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 ↓


Interference in wave optics


Interference is a core concept in the field of wave optics. It is a phenomenon that occurs when two or more light waves overlap and combine to form a new light intensity pattern. It is a natural consequence of the wave nature of light and is one of the many phenomena that clearly demonstrate the wave-like behavior of light.

What is interference? In simple terms, when two waves meet, they interact with each other. This interaction is known as interference. The principle of superposition states that when two or more waves overlap at a point, the resulting wave displacement is the sum of the displacements of the individual waves. There are two primary types of interference: constructive interference and destructive interference.

Constructive interference

Constructive interference occurs when two wave crests (peaks) meet, producing a new wave of increased amplitude. In other words, the waves add together. Here is a simple code formula explaining constructive interference:

( I = I_1 + I_2 + 2sqrt{I_1 cdot I_2}cosPhi )

Where:

  • I is the resultant intensity of light
  • I_1 and I_2 are the intensities of the individual waves
  • Phi is the phase difference between the two waves

Constructive interference usually occurs when the phase difference is an even multiple of Phi pi complete wave cycles overlap).

Destructive interference

Destructive interference occurs when a wave peak hits a wave trough. This results in a reduction in the amplitude of the waves or even their complete cancellation. It can be represented mathematically as follows:

( I = I_1 + I_2 - 2sqrt{I_1 cdot I_2}cosPhi )

Destructive interference occurs when the phase difference is an odd multiple of Phi pi the half wave cycles are out of phase).

Exploring intervention visually

Consider two waves which are depicted as follows:

Wave 1: y(_1) = A(_1)sin(ωt + kx)
Wave 2: y(_2) = A(_2)sin(ωt + kx + (Phi))

The resultant wave at any point can be expressed as:

y = y(_1) + y(_2)

Represented visually, interfering waves look something like this:

In the above svg, the blue and red lines represent the two waves, while the green dashed line shows the resultant wave due to interference.

Young's double-slit experiment

Perhaps the most famous demonstration of light interference is Young's double-slit experiment, performed by Thomas Young in the early 19th century. In this experiment, light is shone through two closely spaced slits, and the resulting light pattern is observed on a screen.

Young found that instead of two spots of light on the screen, there were many bright and dark fringes. This pattern is evidence of light wave interference. The bright areas were areas of constructive interference, while the dark areas showed destructive interference.

This experiment can be summarized by the interference formula for bright fringes:

d sin theta = mlambda

And for the dark edges:

d sin theta = (m + 0.5)lambda

Where:

  • d is the distance between the slits
  • theta is the angle of the fringes from the central maximum
  • m is the fringe order (integer)
  • lambda is the wavelength of light

Real-world applications

Interference is not just a laboratory phenomenon. It has many practical applications:

  • Anti-reflective coatings: Thin films applied to lenses and eyeglasses use destructive interference to reduce reflections.
  • Holography: This technique relies on light interference to create highly detailed three-dimensional images.
  • Thin film interference: The colored patterns seen in oil spills or soap bubbles result from light interference.

In this illustration, the two colored waves represent multiple reflections from the thin films, and the dashed wave pattern shows the possible resulting interference pattern.

Conclusion

Understanding interference in wave optics is important for anyone studying physics or related fields. This concept shows how waves interact and how these interactions give rise to various observable patterns. Interference not only enhances our understanding of the nature of light but also drives technology and innovation in industries such as optics, photography, filmmaking, etc.


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Interference in wave optics

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