Physics Reports 408 (2005) 131–314 www.elsevier.com/locate/physrep
Nano-optics of surface plasmon polaritons Anatoly V. Zayatsa,∗, Igor I. Smolyaninovb, Alexei A. Maradudinc
aSchool of Mathematics and Physics, The Queen’s University of Belfast, Belfast BT7 1NN, UK bDepartment of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA cDepartment of Physics and Astronomy and Institute for Surface and Interface Science, University of California, Irvine, CA 92697, USA Accepted 1 November 2004 editor: D.L. Mills
Abstract A surface plasmon polariton (SPP) is an electromagnetic excitation existing on the surface of a good metal. It is an intrinsically two-dimensional excitation whose electromagnetic field decays exponentially with distance from the surface. In the past, it was possible to study only the (far-field) scattered light produced by the interaction of surface polaritons with surface features. Only with the development of scanning near-field optical microscopy did it become possible to measure the surface polariton field directly in close proximity to the surface where the SPP exists. Here we overview the near-field studies of surface polaritons on randomly rough and nanostructured surfaces, theoretical models of SPP interaction with surface features, and SPP applications in novel photonic technologies. Surface polariton scattering, interference, backscattering, and localization will be discussed, as well as concepts of surface polariton optics and polaritonic crystals. Surface polaritons are finding an ever increasing number of applications in traditional domains of surface characterization and sensors as well as in newly emerging nano-photonic and optoelectronic technologies. © 2004 Elsevier B.V. All rights reserved.
PACS: 73.20.Mf; 78.66.−w; 07.79.Fc
Keywords: Surface plasmon polaritons; Scanning near-field optical microscopy; Nano-optics
∗ Corresponding author. E-mail address: [email protected] (A.V. Zayats).
0370-1573/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2004.11.001 132 A.V. Zayats et al. / Physics Reports 408 (2005) 131–314
Contents
1. Introduction ...... 133 1.1. Surface electromagnetic waves at a dielectric–metal interface ...... 134 1.1.1. Surface plasmon polaritons ...... 135 1.1.2. Surface plasmons ...... 138 1.1.3. A metal film on a dielectric substrate ...... 139 1.1.4. Localized surface plasmons...... 142 1.1.5. Optical excitation of surface plasmon polaritons ...... 144 1.2. Scanning near-field optical microscopy ...... 146 2. Surface polariton scattering ...... 149 2.1. Single scattering regime: the SPP interaction with individual defects ...... 151 2.1.1. The interaction of a surface plasmon polariton with a single surface defect ...... 151 2.1.2. The propagation of a surface plasmon polariton near an index step ...... 163 2.1.3. The interaction of a surface plasmon polariton with a subsurface defect ...... 169 2.1.4. Near-field imaging of the SPP interaction with a single surface defect ...... 175 2.1.5. Interference of surface plasmon polariton beams ...... 179 2.2. Multiple scattering regime: backscattering of surface plasmon polaritons ...... 185 2.2.1. Enhanced backscattering of surface polaritons on a randomly rough surface ...... 186 2.2.2. Direct observation of SPP backscattering with SNOM ...... 192 2.3. Multiple scattering regime: surface polaritons localization on rough surfaces ...... 196 3. Two-dimensional optics of surface polaritons ...... 202 3.1. The interaction of surface plasmon polaritons with material and/or geometrical discontinuities in their propagation path ...... 202 3.1.1. The impedance boundary condition ...... 203 3.1.2. Modal expansions (discrete) ...... 205 3.1.3. Modal expansions (continuous) ...... 211 3.2. Elements of SPP optics ...... 214 4. Near-field microscopy of surface plasmon polaritons and characterization of metal surfaces ...... 218 4.1. Characterization of metallic nanostructures via local excitation of surface polaritons ...... 219 4.2. Imaging of thin film interfaces with air–metal and glass–metal surface polaritons ...... 221 4.3. Fractal properties of rough surfaces and SPP scattering ...... 222 5. Surface polaritons on periodically structured surfaces ...... 226 5.1. Surface polaritonic crystals ...... 228 5.1.1. One-dimensional periodic surface profile ...... 228 5.1.2. Two-dimensional periodic surface profile ...... 234 5.2. Imaging of the SPP Bloch waves ...... 237 5.3. Waveguiding in line-defects of a SPP crystal ...... 239 6. Enhanced optical properties mediated by surface plasmon polaritons ...... 243 6.1. Surface plasmon polaritons and the enhanced backscattering of light from a randomly rough metal surface ...... 243 6.2. Surface plasmon polaritons and the enhanced optical transmission of periodically structured films ...... 257 6.2.1. Optical transmission of a metal film with a periodic hole array: a near field view ...... 257 6.2.2. The origin of the enhanced transmission through periodically nanostructured metal films ...... 262 6.2.3. Polarization effects in the light transmission through a polaritonic crystal ...... 265 6.2.4. Light-controlled transmission of polaritonic crystals ...... 269 7. Nonlinear optics of surface plasmon polaritons ...... 274 7.1. Second harmonic generation in reflection from rough metal surfaces ...... 275 7.1.1. Second harmonic generation from metallic diffraction gratings ...... 281 7.1.2. Second harmonic generation at randomly rough metal surfaces ...... 286 7.2. Near-field second-harmonic generation at metal surfaces ...... 291 A.V. Zayats et al. / Physics Reports 408 (2005) 131–314 133
7.3. Near-field SHG from metallic diffraction gratings ...... 296 8. SPP interrogation using other scanning probe microscopy techniques ...... 298 8.1. Influence of surface polaritons on the tunneling current of STM ...... 298 8.2. Light emission induced by the electron tunneling in STM: a role of surface plasmon polaritons ...... 304 9. Conclusion ...... 306 Acknowledgements ...... 307 References ...... 308
1. Introduction
In its simplest form a surface plasmon polariton (SPP) is an electromagnetic excitation that propagates in a wave like fashion along the planar interface between a metal and a dielectric medium, often vacuum, and whose amplitude decays exponentially with increasing distance into each medium from the interface [1–3]. Thus, a SPP is a surface electromagnetic wave, whose electromagnetic field is confined to the near vicinity of the dielectric–metal interface. This confinement leads to an enhancement of the electromagnetic field at the interface, resulting in an extraordinary sensitivity of SPPs to surface conditions. This sensitivity is extensively used for studying adsorbates on a surface, surface roughness, and related phenomena. Surface plasmon polariton-based devices exploiting this sensitivity are widely used in chemo- and bio-sensors [4]. The enhancement of the electromagnetic field at the interface is responsible for surface-enhanced optical phenomena such as Raman scattering, second harmonic generation (SHG), fluorescence, etc. [1,5]. The intrinsically two-dimensional nature of SPPs provides significant flexibility in engineering SPP- based all-optical integrated circuits needed for optical communications and optical computing [6,7]. The relative ease of manipulating SPPs on a surface opens an opportunity for their application to photonics and optoelectronics for scaling down optical and electronic devices to nanometric dimensions. Most importantly, active photonic elements based on nonlinear surface plasmon polariton optics, which allow controlling optical properties with light, are much easier to realize with suitably patterned metal surfaces, due to the SPP-related electromagnetic field enhancement near a metal surface [7]. Since the electromagnetic field of a SPP decays exponentially with distance from a surface, it cannot be observed in conventional (far-field) experiments unless the SPP is transformed into light, e.g., by its interaction with surface inhomogeneities. In the past, it was possible to measure only the (far-field) scattered light produced by the interaction of a SPP with surface features, and then to derive the information on the surface polariton behavior from further assumptions about the generally unknown scattering processes [1,2]. It was possible to change in one or another way the average defect structure of a surface and to judge the SPP behavior on the surface from the changes in the angular spectrum and intensity of the scattered light. The microscopic structure of the surface was largely unknown in such experiments. Even if a surface topography is visualized by electron microscopy or later, by scanning atomic force and electron tunneling microscopy, the optical signal measured in the far-field is averaged over an ensemble of surface features due to diffraction effects, and it is therefore possible only to correlate the behavior of the scattered light with the average surface topography. The problem of local measurement of the SPP field close to a surface was addressed using a photosensitive polymer layer covering the metal surface under investigation [8]. The electromagnetic field distribution was recorded on the photoresist and, after it was developed, examined with an electron microscope. However, the polymer layer introduces significant 134 A.V. Zayats et al. / Physics Reports 408 (2005) 131–314 perturbations of the SPP resonant conditions and the SPP field over the metal surface, which limits the applicability and possibilities of this approach. Only with the development of scanning probe techniques did it become possible to study SPPs locally on a surface. In particular, scanning near-field optical microscopy (SNOM) [9] provides an opportunity to probe the SPP field directly over the surface where the SPP exists, with a resolution in the nanometer range. The implementation of SNOM has led to a breakthrough in surface polariton studies [10–18]. Surface plasmon polariton scattering, interference, backscattering and localization have been visualized and investigated directly on the surface [16–20]. The idea of two-dimensional surface polariton optics has been proposed and realized experimentally, resulting in the development of optical elements for surface polaritons that allow manipulating and directing SPP beams in the same way as optical beams are directed in three dimensions [17]. Very recently a new class of two-dimensional photonic crystals based on surface plasmon polaritonic band-gap structures has been created on a metal surface [21–26]. In this article we present an overview of the theory of the behavior of SPP on structured surfaces, and of recent advances in experimental surface plasmon polariton studies. SNOM has allowed vi- sualizing SPP related processes on a surface, correlating the SPP behavior to surface structure, and confronting the experimental studies with model calculations. Scattering of surface plasmon polari- tons by surface roughness is one of the subjects that can be significantly complemented with local studies of the electromagnetic field distribution over a surface provided by scanning near-field op- tical microscopy. These will be discussed in Section 2. Imaging of SPP scattering from individual surface defects as well as from ensembles of defects will also be considered. Different types of SPP behavior on rough surfaces, such as scattering, reflection, interference, backscattering, and localiza- tion will be analyzed, and related theoretical models will be presented. A detailed theoretical descrip- tion of the interaction of SPP with surface features of different topology, with surface and subsurface defects, as well as with topographical and index-step defects will be discussed. This interaction is important for an understanding of near-field SPP images and the development of SPP optics. Exam- ples of the elements of two-dimensional SPP optics will be described, and possible applications will be discussed, in Section 3. Section 4 is devoted to the applications of near-field studies of SPPs on rough surfaces to the characterization of metal surfaces, including SNOM measurements of the dielec- tric constants of metallic micro-and nanostructures, imaging of inner interfaces of thin metal films, and fractal properties of surfaces. The effect of periodically structured surfaces on SPP behavior will be discussed in Section 5, where properties of 1D and 2D surface polariton band-gap structures will be considered. The dependence of “bulk” optical properties of rough and periodically nanostructured metallic films on SPP effects will be described in Section 6, where polarization and nonlinear con- trol of optical properties of nanostructures will be discussed. In Section 7, the influence of SPP on nonlinear optical processes occurring on a metal surface will be considered on the example of second- harmonic generation at rough metal surfaces. Finally, in Section 8, applications of scanning tunneling microscopy in surface polariton measurements are presented, and associated mechanisms are consid- ered that are related to the interplay between surface polaritons and localized surface electromagnetic resonances.
1.1. Surface electromagnetic waves at a dielectric–metal interface
Although the emphasis in this article is on the interaction of surface plasmon polaritons with rough and nanostructured surfaces and films, it is useful to preface the discussion of these interactions with a A.V. Zayats et al. / Physics Reports 408 (2005) 131–314 135
|E | E3 x sp 3 k E1 sp ε x 1 1 H2
ε (ω)
Fig. 1. Surface plasmon polariton at a dielectric–metal interface.
brief summary of those properties of surface plasmon polaritons on smooth surfaces and in smooth films that will be needed in what follows. In this way the results presented in this review can be compared or contrasted with the better known results for a planar surface, and in the case of weak surface roughness can serve as the basis of perturbative treatments of these interactions.
1.1.1. Surface plasmon polaritons The electromagnetic field of a surface plasmon polariton at a dielectric–metal interface is obtained from the solution of Maxwell’s equations in each medium, and the associated boundary conditions. The latter express the continuity of the tangential components of the electric and magnetic fields across the interface, and the vanishing of these fields infinitely far from the interface. To introduce the main parameters characterizing surface plasmon polaritons, let us consider a system consisting of a dielectric material, characterized by an isotropic, real, positive dielectric constant 1, in the half-space x3 > 0, and a metal, characterized by an isotropic, frequency-dependent, complex dielectric function ( ) = 1( ) + i2( ), in the half-space x3 < 0 (Fig. 1). We first consider a p-polarized (transverse magnetic or TM) wave in this structure that propagates in the x1-direction. (Due to the optical isotropy of the two media there is no loss of generality in choosing this direction of propagation.) In a wave of this polarization the magnetic vector is perpendicular to the plane of incidence—the plane defined by the direction of propagation and the normal to the surface. The solutions of Maxwell’s equations that are wavelike in the x1-direction and whose amplitudes de- cay exponentially with increasing distance into each medium from the interface x3 = 0 can be written as ( ) > ikx −k 1 x −i t H (x; t) = (0,A,0)e 1 3 3 (1.1a) ( ) > c (1) kx −k 1 x − t ( ; t) =−A (k , , k) i 1 3 3 i E x 3 0 i e (1.1b) i 1
in the region x3 > 0, and as (m) < ikx +k x −i t H (x; t) = (0,B,0)e 1 3 3 (1.2a) c (m) < (m) ikx +k x −i t E (x; t) =−B (−k , 0, ik)e 1 3 3 (1.2b) i ( ) 3 136 A.V. Zayats et al. / Physics Reports 408 (2005) 131–314
x < k(1,m) in the region 3 0, where 3 determine the decay of the electromagnetic field with increasing distance from the surface: k(1) = (k2 − ( /c)2)1/2 , 3 1 (1.3a) k(m) = (k2 − ( )( /c)2)1/2 . 3 (1.3b) k(1) k(m) The real parts of 3 and 3 must be positive in order that Eqs. (1.1) and (1.2) describe an electromag- netic wave localized to the dielectric–metal interface at x3 = 0. The boundary conditions at the plane x3 = 0 yield the pair of equations k(1) k(m) A ==B, A 3 =−B 3 . (1.4) 1 ( ) The condition that this system of equations have a nontrivial solution provides the dispersion relation connecting the frequency of the p-polarized wave and its wavenumber k, k(m) ( ) 3 =− . (1.5) (1) k 1 3 ( ) k(1) k(m) If, for the moment, we assume that the dielectric function of the metal, is real, then 3 and 3 must be real and positive in order that Eqs. (1.1) and (1.2) describe a surface electromagnetic wave. It follows, therefore, from Eq. (1.5) that ( ) must be negative for this surface electromagnetic wave to exist. In this frequency range the wave vector of a volume electromagnetic wave is pure imaginary. Thus such surface electromagnetic waves exist only in the frequency range where volume electromagnetic waves cannot propagate in a metal. The surface electromagnetic wave whose field vectors are given by Eqs. (1.1) and (1.2) with A=B, and whose frequency is obtained from the dispersion relation, Eq. (1.5) is called a surface plasmon polariton. Physically, one can understand it as photons coupled to collective excitations of conduction electrons near a metal surface. If we square both sides of Eq. (1.5), and rearrange terms, we obtain an explicit expression for the wavenumber ksp of the surface plasmon polariton as a function of its frequency , 1/2 1( ) ksp = , (1.6) c 1 + ( ) a relation that is valid even if ( ) is complex. To illustrate the preceding results let us assume for the dielectric function of the metal the too often used, and never completely valid, free electron expression 2 p ( ) = 1 − , (1.7) 2
where p is the frequency of bulk longitudinal electron excitations, the plasma frequency. The corre- k(1) sponding dispersion curve is shown in Fig. 2. The requirement that 3 be real and positive in order that the electromagnetic field of the surface plasmon polariton decay exponentially into the dielectric medium in contact with the metal has the consequence that the SPP dispersion curve must lie to the A.V. Zayats et al. / Physics Reports 408 (2005) 131–314 137
0.8
1/2
1 ε ω=ω ε 1/2 0.6 =ck/ p/(1+ 1) ω p 0.4 ω /ω
0.2
0.0 01234 ω kc/ p
Fig. 2. Dispersion of SPP at a dielectric–metal interface. Also plotted is the dispersion of light in the dielectric medium, and the 15 −1 corresponding surface plasmon frequency. The parameters assumed are p = 11.9989 × 10 s (silver) and ε1 = 2.25.