Abstract

A new method is presented for obtaining intrinsic absorption spectra (i.e., index of refraction and extinction coefficient versus wavelength) from reflection spectra of stratified media, which usually are complicated by substrate absorption, thin-layer interference patterns, etc. A typical application is to obtain intrinsic absorption spectra of an organic monolayer chemisorbed on the surface of a very thin metal film, which is in turn supported by a bulk substrate. The basis of the method is the proven causality of a special reflection function that we introduce, and the application of the transformation of Peterson and Knight to this function. The method is general, and has been found to give accurate results for various multiphase systems without restriction as to number of phases, angle of incidence, or polarization. While the method is valid for general stratified media, its accuracy depends upon the information content of the particular reflection spectrum taken. The sensitivity available can be calculated and the adjustable optical parameters chosen accordingly.

© 1977 Optical Society of America

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References

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  1. N. J. Harrick, Internal Reflection Spectroscopy (Interscience, New York, 1967).
  2. W. N. Hansen, in Symposium of the Faraday Society, No. 4, 1970 (unpublished).
  3. W. N. Hansen, in Advances in Electrochemistry and Electrochemical Engineering, edited by R. H. Muller (Wiley-Inter-science, New York, 1973), Vol. 9, pp. 1–60.
  4. J. D. E. McIntyre, in Ref. 3, pp. 61–166.
  5. J. D. E. McIntyre, in Optical Properties of Solids—New Developments, edited by B. O. Seraphin (American Elsevier, New York, 1976).
  6. W. N. Hansen, J. Opt. Soc. Am. 63, 793 (1973).
    [Crossref]
  7. T. S. Robinson, Proc. Phys. Soc. Lond. B 65, 910 (1952).
    [Crossref]
  8. T. S. Robinson and W. C. Price, Proc. Phys. Soc. Lond. B 66, 969 (1953).
    [Crossref]
  9. J. S. Toll, Phys. Rev. 104, 1760 (1956).
    [Crossref]
  10. H. R. Phillip and E. A. Taft, Phys. Rev. 136, A1445 (1964).
    [Crossref]
  11. Dwight W. Berreman, Appl. Opt. 6, 1519 (1967).
    [Crossref] [PubMed]
  12. M. Gottlieb, J. Opt. Soc. Am. 50, 343 (1960).
    [Crossref]
  13. G. Andermann, A. Caron, and D. A. Dows, J. Opt. Soc. Am. 55, 1210 (1965).
    [Crossref]
  14. C. W. Peterson and B. W. Knight, J. Opt. Soc. Am. 63, 1238 (1973).
    [Crossref]
  15. W. J. Plieth and K. Naegele, Surf. Sci. 50, 53 (1975).
    [Crossref]
  16. K. Naegele and W. J. Plieth, Surf. Sci. 50, 64–76 (1975).
    [Crossref]
  17. W. N. Hansen, J. Opt. Soc. Am. 58, 380 (1968).
    [Crossref]
  18. W. M. Hansen, in Progress in Nuclear Energy, Series IX, edited by H. A. Elion and D. C. Steward (Pergamon, New York, 1972), Vol. 11.
  19. G. Andermann and L. Brantley, J. Opt. Soc. Am. 58, 171 (1968).
    [Crossref]
  20. F. Wooten, Optical Properties of Solids (Academic, New York, 1972).
  21. A. C. Gilby and et al., J. Phys. Chem. 70, 1525 (1966).
    [Crossref]
  22. K. Kozima, W. Suëtka, and P. N. Schatz, J. Opt. Soc. Am. 56, 181 (1966).
    [Crossref]
  23. J. S. Plaskett and P. N. Shatz, J. Chem. Phys. 38, 612 (1963).
    [Crossref]
  24. (a)H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), p. 21;(b)p. 27;(c)p. 19.
  25. Z. Nehari, Introduction to Complex Analysis (Allyn and Bacon, Boston, 1961), Chap. VII.
  26. J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297 (1965).
    [Crossref]
  27. E. Oran Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N. J., 1974).

1975 (2)

W. J. Plieth and K. Naegele, Surf. Sci. 50, 53 (1975).
[Crossref]

K. Naegele and W. J. Plieth, Surf. Sci. 50, 64–76 (1975).
[Crossref]

1973 (2)

1968 (2)

1967 (1)

1966 (2)

1965 (2)

1964 (1)

H. R. Phillip and E. A. Taft, Phys. Rev. 136, A1445 (1964).
[Crossref]

1963 (1)

J. S. Plaskett and P. N. Shatz, J. Chem. Phys. 38, 612 (1963).
[Crossref]

1960 (1)

1956 (1)

J. S. Toll, Phys. Rev. 104, 1760 (1956).
[Crossref]

1953 (1)

T. S. Robinson and W. C. Price, Proc. Phys. Soc. Lond. B 66, 969 (1953).
[Crossref]

1952 (1)

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

Andermann, G.

Berreman, Dwight W.

Brantley, L.

Caron, A.

Cooley, J. W.

J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297 (1965).
[Crossref]

Dows, D. A.

Gilby, A. C.

A. C. Gilby and et al., J. Phys. Chem. 70, 1525 (1966).
[Crossref]

Gottlieb, M.

Hansen, W. M.

W. M. Hansen, in Progress in Nuclear Energy, Series IX, edited by H. A. Elion and D. C. Steward (Pergamon, New York, 1972), Vol. 11.

Hansen, W. N.

W. N. Hansen, J. Opt. Soc. Am. 63, 793 (1973).
[Crossref]

W. N. Hansen, J. Opt. Soc. Am. 58, 380 (1968).
[Crossref]

W. N. Hansen, in Symposium of the Faraday Society, No. 4, 1970 (unpublished).

W. N. Hansen, in Advances in Electrochemistry and Electrochemical Engineering, edited by R. H. Muller (Wiley-Inter-science, New York, 1973), Vol. 9, pp. 1–60.

Harrick, N. J.

N. J. Harrick, Internal Reflection Spectroscopy (Interscience, New York, 1967).

Knight, B. W.

Kozima, K.

McIntyre, J. D. E.

J. D. E. McIntyre, in Ref. 3, pp. 61–166.

J. D. E. McIntyre, in Optical Properties of Solids—New Developments, edited by B. O. Seraphin (American Elsevier, New York, 1976).

Naegele, K.

W. J. Plieth and K. Naegele, Surf. Sci. 50, 53 (1975).
[Crossref]

K. Naegele and W. J. Plieth, Surf. Sci. 50, 64–76 (1975).
[Crossref]

Nehari, Z.

Z. Nehari, Introduction to Complex Analysis (Allyn and Bacon, Boston, 1961), Chap. VII.

Nussenzveig, H. M.

(a)H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), p. 21;(b)p. 27;(c)p. 19.

Oran Brigham, E.

E. Oran Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N. J., 1974).

Peterson, C. W.

Phillip, H. R.

H. R. Phillip and E. A. Taft, Phys. Rev. 136, A1445 (1964).
[Crossref]

Plaskett, J. S.

J. S. Plaskett and P. N. Shatz, J. Chem. Phys. 38, 612 (1963).
[Crossref]

Plieth, W. J.

K. Naegele and W. J. Plieth, Surf. Sci. 50, 64–76 (1975).
[Crossref]

W. J. Plieth and K. Naegele, Surf. Sci. 50, 53 (1975).
[Crossref]

Price, W. C.

T. S. Robinson and W. C. Price, Proc. Phys. Soc. Lond. B 66, 969 (1953).
[Crossref]

Robinson, T. S.

T. S. Robinson and W. C. Price, Proc. Phys. Soc. Lond. B 66, 969 (1953).
[Crossref]

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

Schatz, P. N.

Shatz, P. N.

J. S. Plaskett and P. N. Shatz, J. Chem. Phys. 38, 612 (1963).
[Crossref]

Suëtka, W.

Taft, E. A.

H. R. Phillip and E. A. Taft, Phys. Rev. 136, A1445 (1964).
[Crossref]

Toll, J. S.

J. S. Toll, Phys. Rev. 104, 1760 (1956).
[Crossref]

Tukey, J. W.

J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297 (1965).
[Crossref]

Wooten, F.

F. Wooten, Optical Properties of Solids (Academic, New York, 1972).

Appl. Opt. (1)

J. Chem. Phys. (1)

J. S. Plaskett and P. N. Shatz, J. Chem. Phys. 38, 612 (1963).
[Crossref]

J. Opt. Soc. Am. (7)

J. Phys. Chem. (1)

A. C. Gilby and et al., J. Phys. Chem. 70, 1525 (1966).
[Crossref]

Math. Comput. (1)

J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297 (1965).
[Crossref]

Phys. Rev. (2)

J. S. Toll, Phys. Rev. 104, 1760 (1956).
[Crossref]

H. R. Phillip and E. A. Taft, Phys. Rev. 136, A1445 (1964).
[Crossref]

Proc. Phys. Soc. Lond. B (2)

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

T. S. Robinson and W. C. Price, Proc. Phys. Soc. Lond. B 66, 969 (1953).
[Crossref]

Surf. Sci. (2)

W. J. Plieth and K. Naegele, Surf. Sci. 50, 53 (1975).
[Crossref]

K. Naegele and W. J. Plieth, Surf. Sci. 50, 64–76 (1975).
[Crossref]

Other (10)

F. Wooten, Optical Properties of Solids (Academic, New York, 1972).

W. M. Hansen, in Progress in Nuclear Energy, Series IX, edited by H. A. Elion and D. C. Steward (Pergamon, New York, 1972), Vol. 11.

N. J. Harrick, Internal Reflection Spectroscopy (Interscience, New York, 1967).

W. N. Hansen, in Symposium of the Faraday Society, No. 4, 1970 (unpublished).

W. N. Hansen, in Advances in Electrochemistry and Electrochemical Engineering, edited by R. H. Muller (Wiley-Inter-science, New York, 1973), Vol. 9, pp. 1–60.

J. D. E. McIntyre, in Ref. 3, pp. 61–166.

J. D. E. McIntyre, in Optical Properties of Solids—New Developments, edited by B. O. Seraphin (American Elsevier, New York, 1976).

E. Oran Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N. J., 1974).

(a)H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), p. 21;(b)p. 27;(c)p. 19.

Z. Nehari, Introduction to Complex Analysis (Allyn and Bacon, Boston, 1961), Chap. VII.

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Figures (21)

FIG. 1
FIG. 1

Simulated 10-DHO spectrum for the refractive index n (parameters used are shown in Table I). The solid lines represent ns and the crosses represent nb (when the second band is bleached). (Tenth peak is off scale to right).

FIG. 2
FIG. 2

Simulated 10-DHO spectrum for the absorption coefficient k (Table I shows the parameters used). The solid lines represent ks and the crosses represent kb (second band is bleached).

FIG. 3
FIG. 3

Representation of lnρs (solid line) and lnρb (crosses) calculated using the values of n and k shown in Figs. 1 and 2. In these calculations the sample is considered as the ninth phase in a 10-layer stratified medium as illustrated.

FIG. 4
FIG. 4

Synthetic spectra for θs (solid line) and θb (crosses) for the same situation represented in Fig. 3.

FIG. 5
FIG. 5

Representation of the difference between lnρs and lnρb. Note that this difference is zero everywhere outside the bleached region except for small deviations from zero at the maxima of the neighboring bands. The situation here represents the 10-DHO and the 10-layer stratified medium illustrated in Fig. 3.

FIG. 6
FIG. 6

lnρs (solid line) and lnρb (crosses) for a 3-DHO for a four-phase stratified medium.

FIG. 7
FIG. 7

Synthetic spectra of θs (solid lines) and θb (crosses) for the same situation represented in Fig. 6.

FIG. 8
FIG. 8

Δθ as calculated by the PK transformation of ln(ρsb) (shown in Fig. 7). The mesh size used in these calculations is 16 cm−1.

FIG. 9
FIG. 9

Solid lines represent the synthetic values of θs for the 10-DHO 10-layer situation. The crosses represent (θs)c calculated by the method. The spacing between the crosses does not represent the mesh size which was chosen at 16 cm−1.

FIG. 10
FIG. 10

Solid lines represent the synthetic values of θs for an intermediate layer in a 10-phase stratified medium (illustrated in Fig. 3). Here 3-DHO’s were considered (Fig. 6). The parameters used are ωj = 100, 350, and 850 cm−1, λj = 5, 8, and 6 cm−1, and Bj = 10 000, 12 000, and 10 000 cm−2. The crosses represent (θs)c calculated by the method using the PK transformation.

FIG. 11
FIG. 11

ln(ρsb) in the experimental range (the bleached region). The sample here is a 1000 Å film of CCl4. This curve is transformed by the method, using the PK equation, into (θs)c shown in Fig. 15.

FIG. 12
FIG. 12

ln(ρsb) for a 3-phase system in which a 1000 Å film of CCl4 represents the second medium. This curve is transformed by the method, using the PK equation into (θs)c, shown in Fig. 16.

FIG. 13
FIG. 13

ln(ρsb) for 1000 Å CCl4 film as the fourth layer in the illustrated stratified medium. Here ρb was calculated by guessing two values for nb at the band edges (kb = 0).

FIG. 14
FIG. 14

Solid lines represent θs for the 1000 Å CCl4 sample. The crosses represent (θs)c calculated by the method using the PK transformation. The mesh size used in the calculations was 2 cm−1.

FIG. 15
FIG. 15

Same as Fig. 14, but with phases as shown.

FIG. 16
FIG. 16

Same as Fig. 14, but with phases as shown.

FIG. 17
FIG. 17

Δθ as calculated by the PK transformation always has similar shape to the actual values of θs (the solid lines). The crosses represent (θs)c calculated by the method. The sample here is semi-infinite medium of liquid benzene. Polarization and normal incidence were considered in the calculations.

FIG. 18
FIG. 18

Integration contour in the complex frequency plane. The radius of C1 → ∞.

FIG. 19
FIG. 19

(a) Falsified data point at frequency 760 cm−1 (indicated by the arrow) results in a zig-zag shape for (θs)c when calculated by PK equation. Note that half the points coincide with θs while the rest of the calculated values of (θs)c are shifted either above the θs curve (ν < 760 cm−1) or below the θs curve (ν > 760 cm−1), (b) When (θs)c is calculated by the KK equation an incorrect datum at 760 cm−1 results in a shift either above (ν < 760 cm−1) or below (ν < 760 cm−1) the actual values of θs. This shift could not be detected as due to an incorrect experimental datum as it is obviously indicated by the unusual zigzag shape obtained by the PK equation.

FIG. 20
FIG. 20

Due to the symmetry and antisymmetry of the sin(2πn0k/N) and cos(2πnk/N) (depending on the values of n and n0) the weighting factor 4 N k = 0 N / 2 sin 2 π n 0 k N cos 2 π n k N vanishes when both n and n0 have the same parity as in (a) and (c). (b) and (d) show that the weighting factor has a value different from zero only if n and n0 have different parities.

FIG. 21
FIG. 21

Weighting factors of the real function Re(n) in Eq. (B3). Here n0 = 380 (corresponding to frequency ν = 760 cm−1, N = 1024. All the weighting factors corresponding to the even values of n are equal to zero and the curve shows only those corresponding to the odd values of n.

Tables (1)

Tables Icon

TABLE I Parameters used in the DHO model to simulate n, k, ρ, and θ in Figs. 14, respectively.

Equations (34)

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G ( ω ) ln [ r ̂ s ( ω ) / r ̂ b ( ω ) ] = ln ( ρ s / ρ b ) + i ( θ s θ b ) ,
̂ = 1 + j B j ω j 2 ω 2 i λ j ω ,
= 1 + j B j ( ω j 2 ω 2 ) ( ω j 2 ω 2 ) 2 + λ j 2 ω 2
= j B j λ j ω ( ω j 2 ω 2 ) 2 + λ j 2 ω 2 ,
n = { 1 2 [ ( 2 + 2 ) 1 / 2 + ] } 1 / 2 , k = { 1 2 [ ( 2 + 2 ) 1 / 2 ] } 1 / 2 .
i Δ θ ( ω ) = 1 2 π 0 e i ω t d t ln ( ρ s / ρ b ) e i ω ̂ t d ω ̂ + 1 2 π 0 e i ω t d t ln ( ρ s / ρ b ) e i ω ̂ t d ω ̂ .
4 N k = 0 N / 2 sin 2 π n 0 k N cos 2 π n k N
r ̂ ( ω ) = r ̂ * ( ω ) and G ( ω ) = G * ( ω ) .
C 1 + C 2 f ( ω ̂ ) d ω ̂ = 0 .
C 1 G ω ̂ ω 0 d ω ̂ 0 π i | ω ̂ | G e i θ ω ̂ ω 0 d θ .
| C 1 f ( ω ̂ ) d ω ̂ | C 1 | f ( ω ̂ ) | d ω ̂ C 1 | G | | i | ω ̂ | e i θ ω ̂ ω 0 | d θ .
| ω ̂ | , | G | 0 and | i | ω ̂ | e i θ ω ̂ ω 0 | 1 .
| C 1 f ( ω ̂ ) d ω ̂ | 0 and C 1 f ( ω ̂ ) d ω ̂ = 0 .
C 2 G ( ω ̂ ) ω ̂ ω 0 d ω ̂ = G ( ω ) d ω ω ω 0 + sc G ( ω ̂ ) ω ̂ ω 0 d ω ̂ ,
sc = ϕ = 0 π G ( ω ̂ ) e i θ i e i ϕ d ϕ = i 0 π G ( ω ̂ ) d ϕ = G ( ω 0 ) i π ,
G ( ω ) ω ω 0 d ω G ( ω 0 ) π i = 0
G ( ω 0 ) = 1 π i G ( ω ) ω ω 0 d ω ,
Re G ( ω 0 ) = 1 π Im G ( ω ) ω ω 0 d ω , Im G ( ω 0 ) = 1 π Re G ( ω ) ω ω 0 d ω .
̂ ( ω ) = 1 + j B j ω j 2 ω 2 i λ j ω .
G ( ω ) = H ( , ξ ) .
| H ξ | M for a | ̂ | b , 0 | ξ | C .
| H | = | G | M | ξ | for | ξ | a
ξ k ( ̂ ̂ b ) k = B k ( ω k 2 ω 2 ) i λ k ω
ω , ξ k 0 as const / ω 2 .
| G ( ω ) | 2 d ω = 2 0 | G ( ω ) | 2 d ω .
0 | G | 2 d ω 0 M 2 | ξ | 2 d ω
̂ ( ω ̂ ) = 1 + j B j ω j 2 ω ̂ 2 i λ j ω ̂
ξ ̂ k = B k / [ ( ω k 2 ω ̂ 2 ) i λ k ω ̂ ]
Im ( ω 0 ) = 2 π 0 0 Re ( ω ) cos ω t sin ω 0 t d ω d t , Re ( ω 0 ) = 2 π 0 0 Im ( ω ) sin ω t cos ω 0 t d ω d t ,
Im ( n 0 ) = 4 N n = 0 N / 2 Re ( n ) k = 0 N / 2 cos 2 π n k N sin 2 π n 0 k N , Re ( n 0 ) = 4 N n = 0 N / 2 Im ( n ) k = 0 N / 2 sin 2 π n k N cos 2 π n 0 k N ,
t = K Δ t , ω = 2 π f = 2 π n Δ F , T = N Δ T = 1 / Δ F , F = N Δ f = 1 / Δ T , Δ F Δ T = 1 / N ,
k = 0 N / 2 cos 2 π n k N sin 2 π n 0 k N
Im ( n 0 ) = 4 N n = 0 N / 2 Re ( n ) sin ( 2 π n 0 / N ) cos ( 2 π n 0 / N ) cos ( 2 π n 0 / N ) .
4 N sin ( 2 π n 0 / N ) cos 2 π n / N cos 2 π n 0 / N