Diffraction limits resolution.
The Rayleigh criterion states that two images are just resolvable when the centre of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other.
Diffraction
Diffraction is the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture.
Single-slit diffraction
A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity, in accordance with the Huygens–Fresnel principle.
The far-field diffraction pattern of a plane wave incident on a circular aperture, known as the Airy disk, describes how the intensity of light varies with angle. The intensity distribution $I(\theta) as a function of the angle $\theta$ from the optical axis is given by:
$$
I(\theta) = I_0 \left(\frac{2J_1(k a \sin\theta)}{k a \sin\theta}\right)^2
$$
where $a$ is the radius of the circular aperture, $k$ is equal to $2\pi /\lambda $ and $J_{1}$ is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.
Resolution
Light diffracts as it moves through space, bending around obstacles, interfering constructively and destructively. Diffraction limits the detail we can obtain in images. The figure below shows the effect of passing light through a small circular aperture. Instead of a bright spot with sharp edges, a spot with a fuzzy edge surrounded by circles of light is obtained. This pattern is caused by diffraction similar to that produced by a single slit. Light from different parts of the circular aperture interferes constructively and destructively. The effect is most noticeable when the aperture is small, but the effect is there for large apertures, too.
Figure 3: Airy diffraction patterns generated by light from two point sources passing through a circular aperture. Points far apart (top) or meeting the Rayleigh criterion (middle) can be distinguished. Points closer than the Rayleigh criterion (bottom) are difficult to distinguish.
Figure 3(b) shows the diffraction pattern produced by two point light sources that are close to one another. The pattern is similar to that for a single point source, and it is just barely possible to tell that there are two light sources rather than one. If they were closer together, as in Figure 3(c), we could not distinguish them, thus limiting the detail or resolution we can obtain. This limit is an inescapable consequence of the wave nature of light. Thus light passing through a lens with a diameter D shows this effect and spreads, blurring the image, just as light passing through an aperture of diameter D does. So diffraction limits the resolution of any system having a lens or mirror. Telescopes are also limited by diffraction, because of the finite diameter D of their primary mirror.
The Rayleigh criterion
The imaging system’s resolution can be limited either by diffraction causing blurring of the image. Diffraction is determined by the finite aperture of the optical elements.
consider the diffraction pattern for a circular aperture, which has a central maximum that is wider and brighter than the maxima surrounding it (similar to a slit) [see Figure 4(a)]. It can be shown that, for a circular aperture of diameter D, the first minimum in the diffraction pattern occurs at $\theta = 1.22 {\lambda}/D$ (providing the aperture is large compared with the wavelength of light, which is the case for most optical instruments). The Rayleigh criterion for the diffraction limit to resolution states that two images are just resolvable when the centre of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other. See Figure 4(b). The first minimum is at an angle of $\theta = 1.22 {\lambda}/D$, so that two point objects are just resolvable if they are separated by the angle
$$
\theta_{\text{min}} \approx \frac{1.22\lambda}{D}
$$
where $\lambda$ is the wavelength of light (or other electromagnetic radiation) and D is the diameter of the aperture, with which the two objects are observed. In this expression, $\theta$ has units of radians.
Figure 4: (a) Note that, similar to a single slit, the central maximum is wider and brighter than those to the sides. (b) Two point objects produce overlapping diffraction patterns. Shown here is the Rayleigh criterion for being just resolvable. The central maximum of one pattern lies on the first minimum of the other.
