Volume 54, Number 6, 2000 APPLIED SPECTROSCOPY 7730003-7028 / 00 / 5406-0773$2.00 / 0

q 2000 Society for Applied Spectroscopy

Modeling and Measuring the Effect of Refraction on theDepth Resolution of Confocal Raman Microscopy

NEIL J. EVERALLICI plc, Wilton Research Centre, P.O. Box 90, Wilton, Middlesbrough, Cleveland, TS90 8JE, UK

A simple ray-tracing analysis has been used to predict the effect ofrefraction at the sample/air interface, on the depth resolution of

confocal Raman microscopy. This analysis applies to the ``z-scan-

ning’’, or ``optical sectioning’’, approach to obtaining depth pro ® les,where the laser beam is incident normal to the sample surface, and

spectra are recorded sequentially as the focus is moved deeper into

the material. It is shown that when a ``dry’’ metallurgical objective(the most common con® guration for commercial Raman micro-

scopes) is used, both the position and the depth of focus increase

dramatically as the beam is focused deeper into the sample. Itquickly becomes impossible to obtain ``pure’’ spectra of thin layers

that are buried more than a few micrometers below the air inter-

face. Equations are presented which model the intensity responseexpected when focusing through a coating into a substrate. The

model requires knowledge of the sample refractive index, the nu-

merical aperture, and the laser beam intensity distribution at thelimiting aperture, of the objective. Given these values, one can pre-

dict the substrate Raman intensity as a function of the nominal focal

point within the sample. For a 36 m m coating on a thick substrate,we predict that even for a perfectly sharp interface ( 1 m m), sub-

strate bands rise slowly (over an apparent distance of 10 m m or

more), and are strong when the focus is apparently only ; 18 m mbelow the air/coating interface. This prediction was con® rmed

through experimental observation. The model was also used to an-

alyze literature data that had been interpreted previously as show-

ing interfacial diffusion in polymer laminates. The model correctly

reproduced the main features of the observed data without invoking

interfacial penetrationÐ the optical aberrations alone accounted foralmost all the observed broadening and the fact that the apparent

thickness of the buried layer is also distorted. It was concluded that,

with the use of this illumination geometry, it is very dif ® cult todetect or quantify interfacial broadening unless it occurs on a very

large scale indeed (tens of micrometers). It is concluded that ``op-

tical sectioning’’ cannot be recommended for quantitative depthpro® ling at signi® cant depths using metallurgical objectives. The

optimum practical solution is to cut a cross section and map lat-

erally across the sample, thereby utilizing and maintaining the ex-

cellent (lateral) resolution of the Raman microprobe. An alternative

solution is to use an immersion objective to minimize refraction at

the sample surface.

Index Headings: Raman microscopy; Confocal; Refraction; Depthpro® ling; Mapping; Imaging; Polymers.

INTRODUCTION

Raman microscopy offers a unique combination ofspatial resolution ( ; 1 m m) and chemical/physical char-

acterization. Although its spatial resolution is worse, say,than that of electron microscopy, it enables measurementof a host of chemical and physical properties, includingchemical composition, molecular orientation, conforma-

tion, crystallinity, strain, temperature, and so on. No othermicroprobe technique offers this combination of spatial

Received 18 December 1999; accepted 25 February 2000.

resolution and information content. However, it has longbeen recognized that simply focusing the laser beam toa small spot does not, of itself, secure a good spatialresolution. Often, turbidity spreads the laser intensityover a large sample volume, and the resultant Ramansignal source is not ``tightly’ ’ located in space. Many au-

thors have pointed out (and constructed spectrometers onthe basis of the fact) that this effect can be suppressedby placing a confocal aperture at a back-focal imageplane, to block all but the light that originates from thediffraction-limited laser focal volume.1±5 This arrange-

ment improves both the lateral and ``depth’ ’ resolutionthat can be attained, and considerable effort has been ex-

pended on modeling the improvement in spatial resolu-

tion as a function of aperture size.4,6 While most workershave used a physical aperture such as an adjustable pin-

hole, Batchelder and co-workers5 have cleverly shownhow the same bene® ts can be obtained by using a charge-

coupled device (CCD) detector as an ``electronic’ ’ aper-

ture. Their design uses the entrance slit as one ``dimen-

sion’ ’ of the aperture and pixel binning of the CCD asthe orthogonal aperture axis.

However the aperture is designed, it becomes possible,in principle, to collect Raman radiation that originatesonly from within a diffraction-limited laser focal volume.The dimensions of the focal region depend on the nu-

merical aperture (NA) of the focusing lens and the degreeto which it is ® lled by the laser beam. Typically, for ahigh NA objective (0.9±0.95) that is ® lled by the laserbeam so as truncate the beam at the ``1/e’ ’ ® eld points(thereby transmitting ; 86% of the intensity), one obtainsa focal ``tube’ ’ with waist diameter of ; 1.22 l /NA anddepth of focus of ; 4 l /(NA) 2. The lateral diameter canbe measured by scanning the laser spot over a materialwith a very sharp edge or feature, and measuring theresponse. 2 The depth of focus can be measured by fo-

cusing the laser beam just above a silicon wafer, and mea-

suring the 520 cm 2 1 band intensity as the wafer is raisedvertically though the beam focus (termed z-scanningthroughout this paper, assuming the z-axis is normal tothe sample surface). The full width at half-maximum(FWHM) of the silicon signal gives an indication of thedepth of the laser focus in air. Its shape and volume will,however, be distorted if the beam is focused into a ma-

terial with a different refractive index.Confocal Raman microscopy has been widely used to

analyze, map, and image chemical and physical proper-

ties in one, two, and three dimensions. One approach isto place a sample on an automated x,y,z mapping stageand obtain Raman spectra at different points in the sam-

ple. In this way, full spectra can be obtained throughouta sample, and properties can be reconstructed from the

774 Volume 54, Number 6, 2000

spectra and displayed as line, area, or volume images. 7±9

Thus we can obtain maps or images in which contrast isbased on chemical or physical properties derived fromRaman features. Considerable effort has been expendedon developing data analysis techniques to extract mean-

ingful chemical or physical information from the hugedata sets which result,7±9 but surprisingly little work hasbeen reported on deriving accurate ``spatial’ ’ propertiesfrom the data. If a laser spot is simply rastered over asample surface, it is simple to correlate the spectra withthe physical position of the laser spot on the surface.However, if we are mapping in three dimensions, wemust necessarily focus below the surface of the sample.Commercial Raman microscopes are usually con® guredwith metallurgical objectives that are designed to workin air; if this is the case, the tightly convergent laser beamwill suffer refraction at the sample surface, and it be-

comes more dif® cult to de® ne the z-coordinate of theregion under investigation. Unfortunately, this effect di-

minishes one of the key advantages of confocal RamanmicroscopyÐ namely, the ability to obtain spectrathroughout the volume of a transparent sample withoutany sample pretreatment. For example, suppose one hasa laminated multilayer ® lm; it is possible to obtain spectraof the individual layers by microtoming a cross sectionand then moving the laser probe from one side to theother, examining each layer individually. Similarly, con-tinuous property gradients such as polymer crystallinity,cure, or orientation can be measured in the same way.10±13

However, obtaining good sections without disrupting thestructure requires specialist equipment and can be dif® -

cult. At ® rst sight, the confocal Raman microscopeshould circumvent this problem, since we can simply fo-cus on the sample surface, record a spectrum, and then``z-scan’ ’ down through the sample, recording spectra in-

crementally as we procede. Since the depth resolution asde® ned by the silicon experiment is on the order of 2m m, it might appear that we would be able to resolvequite thin layers in this way. Several authors have usedthis approach, particularly for the analysis of poly-

mers4,5,13±17 and emulsions.7 It is often noted in such stud-

ies that at the air/sample interface the laser focus is dis-

torted and makes localization along the z-axis dif® cult,but the problem is rarely treated quantitatively.

A few authors have considered the refraction-aberra-tion effect in some detail. Rosasco 1 discussed the problemand predicted the magnitude of the effect for an ellipsoi-

dal collection mirror with uniform laser illumination,while Delhaye et al.18 gave an expression for the maxi-

mum depth of focus. They also pointed out that if a``dry’ ’ objective is used to analyze samples either beneatha cover slip or embedded deep in a matrix, the confocaldepth resolution is severely degraded. However, neitherarticle compared theory with experiment or consideredthe effect of different laser illumination conditions on theobjective. With this in mind, the objective of this paperis to derive simple expressions for predicting the spatialdepth, shape, and position of the Raman response alongthe z-axis, and to show how well these expressions modelexperimental data. It will be shown how, unless the ef-

fects of refraction are accounted for quantitatively, highlyinaccurate conclusions regarding interface and layer po-

sition/thickness can be inferred from raw z-scan intensity

data. The expressions will also be used to interpret pre-

viously published data regarding interfacial diffusion inpolymer laminates. We will also consider the effect ofthe laser intensity distribution on the microscope objec-

tive upon the depth resolution of a confocal system.

EXPERIMENTAL

Instrumentation. All Raman data discussed belowwere obtained with a Dilor ``LabRam’ ’ confocal Ramanmicroscope, with He±Ne excitation (633 nm), a 100 3 ,0.9 NA objective, and ; 3 mW laser power incident onthe sample. Samples were mounted on an x,y motorizedstage, with z-displacement controlled with a piezo-trans-

ducer on the objective. The confocal pinhole diameterwas 200 m m, and the slit width was 100 m m. It is as-

sumed (for now) that the laser beam diameter was suchthat it was truncated at the ``1/e’ ’ ® eld strength points inthe objective (i.e., optimum ® lling). Under these condi-

tions, the confocal ``depth of focus’ ’ , assessed by z-scan-

ning a silicon wafer, was measured to be ; 2 m m FWHM.Sample Preparation. The sample used for this work

was a poly(ethylene terephthalate) (PET) ® lm that hadbeen coated with a UV-cured, acrylate-based coating. Theexact coating formulation is proprietary. The coat thick-

ness was uniform at ; 36 m m ( 6 1 m m), as assessed bymicroscope measurements of cross sections. Over shortdistances (100 m m) the interface was very sharp ( , 1 m m)compared with the confocal depth resolution measuredwith a silicon wafer.

THEORETICAL ANALYSIS

Ray Tracing. In this analysis we assume that the ef-

fects of diffraction are small compared with the in¯ uenceof refraction, so we only consider refraction-induced ab-

errations, which can be modeled by ray tracing. This ap-proach will give a lower bound for the ``defocusing’ ’ thatoccursÐ diffraction will presumably make matters worse.Figure 1 de® nes the relevant parameters. The objectiveis in a medium of refractive index n2 (assumed to be 1for air) and has a numerical aperture of sin u max and aworking distance f . The ® nal lens in the objective is as-sumed to have a limiting aperture radius of rmax micro-

meters. The latter three parameters are related throughEq. 1.

f ´NAr 5 (1)max 2 1/2(1 2 NA )

When focused through a material boundary, a ray origi-

nating at a radius r measured from the center of the ob-

jective is brought to a focus at point P2, which lies adistance z below the sample/air interface. If the materialhad a refractive index of n1 5 1, the same ray would befocused at P1, a distance D below the interface. Note thatD is actually the distance moved by the sample whenshifting focus from the sample surface to point P2. Werefer to D as the ``apparent’ ’ or ``nominal’ ’ focal point,since it is the vertical distance moved according to thescale on the microscope stage adjustment.

The object of this analysis is to calculate the true focalposition (z) and the intensity distribution within the ma-

terial, as a function of n ( 5 n1 /n2), NA, and D , and torelate the position and depth of focus to D . This approach

APPLIED SPECTROSCOPY 775

FIG. 1. De® nition of parameters used in ray-tracing analysis. For con-

venience the objective lens is depicted as a hemisphereÐ the exactshape is not important. In the absence of refraction, all rays are focusedat point P1, but refraction causes rays to be focused deeper into thesample (P2). All terms are de® ned in the main text.

FIG. 2. Distance of point of focus (zm) below air interface, as a functionof origin on objective (m). Curves derived from Eq. 6. Note how in-

creasing D causes rays to be focused deeper in the sample. Rays orgin-

ating near the maximum objective aperture (m 5 1) are also focuseddeeper.

enables estimation of the true illuminated region whenfocusing a nominal distance D into a sample. Note, weassume that all rays are focused along the z-axis, andneglect the orthogonal spreading.

The value of z for a given ray can be calculated byusing Snell’s law (sin u t 5 sin u i /n) and the values of fand r. From Fig. 1, considering the kth ray (i.e., oneoriginating from radius rk), we have

rk 2sin u 5 , cos u 5 Ï 1 2 sin u (2)i i i2 2Ï r 1 fk

2r sin uk isin u 5 , cos u 5 1 2 (3)t t 22 2 ! nn ´ Ï r 1 fk

y D tan uk iz 5 5 (4)k tan u tan ut t

These equations lead to a simple expression for zk (Eq.5).

D2 2 2 2 1/2z 5 [n (r 1 f ) 2 r ] (r , r ) (5)k k k k maxf

Equation 5 gives the true point of focus (z) within a me-

dium for any ray, as a function of the apparent focal pointD , the lens working distance, and the refractive index.With the use of Eq. 1 to substitute for f , Eq. 5 can berecast as Eq. 6.

1/22 2 2r NA (n 2 1)k 2z 5 D 1 nm 2 2[ ]r (1 2 NA )max

1/22 2NA (n 2 1)

2 25 D m ´ 1 n (6)2[ ](1 2 NA )

Here we de® ne m as the normalized radius (m 5 r /rmax),i.e., the fractional distance across the objective. Equation6 requires only the lens NA to compute the focal pointof any ray according to its point of origin on the objec-

tive. When m 5 1 we have a marginal ray, i.e., one withthe highest possible angle of incidence [ u 1 5 sin 2 1 (NA)],whereas m 5 0 implies a ray normal to the sample sur-

face. Figure 2 illustrates the variation of zm with m forthree nominal focal points ( D 5 2, 5, and 10 m m), as-

suming a 0.9 NA objective.Several points are immediately apparent from Eq. 6

and Fig. 2.

1. If we use an oil-immersion objective to match the in-

dices, we have n ; 1 and zm ; D ; hence the apparentand actual focal points coincide at P1 irrespective ofthe value of D , as expected. Use of an immersion ob-jective w ill min imize the aberrations discussedthroughout this paper.

2. If we grossly under ® ll the objective (m ; 0), Eq. 6reduces to zm 5 D ´n, and the true focal point is simplyshifted by a factor ``n’ ’ relative to the apparent focus.This is the minimum deviation between apparent andtrue focii that can be obtained with a dry objectiveÐand even so, for a nominal focus 10 m m below thesample surface, the true (point) focus would actuallybe 15 m m. Also, this con® guration is not optimizedfor Raman microscopy as the diffraction-limited spotsize would be too large.

3. For the marginal ray, m 5 1, and Eq. 6 then yieldsthe maximum depth of a focal point in the sample fora given D .

Perhaps the most important point arising from thisanalysis is that the range of laser focal positions withinthe sample (i.e., the depth of focus, d.o.f.) is given byzm 5 1 2 zm 5 0 (Eq. 7).

776 Volume 54, Number 6, 2000

FIG. 3. Ratio of depth of focus (d.o.f.)/D as a function of NA, assumingthat diffraction-induced broadening is relatively unimportant. Note, forhigh NA and large D , the d.o.f. becomes very large (e.g., ; 22 m m for0.95 NA and D 5 10 m m). However, diffraction effects will dominateat low NA and small D , so in practice there will be an optimum NA( . 0), which minimizes the d.o.f.; see text for details.

1/22 2NA (n 2 1)2d.o.f. 5 D 1 n 2 n (7)

2[ ][ ](1 2 NA )

This important result means that the depth of focusincreases linearly with D ; not only does the point offocus change with D , but also the depth of focus. Thisconsideration makes interpretation of depth pro® ling re-

sults rather complicated to say the leastÐ the breadth ofan interface, as deduced from raw signal changes duringz-scanning, will be broader, the deeper the interface isburied in the sample. The effect can be very signi® cantÐa typical high-power ``dry’ ’ objective might have an NAof 0.95, for which Eq. 7 would predict a d.o.f. of ; 2.2 D .So, even if we only tried to focus 5 m m below the surface,the depth of focus would actually be 11 m m, extendingfrom 7.5 to 18.5 m mÐ which somewhat exceeds the ; 2m m spread that a depth pro® le from a silicon wafer wouldimply! It becomes impossible to obtain ``pure’ ’ spectraof thin layers once they lie several micrometers belowthe surface, unless one uses a special objective to mini-

mize refraction.Figure 3 summarizes the effect of NA on depth of

focus of the laser beam by plotting the ratio (d.o.f./ D ) asa function of NA; once the numerical aperture exceeds0.9, the relative depth of focus increases very rapidly.Note, however, that Fig. 3 completely neglects the effectsof diffraction, which tends to increase the depth of focusas NA decreases (in proportion to 1/NA 2). The overallresult is that we cannot monotonically improve the depthresolution by using lower NA objectives; at some pointdiffraction will dominate and the depth of focus will be-

gin to rise again. This point depends upon the value ofD ; refraction effects must dominate for suf® ciently highD . We therefore anticipate that the value of NA whichminimizes the depth of focus will depend on both n andD , and so will be system-speci® c. This point will be ad-

dressed quantitatively in a future publication; however,very rough calculations imply that for n 5 1.5 and D 510 m m, diffraction becomes important once the NA isreduced much below 0.9, and that the depth of focus

should be minimized for NA ; 0.8. For the experimentalsystems described in this paper, the refraction-only modelis perfectly adequate, and we neglect diffraction effectsthroughout.

In order to calculate the laser intensity distributionthroughout the focal region, we note that the integratedintensity of all rays originating from points on the objec-

tive with normalized radius m is proportional to m. If theradial intensity distribution of the laser beam is I (m), thenthe overall intensity of all rays originating from radius mis proportional to m´I (m), assuming circular symmetry.Thus we must weight the contribution of each ray by thisfactor. A plot of m´I (m) vs. zm gives the laser intensitydistribution along the focal axis and can be used to assessthe degree of ``defocusing’ ’ of the beam due to refraction.Furthermore, the ``average’ ’ focal point can then be as-

sessed from the ``center of gravity’ ’ (c.o.g.) of the inten-

sity distribution (Eq. 8).

m ´z ´I(m) dmE m

c.o.g. 5 (8)

m ´I(m) dmEObviously, the laser intensity distribution on the objectivein¯ uences the intensity distribution in the sample. As-

suming that the laser beam has a Gaussian intensity dis-

tribution, one normally tries to ® ll the entrance pupil ofthe objective so that the beam is apertured at the 1/e 2

intensity points. This procedure results in an Airy patternat the focal plane and produces a focal ``tube’ ’ with waistdiameter of ; 1.22 l /NA and depth of focus of ; 4 l /(NA)2. If this result is achieved, the objective is onlyslightly over ® lled, and the intensity distribution is a trun-

cated Gaussian [i.e., I (r ) 5 I0exp( 2 2r 2 /rmax2)], where the

waist radius containing ; 86% of the total beam intensitymatches rmax, the limiting objective radius. This distribu-

tion can be expressed in terms of the normalized radius m.

I (m) 5 I0 exp( 2 2m 2 / f 2) (9)

In Eq. 9 we have also included a ``® ll factor’ ’ f ; if f 51, we have optimum ® lling as described above. If f .1, the objective is over® lled and the distribution is trun-

cated at higher intensity values (less energy is transmit-ted), while the converse holds for f , 1. Figure 4 illus-

trates the laser intensity distributions [I (m)] for three dif-

ferent ® ll factors, and the corresponding radial distribu-

tion functions [m´I (m)]. The effect of truncation of thebeam pro® le for large f is obvious.

Figure 5 shows plots of m´I (m) vs. zm for three differ-ent values of D (2, 5, and 10 m m) assuming a 0.9 NAobjective, I0 5 1, and f 5 1. These plots, which havebeen normalized to constant integrated intensity, indicatethe expected intensity distribution as a function of depthbelow the air interface. For large D , a signi® cant intensityexists throughout the depth of focus de® ned by Eq. 7,i.e., a spread of ; 12 m m when D 5 10. For D 5 2, thedepth of focus is comparable to the diffraction limit( ; 2.5 m m), and so diffraction will cause signi® cant ad-

ditional defocusing.Raman Response Pro® le. Raman scattering can occur

at any point within the illuminated regions described by

APPLIED SPECTROSCOPY 777

FIG. 4. Variation of laser intensity pro® les on objective as a functionof ® ll factor f . Upper traces (a) are the raw intensity distributions I(m)vs. m; traces b account for the total intensity all of rays at normalizedradius m, i.e., m I (m); see text for details. Clearly, the transmitted in-

tensity is very sensitive to the ® ll factor.

FIG. 5. Laser intensity m I(m) vs. focal position in sample (zm) as afunction of D . Note how intensity pro® le broadens as one focuses deeperinto the sample. The integrated intensity is normalized to constant area.

FIG. 6. Comparison of Raman response pro® les as a function of zm fortwo different weighting schemes, assuming NA 5 0.9 and D 5 10 m m.The in¯ uence of the confocal aperture is minimal; see text for the func-

tional forms of the responses.

Fig. 5, with a magnitude proportional to the laser inten-sity at each point. The probability of a Raman photonbeing collected by the objective and passing through theconfocal aperture to reach the detector will depend uponthe point of origin (i.e., zm), which in turn depends on mfor the incident ray. How should we assess the relativecontributions of Raman scatter from each point? Thereare two obvious extremes for weighting the Raman re-

sponse as a function of zm. The ® rst ignores the in¯ uenceof the confocal aperture, and simply weights each pointaccording to the square of the numerical aperture (withinthe sample) into which a photon can be emitted and stillcaptured by the objective. This ``effective’ ’ NA is givenby Eq. 10.

2 21 m NA2(NA ) 5 ´ (10)eff 2 2 2 2[ ]n 1 2 NA 1 m NA

The second approach assumes that the confocal apertureworks perfectly, in which case only photons scatteredwith a speci® c angle will be collected and imaged by theobjective to pass through the aperture. Neglecting chro-

matic aberration, this angle is the same as that of theincident laser rays that are focused at point zmÐ and wehave already seen that the weighting factor for the prob-ability of a ray traversing this path is simply m.

We can now compute our ``Raman spatial response’ ’[R(zm)] by multiplying the laser intensity distribution[m I (m)] by the Raman ``weighting’ ’ factor. This proce-

dure yields two possible distributions, according to theweighting method chosen.

2R(z ) 5 m ´I(m) ´(NA )m eff

(numerical aperture weighting) (11a)

2R(z ) 5 m ´I (m) (confocal weighting) (11b)m

These two Raman response pro ® les are compared in Fig.6 for a nominal focal depth of D 5 10 m m. It turns outthat the distributions are fairly similar, with the main dif-ference being that the ``confocal’ ’ weighting schemeslightly favors signals originating deeper in the sample.It will be shown later that both schemes provide a rea-

sonable ® t to the experimentally observed data, and wehave no evidence for preferring one scheme over the oth-

er. With this in mind, Fig. 7 compares the Raman re-sponse pro ® les as a function of D , calculated under theassumption that the weighting scheme given by Eq. 11apertains. Inevitably, we ® nd that as D is increased, Ramanintensity is detected with an increasing depth of focus,and generation of ``pure’ ’ spectra from deeply buried lay-

ers or interfaces becomes impossible unless they are rel-atively thick ( . 10 m m). The inset in Fig. 7 compares theRaman response pro ® le with the laser intensity distribu-

tion, assuming D 5 10. Note how the Raman response ismaximized several micrometers deeper than the laserpeak intensity.

Comparison of Theory with Experimental Results.Before going too far with predicting the effect of various

778 Volume 54, Number 6, 2000

FIG. 7. ``NA’ ’ weighted Raman responses as a function of D , assumingNA 5 0.9. The response is broad even for D 5 10 m m. The insetcompares the laser intensity and Raman response pro® les for D 5 10m m.

FIG. 8. Schematic diagram illustrating the calculation of the substratesignal as a function of D . The signal is proportional to the area to theright of the coating/substrate interface. See text for details.

FIG. 9. Observed Raman spectra of acrylate-coated PET sample as afunction of D . At no point were pure acr ylate or PET spectra observed.

parameters on depth of focus and Raman response pro-

® le, it is sensible to compare predicted results with thoseobserved from a simple, well-de® ned sample. The systemchosen for the comparison was described in the experi-mental section, i.e., a 36 m m layer of an acrylate polymercoated on top of a thick ( . 200 m m) PET ® lm. This is auseful sample for this study since the coating and sub-

strate show good spectral contrast, have well-de® ned,uniform thicknesses, and do not signi® cantly interpene-

trate (sharp interface); also, the substrate is a very strongRaman scatterer. This means that intense signals can beobtained even from a point some tens of micrometersbelow the surface.

It is important to note than we cannot directly measurethe Raman response pro® les as shown in Fig. 7. In fact,only the change in Raman signal as the laser is focusedat different points, and the Raman pro ® le moves acrossa boundary, can be measured. The measured response isthe convolution of the (changing) Raman response pro® lewith the spatial composition pro® le of the sample. Weevaluate this overlap by calculating the variation in thesubstrate signal as we focus down through the coatinginto the polyester (i.e., predict the PET signal as a func-

tion of D ). This procedure entails calculating the overlapintegral (Eq. 12) depicted schematically by Fig. 8; i.e.,we must calculate the area beneath the Raman responsecurve to the right of the dashed line, representing theboundary between coating and substrate, as a function ofD . If we neglect attenuation of the signal due to absorp-

tion and turbidity, the variation in this area with D shouldmatch that of the observed Raman signal.

m 5 1

I(substrate) 5 R(z ) dm (12)E m

m 5 mb

In this integral, which is performed in m space rather thanzm space, mb is the value of m for which a Raman rayoriginates precisely from the coating/substrate boundary.For values m . mb, the rays originate from within thesubstrate; it is these rays which contribute to the detectedintensity. The value of mb is a function of D , because as

D increases the pro ® le broadens and shifts to higher zm,while the boundary position remains ® xed (at 36 m m inthis case). Therefore, to evaluate the integral we calculatethe value of m ( 5 mb) which corresponds to zm 5 36 m m(by inverting Eq. 6), and evaluate the integral using theappropriate weighting scheme from Eq. 11.

As mentioned above, the ``depth of focus’ ’ as mea-

sured with a silicon wafer was ; 2 m m, indicating thatthe system is well aligned and has the expected diffrac-

tion-limited depth of focus in air. However, we will showbelow that this is an unrealistic parameter by which tointerpret the response of thick, transparent samples withburied interfaces.

Figure 9 compares several spectra obtained on focus-

ing onto the coating surface and then moving the pointof focus down through into the PET. The evolution ofthe strong ring-stretching mode near 1612 cm 2 1 is ap-parent and can be quanti® ed by the band intensity[I (1612)], integrated from ; 1600 cm 2 1 to 1623 cm 2 1.This integrated intensity was used to test the predictedvariation of I (substrate) (Eq. 12). The spectrum of theacrylate coating is best monitored by using the intensityof an aromatic (additive) band near 997 cm 2 1. This ® gurealso illustrates the severe degradation in the depth of fo-

cus that occurs when focusing deep into a sampleÐ at nopoint do we observe a ``pure’ ’ PET or acrylate spectrum.

APPLIED SPECTROSCOPY 779

FIG. 10. Comparison of calculated (line) vs. observed (cross) data forPET signal as a function of D . Note the good agreementÐ in particular,the early onset of the substrate signal even though the coating was 36m m thick, and also the breadth of the interface ( ; 10 m m). The PETsignal saturated above D 5 20 m m; hence no experimental data areshown beyond this point.

FIG. 11. Repeat of experiment shown in Fig. 11. Calculations usingthe two weighting methods are shownÐ the predictions are essentiallyidentical. Note how well the predicted response matches the observedsubstrate signal; the model appears to reproduce the main observedfeatures.

FIG. 12. Improving the ® t by adjusting the ® ll factor f . The data implythat the objective might have been slightly under ® lled, compared to theoptimum for minimum diffracted spot size.

The PET band is particularly persistent even at small D ;this observation is in part due to the fact that the PETmode is much stronger than the acrylate spectrum, so itis intrinsically dif® cult to reject completely.

We tested the predicted response (Eq. 12) by measur-

ing I (1612) as a function of D . Two pieces of the coated® lm were examined to assess reproducibility. Figure 10compares the predicted response (solid line) with the ob-

served data, using ``numerical aperture’ ’ weighting (Eq.11a). Unfortunately, the acquisition time was set too highfor the ® rst experiment, so the PET band began to satu-

rate at D . 20 m m. Even so, the main features werecorrectly predicted, as follows:

1. The PET bands reached signi® cant intensity for D ;17 m m, even though the coating was 36 m m thick. Thisobservation clearly illustrates how refraction causes usto penetrate much deeper into the sample than the``nominal’ ’ focal point (a factor of ; 2 on average).

2. Even though the coating/substrate interface was sharp( , 1 m m), the PET signal grows much more slowly(over a distance of more than 8 m m). Thus the ob-served blurring is much greater than the diffractionlimit (but signi® cantly less than the maximum laserdepth of focus predicted by Eq. 7).

Figure 11 shows data from the repeat run, for whichband saturation was avoided. It also compares the theo-

retical ® ts from the two Raman weighting schemes (Eq.11) and shows that neither scheme is preferred in ® ttingthe observed resultsÐ both lines overlap almost perfectly.Figure 12 shows how the ® t can perhaps be improvedslightly if we assume that the objective is in fact under-

® lled ( f 5 0.8). The initial conclusions are therefore con-

® rmed by the repeat measurement; in particular, thatgrowth of the PET signal is blurred over a range of ; 10m m even though the diffraction-limited depth of focus isonly about 2 m m and the true interface is very sharp.Naive interpretation of the raw data could lead one tothe conclusion that the interface is broadened signi® -cantly; this would be wholly incorrect. Overall, consid-

ering the approximations that have been made, the pre-

dicted variations well match the observed data, and theray-trace analysis provides a valid approach for the broadinterpretation of confocal depth-pro ® ling results.

Although detailed analysis of depth-pro ® le data is fair-

ly complex, it is possible to carry out a ``back of theenvelope’ ’ calculation to assess the position of an inter-

face in a system, given its refractive index. Figure 13illustrates the variation in the ``center of gravity’ ’ of theRaman response as a function of NA and D , assuming n

5 1.5. This ® gure was derived by substituting R(zm) form I (m) in Eq. 8. When the c.o.g. is coincident with thecoat/substrate interface, the substrate signal should havereached about half its maximum value. For the observeddata shown in Fig. 11, this point occurs at about D ; 18m m, for which Fig. 13 places the c.o.g. at zm 5 36 m mfor a 0.9 NA objective. This result is in excellent agree-

ment with the known coat thickness.The Relationship between R(zm) and the z-Scan

Pro® le. It is worth considering why the width of the Ra-man response pro ® le R(zm), given by Eq. 7, is greaterthan the ``blurring’ ’ observed when monitoring the signalfrom the substrate during z-scanning. This is because dur-

ing z-scanning we observe the convolution of R(zm) with

780 Volume 54, Number 6, 2000

FIG. 13. Predicted variation in center of gravity (c.o.g.) of the Ramanresponse pro® le vs. objective NA, assuming n 5 1.5. For a ``® lled’ ’ 0.9NA objective, the center of the Raman response is located about twiceas deep as the nominal focal point D .

FIG. 14. Calculated effect of the depth of an interface on its apparentbreadth, assuming NA 5 0.9. Deeply buried interfaces will appear tobe inherently broader unless the data are corrected to account for re-

fraction.

FIG. 15. Effect of gross changes in ® ll factor on the predicted variationin substrate intensity. Note, for a f 5 0.5 objective, diffraction effectswill probably dominate in terms of spot size and depth of focus, so thepredicted response may be unrealistically sharp.

the compositional pro ® le, rather than R(zm) itself. Theconvolved response depends on not only the width of theresponse but also the ``speed’ ’ with which it moves acrossthe interface as D increases. The latter is quite high, sothe apparent interfacial region is sharper than R(zm) itself.Furthermore, the slope of this convolution pro® le (i.e.,the apparent ``sharpness’’ of the interface) depends onhow deeply the interface is buried in the sample. As wemove to deeper focal positions, the interface appears tobe broader owing to the broadening of R(zm). This effectis demonstrated in Fig. 14, which illustrates the predictedsubstrate signal, as a function of D , for interfaces buriedat 36 m m and 60 m m, respectively. Obviously, this effectmakes interpretation of depth pro® ling results complex;the raw intensity data can be very misleading. In short,we have to use Eq. 12 to assess whether an observedpro® le is due to a true interfacial effect, or simply anoptical artifact. The raw data convey little useful infor-mation.

Re® ning the Model. It is interesting to explore theeffect of changing certain parameters on the predictionsof the model. First, perhaps the most poorly de® ned ex-

perimental parameter is the ® ll factor, f . Figure 15 showsthe effect of changing f from 0.5 to 2. The predictedpro® le is particularly sensitive to under ® lling of the ob-

jective, while over® lling has a relatively small effect.Note, we cannot monotonically sharpen the focus by sim-

ply under ® lling the objective (or using a lower NA lens),since the diffraction-limited depth of focus will quicklybecome comparable to the refraction-induced defocusing.(As a rule of thumb, reducing the ® ll factor by ½ willquadruple the diffraction-limited depth of focus, and inthe case of a 0.9 NA objective give a depth of . 10 m mÐcomparable to the refraction-induced defocusing.) Forthis reason it is suggested that the curve representing f5 0.5 in Fig. 15 is not physically realistic.

Another effect that has been neglected so far is thedependence of re¯ ectivity on incident angle. This consid-eration is dif® cult to analyze rigorously, since the re¯ ec-

tivity is strongly in¯ uenced by polarization. However, asimpli® ed calculation, assuming fully depolarized radia-

tion, indicated that the effect is minimal, so it was ig-

nored for the purposes of this work.De® ciencies of the Ray Trace Analysis. The analysis

is by no means perfect, as can be seen from Figs. 10 and11. First, the PET signal was actually detected at an ear-

lier stage (lower D ) than was predicted. This result ispresumably due in part to scattering within the sample,broadening still further the depth of focus, coupled withthe effects of diffraction. It could also mean that there isa small contribution to the 1612 cm 2 1 band intensity fromthe aromatic component in the acrylate coat. Second, thePET signal in the ``interfacial’ ’ region (15±20 m m, Fig.11) shows systematic differences from the predicted sig-nal. This observation could be due to the fact that weneglected the second refraction that will occur at the coat/substrate interface, which will further increase the pene-

tration depth and depth of focus. We have also ignoredthe fact that not all radiation will be focused perfectlyalong the z-axis. There will be a ® nite spot radius per-pendicular to the z-axis which we have not modeled.Therefore, the analysis cannot be expected, in its currentform, to model the subtleties of the interfacial region onthe micrometer scale. However, the analysis can be usedto assess, in broad terms, whether an apparent change insignal on passing through an interface is due to a realcompositional variationt, or merely an optical aberration

APPLIED SPECTROSCOPY 781

FIG. 16. Predicting the signal of a substrate (PAN), which is thinnerthan the Raman response width. This analysis relates to previously pub-

lished depth-pro® ling studies.16 See text for details.

FIG. 17. Comparison of (a) raw data with predicted response and (b)data shifted by 2 m m, for the laminate described in Fig. 16 and Ref.16. A 2 m m error, in either ® lm thickness or de® nition of the D 5 0starting point, would give a good overlay between observed and pre-

dicted data.

due to refraction. It indicates the magnitude of the ``blur-

ring’ ’ that will occur even for a sharp interface, so unlessan observed variation occurs on a scale signi® cantlygreater than the predicted blurring, we must presume itto be an artifact. An example of its use in this mode isgiven below.

Example: Analysis of Raman Data on InterfacialDiffusion in Polymer Laminates. It is interesting to ex-

amine whether the simple approach outlined above canbe used to assist interpretation of previously publisheddepth-pro® le data. Much of the published work has en-

tailed analysis of relatively thin coats on thick substrates,with the aim of producing ``pure’ ’ top-layer spectra. Un-

der these conditions the in¯ uence of refraction is mini-

mized and diffraction effects should dominate. However,some workers have endeavored to analyze quite deeplyburied interfaces and have favored the ``optical section-

ing’ ’ confocal Raman geometry, since this approachavoids an physical sectioning of the sample. 14,15 However,this is just the sort of situation where refraction effectswill come into play.

As an example, we consider a study of interfacial dif-

fusion in poly(vinylalcohol)/poly(acrylonitrile) (PVOH/PAN) laminates carried out by Hajatdoost and Yar-

wood.15 In this work, ; 13 m m of PAN was coated ontoa glass slide, and then ; 20 m m of PVOH was coated ontop of the PAN. A 0.95 NA, 100 3 objective was used toobtain confocal Raman spectra as the laser focus wasmoved down from the air/PVOH interface into the PANsubstrate, and the integrated nitrile stretching band inten-

sity was used as a measure of the PAN concentration.The nominal depth of focus of the system (silicon re-

sponse) was about 2.5 m m FWHM. The authors notedthat refraction would shift the laser focus deeper into thesample, but did not comment on the effect on the depthof focus or its variation with D .

The authors clearly observed broadening of the nitrileband, which was logically attributed to hydrogen bondingof the polymers resulting from interdiffusion at the in-

terface. They also observed that the PAN signal growsrelatively slowly as one moves into the sample, and thispattern was attributed, in part, to diffusion and broaden-ing of the interface, as well as to detection of signal aris-

ing from outside the nominal depth of focus (see ® gure3 of Ref. 15). However, can any or all of the variationof PAN signal be attributed directly to refraction effects?To test this possibility, we calculated the expected PANresponse assuming f 5 1, a PVOH refractive index of; 1.5, and a thickness of 20 m m. This calculation is alittle more complex than our acrylate/PET calculation,since the PAN substrate is quite thin (13 m m) comparedwith the width of the Raman response pro® le ( . 20 m m)for a position D 5 10 m m (Fig. 16). We therefore needto calculate, as a function of D , the area in m space be-tween the points represented by m1 and m2, which cor-

respond to the PVOH/PAN and PAN/quartz interfaces,respectively. These interfaces should occur at zm 5 20and 33 m m, respectively, assuming ® lm thicknesses of 20and 13 m m.

Figure 17a compares the predicted variation (solidline) with the PAN intensities observed by Hajatdoostand Yarwood.15 Although the breadth of predicted andobserved responses is similar ( ; 6.7 and 8 m m), we pre-

dict that the PAN signal maximum should have been ob-

served at D 5 12 m m rather than D 5 14 m m. We cannotrationalize, using our model, the observed response un-less (1) the PVOH thickness was greater than 20 m m or(2) the PAN thickness was thicker than 13 m m. We cer-

tainly cannot rationalize the observed data in terms oflarge-scale diffusion of the PAN into the PVOH, sincethis would cause the PAN signal to be observed at lower,not higher, D .

If the PVOH thickness were actually about 22 m m, theobserved and predicted data would overlay quite well(Fig. 17b). This result may be a coincidence, but a smallexperimental error in layer thickness or in the de® nitionof the D 5 0 scanning position clearly has implicationsfor the analysis, and extreme care must be taken to rulethis possibility out. Furthermore, if the experimental dataare baseline corrected (to remove the offset due to PETsignal originating well outside the diffraction-limiteddepth of focus), the ® t is excellent given the simplicityof the model (Fig. 18). It is not wise to read too muchsigni® cance into this ® t, since we do not know the ® ll

782 Volume 54, Number 6, 2000

FIG. 18. If the raw data baseline offset is removed and a 2 m m shiftimposed on the raw data from Ref. 16, an excellent ® t is obtainedbetween observed and predicted data. This observation suggests thatthe effect of refraction accounts for the apparent interfacial broadening,and there is no need to invoke interfacial diffusion.

factor f for the author’ s equipmentÐ the quality of ® tcould change considerably if f is not approximately uni-

ty. Even so, the effects of refraction seem to be signi® cantand must be taken into account when interpreting thistype of data. Otherwise there must be a concern that theapparent interface broadening can be rationalized simplyas an optical artifact.

This example also shows how the apparent thicknessof a buried layer can be perturbed by refraction. Hencethe FWHM of the PAN signal was about 8 m m, eventhough the ® lm thickness was ; 13 m m! Thus the resultscan be doubly confusing, with interfaces apparentlybroadened but layer thicknesses reduced. The apparentlayer thickness will depend on its depth and can be thin-

ner, or thicker, than the true thickness. This considerationwill be discussed in a future publication.

Discussion: Minimizing Refraction. The analysis pre-

sented above clearly demonstrates that one cannot use adry metallurgical objective to accurately depth pro® lesystems by z-scanning; the aberrations due to refractionare just too severe. This is not to say that depth pro® lingcannot be achieved by z-scanning; provided that a suit-

able objective is employed, this approach is perfectlypossible. For example, using an oil-immersion objectivewill minimize refraction at the liquid/sample interface,provided that a suitable ¯ uid can be found which matchesthe refractive index without damaging the sample or dis-torting the Raman spectrum. However, if these objectivesare not available, the preferred solution for quantitativestudies is cross sectioning and lateral scanning.

CONCLUSION

A relatively straightfoward analysis, based on ray trac-

ing, allows the calculation of the micro-Raman responsepro® le as a function of the nominal focal point and theNA of the objective, assuming that a ``dry’ ’ , metallurgicalobjective is in use (the standard micro-Raman con® gu-

ration). With this approach the true positions of buriedlayers/features can be calculated from the raw data. Boththe position and the depth of focus increase dramatically

on moving deeper into transparent media, and it becomesimpossible to obtain ``pure’ ’ spectra of thin layers burieddeeper than a few micrometers. Interpretation of inten-sity±distance pro ® les obtained from ``z-scanning’ ’ iscomplexÐ even sharp interfaces appear to broaden verysigni® cantly, and their apparent position is much ``shal-

lower’ ’ than their true point in the sample. Apparent layerthicknesses are also distorted. Consequently, it is con-

cluded that analysis of phenomena at buried interfaces isvery dif® cult with this approach, and if it is to be at-

tempted, any apparent interface broadening or shiftingmust be compared with predicted data. This condition, inturn, requires that the optical system be well character-

ized in order to yield valid predictions.In short, physical sectioning and lateral mapping

through a cross section are strongly recommended as analternative approach to z-scanning, since there is then nodoubt about the position of the microprobe beam, and thespatial resolution remains constant throughout the pro ® le.If z-scanning is the only practical solution (for example,if a cross section cannot be obtained), an objective de-signed to minimize refraction at the sample interfaceshould be used. This requirement typically will necessi-

tate use of an immersion objective.

ACKNOWLEDGMENTS

The author would like to thank Dave Higgins (formerly of ICI Films)for providing the polymer laminate sample. ICI plc is thanked for per-

mission to publish this work.

1. G. Rosasco, ``Raman Microprobe Spectroscopy’ ’ , in Advances inIR and Raman Spectroscopy, R. Clark and R. Hester, Eds. (Heydon,London, 1980), Vol. 7, Chap. 4, pp. 223±283.

2. D. J. Gardiner, M. Bowden, and P. R. Graves, Phil. Trans. Roy. Soc.Lond. A320, 295 (1986).

3. G. J. Puppels, W. Colier, J. H. F. Olminkhof, F. F. H. de Mul, andJ. Greve, J. Raman. Spectrosc. 22, 217 (1991).

4. R. Tabaksblat, R. J. Meier, and B. J. Kip, Appl. Spectrosc. 46, 60(1992).

5. K. P. J. Williams, G. D. Pitt, D. N. Batchelder, and B. J. Kip, Appl.Spectrosc. 48, 232 (1994).

6. G. Turrell, M. Delhaye, and P. Dhamelincourt, in Raman Micros-copy: Developments and Applications, G. Turrell and J. Corset, Eds.(Academic Press, London, 1996), pp. 39±49.

7. S. Zhang, D. Palatini, T. Hancewicz, and J. Andrew, Abstracts fromFACSS Conference, Vancouver (1999), abstract No. 591, p. 240.

8. J. J. Andrew and T. H. Hancewicz, Appl. Spectrosc. 52, 797 (1998).9. N. L. Jestel, J. M. Shaver, and M. D. Morris, Appl. Spectrosc. 52,

64 (1998).10. N. Everall, ``Raman Spectroscopy of Synthetic Polymers’ ’ , in An-

alytical Applications of Raman Spectroscopy, M. J. Pelletier, Ed.(Blackwell Science, Oxford, 1999), pp. 127±192.

11. S. L. Zhang, J. A. Pezzuti, M. D. Morris, A. Appadwedula, C.-M.Hsiung, and A. Leugers, Appl. Spectrosc. 52, 1264 (1998).

12. N. Everall, K. Davis, H. Owen, M. J. Pelletier, and J. Slater, Appl.Spectrosc. 50, 388 (1996).

13. N. J. Everall, Appl. Spectrosc. 52, 1498 (1998).14. S. Hajatdoost and J. Yarwood, Appl. Spectrosc. 50, 558 (1996).15. S. Hajatdoost and J. Yarwood, Appl. Spectrosc. 51, 1784 (1997).16. L. Markwort, B. Kip, E. da Silva, and B. Roussel, Appl. Spectrosc.

49, 1411 (1995).17. R. J. Meier and B. J. Kip, Microbeam Analysis 3, 61 (1994).18. M. Delhaye, J. Barbillat, J. Aubard, M. Bridoux, and E. da Silva,

in Raman Microscopy: Developments and Applications, G. Turrelland J. Corset, Eds. (Academic Press, London, 1996), pp. 66±69.