Classical optical lithography is diffraction limited to writing features of a size lambda/2 or greater, where lambda is the optical wavelength. Using nonclassical photon-number states, entangled N at a time, we show that it is possible to write features of minimum size lambda/(2N) in an N-photon absorbing substrate. This result allows one to write a factor of N2 more elements on a semiconductor chip. A factor of N = 2 can be achieved easily with entangled photon pairs generated from optical parametric down-conversion. It is shown how to write arbitrary 2D patterns by using this method.

As demonstrated by Boto et al. [Phys. Rev. Lett. 85, 2733], quantum lithography offers an increase in resolution below the diffraction limit. Here, we generalize this procedure in order to create patterns in one and two dimensions. This renders quantum lithography a potentially useful tool in nanotechnology.PACS numbers: 42.50. Hz, 42.25.Hz, 85.40.Hp Optical lithography is a widely used printing method. In this process light is used to etch a substrate. The exposed or unexposed areas on the substrate then define the pattern. In particular, the micro-chip industry uses lithography to produce smaller and smaller processors. However, classical optical lithography can only achieve a resolution comparable to the wavelength of the light used [1][2][3]. It therefore minimizes the scale of the patterns. To create smaller patterns we need to venture beyond this classical boundary [4]. In Ref.[5] we introduced a procedure called quantum lithography that offers an increase in resolution beyond the diffraction limit. This process allows us to write closely spaced lines in one dimension. However, for practical purposes (e.g., optical surface etching) we need to create more complicated patterns in both one and two dimensions. Here, we study how quantum lithography can be extended to create these patterns.This paper is organized as follows: first, for completeness, we present a derivation of the Rayleigh diffraction limit. Then, in Sec. II we reiterate the method introduced in Ref. [5]. Then, in Sec. III we give a generalized version of the states used in this procedure. We show how we can tailor arbitrary one-dimensional patterns with these states. In Sec. IV we show how four-mode entangled states lead to patterns in two dimensions. Sec. V addresses the physical implementation of quantum lithography. I. CLASSICAL RESOLUTION LIMITWhen we talk about optical resolution, we can mean two things: it may denote the minimum distance between two nearby points which can still be resolved with microscopy. Or it can denote the minimum distance separating two points which are printed using lithography. In the limit of geometric optics these resolutions would be identical. In this section we derive the classical resolution limit for interferometric lithography using the so-called Rayleigh criterion [6].Suppose two plane waves characterised by k 1 and k 2 hit a surface under an angle θ from the normal vector. The wave vectors are given by k 1 = k(cos θ, sin θ) and k 2 = k(cos θ, − sin θ) , (1) where we used | k 1 | = | k 2 | = k. The wave number k is related to the wavelength of the light according to k = 2π/λ.In order to find the interference pattern in the intensity I, we sum the two plane waves at position r at the amplitude level:When we calculate the inner product ( k 1 − k 2 ) · r/2 from Eq.(1) we obtain the expressionfor the intensity along the substrate in direction x. The Rayleigh criterion states that the minimal resolvable feature size ∆x corresponds to the distance between an intensity maximum and an adjacent minimum. From Eq...

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