Luminescent
solar
concentrators.
1:
Theory
of
operation
and
techniques
for
performance
evaluation
J.
S.
Batchelder,
A.
H.
Zewail,
and
T.
Cole
Techniques
and
calculations
are
presented
that
give
explicit
expressions
for
the
over-all
performance
of
a lu-
minescent
solar
concentrator
(LSC)
in terms
of
the
intrinsic
spectral
response
and
quantum
efficiency
of
its
constituents.
We
examine
the
single
dye
(or
inorganic
ion)
LSC
with
emphasis
on the
planar
geometry.
Preliminary
data
on
the
degradation
of
candidate
LSC
dyes
under
severe
weathering
conditions
are
also
given.
Armed
with
our
experimental
results
and
analysis
of
solar
absorption,
self-absorption,
and
solar
cell
efficiency,
we
present
a
new
genre
of solar
concentrator
with
a theory
of
operation
for
the
device.
1.
Introduction
A
new
concept
in
solar
energy
concentration
tech-
nology
has
been
evolving
over
the
past
several
years,
which
has
given
rise
to
a device
called
a
luminescent
solar
concentrator,
or
LSC
(see
Appendix
A
for
an
his-
torical
account).
In
our
earlier
paper'
we
referred
to
the
generic
device
as
a planar
solar
concentrator,
or
PSC.
However,
since
the
optimal
geometry
for
an
LSC
may
well
involve
a nonplanar
device,
we
shall
use
the
general
name
of
LSC
for
this
entire
classification
of
solar
concentrator.
The
operation
of
an
LSC
is
based
on
the
idea
of
light
pipe
trapping
of
molecular
or
ionic
lumi-
nescence.
This
trapped
light
can
be
coupled
out
of the
LSC
into
photovoltaic
cells
(PVC)
in
such
a
way
that
the
LSC
provides
a
concentrated
flux
that
is
spectrally
matched
to
the
PVC
so
as
to
reduce
the
radiation
heating
and
increase
the
electrical
output
of
the
PVC.
An
LSC
does
not
need
to
track
the
sun
and
in
fact
can
produce
highly
concentrated
light
output
under
either
diffuse
or
direct
insolation.
The
conceptual
operation
of
an
LSC
is
illustrated
by
the
diagram
of
a planar
solar
concentrator
(PSC)
shown
in
Fig.
1;
a
transparent
material
(e.g.,
polymethyl
methacrylate,
PMMA)
is
impregnated
with
guest
lu-
minescent
absorbers
(e.g.,
organic
dye
molecules)
having
strong
absorption
bands
in
the
visible
and
UV
regions
A.
H.
Zewail
(to whom
correspondence
should
be addressed)
and
J.
S.
Batchelder
are
with
California
Institute
of
Technology,
Department
of Applied
Physics
and
A.
A. Noyes
Laboratory
of
Chemical
Physics,
Pasadena,
California
91125.
T.
Cole
is with
Ford
Motor
Company,
Engineering
&
Research
Staff,
Dearborn,
Michigan,
48121.
Received
13
April
1979.
0003-6935/79/183090-02$00.50/0.
Oc
1979
Optical
Society
of
America.
of the
spectrum,
and
also
having
an
efficient
quantum
yield
of
emission.
Solar
photons
entering
the
upper
face
of
the
plate
are
absorbed,
and
luminescent
photons
are
then
emitted.
Snell's
law
dictates
that
a
large
fraction
of
these
luminescent
photons
are
trapped
by
total
internal
reflection;
for
example,
about
74%
of
an
isotropic
emission
will
be
trapped
in
a
PMMA
plate
with
an
index
of
refraction
of
1.49.
Successive
reflec-
tions
transport
the
luminescent
photons
to
the
edge
of
the
plate
where
they
can
enter
an
edge-mounted
array
of
PVCs.
The
photon
flux
at
the
edge
of an
idealized
LSC
is the
product
of
the
absorbed
solar
flux,
the
fraction
of the
resulting
luminescence
that
is
trapped,
and
the
geo-
metric
ratio
of
the
area
of
the
face
directly
exposed
to
sunlight
divided
by
the
area
of
the
edge
that
is
covered
by
solar
cells.
Using
the
PSC
of
Fig.
1
as
an
example,
a unit
length
of
the
plate
which
is
L
units
wide
and
D
units
thick
will
have
a
geometric
gain
Ggeom,
which
is
given
by
Ggeom
=
LID
=
Af/Ae,
(1)
where
Af
is
the
area
of
a
face,
and
Ae
is
the
area
of
an
edge.
A
typical
PMMA
plate
is
3
mm
thick,
so
that
a
square
meter
PSC
section
will
have
a geometric
gain
of
Ggeom
=
333.
Such
a
gain
exceeds
the
concentration
of
other
known
nontracking
collectors
using
lenses
or
mirrors.
Thus
a high-cost
high-efficiency
solar
cell
can
be
coupled
to
this
high-gain
low-cost
concentrator
for
a potentially
low-cost
system.
However,
as
nature
usually
dictates,
a
practical
LSC
will
have
a
number
of
parasitic
losses
that
limit
the
ac-
tual
concentration
to
values
lower
than
Ggeom.
Among
these
losses
are
inadequate
absorption
bandwidth,
imperfect
quantum
efficiency,
self-absorption
of
lu-
minescence,
absorption
by
the
matrix
material,
reflec-
tive
mismatches,
geometrical
trapping
effects,
and,
of
3090
APPLIED
OPTICS
/
Vol.
18,
No.
18
/
15 September
1979
x fir
MIRRORED
EOGE
0
C
MIRROREO /
BRCK
Fig.
1.
A planar
solar concentrator,
or
PSC, which
is
the particular
embodiment
of an LSC.1
0
Sunlight
enters
from
above
and passes
twice
through
the plate
thickness
D,
during
which
a dye of
inorganic
ion absorbs
a certain
portion
of
the solar
flux. The
ensuing
lumi-
nescence
can either
escape
back out
of the face
(A) or
be trapped
by
total internal
reflection
(B).
This trapped
light will
then propagate
to the
photovoltaic
cells
(PVC) where
it is
absorbed
and converted
into electricity.
course, the
lifetime of the
materials
used. Clearly
system
optimization
means
close attention
to
mini-
mizing
these
various losses.
In a previous
publications we
have demonstrated
a
method
for overcoming
one of
the above losses-namely
inadequate
absorption
bandwidth.
For example,
by
including
several
dyes with
successively
overlapping
absorption
and emission
bands,
solar photons
can be
absorbed
over the integrated
absorption
spectrum of all
the
dyes, with
a cascade being
formed
by excitations
being
transferred
from one dye
to the next. In
such a
multiple-dye
system
the over-all
efficiency will
also
depend
on the mechanisms
by
which energy
is trans-
ferred from
one molecule to
the next.
In this
paper we shall
present
a formalism
for ex-
pressing
the operating
characteristics
of an
LSC-PVC
system
in terms
of measured molecular
spectral
re-
sponses and PVC
characteristics.
Preliminary
exper-
iments on LSC
dye stability
and self-absorption
are
reported.
We will emphasize
the methods
used in cal-
culating
the performance
of an
LSC containing
a single
luminescent
species
and will
typically use
the PSC of
Fig. 1
for illustrative
calculations,
although
the for-
malism
is developed
in such a way
so as to be readily
adaptable
to LSCs
with more
complex
geometries
containing multiple
luminescent
dyes or inorganic
ions.
In a subsequent
paper we
will treat the
formalism of
multiple-dye
LSCs and
nonplanar geometries,
together
with
our experiments
on prototype
systems.
II. Single
Dye LSC Formalism
Few processes
in an LSC
are subtle; an
accurate cal-
culation
of the LSC output
requires
that the various
channels
that the photon
can take
be identified
and
weighted
appropriately.
A general
description
of these
channels is given
in the following
section, and
a pictorial
flow
chart is given
in Fig. 2. Subsequently
we develop
relationships
for the coefficients
weighting each
chan-
nel. For
clarity we will
typically discuss
an LSC using
AIR
LSC
PVC I LSC
Loss
l
Fig. 2. A
photon flow
diagram depicting
the predominant
channels
available in an
LSC. Dotted lines
represent changes
in index of re-
fraction,
and squares represent
photon sinks.
Light from the sun
enters the
dye ensemble, resulting
in lumine cence
that is converted
in the
PVC. The two feedback
loops around
the dye ensemble
rep-
resent
the effects of self-absorption
inside and
outside of the critical
cones.
organic
dyes in the PSC
geometry of
Fig. 1. However,
unless
explicitly stated
the results
are applicable
to any
absorber
or geometry.
A.
Photon Flow
Diagram for
i Single Dye System
We can
trace the flow
of excitations
in an LSC with
the aid
of the flow chart
in Fig. 1. Above
all is the
sun.
Part
of its incident
flux will be lost
directly by reflection
from the LSC
surface, and
part is lost because
its
wavelength does
not correspond
to the absorption
band
of the dye used.
What is left
is the absorbed
solar flux
in the dye
ensemble,
denoted by
S. There is a
net rate
of excitation
J of the
dye ensemble,
which in steady
state
must correspond
to the rate of
deexcitation.
The
photon
output of the
dye ensemble
is the quantum
ef-
ficiency
of luminescence
n times
J. This luminescence
is
geometrically divided
into the
fraction JPq, which
is
emitted within
any of the critical
escape cones,
and the
fraction J(1
- P), which
is trapped. For
light within
the critical
cones there
is an average probability
T that
self-absorption
will
take place before
the light can
es-
cape
out of the LSC,
so that there
is a feedback
loop of
magnitude
PTJi7 of emissions
in the
critical cones
that
are recovered
as excitations
in the
dye ensemble.
A similar
feedback loop
occurs with
the self-absorp-
tion
of light, which
is trapped by total
internal reflec-
tion; in this case
the probability
that a trapped
photon
will be self-absorbed
before
it reaches the
LSC-PVC
interface is
r. An additional
lumped parameter
3 de-
scribes the fraction
of the trapped
luminescence
which
is lost
due to matrix
absorption
or imperfect
reflections.
15 September
1979 / Vol. 18,
No. 18 / APPLIED
OPTICS 3091
A
hardy
fraction
Q
of
the
originally
absorbed
solar
photons
arrives
at
the
LSC-PVC
interface
where
a
re-
flection
takes
place
of
magnitude
Rpm-
In
this
analysis
we
will
assume
that
such
reflected
flux
is lost;
to
the
extent
that
this
assumption
is
not
true
we
will
under-
estimate
the
final
output.
B.
Solar
Absorption-S
The
solar
spectrum
is
a
variable
quantity.
The
spectrum
that
we
have
used
in
our
calculations
is
the
Air
Mass
1 (AMi)
total
incident
radiation,
measured
on
a
clear
cool
summer
day
in
Delaware.
2
The
spectral
flux
per
wavenumber
will
be
noted
by
N(v)
and
the
total
flux
by
I,
where
I=
f
N(v)dv.
(2)
Since
both
the
spectral
distribution
and
the
total
flux
vary
considerably
with
atmospheric
conditions,
an
ac-
curate
performance
prediction
requires
that
a
solar
spectrum
be
used
that
duplicates
as
closely
as
possible
the
sunlight
that
will
be
found
under
typical
operating
conditions.
3
We
will
also
use
the
normalized
function
U(Oi,V)
for
the
angular
distribution
of
the
incident
light,
so
that
any
combination
of
diffuse
and
direct
sunlight
can
be
modeled.
In
this
case
the
total
incident
flux
be-
comes
I
= f
di
,f
dOiN(V)U(Oi,)
sin(Oi)
cos(Oi)
(3)
with
the
normalization
condition
for
U(Oi,i)
given
by
1
=
d
,
dOjU(Ojv)
sin(Oi)
cos(Oi).
(4)
A
fraction
of
the
sunlight
will
inevitably
be
reflected
by
the
LSC
surface.
If
no
antireflection
coating
is
used,
the
reflection
coefficient
R
(0i)
for
unpolarized
light
is
given
by
the
Fresnel
equation
4
R(O,)
=
ftan
2
(Oi
-
t)
+
sin
2
(0,
- Ot)
2
tan
2
(
+
Ot)
sin
2
(0
+
t)j
(5a)
sin(Oi)
=
n
sin(Ot),
(5b)
where
n
is the
index
of
refraction
of
the
matrix
material.
The
transmission
coefficient
T(0)
is
given
by
T(Oi
)
=
1 -
R(0
) .
(5c)
Increasing
the
index
of
refraction
of
the
matrix
material
increases
the
loss
of
sunlight
due
to
surface
reflections,
but
it
also
increases
the
fraction
of
the
luminescence
that
is
trapped
by
total
internal
reflection.
The
critical
angle
0,
for
total
internal
reflection
is
0
=
sin-'
(1/n).
(6)
In
Sec.
II.
D
we
show
that
in
a planar
LSC,
such
as
in
Fig.
1,
this
leads
to
the
geometrical
fraction
P
of
the
luminescence
that
is
emitted
at
an
angle
0
<
0
c,
i.e.,
not
trapped
by
total
internal
reflection,
which
is given
by
P
=
1 -
(1
-
1/n
2
)
1
/2.
(7)
that
T(Oi)(1
-
P)
be
maximized
with
respect
to
the
index
of
refraction
n.
If
no
antireflection
coating
is
used,
this
occurs
at
n =
2.5
The
addition
of
an
antireflection
coating
might
be
desirable
to
remove
some
of
the
reflective
loss,
especially
of
higher
index
matrix
materials
are
used.
In
this
case
the
reflection
coefficient
for
unpolarized
light
is
R(6i,i),
6
where
1
r
2
+ r
+
2r
12
r
23
cos(2)
R (0i,
T)
2
1 +
r 2r2
3
+
2r
1
2r
23
cos(20)
+
R12
+
R2
3
+
2R
1
2
R
2
3
cos(2)]
R1
2
R2
3
+ 2R
12
R
23
cos(2,13)
cos(0)
- n
cos(0
1
)
cos(Oi)
+
nj
cos(0
1
)
nj
cos(0i)
-
n
cos(Ot)
n
1
cos(01)
+ n
cos(0t)
R
12
nj
cos(01)
-
cos(0)
n
1
cos(Oi)
+ cos(0
1
)
R
23
n
cos(Ot)
-
n
cos(01)
n cos(01)
+ n
cos(Ot)
=
27rnl
h
v cos(0
1
),
sin(Oi)
=
n
1
sin(6
1
=
n
sin(Ot).
(8a)
(8b)
(8c)
(8d)
(8e)
(8f)
(8g)
ni
is
the
index
of
refraction
of
the
antireflection
coating,
and
h
is
its
thickness.
Since
the
critical
cone
angle
depends
only
on
the
index
of
the
matrix
material,
add-
ing
an
AR
coating
does
not
change
the
fraction
of
lu-
minescence
that
is
trapped,
so
that
in
this
case
the
matrix
index
should
be
made
as
high
as
possible.
Numerical
Example:
Typical
matrix
materials
such
as
PMMA
have
an
index
of
refraction
of
about
1.5.
The
critical
angle
is found
by
Eq.
(6)
to
be
= sin-
1
(1/1.5)
=
420,
and
the
fraction
of
the
luminescence
that
can
escape
out
of
the
critical
cones
is
P
=
1 -
[1
-
(1/1.52]1/2
=
0.255
for
a
planar
device.
Without
an
antireflection
coating,
unpolarized
light
will
be
96%
transmitted
at
vertical
incidence,
and
94.2%
is
transmitted
at
an
angle
of
incidence
of
50°.
If a
MgF
2
antireflection
coating
is
used,
with
an
index
of
1.38,
the
normal
transmission
is
98.5%,
and
at
500
the
transmission
is
97.4%
for
6000-A
light
and
a 1200-A
coating
thickness.
Passing
the
air-LSC
interface,
the
light
will
be
par-
tially
absorbed
by
both
the
dye
molecules
and
the
ma-
trix
material.
We
define
an
absorption
coefficient
for
the
dye
(v):
a(i)
=
CE(1)
ln(10),
(9a)
where
C
is the
molar
dye
concentration
in
the
matrix
material,
and
E(v)
is
the
molar
extinction
coefficient
of
the
dye.
The
ratio
of
the
transmitted
intensity
It ()
to
the
initial
intensity
I
()
over
a
path
length
x is
then
It(T)/Ii(v)
=
exp[-a(T)x]
=
10
_-Cv)x.
(9b)
(9c)
Similarly
let
cxm(v)
be
the
matrix
absorption
coefficient.
so
that
the
total
combined
absorption
coefficient
cet
(-j
is
Maximizing
the
amount
of trapped
light
then
requires
at(V)
=
a(V)
+
am().
3092
APPLIED
OPTICS
/ Vol.
18,
No.
18
/
15 September
1979
(10)
We
now
have
sufficient
information
to determine
the
amount
of
solar
flux
S
that
is absorbed
by
the
dye
per
unit
area
of
the LSC.
In
its most
general
form,
S
is
defined
as follows:
S
=
dT gJ
dOiT(0i,T)N(i)U(0i,V)
()
at
(i
X $1 -
exp[-at(v)ls]
sin(6i)
cos(O)
(11)
This
is just
the
incident
solar
flux
of Eq.
(3),
N(v),
times
the
transmission
coefficient
for
entering
the
LSC,
T(Oi,v
times
the
fraction
a(V)/at(v),
which
specifies
how much
of
the absorbed
flux was
absorbed
by
the dye,
times
the
total
absorption
probability
1
- exp[-at
(v)-
1JJ,
integrated
over
all angles
of
incidence
and wave-
numbers.
is
the path
length
traveled
by the
solar
flux
inside
the
LSC,
which
for the
PSC
geometry
of Fig.
1
is
given by
4 = 2D/cos(Ot)
=
2D/[1
- sin
2
(O)/n
2
]1/2,
(12)
where
D is the
thickness
of
the
plate.
The
factor
of
two
appears
in
Eq.
(12)
due
to a
backing
mirror,
which
causes
the
solar
flux
to pass
twice
through
the
plate.
We
have
ignored
the
effect
of multiple
internal
reflec-
tions
of
the sunlight
by
the LSC
faces,
which
is shown
in Appendix
B to be
a good
approximation
for moderate
angles
of
incidence.
It is
worthwhile
now
to ask
what is
the angular
de-
pendence
of the
solar
absorption
S
with respect
to
the
incident
light.
From
Eq.
(11) it follows
that
S is
nearly
proportional
to the
cosine
of the
angle
of incidence,
which
means
that the
LSC behaves
as
a selectively
ab-
sorbing
blackbody.
This
tendency
to
imitate
a
black-
body
is
shown
in
the following
example.
BLACK BODY
G
IDERLIZED SINGLE DYE PSC
c:
SINGLE
DYE PSC
oRH-GD
IN PMMR
1 I 8
36 54
72 90
RNGLE
OF INCIDENCE
Fig. 3.
The lowest
curve shows
the result
of a numerical
integration
of
Eq. (11) for
an AM1
solar spectrum
incident
on a 3-mm
thick
LSC
containing
the
laser dye
rhodamine-6G
at
a concentration
of
0.001
moles/liter.
The total
absorbed
flux
is plotted
as a
function
of the
angle
of incidence
of the sunlight
and
is found
to be remarkably
similar
to just
the decrease
in subtended
area described
by
the cosine
function.
We
can say that
to a good
approximation
an idealized
LSC
will
have the
cosine
dependence
of a blackbody
absorber
but with
a
reduced
total absorption
(middle
curve).
Numerical
Example:
The
bottom
curve in
Fig. 3
represents
the numerical
integration
of Eq.
(11) for a
3-mm
LSC plate
containing
a
0.001-M
concentration
of rhodamine-6G
(Ref.
7) under
AM1 insolation.
The
angle
of incidence
of the
sunlight
is varied
using
the
angular
distribution
function
U(0i,iV),
treating
the
sun
as a point
source
at a variable
polar
angle.
The decrease
in S
at large
angles
of incidence
is due
mainly
to
the
decrease
in the
effective
area
exposed
to the sun,
given
by the
actual
area times
cos(Oi).
For comparison,
the
center
curve is
a simple
cosine function
that
is tangent
to
the calculated
S
curve at
about the
Brewster
angle,
and the
top curve
shows the
absorption
by
a perfect
blackbody.
This
particular
dye
concentration
and
plate
thickness
combination
absorb
about
20%
of the solar
flux,
which
is the
typical
limit
of a single
dye
device.
The
fraction
of the
solar flux
that
is absorbed
is
quite
sensitive
to the
concentration
of
the particular
dye
used.
In
the following
example
we calculate
how S
varies
with
concentration
in
a typical
system.
Numerical
Example:
Figure
4 shows
the result
of a
numerical
integration
of Eq.
(11) for
a single pass
2-mm
thick PMMA
plate
containing
a variable
concentration
of rhodamine-6G
for an
AM1 spectrum
at vertical
in-
cidence.
This
function
can
be approximated
by a
simple
analytic
expression
containing
two adjustable
parameters
which
characterize
the dye
used, K
1
and K
2
.
This
analytic
approximation
is
given by
S
K, - I[1 - exp(-
K2CU)],
(13)
where
C is the
molar
dye concentration,
and Is
is the
path
length
traversed
by the
sunlight
in the LSC.
I is
the total
integrated
solar
flux. K
1
0.20,
and K
2
=
6440. liters
mole-
1
cm-'
for a rhodamine-6G
LSC.
FITTED
-; _ FUNCTION_
NUMERIC7L
INTEGRATION
lo-
los
lo-, lo-,
MOLRR CONCENTRATION
2MM
THI CK PLRTE. VERTICAL
INCIDENCE)
Fig.
4. The
numerically
integrated
curve shows
the result
of solving
Eq. (11)
for the
case of
vertically
incident
sunlight
on
a single
pass
2-mm
PMMA
plate containing
a variable
concentration
of rhoda-
mine-6G.
A reasonable
approximation
to this result
can be
made
using
the form
S/I = K, [1
- exp(-K
2
C.1)], where
C is the dye
con-
centration
in moles/liter,
and 15 is the
path length
of the sunlight
in
the
LSC in centimeters.
K, and K
2
are fitted constants,
which
for
a
rhodamine-6G
single
dye LSC are
given by
K, = 0.20
and K
2
=
6440.0
liter mole-'
cm-1.
15 September
1979 /
Vol. 18, No.
18 / APPLIED
OPTICS
3093
C.
Dye
Quantum
Efficiency-7j
Upon
the
absorption
of
a photon,
a
dye
molecule
will
vibrationally
relax
to
an
excited
singlet
state
on
the
time
scale
of picoseconds,
or
more
slowly
to
the
excited
triplet
state
via
intersystem
crossing.
Four
major
channels
are
then
available
by
which
the
molecule
can
relax
to
its
ground
state:
fluorescence
from
the
excited
singlet
state;
phosphorescence
from
the
excited
triplet
state;
direct
or
nonradiative
transfer
of
the
excitation
to
a
nearby
molecule;
and
finally
by
internal
conversion
of
the
excitation
to
molecular
vibrations
or
phonons,
which
are
dispersed
in
the
lattice.
In
a
single
dye
LSC
the
only
direct
transfer
that
can
take
place
is to
a
similar
dye
molecule
or
to the
matrix
material,
both
of which
effects
are
relatively
negligible
due
to
the
dominance
of
intra-
molecular
effects.
So
in a
single
dye
LSC
the
important
types
of
energy
transfer
to
consider
are
fluorescence
and
phosphorescence
(which
are
combined
in
the
term
lu-
minescence)
and
internal
conversion.
In
this
paper
we
define
the
quantum
efficiency
of
luminescence
as
follows:
if
an
isolated
dye
molecule
in
the
matrix
material
of
an
LSC
absorbs
a
photon
of
sufficient
energy
to
excite
the
electronic
state
of
the
molecule
to
or
above
the
first
excited
singlet
state
(but
of
insufficient
energy
to
cause
photodissociation),
is
the
probability
that
the
molecule
will
subsequently
emit
a photon.
For
dyes
typically
used
in
an
LSC,
al will
typically
range
from
0.85
to
0.95.
When
an
excited
dye
molecule
luminesces,
there
is
a
probability
distribution
f(i)
describing
the
wave-
MEASURED
MEASURED
RH-SG
BSORPTION
NON-SELF-RBSORBED
OVER
2MM
PRTH
EMISSION
VMERSURED
MAUE
EMISSION
FOR
EMISSION
FOR
1
6MM
PHLENGTH
311MM
RTHLENGTH
1J1
CALCULATED
CRLCULATED
EMISSION
FOR
EMISSION
FOR
z
16MM
PRTHLENGTH
31MM
PRTHLENGTH
28667
2Do00
21333
8687
16000
13333
WAVE
NUMBERS
Fig.
5.
The
effect
of
self-absorption
in
a semi-infinite
rod.
A
dye
with
the
absorption
and
emission
spectra
shown
in
the
upper
graph
is placed
in a
PMMA
rod
2
mm
in
diameter
and
400
mm
long.
Lu-
minescent
photons
from
molecules
in
the
middle
of the
rod
will
un-
dergo
self-absorption
on
their
way
to
the
end
of
the
rod.
The
cylin-
drical
surface
of the
rod
is
roughened
and
blackened
to
eliminate
in-
ternal
reflections.
The
center
graph
shows
the
measured
self-ab-
sorbed
emission
from
the
end
of
the
rod
for
excitations
originating
16
mm
and
314
mm
away
from
the
end
of
the
rod.
The
lower
graph
shows
these
same
spectra
as
predicted
by
the
self-absorption
calcu-
lation
of
Sec.
II.E.
number
of
the
emitted
photon.
f (v)
is called
the
nor-
malized
luminescence
spectrum
of
the
dye,
and
the
normalization
condition
is
given
by
f
f()d=
1
(14)
D.
Losses
via
the
Critical
Cone-P
Luminescence
incident
to
the
LSC
surface
at
an
angle
of
incidence
greater
than
0c
is
totally
internally
re-
flected.
The
cone
formed
by
all
rays
originating
at
the
point
of
luminescence
and
forming
an
angle
0
c with
the
surface
is
called
the
critical
cone.
Typically
in
an
LSC
there
will
be
two
such
cones,
one
pointing
toward
the
top
face
and
one
towards
the
bottom.
For
an
LSC
which
is not
planar,
the
polar
angle
formed
by
these
rays
will
be
a
function
of
depth
z
and
azimuthal
angle
0,
so
that,
neglecting
reflections
at
the
air-LSC
interface
within
the
critical
cones,
the
fraction
of
the
luminescence
lost
out
of
the
critical
cone
P
is
P(z)
=
1-
(470-1
2Jo dp
f
dO
sinO.
f.
O(2k)
(15)
In
words
this
says
that
the
fraction
of
the
4r
sphere
of
emission
that
escapes
is one
minus
the
probability
that
the
angle
of
incidence
will
be
greater
than
the
critical
angle.
If
either
the
top
or
bottom
surface
of
the
LSC
is nonplanar,
the
polar
angle
of
emission
which
will
form
a
critical
angle
of
incidence
is
not
necessarily
the
critical
angle,
so
that
both
the
upper
and
lower
critical
cones
require
a
variable
limit
on
the
0 integration.
These
limits
are
0,
(z,o)
and
O'c
(z,)
for
the
upper
and
lower
critical
cones,
respectively.
In
the
case
of a
planar
geometry,
the
critical
cones
are
independent
of
the
depth
and
azimuthal
angle,
c
(z,k)
=
0C(z,')
=
0,,
and
so
the
above
integral
becomes
P
=
1 -
cosO
6
.
(16)
If
the
index
of
refraction
of
air
is
assumed
to
be
1,
then
0
=
sin-
1
(1/n)
from
Eq.
(6),
and
P
becomes
P =
1
- (1
-1/n
2
)/
2
.
(17)
For
example,
P
=
0.26
for
PMMA,
which
has
an
index
of
refraction
of
1.49.
Even
within
the
critical
cones,
part
of
the
lumines-
cence
can
be
retained
via
incomplete
transmission
at
the
air-LSC
interface.
As
computed
in
Appendix
B,
this
effect
typically
reduces
P by
no
more
than
0.01%.
E.
Self-Absorption
Effects-r,?
A
phenomenon
with
which
we
now
have
to
concern
ourselves
is
the
fact
that
there
is
some
overlap
between
the
absorption
and
emission
spectra
for
most
of
the
dyes
or
ions
to
be
used
in
an
LSC.
The
upper
graph
of
Fig.
5
shows
the
absorption
spectrum
of
rhodamine-6G
on
the
left,
superimposed
on
its
emission
spectrum
on
the
right.
The
observed
overlap
allows
a fluorescent
pho-
ton
to
be
reabsorbed
by
another
dye
molecule
of
the
same
type.
This
effect
has
been
seen
in
liquids
in
the
form
of
secondary
emission.
8
Such
a
reabsorption
is
termed
self-absorption
and
is actually
a dominant
effect
over
the
long
path
lengths
traveled
by
light
trapped
in
3094
APPLIED
OPTICS
/
Vol.
18,
No.
18
/ 15
September
1979