% miko:type: complete
% miko:title: Complete Optics
% miko:summary: A structured path from rays and lenses to interference, diffraction, polarization, and Gaussian beams.
% miko:tags: optics
% miko:difficulty: undergrad
% miko:related: complete-electromagnetism, wave-equation
% miko:keywords: light, lens, interference, diffraction, polarization

\documentclass{article}
\usepackage{amsmath}
\title{Complete Optics}
\begin{document}
\maketitle

\section{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.

\section{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.

\section{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.

\section{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 $2\pi$.

\section{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

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

\section{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.

\section{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.

\end{document}
