Complete Optics

Geometric Optics

Geometric optics treats light as rays. It is valid when wavelengths are much smaller than the length scales of apertures, lenses, and obstacles.

Snell Law

At an interface between refractive indices $n_1$ and $n_2$, transmitted rays obey

$$n_1\sin\theta_1=n_2\sin\theta_2.$$

This follows from phase matching along the boundary.

Lens Equation

For a thin lens with focal length $f$, object distance $s$, and image distance $s'$, the paraxial relation is

$$\frac{1}{f}=\frac{1}{s}+\frac{1}{s'}.$$

It is the workhorse formula for elementary imaging systems.

Interference

Coherent waves superpose with phase-sensitive intensity. For two waves of equal amplitude, constructive interference occurs when the phase difference is a multiple of $2\pi$.

Diffraction

Diffraction is the spreading and structure produced when waves encounter finite apertures or obstacles. For a single slit of width $a$, the first minimum occurs near

$$a\sin\theta=\lambda.$$

Polarization

Polarization describes the direction and phase relation of the transverse electric field. Linear, circular, and elliptical polarization are different states of transverse wave motion.

Gaussian Beam

A Gaussian beam is a paraxial optical mode with transverse intensity profile approximately

$$I(r)\propto e^{-2r^2/w^2(z)}.$$

The beam waist and Rayleigh range control focusing and divergence.