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## Single Particle Plasmon Resonances – The Quasi-Static Approximation

The oscillations of localized surface plasmons originate from the characteristic dielectric response of metals at optical frequencies. At resonance the polarizability is resonantly increased resulting in enhanced scattering, absorption and near-field intensities.

A full theoretical treatment of the interaction of a nanoparticle with an electromagnetic wave is beyond our scope. For a better understanding of the underlying physics let us introduce the quasi-static approximation, which is a simple model describing well the main results from such an interaction in first approximation.

The response of sub-wavelength sized particles to a plane-wave illumination can be analysed using the quasi-static approximation which is illustrated in Fig. 1.1. This model neglects retardation, i.e. eﬀects due to self-induction of electromagnetic fields, by consid-ering only a region in space which is much smaller than the wavelength of light, so that the electromagnetic phase is constant throughout the region of interest. Consequently, the quasi-static approximation is only valid for particles that are smaller in size than the skin depth ds of the metal. 1 For small spherical particles with a radius much smaller than the wavelength, this condition is fulfilled. We consider now a sphere of polarizable material with a radius R and a dielectric constant ε, embedded in a medium with di-electric constant εmedium, under the influence of a static electric field Eext. The external field induces a dipole moment in the sphere. The polarizability is written as (Bohren and Huﬀman 1983):α = 4πR3ε0 ε − εmedium (1.2) ε + 2εmedium.

It is evident from Eq. (1.2) that in vacuum (i.e. εmedium = 1) this expression exhibits a resonance when the real part of ε approaches −2. Replacing the static dielectric constant 1 The skin depth is the extent to which a field penetrates the metal surface and is given by ds = λ/(4π√ε).

ε with the wavelength dependent form ε(ω) = ε1(ω) + iε2(ω), the dielectric function of a noble metal, for example gold and silver, fulfils the resonance condition ε1(ω) → −2 at visible frequencies. Hence, at resonance the real part of the denominator in Eq. (1.2) almost vanishes and causes a strongly increased induced dipole moment µP = α · Eext. This in turn leads to enhanced scattering and local fields.

For nanoparticles with an elongated shape, the electron oscillation is non-isotropic and localized either along the principle axes or at the edges and corners of the nanoparti-cle, leading to an additional shape-dependent depolarization and splitting of the sur-face plasmon resonance into several modes (such as longitudinal or transverse modes for nanorods) (Lal et al. 2007).

The electrostatic theory for light scattering by sub-wavelength sized spherical particles has a relatively simple extension to ellipsoidal particles. In particular, we are interested in the scattering by disks and rods which can be described as spheroidal particles in a first approximation. A rod can de described by a prolate (cigar-shaped) particle, whereas a disk is approximately an oblate (pancake-shaped) particle. The polarizability αi of such an ellipsoidal particle along the axis i is given by (Bohren and Huﬀman 1983): αi = V ε0 ε(ω) − εmedium (1.3) 3 εmedium + 3 Li[ε(ω) − εmedium].

### Theoretical Considerations beyond the Quasi-Static Limit: Retardation Eﬀects

For large particles with sizes not negligible compared to the wavelength, the quasi-static approximation breaks down due to retardation eﬀects. The eﬀect of retardation of the exciting field over the volume in the particle leads to a shift in the plasmon resonance. Further, the retardation of the depolarization field inside the particle also leads to a shift in resonance (Meier and Wokaun 1983). For noble metals, the overall shift is towards lower energies, i.e. the dipole resonance position red-shifts with increasing particle size. Intuitively, this eﬀect can be understood by considering that the distance between the charges at opposite interfaces of the particle increases with the particle size. This leads to a smaller restoring force and therefore a lowering of the resonance frequency (Maier 2007).

Another retardation eﬀect that has to be taken into account is the reduction of ab-sorption (which is directly related to the imaginary part of ε(ω), i.e. I m{ε(ω)}), due to the increase of polarization. In order to account qualitatively for this eﬀect, we have to consider the influence of radiation damping which leads to an attenuation in polariza-tion. Radiation damping is caused by a direct radiative decay of the coherent electron oscillations into photons (Kokkinakis and Alexopoulos 1972), and it is the main cause of the weakening of the strength of the dipole plasmon resonance as the particle volume in-creases (Wokaun et al. 1982). As a result, the influence of radiation damping countervails the increase of polarization in strength due to the reduction of absorption.

In summary, the plasmon resonance of particles beyond the quasistatic regime is damped by two competing processes: a radiative decay process into photons, dominating for larger particles, and a non-radiative process due to absorption.

#### Principles of Holography

Holography consists of two processes: the recording of the hologram and the hologram reconstruction. Since holography is based on the interference of two light waves, the light source must have a suﬃcient coherence length.

The general setup of holography is shown in Fig. 2.1. The coherent light source is split into two partial waves by a beam splitter. To record a hologram the object is illuminated by the first wave. The object scatters light that is reflected to the recording medium, e.g. a photographic plate, that is placed at a distance d from the object. The second wave, the so called reference wave, illuminates the light sensitive medium directly where it interferes with the scattered light from the object. The interference pattern is recorded, e.g. by chemical development of the photographic plate. The recorded interference pattern is called hologram.

Then, the original object wave is reconstructed by illuminating the hologram with the reference wave, see right image in Fig. 2.1. The observer sees the three-dimensional reconstructed object at a distance d, which is the position where the object had been before. This image is named virtual image.

**Phase Shifting Holography**

In 1997 Yamaguchi and Zhang proposed the phase-shifting holography technique (Ya-maguchi and Zhang 1997). This methods allows the suppression of the zero-order and twin-images in an in-line holographic setup by recording several images with a diﬀerent phase for the object beam. In their experiment the phase shift was created by a modu-lation of the optical path length. Therefore, a mirror is introduced in the reference arm and moved by a piezoelectric transducer actuator. Le Clerc et al. presented a hetero-dyne holographic scheme in which the reference beam is dynamically phase shifted by frequency shifting (∆f ) the reference field (Le Clerc et al. 2000). This can be done with acousto-optic modulators (AOM). However, the use of AOMs requires typical working frequencies in the range of MHz. Such a high frequency modulation is nowadays not de-tectable even with a fast digital camera. Therefore, two AOMs, working at 80 MHz and 80 MHz+∆f , respectively, are used. The oﬀset in frequency has the advantage that the resulting phase shift is highly accurate. This technique is called heterodyne holography because it is based on heterodyne techniques which are classical in interferometry, where high frequency signals are converted to lower frequencies by combining two frequencies.

When the reference beam, phase shifted either by a modulation in the optical path or by a frequency modulation, interferes with the object wave, the CCD camera records the following intensity: IH = (EO + ER) · (EO + ER)∗.

**Table of contents :**

**1 Plasmonics and Optical Antennas **

1.1 A Short Introduction to Plasmonics

1.1.1 Surface Plasmon Polaritons

1.1.2 Localized Surface Plasmons

1.1.3 Single Particle Plasmon Resonances – The Quasi-Static Approximation

1.1.4 Theoretical Considerations beyond the Quasi-Static Limit: Retardation Effects

1.2 Optical Antennas

1.2.1 Properties of Optical Antennas

1.2.2 Nanoantenna Geometries

1.2.3 Applications of Optical Antennas – State of the Art

1.3 Conclusion

**2 Digital Heterodyne Holography **

2.1 Historical Overview

2.2 Principles of Holography

2.2.1 Off-Axis Holography

2.3 Digital Holography

2.3.1 Numerical Holographic Reconstruction

2.3.2 Phase Shifting Holography

2.3.3 Digital Heterodyne Off-Axis Holography

2.3.4 Experimental Setup of DHH

2.4 Photothermal Heterodyne Holography

2.4.1 Experimental Setup of Photothermal DHH

2.5 Frequency Domain Detection by Heterodyne Holography

2.5.1 Application 1: Frequency-Resolved Temperature Imaging of Integrated Circuits

2.5.2 Application 2: Frequency Detection in the Brownian Regime of Gold Nanorods

2.5.3 Conclusion on Frequency Detection

2.6 Conclusion on Digital Heterodyne Holography

**3 The Nanostructures under Study **

3.1 Design of the Nanostructures

3.1.1 Nanostructure Fabrication

3.2 Spectroscopy of Single Nanoobjects

3.3 FEM Simulation of Plasmonic Nanoobjects

3.3.1 Scattering in the Far-Field

3.3.2 Scattering in the Near-Field

3.3.3 Tests for Validation

3.4 Conclusion

**4 Validation of Experimental Techniques on Elementary Nanoobjects **

4.1 Single Nanodisks

4.1.1 Light Scattering by a Single Disk

4.1.2 Near-Field of Single Disks

4.1.3 Holography of Single Disks

4.1.4 Conclusion on the Scattering Behaviour of Single Disks

4.2 Coupling of Two Nanodisks

4.2.1 Introduction to the Study of Two Coupled Disks

4.2.2 Plasmon Hybridization Model

4.2.3 Scattering Spectra of Two Coupled Disks

4.2.4 Study of Different Modes Excited in Two Coupled Disks

4.2.5 Conclusion on Two Coupled Nanodisks

4.3 Two Coupled Nanorods

4.3.1 Light Scattering of Coupled Rods

4.3.2 Near-Field of Single and Coupled Rods

4.3.3 3D Far-Field Images of Light Scattered by Coupled Rods

4.3.4 Conclusion on Coupled Rods

4.4 Conclusion

**5 Extensive Study of Plasmonic Nanostructures **

5.1 Probing the Coupling of Nanodisk Chains by Spectroscopy

5.1.1 Longitudinal and Transverse Modes in Nanodisk Chains

5.1.2 Far-Field Scattering Revealing Near-Field Coupling in Nanodisk Chains

5.2 Far- and Near-Field Maps of Scattered Light by Nanodisk Chains

5.2.1 The Influence of the Chain Length on the Far-Field Maps

5.2.2 TE Wave and TM Wave Excitation

5.2.3 Imaging of Longitudinal and Transverse Modes in a Chain

5.2.4 Influence of the Exciting Wavelength

5.2.5 Probing the Coupling of Nanodisk Chains by Holography

5.2.6 Imaging of Directional Scattering

5.2.7 Conclusion on the Far- and Near-Field Maps of Nanodisk Chains .

5.3 Probing the Plasmonic Coupling of Disks by Heating

5.3.1 Comparison of Holographic and Photothermal Images

5.3.2 Photothermal Imaging of Nanostructures – An Analytical Analysis of the Photothermal Signal

5.3.3 Photothermal Signal and Absorption Cross Section

5.3.4 Photothermal Holography Reveals Coupling of Nanodisk Chains .

5.3.5 Conclusion on Photothermal Imaging of Nanodisk Chains

5.4 Coupled Triangles

5.5 Conclusion on the Application of Holography to Plasmonics

**Conclusions and Prospects **

**A FEM Simulation Parameters – Fresnel Coefficients **

A.1 Excitation Field in Reflection

A.2 Excitation Field in Transmission

**B Square-Wave Function in Matlab **

**Bibliography **