Abstract

The reflectance R and transmittance T of thin absorbing films deposited on a transparent substrate are calculated for normal and oblique incidence, s and p polarization, and different film thicknesses The results are presented as contours of constant R and T on the ñ plane, where ñ is the complex refractive index. The conditions for sensitive dependence of measured quantities on ñ are examined. A computer-based method of finding ñ from chosen combinations of measured R or T values is described. Oblique incidence measurements on thin films can give accurate results in some regions where other methods may be less sensitive. Accurate film-thickness value can be obtained from the optical measurements.

© 1972 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).
  2. A. Vašìček, Optics of Thin Films (North-Holland, Amsterdam, 1960).
  3. See, for example, our discussion of this problem in Am. J. Phys. 39, 313 (1971).
  4. P. O. Nilsson, Appl. Opt. 7, 435 (1968).
    [CrossRef] [PubMed]
  5. D. W. Juenker, J. Opt. Soc. Am. 55, 295 (1965).
    [CrossRef]
  6. M.-G. Bouchard, in Basic Problems in Thin Film Physics, R. Niedermayer, H. Mayer, Eds. (Vandenhoeck & Ruprecht, Göttingen, 1966), p. 301.
  7. H. E. Bennett, J. M. Bennett, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic Press, New York, 1967), Vol. 4, p. 1.
  8. T. S. Robinson, Proc. Phys. Soc. B 65, 910 (1952).
    [CrossRef]
  9. A. P. Lenham, D. M. Treherne, A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (Wiley, New York, 1966), p. 40.
  10. R. Tousey, J. Opt. Soc. Am. 29, 235 (1939).
    [CrossRef]
  11. The transmittance is defined here as a ratio of power per unit area normal to the propagation direction, which differs in medium 1 and medium 3. Usually one measures the ratio of total power in a finite beam, which is (cosθ3/cosθ1)T. Experimentally, a correction must also be made in Tand Rfor reflections at the back surface of the substrate.
  12. F. Forstmann, Z. Physik 203, 495 (1967).
    [CrossRef]
  13. R. Fuchs, K. L. Kliewer, Phys. Rev. 185, 905 (1969).
    [CrossRef]
  14. R. B. Dingle, Physica 19, 311, 729, 1187 (1953).
    [CrossRef]
  15. M.-L. Thèye, Phys. Lett. 25A, 764 (1967).
  16. K. L. Kliewer, R. Fuchs, Phys. Rev. B 2, 2923 (1970).
    [CrossRef]
  17. D. Malé, cited in Ref. 1, p. 137.
  18. These were plotted using a Timeshare Devices, Inc., C/P 701 plotter, by picking a value of nand stepping k(or vice versa).
  19. P. B. Johnson, R. W. Christy, unpublished.
  20. F. Abèles, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1963), Vol. 2, p. 249.
    [CrossRef]
  21. J. M. Bennett, M. J. Booty, Appl. Opt. 5, 41 (1966).
    [CrossRef] [PubMed]
  22. Copies of the inversion programs in BASIC are available on request, as well as copies of the contours we have plotted.
  23. F. Abelès, M.-L. Thèye, Surface Sci. 5, 325 (1966).
    [CrossRef]
  24. H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1947), p. 476.
  25. N. Draper, H. Smith, Applied Regression Analysis (Wiley, New York, 1969).
  26. S. P. F. Humphreys-Owen, Proc. Phys. Soc. 77, 949 (1961).
    [CrossRef]
  27. W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965).
    [CrossRef]
  28. H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945), Chap. 14.
  29. F. Stern, Solid State Physics, F. Seitz, D. Turnbull, Eds. (Academic, New York, 1963), Vol. 15, p. 299.
    [CrossRef]
  30. E. L. Green, L. Muldawer, Phys. Rev. B 2, 330 (1970).
    [CrossRef]
  31. G. W. Rubloff, Phys. Rev. B 3, 285 (1971).
    [CrossRef]
  32. D. L. Decker, J. L. Stanford, J. Opt. Soc. Am. 61, 679A (1971).
  33. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), Sec. 13.2. Equations (12)–(14) contain misprints: the subscripts for perpendicular and parallel polarization should be interchanged.
  34. P. Drude, Ann. Physik 39, 504 (1890).
  35. G. K. T. Conn, G. K. Eaton, J. Opt. Soc. Am. 44, 546 (1954); J. R. Beattie, G. K. T. Conn, Philos. Mag. 46, 223 (1955).
    [CrossRef]
  36. N. V. Smith, Phys. Rev. 183, 634 (1969).
    [CrossRef]
  37. D. Malé, Compt. Rend. 230, 1349 (1950).

1971 (3)

See, for example, our discussion of this problem in Am. J. Phys. 39, 313 (1971).

G. W. Rubloff, Phys. Rev. B 3, 285 (1971).
[CrossRef]

D. L. Decker, J. L. Stanford, J. Opt. Soc. Am. 61, 679A (1971).

1970 (2)

K. L. Kliewer, R. Fuchs, Phys. Rev. B 2, 2923 (1970).
[CrossRef]

E. L. Green, L. Muldawer, Phys. Rev. B 2, 330 (1970).
[CrossRef]

1969 (2)

R. Fuchs, K. L. Kliewer, Phys. Rev. 185, 905 (1969).
[CrossRef]

N. V. Smith, Phys. Rev. 183, 634 (1969).
[CrossRef]

1968 (1)

1967 (2)

F. Forstmann, Z. Physik 203, 495 (1967).
[CrossRef]

M.-L. Thèye, Phys. Lett. 25A, 764 (1967).

1966 (2)

F. Abelès, M.-L. Thèye, Surface Sci. 5, 325 (1966).
[CrossRef]

J. M. Bennett, M. J. Booty, Appl. Opt. 5, 41 (1966).
[CrossRef] [PubMed]

1965 (2)

1961 (1)

S. P. F. Humphreys-Owen, Proc. Phys. Soc. 77, 949 (1961).
[CrossRef]

1954 (1)

1953 (1)

R. B. Dingle, Physica 19, 311, 729, 1187 (1953).
[CrossRef]

1952 (1)

T. S. Robinson, Proc. Phys. Soc. B 65, 910 (1952).
[CrossRef]

1950 (1)

D. Malé, Compt. Rend. 230, 1349 (1950).

1939 (1)

1890 (1)

P. Drude, Ann. Physik 39, 504 (1890).

Abelès, F.

F. Abelès, M.-L. Thèye, Surface Sci. 5, 325 (1966).
[CrossRef]

Abèles, F.

F. Abèles, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1963), Vol. 2, p. 249.
[CrossRef]

Bennett, H. E.

H. E. Bennett, J. M. Bennett, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic Press, New York, 1967), Vol. 4, p. 1.

Bennett, J. M.

J. M. Bennett, M. J. Booty, Appl. Opt. 5, 41 (1966).
[CrossRef] [PubMed]

H. E. Bennett, J. M. Bennett, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic Press, New York, 1967), Vol. 4, p. 1.

Bode, H. W.

H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945), Chap. 14.

Booty, M. J.

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), Sec. 13.2. Equations (12)–(14) contain misprints: the subscripts for perpendicular and parallel polarization should be interchanged.

Bouchard, M.-G.

M.-G. Bouchard, in Basic Problems in Thin Film Physics, R. Niedermayer, H. Mayer, Eds. (Vandenhoeck & Ruprecht, Göttingen, 1966), p. 301.

Christy, R. W.

P. B. Johnson, R. W. Christy, unpublished.

Conn, G. K. T.

Decker, D. L.

D. L. Decker, J. L. Stanford, J. Opt. Soc. Am. 61, 679A (1971).

Dingle, R. B.

R. B. Dingle, Physica 19, 311, 729, 1187 (1953).
[CrossRef]

Draper, N.

N. Draper, H. Smith, Applied Regression Analysis (Wiley, New York, 1969).

Drude, P.

P. Drude, Ann. Physik 39, 504 (1890).

Eaton, G. K.

Forstmann, F.

F. Forstmann, Z. Physik 203, 495 (1967).
[CrossRef]

Fuchs, R.

K. L. Kliewer, R. Fuchs, Phys. Rev. B 2, 2923 (1970).
[CrossRef]

R. Fuchs, K. L. Kliewer, Phys. Rev. 185, 905 (1969).
[CrossRef]

Green, E. L.

E. L. Green, L. Muldawer, Phys. Rev. B 2, 330 (1970).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

Humphreys-Owen, S. P. F.

S. P. F. Humphreys-Owen, Proc. Phys. Soc. 77, 949 (1961).
[CrossRef]

Hunter, W. R.

Johnson, P. B.

P. B. Johnson, R. W. Christy, unpublished.

Juenker, D. W.

Kliewer, K. L.

K. L. Kliewer, R. Fuchs, Phys. Rev. B 2, 2923 (1970).
[CrossRef]

R. Fuchs, K. L. Kliewer, Phys. Rev. 185, 905 (1969).
[CrossRef]

Lenham, A. P.

A. P. Lenham, D. M. Treherne, A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (Wiley, New York, 1966), p. 40.

Malé, D.

D. Malé, Compt. Rend. 230, 1349 (1950).

D. Malé, cited in Ref. 1, p. 137.

Margenau, H.

H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1947), p. 476.

Muldawer, L.

E. L. Green, L. Muldawer, Phys. Rev. B 2, 330 (1970).
[CrossRef]

Murphy, G. M.

H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1947), p. 476.

Nilsson, P. O.

Robinson, T. S.

T. S. Robinson, Proc. Phys. Soc. B 65, 910 (1952).
[CrossRef]

Rubloff, G. W.

G. W. Rubloff, Phys. Rev. B 3, 285 (1971).
[CrossRef]

Smith, H.

N. Draper, H. Smith, Applied Regression Analysis (Wiley, New York, 1969).

Smith, N. V.

N. V. Smith, Phys. Rev. 183, 634 (1969).
[CrossRef]

Stanford, J. L.

D. L. Decker, J. L. Stanford, J. Opt. Soc. Am. 61, 679A (1971).

Stern, F.

F. Stern, Solid State Physics, F. Seitz, D. Turnbull, Eds. (Academic, New York, 1963), Vol. 15, p. 299.
[CrossRef]

Thèye, M.-L.

M.-L. Thèye, Phys. Lett. 25A, 764 (1967).

F. Abelès, M.-L. Thèye, Surface Sci. 5, 325 (1966).
[CrossRef]

Tousey, R.

Treherne, D. M.

A. P. Lenham, D. M. Treherne, A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (Wiley, New York, 1966), p. 40.

Vašìcek, A.

A. Vašìček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), Sec. 13.2. Equations (12)–(14) contain misprints: the subscripts for perpendicular and parallel polarization should be interchanged.

Woodall, A. J.

A. P. Lenham, D. M. Treherne, A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (Wiley, New York, 1966), p. 40.

Am. J. Phys. (1)

See, for example, our discussion of this problem in Am. J. Phys. 39, 313 (1971).

Ann. Physik (1)

P. Drude, Ann. Physik 39, 504 (1890).

Appl. Opt. (2)

Compt. Rend. (1)

D. Malé, Compt. Rend. 230, 1349 (1950).

J. Opt. Soc. Am. (5)

Phys. Lett. (1)

M.-L. Thèye, Phys. Lett. 25A, 764 (1967).

Phys. Rev. (2)

R. Fuchs, K. L. Kliewer, Phys. Rev. 185, 905 (1969).
[CrossRef]

N. V. Smith, Phys. Rev. 183, 634 (1969).
[CrossRef]

Phys. Rev. B (3)

E. L. Green, L. Muldawer, Phys. Rev. B 2, 330 (1970).
[CrossRef]

G. W. Rubloff, Phys. Rev. B 3, 285 (1971).
[CrossRef]

K. L. Kliewer, R. Fuchs, Phys. Rev. B 2, 2923 (1970).
[CrossRef]

Physica (1)

R. B. Dingle, Physica 19, 311, 729, 1187 (1953).
[CrossRef]

Proc. Phys. Soc. (1)

S. P. F. Humphreys-Owen, Proc. Phys. Soc. 77, 949 (1961).
[CrossRef]

Proc. Phys. Soc. B (1)

T. S. Robinson, Proc. Phys. Soc. B 65, 910 (1952).
[CrossRef]

Surface Sci. (1)

F. Abelès, M.-L. Thèye, Surface Sci. 5, 325 (1966).
[CrossRef]

Z. Physik (1)

F. Forstmann, Z. Physik 203, 495 (1967).
[CrossRef]

Other (16)

H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945), Chap. 14.

F. Stern, Solid State Physics, F. Seitz, D. Turnbull, Eds. (Academic, New York, 1963), Vol. 15, p. 299.
[CrossRef]

H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1947), p. 476.

N. Draper, H. Smith, Applied Regression Analysis (Wiley, New York, 1969).

Copies of the inversion programs in BASIC are available on request, as well as copies of the contours we have plotted.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), Sec. 13.2. Equations (12)–(14) contain misprints: the subscripts for perpendicular and parallel polarization should be interchanged.

A. P. Lenham, D. M. Treherne, A. J. Woodall, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abelès, Ed. (Wiley, New York, 1966), p. 40.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

A. Vašìček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

M.-G. Bouchard, in Basic Problems in Thin Film Physics, R. Niedermayer, H. Mayer, Eds. (Vandenhoeck & Ruprecht, Göttingen, 1966), p. 301.

H. E. Bennett, J. M. Bennett, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic Press, New York, 1967), Vol. 4, p. 1.

The transmittance is defined here as a ratio of power per unit area normal to the propagation direction, which differs in medium 1 and medium 3. Usually one measures the ratio of total power in a finite beam, which is (cosθ3/cosθ1)T. Experimentally, a correction must also be made in Tand Rfor reflections at the back surface of the substrate.

D. Malé, cited in Ref. 1, p. 137.

These were plotted using a Timeshare Devices, Inc., C/P 701 plotter, by picking a value of nand stepping k(or vice versa).

P. B. Johnson, R. W. Christy, unpublished.

F. Abèles, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1963), Vol. 2, p. 249.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Contours of const. n and k for normal incidence and d/λ = 0.1. (a) For each curve n has the value indicated and 0 ≤ k ≤ 4. (b) For each curve k has the value indicated and 0 ≤ n ≤ 3.

Fig. 2
Fig. 2

Contours of const. R and T for normal incidence and d/λ = 0.1. The contour width indicates a nominal error of measurement, to show the uncertainty in the derived n and k values.

Fig. 3
Fig. 3

Contours of const. R for normal incidence and d/λ = 0.05, 0.2.

Fig. 4
Fig. 4

Experimental values for some representative materials. The cross-hatched area is where the normal-incidence R and T contours are tangent for values of d/λ from 0.05 to 0.2.

Fig. 5
Fig. 5

Contours of const. R for s and p polarization with 60° incident angle and d/λ = 0.1.

Fig. 6
Fig. 6

Contours of const. T for s and p polarization with 60° incident angle and d/λ = 0.1.

Fig. 7
Fig. 7

Example of intersection of normal-incidence R and T contours with p-polarization T contour for 60° incident angle. d/λ = 0.118.

Fig. 8
Fig. 8

Loci of the intersections of some particular normal-incidence R and T contours for d/λ values from 0.05 to 0.2.

Fig. 9
Fig. 9

Contours of const. R for normal incidence and p polarization at 60° incidence for infinite thickness.

Fig. 10
Fig. 10

Contours of const. R and const. phase ϕ for normal incidence and infinite thickness (Kramers-Kronig method).

Fig. 11
Fig. 11

Contours of const. ratio of amplitudes tanψ and const. phase difference Δ for p and s polarization (Drude method). (a) 60° incidence; (b) 75° incidence.

Fig. 12
Fig. 12

Dependence on incident angle of R for infinite thickness and T for d/λ = 0.1, both with p polarization, in the case n = k = 1.

Equations (16)

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R = | r ˜ | 2 , T = ( n 3 / n 1 ) | t ˜ | 2 ,
r ˜ = [ r ˜ 12 + r ˜ 23 exp ( i δ ˜ ) ] / [ 1 + r ˜ 12 r ˜ 23 exp ( i δ ) ] , t ˜ = [ t ˜ 12 t ˜ 23 exp ( i δ ˜ / 2 ) ] / [ 1 + r ˜ 12 r ˜ 23 exp ( i δ ˜ ) ] .
r ˜ 12 s = ( n 1 cos θ 1 n ˜ 2 cos θ ˜ 2 ) / ( n 1 cos θ 1 + n ˜ 2 cos θ ˜ 2 ) , t ˜ 12 s = 2 n 1 cos θ 1 / ( n 1 cos θ 1 + n ˜ 2 cos θ ˜ 2 ) ,
r ˜ 12 p = ( n ˜ 2 cos θ 1 n 1 cos θ ˜ 2 ) / ( n ˜ 2 cos θ 1 + n 1 cos θ ˜ 2 ) , t ˜ 12 p = 2 n 1 cos θ 1 / ( n ˜ 2 cos θ 1 + n 1 cos θ ˜ 2 ) .
δ ˜ = 4 π ( d / λ ) n 2 cos θ ˜ 2 ,
n ˜ 2 cos θ ˜ 2 u + i υ .
u 2 = 1 2 { ( n 2 k 2 n 1 2 sin 2 θ 1 ) + [ 4 n 2 k 2 + ( n 2 k 2 n 1 2 sin 2 θ 1 ) 2 ] 1 2 } , υ 2 = 1 2 { ( n 2 k 2 n 1 2 sin 2 θ 1 ) + [ 4 n 2 k 2 + ( n 2 k 2 n 1 2 sin 2 θ 1 ) 2 ] 1 2 } , cos θ ˜ 2 = ( u + i υ ) / ( n + i k ) .
n ˜ 3 cos θ 3 = ( n 3 2 n ˜ 1 2 sin 2 θ 1 ) 1 2 .
( Δ ψ ) 0 ψ ¯ ψ 0 = ( ψ n ) 0 Δ n + ( ψ k ) 0 Δ k , ( Δ ϕ ) 0 ϕ ¯ ϕ 0 = ( ϕ n ) 0 Δ n + ( ϕ k ) 0 Δ k .
ψ ( n , k ) a 0 + a 1 n + a 2 k , ϕ ( n , k ) b 0 + b 1 n + b 2 k ,
a 1 = ( ψ n ) 0 , a 2 = ( ψ k ) 0 , b 1 = ( ϕ n ) 0 , b 2 = ( ϕ k ) 0 .
X ( 1 ( n 0 + Δ n 0 ) ( k 0 + Δ k 0 ) 1 ( n 0 + Δ n 0 ) ( k 0 Δ k 0 ) 1 ( n 0 Δ n 0 ) ( k 0 + Δ k 0 ) 1 ( n 0 Δ n 0 ) ( k 0 Δ k 0 ) ) , Ψ = ( ψ ( n 0 + Δ n 0 , k 0 + Δ k 0 ) ψ ( n 0 + Δ n 0 , k 0 Δ k 0 ) ψ ( n 0 Δ n 0 , k 0 + Δ k 0 ) ψ ( n 0 Δ n 0 , k 0 Δ k 0 ) ) , Φ = ( ϕ ( n 0 + Δ n 0 , k 0 + Δ k 0 ) ϕ ( n 0 + Δ n 0 , k 0 Δ k 0 ) ϕ ( n 0 Δ n 0 , k 0 + Δ k 0 ) ϕ ( n 0 Δ n 0 , k 0 Δ k 0 ) ) .
a = ( X T X ) 1 X T Ψ , b = ( X T X ) 1 X T Φ ,
a ( a 0 a 1 a 2 ) , b ( b 0 b 1 b 2 )
ϕ ( ω ) = 2 ω π 0 ln r ( ω ) d ω ω 2 ω 2 ,
n = ( 1 r 2 ) / ( 1 + r 2 2 r cos ϕ ) , k = 2 r sin ϕ / ( 1 + r 2 2 r cos ϕ ) .

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