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EDL-G266
463413
Directional Reflectance
and Emissivity of
=
Opaque Surface
_an
FRED E. NICODEMUS
i
")Y11 L\NI\, ELECTR NIC SYSTEMS
.'ELGENERAL
TELEPHONE &ELENTRONIC.
II
7! r,
PREP A RE D
FOR
THE
UNITED
STATES
ARMY
EDL-G? 66
ELECTRONIC
Mountain
)
DEFENSE
View,
S
LABOR-P9-%
Califor--i-a
TECHNICAL MEMORWIW
.
i'19 May I_64,
DIRECTIONAL REFLECTANCE AND EMISSIVITY
OF AN OPAQUE SURFACE
Fred E. Nicodemus8
Approved for Publication
....
John Barton
Head
OR Analysis Group
R. H. Krolick
Manager
Applied Science
Laboratory
Prepared for the .J.S. Army Electronics Research
and Development Laboratories.
SYLVANIA
ELECTRIC
PRODUCTS
INC,
EDL-G266
C ONT E N T S
Section
Title
Page
1.
ABSTRACT ..........
..........................
1
2.
INTRODUCTION
.......................
2
3.
DIRECTIONAL REFLECTANCE
4.
DIRECTIONAL EMISSIVITY (AND
ABSORPTANCE)
............
...................
5
13
5.
EXAMPLES
6.
SUM M ARY .
7.
APPENDIX A -- Reciprocity in an Isothermal
E nclosure . . . . . . . . . . . . . . . .. . . . . . . .
23
REFERENCES
24
8.
..........................
.........................
...............................
-ii1
-
16
20
EDL -G(66
ILLUSTR ATIONS
Figure
Title
Page
Geometry of Incident and Reflected Elementary
Beams. (Z-Axis is Chosen Along the Normal
to the Surface Element at 0) .......................
7
TABLES
Table
!
Title
Radiometric Quantities,
and Units.
-
Page
Symbols, Definitions
.................................
1ii -
4
DIRECTIONAL REFLECTANCE AND EMISSIVITY
OF AN OPAQUE SURFACE
Fred E. Nicodemus
1.
ABSTRACT.
\
Concepts,
terminology, and symbols are presented
for specifying and relating directional variations in
reflectance and emissivity of an opaque surface elernent.
Their relationship to more familiar concepts,
including
those of perfectly diffuse and specular rcflectancc,
is
given, and they are applied to illustrative examples.
is shown that, when the usual reciprocity relationship
holds,
the reilectance for a ray incident on an opaque
surface element is related by Kirchhoff's Law to the
emissivity of that element for a ray emitted along the
same line in the opposite sense.
It
~IN
Z.
UROLtJC)D•
L'ION.
Reflectance and emissivity of the surface of an opaque body
are considered as properties of the surface material and of its microscopic configuration (roughness) but not of its gross configuration
(curvature).
This distinction between microscopic and gross details
of the surface configuration,
which is to some extent an arbitrary one,
will be discussed fuither below.
But reflectance and emissivity are
commonly defined or specified in ways which include an implicit
(and often overlooked) dependence on the geometry of the radiation
beam (including incident,
emitted, and reflected rays and the effects
on those rays of the gross surface features).
Even when this depen-
dence is recognized, the specified reflectance or emissivity is usually
applicable only to situations which reproduce the same geometry.
On
the other hand, it is possible to specify the reflectance and emissivity
of an opaque surface (i.e.,
of any planar surface element) concisely
and unambiguously as functions of direction (with reference to the
orientation of the surface element) which can be applied quite generally.
The purpose of th!s paper is,
first, to describe such a way of
specifying the directional reflectance and emissivity of an opaque surface,
to recommend appropriate terminology and symbols,
and to relate them
to those in common use.
Second,
a relationship will be established
between the directional reflectance of a surface element (I e , its
reflectance for a ray incident from a particular direction) and the
directional ernissivity of the surface element for radiation emitted
in that same direction.
In other words,
the related quantities are
the reflectance for a ray incident along a line which intersects the
surface elerrent and the emissivity for a ray emitted along the same
line in the opposite sense.
The radiometric quantities used are listed in Table I, repro1
duced from an earlier paper
The radiometric relations will be
analyzed below primarily in terms of the basic quantity radiance (N).
1
In the earlier paper
in Table I,
it was shown that when radiance is defined,
as the radiant flux or power (P)
as
per unit solid-angle ( a ) -
in-the-direction-of-a-ray per unit projected-area (Acos6)-perpendicular-to-the-ray,
it has the same value at any point along this ray with-
in an Isotropic medium,
scattering,
or reflection.
in the absence of losses by absorption,
More generally, the quantity N/n
2
(where
n is the index of refraction) in the direction of a ray was shown to be
invariant along that ray,
even across a smooth boundary between
different lossless media.
1
Se
list of references
in ,½ection ,
-3-
l
i
IInI
3ad
nWc
It I
-L(I
QnaroitI
Ss mnbo.
I
nlc'rgv
U
Raiant
IAltti'51
SN.I'll
k1,d(Iet tiit'i ...
s.and
ullli ts.
Deftinig relations
eits
J
) UI
t l.cdi
i I i•cI,•,g
5i
a
t,
Jt .cIn
aU
Radiadnt posn er
P
P
-,waott
-
(IN)
0t
hOintenit,
ta•,dmant
J
JI---
oil
Radiant estittance
If,
lv
o.'V
Irradiance
H
Hj
0.4
Radiansce
X
Wavelength
X
Spectral radiant power
Px
NV
XV•:
'-S
i
al'l
-W.cm-XV
eosOOA 80
sr-I
micron (y)
op
Pi-
W.ON
Spectral radiant intensity
Ax
Spectral radiant enshttance
WX
A-, -Vsr-',Ct-
ax
oW
w'x --
W
Acm--1
ax
Spectral irradiance
HX
Spectral radiance
NX
HmON
IVcm-I.sr"
Radiant enissivity
R ,.liant absorpra,,ce
0
Radiant tranismittanle
t
u-I
Ratio of "emitted" radiant power to that
from an ideal blackbody at the same temperature.
of "absorbed" radiant power to incident
radiant power.
Ratio of "reflected" radiant power to incident
radiant power.
Ratio of "transmitted" radianl power to
incident radiant power.
aRatio)
-A-e. •acn
OH
e
nre
The ipe, tral radiant emissivity .eX) !{x 'Lea#0eilaX. Hence, the subscript notation ef, which
,:,!,;I1 be c-fused with a'/aX, is tnt recommended, although itis often used. Similarly, it is recommnended
ihaa the spectral absirptance, spectral reflectance, and spectral tranesnittanc- be written as nI.X, p(M,
a,,x r
respectivel-.
-.
ci,,-I ...
4
--
n*
-
ED L- G2 ,(,
In Table I. the definitions given for radiant err. ssivity
dnd
radiant reflectance take no account of the effects of the geometry of
the radiation beam.
The following treatment will refine these
definitions to recognize explicitly the way in which these quantities
Only opaque
may vary with orientation (relative to the surface).
surfaces (of zero transmittance) and the geometrical ray optics of
incoherent radiation will be considered.
3.
DIRECTIONXAL REFLECTANCE
Consider a radiation field, where the radiance N.1 is a function
of both position and direction, incident on the surface of an opaque body
where some of the radiation is absorbed and the rest is reflected (as
used here, "reflected" includes diffuse reflectance or scattering) to
iorm a second radiation field, where the radiance N
radiation is also a function of position and direction.
r
of the reflected
N r is directly
proportional to N. in the sense that, if the value of N. is multiplied by
a constant that is independent of position and direction, the resulting
values of N
will all be multiplied by the same constant factor.
It will
be seen below that the interdependence of the spatial and directional
distributions of N
Next,
and N . is more complex.
consider only the radiant power incident on a particular'
1."
1) L-GC',
element 8A of a reflecting surface through an elementary beam of
solid angle 6 i
from a direction (,
1 p0),
where 61is the angle from
the normal to 8A and y i is the azimuth about that normal (see Figure
1).
This incident radiant power is given by 1
6P (
t , y
= N1 (0i
= N1 (0t,
where
61f'i
=
T)
cos 6i 6A 60n
oti)
61't 6A
I
],
(1)
cos 86 60
= sin e
cos
eid
is the "projected solid angle" 1, 4
1dicp1
of the elementary beam.
Corre-
spondingly, the irradiance at 6A is
6H11 (0,
Pi )
= r" (oiYI) 6
[w. cm- 2 ].
'i
Then the radiant intensity of the surface element 6A,
(2)
due to reflection
(scattering) of radiatlon from this incident elementary beanm,
in the
direction (e r, Cr) is
8J1
or,
(6r Tcp)
r
r
r
= p'(i, PVToi
'
'rr
)cos 6 6Pi (61,cT)
r
Ii
by dividing both sides of Equation (3) by IAcos8r,
1
[w.sr-a
(3)
we obtain the
reflected (scattered) radiance
6N r (9r' 'Pr)
= P'(il
i. 0 r.
8Hr)
6H(8,
- 6-
)
w. cm-2 .r-]
(4)
z
6q-Q
-- er
x
Figure 1 Geometry of incident and reflected elementary
beams.
(Z-axis is chosen along the normal to the
surface element at 0.
whre n
(91¢I
'
eNr(9Spt)
NrI(•r~.L.W)
sr
and reflected (scattered in the direction (@r,
Pr).
(i
of the
is the partial retfect,tice or "reflection- (distribution function'
surface element5A for radiation incident from the direction
1
(
ni, o
Furthermore,
by
6
a reciprocity theorem of wide generality ',7 first enunciated by Helmholtz,
wc may write
P,
Thus p'
(e1,
(e
,
O
p, ) = P' (e
C
2 '',
'
)
[l
,.
(6)
R, z, cpo) is ordinarily the partial reflectance between the two
directions (9, pT ) and (8•', T)
where either direction. may be that of
the incident elementary beam and the other that of the reflected (scattered)
elementary beam.
Hence,
we can write the expression for the radiance at a point
of the reflecting surface (taken as the origin for spherical coordinates)
in the direction (Gr' Tr) due to reflection (scattering) of all beams of
incident radiation as
* A search for a proof
(in English) of this important theorem also turned
up a number of authors who referred to,or made use of,the theorem in
various ways without giving a proof'
although Planck14 states,
9, 10, ii,
12 Including Von Helmholtz
without specific citation, that Von Helmholtz
7
"proved" the theorem.
DeHoop
not only gives a proof (essentially the
same as that of Kerr 6).but also includes an explicit statement of the
requisite conditions.
-8-
N r
rr
' Ti, 9r'
0
p'(ecp
J
rP Nii
i1d'i,
i)'h
i
1
1
d
0
, yr) N, ( 9,,c)
[w-cm ".sr-s
d o
l
(7)
hr
where we adopt the following notation to designate integration over a
hemisphere:
Ff
fJ
(6,c) dO
"h
STT
f(e,
and
h
This relation --
) do' =
f
Equation (7) --
de d0
o
f Me,
J
o
Ce, ) sin
cp)
sin a
cos 6 d 6 d cp.
o
is for a particular point,
surface element 6A at that point.
or for the
For a more general expression, we
must also establish the reflected radiance from other points.
When p'
and N are expressed as functions of spatial location (as well as direction)
for all points on the reflecting surface, Equation (7) gives the reflected
radiance N r as a function of position for these points on the reflecting
surface,
as well as for direction ( r'1 r ) at each such point.
However,
it is important in that case to recognize that Equation (7) is written above
in coordinates which, for convenience,
respect to the surface element 8 A.
are specially oriented with
Appropriate adjustments must be
made when dealing with Irregular surfaces where the direction of the
normal, with respect to fixed coordinates,
surface element to another.
-9-
changes in going from one
Whether su rfac e I rreg ula rities are treated as rll'
ros copic,
(in the sense that their effects are integrated or averaged in the distribution function or partial reflectance
macroscopic
(in the sense that they may be
C, 1::.9g.,
y2)), or as
analyzed into smaller
surface elements 6A for treatment as In the preceding paragraph) can
be arbitrary,
depending on the degree of resolution desired,
dependent on circumstances limiting achievable resolution.
or can be
For example,
in examining the reflectance of a highly irregular surface containing
deep cavities,
such as a piece of volcanic scoria, or a coarse, blackened
cellulose sponge in the laboratory,
it may be possible to consider the
reflectance of different portions of the walls of single cavities (which
are ther regarded as macroscopic Irregularities).
But when studying
the possible effects of similar surfaces which may exist on the moon,
where such flne detail cannot possibly be resolved by the best telescopes
on earth, these are necessarily treated as microscopic
Irregularities. 15, 16
Still more complicated considerations are introduced when microscopic
irregularities are small enough to have dimensions of the order of, or
less than,
the wavelength of the incident light or other electromagnetic
radiation. 17, 18, 19, 20
The total reflectance. p of a surface element 6A is defined in
general as
p
-
8P r /
[dimensionless],
PI
-10-
(8)
bI)[ - ;2.€
A.
the total ra diant power iticidt'elt ( tr [- all directions) on
FPis
whe re
and
P
£
the total re'iltittng vet'lected radiant power (in all
""
directionsi.
As stated above, the value of p depends upon the geometry
and spectrum of the incident beam of radiation, which may be different
in each particular case.
Here for the moment we are concerned
prrniarllv with the geometrical relations.
of this paper,
Hence,
except where otherwise stated,
for the remainder
we will eliminate spectral
considerations by restricting the spectrum of the incident radiation
to a region over which
length.
p does not change significantly with wave-
It is then useful to consider some special cases of incident-
beam geometrv.
If the incident radiation is well collimated,
within a small
element of solid angle 6,2i = sin 8 deP dTi from the direction(O8 i,0
the
total radiant power incident on 6A is
8P,
Then.
51HI (C i'IA
=
[w] .
(9)
from Equation (5),
IN
r
(v
r
r
)'
Q
Ii
SP (
T"-, er Ic ) 5-H1 (8
r
r
i
ii
8a rt r
-
iil-
6PiF/6A
(q)
[w. cm--.sr-1]
(10)
1,:1) L ~-(G
But
,P
V
= 5A
$N
= 5Pt'
P
.
(? ,
) dO
r r
r
(9t?
Or'
Qr
d2'r
I di
where
d
(
e)
L J)
[w],
is the (total) directional reflectance,
collimated incident beam,
-
ýdi (i)
Ec-uaton (12)
Pdi (9i' 1)
given by
Fh p. (9 1, Ti, Gr' (P
For isotropic surfaces,
there is no dependence on the azimuth CPand
If the well-collimated beam is incident perpen-
dicularly on a plane surface,
pn = Pdi (0).
we have the commonly-reported normal
If a point on the surface of a solid is uniformly
irradiated from all external directions,
i. e., if N
reflected radiance in the direction (Or I ýPr),
Nr(0r'
(
r)
0
- Ni fh p, (
Di,
V
0
Pdr (ýr'Tr) =- h P'
r' Tr) dC
But, from the reciprocity relation,
the
I
[w.
i' 0r'
rr)
(1
is a constant,
from Equation (7) is given by
= Pdr (9r'pr) Ni
where
(12)
[dimensionless]
r
simplifies to the frequently recognized dependence on 0:
= Pdi (P).
reflectance,
for a well-
d' r
Equation (6),
(13)
cm-2.sr1];
[dimensionless]
(14)
.
and Equations (12)
and (14) we can write
Pdif
G
1t ' 1
Pd, (
= Pdr
'
D
Pd (01,
=
-
12 -
P)
[dimensionlessl .
(15)
Thus.
the (tota I) dir ect iona I r eflec ta nc e Pd (0
'.)
for a welll-colimated
beam iocrdent from the direction (01, •I) is also the ratio hetween the
reflected radiance N
(@l,
ý, ) in that same direction ant. the incident
radiance N. when the surface is uniformlv irradiated from all directions
(Ihemisrpherical irradiation).
This relation --
Equations (13)
and (15) -8
is the basis for a reflectometry technique described by McNicholas.
4.
DIRECTIONAL EMISSIVITY (AND ABSORPTANCE)
More important,
Equations (11),
(13)
and (15) are the basis for
evaluating and equating the directional absorptance and directional
emissivity of the surface element 6A in a simple relation which has the
same form as the Kirchhoff's-Law relation
If, in Equation (13),
--
see Equation (18) below.
the uniform incident radiance N.i is equal to Nb (T),
the blackbody radiance (either total or spectral, i. e.,
in a small wave-
length interval at a given wavelength) in an isothermal enclosure at
T°K, and if,
in fact, the reflecting surface forms the wall of such an
enclosure so that it,
in the direction (8_, ,4
too,
is at this same temperature,
then the radiance
) from the element of wall surfP :e 6A is made up
of an emitted radiance and a reflected radiance, as follows:
Ne + N
= E d (891'
SNb
Similarly,
(elI',1'I)
1
) Nb (T) 4. Pdr (81
(T)
,pl) Nb (T)
[w. cm-2,
sr-]
.
(16a)
of the rad.ance Nb (T) incident on the element 6A from a direction
, a portion N
is absorbed and the remainder N.
(scattered) in all directions (into a hemisphere):
-
13 -
is
reflected
121)1,-(2366
N'
Nit
d
q' 06 Nb (T)
1 edi (01, ý I) NT (T)
Nb (T)
Here,
Zd (
i
Iw
c
-•
-
sr-'
(16b)
the directional emissivity (at temperature T) of the
element IA for radiEation emitted in the direction (01
(P, ) and od.Y(
CO
,
is the absorptance (at T) for radiation incident from that direction.
Consequently,
from Equation (15),
=
-
Odi (9',, 4)
= ad (e8,1'
[dimensionless].
(17)
Note that equilibrium maintenance with conservation of energy (Kirchhoff's
Law) by itself would justify only each line of Equation (17) independently,
and the Helmhotz Reciprocity Law
(which is the basis for Equation (6)
and, in turn, Equation (15)) must also be invoked in order to equate
them to each other and so to relate emissivity for radiation emitted into
a given direction to the absorptance for radiation incident from that same
direction
(See Appendix A concerning a contrary position.)
In the more familiar form of Kirchhoff's Law,
= 1I - P = C
[ dimensionless],
directional quantities are not considered.
(18)
Instead, the total emissivity
for radiation emitted in all directions (into a hemisphere) is related to
the total reflectance (in
all directions into a hemisphere) for uniform
incident radiance (from all directions,
i. e. , from a hemisphere) and to
the total absorptance for uniform incident radiance (from all directions,
i.e. , from a hemisphere).
uniform
The total reflectance a in Equation (18),
incident radiance (N.
for
= a constant independent of direction) is then
- 14-
Ai,
Qp
=P
=
Nr
rA h N
.h Pd
P
N.
2T
_________
rhd(0, C) di
d2'
, ,) d2'
[dimensionless]
(19)
The quantities in Equation (18) are those involved in heat-transfer
computations where the interest is in the not flow of energy across a
bounding surface,
involving radiation received,
emitted, or reflected
in all directions.
Equations (17) and (18) apply In all cases to spectral radiation
(I. e.,
the radiation in a very small wavelength Interval about a specified
wavelength) and hence also to any spectral interval In which p or Pd (and
therefore also C or
wavelength.
ýI
;
d and o, or ord) do not change significantly with
When thermal equilbrium exists (I.e..
when N
= Nb (T) =
Nxb(T, X) d X, where N1b (T, %) is the spectral radiance of a blackbody
0
a
T K),
they also apply to total radiation (all wavelengths),
the spectral reflectance varies with wavelength.
However,
even though
if the spectral
reflectance is not a constant and the spectral distribution of the incident
radiation is arbitrary (non-equilibrium condition),
Equations (17) and (18)
do not necessarily hold for the total (all wavelengths) reflectance,
absorptance,
and emissivity.
-15-
%I P11. ES.
_____
'--de
:o ."t.- i v the ioregoing treatment of reflectance,
:nav be helpful to apply it to some f"
in reflectance measurements.
it
quently encountered situations.
it is a common practice to make compar-
!Sons with standard surfaces which approximate the limiting cases of
perfectly diffuse reflectance (MgO is often used) and specular reflectance
(a highly polished mirror).
irst, a perfectlv diffuse reflector Is characterized bf a
constant value o: partial reflectance p' in all directions .
-ur-ace
If such a
s di..usely irradiated (N"I constant over a hemisphere) and the
.eflected radiation in a well-collimated beam in any particular direction
.s measured,
or. in the reverse situation, if well-collimated incident
radiation is reflected into a hemispherical receiver (e. g..
sohere).
an integrating
the ratio of- reflected power (flux) from a given surface area
to the incident power on that area is given in either case by the directional ref-lectance pd which,
Pe = P'd'
h
Hence,
by Equation (12).
[dimensionless].
=
from Equation (13),
is also a constant:
the total reflectance for any arbitrary
configuration of incident radiation is given by
-16-
(20)
Pr
-
P
= Pd
d2'dA
r L1,"N
•1
N
p ci
-"P
h NdI'd
r
d2'dA
L,
A
NI d•'dA
fd•lcnsionlce]
--
(u1)
,
where the integration with respect to dA is carried out over the same
area in
both numnerator and denominator,
of direction and position.
and N 1 may be any function
When the incident radiation is uniformly
distributed over the surface (even though it is not necessarily uniform
with respect to incident direction), the reflected radiance N r in any
direction is related to the irradiance HI by the partial reflectance p'
defined in Equation (5),
P' = Nr/HN
as follows:
= p/v
[sr-
1
Second, a perfectly specular refler:r
].
(22)
,
.aracterized by the
relation
Nr(9,
.
"
- Pd ( , p) N, (e,8 )
1w.
sr-]
(23)
By comparing this with the general relationship between incident and
reflected radiances,
it can be seen that Equation (23) will result if the
partial reflectance p' in Equation (7) has the form
Pt (91, (;, ar', r) = 2 Pd (ei'ct) 6(sins Or - sin'2
Pi ±)
W(Trwhere t (sIn 2 i
-sin~g
) and ,((pr
-CIi
[sr-]
(24)
a) are Dirac delta-functions which
satisfy the defining relations
-
9)
17-
(it) = 0 for U
Siu) du
t
0,
= 1, ,rnd
iu) du = f (0),
f kit)
when the integration is carried out over the full range of the variable,
0 < a
-/•
,
2-
and 0 <p
in each case.
Sometimes attempts are made to state apparently simple
relationships between the output of a given reflectometer for a diffuse
standard surface and a specular standard surface.
matter.
This is not a simple
It depends critically upon the configuration,
and a wide variety
employed.
of configu rations are
As an illustration, assume that a sample surface is uniformly
irradiat'd by a well-collimated beam of uniform radiance Ni within a
small solid angle Li
incident from the direction (p, p)
.
Assume also that
a detector is placed with appropriate optics (stops and, if necessary,
focussing elements) to insure that it receives radiation only from a welldefined portion of the irradiated surface, of area A A,
defined solid angle,
AZ'r < A•C, in the direction (0, cr± ,).
through a wellFirst, if the
reflecting surface is perfectly diffusing, the reflected radiance is constant in all directions and is related to the incident irradiance.
cos 9,= "N
.
and (21).
by Equations (5),
H, = N A
The total power (flux)
received by the detector is then
Pd =
cr 'r
-p/)
Ni
= (P-)
N1 A
r
1
cos
9
r' do r' dA
A'r I?' AA
- 18-
(25)
Note that if there is vitnettin,.z,
- o that the solid aingle A 0
through which
the detector receives radiation is not exactly the sane for each point of
the surface A A.
Nex.•,
it may be difficult to evaluate the integrals.
if a specular standard surface is substituted for the
diffuse surface (and if it is carefully aligned to insure that the solid angle
I2s completely filled with reflected radiation),
then, from Equation
r
(23),
the total power (flux) received by the detector can be written as
P
rj
jNrd . dA
= Pd (' ,
) N1
"
(26)
[w] .
A
If these were ideal standards, with reflectance values of unity in each
case
(P = Pd = 1),
the ratio of the detector outputs (proportional to
received power) for the two surfaces under the described conditions
would then be
P
7
=Pd
=
e n? I
[ dimensionless]
.
(27)
QLCos 0i
Itis obvious that this relation depends directly (inversely) on the solidangle spread of the incident collimated ieam.
other factors in the configuration -acceptance of the detector,
etc,
The dependence on the
the alignment, the solid angle of
-- is clear from the foregoing discussion
and specification of the conditions for which this relation was derived.
-19-
,ilv
what nmiht be terned the evternal radionietric relations
have been considered in the foregoinE treatment and no attempt has
been made to deal with the deeper theory relating reflectance,
emissiv-
ity. and absorptance to the optical constants of the materials,
A good
summary of the most important aspects of that approach is given In
r.
SUMMARY.
The partial reflectance of a surface element p' (6k,
@P1
',
82'
Va)
defined in Equation (5) as the ratio between the reflected radiance
in the direction (a=, a)
6.
,
and the incident irradiance from the direction
a) •which produces it.
Integration of this quantity over the solid
angle of a hemisphere in Equation(iZ) yields the directional reflectance
Pd ý I'
I ),
which is the fraction of the radiant power incident from the
dcrection (
) that
t
is reflected In all directions (into a hemisphere).
Furthermore,
if the reciprocity theorem--Equation (6) --
as it ordinarily is,
at least to a good approximation,
is applicable,
then this direc-
tional reflectance is also the ratio between the radiance in the given
direction and the incident radiance when the surface element is
uniformly irradiated from all directions,
-20
-
as indicated in Equation (15).
"hoheeo
,
1.
,i jiitv o-'
A. t eC 1
t
III
opaI.que sutrf1t-e eh, m:iIt In the dire't oii
ot I10 def r e.C t i 0( J, I
IS S.hown
0,V)
-u: 7quaticon 4 171.
\hen the reocprocity theorent is applicable (as Is
usuallv the case),
the emnis ivttv tn a given direction Is
absor'tivitv tfot
radiation incident from that direction,
ec.a 1 . to one nt an
s
equal to the
which is also
the directional reflectance for that same direction.
A perfectly diffuse reflector is
characterized by uniform
re, ectance in all directions.
It is shown In Equation (21) that this
;5 ea.u al to pi times the partial
reflectance.
The relationship
retween the partial and directional reflectances for a perfectly
specular reflector involves Dirac delta-functions,
The completely general expression,
as given in Equation
relating the reflected
radiance of a surface element in a given direction Nr (9r, yr) to the
incident field of radiation,
specified by expressing the incident
radiance as a function of direction N1 (9l
partial reflectance p'(81 , PC ep)by
) is given in terms of the
Equation (7).
This holds true
regardless of the geometrical configuration of the Incident beam.
The
radiant power in a reflected beam Is then computed by integrating
the resulting value of reflected radiance, as a function of direction,
over the appropriate projected area and solid angle as indicated in
-21
-
!
E•1)1,- G..,t
,-e first line of Equation (25) and discussed in greater detail in
Reference 1.
As with many idealized physical quantities. partial and
d.rectional reflectances can never be measured exactly. even with
perfect instrumentation.
Since a measurement reqLuires a beam
o. radiation of non-zero cross section and solid angle, the measurement at best can only yield average values over these intervals of
However, the concepts,
projected area and solid angle.
terminology,
and symbols presented here make it possible to specify explicitly
and unambiguously the interrelationships and approximations
involved in dealing with real situations.
Kirchhoff's Law --
Equation (18)
--
to the directional quantities can
be stated explicitly, as in Equation (17).
-
22
Also, the application of
-
7
APPENDIX A --
A paper by Bauer
Reciprocitv in an Isothermal Enclosure
presents an alleged proof of the Helnrihoh.o
rec~proc.:.- law for diffuse reflection as a consequence of equilibrium
The argument hinges on the
condctions '-- an -sothermal enclosure.
statement that the second law of thermodynamics requires that there
be no net exchange of ener-" by radiation between any two individual
elements of the itern-al (opaque) surface of an isothermal enclosure.
H!Zowever. it seems to me :hat the requirements of the second law apply
to the total flow of energy, taking in:.'
account radiation emitted into,
and absorbed from, all directions (full hemisphere) by such a surface
element,
as the basis for Kirc!-hoff'- Law( E:*quation 18).
-23-
EDI,-G.6
S
REFERENCES.
R
Fr
May
2.
Ed Nicodr::.
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-3 (UNCL-ASSIFIED
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v 3 1. n
.
p {&S;
e: a!..
"Infrared Target and iBackground Radionetric
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A!.-:,,
by~-'
U.
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(RNCLASS=.T.--D . ..ti-.
'ejs-:rcen:ts
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el.
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1432: September 1050. (UNCLASSIFIED
RE. v -
u":a:-on)
4
R
•".
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KD.
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..
T.
K. E. Nelson and R. D. Roddick
Gier.
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oDna1.d
Kerr.
"
J Opt Soc
(UNCLASSIFIED
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i i.
C
n.--.r.ttee :n Colori.etr,,
Optical Society oi America,
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7E;
5
(UNCLASSIFIED p•blication)
-
24
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:2
13.Vo
T ~~de Ia Perreie. T. S. Moss~. and H. HeItrbert.
T1
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W.
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Bruce W. Hap'-e and Hu!;; Va- Horn. 'Photomnetric Studies ,.f
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%ic-:o. Twers'ky. "On -MultipleScattering of Waves," J Res
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:z -r
ennet-, and
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-chard K
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!.:0t' ,-G' 2.(~'0
in SECRET V\0''C .
t, iI'shci ,t1d dtst rttibtCd by
Re srch.', 4t15 Suý.11ninir Strcet. Bostotn)
en:r:v H
21
Bl'au.
)r
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"R d'.+t'n
Miles and Lelhnd E.
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of Solid
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KUNCOLASSIFIED publication)
27
0.
"vIl
Da'.cer
IS,
Reflectiou Nieasurenient'
no 12
3
P V0 ,
De"cnt,
6
--
'r
on Open Cavities" Optik,
1061.
(tin +
Ž;6+)
,.-
tz
4
f
:- 'z •-
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•
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-
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-
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cc,
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1>-.
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1
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