Abstract

A generalized matrix method to treat multilayer systems with mixed coherent and incoherent optical behavior is presented. The method is based on the calculation of the light energy flux inside the multilayer, whose internal light absorption is straightforwardly derived. The Poynting vector is used to derive the light energy flux in the case of a layer with coherent behavior. Multilayer structures with any distribution of layers with coherent or incoherent behavior can be treated, including the case of oblique incidence. Use of the light energy flux instead of the more commonly used light intensity permits the calculation of light absorption with a better accuracy and a much shorter computation time.

© 2005 Optical Society of America

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References

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  1. C. C. Katsidis, D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41, 3978–3987 (2002).
    [CrossRef] [PubMed]
  2. J. S. C. Prentice, “Optical generation rate of electron–hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D 32, 2146–2150 (1999).
    [CrossRef]
  3. K. Ohta, H. Ishida, “Matrix formalism for calculation of the light beam intensity in stratified multilayered films, and its use in the analysis of emission spectra,” Appl. Opt. 29, 2466–2473 (1990).
    [CrossRef] [PubMed]
  4. R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977), pp. 332–340.
  5. J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D 33, 3139–3145 (2000).
    [CrossRef]
  6. K. Ohta, H. Ishida, “Matrix formalism for calculation of electric field intensity of light in stratified multilayered films,” Appl. Opt. 29, 1952–1959 (1990).
    [CrossRef] [PubMed]
  7. L. A. A. Pettersson, L. S. Roman, O. Inganas, “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” J. Appl. Phys. 86, 487–496 (1999).
    [CrossRef]
  8. H. Hoppe, N. Arnold, N. S. Sariciftci, D. Meissner, “Modeling the optical absorption within conjugated polymer/fullerene-based bulk-heterojunction organic solar cells,” Sol. Energy Mater. Sol. Cells 80, 105–113 (2003).
    [CrossRef]
  9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Chap. 12.
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, 1970), pp. 33.
  11. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1991), Vol. II, pp. 38–41.
  12. E. Centurioni, “A GPL optical simulation program for mixed coherent/incoherent multilayer systems,” available at www.bo.imm.cnr.it/∼centurio/optical.html .

2003 (1)

H. Hoppe, N. Arnold, N. S. Sariciftci, D. Meissner, “Modeling the optical absorption within conjugated polymer/fullerene-based bulk-heterojunction organic solar cells,” Sol. Energy Mater. Sol. Cells 80, 105–113 (2003).
[CrossRef]

2002 (1)

2000 (1)

J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D 33, 3139–3145 (2000).
[CrossRef]

1999 (2)

L. A. A. Pettersson, L. S. Roman, O. Inganas, “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” J. Appl. Phys. 86, 487–496 (1999).
[CrossRef]

J. S. C. Prentice, “Optical generation rate of electron–hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D 32, 2146–2150 (1999).
[CrossRef]

1990 (2)

Arnold, N.

H. Hoppe, N. Arnold, N. S. Sariciftci, D. Meissner, “Modeling the optical absorption within conjugated polymer/fullerene-based bulk-heterojunction organic solar cells,” Sol. Energy Mater. Sol. Cells 80, 105–113 (2003).
[CrossRef]

Azzam, R. M.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977), pp. 332–340.

Bashara, N. M.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977), pp. 332–340.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1970), pp. 33.

Hoppe, H.

H. Hoppe, N. Arnold, N. S. Sariciftci, D. Meissner, “Modeling the optical absorption within conjugated polymer/fullerene-based bulk-heterojunction organic solar cells,” Sol. Energy Mater. Sol. Cells 80, 105–113 (2003).
[CrossRef]

Inganas, O.

L. A. A. Pettersson, L. S. Roman, O. Inganas, “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” J. Appl. Phys. 86, 487–496 (1999).
[CrossRef]

Ishida, H.

Katsidis, C. C.

Meissner, D.

H. Hoppe, N. Arnold, N. S. Sariciftci, D. Meissner, “Modeling the optical absorption within conjugated polymer/fullerene-based bulk-heterojunction organic solar cells,” Sol. Energy Mater. Sol. Cells 80, 105–113 (2003).
[CrossRef]

Ohta, K.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1991), Vol. II, pp. 38–41.

Pettersson, L. A. A.

L. A. A. Pettersson, L. S. Roman, O. Inganas, “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” J. Appl. Phys. 86, 487–496 (1999).
[CrossRef]

Prentice, J. S. C.

J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D 33, 3139–3145 (2000).
[CrossRef]

J. S. C. Prentice, “Optical generation rate of electron–hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D 32, 2146–2150 (1999).
[CrossRef]

Roman, L. S.

L. A. A. Pettersson, L. S. Roman, O. Inganas, “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” J. Appl. Phys. 86, 487–496 (1999).
[CrossRef]

Sariciftci, N. S.

H. Hoppe, N. Arnold, N. S. Sariciftci, D. Meissner, “Modeling the optical absorption within conjugated polymer/fullerene-based bulk-heterojunction organic solar cells,” Sol. Energy Mater. Sol. Cells 80, 105–113 (2003).
[CrossRef]

Siapkas, D. I.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Chap. 12.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1970), pp. 33.

Appl. Opt. (3)

J. Appl. Phys. (1)

L. A. A. Pettersson, L. S. Roman, O. Inganas, “Modeling photocurrent action spectra of photovoltaic devices based on organic thin films,” J. Appl. Phys. 86, 487–496 (1999).
[CrossRef]

J. Phys. D (2)

J. S. C. Prentice, “Optical generation rate of electron–hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D 32, 2146–2150 (1999).
[CrossRef]

J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D 33, 3139–3145 (2000).
[CrossRef]

Sol. Energy Mater. Sol. Cells (1)

H. Hoppe, N. Arnold, N. S. Sariciftci, D. Meissner, “Modeling the optical absorption within conjugated polymer/fullerene-based bulk-heterojunction organic solar cells,” Sol. Energy Mater. Sol. Cells 80, 105–113 (2003).
[CrossRef]

Other (5)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Chap. 12.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1970), pp. 33.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1991), Vol. II, pp. 38–41.

E. Centurioni, “A GPL optical simulation program for mixed coherent/incoherent multilayer systems,” available at www.bo.imm.cnr.it/∼centurio/optical.html .

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977), pp. 332–340.

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Figures (10)

Fig. 1
Fig. 1

Schematic representation of a multilayer with forward- and backward-propagating electric field components shown.

Fig. 2
Fig. 2

Schematic representation of a multilayer with U = |E|2 forward- and backward-propagating components shown.

Fig. 3
Fig. 3

Schematic representation of a mixed coherent–incoherent multilayer structure with U = |E|2 forward- and backward-propagating components shown.

Fig. 4
Fig. 4

Detail of a mixed coherent–incoherent multilayer structure with U = |E|2 and E forward- and backward-propagating components shown.

Fig. 5
Fig. 5

Refractive index spectra used for the simulations.

Fig. 6
Fig. 6

Extinction coefficient spectra used for the simulations.

Fig. 7
Fig. 7

Calculated normalized internal light absorption at normal incidence for the pin solar cell described in the text. Total absorption (100 − RT) is also shown.

Fig. 8
Fig. 8

Calculated quantum efficiencies for different incidence angles.

Fig. 9
Fig. 9

Calculated quantum efficiencies for different ITO thicknesses.

Fig. 10
Fig. 10

Calculated transmittance for the structure of ITO on glass (1 mm) for different ITO thicknesses (as in Fig. 9).

Equations (60)

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A z 1 z z 2 = z 1 z 2 α ( z ) I ( z ) d z ,
A z 1 z z 2 = Φ ( z 1 ) Φ ( z 2 ) ,
[ E 0 R + E 0 R ] = S [ E ( m + 1 ) L + E ( m + 1 ) L ] ,
S = I 01 L 1 I 12 L m I ( m + 1 ) ,
I i , j = 1 t i j [ 1 r i j r i j 1 ] ,
r i j , p = N j cos ϕ i N i cos ϕ j N j cos ϕ i + N i cos ϕ j ,
r i j , s = N i cos ϕ i N j cos ϕ j N i cos ϕ i + N j cos ϕ j ,
t i j , p = 2 N i cos ϕ i N j cos ϕ i + N i cos ϕ j ,
t i j , s = 2 N i cos ϕ i N i cos ϕ i + N j cos ϕ j ,
L j = L ( β j ) = [ exp ( ι β j ) 0 0 exp ( ι β j ) ] ,
β j = 2 π d j N λ cos ϕ j ,
r = E 0 R E 0 R + = S 21 S 11 ,
t = E ( m + 1 ) L + E 0 R + = 1 S 11 ,
r = E ( m + 1 ) L + E ( m + 1 ) L = S 12 S 11 ,
t = E 0 R E ( m + 1 ) L = det S S 11 .
Φ = C | E s | 2 Re ( N cos ϕ ) ,
Φ = C | E p | 2 Re ( N * cos ϕ ) ,
R = | r | 2
T s = | t s | 2 Re ( N m + 1 cos ϕ m + 1 ) N 0 cos ϕ 0 ,
T p = | t p | 2 Re ( N m + 1 * cos ϕ m + 1 ) N 0 cos ϕ 0 .
R = R p + R s 2 , T = T p + T s 2
[ U 0 R + U 0 R ] = S ¯ [ U ( m 1 ) L + U ( m 1 ) L ] ,
S ¯ = Ī 0 1 L ¯ 1 Ī 1 2 L ¯ m , Ī m ( m + 1 ) ,
Ī j ( j + 1 ) = 1 | t | 2 [ 1 | r | 2 | r | 2 | t t | 2 | r r | 2 ] ,
r = E j R E j R + , t = E ( j 1 ) L + E j R + ,
r = E ( j + 1 ) L + E ( j + 1 ) L , t = E j R E ( j + 1 ) L .
L ¯ j = L ¯ ( β j ) = [ | exp ( ι β j ) | 2 0 0 | exp ( ι β j ) | 2 ] .
r ¯ = U 0 R U 0 R + = S ¯ 21 S ¯ 11 ,
t ¯ = U ( m + 1 ) L + U 0 R + = 1 S ¯ 11 ,
r ¯ = U ( m + 1 ) L + U ( m + 1 ) L = S ¯ 12 S ¯ 11 ,
t ¯ = U 0 R U ( m + 1 ) L = det S ¯ S ¯ 11 .
R = r ¯
T s = t ¯ s Re ( N m + 1 cos ϕ m + 1 ) N 0 cos ϕ 0 ,
T p = t ¯ p Re ( N m + 1 * cos ϕ m + 1 ) N 0 cos ϕ 0 ,
Ī i ( i + 1 ) = [ | S 11 | 2 | S 12 | 2 | S 21 | 2 | det S | 2 | S 12 S 21 | 2 | S 11 | 2 ] .
[ U i R + U i R ] = S ¯ i [ U ( m + 1 ) L + U ( m + 1 ) L + ] ,
S ¯ i = Ī i ( i + 1 ) L ¯ i + 1 L ¯ m Ī m ( m + 1 ) .
[ U i R + U i R ] = S ¯ i [ 1 S ¯ 11 0 ] U i R + .
[ Φ i R + Φ i R ] = S ¯ i [ 1 S ¯ 11 0 ] γ Φ 0 R + ,
γ s = Re ( N i cos ϕ i ) N 0 cos ϕ 0 , γ p = Re ( N i * cos ϕ i ) N 0 cos ϕ 0
[ Φ i + ( z ) Φ i ( z ) ] = L ¯ ( β i d i z d i ) [ Φ i R + Φ i R ] ,
Φ i ( z ) = Φ i + ( z ) Φ i ( z ) .
[ E j R + E j R ] = S j [ E ( m + 1 ) L + 0 ] = S j [ 1 S 11 0 ] E 0 R + ,
S j = I j ( j + 1 ) L j + 1 L m I m ( m + 1 ) .
E 0 R + E 0 R + = ( U i R + U 0 R + ) 1 / 2 .
E 0 R + = { ( 1 , 0 ) S ¯ i [ 1 S ¯ 11 0 ] } 1 / 2 E 0 R + ,
[ E j + ( z ) E j ( z ) ] = L ( β j d j z d j ) [ E j R + E j R ] ,
S ( z ) = c Re { E ( z ) × [ B ( z ) ] * } N 0 cos ϕ 0 ,
Φ ( z ) = S ( z ) · [ 0 , 0 , 1 ] .
E f ( z ) = [ 1 , 0 , 0 ] E j + ( z ) ,
E b ( z ) = [ 1 , 0 , 0 ] E j ( z ) ,
E f ( z ) = [ 0 , cos ϕ , sin ϕ ] E j + ( z ) ,
E b ( z ) = [ 0 , cos ϕ , sin ϕ ] E j ( z ) .
E ( z ) = E f ( z ) + E b ( z ) .
B ( z ) = N j u f × E f ( z ) + N j u b × E b ( z ) c ,
u f = [ 0 , sin ϕ , cos ϕ ] , u b = [ 0 , sin ϕ , cos ϕ ] .
[ E j R + E j R ] = S j [ E ( m + 1 ) L + E ( m + 1 ) L ] = S j [ S 12 S 11 1 ] E ( m + 1 ) L .
E ( m + 1 ) L E 0 R + = [ U ( i + 1 ) L U 0 R + ] 1 / 2 .
E ( m + 1 ) L = { ( 0 , 1 ) L ¯ i + 1 S ¯ i + 1 [ 1 S ¯ 11 0 ] } 1 / 2 E 0 R + .
Φ j ( x ) = Φ j l ( x ) + Φ j r ( x ) .

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