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Optical Microlithography XVIII

Quantum image-forming theory beyond diffraction limit

Naoki Fukutake

Published May 12, 2005• 2 citations
Publisher

Abstract

We formulate an image-forming optical theory of quantum lithography in which Entangled-photon pairs generated by spontaneous parametric down-conversion play an important role. Our optical system consists of an image-forming system, an illumination system with a second-order nonlinear medium, and two-photon absorbing materials. We evaluate the resolution of the quantum lithography system by using the optical transfer function and show a super-resolving condition which is, however, difficult to achieve.

DOI
10.1117/12.598590

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We formulate an image-forming optical theory of quantum lithography in which Entangled-photon pairs generated by spontaneous parametric down-conversion play an important role. Our optical system consists of an image-forming system, an illumination system with a second-order nonlinear medium, and two-photon absorbing materials. We evaluate the resolution of the quantum lithography system by using the optical transfer function and show a super-resolving condition which is, however, difficult to achieve.
It is well known that entangled-photon pairs generated by spontaneous parametric down-conversion exhibit a property that is analogous to that of ordinary incoherent lights. We formulate an image-forming optical theory of quantum lithography that comprehends the both conceptions of partial coherence and partial entanglement by extending a classical image-forming optical theory. Our optical system consists of an image-forming system, an illumination system with a second-order nonlinear medium, and two-photon absorbing materials. We evaluate the resolution of the quantum lithography system by using the optical transfer function and show a special case to be a super-resolving system.
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.
Relating to magnified image formation systems, we consider two processes in this work: quantum and classical incoherent imaging. There are similarities and differences between them. A similarity is that the optical transfer functions (OTFs) of both systems are obtained by calculating a correlation function. Concerning the differences, only two were found. The first difference is that, unlike classical incoherent imaging, quantum imaging with photon pairs generated by spontaneous parametric down-conversion is sensitive to the phase in the function which describes the object. The second difference is that the OTF of the quantum imaging system is a cross-correlation function of the OTFs of the signal and idler arms, while the OTF of the classical incoherent imaging system is an autocorrelation of the only pupil function existing in this case. We explore experimentally the second difference. We report the observation of a highly resolved image produced by a quantum light source in a setup where an incoherent light source is unable to spatially resolve the image of the object. The spatial resolution of the image is studied in a situation where the size of the object is magnified at the image.
One of the most surprising consequences of quantum mechanics is the entanglement of two or more distance particles. The 'ghost' image experiment demonstrated the astonishing nonlocal behavior of an entangled photon pair. Even though we still have questions in regard to fundamental issues of the entangled quantum systems, quantum entanglement has started to play important roles in practical applications. Quantum lithography is one of the hot topics. We have demonstrated a quantum lithography experiment recently. The experimental results have beaten the classical diffraction limit by a factor of two. This is a quantum mechanical two-photon phenomenon but not a violation of the uncertainty principle.
We present a theoretical framework for super-resolution imaging based on quantum coincidence detection. By comparing the classical Airy point spread function with the two-photon wavefunction of entangled photon pairs, we derive that coincidence detection yields an effective narrowing of the point spread function by a factor of two or the square root of two, depending on the maintenance of correlation. This model highlights the role of transverse phase accumulation in the image plane and predicts a quantum resolution limit that is half of the classical diffraction limit when correlation is maintained. The results offer clear insight into recent experimental demonstrations of quantum imaging beyond the classical diffraction limit.