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S Parameter extraction

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This page describes how to calculate the meta-material S-parameters of a metamaterial device.

 

See also

Methodology

Effective bulk properties

Effective parameters - Smith

Related publications

D. R. Smith et al., "Electromagnetic parameter retrieval from inhomogeneous metamaterials", Phys Rev E 71, 036617 (2005)

Szabó, Z., G.-H. Park, R. Hedge, and E.-P. Li, "A unique extraction of metamaterial parameters based on kramers-kronig relationship," IEEE MTT, vol. 58, no. 10, 2010, pp. 2646-2653

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S parameters measurements

Understanding S-parameters

S parameters describe behaviors of a 2 by 2 network or transmission lines (see the figure at the right below):

 

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S parameters for a 2 port network

Phase compensation for source and monitors

 

for two given input signals a1 and a2, the outputs b1 and b2 can be calculated as

 

$$ \left(\begin{array}{l}{b 1} \\ {b 2}\end{array}\right)=\left(\begin{array}{ll}{S11} & {S12} \\ {S21} & {S22}\end{array}\right)\left(\begin{array}{l}{a 1} \\ {a 2}\end{array}\right) $$

 

Measuring S-parameters from your FDTD simulation

S parameters are complex amplitude reflection and transmission coefficients (in contrast to the power reflection and transmission coefficients). For example, S11 is the reflection coefficient and S21 is the transmission coefficient for a1 incidence; and S22 is the reflection coefficient and S12 is the transmission coefficient for a2 incidence.

 

Recognizing that the a,b coefficients are directly proportional to the electric fields, the S parameters can be calculated from the definition above, eg, S11=b1/a1=E_r/E_i and S21=b2/a1=E_t/E_i where E_i, E_r, E_t are the incident, reflected and transmitted electric fields. This technique makes it very simple to obtain S-parameters for a given device, but it does make a number of assumptions. It's important to ensure that all of the assumptions are valid for your simulation.

 

This analysis assumes that the structure does not significantly affect the polarization of the incident fields, and that the fields are polarized along one of the major axes (X,Y,Z). For example, if the incident fields are X polarized, this analysis assumes the reflected and transmitted fields are primarily X polarized. This analysis selects the largest field component to do the parameter extraction.  
If the polarization is important, this analysis must be generalized to treat each polarization separately.

To measure the S-parameters in the near field (as done in the S-parameter analysis object), the transmitted and reflected fields must be propagating like a simple plane wave at the point of measurement.  The measurements will not be correct if evanescent fields are present, or if the structure supports multiple grating orders. To avoid the evanescent fields, make sure the monitor is sufficiently far from the structure. Multiple grating orders are rare, since the structures are usually sub-wavelegth, but if they do exist, the S-parameters must be extracted from far field measurements. Visit the Support Center for more assistance on this calculation.

In your simulation, the sources and monitors are always some distance from the surface of the metamaterial (for example, the monitors must be far enough from the structure to avoid evanescent fields). The fields will accumulate additional phase as they propagate from the source to the metamaterial, and from the metamaterial to the monitors.  The S parameters are intended to characterized only the metamaterial, without this additional propagation phase, thus we must compensate this additional phase. Suppose the wavenumber in the incidence space is ki, and the wavenumber in transmission space is  kt, the extra phase in Monitor T from source S is  kirs+ktrt where rs and rt  are distances (so they are positive). For reflection Monitor R, the extra phase from source S is  kirs+kirr .

 

Based on the above discussion, a ready-to-use S Parameter analysis group is available in the Object Library. In most cases you only need to set the slab thickness, the locations of the slab center and source position along x axis, and the background refractive index in Analysis--Variables. The phase compensation is done in the Script. The phase compensation assumes that the background index is the same on both sides of the metamaterial. Also note that because the reflected and transmitted waves must be propagating like plane waves, we use a point frequency-domain monitor to record the electric field component. The analysis group also contains two 2D surface-monitors, which are used to measure the power transmission and to check that the fields are in fact propagating like a single plane wave.

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