Direct deflection method for determining refractive-index profiles of

Direct deflection method for determining refractive-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 preforms through a direct-deflection measurement. The method is simple to use, compact, and has good resolution. The profile is obtained from the deflection data by numerically integrating the differential-ray equation for a radial refractive-index gradient. Corrections for topographical deviations are also discussed. Results for both graded-index and step-index fibers are presented. © 2002 Optical Society of America OCIS codes: 060.2270, 120.5710, 160.5470, 290.3030.

1. Introduction

Interest in polymer fibers is accelerating because of present efforts aimed at building graded-index 共GRIN兲 fiber into local area networks. GRIN fibers of high bandwidth and low loss have been developed,1,2 and a process for making step-index fibers3,4 has been used for making fibers with large nonlinearities5 that can be used for devices, such as all-optical switches. Furthermore, polymer fiber amplifiers,6 switches7 and photomechanical actuators,8,9 and directional couplers10,11 have been reported. For all of these applications it is important to know the fiber’s refractiveindex profile. Various techniques for measuring the refractiveindex profile of an optical waveguide exist. These techniques include ellipsometry,12 light scattering approximations,13,14 transverse interferometry,15–17 interference microscopy and holographic shearing interferometry,18 –20 multiple-beam Fizeau interferometry,21–24 measuring the mode indices of a guiding fiber,25–28 and the refracted near-field technique for fibers29 –34 or generalized for one- and twoThe authors were with Washington State University, Department of Physics, Pullman, Washington 99164-2814 when this paper was written. C. S. Kwiatkowski’s current address is Los Alamos National Laboratory, Electronic and Electrochemical Materials and Devices Group, Los Alamos, New Mexico 87545. M. G. Kuzyk is also with Washington State University Materials Science Program. Author’s e-mail addresses are [email protected], [email protected], and [email protected]. Received 16 February 2001; revised manuscript received 4 February 2002. 0003-6935兾02兾173404-08$15.00兾0 © 2002 Optical Society of America 3404

APPLIED OPTICS 兾 Vol. 41, No. 17 兾 10 June 2002

dimensional waveguides.35–38 Comparisons of different profiling methods are to be found in the literature as well.39 – 41 Several methods for determining the refractiveindex profiles of optical fiber preforms have also been developed. Atomic-force microscopy has recently been applied,42,43 but requires specialized equipment. Other methods employ variations on direct-deflection measurements, wherein a deflection function is obtained, and followed by transformation and numerical integration of the paraxial ray equation.44 – 48 Disadvantages of these techniques are that they require foreknowledge of the general form of the profile and a corresponding form of the paraxial equation, necessitating large amounts of computation. In this paper we present a technique we call the directdeflection method 共DDM兲, which requires relatively simple analysis based on deflection due to refractiveindex gradients to obtain the refractive-index profile of polymer optical fiber preforms. This technique has several advantages: The required equipment is simple, inexpensive, and compact; the experiment is highly automated; the procedure yields good resolution and is not limited to only a certain type of fiber preform 共such as GRIN or step-index兲; and foreknowledge of the index-profile form is not required. 2. Preform Fabrication

We have profiled various preforms with DDM, both graded index and step index. The GRIN preforms were provided by Boston Optical Fiber, Inc. They were fabricated using the interfacial gel polymerization technique, where a solution of methyl methacrylate 共MMA兲 and polymerizing agents are placed in a hollow cylindrical poly共methyl methacrylate兲

Fig. 3. Profiling method geometry. Fig. 1. Graded-index preform fabrication by interfacial gel polymerization.

3. Direct-Deflection Method

共PMMA兲 cladding 共Fig. 1兲. Because the monomer is a solvent, it partially dissolves the cladding at the interface, forming a gel. Some of the polymerizing agents then diffuse into the gel, resulting in a nonuniform polymerizer concentration throughout the preform cross section. The nonuniform concentration causes the preform cross section to polymerize with different chain lengths, with the highest concentration, and hence longest chains, being at the center and concentration 共and thus chain length兲 dropping off radially. Because the refractive index is related to polymer chain length, the index profile will show a nonlinear dependence on cross-sectional position in the preform. Step-index fiber preforms are fabricated in the Nonlinear Optics Laboratory at Washington State University.5 First, a core preform is made by doping 10 ml of MMA with a dye, such as DR1 shown in Fig. 2. After polymerizing and degassing at 90 °C, the core preform is pulled into a large-diameter core fiber of approximately 800 ␮m. A suitable section of the core fiber is cut and placed in a pair of grooved, neat PMMA half rounds of a half-inch diameter. The fiber preform is then placed in a squeezer and squeezed at 120 °C until the three parts fuse together. The preform is then allowed to degas once again in the 90 °C oven. It is then ready for pulling into fiber.

A.

Direct-Deflection Method Theory

In the DDM a laser beam passes through a thin slice of the fiber preform parallel to the preform axis. If there is a transverse refractive-index gradient at the point where the beam enters the slice, the beam will bend in the direction of the gradient. A measurement of the deflection angle, then, is related to the gradient. Figure 3 illustrates how this deflection is measured with a CCD camera array. The refractive-bending differential equation49 for a radial-index gradient yields ␳⫽

1 n ⫽ , 兩ⵜ共Log n兲兩兩ⵜn兩

(1)

where ␳ is the 共large兲 radius of curvature described by the path of a light ray through the slice. Because typical variations in the refractive index between the core and cladding are ⬃10⫺3 or less, the deflections will be small, validating the small-angle approximation. We can find the deflection angle within the slice, ␪, in terms of ␳ and t, the thickness of the slice, by noting that the ray travels a vertical distance ⌬y through the slice. From the equation of a circle with radius ␳, we find ␳ 2 ⫽ t 2 ⫹ 共␳ ⫺ ⌬y兲 2.

(2)

When 共⌬y兲兾␳ ⬍⬍ 1, Eq. 共2兲 yields ⌬y ⫽

t2 . 2␳

(3)

If we divide Eq. 共3兲 by t, we can write ␪ in the smallangle approximation 共tan ␪ ⬇ ␪兲 in terms of t and ␳: ␪⫽

t , 2␳

(4)

and, using Snell’s Law, in terms of the exit angle of the ray, ␣: ␪⫽

Fig. 2. Step-index preform fabrication.

␣ , n

(5)

where we have used the small-angle approximation for ␣, that is sin ␣ ⬇ ␣. 10 June 2002 兾 Vol. 41, No. 17 兾 APPLIED OPTICS

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Fig. 4. DDM experimental setup.

Combining Eqs. 共1兲, 共4兲, and 共5兲, we have 兩ⵜn兩 ⫽

2␣ . t

(6)

If we define f共r兲 as the spot position on the CCD camera 关f共r兲 ⬅ 0 with no deflection兴 as a function of the radial position in the preform slice and z as the distance from the slice to the camera, then ␣共r兲 ⬇ f共r兲兾z. Because the beam deflection is described by two independent angles, we represent it as a vector: ␣共r兲 ⫽ ␣ 共r兲 ␣ˆ , where the unit vector ␣ˆ lies in the plane of the CCD array. Because ␣ˆ is in the direction of the refractive-index gradient, Eq. 共6兲 then becomes ⵜn ⫽

2␣共r兲 . t

(7)

Dotting dr into both sides of Eq. 共7兲 and integrating, we arrive at the general relationship between the refractive-index difference and the spot deflection: n共r2兲 ⫺ n共r1兲 ⫽

2 zt

兰

r2

f 共r兲␣ˆ 䡠 dr.

(8)

r1

Note that the coordinate system in the sample is arbitrary, so dr represents any arbitrary infinitesimal displacement in the sample. Equation 共8兲 therefore provides the means for finding the refractive-index difference between two arbitrary points in the sample. However, because only a finite number of data points is taken, the integral is approximated by a sum, and the step size dr corresponds to the spacing between data points. This approximation is valid when the deflection angles for any two adjacent data points are approximately equal. Also, if the refractive-index value is known for one point in the sample, the absolute index can be obtained from Eq. 共8兲 at any other point. B. Direct-Deflection Method Experiment

Figure 4 shows the original experimental setup used to measure the deflection data for a thin slice of a fiber preform. The slice has been polished according to the following process to render it optically flat. An approximately 1-mm thick slice is first excised from the end of the preform. Next, the slice is hand polished with a fine sandpaper and then with a successively 5- and 3-␮m lapping film. Finally, the finishing touch is applied using 3-, 1-, and then .1-␮m liquid alumina polishing suspensions. The slice is 3406


rotated frequently during the polishing stages to maintain parallel, plane faces and to randomize any resultant scratch patterns. 共Corrections to the data for topographical deviations, such as wedges or paraboloids can be made provided that the deviations from the plane are small. These geometric issues are discussed in Subsection 3.C.兲 A HeNe laser 共␭ ⫽ 632.8 nm兲 passes through a polarizer and is focused and recollimated to a smaller-diameter spot 共1兾e value of 21 ␮m兲 with a 10⫻ and 20⫻ microscope objectives. The spot diameter was obtained by performing a knife-edge experiment. While chopping the beam, a razor blade was translated across the beam at the sample location, and the beam intensity was measured with a photodetector and a lock-in amplifier. The resulting data were then fitted to a complementary error function of the form: erfc共N兲 ⫽ k

兰

N

exp共 ⫺ ax 2兲 dx,

(9)

⫺⬁

where k is a numerical constant, N is the number of data points taken, and a gives the inverse square of the 1兾e width. Trial and error showed that this combination of objectives provided suitable collimation between the sample and the camera. We want to optimize the spot size based on the competing effects of having a small spot at the sample to probe a single index-gradient value while limiting defocusing at the camera. The camera needs to be relatively far from the sample so that very small deflections can be measured. Lenses after the sample are avoided because small deflections 共paraxial rays兲 would be refocused onto the optic axis, negating deflection information. The beam is normally incident upon the sample, which is held in a self-centering mount attached to an x–y translation stage driven by stepper motors. A 200-␮m pinhole is placed immediately behind the sample. The pinhole acts as both a spatial filter, blocking extraneous light and multiple reflections from inside the sample, and as an aid in centering the spot. The angle subtended by the pinhole and the incident laser beam is large enough not to affect the beam through its full deflection range. The beam then proceeds through a second polarizer and onto the CCD array of the camera 共PoleStar Model CDT500兲共the lens and infrared LED illumination ring in the camera were removed兲. The second polarizer can be used in conjunction with the first as an analyzer for determining sample birefringence. The entire setup 共except the computer兲 can fit on a small breadboard, so the experiment is both compact and portable. The position of the beam-spot center is determined by measuring its intensity profile with the CCD array. A framegrabber digitizes the data so that a data-acquisition program can determine the spot center by spatially averaging the brightest pixels in the frame and recording this intensity-averaged pixel location as the center of the spot. To obtain pure de-

flection data, we first measure the spot center 10 times without the sample present over a period of approximately 10 s. This procedure allows us to account for small temperature or beam-intensity fluctuations or mechanical vibrations while measuring the undeflected spot center. We then calculate the average x and y values of the undeflected spot position and subsequently subtract them from every deflection data point. The x and y coordinates of the data so obtained are saved in a two-column file, thus preserving deflection-vector information. An array of data points corresponding to minimum spatial separations in the sample of .85 ␮m 共the minimum step size of the stepper motor兾stage combination兲 can be taken using the x–y stage. The dataacquisition program runs as an x–y raster scan, running through all x positions in the programmed range for a given y position, then returning to the initial x position before moving on to the next y position. This protocol prevents backlash in the mechanical stages from biasing the data. The program also sets a short time delay between translation and imaging by the frame-grabber routine to allow vibrations to subside. A one-dimensional scan of 100 data points takes approximately 12 min, while a twodimensional scan of 100 ⫻ 100 data points may take 20 h. The actual run time also depends on the x distance the stage has to move during a scan, because it must return to its initial x position for each new y position. However, the operator need not continuously monitor data acquisition because the system is automated, and once started, it can be left unattended until the run is finished. C. Analysis and Results

The data file can be analyzed with any software that treats matrices, such as Mathematica or Matlab, as an R ⫻ C matrix of 共x, y兲 vector positions 共R rows by C columns, corresponding to the number of x and y positions mapped兲. It should be noted that the CCD camera inverts the y coordinate of data points owing to the fact that its imaging lens has been removed, so this change of sign must be included when analyzing the raw data. A vector-field plot of the deflection angle for a typical GRIN sample two-dimensional data scan 共Fig. 5兲 shows the refractive-index gradient as a function of its position in the sample. For illustrative purposes this data scan includes regions outside the preform 共the corners兲. Note that the air兾cladding boundary gives rise to large, discontinuous deflections, as expected by the large discontinuity in the refractive index at the boundary 共n ⫽ 1 to n ⬇ 1.5兲, and should not be considered as yielding meaningful data. Because the laser spot has a finite diameter, part of it is deflected differently than the rest as it passes sequentially through an interfacial boundary. As a result, the spot on the CCD array may subsequently range from elongated to heavily distorted, and may even split into two or more sections depending on the spot diameter and type of interface. While the dataacquisition program still spatially averages the

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

bright pixels even in these configurations, such distortions lead to discontinuities in the deflection data and also to artificial interface broadening, making the interface boundary appear wider and more rounded than it actually is. Those vectors at the air兾cladding interface that extended far beyond the boundary of the scan area were removed to clarify the plots. A close inspection of Fig. 5共a兲 shows a net linear bias of the vectors toward the right. In this case, the sample happens to be wedge shaped, measuring 2.671 mm on one edge and 2.759 mm on the opposite side, corresponding to a wedge angle of 0.2°. The amount of bias is consistent with the wedge angle. This wedge results in a constant deflection that mimics a refractive-index gradient toward the right of the figure. If the preform is known to be cylindrically symmetric, the sum of all vector deflections must sum to zero. The average of the deflection-vector sum over an acceptably uniform region of the sample must then be subtracted from each vector to yield a corrected data set 关Fig. 5共b兲兴. For a two-dimensional scan, the core region is thus chosen for averaging because it comprises a comparatively large portion of the sample, and the deflection within it is highly coherent in direction—that is, very uniform compared to transition regions, such as the core兾cladding or air兾cladding interfaces, which strongly scatter the beam. The wedge bias-corrected data can now be used to find the magnitude and angle of deflection for each point. We obtain f共r兲 by taking the square root of the sum of the squares of the x and y coordinates for each deflection data point in the matrix. The conversion factor between CCD pixel location and actual distance 共pixels are 9.6 ⫻ 6.3 ␮m兲 must also be considered to convert the deflection data recorded by the CCD array to absolute deflection displacements f共r兲. Next, the absolute deflection angle is calculated 共recall that ␣共r兲 ⬇ f共r兲兾z兲. This angle is used in Eq. 共8兲, where the dot product in the integrand reduces to cos ␾ cos ␣ if the numerical integration is performed in the x direction, or sin ␾ cos ␣ if it is performed in the y direction. Figure 6 illustrates the coordinate system used. Angle ␾ corresponds to the usual rotation 10 June 2002 兾 Vol. 41, No. 17 兾 APPLIED OPTICS

3407

Fig. 6. Coordinate system and angle definitions.

in the x–y plane. The value cos ␾ 共or sin ␾兲 is found by dividing the x coordinate 共or y coordinate兲 of deflection by f共r兲, with the condition that cos ␾ ⫽ 1 共or sin ␾ ⫽ 1兲 if f共r兲 ⫽ 0 to avoid singularities. The refractive-index difference between two points can now be determined through numerical integration, and the index profile can be obtained by tabulating this difference for each successive point. The discrete-difference equation used to approximate Eq. 共8兲共for x-direction integration兲 is then ⌬n关R, C兴 ⬅ n关R, C兴 ⫺ n关R, 1兴 ⫽

2dr zt

C

兺 f 关R, j兴cos ␾关R, j兴cos ␣关R, j兴, j⫽1

(10) where C and R run over the columns and rows of the matrix and 关R, j兴 denotes a matrix element. For integration along the y axis, sin ␾ replaces cos ␾ and 关R, j兴 becomes 关j; C兴. The summation is over R. Figure 7 shows the resulting three-dimensional refractive index profile for a two-dimensional GRIN preform scan with the absolute refractive index 共the cladding index value is 1.491兲 along the vertical axis. The corners are within the cladding, while the central peak spans the core. The small-amplitude ridges visible are the result of minor discontinuities in individual deflection data points that, once entered into the summation, propagate throughout the rest of the row. These discontinuties may arise from imperfec-

Fig. 7. One-dimensional GRIN preform refractive index profile. Top graph shows original biased data and linear fit; middle graph shows wedge-corrected data and paraboloid bias fit; bottom graph shows final corrected profile and parabolic profile fit. 3408


Fig. 8. Three-dimensional GRIN preform profile.

tions in the sample, such as surface scratches or material impurities. If the effect of these discontinuities leads to a large-amplitude feature, the amplitude can often be greatly reduced by repeating the numerical integration in the reverse direction 共i. e., summing from C to 1 instead of 1 to C兲 and averaging the two results. For a one-dimensional scan bias corrections are simpler. Figure 8 shows a one-dimensional index profile along the preform diameter 共in this case, along the y axis兲 for a sample with a slight wedge shape. The top graph shows the original, biased data where the constant gradient due to the wedge angle is clearly visible. The dotted line represents the result of a linear fit to the data, and approximates the bias due to the wedge angle. Once the wedge bias has been subtracted, the profile appears much improved, as shown in the middle graph. However, another bias that occurs is a very small paraboloid deviation in the surface of the preform sample slice, which may result from internal relaxation of the polymer after polishing. This bias appears as a parabolic contribution to the index profile and is most apparent as the non-zero slopes in the cladding-section data 共on either side of the core兲 that should have a flat profile. In other words, the index profile within the cladding should remain constant because there is no gradient in the cladding beyond the interfacial regions. We can also correct for this bias by fitting a parabola to these cladding sections of the profile and subtracting

along the preform slice diameter. The core diameter is indicated by the step. This profile yields a clear example of the artificial interface broadening mentioned earlier. The rounding of the profile from a pure step, and the peaks and the dip in the core region, result mostly from the finite size of the laserbeam spot, which is also shown 共this scan was obtained with 4⫻ and 5⫻ objectives in place of the 10⫻ and 20⫻ objectives, resulting in a larger beam-spot diameter兲. Once again, because the spot has a finite width, the beam will be partially deflected to varying degrees as its cross-section transits the interfacial region, resulting in the rounded peaks of the step profile. The data shown in Fig. 10 were corrected for a slight paraboloid bias. Fig. 9. Raw deflection magnitude data for a step-index preform.

the fit from the data. The resulting profile appears in the bottom graph of Fig. 8. Although the magnitude of these biases may seem insignificant, they can have an enormous effect on the refractive-index profile measured. The sample investigated here was fabricated to have a parabolic refractive-index profile. A parabolic fit to the corrected data, also included on the bottom graph, shows that the profile is not exactly parabolic. Our method also works for step-index profiles. A step-index polymer optical fiber preform was fabricated in the Nonlinear Optics Laboratory at Washington State University 共WSU兲.4,5 The core consists of dye-doped PMMA from a core preform of 10 mg & DR1 dye dissolved in 10 ml of MMA 共.07% by weight兲, and had a diameter of 723 ␮m at the time of fabrication. The preform was squeezed at 120 °C for 96 h before the sample slice was obtained and the profile measured. The magnitude of the raw deflection data obtained by the data-acquisition program for a two-dimensional scan is shown in Figure 9. Note that the interface between the two PMMA cladding half-rounds is clearly visible. In this case the two half-rounds did not fuse completely during squeezing, leaving a noticeable discontinuity the DDM experiment easily measured. Figure 10 shows a highresolution 共1.7 ␮m兾point兲 one-dimensional scan

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

4. Conclusion

We presented a novel method 共to the best of our knowledge兲 for determining the refractive-index profile of polymer optical fiber preforms. While the DDM process requires removing part of the preform, the sample required is but a small fraction of the entire preform and thus the method has a negligible detrimental effect on the preform itself, which can still be pulled into plenty of fiber. The sample may also be taken from the remnant of a preform that has already been pulled. The method provides relatively quick results with good resolution and fairly simple analysis. Moreover, it can be applied to various types of preforms and variations in the refractive index of samples. The effects of pulling the preform into a fiber can be investigated to see if the index profile merely scales down with the fiber or if unknown changes take place in the process. Once the index profile has been determined, the mode profile for the fiber can be calculated, and useful devices may be fabricated from the fiber. An interesting application of DDM is to study the diffusion effects of prolonged baking at high temperatures on the dye-doped core of a fiber preform. Another diffusion measurement that can be performed is investigating the effect of pulling the preform into a fiber. Cross-sectional slices at successive diameters of the pulled preform can be measured to see how the pulling process affects diffusion of the dye from the core. Another possible application is measuring the birefringence of samples by placing a crossed polarizer–analyzer pair around the sample. If the sample exhibits birefringence, it will rotate the polarization of the beam slightly so that part of it will then pass through the analyzer to be recorded by the camera. Mutual rotation of the polarizer and analyzer can thus be used to map the index ellipse. Index changes induced by photobleaching, such as when writing gratings, could also be measured with DDM. Certain improvements to the DDM experiment may be suggested. It is possible to employ a diode laser with a focusing lens that can replace the HeNe and microscope objectives. With appropriate filtering, if the focal length is made to be relatively long 共thereby increasing the sample– camera distance be10 June 2002 兾 Vol. 41, No. 17 兾 APPLIED OPTICS

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cause the beam will not defocus as quickly as with the objectives兲, then the resolution of the index profile may be increased by allowing even smaller deflections to be measured. A translation stage system with even finer resolution could be used. In addition, a higher-resolution CCD array may be employed. Also, the beam-spot diameter could be deconvolved from the integral to remove the effects of its finite size, reducing or even eliminating rounded, peaked interfaces that should be sharp and flat. Finally, more accurate polishing methods, such as polishing machines would eliminate wedge biases and may even lower the likelihood of paraboloid bias. This work was funded by The Air Force Office of Scientific Research and Sentel Technologies, L.L.C. We also thank Boston Optical Fiber, Inc., for providing graded-index preform samples. References 1. Y. Ohtsuka, E. Nihei, and Y. Koike, “Graded-index optical fibers of methyl methacrylate—vinyl benzoate copolymer with low loss and high bandwidth,” Appl. Phys. Lett. 57, 120 –122 共1990兲. 2. E. Nihei, T. Ishigure, and Y. Koike, “High-bandwidth, gradedindex polymer optical fiber for near-infrared use,” Appl. Opt. 35, 7085–7090 共1996兲. 3. M. G. Kuzyk, U. C. Paek, and C. W. Dirk, “Guest-host polymer fibers for nonlinear optics,” Appl. Phys. Lett. 59, 902–904 共1991兲. 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 共1996兲. 5. D. W. Garvey, Q. Li, M. G. Kuzyk, C. W. Dirk, and S. Martinez, “Sagnac interferometric intensity-dependent refractive-index measurements of polymer optical fiber,” Opt. Lett. 21, 104 –106 共1996兲. 6. A. Tagaya, Y. Koike, T. Kinoshita, E. Nihei, T. Yamamoto, and K. Sasaki, “Polymer optical fiber amplifier,” Appl. Phys. Lett. 63, 883– 884 共1993兲. 7. D. J. Welker and M. G. Kuzyk, “All-optical switching in a dye-doped polymer fiber Fabry–Perot waveguide,” Appl. Phys. Lett. 69, 1835–1836 共1996兲. 8. D. J. Welker and M. G. Kuzyk, “Photomechanical stabilization in a polymer fiber-based all-optical circuit,” Appl. Phys. Lett. 64, 809 – 811 共1994兲. 9. D. J. Welker and M. G. Kuzyk, “Optical and mechanical multistability in a dye-doped polymer fiber Fabry–Perot waveguide,” Appl. Phys. Lett. 66, 2792–2794 共1995兲. 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 共1998兲. 11. Z. Xiong, G. D. Peng, and P. L. Chu, “Nonlinear coupling and optical switching in a ␤-carotene-doped twin-core polymer optical fiber,” Opt. Eng. 39, 624 – 627 共2000兲. 12. V. N. Van, A. Brunet-Bruneau, S. Fisson, J. M. Frigerio, G. Vuye, Y. Wang, F. Abele`s, J. Rivory, M. Berger, and P. Chaton, “Determination of refractive-index profiles by a combination of visible and infrared ellipsometry measurements,” Appl. Opt. 35, 5540 –5544 共1996兲. 13. C. Saekeang and P. L. Chu, “Backscattering of light from optical fibers with arbitrary refractive-index distributions: uniform approximation approach,” J. Opt. Soc. Am. 68, 1298 – 1305 共1978兲. 3410


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