Characterization of graded index optical fibers by digital holographic ...

Characterization of graded index optical fibers by digital holographic interferometry Hamdy H. Wahba and Thomas Kreis* BIAS–Bremer Institut für angewandte Strahltechnik Klagenfurter Strasse 2, D 28359 Bremen, Germany *Corresponding author: [email protected] Received 27 October 2008; accepted 25 January 2009; posted 9 February 2009 (Doc. ID 102487); published 6 March 2009

For the first time to our knowledge, digital holography is used to determine the distribution of parabolic or nonparabolic refractive index fields in graded index optical fibers. The fiber is embedded in an index matching fluid whose refractive index can be varied to a matching and mismatching index with respect to that of the cladding. In experiments for both cases high precision phase-shifting digital holographic interferometry is applied with numerical evaluation employing the multilayer model for recognition of the refraction. Due to the higher redundancy in the multiple phase-shifted holograms better accuracy can be obtained compared to classical two-beam interferometry. Therefore the holographic method is recommended as a nondestructive and noncontacting method for characterizing graded index optical fibers. © 2009 Optical Society of America OCIS codes: 060.2270, 090.1995, 090.2880, 100.2650.

1. Introduction

Graded index optical fibers are widely used in short distance communications as well as in local area networks. Therefore it is important to obtain knowledge about the parameters of these fibers, especially the refractive index profile, the numerical aperture (N.A.), and the shape parameters. Each optical fiber consists of a core and a cladding. Contrary to the case for step index fibers, the core of graded index optical fibers has a parabolic or other nonuniform distribution of the refractive index, with a maximum at the fiber center [1,2] (Fig. 1). The refractive index profile nðrÞ of a graded index optical fiber can be described by the formula nðrÞ ¼ nc − Δn

α r ; R

0 ≤ r ≤ R:

ð1Þ

In this equation α is the exponential order of the refractive index profile and acts as a shape parameter, nc is the refractive index at the fiber core center, 0003-6935/09/081573-10$15.00/0 © 2009 Optical Society of America

and r is the distance measured from the core center. Δn ¼ nc − ncl with ncl the refractive index of the cladding and R the radius of the core. The N.A. of the fiber defines the maximum angle of an incident ray to the fiber axis, so that the ray remains confined inside the fiber core. The N.A. is given as [3] N:A: ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2c − n2cl :

ð2Þ

For a long time the determination of refractive index distributions in fibers or other transparent solids has been performed by interferometric methods, where two-beam interferometers as well as multiple-beam interferometers are used [4–6]. In [6] a two-beam interferometric system determined the refractive index distribution of a graded index optical fiber with a quadratic refractive index profile in the core. So Δn and α of the fiber have been determined nondestructively. This approach is feasible not only for quadratic profiles (α ¼ 2) but also for general values of α [7]. Also classical multiple-beam Fizeau interferometry was successfully used to determine Δn and α of graded index optical fibers [8]. The mathematics used in [6–8] neglect the nonstraightforward refraction of the light beam inside the fiber. A more 10 March 2009 / Vol. 48, No. 8 / APPLIED OPTICS

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Fig. 1. Refractive index distribution in (a) step-index fibers and (b) graded index fibers.

refined mathematical approach is based on the area under the graph of the interference phase shift due to the integrated refractive index. This method and modifications of it have been applied to fibers immersed in index matching fluid as well as in fluids not matching the index [9,10]. Hamza et al. [11,12] derived an accurate mathematical model that considered the exact local refraction of the incident beam on its way through the graded index optical fiber, which is divided into a number of thin concentric layers of constant refractive index. This model was verified with two-beam and multiple-beam interferometers. The consideration of the variation in the refraction gave better accuracy in the determination of the parameters of graded index optical fibers [12,13], thick optical fibers [14], and mechanically stressed optical fibers [15]. Multiple-beam Fizeau interference patterns have been analyzed using the Fourier transform method to obtain their contour lines [16]. Additionally a phase analysis was combined with the Fourier transform method to analyze the interference patterns [17]. The automated Fizeau interferometric techniques were applied in studies of the optical and optothermal properties of graded index optical fibers [18]. These cited developments substantially increased the accuracy of the measured optical fiber parameters. Holography is a method for recording and reconstructing whole optical wavefields, which means intensity and phase, while ordinary photography deals with only intensities [19,20]. Thus it exhibits 3D characteristics such as depth of field or parallax. The wavefields recorded by holography can be specularly or diffusely reflected by an opaque object, scattered by dispersed media like particles, or can be pure phase objects [21]. The availability of the full complex wavefield enables the interferometric comparison of two such fields with minor mutual changes. This information is obtained by holographic interferometry [22], which is a very effective nondestructive, contact-free tool to measure shape, deformation, or refractive index distributions [23]. The recent replacement of holographic plates and sheets 1574

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based on silver halides by CCD targets and the numerical reconstruction of the complex wavefields using a computer is known as digital holography (DHI), while the numerical comparison of wave fields reconstructed from digital holograms is digital holographic interferometry (DHI) [24]. The optically generated hologram is captured by a CCD target, digitized and quantized, and stored as a digital hologram file in a computer. There are a lot of applications of DHI to the diagnosis of refractive index profiles. Digital holographic microscopy was used to calculate the mean integral refractive index and thickness of living cells [25]. The absolute accuracy of the mean refractive index measurement was about 0.0003. Kebbel et al. applied digital holography to refractive index variations within transparent media in microgravity experiments [26]. Two-dimensional refractive index profiles of phase gratings have been investigated using DHI [27] as well as the refractive indices of liquids using lensless Fourier DHI [28]. The high performance of direct interferometry as an aid to digital holography was tested by measurements of low variation refractive indices of fluids in a comparative study with other techniques [29] such as traditional Mach–Zehnder interferometers. Owen and Zozulya [30] performed comparative field studies of DHI and Shack–Hartmann sensors. DHI with combined phase contrast imaging and amplitude imaging in digital holographic microscopy was presented by Cuche et al. [31]. The coupling of digital holographic microscopy and polarization imaging digital holography was demonstrated in an investigation of unbent and bent fibers and presented in [31– 33]. Reconstructed phase and amplitude distributions of unbent and bent fibers were presented. The mathematics describing the refractive index of transparent materials used in [26–30] cannot be used directly to measure and configure the refractive index profile of graded index fibers, since this assumes a constant refractive index along its path in a material. So it can be used only to determine the mean refractive index of the fiber, but it does not consider the varying refraction of the beam along

its path inside the fiber. Large scale strongly refracting fields produce a bending of the rays. This effect was recognized in holographic interferometric investigations combined with iterative calculations [34] as well as tomographic methods [35]. In this paper we apply for what is believed to be the first time digital holographic microscopic interferometry in combination with the multilayer model [12] to analyze and present the refractive index profile of a graded index optical fiber. We demonstrate the superior performance of digital holography for this purpose in experimental results, discuss the refraction models, and determine the shape parameters and optical properties of the investigated graded index fibers. 2. Theory A. Digital Holography

In conventional optical holography the recorded wavefield is reconstructed by illuminating the hologram with the reference wave. In digital holography instead we multiply the stored hologram values with the complex conjugate of a numerical model of the reference wave r ðξ; ηÞ and calculate the resulting diffraction field b0 ðx0 ; y0 Þ in the image plane; see Fig. 2. This is theoretically performed by the diffraction integral ZZ 1 expfikρg b ðx ; y Þ ¼ hðξ; ηÞr ðξ; ηÞ dξ dη ð3Þ iλ ρ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with ρ ¼ d02 þ ðξ − x0 Þ2 þ ðη − y0 Þ2 and k ¼ 2π=λ the wavenumber. The coordinates x, y, ξ, η, x0 , and y0 are as shown in Fig. 2. The finite discrete form of the Fresnel approximation to the diffraction integral is [36] 0

0

0

N−1 M−1

b0 ðnΔx0 ; mΔy0 Þ ¼ A Σ Σ hðjΔξ; lΔηÞr ðjΔξ; lΔηÞ

j¼0 l¼0

iπ × exp 0 j2 Δξ2 þ l2 Δη2 dλ

jn lm exp 2iπ þ : N M ð4Þ

The parameters used in this formula for calculating the complex field in the image plane are given by the CCD array used, having N × M pixels of pixel pitch Δξ and Δη in the two orthogonal directions; see Fig. 2. The stored hologram is hðjΔξ; lΔηÞ. The distance between object and CCD is denoted by d, and normally d0 ¼ d. Complex factors not depending on the hologram under consideration are contained in A. Given a specific CCD the pixel spacing in the reconstructed field is Δx0 ¼

d0 λ ; NΔξ

Δy0 ¼

d0 λ : MΔη

ð5Þ

An alternative to the Fresnel approximation uses the fact that Eq. (3) describes a convolution of hðξ; ηÞr ðξ; ηÞ, with the impulse response gðx0 ; y0 ; ξ; ηÞ ¼ ðexpfikρgÞ=ðiλρÞ. The convolution theorem now states that b0 can be calculated by b0 ¼ A0 F −1 fℱfh · r g · F fggg;

ð6Þ

where F denotes the Fourier transform and F −1 its inverse. In practice both are calculated by the fast Fourier transform algorithm. The resulting pixel spacing for this convolution approach is [37] Δx0 ¼ Δξ; B.

Δy0 ¼ Δη:

ð7Þ

Phase-Shifting Digital Holography

If we use a real hologram in the Fresnel reconstruction or convolution reconstruction, then we will obtain a strong d.c. term, a focused real image, and an virtual image that is not sharp. This can be avoided if in the hologram plane of Fig. 2 we have the complex wavefield instead of the real hologram. The complex field can be recorded and calculated by phase-shifting digital holography. For this purpose several holograms—at least three—with known mutual phase shifts are recorded. These holograms are I n ¼ aðx; yÞ þ bðx; yÞ cosðϕðx; yÞ þ ϕRn Þ; n ¼ 1; …; 5;

ð8Þ

Fig. 2. Geometry of digital Fresnel holography. 10 March 2009 / Vol. 48, No. 8 / APPLIED OPTICS

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Fig. 3. Arrangement for digital holography interferometry.

where aðx; yÞ and bðx; yÞ are the additive and the multiplicative distortions and ϕRn is the phase shift performed in the reference wave during recording of the holograms. In our case the phase shift is 90°, and it starts with ϕR1 ¼ 0°. We get a nonlinear set of five equations that is pointwisely solved by a Gaussian least squares method [23]. Then the real part of the field in the hologram plane is Re½Hðx; yÞ ¼4I 1 ðx; yÞ − I 2 ðx; yÞ − 6I 3 ðx; yÞ − I 4 ðx; yÞ þ 4I 5 ðx; yÞ;

ð9Þ

and the imaginary part is Im½Hðx; yÞ ¼ 7½I 4 ðx; yÞ − I 2 ðx; yÞ:

ð10Þ

Clearly the intensity distribution in the hologram plane is Iðx; yÞ ¼ jHðx; yÞj2, and the phase distribution is ϕðx; yÞ ¼ arctanfIm½Hðx; yÞ=Re½Hðx; yÞg. C. Calculation of the Refractive Index Profile Using the Multilayer Model

In the multilayer model the core of the graded index optical fiber is assumed to consist of a large number of thin layers. The cross-sectional area of each layer is circular. Each layer is considered to have same thickness a and refractive index nj , where the layers are numbered by j ¼ 1; …; N ¼ R=a with R the core 1576


Fig. 4. Phase-shifted digital holograms of the graded index optical fiber immersed in liquid of refractive index nL ¼ 1:46, with additional phase (a) 0; (b) π=2; (c) π; (d) 3π=2; and (e) 2π.

Fig. 5. Reconstructed interference phase modulo 2π from the phase-shifted digital holograms of Fig. 4.

radius. The core is surrounded by a homogeneous clad of refractive index ncl . If this optical fiber is immersed in a liquid of refractive index nL ¼ ncl there will be no refraction of the incident beam of wavelength λ at the boundary between the liquid and the cladding. Hamza et al. [12] obtained a recurrence relation that predicts the optical path difference of the refracted beam due to an optical path through Q layers of the fiber. The fiber is illuminated by a collimated beam, with the ray crossing the center of the core defining the optical axis. An arbitrary beam crosses the fiber at a distance dQ from the optical axis and leaves the core at a distance xQ . The following recurrence relation also describes the shape of the fringes that are produced as a result of the refraction of the incident beam through the Q layers of the fiber core. Assume a coordinate system with origin in the fiber core center, then the corresponding fringe shift is defined as ZQ and the optical path difference is given by the recurrence formula

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λZQ Q−1 ¼ Σ 2nj ðR − ðj − 1ÞaÞ2 − d2Q n2L =n2j j¼1 h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − ðR − jaÞ2 − d2Q n2L =n2j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðR − ðQ − 1ÞaÞ2 − d2Q n2L =n2Q þ 2nQ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 − d2Q þ R2 − x2Q ; − nL

Fig. 7. Mean interference phase difference across the graded index optical fiber core relative to the cladding/liquid refractive index.

with Q running from 1 to N and h the interfringe spacing. Without restriction of generality we can assume an incident light beam that passes through the middle of the Qth layer. This can be obtained using the following equation, which is used to predict the value of dQ in Eq. (11):

dQ ¼

nQ ½R − ðQ − 0:5Þa : nL

ð12Þ

Equation (11) now can be used to get the refractive index profile for any circular optical fiber such as homogeneous (N ¼ 1), skin core (N ¼ 2), multilayer (N), and graded index fibers (N → ∞). However in practice the graded index fibers are approximated by multilayer fibers. Hamza et al. [11,12] have shown that the inhomogeneous refraction must be considered in the measurement and calculation of optical fiber parameters, otherwise severe errors occur.

ð11Þ

Fig. 6. Unwrapped interference phase distribution with normalized background.

Fig. 8. Mean fringe shift across the graded index optical fiber core. 10 March 2009 / Vol. 48, No. 8 / APPLIED OPTICS

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Fig. 10. Refractive index profiles measured across the graded index optical fiber core with digital holographic interferometry and two-beam interferometry.

holographic arrangement; see Fig. 3. The collimated object beam crosses the sample and passes through the microscope objective MO1 with magnification 10× and N.A. 0.25. An identical microscope objective MO2 is installed in the reference arm to eliminate the curvature of the optical field. The position of MO2 is precisely adjusted. A piezoelectric transducer (PZT) acting as the phase-shifting tool is applied in the reference arm and helps us to obtain the five phase-shifted holograms. Starting with 0° the holograms are shifted mutually by π=2. The holograms are recorded by an Allied Vision Marlin F145B2 CCD camera with pixel pitch 4:65 μm × 4:65 μm

Fig. 9. (a) Two-beam interferogram; (b) contour lines; and (c) interferogram with overlaid contour lines of graded index optical fiber immersed in liquid of refractive index nL ¼ 1:46.

3. Experimental Results A. Experimental Arrangement

The method of digital holography in combination with the multilayer model was used to estimate the refractive index profile of a graded index optical fiber. The fiber sample was immersed in a liquidfilled optical cavity. The liquid used here was a mixture of butyl stearate and paraffin oil with concentrations such that first the refractive index of the cladding was matched perfectly and second it was slightly mismatched. Fresnel off-axis holograms have been produced by using a Mach–Zehnder-like 1578


Fig. 11. Phase-shifted digital holograms of graded index optical fiber immersed in liquid of refractive index nL ¼ 1:4565, with additional phase (a) 0; (b) π=2; (c) π; (d) 3π=2; and (e) 2π.

Fig. 12. Reconstructed interference phase modulo 2π from the phase-shifted holograms of Fig. 11.

and pixel numbers 1392 in the horizontal and 1040 in the vertical direction. The graded index optical fiber was a DRAKA MM fiber with core diameter 62:5 2:5 μm and cladding diameter 125 1 μm according to the specifications of the supplier. The N.A. is 0:275 0:015. B. Experimental Results with Matching the Refractive Index of the Cladding

Figure 4(a) represents the digital hologram for the graded index fiber immersed in liquid of refractive index 1.46, which perfectly matches the cladding index. The additional phase of this hologram, Fig. 4(a), is assumed to be zero, then the holograms of Figs. 4 (b) to 4(e) are the holograms with additional phase shifts π=2, π, 3π=2, and 2π. According to the phaseshift algorithm these five holograms are used to produce the complex field in the hologram plane. From this field the interference phase distribution is reconstructed by the convolution algorithm, and the result is shown in Fig. 5. The distance between hologram plane and image plane is −150 mm, and the pixel pitch and magnification of the reconstructed phase image are the same as those of the original holograms. While in Fig. 5 we have the wrapped interference phase, the phase first has been unwrapped. Then the linear increase of the background phase from the left to the right has been approximated by linear regression and subtracted from the unwrapped interference phase. The result with normalized background is displayed in Fig. 6. This image represents the mean phase of the fiber, which varies

Fig. 13. Unwrapped interference phase distribution with normalized background.

Fig. 14. Mean interference phase difference across the optical fiber core and cladding in the mismatching case.

across the fiber but remains nearly constant along the fiber. This is due to the fact that the refractive index of every layer of the fiber remains constant along the fiber. The mean phase across the fiber is presented in Fig. 7. Now it is easy to convert the measured interference phase difference ΔϕQ at any distance xQ from the core center to a shift ZQ =h, e.g., by the following formula, which converts the phase diffrence ΔϕQ into an optical path difference (OPD): OPD ¼

λZQ λ ¼ ΔϕQ : h 2π

ð13Þ

Figure 8 shows the fringe shift ZQ =h across the fiber diameter. These data now are used within the multilayer model to calculate the refractive index profile of the measured graded index optical fiber, which is realized in a computer program. For comparison reasons the refractive index profile was also measured by ordinary two-beam interferometry as a standard method. The resulting interferogram of two-beam interferometry obtained with the same Mach–Zehnder arrangement is given in Fig. 9(a),

Fig. 15. Mean fringe shift across the optical fiber regions. 10 March 2009 / Vol. 48, No. 8 / APPLIED OPTICS

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Fig. 17. Refractive index profiles measured across the graded index optical fiber regions (core and cladding) with digital holographic interferometry and two-beam interference in the mismatching case.

structed interference phase distribution in a distance of d ¼ 221 mm after applying the phase-shift formula and the convolution reconstruction algorithm is

Fig. 16. (a) Two-beam interferogram; (b) contour lines; and (c) interferogram with overlaid contour lines of graded index optical fiber immersed in liquid of refractive index nL ¼ 1:4565.

with the contour lines estimated by a Fouriertransform-based algorithm in Figs. 9(b) and 9(c). Again the refractive index profile is calculated by the multilayer model. The refractive index profiles of a graded index optical fiber thus are shown in Fig. 10 as measured by digital holography and normal interferometry. The curves indicate that the refractive index profiles obtained by the two different methods are nearly the same. C. Experimental Results with Index Mismatching

The same approach of digital holography and the multilayer model was used when the liquid was not matching the index of the cladding. Figure 11 shows the phase-shifted holograms, where now the refractive index of the liquid is 1.4565. The recon1580


Fig. 18. Refractive index profiles across the graded index optical fiber core with digital holographic interferometry and two-beam interferometry in the (a) matching and (b) mismatching case after fitting α.

Table 1.

Statistical Summary of Characterization of Graded Index Optical Fiber in Matching and Mismatching Cases with Digital Holography and Two-Beam Interferometry

Digital Holography nL R2 χ 2 =DoF α αmean Δn Δnmean N.A. N:A: mean

Two-Beam Interferometry

1.46 1.4565 0.99898 0.99873 0:793 × 10−7 1:226 × 10−7 1:966 0:017 2:087 0:029 2:0265 0:0696 0.02876 0.02879 0:02878 2:12132 × 10−5 0.29121 0.29137 0:29129 0:00011

shown in Fig. 12. The effect of the mismatching liquid appears in the cladding region. One can note that the interference phase across the graded index optical fiber can be divided into two regions. The outer one is due to the difference between the refractive indices of the liquid and the cladding used. The central region is a result of the gradient present in the core refractive index relative to the liquid’s and cladding’s refractive indices. Figure 13 gives the unwrapped phase with normalized background phase. As before, we have the mean interference phase across the fiber, Fig. 14, and the resulting fringe shift in Fig. 15. Now again the refractive index profile is determined with the aid of the multilayer model. For comparison normal interferometry is used also in the nL ¼ 1:4565 case. Figure 16 shows the two-beam interferogram as well as the contour lines of the fringes. Figure 17 represents the refractive index profiles of the graded index optical fiber using both methods. Both profiles nearly fit together. So altogether the feasibility of digital holography is confirmed in the matching as well as in the mismatching case. D. Graded Index Fiber Parameter Estimation

From the refractive index profiles measured and calculated as described in the preceding subsections we can determine the fiber parameters, e.g., α of Eq. (1). Figure 18 gives the refractive index profiles of the core, measured with the two cited interferometric methods with α fitted in a way that results in the lowest standard deviation. Figure 18(a) is obtained with a matching liquid, while Fig. 18(b) shows the result for mismatching liquid. From these the most important fiber parameters are estimated, such as the shape parameter α, the refractive index difference Δn, and the N.A. In a statistical analysis for the cases of matching (nL ¼ 1:46) and mismatching (nL ¼ 1:4565) the refractive index of the surrounding liquid to the cladding’s index and for digital holography as well as two-beam interferometry the quality of the results was checked. The results of this analysis are shown in Table 1. The most interesting statistical parameter is the chi square divided by the degrees of freedom, χ 2 =DoF, which is generally lower for digital holography than for two-beam interferometry, indicating better accuracy of the aforementioned method. The calculated N.A. is in agreement with that given by the supplier, which

1.46 0.99855 1:228 × 10−7 1:801 0:018

1.4565 0.99694 2:778 × 10−7 1:919 0:043 1:86 0:07104 0.0303 0.02925 0:02978 7:4 × 10−4 0.29911 0.2939 0:29651 0:00368

was N:A: ¼ 0:275 0:015. The determined core and cladding diameters are 62:775 0:93 μm and 125:085 0:93 μm, respectively. The results are in the bounds provided by the supplier but with a drastically shorter error interval. However the figures of the supplier are valid for the whole length of the fiber, while our results apply to a single cross section of the fiber. Nevertheless our results approve the validity of digital holography to study graded index optical fibers with high accuracy. 4.

Conclusions

For the first time to our knowledge the nonhomogeneous refractive index distribution in graded index optical fibers is measured by digital holographic interferometry, here employing phase shifting and reconstruction by convolution. The determined interference phase distributions then have been evaluated by the multilayer model to yield the exact parameters of the fiber core. The results have been compared to those gained by classical two-beam Mach–Zehnder interferometry. A statistical analysis exhibited a better accuracy for the digital holography method. Thus it is proved that this new approach can be recommended as a method for nondestructive and noncontacting measurement of the important fiber characteristics. H. H. Wahba gratefully acknowledges financial sponsorship by the Channel System of the Egyptian government for his stay at BIAS. Furthermore the authors thank M. A. Agour for fruitful discussions and help. H. H. Wahba is on leave from the Department of Physics, Faculty of Science, University of Mansoura, Damietta, Egypt. References 1. N. Barakat and A. A. Hamza, Interferometry of Fibrous Materials (Hilger, 1990). 2. G. E. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002). 3. S. Yin, P. B. Ruffin, and F. T. S. Yu, eds., Fiber Optic Sensors, 2nd ed. (CRC, 2008). 4. R. C. Faust, “An interferometric method of studying local variations in the refractive indices of a solid,” Proc. Phys. Soc., London, Sect. B 65, 48–62 (1952). 5. R. C. Faust, “The determination of the refractive indices of inhomogeneous solids by interference microscopy,” Proc. Phys. Soc., London, Sect. B 67, 138–148 (1954). 10 March 2009 / Vol. 48, No. 8 / APPLIED OPTICS

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