Quantum-Inspired Method Reveals Details Hidden in Noise

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Researchers at the University of Warsaw's Faculty of Physics with colleagues from Stanford University and Oklahoma State University have introduced a quantum-inspired phase-imaging method based on light intensity correlation measurements that is robust to phase noise.

The new imaging method can operate even with extremely dim illumination and can prove useful in emerging applications such as infrared and x-ray interferometric imaging and quantum and matter-wave interferometry.

Whether taken with a smartphone or an advanced microscope, photographs measure the intensity of light, pixel by pixel. Light is characterized not only by its intensity, but also its phase. Transparent objects can become visible if the phase delay of light they introduce can be measured.
Researchers from the University of Warsaw and their collaborators developed a noise-resistant phase imaging technique with intensity correlation. Courtesy of the University of Warsaw.
Researchers from the University of Warsaw and their collaborators developed a noise-resistant phase imaging technique with intensity correlation. Courtesy of the University of Warsaw. 

Phase contrast microscopy, for which Frits Zernike received a Nobel Prize in 1953, brought about a revolution in biomedical imaging due to the possibility of obtaining high-resolution images of various transparent and optically thin samples. The research field that emerged from Zernike's discovery includes modern imaging techniques such as digital holography and quantitative phase imaging.

“It enables label-free and quantitative characterization of living specimens, such as cell cultures, and can find applications in neurobiology or cancer research,” said Radek Lapkiewicz, head of the Quantum Imaging Laboratory at the University of Warsaw's Faculty of Physics.

However, there is still room for improvement. “For example, interferometry, a standard measurement method for precise thickness measurements at any point of the examined object, only works when the system is stable, not subject to any shocks or disturbances. It is very challenging to carry out such a test, for example, in a moving car or on a shaking table,” said Jerzy Szuniewcz, a doctoral student at the University of Warsaw's Faculty of Physics.

Researchers from the Faculty of Physics at the University of Warsaw with colleagues from Stanford University and Oklahoma State University decided to tackle this problem and develop a new method of phase imaging that is immune to phase instability.

The work takes inspiration from the experiments of Leonard Mandel in the 1960s. Mandel and his group demonstrated that even when interference is not detectable in intensity, correlations can reveal its presence. With this in mind, Lapkiewicz and his collaborators sought to investigate how intensity correlation measurements could be used for phase imaging.

In a correlation measurement they looked at pairs of pixels and observed whether they became brighter or darker at the same time. The researchers showed that these measurements contain additional information that can’t be obtained with a single photo, i.e., intensity measurement. With that in mind, they demonstrated that in interference-based phase microscopy, observations are possible even when standard interferograms average out losing all phased information and without fringes recorded in the intensity, Lapkiewicz said.

“With a standard approach, one would assume that there is no useful information in such an image. However, it turns out that the information is hidden in the correlations and can be recovered by analyzing multiple independent photos of an object allowing us to obtain perfect interferograms, even though the ordinary interference is undetectable due to the noise,” said Lapkiewicz.

In the experiment, the light passing through the target phase object is superposed with a reference light. Then, a random phase delay is introduced between the object and the reference light beams to simulate a disturbance obstructing the standard phase imaging methods. Consequently, no interference is observed when the intensity is measured, meaning that no information about the phase object can be obtained from intensity measurements.

“However, the spatially dependent intensity-intensity correlation displays a fringe pattern that contains the complete information about the phase object. This intensity-intensity correlation is unaffected by any temporal phase noise varying slower than the speed of the detector (~10 nanoseconds in the performed experiment) and can be measured by accruing data over an arbitrarily long period of time,” said Jerzy Szuniewicz, the first author of the work.

That, he said, is a game changer. Longer measurements mean more photons, translating to higher accuracy. Put simply, recording a single frame would produce no useful information about what the object being studied looks like.

“Therefore, first we recorded a whole series of such frames using a camera and then multiplied the measurement values at each pair of points from every frame. We averaged these correlations, and recorded a full image of our object,” said Szuniewicz.

According to second author Stanislaw Kurdzialek, there are many ways to recover the phase profile of an observed object from a sequence of images. “However, we proved that our method based on intensity-intensity correlation and a so-called off-axis holography technique provides an optimal reconstruction precision.”

A phase imaging approach based on intensity correlation can be widely used in very noisy environments. The new method works with both classical (laser and thermal) and quantum light. It can also be implemented in the photon counting regime, for example using single photon avalanche diodes.

“We can use it in cases where there is little light available or when we cannot use high light intensity so as not to damage the object, for example a delicate biological sample or a work of art,” said Szuniewicz.

Lapkiewicz expects the technique to broaden the prospects of phase measurements, including emerging applications such as infrared and x-ray imaging and quantum and matter-wave interferometry.

The research was published in Science Advances (

Published: January 2024
In optics and photonics, "phase" refers to a property of electromagnetic waves, such as light, that describes the position of a wave at a given point in time within its oscillation cycle. More specifically, it indicates the position of a wave relative to a reference point, typically the starting point of a cycle. When discussing phase in optics, it's often described in terms of the phase difference between two waves or the phase of a single wave. The phase difference between two waves is the...
Infrared (IR) refers to the region of the electromagnetic spectrum with wavelengths longer than those of visible light, but shorter than those of microwaves. The infrared spectrum spans wavelengths roughly between 700 nanometers (nm) and 1 millimeter (mm). It is divided into three main subcategories: Near-infrared (NIR): Wavelengths from approximately 700 nm to 1.4 micrometers (µm). Near-infrared light is often used in telecommunications, as well as in various imaging and sensing...
The term quantum refers to the fundamental unit or discrete amount of a physical quantity involved in interactions at the atomic and subatomic scales. It originates from quantum theory, a branch of physics that emerged in the early 20th century to explain phenomena observed on very small scales, where classical physics fails to provide accurate explanations. In the context of quantum theory, several key concepts are associated with the term quantum: Quantum mechanics: This is the branch of...
The study and utilization of interference phenomena, based on the wave properties of light.
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