Abstract

A new algorithm has been proposed for the calculation of the electric field intensity in stratified multilayered films when light is incident on the system. The algorithm utilizes matrix formulas based on Abeles’s formulas for the calculation of reflectance and transmittance. Equations for calculating patial absorptance due to a certain depth in the films are also derived. Some examples of the application of the electric field description are given for the analysis of three kinds of reflection spectroscopic methods which use metal surfaces: reflection–absorption, surface electromagnetic wave, and metal overlayer ATR methods. The algorithm given here offers a useful tool in understanding the mechanism of light absorption in various spectroscopic methods, and is convenient to use where intensity of the IR spectrum is of interest.

© 1990 Optical Society of America

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References

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  1. T. C. Fry, “Plane waves of light. III: Absorption by Metals,” J. Opt. Soc. Am. 22, 307–332 (1932).
    [CrossRef]
  2. S. A. Francis, A. H. Ellison, “Infrared Spectra of Monolayers on Metal Mirrors,” J. Opt. Soc. Am. 49, 131–138 (1959).
    [CrossRef]
  3. N. J. Harrick, “Electric Field Strengths at Totally Reflecting Interface,” J. Opt. Soc. Am. 55, 851–857 (1965).
    [CrossRef]
  4. K. Ohta, R. Iwamoto, “Experimental Proof of the Relation Between Thickness of the Probed Surface Layer and Absorbance in FT-IR-ATR Spectroscopy,” Appl. Spectrosc. 39, 418–425 (1985).
    [CrossRef]
  5. W. N. Hansen, “Internal Reflection Spectroscopy in Electrochemistry,” in Advances in Electrochemistry and Electrochemical Engineering, Volume 9, P. Delahay, C. W. Tobias, Eds. (Wiley, New York, 1973), pp. 1–60.
  6. J. D. E. McIntyre, “Specular Reflection Spectroscopy of the Electrode–Solution Interphase,” in Advances in Electrochemistry and Electrochemical Engineering, Volume 9, P. Delahay, C. W. Tobias Eds. (Wiley, New York, 1973), pp. 61–166.
  7. N. M. Bashara, D. W. Peterson, “Ellipsometer Study of Anomalous Absorption in Very Thin Dielectric Films on Evaporated Metals,” J. Opt. Soc. Am. 56, 1320–1331 (1966).
    [CrossRef]
  8. M. R. Philpott, J. D. Swalen, “Exciton Surface Polaritons on Organic Crystals,” J. Chem. Phys. 69, 2912–2921 (1978).
    [CrossRef]
  9. A. Brillante, M. R. Philpott, I. Pockrand, “Experimental and Theoretical Study of Exciton Surface Polaritons on Organic Crystals. I. (010) Face of TCNQ0 Single Crystals,” J. Chem. Phys. 70, 5739–5746 (1979).
    [CrossRef]
  10. M. Ohsawa, W. Kusakari, W. Suetaka, “Raman Spectra and Molecular Orientation of 7,7′, 8,8′-Tetracyanoquinodimethane in Thin Films Evaporated on Aluminum,” Spectrochim. Acta 36A, 389–396 (1980).
  11. M. Osawa, M. Kuramitsu, A. Hatta, W. Suetaka, H. Seki, “Electromagnetic Effect in Enhanced Infrared Absorption of Adsorbed Molecules on Thin Metal Films,” Sur. Sci. 175, L787–L793 (1986).
    [CrossRef]
  12. Y. Ishino, H. Ishida, “Grazing Angle Metal Overlayer Infrared ATR Spectrscopy,” Appl. Spectrosc. 42, 1296–1302 (1988).
    [CrossRef]
  13. Y. Ishino, H. Ishida, “Fourier Transform Infrared Surface Electromagnetic Spectroscopy of Polymer Thin Films on Metallic Substrate,” Anal. Chem. 58, 2448–2453 (1986).
    [CrossRef]
  14. Y. Ishino, H. Ishida, “Spectral Simulation of Infrared Surface Electromagnetic Wave Spectroscopy,” Sur. Sci., in press (1988).
  15. W. N. Hansen, “Electric Fields Produced by the Propagation of Plane Coherent Electromagnetic Radiation in a Stratified Medium,” J. Opt. Soc. Am. 58, 380–390 (1968).
    [CrossRef]
  16. F. Abeles, “Recherches sur la Propagation des Ondes Electromagnetiques Sinusoidales dans les Milieux Stratifies. Application aux Couches Minces,” Ann. Phys. (Paris) 5, 596–640 (1950).
  17. F. Abeles, “Sur la Propagation des Ondes Electromagnetiques dans les Milieux Stratifies,” Ann. Phys. (Paris) 3, 504–520 (1948).
  18. K. Ohta, H. Ishida, “Matrix Formulation for Calculation of Light Beam Intensity in Stratified Multilayered Films and its Application to the Analysis of Emission Spectra,” Appl. Opt. (in press).
  19. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4.
  20. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.
  21. J. D. Swalen, J. F. Rabolt, “Characterization of Orientation and Lateral Order in Thin Films by Fourier Transform Infrared Spectroscopy,” in Fourier Transform Infrared Spectroscopy, Volume 4, J. R. Ferraro, L. J. Basile Eds. (Academic, Orlando, FL, 1985), pp. 283–314.
  22. J. A. Johnson, D. W. Peterson, “The Relative Electric Fields in a Thin Film on an Opaque Substrate,” Sur. Sci. 16, 217–220 (1969); and P. H. Lissberger, “The Relationship Between Optical Absorptance and Electric Field of the Radiation in Multilayer Thin Films,” Opt. Acta 28, 187–200 (1981).
    [CrossRef]
  23. D. J. Carlsson, D. M. Wiles, “Photooxidation of Polypropylene Films. IV. Surface Changes Studied by Attenuated Total Reflection Spectroscopy,” Macromolecules 4, 174–179 (1971).
    [CrossRef]
  24. H. G. Tompkins, “The Physical Basis for Analysis of the Depth of Absorbing Species Using Internal Reflection Spectroscopy,” Appl. Spectrosc. 28, 335–341 (1974).
    [CrossRef]
  25. S. D. Stuchebryukov, A. A. Vavkushevskii, V. M. Rudoi, “Frustrated Total Internal Reflection Spectroscopy of Films with Absorption Gradients,” Sov. Phys. Dokl. 27, 931–932 (1982).

1988 (1)

1986 (2)

M. Osawa, M. Kuramitsu, A. Hatta, W. Suetaka, H. Seki, “Electromagnetic Effect in Enhanced Infrared Absorption of Adsorbed Molecules on Thin Metal Films,” Sur. Sci. 175, L787–L793 (1986).
[CrossRef]

Y. Ishino, H. Ishida, “Fourier Transform Infrared Surface Electromagnetic Spectroscopy of Polymer Thin Films on Metallic Substrate,” Anal. Chem. 58, 2448–2453 (1986).
[CrossRef]

1985 (1)

1982 (1)

S. D. Stuchebryukov, A. A. Vavkushevskii, V. M. Rudoi, “Frustrated Total Internal Reflection Spectroscopy of Films with Absorption Gradients,” Sov. Phys. Dokl. 27, 931–932 (1982).

1980 (1)

M. Ohsawa, W. Kusakari, W. Suetaka, “Raman Spectra and Molecular Orientation of 7,7′, 8,8′-Tetracyanoquinodimethane in Thin Films Evaporated on Aluminum,” Spectrochim. Acta 36A, 389–396 (1980).

1979 (1)

A. Brillante, M. R. Philpott, I. Pockrand, “Experimental and Theoretical Study of Exciton Surface Polaritons on Organic Crystals. I. (010) Face of TCNQ0 Single Crystals,” J. Chem. Phys. 70, 5739–5746 (1979).
[CrossRef]

1978 (1)

M. R. Philpott, J. D. Swalen, “Exciton Surface Polaritons on Organic Crystals,” J. Chem. Phys. 69, 2912–2921 (1978).
[CrossRef]

1974 (1)

1971 (1)

D. J. Carlsson, D. M. Wiles, “Photooxidation of Polypropylene Films. IV. Surface Changes Studied by Attenuated Total Reflection Spectroscopy,” Macromolecules 4, 174–179 (1971).
[CrossRef]

1969 (1)

J. A. Johnson, D. W. Peterson, “The Relative Electric Fields in a Thin Film on an Opaque Substrate,” Sur. Sci. 16, 217–220 (1969); and P. H. Lissberger, “The Relationship Between Optical Absorptance and Electric Field of the Radiation in Multilayer Thin Films,” Opt. Acta 28, 187–200 (1981).
[CrossRef]

1968 (1)

1966 (1)

1965 (1)

1959 (1)

1950 (1)

F. Abeles, “Recherches sur la Propagation des Ondes Electromagnetiques Sinusoidales dans les Milieux Stratifies. Application aux Couches Minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

1948 (1)

F. Abeles, “Sur la Propagation des Ondes Electromagnetiques dans les Milieux Stratifies,” Ann. Phys. (Paris) 3, 504–520 (1948).

1932 (1)

Abeles, F.

F. Abeles, “Recherches sur la Propagation des Ondes Electromagnetiques Sinusoidales dans les Milieux Stratifies. Application aux Couches Minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

F. Abeles, “Sur la Propagation des Ondes Electromagnetiques dans les Milieux Stratifies,” Ann. Phys. (Paris) 3, 504–520 (1948).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.

Bashara, N. M.

Brillante, A.

A. Brillante, M. R. Philpott, I. Pockrand, “Experimental and Theoretical Study of Exciton Surface Polaritons on Organic Crystals. I. (010) Face of TCNQ0 Single Crystals,” J. Chem. Phys. 70, 5739–5746 (1979).
[CrossRef]

Carlsson, D. J.

D. J. Carlsson, D. M. Wiles, “Photooxidation of Polypropylene Films. IV. Surface Changes Studied by Attenuated Total Reflection Spectroscopy,” Macromolecules 4, 174–179 (1971).
[CrossRef]

Ellison, A. H.

Francis, S. A.

Fry, T. C.

Hansen, W. N.

W. N. Hansen, “Electric Fields Produced by the Propagation of Plane Coherent Electromagnetic Radiation in a Stratified Medium,” J. Opt. Soc. Am. 58, 380–390 (1968).
[CrossRef]

W. N. Hansen, “Internal Reflection Spectroscopy in Electrochemistry,” in Advances in Electrochemistry and Electrochemical Engineering, Volume 9, P. Delahay, C. W. Tobias, Eds. (Wiley, New York, 1973), pp. 1–60.

Harrick, N. J.

Hatta, A.

M. Osawa, M. Kuramitsu, A. Hatta, W. Suetaka, H. Seki, “Electromagnetic Effect in Enhanced Infrared Absorption of Adsorbed Molecules on Thin Metal Films,” Sur. Sci. 175, L787–L793 (1986).
[CrossRef]

Heavens, O. S.

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

Ishida, H.

Y. Ishino, H. Ishida, “Grazing Angle Metal Overlayer Infrared ATR Spectrscopy,” Appl. Spectrosc. 42, 1296–1302 (1988).
[CrossRef]

Y. Ishino, H. Ishida, “Fourier Transform Infrared Surface Electromagnetic Spectroscopy of Polymer Thin Films on Metallic Substrate,” Anal. Chem. 58, 2448–2453 (1986).
[CrossRef]

K. Ohta, H. Ishida, “Matrix Formulation for Calculation of Light Beam Intensity in Stratified Multilayered Films and its Application to the Analysis of Emission Spectra,” Appl. Opt. (in press).

Y. Ishino, H. Ishida, “Spectral Simulation of Infrared Surface Electromagnetic Wave Spectroscopy,” Sur. Sci., in press (1988).

Ishino, Y.

Y. Ishino, H. Ishida, “Grazing Angle Metal Overlayer Infrared ATR Spectrscopy,” Appl. Spectrosc. 42, 1296–1302 (1988).
[CrossRef]

Y. Ishino, H. Ishida, “Fourier Transform Infrared Surface Electromagnetic Spectroscopy of Polymer Thin Films on Metallic Substrate,” Anal. Chem. 58, 2448–2453 (1986).
[CrossRef]

Y. Ishino, H. Ishida, “Spectral Simulation of Infrared Surface Electromagnetic Wave Spectroscopy,” Sur. Sci., in press (1988).

Iwamoto, R.

Johnson, J. A.

J. A. Johnson, D. W. Peterson, “The Relative Electric Fields in a Thin Film on an Opaque Substrate,” Sur. Sci. 16, 217–220 (1969); and P. H. Lissberger, “The Relationship Between Optical Absorptance and Electric Field of the Radiation in Multilayer Thin Films,” Opt. Acta 28, 187–200 (1981).
[CrossRef]

Kuramitsu, M.

M. Osawa, M. Kuramitsu, A. Hatta, W. Suetaka, H. Seki, “Electromagnetic Effect in Enhanced Infrared Absorption of Adsorbed Molecules on Thin Metal Films,” Sur. Sci. 175, L787–L793 (1986).
[CrossRef]

Kusakari, W.

M. Ohsawa, W. Kusakari, W. Suetaka, “Raman Spectra and Molecular Orientation of 7,7′, 8,8′-Tetracyanoquinodimethane in Thin Films Evaporated on Aluminum,” Spectrochim. Acta 36A, 389–396 (1980).

McIntyre, J. D. E.

J. D. E. McIntyre, “Specular Reflection Spectroscopy of the Electrode–Solution Interphase,” in Advances in Electrochemistry and Electrochemical Engineering, Volume 9, P. Delahay, C. W. Tobias Eds. (Wiley, New York, 1973), pp. 61–166.

Ohsawa, M.

M. Ohsawa, W. Kusakari, W. Suetaka, “Raman Spectra and Molecular Orientation of 7,7′, 8,8′-Tetracyanoquinodimethane in Thin Films Evaporated on Aluminum,” Spectrochim. Acta 36A, 389–396 (1980).

Ohta, K.

K. Ohta, R. Iwamoto, “Experimental Proof of the Relation Between Thickness of the Probed Surface Layer and Absorbance in FT-IR-ATR Spectroscopy,” Appl. Spectrosc. 39, 418–425 (1985).
[CrossRef]

K. Ohta, H. Ishida, “Matrix Formulation for Calculation of Light Beam Intensity in Stratified Multilayered Films and its Application to the Analysis of Emission Spectra,” Appl. Opt. (in press).

Osawa, M.

M. Osawa, M. Kuramitsu, A. Hatta, W. Suetaka, H. Seki, “Electromagnetic Effect in Enhanced Infrared Absorption of Adsorbed Molecules on Thin Metal Films,” Sur. Sci. 175, L787–L793 (1986).
[CrossRef]

Peterson, D. W.

J. A. Johnson, D. W. Peterson, “The Relative Electric Fields in a Thin Film on an Opaque Substrate,” Sur. Sci. 16, 217–220 (1969); and P. H. Lissberger, “The Relationship Between Optical Absorptance and Electric Field of the Radiation in Multilayer Thin Films,” Opt. Acta 28, 187–200 (1981).
[CrossRef]

N. M. Bashara, D. W. Peterson, “Ellipsometer Study of Anomalous Absorption in Very Thin Dielectric Films on Evaporated Metals,” J. Opt. Soc. Am. 56, 1320–1331 (1966).
[CrossRef]

Philpott, M. R.

A. Brillante, M. R. Philpott, I. Pockrand, “Experimental and Theoretical Study of Exciton Surface Polaritons on Organic Crystals. I. (010) Face of TCNQ0 Single Crystals,” J. Chem. Phys. 70, 5739–5746 (1979).
[CrossRef]

M. R. Philpott, J. D. Swalen, “Exciton Surface Polaritons on Organic Crystals,” J. Chem. Phys. 69, 2912–2921 (1978).
[CrossRef]

Pockrand, I.

A. Brillante, M. R. Philpott, I. Pockrand, “Experimental and Theoretical Study of Exciton Surface Polaritons on Organic Crystals. I. (010) Face of TCNQ0 Single Crystals,” J. Chem. Phys. 70, 5739–5746 (1979).
[CrossRef]

Rabolt, J. F.

J. D. Swalen, J. F. Rabolt, “Characterization of Orientation and Lateral Order in Thin Films by Fourier Transform Infrared Spectroscopy,” in Fourier Transform Infrared Spectroscopy, Volume 4, J. R. Ferraro, L. J. Basile Eds. (Academic, Orlando, FL, 1985), pp. 283–314.

Rudoi, V. M.

S. D. Stuchebryukov, A. A. Vavkushevskii, V. M. Rudoi, “Frustrated Total Internal Reflection Spectroscopy of Films with Absorption Gradients,” Sov. Phys. Dokl. 27, 931–932 (1982).

Seki, H.

M. Osawa, M. Kuramitsu, A. Hatta, W. Suetaka, H. Seki, “Electromagnetic Effect in Enhanced Infrared Absorption of Adsorbed Molecules on Thin Metal Films,” Sur. Sci. 175, L787–L793 (1986).
[CrossRef]

Stuchebryukov, S. D.

S. D. Stuchebryukov, A. A. Vavkushevskii, V. M. Rudoi, “Frustrated Total Internal Reflection Spectroscopy of Films with Absorption Gradients,” Sov. Phys. Dokl. 27, 931–932 (1982).

Suetaka, W.

M. Osawa, M. Kuramitsu, A. Hatta, W. Suetaka, H. Seki, “Electromagnetic Effect in Enhanced Infrared Absorption of Adsorbed Molecules on Thin Metal Films,” Sur. Sci. 175, L787–L793 (1986).
[CrossRef]

M. Ohsawa, W. Kusakari, W. Suetaka, “Raman Spectra and Molecular Orientation of 7,7′, 8,8′-Tetracyanoquinodimethane in Thin Films Evaporated on Aluminum,” Spectrochim. Acta 36A, 389–396 (1980).

Swalen, J. D.

M. R. Philpott, J. D. Swalen, “Exciton Surface Polaritons on Organic Crystals,” J. Chem. Phys. 69, 2912–2921 (1978).
[CrossRef]

J. D. Swalen, J. F. Rabolt, “Characterization of Orientation and Lateral Order in Thin Films by Fourier Transform Infrared Spectroscopy,” in Fourier Transform Infrared Spectroscopy, Volume 4, J. R. Ferraro, L. J. Basile Eds. (Academic, Orlando, FL, 1985), pp. 283–314.

Tompkins, H. G.

Vavkushevskii, A. A.

S. D. Stuchebryukov, A. A. Vavkushevskii, V. M. Rudoi, “Frustrated Total Internal Reflection Spectroscopy of Films with Absorption Gradients,” Sov. Phys. Dokl. 27, 931–932 (1982).

Wiles, D. M.

D. J. Carlsson, D. M. Wiles, “Photooxidation of Polypropylene Films. IV. Surface Changes Studied by Attenuated Total Reflection Spectroscopy,” Macromolecules 4, 174–179 (1971).
[CrossRef]

Anal. Chem. (1)

Y. Ishino, H. Ishida, “Fourier Transform Infrared Surface Electromagnetic Spectroscopy of Polymer Thin Films on Metallic Substrate,” Anal. Chem. 58, 2448–2453 (1986).
[CrossRef]

Ann. Phys. (Paris) (2)

F. Abeles, “Recherches sur la Propagation des Ondes Electromagnetiques Sinusoidales dans les Milieux Stratifies. Application aux Couches Minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

F. Abeles, “Sur la Propagation des Ondes Electromagnetiques dans les Milieux Stratifies,” Ann. Phys. (Paris) 3, 504–520 (1948).

Appl. Spectrosc. (3)

J. Chem. Phys. (2)

M. R. Philpott, J. D. Swalen, “Exciton Surface Polaritons on Organic Crystals,” J. Chem. Phys. 69, 2912–2921 (1978).
[CrossRef]

A. Brillante, M. R. Philpott, I. Pockrand, “Experimental and Theoretical Study of Exciton Surface Polaritons on Organic Crystals. I. (010) Face of TCNQ0 Single Crystals,” J. Chem. Phys. 70, 5739–5746 (1979).
[CrossRef]

J. Opt. Soc. Am. (5)

Macromolecules (1)

D. J. Carlsson, D. M. Wiles, “Photooxidation of Polypropylene Films. IV. Surface Changes Studied by Attenuated Total Reflection Spectroscopy,” Macromolecules 4, 174–179 (1971).
[CrossRef]

Sov. Phys. Dokl. (1)

S. D. Stuchebryukov, A. A. Vavkushevskii, V. M. Rudoi, “Frustrated Total Internal Reflection Spectroscopy of Films with Absorption Gradients,” Sov. Phys. Dokl. 27, 931–932 (1982).

Spectrochim. Acta (1)

M. Ohsawa, W. Kusakari, W. Suetaka, “Raman Spectra and Molecular Orientation of 7,7′, 8,8′-Tetracyanoquinodimethane in Thin Films Evaporated on Aluminum,” Spectrochim. Acta 36A, 389–396 (1980).

Sur. Sci. (2)

M. Osawa, M. Kuramitsu, A. Hatta, W. Suetaka, H. Seki, “Electromagnetic Effect in Enhanced Infrared Absorption of Adsorbed Molecules on Thin Metal Films,” Sur. Sci. 175, L787–L793 (1986).
[CrossRef]

J. A. Johnson, D. W. Peterson, “The Relative Electric Fields in a Thin Film on an Opaque Substrate,” Sur. Sci. 16, 217–220 (1969); and P. H. Lissberger, “The Relationship Between Optical Absorptance and Electric Field of the Radiation in Multilayer Thin Films,” Opt. Acta 28, 187–200 (1981).
[CrossRef]

Other (7)

K. Ohta, H. Ishida, “Matrix Formulation for Calculation of Light Beam Intensity in Stratified Multilayered Films and its Application to the Analysis of Emission Spectra,” Appl. Opt. (in press).

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

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.

J. D. Swalen, J. F. Rabolt, “Characterization of Orientation and Lateral Order in Thin Films by Fourier Transform Infrared Spectroscopy,” in Fourier Transform Infrared Spectroscopy, Volume 4, J. R. Ferraro, L. J. Basile Eds. (Academic, Orlando, FL, 1985), pp. 283–314.

W. N. Hansen, “Internal Reflection Spectroscopy in Electrochemistry,” in Advances in Electrochemistry and Electrochemical Engineering, Volume 9, P. Delahay, C. W. Tobias, Eds. (Wiley, New York, 1973), pp. 1–60.

J. D. E. McIntyre, “Specular Reflection Spectroscopy of the Electrode–Solution Interphase,” in Advances in Electrochemistry and Electrochemical Engineering, Volume 9, P. Delahay, C. W. Tobias Eds. (Wiley, New York, 1973), pp. 61–166.

Y. Ishino, H. Ishida, “Spectral Simulation of Infrared Surface Electromagnetic Wave Spectroscopy,” Sur. Sci., in press (1988).

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Figures (8)

Fig. 1
Fig. 1

Coordinate system and notation of the number of the films for the present stratified multilayered film system. The film structure is sandwiched between the incident (0) and substrate (m + 1) phases.

Fig. 2
Fig. 2

Angular dependence of the electric field intensity Fz(z) at the surface of the metal for the case of reflection-absorption spectroscopy. The parameters used in the calculation are as follows: n ^ 0 = 1.0, n ^ 1 = 1.0 (a), 1.5 (b), and 1.5 + 0.5i (c), n ^ 2 = 3.0 + 30.0i, h1 = 0.01 μm, ν = 1000 cm−1.

Fig. 3
Fig. 3

Depth profile of the electric field intensity Fz(z) at the incident angle of 75° for the case of reflection-absorption spectroscopy. The parameters used in the calculation are as follows: n ^ 0 = 1.0, n ^ 1 = 1.0 (a), 1.5 (b), and 1.5 + 0.5i (c), n ^ 2 = 3.0 + 30.0i, h1 = 0.01 μm, ν = 1000 cm−1.

Fig. 4
Fig. 4

Angular dependence of the electric field intensity Fz(z) at the surface of the metal for the case of surface electromagnetic wave spectroscopy. The parameters used in the calculation are as follows: n ^ 0 = 2.4, n ^ 1 = 1.0, n ^ 2 = 1.0 (a), 1.5 (b), and 1.5 + 0.5i (c), n ^ 3 = 3.0 + 30.0i, h1 = 45.76 μm (a), 44.22 μm (b), and 38.60 μm (c), h2 = 0.01 μm, ν = 1000 cm−1.

Fig. 5
Fig. 5

Depth profile of the electric field intensity Fz(z) at the incident angle of 24.622° (a) 24.626° (b), and 24.619° (c) for the case of surface electromagnetic wave spectroscopy. The parameters used in the calculation are as follows: n ^ 0 = 2.4, n ^ 2 = 1.0 (a), 1.5 (b), and 1.5 + 0.5i (c), n ^ 3 = 3.0 + 30.0i, h1 = 45.76 μm (a), 44.22 μm (b), and 38.60 μm (c), h2 = 0.01 μm, ν = 1000 cm−1. In this figure, the film thickness for (b) or (c) is so small that the electric field profile within the film cannot be seen. See Fig. 6.

Fig. 6
Fig. 6

Expanded depth profile of the electric field intensity Fz(z) around the sample film. The parameters used in the calculation are the same as those in Fig. 5.

Fig. 7
Fig. 7

Angular dependence of the electric field intensity Fz(z) at the surface of the metal for the case of metal overlayer ATR spectroscopy. The parameters used in the calculation are as follows: n ^ 0 = 4.0, n ^ 1 = 1.0 (a), 1.5 (b), and 1.5 + 0.5i (c), n ^ 2 = 3.0 + 30.0i, h1 = 0.01 μm, ν = 1000 cm−1.

Fig. 8
Fig. 8

Depth profile of the electric field intensity Fz(z) at the incidence angle of 75° for the case of metal overlayer ATR spectroscopy. The parameters used in the calculation are as follows: n ^ 0 = 4.0, n ^ 1 = 1.0 (a), 1.5 (b), and 1.5 + 0.5i (c), n ^ 2 = 3.0 + 30.0i, h1 = 0.01 μm, ν = 1000 cm−1.

Tables (3)

Tables Icon

Table I Explicit Formulas of Di Matrices for the Cases of Three-Phase System.

Tables Icon

Table II Explicit Formulas of the Relationships Between E 0 + and E j ± for the Case of Three-Phase System.

Tables Icon

Table III Explicit Formulas of Each Component of Electric Field F(z) for the Case of Three-Phase System.

Equations (53)

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n ^ j = n j + i k j .
( E 0 + E 0 - ) = C 1 C 2 C m + 1 t 1 t 2 t m + 1 ( E m + 1 + E m + 1 - ) .
C j = ( exp ( - i δ j - 1 ) r j exp ( - i δ j - 1 ) r j exp ( i δ j - 1 ) exp ( i δ j - 1 ) ) ,
r j p = n ^ j - 1 cos θ j - n ^ j cos θ j - 1 n ^ j - 1 cos θ j + n ^ j cos θ j - 1
t j p = 2 n ^ j - 1 cos θ j - 1 n ^ j - 1 cos θ j + n ^ j cos θ j - 1
r j s = n ^ j - 1 cos θ j - 1 - n ^ j cos θ j n ^ j - 1 cos θ j - 1 + n ^ j cos θ j
t j s = 2 n ^ j - 1 cos θ j - 1 n ^ j - 1 cos θ j - 1 + n ^ j cos θ j .
n ^ j - 1 sin θ j - 1 = n ^ j sin θ j             ( j = 1 , 2 , , m + 1 )
δ 0 = 0
δ j - 1 = 2 π ν n ^ j - 1 cos θ j - 1 h j - 1
r = E 0 - E 0 + = c a
t = E m + 1 + E 0 + = t 1 t 2 t m + 1 a ,
C 1 C 2 C 3 C m + 1 = ( a b c d ) .
R = r 2
T s = Re ( n ^ m + 1 cos θ m + 1 n ^ 0 cos θ 0 ) t s 2 ,
T P = Re ( cos θ m + 1 / n ^ m + 1 cos θ 0 / n ^ 0 ) | n ^ m + 1 n ^ 0 t p | 2 , = Re ( n ^ m + 1 * cos θ m + 1 n ^ 0 * cos θ 0 ) t p 2 ,
( E 0 + E 0 - ) = C 1 C 2 C j t 1 t 2 t j ( E j + E j - )
( E j + E j - ) = C j + 1 C j + 2 C m + 1 t j + 1 t j + 2 t m + 1 ( E m + 1 + E m + 1 - ) .
D j = C j + 1 C j + 2 C m + 1 = ( a j b j c j d j ) .
E j + = a j t j + 1 t j + 2 t m + 1 E m + 1 +
E j - = c j t j + 1 t j + 2 t m + 1 E m + 1 + .
E j + = t 1 t 2 t j a j a E 0 +
E j - = t 1 t 2 t j c j a E 0 +
D 0 = C 1 C 2 C m + 1 = ( a b c d ) .
D m + 1 = ( 1 0 0 1 ) .
E m + 1 - = 0 ,
E + ( z ) = E j + exp ( i K z j Δ z ) ,
E - ( z ) = E j - exp ( - i K z j Δ z ) ,
K z j = 2 π ν n ^ j cos θ j
Δ z = z - i = 1 j - 1 h i .
E x ( z ) = [ E p + ( z ) - E p - ( z ) ] cos θ j
E y ( z ) = E s + ( z ) + E s - ( z )
E z ( z ) = [ E p + ( z ) + E p - ( z ) ] sin θ j .
F x ( z ) = E x ( z ) 2 / E op + 2
F y ( z ) = E y ( z ) 2 / E os + 2
F y ( z ) = E z ( z ) 2 / E op + 2
F s ( z ) = F y ( z )
F p ( z ) = F x ( z ) + F z ( z ) .
X = E op + 2 / ( E op + 2 + E os + 2 ) ,
1 - X = E os + 2 / ( E op + 2 + E os + 2 ) ,
F ( z ) = [ E x ( z ) 2 + E y ( z ) 2 + E z ( z ) 2 ] / [ E op + 2 + E os + 2 ) = X F p ( z ) = ( 1 - X ) F s ( z ) = X [ F x ( z ) + F z ( z ) ] + ( 1 - X ) F y ( z ) .
F ( z ) = [ F x ( z ) + F y ( z ) + F z ( z ) ] / 2.
β 2 = 2 Im ( K z 2 ) = 4 π ν Im ( n ^ 2 cos θ 2 ) ,
A = 1 - R - T ,
A = 1 - R ,
d A = q j F ( z ) d z ,
q j = Re ( n ^ j cos θ j n ^ 0 cos θ 0 ) β j
β j = 2 Im ( K z j ) = 4 π ν Im ( n ^ j cos θ j ) .
A ( z 1 < z < z 2 ) = z 1 z 2 q j F ( z ) d z .
q j = n j α j n 0 cos θ 0
α j = 4 π ν k j ,
A ( z 1 < z < z 2 ) = 4 π ν n 0 cos θ 0 z 1 z 2 n ( z ) k ( z ) F 0 ( z ) d z ,
I ( z ) = Re ( n ^ j cos θ j n ^ 0 cos θ 0 ) F ( z ) .

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