Rayleigh understood, however, what he put forward was not the absolute resolution limit. If the vertical dimension of the aperture is much smaller than its horizontal dimension, then the intensity of the diffraction pattern close to the horizontal axis is the square of the sinc function given by Rayleigh. The Fraunhofer diffraction of a rectangular aperture can also be easily evaluated to be the product of two sinc functions in the horizontal and the vertical directions, respectively (see e.g., Ref. 3, pp. 76–79), to find that the light intensity of the Airy disk which is the square of the point-spread function (the Fraunhofer diffraction of a circular aperture) is proportional to One can look up, in a number of textbooks (see e.g., Ref. Light disturbance in the image plane, produced by a distant star, is simply the point-spread function of the optical system of the telescope, since the distant star can be regarded as a δ-function object. What Rayleigh stated in his article can be easily explained. If we allow the maximum of the first Airy pattern to coincide with the edge of the bright central disk of the second pattern, then the light intensity at the saddle point in the middle of the two intensity peaks is 0.7348 times the intensity at either peak, and the minimum discernable distance in this case is 0.61 λ N A, as has been stated in many textbooks. The same criterion can also be applied to the Airy patterns. Hence, the Rayleigh criterion simply implies that the discernable separation of two neighboring lines is 0.5 λ NA. If we translate this to our language, a / f is twice the NA of a one-dimensional lens in air. ABCD is ( sin u u ) 2 OA′C′ is ( sin ( u − π ) u − π ) 2 and E′BEF is half of. Rayleigh first stated, quoting Airy and Verdet, that the intensity (which he called brightness) of a luminous spectral line was proportional to the square of the sinc function Such luminous lines were generated in prism or grating spectroscopes by light sources with two spectral lines very close in wavelength. Under these conditions there can be no doubt that the star would appear to be fairly resolved, since the brightness of the external ring-systems is too small to produce any material confusion, unless indeed the components are of very unequal magnitude.” He then went on to discuss two neighboring luminous lines and proposed his resolution criterion that is more lenient than above. If the angular interval between the components of the star were equal to 2 θ, the central disks would be just in contact. Θ = 1.2197 λ 2 R ,where θ is the angular radius of the bright central disk, λ represents the wavelength of the light, and 2 R is the diameter of the circular aperture in front of a perfect lens, and went on to state that “in estimating theoretically the resolving-power of a telescope on a double star, we have to consider the illumination of the field due to the superposition of the two independent images. In the beginning part of this article, he put forward the formula obtained by Airy in 1834, The Rayleigh criterion for resolution originates from Lord Rayleigh’s 1879 article 2 (see Fig. 1), though Helmholtz had already come up with the 0.5 λ NA resolution limit using similar arguments in 1874 (see caption of Fig. 5 this fact was also acknowedged by Rayleigh in a later article of his). Others have already used this name in their various publications.) (To be clear, this is not the first time the name “Abbe formula” is mentioned. In this letter, we argue that it was Abbe who definitively stated the 0.5 λ NA resolution limit (for pitch instead of half-pitch) first, using an approach more relevant to projection imaging, and hence, the above expression should be more appropriately referred to as the Abbe formula for the resolution of a projection imaging system. He or she also knows that k 1 has a lower bound of 0.25. 1, in which the formula here and the one for the depth of focus were referred to as the Rayleigh criteria.). Every semiconductor lithographer seems to be aware that the resolution of a projection-imaging lithographic system can be described by what is commonly called Rayleigh’s equation, which says that the printable minimum half-pitch in the photoresist is k 1 λ NA where λ is the exposing wavelength, NA is the numerical aperture of the projection optics, and k 1 depends on several factors such as the configuration of the illuminator and the resolution of the photoresist in which a relief image of the pattern on the photomask is printed (the earliest reference to this name that we could find is Ref.
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