Lecture Notes: Nano Photonics and Wave Equations
Recap of Previous Lecture
- Discussed scientific problems in nano photonics.
- Focused on solving electromagnetic responses with varying permittivities.
Maxwell's Equations and Wave Equation
- Solving for electromagnetic response involves Maxwell's equations.
- Two curl equations from Maxwell's equations lead to the wave equation for free space.
Wave Equation Derivation
- The wave equation is a second-order partial differential equation relating time and space derivatives.
- Form: ( \frac{\partial^2 E}{\partial t^2} = v^2 \nabla^2 E )
- ( \nabla^2 ) is the Laplacian operator (( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} )).
Steps to Derive Wave Equation
- Start with curl of electric field (E) and magnetic field (H).
- Assume magnetization is zero: ( B = \mu H ).
- Apply curl to the first equation to yield wave equation terms:
- Include terms for displacement (D), polarization (P), and current density (J).
- Simplify using vector calculus identities.
- In free space (no charges, no currents), equation reduces to:
- ( \nabla^2 E = \mu_0 \epsilon_0 \frac{\partial^2 E}{\partial t^2} )
Solutions to Wave Equation
- Solutions are plane waves where:
- Electric and magnetic fields oscillate perpendicularly to each other and to the direction of propagation.
- Wave speed in vacuum is defined as:
- ( v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = c \approx 3 \times 10^8 m/s )
Electromagnetic Waves in Mediums
- For a dielectric medium (e.g., glass):
- No current (J = 0).
- Polarization (P) is non-zero and related to susceptibility (( \chi )).
- Wave equation becomes:
- ( \nabla^2 E = \mu_0 \epsilon_0 (1 + \chi) \frac{\partial^2 E}{\partial t^2} )
- Velocity of wave in a medium is modified:
- ( v = \frac{c}{\sqrt{\epsilon_r}} )
- Refractive index ( n = \sqrt{\epsilon_r} ).
Nonlinear Optics
- Assumptions made include linear polarization.
- Non-linear responses can occur, leading to phenomena like harmonic generation.
Refractive Index and Dispersion
- Refractive index varies with wavelength (dispersion).
- Example materials and their band gaps:
- Germanium: 0.8 eV
- Gallium Arsenide: 1.42 eV
- Gallium Nitride: 3.4 eV
- Silicon Dioxide: 9 eV
Relation Between Photons and Electrons
- Electromagnetic waves are generated by moving charges (e.g., antennas).
- Electric fields induce charge movement (light detection).
- Key similarities include:
- Wave-particle duality of electrons and photons.
- Electron energy: ( \frac{p^2}{2m} ) (parabolic dispersion).
- Photon energy: linear relation (energy proportional to momentum).
Conclusion and Questions
- Discussed implications of different energy levels for electrons and photons.
- Questions about parameter comparisons between electrons and photons.
- Importance of understanding both fields as they are interconnected.
Note: The lecture emphasizes the fundamental connections between optics, electromagnetics, and electronics, highlighting the physical laws governing the behavior of light and electrons.