Microscope Resolution and Numerical Aperture

What is modelled

A lens collects only a cone of the light coming from each point of the specimen, and the wider the cone, the sharper the image. A point is therefore never imaged as a point but as a bright disc surrounded by faint rings, the Airy pattern. Two points can be told apart when the centre of one disc falls on the first dark ring of the other. The sine of the half-angle of the cone times the refractive index of the medium in front of the lens is the numerical aperture, so oil or water in front of the lens raises the limit, and shorter light shrinks the discs. Magnification beyond about a thousand times the numerical aperture only makes a bigger blur.

Equations

• Numerical aperture NA = n sinα; Airy intensity I = (2 J1(v) / v)2 with v = 2π NA r / λ.
• Rayleigh distance 0.61 λ / NA; Abbe limit λ / (2 NA), found from the first diffraction order of a grating.
• Useful magnification between about 500 NA and 1000 NA; depth of field (wave term) λ n / NA2.

Checks run in the page

• The first zero of the Bessel function and the Rayleigh formula.
• The energy inside the first dark ring.
• The dip between two points at the Rayleigh distance against 73.5 percent.
• The finest grating passed with straight and with tilted light.

Honest simplifications

Two incoherent point sources of equal brightness, a lens with no aberrations and scalar diffraction, which is only an approximation for numerical apertures above about 0.7, where the polarisation of light matters. Real images also depend on the camera, the noise and the contrast, and super-resolution methods go beyond these limits.

• Press Dry 40x objective and drag the separation until the dip between the two points disappears.
• Press Oil immersion 100x and compare the Rayleigh distance with the dry lens.
• Raise the eyepiece magnification and watch the label change to empty magnification.