Abstract

We show that it is possible to determine the refractive index distribution of a fiber or preform with slight index variation by observing the power distribution of the light field that is focused by the core acting as a lens. This method requires index matching of the cladding and illumination of the core at right angles to its axis with a broad beam of incoherent collimated light. The refractive index distribution is obtained after two numerical integrations to be performed by computer. The first integration establishes the relation between the output and input ray positions from the observed power distribution, the second uses this information to determine the refractive index distribution. However, it is not necessary to solve a large system of simultaneous equations. The sensitivity of the method to measurement inaccuracies was tested by computer simulation. It was found that the method has a builtin smoothing effect that attenuates rather than amplifies measurement errors.

© 1979 Optical Society of America

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References

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  1. P. L. Chu, Electron. Lett. 13, 736 (1977).
    [CrossRef]
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 99.
  3. B. C. Wonsiewicz, W. G. French, P. D. Lazay, J. R. Simpson, Appl. Opt. 15, 1048 (1976).
    [CrossRef] [PubMed]
  4. H. M. Presby, D. Marcuse, H. W. Astle, Appl. Opt. 17, 2209 (1978).
    [CrossRef] [PubMed]
  5. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 419.
  6. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 230.
  7. Ref. 5, p. 730.

1978 (1)

1977 (1)

P. L. Chu, Electron. Lett. 13, 736 (1977).
[CrossRef]

1976 (1)

Astle, H. W.

Chu, P. L.

P. L. Chu, Electron. Lett. 13, 736 (1977).
[CrossRef]

French, W. G.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 419.

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 230.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 230.

Lazay, P. D.

Marcuse, D.

H. M. Presby, D. Marcuse, H. W. Astle, Appl. Opt. 17, 2209 (1978).
[CrossRef] [PubMed]

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 99.

Presby, H. M.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 419.

Simpson, J. R.

Wonsiewicz, B. C.

Appl. Opt. (2)

Electron. Lett. (1)

P. L. Chu, Electron. Lett. 13, 736 (1977).
[CrossRef]

Other (4)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 99.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 419.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 230.

Ref. 5, p. 730.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the geometry of the focusing method.

Fig. 2
Fig. 2

Computer simulated experiment. The dotted line represents the square-law distribution used as the input, the solid line represents the output distribution obtained by the theory described in this paper.

Fig. 3
Fig. 3

The power distribution in the observation plane is disturbed by adding random numbers so that the rms deviation of P is ΔP = 0.06. The dotted line indicates the input square law distribution, the solid curve traces the reconstructed index distribution.

Fig. 4
Fig. 4

Same as Fig. 3 with ΔP = 0.12.

Fig. 5
Fig. 5

Same as Fig. 4 also with ΔP = 0.12 but a different set of random numbers.

Fig. 6
Fig. 6

Ideal power density distribution in the observation plane (dotted line) and the randomly perturbed distribution with ΔP = 0.06 (solid line).

Fig. 7
Fig. 7

Same as Fig. 6 with ΔP = 0.12.

Equations (21)

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d 2 y d x 2 = 1 n c n y .
d y d x = 1 n c [ - a r 0 n r y ( r 2 - y 2 ) 1 / 2 d r + r 0 a n r y ( r 2 - y 2 ) 1 / 2 d r ] .
y ( t ) = t + 2 L t n c t a n r d r ( r 2 - t 2 ) 1 / 2 .
n ( r ) - n c = n c π L r a t - y ( t ) ( t 2 - r 2 ) 1 / 2 d t .
1 / P = ( d y / d t ) .
t ( y ) = 0 y P ( y ) d y .
n ( r ) - n c = 1 π L r a t - y ( t ) ( t 2 - r 2 ) 1 / 2 d t .
t ( y ) - y = 0 y [ P ( y ) - 1 ] d y .
n ( r ) = n 0 [ 1 - ( r / a ) 2 Δ ] .
y = t [ 1 - ( z / a ) 2 Δ ] .
y m = t [ 1 - 4 ( 1 - t 2 a 2 ) Δ ] .
t - y m t = 4 ( 1 - t 2 a 2 ) Δ .
t - y m t = 4 3 ( 1 - t 2 a 2 ) Δ .
F ( t ) = t G r d r ( r 2 - t 2 ) 1 / 2
G ( r ) = n ( r ) - n c ,
F ( t ) = n c 2 L y ( t ) - t t .
G ( r ) = 0 g ( u ) cos ( u r ) d u .
F ( t ) = - 0 u g ( u ) [ t sin u r ( r 2 - t 2 ) 1 / 2 d r ] d u = - π 2 0 u g ( u ) J 0 ( u t ) d u .
g ( u ) = 2 π 0 t F ( t ) J 0 ( u t ) d t .
G ( r ) = 2 π 0 t F ( t ) [ 0 cos ( u r ) J 0 ( u t ) d u ] d t .
G ( r ) = - 2 π r t ( t 2 - r 2 ) 1 / 2 F ( t ) d t .

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