Extended interference formula foroptical fiber characterization

Mohamed A. El-MorsyAdel M. Sadik

Mohamed A. El-Morsy, Adel M. Sadik, “Extended interference formula for optical fiber characterization,”Opt. Eng. 55(6), 066104 (2016), doi: 10.1117/1.OE.55.6.066104.

Extended interference formula for optical fibercharacterization

Mohamed A. El-Morsya,*,† and Adel M. Sadikb,‡

aPrince Sattam bin Abdulaziz University, College of Science and Humanitarian Studies, Physics Department, Al-kharj, Saudi ArabiabKing Khalid University, Physics Department, Faculty of Science for Girls, 960-61421 Abha, Saudi Arabia

Abstract. An extended interference formula has been derived permitting simultaneous determination of geo-metrical and optical parameters of optical fibers whether the cores or stress-applying parts are nonconcentric tocladding. This formula can be applied for normal and oblique transverse interferometric techniques. In this paper,this interference formula is combined with computer-aided Mach–Zehnder microinterferometer for characteriza-tion of these optical fibers. The advantage of the presented interference formula is that it can be applied for alltypes of optical fibers. Also, it is used to minimize any uncertainty of the thickness measurement. In comparisonwith the well-known conventional interference formula, the measurement accuracy of this extended formula isdiscussed. © 2016 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.55.6.066104]

Keywords: interference formula; homogenous; multicore; polarization-maintaining optical fiber; variable incidence angle; fringedeflection; refractive index; thickness.

Paper 160214 received Feb. 22, 2016; accepted for publication May 10, 2016; published online Jun. 3, 2016.

1 IntroductionFabrication of long low-loss organic or inorganic standardsingle mode fiber (SSMF) and multicore fiber (MCF) for im-aging or telecommunication purposes has prompted thedevelopment of sensitive methods to characterize the opticalparameters of such fibres.1 The measurement of the maxi-mum refractive index difference and the index profile canbe used to find out optimal object structures for certain appli-cations. The measuring accuracy of these parameters isaffected strongly by the mathematical model of the outputinterference pattern.2–8 Sadik8 was applied a general interfer-ence formula for optical characterization of MCF on the con-dition that the thickness is known. Wahba9 was applied themultilayer mathematical model for index profiling measure-ment of polarization-maintaining (PM) PANDA fiber undercondition of using a matching liquid with the cladding. Thismeant that this model cannot apply for determination of opti-cal parameters of fiber whether the cores (MCF) or stress-applying parts (SAPs) are not concentric with claddingwhen the fiber submerged in a mismatching liquid withcladding.

The measured optical parameters are also affectedstrongly by the method of thickness measurement.8–14 Inmost transverse interferometic techniques, the refractiveindices can be calculated from simple formulae on the con-dition that the object thickness is known.8,15,16 The smalldiscrepancy observable in optical parameters may be dueto uncertainties in the measurement of the thickness ofthe fiber core and cladding. Unfortunately, accurate

measurement of the thickness is not trivial even whenmodern image processing techniques are employed. In theconventional method, the thickness measurement wasaffected by a small defocusing on the fiber under study(fiber edges were blurred).17 This undesirable effect andthe precise measurement of thickness required reconfiguringof the optical system each time, which is inconvenient andsignificantly slows the measurement procedure. The globalerror in the thickness measurement is estimated and found tobe �48 × 10−2 μm when the image processing system isused. Therefore, the global error in determination of therefractive index is �4 × 10−4.17

Sadik17 and Sadik et al.14,18 were applied the computer-aided variable-incidence-angle (VIA) interference methodwith mathematical interference formulas to overcome theproblem of the thickness determination uncertainty of thesingle- and coaxial multimedium objects. In this method,the optical path difference produced by the fiber understudy at different incidence angles is the only quantity mea-sured directly. Using these formulas, the global errors indetermination of the refractive index and the thickness areestimated and found to be �21 × 10−5 and �19 × 10−2,respectively. Unfortunately, these interference formulas can-not apply for the optical fiber characterization whether thecores or SAPs are nonconcentric to cladding.

The aim of the presented study is to overcome the abovementioned problems. In this paper, an extended derivedinterference formula can describe the fringe field interferencepattern of fiber whether the cores (MCF) or SAPs are non-concentric to cladding in a transverse interferometer. Thisformula with computer-aided VIA Mach–Zehnder interfer-ometer is proposed for simultaneous determination of geo-metrical and optical parameters of the nine-elliptical core andthe PM PANDA optical fibers.

*Address all correspondence to: Mohamed A. El-Morsy, E-mail: elmorsym@yahoo.com

†Permanent address: University of Damietta, Physics Department, Faculty ofScience, New Damietta, Egypt.

‡Permanent address: University of Mansoura, Physics Department, Faculty ofScience, 35516 Mansoura, Egypt. 0091-3286/2016/$25.00 © 2016 SPIE

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Optical Engineering 55(6), 066104 (June 2016)

2 TheoryThe schematic diagram of the Mach–Zehnder interferometeris shown in Fig. 1(a). A homogenous MCF [see Fig. 1(b)]under study is immersed in a liquid, which has a refractiveindex nL and introduced in one of the optical paths of theMach–Zehnder interferometer.

The interference pattern of homogenous MCF has beendescribed by an extended derived interference formula asthe following subsections (Secs. 2.1 and 2.2).

2.1 Anisotropic Homogenous Multielliptical CoreOptical Fiber

Assume that an anisotropic homogenous multicore opticalfiber with m’th identical elliptical cores of radii r1 (majoraxis) and r2 (minor axis) and directional refractive indicesnCe and nCo are surrounded by a homogeneous claddingof refractive index nS and radius rf .

At the beginning, we will deal with the simplest case asfollows:

1. Let anisotropic homogeneous optical fiber with clad-ding-core structure be illuminated by a linear polarizedlight, as shown in Fig. 2(a). ~nL, ~nS, and ~nC representthe incidence and transmitted wave vectors in the sur-rounding medium, cladding, and core layer of theobject, respectively. Let p and s be unit vectors paral-lel and normal to the object axis, respectively. Theequation of the index ellipsoid for the extraordinarywave in an anisotropic optical fiber with cladding-core structure and the general form of the Snell’slaw can be expressed asEQ-TARGET;temp:intralink-;e001;63;111j~nCj2 ¼ n2Ce − NCð~nC · pÞ2; ~nC ¼ ~nS þ ΓCSs;

j~nSj2 ¼ n2Se − NSð~nS · pÞ2; ~nS ¼ ~nL þ ΓSs; (1)

where ΓS and ΓCS are the scaling constants in the fibercladding and core, respectively, N ¼ ½ðn2e − n2oÞ∕n2o�,ne and no are the extraordinary and ordinary refractiveindices. In the present paper, the cladding and the sur-rounding medium are anisotropic media; i.e., NS andNL are equal to zero ðne ¼ no ¼ nÞ. Therefore, Eq. (1)can be rewritten as follows:

EQ-TARGET;temp:intralink-;e002;326;675n2S ¼ n2L þ 2nLΓS cos θ þ Γ2S (2)

and

EQ-TARGET;temp:intralink-;e003;326;632n2Ce ¼ NCn2L sin2 θ þ n2S þ 2nLΓCS cos θ þ 2ΓSΓCS

þ Γ2CS;

(3)

where θ is incidence angle, n⇀L · s ¼ nL cos θ and

p · s ¼ 0.The transversal dimensions and the refractive indi-

ces of the fiber cladding and core can be derived asfollows:

a. The refractive index (ns) and the transversal dimen-sion (dS) of the optical fiber cladding can be cal-culated as a function of the known refractive index(nL) of the surrounding medium. Let the beam (I)be incident at distance ðr2 ≤ x < rfÞ from the fibercenter, Eq. (2) can be rewritten as follows:

EQ-TARGET;temp:intralink-;e004;326;445n2S ¼ n2L þ 2nL½ΓSðxÞ�θ cos θ þ ½Γ2SðxÞ�θ: (4)

The scaling constant ½ΓSðxÞ�θi is given by

EQ-TARGET;temp:intralink-;sec2.1;326;402ðΓSjxjÞθ ¼½ZSðxÞ�θhdSðxÞ

λ;

where h is the interfringe spacing, ZSðxÞ is thefringe deflection in the cladding image at distancex from the center, and λ is the wavelength of mono-chromatic light used. dSðxÞ½¼ ðr2f − x2Þ1∕2� is thetransversal dimension of the fiber cladding.

b. The directional refractive indices (nCe and nCo) andthe transversal dimension (dC) of the fiber core canbe calculated as a function of the known refractiveindex (nL) of the surrounding medium. Let beam(II) be incident at distance ð0 ≤ x < r2Þ from thefiber center, Eq. (3) can be rewritten as

EQ-TARGET;temp:intralink-;e005;326;234

n2Ce ¼ n2L þ NCn2L sin2 θ

þ 2nL cos θ½ΓSðxÞ þ ΓCSðxÞ�θþ 2½ΓSðxÞΓCSðxÞ�θ þ ½Γ2

SðxÞ�θ þ ½Γ2CSðxÞ�θ:

(5)

The scaling constant ½ΓCSðxÞ�θ is given by

EQ-TARGET;temp:intralink-;sec2.1;326;139½ΓCSðxÞ�θ ¼½ZCSðxÞ − ZSðxÞ�θ

hdCðxÞλ;

where ZCSðxÞ is the extraordinary fringe deflectionin the fiber core image at distance x from the fiber

Fig. 1 (a) Schematic diagram of computer-aided Mach–Zehnderinterferometer; laser beam source S, two lenses L1 and L2, tworeflected mirrors M1 and M2, two beam splitters D1 and D2, rotatingdisk RD, prismatic device PD, CCD camera, and PC compatible com-puter; (b) photography of a nine-elliptical core fiber (MCF) crosssection.

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center. dCðxÞ½¼ 2r1∕r2ðr22 − x2Þ1∕2� is the transver-sal dimension of the fiber core.

Having determined the fringe deflections[ZSðxÞ and ZCSðxÞ] in the image of the MCFunder study at three different incidence angles(θ) and by using Eq. (5), the refractive indices(nCe and nCo) and the transversal dimension (dC)of the fiber core can be calculated. Finally, theradii (major and minor) of the elliptical fibercore can be calculated.

2. In this case, we assume that the major axis of ellipticalcore made an angle α with the Y-axis and the corecenter is shifted away from the fiber center by C asshown in Fig. 2(b). The refractive indices of the fibercladding and core of the optical fiber (nS, nCe, and nCo)can be calculated by using the Eqs. (4) and (5) but thefiber core transversal dimension dCðxÞ*** can be cal-culated as follows:

EQ-TARGET;temp:intralink-;e006;63;232dCðxÞ¼2jr1r2j½ðr22−r21Þcos2ðαÞ−ðx−C sinαÞ2þr21�

12

r21−ðr21−r22Þcos2ðαÞ:

(6)

Equations (4) and (5) are applicable in the range ðA ≤x < rfÞ and ðA ≤ x ≤ BÞ, respectively, where A and Bare constants and can be written as follows:

EQ-TARGET;temp:intralink-;sec2.1;63;141A ¼ C sin α − r2

�cos2 αþ r21

r22sin2 α

�1∕2

;

and

EQ-TARGET;temp:intralink-;sec2.1;326;446B ¼ C sin αþ r2

�cos2 αþ r21

r22sin2 α

�1∕2

.

3. We can generalize this formula to be applicable for ahomogeneous m’th- cores MCF with cores are non-concentric to cladding. The refractive index (ns) andthe transversal dimension (dS) of the fiber claddingare calculated using Eq. (4). Whereas the directionalrefractive indices (nCe and nCo) and the transversaldimension (dC) of each core can be calculated usingthe following equation:EQ-TARGET;temp:intralink-;e007;326;318

n2Ce¼n2LþNCn2L sin2θ

þ2nL cos θ½ΓSðxÞþΓCSiðxÞ�θ

þ2½ΓSðxÞΓCSiðxÞ�θþ½Γ2

SðxÞ�θþ½Γ2CSi

ðxÞ�θ; (7)

where the scaling constant ½ΓCSiðxÞ�θ is given by

EQ-TARGET;temp:intralink-;sec2.1;326;240½ΓCSiðxÞ�θ ¼

½ZCSiðxÞ − ZSðxÞ�θhdCiðxÞ

λ;

where dCi is the fiber core transversal dimensionwhich is given by

EQ-TARGET;temp:intralink-;e008;326;176dCiðxÞ

¼2jr1r2jXmi¼1

½ðr22−r21Þcos2ðαiÞ−ðx−C sinαiÞ2þr21�12

r21−ðr21−r22Þcos2ðαiÞ.

(8)

This equation is applicable in the range ðAi ≤ x ≤ BiÞ,where Ai and Bi are constants and can be written asfollows:

Fig. 2 (a) Diagram of the vectors that describe the reflection of rays of light in anisotropic optical fiber withcore is concentric to cladding; (b) schematic diagram of optical fiber with core is nonconcentric to claddingand the major axis of elliptical core made an angle α with the Y -axis.

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EQ-TARGET;temp:intralink-;sec2.1;63;752Ai ¼ C sin αi − r2

�cos2 αi þ

r21r22

sin2 αi

�1∕2

;

and

EQ-TARGET;temp:intralink-;sec2.1;63;705Bi ¼ C sin αi þ r2

�cos2αi þ

r21r22

sin2 αi

�1∕2

.

Having determined the fringe deflections½ZSðxÞ andZCSðxÞ� in the image of the MCF understudy at three different incidence angles (θ) and byusing Eqs. (4), (7), and (8), the refractive indices (nS,nCe, and nCo) and the transversal dimensions (dS anddC) of the optical fiber cladding and core can becalculated.

2.2 Isotropic Homogenous Multicore Fiber

If the cores of MCF under study are homogeneous isotropicmedia (NC ¼ 0; nCe ¼ nCo ¼ nC) and have circular crosssections (r1 ¼ r2 ¼ rC), Eqs. (7) and (8) can be rewrittenas follows:EQ-TARGET;temp:intralink-;e009;63;508

n2C ¼ n2L þ 2nL cos θ½ΓSðxÞ þ ΓCSiðxÞ�θ

þ 2½ΓSðxÞΓCSiðxÞ�θ þ ½Γ2

SðxÞ�θ þ ½Γ2CSi

ðxÞ�θ

(9)

and

EQ-TARGET;temp:intralink-;e010;63;446dCiðxÞ ¼ 2

Xmi¼1

½r2C − ðx − C sin αiÞ2�12: (10)

In this case having the fringe deflections at two differentincidence angles (θ) the refractive indices (nS and nC) andthe transversal dimensions (dS and dCi) of the opticalfiber cladding and core can be calculated using Eqs. (4),(9), and (10).

3 Measuring TechniqueIn this paper, all experiments have been performed with com-puter-aided normal and VIA Mach–Zehnder interferometictechnique [see Fig. 1(a)]; i.e., the conventional Mach–Zehnder interferometer is equipped with the prismatic

device, i.e., designed to change the angle of the incidenceof the polarized monochromatic light (laser beam) withrespect to the Z-axis from θ ¼ 0.0 deg to 70 deg�1 s.Also, a rotating disk (RD) is used to provide smooth rotationof the investigated optical fiber around Y-axis fromα ¼ 0.0 deg to 360 deg�1 s. Additionally, RD is used tomeasure the fringe deflection in a selected core. Fulfillingthe selecting region can be done by turning the fiber aroundits main axis through an angle arising from its geometry.

An investigated optical fiber is submerged in a certainimmersion oil of refractive index nL and put in one armof the Mach–Zehnder interferometer. This fiber will be illu-minated transversally to its axis (incidence angle θ ¼ 0) by alinear polarized monochromatic light. The interferogram isproduced through interference of the wave passing throughthe multicore optical fiber and the wave propagated in thesecond arm of the interferometer. The distance betweenfringes depends on the wavelength and the angle betweenboth beams. The interference pattern of the output field isgrabbed by a CCD camera for further automatic processingand analysis by the computer system. The fringe deflectionsin the fiber images and the liquid interfringe spacing areextracted and processed by PC at different incidence angles.

The present interference formula is combined with thecomputer-aided VIA Mach–Zehnder interferometer forsimultaneous determination of the optical and geometricalparameters of the investigated optical fiber.

4 Materials

4.1 The Polarization-Maintaining PANDA OpticalFiber

The PM PANDA optical fiber is designed with the bestpolarization maintaining properties and offers low attenua-tion and excellent birefringence for high performance appli-cations. This fiber has a core and two identical and isolatedSAPs positioned on opposite sides of the core as shown inFig. 3(a). The core of radius rcð¼ 3 μmÞ and SAPs of radiusrSAPð¼ 17.5 μmÞ are surrounded by a homogeneous clad-ding of radius rfð¼ 62.5 μmÞ (manufacturing data). Thepropagation constants of the PM PANDA optical fibersstrongly depend on the directional refractive indices of the

Fig. 3 (a) is the schematic diagram of the PM PANDA optical fiber. (b), (c), and (d) are the fringe fieldinterference patterns of the PM PANDA optical fiber when the polarized light vibrating parallel to its axisand the common axis of both core and SAPs made an angle α ¼ 90 deg, α ¼ 15 deg, α ¼ 0 deg withY -axis, respectively, and (e) is the skeletoning of interferogram (c).

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fiber core. The relative propagation delay difference betweenthe two orthogonally polarized modes depends on the stressbirefringence in these directions.

4.2 Homogenous Nine-Elliptical Core Optical Fiber

MCF is a fiber that can have multiple cores inside a singlecladding. This MCF has greater transmission capacity ascompared with SSMF. MCF offers a possible solution toincrease the fiber density and overcome cable size limitationsand duct congestion problems.1 Design and fabrication ofseveral types of MCFs have been reported to address thisneed for high density while maintaining low loss and lowcrosstalk.1 Heterogeneous MCF consists of several kindsof step-index single mode cores whose propagation con-stants and refractive indices are different from each other.In the homogenous MCF, all of the cores are identical toeach other; i.e., they have the same refractive index andradii. In this paper, the multicore optical fiber under inves-tigation is a homogenous nine-elliptical core optical fiber asshown in Figs. 5(a)–5(i). The fiber core of radii r1 (¼16 μm;major axis) and r2 (¼8 μm; minor axis) are surrounded by ahomogeneous cladding of radius rfð¼ 190 μmÞ (manufac-turing data).

5 Results and Discussions

5.1 Determination of the Optical and GeometricalParameters of Polarization-Maintaining PANDAOptical Fiber

Figure 3(a) shows the schematic diagram of the cross sectionof the PM PANDA optical fiber. This fiber consists of cir-cular birefringent core of radius rc and directional refractiveindices ne and no and two identical and isolated SAPs ofradius rSAP and refractive index nSAP positioned on oppositesides of the core. The core and SAPs are surrounded by ahomogeneous cladding of refractive index ns and radiusrs. Also, this figure shows that the SAPs are nonconcentricto cladding. This meant that the SAPs center is shifted awayfrom the PM fiber center by C (¼46.5 pixels) as shown inFig. 3(a). Also, Fig. 3(a) shows that the common axis of boththe core and SAPs made an angle α ¼ 90 degwith respect toY-axis. The optical and the geometrical parameters of the PMPANDA optical fiber can be determined using the currentextended interference formula [Eqs. (4), (5), and (6)] pre-cisely. The PM fiber is immersed in a suitable liquid ofrefractive index 1.45700� 2 × 10−5 and put in one arm ofthe Mach–Zehnder interferometer. It is important to notethat the optical fiber is usually submerged in a liquid ofrefractive index close to the fiber cladding. This is meantto minimize the optical path differences and to avoid addi-tional reflections at the liquid-fiber cladding boundary. Thisfiber is trans illuminated normally to its axis by a linearpolarized monochromatic light of wavelength 632.8 nm.

Figure 3(b), 3(c), and 3(d) demonstrates the fringe fieldinterference patterns of the PM PANDA optical fiber for lin-ear polarized light vibrating parallel to the fiber axis and thecommon axis of both the core and SAPs made an angleα ¼ 90 deg, 15 deg, and 0.0 deg with respect to Y-axis,respectively. Figure 3(e) shows the skeletoning of interfero-grams, Fig. 3(c) by using a thinning procedure. All experi-mental data (fringe liquid spacing h and fringe deflectionZðxÞ in the image of the PM PANDA fiber at three different

incidence angles) are extracted and processed by PC compat-ible computer was directly coupled to the image processingsystem via a CCD camera. The measurements are performedamong as many fringes as possible and the result averaged toimprove the precision.

The boundaries position between the cladding and both ofthe core and SAPs can be detected via the maximal and theminimal values of the fringe deflection. Subsequently, themaximal and the minimal values of the fringe deflectionZðxÞ in the PM PANDA optical fiber microinterferogramcan be automatically measured (in pixels) along theX-axis. A single fringe interferogram with interference pat-tern perpendicular to the fiber axis can be used to extract allinformation by performing the intensity scanning along asmany lines parallel to the interference fringes. Figure 4shows the average fringe deflection profile of the interfero-gram along the fiber width when the common axis of boththe core and SAPs made an angle α ¼ 45 degwith respect toY-axis, as an example. The measured experimental data inthe range C sin αþ rSAP ≤ x < rf are substituted into thisextended formula [Eq. (4)] and using a simple computer pro-gram, the refractive index (ns) and radius rf of the PM fibercladding can be calculated as a function of the known refrac-tive index (nL) of the surrounding medium and the results aretabulated in Table 1. Furthermore, the geometrical and opti-cal parameters of isotropic SAPs of the PM fiber can be cal-culated in the range C sin α − rSAP ≤ x < rSAP using thecurrent extended formula [Eqs. (9) and (10)] and the resultsare shown in Table 1. Finally, the radius rc and the directionalrefractive indices (nCo and nCe) of the PM fiber core can becalculated in the range 0 ≤ x < rc using Eq. (5) (r1 ¼ r2 ¼rC), as shown in Table 1. Having determined the directionalrefractive indices the birefringence ðBc ¼ nCe − nCoÞ and thebeat length ½Lb ¼ ðλ∕BCÞ� of the PM fiber are calculated, asshown in Table 1.

In comparison with the well-known conventional interfer-ence formula, the same part of length of the same fibershould be investigated to minimize any uncertainty of mea-surements of the fiber optical parameters (see Table 1). Theprocedures of this conventional interference formula aregiven elsewhere.15,16 The refractive indices (ns, nCo, nCe, andnSAP) of the PM PANDA fiber can be determined using this

-125 -100 -75 -50 -25 0 25 50 75 100 125

Fiber width (pixel)

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Z (

x ) /

h

Perpendicular

Parallel

Fig. 4 Profile of the mean fringe deflection along the PM PANDA opti-cal fiber width when the polarized light vibrating parallel andperpendicular to the fiber axis and the common axis of both coreand SAPs made an angle α ¼ 45 deg.

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conventional interference formula on condition that the geo-metrical parameters (rf , rSAP, and rc) are known. The digitalimage tracing method is applied for fiber geometrical param-eters measurement. The average values of the real rf , rSAP,and rc are measured when the intersection line tracing meth-ods to be used at different areas along the same part of theinvestigated fiber length, as shown in Table 1. Having deter-mined these geometrical parameters and the fringe deflectionalong the fiber width, the refractive indices of the PMPANDA fiber could be determined and the results areshown in Table 1. From Table 1, it seems that the resultsof the optical parameters of the PM PANDA fiber byusing the extended and well-known conventional interfer-ence formulas were approximately identical, but the accu-racy of the former is significantly higher. Using the currentextended interference formula, the global errors in determi-nation of the refractive index ΔnðnL; θ; ZÞ and the transver-sal dimension ΔdðnL; θ; ZÞ are estimated and found to be�14 × 10−5 and �19 × 10−2, respectively.

5.2 Determination of the Optical and GeometricalParameters of Homogenous Nine-Elliptical CoreOptical Fiber

Figure 1(b) shows the microscopic cross-section image ofnoncoaxial homogenous multicore fiber. This fiber consistsof identical isotropic nine elliptical cores of refractive indexnc and radii r1 and r2. These cores are surrounded by ahomogenous cladding of refractive index ns and radius rf .The cores’ center is shifted away from the fiber center byC (¼50.68 pixels). Also, the major axis of these cores ismade an angles α ¼ 0 deg, �40 deg, �80 deg,�120 deg, and �160 deg as shown in Figs. 5(a)–5(i). Thisfiber is immersed in a suitable liquid of refractive index(nL ¼ 1.52896� 2 × 10−5) and illuminated perpendicularlyto its axis by linear polarized monochromatic light of wave-length 632.8 nm. This investigated fiber is oriented perpen-dicularly to the direction of the empty interference fringes.The fringe field interference patterns of the nine-elliptical

Table 1 The values of the refractive indices and radii of core, SAPs, and cladding of PM PANDA optical fiber using the well-known conventionalinterference and the current extended interference formulas.

r f (μm) r c (μm) rSAP (μm) ns nSAP nCe nCo

Manufacturing data 62.50 3.00 17.50 — — — —

Computer-aided well-knownconventional interferenceformula

62.78 3.17 17.89 1.4601 1.4494 1.4659 1.4652

Δd ¼ �48 × 10−2 Bc ¼ 0.0007 and Lb ¼ ðλ∕BCÞ ¼ 0.904 mm

ΔnðnL; d ; δÞ ¼ �4 × 10−4

Computer-aided currentextended interference formula[Eqs. (4), (5), and (6)]

62.46 3.05 17.47 1.45987 1.44931 1.46584 1.46518

Δd ¼ �19 × 10−2 Bc ¼ 0.00066 and Lb ¼ ðλ∕BCÞ ¼ 0.958 mm

ΔnðnL; θ; δÞ ¼ �14 × 10−5

Fig. 5 (a) and (b) demonstrate two different angles, α ¼ 0 deg and90 deg, between Y -axis and the direction of incident light, respec-tively. Parts (i) display the position angle of the nine-elliptical core opti-cal fiber (MCF). Parts (ii) illustrate the fringe field interference patternsof this MCF.

0 25 50 75 100 125 150 175 200 225

Fibre radius (pixel)

0

2

4

6

Z ( x

) / h

Cladding regionNine-core region

B3

Fig. 6 Profile of the mean fringe deflection along the nine-ellipticalcore optical fiber (MCF) radius when the direction of incident lightmade an angle α ¼ 0 deg and θ ¼ 0 deg with Y -and Z -axis (fiberaxis), respectively, h ¼ 31 pixels.

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core fiber are captured by a CCD camera for further auto-matic processing system.

Figures 5(a-ii) and 5(b-ii) shows the microinterferogramsof the nine-elliptical core MCF at two different angles α ¼0.0 deg and 90 deg between Y-axis and the direction of theincident light, respectively. Intensity scanning along a linevector and perpendicular to the liquid fringe is applied tofind the minimum and maximum points ½x; ZðxÞ� of the vec-tor. This procedure is repeated for each line vector in both theliquid fringe (the absence of the fiber) and the fiber image tomeasure the liquid interfringe spacing (h) and the fringedeflections ZðxÞ, respectively. All information can beextracted by performing the intensity scanning method at dif-ferent incidence angles (θ). Figure 6 shows the averagefringe deflection profile of the interferogram Fig. 5(a-ii)along the fiber radius, as an example. The minimal and maxi-mal values are corresponding to the boundaries positionbetween each two consecutive media and the fiber center.

In the range B3 ≤ x < rf (B3 ¼ 67.73 pixels andrf ¼ 214 pixels) the liquid interfringe spacing (h) and themaximal or minimal value of the fringe deflection ZðxÞ attwo different incidence angles are measured and substitutedinto this extended equation [Eq. (4)]. By using a simple com-puter program, the refractive index ns and radius rs of theMCF cladding can be calculated as a function of theknown refractive index (nL) of the surrounding mediumand the results are tabulated in Table 2. Also the transversaldimensional dc and the refractive index nc of the fiber can becalculated using the current extended formula [Eqs. (8) and(9)] at two different incidence angles. Having determined thetransversal dimensional the radii of the elliptical core can becalculated as shown in Table 2. It is possible to determine thegeometrical and optical parameters in a selected ellipticalcore of a major axis r2 that makes an angle α ¼ 0 degwith Y-axis in the range 0.0 ≤ x < r2 fdCi

ðxÞ ¼ 2P

mi¼1½r2C − ðx − C sin αiÞ2�1∕2; i ¼ 1; 5; α1 ¼ 0.0°; α5 ¼ 160°g.

This can be done by fiber turn around its axis (Z-axis).The optical parameters of the nine-elliptical core MCF

can be determined using the well-known conventional inter-ference formula15,16 and the results are shown in Table 2. Incomparison to the well-known conventional formula, thepresent extended interference formula is more accurate.

6 SummaryIn this paper, an extended interference formula gives a pos-sibility to measure simultaneously the refractive index and

the radii of optical fibers whether the cores or SAPs are non-concentric to cladding. Using this extended interference for-mula, the measuring accuracy of optical and geometricalparameters of the PM PANDA and MCF is increased andis found to be �14 × 10−5 and �19 × 10−2, respectively.This accuracy is significantly higher compared with thewell-known conventional interferometric method of indexand the thickness measurements.

Furthermore, using the present extended interference for-mula, the optical system does not need to calibrate and trans-late the pixels to real values in microns. This means that thepresent interference formula overcomes the problem of thethickness determination uncertainty. Finally, all types ofoptical fibers can be investigated using this extended inter-ference formula.

AcknowledgmentsThe authors would like to thank Prince Sattam bin AbdulazizUniversity, KSA, and Deanship of Scientific Research fortheir support. The paper was supported by the Deanshipof Scientific Research at Prince Sattam bin AbdulazizUniversity, Saudi Arabia, under Grant No. 2254/01/2014.

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Table 2 The values of the refractive indices and radii of core and cladding the nine-elliptical core optical fibre (MCF) using the well-known conven-tional interference and the current extended interference formulas.

r f (μm) r 1 (μm) r 2 (μm) nCe nCo

Manufacturing data 190 16 8 — —

Computer-aided well-knownconventional interference formula

189.39 16.41 7.86 1.5178 1.5389

Δd ¼ �48 × 10−2 ΔnðnL; d ; δÞ ¼ �4 × 10−4

Computer-aided current extendedinterference formula [Eqs. (4), (9),and (10)]

190.21 15.96 8.03 1.51800 1.53933

Δd ¼ �19 × 10−2 ΔnðnL; θ; δÞ ¼ �14 × 10−5

Optical Engineering 066104-7 June 2016 • Vol. 55(6)

El-Morsy and Sadik: Extended interference formula for optical fiber characterization

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Biographies for the authors are not available.

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El-Morsy and Sadik: Extended interference formula for optical fiber characterization