🌀Principles of Physics III Unit 4 – Geometric Optics

Key Concepts and Foundations

• Geometric optics studies light propagation and image formation assuming light travels in straight lines (rays)
• Light rays are perpendicular to wavefronts and point in the direction of wave propagation
• Fermat's principle states light follows the path of least time between two points
• Optical media can be transparent (transmits light), opaque (absorbs or reflects light), or translucent (partially transmits light)
• Refractive index ($n$) measures the speed of light in a medium relative to the speed of light in vacuum ($c$)
  • Defined as $n = c/v$, where $v$ is the speed of light in the medium
  • Higher refractive index indicates slower light propagation in the medium (water, glass)
• Optical path length is the product of the geometric path length and the refractive index of the medium
• Wavefront is a surface of constant phase, perpendicular to light rays

Laws of Reflection and Refraction

• Law of reflection states the angle of incidence equals the angle of reflection ($\theta_i = \theta_r$)
  • Incident ray, reflected ray, and normal to the surface lie in the same plane
• Specular reflection occurs on smooth surfaces, producing a clear reflected image (mirrors)
• Diffuse reflection occurs on rough surfaces, scattering light in various directions (matte surfaces)
• Snell's law describes refraction: $n_1 \sin \theta_1 = n_2 \sin \theta_2$
  • $n_1$ and $n_2$ are refractive indices of the two media, $\theta_1$ and $\theta_2$ are angles with the normal
• Refraction bends light towards the normal when entering a higher refractive index medium (air to water)
• Total internal reflection occurs when light in a higher refractive index medium reaches the critical angle ($\theta_c$)
  • Defined as $\theta_c = \arcsin(n_2/n_1)$, where $n_1 > n_2$
  • Enables light transmission through optical fibers and prisms

Mirrors: Flat and Curved

• Flat mirrors produce virtual, upright, and laterally inverted images
  • Image distance equals object distance, and image height equals object height
• Spherical mirrors can be concave (converging) or convex (diverging)
• Concave mirrors form real, inverted images when object distance is greater than focal length
  • Used in telescopes, solar cookers, and car headlights
• Convex mirrors form virtual, upright, and smaller images
  • Provide a wider field of view (security mirrors, rearview mirrors)
• Mirror equation relates object distance ($s$), image distance ($s'$), and focal length ($f$): $1/f = 1/s + 1/s'$
• Magnification ($m$) is the ratio of image height to object height: $m = -s'/s = h_i/h_o$
• Parabolic mirrors eliminate spherical aberration and are used in precision optics (telescopes, antennas)

Lenses and Image Formation

• Lenses refract light to converge (convex) or diverge (concave) rays
• Thin lens equation: $1/f = 1/s + 1/s'$, similar to the mirror equation
• Converging lenses form real, inverted images when object distance is greater than focal length
  • Used in cameras, projectors, and the human eye
• Diverging lenses form virtual, upright, and smaller images
  • Used in combination with converging lenses to correct vision (eyeglasses)
• Lens maker's equation relates focal length to lens surfaces and refractive index: $1/f = (n-1)(1/R_1 - 1/R_2)$
  • $R_1$ and $R_2$ are radii of curvature of the lens surfaces
• Combination of lenses follows the thin lens equation, using the effective focal length
• Spherical aberration, chromatic aberration, and astigmatism are common lens aberrations
  • Corrected using aspherical lenses, achromatic lenses, and cylindrical lenses, respectively

Optical Instruments and Applications

• Human eye focuses light using a lens and forms an inverted image on the retina
  • Accommodation is the eye's ability to change focal length to focus on objects at different distances
• Simple magnifier (magnifying glass) uses a converging lens to produce a magnified, virtual image
  • Angular magnification is the ratio of the angle subtended by the image to the angle subtended by the object
• Compound microscope uses an objective lens and an eyepiece to achieve high magnification
  • Total magnification is the product of objective and eyepiece magnifications
• Telescopes use a large objective (lens or mirror) to collect light and an eyepiece to magnify the image
  • Refracting telescopes use lenses, while reflecting telescopes use mirrors (Newtonian, Cassegrain)
• Cameras use a converging lens to form a real image on a sensor or film
  • Aperture, shutter speed, and ISO control exposure and depth of field
• Projectors use a converging lens to project a real image onto a screen
• Fiber optics rely on total internal reflection to transmit light signals over long distances
  • Used in telecommunications, medical imaging (endoscopes), and lighting

Problem-Solving Techniques

• Identify the type of problem (reflection, refraction, mirrors, lenses) and the given information
• Draw a diagram showing the object, image, mirrors, or lenses, and relevant rays
  • Use principal rays for mirrors and lenses (parallel to the axis, through the focus, through the center)
• Apply the appropriate equations (mirror equation, thin lens equation, Snell's law) to solve for unknowns
• Check the sign conventions for object and image distances (real is positive, virtual is negative)
• Verify the units and the reasonableness of the answer
• For more complex problems, break them down into smaller sub-problems and solve each part separately
• Use trigonometry for problems involving angles and distances
• Apply the small-angle approximation for paraxial rays (close to the optical axis)

Real-World Examples and Demonstrations

• Periscope uses two flat mirrors to see around obstacles
  • Used in submarines, tanks, and trench warfare
• Mirages are caused by refraction due to temperature gradients in the atmosphere
  • Inferior mirages (cold air above hot ground) and superior mirages (hot air above cold ground)
• Dispersion of white light into colors by a prism demonstrates wavelength dependence of refractive index
• Polarized sunglasses reduce glare by filtering out light reflected at Brewster's angle
• Soap bubbles and oil slicks display iridescent colors due to thin-film interference
• Optical illusions exploit the brain's interpretation of visual information
  • Ames room, impossible trident, and Müller-Lyer illusion
• Laser light shows and holograms use coherent light to create stunning visual effects
• Adaptive optics in telescopes corrects for atmospheric distortions using deformable mirrors

Common Misconceptions and FAQs

• Misconception: Mirrors flip images left-to-right
  • Reality: Mirrors flip images front-to-back (lateral inversion)
• Misconception: Wearing glasses weakens the eyes
  • Reality: Glasses correct vision and do not weaken eye muscles or cause vision to deteriorate
• Misconception: Lasers are always visible and can travel indefinitely
  • Reality: Lasers can be infrared or ultraviolet, and their range is limited by divergence and absorption
• FAQ: Why do we see a rainbow?
  • Rainbows form when sunlight is refracted and reflected inside water droplets, dispersing colors
• FAQ: How do 3D movies work?
  • 3D movies use polarized glasses or active shutter glasses to present different images to each eye
• FAQ: Why does a pool appear shallower than it actually is?
  • Refraction at the air-water interface bends light, making the pool appear shallower
• FAQ: How do one-way mirrors work?
  • One-way mirrors have a partially reflective coating that reflects more light from the brightly lit side
• FAQ: Why do stars twinkle?
  • Atmospheric turbulence causes rapid refraction of starlight, making stars appear to twinkle