Light and Optics

• Includes radio waves on long λ, low f/E end, and gamma rays on short λ, high f/E end.
• Electromagnetic waves:
  • Changing magnetic fields induce changing electric fields and vice versa.
  • Oscillating fields generate oscillations in the other field.
  • Transverse waves: electric and magnetic field vectors perpendicular to propagation.
• Spectrum: Full range of frequencies and wavelengths.
• Wave speed: All waves travel at $c = 3 \times 10^8\ \mathrm{m/s}$.
  • $v = f \lambda → c = f \lambda$
• Color and visible spectrum:
  • Visible range: violet (400 nm) → red (700 nm)
  • Light of all colors appears white.
  • Reflected light shows as that color.
• Blackbody:
  • Ideal absorber of all wavelengths.
  • Appears black if cooler than surroundings.

• Rectilinear propagation: Light travels in straight lines in a medium.

Reflection

• Rebounding of incident waves at medium boundary.
• Reflected waves bounce back; Θ1 = Θ2.
• Normal line perpendicular to boundary.

Plane mirrors

• Images: real or virtual
  • Real: light converges at image, can project onto screen.
  • Virtual: appears to come from position, does not converge.
• Parallel incident rays remain parallel after reflection.
• Plane mirrors flat; image distance behind mirror = object distance in front.

Spherical mirrors

• Center of curvature C, radius r
  • Inside sphere → concave (converging)
  • Outside sphere → convex (diverging)

• Focal length: $f = r/2$
• Distances:
  • $o$ = object distance
  • $i$ = image distance
  • Mirror equation: $(1/f) = (1/o) + (1/i) = 2/r$
• Sign conventions:
  • $i > 0$→ real, in front
  • $i < 0$ → virtual, behind
• Plane mirror: $F = r = \infty → i = -o$
• Magnification: $m = -i/o$
  • Negative: inverted, positive: upright
  • |m|<1 → smaller, =1 → same size, >1 → larger
• Ray diagrams for approximating image type:
  • Concave mirror rays:
    • Parallel → through focal point
    • Through focal point → parallel
    • Strikes vertex → reflects symmetrically
  • Examples:
    • A = real, inverted, magnified
    • B = no image, rays parallel
    • C = virtual, upright, magnified
  • Small diverging mirror → smaller image

Refraction

• Bending of light between media
• Index: $n = c/v$
  • Vacuum: 1, air ≈ 1
• Snell's law: $n_1 \sin \Theta_1 = n_2 \sin \Theta_2$
• Moving to higher $n$ → smaller Θ
• Total internal reflection:
  • High n → low n
  • Refracted angle > incident
  • Critical angle: $\Theta_c = \sin^{-1}(n_2/n_1)$
  • Angle > Θc → total internal reflection

Lenses

• Refract light through two surfaces; can be thin spherical.
• Two focal points, measured from center.
• Converging lens: thicker at center
• Diverging lens: thinner at center
• Real lenses: thickness matters
  • $1/f = (n-1)(1/r_1 - 1/r_2)$
• Eye uses real lens: cornea + lens → adjust focal length
• Sign conventions differ slightly from mirrors.

• Power:
  • $P = 1/f$ (diopters)
  • Nearsighted → diverging lens
  • Farsighted → converging lens
  • Eye power ~60 D; contacts 0.25–8 D
• Multiple lens systems:
  • $P = P_1 + P_2 + P_3$
  • Magnification: $m = m_1 × m_2 × m_3$
• Aberrations:
  • Spherical: peripheral blur from parallel rays
  • Chromatic: dispersion causes rainbow halo
• Dispersion: Different wavelengths travel at different speeds → separate
  • Violet λ < red λ
  • Red light refracted less

• Single slit:
  • Narrow opening → spreading of light
  • Slit + lens → bright central fringe, alternating dark fringes
  • Central fringe twice as wide as side fringes
  • Minima: $a \sin \Theta = n \lambda$ (a = slit width)
• Multiple slits:
  • Interference: displacement adds → maxima (bright), minima (dark)
  • Young's experiment: parallel slits interfere
  • $d \sin \Theta = (n + 1/2)\lambda$
  • Small angles: sin ≈ tan

• Diffraction gratings:
  • Multiple slits in pattern → colorful patterns (CD rainbow)
  • Thin films → interference patterns (soap bubbles)
• X-ray diffraction:
  • Bending light to model molecules
  • Combined with protein crystallography
  • Dark/light fringes → 2D images

• Plane polarized light:
  • Electric fields aligned
  • Used in diagnostics
• Unpolarized light:
  • Fields in multiple directions
  • Occurs in stereoisomers
  • Rotates CCW or CW depending on chiral centers
• Polarizers:
  • Allow specific field orientation
  • Second polarizer angle controls transmitted light
  • Perpendicular polarizers → no light
• Circular polarization:
  • Rare, caused by pigments/filters
  • Uniform amplitude, changing direction
  • Helical E and B field vectors perpendicular