Matthew J. Berg - Solving the Inverse Problem

Multi-Wavelength Digital Holography and the Inverse Problem

There are two elements to light scattering analysis for aerosol-particle characterization First, the one- or two-dimensional (2D) scattered intensity pattern produced by an illuminated particle is measured. This can now be done very well on the single and multi-particle level using a variety of techniques. Second, the pattern must be interpreted to infer the desired particle characteristics, and herein lies much difficulty. This is because there is no general mathematical relationship between a measured pattern and the (unknown) particle characteristics. Thus, the interpretation step must involve a priori information or questionable assumptions that often relate to a presumption of the particle shape. There is generally no guarantee that the particle size and shape one associates with a pattern is in fact the correct one, which is the essence of the inverse problem.

With digital holography, however, the particle's size, shape, and orientation may be known unambiguously provided that the image resolution of the sensor is sufficient. The drawback is that the scattering pattern is not readily known in holography since what is measured derives from the superposition of the unscattered and scattered light rather than the scattered light alone. Knowledge of the scattering pattern is nevertheless important as it describes how the particle redistributes light, which, e.g., is key to quantifying the radiative impacts of atmospheric aerosols. The pattern also contains information about the particle's material composition, which is also not directly available from the holographic image at this time.

Figure 1: Optical arrangement used to measure 2D light scattering patterns of single free-flowing aerosol particles simultaneous with holographic images of the particles. See [here] for more.

In recent [work], we have achieved a proof-of-principle experiment where the digital hologram of a single particle is measured simultaneous with the particle's 2D light-scattering pattern around the forward direction. This is done by illuminating the a free-flowing aerosol particle with light at two wavelengths, red and green, and recording the hologram and pattern on a color CCD sensor. See Fig. 1 above. Following generation of the particle image, the pattern can then be unambiguously and quantitatively associated with the true size, shape, and orientation of the particle as shown in Fig. 2 below. Thus, in a sense, this work constitutes a laboratory-based solution to the inverse problem at least as it relates to size and shape determination. While there are examples of seemingly similar experiments in the literature, this work is unique in that the pattern and image are obtained from the same particle at the same time. With this capability, the validity of scattering-pattern-only characterization techniques can be directly assessed and their capabilities improved. For example, many optical particle-counters/sizers assume that the observed particles are spherical so that Mie theory may be used to retrieve particle size despite the fact that the particles under study are likely not spherical.

Figure 2: An example of data from the experiments [here]. The top row shows two-dimensional light scattering patterns in log scale and false color for single particles. The bottom row shows the image of each particle derived from the hologram measurement at the same time as the pattern is measured.

More recently, we tested the idea that a particle's scattering pattern could be extracted from the holographic image via application of Huygens’s principle. If possible, there would be no need to measure the scattering pattern if what one wants to know is the scattering in the near-forward scattering directions. With the benefit of having measured scattering patterns from the experiments in Figs. 1 and 2 above, we apply Huygens' principle to a binarized version of the particle's image resulting from the digital hologram. An example is presented in Fig. 3 beolw where the aerosol particle is a single sphere. Further examples of the concept applied to other particle types can be found [here] where the efficacy of the approach is demonstrated.

Figure 3: Generation of an approximate scattering pattern for a spherical aerosol particle from a digital hologram via Huygens’s principle. The aerosol particle is a single 50 um diameter glass sphere from the experiment in Fig.2. The hologram is shown in (a), and the gray-level and binarized particle silhouettes are shown in (b) and (c), respectively. (d) The approximated 2D pattern, shown in false-color in log scale, is azimuthally averaged and compared to (e) the azimuthal average of the measured pattern. Here, the hole in the SFM causes aberration and loss of the measured pattern over the smallest angular range in (e), for angles < 0.6 degrees. Note that the bright spot in (a) is “bleed through” from the sensor’s green channel, and the Huygens data is normalized in (d) but not in (e).