2.1 Wave equation

Electromagnetic waves are a fundamental concept in electromagnetism, describing how electric and magnetic fields propagate through space and time. The wave equation, derived from Maxwell's equations, mathematically describes this propagation.

Understanding wave equations and plane electromagnetic waves is crucial for grasping how electromagnetic energy travels. This knowledge forms the basis for studying more complex phenomena like reflection, transmission, and guided waves in various media and structures.

Wave equation in electromagnetics

• The wave equation is a fundamental equation in electromagnetics that describes the propagation of electromagnetic waves through space and time
• It is a second-order partial differential equation that relates the electric and magnetic fields to their spatial and temporal derivatives
• The wave equation can be derived from Maxwell's equations, which are a set of four equations that form the foundation of classical electromagnetism

Derivation of wave equation

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• The wave equation can be derived by combining Faraday's law and Ampere's law from Maxwell's equations
• Faraday's law states that a time-varying magnetic field induces an electric field, while Ampere's law states that a time-varying electric field and an electric current density induce a magnetic field
• By taking the curl of Faraday's law and substituting Ampere's law, the wave equation for the electric field can be obtained
• Similarly, by taking the curl of Ampere's law and substituting Faraday's law, the wave equation for the magnetic field can be derived

Wave equation for electric field

• The wave equation for the electric field is given by: $\nabla^2 \vec{E} - \mu_0\epsilon_0\frac{\partial^2 \vec{E}}{\partial t^2} = 0$
• $\nabla^2$ is the Laplacian operator, $\vec{E}$ is the electric field vector, $\mu_0$ is the permeability of free space, $\epsilon_0$ is the permittivity of free space, and $t$ is time
• This equation describes how the electric field varies in space and time in the absence of charges and currents

Wave equation for magnetic field

• The wave equation for the magnetic field is given by: $\nabla^2 \vec{B} - \mu_0\epsilon_0\frac{\partial^2 \vec{B}}{\partial t^2} = 0$
• $\vec{B}$ is the magnetic field vector, and the other symbols have the same meaning as in the electric field wave equation
• This equation describes how the magnetic field varies in space and time in the absence of charges and currents

Solutions to wave equation

• The solutions to the wave equation are electromagnetic waves that propagate through space at the speed of light $c = 1/\sqrt{\mu_0\epsilon_0}$
• The general solution to the wave equation can be expressed as a linear combination of plane waves, which are waves with constant frequency, wavelength, and amplitude
• The plane wave solutions can be written in the form $\vec{E}(\vec{r},t) = \vec{E}_0 e^{i(\vec{k}\cdot\vec{r} - \omega t)}$ and $\vec{B}(\vec{r},t) = \vec{B}_0 e^{i(\vec{k}\cdot\vec{r} - \omega t)}$, where $\vec{E}_0$ and $\vec{B}_0$ are the amplitudes, $\vec{k}$ is the wave vector, $\omega$ is the angular frequency, and $\vec{r}$ is the position vector

Plane electromagnetic waves

• Plane electromagnetic waves are the simplest type of electromagnetic waves, characterized by constant frequency, wavelength, and amplitude in a given direction
• They are called "plane" waves because their wavefronts (surfaces of constant phase) are infinite parallel planes perpendicular to the direction of propagation
• Plane waves are an idealization of electromagnetic waves in the far-field region of a source, where the curvature of the wavefronts can be neglected

Plane wave solutions

• The plane wave solutions to the wave equation for the electric and magnetic fields are given by:
  • $\vec{E}(\vec{r},t) = \vec{E}_0 e^{i(\vec{k}\cdot\vec{r} - \omega t)}$
  • $\vec{B}(\vec{r},t) = \vec{B}_0 e^{i(\vec{k}\cdot\vec{r} - \omega t)}$
• $\vec{E}_0$ and $\vec{B}_0$ are the complex amplitudes, $\vec{k}$ is the wave vector, $\omega$ is the angular frequency, and $\vec{r}$ is the position vector
• The wave vector $\vec{k}$ points in the direction of propagation and has a magnitude equal to the wavenumber $k = 2\pi/\lambda$, where $\lambda$ is the wavelength
• The angular frequency $\omega$ is related to the frequency $f$ by $\omega = 2\pi f$

Polarization of plane waves

• Polarization refers to the orientation of the electric and magnetic field vectors in a plane perpendicular to the direction of propagation
• In a plane wave, the electric and magnetic fields are perpendicular to each other and to the direction of propagation
• The polarization of a plane wave can be linear, circular, or elliptical, depending on the relative phase and amplitude of the electric field components
• Linear polarization occurs when the electric field oscillates in a single plane, while circular and elliptical polarizations involve rotating electric field vectors

Poynting vector and energy flux

• The Poynting vector $\vec{S}$ represents the directional energy flux (power per unit area) of an electromagnetic wave
• It is defined as the cross product of the electric and magnetic fields: $\vec{S} = \vec{E} \times \vec{H}$, where $\vec{H}$ is the magnetic field intensity related to the magnetic field by $\vec{H} = \vec{B}/\mu_0$
• The magnitude of the Poynting vector gives the intensity of the electromagnetic wave, and its direction indicates the direction of energy flow
• For a plane wave, the Poynting vector is parallel to the wave vector $\vec{k}$ and has a magnitude of $|\vec{S}| = |\vec{E}||\vec{H}| = |\vec{E}|^2/\eta$, where $\eta = \sqrt{\mu_0/\epsilon_0}$ is the characteristic impedance of the medium

Boundary conditions at interfaces

• When an electromagnetic wave encounters an interface between two different media, it undergoes reflection, transmission, or both
• The behavior of the wave at the interface is governed by the boundary conditions, which ensure the continuity of the tangential components of the electric and magnetic fields across the interface
• The boundary conditions lead to the Fresnel equations, which describe the relationship between the incident, reflected, and transmitted waves

Reflection and transmission coefficients

• The reflection and transmission coefficients quantify the fraction of the incident wave's amplitude that is reflected or transmitted at an interface
• The reflection coefficient $r$ is defined as the ratio of the reflected wave's electric field amplitude to the incident wave's electric field amplitude: $r = E_r/E_i$
• The transmission coefficient $t$ is defined as the ratio of the transmitted wave's electric field amplitude to the incident wave's electric field amplitude: $t = E_t/E_i$
• The reflection and transmission coefficients depend on the polarization of the incident wave (parallel or perpendicular to the plane of incidence) and the angle of incidence

Fresnel equations

• The Fresnel equations relate the reflection and transmission coefficients to the material properties (refractive indices) and the angle of incidence
• For a wave with electric field perpendicular to the plane of incidence (s-polarization or TE polarization), the Fresnel equations are:
  • $r_s = (n_1\cos\theta_i - n_2\cos\theta_t) / (n_1\cos\theta_i + n_2\cos\theta_t)$
  • $t_s = (2n_1\cos\theta_i) / (n_1\cos\theta_i + n_2\cos\theta_t)$
• For a wave with electric field parallel to the plane of incidence (p-polarization or TM polarization), the Fresnel equations are:
  • $r_p = (n_2\cos\theta_i - n_1\cos\theta_t) / (n_2\cos\theta_i + n_1\cos\theta_t)$
  • $t_p = (2n_1\cos\theta_i) / (n_2\cos\theta_i + n_1\cos\theta_t)$
• $n_1$ and $n_2$ are the refractive indices of the media, $\theta_i$ is the angle of incidence, and $\theta_t$ is the angle of transmission

Brewster's angle

• Brewster's angle is a special angle of incidence at which the reflected wave vanishes for p-polarized light
• It occurs when the refracted and reflected rays are perpendicular to each other
• Brewster's angle $\theta_B$ is given by $\tan\theta_B = n_2/n_1$, where $n_1$ and $n_2$ are the refractive indices of the media
• At Brewster's angle, the reflected wave is entirely s-polarized, and the transmitted wave is partially p-polarized

Total internal reflection

• Total internal reflection occurs when light travels from a medium with a higher refractive index to a medium with a lower refractive index, and the angle of incidence exceeds the critical angle
• The critical angle $\theta_c$ is given by $\sin\theta_c = n_2/n_1$, where $n_1$ and $n_2$ are the refractive indices of the media (with $n_1 > n_2$)
• When the angle of incidence is greater than the critical angle, the transmitted wave becomes evanescent, meaning it decays exponentially with distance from the interface and does not propagate in the second medium
• Total internal reflection is used in optical fibers and prisms to guide light with minimal loss

Electromagnetic wave propagation

• Electromagnetic wave propagation describes how electromagnetic waves travel through different media and how their properties change as a result of the interaction with the medium
• The behavior of electromagnetic waves in a medium depends on the medium's electrical and magnetic properties, such as permittivity, permeability, and conductivity
• Understanding wave propagation is crucial for designing and analyzing electromagnetic systems, such as communication links, radar, and imaging devices

Propagation in lossless media

• In lossless media, electromagnetic waves propagate without attenuation, meaning their amplitude remains constant as they travel through the medium
• Examples of lossless media include vacuum, air (at low frequencies), and perfect dielectrics
• The wave equation in lossless media is given by $\nabla^2 \vec{E} - \mu\epsilon\frac{\partial^2 \vec{E}}{\partial t^2} = 0$, where $\mu$ and $\epsilon$ are the permeability and permittivity of the medium
• The phase velocity of the wave in a lossless medium is $v_p = 1/\sqrt{\mu\epsilon}$, which is equal to the speed of light in vacuum ($c$) divided by the refractive index of the medium ($n = \sqrt{\mu_r\epsilon_r}$)

Propagation in lossy media

• In lossy media, electromagnetic waves experience attenuation as they propagate, meaning their amplitude decreases with distance
• Attenuation is caused by the conversion of electromagnetic energy into heat due to the medium's conductivity or other dissipative mechanisms
• Examples of lossy media include conductors, seawater, and biological tissues
• The wave equation in lossy media includes an additional term related to the conductivity $\sigma$: $\nabla^2 \vec{E} - \mu\epsilon\frac{\partial^2 \vec{E}}{\partial t^2} - \mu\sigma\frac{\partial \vec{E}}{\partial t} = 0$
• The attenuation constant $\alpha$ describes the rate at which the wave amplitude decreases with distance and is given by $\alpha = \omega\sqrt{\mu\epsilon/2}(\sqrt{1 + (\sigma/\omega\epsilon)^2} - 1)^{1/2}$

Dispersion in electromagnetic waves

• Dispersion is the phenomenon where the phase velocity of a wave depends on its frequency
• In dispersive media, waves with different frequencies travel at different speeds, causing the wave packet to spread out over time
• Dispersion can be caused by the frequency-dependent response of the medium's permittivity, permeability, or conductivity
• Examples of dispersive media include glass, water, and plasmas
• The dispersion relation $\omega(k)$ describes the relationship between the angular frequency $\omega$ and the wavenumber $k$ in a dispersive medium

Phase and group velocity

• The phase velocity $v_p$ is the speed at which the phase of a single-frequency wave propagates and is given by $v_p = \omega/k$
• The group velocity $v_g$ is the speed at which the envelope of a wave packet (a superposition of waves with different frequencies) propagates and is given by $v_g = d\omega/dk$
• In non-dispersive media, the phase and group velocities are equal, but in dispersive media, they can differ
• The group velocity determines the speed at which information or energy is conveyed by the wave
• When the group velocity exceeds the speed of light in vacuum, it does not violate special relativity because the energy and information still travel at or below the speed of light

Guided electromagnetic waves

• Guided electromagnetic waves are waves that are confined to propagate along a specific path, such as a transmission line, waveguide, or optical fiber
• Guiding structures allow electromagnetic energy to be efficiently transmitted over long distances with minimal loss and provide control over the wave's propagation characteristics
• Guided waves have discrete modes, which are specific field configurations that satisfy the boundary conditions imposed by the guiding structure

Waveguides and modes

• A waveguide is a hollow metallic structure (usually rectangular or circular) that guides electromagnetic waves along its length
• The dimensions of the waveguide determine the cutoff frequencies for different modes, which are the lowest frequencies at which a particular mode can propagate
• Modes in a waveguide are classified as transverse electric (TE) or transverse magnetic (TM) modes, depending on whether the electric or magnetic field is entirely transverse to the direction of propagation
• The mode with the lowest cutoff frequency is called the dominant mode (TE10 for rectangular waveguides, TE11 for circular waveguides)

Transverse electric (TE) modes

• In TE modes, the electric field is entirely transverse to the direction of propagation ($E_z = 0$), while the magnetic field has a longitudinal component ($H_z \neq 0$)
• The field components for TE modes in a rectangular waveguide are given by:
  • $E_x = -\frac{i\omega\mu}{k_c^2}H_0\frac{n\pi}{b}\sin(\frac{m\pi x}{a})\cos(\frac{n\pi y}{b})e^{i(k_z z - \omega t)}$
  • $E_y = \frac{i\omega\mu}{k_c^2}H_0\frac{m\pi}{a}\cos(\frac{m\pi x}{a})\sin(\frac{n\pi y}{b})e^{i(k_z z - \omega t)}$
  • $H_x = -\frac{k_z}{k_c^2}H_0\frac{m\pi}{a}\cos(\frac{m\pi x}{a})\sin(\frac{n\pi y}{b})e^{i(k_z z - \omega t)}$
  • $H_y = -\frac{k_z}{k_c^2}H_0\frac{n\pi}{b}\sin(\frac{m\pi x}{a})\cos(\frac{n\pi y}{b})e^{i(k_z z - \omega t)}$
  • $H_z = H_0\sin(\frac{m\pi x}{a})\sin(\frac{n\pi y}{b})e^{i(k_z z - \omega t)}$
• $a$ and $b$ are the waveguide dimensions, $m$ and $n$ are the mode indices, $k_c = \sqrt{(m\pi/a)^2 + (n\pi/b)^2}$ is the cutoff wavenumber, and $k_z = \sqrt{k^2 - k_c^2}$ is the longitudinal wavenumber

Transverse magnetic (TM) modes

• In TM modes, the magnetic field is entirely transverse to the direction of propagation ($H_z = 0$), while the electric field has a longitudinal component ($E_z \neq 0$)
• The field components for TM modes in a rectangular waveguide are given by:
  • $E_x = -\frac{k_z}{k_c^2}E_0\frac{m\pi}{a}\cos(\frac{m\pi x}{a})\sin(\frac{n