Abstract

A modified Kramers–Kronig integral is derived, which offers greater convergence than the conventional expression when reflectance data are available over a limited range. The modified expression produces good convergence when applied to synthetic spectra.

© 1971 Optical Society of America

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References

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  1. T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).
  2. G. Andermann, A. Caron, and D. A. Dows, J. Opt. Soc. Am. 55, 1210 (1965).
    [Crossref]
  3. D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
    [Crossref]
  4. R. Z. Bachrach and F. C. Brown, Phys. Rev. B1, 818 (1970).
  5. T. S. Moss, Optical Properties of Semiconductors (Butterworths, London, 1959), Appendix B.
  6. R. K. Ahrenkiel, F. Moser, S. Lyu, and C. R. Pidgeon, J. Appl. Phys. 42, 1452 (1971).
    [Crossref]
  7. R. A. Roberts, Ph.D. thesis, University of California, Santa Barbara, 1967.

1971 (1)

R. K. Ahrenkiel, F. Moser, S. Lyu, and C. R. Pidgeon, J. Appl. Phys. 42, 1452 (1971).
[Crossref]

1970 (1)

R. Z. Bachrach and F. C. Brown, Phys. Rev. B1, 818 (1970).

1965 (2)

1952 (1)

T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).

Ahrenkiel, R. K.

R. K. Ahrenkiel, F. Moser, S. Lyu, and C. R. Pidgeon, J. Appl. Phys. 42, 1452 (1971).
[Crossref]

Andermann, G.

Bachrach, R. Z.

R. Z. Bachrach and F. C. Brown, Phys. Rev. B1, 818 (1970).

Brown, F. C.

R. Z. Bachrach and F. C. Brown, Phys. Rev. B1, 818 (1970).

Caron, A.

Dows, D. A.

Lyu, S.

R. K. Ahrenkiel, F. Moser, S. Lyu, and C. R. Pidgeon, J. Appl. Phys. 42, 1452 (1971).
[Crossref]

Moser, F.

R. K. Ahrenkiel, F. Moser, S. Lyu, and C. R. Pidgeon, J. Appl. Phys. 42, 1452 (1971).
[Crossref]

Moss, T. S.

T. S. Moss, Optical Properties of Semiconductors (Butterworths, London, 1959), Appendix B.

Pidgeon, C. R.

R. K. Ahrenkiel, F. Moser, S. Lyu, and C. R. Pidgeon, J. Appl. Phys. 42, 1452 (1971).
[Crossref]

Roberts, R. A.

R. A. Roberts, Ph.D. thesis, University of California, Santa Barbara, 1967.

Robinson, T. S.

T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).

Roessler, D. M.

D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
[Crossref]

Brit. J. Appl. Phys. (1)

D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
[Crossref]

J. Appl. Phys. (1)

R. K. Ahrenkiel, F. Moser, S. Lyu, and C. R. Pidgeon, J. Appl. Phys. 42, 1452 (1971).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Rev. (1)

R. Z. Bachrach and F. C. Brown, Phys. Rev. B1, 818 (1970).

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

T. S. Robinson, Proc. Phys. Soc. (London) B65, 910 (1952).

Other (2)

T. S. Moss, Optical Properties of Semiconductors (Butterworths, London, 1959), Appendix B.

R. A. Roberts, Ph.D. thesis, University of California, Santa Barbara, 1967.

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Figures (7)

Fig. 1
Fig. 1

Synthetic reflectance spectra used to estimate convergence of SKK procedure relative to that of ordinary KK method.

Fig. 2
Fig. 2

Synthetic spectrum evolved from three damped, harmonic oscillators: —, reflectance; ⋯, κ.

Fig. 3
Fig. 3

Values of κ computed by the SKK method using the three designated integration ranges: ○, 0.5–2.7 eV; ■, 0.5–3.0 eV; △, 0.5–3.5 eV.

Fig. 4
Fig. 4

Measured reflectance (curve R) of CdCr2Se4 at 4.2 K and the extinction coefficient (curve K) calculated by SKK technique.

Fig. 5
Fig. 5

Calculation of ϕ0 for CdCr2Se4 as explained in the text. Curve 1 is the SKK calculation of ϕ(E) assuming ϕ0 = 0 at 2.2 eV. The solid line gives the slope of ϕ(E) below the band edge from which we get ϕ0 = 0.383 rad. Finally, the corrected ϕ(E) is plotted in curve 2.

Fig. 6
Fig. 6

The reflectance of diamond. Lines A and B indicate the low- and high-energy cutoffs mentioned in the text.

Fig. 7
Fig. 7

The values of 2 computed by Roberts is given by curve 1. The SKK computation of 2 is given by curve 2.

Tables (2)

Tables Icon

Table I Error in κ for various integration ranges.

Tables Icon

Table II Error in κ for various expansion points. Integration range 0.50–3.50 eV.

Equations (12)

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ϕ KK ( E ) = - ( E Π ) P 0 ln R ( X ) d X X 2 - E 2 ,
ϕ ( E ) = lim Δ 0 { - ( E Π ) 0 E - Δ ln R ( X ) d X X 2 - E 2 - ( E Π ) E + Δ ln R ( X ) X 2 - E 2 d X - ( 2 Δ E Π ) [ ln R ( X ) X 2 - E 2 | E - Δ E + Δ ] } .
ϕ 0 = - ( E 0 Π ) P 0 ln R ( X ) d X X 2 - E 0 2 .
ϕ E - ϕ 0 E 0 = 1 Π P 0 ln R ( X ) d X X 2 - E 0 2 - 1 Π P 0 ln R ( X ) d X X 2 - E 2 .
ϕ E - ϕ E 0 = 1 Π ( E 0 2 - E 2 ) P 0 ln R ( X ) d X ( X 2 - E 0 2 ) ( X 2 - E 2 ) ,
ϕ SKK ( E ) = E E 0 ϕ 0 + E ( E 0 2 - E 2 ) Π P 0 ln R ( X ) d X ( X 2 - E 0 2 ) ( X 2 - E 2 ) .
Δ [ ln R ( X ) ] | E 1 - δ E 1 + δ = A δ ( E 1 ) .
Δ ϕ KK ( E ) = - ( E Π ) E b Δ [ ln R ( X ) ] X 2 - E 2 d X = - ( E Π ) · A E 1 2 - E 2 .
Δ ϕ SKK ( E ) = E ( E 0 2 - E 2 ) Π E b Δ [ ln R ( X ) ] d x ( X 2 - E 0 2 ) ( X 2 - E 2 ) E ( E 0 2 - E 2 ) Π A ( E 1 2 - E 0 2 ) ( E 1 2 - E 2 ) .
R = Δ ϕ SKK ( E ) Δ ϕ KK ( E ) = E 2 - E 0 2 E 1 2 - E 2 .
ϕ SKK ( E ) = [ E ( E 0 2 - E 2 ) / π ] I ,
E ( E 0 2 - E 2 ) Π · I = - E E 0 ϕ 0