Incidence angles for optimized ATR excitation ofsurface plasmons

Eric F. Y. Kou and Theodor Tamir

By quantitatively examining dispersion curves of surface plasmons along Ag, Al, Ni, and Fe films, we find thatthe incidence angle 00 that produces a reflection minimum in a variety of ATR geometries is actually differentfrom the angle 0p that corresponds to the phase-matching condition. The difference between Op and 00 is mostsignificant in the Otto and Kretschmann geometries using metal films with large damping loss, such as Ni orFe. In addition, we determine the angle 0 m at which maximum power density is coupled into the metal surfaceand show that in general the value of 0,,, is also different from both 00 and Op, the difference being again largestin the Otto or Kretschmann geometries. The differences between 0O, ,,p, and 0

m are much smaller, but stilldetectable, in long-range surface plasmon and extended-range surface plasmon geometries. In all cases, thepreferred incidence angle depends on whether one needs to optimize either the plasmon field composition orthe power density in the metal.

1. Introduction

When exciting plasmon waves along metal surfacesin attenuated total reflection (ATR) geometries, theincidence angle 00 that produces a dip in the reflectedfield has usually been assumed to optimize the plas-mon field. In particular, it was often understood thata beam incident at the angle 0 would be phase-matched to the plasmon wave, on the one hand, andwould then couple a maximum power density at themetal surface, on the other hand. Careful examina-tion of these conditions in a variety of ATR geometriesshows, however, that the incidence angles Oi that opti-mize the plasmon field are generally different from 0,.The aim of this paper is to present an analytical expla-nation for such differences and to provide quantitativedata for optimizing the excitation of plasmon wavesalong Ag, Al, Ni, and Fe surfaces in four basic ATRgeometries.

In the earliest studies of plasmons in ATR configu-rations,1-8 it was found that reflectometry measure-ments yield data that agreed reasonably well with thetheoretically predicted dispersion curves of plasmonwaves at visible-light frequencies. Thus Otto1 usedthe geometry shown in Fig. 1(a) to demonstrate that alight beam incident from the prism couples energy to asurface plasmon along the interface between the metal

The authors are with Polytechnic University, Department of Elec-trical Engineering & Computer Science, Weber Research Institute,Brooklyn, New York 11201.

Received 15 January 1988.

0003-6935/88/194098-06$02.00/0.© 1988 Optical Society of America.

substrate and dielectric gap. He observed that, owingto the intrinsic damping of the metal, the excitation ofthe surface plasmon was accompanied by an absorp-tion dip in the reflectance curve plotted as a function ofthe incidence angle. Kretschmann and Raether2 thenapplied this approach to a different geometry, asshown in Fig. 1(b), to show that a surface plasmon iscoupled along the lower boundary of a metal film by alight beam incident from the prism. Subsequently,both Otto and Kretschmann geometries have beenused extensively for studying the dispersion relation ofsurface plasmons and for measuring the dielectric con-stant of metals.3 -8

Later studies have recognized4' 910 that the absorp-tion minimum in the reflected field does not yield theexact value for determining the dispersion relation ofthe surface plasmon. The difference from the exactvalue is very small for surface plasmons along low-lossmetal films and may, therefore, be neglected whenconstructing the dispersion curves. However, for sur-face plasmons along very lossy metals, this differencebecomes so significant that the reflectance minimumcannot be used for that purpose. Consequently, vari-ous schemes were developed to improve the accuracyof the ATR method. 9 -13 One is based on the detectionof light emitted through the roughness of the metalsurface in the ATR geometry.9-11 Another one makesuse of the photoelectronic effect which occurs if theenergy is larger than the work function of the met-al.1 0 12 13 Both procedures are based on the assump-tion that the phase-matching condition for exciting thesurface plasmon is accompanied by a maximum in thefield intensity along the metal surface. However, ourresults show that this maximum does not occur at the

4098 APPLIED OPTICS / Vol. 27, No. 19 / 1 October 1988

phase-matching incidence angle, i.e., the angle thatcorresponds to the plasmon wavenumber.

To clarify why both the reflection dip and the maxi-mum field intensity are different from the phase-matching condition, we specifically examine surfaceplasmons in Ag, Al, Ni, and Fe films in the Otto andKretschmann geometries as well as in the long-rangesurface plasmon (LRSP) and the extended-range sur-face plasmon (ERSP) geometries. The LRSP geome-tries, which are shown in Fig. 1(c), were developed bySarid14 and Stegeman et al. 15 to provide plasmonwaves having longer propagation ranges than thosealong the Otto or Kretschmann configurations. TheERSP geometry was proposed by us16 and involves anadditional dielectric layer, as shown in Fig. 1(d); thislayer can extend the propagation range by more than, 1order of magnitude larger than that in the LRSP geom-etry. By using a careful evaluation of the plasmondispersion curves, we find that the phase-matchingcondition, the maximum power density in the metalfilm, and the dips observed in the reflectance functioncan all occur at different incidence angles. In fact, wefind that these differences can be substantial, especial-ly in poorly absorbing metal film in the Otto andKretschmann geometries. It, therefore, follows thatthe incidence angle needs to be carefully determined ineach situation. Specifically, its value may be differentdepending on whether it is required to optimize theplasmon field by working at the exact phase-matchingcondition or to optimize the energy coupling process bymaximizing the power density at the metal surface.

II. Reflectance of ATR Geometries and SurfacePlasmon Excitation

As shown in Fig. 1, the four pertinent ATR geome-tries consist of a prism with dielectric constant uplaced on a structure consisting of layered media ( =

aXa

I1

_ S - -- - _-- -- - - -- -

(a) Otto

/ ei\(c) IS

(c) LRSP

e,- e- -trt

(b) Kretschmann

at~~~~~~t- a-

a3 t3

a4 t,64 '4t

(d) ERSP

Fig. 1. Typical ATR configurations for exciting surface plasmons:(a) Otto; (b) Kretschmann; (c) long-range surface plasmon; and (d)extended-range surface plasmon geometries. In all cases, em de-

notes the dielectric constant of the metal film.

1, 2, or 4) having dielectric constants Ej (j = 1,2, . . ,1)and a substrate with dielectric constant Es. Thesedielectric constants are generally complex numbers,and the thickness of the respective layer is denoted bytj. We choose a coordinate system whose x axis isparallel to the upper boundary of the ATR configura-tion and whose +z direction points into the substrate.For simplicity, we assume that the fields and media areinvariant with respect to the y direction.

For a parallel polarized (TM) plane wave, the mag-netic vector has a single component H along the ydirection. Assuming then that a plane wave is inci-dent in the prism in the form

Hi = exp[ik(Kx + 'rZ)], (1)

where a time dependence exp(-iwt) is implied andsuppressed, the reflected field in the prism can beexpressed as

H. = r(K), exp[ik(Kx - Tu)] (2)

where k = 27r/X, X is the wavelength in vacuum, and K

and mrt are normalized longitudinal and transversepropagation constants in the superstrate, respectively.The reflection coefficient r(K), which is defined as theratio of the amplitude of the reflected field to that ofthe incident field at the prism base, is given by171 8r(K) = 12- (Zm 22- ZM11) -ZuZm21 N(K) (3)

M 12 - (Z.m22 + Zum11) + ZuZm2 - D(K)

where the coefficients mqr(q,r = 1,2) are defined by

FM11 M121[ I2] = MIMIi ... M 2M 1, (4)LM21 M22J

and Mj is a 2 X 2 matrix given by

Mj cos(kTjzj) iZj sin(krjzj) 1iYj sin(kTjzj) cos(krjzj)

with

Zj = rj/ej, (6)

Tj = +(j- K2

)'1

2, j u,1,2, - 1,,, (7)where u and s designate the superstrate (prism) andsubstrate regions, respectively.

In Eq. (3), the zeros of the denominator D(K) yieldthe propagation factors KP of the plasmon field; i.e., thedispersion equation of surface plasmons is given by17"18

D(KP) = M1 2 - (Zm 2 2 + ZumI) + ZuZsm21 = 0. (8)From the expressions of mqr in Eqs. (4) and (5), we inferthat D(KP) is generally a complex transcendental func-tion. Its roots are, therefore, complex numbers Kp = KP+ ZKp, which depend on the values of the parameters X,tj, and Ej. Here K and K designate the phase factorand the decay constant of the plasmon field, respec-tively.

The phase-matching condition for the incident fieldat the prism base is given by

Ki = , sin0i = Cc,, sin0p = KP, (9)

1 October 1988 / Vol. 27, No. 19 / APPLIED OPTICS 4099

where As is the incidence angle, Ki is the phase factor ofthe incident field along x, and the pole angle 0 isassociated with the real part of KP.

Inside every jth layer, the field coupled by a planewave incident at the phase-matching condition, i.e., K

K' in Eq. (1), has the form

Hj = [Hj+ exp(ikTjpzj) - Hj. exp(-ikrjPzj)] exp(ikKpx), (10)

where Hj+ and Hj- are the amplitudes of forward- andbackward-traveling field components, respectively, ,rpis obtained by substituting K = K' in Eq. (7), and

j-1Zji=z E (11)

n=1

is the transverse distance measured from the interfacebetween the j - 1 and j media. The amplitudes Hj+and Hj. satisfy the recursive relation

(Hj+ + Hj)] Mj 1 { [H()+ + H(l)]Z('l))1 (12)H - Hi.. M. H~j-)+ - H~j-)+ J

so that

[ H- ]= MjlMj-2.. .M2M1 1 r(K) 1* (13)(H+ H.jZ =IM 1 - r()Z 1

The power density can then be evaluated by usingthe Poynting vector

P = 1/2 Re(E X H*), (14)

where the electrical field E is obtained through Max-well's equations, and the asterisk is used to denote thecomplex conjugate.

Ill. Results and Discussion

In the following, we discuss the implication of theresults obtained by calculating the reflection coeffi-cient r(K) and the plasmon power density P that areassociated with the coupling of a plane wave to a sur-

face plasmon in the various ATR geometries of Fig. 1.Both the reflectance r(K) defined in Eq. (3) and thepower density P given by Eq. (14) are calculated eitheras functions of the wavelength X at a fixed incidenceangle Oi or vice versa. When using X as a variable, theoptical constant of the dielectric materials is assumedconstant, but that of metals is derived through theDrude model, as discussed in the Appendix. We willconsider r(K) and P in the visible light frequenciesrange and only incidence angles 0i that are larger thanthe critical angle of total reflection in each respectivegeometry.

We first observe that the behavior of r(K) is quitedifferent quantitatively if the metal film is highly con-ductive. To illustrate the basic differences, we show inFig. 2 the case of a very good conductor (silver) film inan Otto geometry having eu = 4, Ed = 2.25, and Em =-15.966 + iO.5256 (Ag at X = 0.6328Mm). By contrast,in a poorer conductor such as aluminum, the variationof r(K) is shown in Fig. 3 for a Kretschmann geometryinvolving em = -29.8 + ill.6 (Al at X = 0.6328 ,um). Inboth cases, the magnitude of r(K) is shown for the total-reflection range d S Ki S F. We note that the profileof lr(K)I always reveals a dip at some point KO = FEZ sino,,along the Ki = FEj sini axis. Because both silver andaluminum are lossy metals, the energy of the incidentfield is partially absorbed in the process of coupling tothe surface plasmon; consequently, the magnitude ofthe reflected field can be sharply reduced even in thetotal-reflection regime. At K, i.e., for O = 0,,, theabsorption in the metal is largest and r(K)I is smallest.By assuming that the coupling to the plasmon field isstrongest when the absorption into the metal is largest,the angle 0 was often assumed to be the one thatsatisfies the phase-matching condition. To examinethis assumption, we recognize from the expression ofr(K) in Eq. (3) that, at KO, the magnitude of the denomi-nator D(K) must be .much greater than that of thenumerator N(K) to provide the dips in Figs. 2 and 3, i.e.,

0.5

o _1.5 , = vs i n - % F. sino%

Fig. 2. Variation of the magnitude of the reflection coefficientIr(KI)l and the normalized power density P at the metal surface inthe Otto geometry shown in the inset. P,, is a normalization con-

stant. In this case, K Kn r KP Km.

cp /c} cN IC = V7-Siini -

Fig. 3. Same as in Fig. 2, except that the curves are calculated forthe Kretschmann geometry shown in the inset.

4100 APPLIED OPTICS / Vol. 27, No. 19 / 1 October 1988

ID(Kx)I >> IN(K)I. (15)

We, therefore, introduce a new complex quantity Kn =K' + iK', which is a null (zero) of the numerator N(K),i.e.,

IN(Kn)l = 0. (16)

For convenience, we may assume that, of all possiblenulls of N(K), the zero at Kn is the one closest to K 0. Ifthe pole at Kp defined in Eq. (8) satisfies the conditionKP << K, it is clear that K could not easily satisfy aninequafity such as that given in Eq. (15). Equations(15) and (16) then suggest that K is closer to the realpart K' of Kn than to the real part K' of the pole KP, i.e.,

KO _ Kn = Cc, sinan / K (17)

We have calculated the locations of both K' and K' forthe cases illustrated in Figs. 2 and 3. Their values arevery close to each other and coincide with K on thescale of Fig. 2 for silver, which has very low absorptionlosses; however, these values are quite different andthey appear distinctly in Fig. 3 for aluminum, whichhas larger losses. To amplify further the above argu-ment, we present in Table I values of AO = p - i,where Op and Oi were defined in Eq. (9). This table wascalculated for typical situations involving coupling toplasmon modes in Ag, Al, Ni, and Fe films in variousATR geometries. The values in Table I were obtainedby using physical parameters for which r(K0)I is alwaysless than 1/2, so that the dip is clearly discerned incurves for r(K) such as those in Figs. 2 and 3. We notethat the magnitude of AO increases with the imaginarypart of the dielectric constant Em in the metal film, andit has much larger values in the Otto and the Kretsch-mann geometries than those in the LRSP and ERSPgeometries. In extreme cases, such as Fe in theKretschmann geometry, the magnitude of AO can be aslarge as 200. On the other hand, for a low-loss metalsuch as Ag, AO is trivially small in both the LRSP andthe ERSP geometries.

As is also implied in Eq. (17), we recognize thatfunctional analysis shows that K = Fe,, sinGn is general-

Table 1. Typical Values of Aby = 0, - Gn; In All Cases eu = 4 and theDielectric Materials In Evanescence (Nongulding) Layers have a Constant

e = 2.25

Material Ag Al Ni Fe

Cm -16 + i 0.53 -29.8 + i 11.6 -12.9 + i 16.4 -3.07 + i 19.3

Otto 0.026° 0.50° 3.50 5.1°(t1 in pm) (0.380) (0.296) (0.145) (0.148)

Kretschmann -0.0430 -1.T -9.40 -20°(tm in pm ) (0.054) (0.020) (0.020) (0.025)

LRSP 0.00023' 0.00830 0.020° 0.00550(tl in lpm ) (1.000) (0.611) (0.433) (0.382)

tm = 200A

ERSP -0.000160 -0.001T -0.0069° -0.00640(t1 in m ) (0.933) (0.514) (0.368) (0.306)

64 = 2.9584

ly not exactly equal to K = XEj sinG0. However, wefound the difference lK, - K to be negligibly small inall cases, so that we shall assume Kn = K henceforth.However, we recall that K £ xP, and, because thisdifference can be large in certain cases (as illustrated inTable I), we conclude that measuring the dip at Ko willgenerally not yield the correct propagation constant Kpfor plasmon waves.

To compare the difference between the accuratedispersion curve, which yields exact values of Kp via Eq.(8), and the minimum of the reflection coefficient,which is obtained from Eq. (16), we take Ni as a partic-ular example to examine the curves of KP(X) and Ki(X)in the various ATR geometries. In Fig. 4, we consider

l.75F

1.71-

1.65 .

1.6 F

1.55

0.4 0.5 0.6

A(pm) -

0.7 0.8 0.9

T62.6

1.95

1.3

0.65

Fig. 4. Curves of K, Kn, Kn, and as functions of the free-spacewavelength X for the Otto geometry shown in the inset. The loss

factor 6 is given by = Iem/cml-

A (p ) -

Fig. 5. Same as in Fig. 4, except that the curves are obtained for theKretschmann geometry shown in the inset.

1 October 1988 / Vol. 27, No. 19 / APPLIED OPTICS 4101

e = 4

a1 = 2.25 | 0145pm

, (A)

---------------

% 0)

0.6 0.7A (n )-

Fig. 6. Same as in Fig. 4, except that the curves are obtained for theLRSP geometry shown in the inset.

the case of a surface plasmon supported in an Ottogeometry. We note that Kp is always larger than K' andthat the difference is smaller at the shorter wave-lengths, where the loss coefficient 1 e,,/EI is also larger.Figure 5 shows analogous curves in a Kretschmanngeometry. In this case, we note that K' is always small-er than Kn and their difference is also smaller at theshorter wavelengths. The curves in Fig. 6 were calcu-lated for the LRSP geometry shown in the inset. Inthis case, K' is always slightly larger than K', but thedifferences are much smaller than those in Figs. 4 and5. The deviation of K(X) from K(X) is still wider atlarger values of X in Fig. 6.

It follows that we cannot generally use a reflection-coefficient measurement to construct dispersioncurves for surface plasmons because the location of thedip in the Ir(K)I vs K curve does not yield directly thevalue of K; but is more closely related to that of K. Thedifferences between K' and K' generally increase withabsorption in the metal, and, for the same metal sam-ple, those differences are larger in the Otto andKretschmann geometries than those in the LRSP orERSP geometries.

Other methods have been proposed to measure thepropagation constant Kp of surface plasmons in ATRgeometries. Most of them have measured the re-sponse function (e.g., P in the substrate) so as to locatethe maximum of the power density coupled along themetal surface.9-13 They then assumed that the inci-dence angle that couples maximum power density intothe metal is also the angle that satisfies the phase-matching condition. However, we have used Eq. (14)to calculate the value Pm of that power density at themetal surface and found that such an assumption maybe quite erroneous. Thus, as shown in Fig. 3, themaximum at Km of the response function Pm is alsodifferent from the value of K'. To further emphasizethis point, we show in Table I the difference between

the angle Am that corresponds to the maximum powerdensity on the metal surface and the pole angle Op inthe geometries used for Table I. From Table II, we seethat the magnitude of AOm = 0,,, - Op is largest for Niand Fe in the Otto and the Kretschmann geometries,but it is negligible for Ag in the LRSP and ERSPgeometries.

Using Eq. (14), we can find the dependence of themaximum location Km on the wavelength X. The varia-tion of Km(X) is also indicated in Figs. 4, 5, and 6 for Niin the various ATR geometries. In an Otto geometry,as shown in Fig. 4, the value of Km is closer to K' than toKP. However, as shown for the Kretschmann geometryin Fig. 5, the value of Km is indeed closer to Kp than to K,,.

Figure 6 also shows that the difference between Km andKis larger than that between K' and KP. Therefore, byusing the measurement of the maximum at Km of thetransmitted power, as suggested in Ref. 9, the approxi-mation of the dispersion curve K' (X) is good only for theKretschmann geometry, but it becomes quite poor forboth the Otto and LRSP geometries.

IV. Conclusions

We have shown that the location of the reflection dipKn may be appreciably different from that of the phasefactor K. This difference is largest for Ni and Fe in theOtto and Kretschmann geometries, and it is smallestfor Ag in the LRSP and ERSP geometries. Conse-quently, the variation of K,(X) yields a poor approxima-tion of the dispersion curves for surface plasmons inthe Otto and Kretschmann geometries having metalfilms with large damping loss.

The location Km of the maximum in the power cou-pled into the metal is also different from K'. When Nior Fe films are used in the Otto or Kretschmann geom-etries, the variation of K(X) deviates significantlyfrom the dispersion curve K(X). In the Otto geometry,owing to the fact that Km is closer to K', the minimum ofthe reflection coefficient becomes an important indi-cation of the coupling of a larger plasmon field intensi-ty in this type of geometry. On the other hand, K iscloser to Kp than to Kn in the Kretschmann geometry.Therefore, to measure the dispersion relation of sur-face plasmons in an ATR geometry, the method thatdetects the maximum of the transmitted power canimprove the accuracy only in the Kretschmann geome-try; however, this method may be quite misleading inthe other geometries.

On the other hand, we have found that the differencebetween the values of Kp, kn, and Km are usually verysmall in the LRSP and ERSP geometries. Therefore,in addition to the advantages offered by their corre-

Table II. Values of AO, = 0,- 0, In the ATR Geometries Used for Table I

Ag Al Ni Fe

Otto 0.016° 0.50° 4.2° 4.9

Kretschmann -0.00180 -0.320 -1.2° -1.2°

LRSP -0.00300 0.0087° -0.00520 -0.038°

ERSP 0.00630 0.0210 0.0150 0.0130

4102 APPLIED OPTICS / Vol. 27, No. 19 / 1 October 1988

sponding surface plasmons,6 19 reflectometry mea-surements in these two geometries yield very reliableinformation on the pertinent dispersion curves of theseplasmons. For application purposes, however, thechoice of Op = sin-(K / ) 0, = sin-'(K/FI), or Om =sin'1(Km//E,) for the incidence condition is dictated bythe specific parameter to be optimized. In practice,this implies that Op will be preferred to emphasize the(phase-match) plasmon field condition, but Om will bechosen to enhance the power density in the metal.

The authors thank Dror Sarid, Optical SciencesCenter, University of Arizona, for alerting them to theimportance of the subject matter presented here andfor providing background material. This research wassupported by the National Science Foundation undergrant ECS-8711004, the Joint Services ElectronicsProgram under contract F49620-82-C-0084, and theNew York State Center for Advanced Technology inTelecommunications.

Appendix: Frequency Dependence of the DielectricConstant in Metals

Based on the free electron model,20 the dielectricconstant of metal is given in terms of the plasma fre-quency wp and the electron mean scattering time t as

Em em + em' (Al)2

E1- 2S (Ala)w 2t2 + 1

em 2t + 1) (Alb)

Using the real and imaginary parts of the dielectricconstant given at a particular wavelength SO we canalso rewrite em and em as functions of the wavelength Xas

E.(X) [ X- r(o)](l + 1 ' (A2a)(X)2+ 0X1'(X]

Em = X0m] (A2b)

where

= Em(X) (A2c)em(^O)

For example, the variation of the dielectric constant ofNi used in Figs. 4-6 are calculated through

E =1- 23.89 (A3a)Em 1-~+070.6328)2 (Aa

o (1-E' )Em = 0.5379 (A3b)

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