n a companion paper1 �referred to hereafter as T1�he way in which a spectrograph mixes spatial andpectral information for a single light source of finitenonzero� dimensions was described. Our main aimn that paper was to show that the spatial informa-ion can be extracted from the entangled data by usef a dedicated linear Bayesian deconvolution filter.n this paper we demonstrate this procedure on aarticularly interesting example, viz., determinationf a two-dimensional �2-D� stoichiometry map of aombustible �but nonburning� methane–air mixturey Raman imaging through a spectrograph. Theechnique described in this paper was also applied tohe spectroscopically easier case of a dry-air flow.2
The stoichiometry of a fuel–oxidizer mixture is onef the key parameters that characterize a combustionrocess. It can be used to predict reaction pathwaysnd the gross behavior of the mixture after igni-ion.3,4 The practical measurement of the local stoi-hiometry tends to be complicated because the localole fractions of at least two different chemical spe-
ies �fuel and oxidizer; both may be one or severalompounds� have to be determined simultaneously.ptical techniques are attractive candidates for ac-
omplishing this because of their nonintrusive naturend their potentially good spatial and temporal res-
The authors are with the Applied Physics Group, Radboud Uni-ersity of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Neth-rlands. N. J. Dam’s e-mail address is [email protected] 24 December 2003; revised manuscript received 16
uly 2004; accepted 27 July 2004.0003-6935�04�305682-09$15.00�0
lution.5,6 For quantitative purposes, Raman-cattering diagnostics has the advantage of atraightforward interpretation of the scattered-lightntensities in terms of molecular number densitiesno quenching correction is required, as it is for laser-nduced fluorescence�.7 Because of its low signaltrength, however, Raman scattering requires highlyelective spectral filtering. A spectrograph is argu-bly the best choice for filtering, because it simulta-eously provides a check on the spectral purity of theecorded data. Direct imaging through a spectro-raph, however, produces a convolution of spectralnd spatial information on the entrance slit andherefore requires use of a postprocessing step to de-onvolve the two. This postprocessing step was theubject of companion paper T1.In Section 2 of this paper we adapt the
onvolution–deconvolution algorithm for imaginghrough an optical multichannel analyzer �OMA; anmaging grating spectrograph equipped with a CCDamera on its exit port�, as described in paper T1, tohe case of multispecies Raman-scattered light. Inection 3 we describe the experimental setup, fol-
owed by a discussion of the Raman spectrum of theethane–air mixture. Subsequently we provide re-
ults on determination of the density field of the rel-vant chemical species and on the stoichiometry fielderived from them. The paper concludes with arief discussion of the prospects for single-shot OMAaman imaging.
. Two-Dimensional Multispecies Raman Imaginghrough a Spectrograph
he measurements discussed in this paper all con-ern quantitative density field determinations based
n Raman scattering in gaseous media. For this
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eason we extend the discussion of paper T1 to thepecific case of OMA Raman imaging. In particular,he identification of the various quantities that ap-ear in the convolution–deconvolution formalism ofaper T1 will receive attention.The specific experimental configuration that we
iscuss is schematically shown in Fig. 1�a�. A thinheet of light �light gray in the figure� illuminates agas phase� sample. Scattered light is imaged ontohe entrance slit of a spectrograph �the field of view isnclosed by a black box rule� and detected by a CCDhip in its exit plane. The system is aligned suchhat the coordinate system that is adopted �x, y� isligned with �pixel columns, pixel rows� of the CCDnd with the �height, propagation direction� of theaser sheet, respectively. Spectral dispersion in thepectrograph takes place along the x direction, ande assume perfect imaging along the y direction.Under monochromatic illumination, the spectro-
raph projects exactly one faithful image of the en-rance slit onto the exit plane, on a location �along x�hat depends on the wavelength of the incident light.nder polychromatic illumination, each wavelength
omponent produces such an image �Fig. 1�b��, whichay or may not overlap others, depending on the
pectral structure of the input �and on the technicalpecifications and settings of the spectrograph, ofourse�. It has been shown that the �spectrally in-egrated� intensity distribution in the spectrograph’sxit plane takes the form of a one-dimensional con-olution of the spectral and the spatial intensity dis-ributions on the entrance slit �see paper T1�; that is,
ig. 1. Schematic experimental configuration. �a� A thin ribbonf light �light gray� illuminates a sample. The probe volume,nclosed by the black box rule, is imaged onto the entrance slit ofn imaging spectrograph with fixed length Ls and adjustable widths. The line of sight is perpendicular to the plane of the figure.
b� Each wavelength component in the scattered light produces oneubimage of the probe volume on the exit plane of the spectro-raph, some of them on the CCD detector chip. �Here we indicatechematically the hypothetical case of five discrete wavelengthomponents, two of which produce overlapping subimages.�
T� x, y� � S�� y, �� � S� xin, y�, (1) l
n which T is the power incident upon pixel �x, y� inhe exit plane and S� and S denote spectral andpatial intensity distributions, respectively, on thentrance slit. The asterisk denotes the convolu-ion of both signals, which involves the x directionnly. In companion paper T1 we described theayesian deconvolution procedure that recon-tructs S�xin, y� from T�x, y�, given spectral struc-ure S�, in the presence of noise in the recorded datanoise is neglected in Eq. �1��. This formalism isailored to the particular situation of Fig. 1 in Sub-ection 2.D below.When several chemical species contribute differentavelength components to the Raman-scattered
ight, a main point of concern is the factorizabilityssumption of Eq. �4� of paper T1, that is, on whetherhe wavelength- and position-dependent scattered-ight intensity distribution can be factored into com-onents that, for each strip, depend only onavelength or position:
S� x, y; �� � S�� y; ��S� x, y�. (2)
ote that the factorizability assumption involvesnly the coordinate in the plane of spectral diffrac-ion. Individual strips can be treated separatelyassuming sufficiently high-quality imaging optics,uch that the point-spread function can be neglect-d�, and we will suppress the y index in what fol-ows. For the time being, the �x, y� coordinates areaken to label both pixels on the CCD and the cor-esponding probe volume element �voxel� in the il-uminated sample. The thickness of the sheet ofight �z direction, along the line of sight of the de-ection system� is assumed to be well within theepth of field of the imaging optics.Even with monochromatic incident light, theaman-scattered light in general contains severalavelength components, depending on the chemical
omposition of the scattering volume. As dis-ussed in paper T1, the spectrograph will producen individual image in the exit plane for everyavelength component. The factorizability as-
umption is justified if the spectrum is sufficientlyparse that the images that are due to individualavelength components do not overlap, or if the
mages do overlap but all the components arise fromhe same spectral source. �These conditions areufficient but not exclusive; other conditions inhich the factorizability assumption is justified ex-
st but are less general, and they are not consideredere.�The spectral power scattered by a small voxel in
he field of view can be written as
P��, x� � �s,i
Ni�s�� x�IL� x�i
�s���� (3)
or every strip y. In relation �3� the first two factorsenote the population �number density� Ni
�s� of a par-icular scatterer �molecule of a specific chemical spe-ies s in a specific quantum state i� and of the local
aser intensity IL, respectively. Both may depend on
osition. The last factor of relation �3� is the totalaman-scattering cross section, which �for everyhemical species and every initial quantum state�ncludes contributions from all possible final quan-um states and contains all wavelength dependence.he summation extends over all chemical species andll initial quantum states, and it is because of thisummation that relation �3� is not automatically sep-rable.In this paper we deal with the specific case of a
ombustible mixture of methane �CH4� and air �N2–2�. The Raman spectra of Fig. 2 �main constituents
isted in Table 1� show that O2 and N2 behave quiteifferently from CH4. Whereas O2 and N2 each giveise to essentially just one scattered wavelength �Qranch5; their width is due to the finite resolvingower of the spectrograph plus a small effect causedy nonzero rotational distortion�, the Raman spec-rum of CH4 is much more complex, with contribu-ions from various vibrational bands as well asotational envelopes.
Two issues now require special attention, viz. �i�he overlap of the fundamental O2 Raman band withne of the CH4 lines �see Fig. 2 at � 375 nm� and �ii�he spectral structure of the main CH4 Raman linesat � 395–400 nm�.
ig. 2. Raman spectra of ambient air and of pure methane �1 bar105 Pa��. Note the logarithmic ordinate; the curves have beenffset for clarity. The spectra were recorded through a spectro-raph with a 1200-groove�mm grating using 355-nm light �tripledd:YAG laser� for illumination. The N2 band and the strongestH4 ��1, �3, 2�2� bands are isolated, but the O2 band overlaps aeak ��2� CH4 band. There is also a relatively small contribution
f water vapor in the ambient air spectrum.
Table 1. Assignments,a Raman Shifts, and Peak Positions ��248�355� ofthe Raman Bands Observed in a Methane–Air Mixture on 248- and
toichiometry � of a methane–air mixture is definedere as
�� x, y� ��CH4�� x, y���O2�� x, y�
��CH4���O2���stoich�
2�CH4�� x, y�
�O2�� x, y�,
(4)
n which the brackets indicate number densities.hus � corresponds to the fuel equivalence ratio.4o determine � we must measure �CH4� and �O2�eparately. From Fig. 2 it follows that CH4 makesn unambiguous contribution to the Raman spec-rum �broad band system centered at 397 nm� but O2oes not: The �2 band of CH4 cannot be spectrallyeparated from the fundamental O2 band. As a re-ult, the determination of �O2� on the basis of theaman intensity at 375 nm becomes ambiguous, es-ecially in fuel-rich regions.There are two solutions to this problem. One is to
orrect the combined O2–CH4 Raman signal at 375m for the CH4 contribution by using the CH4 con-entration that can be determined from the 397-nmand. The other solution is to exploit the fact that,n the nonburning mixture, the �N2�:�O2� ratio is con-tant, say, , so � can also be written as
�� x, y� �2�CH4�� x, y�
�O2�� x, y��
2 �CH4�� x, y�
�N2�� x, y�. (5)
he contribution of N2 to the Raman spectrum is welleparated �with 355-nm illumination� from that ofhe other components, so �N2� can be determined di-ectly from the Raman intensity. We followed theecond approach for the research described in thisaper.
. Temperature
here is one more caveat that needs to be discussed,nd that is the temperature distribution over the fieldf view. Temperature affects the population distri-ution over available quantum states, and, inasmuchs different states possess different Raman-cattering cross sections and give rise to spectralines at different Raman shifts, the shape of the ro-ational envelope of any Raman band will depend onemperature. For Q-branch Raman scattering, as inhe cases of O2 and N2, this will be only a marginalffect �their widths are at the resolution limit of thequipment anyway�, but for extended spectral struc-ures like that of CH4 it may preclude factorization ofelation �3� �depending on the range of temperatureariation, of course�.To estimate the influence of temperature on the
pectral structure we made a �fairly crude� simula-ion of the methane Raman bands in the 400-nmange of Fig. 2 ��1, �3, and 2�2 at an �3000-cm�1
aman shift�. Band strengths were scaled to matchhe experimentally found relative intensities, and in-ividual rotational line strengths were scaled withhe appropriate statistical weights and a Boltzmann
actor. We calculated rotational energy levels in the
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igid spherical top approximation, neglecting the dif-erence in B constants for various vibrational states.
� 5.25 cm�1 was taken throughout, and only thehree strongest branches of the �3 band were takennto account.8 A Gaussian instrumental linewidthf 10 cm�1 �0.16 nm� was assumed. Figure 3 showshe calculated band contours for temperatures rang-ng from 200 to 400 K. Evidently there are only
inor differences in the spectral envelopes over thisemperature range �note the logarithmic ordinate�ecause the strongest contributions to the spectrumll arise from Q branches. We conclude, therefore,hat the assumption of constant temperature isardly restrictive for vibrational OMA Raman-cattering experiments of flows near room tempera-ure.
The factorization of Eq. �2� can thus be completedor all individual Raman bands, resulting in
S���� � �i
i���gi exp���Ei�kB T��
Z, (6a)
S� x� � N� x�IL� x��t, (6b)
n which �t denotes the recording time �gate width�,is the partition function, gi is a degeneracy factor,
B is Boltzmann’s constant, and T is the absoluteemperature. Ei is the energy of initial �rovibra-ional� quantum state i.
. Spectral Width of the Main CH4 Band
he strongest feature in the Raman spectrum of CH4,397 nm, is in fact a complex structure that involves
hree Raman-active vibrational modes �Table 1�.ffectively, therefore, this Raman band has a widthf approximately 5 nm �with 355-nm incident light�.hen the entrance slit of the spectrograph is broad-
ned, the spatial structure is entangled with this
ig. 3. Simulated methane Raman spectrum near a 3000-cm�1
aman shift under 355-nm illumination. See text for details ofhe calculation. The three spectra correspond to three differentemperatures. Note the logarithmic ordinate.
pectral profile according to the convolution integral fi
f Eq. �8� of paper T1. For the calculation of thetoichiometry distribution we must reconstruct thepatial distribution from such a convolved image.his corresponds to a deconvolution of the raw dataith the spectral distribution.
. Formalism
o write the equations derived in paper T1 foronvolution–deconvolution, we must in the remain-er of this section distinguish between coordinates inhe entrance plane �xin� and in the exit plane �xout�.he entrance slit width of the spectrograph ispanned by coordinate xin, and the y coordinate isarallel to the grooves of the grating. Including this-D extension and the factorization, the convolutionEq. �8� of T1� for the specific case of 2-D Ramanmaging reads as
T� xout, y� � �xin
�� x�, y�
� ��i
i� x��gi exp���Ei�kB T��
Z �� N� xin, y�IL� xin, y��tdxin, (7)
n which x� � ��xout � xout,0� � Ms�xin � xin,0����.ote that, although is a function of wavelength
nly, the spectrograph introduces an implicit depen-ence on xin �Eq. �7� of paper T1�. To simplify thisquation we can reintroduce the �single-species� spec-ral reference function �Eq. �11� of paper T1� as theesult of a narrow-slit reference measurement inhich a known number density Nref�y� is illuminatedith laser intensity IL,ref�y� during �tref, yielding
R� x�, y� fs � �� x�, y�
� ��i
i� x��gi exp���Ei�kB T��
Z �� Nref� y�IL,ref� y��tref, (8)
here fs accounts for the fact that the infinitesimallyarrow entrance slit has a finite width in practice, asiscussed below. �The extent to which the temper-ture of the reference measurement may deviaterom that of the actual imaging experiment dependsn the sensitivity of the spectral contour to temper-ture, as discussed above.� For every individualpecies, therefore, one can write the correspondingaman signal, by combining Eqs. �7� and �8�, as
T� xout, y� � �xin
R� xout � Ms� xin � xin,0�, y�
� ��R�fs
N� xin, y�
Nref� y�
IL� xin, y�
IL,ref� y�
�t�tref
�dxin.
(9)
n Eq. �9� we split the measured spectral referenceunction into a normalized part �R, the spectral pro-
le�, and its norm ��R�, the integrated intensity�. If
aw data T are deconvolved with normalized part R,he result �the whole term in the second set of brack-ts� will obviously be given in the same units as theeasured data, T. On postnormalization of this re-
ult with �R�IL�t�� fsLL,ref�tref�, we express the de-ired number density distribution N in units of theeference distribution Nref. Evidently, cross sec-ions i and the Boltzmann factor have disappearedrom this formula. As previously for grating effi-iency � and grating constant �, they have been in-orporated into spectral reference distribution R. Inhis paper deconvolutions are performed with theormalized spectral reference function.Factor fs is a property of the specific OMA that is
sed in any OMA Raman imaging experiment. Itepends on ds, the width of the entrance slit. In aractical situation the image of the entrance slit isimited by the resolution of the CCD. The hypothet-cal infinitesimally narrow entrance slit of Eq. �9� ofaper T1 is obtained experimentally for slit widthsor which the image of the entrance slit no longerepends on ds. This is the regime in which the im-ge is below the resolution of the CCD, for whichurther decrease of ds leads to a loss of intensity only.herefore it is important for what ds the spectraleference function is recorded. This dependence ons is accounted for explicitly by factor fs. Note that
s�ds� is the same for all species if their referencepectra are recorded at the same setting of ds.For a certain measurement T � t the best recon-
truction of the molecular distribution as it was im-ged onto the entrance slit of the spectrograph isiven by
N� xin, y� � �c� y�����2 � bRk�0� y�
Rk�02 � y� � ����2
� FT�1� R*k,n0� y�tk� y�
�Rk,n0� y��2 � ����2��gref, (10)
ith
gref �Nref� y� fs
�R� y��IL,ref� y�
IL� xin, y�
�tref
�t,
ccording to the linear Bayesian deconvolution filterf Eq. �32� of paper T1. As in that paper, boldfaceymbols represent entire pixel rows of data. Equa-ion �10� provides an algorithm with which to recon-truct N�y� in units of gref. If the referenceolecular distribution is a uniform one, Nref�y� is
qual for all rows.Whether the deconvolved result is automatically
orrected for laser sheet inhomogeneities depends onhe experimental geometry. If the laser beam prop-gates parallel to the xin coordinate �i.e., in the planef spectral diffraction�, the xin dependency of the laserntensity, IL�xin, y�, will disappear in the absence ofttenuation. Consequently, laser intensity ratioL,ref�y��IL�xin, y� cancels any illumination inhomo-eneity in the reconstructed image. If, however, the
aser radiation propagates parallel to the grooves of t
he grating, that is, perpendicular to the dispersionoordinate, laser-sheet inhomogeneities expresshemselves as the dependence of IL on xin. In thexperiments discussed in this paper the experimentalonfiguration is such that the sheet of light propa-ates parallel to the grooves of the grating, and scat-ering and absorption losses in the medium areegligible. Because, however, reference spectra forethane could not be determined over the whole
ange of y, only one generic reference spectrum wassed to deconvolve all pixel columns. Thus the de-onvolved Raman images are not automatically cor-ected for laser-sheet inhomogeneities.
. Experiment
he experimental setup is shown schematically inig. 4. A laminar flat flame burner developed at theniversity of Eindhoven10 was used as the source ofpremixed methane–air mixture. The flame was
tabilized on a metal mesh at a distance of �12 mmbove the burner surface. Methane and dry air werebtained from gas bottles; the dry air was a 78.1:20.9ixture of N2 and O2 �that is, � 3.73 in Eq. �5��.he stoichiometry could be adjusted by means of cal-
brated mass flow controllers �MFC; Bronkhorst Hi-ec� in both gas supply leads.The most appropriate illumination wavelength is a
rade-off among Raman scattering cross sectionwhich scales with ��4�, detection efficiency �whichepends on the detector�, and, for this case, the sep-ration of the Raman bands of the relevant chemicalpecies. As the spectrograph available for these ex-eriments could not sufficiently separate the spectraltructure of N2 from the �weak� 2�4 CH4 line on48-nm illumination, we used a tripled multimoded:YAG laser �Spectra-Physics QuantaRay 250-10: 355 nm, 0.32-J�pulse, 5-ns pulse duration�. Its
ircular beam was collimated by two positive cylin-rical lenses into a sheet of approximately 10 mm �.1 mm �height � thickness �depth of field��. Scat-ered light was detected perpendicular to the plane ofhe sheet of light and imaged by a camera lens �NikonV-Nikkor 105 mm, f�4.5� onto the entrance slit of an
maging spectrograph �ARC SpectraPro 300i, with a400-grooves�mm UV-blazed grating�, as sketched at
ig. 4. Experimental setup; see text for details. �c� The rectan-le shows an artist’s impression of the expected stoichiometryistribution over the field of view of the detection system. Theoordinate system used in the discussion is shown.
he left in Fig. 4.
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The polarization of the incident laser light wasotated to be in the plane of the sheet of light toaximize the Raman-scattering efficiency. The
pectrograph was oriented such that the grooves of itsrating were parallel to the propagation vector of theaser sheet. It has an entrance slit of length Ls � 30
m �Fig. 1� and a variable width �ds along xin, theimension of diffraction� up to 3 � 10 mm. In thexperiments the laser-sheet height �10 mm� was im-ged to more-or-less match the maximum width ofhe entrance slit �upper right part of Fig. 4�. Theetup was found to be sensitive to the polarization ofhe scattered light,11 and the detection efficiencyould in principle be improved by use of a ��2 plate inront of the imaging optics to optimize the polariza-ion of the incident light. Because, however, a ��2late with sufficiently large aperture was not avail-ble, we did not use this opportunity in the experi-ents reported here.An intensified CCD �ICCD� camera �Fig. 4; Prince-
on Instruments ICCD-512T� collected the images athe exit port of the spectrograph. The OMA con-tructed in this way was aligned such that its field ofiew comprised both ambient air �� � 0� and the pre-ixed flow �� as set by the mass flow controllers�; an
rtist’s impression of the stoichiometry distributionhat would be expected for this case is included in Fig.
�bottom right�. With a narrow entrance slit �100m� the spectral resolution was approximately 0.033m�pixel �xout dimension� and the spatial resolutionmounted to 209 �m�pixel �in the y dimension� and,fter deconvolution, 184 �m�pixel in the x dimension.
. Results and Discussion
. Average Stoichiometry
o test the performance of the deconvolution proce-ure we determined the stoichiometry distribution onhe basis of an average over 25 accumulations of 250aser pulses each �well over 10-min measurementime in total�. The raw data, taken for a stoichio-etric methane–air mixture, are shown in Fig. 5�b�;
he coordinate system used is indicated in Fig. 4. Asntroduced in Eq. �1�, an asterisk refers to the convo-
ig. 5. Raw data Raman OMA graphs: �a� narrow slit measure-ent of ambient air; �b� broad slit measurement of a methane–
ry-air flow at nominally stoichiometric conditions �for the field ofiew see Fig. 2�; �c�, �d� narrow slit measurements of a pure meth-ne flow. All spectra are on different linear scales; image �b� is onlinear gray scale �white, low; black, high intensity�.
ution of spatial and spectral information and x � xout t
enotes the spatial axis at the exit port of the spec-rograph. Two bands are distinguished in the rawata of Fig. 5�b�; the upper one is due to N2 �presentn the flow as well as in ambient air� and the lowerne is due to CH4 �present only in the flow�. Bothands are truncated at the right �the N2 band also athe left� by finite entrance slit length Ls of the spec-rograph; the slit width was set to ds � 3.10 mm.he signal-to-noise ratio �S�N� can be defined as theatio of the average pixel value �in a certain region� tohe standard deviation �1�, assuming uniform gasensity and illumination intensity. It was deter-ined for the two bands over a probe area of 100 � 20
ixels located within the main flow, resulting in a�N of 11 in the N2 band and a S�N of 12 in the CH4and. Part of the structure along the xout directions due to laser-sheet inhomogeneity. This shows upn both the N2 and the CH4 bands but will cancel inhe calculation of � �Eq. �5�� because of the linearependency of the Raman signal on laser power den-ity IL �Section 2�. Besides this, the CH4 band showsdistinct wing extending to the red �bottom of theMA graph in Fig. 5�b��, which is due to the structuref the Raman spectrum �Fig. 2�.Both the N2 and the CH4 images need to be decon-
olved with a proper spectral reference function, asiscussed at the end of Section 2. To this end,arrow-slit spectra �ds � 0.10 mm� were taken formbient air and for a pure methane flow �Figs. 5�a�,5 � 71 laser pulses, and 5�c� and 5�d�, 25 � 5 laserulses, respectively� for the same setting of the grat-ng, that is, equivalent xout axes. As only the right-and part of the imaged region was covered with aow out of the burner, it was not possible to record aH4 spectrum for every longitude. Therefore the
eference spectra RN2fs �ds � 0.10 mm� �Fig. 5�a�� andCH4fs �ds � 0.10 mm� �Fig. 5�c�� are averages over aarrow range in the flow. These spectra serve as thenonnormalized� species-specific spectral referenceunctions for deconvolution of the OMA graph in Fig.�b�. If the species-specific parts of this figure, indi-ated by the I-bars in Fig. 5�b�, are isolated and de-onvolved with their corresponding spectral referenceunctions, the reconstruction shown in Fig. 6�c� re-ults. This image shows the best reconstructions ofhe individual N2 and CH4 intensity distributionsbtained with recentered, normalized spectral refer-nce functions. Therefore the reconstructed data re-ain in the position of the raw data �for N2 as well as
or CH4�. Because the reconstructions were notostnormalized for reference conditions, the graycale is the same intensity scale as that for the rawata �see the discussion below Eq. �9��. Note thattructure that is due to the 2�4 CH4 band �betweenhe N2 and the main CH4 images� is discarded byudicious choice of the processed regions �I-bars� inig. 5�b�.In comparison to the raw data �depicted also in Fig.
�a� for comparison�, in the reconstructed data theontrast in both bands has increased. The bound-ries have become sharper �as they should, because
pectral tail below �to the red of � the CH4 band has allut disappeared, as can be seen more clearly fromig. 6�b�, which shows a vertical cross section that isne pixel wide through Fig. 6�c� �location of theighter arrow at the top of Fig. 6�c� and lighter curven Fig. 6�b��. For comparison, the correspondingross section through the raw data of Fig. 6�a� isncluded as well �darker curve in Fig. 6�b��. Thetructure along the xin direction that remains in theeconvolved N2 image is due largely to intensity vari-tions in the laser-light sheet. The CH4 band showshe same variation �one of the advantages of using apectrograph: both bands are really measured si-ultaneously�, but in addition there is some residual
tructure that is due to incompletely removed spec-ral features of CH4, similar to the ghost images inig. 6�b� of paper T1.Postnormalization of the deconvolution result au-
omatically takes into account the different Ramancattering cross sections of N2 and CH4. Conse-uently, the pixel values in Fig. 6�c� can be consideredelative partial number densities with respect toref
s�y�. A minor complication arises from the facthat the reference spectral distributions for N2 wereeasured for dry air rather than for pure N2. Most
onveniently, all number densities in the variousows can be expressed in terms of the number den-ity of N2 in dry ambient air, �N2�amb. The flow, withn exit speed of the order of 24 cm�s, is treated asncompressible; that is,
�O2�� � �N2�� � �CH4�� � �O2�amb � �N2�amb
� � 1
�N2�amb, (11)
or any composition of the methane–dry-air mixture,
ig. 6. �a� Raw data and �c� deconvolved Raman OMA graphs ofmethane–dry-air flow at nominally stoichiometric conditions �for
he field of view see Fig. 3�. Both images contain a N2 �top� and aethane �bottom� contribution, which are truncated at the left and
he right by the spectrograph entrance slit length. The verticalxis of �c� contains the label �wavelength ��, as it is purely spatialnly within the N2 and CH4 reconstructions �see text for compu-ational details�. Vertical cross sections through the images of �a�raw data� and �c� �reconstruction� at the locations indicated by therrows are plotted in �b�. All images are on the same linearntensity scale.
ncluding the limiting cases of pure methane ��3 �� p
nd dry air �� � 0�. For a premixed flow of stoichi-metry �, the N2, O2, and CH4 concentrations read as
�N2�� �2� � 1�
2 � 2 � ��N2�amb, (12a)
�O2�� �2� � 1�
�2 � 2 � ���N2�amb, (12b)
�CH4�� �� � 1��
�2 � 2 � ���N2�amb. (12c)
n particular, the reference concentrations for thepectral reference function read as
�CH4�ref � � 1
�N2�amb, �N2�ref � �N2�amb (13)
or pure methane and for ambient air, respectively.fter normalization to this ambient nitrogen concen-
ration and correction for the number of laser pulses,he �N2� and 2 �CH4� �with � 3.73� images are ashown in Fig. 7 �the deconvolution is performed withhe normalized spectral reference functions at theirriginal positions, i.e., not recentered�. Both nowurely spatial patterns are centered vertically in themages: The deconvolution automatically ensureshe correct spatial alignment �a property of Fourierransforms�. Furthermore, both images are on theame gray scale in units of ��N2�amb fs�ds � 0.10 mm�
L,ref�y��IL�y��. This unit cancels on a pixel-by-pixelivision of the two images, resulting in the dimen-ionless 2-D stoichiometry distribution, according toq. �5�.Figure 8 shows stoichiometry distributions derived
rom 625 single-shot accumulations at various presetixing ratios. At the left the experimental condi-
ions are indicated �set value for �; number of laserhots �#� for the raw data acquisition�, and at theight the spatially averaged stoichiometry as deter-ined from the images is listed �average and one
tandard deviation over a rectangular region of 104
ig. 7. Results of deconvolution of the raw data �Fig. 4�b�� accord-ng to Eq. �10� with RN2 �Fig. 4�a�� for �N2� and with RCH4 �Fig. 4�c��or �CH4�. The latter was postmultiplied by 2 . Both images arei� on a purely spatial scale in both dimensions; �ii� on the sameray scale; and �iii� in units of �N2�ambIL,ref �y� fs�ds � 0.10 mm��L�y�, that is, still dependent on the laser-sheet inhomogeneity.or clarity the error bars indicate the parts that were selected inig. 4�b� for species-specific deconvolution as well as the actualidth of the entrance slit.
ixels in the central part of the flow, indicated by the
dr1rcitnetfimtfi
B
Twpsao�mb
tt
imfslsbpormiS1ss1otflpv
tbctciluitaesl2s
Fllad
Fc�
Fnat
ashed rectangle in Fig. 8�a��. Clearly, the averageseproduce the nominal values very well, with about0% statistical error. Part of the error is due to theesidual �spectral� structure in the images, as dis-ussed above �the horizontal streak pattern in themages�. For this reason, increasing the data collec-ion time by a factor of 10 �cf. Figs. 8�b� and 8�c�� doesot improve the S�N by a factor of �10, as would bexpected for shot-noise-limited data. Note also thathe fanning out of the flow near the upper limit of theeld of view, where the flow bumps into the metalesh, can clearly be seen. �The corresponding fea-
ure at the right-hand side of the flow falls outside theeld of view; see Fig. 4.�
. Snapshot Stoichiometry
he full advantage of imaging through a spectrographould be exploited if signal levels were high enough toermit accurate stoichiometry determinations withingle laser shots. An example of such a single-shotcquisition is shown in Fig. 9 for nominally stoichi-metric conditions ��N2���1 � 0.90�N2�amb andCH4���1 � 0.12�N2�amb�. Although the S�N of Fig. 9
ay be less than what is achievable with 2-D opticalandpass filtering, the merit of 2-D OMA imaging is
ig. 8. Average stoichiometry distributions, derived from imagesike those of Fig. 6. Experimental settings are indicated at theeft; the spatially averaged stoichiometries derived from the im-ges are listed at the right �region involved indicated in �a��. Alleconvolutions were performed for �� � 6 counts�1.
ig. 9. Single-shot Raman image from a flow in nominally stoi-hiometric conditions. Linear, inverted gray scale from 0 � I
tcounts� � 150.
hat all species are imaged simultaneously for exactlyhe same spatial region.
For the current experimental setup �illuminationntensity as much as 60 MW�cm2�, we have deter-
ined the statistics of stoichiometry distributionsor averages over a number of single-shot imagesuch as Fig. 9. After this averaging, the deconvo-ution algorithm was applied as discussed in Sub-ection 4.A, followed by division of the CH4 imagey the N2 image. Each of the stoichiometry imagesresented below is in fact the worst-case averagever a number of laser shots �owing to the CCDead-out noise� recorded on a nominally stoichio-etric �� � 1� flow. Part of the results are shown
n Fig. 10, which shows the distributions and the�N values that result from averages over 1, 10,00, and 625 single-shot images. In the presentetup, obtaining a stoichiometry distribution with apatially averaged S�N of 1 requires fewer than0 laser shots. This performance increases a S�Nf 10 when more than 250 laser shots are used. Inhe single-shot image the regions with and withoutow are recognizable, but the statistics are veryoor and do not allow one to estimate stoichiometryalues accurately.There is, however, room for improvement now that
he major obstacle �the deconvolution procedure� haseen tackled. One can increase the count rate by in-reasing the laser fluence and by increasing the detec-ion efficiency. Obviously, the laser pulse energyould be increased but such an increase would be lim-ted by the occurrence of optical breakdown. Also, theaser could be multipassed through the probe vol-me.12,13 On the detection side, the efficiency is lim-
ted by the collection angle of the spectrograph � f�4 forhe ARCSpectraPro 300i�. Improvement could bechieved by binning pixels on the CCD chip, at thexpense of spatial resolution. Optimization of theetup with respect to the polarization of the scatteredight promises a potential gain of at most a factor of.5.11 Finally, there are prospects �currently undertudy� to improve the deconvolution procedure further,
ig. 10. Stoichiometry distributions determined from differentumbers of laser shots. Spatial averages and standard deviationsre determined in the dashed rectangular region �4050 pixels� ofhe stoichiometry for 625 laser pulses. See text for discussion.
aman imaging through a grating spectrograph isseful for determination of two-dimensional stoichi-metry distributions. The main advantage of theethod is the really simultaneous determination of
uel and oxidizer distributions over exactly the sameeld of view and under exactly the same experimen-al conditions. Moreover, the spectrograph is a per-anent monitor of the spectral purity of the data.he raw data, which consist of a convolution of thepectral and spatial intensity distributions in thepectrograph entrance plane, can be quantified aseparate, purely spatial distributions of both fuel andxidizer through the application of a dedicated Bayes-an deconvolution filter. These patterns directlyield the stoichiometry field. The signal-to-noise ra-io is limited because of the low Raman scatteringross sections. Spatial averages of the stoichiometryn uniform parts of a methane–air flow reproduce theet values with a statistical error below 10% when50 or more single-shot images are averaged.
We appreciate the stimulating discussion of thisubject with Michael Golombok. This research wasade possible by financial support from the Technol-
gy Foundation, the Applied Science Division of theetherlands Organisation for Scientific Research,nd the technology program of the Ministry of Eco-omic Affairs.
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