Monte-Carlo Radiative Transfer for a Suspension of
Homogeneous Spherical Particles.
Radiative transfer calculations are a versatile and efficient tool for
the calculation of the optical properties of inhomogeneous media. The play
an important role particularly in astrophysics (interstellar dust clouds)
and atmospheric physics (clouds) where the basic assumptions underlying
the formalism are excellently fulfilled. Applications to other media such
as biological tissue, powders, paints etc. also have a long history (namely,
the Kubelka-Munk theory). With the increasing popularity of laser applications,
radiative transfer calculations plays a more and more important role in
applied sciences. The applet presented on this Web-page demonstates raditive
transfer for the very simple (yet frequently encountered) case of a not
too thick layer consisting of a suspension of homogeneous spherical particles.
More than one particle species may be included in the calculations. Reflection
of radiation from the surface of the layer is not considered.
Basic assumptions of radiative transfer theory are:
Neglection of interferences requires that the typical inter-particle separation
is much larger than the wavelength and that the volumetric concentration
of the particles is small. These conditions can be relaxed in practical
applications: radiative transfer theory even works in powders, where the
particles are touching. With appropriate corrections, it produces good
results even for densely packed materials with volume concentrations of
the particles up to 50% (A. Kokhanovsky, Optics of Light Scattering Media,
Wiley-Praxis Series, Chichester, 1999). The particles should be substantially
larger than the wavelength in the latter case. If the particles' sizes
and inter-particle separations are comparable to or smaller than the wavelength,
radiatiative transfer theory is not applicable. Random medium formalisms
such as the Strong Property Fluctuation Theory (L. Tsang, J. A. Kong, T.
Shin, Theory of Microwave Remote Sensing, Wiley, New York, 1986; B. Michel,
A. Lakhtakia, J. Phys. D: Appl. Phys 32, 404-406, 1999) should be used
instead. For even smaller particles/separation distances, effective medium
formalisms are recommended (A. Lakhtakia et., Selected Papers on Linear
Optical Composite Materials, SPIE Milestone Series, Vol. 120, SPIE Press,
The particles in the suspension are well-separated
interference of light radiated from neighboring particles can be neglected
In Monte-Carlo radiative transfer calculations, a large number of "model
photons" (also called "weighted photons") propagate through the layer in
a random process which consists of a sequence of two steps:
Note that the "model photons" do not represent real photons. They are just
a numerical trick to solve the radiative transfer equation by means of
a random process.
the photon is scattered by a single particle. Additionally
the intensity (energy carried by the photon) is reduced by the amount absorbed
by the particle. The direction of the scattered photon is determined from
a random process, taking the phase function of the particle as probability
Straight propagation of the photon through the medium. The pathlength is
determined randomly, based on the extinction coefficient of the inhomogeneous
About the applet
With the applet you can trace the path of single model photons propagating
through an inhomogeneous layer. It is assumed that the layer consists of
of monodisperse spherical particles surrounded by vacuum (or air). This
means that no refraction and specular reflection occurs at the surface
of the layer.
The applet uses a vector radiative transfer model based on Mie scattering,
i.e., the full Stokes vector of the model photons is propagated through
the layer. The layer is parallel to the xy-plane and the light is assumed
to be initially unpolarized and perpendicularly incident on the layer.
The code is kept quite general for sake of upward-compatibility. With not
too much extra effort one can do full radiative transfer calculations.
How to use the applet
Enter the following parameters into the text fields:
Length units are arbitrary, but must be used consistently for the radius,
thickness and wavelength. Wrong input (non-numeric input, unphysical values
such as a zero wavelength or a negative thickness etc.) are detected by
radius of the particles
filling factor (volume portion) of the particles in the layer
real part n' of the refractive index of the particles
imaginary part n'' of the refractive index of the particles
thickness of the layer
wavelength of the incident radiation
Interested in radiative transfer? Don't hesitate to
Use the Run-button to display the path of a single photon in the x-z plane.
Press this button several times to display the paths of more than one photon.
The displayed traces of the photons' paths contain quite a bit of physical
Note that all photons start at the origin propagating in positive z-direction
(upwards on the screen). Scattering events are marked by crosses.
In addition to the photon path the optical depth tauExt and the scattering
depth tauSca are displayed on the plot window. These quantities are measures
for the strength of multiple extinction/scattering processes.
how often the photons are scattered,
whether mainly forward scattering takes place,
how much the incident beam is broadened
You may delete the plot window without harm. The next click on Run will
then cause a new plot window to be opened.