Abstract

We describe a method to determine the refractive index and extinction coefficient of thin film materials without prior knowledge of the film thickness and without the assumption of a dispersion model. A straightforward back calculation to the optical parameters can be performed starting from simple measurements of reflection and transmission spectra of a 100250nm thick supported film. The exact film thickness is found simultaneously by fulfilling the intrinsic demand of continuity of the refractive index as a function of wavelength. If both the layer and the substrate are homogeneous and isotropic media with plane and parallel interfaces, effects like surface roughness, scattering, or thickness inhomogenei ties can be neglected. Then, the accuracy of the measurement is approximately 102 and 103 for the refractive index and the extinction coefficient, respectively. The error of the thin film thickness determination is well below 1nm. Thus this technique is well suited to determine the input parameters for optical simulations of organic thin film devices, such as organic light-emitting diodes (OLEDs) or organic photovoltaic (OPV) cells. We apply the method to the electroluminescent polymer poly(2,5-dioctyl-p-phenylene vinylene) (PDO-PPV) and show its applicability by comparing the measured and calculated reflection and transmission spectra of OLED stacks with up to five layers.

© 2009 Optical Society of America

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  30. Note, that Eq. is the general error estimation of all parameters affecting nf and kf. For a different measurement setup, e.g., utilizing oblique incidence of light, the errors of φ may have a major effect on the uncertainties Δnf and Δkf and have to be considered explicitly.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. L. A. A. Pettersson, S. Ghosh, and O. Inganäs, “Optical anisotropy in thin films of poly (3,4-ethylenedioxythiophene)-poly (4-styrenesulfonate),” Org. Electron. 3, 143-148(2002).
    [CrossRef]
  39. M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
    [CrossRef]

2008 (3)

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

J. Sancho-Parramon, M. Modreanu, S. Bosch, and M. Stchakovsky, “Optical characterization of HfO2 by spectroscopic ellipsometry: Dispersion models and direct data inversion,” Thin Solid Films 516, 7990-7995 (2008).
[CrossRef]

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

2007 (3)

M. C. Gather, A. Köhnen, A. Falcou, H. Becker, and K. Meerholz, “Solution-processed full-color polymer organic light-emitting diode displays fabricated by direct photolithography,” Adv. Funct. Mater. 17, 191-200 (2007).
[CrossRef]

P. Zacharias, M. C. Gather, M. Rojahn, O. Nuyken, and K. Meerholz, “New crosslinkable hole conductors for blue phosphorescent OLEDs,” Angew. Chem., Int. Ed. Engl. 46, 4388-4392 (2007).
[CrossRef]

A. J. Moulé and K. Meerholz, “Interference method for the determination of the complex refractive index of thin polymer layers,” Appl. Phys. Lett. 91, 061901 (2007).
[CrossRef]

2006 (1)

T. Tsuboi, Y. Wasai, and N. Nabatova-Gabain, “Optical constants of platinum octaethyl porphyrin in single-layer organic light emitting diode studied by spectroscopic ellipsometry,” Thin Solid Films 496, 674-678 (2006).
[CrossRef]

2005 (3)

M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, and D. D. C. Bradley, “Ellipsometric characterization of the optical constants of polyfluorene gain media,” Adv. Funct. Mater. 15, 925-933 (2005).
[CrossRef]

K. Meerholz, “Device physics: enlightening solutions,” Nature 437, 327-328 (2005).
[CrossRef] [PubMed]

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[CrossRef]

2003 (2)

X. Wu, G. Shi, L. Qu, J. Zhang, and F. Chen, “Novel route to poly(p-phenylene vinylene) polymers,” J. Polym. Sci., Part A: Polym. Chem. 41, 449-455 (2003).
[CrossRef]

P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” J. Appl. Phys. 93, 3693-3723 (2003).
[CrossRef]

2002 (2)

L. A. A. Pettersson, S. Ghosh, and O. Inganäs, “Optical anisotropy in thin films of poly (3,4-ethylenedioxythiophene)-poly (4-styrenesulfonate),” Org. Electron. 3, 143-148(2002).
[CrossRef]

C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interface,” Appl. Opt. 41, 3978-3987 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

A. B. Djurisic, T. Fritz, and K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35-42 (1999).
[CrossRef]

1998 (2)

K. Vedam, “Spectroscopic ellipsometry: a historical overview,” Thin Solid Films 313-314, 1-9 (1998).
[CrossRef]

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[CrossRef]

1997 (1)

S. V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, and Z. V. Vardeny, “Cooperative emission in pi-conjugated polymer thin films,” Phys. Rev. Lett. 78, 729-732 (1997).
[CrossRef]

1993 (1)

G. E. Jellison, Jr., “Data analysis for spectroscopic ellipsometry,” Thin Solid Films 234, 416-422 (1993).
[CrossRef]

1992 (2)

O. Stenzel, R. Petrich, W. Scharff, V. Hopfe, and A. V. Tikhonravov, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324-329 (1992).
[CrossRef]

J. M. del Pozo and L. Diaz, “Method for the determination of optical constants of thin films: dependence on experimental uncertainties,” Appl. Opt. 31, 4474-4481 (1992).
[CrossRef] [PubMed]

1991 (2)

V. Panayotov and I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: an algebraic solution for the (T,Rf,Rb) method,” Appl. Opt. 30, 2795-2800 (1991).
[CrossRef] [PubMed]

O. Stenzel, V. Hopfe, and P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates form transmittance and reflectance measurements,” J. Phys. D: Appl. Phys. 24, 2088-2094 (1991).
[CrossRef]

1990 (1)

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

1989 (1)

T. Fritz, J. Hahn, and H. Böttcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249-257 (1989).
[CrossRef]

1988 (1)

1987 (1)

C. W. Tang and S. A. VanSlyke, “Organic electroluminescent diodes,” Appl. Phys. Lett. 51, 913-915 (1987).
[CrossRef]

1986 (1)

H. Arwin and D. E. Aspnes, “Determination of optical properties of thin organic films by spectroellipsometry,” Thin Solid Films 138, 195-207 (1986).
[CrossRef]

1984 (2)

1983 (1)

R. T. Phillips, “A numerical method for determining the complex refractive index from reflectance and transmittance of supported thin films,” J. Phys. D: Appl. Phys. 16, 489-497 (1983).
[CrossRef]

1982 (1)

L. Ward, “The accuracy of photometric methods for determining the optical constants of thin absorbing films,” J. Phys. D: Appl. Phys. 15, 1361-1371 (1982).
[CrossRef]

1972 (1)

R. E. Denton, R. D. Campbell, and S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D: Appl. Phys. 5, 852-863(1972).
[CrossRef]

Ariu, M.

M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, and D. D. C. Bradley, “Ellipsometric characterization of the optical constants of polyfluorene gain media,” Adv. Funct. Mater. 15, 925-933 (2005).
[CrossRef]

Arndt, D. P.

Arwin, H.

H. Arwin and D. E. Aspnes, “Determination of optical properties of thin organic films by spectroellipsometry,” Thin Solid Films 138, 195-207 (1986).
[CrossRef]

Aspnes, D. E.

H. Arwin and D. E. Aspnes, “Determination of optical properties of thin organic films by spectroellipsometry,” Thin Solid Films 138, 195-207 (1986).
[CrossRef]

Azzam, R. M. A.

Becker, H.

M. C. Gather, A. Köhnen, A. Falcou, H. Becker, and K. Meerholz, “Solution-processed full-color polymer organic light-emitting diode displays fabricated by direct photolithography,” Adv. Funct. Mater. 17, 191-200 (2007).
[CrossRef]

Bennett, J. M.

Bernitzki, H.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[CrossRef]

Borgogno, J. P.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1964).

Bosch, S.

J. Sancho-Parramon, M. Modreanu, S. Bosch, and M. Stchakovsky, “Optical characterization of HfO2 by spectroscopic ellipsometry: Dispersion models and direct data inversion,” Thin Solid Films 516, 7990-7995 (2008).
[CrossRef]

Böttcher, H.

T. Fritz, J. Hahn, and H. Böttcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249-257 (1989).
[CrossRef]

Botten, L. C.

Bradley, D. D. C.

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[CrossRef]

M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, and D. D. C. Bradley, “Ellipsometric characterization of the optical constants of polyfluorene gain media,” Adv. Funct. Mater. 15, 925-933 (2005).
[CrossRef]

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

Bräuer, A. H.

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

Brown, A. R.

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

Burns, P. L.

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

Burroughes, J. H.

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

Campbell, R. D.

R. E. Denton, R. D. Campbell, and S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D: Appl. Phys. 5, 852-863(1972).
[CrossRef]

Campoy-Quiles, M.

M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, and D. D. C. Bradley, “Ellipsometric characterization of the optical constants of polyfluorene gain media,” Adv. Funct. Mater. 15, 925-933 (2005).
[CrossRef]

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[CrossRef]

Carniglia, C. K.

Case, W. E.

Chen, F.

X. Wu, G. Shi, L. Qu, J. Zhang, and F. Chen, “Novel route to poly(p-phenylene vinylene) polymers,” J. Polym. Sci., Part A: Polym. Chem. 41, 449-455 (2003).
[CrossRef]

Danz, N.

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

del Pozo, J. M.

Denton, R. E.

R. E. Denton, R. D. Campbell, and S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D: Appl. Phys. 5, 852-863(1972).
[CrossRef]

Diaz, L.

Djurisic, A. B.

A. B. Djurisic, T. Fritz, K. Leo, and E. H. Li, “Improved method for determination of optical constants of organic thin films from reflection and transmission measurements,” Appl. Opt. 39, 1174-1182 (2000).
[CrossRef]

A. B. Djurisic, T. Fritz, and K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35-42 (1999).
[CrossRef]

Dobrowolski, J. A.

Etchegoin, P.

M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, and D. D. C. Bradley, “Ellipsometric characterization of the optical constants of polyfluorene gain media,” Adv. Funct. Mater. 15, 925-933 (2005).
[CrossRef]

Etchegoin, P. G.

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[CrossRef]

Falcou, A.

M. C. Gather, A. Köhnen, A. Falcou, H. Becker, and K. Meerholz, “Solution-processed full-color polymer organic light-emitting diode displays fabricated by direct photolithography,” Adv. Funct. Mater. 17, 191-200 (2007).
[CrossRef]

Flämmich, M.

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

Forrest, S. R.

P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” J. Appl. Phys. 93, 3693-3723 (2003).
[CrossRef]

Friend, R. H.

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

Fritz, T.

A. B. Djurisic, T. Fritz, K. Leo, and E. H. Li, “Improved method for determination of optical constants of organic thin films from reflection and transmission measurements,” Appl. Opt. 39, 1174-1182 (2000).
[CrossRef]

A. B. Djurisic, T. Fritz, and K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35-42 (1999).
[CrossRef]

T. Fritz, J. Hahn, and H. Böttcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249-257 (1989).
[CrossRef]

Frolov, S. V.

S. V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, and Z. V. Vardeny, “Cooperative emission in pi-conjugated polymer thin films,” Phys. Rev. Lett. 78, 729-732 (1997).
[CrossRef]

Gather, M. C.

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

M. C. Gather, A. Köhnen, A. Falcou, H. Becker, and K. Meerholz, “Solution-processed full-color polymer organic light-emitting diode displays fabricated by direct photolithography,” Adv. Funct. Mater. 17, 191-200 (2007).
[CrossRef]

P. Zacharias, M. C. Gather, M. Rojahn, O. Nuyken, and K. Meerholz, “New crosslinkable hole conductors for blue phosphorescent OLEDs,” Angew. Chem., Int. Ed. Engl. 46, 4388-4392 (2007).
[CrossRef]

Gellermann, W.

S. V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, and Z. V. Vardeny, “Cooperative emission in pi-conjugated polymer thin films,” Phys. Rev. Lett. 78, 729-732 (1997).
[CrossRef]

Ghosh, S.

L. A. A. Pettersson, S. Ghosh, and O. Inganäs, “Optical anisotropy in thin films of poly (3,4-ethylenedioxythiophene)-poly (4-styrenesulfonate),” Org. Electron. 3, 143-148(2002).
[CrossRef]

Gibson, U. J.

Götzelmann, R.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[CrossRef]

Gupta, M. C.

Hahn, J.

T. Fritz, J. Hahn, and H. Böttcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249-257 (1989).
[CrossRef]

Heliotis, G.

M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, and D. D. C. Bradley, “Ellipsometric characterization of the optical constants of polyfluorene gain media,” Adv. Funct. Mater. 15, 925-933 (2005).
[CrossRef]

Hertel, D.

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

Ho, F. C.

Hodgkin, V. A.

Holmes, A. B.

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

Hopfe, V.

O. Stenzel, R. Petrich, W. Scharff, V. Hopfe, and A. V. Tikhonravov, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324-329 (1992).
[CrossRef]

O. Stenzel, V. Hopfe, and P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates form transmittance and reflectance measurements,” J. Phys. D: Appl. Phys. 24, 2088-2094 (1991).
[CrossRef]

Inganäs, O.

L. A. A. Pettersson, S. Ghosh, and O. Inganäs, “Optical anisotropy in thin films of poly (3,4-ethylenedioxythiophene)-poly (4-styrenesulfonate),” Org. Electron. 3, 143-148(2002).
[CrossRef]

Jellison, G. E.

G. E. Jellison, Jr., “Data analysis for spectroscopic ellipsometry,” Thin Solid Films 234, 416-422 (1993).
[CrossRef]

Kaiser, N.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[CrossRef]

Karthe, W.

W. Karthe and R. Müller, Integrierte Optik (Akademische Verlagsgesellschaft, 1991).

Katsidis, C. C.

Klapp, W. P.

Klobes, P.

O. Stenzel, V. Hopfe, and P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates form transmittance and reflectance measurements,” J. Phys. D: Appl. Phys. 24, 2088-2094 (1991).
[CrossRef]

Köhnen, A.

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

M. C. Gather, A. Köhnen, A. Falcou, H. Becker, and K. Meerholz, “Solution-processed full-color polymer organic light-emitting diode displays fabricated by direct photolithography,” Adv. Funct. Mater. 17, 191-200 (2007).
[CrossRef]

Konstantinov, I.

Kudaev, S.

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

Lauth, H.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[CrossRef]

Laux, S.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[CrossRef]

Leo, K.

A. B. Djurisic, T. Fritz, K. Leo, and E. H. Li, “Improved method for determination of optical constants of organic thin films from reflection and transmission measurements,” Appl. Opt. 39, 1174-1182 (2000).
[CrossRef]

A. B. Djurisic, T. Fritz, and K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35-42 (1999).
[CrossRef]

Li, E. H.

Mackay, K.

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

Macleod, H. A.

Marks, R. N.

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

McKenzie, D. R.

McPhedran, R. C.

Meerholz, K.

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

A. J. Moulé and K. Meerholz, “Interference method for the determination of the complex refractive index of thin polymer layers,” Appl. Phys. Lett. 91, 061901 (2007).
[CrossRef]

M. C. Gather, A. Köhnen, A. Falcou, H. Becker, and K. Meerholz, “Solution-processed full-color polymer organic light-emitting diode displays fabricated by direct photolithography,” Adv. Funct. Mater. 17, 191-200 (2007).
[CrossRef]

P. Zacharias, M. C. Gather, M. Rojahn, O. Nuyken, and K. Meerholz, “New crosslinkable hole conductors for blue phosphorescent OLEDs,” Angew. Chem., Int. Ed. Engl. 46, 4388-4392 (2007).
[CrossRef]

K. Meerholz, “Device physics: enlightening solutions,” Nature 437, 327-328 (2005).
[CrossRef] [PubMed]

Michaelis, D.

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

Modreanu, M.

J. Sancho-Parramon, M. Modreanu, S. Bosch, and M. Stchakovsky, “Optical characterization of HfO2 by spectroscopic ellipsometry: Dispersion models and direct data inversion,” Thin Solid Films 516, 7990-7995 (2008).
[CrossRef]

Moulé, A. J.

A. J. Moulé and K. Meerholz, “Interference method for the determination of the complex refractive index of thin polymer layers,” Appl. Phys. Lett. 91, 061901 (2007).
[CrossRef]

Müllen, K.

K. Müllen and U. Scherf, Organic Light Emitting Devices: Synthesis, Properties and Applications (Wiley-VCH, 2006).

Müller, R.

W. Karthe and R. Müller, Integrierte Optik (Akademische Verlagsgesellschaft, 1991).

Nabatova-Gabain, N.

T. Tsuboi, Y. Wasai, and N. Nabatova-Gabain, “Optical constants of platinum octaethyl porphyrin in single-layer organic light emitting diode studied by spectroscopic ellipsometry,” Thin Solid Films 496, 674-678 (2006).
[CrossRef]

Netterfield, R. P.

Nuyken, O.

P. Zacharias, M. C. Gather, M. Rojahn, O. Nuyken, and K. Meerholz, “New crosslinkable hole conductors for blue phosphorescent OLEDs,” Angew. Chem., Int. Ed. Engl. 46, 4388-4392 (2007).
[CrossRef]

Ozaki, M.

S. V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, and Z. V. Vardeny, “Cooperative emission in pi-conjugated polymer thin films,” Phys. Rev. Lett. 78, 729-732 (1997).
[CrossRef]

Panayotov, V.

Pelletier, E.

Petrich, R.

O. Stenzel, R. Petrich, W. Scharff, V. Hopfe, and A. V. Tikhonravov, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324-329 (1992).
[CrossRef]

Pettersson, L. A. A.

L. A. A. Pettersson, S. Ghosh, and O. Inganäs, “Optical anisotropy in thin films of poly (3,4-ethylenedioxythiophene)-poly (4-styrenesulfonate),” Org. Electron. 3, 143-148(2002).
[CrossRef]

Peumans, P.

P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” J. Appl. Phys. 93, 3693-3723 (2003).
[CrossRef]

Phillips, R. T.

R. T. Phillips, “A numerical method for determining the complex refractive index from reflectance and transmittance of supported thin films,” J. Phys. D: Appl. Phys. 16, 489-497 (1983).
[CrossRef]

Pintani, M.

M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, and D. D. C. Bradley, “Ellipsometric characterization of the optical constants of polyfluorene gain media,” Adv. Funct. Mater. 15, 925-933 (2005).
[CrossRef]

Purvis, M. K.

Qu, L.

X. Wu, G. Shi, L. Qu, J. Zhang, and F. Chen, “Novel route to poly(p-phenylene vinylene) polymers,” J. Polym. Sci., Part A: Polym. Chem. 41, 449-455 (2003).
[CrossRef]

Quinn, D. M.

Rehmann, N.

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

Rojahn, M.

P. Zacharias, M. C. Gather, M. Rojahn, O. Nuyken, and K. Meerholz, “New crosslinkable hole conductors for blue phosphorescent OLEDs,” Angew. Chem., Int. Ed. Engl. 46, 4388-4392 (2007).
[CrossRef]

Sancho-Parramon, J.

J. Sancho-Parramon, M. Modreanu, S. Bosch, and M. Stchakovsky, “Optical characterization of HfO2 by spectroscopic ellipsometry: Dispersion models and direct data inversion,” Thin Solid Films 516, 7990-7995 (2008).
[CrossRef]

Scharff, W.

O. Stenzel, R. Petrich, W. Scharff, V. Hopfe, and A. V. Tikhonravov, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324-329 (1992).
[CrossRef]

Scherf, U.

K. Müllen and U. Scherf, Organic Light Emitting Devices: Synthesis, Properties and Applications (Wiley-VCH, 2006).

Schubert, U. S.

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

Shi, G.

X. Wu, G. Shi, L. Qu, J. Zhang, and F. Chen, “Novel route to poly(p-phenylene vinylene) polymers,” J. Polym. Sci., Part A: Polym. Chem. 41, 449-455 (2003).
[CrossRef]

Siapkas, D. I.

Stchakovsky, M.

J. Sancho-Parramon, M. Modreanu, S. Bosch, and M. Stchakovsky, “Optical characterization of HfO2 by spectroscopic ellipsometry: Dispersion models and direct data inversion,” Thin Solid Films 516, 7990-7995 (2008).
[CrossRef]

Stenzel, O.

O. Stenzel, R. Petrich, W. Scharff, V. Hopfe, and A. V. Tikhonravov, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324-329 (1992).
[CrossRef]

O. Stenzel, V. Hopfe, and P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates form transmittance and reflectance measurements,” J. Phys. D: Appl. Phys. 24, 2088-2094 (1991).
[CrossRef]

Strome, D. H.

Swenson, R.

Tang, C. W.

C. W. Tang and S. A. VanSlyke, “Organic electroluminescent diodes,” Appl. Phys. Lett. 51, 913-915 (1987).
[CrossRef]

Temple, P. A.

Thonn, T. F.

Tikhonravov, A. V.

O. Stenzel, R. Petrich, W. Scharff, V. Hopfe, and A. V. Tikhonravov, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324-329 (1992).
[CrossRef]

Tomlin, S. G.

R. E. Denton, R. D. Campbell, and S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D: Appl. Phys. 5, 852-863(1972).
[CrossRef]

Tsuboi, T.

T. Tsuboi, Y. Wasai, and N. Nabatova-Gabain, “Optical constants of platinum octaethyl porphyrin in single-layer organic light emitting diode studied by spectroscopic ellipsometry,” Thin Solid Films 496, 674-678 (2006).
[CrossRef]

Tuttle Hart, T.

Ulbricht, C.

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

VanSlyke, S. A.

C. W. Tang and S. A. VanSlyke, “Organic electroluminescent diodes,” Appl. Phys. Lett. 51, 913-915 (1987).
[CrossRef]

Vardeny, Z. V.

S. V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, and Z. V. Vardeny, “Cooperative emission in pi-conjugated polymer thin films,” Phys. Rev. Lett. 78, 729-732 (1997).
[CrossRef]

Vedam, K.

K. Vedam, “Spectroscopic ellipsometry: a historical overview,” Thin Solid Films 313-314, 1-9 (1998).
[CrossRef]

Wächter, C. A.

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

Ward, L.

L. Ward, “The accuracy of photometric methods for determining the optical constants of thin absorbing films,” J. Phys. D: Appl. Phys. 15, 1361-1371 (1982).
[CrossRef]

Wasai, Y.

T. Tsuboi, Y. Wasai, and N. Nabatova-Gabain, “Optical constants of platinum octaethyl porphyrin in single-layer organic light emitting diode studied by spectroscopic ellipsometry,” Thin Solid Films 496, 674-678 (2006).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1964).

Wu, X.

X. Wu, G. Shi, L. Qu, J. Zhang, and F. Chen, “Novel route to poly(p-phenylene vinylene) polymers,” J. Polym. Sci., Part A: Polym. Chem. 41, 449-455 (2003).
[CrossRef]

Xia, R.

M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, and D. D. C. Bradley, “Ellipsometric characterization of the optical constants of polyfluorene gain media,” Adv. Funct. Mater. 15, 925-933 (2005).
[CrossRef]

Yakimov, A.

P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” J. Appl. Phys. 93, 3693-3723 (2003).
[CrossRef]

Yoshino, K.

S. V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, and Z. V. Vardeny, “Cooperative emission in pi-conjugated polymer thin films,” Phys. Rev. Lett. 78, 729-732 (1997).
[CrossRef]

Zacharias, P.

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

P. Zacharias, M. C. Gather, M. Rojahn, O. Nuyken, and K. Meerholz, “New crosslinkable hole conductors for blue phosphorescent OLEDs,” Angew. Chem., Int. Ed. Engl. 46, 4388-4392 (2007).
[CrossRef]

Zhang, J.

X. Wu, G. Shi, L. Qu, J. Zhang, and F. Chen, “Novel route to poly(p-phenylene vinylene) polymers,” J. Polym. Sci., Part A: Polym. Chem. 41, 449-455 (2003).
[CrossRef]

Zöller, A.

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[CrossRef]

Adv. Funct. Mater. (2)

M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, and D. D. C. Bradley, “Ellipsometric characterization of the optical constants of polyfluorene gain media,” Adv. Funct. Mater. 15, 925-933 (2005).
[CrossRef]

M. C. Gather, A. Köhnen, A. Falcou, H. Becker, and K. Meerholz, “Solution-processed full-color polymer organic light-emitting diode displays fabricated by direct photolithography,” Adv. Funct. Mater. 17, 191-200 (2007).
[CrossRef]

Adv. Mater. (1)

N. Rehmann, C. Ulbricht, A. Köhnen, P. Zacharias, M. C. Gather, D. Hertel, U. S. Schubert, and K. Meerholz, “Advanced device architecture for highly efficient organic light-emitting diodes with an orange emitting crosslinkable iridium (III) complex,” Adv. Mater. 20, 129-133 (2008).
[CrossRef]

Angew. Chem., Int. Ed. Engl. (1)

P. Zacharias, M. C. Gather, M. Rojahn, O. Nuyken, and K. Meerholz, “New crosslinkable hole conductors for blue phosphorescent OLEDs,” Angew. Chem., Int. Ed. Engl. 46, 4388-4392 (2007).
[CrossRef]

Appl. Opt. (7)

R. C. McPhedran, L. C. Botten, D. R. McKenzie, and R. P. Netterfield, “Unambiguous determination of optical constants of absorbing films by reflectance and transmittance measurements,” Appl. Opt. 23, 1197-1205 (1984).
[CrossRef] [PubMed]

D. P. Arndt, R. M. A. Azzam, J. M. Bennett, J. P. Borgogno, C. K. Carniglia, W. E. Case, J. A. Dobrowolski, U. J. Gibson, T. Tuttle Hart, F. C. Ho, V. A. Hodgkin, W. P. Klapp, H. A. Macleod, E. Pelletier, M. K. Purvis, D. M. Quinn, D. H. Strome, R. Swenson, P. A. Temple, and T. F. Thonn, “Multiple determination of the optical constants of thin-film coating materials,” Appl. Opt. 23, 3571-3596 (1984).
[CrossRef] [PubMed]

M. C. Gupta, “Optical constant determination of thin films,” Appl. Opt. 27, 954-956 (1988).
[CrossRef] [PubMed]

V. Panayotov and I. Konstantinov, “Determination of thin film optical parameters from photometric measurements: an algebraic solution for the (T,Rf,Rb) method,” Appl. Opt. 30, 2795-2800 (1991).
[CrossRef] [PubMed]

J. M. del Pozo and L. Diaz, “Method for the determination of optical constants of thin films: dependence on experimental uncertainties,” Appl. Opt. 31, 4474-4481 (1992).
[CrossRef] [PubMed]

A. B. Djurisic, T. Fritz, K. Leo, and E. H. Li, “Improved method for determination of optical constants of organic thin films from reflection and transmission measurements,” Appl. Opt. 39, 1174-1182 (2000).
[CrossRef]

C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interface,” Appl. Opt. 41, 3978-3987 (2002).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

C. W. Tang and S. A. VanSlyke, “Organic electroluminescent diodes,” Appl. Phys. Lett. 51, 913-915 (1987).
[CrossRef]

A. J. Moulé and K. Meerholz, “Interference method for the determination of the complex refractive index of thin polymer layers,” Appl. Phys. Lett. 91, 061901 (2007).
[CrossRef]

J. Appl. Phys. (1)

P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” J. Appl. Phys. 93, 3693-3723 (2003).
[CrossRef]

J. Phys. D: Appl. Phys. (4)

L. Ward, “The accuracy of photometric methods for determining the optical constants of thin absorbing films,” J. Phys. D: Appl. Phys. 15, 1361-1371 (1982).
[CrossRef]

O. Stenzel, V. Hopfe, and P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates form transmittance and reflectance measurements,” J. Phys. D: Appl. Phys. 24, 2088-2094 (1991).
[CrossRef]

R. E. Denton, R. D. Campbell, and S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D: Appl. Phys. 5, 852-863(1972).
[CrossRef]

R. T. Phillips, “A numerical method for determining the complex refractive index from reflectance and transmittance of supported thin films,” J. Phys. D: Appl. Phys. 16, 489-497 (1983).
[CrossRef]

J. Polym. Sci., Part A: Polym. Chem. (1)

X. Wu, G. Shi, L. Qu, J. Zhang, and F. Chen, “Novel route to poly(p-phenylene vinylene) polymers,” J. Polym. Sci., Part A: Polym. Chem. 41, 449-455 (2003).
[CrossRef]

Nature (2)

K. Meerholz, “Device physics: enlightening solutions,” Nature 437, 327-328 (2005).
[CrossRef] [PubMed]

J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539-541 (1990).
[CrossRef]

Opt. Commun. (1)

A. B. Djurisic, T. Fritz, and K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35-42 (1999).
[CrossRef]

Org. Electron. (1)

L. A. A. Pettersson, S. Ghosh, and O. Inganäs, “Optical anisotropy in thin films of poly (3,4-ethylenedioxythiophene)-poly (4-styrenesulfonate),” Org. Electron. 3, 143-148(2002).
[CrossRef]

Phys. Rev. B (1)

M. Campoy-Quiles, P. G. Etchegoin, and D. D. C. Bradley, “On the optical anisotropy of conjugated polymer thin films,” Phys. Rev. B 72, 045209 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

S. V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, and Z. V. Vardeny, “Cooperative emission in pi-conjugated polymer thin films,” Phys. Rev. Lett. 78, 729-732 (1997).
[CrossRef]

Proc. SPIE (1)

C. A. Wächter, N. Danz, D. Michaelis, M. Flämmich, S. Kudaev, A. H. Bräuer, M. C. Gather, and K. Meerholz, “Intrinsic OLED emitter properties and their effect on device performance,” Proc. SPIE 6910, 691006 (2008).
[CrossRef]

Thin Solid Films (8)

H. Arwin and D. E. Aspnes, “Determination of optical properties of thin organic films by spectroellipsometry,” Thin Solid Films 138, 195-207 (1986).
[CrossRef]

K. Vedam, “Spectroscopic ellipsometry: a historical overview,” Thin Solid Films 313-314, 1-9 (1998).
[CrossRef]

G. E. Jellison, Jr., “Data analysis for spectroscopic ellipsometry,” Thin Solid Films 234, 416-422 (1993).
[CrossRef]

J. Sancho-Parramon, M. Modreanu, S. Bosch, and M. Stchakovsky, “Optical characterization of HfO2 by spectroscopic ellipsometry: Dispersion models and direct data inversion,” Thin Solid Films 516, 7990-7995 (2008).
[CrossRef]

T. Tsuboi, Y. Wasai, and N. Nabatova-Gabain, “Optical constants of platinum octaethyl porphyrin in single-layer organic light emitting diode studied by spectroscopic ellipsometry,” Thin Solid Films 496, 674-678 (2006).
[CrossRef]

T. Fritz, J. Hahn, and H. Böttcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249-257 (1989).
[CrossRef]

O. Stenzel, R. Petrich, W. Scharff, V. Hopfe, and A. V. Tikhonravov, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324-329 (1992).
[CrossRef]

S. Laux, N. Kaiser, A. Zöller, R. Götzelmann, H. Lauth, and H. Bernitzki, “Room-temperature deposition of indium tin oxide thin films with plasma ion-assisted evaporation,” Thin Solid Films 335, 1-5 (1998).
[CrossRef]

Other (5)

Note, that Eq. is the general error estimation of all parameters affecting nf and kf. For a different measurement setup, e.g., utilizing oblique incidence of light, the errors of φ may have a major effect on the uncertainties Δnf and Δkf and have to be considered explicitly.

K. Müllen and U. Scherf, Organic Light Emitting Devices: Synthesis, Properties and Applications (Wiley-VCH, 2006).

E.D.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1991).

W. Karthe and R. Müller, Integrierte Optik (Akademische Verlagsgesellschaft, 1991).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1964).

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Figures (6)

Fig. 1
Fig. 1

Sketch of the setup used for measurements of reflection ( R exp ) and transmission ( T exp ). The angle of incidence is denoted by φ. n i and k i denote the refractive index and the extinction coefficient of the different materials and t i their thicknesses.

Fig. 2
Fig. 2

Calculated R and T spectra (a) for the thin supported film ( t f = 150 nm ) of the artificial material ( n f = 1.8 and k f = 0.05 ), (b) solutions of the refractive index obtained assuming t f = 149.7 nm , (c) t f = 150.0 nm  and (d) t f = 150.3 nm , respectively, and (e) computed refractive index error and indicated spectral CP and IP positions.

Fig. 3
Fig. 3

Measured R and T spectra of (a) the supported PDO-PPV thin film and (b) solutions of the refractive index obtained assuming t f = 161.0 nm , (c)  t f = 161.3 nm , and (d) t f = 161.6 nm , respectively, and (e) computed refractive index error.

Fig. 4
Fig. 4

(a) Determined refractive index and (b) extinction coefficient of PDO-PPV.

Fig. 5
Fig. 5

Measured (squares) and computed (line) reflection spectrum, and measured (circles) and computed (line) transmission spectrum of the OLED stack sketched in the inset.

Fig. 6
Fig. 6

Measured (squares) and computed (line) reflection spectrum of the OLED stack sketched in the inset.

Equations (4)

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R calc ( λ , n f , k f ) R exp ( λ ) = 0 , T calc ( λ , n f , k f ) T exp ( λ ) = 0
d R = R n f d n f + R k f d k f + i R γ i d γ i , d T = T n f d n f + T k f d k f + i T γ i d γ i .
Δ n f = | T k f | Δ R + | R k f | Δ T + i | T γ i R k f R γ i T k f | Δ γ i | T k f R n f R k f T n f | , Δ k f = | T n f | Δ R + | R n f | Δ T + i | T γ i R n f R γ i T n f | Δ γ i | T k f R n f R k f T n f | .
Δ n f = | T k f | Δ R + | R k f | Δ T + | T t f R k f R t f T k f | Δ t f | T k f R n f R k f T n f | , Δ k f = | T n f | Δ R + | R n f | Δ T + | T t f R n f R t f T n f | Δ t f | T k f R n f R k f T n f | .

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