Transport of Intensity Imaging

The transport of intensity equation (TIE) describes the flow of intensity along the optical axis as light waves propagate. By capturing just two intensity images at different z planes and using them to compute dI/dz, one can solve for the phase and amplitude of the wave.

We are developing new methods and experimental realizations for measuring quantitative phase profiles from intensity measurements.

(1) Phase from chromatic aberrations: Here we recognize that wavelength and distance z appear always together in the propagation equation, and thus are interchangable. Thus, one can take a single color image in lieu of multiple images at different distances. This means that phase can be computed in a single shot, without moving the camera. This allows real-time, fast and simple phase imaging that uses the aberrations of the imaging system to obtain phase contrast.

This technique is useful for real-time surface measurements of reflective surfaces, such as the deformable mirror shown above, as well as for transparent biological samples that often exhibit significant phase contrast, but very low amplitude contrast.

(2) Transport of Intensity imaging with higher order intensity derivatives: Alternatively, if one moves the camera to take multiple images, it is not much more difficult to take an entire stack of images at different planes, and the more images, the better the phase result should be. By using this stack of images, we have shown that one can cancel the higher order artifacts that plague the TIE method and are a result of using finite difference approaches to computing the intensity derivative. In addition, by using many images, one gets better phase contrast across all spatial frequencies and is able to reduce the effects of noise.

(3) Quantitative phase imaging in a Volume Holographic Microscope (VHM): Combining work done on VHM and the TIE, we are able to make quantitative phase measurements in a VHM system from it’s intensity sub-images. Each sub-image is an intensity measurement at a different z plane. And therefore, VHM systems with many focal planes are able to capture large intensity data sets in a single-shot, which can be used with higher order TIE (described above) to recover higher-order accuracy phase results.

(4) Phase imaging with noisy intensity measurements using a Kalman filter: When the measured intensity images are very noisy, most phase imaging methods fail to work. Here we have developed a version of the Kalman filter that is able to find the near-optimal result when given a lot of intensity measurements at different planes, that are corrupted by severe noise.