🌀Principles of Physics III Unit 5 – Wave Optics

Key Concepts and Definitions

• Wave-particle duality suggests light exhibits both wave and particle properties
  • Photoelectric effect demonstrates particle nature of light (photons)
  • Young's double-slit experiment reveals wave nature of light (interference)
• Electromagnetic waves are transverse waves consisting of oscillating electric and magnetic fields perpendicular to each other and the direction of propagation
• Wavelength $(\lambda)$ represents the distance between two consecutive crests or troughs of a wave
• Frequency $(f)$ is the number of wave cycles passing a fixed point per unit time, measured in hertz (Hz)
• Amplitude $(A)$ is the maximum displacement of a wave from its equilibrium position
• Phase $(\phi)$ describes the position of a point on a wave cycle relative to its origin
• Coherence refers to the ability of two or more waves to maintain a constant phase difference over time and space
  • Coherent sources are necessary for observable interference patterns

Wave Nature of Light

• Light propagates as electromagnetic waves with both electric and magnetic field components
• The speed of light in vacuum $(c)$ is approximately $3 \times 10^8$ m/s
• Light waves exhibit properties such as reflection, refraction, interference, diffraction, and polarization
• Huygens' principle states that every point on a wavefront acts as a source of secondary wavelets, which combine to form the new wavefront
• The wavelength of visible light ranges from about 380 nm (violet) to 700 nm (red)
• The energy of a photon $(E)$ is directly proportional to its frequency $(f)$ and inversely proportional to its wavelength $(\lambda)$, given by the equation $E = hf = hc/\lambda$, where $h$ is Planck's constant
• The wave nature of light explains phenomena such as interference and diffraction, which cannot be accounted for by the particle model alone

Interference of Light Waves

• Interference occurs when two or more waves superpose, resulting in a new wave pattern
  • Constructive interference happens when waves are in phase, leading to an increased amplitude
  • Destructive interference occurs when waves are out of phase, resulting in a decreased amplitude or complete cancellation
• Young's double-slit experiment demonstrates the interference of light waves
  • Light passing through two closely spaced slits creates an interference pattern of bright and dark fringes on a screen
• The condition for constructive interference is given by $d \sin \theta = m \lambda$, where $d$ is the slit separation, $\theta$ is the angle to the fringe, $m$ is an integer, and $\lambda$ is the wavelength
• The condition for destructive interference is given by $d \sin \theta = (m + 1/2) \lambda$
• Thin-film interference occurs when light reflects from the top and bottom surfaces of a thin film, leading to colorful patterns (soap bubbles, oil slicks)
• The Michelson interferometer uses interference to measure small displacements and changes in the refractive index of materials

Diffraction Phenomena

• Diffraction is the bending and spreading of waves when they encounter an obstacle or aperture
• Fraunhofer diffraction occurs when the light source and screen are effectively at infinity relative to the diffracting object
  • Single-slit diffraction produces a central bright fringe and alternating dark and bright fringes of decreasing intensity
  • The angular width of the central bright fringe is given by $\theta = 2 \arcsin (\lambda / a)$, where $a$ is the slit width
• Fresnel diffraction occurs when the light source or screen is close to the diffracting object
  • Fresnel diffraction patterns exhibit a series of concentric bright and dark rings
• Diffraction gratings are composed of many equally spaced slits or grooves that produce a series of sharp, intense interference maxima
  • The grating equation, $d \sin \theta = m \lambda$, relates the slit spacing $(d)$, angle $(\theta)$, order $(m)$, and wavelength $(\lambda)$
• The Rayleigh criterion determines the minimum angular separation $(\theta)$ between two point sources that can be resolved by an optical system, given by $\theta \approx 1.22 \lambda / D$, where $D$ is the aperture diameter
• Diffraction limits the resolution of optical instruments, such as telescopes and microscopes

Polarization of Light

• Polarization refers to the orientation of the electric field vector in an electromagnetic wave
• Unpolarized light has electric field vectors oscillating in all directions perpendicular to the direction of propagation
• Linearly polarized light has electric field vectors oscillating in a single plane
  • Polarizing filters (Polaroid) can produce linearly polarized light by absorbing one component of the electric field
• Circularly polarized light has electric field vectors rotating in a circular path
  • Created by passing linearly polarized light through a quarter-wave plate
• Elliptically polarized light has electric field vectors tracing an elliptical path
• Brewster's angle $(\theta_B)$ is the angle of incidence at which reflected light is completely linearly polarized, given by $\tan \theta_B = n_2 / n_1$, where $n_1$ and $n_2$ are the refractive indices of the two media
• Polarization by scattering occurs when light is scattered by particles much smaller than the wavelength (Rayleigh scattering), resulting in partially polarized light perpendicular to the direction of propagation (blue sky)

Optical Instruments and Applications

• Lenses and mirrors are used to manipulate light in various optical instruments
  • Converging (convex) lenses focus light to a point, while diverging (concave) lenses spread light
  • Concave mirrors focus light, while convex mirrors diverge light
• The thin lens equation, $1/f = 1/d_o + 1/d_i$, relates the focal length $(f)$, object distance $(d_o)$, and image distance $(d_i)$
• Magnification $(M)$ is the ratio of the image size to the object size, given by $M = -d_i / d_o = h_i / h_o$, where $h_i$ and $h_o$ are the image and object heights, respectively
• Telescopes use a combination of lenses or mirrors to collect and focus light from distant objects
  • Refracting telescopes use lenses, while reflecting telescopes use mirrors
• Microscopes use lenses to magnify small objects
  • Compound microscopes have an objective lens and an eyepiece lens to achieve higher magnification
• Interferometers, such as the Michelson and Fabry-Pérot interferometers, use interference for precise measurements and spectroscopy
• Diffraction gratings are used in spectrometers to separate and analyze the wavelengths of light
• Polarizing filters are used in sunglasses, camera lenses, and liquid crystal displays (LCDs) to control light intensity and reduce glare

Mathematical Models and Equations

• The wave equation, $\nabla^2 \psi = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2}$, describes the propagation of waves in a medium, where $\psi$ is the wave function and $v$ is the wave speed
• The electric field of an electromagnetic wave is given by $\vec{E}(x, t) = \vec{E}_0 \cos(kx - \omega t + \phi)$, where $\vec{E}_0$ is the amplitude, $k$ is the wavenumber, $\omega$ is the angular frequency, and $\phi$ is the phase constant
• The magnetic field of an electromagnetic wave is given by $\vec{B}(x, t) = \vec{B}_0 \cos(kx - \omega t + \phi)$, where $\vec{B}_0$ is the amplitude
• The Poynting vector, $\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}$, represents the energy flux density and direction of propagation of an electromagnetic wave
• Snell's law, $n_1 \sin \theta_1 = n_2 \sin \theta_2$, relates the angles of incidence $(\theta_1)$ and refraction $(\theta_2)$ when light passes between two media with refractive indices $n_1$ and $n_2$
• The Fresnel equations describe the reflection and transmission coefficients for light incident on a boundary between two media
• Malus's law, $I = I_0 \cos^2 \theta$, gives the intensity $(I)$ of linearly polarized light after passing through a polarizer at an angle $\theta$ relative to the initial polarization direction, where $I_0$ is the initial intensity

Experimental Techniques and Observations

• Young's double-slit experiment (1801) provided strong evidence for the wave nature of light by demonstrating interference
• Fresnel and Arago's experiments (1816) on the interference of polarized light showed that light waves are transverse
• Michelson and Morley's interferometer experiment (1887) attempted to detect the "luminiferous ether" and ultimately led to the development of special relativity
• Hertz's experiments (1886-1888) confirmed the existence of electromagnetic waves and their properties, as predicted by Maxwell's equations
• Photoelectric effect experiments by Lenard (1902) and Millikan (1916) demonstrated the particle nature of light and led to the concept of photons
• Compton scattering experiments (1923) provided further evidence for the particle nature of light and the existence of photons
• Davisson and Germer's electron diffraction experiment (1927) showed that electrons exhibit wave-like properties, confirming the wave-particle duality of matter
• Modern experiments using advanced techniques, such as attosecond spectroscopy and quantum optics, continue to explore the fundamental properties of light and its interaction with matter