7.1 Surface plasmon polaritons

Surface plasmon polaritons (SPPs) are electromagnetic waves that propagate along metal-dielectric interfaces, coupling with electron oscillations in the metal. They exhibit unique properties like subwavelength confinement and field enhancement, making them promising for various applications in metamaterials and photonic crystals.

Understanding SPPs is crucial for designing plasmonic devices and structures. This topic covers their fundamental concepts, excitation methods, applications, and advanced topics like nonlinear and quantum plasmonics. It also explores numerical modeling and experimental characterization techniques used to study and optimize SPP-based systems.

Fundamentals of surface plasmon polaritons

• Surface plasmon polaritons (SPPs) are electromagnetic waves that propagate along the interface between a metal and a dielectric, coupling with collective oscillations of free electrons in the metal
• SPPs exhibit unique properties such as subwavelength confinement and field enhancement, making them promising for various applications in metamaterials and photonic crystals
• Understanding the fundamental concepts of SPPs is crucial for designing and optimizing plasmonic devices and structures

Definition and properties

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• SPPs are TM-polarized surface waves that arise from the interaction between electromagnetic fields and free electron oscillations at metal-dielectric interfaces
• Characterized by evanescent fields that decay exponentially away from the interface, leading to subwavelength confinement of electromagnetic energy
• SPPs can propagate along the interface with a wavelength shorter than that of the excitation light, enabling the manipulation of light at the nanoscale

Dispersion relation

• The dispersion relation describes the relationship between the frequency ($\omega$) and the wave vector ($k$) of SPPs
• For a simple metal-dielectric interface, the dispersion relation is given by: $k_{spp} = \frac{\omega}{c} \sqrt{\frac{\varepsilon_m \varepsilon_d}{\varepsilon_m + \varepsilon_d}}$
  • $\varepsilon_m$ and $\varepsilon_d$ are the dielectric functions of the metal and dielectric, respectively
• The dispersion curve of SPPs lies to the right of the light line, indicating that SPPs have a higher momentum than free-space photons at the same frequency

Propagation length

• The propagation length ($L_{spp}$) is the distance over which the intensity of SPPs decays to 1/e of its initial value due to ohmic losses in the metal
• Determined by the imaginary part of the SPP wave vector: $L_{spp} = \frac{1}{2 \text{Im}(k_{spp})}$
• Propagation length is typically on the order of tens to hundreds of micrometers in the visible and near-infrared range, depending on the metal and wavelength

Penetration depth

• The penetration depth ($\delta$) is the distance over which the evanescent field of SPPs decays to 1/e of its value at the interface
• Penetration depth into the dielectric ($\delta_d$) and metal ($\delta_m$) are given by: $\delta_d = \frac{1}{\text{Re}\left(\sqrt{\frac{\omega^2}{c^2} - k_{spp}^2}\right)}$ $\delta_m = \frac{1}{\text{Re}\left(\sqrt{k_{spp}^2 - \frac{\omega^2}{c^2}\varepsilon_m}\right)}$
• Typically, the penetration depth into the dielectric is larger than that into the metal, ranging from tens to hundreds of nanometers

Excitation methods for surface plasmons

• To excite SPPs, the momentum mismatch between free-space photons and SPPs must be overcome, as SPPs have a higher momentum than light at the same frequency
• Various excitation methods have been developed to couple light to SPPs efficiently, each with its advantages and limitations
• The choice of excitation method depends on factors such as the desired coupling efficiency, spatial resolution, and experimental constraints

Prism coupling techniques

• Prism coupling relies on the evanescent wave generated by total internal reflection at the prism-metal interface to excite SPPs
• Two common configurations: Kretschmann (thin metal film on a prism) and Otto (prism separated from the metal by a thin dielectric gap)
• Prism coupling offers high coupling efficiency (up to 90%) and is widely used for SPP excitation and sensing applications

Grating coupling

• Periodic grating structures can provide the additional momentum required to couple light to SPPs by diffraction
• The grating period and geometry determine the coupling efficiency and the range of excitation wavelengths and angles
• Grating coupling allows for more compact and integrated plasmonic devices compared to prism coupling

Near-field excitation

• Near-field excitation uses subwavelength sources, such as scanning near-field optical microscopy (SNOM) probes or quantum dots, to directly couple light to SPPs
• Enables high spatial resolution and localized excitation of SPPs, suitable for studying nanoscale plasmonic phenomena and devices
• Near-field excitation typically has lower coupling efficiency compared to prism or grating coupling

Comparison of excitation methods

• Prism coupling: high efficiency, bulky setup, limited to planar geometries
• Grating coupling: compact, integrated, lower efficiency than prism coupling, wavelength and angle-dependent
• Near-field excitation: high spatial resolution, localized excitation, lower efficiency, suitable for nanoscale studies

Applications of surface plasmon polaritons

• SPPs have found numerous applications in various fields due to their unique properties, such as subwavelength confinement, field enhancement, and sensitivity to the surrounding environment
• Plasmonic devices and structures based on SPPs have the potential to revolutionize areas such as sensing, imaging, and information processing
• The following sections highlight some of the key applications of SPPs in metamaterials and photonic crystals

Surface-enhanced Raman spectroscopy (SERS)

• SERS exploits the strong field enhancement near plasmonic nanostructures to dramatically increase the Raman scattering signal of molecules adsorbed on the surface
• Enhancement factors of up to 10^10 have been reported, enabling single-molecule detection and ultrasensitive chemical analysis
• SERS substrates based on engineered plasmonic metamaterials and photonic crystals can provide uniform and reproducible enhancement, crucial for practical applications

Biosensing and chemical sensing

• SPPs are highly sensitive to changes in the refractive index of the surrounding medium, making them ideal for biosensing and chemical sensing applications
• Plasmonic sensors based on SPP resonance shifts or intensity changes can detect the presence of specific analytes (proteins, DNA, gases) with high sensitivity and specificity
• Plasmonic metamaterials and photonic crystals can be designed to enhance the sensitivity and selectivity of SPP-based sensors

Subwavelength imaging and lithography

• The subwavelength confinement of SPPs can be harnessed for imaging and lithography beyond the diffraction limit
• Plasmonic superlenses and hyperlenses based on metamaterials can achieve nanoscale resolution by amplifying and propagating evanescent waves
• SPP-assisted lithography techniques, such as plasmonic nanolithography and plasmonic photoresist, enable the fabrication of nanoscale patterns and structures

Plasmonic waveguides and circuits

• SPPs can be guided along metal-dielectric interfaces or nanostructures, enabling the development of plasmonic waveguides and circuits for nanoscale information processing
• Plasmonic metamaterials and photonic crystals can be engineered to control the propagation, confinement, and dispersion of SPPs
• Plasmonic components such as splitters, routers, and modulators have been demonstrated, paving the way for integrated plasmonic circuits and devices

Localized surface plasmons vs propagating plasmons

• Surface plasmons can be classified into two main categories: localized surface plasmons (LSPs) and propagating surface plasmons (PSPs)
• LSPs and PSPs exhibit distinct properties and find applications in different areas of metamaterials and photonic crystals
• Understanding the differences between LSPs and PSPs is essential for designing plasmonic structures and devices tailored to specific requirements

Definitions and properties

• LSPs are non-propagating excitations of conduction electrons in metallic nanoparticles or nanostructures, coupled to electromagnetic fields
• LSPs are characterized by strong field confinement and enhancement near the nanoparticle surface, with resonance frequencies determined by the particle geometry and material
• PSPs, also known as SPPs, are propagating electromagnetic waves coupled to electron oscillations along metal-dielectric interfaces, as discussed in previous sections

Resonance conditions

• LSP resonances occur when the frequency of the incident light matches the natural oscillation frequency of the conduction electrons in the nanoparticle
• The resonance frequency of LSPs depends on factors such as the particle size, shape, and dielectric environment
• PSP resonances, on the other hand, are determined by the dispersion relation and the phase-matching conditions required for efficient excitation

Near-field enhancement

• Both LSPs and PSPs can generate strong near-field enhancement, but the spatial extent and distribution of the enhanced fields differ
• LSPs exhibit highly localized field enhancement near the nanoparticle surface, with the field decaying rapidly away from the particle
• PSPs generate field enhancement along the metal-dielectric interface, with the field extending further into the dielectric medium

Radiative vs non-radiative decay

• LSPs can decay radiatively by emitting photons or non-radiatively through absorption and generation of hot electrons
• The balance between radiative and non-radiative decay depends on the particle size and material, with larger particles favoring radiative decay
• PSPs primarily decay non-radiatively due to ohmic losses in the metal, limiting their propagation length

Advanced topics in surface plasmon polaritons

• As the field of plasmonics continues to evolve, new concepts and phenomena are being explored to extend the capabilities of SPP-based devices and structures
• Advanced topics in SPPs encompass nonlinear effects, quantum phenomena, active control, and novel material platforms
• These emerging areas offer exciting opportunities for the development of next-generation plasmonic metamaterials and photonic crystals

Nonlinear plasmonics

• Nonlinear optical processes, such as second-harmonic generation (SHG) and four-wave mixing (FWM), can be enhanced by the strong field confinement and enhancement in plasmonic structures
• Plasmonic metamaterials and photonic crystals can be designed to optimize nonlinear optical interactions and enable efficient frequency conversion and all-optical signal processing
• Nonlinear plasmonic devices have potential applications in sensing, imaging, and quantum information processing

Quantum plasmonics

• Quantum plasmonics explores the interaction between SPPs and quantum emitters, such as quantum dots or single molecules
• Plasmonic nanostructures can be used to control the emission and absorption properties of quantum emitters, enabling the development of single-photon sources and quantum sensors
• Quantum plasmonic metamaterials and photonic crystals can be engineered to create entangled states and perform quantum operations

Active control of plasmonic properties

• Active control of SPP properties, such as propagation, dispersion, and coupling, can be achieved through external stimuli (electrical, optical, thermal, or magnetic)
• Plasmonic metamaterials and photonic crystals incorporating active materials, such as phase-change materials or graphene, can enable dynamic tuning and modulation of SPP behavior
• Active plasmonic devices have applications in adaptive optics, reconfigurable metamaterials, and optical computing

Chiral and hyperbolic plasmonics

• Chiral plasmonic structures exhibit different responses to left- and right-circularly polarized light, enabling the control of light-matter interactions at the nanoscale
• Hyperbolic plasmonic metamaterials possess anisotropic dielectric functions, leading to unusual dispersion relations and the emergence of high-k modes
• Chiral and hyperbolic plasmonic systems offer new opportunities for polarization control, directional emission, and negative refraction

Numerical methods for modeling surface plasmons

• Numerical modeling plays a crucial role in the design, optimization, and understanding of plasmonic structures and devices
• Various computational methods have been developed to simulate the propagation and interaction of SPPs in complex geometries and materials
• Each numerical method has its strengths and limitations, and the choice of method depends on factors such as the problem size, desired accuracy, and computational resources

Finite-difference time-domain (FDTD) method

• FDTD is a popular method for modeling electromagnetic wave propagation in plasmonic structures
• The method discretizes the spatial and temporal domains and solves Maxwell's equations iteratively using finite differences
• FDTD is well-suited for modeling broadband and transient phenomena, as well as complex geometries and dispersive materials

Finite element method (FEM)

• FEM is a versatile method for solving partial differential equations, including Maxwell's equations, in complex geometries
• The method divides the computational domain into smaller elements and approximates the solution using basis functions
• FEM is particularly effective for modeling plasmonic structures with irregular shapes and inhomogeneous materials

Boundary element method (BEM)

• BEM is a computational method that solves electromagnetic scattering problems by discretizing the boundaries of the scatterers
• The method is based on the integral form of Maxwell's equations and the Green's function for the background medium
• BEM is efficient for modeling plasmonic structures with homogeneous domains and piecewise constant dielectric functions

Comparison of numerical techniques

• FDTD: broadband, time-domain, suitable for complex geometries and dispersive materials, computationally intensive for large problems
• FEM: frequency-domain, handles complex geometries and inhomogeneous materials, efficient for resonant phenomena, requires meshing of the entire domain
• BEM: efficient for homogeneous domains and piecewise constant dielectrics, less suitable for inhomogeneous materials, limited to linear problems

Experimental characterization of surface plasmons

• Experimental techniques for characterizing SPPs are essential for validating theoretical predictions, optimizing plasmonic devices, and exploring new phenomena
• Various methods have been developed to probe the spatial, spectral, and temporal properties of SPPs with high resolution and sensitivity
• The choice of characterization technique depends on factors such as the desired information, sample geometry, and available instrumentation

Near-field scanning optical microscopy (NSOM)

• NSOM is a scanning probe technique that uses a subwavelength aperture or tip to map the near-field distribution of SPPs with nanoscale resolution
• The method can provide direct visualization of SPP propagation, confinement, and interference patterns
• NSOM is particularly useful for studying localized plasmonic modes and near-field enhancement in nanostructures

Electron energy loss spectroscopy (EELS)

• EELS is a spectroscopic technique that measures the energy loss of electrons interacting with a sample, including the excitation of SPPs
• The method provides high spatial and energy resolution, enabling the mapping of SPP modes and their dispersion relations
• EELS is well-suited for studying plasmonic excitations in nanoparticles, thin films, and metamaterials

Cathodoluminescence imaging

• Cathodoluminescence (CL) imaging uses an electron beam to excite SPPs and collect the resulting light emission
• The technique offers high spatial resolution and can provide information on the local density of optical states and radiative decay channels
• CL imaging is particularly useful for studying plasmonic modes in nanoparticles and nanostructures, as well as the coupling between SPPs and quantum emitters

Far-field techniques for plasmon characterization

• Far-field techniques, such as leakage radiation microscopy and Fourier plane imaging, can be used to study the propagation and dispersion of SPPs
• These methods rely on the detection of light scattered or radiated by SPPs, providing indirect information on their properties
• Far-field techniques are less invasive than near-field methods and can be applied to a wide range of plasmonic structures and devices