Photonics Formula Guide

This guide connects the principal ray, wave, and detector models used in the worked exercises. Use Mathematical Formula Reference for the mathematical identities behind the detailed derivations.

Conventions

Use one consistent length unit within each equation. State the optical axis, radius and angle signs, ray-vector order, and complex-phase convention. Distinguish field amplitude, intensity, irradiance, and power.

Geometrical optics

Optical path length and refraction are described by

\[\mathrm{OPL}=\int n\,\mathrm{d}s, \qquad n_1\sin\theta_1=n_2\sin\theta_2.\]

Angles in Snell’s law are measured from the interface normal. In a homogeneous medium, rays travel along straight segments between surfaces.

Paraxial systems

For the ray vector \((y,\theta)^T\) in air,

\[\begin{split}\begin{pmatrix}y_2\\\theta_2\end{pmatrix} = \begin{pmatrix}A&B\\C&D\end{pmatrix} \begin{pmatrix}y_1\\\theta_1\end{pmatrix}, \qquad T(d)=\begin{pmatrix}1&d\\0&1\end{pmatrix}, \qquad L(f)=\begin{pmatrix}1&0\\-1/f&1\end{pmatrix}.\end{split}\]

Apply matrices in physical traversal order, with the first encountered element on the right. Paraxial models assume small angles.

Gaussian beams

For an ideal fundamental Gaussian beam in free space,

\[z_R=\frac{\pi w_0^2}{\lambda}, \qquad w(z)=w_0\sqrt{1+(z/z_R)^2}, \qquad \frac{1}{q}=\frac{1}{R}-i\frac{\lambda}{\pi w^2}.\]

Here \(w\) is the intensity \(1/e^2\) radius. Under the stated phase convention and an appropriate paraxial matrix,

\[q_2=\frac{Aq_1+B}{Cq_1+D}.\]

Diffraction and image formation

For scalar Fraunhofer propagation, the far-field amplitude is related to the Fourier transform of the aperture field. For incoherent imaging,

\[\mathrm{PSF}\propto \left|\mathcal{F}\{P(x,y)\}\right|^2, \qquad P(x,y)=a(x,y)\exp\!\left(i\frac{2\pi W(x,y)}{\lambda}\right).\]

Here \(W\) is optical-path error in length units. State sampling, normalization, and frequency-coordinate scaling when calculating a PSF.

Absorption and detection

For a uniform absorbing medium and a simple quantum-efficiency model,

\[I(d)=I(0)e^{-\alpha d}, \qquad R_\lambda=\eta\frac{q_e\lambda}{hc}.\]

Absorption coefficient and thickness must use reciprocal units. Responsivity is in A/W when SI units are used. This relation does not supply bandwidth, saturation, dark current, or noise specifications.

The browser lab includes a standalone photodiode module and worked transport examples in Photonics Essentials: Chapter 3 Interactive Physics Lab.

Choose the appropriate model

  • Use rays for geometric paths, apertures, and paraxial sanity checks.

  • Use Gaussian beams for ideal paraxial mode size and curvature.

  • Use wave propagation for diffraction and coherent interference.

  • Use calibrated radiometry and detector models for absolute signals.

  • Verify convergence and compare with independent measurements or limits.