Material Parameter Measurements for Microwave Antireflection Coating Development

Abstract

The main steps for characterization and measurement of microwave absorbent materials in the 1–10 GHz range are introduced. The coaxial reflection-transmission type of material parameter measurement is analyzed in detail and the main measurement error is corrected. The microscopic material particle parameter measurement concept is also presented using different mixing rule laws to determine the material parameters of the single particles from the macroscopic parameters. Two-dimensional FDTD simulations have been used to model the behavior of mixed electric and magnetic type of material.

1. Introduction

Antireflection coatings are widely used for optical and high-frequency applications such as microwave absorbing materials for electromagnetic compatibility or decreasing RCS for stealth applications. Even though a single-layer coating may be sufficient to obtain low reflection coefficients, its efficiency is usually reduced to a narrow bandwidth and to a specific polarization and incidence angle.

In order to extend the performance of antireflection coatings to wider ranges of frequency and incidence, a multilayered structure has to be considered.

The optimization of such complex architectures may consist in measurement of macroscopic and microscopic material parameters. This article presents a new calibration flow graph analysis and analytical equation to evaluate the correcting terms. Measurement and simulated results are presented to validate the calibration proposed and to correct the sample size deviation [1].

2. Measurements

2.1. Macroscopic Material Parameter Measurements

The material samples are placed in a coaxial line and two complex parameters are measured, the reflection and the transmission coefficients.

The widely used measurement method of macroscopic material parameters is based on the Nicolson and Ross system. A sample of material which has complex relative permittivity ε_R and permeability μ_R fills a coaxial waveguide section, as shown in Figure 1. TEM electromagnetic wave is propagated along the coaxial line with the material sample and the scattering parameters (S₁₁ and S₁₂) are measured using reference planes on the two faces of the sample. The closed form equations derived by Nicolson and Ross are the following: 𝑉₁=𝑆₂₁+𝑆₁₁,𝑉₂=𝑆₂₁−𝑆₁₁,Γ=1−𝑉₁𝑉₂𝑉₁−𝑉₂±(1−𝑉₁𝑉₂𝑉₁−𝑉₂)²𝑐−1,₁=(1+Γ)1−Γ²,𝑐₂𝑐=−[𝜔𝑑ln(1−𝑉₁Γ𝑉₁−Γ)]², (1) from which the electrical and magnetic material parameters follows:

Figure 1 

Coaxial line with annular disk of material.

Figure 2 

Measurement setup.

(a)

(b)

Figure 3 

Coaxial line sample holder.

𝜇_𝑅=√𝑐₁𝑐₂,𝜀_𝑅=𝑐₁𝑐₂. (2)The sample holder used with the support of teflon beads and the removable connector is shown in Figure 4.

Figure 4 

Signal flow graph of sample holder.

Figure 5 

Reflection measurements for Sa.

Figure 4 shows the signal flow graph for the measurement system with connectors and the sample holder in the middle. The S^a and S^b represent the reflections and transmissions of the connectors and joints, but not including the sample holder itself which is an air-filled coaxial line in empty stage and has S^o scattering matrix. The resulting S-matrix is S^e.

The S^a and S^b only differ in terms of S₁₂ and S₂₁; this leads to the length difference of the two conductors inside the support beads and connectors:

𝑆^𝑎₁₁=𝑆^𝑏₁₁,𝑆^𝑎₂₂=𝑆^𝑏₂₂. (3)The sample holder is assumed to be symmetric, reciprocal, lossless, and nonreflecting; therefore,

𝑆^𝑜₁₂=𝑆^𝑜₂₁=𝑒^{−𝑗𝑘𝑑},𝑆^𝑜₁₁=𝑆^𝑜₂₂=0. (4)The resulting S-matrix S^e and S^a, and S^b is assumed to be reciprocal; therefore,

𝑆^𝑎₁₂=𝑆^𝑎₂₁,𝑆^𝑏₁₂=𝑆^𝑏₂₁,𝑆^𝑒₁₂=𝑆^𝑒₂₁. (5)The sample holder has a removable connector only at one side, therefore, directly only S^a can be measured using a short load at various positions in the air-filled sample holder.

There are 4 unknowns in (3)–(5) which can be determined by using 4 measurements, from which there are two reflection measurements for S^a using short load at two different positions and there is the measurement of S^e.

The calibration measurements result in the following four equations:

Γ_𝑚1=𝑆^𝑎₁₁−𝑆^𝑎₂₁⋅𝑆^𝑎₁₂11+𝑆^𝑎₂₂,Γ_𝑚2=𝑆^𝑎₁₁−𝑆^𝑎₂₁⋅𝑒^{−𝑗2𝑘𝑙}⋅𝑆^𝑎₁₂11+𝑆^𝑎₂₂⋅𝑒^{−𝑗2𝑘𝑙}, (6) where Γ_m1 and Γ_m2 are the two measured reflection coefficients for S^a using short load at two different positions: 𝑆^𝑒₁₁=𝑆^𝑎₁₁+𝑆^𝑎₂₁⋅𝑒^{−𝑗2𝑘𝑑}⋅𝑆^𝑏₂₂11−𝑆^𝑎₂₂⋅𝑆^𝑏₂₂⋅𝑒^{−𝑗2𝑘𝑑}𝑆^𝑎₁₂,𝑆^𝑒₂₁=𝑆^𝑎₂₁⋅𝑒^{−𝑗𝑘𝑑}11−𝑆^𝑎₂₂⋅𝑆^𝑏₂₂⋅𝑒^{−𝑗2𝑘𝑑}𝑆^𝑏₂₁. (7)The solutions for the four unknowns are

(𝑆^𝑎₂₂)_1,2=−𝐾₁(𝑒^{−𝑗2𝑘𝑙}+𝑒^{−𝑗2𝑘𝑑})±𝑄−𝑍2⋅(𝐾₂𝑒^{−𝑗2𝑘𝑑}+𝐾₁𝑒^{−𝑗2𝑘(𝑙+𝑑)})𝑆_𝑎12,=(𝑆^𝑒₁₁−Γ_𝑚1)(1+𝑆^𝑎₂₂)(1−𝑒^{−𝑗2𝑘𝑑}⋅(𝑆^𝑎₂₂)²)1+𝑆^𝑎₂₂⋅𝑒^{−𝑗2𝑘𝑑}𝑆_𝑎11,=Γ_𝑚1+(𝑆^𝑎₁₂)²1+𝑆^𝑎₂₂𝑆_𝑏12,=𝑆^𝑒₂₁(1−𝑒^{−𝑗2𝑘𝑑}(𝑆^𝑎₂₂)²)𝑆^𝑎₁₂⋅𝑒^{−𝑗𝑘𝑑}, (8)(9)(10)(11)where 𝑄=𝐾²₁(𝑒^{−𝑗2𝑘𝑙}+𝑒^{−𝑗2𝑘𝑑})² and 𝑍=4(𝐾₁−𝐾₂)(𝐾₂𝑒^{−𝑗2𝑘𝑑}+𝐾₁𝑒^{−𝑗2𝑘(𝑙+𝑑)}),𝐾₁=Γ_𝑚2−Γ_𝑚11−𝑒^{−𝑗2𝑘𝑙},𝐾₂=𝑆^𝑒₁₁−Γ_𝑚1,|𝑆^𝑎₂₂|≤1. (12)

Figure 6 shows the angle of S₁₂^a and S₁₂^b evaluated from (9) and (11) and based on calibration measurements. There is only a slight difference between them which gives proofs to the previous assumptions (3)–(5) but also confirms the small deviation in length and geometry between the two connector parts.

(a)

(b)

Figure 6 

S12a and S12b used for compensation of measurements.

2.2. Measurement Errors

The main sources of errors are the higher-order modes propagating on the coaxial sample holder, measuring line and the sample size with placement errors.

2.2.1. Higher Mode Propagation

The higher modes can be suppressed using smaller diameter coaxial line. This effect is illustrated in Figure 7 using 7/3 mm coaxial line outer/inner diameters. The relative measurement error for typical high-frequency ferrite material (𝜀=2−0.8𝑗,𝜇=5−1𝑗) is below 1.5%. The error reduction or the measurement frequency extension is possible by reducing the measurement line diameter.

Figure 7 

Higher mode propagation effect.

Figure 8 

Sample diameters for air gap correction.

The main effect was investigated and analyzed, which is related to the sample size and causes the main errors in measurements.

2.2.2. The Effect of Air Gap between the Sample and the Coaxial Line

The effect of a small air gap caused by manufacturing tolerances was investigated. The measuring error is more significant in case if the air gap is at the inner conductor of the coaxial line so this effect was analyzed.

The effect causes error in permittivity and permeability. The error in permittivity can be corrected with a simple correction routine. The corrected values are

Re𝜀_corr.=Re𝜀_meas.ln(𝐷₃/𝐷₂)ln(𝐷₃/𝐷₁)−Re𝜀_meas.ln(𝐷₂/𝐷₁),Im𝜀_corr.=Re𝜀_corr.Im𝜀_meas.Re𝜀_meas.(1+Re𝜀_corr.ln(𝐷₂/𝐷₁)ln(𝐷₃/𝐷₂)). (13)Similar expressions can be derived for the permeability correction.

For proving the correction equations, simulations were performed using the HFSS 3D simulator and using the arrangement in Figures 9-10.

Figure 9 

HFSS model of sample holder.

Figure 10 

Modeling of sample air gap.

The effect of 0.1 mm air gap on the electrical parameters is shown in Figure 11, where the imperfect sample size is modeled and from the HFSS simulated S₁₁ and S₁₂ values the ε and μ were calculated using (1)-(2). Then the correction equations (13) are used and the remaining errors are presented in Figure 12.

Figure 11 

Material parameter measurement error caused by 0.1 mm air gap.

Figure 12 

Measurement error corrected.

(a)

(b)

(c)

Figure 13 

Plane wave propagating between randomly distributed small particles.

Figure 14 

Mixing rules and FDTD simulation results of material mixture.

With the above correction, the measuring error can be decreased to 4% in the frequency range of 1–10 GHz for typical materials for air gap less than 0.1 mm.

3. Microscopic Material Parameters

The material parameters of small sizes of particles in nanomaterials cannot be measured directly by using the conventional measuring methods, thus indirect procedure is suggested. The nanoparticles are solved in known fluid material and in known volume fraction.

Estimating of material parameters of particles in host materials there exist different mixing rules depending on particle interaction effects. The main mixing rules are

Maxwell Garnett mixing rule 𝜀_eff=𝜀_𝑒+2𝑓𝜀_𝑒𝜀_𝑖−𝜀_𝑒𝜀_𝑖+𝜀_𝑒−𝑓(𝜀_𝑖−𝜀_𝑒), (14) and Bruggeman mixing rule [2] 𝜀(1−𝑓)^𝑒−𝜀_eff𝜀_𝑒+2𝜀_eff𝜀+𝑓^𝑖−𝜀_eff𝜀_𝑖+2𝜀_eff=0, (15) where ε_eff is the effective permittivity of the mixture, where spherical inclusions with permittivity ε_i occupy a volume fraction f in a host material ε_e.

The Maxwell Garnett and Bruggeman are the special cases of a general mixing rule

𝜀_eff−𝜀_𝑒𝜀_eff+2𝜀_𝑒+𝜈(𝜀_eff−𝜀_𝑒)𝜀=𝑓^𝑖−𝜀_𝑒𝜀_𝑖+2𝜀_𝑒+𝜈(𝜀_eff−𝜀_𝑒). (16)The ν dimensionless parameter ν = 1 gives the Maxwell Garnett rule, ν = 2 gives the Bruggeman formula, and ν = 3 gives the coherent potential approximation.

A similar mixing formula (Bruggeman) can be obtained for the permeability

𝑓𝜇_𝑖−𝜇_eff𝜇_𝑖+𝜇_eff𝜇+(1−𝑓)^𝑒−𝜇_eff𝜇_𝑒+𝜇_eff=0, (17)where μ_eff is the effective permeability of the mixture, and spherical inclusions with permeability μ_i occupy a volume fraction f in a host material μ_e.

Two-dimensional simulation using an FDTD solver is applied to prove and illustrate the indirect nanoparticle material parameter determination method. Small particles are randomly distributed in two-dimensional areas and an incident wave is propagating on this area, influenced by them. From simulated macroscopic material parameters (16), (17) are verified and the ν is determined for practical application [3, 4].

The FDTD simulation results show an effective permittivity, which is close to Bruggeman’s approximation but using an error minimization ν = 1.7 gives an optimum curve (for 𝜀_𝑖=50−𝑗^∗4.0,𝜀_𝑒=2.0−𝑗^∗0).

4. Conclusion

The main steps of characterization and measurement of microwave absorbent materials in the 1–10 GHz range are introduced for macroscopic and microscopic material parameter determination. The coaxial reflection-transmission type of material parameter measurement is analyzed in detail and measurement error is corrected. Genetic algorithm concept of optimization multilayer antireflection coating is summarized. Two-dimensional FDTD simulations were presented to model the behavior of mixed electric and magnetic type of material.

Acknowledgments

This work was supported by the ACE2 European FP6 framework program and OMFB-00252/2007 Project.