kb/data/en.wikipedia.org/wiki/Diffraction-limited_system-1.md

title	chunk	source	category	tags	date_saved	instance
Diffraction-limited system	2/3	https://en.wikipedia.org/wiki/Diffraction-limited_system	reference	science, encyclopedia	2026-05-05T09:47:17.785058+00:00	kb-cron

In the case where the spread of the IRF is small with respect to the spread of the diffraction PSF, in which case the system may be said to be essentially diffraction limited (so long as the lens itself is diffraction limited). In the case where the spread of the diffraction PSF is small with respect to the IRF, in which case the system is instrument limited. In the case where the spread of the PSF and IRF are similar, in which case both impact the available resolution of the system. The spread of the diffraction-limited PSF is approximated by the diameter of the first null of the Airy disk,

    d
    
      /
    
    2
    =
    1.22
    λ
    N
    ,
    
  

{\displaystyle d/2=1.22\lambda N,\,}

where

    λ
  

{\displaystyle \lambda }

is the wavelength of the light and

    N
  

{\displaystyle N}

is the f-number of the imaging optics, i.e.,

    2
    
      N
      A
    
    →
    (
    2.44
    N
    
      )
      
        −
        1
      
    
  

{\displaystyle 2\mathrm {NA} \rightarrow (2.44N)^{-1}}

in the Abbe diffraction limit formula. For instance, for an f/8 lens (

    N
    =
    8
  

{\displaystyle N=8}

and

      N
      A
    
    ≈
    2.5
    %
  

{\displaystyle \mathrm {NA} \approx 2.5\%}

) and for green light (

      λ
      
        g
      
    
    =
  

{\displaystyle \lambda _{g}=}

0.5 μm wavelength) light, the focusing spot diameter will be d = 9.76 μm or 19.5

      λ
      
        g
      
    
  

{\displaystyle \lambda _{g}}

. This is similar to the pixel size for the majority of commercially available 'full frame' (43mm sensor diagonal) cameras and so these will operate in regime 3 for f-numbers around 8 (few lenses are close to diffraction limited at f-numbers smaller than 8). Cameras with smaller sensors will tend to have smaller pixels, but their lenses will be designed for use at smaller f-numbers and it is likely that they will also operate in regime 3 for those f-numbers for which their lenses are diffraction limited. Given the same field of view, pixel count, shutter speed and shot noise SNR (i.e. the same amount of light collected per pixel), a small sensor and a large sensor of equivalent quality will produce the same digital image, with the same amount of blur due to both diffraction and depth of field. The larger sensors do have an advantage in that the lenses for them tend to have larger maximum entrance pupils, which allows, to a larger extent, trading depth of field for less diffraction blur and either more SNR or higher shutter speeds.

== Obtaining higher resolution ==

Using special optical systems and digital image processing, it is possible to produce images that have higher resolution (in some specific aspects of the subject, such as color or shape) than would be allowed by simple use of diffraction-limited optics. These largely computational methods offer advantages over other workarounds such as electron microscopy, and have been used to produce reasonably accurate images of individual molecules. Although these techniques improve some aspect of resolution, they generally come at an enormous increase in cost and complexity compared to using a simple light microscope. Usually the technique is only appropriate for a small subset of imaging problems, with several general approaches outlined below.

=== Extending numerical aperture === The effective resolution of a microscope can be improved by illuminating from the side. In conventional microscopes such as bright-field or differential interference contrast, this is achieved by using a condenser. Under spatially incoherent conditions, the image is understood as a composite of images illuminated from each point on the condenser, each of which covers a different portion of the object's spatial frequencies. This effectively improves the resolution by, at most, a factor of two. Simultaneously illuminating from all angles (fully open condenser) drives down interferometric contrast. In conventional microscopes, the maximum resolution (fully open condenser, at N = 1) is rarely used. Further, under partially coherent conditions, the recorded image is often non-linear with object's scattering potential—especially when looking at non-self-luminous (non-fluorescent) objects. To boost contrast, and sometimes to linearize the system, unconventional microscopes (with structured illumination) synthesize the condenser illumination by acquiring a sequence of images with known illumination parameters. Typically, these images are composited to form a single image with data covering a larger portion of the object's spatial frequencies when compared to using a fully closed condenser (which is also rarely used). Another technique, 4Pi microscopy, uses two opposing objectives to double the effective numerical aperture, effectively halving the diffraction limit, by collecting the forward and backward scattered light. When imaging a transparent sample, with a combination of incoherent or structured illumination, as well as collecting both forward, and backward scattered light it is possible to image the complete scattering sphere. Unlike methods relying on localization, such systems are still limited by the diffraction limit of the illumination (condenser) and collection optics (objective), although in practice they can provide substantial resolution improvements compared to conventional methods.