Optics Cheatsheet

Cheatsheet Content

1. What is Light? (Wave Nature) Light is an electromagnetic wave , meaning it's made of oscillating electric and magnetic fields. It travels very fast! The speed of light in vacuum is $c \approx 3 \times 10^8 \text{ m/s}$. Wavelength ($\lambda$): The distance between two consecutive crests or troughs of a wave. Frequency ($f$): The number of wave cycles passing a point per second. Relationship: $c = f\lambda$. For light, shorter wavelengths mean higher frequencies and more energy. Visible Spectrum: Our eyes can only see a small range of wavelengths (colors from red to violet). Red has a longer wavelength, violet has a shorter wavelength. Energy of Light: Light also behaves like tiny particles called photons . Each photon has energy $E = hf$, where $h$ is Planck's constant. Coherent Light: Waves have a constant phase relationship (like laser light). This is important for interference. 2. Interference: Waves Interacting When two or more light waves meet, they combine. This is called superposition . Constructive Interference: When waves meet "in-phase" (crest meets crest, trough meets trough), they add up to create a brighter spot. Condition: Path difference $\Delta L = m\lambda$ (where $m = 0, 1, 2, ...$) Destructive Interference: When waves meet "out-of-phase" (crest meets trough), they cancel each other out, creating a dark spot. Condition: Path difference $\Delta L = (m + \frac{1}{2})\lambda$ (where $m = 0, 1, 2, ...$) 2.1 Young's Double-Slit Experiment This famous experiment shows light's wave nature. Light passes through two narrow slits, creating an interference pattern on a screen. Setup: $d$: Distance between the two slits. $L$: Distance from the slits to the screen. $\lambda$: Wavelength of the light. Bright Fringes (Constructive Interference): Occur at angles $\theta$ where $d \sin\theta = m\lambda$. For small angles, the position $y_m$ from the center of the screen is $y_m = \frac{m\lambda L}{d}$. Dark Fringes (Destructive Interference): Occur at angles $\theta$ where $d \sin\theta = (m + \frac{1}{2})\lambda$. For small angles, the position $y_m$ from the center of the screen is $y_m = \frac{(m + \frac{1}{2})\lambda L}{d}$. The central spot ($m=0$) is always a bright fringe. 2.2 Thin Film Interference (e.g., Soap Bubbles, Oil Slicks) Light reflects off both the top and bottom surfaces of a thin layer of material (like a soap film). These two reflected waves can interfere. Key Concept: Phase Change upon Reflection: If light reflects from a boundary where the second medium has a higher refractive index ($n$), it undergoes a $180^\circ$ phase shift (like a wave hitting a fixed end). If light reflects from a boundary where the second medium has a lower refractive index, there is no phase shift. The conditions for constructive/destructive interference depend on the film thickness ($t$), the refractive index of the film ($n_{film}$), and any phase shifts. This is why you see colors in soap bubbles – different wavelengths interfere constructively at different film thicknesses and angles. 3. Diffraction: Light Bending Around Obstacles Diffraction is the spreading out of waves as they pass through an opening or around an obstacle. It's why you can hear sound around a corner, and why light doesn't cast perfectly sharp shadows. 3.1 Single-Slit Diffraction When light passes through a single narrow slit, it spreads out and creates a pattern of bright and dark bands. The central bright band is the widest and brightest. Dark Fringes (Minima): Occur at angles $\theta$ where $a \sin\theta = m\lambda$ (where $m = 1, 2, 3, ...$). $a$: Width of the slit. The central maximum spans from $m=-1$ to $m=1$. 3.2 Diffraction Grating A diffraction grating has many equally spaced parallel slits (like a CD or DVD surface). It produces much sharper and brighter interference patterns than a double slit. Bright Fringes (Principal Maxima): Occur at angles $\theta$ where $d \sin\theta = m\lambda$ (where $m = 0, 1, 2, ...$). $d$: Distance between adjacent slits (grating spacing). Diffraction gratings are used to separate light into its different wavelengths (like a prism). 3.3 Resolution (Seeing Fine Details) Diffraction limits how clearly we can distinguish between two close objects. Rayleigh Criterion: Two objects are just resolvable when the center of the diffraction pattern of one object is directly over the first minimum of the diffraction pattern of the other. For a circular aperture (like an eye or telescope lens), the minimum resolvable angle is $\theta_{min} = 1.22 \frac{\lambda}{D}$. $D$: Diameter of the aperture. This means larger apertures and shorter wavelengths give better resolution. 4. Polarization: The Direction of Oscillation Light waves are transverse waves , meaning the electric and magnetic fields oscillate perpendicular to the direction the light travels. Unpolarized Light: The electric field oscillates in all possible directions perpendicular to the travel direction (e.g., sunlight, light from a light bulb). Polarized Light: The electric field oscillates in only one specific plane (e.g., light from an LCD screen, or light after passing through a polaroid filter). Polarizers: Filters that only allow light oscillating in a specific plane (its transmission axis ) to pass through. Malus's Law: If polarized light of intensity $I_0$ passes through a polarizer whose transmission axis is at an angle $\theta$ to the light's polarization direction, the transmitted intensity is $I = I_0 \cos^2\theta$. Polarization by Reflection: When unpolarized light reflects off a surface, the reflected light can become partially or fully polarized. This is why sunglasses often have polarizing filters to reduce glare. Brewster's Angle: At a specific angle of incidence (called Brewster's angle, $\theta_B$), the reflected light is completely polarized parallel to the surface. $\tan\theta_B = \frac{n_2}{n_1}$.