Physics of Surface Plasmon Resonance - Columbia …

1 CHAPTER 2. Physics of Surface Plasmon Resonance ROB KOOYMAN. Biophysical Engineering Group, Faculty of Science and Technology, University of Twente, Box 217, 7500 AE Enschede, The Netherlands Introduction In the last two decades, Surface Plasmon Resonance (SPR) has evolved from a fairly esoteric physical phenomenon to an optical tool that is widely used in physical, chemical and biological investigations where the characterization of an interface is of interest. Recently, the eld of SPR nano-optics has been added where metallic structures on a nanoscale can be designed such that they can perform certain optical functions. This chapter will be mainly concerned with the more conventional, well-understood SPR theory used in sensor applications and 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 always implies that one of these materials is a metal where free conduction electrons are abundant.

2 This condition follows naturally from the analysis of a metal- dielectric interface by Maxwell's equations. From this analysis, the picture emerges that Surface plasmons can be considered as propagating electron density waves occurring at the interface between metal and dielectric. Alternatively, Surface plasmons can be viewed as electromagnetic waves strongly bound to this interface; it is found that the Surface Plasmon eld intensity at the interface can be made very high, which is the main reason why SPR is such a powerful tool 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 the focus will be on the optics of SPR. The following is by no means intended as an 15.

3 16 Chapter 2. 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 actually involved in SPR work and want to understand a bit more than measuring the shift of the SPR dip''. 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 of SPR sensing. This is conveniently done by considering the phenomenon of total internal re ection. An electromagnetic plane wave that propagates in a medium with refractive index n can mathematically be described by an electric eld E: . E E0 exp jot jk r E0 exp jot jkx x jky y jkz z 2:1 . where E0 is the amplitude of the electric eld, o is the angular frequency, k is the wavevector, r (x,y,z) is the position vector and j O 1. Note that eq.

4 ( ) 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 q 2p o k k2x k2y k2z n n 2:2 . l c 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 ). Without loss of generality, we can choose the direction of the light beam such that kz 0 and our problem becomes essentially two-dimensional. From elementary Physics we know that for this situation Snell's law holds: n1 sin a n2 sin b 2:3a . or, equivalently, kx1 kx2 kx 2:3b . By using eqs. ( ) and ( ), we can nd an expression for the component of the wavevector ky perpendicular to the interface1: 2 2.

5 2 2 2p n2 2. k y 2 n1 sin a 2:4 . l n21. Now, let us assume that n1 4 n2. From eq. ( ), it is seen that for sin a 4 n2/n1. the right part is negative, and, consequently, ky is purely imaginary. Returning to 1. Note that the direction y is in this chapter always perpendicular to the Surface . Physics of Surface Plasmon Resonance 17. Figure Refraction of light at an incident angle a, at an interface of two materials with refractive indices n1 and n2. Definition of axis system and quantities. eq. ( ), we conclude that for this case in medium 2 there is only a traveling wave parallel to the interface: E2 E0 e ky2 y exp jot jkx x 2:5 . with the amplitude of the electric eld exponentially decaying along the y-direction with a characteristic distance 1/ky2 1/jky2. For obvious reasons, this eld in medium 2 is denoted as the evanescent eld. Eq. ( ) can be used to calculate its penetration depth, which is of the order of half a wavelength.

6 This explains the interface sensitivity of the evanescent eld: only close to the interface is an electromagnetic eld present; therefore, only a changing dielectric property ( a changing refractive index) in the vicinity of the interface will in uence this eld. We will see that also in SPR an evanescent eld is generated. Surface Plasmons 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 his last standard treatise on SPs, Raether [3] calculated the SP dispersion relation from rst principles, viz. Maxwell's equations. A particularly elegant approach was suggested by Cardona [4] and we will adopt it here. For reasons that will become clear in the course of Section , we will only discuss p-polarized2. 2. p-Polarized light has its electric eld vector in the plane of incidence.

7 18 Chapter 2. light interacting with an interface. For any interface between two media, the complex re ection coef cient rp for p-polarized incident light electric eld is described by Fresnel's equations (see, , ref. [5] for a derivation on the basis of Maxwell's equations): . Ei jj tan a b jj rp rp e e 2:6a . Er tan a b . where Ei and Er are the incident and re ected electric elds, respectively, and the angles a and b are de ned as shown in Figure Of course, the angles a and b are again related by Snell's law [eq. ( )]; in addition, a phase change j of the re ected eld relative to the incident eld occurs, depending on the refractive indices of the materials involved. For the re ectance, de ned as the ratio of the re ected intensities, the following relation holds: 2. R p rp 2:6b . Now, following Cardona [4], two special cases exist: if a+b p/2, then the denominator of eq.

8 ( ) becomes very large and thus Rp becomes zero. This situation describes the Brewster angle, where there is no re ection for p-polarized light. The other special case occurs when a b p/2: we see from eqs. ( ) and ( ) that Rp becomes in nite: there is a nite Er for a very small Ei. This circumstance corresponds to Resonance . From this relation between a and b we can deduce the dispersion relation if a b p/2, then cosa sinb and tana k1x/k1y n2/n1. For the components of the wave- vector k (kx, ky), we can write e1. k2x k21 k2y1 k21 k2x 2:7 . e2. r s . o e1 e2 o e2i kx and kyi 2:8 . c e1 e2 c e1 e2. where e1 and e2 are the dielectric constants4 of materials 1 and 2, respectively, and i 1 or 2. Equation ( ) is the sought SPR dispersion equation for an interface between two half-in nite 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 angular frequency ooop its dielectric constant e2 will be negative (see, , ref.)

9 [5]): o2p e2 o 1 2:9a . o2. p . op 4pne e2 =me 2:9b . 3. Note that in Figure the direction y (instead of z) is perpendicular to the Surface . 4. Dielectric constant and refractive index are related by e n2. Physics of Surface Plasmon Resonance 19. where op is the so-called plasma frequency, ne is the free electron density and e and me are the electron charge and mass, respectively. Generally, this implies that for ooop no electromagnetic eld can propagate in a metal [cf. eqs. ( ) and ( )]. More specifically, provided that e2 4 e1, we nd for the interface that kyi is imaginary, whereas kx remains real. Thus an electromagnetic wave exists, propagating strictly along the interface, with evanescent tails extending into both sides of the interface [cf. eq. ( )]. To get a feeling for the quantities involved, it is instructive to calculate penetration depths for a real case, on the basis of eq.

10 ( ). We take l 700 nm, thus o 1015 s 1 and a gold/water interface. At this wavelength egoldE 16. and We calculate for the penetration depths 1/ky,water 238 nm and 1/ky,gold 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 are allowed to adsorb to the water/metal interface. We can view this as a process where water molecules are replaced by molecules X. Because, generally, eXaewater, the average dielectric constant close to the interface will change. Equation ( ) then describes the concomitant change of the wavevector kx. Because the SP eld 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 eld: an SPR. sensor will only be sensitive to molecular processes (binding, adsorption, etc.)