PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Electrodynamics → Electromagnetic waves ↓


Waveguides and resonators


Waveguides and resonators are key concepts in the study of electromagnetic waves, particularly in the field of electrodynamics. They are important in understanding how electromagnetic waves propagate and are controlled. In short, waveguides guide the path of electromagnetic waves, and resonators store and control their frequencies.

Waveguides

A waveguide is a structure that directs electromagnetic waves from one point to another. This is usually done in the context of beamforming or to prevent loss over long distances. Common applications include fiber optics, microwave transmission, and even audio applications in acoustics.

Types of waveguides

The main types of waveguides are:

  • Rectangular waveguides: These are most commonly used at microwave frequencies. They are simple in design and can guide waves with minimal dispersion.
  • Cylindrical waveguides (also called circular waveguides): These are suitable for conducting TE and TM modes if they are properly shaped.
  • Dielectric waveguides: These are often used in optical fibers. They rely on the refractive index contrast between the core and cladding to guide light through total internal reflection.

Wave propagation in waveguides

When analyzing waveguides, it is important to understand how waves propagate within them. Wave propagation can be characterized by modes, each of which has distinct patterns and characteristics.

Wave transmission methods

Transverse electric (TE) modes: These modes have no electric field component in the direction of propagation. The electric field and magnetic field are perpendicular to the direction of propagation. The primary equation governing TE modes is:

e_z = 0

Transverse Magnetic (TM) Modes: These modes have no magnetic field component in the direction of wave propagation. Thus:

h_z = 0

Hybrid modes: These are the EH and HE modes where both the electric and magnetic field components have a component along the direction of propagation.

Mathematical treatment

For waveguides with simple geometries such as rectangular waveguides, Maxwell's equations can be solved to find the field pattern within the waveguide. For example, the fundamental TE mode solution can be obtained using:

E_y = E_0 sin((mπx/a)) sin(nπy/b) e^(-jβz)

where m and n are the mode numbers, a and b are the dimensions of the waveguide, and β is the propagation constant.

Resonator

A resonator is a structure designed to trap electromagnetic waves in such a way that they resonate at certain frequencies. These devices are important in applications such as filters, oscillators, and frequency stabilization devices.

Types of resonators

Resonators come in a variety of forms depending on the application, including:

  • Cavity resonators: Often used in the microwave region, these structures confine electromagnetic waves within a fixed volume, usually made of metal walls.
  • Dielectric resonators: These use dielectric materials to achieve resonance at microwave frequencies.
  • Crystal resonators: These are often used for frequency control in clocks and communications equipment because of their high precision.

Fundamentals of resonators

The resonance frequency of a resonator is determined by its physical structure and the boundary conditions imposed by the materials used. The principle may be stated as follows:

f_0 = frac{1}{2πsqrt{LC}}

Where L and C are the inductance and capacitance of the system.

Electromagnetic standing waves

In resonators, electromagnetic standing waves are set up. These waves lead to conditions where some frequencies are strengthened while others may be cancelled. For a rectangular cavity resonator, the resonance mode can be expressed as:

f_{mn} = frac{1}{2}sqrt{(frac{m}{L_x})^2 + (frac{n}{L_y})^2}cdot c

where m and n are integers, and L_x , L_y represent the dimensions of the cavity.

Example of a cavity resonator

Electromagnetic standing waves

Application

Both waveguides and resonators have wide use in various applications. For example, waveguides are invaluable in telecommunications for guiding signals with minimal loss. On the other hand, resonators are important in ensuring stability in filters, oscillators, and frequency-dependent devices.

Waveguides in telecommunications

In fiber optic systems, waveguides play the role of transmitting light signals over long distances with minimal loss. The principle of total internal reflection allows effective signal transmission within the optical fiber core.

Resonators in oscillators

Resonators are important in oscillators where the maintenance of a stable frequency is essential. For example, quartz crystal resonators are extremely efficient at providing precise frequency control.

Filter

In filters, resonators help to selectively allow or block frequencies. This makes them indispensable in applications such as radio communications, where multiple channels need to be separated.

Conclusion

Understanding waveguides and resonators is essential for advancing applications in traditional and modern electrodynamic frameworks. Whether routing signals in telecommunication networks or tuning resonance in sophisticated electronic devices, these devices remain at the core of many innovative technologies. Their adaptability and efficacy underline their value across a diverse spectrum of science and engineering.


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Waveguides and resonators

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