Nicolson-Ross-Weir method Print E-mail
Nicolson-Ross-Weir (NRW) retrieval method is based on the inversion of the Fresnel-Airy formulas expressing the normal reflection and transmission coefficients of a material layer through the wave impedance of the medium and its refraction index. Through the wave impedance and the refraction index one can find medium permittivity and permeability (for anisotropic media – tangential components of the permittivity and permeability tensors).
Under the alternative name of the distributed impedance method the NRW method is known in both numerical and experimental characterization of natural materials [5,6] as well as of composite (granular) materials with very optically dense arrangement ofgrains[10,11]. However in these works only non-resonant grains were considered. Asitwas theoretically shown in works [12,13,14,15] and experimentally confirmed in [16] this techniqueis not applicable when the sample material experiences the resonance. Another limitation of this method is poor resolution for losses [5]. The advantage is broadband description of the dielectric (and non-resonant magnetic) properties[10].

In the NRW method the transmission and reflection signals are tested to calculate the dielectric properties of the layer. In the microwave frequency range one uses network analyzers which combine the tester and signal source [5,10]. In the optical frequency range the reflected signal amplitude shouldbe measuredby a spectroscope. To spatially separate the reflected wave from the incident one one commonly uses in optical measurements the semi-transparent mirror which is tilted under the angle45â—¦ to the incident wave (see e.g. [17]). The phase of the optical signalis measured using the Mach-Zender interferometer (see e.g. [18]). Since non-resonantbulk nanostructures do not possess magnetic permeability in the optical range the interferometric measurements can be avoided. The complex permittivity (real and imaginary part) of the dielectricl ayer or film can be retrieved only from spectroscopic measurements, i.e. from |R|and |T| [9]. However, the film is always prepared on the substrate which ensures the mechanical robustness of the film. Therefore the retrieval procedure obviously takes into account the permittivity ofthe substrate [9]. For nanofilms obtainedby epitaxial orlithographic methods it is not a problem since the permittivity of the substrate does not change after the fabrication of the film. However, many nanostructured materials are chemically grown on the substrate and the fabrication process is related with high temperatures. Then the substratepermittivity will be slightly modified. Since the substrate is always optically thick this slight modification leads to a serious mistake in the transmittance |T| calculated for a non-perturbed substrate. In this case a more accurate experimental retrieval of complex permittivity is achieved by the measurements of |R|(spectroscopic) and phase (R) (interferometric) [18].



Figure 1: The configurations used in the NRW method at optical frequencies. (a) – Measurements of the amplitude and phase of the transmission coefficient T. The amplitude of T is measured by a spectroscope (SS). The spectroscope signal contains also a1/4part of the incident wave. Since the optical path of this part is known this parasitic signal can be eliminated by an additionalbranch (not shown) which adds the signal E0 /4 with the opposite phase. Alternatively, this can be taken into account in the software. Semitransparent mirror 1 is needed to split the incident beam to that illuminating the sample and that used for the measurement of the phase. Semitransparent mirror 4 is needed to split the transmitted beam to that directed to SS and that used in the phase measurement. The phase is measured by the Mach-Zender Interferometer (MZI) formed by fully reflecting mirrors 3 and 5 with the help of semi transparent mirror 4. (b) – Measurements of the amplitude and phase of the reflection coefficient R. The amplitude of T is measured by a spectroscope (SS) where as semitransparent mirror 1 is used to split the incident beam. This spectroscope signal contains the known parasitic part to be compensated. The MZI is formed by semitransparent mirror 2 and fully reflecting mirror 3.

The NRW technique is more troublesome at optical frequencies than at microwaves as difficult and precise phase measurements must be taken from the signal. This is achieved through the use of a Mach-Zender interferometer which splits the beam into two (one wave passes through the sample, the other is used as a reference) before each being picked up by a detector [43]. The measurement schemes are rather cumbersome as one can seein Fig. 1. In the transmission scheme the spectroscopic device SS measures the signal with absolute value of the complex amplitude |E0 (T +1)/4| and the Mach-Zender interferometer (MZI) measures the phase shift between two signals, one has the complex amplitude E0 T/4 and another – E0 /4. From these data one extracts the amplitude and phase of T. In the reflection scheme SS measures |E0 (R+1)/4|and the MZI measures the phase shift between signals E0 R/8 and E0 /2.

In fact, interferometric measurements need not be taken with non-resonant bulk nanostructures as they do not exhibit magnetic permeability at optical frequencies. The natural magnetism in the optical range is absent, only resonant structures demonstrate the artificial magnetism. To extract complex permittivity it is sufficient to measure two real values – |R| and |T|.

It should alwaysbe remembered that the thin nanofilm ordielectric being characterized sits atop of an optically thick substrate. Thus, the substrate too should be measured. The values retrieved for both the real and imaginary parts of the permittivity are influenced by some degreeby the sample substrate [44], the extent of which is dependent on the fabrication process used to create the nanofilm. If epitaxial or lithographic techniques are used the effect will be non-existent because the substrate permittivity will remain the same after the fabrication of the nanostructure. If the nanostructured materials are however fabricated using chemical growth, a process that crucially involves the sample being subjected to high temperatures, the substrate permittivity will be irreparably altered. The effects of this modification will be augmented by the fact that the substrate is comparatively thick in size. Consequently when the transmittance coefficients are measured they and the material parameters calculated using them will be completely incorrect for a non-perturbed substrate. An alternative method in this instance would be retrieval of parameters using both the spectroscopic and interferometric reflectance [43].

Let us conclude this subsection by three important comments. The NRW method implies the normal propagation of the wave in the material. Therefore, the obvious condition of optical smallness of the structure period a refers only to the period across the slab. To consider the medium as effectively continuous is possible not only if the slab is formed by optically small inclusions. It can be an array of long inclusions (e.g. wires) if they are parallel to the boundary. This possibility seems to broaden the scope of applicability for this method. However, it is not so simple. In fact, the retrieved effective material parameters can be treated as characteristic parameters only in the case when the particles are optically small and isotropic. Then once retrieved from the measurements of r and t coefficients for the normalincidence they can be applied for condensed description of the materials. For a slab of the wire medium or for a structure of alternating metal and dielectric layers it is not so.

In them the oblique propagation obeys to different laws than the normal propagation. The interaction of the obliquely propagating wave with such media cannot be considered in terms of the permittivity which was retrieved for the normal incidence. Moreover, in the case of the oblique propagation of the wave such media are spatially dispersive and to relate their effective material parameters to r and t coefficients so-called additional boundary conditions are needed (see e.g. in[41]). However, if there is a reliable theoretical model of the structure and a minimal knowledge on its geometrical parameters (e.g. the period a across the slab) the NRW retrieval will be not useless. For example, for a wire medium from the retrieved permittivity one can find the so-called plasma frequency which can be further used for calculating the spatially dispersive material parameters. This case can be referred as partial or conditional electromagnetic characterization.

The second comment refers to the obvious condition of the absence of the resonances. In fact, the NRW method is applicable beyond the frequency range of the Fabry-Perot resonances (thickness resonances ofthe slab) [40]. At these resonances the method if applied leads to the violation of physical laws in the retrieved materialparameters.

The third commentis about the NRW retrieval in comparison with the alternative approach to the material parameters retrieval, i.e. the ellipsometric methods. The drawback of the NRW method is that it does not have a sufficient resolution to measure low losstangents [45]. Its main advantageis that canbe applied in a wide frequency range where provides the broadband description of the medium dispersive properties.


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