Optics fundamentals

Quarter-wave anti-reflection design: film thickness, ideal index, and residual reflectance with a live spectrum plot.

Reflectance and transmittance of any dielectric layer stack by the characteristic-matrix method, s and p polarization at any angle, with presets for anti-reflection coatings, quarter-wave Bragg mirrors, laser facet coatings, and etalons, live spectrum and angle plots, and a 16-row validation table.

Airy disk radius and diameter, FWHM, dark and bright rings, encircled energy, and the Rayleigh, Sparrow and Abbe resolution limits for any wavelength and f-number, aperture and focal length, or numerical aperture, with angular resolution for telescopes, pixel sampling for cameras, a rendered pattern, and an 18-row validation table.

Free-carrier index and absorption change in silicon from the Soref and Bennett relations at 1310 and 1550 nm and the Nedeljkovic fits at 1550 and 2000 nm: Δn and Δα for any electron and hole density, phase shift and loss for a given length and mode overlap, the length for π and its cost in dB, doped-waveguide loss, and a 17-row validation table.

Solve slab waveguide modes exactly: mode profiles with evanescent tails, an neff-versus-thickness cutoff map, confinement, birefringence, the single-mode window, and a propagating field view for TE and TM.

Free spectral range, finesse, linewidth, Q, photon lifetime, mirror loss, peak transmission, fringe contrast and intracavity power for a plane-parallel or confocal two-mirror cavity of any length, index, reflectivities and internal loss, with exact Airy transmission and reflection spectra, presets from etalons and laser diode chips to ULE reference cavities, guided experiments and a validation table.

Chromatic dispersion of single-mode fiber from D and slope or from the zero-dispersion wavelength with the ITU-T G.652 formula: β₂ and β₃, dispersion length, broadening and compression of a chirped Gaussian pulse, accumulated dispersion, compensating-fiber length and residual slope, the dispersion-limited bit rate and length, fiber presets, and a 15-row validation table.

Optical power between dBm, mW, W and dBW; gain and loss as dB, linear and field ratios, percent and optical density; wavelength to frequency, photon energy and wavenumber; and linewidth between Δν, Δλ and Δσ, with photon flux, ideal responsivity, telecom band, nearest DWDM channel, coherence length and a validation table.

Rayleigh range, depth of focus, divergence and beam parameter product of a Gaussian or M² beam; beam radius, wavefront curvature, Gouy phase, peak intensity and aperture transmission at any distance; and the focused waist size and position behind a thin lens, with beam plots, presets, guided experiments and a validation table.

Diffraction angles of every propagating order, the Littrow angle, angular and linear dispersion, resolving power, spectrometer resolution for a given slit, free spectral range and beam width ratio for any groove density, wavelength, incidence and order, with a ray diagram, angle-versus-wavelength curves, presets, guided experiments and a validation table.

Split ratio, crossover length and wavelength range of a directional coupler from the supermode index difference by coupled-mode theory, and the free spectral range, output and extinction ratio of a Mach-Zehnder interferometer built from those couplers, with plots, presets, guided experiments and a validation table.

Refractive index, group index and material dispersion of fused silica, N-BK7, N-SF11, calcium fluoride, sapphire and silicon from published Sellmeier equations, or from your own coefficients, with the Abbe number, zero-dispersion wavelength, plots, guided experiments and a validation table.

Near and far limits of sharpness, hyperfocal distance and the diffraction limit for a camera lens, and the depth of field of a microscope objective from its NA, magnification and camera pixel size, with plots, presets, guided experiments and a validation table.

Pulse energy, peak power, duty cycle, photons per pulse, fluence and peak intensity of a pulsed laser from its average power, repetition rate, pulse duration and shape, and spot size, with presets from Q-switched to femtosecond sources, guided experiments and a validation table.

Wavelength, frequency, photon energy and wavenumber from any one of them, the wavelength and speed in a medium of given refractive index, the region of the electromagnetic spectrum, and λ = v/f for sound or any other wave, with presets from radio to X-rays, guided experiments and a validation table.

Single-slit, double-slit and multi-slit Fraunhofer patterns: first minimum angle, central maximum width on a screen, fringe spacing, fringes inside the envelope, missing orders and the Fresnel number, with the intensity plotted and drawn as it looks on the screen, presets, guided experiments and a validation table.

Any input polarization through up to four polarizers, half-wave and quarter-wave plates, retarders and rotators: transmitted intensity, output Jones vector, Stokes parameters, azimuth, ellipticity and handedness, with the polarization ellipse, an intensity-versus-angle plot, presets, guided experiments and a validation table.

Image distance, magnification, image height and image type from the thin lens equation for converging and diverging lenses, or a pair of lenses with its effective focal length, with a live ray diagram, presets from a magnifier to a microscope, guided experiments and a validation table.

Refracted angle, critical angle and Brewster angle for any two refractive indices, with Fresnel reflectance and transmittance for s, p and unpolarized light and the evanescent depth under total internal reflection, with a ray diagram, reflectance curves, presets, guided experiments and a validation table.

Numerical aperture and acceptance angle of an optical fiber from its core and cladding indices, the V number, single-mode or multimode guidance and the guided LP modes, the cutoff wavelength, and the mode field diameter by Marcuse's formula, with an acceptance-cone diagram, V and MFD against wavelength, presets, guided experiments and a validation table.

Optical density, transmission, decibels and attenuation factor from any one of them, the total of a stack of neutral-density filters and the power it passes, the OD needed to reach a target power, and the absorbance and transmittance of an absorbing sample with its surface reflections, with plots, presets, guided experiments and a validation table.

Complex Fresnel reflection coefficients for s and p polarization at any angle, the reflectance and its phase, the phase difference and ellipsometric angles, the Brewster, pseudo-Brewster and critical angles, and absorbing media with a complex refractive index, with reflectance and phase plots, presets, guided experiments and a validation table.

One-way and round-trip delay of a fiber link from its length and group index, the data in flight, the saving from hollow-core fiber, the comparison with a straight line at the speed of light, the length an OTDR reports with a given index setting and the dispersion skew between two wavelengths, with a delay plot, presets, guided experiments and a validation table.

Coherence time and coherence length from a linewidth in frequency or wavelength for Lorentzian, Gaussian and rectangular spectra, the fringe visibility at a given path difference, the path difference where it halves and the OCT axial resolution, with a visibility plot, presets, guided experiments and a validation table.

The quiet foundation of much of photonics: the fiber that spans oceans at 0.2 dB/km, the cladding of silicon and nitride waveguides, and a common substrate for free-space optics.