1.2 Electromagnetic wave propagation

Electromagnetic waves are the foundation of metamaterials and photonic crystals. These oscillating electric and magnetic fields propagate through space, carrying energy. Understanding their behavior is crucial for designing structures that can manipulate light in unprecedented ways.

Maxwell's equations describe electromagnetic waves mathematically. From these, we can derive wave equations and solutions like plane waves. Concepts like permittivity, permeability, and refractive index explain how waves interact with different media, forming the basis for engineering novel optical properties.

Electromagnetic wave fundamentals

• Electromagnetic waves are oscillating electric and magnetic fields that propagate through space and carry energy
• Understanding the fundamental principles of electromagnetic waves is crucial for designing and analyzing metamaterials and photonic crystals
• Key concepts in this section include Maxwell's equations, wave equation derivation, and plane wave solutions

Maxwell's equations

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• Maxwell's equations are a set of four partial differential equations that describe the behavior of electric and magnetic fields
• Gauss's law relates the electric field to the electric charge density: $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$
• Gauss's law for magnetism states that magnetic monopoles do not exist: $\nabla \cdot \mathbf{B} = 0$
• Faraday's law describes how a changing magnetic field induces an electric field: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
• Ampère's law with Maxwell's correction relates the magnetic field to the electric current density and the changing electric field: $\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0\frac{\partial \mathbf{E}}{\partial t}\right)$

Wave equation derivation

• The wave equation describes the propagation of electromagnetic waves in space and time
• Can be derived from Maxwell's equations by taking the curl of Faraday's law and substituting Ampère's law
• In a source-free region, the wave equation for the electric field is: $\nabla^2 \mathbf{E} - \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$
• The wave equation for the magnetic field has a similar form: $\nabla^2 \mathbf{B} - \frac{1}{c^2}\frac{\partial^2 \mathbf{B}}{\partial t^2} = 0$

Plane wave solutions

• Plane waves are the simplest solutions to the wave equation in a homogeneous, isotropic medium
• Electric and magnetic fields in a plane wave are perpendicular to each other and to the direction of propagation (transverse waves)
• The general form of a plane wave solution for the electric field is: $\mathbf{E}(\mathbf{r}, t) = \mathbf{E}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}$
  • $\mathbf{E}_0$ is the complex amplitude
  • $\mathbf{k}$ is the wave vector
  • $\omega$ is the angular frequency
• The magnetic field in a plane wave is related to the electric field by: $\mathbf{B}(\mathbf{r}, t) = \frac{1}{c} \hat{\mathbf{k}} \times \mathbf{E}(\mathbf{r}, t)$

Wave propagation in media

• When electromagnetic waves propagate through different media, their behavior is influenced by the material properties
• Understanding how permittivity, permeability, and refractive index affect wave propagation is essential for designing metamaterials and photonic crystals with desired properties
• Dispersion relations describe the relationship between the wave vector and the frequency in a given medium

Permittivity and permeability

• Permittivity ($\varepsilon$) is a measure of how an electric field affects and is affected by a medium
  • Related to a material's ability to polarize in response to an applied electric field
  • Vacuum permittivity: $\varepsilon_0 \approx 8.85 \times 10^{-12} \ \text{F/m}$
• Permeability ($\mu$) is a measure of how a magnetic field affects and is affected by a medium
  • Related to a material's ability to magnetize in response to an applied magnetic field
  • Vacuum permeability: $\mu_0 = 4\pi \times 10^{-7} \ \text{H/m}$
• Relative permittivity ($\varepsilon_r$) and relative permeability ($\mu_r$) are dimensionless quantities that compare a material's properties to those of vacuum: $\varepsilon_r = \frac{\varepsilon}{\varepsilon_0}, \quad \mu_r = \frac{\mu}{\mu_0}$

Refractive index

• The refractive index ($n$) is a dimensionless number that describes how light propagates through a medium
• Related to the permittivity and permeability of the medium: $n = \sqrt{\varepsilon_r \mu_r}$
• Determines the phase velocity of light in the medium: $v_p = \frac{c}{n}$
• Also affects the angle of refraction when light passes from one medium to another (Snell's law): $n_1 \sin \theta_1 = n_2 \sin \theta_2$

Dispersion relations

• Dispersion relations describe how the wave vector ($\mathbf{k}$) and the angular frequency ($\omega$) are related in a given medium
• In a non-dispersive medium, the dispersion relation is linear: $\omega = c|\mathbf{k}|$
• In a dispersive medium, the refractive index depends on the frequency, leading to a nonlinear dispersion relation: $\omega = \frac{c}{n(\omega)}|\mathbf{k}|$
• Dispersion can cause pulse broadening and distortion as different frequency components travel at different velocities
• Metamaterials and photonic crystals can be designed to engineer desired dispersion relations for various applications

Propagation in periodic structures

• Periodic structures, such as photonic crystals, exhibit unique wave propagation properties due to their repeating patterns
• The Bloch theorem, Brillouin zones, and band structures are essential concepts for understanding wave propagation in periodic structures
• Photonic crystals can be designed to control the flow of light, enabling applications such as waveguiding, filtering, and localization

Bloch theorem

• The Bloch theorem states that the eigenfunctions of a wave equation in a periodic potential can be written as the product of a plane wave and a periodic function: $\psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} u_{\mathbf{k}}(\mathbf{r})$
  • $u_{\mathbf{k}}(\mathbf{r})$ has the same periodicity as the potential
• The wave vector $\mathbf{k}$ is restricted to the first Brillouin zone
• Bloch waves are the solutions to the wave equation in a periodic structure

Brillouin zones

• Brillouin zones are primitive cells in the reciprocal lattice of a periodic structure
• The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice
  • Contains all the unique wave vectors
  • Boundaries are defined by planes perpendicular to the reciprocal lattice vectors at their midpoints
• Higher-order Brillouin zones are defined by translating the first Brillouin zone by reciprocal lattice vectors
• Brillouin zones are important for understanding the dispersion relations and band structures of periodic structures

Band structures and gaps

• Band structures show the dispersion relations (frequency vs. wave vector) for a periodic structure
• Calculated by solving the eigenvalue problem for the wave equation in the periodic potential
• Bands represent allowed frequencies for a given wave vector
• Band gaps are frequency ranges where no propagating modes exist
  • Result from destructive interference of waves scattered by the periodic structure
  • Can be used to control the flow of light (photonic band gaps) or electrons (electronic band gaps)
• The width and position of band gaps depend on the geometry and material properties of the periodic structure
• Metamaterials and photonic crystals can be designed to have desired band structures and gaps for various applications

Anisotropic media

• Anisotropic media have properties that depend on the direction of the applied field or the direction of wave propagation
• Permittivity and permeability in anisotropic media are described by tensors rather than scalar quantities
• Anisotropy can lead to interesting phenomena such as birefringence, dichroism, and polarization control

Permittivity and permeability tensors

• In anisotropic media, the permittivity and permeability are represented by 3×3 tensors: $\overline{\overline{\varepsilon}} = \begin{pmatrix} \varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\ \varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\ \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \end{pmatrix}, \quad \overline{\overline{\mu}} = \begin{pmatrix} \mu_{xx} & \mu_{xy} & \mu_{xz} \\ \mu_{yx} & \mu_{yy} & \mu_{yz} \\ \mu_{zx} & \mu_{zy} & \mu_{zz} \end{pmatrix}$
• The tensors relate the electric displacement field ($\mathbf{D}$) to the electric field ($\mathbf{E}$) and the magnetic field ($\mathbf{H}$) to the magnetic flux density ($\mathbf{B}$): $\mathbf{D} = \overline{\overline{\varepsilon}} \mathbf{E}, \quad \mathbf{B} = \overline{\overline{\mu}} \mathbf{H}$
• The tensors can be diagonalized by choosing a suitable coordinate system aligned with the principal axes of the material

Birefringence and dichroism

• Birefringence is the property of a material having a refractive index that depends on the polarization and propagation direction of light
  • Occurs in anisotropic materials with different principal refractive indices
  • Causes double refraction, where a single ray of light splits into two rays (ordinary and extraordinary) with different polarizations and velocities
• Dichroism is the property of a material having different absorption coefficients for different polarizations of light
  • Can be linear (absorption depends on linear polarization) or circular (absorption depends on circular polarization)
  • Leads to polarization-dependent attenuation of light
• Both birefringence and dichroism are used in various optical devices (wave plates, polarizers) and can be engineered in metamaterials

Polarization control

• Anisotropic metamaterials can be designed to control the polarization of electromagnetic waves
• Examples include:
  • Polarization rotators: rotate the polarization of an incident wave by a desired angle
  • Polarization converters: convert between linear and circular polarizations or between different linear polarization states
  • Polarization filters: selectively transmit or reflect certain polarizations
• Achieved by tailoring the geometry and arrangement of the metamaterial elements to create anisotropic effective permittivity and permeability tensors
• Enables compact and integrated polarization control devices for various applications (imaging, sensing, communication)

Negative index materials

• Negative index materials (NIMs) are a class of metamaterials with simultaneously negative permittivity and permeability
• Exhibit unusual properties such as negative refraction, backward wave propagation, and perfect lensing
• Potential applications include super-resolution imaging, cloaking, and novel optical devices

Negative refraction

• Negative refraction occurs when light bends in the opposite direction than expected at an interface between a positive and a negative index material
• Follows a modified Snell's law: $n_1 \sin \theta_1 = -|n_2| \sin \theta_2$
• Results from the negative phase velocity of light in the NIM
• Enables novel applications such as flat lenses and superlenses that can overcome the diffraction limit

Backward wave propagation

• In NIMs, the phase velocity and group velocity of light are antiparallel
• The wave vector ($\mathbf{k}$), electric field ($\mathbf{E}$), and magnetic field ($\mathbf{H}$) form a left-handed triad, in contrast to the right-handed triad in conventional materials
• Backward waves can be used for phase compensation and dispersion control in metamaterials
• Enables novel devices such as backward-wave antennas and phase conjugators

Perfect lensing

• A slab of NIM with a refractive index of -1 can act as a perfect lens
• Focuses both propagating and evanescent waves, overcoming the diffraction limit
• Requires careful design to minimize losses and impedance mismatch at the interfaces
• Potential applications in super-resolution imaging, lithography, and data storage

Guided wave propagation

• Guided wave propagation refers to the confinement and control of electromagnetic waves in structures such as waveguides and fibers
• Metamaterials and photonic crystals can be used to engineer novel waveguiding structures with unique properties and functionalities
• Key concepts include waveguide modes, photonic crystal fibers, and slow and stopped light

Waveguide modes

• Waveguide modes are the allowed electromagnetic field distributions that can propagate along a waveguide structure
• Determined by solving Maxwell's equations with appropriate boundary conditions
• Can be classified as transverse electric (TE), transverse magnetic (TM), or hybrid modes, depending on the field components
• Each mode has a specific cutoff frequency, below which it cannot propagate
• Metamaterials can be used to design waveguides with unusual mode properties (negative index modes, mode conversion)

Photonic crystal fibers

• Photonic crystal fibers (PCFs) are optical fibers with a periodic arrangement of air holes in the cladding
• The periodic structure creates a photonic bandgap that confines light in the core
• Can be designed to have unique properties, such as:
  • Endlessly single-mode operation: support only the fundamental mode over a wide wavelength range
  • Large mode area: reduce nonlinear effects and increase power handling
  • High birefringence: maintain polarization state of light
  • Dispersion engineering: control the dispersion profile for various applications (supercontinuum generation, pulse compression)
• PCFs have applications in sensing, imaging, and telecommunications

Slow light and stopped light

• Slow light refers to the phenomenon of light propagation with a greatly reduced group velocity
• Can be achieved in metamaterials and photonic crystals by exploiting resonances or band structure engineering
• Enables enhanced light-matter interactions and nonlinear effects
• Stopped light is an extreme case where the group velocity is reduced to zero
  • Can be achieved using electromagnetically induced transparency (EIT) or dynamic modulation of the structure
  • Allows for light storage and retrieval, with potential applications in quantum information processing and optical buffering
• Slow and stopped light have applications in optical delay lines, sensing, and signal processing

Nonlinear effects

• Nonlinear effects in metamaterials and photonic crystals arise from the interaction between the electromagnetic field and the nonlinear properties of the constituent materials
• Can be used to generate new frequencies, control the phase and amplitude of waves, and create novel functionalities
• Key concepts include second and third-order nonlinearities, phase matching conditions, and soliton propagation

Second and third-order nonlinearities

• Second-order nonlinear effects, such as second-harmonic generation (SHG) and sum-frequency generation (SFG), occur in non-centrosymmetric materials
  • Described by the second-order nonlinear susceptibility tensor $\chi^{(2)}$
  • Enable frequency doubling, parametric amplification, and optical rectification
• Third-order nonlinear effects, such as third-harmonic generation (THG) and four-wave mixing (FWM), occur in all materials
  • Described by the third-order nonlinear susceptibility tensor $\chi^{(3)}$
  • Enable frequency tripling, self-phase modulation, and Kerr effect (intensity-dependent refractive index)
• Metamaterials can be designed to enhance or engineer nonlinear effects by tailoring the local field distribution and the nonlinear properties of the constituent materials

Phase matching conditions

• Phase matching is a condition that ensures efficient nonlinear interactions by maintaining a fixed phase relationship between the interacting waves
• In second-order processes, the phase matching condition is: $\mathbf{k}_1 + \mathbf{k}_2 = \mathbf{k}_3$
• In third-order processes, the phase matching condition is: $\mathbf{k}_1 + \mathbf{k}_2 + \mathbf{k}_3 = \mathbf{k}_4$
• Phase mismatch leads