Abstract

We show that modified Kramers–Kronig relations provide a useful tool to test the validity of the complex refractive index extracted from transmission terahertz spectra of porous matrices containing pharmaceutical materials. The role of scattering of terahertz radiation is qualitatively considered as a reason for the observed discrepancy between experimental data and the values extracted from the inverted complex refractive index. As an example we present an analysis of the terahertz spectra of carbamazepine and lactose α-monohydrate.

© 2010 Optical Society of America

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  1. R. Kronig, J. Opt. Soc. Am. 12, 547 (1926).
    [CrossRef]
  2. H. A. Kramers, Phys. Z. 30, 522 (1929).
  3. V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).
  4. E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, Appl. Phys. Lett. 89, 142903 (2006).
    [CrossRef]
  5. K.-E. Peiponen and J. J. Saarinen, Rep. Prog. Phys. 72, 056401 (2009).
    [CrossRef]
  6. J. A. Zeitler, P. F. Taday, D. A. Newnham, M. Pepper, K. C. Gordon, and T. Rades, J. Pharm. Pharmacol. 59, 209 (2007).
    [CrossRef] [PubMed]
  7. J. A. Zeitler, P. F. Taday, K. C. Gordon, M. Pepper, and T. Rades, ChemPhysChem 8, 1924 (2007).
    [CrossRef] [PubMed]
  8. J. A. Zeitler, P. F. Taday, M. Pepper, and T. Rades, J. Pharm. Sci. 96, 2703 (2007).
    [CrossRef] [PubMed]
  9. I. Pupeza, R. Wilk, and M. Koch, Opt. Express 15, 4335 (2007).
    [CrossRef] [PubMed]
  10. M. Franz, B. M. Fischer, and M. Walther, Appl. Phys. Lett. 92, 021107 (2008).
    [CrossRef]
  11. E. P. J. Parrott, J. A. Zeitler, and L. F. Gladden, Opt. Lett. 34, 3722 (2009).
    [CrossRef] [PubMed]
  12. X. Xu, X. Wang, and L. Wang, Appl. Opt. 45, 648 (2006).
    [CrossRef] [PubMed]
  13. J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
    [CrossRef]

2009 (2)

2008 (1)

M. Franz, B. M. Fischer, and M. Walther, Appl. Phys. Lett. 92, 021107 (2008).
[CrossRef]

2007 (5)

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

J. A. Zeitler, P. F. Taday, D. A. Newnham, M. Pepper, K. C. Gordon, and T. Rades, J. Pharm. Pharmacol. 59, 209 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, P. F. Taday, K. C. Gordon, M. Pepper, and T. Rades, ChemPhysChem 8, 1924 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, P. F. Taday, M. Pepper, and T. Rades, J. Pharm. Sci. 96, 2703 (2007).
[CrossRef] [PubMed]

I. Pupeza, R. Wilk, and M. Koch, Opt. Express 15, 4335 (2007).
[CrossRef] [PubMed]

2006 (2)

X. Xu, X. Wang, and L. Wang, Appl. Opt. 45, 648 (2006).
[CrossRef] [PubMed]

E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, Appl. Phys. Lett. 89, 142903 (2006).
[CrossRef]

1929 (1)

H. A. Kramers, Phys. Z. 30, 522 (1929).

1926 (1)

Aaltonen, J.

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

Fischer, B. M.

M. Franz, B. M. Fischer, and M. Walther, Appl. Phys. Lett. 92, 021107 (2008).
[CrossRef]

Franz, M.

M. Franz, B. M. Fischer, and M. Walther, Appl. Phys. Lett. 92, 021107 (2008).
[CrossRef]

Gladden, L. F.

Gordon, K. C.

J. A. Zeitler, P. F. Taday, K. C. Gordon, M. Pepper, and T. Rades, ChemPhysChem 8, 1924 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, P. F. Taday, D. A. Newnham, M. Pepper, K. C. Gordon, and T. Rades, J. Pharm. Pharmacol. 59, 209 (2007).
[CrossRef] [PubMed]

Gornov, E.

E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, Appl. Phys. Lett. 89, 142903 (2006).
[CrossRef]

Ino, Y.

E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, Appl. Phys. Lett. 89, 142903 (2006).
[CrossRef]

Koch, M.

Kogermann, K.

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

Kramers, H. A.

H. A. Kramers, Phys. Z. 30, 522 (1929).

Kronig, R.

Kuwata-Gonokami, M.

E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, Appl. Phys. Lett. 89, 142903 (2006).
[CrossRef]

Lucarini, V.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).

Newnham, D. A.

J. A. Zeitler, P. F. Taday, D. A. Newnham, M. Pepper, K. C. Gordon, and T. Rades, J. Pharm. Pharmacol. 59, 209 (2007).
[CrossRef] [PubMed]

Parrott, E. P. J.

Peiponen, K. -E.

K.-E. Peiponen and J. J. Saarinen, Rep. Prog. Phys. 72, 056401 (2009).
[CrossRef]

E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, Appl. Phys. Lett. 89, 142903 (2006).
[CrossRef]

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).

Pepper, M.

J. A. Zeitler, P. F. Taday, D. A. Newnham, M. Pepper, K. C. Gordon, and T. Rades, J. Pharm. Pharmacol. 59, 209 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

J. A. Zeitler, P. F. Taday, K. C. Gordon, M. Pepper, and T. Rades, ChemPhysChem 8, 1924 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, P. F. Taday, M. Pepper, and T. Rades, J. Pharm. Sci. 96, 2703 (2007).
[CrossRef] [PubMed]

Pupeza, I.

Rades, T.

J. A. Zeitler, P. F. Taday, K. C. Gordon, M. Pepper, and T. Rades, ChemPhysChem 8, 1924 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, P. F. Taday, D. A. Newnham, M. Pepper, K. C. Gordon, and T. Rades, J. Pharm. Pharmacol. 59, 209 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, P. F. Taday, M. Pepper, and T. Rades, J. Pharm. Sci. 96, 2703 (2007).
[CrossRef] [PubMed]

Rantanen, J.

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

Rodes, T.

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

Saarinen, J. J.

K.-E. Peiponen and J. J. Saarinen, Rep. Prog. Phys. 72, 056401 (2009).
[CrossRef]

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).

Strachan, C. J.

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

Svirko, Y.

E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, Appl. Phys. Lett. 89, 142903 (2006).
[CrossRef]

Taday, P. F.

J. A. Zeitler, P. F. Taday, M. Pepper, and T. Rades, J. Pharm. Sci. 96, 2703 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

J. A. Zeitler, P. F. Taday, K. C. Gordon, M. Pepper, and T. Rades, ChemPhysChem 8, 1924 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, P. F. Taday, D. A. Newnham, M. Pepper, K. C. Gordon, and T. Rades, J. Pharm. Pharmacol. 59, 209 (2007).
[CrossRef] [PubMed]

Vartiainen, E. M.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).

Walther, M.

M. Franz, B. M. Fischer, and M. Walther, Appl. Phys. Lett. 92, 021107 (2008).
[CrossRef]

Wang, L.

Wang, X.

Wilk, R.

Xu, X.

Zeitler, J. A.

E. P. J. Parrott, J. A. Zeitler, and L. F. Gladden, Opt. Lett. 34, 3722 (2009).
[CrossRef] [PubMed]

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

J. A. Zeitler, P. F. Taday, K. C. Gordon, M. Pepper, and T. Rades, ChemPhysChem 8, 1924 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, P. F. Taday, D. A. Newnham, M. Pepper, K. C. Gordon, and T. Rades, J. Pharm. Pharmacol. 59, 209 (2007).
[CrossRef] [PubMed]

J. A. Zeitler, P. F. Taday, M. Pepper, and T. Rades, J. Pharm. Sci. 96, 2703 (2007).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, Appl. Phys. Lett. 89, 142903 (2006).
[CrossRef]

M. Franz, B. M. Fischer, and M. Walther, Appl. Phys. Lett. 92, 021107 (2008).
[CrossRef]

ChemPhysChem (1)

J. A. Zeitler, P. F. Taday, K. C. Gordon, M. Pepper, and T. Rades, ChemPhysChem 8, 1924 (2007).
[CrossRef] [PubMed]

Int. J. Pharm. (1)

J. A. Zeitler, K. Kogermann, J. Rantanen, T. Rodes, P. F. Taday, M. Pepper, J. Aaltonen, and C. J. Strachan, Int. J. Pharm. 334, 78 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Pharm. Pharmacol. (1)

J. A. Zeitler, P. F. Taday, D. A. Newnham, M. Pepper, K. C. Gordon, and T. Rades, J. Pharm. Pharmacol. 59, 209 (2007).
[CrossRef] [PubMed]

J. Pharm. Sci. (1)

J. A. Zeitler, P. F. Taday, M. Pepper, and T. Rades, J. Pharm. Sci. 96, 2703 (2007).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Z. (1)

H. A. Kramers, Phys. Z. 30, 522 (1929).

Rep. Prog. Phys. (1)

K.-E. Peiponen and J. J. Saarinen, Rep. Prog. Phys. 72, 056401 (2009).
[CrossRef]

Other (1)

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).

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Figures (2)

Fig. 1
Fig. 1

Dispersion data obtained from Fresnel’s theory (solid curve) and from modified K–K analysis (dashed curve).

Fig. 2
Fig. 2

Similar notations are used as in Fig. 1 but for the sample of lactose α-monohydrate.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

E sample E reference = | T ( ω )   exp ( i ϕ ( ω ) ) | ,
E sample ( ω ) E reference ( ω ) = t reference sample ( ω ) t sample air ( ω ) exp ( i ( N sample 1 ) ω d c ) ,
n ( ω ) 1 2 π P ω a ω b ω κ ( ω ) ω 2 ω 2 d ω ,
κ ( ω ) 2 ω π P ω a ω b n ( ω ) 1 ω 2 ω 2 d ω ,
Re { f ( ω ) } = 2 π P ω a ω b ω   Im { f ( ω ) } ω 2 ω 2 d ω ,
Im { f ( ω ) } = 2 ω π P ω a ω b Re { f ( ω ) } ω 2 ω 2 d ω ,
f ( ω ) = [ N ( ω ) N ( ω a ) ] [ N ( ω ) N ( ω a ) ] [ N ( ω ) N ( ω b ) ] [ N ( ω ) N ( ω b ) ] ,

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