View source for Nicolson–Ross–Weir method

{{Short description|Measurement technique in microwave engineering}}
[[File:VNWA3 DG8SAQ.jpg|thumb|A [[vector network analyzer]], which can be used to extract [[scattering parameters]] for Nicolson–Ross–Weir calculations]]
'''Nicolson–Ross–Weir method''' is a [[measurement]] technique for determination of complex [[Permittivity|permittivities]] and [[Permeability (electromagnetism)|permeabilities]] of material samples for [[microwave]] frequencies. The method is based on insertion of a material sample with a known thickness inside a [[waveguide]], such as a [[coaxial cable]] or a rectangular waveguide, after which the [[Dispersion (optics)|dispersion]] data is extracted from the resulting [[scattering parameters]]. The method is named after A. M. Nicolson and G. F. Ross,<ref name="NicolsonRoss1970"/> and W. B. Weir,<ref name="Weir1974"/> who developed the approach in 1970 and 1974, respectively.

The technique is one of the most common procedures for material characterization in [[microwave engineering]].<ref name="Costa2017"/>

==Method==
The method uses [[scattering parameters]] of a material sample embedded in a waveguide, namely <math>S_{11}</math> and <math>S_{21}</math>, to calculate permittivity and permeability data. <math>S_{11}</math> and <math>S_{21}</math> correspond to the cumulative [[Reflection coefficient|reflection]] and [[transmission coefficient]] of the sample that are referenced to the each sample end, respectively: these parameters account for the multiple internal reflections inside the sample, which is considered to have a thickness of <math>d</math>. The reflection coefficient of the bulk sample is:<ref name="NicolsonRoss1970"/><ref name="Weir1974"/>

:<math>\Gamma=X \pm \sqrt{X^2-1}</math>

where

:<math>X=\frac{1-(S_{21}^2-S_{11}^2)}{2 S_{11}}</math>

The sign of the root for the reflection coefficient is chosen appropriately to ensure its [[Passivity (engineering)|passivity]] (<math>|\Gamma| \leq 1</math>). Similarly, the transmission coefficient of the bulk sample can be written as:

:<math>T=\frac{S_{11}+S_{21}-\Gamma}{1-(S_{11}+S_{21})\Gamma}</math>

Thus, the effective permeability (<math>\mu^*</math>) and permittivity (<math>\varepsilon^*</math>) of the material can be written as:

:<math>\mu^*=\frac{\lambda_{0g}}{\Lambda} \left( \frac{1+\Gamma}{1-\Gamma} \right)</math>

:<math>\varepsilon^*=\frac{\lambda^2_0 \left( \frac{1}{\Lambda^2}+\frac{1}{\lambda_c^2} \right)}{\mu^*}</math>

where
:<math>\frac{1}{\Lambda^2}=-\left[ \frac{1}{2\pi d} ln\frac{1}{T} \right]^2 </math>

and 

*<math>\lambda_0</math> is the free-space [[wavelength]]. 
*<math>\lambda_{0g}</math> is the guided mode wavelength of the unfilled transmission line.
*<math>\lambda_{c}</math> is the [[Cutoff frequency|cutoff wavelength]] of the unfilled transmission line

The constitutive relation for <math>\Lambda</math> admits an infinite number of solutions due to the [[Principal branch|branches]] of the [[complex logarithm]]. The ambiguity regarding its result can be resolved by taking the [[Group delay and phase delay|group delay]] into account.<ref name="Weir1974"/>

==Limitations and extensions==
In the case of low material loss, the Nicolson–Ross–Weir method is known to be unstable for sample thicknesses at integer multiples of one half wavelength due to [[resonance]] phenomenon. Improvements over the standard algorithm have been presented in engineering literature to alleviate this effect.<ref name="BakerJavis1990"/><ref name="Boughriet1997"/><ref name="Luukkonen2011"/> Furthermore, complete filling of a waveguide with sample material may pose a particular challenge: presence of gaps during the filling of the waveguide section would excite higher-order modes, which may yield errors in scattering parameter results.<ref name="Champlin1966"/> In such cases, more advanced methods based on the rigorous modal analysis of partially-filled waveguides<ref name="CatalaCivera2003"/><ref name="Wilson1988"/> or [[Mathematical optimization|optimization]] methods<ref name="RequenaPerez2006"/> can be used. A modification of the method for single-port measurements was also reported.<ref name="Sahin2020"/>

In addition to homogenous materials, the extension of the method was developed to obtain constitutive parameters of isotropic and [[Anisotropy|bianisotropic]] [[metamaterial]]s.<ref name="Smith2001"/><ref name="Chen2004"/><ref name="Li2009"/>

==See also==
* [[Fourier-transform spectroscopy]]
* [[Microwave radiometer]]
* [[Reflection seismology]]
* [[Spectroscopy]]
* [[Time-domain reflectometer]]
* [[Vector network analyzer]]

==References==
{{Reflist|refs=

<ref name="BakerJavis1990">{{cite journal |last1=Baker-Jarvis |first1=J. |last2=Vanzura |first2=E. J. |last3=Kissick |first3=W. A. |title=Improved technique for determining complex permittivity with the transmission/reflection method |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=August 1990 |volume=38 |issue=8 |pages=1096–1103 |doi=10.1109/22.57336|bibcode=1990ITMTT..38.1096B |url=https://zenodo.org/record/1261115 |author-link1=James Baker-Jarvis}}</ref>

<ref name="Boughriet1997">{{cite journal |last1=Boughriet |first1=A.-H. |last2=Legrand |first2=C. |last3=Chapoton |first3=A. |title=Noniterative stable transmission/reflection method for low-loss material complex permittivity determination |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=January 1997 |volume=45 |issue=1 |pages=52–57 |doi=10.1109/22.552032|bibcode=1997ITMTT..45...52B }}</ref>

<ref name="CatalaCivera2003">{{cite journal |last1=Catala-Civera |first1=J. M. |last2=Canos |first2=A. J. |last3=Penaranda-Foix |first3=F. L. |last4=de los Reyes Davo |first4=E. |title=Accurate determination of the complex permittivity of materials with transmission reflection measurements in partially filled rectangular waveguides |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=14 January 2003 |volume=51 |issue=1 |pages=16–24 |doi=10.1109/TMTT.2002.806940|bibcode=2003ITMTT..51...16C }}</ref>

<ref name="Champlin1966">{{cite journal |last1=Champlin |first1=K. S. |author2-link=Gary H. Glover |last2=Glover |first2=G. H. |title="Gap Effect" in Measurement of Large Permittivities (Correspondence) |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=August 1966 |volume=14 |issue=8 |pages=397–398 |doi=10.1109/TMTT.1966.1126280|bibcode=1966ITMTT..14..397C }}</ref>

<ref name="Chen2004">{{cite journal |last1=Chen |first1=Xudong |last2=Grzegorczyk |first2=Tomasz M. |last3=Wu |first3=Bae-Ian |last4=Pacheco, Jr. |first4=Joe |last5=Kong |first5=Jin Au |title=Robust method to retrieve the constitutive effective parameters of metamaterials |journal=[[Physical Review E]] |date=26 July 2004 |volume=70 |issue=1 Pt 2 |page=016608 |doi=10.1103/PhysRevE.70.016608|pmid=15324190 |bibcode=2004PhRvE..70a6608C }}</ref>

<ref name="Costa2017">{{cite journal |last1=Costa |first1=Filippo |last2=Borgese |first2=Michele |last3=Degiorgi |first3=Marco |last4=Monorchio |first4=Agostino |title=Electromagnetic Characterisation of Materials by Using Transmission/Reflection (T/R) Devices |journal=Electronics |date=2017 |volume=6 |issue=4 |page=95 |doi=10.3390/electronics6040095|doi-access=free }}</ref>

<ref name="Li2009">{{cite journal |last1=Li |first1=Zhaofeng |last2=Aydin |first2=Koray |last3=Ozbay |first3=Ekmel |title=Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients |journal=[[Physical Review E]] |date=2009 |volume=79 |issue=2 |page=026610 |doi=10.1103/PhysRevE.79.026610 |pmid=19391866 |bibcode=2009PhRvE..79b6610L |hdl=11693/22835 |author-link3=Ekmel Özbay|hdl-access=free }}</ref>

<ref name="Luukkonen2011">{{cite journal |last1=Luukkonen |first1=Olli |last2=Maslovski |first2=Stanislav I. |last3=Tretyakov |author-link3=Sergei Tretyakov (scientist)|first3=Sergei A. |title=A Stepwise Nicolson–Ross–Weir-Based Material Parameter Extraction Method |journal=[[ IEEE Antennas and Wireless Propagation Letters]] |date=11 November 2011 |volume=10 |pages=1295–1298 |doi=10.1109/LAWP.2011.2175897|bibcode=2011IAWPL..10.1295L |s2cid=20530272 }}</ref>

<ref name="NicolsonRoss1970">{{cite journal |last1=Nicolson |first1=A. M. |last2=Ross |first2=G. F. |title=Measurement of the intrinsic properties of materials by time-domain techniques |journal=[[IEEE Transactions on Instrumentation and Measurement]] |date=November 1970 |volume=19 |issue=4 |pages=377–382 |doi=10.1109/TIM.1970.4313932|bibcode=1970ITIM...19..377N }}</ref>

<ref name="RequenaPerez2006">{{cite journal |last1=Requena-Perez |first1=M. E. |last2=Albero-Ortiz |first2=A. |last3=Monzo-Cabrera |first3=J. |last4=Diaz-Morcillo |first4=A. |title=Combined use of genetic algorithms and gradient descent optmization methods for accurate inverse permittivity measurement |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=615–624 |volume=54 |issue=2 |pages=615–624 |doi=10.1109/TMTT.2005.862671|bibcode=2006ITMTT..54..615R |hdl=10317/1427 |s2cid=18731169 |hdl-access=free }}</ref>

<ref name="Sahin2020">{{cite journal |last1=Sahin |first1=Seckin |last2=Nahar |first2=Niru K. |last3=Sertel |first3=Kubilay |title=A Simplified Nicolson–Ross–Weir Method for Material Characterization Using Single-Port Measurements |journal=[[ IEEE Transactions on Terahertz Science and Technology]] |date=July 2020 |volume=10 |issue=4 |pages=404–410 |doi=10.1109/TTHZ.2020.2980442|bibcode=2020ITTST..10..404S |s2cid=216481222 |doi-access=free }}</ref>

<ref name="Smith2001">{{cite journal |last1=Smith |first1=D. R. |last2=Schultz |first2=S. |last3=Markoš |first3=P. |last4=Soukoulis |first4=C. M. |author-link1=David R. Smith (physicist) |author-link4=Costas Soukoulis |title=Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients |journal=[[Physical Review B]] |year=2002 |volume=65 |issue=19 |page=195104 |doi=10.1103/PhysRevB.65.195104|arxiv=physics/0111203 |bibcode=2002PhRvB..65s5104S |s2cid=1637551 }}</ref>

<ref name="Weir1974">{{cite journal |last1=Weir |first1=W. B. |title=Automatic measurement of complex dielectric constant and permeability at microwave frequencies |journal=[[Proceedings of the IEEE]] |date=January 1974 |volume=62 |issue=1 |pages=33–36 |doi=10.1109/PROC.1974.9382}}</ref>

<ref name="Wilson1988">{{cite journal |last1=Wilson |first1=S. B. |title=Modal analysis of the 'gap effect' in waveguide dielectric measurements |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=April 1988 |volume=36 |issue=4 |pages=752–756 |doi=10.1109/22.3581|bibcode=1988ITMTT..36..752W }}</ref>

}}

==Further reading==
* {{cite book |last1=Chen |first1=L. F. |last2=Ong |first2=C. K. |last3=Neo |first3=C. P. |last4=Varadan |first4=V. V. |last5=Varadan |first5=V. K. |title=Microwave electronics: measurement and materials characterization |date=2004 |publisher=[[Wiley (publisher)|John Wiley & Sons]] |isbn=9780470020456}}

[[Category:Microwave technology]]
[[Category:Spectroscopy]]
[[Category:Electric and magnetic fields in matter]]
{{DEFAULTSORT:Nicolson-Ross-Weir method}}