Abstract

To simulate and optimize optical designs regarding perceived color and homogeneity in commercial ray tracing software, realistic light source models are needed. Spectral rayfiles provide angular and spatial varying spectral information. We propose a spectral reconstruction method with a minimum of time consuming goniophotometric near field measurements with optical filters for the purpose of creating spectral rayfiles. Our discussion focuses on the selection of the ideal optical filter combination for any arbitrary spectrum out of a given filter set by considering measurement uncertainties with Monte Carlo simulations. We minimize the simulation time by a preselection of all filter combinations, which bases on factorial design.

© 2015 Optical Society of America

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References

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  1. I. Ashdown, “Near-field photometry: a new approach,” J. Illum. Eng. Soc. 22(1), 163–180(1993).
    [Crossref]
  2. I. Ashdown and M. Salsbury, “A near-field goniospectroradiometer for LED measurements,” Proc. SPIE 6342, 634215(2007).
    [Crossref]
  3. M. Lépez, K. Bredemeier, N. Rohrbeck, C. Véron, F. Schmidt, and A. Sperling, “LED near-field goniophotometer at PTB,” Metrologia 49(2), 141–145(2012).
    [Crossref]
  4. R. Rykowski, “Spectral ray tracing from near field goniophotometer measurements,” Light. Eng 19(1), 23–29(2011).
  5. OSRAM Opto Semiconductors application note, “Importing rayfiles of LEDs from OSRAM Opto Semiconductors”, (OSRAM Opto Semiconductors, 2013), http://www.osram-os.com/Graphics/XPic5/00165120_0.pdf , (accessed, July, 2015).
  6. V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “Rayfiles including spectral and colorimetric information,” Opt. Express 23(7), A361–A370(2015).
    [Crossref] [PubMed]
  7. V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “On spectral ray files of light sources using principal component analysis,” in Proceedings of 28th CIE Session 2015 (CIE, 2015), pp. 543–547.
  8. IES, “Ray File Format for the Description of the Emission Property of Light Sources,” TM25-13 (2013).
  9. F. Reifergeiste and J. Lienig, “Modelling of the temperature and current dependence of LED spectra,” J. Light Vis. Environ. 32(3), 288–294(2008).
    [Crossref]
  10. CVI Laser Optics, “Datasheet selection guide,” pp. 232–233, https://marketplace.idexop.com/store/SupportDocuments/12_FiltersandEtalons_2013_final-e.pdf , (accessed, July, 2015).
  11. Newport, “Newport resource e-catalog,” pp. 655–658, http://www.nxtbook.com/nxtbooks/newportcorp/resource2011/#/684 , (accessed, July, 2015).
  12. Schott Glass Technologies catalog, Catalog optical filters, (Schott Glass Technologies, 2015), http://www.schott.com/advanced_optics/german/download/schott-optical-filters-2015-catalog-complete-en.pdf , (accessed, July, 2015).
  13. U. Krüger and F. Schmidt, “The impact of cooling on CCD-based camera systems in the field of image luminance measuring devices,” Metrologia 46(4), 252–259(2009).
    [Crossref]
  14. TechnoTeam, “LMK 50 color datasheet,” (TechnoTeam, 2012), http://www.technoteam.de/product_overview/lmk/products/lmk_50_color/index_eng.html , (accessed, July, 2015).
  15. H. M. Antia, Numerical Methods for Scientists and Engineers(Springer Science & Business Media, 2002).
  16. Schott Glass Technologies Technical Information, “TIE-35: Transmittance of optical glass,” (Schott Glass Technologies, 2005), http://fp.optics.arizona.edu/optomech/references/glass/Schott/tie-35_transmittance_us.pdf , (accessed, July, 2015).
  17. Z. M. Zhang, T. R. Gentile, A. L. Migdall, and R. U. Datla, “Transmittance measurements for filters of optical density between one and ten,” Appl. Opt. 36(34), 8889–8895(1997).
    [Crossref]
  18. Oriel Instruments, “Filter characteristics”, pp. 10-30–10-32, http://ecee.colorado.edu/mcleod/pdfs/AOL/labs/10030.pdf , (accessed, July, 2015).
  19. I. H. Blifford, “Factors affecting the performance of commercial interference filters,” Appl. Opt. 5(1), 105–111(1966).
    [Crossref] [PubMed]
  20. CIE Technical Report, CIE 198:2011: Determination of Measurement Uncertainties in Photometry(Commission internationale de l’Eclairage, 2011).
  21. C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
    [Crossref]
  22. C.-C. Sun, I. Moreno, Y.-C. Lo, B.-C. Chiu, and W.-T. Chien, “Collimating lamp with well color mixing of red/green/blue LEDs,” Opt. Express 20(S1), A75–A84(2012).
    [Crossref] [PubMed]
  23. D. C. Montgomery, Design and Analysis of Experiments, 7th Edition(Wiley, 2009).
  24. R. F. Gunst and R. L. Mason, “Fractional factorial design,” Wiley Interdiscip Rev: Comput Stat 1(2), 234–244 (2009).
    [Crossref]

2015 (1)

2012 (2)

C.-C. Sun, I. Moreno, Y.-C. Lo, B.-C. Chiu, and W.-T. Chien, “Collimating lamp with well color mixing of red/green/blue LEDs,” Opt. Express 20(S1), A75–A84(2012).
[Crossref] [PubMed]

M. Lépez, K. Bredemeier, N. Rohrbeck, C. Véron, F. Schmidt, and A. Sperling, “LED near-field goniophotometer at PTB,” Metrologia 49(2), 141–145(2012).
[Crossref]

2011 (1)

R. Rykowski, “Spectral ray tracing from near field goniophotometer measurements,” Light. Eng 19(1), 23–29(2011).

2009 (2)

U. Krüger and F. Schmidt, “The impact of cooling on CCD-based camera systems in the field of image luminance measuring devices,” Metrologia 46(4), 252–259(2009).
[Crossref]

R. F. Gunst and R. L. Mason, “Fractional factorial design,” Wiley Interdiscip Rev: Comput Stat 1(2), 234–244 (2009).
[Crossref]

2008 (1)

F. Reifergeiste and J. Lienig, “Modelling of the temperature and current dependence of LED spectra,” J. Light Vis. Environ. 32(3), 288–294(2008).
[Crossref]

2007 (1)

I. Ashdown and M. Salsbury, “A near-field goniospectroradiometer for LED measurements,” Proc. SPIE 6342, 634215(2007).
[Crossref]

2005 (1)

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

1997 (1)

1993 (1)

I. Ashdown, “Near-field photometry: a new approach,” J. Illum. Eng. Soc. 22(1), 163–180(1993).
[Crossref]

1966 (1)

Ansems, J.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

Antia, H. M.

H. M. Antia, Numerical Methods for Scientists and Engineers(Springer Science & Business Media, 2002).

Ashdown, I.

I. Ashdown and M. Salsbury, “A near-field goniospectroradiometer for LED measurements,” Proc. SPIE 6342, 634215(2007).
[Crossref]

I. Ashdown, “Near-field photometry: a new approach,” J. Illum. Eng. Soc. 22(1), 163–180(1993).
[Crossref]

Audenaert, J.

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “Rayfiles including spectral and colorimetric information,” Opt. Express 23(7), A361–A370(2015).
[Crossref] [PubMed]

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “On spectral ray files of light sources using principal component analysis,” in Proceedings of 28th CIE Session 2015 (CIE, 2015), pp. 543–547.

Bleumers, J.

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “Rayfiles including spectral and colorimetric information,” Opt. Express 23(7), A361–A370(2015).
[Crossref] [PubMed]

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “On spectral ray files of light sources using principal component analysis,” in Proceedings of 28th CIE Session 2015 (CIE, 2015), pp. 543–547.

Blifford, I. H.

Bredemeier, K.

M. Lépez, K. Bredemeier, N. Rohrbeck, C. Véron, F. Schmidt, and A. Sperling, “LED near-field goniophotometer at PTB,” Metrologia 49(2), 141–145(2012).
[Crossref]

Calon, G.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

Chao, O.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

Chien, W.-T.

Chiu, B.-C.

Datla, R. U.

Deurenberg, P.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

Durinck, G.

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “Rayfiles including spectral and colorimetric information,” Opt. Express 23(7), A361–A370(2015).
[Crossref] [PubMed]

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “On spectral ray files of light sources using principal component analysis,” in Proceedings of 28th CIE Session 2015 (CIE, 2015), pp. 543–547.

Gentile, T. R.

Gunst, R. F.

R. F. Gunst and R. L. Mason, “Fractional factorial design,” Wiley Interdiscip Rev: Comput Stat 1(2), 234–244 (2009).
[Crossref]

Hanselaer, P.

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “Rayfiles including spectral and colorimetric information,” Opt. Express 23(7), A361–A370(2015).
[Crossref] [PubMed]

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “On spectral ray files of light sources using principal component analysis,” in Proceedings of 28th CIE Session 2015 (CIE, 2015), pp. 543–547.

Hoelen, C.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

Jacobs, V.

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “Rayfiles including spectral and colorimetric information,” Opt. Express 23(7), A361–A370(2015).
[Crossref] [PubMed]

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “On spectral ray files of light sources using principal component analysis,” in Proceedings of 28th CIE Session 2015 (CIE, 2015), pp. 543–547.

Krüger, U.

U. Krüger and F. Schmidt, “The impact of cooling on CCD-based camera systems in the field of image luminance measuring devices,” Metrologia 46(4), 252–259(2009).
[Crossref]

Lépez, M.

M. Lépez, K. Bredemeier, N. Rohrbeck, C. Véron, F. Schmidt, and A. Sperling, “LED near-field goniophotometer at PTB,” Metrologia 49(2), 141–145(2012).
[Crossref]

Lienig, J.

F. Reifergeiste and J. Lienig, “Modelling of the temperature and current dependence of LED spectra,” J. Light Vis. Environ. 32(3), 288–294(2008).
[Crossref]

Lijten, G.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

Lo, Y.-C.

Mason, R. L.

R. F. Gunst and R. L. Mason, “Fractional factorial design,” Wiley Interdiscip Rev: Comput Stat 1(2), 234–244 (2009).
[Crossref]

Mercier, V.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

Migdall, A. L.

Montgomery, D. C.

D. C. Montgomery, Design and Analysis of Experiments, 7th Edition(Wiley, 2009).

Moreno, I.

Reifergeiste, F.

F. Reifergeiste and J. Lienig, “Modelling of the temperature and current dependence of LED spectra,” J. Light Vis. Environ. 32(3), 288–294(2008).
[Crossref]

Rohrbeck, N.

M. Lépez, K. Bredemeier, N. Rohrbeck, C. Véron, F. Schmidt, and A. Sperling, “LED near-field goniophotometer at PTB,” Metrologia 49(2), 141–145(2012).
[Crossref]

Rombauts, P.

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “Rayfiles including spectral and colorimetric information,” Opt. Express 23(7), A361–A370(2015).
[Crossref] [PubMed]

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “On spectral ray files of light sources using principal component analysis,” in Proceedings of 28th CIE Session 2015 (CIE, 2015), pp. 543–547.

Rykowski, R.

R. Rykowski, “Spectral ray tracing from near field goniophotometer measurements,” Light. Eng 19(1), 23–29(2011).

Salsbury, M.

I. Ashdown and M. Salsbury, “A near-field goniospectroradiometer for LED measurements,” Proc. SPIE 6342, 634215(2007).
[Crossref]

Schmidt, F.

M. Lépez, K. Bredemeier, N. Rohrbeck, C. Véron, F. Schmidt, and A. Sperling, “LED near-field goniophotometer at PTB,” Metrologia 49(2), 141–145(2012).
[Crossref]

U. Krüger and F. Schmidt, “The impact of cooling on CCD-based camera systems in the field of image luminance measuring devices,” Metrologia 46(4), 252–259(2009).
[Crossref]

Sondag-Huethorst, J.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

Sperling, A.

M. Lépez, K. Bredemeier, N. Rohrbeck, C. Véron, F. Schmidt, and A. Sperling, “LED near-field goniophotometer at PTB,” Metrologia 49(2), 141–145(2012).
[Crossref]

Sun, C.-C.

Treurniet, T.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

van Lier, E.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

van Os, K.

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

Véron, C.

M. Lépez, K. Bredemeier, N. Rohrbeck, C. Véron, F. Schmidt, and A. Sperling, “LED near-field goniophotometer at PTB,” Metrologia 49(2), 141–145(2012).
[Crossref]

Zhang, Z. M.

Appl. Opt. (2)

J. Illum. Eng. Soc. (1)

I. Ashdown, “Near-field photometry: a new approach,” J. Illum. Eng. Soc. 22(1), 163–180(1993).
[Crossref]

J. Light Vis. Environ. (1)

F. Reifergeiste and J. Lienig, “Modelling of the temperature and current dependence of LED spectra,” J. Light Vis. Environ. 32(3), 288–294(2008).
[Crossref]

Light. Eng (1)

R. Rykowski, “Spectral ray tracing from near field goniophotometer measurements,” Light. Eng 19(1), 23–29(2011).

Metrologia (2)

U. Krüger and F. Schmidt, “The impact of cooling on CCD-based camera systems in the field of image luminance measuring devices,” Metrologia 46(4), 252–259(2009).
[Crossref]

M. Lépez, K. Bredemeier, N. Rohrbeck, C. Véron, F. Schmidt, and A. Sperling, “LED near-field goniophotometer at PTB,” Metrologia 49(2), 141–145(2012).
[Crossref]

Opt. Express (2)

Proc. SPIE (2)

C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, “Multi-chip color variable LED spot modules,” Proc. SPIE 5941, 59410A(2005).
[Crossref]

I. Ashdown and M. Salsbury, “A near-field goniospectroradiometer for LED measurements,” Proc. SPIE 6342, 634215(2007).
[Crossref]

Wiley Interdiscip Rev: Comput Stat (1)

R. F. Gunst and R. L. Mason, “Fractional factorial design,” Wiley Interdiscip Rev: Comput Stat 1(2), 234–244 (2009).
[Crossref]

Other (12)

D. C. Montgomery, Design and Analysis of Experiments, 7th Edition(Wiley, 2009).

V. Jacobs, J. Audenaert, J. Bleumers, G. Durinck, P. Rombauts, and P. Hanselaer, “On spectral ray files of light sources using principal component analysis,” in Proceedings of 28th CIE Session 2015 (CIE, 2015), pp. 543–547.

IES, “Ray File Format for the Description of the Emission Property of Light Sources,” TM25-13 (2013).

OSRAM Opto Semiconductors application note, “Importing rayfiles of LEDs from OSRAM Opto Semiconductors”, (OSRAM Opto Semiconductors, 2013), http://www.osram-os.com/Graphics/XPic5/00165120_0.pdf , (accessed, July, 2015).

Oriel Instruments, “Filter characteristics”, pp. 10-30–10-32, http://ecee.colorado.edu/mcleod/pdfs/AOL/labs/10030.pdf , (accessed, July, 2015).

CIE Technical Report, CIE 198:2011: Determination of Measurement Uncertainties in Photometry(Commission internationale de l’Eclairage, 2011).

TechnoTeam, “LMK 50 color datasheet,” (TechnoTeam, 2012), http://www.technoteam.de/product_overview/lmk/products/lmk_50_color/index_eng.html , (accessed, July, 2015).

H. M. Antia, Numerical Methods for Scientists and Engineers(Springer Science & Business Media, 2002).

Schott Glass Technologies Technical Information, “TIE-35: Transmittance of optical glass,” (Schott Glass Technologies, 2005), http://fp.optics.arizona.edu/optomech/references/glass/Schott/tie-35_transmittance_us.pdf , (accessed, July, 2015).

CVI Laser Optics, “Datasheet selection guide,” pp. 232–233, https://marketplace.idexop.com/store/SupportDocuments/12_FiltersandEtalons_2013_final-e.pdf , (accessed, July, 2015).

Newport, “Newport resource e-catalog,” pp. 655–658, http://www.nxtbook.com/nxtbooks/newportcorp/resource2011/#/684 , (accessed, July, 2015).

Schott Glass Technologies catalog, Catalog optical filters, (Schott Glass Technologies, 2015), http://www.schott.com/advanced_optics/german/download/schott-optical-filters-2015-catalog-complete-en.pdf , (accessed, July, 2015).

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Figures (7)

Fig. 1
Fig. 1 Visualization of our four LED test spectra R(λ) and their derived basis spectra Si(λ). All semiconductors’ basis spectra are described by phenomenological models, such as asymmetric logistic power peak and second order Lorentzian. The phosphor as basis spectrum is described as smoothed spline for each test spectrum: a) Blue/Yellow (BY) b) Red/Blue/Yellow (RBY) c) Blue/Blue/Yellow (BBY) b) Red/Green/Blue/Yellow (RBGY).
Fig. 2
Fig. 2 Normalized filter transmission profiles τ(λ). a) Optical glasses as band pass and edge filters b) Interference filters with different FWHMs (Full Width at Half Maximum).
Fig. 3
Fig. 3 Principle of reconstruction: A spectra consisting of n known basis spectra can generally be reconstructed with n arbitrary filter measurements according to Eq. (3).
Fig. 4
Fig. 4 Visualization of the workflow to create spectral rayfiles.
Fig. 5
Fig. 5 Model parameters of the uncertainties. The shape of the probability density function (PDF) is chosen in agreement with [20].
Fig. 6
Fig. 6 Histograms visualizing the reconstruction error derived from 105 Monte Carlo simulations for test spectrum RGBY with the large angular distribution and ηN = 0.4% for three different filter combinations.
Fig. 7
Fig. 7 Validation of preselection: The graphs visualize four of the performed Monte Carlo simulations. The reconstruction performance RSSmean of different filter combinations is plotted on a logarithmic scale against their preselection number N. The highlighted area corresponds to the first two percent of the preselection. The minimum equals the optimal combination.

Tables (1)

Tables Icon

Table 1 Validation of preselection: The table shows the preselection number N of the optimal combination, its relative position in percent and its associated Δuvmean of all simulations.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

R ( x , y , z , ϕ , θ , λ ) = i = 1 n A i ( x , y , z , ϕ , θ ) × S i ( λ ) .
R S S = j = 1 k λ = λ 1 λ 2 w j ( λ ) × [ R ( λ ) R ˜ ( λ ) ] 2 .
[ M 1 M n ] = [ τ 1 ( λ ) S 1 ( λ ) τ sys ( λ ) d λ τ 1 ( λ ) S n ( λ ) τ sys ( λ ) d λ τ n ( λ ) S 1 ( λ ) τ sys ( λ ) d λ τ n ( λ ) S n ( λ ) τ sys ( λ ) d λ ] M S τ [ A 1 A n ] .
[ M 1 + η 1 M n + η n ] = M S τ [ A 1 + Δ A 1 A n + Δ A n ] .
Δ A 2 A 2 ( M S τ 1 2 M S τ 2 ) cond 2 ( M S τ ) η 2 M 2 .
τ ( λ ) = τ Fr ( α , n r ) × τ in ( λ ) d d 0 .
τ ( λ ) = τ Fr ( α + u ( α ) ) × [ u ( τ in ) + τ in ( λ + u ( λ ) + d λ d T [ u ( T ) + u ( T a b s ) ] ) ] exp ( d + u ( d ) d 0 c o s ( β ) )
τ ( λ ) = u ( τ in ) + τ in ( λ + u ( λ ) + d λ d T u ( T ) + u ( λ α ) )

Metrics