It was long believed that spatial resolution of light microscopes is ultimately limited by diffraction. Ernst Abbe is perhaps the one who is most often cited for the notion that the resolution in microscopes would always be limited to half the wavelength of blue light, because he stated it saliently. But also many others were aware of the resolution limitation due to diffraction and contributed to its understanding.

The limiting role of diffraction for microscopy is extensively discussed in the famous work by Ernst Abbe in 1873 (Abbe 1873). Here, the resolution limit for microscopic images of half the wavelength (of blue light) is explicitly stated for the first time^{a}.

Abbe considers therefore replacing the eye by a detector that is sensitive to even shorter wavelengths – at his time photographic plates – to increase resolution^{b}.

Abbe describes in words also his famous formula:

${d}_{\mathrm{min}}=\lambda /\left[2\phantom{\rule{0.3em}{0ex}}sin\left(\alpha \right)\right],$

where *d*
_{min} is the minimal resolvable distance, *λ* the wavelength of the light, and *α* the half aperture angle of the microscope’s objective^{c}; it is left open whether *λ* refers to the wavelength in the immersion medium or in air. Abbe does not discuss explicitly the influence of the refractive index in the sample and the immersion medium, though he does consider immersion objectives.

It is interesting to note that Abbe’s 56-page article does not contain any formula in mathematical notation. Abbe sees the microscopic object as consisting of diffraction gratings. The object diffracts the illuminating light and only if a sufficient number of diffraction orders passes the finite-sized objective, the object can be resolved. As becomes apparent in a later article (Abbe 1880), he did therefore not recognize that the same resolution limits also apply to self-luminous objects^{d} (as used in fluorescence microscopy, which was developed much later). Nevertheless, in his article from 1873 (Abbe 1873), he already acknowledges the possibility of new developments that are not covered by his theory and that might enhance the possibilities of optical microscopes beyond the limits that he derived^{e}.

Only one year after Abbe’s first article about the resolution limit (Abbe 1873) appeared, Hermann von Helmholtz published the same results^{f} (von Helmholtz 1874). In contrast to Abbe, von Helmholtz gives a detailed mathematical derivation of his findings. In the last paragraph of his article he states that he had finished his work before he became aware of Abbe’s publication and that it seems acceptable for him to publish his findings in addition to Abbe’s work for they contained the mathematical proofs, which were missing in Abbe’s article.

In addition, von Helmholtz tries to illuminate the object in a way that avoids phase relations at different object points (i. e. incoherently) by imaging the light source onto the object. From his theory he concludes that diffraction effects should then vanish. He denotes the persistence of diffraction to the remaining phase relationships in the object plane. Like Abbe he does not recognize that diffraction effects would remain even with self-luminous objects and would hence limit the resolution.

Although the articles from Abbe and von Helmholtz are the first ones dealing in detail with the resolution limitations of microscopes, the effects of diffraction and its implication for resolution were known earlier. In 1869 Émile Verdet (1869) seems to be one of the first who explicitly mention that microscopes are limited in their resolution by diffraction^{g,h}.

He uses a slightly different separation criterion and arrives at similar results for resolution as later Abbe and von Helmholtz, which he derives for the case of telescopes (i. e. in terms of viewing angle and aperture diameter). He finds that for circular apertures sin *ω*=0.819*λ*/*R*, where *ω* denotes the viewing angle of the first bright ring, *λ* the wavelength of the light used and *R* the radius of the aperture. He considers 1/(2*ω*) as the resolution limit.

Detailed experimental tests of Abbe’s theory including the demonstration of artifacts in the microscopic images are published by J. W. Stephenson in 1877 (Stephenson 1877).

Some years later, in 1896, Lord Rayleigh (1896) discusses extensively the resolution of microscopes. He is the first to deal with illuminated objects as well as with self-luminous objects. He also distinguishes between different phase relationships of the illuminated objects. Lord Rayleigh extends his investigations to different objects (points, lines, gratings) and different aperture shapes. He emphasizes the similarities of microscopes and telescopes and complains about insufficient communication between physicists and microscopists^{i}. Already in 1872, he deals – still under his former name J. W. Strutt – with the diffraction in telescopes and extends known results to annular apertures (Strutt 1872), being unaware of an earlier publication by Airy (1841), which also deals with diffraction at annular apertures, as he states in a post scriptum. In 1874 Lord Rayleigh investigates the resolution – also in terms of the “Rayleigh criterion”^{j} – when imaging gratings (Rayleigh 1874). Here, he states that the theoretical resolution cannot be obtained for large areas due to imperfections (spherical and chromatic aberrations) of the available lenses but that it would be possible with microscope objectives^{k}.

It is Airy in 1835 (Airy 1835) who calculates for the first time the diffraction image of a point source when the limiting aperture is circular in shape. As an example, he states a star seen through a good telescope. Apparently, Airy considers the case of other aperture shapes so well known that he only states that the calculation of their diffraction patterns is never difficult but does not give further references^{l}. Airy does not explicitly state that the diffraction limits resolution (i. e. the possibility to separate different stars), but it can be assumed that he was aware of this fact.

Later, in 1867, W. R. Dawes (1867) addresses the problem of separating double stars. From his observations he derives empirically that the angular separating power scales as 4^{
″
}.56/*a*, where *a* is the aperture size in inches. He points out that he had found by observation the inverse scaling of diameters of star-disks with aperture diameter about 35 years ago^{m}. He, too, does not mention the earlier work of Airy.

As will be shown below, ways to shift, circumvent and break the diffraction limit were found later.

Recognizing that broadening of imaged structures is inevitable due to diffraction is the first step in understanding the resolution of microscopes. The second step is the finding of criteria to define a structure as “resolved”. These criteria will be discussed in the following section.