Abstract

We propose a novel measurement system to simultaneously measure surface and thickness profiles of thin film structures, which cannot be realized in typical measurement techniques. This measurement is accomplished by combining spectral interferometry and ellipsometry. These two distinct measurement techniques are involved in a single system by the abnormal optical configuration. Further, the measurement results are complementary in order to characterize film structures. Film thickness profiles are measured by spectroscopic imaging ellipsometry and surface profiles are obtained from the spectral phase by spectrally resolved interferometry. This method eliminates the theoretical spectral phase by film thicknesses. The proposed system can determine the dimensional film structures at once, even though they have multi-layered films, substrate textures, and even thin film layers. In the experiments, each measurement principle was fundamentally verified with standard specimens. Further, a 4-layered film specimen was measured in order to reconstruct its 3D film structure. As the result, the repeatability of spectroscopic imaging ellipsometry was less than 1 nm and that of spectrally resolved interferometry was a few nanometers, which dominantly affected the performance of the whole system. Several issues for improving accuracy and precision of the proposed system are also discussed in this paper.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  9. C. Zhang, K. Lin, Y. Huang, and J. Zhang, “Graphene-Ag Hybrids on Laser-Textured Si Surface for SERS Detection,” Sensors (Basel) 17(7), 1462 (2017).
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    [Crossref]

2017 (5)

C. Zhang, K. Lin, Y. Huang, and J. Zhang, “Graphene-Ag Hybrids on Laser-Textured Si Surface for SERS Detection,” Sensors (Basel) 17(7), 1462 (2017).
[Crossref] [PubMed]

Y. S. Ghim, H. G. Rhee, and A. Davies, “Simultaneous measurements of top surface and its underlying film surfaces in multilayer film structure,” Sci. Rep. 7(1), 11843 (2017).
[Crossref] [PubMed]

L. Jin, T. Tanaka, E. Kondoh, B. Gelloz, and M. Uehara, “Extracting calibrated parameters from imaging ellipsometric measurement,” Jpn. J. Appl. Phys. 56(11), 116602 (2017).
[Crossref]

L. Kewu, Z. Rui, J. Ning, C. Youhua, Z. Minjuan, W. Liming, and W. Zhibin, “Fast and full range measurements of ellipsometric parameters using a 45° dual-drive symmetric photoelastic modulator,” Opt. Express 25(5), 5725–5733 (2017).
[Crossref] [PubMed]

J. Bae, J. Park, H. Ahn, and J. Jin, “Total physical thickness measurement of a multi-layered wafer using a spectral-domain interferometer with an optical comb,” Opt. Express 25(11), 12689–12697 (2017).
[Crossref] [PubMed]

2012 (1)

J. You and K.-N. Joo, “Minimization of spectral phase errors in spectrally resolved white light interferometry by the iterative least-squared phase-shifting method,” Meas. Sci. Technol. 23(12), 125203 (2012).
[Crossref]

2011 (1)

2010 (1)

2008 (1)

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Spectral interferometry and reflectometry used to measure thin films,” Appl. Phys. B 92(2), 203–207 (2008).
[Crossref]

2007 (4)

H. Jha, T. Kikuchi, M. Sakairi, and H. Takahashi, “Microfabrication of an anodic oxide film by anodizing laser-textured aluminium,” J. Micromech. Microeng. 17(10), 1949–1955 (2007).
[Crossref]

Y.-S. Ghim and S.-W. Kim, “Fast, precis, tomographic measurement of thin films,” Appl. Phys. Lett. 91(9), 091903 (2007).
[Crossref]

J. D. Froehlich, R. Young, T. Nakamura, Y. Ohmori, S. Li, A. Mochizuki, M. Lauters, and G. E. Jabbour, “Synthesis of multi-functional POSS emitters for OLED applications,” Chem. Mater. 19(20), 4991–4997 (2007).
[Crossref]

P. de Groot and X. C. de Lega, “Angle-resolved three-dimensional analysis of surface films by coherence scanning interferometry,” Opt. Lett. 32(12), 1638–1640 (2007).
[Crossref] [PubMed]

2006 (3)

2004 (1)

M. Ichiki, R. Maeda, Y. Morikawa, Y. Mabune, T. Nakada, and K. Nonaka, “Photovoltaic effect of lead lanthanum zirconate titanate in a layered film structure design,” Appl. Phys. Lett. 84(3), 395–397 (2004).
[Crossref]

1999 (1)

1998 (2)

T. Wang, H. Wang, P. Xu, X. Zhao, Y. Liu, and S. Chao, “The effect of properties of semiconductor oxide thin films on photocatalytic decomposition of dyeing waste water,” Thin Solid Films 334(1-2), 103–108 (1998).
[Crossref]

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313, 33–39 (1998).
[Crossref]

1992 (1)

1987 (1)

Ahn, H.

Bae, J.

Brock, N. J.

Byun, Y. T.

C. W. Son, S. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, S. H. Kim, and T.-H. Yoon, “Realisation of all-optical multi-functional logic gates using semiconductor optical amplifiers,” Electron. Lett. 42(18), 1057 (2006).
[Crossref]

Carr, S.

Chao, S.

T. Wang, H. Wang, P. Xu, X. Zhao, Y. Liu, and S. Chao, “The effect of properties of semiconductor oxide thin films on photocatalytic decomposition of dyeing waste water,” Thin Solid Films 334(1-2), 103–108 (1998).
[Crossref]

Chlebus, R.

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Spectral interferometry and reflectometry used to measure thin films,” Appl. Phys. B 92(2), 203–207 (2008).
[Crossref]

Ciprian, D.

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Spectral interferometry and reflectometry used to measure thin films,” Appl. Phys. B 92(2), 203–207 (2008).
[Crossref]

P. Hlubina, D. Ciprian, J. Lunácek, M. Lesnák, and M. Lesňák, “Dispersive white-light spectral interferometry with absolute phase retrieval to measure thin film,” Opt. Express 14(17), 7678–7685 (2006).
[Crossref] [PubMed]

Davies, A.

Y. S. Ghim, H. G. Rhee, and A. Davies, “Simultaneous measurements of top surface and its underlying film surfaces in multilayer film structure,” Sci. Rep. 7(1), 11843 (2017).
[Crossref] [PubMed]

Y. S. Ghim, A. Suratkar, and A. Davies, “Reflectometry-based wavelength scanning interferometry for thickness measurements of very thin wafers,” Opt. Express 18(7), 6522–6529 (2010).
[Crossref] [PubMed]

Davies, D. E. N.

de Groot, P.

de Lega, X. C.

Froehlich, J. D.

J. D. Froehlich, R. Young, T. Nakamura, Y. Ohmori, S. Li, A. Mochizuki, M. Lauters, and G. E. Jabbour, “Synthesis of multi-functional POSS emitters for OLED applications,” Chem. Mater. 19(20), 4991–4997 (2007).
[Crossref]

Fujimoto, J. G.

Gelloz, B.

L. Jin, T. Tanaka, E. Kondoh, B. Gelloz, and M. Uehara, “Extracting calibrated parameters from imaging ellipsometric measurement,” Jpn. J. Appl. Phys. 56(11), 116602 (2017).
[Crossref]

Ghim, Y. S.

Ghim, Y.-S.

Y.-S. Ghim and S.-W. Kim, “Fast, precis, tomographic measurement of thin films,” Appl. Phys. Lett. 91(9), 091903 (2007).
[Crossref]

Hee, M. R.

Hlubina, P.

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Spectral interferometry and reflectometry used to measure thin films,” Appl. Phys. B 92(2), 203–207 (2008).
[Crossref]

P. Hlubina, D. Ciprian, J. Lunácek, M. Lesnák, and M. Lesňák, “Dispersive white-light spectral interferometry with absolute phase retrieval to measure thin film,” Opt. Express 14(17), 7678–7685 (2006).
[Crossref] [PubMed]

Huang, D.

Huang, Y.

C. Zhang, K. Lin, Y. Huang, and J. Zhang, “Graphene-Ag Hybrids on Laser-Textured Si Surface for SERS Detection,” Sensors (Basel) 17(7), 1462 (2017).
[Crossref] [PubMed]

Ichiki, M.

M. Ichiki, R. Maeda, Y. Morikawa, Y. Mabune, T. Nakada, and K. Nonaka, “Photovoltaic effect of lead lanthanum zirconate titanate in a layered film structure design,” Appl. Phys. Lett. 84(3), 395–397 (2004).
[Crossref]

Jabbour, G. E.

J. D. Froehlich, R. Young, T. Nakamura, Y. Ohmori, S. Li, A. Mochizuki, M. Lauters, and G. E. Jabbour, “Synthesis of multi-functional POSS emitters for OLED applications,” Chem. Mater. 19(20), 4991–4997 (2007).
[Crossref]

Jellison, G. E.

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313, 33–39 (1998).
[Crossref]

Jha, H.

H. Jha, T. Kikuchi, M. Sakairi, and H. Takahashi, “Microfabrication of an anodic oxide film by anodizing laser-textured aluminium,” J. Micromech. Microeng. 17(10), 1949–1955 (2007).
[Crossref]

Jhon, Y. M.

C. W. Son, S. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, S. H. Kim, and T.-H. Yoon, “Realisation of all-optical multi-functional logic gates using semiconductor optical amplifiers,” Electron. Lett. 42(18), 1057 (2006).
[Crossref]

Jin, J.

Jin, L.

L. Jin, T. Tanaka, E. Kondoh, B. Gelloz, and M. Uehara, “Extracting calibrated parameters from imaging ellipsometric measurement,” Jpn. J. Appl. Phys. 56(11), 116602 (2017).
[Crossref]

Joo, K.-N.

J. You and K.-N. Joo, “Minimization of spectral phase errors in spectrally resolved white light interferometry by the iterative least-squared phase-shifting method,” Meas. Sci. Technol. 23(12), 125203 (2012).
[Crossref]

Kewu, L.

Kikuchi, T.

H. Jha, T. Kikuchi, M. Sakairi, and H. Takahashi, “Microfabrication of an anodic oxide film by anodizing laser-textured aluminium,” J. Micromech. Microeng. 17(10), 1949–1955 (2007).
[Crossref]

Kim, G.-H.

Kim, S. H.

C. W. Son, S. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, S. H. Kim, and T.-H. Yoon, “Realisation of all-optical multi-functional logic gates using semiconductor optical amplifiers,” Electron. Lett. 42(18), 1057 (2006).
[Crossref]

C. W. Son, S. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, S. H. Kim, and T.-H. Yoon, “Realisation of all-optical multi-functional logic gates using semiconductor optical amplifiers,” Electron. Lett. 42(18), 1057 (2006).
[Crossref]

Kim, S. W.

Kim, S.-W.

Kimbrough, B.

Kondoh, E.

L. Jin, T. Tanaka, E. Kondoh, B. Gelloz, and M. Uehara, “Extracting calibrated parameters from imaging ellipsometric measurement,” Jpn. J. Appl. Phys. 56(11), 116602 (2017).
[Crossref]

Lauters, M.

J. D. Froehlich, R. Young, T. Nakamura, Y. Ohmori, S. Li, A. Mochizuki, M. Lauters, and G. E. Jabbour, “Synthesis of multi-functional POSS emitters for OLED applications,” Chem. Mater. 19(20), 4991–4997 (2007).
[Crossref]

Lee, C. C.

Lee, S.

C. W. Son, S. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, S. H. Kim, and T.-H. Yoon, “Realisation of all-optical multi-functional logic gates using semiconductor optical amplifiers,” Electron. Lett. 42(18), 1057 (2006).
[Crossref]

Lesnák, M.

Li, S.

J. D. Froehlich, R. Young, T. Nakamura, Y. Ohmori, S. Li, A. Mochizuki, M. Lauters, and G. E. Jabbour, “Synthesis of multi-functional POSS emitters for OLED applications,” Chem. Mater. 19(20), 4991–4997 (2007).
[Crossref]

Liming, W.

Lin, C. P.

Lin, K.

C. Zhang, K. Lin, Y. Huang, and J. Zhang, “Graphene-Ag Hybrids on Laser-Textured Si Surface for SERS Detection,” Sensors (Basel) 17(7), 1462 (2017).
[Crossref] [PubMed]

Liu, Y.

T. Wang, H. Wang, P. Xu, X. Zhao, Y. Liu, and S. Chao, “The effect of properties of semiconductor oxide thin films on photocatalytic decomposition of dyeing waste water,” Thin Solid Films 334(1-2), 103–108 (1998).
[Crossref]

Lunácek, J.

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Spectral interferometry and reflectometry used to measure thin films,” Appl. Phys. B 92(2), 203–207 (2008).
[Crossref]

P. Hlubina, D. Ciprian, J. Lunácek, M. Lesnák, and M. Lesňák, “Dispersive white-light spectral interferometry with absolute phase retrieval to measure thin film,” Opt. Express 14(17), 7678–7685 (2006).
[Crossref] [PubMed]

Mabune, Y.

M. Ichiki, R. Maeda, Y. Morikawa, Y. Mabune, T. Nakada, and K. Nonaka, “Photovoltaic effect of lead lanthanum zirconate titanate in a layered film structure design,” Appl. Phys. Lett. 84(3), 395–397 (2004).
[Crossref]

Maeda, R.

M. Ichiki, R. Maeda, Y. Morikawa, Y. Mabune, T. Nakada, and K. Nonaka, “Photovoltaic effect of lead lanthanum zirconate titanate in a layered film structure design,” Appl. Phys. Lett. 84(3), 395–397 (2004).
[Crossref]

Minjuan, Z.

Mochizuki, A.

J. D. Froehlich, R. Young, T. Nakamura, Y. Ohmori, S. Li, A. Mochizuki, M. Lauters, and G. E. Jabbour, “Synthesis of multi-functional POSS emitters for OLED applications,” Chem. Mater. 19(20), 4991–4997 (2007).
[Crossref]

Morikawa, Y.

M. Ichiki, R. Maeda, Y. Morikawa, Y. Mabune, T. Nakada, and K. Nonaka, “Photovoltaic effect of lead lanthanum zirconate titanate in a layered film structure design,” Appl. Phys. Lett. 84(3), 395–397 (2004).
[Crossref]

Nakada, T.

M. Ichiki, R. Maeda, Y. Morikawa, Y. Mabune, T. Nakada, and K. Nonaka, “Photovoltaic effect of lead lanthanum zirconate titanate in a layered film structure design,” Appl. Phys. Lett. 84(3), 395–397 (2004).
[Crossref]

Nakamura, T.

J. D. Froehlich, R. Young, T. Nakamura, Y. Ohmori, S. Li, A. Mochizuki, M. Lauters, and G. E. Jabbour, “Synthesis of multi-functional POSS emitters for OLED applications,” Chem. Mater. 19(20), 4991–4997 (2007).
[Crossref]

Ning, J.

Nonaka, K.

M. Ichiki, R. Maeda, Y. Morikawa, Y. Mabune, T. Nakada, and K. Nonaka, “Photovoltaic effect of lead lanthanum zirconate titanate in a layered film structure design,” Appl. Phys. Lett. 84(3), 395–397 (2004).
[Crossref]

Ohmori, Y.

J. D. Froehlich, R. Young, T. Nakamura, Y. Ohmori, S. Li, A. Mochizuki, M. Lauters, and G. E. Jabbour, “Synthesis of multi-functional POSS emitters for OLED applications,” Chem. Mater. 19(20), 4991–4997 (2007).
[Crossref]

Park, J.

Puliafito, C. A.

Rhee, H. G.

Y. S. Ghim, H. G. Rhee, and A. Davies, “Simultaneous measurements of top surface and its underlying film surfaces in multilayer film structure,” Sci. Rep. 7(1), 11843 (2017).
[Crossref] [PubMed]

Rui, Z.

Sakairi, M.

H. Jha, T. Kikuchi, M. Sakairi, and H. Takahashi, “Microfabrication of an anodic oxide film by anodizing laser-textured aluminium,” J. Micromech. Microeng. 17(10), 1949–1955 (2007).
[Crossref]

Son, C. W.

C. W. Son, S. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, S. H. Kim, and T.-H. Yoon, “Realisation of all-optical multi-functional logic gates using semiconductor optical amplifiers,” Electron. Lett. 42(18), 1057 (2006).
[Crossref]

Suratkar, A.

Swanson, E. A.

Takahashi, H.

H. Jha, T. Kikuchi, M. Sakairi, and H. Takahashi, “Microfabrication of an anodic oxide film by anodizing laser-textured aluminium,” J. Micromech. Microeng. 17(10), 1949–1955 (2007).
[Crossref]

Tanaka, T.

L. Jin, T. Tanaka, E. Kondoh, B. Gelloz, and M. Uehara, “Extracting calibrated parameters from imaging ellipsometric measurement,” Jpn. J. Appl. Phys. 56(11), 116602 (2017).
[Crossref]

Uehara, M.

L. Jin, T. Tanaka, E. Kondoh, B. Gelloz, and M. Uehara, “Extracting calibrated parameters from imaging ellipsometric measurement,” Jpn. J. Appl. Phys. 56(11), 116602 (2017).
[Crossref]

Wang, H.

T. Wang, H. Wang, P. Xu, X. Zhao, Y. Liu, and S. Chao, “The effect of properties of semiconductor oxide thin films on photocatalytic decomposition of dyeing waste water,” Thin Solid Films 334(1-2), 103–108 (1998).
[Crossref]

Wang, T.

T. Wang, H. Wang, P. Xu, X. Zhao, Y. Liu, and S. Chao, “The effect of properties of semiconductor oxide thin films on photocatalytic decomposition of dyeing waste water,” Thin Solid Films 334(1-2), 103–108 (1998).
[Crossref]

Woo, D. H.

C. W. Son, S. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, S. H. Kim, and T.-H. Yoon, “Realisation of all-optical multi-functional logic gates using semiconductor optical amplifiers,” Electron. Lett. 42(18), 1057 (2006).
[Crossref]

Wu, K.

Xu, P.

T. Wang, H. Wang, P. Xu, X. Zhao, Y. Liu, and S. Chao, “The effect of properties of semiconductor oxide thin films on photocatalytic decomposition of dyeing waste water,” Thin Solid Films 334(1-2), 103–108 (1998).
[Crossref]

Yoon, T.-H.

C. W. Son, S. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, S. H. Kim, and T.-H. Yoon, “Realisation of all-optical multi-functional logic gates using semiconductor optical amplifiers,” Electron. Lett. 42(18), 1057 (2006).
[Crossref]

You, J.

J. You and K.-N. Joo, “Minimization of spectral phase errors in spectrally resolved white light interferometry by the iterative least-squared phase-shifting method,” Meas. Sci. Technol. 23(12), 125203 (2012).
[Crossref]

Youhua, C.

Young, R.

J. D. Froehlich, R. Young, T. Nakamura, Y. Ohmori, S. Li, A. Mochizuki, M. Lauters, and G. E. Jabbour, “Synthesis of multi-functional POSS emitters for OLED applications,” Chem. Mater. 19(20), 4991–4997 (2007).
[Crossref]

Youngquist, R. C.

Zhang, C.

C. Zhang, K. Lin, Y. Huang, and J. Zhang, “Graphene-Ag Hybrids on Laser-Textured Si Surface for SERS Detection,” Sensors (Basel) 17(7), 1462 (2017).
[Crossref] [PubMed]

Zhang, J.

C. Zhang, K. Lin, Y. Huang, and J. Zhang, “Graphene-Ag Hybrids on Laser-Textured Si Surface for SERS Detection,” Sensors (Basel) 17(7), 1462 (2017).
[Crossref] [PubMed]

Zhao, X.

T. Wang, H. Wang, P. Xu, X. Zhao, Y. Liu, and S. Chao, “The effect of properties of semiconductor oxide thin films on photocatalytic decomposition of dyeing waste water,” Thin Solid Films 334(1-2), 103–108 (1998).
[Crossref]

Zhibin, W.

Appl. Opt. (1)

Appl. Phys. B (1)

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Spectral interferometry and reflectometry used to measure thin films,” Appl. Phys. B 92(2), 203–207 (2008).
[Crossref]

Appl. Phys. Lett. (2)

M. Ichiki, R. Maeda, Y. Morikawa, Y. Mabune, T. Nakada, and K. Nonaka, “Photovoltaic effect of lead lanthanum zirconate titanate in a layered film structure design,” Appl. Phys. Lett. 84(3), 395–397 (2004).
[Crossref]

Y.-S. Ghim and S.-W. Kim, “Fast, precis, tomographic measurement of thin films,” Appl. Phys. Lett. 91(9), 091903 (2007).
[Crossref]

Chem. Mater. (1)

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Figures (10)

Fig. 1
Fig. 1 Optical configuration of the combined measurement system; P, polarizer; CR, rotating compensator; BS, beam splitter; BSH, beam shutter; A, analyzer; IL imaging lens; IS, imaging spectrometer; MR and MM, a reference and measurement mirrors; PZT, piezoelectric transducer. The inlet indicates the image obtained by the imaging device.
Fig. 2
Fig. 2 Measurement principle and procedure of the system to extract the surface height and film thickness of the specimen.
Fig. 3
Fig. 3 Film theory of (a) single layered film and (b) multi-layered film structures.
Fig. 4
Fig. 4 (a) Photograph of the stepped gauge block specimen and the measurement results of (b) 5 μm and (c) 10 μm step heights.
Fig. 5
Fig. 5 Optical configurations to measure (a) ( B ˜ )t, (b) ( B ˜ )r and (c) ( B ˜ )t M ˜ ( B ˜ )r.
Fig. 6
Fig. 6 Determined Jones matrices of (a) ( B ˜ ) t and (b) ( B ˜ ) r .
Fig. 7
Fig. 7 Comparison between experimental and theoretical p-polarization component of ( B ˜ ) t M ˜ ( B ˜ ) r with (a) amplitude and (b) phase.
Fig. 8
Fig. 8 Comparison between experimental and theoretical images containing spectral and polarization changes corresponding to a single point on the line of each CRM; (a), (b) and (c) are the measured, theoretical and difference images for the CRM with 14.3 nm film thickness. (d), (e) and (f) are the measured, theoretical and difference images for the CRM with 55.8 nm film thickness.
Fig. 9
Fig. 9 (a) Film structure of the 4 film layered specimen, (b) image obtained by the CCD camera at a certain rotating angle of the compensator, (c) measured film thickness profiles by ellipsometry mode; (d), (e) and (f) are the measured, theoretical and difference images at a certain point of the specimen.
Fig. 10
Fig. 10 (a) Spectral phase extracted from the interferometry mode, (b) line profile of substrate, (c) surface and thickness line profiles and (d) 3D film structure of the 4 layered film specimen.

Tables (1)

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Table 1 Summary of the measurement results compared to reference values designed and provided by manufacturer.

Equations (10)

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E ˜ out = R ˜ 45 1 A ˜ R ˜ 45 ( B ˜ ) r S ˜ M ˜ S ˜ ( B ˜ ) t C ˜ R R ˜ 45 1 P ˜ R ˜ 45 E ˜ in = ( cos( 45° ) sin( 45° ) sin(45°) cos( 45° ) )( 1 0 0 0 )( cos( 45° ) sin( 45° ) sin( 45° ) cos( 45° ) ) ( B ˜ ) r ( r p 0 0 r s )( 1 0 0 1 )( r p 0 0 r s ) ( B ˜ ) t ( cos θ R sin θ R sin θ R cos θ R )( 1 0 0 e iε )( cos θ R sin θ R sin θ R cos θ R ) ( cos45° sin45° sin45° cos45° )( 1 0 0 0 )( cos45° sin45° sin45° cosP45° ) E ˜ in
r= r 12 + r 23 exp( i2β ) 1+ r 12 r 23 exp( i2β )
T ij = D i + V i cos( φ i ( k ) δ j )
φ i ( k )=2k h i ¯ + ψ i,film ( k )
ξ ij = V i cos φ i ( cos δ j 1 )+ V i sin φ i sin δ j = C i ( cos δ j 1 )+ S i sin δ j
φ i = tan 1 ( S i C i )
η i = j=1 m ( ξ ij ξ ij ) ˇ 2 = j=1 m ( C i ( cos δ j 1 )+ S i sin δ j ξ ij ) ˇ 2 η j = i=1 n ( ξ ij ξ ij ) ˇ 2 = i=1 n ( C i ( cos δ j 1 )+ S i sin δ j ξ ij ) ˇ 2
η i C i = η i S i =0 , and η j ( cos δ j ) = η j ( cos δ j ) =0
σ( h ¯ )= [ φ m ( k ) φ th ( k; h ¯ ) ] 2
h ¯ =hcos( θ in )

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