Lasers and fibre optics

Gaussian beam. Enter the wavelength (or pick a laser line), the 1/e² waist radius w₀, the beam quality M² and a distance z from the waist. You get the Rayleigh range z_R = πw₀²/(M²λ) — the distance over which the beam’s area doubles — the far-field divergence θ = M²λ/(πw₀), the beam parameter product, the radius w(z) = w₀√(1 + (z/z_R)²), the wavefront radius of curvature R(z) = z(1 + (z_R/z)²), the Gouy phase arctan(z/z_R) and the peak irradiance. The plot shows the envelope with its wavefronts moving through the focus: flat at the waist, most curved at ±z_R. Hover (or drag a finger across it) for w and R at any point. ❚❚ pauses the animation.

Focus & expand. A beam of radius w at a lens of focal length f focuses to a waist of about M²λf/(πw); the tab uses the exact Gaussian result (Self 1983), so a beam whose waist is some way before the lens, or a weak lens, is handled too. Depth of focus is twice the new Rayleigh range. Below, a two-lens beam expander: magnification |f₂/f₁|, separation f₁ + f₂ (Galilean if one lens is negative), and the divergence divided by the magnification.

ABCD train. Build an optical train of free-space gaps and thin lenses (or start from a preset: focusing lens, Galilean and Keplerian expanders, coupling into single-mode fibre, a multimode diode, a 4f relay). The complex beam parameter q goes through each element as q′ = (Aq + B)/(Cq + D) and the envelope is drawn along the whole train, with the waist after each lens marked and listed. ⛶ Full screen (or F) gives the plot the whole screen; ◫ Panels hides the element list.

Cavity. Two mirrors of radius R₁ and R₂ (positive when concave towards each other, inf for flat) a distance L apart. The cavity is stable when 0 < g₁g₂ < 1 with g = 1 − L/R. Tap the stability diagram to choose a point; the planar, confocal, concentric and hemispherical cavities sit on its edge. For a stable cavity: the TEM₀₀ mode’s waist and its size on each mirror, the free spectral range c/2L and the transverse mode spacing.

Fibre. Wavelength, core radius, the core index and either the NA or the cladding index, and step or (parabolic) graded index. You get the NA and acceptance angle, Δ, the V number, the number of guided modes (≈ V²/2 step, V²/4 graded), the single-mode cutoff wavelength and, near single mode, the mode field diameter from Marcuse’s fit. The animation shows rays (step index: steep rays lag) or curving paths that arrive together (graded index).

Link budget. Transmitter power and receiver sensitivity in dBm, fibre attenuation and length, connectors, splices and other losses, and a margin: the received power, what is left over, and the longest fibre the budget allows, with a chart of power along the link.

Dispersion. Modal spread for step and graded multimode fibre, chromatic spread |D| L σ_λ (D entered or from the zero-dispersion wavelength and slope of standard single-mode fibre), the total, the maximum bit rate ≈ 0.2/σ and the bandwidth–distance product, with D(λ) plotted.

Safety. A rough class from the power, the maximum permissible exposure for a CW beam (400–1150 nm, 0.25–10 s), the nominal ocular hazard distance and irradiance against distance, plus a summary of the IEC 60825-1 classes. These are approximate estimates for learning — not a safety assessment.

Photons. Photon energy, photons per second, momentum and radiation force for a power, average and Gaussian peak irradiance, and pulse arithmetic (peak power, photons per pulse).

Long distances (fibre lengths, the hazard distance) can be shown in miles and feet with the switch at the top; wavelengths are in nm and beam sizes in mm and µm throughout.

The tests check: z_R = πw₀²/λ (0.794 m for a 0.4 mm HeNe waist), w(z_R) = √2 w₀, R(z_R) = 2z_R and a Gouy phase of π/4 there; ABCD propagation of q against the closed form, and the focused waist and its position against Self’s formula (S. A. Self, "Focusing of spherical Gaussian beams", Applied Optics 22, 658, 1983); cavity edge cases (planar, confocal, concentric, hemispherical, unstable on both sides) and the symmetric-cavity waist (Siegman, Lasers, ch. 19); fibre NA, V and mode counts; Marcuse’s mode field diameter (D. Marcuse, Bell Syst. Tech. J. 56, 703, 1977) against Corning SMF-28 (9.2 ± 0.4 µm at 1310 nm, 10.4 ± 0.5 µm at 1550 nm, with a step-equivalent NA of 0.12); Senior’s worked example (Optical Fiber Communications, 3rd ed.: 6 km of step-index fibre with n₁ = 1.5 and Δ = 1% gives 300 ns, an rms spread of 86.7 ns, about 2.3 Mbit/s and 13.8 Mbit/s·km; graded index 14.4 ps/km); D(1550 nm) = 17.5 ps/(nm·km) for λ₀ = 1310 nm, S₀ = 0.092 (Corning SMF-28 data sheet, ITU-T G.652); the MPE of 2.55 mW/cm² at 0.25 s and about 1 mW/cm² at 10 s for visible light, and a 14.8 m NOHD for a 5 mW, 1 mm, 1 mrad beam (ANSI Z136.1-2014; IEC 60825-1:2014 Table A.1); and 3.19 × 10¹⁵ photons/s in 1 mW at 632.8 nm. Typical losses are from ANSI/TIA-568.3-D and fibre data sheets.

Related: lens design bench (ray tracing, aberrations, achromats), telescopes and optics (thin-lens ray bench, Snell’s law, total internal reflection), links, optics and energy per bit (transceivers, pJ/bit), laser cutter test patterns, diffraction and pixel pitch, spectroscopy, decibels.

Share link copies a link with all the tabs' settings (the ABCD train is packed into ?s=). Links can be written by hand; add the tab after #: beam, focus, train, cavity, fiber, budget, dispersion, safety or photons.