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Wiener Bounds for Complex Permittivity in Terahertz Spectroscopy: Case Study of Two-Phase Pharmaceutical Tablets HEIKKI TUONONEN,* KAORI FUKUNAGA, MARKO KUOSMANEN, JARKKO KETOLAINEN, and KAI-ERIK PEIPONEN Department of Physics and Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland (H.T., K.-E.P.) National Institute of Information and Communication Technology, Koganei, Tokyo 184-8795, Japan (K.F.) and Department of Pharmaceutics, University of Kuopio, P.O. Box 1627, FI-70211 Kuopio, Finland (M.K., J.K.) The terahertz measurement technique has become popular in the field of pharmaceutical technology for tablet quality inspection. Spectral data obtained from the tablets is based on the utilization of Fresnel���s formulas for an ideal slab. However, a tablet is a porous medium. Hence, in the THz gap one has to assume that a tablet constitutes at least an effective medium if the Fresnel theory is applied in quantitative permittivity spectra analysis. Hence, it is suggested that one should consider instead of the permittivity of homogeneous media the concept of effective permittivity in the THz terminology of porous tablets. Usually the fill factor of a component of a tablet is known but not the detailed bulk structure. Nevertheless, it is possible to estimate the complex effective permittivity of a tablet with the aid of so-called Wiener bounds. The idea of this article is to present a modification of Wiener bounds applied to the estimation of the real and imaginary part of the permittivity of the pure component of a tablet. As an example, the effective complex permittivity of a starch acetate tablet is considered. Index Headings: Pharmaceutical tablet Terahertz spectroscopy Complex permittivity. INTRODUCTION Inspection of porous media by pulsed terahertz (THz) waves,1 especially in the field of pharmaceutical sciences, has drawn much attention. The technique of THz measurement of pharmaceutical tablets has become popular due to the fact that it is possible to attain image and spectral data for structural studies and chemical mapping of relevant species.2���4 Process analytical technology (PAT), which is gaining much interest in the sector of pharmaceutical industry, could make use of THz technology in tablet inspection.5 The spectral data from a pharmaceutical compact yields, e.g., the complex refractive index of the tablet. In principle, such data can be used for the identification and monitoring of the concentration of a component of the tablet. Information on the complex refractive index can be obtained in the reflection measurement mode,6 but this measurement mode and the relevant spectral data is highly susceptible to error because of possible sample displacement with respect to the reference. The error can be revealed and corrected,7 or one can avoid the error by using the principle of ellipsometric measurement.8 Errors in THz transmission data may also be present,9 and the method of data correction7 can be used as well in the transmission studies. Even though the measurement uncertainties can be solved, there remains one important factor that has not been, as far as we know, stressed enough in the THz studies on pharmaceutical tablets. Namely, a tablet should be treated always at least as an effective medium10 if scattering of THz radiation is neglected. This has some consequences in the interpretation of the complex permittivity (or complex refractive index) that is obtained from the porous pharmaceu- tical tablet with the aid of the Fresnel theory,11 which is valid for isotropic and homogeneous media, either from transmission or reflection data. The problem arises from the fact that the voids due to air or other substances can sometimes have an important role in the magnitude and even the location of the spectral features in the frame of the model of effective complex permittivity. Even the effective model itself for a tablet is rather problematic because it typically assumes regular shape of inclusions. Regular shape of inclusions is usually not the case when pharmaceutical tablets are concerned. Hence, one must seek an approximate model and use, e.g., the Wiener bounds12 for the prediction of the intrinsic permittivity of the powder materials used in tablet making. In this paper it is shown with the aid of a modification of the Wiener bounds that there may be a rather large uncertainty in the magnitude of the complex permittivity of a component of a tablet therefore, some care has to be taken when interpreting quantitative measures of a tablet���s components that affect estimation of density or porosity by the THz technique. THEORY In this study the transmission measurement mode was used. In the case of recording a transmission spectrum in the THz gap, two waveforms Esample(t) and Ereference(t), with and without the tablet, are measured in the time domain. The reference waveform contains information about the shape of the THz pulse and the geometry of the experimental setup. The measured waveforms are transformed by means of Fourier transform to the frequency domain. The ratio of the signal and reference fields is obtained from Fresnel theory for a homogeneous medium as follows: Esample Ereference �� T a 1 eeff��x�� p������������������������������������������ " # exp i 1 eeff��x�� p������������������������������������������ x d c " # ��1�� where x is the circular frequency, a is the angle of incidence, T is the complex (number) transmission coefficient at the air��� sample boundary, eeff is the effective complex permittivity of the tablet, d is the thickness of the tablet, and c is the velocity of electromagnetic radiation in vacuum. The explicit concept of the effective permittivity is here introduced in Eq. 1. In optical physics the issue of an effective medium model of nanocomposites is well known.13,14 The issue is that in the case of a two-phase or multiphase composite the optical spectra both in transmission and reflection mode depend on the complex permittivity of the ������host������, ������inclusion������, Received 1 December 2008 accepted 2 October 2009. * Author to whom correspondence should be sent. E-mail: heikki. tuononen@joensuu.fi. Volume 64, Number 1, 2010 APPLIED SPECTROSCOPY 0003-7028/10/6401-000000$2.00/0 �� 2010 Society for Applied Spectroscopy