Chapter Electromagnetic Theory of 3D Nanofocusing of Surface Plasmon Polariton and its Application to Surface Enhanced Raman Scattering (SERS) In Chapter 2, confined and intense electric fields (the SERS hot-spots) on a spherical metallic nano-particle were shown to be a result of the redistribution of optical energy by the surface plasmon (SP) In this chapter, a coherent fluctuation of surface charges on a flat airmetal interface is discussed (147) Such a “planar” SP or propagating surface plasmon polariton (SPP) has been demonstrated in the electron energy-loss experiments by Powell and Swan (148), and shown to be localized or bound to the interface within the Thomas-Fermi screening thickness of about Å (149) One interesting consequence of an oscillating surface-charge is the resultant transversal and longitudinal electromagnetic field which disappears at |z| →∞, and has its maximum on the surface z = (see Figure 1) Just as the SP fields on a metal particle are capable of Raman enhancement, the same is also true for the SPP fields on a smooth metal surface as will be discussed below 99 SPP Dielectric z Au x Figure Intensity distribution showing the field confinement of the propagating surface plasmon polariton on a the smooth surface of a semi-infinite Au 4.1 General Theory of Propagating Surface Plasmon Polariton on Smooth Surfaces Unlike nano-particle-bound SP fields, which are confined to the north and south poles of the particle (see § 2.3.2 of Chapter 2), a SPP field is non-localized and spread out across the entire metal-dielectric interface The field distribution of a surface-bound SP field can be v v easily found through the Maxwell equation (∇ + κ )E = , which gives E the general form, v ˆ ˆ Ei = (Eix a x + Eiz a z )ei (k ix x + k iz z ) (1) v in which the projected direction of k on the dielectric-metal boundary is assumed to be along the positive x-axis The subscript, i, indicates whether the equation is applicable to the dielectric (i = 1) or the metallic (i = 2) medium One can also see from the Maxwell equation that both kix and k iz are generally complex, satisfying, 100 2 kix + kiz = ω µ iε i (2) ' " where ε is real, and ε , is complex, ε = ε + iε v v Whether the E1 and E2 field is bound to the surface, i.e decays away from the surface |z| > 0, depends primarily on the values of kix and k iz By imposing the boundary conditions, E1 x = E2 x , H1x = H x (3) v v v v and making use of the fact that, ∇ × H = i ωε E and ì E = iàH , it follows that, E1x = H1 y k1z ωε1 , E2 x = H y k2 z ωε (4) thus, k1z ε1 − k2 z ε2 =0 (5) Owing to the continuity of the tangential fields across the interface, one has k1 x = k2 x = k x (6) Since all the k values (i.e k ix and k iz ) are totally described by Eqn 2, and 6, these v v equations thus collectively determine whether the E1 and E2 fields are bound to the interface 101 Now by re-writing Eqn as, ω  k iz = ± ε i   − k ix c (i = 1, 2) (7) and in combination with Eqn 5, one can obtain a dispersion equation that relates k x to ω , kx = ω  ε1ε    c  ε1 + ε    (8) Note that in deriving Eqn 8, the upper sign in Eqn has been chosen for i = 1, and the lower v v sign for i = This is to ensure that the E1 and E2 fields will not increase away from the interface, and thereby result in an unrealistic field Since the current thesis concerns mainly with low-damping noble metals such as Au ' " ' ' and Ag with ε < , ε > ε and ε > ε - true for the metals commonly employed in the majority of SERS studies, k x can thus be approximated as, kx ≈ ' ω  ε1ε    ' c  ε1 + ε    (9) Plotting ω versus k x produces the dispersion curve for SPP (see Figure 2), which describes the allowable SPP modes that can be supported on a planar metal surface for a range of ω The shaded region bound by the light line ε ω / c and the ω -axis in Figure represents v points where both k x ( = k x ) and k1z are real, i.e where E1 is propagating away from the 102 surface The fact that the SPP dispersion curve is not included in this particular region indicates that the k x (or k x ) for the SPP modes must be greater than ε ω / c , and thus the corresponding k1z must be a pure positive imaginary according to Eqn Hence, as expected, v v the E1 field of the SPP is decaying away (i.e E1 field is surface bound) from the surface rather than propagating away Inside the metal (i = 2), on the other hand, k z of the SPP will v possess a negative imaginary part (see Eqn and 7), which suggests that the E2 field should reduce as z → -∞ Thus, the SPP dispersion curve represents “non-radiative” surface-bound field with exponentially decreasing intensities above and below the dielectric-metal boundary z > (Figure 1) Note that the dispersion curve does not increase monotonically with k x , but levels at ω sp = ωp ' 1+ ε , where ωsp and ω p is respectively the SPP frequency and the ' bulk plasma frequency of the metal With increasing ε , the value of ω sp reduced One can E also conclude that, according to Eqn 1, ∠ iz E  ix  k  = ∠ ix  k   iz  π ≈  for a supported SPP field on  the interface, suggesting the longitudinal and transverse fields, i.e Eix and Eiz , are always 90 º out of phase 103 Dispersion Curve for SPP on semi-infinite Au Light-line Freq (1e15 rad/s) ω sp Dispersion 1 10 k (10e7) Figure Dispersion curve for a planar surface plasmon Finally, through the divergence theorem, expression for the SPP charges across the interface can be obtained, v ρ (s ) = ε o (ε1E1+z − ε E2+z )  k  = ε o  ε1 − ε z1  E1+z  kz2     k v ˆ = ε o  ε1 − ε z1  E2 ( x, y )a z  kz2    (10) where the plus sign indicates the field is measured immediately above the metal surface, and v s is a surface vector ( x, y ) 104 4.2 Excitation and Evidence of Propagating Surface Plasmon v An electron penetrating a solid can transfer its momentum hq and energy ∆Eo to the v v electrons of the solid The projection of q upon the surface kx determines the plasmon wave vector and, together with the dispersion relation, the energy loss of the scattered electron ∆Eo = hω (see Figure 3) can be determined Since the electrons scatter at different angles θ , v different momenta hk x are transferred to the metal surface Thus measurements of the energy loss ∆Eo over a range of θ , allow the dispersion relation of the SPPs to be measured up to v large kx , beyond the Brillouin zone (149) The physics of SPPs has thus been studied extensively with electrons, especially with fast electrons, and the fundamental properties of SPP have been found in good agreement with theory (150) v kel Incident electron v Projection of hq onto the metal surface v kx Thin metal film Scattered v electron k ' v hq el - + Energy loss measurement Figure An energy loss measurement for determining the dispersion relation of SPP on a v' v thin metal film kel is the final momentum of the incident electron after collision, while hq is the momentum transferred to an electron within the metal 105 While fast electrons are good tools for the study of the dispersion relation at very v large kx (~0.3 Å-1) as has been performed previously in order to study the slow increase of the dispersion relation in Al (150-153), it is however not convenient to reach the region of v small kx because of the difficulties in producing an extremely narrow incident electron beam v with a low kel required for such a measurement This limits the experimental uncertainty in v the measurable kx value via electron loss to about × 10-3 Å-1, which is comparable with the extension of the whole linear region of the dispersion relation Hence, either a fast- or a slow- v electrons beam is not suitable for exciting SPP in the low kx value range of interest to the current study Visible light, on the other hand, possesses a low momentum in the × 10-3 Å-1, range, and can thus allow for the excitation of SPPs on a smooth metal surface However, this is not straightforward since it is not possible for a radiative beam to launch a SPP on a metal v surface via a direct incidence, simply because the projection of k of the light onto the surface (i.e k x = ω c ε sin θ o , where θ o is the incident angle, and ε is the dielectric constant of the medium above the metal surface) is always less than the SPP momentum ' ω  ε 1ε    (see ' c  ε1 + ε    ' ' Eqn and recalling that ε < and ε > ε ) for any incident angle between º to 90 º; the incident photon must at least gain an extra momentum ω  c  '  ε 1ε  ε +ε '  ∆k in the order of   ω − ε sin θ o  in order to induce a SPP  c    Conventionally, this is achieved through a three-layer configuration as shown in Figure First, the exciting beam is launched into the top medium (i.e region 0), normally a glass semi-cylindrical prism, which has a high refractive index (normally a glass) to up the k x 106 to ω c ε sin θ o (where ε o > ) before impinging upon the 0/1 interface Due to electromagnetic theory and the theory of conservation of momentum, this momentum ( k x ) is carried across region and reaches the 1/2 interface Depending on whether a KrestchmanRaether (1 = metal, = dielectic) or an Otto configuration (1 = dielectric, = metal) is used, one of the regions and can be a metal, while the other a dielectric (149, 154) In any case, it is possible to choose an appropriate ε so that k x = ω c ε sin θ o = ω  ε 1ε    , which c  ε1 + ε    matches the SPP wave vector at the 1/2 interface The incident angle at which this condition is satisfied is thus simply,  ε 1ε   ε + ε  ε o   θ o, sp = sin −1       (11) Note that ε must always be sufficiently large compared to, | ε1 | in the case of a Krestchman-Raether setup, or | ε | in the case of Otto setup, in order for θ o , sp to be a real value, i.e in order to have a realistic incident angle 107 High refractive index θ Incident beam kx Reflected beam ε0 0/1 ε1 kx 1/2 ε2 Figure A three-layer configuration for exciting surface plasmons 4.3 Exciting Propagating Surface Plasmon Optically 4.3.1 Exciting Surface Plasmons via Krestchman-Raether Configuration Figure shows a typical Krestchman-Raether configuration with the three layers respectively being glass (or quartz) (0), metal (1), and a dielectric medium (2) exhibiting a refractive index lower than that in In this particular configuration, the incident field gains its momentum while traversing through region before reaching the metal film layer in which the field decreases exponentially in amplitude prior to impinging on the 1/2 interface If the field is p-polarized and the incident angle θ o is such that the k x of the field matches ksp ( R ω  ε 1' ε  c  ε 1' + ε    ) at the 1/2 interface, a SPP is excited in which case the reflected intensity   (see Figure 5) reduces to a minimum as a result of the incident energy being completely converted to the excited SPP mode Such a dependence of the reflected light with the incident 108 The electric field on the metal surface above the nano-cavity can be derived once (s) v Ect is known; ∞ v v v ˆ ~ ˆv E c = E1i + ∑ ∫∫∫ e −ik + ⋅ha z Ts e ks+ m ,n with ~ Ts ( p ) ( ( (s) v ~ ˆv ˆ E ct e ks+ + T p e kvp+ ) ( being a tensor that relates the wave surface to the transmitted wave v ~ ˆv E s ( p ) = Ts ( p ) eks (+p )   v s( p) E = (a − b )   , where v E s( p) v E s( p) v 2 k1 − k1 x v k1 ˆs ekv (+p ) (s) v ˆ E ct e kvp+ )) (VII.56) directing from the cavity toward the in the dielectric medium above the metal, i.e has the form,  k  k   −ik x − ik z ik z 1 ˆ ˆ ˆ  e x a x + a  v x  e z a z + b v x  e z a z k  k        v with a and b being some complex constants, and k1 being the wave vector in the dielectric at the light frequency ( ω ) of interest Reference C Flammer, in Spheroidal Wave Functions, Dover Pubns, New York (2005) xl Publications and Conference Abstracts Publications Kho KW, Koh ZY, Shen ZX, Watt F, Olivo MC, “3D Nanofocusing and Confined Guiding of Optical Energy by Nonspherical Nanocavity”, Physical Review A 2009 (submitted) Kho KW, Koh ZY, Shen ZX, Mhaisalkar S, White TJ, Watt F, Olivo MC, “Fabrication of a Smooth Metal Substrate with Embedded Spherical and Non-spherical Nanocavities”, Langmuir 2009 (submitted) Kho KW, Shen ZX, Lei Z, Watt F, Soo KC, Olivo M, “Investigation into a surface plasmon related heating effect in surface enhanced Raman spectroscopy”, Anal Chem 2007 Dec 1; 79(23): 8870-82 Kho KW, Shen ZX, Zeng HC, Soo KC, Olivo M, “Deposition method for preparing SERSactive gold nanoparticle substrates”, Anal Chem 2005 Nov 15; 77(22): 7462-71 Conference Abstracts Kho KW, Koh ZY, Shen ZX, Watt F, Olivo MC, Surface Enhanced Raman Spectroscopy xli Using a Smooth Planar Metal Surface with Embedded Nano-Cavities, XX1st International Conference on Raman Spectroscopy, 19 August, 2008, London, UK Kho KW, Olivo MC, Colloguium Spectroscopium Internationale XXXV, Xiamen, China, September 22 – 28, 2007 Kho KW, Shen ZX, Watt F, Soo KC, Olivo M 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