Characterization of strong fiber Bragg gratings using an applied thermal chirp and iterative algorithm

Characterization of strong fiber Bragg gratings usingan applied thermal chirp and iterative algorithm

Gary A. Miller,* John R. Peele, Charles G. Askins, and Geoffrey A. CranchOptical Sciences Division, Naval Research Laboratory, 4555 Overlook Avenue,

SW, Washington, D.C. 20375, USA

*Corresponding author: [email protected]

Received 27 June 2011; revised 2 September 2011; accepted 2 September 2011;posted 7 September 2011 (Doc. ID 149590); published 15 December 2011

Coupling coefficients of various grating types and strengths are calculated from measurements of thecomplex reflectivity using an applied thermal chirp and optical frequency domain reflectometry (OFDR).The complex reflectivity is then utilized by a layer peeling algorithm to determine the coupling coefficientof the thermally chirped grating. A guess of the temperature profile enables the coupling coefficient of theunchirped grating to be estimated. An iterative algorithm is then used to converge on the exact couplingcoefficient, employing an error minimization method applied to the reflectivity spectra. This techniqueremoves the need for a reference grating while preserving the spatial resolution obtained with the initialOFDR measurement. Successful reconstruction of gratings with integrated jκjL ∼ 9:0 are demonstratedwith a spatial resolution of less than 100 μm. © 2011 Optical Society of AmericaOCIS codes: 060.2270, 060.3735, 120.4290, 120.5050, 120.6810.

1. Introduction

Fiber Bragg gratings (FBGs) have become an inte-gral part of many telecommunication systems andfiber optic sensors. Central to their performance inthese applications is the quality of the gratingformed during the inscription process. The presenceof errors in the amplitude and phase of the refractiveindex modulation can produce considerable depar-tures from idealized spectral properties [1]. Thus,it is often desirable to characterize a grating to un-derstand the origin of these errors and how the de-sign affects system performance. Characterizationof FBGs can also provide useful information when de-veloping advanced grating writing systems capableof compensating systematic phase errors [2].

In a previous publication, a technique was demon-strated to characterize strong gratings using a heatperturbation method [3]. One disadvantage of thisapproach is its limited spatial resolution (∼1mm).Optical frequency domain reflectometry (OFDR)and layer peeling can achieve a very high spatial

resolution (<100 μm) [4]; however, it is well knownthat layer peeling fails with strong gratings due tonoise accumulation [5]. Although advanced techni-ques, such as integral layer peeling [6] and regular-ization [7], achieve better performance than directapplication of the discrete layer peeling algorithm(DLP) [8], they cannot overcome this fundamentallimitation.

One method that can improve the performanceof OFDR and DLP for measuring strong FBGs uti-lizes a thermal gradient to increase the grating’stransmission to a point where it can be successfullyreconstructed [9]. Here, a weak grating (called thereference grating) is either superimposed onto thetest grating (a destructive process) or colocated with-in the experimental apparatus. Using OFDR, thecomplex reflection spectra of the gratings are ob-tained before and after the thermal gradient isapplied. Whereas the test grating can be recon-structed only in the presence of the temperaturegradient, the weak reference grating’s couplingcoefficient can be accurately reconstructed at anythermal distribution using an inverse scatteringtechnique, such as the DLP [8]. The phase related

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to the thermal chirp is then subtracted from the testgrating’s phase, yielding the unchirped phase of thetest grating. Note that the amplitude of the couplingcoefficient is largely unaffected by the thermal gra-dient since it is related to the index modulation am-plitude rather than the average index.

While this method is capable of measuring awide variety of grating types, including distributedfeedback (DFB) fiber lasers [10], there are somelimitations to the technique. First, the method is re-stricted by the amount of thermal chirp that can beinduced in the grating. Since many gratings are TypeI and, therefore, begin to anneal around 150 °C, thisplaces an upper limit on the thermal gradient. In thecurrent work, we demonstrate a temperature differ-ence, ΔT, measured between each end of a stainlesssteel beam, of 190 °C. This is accomplished by coolingone end to −30 °C with a thermoelectric cooler (TEC)and heating the other end to 160 °C with a high im-pedance wire. The maximum strength of the testgrating remains limited by the ability to perform avalid reconstruction from the chirped spectrum.

Another limitation of the method is the necessityto quantify the thermal chirp. An ideal characteriza-tion method would allow the user to place an un-known, coated fiber into the system and measureits grating properties without the use of a referencegrating. As noted in [10], there are two ways a refer-ence grating can be employed to find the thermallyinduced phase response: (1) superimpose a weakFBG onto the test grating and (2) colocate a weakFBG with the test grating. The former technique willperturb the test grating as it must be re-exposed toultraviolet (UV) light to imprint the weak grating.Furthermore, the extra handling of the fiber placesthe grating at a greater risk for breakage. In thelatter case, the colocated FBG is assumed to experi-ence the same temperature profile and exhibit thesame temperature-induced phase response as thetest grating. Even for gratings written in the sametype of fiber, it is unlikely that both gratings will ex-perience identical thermal chirps. Because of thecomplex heat transfer processes involved in the tech-nique, the cross-sectional profile of temperature willgenerally not be uniform. Thus, two spatially sepa-rated FBGs may experience a different temperaturedistribution. There is also the additional problem ofregistering the gratings so that the spatial variationof the thermal chirp can be appropriately subtractedfrom the test grating’s phase. Simple methods likeapplying local pressure or situating a hot filamentcan damage the fiber or permanently affect the re-fractive index profile.

In this paper, we present a method to characterizethe spatial profile of strong FBGs that addressesthese limitations. As with [10], the experimental set-up incorporates an applied thermal chirp and OFDRto obtain the coupling coefficient of a FBG. Howeverfor this experiment, the thermal gradient is drivenby a thermoelectric cooler and heating wire, extend-ing the amount of chirp that can be applied. Heat

transfer from the beam to the test fiber is controlledusing a stainless steel beam containing a channelfilled with a thermal silicone fluid. We also show howan iterative algorithm (IA) can be used to reconstructthe coupling coefficient of the test grating from itsthermally chirped spectrum. This removes the needfor a separate reference FBG to characterize thethermal gradient. We demonstrate the effectivenessof the technique by reconstructing four FBGs: oneweak grating that can be reconstructed using layerpeeling alone, and three strong FBGs with inte-grated coupling coefficients between 6 and 9.

2. Experiment

The experimental setup is depicted in Fig. 1 andbased largely on the design suggested in [10]. Suc-cessful reconstructions of the gratings are verifiedusing a reference grating to quantify the thermalgradient in situ. Thus, the experimental system is de-signed so that it can accommodate two operationalconfigurations, with (configuration A) or without(configuration B) a reference grating. In configura-tion A, a weak, uncoated reference grating is con-nected in series to an uncoated test grating andinterrogated with a polarization-resolved OFDR(Luna Technologies, Optical Vector Analyzer, orOVA). The test grating is placed alongside the refer-ence grating and centered. Choosing a long referenceFBG provides added flexibility in the placement ofthe test grating. To spatially register the gratings,a hot filament is placed in close proximity to thegratings, inducing a distinct perturbation in the per-iod of the FBG. In some instances, this local heatingpermanently alters the coupling coefficient. Thefibers are held under constant tension (∼15 g) andclamped to a support structure (shown in Fig. 2).Pulleys ensure the fibers are not subjected to anyadditional strain.

Once secured, the fibers are lowered to rest withina small channel of a stainless steel beam (type 316).The beam is 160mm long and 5mm by 5mm in cross

Fig. 1. Top: operating configurations for the thermal chirp setup.Bottom: schematic of the experimental apparatus.

6618 APPLIED OPTICS / Vol. 50, No. 36 / 20 December 2011

section. A 1mm wide and 0:5mm deep channel ex-tends the length of the beam, into which the fibersare placed. The channel is filled with a thermal sili-cone fluid (Gelest, Inc., PDM-1992). This promotesefficient thermal conduction between the beam andfibers, reduces convective losses in the vicinity ofthe fibers, and minimizes cross-sectional variationsin temperature.

The thermal gradient is supplied by a heatingcoil and TEC. The heating coil is comprised of insu-lated thermocouple wire (Omega, Chromega Nextelcoated wire) wrapped around an aluminum spool andpotted with thermal cement (Omega, Omegabond400). The spool is suspended from the distal end ofthe beam, and vibrations are minimized using adashpot. The proximal end of the beam is housed inan aluminum block that rests on a potted TEC [TETechnologies, TE-2-(127-127)-1.15]. The heat fromthe TEC is dissipated with a water-cooled heat sink(Koolance, GPU-180-L06) connected to a Neslab, En-docal refrigerated circulating bath. The heat sinkdirectly contacts the TEC via a feed-through holemilled into the base plate of the system. Temperaturereadings at each end of the beam are provided byUSB-powered thermocouple acquisition devices(National Instruments, USB-TC01, and Omega, typeK thermocouple). Both the TEC and the heating coilare powered by separate constant current powersupplies. Finally, the entire system is housed in anitrogen-purged enclosure to limit external air cur-rents and reduce humidity in the chamber.

The experiment begins by registering the gratinglocations with the hot filament. While the fibers aresituated above the beam, a Nichrome wire is broughtinto close proximity. When current is applied to thewire (∼0:3A), enough heat is generated to create aphase perturbation in the gratings. The complexreflection spectrum of each FBG is then obtainedin serial by the OFDR, scanning each grating overa wavelength span of 10nm, centered about the un-chirped peak wavelength. After registration is com-plete, the fibers are lowered into the channel and

immersed in silicone fluid. The enclosure is sealedand purged with nitrogen for ∼5 min. Next, a base-line OFDR measurement of each FBG in reflectionand transmission is taken. Then, depending on thestrength of the test grating, the TEC and heating coilcurrents are adjusted to the appropriate level neces-sary to produce the desired temperature gradient.In general, the heating coil needs between 1.1 and1:6A to yield a temperature range of 100 to 160 °C,respectively. The TEC requires 0.85 to 5A to achieveoperation at 0 to −30 °C, respectively. The recircula-tor is set to a constant temperature of 10 °C. Once thetemperature gradient has reached an equilibrium,the grating spectra are once again obtained by theOFDR. While a feedback loop was not used to controlthe temperature operation of the experiment, activecontrol, such as PID, should improve the tempera-ture stability. Data is acquired using a customizedMATLAB interface, and both the thermocouplesand the OFDR are interrogated via MATLAB. Afterthe spectra are recorded, the TEC and heating coilare turned off, allowing the system to return to roomtemperature. Then, the data is analyzed, and thecoupling coefficient is determined using the DLPand IAs.

3. Grating Reconstruction and Iterative Algorithm

A grating is typically formed by exposing opticalfiber to a UV interference pattern. The UV lightchanges the chemistry of the glass and imprints aperiodic perturbation in the refractive index of thecore of an optical fiber. FBGs can be characterizedby examining this perturbation along the axis of thefiber. For a sinusoidal index change, the refractiveindex profile is given by

δnðzÞ ¼ δndcðzÞ þ δnacðzÞ cos�2πΛ zþ θðzÞ

�; ð1Þ

where δnðzÞ is the total refractive index change,δndcðzÞ is the slowly varying index change, δnacðzÞis the index modulation amplitude, Λ is the periodof the grating, z is the spatial coordinate along thefiber axis, and θðzÞ is the excess phase due to gratingperiod chirp and phase perturbations. Within thecontext of coupled-mode theory, the spatial indexprofile is often described by its complex coupling coef-ficient [11], κ, where

jκðzÞj ¼ πηδnacðzÞλB

;

argðκðzÞÞ ¼ θðzÞ − 4πηλB

Zz

0δndcðz0Þdz0 þ

π2: ð2Þ

Here, λB is the nominal Bragg wavelength, η is themode overlap factor, and L is the length of the grat-ing. The integrated coupling coefficient is defined asjκjL ¼ R

L0 jκðzÞjdz. If the grating experiences pure

chirp, θðzÞ can be described by

Fig. 2. Annotated photograph of the experimental setup.


θðzÞ ¼ 4πηneff

λB

Zz

0

� λBλDðz0Þ

− 1

�dz0; ð3Þ

where neff is the effective index of the waveguide, andλDðzÞ is the local Bragg wavelength.

To determine how the temperature gradient af-fects the phase of the grating, first consider thatthe Bragg wavelength of an infinitesimally weakgrating is given by λB ¼ 2neffΛ, where Λ representsthe grating period. Assuming no external strain ispresent, a temperature distribution, ΔT, will affectthe grating both through a change in the average in-dex as given by the thermo-optic coefficient, dn=dT(∼8:5 × 10−6 °C−1), and a change in grating pitchas related by the coefficient of thermal expansion,α (∼5:5 × 10−7 °C−1). For a grating subjected to auniform temperature profile, the resultant Braggwavelength will be given by λ0B ¼ 2n0

effΛ0, wheren0eff ¼ neff þ dn=dT ·ΔT, and Λ0 ¼ ΛþΛ · αΔT. If

the temperature distribution is spatially varying,the average index can then be expressed as δndcðzÞ ¼dn=dT ·ΔTðzÞ, and the local Bragg wavelengthwill be λDðzÞ ¼ λB þ λB · αΔTðzÞ. Combining Eqs. (2)and (3) and rewriting them in terms of the tempera-ture gradient reveals the excess phase due totemperature:

argðκðzÞÞΔT ¼ 4πηneff

λB

Zz

0

�1

1þ αΔTðz0Þ − 1

−

dn=dTneff

ΔTðz0Þ�dz0 þ π

2: ð4Þ

This expression relates the temperature gradientgenerated in the experiment to the excess phaseincluded in the reconstruction of the grating by theDLP algorithm. In other words, the phase as calcu-lated by the DLP from the experimental data is acombination of Eqs. (2) and (4). If the thermal gradi-ent is known, or can be estimated, the phase due tothe temperature distribution can be calculated andsubtracted from the measured phase. The resultthen represents the original, unchirped phase of thegrating.

A key component of the IA is the estimation ofthe thermal gradient experienced by the test grating.An initial guess for the temperature gradient can befound by analyzing the coupling coefficient of thechirped grating. Taking the derivative of both sidesof Eq. (4) and using a small number approximationleads to a simple expression relating the tempera-ture distribution to the excess phase in the couplingcoefficient:

ΔTðzÞ ¼ −

d argðκÞdz

λB4πηneff

�αþ dn=dT

neff

�−1: ð5Þ

Assuming the total grating phase is dominatedby the thermal chirp, Eq. (5) can be used to providean initial guess of the temperature distribution.Gratings that contain discrete phase shifts can be

accommodated by performing a polynomial or splinefit through the slowly varying component of thephase, effectively ignoring the discontinuities fromthe derivative. In general, a third-order polynomialfit to the temperature distribution is sufficient. Thismeans that the excess phase due to temperature isrepresented by a fourth-order polynomial becauseof the integral relationship in Eq. (4). For gratingsthat contain intrinsic chirp with a magnitude similarto the thermal chirp, this regression may fail. Inthese instances, the functional form of the tempera-ture gradient will need to be determined prior to test-ing strong gratings (with a weak, unchirped uniformgrating). It is also possible that apodized gratingswill experience some additional phase errors at theedges of the grating, owing to the diminished influ-ence of the phase on the reflectivity as the indexamplitude decreases. This may cause the algorithmto incorrectly estimate the thermally induced phasecontributions. While weighting the phase in thewings of the gratings could help converge on a moreappropriate solution, for highly photosensitive fiberswith strong gratings, the effect is generally not asapparent because the index change in the tapered re-gions is still relatively large.

The IA used to estimate the thermal gradientand reconstruct the test FBG is based on minimizingthe difference between a target spectrum and a cal-culated spectrum. Iterative routines, such as thegenetic algorithm [12,13] and the Nelder–Mead sim-plex algorithm [14], have been utilized in the past tosynthesize a coupling coefficient directly from spec-tral data. These routines have also been used to ex-tract external strain fields from experimental data[13]. Unfortunately, these methods often compromisetheir spatial resolution for computational speed.This is accomplished by either subsampling the cou-pling coefficient [12,14] or by reducing the complexityof the grating profile to simple functions with fewparameters [13]. The method presented in this paperuses a combination of layer peeling and optimizationto arrive at the coupling coefficient of the test grating[13,14], without sacrificing the spatial resolution ofthe DLP algorithm and without the use of a referencegrating to quantify the thermal gradient.

The proposed method is illustrated in Fig. 3. Thetarget reflectivity, R1ðλÞ, represents the unchirped,test grating power spectrum scaled by its transmis-sion. For each measurement, the complex spectrumis obtained via polarization-resolved OFDR. Withthe OVA, orthogonal polarizations of light launchedinto the fiber are used to generate the four impulseresponses (two for each polarization axis) needed todescribe the complete Jones matrix. The complexspectrum is then found by averaging the polarizationdata in the time or frequency domain. After the base-line measurement is taken, a temperature gradientis applied to the test grating using the experimentalsetup described earlier. The chirped spectrum is thenobtained via OFDR and scaled by another transmis-sion measurement. Next, the DLP algorithm is used


to derive the coupling coefficient of the thermallychirped spectrum. Assuming that the thermallyinduced phase dominates the total phase, no priorknowledge of the temperature profile is needed tomake an estimation of the thermal gradient and itscorresponding optical phase. The slowly varyingcomponent of the derivative of phase, a low-orderpolynomial, provides an adequate starting point forthe algorithm. This temperature distribution is usedin Eq. (4) to calculate the thermally induced phase,which is then subtracted from Eq. (2), the total phaseas measured by the OFDR and reconstructed usingDLP. The grating phase is further modified by addinga linear slope to adjust the calculated spectrum’s de-tuning. Finally, the grating spectrum is computedusing the inverse DLP (i-DLP) [11] and comparedto the original, unchirped spectrum. Note that thei-DLP is utilized here due to its computational effi-ciency (3–4 times faster than the traditional transfermatrix approach [15]). In subsequent iterations ofthe algorithm, the polynomial coefficients are opti-mized by minimizing the difference between theunchirped grating spectrum and the reconstructedspectrum.

The estimation of the temperature profile is opti-mized using the Nelder–Mead simplex search algo-rithm found in MATLAB [16]. The error functionused in the optimization routine is given by

error ¼Xall λ

ðR1ðλÞ −R2ðλÞÞ2; ð6Þ

where R2ðλÞ is the power spectrum derived from thealgorithm. The parameters of the fit are updated ateach iteration based on the result of this error func-tion, iterating until the two spectra converge to with-in some tolerance or until an exit criteria is met.While optimizing using the complex spectrum mayseem a more appropriate choice, we have found thatthe algorithm converges to the global minimum bestif the power spectrum is used. Other refinements,such as incorporating the derivative of phase (thegrating’s time delay) or implementing a simpleweighting function (giving low weight within thepassband), also do not appear to improve the algo-rithm’s convergence. Our choice of convergencealgorithm may be related to the performance of thedifferent merit functions, which, in turn, may makeit easier for the optimization routine to get stuckin a local minimum. Thus, it is possible that more so-phisticated combinations of algorithms and meritfunctions will improve convergence. The IA can besummarized as follows.

1. Measure the complex reflectivity of the testgrating using OFDR and calculate power reflectivity,R1ðλÞ. Scale to the transmission measurement.

2. Apply the thermal gradient to the test gratingand measure complex reflectivity using OFDR. Scaleaccording to the transmission measurement.

3. Calculate Eq. (2) using the DLP algorithm.4. Fit Eq. (5) to a third-order polynomial.5. Calculate the thermally induced optical phase

using Eq. (4).6. Subtract Eq. (4) from Eq. (2) and add a linear

slope to account for grating detuning.7. Calculate the grating spectrum using i-DLP

with the coupling coefficient in step 6.8. Compare the target and calculated power spec-

tra using Eq. (6).9. Iterate steps 5–8 until convergence, using

the Nelder–Mead simplex algorithm provided inMATLAB to optimize the thermal gradient fitparameters.

As a final remark, we note that the spatial re-solution of the reconstruction is determined by thewavelength span of the OFDRmeasurement. Assum-ing that sideband information in the FBG spectraexceeds the OFDR measurement noise over thefull wavelength span, the spatial resolution can bedefined as

Δz ¼ π2πneff

Δλλ2; ð7Þ

where λ is the center wavelength of the sweep,and Δλ is wavelength span. For the following mea-surements, Δλ ¼ 10nm at ∼1550nm, resulting in aspatial resolution of Δz ∼ 80 μm.

Fig. 3. Diagram of the IA. The process flows clockwise from thetop-left corner.


4. Results

To test the validity of using the IA to reconstruct thegrating profile, we first apply the measurement tech-nique to a weak FBG. Because the grating is weak,the FBG can be reconstructed at all temperature dis-tributions. Thus, a direct comparison of the result ob-tained using the IA with one obtained from the DLPand without a thermal chirp (i.e., direct DLP) ispossible. The first grating tested is a 90mm long uni-form grating written in Corning SMF28, with a peakreflectivity of 13%. For this experiment, the tempera-ture gradient spans 100 °C, and configuration A inFig. 1 is utilized. The temperature profile along thestainless steel beam, as derived from the phase of theweak grating and Eq. (5), is illustrated at the top ofFig. 4. The high-frequency component is due to mea-surement noise and error brought about by takingthe derivative of the grating phase. The temperatureprofile is nonlinear, and considering that the experi-ment is operated in air, it is likely that convectivelosses drive the profile’s shape.

Figure 5 shows the grating reconstructed usingthe DLP algorithm and the IA. The gray curves inthe top and middle graphs represent the amplitudeand phase of the coupling coefficient without the

temperature gradient applied. The black curves de-note the reconstructed grating profile using the IA.Noise has been reduced in the IA trace by averagingthe coupling coefficient over several measurements.Averaging is implemented by taking multiple mea-surements from the OFDR (5 to 10), reconstructingthem individually, then averaging the results. In thisexample, the direct DLP result is not averaged. Asshown at the top of the figure, the amplitude is un-affected by the thermal gradient. The notch in theamplitude at ∼58mm is a result of the registrationwire coming too close to the fiber. The middle plotshows the phase of the grating before and aftersubtraction of the temperature-induced phase. Thethermally chirped phase (right axis) exhibits theexpected quartic behavior resulting from the cubictemperature distribution. After subtraction, the re-constructed phase demonstrates a slowly varyingcomponent, which is much smaller than the ther-mally induced phase variation and is clearly resolvedwith the IA. More rapid phase fluctuations can alsobe identified, illustrating that the high spatial reso-lution achieved by OFDR is not compromised withthe IA. The bottom graph in Fig. 5 shows the residualphase error generated by the algorithm (differencebetween the black and gray curves). The total phaseerror introduced by the algorithm is roughly�0:25 rad, constituting 12.5% of the total phase.

Fig. 4. Top: temperature profile along the stainless steel beam asmeasured by reference grating (gray) and third-order polynomialfit (black). Bottom: transmission spectrum of triangular apodizedgrating before (black) and after (gray) thermal chirp.

Fig. 5. Top and middle: amplitude and phase of coupling coeffi-cient, before (right axis) and after (left axis) subtraction of the ex-cess phase due to temperature, for the weak grating using directDLP (gray) and using the IA (black). Bottom: phase error attrib-uted to the IA.


We now examine three stronger gratings: oneapodized and two apodized with a phase shift. Thestrong FBGs are written in Nufern PS-GSF-3/125,a highly doped germanosilicate fiber. The referencegrating used in the experiments is the weak FBGpreviously described. As with the previous example,the coupling coefficients are averaged over severalmeasurements. The IA is first applied to the recon-struction of a strong grating with triangular apodiza-tion, using aΔT ∼ 100 °C. The bottom of Fig. 4 showsthe transmission spectrum of the test grating beforeand after the application of the thermally inducedchirp. The reflectivity of the FBG has reduced from99.95% to 86%, and the bandwidth significantlyincreased.

The top graph in Fig. 6 depicts the amplitude of thecoupling coefficient of the test grating. The graycurve represents a reconstruction attempt withoutthe use of the thermal gradient. It is immediatelyapparent that the standard DLP algorithm (withoutregularization) is able to reconstruct only a smallfraction of the grating length. The black curve de-notes the amplitude as derived by the IA. In exam-ining the coupling coefficient, the triangular shapeof the amplitude profile is readily apparent. Further-more, the notch previously seen in the reference

grating is clearly visible at 36:5mm. Also clearly re-solved are high-frequency oscillations in the ampli-tude of the coupling coefficient. These are possiblydue to additional interference patterns present dur-ing the inscription process. For gratings produced viaa direct write system [2], higher diffraction ordersfrom the phase mask are not completely suppressed,resulting in a complex UV fringe pattern incident onthe fiber. The gradual chirping of the ripple periodalso supports this conjecture, perhaps deriving froman angular misalignment of the fiber with the writ-ing beam. For weak gratings, the index modulationproduced by these auxiliary patterns is small com-pared to the index change produced by the primaryinterference pattern used to write the FBG. However,as the grating is made stronger, the auxiliary indexmodulation becomes appreciable and affects the FBGspectrum. These oscillations in the coupling coeffi-cient result in slightly elevated sidelobe levels atwavelengths around λB�0:8nm.

The middle graph in Fig. 6 shows the phasereconstruction using the temperature profile ob-tained from the reference grating (black) and byusing the IA (dotted). The two phases match closely(�0:25 rad), and since the grating cannot be recon-structed without the thermal chirp, it is difficult tosay which is more representative of the actual phase.However, an alternative means of assessment can bemade by comparing the power reflectivity calculatedfrom the coupling coefficient and the one measureddirectly by the OFDR. The bottom graph in Fig. 6shows the unchirped (gray) reflection spectrum forthe triangular apodized grating and the spectra cal-culated using the temperature gradient as definedby the reference grating (black) and as derived bythe IA (dotted). One metric for determining thecloseness of fit is the residual spectral error as ex-pressed in Eq. (6). While both methods demonstratean adequate fit, the IA-derived spectrum has a factorof 2 lower residual error. This can be seen by exam-ining the sidebands in Fig. 6, where the dotted curveappears to more closely match the original spectrum,especially on the short wavelength side.

To check the repeatability of the IA, a phase-shifted apodized grating is subjected to three tem-perature profiles, ΔT ∼ 100 °C, 130 °C, and 160 °C.The thermal gradients decrease the peak reflectiv-ities from 99.97% to 94.9%, 90.6%, and 82.1% for100 °C, 130 °C, and 160 °C, respectively. Additionally,for the measurement at ΔT ∼ 130 °C, the grating isrecoated with a UV-curable acrylate from DSM Des-otech and placed in the system using configuration Bof Fig. 1. The measurements are illustrated in Fig. 7.The top and middle plot show the amplitude andphase of the coupling coefficient obtained from theIA. The apodization profile is consistent with thegrating design applied during inscription, exhibitingan integrated jκjL ¼ 6:2. The ends of the gratings aretapered, and a smooth notch is located at the centerof the profile. Because the fiber is highly photosensi-tive, an interesting feature is observed at the edges of

Fig. 6. Top and middle: amplitude and phase of coupling coeffi-cient for a triangular apodized grating calculated using the un-chirped spectrum (gray), the reference grating (black), and theIA (dotted). Bottom: reflection spectra of the unchirped grating(gray) and that calculated from the coupling coefficient obtainedusing the reference grating (black) and IA (dotted).


the grating. This pedestal, or step, is related to therapid index change induced in the fiber for low flu-ences and the imperfect extinction of fringes fromthe dithered phase mask used to write the gratings[2]. As expected, all three temperature gradientsyield the same result, while the gray line from thedirect DLP calculation severely departs from the cor-rect grating profile. This implies that the DLP and IAare accurately reconstructing the coupling coefficientof the test grating, and that the recoat does not, inthis case, inhibit the performance of the characteri-zation system.

The derived phase distributions are shown in themiddle of Fig. 7. The phase shift (∼3:3 rad) is locatedat 27mm, coinciding with the amplitude notch inthe top graph. The reproducibility in the estimationof the phase is exceptional, showing a variation of�0:05 rad. The lower graph shows the correspondingreflectivity spectra calculated for the each experi-ment, illustrating the excellent agreement with theunchirped spectrum down to the fine sideband struc-ture. The residual errors for each coupling coefficientare similar, as evidenced by the minimal differencebetween the three spectra.

Figure 8 shows an expanded view of the reflectionspectra for ΔT ∼ 100 °C and 130 °C. This figure de-monstrates that the computed spectra correctly re-plicate the sideband structure over the full lengthof the acquired OFDR trace. The right inset illus-trates the agreement between the original unchirpedspectrum and the spectra calculated from the IA-derived coupling coefficients. The left inset showsthe advantage of regularizing the spectrum priorto reconstruction. Before implementing the IA, thespectra corresponding to the recoated grating(ΔT ∼ 130 °C) are regularized using transmissionmeasurements obtained immediately following thereflection measurements [7]. The algorithm then

Fig. 7. Top and middle: amplitude and phase of coupling coeffi-cient for a phase-shifted apodized grating calculated using the un-chirped spectrum (gray), and the IA with 100 °C (dotted), 130 °C(×), and 160 °C (þ) temperature gradients. Bottom: reflection spec-tra of the unchirped grating (gray) and that calculated from thecoupling coefficient obtained using the IA with 100 °C (dotted),130 °C (×), and 160 °C (þ) temperature gradients.

Fig. 8. Reflection spectra of the unchirped grating (gray) and that calculated from the coupling coefficient obtained using the IA with100 °C (dotted) and 130 °C (×) temperature gradients. The inset graphs show expanded views of spectral regions.


proceeds normally, utilizing the DLP to reconstructthe chirped coupling coefficients, averaging those re-sults, and using the IA to calculate the unchirpedgrating profile. The variation in the coupling coeffi-cient derived from the regularized spectrum andthose that are not is extremely subtle. The noticeablelack of disparity among the coupling coefficientsshown in Fig. 7 supports this assertion. For theregularized spectrum, the amplitude of the couplingcoefficient has “softer” shoulders, i.e., stronger taper-ing at the edges of the grating profile. As shown inFig. 8, the regularized spectrum matches the side-band structure of the original spectrum in this regionmuch better than without regularization. Outsidethis spectral region, the sideband reconstructions arecomparable.

The final FBG tested is a stronger version ofthe grating studied in the previous example. Thisgrating has an unchirped reflectivity of 99.9999%.A 190 °C temperature gradient is necessary to reducethe reflectivity to such a level (95.42%) that thegrating can be reconstructed with the DLP algo-rithm. The reconstructed amplitude, with integratedjκjL ¼ 8:6, is illustrated in the top graph of Fig. 9.As before, the gray curve represents a reconstructionattempt without the use of the thermal chirp, and the

black curve denotes the amplitude obtained byemploying a thermal chirp (for both the referencegrating and the IA). The grating profile is similarto the last example; it has tapered edges and a notchin the center. A registration mark can be seen at47mm. The step feature is also present in this FBG,although here it is more prominent because of thegrating’s strength. The middle plot shows the phaseof the grating as derived using the reference grating(black) and the IA (dotted). Both phase profiles exhi-bit a 3 rad phase shift in the center of the grating,coinciding with the notch in the amplitude profile,and match to within �0:25 rad. Their correspondingspectra are compared to the unchirped spectrum atthe bottom of Fig. 9. While both spectra match theoriginal spectrum well, the IA-derived coupling coef-ficient produces the smallest residual error.

5. Conclusions

A simplified technique for extracting the couplingcoefficients of strong gratings has been demon-strated. The system employs a thermal gradient toincrease the bandwidth of a grating and decreaseits peak reflectivity. Using the discrete layer peelingalgorithm and an iterative routine based on an esti-mation of the thermally induced chirp, the gratingprofile can be determined. Because the index modu-lation amplitude is unaffected by the thermal gradi-ent, the amplitude is readily derived from the DLPalgorithm. The phase of the grating is calculated byexamining the derivative of the phase of the chirpedgrating and implementing an optimization routinebased on the Nelder–Mead simplex method. Thetechnique is capable of characterizing uniform, apo-dized, and chirped gratings. For the tested gratings,the error in the phase is within �0:25 rad, with arepeatability of �0:05 rad. The technique is alsoamenable to characterizing recoated gratings, de-monstrating similar residual spectral errors whencompared to uncoated fiber samples.

Iteration times depend on several factors. Mostnotable is the size of the arrays used in the algo-rithm. A grating profile with 1000 discrete layersand 2000 wavelength points can take approximately10 min to converge on a Pentium 4, Windows XP ma-chine with 3Gbytes of memory. While reducing thenumber of points will speed the convergence, it willalso reduce the spatial and spectral resolution. TheIA presented in this paper is the simplest implemen-tation of the method. It is possible that a differentoptimization routine, different error function (onethat includes spectral and spatial weighting), anddifferent initial guess of the thermal gradient willimprove the convergence time.

In the course of an experiment, it is often difficultto control all external factors that might influencethe temperature gradient. As demonstrated by theresults reported here, there are several advantagesto using configuration B depicted in Fig. 1. Namely,there is no assumption that the reference and testgratings experience the same thermal chirp. While

Fig. 9. Top and middle: amplitude and phase of coupling coeffi-cient for a phase-shifted apodized grating calculated using the un-chirped spectrum (gray), the reference grating (black), and the IA(dotted). Bottom: reflection spectra of the unchirped grating (gray)and that calculated from the coupling coefficient obtained usingthe reference grating (black) and IA (dotted).


the use of silicone fluid helps tomitigate these factorsin a two-fiber system, it is still possible that thereference and test gratings experience slightly differ-ent temperature distributions. By eliminating the re-ference grating and instead estimating the thermalgradient, the experiment becomes more robust andless susceptible to environmental conditions.

This research was funded by Naval ResearchLaboratory (NRL) 6.2 base funds. The authors thankClay Kirkendall for his programmatic support.

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Characterization of strong fiber Bragg gratings using an applied thermal chirp and iterative algorithm