Abstract

Techniques and calculations are presented that give explicit expressions for the over-all performance of a luminescent 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.

© 1979 Optical Society of America

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  1. B. A. Swartz, T. Cole, A. H. Zewail, Opt. Lett. 1, 73 (1977).
    [CrossRef] [PubMed]
  2. K. W. Boer, Solar Energy 19, 525 (1977).
    [CrossRef]
  3. B. Goldberg, W. H. Klein, Solar Energy 19, 3 (1977).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 63.
  5. A. Goetzberger, W. Greubel, Appl. Phys. 14, 123 (1977).
    [CrossRef]
  6. H. J. Hovel, Solar Cells, Vol. 11, Semiconductors and Semimetals (Academic, 1975), New York.
  7. K. H. Drexhage, “Structure and Properties of Laser Dyes,” in Dye Lasers, Vol. 1, Topics in Applied Physics (Springer, New York, 1977), p. 168.
  8. A. Budo, I. Ketskemety, J. Chem. Phys. 25, 595 (1956).
    [CrossRef]
  9. J. B. Birks, Photophysics of Aromatic Molecules (Wiley, New York, 1970), p. 42.
  10. W. H. Weber, J. Lambe, Appl. Opt. 15, 2299 (1976).
    [CrossRef] [PubMed]
  11. L. G. Rainhart, W. P. Schimmel, Solar Energy 17, 259 (1975).
    [CrossRef]
  12. M. Wolf, Proc. IRE 48, 1246 (1960).
    [CrossRef]
  13. W. Shockley, Electrons and Holes in Semiconductors (Van Nostrand, New York, 1954), p. 309.
  14. C. J. Sah, R. N. Noyce, W. Shockley, Proc. IRE 45, 1228 (1957).
    [CrossRef]
  15. A. G. Chynoweth, K. G. McKay, Phys. Rev. 106, 418 (1957).
    [CrossRef]
  16. M. Uman, Introduction to the Physics of Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1957), p. 418.
  17. J. Weber, Phys. Lett. A, 57, 465 (1976).
    [CrossRef]
  18. D. Beer, J. Weber, Opt. Commun. 5, 307 (1972).
    [CrossRef]
  19. J. Weber, Opt. Commun. 7, 420 (1973).
    [CrossRef]
  20. Acrilex Inc., 8 Hope Street, Jersey City, N.J., 07307.
  21. Q-Panel Company, 15610 Industrial Parkway, Cleveland, Ohio 44135.
  22. G. Grossman, J. Coating Technol. 49, 45 (1977).
  23. A. Goetzberger, Appl. Phys. 16, 399 (1978).
    [CrossRef]
  24. Solar Cell Array Design Handbook, Vol. 1, Jet Propulsion Laboratory Publication JPL SP 43-38, p. 3.1.2 (1976).
  25. R. M. Lerner, private communication.
  26. J. A. Levitt, W. H. Weber, Appl. Opt. 16, 2684 (1977).
    [CrossRef] [PubMed]
  27. G. Bauer, W. Greubel, Appl. Phys. Lett. 31, 4 (1977).
    [CrossRef]
  28. W. A. Shurcliff, J. Opt. Soc. Am. 41, 209, (1951).
    [CrossRef]
  29. W. A. Shurcliff, R. C. Jones, J. Opt. Soc. Am. 39, 912 (1949).
    [CrossRef]
  30. R. L. Garwin, Rev. Sci. Instrum. 31, 1010 (1960).
    [CrossRef]
  31. G. Keil, J. Appl. Phys. 40, 3544 (1969).
    [CrossRef]
  32. T. C. Weekes, private communication.

1978 (1)

A. Goetzberger, Appl. Phys. 16, 399 (1978).
[CrossRef]

1977 (7)

B. A. Swartz, T. Cole, A. H. Zewail, Opt. Lett. 1, 73 (1977).
[CrossRef] [PubMed]

K. W. Boer, Solar Energy 19, 525 (1977).
[CrossRef]

B. Goldberg, W. H. Klein, Solar Energy 19, 3 (1977).
[CrossRef]

A. Goetzberger, W. Greubel, Appl. Phys. 14, 123 (1977).
[CrossRef]

G. Grossman, J. Coating Technol. 49, 45 (1977).

J. A. Levitt, W. H. Weber, Appl. Opt. 16, 2684 (1977).
[CrossRef] [PubMed]

G. Bauer, W. Greubel, Appl. Phys. Lett. 31, 4 (1977).
[CrossRef]

1976 (3)

J. Weber, Phys. Lett. A, 57, 465 (1976).
[CrossRef]

W. H. Weber, J. Lambe, Appl. Opt. 15, 2299 (1976).
[CrossRef] [PubMed]

Solar Cell Array Design Handbook, Vol. 1, Jet Propulsion Laboratory Publication JPL SP 43-38, p. 3.1.2 (1976).

1975 (1)

L. G. Rainhart, W. P. Schimmel, Solar Energy 17, 259 (1975).
[CrossRef]

1973 (1)

J. Weber, Opt. Commun. 7, 420 (1973).
[CrossRef]

1972 (1)

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

1969 (1)

G. Keil, J. Appl. Phys. 40, 3544 (1969).
[CrossRef]

1960 (2)

R. L. Garwin, Rev. Sci. Instrum. 31, 1010 (1960).
[CrossRef]

M. Wolf, Proc. IRE 48, 1246 (1960).
[CrossRef]

1957 (2)

C. J. Sah, R. N. Noyce, W. Shockley, Proc. IRE 45, 1228 (1957).
[CrossRef]

A. G. Chynoweth, K. G. McKay, Phys. Rev. 106, 418 (1957).
[CrossRef]

1956 (1)

A. Budo, I. Ketskemety, J. Chem. Phys. 25, 595 (1956).
[CrossRef]

1951 (1)

1949 (1)

Bauer, G.

G. Bauer, W. Greubel, Appl. Phys. Lett. 31, 4 (1977).
[CrossRef]

Beer, D.

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

Birks, J. B.

J. B. Birks, Photophysics of Aromatic Molecules (Wiley, New York, 1970), p. 42.

Boer, K. W.

K. W. Boer, Solar Energy 19, 525 (1977).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 63.

Budo, A.

A. Budo, I. Ketskemety, J. Chem. Phys. 25, 595 (1956).
[CrossRef]

Chynoweth, A. G.

A. G. Chynoweth, K. G. McKay, Phys. Rev. 106, 418 (1957).
[CrossRef]

Cole, T.

Drexhage, K. H.

K. H. Drexhage, “Structure and Properties of Laser Dyes,” in Dye Lasers, Vol. 1, Topics in Applied Physics (Springer, New York, 1977), p. 168.

Garwin, R. L.

R. L. Garwin, Rev. Sci. Instrum. 31, 1010 (1960).
[CrossRef]

Goetzberger, A.

A. Goetzberger, Appl. Phys. 16, 399 (1978).
[CrossRef]

A. Goetzberger, W. Greubel, Appl. Phys. 14, 123 (1977).
[CrossRef]

Goldberg, B.

B. Goldberg, W. H. Klein, Solar Energy 19, 3 (1977).
[CrossRef]

Greubel, W.

A. Goetzberger, W. Greubel, Appl. Phys. 14, 123 (1977).
[CrossRef]

G. Bauer, W. Greubel, Appl. Phys. Lett. 31, 4 (1977).
[CrossRef]

Grossman, G.

G. Grossman, J. Coating Technol. 49, 45 (1977).

Hovel, H. J.

H. J. Hovel, Solar Cells, Vol. 11, Semiconductors and Semimetals (Academic, 1975), New York.

Jones, R. C.

Keil, G.

G. Keil, J. Appl. Phys. 40, 3544 (1969).
[CrossRef]

Ketskemety, I.

A. Budo, I. Ketskemety, J. Chem. Phys. 25, 595 (1956).
[CrossRef]

Klein, W. H.

B. Goldberg, W. H. Klein, Solar Energy 19, 3 (1977).
[CrossRef]

Lambe, J.

Lerner, R. M.

R. M. Lerner, private communication.

Levitt, J. A.

McKay, K. G.

A. G. Chynoweth, K. G. McKay, Phys. Rev. 106, 418 (1957).
[CrossRef]

Noyce, R. N.

C. J. Sah, R. N. Noyce, W. Shockley, Proc. IRE 45, 1228 (1957).
[CrossRef]

Rainhart, L. G.

L. G. Rainhart, W. P. Schimmel, Solar Energy 17, 259 (1975).
[CrossRef]

Sah, C. J.

C. J. Sah, R. N. Noyce, W. Shockley, Proc. IRE 45, 1228 (1957).
[CrossRef]

Schimmel, W. P.

L. G. Rainhart, W. P. Schimmel, Solar Energy 17, 259 (1975).
[CrossRef]

Shockley, W.

C. J. Sah, R. N. Noyce, W. Shockley, Proc. IRE 45, 1228 (1957).
[CrossRef]

W. Shockley, Electrons and Holes in Semiconductors (Van Nostrand, New York, 1954), p. 309.

Shurcliff, W. A.

Swartz, B. A.

Uman, M.

M. Uman, Introduction to the Physics of Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1957), p. 418.

Weber, J.

J. Weber, Phys. Lett. A, 57, 465 (1976).
[CrossRef]

J. Weber, Opt. Commun. 7, 420 (1973).
[CrossRef]

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

Weber, W. H.

Weekes, T. C.

T. C. Weekes, private communication.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 63.

Wolf, M.

M. Wolf, Proc. IRE 48, 1246 (1960).
[CrossRef]

Zewail, A. H.

Appl. Opt. (2)

Appl. Phys. (2)

A. Goetzberger, Appl. Phys. 16, 399 (1978).
[CrossRef]

A. Goetzberger, W. Greubel, Appl. Phys. 14, 123 (1977).
[CrossRef]

Appl. Phys. Lett. (1)

G. Bauer, W. Greubel, Appl. Phys. Lett. 31, 4 (1977).
[CrossRef]

J. Appl. Phys. (1)

G. Keil, J. Appl. Phys. 40, 3544 (1969).
[CrossRef]

J. Chem. Phys. (1)

A. Budo, I. Ketskemety, J. Chem. Phys. 25, 595 (1956).
[CrossRef]

J. Coating Technol. (1)

G. Grossman, J. Coating Technol. 49, 45 (1977).

J. Opt. Soc. Am. (2)

Opt. Commun. (2)

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

J. Weber, Opt. Commun. 7, 420 (1973).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

J. Weber, Phys. Lett. A, 57, 465 (1976).
[CrossRef]

Phys. Rev. (1)

A. G. Chynoweth, K. G. McKay, Phys. Rev. 106, 418 (1957).
[CrossRef]

Proc. IRE (2)

C. J. Sah, R. N. Noyce, W. Shockley, Proc. IRE 45, 1228 (1957).
[CrossRef]

M. Wolf, Proc. IRE 48, 1246 (1960).
[CrossRef]

Rev. Sci. Instrum. (1)

R. L. Garwin, Rev. Sci. Instrum. 31, 1010 (1960).
[CrossRef]

Solar Cell Array Design Handbook (1)

Solar Cell Array Design Handbook, Vol. 1, Jet Propulsion Laboratory Publication JPL SP 43-38, p. 3.1.2 (1976).

Solar Energy (3)

L. G. Rainhart, W. P. Schimmel, Solar Energy 17, 259 (1975).
[CrossRef]

K. W. Boer, Solar Energy 19, 525 (1977).
[CrossRef]

B. Goldberg, W. H. Klein, Solar Energy 19, 3 (1977).
[CrossRef]

Other (10)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 63.

J. B. Birks, Photophysics of Aromatic Molecules (Wiley, New York, 1970), p. 42.

H. J. Hovel, Solar Cells, Vol. 11, Semiconductors and Semimetals (Academic, 1975), New York.

K. H. Drexhage, “Structure and Properties of Laser Dyes,” in Dye Lasers, Vol. 1, Topics in Applied Physics (Springer, New York, 1977), p. 168.

W. Shockley, Electrons and Holes in Semiconductors (Van Nostrand, New York, 1954), p. 309.

M. Uman, Introduction to the Physics of Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1957), p. 418.

R. M. Lerner, private communication.

T. C. Weekes, private communication.

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

Q-Panel Company, 15610 Industrial Parkway, Cleveland, Ohio 44135.

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Figures (16)

Fig. 1
Fig. 1

A planar solar concentrator, or PSC, which is the particular embodiment of an LSC.10 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 luminescence 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.

Fig. 2
Fig. 2

A photon flow diagram depicting the predominant channels available in an LSC. Dotted lines represent changes in index of refraction, and squares represent photon sinks. Light from the sun enters the dye ensemble, resulting in luminescence that is converted in the PVC. The two feedback loops around the dye ensemble represent the effects of self-absorption inside and outside of the critical cones.

Fig. 3
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).

Fig. 4
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 rhodamine-6G. A reasonable approximation to this result can be made using the form S/I = K1 [1 − exp(−K2Cls)], where C is the dye concentration in moles/liter, and ls is the path length of the sunlight in the LSC in centimeters. K1 and K2 are fitted constants, which for a rhodamine-6G single dye LSC are given by K1 = 0.20 and K2 = 6440.0 liter mole−1 cm−1.

Fig. 5
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. Luminescent photons from molecules in the middle of the rod will undergo self-absorption on their way to the end of the rod. The cylindrical surface of the rod is roughened and blackened to eliminate internal reflections. The center graph shows the measured self-absorbed 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 calculation of Sec. II.E.

Fig. 6
Fig. 6

The upper graph shows the approximate collection efficiency for a PSC in the limit that the reemission of self-absorbed luminescence is ignored. This collection efficiency Q PSC ( 1 ) is calculated as a function of the width L of the PSC for four different concentrations: 10−2, 10−3, 10−4, and 10−5 moles/liter of rhodamine-6G. The lower plot combines the information in the upper plot with that of Fig. 4 to find what the output flux is at the edge of the PSC in units of the total solar flux I for the four different concentrations. The collected flux here is the number of photons per unit area that arrive at the edge in units of the total integrated solar flux per unit area.

Fig. 7
Fig. 7

The semi-infinite LSC rod geometry used in the sample self-absorption calculation given in the text. The rod is composed of a cylinder of diameter d whose surface is completely absorbing and which terminates on a PVC disk at the origin. The initial excitation distribution is assumed to be contained in a disk located a distance x0 down the rod.

Fig. 8
Fig. 8

The fraction of the trapped luminescence in an idealized PSC that is lost during transport to the PVC via matrix absorption. Assuming uniform initial illumination and ignoring the possibility of self-absorption and internal reflection losses from surface roughness, a finite element analysis is used to compute the emission that arrives at the edge of the PSC for a number of different optical densities of the matrix material. The details of the calculation are given in Appendix C.

Fig. 9
Fig. 9

The effect of surface undulations on the trapped light. If the surface of a planar LSC is not entirely flat, but has undulations over a characteristic distance DD, there will be convex portions of the surface where light which would have otherwise been trapped can escape. If the undulation is slow compared to the thickness of the plate, the altered critical cone loss Pund is given by Eq. (43) for a plate of thickness D. Typical manufacturing tolerances for the flatness of these plates can be such that this effect should be negligible.

Fig. 10
Fig. 10

The results of a finite element analysis developed in Appendix C for the intensity arriving at the edge of a PSC as a function of the angle of incidence to the plane of the edge. The intensity peaks at the compliment of the critical angle and extends to very large angles of incidence due to the infinite strip geometry of the PSC. The optical densities shown are for the matrix material measured across the width of the PSC.

Fig. 11
Fig. 11

Here we plot the experimentally measured deterioration of the laser dyes rhodamine-6G and coumarin-6 under xenon lamp illumination in a QUV21 test chamber. Typically an hour of exposure in such a test chamber represents between 8 h and 25 h of exposure in normal environmental conditions. After an initial period of rapid deterioration, both dyes reached slower rates of deterioration, which the coumarin dye then maintained for the duration of the test.

Fig. 12
Fig. 12

The geometry used in calculating the effect of Fresnel reflections inside of the critical cones. We begin with a ray of intensity I in a flat LSC plate of thickness D. This ray is incident to the surface at an angle θ, and the fraction of the ray that is reflected is R(θ).

Fig. 13
Fig. 13

(a) A typical photon trajectory in a planar LSC; (b) this trajectory can be unfolded by the method of images to form a straight line from the point of emission to the PVC.

Fig. 14
Fig. 14

This figure shows how the PSC geometry is unfolded from its original form on the left to the symmetric wedge form on the right. Any trajectory in the original geometry which is outside of the critical cones will intercept the PVC after some number of reflections, and for that trajectory there is a corresponding trajectory on the right which intersects the PVC at the same angle and has the same path length, but lies in a straight line.

Fig. 15
Fig. 15

This figure shows how the symmetric wedge of Fig. 14 is divided into finite elements. First the wedge is separated into plates of thickness dy, each plate being parallel to the PVC. Then each plate is broken into a series of concentric rings, each ring being ydθ/cos(θ) across, and the rings are subdivided into sections ydϕ/cos(θ) wide, giving the finite element volume y2dydθdϕ/CoS2(θ).

Fig. 16
Fig. 16

The polar coordinate system in the frame of the PVC. The dotted semicircles represent sections of the half-sphere where there will be no incident light (because it has been lost out of the critical cones). For θ < π/2 − θc there are no restrictions on the allowed values of ϕ. For larger values of θ we find that ϕ cannot have values between −cos−1 [cos(θc)/sin(θ)] and cos−1 [cos(θc)/sin(θ)] and also between π − cos−1 [cos(θc)/sin(θ)] and π + cos−1 [cos(θc)/sin(θ)].

Equations (104)

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G geom = L / D = A f / A e ,
I = 0 N ( ν ¯ ) d ν ¯ .
I = 0 d ν ¯ 0 π / 2 d θ i N ( ν ¯ ) U ( θ i , ν ¯ ) sin ( θ i ) cos ( θ i )
1 = 0 d ν ¯ 0 π / 2 d θ i U ( θ i , ν ¯ ) sin ( θ i ) cos ( θ i ) .
R ( θ i ) = 1 2 [ tan 2 ( θ i - θ t ) tan 2 ( θ i + θ t ) + sin 2 ( θ i - θ t ) sin 2 ( θ i + θ t ) ] ,
sin ( θ i ) = n sin ( θ t ) ,
T ( θ i ) = 1 - R ( θ i ) .
θ c = sin - 1 ( 1 / n ) .
P = 1 - ( 1 - 1 / n 2 ) 1 / 2 .
R ( θ i , ν ¯ ) = 1 2 [ r 12 2 + r 23 2 + 2 r 12 r 23 cos ( 2 β ) 1 + r 12 2 r 23 2 + 2 r 12 r 23 cos ( 2 β ) + R 12 2 + R 23 2 + 2 R 12 R 23 cos ( 2 β ) 1 + R 12 2 R 23 2 + 2 R 12 R 23 cos ( 2 β ) ] ,
r 12 = cos ( θ i ) - n 1 cos ( θ 1 ) cos ( θ i ) + n 1 cos ( θ 1 ) ,
r 23 = n 1 cos ( θ 1 ) - n cos ( θ t ) n 1 cos ( θ 1 ) + n cos ( θ t ) ,
R 12 = n 1 cos ( θ 1 ) - cos ( θ i ) n 1 cos ( θ i ) + cos ( θ 1 ) ,
R 23 = n cos ( θ t ) - n 1 cos ( θ 1 ) n cos ( θ 1 ) + n 1 cos ( θ t ) ,
β = 2 π n 1 h ν ¯ cos ( θ 1 ) ,
sin ( θ i ) = n 1 sin ( θ 1 ) = n sin ( θ t ) .
α ( ν ¯ ) = C ( ν ¯ ) ln ( 10 ) ,
I t ( ν ¯ ) / I i ( ν ¯ ) = exp [ - α ( ν ¯ ) x ]
= 10 - C ( ν ¯ ) x .
α t ( ν ¯ ) = α ( ν ¯ ) + α m ( ν ¯ ) .
S = 0 d ν ¯ 0 π / 2 d θ i T ( θ i , ν ¯ ) N ( ν ¯ ) U ( θ i , ν ¯ ) α ( ν ¯ ) α t ( ν ¯ ) × { 1 - exp [ - α t ( ν ¯ ) l s ] } sin ( θ i ) cos ( θ i )
l s = 2 D / cos ( θ t ) = 2 D / [ 1 - sin 2 ( θ i ) / n 2 ] 1 / 2 ,
S K 1 · I [ 1 - exp ( - K 2 C l s ) ] ,
0 f ( ν ¯ ) d ν ¯ = 1
P ( z ) = 1 - ( 4 π ) - 1 0 2 π d ϕ θ c ( z , ϕ ) π - θ o ( z , ϕ ) d θ sin θ .
P = 1 - cos θ c .
P = 1 - ( 1 - 1 / n 2 ) 1 / 2 .
Q idealized = ( 1 - P ) η .
Q PSC ( 1 ) = η ( 2 π L ) - 1 0 f ( ν ¯ ) d ν ¯ 0 L d y 0 π d ϕ θ c π / 2 sin ( θ ) d θ { exp [ - a ( ν ¯ ) ( L - y ) / sin ( θ ) sin ( ϕ ) ] + exp [ - α ( ν ¯ ) ( L + y ) / sin ( θ ) sin ( ϕ ) ] } .
Q PSC ( 1 ) = η π - 1 0 f ( ν ¯ ) d ν ¯ θ c π / 2 sin ( θ ) d θ × 0 π / 2 d ϕ sin ( θ ) sin ( ϕ ) / α ( ν ¯ ) L { 1 - exp [ - 2 α ( ν ¯ ) L / sin ( θ ) sin ( ϕ ) ] } .
Q = Q ( 1 ) + Q ( 2 ) + Q ( 3 ) + .
N e = i i Q ( i ) / Q .
Q = ( 1 - P ) ( 1 - r ) η + ( 1 - P ) ( 1 - r ) [ r ¯ P + ( 1 - P ) r ] η 2 + ( 1 - P ) ( 1 - r ) [ r ¯ P + ( 1 - P ) r ] 2 η 3 + , Q = ( 1 - r ) ( 1 - P ) η 1 - η [ r ¯ P + ( 1 - P ) r ] .
r = ( 1 - P ) η - Q η ( 1 - P ) ( 1 - Q ) .
r ¯ = Q [ 1 - η r ( 1 - P ) ] - ( 1 - r ) ( 1 - P ) η Q P η .
S A f = 0 d ν ¯ LSC d 3 x Ξ ( 1 ) ( x ¯ , ν ¯ ) .
f ( 1 ) ( x ¯ , ν ¯ ) = η f ( ν ¯ ) ν ¯ d ν ¯ Ξ ( 1 ) ( x ¯ , ν ¯ ) / 0 ν ¯ d ν ¯ f ( ν ¯ ) .
Ξ ( 2 ) ( x ¯ , ν ¯ ) = d 3 y Λ ( x ¯ , y ¯ , ν ¯ ) f ( 1 ) ( y ¯ , ν ¯ ) .
Ξ ( n ) ( x ¯ , ν ¯ ) = f ( ν ¯ ) η ν ¯ d ν ¯ [ 0 ν ¯ f ( ν ¯ ) d ν ¯ ] - 1 d 3 y Λ ( x ¯ , y ¯ , ν ¯ ) Ξ ( n - 1 ) ( y ¯ , ν ¯ )
Q ( n ) = 0 d ν ¯ PVC d A ( y ¯ ) × LSC d 3 x f ( n ) ( x ¯ , ν ¯ ) exp [ - α ( ν ¯ ) x ¯ - y ¯ ] [ A ¯ ( y ¯ ) · ( x ¯ - y ¯ ) ] / [ S x ¯ - y ¯ 3 A e A f ] .
Q = η 0 d ν ¯ f ( ν ¯ ) PVC d A ( y ¯ ) LSC d 3 x exp [ - α ( ν ¯ ) x ¯ - y ¯ ] { [ A ¯ ( y ¯ ) · ( x ¯ - y ¯ ) ] / ( S x ¯ - y ¯ 3 A e A f ) } · { ν ¯ d ν ¯ [ n = 1 Ξ ( n ) ( x ¯ , ν ¯ ) / 0 ν ¯ f ( ν ¯ ) d ν ¯ ] } .
Λ ( x , y , ν ¯ ) = α ( ν ¯ ) d 2 exp ( - α ( ν ¯ ) · x - y ) / 8 ( ( x - y ) 2 + d 2 / 4 )
Ξ ( 1 ) ( x , ν ¯ ) = S δ ( x - x 0 ) δ ( ν ¯ - ν ¯ 0 ) ,
f ( 1 ) ( x , ν ¯ ) = η f ( ν ¯ ) ν ¯ d ν ¯ S δ ( x - x 0 ) δ ( ν ¯ - ν ¯ 0 ) / 0 ν ¯ d ν ¯ f ( ν ¯ ) = η f ( ν ¯ ) S δ ( x - x 0 )
Ξ ( 2 ) ( x , ν ¯ ) = 0 d y Λ ( x , y , ν ¯ ) f ( ν ¯ ) δ ( y - x 0 ) S η = Λ ( x , x 0 , ν ¯ ) f ( ν ¯ ) S η .
Ξ ( n ) ( x , ν ¯ ) = f ( ν ¯ ) η 0 d y Λ ( x , y , ν ¯ ) · ν ¯ Ξ ( n - 1 ) ( y , ν ¯ ) d ν ¯ / 0 ν ¯ f ( ν ¯ ) d ν ¯ .
Q = η 0 d ν ¯ f ( ν ¯ ) 0 d x exp ( - α ( ν ¯ ) x ) a ( x ) / S · [ ν ¯ d ν ¯ n = 1 Ξ ( n ) ( x , ν ¯ ) / 0 ν ¯ f ( ν ¯ ) d ν ¯ ] .
Q Q ( 1 ) .
Λ ( x , y , ν ¯ ) = α ( ν ¯ ) 2 π 0 π d ϕ sin ( ϕ ) θ c π / 2 d Θ · [ exp ( - α ( ν ¯ ) x - y / sin ( θ ) sin ( ϕ ) ) + exp ( - α ( ν ¯ ) x + y / sin ( θ ) sin ( ϕ ) ) ] .
Ξ ( n ) ( x , ν ¯ ) = n f ( ν ¯ ) 0 L d y Λ ( x , y , ν ¯ ) ν ¯ Ξ ( n - 1 ) ( y , ν ¯ ) d ν ¯ / 0 ν ¯ f ( ν ¯ ) d ν ¯ .
Q = η 2 π { 0 f ( ν ¯ ) d ν ¯ 0 L d x 0 π d ϕ θ c π 2 d θ sin ( θ ) · [ exp ( - α ( ν ¯ ) ( L - x ) / sin ( θ ) sin ( ϕ ) ] + exp ( - α ( ν ¯ ) ( L + x ) / sin ( θ ) sin ( ϕ ) ) ] · ν ¯ d ν ¯ n = 1 Ξ ( n ) ( x , ν ¯ ) / 0 ν ¯ f ( ν ¯ ) d ν ¯ } / 0 L d x 0 d ν ¯ Ξ ( 1 ) ( x , ν ¯ ) .
Q = ( 1 - r ) ( 1 - P ) η 1 - η [ r ¯ P + ( 1 - P ) r ] .
Q = ( 1 - δ ) ( 1 - r ) ( 1 - P ) η 1 - η [ r ¯ P + ( 1 - P ) r ] .
P und . = 1 - cos ( θ c ) / ( 1 + D D / D ) ,
N = n 1 + n 2 .
n 1 W 12 = n 2 W 21 .
η W 21 n 2 = η W 12 ( N - n 2 ) = η W 12 N .
J = W 12 N .
J = S A f + J η r ( 1 - P ) + J η r ¯ P ; J = S A f / { 1 - η [ r ¯ P + ( 1 - P ) r ] } .
W = J η ( 1 - P ) ( 1 - r ) ( 1 - δ ) , W = S A f η ( 1 - P ) ( 1 - r ) ( 1 - δ ) / { 1 - η [ r ¯ P + ( 1 - P ) r ] } .
W = S A f Q .
W = A e 0 d ν ¯ 0 π / 2 d θ I ( θ , ν ¯ ) .
R pvc = 0 d ν ¯ [ 0 π / 2 d θ R ( θ , ν ¯ ) I ( θ , ν ¯ ) ] / 0 d ν ¯ 0 π / 2 d θ I ( θ , ν ¯ ) .
T pvc = 1 - R pvc .
η pvc = ( V . F . ) ( F . F . ) q E g / h c P in ( ν ¯ ) d ν ¯ / ν ¯ h c 0 P in ( ν ¯ ) d ν ¯ E g ,
P in ( ν ¯ ) = h c 0 π / 2 d θ T ( θ , ν ¯ ) I ( θ , ν ¯ ) ν ¯ .
η pvc = ( V . F . ) ( F . F . ) q E g / h c d ν ¯ 0 π / 2 d θ T ( θ , ν ¯ ) I ( θ , ν ¯ ) h c 0 d ν ¯ · ν ¯ 0 π / 2 d θ T ( θ , ν ¯ ) I ( θ , ν ¯ ) E g .
V . F . = V oc / E g = A k T q E g ln ( I sc I 0 + 1 ) .
I 0 = e ( L h p n 0 τ h + L e n p 0 τ e ) ,
V . F . = V . F . AM 1 ln [ ( I sc ) LSC - ( I sc ) AM 1 I 0 ] ,
( I sc ) LSC = W T pvc q / A e .
G f = W T pvc A e T 0 pvc E g / h c d ν ¯ N ( ν ¯ ) .
G f = G geom T pvc S η ( 1 - P ) ( 1 - r ) ( 1 - δ ) T 0 pvc ( 1 - η [ r ¯ P + r ( 1 - P ) ] ) E g / h c d ν ¯ N ( ν ¯ ) .
η lsc = A e E s / h c d ν ¯ · ν ¯ 0 π / 2 d θ I ( θ , ν ¯ ) T ( θ , ν ¯ ) A f 0 d ν ¯ N ( ν ) · ν ¯ η pvc .
η lsc = ( F . F ) V o c q A e E s / h c d ν ¯ 0 π / 2 d θ I ( θ , ν ¯ ) T ( θ , ν ¯ ) h c A f · 0 d ν ¯ · ν ¯ N ( ν ) .
η 1 sc = V oc q ( F . F ) W T pvc h c A f · 0 d ν ¯ · ν ¯ N ( ν ¯ ) .
η lsc = V oc q ( F . F ) T pvc η ( 1 - P ) ( 1 - r ) ( 1 - δ ) S h c { 1 - η [ r ¯ P + ( 1 - P ) r ] } · 0 d ν ¯ · ν ¯ N ( ν ¯ ) .
P = 2 0 θ c sin 3 ( θ ) d θ 0 π sin 3 ( θ ) d θ .
P = 1 - 3 cos ( θ c ) / 2 + cos 3 ( θ c ) / 2 = 1 - ( 1 + / 2 n 2 ) ( 1 - 1 / n 2 ) 1 / 2 ,
ξ = E g / h c N ( ν ¯ ) d ν ¯ / 0 N ( ν ¯ ) d ν ¯ ,
G f = G geom [ η ( 1 - P ) ] N e S / I ξ .
η lsc = η pvc [ η ( 1 - P ) ] N e S / I ξ .
I ( x ) = { I 0 x < D tan ( θ ) , I R ( θ ) D tan ( θ ) x < 2 D tan ( θ ) , I R 2 ( θ ) 2 D tan ( θ ) x < 3 D tan ( θ ) ,
I ( x ) = I R x / D ( θ ) = I exp { x ln [ R ( θ ) ] / D } .
1 / 2 = exp { 10 ln [ R ( θ ) ] } .
D = D + D R ( θ ) + D R 2 ( θ ) + D R 3 ( θ ) + D = D / [ 1 - R ( θ ) ] = D / T ( θ ) } ,
δ = 1 - 0 π / 2 I ( θ ) d θ / 0 π / 2 I ( θ ) d θ .
I ( θ ) = { 0 2 L d y 0 2 π d ϕ sin ( θ ) exp [ - y α / cos ( θ ) ] / 2 π             0 θ π / 2 - θ c 0 2 L d y { 0 2 π d ϕ sin ( θ ) exp [ - y α / cos ( θ ) ] / 2 π             π / 2 - θ c θ π / 2 - 4 0 cos - 1 ( cos θ c / sin θ ) d ϕ sin ( θ ) exp [ - y α / cos ( θ ) ] / 2 π } ,
I ( θ ) = { 0 2 L d y sin ( θ ) exp [ - y α / cos ( θ ) ]             0 θ π / 2 - θ c 0 2 L d y sin ( θ ) exp [ - y α / cos ( θ ) ] ( 1 - 2 π cos - 1 [ cos ( θ c ) / sin ( θ ) ]             π / 2 - θ c θ π / 2
r = 0 f ( ν ¯ ) { 1 - exp [ - α ( ν ¯ ) l ] } d ν ¯ ,
Q = ( 1 - P ) ( 1 - r ) η + ( 1 - P ) ( 1 - r ) [ r ¯ P + ( 1 - P ) r ] η 2 + ( 1 - P ) ( 1 - r ) [ r ¯ P + ( 1 - P ) r ] 2 η 3 + .
Q ( 1 ) = ( 1 - P ) ( 1 - r ) η .
Q ( 1 ) = η 0 f ( ν ¯ ) exp [ - α ( ν ) l ] d ν ¯ .
( 1 - r ) = 0 f ( ν ¯ ) exp [ - α ( ν ¯ ) l ] d ν ¯ .
0 f ( ν ¯ ) d ν = 1 ,
Q ( i ) = l i η i .
Q = i Q ( i ) = i l i η i .
η Q η = i i l i η i = i i Q ( i ) .
N e Q = i i Q ( i ) .
η Q η = N e Q ,             Q = ( η ) N e .
Q = [ η ( 1 - P ) ] N e ,
r = η - Q η ( 1 - Q ) .
N e = 1 + ln ( 1 - r ) - ln ( 1 - η r ) ln ( η ) .
N e = 1 + r .

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