Abstract

The consequences of causality and analyticity are commonly invoked in procedures for determining optical constants from reflectance data using the Kramers–Kronig relation. Here an entirely elementary argument is advanced, which exploits only the parity of Fourier transforms and the vanishing of the impulse response for negative times, and which avoids the concept of analyticity. This leads to a simply understood algorithm for such computations. The new procedure shows large gains of computational efficiency over the classical Kramers–Kronig approach. The method is applied first to model data and compared with exact results; it is then applied to real data and compared with the result obtained by the standard method. Excellent agreement is obtained in all cases.

© 1973 Optical Society of America

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References

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  1. T. S. Robinson, Proc. Phys. Soc. Lond. B 65, 910 (1952); J. S. Toll, Phys. Rev. 104, 1760 (1956); F. C. Jahoda, Phys. Rev. 107, 1261 (1957); M. Gottlieb, J. Opt. Soc. Am. 50, 343 (1960); H. Philipp and E. Taft, Phys. Rev. 136, A1445 (1964).
    [Crossref]
  2. R. de L. Kronig, Physica (Utr.) 3, 1009 (1936); H. A. Kramers, Atti Congr. Intern. Fisici, Como 2, 545 (1927).
    [Crossref]
  3. B. G. Anex and W. T. Simpson, Rev. Mod. Phys. 32, 466 (1960); B. Franconi, G. A. Gerhold, and W. T. Simpson, Mol. Cryst. Liq. Cryst. 6, 41 (1969).
    [Crossref]
  4. C. W. Peterson (private communication, 1970).
  5. J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297 (1965).
    [Crossref]
  6. See, e.g., J. D. Jackson, in Dispersion Relations, edited by G. R. Screaton (Wiley–Interscience, New York, 1961), p. 1.
  7. O. A. Ershov and V. M. Burtseva, Opt. Spektrosk. 27, 167 (1970) [Opt. Spectrosc. 27, 84 (1970)].

1970 (1)

O. A. Ershov and V. M. Burtseva, Opt. Spektrosk. 27, 167 (1970) [Opt. Spectrosc. 27, 84 (1970)].

1965 (1)

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

1960 (1)

B. G. Anex and W. T. Simpson, Rev. Mod. Phys. 32, 466 (1960); B. Franconi, G. A. Gerhold, and W. T. Simpson, Mol. Cryst. Liq. Cryst. 6, 41 (1969).
[Crossref]

1952 (1)

T. S. Robinson, Proc. Phys. Soc. Lond. B 65, 910 (1952); J. S. Toll, Phys. Rev. 104, 1760 (1956); F. C. Jahoda, Phys. Rev. 107, 1261 (1957); M. Gottlieb, J. Opt. Soc. Am. 50, 343 (1960); H. Philipp and E. Taft, Phys. Rev. 136, A1445 (1964).
[Crossref]

1936 (1)

R. de L. Kronig, Physica (Utr.) 3, 1009 (1936); H. A. Kramers, Atti Congr. Intern. Fisici, Como 2, 545 (1927).
[Crossref]

Anex, B. G.

B. G. Anex and W. T. Simpson, Rev. Mod. Phys. 32, 466 (1960); B. Franconi, G. A. Gerhold, and W. T. Simpson, Mol. Cryst. Liq. Cryst. 6, 41 (1969).
[Crossref]

Burtseva, V. M.

O. A. Ershov and V. M. Burtseva, Opt. Spektrosk. 27, 167 (1970) [Opt. Spectrosc. 27, 84 (1970)].

Cooley, J. W.

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

Ershov, O. A.

O. A. Ershov and V. M. Burtseva, Opt. Spektrosk. 27, 167 (1970) [Opt. Spectrosc. 27, 84 (1970)].

Jackson, J. D.

See, e.g., J. D. Jackson, in Dispersion Relations, edited by G. R. Screaton (Wiley–Interscience, New York, 1961), p. 1.

Kronig, R. de L.

R. de L. Kronig, Physica (Utr.) 3, 1009 (1936); H. A. Kramers, Atti Congr. Intern. Fisici, Como 2, 545 (1927).
[Crossref]

Peterson, C. W.

C. W. Peterson (private communication, 1970).

Robinson, T. S.

T. S. Robinson, Proc. Phys. Soc. Lond. B 65, 910 (1952); J. S. Toll, Phys. Rev. 104, 1760 (1956); F. C. Jahoda, Phys. Rev. 107, 1261 (1957); M. Gottlieb, J. Opt. Soc. Am. 50, 343 (1960); H. Philipp and E. Taft, Phys. Rev. 136, A1445 (1964).
[Crossref]

Simpson, W. T.

B. G. Anex and W. T. Simpson, Rev. Mod. Phys. 32, 466 (1960); B. Franconi, G. A. Gerhold, and W. T. Simpson, Mol. Cryst. Liq. Cryst. 6, 41 (1969).
[Crossref]

Tukey, J. W.

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

Math. Comput. (1)

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

Opt. Spektrosk. (1)

O. A. Ershov and V. M. Burtseva, Opt. Spektrosk. 27, 167 (1970) [Opt. Spectrosc. 27, 84 (1970)].

Physica (Utr.) (1)

R. de L. Kronig, Physica (Utr.) 3, 1009 (1936); H. A. Kramers, Atti Congr. Intern. Fisici, Como 2, 545 (1927).
[Crossref]

Proc. Phys. Soc. Lond. B (1)

T. S. Robinson, Proc. Phys. Soc. Lond. B 65, 910 (1952); J. S. Toll, Phys. Rev. 104, 1760 (1956); F. C. Jahoda, Phys. Rev. 107, 1261 (1957); M. Gottlieb, J. Opt. Soc. Am. 50, 343 (1960); H. Philipp and E. Taft, Phys. Rev. 136, A1445 (1964).
[Crossref]

Rev. Mod. Phys. (1)

B. G. Anex and W. T. Simpson, Rev. Mod. Phys. 32, 466 (1960); B. Franconi, G. A. Gerhold, and W. T. Simpson, Mol. Cryst. Liq. Cryst. 6, 41 (1969).
[Crossref]

Other (2)

C. W. Peterson (private communication, 1970).

See, e.g., J. D. Jackson, in Dispersion Relations, edited by G. R. Screaton (Wiley–Interscience, New York, 1961), p. 1.

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Figures (4)

Fig. 1
Fig. 1

Reflectance phase for single oscillator solid versus frequency separation from peak f0 (in units of Γ, the damping constant) times 100. Continuous curve is the exact phase, crosses show the time-domain result, for two cases of f0/Γ: On the left-hand side, f0 equals 2.0 and Γ equals 1.0, and on the right-hand side, f0 equals 2.0 and Γ equals 2.0. At the top is shown the corresponding first Fourier transform or impulse response versus time in units of (mesh)−1; the vertical stroke indicates the point halfway through the mesh.

Fig. 2
Fig. 2

Reflectance curves for one of the principal directions in the (111) plane for 7-7 dihydro-β-carotene (Ref. 4). Figure 2(a) is the laboratory polarized reflectance spectrum as a function of energy in the visible and near uv, and 2(b) is a model spectrum synthesized as the response of a number of damped oscillators, each characterized by an amplitude, peak frequency, and damping parameter. This model response was then treated by the time-domain method for purposes of comparison with the exactly calculable phase, hence extinction coefficient.

Fig. 3
Fig. 3

Comparison of molar extinction coefficients for model 7-7 dihydro-β-carotene: exact (curve) and calculated by the time-domain method (circles). A mesh of 1024 data points was used for the calculation. Corresponding model amplitude is shown in Fig. 2(b).

Fig. 4
Fig. 4

Analysis of laboratory data [as shown in Fig. 2(a)] by both the standard Kramers–Kronig technique (circles) and by the time-domain method (curve), for one of the principal directions on (111) for 7-7 dihydro-β-carotene. Total number of mesh points in the time-domain method was 1024.

Equations (21)

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δ ( t ) = 1 2 π d ω e i ω t ,
I ( t ) = 1 2 π d ω F ( ω ) e i ω t ,
F ( ω ) = d t e i ω t I ( t ) .
I ( t ) = 0 for t < 0 .
F ( ω ) = F ( ω ) * ,
F ( ω ) = S ( ω ) + i A ( ω ) ,
e i ω t = cos ω t + i sin ω t ,
I ( t ) = I S ( t ) + I A ( t ) .
I A ( t ) = I S ( t ) for t < 0 .
I A ( t ) = I S ( t ) for t > 0 .
i A ( ω ) = d t e i ω t I A ( t )
= 0 d t e i ω t I S ( t ) + 0 d t e i ω t I S ( t )
= 1 2 π 0 d t e i ω t d ω S ( ω ) e i ω t + 1 2 π 0 d t e i ω t d ω S ( ω ) e i ω t .
h ( t ) = { + 1 for t > 0 1 for t < 0 .
H ( ω ) = 1 / π i ω .
i A ( ω ) = H ( ω ) * S ( ω ) .
F ( ω ) = R ( ω ) e i θ ( ω ) ,
G ( ω ) = log F ( ω ) = log R ( ω ) + i θ ( ω ) .
x ( ω ) = 1 3 ω p 2 / ( ω 0 2 ω 2 + i Γ ω ) .
n = [ ( 1 + 2 x ) / ( 1 x ) ] 1 2 ,
R = ( n 1 ) / ( n + 1 ) .