Optics & beams

Glossary

ABCD matrix (ray transfer matrix)

A 2×2 matrix that describes how an optical element or a stretch of free space changes a paraxial ray's height and angle. Matrices multiply along a system, and the same four numbers propagate a Gaussian beam's q parameter and decide whether a laser resonator is stable.

Adaptive optics

Correcting a distorted wavefront in real time by measuring it with a wavefront sensor and reshaping a deformable mirror to cancel the error. Restores near diffraction-limited imaging through the atmosphere, the eye or biological tissue.

Beam divergence

The angle at which a beam spreads in the far field, quoted as a half angle or a full angle and usually at the 1/e² intensity points. A Gaussian beam of waist radius w₀ diverges with half angle λ/(πw₀); a 0.4 mm waist at 632.8 nm gives 0.50 mrad.

Beam expander

A pair of lenses spaced so that a collimated beam enters and a wider collimated beam leaves, magnified by M = f₂/f₁. Expanding a beam by M reduces its divergence by M and lets it focus to a spot M times smaller.

Beam parameter product (BPP)

The product of a beam's waist radius and its far-field half-angle divergence, usually in mm·mrad. It equals M²λ/π, so its minimum is set by the wavelength: 0.339 mm·mrad for a perfect Gaussian beam at 1064 nm. Lossless passive optics cannot reduce it.

Collimation

Making a beam's rays parallel, so its width stays nearly constant over the working distance, usually by placing the source at the focal point of a lens. A fiber collimator with f = 11 mm turns single-mode fiber output at 1550 nm into a 2.1 mm beam diverging at 0.47 mrad.

Étendue

The product of the area a light beam passes through and the solid angle it fills, G = πn²A sin²θ for a uniform cone. Lossless passive optics cannot reduce it, which limits how much light from a large, wide-angle source a small fiber or detector can collect.

f-number

The ratio of a lens's focal length to the diameter of its entrance pupil, N = f/D, written f/2, f/4 and so on. It sets the cone angle of the focused light, NA ≈ 1/(2N), and with it the diffraction-limited spot size, 2.44λN in diameter.

Fluence

Optical energy delivered per unit area, usually in J/cm², the quantity that governs pulsed-laser damage and ablation. A Gaussian pulse of energy E and 1/e² radius w has peak fluence 2E/(πw²); 1 mJ in a 100 µm radius spot gives 6.4 J/cm².

Focal length

The distance from a lens or mirror to the point where it brings an incoming collimated beam to focus. For a thin lens, 1/f = 1/s + 1/s′ links object and image distances; a plano-convex N-BK7 lens with a 25.84 mm radius has f = 50.0 mm at 587.6 nm and 51.6 mm at 1550 nm.

Irradiance

Optical power per unit area falling on a surface, in W/m² or, in laser work, W/cm². The peak irradiance of a Gaussian beam is 2P/(πw²), twice the power divided by the 1/e² beam area; 5 mW in a 0.4 mm radius beam gives 2.0 W/cm².

Modulation transfer function (MTF)

How much of an object's contrast an optical system passes to the image, as a function of spatial frequency in line pairs per millimetre. For a diffraction-limited circular aperture it falls from 1 to zero at the cutoff 1/(λN): 227 lp/mm at 550 nm and f/8.

Rayleigh criterion

Two point sources are just resolved when the central maximum of one image falls on the first dark ring of the other: an angular separation of 1.22λ/D, or 0.61λ/NA in a microscope. At 550 nm and NA 1.4 that is 240 nm.

Spatial filter

A lens that focuses a laser beam onto a pinhole, which passes the smooth Gaussian core and blocks the scattered light that makes a beam noisy. A pinhole about 1.5 times the focused 1/e² diameter passes 98.9% of an ideal Gaussian beam.

Wavefront error

The departure of a real wavefront from the ideal sphere or plane, quoted as a peak-to-valley or an rms value in waves. The Rayleigh quarter-wave limit (λ/4 peak-to-valley) and the Maréchal limit (λ/14 rms) both correspond to a Strehl ratio near 0.8.

Zernike polynomials

A set of polynomials orthogonal over a circular pupil, used to describe wavefront error term by term: piston, tilt, defocus, astigmatism, coma, spherical aberration and higher orders. With orthonormal normalization the total rms error is the root-sum-square of the coefficients.