Abstract

Index-determination methods based on reflectance and transmittance measurements are developed for both opaque and semitransparent metallic films. Results are given concerning chromium and nickel layers manufactured by electron-beam deposition. To take account of the evolution of the optical constants versus layer thickness, an inhomogeneous layer model is used, which permits us to obtain a good agreement between measurements and calculations. Results are applied to the design and manufacture of light absorbers for which accurate index knowledge is required. Measured absorption is higher than 0.999 on both broadband and monochromatic components.

© 2002 Optical Society of America

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References

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  1. H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Hilger, Bristol, UK, 1986).
    [CrossRef]
  2. A. Thelen, Design of Optical Interference Coating (McGraw-Hill, New York, 1989).
  3. Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Frontières, Gif-sur-Yvette, France, 1992).
  4. A. V. Tikhonravov, M. K. Trubetskov, G. W. DeBell, “Application of the needle optimization technique of the design of optical coatings,” Appl. Opt. 35, 5493–5508 (1996).
    [CrossRef] [PubMed]
  5. H. A. Macleod, “A new approach to the design of metal-dielectric thin film optical coatings,” Opt. Acta 25, 93–106 (1978).
    [CrossRef]
  6. P. H. Berning, A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” J. Opt. Soc. Am. 47, 230–239 (1957).
    [CrossRef]
  7. B. V. Landau, P. H. Lissberger, “Theory of induced transmission filters in terms of the concept of equivalent layers,” J. Opt. Soc. Am. 62, 1258–1264 (1972).
    [CrossRef]
  8. P. H. Lissberger, “Coatings with induced transmission,” Appl. Opt. 20, 95–104 (1981).
    [CrossRef] [PubMed]
  9. Y. Zheng, K. Kikuchi, M. Yamasaki, K. Sonoi, K. Uehara, “Two-layer wideband antireflection coatings with an absorbing layer,” Appl. Opt. 36, 6335–6338 (1997).
    [CrossRef]
  10. J. A. Dobrowolski, L. Li, R. A. Kemp, “Metal/dielectric transmission interference filters with low reflectance. 1. Design,” Appl. Opt. 34, 5673–5683 (1995).
    [CrossRef] [PubMed]
  11. B. T. Sullivan, K. L. Byrt, “Metal/dielectric transmission interference filters with low reflectance. 2. Experimental results,” Appl. Opt. 34, 5684–5694 (1995).
    [CrossRef] [PubMed]
  12. V. T. Bly, J. T. Cox, “Infrared absorbers for ferroelectric detectors,” Appl. Opt. 33, 26–30 (1994).
    [CrossRef] [PubMed]
  13. J. J. Monzon, L. L. Sanchez-Soto, “Optical performance of absorbers structures for thermal detectors,” Appl. Opt. 33, 5137–5141 (1994).
    [CrossRef]
  14. F. Lemarquis, G. Marchand, “Analytical achromatic design of metal–dielectric absorbers,” Appl. Opt. 38, 4876–4884 (1999).
    [CrossRef]
  15. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985, 1991), Vols. 1 and 2.
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    [CrossRef]
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    [CrossRef]
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  19. E. Quesnel, P. Chaton, O. Lartigue, F. Baume, “A very thin coating technology for the production of broad band absorbers,” in Optical Interference Coatings, Vol. 9 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 362–364.
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    [CrossRef]
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    [CrossRef]
  22. J. M. Siqueiros, L. E. Regalado, R. Machorro, “Determination of (n, k) for absorbing thin films using reflectance measurements,” Appl. Opt. 27, 4260–4264 (1988).
    [CrossRef] [PubMed]
  23. J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Thin Solid Films 102, 209–220 (1983).
    [CrossRef]
  24. J. C. Maxwell Garnett, “Colours in metal glasses and metallic films,” Phil. Trans. A 203, 385–420 (1904).
    [CrossRef]
  25. R. M. Hill, C. Weaver, “The optical properties of evaporated chromium films,” Trans. Faraday Soc. 54, 1464–1476 (1958).
    [CrossRef]
  26. J. E. Nestell, R. W. Christy, “Derivation of optical constants of metals from thin-film measurements at oblique incidence,” Appl. Opt. 11, 643–651 (1972).
    [CrossRef] [PubMed]
  27. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Review B 6, 4370–4379 (1972).
    [CrossRef]
  28. R. C. McPhedran, L. C. Botten, D. R. McKenzie, R. P. Netterfield, “Unambiguous determination of optical constants of absorbing films by reflectance and transmittance measurements,” Appl. Opt. 23, 1197–1205 (1984).
    [CrossRef] [PubMed]
  29. A. Mestreau-Garreau, Ch. Pezant, B. Cousin, P. Etcheto, G. Otrio, “Development of black scattering coatings for space application,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 129–136.
  30. J. Loesel, J. Berthon, M. Saccoccio, “Optical design of PHARAO,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 289–298.

2000 (1)

N. Jiang, J. Silcox, “Observation of reaction zones at chromium/oxide glass interfaces,” J. Appl. Phys. 87, 3768–3776 (2000).
[CrossRef]

1999 (1)

1997 (1)

1996 (1)

1995 (2)

1994 (2)

1988 (1)

1984 (1)

1983 (1)

J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

1981 (1)

1978 (1)

H. A. Macleod, “A new approach to the design of metal-dielectric thin film optical coatings,” Opt. Acta 25, 93–106 (1978).
[CrossRef]

1974 (1)

P. B. Johnson, R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni and Pb,” Phys. Rev. B 9, 5056–5070 (1974).
[CrossRef]

1972 (3)

1958 (2)

R. M. Hill, C. Weaver, “The optical properties of evaporated chromium films,” Trans. Faraday Soc. 54, 1464–1476 (1958).
[CrossRef]

R. M. Hill, C. Weaver, “The optical properties of chromium,” Trans. Faraday Soc. 54, 1141–1146 (1958).

1957 (1)

1954 (1)

1950 (1)

1904 (1)

J. C. Maxwell Garnett, “Colours in metal glasses and metallic films,” Phil. Trans. A 203, 385–420 (1904).
[CrossRef]

Baume, F.

E. Quesnel, P. Chaton, O. Lartigue, F. Baume, “A very thin coating technology for the production of broad band absorbers,” in Optical Interference Coatings, Vol. 9 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 362–364.

Berning, P. H.

Berthon, J.

J. Loesel, J. Berthon, M. Saccoccio, “Optical design of PHARAO,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 289–298.

Bly, V. T.

Borgogno, J. P.

J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

Botten, L. C.

Byrt, K. L.

Chaton, P.

E. Quesnel, P. Chaton, O. Lartigue, F. Baume, “A very thin coating technology for the production of broad band absorbers,” in Optical Interference Coatings, Vol. 9 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 362–364.

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni and Pb,” Phys. Rev. B 9, 5056–5070 (1974).
[CrossRef]

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Review B 6, 4370–4379 (1972).
[CrossRef]

J. E. Nestell, R. W. Christy, “Derivation of optical constants of metals from thin-film measurements at oblique incidence,” Appl. Opt. 11, 643–651 (1972).
[CrossRef] [PubMed]

Cousin, B.

A. Mestreau-Garreau, Ch. Pezant, B. Cousin, P. Etcheto, G. Otrio, “Development of black scattering coatings for space application,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 129–136.

Cox, J. T.

DeBell, G. W.

Dobrowolski, J. A.

Etcheto, P.

A. Mestreau-Garreau, Ch. Pezant, B. Cousin, P. Etcheto, G. Otrio, “Development of black scattering coatings for space application,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 129–136.

Furman, Sh. A.

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Frontières, Gif-sur-Yvette, France, 1992).

Hill, R. M.

R. M. Hill, C. Weaver, “The optical properties of chromium,” Trans. Faraday Soc. 54, 1141–1146 (1958).

R. M. Hill, C. Weaver, “The optical properties of evaporated chromium films,” Trans. Faraday Soc. 54, 1464–1476 (1958).
[CrossRef]

Jiang, N.

N. Jiang, J. Silcox, “Observation of reaction zones at chromium/oxide glass interfaces,” J. Appl. Phys. 87, 3768–3776 (2000).
[CrossRef]

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni and Pb,” Phys. Rev. B 9, 5056–5070 (1974).
[CrossRef]

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Review B 6, 4370–4379 (1972).
[CrossRef]

Kemp, R. A.

Kikuchi, K.

Landau, B. V.

Lartigue, O.

E. Quesnel, P. Chaton, O. Lartigue, F. Baume, “A very thin coating technology for the production of broad band absorbers,” in Optical Interference Coatings, Vol. 9 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 362–364.

Lazarides, B.

J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

Lemarquis, F.

Li, L.

Lissberger, P. H.

Loesel, J.

J. Loesel, J. Berthon, M. Saccoccio, “Optical design of PHARAO,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 289–298.

Machorro, R.

Macleod, H. A.

H. A. Macleod, “A new approach to the design of metal-dielectric thin film optical coatings,” Opt. Acta 25, 93–106 (1978).
[CrossRef]

H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Hilger, Bristol, UK, 1986).
[CrossRef]

Marchand, G.

Maxwell Garnett, J. C.

J. C. Maxwell Garnett, “Colours in metal glasses and metallic films,” Phil. Trans. A 203, 385–420 (1904).
[CrossRef]

McKenzie, D. R.

McPhedran, R. C.

Mestreau-Garreau, A.

A. Mestreau-Garreau, Ch. Pezant, B. Cousin, P. Etcheto, G. Otrio, “Development of black scattering coatings for space application,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 129–136.

Monzon, J. J.

Nestell, J. E.

Netterfield, R. P.

Otrio, G.

A. Mestreau-Garreau, Ch. Pezant, B. Cousin, P. Etcheto, G. Otrio, “Development of black scattering coatings for space application,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 129–136.

Pelletier, E.

J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

Pezant, Ch.

A. Mestreau-Garreau, Ch. Pezant, B. Cousin, P. Etcheto, G. Otrio, “Development of black scattering coatings for space application,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 129–136.

Quesnel, E.

E. Quesnel, P. Chaton, O. Lartigue, F. Baume, “A very thin coating technology for the production of broad band absorbers,” in Optical Interference Coatings, Vol. 9 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 362–364.

Regalado, L. E.

Saccoccio, M.

J. Loesel, J. Berthon, M. Saccoccio, “Optical design of PHARAO,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 289–298.

Sanchez-Soto, L. L.

Schulz, L. G.

Scott, G. D.

Sennett, R. S.

Silcox, J.

N. Jiang, J. Silcox, “Observation of reaction zones at chromium/oxide glass interfaces,” J. Appl. Phys. 87, 3768–3776 (2000).
[CrossRef]

Siqueiros, J. M.

Sonoi, K.

Sullivan, B. T.

Thelen, A.

A. Thelen, Design of Optical Interference Coating (McGraw-Hill, New York, 1989).

Tikhonravov, A. V.

Trubetskov, M. K.

Turner, A. F.

Uehara, K.

Weaver, C.

R. M. Hill, C. Weaver, “The optical properties of evaporated chromium films,” Trans. Faraday Soc. 54, 1464–1476 (1958).
[CrossRef]

R. M. Hill, C. Weaver, “The optical properties of chromium,” Trans. Faraday Soc. 54, 1141–1146 (1958).

Yamasaki, M.

Zheng, Y.

Appl. Opt. (11)

J. E. Nestell, R. W. Christy, “Derivation of optical constants of metals from thin-film measurements at oblique incidence,” Appl. Opt. 11, 643–651 (1972).
[CrossRef] [PubMed]

P. H. Lissberger, “Coatings with induced transmission,” Appl. Opt. 20, 95–104 (1981).
[CrossRef] [PubMed]

R. C. McPhedran, L. C. Botten, D. R. McKenzie, R. P. Netterfield, “Unambiguous determination of optical constants of absorbing films by reflectance and transmittance measurements,” Appl. Opt. 23, 1197–1205 (1984).
[CrossRef] [PubMed]

J. M. Siqueiros, L. E. Regalado, R. Machorro, “Determination of (n, k) for absorbing thin films using reflectance measurements,” Appl. Opt. 27, 4260–4264 (1988).
[CrossRef] [PubMed]

V. T. Bly, J. T. Cox, “Infrared absorbers for ferroelectric detectors,” Appl. Opt. 33, 26–30 (1994).
[CrossRef] [PubMed]

J. J. Monzon, L. L. Sanchez-Soto, “Optical performance of absorbers structures for thermal detectors,” Appl. Opt. 33, 5137–5141 (1994).
[CrossRef]

Y. Zheng, K. Kikuchi, M. Yamasaki, K. Sonoi, K. Uehara, “Two-layer wideband antireflection coatings with an absorbing layer,” Appl. Opt. 36, 6335–6338 (1997).
[CrossRef]

J. A. Dobrowolski, L. Li, R. A. Kemp, “Metal/dielectric transmission interference filters with low reflectance. 1. Design,” Appl. Opt. 34, 5673–5683 (1995).
[CrossRef] [PubMed]

B. T. Sullivan, K. L. Byrt, “Metal/dielectric transmission interference filters with low reflectance. 2. Experimental results,” Appl. Opt. 34, 5684–5694 (1995).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, G. W. DeBell, “Application of the needle optimization technique of the design of optical coatings,” Appl. Opt. 35, 5493–5508 (1996).
[CrossRef] [PubMed]

F. Lemarquis, G. Marchand, “Analytical achromatic design of metal–dielectric absorbers,” Appl. Opt. 38, 4876–4884 (1999).
[CrossRef]

J. Appl. Phys. (1)

N. Jiang, J. Silcox, “Observation of reaction zones at chromium/oxide glass interfaces,” J. Appl. Phys. 87, 3768–3776 (2000).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Acta (1)

H. A. Macleod, “A new approach to the design of metal-dielectric thin film optical coatings,” Opt. Acta 25, 93–106 (1978).
[CrossRef]

Phil. Trans. A (1)

J. C. Maxwell Garnett, “Colours in metal glasses and metallic films,” Phil. Trans. A 203, 385–420 (1904).
[CrossRef]

Phys. Rev. B (1)

P. B. Johnson, R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni and Pb,” Phys. Rev. B 9, 5056–5070 (1974).
[CrossRef]

Phys. Review B (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Review B 6, 4370–4379 (1972).
[CrossRef]

Thin Solid Films (1)

J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Thin Solid Films 102, 209–220 (1983).
[CrossRef]

Trans. Faraday Soc. (2)

R. M. Hill, C. Weaver, “The optical properties of chromium,” Trans. Faraday Soc. 54, 1141–1146 (1958).

R. M. Hill, C. Weaver, “The optical properties of evaporated chromium films,” Trans. Faraday Soc. 54, 1464–1476 (1958).
[CrossRef]

Other (7)

E. Quesnel, P. Chaton, O. Lartigue, F. Baume, “A very thin coating technology for the production of broad band absorbers,” in Optical Interference Coatings, Vol. 9 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 362–364.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985, 1991), Vols. 1 and 2.

H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Hilger, Bristol, UK, 1986).
[CrossRef]

A. Thelen, Design of Optical Interference Coating (McGraw-Hill, New York, 1989).

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Frontières, Gif-sur-Yvette, France, 1992).

A. Mestreau-Garreau, Ch. Pezant, B. Cousin, P. Etcheto, G. Otrio, “Development of black scattering coatings for space application,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 129–136.

J. Loesel, J. Berthon, M. Saccoccio, “Optical design of PHARAO,” in Proceedings of the International Conference on Space Optics, ICSO 2000 (Centre National d’Etudes Spatiales, Toulouse, France, 2000), pp. 289–298.

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Figures (15)

Fig. 1
Fig. 1

Comparison of chromium optical constants given in Ref. 15 (unmarked), Ref. 16 (pluses), and determined from reflectance measurements on both sides of an opaque chromium layer deposited by electron beam (circles). The solid and the dotted curves correspond to the refractive index n m and the extinction coefficient k m , respectively. The results discrepancy is likely due to oxidation and film-substrate interface quality.

Fig. 2
Fig. 2

Comparison of nickel optical constants given in Ref. 15 (unmarked), Ref. 16 (pluses), and determined from reflectance measurements on both sides of an opaque nickel layer deposited by electron beam (circles). The solid and the dotted curves correspond to the refractive index n m and the extinction coefficient k m , respectively.

Fig. 3
Fig. 3

Calculated reflectance of an opaque metallic layer before and after the deposition of a thick dielectric layer. In this example the index layer is assumed to be n m - ik m = 2 - i3.5. The dielectric layer index is 2.35, and the dielectric layer thickness is 380 nm. Note that half the intersection points of these two curves correspond to wavelengths for which the dielectric layer is half-wave or multiple. For these particular wavelengths, the determination index cannot be solved.

Fig. 4
Fig. 4

Optical constant envelopes determined from reflectance curves given in Fig. 3, with a systematic relative error of ±0.2% on both reflectance curves. The solid and the dotted curves correspond to the refractive index n m and the extinction coefficient k m , respectively. The problem cannot be solved at a wavelength for which the dielectric layer is half-wave. Note that the index determination method is sensitive to noise at the minimum of the dielectric reflectance curve. In contrast, some wavelengths give accurate results. At 715 nm we obtain n m = 2 ± 0.025 and k m = 3.5 ± 0.035.

Fig. 5
Fig. 5

Index determination for an opaque chromium layer deposited by electron beam. R 0 and R d , respectively, represent the reflectances measured before and after the deposition of a zinc sulfide dielectric layer 379 nm thick. n m and k m represent the real and the imaginary parts, respectively, of the chromium index.

Fig. 6
Fig. 6

Index determination for an opaque nickel layer deposited by electron beam. R 0 and R d , respectively, represent the reflectances measured before and after the deposition of a zinc sulfide dielectric layer 373 nm thick. n m and k m represent the real and the imaginary parts, respectively, of the nickel index.

Fig. 7
Fig. 7

Comparison between measured and calculated reflectance and transmittance, before (R, T) and after (R d , T d ) the deposition of the zinc sulfide dielectric layer, for a semitransparent nickel layer (T = 0.2 at 600 nm). The dielectric and the metallic layer thicknesses are 385.4 and 24.1 nm, respectively. Dotted curves correspond to measured optical properties.

Fig. 8
Fig. 8

Refractive index n m determined for semitransparent nickel layers with 0, 0.7, 0.5, and 0.2 transmittance levels at 600 nm.

Fig. 9
Fig. 9

Extinction coefficient k m determined for semitransparent nickel layers with 0, 0.7, 0.5, and 0.2 transmittance levels at 600 nm.

Fig. 10
Fig. 10

Evolution of the optical constants versus packing density for a mixture of voids and nickel, according to Maxwell Garnett theory. For p = 0 and p = 1, opticals constants correspond to the void and the metal index, respectively, determined for the opaque layer at 600 nm.

Fig. 11
Fig. 11

Comparison between measured and calculated reflectance and transmittance, before (R, T) and after (R d , T d ) the deposition of the zinc sulfide dielectric layer, for a semitransparent nickel layer (T = 0.2 at 600 nm). The dielectric and the metallic layer thicknesses are 391 and 23.3 nm, respectively. Dotted curves correspond to measured optical properties. Contrary to the calculation performed in Fig. 7, the metallic layer is now assumed to be inhomogeneous.

Fig. 12
Fig. 12

Evolution of the optical constants of a nickel layer versus thickness at 600 nm. The solid and the dotted curves correspond to the refractive index n m and the extinction coefficient k m , respectively. With increase of the layer thickness, optical constants tend to the index determined for the opaque nickel layer. The curve with circles corresponds to the evolution of the packing density versus thickness used in calculations.

Fig. 13
Fig. 13

Measured reflectance of a broadband nickel-cryolite absorber manufactured by electron-beam deposition. The design contains both opaque and semitransparent metallic layers. Since transmittance is zero, absorption is higher than 0.999 in the visible spectral range.

Fig. 14
Fig. 14

Measured reflectance of a monochromatic chromium-SiO2–Ta2O5 absorber. The design is formed by an opaque chromium layer covered by a dielectric stack. The chromium layer was deposited by electron beam and the dielectric stack by ion plating. Absorptance is higher than 0.999 at 852 nm.

Fig. 15
Fig. 15

Notation used in Appendix A to calculate the optical properties of an absorbing coating deposited on a transparent substrate. T S and R S correspond to the transmittance and reflectance, respectively, of the bare substrate interface. T C and R C , R C ′, respectively, correspond to the coating transmittance and reflectance (on the front and the back sides).

Equations (15)

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

R0=nm-n02+km2nm+n02+km2,
Rs=nm-ns2+km2nm+ns2+km2.
nm=ns2-n02R0-1Rs-12ns-n01-R0Rs+ns+n0Rs-R0,
km2=nm-n02-R0nm+n02R0-1.
nZnS=2.263+1.371.104/λ2+0.507510/λ4,
1+RdTd+1+R0T0+1-R0T0,
Rd+R0+Td+T0,
1+RdTd+1+R0T0+1-R0T0+Rd+R0+Td+T0.
nm-ikm2=n021+2pr1-pr,
r=Nm-iKm2-n02Nm-iKm2+n02.
p=2πarctandq,
ΦRs=Φ0RS+RSTS2+RSTS2RS2+RSTSRS4+=Φ0RS1+TS21-RS2.
ΦRc=Φ0RC+RSTC2+RSTC2RCRS+RSTC2RCRS2+=Φ0RC+RSTC21-RCRS.
ΦTc=Φ0TCTS+TCTSRCRS+TCTSRCRS2+=Φ0TCTS1-RCRS.
ΦRc=Φ0RS+RCTS2+RCTS2RCRS+RCTS2RCRS2+=Φ0RS+RCTS21-RCRS.

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