Knowledge of the refractive-index profiles of opticalfibers and fiber devices is of critical importance fordetermining their subsequent performance. For ex-ample, the refractive-index profile of dispersion-compensating optical fiber is tailored to achievespecific levels of dispersion at telecommunicationwavelengths.1 Polarization-maintaining optical fiberrelies on circular asymmetry present in the fiberstructure to decouple orthogonal polarization states.Small, irregular index variations can also affect op-tical fibers and fiber devices; this is especially true ifsuch variations lead to asymmetry in the transverserefractive-index profile. Birefringence in optical fibergratings alters transmission spectra and introducespolarization-dependent loss.2–4 Correct modeling oftransmission spectra of fiber gratings that possessarbitrary azimuthal–radial refractive-index varia-
tions requires knowledge of the transverse refractive-index profile.5,6 The form of the index asymmetrymust be known if one is trying to reduce birefringenceduring grating fabrication.7 To understand and pre-dict the effects of small, asymmetric index variations��1 � 10�4�, it is necessary to measure accurately therefractive-index profiles of optical fiber and fiber de-vices. It is also desirable to be able to measure indexprofiles nondestructively to facilitate testing of fiberdevices.
Numerous techniques exist for measuringrefractive-index profiles of optical fibers and fiber de-vices. However, many of these techniques require theassumption that the fiber being tested is circularlysymmetric. For example, traditional transverse in-terferometry, although it is nondestructive, assumescircular symmetry when it is profiling optical fi-bers.8,9 Similarly, index profiling with the focusingmethod yields accurate one-dimensional profiles onlyfor circularly symmetric fibers.10 The implicit as-sumption of circular symmetry prevents these tech-niques from being used to characterize irregular,asymmetric index variations in optical fibers.
Additional techniques have been developed forcharacterizing asymmetry in optical fiber index pro-files. Etching combined with atomic-force microscopyprovides topographical detail over small regions, but
B. L. Bachim and T. Gaylord ([email protected]) are withthe School of Electrical and Computer Engineering, Georgia Insti-tute of Technology, Atlanta, Georgia 30332-0250. T. K. Gaylord’se-mail address is [email protected].
Received 9 February 2004; revised manuscript received 6 Octo-ber 2004; accepted 8 October 2004.
quantitative interpretation requires calibration, andthe etching process is destructive.11,12 A variation onthe refracted-near-field scanner can measure two-dimensional index profiles but also requires access toan end face and is therefore destructive.13 Severalbasic (one-dimensional) profiling techniques havebeen combined with computed tomography to permitnondestructive measurement of asymmetric indexprofiles. Profiling of optical fibers in combination withtomography has been demonstrated by use of focus-ing, multidirectional scattering-pattern, and quanti-tative phase microscopy approaches. Whereas thesecombined techniques are effective for profiling typicaloptical fibers and are nondestructive, they can lacksufficient resolution to detect small, irregular varia-tions in fiber profiles, such as those that could beinduced by one-sided exposure to ultraviolet light(typically on the level of 1 � 10�4). Considering theneed to measure accurately small, irregular indexvariations and the currently available profiling tech-niques, there is thus a need for a nondestructive mea-surement technique that permits high-resolution,high-accuracy measurements of small, asymmetricvariations in the index profiles of optical fibers andfiber devices.
In this paper we present a measurement techniquebased on microinterferometry and tomography foruse in profiling optical fibers and fiber devices withsmall, asymmetric index variations over the cross-sectional profile. This technique, microinterferomet-ric optical phase tomography (MIOPT), combines thehigh-resolution, high-accuracy measurement capa-bilities of interferometry with the ability to profileirregular objects provided by computed tomography.Using microscopy-based fringe-field interferometrypermits detailed inspection of objects such as opticalfibers under increased magnification. Characteriza-tion of small, asymmetric index changes is importantin a number of optical fibers and fiber devices, includ-ing elliptical-core polarization-maintaining fiber,twin-core optical fiber, fiber exposed to ultraviolet orcarbon dioxide laser light, fiber couplers, and fiberfusion splices.
Interferometry and tomography were combined forindex profiling measurements of optical fibers in twoprevious efforts, neither of which focused primarilyon detecting small, asymmetric index variations. Thefirst effort involved characterizing graded-indexwaveguides that possessed known profile forms(power law) and in which significant ray refractionoccurred over sample cross sections.17 The observedsignificant ray refraction can be attributed to rela-tively large refractive-index gradients present ingraded-index fibers. The second effort, by Górski, in-volved profiling optical fibers under conditions of rel-atively large index differences (greater than 0.015)between fiber cladding and surrounding matchingoil.18 Under such conditions, it becomes difficult tocharacterize small index changes over an entire crosssection, in part because of enhanced diffraction ef-fects. Simulations and measurements made with a
bulk interferometer system were conducted for asymmetric multimode optical fiber.
The measurement approach that we present in thispaper is concerned with characterizing small indexvariations ��1 � 10�4� in small objects (�125��m di-ameter). A number of additions and alterations canbe made to the combined interferometry and tomog-raphy methodology to enhance detection of small in-dex variations. The presence of only small indexdifferences over a cross-sectional profile permits theuse of a ray-based, no-deviation formulation that iscompatible with established parallel projection com-puted tomography. Use of commercial interferencemicroscopes to conduct measurements enables con-trol and optimization of interference images to en-hance detection of small index differences. Developedinterference microscopes also reduce wave-front de-viation and diffraction errors and can easily beadapted to perform automated measurements. In thetomography reconstruction process, the acquisitionprocedure and reconstruction algorithm can bechanged to lower noise and enhance detection ofsmall index differences. Through numerical simula-tions with example optical fiber profiles, we demon-strate that is possible to characterize fibers withsmall, asymmetric index variations beginning from aset of interference images. The average error in thereconstructed profiles is less than 0.1% for three sim-ulated profiles and results, in part, from implemen-tation of practices to enhance reconstructionaccuracy.
We begin the discussion of MIOPT in Section 2 bypresenting the ray-based interpretation of the mea-surement process and include details on what typesof measurement must be conducted and on how theinformation is interpreted and analyzed. Restrictionson using this measurement approach are also dis-cussed. Implementation of interference image analy-sis and tomographic reconstruction is presented inSection 3. A configuration for collecting the data re-quired for conducting MIOPT by use of an interfer-ence microscope is presented in Section 4. Specificways to improve the detection of small index differ-ences are detailed in Section 5. The results of numer-ical simulations, used for testing and exploring theanalysis portion of the technique, for three differentoptical fiber profiles are presented in Section 6. Spa-tial and refractive-index resolution and accuracy is-sues are addressed in Section 7.
Computed tomography permits multidimensionalprofiling of irregular objects. Such profiling is accom-plished through the measurement of a set of projec-tions. For the type of profiling considered here, theprojections must be related to the refractive-indexvalues over the optical fiber’s transverse cross-section. In this section we discuss the relationshipamong measured interference images of optical fibertest objects, projections, computed tomography recon-
struction, and the general process for conducting non-destructive characterization. The ray-based approachused for developing the theory is valid for the smallindex differences considered here and also provides aclear physical understanding of the measurementprocess.19
In the context of measuring two-dimensional trans-verse refractive-index profiles of optical fibers by useof computed tomography, a set of one-dimensionalprojection measurements must be collected. An indi-vidual projection is a one-dimensional representationof an object that contains both intrinsic property andspatial information.20 Projections are then used toreconstruct the two-dimensional transverse indexprofile. Three-dimensional measurements (additionof longitudinal direction) are achieved by stackingtwo-dimensional reconstructed profiles. In the con-text of measuring refractive-index profiles, a projec-tion is a line integral of the object’s refractive indextaken at a specific angle about the object and over itsspatial extent. Such a projection can be interpreted asthe optical path length over its spatial extent whenthe object is viewed at a particular angle. The conceptof a projection is illustrated in Fig. 1(a) and the no-tation used in discussing them is shown in Fig. 1(b).The d and L axes represent the rotated coordinatesystem of the projection and are related to the fixedcoordinate system of the object (x and y axes) byprojection angle �. In mathematical terms, a projec-tion, p�d, ��, taken at a particular angle (�) from thex axis is
p(d, �) ����
�
n(d, L)dL, (1)
where n�d, L� is the refractive-index profile of theobject in the rotated coordinate system. Profilen�d, L� is related to n�x, y� by a transformation in-volving angle �. In practice, the projection integral istaken only over the object’s spatial extent.
In developing the theory we consider parallel pro-jections only for use in measuring two-dimensionalprofiles that possess small index differences. The as-sumption regarding parallel projections places a re-striction on rays traveling through the test object,namely, that no refraction occur. The absence of re-fraction implies that rays traveling through the sam-ple cross section will always be perpendicular to thed axis in the rotated coordinate system at every pro-jection angle. This restriction cannot be met even inan ideal situation, as some form of index differencealways exists in optical fibers (at least at the core–cladding interface) and this causes some rays to berefracted. However, if the refractive index of the sur-rounding matching oil is closely matched to the sam-ple cladding (within 1 � 10�3) in an interferencemicroscope system, refraction at the outer boundariesis limited. Other measures can be taken to limit re-
fraction effects and are discussed in Section 5. Ifproper practices are adopted, the parallel projectionapproximation for ray travel is valid for use in char-acterizing fiber samples with small index differences.
A set of projections consists of individual projectionmeasurements taken at various angles about the testobject. From a set of projections, the object’srefractive-index profile can be reconstructed:
n(x, y) ��0
2�
d� �0
�
P(, �) exp(i2�d)d, (2)
Fig. 1. (a) Illustration of refractive-index projections [optical pathlength (OPL)] of a twin-core optical fiber taken 90° apart. (b) Re-lationship between the fixed coordinate system �x, y� of the opticalfiber and the rotated coordinate system �d, L� of the projection,p�d, ��, at angle �. The projections go to zero outside the spatiallimits of the fiber cross sections.
where � is the spatial frequency and P�, �� is theFourier transform of projection p�d, ��.20
One can determine the optical path-length valuesrequired for the projection set by measuring phase, asthe two are simply related by the free-space wave-vector’s magnitude. Several methods exist to mea-sure phase, but one of the most accurate involvesinterfering an optical wave that has passed throughan object with a reference wave. Interference mea-surement schemes routinely detect optical path dif-ferences of less than �100 and thus can detect smallchanges in index for the same path length. Numeroustechniques exist for generating interference imagesof phase objects,21–23 but static fringe-field inter-ferometry is considered in the present approach.
Figure 2(a) shows a ray passing through an opticalfiber sample. The figure depicts a transverse crosssection of a typical single-mode optical fiber but couldeasily represent another object or device with a morecomplicated profile. Only one ray is shown in theillustration; a collection of rays at points along the daxis is necessary to produce one projection at eachangle �. The following equations were developed for aray passing perpendicularly (to the d axis) throughthe object, as occurs in the rotated projection coordi-nate system, and are correct for any angle. The math-ematical relationship between the measured phasefrom the interferogram and the projection integral isdeveloped below through examination of the accumu-lated phase of rays in the sample and reference armsof an interferometer.
The accumulated phase of a ray passing throughthe optical fiber in the sample arm of the interferom-eter, �samp, is given by
�samp � k0noil[Lr � Lf] � k0 �Lf
n(d, L)dL, (3)
where k0 is the free-space wave-vector magnitude, noilis the refractive index of the matching oil, Lr is anarbitrary reference length, Lf is the length of thesample that the ray traverses (with Lr Lf), andn�d, L� is the two-dimensional refractive-index pro-file of the optical fiber sample in the rotated coordi-nate system. The accumulated phase of a matchingray in the reference arm of the interferometer, �ref, issimply
�ref � k0noilLr � k0noil[Lr � Lf] � k0noilLf. (4)
As the waves in the two interferometer arms inter-fere, the phase differences between the sample andthe reference result in relative shifts in theminimum–maximum intensity peaks of the static in-terferogram. The phase difference between rays inthe reference and sample beams that pass throughthe matching oil equals zero and the interferencepeaks of the rays serve as the baseline for calculatingthe phase shift that is due to the presence of thesample. The phase difference between reference rays
and rays traveling through the sample are calculatedby subtracting Eqs. (3) and (4). This difference isinterpreted as the relative phase shift ����, with
�� � �samp � �ref � k0noil[Lr � Lf] � k0 �Lf
n(d, L)dL
� k0noil[Lr � Lf] � k0noilLf, (5)
�� � k0 �Lf
n(d, L)dL � k0noilLf. (6)
Fig. 2. (a) Diagram of a typical ray passing through the opticalfiber sample. The quantities d and L are the rotated coordinatesystem axes, n�d, L� is the two-dimensional transverse refractive-index profile of the sample, noil is the index of the matching oil, �ref
is the phase of a ray traveling through the oil in the reference arm,�samp is the accumulated phase of a ray traveling through thesample, dr is the distance from the fiber core to the sample ray, Lf
is the length of the sample through which the ray passes, and Lr isan arbitrary reference length. (b) Interference image of opticalfiber. D is the fringe separation distance and Qd is the relativefringe shift.
The resultant integral term in Eq. (6) is the pro-jection required for directly implementing computedtomography reconstruction to retrieve the index pro-file. However, a different form of the integral is moreconducive to performing the reconstruction. One canderive the alternative form by rewriting the noil termin Eq. (6) as
k0noilLf � k0 �Lf
noildL. (7)
The integral term in Eq. (7) can then be substitutedinto Eq. (6) to yield
�� � k0 �Lf
n(d, L)dL � k0�Lf
noildL
� k0 �Lf
[n(d, L) � noil]dL. (8)
The integral term containing the difference betweenn�d, L� and noil can be calculated directly from a re-corded interference image and confers the advantageof eliminating the need to calculate Lf during analysisand reconstruction. Although the relative refractiveindex is now being reconstructed, simply adding therefractive index of the matching oil after the recon-struction is completed yields the desired sample re-fractive index, n�x, y�.
Calculating the relative refractive-index projectionintegral from interference image data requires mea-suring the relative fringe shift from the baselinefringes that do not pass through the sample and thefringe separation distance. The relative phase shift,��, at some distance dr from the fiber core is calcu-lated from the interferogram by
�� �2�Qd
D (9)
where Qd is the distance from the baseline fringereference and D is the separation distance betweenfringe minima (or maxima) and represents a 2�phase difference.8 The two values are illustrated inthe example interference image shown in Fig. 2(b).
Equating Eqs. (6) and (7) and then rearranginggive the relative refractive-index projection integralin terms of the quantities measured from the inter-ference images taken at each projection angle:
pr(d, �) ��Lf
[n(d, L) � noil]dL �2�Qd
k0D�
Qd
D 0,
(10)
where pr�d, �� is now the relative index projection. Asthe physical path is the same, the integral also rep-
resents the optical path difference along the d axis. Aset of relative projections taken at various anglesabout the test object can be used to reconstruct therelative-index profile, from which one can determinethe actual index profile by adding the matching oil’srefractive-index value. Equation (10) has a funda-mental relationship to the corresponding equationused for determining the one-dimensional refractive-index profile in traditional transverse interferome-try.8
From the analysis presented above, it can be seenthat MIOPT consists of (1) measuring interferenceimages at a set of projection angles, (2) analyzing theimages to extract the phase information, (3) convert-ing the phase information into projection data, (4)collecting the analyzed projection data at all angles,(5) performing computed tomography reconstruction,and (6) adding the matching oil’s index to the recon-structed profile to retrieve index profile n�x, y�.
3. Analysis Implementation
The general measurement procedure described inSection 2 consists of acquiring interference images atmany angles about a sample and analyzing the im-ages to perform reconstruction. In this section wediscuss specific implementation of the fringe analysisand computed tomography reconstruction portions ofthe measurements procedure; the experimental con-figuration for recording interference images is dis-cussed in Section 4.
Implementation of the analysis portion of MIOPTcan be separated into two parts: interference fringeanalysis and tomographic reconstruction. Splittingthe analysis task into two parts allows for flexibilityduring reconstruction. For instance, the data thatresult from analyzing interference images can bestored and a variety of reconstruction approaches in-dependently attempted. The particular approachesselected for implementing fringe analysis and com-puted tomography reconstruction are discussed be-low. Both the interferogram analysis and thetomographic reconstruction algorithm were imple-mented discretely (as opposed to the continuous formpresented in Section 2).
The static interferogram analysis technique se-lected for use employs a direct polynomial fitting rou-tine based on parabolic approximation of fringeminima.21 This approach was used previously inother optical fiber index profiling systems and has theadvantage of requiring only one interference imageper projection for calculating the phase.24 A thresholdis first applied to the images to locate approximatelythe fringe minima. Data below the threshold level areretained for use in polynomial fitting. Each pixel col-umn of an interference image is treated as an indi-vidual ray for calculation purposes. Therefore thefitting routine is used to identify fringe minima pixellocations along each column. Once the minima loca-tions are known, the relative-index projection can becalculated [from the right-hand side of Eq. (10)]. Allimages captured during measurement are analyzed
to extract their phase, which is used to calculate theprojection.
The second portion of the analysis procedure, to-mographic reconstruction, was implemented by useof the filtered backprojection algorithm.20 Again, anumber of reconstruction algorithms exist, but thefiltered backprojection algorithm is a well-established technique in tomography that providesflexibility for optimizing the reconstruction process toincrease accuracy. The projections derived from in-terference images taken at various angles serve asthe input to the algorithm. Basically, the algorithminvolves taking the Fourier transform of each projec-tion, applying a reconstruction filter to it, and thenperforming an inverse Fourier transform. The fil-tered projection is then backprojected to form asquare matrix; the matrix is rotated by the corre-sponding projection angle (�) and then added to pre-viously processed backprojection matrices. Objectreconstruction is complete when all projections havebeen processed and the matching oil’s index value isadded.
4. Experimental Configuration
A microinterferometer arrangement, with associatedhardware, is necessary for obtaining interference im-ages of optical fiber and fiber devices to perform pro-filing. Although it is possible to construct an apparatusfor conducting measurements from bulk optical ele-ments, several interference microscopes already existthat are suitable for use in the system.8,25,26 Using acommercial interference microscope confers many ad-vantages, which we addressed in Section 5 below. Aninterference microscope suitable for conducting MI-OPT measurements is described below, along with theoverall experimental configuration.
The Mach–Zehnder two-objective, transmitted-light interference microscope that is traditionallyused for profiling symmetric optical fibers can beadapted to facilitate interference measurements atvarious angles.8,25 Adapting the microscope requiresonly the addition of a rotary stage, a sample holder–positioner, and a motion controller to the traditionalarrangement, as the system already includes a cam-era and a frame grabber for capturing interferenceimages. A diagram of the overall experimental con-figuration is shown in Fig. 3. The measurement pro-cess, in this arrangement, can be automated toreduce the amount of time required for taking thelarge number of projections needed for low-noise re-construction. The microscope (originally manufac-tured by Ernst Leitz GmbH, Wetzlar, Germany), isfound in many optical fiber characterization labora-tories because it is currently used for profiling circu-larly symmetric optical fiber.
5. Measurement Optimization for Characterizing Small,Asymmetric Index Differences
Characterizing small asymmetric index differencesin the refractive-index profiles of optical fibers andfiber devices requires consideration of the measure-ment procedures associated with fringe-field inter-
ferometry and tomography. Additions and alterationsto the basic interferometry and tomography ap-proaches can lower noise levels and enhance detec-tion of small index differences. Methods forimproving detection are identified and discussed inthis section.
Use of a developed commercial interference micro-scope, as opposed to implementation of a bulk opticinterferometer,18 offers several advantages for con-ducting this type of measurement. Interference mi-croscopes, such as the Mach–Zehnder transmitted-light system discussed in Section 4, are designed tohave precise, stable optical elements that minimizewave-front distortion and maintain path balance andthereby increase interference-image stability. Opticalplates and wedges incorporated within the micro-scope permit precise adjustment of fringe spacing,orientation, and width. The ability to conduct preciseadjustments means that fringe properties can be op-timized for detecting small index differences. Spuri-ous fringes and speckle noise are reduced by use of abright, bandpass-filtered mercury lamp instead of alaser-based illumination system commonly employedin bulk systems. With the automated measurementconfiguration shown in Fig. 3, multiple images can becaptured at each projection angle and averaged toreduce noise effects.
Using an interference microscope to conduct mea-surements has the additional advantage of reducingrefraction effects. As mentioned above, optical fibersamples must be surrounded with an accuratelyknown index-matching oil whose refractive-indexvalue is close to (but not equal to) that of the sampleouter cladding. Matching the indices of the oil andcladding lowers the deviation of the rays at the sur-face boundaries.27 Direct use of high-magnificationoil-immersion objectives ensures that the matchingcriteria will be met and eliminates the need for mi-croscope slides and coverslips that can introducewave-front distortion. In situations when the index
Fig. 3. Experimental configuration for measuring interferenceimages of an optical fiber test object at various projection angles.An optical fiber sample, secured in the holder, can be rotated aboutits axis to enable interference images to be recorded at any angle.The measurement system is automated easily by incorporation ofa motion controller and a frame grabber.
value of the cladding is not known, oils with differentrefractive indices can be tried until a suitable fringefield is observed. In addition to facilitating precisematching, the microscopy approach presented cor-rects for some refraction effects when the system isproperly focused on the center of the fiber.27 Evenwith the two corrective measures suggested, samplessuch as graded-index and air–silica microstructureoptical fibers would not meet the parallel projectioncriteria owing to excessive ray refraction over theirtransverse cross sections. However, as our concern isprimarily with measuring small perturbations in in-dex profiles of commercial telecommunications fiber,refraction effects owing to asymmetry are expected tobe below those that are due to interfaces (oil–cladding and core–cladding). In cases of excessiverefraction, a different form from parallel projectionsmay be adopted in describing ray paths through thesample (for example, a fan-beam projection20). Raytracing offers one method for investigating whether aparticular optical fiber sample would introduce toomuch deviation.18,27
Consistent with the primary purpose of detectingsmall variations in refractive index, several portionsof the measurement and analysis procedures can bechanged within the reconstruction process to lowernoise levels and subsequently improve detection ofsmall variations in refractive index. Taking projec-tions over a full 360°, taking additional projections,and employing various reconstruction filters all act tolower noise levels in certain regions of reconstructedobjects. Taking projections 360° around the sampleobject, instead of over just 180°, increases averagingof noisy data and reduces asymmetric ringing ef-fects.28 Increasing the total number of projections (de-creasing the angle between projections) also leads toincreased averaging of noisy data.28 Altering the re-construction filter (part of the filtered backprojectionalgorithm) to introduce averaging and attenuation ofhigher frequencies lowers the noise level and im-proves the changes of detecting small variationswithin interior regions but not near edges or sharptransitions.19,20,28 Various effects of the type of filterused in reconstruction are demonstrated in Section 6below.
By incorporating all the additions and alterationsdiscussed in this section into the MIOPT measure-ment process, the ability to detect small, asymmetricindex changes is improved. The effect of some of thesechanges can be illustrated through numerical simu-lations.
6. Simulations and Simulation Results
To evaluate the proposed measurement methodology,to test the analysis programs, and to verify the abilityto profile asymmetric objects we conducted a series ofdiscrete numerical simulations. Three optical fibertransverse cross sections were generated, from whichinterference images were created. The generated in-terference images served as simulated inputs to thefringe analysis and reconstruction programs. Thespecific details of interferogram analysis and tomo-
graphic reconstruction implementation are given be-low, along with results for the three different types ofsimulated profile. The method for generating the in-terference images by use of the MATLAB programminglanguage is also given. The average error in the re-constructed refractive-index profiles was less than0.1%, depending on the type of reconstruction filteremployed and on whether there is any postrecon-struction image processing. Using the simulations,we demonstrate that it is possible to reconstruct ac-curately refractive-index profiles with small, asym-metric index differences from a set of measuredinterference images taken at various angles aboutthe test object.
To begin the simulation, a desired cross-sectionalrefractive-index profile of an optical fiber is generatedas a 512 � 512 matrix, with the fiber surrounded byan index-matching oil. The optical fiber index profilematrix is subtracted from another constant-valued�noil� matrix of the same dimensions, and the result ismultiplied by the image pixel spacing ��L�. The re-sultant matrix serves as the basis for calculating thephase used in generating the interference image foreach projection angle. An example relative refractive-index profile is shown in Fig. 4. The difference matrixis then rotated by the current projection angle and itscolumns summed to generate an array containing theoptical path difference that is due to the object. For allthree simulations, projections were taken every 0.5°about the generated profile (720 projections total foreach simulation). The large number of projectionsreduces noise levels in the reconstructed image.
The interference images used for evaluating the to-mographic process were generated from Kingslake’sformulation, which is given by
I(p, q) � {A � B cos[k0W(p, q)]} � N(p, q), (11)
Fig. 4. Gray-scale plot of a transverse optical fiber refractive-index profile relative to the matching oil’s index. Simulated profileslike this one are used for generating interference images and test-ing the fringe analysis reconstruction programs. This particularsimulated profile is circularly symmetric and possesses outer clad-ding, inner cladding, and core regions.
where p and q are, respectively, the columns androws of the image, I�p, q� is the irradiance in theinterferogram plane, A is the static bias, B is theamplitude, W�p, q� is the optical path difference, andN�p, q� is the noise added to the interference im-age.29 The irradiance, static bias, and amplitude aregiven in terms of an 8-bit gray scale (0–255), as wouldbe captured from a typical CCD camera. We calcu-lated the optical path difference by first generating amatrix with the dimensions of the image and thatpossesses a linear optical path variation along itscolumns but an optical path that is constant across itsrows. This configuration mimics the interference oftwo waves that are tilted with respect to each other tocreate carrier fringes. Another matrix, with the samedimensions as the carrier fringe matrix, is createdwith the projection optical path-difference array (rep-resenting the phase effect of the sample) and is addedto the carrier fringe matrix. After the complete inter-ference image is calculated, Gaussian noise is addedthat possesses a zero mean and standard deviation�m.29,30 A noisy interference image is generated foreach projection taken of the test index profile andsaved in an image file format similar to that of ex-perimentally acquired images. The entire image setfor a particular test profile served as the input to thefringe analysis program. A typical interference imagegenerated from the index profile in Fig. 4 is shown inFig. 5. The values of parameters used in generatingthe interference images for all three simulated crosssections are as follows: A, 128; B, 108; 0, 546 nm; �,0; �m, 8; and �L, 312.5 nm. The mean noise (�) andstandard deviation (�m), similar to the static bias (A)and amplitude (B), are in terms of gray-scale digitalnumbers that represent intensity levels.
In generating the interference images we followedthe parallel projection (no refraction) assumption.
Therefore any variations in the gray-scale intensityare due only to changes in phase. It was also assumedthat sample rotation occurred exactly axially, imply-ing that tilt and shifting of the sample cross-sectioncenter location (center of fiber core) are not factors. Inpractice, tilting and shifting of sample objects in im-ages can be compensated for if they are found tooccur.31
The first simulated profile was that of a circularlysymmetric optical fiber with an inner cladding, anouter cladding, and a core, the same as that shown inFig. 4. A symmetric profile was simulated initially totest the programs, as it represents a simple, knownprofile. The reconstructed object profile, after the sim-ulated interference images have been processed byuse of the interferogram analysis and reconstructionprograms, is shown in Fig. 6. The maximum error inthe reconstructed profile over the entire cross sectionwas 0.12%, and the average, 0.002%. Specific resultsare shown in Fig. 7(a) for a line section of the profilealong its length at the center width. Figure 7(b)shows the absolute difference between the test andthe reconstructed profiles. The reconstructed profilematches the generated test profile closely. All profilesfrom Fig. 4 to Fig. 11 have been plotted relative to thematching-oil index to enhance illustration of indexvariations. Only the refractive-index value of thematching oil (1.4571) must be added to yield the ab-solute index profiles. A modified reconstruction filter(typical ramp-type combined with a Hanning filter)was used to enhance the accuracy in interior regionsof the profile, with a corresponding decrease in accu-racy near edges and sharp transitions.20,28
Next, a twin-core optical fiber was simulated witha profile similar to that in the research of Barty et al.(measured by quantitative phase microscopy).16 Thisfiber is not circularly symmetric, as can be seen fromFig. 8(a), and is therefore useful for evaluating theability of the present method to characterize asym-metric objects. The reconstructed object profile isshown in Fig. 8(b). The maximum error in the recon-
Fig. 5. Example interference image generated by use of Eq. (11)from the test profile shown in Fig. 3. As the profile is symmetric, allthe projections are identical (except for additive noise).
Fig. 6. Gray-scale plot of the reconstructed index profile of thecircularly symmetric optical fiber.
structed profile over the entire cross section was0.21%; and the average, 0.083%. Specific results, fol-lowing processing, are shown in Fig. 9(a) for a linesection of the profile along its length at the centerwidth. Figure 9(b) shows the absolute difference be-tween the test and reconstructed profiles. The error islarger over the entire profile for this reconstruction,in contrast to the profile shown in Fig. 7(b), becauseonly the required ramp-type filter (unmodified) wasused for the reconstruction. The error is larger,though more nearly uniform, over the line section,and the core features are better preserved.
The third profile, a single-mode optical fiber withan exponential variation over the cross section, isshown in Fig. 10(a). The index profile is circularlyasymmetric and has one side of the cladding at aslightly higher index value than the other side�1.5 � 10�4�, with exponential variation between.The asymmetry in the profile is similar to that ob-served in optical fibers exposed to ultraviolet light butapplied over the entire cross section and not simply in
the core. Optical fibers exposed to carbon dioxide la-ser light have approximately this form of small indexasymmetry. The reconstructed object profile is shownin Fig. 10(b). The maximum error in the recon-structed profile over the entire cross section was0.08%; and the average, 0.002%. Specific results, fol-lowing processing, are shown in Fig. 11(a) for a linesection of the profile along its length at the centerwidth. Figure 11(b) shows the absolute difference be-tween the test and reconstructed profiles. Becausethe exponential variation is concentrated predomi-nantly within the cladding, a modified reconstructionfilter (the same as was used for the symmetric profilesimulation) was used to achieve lower error in thecladding regions.
The results of the simulations demonstrate that itis indeed possible to reconstruct the index profiles ofoptical fibers with index asymmetry and small indexdifferences by analyzing interference images taken at
Fig. 7. Symmetric optical fiber simulation results. (a) Compari-son of test and reconstructed profiles taken along the length at thecenter of the width. (b) Absolute index difference between test andreconstructed profiles shown in (a). The noise in the interior clad-ding regions is lower than that near the edges and in the corebecause of the modified filter used in reconstruction.
Fig. 8. (a) Gray-scale plot of the generated transverse refractive-index profile of a twin-core optical fiber relative to the matchingoil’s index. The profile is not circularly symmetric because of thetwo offset (from center) cores. (b) Reconstructed index profile.
multiple angles and using computed tomography.They also show how one can change reconstruction toemphasize certain aspects (such as interior regions)by modifying the basic ramp-type reconstruction fil-ter used in the filtered backprojection algorithm toattenuate selected spatial frequencies. Modifying thefilter can be useful when one is attempting to profilefibers with small index variations in, for instance, thecladding region.
7. Resolution and Accuracy
High resolution and high accuracy, in both spatialand refractive-index terms, are two of the advantagesoffered by using microinterferometry to conduct pro-filing. The factors that influence resolution and accu-racy are different for the refractive-index and spatialdomains; the issues are far more complicated for theindex case.
Spatial resolution is determined by distinct factorsin the transverse (cross-sectional) and axial direc-tions. The transverse spatial resolution is set by a
combination of the microscope’s lateral resolvingpower and the equivalent pixel spacing of the CCDcamera at a given magnification. For the Mach–Zehnder transmitted light microscope discussed inSection 4 the lateral resolvable power is approxi-mately 0.5 �m at 50� magnification.32 For example,if the equivalent pixel spacing of a typical CCD cam-era at that magnification is approximately 0.6 �m,then the pixel spacing limits the transverse spatialresolution because it is the larger value. Axial spatialresolution is set by the fringe separation distance. Incalculating the relative phase shift from interferenceimages, the phase difference between fringe minimais assumed to be the same. The assumption is neces-sary for scaling relative shifts to a known phase value�2��. For the assumption to be true, the transverse
Fig. 9. Twin-core optical fiber simulation results. (a) Comparisonof test and reconstructed profiles taken along the length at thecenter of the width. (b) Absolute index difference between test andreconstructed profiles shown in (a). Noise levels are roughly sim-ilar in the cladding and cores and near the edges because only thebasic ramp-type filter was used.
Fig. 10. (a) Gray-scale plot of the generated transverse refractive-index profile of a single-mode optical fiber relative to the matchingoil’s index. The exponential variation originates from one side andwas calculated from an equation of Dossou et al.5, but applied overthe entire cross section. (b) Reconstructed index profile. A shorterrelative index range is used to highlight index variations in thecladding region (core features are not shown).
index profile must be constant along the axial direc-tion. In the example interferogram shown in Fig. 5,the fringe separation distance is approximately 100�m, and that value represents the axial spatial res-olution limit. The fringe separation can be decreasedto improve axial resolution but only with an accom-panying reduction in the ability to detect smallchanges in phase. The spatial resolution gained byusing microinterferometry permits profiling of sam-ples with more rapidly varying radial changes, asoccurs in dispersion-compensating fiber, than in typ-ical optical fiber.
Refractive-index resolution and accuracy are morecomplicated than their spatial equivalents. Noise lev-els in the reconstructed image depend not only on thesmallest detectable fringe shift difference but also ontomography reconstruction practices. The smallestdetectable fringe shift difference is influenced by en-vironmental factors, system noise, and the fringeanalysis program. Factors that influence index reso-lution originating from computed tomography in-
clude the type of algorithm used for reconstruction,filters, the number of projections taken, and the num-ber of samples per projection.28 In addition, the accu-racy of the reconstructed object can vary in differentregions (interior versus edge). No models exist forpredicting the signal-to-noise ratio for a measure-ment approach such as MIOPT, though simplifiednoise models are available for traditional (x-ray) com-puted tomography.28,33 Quantitative signal-to-noiseratio values are usually established through mea-surements of a uniform phantom.20 Despite the com-plexity inherent in the process, the simulation resultsgive a general idea of the noise that arises from theanalysis portion of the measurement process.
8. Summary
Microinterferometric optical phase tomography com-bines the ability to detect small refractive-indexchanges found in fringe-field interferometry withthe ability to characterize irregular objects offeredby computed tomography. In this paper we havepresented and discussed the underlying theory ofMIOPT, analysis implementation, numerical simu-lation results, experimental configurations, andresolution and accuracy issues. MIOPT is uniqueamong the various profiling techniques in that itwas intentionally designed to characterize small,asymmetric index perturbations in optical fibersand fiber devices.
The research of B. L. Bachim was sponsored in partby a National Science Foundation graduate researchfellowship.
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