Abstract

Multilayer coatings with a small number of layers were designed and prepared to provide an increase in normal-incidence reflectance in the extreme ultraviolet compared with the reflectance of available single-layer coatings, namely, SiC, B4C, and Ir. Multilayers were designed to produce coatings with the highest possible reflectance at 91.2 and at 58.4 nm. At these wavelengths all the materials absorb radiation strongly, but still a reflectance enhancement can be obtained by means of sub-quarter-wave multilayer coatings with two or more different materials. Sub-quarter-wave multilayer coatings based on Al, MgF2, diamondlike carbon, B4C, SiC, and Ir showed higher reflectance than single-layer coatings of SiC and B4C not only at the target wavelength but in a wide band ranging from 50 nm to the 121.6-nm H Lyman-α line. Multilayer coatings suffered some reflectance degradation over time. However, after approximately 80–100 days of aging in a desiccator, the reflectance for the multilayer coatings was greater than for the single-layer coatings.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Bajt, J. B. Alameda, T. W. Barbee, W. M. Clift, J. A. Folta, B. P. Kaufmann, E. A. Spiller, “Improved reflectance and stability of Mo/Si multilayers,” in Soft X-Ray and EUV Imaging Systems II, D. A. Tichenor, J. A. Folta, eds., Proc. SPIE4506, 65–75 (2001).
    [CrossRef]
  2. Y. U. Uspenskii, V. E. Levashov, A. V. Vinogradov, A. I. Fedorenko, V. V. Kondratenko, Y. P. Pershin, E. N. Zubarev, V. Y. Fedotov, “High-reflectivity multilayer mirrors for a vacuum-ultraviolet interval of 35–50 nm,” Opt. Lett. 23, 771–773 (1998).
    [CrossRef]
  3. M. Zukic, D. G. Torr, J. F. Spann, M. R. Torr, “Vacuum ultraviolet thin films. 2: Vacuum ultraviolet all-dielectric narrowband filters,” Appl. Opt. 29, 4293–4302 (1990).
    [CrossRef] [PubMed]
  4. J. I. Larruquert, “Reflectance enhancement with sub-quarterwave multilayers of highly absorbing materials,” J. Opt. Soc. Am. A 18, 1406–1414 (2001).
    [CrossRef]
  5. J. I. Larruquert, “General theory of sub-quarterwave multilayers with highly absorbing materials,” J. Opt. Soc. Am. A 18, 2617–2627 (2001).
    [CrossRef]
  6. J. I. Larruquert, “Reflectance enhancement in the extreme ultraviolet and soft x rays by means of multilayers with more than two materials,” J. Opt. Soc. Am. A 19, 391–397 (2002).
    [CrossRef]
  7. J. I. Larruquert, “Sub-quarterwave multilayers with enhanced reflectance at 13.4 and 11.3 nm,” Opt. Commun. 206, 259–273 (2002).
    [CrossRef]
  8. J. B. Kortright, D. L. Windt, “Amorphous silicon carbide coatings for extreme ultraviolet optics,” Appl. Opt. 27, 2841–2846 (1988).
    [CrossRef] [PubMed]
  9. R. A. M. Keski-Kuha, J. F. Osantowski, H. Herzig, J. S. Gum, A. R. Toft, “Normal incidence reflectance of ion beam deposited SiC films in the EUV,” Appl. Opt. 27, 2815–2816 (1988).
    [CrossRef] [PubMed]
  10. G. Hass, F. G. Jacobus, W. R. Hunter, “Optical properties of evaporated iridium in the vacuum ultraviolet from 500 Å to 2000 Å,” J. Opt. Soc. Am. 57, 758–762 (1967).
    [CrossRef]
  11. G. M. Blumenstock, R. A. M. Keski-Kuha, “Ion-beam-deposited boron carbide coatings for the extreme ultraviolet,” Appl. Opt. 33, 5962–5963 (1994).
    [CrossRef] [PubMed]
  12. J. F. Osantowski, “Reflectance and optical constants for Cer-Vit from 250 to 1050 Å,” J. Opt. Soc. Am. 64, 834–838 (1974).
    [CrossRef]
  13. Lyman, Assessment Study (European Space Agency, Paris, 1985), Vol. SCI(85) 4.
  14. R. S. Polidan, J. B. Holberg, “The status of the absolute calibration of stellar fluxes between 912 and 1200 Å,” in Calibration of Fundamental Stellar Quantities, D. S. Hayes, L. E. Pasinetti, A. G. Davis Philip, eds., IAU Symp. III (Reidel, Dordrecht, The Netherlands1985), pp. 479–483.
  15. J. I. Larruquert, R. A. M. Keski-Kuha, “Multilayer coatings with high reflectance in the EUV spectral region from 50 to 121.6 nm,” Appl. Opt. 38, 1231–1236 (1999).
    [CrossRef]
  16. J. I. Larruquert, “New layer-by-layer design method for multilayers,” J. Opt. Soc. Am. A. 19, 385–390 (2002).
    [CrossRef]
  17. J. I. Larruquert, J. A. Méndez, J. A. Aznárez, “Far-ultraviolet reflectance measurements and optical constants of unoxidized aluminum films,” Appl. Opt. 34, 4892–4899 (1995).
    [CrossRef] [PubMed]
  18. G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,” in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
    [CrossRef]
  19. J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet for thin films of ion-beam-deposited SiC, Mo, Mg2Si, and InSb and of evaporated Cr,” Appl. Opt. 38, 2772–2781 (2000).
    [CrossRef]
  20. J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet of thin films of ion-beam-deposited carbon,” Opt. Commun. 183, 437–443 (2000).
    [CrossRef]
  21. J. I. Larruquert, R. A. M. Keski-Kuha, “Far-ultraviolet optical properties of MgF2 films deposited by evaporation and by ion-beam-sputtering and their application as protective coatings for Al,” submitted to Opt. Commun.
  22. W. J. Choyke, W. D. Partlow, E. P. Supertzi, F. J. Venskytis, G. B. Brandt, “Silicon-carbide diffraction grating for the vacuum ultraviolet: feasibility,” Appl. Opt. 16, 2013–2014 (1977).
    [CrossRef] [PubMed]
  23. For example, M. Bzowski, H. J. Fahr, D. Rucinski, “Interplanetary neutral particle fluxes influencing the Earth’s atmosphere and the terrestrial environment,” Icarus 24, 209–212 (1996).
    [CrossRef]
  24. E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1998).

2002

J. I. Larruquert, “Reflectance enhancement in the extreme ultraviolet and soft x rays by means of multilayers with more than two materials,” J. Opt. Soc. Am. A 19, 391–397 (2002).
[CrossRef]

J. I. Larruquert, “Sub-quarterwave multilayers with enhanced reflectance at 13.4 and 11.3 nm,” Opt. Commun. 206, 259–273 (2002).
[CrossRef]

J. I. Larruquert, “New layer-by-layer design method for multilayers,” J. Opt. Soc. Am. A. 19, 385–390 (2002).
[CrossRef]

2001

2000

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet for thin films of ion-beam-deposited SiC, Mo, Mg2Si, and InSb and of evaporated Cr,” Appl. Opt. 38, 2772–2781 (2000).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet of thin films of ion-beam-deposited carbon,” Opt. Commun. 183, 437–443 (2000).
[CrossRef]

1999

1998

1996

For example, M. Bzowski, H. J. Fahr, D. Rucinski, “Interplanetary neutral particle fluxes influencing the Earth’s atmosphere and the terrestrial environment,” Icarus 24, 209–212 (1996).
[CrossRef]

1995

1994

1990

1988

1977

1974

1967

Alameda, J. B.

S. Bajt, J. B. Alameda, T. W. Barbee, W. M. Clift, J. A. Folta, B. P. Kaufmann, E. A. Spiller, “Improved reflectance and stability of Mo/Si multilayers,” in Soft X-Ray and EUV Imaging Systems II, D. A. Tichenor, J. A. Folta, eds., Proc. SPIE4506, 65–75 (2001).
[CrossRef]

Aznárez, J. A.

Bajt, S.

S. Bajt, J. B. Alameda, T. W. Barbee, W. M. Clift, J. A. Folta, B. P. Kaufmann, E. A. Spiller, “Improved reflectance and stability of Mo/Si multilayers,” in Soft X-Ray and EUV Imaging Systems II, D. A. Tichenor, J. A. Folta, eds., Proc. SPIE4506, 65–75 (2001).
[CrossRef]

Barbee, T. W.

S. Bajt, J. B. Alameda, T. W. Barbee, W. M. Clift, J. A. Folta, B. P. Kaufmann, E. A. Spiller, “Improved reflectance and stability of Mo/Si multilayers,” in Soft X-Ray and EUV Imaging Systems II, D. A. Tichenor, J. A. Folta, eds., Proc. SPIE4506, 65–75 (2001).
[CrossRef]

Blumenstock, G. M.

G. M. Blumenstock, R. A. M. Keski-Kuha, “Ion-beam-deposited boron carbide coatings for the extreme ultraviolet,” Appl. Opt. 33, 5962–5963 (1994).
[CrossRef] [PubMed]

G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,” in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
[CrossRef]

Brandt, G. B.

Bzowski, M.

For example, M. Bzowski, H. J. Fahr, D. Rucinski, “Interplanetary neutral particle fluxes influencing the Earth’s atmosphere and the terrestrial environment,” Icarus 24, 209–212 (1996).
[CrossRef]

Choyke, W. J.

Clift, W. M.

S. Bajt, J. B. Alameda, T. W. Barbee, W. M. Clift, J. A. Folta, B. P. Kaufmann, E. A. Spiller, “Improved reflectance and stability of Mo/Si multilayers,” in Soft X-Ray and EUV Imaging Systems II, D. A. Tichenor, J. A. Folta, eds., Proc. SPIE4506, 65–75 (2001).
[CrossRef]

Fahr, H. J.

For example, M. Bzowski, H. J. Fahr, D. Rucinski, “Interplanetary neutral particle fluxes influencing the Earth’s atmosphere and the terrestrial environment,” Icarus 24, 209–212 (1996).
[CrossRef]

Fedorenko, A. I.

Fedotov, V. Y.

Folta, J. A.

S. Bajt, J. B. Alameda, T. W. Barbee, W. M. Clift, J. A. Folta, B. P. Kaufmann, E. A. Spiller, “Improved reflectance and stability of Mo/Si multilayers,” in Soft X-Ray and EUV Imaging Systems II, D. A. Tichenor, J. A. Folta, eds., Proc. SPIE4506, 65–75 (2001).
[CrossRef]

Ginter, M. L.

G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,” in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
[CrossRef]

Gum, J. S.

Hass, G.

Herzig, H.

Holberg, J. B.

R. S. Polidan, J. B. Holberg, “The status of the absolute calibration of stellar fluxes between 912 and 1200 Å,” in Calibration of Fundamental Stellar Quantities, D. S. Hayes, L. E. Pasinetti, A. G. Davis Philip, eds., IAU Symp. III (Reidel, Dordrecht, The Netherlands1985), pp. 479–483.

Hunter, W. R.

Jacobus, F. G.

Kaufmann, B. P.

S. Bajt, J. B. Alameda, T. W. Barbee, W. M. Clift, J. A. Folta, B. P. Kaufmann, E. A. Spiller, “Improved reflectance and stability of Mo/Si multilayers,” in Soft X-Ray and EUV Imaging Systems II, D. A. Tichenor, J. A. Folta, eds., Proc. SPIE4506, 65–75 (2001).
[CrossRef]

Keski-Kuha, R. A. M.

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet for thin films of ion-beam-deposited SiC, Mo, Mg2Si, and InSb and of evaporated Cr,” Appl. Opt. 38, 2772–2781 (2000).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet of thin films of ion-beam-deposited carbon,” Opt. Commun. 183, 437–443 (2000).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha, “Multilayer coatings with high reflectance in the EUV spectral region from 50 to 121.6 nm,” Appl. Opt. 38, 1231–1236 (1999).
[CrossRef]

G. M. Blumenstock, R. A. M. Keski-Kuha, “Ion-beam-deposited boron carbide coatings for the extreme ultraviolet,” Appl. Opt. 33, 5962–5963 (1994).
[CrossRef] [PubMed]

R. A. M. Keski-Kuha, J. F. Osantowski, H. Herzig, J. S. Gum, A. R. Toft, “Normal incidence reflectance of ion beam deposited SiC films in the EUV,” Appl. Opt. 27, 2815–2816 (1988).
[CrossRef] [PubMed]

G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,” in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha, “Far-ultraviolet optical properties of MgF2 films deposited by evaporation and by ion-beam-sputtering and their application as protective coatings for Al,” submitted to Opt. Commun.

Kondratenko, V. V.

Kortright, J. B.

Larruquert, J. I.

J. I. Larruquert, “Reflectance enhancement in the extreme ultraviolet and soft x rays by means of multilayers with more than two materials,” J. Opt. Soc. Am. A 19, 391–397 (2002).
[CrossRef]

J. I. Larruquert, “Sub-quarterwave multilayers with enhanced reflectance at 13.4 and 11.3 nm,” Opt. Commun. 206, 259–273 (2002).
[CrossRef]

J. I. Larruquert, “New layer-by-layer design method for multilayers,” J. Opt. Soc. Am. A. 19, 385–390 (2002).
[CrossRef]

J. I. Larruquert, “Reflectance enhancement with sub-quarterwave multilayers of highly absorbing materials,” J. Opt. Soc. Am. A 18, 1406–1414 (2001).
[CrossRef]

J. I. Larruquert, “General theory of sub-quarterwave multilayers with highly absorbing materials,” J. Opt. Soc. Am. A 18, 2617–2627 (2001).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet for thin films of ion-beam-deposited SiC, Mo, Mg2Si, and InSb and of evaporated Cr,” Appl. Opt. 38, 2772–2781 (2000).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet of thin films of ion-beam-deposited carbon,” Opt. Commun. 183, 437–443 (2000).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha, “Multilayer coatings with high reflectance in the EUV spectral region from 50 to 121.6 nm,” Appl. Opt. 38, 1231–1236 (1999).
[CrossRef]

J. I. Larruquert, J. A. Méndez, J. A. Aznárez, “Far-ultraviolet reflectance measurements and optical constants of unoxidized aluminum films,” Appl. Opt. 34, 4892–4899 (1995).
[CrossRef] [PubMed]

J. I. Larruquert, R. A. M. Keski-Kuha, “Far-ultraviolet optical properties of MgF2 films deposited by evaporation and by ion-beam-sputtering and their application as protective coatings for Al,” submitted to Opt. Commun.

Levashov, V. E.

Méndez, J. A.

Osantowski, J. F.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1998).

Partlow, W. D.

Pershin, Y. P.

Polidan, R. S.

R. S. Polidan, J. B. Holberg, “The status of the absolute calibration of stellar fluxes between 912 and 1200 Å,” in Calibration of Fundamental Stellar Quantities, D. S. Hayes, L. E. Pasinetti, A. G. Davis Philip, eds., IAU Symp. III (Reidel, Dordrecht, The Netherlands1985), pp. 479–483.

Rucinski, D.

For example, M. Bzowski, H. J. Fahr, D. Rucinski, “Interplanetary neutral particle fluxes influencing the Earth’s atmosphere and the terrestrial environment,” Icarus 24, 209–212 (1996).
[CrossRef]

Spann, J. F.

Spiller, E. A.

S. Bajt, J. B. Alameda, T. W. Barbee, W. M. Clift, J. A. Folta, B. P. Kaufmann, E. A. Spiller, “Improved reflectance and stability of Mo/Si multilayers,” in Soft X-Ray and EUV Imaging Systems II, D. A. Tichenor, J. A. Folta, eds., Proc. SPIE4506, 65–75 (2001).
[CrossRef]

Supertzi, E. P.

Toft, A. R.

Torr, D. G.

Torr, M. R.

Uspenskii, Y. U.

Venskytis, F. J.

Vinogradov, A. V.

Windt, D. L.

Zubarev, E. N.

Zukic, M.

Appl. Opt.

M. Zukic, D. G. Torr, J. F. Spann, M. R. Torr, “Vacuum ultraviolet thin films. 2: Vacuum ultraviolet all-dielectric narrowband filters,” Appl. Opt. 29, 4293–4302 (1990).
[CrossRef] [PubMed]

J. B. Kortright, D. L. Windt, “Amorphous silicon carbide coatings for extreme ultraviolet optics,” Appl. Opt. 27, 2841–2846 (1988).
[CrossRef] [PubMed]

R. A. M. Keski-Kuha, J. F. Osantowski, H. Herzig, J. S. Gum, A. R. Toft, “Normal incidence reflectance of ion beam deposited SiC films in the EUV,” Appl. Opt. 27, 2815–2816 (1988).
[CrossRef] [PubMed]

G. M. Blumenstock, R. A. M. Keski-Kuha, “Ion-beam-deposited boron carbide coatings for the extreme ultraviolet,” Appl. Opt. 33, 5962–5963 (1994).
[CrossRef] [PubMed]

J. I. Larruquert, R. A. M. Keski-Kuha, “Multilayer coatings with high reflectance in the EUV spectral region from 50 to 121.6 nm,” Appl. Opt. 38, 1231–1236 (1999).
[CrossRef]

J. I. Larruquert, J. A. Méndez, J. A. Aznárez, “Far-ultraviolet reflectance measurements and optical constants of unoxidized aluminum films,” Appl. Opt. 34, 4892–4899 (1995).
[CrossRef] [PubMed]

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet for thin films of ion-beam-deposited SiC, Mo, Mg2Si, and InSb and of evaporated Cr,” Appl. Opt. 38, 2772–2781 (2000).
[CrossRef]

W. J. Choyke, W. D. Partlow, E. P. Supertzi, F. J. Venskytis, G. B. Brandt, “Silicon-carbide diffraction grating for the vacuum ultraviolet: feasibility,” Appl. Opt. 16, 2013–2014 (1977).
[CrossRef] [PubMed]

Icarus

For example, M. Bzowski, H. J. Fahr, D. Rucinski, “Interplanetary neutral particle fluxes influencing the Earth’s atmosphere and the terrestrial environment,” Icarus 24, 209–212 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

J. I. Larruquert, “New layer-by-layer design method for multilayers,” J. Opt. Soc. Am. A. 19, 385–390 (2002).
[CrossRef]

Opt. Commun.

J. I. Larruquert, “Sub-quarterwave multilayers with enhanced reflectance at 13.4 and 11.3 nm,” Opt. Commun. 206, 259–273 (2002).
[CrossRef]

J. I. Larruquert, R. A. M. Keski-Kuha, “Reflectance measurements and optical constants in the extreme ultraviolet of thin films of ion-beam-deposited carbon,” Opt. Commun. 183, 437–443 (2000).
[CrossRef]

Opt. Lett.

Other

S. Bajt, J. B. Alameda, T. W. Barbee, W. M. Clift, J. A. Folta, B. P. Kaufmann, E. A. Spiller, “Improved reflectance and stability of Mo/Si multilayers,” in Soft X-Ray and EUV Imaging Systems II, D. A. Tichenor, J. A. Folta, eds., Proc. SPIE4506, 65–75 (2001).
[CrossRef]

G. M. Blumenstock, R. A. M. Keski-Kuha, M. L. Ginter, “Extreme ultraviolet optical properties of ion-beam-deposited boron carbide thin films,” in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2515, 558–564 (1995).
[CrossRef]

Lyman, Assessment Study (European Space Agency, Paris, 1985), Vol. SCI(85) 4.

R. S. Polidan, J. B. Holberg, “The status of the absolute calibration of stellar fluxes between 912 and 1200 Å,” in Calibration of Fundamental Stellar Quantities, D. S. Hayes, L. E. Pasinetti, A. G. Davis Philip, eds., IAU Symp. III (Reidel, Dordrecht, The Netherlands1985), pp. 479–483.

J. I. Larruquert, R. A. M. Keski-Kuha, “Far-ultraviolet optical properties of MgF2 films deposited by evaporation and by ion-beam-sputtering and their application as protective coatings for Al,” submitted to Opt. Commun.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Optical constants n and k of various materials at 91.2 nm (solid lines) and at 58.4 nm (dashed lines). ΔN i are segments connecting the refractive index of a material with that of the material underneath in the multilayer.

Fig. 2
Fig. 2

Near-normal reflectance in the EUV of fresh multilayer coatings of SiC/B4C, SiC/DLC, and SiC/B4C/DLC compared with reflectance of a fresh single-layer coating of SiC.

Fig. 3
Fig. 3

Near-normal reflectance in the EUV of a fresh B4C/DLC multilayer coating and a coating aged for 1 h in a desiccator compared with reflectance of a fresh single-layer coating of B4C.

Fig. 4
Fig. 4

Near-normal reflectance in the EUV of fresh multilayer coatings of SiC/B4C/MgF2/Al and SiC/B4C/DLC compared with reflectance of a fresh single-layer coating of SiC.

Fig. 5
Fig. 5

Near-normal reflectance in the EUV of a B4C/Ir multilayer coating aged in air for 50 min compared with reflectance of fresh single-layer coatings of B4C and of Ir. Palik refers to Ref. 24.

Tables (3)

Tables Icon

Table 1 Optical Constants n and k of the Materials Used in the Multilayer Coatings Design

Tables Icon

Table 2 Layer Thicknesses of the Multilayer Coatings

Tables Icon

Table 3 Reflectance of Samples Either Freshly Deposited or Aged in a Desiccator

Equations (2)

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

Δn11+n12+k12k1+Δk11-n12-k12n1>0,
ImΔNiΔNi-1 < 0  Δni-1Δki> ΔniΔki-1,  i=2-m.

Metrics