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 $k_1\frac{\lambda }{\mathrm{NA}}$ where $\lambda$ 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. 1, in which the formula here and the one for the depth of focus were referred to as the Rayleigh criteria.). He or she also knows that $k_1$ has a lower bound of 0.25. In this letter, we argue that it was Abbe who definitively stated the $0.5\frac{\lambda }{\mathrm{NA}}$ 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. (To be clear, this is not the first time the name “Abbe formula” is mentioned. Others have already used this name in their various publications.)

The Rayleigh criterion for resolution originates from Lord Rayleigh’s 1879 article² (see Fig. 1), though Helmholtz had already come up with the $0.5\frac{\lambda }{\mathrm{NA}}$ 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). In the beginning part of this article, he put forward the formula obtained by Airy in 1834,

where $\theta$ is the angular radius of the bright central disk, $\lambda$ represents the wavelength of the light, and $2R$ 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. If the angular interval between the components of the star were equal to $2\theta$, the central disks would be just in contact. 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. Such luminous lines were generated in prism or grating spectroscopes by light sources with two spectral lines very close in wavelength. 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 where $\xi$ is the horizontal axis, $a$ is the horizontal dimension of the rectangular aperture (placed after the prism but before the focusing lens), and $f$ is the focal length of the lens. He then tabulated the values of the above function and pronounced that if the two neighboring lines were so separated that the maximum intensity of one line fell onto the first minimum of that of the other line, then the two lines could be discerned, because the combined brightness in the middle of the two peaks (which have the brightness of 1) dipped down to 0.8106 (see Fig. 2). Hence, the smallest discernable separation $d$ of the two lines was

If we translate this to our language, $a/f$ is twice the NA of a one-dimensional lens in air. Hence, the Rayleigh criterion simply implies that the discernable separation of two neighboring lines is $0.5\frac{\lambda }{\mathrm{NA}}$. The same criterion can also be applied to the Airy patterns. 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 $0.61\frac{\lambda }{NA}$, as has been stated in many textbooks.

What Rayleigh stated in his article can be easily explained. 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 $\delta$-function object. One can look up, in a number of textbooks (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

where $J_1$ is the Bessel function of the first kind, order 1, whose first zero occurs at the argument of $1.22\pi$, and NA, the numerical aperture of the optical system, equals Rayleigh’s $R/d$ with $R$ being the radius of the aperture and $d$ the distance from the aperture to the image plane. Setting the argument of the above Bessel function to $1.22\pi$, the diameter of the Airy disk is then or as Rayleigh stated, the angular radius is

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, p. 76). 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.

Rayleigh understood, however, what he put forward was not the absolute resolution limit. He stated in the article that “this rule is convenient on account of its simplicity.” Born and Wolf⁴ also stated in their book that “no special physical significance is to be attached to the Rayleigh criterion, and from time to time other criteria of resolution have been proposed.” Rayleigh dealt with incoherent illumination. Under incoherent illumination, light intensity of the final image is the sum of the intensities produced by each point or line alone (see e.g., Ref. 3, p. 135). For two neighboring lines, we may argue that their minimum discernable separation can even be $0.45\frac{\lambda }{\mathrm{NA}}$, as the intensity in the mid-point between the two peaks dips down to 0.954 times the intensity at either peak. To be extreme, one can even argue that a one-percent intensity dip at the mid-point should be considered discernment of the two features. In fact, more than a century ago, Sparrow⁵ stated that he was able to discern the two lines, by direct vision and in positive and negative film, all the way down to where the second derivative of the combined intensity curve at point B in Fig. 2 reached zero, meaning no intensity dip at all, at 0.83 times the Rayleigh separation. Therefore, a criterion based on a two-point or two-line structure is ambiguous. Also, the locations of the two peaks in the image intensity for the $0.45\frac{\lambda }{\mathrm{NA}}$ case are not $0.45\frac{\lambda }{\mathrm{NA}}$ but $0.365\frac{\lambda }{\mathrm{NA}}$ from each other. The root cause of all this ambiguity lies in the continuous nature of the spatial frequencies of a two-point (or two-line) object; they are not sharp peaks ($\delta$-functions or near $\delta$-functions in the spatial frequency domain) associated with periodic or regular structures which are either passed or eliminated without ambiguity by the pupil aperture. Abbe investigated exactly such periodic or regular structures.

In April of 1882, Abbe⁶ submitted a paper, written in English, to the Royal Microscopical Society (see Fig. 3), in which he stated that, for periodic and regular features, “the minimum distance apart at which given elements can be delineated separately with the feature in question” was

where $\lambda$ is the wavelength of the illuminating light and $a$ is the NA of the system. We are not sure whether Abbe knew of Rayleigh’s prior publication. Even if he knew, he made no reference to that work in this paper. Instead, Abbe stated that he had worked on this topic “for more than ten years,” both theoretically and experimentally. Indeed, the first paper published by Abbe on the theory of microscopes appeared in 1873.⁷ It is fifty-three pages of pure text without a single equation. One of the most succinct remarks in that paper is shown in Fig. 4.

One might argue that such a verbal description is no substitution for a derivation of the formula. True, while Abbe stated in 1882 that he had worked on the subject both theoretically and experimentally for more than ten years, he did not publish a derivation of his resolution formula during his lifetime. However, an extended version of Abbe’s mathematically oriented lecture notes was later published by Lummer, who attended Abbe’s lectures in 1887, and Reiche after Abbe’s death.⁸ (An English translation of this book, to be published by SPIE Press, is in preparation.) In it, the $\frac{1}{2}\frac{\lambda }{\mathrm{NA}}$ formula was given (see Fig. 5) in connection with an extensive write-up on the limit of resolution of imaging. The book and the remark on the timing of Helmholtz’s work (also shown in Fig. 5) is the best evidence there is to substantiate the argument that Abbe had a full understanding of this subject and had arrived at his resolution formula by 1873.

A simple derivation of the Abbe resolution formula may proceed as follows. If we illuminate the object which is a grating of pitch $p$, the directions of diffracted beams obey the following grating equation (for a simple derivation of the grating equation, see e.g. Ref. 3, pp. 463–464)

where $m$ is the order of the diffracted beam, $n$ is the index of refraction of the medium, ${\theta }_i$ is the angle of the incident beam, and ${\theta }_m$ is the angle of the $m$’th order diffracted beam. Following Abbe, to form an image, at least two of the diffracted beams from the object have to be collected and brought to interfere by the imaging optics. The pitch of the standing wave as a result of the interference of these two beams is where $\theta$ is half the angle formed by the two interfering beams. This formula is easily obtained since the normalized intensity of the standing wave evaluates to be $1+C\cos (\frac{2\pi x}{p^{\prime }})$ where $C$ is the contrast. And finer pitches are realized if $\theta$ is increased.

Let us now consider a symmetric configuration shown in Fig. 6, where only the zeroth and the minus first-order beams are admitted. [This simple configuration for demonstrating maximum achievable resolution of a projection imaging system was first shown to me by Hank Smith in his Submicron Structures class (MIT course 6.781) in the spring of 1986.] To simplify our analysis, we let the system have unit magnification. (The conclusion $p_{\min }=\frac{1}{2}\frac{\lambda }{\mathrm{NA}}$ remains valid if the system magnification is different from unity. In that case, we must distinguish the two NA’s: the object-side NA and the image-side NA. Their ratio is the system magnification.) Due to symmetry and because $p^{\prime }=\lambda /(2n\sin \theta )$, we have ${\theta }_i={\theta }_0=\theta$, ${\theta }_{-1}=-\theta$, and $p=p^{\prime }$. The imaging system attains its maximum resolution when the two diffracted beams just reach the aperture of the imaging optics. This maximum resolution is then

as was shown by Abbe, since $\mathrm{NA}\equiv n\sin {\theta }_{\max }$.

Abbe’s formulation has several advantages over Rayleigh’s for us lithographers. First, working with periodic features rather than the more ambiguous two-point or two-line object, Abbe clarified for us that the $\frac{1}{2}\frac{\lambda }{\mathrm{NA}}$ resolution limit on pitch is not for convenience but is absolute, beyond which there is no resolution at all; he also pointed out how this resolution limit could be obtained in practice. Second, Abbe’s explicit use of NA means the index of refraction is included, and hence, NA can be made greater than unity which Abbe and others had already put into practice in microscopy by then and which was also put into practice in microlithography early in this century. Finally, the resolution limit is always about pitch and not the linewidth of a feature. We lithographers know well that the linewidth of a feature can be made (theoretically infinitely) small by e.g., overexposing a positive-tone photoresist.

Lord Rayleigh was a master physicist (J.D. Jackson’s words) who worked on diverse topics of physics and his name deserves our utmost admiration. In this case, however, Abbe was there first, and his investigations were more relevant to microlithography that we practice today. Hence perhaps the resolution formula in microlithography ought to more appropriately be named the Abbe formula rather than Rayleigh’s equation.