Abstract

In spite of the fact that solutions to Maxwell’s equations in stratified isotropic optical media are well known, it appears that an explicit expression of the Poynting vector flux spatial evolution inside such a medium has not been derived so far. Based on exact electromagnetic field solutions in the transfer-matrix formalism, I derive such an expression and show that, due to the presence of counterpropagating waves in the medium, an additional contribution to the flux appears that exists only in optically absorbing layers and arises from the interference between these waves. Based on this theory, the concept of incremental absorption is introduced for the calculation of the light absorption profile along the stratification direction. As an illustration of this concept, absorption profiles in a Si-based thin-film tandem solar cell are predicted at typical wavelengths.

© 2011 Optical Society of America

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References

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  1. S. Vukovic, R. Dragila, and A. M. Smith, Opt. Lett. 13, 164 (1988).
    [CrossRef] [PubMed]
  2. D. K. Gramotnev, Opt. Lett. 23, 91 (1998).
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  3. S. D. Gupta, Opt. Lett. 32, 1483 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos, Opt. Express 15, 16986 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).
  8. P. Yeh, Optical Waves in Layered Media, 2nd ed. (Wiley, 2005).
  9. C. Tsao, Optical Fibre Waveguide Analysis (Oxford U. Press, 1992).
  10. Complex refractive index values at 550 nm (700 nm) used in simulations: (1) cover glass, n˜glass=1.5 (1.5); (2) amorphous silicon, n˜aSi=4.54+i0.33 (4.00+i0.01); (3) polycrystalline silicon, n˜mcSi=4.19+i0.19 (3.83+i0.06); (4) steel substrate, n˜steel=2.1+i3.86 (2.68+i4.47).

2010 (1)

2009 (1)

2007 (2)

1998 (1)

1988 (1)

Avniel, Y.

Bermel, P.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

Chang, F.

Dragila, R.

Ghebrebrhan, M.

Gramotnev, D. K.

Gupta, S. D.

Joannopoulos, J. D.

Johnson, S. G.

Kimerling, L. C.

Luo, C.

Smith, A. M.

Soret, R. A.

Sun, G.

Tsao, C.

C. Tsao, Optical Fibre Waveguide Analysis (Oxford U. Press, 1992).

Vukovic, S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

Yeh, P.

P. Yeh, Optical Waves in Layered Media, 2nd ed. (Wiley, 2005).

Zeng, L.

Opt. Express (3)

Opt. Lett. (3)

Other (4)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

P. Yeh, Optical Waves in Layered Media, 2nd ed. (Wiley, 2005).

C. Tsao, Optical Fibre Waveguide Analysis (Oxford U. Press, 1992).

Complex refractive index values at 550 nm (700 nm) used in simulations: (1) cover glass, n˜glass=1.5 (1.5); (2) amorphous silicon, n˜aSi=4.54+i0.33 (4.00+i0.01); (3) polycrystalline silicon, n˜mcSi=4.19+i0.19 (3.83+i0.06); (4) steel substrate, n˜steel=2.1+i3.86 (2.68+i4.47).

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Figures (3)

Fig. 1
Fig. 1

Stratified isotropic medium in transfer-matrix formalism. The complex field amplitudes A j + and A j in layer j take fully into account the effect of multiple scattering at all layer interfaces.

Fig. 2
Fig. 2

Evolution of the normalized Poynting vector (top) and incremental absorption (bottom) as functions of depth inside an a-Si/mc-Si double-layer structure at wavelengths of 550 (magenta/lighter curves) or 700 nm (blue/darker curves).

Fig. 3
Fig. 3

Evolution of the three normalized contributions to the Poynting vector as a function of depth inside an a-Si/mc-Si double-layer structure at a wavelength of 700 nm .

Equations (10)

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S = 1 4 ( E × H + E * × H * ) + 1 4 ( E × H * + E * × H ) .
E x ( z , t ) = A j + e i ϕ j + ( z , t ) + A j e i ϕ j ( z , t ) ,
ϕ j ± ( z , t ) = k j , z ( z z j ) ω t ,
H y ( z , t ) = 1 i ω μ 0 E x z , H z ( z , t ) = k y ω μ 0 E x .
S j , z ( z , t ) = 1 2 ω μ 0 { R k j , z | A j + | 2 e 2 I k j , z ( z z j ) R k j , z | A j | 2 e 2 I k j , z ( z z j ) 2 I k j , z × I [ A j + A j * e i 2 R k j , z ( z z j ) ] + R [ k j , z ( A j + ) 2 e i ( 2 k j , z ( z z j ) 2 ω t ) ] R [ k j , z ( A j ) 2 e i ( 2 k j , z ( z z j ) + 2 ω t ) ] } ,
S j , z ( z ) = S j , z + ( z ) S j , z ( z ) + S j , z ± ( z ) ,
S j , z + ( z ) = 1 2 ω μ 0 R k j , z | A j + | 2 e 2 I k j , z ( z z j ) ,
S j , z ( z ) = 1 2 ω μ 0 R k j , z | A j | 2 e 2 I k j , z ( z z j ) ,
S j , z ± ( z ) = 1 ω μ 0 I k j , z × I [ A j + A j * e i 2 R k j , z ( z z j ) ] .
a j , z ( z ) = [ S 0 , z + S 0 , z ] [ S j , z + ( z ) S j , z ( z ) + S j , z ± ( z ) ] S 0 , z + .

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