



Bow
from 0.75mm diameter drops illuminated by a distant
point source. Supernumeraries like these are not
seen in nature because they are blurred by the finite
angular size of the sun and variations in drop size.
AirySim simulation.





When
a plane light wave interacts with a water drop the outgoing wave after
a single internal reflection is curved. If the shape of the wave is
somehow known then phase differences along it can be calculated and
thus the intensity of the resulting rainbow and its supernumeraries.
The English Astronomer Royal, George Biddell Airy (18011892), approximated
the scattered wavefront shape with a cubic form and developed an analytic
expression for the rainbow intensities in terms of what are now called
Airy integrals or functions. Airy's theory gives satisfactory predictions
of the observable features of white light rainbows(1) and is computationally
very considerably faster than the exact predictions of Mie theory.
AirySim precomputes
and stores Airy functions for a whole range of arguments using an
ascending series expansion(2). To compute a rainbow for a particular
drop size, wavelength and refractive index(3), appropriate values
of the Airy functions for each scattering angle are derived by interpolation
of the stored values or, where necessary, additional direct computation.
White light rainbows are obtained by repeating the calculation for
closely spaced wavelengths between 380 and 700 nm and summing the
intensities at each angle after weighting them by a spectral solar
radiance(4).
When simulations
from non monodisperse droplets are required AirySim uses a droplet
population function normally distributed in radius. All the above
calculations have to be repeated for the different droplet radii
and the intensities summed.
Rainbows from
the sun rather than plane parallel light are derived by convolving
the angular intensities with a disk intensity function.
Representation
of colours is always problematic. AirySim uses the CIE and Bruton(5)
colour models written for IRIS.
Regrettably, AirySim is not available for download. The intentions were good bit I did not have the time to construct the necessary user friendly interface.

(1) 
Lee,
R. L., "Mie theory, Airy theory, and the natural rainbow,"
Applied Optics 37, 15061519 (1998).
http://www.usna.edu/Users/oceano/raylee/papers/RLee_papers.html

(2) 
Abramowitz
M. & Stegun I.A., Handbook of Mathematical Functions, Dover.

(3) 
Water
refractive indices are from IAPWS (International Association for the
Properties of Water and Steam) which are in turn based on P. Schiebener,
J. Straub, J. M. H. L. Sengers, J. S. Gallagher, "Refractive
index of water and steam as function of wavelength, temperature and
density," J. Phys. Ch. R.,19, 677717, (1990).

(4) 
Spectral
solar radiances were those used by Raymond Lee(1) and kindly supplied
by him in detailed tabular form.

(5) 
Dan
Bruton of Stephen F. Austin State University, "Color Science",
http://www.physics.sfasu.edu/astro/color.html




