Surface plasmons (SPs) are surface electromagneticwaves that can propagate along metal surfaces. Inview of the superb dielectric properties of gold andsilver, many studies were performed with thesemetals. The effects related to SPs were discoveredmore than a century ago, starting with reflectionanomalies on gratings, observed by Wood [1], someof which were later explained in terms of surfaceplasmons, following Fano [2] and Ritchie [3]. Sincethe works of Kretschmann and Raether [4,5], the at-tenuated total internal reflection (ATIR) geometry,employing the matching of the phase velocities ofthe incident light and SPs along the metal surface(SP resonance condition) via a prism, is one of themost commonly used, and its different applicationswere reviewed by Raether [6]. Several new directionsemerged more recently. The sensitivity of the propa-gation velocity of SPs to the dielectric properties ofthe adjacent medium makes them an efficient toolin biosensing [7]. The attenuation of SPs determinesthe width of the SP resonance, which affects the sen-sitivity of the sensor. The attenuation length of SPs
also determines the spatial resolution of an imagingtechnique in the ATIR configuration [8]. A planaroptical geometry is attractive for compact opticaldevices interconnected with electronic components,therefore the development of the optics of SPs [9]is also one of the promising directions of research.It was shown that, for a symmetrical structure withthe metal film sandwiched between two dielectriclayers, a mode of long-range SPs can propagate [10].In view of the special requirements of the geometry ofsuch a system, long-range plasmons are not consid-ered in this paper.
For the above-mentioned developments, the propa-gation of the SPs is crucial, and it was studied usingdifferent approaches. The attenuation of SPs is re-lated to the angular width of the minimum in thereflected light, and it strongly depends on the lightwavelength [4,5]. The portion of light converted intoSPs eventually produces heat. Consequently, SPresonance can be studied with photo-acoustic techni-ques [11]. When excited with white light, SPs of dif-ferent frequencies propagate different distancesgiving rise to “color jets” that can be viewed in amicroscope [12]. Propagating SPs can be visualizeddue to effects of their near field. For instance, whenthe medium adjacent to the metal film is fluorescent,a decaying trace of fluorescence can be observed along
the direction of SPs propagation [13]. Also other opti-cal near-field techniques have been developed, suchas the photon scanning tunneling microscope, whichrelies on the near-field coupling of the SP field to thesharpened optical fiber tip of a probe [14].The attenuation length is affected by the scatter-
ing of SPs on the surface roughness, scratches, andinhomogeneities of the metal film or the adjacentmedium [15,16]. Much work was dedicated to inves-tigating the possibility of extracting roughness pa-rameters from measurements of the scattered light[17,18] and relating them with the observed SP reso-nance angular dependences [19]. The absorption ofthe adjacent medium provides additional atten-uation, and this effect can be used to measure itsabsorption [20,21].In the present work, we directly observe the pro-
pagation of SPs on a gold film with roughness bydetecting the scattered light in the far field with amicroscope and an attached CCD camera. The ex-perimental results are compared with model calcula-tions that take into account different factors affectingattenuation, including surface roughness. The ex-perimental study was performed at two optical wave-lengths for 633 and 805nm. For reference, thecalculations of the SP attenuation length were per-formed in a broad wavelength interval.
2. Theoretical Description
The attenuation length of SPs in a planar systemconsisting of an arbitrary number of smooth layers(without roughness), some of which are metallic forguiding SPs, can be found with the following proce-dure that is based on the expressions for the inten-sity reflection coefficient, or reflectivity for short, R.The problem of reflection of light from a layered sys-tem leads to a system of algebraic equations. For agiven frequency the complex wavenumber of a SPcorresponds to the zero of the determinant of thissystem. Consequently, the solution for zero of the de-nominator of the reflection coefficient, or the require-ment jRj → ∞, determines the dispersion equation ofSPs [5,22]. For a multilayered structure, the reflec-tion coefficient can be determined following the itera-tive procedures [23] (described below) and [22] (thelatter is a generalization of the approach used in[5]). The dispersion relation was also derived expli-citly for an arbitrary number of layers [24], and fora four layer system an explicit form of the determi-nant is presented in [25]. Thus, knowing R ¼ Rðk;ωÞas a function of the parallel to the interface wave vec-tor component, k, and the light frequency, ω, one canfind the dispersion relation of SPs by solving theequation jR−1j ¼ 0.We calculated the reflectivity R for a layered sys-
tem with smooth planar interfaces and an arbitrarynumber M of layers using recursive equations [23]:
R ¼����Zin;1 − Z0
Zin;1 þ Z0
����2: ð1Þ
In this recursive procedure for an intermediate layerm, each input impedance Zin;m, as well as each layerimpedance Zm, are calculated in descending se-quence of layer numbers. It is assumed that thetwo outer media are semi-infinite, and the mediumm ¼ 0 corresponds to z > 0. The laser beam is inci-dent on plane z ¼ 0 at an angle θ with respect tothe positive z direction, which also coincides withthe direction of the normal to the interface. The inputimpedance and layer impedance at a particular layerm are calculated from the relations
light wavelength in vacuum, i is the imaginary unit,and the integer index m runs from 0 to M − 1. Thefield in each medium can be presented as a sumof a direct wave ∝ expð−ikm;zzÞ and (except for thelast medium with m ¼ M − 1) a backward wave∝ expðikm;zzÞ, propagating correspondingly into(−z) and (þz) directions. Consequently, the require-ment Imðkm;zÞ ≥ 0 must be fulfilled for these wavesto have finite amplitudes.
The condition jR−1j ¼ 0 gives an equation for k,which in general has a complex solution kSPðωÞ.Although kSP enters the denominator of Eq. (1) inthe same way as k, it has a different meaning. Weintroduced k as a projection of the wave vector ofthe incident light on the metal surface boundary.For the solution kSP, its real part determines thewavenumber, and its imaginary part determinesthe amplitude attenuation coefficient of the SP.The resonance excitation corresponds to the condi-tion k ¼ ReðkSPÞ, which takes place at the incidenceangle of the reflectivity minimum that also corre-sponds to the maximum conversion of light into SPs.
To take into account the influence of the surfaceroughness on the propagation of SPs, we examineda three layer system consisting of a glass prism(m ¼ 0), ametal film (m ¼ 1)with a complex dielectricpermittivity ε1 ¼ ε1;r þ ε1;i and a thickness d1, and anadjacent dielectric medium (m ¼ 2), which in generalcan also have losses, so that ε2 ¼ ε2;r þ iε2;i. We as-sume that the metal film at the interface with themediumm ¼ 2 has certain roughness, and then referto themodels that neglect roughness as “0-ordermod-els”. The presence of roughness will be taken intoaccount in the first order perturbation approach;see further Eqs. (8) and (13).
In the approximation ε1;r ≫ ε1;i and ε2;r ≫ ε2;i,which is usually fulfilled for gold and silver filmsin the optical and IR spectral regions, the reflectivitycan be expressed [5] in a form with Lorentziandenominator, explicitly showing the resonance char-acter of the interaction:
is the complex wavenumber of the SP that takes intoaccount surface roughness. The star ð�Þ denotescomplex conjugation, Γi ¼ Γi1 þ Γi2; kp ¼ ð2π np=λ0Þis the wavenumber of the SP corresponding to thecase of infinitely thick metal film, and Δkp ¼½ða2 − ε02Þ=ða2 þ ε02Þ�g is the correction to this num-ber accounting for the finite thickness of the film. Thefollowing notations were introduced in the aboveformulas:
g ¼ 2kpnp2 exp½−2kpðε1r=ε2rÞ1=2d1�=ðReε2r þ jε1rjÞ;
The quantities Γi1 and Γi2 describe the damping thatoriginates from the internal losses in the metal andthe adjacent medium, respectively, and Γrad de-scribes the radiative loss due to the transmissionof light through the metal film. Quantities Δkrand Γr in Eq. (4) describe the changes of the realand imaginary part of the wavenumber due to sur-face roughness. The term Γr includes additional con-tributions to attenuation of SPs related to thechanges of the dielectric properties due to the surfaceroughness, rescattering losses, as well as the lossesresulting from the conversion of SPs into radiativebulk waves.The SP resonance is most pronounced for the opti-
mal metal film thickness d1;opt, which is about 47nmfor gold. The inverse of the imaginary part of kSP;rdetermines the attenuation length (at e−1 level) ofthe SPs in terms of the SP field amplitude. However,the decay of the SP intensity, which is proportional tothe square of the field, is measured in most experi-ments. As a result, for the intensity attenuationlength a factor of 0.5 must be introduced:
Lsp;r ¼ 0:5ðΓi1 þ Γi2 þ Γr þ ΓradÞ−1: ð6Þ
As it follows from Eq. (3), this length can also beapproximately determined from the FWHM widthΔθ of the resonance curve:
Lsp ≈ ð ffiffiffiffiffiε0pk0 cos θresΔθÞ−1: ð7Þ
The incident light with a power P0 scatters from arough surface on the back side of themetal film yield-ing the power dP that goes into a solid angle elementdΩ [17]:
dP ¼ 4P0
�πλ
�4 jε2j0:5cos θ jt012ðθÞj2jWðθÞj2 ~Gðk0 − kÞdΩ ;
ð8Þ
where θ0 is the incidence angle of the excitation light,jt012ðθÞj2 is the transmission function for a two-boundary system with a metal film [specifically, thisfunction is the square of the ratio of strengths of themagnetic field at the second boundary (metal-air)and in the incident wave]. jWðθÞj2 is the dipole radia-tion function of the surface (for an explicit expressionsee [17], Eq. (2)). ~GðkÞ is the two-dimensional powerspectral density (PSD) function
~GðkÞ ¼ 1
ð2πÞ2Z
GðrÞ expð−ikrÞdr ð9Þ
of the correlation function of the surface roughnessςðrÞ:
GðrÞ ¼ 1S
ZS
ςðr0Þςðr0 þ rÞdr0; ð10Þ
where the integral is taken over the illuminated areaof the rough surface S. In Eq. (8) the argument of thePSD function is the difference of the interface com-ponents of the wave vectors of the scattered andincident light. We assume, following the often usedapproximation, that the roughness correlation func-tion is isotropic and Gaussian, i.e.,
GðrÞ ¼ δ2 exp�−r2
σ2�; ð11Þ
where σ is the correlation length and δ is the averageheight of the roughness. In this particular case, thetwo-dimensional PSD function is proportional to theone-dimensional PSD function, ~G1ðkÞ, namely
~GðkÞ ¼ σ2
ffiffiffiπp ~G1ðkÞ; ~G1ðkÞ ¼δ2σ2
ffiffiffiπp exp�−k2σ24
�:
ð12Þ
The roughness changes the complex wavenumber ofSPs as well as the SP dispersion equation [26–29].We calculated this wavenumber change due toroughness
ΔkSP ¼ kSP;r − kSP ¼ Δkr þ iΓr; ð13Þ
using Eq. (A42) of [29]. The real part of Δk is respon-sible for the angular displacement of the SP reso-nance and the imaginary part determines the
change in attenuation of SPs and also the change ofthe width of the resonance curve due to roughness.Using the approximation of Eqs. (4) and (5), onecan also calculate an effective dielectric constant ofthe metal, ε1;eff , which gives the same change ΔkSPas the surface roughness (assuming that its contribu-tion is relatively small), i.e., ε1;eff can be found fromthe equation
½kp þΔkp þ iðΓi þ ΓradÞ�jε1−>ε1;eff
¼ ½kp þΔkp þΔkr þ iðΓi þ Γr þ ΓradÞ�jε1 : ð14Þ
In the left-hand side of this equation ε1 is substitutedby ε1;eff, while the right-hand side is calculated for ε1,and it takes into account the corrections Δkr and Γrdue to roughness. The obtained value of ε1;eff canthen be used to calculate the modified-by-roughnessSPR curve, using the exact formulas of Eqs. (1) and(2) or the approximate expressions of Eqs. (3)–(5).The results of such calculations are presented inSection 4.The transmission function is determined as [6]
jt012ðθÞj2 ¼����ð1þ r21Þð1þ r10Þ expðik1d1Þ
1þ r21r10 expð2ik1d1Þ����2; ð15Þ
where rij ¼ ðεjki − εikjÞ=ðεjki þ εikjÞ with i ¼ 1, j ¼ 0and i ¼ 2, j ¼ 1 are the Fresnel’s reflection coeffi-cients at two interfaces for p-polarized light. For suchlight the function jt012ðθÞj2 has a resonance character,and it strongly increases at ϑ ≈ ϑSPR, acquiring itsmaximum in proximity of the optimal thickness ofthe metal film. Thus, the intensity of the scatteredlight can be strongly enhanced at the SP resonance.In the experiment, we inferred the attenuation
length of SPs from the decay of the scattered lightintensity with the distance from the illuminationstripe. To evaluate this attenuation length we usedthe following procedure. We assume that the laserbeam with a power P is focused on the surface ina stripe with a Gaussian intensity distributionIðx; yÞ ¼ ðP=πabÞ expð−x2=a2 − y2=b2Þ, where b and aare the length and the width, respectively. For a ty-pical attenuation length of surface plasmons Lp, thedimensionless diffraction parameter ðλpLp=πb2Þ issmall, where λp is the wavelength of the SPs; thusthe propagation of surface plasmons from a stripcan be considered one dimensional. We assume that,on average, the surface roughness is uniform alongthe film, and consequently the scattering of SPs intoradiative bulk waves is, on average, the same overthe surface. At the SP resonance, light due to scatter-ing of SPs prevails over the scattering produced bythe directly incident light because of the enhance-ment factor of Eq. (15) for the SP field. The SP inten-sity at a certain position x is proportional to theintegral of all partial contributions from differentcross sections of the irradiation spot, taking into ac-count also the decay of SPs during their propagation.Then the observed density F of the light radiation
from the surface in the far field is proportional tothe integral
FðxÞ ∝ expð−x2=a2ÞZ∞
0
exp½−x21=a2
� x1ð2x=a2 þ 1=LspÞ�dx1; ð16Þ
representing the intensity distribution of the SP fieldalong the surface, where signs (�) correspond to theSPs propagating in the directions x → ∓∞. For largedistances, jxj ≫ a, Eq. (16) reduces to an exponentialdecay∝ expð�x=LspÞ. The distribution of Eq. (16) hasasymmetry, because for the ATIR configuration SPspropagate on the surface in the direction of the pro-jection of the wave vector of the incident light on theinterface plane. In the following, Eq. (16) and itsasymptotic exponential function is used for fittingand extraction of the SP attenuation length frommeasurements of the distribution of the scatteredlight.
3. Experimental Setup
To excite and observe SPs and their propagationproperties the following setup was employed (Fig. 1).For the excitation we used the Kretschmann–Raether configuration, with a prism enabling thecoupling of the incident laser light to the surfaceplasmon mode [5,6]. Two different lasers were uti-lized for the excitation of SPs at two different wave-lengths: a He–Ne laser with a power of 3mW and awavelength of 633nm and a cw Ti:sapphire laserwith a wavelength of 805nm and a power attenuatedto about 20mW. For comparison, the measurementswere done for p and s polarizations of the incidentlight. For this purpose, the initial polarization of thebeam was adjusted with a half-wave plate, so thatthe intensities of the p and s components of the inci-dent light were of the same order of magnitude.
Fig. 1. Experimental setup for studies of surface plasmons.
The laser beam was initially expanded with a col-limator to a 10mm diameter and then focused with acylindrical lens through a hemicylindrical prism intoa stripe of about 8 μmwidth on the surface of a sensorchip (research grade CM5 chip, BIAcore AB). Thesensor chip had a 47nm thick gold film on a 0:3mmthick slide of BK7 glass. The carboxymethyl dextranlayer was removed from the surface of the film, whichwas roughened in the process. The light passingthrough the prism was coupled to the clear side ofthe glass slide by using a matching fluid (CargilleLabs Incorporated). The SP resonance excitationtook place when the light was p polarized (witha calcite polarizer, p − s intensity ratio 105 : 1). Thelight scattered by propagating SPs close to the nor-mal direction on the back side of the gold film (oppo-site to the side of the incident laser beam) wascollected by a microscope (30× magnification) and de-tected with a CCD camera. To avoid saturation of theCCD camera, the light intensity was adjusted byneutral density filters. The imaged surface region en-compassed the area around the strip illuminated bythe focused laser beam. The CCD image was formedby 2048 × 1500 pixels, and a typical acquisition timewas about 60 μs. The resulting digitized 2D image re-flected the distribution of the intensity of scatteredlight over the surface. The SP resonance phenomen-on manifested itself as a dark area in the reflectedbeam projected on the screen. To register the inten-sity distribution of the reflected light, the CCD cam-era was appropriately positioned in place of thescreen. From the CCD camera the image was trans-ferred to a PC for further processing with XCAP soft-
ware (EPIX Incorporated). Roghness measurementsof the gold film were done using tapping mode AFM(Veeco, CP II) in ambient air at driving frequenciesranging from 70–90kHz. Antimony-doped siliconcantilevers with a spring constant of 15N=m andnominal tip radius of 8nm were used.
4. Results
The angular distribution of the reflected light inten-sity in the vicinity of the SPR angle for 633 and805nm is shown in Fig. 2. The minima indicatethe angular positions of the SP resonance, which oc-curred at the light incidence angle onto the gold filminterface of 44:5° for 633nm and 42:8° for 805nm.The dashed lines show calculated SPR curves with-out surface roughness. Thin solid curves are obtainedtaking into account roughness correction [Eq. (13)],then calculating the value of the effective dielectricconstant ε1;eff from Eq. (14) and substituting it intothe formulas of Eqs. (1) and (2). For the calculationswe used the dielectric constants compiled in Table 1.The roughness parameters were determined frommeasurements with an AFM; see Fig. 3(a). Fromthe profile measurements, the 1D PSD function ofthe surface roughness was calculated Fig. 3(b). By fit-ting the 1D PSD function with a Gaussian of the formof Eq. (12) [see the inset in Fig. 3(b)], we determinedδ ¼ 2:0nm and σ ¼ 36nm.
The plots of the decay of the light intensity awayfrom the focal region are shown in Fig. 4 for the ex-citation by both p and s polarizations. The light in-tensity decays from the center of the excited SPRstripe.
Fig. 2. (Color online) Measured and calculated SPR curves for (a) 633nm and (b) 805nm.
Table 1. Dielectric Constants of Gold ε1 and Glass ε2, SP Attenuation Length Without Roughness Lsp; Effective Dielectric Constant ε1;eff;Attenuation Length Lsp; r Calculated with Roughness (δ ¼ 2:0nm, σ ¼ 36nm) and Experimentally Determined SP Attenuation Lengths
The small peaks in Fig. 4(b) appearing on the slopeof the dependences originate from scattering onscratches and on local inhomogeneities of the goldfilm. For coupling with a prism, as in our case, theSPs propagate in the same direction as the projectionof the wave vector of the incident light on the surface.Consequently, the light intensity falls off on a rela-tively long distance along the direction of the SP pro-pagation (negative x direction in Fig. 5) and on ashorter distance in the positive x direction.The attenuation length is determined from the
decaying tail in thedistribution of the scattered inten-sity beyond the laser illumination stripe on thesurface of the gold film. Fitting the curve for p polar-ization of Fig. 4(a) with Eq. (16) was performed for thewhole range presented. For Fig. 4(b), the fitting wasdone only in the range shown by the fitting curve,since the presence of a flat portion shows a saturationof the CCD camera close to the excitation stripe.As the result of fitting, we found the following
parameters for the optical wavelength 633nm: widtha ¼ 4 μm and the attenuation length Lsp ¼ 3 μm. For805nm, the fitting produced the value Lsp ¼ 15 μm.The large (by a factor of about 20) difference in theintensity of the images produced by p and s polariza-tions is due to the fact that SPs are excited only for p-polarized light. Consequently, on the back side of thegold film, where the scattered light is registered, for
p polarization the scattering is enhanced by the fac-tor of Eq. (15). To illustrate the effect of this enhance-ment, the factor jt012ðθÞj2 was calculated and plottedin Fig. 5 for the parameters of our system and thedielectric constants of gold at 633nm and 805nm (so-lid lines). Dashed lines show similar calculationsperformed for the determined effective dielectric con-stants, accounting for the roughness of the film. Itshould be noted that the experimental enhancementfactor is somewhat less, because it was observed for aconvergent incident wave, while the calculation wasperformed for a plane wave.
The attenuation length was also calculated in abroad wavelength interval (see Fig. 6) for smoothgold and silver films of thickness 47nm, borderingglass and air in the Kretschmann–Raether geometry.These calculations were done by using the conditionjR−1j → 0 for the reflection coefficient of Eqs. (1) and(2) (exact 0-order model) and also with formulas ofEqs. (5) and (6) (approximate 0-order model). The di-electric constants for gold and silver from [30–32]and for BK7 glass from [33] were used, as shownin Table 1. The calculated and experimental valuesof the attenuation length for 633 and 805nm are alsopresented in Table 1.
As can be seen from Fig. 6, the attenuation lengthstrongly increases with the optical wavelength,reaching values on the order of a millimeter at
Fig. 3. (Color online) Surface roughness measured with AFM: (a) false color surface relief and (b) the PSD function of the surface rough-ness in logarithmic scale. The inset shows the fitting of the PSD function by ~G1ðkÞ of Eq. (12) with δ ¼ 2:0nm and σ ¼ 36nm [axes have thesame units as in Fig. 3(b)].
Fig. 4. (Color online) Measured distribution of the scattered light intensity near the laser illumination stripe for (a) 633nm and(b) 805nm.
λ ¼ 2:4 μm. The thickness of the film also stronglyaffects the SP losses. Figure 7(a) presents the angu-lar dependences of the reflectivity, described byEq. (1), in the vicinity of the resonance for differentthicknesses of the gold film for an optical wavelength633nm. For thin films the resonance becomes verybroad, while for thick films the dip in the reflectivityreduces, which means that the coupling of light withthe SP mode decreases.The dependence of the attenuation length on the
thickness of the gold film [see Fig. 7(b)] shows thatthis length increases with the thickness of the filmand experiences flattening of the dependence at lar-ger thicknesses, which starts at around d ¼ 70nm.
5. Discussion
For observing the propagation of SPs, we used theirscattering on the surface roughness, which is alwayspresent to a certain extent. Although the scatteringof SPs was studied in many papers (see, for instance[15–19,26–29]), the question about the influence ofthe surface roughness on the propagation of SPs isstill not completely understood, and the presentpaper is intended to contribute to answering this
question. The calculations of the SP attenuation,based on the expressions for rescattering and conver-sion of SPs into light waves [16] give values for lossesthat are too low [29]. To estimate the influence ofroughness, we followed [29], which described a quan-titative model producing a good agreement with ex-perimental observations. This model is an extensionof previous theories [17,27,28].
As Fig. 2 and the values of ε1;eff in Table 1 show, theroughness introduces noticeable changes for bothstudied wavelengths, 633 and 805nm. Therefore, thedetermined attenuation lengths were also affected bythe roughness. The surface roughness leads to an in-crease of the SP losses, which is reflected in the broad-ening of the SP resonance curve and the displacementof the resonance angle to larger angular values. Con-sequently, the dielectric constants of ametal obtainedfrom the measurements of the SP resonance curvescan be dependent on the surface roughness and alsoon internal inhomogeneities of the film that play a si-milar role.Themeasured roughness height δ ≈ 2nm isstill within the range of validity of the first-order ap-proximation [34]. Calculated positive angular shiftsand broadening of the resonance curves due to rough-ness are in qualitative agreement with conclusions ofprevious studies on the effect of roughness [19]. Itshould be noted that the reduction of the correlationlength (provided that the averaged height of theroughness is the same) leads to an increase of the var-iations of the surface slope and also to a somewhathigher SP losses. The calculations for roughness pa-rameters δ ≤ 0:5nm and σ ≥ 30nm show that thereduction of the attenuation length due to roughnessis less than 2% for λ ≤ 0:9 μm. This correction margingradually increases with the optical wavelength toabout 5% at λ ¼ 2:4 μm. An increase of the relative in-fluence of roughness for longer wavelength can be ex-plainedas follows.Althoughwitha longerwavelengththe ratio δ=λ decreases, leading to a smaller contribu-tion of roughness to attenuation, the reduction of theintrinsic losses in themetal owing to improved dielec-tric constants makes the relative role of roughnessmore important. When the influence of roughnessis negligible, the attenuation length of SPs can be
Fig. 5. (Color online) Dependence of the factor jt012ðθÞj2 on theangle for two wavelengths 633 and 805nm, calculated for ε1 from[30] and the corresponding effective dielectric constant of the goldfilm, ε1;eff .
Fig. 6. (Color online) Surface plasmon attenuation lengths for (a) gold and (b) silver films of 47 nm thickness, calculated for a broadspectral range without account for roughness [the exact 0-order model, Eqs. (1) and (2); the results are shown by solid lines and solidlines with triangles] and approximate 0-order model [Eqs. (3)–(6) with Δkr ¼ Γr ¼ 0; the results are shown by dashed lines]. The stepsin the dependences, calculated with the data from [32], are due to the shifts present in this data.
calculated directly by applying the condition jR−1j →0 to the reflection coefficient of light in Eq. (1).The scattering on sub-wavelength-scale inhomo-
geneities has a broad and relatively smooth angulardistribution [17], and measuring the distribution ofthe scattered light intensity along the direction ofthe SP propagation allows to infer the attenuationlength Lsp. The presence of scratches and other inho-mogeneities, such as grain boundaries, gives spikesin the scattered light and leads to additional attenua-tion of SPs.The measurements of the widths of the SP reso-
nance gives an alternative approach for the Lsp deter-mination. In this case the focusing of light by acylindrical lens produces a set of plane waves propa-gating within a certain angular interval. The lightincident on the interface exactly at the resonance an-gle corresponds to the most efficient conversion oflight into SPs, and therefore in the reflected light thisarea looks dark and corresponds to a dip in the inten-sity (see Fig. 2). Our results have demonstrated thatboth approaches, by a direct observation of the SPsattenuation and by measuring the SP resonancecurve, give close values for the Lsp as presented inTable 1. For conditions similar to ours, the attenua-tion length at an optical wavelength of 800nm wasconsidered in [35], and a close value of 25 μm wasobtained.There are several applications of the obtained re-
sults. In particular, the knowledge of attenuationproperties of SPs helps selecting the rightwavelengthfor the SP microscopy [36]. Since the resolution in SPmicroscopy is determined by the attenuation length ofthe SPs, which is smaller for shorter wavelength, anemployment of shorter optical wavelengths for theexcitation of SPs is preferable for high-resolution mi-croscopy. Attenuation characteristics of SPs are alsoimportant for direct [20,21] and ellipsometric [37]absorption spectroscopy with SPs. In these tech-niques it is desirable that attenuation of SPs due tometal is low, and this is achieved for longer opticalwavelengths and the optimal and larger thicknessesof the metal film, although it should be noted that forthicker films the excitation efficiency of SPs is lower
due to a reduction of the coupling of light with thesurface plasmon mode.
6. Conclusions
In this work the attenuation length of SPs with agold film at 633 and 805nm was experimentally stu-died in the Kretschmann–Raether configuration byobserving the distribution of the intensity of the scat-tered light. For a gold film of 47nm thickness withroughness parameters δ ¼ 2:0nm, and σ ¼ 36nm, wedetermined the SP attenuation length at 633nm of3:4 μm (the calculations with different sets of the di-electric constants for gold [30–32] produced values2:5–3:6 μm) and at 805nm this length was 15 μm(the theory gave values 11–17 μm). For comparison,the calculations of the attenuation length of SPswithout accounting for roughness gave values2:7–4:4 μm at 633nm and 14–24 μm at 805nm. Theattenuation lengths were also determined from thewidths of the angular dependences of the SP reso-nance and gave similar values of 3:0 μm for 633nmand 17 μm for 805nm. The roughness of the filmwas measured with an AFM, and the contributionof the roughness to the losses of SPs was evaluated,using the model for the SP resonance taking rough-ness into account. We proposed and implemented aprocedure to calculate the effect of the surface rough-ness on the effective dielectric constant of the metalfilm. The SP attenuation length was also calculatedfor smooth gold and silver films in a broad opticalspectral range, showing a strong increase of the pro-pagation length with increasing optical wavelength.The coupling of light to the SP mode and the SP at-tenuation also strongly depends on the metal filmthickness: the coupling is maximal for the optimalfilm thickness, and the losses strongly increase forthinner films.
We are thankful to Wonmuk Hwang for providingAFM for roughness measurements. This work waspartially supported by the Robert A. Welch Founda-tion (grant A1546) and by the National Science Foun-dation (NSF) (grant 0722800).
Fig. 7. (Color online) Influence of the thickness of the gold film on the properties of SPs: (a) SP resonance curves at 633nm for differentfilm thicknesses, (b) the dependence of the attenuation length on the film thickness for 633 and 805nm. The dielectric constants from [30]are used.
References1. R. W. Wood, “On a remarkable case of uneven distribution
of light in a diffraction grating spectrum,” Philos. Mag. 4,396–402 (1902).
2. U. Fano, “The theory of anomalous diffraction gratings and ofquasi-stationary waves on metallic surfaces (Sommerfeld’swaves),” J. Opt. Soc. Am. 31, 213–221 (1941).
3. R. H. Ritchie, “Plasma losses by fast electrons in thin films,”Phys. Rev. 106, 874–881 (1957).
4. H. Raether and E. Kretschmann, “Radiative decay of nonradiative surface plasmons excited by light,” Z. Naturforsch.23a, 2135–2136 (1968).
5. E. Kretschmann, “Die bestimmung optischer konstanten vonmetallen durch anregung von oberflaechenplasmaschwingun-gen,” Z. Phys. 241, 313–324 (1971).
6. H. Raether, Surface Plasmons on Smooth and Rough Surfacesand on Gratings (Springer, 1988).
7. U. Jönsson, L. Fägerstam, B. Ivarsson, B. Johnsson, R.Karlsson, K. Lundh, S. Löfås, B. Persson, H. Roos, I. Rönnberg,S. Sjölander, E. Stenberg, R. Ståhlberg, C. Urbaniczky,H. Östlin, and M. Malmqvist, “Real-time biospecific interac-tion analysis using surface plasmon resonance and a sensorchip technology,” BioTechniques 11, 620–627 (1991).
8. C. E. Jordan, A. G. Frutos, A. J. Thiel, and R.M. Corn, “Surfaceplasmon resonance imagingmeasurements of DNA hybridiza-tion adsorption and streptavidin/DNA multilayer formationat chemically modified gold surfaces,” Anal. Chem. 69,4939–4947 (1997).
9. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314(2005).
10. A. Degiron and D. R. Smith, “Numerical simulations of long-range plasmons,” Opt. Express 14, 1611–1625 (2006).
11. S. Negm and H. Talaat, “Surface plasmon resonance half-widths as measured using attenuated total reflection, forwardscattering and photoacoustics,” J. Phys. Condens. Matter 1,10201–10205 (1989).
12. A. Bouhelier and G. P.Wiederrecht, “Surface plasmon rainbowjets,” Opt. Lett. 30, 884–886 (2005).
13. H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider,M. Salerno, A. Leitner, and F. R. Aussenegg, “Fluorescenceimaging of surface plasmon fields,” Appl. Phys. Lett. 80,404–405 (2002).
14. P. Dawson, F. de Fornel, and J-P. Goudonnet, “Imaging of sur-face plasmon propagation and edge interaction using a photonscanning tunneling microscope,” Phys. Rev. Lett. 72, 2927(1994).
15. E. Kretschmann, “Decay of non radiative surface plasmonsinto light on rough silver films. Comparison of experimentaland theoretical results,” Opt. Commun. 6, 185–187(1972).
16. D. L. Mills, “Attenuation of surface polaritons by surfaceroughness,” Phys. Rev. B 12, 4036–4046 (1975).
17. E. Kretschmann, “The angular dependence and the polariza-tion of light emitted by surface plasmons on metals due toroughness,” Opt. Commun. 5, 331–336 (1972).
18. A. A. Maradudin and D. L. Mills, “Scattering and absorption ofelectromagnetic radiation by a semi-infinite medium in the
presence of surface roughness,” Phys. Rev. B 11, 1392–1415(1975).
19. A. Hoffmann, Z. Lenkefi, and Z. Szentirmay, “Effect of rough-ness on surface plasmon scattering in gold films,” J. Phys.Condens. Matter 10, 5503–5513 (1998).
20. H. Kano and S. Kawata, “Surface-plasmon sensor forabsorption-sensitivity enhancement,” Appl. Opt. 33, 5166–5170 (1994).
21. A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, “Sur-face-plasmon resonance spectrometry and characterization ofabsorbing liquids,” Appl. Opt. 39, 3314–3320 (2000).
22. G. Kovacs, “Optical excitation of surface plasmon-polaritons inlayered media,” in Electromagnetic Surface Modes, A. D.Boardman, ed. (Wiley, 1982), pp. 143–197.
23. L. M. Brekhovskikh, Waves in Layered Media, 2nd ed.(Academic, 1980).
24. C. A. Ward, K. Bhasin, R. J. Bell, R. W. Alexander, and I. Tyler,“Multimedia dispersion relation for surface electromagneticwaves,” J. Chem. Phys. 62, 1674–1676 (1975).
25. P. Dawson, B. A. F. Puygranier, and J-P. Goudonnet, “Surfaceplasmon polariton propagation length: a direct comparisonusing photon scanning tunneling microscopy and attenuatedtotal reflection,” Phys. Rev. B 63, 205410 (2001).
26. E. Kröger and E. Kretschmann, “Surface plasmon and polar-iton dispersion at rough boundaries,” Phys. Stat. Sol. (B) 76,515–523 (1976).
27. F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical propertiesof rough surfaces: general theory and the small roughnesslimit,” Phys. Rev. B 15, 5618–5626 (1977).
28. S. O. Sari, D. K. Coben, and K. D. Scherkoske, “Study of sur-face plasma-wave reflectance and roughness-induced scatter-ing in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
29. E. Fontana and R. H. Pantell, “Characterization of multilayerrough surfaces by use of surface-plasmon spectroscopy,” Phys.Rev. B 37, 3164–3182 (1988).
30. American Institute of Physics Handbook, D. E. Gray, ed.(McGraw-Hill, 1972), p. 105.
31. U. Schröder, “Der einfluss dünner metallischer deckschichtenauf die dispersion von oberflaechenplasmaschwingungenin gold-silber-schichtsystemen,“ Surf. Sci. 102, 118–130(1981).
32. Handbook of Optical Constants of Solids, E. D. Palik, ed.(Academic, 1985).
33. The Properties of Optical Glass, H. Bach and N. Neuroth, eds.(Springer, 1995), pp. 4–9.
34. H. Raether, “The dispersion relation of surface plasmons onrough surfaces; a comment on roughness data,” Surf. Sci.125, 624–634 (1983).
35. M. U. Gonzalez, J.-C. Weeber, A. L. Baudrion, A. Dereux,A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen,“Design, near-field characterization, and modeling of 45°surface-plasmon Bragg mirrors,” Phys. Rev. B 73, 155416(2006).
36. W. Hickel, D. Kamp, and W. Knoll, “Surface-plasmon micro-scopy,” Nature 339, 186 (1989).
37. T. Iwata and S. Maeda, “Simulation of an absorption-basedsurface-plasmon resonance sensor by means of ellipsometryAppl. Opt. 46, 1575–1582 (2007).
