3.3.1 Requirements for free-space measurements
To achieve accurate measurement results using free-space method, several requirements should be satisfied, which mainly include far-field require- ment, sample size, and measurement environment.
3.3.1.1 Far-field requirement
In free-space measurement, to ensure that the wave incident to the sample from the antenna can be taken as a plane wave, the distance d between the antenna and the sample should satisfy the following far-field requirement:
d> 2D2
λ (3.79)
where λ is the wavelength of the operating elec- tromagnetic wave and D is the largest dimension of the antenna aperture. For an antenna with circu- lar aperture,Dis the diameter of the aperture, and for an antenna with rectangular aperture, D is the diagonal length of the rectangular aperture. When the far-field requirement is fulfilled, free space can be taken as a uniform transmission line, and most of the measurement schemes discussed above for
coaxial reflection measurements can be realized by free space.
3.3.1.2 Sample size
In the measurement of permittivity and permeabil- ity of planar samples using free-space method, if the sample size is much smaller than the wave- length, the responses of the sample to electro- magnetic waves are similar to those of a particle object. To achieve convincing results, the size of the sample should be larger than the wavelength of the electromagnetic wave. To further minimize the effects of the scatterings from the sample bound- ary, the sample size should be twice larger than the wavelength.
3.3.1.3 Measurement environment
In a free-space transmission structure, as the electromagnetic wave is not limited by a sharp boundary, the measurement results may be affected by the environments. At lower frequencies, the effects of environments are more serious. To minimize the effects of the environments, it is recommended to conduct free-space measurements in an anechoic chamber. Meanwhile, we can also use time-domain gating to eliminate the unwanted signal caused by environment reflections and multireflections, as discussed in Section 2.4.6.
3.3.2 Short-circuited reflection method
Figure 3.30 shows a typical setup for short- circuited free-space reflection measurement. The sample backed by a metal plate is placed in front of an antenna with a distance satisfying the far-field requirement.
The complex reflectivity S11 at the interface between the free space and the sample is given by
S11= jztan(βd)−1
jztan(βd)+1, (3.80) where z is the wave impedance of material under test normalized to the wave impedance of free space and β is the phase constant in the
d
Sample Metal plate
Γ
Figure 3.30 Free-space reflection method
material under test. For nonmagnetic materials, the normalized wave impedance is given by
z= 1
√εr
, (3.81)
and the phase constantβ is given by β= 2π
λ
√εr (3.82)
where λis the free-space wavelength andd is the thickness of the sample. Therefore, the dielectric permittivity of the sample can be obtained from the complex reflectivity.
However, Eqs. (3.80)–(3.82) indicate that the permittivity of the sample cannot be expressed explicitly in terms of S11 and d, and numerical methods are often used in the calculation of dielectric permittivity.
3.3.3 Movable metal-backing method
Kalachevet al. proposed a movable metal-backing method, whose measurement configuration is shown in Figure 3.31 (Kalachevet al. 1991). The sample under test is placed directly against the aperture of the horn, so the sample size does not need to be very large, and the microwave anechoic chamber is not required.
The measurement structure consisting of the sample under test and the metal backing is quite similar to an open resonator. As the metal backing is movable, the resonant frequency can be adjusted. The dielectric sample is semitransparent
t d
Horn antenna
Dielectric sample Movable metal backing Figure 3.31 Measurement configuration of dielectric permittivity for semitransparent samples
to electromagnetic wave. When an electromagnetic wave is incident to the dielectric sample, some of the energy is reflected back and some of the energy passes through the sample. In the resonant structure, the dielectric sample serves as a reflection mirror, and meanwhile it also provides coupling to the antenna.
This method is based on the reflectivity mea- surement near the resonant frequency of the open resonator, and so it is a method between a nonres- onant method and a resonant method. It should be noted that, in this method, as the sample directly contacts the antenna aperture, the far-field require- ment is not satisfied, and the residual reflections of the horn should be taken into consideration.
The residual reflections can be eliminated by time- domain techniques, and they can also be corrected by an additional measurement when the distance between the sample and the horn is changed by a quarter wavelength (Kalachevet al. 1991).
To determine the permittivity of the dielectric sample, we measure the reflection coefficient of the system consisting of the sample and the metal backing. The distancedbetween the sample and the metal backing can be varied, and the distance dependence of the reflection coefficient Ŵ(d) exhibits a resonance form. For an optically thin sample satisfying
2π t√
|εr|
λ ≪1 (3.83)
where t is the thickness of the sample, we can get (Kalachevet al. 1991)
cot2πd0
λ = 2π tε′r
λ (3.84)
2π tε′′r
λ = 1±Ŵ0
1∓Ŵ0 (3.85) where d0 is the distance satisfying the resonance conditions and Ŵ0 is the reflection coefficient at resonance.
So the dielectric properties of the sample can be calculated from Eqs. (3.84) and (3.85). However, there are two solutions for Eq. (3.85). This is a common disadvantage of using reflection method to measure resonant structures. One convenient method to resolve this ambiguity is to introduce an additional source of low dielectric loss into the structure and find out whether the additional loss results in an increase or decrease in the reflection coefficient.
To avoid the limitation on the optical thick- ness of the sample, and to reduce the random experimental errors, we can measure the reflection coefficient Ŵ(d) at several values of d near the resonance. The reflection coefficient of the reso- nant structure can be calculated using the multi- layer interferometer model (Brekhovskikh 1980).
The value of complex permittivity is selected so that the agreement between the measured reflection constants Ŵm(d) and the calculated ones Ŵc(d) is the best. The coincidence between the calculated and the measured values of reflection coefficients can be described by
F =
n
i=1
[Ŵm(di)−Ŵc(di)]2 (3.86) The dielectric permittivity of the sample corre- sponding to the minimum value ofF can be found by two-dimensional simplex minimization routine with the use of Eqs. (3.84) and (3.85) as a start- ing point.
The applicability of this method depends on the value of Ŵ0 of the sample. When the dielectric losses are absent (εr′′=0) or very high (εr′′→
∞), the value of Ŵ0 is close to unity and there are secondary wave reflections between the sample and the horn, which increase the errors
of reflection measurement as well as the errors of the determination of the dielectric permittivity values. Supposing that accurate measurement of reflection coefficient is possible whenŴ <−5 dB, from Eq. (3.85) we can find the range ofεr′′suitable for this method (Kalachev 1991):
0.5< 2π tεr′′
λ <4 (3.87) If the above requirements cannot be satisfied, this method should be modified. For a low-loss sample with
2π tεr′′
λ ≪1, (3.88)
to obtain accurate results, it is recommended to characterize the sample together with an additional dielectric-sheet material with known properties.
The additional sample should provide the dielectric loss that is sufficient for obtaining low reflectivity.
For an extremely high-loss material with 2π tεr′′
λ ≫1, (3.89)
the sample is practically nontransparent and the reflection coefficient does not depend on the pres- ence and position of the metal backing behind the sample. To measure such nontransparent sam- ples, Kalachev et al. modified the measurement technique discussed above (Kalachevet al. 1991).
As the loss factor of a material is proportional to the conductivity of the material, we may use surface impedance to describe the electromag- netic properties of an extremely high-loss mate- rial. The reflection methods used for the measure- ment of surface impedance will be discussed in Section 3.5.
3.3.4 Bistatic reflection method
In most of the reflection methods, only one probe is used, and this probe transmits signal and meanwhile receives signal. In such a configuration, it is difficult to change the incident angle of the transmitted signal. Figure 3.32 shows a bistatic system for free-space reflection measurement. In this configuration, two antennas are used for transmitting and receiving signals respectively,
Sample terminated by metal plate Transmit
antenna
Receive antenna
Figure 3.32 Bistatic reflection measurement
and the reflections at different incident angles can be measured. Using this configuration, the properties of materials at different directions can be characterized.
It should be noted that in bistatic reflection mea- surements, the reflection is dependent on the polar- ization of the incident wave. Incident waves with parallel and perpendicular polarization usually result in different reflection coefficients. Besides, special calibration is needed for free-space bistatic reflection measurements (Umariet al. 1991).