Abstract

We demonstrate selection of reliable approaches for post-production characterization of oblique incidence multilayer optical coatings. The approaches include choice of input information, selection of adequate coating model, corresponding numerical characterization algorithm, and verification of the results. Applications of the approaches are illustrated with post-production characterization of oblique incidence edge filter, oblique incidence beam splitter and oblique incidence 43-layer quarter-wave mirror.

© 2013 OSA

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Design, production, and reverse engineering of two-octave antireflection coatings

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Oscillations in spectral behavior of total losses (1 − R − T) in thin dielectric films

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References

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    [Crossref] [PubMed]
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    [Crossref]
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2012 (4)

2011 (6)

2010 (1)

H. E. Ehlers, S. S. Schlichting, C. S. Schmitz, and D. R. Ristau, “Adaptive manufacturing of high-precision optics based on virtual deposition and hybrid process control techniques,” Chin. Opt. Lett. 8, 62–66 (2010).
[Crossref]

2009 (1)

2008 (2)

S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “In situ optical characterization and reengineering of interference coatings,” Appl. Opt. 47(13), C49–C54 (2008).
[Crossref] [PubMed]

D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of the art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C, 71010C-14 (2008).
[Crossref]

2007 (1)

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87(1), 5–12 (2007).
[Crossref]

2006 (1)

2003 (1)

P. A. van Nijnatten, “An automated directional reflectance/transmittance analyser for coating analysis,” Thin Solid Films 442(1-2), 74–79 (2003).
[Crossref]

1979 (1)

Ahmad, I.

Amotchkina, T. V.

A. V. Tikhonravov, T. V. Amotchkina, M. K. Trubetskov, R. J. Francis, V. Janicki, J. Sancho-Parramon, H. Zorc, and V. Pervak, “Optical characterization and reverse engineering based on multiangle spectroscopy,” Appl. Opt. 51(2), 245–254 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, B. Romanov, and A. V. Tikhonravov, “On the reliability of reverse engineering results,” Appl. Opt. 51(22), 5543–5551 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a key element in the design-production chain for modern multilayer coatings,” Appl. Opt. 51(31), 7604–7615 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a tool for the selection of the most manufacturable design,” Appl. Opt. 51(36), 8677–8686 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, S. Schlichting, H. Ehlers, D. Ristau, and A. V. Tikhonravov, “Comparison of algorithms used for optical characterization of multilayer optical coatings,” Appl. Opt. 50(20), 3389–3395 (2011).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Design, production, and reverse engineering of two-octave antireflection coatings,” Appl. Opt. 50(35), 6468–6475 (2011).
[Crossref] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the error self-compensation effect associated with broadband optical monitoring,” Appl. Opt. 50(9), C111–C116 (2011).
[Crossref] [PubMed]

Apolonski, A.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87(1), 5–12 (2007).
[Crossref]

Biskupek, J.

Ehlers, H.

Ehlers, H. E.

H. E. Ehlers, S. S. Schlichting, C. S. Schmitz, and D. R. Ristau, “Adaptive manufacturing of high-precision optics based on virtual deposition and hybrid process control techniques,” Chin. Opt. Lett. 8, 62–66 (2010).
[Crossref]

Fornier, A.

Francis, R. J.

Fulop, J.

Gross, T.

Held, M.

Janicki, V.

Kaiser, N.

Kaiser, U.

Krausz, F.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87(1), 5–12 (2007).
[Crossref]

Lappschies, M.

D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of the art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C, 71010C-14 (2008).
[Crossref]

D. Ristau, H. Ehlers, T. Gross, and M. Lappschies, “Optical broadband monitoring of conventional and ion processes,” Appl. Opt. 45(7), 1495–1501 (2006).
[Crossref] [PubMed]

Naumov, S.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87(1), 5–12 (2007).
[Crossref]

Pelletier, E.

Pervak, V.

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, B. Romanov, and A. V. Tikhonravov, “On the reliability of reverse engineering results,” Appl. Opt. 51(22), 5543–5551 (2012).
[Crossref] [PubMed]

A. V. Tikhonravov, T. V. Amotchkina, M. K. Trubetskov, R. J. Francis, V. Janicki, J. Sancho-Parramon, H. Zorc, and V. Pervak, “Optical characterization and reverse engineering based on multiangle spectroscopy,” Appl. Opt. 51(2), 245–254 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Design, production, and reverse engineering of two-octave antireflection coatings,” Appl. Opt. 50(35), 6468–6475 (2011).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, S. Schlichting, H. Ehlers, D. Ristau, and A. V. Tikhonravov, “Comparison of algorithms used for optical characterization of multilayer optical coatings,” Appl. Opt. 50(20), 3389–3395 (2011).
[Crossref] [PubMed]

V. Pervak, M. K. Trubetskov, and A. V. Tikhonravov, “Robust synthesis of dispersive mirrors,” Opt. Express 19(3), 2371–2380 (2011).
[Crossref] [PubMed]

V. Pervak, “Recent development and new ideas in the field of dispersive multilayer optics,” Appl. Opt. 50(9), C55–C61 (2011).
[Crossref] [PubMed]

V. Pervak, I. Ahmad, J. Fulop, M. K. Trubetskov, and A. V. Tikhonravov, “Comparison of dispersive mirrors based on the time-domain and conventional approaches, for sub-5-fs pulses,” Opt. Express 17(4), 2207–2217 (2009).
[Crossref] [PubMed]

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87(1), 5–12 (2007).
[Crossref]

Ristau, D.

Ristau, D. R.

H. E. Ehlers, S. S. Schlichting, C. S. Schmitz, and D. R. Ristau, “Adaptive manufacturing of high-precision optics based on virtual deposition and hybrid process control techniques,” Chin. Opt. Lett. 8, 62–66 (2010).
[Crossref]

Romanov, B.

Sancho-Parramon, J.

Schlichting, S.

Schlichting, S. S.

H. E. Ehlers, S. S. Schlichting, C. S. Schmitz, and D. R. Ristau, “Adaptive manufacturing of high-precision optics based on virtual deposition and hybrid process control techniques,” Chin. Opt. Lett. 8, 62–66 (2010).
[Crossref]

Schmitz, C. S.

H. E. Ehlers, S. S. Schlichting, C. S. Schmitz, and D. R. Ristau, “Adaptive manufacturing of high-precision optics based on virtual deposition and hybrid process control techniques,” Chin. Opt. Lett. 8, 62–66 (2010).
[Crossref]

Stenzel, O.

Tikhonravov, A. V.

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, B. Romanov, and A. V. Tikhonravov, “On the reliability of reverse engineering results,” Appl. Opt. 51(22), 5543–5551 (2012).
[Crossref] [PubMed]

A. V. Tikhonravov, T. V. Amotchkina, M. K. Trubetskov, R. J. Francis, V. Janicki, J. Sancho-Parramon, H. Zorc, and V. Pervak, “Optical characterization and reverse engineering based on multiangle spectroscopy,” Appl. Opt. 51(2), 245–254 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a key element in the design-production chain for modern multilayer coatings,” Appl. Opt. 51(31), 7604–7615 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a tool for the selection of the most manufacturable design,” Appl. Opt. 51(36), 8677–8686 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Design, production, and reverse engineering of two-octave antireflection coatings,” Appl. Opt. 50(35), 6468–6475 (2011).
[Crossref] [PubMed]

V. Pervak, M. K. Trubetskov, and A. V. Tikhonravov, “Robust synthesis of dispersive mirrors,” Opt. Express 19(3), 2371–2380 (2011).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, S. Schlichting, H. Ehlers, D. Ristau, and A. V. Tikhonravov, “Comparison of algorithms used for optical characterization of multilayer optical coatings,” Appl. Opt. 50(20), 3389–3395 (2011).
[Crossref] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the error self-compensation effect associated with broadband optical monitoring,” Appl. Opt. 50(9), C111–C116 (2011).
[Crossref] [PubMed]

V. Pervak, I. Ahmad, J. Fulop, M. K. Trubetskov, and A. V. Tikhonravov, “Comparison of dispersive mirrors based on the time-domain and conventional approaches, for sub-5-fs pulses,” Opt. Express 17(4), 2207–2217 (2009).
[Crossref] [PubMed]

S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “In situ optical characterization and reengineering of interference coatings,” Appl. Opt. 47(13), C49–C54 (2008).
[Crossref] [PubMed]

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87(1), 5–12 (2007).
[Crossref]

Trubetskov, M. K.

A. V. Tikhonravov, T. V. Amotchkina, M. K. Trubetskov, R. J. Francis, V. Janicki, J. Sancho-Parramon, H. Zorc, and V. Pervak, “Optical characterization and reverse engineering based on multiangle spectroscopy,” Appl. Opt. 51(2), 245–254 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, B. Romanov, and A. V. Tikhonravov, “On the reliability of reverse engineering results,” Appl. Opt. 51(22), 5543–5551 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a key element in the design-production chain for modern multilayer coatings,” Appl. Opt. 51(31), 7604–7615 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a tool for the selection of the most manufacturable design,” Appl. Opt. 51(36), 8677–8686 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Design, production, and reverse engineering of two-octave antireflection coatings,” Appl. Opt. 50(35), 6468–6475 (2011).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, S. Schlichting, H. Ehlers, D. Ristau, and A. V. Tikhonravov, “Comparison of algorithms used for optical characterization of multilayer optical coatings,” Appl. Opt. 50(20), 3389–3395 (2011).
[Crossref] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the error self-compensation effect associated with broadband optical monitoring,” Appl. Opt. 50(9), C111–C116 (2011).
[Crossref] [PubMed]

V. Pervak, M. K. Trubetskov, and A. V. Tikhonravov, “Robust synthesis of dispersive mirrors,” Opt. Express 19(3), 2371–2380 (2011).
[Crossref] [PubMed]

V. Pervak, I. Ahmad, J. Fulop, M. K. Trubetskov, and A. V. Tikhonravov, “Comparison of dispersive mirrors based on the time-domain and conventional approaches, for sub-5-fs pulses,” Opt. Express 17(4), 2207–2217 (2009).
[Crossref] [PubMed]

S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “In situ optical characterization and reengineering of interference coatings,” Appl. Opt. 47(13), C49–C54 (2008).
[Crossref] [PubMed]

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87(1), 5–12 (2007).
[Crossref]

Tünnermann, A.

van Nijnatten, P. A.

P. A. van Nijnatten, “An automated directional reflectance/transmittance analyser for coating analysis,” Thin Solid Films 442(1-2), 74–79 (2003).
[Crossref]

Vidal, B.

Wilbrandt, S.

Yulin, S.

Zorc, H.

Appl. Opt. (11)

B. Vidal, A. Fornier, and E. Pelletier, “Wideband optical monitoring of nonquarterwave multilayer filters,” Appl. Opt. 18(22), 3851–3856 (1979).
[PubMed]

D. Ristau, H. Ehlers, T. Gross, and M. Lappschies, “Optical broadband monitoring of conventional and ion processes,” Appl. Opt. 45(7), 1495–1501 (2006).
[Crossref] [PubMed]

S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “In situ optical characterization and reengineering of interference coatings,” Appl. Opt. 47(13), C49–C54 (2008).
[Crossref] [PubMed]

V. Pervak, “Recent development and new ideas in the field of dispersive multilayer optics,” Appl. Opt. 50(9), C55–C61 (2011).
[Crossref] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the error self-compensation effect associated with broadband optical monitoring,” Appl. Opt. 50(9), C111–C116 (2011).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, S. Schlichting, H. Ehlers, D. Ristau, and A. V. Tikhonravov, “Comparison of algorithms used for optical characterization of multilayer optical coatings,” Appl. Opt. 50(20), 3389–3395 (2011).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Design, production, and reverse engineering of two-octave antireflection coatings,” Appl. Opt. 50(35), 6468–6475 (2011).
[Crossref] [PubMed]

A. V. Tikhonravov, T. V. Amotchkina, M. K. Trubetskov, R. J. Francis, V. Janicki, J. Sancho-Parramon, H. Zorc, and V. Pervak, “Optical characterization and reverse engineering based on multiangle spectroscopy,” Appl. Opt. 51(2), 245–254 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, M. K. Trubetskov, V. Pervak, B. Romanov, and A. V. Tikhonravov, “On the reliability of reverse engineering results,” Appl. Opt. 51(22), 5543–5551 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a key element in the design-production chain for modern multilayer coatings,” Appl. Opt. 51(31), 7604–7615 (2012).
[Crossref] [PubMed]

T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a tool for the selection of the most manufacturable design,” Appl. Opt. 51(36), 8677–8686 (2012).
[Crossref] [PubMed]

Appl. Phys. B (1)

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87(1), 5–12 (2007).
[Crossref]

Chin. Opt. Lett. (1)

H. E. Ehlers, S. S. Schlichting, C. S. Schmitz, and D. R. Ristau, “Adaptive manufacturing of high-precision optics based on virtual deposition and hybrid process control techniques,” Chin. Opt. Lett. 8, 62–66 (2010).
[Crossref]

Opt. Express (2)

Opt. Mater. Express (1)

Proc. SPIE (1)

D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of the art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C, 71010C-14 (2008).
[Crossref]

Thin Solid Films (1)

P. A. van Nijnatten, “An automated directional reflectance/transmittance analyser for coating analysis,” Thin Solid Films 442(1-2), 74–79 (2003).
[Crossref]

Other (8)

A. V. Tikhonravov and M. K. Trubetskov, “OptiLayer software,” http://www.optilayer.com .
[Crossref]

A. V. Tikhonravov, M. K. Trubetskov, I. V. Kochikov, J. B. Oliver, and D. J. Smith, “Real-time characterization and optimization of E-beam evaporated optical coatings,” in Optical Interference Coatings, OSA Technical Digest Series (2001), ME8.

T. V. Amotchkina, M. K. Trubetskov, A. V. Tikhonravov, and V. Pervak, “Reverse engineering of an output coupler using broadband monitoring data and group delay measurements,” in Optical Interference Coatings, OSA Technical Digest Series (2013), WB.2.

A. N. Tikhonov and V. I. Arsenin, Solutions of Ill-posed Problems (Winston, 1977).

D. Death, R. J. Francis, C. Bricker, T. Burt, and C. Colley, “The UMA: A new tool for multi-angle photometric spectroscopy,” in Optical Interference Coatings, OSA Technical Digest Series (2013), ThC.3.

S. A. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontières, 1992).

P. Baumeister, Optical Coating Technology (SPIE Optical Engineering Press, 2004).

H. A. Macleod, Thin-film Optical Filters, 4th ed, Series in Optics and Optoelectronics (CRC Press/Taylor & Francis, 2010).

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Figures (13)

Fig. 1
Fig. 1

Comparison of near-normal incidence experimental and theoretical reflectance data related to HR400-BK7 sample (a). Fitting of experimental reflectance data by model reflectance data achieved by RE algorithm (b).

Fig. 2
Fig. 2

Estimated relative errors in layer thicknesses of 21-layer quarter-wave mirror.

Fig. 3
Fig. 3

Comparison of oblique incidence experimental and model reflectance data related to HR400-BK7 sample.

Fig. 4
Fig. 4

Comparison of experimental and theoretical near-normal incidence data (EF-Suprasil sample) (a). Fitting of experimental normal incidence transmittance data by model data (b).

Fig. 5
Fig. 5

Estimated relative errors in layer thicknesses of the edge filter coating (a). Comparison of oblique incidence experimental transmittance data related to EF-Suprasil sample with model transmittance (b).

Fig. 6
Fig. 6

Fittings of in situ experimental transmittance data (red crosses) by model transmittances (black solid curves) related to sample BS-Glass after the deposition of layer 13, 26, 39, and 52.

Fig. 7
Fig. 7

Fittings of normal incidence experimental transmittance data by model transmittance related to the sample AR-Glass (a). Estimated relative errors in layer thicknesses of the samples BS-Glass and AR-Glass (b).

Fig. 8
Fig. 8

Comparison of oblique incidence experimental transmittance data of the sample BS-AR-Glass with model transmittance: (a) non-polarized light at 45°, (b) s- and p-polarizations at 30°.

Fig. 9
Fig. 9

Comparison of in situ experimental transmittance data, ex situ experimental data and theoretical transmittance related to HR800-Glass sample (a). Fitting of in situ experimental transmittance data by model transmittances related to HR800-Glass sample (b).

Fig. 10
Fig. 10

Refractive indices of HfO2 layers: nominal, average in situ and ex situ (a). Estimated relative errors in layer thicknesses of 43-layer quarter-wave mirror (b).

Fig. 11
Fig. 11

Comparison of normal (a) and AOI = 45° (b) experimental transmittance data related to HR800-FusedSilica sample with model transmittances calculated for intermediate design.

Fig. 12
Fig. 12

Final fitting of normal (a) and oblique incidence AOI = 45° (b) experimental transmittance data related to HR800-FusedSilica sample by model transmittances.

Fig. 13
Fig. 13

Comparison of normal incidence experimental data related to the second deposition run and theoretical transmittance data (a). Model reflectance at AOI = 45° after additional RE procedure (b).

Tables (2)

Tables Icon

Table 1 Refractive indices of layer materials and substrates (λ should be expressed in µm).

Tables Icon

Table 2 Design structures considered in Sections 3−5, layer numbers LN starts from the substrate, physical thicknesses and layer materials are listed.

Equations (11)

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D F 2 = 1 L j=1 L [ R(X; λ j ) R ^ ( λ j ) ] 2 ,
R(X,λ)=R((1+ δ 1 ) d 1 ,,(1+ δ N ) d N ; n H (λ), n L (λ);λ),
MD F 2 = 1 NL i=1 N j=1 L [ T (i) ( X i ; λ j ) T ^ (i) ( λ j ) ] 2 .
T( X i ;λ)=T( d 1 ,, d i ; n H (λ)(1+ h H ), n L (λ)(1+ h L );λ).
T( X i ;λ)=T( d 1 (1+ δ 1 ),, d i (1+ δ i ); n ˜ H (λ), n ˜ L (λ);λ).
GMD F 2 = 1 NL i=1 N j=1 L [ T (i) ( X i ; λ j ) T ^ (i) ( λ j ) ] 2 +α 1 N i=1 N δ i 2 ,
T (i) ( X i ;λ)= T (i) ( d 1 (1+ δ 1 ),, d i (1+ δ i ); n H (λ), n L (λ);λ).
T (i) ( X i ;λ)= T (i) ( d 1 (1+ δ 1 ),, d i (1+ δ i ); n H (1+ h H,1 ), n L ,..., n H (1+ h H,i );λ).
δ=( n + n )/ n av ,
D F 2 = 1 2L j=1 L [ T(X; λ j ) T ^ ( λ j ) ] 2 + 1 2L j=1 L [ R(X; λ j ) R ^ ( λ j ) ] 2 ,
T(X;λ)=T( d ˜ 1 ,, d ˜ N ; n H + h H , n L ; δ H ;λ).

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