Abstract

Approximate equations are derived for calculating the transmittance and reflectance of a slightly absorbing film when radiation is incident at an arbitrary angle. These formulas are compared with those derived from wave optics. Examination of the real and the imaginary parts of the complex phase change and the complex angle of refraction shows the simple equations to be consistent with the wave-optics formulation under the assumption that the imaginary part of the refractive index of the film is much smaller than its real counterpart.

© 1999 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 46–95.
  2. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), Chap. 13, pp. 611–664.
  3. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 195.
  4. M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, New York, 1992), Chap. 4, pp. 114–156.
  5. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 4, pp. 83–101.
  6. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 2, pp. 12–56.
  7. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Hemisphere, Washington, D.C., 1992), Chap. 18, pp. 923–967.
  8. M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993), Chap. 2, pp. 37–74.
  9. Y. P. Zhang, C. Zhou, X. S. Ge, X. G. Liang, “A precise and simple method, the relative transmittance fringes depth method, of determining the optical constants and thickness and thin semitransparent films,” J. Phys. D. 25, 1004–1009 (1992).
    [CrossRef]
  10. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, FL, 1985 and 1991), Vols. 1 and 2.
  11. Z. M. Zhang, T. R. Gentile, A. L. Migdall, R. U. Datla, “Transmittance measurements for filters of optical density from one to ten,” Appl. Opt. 36, 8889–8895 (1997).
    [CrossRef]
  12. Z. M. Zhang, L. M. Hanssen, R. U. Datla, “Polarization-dependent angular reflectance of silicon and germanium in the infrared,” Infrared Phys. Technol. 37, 539–546 (1996).
    [CrossRef]
  13. Z. M. Zhang, G. Lefever-Button, F. R. Powell, “Infrared refractive index and extinction coefficient of polyimide films,” Int. J. Thermophys. 19, 905–916 (1998).
    [CrossRef]
  14. J.-P. Borgogno, E. Pelletier, “Determination of the extinction coefficient of dielectric thin films from spectrophotometric measurements,” Appl. Opt. 28, 2895–2901 (1989).
    [CrossRef] [PubMed]
  15. M. Commandré, P. Roche, J.-P. Borgogno, G. Albrand, “Absorption mapping for characterization of glass surfaces,” Appl. Opt. 34, 2372–2379 (1995).
    [CrossRef] [PubMed]

1998 (1)

Z. M. Zhang, G. Lefever-Button, F. R. Powell, “Infrared refractive index and extinction coefficient of polyimide films,” Int. J. Thermophys. 19, 905–916 (1998).
[CrossRef]

1997 (1)

1996 (1)

Z. M. Zhang, L. M. Hanssen, R. U. Datla, “Polarization-dependent angular reflectance of silicon and germanium in the infrared,” Infrared Phys. Technol. 37, 539–546 (1996).
[CrossRef]

1995 (1)

1992 (1)

Y. P. Zhang, C. Zhou, X. S. Ge, X. G. Liang, “A precise and simple method, the relative transmittance fringes depth method, of determining the optical constants and thickness and thin semitransparent films,” J. Phys. D. 25, 1004–1009 (1992).
[CrossRef]

1989 (1)

Albrand, G.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 2, pp. 12–56.

Borgogno, J.-P.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), Chap. 13, pp. 611–664.

Brewster, M. Q.

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, New York, 1992), Chap. 4, pp. 114–156.

Commandré, M.

Datla, R. U.

Z. M. Zhang, T. R. Gentile, A. L. Migdall, R. U. Datla, “Transmittance measurements for filters of optical density from one to ten,” Appl. Opt. 36, 8889–8895 (1997).
[CrossRef]

Z. M. Zhang, L. M. Hanssen, R. U. Datla, “Polarization-dependent angular reflectance of silicon and germanium in the infrared,” Infrared Phys. Technol. 37, 539–546 (1996).
[CrossRef]

Ge, X. S.

Y. P. Zhang, C. Zhou, X. S. Ge, X. G. Liang, “A precise and simple method, the relative transmittance fringes depth method, of determining the optical constants and thickness and thin semitransparent films,” J. Phys. D. 25, 1004–1009 (1992).
[CrossRef]

Gentile, T. R.

Hanssen, L. M.

Z. M. Zhang, L. M. Hanssen, R. U. Datla, “Polarization-dependent angular reflectance of silicon and germanium in the infrared,” Infrared Phys. Technol. 37, 539–546 (1996).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 46–95.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Hemisphere, Washington, D.C., 1992), Chap. 18, pp. 923–967.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 2, pp. 12–56.

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 195.

Lefever-Button, G.

Z. M. Zhang, G. Lefever-Button, F. R. Powell, “Infrared refractive index and extinction coefficient of polyimide films,” Int. J. Thermophys. 19, 905–916 (1998).
[CrossRef]

Liang, X. G.

Y. P. Zhang, C. Zhou, X. S. Ge, X. G. Liang, “A precise and simple method, the relative transmittance fringes depth method, of determining the optical constants and thickness and thin semitransparent films,” J. Phys. D. 25, 1004–1009 (1992).
[CrossRef]

Migdall, A. L.

Modest, M. F.

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993), Chap. 2, pp. 37–74.

Pelletier, E.

Powell, F. R.

Z. M. Zhang, G. Lefever-Button, F. R. Powell, “Infrared refractive index and extinction coefficient of polyimide films,” Int. J. Thermophys. 19, 905–916 (1998).
[CrossRef]

Roche, P.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Hemisphere, Washington, D.C., 1992), Chap. 18, pp. 923–967.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), Chap. 13, pp. 611–664.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 4, pp. 83–101.

Zhang, Y. P.

Y. P. Zhang, C. Zhou, X. S. Ge, X. G. Liang, “A precise and simple method, the relative transmittance fringes depth method, of determining the optical constants and thickness and thin semitransparent films,” J. Phys. D. 25, 1004–1009 (1992).
[CrossRef]

Zhang, Z. M.

Z. M. Zhang, G. Lefever-Button, F. R. Powell, “Infrared refractive index and extinction coefficient of polyimide films,” Int. J. Thermophys. 19, 905–916 (1998).
[CrossRef]

Z. M. Zhang, T. R. Gentile, A. L. Migdall, R. U. Datla, “Transmittance measurements for filters of optical density from one to ten,” Appl. Opt. 36, 8889–8895 (1997).
[CrossRef]

Z. M. Zhang, L. M. Hanssen, R. U. Datla, “Polarization-dependent angular reflectance of silicon and germanium in the infrared,” Infrared Phys. Technol. 37, 539–546 (1996).
[CrossRef]

Zhou, C.

Y. P. Zhang, C. Zhou, X. S. Ge, X. G. Liang, “A precise and simple method, the relative transmittance fringes depth method, of determining the optical constants and thickness and thin semitransparent films,” J. Phys. D. 25, 1004–1009 (1992).
[CrossRef]

Appl. Opt. (3)

Infrared Phys. Technol. (1)

Z. M. Zhang, L. M. Hanssen, R. U. Datla, “Polarization-dependent angular reflectance of silicon and germanium in the infrared,” Infrared Phys. Technol. 37, 539–546 (1996).
[CrossRef]

Int. J. Thermophys. (1)

Z. M. Zhang, G. Lefever-Button, F. R. Powell, “Infrared refractive index and extinction coefficient of polyimide films,” Int. J. Thermophys. 19, 905–916 (1998).
[CrossRef]

J. Phys. D. (1)

Y. P. Zhang, C. Zhou, X. S. Ge, X. G. Liang, “A precise and simple method, the relative transmittance fringes depth method, of determining the optical constants and thickness and thin semitransparent films,” J. Phys. D. 25, 1004–1009 (1992).
[CrossRef]

Other (9)

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

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 46–95.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), Chap. 13, pp. 611–664.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 195.

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, New York, 1992), Chap. 4, pp. 114–156.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 4, pp. 83–101.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 2, pp. 12–56.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Hemisphere, Washington, D.C., 1992), Chap. 18, pp. 923–967.

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1993), Chap. 2, pp. 37–74.

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Figures (1)

Fig. 1
Fig. 1

Schematic of multiple reflections.

Equations (14)

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

n1 sin θ1=n2 sin θ2=n3 sin θ3.
τ=exp-4πk2λ d2/cos θ2.
T=n3 cos θ3n1 cos θ1×1+r121+r22τ1+r12r22τ2+2r1r2τ cos ϕ,
R=r12+r22τ2+2r1r2τ cos ϕ1+r12r22τ2+2r1r2τ cos ϕ,
ϕ=4πn2λ d2 cos θ2
T=1-r122τ1+r14τ2-2r12τ cos ϕ.
T=n3 cos θ3n1 cos θ1×1+r˜11+r˜2expiδ˜1+r˜1r˜2 expi2δ˜2,
R=r˜1+r˜2 expi2δ˜1+r˜1r˜2 expi2δ˜2.
δ˜=2πñ2λ d2 cos θ˜2
n1 sin θ1=ñ2 sin θ˜2.
sin θ˜2=n1 sin θ1n22+k22n2-ik2n1n2sin θ11-i k2n2=sin θ21-i k2n2.
cos θ˜2=1-sin2 θ˜21/2cos θ21+i tan2 θ2k2n2.
Re δ˜2πn2λ d2 cos θ2,
Im δ˜=2πd2λn2 Imcos θ˜2+k2 Recos θ˜22πk2λ d2/cos θ2.

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