J~nlald
PfI]IIBCHEM~I~
tFFI31UIIEI~F/
B:BIOLO(P/
ELSEVIER
Journal of Photochemistry and Photobiology B: Biology 29 ( 1995 ) 179-183
Estimation of fluence rate from irradiance measurements with a cosinecorrected sensor
L.O. Bjt~rn
Section of Plant Physiology, Lund University, PO Box 117, S-221 O0 Lund, Sweden
Received 2 December 1994; accepted 22 February 1995
Abstract
A method is described by which it is possible to measure fluence rate (scalar irradiance) using a cosine-sensitive instrument intended for
measurement of irradiance (vector irradiance).
Keywords:Fluencerate measurement;Irradiancemeasurement;Lightmeasurement
1. Need for a simple way to measure fluence rate
In photobiology most measurements of visible light and
UV radiation are taken with a cosine-corrected sensor, i.e.
they are measurements of irradiance (or illuminance in the
case of visible light). This means that the flux intercepted by
a flat surface of unit area is recorded. This is appropriate if
the biological object under investigation is fiat, e.g. a plant
leaf or a small patch of human skin. In many cases, however,
the biological specimen is three dimensional and it would be
more relevant to relate the effect of light on it to fluence rate,
i.e. the flux intercepted by a sphere of unit cross-section [ 1,2].
There are a few instruments for fluence rate measurements
on the market, but most instruments, including almost all
spectroradiometers, have direction-sensitive input optics. As
a journal editor, the author has experienced that in many cases
investigators measure irradiance but erroneously report it as
fluence rate.
One example may serve to demonstrate how misleading it
may be to quantify light as irradiance. Imagine a water body
with a fiat water surface illuminated by a cloudy sky with
approximately the same brightness in all directions. As light
strikes the surface, some of it will be reflected back. The
reflected light will not be available for photosynthesis by the
phytoplankton in the water. Thus it seems likely that the
photosynthetically active radiation (PAR) must be lower in
the water than above the surface, but in fact this depends on
how it is quantified. PAR vectorial irradiance will be higher
1011-1344/95/$09.50 ~ 1995 Elsevier Science S.A. All rights reserved
SSDI 101 1 - 1 3 4 4 ( 9 5 ) 0 7 1 3 5 - 0
below the surface, because the light becomes more vertical
by refraction at the surface. PAR fluence rate, on the other
hand, will be lower by a fraction corresponding to the fraction
of light reflected.
We thought it would be valuable to be able to estimate
fluence rate using the readily available cosine-corrected
receivers. Intuitively it can be suspected that this would be
possible by taking measurements in several different directions and adding them up in some way.
2. Representative light sampling and absolute
calibration
Two problems have to be solved in order to measure scalar
irradiance. First, the light from the total solid angle of 4 ~rsr
has to be sampled, by taking several measurements, in a way
that gives equal weight to all light irrespective of direction.
Second, the various measurements have to be added in such
a way that the sum, in absolute units, is equal to the fluence
rate.
To sample light from various directions in such a way that
all light is given equal weight requires that the solid angle of
4 lrsr be divided symmetrically into equal parts. This can be
done in only five different ways, corresponding to the five
kinds of regular polyhedra (platonic bodies).
At first thought it might be assumed that a good way would
be to take readings in six different, mutually perpendicular
or antiparallel directions corresponding to the faces of a cube
L.O. BjOrn/Journal of Photochemistryand Photobiology B: Biology 29 (1995) 179-183
180
(e.g. north, east, south, west, up, down). Closer examination
reveals that this is a false way, since such a combination
would have extremely variable sensitivity to light coming
from various directions between the directions mentioned (up
to 73% deviation). A very accurate way would be to take 20
readings in directions corresponding to the faces of a regular
icosahedron (less than 3% deviation), but this would be
rather impractical. We have also considered eight or 12 directions corresponding to the faces of a regular octahedron or
dodecahedron (44 % and 14% deviations respectively). ( The
values for the various polyhedra were obtained using a computer program by which the outline of the bodies could be
projected in various directions. The "sensitivities", i.e. the
amount of light intercepted, in the various directions are proportional to the areas of the projections.)
Neither of these ways is suitable and we have settled for
four measurements corresponding to the faces of a regular
tetrahedron, which we will examine in more detail in the
following. Even three measurements may be sufficient if light
from one direction (e.g. from below) can be neglected; however, six measurements may give a better result in some cases.
3. T e t r a h e d r a l light s a m p l i n g
Consider a plane through a regular tetrahedron that is perpendicular to one of the edges of the tetrahedron and contains
the opposite edge (the plane containing points A, B and C in
Fig. 1). Taking the line through the midpoints of the two
mentioned tetrahedron edges as reference direction, we can
plot the relative interception of light by the tetrahedron as the
angle (or) between the light direction and the reference is
varied in the plane. This will include the highest and lowest
interception values that are possible for the tetrahedron. The
relative interception can be shown to be
~
+
1
1
7r
t
1
( 1/cos2a + 1 - tan2a) cosa
sqr ( 3 )
with only positive main terms considered (see derivation in
Appendix). Here "sqr" denotes square root. The expression
l
..................................................
o 0
0.8°v--4
>
°v--~
°v--I
f./3
0.60.40.2OoO
~
2
0
,
4
6
angle, radians
Fig. 2. Relative light interception of a regdar tetrahedron for various angles
in the plane of ABC (Fig. 1 ) from the reference direction. This can also be
interpreted as the relative sensitivity of a device with similar cosine-corrected
sensors on each face of a regular tetrahedron.
has three main terms. The first two terms describe light interception by the lower and far left faces in Fig. 1 (faces ADE
and BDE in Fig. A1) and the last term describes the interception by the front and right-side faces in Fig. 1 (faces ABD
and ABE in Fig. A1 ). Only values of each main term greater
than zero are considered (formally, negative terms set equal
to zero) to allow light to be intercepted only by the outside
of the face. The expression is plotted in Fig. 2 as a function
of a. The variation can be regarded as the typical relative
sensitivity for light from various directions for a device measuring light with cosine-sensitive receivers in directions corresponding to the faces of a tetrahedron. The maximum
sensitivity (16% above the average) occurs when the light
enters along the reference direction (or the opposite direction), i.e. normal to two edges, while the minimum sensitivity
(19% below the average) occurs when the light enters parallel to one edge. The horizontal line indicates the average
sensitivity in the plane (compared with that for light striking
a face perpendicularly).
A
4. Practical c o n s i d e r a t i o n s
B
C
B
Fig. 1. The left sketch shows the whole tetrahedron (visible edges in bold
lines). The right sketch shows only the triangle ABC. Fig. 2 depicts semitivity in the plane of ABC. The reference direction mentioned in the text is
the direction from the midpoint of AB to C.
In practice, light from below can often be neglected. If we
position our tetrahedron such that one side faces straight
down, we will then restrict ourselves to taking three light
measurements in directions normal to the remaining three
sides, i.e. turning the sensor in 120° steps around a vertical
axis.
Choosing now a new reference direction, i.e. the vertical
direction, we can plot (Fig. 3 ) the angular sensitivity function
for the imagined tetrahedron from horizontal to horizontal.
L. O. BjOrnI Journal of Photochemistry and Photobiology B: Biology 29 (1995) 179-183
1.2
1.2
1.0
1
0.8
0.8
>
>
°¢-,I
¢/)
°~-~
0.6-
.~,-4
0.6
0.4-
0.4
0.2-
0.2
0.0
-1.5
(a)
181
I
I
I
I
I
-1
-0.5
0
0.5
1
0
0
1.5
I
I
I
0.5
1
1.5
angle, radians
angle, radians
Fig. 4. As Fig. 3(a), but showing the average for two tetrahedra displaced
60 ° (Ir/3 radians) around a vertical axis relative to one another. Only the
span from horizontal to vertical is shown, since the span to the next horizontal
direction is a mirror image.
B
(b)
Fig. 3. (a) Light interception of a regular tetrahedron. The angle indicated
is the inclination of the light direction to the vertical. (b) The diagram in
(a) pertains to the plane indicated in this sketch. The rays converging on
the centre of the tetrahedron are examples of the directions along the horizontal axis in (a). The two lowest rays, parallel to BC, are horizontal,
corresponding to angles of - *r/2 and 7r/2 radians with the vertical.
(and dividing the sum by two), we can obtain a very accurate
measure of the fluence rate.
It was shown above (see Fig. 2) that the average sensitivity
is very close to unity. This can also be shown in another,
exact way. Suppose we have only one cosine-responsive
receiver pointing straight up. What is its average sensitivity
over the 2qr solid angle that it is facing (the upper hemisphere) ? This average can be computed as J sinacosotda/
sint~dot, where the integrals should be taken from 0 to Ir/2
radians [3]. The ratio of the integrals is ½. Thus the average
sensitivity over the upper 27r solid angle is ½ and that over
o*t" • " " !
1.2
For light coming from straight above such a device has the
same sensitivity as an irradiance meter. For light coming
exactly in a horizontal direction the sensitivity is systematically lower, varying between 2sqr(2)/3=0.943 and
sqr(3)/2 of this value, or 0.816. Already for light striking
the horizontal from above at an angle of 19.5 ° the sensitivity
has increased to between 0.943 and 1,
If we repeat the measurements with the imagined tetrahedron turned 60° (7r/3 radians) around the vertical line and
take the average with the previous measurements, we obtain
a curve (Fig. 4; we show here only the section from 0 to
7r/2 radians, i.e. from the horizontal to the vertical, since the
rest is just the mirror image) which is much less angle
dependent, with a maximum only 7% above the average and
a minimum only 6% below the average. Thus, by taking a
total of six measurements with a cosine-responsive receiver
1,0
0.8
o ~,,,q
°~,,~
ra~
~D
positive error
0.6
exact cosine response
0.4-
negative error
0.2
0.0
0
I
I
|
2
4
6
angle, radians
Fig. 5. Effect of deviation from cosine sensitivity. The curves "with error"
are computed by multiplying the cosine factor cos x by 1 + 0.1 sinZx.
182
L.O. Bji~rn/Journal of Photochemistry and Photobiology B: Biology 29 (1995) 179-183
the total 4~'solid angle is ¼. This means that if four measurements are added together, the average sensitivity is unity.
5. Effect of deviation from ideal cosine response of
sensor
Real light sensors never have an ideal cosine response.
Often deviations become appreciable only for incidence
angles larger than 60 °. For instance, the Optronics OL754
spectroradiometer has a - 4 % error at 60 ° and about - 10%
error at 70 ° incidence angle. To model this effect with a
maximum cosine error of 10%, we multiplied the cosine
factors (we call them cosx since the angle is not identical
with the angle a used above) by a quadratic sine expression
1 __+0.1 sin2x. The effect of this is depicted in Fig. 5. It can be
seen that for fluence rate measurements, as for ordinary irradiance measurements, it is important that the sensor has a
good cosine response. The sections in Fig. 5 where the curves
coincide is where only one face of the tetrahedron is illuminated, with near-normal incidence.
6. Experimental details
To test whether sampling of three light directions is sufficient under natural light conditions, an International Light
(IL) SU033 PAR sensor was mounted at a 70.5 ° angle to the
horizontal in such a way that it could be aimed at various
azimuths (compass directions) in 30 o steps, i.e. 12 readings
around the compass. In a few experiments a LiCor Quantum
sensor was also used. Both these sensors are cosine corrected,
Readings were taken outdoors at a place which was relatively
unobstructed, but with buildings slightly influencing the light
field, both under sunny conditions and with the sun visible
through clouds or completely hidden by an overcast sky.
Readings were also taken in a climate chamber.
The original readings, 12 for each light condition, were
normalized such that for each light condition the average of
the 12 readings was unity. The standard deviation (s~) was
computed. Then the readings for each light condition were
grouped such that the first group contained the readings for
azimuths 0 °, 120 ° and 240 ° to a reference direction, the next
group the readings for azimuths 30 ° , 150 ° and 270 ° , the third
group those for 60 °, 180 ° and 300 ° and the last group those
for 90 ° , 210 ° and 330 ° . The average for each group was
computed as well as the standard deviation (s2) for the averages. A low standard deviation in the latter case (s2) indicates
that the method of including every 120 ° in the calculation of
fluence rate gives a representative sampling of the azimuth
directions irrespective of reference direction. Mostly the s2
values (Table 1) are 0.05 (i.e. 5% of the mean) or below.
This shows that averaging three measurements taken 120 °
apart removes most of the direction-dependent variation
(note that it is the sum, not the average, that is an estimate of
Table 1
Standard deviations
Light conditions
Sensor
.fl
s2
Sunny
Sunny
Cloudy
Cloudy
Light cloud
Cloudy
Cloudy
Growth chamber
Growth chamber
Light cloud
1L
IL
1L
IL
IL
IL
LiCor
LiCor
IL
LiCor
0.797
0.867
0.244
0.149
0.585
0.402
0.118
0.129
0.152
0.117
0.039
0.048
0.011
0.051
0.042
0.077
0.006
0.019
0.011
0.009
fluence rate; the average has been used throughout here for
easier comparison).
7. Conclusions
The principle of tetrahedral light measurement allows an
estimation of fluence rate using an irradiance sensor. For
natural light and most biological purposes three irradiance
measurements are adequate for one fluence rate determination. The principle can also be used for artificial light if this
is not highly directional, in which case ordinary irradiance
measurement is adequate.
Acknowledgements
The author thanks Dr. Weine Josefsson (Swedish Meteorological and Hydrological Institute) for critical remarks
and Dr. Janet F. Bornman for linguistic revision.
Appendix: Derivation of the dependence of light
interception by a tetrahedron on light direction
The angle between faces ABD and ABE in Fig. A1 is
2 t a n - l [ 1/sqr(2)]. This is easily seen by considering the
triangle DEM (not drawn), which has one side equal to the
tetrahedron edge and two sides that are the fraction s q r ( 3 ) /
2 thereof.
The angle ACB is also 2tan- ~[ 1/sqr(2) ] and is bisected
by the reference direction, hence the appearance of t a n - ~[ 1/
sqr(2) ] in the first two terms of the expression.
For the last term we proceed from the information in the
caption of Fig. A1.
From the value t a n - ~ [ 1 / s q r ( 2 ) ] for half the angle
between the faces at M, it follows that the length of segment
MN is 1 / c o s { T r / 2 - t a n - l [ 1 / s q r ( 2 ) ] } . This can also be
expressed as 1/sin{tan- 1[ 1/sqr(2) ] } = sqr(3). NO equals
L. O. Bj6rn / Journal of Photochemistry and Phowbiology B: Biology 29 (1995) 179-183
,o,
B
183
side of the triangle M N P c o n t a i n i n g the angle P M N . This
angle is the angle b e t w e e n the light and the n o r m a l to face
ABE, the cosine of which is the proportionality factor for
computing the light interception. The angle P M N is designated/3. W e have the relation
NP 2 =
MN 2 +
M P 2 - 2" M N . M P ' cos/3
or
MN2+MP2-Np
COS/3=
2
2-MN'MP
or
3 + 1 / c o s Z a - [ tan2a + 2]
COS~ =
2sqr(3)/cosa
( l/cos2a+ 1 -tan2a) cosa
2sqr(3)
D
Fig. A1. Sketch of tetrahedron with additions to aid the derivation of the
expression for light interception by a regular tetrahedron ( the relation plotted
m Fig. 2 as a function of a, the angle between the light and the reference
direction shown in Fig. 1). "Hidden" lines are drawn thinner. To aid the
derivation, we have drawn the following line segments (M is the midpoint
of AB). MO: a line segment of length one, normal to AB and in the plane
ABC (C is the midpoint of edge DE). MP: a line segment along the light
direction to be considered for the moment. Thus the angle OMP is a. It
follows that the length of the line segment MP is 1/cosa. OP: the length of
this segment is tana. MN: the normal to the face ABE, drawn to such a
length that the angle MON is a right angle.
tan ~- -- t a n -
t a n { t a n - t[ 1/sqr(2) ] }
N O P is a right angle, because O P is parallel to AB and the
triangle N M O is n o r m a l to AB. Thus it follows that N P is
s q r ( O P 2 + N O 2) = sqr [ t a n 2 a + 2]. However, N P is also one
This expression, cos/3 is the relative light interception by one
of the faces A B D or A B E for unit face area, i.e. on the same
scale as calculated for faces A D E and BDE. T o get the light
interception by A B D a n d ABE, we have to m u l t i p l y by two,
so the final term will be
( 1 / c o s E a + 1 - tanEct) cosot
sqr(3)
References
[1] C.S. Rupert, Dosimetric concepts in photobiology, Photochem.
Photobiol., 20 (1974) 203-212.
[2] L.O. Bj6m and T.C. Vogelmann, Quantification of light, in R.E.
Kendrick and G.H.M. Kronenberg (eds.), Photomorphogenesis in
Plants, 2nd edn., Martinus Nijhoff, Dordrecht, 1994, pp. 17-25.
[3] T.C. Vogeimann and L.O. BjSm, Measurement of light gradients and
spectral regime in plant tissue with a fiber optic probe, Physiol. Plant.,
60 (1984) 361-368.