Criteria for success
The mathematical description of optical resolution was laid down separately by Lord Rayleigh and Ernst Abbe in the latter part of the 19th century. Although both approaches lead to essentially similar conclusions, each has a somewhat different way of describing the underlying rules of resolution. The Rayleigh criterion is very simple: it states that the image of a point object is given by a distribution called the Airy disc or “jinc” distribution (top left), which has a peak at the centre and series of sidelobes. For a uniformly illuminated object, the width of the main lobe is given by 0.61λ/NA (where λ is the wavelength of the light and NA is its numerical aperture). If two point objects are illuminated, then the image produced is the sum of the distribution each object would produce individually, and the two points are deemed resolved when the first minimum of one distribution coincides with the maximum of the other. So two points separated by a distance equal to half the Rayleigh resolution distance (top middle) are not resolved, while two points separated by exactly the Rayleigh distance (top right) are resolved.
The Abbe criterion takes a complementary approach to describe the performance of a non-fluorescence microscope. Consider a diffraction grating (bottom) that splits the incident light (black) into three diffracted orders: the (undiffracted) zeroth order (red) and the +1 (blue) and –1 (green) diffracted orders. If the grating is coarse, then the diffracted orders are diffracted through a small angle and pass through the objective lens L1, forming diffraction-limited points in the so-called Fourier plane. These are then re-imaged by lens L2 to form an image on the eye or camera. If the grating is very fine, however, these diffracted orders are diffracted through such a large angle that only the zeroth order passes through L2 and the image that is formed shows no trace of the grating. The Abbe criterion states that the finest grating that can be imaged (which corresponds to the diffracted orders just passing through L2, as shown above) has a period of 0.5λ/NA. Both the Rayleigh and Abbe criteria are often referred to as the “diffraction limit” of microscopic imaging, and they provide convenient ways of thinking about many of the techniques used to achieve resolution beyond the diffraction limit.