Abstract

Experimental techniques are developed to determine the applicability of a particular luminescing center for use in a luminescent solar concentrator (LSC). The relevant steady-state characteristics of eighteen common organic laser dyes are given. The relative spectral homogeneity of such dyes are shown to depend upon the surrounding material using narrowband laser excitation. We developed three independent techniques for measuring self-absorption rates; these are time-resolved emission, steady-state polarization anisotropy, and spectral convolution. Preliminary dye degradation and prototype efficiency measurements are included. Finally, we give simple relationships relating the efficiency and gain of an LSC to key spectroscopic parameters of its constituents.

© 1981 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. S. Batchelder, A. H. Zewail, T. Cole, Appl. Opt. 18, 3090 (1979) and references therein.
    [CrossRef] [PubMed]
  2. Exciton Chemical Co., Inc., P.O. Box 3204, Overlook Station, Dayton, Ohio 45431.
  3. Acrilex, Inc., 8 Hope Street, Jersey City, N.J. 07307.
  4. R. G. Gordon, J. Chem. Phys. 45, 1643 (1966).
    [CrossRef]
  5. R. J. Robbins, D. M. Millar, A. H. Zewail, J. Chem. Phys. 75, 3649 (1981). [A brief description of the apparatus appears in R. J. Robbins, D. P. Millar, and A. H. Zewail, Picosecond Phenomena II, Springer Topics in Chemical Physics 14, R. M. Hochstrasser, W. Kaiser, and C. V. Shank, Eds. (Springer, Berlin, 1980).]
    [CrossRef]
  6. D. Beer, J. Weber, Opt. Commun. 5, 307 (1972).
    [CrossRef]
  7. R. E. Sah, G. Baur, H. Keller, Appl. Phys. 23, 369 (1980).
    [CrossRef]
  8. For a planar LSC a characteristic path length can be defined that reduces the 3-D problem to a pseudo-1-D problem.
  9. J. N. Demas, G. A. Crosby, J. Phys. Chem. 75, 991 (1971).
    [CrossRef]
  10. T. Tao, Biopolymers 8, 609 (1969).
    [CrossRef]
  11. A. Von Jena, H. E. Lessing, Chem. Phys. 40, 245 (1979).
    [CrossRef]
  12. J. S. Batchelder, Ph.D. thesis, Calif. Inst. Tech. (1981).

1981 (1)

R. J. Robbins, D. M. Millar, A. H. Zewail, J. Chem. Phys. 75, 3649 (1981). [A brief description of the apparatus appears in R. J. Robbins, D. P. Millar, and A. H. Zewail, Picosecond Phenomena II, Springer Topics in Chemical Physics 14, R. M. Hochstrasser, W. Kaiser, and C. V. Shank, Eds. (Springer, Berlin, 1980).]
[CrossRef]

1980 (1)

R. E. Sah, G. Baur, H. Keller, Appl. Phys. 23, 369 (1980).
[CrossRef]

1979 (2)

1972 (1)

D. Beer, J. Weber, Opt. Commun. 5, 307 (1972).
[CrossRef]

1971 (1)

J. N. Demas, G. A. Crosby, J. Phys. Chem. 75, 991 (1971).
[CrossRef]

1969 (1)

T. Tao, Biopolymers 8, 609 (1969).
[CrossRef]

1966 (1)

R. G. Gordon, J. Chem. Phys. 45, 1643 (1966).
[CrossRef]

Batchelder, J. S.

Baur, G.

R. E. Sah, G. Baur, H. Keller, Appl. Phys. 23, 369 (1980).
[CrossRef]

Beer, D.

D. Beer, J. Weber, Opt. Commun. 5, 307 (1972).
[CrossRef]

Cole, T.

Crosby, G. A.

J. N. Demas, G. A. Crosby, J. Phys. Chem. 75, 991 (1971).
[CrossRef]

Demas, J. N.

J. N. Demas, G. A. Crosby, J. Phys. Chem. 75, 991 (1971).
[CrossRef]

Gordon, R. G.

R. G. Gordon, J. Chem. Phys. 45, 1643 (1966).
[CrossRef]

Keller, H.

R. E. Sah, G. Baur, H. Keller, Appl. Phys. 23, 369 (1980).
[CrossRef]

Lessing, H. E.

A. Von Jena, H. E. Lessing, Chem. Phys. 40, 245 (1979).
[CrossRef]

Millar, D. M.

R. J. Robbins, D. M. Millar, A. H. Zewail, J. Chem. Phys. 75, 3649 (1981). [A brief description of the apparatus appears in R. J. Robbins, D. P. Millar, and A. H. Zewail, Picosecond Phenomena II, Springer Topics in Chemical Physics 14, R. M. Hochstrasser, W. Kaiser, and C. V. Shank, Eds. (Springer, Berlin, 1980).]
[CrossRef]

Robbins, R. J.

R. J. Robbins, D. M. Millar, A. H. Zewail, J. Chem. Phys. 75, 3649 (1981). [A brief description of the apparatus appears in R. J. Robbins, D. P. Millar, and A. H. Zewail, Picosecond Phenomena II, Springer Topics in Chemical Physics 14, R. M. Hochstrasser, W. Kaiser, and C. V. Shank, Eds. (Springer, Berlin, 1980).]
[CrossRef]

Sah, R. E.

R. E. Sah, G. Baur, H. Keller, Appl. Phys. 23, 369 (1980).
[CrossRef]

Tao, T.

T. Tao, Biopolymers 8, 609 (1969).
[CrossRef]

Von Jena, A.

A. Von Jena, H. E. Lessing, Chem. Phys. 40, 245 (1979).
[CrossRef]

Weber, J.

D. Beer, J. Weber, Opt. Commun. 5, 307 (1972).
[CrossRef]

Zewail, A. H.

R. J. Robbins, D. M. Millar, A. H. Zewail, J. Chem. Phys. 75, 3649 (1981). [A brief description of the apparatus appears in R. J. Robbins, D. P. Millar, and A. H. Zewail, Picosecond Phenomena II, Springer Topics in Chemical Physics 14, R. M. Hochstrasser, W. Kaiser, and C. V. Shank, Eds. (Springer, Berlin, 1980).]
[CrossRef]

J. S. Batchelder, A. H. Zewail, T. Cole, Appl. Opt. 18, 3090 (1979) and references therein.
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. (1)

R. E. Sah, G. Baur, H. Keller, Appl. Phys. 23, 369 (1980).
[CrossRef]

Biopolymers (1)

T. Tao, Biopolymers 8, 609 (1969).
[CrossRef]

Chem. Phys. (1)

A. Von Jena, H. E. Lessing, Chem. Phys. 40, 245 (1979).
[CrossRef]

J. Chem. Phys. (2)

R. G. Gordon, J. Chem. Phys. 45, 1643 (1966).
[CrossRef]

R. J. Robbins, D. M. Millar, A. H. Zewail, J. Chem. Phys. 75, 3649 (1981). [A brief description of the apparatus appears in R. J. Robbins, D. P. Millar, and A. H. Zewail, Picosecond Phenomena II, Springer Topics in Chemical Physics 14, R. M. Hochstrasser, W. Kaiser, and C. V. Shank, Eds. (Springer, Berlin, 1980).]
[CrossRef]

J. Phys. Chem. (1)

J. N. Demas, G. A. Crosby, J. Phys. Chem. 75, 991 (1971).
[CrossRef]

Opt. Commun. (1)

D. Beer, J. Weber, Opt. Commun. 5, 307 (1972).
[CrossRef]

Other (4)

Exciton Chemical Co., Inc., P.O. Box 3204, Overlook Station, Dayton, Ohio 45431.

Acrilex, Inc., 8 Hope Street, Jersey City, N.J. 07307.

J. S. Batchelder, Ph.D. thesis, Calif. Inst. Tech. (1981).

For a planar LSC a characteristic path length can be defined that reduces the 3-D problem to a pseudo-1-D problem.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (27)

Fig. 1
Fig. 1

Composite of extinction coefficient spectra for a variety of representative organic laser dyes. Vertical axis is extinction coefficient (liter/mole/cm), and horizontal axis is in wave numbers. All spectra are from low concentration methanol solutions.

Fig. 2
Fig. 2

Schematic of the apparatus for emission and excitation measurements. Regulated mercury or tungsten source was focused down, chopped, and monochromated prior to exciting the sample. Resulting emission was monochromated and detected by a PMT, and the signal was amplified with phase-sensitive detection. Remote computer controlled both monochromators and recorded the measured spectra.

Fig. 3
Fig. 3

Three emission spectra of methanol dye solutions resulting from 22,220-wave number (4500-Å) excitation. Top spectrum is from a micromolar oxazine-720 solution, and the bottom two are micromolar and hundred micromolar solutions, respectively, of coumarin-540, rhodamine-640, and oxazine-720.

Fig. 4
Fig. 4

Three excitation spectra of methanol dye solutions due to 15,630-wave number (6400-Å) detection. Three spectra correspond to the same three solutions described in Fig. 3.

Fig. 5
Fig. 5

Diagram of the apparatus used to measure the luminescence spectra of liquid, cast plastic, and diffused plastic samples as a function of excitation energy and temperature. Either the argon-ion laser or the tunable dye laser was used as the excitation source. Apparent sample temperature was maintained at either 77 or 300K. Resulting emission was analyzed by a Spex 14018 double monochtomator.

Fig. 6
Fig. 6

Structure of rhodamine-575.

Fig. 7
Fig. 7

Emission spectra of rhodamine-575. Lower three plots show the room temperature emission of rhodamine-575 in methanol due to excitation at 16,509, 17,164, and 20,492 wave numbers (6075, 5826, and 4880 Å, respectively). Upper three plots show similar emission spectra taken at 77 K using excitations at 16,877, 17,163, and 20,492 wave numbers (5925, 5826, and 4880 Å, respectively). Room temperature spectra show that the dye emits with essentially the same spectrum for all the excitation energies.

Fig. 8
Fig. 8

Emission spectra of rhodamine-575. Lower three plots show the luminescence of rhodamine-575 cast in PMMA at room temperature for excitations at 16,273, 17,060, 19,436, and 20,492 wave numbers (6145, 5826, 6145, and 4880 Å, respectively). Notice the pronounced skewing of the emission toward lower energies by the low energy excitations. Upper plot shows luminescence at 77 K for excitations at 17,423, 19,436, and 20,492 wave numbers (5740, 5145, and 4880 Å, respectively).

Fig. 9
Fig. 9

Emission of rhodamine-575 diffused into PMMA. Emission spectra were taken at room temperature. Excitation energies were 16,116, 17,449, 19,436, and 20,492 wave numbers (6205, 5731, 5145, and 4880 Å, respectively).

Fig. 10
Fig. 10

Apparatus for measuring the intensity and spectral shifts of the sample emission as a function of the path length through the sample traveled by the emission. The 22,940-wave number (4359-Å) line from a mercury lamp was monochromated, chopped, and focused onto an optical fiber. The opposite end of the fiber was scanned horizontally with a micrometer stage along the rod-shaped sample of LSC material. Output was detected by a scanning monochromator.

Fig. 11
Fig. 11

Emission spectra sequence from a 92-μM methanol solution of rhodamine-575. Sample path lengths for each spectrum in order of the most to least intense were 0.3, 1.0, 3.0, 10.0, and 30.0 cm. Spectra have been corrected for the system response of the detection monochromator and the PMT, so that the amplitude of each spectrum corresponds to the intensities emerging from the end of the rod.

Fig. 12
Fig. 12

Principal direction for polarization anisotropy measurements. Monochromated light excites rhodamine-575 in an ethylene glycol solution, and the resulting emission is detected 90° to the axis of excitation. Excitation is polarized with its electric field either horizontal or vertical, and the emission is similarly polarized horizontally or vertically (perpendicular or parallel, respectively).

Fig. 13
Fig. 13

Reduced polarization anisotropy vs sample concentration. Polarization anisotropy of the emission from rhodamine-575 in ethylene glycol as a function of concentration for parallel polarized excitation is plotted. Reduced polarization anisotropy is defined to be the difference between the parallel and perpendicular emission intensities divided by the sum of the parallel and twice the perpendicular intensities. Solid line connects data points.

Fig. 14
Fig. 14

System response and transient lifetime measurement. Upper plot shows a typical histogram of the system response to a 5145-Å pulse from a mode-locked argon-ion laser with a duration of <200 psec scattered off of a dilute coffee creamer solution. Width and symmetry of the response were predominantly due to the PMT response. Lower plot shows the observed emission of a 4.6-μM rhodamine-575 methanol solution superimposed on a best fit single exponential convoluted with the system response. Exponential fit gives a lifetime of 4.1 nsec.

Fig. 15
Fig. 15

Measured lifetime of rhodamine-575 in methanol as a function of dye concentration is plotted. Solid line connects data points.

Fig. 16
Fig. 16

Position of the first generation emission mask. The 5145-Å laser excitation is 99% absorbed in the first 0.05 cm of the 460-μM rhodamine-575 methanol solution. Blocking just the leftmost portion of the cell, as shown, prevents the emission due to the initial excitation from being directly detected.

Fig. 17
Fig. 17

Unmasked and masked transient emission measurements. Upper plot shows the transient emission spectrum from 460-μM rhodamine-575 methanol solution for the case where a mask is positioned between the region of the sample excited by the laser and detector. When the mask is removed, we obtain the spectrum shown in the lower half of the figure.

Fig. 18
Fig. 18

Solar bleaching of a multiple-dye LSC plate. These are the absorption spectra for a PMMA plate containing 210-, 81-, and 22-μM concentrations of coumarin-540, rhodamine-590, and sulforhodamine-640, respectively. Four spectra correspond to 243 continuous h of exposure through a 4000-Å cutoff filter, a soda-lime glass plate with no filter, and a control which was unexposed.

Fig. 19
Fig. 19

Peak optical densities of methanol dye solutions under continuous solar exposure: 2.5-cm i.d. soda-lime glass bottles were filled with solutions of oxazine-720 perchlorate, LD-700 perchlorate, cresyl violet-670 perchlorate, rhodamine-590 perchlorate, coumarin-540, coumarin-535, or coumarin-500. Screw-on caps were sealed with RTV. Absorption spectra were taken without disturbing the seals.

Fig. 20
Fig. 20

Absorption, emission, and excitation spectra for DCM in PMMA. Arrows A and E indicate the absorption and emission maxima, respectively, for DCM in methanol. Adding DMSO to the monomer prior to polarimerization should substantially increase the Stokes shift in PMMA.7

Fig. 21
Fig. 21

Calculated spectral and intensity changes due to self-absorption. Experimental absorption and emission spectra of rhodamine-575 were approximated by the sum of two Gaussians. Stokes shift of 1000 wave numbers reproduces the typical overlap between these two spectra. Using Eqs. (6)(8), we have analytically imitated the experiment shown in Fig. 11 using the same geometry, concentration, and path lengths.

Fig. 22
Fig. 22

Calculated spectral and intensity changes due to self-absorption. Same calculation was made as in Fig. 21, except that the Stokes shift was increased to 2500 wave numbers.

Fig. 23
Fig. 23

Average path length of collected emission vs width of a PSC plate. Using the full PSC collection efficiency calculation from LSC-1, we have calculated the average distance through the plate traversed by emission which is collected at the edge of the plate for four different concentrations of rhodamine-575. Cells were assumed to be mounted on both edges of the ribbon. Plate thickness was assumed to be 1 cm so that the 10−5-M concentration corresponds to a peak o.d. of 1. Dashed line indicates the characteristic length approximation that the average path length is equal to the plate width.

Fig. 24
Fig. 24

Collection efficiency vs self-absorption probability. Assuming that P = 0.26, we have used Eq. (1) to calculate the collection efficiencies for the cases where there is no self-absorption in the critical cones (solid lines) and where the self-absorption rates inside and outside of the critical cones are the same (dotted lines). Upper two curves assume a quantum efficiency of luminescence of 0.9, while the lower two assume 0.7. Vertical line at r = 0.5 indicates the self-absorption rate at the CODE.

Fig. 25
Fig. 25

Self-absorption probabilities for rhodamine-575. This shows a juxtaposition of the predicted self-absorption probabilities for the three measurement methods: spectral overlap convolution (solid curve); emission depolarization (error boxes); and transient lifetime (error bars). Lifetime measurements have the highest precision, especially at low concentrations, and the lifetime and polarization experiments are in good agreement. These two techniques actually measure the self-absorption probability times the quantum efficiency of luminescence, which we have approximated as being unity.

Fig. 26
Fig. 26

Optimal efficiency and dye concentration for a single-dye LSC. Geometry assumed is an infinite ribbon with 18% AM1 silicon cells mounted on both edges. Plate has an index of 1.49 and contains rhodamine-575 with an assumed quantum efficiency of luminescence of 0.9. Dotted line represents the performance of a scattering plate with an identical geometry.

Fig. 27
Fig. 27

Schematic diagram of a scattering plate.

Tables (2)

Tables Icon

Table I Optical Properties of Dyes used In the LSCs

Tables Icon

Table II Performance Parameters of LSCs

Equations (59)

Equations on this page are rendered with MathJax. Learn more.

Q = ( 1 - r ) ( 1 - P ) η 1 - η [ r ¯ P + ( 1 - P ) · r ] ,
F = 2 T W π · { W T tan - 1 ( T / W ) = ½ ln [ 1 + ( W / T ) 2 ] } .
Q ( 1 ) = η scat · F .
Q ( 2 ) = η scat 2 · F ( 1 - F ) ( 1 - P ) .
Q ( i ) = η scat · F [ η scat ( 1 - F ) ( 1 - P ) ] i - 1 .
Q = η scat · F 1 - ( 1 - F ) ( 1 - P ) η scat .
N e = i = 1 i Q ( i ) / Q .
η c = η pvc η scat F 1 - ( 1 - F ) ( 1 - P ) η scat .
Λ ( x , y ) = ln ( 10 ) · C 2 0 d ν ¯ f ( ν ¯ ) ( ν ¯ ) A B d z exp ( - z ) z ,
z = x - y · [ ln 10 · C · ( ν ¯ ) + S C ] , A = [ ln ( 10 ) · C · ( ν ¯ ) + S C ] · x - y , B = { [ ln ( 10 ) · C · ( ν ¯ ) + S C ] · x - y / sin θ c x - y > d 2 tan θ c , [ ln ( 10 ) · C · ( ν ¯ ) + S C ] d 2 4 + ( x - y ) 2 x - y < d 2 tan θ c .
Ξ ( 1 ) ( x ) = I 0 δ ( x ) .
Ξ ( 2 ) ( x ) = η 0 d y Ξ ( 1 ) ( y ) Λ ( x , y ) ,
Ξ ( i ) ( x ) = η 0 d y Ξ ( i - 1 ) ( y ) Λ ( x , y ) ,
Q ( ν ¯ ) = η 0 d x 0 π / 2 - θ c × sin θ d θ exp { - x [ ln ( 10 ) · C · ( ν ¯ ) + S C ] / cos θ } × f ( ν ¯ ) · i = 1 Ξ ( i ) ( x ) .
r = 0 d ν ¯ f ( ν ¯ ) [ 1 - 10 - x C ( ν ¯ ) ] ,
I = [ ^ i · μ ¯ a ] 2 [ ^ f · μ ¯ e ] 2 .
I = [ ^ i · μ ¯ a ] 2 [ ^ i · μ ¯ e ] 2 = μ a 2 μ e 2 4 π 0 2 π d ϕ 0 π d θ sin θ cos 4 θ = μ a 2 μ e 2 / 5.
I = ½ [ ^ i · μ ¯ a ] 2 [ ^ i × μ ¯ e ] 2 = μ a 2 μ e 2 8 π 0 2 π d ϕ 0 π d θ sin 3 θ cos 2 θ = μ a 2 μ e 2 15 .
R A = I - I I + 2 I .
R A = 2 e / 5             0 e 1.
R A ( 2 ) = ( 2 / 5 ) 3
R A ( i ) = ( 2 / 5 ) ( 2 i - 1 ) .
R A exp = i = 1 R A ( i ) Q ( i ) / Q .
R A exp = [ 2 e 5 η ( 1 - r ) ( 1 - P ) + ( 2 e 5 ) 3 × η 2 ( 1 - r ) ( 1 - P ) [ r ¯ P + r ( 1 - P ) ] + ( 2 e 5 ) 5 η 3 ( 1 - r ) ( 1 - P ) [ r ¯ P + r ( 1 - P ) ] 2 + ] / Q = 2 e 5 · 1 - η ( r ¯ P + r ( 1 - P ) ) 1 - η ( 2 e 5 ) 2 [ r ¯ P + r ( 1 - P ) ] .
R A exp = 2 e 5 1 - n r 1 - ( 2 e 5 ) 2 η r 2 e 5 ( 1 - η r ) .
t n ( 1 ) ( t ) = - n ( 1 ) ( t ) τ             n ( 1 ) ( 0 ) = n 0 ,
t n ( 2 ) ( t ) = - n ( 2 ) ( t ) τ + η n ( 1 ) ( t ) · [ r ¯ P + r ( 1 - P ) ] / τ .
t n ( 3 ) ( t ) = - n ( 3 ) ( t ) / τ + η n ( 2 ) ( t ) [ r ¯ P + r ( 1 - P ) ] / τ .
n ( 1 ) ( t ) = n 0 exp ( - t / τ ) n ( 2 ) ( t ) = n 0 t τ η [ r ¯ P + r ( 1 - P ) ] exp ( - t / τ ) , n ( i ) ( t ) = n 0 { t τ η [ r ¯ P + r ( 1 - P ) ] } i - 1 exp ( - t / τ ) / ( i - 1 ) !
n ( t ) = i = 1 n ( i ) ( t ) = n 0 exp ( - t τ { 1 - η [ r ¯ P + r ( 1 - P ) ] } ) .
n m ( t ) = n 0 · { exp [ - t ( 1 - r η ) / τ ] - exp ( - t / τ ) } = n ( t ) - n ( 1 ) ( t ) .
Q PSC = Q PSC ( 1 ) / { 1 + Q PSC ( 1 ) - η [ 1 - P ( 1 - r ¯ ) ] } ,
CODE = ( ν ¯ m ) C L ,
½ = 0 f ( ν ¯ ) d ν ¯ 10 - C L ( ν ¯ ) .
r η = 1 - ( R A exp / R A max ) 1 - ( R A exp · R A max ) 1 - ( R A exp / R A max ) .
½ = 0 f ( ν ¯ ) d ν ¯ 10 - ( ν ¯ ) C X .
S / I = ν ¯ a I ( ν ¯ ) d ν ¯ [ 1 - 10 - C ( ν ¯ ) T ] / ν ¯ a I ( ν ¯ ) d ν ¯ ,
Eff . = ½ η η AM 1 S / I ( S / I ) Q η cell ,
F ( x ) = 1 4 π - π / 2 π / 2 d ϕ a - a + sin θ d θ = T 2 π x - π / 2 π / 2 d ϕ cos ϕ 1 + ( T cos ϕ x ) 2 = 1 π tan - 1 ( T x ) ,
Q s ( 1 ) = 2 W η scat 0 W d x F ( x ) = 2 W π η scat 0 W d x tan - 1 ( T x ) = 2 T π W η scat { W T tan - 1 ( T W ) + ½ ln [ 1 + ( W T ) 2 ] } ,
Q s ( 2 ) = ( 1 - P ) [ η scat - Q s ( 1 ) ] Q s ( 1 ) , Q s ( 3 ) = ( 1 - P ) 2 [ η scat - Q s ( 1 ) ] 2 Q s ( 1 ) .
Q s = i Q s ( i ) = Q s ( 1 ) 1 - ( 1 - P ) [ η s - Q s ( 1 ) ] .
I ( 2 ) = K · [ ^ · μ ^ a 1 ] 2 [ μ ^ e 1 · ^ 1 ] 2 [ ^ 1 · μ ^ a 2 ] 2 [ μ ^ e 2 · ^ ] 2 , I ( 2 ) = K 2 · [ ^ · μ ^ a 1 ] 2 [ μ ^ e 1 · ^ 1 ] 2 [ ^ 1 · μ ^ a 2 ] [ μ ^ e 2 × ^ ] 2 .
I ( 2 ) = K ( 4 π ) 3 0 2 π d ϕ 0 π d θ sin θ 0 2 π d ϕ 1 × 0 π d θ 1 sin θ 1 0 2 π d ϕ 2 0 π d ϕ 2 sin θ 2 × cos 2 θ · [ cos θ cos θ 1 + sin θ sin θ 1 cos ( ϕ - ϕ 1 ) ] 2 × [ cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos ( ϕ 1 - ϕ 2 ) ] 2 cos 2 θ 2 = K · 47 / 15 3 ;
I ( 2 ) = K 2 ( 4 π ) 3 0 2 π d ϕ 0 π d θ sin θ 0 2 π d ϕ 1 × 0 π d θ 1 sin θ 1 0 2 π d ϕ 2 0 π d θ 2 sin θ 2 × cos 2 θ [ cos θ cos θ 1 + sin θ sin θ 1 cos ( ϕ - ϕ 1 ) ] 2 × [ cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos ( ϕ 1 - ϕ 2 ) ] 2 ( 1 - cos 2 θ 2 ) = K · 39 / 15 3 .
r ( 2 ) = I ( 2 ) - I ( 2 ) I ( 2 ) + 2 I ( 2 ) = ( 2 5 ) 3 .
r ( i ) = ( 2 5 ) ( 2 i - 1 ) .
P = π / 2 - θ c π / 2 + θ c sin θ d θ / 0 π sin θ d θ = sin θ c = 1 / n ,
T ( a , ν ¯ ) = exp { - a [ ln ( 10 ) · C · ( ν ¯ ) + S C ] } .
T ( a + Δ , ν ¯ ) = T ( a , ν ¯ ) exp [ - Δ · ln ( 10 ) · C · ( ν ¯ ) ] = T ( a , ν ¯ ) [ 1 - Δ · ln ( 10 ) · C · ( ν ¯ ) ] .
T ( a , ν ¯ ) - T ( a + Δ , ν ¯ ) = Δ · ln ( 10 ) · C · ( ν ¯ ) · exp { - a [ ln ( 10 ) · C · ( ν ¯ ) + S C } .
Λ ( x , y , ν ¯ ) = 0 B T ( a cos θ , ν ¯ ) - T ( a + Δ cos θ , ν ¯ ) Δ sin θ d θ = 0 B d θ sin θ cos θ ln ( 10 ) · C · ( ν ¯ ) × exp { - x - y [ ln ( 10 ) · C · ( ν ¯ ) + S C } , B = { π / 2 - θ c for x - y > d 2 tan θ c , tan - 1 ( d 2 x - y ) for x - y < d 2 tan θ c .
z = x - y · [ ln ( 10 ) · C · ( ν ¯ ) + S C ] / cos θ , d z z = sin θ cos θ d θ .
Λ ( x , y , ν ¯ ) = A B d z exp ( - z ) z ln ( 10 ) · C · ( ν ¯ ) , A = x - y · [ ln ( 10 ) · C · ( ν ¯ ) + S C ] , B = { x - y · [ ln ( 10 ) · C · ( ν ¯ ) + S C ] / sin θ c when x - y > d 2 tan θ c , [ ln ( 10 ) · C · ( ν ¯ ) + S C ] d 2 4 + ( x - y ) 2 when x - y < d 2 tan θ c .
Λ ( x , y ) = 0 d ν ¯ f ( ν ¯ ) Λ ( x , y , ν ¯ ) .
η f ( ν ¯ ) i = 1 Ξ ( i ) ( x ) .
T ( x cos θ , ν ¯ ) = exp { - x [ ln ( 10 ) · C · ( ν ¯ ) + S C ] / cos θ } .
Q ( ν ¯ ) = 0 π / 2 - θ c d θ sin θ 0 d x · exp { - x [ ln ( 10 ) · C · ( ν ¯ ) + S C ] / cos θ } · η · f ( ν ¯ ) · i = 1 Ξ ( i ) ( x ) .
Q = 0 d ν ¯ Q ( ν ¯ ) .

Metrics