High-resolution refractive index anisotropy measurement in optical fibers through phase retardation modulation

$Page 1: High-resolution refractive index anisotropy measurement in optical fibers through phase retardation modulation$
High-resolution refractive index anisotropymeasurement in optical fibers through phase

retardation modulation

Benoit Sévigny,1,* François Busque,1 Nicolas Godbout,1 Suzanne Lacroix,1

and Mathieu Faucher2

1Centre d’optique, photonique et laser, Fiber Optics Laboratory, Engineering Physics Department, École Polytechniquede Montréal, P.O. Box 6079, Station Centre-ville, Montréal (Québec), H3C 3A7, Canada

2ITF Laboratories, 400 Montpellier Boulevard, Montréal (Québec), H4N 2G7 Canada

*Corresponding author: [email protected]

Received 15 June 2007; revised 21 December 2007; accepted 22 January 2008;posted 1 February 2008 (Doc. ID 84122); published 20 March 2008

We present an improved, high-resolution method for the measurement of phase retardation induced bythe material birefringence of optical fibers. Such a method can be used to retrieve information about thespatial distribution of refractive index anisotropy in the fiber by comparing the accumulated phase of apolarization component oriented along the fiber transmission axis and another located in the transverseplane. Themethod is based on the nonlinear regression of a phasemodulated signal of knownmodulationamplitude altered by the sample. Experimental results obtained with our method for a standard tele-communications fiber (the Corning SMF-28) as well as photosensitive fibers (Fibercore PS1250 andPS1500) are presented with a noise-limited phase resolution below 10−4 radians and a spatial resolutionbelow 1 μm. An analysis of the limitation of such measurement methods is also presented including dif-fraction by the fibers. © 2008 Optical Society of America

OCIS codes: 060.0060, 060.2270, 060.2280, 060.2300, 060.2400.

1. Introduction

Since the late 1990s, there has been a growing inter-est in small material birefringence measurementsfor optical fiber characterization. Due to fabricationprocesses of optical fibers and all-fiber components,a refractive index anisotropy is induced through var-ious mechanisms: thermal stresses caused by thethermal expansion coefficient mismatch between dif-ferent layers of the optical fiber [1], the viscoelasticbehavior of glass [2,3], and possible flow-relatednonlinear effects [4]. Although material refractiveindex anisotropy is not directly linked to modal bire-fringence (which is not measured here but wouldconstitute interesting complementary data), it is sus-pected to be in part responsible for polarization mode

dispersion[5] that can lead to signal degradation thatis of utmost concern for high bit rate (>40Gbits=s)and long-haul communication. In this paper, wepresent an improved approach to measure the refrac-tive index anisotropy in optical fibers with a veryhigh spatial (<1 μm) and signal (12 bit) resolution.The standardmeasurementmethod to characterize afiber refractive index anisotropy consists of measur-ing the phase retardation between light polarizedalong the fiber axis direction and light polarized inthe transverse plane. This measurement can be ac-complished by sending a collimated light beam trans-versally through the optical fiber with a polarizationstate oriented at 45° from the fiber axis. The modifi-cation of the ellipticity of the light passing throughthe fiber can thus be analyzed to extract phase re-tardation information and, eventually, under spe-cific conditions, refractive index anisotropy data.Such measurements can provide useful information

0003-6935/08/091215-08$15.00/0© 2008 Optical Society of America

20 March 2008 / Vol. 47, No. 9 / APPLIED OPTICS 1215

on optical fiber fabrication processes provided thedifferent contributions to the phase retardation pro-file can be properly isolated. Up to date, publishedresults also suggest the presence of so-called “frozen-in” strains [2] that would contribute to the phaseretardation profiles measured and explain the appar-ent violation of the mechanical equilibrium of thefiber observed by many authors when assuming thatbirefringence is exclusively due to stress [6,7]. Thosefrozen-in strains are thought to take origin in viscoe-lastic effects, occurring during the fiber drawingprocess [3,8]. Significant progress has been madein phase retardation measurement in the past fewyears using various approaches ranging from thewell-known half-shade device method [9,10] to dir-ect retardation compensation techniques [5,11–13].Both methods require accurate measurement ofangles, which can be difficult to perform. In addition,retardation compensation often requires finding aminimum in transmitted intensity, therefore limit-ing the accuracy to the input power and signal reso-lution. The measurement technique presented inthis paper, however, can provide high-resolutionphase retardation profiles without the need for pre-cise angular data or limitations with respect to thesignal resolution. In the following sections, we pre-sent the retardation profile acquisition setup, a de-tailed analysis of the phase modulation techniqueused, as well as experimental results.

2. Experimental Setup

The measurement system described in Fig. 1 is basedon a crossed polarizers setup. A rotating diffuseris used in order to decrease the spatial coherenceof the Arþ laser (λ ¼ 488nm), thus reducing thecontrast of parasitic interference fringes created byvarious optical surfaces present in the system. Sucha laser has been chosen for its relatively high poweroutput (2:5W) and relatively short wavelength, en-abling high spatial resolution. The half-wave platecombined with the polarizer act as a tunable at-tenuator. The sample under test is immersed inindex-matching oil between the polarizers. A phasemodulator is also introduced in order to modulatethe overall birefringence of the setup in a knownway. This is similar to the quarter-wave plate methodin the fact that a reference is compared to the sampleunder test in order to extract birefringence data.Phase modulation has the advantage of providingmany different known references (each modulatedstate) for comparison, which allows more precisemeasurements. Figure 2 illustrates the polarizationstates after passing through the modulator and sam-ple. The nature of the phase modulator is not of muchimportance for this system, but we chose to designour own modulator consisting of a rotating arrange-ment of two almost compensating quarter-waveplates, as illustrated in Fig. 3. This arrangementis approximately equivalent to an electro-optic cell.It is reminiscent of the two-wave-plate phase

Fig. 1. Illustration of the small birefringence measurement setup. This setup is based on a crossed polarizers scheme combined witha phase modulator and an imaging system. The probe beam is sent transversally through the fiber with a polarization state oriented at45° from the fiber axis z.

1216 APPLIED OPTICS / Vol. 47, No. 9 / 20 March 2008

compensator presented by Montarou et al. [13,14]but is used here as a modulator. The modulator de-sign is described in detail in the next section. Thecondenser lens has a focal length of 15 cm and is2:54 cm in diameter, yielding a beam numerical aper-ture smaller than 0.035, considering the fact thatthe beam is smaller than the lens aperture. The con-denser is necessary to take full advantage of theCCD’s dynamic range when measuring samples withvery small birefringence since, in this situation, mostof the light is lost in the diffuser or blocked by theanalyzer. The imaging lens is a plano-convex 25× mi-croscope objective (N:A:∼ 0:4). The method for ade-quately focusing the sample image onto the cam-era CCD array is described in Section 4. The CCDis a 1024 × 1024, 12-bit linear digital camera with

adjustable exposure time (and frame rate), whichis our primary means of setting the output gain.

3. Phase Modulation and Signal Analysis

Our phase modulator was designed to mimic the be-havior of an adjustable linear retarder. Although anelectro-optic cell could be used in our setup, due tothe long camera integration times needed, it ismore convenient to use a rotating arrangement withconstant angular increments in a frame-by-framesequential acquisition scheme. An electro-optic de-vice would be better used in a continuous acquisitionscheme since it can be driven by a wave generator.The equivalent Jones matrix of the modulator struc-ture is as follows:

R�π=4RθWπ=2RαW−π=2R−1α R−1

θ Rπ=4 ¼ RθRαW2αR−1θ ;

ð1Þ

where Ri is a rotation matrix of angle i, and Wi is alinear retarder matrix of phase retardation i. Asshown in Fig. 3, θ represents the overall rotationof the modulator structure, and α is the angular off-set between the quarter-wave plates, neutral axes.Figure 4 shows the path described by the output po-larization the modulator on the Poincaré sphere fordifferent angular offsets α and modulation anglesθ ∈ ½0; π�. For small values of α, the path is quite si-milar to that of an electro-optic cell with maximumphase modulation amplitude Ω ¼ 2α. The polariza-tion rotation Rα around axis S3 (Fig. 4) induced bythe modulator can be compensated by rotating theentrance polarizer in the opposite way. This is impor-tant in order to maintain the average orientation ofthe polarization ellipse at 45° from the fiber axis.Note that the modulator shown on Fig. 3 providesa low-cost solution for phase measurement applica-tions. Of course, knowing the exact transmissionfunction of the whole system allows one to use themodulator for retardation values of almost anymagnitude. The exact intensity transmission of the

Fig. 2. Examples of the polarization state after passing through the modulator and sample. The modulator produces an ellipticallypolarized state with known ellipticity. When the light subsequently passes through the sample, its polarization state is altered. The solidcurve illustrates a modulated polarization state in the case where the phase retardation induced by the modulator and sample is sub-tractive. The dashed curve indicates a modulated state for the case where the phase retardation induced by the sample and by the mod-ulator is additive. The output elliptical state of light is measured to deduce the phase retardation induced by the sample. This operation isperformed for every coordinate y.

Fig. 3. Arrangement of two almost compensating quarter wave-plates used as a phase modulator. The modulation amplitude Ω isequal to 2α, α being the angular offset between the compensatedrefractive index axes. The angle θ is the rotation angle of the wholestructure with respect to an arbitrary reference axis.


system for a ray passing through the fiber at y isgiven by

ItðyÞ ¼ I0ðyÞ��

sin�ϕðyÞ

2

�sin

�Ω2

�cosð2θÞ

�2

þ�sin

�ϕðyÞ2

�cos

�Ω2

�

þ sin�Ω2

�cos

�ϕðyÞ2

�sinð2θÞ

�2�; ð2Þ

where Ω ¼ 2α is the modulation amplitude, ϕðyÞis the sample retardation that we are measuring,and θ is the angular position of the modulator. Anexample of the signal generated at each pixel is givenin Fig. 5. For each pixel, we fit Eq. (2) with I0 and ϕas free parameters and thus retrieve the phase retar-dation information [15]. The regression is performedby the use of a Levenberg–Marquardt least squarealgorithm [16].

4. Experimental Results

Optical fibers make excellent candidates for thevalidation of a phase retardation measurements set-up because of their low birefringence. We measuredphase retardation data for three fibers (SMF-28,PS1250 and PS1500) using our setup. For phaseretardation measurements, the exposure time ofthe camera is adjusted in order to take advantageof the whole 12-bit dynamic range. The adequateperiodicity of the signal ensures that the laser outputpower is stable throughout the duration of the dataacquisition. In order to suppress noise, a number oflines on the CCD array data (typically one hundred)

are averaged along the sample invariant axis z.The average signal is then fit to retrieve the value ofthe phase retardation parameter ϕðyÞ. Direct tomo-graphic inversion of the phase retardation profilesinto axial stress profiles though is often inaccuratebecause of an additional viscoelastic term in thephase retardation integral:

ϕðyÞ ¼ 2πλ

Z ffiffiffiffiffiffiffiffiffiffiR2

f −y2

p−

ffiffiffiffiffiffiffiffiffiffiR2

f−y2

p ½Cσzðx; yÞ þ PΔϵresðx; yÞ�dx; ð3Þ

where λ is the wavelength of the probe beam, Rf isthe fiber radius, σz is the total axial stress profile[2,9], C is the stress-optic coefficient [17] (consideredconstant over a wide dopant range), P is the strain-optic coefficient [11], andΔϵres is the residual viscoe-lastic strain anisotropy. Thus, the inversion of thephase retardation profiles must be carefully ana-lyzed for each separate case in light of the differentcontributions to both the stress and residual viscoe-lastic strain profiles (see, for example, Ref. [18]). Byintegrating the tomographically inversed data ob-tained by other authors for the SMF-28 fiber (assum-ing a stress-only case) [19], the result is the samephase retardation obtained in our study.

For cylindrically symmetric optical fibers, if thedominant source of residual stress is due to thermalexpansion coefficient (TEC) mismatch between theregions with different dopant compositions, the resi-dual axial stress can be approximated by the follow-ing relation [1,9]:

σzðrÞ ¼EΔT1 − ν

�2

R2f

ZRf

0αðrÞrdr − αðrÞ

�: ð4Þ

Here, αðrÞ is the TEC distribution over the fiber crosssection, E is the Young’s modulus of the material(considered constant over the cross section and equalto 70GPa for pure silica), ν is the Poisson ratio of the

Fig. 4. Polarization path of the modulator on the Poincaré spherefor different angular offsets α (in radians) and θ ∈ ½0; π�. The inputpolarization is S ¼ ½0; 1;0�. We notice that the modulation ampli-tude is indeed Ω ¼ 2α and that the polarization state rotates byangle α. To compensate for that rotation and to preserve the 45°polarization orientation with respect to the fiber axis, the polarizercan be rotated by an angle −α.

Fig. 5. Simulation of the signal captured for each pixel. Here,fiber phase retardation of 0 and 5 degrees is illustrated. The phasemodulation amplitude Ω ¼ 2α is set to 10 degrees.


material (about 0.2 for pure silica, also consideredconstant over the fiber cross section) and ΔT ¼Tamb − Tg is the difference between the room tem-perature and the glass transition temperature. Ne-glecting other contributions to refractive indexanisotropy, we can integrate the axial stress profiledirectly through the use of photoelasticity theory:

ϕðyÞ ≅ 2πλ

Z ffiffiffiffiffiffiffiffiffiffiR2

f −y2

p−

ffiffiffiffiffiffiffiffiffiffiR2

f −y2

p Cσzðx; yÞdx: ð5Þ

For a step index profile fiber, which is the case of allfibers under study in this paper, the dopant distribu-tion as well as the TEC distribution are also step dis-tributions. In this case, the axial stress should beconstant in each layer because of the uniform dopantcomposition and the phase retardation in the clad-ding should follow an ellipse given by

ϕðyÞ ¼ 4πCσz;cladλ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

f − y2q

; ð6Þ

so the phase retardation profile should vary mono-tonically from the fiber edge to the next dopantregion. Such a behavior is visible on the PS1250measurement (Fig. 7) and somewhat apparent onthe PS1500 measurement (Fig. 8), whereas it can-not properly describe the phase retardation of theSMF-28 fiber (Fig. 6). Other contributions to refrac-tive index anisotropy, whether they are elastic or vis-coelastic in origin, are mostly side effects of the fiberdrawing process [2,9]. Thus, considering that the do-pant concentration in the core of the photosensitivefibers is significantly more important than that of theSMF-28 fiber, it can be expected that the thermalstress signature be relatively more present in thephase retardation profile of both the PS1250 andthe PS1500, whereas the phase retardation profileof the SMF-28 fiber displays features that can pre-sumably be dominated by draw-induced effects. Inaddition to the fact that the SMF-28 phase retarda-tion profile does not clearly exhibit the behaviordescribed by Eq. (6), its phase retardation profileintegrated over y is clearly positive and can beexpressed by the following equation [18]:ZZ

A⊥

σzdA ¼ 0 ⇒Z

Rf

−Rf

λ2π ϕðyÞdy ¼

ZZA⊥

PΔϵresdA;

ð7Þwhere A⊥ denotes the fiber cross section area. A non-zero phase retardation profile integral is thus theindication of the presence of residual viscoelas-tic strain. Apart from the SMF-28, the PS1250 andPS1500 also exhibit this feature that presumably re-sults from the drawing process. Conversely, retriev-ing σz from ϕðyÞ while assuming Δϵres ¼ 0 yields anonphysical result since the mechanical equilibriumcondition is then not respected, as observed by sev-eral authors [2,9,20].From the measurements in Figs. 6–8, we note that

the system is sensitive and provides high spatial

resolution and that the noise is very low. It is esti-mated to be smaller than a few 10−4 rad. We also no-tice fringes near the oil-cladding interface of the fiberand the core–cladding interface in Fig. 6(a), whichseem to indicate the presence of diffraction. Althoughfluctuations in the dopant profile could, in part, ac-count for the fringes near the core–cladding inter-face, diffraction fringes caused by the index stepbetween the oil and cladding can also be seen butwith less resolution in some other published mea-surements [8,10]. High resolution is probably theroot cause as to why diffraction is so clearly visiblein our measurements. The system high magnifica-tion gives a spatial resolution of approximately0:15 μm for each pixel. This value is smaller thanthe physical resolution of the system which is close

Fig. 6. Phase retardation of the SMF-28 telecommunicationsfiber: (a) the oil refractive index is not matched with the indexof the cladding, (b) the oil refractive index is matched with theindex of the cladding. We see that fringes are still visible in thefiber core region. Note that this fiber has a 125 μm diameter clad-ding and an approximately 9 μm diameter, Ge-doped core (N:A: ¼0:12). Regions with different dopant compositions are separated byvertical dashed lines.


to 0:8 μm considering the objective N.A. We can no-tice in Fig. 6(a) that the fringes have a spacing ofa few microns, which indicates that they have beenresolved by the system, and that rules out the possi-bility that they are an imaging artifact. The signalresolution is very high compared to 8-bit analog-to-digital conversion. This, combined to the fact that weuse the camera full dynamic range for the measure-ments (instead of resolving intensity minima) provid-

ing our low noise level, could explain why we canobserve the fringes where other systems could not.

To complement the study of diffraction by the fiber,this phenomenon has been simulated with a scalarbeam propagation method (BPM) [21] as shown inFigs. 9(a) and 9(b). Diffraction by the cladding–oil

Fig. 7. Phase retardation of the photosensitive fiber FibercorePS1250. This fiber has an intermediate buried cladding that isclearly visible on the retardation profile. Notches in the cladding,visible around 45 μm, are presumably due to the interface of thehollow silica tube used to manufacture the preform. Note that thisfiber has a 125 μm diameter cladding and an approximately 9 μmdiameter core (N:A: ¼ 0:14) surrounded by the low-index inter-mediate cladding about 32 μm in diameter. Regions with differentdopant compositions are separated by vertical dashed lines.

Fig. 8. Phase retardation of the photosensitive fiber FibercorePS1500. Notches in the cladding, visible around 35 μm, are pre-sumably due to the interface of the hollow silica tube used to man-ufacture the preform. One can also notice diffraction fringes dueto the refractive index difference between the fiber core and clad-ding. Note that this fiber has a 125 μm diameter cladding and anapproximately 9 μm diameter core (N:A: ¼ 0:11). Regions with dif-ferent dopant compositions are separated by vertical dashed lines.

Fig. 9. (Color online) Plane-wave BPM simulation of diffractionby a standard telecommunications fiber (Corning SMF-28): (a) dif-ference between the oil refractive index and the fiber claddingrefractive index is Δn ¼ nclad − noil ¼ −3 × 10−3, (b) the refractiveindex difference is one order of magnitude lower, Δn ¼ −3×10−4. The fringe contrast is diminished at the cladding–oil inter-face when the refractive index difference is smaller. Diffractionby the fiber core–cladding interface is not significantly alteredwhether the oil–cladding index difference is small or not.


interface can be avoided, as seen in Figs. 6 and 9, byusing a temperature control that allows us to pre-cisely tune the oil index to match the fiber cladding.This index matching technique has been used for themeasurements featured in Figs. 6(b), 7, and 8. This isan additional proof that the fringes are mostlycaused by diffraction due to the refractive index stepbetween the oil and the cladding since fringe contrastshould theoretically disappear for perfectly matchedindices [22]. Diffraction by the fiber core, however,should exist independently of possible dopant fluc-tuations in the core vicinity but cannot be eliminatedand certainly has an effect on retardation measure-ments since the diffraction pattern depends on polar-ization. This dependence is related to the boundaryconditions for the fields at the core–cladding inter-face being different if the electric field is polarizedalong or perpendicular to the fiber axis. The use ofspatial filtering to eliminate the fringes would be ir-relevant here since it would only artificially impairspatial resolution. These diffraction fringes cannotbe averaged by use of a rotating diffuser. As a proof,we conducted multiple BPM simulations withsmooth random phase fronts (Fig. 10). We notice fromthe result obtained for the average (in intensity) of60 BPM runs with different random phase profiles(Fig. 11) that the diffuser has no effect on the diffrac-tion from the fiber. Note, however, that the diffuser isuseful to attenuate parasitic interference patternspresent in the system, an effect that has proven to beof utmost importance for the quality of our measure-ments. This is in agreement with experiments sincediffraction by the fiber is present in our setup eventhough we are using a rotating diffuser. Thus, theonly way to efficiently eliminate diffraction from themeasurements would be to elaborate a deconvolution

procedure from analytical solutions of the near-fielddiffraction considering the true refractive index pro-file measured by other means. Diffraction is thusdetrimental to phase retardation measurements.However, it proves to be a useful tool for adequatelyfocusing the image of the sample on the camera withthe object plane properly located since diffractionfringes by the fiber cladding converge toward thecentral plane of the fiber parallel to the y direction.This technique is analogous to the one presentedby Park et al. [5] but considering the sharpness ofthe cladding borders as a focusing criteria insteadof using the core. The core diffraction pattern ismore complex and could possibly lead to errors. Themethod proposed by Park, based on ray tracing, doesnot take diffraction into account.

An important issue is the determination of the signof the phase retardation ϕðyÞ. This can be easily doneby measuring the same sample a second time whileadding a compressed glass plate right after the mod-ulator, which introduces a retardation of known sign,or by remeasuring the sample under tensile stress.

5. Conclusion

We have reported an improved, high-resolutionmethod for measuring small birefringence in opti-cal fibers. Our method is based on the analysis of aphase modulated signal of known modulation ampli-tude. Retardation is modulated by an innovativephase modulator of very simple design. Phase infor-mation is extracted by fitting the transmission signalwith the theoretical transmission function. We also

Fig. 10. Example of phase noise added to the wavefront in theBPM analysis in an attempt to model the contribution of the dif-fuser to the lowering of the visibility of the diffraction fringes fromthe fiber. This profile was generated from 10 uniformly distributedrandom phase points in the range ½0; 2π�. Those phase points areuniformly spaced along the wavefront and are interconnected witha cubic spline for smooth variation.

Fig. 11. Illustration of the average (in intensity) of multiple BPMruns, each affected with a random phase pattern as shown inFig. 10. This result represents the contribution of the rotatingdiffuser in the lowering of the visibility of the diffraction fringesfrom the fiber. As we can see, it has little or no effect, which isin accordance with our experimental data.


showed that, whereas diffraction by the fiber clad-ding can be eliminated by carefully tuning the oilrefractive index to match that of the fiber cladding,diffraction by the core is always present and can leadto errors in measurement of the phase retardationprofile. The resolution of the measured phase retar-dation ϕðyÞ is limited in our system to an ∼10−4 radnoise. Our system has sufficient spatial resolutionto resolve diffraction fringes caused by the fiber.Those diffraction patterns interfere with the phaseretardation measurement because of their depen-dence on polarization. In order to isolate diffraction,a deconvolution method considering the refractiveindex profile would have to be elaborated since itcannot be eliminated by currently used devicessuch as the rotating diffuser. High-resolution phaseretardation measurement methods, such as ours,constitute a premium tool to characterize process-in-duced stresses, strains and structural modificationsin optical fibers, all-fiber components (tapers, Bragggratings) as well as integrated optics. Refractiveindex anisotropy information can provide a usefulinsight on the effects of fabrication processes thatcan be of critical importance for the manufacturingof high-quality optical devices.

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High-resolution refractive index anisotropy measurement in optical fibers through phase retardation modulation