The optical power absorbed by a slab at the focus of a parabolic dish concentrator is calculated. The calculations are plotted versus maximum angle of incidence of irradiation (which corresponds to solar concentration) with absorption coefficient as a parameter for several different indices of refraction that represent real materials

Williams, M. D.

English

NASA Technical Reports Server

NASA Technical Memorandum 4056 
 
Influence of Refractive Index 
and Solar Concentration on 
Optical Power Absorption in Slabs 
LlBRkRY C O P Y  
M. D. Williams 
JULY 1988 
NASA 
https://ntrs.nasa.gov/search.jsp?R=19880016806 2020-03-20T06:49:48+00:00Z
NASA Technical Memorandum 4056 
Influence of Refractive Index 
and Solar Concentration on 
Optical Power Absorption in Slabs 
M .  D. Williams 
Langley Research Center 
Hampton, Virginia 
National Aeronautics 
and Space Administration 
Scientific and Technical 
Information Division 
Introduction 
Since the late 1970's, NASA has been evaluating 
solar-pumped lasers for beamed power transmission 
in space. (See refs. 1 and 2.) Initially, efforts fo- 
cused on gaseous lasing media because they facili- 
tate cooling. More recently, solid and liquid media 
have been included in the research. Solid materials 
rely on heat conduction for cooling, and heat con- 
duction is increased by geometries that feature large 
surface/volume ratios. Such a geometry is a slab. As 
presented here, a slab is a transparent material with 
two large, flat, parallel surfaces separated by a dis- 
tance that is small relative to any other dimension. 
(The material may be solid, liquid, or gaseous-this 
report is quite general in its applicability-but the 
focus here (lasers) makes the results more important 
for solids.) 
Practical solar-pumped lasers cannot operate on 
the normal power density (irradiance) provided by 
sunlight in near-Earth space (about 1365 w/m2).  
The irradiance must be increased by concentration. 
That is, solar power must be collected over a large 
area and focused into a much smaller area that 
approximates the area of the laser. Typically, this 
is done with a reflective parabolic dish. The dish 
aperture is the large area. The small area is the 
image of the Sun at the focal distance. Average 
concentration is the ratio of the large area to the 
small area. Increasing aperture size by scaling all 
dimensions of the dish will not increase concentration 
because the focal area will increase proportionately. 
Concentration increases by extending the extremities 
of the dish while leaving the focal length unchanged. 
That is, concentration depends on the solid angle of 
incidence at  the small focal area-not on the dish 
size. 
The use of a parabolic dish concentrator to couple 
power to a slab laser introduces some interesting 
phenomena. The phenomena are caused by the 
refractive index of the slab and the reflector geometry 
required for solar concentration. 
Any refractive index of slabs that mismatches 
that of free space causes reflections. Light is reflected 
from the surfaces, transmitted through the surfaces, 
and reflected and absorbed between the surfaces of 
the slab. Less light enters slab media (especially 
solids and liquids) because of reflections, but more 
of what enters is absorbed because of reflections. 
Dish geometry necessarily introduces a solid an- 
gle of incidence on the slab and a variation of radiant 
power with angle of incidence. More power is inci- 
dent at large angles of incidence, but more is reflected 
(compared with that at small angles of incidence). 
Also, large angles of incidence create longer absorp- 
tion paths in the solid. 
A calculation that includes these phenomena is 
needed to determine power absorption. A separate 
calculation is made for light that is not incident 
through a solid angle (collimated light). A compar- 
ison of the two calculations reveals the influence of 
refractive index and solar concentration on absorbed 
power. The results of the two calculations can be 
applied to more than laser technology. 
Analysis 
In this analysis, a slab is located at the focal 
point of a parabolic dish concentrator. The plane of 
the slab is perpendicular to a line joining the focus 
and the vertex of the dish. The dish is uniformly 
irradiated over its aperture, except where the slab 
blocks 1 percent of the light (see the appendix) 
and all that radiation is incident on the slab. For 
the purposes of this calculation, the slab is large 
enough and thin enough that loss of radiation at  its 
extremities is negligible. 
At points on the slab, light is incident at  angles 
from 0' to some large angle that depends on the 
focal length and diameter of the parabolic dish. In 
this geometry, radiation at small angles of incidence 
comes from more central regions of the dish aperture, 
whereas radiation at large angles of incidence comes 
from peripheral regions of the aperture. Since the 
peripheral regions of the aperture encompass more 
area than the central regions, more radiant power is 
incident at  large angles of incidence than is incident 
at  small angles. Also, at large angles of incidence, the 
absorption path length in the slab is increased. On 
the other hand, reflection at the slab surfaces tends 
to increase. From figure 1, the differential power 
incident on the slab is given by 
d P  = I dA = 2 r I r  sin 0 d(r sin 0) 
= 2nIr sin 0 (r cos 0 dO + sin 0 dr) (1) 
where 
I aperture irradiance 
A aperture area 
r distance of parabolic surface from focus 
0 angle of incidence 
For a parabola, 
r  = 
R 
1 + cos 0 
R sin 0 d0 dr = (1 + cos 0)2 (2) 
where R is the radius of the parabola for 0 = 
90'. Substituting equation (2) into equation (1) and 
integrating gives 
P = 2 n R 2 1 i o '  [ (1 sin + 0 cos cos 0)2 0 + ( 1 + sin COS v3] 0 d0 
(3) 
where 0' is the maximum angle of incidence. Smaller 
diameter dishes with the same focal length subtend 
smaller angles. In that case the dish radius is ex- 
pressed by 
R sin 0' q = r  sin 0 = 
1 + cos 0' 
q(1 + cos 0') R = 
sin 0' (4) 
Hence, substituting equation (4) into equation (3) 
gives 
1 + cos 0' A(O1) = 
sin 0' 
sin 0 cos 0 + ( sin 0 ) 3  
B(0) = (1 + cos 0)2 1 + cos 0 
The portions of incident s and p polarized power 
absorbed by the slab are given by (ref. 3, pp. 36-41) 
and 
where 
-nut E, = exp (-) cos 4 = exp (2) cos 4 
and 
2 
n density of absorbing particles in slab 
u absorption cross section of particle at  a 
particular wavelength 
t thickness of slab 
CY = nut 
N index of refraction of slab at  a particular 
wavelength 
Rs, Rp Fresnel power reflection coefficients at 
slab surfaces (parallel and perpendicular 
components of polarization) 
Power absorbed in the slab then is given by 
Since Ap and As are functions of the angle of in- 
cidence, the portion of power incident on the dish 
aperture that is absorbed by the slab is given by 
where 
en = nq21 
is the power incident on the parabolic aperture. Solar 
concentration can be related to the solid angle sub- 
tended by the dish (if dish reflectivity is neglected) 
by the relation (ref. 4, p. 243) 
where a is the half-angle of solar subtense near Earth 
(approximately 0.25'). Equation (7) could be substi- 
tuted into equation (6) to obtain the absorbed power 
as a function of solar concentration. The results are 
clearer however if Pa/Pin remains a function of the 
angle of incidence. 
Results 
Equation (6) was evaluated on a computer for var- 
ious values of N ,  0, and 0'. The solid curves of fig- 
ures 2 to 5 show the results for several real substances 
and are plotted to feature the effect of concentration 
(solid angle). Figure 2 depicts slabs that can be ap- 
proximated by gases or with broadband antireflection 
coatings on solids. Results for a slab of water and 
some plastics are shown in figure 3. Figure 4 shows 
the results for most commonly encountered glasses 
(index of refraction (N)  of 1.5). Figure 5 shows the 
absorption at  N = 1.823 of a Nd:YAG slab (i.e., a 
neodymium doped yttrium aluminum garnet slab). 
The data of figures 2 to 5 are plotted for half-degree 
increments of the abscissa. Some information on the 
variation of absorbed power with refractive index can 
be obtained by comparing figures 2 to 5. However, 
that information is plotted more explicitly in figure 6. 
The curves of figure 6 are the loci of the maxima of 
the solid curves in figures 2 to 5. 
If the power incident on the slab were incident at 
only one angle instead of being distributed through 
a range of angles, the fraction of absorbed power 
is represented by the dotted curves on figures 2 
to 5. The difference of the dotted- and solid-line 
curves represents the difference between absorption 
of concentrated light (incident at all angles up to 0') 
and absorption of collimated light (all of it incident 
at angle 0'). The comparison that best shows the 
effect of the solid angle of incidence, however, is the 
comparison between absorption of concentrated light 
(corresponding to a maximum angle 0') with that of 
collimated light at an angle of incidence of 0'. (In 
both of these cases, slab orientation is the same.) 
Discussion 
The influence of solar concentration is evident 
mainly at  large values of concentration (large angles 
of incidence) in figures 2 to 5. At these large an- 
gles there is a slight increase in the fractional power 
absorptance over that at small angles of incidence. 
The solid angle of peak absorptance decreases ap- 
proximately 20' in going from the less absorptive to 
the more absorptive slabs ( N  > 1.0). At very large 
solid angles absorbed power decreases somewhat, de- 
pending on the absorption coefficient. The decrease 
is caused primarily by increased reflection at graz- 
ing incidence, but it is counteracted to some degree 
by increased absorption path length in the slab and 
greater power collection at  larger angles of incidence. 
The most significant influence of refractive index 
is best shown by figure 6. There is a decrease in 
the fraction of absorbed power between indices of 1.0 
and 1-.l, and it is much more significant in the less 
absorptive slabs. In terms of materials, this means 
that solids and liquids absorb amounts of power (ex- 
pressed as a fraction of incident power) that are 
comparable to each other and are significantly less 
than the fractional power absorption of gas and anti- 
reflection coated slabs (all having the same absorp- 
tion coefficient). (Because of their density, however, 
thin gas slabs compare only with weakly absorbing, 
thin, solid or liquid slabs.) 
Figure 2 represents the extreme case ( N  = 1.0) in 
which there is no reflection from the front surface of 
the slab. Therefore, at  grazing angles of incidence, 
absorption path lengths in the slab increase toward 
infinity and absorption approaches totality for col- 
limated light. Concentrated light, conversely, never 
reaches total absorption because most of the rays are 
not at grazing incidence. 
The absorption coefficient is usually dependent on 
the wavelength of light. Thus, these results apply 
strictly to light within a wavelength band where 
the absorption cross section of the absorbing species 
does not vary significantly. On the other hand, 
the results do not apply to coherent radiation (very 
narrow wavelength band) because the reflectance 
formulas apply only to incoherent light. To apply the 
results to wavelength bands that include a significant 
variation in absorption coefficient, summations of 
smaller bands must be made. 
The index of refraction and the absorption coeffi- 
cient are related through material permittivity such 
that at a wavelength where most absorption occurs, 
reflection maximizes also. In laser media the density 
of absorbing particles is usually small enough that 
the overall effect on the refractive index is small. In 
applications where the density of resonant absorbing 
particles is large, the absorption coefficient and the 
refractive index should both be measured at  the same 
wavelength. 
Concluding Remarks 
For parabolic dish concentrators, the main effect 
of solar concentration is a slight increase in the frac- 
tion of absorbed power at large solid angles (maxi- 
mum angles of incidence (0') from 60' to 80'). At 
a given index of refraction, the solid angle of maxi- 
mum absorptance decreases as absorption coefficient 
increases. The fraction of power absorbed from nor- 
mally incident light (0' = 0') is a good approxima- 
tion to the fraction absorbed through solid angles less 
than 27r steradians. 
The influence of refractive index is primarily a re- 
duction in the fraction of absorbed power with an in- 
crease in refractive index. The reduction is dramatic 
for weakly absorbing materials with a refractive in- 
dex in the range from 1.0 to 1.1. For more absorptive 
materials, the variation is smaller and spreads over a 
larger range of refractive indices. 
Calculations of absorbed power (expressed as a 
fraction of incident power) have been made for sev- 
eral representative materials and can be used to de- 
termine power absorption of slabs accurately if all 
the concentrated power is incident on the slab and 
spectral parameters are known at  the wavelength of 
interest. If only a fraction of the concentratcd power 
irradiates the slab, absorbed power must be reduced 
by that fraction. 
Appendix 
Solar Irradiance 
r sin a e = 
cos (0' + a )  
The Sun provides a radiance of I watts per square 
meter of surface area per unit solid angle. Since each 
unit surface area radiates isotropically, the Sun may 
be treated as a radiating flat disk at  distances far 
from it. In the vicinity of Earth, the power received 
from solar radiation is 
where 
A, area of solar disk, T R ~  
Rs radius of solar disk 
R solid angle subtended by area A near Earth, 
AIR; 
Re distance from solar disk 
Near Earth, the bracketed term is a fixed quantity 
called the solar constant. Thus, the power received 
from the Sun near Earth is proportional to the area 
that intercepts the sunlight. 
At any area element dA, solar power is received 
through the solid angle T (R,/R[)~ and can be trans- 
mitted, reflected, and/or absorbed through the same 
solid angle. Any such area element on a parabolic 
dish reflects power through the same solid angle to 
the focal region. 
At the focal region the radiation is received over 
an elliptical area. The farthest extent of the ellipse 
from the focal point of the parabolic dish is given by 
and is illustrated in sketch A. Using equation (2) of 
the text gives 
q sin a e = 
cos(0' + a )  sin 0' 
Parabolic 
surface 
Focal point 1 
Sketch A 
The distance e determines the radius of a slab that 
can intercept all radiation reflected from the dish. If 
the area of the slab is arbitrarily set to 1 percent of 
the aperture area of the dish, then 
4 - 
-
cos(0' + a) sin 0' 
e - 
= 10 
sin a 
a = 0.25' and 0' = 87.24' 
References 3. Bohren, Craig F.; and Huffman, Donald R.: Absorption 
1. Williams, M. D.; and Zapata, L.: Solar-Pumped Solid and scattering of Light b y  Small Particles. John wiley & 
State Nd Lasers. NASA TM-87615, 1985. Sons, Inc., c.1983. 
2. Williams, M. D.; and Conway, E. J., eds.: Space Laser 4. S a ~ i g h ,  A. A. M., ed.: Solar Energy Engineering. Aca- 
Power Transmission System Studies. NASA CP-2214, demic Press, 1977. 
1982. 
Figure 1. Geometric variables. 
Concentrated 
....................... Collimated 
0 10 20 30 40 50 60 70 80 90 
Maximum angle of incidence, 8 ', deg 
Figure 2. Optical power absorbed in coated slab. N = 1.0. 
t Concentrated 'I 
....................... Collimated 
1 I I I 
0 
I t 
10 
I 
20 
I 
30 40 5 0 6 0 70 80 90 
Maximum angle of incidence, e ', deg 
Figure 3. Optical power absorbed in liquid slab. N = 1.32. 
0 .8 
a, 
e 
................. 2 .7 - ..__ 
_._ "".....,.. - ....... ................... m ....,I _ .............. 
..... 
L 
..- $ . 6 -  ,,_..... _." .....-.......... ......... -...... ... 
0 
a 
E .5 - _ -..._... .( ....... ....... ........ 
,._-._- ... c ~ ~ ' ~ " " . ' ' '  .... a, cc = 0.6 
z 
," .4 - 
.- 
- 
o a = 0.4 
s .3 - 
.- + 
0 
s a = 0.2 
LL .2 - 
.1 - Concentrated 
....................... Collimated 
I I t I I I 
0 
I 
10 
I 
20 3 0 40 50 60 70 80 90 
!vkximum angle of incidence, 8 ', deg 
Figure 4. Optical power absorbed in glass dab. N = 1.5. 
Figure 5. Optical power absorbed in Nd:YAG slab. N = 1.823. 
1 .o 
Index of refraction, N 
.9 
Figure 6. Optical power absorbed from concentrated light. 
- 
0 = 0.4 
C 
0 .3 - 
.- + 
0 
9 a = 0.2 
LL .2 - 
.1 - 
....................... Collimated 
I I I I I I I I 
0 10 20 30 40 50 60 70 80 90 
Maximum angle of incidence, 0 ', deg 
NASA FORM 1626 OCT 86 NASA-Langley, 1988 
For sale by the National Technical Information Service, Springfield, Virginia 22161-2171 
NASA 
Nalonal Aeronaulics and Report Documentation Page 
Soace Admnlslrat~on 
3. Recipient's Catalog No. 
5. Report Date 
July 1988 
6. Performing Organization Code 
1. Report No. 
NASA TM-4056 
2. Government Accession No. 
4. Title and Subtitle 
Influence of Refractive Index and Solar Concentration on Optical 
Power Absorption in Slabs 
7. Author(s) 
M. D. Williams 
9. Performing Organization Name and Address 
NASA Langley Research Center 
Hampton, VA 23665-5225 
12. Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Washington, DC 20546-0001 
8. Performing Organization Report No. 
L-16442 
10. Work Unit No. 
506-41-41-01 
11. Contract or Grant No. 
13. Type of Report and Period Covered 
Technical Memorandum 
14. Sponsoring Agency Code 
15. Supplementary Notes 
16. Abstract 
The optical power absorbed by a slab at the focus of a parabolic dish concentrator is calculated 
as a fraction of the power incident on the dish. The calculations are plotted versus maximum 
angle of incidence of irradiation (which corresponds to different magnitudes of solar concentration) 
with absorption cocfficicnt as a parameter for several different indices of refraction that represent 
real materials. Results show that the effect of the solid angle of incidence is a small increase in the 
fraction of absorbed power over that of a similarly oriented slab irradiated by collimated light. Large 
indices of refraction allow less fractional absorptance and affect the solid angle at which maximum 
fractional absorptance occurs. 
17. Key Words (Suggested by Authors(s)) 
Solar concentration 
Slab lasers 
Optical absorption 
Refractive index 
18. Distribution Statement 
Unclassified-Unlimited 
Subject Category 74 
19. Security Classif.(of this report) 
Unclassified 
21. No. of Pages 
9 
20. Security Classif.(of this page) 
Unclassified 
22. Price 
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Influence of refractive index and solar concentration on optical power absorption in slabs

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