Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Optics → Wave optics ↓


Coherence and holography


In the field of wave optics, two fascinating and interrelated concepts are coherence and holography. These topics not only help us understand light as a wave phenomenon, but also have practical applications that have revolutionized photography, imaging, and data storage. In this lesson, we will explore the basic principles behind coherence and holography, using simple language and visual aids to untangle these complex concepts.

Understanding coherence

Coherence refers to the degree of correlation between the phases of light waves at different points in space and time. It is a measure of the ability of light waves to exhibit interference, a phenomenon where waves superimpose on each other to form a resultant wave of greater, lesser, or the same amplitude.

Types of coherence

There are two main types of coherence: spatial coherence and temporal coherence.

Spatial coherence

Spatial coherence describes the correlation between waves at different points in space, perpendicular to the direction of wave propagation. In simple terms, it represents how well a plane wave approximates a point across a beam of light.

For perfect spatial symmetry, light coming from a point source will show perfect coherence in frequency and phase at every point in space. Mathematically, spatial symmetry can be represented as follows.

I(x) = E_0^2 cos^2(kx)

The diagram below illustrates the concept of spatial coherence:

Point A Point B

Here, the waves at point A and point B are in sync, indicating spatial coherence.

Temporal coherence

Temporal coherence considers the correlation of the phase of a wave with itself at different times. It represents the predictability of a wave over time and is closely related to the concept of monochromatic light.

A high level of temporal coherence implies that the light source emits waves of a single, unchanging frequency over a long period of time. This is important for applications such as interferometry and holography.

The mathematical expression for temporal coherence can be represented as follows:

 = frac{1}{Delta f}

where τ is the coherence time, and Δf is the bandwidth of the light source.

Introduction to holography

Holography is a technique that records and reconstructs the light field scattered by an object, creating a three-dimensional image. It is a step beyond conventional photography, which only captures the amplitude of light, not its phase.

Holographic process

Hologram recording

Holography involves the interference of two light waves: a reference wave and an object wave. A coherent light source, such as a laser, is split into two beams. One beam illuminates the object, while the other serves as a reference.

The superposition of the rays creates an interference pattern, which is then recorded, usually on a photographic plate. This pattern encodes information about both the amplitude and phase of the object wave.

Reconstruction of the image

To reconstruct the image, the reference beam is shone onto the recorded interference pattern. This interaction recreates the original light field as if the object were present, allowing the viewer to see a 3D image.

Applications of holography

Holography has diverse applications in fields such as art, data storage, microscopy, and metrology. It enables innovative solutions in various industries.

Interplay between coherence and holography

Coherence is important for effective holography. A coherent light source is necessary to obtain detailed interference patterns and to accurately reconstruct the image during hologram recording.

Importance of coherence in holography

Coherent light ensures that the recorded interference pattern accurately reflects the phase and amplitude of the object. Any reduction in coherence degrades the quality of the hologram, resulting in blurred or distorted images.

Visualization of holographic principles

Consider a simple holographic system:

object Reference Beam Object Beam Hologram

The laser emits a beam that splits into two paths: object rays and reference rays. Their interference is recorded on the hologram.

Mathematics of interference in holography

Understanding the mathematical basis can help to understand the beauty behind holography. The interference of object and reference waves is described by the following equation:

I(x) = |E_o + E_r|^2 = |E_o|^2 + |E_r|^2 + 2|E_o||E_r| cos(phi_o - phi_r)

where E_o is the amplitude of the object wave, E_r is the amplitude of the reference wave, and φ are their respective phases.

Conclusion

Coherence and holography reveal the complex nature of light and its importance in optical applications. Coherence plays a fundamental role in enabling holography, making it possible to create and reconstruct three-dimensional images. The theory and mathematics behind these phenomena pave the way for further advances in optical technologies, impacting many fields and broadening our understanding of the universe.


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Coherence and holography

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