Chapter Review
Optics
Interference Phenomena · Diffraction and Polarization · Optical Instruments · Fiber Optics
Young's Double Slit Experiment
Two coherent sources (created by splitting a single wavefront through two slits) produce a stable interference pattern of equally spaced bright and dark fringes on a screen.
Key Points
- •Coherent sources maintain a constant phase difference — independent sources fail because their phase changes ~10⁸ times per second
- •Constructive interference (bright): path difference = $m\lambda$; Destructive (dark): path difference = $(m + \frac{1}{2})\lambda$
- •Phase–path relation: $\phi = \frac{2\pi}{\lambda}\Delta L$
- •Fringe width $\Delta y \propto \lambda L / d$ — longer wavelength or farther screen widens fringes; wider slit separation compresses them
- •Bright and dark fringes have equal spacing; the $m$th bright fringe is at $y_m = m\lambda L / d$
- •In a medium of refractive index $n$, fringe width shrinks by factor $n$
Formula
$$\Delta y = \frac{\lambda L}{d}$$
Thin Film Interference & Newton's Rings
Interference between light reflected from the two surfaces of a thin film produces colour patterns; Newton's rings are a special case using the air wedge between a plano-convex lens and a flat plate.
Key Points
- •Two reflected rays from top and bottom surfaces of a film interfere — path difference depends on thickness, refractive index, and angle of incidence
- •Phase reversal of $\pi$ (extra $\lambda/2$) occurs when light reflects from a denser medium (low-n → high-n)
- •White light produces rainbow colours because different thicknesses satisfy destructive interference for different wavelengths
- •Newton's rings: air gap increases outward from the contact point, producing concentric dark and bright circles
- •Centre of Newton's rings is always dark due to the $\lambda/2$ phase shift from reflection
Michelson Interferometer
Splits a beam into two perpendicular paths with a half-silvered mirror, reflects them back, and recombines them — moving one mirror by $\lambda/2$ shifts exactly one fringe.
Key Points
- •Beam splitter ($G_1$) divides incident light; compensator plate ($G_2$) equalises optical path through glass
- •Round-trip factor of 2: mirror displacement $L$ changes the path by $2L$
- •One fringe shift corresponds to $\lambda/2$ mirror displacement
- •Precision on the order of $\lambda/4$ (~100 nm for visible light)
Formula
$$L = m\frac{\lambda}{2}$$
Single Slit Diffraction
Light passing through a narrow slit spreads into a broad central maximum flanked by fainter secondary maxima, explained by Huygens' principle treating each slit point as a secondary source.
Key Points
- •Minima (dark fringes) occur at $b\sin\theta = m\lambda$ for $m = \pm 1, \pm 2, ...$
- •The central maximum is twice as wide as any secondary maximum and contains most of the energy
- •Diffraction is significant when slit width $b \approx \lambda$; for $b \gg \lambda$, light travels in straight lines
- •Doubling wavelength doubles the spread; doubling slit width halves it
Formula
$$b\sin\theta = m\lambda$$
Diffraction Gratings & Bragg's Law
A diffraction grating with thousands of slits produces sharp, bright maxima; crystal planes act as natural gratings for X-rays, governed by Bragg's law.
Key Points
- •Grating equation: $d\sin\theta = n\lambda$, where $d = 1/N$ (reciprocal of line density — convert lines/cm to lines/m first)
- •Maximum observable order: $n_{max} = \lfloor d/\lambda \rfloor$ since $\sin\theta \leq 1$
- •Bragg's law: $2d\sin\theta = n\lambda$ — the factor of 2 arises from reflection off parallel crystal planes
- •Bragg angle $\theta$ is measured from the crystal surface (glancing angle), not from the normal
Formula
$$2d\sin\theta = n\lambda$$
Polarization & Malus's Law
Polarization restricts light oscillations to a single plane, proving light is a transverse wave. Malus's law governs intensity through successive polarizers.
Key Points
- •Unpolarized light through a polarizer loses half its intensity: $I_0/2$
- •Malus's law: $I = I_0\cos^2\theta$ where $\theta$ is the angle between polarizer and analyzer axes
- •Crossed polarizers ($\theta = 90°$) block all light; inserting a third at 45° between them allows light through
- •Brewster's angle: reflected light is fully polarized when $n = \tan\theta_p$ (reflected and refracted rays are perpendicular)
- •Polaroid sunglasses exploit Brewster's law to block horizontally polarized glare
Formula
$$I = I_0\cos^2\theta$$
Simple and Compound Microscopes
A simple microscope (single convex lens) magnifies by creating a virtual image; a compound microscope uses an objective for linear magnification and an eyepiece for angular magnification.
Key Points
- •Simple microscope: $M = 1 + D/f$ (image at near point) or $M = D/f$ (image at infinity, relaxed eye)
- •Near point $D \approx 25$ cm; magnification requires $f \ll D$
- •Compound microscope: total $M = (q/p)(1 + D/f_e)$ — objective linear magnification × eyepiece angular magnification
- •Shorter focal lengths for both objective and eyepiece increase total magnification
- •The intermediate image must fall inside $f_e$ for the eyepiece to produce a useful virtual image
Formula
$$M = \frac{q}{p}\left(1 + \frac{D}{f_e}\right)$$
Astronomical Telescope & Resolving Power
A telescope uses a long focal length objective and short focal length eyepiece; resolving power is limited by diffraction through the aperture (Rayleigh criterion).
Key Points
- •Normal adjustment: $M = f_o/f_e$, tube length $L = f_o + f_e$, final image at infinity
- •Light-gathering power $\propto D^2$ (objective area) — larger aperture captures more light from faint objects
- •Rayleigh criterion: $\alpha_{min} = 1.22\lambda/D$ — minimum angle to resolve two point sources
- •Shorter wavelength and larger aperture both improve resolution
- •Given $M$ and $L$, solve simultaneous equations to find both $f_o$ and $f_e$
Formula
$$\alpha_{min} = 1.22\frac{\lambda}{D}$$
Total Internal Reflection & Fiber Optics
Optical fibers guide light via total internal reflection at the core-cladding boundary; the critical angle and numerical aperture determine which rays are captured.
Key Points
- •Critical angle: $\sin\theta_c = n_2/n_1$ — TIR only occurs going from higher to lower refractive index
- •Numerical aperture: $NA = \sqrt{n_{core}^2 - n_{cladding}^2}$ — measures light-gathering ability
- •Acceptance angle changes with external medium: $\theta_a = \arcsin(NA/n_0)$
- •Three fiber types: single-mode step-index (~5 µm core, minimal dispersion), multimode step-index (~50 µm, 33 ns/km dispersion), multimode graded-index (50–1000 µm, ~1 ns/km dispersion)
- •Graded-index fibers reduce dispersion because outer rays travel faster through lower-index material
Formula
$$NA = \sqrt{n_{core}^2 - n_{cladding}^2}$$
Speed of Light & Fiber Communication
Michelson measured $c$ using a rotating octagonal mirror; fiber communication converts electrical signals to light pulses transmitted over long distances with minimal loss.
Key Points
- •Michelson's method: $c = 16fd$ where $f$ is the rotation frequency of the 8-sided mirror and $d$ is the one-way distance
- •Accepted value: $c \approx 3.00 \times 10^8$ m/s in vacuum
- •Fiber communication: laser/LED transmitter → fiber → photodiode receiver; digital modulation (on/off pulses)
- •Advantages over copper: wider bandwidth, electromagnetic interference immunity, thinner/lighter cables, greater security
- •Attenuation from Rayleigh scattering and impurity absorption; repeaters regenerate signals every 30–100 km
Formula
$$c = 16fd$$
Formulas
YDSE Fringe Width
Spacing between adjacent bright (or dark) fringes on the screen.
Formula
$\Delta y = \frac{\lambda L}{d}$
Bright Fringe Condition
Path difference for constructive interference.
Formula
$d\sin\theta = m\lambda$
Michelson Interferometer
Mirror displacement from counted fringe shifts.
Formula
$L = m\frac{\lambda}{2}$
Single Slit Minima
Dark fringes in single-slit diffraction.
Formula
$b\sin\theta = m\lambda$
Bragg's Law
X-ray constructive reflection from crystal planes.
Formula
$2d\sin\theta = n\lambda$
Malus's Law
Intensity through a polarizer-analyzer pair.
Formula
$I = I_0\cos^2\theta$
Brewster's Angle
Angle of incidence for complete polarization by reflection.
Formula
$n = \tan\theta_p$
Telescope Magnification
Angular magnification in normal adjustment.
Formula
$M = \frac{f_o}{f_e}$
Rayleigh Criterion
Minimum resolvable angle through a circular aperture.
Formula
$\alpha_{min} = \frac{1.22\lambda}{D}$
Numerical Aperture
Light-gathering ability of an optical fiber.
Formula
$NA = \sqrt{n_{core}^2 - n_{cladding}^2}$
Critical Angle
Threshold for total internal reflection at core-cladding boundary.
Formula
$\sin\theta_c = \frac{n_2}{n_1}$