Abstract

This paper describes a new nondestructive method to measure cylindrically symmetric refractive-index profiles of transparent cylinders. The technique is based on the measurement of the axial displacement of rays that are refracted within the cylinder. Three different types of index profile were experimentally determined. Profile errors of better than one part in 103 were achieved using very modest equipment. The effects of certain experimental parameters on the profile accuracy are noted. The technique may be applied to the characterization of optical fiber preforms and graded-index rod lenses.

© 1983 Optical Society of America

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References

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  1. H. M. Presby, D. Marcuse, Appl. Opt. 18, 671 (1979).
    [CrossRef] [PubMed]
  2. P. Chu, T. Whitbread, Electron. Lett. 15, 295, (1979).
    [CrossRef]
  3. D. Peri, P. L. Chu, T. Whitbread, Appl. Opt. 21, 809 (1982).
    [CrossRef] [PubMed]
  4. H. M. Presby, D. Marcuse, Appl. Opt. 13, 2882 (1974).
    [CrossRef] [PubMed]
  5. Y. Kokubun, K. Iga, Appl. Opt. 21, 1030 (1982).
    [CrossRef] [PubMed]
  6. W. E. Martin, Appl. Opt. 13, 2112 (1974).
    [CrossRef] [PubMed]
  7. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, 1964).
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 122.
  9. C. M. Vest, Appl. Opt. 14, 1601 (1975).
    [CrossRef] [PubMed]
  10. R. P. Kanwal, Linear Integral Equations (Academic, New York, 1971), pp. 167–173.
  11. R. M. Allen, Practical Refractometry (R. P. Cargille Laboratories, 1962), pp. 1–28.
  12. D. Gregoris, M.A.Sc. Thesis, U. Toronto (1982).

1982 (2)

1979 (2)

H. M. Presby, D. Marcuse, Appl. Opt. 18, 671 (1979).
[CrossRef] [PubMed]

P. Chu, T. Whitbread, Electron. Lett. 15, 295, (1979).
[CrossRef]

1975 (1)

1974 (2)

Allen, R. M.

R. M. Allen, Practical Refractometry (R. P. Cargille Laboratories, 1962), pp. 1–28.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 122.

Chu, P.

P. Chu, T. Whitbread, Electron. Lett. 15, 295, (1979).
[CrossRef]

Chu, P. L.

Gregoris, D.

D. Gregoris, M.A.Sc. Thesis, U. Toronto (1982).

Iga, K.

Kanwal, R. P.

R. P. Kanwal, Linear Integral Equations (Academic, New York, 1971), pp. 167–173.

Kokubun, Y.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, 1964).

Marcuse, D.

Martin, W. E.

Peri, D.

Presby, H. M.

Vest, C. M.

Whitbread, T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 122.

Appl. Opt. (6)

Electron. Lett. (1)

P. Chu, T. Whitbread, Electron. Lett. 15, 295, (1979).
[CrossRef]

Other (5)

R. P. Kanwal, Linear Integral Equations (Academic, New York, 1971), pp. 167–173.

R. M. Allen, Practical Refractometry (R. P. Cargille Laboratories, 1962), pp. 1–28.

D. Gregoris, M.A.Sc. Thesis, U. Toronto (1982).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, 1964).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 122.

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Figures (11)

Fig. 1
Fig. 1

Geometry of the launching conditions and refraction of an incident ray.

Fig. 2
Fig. 2

Geometry of ray path in a cylindrically symmetric refractive-index profile.

Fig. 3
Fig. 3

Schematic of experimental apparatus.

Fig. 4
Fig. 4

Axial ray displacement ΔZ(c2) for homogeneous profile.

Fig. 5
Fig. 5

Reconstructed radial-index profile for homogeneous cylinder.

Fig. 6
Fig. 6

Axial ray displacement Δz(c2) for Selfoc microlens.

Fig. 7
Fig. 7

Reconstructed radial-index profile of Selfoc microlens.

Fig. 8
Fig. 8

Axial ray displacement Δz(c2) for step-index profile.

Fig. 9
Fig. 9

Refractive-index profiles of the step-index cylinder when the entire Δz(c2) is processed directly and when the additional processing is performed on the Δz(c2) data.

Fig. 10
Fig. 10

Conditions for the existence of the cutoff radius, rc.

Fig. 11
Fig. 11

Effect of the index step Δn = n2n1 on the reconstructed profiles (—, refractive-index profile; ---, reconstructed profile).

Tables (1)

Tables Icon

Table I Experimental Conditions

Equations (32)

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L = p n d s ,
L = z 0 z n ( 1 + r ˙ 2 + r 2 θ ˙ 2 ) 1 / 2 d z ,
d d s ( n d r ¯ d s ) = n ,
[ d d s ( n d r d s ) n r ( d θ d s ) 2 ] r ˆ = n r r ˆ
d d s [ n ( r ) r 2 d θ d s ] θ ˆ = n θ θ ˆ ,
[ d d s ( n d z d s ) ] z ˆ = n z z ˆ .
n ( r ) r 2 d θ d s = n ( r ) r cos δ c ,
c = n l R cos δ 0 ,
c = n l R cos γ sin ϕ .
n ( r ) d z d s = n ( r ) 1 + r ˙ 2 + r 2 θ ˙ 2 l 0 .
l 0 = n l sin γ .
θ ˙ 2 = c 2 r 4 l 0 2 .
d r d z = ± 1 l 0 [ n 2 ( r ) l 0 2 c 2 r 2 ] 1 / 2 .
Δ z ( r ) = ± l 0 R r r d r [ n 2 ( r ) l 0 2 ] r 2 c 2 ,
Δ z ( r p ) = ± 2 l 0 R r p r d r [ n 2 ( r ) l 0 2 ] r 2 c 2 .
c 2 = [ n 2 ( r p ) l 0 2 ] r p 2 .
u 2 ( r ) = [ n 2 ( r ) l 0 2 ] r 2 ,
c = u ( r p ) .
Δ z ( r p ) = ± 2 l 0 R r p r d r u 2 ( r ) u 2 ( r p ) .
r = 1 π l 0 d d r r R Δ z ( r p ) u ( r p ) [ d u ( r p ) d r p ] d r p u 2 ( r p ) u 2 ( r ) .
r 2 = R 2 1 π l 0 u 2 ( r ) u 2 ( R ) Δ z [ u 2 ( r p ) ] d u 2 ( r p ) u 2 ( r p ) u 2 ( r ) .
r 2 = R 2 1 π l 0 u 2 ( r ) u 2 ( R ) Δ z ( t ) d t t u 2 ( r ) .
r p 2 = R 2 1 π l 0 c 2 c max 2 Δ z ( t ) d t t c 2 ,
c 2 = [ n 2 ( r p ) l 0 2 ] r p 2 and c max 2 = [ n 2 ( R ) l 0 2 ] R 2 .
n ( r ) = n c ( 1 α 2 r 2 ) ,
n = ( c 2 r p 2 + l 0 2 ) 1 / 2 ,
d n n = c 2 n 2 r p 2 d r p r p + l 0 2 n 2 d l 0 l 0 .
c 2 n 2 r p 2 = [ n 2 ( r p ) l 0 2 n 2 ( r p ) ] .
n 2 ( r p ) = d c 2 d r p 2 + l 0 2 r p 2 d n 2 ( r p ) d r p 2 .
n 2 ( r p ) = d c 2 d r p 2 + l 0 2 .
n ( r ) = n 1 [ 1 Δ ( r R ) 2 ] ,
r p 0.02 R 3 Δ .

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