# A Hiccup in Optical Position Estimation?

In an optical imaging system, the spiral wavefront of elliptically polarized light from a point emitter hits the lens at a slight angle, leading to potential errors in the estimated position of the emitter. Researchers from the University of Innsbruck and TU Wien mathematically estimated the effect’s magnitude and demonstrated it for several point emitters. [Image: © IQOQI Innsbruck/Harald Ritsch]

As optical imaging techniques have gotten more and more sophisticated, they’ve given researchers the ability to measure the position of atoms, molecules and nanoparticles with an almost unearthly precision—on the order of nanometers. And for some applications, such as super-resolution microscopy and quantum experiments, position errors even on the scale of the wavelength of light can make a big difference.

Now, a team of Austrian scientists has demonstrated that, for tiny point emitters that give off elliptically polarized light, optical estimates of particle position can be off by hundreds of nanometers (Nat. Phys., 10.1038/s41567-018-0301-y). The effect, the team argues, is large enough to have implications for interpreting images from super-resolution microscopy, and for the measurement and manipulation of certain quantum systems, such as trapped ions.

## Gaming the point-spread function

The maximum, diffraction-limited angular resolution of an optical imaging system is λ/D, where λ is the wavelength of the imaging light and D is the system’s aperture diameter. But the position of particles with angular diameter even smaller than this limit can still be determined. That bit of magic is accomplished by fitting the imaging system’s point-spread function (PSF)—the mathematical expression describing the system’s response to a diffracting point source—to the indistinct image of the subwavelength emitter. In principle, the subwavelength particle’s position can be estimated as the centroid, or weighed center, of the blob-like PSF in the image plane.

The problem, the Austrian research team notes, is that the light coming from the emitter may be elliptically polarized, containing coupled spin and orbital angular momentum. (The spin angular momentum is tied to the light’s circular polarization, and the orbital angular momentum is associated with light with helical or “twisting” wavefronts.) Because of the details of the spin-orbit coupling, light from an elliptically polarized emitter hits the imaging aperture not head-on but in a sort of glancing blow—leading, mathematically, to a potential positioning error as high as λ/2π.

“The elliptical polarization causes the wavefronts of the light to have a spiral shape and to hit the imaging optics at a slight angle,” Yves Colombe, a member of the research team from the University of Innsbruck, said in a press release. “This leads to the impression that the source of the light is somewhat off its actual position.”

## Barium ions and gold nanoparticles

The notion that photons might seem to originate from a point offset from the actual emitter was first suggested by Charles G. Darwin (the grandson of the 19th-century naturalist) more than 80 years ago. But it’s only recently that optical systems have arisen with sufficient precision to test the prediction.

To perform that test, the Austrian scientists, including researchers both from Innsbruck and from Technische Universität (TU) Wien, measured the apparent position of two objects—a fluorescent barium ion captured in a radio-frequency (Paul) trap; and a 100-nm-diameter gold nanoparticle carefully positioned at the center of a glass sphere, and illuminated by a 685-nm laser beam. In particular, they assessed how the apparent positions changed with changes in the dipole polarization, which includes coupled spin and orbital components. They used polarization beamsplitters or wave plates to select or adjust polarization of the imaging signal, and captured the images on CCD cameras.

The team found that, when the sign of the polarization was changed, the apparent positions of the particles derived from fitting the system PSF to the image were displaced by as much as 158 nm for the barium ion (for an imaging system with a numerical aperture of 0.4) and 146 nm for the nanoparticle (for an imaging system with a numerical aperture of 0.61). Those displacement magnitudes, the researchers noted, are “comparable to the optical wavelength.”

## Implications for super-resolution imaging—and more

Systematic deviations of that magnitude could “add up to a considerable measurement error in many applications,” Stefan Walser, another team member from TU Wien, argued in a press release accompanying the work. “Super-resolution light microscopy, for example, has already penetrated far into the nanometer range, whereas this effect can lead to errors of several hundred nanometers.” The fuzzy nature of positioning could thus, in principle, cause distortions in the imaging of biological structures using super-resolution techniques, according to the researchers.

Intriguingly, the authors conclude by noting that the demonstrated effect is relevant not just for optical waves, but for “any kind of wave carrying transverse orbital angular momentum.” That means that position measurements via radar or sonar might also have their own systematic errors associated with analogous effects. Indeed, the team notes, the polarization-related shift could “even alter the apparent position of astronomical objects detected through their emission of gravitational waves.”

Publish Date: