Physics of Surface Plasmon Resonance - Turro Home Page

CHAPTER 2

Physics of Surface PlasmonResonance

ROB P.H. KOOYMAN

Biophysical Engineering Group, Faculty of Science and Technology,University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

2.1 Introduction

In the last two decades, surface plasmon resonance (SPR) has evolved from afairly esoteric physical phenomenon to an optical tool that is widely used inphysical, chemical and biological investigations where the characterization ofan interface is of interest. Recently, the field of SPR nano-optics has been addedwhere metallic structures on a nanoscale can be designed such that they canperform certain optical functions. This chapter will be mainly concerned withthe more conventional, well-understood SPR theory used in sensor applicationsand it will touch upon some of the newer developments relevant for this area.Essential for the generation of surface plasmons (SPs) is the presence of free

electrons at the interface of two materials – in practice this almost alwaysimplies that one of these materials is a metal where free conduction electronsare abundant. This condition follows naturally from the analysis of a metal-dielectric interface by Maxwell’s equations. From this analysis, the pictureemerges that surface plasmons can be considered as propagating electron densitywaves occurring at the interface between metal and dielectric. Alternatively,surface plasmons can be viewed as electromagnetic waves strongly bound tothis interface; it is found that the surface plasmon field intensity at the interfacecan be made very high, which is the main reason why SPR is such a powerfultool for many types of interface studies.Experimental research on SPs started with electron beam excitation; in 1968,

optical excitation was demonstrated by Otto [1] and Kretschmann and Raether [2].This last approach turned out to be much more versatile, so in this chapter thefocus will be on the optics of SPR. The following is by no means intended as an

15

in-depth treatment of surface plasmons, rather it is an attempt to provide a low-threshold introduction to the physics of SPR for those who are actuallyinvolved in SPR work and want to understand a bit more than ‘‘measuringthe shift of the SPR dip’’.

2.2 The Evanescent Wave

Before we discuss SPs in more detail, it may be appropriate to provide a mathe-matical description of the evanescent wave, which is so central in the concept ofSPR sensing. This is conveniently done by considering the phenomenon of totalinternal reflection.An electromagnetic plane wave that propagates in a medium with refractive

index n can mathematically be described by an electric field E:

E ¼ E0 exp jot� jk � rð Þ ¼ E0 exp jot� jkxx� jkyy� jkzz� �

ð2:1Þ

where E0 is the amplitude of the electric field, o is the angular frequency, k isthe wavevector, r¼ (x,y,z) is the position vector and j¼O�1. Note that eq.(2.1) only represents a traveling wave if the exponent is complex.In the present context, we will mainly be concerned with the wavevector k: its

direction is parallel to that of the wave propagation; its magnitude is given by

k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k2y þ k2z

q¼ n

2pl

¼ noc

ð2:2Þ

where l and c are the wavelength and propagation velocity in vacuum,respectively.Next we consider the refraction of such a wave at an interface between two

media 1 and 2 with refractive indices n1 and n2, respectively (see Figure 2.1).Without loss of generality, we can choose the direction of the light beam suchthat kz¼ 0 and our problem becomes essentially two-dimensional. Fromelementary physics we know that for this situation Snell’s law holds:

n1 sin a ¼ n2 sin b ð2:3aÞ

or, equivalently,

kx1 ¼ kx2 � kx ð2:3bÞ

By using eqs. (2.2) and (2.3b), we can find an expression for the component ofthe wavevector ky perpendicular to the interface1:

k2y2 ¼ n212pl

� �2n22n21

� sin2 a� �

ð2:4Þ

Now, let us assume that n14 n2. From eq. (2.4), it is seen that for sin a4 n2/n1the right part is negative, and, consequently, ky is purely imaginary. Returning to

1Note that the direction y is in this chapter always perpendicular to the surface.

16 Chapter 2

eq. (2.1), we conclude that for this case in medium 2 there is only a traveling waveparallel to the interface:

E2 ¼ E0e�ky2y exp jot� jkxxð Þ ð2:5Þ

with the amplitude of the electric field exponentially decaying along the y-directionwith a characteristic distance 1/ky2 � 1/jky2. For obvious reasons, this field inmedium 2 is denoted as the evanescent field. Eq. (2.4) can be used to calculate itspenetration depth, which is of the order of half a wavelength. This explains theinterface sensitivity of the evanescent field: only close to the interface is anelectromagnetic field present; therefore, only a changing dielectric property (e.g.a changing refractive index) in the vicinity of the interface will influence this field.We will see that also in SPR an evanescent field is generated.

2.3 Surface Plasmons

2.3.1 Surface Plasmon Dispersion Equations, Resonance

There are several approaches that all result in the dispersion relation for an SP,that is, a relation between the angular frequency o and the wavevector k. In hislast standard treatise on SPs, Raether [3] calculated the SP dispersion relationfrom first principles, viz. Maxwell’s equations. A particularly elegant approachwas suggested by Cardona [4] and we will adopt it here. For reasons that willbecome clear in the course of Section 2.3.2, we will only discuss p-polarized2

Figure 2.1 Refraction of light at an incident angle a, at an interface of two materialswith refractive indices n1 and n2. Definition of axis system and quantities.

2p-Polarized light has its electric field vector in the plane of incidence.

17Physics of Surface Plasmon Resonance

light interacting with an interface. For any interface between two media, thecomplex reflection coefficient rp for p-polarized incident light electric field isdescribed by Fresnel’s equations (see, e.g., ref. [5] for a derivation on the basisof Maxwell’s equations):

rp ¼Ei

Er¼ rp

�� ejj ¼ tanða� bÞtanðaþ bÞ

��ejj ð2:6aÞ

where Ei and Er are the incident and reflected electric fields, respectively, andthe angles a and b are defined as shown in Figure 2.1.3

Of course, the angles a and b are again related by Snell’s law [eq. (2.3)]; inaddition, a phase change j of the reflected field relative to the incident fieldoccurs, depending on the refractive indices of the materials involved.For the reflectance, defined as the ratio of the reflected intensities, the

following relation holds:

Rp ¼ rp�� 2 ð2:6bÞ

Now, following Cardona [4], two special cases exist: if a+b¼ p/2, then thedenominator of eq. (2.6a) becomes very large and thus Rp becomes zero. Thissituation describes the Brewster angle, where there is no reflection forp-polarized light. The other special case occurs when a�b¼ p/2: we see fromeqs. (2.6a) and (2.6b) that Rp becomes infinite: there is a finite Er for a verysmall Ei. This circumstance corresponds to resonance. From this relationbetween a and b we can deduce the dispersion relation if a�b¼ p/2, thencosa¼�sinb and tana¼ k1x/k1y¼�n2/n1. For the components of the wave-vector k¼ (kx, ky), we can write

k2x ¼ k21 � k2y1 ¼ k21 � k2xe1e2

ð2:7Þ

kx ¼ oc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffie1e2

e1 þ e2

rand kyi ¼

oc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffie2i

e1 þ e2

sð2:8Þ

where e1 and e2 are the dielectric constants4 of materials 1 and 2, respectively,and i¼ 1 or 2. Equation (2.8) is the sought SPR dispersion equation for aninterface between two half-infinite media.Next, we investigate the case where medium 2 is a metal. This medium then

contains a large number of free electrons and the consequence is that at an angularfrequency ooop its dielectric constant e2 will be negative (see, e.g., ref. [5]):

e2ðoÞ ¼ 1�o2

p

o2ð2:9aÞ

op ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pnee2=me

pð2:9bÞ

3Note that in Figure 2.1 the direction y (instead of z) is perpendicular to the surface.4Dielectric constant and refractive index are related by e� n2.

18 Chapter 2

where op is the so-called plasma frequency, ne is the free electron density ande and me are the electron charge and mass, respectively.Generally, this implies that for ooop no electromagnetic field can propagate

in a metal [cf. eqs. (2.1) and (2.2)]. More specifically, provided that e24�e1, wefind for the interface that kyi is imaginary, whereas kx remains real. Thus anelectromagnetic wave exists, propagating strictly along the interface, withevanescent tails extending into both sides of the interface [cf. eq. (2.5)]. Toget a feeling for the quantities involved, it is instructive to calculate penetrationdepths for a real case, on the basis of eq. (2.8). We take l¼ 700 nm, thuso¼ 2.69� 1015 s�1 and a gold/water interface. At this wavelength egoldE�16and ewaterE1.77. We calculate for the penetration depths 1/ky,water¼ 238 nmand 1/ky,gold¼ 26 nm.Now all ingredients are available to appreciate the use of SPR in sensor

applications. Let us assume that we have a situation where molecules X areallowed to adsorb to the water/metal interface. We can view this as a processwhere water molecules are replaced by molecules X. Because, generally,eXaewater, the average dielectric constant close to the interface will change.Equation (2.8) then describes the concomitant change of the wavevector kx.Because the SP field is evanescent in the direction perpendicular to the inter-face, a change of the dielectric constant e2 is only detectable in SP character-istics if this change occurs within the penetration depth of the SP field: an SPRsensor will only be sensitive to molecular processes (binding, adsorption, etc.)that occur at a distance to the metal surface that is roughly half the wavelengthof the used light.

2.3.2 Excitation of Surface Plasmons

By substitution of eqs. (2.9a) and (2.9b) into eq. (2.8), we obtain a graphicalrepresentation of the SPR dispersion relation as shown in Figure 2.2 (line I). Inthe same figure, the dispersion relation for ‘‘normal’’ light is depicted (line a).We immediately see that, apart from the origin, there is no point where the SPRcurve and the light curve intersect, implying that in this geometry ‘‘normal’’light cannot simultaneously provide the correct wavevector and angularfrequency to excite a surface plasmon.One way to circumvent this problem is to introduce a second interface, as

depicted in the inset of Figure 2.2. Here a thin metal layer (dielectric constant em)is sandwiched between two dielectric materials 1 and 3 with different dielectricconstants e1 and e3, with e1 4 e3. By applying Fresnel’s equations to the twointerfaces, more complicated dispersion equations are found than eq. (2.8);however the essential physics remains unchanged. We now find two dispersionequations for kx, one for each interface, and we see that the line representing thedispersion relation for ‘‘normal’’ light in medium 1 (line b) intersects the SPdispersion line for the metal/medium 3 interface. This indicates that lightincident from medium 1 can excite SPs: by proper adjustment of the incomingangle a (Figure 2.2, inset), we can tune the incoming wavevector kx¼ kn1sina to


match the wavevector necessary for SP excitation. In this way, any kx betweenthe two lines, labeled a and b in Figure 2.2, can be set. As an example, one suchline, labeled c, is indicated. This so-called attenuated total reflection (ATR)technique was first demonstrated5 by Kretschmann and Raether [2] and hassince then almost become the standard technique for SP excitation.Another way of providing a wavevector appropriate for SP excitation is the

use of a metal layer on which a periodic structure is prepared [6] as illustrated inFigure 2.3.When light with wavevector kx¼ 2p/l.nisiny falls on such a structure, this

acts as a diffraction grating and diffraction orders m¼ 0, �1, �2, . . . aregenerated in the reflected light (see, e.g., ref. [7]). The generated wavevectorkx,net parallel to the interface can be written as

kx;net ¼ kx þm2pL

ð2:10Þ

Figure 2.2 Dispersion relation for surface plasmons. Curves I and II represent the SPdispersion for the interfaces e3/em and e1/em, respectively. The lines a and bare the dispersion relations for ‘‘normal’’ light in medium e3 and e3,respectively, which are dependent on the angle of incidence a in theexperimental setup as indicated in the inset. By varying a, any line cbetween the lines a and b can be realized.

5 In fact, Otto was the very first to demonstrate this in a somewhat less versatile form.

20 Chapter 2

where L is the periodicity of the grating. Again, the wavevector kx,net can betuned to the SPR wavevector, given by eq. (2.8),6 by changing the incidentangle.Up to now the required polarization direction of the incoming light remained

unmentioned. As already pointed out, SPs are conductivity fluctuationsbrought about by collective surface charge density oscillations. These chargedensity waves have to be excited by an external electric field. Only an electricfield with a component perpendicular to the interface can induce a surfacecharge density; only p-polarized light has a perpendicular electric fieldcomponent.

2.3.3 Surface Plasmon Properties

With SPs, a number of specific properties are associated that are particularlyrelevant to sensor applications: (1) the field enhancement, (2) the phase jump ofthe reflected field upon SP excitation and (3) the SP coherence length.Field enhancement. A calculation of the electric field transmission coefficient

on the basis of Fresnel’s equations for the interface reveals that the electric fieldat the low index side of the metal can be much larger than that at the other sideof the metal layer.In Figure 2.4, the intensity enhancement is depicted as a function of the angle

of incidence of incoming light for a number of different thicknesses of a goldlayer. It is found that very close to the SPR angle the intensity can be enhancedby a factor of more than 30. This circumstance accounts for much of the

Figure 2.3 Schematic view of a grating coupler. By diffraction of an incident lightbeam, the grating produces kx values larger than that of the incident light.By adjusting the incident angle a the wavevector can be tuned to kxrequired to produce a surface plasmon.

6The dispersion equation eq. (2.8) is hardly affected if the metal surface has a shallow corrugation.


remarkable sensitivity that the SPR condition has for a changing dielectricenvironment.Phase jump. As already mentioned in Section 2.3.1 and expressed in eq. (2.6),

a reflection event at an interface is generally accompanied by a phase jump ofthe reflected field. This is illustrated in Figure 2.5a for a prism–gold–watersystem.For comparison, the ‘‘conventional’’ SPR dip is shown in Figure 2.5b for the

same layer system. We see that around the SPR dip the phase of the reflectedelectric field undergoes a relatively large change. The significance of this phe-nomenon for sensing purposes is more clear when we plot the reflectance andphase changes as a function of incident angle for a certain change in dielectricconstant of the water. This is depicted in Figure 2.5c and d. In the followingrough calculation, we assume that both the change in reflection coefficient DRand the phase change Dj are proportional to De. From Figure 2.5c we estimatethat DR/De E30, whereas from Figure 2.5d we find that Dj/De E250.Experimentally, a minimum DRE10�3 can be measured, whereas a mini-mum Dj E 10�3 is feasible, using interferometric techniques. The conclu-sion is that on the basis of reflectance measurements a minimum DeE4� 10�5

can be detected, whereas a phase measurement provides a sensitivity ofDe E4� 10–6. In view of the very approximate character of this calculation,

Figure 2.4 Field enhancement for various values of the thickness of the gold layer.Wavelength of the excitation light is 700 nm; the low-index side of themetal layer consists of water (e3¼ 1.77).

22 Chapter 2

the absolute values found are of limited validity; however, the finding that aphase measurement provides an order of magnitude better sensitivity is a hardconclusion and, indeed, this was demonstrated by Nikitin and co-workers [8,9].The only drawback of this approach seems to be the much more complicatedexperimental setup.SP coherence length. Generally, the metal’s dielectric constant e2 is complex

and this circumstance results in a complex propagation constantkx¼ kx0 0+ jkx0 0 [cf. eq. (2.8)], where kx0 0 and kx0 0 are real and imaginary parts,respectively. For a surface plasmon, traveling along the interface with wave-vector kx, this implies that the field intensity decays with a characteristicdistance 1/2kx0 0. For gold and silver, the standard metals in sensor applications,

Figure 2.5 Comparison of the angle-dependent phase changes (a, c) and reflectancechanges (b, d) for variation of the dielectric constant at the low-index sideof the metal layer. A gold layer is used, SPs are excited at l¼ 700 nm.(c) and (d) depict the differential phase and reflectance, respectively, for achange in the medium’s dielectric constant of 0.01.


the imaginary part of the dielectric constant increases with decreasing wave-length and the SP propagation length decreases accordingly.This is illustrated in Figure 2.6: here a layer system was prepared where a

30nm SiO2 strip was deposited on a 50nm silver layer. For a series of wave-lengths the angle of incidence was chosen such that SPs were excited in the area

Figure 2.6 SPR response to a dielectric step at several wavelengths. For each wave-length the light angle of incidence is set such that outside the strip(extending from 0 to 125 mm) the interrogating kx is resonant with thesurface plasmon wavevector. The surroundings of the strip has dielectricconstant e3¼ 1.

24 Chapter 2

outside the strip and for each wavelength the whole area was illuminated with acollimated light beam under a constant angle of incidence. Because of the con-trast in dielectric constant between the strip and its surroundings (air). The SPresonance condition is not fulfilled in the area below the strip and we see thedecaying SP (increasing reflectance) at the left edge of the strip. Beyond the rightedge of the strip the SPR condition is again fulfilled and the SP resonance buildsup. The figure nicely demonstrates that with decreasing wavelength the SPpropagation length becomes shorter: the blurring on the left side of the stripbecomes less prominent for shorter wavelengths. It turns out that in the wave-length range 500–00nm the propagation length varies between o10 and 40mm.For a quantitative description of the findings depicted in Figure 2.6, we have

to analyze the interference between the several fields that are present in the layer.In Figure 2.7, the layer system and the fields involved are indicated: the

resonant SP field, the non-resonant SP field and the external exciting field withamplitudes E1, E2, E3, respectively. For a resonant SP that enters the SiO2 strip-covered layer, the total field reaching the detector can be written as [10,11]

EtotðxÞ ¼ E1 � E2ð Þej k01þjk

001

� �x þ E2e

jk00x � E3e

jk00x ð2:11Þ

where k01 is the wavevector corresponding to resonance in the covered area and

k00 is the wavevector that excites SPs in the uncovered area.Defining A¼E1�E2 and B¼E2�E3, the resulting intensity at the detector

becomes

ItotðxÞ ¼ B2 þ A2e�2k001x þ 2ABe�k

001x cos k

01 � k

00

� �x ð2:12Þ

When a non-resonant SP leaves the covered area, the resonant SP builds up andthe intensity at the detector decreases accordingly [11]:

ItotðxÞ ¼ Bþ A 1� e�k000x

� �h i2ð2:13Þ

From Figure 2.6, we see that this model gives a very accurate description of theexperiments.

Figure 2.7 Definition of wave vectors and fields for the system consisting of adielectric strip on top of a metal layer.


Both this model and the experiments indicate that a plasmon needs roughlyfour times the propagation length Lx for a full decay or for a full build-up; thispropagation length can be loosely defined as

Lx ¼ 1

2k00x

ð2:14Þ

This implies that SPs with mutual distances significantly larger than Lx areindependent. This is a very important conclusion because it is the fundament ofsurface plasmon microscopy [12,13], with its many applications in SPR imagingand SPR multisensing: on a substrate we can define areas that in an SPRexperiment will behave mutually independently, provided that these areas aresignificantly larger than Lx

2. For SPs on gold, excited at l¼ 632 nm, LxE7 mmand on a total sensor area of 1 cm2 more than 104 independent sensor ‘‘patches’’that each have an area of somewhat smaller than 100� 100 mm2 can in principlebe defined, of which the optical responses can be simultaneously read out byusing an imaging system. As practical aspects are outside the scope of thischapter, the interested reader should consult Chapter 7 for more details.

2.3.4 Choice of Experimental Parameters

It is impossible to define a general set of optimum SPR parameters, forinstance, optimal spatial resolution in an SPR microscopy/imaging setuprequires values of the experimental parameters other than those to obtainmaximum sensitivity for dielectric changes. Therefore, this section providesonly some general guidelines, based on consideration of the properties of themetal layer.To obtain maximum sensitivity, it is advantageous to maximize the steepness

of the reflectance as a function of the angle of incidence, because this allows for amore accurate determination of the angle of minimum reflectance (cf. Figure 2.8).This implies optimization of the reflectance minimum Rmin and minimizing the

width of the resonance curve. Rmin can be made very close to zero by selectingthe appropriate thickness of the metal layer; as can be seen in Figure 2.8,optimum thicknesses are somewhat dependent on the applied wavelength andare between 40 and 50nm. The width of the resonance curve is mainly deter-mined by the complex value of the metal’s dielectric constant. Generally, a large(negative) real part, together with a small imaginary part, results in narrowresonance curves. In practice only two options are available for the choice of themetal layer: gold or silver. As seen in Figure 2.8, silver has the better SPRcharacteristics in view of the larger real part of its dielectric constant; however, itis chemically less inert. In Figure 2.8, it is also seen that the use of higherexcitation wavelengths has an appreciable effect on the width of the resonancecurve. This is one of the reasons why (near-) infrared SPR experiments areattracting attention [14,15]. However, it should be realized that narrowing thereflectance curve necessarily implies increasing the SPR propagation length [eq.(2.13)], which can be a disadvantage in certain SPR imaging applications. For a

26 Chapter 2

gold layer, it can be calculated that an increase in wavelength from 450 to1500nm results in a change in the propagation length from 100 to almost 1 mm.Finally, it should be mentioned that an increase in wavelength results in anincrease in the penetration depth 1/ky [cf. eqs. (2.4) and (2.5)], with the conse-quence that the reflectance minimum will become more sensitive to dielectricchanges relatively far from the metal/dielectric interface; hence the surface-sensitive character of SPR becomes less prominent. This implies that for detec-tion of the growth of thin layers the optimum choice of wavelength will bedifferent from that in a situation where a more bulk-like change in refractiveindex has to be detected [33].

2.4 Analysis of Multi-layered Systems

In most SPR-based sensor applications, the system of interest consists of a goldor silver layer on which one or more thin layers are deposited in an aqueousenvironment. Often the desired parameters are the thicknesses of the severallayers, which can be converted into surface concentrations of the layer-composingmolecules (cf. Chapters 4 and 5).One way to obtain these parameters is a repeated application of the Fresnel

equation [eq. (2.6a)]. The following relation holds for a system consisting of N

Figure 2.8 The SPR ‘‘dip’’ for 46 nm of silver (dashed) and gold (solid) with wateron the low-index side, for several excitation wavelengths. Dielectric datafor the metal layer obtained from refs. [3] and [33].


layers with dielectric constants and thicknesses ei and di, respectively, placedbetween a prism with dielectric constant ep and a medium (e.g. water) withdielectric constant ew (see. e.g., ref. [5]):

rpðaÞ ¼M11 þM12

ky;wew

� �ky;pep

� M21 þM22ky;wew

� �M11 þM12

ky;wew

� �ky;pep

þ M21 þM22ky;wew

� � ð2:15aÞ

where M is the so-called transfer matrix:

M ¼ M1 �M2 � . . . �MN ð2:15bÞ

with

Mi ¼cos ky;idi

� � �jeiky;i

sin ky;idi� �

�jky;iei

sin ky;idi� �

cos ky;idi� �

" #ð2:15cÞ

The angular dependence of rp is contained in the wavevectors ky,i, perpen-dicular to the layer system; these can be calculated using eq. (2.4). Thereflectance can be obtained by application of eq. (2.6b).Provided all thicknesses di and dielectric constants ei are known, eq. (2.15)

gives an accurate description of the SPR experiment. Of course, in practice oneis concerned with the inverse problem and a priori knowledge, such as thedielectric constant and dimensions of the molecules composing a certain layer,is required for a satisfactory analysis of experimental SPR results. Thisprecludes an unambiguous analysis of more than two or three layers.Another, more intuitive, approach is the introduction of an effective dielec-

tric constant, eeff. Here, the actual multi-layered system is replaced by a two-layer system, where e1 in eq. (2.8) is replaced by the effective dielectric constanteeff, given by the average of all dielectric constants in the layer system, weightedby the penetration depth y0 of the SPR evanescent field [16,17]:

eeff ¼2

y0

ZN0

eðyÞe�2y=y0dy ð2:16Þ

Of course, with the use of this equation. we face the same problems as thosewhen we use eq. (2.15).

2.5 SPR Spectroscopy

2.5.1 Enhancement of Fluorescence and Absorbance

Up to now we have only considered, apart from the metal layer, transparentlayers, i.e. layers that are characterized by a positive, real dielectric constant.When one or more layers contain a light-absorbing compound, SPs can boostthe fluorescence intensity from a thin layer more than 40-fold [18]. This effect isdue solely to the large field enhancement that occurs on the low index side of

28 Chapter 2

the metal layer when an SP resonance condition is established (cf. Figure 2.4).More information on this phenomenon can be found in Chapter 9. Another,more subtle, feature of the interaction between SPs and light-absorbing mol-ecules is the increased sensitivity for the detection of absorbances in thin layers.The addition of a light-absorbing layer results in two effects on the SPRangular-dependent curve, which are quantitatively described by eq. (2.15): (1)the SPR dip shifts to a larger angle of incidence and (2) the value of thereflectance minimum increases. This last effect can readily be understood aslight absorption necessarily results in a decreased reflectance. In addition, theSPR field enhancement on the low-index side of the metal layer will result inincreased sensitivity for dielectric changes [19] and therefore also for changes inthe absorbance. The first effect is a consequence of the Kramers–Kronigrelation (see, e.g., ref. [20]), which in the present context can be expressed asthe statement that any change in the imaginary part of the dielectric constantwill be accompanied by a change in the real part; in SPR it is mainly the realpart of the dielectric profile on top of the metal layer that determines theangular position of the SPR dip. It has been demonstrated [21] that an SPR-assisted monolayer absorbance measurement can result in a 40-fold reflectanceincrease as compared with a metal-lacking ATR system. In addition, it ispossible to extract unambiguously the thickness and the dielectric constant ofan absorbing layer from a single SPR experiment [22].

2.5.2 SPR and Metal Nanoparticles

SPR phenomena are not restricted to planar multilayers as discussed so far; itturns out that for metal particles with dimensions much smaller than thewavelength of the interacting light, SP effects can be much more prominent.Generally, the net electric field Etot around a dielectric particle is composed ofthe superposition of an external applied field E0 and the induced (dipole) field inthe particle. For a polarizable spherical particle with radius rm and dielectricconstant e, placed in a medium with dielectric constant e1, the followingexpression is found (see, e.g., ref. [5]) for the field gain G:

GðoÞ ¼ EtotðoÞE0ðoÞ

E1þ eðoÞ � e1eðoÞ þ 2e1

rm

rþ rm

� �3

ð2:17Þ

It is seen that G can reach enormous values for e close to �2e1; in a ‘‘normal’’dielectric medium where e1 4 0, this condition points to the use of a metal,where e can be negative; additionally, the imaginary part of e should be as smallas possible. It turns out that this condition corresponds to the excitation of asurface plasmon in the metallic nanoparticle [23]. Particularly in the field ofRaman spectroscopy this can result in enormous sensitivity enhancements7 (fora review, see ref. [24]).

7Conventional Raman spectroscopy suffers from a very low scattering efficiency which can be 12orders of magnitude lower than that of fluorescence.


Now consider a Raman-active molecule near a metal nanoparticle. Thedetected Raman intensity I(o,osc) can be expressed as

IRamanðo;oscÞ ¼ gE2exc � E2

sc ¼ gG2ðoÞG2ðoscÞI0ðoÞI0;scðoscÞ ð2:18Þ

where Eexc is the total excitation electric field to which the molecule is exposedand Esc is the total Raman-scattered field. The constant g is an experimentalconstant that is unimportant in the present discussion.By choosing an angular frequency o that excites surface plasmons in the

metal (usually gold or silver) and detecting scattering frequencies osc not toofar from the excitation frequency, both the excitation and the scattered field areenhanced by the presence of the metal particle. By substituting eq. (2.17) intoeq. (2.18), we see that the distance dependence of the net Raman scatteringintensity changes with the power –12 of the molecule–nanoparticle distance!Indeed, it has been found experimentally that a surface-enhanced Ramanspectroscopy (SERS) experiment can result in experimental Raman signals thatare enhanced 1012–1014 times compared with those obtained from non-surface-enhanced experiments.It should be added that apart from this SPR enhancement mechanism,

another chemical enhancement effect is operational, which accounts for a10–100-fold amplification of the bare Raman signal [24].It has been demonstrated [25,26] that SERS is able to detect single molecules.

Together with its very high molecular specificity, this offers great promise as adetection tool for very low concentrations of biomolecules, such as DNAstrands or proteins.In principle, these field enhancements should also be important in the

detection of fluorescent molecules near a metal nanoparticle. However, thenearby presence of a metal layer leads to additional non-radiative decay pathsof the electronic excited states of a nearby molecule, with the net result that inmany cases the fluorescence will be largely quenched.So far, metal nanoparticles were considered as surface plasmon-assisted field

amplifiers. However, these particles can also be exploited as intrinsic refractiveindex sensors, analogous to the more familiar planar SPR experiments (forreviews, see refs. [27] and [28]). The physical basis of this application is the lightextinction (absorption and scattering), which is heavily dependent on thenanoparticle’s dielectric constant, size and geometry and also on the dielectricconstant e1 of the surrounding medium.Mie theory gives a reasonably adequatedescription of the extinction coefficient Aext and for spherical particles withdiameter less than about 20 nm the following expression is found [23]:

Aext ¼18pNpVe3=21

lImðeÞ

ReðeÞ þ 2e1½ �2þ ImðeÞ½ �2ð2:19Þ

where Np is the number of nanoparticles, each of which has a volume V, and lis the wavelength of the applied light.

30 Chapter 2

Again we see the pronounced influence of the occurrence of SPs: at Re(e)¼ –2e1we find a maximum in the extinction coefficient, which can reach large valuesfor low values of Im(e). Hence also in this situation we are led to the use of goldor silver as a metal nanoparticle.More sophisticated models (see, e.g., ref. [29]) also account for the size and

shape of the nanoparticles and computer programs are available in the publicdomain that can predict the extinction spectrum of nanoparticles of any shape[30], by modeling the particle as a series of dipoles placed in an oscillatingelectric field. However, the main features of nanoparticle extinction remaincontained in eq. (2.19).For one nanoparticle with a diameter around 25 nm, excited close to its SP

resonance, eq. (2.19) results in Aext of the order of 10–16 m2, which corresponds

to the more familiar molar extinction coefficient in the order of 109

lmol�1 cm�1. This value, which indeed was observed experimentally [29], ismore than three orders of magnitude larger than that of strong light-absorbingorganic dye molecules, allowing for relatively simple optical detection andcharacterization of individual nanoparticles [31,32]. In another series of exper-iments, the shift of the extinction maximum as a function of the refractive indexof the surrounding medium was investigated [28]. It was found experimentallythat for silver nanoparticles the spectrum could shift as much as 20 nm for achange of 0.1 in the refractive index. Because molecules that adsorb to ananoparticle change the refractive index around the particle, it is obvious thatthis, analogous to conventional SP resonance, can be used as a sensor principle.Indeed, it has been demonstrated experimentally that the full coverage of asilver nanoparticle with low molecular mass molecules resulted in a spectrumshift of approximately 40 nm. The full coverage corresponded to only 4� 104

molecules. Together with the single particle detection capability, this promisesenormous sensitivity, allowing for near single molecule detection [32].

2.6 Concluding Remarks

The phenomenon of SPR is one of the many examples where an interestingphysical phenomenon leads to applications that are highly important to bothapplied science and society. In a planar SPR system, it is particularly thecombination of field enhancement and relatively short coherence length thatallows for a unique sensor concept that provides both multiplexing capabilitiesand very high sensitivity. The general physical picture is well understood;however, some areas are still in vivid scientific debate (SERS, optics ofnanoparticles).From a technological point of view, the emerging field of nanotechnology

will enable us to exploit to its full potential the SP phenomena of tailorednanoparticles. It is the author’s firm conviction that merging of (bio-)nano-technology and SP phenomena of nanoparticles will ultimately lead to sensorconcepts and sensor realizations that will really be important in numerous


aspects of society, varying from food safety monitoring and high-throughputscreening to early in vivo detection of tumor growth.

2.7 Questions

1. Derive eq. (2.4) from eqs. (2.2) and (2.3).2. Calculate for a gold/water interface at l¼ 700 nm (�gold ¼ �16;

�water ¼ 1:770) the angular shift of the SPR dip when �water increases to1.775. For the light-incoupling we use a semi-circular glass piece withrefractive index nglass ¼ 1.5.

3. Estimate the effective dielectric constant for the following system, whenthis interface is probed with a wavelength l¼ 700 nm (for dielectricconstants of gold and water, see previous question)

gold

water

The squares in the figure represent cubes of protein molecules in anaqueous environment, adsorbed to the gold surface. Each protein mole-cule has a volume of 5*5*5 nm3 and a dielectric constant �protein ¼ 2.30.The average distance between the edges of the cubes is 7 nm.

4. A particular SPR application could be the detection of micro-organismsin, e.g., waste water, by detecting changes in bulk refractive index. WhichSPR excitation wavelength region would be more favourable, the blue/UVor the infrared region?

5. Express the extinction coefficient Aext (eq. 2.16) in units M�1cm�1.

2.8 Symbols

A A ¼ E1 � E2

Aext extinction coefficientB B ¼ E2 � E3

const experimental constant in surface-enhanced Raman spectroscopyc propagation velocity in vacuumdi film or layer thicknessE electric field strengthe charge of electronE0 amplitude of electric fieldE0 external applied field E0 (Section 2.5.2)Eexc total excited electric field (Raman)Ei incident electric fieldEr reflected electric field

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Esc total scattered field (Raman)Etot net electric field around nanoparticlesG field gain of nanoparticle SPRI intensityIRaman Raman intensity near an SPR nanoparticlek wavevectorky y component of wavevectorkx, net x component of wave vectorLx propagation length of a full decay or build-upme mass of electronm diffraction orderN number of layers in systemne free electron densityn refractive indexNp number of nanoparticlesni refractive index of material ir position vectorrm radius of polarizable spherical (nano)particlerp reflection coefficient (complex)Rp reflectanceV volume of nanoparticlesy0 penetration depth of SPR evanescent fielda incident angle of lightb refraction angle of lighte dielectric constante0 dielectric constant of mediumeeff effective dielectric constantep dielectric constant of prismew dielectric constant of waterk0 wavevector excited in uncovered areak1 wavevector corresponding to resonance of strip-covered areal wavelength in vacuumL periodicity of gratingj phase change of the reflected field relative to the incident fieldo angular frequencyop plasma frequencyosc scattering frequency

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