## 2 Literature Review  
### Super-Resolution  
#### 2.1.1 Resolution Criteria and Resolution Limits  
Before we can discuss super-resolution, we must consider what we mean by resolution. There are many different alternative resolution criteria. The simplest is based on the width of the central lobe of the point spread function (PSF), often defined as the full-width at half maximum (FWHM). While this can give a reasonable estimate in many cases, it is not a reliable measure. The two most well-known criteria are those due to [PERSON] and to [PERSON]. [PERSON]'s criterion is based on an object consisting of two self-luminous (i.e., incoherent) points of equal strength [1]. He claimed that the image of the points is resolved if the points are separated by a distance \(1.22\lambda/NA\), where \(\lambda\) is the wavelength of the light and \(NA=n\sin\alpha\) is the numerical aperture, with \(n\) the refractive index of the immersion medium and \(\alpha\) the angle subtended by the edge of the lens pupil with respect to the optical axis. [PERSON] himself explained that this criterion is arbitrary and not a hard limit to resolution. [PERSON]'s criterion was proposed for the special case of a circular aperture with no aberrations in an incoherent imaging system. As the intensity midway between the points when they are just resolved in this case is 0.735 times the intensity at the points, the criterion has been generalised by taking the points to be just resolved when this same ratio is achieved in the general case. The FWHM of the PSF is often consistent with the [PERSON] criterion, but not always. For example, pupil filters can sharpen the PSF but at the expense of an increased side-lobe level, which can worsen the Rayleigh two-point resolution. Additionally, coherent and incoherent systems have the same intensity PSF, but the Rayleigh resolution is better for the incoherent case. [PERSON]'s resolution limit is a well-defined transition from being resolved to being not resolved [2]. It is based on the image of a grating of lines. [PERSON] assumed the object to be illuminated by a plane wave. The grating diffracts light into different orders, which are either transmitted through the optical system or not. The angles of propagation of these orders are given by a Fourier series expansion of the grating structure, so there is a well-defined cut-off spatial frequency. In order to get an image of the grating, two diffracted orders must travel through the system. The cut-off frequency if the object is illuminated along the optical axis is \(NA/\lambda\). [PERSON] was aware that illuminating at an oblique angle increases the cut-off frequency. If the illumination angle coincides with the edge of the aperture of the objective lens, then the cut-off is \(2 NA/\lambda\). However, [PERSON] also recognised that an image depends on the strength of the spatial frequency response in addition to the cut-off frequency, which is now described by a transfer function. For a coherent imaging system, we use a coherent transfer function (CTF), which operates on amplitudes, to describe the imaging properties. For incoherent imaging, we use an optical transfer function (OTF), which operates on intensities. A major limitation of [PERSON]'s resolution limit is that it is based on coherent illumination, whereas in practice, microscopes used illumination with an incoherent source. Later, [PERSON] summed the illumination over a range of incident angles and showed that the result for the cut-off frequency can be up to \(2 NA/\lambda\) and that this is also the cut-off for a self-luminous (incoherent) object [3]. A more complete understanding of the effect of condenser aperture was achieved after the development of the theory of partial coherence [4]. Instead of a CTF or OTF, we have a transmission cross-coefficient (TCC) (sometimes called a partially-coherent transfer function) that describes the transfer of pairs of spatial frequencies into the intensity image. Different diffraction orders interfere and generate sum and difference frequencies in the image.