paraxial

Paraxial refers to rays or light paths that travel close to the optical axis of an optical system. In the paraxial region, angles with the axis are small and transverse distances from the axis are modest, which allows simplifying assumptions that make analysis tractable.

The paraxial approximation is the foundation of paraxial optics. It assumes small-angle approximations for trigonometric functions, namely sin θ ≈ θ and tan θ ≈ θ when angles are measured in radians. This leads to simple linear relations in ray tracing, such as the lens maker equation in its first-order form and the use of Gaussian optics to describe focusing and image formation. The approach is often implemented with ray-transfer (ABCD) matrices to model how rays propagate through sequences of optical elements.

Applications of paraxial optics include the design and analysis of lenses, telescopes, microscopes, and cameras. It provides a practical framework for initial optical system design, alignment, and performance estimation before more exact, nonparaxial methods are employed. The approach underpins many engineering tools and educational treatments of imaging.

Limitations arise because the paraxial approximation neglects higher-order terms. It becomes inaccurate for wide-angle or high numerical aperture systems, off-axis fields, or strong aberrations, where coma, astigmatism, and distortion dominate. In such cases, nonparaxial ray tracing, wave optics, or full numerical simulations are required.

Beyond geometric optics, the term also appears in wave and quantum contexts. The paraxial wave equation describes slowly varying envelopes of wave fields, and the paraxial approximation appears in electron optics and laser beam propagation (notably Gaussian beams). The concept derives from para- “beside” and axis, reflecting its focus on rays near the optical axis.