Abstract

The angular properties of various wideband high reflectors are investigated. The theory is developed for the design of high reflectors based on contiguous quarter-wave stacks for use at one oblique angle or for a range of angles of incidence. Numerical results are presented for several high reflectors designed to have a high reflectance in the 0.4–0.8-µm spectral region for use at 50° and with angles of incidence ranging between 0° and 50°. A random error perturbation analysis shows that such layer systems can be produced experimentally.

© 1997 Optical Society of America

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References

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  1. K. Ferencz, R. Szipocs, “Recent developments of laser optical coatings in Hungary,” Appl. Opt. 32, 2525–2538 (1993).
  2. P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta 8, 105–119 (1961).
    [CrossRef]
  3. H. F. Mahlein, “Properties of laser mirrors at non-normal incidence,” Opt. Acta 20, 687–697 (1973).
    [CrossRef]
  4. A. V. Tikhonravov, “Multilayer dielectric mirrors with oblique light incidence,” Opt. Spectrosc. (USSR) 54, 216–219 (1983).
  5. D. W. Lynch, W. R. Hunter, “Comments on the optical constants of metals and an introduction to the data for several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 275–367.
    [CrossRef]
  6. G. Hass, J. B. Heaney, J. F. Osantowski, J. J. Triolo, “Reflectance and durability of Ag mirrors coated with thin layers of Al2O3 plus reactively deposited silicon oxide,” Appl. Opt. 14, 2639–2644 (1975).
    [CrossRef] [PubMed]
  7. E. A. Volgunova, G. I. Golubeva, A. M. Klochkov, “Highly stable silver mirrors,” Sov. J. Opt. Technol. 50, 128–129 (1983).
  8. G. Hass, “Filmed surfaces for reflecting objects,” J. Opt. Soc. Am. 45, 945–952 (1955).
    [CrossRef]
  9. D. Y. Smith, E. Shiles, M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 369–406.
    [CrossRef]
  10. W. Geffcken, E. Berger, “Lichtfilter aus einer Mehrzahl lichtdurchlässiger, nichtmetallischer Schichten, von denen immer je zwei voneinander durch eine ebensolche Schicht, jedoch von anderer Brechungszahl, getrennt sind,” German patent904357 (11April1943).
  11. Z. Knittl, Optics of Thin Films (Wiley, London, 1976), p. 548.
  12. H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
    [CrossRef]
  13. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, Company, New York, 1988).
  14. S. A. Furman, A. V. Tikhonravov, Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).
  15. S. Penselin, A. Steudel, “Fabry–Perot-Interferometer-verspiegelungen aus dielektrischen Vielfachschichten,” Z. Phys. 142, 21–41 (1955).
    [CrossRef]
  16. O. S. Heavens, H. M. Liddell, “Staggered broad-band reflecting multilayers,” Appl. Opt. 5, 373–376 (1966).
    [CrossRef] [PubMed]
  17. F. A. Korolev, A. Y. Klementeva, T. F. Meshcheryakova, I. A. Ramazina, “Wide-band reflectors with multilayer dielectric coatings,” Opt. Spectrosc. (USSR) 28, 416–419 (1970).
  18. M. A. Khashan, “A Fresnel formula for dielectric multilayer mirrors,” Optik 54, 363–371 (1979).
  19. D. L. Perry, “Broadband dielectric mirrors for multiple wavelength laser operation in the visible,” Proc. IEEE 53, 76–77 (1965).
    [CrossRef]
  20. A. F. Turner, P. W. Baumeister, “Multilayer mirrors with high reflectance over an extended spectral region,” Appl. Opt. 5, 69–76 (1966).
    [CrossRef] [PubMed]
  21. Z. Knittl, “Applications of thin films in optics and the principles and methods of their design,” in Thin Film Technologies I, J. R. Jacobsson, ed., Proc. SPIE401, 2–18 (1983).
    [CrossRef]
  22. W. H. Southwell, “Extended bandwidth reflector designs using wavelets,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 11–13.
  23. A. Thelen, “Design of optical minus filters,” J. Opt. Soc. Am. 61, 365–369 (1971).
    [CrossRef]
  24. K. M. Yoo, R. R. Alfano, “Broad bandwidth mirror with random layer thicknesses,” Appl. Opt. 28, 2456–2458 (1989).
    [CrossRef] [PubMed]
  25. K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
    [CrossRef]
  26. V. A. Antonov, V. I. Pshenitsyn, “Transmission of an electromagnetic wave through randomly organized layered structures,” Opt. Spectrosc. (USSR) 69, 394–396 (1990).
  27. T. J. Alfrey, E. F. Gurnee, W. J. Schrenk, “Physical optics of iridescent multilayered plastic films,” Polym. Eng. Sci. 9, 400–404 (1969).
    [CrossRef]
  28. L. Edmonds, P. Baumeister, M. Krisl, N. Boling, “Spectral characteristics of a narrowband rejection filter,” Appl. Opt. 29, 3203–3204 (1990).
    [CrossRef] [PubMed]
  29. B. T. Sullivan, J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings: I. Theoretical description,” Appl. Opt. 31, 3821–3835 (1992).
    [CrossRef] [PubMed]
  30. B. T. Sullivan, J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. II. Experimental results—sputtering system,” Appl. Opt. 32, 2351–2360 (1993).
    [CrossRef] [PubMed]
  31. A. V. Tikhonravov, “On the synthesis of multilayer optical systems,” Ph.D. dissertation (Moscow State University, Moscow, 1973).

1993 (2)

1992 (1)

1990 (2)

L. Edmonds, P. Baumeister, M. Krisl, N. Boling, “Spectral characteristics of a narrowband rejection filter,” Appl. Opt. 29, 3203–3204 (1990).
[CrossRef] [PubMed]

V. A. Antonov, V. I. Pshenitsyn, “Transmission of an electromagnetic wave through randomly organized layered structures,” Opt. Spectrosc. (USSR) 69, 394–396 (1990).

1989 (2)

K. M. Yoo, R. R. Alfano, “Broad bandwidth mirror with random layer thicknesses,” Appl. Opt. 28, 2456–2458 (1989).
[CrossRef] [PubMed]

K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
[CrossRef]

1983 (2)

A. V. Tikhonravov, “Multilayer dielectric mirrors with oblique light incidence,” Opt. Spectrosc. (USSR) 54, 216–219 (1983).

E. A. Volgunova, G. I. Golubeva, A. M. Klochkov, “Highly stable silver mirrors,” Sov. J. Opt. Technol. 50, 128–129 (1983).

1979 (1)

M. A. Khashan, “A Fresnel formula for dielectric multilayer mirrors,” Optik 54, 363–371 (1979).

1975 (1)

1973 (1)

H. F. Mahlein, “Properties of laser mirrors at non-normal incidence,” Opt. Acta 20, 687–697 (1973).
[CrossRef]

1971 (1)

1970 (1)

F. A. Korolev, A. Y. Klementeva, T. F. Meshcheryakova, I. A. Ramazina, “Wide-band reflectors with multilayer dielectric coatings,” Opt. Spectrosc. (USSR) 28, 416–419 (1970).

1969 (1)

T. J. Alfrey, E. F. Gurnee, W. J. Schrenk, “Physical optics of iridescent multilayered plastic films,” Polym. Eng. Sci. 9, 400–404 (1969).
[CrossRef]

1966 (2)

1965 (1)

D. L. Perry, “Broadband dielectric mirrors for multiple wavelength laser operation in the visible,” Proc. IEEE 53, 76–77 (1965).
[CrossRef]

1961 (1)

P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta 8, 105–119 (1961).
[CrossRef]

1955 (2)

G. Hass, “Filmed surfaces for reflecting objects,” J. Opt. Soc. Am. 45, 945–952 (1955).
[CrossRef]

S. Penselin, A. Steudel, “Fabry–Perot-Interferometer-verspiegelungen aus dielektrischen Vielfachschichten,” Z. Phys. 142, 21–41 (1955).
[CrossRef]

Alfano, R. R.

K. M. Yoo, R. R. Alfano, “Broad bandwidth mirror with random layer thicknesses,” Appl. Opt. 28, 2456–2458 (1989).
[CrossRef] [PubMed]

K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
[CrossRef]

Alfrey, T. J.

T. J. Alfrey, E. F. Gurnee, W. J. Schrenk, “Physical optics of iridescent multilayered plastic films,” Polym. Eng. Sci. 9, 400–404 (1969).
[CrossRef]

Antonov, V. A.

V. A. Antonov, V. I. Pshenitsyn, “Transmission of an electromagnetic wave through randomly organized layered structures,” Opt. Spectrosc. (USSR) 69, 394–396 (1990).

Baumeister, P.

L. Edmonds, P. Baumeister, M. Krisl, N. Boling, “Spectral characteristics of a narrowband rejection filter,” Appl. Opt. 29, 3203–3204 (1990).
[CrossRef] [PubMed]

P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta 8, 105–119 (1961).
[CrossRef]

Baumeister, P. W.

Berger, E.

W. Geffcken, E. Berger, “Lichtfilter aus einer Mehrzahl lichtdurchlässiger, nichtmetallischer Schichten, von denen immer je zwei voneinander durch eine ebensolche Schicht, jedoch von anderer Brechungszahl, getrennt sind,” German patent904357 (11April1943).

Boling, N.

Dobrowolski, J. A.

Edmonds, L.

Ferencz, K.

K. Ferencz, R. Szipocs, “Recent developments of laser optical coatings in Hungary,” Appl. Opt. 32, 2525–2538 (1993).

Furman, S. A.

S. A. Furman, A. V. Tikhonravov, Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

Geffcken, W.

W. Geffcken, E. Berger, “Lichtfilter aus einer Mehrzahl lichtdurchlässiger, nichtmetallischer Schichten, von denen immer je zwei voneinander durch eine ebensolche Schicht, jedoch von anderer Brechungszahl, getrennt sind,” German patent904357 (11April1943).

Golubeva, G. I.

E. A. Volgunova, G. I. Golubeva, A. M. Klochkov, “Highly stable silver mirrors,” Sov. J. Opt. Technol. 50, 128–129 (1983).

Gurnee, E. F.

T. J. Alfrey, E. F. Gurnee, W. J. Schrenk, “Physical optics of iridescent multilayered plastic films,” Polym. Eng. Sci. 9, 400–404 (1969).
[CrossRef]

Hass, G.

Heaney, J. B.

Heavens, O. S.

Hunter, W. R.

D. W. Lynch, W. R. Hunter, “Comments on the optical constants of metals and an introduction to the data for several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 275–367.
[CrossRef]

Inokuti, M.

D. Y. Smith, E. Shiles, M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 369–406.
[CrossRef]

Khashan, M. A.

M. A. Khashan, “A Fresnel formula for dielectric multilayer mirrors,” Optik 54, 363–371 (1979).

Klementeva, A. Y.

F. A. Korolev, A. Y. Klementeva, T. F. Meshcheryakova, I. A. Ramazina, “Wide-band reflectors with multilayer dielectric coatings,” Opt. Spectrosc. (USSR) 28, 416–419 (1970).

Klochkov, A. M.

E. A. Volgunova, G. I. Golubeva, A. M. Klochkov, “Highly stable silver mirrors,” Sov. J. Opt. Technol. 50, 128–129 (1983).

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), p. 548.

Z. Knittl, “Applications of thin films in optics and the principles and methods of their design,” in Thin Film Technologies I, J. R. Jacobsson, ed., Proc. SPIE401, 2–18 (1983).
[CrossRef]

Korolev, F. A.

F. A. Korolev, A. Y. Klementeva, T. F. Meshcheryakova, I. A. Ramazina, “Wide-band reflectors with multilayer dielectric coatings,” Opt. Spectrosc. (USSR) 28, 416–419 (1970).

Krisl, M.

Liddell, H. M.

Lynch, D. W.

D. W. Lynch, W. R. Hunter, “Comments on the optical constants of metals and an introduction to the data for several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 275–367.
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
[CrossRef]

Mahlein, H. F.

H. F. Mahlein, “Properties of laser mirrors at non-normal incidence,” Opt. Acta 20, 687–697 (1973).
[CrossRef]

Meshcheryakova, T. F.

F. A. Korolev, A. Y. Klementeva, T. F. Meshcheryakova, I. A. Ramazina, “Wide-band reflectors with multilayer dielectric coatings,” Opt. Spectrosc. (USSR) 28, 416–419 (1970).

Osantowski, J. F.

Penselin, S.

S. Penselin, A. Steudel, “Fabry–Perot-Interferometer-verspiegelungen aus dielektrischen Vielfachschichten,” Z. Phys. 142, 21–41 (1955).
[CrossRef]

Perry, D. L.

D. L. Perry, “Broadband dielectric mirrors for multiple wavelength laser operation in the visible,” Proc. IEEE 53, 76–77 (1965).
[CrossRef]

Pshenitsyn, V. I.

V. A. Antonov, V. I. Pshenitsyn, “Transmission of an electromagnetic wave through randomly organized layered structures,” Opt. Spectrosc. (USSR) 69, 394–396 (1990).

Ramazina, I. A.

F. A. Korolev, A. Y. Klementeva, T. F. Meshcheryakova, I. A. Ramazina, “Wide-band reflectors with multilayer dielectric coatings,” Opt. Spectrosc. (USSR) 28, 416–419 (1970).

Schrenk, W. J.

T. J. Alfrey, E. F. Gurnee, W. J. Schrenk, “Physical optics of iridescent multilayered plastic films,” Polym. Eng. Sci. 9, 400–404 (1969).
[CrossRef]

Shiles, E.

D. Y. Smith, E. Shiles, M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 369–406.
[CrossRef]

Smith, D. Y.

D. Y. Smith, E. Shiles, M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 369–406.
[CrossRef]

Southwell, W. H.

W. H. Southwell, “Extended bandwidth reflector designs using wavelets,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 11–13.

Steudel, A.

S. Penselin, A. Steudel, “Fabry–Perot-Interferometer-verspiegelungen aus dielektrischen Vielfachschichten,” Z. Phys. 142, 21–41 (1955).
[CrossRef]

Sullivan, B. T.

Szipocs, R.

K. Ferencz, R. Szipocs, “Recent developments of laser optical coatings in Hungary,” Appl. Opt. 32, 2525–2538 (1993).

Thelen, A.

A. Thelen, “Design of optical minus filters,” J. Opt. Soc. Am. 61, 365–369 (1971).
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, Company, New York, 1988).

Tikhonravov, A. V.

A. V. Tikhonravov, “Multilayer dielectric mirrors with oblique light incidence,” Opt. Spectrosc. (USSR) 54, 216–219 (1983).

S. A. Furman, A. V. Tikhonravov, Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

A. V. Tikhonravov, “On the synthesis of multilayer optical systems,” Ph.D. dissertation (Moscow State University, Moscow, 1973).

Triolo, J. J.

Turner, A. F.

Volgunova, E. A.

E. A. Volgunova, G. I. Golubeva, A. M. Klochkov, “Highly stable silver mirrors,” Sov. J. Opt. Technol. 50, 128–129 (1983).

Yoo, K. M.

K. M. Yoo, R. R. Alfano, “Broad bandwidth mirror with random layer thicknesses,” Appl. Opt. 28, 2456–2458 (1989).
[CrossRef] [PubMed]

K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
[CrossRef]

Appl. Opt. (8)

J. Opt. Soc. Am. (2)

Opt. Acta (2)

P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta 8, 105–119 (1961).
[CrossRef]

H. F. Mahlein, “Properties of laser mirrors at non-normal incidence,” Opt. Acta 20, 687–697 (1973).
[CrossRef]

Opt. Spectrosc. (USSR) (3)

A. V. Tikhonravov, “Multilayer dielectric mirrors with oblique light incidence,” Opt. Spectrosc. (USSR) 54, 216–219 (1983).

F. A. Korolev, A. Y. Klementeva, T. F. Meshcheryakova, I. A. Ramazina, “Wide-band reflectors with multilayer dielectric coatings,” Opt. Spectrosc. (USSR) 28, 416–419 (1970).

V. A. Antonov, V. I. Pshenitsyn, “Transmission of an electromagnetic wave through randomly organized layered structures,” Opt. Spectrosc. (USSR) 69, 394–396 (1990).

Optik (1)

M. A. Khashan, “A Fresnel formula for dielectric multilayer mirrors,” Optik 54, 363–371 (1979).

Phys. Rev. B (1)

K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
[CrossRef]

Polym. Eng. Sci. (1)

T. J. Alfrey, E. F. Gurnee, W. J. Schrenk, “Physical optics of iridescent multilayered plastic films,” Polym. Eng. Sci. 9, 400–404 (1969).
[CrossRef]

Proc. IEEE (1)

D. L. Perry, “Broadband dielectric mirrors for multiple wavelength laser operation in the visible,” Proc. IEEE 53, 76–77 (1965).
[CrossRef]

Sov. J. Opt. Technol. (1)

E. A. Volgunova, G. I. Golubeva, A. M. Klochkov, “Highly stable silver mirrors,” Sov. J. Opt. Technol. 50, 128–129 (1983).

Z. Phys. (1)

S. Penselin, A. Steudel, “Fabry–Perot-Interferometer-verspiegelungen aus dielektrischen Vielfachschichten,” Z. Phys. 142, 21–41 (1955).
[CrossRef]

Other (10)

A. V. Tikhonravov, “On the synthesis of multilayer optical systems,” Ph.D. dissertation (Moscow State University, Moscow, 1973).

Z. Knittl, “Applications of thin films in optics and the principles and methods of their design,” in Thin Film Technologies I, J. R. Jacobsson, ed., Proc. SPIE401, 2–18 (1983).
[CrossRef]

W. H. Southwell, “Extended bandwidth reflector designs using wavelets,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 11–13.

D. W. Lynch, W. R. Hunter, “Comments on the optical constants of metals and an introduction to the data for several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 275–367.
[CrossRef]

D. Y. Smith, E. Shiles, M. Inokuti, “The optical properties of metallic aluminum,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 369–406.
[CrossRef]

W. Geffcken, E. Berger, “Lichtfilter aus einer Mehrzahl lichtdurchlässiger, nichtmetallischer Schichten, von denen immer je zwei voneinander durch eine ebensolche Schicht, jedoch von anderer Brechungszahl, getrennt sind,” German patent904357 (11April1943).

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), p. 548.

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, Company, New York, 1988).

S. A. Furman, A. V. Tikhonravov, Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

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Figures (10)

Fig. 1
Fig. 1

Calculated performance at various angles of incidence of a Ag–air interface. Row A depicts the interface between opaque Ag and the incident medium. Rows B–G represent the reflectance of Ag for p- and s-polarized light incident at angles of 0° to 50°.

Fig. 2
Fig. 2

Calculated performance at various angles of incidence of an Al surface enhanced with an 11-layer all-dielectric system. The system has been designed to have a high reflectance at 0° in the 0.4–0.8 µm spectral region. Row A depicts the refractive index profile of this metal-dielectric system. Rows B–G represent the reflectances for p- and s-polarized light of the system for 0° to 50° angles of incidence.

Fig. 3
Fig. 3

Calculated performance at various angles of incidence of a 43-layer two-material system whose layer thicknesses increase in an arithmetic progression. The system has been designed to have a high reflectance at 50° in the 0.4–0.8-µm spectral region. Row A depicts the refractive index profile of the multilayer system; rows B–G represent the reflectances for p- and s-polarized light of the system for 0° to 50° angles of incidence.

Fig. 4
Fig. 4

Calculated performance at various angles of incidence of a 199-layer two-material system with random layer thicknesses. The system has been designed to have a high reflectance at 50° in the 0.4–0.8 µm spectral region. Row A depicts the refractive index profile of the multilayer system.

Fig. 5
Fig. 5

Calculated performance at various angles of incidence of broadband reflectors consisting of several contiguous quarter-wave stacks. Column 1 represents the performance of a 43-layer system consisting of three quarter-wave stacks designed to have a high reflectance in the 0.4–0.8 µm spectral region for an angle of incidence of 50°. Column 2 depicts the performance of a four quarter-wave stack 57-layer system designed to have a high reflectance in the same spectral region for all angles between 0° and 50°.

Fig. 6
Fig. 6

Analogous to Fig. 5, except that the number of layers in the two multilayer systems has been increased to 55 and 97, respectively. It is seen that the ripples in the s-polarized reflectance curves are thereby considerably reduced.

Fig. 7
Fig. 7

Calculated performance at various angles of incidence of the two multilayer systems that resulted from the refinement of all layer thicknesses of the systems depicted in Fig. 5, row A.

Fig. 8
Fig. 8

Effect of 3% random error perturbations of the layer thicknesses on the calculated reflectance at an angle of incidence of 50° for p- and s-polarized light (columns 1 and 2, respectively). Rows A, B, C, and D correspond to the layer systems of Figs. 3, 5, 7, and 6, respectively. The heavy curves represent the reflectance of the unperturbed systems. The reflectance curves of 67% of experimentally produced systems should fall within the shaded areas shown.

Fig. 9
Fig. 9

Effect of finite extinction coefficients of the coating materials on the calculated absorptance for the system depicted in column 2 of Fig. 7 for s- and p-polarized light incident at an angle of 50°. The extinction coefficients of the materials used for these calculations were kH = kL = 0.005. Also shown is the derivative of the phase change on reflection with respect to wavelength.

Fig. 10
Fig. 10

Variation of functions f(b), g(b) [Eq. (A5)] with b=ηLs, p/ηHs, p (see Appendix A).

Equations (28)

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

T=T1T21+1-T11-T2-2 cos2ψ1-T11-T2,
cos 2ψ1
dLnL2-αq=dHnH2-αq=λ04
Fs, pλ, α=-1,
Fs, pλ, α=cos2πλdHnH2-αcos2πλdLnL2-α-12Φs, pαsin2πλdHnH2-α×sin2πλdLnL2-α.
Φs, pα=ηLs, pαηHs, pα+ηHs, pαηLs, pα,
ηL,Hsα=nL,H2-α, ηL,Hpα=nL,H2nL,H2-α
λus, p=λ0ππ-Bs, p, λls, p=λ0ππ+Bs, p,
ξs, p=ηLs, pαq2+ηHs, pαq2-6ηLs, pαqηHs, pαqηLs, pαq+ηHs, pαq2.
dλu, ls, pdα=-Fs, p/αFs, p/λ at λ=λu, ls, p, α=αq.
1λu, lsdλu, lsdα=-141ηLs2+1ηHs2±λu, lsηLsηHsπλ0ηLsηHs2ηHs-ηLs,
1λu, lpdλu, lpdα=-141ηLs2+1ηHs2±signηHp-ηLpλu, lpηLpηHpπλ0ηLsηHs2ηLs2-ηHs2ηHp+ηLp.
signηHp-ηLp=1 if ηHp>ηLp-1 if ηHp<ηLp.
λu, jpλl, jp=π+Bpπ-Bp,
NlnλU/λLlnπ+Bp/π-Bp.
Dj=λU4ππ-Bpjπ+Bpj-1.
dλu,2p-λl,1pdα=signηHp-ηLpλu,2p2ηLpηHpπηLsηHs2×ηLs2-ηHs2ηHp+ηLp1λ0,2+1λ0,1.
θb=arcsinnHnLnanH2+nL2.
nHnLnanH2+nL2<1
dλu,2p-λl,1pdα<0 if θq<θb, dλu,2p-λl,1pdα>0 if θq>θb
ηLsηHsηLsηHs2ηHs-ηLs1141/ηLs2+1/ηHs2<πλ0λus.
4ηLsηHsηLs+ηHsηHs2-ηLs2ηHs2+ηLs2<πλ0λus.
4ηLsηHsηLs+ηHs<πλ0λus.
fb<gb,
fb=4b1+bgb=π-arccos6b-b2-11+b2.
ηLpηHp0.5ηLp+ηHpηHp2-ηLp2ηHp2+ηLp2<πλ02λlp.
πλ02λlp>π2>1.
4ηLpηHpηLp+ηHp<πλ0λup.

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