The Sparrow limit is a less commonly encountered resolution limit proposed by the American physicist Carrol Mason Sparrow (1880 – 1941) in 1916. However, this does not make it any less useful and the Rayleigh criterion has become one of the most ubiquitous definitions of microscope resolution. This rule is convenient on account of its simplicity and it is sufficiently accurate in view of the necessary uncertainty as to what exactly is meant by resolution. This was clearly stated by Rayleigh himself in 1879: 2 The Rayleigh criterion is therefore not a fundamental physical law and instead a somewhat arbitrarily defined value. Rayleigh chose his criterion based on the human visual system and to provide sufficient contrast for a human observer to distinguish two separate objects in the image. 1,2 In other words, the minimum resolvable separation between the points is the radius of the Airy disc which is given by:įigure 3 Two points separated by the Rayleigh resolution limit. Rayleigh defined the resolution limit as the separation where the central maximum of the Airy pattern of one point emitter is directly overlapping with the first minimum of the Airy pattern of the other. The Rayleigh criterion is named after English physicist John William Strutt, 3 rd Baron Rayleigh (1842-1919) who investigated the image formation of telescopes and microscopes in the late 19 th century. The various microscopy lateral resolution limits, of which the Rayleigh criterion is but one, are essentially just different definitions of what constitutes a sufficient level of contrast between the objects for them to be resolved.įigure 2: Overlap of PSFs and resolution. When the objects are sufficiently far apart there is a dip in the intensity of the total PSF between the objects and they can be distinguished as separate entities and said to be resolved. When two objects are brought together their PSFs combine additively and the total PSF of both objects is what is imaged by the microscope. Resolution can be defined as the minimum separation between two objects that results in a certain level of contrast between them. NA is a measure of the objectives ability to capture light and is the product of the sine of the half-angle of the objective’s acceptance cone, α, and the refractive index, n, of the medium between the sample and objective lens:įigure 1 2D-Point Spread Function / Airy Pattern of a point emitter. Where λ is the wavelength of the emitted or scattered light and NA is the numerical aperture of the microscope’s objective lens.
#Abbe diffraction limit derivation series
It consists of a bright central circle, the Airy disc, which contains 84% of the total light intensity, with the remaining 16% distributed across a series of progressively less intense concentric rings. For a perfect imaging system with no aberrations, this pattern is also known as the Airy pattern and is shown in Figure 1. The diffraction pattern of a point emitter in the image plane of a microscope is described by the 2D point spread function (PSF). When a point emitter (such as a quantum dot) is imaged by a microscope, its image is blurred due to diffraction.
This definition of microscope resolution is also often referred to as the Rayleigh Criterion.The lateral (X-Y) resolution of fluorescence and Raman microscopes is frequently calculated using the famous Rayleigh Criterion for resolution, 0.61 λ/NA, but where does this resolution limit arise from and how does it relate to the other resolution limits encountered in microscopy? As a result of the diffraction limit, two emitting points cannot be optically resolved if the distance between them is smaller than the diffraction limit, which is illustrated in figure 1 (b). In a perfect optical system without any aberrations, the PSF is well-described by the so called Airy function. The PSF is the response of an optical system to a point emitter due to the diffraction limit and imperfections in the optical system. In more mathematical sense it can also be said that the resulting image is a convolution of the actual object with the so-called point spread function (PSF) of the optical system. This results in a blurry appearance of the captured image. The high frequency components that give an image its sharpness are lost by the finite numerical aperture of the lens that collects the light.
The limit is basically a result of diffraction processes and the wave nature of light. The numerical aperture (NA) and the resolution limit is schematically illustrated in figure 1.
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