Abstract

A statistical approach is taken toward the ray optics of optical media with complicated nonspherical and nonplanar surface shapes. As a general rule, the light in such a medium will tend to be randomized in direction and of 2n2(x) times greater intensity than the externally incident light, where n(x) is the local index of refraction. A specific method for doing optical calculations in statistical ray optics will be outlined. These optical enhancement effects can result in a new type of antireflection coating. In addition, these effects can improve the efficiency as well as reduce the cost of solar cells.

© 1982 Optical Society of America

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References

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  1. A. E. St. John, “Multiple internal reflection structure in a silicon detector which is obtained by sandblasting,” U.S. Patent No. 3,487,223 (1969).
  2. O. Krumpholz and S. Maslowski, “Schnelle Photodioden mit wellenlangenunabhangigen Demodulatonseigenschaften,” Z. Angew. Phys. 25, 156 (1968).
  3. D. Redfield, “Multiple-pass thin-film silicon solar cell,” Appl. Phys. Lett. 25, 647 (1974).
    [CrossRef]
  4. M. Spitzer, J. Shewchun, E. S. Vera, and J. J. Loferski, “Ultra-high efficiency thin silicon p-n junction solar cells using reflective surfaces,” in Proceedings of the Fourteenth IEEE Photovoltaic Specialists Conference (Institute of Electrical and Electronics Engineers, New York, 1980), p. 375.
  5. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1969).
  6. Another situation in which Eq. (4) is not valid is an optically thick turbid sheet illuminated from only one side. In this case, angular averaging is no problem, but spatial averaging will be incomplete. The side of the sheet away from the source of illumination will be dark.
  7. M. Born and E. Wolf, Principles of Optics (Macmillan, New York, 1964).
  8. M. Neuberger and S. J. Welles, Silicon (National Technical Information Service, Springfield, Va., 1969), p. 113.
  9. J. S. Kilby, J. W. Lathrop, and W. A. Porter, “Light energy conversion,” U.S. Patent No. 4,136,436 (1979); T. S. T. Velde “Electrical monogram layers and method for making same,” U.S. Patent No. 3,625,688 (1971).

1974 (1)

D. Redfield, “Multiple-pass thin-film silicon solar cell,” Appl. Phys. Lett. 25, 647 (1974).
[CrossRef]

1968 (1)

O. Krumpholz and S. Maslowski, “Schnelle Photodioden mit wellenlangenunabhangigen Demodulatonseigenschaften,” Z. Angew. Phys. 25, 156 (1968).

Born, M.

M. Born and E. Wolf, Principles of Optics (Macmillan, New York, 1964).

John, A. E. St.

A. E. St. John, “Multiple internal reflection structure in a silicon detector which is obtained by sandblasting,” U.S. Patent No. 3,487,223 (1969).

Kilby, J. S.

J. S. Kilby, J. W. Lathrop, and W. A. Porter, “Light energy conversion,” U.S. Patent No. 4,136,436 (1979); T. S. T. Velde “Electrical monogram layers and method for making same,” U.S. Patent No. 3,625,688 (1971).

Krumpholz, O.

O. Krumpholz and S. Maslowski, “Schnelle Photodioden mit wellenlangenunabhangigen Demodulatonseigenschaften,” Z. Angew. Phys. 25, 156 (1968).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1969).

Lathrop, J. W.

J. S. Kilby, J. W. Lathrop, and W. A. Porter, “Light energy conversion,” U.S. Patent No. 4,136,436 (1979); T. S. T. Velde “Electrical monogram layers and method for making same,” U.S. Patent No. 3,625,688 (1971).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1969).

Loferski, J. J.

M. Spitzer, J. Shewchun, E. S. Vera, and J. J. Loferski, “Ultra-high efficiency thin silicon p-n junction solar cells using reflective surfaces,” in Proceedings of the Fourteenth IEEE Photovoltaic Specialists Conference (Institute of Electrical and Electronics Engineers, New York, 1980), p. 375.

Maslowski, S.

O. Krumpholz and S. Maslowski, “Schnelle Photodioden mit wellenlangenunabhangigen Demodulatonseigenschaften,” Z. Angew. Phys. 25, 156 (1968).

Neuberger, M.

M. Neuberger and S. J. Welles, Silicon (National Technical Information Service, Springfield, Va., 1969), p. 113.

Porter, W. A.

J. S. Kilby, J. W. Lathrop, and W. A. Porter, “Light energy conversion,” U.S. Patent No. 4,136,436 (1979); T. S. T. Velde “Electrical monogram layers and method for making same,” U.S. Patent No. 3,625,688 (1971).

Redfield, D.

D. Redfield, “Multiple-pass thin-film silicon solar cell,” Appl. Phys. Lett. 25, 647 (1974).
[CrossRef]

Shewchun, J.

M. Spitzer, J. Shewchun, E. S. Vera, and J. J. Loferski, “Ultra-high efficiency thin silicon p-n junction solar cells using reflective surfaces,” in Proceedings of the Fourteenth IEEE Photovoltaic Specialists Conference (Institute of Electrical and Electronics Engineers, New York, 1980), p. 375.

Spitzer, M.

M. Spitzer, J. Shewchun, E. S. Vera, and J. J. Loferski, “Ultra-high efficiency thin silicon p-n junction solar cells using reflective surfaces,” in Proceedings of the Fourteenth IEEE Photovoltaic Specialists Conference (Institute of Electrical and Electronics Engineers, New York, 1980), p. 375.

Vera, E. S.

M. Spitzer, J. Shewchun, E. S. Vera, and J. J. Loferski, “Ultra-high efficiency thin silicon p-n junction solar cells using reflective surfaces,” in Proceedings of the Fourteenth IEEE Photovoltaic Specialists Conference (Institute of Electrical and Electronics Engineers, New York, 1980), p. 375.

Welles, S. J.

M. Neuberger and S. J. Welles, Silicon (National Technical Information Service, Springfield, Va., 1969), p. 113.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Appl. Phys. Lett. (1)

D. Redfield, “Multiple-pass thin-film silicon solar cell,” Appl. Phys. Lett. 25, 647 (1974).
[CrossRef]

Z. Angew. Phys. (1)

O. Krumpholz and S. Maslowski, “Schnelle Photodioden mit wellenlangenunabhangigen Demodulatonseigenschaften,” Z. Angew. Phys. 25, 156 (1968).

Other (7)

A. E. St. John, “Multiple internal reflection structure in a silicon detector which is obtained by sandblasting,” U.S. Patent No. 3,487,223 (1969).

M. Spitzer, J. Shewchun, E. S. Vera, and J. J. Loferski, “Ultra-high efficiency thin silicon p-n junction solar cells using reflective surfaces,” in Proceedings of the Fourteenth IEEE Photovoltaic Specialists Conference (Institute of Electrical and Electronics Engineers, New York, 1980), p. 375.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1969).

Another situation in which Eq. (4) is not valid is an optically thick turbid sheet illuminated from only one side. In this case, angular averaging is no problem, but spatial averaging will be incomplete. The side of the sheet away from the source of illumination will be dark.

M. Born and E. Wolf, Principles of Optics (Macmillan, New York, 1964).

M. Neuberger and S. J. Welles, Silicon (National Technical Information Service, Springfield, Va., 1969), p. 113.

J. S. Kilby, J. W. Lathrop, and W. A. Porter, “Light energy conversion,” U.S. Patent No. 4,136,436 (1979); T. S. T. Velde “Electrical monogram layers and method for making same,” U.S. Patent No. 3,625,688 (1971).

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Figures (10)

Fig. 1
Fig. 1

Textured optical sheet bathed in blackbody radiation. The intensity inside the sheet is greater than that outside by the factor n2(x).

Fig. 2
Fig. 2

In the textured sheet (a) the light fills the internal-phase space with radiation of enhanced intensity. In (b) the behavior of the light is nonergodic, and no intensity enhancement occurs.

Fig. 3
Fig. 3

A white rear reflector reduces the number of escape channels for light by half and therefore increases the internal intensity by a factor of 2.

Fig. 4
Fig. 4

The angular texturing required at the rear surface to prevent escape on the second bounce at the front surface is relatively small. This may be regarded as a rough guide to the degree of texturing required to ensure that randomization will compete effectively with the escape of light rays through the loss cone.

Fig. 5
Fig. 5

Detailed balancing requires that the light escaping in the loss cone through area element dA equal the amount of incoming light through the same area element dA.

Fig. 6
Fig. 6

A thin plastic film on a textured silicon surface combines an index match with light trapping to produce a superior antireflection coating [curve (b)]. Texturing alone, without the plastic layer, produces only ~ 2% reduction in surface reflectivity [curve (a)].

Fig. 7
Fig. 7

Geometry used to measure absorption enhancement in a silicon wafer, textured in the rear.

Fig. 8
Fig. 8

Comparison of the light absorption in a plane-parallel slab with a textured sheet. The two experimental configurations are almost identical, but the results are quite different. The effective position of the band edge is shifted to the infrared in the light-trapping case. The dashed lines are the theory given by Eq. (14) for the statistical case and by a simple double pass absorption formula in the nonergodic case.

Fig. 9
Fig. 9

Monolayer sheet of silicon granules embedded in plastic on a white filter paper. Light is trapped both in the silicon and the plastic binder, resulting in favorable conditions for light collection in a solar cell.

Fig. 10
Fig. 10

Reflectivity of the structure shown in Fig. 9. The absorption was ~ 89%, in spite of an area coverage of less than 60%. In addition, band-edge absorption is greatly enhanced. The upper curve shows that the reflectivity of the plastic-covered paper is near 99% in the absence of the silicon grains.

Equations (33)

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U = ω exp ( ω k T ) - 1 2 d Ω k 2 d k ( 2 π ) 3 .
I U v g = ω exp ( ω k T ) - 1 2 d Ω n 2 ω 2 ( 2 π ) 3 c 2 d ω .
I int ( ω , x ) = n 2 ( ω , x ) I ext bb ( ω ) .
I int ( ω , x ) = n 2 ( ω , x ) I ext ( ω ) .
I int ( ω , x ) = 2 n 2 ( ω , x ) I inc ( ω ) ,
sin θ c = 1 n ,
Ω c 1 2 n 2 × 4 π .
2 θ = arctan ( 1 / n ) .
I int = B int cos θ d ω ,
I int = 2 × 2 π 0 π / 2 B int cos θ sin θ d θ , I int = 2 π B int .
I esc = 2 π 0 θ c I int 2 π T esc ( θ ) cos θ sin θ d θ ,
I esc = I int T ¯ esc 2 n 2 .
T inc ( ϕ ) I inc = I int × T ¯ esc 2 n 2 .
I int = 2 n 2 × T inc ( ϕ ) T ¯ esc × I inc .
I int = 2 n 2 × I inc .
A esc I int T ¯ esc 2 n 2 ,
0 π / 2 η A refl I int cos θ sin θ d θ = η A refl I int 2 ,
α I int 2 π d V d Ω = α l I int A inc 0 π sin θ d θ = 2 α l I int A inc ,
A inc T inc I inc = ( A esc T ¯ esc 2 n 2 + η A refl 2 + 2 α l A inc ) I int .
I inc = T inc I inc ( A esc A inc T ¯ esc 2 n 2 + η A refl 2 A inc + 2 α l ) .
2 α l A inc I int = 2 α l A inc T inc I inc ( A esc A inc T ¯ esc 2 n 2 + η A refl 2 A inc + 2 α l ) .
f vol 2 α l A inc I int A inc I inc = 2 α l T inc ( A esc A inc T ¯ esc 2 n 2 + η A refl 2 A inc + 2 α l ) .
f tot = 2 α l + η A refl 2 A inc ( A esc A inc T ¯ esc 2 n 2 + η A refl 2 A inc + 2 α l ) T inc .
Absorption = T inc η + T inc ( 1 - η ) η T ¯ esc n 2 + η ,
Absorption = T inc η 1 - ( 1 - η ) / ( 1 + T ¯ esc n 2 ) .
Absorption = T inc η 1 - ( 1 - η ) ( 1 - T ¯ esc n 2 ) .
A refl / A inc = 2 π r l / π r 2 = 2 l / r .
A 23 T ¯ 23 I 2 2 ( n 2 / n 3 ) 2 + A 13 T 13 I 1 = ( A 13 T ¯ 13 2 n 3 2 + A 23 T ¯ 23 2 ) I 3 ,
A 23 T ¯ 23 I 3 2 + A 12 T 12 I 1 = ( A 12 T ¯ 12 2 n 2 2 + A 23 T ¯ 23 2 ( n 2 / n 3 ) 3 + 2 α l A 12 ) I 2 .
I 2 = ( 0.8316 α l + 0.0319 ) 0.96 I 1 ,
I 2 = 0.8316 × 0.96 0.0319 I 1 ,
2 α l I 2 = 1.6632 α l 0.0319 + α l × 0.96 I 1 ,
lim α 2 α l I 2 = 1.66632 × 0.96 I 1 .