Interference-enhanced Raman scattering from T102/S10 2

multilayers: measurement and theory

Richard A. Craig, Gregory J. Exarhos, Walter T. Pawlewicz, and Ralph E. Williford

Raman-band intensities for sputter-deposited TiO2/SiO2 multilayer coatings are found to exhibit a strongdependence on excitation wavelength of the probe laser. A model based on standing waves within the stackcorrectly predicts the magnitude and wavelength dependence of the observed enhancements.

1. Introduction

Rapid nondestructive characterization of dielectriccoatings is paramount in studying the factors whichcontrol optical performance and coating stability inextreme conditions. Raman spectroscopy, whichprobes the local vibrational environment, has beensuccessfully applied to the characterization of crystal-line phase, stoichiometry, and homogeneity of sputter-deposited TiO2. 1 2 Previous work on TiO2 /SiO2 mul-tilayers demonstrated that significant signalenhancement was possible by either designing thecoating for a specific probe wavelength 3 or using atunable laser to find the most sensitive wavelength.4

It is the purpose of this paper to demonstrate apredictive model for the Raman enhancement in mul-tilayer stacks. A successful model for multilayer in-terference enhancement allows prediction, beforemeasurement, of which probe wavelengths will give themost intense Raman signal for a given multilayer-coating design. This avoids the tedious trial-and-er-ror switching of lasers and laser wavelengths for themeasurement. Enhancement of Raman-band intensi-ties allows structurally significant spectral featuressuch as peak frequencies, widths, relative heights, andsmall minor-phase peaks to be more accurately mea-sured.

A model which predicts the wavelength-dependentRaman-scattering-intensity enhancement was devel-oped based on time-averaged electric fields. Thismodel incorporates the effect of probe electric-field

The authors are with Pacific Northwest Laboratory, Richland,Washington 99352.

Received 22 January 1986.0003-6935/87/194193-05$02.00/0.© 1987 Optical Society of America.

enhancement, Raman-signal transmission coeffi-cients, and Raman-scattering wavelength dependence.The generalized model, based on layer thickness andrefractive index, predicts the probe-laser wavelengthdependence of the Raman signal in good agreementwith Raman-band intensity variations measured usingcw probe radiation from a tunable dye laser. Thephysical basis for this model and a comparison be-tween predicted Raman enhancement and experimen-tally determined Raman-excitation profiles for a mul-tilayer TiO2/SiO2 coating are reported here. Thephysical process of interference enhancement of Ra-man scattering reported here for multilayer stacks isconceptually similar to that reported by Kerker et al.5

and Chew and Wang6 for spherical geometry.

II. Experimental

Two similarly fabricated TiO2/SiO2 multilayer coat-ings were examined in these experiments. These weresingle-cavity and three-cavity Fabry-Perot transmis-sion filters. These designs were selected because cal-culations for the three-cavity multilayer predict anorder-of-magnitude enhancement of Raman scatter-ing inside the transmission peak compared with that inthe neighboring high-reflectance region. The designsof the single- and three-cavity filters are described asDS and DLDLDS with D = (HL)2H(LH). Here H andL represent the usual quarterwave optical thickness ofhigh-index (TiO2, n 2.40) and low-index (SiO 2 , n -1.46) material, and S represents the substrate (fusedsilica, n 1.46). The design wavelength for bothstructures (610 nm) was chosen to coincide approxi-mately with the center of the emission band of a rhoda-mine 6G dye laser pumped with multiline radiationfrom an argon-ion laser. Coatings were fabricated byreactive diode sputtering; conditions for sputtering ofthe TiO2 layers were selected to yield 100% anatase-phase composition.7 The anatase phase exhibits avery intense Raman mode at 143 cm-1; this mode was

1 October 1987 / Vol. 26, No. 19 / APPLIED OPTICS 4193

used to determine the wavelength dependence of theRaman intensity for a specific vibrational mode.1 8

Spectra were recorded in a backscattering geometry,Z(XY)Z, using tunable output from a Coherent model590 cw dye laser having output power between 30 and300 mW over the tuning range of the dye. Scatteredlight was imaged onto the slits of a SPEX 0.85 doublemonochromator using all-reflective optics and wasmeasured photometrically. Detector response wasfound to be linear over several orders of magnitude.The detected signal was compared to a fraction of thedye-laser output to correct for intensity fluctuations.Slit widths were maintained at 400 Asm. A quartzcuvette filled with 1,2-trichloroethane served as thereference standard. This material exhibits a mediumintensity Raman line near the 143-cm-1 Eg for anataseand does not undergo resonance enhancement in thevisible region of the spectrum. Relative Raman-bandintensities were obtained by comparing the Ramansignal with that from the reference standard. In thismanner, the response of the instrument (grating effi-ciency and detector response) to different wavelengthsand intensities of light was standardized.

111. Theory

The physical model adopted for predicting the Ra-man-band intensities in the Z(XY)Z direction is basedon the following assumptions: (1) The electric-fieldstrength of the Raman signal is proportional to theelectric-field strength of the probe source at that point;the constant of proportionality depends on the materi-al at that point and is otherwise independent of posi-tion. (2) The forward-scattered and back-scatteredfield strengths are equal for equal probe fields. Thefirst of these is a standard assumption of Raman the-ory. The second is not a requirement for the physicalmodel but is a simplification appropriate to the Ra-man-active medium (anatase-phase TiO2) being stud-ied.

Three cases were studied: (a) Raman signals fromdifferent places in the stack add coherently; (b) Ramansignals from different places in the stack add incoher-ently; (c) Raman signals arising from a single TiO2layer add coherently but add incoherently to signalsfrom other layers.

Prediction of the Raman signal strength is a three-part problem: (1) calculation of the electric-fieldstrength at arbitrary points within the stack, (2) calcu-lation of the Raman signal as a function of the probeelectric-field strength; and (3) calculation of the re-lease of the Raman signal from within the stack.

The method of Arnon and Baumeister9 was used tocalculate the electric-field strength throughout thestack. That method relates the electric-field strengthand phase at an arbitrary point to that in the substrateand in turn relates the electric-field strength in thesubstrate to the incident field; both relationships areeffected using the method of the characteristic matrix.Application of this approach gives the electric-fieldstrength at an arbitrary point z in the stack:

E(z) = 2E0 (mj1 + m{2n,)/[(m11 + M12 ) + ( 12 + M22)n]

= CE(Z)EO. (1)In Eq. (1), the mij are the components of the character-istic matrix for the entire stack, the mf are the compo-nents of the characteristic matrix for the part of thestack forward of the point z, n8 is the substrate refrac-tive index, and E0 is the electric-field strength of theincident light. The coefficient CE(z) is a function of zby virtue of the dependence of mf on z.

By the first of the assumptions above, the Ramanelectric-field strength is proportional to the electric-field strength of the probe source at each point, theconstant of proportionality depending only on the na-ture of the material at that point.

Release of Raman-scattered radiation from the inte-rior of the multilayer stack is addressed here as a self-consistent cavity problem. At each point within thestack the Raman signal can be scattered either in theforward direction or the backward direction. For eachdirection of propagation, two things can happen to theradiation: (1) the radiation can be transmittedthrough the remainder of the stack exiting from thesystem or (2) the radiation can be reflected by theremainder of the multilayer stack.

In turn, the reflected part of the signal can be trans-mitted or backreflected repeatedly through the multi-layer stack. The net result is a set of standing wavesfor the Raman signal in the same manner as for theincident probe with the emission of the Raman-shiftedsignal out of the multilayer stack in the forward andbackward directions. The standing waves (i.e., thesum of all the reflected components) constitute thetotal Raman-scattered signal. The mathematicalanalysis of this physical picture (Appendix) leads to abackscattered Raman-signal electric-field strength re-sulting from the material in an infinitesimal elementdz at the point z of

dER = tB(f/32)E(z)dz. (2)In Eq. (2), tB is the transmission coefficient for the partof the stack backward of the point z, 3 is the Ramanproportionality factor expressing the first assumptionabove (E is the Raman-scattered electric-fieldstrength resulting from an incident signal strength E),and E(z) is the probe electric-field strength at z and isgiven by Eq. (1). The Raman coefficient fi varies asthe second power of the Raman frequency.2 This de-pendence was included in our analysis by writing =2coRa (Ref. 3), where a is, approximately frequency-independent. Equation (2) was numerically integrat-ed over all the Raman-active layers of the multilayerstack to obtain the total Raman-signal electric-fieldstrength for the three coherence conditions describedabove. Dispersion and absorption were included forthe TiO2. Representative values of index and extinc-tion coefficient were used for the analysis because sam-ple-to-sample variations of -3% in index and of anorder of magnitude in extinction coefficient are to beexpected in similarly prepared samples. Neither dis-persion nor absorption was included for the io2 lay-ers.

4194 APPLIED OPTICS / Vol. 26, No. 19 / 1 October 1987

(NM) (NM)

Three-Cavity Fabry-Perot Single-Cavity Fabry-Perot

(a) (b)

Fig. 1. Transmission spectra for (a) single- and (b) three-cavityFabry-Perot structures.

IV. Results

The multilayer stacks fabricated for this study havebroad high-transmission bands centered at -620 nm(622 nm for the single-cavity multilayer stack and 617for the three-cavity stack). These exhibit high reflec-tivity in the near IR and green spectral regions asshown in Figs. 1(a) and (b). The single-cavity stackexhibits a single-humped transmission peak; thethree-cavity stack exhibits a three-humped transmis-sion spectrum. It is noteworthy that the transmissionfor the middle peak of the three-cavity spectrum isgreater than that for the long-wavelength peak. Thisis indicative of a deviation from the design structure.A perfect structure would exhibit a monotonic increasein transmission from the short-wavelength to the long-wavelength peak. Such a deviation is not surprisingand is consistent with the expected variation in opticalproperties of TiO2 noted above. However, the precisenature of this deviation in structure was not identified.

Raman spectra were recorded for a number of exci-tation frequencies around the red transmission band.Figure 2 shows several representative Raman spectraand depicts the strong 143-cm-1 Eg line of the anataseform of TiO2. The SiO2 layers contribute little to theRaman signal in the scattering geometry chosen for themeasurements.2 These spectra have all been normal-ized with respect to probe-laser intensity and instru-ment response and show marked band-intensity varia-tions with probe wavelength. The optical propertiesof the coating are manifested in Raman-band intensi-ties that change dramatically with probe wavelength assummarized in Table I. Normalized band intensitiesvary by an order of magnitude as a result of interfer-ence phenomena. These results are compared withthe theoretical analysis in the following section.

V. Discussion

Comparisons of experimental data for the three-cavity Fabry-Perot filter with theoretical relative Ra-man intensities for the three coherence conditionsconsidered are shown in Fig. 3. Normalization waseffected at the respective peaks. In general, the agree-ment between theory and experiment is seen to bequite satisfactory. In detail, the case of mixed coher-ence demonstrates somewhat better agreement thaneither the totally incoherent or totally coherent case.

WAVENUMBERS

Fig. 2. Raman spectrum of 143-cm-1 line.

Table I. Normalized Band Intensities for the 143-cm 1 Anatase Mode

Probewavelength Three-cavity One-cavity

(nm) Fabry-Perot Fabry-Perot

574.1 - 145577.0 404 -580.8 556 263583.3 1108 273586.5 1077 292590.2 1297 330592.8 1370 383595.1 1281 -

597.0 817 450598.5 736 442600.4 658 481605.0 464 550612.3 460 552621.2 503 514624.4 637 507628.9 319 -632.8 344 297636.5 448 -

Figure 4 compares the experimental data for thesingle-cavity Fabry-Perot filter with theoretical rela-tive Raman intensities for the three coherence condi-tions considered. The general agreement betweentheory and experiment is seen to be satisfactory withthe mixed coherence case providing somewhat betteragreement than the other two cases.

VI. Conclusions

The frequency dependence of normalized Ramanintensities for the Eg mode in a multilayer anatasecoating has been determined, and contributions to theenhancement have been identified. A model based onstanding-wave electric fields in single- and multiple-interference cavities successfully explains the data.

The authors wish to thank D. M. Friedrich for hisreview and helpful comments. They also wish to thanka referee for suggesting that the various coherencecases be discussed and for suggesting consideration ofthe mixed coherence condition.

This work was supported by the Materials Sciences

1 October 1987 / Vol. 26, No. 19 / APPLIED OPTICS 4195

600 610 620 630 61

0.5 - a0.

0.3 0.2 _ U0.1 I I I I I

560 570 580 590 600 610 620 630 6

1.0 Mixed Coherence

0.9 _

07 - +f 0.7 -

0.6-

0.5_

0.4-a

0.3 - a

0.2- 4

1.0

0.9

0.8

0.7

0.6

0.5

0.4

d.3

0.2

57

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2o5E

0

560 570 580 590 600 610 620 630 640

Fig. 3. Raman enhancement data for three-cavity Fabry-Perotstructure compared to calculations for three coherence conditions.

Division of the Office of Basic Energy Sciences, U.S.Department of Energy. Pacific Northwest Laborato-ry is operated by Battelle Memorial Institute for theU.S. Department of Energy under contract DE-AC06-76RLO 1830.

Appendix: Cavity Calculation

This Appendix examines the relationship betweenincident laser electric-field strength at a point withinthe laser-active material and the backscattered Ra-man-shifted electric-field strength at a point outsidethe multilayer. The issue is that the source of theRaman signal is located in a resonant cavity-albeit avery leaky cavity-thereby affecting the density ofstates for scattering of the Raman-shifted signal.

To see how this comes about, consider a unit-ampli-tude plane wave emitted in the forward direction fromthe point z within the multilayer stack. Point b rep-resents the boundary of the remainder of the stackbehind the point z, and f represents the boundary ofthe remainder of the stack forward of the point zo.The quantities r, r, t, and tB represent, respectively,the reflection coefficient of the remainder of the stackforward of the point z, the reflection coefficient of theremainder of the stack backward of the point z, thetransmission coefficient of the remainder of the stackforward of the point z, and the transmission coeffi-cient of the remainder of the stack backward of thepoint z. The quantities 4' and are the phase from

3 i-. o

C; .

T _,

I a

1.0

0.9

0.8k

0.7F

0.6

0.5

350

340

0.

O _0.2' I 1 1 I 1 1 1 1 540 550 560 570 580 590 600 610 620 630 640 650

Fig. 4. Raman enhancement data for single-cavity Fabry-Perotstructure compared to calculations for three coherence conditions.

point b to point f and the phase from point b to zo,respectively.

The plane wave, reaching the forward boundary ofthe cavity at f, is partially transmitted and partiallyreflected. The transmitted fraction exits from thecavity in the forward direction and contributes to asignal that might be observed in the forward direction.The reflected wave in turn suffers partial transmissionand partial reflection at the backward boundary of thecavity at b. The transmitted fraction exits from thecavity in the backward direction and contributes to asignal that might be observed in the backward direc-tion. This process occurs repeatedly, the amplitudesof the forward and backward transmitted fractionsadding. The result is a wave emitted out of the multi-layer stack in the forward direction, a wave emitted outof the multilayer stack in the backward direction, and amix of forward and backward propagating waves be-tween f and b.

Similarly, a plane wave initially emitted in the back-ward direction undergoes repeated partial transmis-sion and partial reflection at the front and back sur-faces. The result is qualitatively as described above, awave emitted out of the multilayer stack in the forwarddirection, a wave emitted out of the multilayer stack inthe backward direction, and a mix of forward andbackward propagating waves between f and b. (The

4196 APPLIED OPTICS / Vol. 26, No. 19 / 1 October 1987

K Coherent

U -

I I

1.0

0.9

0.

0.7

0.6

0.5

0.4

0.3

2.00.1

1.00.9

0.'8

0.'70.6

30 570 580 590

U

r Incoherent

3 - + it\~~

Coherent

) _

0 580 590 600 610 620 630 640Incoherent Direct

,~~~ _U0 570 580 590 600 610 620 630

Mixed Coherence

_~~~~ U

- - - --P n

tO

relative amplitudes of the waves are, of course, differ-ent.)

To describe the emission process, we, therefore, usea basis set different from the plane-wave basis and thatincorporates these repeated partial transmissions andreflections. We introduce two basis functions, APF and

4 B; 41F is derived from the plane wave initially propa-gating in the forward direction, while ,B is derivedfrom the plane wave initially propagating in the back-ward direction. Thus

VtF exp(inkz) exp[i(k - )]1/D in the substrate

t/7(z) = exp[ink(z - z)] (Ala)

+ rF exp[-ink(z - zo)] exp[2i(I - )]1/D between b and f

ItBrF exp[-ikz) exp[i(20 - O)]1/D out of the multilayer

stack in the backward direction,

ItF exp[inkz) exp[i(o + 0)]/D in the substrate

A(z) = exp[-ink(z - zO)]

+ rB exp[ink(z - z0 )] exp(2iO))/D (Alb)

between b and f

[tBrF exp(-ikz) exp(iO)]/D out of the multilayer stack in the

backward direction,

where D = 1 - rFrB exp(2io) and 0 = nkzo. Here n isthe complex index of refraction in the active medium,nS is the index of refraction of the substrate, and k isthe vacuum wave vector.

The Raman-shifted field at z originating from thepoint z0 can be written in terms of these basis functions4'F(z) and 0B(z), namely,

ER(z) = AtF(z) + B/A(z). (A2)

At z0, the Raman field is directly related to the incidentfield via

ER(ZO) = flE(zo). (A3)

Thus Eq. (A2) requires

ER(zO) = Atg(z,) + Bpr(zo). (A4)

One additional condition is needed to specify A and B.This comes from the symmetry of the Raman-scattingprocess. For the materials considered here, the matrix

element for scattering into a plane wave in the forwarddirection is equal to that for scattering into the back-ward direction. Thus, at z, the coefficient of theforward-propagating Raman-scattered plane wave isequal to the coefficient of the backward-propagatingRaman-scattered plane wave in Eq. (2). (For Raman-scattering centers for which this symmetry does notprevail, use of the Arnon and Baumeister methodwould provide no simplicity because the incident wavewould have to be decomposed into forward and back-ward propagating plane waves for scattering analysis.)Applying these two conditions gives

A = ER(zo)[1 - rB exp(2i0)]/2,

B = ER(zo)l - rF exp[2i(q - 0)]1/2.

(A5a)

(A5b)

Thus, outside the multilayer stack in the backwarddirection, ER(Z) = [ER(ZO)tB exp(iO) exp(-ikz)]/2.Thus the differential contribution to the Raman elec-tric-field strength at a point outside the multilayerstack arising from an element dz centered on a point z0within the stack is dER(z) = dz{ER(zo)tB exp[i0(zo)]}/2.

References1. W. T. Pawlewicz, G. J. Exarhos, and W. E. Conaway, "Structural

Characterization of TiO 2 Optical Coatings by Raman Spectrosco-py," Appl. Opt. 22, 1837 (1983).

2. G. J. Exarhos, "Substrate Signal Suppression in Raman Spectra

of Sputter Deposited TiO 2 Films," J. Chem. Phys. 81,5211 (1984).3. R. J. Nemanich, C. C. Tsai, and G. A. N. Connell, "Interference-

Enhanced Raman Scattering of Very Thin Titanium and Titani-um Oxide Films," Phys. Rev. Lett. 44, 273 (1980).

4. G. H. Exharos and W. T. Pawlewicz, "Raman Characterization of

All-Dielectric Multilayer SiO2/TiO2 Optical Coatings," Appl.

Opt. 23, 1986 (1984).5. H. Chew and D.-S. Wang, "Double Resonance in Fluorescent and

Raman Scattering by Molecules in Small Particles," Phys. Rev.

Lett. 49, 490 (1982).6. M. Kerker, H. Chew, and P. J. McNulty, "Model for Raman and

Fluorescent Scattering by Molecules Imbedded in Small Parti-cles," Phys. Rev. A 13, 396 (1976).

7. W. T. Pawlewicz, P. M. Martin, D. D. Hays, and I. B. Mann,"Recent Developments in Reactively Sputtered Optical ThinFilms," Proc. Soc. Photo-Opt. Instrum. Eng. 325, 105 (1982).

8. R. J. Capwell, K. S. Pagnolo, and M. A. DeSeva, "A Rapid Deter-mination of Low Concentrations of Anatase in Rutile TiO 2 Pig-ments by Raman Spectroscopy," Appl. Spectrosc. 26, 537 (1972).

9. 0. Arnon and P. Baumeister, "Electric Field Distribution and theReduction of Laser Damage in Multilayers," Appl. Opt. 19, 1853(1980).

Pacific Northwest Laboratory is operated by Battelle MemorialInstitute for the U.S. Department of Energy.

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