photonic bandgap fiber bundle spectrometerphotonic bandgap fiber bundle spectrometer qu hang, bora...

Photonic bandgap fiber bundle spectrometer

Qu Hang, Bora Ung, Imran Syed, Ning Guo, and Maksim Skorobogatiy*Génie physique, Ecole Polytechnique de Montreal, C.P. 6079, succ.

Centre-ville, Montréal, Quebec, Canada, H3C 3A7

*Corresponding author: [email protected]

Received 4 March 2010; revised 19 July 2010; accepted 19 July 2010;posted 22 July 2010 (Doc. ID 124961); published 27 August 2010

By using a photonic bandgap (PBG) fiber bundle and a monochrome CCD camera, we experimentallydemonstrate an all-fiber spectrometer. A total of 100 Bragg fibers that have complementary and over-lapping bandgaps are chosen to compose the fiber bundle. AmonochromeCCD is then used to capture thebinned image. To reconstruct the test spectrum from a single CCD image, we develop an algorithm basedon pseudoinversion of the spectrometer transmission matrix. We demonstrate that the peak center wave-length can always be reconstructed within several percent of its true value regardless of the peak widthor position, and that, although the widths of the individual Bragg fiber bandgaps are quite large(60–180 nm), the spectroscopic system has a resolution limit of ∼30 nm. Moreover, we conclude that,by minimizing system errors, the resolution can be further improved down to several nanometers inwidth. Finally, we report fabrication of PBG fiber bundles containing hundreds of fibers using a two-stagedrawing technique. This method constitutes a very promising approach toward industrial-strength fab-rication of all-fiber spectrometers. © 2010 Optical Society of AmericaOCIS codes: 060.5295, 120.6200, 060.2350.

1. Introduction

Development of multispectral imaging systems pre-sents an active research field in rapid development.Inherent advantages of such systems include real-time monitoring, cost-effectiveness, and high accu-racy. Imaging spectrometers have been demonstratedin many applications, including chemical componentanalysis [1], remote sensing [1,2], astronomy [3], andbiochemistry [4]. Moreover, multispectral imagingsystems have been successfully demonstrated bothin the visible and the near-infrared spectral ranges[5–11].

An implementation of a multispectral imaging sys-tem can be realized by using a high-sensitivity, highsignal-to-noise ratio photoelectric conversion device,such as a CCD array; optical filters or dispersive ele-ments, which are either used to provide the referencespectra or to filter out a specific spectral component;as well as a broadband illuminant (light source). Insuch an imaging spectrometer, spectral information

is acquired indirectly by taking several images inthe complementary, while possibly overlapping,spectral ranges. A set of intensity images is then con-verted into spectra by utilizing various interpreta-tion algorithms, such as pseudoinverse estimation,principal-component analysis, and nonlinear fitting[5,6]. In addition, the imaging Fourier transformspectrometer (IFTS), different from the above-mentioned implementation, is also widely used dueto its high resolution. A traditional IFTS is realizedby relaying a mechanically scanned Michelson inter-ferometer to an imaging system. By scanning one ofthe mirrors in the interferometer, the optical pathdifference between the interferometer’s two arms isvaried. Thus, the interferograms are generated andinterrogated by the imaging system. Applying Four-ier transform on these interferograms will recon-struct the original spectrum.

Recently, several optical-fiber-based imaging spec-trometers have been reported [7,8]. In [7], for exam-ple, the authors used, at the input end, a 10 × 10bundle of mid-infrared fibers, which were then sepa-rated into a linear 1 × 100 array at the outputend. The output beams were then simultaneously

0003-6935/10/254791-10$15.00/0© 2010 Optical Society of America

1 September 2010 / Vol. 49, No. 25 / APPLIED OPTICS 4791

dispersed using a single Bragg grating and imagedon a CCD array. This system, thus, functions as a10 × 10 spectral imaging array. In another imple-mentation [8], a multiple range spectrophotometerwas implemented using an optical fiber probe. Inthat system, linear displacement of the light sourcewas projected onto a fiber array (bundle) so that onlya single fiber was lit for a given position of the lightsource. The fiber bundle was then interrogated usinga spectrometer, thus enabling spatially resolved real-time data acquisition. Note that, in both examples,optical fibers were used exclusively as broadband op-tical guides, while a separate spectrometer was stillrequired for the spectra acquisition.

Various fundamental properties of a spectrometerare encoded is its instrumental line shape function(ILSF), which can be used to deduce such perfor-mance parameters as spectral resolution, optical ef-ficiency, and noise level. The spectrum acquired bythe spectrometer is actually a convolution of theILSF and the real spectra. The ILSF is affected bymany factors, including the spectral acquisition ap-proach and the mechanical alignment. For instance,the ILSF of a grating or a prism monochromator de-pends strongly on the widths of entrance and exitslits, the subcomponent dispersion, optical couplingefficiencies, aberrations in the setup optics, etc. Asto the Fourier transform spectrometer, except for theinstrument mechanical factors, the ILSF is also de-pendent on the Fourier transform algorithm used toresolve the interferograms [12], as well as on theother mathematical procedures, such as “apodiza-tion” or Fourier deconvolution [12–14], involved inthe spectral calculations.

As we detail in what follows, our photonic bandgap(PBG) fiber bundle spectrometer realizes spectralreconstructions by interrogating the fiber bundletransmittances and then using a matrix pseudo in-version algorithm to reconstruct the original spectra.The ILSF of our fiber bundle spectrometer is, thus,mainly determined by the spectral reconstructionalgorithm. Particularly, as we will see, the errorscaused by the experimental noise can be multipliedby an ill-conditioned inversion procedure, thus af-fecting the ILSF of the fiber bundle spectrometer.

The goal of this paper is to demonstrate that thefunction of a spectrometer can be implemented di-rectly inside a single fiber bundle. Then a completeimaging spectrometer can be realized by using an ar-ray of such fiber bundles interrogated by a linear or atwo-dimensional (2D) CCD sensor. Such an approachwould allow getting rid of costly and slow traditionalspectrometers based on moving gratings, thus lead-ing to significant cost savings and increased acquisi-tion speed. Another advantage of our spectrometer isthat its resolution is not dependent on the length of afiber bundle. In fact, fiber bundles can be designedthat are able to provide optical filtering over evencentimeters-long lengths. This is in contrast withthe grating-based spectrometer, whose resolutionimproves for longer optical paths traveled by light

inside of a spectrometer. In this paper we reportfor the first time, to our knowledge, a PBG-fiber-based spectrometer that consists of a photonic crys-tal fiber bundle and a monochrome CCD camera.

2. Characteristics of the Subcomponents: BraggFibers, Fiber Bundle, and CCD Camera

A. Photonic Bandgap Bragg Fibers

PBG fiber is a key element of our spectroscopic sys-tem. The PBG Bragg fibers used in the research arefabricated by our group and have been reported pre-viously in [15–17]. An individual Bragg fiber featuresa large 300–700 μm diameter core made of a poly-methyl methacrylate (PMMA) plastic. The coreregion is surrounded with a periodic multilayer re-flector featuring ∼100 submicrometer-thick layersof low and high refractive index PMMA/polystyrene(PS) plastics with refractive indices of 1:49=1:59.Such a multilayer (Bragg reflector) is responsible forthe appearance of a spectrally narrow transmissionband (reflector bandgap) within which the light isstrongly confined inside the fiber core. For the wave-lengths outside of the reflector bandgap, the light pe-netrates deeply into the multilayer region, exhibitingstrong propagation loss due to scattering on theimperfections inside the multilayer region. A typicalfiber loss within the reflector bandgap region is∼10 dB=m and is mostly determined by the bulk ab-sorption loss of a low-purity PMMA plastic. Outsideof a bandgap region, scattering loss (on the imper-fections in the multilayer structure) dominates,resulting in >60 dB=m propagation loss. In our ex-periments, we used 30 cm long fibers so that the lossof guided light was below 3 dB, while the loss of non-guided light was >20 dB. The numerical aperture ofall the Bragg fibers was in the range of 0.17–0.22.

To construct a fiber bundle spectrometer, we chose100 Bragg fibers with complementary and partiallyoverlapping bandgaps, as shown in Fig. 1(a). All thefibers in a bundle were drawn from the same pre-form, with the only difference among them being thefinal diameter. The smaller diameter fibers featurebandgaps shifted toward the blue part of a spectrum.This is easy to rationalize from the basic theory ofBragg fibers, which predicts that the bandgap centerwavelength is proportional to the thickness of themultilayer in the Bragg reflector [18].

In Fig. 1(b), we also show distribution of the fibertransmission bandwidth as a function of the fibertransmission central wavelength. We note that thefiber transmission bandwidth increases with its cen-ter frequency. Thus, in the blue part of a spectrum,the fiber bandgap width is ∼60 nm, while increasingto ∼180 nm in the red part of a spectrum. There aretwo factors that contribute to this phenomenon.First, from the basic theory of Bragg fibers, it followsthat the relative width of a PBG (ratio of the bandgapsize to the bandgap center frequency) is mostly afunction of the refractive index contrast in the Braggfiber reflector [18,19]. As all the fibers used in this

4792 APPLIED OPTICS / Vol. 49, No. 25 / 1 September 2010

work are fabricated from the same preform, it meansthat the relative bandwidth is approximately con-stant for all the fibers, hence the absolute bandgapwidth should increase with the bandgap centerfrequency. Second, as we have mentioned in the pre-vious paragraph, fibers featuring bandgaps at longerwavelengths have larger core size. In Bragg fibers,modal losses due to radiation and scattering onthe reflector imperfections scale as 1=R3 with thecore size. Therefore, larger core fibers exhibit smallerloss than smaller core fibers. Note that, in Fig. 1, weshow the transmission intensities and not the actuallosses. Clearly, for the fibers of the same length,smaller core fibers will exhibit spectrally narrowertransmission, as wavelengths near the bandgap edgewill be attenuated much more strongly than those inthe larger core fibers.

B. Photonic Bandgap Fiber Bundle

The fiber bundle used in our experiments featured a5:6 mm inner diameter plastic tube that hosted 100Bragg fibers of 30 cm of length. Fibers at the inputend of a bundle were rigidly attached with epoxy

to each other and to the confining tube, and the wholeassembly was then polished using optical films ofvarious granularity. On the other end of a bundle,100 fibers were inserted into a custom-made blockfeaturing holes of diameter 0:8 mm placed in a per-iodic 10 × 10 square array, see Fig. 2. The block, thetube, and all the fibers were bonded together withepoxy, and the output end was then polished.

The principle of operation of a PBG fiber bun-dle spectrometer can be clearly understood fromFigs. 1 and 2. In principle, if all the fiber bandgapswere to be spectrally narrow and nonoverlapping,then, at the end of a fiber bundle, the relative inten-sities of light coming out of the individual fiberswould be unambiguously related to the correspond-ing spectral components of an incoming light. Inpractice, individual fiber bandgaps are always over-lapping and, in our particular implementation, thebandgaps are quite broad. Therefore, to reconstructthe intensity of an incoming light from the intensitiesof light coming out of the individual fibers of a bun-dle, we have to use a certain deconvolution algo-rithm. Note that spectrometer resolution is notdirectly limited by the width of fiber bandgaps. Westudy the resolution by testing a set of spectral peakswith bandwidth from 5 to 40 nm. As we will demon-strate in Section 5, even with all the individual band-gaps as wide as 60 nm, one could, in principle,reconstruct the peaks as narrow as 5–10 nm. How-ever, to achieve such a high resolution, it is necessaryto minimize the experimental errors and noisecontributions.

Another important comment is about the intensitythroughput of our spectrometer, and its relation tothe spectrometer resolution. First of all, if all thefibers used in the spectrometer were to have strictlycomplementary (spectrally nonoverlapping) band-gaps of width δλ, then, to enable a certain spectro-meter range Δλ ¼ ðλmax − λminÞ, one would need to

Fig. 1. (Color online) (a) Transmission spectra of five typicalBragg fibers used in the fiber bundle. (b) Distribution of the fibertransmission bandwidth as a function of the fiber bandgap centerwavelength for all the Bragg fibers in a bundle. Inset, photo of aBragg fiber cross section showing a solid core surrounded by aperiodic multilayer reflector.

Fig. 2. (Color online) Fiber bundle spectrometer. Top, schematicof the spectrometer. Light from the illuminant is launched into thefiber bundle; the image is taken by the monochrome CCD. Bottom,when the broadband light is launched into the fiber bundle, theoutput is a mosaic of colors selected by the individual Bragg fibers.


use N ¼ Δλ=δλ fibers. Assuming a broadband lightwith a unit integral power in the spectral rangeΔλ, we conclude that the intensity throughputthrough such a spectrometer would be ∼1=N2.Indeed, the light beam first has to be physically di-vided into N parts to be launched into the individualfibers of a bundle; then, an individual fiber would cutout a spectrally narrow (∼1=N) part of the broad-band light.

It is important to realize, that our implemen-tation uses N Bragg fibers with relatively large andstrongly overlapping bandgaps of spectral widthδλ ≫ Δλ=N. Spectral resolution of such a spectro-meter can be still as small as Δλ=N, while the inten-sity throughput would be much higher andproportional to 1=N, rather than 1=N2. Such inten-sity throughput is comparable to the throughput ofa grating-based spectrometer with resolution ofΔλ=N. Indeed, as we will see in what follows, becauseof the use of a pseudoinversion algorithm, the resolu-tion of a fiber-based spectrometer is fundamentallylimited only by the number of the spectroscopic ele-ments, such as individual Bragg fibers, regardless ofwhether they feature overlapping or strictly comple-mentary bandgaps. On the other hand, a throughputthrough an individual Bragg fiber with a relativelywide bandgap is δλ=Δλ ≫ 1=N, and is no longer lim-ited by the resolution. The overall 1=N scaling of thethroughput intensity, therefore, mostly comes fromthe necessity of subdivision of a test beam into N in-dividual fibers, each of which individually transmitsas much as 30%–50% of the in-coupled intensity. Inpractice, achieving Δλ=N resolution with N fibersfeaturing overlapping bandgaps is somewhat chal-lenging, as a pseudoinverse algorithm used in thespectrum reconstruction is sensitive to the experi-mental errors.

C. Sensitivity and Linear Response of a CCD Sensor

The output of the fiber bundle is a 10 × 10 matrix ofcolored fibers, which is then projected with a beamcompressor onto a monochrome CCD array (seeFig. 2). The sensitivity of a CCD sensor would di-rectly affect the measuring range, the signal-to-noiseratio, and the algorithm that we use to reconstructthe spectrum of an illuminant. The CCD sensor inour experiment is an Opteon Depict 1, E and S SeriesB1A 652 × 494 black and white CCD. The normalizedsensitivity for different wavelengths is displayed inFig. 3(a), from which it is clear that the sensor coverswell all the visible and a part of a near-infrared spec-tral range. The sensitivity curve, however, is a highlyvariable function of wavelength, which must be ta-ken into account in the reconstruction algorithm.Figure 3(a) was obtained by launching the light froma supercontinuum source into a monochromatorwhose output would produce a 2 nm wide (FWHM)peak at a desired center wavelength. Light from sucha tunable source would then be launched into acommercial multimode fiber of ∼1 mm diameter.The output power from such a fiber would then be

measured by both the CCD array and a calibratedpowermeter; the sensitivity curve is then achievedby dividing one by the other.

As we will see in Subsection 3.A, one of the key re-quirements of a CCD array is linearity of its mono-chromatic response as a function of the intensity ofincident light. The near-linear response of a CCDsensor was confirmed at several wavelengths of in-terest; a typical CCD response (at λ ¼ 560 nm) is pre-sented in Fig. 3(b). To obtain Fig. 3(b), two polarizerswere placed between a CCD array and the output of acommercial fiber in a setup described in the previousparagraph. By varying the angle between the two po-larizers, we also varied the intensity of light comingonto a CCD array. A reference measurement is per-formed with a calibrated powermeter for the sameangles between polarizers; the results are comparedto establish a near-linear sensor response.

3. Calibration of the Fiber Bundle Spectrometer andSpectrum Reconstruction Algorithm

In its operation mode, the fiber bundle spectrometeris illuminated with a test light, which is then spec-trally and spatially decomposed by the fiber bundle(see Fig. 4). The test spectra used in our experimentswere either broadband (halogen lamp source), ornarrow-band (tunable monochromator source). Theimage of the fiber bundle output end was then pro-jected onto a monochrome CCD array using a 2∶1beam compressor. Note that a CCD array does nothave to be a 2D camera; instead, one can use an

Fig. 3. (a) Normalized spectral response of a CCD array.(b) Typical monochromatic near-linear response of a CCD array(λ ¼ 560 nm).

Fig. 4. Setup for the spectrometer calibration measurement.


economical linear array with the number of pixelsmatching the number of Bragg fibers in a bundle.The output of each fiber carries information aboutthe intensity of a certain fraction of an illuminantspectrum as filtered by the individual Bragg fibers.It is reasonable to assume, then, that if the Bragg fi-bers of a bundle feature complementary, while possi-bly overlapping, bandgaps that together cover thespectral range of interest, then the spectrum of anilluminant could be reconstructed.

A. Transmission Matrix Method

To interpret a monochrome CCD image, we use a so-called luminance adaptation model of the fiber spec-trometer. Particularly, for a fixed exposure time of aCCD array, total intensity Cn registered by an as-signed region of a CCD sensor from the output ofthe nth fiber can be presented as

Cn ¼Z λmax

λmin

IðλÞAnFnðλÞSðλÞOnðλÞdλ; ð1Þ

where IðλÞ is the illuminant spectral flux density atthe fiber bundle input end, An is the area of the nthfiber input cross section, FnðλÞ is the transmissionfunction of the nth fiber, SðλÞ is the spectral sensitiv-ity of a CCD array, and OnðλÞ is a fiber-position-dependent transmission function of various optics(diffuser, beam compressor, and a CCD objective).Measuring such transmission functions individuallyis a daunting task. Instead, we use a calibrationprocedure that measures a compounded transmis-sion matrix of our spectrometer, defined as TnðλÞ ¼AnFnðλÞSðλÞOnðλÞ. The model in Eq. (1) could thenbe rewritten as

Cn ¼Z λmax

λmin

IðλÞTnðλÞdλ

¼XN¼ðλmax−λminÞ=Δλ

i¼0;λi¼λminþiΔλTnðλiÞ

ZλiþΔλ

λi

IðλÞdλ ¼XNi¼0

Ii · Tin:

ð2Þ

Experimentally, the integral in Eq. (2) is rather asum over the intensity contributions from smallspectral “bins” of size Δλ. Equation (2) presentedin matrix form becomes

2664C1

..

.

C100

3775 ¼

2664

T1;1 � � � T1;N

..

. . .. ..

.

T100;1 � � � T100;N

3775

2664I1...

IN

3775; ð3Þ

where vector ðCÞ100×1 represents the intensities oflight coming out of the individual fibers as measuredby the CCD sensor, ðTÞ100×N is a spectrometer trans-mission matrix, and ðIÞN×1 is a discretized spectrumof the illuminant. From Eq. (3) it follows that adiscretized spectrum of the illuminant can be

reconstructed from the corresponding CCD image(C vector) by inverting the transmission matrix ofa spectrometer.

B. Calibration Measurement, Building a TransmissionMatrix

To construct transmission matrix experimentally wenote that, if the illuminant spectrum is monochro-matic [Ii ¼ 0, ∀i ≠ iλ in Eq. (3)], then the measuredCiλ vector is proportional to the iλ column of thetransmission matrix:

Ciλð1∶100Þ ¼ Tð1∶100; iλÞ · Iiλ : ð4Þ

To construct the transmission matrix experimentallywe use a tunable monochromator-based narrowband(2 nm FWHM) source to generated “monochromatic”spectra (see Fig. 4). Particularly, we vary the sourcecenter wavelength in 2 nm increments, thus effec-tively subdividing the 400–840 nm spectral intervalunder consideration into N ¼ 221 equivalent 2 nmwide bins. For every new position of the source centerwavelength, we acquire aC vector using a CCD arrayand consider it as the next column of the spectro-meter transmission matrix. Finally, to finish cali-bration, we measure the wavelength-dependentintensities Ii of a tunable source by placing a cali-brated powermeter directly at the output of a mono-chromator. By dividing every C vector by thecorresponding Ii value, the transmission matrix isconstructed.

In our experiments, an S&Y halogen lamp sourcewas used with a Newport Oriel 1=8 m monochroma-tor to build a narrowband tunable source of 2:0 nmFWHM (see Fig. 4). The light beam from the sourcewas then directed onto a diffuser placed right beforethe fiber bundle to guarantee a uniform illuminationof its input end. At the output end of a fiber bundle, a2∶1 beam compressor was used to image the fiberbundle output facet onto the 8 bit monochrome CCDsensor array. Individual images were then inter-preted using a singular value decomposition (SVD)pseudoinversion algorithm to construct the corre-sponding C vectors. The wavelength-dependent in-tensity of a tunable source was measured using acalibrated Newport 841-PE powermeter.

C. Spectral Reconstruction Algorithm

As it was noted in Subsection 3.A, a discretized spec-trum of the illuminant can be reconstructed from thecorresponding CCD image (C vector) by inverting thetransmission matrix of a spectrometer in Eq. (3).However, an immediate problem that one encounterswhen trying to invert the transmission matrix is thatthe matrix is nonsquare. Even if the number ofspectral bins is chosen to match the number of fibersin the bundle, thus resulting in a square transmis-sion matrix, one finds that such a matrix is ill-conditioned. One, therefore, has to resort to anapproximate inverse of a transmission matrix. Tofind a pseudoinverse of a transmission matrix in


Eq. (3), we employ a SVD algorithm. Particularly,from linear algebra, we know that any (100 ×N) ma-trix (suppose that N > 100) can be presented in theform

ðTÞ100×N ¼ ðUÞ100×100ðSÞ100×NðVTÞN×N ; UTU ¼ 1;

VTV ¼ 1; S ¼ diagðσ1; σ2; σ3;…σ100Þ;σ1 > σ2 > σ3 > … > σ100 > 0; ð5Þ

where matrices U and V are unitary, and matrix S isa diagonal matrix of the real positive singular values.For the ill-conditioned matrices, most of the singularvalues are small and can be taken as zero. By limit-ing the number of nonzero singular values to Nσ,Eq. (5) can be rewritten as

ðTÞ100×N ¼ ðUÞ100×NσðSÞNσ×Nσ

ðVTÞNσ×N ;

σ1; σ2;…; σNσ ≠ 0; ð6Þ

and a corresponding pseudoinverse of the transmis-sion matrix is then found as

ðTÞ−1N×100 ¼ ðVÞN×NσðSÞ−1Nσ×Nσ

ðUTÞNσ×100: ð7Þ

4. Experimental Results

To test our fiber bundle spectrometer, we perform re-construction of several test spectra. In one set of ex-periments, the test spectra is a set of the 25 nm widepeaks centered at 450, 500, 550, 600, and 700 nm.Such peaks were created using the same monochro-mator-based tunable source as was used for the spec-trometer calibration; however, it was adjusted tohave a 25 nm bandwidth of the outgoing light. In thesecond set of experiments, we measure the spectra offour 40 nm wide bell-shaped curves centered at 450,500, 550, and 600 nm created by using four commer-cial bandpass filters. In Fig. 5 we demonstrate, withdashed curves, the test spectra of light beams at theinput of a fiber bundle (as resolved by another Orielmonochromator), while with solid curves, we showthe corresponding reconstructed spectra using thetransmission matrix inversion algorithm. In the fig-ures, we also indicate the optimal number of nonzerosingular values used in the matrix inversion proce-dure (7). The choice of the number of nonzero sin-gular values used in the spectrum reconstructionalgorithm affects strongly the quality of the recon-structed spectrum. Note, in particular, that althoughthe spectral intensity function should be strictly non-negative, the reconstructed spectral intensity cantake negative values. Therefore, spectrum recon-struction error can be defined as the ratio of the mostnegative value of the reconstructed spectral intensityto its most positive value:

ErrorðNσÞ ¼ −minλ

ðIreconstr:NσÞ=max

λðIreconstr:Nσ

Þ; ð8Þ

and the optimal Nσ to be used in Eq. (7) is the onethat minimizes the error (8).

From Fig. 5 we conclude that the test peak position(the value of the peak center wavelength) can be re-constructed within several percent of its true value inthe whole spectral range covered by the spectro-meter. Moreover, we note that, although the widthsof the individual Bragg fiber bandgaps are quitelarge (>60 nm), the fiber-bundle-based spectroscopicsystem can resolve peaks of much smaller spectralwidth. Indeed, in Fig. 5(a), the widths of the recon-structed spectra are in the 30–50 nm range, withthe exception of a peak centered at 600 nm. We men-tion in passing that we have also conducted measure-ments using 5, 10, 15, and 20 nm wide peaks. In allthe experiments, we saw that the center wavelengthof even the narrowest peak can be reconstructed withhigh precision; however, the reconstructed peakwidth always stayed around 30 nm. Finally, 40 nmwide test peaks could be well reconstructed, in termsof both their spectral positions and widths [seeFig. 5(b)].

Fig. 5. (Color online) Spectra reconstruction using PBG fiber-bundle-based spectrometer. (a) Reconstructed spectra of six25 nm wide peaks. (b) Reconstructed spectra of four 40 nm widebell-shaped spectra. The black dashed lines are the test spectraresolved by another monochromator; the red solid curves arethe spectra reconstructed by the fiber bundle spectrometer. Thegray area indicates the error level.


5. Spectral Resolution Limit for the Fiber-Bundle-Based Spectrometer

To understand the resolution limit of our spectro-meter, we first study the effect of the choice of thenumber of nonzero singular valuesNσ used in the in-version algorithm of Eq. (7). At first, we assume thatthere is no noise in the system. As a test spectrumIðλÞ, we consider a 20 nm wide peak centered atthe wavelength of 520 nm [dashed curve in Fig. 6(a)].

We then multiply the discretized test spectrum bythe transmission matrix of our spectrometer (mea-sured experimentally), and then reconstruct the testspectrum by using a transmission matrix pseudoin-verse with Nσ singular values:

ðCÞ100×1 ¼ ðTÞ100×NðIÞN×1;

ðIreconstr:NσÞN×1 ¼ ðVÞN×Nσ

ðSÞ−1Nσ×NσðUTÞNσ×100ðCÞ100×1:

ð9Þ

The thus reconstructed spectrum is then compared tothe original test spectrum. In Fig. 6(a), we presentseveral reconstructed spectra for the differentnumbers of singular values used in the inversionalgorithm. If all the 100 singular values are used[curve labeled Nσ ¼ 100 in Fig. 6(a)], then the 20 nmwide peak is very well reconstructed, featuring a less

than 5% intensity difference with the reconstructedspectrum. The error manifests itself in the form ofripples at the peak tails. When a smaller number ofsingular values are used [for example,Nσ ¼ 10 curvein Fig. 6(a)], the center wavelength of the recon-structed peak still coincides very well with that ofa test peak; however, the reconstructed peak widthis larger than that of a narrow test peak. This resultis easy to understand by remembering that the ex-pansion basis set used in the reconstruction algo-rithm is formed by the relatively broad (>60 nm;see Fig. 1) transmission spectra of the Bragg fibers;therefore, one needs a large number of such basisfunctions (singular values) to reconstruct a narrowspectral feature. In Fig. 6(b), the FWHM of the recon-structed peak is shown as a function of the number ofsingular values used in the matrix inversion. Notethat more than 40 singular values have to be usedto achieve the width of a reconstructed peak to beclose to the 20 nm width of a test peak. When onlyten singular values are used, the width of a recon-structed peak increases to 40 nm. Additionally, thepeak reconstruction error as defined by Eq. (8) in-creases to 20% when a small number of singularvalues Nσ < 20 is used. From this we conclude that,with no noise present in the system, the peak recon-struction error and the quality of a reconstructedspectrum improves when a larger number of singularvalues is used.

We now study the effect of experimental noise onthe quality of a reconstructed spectrum. There areseveral sources of noise in our spectrometer. One isthe discrete intensity resolution of a CCD array; withthe 8 bit encoding, the fundamental noise is ∼0:1%.The ambient light in the experimental environmentalso contributes to noise on a CCD sensor. Moreover,because of the relatively large size of a fiber bundle(∼6 mm), it is difficult to ensure the same illumina-tion conditions of the bundle input facet for all thesources used in the experiments.

To model experimental noise numerically, we firstmultiply the test spectrum by the transmissionmatrix of our spectrometer, then add a uniformly dis-tributed random noise of the relative intensity δ tothe C vector (monochrome image), and finally, we re-construct the test spectrum by using a transmissionmatrix pseudoinverse with Nσ singular values:

ðCÞ100×1¼ðTÞ100×NðIÞN×1;

ðCnoiseÞ100×1¼ðCÞ100×1 ·ð1þδ ·ðηÞ100×1Þ;η−random⊂½−0:5;0:5�;ðIreconstr:Nσ

ÞN×1¼ðVÞN×NσðSÞ−1Nσ×Nσ

ðUTÞNσ×100ðCnoiseÞ100×1:ð10Þ

With these definitions, we first revisit the case whenall the 100 singular values are used for spectrum re-construction. As noted above, in the absence of noise[δ ¼ 0 in Fig. 7(i)], the 20 nm peak can be well recon-structed with only ∼5% error. However, the addition

Fig. 6. (Color online) Properties of the reconstructed spectra as afunction of the number of singular values used in the inversionalgorithm. No noise is present in the system. (a) Dependence ofthe spectral shape of a reconstructed peak on Nσ . (b) Width of areconstructed peak as a function of Nσ. (c) Reconstruction erroras a function of Nσ.


of even a small amount of noise [δ ¼ 0:005 in Fig. 7(ii)] results in a highly noisy reconstructed image,with errors as large as 50%. Note that, by using asmaller number of singular values Nσ⊂½20; 40� (seethe error plot in the top part of Fig. 7), the error ofpeak reconstruction can be greatly reduced, to 10%,while still allowing a fair estimate of the peak width∼30 nm. When the noise level is increased further, toavoid large reconstruction errors, one has to use a re-latively small number of singular values Nσ < 20[see Figs. 7(iii,iv)]. Notably, reconstruction error sa-turates at 20% even for large noise levels; however,the reconstructed peak width stays always largerthan 30 nm due to a small number of singular valuesused.

To put the noise value of δ ¼ 0:005 into perspec-tive, we note that this correspond to a 0.5% total ex-perimental error in the amplitude measurement ofthe C vector components. The noises in measuringa C vector include the quantization noise of theCCD sensor, the ambient light contribution fromthe experimental environment, the measurement-to-measurement variation in the illumination condi-tions on the input facet of the fiber bundle, etc. Asnoted before, the contribution of quantization noiseis ∼0:1%. We have verified that the ambient lightnoise is below 1 unit in the 8 bit coding, while theCCD sensor was configured so that the maximumof the fiber output is close to (but smaller than)256 units per pixel. Thus, the ambient light noise

is limited by 1=256 ¼ 0:39%. Thus, δ ¼ 0:005 is basedon the assumption that only ambient light noise andquantization noise exist in the measurement of the Cvector, while for other, higher δ, the other types ofnoise gradually become dominant. We find that thebiggest contribution to noise is caused by the diffi-culty in reproducing the same illumination condi-tions of the fiber bundle facet when constructingthe transfer matrix, and when characterizing the testspectra. The main reason for this is that, to constructthe spectrometer transfer matrix, we use a mono-chromator-based source, while for the test spectra,we use a halogen lamp source. Because of the rela-tively large size of a fiber bundle, it is then difficultto guarantee consistent illumination of the fiber bun-dle surface, even when using a diffuser in front of thefiber bundle. An obvious remedy to this problem is toreduce the size of the fiber bundle and to use a betterdiffuser, or mode scrambler, at the input. To addressthe first problems, we report in Section 6 a new fab-rication technique where a smaller size fiber bundleis fabricated by direct drawing. Additionally, several-meter-long large-core multimode fiber can be usedto first scramble and equalize polarization and inci-dent angle distributions of the input light, and thenlaunch the scrambled light into the fiber bundle.

Finally, Fig. 8 presents statistical averages ofvarious parameters (thick solid curves), as well astheir statistical deviation from the average (thindashed curves) as a function of the noise level. Toconstruct these curves, for every value of the noiseamplitude δ, we first generate 200 realizations ofnoise. Then, for every realization of noise, we plotthe error of peak reconstruction as a function of thenumber of singular values (similar to Fig. 7). Last,the smallest error, the corresponding Nσ, and thewidth of a reconstructed peak are recorded andstatistical averages are performed. In Fig. 8(a), wepresent the average peak reconstruction error as afunction of the noise amplitude. Note that, althoughthe error grows with the noise amplitude, it never-theless saturates at ∼20%, even for large noise le-vels. In the inset of Fig. 8(a), we see that, at verysmall noise levels (δ < 0:02), to obtain the smallestreconstruction error one can use almost all 100 sin-gular values, while at higher noise levels, fewer than20 singular values should be used. As a consequence,for larger noise amplitudes, the width of a recon-structed peak could become considerably larger thanthat of a test peak (20 nm).

6. Novel Technique for Drawing PBG Fiber Bundles

Finally, we report a two-stage drawing technique fordirect fabrication of the PBG fiber bundles. This tech-nique constitutes an industrial-strength alternativeto the manual bundling of the individual fibers, as itwas presented in Subsection 2.B. The proposed tech-nique comprises “two stages.” First, we produce aBragg fiber preform by a co-rolling technique [20]and draw it into intermediate preforms withrelatively large and variable diameters (1–1:5 mm).

Fig. 7. (Color online) Effect of noise on the quality of reconstruc-tion. Examples of the reconstructed spectra for several particularrealizations of noise with amplitudes (i) δ ¼ 0, (ii) δ ¼ 0:005, (iii)δ ¼ 0:05, and (iv) δ ¼ 0:1.


Second, we bundle these intermediate preforms,place them into a thin supercladding tube, and, final-ly, draw this structure into a PBG fiber bundle. Anexample of a cross section of a PBG fiber bundle ispresented in Fig. 9. It is a 3:8 mm diameter fiberbundle that contains ∼100 Bragg fibers featuringcomplementary bandgaps. At the input end ofthe bundle we use white light, which is then splitby the individual fibers into the respective spectral

components, as seen in Fig. 9. We are currently work-ing on the perfection of this fiber bundle fabricationtechnique to include more fibers, and to reduce thebundle outer diameter in order to demonstrate com-pact and high-resolution spectrometers that are alsoresistant to experimental noise.

7. Conclusion

We have demonstrated experimentally a novel all-fiber spectrometer comprising a PBG fiber bundleand amonochrome CCD camera. One hundred Braggfibers with complementary bandgaps covering a400–840 nm spectral range were placed together tomake a ∼6 mm in diameter, 30 cm long fiber bundle.The spectrometer operates by launching the lightinto a fiber bundle and then recording the color-separated image at the fiber bundle output facetusing a CCD camera. An inversion algorithm was de-veloped to reconstruct the test spectrum by usingblack and white intensity images. Among the clearadvantages of our spectrometer is the lack of movingparts, a near-instantaneous and parallel acquisitionof all the spectral components, compactness, a highdegree of integration, and simplicity of operation,as well as a relatively high throughput.

The PBG fiber bundle spectrometer was demon-strated experimentally to resolve relatively narrowspectral peaks (5–40 nm) with high precision indetermining the position of the peak center wave-length. Even when the peak width was much nar-rower than the bandwidth of the individual Braggfibers making up the fiber bundle, the peak widthcould still be resolved correctly; however, the recon-struction algorithm was found to be sensitive to ex-perimental noise. The experimental resolution limitof the fiber bundle spectrometer presented in thispaper is found to be ∼30 nm. Theoretical analysispredicts that the resolution limit of the existing set-up could be greatly improved by (1) minimizing theexperimental errors related to repeatability of lightinjection into the bundle, (2) optical isolation of thebundle from the environment, and (3) minimizingdiscretization errors introduced by the camera. Fi-nally, we demonstrate a two-stage industrial-strength drawing technique suitable for the massproduction of the PBG fiber bundles.

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