ) / a ] × rec [ θ 0 ( L - z 2 ) / a ] f ( z 1 - z 2 ) d z 1 d z 2 .
v = 1 2 ( z 1 + z 2 )
u = z 1 - z 2 ,
θ 2 f = θ 0 2 + 2 0 f ( u ) d u 0 L rec [ θ 0 ( L - v - u 2 ) / a ] × rec [ θ c ( L - v + u 2 ) / a ] d v .
θ 2 f = θ 0 2 + L 0 f ( u ) tri ( θ 0 u / a ) d u ,
F ( ν ) = - + f ( u ) exp ( i 2 π ν u ) d u
θ 2 f = θ 0 2 + 8 L π 2 m = 0 ( - 1 ) m ( 2 m + 1 ) 2 F [ ( 2 m + 1 ) θ 0 4 a ] .
θ 2 f = θ 0 2 + ( 8 L / π 2 ) F ( θ 0 / 4 a ) .
θ 2 g = θ 0 2 + ( L / 2 ) F ( θ c / 2 π a ) .
y f = ( 4 L / π 2 Δ ) F [ ( 2 Δ ) 1 2 / 4 a ]
y g = ( L / 4 Δ ) F [ ( 2 Δ ) 1 2 / 2 π a ] .
F ( ν 0 ) = 8 π 4 ν 0 2 η δ 2 / L .
y f = 4 π 2 η ( δ 2 / a 2 )
y g = π 2 η ( δ 2 / a 2 ) .
( θ 2 - θ 0 2 ) f 1 2 = 2 π ( δ / a ) θ c ( η ) 1 2
( θ 2 - θ 0 2 ) g 1 2 = π ( δ / a ) θ c ( η ) 1 2
δ ( η ) 1 2 = a / ( 8 ) 1 2 π
θ d θ = ( 4 / π 2 ) d L F ( θ / 4 a ) ,
θ 0 θ L θ d θ F ( θ / 4 a ) = 4 π 2 L .
F ( ν ) = c 2 2 u 0 / [ 1 + ( 2 π ν u 0 ) 2 ]
L f = π 2 16 1 u 0 c 2 ( θ L 2 - θ 0 2 ) [ 1 + π 2 8 u 0 2 a 2 ( θ L 2 + θ 0 2 ) ] .
L g = ( 1 / u 0 c 2 ) ( θ L 2 - θ 0 2 ) [ 1 + θ c 2 ( u 0 2 / a 2 ) ] .
L f = 0.76 θ c 4 u 0 / c 2 a 2
L g = θ c 4 u 0 / c 2 a 2 ,
c 2 = 2 0 F ( ν ) d ν .

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