Direct deflection method for determiningrefractive-index profiles of polymer optical fiber preforms

Brian K. Canfield, Christopher S. Kwiatkowski, and Mark G. Kuzyk

We present a method for determining the refractive-index profile of polymer optical fiber preformsthrough a direct-deflection measurement. The method is simple to use, compact, and has good resolu-tion. The profile is obtained from the deflection data by numerically integrating the differential-rayequation for a radial refractive-index gradient. Corrections for topographical deviations are also dis-cussed. Results for both graded-index and step-index fibers are presented. © 2002 Optical Society ofAmerica

OCIS codes: 060.2270, 120.5710, 160.5470, 290.3030.

1. Introduction

Interest in polymer fibers is accelerating because ofpresent efforts aimed at building graded-index �GRIN�fiber into local area networks. GRIN fibers of highbandwidth and low loss have been developed,1,2 and aprocess for making step-index fibers3,4 has been usedfor making fibers with large nonlinearities5 that can beused for devices, such as all-optical switches. Fur-thermore, polymer fiber amplifiers,6 switches7 andphotomechanical actuators,8,9 and directional cou-plers10,11 have been reported. For all of these appli-cations it is important to know the fiber’s refractive-index profile.

Various techniques for measuring the refractive-index profile of an optical waveguide exist. Thesetechniques include ellipsometry,12 light scatteringapproximations,13,14 transverse interferometry,15–17

interference microscopy and holographic shearinginterferometry,18–20 multiple-beam Fizeau inter-ferometry,21–24 measuring the mode indices of aguiding fiber,25–28 and the refracted near-field tech-nique for fibers29–34 or generalized for one- and two-

The authors were with Washington State University, Depart-ment of Physics, Pullman, Washington 99164-2814 when this pa-per was written. C. S. Kwiatkowski’s current address is LosAlamos National Laboratory, Electronic and Electrochemical Ma-terials and Devices Group, Los Alamos, New Mexico 87545. M. G.Kuzyk is also with Washington State University Materials ScienceProgram. Author’s e-mail addresses are bcanfie@mail.wsu.edu,kuz@wsu.edu, and chrisk@lanl.gov.

Received 16 February 2001; revised manuscript received 4 Feb-ruary 2002.

0003-6935�02�173404-08$15.00�0© 2002 Optical Society of America

3404 APPLIED OPTICS � Vol. 41, No. 17 � 10 June 2002

dimensional waveguides.35–38 Comparisons ofdifferent profiling methods are to be found in theliterature as well.39–41

Several methods for determining the refractive-index profiles of optical fiber preforms have also beendeveloped. Atomic-force microscopy has recentlybeen applied,42,43 but requires specialized equipment.Other methods employ variations on direct-deflectionmeasurements, wherein a deflection function is ob-tained, and followed by transformation and numeri-cal integration of the paraxial ray equation.44–48

Disadvantages of these techniques are that they re-quire foreknowledge of the general form of the profileand a corresponding form of the paraxial equation,necessitating large amounts of computation. In thispaper we present a technique we call the direct-deflection method �DDM�, which requires relativelysimple analysis based on deflection due to refractive-index gradients to obtain the refractive-index profileof polymer optical fiber preforms. This techniquehas several advantages: The required equipment issimple, inexpensive, and compact; the experiment ishighly automated; the procedure yields good resolu-tion and is not limited to only a certain type of fiberpreform �such as GRIN or step-index�; and foreknowl-edge of the index-profile form is not required.

2. Preform Fabrication

We have profiled various preforms with DDM, bothgraded index and step index. The GRIN preformswere provided by Boston Optical Fiber, Inc. Theywere fabricated using the interfacial gel polymeriza-tion technique, where a solution of methyl metha-crylate �MMA� and polymerizing agents are placedin a hollow cylindrical poly�methyl methacrylate�

�PMMA� cladding �Fig. 1�. Because the monomer isa solvent, it partially dissolves the cladding at theinterface, forming a gel. Some of the polymerizingagents then diffuse into the gel, resulting in a non-uniform polymerizer concentration throughout thepreform cross section. The nonuniform concentra-tion causes the preform cross section to polymerizewith different chain lengths, with the highest concen-tration, and hence longest chains, being at the centerand concentration �and thus chain length� droppingoff radially. Because the refractive index is relatedto polymer chain length, the index profile will show anonlinear dependence on cross-sectional position inthe preform.

Step-index fiber preforms are fabricated in theNonlinear Optics Laboratory at Washington StateUniversity.5 First, a core preform is made by doping10 ml of MMA with a dye, such as DR1 shown in Fig.2. After polymerizing and degassing at 90 °C, thecore preform is pulled into a large-diameter core fiberof approximately 800 �m. A suitable section of thecore fiber is cut and placed in a pair of grooved, neatPMMA half rounds of a half-inch diameter. The fi-ber preform is then placed in a squeezer and squeezedat 120 °C until the three parts fuse together. Thepreform is then allowed to degas once again in the90 °C oven. It is then ready for pulling into fiber.

3. Direct-Deflection Method

A. Direct-Deflection Method Theory

In the DDM a laser beam passes through a thin sliceof the fiber preform parallel to the preform axis. Ifthere is a transverse refractive-index gradient at thepoint where the beam enters the slice, the beam willbend in the direction of the gradient. A measure-ment of the deflection angle, then, is related to thegradient. Figure 3 illustrates how this deflection ismeasured with a CCD camera array.

The refractive-bending differential equation49 for aradial-index gradient yields

� �1

���Log n���

n��n�

, (1)

where � is the �large� radius of curvature described bythe path of a light ray through the slice. Becausetypical variations in the refractive index between thecore and cladding are �10�3 or less, the deflectionswill be small, validating the small-angle approxima-tion. We can find the deflection angle within theslice, �, in terms of � and t, the thickness of the slice,by noting that the ray travels a vertical distance ythrough the slice. From the equation of a circle withradius �, we find

�2 � t2 � �� � y�2. (2)

When �y��� 1, Eq. �2� yields

y �t2

2�. (3)

If we divide Eq. �3� by t, we can write � in the small-angle approximation �tan � � �� in terms of t and �:

� �t

2�, (4)

and, using Snell’s Law, in terms of the exit angle ofthe ray, �:

� ��

n, (5)

where we have used the small-angle approximationfor �, that is sin � � �.

Fig. 1. Graded-index preform fabrication by interfacial gel poly-merization.

Fig. 2. Step-index preform fabrication.

Fig. 3. Profiling method geometry.

10 June 2002 � Vol. 41, No. 17 � APPLIED OPTICS 3405

Combining Eqs. �1�, �4�, and �5�, we have

��n� �2�

t. (6)

If we define f�r� as the spot position on the CCDcamera f�r� � 0 with no deflection� as a function ofthe radial position in the preform slice and z as thedistance from the slice to the camera, then ��r� �f�r��z. Because the beam deflection is described bytwo independent angles, we represent it as a vector:��r� � � �r� �, where the unit vector � lies in the planeof the CCD array. Because � is in the direction ofthe refractive-index gradient, Eq. �6� then becomes

�n �2��r�

t. (7)

Dotting dr into both sides of Eq. �7� and integrating,we arrive at the general relationship between therefractive-index difference and the spot deflection:

n�r2� � n�r1� �2zt �

r1

r2

f �r�� � dr. (8)

Note that the coordinate system in the sample isarbitrary, so dr represents any arbitrary infinitesi-mal displacement in the sample.

Equation �8� therefore provides the means for find-ing the refractive-index difference between two arbi-trary points in the sample. However, because only afinite number of data points is taken, the integral isapproximated by a sum, and the step size dr corre-sponds to the spacing between data points. This ap-proximation is valid when the deflection angles forany two adjacent data points are approximatelyequal. Also, if the refractive-index value is knownfor one point in the sample, the absolute index can beobtained from Eq. �8� at any other point.

B. Direct-Deflection Method Experiment

Figure 4 shows the original experimental setup usedto measure the deflection data for a thin slice of afiber preform. The slice has been polished accordingto the following process to render it optically flat.An approximately 1-mm thick slice is first excisedfrom the end of the preform. Next, the slice is handpolished with a fine sandpaper and then with a suc-cessively 5- and 3-�m lapping film. Finally, the fin-ishing touch is applied using 3-, 1-, and then .1-�mliquid alumina polishing suspensions. The slice is

rotated frequently during the polishing stages tomaintain parallel, plane faces and to randomize anyresultant scratch patterns. �Corrections to the datafor topographical deviations, such as wedges or pa-raboloids can be made provided that the deviationsfrom the plane are small. These geometric issuesare discussed in Subsection 3.C.�

A HeNe laser �� � 632.8 nm� passes through apolarizer and is focused and recollimated to asmaller-diameter spot �1�e value of 21 �m� with a10� and 20� microscope objectives. The spot diam-eter was obtained by performing a knife-edge exper-iment. While chopping the beam, a razor blade wastranslated across the beam at the sample location,and the beam intensity was measured with a photo-detector and a lock-in amplifier. The resulting datawere then fitted to a complementary error function ofthe form:

erfc�N� � k ���

N

exp� � ax2� dx, (9)

where k is a numerical constant, N is the number ofdata points taken, and a gives the inverse square ofthe 1�e width. Trial and error showed that this com-bination of objectives provided suitable collimationbetween the sample and the camera. We want tooptimize the spot size based on the competing effectsof having a small spot at the sample to probe a singleindex-gradient value while limiting defocusing at thecamera. The camera needs to be relatively far fromthe sample so that very small deflections can be mea-sured. Lenses after the sample are avoided becausesmall deflections �paraxial rays� would be refocusedonto the optic axis, negating deflection information.The beam is normally incident upon the sample,which is held in a self-centering mount attached to anx–y translation stage driven by stepper motors. A200-�m pinhole is placed immediately behind thesample. The pinhole acts as both a spatial filter,blocking extraneous light and multiple reflectionsfrom inside the sample, and as an aid in centering thespot. The angle subtended by the pinhole and theincident laser beam is large enough not to affect thebeam through its full deflection range. The beamthen proceeds through a second polarizer and ontothe CCD array of the camera �PoleStar Model CDT-500� �the lens and infrared LED illumination ringin the camera were removed�. The second polar-izer can be used in conjunction with the first as ananalyzer for determining sample birefringence.The entire setup �except the computer� can fit on asmall breadboard, so the experiment is both com-pact and portable.

The position of the beam-spot center is determinedby measuring its intensity profile with the CCD ar-ray. A framegrabber digitizes the data so that adata-acquisition program can determine the spot cen-ter by spatially averaging the brightest pixels in theframe and recording this intensity-averaged pixel lo-cation as the center of the spot. To obtain pure de-

Fig. 4. DDM experimental setup.

3406 APPLIED OPTICS � Vol. 41, No. 17 � 10 June 2002

flection data, we first measure the spot center 10times without the sample present over a period ofapproximately 10 s. This procedure allows us to ac-count for small temperature or beam-intensity fluc-tuations or mechanical vibrations while measuringthe undeflected spot center. We then calculate theaverage x and y values of the undeflected spot posi-tion and subsequently subtract them from every de-flection data point. The x and y coordinates of thedata so obtained are saved in a two-column file, thuspreserving deflection-vector information.

An array of data points corresponding to minimumspatial separations in the sample of .85 �m �the min-imum step size of the stepper motor�stage combina-tion� can be taken using the x–y stage. The data-acquisition program runs as an x–y raster scan,running through all x positions in the programmedrange for a given y position, then returning to theinitial x position before moving on to the next y posi-tion. This protocol prevents backlash in the me-chanical stages from biasing the data. The programalso sets a short time delay between translation andimaging by the frame-grabber routine to allow vibra-tions to subside. A one-dimensional scan of 100 datapoints takes approximately 12 min, while a two-dimensional scan of 100 � 100 data points may take20 h. The actual run time also depends on the xdistance the stage has to move during a scan, becauseit must return to its initial x position for each new yposition. However, the operator need not continu-ously monitor data acquisition because the system isautomated, and once started, it can be left unat-tended until the run is finished.

C. Analysis and Results

The data file can be analyzed with any software thattreats matrices, such as Mathematica or Matlab, asan R � C matrix of �x, y� vector positions �R rows byC columns, corresponding to the number of x and ypositions mapped�. It should be noted that the CCDcamera inverts the y coordinate of data points owingto the fact that its imaging lens has been removed, sothis change of sign must be included when analyzingthe raw data.

A vector-field plot of the deflection angle for a typ-ical GRIN sample two-dimensional data scan �Fig. 5�shows the refractive-index gradient as a function ofits position in the sample. For illustrative purposesthis data scan includes regions outside the preform�the corners�. Note that the air�cladding boundarygives rise to large, discontinuous deflections, as ex-pected by the large discontinuity in the refractiveindex at the boundary �n � 1 to n � 1.5�, and shouldnot be considered as yielding meaningful data. Be-cause the laser spot has a finite diameter, part of it isdeflected differently than the rest as it passes sequen-tially through an interfacial boundary. As a result,the spot on the CCD array may subsequently rangefrom elongated to heavily distorted, and may evensplit into two or more sections depending on the spotdiameter and type of interface. While the data-acquisition program still spatially averages the

bright pixels even in these configurations, such dis-tortions lead to discontinuities in the deflection dataand also to artificial interface broadening, makingthe interface boundary appear wider and morerounded than it actually is. Those vectors at theair�cladding interface that extended far beyond theboundary of the scan area were removed to clarify theplots.

A close inspection of Fig. 5�a� shows a net linearbias of the vectors toward the right. In this case, thesample happens to be wedge shaped, measuring2.671 mm on one edge and 2.759 mm on the oppositeside, corresponding to a wedge angle of 0.2°. Theamount of bias is consistent with the wedge angle.This wedge results in a constant deflection that mim-ics a refractive-index gradient toward the right of thefigure. If the preform is known to be cylindricallysymmetric, the sum of all vector deflections must sumto zero. The average of the deflection-vector sumover an acceptably uniform region of the sample mustthen be subtracted from each vector to yield a cor-rected data set Fig. 5�b��. For a two-dimensionalscan, the core region is thus chosen for averagingbecause it comprises a comparatively large portion ofthe sample, and the deflection within it is highlycoherent in direction—that is, very uniform com-pared to transition regions, such as the core�claddingor air�cladding interfaces, which strongly scatter thebeam.

The wedge bias-corrected data can now be used tofind the magnitude and angle of deflection for eachpoint. We obtain f�r� by taking the square root of thesum of the squares of the x and y coordinates for eachdeflection data point in the matrix. The conversionfactor between CCD pixel location and actual dis-tance �pixels are 9.6 � 6.3 �m� must also be consid-ered to convert the deflection data recorded by theCCD array to absolute deflection displacements f�r�.Next, the absolute deflection angle is calculated �re-call that ��r� � f�r��z�. This angle is used in Eq. �8�,where the dot product in the integrand reduces to cos� cos � if the numerical integration is performed inthe x direction, or sin � cos � if it is performed in they direction. Figure 6 illustrates the coordinate sys-tem used. Angle � corresponds to the usual rotation

Fig. 5. Vector field plots showing refractive index gradient: �a�original data, �b� wedge-corrected data.

10 June 2002 � Vol. 41, No. 17 � APPLIED OPTICS 3407

in the x–y plane. The value cos � �or sin �� is foundby dividing the x coordinate �or y coordinate� of de-flection by f�r�, with the condition that cos � � 1 �orsin � � 1� if f�r� � 0 to avoid singularities.

The refractive-index difference between two pointscan now be determined through numerical integra-tion, and the index profile can be obtained by tabu-lating this difference for each successive point. Thediscrete-difference equation used to approximate Eq.�8� �for x-direction integration� is then

n R, C� � n R, C� � n R, 1�

�2drzt �

j�1

C

f R, j�cos � R, j�cos � R, j�,

(10)

where C and R run over the columns and rows of thematrix and R, j� denotes a matrix element. Forintegration along the y axis, sin � replaces cos � and R, j� becomes j; C�. The summation is over R.Figure 7 shows the resulting three-dimensional re-fractive index profile for a two-dimensional GRINpreform scan with the absolute refractive index �thecladding index value is 1.491� along the vertical axis.The corners are within the cladding, while the centralpeak spans the core. The small-amplitude ridgesvisible are the result of minor discontinuities in in-dividual deflection data points that, once entered intothe summation, propagate throughout the rest of therow. These discontinuties may arise from imperfec-

tions in the sample, such as surface scratches ormaterial impurities. If the effect of these disconti-nuities leads to a large-amplitude feature, the ampli-tude can often be greatly reduced by repeating thenumerical integration in the reverse direction �i. e.,summing from C to 1 instead of 1 to C� and averagingthe two results.

For a one-dimensional scan bias corrections aresimpler. Figure 8 shows a one-dimensional indexprofile along the preform diameter �in this case, alongthe y axis� for a sample with a slight wedge shape.The top graph shows the original, biased data wherethe constant gradient due to the wedge angle isclearly visible. The dotted line represents the resultof a linear fit to the data, and approximates the biasdue to the wedge angle. Once the wedge bias hasbeen subtracted, the profile appears much improved,as shown in the middle graph. However, anotherbias that occurs is a very small paraboloid deviationin the surface of the preform sample slice, which mayresult from internal relaxation of the polymer afterpolishing. This bias appears as a parabolic contri-bution to the index profile and is most apparent asthe non-zero slopes in the cladding-section data �oneither side of the core� that should have a flat profile.In other words, the index profile within the claddingshould remain constant because there is no gradientin the cladding beyond the interfacial regions. Wecan also correct for this bias by fitting a parabola tothese cladding sections of the profile and subtracting

Fig. 6. Coordinate system and angle definitions.

Fig. 7. One-dimensional GRIN preform refractive index profile.Top graph shows original biased data and linear fit; middle graphshows wedge-corrected data and paraboloid bias fit; bottom graphshows final corrected profile and parabolic profile fit.

Fig. 8. Three-dimensional GRIN preform profile.

3408 APPLIED OPTICS � Vol. 41, No. 17 � 10 June 2002

the fit from the data. The resulting profile appearsin the bottom graph of Fig. 8. Although the magni-tude of these biases may seem insignificant, they canhave an enormous effect on the refractive-index pro-file measured. The sample investigated here wasfabricated to have a parabolic refractive-index profile.A parabolic fit to the corrected data, also included onthe bottom graph, shows that the profile is not exactlyparabolic.

Our method also works for step-index profiles. Astep-index polymer optical fiber preform was fabri-cated in the Nonlinear Optics Laboratory at Wash-ington State University �WSU�.4,5 The core consistsof dye-doped PMMA from a core preform of 10 mg &DR1 dye dissolved in 10 ml of MMA �.07% by weight�,and had a diameter of 723 �m at the time of fabrica-tion. The preform was squeezed at 120 °C for 96 hbefore the sample slice was obtained and the profilemeasured. The magnitude of the raw deflectiondata obtained by the data-acquisition program for atwo-dimensional scan is shown in Figure 9. Notethat the interface between the two PMMA claddinghalf-rounds is clearly visible. In this case the twohalf-rounds did not fuse completely during squeezing,leaving a noticeable discontinuity the DDM experi-ment easily measured. Figure 10 shows a high-resolution �1.7 �m�point� one-dimensional scan

along the preform slice diameter. The core diameteris indicated by the step. This profile yields a clearexample of the artificial interface broadening men-tioned earlier. The rounding of the profile from apure step, and the peaks and the dip in the coreregion, result mostly from the finite size of the laser-beam spot, which is also shown �this scan was ob-tained with 4� and 5� objectives in place of the 10�and 20� objectives, resulting in a larger beam-spotdiameter�. Once again, because the spot has a finitewidth, the beam will be partially deflected to varyingdegrees as its cross-section transits the interfacialregion, resulting in the rounded peaks of the stepprofile. The data shown in Fig. 10 were corrected fora slight paraboloid bias.

4. Conclusion

We presented a novel method �to the best of ourknowledge� for determining the refractive-index pro-file of polymer optical fiber preforms. While theDDM process requires removing part of the preform,the sample required is but a small fraction of theentire preform and thus the method has a negligibledetrimental effect on the preform itself, which canstill be pulled into plenty of fiber. The sample mayalso be taken from the remnant of a preform that hasalready been pulled. The method provides rela-tively quick results with good resolution and fairlysimple analysis. Moreover, it can be applied to var-ious types of preforms and variations in the refractiveindex of samples. The effects of pulling the preforminto a fiber can be investigated to see if the indexprofile merely scales down with the fiber or if un-known changes take place in the process. Once theindex profile has been determined, the mode profilefor the fiber can be calculated, and useful devices maybe fabricated from the fiber.

An interesting application of DDM is to study thediffusion effects of prolonged baking at high temper-atures on the dye-doped core of a fiber preform. An-other diffusion measurement that can be performedis investigating the effect of pulling the preform intoa fiber. Cross-sectional slices at successive diame-ters of the pulled preform can be measured to see howthe pulling process affects diffusion of the dye fromthe core. Another possible application is measuringthe birefringence of samples by placing a crossedpolarizer–analyzer pair around the sample. If thesample exhibits birefringence, it will rotate the po-larization of the beam slightly so that part of it willthen pass through the analyzer to be recorded by thecamera. Mutual rotation of the polarizer and ana-lyzer can thus be used to map the index ellipse. In-dex changes induced by photobleaching, such aswhen writing gratings, could also be measured withDDM.

Certain improvements to the DDM experimentmay be suggested. It is possible to employ a diodelaser with a focusing lens that can replace the HeNeand microscope objectives. With appropriate filter-ing, if the focal length is made to be relatively long�thereby increasing the sample–camera distance be-

Fig. 9. Raw deflection magnitude data for a step-index preform.

Fig. 10. One-dimensional step-index preform profile.

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cause the beam will not defocus as quickly as with theobjectives�, then the resolution of the index profilemay be increased by allowing even smaller deflec-tions to be measured. A translation stage systemwith even finer resolution could be used. In addi-tion, a higher-resolution CCD array may be em-ployed. Also, the beam-spot diameter could bedeconvolved from the integral to remove the effects ofits finite size, reducing or even eliminating rounded,peaked interfaces that should be sharp and flat. Fi-nally, more accurate polishing methods, such as pol-ishing machines would eliminate wedge biases andmay even lower the likelihood of paraboloid bias.

This work was funded by The Air Force Office ofScientific Research and Sentel Technologies, L.L.C.We also thank Boston Optical Fiber, Inc., for provid-ing graded-index preform samples.

References1. Y. Ohtsuka, E. Nihei, and Y. Koike, “Graded-index optical

fibers of methyl methacrylate—vinyl benzoate copolymer withlow loss and high bandwidth,” Appl. Phys. Lett. 57, 120–122�1990�.

2. E. Nihei, T. Ishigure, and Y. Koike, “High-bandwidth, graded-index polymer optical fiber for near-infrared use,” Appl. Opt.35, 7085–7090 �1996�.

3. M. G. Kuzyk, U. C. Paek, and C. W. Dirk, “Guest-host polymerfibers for nonlinear optics,” Appl. Phys. Lett. 59, 902–904�1991�.

4. D. W. Garvey, K. Zimmerman, P. Young, J. Tostenrude, J. S.Townsend, Z. Zhou, M. Lobel, M. Dayton, R. Wittorf, and M. G.Kuzyk, “Single-mode nonlinear-optical polymer fibers,” J. Opt.Soc. Am. B 13, 2017–2023 �1996�.

5. D. W. Garvey, Q. Li, M. G. Kuzyk, C. W. Dirk, and S. Martinez,“Sagnac interferometric intensity-dependent refractive-indexmeasurements of polymer optical fiber,” Opt. Lett. 21, 104–106�1996�.

6. A. Tagaya, Y. Koike, T. Kinoshita, E. Nihei, T. Yamamoto, andK. Sasaki, “Polymer optical fiber amplifier,” Appl. Phys. Lett.63, 883–884 �1993�.

7. D. J. Welker and M. G. Kuzyk, “All-optical switching in adye-doped polymer fiber Fabry–Perot waveguide,” Appl. Phys.Lett. 69, 1835–1836 �1996�.

8. D. J. Welker and M. G. Kuzyk, “Photomechanical stabilizationin a polymer fiber-based all-optical circuit,” Appl. Phys. Lett.64, 809–811 �1994�.

9. D. J. Welker and M. G. Kuzyk, “Optical and mechanical mul-tistability in a dye-doped polymer fiber Fabry–Perotwaveguide,” Appl. Phys. Lett. 66, 2792–2794 �1995�.

10. S. R. Vigil, Z. Zhou, B. K. Canfield, J. Tostenrude, and M. G.Kuzyk, “Dual-core single-mode polymer fiber coupler,” J. Opt.Soc. Am. B 15, 895–900 �1998�.

11. Z. Xiong, G. D. Peng, and P. L. Chu, “Nonlinear coupling andoptical switching in a �-carotene-doped twin-core polymer op-tical fiber,” Opt. Eng. 39, 624–627 �2000�.

12. V. N. Van, A. Brunet-Bruneau, S. Fisson, J. M. Frigerio, G.Vuye, Y. Wang, F. Abeles, J. Rivory, M. Berger, and P. Chaton,“Determination of refractive-index profiles by a combination ofvisible and infrared ellipsometry measurements,” Appl. Opt.35, 5540–5544 �1996�.

13. C. Saekeang and P. L. Chu, “Backscattering of light fromoptical fibers with arbitrary refractive-index distributions:uniform approximation approach,” J. Opt. Soc. Am. 68, 1298–1305 �1978�.

14. D. Marcuse, “Refractive index determination by the focusingmethod,” Appl. Opt. 18, 9–13 �1979�.

15. J. A. Ferrari, E. Frins, A. Rondoni, and G. Montaldo, “Retrievalalgorithm for refractive-index profile of fibers from transverseinterferograms,” Opt. Commun. 117, 25–30 �1995�.

16. H. El-Ghandoor, E. A. El-Ghafar, and R. Hassan, “Refractiveindex profiling of a GRIN optical fiber using a modulatedspeckled sheet of light,” Opt. Laser Technol. 31, 481–488�1999�.

17. J. Sochacki, “Accurate reconstruction of the refractive-indexprofile of fibers and preform rods from transverse interfero-metric data,” Appl. Opt. 25, 3473–3482 �1986�.

18. M. Hai, X. Jianping, Z. Hui, S. Yuensheng, and C. Nong, “Mea-suring abberations of a gradient-index rod lens with holo-graphic shearing interferometry,” Int. J. Optoelectron. 7,53–56 �1992�.

19. Z. Z. Wu, H. Davis, and S. K. Batra, “Correct ray-tracinganalysis for interference microscopy of fibers,” Proc. R. Soc.London Ser. A 450, 23–36 �1995�.

20. R. Posey, L. Philips, D. Diggs, and A. Sharma, “LP01-LP02interference using a spectrally extended light source: mea-surement of the non-step-refractive-index profile of optical fi-bers,” Opt. Lett. 21, 1357–1359 �1996�.

21. M. A. Mabrouk and H. F. El-Bawab, “Refractive index profileof GRIN optical fibre considering the area under the interfer-ence fringe shift. I. The matching case,” Pure Appl. Opt. 6,247–256 �1997�.

22. T. Z. N. Sokkar, M. A. Mabrouk, and H. F. El-Bawab,“Refractive-index profile of GRIN optical fiber considering thearea under the interference fringe shift: II. The mismatch-ing case,” J. Opt. A 1, 64–72 �1999�.

23. A. A. Hamza and A. M. Nasr, “Interferometric studies onmulti-mode step-index optical fibres,” Pure Appl. Opt. 7, 449–456 �1998�.

24. A. A. Hamza, T. Z. N. Sokkar, M. A. Mabrouk, and M. A.El-Morsy, “Refractive index profile of polyethylene fiber usinginteractive multiple-beam Fizeau fringe analysis,” J. Appl.Polym. Sci. 77, 3090–3106 �2000�.

25. B. X. Chen, H. Hamanaka, and K. Iwamura, “Recovery ofrefractive-index profiles of planar graded-index waveguidesfrom measured mode indices: an iteration method,” J. Opt. Soc.Am. A 9, 1301–1305 �1992�.

26. R. Oven, S. Batchelor, D. G. Ashworth, D. Gelder, and J. M.Bradshaw, “Iterative refinement technique for reconstructingrefractive index profiles from mode indices,” Electron. Lett. 31,229–231 �1995�.

27. F. Gonella, A. Quaranta, A. Sambo, F. Caccavale, and I. Man-sour, “Construction of glass waveguide refractive index profilesby the effective-index finite-difference method,” Opt. Mat. 5,321–326 �1996�.

28. S. Batchelor, R. Oven, and D. G. Ashworth, “Reconstruction ofrefractive index profiles from multiple wavelength mode indi-ces,” Opt. Commun. 131, 31–36 �1996�.

29. K. I. White, “Practical application of the refracted near-fieldtechnique for the measurement of optical fibre refractive indexprofiles,” Opt. Quant. Electron. 11, 185–196 �1978�.

30. M. Young, “Calibration technique for refracted near-field scan-ning of optical fibers,” Appl. Opt. 19, 2479–2480 �1980�.

31. M. Young, “Optical fiber index profiles by the refracted-raymethod �refracted near-field scanning�,” Appl. Opt. 20, 3415–3422 �1981�.

32. T. Muller, “Resolution improvement in refracted near-field in-dex measurement by a lens-shaped liquid cell,” Electron. Lett.19, 580–582 �1983�.

33. N. Gisin, “Correcting refracted near field refractive index pro-file measurements for Gaussian intensity distributions,” Opt.Commun. 83, 295–299 �1991�.

34. I. Mansour and F. Caccavale, “An improved procedure to cal-

3410 APPLIED OPTICS � Vol. 41, No. 17 � 10 June 2002

culate the refractive index profile from the measured near-fieldintensity,” J. Lightwave Technol. 14, 423–428 �1996�.

35. N. Gisin, R. Passy, P. Stamp, N. Hori, and S. Nagano, “Newoptical configuration for nondestructive measurements ofrefractive-index profiles of LiNbO3 waveguides,” Appl. Opt. 33,1726–1731 �1994�.

36. B. Groebil, B. Gisin, N. Gisin, and H. Zbinden, “Measuringrefractive index profiles of integrated LiNbO3 waveguides,”Opt. Eng. 34, 2309–3214 �1995�.

37. T. Yabu, S. Sawa, and M. Geshiro, “Measurement of refractiveindex distribution of optical waveguides by the propagationmode near-field method employing an improved inverse anal-ysis,” Electron. Commun. Jpn. 79, 21–29 �1996�.

38. P. Oberson, B. Gisin, B. Huttner, and N. Gisin, “Refractednear-field measurements of refractive index and geometry ofsilica-on-silicon integrated optical waveguides,” Appl. Opt. 37,7268–7272 �1998�.

39. M. J. Saunders, “Optical fiber profiles using the refracted near-field technique: A comparison with other methods,” Appl.Opt. 20, 1645–1651 �1981�.

40. W. J. Stewart, “Optical fiber and preform profiling technology,”IEEE J. Quantum Electron. 18, 1451–1465 �1982�.

41. A. A. Hamza, T. Z. N. Sokkar, K. A. El-Farahaty, and H. M.El-Dessouky, “Comparative study on interferometric tech-niques for measurement of the optical properties of a fiber,” J.Opt. 1, 41–50 �1999�.

42. S. T. Huntington, P. Mulvaney, A. Roberts, K. A. Nugent, andM. Bazylenko, “Atomic force microscopy for the determinationof refractive index profiles of optical fibers and waveguides: aquantitative study,” J. Appl. Phys. 820, 2730–2734 �1997�.

43. S. T. Huntington, A. Roberts, K. A. Nugent, P. Mulvaney, andM. Bazylenko, “Fibre and waveguide refractive index measure-ments with AFM resolution,” Jpn. J. Appl. Phys. 37, 62–64�1998�.

44. P. L. Chu, “Nondestructive measurement of index profile of anoptical-fibre preform,” Electron. Lett. 13, 736–738 �1977�.

45. D. Peri and P. L. Chu, “Measurement of refractive-index profileof optical-fibre preform by means of spatial filtering,” Electron.Lett. 17, 371–372 �1981�.

46. J. Sochacki, M. Sochaka, and C. Gomez-Reino, “Reconstructionof axially nonsymmetric refractive index profiles from ray de-flection by means of the Abel integral transform,” Opt. Com-mun. 71, 20–22 �1989�.

47. P. Skok and M. Miller, “Spatial filtering method of oblique slitfor measurement of refractive-index profile of optical-fibre pre-forms,” Electron. Lett. 23, 859–860 �1987�.

48. W. Urbanczyk, K. Pietraszkiewicz, and W. A. Wozniak, “Novelbifunctional systems for measuring the refractive index profileand residual stress birefringence in optical fibers and pre-forms,” Opt. Eng. 31, 491–499 �1992�.

49. M. Born and E. Wolf, Principles of Optics, 6th ed. �PergamonPress, Oxford, UK 1980�.

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