A Coaxial Cable Fabry-Perot Interferometer for Sensing Applications

2.1. Fundamental Physics

As shown in Figure 2, a CCFPI consists of a pair of partial reflectors separated by millimeters to centimeters. The EM wave traveling inside the cable is partially reflected at the first reflector while the remaining energy transmits through to reach the second reflector. At the second reflector, the EM wave is again partially reflected. The two reflected waves travel backwards and interfere coherently to generate an interference signal. When observed in the spectrum domain, the interference signal manifests itself as an interferogram.

The two reflectors can be engineered to have a low reflectivity. As a result, multiple reflections between two reflectors can be negligible in the calculation. Assuming the amplitude reflection coefficients of the two reflectors are the same, the two reflected waves (U₁ and U₂) can be written as:

The two reflected waves have a time delay and the delay is associated with the distance between the two reflectors and the phase velocity of the wave. The interference signal (U) is the summation of the two reflected waves, which can be written as:

Equation (2) describes a wave with its amplitude given by $2\cdot \mathrm{\Gamma }(f)e^{-az}cos(2\pi f\tau )$ and its phase of 2πfτ. The amplitude and phase vary as functions of frequency and the delay. In essence, the amplitude of spectrum varies sinusoidally as the frequency of wave is scanned.

In Equation (2), the only unknown parameter is the reflection coefficient Γ(f). A partial reflector can be generated by introducing an impedance discontinuity in a coaxial cable. There are many methods to implant the impedance discontinuity. In our preliminary research, we used a simple method by drilling a cylindrical air hole into a coaxial cable without touching the inner conductor to avoid significant signal loss. The depth and size of the air hole can be varied to change the reflectivity. The reflection coefficient can be numerically simulated by a commercial full-wave solver including magnitude and phase at discrete frequency. Figure 3 plots that the reflection coefficient (in magnitude) increased as the interrogated frequency increased.

To numerically calculate the interferogram of a CCFPI, the relative permittivity of the dielectric material was set to be 2.25 and α was 0.04 dB/m in the bandwidth of 0 to 6 GHz. The distance between two reflectors (d) was 60 mm. By substituting the calculated reflection coefficient in Figure 3, the interferogram of a typical CCFPI was plotted in Figure 4 (red curve). Several resonant dips can be observed including fundamental frequency and its harmonics. The amplitude of the constructive interferences from 0 to 6 GHz followed the trends shown in Figure 2. The signal-to-noise ratio (SNR) was over 40 dB, indicating that it can be used for sensing after proper calibration. The quality factor (Q-factor) was about 5. Typically, the Q-factor for a FPI is associated with the reflectivity of each reflector and the transmission loss inside the cavity. The Q-factor increases as reflectivity increases or the loss decreases. People can make the Q-factor up to several thousand due to higher reflectivity and low loss. However, the reflection coefficient of each reflector in a CCFPI needs to be designed as low as possible (∼−30 dB in Figure 2) due to the purpose of multiplexing capability.

The above investigation into device physics reveals that the resonant behavior mainly results from the coherent interference. The reflections are generated by impedance discontinuities as a result of interruption in material properties such as the permittivity and permeability or in cable parameters such as the resistance, capacitance or inductance. As such, there are many potential methods to create impedance discontinuity in a coaxial cable besides hole-drilling method. In addition, the reflection coefficient of discontinuity can be designed to obtain more complex reflection profile and this would require more involved analysis.

2.2. Measurement Error Analysis

In practical measurement, measurement errors often occur due to the lack of data information or noisy measurement, and are often difficult to detect since the true value of the parameter under test is unknown. However, it can be numerically analyzed through proper assumption.

According to Equation (2), the environmental parameters change (e.g., strain or temperature) will cause a change in time delay. Subsequently, the interference pattern will change. By tracking the shift in the interferogram, the changes of parameters can be found. In real operation, it is easy to follow the shift in the resonant frequency. From Equation (2) the Nth resonant frequency can be deduced, which is:

In general, the stretch of the cable will cause an elongation to the cable and a decreasing to the dielectric constant due to the photoelasticity effect. These are the dominant factors that will further influence the interferogram. As a result, the applied strain (ε) can be expressed in terms of changes in distance and relative permittivity of the material (Δε_r) using the following equation:

The applied strain (ε) in terms of the Nth resonant frequency change (Δf_N) can be further derived from Equation (5):

Equation (6) describes that the applied strain can be calculated by simply measuring the Nth resonant frequency shift (Δf_N) while assuming that the other parameters in the equation are initially known. The measurement error of strain (Δε_error) can be deduced from Equation (6) as follows:

When the CCFPI is subjected to temperature variation, both the relative permittivity of the material and the length (d) will change due to the effect of temperature on the dielectric constant and the thermal expansion of the material, respectively. By using the same derivation method for strain measurement error, the temperature measurement error (ΔT_error) can be expressed as follows:

3.1. Hole-Drill Method

In order to precisely control the shape and depth of the drilled hole, a computer numerical controlled (CNC) drilling operator (Model 2000, P/N 8020A, Sherline, Carlsbad, CA, USA) was used, where the minima feeding step of the three axes (x, y, z) is 10 μm. A vector network analyzer (VNA HP 8753ES, Santa Clara, CA, USA) was used to monitor in situ the reflection spectrum during fabrication process. One end of the coaxial cable (50 Ω, RG-58, Jamco Electronics, Belmont, CA, USA) was launched to one port of VNA and the other end was matched with a 50 Ω terminator. A drilling bit with diameter of 1/12 inch was used. All the machines including VNA were controlled by a computer. The distance between two holes was 60 mm. The drilling depth was 2.1 mm and the out diameter of the cable was 5 mm. The coaxial cable was properly calibrated by VNA before fabrication. The VNA was configured to with an observation bandwidth from 100 kHz to 6 GHz, a total of 1,601 sampling points and intermediate frequency bandwidth (IFBW) of 10 kHz. A band-pass gating was applied in time domain (after Fourier transfer to the microwave spectrum, a built-in function in VNA) to select the two reflections of drilled holes and suppress multiple reflections from two ends of the coaxial cable.

Figure 4 plots the measured interferogram (black curve) of a CCFPI within the frequency range of 100 kHz to 6 versus the calculated interferogram (red curve). Several resonant frequencies can be observed including fundamental and harmonics. The SNR is over 40 dB. The Q-factor is about 5. The SNR and Q-factor match exactly well with the calculated data, indicating that the hole-drilling method did not incur in any extra loss to the cable. The resonant frequencies cannot be exactly matched because the relative permittivity of the inner dielectric material is frequency dependent, which is the similar with the chromatic dispersion in optics.

3.2. Temperature and Strain Measurement

To demonstrate the capability of using CCFPI as a sensing device in SHM, temperature and strain measurements were conducted. The CCFPI used for temperature measurement had a distance of 60 mm. The VNA was configured to acquire the resonant frequency of ∼4.2 GHz with an observation bandwidth from 3.6 to 4.8 GHz. The CCFPI was placed in a tubular furnace. The temperature was raised from 30 °C to 90 °C with an increasing step of 10 °C. The rising time for each step was set to be 5 °C/min. For each temperature point, the reflection spectrum was measured multiple times consecutively, and the averaged spectrum was applied to find the center frequency of the resonant peak. Fourth-order polynomial curve-fitting was applied to smooth the resonant peak for further improvement of the measurement accuracy.

Figure 5 plots the change in resonant frequency as a function of the ambient temperature and the inset in Figure 5 plots the shift in reflection spectra as the ambient temperature increases. The spectra shift to higher frequency region indicating that the effective length decreases as the temperature increases. The effective length is associated with the physical length between two reflectors and the relative permittivity of the inner dielectric layer. The decreasing in effective length indicates that the relative permittivity decreases as temperature increases. As a result, the change in relative permittivity is the dominant factor when CCFPI is subjected to temperature variation. The Q-factor of the dips decreased as the temperature increased indicating that the propagation loss between two reflectors increased. The resonant frequency increases almost linear with a slope of 1.58 MHz/°C as the temperature increased. The linear temperature-frequency shift relation indicates that CCFPI can be used as a sensor for temperature after it is properly calibrated.

It also has the potential in large strain measurement because the stretch of the CCFPI will directly increase the distance between two reflectors, resulting in a linearly decrease of the resonant frequency. In the strain test, the CCFPI had a distance of 70 mm and was fixed onto two translation stages. The gauge length (distance between two stages) was 500 mm. A pre-strain was initially applied to the cable before testing. After elongating the gauge length at a step of 1 mm, corresponding to a strain increase of 2,000 με (0.2%), the reflection spectrum was acquired through VNA. Eighteen increasing steps or a total strain of about (34,000 με) 3.4% were applied to the cable. Figure 6 plots the change in resonant frequency as a function of the applied strain. The inset plots the shift in reflection spectra as the axial strain increased. The spectra shifted to the lower frequency range, which can be predicted in Equation (2). The quasi-linear strain-frequency shift relation (∼3.3 kHz/με) indicates that CCFPI can be used as a sensor for large strain measurement after it is properly calibrated.