Abstract

The relations between the far-field intensity distribution and the corresponding modal power distribution resulting from differential excitation are employed to determine the index profile of a multimode fiber. The modal propagation constant and the modal time delay are determined from the far-field distribution, eliminating the need for ultrafast sources and detectors, required in the direct time-delay measurements. The temporal impulse response is reconstructed numerically from the modal time-delay distribution for specific excitation conditions.

© 1983 Optical Society of America

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References

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  1. E. Olshansky, S. M. Oaks, Appl. Opt. 17, 1830 (1978).
    [CrossRef] [PubMed]
  2. L. Jeunhomme, J. P. Pocholle, Appl. Opt. 17, 463 (1978).
    [CrossRef] [PubMed]
  3. A. Ankiewicz, M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 14, 811 (1978).
    [CrossRef]
  4. M. J. Buckler, J. W. Shiever, F. P. Partus, “Optimization of Multimode Fiber Bandwidth Via Differential Group Delay Analysis,” 6th Eur. Conf. on Opt. Coram., Pub. 190, p. 33, York, Sept. 16–19 (1980).
  5. K. Petermann, Electron. Lett. 14, 793 (1978).
    [CrossRef]
  6. K. Nageno, S. Kawakami, Appl. Opt. 19, 2426 (1980).
    [CrossRef]
  7. G. K. Grau, O. G. Leminger, Appl. Opt. 20, 457 (1981).
    [CrossRef] [PubMed]
  8. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).
  9. R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
    [CrossRef] [PubMed]
  10. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  11. R. Olshansky, Appl. Opt. 14, 935 (1975).
    [PubMed]
  12. H. G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977).
  13. E. A. J. Marcatili, Bell Syst. Tech. J. 56, 49 (1977).
  14. M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
    [CrossRef]
  15. D. Marcuse, Appl. Opt. 18, 2073 (1979).
    [CrossRef] [PubMed]

1981 (1)

1980 (1)

1979 (1)

1978 (4)

E. Olshansky, S. M. Oaks, Appl. Opt. 17, 1830 (1978).
[CrossRef] [PubMed]

L. Jeunhomme, J. P. Pocholle, Appl. Opt. 17, 463 (1978).
[CrossRef] [PubMed]

A. Ankiewicz, M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 14, 811 (1978).
[CrossRef]

K. Petermann, Electron. Lett. 14, 793 (1978).
[CrossRef]

1977 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 56, 49 (1977).

1976 (2)

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[CrossRef] [PubMed]

1975 (1)

1973 (1)

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

1971 (1)

Adams, M. J.

A. Ankiewicz, M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 14, 811 (1978).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

Ankiewicz, A.

A. Ankiewicz, M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 14, 811 (1978).
[CrossRef]

Buckler, M. J.

M. J. Buckler, J. W. Shiever, F. P. Partus, “Optimization of Multimode Fiber Bandwidth Via Differential Group Delay Analysis,” 6th Eur. Conf. on Opt. Coram., Pub. 190, p. 33, York, Sept. 16–19 (1980).

Gloge, D.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

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

Grau, G. K.

Jeunhomme, L.

Kawakami, S.

Keck, D. B.

Leminger, O. G.

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 56, 49 (1977).

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Marcuse, D.

Nageno, K.

Oaks, S. M.

Olshansky, E.

Olshansky, R.

Partus, F. P.

M. J. Buckler, J. W. Shiever, F. P. Partus, “Optimization of Multimode Fiber Bandwidth Via Differential Group Delay Analysis,” 6th Eur. Conf. on Opt. Coram., Pub. 190, p. 33, York, Sept. 16–19 (1980).

Payne, D. N.

A. Ankiewicz, M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 14, 811 (1978).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

Petermann, K.

K. Petermann, Electron. Lett. 14, 793 (1978).
[CrossRef]

Pocholle, J. P.

Shiever, J. W.

M. J. Buckler, J. W. Shiever, F. P. Partus, “Optimization of Multimode Fiber Bandwidth Via Differential Group Delay Analysis,” 6th Eur. Conf. on Opt. Coram., Pub. 190, p. 33, York, Sept. 16–19 (1980).

Sladen, F. M. E.

A. Ankiewicz, M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 14, 811 (1978).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

Unger, H. G.

H. G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977).

Appl. Opt. (8)

Bell Syst. Tech. J. (2)

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

E. A. J. Marcatili, Bell Syst. Tech. J. 56, 49 (1977).

Electron. Lett. (3)

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

K. Petermann, Electron. Lett. 14, 793 (1978).
[CrossRef]

A. Ankiewicz, M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 14, 811 (1978).
[CrossRef]

Other (2)

M. J. Buckler, J. W. Shiever, F. P. Partus, “Optimization of Multimode Fiber Bandwidth Via Differential Group Delay Analysis,” 6th Eur. Conf. on Opt. Coram., Pub. 190, p. 33, York, Sept. 16–19 (1980).

H. G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977).

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Figures (9)

Fig. 1
Fig. 1

Differential excitation geometry.

Fig. 2
Fig. 2

Calculated far-field intensity distribution. These three curves are plotted from Eq. (12) for a parabolic profile [g(r/a) = (r/a)2].

Fig. 3
Fig. 3

Experimental arrangement for the recordings of the far-field intensity distributions from a differentially excited fiber.

Fig. 4
Fig. 4

Typical far-field distributions as recorded for a 1-m graded-index fiber. The angles θm(r0) at which the distributions attain their off-axis maximum values serve to determine the fiber-index profile from Eq. (18).

Fig. 5
Fig. 5

Normalized refractive-index profile. The index profile was calculated from Eq. (18) and the measured θm of Fig. 4. The solid line represents least-squares fit to the values obtained from three different spot sizes. The dashed line represents a parabolic profile for reference.

Fig. 6
Fig. 6

Normalized propagation constant δ as a function of the radius r. δ is defined by Eq. (4) and determined from Eqs. (7) and (13) and the measured values of 0m(r0).

Fig. 7
Fig. 7

Time delay as a function of the radius r relative to the zero-order delay [τ(0) = Ln1/c]. The solid line represents the time delay as calculated from Eq. (16), and the dashed line represents the direct time-delay measurements.

Fig. 8
Fig. 8

Impulse response as determined from Fig. 7 and Eq. (17) assuming uniform excitation. The mean arrival time t ̅ and rms width σ of the impulse response are calculated from Eq. (19).

Fig. 9
Fig. 9

Impulse response as determined from Fig. 7 and Eq. (17) assuming more energy is distributed to lower-order modes according to the function E(m). Note that this weighting factor and the delay distribution from Fig. 7 dictate an impulse response with shorter mean arrival time than that obtained for uniform excitation, Fig. 8.

Equations (24)

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n 2 ( r ) = n 1 2 [ 1 2 Δ g ( r a ) ] ,
q π = R 1 R 2 [ k 2 ( r ) β 2 l 2 / r 2 ] 1 / 2 d r ,
n = 2 q + l
V = k 0 a 2 Δ ; δ = 1 2 ( 1 β 2 k 0 2 ) ,
δ Δ = g ( r a ) + sin 2 θ sin 2 θ c ; l V = r sin θ a sin θ c | sin ϕ | ,
P ( δ , l ) = { 1 { δ min δ δ max l min l l max 0 otherwise .
δ min = { Δ g ( r 0 γ a ) r 0 γ 0 r 0 γ , δ max = { Δ g ( r 0 + γ a ) + Δ sin 2 θ 0 sin 2 θ c for g ( r 0 + γ a ) + sin 2 θ 0 sin 2 θ c 1 Δ for g ( r 0 + γ a ) + sin 2 θ 0 sin 2 θ c 1 ,
l min = 0 , l max = V r 0 a sin θ sin θ c γ ( γ 2 + r 0 2 ) 1 / 2 .
Q ( θ ) = 2 ( k 0 a sin θ c ) 2 cos θ sin 2 θ sin 2 θ c Δ Δ T ( θ , l ) d δ × 0 S ( θ , δ ) [ S 2 ( θ , δ ) l 2 ] 1 / 2 P ( δ , l ) d l
T ( θ , δ ) = 2 x ( θ , δ ) g [ x ( θ , δ ) ] , S ( θ , δ ) = V sin sin c x ( θ , δ ) , x ( θ , δ ) = g 1 ( δ Δ sin 2 θ sin 2 θ c ) , g [ g 1 ( x ) ] = x .
P tot = 2 π Q ( θ ) sin θ d θ .
Q ( θ ) = B g 2 ( δ max Δ sin 2 θ sin 2 θ c )
Q ( θ ) = A { g 1 ( δ max Δ sin 2 θ sin 2 θ c ) g 1 [ g ( r 0 γ a ) sin 2 θ sin 2 θ c ] }
for γ < and sin 2 θ sin 2 θ c g ( r 0 γ a ) ; Q ( θ ) = A g 1 ( δ max Δ sin 2 θ sin 2 θ 0 )
for γ < r 0 and g ( r 0 γ a ) sin 2 θ sin 2 θ c δ max Δ ; Q ( θ ) = 0
sin θ m ( r 0 ) = sin θ c g 1 / 2 ( r 0 γ a ) .
τ ( l , q ) = L n 1 c d β ( l , q ) d k 0 ,
m ( β , k 0 ) = 0 R 2 ( 0 ) [ k 2 ( r ) β 2 ] rdr ,
τ ( δ ) = L n 1 c m k 0 / m β = = L n 1 / c ( 1 2 δ ) 1 / 2 { 1 4 Δ 0 a g 1 ( δ / Δ ) g ( r / a ) rdr a 2 [ g 1 ( δ Δ ) ] 2 } .
p ( t ) = E ( m ) | d m d t | ,
γ 2 λ f 1 d and θ 0 d 2 f 1 ,
n ( r 0 γ ) n 1 = ( 1 sin 2 θ m n 1 2 ) 1 / 2 .
t ̅ = τ p ( τ ) d τ p ( τ ) d τ ; σ = ( τ t ̅ ) 2 p ( τ ) d τ p ( τ ) d τ .
E ( m ) = exp [ 1 2 ( m / M 0.2 ) 2 ] ,

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