Long Period Bragg Grating in Coaxial Transmission Lines

This work shows the utilization of a coaxial cable for the fabrication of a long period Bragg grating. The grating is fabricated removing the dielectric in short pieces of the cable so that the discontinuities account for the variation in the medium refractive index. Simulated and experimental results of the grating resonances are shown as a function of the scattering parameters S, demonstrating the feasibility of the technique. By using the sensor in the MHz frequency range cheaper electronics can be employed, which reduce the overall cost of the sensing system.


I. INTRODUCTION
Transmission lines, by definition, are guided systems used to transmit data and energy in the electromagnetic form [1]. The energy propagates in a longitudinal way, with two or more conductors immersed in a homogenous dielectric connecting a source to a load.For low power applications, transmission lines are used to transmit data in telecommunication systems, whereas they are also used to transport energy in high voltage systems [2][3].Examples of transmission lines are lines with two parallel or tracing conductors, planar lines or parallel plates, a parallel wire connected to a conductor plane, microstrip lines, optical fiber, and a coaxial cable [4].Some of these lines are used as sensors for detecting physical quantities such as temperature and strain.For instance, optical fibers have been used with great advantages for this purpose due to its low cost, low signal attenuation and electromagnetic immunity [5][6].However, they present disadvantages such as the mechanical fragility to transverse stresses and the high cost of the required electro-optical equipment.Although coaxial cables show higher attenuation, they turn out to be more resistant and, depending on the frequency range, they also offer good insulation to electromagnetic interference.At the same time, interrogators and lower cost electronic circuits are available, eliminating the need for expensive electro-optical conversion and contributing to the construction of low-cost sensing systems [7].Different devices and techniques commonly used in optics for sensing applications, such as Bragg Gratings [8] and Fabry-Perot interferometers [9], have been replicated at lower frequencies in coaxial cables.

Long Period Bragg Grating in Coaxial Transmission Lines
Sergio Luiz Stevan Jr. 1 , José Jair Alves Mendes Júnior 1 , Frederich Conrad Janzen 1 , Murilo Leme Oliveira 1 , 1 Universidade Tecnológica Federal do Paraná -UTFPR -Ponta Grossa, Brazil, sstevanjr@utfpr.edu.br,mendes.junior13@yahoo.com.br,fcjanzen@utfpr.edu.br,muriloleme@utfpr.edu.br,Alexandre de Almeida Prado Pohl 2 With this in mind, this work reports on the fabrication and characterization of Long Period Bragg Gratings (LPBG) in coaxial cables, which employ the same concepts of gratings written in optical fibers.Thus, the implementation of Long Period Bragg Gratings in coaxial transmission lines is proposed and described through a mathematical model, simulation and experimental results, demonstrating that they can be used as an option for applications in which gratings fabricated in optical fibers present limitations.

II. COAXIAL TRANSMISSION LINES
A coaxial cable is a transmission line made of a central conductor and a concentric cylindrical ground loop, separated by a dielectric material and extruded to an external polymer-based protection cap [10].One of the main parameters of coaxial cables is the characteristic impedance, which for commercial cables presents values of either 50 Ω or 75 Ω.The 50 Ω-cable is commonly used for data transmission, while 75 Ω-cables are used in antenna reception systems and broadband networks for transporting digital and analog TV signals [10].
The electric equivalent of the coaxial transmission line of an infinitesimal length Δz is represented by the circuit shown in Figure 1, in which the parameters L, R, C, and G represent its Inductance  Applying Kirchhoff Laws to the circuit shown in Figure 1, one can assess the electric voltage, v(t, z) and the current, i(t, z), along the coaxial line in the time domain.The resulting equations that describe the behavior of v(t, z) and i(t, z) are shown in (1) and ( 2) [11] - - (2) Through this system of coupled equations, one derives the characteristic impedance, Z, of the coaxial transmission line, which is given as [12] Z=√ R+jωL G+jωC . ( Solving equations ( 1) and ( 2) by means of phasor terms one obtains equations ( 4) and (5), which refer to the voltage and current in the transmission line along the distance z of the line [11].
= γ²i(z) . ( In such equations γ is the complex propagation constant of the line, given as γ= √(R+jωL).(G+jωC)≡α+ jβ .(6) The relationship between the cable parameters and the frequency of the signal to be transmitted can also be described by the real part α of (6), which is the attenuation constant given in neper/ meter, and by the imaginary part β, which represents the phase constant in radians/meter.Solving the system of coupled equations given in ( 4) and ( 5) one obtains [11] V(z)=V 0 + e -γz + V 0 -e γz (7) I(z)= (V 0 + e -γz + V 0 -e γz ) Z =I 0 + e -γz + I 0 -e γz , (8) in such equations V 0 + and I 0 + represent the amplitude of the signal wave propagating in the positive direction of z (transmission wave) and V 0 -e I 0 -represent the amplitude of the signal wave propagating in the negative direction of z (reflected wave), respectively.
Another method of analyzing the transmission line considers the input and output behavior of the signals in a short path of coaxial cable, along which the fractions of reflected and transmission waves are taken into account.This technique is known in the literature as the Scattering Matrix approach [13][14].In this case, the path of cable can be represented as a quadrupole (see Figure 2) and the relationship between the input and output signal waves is obtained by the product of matrices, given in ( 9) respectively, in which V 1 -is the voltage amplitude that is reflected on the left side of the cable patch seen in Figure 2 and V 1 + is the voltage amplitude on the input to the left side.Similarly, V 2 -represent the value of the voltage reflected on the right side of the patch and V 2 + is the value of voltage impinging on the right side.Thus, S 11 and S 21 are the main parameters used to assess the behavior of the reflected and transmitted waves, respectively, propagating in the transmission line.

III. BRAGG GRATING AND LONG PERIOD BRAGG GRATING
A grating in an optical fiber is a structure fabricated by means of a periodic modulation of the refractive index of the fiber core along its length.They are classified either as Bragg or Long Period grating, depending on the size of the period.For instance, Bragg gratings present very short periods, while Long Period Bragg Gratings (LPBG) show periods in the order of hundreds of micrometers [15][16][17].Figure 3 shows the behavior of a Fiber Bragg Grating (FBG).Whenever an optical signal of large spectral width impinges on the grating, a small range of frequencies is reflected back, whenever this range matches the Bragg condition determined by the period of the grating [18].On the other hand, a LPBG presents no reflected band and behaves as a band-stop filter in transmission [17,19], as a result of the coupling between the forward propagating and cladding modes of the fiber.

AoP32
Fig. 3: Interaction of an optical signal band with the Bragg Grating [15] The fact that the optical properties of gratings are affected by temperature variations and/or mechanical deformations makes them an important and useful sensor [20,21], which can be applied in different areas, such as biomechanics [22], civil engineering, [23] and aeronautics [24,25].The same working principle of gratings, when used in coax cables, presents many advantages, mainly when the final application is devoted to the continuous monitoring of buildings, bridges and other civil engineering constructions.
The behavior of Bragg gratings in optical fibers are described by the Maxwell system of differential equations [11].In this way, a parallel can be traced between the optical fiber and the coaxial cable, as the signal propagating in them are both governed by the same laws.For instance, when an electromagnetic wave traverses an interface between two different materials, part of it is transmitted and the other is reflected back.The reflectivity accounts for the part that is reflected and is given as in which n 1 is the refractive index of the incidence medium and n 2 is the refractive index of the transmission medium, respectively.When the discontinuity of the refractive index between the two media is small (n 2 ≈ n 1 -Δn) the reflectivity can be written as A Bragg grating fabricated in a fiber can be seen as a medium that presents several small discontinuities in the refractive index along the fiber core.Therefore, such discontinuities result in many reflections.The spacing between the discontinuities may result in destructive or constructive interference of the electromagnetic wave along the guide.In the case of constructive interference, the sum of the reflected power is concentrated in a small band of specific wavelengths, whose maximum value is given by the Bragg relationship λ B =2n eff Λ (14) in which λ B is the peak wavelength of the reflection band, n eff is the effective index of the propagating mode in the fiber, and Λ is the spatial period of the refractive index discontinuity.Therefore, these two parameters (effective refractive index and period) have paramount importance on the employment of the Bragg grating in optical fibers and coaxial cables.
Still, the grating behavior [18] can also be explained using the Bragg condition for two interacting modes in the fiber, which propagate in opposite directions, given as In (15) β is the propagation constant of a guided mode, the same parameter that appears in (6); β + and β -represent the constants of the forward propagating and the backward propagating modes, respectively; Λ is the period of the refractive index discontinuity that characterizes the Bragg grating and Δβ is the difference of the propagation constants involved in the mode coupling.Since the propagation constants are a function of the wavelength, the corresponding phase is highly selective to it [18].
The propagation constant of the fundamental forward propagating mode is given as in which β 01 is the fundamental forward constant, λ is the wavelength and n eff is the effective refractive index of that mode.For the reverse (backward) propagating mode, the propagation constant has the same magnitude but opposite sign, as shown in (15).Replacing (15) in ( 16) one finds the phase matching condition for these two interacting modes given as [18] λ B = which leads to the expression in (14) when N = 1.The wavelength λ B is the Bragg wavelength in which the mode of forward propagation couples with the mode of reverse propagation.N is the order of interaction between the two modes (typically, N is equal to one in the case of coupling of the forward to the backward fundamental mode).Therefore, one observes that the Bragg wavelength is a AoP34 function of the period of the grating, Λ, and the mode refractive index, n eff .In this way, by submitting the grating to external effects (change of temperature and/or application of strain), both parameters can be changed and the Bragg wavelength will be shifted [26].
These concepts are valid for modes in optical fibers as as for modes propagating in coaxial cables [27].The expression 2β= 2πm Λ (18) is similar to (15) and can be used to describe the mode coupling in coaxial cables in which discontinuities are present with m as an integer representing the interaction order [28].For coaxial cables the propagation constant is evaluated as [29] β=2πf√LC (19) The expression above describes the propagation constant β, in which f is the frequency related to the propagating mode and L and C are the inductance and capacitance parameters of the coaxial cable.
The capacitance can be obtained from the cable datasheet and L can be calculated, for instance, from equation ( 3) as long as the impedance and attenuation of the coaxial cable are known.
Replacing (19) in (18) gives Making vp ≡ 1/√LC, in which vp is the phase velocity of the wave in the continuous medium, one obtains  0 =vp/2Λ, for m = 1.That is, the fundamental resonance frequency of the grating structure is directly proportional to the phase velocity and inversely proportional to the grating period.
Wei et al [27] derived such equation, which exhibits an efficient form to realize the mode conversion in coaxial cables with periodic discontinuities.
Hence, the resonances of the Long Period Gratings depend on the parameters L and C of the cable, which determine its characteristic impedance as well as on the period Λ of the grating.One way of changing the characteristic impedance is to impose changes in the dielectric structure of the cable.
Particularly, if one removes the dielectric in a controlled and regular way it is possible to create a number of discontinuities along a patch of cable, which are necessary to form the grating.Considering the distributed parameters L=0.262 µH/m and C=93.5 pF/m for the RG58 cable and using (20), one estimates the phase velocity approximately as 2.02×10 8 m/s.In this case, the estimated fundamental resonance frequency is f o =1.141 GHz for =8.7 cm, 1.171 GHz for =8.5 cm and f o =1.291 GHz for =7.7 cm, respectively.From Figure 5 one notices that the reflection becomes stronger at higher-order harmonics.However, due to the cable loss, the increase is not linear [30].

V. FABRICATION AND EXPERIMENTAL RESULTS
The LPBG was fabricated removing the dielectric in a stretch of cable so that the discontinuities always have the same width and depth.For this purpose, a circular saw and mold were used, as illustrated in Figure 6.Figures 7a) and

Λ
Especially regarding the period =8.7 cm, measurements were conducted for different numbers of discontinuities.Figure 8 shows the evolution of the grating rejection peaks considering the period of 8.7 cm, within the same frequency range, as the number of discontinuities inserted in the cable is increased: 10, 22, 34, 46 and 58, respectively.Thus, one observes that the higher the number of discontinuities, the deeper the resonance amplitude.The portion of the wave matching the reflected part is found as multiple frequencies of 1.141 GHz.
0,0 0,5 Figure 9 shows the comparison between the simulated (continuous line) and experimental (dotted line) values of S 12 for the period of 25 cm.One observes that, even if the resonance strength is not high, multiple frequencies of 400 MHz are seen over the frequency range, whose amplitude are more intense for the higher order harmonics.For instance, by fabricating a grating of period 8.7 cm in a RG58 coax type cable, the fundamental 1.141 GHz resonance with 4 dB intensity and resonances deeper than 10 dB for the first harmonic (at 2.281 GHz) have been experimentally observed.It has been also observed that the larger the period of the grating, the lower is the fundamental resonance for sensing purposes.Increasing the number of discontinuities in the stretch of cable also leads to deeper resonance peaks as shown in Figure 8.
Future work on this topic can approach the modeling of discontinuities (such as its number and length) in order to verify their influence on the intensity of the resonance peaks.Moreover, one can improve the design for the sensing element, as the present technique of removing portions of the dielectric is invasive and imposes mechanical variations in the cable structure.Furthermore, one can also investigate the design of cheaper electronic circuits for applications concerning frequencies in the megahertz range, leading to sensing systems that can be implemented as simple and cheap solutions.

Fig. 1 :
Fig. 1: Electric equivalent model of the coaxial transmission line.

Fig. 2 :
Fig. 2: Coaxial Transmission Line represented by a quadrupole the transmission line with discontinuities was performed using the software Agilent Advanced Design System, version 2013.6.Figure4shows a schematic diagram of the coaxial transmission line (RG58; 50 ) simulated in 5 meters length, in which discontinuities of 2 mm length intercalated along 18 stretches of coaxial cable determine the period Λ of the long period grating.A larger stretch of cable finishes the 5 m length.Simulations with Λ = 7.7 cm, 8.5 cm and 8.7 cm were performed.The discontinuities are represented by the absence of the polyethylene dielectric, and the dielectric constant is set at 1.0006 (air) and the loss tangent is zero.The main parameters of coax cable used in the simulation are: A is the radius of inner conductor; Ri is the inner radius of outer conductor; Ro is the outer radius of outer conductor; L is the length; Cond1 is the plating metal conductivity; Cond2 is the base metal conductivity; Mur is the relative permeability of dielectric; Er is the dielectric constant of the dielectric between the inner and outer conductors; and TanD is the dielectric loss tangent; and the attributed values are presented in figure4.The parameter comp represents the length between discontinuities and the parameter hole, the length of the discontinuity.The simulation aims to analyze the propagation of a signal whose frequency varies between 100 MHz to 6 GHz and to observe the transmission and reflection behavior by means of the Scattering Parameters S 11 and S 12.

Figure 5
shows the results of the S 12 coefficient for Λ = 7.7 cm (doted curve), 8.5 cm (continuous curve) and 8.7 cm (traced curve), in which an alteration in the discontinuities results in the alteration of the resonance frequency by the grating.One notices that resonances appear at a fundamental frequency and at subsequent harmonics.

Fig. 4 :Fig. 5 :
Fig. 4: Model for simulation of the coaxial transmission line with discontinuities 7b) show, respectively, a top and side view of the discontinuity in the cable.Using a Vector Network Analyzer (VNA), Agilent Model ENA E5071C, set to scan the frequency range between 100 MHz and 3.3 GHz, the parameters S 11 , S 12 , S 21 , and S 22 were measured considering different discontinuity periods in the RG58 (50 Ω) coax cable.Experiments were conducted considering periods of 7.7 cm, 8.7 cm and 25 cm, respectively.

Fig. 8 :
Fig. 8: Behavior of parameter S 12 as a function of the frequency and the number of discontinuities (grating period =8.7 cm).

Fig. 9 :
Fig. 9: Graphic of the parameter S 12 with the values of experimental and simulated values with period of 25 cm