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26 formulas

Waves & Optics

Interference, refraction. Every formula below opens into a live, hands-on simulation.

refraction
n1sin⁡θ1=n2sin⁡θ2n_1 \sin\theta_1 = n_2 \sin\theta_2

Snell's Law

Light bends toward the normal when entering a denser medium.

reflection
θi=θr\theta_i = \theta_r

Law of Reflection

Light bounces off a mirror at the same angle it arrives.

wave mechanics
v=fλv = f\lambda

Wave Speed Equation

Wave speed equals how many wavelengths pass a point each second.

geometric optics
1f=1do+1di\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Thin Lens Equation

Reciprocals of object and image distances always add up to the lens power.

interference
dsin⁡θ=mλd \sin\theta = m\lambda

Young's Double Slit Interference

Two overlapping wave sources create bright and dark bands.

diffraction
asin⁡θ=mλa \sin\theta = m\lambda

Single Slit Diffraction

A narrow slit spreads light into a pattern of bright and dark bands.

polarization
tan⁡θB=n2n1\tan\theta_B = \frac{n_2}{n_1}

Brewster's Angle

At one special angle, reflected light becomes perfectly polarized.

wave mechanics
f′=fv+vov−vsf' = f \frac{v + v_o}{v - v_s}

Doppler Effect

Moving toward a wave source compresses waves; moving away stretches them.

polarization
I=I0cos⁡2θI = I_0 \cos^2\theta

Malus's Law

Only the component of the electric field aligned with the polarizer axis gets through; the rest is absorbed.

diffraction
θmin=1.22λD\theta_{min} = 1.22 \frac{\lambda}{D}

Rayleigh Criterion

Two point sources are just resolvable when the central maximum of one falls on the first dark ring of the other.

diffraction
dsin⁡θ=mλd \sin\theta = m \lambda

Diffraction Grating Equation

Light constructively interferes whenever the path difference between adjacent slits equals an integer number of wavelengths.

geometric optics
1f=(n−1)(1R1−1R2)\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)

Lensmaker's Equation

Focal length depends on how strongly light bends at each curved surface, governed by the index step and surface curvature.

geometrical optics
1f=1v+1u\frac{1}{f} = \frac{1}{v} + \frac{1}{u}

Mirror Equation

Object distance, image distance, and focal length lock together — change one and the others must rearrange.

electromagnetic waves
n=cvn = \frac{c}{v}

Refractive Index Definition

The refractive index tells you the factor by which light slows down inside a material.

electromagnetic waves
I=12cϵ0E02I = \frac{1}{2} c \epsilon_0 E_0^2

Intensity of an Electromagnetic Wave

Intensity scales with the square of the electric field amplitude — doubling the field quadruples the power flow.

quantum optics
E=hf=hcλE = h f = \frac{h c}{\lambda}

Photon Energy (Planck-Einstein Relation)

Higher frequency means each light particle carries more punch — color literally equals energy.

wave mechanics
∂2y∂t2=v2∂2y∂x2\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}

Wave Equation

The curvature of the wave in space drives its acceleration in time — sharper bends snap back faster.

wave mechanics
fn=nv2Lf_n = \frac{n v}{2 L}

Standing Wave Frequencies

Only wavelengths that fit a whole number of half-wave humps between the two fixed ends survive — those are the notes.

refraction
θc=arcsin⁡(n2n1)\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)

Critical Angle (Total Internal Reflection)

Past a certain angle, light heading from dense to rare can't refract out — it gets reflected back perfectly.

superposition
fbeat=∣f1−f2∣f_{beat} = |f_1 - f_2|

Beat Frequency

Two waves of nearly equal frequency drift in and out of phase. When they line up they add (loud); when opposed they cancel (silent). The loudness pulses at the difference frequency.

geometric optics
m=−didom = -\frac{d_i}{d_o}

Lateral Magnification

Magnification compares image height to object height. A negative sign means the image is inverted; its magnitude tells you how many times larger or smaller the image is.

interference
2nt=(m+12)λ2 n t = \left(m + \tfrac{1}{2}\right)\lambda

Thin-Film Interference

Light reflecting off the top and bottom surfaces of a film travels different path lengths. A half-wave phase flip at the top surface makes the half-integer condition give bright reflection — and different colors satisfy it at different thicknesses.

diffraction
nλ=2dsin⁡θn\lambda = 2 d \sin\theta

Bragg's Law

Atomic planes act like a stack of partial mirrors. Reflections from successive planes add up only when their extra path length is a whole number of wavelengths — that condition pins the angle for each wavelength.

geometric optics
NA=nsin⁡θmaxNA = n \sin\theta_{max}

Numerical Aperture

Numerical aperture is the sine of the widest cone of light an optical system can gather or emit, scaled by the medium's index. Bigger NA means more light collected and finer resolving power.

wave propagation
vg=dωdkv_g = \frac{d\omega}{dk}

Group Velocity

A localized wave packet is a sum of many frequencies. The crests move at the phase velocity, but the packet's envelope — where the energy and information ride — moves at the group velocity, the slope of the dispersion curve.

interferometry
ΔN=2 Δdλ\Delta N = \frac{2\,\Delta d}{\lambda}

Michelson Interferometer Fringe Shift

Light in one arm makes a round trip, so moving the mirror by Delta d changes the path by 2*Delta d. Each whole wavelength of extra path slides the fringe pattern by exactly one fringe — turning displacement into a count.