Numerical aperture

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Numerical aperture (NA) is a simple, dimensionless number that tells you how much light an optical system can accept or emit. It depends on the refractive index of the surrounding material and the largest angle of light that can pass through the system. The basic formula is NA = n sin theta, where n is the refractive index of the medium around the lens and theta is half the widest cone of light that can enter or leave the lens.

In microscopy, NA is a key measure of light gathering and resolution. A higher NA means brighter images and the ability to see finer details. The finest visible detail scales roughly with wavelength divided by 2NA, so increasing NA improves resolution but typically reduces depth of field.

In photography, the f-number (f/N) is the main indicator of light gathering. NA is related to the f-number, but cameras use many other factors, especially for close objects. For distant objects, a useful link is that the image-side NA is roughly inversely related to the working f-number, though exact relations can be complex.

In lasers, NA is defined similarly (NA = n sin theta), but theta is the beam’s divergence rather than a lens’s marginal ray. For a Gaussian laser beam, the NA is related to how tightly the beam can be focused: a smaller focus spot (smaller w0) gives a larger NA, but causes faster beam spread beyond the focus.

In optical fibers, NA describes which light angles can be guided through a fiber. For multimode step-index fibers, NA ≈ sqrt(n_core^2 − n_clad^2). The corresponding acceptance angle satisfies sin theta_max ≈ NA / n_medium. This approximation works well for many practical cases, but single-mode fibers don’t follow this simple relation.

In short, numerical aperture connects the range of light angles a device can handle with how much light it can collect and how sharply it can resolve details. It’s a practical, cross-cutting concept used across microscopy, photography, lasers, and fiber optics.