A
METHOD FOR CONTROLLING
THE DIVERGENCE OF LIGHT
BY
RICK OSTROV
It is a fundamental axiom of Optics that
when light hits a mirror the angle of reflection will
be equal to the angle of incidence. Thus, when light
comes from an extended source, the divergence angle
(α) of the light reflected from a plane mirror
will be equal to the angle subtended by the source.
When this reflected light hits a "target", the size
of the image spot produced is determined by the distance
through which the light has traveled and the rate at
which it was diverging (see Fig.1).
In many applications it would
be useful to have a beam of light that was less divergent
even at the sacrifice of an optical image. This can
be accomplished by dual or multiple reflections. A second
mirror can be positioned so that it reflects only a
portion of the entire beam coming from the primary mirror.
In this case, the subtended angle of the source, as
seen from the secondary mirror, is only a portion of
the subtended angle of the source as seen from the primary
mirror. As a result, the divergence angle of the beam
reflected from the secondary mirror is also less than
the divergence angle of the primary beam reflected from
the primary mirror (see Fig.2).
The secondary reflected beam has a divergence
angle that is smaller than a in proportion to the diameter
of the secondary mirror compared to the diameter of
the primary image spot at the same distance.
The radiant flux in the secondary reflected beam
is proportionally less than the flux in the primary
beam as the area of the secondary mirror is to the area
of the primary image spot.
If there are enough secondary mirrors
to cover the entire area of the primary image spot,
assuming a perfect optical system with no losses, and
all of them reflect their beams at the same "target"
or secondary spot, then their total energy will be concentrated
at this spot.
The secondary beam will be diffraction
limited by the size of the secondary mirror according
to Rayleigh's criterion ( α = 1.22 x λ [Wavelength
of the light] / d [diameter of secondary mirror]).
Since the distance between the primary and secondary
mirrors is so large in comparison to the diameter of
the secondary mirror and the radius of the primary image
spot, the cosine to the fourth law, which describes
the radiance from a Lambertian source such as the sun,
produces the same calculated results.
Versions of this method have been proposed
in the past for use in space based energy systems. Beginning
in the 1970's, Krafft A. Ehricke proposed a system utilizing
large, paired satellites that would take turns functioning
as primary and secondary mirrors depending on whether
they were moving toward or away from the sun in their
orbit around the earth. Half or more of the light coming
from the primary would bypass the secondary and thus
be wasted. This system was proposed and accordingly
sized for various uses ranging from night illumination
for urban areas, to extended hours of agricultural production
on land (for growth, sowing and harvesting) and for
aquaculture applications at sea, for emergency and natural
disaster rescue operations, for large scale power production
and for potential weather modification. While these
systems would be huge, and the cost of putting the materials
into space and assembling the structures would be large,
the environmental benefits and financial profits would
be even larger.4
Since then others have worked in this
area.5 In
addition, at least one patent has been granted to A.K.
Heitzmann for a satellite to collect solar energy and
beam it to the earth.6
This satellite is a large curved dish that focuses
light on a small dish held out at the focal distance
by structural "arms" connected to the large dish. The
large dish could be parabolic and perhaps a km in diameter;
the small dish would be comprised of many small, 4cm
mirrors, 320 across for a total diameter of about 13m.
The beam from each 4cm secondary mirror would spread
to 1200m wide at the surface of the earth. Each small
mirror would be fixed in place, in the small dish, so
that the beam from each one overlapped the beams of
the others at the surface of the earth. The small dish
could alternatively be made of small collimating lenses,
similarly angled so their beams overlapped, all of whose
beams would then be reflected at once to the earth by
a mirror. The advantage of this system is its smaller
size and mass. However, it would still require a costly
transport to and assembly in space and the beam would
not be easily adjustable in size or concentration.
The system described here, called Starbeam,
utilizes adaptive optics rather than fixed structures.
One possible configuration might be an off-axis parabolic
collector that focuses light on a properly oriented
small parabolic concentrator at the correct focal distance;
this beam would be reflected to a primary plane mirror
which would direct it through an expansion field to
the array of small secondary mirrors which would direct
the separate, smaller and less divergent beams to the
target (see Fig.3).
A Starbeam system can consist completely
or partially of individual, computer controlled, adaptive
optics elements. These elements would be optimally sized
depending on the purpose, function and geometry of the
entire system. The parabolic collector might consist
of flat, 1 m^2 units. In space, each unit could have
smaller steering vanes (or movable rim units); these
would be used to control the orientation of the unit
and could also be used to maneuver the unit (or a replacement)
into place. These flat units would typically have a
mass of 10-60 g/m^2 (or more if desired) and could be
“solar sailed” into place.7
All the elements in a space based Starbeam adaptive
optics system would be oriented and maintained by simple
parallel processing computer controls that would guide
the system throughout in order to maximize the beams
on the target.8
A space based Starbeam energy production
system in geostationary earth orbit (GEO), might have
the following approximate dimensions: parabolic collector
= 1 km^2; parabolic concentrator = 7m diameter; plane
mirror reflector = 10m diameter; and the secondary mirror
array would consist of 330 units, each 12cm across,
for a total diameter of 40m. The expansion field distance
between the primary plane mirror reflector and the secondary
mirror array would be 4301m. The total mass in space
would be about 11,000 KG (includes a 10% overallowance).
The beam reflected to the earth would be 1 km in diameter,
930 W/m^2 in power and consist primarily of visible
light and infrared wavelengths from about .4 to 2 micrometers
(from using UV filter coating and/or silver coated mirrors).
This represents approximately 85% of the solar constant
and that portion least harmful to the atmosphere and
the environment. The stable beam, from GEO, can be collected
and concentrated (e.g. by compound parabolic collectors
on the ground) and thereby utilized in a highly efficient
electric power generation system (e.g. 55% efficient
using a gas turbine, combined-cycle system).
The cost of the electricity produced from
this Starbeam system would be 1.5 cents per KWH.9
This system would produce 512 MW of power 22 hours
a day (with 10% downtime for maintenance). This system
would pay for itself in 3 years and still produce about
a $650 million profit during the 3 years. Over a 35
year timespan, this system would produce a profit of
almost $15 billion. If the space based optics portion
of the Starbeam system were twice as large or expensive,
the system would still pay for itself in 3 years and
produce a $330 million profit during this time and a
total $14.6 billion profit over 35 years. If the electric
power was used at night to produce fuel (e.g. hydrogen)
for a second plant to generate electricity during the
day, the cost of electricity would be about 4 cents
per KWH.10
If the space based portion of the system cost
three times as much as estimated and if the fuel conversion
plant cost twice as much, the electricity cost would
be about 5 cents per KWH; this would still produce a
profit of about $7 billion over 35 years. It is also
important to note that the space based portion of the
system could function for perhaps a hundred years or
more while elements of it could be upgraded or replaced
if desired.
All of the basic elements in the Starbeam
system are within the compass of existing technology
although research and development would be necessary
to design and refine an actual system. Such a system
could be available within a relatively short timeframe.
Little or no human assembly in space is required eliminating
one of the largest expenses and impediments to most
space utilization concepts. Starbeam systems can beam
energy down to urban areas around the world with little
or no pollution at costs that are competitive with or
cheaper than current 2005 market rates for electricity.
Starbeam systems can be designed in various
sizes for different purposes. The Starbeam system has
potential applications, as described above, for night
lighting in various situations (e.g. urban areas or
emergencies), for possible weather modification by creating
low pressure areas or cloud dispersal, etc. and for
agricultural applications. These are just some of the
possible Starbeam applications.
Patent
applied for by Rick Ostrov 2005
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