Contemporary Physics, 1991, volume 32, number 3, pages 173-183 Optical excitation of surface plasmons: an introduction J. R. SAMBLES, G. W. BRADBERY and FUZI YANG Beginning from low level concepts the basic understanding for the optical excitation of surface plasmons is developed. Prism coupling using the attenuated total reffection technique is discussed as well as the less traditional grating coupling technique. A brief discussion of some recent developments using twisted gratings is also presented. Finally a short summary of the potential device applications is given 1.Introduction 2. Simple theory The interaction of electromagnetic radiation with an In order that we may study this interface and the interface can generate interesting surface excitations interesting electromagnetic phenomenon which oc ccurs ere are, in electromagnetic terms, a range of interfaces there we need first to examine some relatively simple of interest, for example dielectric to dielectric, dielectric concepts of solid state physics and electromagnetism. Electromagnetic radiation in isotropic media consists of ong orthogonal oscillating electric and magnetic fields rmal transverse to the direction of propagation. If, as is often h are both the case, we pass such radiation through a linear ansm polarizer then the radiation being transmitted will be plane polarized. This means there is a well specified plane ir and in which E or B oscillate, this plane containing the gnetic field vector and the propagation. Now if we consider such ident angle 0. upon a smooth anar interface then we have to consider two important d in figure 1, the incident contains orthogonal to the 0010-7514/91
J.R. Sambles et al. Let us examine in detail an important limit of Snell's law. Suppose that the radiation is incident from a high index medium, n,=VE, on to a low index medium n2=VE2,(where E1, E2 are the relative permittivities) with n2 <n,. Then Snell's law, the conservation of in-surface-plane momentum condition, gives medium 2 sinO2=√e1sin0 Since the greatest in-surface-plane component available in medium 2 is when 02=90, there is a limiting angle Incidence Be, given by Figure 1. Representation of p-polarized electromagnetic 02=√a2/√a (2) radiation incident upon a planar interface between two media at beyond which, for radiation incident from medium an angle of incidenceθ 1, there can be no propagating wave in medium 2. This limiting angle is called the critical angle. Radiation linearly polarized radiation may be readily represented incident beyond the critical angle has more momentum by a sum of the above two cases along the surface plane than can be supported by Now consider that the second medium is a medium 2. For such radiation incident from medium I non-magnetic material, that is at the frequency of the the oscillating E field will cause the charges in medium incident radiation the relative permeability is unity. Then 1, including those at the 1-2 interface, to oscillate. Thus as far as the B part of the electromagnetic oscillation is even though the radiation is now totally reflected at the concerned there is no discontinuity at the interface. In interface there are oscillating charges here which have this case, which represents the majority of materials, it associated radiation fields penetrating into medium 2 govern the behaviour of the radiation on encountering they are spatially decaying fields (evanescent) which the interface. For simplicity we shall throughout this article ignore optical activity, that is the property (the radiation, decaying in amplitude in medium 2 in a chirality)of a material which allows it to rotate the plane direction normal to the interface. At the critical angle the propagating along an axis of symmetry in the syste s decay length is infinte but this falls rapidly to the order of polarization of an incident photon even if it of the wavelength of light as the angle of incidence is Photons, with momentum hk. when in a medium of further increased. This evanescent field for radiation refractive index n,, are regarded as having momentum incident beyond the critical angle is useful for coupling (strictly pseudomomentum)hkn,=hk ,(where k= 2x/a). radiation to surface plasmons as we shall see later. If these arrive at a planar interface they may impart For the moment let us return to the boundary momentum in a direction normal to the interface and so conditions on the E and B field components of our there is no need to conserve the normal component of incident radiation. Since there is no boundary ortho- hoton momentum, hk, For the reflected signal, since gonal to Ex this component is conserved across the hIk,I is conserved, unless the photon frequency is boundary. However this is not the case for E,, the normal changed, and hkx is conserved for a smooth planar component of E. It is the normal component of D, D interface, then it follows that k,, of the reflected signal which is continuous (there is no free charge) and Ez is simply -kzl, the usual law of reflection at a planar is forced to change if a is changed since Dz interface E,EoEn =E2EoEx2. This discontinuity in E, results in On the other hand inside the second medium the polarization changes at the interface refractive index is n, so the radiation has a new From these simple considerations it is obvious that wavelength, A2= A/nz and a new wavevector k2=n2k. while s-polarized incident radiation will not normally In this medium the radiation propagates in a new cause the creation of charge at a planar interface, direction, conserving k, but allowing k, to change. Now p-polarized radiation will automatically create time k,1=k, sin 6, and k,2= k2 sin 6, where 0, is the angle dependent polarization charge at the interface of refraction. Since the tangential momentum component Suppose now we consider one of the two materials to is conserved, kxI=kx2 and n, sin 0,=n2 sin 0, which is be a metal. A metal may be regarded as a good conductor Snell,s law(resulting from the translational invariance of of electricity and heat and a reflector of radiation. This the system parallel to the interface) is a rather loose definition of a metal which relates to
Optical excitation of surface plasmons the ability of thefree electrons in the metal to respond If we now apply Maxwells equation VE=0 we find to the externally imposed fields. If the electrons are free (still of course constrained inside the metal) then they are able to respond with no scattering to the incident radiation giving an ideal metallic response. Such a E2=-E3 k that is E=0 everywhere inside the metal, must therefore have &=+oo. An ideal metal in which the electrons Then to find the relationship between H, and E respond perfectly to the applied external field, therefore use Maxwell's equation A E=-H:(Faraday's law H cancelling it, is the limit 8--oo. Such a material of course does not exist, for the free electrons inside a metal of electromagnetic induction) which with H= Ho gives the oscillating field. The electrons have a finite mass and they permittivities and the normal component of the suffer scattering with lattice vibrations (phonons), defects wavevectors in the two media and the surface This means that as we increase the frequency of the H (5a) incident radiation the free electrons progressively find =oF (5b) it harder to respond. Ultimately at high enough frequencies, low enough wavelengths, the metal becomes Finally we need to apply the boundary conditions at transparent and behaves more like a dielectric z=0. We know tangential H is continuous and so is tangential E, thus HyI=Hy2 and Ex1= Ex2 leading to the following simple relationship between the relative permittivities and the normal components of the wavevectors in both media 3. More detailed theory From this simplistic treatment of the free electrons in a metal it is easy to show that there is a limiting frequency, k,t k2,z the plasma frequency, (for many metals in the Also we have ultra-violet) above which the metal is no longer metallic In this article we shall concern ourselves only with kzi=i(k:,k2)2, requiring k2>E,k2(7a) frequencies below this limit, that is with long enough wavelengths so that a is largely real and negative. As mentioned for real metals there is resistive scattering and k22=i(k2-6,k2)/, requiring k2>E,k2,(7b) hence damping of the oscillations created by the incident where k= o/c. If the wave is truly a trapped surface E field. This damping causes an imaginary component wave with exponential decays into both media then we E to E. Before, however, concerning ourselves with the need iki >0 and ik22 <0. Thus both k, s are imaginary added complexity of E, let us examine the implications of with opposite signs and so E, and c, are of opposite sign having a dielectric with positive er adjacent to the metal with negative E tells us the surface mode wavevector k, is greater than Because of the requirement of the normal E fields to the maximum photon wavevector available in the create surface charges we need only consider p-polarized dielectric, VE,k. The second condition, for the metal, is electromagnetic waves. Further whatever form the automatically satisfied with E, negative surface wave takes it has to satisfy the electromagnetic We may substitute expressions(7) for k,i and k,, into wave equation in both media. If we take the x-y plane (6)to give to be the interface plane and the positive z half space as medium 2, then for wave propagation in the x direction only, we have And we then see for k to be real, the requirement for E,=(Ex, 0, E,)exp [i(k,x-an)] exp(ikz1z)(3a) a propagating mode, with e2 negative, is that le2l>E, H,=(0, H,i, O)exp [(k, x-or)] exp(ik, z), (3b) Thus we now have satisfied Maxwells equations ar boundary conditions to give a trapped surface wave, with E,=(E,2, 0, E, )exp [i(k, x-or)] exp(ik, z)(3c) real k, and appropriate kx, provided IE2l >&1 and e2 <0 Following the above analysis with purely real g values H2=(, Hy2, O)exp [i(k, x-an)] exp(ik,2 z).(3d) leads to a surface wave having purely real k, which is
J. R. Sambles et al larger than VE,k the maximum value for the medium 1. Thus the shift is inversely proportional to e,,while It is also clear that this surface plasmon resonance is the width of the resonan Ice, which of course is infinitely sharp and has an infinite propagation length. proportional to kxi, is proportional to Exi and inversely As mentioned, for real metals there is resistive proportional to e We therefore see that while at first scattering and hence damping of the oscillations created sight it may appear beneficial to use small Ezi to give a by the incident E field. This damping causes an sharp resonance, this idea has to be balanced with the imaginary component to E, E; Then with, E2= E2r + iE2i, requirement that we need a large negative value of e Indeed if we examine a range of metals it is clear that erally smallest in the visible of the E1(2x+iE2) (9) spectrum becoming larger as we move to infra-red E1+e2r+lei wavelengths there is an even more rapid increase in errl In figure 2 we illu dependence of both th which for k,=ku +ik i gives, provided k il <kx, with and imaginary parts of the relative permittivities of silver E2r|》E1andE2; and aluminium from the ultra-violet to the infra-red. This shows that both parameters increase in magnitude with ke12(1 wavelength. However, the width of the surface plasmon resonance is, remember, dictated by a/e and since a changes faster than e, there is, almost without exception, a narrowing of the resonance and consequential increase in observability as the wavelength increases. In figure 3 (10b) this ratio is shown for several metals over the visible and Eir near infra-red region of the spectrum. A ratio of the order of 0.2 is the limit of sensible observability for a surface Hence we find the shift in wavevector, Akar, of this plasmon resonance. This leads to the general conclusion surface plasmon resonance from the critical val that while only a few metals such as Ag, Au, Al support a sharp surface plasmon resonance in the visible many more metals support a sharp resonance in the near △kx=kx-e12k-k (11) infra-red. This is illustrated for nickel and platinum in -120 Figure 2. Wavelength dependence TTTTTTT imaginary, Ei, parts of the relative permittivity (E=E, +ie, for gold Wavelength/nm Wavelength/nm and aluminium Compiled from data in references [1] and [21
Optical excitation of surface plasmons prsm 1000 1200 \\thin metal film Wavelength/nm Figure 3. This shows how the surface plasmon resonance width (oE /E varies with wavelength. Many other metals can support sharp resonances as the wavelength of the incident radiation is increased. Compiled from data in references [1] and [2. 4. Coupling to the surface (c) thin dielectric Before moving on to discuss some experimental results thin metal film we need finally to examine how best to couple radiation dielectric to the surface plasmon resonance given that we have Fi clearly established that its momentum is beyond that Figure 4. Geometries used uple photons into a surface available in the dielectric medium adjacent to the metal mode:(a)Otto,(b)Kretschmann-Raether, and(e)mixed hybrid arran Recall that, for our original two-dielectric system eyond the critical angle of incidence there will be an evanescent field in the second half space. This evanescent radiation at the prism/dielectric interface we vary the field does not propagate in the z-direction, but it has momentum in the x-direction and this allows for simple h the resonance. The fo form of the reflectivity is obvious that since sin 0;> sin 0.(= n2/n,), then curve for gold and silver at 632.8 nm is shown in figure n, hk sin 0:>nhk. Hence we have an enhancement 5, where we also show for comparison that for of the x-component of momentum in the second s-polarized light which is, of course, not capable of dielectric half space, above the limit value of nahk for creating the surface plasmon. The position of the a propagating wave. minimum of the resonance which is a measure of the This enhancement of momentum given by n, (sin 8: surface plasmon momentum, is no longer dictated simply sin Oe)hk may be used to couple radiation to a surface by the dielecric/metal boundary for it is additionally plasmon provided it is possible to place the metal/di- perturbed by the presence of the coupling prism electric interface which supports the surface plasmon Likewise the linewidth, which is a measure of damping close enough to the totally internally reflecting interface. is also perturbed by the presence of the prism. As the An obvious geometry to consider is that shown in figure coupling gap is increased so the perturbation by the Is conventionally called the Otto geometry, prism diminishes and the resonance moves to the after Otto who first demonstrated this coupling position corresponding to the two media surface technique in 1968[3]. An air gap (or a spacer of low plasmon and it also narrows. Of course in this index)less than a few radiation wavelengths thick (fo process, illustrated for gold in figure 6, the resonane visible< 2 um) provides the evanescent tunnel barrier progressively shallows. If we wish to achieve we c across which the radiation couples, from the totally coupling then for visible radiation the gap has to be of internally reflecting situation, to excite the surface the order of 0.5 um which for an air gap demands plasmon at the air (dielectric) metal interface. By extreme care in sample fabrication. This constraint is not e angle of incidence of the p-polarized so severe if we choose instead to work in the infra-red