Abstract

A method is described for measuring the absolute phase change on reflection of semitransparent films which is both precise and accurate. The films are deposited on portions of two fuzed-quartz optical flats and the shift in the fringes of equal chromatic order between the coated and uncoated portions of the interferometer is measured. Since the phase change is very sensitive to small changes in the optical constants, this method is useful for studying the effects of aging, applied electromagnetic fields, oxide growth, and other factors. Also, since areas of the order of 1.3 by 0.0033 mm are sufficient for each measurement, the phase change can be used to study possible variations in the film structure over the interferometer surface. Measured values of phase change on reflection versus wavelength are smooth to ±0.1°. When systematic errors have been taken into account, the measurements are still accurate to about ±1°.

© 1964 Optical Society of America

Full Article  |  PDF Article

Corrections

J. M. Bennett, "Errata: Precise Method for Measuring the Absolute Phase Change on Reflection," J. Opt. Soc. Am. 56, 409_2-409 (1966)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-56-3-409_2

References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worths Scientific Publications Ltd., London, 1955), p. 174.
  2. O. Wiener, Ann. Physik 31, 629 (1887).
    [Crossref]
  3. R. C. Faust, Phil. Mag. 41, 1238 (1950).
  4. L. G. Schulz and E. J. Scheibner, J. Opt. Soc. Am. 40, 761 (1950).
    [Crossref]
  5. L. G. Schulz, J. Opt. Soc. Am. 41, 261, 1047 (1951);J. Opt. Soc. Am.,  44, 357 (1954).
    [Crossref]
  6. S. Nawata, Sci. Rep. Res. Inst. Tohoku Univ. Ser. A3, 740 (1951).
  7. R. Fleischmann and H. Schopper, Z. Physik 129, 285 (1951).
    [Crossref]
  8. G. Dornenburg and R. Fleischmann, Z. Physik 129, 300 (1951).
    [Crossref]
  9. H. Schopper, Z. Physik 130, 427, 565 (1951);Z. Physik 131, 215 (1952);Z. Physik 135, 516(1953).
    [Crossref]
  10. R. Philip, Compt. Rend. 241, 559, 596 (1955);Compt. Rend. 243, 365 (1956).
  11. H. J. Bolle, Z. Physik 143, 538 (1956).
    [Crossref]
  12. I. N. Shklyarevskii, Soviet Phys.—Tech. Phys. 1, 327 (1956)[Zh. Tekhn. Fiz. 26, 333 (1956)].
  13. I. N. Shklyarevskii and A. N. Ryazanov, Opt. i Spektroskopiya 2, 645 (1957).
  14. M. P. Lisitsa and N. G. Tsvelykh, Opt. i Spcktroskopiya 2, 674 (1957).
  15. C. Weaver, R. M. Hill, and J. E. S. Macleod, J. Opt. Soc. Am. 49, 992 (1959).
    [Crossref]
  16. A. Heisen, Optik 18, 27 (1961).
  17. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 279.
  18. Four other forms of the complex refractive index are in current use in the literature: n+ik (Ref. 4), n(1−ik) (Ref. 19), n(1+iκ) (Ref. 20) and n(1−iκ) (Ref. 21), where k represents the attenuation of the wave per vacuum wavelength and κ is related in the same way to the attenuation per wavelength in the medium. Hence the expressions for r in various references may actually be identical although they appear to be different. Care should be taken to determine which form of the complex refractive index is used when applying equations for the reflectance and the phase change on reflection.
  19. See Ref. 1, p. 49.
  20. See Ref. 17, p. 610.
  21. R. W. Ditchburn, Light (Blackie & Son Ltd., London, 1952), p. 441.
  22. Care should be taken not to confuse this positive β with Heavens’ “phase advance” (Ref. 1, p. 171). His “phase advance” and “phase retard” are measured relative to the negative real axis so that values of Er in the second quadrant are called “retard” and those in the third quadrant “advance.”
  23. See Ref. 1, pp. 55–88.
  24. This expression may be obtained from the relations given on p. 91 of Ref. 1. However, there is a typographical error in Eq. 4(140). The correct expression should read Δ0=arc tan(BC−AD)/(AC+BD) where Δ0 is identical to β.
  25. S. Tolansky, Multiple-Beam Inlerferometry of Surfaces and Films (Clarendon Press, Oxford, England, 1948), p. 9.
  26. W. E. Williams, Applications of Inlerferometry (Methuen and Company Ltd., London, 1930), p. 77.
  27. K. W. Meissner, J. Opt. Soc. Am. 31, 405 (1941).
    [Crossref]
  28. F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1957), 3rd ed., p. 273.
  29. By one reflecting surface is meant the inner-coated surface of one interferometer plate. The light lost by reflection from the outer surfaces of the plates is ignored since it only decreases the total intensity, and the change in the shape of the transmitted energy curves is negligible. It is also assumed that the two inner-coated surfaces have nearly the same values of T and R.
  30. The minus sign in Eq. (10) is absolutely necessary to be consistent with the previous definition of β as a positive angle. Since the optical path is shorter than the geometrical path, this is equivalent to having the wave reflected from a fictitious surface in front of the actual surface.
  31. This apparent paradox was first noted and explained by D. H. Rank and H. E. Bennett, J. Opt. Soc. Am. 45, 69 (1955).
    [Crossref]
  32. See Ref. 26, p. 83.
  33. D. H. Rank, E. R. Shull, J. M. Bennett, and T. A. Wiggins, J. Opt. Soc. Am. 43, 952 (1953).
    [Crossref]
  34. See Ref. 27, p. 423.
  35. W. F. Meggers, Natl. Bur. Std. (U.S.) Bureau of Standards, Bull. 12, 198 (1915).
  36. See Ref. 25, p. 97.
  37. These fringes are similar to channelled spectra (Ref. 38), Edser–Butler fringes, or white light Fabry–Perot fringes (Ref. 36) except that an image of the interferometer (rather than the light source) is focused on the slit of the spectrograph.
  38. See Ref. 26, p.101.
  39. It is assumed that the wavelengths of the two-beam reflection minima are identical to the wavelengths of the maxima in transmission.
  40. Equation (13) is strictly correct for an air–dielectric reflection. It also holds approximately for interferometers coated with some metal films. However, it does not hold for all metal films or for dielectric coatings.
  41. This method is similar to that used by Shklyarevskii (Ref. 42) except that the integers he used for his Mλ products were one less than those used here. Both Koehler (Ref. 43) and Koester (Ref. 44) have shown that τ may be determined exactly regardless of the choice of the integer if the phase change is properly eliminated. Furthermore, Koester proves that by using a graphical method similar to the one described here (his integers differ from the ones in the present paper by an arbitrary integer N), “the difference between the two curves is a constant. Regardless of the selection of N and the variation of phase shift with wavelength the difference between the curves is twice the step height ….” The integers M used in this paper are purposely chosen to give the most nearly constant Mλ product so that the graphical method will have the most accuracy.
  42. I. N. Shklyarevskii, Opt. i Spektroskopiya 5, 617 (1958).
  43. W. F. Koehler, J. Opt. Soc. Am. 45, 934 (1955).
    [Crossref]
  44. C. J. Koester, J. Opt. Soc. Am. 48, 255 (1958).
    [Crossref]
  45. G. D. Scott, T. A. McLauchlan, and R. S. Sennett, J. Appl. Phys. 21, 843 (1950).
    [Crossref]
  46. This shift of approximately one order toward the blue (higher orders of interference) in the multiple-beam fringe system in reflection relative to the two-beam fringe system is observed when the reflecting film on the side of the incident beam is a metal. This same effect has also been observed by Faust (Ref. 3) for silver and aluminum but is not completely understood.
  47. Johnson Matthey JM 50 silver rod obtained from Jarrell-Ash Company.
  48. Obtained from the Aluminum Company of America.
  49. W. F. Koehler, J. Opt. Soc. Am. 43, 743 (1953);J. Opt. Soc. Am. 45, 1015 (1955).
    [Crossref]
  50. H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
    [Crossref]
  51. H. E. Bennett, J. M. Bennett, and E. J. Ashley, J. Opt. Soc. Am. 52, 1245 (1962).
    [Crossref]
  52. Vickers projection microscope manufactured by Cooke, Troughton, and Simms, Ltd., York, England.
  53. Model BK 3643 direct reading spectrograph manufactured by Bausch & Lomb Optical Company, Rochester, New York.
  54. Very fast Type C panchromatic antihalation plates obtained from Eastman Kodak Company, Rochester, New York.
  55. Corning filter No. 2404 is used in photographing the neon spectrum to eliminate the lines which obscure some of the interference fringes.
  56. A commercial model is available from the David W. Mann Company, Lincoln, Massachusetts.
  57. J. M. Bennett and W. F. Koehler, J. Opt. Soc. Am. 49, 466 (1959).
    [Crossref]
  58. W. F. Koehler and F. K. Odencrantz, J. Opt. Soc. Am. 47, 862 (1957).
    [Crossref]
  59. A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley & Sons, Inc., New York, 1943), p. 207.
  60. B. O. Seraphin, J. Opt. Soc. Am. 52, 912 (1962).
    [Crossref]
  61. Note the typesetter’s errors on p. 1246 of Faust’s paper (Ref. 3) which give Δν=12d.
  62. J. Holden, Proc. Phys. Soc. (London) 62, 405 (1949).
    [Crossref]
  63. I. N. Shklyarevskii (Ref. 42) goes through a similar type of derivation to obtain his Eq. (1) [our Eq. (15)]. However, he makes the approximation that β1/π∼β2/π∼1 (our notation) which is a much more drastic assumption than is necessary. Furthermore, he apparently was not successful with the simple formula since he states “It is easy to see that if the dispersion of the phase change is neglected, the thin film thicknesses cannot be measured with a high degree of accuracy.”
  64. T. M. Donovan, E. J. Ashley, and H. E. Bennett, J. Opt. Soc. Am. 53, 1403 (1963).
    [Crossref]

1963 (1)

1962 (2)

1961 (2)

1959 (2)

1958 (2)

C. J. Koester, J. Opt. Soc. Am. 48, 255 (1958).
[Crossref]

I. N. Shklyarevskii, Opt. i Spektroskopiya 5, 617 (1958).

1957 (3)

I. N. Shklyarevskii and A. N. Ryazanov, Opt. i Spektroskopiya 2, 645 (1957).

M. P. Lisitsa and N. G. Tsvelykh, Opt. i Spcktroskopiya 2, 674 (1957).

W. F. Koehler and F. K. Odencrantz, J. Opt. Soc. Am. 47, 862 (1957).
[Crossref]

1956 (2)

H. J. Bolle, Z. Physik 143, 538 (1956).
[Crossref]

I. N. Shklyarevskii, Soviet Phys.—Tech. Phys. 1, 327 (1956)[Zh. Tekhn. Fiz. 26, 333 (1956)].

1955 (3)

1953 (2)

1951 (4)

R. Fleischmann and H. Schopper, Z. Physik 129, 285 (1951).
[Crossref]

G. Dornenburg and R. Fleischmann, Z. Physik 129, 300 (1951).
[Crossref]

H. Schopper, Z. Physik 130, 427, 565 (1951);Z. Physik 131, 215 (1952);Z. Physik 135, 516(1953).
[Crossref]

L. G. Schulz, J. Opt. Soc. Am. 41, 261, 1047 (1951);J. Opt. Soc. Am.,  44, 357 (1954).
[Crossref]

1950 (3)

L. G. Schulz and E. J. Scheibner, J. Opt. Soc. Am. 40, 761 (1950).
[Crossref]

R. C. Faust, Phil. Mag. 41, 1238 (1950).

G. D. Scott, T. A. McLauchlan, and R. S. Sennett, J. Appl. Phys. 21, 843 (1950).
[Crossref]

1949 (1)

J. Holden, Proc. Phys. Soc. (London) 62, 405 (1949).
[Crossref]

1941 (1)

1915 (1)

W. F. Meggers, Natl. Bur. Std. (U.S.) Bureau of Standards, Bull. 12, 198 (1915).

1887 (1)

O. Wiener, Ann. Physik 31, 629 (1887).
[Crossref]

Ashley, E. J.

Bennett, H. E.

Bennett, J. M.

Bolle, H. J.

H. J. Bolle, Z. Physik 143, 538 (1956).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 279.

Ditchburn, R. W.

R. W. Ditchburn, Light (Blackie & Son Ltd., London, 1952), p. 441.

Donovan, T. M.

Dornenburg, G.

G. Dornenburg and R. Fleischmann, Z. Physik 129, 300 (1951).
[Crossref]

Faust, R. C.

R. C. Faust, Phil. Mag. 41, 1238 (1950).

Fleischmann, R.

G. Dornenburg and R. Fleischmann, Z. Physik 129, 300 (1951).
[Crossref]

R. Fleischmann and H. Schopper, Z. Physik 129, 285 (1951).
[Crossref]

Geffner, J.

A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley & Sons, Inc., New York, 1943), p. 207.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worths Scientific Publications Ltd., London, 1955), p. 174.

Heisen, A.

A. Heisen, Optik 18, 27 (1961).

Hill, R. M.

Holden, J.

J. Holden, Proc. Phys. Soc. (London) 62, 405 (1949).
[Crossref]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1957), 3rd ed., p. 273.

Koehler, W. F.

Koester, C. J.

Lisitsa, M. P.

M. P. Lisitsa and N. G. Tsvelykh, Opt. i Spcktroskopiya 2, 674 (1957).

Macleod, J. E. S.

McLauchlan, T. A.

G. D. Scott, T. A. McLauchlan, and R. S. Sennett, J. Appl. Phys. 21, 843 (1950).
[Crossref]

Meggers, W. F.

W. F. Meggers, Natl. Bur. Std. (U.S.) Bureau of Standards, Bull. 12, 198 (1915).

Meissner, K. W.

Nawata, S.

S. Nawata, Sci. Rep. Res. Inst. Tohoku Univ. Ser. A3, 740 (1951).

Odencrantz, F. K.

Philip, R.

R. Philip, Compt. Rend. 241, 559, 596 (1955);Compt. Rend. 243, 365 (1956).

Porteus, J. O.

Rank, D. H.

Ryazanov, A. N.

I. N. Shklyarevskii and A. N. Ryazanov, Opt. i Spektroskopiya 2, 645 (1957).

Scheibner, E. J.

Schopper, H.

R. Fleischmann and H. Schopper, Z. Physik 129, 285 (1951).
[Crossref]

H. Schopper, Z. Physik 130, 427, 565 (1951);Z. Physik 131, 215 (1952);Z. Physik 135, 516(1953).
[Crossref]

Schulz, L. G.

Scott, G. D.

G. D. Scott, T. A. McLauchlan, and R. S. Sennett, J. Appl. Phys. 21, 843 (1950).
[Crossref]

Sennett, R. S.

G. D. Scott, T. A. McLauchlan, and R. S. Sennett, J. Appl. Phys. 21, 843 (1950).
[Crossref]

Seraphin, B. O.

Shklyarevskii, I. N.

I. N. Shklyarevskii, Opt. i Spektroskopiya 5, 617 (1958).

I. N. Shklyarevskii and A. N. Ryazanov, Opt. i Spektroskopiya 2, 645 (1957).

I. N. Shklyarevskii, Soviet Phys.—Tech. Phys. 1, 327 (1956)[Zh. Tekhn. Fiz. 26, 333 (1956)].

Shull, E. R.

Tolansky, S.

S. Tolansky, Multiple-Beam Inlerferometry of Surfaces and Films (Clarendon Press, Oxford, England, 1948), p. 9.

Tsvelykh, N. G.

M. P. Lisitsa and N. G. Tsvelykh, Opt. i Spcktroskopiya 2, 674 (1957).

Weaver, C.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1957), 3rd ed., p. 273.

Wiener, O.

O. Wiener, Ann. Physik 31, 629 (1887).
[Crossref]

Wiggins, T. A.

Williams, W. E.

W. E. Williams, Applications of Inlerferometry (Methuen and Company Ltd., London, 1930), p. 77.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 279.

Worthing, A. G.

A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley & Sons, Inc., New York, 1943), p. 207.

Ann. Physik (1)

O. Wiener, Ann. Physik 31, 629 (1887).
[Crossref]

Compt. Rend. (1)

R. Philip, Compt. Rend. 241, 559, 596 (1955);Compt. Rend. 243, 365 (1956).

J. Appl. Phys. (1)

G. D. Scott, T. A. McLauchlan, and R. S. Sennett, J. Appl. Phys. 21, 843 (1950).
[Crossref]

J. Opt. Soc. Am. (15)

Natl. Bur. Std. (U.S.) Bureau of Standards, Bull. (1)

W. F. Meggers, Natl. Bur. Std. (U.S.) Bureau of Standards, Bull. 12, 198 (1915).

Opt. i Spcktroskopiya (1)

M. P. Lisitsa and N. G. Tsvelykh, Opt. i Spcktroskopiya 2, 674 (1957).

Opt. i Spektroskopiya (2)

I. N. Shklyarevskii and A. N. Ryazanov, Opt. i Spektroskopiya 2, 645 (1957).

I. N. Shklyarevskii, Opt. i Spektroskopiya 5, 617 (1958).

Optik (1)

A. Heisen, Optik 18, 27 (1961).

Phil. Mag. (1)

R. C. Faust, Phil. Mag. 41, 1238 (1950).

Proc. Phys. Soc. (London) (1)

J. Holden, Proc. Phys. Soc. (London) 62, 405 (1949).
[Crossref]

Soviet Phys.—Tech. Phys. (1)

I. N. Shklyarevskii, Soviet Phys.—Tech. Phys. 1, 327 (1956)[Zh. Tekhn. Fiz. 26, 333 (1956)].

Z. Physik (4)

H. J. Bolle, Z. Physik 143, 538 (1956).
[Crossref]

R. Fleischmann and H. Schopper, Z. Physik 129, 285 (1951).
[Crossref]

G. Dornenburg and R. Fleischmann, Z. Physik 129, 300 (1951).
[Crossref]

H. Schopper, Z. Physik 130, 427, 565 (1951);Z. Physik 131, 215 (1952);Z. Physik 135, 516(1953).
[Crossref]

Other (34)

S. Nawata, Sci. Rep. Res. Inst. Tohoku Univ. Ser. A3, 740 (1951).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 279.

Four other forms of the complex refractive index are in current use in the literature: n+ik (Ref. 4), n(1−ik) (Ref. 19), n(1+iκ) (Ref. 20) and n(1−iκ) (Ref. 21), where k represents the attenuation of the wave per vacuum wavelength and κ is related in the same way to the attenuation per wavelength in the medium. Hence the expressions for r in various references may actually be identical although they appear to be different. Care should be taken to determine which form of the complex refractive index is used when applying equations for the reflectance and the phase change on reflection.

See Ref. 1, p. 49.

See Ref. 17, p. 610.

R. W. Ditchburn, Light (Blackie & Son Ltd., London, 1952), p. 441.

Care should be taken not to confuse this positive β with Heavens’ “phase advance” (Ref. 1, p. 171). His “phase advance” and “phase retard” are measured relative to the negative real axis so that values of Er in the second quadrant are called “retard” and those in the third quadrant “advance.”

See Ref. 1, pp. 55–88.

This expression may be obtained from the relations given on p. 91 of Ref. 1. However, there is a typographical error in Eq. 4(140). The correct expression should read Δ0=arc tan(BC−AD)/(AC+BD) where Δ0 is identical to β.

S. Tolansky, Multiple-Beam Inlerferometry of Surfaces and Films (Clarendon Press, Oxford, England, 1948), p. 9.

W. E. Williams, Applications of Inlerferometry (Methuen and Company Ltd., London, 1930), p. 77.

O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worths Scientific Publications Ltd., London, 1955), p. 174.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1957), 3rd ed., p. 273.

By one reflecting surface is meant the inner-coated surface of one interferometer plate. The light lost by reflection from the outer surfaces of the plates is ignored since it only decreases the total intensity, and the change in the shape of the transmitted energy curves is negligible. It is also assumed that the two inner-coated surfaces have nearly the same values of T and R.

The minus sign in Eq. (10) is absolutely necessary to be consistent with the previous definition of β as a positive angle. Since the optical path is shorter than the geometrical path, this is equivalent to having the wave reflected from a fictitious surface in front of the actual surface.

See Ref. 25, p. 97.

These fringes are similar to channelled spectra (Ref. 38), Edser–Butler fringes, or white light Fabry–Perot fringes (Ref. 36) except that an image of the interferometer (rather than the light source) is focused on the slit of the spectrograph.

See Ref. 26, p.101.

It is assumed that the wavelengths of the two-beam reflection minima are identical to the wavelengths of the maxima in transmission.

Equation (13) is strictly correct for an air–dielectric reflection. It also holds approximately for interferometers coated with some metal films. However, it does not hold for all metal films or for dielectric coatings.

This method is similar to that used by Shklyarevskii (Ref. 42) except that the integers he used for his Mλ products were one less than those used here. Both Koehler (Ref. 43) and Koester (Ref. 44) have shown that τ may be determined exactly regardless of the choice of the integer if the phase change is properly eliminated. Furthermore, Koester proves that by using a graphical method similar to the one described here (his integers differ from the ones in the present paper by an arbitrary integer N), “the difference between the two curves is a constant. Regardless of the selection of N and the variation of phase shift with wavelength the difference between the curves is twice the step height ….” The integers M used in this paper are purposely chosen to give the most nearly constant Mλ product so that the graphical method will have the most accuracy.

See Ref. 27, p. 423.

See Ref. 26, p. 83.

I. N. Shklyarevskii (Ref. 42) goes through a similar type of derivation to obtain his Eq. (1) [our Eq. (15)]. However, he makes the approximation that β1/π∼β2/π∼1 (our notation) which is a much more drastic assumption than is necessary. Furthermore, he apparently was not successful with the simple formula since he states “It is easy to see that if the dispersion of the phase change is neglected, the thin film thicknesses cannot be measured with a high degree of accuracy.”

Note the typesetter’s errors on p. 1246 of Faust’s paper (Ref. 3) which give Δν=12d.

Vickers projection microscope manufactured by Cooke, Troughton, and Simms, Ltd., York, England.

Model BK 3643 direct reading spectrograph manufactured by Bausch & Lomb Optical Company, Rochester, New York.

Very fast Type C panchromatic antihalation plates obtained from Eastman Kodak Company, Rochester, New York.

Corning filter No. 2404 is used in photographing the neon spectrum to eliminate the lines which obscure some of the interference fringes.

A commercial model is available from the David W. Mann Company, Lincoln, Massachusetts.

A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley & Sons, Inc., New York, 1943), p. 207.

This shift of approximately one order toward the blue (higher orders of interference) in the multiple-beam fringe system in reflection relative to the two-beam fringe system is observed when the reflecting film on the side of the incident beam is a metal. This same effect has also been observed by Faust (Ref. 3) for silver and aluminum but is not completely understood.

Johnson Matthey JM 50 silver rod obtained from Jarrell-Ash Company.

Obtained from the Aluminum Company of America.

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Figures (11)

F. 1
F. 1

(a) Reflection and transmission of a plane wave at normal incidence on a boundary between two media. (b) Definition of the phase angle β. (c) Reflection and transmission of a plane wave incident on an absorbing film.

F. 2
F. 2

(a) Experimental arrangement for measuring the phase change β. (b) Experimental arrangement for measuring the film thickness τ.

F. 3
F. 3

Interferometer and fringe pattern used for determining the correct N in Regions ➀ and ➂.

F. 4
F. 4

Photograph of the interferometer holder.

F. 5
F. 5

Schematic diagram of the optical system used to image the interference fringe patterns.

F. 6
F. 6

(a) Photograph of the fringe pattern produced by the interferometer shown in Fig. 2(a) for the phase change measurement. (b) Photograph of the fringe pattern produced by the interferometer shown in Fig. 2(b) for the film thickness measurement.

F. 7
F. 7

Phase change on reflection β versus wavelength for a 128-Å aluminum film.

F. 8
F. 8

Phase change on reflection β versus wavelength for a 435-Å silver film.

F. 9
F. 9

Optical thickness versus wavelength plot used to determine τ for a 435-Å silver film.

F. 10
F. 10

Irregularities on Fizeau fringes caused by the integrated surface roughness of the interferometer plates. (a) Irregularities on one Fizeau fringe given in orders of interference. The integrated surface roughness on one interferometer plate which would produce this fringe irregularity is given on the right. (b) Variation of the Fizeau fringe spacing across a 2-mm section of the interferometer. The magnification of the interferometer was 14.5.

F. 11
F. 11

Irregularities on feco fringes caused by the integrated surface roughness of the interferometer plates. The integrated surface roughness on one interferometer plate which would produce the fringe irregularity is given on the left. The magnifications of the interferometer used are (a) 82, (b) 19, and (c) 5. The areas of the interferometer plates producing the fringes are (a) 0.10 ×0.00077 mm, (b) 0.50×0.0031 mm, and (c) 1.7×0.0126 mm.

Tables (3)

Tables Icon

Table I Determination of the absolute phase change on reflection for a 128-Å aluminum film on fuzed quartz. All quantities except N and β are in Å. β is in degrees.

Tables Icon

Table II Illustration of the inadequacy of using Eq. (18) for obtaining β. (a) Bennett, 400-Å silver films and (b) Faust, 529-Å silver films (Ref. 3).

Tables Icon

Table III Calculation of the film thickness τ using Eq. (15). Wavelengths of feco fringes are obtained from (a) Bennett, present paper, (b) Koester, Ref. 44, and (c) Koehler, Ref. 43. All quantities except M are in Å.

Equations (25)

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r = E r / E i = ( n 1 n 0 ) / ( n 1 + n 0 ) .
r = E r E i = ( n 1 n 0 ) i k 1 ( n 1 + n 0 ) i k 1 .
r = ( n 1 2 + k 1 2 n 0 2 ) + 2 i k 1 n 0 n 1 2 + n 0 2 + k 1 2 .
tan β = 2 k 1 n 0 / ( n 1 2 + k 1 2 n 0 2 ) .
tan β = { 2 n 0 k 1 [ k 1 2 + ( n 1 + n 2 ) 2 ] exp ( 4 π k 1 τ / λ ) 2 n 0 k 1 [ k 1 2 + ( n 1 n 2 ) 2 ] exp ( 4 π k 1 τ / λ ) + 8 n 0 n 1 n 2 k 1 [ sin 2 ( 2 π n 1 τ / λ ) cos 2 ( 2 π n 1 τ / λ ) ] 8 n 0 n 1 ( k 1 2 + n 1 2 n 2 2 ) sin ( 2 π n 1 τ / λ ) cos ( 2 π n 1 τ / λ ) ] } / { [ n 0 2 k 1 2 n 2 2 k 1 2 + n 0 2 ( n 1 + n 2 ) 2 ( n 1 2 + k 1 2 + n 1 n 2 ) 2 ] exp ( 4 π k 1 τ / λ ) + [ n 0 2 k 1 2 n 2 2 k 1 2 + n 0 2 ( n 1 n 2 ) 2 ( n 1 2 + k 1 2 n 1 n 2 ) 2 ] exp ( 4 π k 1 τ / λ ) ] 2 [ n 0 2 ( k 1 2 + n 1 2 n 2 2 ) + ( k 1 2 + n 1 2 n 1 n 2 ) ( k 1 2 + n 1 2 + n 1 n 2 ) n 2 2 k 1 2 ] [ sin 2 ( 2 π n 1 τ / λ ) cos 2 ( 2 π n 1 τ / λ ) ] 8 n 2 k 1 ( n 0 2 + n 1 2 + k 1 2 ) sin ( 2 π n 1 τ / λ ) cos ( 2 π n 1 τ / λ ) } ,
r r * = ( n 0 n 1 ) 2 / ( n 0 + n 1 ) 2 .
r r * = [ ( n 0 n 1 ) 2 + k 1 2 ] / [ ( n 0 + n 1 ) 2 + k 1 2 ] .
r r * = ( { n 0 [ n 0 2 n 1 ] [ ( n 1 + n 2 ) 2 + k 1 2 ] + [ ( n 1 2 + k 1 2 + n 1 n 2 ) 2 + n 2 2 k 1 2 ] } ) exp ( 4 π k 1 τ / λ ) + { n 0 [ n 0 + 2 n 1 ] [ ( n 1 n 2 ) 2 + k 1 2 ] + [ ( n 1 2 + k 1 2 + n 1 n 2 ) 2 + n 2 2 k 1 2 ] } exp ( 4 π k 1 τ / λ ) + 2 [ n 0 2 ( n 2 2 n 1 2 k 1 2 ) + ( n 1 2 + k 1 2 + n 1 n 2 ) ( n 1 2 + k 1 2 n 1 n 2 ) ] n 2 2 k 1 2 + 4 n 0 n 2 k 1 2 ] [ sin 2 ( 2 π n 1 τ / λ ) cos 2 ( 2 π n 1 τ / λ ) ] 8 [ n 2 k 1 ( n 0 2 n 1 2 k 1 2 ) + n 0 k 1 ( n 1 2 n 2 2 + k 1 2 ) ] sin ( 2 π n 1 τ / λ ) cos ( 2 π n 1 τ / λ ) ) / ( { n 0 [ n 0 + 2 n 1 ] [ ( n 1 + n 2 ) 2 + k 1 2 ] + [ ( n 1 2 + k 1 2 + n 1 n 2 ) 2 + n 2 2 k 1 2 ] } exp ( 4 π k 1 τ / λ ) + { n 0 [ n 0 2 n 1 ] [ ( n 1 + n 2 ) 2 + k 1 2 ] + [ ( n 1 2 + k 1 2 + n 1 n 2 ) 2 + n 2 2 k 1 2 ] } exp ( 4 π k 1 τ / λ ) + 2 [ n 0 2 ( n 2 2 n 1 2 k 1 2 ) + ( n 1 2 + k 1 2 + n 1 n 2 ) ( n 1 2 + k 1 2 n 1 n 2 ) n 2 2 k 1 2 4 n 0 n 2 k 1 2 ] [ sin 2 ( 2 π n 1 τ / λ ) cos 2 ( 2 π n 1 τ / λ ) ] 8 [ n 2 k 1 ( n 0 2 n 1 2 k 1 2 ) n 0 k 1 ( n 1 2 n 2 2 + k 1 2 ) ] sin ( 2 π n 1 τ / λ ) cos ( 2 π n 1 τ / λ ) ) ] .
I t = T 2 ( 1 R ) 2 ( 1 + 4 R ( 1 R ) 2 sin 2 { 1 2 2 π λ [ 2 n d λ 2 π ( β 1 + β 2 ) ] } ) 1 ,
N λ = 2 n d ( λ / 2 π ) ( β 1 + β 2 )
( N + 1 ) λ = 2 n d + ( λ δ / π ) .
N λ = 2 d ( λ / 2 π ) ( π + π ) ,
N 1 + 1 = λ 2 Δ N / ( λ 1 λ 2 ) .
N λ = 2 ( d 2 τ ) ( λ / 2 π ) ( β + β )
τ = ( M / 2 ) ( λ 2 λ 1 ) ,
λ H = λ 0 + C / ( X X 0 )
λ β π
λ β π
λ N + 1 λ N λ N + 1
β π apparent
τ = M 2 Δ λ
[ N + ( β / π ) ] λ = 2 d ,
[ N + ( β / π ) ] = λ N + 1 / ( λ N λ N + 1 ) = 2 d / λ N ,
M λ 1 = 2 ( d τ ) + ( λ 1 δ 1 / π ) , M λ 2 = 2 d + ( λ 2 δ 2 / π ) ,
τ = [ ( M + 0.013 / 2 ) ] ( λ 2 λ 1 ) .