Multiple Light Scattering. Tables, Formulas, and Applications. Volume 1

317 Pages·3.839 MB·English

by  Hulst Van De H.C.

MULTIPLE LIGHT SCATTERING
Tables, Formulas, and Applications
Volume 1
H. C. VAN DE HULST
Astronomical Observatory
University of Leiden
Leiden, The Netherlands
1980
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Library of Congress Cataloging in Publication Data
Hulst, Hendrik Chris toff el van de.
Multiple light scattering
Includes bibliographies and index.
1. Light—Scattering—Handbooks, manuals, etc.
2. Radiative transfer—Handbooks, manuals, etc.
I. Title.
QC427.6.H84 535\4 79-51687
ISBN 0-12-710701-0 (v. 1)
PRINTED IN THE UNITED STATES OF AMERICA
80 81 82 83 9 8 7 6 5 4 3 2 1
• Preface
The play of radiation by repeated scattering in a cloud layer or any other
slab of particles poses a problem that is common to atmospheric physics,
astronomy, ocean optics, and branches of industrial research. Methods for
solving this problem in diverse situations have been known for decades but
their complexity has given the subject the reputation of being accessible only to
specialists.
This book is aimed at the nonspecialist, e.g., an expert in an applied field,
who needs a result from multiple scattering theory but does not wish to spend
excessive time in solving it himself or searching the very extensive literature.
Numerical results form the core of these two volumes. Since users from
diverse fields should be served, the tabulated quantities are named by their
physical meaning, e.g., reflection function, gain, diffusion pattern, net flux, but
are presented in the form of functions of a few dimensionless parameters. Most
tables have five-figure accuracy in order to enable readers to use them for check­
ing their own computer programs. The graphical illustrations have been chosen
to serve as a quick orientation and also to highlight key phenomena such as
asymptotic behavior.
Special cases such as the limits adopted for each quantity for conservative
scattering (a = 1), or in a semi-infinite atmosphere (b = oo), or at large depth
(τ ρ 1) have been included in each tabulation. The same is true for moments
and bimoments of the functions of the angles of incidence and emergence.
The formulas expressing these results show a similar ramification of special
cases and asymptotic forms. For clarity and ease of access, they have been
arranged, where possible, in a "Display," which is a collection of formulas in
tabular form. Derivations have been kept to a minimum. They are presented in
ix
χ Preface
a form emphasizing the physical content and the use of certain intermediate
results. Only rarely does an intricate derivation require the use of numbered
equations.
Although the author's prime intention is to present known results, new
discoveries or new light shed on the meaning and use of known forms was un­
avoidable. The major findings have been published in scientific journals and
several have come into general use. Subjects like doubling, similarity relations,
reduction to Η functions, and, generally, the interpretation of mathematical
results in physical terms, are presented here in their proper context.
The volumes have a strict organization: Part I on general relations and
Part II on isotropic scattering (Volume 1), Part III on anisotropic scattering
and Part IV on applications to selected fields (Volume 2). The division of parts
into chapters again follows a strictly logical scheme as the table of contents for
each volume shows.
What I started as a sideline has become a major project. This would not
have been possible without the help and encouragement of a great many
people. Among this long list I wish to record my special gratitude to K. G. Gross­
man and J. W. Hovenier for their support throughout the work and to W. M.
Irvine and V. V. Ivanov, whose enthusiasm helped the project gain momentum
in the early years.
• Contents of Volume 2
Part III ANISOTROPIC SCATTERING
10 Phase Functions
11 Results for the Henyey-Greenstein Phase Function,
Unbounded and Semi-Infinite Medium
12 Other Phase Functions, Semi-Infinite Atmospheres
13 Henyey-Greenstein Functions, Results for Finite Layers
14 Results for Other Phase Functions, Finite Layers
15 Polarization and Azimuth-Dependent Terms
16 Rayleigh Scattering
Part IV SAMPLE APPLICATIONS
17 Photon Optical Paths and Absorption Lines
18 Planets
19 Scattered Light in the Earth's Atmosphere
20 Miscellaneous Applications
INDEX
xi
1 • Concepts, Terms, Notation
1.1 DIRECTIONS FOR USE
Those who wish numerical results for homogeneous slabs with anisotropic
scattering should turn to Chapters 11 and 13 (Volume 2).
Those who wish numerical results for homogeneous slabs with isotropic
scattering should turn to Chapters 8 and 9.
Those who wish to avoid certain common traps should glance over this
chapter.
Those who have a particular application in mind should see if Chapters
18 to 20 contain anything of interest to them (Volume 2).
All others should consult the Table of Contents and/or the Index.
1.2 SOME HARD CHOICES
It is unavoidable that in a subject so extensively studied different conventions
on terms and notation have come into use. The following comments are intended
to warn the reader of when different conventions might lead to confusion and to
explain why I have made a particular choice.
Albedo (= single scattering albedo). Confusion may arise from this
notation. The symbol w (curly pi) used by Chandrasekhar is too exotic. Most
authors have, perhaps unintentionally, changed it to ω or ω . Sobolev and his
0
school use λ. The neutron transport people use c. In marine optics it is written
as the ratio b/c. I have chosen a, the first letter of albedo. See also phase function.
3
4 1 Concepts, Terms, Notation
Another source of confusion may be the use of the word albedo in different
meanings. In reactor research it is used for what we call diffuse reflection.
Throughout astrophysics and geophysics it is also used for the fraction of the
incident energy returned from a diffusely reflecting surface. We concur with this
use and call it plane albedo, symbol UR. Similarly, the fraction of energy re­
turned for uniform radiance from the entire hemisphere is called spherical
albedo, symbol URU or A*.
Angles with the normal. The problem is from which side to measure them,
i.e., which sign to give the cosine. The three systems in use are shown side by
side in Display 1.1. Most physicists prefer system lb. The astrophysical tradition
la, whereby optical depth is measured into a stellar atmosphere but angle 0 is
used for the normal direction out of the same atmosphere, is still strong enough
to dominate a good part of the literature but it is awkward. In the recent astro-
physical literature system lb is fairly common.
The use of system II is mostly limited to the radiation field outside slabs.
There it has the advantages of never needing minus signs, having equal symbols
for reciprocal quantities, and having very simple matrix formulations. I have
chosen to use system II with symbol μ wherever possible, and system lb with
symbol u where necessary. The same convention has been adopted by Hansen
and Travis (1974).
Gain. This concept, in particular the point-direction gain, has two ad­
vantages. It is applicable without change of terminology or notation in two
reciprocal experimental situations and it is virtually foolproof against normaliza-
DISPLAY 1.1
Sign Conventions for Cosine of Angle with Normal
System la System lb System II
Range of angle 0-π 0 π 0-π/2
Range of cosine 1 to -1 1 to -1 1 toO
Defined at depth anywhere anywhere top or bottom
surface
Angle 0, cosine 1 is decreasing optical increasing optical normal, in or out
along depth depth
Used in work of Chandrasekhar; most Sobolev; most Chandrasekhar; my
astrophysics texts physics texts earlier papers
Used in this book not at all when discussing when discussing slab
internal radiation properties, or
field when combining
slabs
Symbol of cosine - 1 < u < 1 0 < μ < 1
used in this book
Relation where both μ = \η\
apply
1.2 Some Hard Choices 5
tion errors of a factor 2 or π, which are a frequent source of irritation. A dis­
advantage is that this concept (van de Hulst, 1964) has not become commonly
used.
Phase function. Sometimes it is more convenient to put the albedo inside
and sometimes more convenient to leave it outside the function describing the
local scattering properties. We chose in agreement with most authors to describe
these properties by the product aO(cos a), where a is the albedo and <D(cos a) is
the phase function, so that the phase function is normalized to 1. In some theo­
retical developments (Chapter 6) the product aO(cos a) is expanded in a series
of Legendre polynomials with coefficients ω . This makes a = ω .
η 0
Polarization. The usefulness of the Stokes parameters (/, Q, 17, V) is beyond
doubt. They require the choice of a plane of reference and express the intensity
and state of polarization of a beam of light of intensity /. Other options are
possible (Section 16.1.2), but there is a practical advantage in having intensity
as the first element of the set of four. Other quantities (J, B, 7 , etc.) each have to
0
be similarly completed with three more elements if we wish to express the state
of polarization. I have avoided an avalanche of new symbols by employing the
words including polarization as a technical term indicating the presence of these
three further elements. It works.
Radiance, intensity. The energy stream per unit time per unit area per unit
solid angle is by international convention called radiance. In astrophysics,
where radiance is usually measured per unit frequency band, it was formerly
called specific intensity (of the radiation) or surface brightness (of the emitting
body). For brevity, the terms intensity and brightness are also frequently used.
I have chosen to use the word "intensity" in this book. The writers on the subject
are from such diverse fields that one cannot hope for uniform terminology.
See translation rules.
The integral of radiance over a solid angle is the irradiance. This is an
awkward term, particularly because different integrals have to be distinguished.
The two most important ones are
net flux = vector irradiance
and
_1
(spherical) average intensity = (4π) χ scalar irradiance
Reflection and transmission function. The reflection function R used in this
book is related to Chandrasekhar's function S by
Κ(μ, φ; μ , φ) = Ξ(τ ; μ, φ, μΨο)/4μμο
0 0 ί 0
In order to introduce a short notation (Chapters 5 and 7) it was necessary to
write the reflection formula for arbitrary incident radiation in a fully symmetric
/ Concepts, Terms, Notation
form. With limitation to azimuth-independent terms and omission of the argu­
ments τ,φ,ψ, this might be done in two equivalent ways:
χ 0
(i)
(2)
I have selected the second form [Eq. (2)] because of several advantages. The
intensities themselves appear and ϋ(μ, μ ) assumes finite nonzero values for
0
μ = 0 or μ = 0 (it becomes oo for μ = μ = 0). The function β(μ, μ ) thus
0 0 0
defined is also the ratio of the actual reflection function to the reflection function
of an ideal white (Lambert) surface.
In the transmission function, where both Chandrasekhar and I use the sym­
bol Τ(μ, μ ), a corresponding difference in definition occurs.
0
Translation rules. Quoting a result from one field and applying it to a
different field generally requires a translation of terms and notations. This
job in itself is rarely difficult, but it constitutes a psychological barrier. How hard
it is to overcome this barrier, even for one well versed in the mathematics and
physics required, is candidly expressed by Preisendorfer (1976, p. xxxii):
"These concepts in other branches of radiative transfer, notably astrophysical
optics, were either nonexistent or in the form of unrealizable mathematical
abstractions of no use to one with direct instrumental access to the interior
of the optical medium of interest; in our case, the sea."
There is no unique remedy for this problem. I have tried to cope with it by:
(a) Spelling out translation rules where this seems prudent. See examples
in Sections 18.1.1, 20.1.1, 20.2.1, and 20.4.
(b) Dealing in the three last chapters with a representative set of applied
fields in order to show by examples from that field where the access to the
earlier, more abstract, results lies.
Wavelength dependence. This has been omitted everywhere in order not to
overburden the notation. The reader who works with dimensionless quantities
will find this an advantage. The reader who requires physical quantities will
have to complete each definition with a choice of units and a specification of the
wavelength or frequency interval. The same quantities are applicable to a
wider band or to the entire spectrum, provided the properties are not dependent
on wavelength within the chosen band.
REFERENCES
In this list of general references we have been very restrictive. Works that deal with a small
subset of the problems discussed in this monograph are cited in the chapters where they belong.
The choice is clearly subjective.