Nondestructive interferometric measurement of the delta andalpha of clad optical fibers

M. J. Saunders and W. B. Gardner

It is shown that interferograms generated by passing light perpendicular to the axis of a fiber can be used toobtain the maximum refractive index difference and the index profile shape to better than 10% accuracyin regions of the core where a is constant. This technique avoids the time consuming sample preparationrequired for the slab method, the propagation problems associated with the near-field technique, and thesurface quality problem associated with the reflection technique.

1. Introduction

A convenient representation of a class of refractiveindex profiles which contains many profiles of interestis 1

n(r) = n(0)[1 - 2A(r/a)a]l/2, 0 r <a (1)

where r is the distance from the center of the core, anda is the core radius. Note that A = [n 2(0) -n2(a)]/

2n2 (0) _t An/n(O), where An n(O) - n(a). In thispaper, we deal only with those profiles that can be ap-proximated by Eq. (1). These profiles are characterizedby the two parameters a and A, where a determines theshape of the profile and A (or An) is related to the nu-merical aperture. Both a and An are important in de-termining the bandwidth of the fiber. A procedure fordetermining a and An that is quick and accurate tobetter, than h10% is needed for production use, wherelarge numbers of fibers must be examined.

These two quantities have been determined, pri-marily by the slab method of examining a cross sectioncut from the fiber, via interferometry. 2 -4 The surfacesof the slab must be devoid of curvature, since curvaturecauses errors in both An and a.5 Polishing flat,scratch-free surfaces, is difficult and time consuming,and samples less than 20 Am thick may require a sub-strate6 for reinforcement. The slab geometry alsomakes the accurate measurement of slab thicknesssomewhat difficult.

All these difficulties with the slab method havestimulated a search for a simpler technique that is bettersuited for use in a production environment. Recentlythree methods have been reported for determining An

The authors are with Bell Laboratories, Norcross, Georgia30071.

Received 17 January 1977.

and a. One method7 makes use of the close resem-blance between the near-field intensity distribution andthe refractive index of a fiber in which all bound modesare equally excited, but this method requires a length-dependent correction factor for leaky modes. A secondmethod810 entails the monitoring of the reflected powerof a focused laser beam from a smooth fiber face, therefractive index profile being obtained from the Fresnelformula as the focused beam is caused to scan the fiberface. However, it has been shown10 that inaccurateresults are obtained when the fiber face has been opti-cally polished. Although satisfactory results are ob-tained from fractured fiber ends for germanium andphosphorus doped samples, rapid surface changes dueto atmospheric exposure lead to inaccurate results forborosilicate fibers. Further, the positioning of the fo-cused laser beam at the center of the refractive indexdistribution (to obtain An) poses some difficulties.

A third method involves interferometry with the fiberimmersed in an index matching oil and illuminatedperpendicular to the fiber axis.'" Thus in this method(that we call the Marhic method), no sample prepara-tion is required. Marhic et al. indicate that the re-fractive index profile can be obtained from the fringeshifts by an Abel inversion and show that simple ana-lytical expressions for the optical path difference areobtained for fibers whose index profile is a quadraticfunction of the radius of the core. (Just as with the slabmethod, the fiber is assumed to be a phase object, whichdoes not appreciably distort the emerging wavefront.)In this paper we apply the Marhic method to the moregeneral class of profile shapes given by Eq. (1) with theassumption that a is constant throughout the core. Inthis case, An can be calculated from the maximumfringe shift, and, using a computer program, a can becalculated from any point on the fringe (i.e., at any pointin the core). For seven fibers, values of An and a ob-tained by the slab and Marhic methods are compared.

2368 APPLIED OPTICS / Vol. 16, No. 9 / September 1977

CLADDING

I I

I~~~~~~ c

Fig. 1. Schematic diagram of Marhic fringes for a clad fiber: (a) (nf= ncl) and (b) (nf $ n).

I'

(A) (B)

Fig. 2. Marhic fringes for a clad fiber: (a) (nf = nel) and (b) (nf #no ) -

X(Direction of ray)

Fig. 3. Geometry for optical path difference for Marhic fringes.

Finally, since we have found that many fiber profileslack circular symmetry, we present an experimentalmethod for averaging the asymmetries.

I. Theory

Consider Fig. 1 showing a clad fiber immersed in anoil of refractive index nf. The fiber is illuminated alonga direction normal to the fiber axis. If the fiber is ex-amined in an interferometer and if nf = nc1, the refrac-tive index of the cladding, the appearance of the fringeswill be as shown in Fig. 1(a), where the interferometerhas been adjusted so that the fringes in the oil are per-pendicular to the axis of the fiber. Figure 1(b) showsthe fringes when nf X n,1, so that the core-cladding andcladding-oil interfaces are visible. We are interestedin the fringe displacement, yp at the radial location rp.Figures 2(a) and 2(b) are photographs of the fringeswhen nf = nc and nf X nc, respectively.

For A << 1, Eq. (1) becomes

n(r) _ n(0) - An(r/a)a. (2)

Now consider Fig. 3. We have r 2= x 2 + rp 2, and using

this in Eq. (2),

n(x) = n() -a [(X2 + rP2)1/2a. (3)aa

The optical path difference between the cladding andthe fringe at rp for two passes through the sample is

4 f Ja2-rp/ n(x)dx-(a2 -r 2 )1/ 2 nci = NpA, (4)

where n,1 = refractive index of the cladding and Np =fringe shift between the cladding and the fringe at rp.Using Eq. (3),

N X/4=n (a 2 -r 2)1/2 -An (a2rp2)1/2 (X2 +r 2)alsdX, (5)

where we have used

n(0) = n,1 + An. (6)

From Eq. (5) we have when rp = 0,

An = (a + 1)XN*/4aa, (7)

where N* is the fringe shift between the cladding andthe center of the index distribution. Also,

N = Ymax/AY, Np = Yp/AY, (8)

where Ymax and yp are the fringe displacements at rp =0 and rp, respectively, and Ay is the fringe spacing in thecladding (Fig. 1). We now use the following procedureto obtain An and a:

(1) Ymax, Ay, and a are determined by measure-ment, a value of a is assumed, and Eq. (7) is solved foran initial value of An.

(2) The value of An and the value of Np at rp areused in Eq. (5) to obtain a new value of a numerically,using an iterative computer program.

(3) If this value of a does not agree with the a valueassumed in step (1), the a value from step (2) is used inEq. (7) to calculate a new An, and the procedure in step(2) is repeated.

September 1977 / Vol. 16, No. 9 / APPLIED OPTICS 2369

Xol

VACUUM

HEATING

Fig. 4. Apparatus for rotating fiber about axis.

(4) This procedure is continued until the a valuefrom step (2) agrees to within the desired accuracy withthat used in step (1). The value of An used in step (2)to obtain this value of a is then the estimated value ofAn.

(5) The entire procedure is repeated for new valuesof rp and Np.

In step (2), the iteration is continued until the dif-ference between the two sides of Eq. (5) is less than0.0001 ,um. Because of this equating of the two sides ofEq. (5), we refer to this version of the Marhic techniqueas the right-left method. No instances of a failure toconverge have been encountered.

From Eq. (7) we have for uncertainties a, in a and a,,in a,

2+ 1./2. (9)

An a2(a+ 1)2 a2

As an example, assume a core radius of a = 30 ,im. TheMarhic fringes themselves can be used to determine thecore radius to an uncertainty of aa = 1 gim, providing thecore-cladding interface is unambiguously delineated(as in Fig. 2, for example). If the a of the fiber is knownto fall within the range 1.7 < a < 2.3 (i.e., a = 2.0 and ac-= 0.3), Eq. (9) shows that Eq. (7) may be used to calcu-late An to an uncertainty of a-An/An = 6.0%, withoutever knowing the precise value of a.

Ill. Experimental TechniqueTo obtain Marhic fringes, we place an optical flat on

an electrically heated element which, in turn, is placedon the mechanical stage of a microscope. A smallamount of index oil (nD25C = 1.460) is placed on theoptical flat, and the refractive index of the oil is adjustedby controlling the temperature of the oil. The inter-ferometric element is a Watson interference microscopeobjective,'2 and the light source is a He-Ne laser oper-ating through a moving diffuser.' 3 A sample of fiberabout 3 cm in length, free of coating, is placed in the oilon the optical flat. In order to obtain the core radiusfrom the interferogram, we use a reference fiber, thecore radius having been accurately determined. This

fiber is cemented to the optical flat and is photographedtogether with the fiber to be studied so that the mag-nification of the interferogram can readily be obtained.The interferogram is placed on the stage of a microscopeand the fringe displacements measured as a function ofthe radial position of the fringe. We visually set thecrosshair of an eyepiece micrometer at the center ofthe fringe, and, at each radial position, we make thiscrosshair setting numerous times to increase the pre-cision of the measurement. Last, the spacing of thefringes in the cladding is determined and is used withthe fringe displacements to obtain the fringe shiftsNp.

We have found that many fibers that we have exam-ined by the slab technique contain an asymmetric re-fractive index distribution, so that, in order to obtainan average value of a, numerous interferograms mustbe taken on a given slab at different angular orientationsof the slab. We used this procedure in comparingvalues of a obtained by the Marhic method with the avalues obtained by the slab technique. The Marhictechnique involves some averaging since, from Fig. 3,a ray traversing the fiber at a distance rp from the cen-ter, samples the refractive index over a range of azi-muthal angles. In the slab method, a ray samples theindex of refraction at only one point in the core.

We have done some preliminary work on a techniqueto obtain Marhic fringes that are more thoroughly av-eraged over the azimuthal angles. A brass annularfixture surrounds and is separated from the optical flat(Fig. 4). Two wedge-shaped vacuum grooves in thefixture contain the fiber. The oiled optical flat, at-tached to a heating element, is secured to the substagecondenser rack and pinion so that it can be positionedimmediately below the fiber so that the fiber is im-mersed in the oil. One end of the fiber is clamped in apin vise that is rotated by a motor. We have photo-graphed the rotating fiber with a 2-sec exposure whilethe fiber was rotating at about 4 rps so that the fiberrotated through eight revolutions during the exposure.

2�42'

-, �41

4 2'

---- /2' 2 4

(A) (B)

Fig. 5. Marhic fringes: (a) (fiber stationary) and (b) (2-sec exposureof fiber rotating at 4 rps).

2370 APPLIED OPTICS / Vol. 16, No. 9 / September 1977

Table 1. Comparison of Values of a and An Obtained by the Slab and Marhic Methods

a An

Fiber Slab Marhic % Error Slab Marhic % Error

A 2.43 ± 0.05 2.43 i 0.07 0.0 0.0100 + 0.0001 0.0104 1 0.0001 +4.0B 2.15 ± 0.09 2.08 i 0.07 -3.3 0.0196 + 0.0003 0.0202 3 0.0002 +3.1C 2.00 + 0.04 1.98 + 0.11 -1.0 0.0182 0.0001 0.0191 i 0.0003 +4.9D 1.97 i 0.09 2.14 i 0.05 +8.6 0.0163 i 0.0007 0.0170 + 0.0001 +4.3E 1.82 : 0.23 1.94 i 0.05 +6.6 0.0101 ± 0.0007 0.0112 i 0.0001 +10.9F 2.14 + 0.03 2.31 ± 0.02 +7.9 0.0205 ± 0.0003 0.0199 + 0.0001 -2.9G 2.01 t 0.06 2.13 + 0.06 +6.0 0.0118 4- 0.0002 0.0113 :3 0.0001 -4.2

Figure 5(a) is a photograph of the Marhic fringes withthe fiber stationary, and Fig. 5(b) shows the fringes withthe fiber rotating at 4 rps. Apparently, fiber vibrationposes little or no problem.

IV. Results

We determined values of a and An (averaged overseveral orientations) for seven fibers by the slab methodand used short lengths of these fibers (lengths fromwhich the slabs were obtained) to determine thesequantities by the right-left program.

For many fibers that we have examined, a is constantfrom the core-clad interface down to a radius (rb) thatvaries from about 4 gAm to 10 Aim. The value of rb ob-tained from the right-left program generally agrees withthe value obtained from the slab method. Also, the avalue by both methods is sometimes higher within a fewmicrons of the core-cladding interface. The values forthe two methods are compared only over that radialrange for which the slab method shows that a is con-stant, since the current version of the right-left methodis based on the assumption that a does not vary. Forthe slab method, the uncertainties in a and An resultfrom measurements on at least four (and in one case,twelve) interferograms obtained by rotating the slab toobtain average values of a for fibers that show asym-metries in the index profile. The uncertainties of a andAn obtained by the Marhic method result from aver-aging about thirteen individual values of these quan-tities measured at different values of r. The results forthe seven fibers are given in Table I, the percentageerrors being referred to the slab values. The correlationcoefficients for a and An by the two methods are 0.85and 0.83, respectively. The Marhic interferograms werenot obtained while the fiber was rotating so that, asidefrom the fact that the Marhic fringe inherently involvessome averaging, the Marhic a values are not as repre-sentative of the a values throughout the core as are theslab values.

V. Conclusions

The Marhic technique of fiber interferometry, inconjunction with the right-left program, permits thedetermination of a and An to an accuracy of, generally,better than +10% in those regions of the fiber core wherethe slab method shows that a is constant. The tech-nique is fast (since no sample preparation is required),thus making it attractive for use in a production envi-ronment where a and An must be determined for a largenumber of fibers.

We thank C. S. Jackson for preparing the slab sam-ples and W. L. Parham for writing the computer pro-grams. We also thank P. J. Rich for his help in some ofthe computer programming aspects of this work.

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