## 2 Literature Review
### Super-Resolution
#### 2.1.3 Classes of Super-Resolution  
Three distinct levels, or classes, of super-resolution have been identified [14; 15].  
1. Ultra-resolution: Improved spatial frequency response and two-point resolution, but the cut-off is unchanged;
2. Restricted super-resolution: the cut-off is increased, but less than \(2n/\lambda\) (coherent) or \(4n/\lambda\) (incoherent);
3. Unrestricted super-resolution: the cut-off is increased without a limit.  
The first class is not really super-resolution, as the spatial frequency cut-off is not increased, hence the different name: ultra-resolution. This class includes the use of pupil filters to sharpen the PSF. [PERSON] showed that the width of the PSF can be made arbitrarily narrower by using a pupil mask, but with increased strength of sidelobes. The spatial frequency cut-off is, however, unchanged [16; 17]. The effect is now sometimes described as super-oscillations. Other examples of this class include simple linear digital deconvolution, for example, based on Wiener filtering or the nearest neighbour method.  
The next class is basically those systems where the band-width is doubled. This is achieved by modulation/demodulation (coding/decoding) and multiplexing. These systems combine patterned illumination with demodulation, which doubles the cut-off, and include the structured illumination microscope (SIM) and the confocal microscope. As the apertures of the illuminating system and the detection system are limited to \(\sin\alpha=1\), then for each of illumination and detection, \(NA=n\), giving a maximum cut-off frequency of \(2n/\lambda\). Here, we have assumed that the wavelengths of illumination and detection are equal, but if not, the cut-off frequency is the sum of the illumination and detection frequencies.  
In the third class, the increase in the bandwidth is unlimited. These methods include STED [18], localisation microscopy [19; 20; 21], and fluorescence saturation [22] or switching, which, although they are important and give impressive resolution, are beyond the scope of this review. Other examples of this class use evanescent waves, including near-field microscopy, where a scanned, subwavelength aperture is used to illuminate the object. The first proposal of near-field microscopy was by [PERSON] in 1928 [23]. Then the idea was reinvented in 1956 [24; 25]. [PERSON] demonstrated the method using acoustic waves, and [PERSON] and [PERSON] used microwaves in 1972 [26]. Then three groups all published optical demonstrations in 1984 [27; 28; 29]. Interestingly, this followed very soon after the invention of the scanning tunneling microscope in 1982, which stimulated the instrumental designs. Another approach, the photon-tunneling microscope, uses evanescent waves generated at an interface [30; 31]. The aperture was avoided in apertureless near-field microscopy [32; 33]. Another more recent example using evanescent waves is the perfect lens, using negative refractive index materials. Many methods of super-resolution rely on some form of nonlinearity, including digital deconvolution with constraints and nonlinear microscopy based on multiphoton fluorescence [34; 35; 36] or second harmonic generation (SHG) [37]. Furthermore, included in this class is pupil filters (super-oscillations) combined with a restricted field-of-view. In this case, the sidelobes are pushed outside of the field-of-view, and the spatial frequency cut-off increased. However, the pupil filter alone does not give an increased cut-off and is in the first class.