Glossary entry

Numerical aperture (NA)

The sine of the maximum acceptance half-angle for light entering an optical fiber, lens, or waveguide. Quantifies light-gathering or focusing capability.

For an optical fiber, the numerical aperture is

\text{NA} \;=\; \sin\theta_\text{max} \;=\; \sqrt{n_\text{core}^2 - n_\text{clad}^2},

where $\theta_\text{max}$ is the half-angle of the acceptance cone in air. For weakly-guiding fibers ( $\Delta n \ll n_\text{core}$ ):

\text{NA} \;\approx\; n_\text{core} \sqrt{2 \Delta}, \qquad \Delta = \frac{n_\text{core} - n_\text{clad}}{n_\text{core}}.

For a focusing lens of focal length $f$ illuminated with a beam of diameter $D$ in air:

\text{NA} \;\approx\; \sin(\arctan(D / 2f)) \;\approx\; D / (2f) \quad \text{(paraxial)}.

Higher NA gives stronger light gathering and tighter focusing, at the cost of shorter depth of field and increased aberrations.

Typical fiber values:

Fiber	NA
Single-mode telecom (SMF-28, HI-1060)	0.10 – 0.14
50 / 125 μm multimode	0.20
62.5 / 125 μm multimode	0.275
High-NA double-clad fiber	0.46
Photonic crystal fiber	varies, can exceed 0.6

For lens-based focusing onto a fiber or PIC edge coupler, the lens NA should match the target waveguide NA to maximize coupling. Mismatch produces mode mismatch loss.

NA also sets the diffraction-limited focused spot size: $w_0 \approx \lambda / (\pi \cdot \text{NA})$ . Combined with beam quality, this determines the achievable focus for a real source.