Abstract

In many thin-film photovoltaic devices, the photoactive layer has a spatially varying refractive index in the substrate-normal direction, but measurement of this variation with high spatial resolution is difficult due to the thinness of these layers (typically 200 nm for organic photovoltaics). We demonstrate a new method for reconstructing the depth-dependent refractive-index profile with high spatial resolution (~10 nm at a wavelength of 500 nm) in thin (200 nm) photoactive layers by depositing a relatively thick index-matched layer (1-10 μm) adjacent to the photoactive layer and applying the Inverse Wentzel-Kramers-Brillouin (IWKB) method. This novel technique, which we refer to as index-matched IWKB (IM-IWKB), is applicable to any thin film, including the photoactive layers of a broad range of thin-film photovoltaics.

© 2014 Optical Society of America

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References

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    [CrossRef]
  3. A. Ashraf, D. M. N. M. Dissanayake, D. S. Germack, C. R. Weiland, and M. D. Eisaman, “Confinement-induced reduction in phase segregation and interchain disorder in bulk heterojunction films,” ACS Nano (2013).
  4. B. H. Hamadani, S. Jung, P. M. Haney, L. J. Richter, and N. B. Zhitenev, “Origin of nanoscale variations in photoresponse of an organic solar cell,” Nano Lett. 10(5), 1611–1617 (2010).
    [CrossRef] [PubMed]
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2012 (1)

D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Guided-mode quantum efficiency: A novel optoelectronic characterization technique,” Rev. Sci. Instrum. 83(11), 114704 (2012).
[CrossRef] [PubMed]

2011 (1)

W. S. Koh, M. Pant, Y. A. Akimov, W. P. Goh, and Y. Li, “Three-dimensional optoelectronic model for organic bulk heterojunction solar cells,” IEEE J. Photovoltaics 1(1), 84–92 (2011).
[CrossRef]

2010 (2)

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

B. H. Hamadani, S. Jung, P. M. Haney, L. J. Richter, and N. B. Zhitenev, “Origin of nanoscale variations in photoresponse of an organic solar cell,” Nano Lett. 10(5), 1611–1617 (2010).
[CrossRef] [PubMed]

2009 (1)

R. Kniese, M. Powalla, and U. Rau, “Evaluation of electron beam induced current profiles of Cu(In,Ga)Se2 solar cells with different Ga-cotents,” Thin Solid Films 517(7), 2357–2359 (2009).
[CrossRef]

2007 (2)

R. A. Marsh, C. Groves, and N. C. Greenham, “A microscopic model for the behavior of nanostructured organic photovoltaic devices,” J. Appl. Phys. 101(8), 083509 (2007).
[CrossRef]

M. Khardani, M. Bouaïcha, and B. Bessaïs, “Bruggeman effective medium approach for modelling optical properties of porous silicon: comparison with experiment,” Phys. Status Solidi 4(c), 1986–1990 (2007).

2005 (2)

1999 (1)

1997 (1)

R. Scheer, M. Wilhelm, H. J. Lewerenz, H. W. Schock, and L. Stolt, “Determination of charge carrier collecting regions in chalcopyrite heterojunction solar cells by electron-beam-induced current measurements,” Sol. Energy Mater. Sol. Cells 49(1–4), 299–309 (1997).
[CrossRef]

1995 (3)

R. Scheer, C. Knieper, and L. Stolt, “Depth dependent collection functions in thin film chalcopyrite solar cells,” Appl. Phys. Lett. 67(20), 3007–3009 (1995).
[CrossRef]

G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, “Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions,” Science 270(5243), 1789–1791 (1995).
[CrossRef]

L. P. Shi, E. Y. B. Pun, and P. S. Chung, “Extended IWKB method for determination of the refractive=index profile in optical waveguides,” Opt. Lett. 20(15), 1622–1624 (1995).
[CrossRef] [PubMed]

1985 (1)

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3(2), 385–391 (1985).
[CrossRef]

1973 (1)

1969 (1)

P. K. Tien, R. Ulrich, and R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14(9), 291–294 (1969).
[CrossRef]

Akimov, Y. A.

W. S. Koh, M. Pant, Y. A. Akimov, W. P. Goh, and Y. Li, “Three-dimensional optoelectronic model for organic bulk heterojunction solar cells,” IEEE J. Photovoltaics 1(1), 84–92 (2011).
[CrossRef]

Ashraf, A.

D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Guided-mode quantum efficiency: A novel optoelectronic characterization technique,” Rev. Sci. Instrum. 83(11), 114704 (2012).
[CrossRef] [PubMed]

A. Ashraf, D. M. N. M. Dissanayake, D. S. Germack, C. R. Weiland, and M. D. Eisaman, “Confinement-induced reduction in phase segregation and interchain disorder in bulk heterojunction films,” ACS Nano (2013).

D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Mapping spatially resolved charge collection probability within P3HT:PCBM bulk heterojunction photovoltaics,” Adv. Energy Mater. (2013).

Bessaïs, B.

M. Khardani, M. Bouaïcha, and B. Bessaïs, “Bruggeman effective medium approach for modelling optical properties of porous silicon: comparison with experiment,” Phys. Status Solidi 4(c), 1986–1990 (2007).

Bouaïcha, M.

M. Khardani, M. Bouaïcha, and B. Bessaïs, “Bruggeman effective medium approach for modelling optical properties of porous silicon: comparison with experiment,” Phys. Status Solidi 4(c), 1986–1990 (2007).

Cao, Z.

Chan, C. K.

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

Chen, Y.

Chiang, K. S.

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3(2), 385–391 (1985).
[CrossRef]

Chung, P. S.

DeLongchamp, D. M.

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

Dissanayake, D. M. N. M.

D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Guided-mode quantum efficiency: A novel optoelectronic characterization technique,” Rev. Sci. Instrum. 83(11), 114704 (2012).
[CrossRef] [PubMed]

A. Ashraf, D. M. N. M. Dissanayake, D. S. Germack, C. R. Weiland, and M. D. Eisaman, “Confinement-induced reduction in phase segregation and interchain disorder in bulk heterojunction films,” ACS Nano (2013).

D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Mapping spatially resolved charge collection probability within P3HT:PCBM bulk heterojunction photovoltaics,” Adv. Energy Mater. (2013).

Dou, X.

Edoff, M.

O. Lundberg, M. Edoff, and L. Stolt, “The effect of Ga-grading in CIGS thin film solar cells,” Thin Solid Films 480–481, 520–525 (2005).
[CrossRef]

Eisaman, M. D.

D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Guided-mode quantum efficiency: A novel optoelectronic characterization technique,” Rev. Sci. Instrum. 83(11), 114704 (2012).
[CrossRef] [PubMed]

A. Ashraf, D. M. N. M. Dissanayake, D. S. Germack, C. R. Weiland, and M. D. Eisaman, “Confinement-induced reduction in phase segregation and interchain disorder in bulk heterojunction films,” ACS Nano (2013).

D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Mapping spatially resolved charge collection probability within P3HT:PCBM bulk heterojunction photovoltaics,” Adv. Energy Mater. (2013).

Fischer, D. A.

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

Gao, J.

G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, “Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions,” Science 270(5243), 1789–1791 (1995).
[CrossRef]

Germack, D. S.

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

A. Ashraf, D. M. N. M. Dissanayake, D. S. Germack, C. R. Weiland, and M. D. Eisaman, “Confinement-induced reduction in phase segregation and interchain disorder in bulk heterojunction films,” ACS Nano (2013).

Goh, W. P.

W. S. Koh, M. Pant, Y. A. Akimov, W. P. Goh, and Y. Li, “Three-dimensional optoelectronic model for organic bulk heterojunction solar cells,” IEEE J. Photovoltaics 1(1), 84–92 (2011).
[CrossRef]

Greenham, N. C.

R. A. Marsh, C. Groves, and N. C. Greenham, “A microscopic model for the behavior of nanostructured organic photovoltaic devices,” J. Appl. Phys. 101(8), 083509 (2007).
[CrossRef]

Groves, C.

R. A. Marsh, C. Groves, and N. C. Greenham, “A microscopic model for the behavior of nanostructured organic photovoltaic devices,” J. Appl. Phys. 101(8), 083509 (2007).
[CrossRef]

Gundlach, D. J.

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

Hamadani, B. H.

B. H. Hamadani, S. Jung, P. M. Haney, L. J. Richter, and N. B. Zhitenev, “Origin of nanoscale variations in photoresponse of an organic solar cell,” Nano Lett. 10(5), 1611–1617 (2010).
[CrossRef] [PubMed]

Haney, P. M.

B. H. Hamadani, S. Jung, P. M. Haney, L. J. Richter, and N. B. Zhitenev, “Origin of nanoscale variations in photoresponse of an organic solar cell,” Nano Lett. 10(5), 1611–1617 (2010).
[CrossRef] [PubMed]

Heeger, A. J.

G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, “Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions,” Science 270(5243), 1789–1791 (1995).
[CrossRef]

Hummelen, J. C.

G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, “Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions,” Science 270(5243), 1789–1791 (1995).
[CrossRef]

Jiang, Y.

Jung, S.

B. H. Hamadani, S. Jung, P. M. Haney, L. J. Richter, and N. B. Zhitenev, “Origin of nanoscale variations in photoresponse of an organic solar cell,” Nano Lett. 10(5), 1611–1617 (2010).
[CrossRef] [PubMed]

Khardani, M.

M. Khardani, M. Bouaïcha, and B. Bessaïs, “Bruggeman effective medium approach for modelling optical properties of porous silicon: comparison with experiment,” Phys. Status Solidi 4(c), 1986–1990 (2007).

Kline, R. J.

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

Knieper, C.

R. Scheer, C. Knieper, and L. Stolt, “Depth dependent collection functions in thin film chalcopyrite solar cells,” Appl. Phys. Lett. 67(20), 3007–3009 (1995).
[CrossRef]

Kniese, R.

R. Kniese, M. Powalla, and U. Rau, “Evaluation of electron beam induced current profiles of Cu(In,Ga)Se2 solar cells with different Ga-cotents,” Thin Solid Films 517(7), 2357–2359 (2009).
[CrossRef]

Koh, W. S.

W. S. Koh, M. Pant, Y. A. Akimov, W. P. Goh, and Y. Li, “Three-dimensional optoelectronic model for organic bulk heterojunction solar cells,” IEEE J. Photovoltaics 1(1), 84–92 (2011).
[CrossRef]

Lewerenz, H. J.

R. Scheer, M. Wilhelm, H. J. Lewerenz, H. W. Schock, and L. Stolt, “Determination of charge carrier collecting regions in chalcopyrite heterojunction solar cells by electron-beam-induced current measurements,” Sol. Energy Mater. Sol. Cells 49(1–4), 299–309 (1997).
[CrossRef]

Li, Y.

W. S. Koh, M. Pant, Y. A. Akimov, W. P. Goh, and Y. Li, “Three-dimensional optoelectronic model for organic bulk heterojunction solar cells,” IEEE J. Photovoltaics 1(1), 84–92 (2011).
[CrossRef]

Lundberg, O.

O. Lundberg, M. Edoff, and L. Stolt, “The effect of Ga-grading in CIGS thin film solar cells,” Thin Solid Films 480–481, 520–525 (2005).
[CrossRef]

Marsh, R. A.

R. A. Marsh, C. Groves, and N. C. Greenham, “A microscopic model for the behavior of nanostructured organic photovoltaic devices,” J. Appl. Phys. 101(8), 083509 (2007).
[CrossRef]

Martin, R. J.

P. K. Tien, R. Ulrich, and R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14(9), 291–294 (1969).
[CrossRef]

Pang, Y.

D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Guided-mode quantum efficiency: A novel optoelectronic characterization technique,” Rev. Sci. Instrum. 83(11), 114704 (2012).
[CrossRef] [PubMed]

D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Mapping spatially resolved charge collection probability within P3HT:PCBM bulk heterojunction photovoltaics,” Adv. Energy Mater. (2013).

Pant, M.

W. S. Koh, M. Pant, Y. A. Akimov, W. P. Goh, and Y. Li, “Three-dimensional optoelectronic model for organic bulk heterojunction solar cells,” IEEE J. Photovoltaics 1(1), 84–92 (2011).
[CrossRef]

Powalla, M.

R. Kniese, M. Powalla, and U. Rau, “Evaluation of electron beam induced current profiles of Cu(In,Ga)Se2 solar cells with different Ga-cotents,” Thin Solid Films 517(7), 2357–2359 (2009).
[CrossRef]

Pun, E. Y. B.

Rau, U.

R. Kniese, M. Powalla, and U. Rau, “Evaluation of electron beam induced current profiles of Cu(In,Ga)Se2 solar cells with different Ga-cotents,” Thin Solid Films 517(7), 2357–2359 (2009).
[CrossRef]

Richter, L. J.

B. H. Hamadani, S. Jung, P. M. Haney, L. J. Richter, and N. B. Zhitenev, “Origin of nanoscale variations in photoresponse of an organic solar cell,” Nano Lett. 10(5), 1611–1617 (2010).
[CrossRef] [PubMed]

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

Scheer, R.

R. Scheer, M. Wilhelm, H. J. Lewerenz, H. W. Schock, and L. Stolt, “Determination of charge carrier collecting regions in chalcopyrite heterojunction solar cells by electron-beam-induced current measurements,” Sol. Energy Mater. Sol. Cells 49(1–4), 299–309 (1997).
[CrossRef]

R. Scheer, C. Knieper, and L. Stolt, “Depth dependent collection functions in thin film chalcopyrite solar cells,” Appl. Phys. Lett. 67(20), 3007–3009 (1995).
[CrossRef]

Schock, H. W.

R. Scheer, M. Wilhelm, H. J. Lewerenz, H. W. Schock, and L. Stolt, “Determination of charge carrier collecting regions in chalcopyrite heterojunction solar cells by electron-beam-induced current measurements,” Sol. Energy Mater. Sol. Cells 49(1–4), 299–309 (1997).
[CrossRef]

Shen, Q.

Shi, L. P.

Stolt, L.

O. Lundberg, M. Edoff, and L. Stolt, “The effect of Ga-grading in CIGS thin film solar cells,” Thin Solid Films 480–481, 520–525 (2005).
[CrossRef]

R. Scheer, M. Wilhelm, H. J. Lewerenz, H. W. Schock, and L. Stolt, “Determination of charge carrier collecting regions in chalcopyrite heterojunction solar cells by electron-beam-induced current measurements,” Sol. Energy Mater. Sol. Cells 49(1–4), 299–309 (1997).
[CrossRef]

R. Scheer, C. Knieper, and L. Stolt, “Depth dependent collection functions in thin film chalcopyrite solar cells,” Appl. Phys. Lett. 67(20), 3007–3009 (1995).
[CrossRef]

Tien, P. K.

P. K. Tien, R. Ulrich, and R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14(9), 291–294 (1969).
[CrossRef]

Toney, M. F.

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

Torge, R.

Ulrich, R.

R. Ulrich and R. Torge, “Measurement of thin film parameters with a prism coupler,” Appl. Opt. 12(12), 2901–2908 (1973).
[CrossRef] [PubMed]

P. K. Tien, R. Ulrich, and R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14(9), 291–294 (1969).
[CrossRef]

Weiland, C. R.

A. Ashraf, D. M. N. M. Dissanayake, D. S. Germack, C. R. Weiland, and M. D. Eisaman, “Confinement-induced reduction in phase segregation and interchain disorder in bulk heterojunction films,” ACS Nano (2013).

Wilhelm, M.

R. Scheer, M. Wilhelm, H. J. Lewerenz, H. W. Schock, and L. Stolt, “Determination of charge carrier collecting regions in chalcopyrite heterojunction solar cells by electron-beam-induced current measurements,” Sol. Energy Mater. Sol. Cells 49(1–4), 299–309 (1997).
[CrossRef]

Wudl, F.

G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, “Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions,” Science 270(5243), 1789–1791 (1995).
[CrossRef]

Yu, G.

G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, “Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions,” Science 270(5243), 1789–1791 (1995).
[CrossRef]

Zhitenev, N. B.

B. H. Hamadani, S. Jung, P. M. Haney, L. J. Richter, and N. B. Zhitenev, “Origin of nanoscale variations in photoresponse of an organic solar cell,” Nano Lett. 10(5), 1611–1617 (2010).
[CrossRef] [PubMed]

Zhu, H.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

P. K. Tien, R. Ulrich, and R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14(9), 291–294 (1969).
[CrossRef]

R. Scheer, C. Knieper, and L. Stolt, “Depth dependent collection functions in thin film chalcopyrite solar cells,” Appl. Phys. Lett. 67(20), 3007–3009 (1995).
[CrossRef]

IEEE J. Photovoltaics (1)

W. S. Koh, M. Pant, Y. A. Akimov, W. P. Goh, and Y. Li, “Three-dimensional optoelectronic model for organic bulk heterojunction solar cells,” IEEE J. Photovoltaics 1(1), 84–92 (2011).
[CrossRef]

J. Appl. Phys. (1)

R. A. Marsh, C. Groves, and N. C. Greenham, “A microscopic model for the behavior of nanostructured organic photovoltaic devices,” J. Appl. Phys. 101(8), 083509 (2007).
[CrossRef]

J. Lightwave Technol. (1)

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3(2), 385–391 (1985).
[CrossRef]

J. Opt. Soc. Am. A (1)

Macromolecules (1)

D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010).
[CrossRef]

Nano Lett. (1)

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Figures (4)

Fig. 1
Fig. 1

Refractive index profile n(x) of an index-matched layer and an unknown photoactive layer on a glass substrate, as a function of position x. (Ni, xi) represents the (effective index, turning point) for guided mode i.

Fig. 2
Fig. 2

Prism coupler setup used to measure the effective indices of the sample.

Fig. 3
Fig. 3

Reconstructed RIPs for TE-polarized light performed by the IM-IWKB method using guided mode effective indices obtained via FDTD simulation for the structure: substrate(semi-infinite, n = 1.76)/index-matched layer(thickness tIM, n = 1.93)/photoactive layer(thickness tPA, “Actual Profile” (solid black line): nPA(x) = 1.93−0.13(x/tPA)2 unless otherwise stated)/air. Wavelength λ assumed to be 500 nm unless otherwise stated. The number in parentheses in the legend is the root mean squared difference between the reconstruction and the actual profile. (a) tIM = (1 μm, 5 μm, 10 µm); tPA = 1μm. (b) tIM = 10 μm; tPA = 1μm, nPA(x) = (parabolic, exponential, Gaussian). (c) tIM = 5 μm; tPA = (200 nm, 500 nm, 1 µm) normalized to 1. (d) tIM = 5 μm; tPA = 200 nm; λ = (500 nm, 650 nm, 829 nm). (e) tIM = 5μm, nIM = (1.95, 2.0, 2.03), Δn = (0.02, 0.07, 0.1); tPA = 1 μm. (f) Spatial resolution (defined as the average spacing between successive points in the reconstruction) for the IM-IWKB reconstruction tIM = 1-10 μm; tPA = (1 μm, 200 nm).

Fig. 4
Fig. 4

Experimental reconstruction of the RIP by the IM-IWKB method, using guided mode effective indices measured the prism coupler setup shown in Fig. 2. (a) Reflection spectrum for structure: sapphire(430 μm)/index-matched layer(AlN, tIM = 5.3 μm)/photoactive layer (P3HT:PCBM, 200 nm)/air. (b) RIP reconstruction for the structure in (a). (c) RIP reconstruction of the photoactive layer region for the structure in (a). (d) RIP reconstruction for the structure in (a) with (red) and without (green) thermal annealing. In panels (d), the right-hand axis shows the PCBM volume fraction calculated from the RIP by applying Bruggeman effective medium theory as described in the text.

Equations (8)

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k 0 x i [ v 2 ( x ) N i 2 ] 1 2 d x = ( i 1 ) π + ϕ 0 + ϕ t , i = 1 , 2 , 3...
n ( x i ) = N i
v ( x ) = { n ( x ) ( T E ) n ( x ) [ 1 + n ( x ) n ˙ ( x ) 2 n ¨ 2 ( x ) k 2 n 4 ( x ) ] ( T M )
ϕ 0 = tan 1 { r 0 [ N i 2 n g l a s s 2 N 0 2 N i 2 ] 1 2 }
ϕ t = π 4
x i = (i1)π+ ϕ 0 ( N i )+ ϕ t j=1 i1 k{ x j [ ( N avg,j 2 N i 2 ) 1 2 ( N avg,j+1 2 N i 2 ) 1 2 ]} k ( N avg,i 2 N i 2 ) 1 2 , i=1,2,3
v P 3 H T n P 3 H T N i n P 3 H T + 2 N i + v P C B M n P C B M N i n P C B M + 2 N i = 0 , i = 1 , 2 , 3...
v P 3 H T + v P C B M = 1

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