Optical array probes (OAPs) are classical instrumental means to derive shape, size, and number concentration of cloud and precipitation particles from 2-D images. However, recorded 2-D images are subject to distortion based on the diffraction of light when particles are imaged out of the object plane of the optical device. This phenomenon highly affects retrievals of microphysical properties of cloud particles. Previous studies of this effect mainly focused on spherical droplets. In this study we propose a theoretical method to compute diffraction patterns of all kinds of cloud particle shapes in order to simulate the response recorded by an OAP. To check the validity of this method, a series of experimental measurements have been performed with a 2D-S probe mounted on a test bench. Measurements are performed using spinning glass discs with imprinted non-circular opaque particle shapes.

In Earth's atmosphere, the evolution of clouds is highly dependent on
interactions of a number of dynamical, radiative, and microphysical processes

Diffraction patterns of spherical water droplets recorded by OAPs, using
laser wavelengths that are small compared to droplet sizes, have been
thoroughly studied theoretically and experimentally

In this study, we propose a theoretical method to compute diffraction
patterns of all kinds of cloud particle shapes. The validity of the method is
checked with a series of measurements using one of the newest OAPs – the
two-dimensional stereo (2D-S) probe

In Sect.

When light (a laser beam for OAPs) illuminates a cloud particle, a shadow
image can be observed on a screen at the rear of the particle. The formed
image depends on the diffraction, refraction, and transmission of light by
the particle. As a first approximation, it is convenient to neglect the
refraction and transmission of light by the particle, i.e., considering the
cloud particle as an opaque particle. It should be noted that ice particles
which allow significant light transmission will have additional sources of
error that are not captured in this experiment. A further assumption is that
the diffraction pattern produced by an opaque particle is accurately
described by the diffraction pattern produced by an opaque planar object
representing the cross section of the particle. Laboratory studies showed
that these approximations work well for out-of-focus transparent spherical
particles

The diffraction pattern of an opaque planar shape can be computed with
different theoretical methods.

Fast Fourier transform (FFT) can be utilized for the numerical implementation
of the AST. Also, FFT-based algorithms are easy to implement and effective.
That is why the AST is extensively employed in different domains, including
simulations of diffraction patterns

As noted by

Theoretical diffraction pattern simulated for an opaque disc with
diameter

We notice that a bright spot, called Poisson's spot, appears at the center of the diffraction pattern shadow image. The orange dashed line represents the 50 % intensity threshold generally applied in binary (monoscale) OAP probes. For this specific case, it can be seen that an OAP operating with this threshold will produce a “donut” image with an external diameter that exceeds by 30 % the true disc diameter.

Figure

Intensity level of the diffraction pattern of an opaque disc in

With increasing distance

The two-dimensional stereo (2D-S) probe

Therefore, distance

Considering opaque discs crossing the 2D-S laser beam, Fig.

Depth of field (DoF) limit (distance

A particle with

In the following third section, we compare theoretical diffraction patterns
of different opaque shapes with experimental measurements of the 2D-S probe.
Therefore, several spinning glass discs with various chrome opaque particle
shapes imprinted on the glass disc surfaces were used
(Fig.

Theoretical and measured 2D-S diffracting patterns of

Figure

In each column, on the left-hand side are shown the theoretical 2D-S records
and on the right-hand side are shown the images recorded by the 2D-S
experimentally. The theoretical 2D-S records are obtained by AST-FFT
simulations (Sect.

Figure

As Fig.

Comparison between theory and measurements again shows very good agreement.
Also for this crystal geometry, diffraction can produce patterns that are
very different compared to the initial shape. In addition, we notice that
these patterns look very different than those found in
Fig.

Figure

As Fig.

Once again, the very good agreement between simulated and recorded images is
striking. The diffraction pattern for one individual capped columnar particle
can adopt many different image shapes, as can be seen for example for the

As a particle moves away from the object plane, we notice that its image
becomes more and more roundish regardless of its initial shape. The
information of the real shape of the particle ends up being lost as the
diffraction patterns progressively adopt a more circular form. This is
particularly striking on videos. This means that any particle shape far from
the object plane produces a more and more circular diffraction pattern which
no longer allows us to identify the original shape of the respective
particle. See for example the theoretical

Another interesting remark based on these results is that an out-of-focus
image of a distinct particle shape can closely resemble another particle of a
very different shape. As an example, note that the diffraction pattern of a

In this section, we present some diffraction simulation results for four chosen particle shapes to illustrate the evolution of the particle diameter, including uncertainty evaluation. Our purpose here is neither to present an exhaustive list of results related to each shape nor to quantify the uncertainty of the probe in an absolute manner.

The size of the particle from a 2-D image has no absolute definition. Several
definitions are used in the literature with different pros and cons depending
on the objective of the study. In the study presented here, we illustrate
results with two commonly used particle size definitions: the
surface-equivalent diameter

Figure

Evolution of the theoretical

Dashed lines show the true

At first, we discuss diffraction simulation results of short and elongated
columns presented in Figs.

Secondly, the two capped column-type particles are discussed. With increasing

Furthermore, for both columnar and capped columnar particles, it is evident
that the discrete pixel effect (shadow areas) is almost negligible with
respect to the diameter variability along the

Finally, Table

True

In this section, we compare particle size distributions retrieved
theoretically from diffraction pattern simulations and experimentally
measured by the 2D-S probe. For the measurements, a spinning disc
(Fig.

Figure

Each of the three orientations (0, 45, and 90

Theoretical (measured) relative uncertainty

Figure

We stated in Sect.

We presented in this study a first comparison of theoretical diffraction
simulations of non-spherical cloud particles and respective image responses
of OAP probes. First, the angular spectrum method has been applied to obtain
diffraction patterns of spherical and non-spherical particles when viewed at
specific distances from the object plane. For exemplary cloud particle
shapes, diffraction simulations help in studying how the diameter retrieved
from 2-D binary images is impacted by the distance from the object plane
where the particle crosses the laser beam. Furthermore, we compared
theoretical results with experimental measurements made with a 2D-S probe.
The main results are the following.

The diffraction image formed by an opaque planar particle, illuminated
perpendicularly by a monochromatic coherent homogeneous plane wave, at a
distance

Circular particles with diameters larger than 806

Circular particles with diameters larger than 109

Theoretical diffraction simulations allow us to estimate DoF limits (as
from Figs.

Diffraction images of out-of-focus particles are sometimes very similar to other in-focus particle shapes. As an example, we observe that an out-of-focus elongated columnar ice particle can be interpreted as an in-focus capped columnar ice particle. An out-of-focus capped column can also be viewed as a droplet faintly out-of-focus.

In general, diffraction images of all kinds of particle shapes
consecutively lose their real shape information with increasing distance

Due to the finite pixel size of the probe and the 50 % occultation
threshold, there is an uncertainty in the particle size measurements, even
when

The intercomparison of theoretical and experimental

The good agreement between the simulated and measured diffraction patterns
(see Figs. 5, 6, 7, and B1) suggests that the laser beam of the used 2D-S
probe is well collimated and the use of the plane-wave approximation is well
founded. Future investigations, especially concerning grayscale thresholds,
could take into account properties of the laser beam and the optical system.
For example, angular spectrum theory was used in the work by

This study suggests that the incorrect particle sizing of cloud particles by
OAPs is predominantly due to the diffraction effect in the out-of-focus
region. Reducing the distance between the probe arms allows one to reduce
diffraction effects, but simultaneously reduces the sampling volume. In order
to reduce the sizing uncertainty, it would be extremely useful to get a
direct and independent measure of the distance

The data from this study can be obtained by contacting the corresponding author of this article.

In this Appendix, we use the same terminology as has been used in Sect. 3.10
of the classic textbook by

In this work, the amplitude transmittance function

In the Fourier domain, the angular spectrum of

Finally, a simple low-pass filter can be used to remove spurious noisy high frequencies.

As Fig.

Image sequences of diffraction patterns for particles at
increasing distance from the object plane have been added as a Supplement.
All videos are from the series “Simulations of cloud particle diffraction
pattern” (Vaillant de Guélis, 2019) available on the TIB AV-Portal at

TVdG wrote the manuscript. TVdG and VS performed the theoretical part. CG and BL conceived the test bench. TVdG made the experimental measurements. RD wrote the script to read the 2D-S raw data file. All the authors contributed discussion and feedback essential to the study.

The authors declare that they have no conflict of interest.

We would like to thank the FEMTO-ST Institute (UMR 6174) for conceiving the spinning glass discs. This work has been financially supported by the French government space agency CNES.

This paper was edited by Wiebke Frey and reviewed by Darrel Baumgardner and one anonymous referee.