Abstract

Two beamsplitters operating across the near-infrared (770-1050 nm) and mid-infrared (4-8 µm) spectral ranges are developed. For the first time, the beamsplitters based on thin-film materials combinations of ZnS/YbF3 and Ge/YbF3 are investigated. The multilayers operate at the Brewster angle of ZnSe substrate. There are no special temperature conditions. The dichroic coatings are designed, produced, and carefully characterized. Potentials of the ZnS/YbF3 and Ge/YbF3 thin-film material combinations are discussed based on analytical estimations, as well as on optical and non-optical characterization results. The ZnS/YbF3 pair provides high reflectance in the near-infrared spectral range. The Ge/YbF3 solutions exhibit very broadband reflection zones. The Ge/YbF3 coatings are thinner and comprise fewer layers than ZnS/YbF3 multilayers. Ge/YbF3 pair has high potential for design and production of NIR-MIR coatings.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

A. V. Muraviev, V. O. Smolski, Z. E. Loparo, and K. L. Vodopyanov, “Massively parallel sensing of trace molecules and their isotopologues with broadband subharmonic mid-infrared frequency combs,” Nat. Photonics 12(4), 209–214 (2018).
[Crossref]

2017 (1)

2016 (1)

2014 (2)

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

K. Hendrix, J. D. T. Kruschwitz, and J. Keck, “Optical Interference Coatings Design Contest 2013: angle-independent color mirror and shortwave infrared/midwave infrared dichroic beam splitter,” Appl. Opt. 53(4), A360–A376 (2014).
[Crossref] [PubMed]

2013 (3)

G. Wang, X. Ling, X. Liu, and Z. Fan, “Effects of deposition temperature on characterization and laser-induced damage threshold of YbF3 films,” Opt. Laser Technol. 49, 274–278 (2013).
[Crossref]

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

T. V. Amotchkina, “Analytical estimations for the reference wavelength reflectance and width of high reflection zone of two-material periodic multilayers,” Appl. Opt. 52(19), 4590–4595 (2013).
[Crossref] [PubMed]

2011 (2)

2008 (1)

2007 (1)

W. Su, B. Li, D. Liu, and F. Zhang, “The determination of infrared optical constants of rare earth fluorides by classical Lorentz oscillator model,” J. Phys. Appl. Phys. 40(11), 3343–3347 (2007).
[Crossref]

1998 (1)

M. Kennedy, D. Ristau, and H. Niederwald, “Ion beam-assisted deposition of MgF2 and YbF3 films,” Thin Solid Films 333(1-2), 191–195 (1998).
[Crossref]

1997 (1)

1996 (1)

1976 (1)

K. L. Mittal, “Adhesion Measurement of Thin Films,” Electrocompon. Sci. Technol. 3(1), 21–42 (1976).
[Crossref]

1966 (1)

Amotchkina, T.

Amotchkina, T. V.

Baker, M. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Bassan, P.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Bhargava, R.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Butler, H. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Chen, W. L.

Debell, G. W.

Dorling, K. M.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Ennos, A. E.

Fan, Z.

G. Wang, X. Ling, X. Liu, and Z. Fan, “Effects of deposition temperature on characterization and laser-induced damage threshold of YbF3 films,” Opt. Laser Technol. 49, 274–278 (2013).
[Crossref]

Fielden, P. R.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Fogarty, S. W.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Fullwood, N. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Gardner, P.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Gimževskis, U.

Gu, P. F.

Hendrix, K.

Heys, K. A.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Hodgkinson, J.

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

Hughes, C.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Janicki, V.

Keck, J.

Kennedy, M.

M. Kennedy, D. Ristau, and H. Niederwald, “Ion beam-assisted deposition of MgF2 and YbF3 films,” Thin Solid Films 333(1-2), 191–195 (1998).
[Crossref]

Kicas, S.

Kruschwitz, J. D. T.

Lasch, P.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Li, B.

W. Su, B. Li, D. Liu, and F. Zhang, “The determination of infrared optical constants of rare earth fluorides by classical Lorentz oscillator model,” J. Phys. Appl. Phys. 40(11), 3343–3347 (2007).
[Crossref]

Ling, X.

G. Wang, X. Ling, X. Liu, and Z. Fan, “Effects of deposition temperature on characterization and laser-induced damage threshold of YbF3 films,” Opt. Laser Technol. 49, 274–278 (2013).
[Crossref]

Liu, D.

W. Su, B. Li, D. Liu, and F. Zhang, “The determination of infrared optical constants of rare earth fluorides by classical Lorentz oscillator model,” J. Phys. Appl. Phys. 40(11), 3343–3347 (2007).
[Crossref]

Liu, X.

G. Wang, X. Ling, X. Liu, and Z. Fan, “Effects of deposition temperature on characterization and laser-induced damage threshold of YbF3 films,” Opt. Laser Technol. 49, 274–278 (2013).
[Crossref]

Y. Wang, Y. G. Zhang, W. L. Chen, W. D. Shen, X. Liu, and P. F. Gu, “Optical properties and residual stress of YbF3 thin films deposited at different temperatures,” Appl. Opt. 47(13), C319–C323 (2008).
[Crossref] [PubMed]

Loparo, Z. E.

A. V. Muraviev, V. O. Smolski, Z. E. Loparo, and K. L. Vodopyanov, “Massively parallel sensing of trace molecules and their isotopologues with broadband subharmonic mid-infrared frequency combs,” Nat. Photonics 12(4), 209–214 (2018).
[Crossref]

Martin, F. L.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Martin-Hirsch, P. L.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Melnikas, S.

Mittal, K. L.

K. L. Mittal, “Adhesion Measurement of Thin Films,” Electrocompon. Sci. Technol. 3(1), 21–42 (1976).
[Crossref]

Muraviev, A. V.

A. V. Muraviev, V. O. Smolski, Z. E. Loparo, and K. L. Vodopyanov, “Massively parallel sensing of trace molecules and their isotopologues with broadband subharmonic mid-infrared frequency combs,” Nat. Photonics 12(4), 209–214 (2018).
[Crossref]

Niederwald, H.

M. Kennedy, D. Ristau, and H. Niederwald, “Ion beam-assisted deposition of MgF2 and YbF3 films,” Thin Solid Films 333(1-2), 191–195 (1998).
[Crossref]

Obinaju, B.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Pawlewicz, W. T.

Pervak, V.

Ristau, D.

M. Kennedy, D. Ristau, and H. Niederwald, “Ion beam-assisted deposition of MgF2 and YbF3 films,” Thin Solid Films 333(1-2), 191–195 (1998).
[Crossref]

Sancho-Parramon, J.

Shen, W. D.

Smolski, V. O.

A. V. Muraviev, V. O. Smolski, Z. E. Loparo, and K. L. Vodopyanov, “Massively parallel sensing of trace molecules and their isotopologues with broadband subharmonic mid-infrared frequency combs,” Nat. Photonics 12(4), 209–214 (2018).
[Crossref]

Sockalingum, G. D.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Strong, R. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Su, W.

W. Su, B. Li, D. Liu, and F. Zhang, “The determination of infrared optical constants of rare earth fluorides by classical Lorentz oscillator model,” J. Phys. Appl. Phys. 40(11), 3343–3347 (2007).
[Crossref]

Sulé-Suso, J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Tatam, R. P.

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

Tikhonravov, A.

M. Trubetskov and A. Tikhonravov, OptiLayer Thin Film Software (n.d.).

Tikhonravov, A. V.

Traylor Kruschwitz, J. D.

Trevisan, J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Trubetskov, M.

Trubetskov, M. K.

Vodopyanov, K. L.

A. V. Muraviev, V. O. Smolski, Z. E. Loparo, and K. L. Vodopyanov, “Massively parallel sensing of trace molecules and their isotopologues with broadband subharmonic mid-infrared frequency combs,” Nat. Photonics 12(4), 209–214 (2018).
[Crossref]

Walsh, M. J.

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Figures (9)

Fig. 1
Fig. 1 (a) Refractive indices of thin-film materials and ZnSe substrate; (b) Extinction coefficients of Ge (left axis) and ZnSe substrate (right axis).
Fig. 2
Fig. 2 Target and design reflectance of 14-layer (a) and 20-layer (b) BS-ZnS/YbF3 solutions. Dashed lines plot corresponding spectral characteristics of the 14-layer and 20-layer QWM. (c) Target and design transmittance of 14-layer and 20-layer BS-ZnS/YbF3 solutions.
Fig. 3
Fig. 3 Structures of the 14-layer (a) and 20-layer (b) BS-YbF3/ZnS solutions. Optical thicknesses of the layers are presented in QWOT at the wavelength of 920 nm and AOI = 67.9°.
Fig. 4
Fig. 4 Structures of the 8-layer (a) and 14-layer (b) BS-Ge/YbF3 solutions. Optical thicknesses of the layers are presented in QWOT at the wavelength of 1000 nm and AOI = 67.9°.
Fig. 5
Fig. 5 Target and design reflectance of the 8-layer (a) and 14-layer (b) BS-Ge/YbF3 solutions. Dashed lines plot the corresponding spectral characteristics of the 8-layer and 14-layer QPS. (c) Target and design transmittance of 8-layer and 14-layer BS-Ge/YbF3 solutions.
Fig. 6
Fig. 6 (a) Target and design reflectance of the 14-layer BS-Ge/ZnS solution. Dashed lines plot corresponding spectral characteristics of the 14-layer QPS. (b) Structures of the 14-layer BS-Ge/ZnS solution. Optical thicknesses of layers are calculated at the wavelength of 1000 nm.
Fig. 7
Fig. 7 Comparison of theoretical and experimental spectral characteristics of the produced BS-ZnS/YbF3 sample: (a) transmittance measured in the visible-near-infrared range, (b) quasi-normal incidence reflectance (average polarization), (c) s- and p-polarized reflectance at the AOI = 45°, (d) reflectance at the Brewster AOI = 67.9°. The measurements at the Brewster angle were performed at Agilent technologies.
Fig. 8
Fig. 8 Comparison of the theoretical and experimental transmittance in the MIR spectral range of the produced BS-ZnS/YbF3 (a) and BS-Ge/YbF3 (b) samples. The experimental transmittance was recorded several minutes after exposure to the atmosphere and 6 months later.
Fig. 9
Fig. 9 Comparison of theoretical and experimental spectral characteristics of the produced BS-Ge/YbF3 sample: (a) transmittance measured in the visible-near-infrared range, (b) quasi-normal incidence reflectance (average polarization), (c) s- and p-polarized reflectance at the AOI = 45°, (d) reflectance at the Brewster angle of 67.9°. The measurements at the Brewster angle were performed at Agilent technologies.

Tables (4)

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Table 1 Characteristics of BS-ZnS/YbF3 solutions and corresponding quarter-wave stacks, λ 0 =920nm, ΔHR (s,p) denotes the width of the HR zone.

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Table 2 Characteristics of the BS-Ge/YbF3 solutions and corresponding quasi-periodic structures, λ 0 =1000nm, ΔHR (s,p) denotes the width of the HR zone.

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Table 3 Stresses in designed coatings, experimental designs are marked by bold.

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Table 4 Qualitative comparison of beamsplitters comprising ZnS/YbF3 and Ge/YbF3 pairs

Equations (8)

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M F 2 = j=1 61 ( R ( s ) ( X; λ 1,j )100% Δ 1,j ) 2 + j=1 96 ( R ( p ) ( X; λ 2,j )100% Δ 2,j ) 2 + j=1 201 ( T ( p ) ( X; λ 3,j )100% Δ 3,j ) 2 ,
R av = 1 141 j=1 141 R p ( 860+j ) , T av = 1 4001 j=1 4001 T p ( 4000+j ) , σ R = 1 141 j=1 141 [ R p ( 860+j ) R av ] 2 , σ T = 1 4001 j=1 4001 [ T p ( 4000+j ) T av ] 2
R (s,p) ( λ 0 )14 q a (s,p) q s (s,p) ( q L (s,p) q H (s,p) ) m , ΔH R (s,p) = λ u (s,p) λ d (s,p) , λ u (s,p) = λ 0 π πarccos( ξ (s,p) ) , λ d (s,p) = λ 0 π π+arccos( ξ (s,p) ) ,
ξ (s,p) = ( q L (s,p) ) 2 + ( q H (s,p) ) 2 6 q L (s,p) q H (s,p) ( q L (s,p) + q H (s,p) ) 2 , q L,H,s,a (s) = n L,H,s,a 2 n a 2 sin 2 θ , q L,H,s,a (p) = n L,H,s,a 2 n L,H,s,a 2 n a 2 sin 2 θ
ΔH R (s,p) = 4 λ 0 π C 2 1 sin( πp/2 ) C+1( 1 p 2 )( C1 )cos( πp ) , C= 1 2 ( q H ( s,p ) q L ( s,p ) + q L ( s,p ) q H ( s,p ) ),
R abs (s,p) R (s,p) ( λ 0 ) 2π q a (s,p) k H (s,p) ( q H (s,p) ) 2 ( q L (s,p) ) 2 ,
σ M = Σ H σ H + Σ L σ L Σ M ,
σ= 1 6 ( 1 R 2 1 R 1 ) E Σ s 2 ( 1ν ) Σ f ,

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