Abstract

A mathematical expression is derived for the shape of multiple-beam Fizeau fringes crossing a graded-index optical fiber inserted in a silvered liquid wedge. Formulas are obtained for calculating both α and Δn from fringe shifts. A formula is deduced for step-index fiber. Experimental results have been obtained from microinterferograms.

© 1985 Optical Society of America

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References

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  1. M. E. Marhic, P. S. Ho, M. Epstein, “Nondestructive Refractive-Index Profiling Measurement of Clad Optical Fibers,” Appl. Phys. Lett. 26, 574 (1975).
    [CrossRef]
  2. M. J. Saunders, W. B. Gardner, “Nondestructive Interferometric Measurement of the Delta and Alpha of Clad Optical Fibers,” Appl. Opt. 16, 2368 (1977).
    [CrossRef] [PubMed]
  3. D. Gloge, E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563 (1973).
  4. N. Barakat, “Interferometric Studies on Fibers, Part I. Theory of Interferometric Determination on Indices of Fibers,” Text. Res. J. 41, 167 (1971).
    [CrossRef]

1977 (1)

1975 (1)

M. E. Marhic, P. S. Ho, M. Epstein, “Nondestructive Refractive-Index Profiling Measurement of Clad Optical Fibers,” Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

1973 (1)

D. Gloge, E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563 (1973).

1971 (1)

N. Barakat, “Interferometric Studies on Fibers, Part I. Theory of Interferometric Determination on Indices of Fibers,” Text. Res. J. 41, 167 (1971).
[CrossRef]

Barakat, N.

N. Barakat, “Interferometric Studies on Fibers, Part I. Theory of Interferometric Determination on Indices of Fibers,” Text. Res. J. 41, 167 (1971).
[CrossRef]

Epstein, M.

M. E. Marhic, P. S. Ho, M. Epstein, “Nondestructive Refractive-Index Profiling Measurement of Clad Optical Fibers,” Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

Gardner, W. B.

Gloge, D.

D. Gloge, E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563 (1973).

Ho, P. S.

M. E. Marhic, P. S. Ho, M. Epstein, “Nondestructive Refractive-Index Profiling Measurement of Clad Optical Fibers,” Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563 (1973).

Marhic, M. E.

M. E. Marhic, P. S. Ho, M. Epstein, “Nondestructive Refractive-Index Profiling Measurement of Clad Optical Fibers,” Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

Saunders, M. J.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. E. Marhic, P. S. Ho, M. Epstein, “Nondestructive Refractive-Index Profiling Measurement of Clad Optical Fibers,” Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Gloge, E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,” Bell Syst. Tech. J. 52, 1563 (1973).

Text. Res. J. (1)

N. Barakat, “Interferometric Studies on Fibers, Part I. Theory of Interferometric Determination on Indices of Fibers,” Text. Res. J. 41, 167 (1971).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Cross section in a silvered liquid wedge interferometer with graded-index waveguide fiber of variable index core n(r). A schematic representation of the resulting fringes is shown.

Fig. 2
Fig. 2

Setup for producing multiple-beam Fizeau fringes applied to fibers: A, mercury lamp; B, condenser lens; C, iris diaphragm; D, collimating lens; E, monochromatic filter; F, microscope stage; G, silvered liquid wedge interferometer; and I, to camera attached to the microscope.

Fig. 3
Fig. 3

Microinterferogram of graded-index fiber inserted in a silvered liquid wedge for λ = 5461 Å.

Fig. 4
Fig. 4

Schematic diagram of the microinterferogram shown in Fig. 3. δzz = the value of the fringe shift as a fraction of an order separation between two consecutive straight-line fringes. N and N + 1 are 2 orders of interference. The core and fiber radii are a and tf/2, respectively.

Fig. 5
Fig. 5

Refractive-index profile of the fiber when Δn = 0.028, α = 2.8, and n(a) = 1.464.

Fig. 6
Fig. 6

Comparison between the experimental profile and the theoretical one F*(x) calculated with α = 2.8 and Δn = 0.028.

Tables (1)

Tables Icon

Table I Values of the Fringe Shift δz for Different Values of x for a Graded-index Fiber in Millimeters

Equations (14)

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n ( r ) = n ( 0 ) [ 1 2 Δ ( r / a ) α ] 1 / 2 o r a ,
OPL = ( t 2 y 2 ) n L + 2 ( y 2 y 1 ) n clad + 2 0 y 1 = a 2 x 1 2 n ( r ) d y ,
OPL = ( t 2 y 2 ) n L + 2 ( y 2 y 1 ) n clad + 2 n ( o ) a 2 x 1 2 2 Δ n a α 0 a 2 x 1 2 ( x 1 2 + y 2 ) α / 2 d y .
N λ = 2 ( OPL ) = 2 n L t + 4 y 2 ( n clad 2 L ) + 4 Δ n y 1 4 Δ n a α 0 a 2 x 1 2 ( x 1 2 + y 2 ) α / 2 d y .
( N λ 2 n L z tan ) = 4 y 2 ( n clad n L ) + 4 Δ n y 1 4 Δ n a α 0 a 2 x 1 2 ( x 1 2 + y 2 ) α / 2 d y .
z · 2 n L tan = 4 y 2 ( n clad n L ) + 4 Δ n y 1 4 Δ n a α 0 a 2 x 1 2 ( x 1 2 + y 2 ) α / 2 d y .
( δ z Δ z ) x 1 · λ / 2 = 2 [ y 2 ( n clad n L ) + Δ n y 1 Δ n a α 0 a 2 x 1 2 ( x 1 2 + y 2 ) α / 2 d y ] = 2 [ ( n clad n L ) r f 2 x 1 2 + Δ n a 2 x 1 2 ] Δ n a α 0 a 2 x 1 2 ( x 1 2 + y 2 ) α / 2 d y . ]
( δ z Δ z ) λ 2 = ( n clad n L ) t f + t core · Δ n α ( α + 1 ) ,
( δ z Δ z ) · λ 2 = ( n clad n L ) t f + t core ( n core n clad ) .
( δ z Δ z ) x 1 , ( δ z Δ z ) x 2 , n clad , n L .
I α ( x ) = 0 a 2 x 2 ( x 2 + y 2 ) α / 2 d y
F ( a ) = 4 λ Δ z [ ( n clad n L ) r f 2 a 2 ]
F * ( x ) = F ( x ) F ( a ) · δ z exp | x = a .
2 r f = 125 ± 5 μ m n clad = 1.464 , 2 a = 50 ± 3 μ m n L = 1.476 .

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