# Material Parameter Measurements for Microwave Antireflection Coating Development

Authors: Lajos Nagy
Journal: Active and Passive Electronic Components
 (2008)

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## 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.


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## Body

## 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 permittivityεR and permeability μR 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:
(1)V1=S21+S11,V2=S21−S11,Γ=1−V1V2V1−V2±(1−V1V2V1−V2)2−1,c1=(1+Γ1−Γ)2,c2=−[cωdln⁡(1−V1ΓV1−Γ)]2,
from which
the electrical and magnetic material parameters follows:Figure 1
Coaxial
line with annular disk of material.Figure 2
Measurement setup.Coaxial line sample holder.
(a)(b)(2)μR=c1c2,εR=c1c2.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 forSa.Figure4
shows the signal flow graph for the measurement system with connectors and the
sample holder in the middle. The Sa and Sb 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 So scattering matrix.
The resulting S-matrix is Se.TheSa and Sb only differ in terms of S12 and S21;
this leads to the length difference of the two conductors inside the support
beads and connectors:(3)S11a=S11b,S22a=S22b.The sample holder is assumed to be symmetric, reciprocal, lossless, and nonreflecting;
therefore,(4)S12o=S21o=e−jkd,S11o=S22o=0.The resulting
S-matrix Se and Sa, and Sb is
assumed to be reciprocal; therefore,(5)S12a=S21a,S12b=S21b,S12e=S21e.The sample
holder has a removable connector only at one side, therefore, directly only Sa 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 Sa using short load at two different positions and there is the measurement of Se.The
calibration measurements result in the following four equations:(6)Γm1=S11a−S21a⋅S12a11+S22a,Γm2=S11a−S21a⋅e−j2kl⋅S12a11+S22a⋅e−j2kl,
where Γm1 and Γm2 are the two measured reflection coefficients for Sa using short load at two different positions:
(7)S11e=S11a+S21a⋅e−j2kd⋅S22b11−S22a⋅S22b⋅e−j2kdS12a,S21e=S21a⋅e−jkd11−S22a⋅S22b⋅e−j2kdS21b.The solutions
for the four unknowns are(8)(S22a)1,2=−K1(e−j2kl+e−j2kd)±Q−Z2⋅(K2e−j2kd+K1e−j2k(l+d)),(9)S12a=(S11e−Γm1)(1+S22a)(1−e−j2kd⋅(S22a)2)1+S22a⋅e−j2kd,(10)S11a=Γm1+(S12a)21+S22a,(11)S12b=S21e(1−e−j2kd(S22a)2)S12a⋅e−jkd,where Q=K12(e−j2kl+e−j2kd)2 and Z=4(K1−K2)(K2e−j2kd+K1e−j2k(l+d)),(12)K1=Γm2−Γm11−e−j2kl,K2=S11e−Γm1,|S22a|≤1.Figure6
shows the angle of S12a and S12b 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.S12a and S12b used for compensation of measurements.
(a)(b)


### 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 Figure7 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(13)Re⁡εcorr.=Re⁡εmeas.ln⁡(D3/D2)ln⁡(D3/D1)−Re⁡εmeas.ln⁡(D2/D1),Im⁡εcorr.=Re⁡εcorr.Im⁡εmeas.Re⁡εmeas.(1+Reεcorr.ln(D2/D1)ln(D3/D2)).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 Figures9-10.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 Figure11, 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.Plane wave propagating between randomly distributed small particles.
(a)(b)(c)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.


## 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 permittivityεR and permeability μR 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:
(1)V1=S21+S11,V2=S21−S11,Γ=1−V1V2V1−V2±(1−V1V2V1−V2)2−1,c1=(1+Γ1−Γ)2,c2=−[cωdln⁡(1−V1ΓV1−Γ)]2,
from which
the electrical and magnetic material parameters follows:Figure 1
Coaxial
line with annular disk of material.Figure 2
Measurement setup.Coaxial line sample holder.
(a)(b)(2)μR=c1c2,εR=c1c2.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 forSa.Figure4
shows the signal flow graph for the measurement system with connectors and the
sample holder in the middle. The Sa and Sb 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 So scattering matrix.
The resulting S-matrix is Se.TheSa and Sb only differ in terms of S12 and S21;
this leads to the length difference of the two conductors inside the support
beads and connectors:(3)S11a=S11b,S22a=S22b.The sample holder is assumed to be symmetric, reciprocal, lossless, and nonreflecting;
therefore,(4)S12o=S21o=e−jkd,S11o=S22o=0.The resulting
S-matrix Se and Sa, and Sb is
assumed to be reciprocal; therefore,(5)S12a=S21a,S12b=S21b,S12e=S21e.The sample
holder has a removable connector only at one side, therefore, directly only Sa 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 Sa using short load at two different positions and there is the measurement of Se.The
calibration measurements result in the following four equations:(6)Γm1=S11a−S21a⋅S12a11+S22a,Γm2=S11a−S21a⋅e−j2kl⋅S12a11+S22a⋅e−j2kl,
where Γm1 and Γm2 are the two measured reflection coefficients for Sa using short load at two different positions:
(7)S11e=S11a+S21a⋅e−j2kd⋅S22b11−S22a⋅S22b⋅e−j2kdS12a,S21e=S21a⋅e−jkd11−S22a⋅S22b⋅e−j2kdS21b.The solutions
for the four unknowns are(8)(S22a)1,2=−K1(e−j2kl+e−j2kd)±Q−Z2⋅(K2e−j2kd+K1e−j2k(l+d)),(9)S12a=(S11e−Γm1)(1+S22a)(1−e−j2kd⋅(S22a)2)1+S22a⋅e−j2kd,(10)S11a=Γm1+(S12a)21+S22a,(11)S12b=S21e(1−e−j2kd(S22a)2)S12a⋅e−jkd,where Q=K12(e−j2kl+e−j2kd)2 and Z=4(K1−K2)(K2e−j2kd+K1e−j2k(l+d)),(12)K1=Γm2−Γm11−e−j2kl,K2=S11e−Γm1,|S22a|≤1.Figure6
shows the angle of S12a and S12b 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.S12a and S12b used for compensation of measurements.
(a)(b)


## 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 Figure7 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(13)Re⁡εcorr.=Re⁡εmeas.ln⁡(D3/D2)ln⁡(D3/D1)−Re⁡εmeas.ln⁡(D2/D1),Im⁡εcorr.=Re⁡εcorr.Im⁡εmeas.Re⁡εmeas.(1+Reεcorr.ln(D2/D1)ln(D3/D2)).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 Figures9-10.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 Figure11, 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.Plane wave propagating between randomly distributed small particles.
(a)(b)(c)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.


## 2.2.1. Higher Mode Propagation

The higher
modes can be suppressed using smaller diameter coaxial line. This effect is
illustrated in Figure7 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(13)Re⁡εcorr.=Re⁡εmeas.ln⁡(D3/D2)ln⁡(D3/D1)−Re⁡εmeas.ln⁡(D2/D1),Im⁡εcorr.=Re⁡εcorr.Im⁡εmeas.Re⁡εmeas.(1+Reεcorr.ln(D2/D1)ln(D3/D2)).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 Figures9-10.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 Figure11, 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.Plane wave propagating between randomly distributed small particles.
(a)(b)(c)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 areMaxwell Garnett mixing rule(14)εeff=εe+2fεeεi−εeεi+εe−f(εi−εe),
and Bruggeman mixing rule [2]
(15)(1−f)εe−εeffεe+2εeff+fεi−εeffεi+2εeff=0,
where εeff is the effective permittivity of the mixture, where spherical inclusions with permittivity εi occupy a volume fraction f in a host material εe.The Maxwell
Garnett and Bruggeman are the special cases of a general mixing rule(16)εeff−εeεeff+2εe+ν(εeff−εe)=fεi−εeεi+2εe+ν(εeff−εe).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(17)fμi−μeffμi+μeff+(1−f)μe−μeffμe+μeff=0,where μeff
is the effective permeability of the mixture, and spherical inclusions with permeability μi occupy a volume fraction f in a  host material μe.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 (forεi=50−j*4.0,εe=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.



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*Source: 247958-2008-05-28.xml*