Abstract

An absorbing substrate can be coated with a transparent thin film of refractive index N1 (within a certain range) and thickness d such that the ratio of complex reflection coefficients for the p and s polarizations of the film-covered substrate ρ = Rp/Rs is the inverse of that of the film-free substrate ρ¯=R¯p/R¯s at an angle of incidence ϕ. A method to determine the relationship among ϕ, N1, and d that inverts ρ (i.e., makes ρ = 1/ ρ¯) for a given substrate at a given wavelength is described and is applied to aluminum and silver substrates at 0.6328- and 10.6-μm wavelengths, respectively. Sensitivity of the inversion condition to incidence-angle and film-thickness errors is analyzed. ρ-inverting layers can be applied to one of the two metallic mirrors of a beam displacer or axicon to preserve the polarization state of incident monochromatic radiation.

© 1984 Optical Society of America

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References

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  1. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977) , Sec. 4.4.
  2. This separation-of-variables technique proved useful before. See, e.g., Refs. 3 and 4.
  3. R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, “Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry,” J. Opt. Soc. Am. 65, 252–260 (1975).
    [CrossRef]
  4. R. M. A. Azzam, M. Emdadur, Rahman Khan, “Single-reflection film–substrate half-wave retarders with nearly stationary reflection properties over a wide range of incidence angles,” J. Opt. Soc. Am. 73, 160–166 (1983).
    [CrossRef]
  5. T. H. Allen, “Study of Al with combined Auger electron spectrometer–ellipsometer system,” J. Vac. Sci. Technol. 13, 112–115 (1976).
    [CrossRef]
  6. G. Hass, “Mirror coatings,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1965), Vol. 3, Chap. 8.
  7. W. H. Southwell, “Multilayer coatings producing 90° phase change,” Appl. Opt. 18, 1875 (1979).
    [CrossRef] [PubMed]
  8. R. M. A. Azzam, M. Emdadur, Rahman Khan, “Polarization-preserving single-layer-coated beam displacers and axicons,” Appl. Opt. 21, 3314–3322 (1982).
    [CrossRef] [PubMed]
  9. D. Fink, “Polarization effects of axicons,” Appl. Opt. 18, 581–582 (1979).
    [CrossRef] [PubMed]
  10. R. M. A. Azzam, M. Emdadur, Rahman Khan, “Equalization of the TE and TM complex eigenvalues of 90°-rooftop reflectors and waxicons using thin-film dielectric coatings,” Opt. Commun. 44, 223–228 (1983).
    [CrossRef]
  11. See, for example, Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), Sec. 9.7.
  12. See, for example, E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), p. 230.
  13. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 1.7.
  14. V. H. Rumsey, G. A. Deschamps, M. L. Kales, J. I. Bohnert, “Techniques for handling elliptically polarized waves with special reference to antennas,” Proc. IRE 39, 533–551 (1951).
    [CrossRef]
  15. If θ and ɛ represent the azimuth and ellipticity angle, respectively, of the polarization ellipse of the reflected light from the bare substrate, (π/2) − θ and −ɛ will be the corresponding parameters of the light reflected from the substrate coated with the ρ-inverting layer. (See Ref. 1, p. 40 .)

1983 (2)

R. M. A. Azzam, M. Emdadur, Rahman Khan, “Equalization of the TE and TM complex eigenvalues of 90°-rooftop reflectors and waxicons using thin-film dielectric coatings,” Opt. Commun. 44, 223–228 (1983).
[CrossRef]

R. M. A. Azzam, M. Emdadur, Rahman Khan, “Single-reflection film–substrate half-wave retarders with nearly stationary reflection properties over a wide range of incidence angles,” J. Opt. Soc. Am. 73, 160–166 (1983).
[CrossRef]

1982 (1)

1979 (2)

1976 (1)

T. H. Allen, “Study of Al with combined Auger electron spectrometer–ellipsometer system,” J. Vac. Sci. Technol. 13, 112–115 (1976).
[CrossRef]

1975 (1)

1951 (1)

V. H. Rumsey, G. A. Deschamps, M. L. Kales, J. I. Bohnert, “Techniques for handling elliptically polarized waves with special reference to antennas,” Proc. IRE 39, 533–551 (1951).
[CrossRef]

Allen, T. H.

T. H. Allen, “Study of Al with combined Auger electron spectrometer–ellipsometer system,” J. Vac. Sci. Technol. 13, 112–115 (1976).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, M. Emdadur, Rahman Khan, “Single-reflection film–substrate half-wave retarders with nearly stationary reflection properties over a wide range of incidence angles,” J. Opt. Soc. Am. 73, 160–166 (1983).
[CrossRef]

R. M. A. Azzam, M. Emdadur, Rahman Khan, “Equalization of the TE and TM complex eigenvalues of 90°-rooftop reflectors and waxicons using thin-film dielectric coatings,” Opt. Commun. 44, 223–228 (1983).
[CrossRef]

R. M. A. Azzam, M. Emdadur, Rahman Khan, “Polarization-preserving single-layer-coated beam displacers and axicons,” Appl. Opt. 21, 3314–3322 (1982).
[CrossRef] [PubMed]

R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, “Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry,” J. Opt. Soc. Am. 65, 252–260 (1975).
[CrossRef]

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977) , Sec. 4.4.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 1.7.

Balmain, K. G.

See, for example, E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), p. 230.

Bashara, N. M.

R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, “Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry,” J. Opt. Soc. Am. 65, 252–260 (1975).
[CrossRef]

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977) , Sec. 4.4.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 1.7.

Bohnert, J. I.

V. H. Rumsey, G. A. Deschamps, M. L. Kales, J. I. Bohnert, “Techniques for handling elliptically polarized waves with special reference to antennas,” Proc. IRE 39, 533–551 (1951).
[CrossRef]

Deschamps, G. A.

V. H. Rumsey, G. A. Deschamps, M. L. Kales, J. I. Bohnert, “Techniques for handling elliptically polarized waves with special reference to antennas,” Proc. IRE 39, 533–551 (1951).
[CrossRef]

Emdadur, M.

Fink, D.

Hass, G.

G. Hass, “Mirror coatings,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1965), Vol. 3, Chap. 8.

Jordan, E. C.

See, for example, E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), p. 230.

Kales, M. L.

V. H. Rumsey, G. A. Deschamps, M. L. Kales, J. I. Bohnert, “Techniques for handling elliptically polarized waves with special reference to antennas,” Proc. IRE 39, 533–551 (1951).
[CrossRef]

Khan, Rahman

Knittl, Z.

See, for example, Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), Sec. 9.7.

Rumsey, V. H.

V. H. Rumsey, G. A. Deschamps, M. L. Kales, J. I. Bohnert, “Techniques for handling elliptically polarized waves with special reference to antennas,” Proc. IRE 39, 533–551 (1951).
[CrossRef]

Southwell, W. H.

Zaghloul, A.-R. M.

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

J. Vac. Sci. Technol. (1)

T. H. Allen, “Study of Al with combined Auger electron spectrometer–ellipsometer system,” J. Vac. Sci. Technol. 13, 112–115 (1976).
[CrossRef]

Opt. Commun. (1)

R. M. A. Azzam, M. Emdadur, Rahman Khan, “Equalization of the TE and TM complex eigenvalues of 90°-rooftop reflectors and waxicons using thin-film dielectric coatings,” Opt. Commun. 44, 223–228 (1983).
[CrossRef]

Proc. IRE (1)

V. H. Rumsey, G. A. Deschamps, M. L. Kales, J. I. Bohnert, “Techniques for handling elliptically polarized waves with special reference to antennas,” Proc. IRE 39, 533–551 (1951).
[CrossRef]

Other (7)

If θ and ɛ represent the azimuth and ellipticity angle, respectively, of the polarization ellipse of the reflected light from the bare substrate, (π/2) − θ and −ɛ will be the corresponding parameters of the light reflected from the substrate coated with the ρ-inverting layer. (See Ref. 1, p. 40 .)

See, for example, Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), Sec. 9.7.

See, for example, E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), p. 230.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 1.7.

G. Hass, “Mirror coatings,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1965), Vol. 3, Chap. 8.

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977) , Sec. 4.4.

This separation-of-variables technique proved useful before. See, e.g., Refs. 3 and 4.

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Figures (6)

Fig. 1
Fig. 1

Finding a solution for Eq. (9) for the angle of incidence ϕinv at which the ratio ρ of the complex p and s reflection coefficients of an Al substrate (N2 = 1.212 − j6.924) can be inverted by using a transparent-film coating of refractive index 1.55 (e.g., Al2O3) at wavelength λ = 0.6328 μm. Note that the vertical scale changes at zero.

Fig. 2
Fig. 2

Magnitude error |ρn| − 1 caused by shifting the angle of incidence ϕ from the value required for exact inversion ϕinv = 75.8165°. The Al2O3–Al film–substrate system is assumed at λ = 0.6328 μm with the oxide-layer thickness set equal to the value required for inversion, dinv = 0.12863 μm.

Fig. 3
Fig. 3

Same as in Fig. 2, but for the phase error Δn. Both Δn and ϕ are in degrees.

Fig. 4
Fig. 4

Magnitude error ||ρn| − 1| and phase error in degrees |Δn| caused by shifting the thickness d of an Al2O3 film on an Al substrate from the value required for exact inversion, dinv = 0.12863 μm. ϕ is kept fixed at ϕinv = 75.8165°, and λ = 0.6328 μm. Δd is in angstroms.

Fig. 5
Fig. 5

Characteristics of all possible ρ-inverting transparent layers on an Al substrate (N2 = 1.212 − j6.924) at λ = 0.6328 μm. N1 is the film refractive index, and ζinv and dinv are the normalized and actual (in micrometers) least film thicknesses, respectively, plotted versus the angle of incidence ϕinv (in degrees) at which inversion is accomplished. p and s are the reflectances of the film–substrate system for the p and s polarizations under the conditions of inversion. Notice that ϕ is unrestricted (hence it can be chosen anywhere between 0 and 90°), but N1 is limited to the narrow range 1.5 ≲ N1 ≲ 2, which corresponds to several practical thin-film materials.

Fig. 6
Fig. 6

Same as in Fig. 5, but for a Ag (N2 = 9.5 − j73) substrate at λ = 10.6 μm.

Tables (1)

Tables Icon

Table 1 Characteristicsa of ρ-Inverting Transparent Films on an Al Substrate (N2 = 1.212 − j6.924) at λ = 0.6328 μm

Equations (11)

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ρ ¯ = R ¯ p / R ¯ s = ρ ¯ ( ϕ , N 0 , N 2 ) ,
ρ = R p / R s = ρ ( ϕ , d , λ , N 0 , N 1 , N 2 ) .
ρ n = ρ ¯ ( ϕ ) ρ ( ϕ , ζ , N 1 ) = 1.
ζ = d / D ϕ
D ϕ = ( λ / 2 ) ( N 1 2 - N 0 2 sin 2 ϕ ) - 1 / 2
ρ ¯ ρ = ρ ¯ ( A + B X + C X 2 ) / ( D + E X + F X 2 ) = 1 ,
X = exp ( - j 2 π ζ ) .
X = f ± ( ϕ )
X = f ± ( ϕ ) = 1
ζ inv = ( - 1 / 2 π ) arg X .
d inv = ( ζ inv + m ) D ϕ inv ,

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