Illinois researchers derive new bounds on quantum improvement of Fourier optics

Did you know new research ideas can come from old textbooks? For Yunkai Wang, then an Illinois Physics graduate student, reading a chapter in Introduction to Fourier Optics by Joseph W. Goodman inspired an innovative breakthrough that resulted in a collaborative paper with his advisor, Virginia (Gina) Lorenz. The paper, “Fundamental limit of bandwidth-extrapolation-based superresolution” was published in Physical Review A a few months ago.

As a quantum optics researcher, Wang was studying quantum metrology, which uses quantum technologies to better image objects. At the same time, Wang was looking into Fourier optics, which breaks apart an image into component parts, which together are referred to as the spatial spectrum of the object being imaged. “Fourier optics has a well-developed theory; it has textbooks, and much existing discussion and literature”, Wang explains, “but once we introduce quantum technologies, there are new tools which can be integrated with this field. I felt that there might be an improvement to be made by revisiting many existing theories that have been developed in Fourier optics”. By combining the two ideas, Wang and Lorenz were able to show new limits for how well objects may be imaged.

Specifically, Wang’s interest was piqued when he came across a discussion of “superresolution” in Fourier optics. Wang had previously encountered another concept referred to as “superresolution” while studying quantum metrology. The goal of quantum metrology is to better measure certain physical parameters, while making use of quantum concepts to improve results beyond what is possible classically. One use of quantum metrology is to resolve two point-sources: two objects from far away can seem to be the same object if their images overlap; better resolution can pinpoint the locations better and separate the two images. Wang wanted to find the connection between the two ideas of superresolution. “The basic idea behind [the] method developed in Fourier optics is different from our previous papers,” said Wang in reference to his previous quantum-focused work. “The question then became: How are these two methods related and how do we improve this method developed in Fourier optics?”

In the paper, the older idea is referred to as “bandwidth extrapolation.” This title refers to the fact that superresolution here means using some smaller subset (lower-frequency) of the image’s special spectrum information to extract other (high-frequency) information. This latter information is missing due to a finite-size imaging system which introduces cut-offs in an image’s spectrum. Previous research, which did not consider quantum estimation methods, had set some limits on the efficacy of this method of improving image resolution. The image quality can be quantified by considering how well each of its Fourier components can be estimated. “In exploiting quantum estimation theory for the bandwidth extrapolation method”, which potentially yields a better estimation of these components, “we were able to derive a fundamental limit,” Wang emphasizes.

The paper treats two cases. The first of these is called the ‘small-source limit’, in which the size of a source is much smaller than the resolution of the image. The simplest case of this problem considers a single lens; there, the resolution size of the image is simply determined by the diameter of the lens. However, Wang and Lorenz consider an interferometric imaging set-up, in which different lenses working together yield an effective lens, whose diameter then determines the resolution size. Wang underscores that “this is not a necessary choice”, but that this simplifies the calculation, enabling an analytical result. “The set-up can still capture the main features of this method of treating bandwidth extrapolation”, he adds. In the small-source limits, the researchers derive new bounds and show that considering quantum estimation methods improves the sensitivity of the bandwidth extraction significantly “compared to a naïve measurement strategy.”

The other limit is the "large-source limit", in which the size of the source is similar to, or even larger than, the resolution limit. In this case, although the source is larger, one might hope to still be able to resolve information about the details of the source. Wang uses a sheet of paper as an example: “Even though the size of the paper might be larger than our resolution image, we might still want to figure out what is written on the piece of paper.” Unfortunately, as Wang and Lorenz show, in this limit the direct application of quantum estimation theory does not yield any improvements. “It’s an interesting discussion”, Wang maintains. “It’s something that hadn’t been considered before.”

Wang, now a Joint Postdoctoral Fellow at the Perimeter Institute (PI) and the Institute for Quantum Computing (IQC), based in Waterloo, Ontario, explains that there is lots of interesting and inspiring discussions left to be had regarding the unique approach the paper takes in applying quantum ideas to enhance existing classical research. “There is more we can think of beyond quantum estimation theory,” he concludes, giving several potential connections between Fourier optics and quantum information theory, including image compression, and considering “collective measurement over multiple copies of states, as opposed to separable measurement on single copies of states.”