1.3 Dispersion relations

Dispersion relations are crucial for understanding wave behavior in metamaterials and photonic crystals. They describe how frequency relates to wavelength, revealing key properties like bandgaps and wave velocities. This knowledge is essential for designing structures with unique optical properties.

By engineering dispersion, we can create materials with extraordinary capabilities. From slowing light to enabling negative refraction, dispersion control opens up new possibilities in optics and electromagnetics. Understanding these concepts is vital for advancing metamaterial and photonic crystal technologies.

Dispersion relations overview

• Dispersion relations describe how waves propagate through a medium, relating the frequency or energy of a wave to its wavelength or wavenumber
• Understanding dispersion is crucial for designing metamaterials and photonic crystals with desired properties such as slow light, negative refraction, and bandgap engineering
• Different types of dispersion relations lead to distinct wave behaviors and can be tailored for specific applications in metamaterials and photonic crystals

Frequency vs wavenumber

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• Dispersion relations often plot frequency ($\omega$) against wavenumber ($k$), showing how the frequency of a wave changes with its spatial frequency
• The slope of the $\omega$-$k$ curve determines the phase velocity ($v_p = \omega/k$) and group velocity ($v_g = d\omega/dk$) of the wave
• The shape of the dispersion curve reveals important properties such as the presence of bandgaps, slow light regions, or negative refraction

Phase velocity vs group velocity

• Phase velocity ($v_p$) is the speed at which the phase of a single frequency component propagates, given by $v_p = \omega/k$
• Group velocity ($v_g$) is the speed at which the envelope of a wave packet (superposition of multiple frequencies) propagates, given by $v_g = d\omega/dk$
• In dispersive media, phase velocity and group velocity can differ significantly, leading to phenomena such as pulse broadening or superluminal propagation

Brillouin zone boundaries

• The Brillouin zone is the primitive cell in reciprocal space, containing all unique wavenumbers for a periodic structure
• At the Brillouin zone boundaries ($k = \pm \pi/a$ for a 1D lattice with periodicity $a$), standing waves form due to constructive interference of scattered waves
• Band gaps often occur near the Brillouin zone boundaries, where the dispersion relation flattens and the group velocity approaches zero

Types of dispersion relations

• Different functional forms of dispersion relations lead to distinct wave propagation characteristics
• The shape of the dispersion curve can be engineered by designing the material properties or structure of a metamaterial or photonic crystal
• Common types of dispersion relations include linear, parabolic, hyperbolic, and flat band dispersion

Linear dispersion

• Linear dispersion occurs when frequency is directly proportional to wavenumber: $\omega = vk$, where $v$ is the constant phase and group velocity
• Waves with linear dispersion propagate without distortion, as all frequency components travel at the same speed (non-dispersive)
• Examples of linear dispersion include electromagnetic waves in vacuum and acoustic waves in air

Parabolic dispersion

• Parabolic dispersion has a quadratic relationship between frequency and wavenumber: $\omega = \alpha k^2 + \beta$, where $\alpha$ and $\beta$ are constants
• The group velocity ($v_g = d\omega/dk = 2\alpha k$) changes with wavenumber, leading to pulse broadening and dispersion
• Electron bands in semiconductors and some guided wave structures exhibit parabolic dispersion

Hyperbolic dispersion

• Hyperbolic dispersion arises from anisotropic materials with permittivity or permeability tensor elements of opposite signs
• The dispersion relation has the form: $\frac{k_x^2}{\epsilon_x} + \frac{k_y^2}{\epsilon_y} = \frac{\omega^2}{c^2}$, where $\epsilon_x$ and $\epsilon_y$ have opposite signs
• Hyperbolic metamaterials support high-$k$ modes, enabling subdiffraction imaging and increased photonic density of states

Flat band dispersion

• Flat band dispersion occurs when the frequency remains constant over a range of wavenumbers, resulting in a flat segment in the dispersion curve
• Waves in a flat band have zero group velocity ($v_g = d\omega/dk = 0$) and are localized in space
• Flat bands can arise from specific lattice geometries (Lieb lattice) or in the presence of strong interactions (exciton-polariton condensates)

Dispersion in periodic structures

• Periodic structures, such as metamaterials and photonic crystals, exhibit unique dispersion properties due to their spatial periodicity
• The dispersion relation in periodic structures is determined by solving the wave equation with periodic boundary conditions
• Key concepts in understanding dispersion in periodic structures include Bloch waves, band structures, and bandgaps

Bloch wave solutions

• In a periodic structure, the eigenmodes are Bloch waves, which are plane waves modulated by a periodic function: $\psi(r) = e^{ikr}u(r)$, where $u(r) = u(r+a)$ for lattice periodicity $a$
• The wavenumber $k$ is restricted to the first Brillouin zone, and the periodic function $u(r)$ captures the spatial variation within each unit cell
• Bloch waves form the basis for understanding the dispersion relation and band structure of periodic systems

Band structure calculations

• The band structure is the dispersion relation of a periodic structure, plotting the eigenfrequencies $\omega$ against the wavenumbers $k$ in the Brillouin zone
• Band structure calculations involve solving the eigenvalue problem for the wave equation with periodic boundary conditions
• Numerical methods for band structure calculations include the plane wave expansion method, finite-difference time-domain simulations, and eigenmode solvers

Bandgaps vs transmission bands

• Bandgaps are frequency ranges where no propagating modes exist in a periodic structure, resulting in complete reflection or suppression of waves
• Transmission bands are frequency ranges where propagating modes are allowed, enabling wave transmission through the structure
• The width and position of bandgaps and transmission bands can be engineered by designing the geometry, material properties, and periodicity of the structure

Dispersion engineering applications

• Dispersion engineering involves tailoring the dispersion relation of metamaterials and photonic crystals for specific applications
• By designing the structure and material properties, unique wave propagation effects can be achieved, such as slow light, negative refraction, and superlensing
• Dispersion engineering also enables control over the density of states, which is crucial for enhancing light-matter interactions

Slow light devices

• Slow light refers to the reduction of group velocity in a medium, enabling enhanced light-matter interactions and nonlinear effects
• Dispersion engineering can create flat bands or regions of high group index ($n_g = c/v_g$) in the dispersion relation, leading to slow light propagation
• Applications of slow light include optical buffering, enhanced nonlinear optics, and sensing

Negative refraction

• Negative refraction occurs when light bends in the opposite direction to normal refraction, due to a negative refractive index
• Metamaterials with simultaneously negative permittivity and permeability can exhibit negative refraction over a specific frequency range
• Negative refraction enables novel applications such as flat lenses, subwavelength focusing, and cloaking

Superlensing effects

• Superlensing refers to the ability to focus light beyond the diffraction limit, enabling subwavelength imaging
• Metamaterials with hyperbolic dispersion can support high-$k$ modes, which carry subwavelength information and contribute to superlensing
• Near-field superlensing has been demonstrated using silver films and hyperbolic metamaterials

Density of states control

• The density of states (DOS) quantifies the number of available states per unit frequency and volume in a system
• Dispersion engineering can modify the DOS in metamaterials and photonic crystals, enabling control over spontaneous emission, absorption, and thermal radiation
• Examples include enhancing or suppressing spontaneous emission in quantum emitters and tailoring thermal emission for energy applications

Experimental dispersion characterization

• Experimental techniques are essential for measuring the dispersion relation of metamaterials and photonic crystals
• Different methods probe the frequency-dependent response, phase velocity, or spatial distribution of waves in the structure
• Common experimental techniques include angle-resolved measurements, phase-sensitive methods, and near-field microscopy

Angle-resolved measurements

• Angle-resolved measurements involve varying the incident angle of light on a sample and measuring the transmitted or reflected spectra
• By mapping the transmission or reflection spectra as a function of angle (or parallel wavevector), the dispersion relation can be reconstructed
• Techniques such as angle-resolved photoemission spectroscopy (ARPES) and angle-resolved reflectivity are used for dispersion characterization

Phase-sensitive techniques

• Phase-sensitive techniques measure the phase delay or phase velocity of waves propagating through a sample, enabling direct access to the dispersion relation
• Examples include time-domain spectroscopy (TDS), which measures the temporal delay of pulses, and interferometric methods that detect phase shifts
• Phase-sensitive techniques can reveal the presence of slow light, negative refraction, and other dispersive effects

Near-field microscopy

• Near-field scanning optical microscopy (NSOM) probes the evanescent fields close to the surface of a sample, providing subwavelength spatial resolution
• By measuring the near-field distribution of waves in a metamaterial or photonic crystal, the local dispersion relation can be mapped
• Near-field techniques are particularly useful for characterizing surface waves, such as surface plasmon polaritons or Bloch surface waves

Numerical dispersion modeling

• Numerical simulations play a crucial role in designing and optimizing metamaterials and photonic crystals with desired dispersion properties
• Different computational methods are employed to solve the wave equation in complex structures and extract the dispersion relation
• Common numerical techniques for dispersion modeling include the plane wave expansion method, finite-difference time-domain simulations, and eigenmode solvers

Plane wave expansion method

• The plane wave expansion (PWE) method is a frequency-domain technique for calculating the band structure of periodic structures
• In PWE, the fields and material properties are expanded in terms of plane waves, and the wave equation is transformed into an eigenvalue problem
• By solving the eigenvalue problem for each wavenumber in the Brillouin zone, the dispersion relation and eigenmodes can be obtained

Finite-difference time-domain simulations

• Finite-difference time-domain (FDTD) simulations solve the time-dependent Maxwell's equations on a discretized grid, capturing the temporal evolution of electromagnetic fields
• FDTD can model the propagation of pulses or continuous waves through complex structures, including metamaterials and photonic crystals
• By analyzing the spectral content of the fields at different points in the structure, the dispersion relation can be extracted

Eigenmode solver approaches

• Eigenmode solvers, such as the finite-element method (FEM) or the multiple scattering method (MSM), directly solve for the eigenmodes and eigenfrequencies of a structure
• These methods discretize the wave equation into a matrix eigenvalue problem, which is then solved numerically
• Eigenmode solvers are particularly useful for finding localized modes, defect states, and the dispersion relation near band edges

Dispersion in metamaterials

• Metamaterials are artificial structures designed to have unique dispersion properties not found in natural materials
• The dispersion in metamaterials arises from the collective response of subwavelength resonant elements, such as split-ring resonators or plasmonic nanoparticles
• Key concepts in understanding dispersion in metamaterials include effective medium theory, spatial dispersion, and nonlocal material response

Effective medium theory

• Effective medium theory (EMT) describes the macroscopic properties of a metamaterial in terms of effective permittivity and permeability tensors
• EMT assumes that the metamaterial can be treated as a homogeneous medium with averaged properties, valid when the unit cell size is much smaller than the wavelength
• The effective parameters can be engineered by designing the geometry and arrangement of the subwavelength elements

Spatial dispersion effects

• Spatial dispersion refers to the dependence of the material response on the wavevector, in addition to the frequency
• In metamaterials, spatial dispersion can arise from the finite size of the unit cells or the coupling between neighboring elements
• Spatial dispersion can lead to nonlocal effects, such as the dependence of the effective parameters on the wavevector direction

Nonlocal material response

• Nonlocal material response occurs when the polarization at a given point depends on the electric field at other points in the material
• In metamaterials, nonlocal effects can become significant when the unit cell size is comparable to the wavelength or when there is strong coupling between elements
• Nonlocal response can result in additional waves, such as longitudinal modes, and can modify the dispersion relation

Dispersion in photonic crystals

• Photonic crystals are periodic structures designed to control the propagation of light through Bragg scattering and interference effects
• The dispersion in photonic crystals is determined by the geometry, periodicity, and material properties of the structure
• Important aspects of dispersion in photonic crystals include scalability, dimensionality, and the choice of lattice geometry

Scalability vs operating frequency

• The operating frequency of a photonic crystal is determined by its periodicity, with the bandgaps and other features scaling inversely with the lattice constant
• By adjusting the size of the unit cell, photonic crystals can be designed to operate at different frequency ranges, from microwave to optical frequencies
• Scalability enables the realization of similar dispersion properties across different length scales and applications

1D vs 2D vs 3D

• Photonic crystals can be designed in one, two, or three dimensions, with each dimensionality offering unique dispersion properties
• 1D photonic crystals, such as Bragg mirrors, exhibit bandgaps and slow light effects along the direction of periodicity
• 2D photonic crystals, such as photonic crystal slabs, can guide light in-plane and exhibit complete bandgaps for certain polarizations
• 3D photonic crystals offer the possibility of complete bandgaps in all directions, enabling omnidirectional light confinement

Rod-based vs hole-based lattices

• Photonic crystals can be constructed by arranging high-index rods in a low-index background (rod-based) or by etching holes in a high-index material (hole-based)
• The choice of lattice geometry affects the dispersion relation and the width and position of bandgaps
• Examples of rod-based lattices include square and triangular lattices, while hole-based lattices include honeycomb and inverse opal structures