2.1 Wave equations and solutions

Electromagnetic waves are the foundation of optics. They arise from Maxwell's equations, which describe how electric and magnetic fields interact and propagate through space. Understanding these waves is crucial for grasping how light behaves and interacts with matter.

The wave equation is a mathematical description of how electromagnetic waves move. It leads to solutions like plane waves and spherical waves, which have different properties in terms of amplitude, phase, and propagation direction. These concepts are essential for analyzing optical phenomena.

Wave Equations and Solutions

Derivation of wave equation

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• Maxwell's equations describe electromagnetic fields and their interactions
  • Gauss's law for electric fields relates electric field divergence to charge density ($\rho$) and permittivity of free space ($\varepsilon_0$)
  • Gauss's law for magnetic fields states magnetic field divergence is always zero (no magnetic monopoles)
  • Faraday's law of induction relates electric field curl to time-varying magnetic fields ($\mathbf{B}$)
  • Ampère's circuital law (with Maxwell's correction) relates magnetic field curl to current density ($\mathbf{J}$), permittivity of free space, and time-varying electric fields ($\mathbf{E}$)
• Derivation steps for wave equation in free space
  1. Take curl of Faraday's law to relate second-order spatial derivatives of $\mathbf{E}$ to first-order time derivative of $\mathbf{B}$
  2. Substitute Ampère's law to introduce second-order time derivative of $\mathbf{E}$
  3. Use vector identity to simplify curl of curl operator
  4. In free space, electric field is divergence-free, leading to wave equation relating second-order spatial and temporal derivatives of $\mathbf{E}$
• Derivation steps for wave equation in dielectric media
  • Dielectric media introduce electric displacement field ($\mathbf{D}$) and magnetic field intensity ($\mathbf{H}$) related to $\mathbf{E}$ and $\mathbf{B}$ by permittivity ($\varepsilon$) and permeability ($\mu$)
  • Follow similar steps as free space derivation, replacing $\varepsilon_0$ and $\mu_0$ with $\varepsilon$ and $\mu$
  • Resulting wave equation in dielectric media has same form as free space with modified coefficients

Solutions for plane and spherical waves

• Plane waves are simplest solutions to wave equation
  • Assume solution is product of spatial and temporal functions with complex exponential form ($e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}$)
  • Substitute assumed solution into wave equation to obtain dispersion relation between wavenumber ($k$) and angular frequency ($\omega$)
  • Dispersion relation gives phase velocity ($v$) of plane waves in terms of $\mu$ and $\varepsilon$
• Spherical waves are solutions with radial symmetry
  • Assume solution has $1/r$ dependence multiplied by complex exponential ($e^{i(kr - \omega t)}$)
  • Substitute assumed solution into wave equation in spherical coordinates
  • Solution satisfies wave equation if same dispersion relation as plane waves holds

Properties of wave solutions

• Amplitude is maximum value of wave oscillation
  • For plane waves, amplitude is constant ($|\mathbf{E}_0|$)
  • For spherical waves, amplitude decreases with distance from source ($|\mathbf{E}_0|/r$)
• Phase is argument of complex exponential in wave solution
  • For plane waves, phase is dot product of wavevector ($\mathbf{k}$) and position vector ($\mathbf{r}$) minus product of angular frequency and time
  • For spherical waves, phase is product of wavenumber and radial distance minus product of angular frequency and time
• Wavelength ($\lambda$) is spatial period of wave
  • Related to wavenumber by $\lambda = 2\pi/k$
  • Smaller wavenumber corresponds to larger wavelength
• Propagation direction is determined by wavevector
  • For plane waves, propagation is along $\mathbf{k}$
  • For spherical waves, propagation is radially outward from source

Wave equation vs Helmholtz equation

• Monochromatic waves have single frequency ($\omega$)
  • Can be represented as product of spatial function ($\mathbf{E}(\mathbf{r})$) and time-harmonic function ($e^{-i\omega t}$)
  • Useful for analyzing steady-state behavior of waves
• Helmholtz equation is derived from wave equation for monochromatic waves
  • Substitute monochromatic wave solution into wave equation
  • Resulting equation involves only spatial derivatives of $\mathbf{E}(\mathbf{r})$ and wavenumber ($k$)
  • Describes spatial distribution of wave amplitude and phase
• Wave equation and Helmholtz equation are related
  • Wave equation describes full spatio-temporal behavior of waves
  • Helmholtz equation is time-independent form of wave equation for monochromatic case
  • Solutions to Helmholtz equation can be used to construct solutions to wave equation by multiplying with time-harmonic factor ($e^{-i\omega t}$)