Abstract

A comprehensive treatment of ATR spectra on the basis of the Lorentz-Lorenz law and Fresnel equations is given. The standard equation for the effective thickness is redefined showing how the lossy case deviates from the lossless case. A matrix effect due to the influence of the real part of the refractive index within an absorption band is demonstrated theoretically as well as experimentally. The concentration of a nonabsorbing solute sample, i.e., glucose, is determined by ATR spectroscopy near the critical angle.

© 1981 Optical Society of America

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References

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  1. J. Fahrenfort, Spectrochim. Acta 17, 698 (1961).
    [CrossRef]
  2. E. S. Shakaryan, Sov. J. Opt. Technol. 44 (12), 748 (1977).
  3. J. Fahrenfort, W. M. Visser, Spectrochim. Acta 17, 1103 (1961).
    [CrossRef]
  4. W. N. Hansen, Spectrochim. Acta 21, 815 (1965).
    [CrossRef]
  5. T. Hirschfeld, Appl. Spectrosc. 24, 2 (1970).
    [CrossRef]
  6. N. Korolev, S. A. Muraveva, J. Appl. Spectrosc. USSR 23, 1136 (1976).
    [CrossRef]
  7. B. J. Molochnikov et al., Sov. J. Opt. Technol. 46 (2), 88 (1979).
  8. G. Jungk, Appl. Opt. 19, 508 (1980).
    [CrossRef] [PubMed]
  9. H. Albrecht, G. Müller, M. Schaldach, Biomed. Tech. 23E, 98 (1978).
  10. C S. Blackwell, P. J. Degen, F. Osterholtz, Appl. Spectrosc. 32 (5), 480 (1978).
    [CrossRef]
  11. H. Brauner, G. Müller, Biomed. Tech. 25, 26 (1980).
    [CrossRef]
  12. R. Kellner, G. Gidály, Fresenius Z. Anal. Chem. 298, 32 (1979).
  13. W. N. Hansen, J. A. Horton, Anal. Chem. 39, 1097 (1967).
    [CrossRef]
  14. R. T. Holm, E. D. Palik, Opt. Eng. 17, 512 (1978).
  15. R. Wodick, Neue Auswerteverfahren für Reflexionsspektren und Spektren inhomogener Farbstoffverteilung, dargestellt am Beispiel von Hämoglobinspektren Thesis, Marburg, Federal Republic of Germany (1971).
  16. M. Born, Optik (Springer, Berlin, 1972).
    [CrossRef]
  17. K. Abraham, Diplomarbeit, U. Erlangen, Federal Republic of Germany (1980).
  18. F. Hund, Theoretische Physik B. II (Teubner, Stuttgart, 1957).
  19. N. J Harrick, Internal Reflection Spectroscopy (Harrick Scientific Corp., New York, 1967).
  20. N. J. Harrick, F. K. du Pré, Appl. Opt. 5, 1739 (1966).
    [CrossRef] [PubMed]
  21. N. J. Harrick, J. Opt. Soc. Am. 55, 851 (1965).
    [CrossRef]
  22. D. J. Epstein, Appl. Spectrosc. 34 (2), 233 (1980).
    [CrossRef]
  23. G. Müller, “Spectroscopy with the Evanescent Wave in the Visible Region of the Spectrum,” in Multichannel Image Detectors, Y. Talmi, Ed. (American Chemical Society, ACS Symposium Series 102, 1979).
    [CrossRef]

1980 (3)

1979 (2)

B. J. Molochnikov et al., Sov. J. Opt. Technol. 46 (2), 88 (1979).

R. Kellner, G. Gidály, Fresenius Z. Anal. Chem. 298, 32 (1979).

1978 (3)

C S. Blackwell, P. J. Degen, F. Osterholtz, Appl. Spectrosc. 32 (5), 480 (1978).
[CrossRef]

R. T. Holm, E. D. Palik, Opt. Eng. 17, 512 (1978).

H. Albrecht, G. Müller, M. Schaldach, Biomed. Tech. 23E, 98 (1978).

1977 (1)

E. S. Shakaryan, Sov. J. Opt. Technol. 44 (12), 748 (1977).

1976 (1)

N. Korolev, S. A. Muraveva, J. Appl. Spectrosc. USSR 23, 1136 (1976).
[CrossRef]

1970 (1)

T. Hirschfeld, Appl. Spectrosc. 24, 2 (1970).
[CrossRef]

1967 (1)

W. N. Hansen, J. A. Horton, Anal. Chem. 39, 1097 (1967).
[CrossRef]

1966 (1)

1965 (2)

N. J. Harrick, J. Opt. Soc. Am. 55, 851 (1965).
[CrossRef]

W. N. Hansen, Spectrochim. Acta 21, 815 (1965).
[CrossRef]

1961 (2)

J. Fahrenfort, Spectrochim. Acta 17, 698 (1961).
[CrossRef]

J. Fahrenfort, W. M. Visser, Spectrochim. Acta 17, 1103 (1961).
[CrossRef]

Abraham, K.

K. Abraham, Diplomarbeit, U. Erlangen, Federal Republic of Germany (1980).

Albrecht, H.

H. Albrecht, G. Müller, M. Schaldach, Biomed. Tech. 23E, 98 (1978).

Blackwell, C S.

Born, M.

M. Born, Optik (Springer, Berlin, 1972).
[CrossRef]

Brauner, H.

H. Brauner, G. Müller, Biomed. Tech. 25, 26 (1980).
[CrossRef]

Degen, P. J.

du Pré, F. K.

Epstein, D. J.

Fahrenfort, J.

J. Fahrenfort, Spectrochim. Acta 17, 698 (1961).
[CrossRef]

J. Fahrenfort, W. M. Visser, Spectrochim. Acta 17, 1103 (1961).
[CrossRef]

Gidály, G.

R. Kellner, G. Gidály, Fresenius Z. Anal. Chem. 298, 32 (1979).

Hansen, W. N.

W. N. Hansen, J. A. Horton, Anal. Chem. 39, 1097 (1967).
[CrossRef]

W. N. Hansen, Spectrochim. Acta 21, 815 (1965).
[CrossRef]

Harrick, N. J

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

Harrick, N. J.

Hirschfeld, T.

T. Hirschfeld, Appl. Spectrosc. 24, 2 (1970).
[CrossRef]

Holm, R. T.

R. T. Holm, E. D. Palik, Opt. Eng. 17, 512 (1978).

Horton, J. A.

W. N. Hansen, J. A. Horton, Anal. Chem. 39, 1097 (1967).
[CrossRef]

Hund, F.

F. Hund, Theoretische Physik B. II (Teubner, Stuttgart, 1957).

Jungk, G.

Kellner, R.

R. Kellner, G. Gidály, Fresenius Z. Anal. Chem. 298, 32 (1979).

Korolev, N.

N. Korolev, S. A. Muraveva, J. Appl. Spectrosc. USSR 23, 1136 (1976).
[CrossRef]

Molochnikov, B. J.

B. J. Molochnikov et al., Sov. J. Opt. Technol. 46 (2), 88 (1979).

Müller, G.

H. Brauner, G. Müller, Biomed. Tech. 25, 26 (1980).
[CrossRef]

H. Albrecht, G. Müller, M. Schaldach, Biomed. Tech. 23E, 98 (1978).

G. Müller, “Spectroscopy with the Evanescent Wave in the Visible Region of the Spectrum,” in Multichannel Image Detectors, Y. Talmi, Ed. (American Chemical Society, ACS Symposium Series 102, 1979).
[CrossRef]

Muraveva, S. A.

N. Korolev, S. A. Muraveva, J. Appl. Spectrosc. USSR 23, 1136 (1976).
[CrossRef]

Osterholtz, F.

Palik, E. D.

R. T. Holm, E. D. Palik, Opt. Eng. 17, 512 (1978).

Schaldach, M.

H. Albrecht, G. Müller, M. Schaldach, Biomed. Tech. 23E, 98 (1978).

Shakaryan, E. S.

E. S. Shakaryan, Sov. J. Opt. Technol. 44 (12), 748 (1977).

Visser, W. M.

J. Fahrenfort, W. M. Visser, Spectrochim. Acta 17, 1103 (1961).
[CrossRef]

Wodick, R.

R. Wodick, Neue Auswerteverfahren für Reflexionsspektren und Spektren inhomogener Farbstoffverteilung, dargestellt am Beispiel von Hämoglobinspektren Thesis, Marburg, Federal Republic of Germany (1971).

Anal. Chem. (1)

W. N. Hansen, J. A. Horton, Anal. Chem. 39, 1097 (1967).
[CrossRef]

Appl. Opt. (2)

Appl. Spectrosc. (3)

Biomed. Tech. (2)

H. Brauner, G. Müller, Biomed. Tech. 25, 26 (1980).
[CrossRef]

H. Albrecht, G. Müller, M. Schaldach, Biomed. Tech. 23E, 98 (1978).

Fresenius Z. Anal. Chem. (1)

R. Kellner, G. Gidály, Fresenius Z. Anal. Chem. 298, 32 (1979).

J. Appl. Spectrosc. USSR (1)

N. Korolev, S. A. Muraveva, J. Appl. Spectrosc. USSR 23, 1136 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

R. T. Holm, E. D. Palik, Opt. Eng. 17, 512 (1978).

Sov. J. Opt. Technol. (2)

B. J. Molochnikov et al., Sov. J. Opt. Technol. 46 (2), 88 (1979).

E. S. Shakaryan, Sov. J. Opt. Technol. 44 (12), 748 (1977).

Spectrochim. Acta (3)

J. Fahrenfort, W. M. Visser, Spectrochim. Acta 17, 1103 (1961).
[CrossRef]

W. N. Hansen, Spectrochim. Acta 21, 815 (1965).
[CrossRef]

J. Fahrenfort, Spectrochim. Acta 17, 698 (1961).
[CrossRef]

Other (6)

R. Wodick, Neue Auswerteverfahren für Reflexionsspektren und Spektren inhomogener Farbstoffverteilung, dargestellt am Beispiel von Hämoglobinspektren Thesis, Marburg, Federal Republic of Germany (1971).

M. Born, Optik (Springer, Berlin, 1972).
[CrossRef]

K. Abraham, Diplomarbeit, U. Erlangen, Federal Republic of Germany (1980).

F. Hund, Theoretische Physik B. II (Teubner, Stuttgart, 1957).

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

G. Müller, “Spectroscopy with the Evanescent Wave in the Visible Region of the Spectrum,” in Multichannel Image Detectors, Y. Talmi, Ed. (American Chemical Society, ACS Symposium Series 102, 1979).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Real and imaginary parts of the complex refractive index according to the Lorentz-Lorenz law for a model substance by numerical calculation of Eqs. (5) and (6). The parameters are MG0 = 18 g/mole, ρ0 = 0.998323 kg/l, α0 = 1.5 × 10−27 l/particle, MGx = 691.91 g/mole, ρx = 2 kg/l, αx = 8 × 10−26 l/particle. The dashed line would be valid for an absorption which could be described by Beer’s law.

Fig. 2
Fig. 2

Coordinate system as used in this paper at the interface between an optically dense medium 1 (refractive index n1) and an optically rare medium 2 (refractive index n2). A plane electromagnetic wave (amplitude E) strikes the interface by an angle of incidence θ and is partially refracted (amplitude G) and partially reflected (amplitude r).

Fig. 3
Fig. 3

Reflectivity R as a function of the absorption index ɛ × c of the sample for some angles of incidence θ. The real part of the refractive index n2 of the sample is assumed to remain constant (solid curves).19 The dashed line is an example for a more realistic situation, i.e., taking into account a change in n2.

Fig. 4
Fig. 4

Natural logarithm of the reflectivity R as a function of the concentration (e.g., Eosine B) with the angle of incidence θ as a parameter. Values used were n1 = 1.7, n20 = 1.333, Δn2 = 0, λ = 589 nm, ɛ = 250 l/g cm, θc = 51.6°. The dashed lines would be valid for an absorption which could be described by Beer’s law.

Fig. 5
Fig. 5

Penetration depth dp vs angle of incidence according to Eq. (22a) for some concentrations (the other parameters are the same as in Fig. 4).

Fig. 6
Fig. 6

Effective layer thickness according to Eqs. (25) and (26), respectively. The bold line in (a) corresponds to a true effective layer thickness as defined by Eq. (15). For further discussion see text.

Fig. 7
Fig. 7

Absorption spectrum of an aqueous solution of Eosine. The matrix shifts as induced by various concentrations of proteins is clearly visible. A PARC OMA II-spectrometer system was used as described in Refs. 17 and 23.

Fig. 8
Fig. 8

Principal behavior of the dispersion curves of water, proteins, and Eosine B. For further discussions see text.

Fig. 9
Fig. 9

Influence of the angle of incidence on the matrix shift. Plotted are the reflectivity vs the concentration of proteins within an aqueous solution of Eosine, where the concentration of Eosine was kept constant (cEosine = 40 g/l).

Fig. 10
Fig. 10

Calculated reflectivity R vs angle of incidence for some values of the absorption index.

Fig. 11
Fig. 11

Reflection spectrum of an aqueous solution of various glucose concentrations. As indicated by the dashed line, the measurements can be made sensitive to a concentration variation.

Equations (36)

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α ˆ = 3 4 π M G N L 1 c n ˆ 2 1 n ˆ 2 + 2 ,
n ˆ = n ( 1 + i κ ) .
α ˆ = α + i β .
4 π 3 N L ( α ˆ 1 M G 1 c 1 + α ˆ 2 M G 2 c 2 + α ˆ 3 M G 3 c 3 + ) = n ˆ 2 1 n ˆ 2 + 2 .
4 π 3 N L [ α 0 M G 0 · ρ 0 ( 1 c x ρ x ) + 1 M G x ( α x + i β x ) c x ] = n 2 ( 1 + i κ ) 2 1 n 2 ( 1 + i κ ) 2 + 2 ,
n 2 = 3 A 2 + B 2 ( 3 4 A ) + 1 ( 1 1 2 3 A A 2 + B 2 ) ,
κ = B 2 n 2 · 3 A 2 + B 2 ,
A = 1 4 π 3 N L [ ρ 0 M G 0 ( 1 c x ρ x ) α 0 + 1 M G x c x α x ] , B = 4 π 3 N L c x M G x β x .
n ( c x ) = n 0 + Δ ˜ n c x ,
κ ( c x ) = λ · ε 4 π n c x ,
r = E cos θ n ˆ 21 2 sin 2 θ cos θ + n ˆ 21 2 sin 2 θ ,
r = E n ˆ 21 2 cos θ n ˆ 21 2 sin 2 θ n ˆ 21 2 cos θ + n ˆ 21 2 sin 2 θ ,
n ˆ 21 = n ˆ 2 n 1 , n 21 = n 2 n 1 ,
R = r · r * = | r | 2 , R = r · r * = | r | 2 .
n 21 2 sin 2 θ > 0 ( external reflection ) θ < θ c , n 21 2 sin 2 θ < 0 ( external reflection ) θ > θ c .
R = ξ η ± ξ + η ± η + for θ > θ c , η for θ < θ c ,
R = u υ ± u + υ ± υ + for θ > θ c , υ for θ < θ c ,
ξ = α 2 + ( ν 2 + μ 2 ) 1 / 2 , η ± = ± 2 α [ ( ν 2 + μ 2 ) 1 / 2 ν ] 1 / 2 , u = α 2 [ ( sin 2 θ ν 2 ) 2 + μ 2 ] + ( ν 2 + μ 2 ) 1 / 2 , υ ± = ± 2 α { 1 2 ( n 21 2 n 21 2 κ 2 ) [ ( ν 2 + μ 2 ) 1 / 2 ν ] 1 / 2 ± μ 1 2 [ ( ν 2 + μ 2 ) 1 / 2 ± ν ] 1 / 2 } ν = β + n 21 2 κ 2 μ = 2 n 21 2 κ 2 β = sin 2 θ n 21 2 . α = cos θ
R = R 0 · exp ( a · d eff ) = R 0 · exp ( ε · c d eff ) ln R 0 R = d eff · ε · c ,
ln R = ln [ R ( κ = 0 ) ] + ( ln R κ ) 0 κ + 1 2 ( 2 ln R κ 2 ) 0 κ 2 + .
d eff = ( ln R κ ) 0 ,
d eff = n 21 cos θ 0 E 2 d x = n 21 cos θ 0 E 0 exp [ ( x / d p ) ] d x ,
d * eff = λ n 21 cos θ n 1 π ( 1 n 21 2 ) sin 2 θ n 21 2 ,
d * eff = λ n 21 cos θ ( 2 sin θ n 21 2 ) n 1 π ( 1 n 21 2 ) [ ( 1 + n 21 2 ) sin 2 θ n 21 2 ] sin 2 θ n 21 2 .
E x ( 2 ) = E x 0 ( 2 ) exp [ i 2 π λ ( n ˆ 2 n 0 · x 0 c t ) ] ,
n 0 = ( cos θ sin θ 0 ) ; x 0 = ( x y 0 ) ; n 0 · x 0 = x cos θ + y sin θ , sin θ = 1 n ˆ 21 sin θ ; cos θ = 1 n ˆ 21 n ˆ 21 2 sin 2 θ ;
E x ( 2 ) = E x 0 ( 2 ) exp { ( i π λ ) [ n 1 ( i x sin 2 θ n ˆ 21 2 + y · sin θ ) c t ] } .
E x ( 2 ) = E x 0 ( 2 ) exp [ ( 2 π λ ) n 1 sin 2 θ n 21 2 x ] × exp [ i 2 π λ ( n 1 sin θ · y c t ) ] = E x 0 ( 2 ) exp ( d p x ) exp [ i 2 π λ ( n 1 sin θ · y c t ) ] .
d p = λ 2 π n 1 sin 2 θ n 21 2 .
E x ( 2 ) = E x 0 ( 2 ) exp ( 2 π λ n 1 2 ν 2 + μ 2 + ν x ) × exp [ ( i 2 π λ n 1 ) ( 1 2 ν 2 + μ 2 ν x + sin θ · y c ) ] ,
d p = 2 λ 2 π n 1 ( ν 2 + μ 2 ν ) 1 .
E 0 ( 2 ) = ( E x 0 ( 2 ) E y 0 ( 2 ) E z 0 ( 2 ) ) E ( 2 ) = E z ( 2 ) E ( 2 ) = E x ( 2 ) + E y ( 2 ) , E x 0 ( 2 ) = G sin θ , E y 0 ( 2 ) = G cos θ , E z 0 ( 2 ) = G ,
G = 2 cos θ cos θ + n ˆ 21 2 sin 2 θ , G = 2 n ˆ 21 cos θ n ˆ 21 cos θ + n ˆ 21 2 sin 2 θ .
d eff = 2 n 21 λ cos θ π n 1 ( ξ + η + ) ν 2 + μ 2 + ν ,
d eff = 2 n 21 λ cos θ ( sin θ + ν 2 + μ 2 ) π n 1 ( u + υ + ) ν 2 + μ 2 + ν .
n 2 = ( n 0 + Δ ˜ n 0 · c ) matrix + Δ ˜ n c sample substance

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