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Physical Review A

Super-resolved optical lithography with phase controlled source

Peilong Hong, Guoquan Zhang

Published May 18, 2015• 8 citations
Publisher

Abstract

Recently, we have demonstrated that second-order subwavelength interference could be realized in an optical lithography scheme with an effective entangled source [P. Hong and G. Zhang, Phys. Rev. A 88, 043838 (2013)]. In this paper, by considering the correlation function in both the source plane and observation plane, we show how the coherence property of such a source is controlled via introduction of random-phase correlation, which finally affects the two-photon interference effect observed in the far-field plane. Furthermore, by introducing different but similar random-phase correlations, we generalize the phase controlled source with particular high-order coherence properties to obtain higher-order subwavelength interference, i.e., high-order super-resolved optical lithography. These results show the importance of phase control in generating a light field with particular high-order coherence properties.

DOI
10.1103/physreva.91.053830

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Recently, we have demonstrated that second-order subwavelength interference could be realized in an optical lithography scheme with an effective entangled source [P. Hong and G. Zhang, Phys. Rev. A 88, 043838 (2013)]. In this paper, by considering the correlation function in both the source plane and observation plane, we show how the coherence property of such a source is controlled via introduction of random-phase correlation, which finally affects the two-photon interference effect observed in the far-field plane. Furthermore, by introducing different but similar random-phase correlations, we generalize the phase controlled source with particular high-order coherence properties to obtain higher-order subwavelength interference, i.e., high-order super-resolved optical lithography. These results show the importance of phase control in generating a light field with particular high-order coherence properties.
High-order coherence effects between two first-order incoherent sources with fully independent phases have been well studied in the literature, which shows interference fringes with respect to the position separations among different space points. Here we show that this is not the whole story, and find that the high-order coherence effects depend on the mode of the phase random walk of the first-order incoherent sources, which can be controlled artificially and represented geometrically by vectorial polygons. Interestingly, by scanning the detectors along the same direction with the position separations between them kept constant, a set of high-order coherence fringes, which fingerprint the phase random walk of the first-order incoherent sources, can be observed. Our results show that it is possible to control the high-order coherence of two first-order incoherent sources, which could have important practical applications such as superhigh resolution optical lithography.
The resolution of optical interference is ultimately limited by the wavelength of light, which is known as Rayleigh criterion. Interestingly, this limitation could be surpassed by considering photonic de Broglie wave, and the resulting subwavelength interference is useful for lithography and super-resolving phase measurementsuper-resolving phase measurement. Usually, quantum entangled state is necessary to realize photonic de Broglie wave. Recent advances show that subwavelength interference could also be realized with a classical light through post-selected operation or by considering light-matter interaction. In this work, a new classical source is designed to achieve subwavelength interference without any post-selected operation in a linear scheme.
High-precision measurements implemented with light are desired in all fields of science. However, light acts as a wave, and the Rayleigh criterion in classical optics yields a diffraction limit that prevents obtaining a resolution smaller than the wavelength. Sub-wavelength interference has potential application in lithography because it beats the classical Rayleigh resolution limit. Here, we carefully study second-order correlation theory to establish the physics behind sub-wavelength interference in photon coincidence detection. A Young's double slit experiment with pseudo-thermal light is performed to test the second-order correlation pattern. The results show that when two point detectors are scanned in different ways, super sub-wavelength interference patterns can be obtained. We then provide a theoretical explanation for this surprising result, and demonstrate that this explanation is also suitable for the results found for entangled light. Furthermore, we discuss the limitations of these types of super sub-wavelength interference patterns in quantum lithography.
We propose a two-photon subwavelength interference scheme for classical light in which multiple quantum-like entangled two-photon paths play an essential role. These entangled two-photon paths are introduced through a specially designed source composed of many point sources jjj with jjj's complex amplitude being a superposition of modes ${e}^{i{\ensuremath{\phi}}_{j}}$ and ${e}^{i{\ensuremath{\phi}}_{j}^{(1)}}$, where ${\ensuremath{\phi}}_{j}$ and ${\ensuremath{\phi}}_{j}^{(1)}$ are temporally random phases but satisfying ${\ensuremath{\phi}}_{j}+{\ensuremath{\phi}}_{j}^{(1)}={\ensuremath{\phi}}_{0}$, with ${\ensuremath{\phi}}_{0}$ being either a constant or a random phase in time. Interference between the entangled two-photon paths could lead to second-order subwavelength interference of an object put in front of the source plane. In a proof-of-principle experiment, by using a spatial light modulator to modulate the wave front of a coherent light, we have generated such a source and observed subwavelength interference of a double-slit mask via two-photon measurement.
The high-order intensity correlation function of NNN-photon interference with thermal light observed in a recent experiment [S. Oppel, T. B\"uttner, P. Kok, and J. von Zanthier, Phys. Rev. Lett. 109, 233603 (2012)] is analyzed. The terms in the expansion of the NthN\mathrm{th}Nth-order correlation function are put into three groups. One group contributes a homogeneous background. Both of the other two contribute $(N\ensuremath{-}1)$-fold super-resolving fringes. In principle they correspond to coherent and incoherent superpositions of classical optical fields, respectively. Therefore similar super-resolving fringes can be obtained without intensity-correlation measurements. We report the experimental results of the coherent and incoherent super-resolving diffraction fringes, which are observed directly in the intensity distribution. The $N\ensuremath{-}1$ sources in both the coherent and incoherent cases are set in certain definite positions. In the coherent case, moreover, the phase difference between two adjacent source fields is $\ensuremath{\pi}$. The fringe visibility is unity in the incoherent case, while it decreases as NNN increases in the incoherent case.
By generalizing the phase structure of the random-phase grating we recently designed [Opt. Express21, 14056 (2013)OPEXFF1094-408710.1364/OE.21.014056], we show that non-HBT type (synchronous position) two-photon grating interference can be obtained, which physically relies on groups of multiple indistinguishable two-photon paths modulated by the spatial distributions of phase modes. By properly selecting the random-phase structures, synchronous position subwavelength interference can be obtained, the period of which, in the two-photon interference domain, is decreased by a factor N (=3,4,5,6,…), depending on the slit number and random-phase structure, and the visibility of N-fold subwavelength interference fringes could be improved by increasing the slit number of the grating. The results show that modulation on two-photon paths via spatial arrangements of the phase modes offers the possibility to actively control the optical high-order coherence in the same optical scheme.