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in the last decade surface enhanced raman scattering ( sers)@xcite has made rapid progress towards applications . with a sensitivity rivaling fluorescence in some cases , and a much higher structural specificity , sers is a highly attractive technique , being developed simultaneously with the field of plasmonics . progress towards different uses of sers in practical applications has been steady . new substrates including : arrays of inverted pyramids@xcite , silver pillar and torroid arrays@xcite , adaptive silver films@xcite , and metallic nano - shells@xcite , have been demonstrated . in addition , many molecules relevant to a myriad of applications such as glucose@xcite , proteins@xcite , dna@xcite , a wide range of medicinal drugs@xcite , and substances for forensic science@xcite , have been characterized . on the other hand , the understanding of some fundamental aspects of the phenomenon is still incomplete , and in some cases controversial . although a full understanding is sometimes not essential for the development of applications , there can be no denying the fact that better comprehension is desirable . optical pumping of vibrational modes was first suggested in 1996 based on the observed dependence of the anti - stokes / stokes ( as / s ) ratio with incident laser power@xcite , and it was suggested that this could be a possible tool to estimate sers cross sections . however , the interpretation of the experimental results have been the subject of considerable debate in the literature , with many authors simply denying its existence and attributing the experimental observations to either laser - heating , resonance effects , or combinations thereof@xcite . over a series of previous papers we have suggested the investigation of the as / s - ratio as a function of temperature @xmath0 as an alternative method for the observation of pumping@xcite . in a recent paper , we provided definitive evidence for the existence of vibrational pumping under sers conditions thereby bringing to conclusion one aspect of this complex problem@xcite . here we address another aspect , namely the problems associated with estimating the sers cross section using this method , and we extend our analysis to include different analytes and laser excitations . we also highlight several outstanding issues which were not considered previously@xcite . obtaining good estimates of sers cross sections is in general an extremely hard problem and has been a longstanding ambition of the sers community for reasons that go from the purely applied ( quantification of signals ) to the more fundamental ( comparisons of theoretical electromagnetic enhancement factors with experiment ) . any method that can produce a standard protocol for the estimation of sers cross sections is , accordingly , of great interest and an effort to understand its principles and limitations is required . before we go into the details , we briefly review the necessary concepts for understanding as / s - ratios and vibrational pumping in sers . the method we are going to present has not been widely used and therefore we review the historical background and also its peculiarities and limitations . the discussion has some natural overlap with our previous papers@xcite but it is presented here for the sake of completeness , the convenience of the reader , and future reference for forthcoming work in progress . this will inevitably result in a somewhat lengthy introductory section where the principles of the method are laid down . the next sections are fully devoted to this and are followed by a section at the end with a practical experimental demonstration of sers cross section determination for several standard probes . let us consider the different contributions to the population of a single vibrational level at temperature @xmath1 during a sers measurement ; we can identify two main contributions : @xmath2 the laser itself which pumps vibrations through raman processes with a rate proportional to its intensity @xmath3 and to the raman - stokes cross section @xmath4 , and @xmath5 thermal excitation and relaxation . vibrations remain in the level with a finite lifetime @xmath6 , which encompasses all possible relaxation mechanisms such as intramolecular vibrational relaxation ivr ( anharmonic processes ) or external relaxation mechanisms . there are other possible secondary mechanisms , such as relaxation through an anti - stokes raman processes , or excitation to higher vibrational levels . it is possible to write rate equations for the detailed dynamics of vibration populations in such a system @xcite . in the regime of weak pumping ( which is the only case considered here ) , where the vibrational population remains small , @xmath7 , it is sufficient to consider only the two main mechanisms , and the rate equation for @xmath8 can then be written as : @xmath9 where @xmath10 is the raman - stokes cross section , @xmath11 the intensity ( power per surface area ) of the laser , @xmath12 the energy of an exciting photon ( @xmath13 is the number of incident photon per unit time and surface area ) , and @xmath14 the energy of the vibration . the first term on the right is the number of vibrations per unit time being pumped into the level by the action of the laser , while the second and third terms are the contributions of thermal excitation and population relaxation , respectively . in the steady state @xmath15 . generally , @xmath10 is very small , so when @xmath11 is small the pumping contribution is negligible and the vibrational population is dominated by thermal effects ; i.e. @xmath16 is given by a boltzmann factor . when @xmath11 is increased ( so that there is a significant pumping contribution to the population of the level ) @xmath8 becomes : @xmath17 where the pumping term is clearly distinguished from the thermal contribution . in addition , for an ensemble of @xmath18 molecules , the stokes - raman signal is given by @xmath19 while the anti - stokes signal is given by @xmath20 , leading to : @xmath21 moreover , taking the ratio of the anti - stokes to the stokes intensities @xmath22 , we have : @xmath23 where @xmath24 is the _ asymmetry factor _ ( extensively discussed in refs . @xmath24 includes not only any possible difference between anti - stokes and stokes cross sections arising from resonance effects ( due either to plasmon resonances or to resonant raman scattering for a resonant analyte ) , but also the standard wavelength dependence of raman processes @xmath25 . @xmath26 can be expressed as : @xmath27 . \label{pumpratio}\ ] ]    this simple model shows that the stokes - raman signal always remains linearly dependent on @xmath11 . the anti - stokes signal also shows a linear dependence with power when pumping is negligible . but when pumping dominates over thermal effects , then @xmath28 varies quadratically with power while the as / s - ratio is linearly dependent on @xmath11 . the original work by kneipp et . al.@xcite used rhodamine 6 g ( rh6 g ) and crystal violet ( cv ) ( two commonly used sers active dyes ) under @xmath29 excitation at room temperature ( rt ) . the argument for pumping was primarily based on two observations : ( i ) the as / s - ratios were shown to be larger than expected for a boltzmann factor , and ( ii ) the _ power dependence _ of the signal at rt ( where the thermal contribution is likely to dominate , or be important ) was quadratic . this observation of a non - linear dependence of the anti - stokes intensities with power ( resulting in a linear dependence in the as / s - ratio ) was thought to provide strong evidence for vibrational pumping in sers . further measurements on carbon nanotubes@xcite and a dna - base@xcite supported these initial results . the arguments against this original interpretation are summarized in the following subsection . haslett et . al.@xcite were the first to seriously question the existence of vibrational pumping as revealed in the original studies . extensive measurements were made under similar conditions with a number of both resonant and non - resonant molecules . they observed an anomalous ratio for all the resonant molecules tested which was independent of power until photo - bleaching occurred . no anomaly was observed in the case of the non - resonant molecules . it was concluded that the anomalous ratios observed in the original papers were the result of `` hidden '' resonances rather than pumping@xcite . al.@xcite also concluded that the observed anomalous ratio could be explained by resonances . these resonances are accounted for by the asymmetry factor @xmath24 in the model of the previous section . in the absence of pumping , the as / s - ratio is predicted to be : @xmath30 it is clear that the @xmath24 factor can indeed result in anomalous ratios ( different from the boltzmann factor ) , even in the absence of pumping . furthermore , it was suggested that the observed power dependence was not the effect of pumping but rather laser heating . the effect of heating on @xmath26 in the absence of pumping can be simply understood by including it in eq . ( [ ratio - corrected ] ) . the real temperature of the probed molecule , @xmath31 , may be different to the nominal temperature @xmath1 because of heating effects , either in the sers substrate or in the molecule itself @xcite . by writing @xmath32 and assuming that @xmath33 we can obtain the corresponding ratio , @xmath34 , by expanding eq . [ ratio - corrected ] as : @xmath35 this expression first shows that heating can also result in an anomalous as / s ratio , even when @xmath36 . moreover , it can have an important impact on the power dependence of @xmath26 , because @xmath37 should increase with @xmath11 . we can assume in a first approximation that @xmath38 , since heat diffusion and transfer are linear problems . the above expression shows that laser heating should then result in an exponential increase of @xmath34 with @xmath11 . however , in many cases of interest , the argument in the second exponential is small compared to 1 , and eq . ( [ ratio - x ] ) can then be further expanded to give : @xmath39 a linear dependence of @xmath26 with @xmath11 ( or equivalently a quadratic dependence of @xmath28 with @xmath11 ) _ can then equally be the result of conventional heating effects or pumping_. such an observation is therefore insufficient to demonstrate the presence or not of vibrational pumping . we also note that heating effects can also explain the mode - dependent behavior observed in the original work @xcite , as the argument of the second exponential in eq . ( [ ratio - x ] ) depends on the mode energy @xmath14 . most studies have concentrated on finding evidence for sers pumping at room temperature ( rt ) , in particular by studying the power dependence of the as / s ratio . it is true that the huge enhancements in sers conditions greatly increases the contribution from pumping , but may also contribute to an increased heating ( in particular directly in the probe molecules ) . at room temperature ( rt ) and above , the thermal contribution to @xmath26 is relatively large and in many cases dominates , such that @xmath40 . the study of vibrational pumping at rt therefore involves measuring small departures from an already existing ( large ) thermal population . moreover , as the previous discussion has shown it is also extremely difficult to distinguish the relative contributions of pumping and heating at rt . a more practical approach is to study the temperature dependence of the as / s ratios , and in particular the low - temperature regime where thermal effects are expected to be completely negligible compared to any other mechanisms , and in particular pumping . the temperature dependence can be directly studied using the model presented so far . figure [ schmratios ] summarizes the different scenarios for as / s - ratios as a function of temperature . the solid lines show the variation of the ratio with @xmath1 when pumping is absent ; i.e. an exponential decrease as @xmath1 decreases , following the boltzmann factor @xmath41 . the dashed lines show the case when pumping is present . at high @xmath1 s the ratio is approximately the same as if pumping were absent . this is the _ thermally - dominated regime_. as @xmath1 is decreased , there is a cross - over to a _ pumping - dominated regime _ , where the ratio reaches a plateau . we refer to the cross - over temperature between these two regimes as t@xmath42 ; indicated by arrows in fig . [ schmratios ] . the cross - over occurs when pumping and thermal terms in eq . ( [ pumpratio ] ) are comparable , which occurs for : @xmath43^{-1}\ ] ] the cross - over occurs at a larger temperature for higher powers @xmath11 , for which pumping is stronger , but also for higher energy peaks ( with a larger @xmath14 ) , for which thermal excitation is weaker .    in the pumping - dominated regime , the as / s - ratio is constant and equal to @xmath44 . @xmath45 can be estimated to a high degree of accuracy for a given experimental setup . if @xmath46 ( no asymmetry between @xmath47 and @xmath48 ) we can therefore deduce the product @xmath49 from the plateau in the as / s - ratio below @xmath50 . in general , and in particular under sers conditions , @xmath51 , but its value can be determined from the exponential dependence in the thermally - dominated regime ( above @xmath50 ) . in practice , a fit of the experimental data over the whole temperature range with two parameters enables us to determine both @xmath24 and @xmath52 @xcite . nevertheless , the problem of estimating @xmath6 still needs to be circumvented to determine the sers cross - section @xmath10 itself .    by reducing @xmath1 , the contribution from pumping becomes dominant , making measurements much easier , and more reliable . it also simplifies greatly their interpretation . but the main advantage is that it enables one to unambiguously rule out heating effects as an alternative explanation . as discussed , the appearance of plateaus for @xmath53 in fig . [ schmratios ] is a clear characterization of the pumping - dominated regime . note that it does not mean that heating is absent , but simply that its effect on as / s - ratios is negligible ( relative to the pumping contribution ) . any heating effects may possibly affect the value of @xmath50 , but as long as a plateau is observed , vibrational pumping must occur . moreover , the value of @xmath26 in the plateau region ( from which we will infer sers cross - sections ) is independent of heating . studies of the temperature dependence of @xmath26 therefore provide a more conclusive demonstration of sers pumping . however , the technique is not without its drawbacks . in order to extract sers cross - sections for the method discussed here there are several issues that need special attention as described below . several issues need to be considered in the practical determination of sers cross sections via vibrational pumping . not all of them can be resolved satisfactorily and we have to include some _ ad - hoc _ approximations in order to obtain estimates of the cross sections . while there is nothing intrinsically wrong with this , it is necessary to be aware of the range of validity of the estimates in order to be able to comprehend the current limitations of the method as proposed . it is worth mentioning here that in most situations in raman ( and in sers ) we talk plainly about the cross section @xmath54 while in reality there are at least three possible cross sections we can refer to : @xmath2 radiative , @xmath5 non - radiative , and @xmath55 total . in the case of vibrational pumping , for example , the rate at which vibrations are pumped into the molecule is proportional to the _ total _ cross section . but some of the re - emitted photons might not be observable in the far - field because they are absorbed by plasmon resonances in the vicinity of the probe . this is what makes the distinction between _ radiative _ and _ non - radiative _ cross sections . this could lead potentially to inconsistencies among cross sections for different modes if they happen at frequencies where the non - radiative contributions are different . we will not insist on this distinction in the following , but it is worth keeping in mind that as far as the cross sections is concerned what we measure is the `` radiative '' part of an effect ( pumping ) which depends on the `` total '' ( radiative plus non - radiative ) cross section . this distinction could under certain circumstances be important . we summarize in this subsection the different aspects of the problem of lifetime estimation for the purpose of obtaining sers cross sections .    * ultimately , the observation of a cross - over from a thermally - dominated to a pumping - dominated as / s - ratio leads to an estimation of both the asymmetry factor @xmath24 and the product @xmath56 . an estimation of @xmath47 itself requires the knowledge of @xmath6 , as pointed out before . the latter can not be directly obtained , in general , from the sers spectrum potentially leading to a serious limitation of this technique . in the first report of cross section estimation via vibrational pumping@xcite a lifetime of @xmath57 was simply estimated without any reference to the experimental data .    in order to improve upon this we need to make a series of assumptions . the validity of these assumptions needs to be evaluated on a case - by - case basis and the results obtained for the cross section have to be interpreted within these approximations / assumptions . let us first review the problems associated with trying to extract lifetimes from the sers spectra . * raman peaks in molecules have lineshapes which are seldom pure lorentzians and have several contributions to their natural widths ; a subject of a longstanding history in spectroscopy@xcite . the observed linewidth has typically contributions from : @xmath2 groups of raman modes piled up together in narrow energy regions ; @xmath5 inhomogeneous broadening , @xmath55 phase coherence relaxation , and @xmath58 population relaxation . the last two are also known as the _ off - diagonal _ ( dephasing ) and _ diagonal _ ( population ) relaxation times in density matrix formalism@xcite , respectively , and are part of the _ homogeneous broadening _ of the peak . these relaxation times are different from mode to mode and can not be easily separated from the observed linewidth in a plain sers spectrum . the _ population relaxation _ is the one we need ( @xmath6 ) for the estimation of @xmath10 . * the presence of multiple modes contributing to a peak is a drawback that can be overcome in many situations . in the case of dyes it is not unusual for several raman active modes , which are closely spaced in energy and have different cross - sections , to contribute to the observed peaks . this will naturally result in distortions of the lineshape and problems with any estimate of the lifetime based on the width of the peak . this can be overcome by the use of well characterized vibrations which are relatively isolated from the others and even ( if possible ) complementing the information by density functional theory ( dft ) calculations of the raman spectra@xcite to support the selection of those modes that are the best candidates for a reliable estimation of @xmath6 . * if the broadening is homogeneous@xcite , the only remaining contributions to the linewidth are the diagonal and off - diagonal relaxation times . molecular vibrations have typically very strong anharmonic couplings with other vibrations in the molecule and contributions to inhomogeneous broadenings are mostly negligible . even in sers experiments where the single molecule limit is approached@xcite ( which would be more sensitive to inhomogeneous broadening ) the full width at half maximum ( fwhm ) , @xmath59 , of the peaks does not change by more than @xmath60 @xmath61 in peaks with a typical @xmath59 of @xmath62 @xmath61 . the validity of this assumption ( whether inhomogeneous broadening is important or not ) has to be assessed obviously in each specific situation . * the separation between diagonal and off - diagonal relaxation times can not be achieved by a simple sers spectrum . it is always possible to invoke a different type of spectroscopy ( time resolved)@xcite , but this may not be feasible in most situations . one simple possibility ( as done in ref . @xcite ) is to estimate the population relaxation by using the fwhm @xmath63 of the peaks via @xmath64 ; i.e. essentially ignoring the contribution of the dephasing ( off - diagonal ) relaxation to the width . this will produce an underestimation of @xmath6 and an overestimation of @xmath47 . in addition , this typically produces cross - sections for different peaks which are not entirely consistent with the relative ( integrated ) intensities among peaks ; thus showing an intrinsic inconsistency in the estimation of the @xmath6 s ( which can be of the order of a factor of 2 to 10 for the examples we examined ) . it is generally difficult to make a simple estimation that will be valid for all modes . it is quite clear that for the cross sections to be meaningful ( and if they are not affected by different amounts of non - radiative processes ) the relative cross sections among peaks must be in accordance with the relative integrated intensities observed in a normal sers spectrum . * population relaxation is caused mainly by the anharmonic coupling to the `` thermal bath '' . this bath includes all other modes in the molecule as well as those in the solvent or substrate . generally speaking , higher energy modes have shorter lifetimes due to a greater number of possible decay pathways . intramolecular anharmonic decays are energy conserving , meaning that high energy modes have more possibilities for a decay than low energy ones . population relaxation is also the only dynamical process that contributes to the vibrational linewidth in the limit @xmath65 . * it is important to realize that , in principle , _ if we find a single peak in the spectrum where the population relaxation can be gained directly from the fwhm via _ @xmath64 , _ then all of the cross sections of the other modes follow immediately through the relative integrated intensities of the peaks_. when lacking a time resolved experiment to isolate the contributions properly , _ the problem comes down to a judicious choice of a mode where the width is dominated by relaxation population_. a compromise then is to extract the values of the cross sections by means of the width of the highest ( isolated ) raman active mode . a rule of thumb is that the lower the temperature and the higher the energy of the mode the more the linewidth will be dominated by population relaxation . we can then use one mode to estimate its @xmath47 , and all the others follow automatically from this assumption . the cross sections are now consistent by construction and the method provides , in fact , the means to estimate the relative lifetimes of the modes . we call this procedure ( i.e. estimating the @xmath6 of the raman active mode with the largest raman shift , and making the other cross - sections consistent with this value through the relative integrated intensities ) the _ corrected lifetime method _ ( clm ) . * if we are comparing different substrates with the same analyte we could also directly compare the values of @xmath52 without any further assumption . if there is no reason to believe that relaxation population is different between the two substrates , _ the ratio of @xmath52 for two different substrates provides a direct comparison of sers cross sections_.      * problems with measuring the asymmetry factor `` @xmath24 '' can be as important as those to obtain a reliable lifetime . heating in the thermally - dominated regime can be a _ major _ problem . as demonstrated in sec . ii b , heating can be in part a source for a larger than normal @xmath24 , if it is not properly identified . * ideally , the best estimation of @xmath24 can come from measurements at rt with long integration times and very low powers . the obtained @xmath24 s should be more or less the same for a given analyte in the presence of the same metal ( they are mainly related to the local interaction of the dye with the metal@xcite ) . * the product @xmath66 is measured with high accuracy in the pumping dominated `` plateau '' at low temperatures . accordingly , the best use of the technique is to use one well - characterized analyte ( without photo - bleaching if possible ) as a means to compare the relative sers cross - sections of the dye on different substrates of the same metal , by simple comparison of the plateaus . the product @xmath66 is perhaps the best `` objective '' comparison between the cross sections of two substrates without any assumption . however , it is also important to note that the cross sections we are comparing are not the average cross sections but rather a biased average towards the highest cross sections in the sample , as explained in the next subsection . there are additional complications with the estimation of the cross section . it is generally accepted that sers signals are dominated by the presence of `` hot - spots '' or places with high local enhancements . the formulae presented above made implicitly the assumption that the cross section is the same for all @xmath18 molecules in the sample . the question we want to address now is : do we obtain an estimate of the average cross section with this method ? we shall show in what follows that what we actually obtain from the experiment is an estimate which is heavily biased towards the sites with the highest enhancements ; i.e. the method provides an estimate of the enhancements in hot - spots . this is of course an advantage and a disadvantage at the same time . we show this explicitly in this section . we consider a number of molecules ( @xmath67 from 1 to @xmath18 ) in the sample . each molecule can have a different anti - stokes , ( @xmath68 ) and stokes ( @xmath69 ) sers cross sections ( according to the asymmetry factor @xmath70 for that molecule ) . we assume for simplicity that the incident power @xmath11 is the same for all . the rate equation for the average phonon population @xmath71 of a vibrational mode of _ one molecule _ , @xmath67 , and its stationary state solution are given by expressions similar to eqs . ( [ rateeqn ] ) and ( [ nsteady ] ) : following the other formulae in the previous sections , the stokes signal for this molecule is simply @xmath72 , while the anti - stokes signal is given by : @xmath73 . the anti - stokes / stokes ratio _ for this molecule _ , @xmath74 , is given by an expression similar to eq . ( [ pumpratio ] ) . if all the molecules experienced the same enhancements , then the total stokes and anti - stokes intensities would be @xmath75 and @xmath76 , and the ratio would be @xmath77 . however , if the molecules have different cross sections ( sers enhancements ) , which is a much more realistic assumption , we then have : @xmath78 and @xmath79 all the averages here @xmath80 are the usual statistical ensemble averages . the as / s - ratio is then : @xmath81.\ ] ]    comparing this expression with that obtained previously in eq . ( [ pumpratio ] ) , what we called @xmath24 is now replaced by : @xmath82 note that this is not exactly the average of the @xmath83 s , i.e. @xmath84 . moreover , what we measure ( instead of @xmath85 ) is now : @xmath86    to understand the meaning of @xmath87 ( @xmath88 for ensemble ) , we need to understand the sources of non - uniformity . the main variation is in the cross - sections , because of the large differences in sers enhancements at different locations on the sers substrate . we typically have values which are , say , @xmath89 larger at hot - spots than in other places . however , for a given molecule , the ratio @xmath90 should not vary as much ( at least not by more than one order of magnitude ) . in fact , this ratio can in a first approximation be taken as a constant , which is an intrinsic property of the adsorbed molecule / metal complex@xcite . assuming that @xmath91 is the same for all molecules , we simply get that @xmath92 , i.e. the @xmath24 we measure is the correct one . moreover , using @xmath93 , we can express @xmath87 as a function of @xmath94 only as :    @xmath95    if the distribution of @xmath10 s were fairly uniform ( for example a gaussian around an average value with a small standard deviation ) then @xmath87 would be a good estimate of the average of the distribution ( slightly overestimated ) . but this is far from reality in sers conditions . a more realistic situation is to have a small number of molecules at a hot - spot ( hs ) , and a large number of molecules at non - hs positions . for the sake of argument , let us demonstrate the effect on this average by considering @xmath96 molecules with one molecule at a hs and 999 molecules at places with much lower enhancements . the expression given above would give : @xmath97 if there is a difference of a few orders of magnitude between the cross sections @xmath98 and @xmath99 , the numerator is then easily dominated by the one molecule at the hs ( due to the square ) . for the numerator , it is not as clear - cut , it all depends on how much stronger the hs is and how many molecules are at non - hs positions . but overall , the denominator is likely to be dominated by @xmath98 , possibly slightly larger if the second contribution is not negligible . this shows that @xmath100 , or perhaps slightly smaller . a more quantitative argument is only possible if we had a more realistic distribution of sers enhancements for a given substrate , but the qualitative conclusion is that _ pumping experiments provide a good lower estimate of the cross - section of the few molecules experiencing the highest enhancements_.      the last aspect we want to briefly touch upon in this analysis of cross section estimations via vibrational pumping is photo - bleaching . the exact mechanisms of photo - bleaching under sers conditions are still poorly understood and would deserve a full study in itself . photo - bleaching is particularly important in the method we are describing here because molecules sited at hs s ( i.e. those exposed to the largest enhancements ) are affected to a greater degree . at high power densities , for example , the population of molecules at hs s will be bleached at a faster rate than the average and this will change the measured cross section . studying @xmath26 under photo - bleaching conditions could , in principle , tell us something about this distribution . there is robust experimental evidence that photo - bleaching can complicate the analysis , produce experimental artifacts , and -to a large extent- even decide the answer we obtain from a specific experiment . this is added to several experimental complications in as / s - ratios which include the fact that sometimes ( depending on the dispersion of the spectrometer ) the anti - stokes and stokes sides can not be measured simultaneously . this implies a delay between the two measurements and therefore a difference in exposure times to the laser . by choosing a certain power level we are , in a way , selecting the population of hs s that we are going to measure . we shall come back to this problem in the discussion of the results . a rule of thumb is that larger laser spots with low power densities ( but sufficient to produce detectable pumping ) , short integration times , and non - resonant ( with the dye ) laser excitation , are in general preferable for a more reliable estimate of sers cross sections . all possible measures should be taken ( including the type of scanning used ) to address and minimize the undesirable effects of photo - bleaching . we turn now to an experimental demonstration of the principles underlined above . sers measurements have been performed on substrates formed by dried ag colloids on silicon . the colloids were prepared using the standard lee and meisel technique@xcite . sers active samples were prepared by mixing the colloids with a @xmath101 kcl solution in equal amounts . the analytes were then added to give a concentration of @xmath102 in each case . a small amount of this solution was then dried on to a silicon substrate . the investigated analytes are rhodamine 6 g ( rh6 g ) , crystal violet ( cv ) , and 3,3-diethyloxadicarbocyanine ( dodc ) . figure [ analytes ] shows the basic sers spectra of the analytes . we shall show the explicit temperature dependencies of the as / s - ratios for only a few modes ( labeled in fig . [ analytes ] ) while the tables show additional data for other modes . samples were mounted in a closed - cycle he - cryostat ( cti - cryogenics ) with temperature control in the range @xmath103 . raman measurements were performed using several laser lines of a kr@xmath104 and ar@xmath104-ion lasers which were focused to a 20 @xmath105 m diameter spot . the signal was collected using a high - numerical aperture photographic zoom lens ( canon , @xmath10610 magnification ) onto the entrance slit of a high - dispersion double - additive u1000 jobin - yvon spectrometer coupled to a liquid n@xmath107-cooled ccd detector . peaks were analyzed using standard voigt functions with subtracted backgrounds . the variation in the as / s - ratio with @xmath1 for each mode was then fitted to eq . [ pumpratio ] . for this purpose , it is convenient to modify the expression to :    @xmath108 , \label{lnpumpratio}\ ] ]    where @xmath109 , and @xmath110 which are the ( dimensionless ) fitted parameters . the @xmath6 s of the peaks ( necessary to obtain the cross sections ) are estimated in two different ways following the prescriptions in sec . iii b.    as discussed earlier , it is of critical importance that the sample remains stable over the entire length of the experiment . a range of integration times were used depending on the specific sample and measurement to achieve a good signal to noise ratio for different peaks , but power densities were kept to a minimum compatible with the observation of pumping and exposure to the beam was minimized as much as possible in between changes of @xmath1 . we used @xmath111 spread over the 20 @xmath105 m diameter spot . as the rate of photo - bleaching may change with @xmath1@xcite a suitable power density to ensure reasonable stability must be decided at rt , where the effect is greatest . the relatively large spot and low power densities have the triple advantage of ( @xmath67 ) reducing photo - bleaching to a negligible level , ( @xmath112 ) reducing any indirect laser heating effects and ( @xmath113 ) improving the averaging over cluster geometries . still , even under low power density excitation , the choice of laser line can have an important effect on the measurement . an explicit example is shown in fig . [ photobleach ] for rh6 g under @xmath114 excitation . as explained in the caption , this observation is compatible with a bleaching of the hs s with largest cross sections . laser lines that are strongly resonant with the probe molecule@xcite should be avoided so that the photo - stability of the analyte and the accumulated exposure to the laser do not become an issue in the interpretation and analysis of the data . for that reason , we only concentrate hereafter on results obtained with the @xmath115 and @xmath116 lines of a kr@xmath117-laser . these two lines are close enough to the visible ( to profit fully from sers enhancements ) but , by the same token , they do not produce excessive photo - bleaching for the dyes under consideration here . having all the theoretical tools from the previous sections , we can now easily scan through several experimental results that demonstrate the method in practice . figure [ tdeps ] shows the as / s - ratios ( ln(@xmath26 ) ) vs. @xmath1 , as measured using the @xmath118 line , for four modes of each of the investigated analytes . the solid lines represent the best fit to the experimental values using eq . [ lnpumpratio ] . a series of small imperfections can be seen in the data for some modes , but overall the behavior is very well represented by eq . [ lnpumpratio ] , with rh6 g being the best example . the arrows indicate the cross - over points @xmath119 between the thermally - dominated and pumping - dominated regimes for all the investigated modes . @xmath50 occurs at higher @xmath1 s for higher energy modes , as expected . the fact that pumping is observed for each of the modes indicates that the effect is fairly general and should be observable for a large number of analytes for as long as @xmath10 is high enough to make pumping observable at moderate power densities@xcite .    in order to transform these data into estimates of the cross sections we need to address the different methods of estimating @xmath6 . table [ tab : fitsigmas ] shows the experimental results for rh6 g taken with the @xmath116 line . for the sake of comparison , we display the cross section by using two different estimations of the @xmath6 s . this gives a better idea of how reliable these numbers are and how differences among them should be interpreted . table [ tab : fitsigmas ] shows both : @xmath2 the extraction of @xmath6 directly from the widths via @xmath120 ( this normally renders cross sections which are not fully consistent with the relative integrated intensities of the peaks ) , and @xmath5 the extraction of @xmath6 from the highest raman active mode and the subsequent adjustment of the other cross - sections to agree with the relative integrated intensities ( as described in sec . iii b ) . a column with relative intensities and cross - sections among modes is also provided for completeness . the anti - stokes cross sections are obtained through the stokes ones via the fitted value of the asymmetry parameter @xmath24 from eq . [ lnpumpratio ] . these results prove the point that the estimation of the lifetime is a crucial step in this method to transform the experimental values into an estimation of the cross sections themselves . the values should be interpreted within this assumption , and their validity should be assessed on a case - by - case basis . the preferred ( more consistent ) method is of course the one that ensures the relative cross sections agree with the relative intensities of peaks ( clm in sec . iii b ) . finally , we can discuss the effect of estimating the cross sections for two different excitation wavelengths . the experiment is somewhat restricted in the choice of excitations ( to be compatible with photo - stability ) but still provides a hint of resonance contributions to the cross sections with some limitations . table [ tab : fitsigmasall ] shows the cross - sections obtained for rh6 g , dodc , and cv for 647 and @xmath116 excitations for different modes . all the cross sections were obtained by the clm analysis . it is worth noting that in this latter method we only need ( in principle ) one temperature dependent as / s - ratio for the highest raman active mode ( judged to be dominated by population relaxation ) . this together with the plain sers spectrum of the dye is enough to estimate all the cross sections of the modes via the relative integrated intensities . the results will be consistent ( by construction ) but not necessarily accurate ( depending on the one mode that has been chosen to estimate the initial cross section ) . the main conclusions of these experiments can be summarized as follows : @xmath2 rh6 g shows larger cross - sections at 647 than at @xmath116 . this is consistent with a resonant increase known to exist in rh6 g towards the yellow part of the spectrum@xcite . @xmath5 on the contrary , both dodc and cv experience larger cross - sections at @xmath121 , consistent with the maximum of the absorption of these dyes being at longer wavelengths with respect to rh6 g and , therefore , having a much more red - shifted resonant interaction than rh6g@xcite . @xmath55 although there is some variations from mode to mode , it is generally true that cv experiences the largest cross - sections in this near - ir region . the numbers in tables [ tab : fitsigmas ] and [ tab : fitsigmasall ] deserve a special comment regarding their interpretation . for a start , the main source of error in these numbers ( as pointed out in the caption of table [ tab : fitsigmas ] ) is _ systematic _ rather than statistical , and comes from the different methods used in the estimation of @xmath6 . error bars for these systematic errors are not easy to obtain from the data , for they depend on the relative contribution to the fwhm of population relaxation and dephasing , which is unknown and changes from mode to mode . the number of significant figures is decided then by the range in which these numbers span for a given group of modes and laser excitation . it only makes sense to compare statistical errors ( from the propagation of errors in the parameters of the fits and the experimental magnitudes ) for two cross sections with the same systematic errors . this is the situation , for example , of the comparison between the method presented in this paper and the power - dependence method ( quadratic dependence of the @xmath28 signal as a function of power ) as reported in ref . @xcite . in this latter case the same systematic error coming from the estimation of @xmath6 affects both results and , hence , the statistical error bars are meaningful to claim consistency or inconsistency between the two values . in any other case , the overriding source of error in these estimates of the cross sections is systematic . the order of magnitude of these numbers deserves a special comment too . according to the results in sec . iii d , we expect these numbers to represent something close to the maximum available enhancement in these samples . in addition , the fact that @xmath6 could be in general underestimated by any of the methods presented here leads automatically to an overestimation of @xmath54 . last but not least , the values of @xmath48 are obtained from those of @xmath47 via the asymmetry factor @xmath24 . as a consequence of all these , we expect the values to be : @xmath2 at the higher - end of what is known for hs s in sers for @xmath47 and , @xmath5 with a larger error for the anti - stokes values which use the estimation of the asymmetry parameter @xmath122 two times ; once to obtain @xmath47 itself through the fit of @xmath123 vs. @xmath1 , and twice to obtain @xmath48 from @xmath47 . indeed , table [ tab : fitsigmasall ] shows that cross - sections are at the higher - end of what is commonly accepted for sers enhancements . @xmath48 is bound to be less exact than @xmath47 because it depends crucially on the asymmetry parameter and its associated errors ( treated in sec . the most consistent set of values are obtained for @xmath47 using the clm - analysis ( which should be the preferred option ) and are all of the order of @xmath124 . in exceptional cases , cross sections of the order of @xmath125 are obtained , which are still expected if we take into account that the estimation is for hs s and that the values are possibly overestimated by the choice of @xmath6 . unrealistically large values of @xmath47 beyond that should be interpreted as a limitation of the technique , in general , and should be tracked down to a problem in the assignment of @xmath6 for that mode . even for these cases , it is still possible to compare @xmath126 ( or @xmath127 ) for the same modes in different substrates providing a relative comparison of cross section performances at hs s without any additional assumptions . this comparison is independent of the number of molecules involved in the signal , due to the self - normalizing nature of the method . table [ tab : fitsigmasall ] summarizes , therefore , a typical example where the performance of different dyes can be tested through appropriate estimates of the cross sections derived from the method presented in this paper . we have presented an in - depth discussion of the state - of - the - art of sers cross section estimation via vibrational pumping and have given several experimental examples of it . despite the approximations needed and the intrinsic problems of the technique , it is still an excellent tool ( and in many cases possibly the only tool ) to extract an estimation of the sers cross - sections in situations where the number of molecules and hot - spots present in the sample are not known or are difficult to estimate . the _ self - normalizing _ nature of the method ( with respect to the number of molecules involved ) is certainly a major advantage . in addition , we feel that these latter experimental and theoretical developments have moved forward to establish vibrational pumping as a real phenomenon in sers after almost a decade of controversy about its very existence . one outstanding issue that surely deserves further investigation is the effect of photo - 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section is small relative to the analytes used here . mode & @xmath10 ( @xmath128 ) & @xmath48 ( @xmath128 ) & b & @xmath10 ( @xmath128 ) & @xmath48 ( @xmath128 ) & b + ( @xmath61)&(@xmath129 ) & ( @xmath129 ) & ( @xmath130 ) & ( @xmath129 ) & ( @xmath129 ) & ( @xmath130 ) +   + 610 & 20 & 112 & 44.8 & 1.8 & 7 & 25.6 + 780 & 25 & 136 & 25.9 & 2.2 & 11 & 6.85 + 1360 & 2 & 463 & 1.26 & 2.1 & 58 & 1.32 + 1510 & 4 & 499 & 6.38 & 3.7 & 114 & 2.13 + 1650 & 14 & 232 & 22.5 & 1.2 & 95 & 2.11 +     + 875 & 1.8 & 26 & 11.1 & 0.90 & 33 & 3.04 + 925 & 1.7 & 34 & 8.40 & 0.94 & 33 & 3.34 + 970 & 1.7 & 34 & 7.32 & 1.14 & 55 & 2.17 + 1128 & 2.5 & 51 & 4.93 & 1.74 & 143 & 8.23 + 1375 & 1.3 & 25 & 1.21 & 1.01 & 132 & 0.30 +     + 760 & 1.4 & 1.8 & 93.5 & 0.2 & 0.5 & 53.6 + 805 & 10.9 & 14.5 & 74.7 & 3.4 & 6.7 & 28.6 + 910 & 3.9 & 6.5 & 124 & 1.1 & 3.9 & 6.82 + 1175 & 19.5 & 30.7 & 18.0 & 4.3 & 31.8 & 5.62 + 1292 & 3.7 & 6.3 & 8.24 & 1.6 & 4.8 & 3.94 + 1620 & 6.4 & 14.4 & 9.99 & 2.0 & 32.7 & 2.26 +
we present an in - depth analysis of the experimental estimation of cross sections in surface enhanced raman scattering ( sers ) by vibrational pumping . 
 the paper highlights the advantages and disadvantages of the technique , pinpoints the main aspects and limitations , and provides the underlying physical concepts to interpret the experimental results . 
 examples for several commonly used sers probes are given , and a discussion on future possible developments is also presented .