Abstract

For the online writing of ultra-weak fiber Bragg gratings (FBGs) in the drawing optical fibers, the effects of the intensity profile, pulse fluctuation and pulse width of the excimer laser, as well as the transverse and longitudinal vibrations of the optical fiber have been investigated. Firstly, using Lorentz-Loren equation, Gladstone-Dale mixing rule and continuity equation, we have derived the refractive index (RI) fluctuation along the optical fiber and the RI distribution in the FBG, they are linear with the gradient of longitudinal vibration velocity. Then, we have prepared huge amounts of ultra-weak FBGs in the non-moving optical fiber and obtained their reflection spectra, the measured reflection spectra shows that the intensity profile and pulse fluctuation of the excimer laser, as well as the transverse vibration of the optical fiber are little responsible for the inconsistency of ultra-weak FBGs. Finally, the effect of the longitudinal vibration of the optical fiber on the inconsistency of ultra-weak FBGs has been discussed, and the vibration equations of the drawing optical fiber are given in the appendix.

© 2015 Optical Society of America

1. Introduction

A largescale ultra-weak fiber Bragg gratings (FBG) sensor array is made up of hundreds or thousands of identical-wavelength FBGs with a reflectivity of about −50dB for each FBG. Such a largescale ultra-weak sensor array has attracted much attention in major engineer monitoring because of its low cost, low crosstalk and strong multiplexing capacity [1–3]. To fabricate a largescale ultra-weak sensor array, online writing technique should be utilized, correspondingly the optical fiber sensor array has no fusion loss and high tensile strength, these advances open new possibility for further application of largescale FBG networks [4–6].

Recently, our team has realized and reported the online writing of ultra-weak FBGs array during the drawing process of single mode fibers (SMF). In our online writing system, the UV laser emitted by 248 nm excimer laser irradiates onto the drawing optical fiber through the phase mask with constant period [7,8]. However, our measurements showed that the consistency of FBGs in the drawing optical fiber is not very perfect for the reflection spectra of all FBGs in an array are not the same entirely. The main performances are that the Bragg wavelengths of all FBGs fluctuate in the range of ± 0.1 nm, or the reflectivities of all FBGs vary in a range of several dB, or the reflection spectra of some FBGs have larger side lobes.

Then what reasons lead to the inconsistency of FBGs in an array? It is well-known that the reflection spectrum and the Bragg wavelength of an FBG are decided by the effective refractive index (RI) of optical fiber, the index perturbation in the grating region and the constant period of phase mask. Therefore the inconsistency in the Bragg wavelengths of all FBGs in an array should come from the inconsistency of the total RI of the optical fiber at the writing positions, so it is necessary to find and analyze the factors which may affect the total RI distribution in the grating region.

In this paper, for the online writing of ultra-weak FBGs in the drawing single mode fibers, we have investigated the effects of the intensity profile, pulse fluctuation and pulse width of the excimer laser, as well as the transverse and longitudinal vibrations of the optical fiber.

2. Theory

2.1 The RI perturbation along the drawing optical fiber

In Fig. 1, an optical fiber is pulled out at a constant speed c through the outlet of the furnace and the capstan. Building the OXYZ coordinates, the XOY plane is parallel to the phase mask, the OX axis is along the vertical optical fiber. The outlet of the furnace (O) is at x = 0, and the touching point between the optical fiber and the capstan (P) is at x = L. The moving optical fiber may be modeled as a one-dimensional continuum with the axial stiffness EA, linear density ρ and steady-state tension T0. The phase mask and the UV laser are fixed adequately. When the UV beam writes an ultra-weak FBG in the region [x0, x0 + LB] of a single mode fiber, a total index perturbation along the FBG will be produced.

 

Fig. 1 Optical fiber drawing tower and schematic of online writing ultra-weak FBGs.

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The drawing optical fiber is composed of SiO2 and some dopant substances, such as GeO2, B2O3, P2O5, and so on, then the total RI of doped optical fiber ntot is decided by the RIs (n1, n2, …) and the volume fractions (ϕ1, ϕ2, …) of SiO2 and all doped substances. In practical applications, the simplest empirical formula is Gladstone-Dale mixing rule [9, 10]:

ntot=∑c=1Ncnncϕc
where Ncn is the number of components, nc and ϕc are the RI and volume fraction of pure component c, respectively. Since the volume fractions of all components are constants, the differential form of Gladstone-Dale mixing rule is

δntot=∑c=1Ncnϕcδnc

The RI of pure substance is decided by its molar refraction, molar mass of molecule M, density of material ρv, and can be expressed as their function [9, 10]:

n2−1n2+2Mρv=RLL
where RLL=4πNAαmol(T,ρv)/3 is the molar polarization (refractivity), αmol the molar electronic polarization, and NA Avagadro’s number. At given wavelength and temperature, the derivative of Eq. (3) yields
ρv∂n∂ρv=(n2−1)(n2+2)6n
where the derivative of the molar polarization with respect to the density can be omitted.

The vibration of the moving optical fiber may be divided into the transverse vibration and longitudinal vibration, the former is perpendicular to the optical fiber, the latter is along the optical fiber. The small transverse vibration is parallel to the propagation direction of diffraction ray of UV laser though the phase mask. Our experimental results showed that the inconsistency in the reflection spectra of all FBGs is attributed to the longitudinal vibration of optical fiber. Let u(x, t) denotes the longitudinal displacement at point x and time t. Due to the effect of longitudinal elastic wave, the density along the moving optical fiber can be written as

ρv(x,t)=ρv0+Δρv(x,t)
ρv0 is the average density, Δρv is the density fluctuation along the optical fiber. The RI fluctuation along the optical fiber is

δn(x,t)=∂n∂ρv⋅δρv(x,t)

According to the continuity equation:

∂ρv∂t=−∂∂x(ρvu,t)
we can obtain that the increase of the fiber density is proportional to the pulse width (δt) and the velocity gradient along the x direction (u,xt):

δρv(x,t)≈−ρv0u,xtδt

Substituting Eqs. (4) and (8) into Eq. (6), the RI fluctuation of component c along the optical fiber due to the longitudinal vibration mode is

δnc(x,t)=−δt⋅u,xt(nc2−1)(nc2+2)6nc
so the total RI perturbation is
δntot=−δt⋅u,xt∑c=1Ncϕc(nc2−1)(nc2+2)6nc
and the RI distribution along the optical fiber can be expressed as
n(x,t)=n0+δntot(x,t)
It can be seen that, the amplitude of the RI perturbation of along the optical fiber is proportional to the pulse width δt, the velocity gradient of longitudinal vibration, as well as the RIs and volume fractions of all sensitive components.

2.2 The RI perturbation along the ultra-weak FBG in the drawing optical fiber

All ultra-weak FBGs are written by using only one laser pulse. Let an ultra-weak FBG be written in the region [x0, x0 + LB] of a moving optical fiber in a single pulse time δt, then the total index perturbation along the FBG, when the drawing optical fiber was released and free, can be expressed as

Δn(x)=Ω(x)Π(x)⋅Δn0[1+cos(2πΛ(x)x+φ0)],          x0<x<x0+LB
Where Δn0 is the “dc” index change spatially averaged over a grating period, φ0 the additional phase. Due to the local tension and strain at the position of the FBG arising from the longitudinal vibration, when the drawing optical fiber was released and free, the actual period of FBG can be expressed as Λ(x) = Λ0/[1 + (1−pe)u,x(x)], where Λ0 is the nominal period of phase mask, pe the effective elastooptic parameter and u,x(x) the axial strain function. The modulation function Ω(x) comes from the contribution of optical fiber longitudinal vibration to the RI fluctuation along the optical fiber, the modulation function Π(x) comes from the contribution of pulse intensity fluctuation and optical fiber transverse vibration to the RI fluctuation along the optical fiber, and Ω(x) = Π(x) = 1.0 may be chosen for the static optical fiber.

As the UV beam writes an ultra-weak FBG, only the sensitive component (color center, CC) has the contribution to the index change in the region of [x0, x0 + LB]. When the drawing optical fiber was released and free, coming from the writing of an UV pulse on the longitudinal elastic wave, from Eqs. (10) and (11), the modulation function can be reduced to

Ω(x,t)=1−δt⋅u,xtϕCC(nCC2−1)(nCC2+2)6n0nCC
It can be seen that, the second term is proportional to the pulse width (δt) and the velocity gradient of the longitudinal elastic wave (u,xt). In principle, when one found the function u(x,t), one can obtain the reflection spectrum of any ultra-weak FBG by using Eqs. (12) and (13). But in practice, it is very difficult to ascertain the perturbation function of the collecting capstans and obtain the solution of Eq. (33) in the appendix.

Especially, if using the expansion of the longitudinal displacement shown in Eq. (35), we have

Ω(x,t)=1+∑m=1NmGmsin(2πfumt−2πλumx)
Gm=δt⋅ϕCC(nCC2−1)(nCC2+2)6n0nCC⋅Bm
Then, the total RI perturbation in the grating can be changed as
Δn(x)=Δn0[1+∑m=1NmGmsin(2πfumt−2πλumx)]  [1+ηcos(2πΛx+φ0)]
≈Δn0{1+ηcos(2πΛx+φ0)+∑m=1Nm12ηGm[cos(2πΛm−x+2πfumt+φ0')+cos(2πΛm+x−2πfumt+φ0'')]}
where Λm+=Λ(1+Λ/λum) and Λm−=Λ(1−Λ/λum) when λum>>Λ. According to the coupled mode theory [11], it can be seen that besides the ordinary Bragg wavelengthλB=2πΛ, there are many additive Bragg wavelengths λm+=2πΛm+ and λm−=2πΛm−, their locations in the wavelength axis are related to the parameters of longitudinal modes, their maximum intensities are related to the fluctuation amplitude of the modulation function Gm, and proportional to the pulse width δt.

3. Experimental results and discussion

3.1 Effects of excimer laser pulse

For the typical used excimer laser, its intensity profile, pulse fluctuation and pulse width may lead to the inconsistency of the total RI perturbations at the writing positions of the optical fiber and affect the reflection spectra of ultra-weak FBGs. Figure 2 shows the two dimensional intensity profile of 248 nm excimer laser beam at the distance of 40 cm from the output mirror, the beam size is 10 × 4 mm, along the X and Y direction respectively. It can be seen that there are small fluctuations at the top of the beam intensity profile. In ideal condition, the drawing optical fiber is at the center of the beam and is parallel to the X direction. If the optical fiber undergoes small transverse vibration behind the mask, the pulse intensity irradiated onto the optical fiber may has small fluctuation, so the intensity distribution and the position of the optical fiber behind the mask may have some influence on the ultra-weak FBG. In addition, for the 248 nm excimer laser, there is a pulse fluctuation of about 10% of the pulse energy, this could cause stronger and weaker gratings too.

 

Fig. 2 2-D beam profile of 248 nm excimer laser at the distance of 40 cm from the output mirror.

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To understand the effects of above factors on the reflection spectra of ultra-weak FBGs, keeping the UV laser and phase mask fixed, we have prepared huge amounts of ultra-weak FBGs in the non-moving prepared optical fiber, and obtained their reflection spectra by using light grating interrogator (LGI-100B, Sentek instrument, LLC), the measured reflection spectra showed good consistency. Typically Fig. 3 gives the reflection spectra of three ultra-weak FBGs prepared by 284 nm excimer laser with pulse width of 10 ns and pulse power of 20 mJ, where the three FBGs are respectively at the positions of 24.732, 31.678 and 35.725 m of an optical fiber and their central wavelengths are close to 1551.924 nm. In view of above experimental results based on the non-moving optical fibers, one can conclude that the intensity profile and pulse fluctuation of the excimer laser are little responsible for the inconsistency of ultra-weak FBGs.

 

Fig. 3 Reflection spectra of 248 nm laser.

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3.2 Effects of drawing velocity and steady-state tension

All ultra-weak FBGs are written by using only one laser pulse. Supposing the optical fiber is drawing at velocity c, the moving distance of the optical fiber in a pulse time δt is lf = cδt。As shown in Fig. 4, at the begin of the pulse, the laser irradiates onto the AB piece of optical fiber; due to the drawing, at the end of the pulse, the laser irradiates onto the A’B’ piece of optical fiber, where AB¯=A'B'¯=l0 is the length of a phase mask, AA'¯=B'B'¯=lf. In a pulse, the irradiated region of optical fiber is AB' piece and the irradiating intensity of laser is proportional to the irradiating time and can be expressed as:

 

Fig. 4 Irradiating time of a pulse onto the optical fiber.

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G(x)=t(x)δt={x/lf ,                       0≤x≤lf    1  ,                                     lf≤x≤l01−(x− l0)/lf ,            l0≤x≤l0+lf

Considering the length l0~7 mm and the pulse width of 248 nm laser δt~10 ns. For the drawing velocity of optical fiber c~1 m/s, lf = cδt = (1 m/s)?(10 ns) = 0.01 μm<<l0, so when the drawing velocity is lower than 1 m/s, the modulation function led by non uniform irradiating time may be taken as G(x) = 1.0 and the effects of the drawing velocity on the reflection spectra of the FBGs may be omitted.

In fact, we have finished a large amount of online writing experiments of ultra-weak FBGs during the drawing process of SMF. From hundreds of reflection spectra of ultra-weak FBGs, we found that the main characteristics of the reflection spectra change little when the drawing velocity increases from 5 m/min to 30 m/min. Typically, Fig. 5 shows the reflection spectra of three ultra-weak FBGs written at different drawing speeds (5, 10, 17 m/min), and Fig. 6 gives the reflection spectra of three FBGs written at the same drawing speed (20 m/min). In these spectra, the central wavelength fluctuates in the range of ? 0.1 nm, the maxmium reflectivity fluctuates in a range of several dB, and larger side lobes occur at the longer or/and shorter wavelength.

 

Fig. 5 Reflection spectra of three FBGs written at different speeds.

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Fig. 6 Reflection spectra of three FBGs written at the same speed (20 m/min).

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From the governing Eqs. (A5) and (A6), it can be seen that, in the drawing process of optical fiber, the steady-state tension T0 has direct effect on the transverse vibration of the optical fiber, and only the term w,xw,xx of the transverse vibration is great enough, the steady-state tension T0 has small effect on the longitudinal vibration. By adjusting the pulling force from 20 to 100 gram, we have also finished lots of experiments and found no obvious change in the reflection spectra of the ultra-weak FBGs, it shows the pulling force in the range of 20 to 100 gram has little effect on the reflection spectra of the ultra-weak FBGs.

3.3 Effects of transverse and longitudinal vibrations

The vibration of the moving optical fiber may be divided into the transverse vibration and longitudinal vibration, the former is perpendicular to the optical fiber, the latter is along the optical fiber. In the optical fiber drawing tower, the outlet of the furnace (O), the collecting capstan (P), and two cladding coating units between the points O and P are restricting the drawing optical fiber, the vibrations of these restriction points, aroused by the running power unit and the ground vibration, and the airflow of air-conditioning will cause the transverse vibration of optical fiber. In Section 3.1, we have prepared huge amounts of ultra-weak FBGs in the non-moving optical fiber, where without the drawing process, the non-moving optical fiber only undergoes the transverse vibration, the experimental results shows that the transverse vibration of the optical fiber are little responsible for the inconsistency of ultra-weak FBGs.

The larger side lobes in Figs. 5 and 6 are ascribed to the longitudinal vibration in the optical fiber produced by the drawing of optical fiber collecting system. In the drawing process of optical fiber collecting system, two factors may produce the shock waves and hence lead to the longitudinal vibration in the optical fiber. The first is the frequency instability of drive motor, the second factor is the inhomogeneity of the outer radii of the collecting capstans. In Fig. 1, only one capstan is shown for simplicity. In fact, there are 3-5 capstans in the optical fiber collecting system, and any inhomogeneity of the outer radius of each capstan will bring a longitudinal shock wave to the optical fiber.

In principle, the longitudinal vibration of the optical fiber can be calculated by using the longitudinal vibration (33), initial condition (26) and boundary conditions (27) and (28) in the Appendix. In the drawing process of ordinary single-mode optical fiber, T0 is dozens of grams and c is lower than 1 m/s, so we have v12>>V2, by omitting the cross term in Eq. (33), we found that the longitudinal vibration of the optical fiber, as shown by Eq. (34), is coming from the shock wave produced by the inhomogeneity of the collecting capstans.

Therefore, decreasing the inhomogeneity of the collecting capstans to the maximum extent is the most important way to improve the inconsistency of ultra-weak FBGs in an array. In fact, after improving the collecting capstans of optical fiber drawing tower, we obtained better ultra-weak FBGs array whose typical reflection spectra are shown in Fig. 6.

4. Conclusion

For the online writing of ultra-weak FBGs in the drawing optical fiber, the RI fluctuation along the optical fiber and the RI distribution in the FBG are linear with the gradient of longitudinal vibration velocity. The measured reflection spectra shows that the intensity profile and pulse fluctuation of the excimer laser, as well as the transverse vibration of the optical fiber are little responsible for the inconsistency of ultra-weak FBGs. Decreasing the inhomogeneity of the collecting capstans to the maximum extent is the most important way to obtain the ultra-weak FBGs array with better inconsistency.

5. Appendix: the vibrations of the drawing optical fiber

In Fig. 1, an optical fiber is pulled out at a constant speed c through the outlet of the furnace and the capstan. Building the OXYZ coordinates, the plane of XOY is parallel to the grating mask, the OX axis is along the vertical optical fiber, the outlet eyelet O is at x = 0, and the touching point between the optical fiber and the capstan P is at x=L. The traveling optical fiber may be modeled as a one-dimensional continuum with the axial stiffness EA, linear density ρ and steady-state tension T0. The planar vibration is considered, u(x, t) and w(x, t) denote the longitudinal and transverse displacements at point x and time t, respectively. The kinetic and potential energies of the axially moving string are given by [12]

Ek=ρ2∫0L{[w,t+cw,x]2+[u,t+c(1+u,x)]2}dx
Ep=∫0L{T0(u,x+12w,x2)+12EA(u,x+12w,x2)2}dx
where the comma-subscript notation denotes partial differentiation. By Hamilton's principle, one may obtain the Lagrangian equations as follows
−∂∂t∂L∂w,t−∂∂x∂L∂w,x=0
−∂∂t∂L∂u,t−∂∂x∂L∂u,x=0
where the Lagrangian density L = Ek−Ep. Using Eqs. (19) and (20), considering EA>>T0 for the optical fiber in drawing process, one can obtain the governing equations:

ρu,tt+2ρcu,xt+ρc2u,xx=EA(u,xx+w,xw,xx)
        ρw,tt+2ρcw,xt+(ρc2−T0)w,xx=EA{32w,x2w,xx+w,xxu,x+w,xu,xx}

Since the small transverse vibration brings little effect on the intensity distribution of the diffraction laser at the writing positions, we may not consider the transverse vibration and its effect on the longitudinal vibration, then the longitudinal vibration equation is simplified as

ρu,tt+2ρcu,xt+(ρc2−EA)u,xx=0

The initial displacement and velocity of longitudinal vibration in the optical fiber may be considered as 0:

u(x,0)=u,t(x,0)=0,        0<x<L
At the outlet O, the optical fiber is drawing from the molten glass, it can be considered as a viscoelastic body, so the boundary condition of the longitudinal vibration can be written as
u,x(0,t)−hu(0,t)=0,         0<t<∞
where h is elastic coefficient. At the touching point P between the optical fiber and the capstan, due to the periodic perturbation of the rotary motor collecting the optical fiber, the longitudinal motion of the optical fiber is equal to the velocity fluctuation:

u,t(l,t)=g(t),         0<t<∞

Two factors may lead to the velocity instability, they are the inhomogeneity of the outer radius of the capstan and the instability of the frequency of the drive motor. For simplicity, the frequency of the drive motor fs may be considered as a constant. Supposing the nonuniform of the outer circle of the capstan can be expressed by the radius fluctuation δR at the touching point P:

R(θp)=R0+δR
where R0 is the average outer radius of the capstan. Then, as the point P rotates and touches the optical fiber, it gives the contact point of optical fiber a same velocity:
v(θp)=2πfsR(θp)=2πfsR0+2πfsR0⋅δR
and the total velocity fluctuation at x = L can be expressed as

g(t)=2πfsδR⋅∑i=1∞δ(t−i/fs),         0<t<∞

Introducing the following non-dimensional parameters:

ξ=x/L,     τ=t/ρL2T0,    U=u/LV=c/T0ρ,    v1=EAT0
the following nonlinear equations for the longitudinal vibrations of the drawing optical fiber can be obtained

U,ττ+2VU,ξτ−(v12−V2)U,ξξ=0

In principle, using initial condition (26), boundary conditions (27) and (28), one may obtain the solution of Eq. (33). For ordinary single-mode optical fiber,E=7.0×1010   N⋅m−2, bulk density is 2430 kg/m3, the diameter is 125 μm, so ρ=2.982×10−5 kg/m, let T0=100 g, then v12=EA/T=876.56, T0/ρ=181.28. It can be seen that, when T0 is in the range of 10 g ~ 100 g, v12>>1 and V2=c2/(T0/ρ)<<1 for the drawing velocity lower than 1 m/s. Then the solution of Eq. (A15) can be written as

U(ξ,τ)=f1(ξ−v1τ)+f2(ξ+v1τ)
where the functions f1 and f2 are decided by the initial and boundary conditions. Using Eqs. (32) and (34), one can obtain the longitudinal displacement in the optical fiber u(x,t) and the velocity gradient u,xt(x,t).

Supposing the longitudinal displacement of particles in the optical fiber can be expanded as

u(x,t)=∑m=1NmAumsin(2πfumt−2πλumx)
where fum and λum are the vibration frequency and the wavelength of longitudinal mode m ( = 1, 2 … Nm) respectively, and the wave velocity is Vuw=λumfum.The relative longitudinal velocity of particles in the optical fiber is
u,t(x,t)=∑m=1Nm2πfumAumcos(2πfumt−2πλumx)
and the velocity gradient is

u,tx(x,t)=∑m=1Nm4π2fumAumλumsin(2πfumt−2πλumx)

Acknowledgments

This work was supported by the Major Program of the National Natural Science Foundation of China, NSFC (Grant No. 61290311) and the fundamental research funds for the central universities (WHUT 2013-IV-120).

References and links

1. W. Jin, “Multiplexed FBG sensors and their applications,” Proc. SPIE 3897, 468–479 (1999). [CrossRef]  

2. C. S. Kim, T. H. Lee, Y. S. Yu, Y. G. Han, S. B. Lee, and M. Y. Jeong, “Multi-point interrogation of FBG sensors using cascaded flexible wavelength-division Sagnac loop filters,” Opt. Express 14(19), 8546–8551 (2006). [CrossRef]   [PubMed]  

3. Y. B. Dai, Y. J. Liu, J. S. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009). [CrossRef]  

4. Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011). [CrossRef]  

5. Y. Wang, J. Gong, B. Dong, D. Wang, T. J. Shilig, and A. Wang, “A large serial time-division multiplexed fiber Bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012). [CrossRef]  

6. M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012). [CrossRef]  

7. H. Y. Guo, J. G. Tang, X. F. Li, Y. Zheng, and H. F. Yu, “On-line writing weak fiber Bragg gratings array,” Chin. Opt. Lett. 11(3), 030602 (2013). [CrossRef]  

8. H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012). [CrossRef]  

9. V. V. Sechenyh, J. C. Legros, and V. Shevtsova, “Experimental and predicted refractive index properties in ternary mixtures of associated liquids,” J. Chem. Thermodyn. 43(11), 1700–1707 (2011). [CrossRef]  

10. D. W. Van Krevelen and K. Te Nijenhuis, Properties of Polymers (Elsevier B.V., 2009), pp. 287–296.

11. T. Erdogan, “Cladding mode resonances in short and long fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997). [CrossRef]  

12. M. H. Ghayesh, S. Kazemirad, and M. Amabili, “Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance,” Mechanism Mach. Theory 52, 18–34 (2012). [CrossRef]  

References

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  • Publication
  1. W. Jin, “Multiplexed FBG sensors and their applications,” Proc. SPIE 3897, 468–479 (1999).
    [Crossref]
  2. C. S. Kim, T. H. Lee, Y. S. Yu, Y. G. Han, S. B. Lee, and M. Y. Jeong, “Multi-point interrogation of FBG sensors using cascaded flexible wavelength-division Sagnac loop filters,” Opt. Express 14(19), 8546–8551 (2006).
    [Crossref] [PubMed]
  3. Y. B. Dai, Y. J. Liu, J. S. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).
    [Crossref]
  4. Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011).
    [Crossref]
  5. Y. Wang, J. Gong, B. Dong, D. Wang, T. J. Shilig, and A. Wang, “A large serial time-division multiplexed fiber Bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
    [Crossref]
  6. M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012).
    [Crossref]
  7. H. Y. Guo, J. G. Tang, X. F. Li, Y. Zheng, and H. F. Yu, “On-line writing weak fiber Bragg gratings array,” Chin. Opt. Lett. 11(3), 030602 (2013).
    [Crossref]
  8. H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012).
    [Crossref]
  9. V. V. Sechenyh, J. C. Legros, and V. Shevtsova, “Experimental and predicted refractive index properties in ternary mixtures of associated liquids,” J. Chem. Thermodyn. 43(11), 1700–1707 (2011).
    [Crossref]
  10. D. W. Van Krevelen and K. Te Nijenhuis, Properties of Polymers (Elsevier B.V., 2009), pp. 287–296.
  11. T. Erdogan, “Cladding mode resonances in short and long fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997).
    [Crossref]
  12. M. H. Ghayesh, S. Kazemirad, and M. Amabili, “Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance,” Mechanism Mach. Theory 52, 18–34 (2012).
    [Crossref]

2013 (1)

2012 (4)

H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012).
[Crossref]

Y. Wang, J. Gong, B. Dong, D. Wang, T. J. Shilig, and A. Wang, “A large serial time-division multiplexed fiber Bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012).
[Crossref]

M. H. Ghayesh, S. Kazemirad, and M. Amabili, “Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance,” Mechanism Mach. Theory 52, 18–34 (2012).
[Crossref]

2011 (2)

V. V. Sechenyh, J. C. Legros, and V. Shevtsova, “Experimental and predicted refractive index properties in ternary mixtures of associated liquids,” J. Chem. Thermodyn. 43(11), 1700–1707 (2011).
[Crossref]

Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011).
[Crossref]

2009 (1)

Y. B. Dai, Y. J. Liu, J. S. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).
[Crossref]

2006 (1)

1999 (1)

W. Jin, “Multiplexed FBG sensors and their applications,” Proc. SPIE 3897, 468–479 (1999).
[Crossref]

1997 (1)

Amabili, M.

M. H. Ghayesh, S. Kazemirad, and M. Amabili, “Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance,” Mechanism Mach. Theory 52, 18–34 (2012).
[Crossref]

Asundi, A.

Y. B. Dai, Y. J. Liu, J. S. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).
[Crossref]

Bi, W.

Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011).
[Crossref]

Dai, Y. B.

Y. B. Dai, Y. J. Liu, J. S. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).
[Crossref]

Deng, G.

Y. B. Dai, Y. J. Liu, J. S. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).
[Crossref]

Dong, B.

Y. Wang, J. Gong, B. Dong, D. Wang, T. J. Shilig, and A. Wang, “A large serial time-division multiplexed fiber Bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011).
[Crossref]

Erdogan, T.

Ghayesh, M. H.

M. H. Ghayesh, S. Kazemirad, and M. Amabili, “Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance,” Mechanism Mach. Theory 52, 18–34 (2012).
[Crossref]

Gong, J.

Y. Wang, J. Gong, B. Dong, D. Wang, T. J. Shilig, and A. Wang, “A large serial time-division multiplexed fiber Bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011).
[Crossref]

Guo, H.

H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012).
[Crossref]

Guo, H. Y.

Han, Y. G.

Jeong, M. Y.

Jiang, D.

H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012).
[Crossref]

Jin, W.

W. Jin, “Multiplexed FBG sensors and their applications,” Proc. SPIE 3897, 468–479 (1999).
[Crossref]

Kazemirad, S.

M. H. Ghayesh, S. Kazemirad, and M. Amabili, “Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance,” Mechanism Mach. Theory 52, 18–34 (2012).
[Crossref]

Kim, C. S.

Lee, S. B.

Lee, T. H.

Legros, J. C.

V. V. Sechenyh, J. C. Legros, and V. Shevtsova, “Experimental and predicted refractive index properties in ternary mixtures of associated liquids,” J. Chem. Thermodyn. 43(11), 1700–1707 (2011).
[Crossref]

Leng, J. S.

Y. B. Dai, Y. J. Liu, J. S. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).
[Crossref]

Li, X.

M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012).
[Crossref]

H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012).
[Crossref]

Li, X. F.

Liu, D.

M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012).
[Crossref]

Liu, H.

M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012).
[Crossref]

Liu, Y. J.

Y. B. Dai, Y. J. Liu, J. S. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).
[Crossref]

Sechenyh, V. V.

V. V. Sechenyh, J. C. Legros, and V. Shevtsova, “Experimental and predicted refractive index properties in ternary mixtures of associated liquids,” J. Chem. Thermodyn. 43(11), 1700–1707 (2011).
[Crossref]

Shevtsova, V.

V. V. Sechenyh, J. C. Legros, and V. Shevtsova, “Experimental and predicted refractive index properties in ternary mixtures of associated liquids,” J. Chem. Thermodyn. 43(11), 1700–1707 (2011).
[Crossref]

Shilig, T. J.

Sun, Q. Z.

M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012).
[Crossref]

Tang, J.

H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012).
[Crossref]

Tang, J. G.

Wang, A.

Y. Wang, J. Gong, B. Dong, D. Wang, T. J. Shilig, and A. Wang, “A large serial time-division multiplexed fiber Bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011).
[Crossref]

Wang, D.

Y. Wang, J. Gong, B. Dong, D. Wang, T. J. Shilig, and A. Wang, “A large serial time-division multiplexed fiber Bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011).
[Crossref]

Wang, Y.

Y. Wang, J. Gong, B. Dong, D. Wang, T. J. Shilig, and A. Wang, “A large serial time-division multiplexed fiber Bragg grating sensor network,” J. Lightwave Technol. 30(17), 2751–2756 (2012).
[Crossref]

Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011).
[Crossref]

Wang, Z.

M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012).
[Crossref]

Yu, H.

H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012).
[Crossref]

Yu, H. F.

Yu, Y. S.

Zhang, M. L.

M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012).
[Crossref]

Zheng, Y.

H. Y. Guo, J. G. Tang, X. F. Li, Y. Zheng, and H. F. Yu, “On-line writing weak fiber Bragg gratings array,” Chin. Opt. Lett. 11(3), 030602 (2013).
[Crossref]

H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012).
[Crossref]

Chin. Opt. Lett. (1)

IEEE Photonic. Tech. L. (1)

Y. Wang, J. Gong, D. Wang, B. Dong, W. Bi, and A. Wang, “A quasi-distributed sensing network with time-division-multiplexed fiber Bragg gratings,” IEEE Photonic. Tech. L. 23(2), 70–72 (2011).
[Crossref]

J. Chem. Thermodyn. (1)

V. V. Sechenyh, J. C. Legros, and V. Shevtsova, “Experimental and predicted refractive index properties in ternary mixtures of associated liquids,” J. Chem. Thermodyn. 43(11), 1700–1707 (2011).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (1)

J. Wuhan Univ. Technol. (1)

H. Guo, Y. Zheng, J. Tang, X. Li, H. Yu, and D. Jiang, “Reflectivity measurement of weak fiber Bragg grating,” J. Wuhan Univ. Technol. 27(6), 1–3 (2012).
[Crossref]

Mechanism Mach. Theory (1)

M. H. Ghayesh, S. Kazemirad, and M. Amabili, “Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance,” Mechanism Mach. Theory 52, 18–34 (2012).
[Crossref]

Opt. Commun. (1)

M. L. Zhang, Q. Z. Sun, Z. Wang, X. Li, H. Liu, and D. Liu, “A large capacity sensing network with identical weak fiber Bragg gratings multiplexing,” Opt. Commun. 285(13-14), 3082–3087 (2012).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (1)

Y. B. Dai, Y. J. Liu, J. S. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).
[Crossref]

Proc. SPIE (1)

W. Jin, “Multiplexed FBG sensors and their applications,” Proc. SPIE 3897, 468–479 (1999).
[Crossref]

Other (1)

D. W. Van Krevelen and K. Te Nijenhuis, Properties of Polymers (Elsevier B.V., 2009), pp. 287–296.

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Figures (6)

Fig. 1
Fig. 1 Optical fiber drawing tower and schematic of online writing ultra-weak FBGs.

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Fig. 2
Fig. 2 2-D beam profile of 248 nm excimer laser at the distance of 40 cm from the output mirror.

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Fig. 3
Fig. 3 Reflection spectra of 248 nm laser.

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Fig. 4
Fig. 4 Irradiating time of a pulse onto the optical fiber.

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Fig. 5
Fig. 5 Reflection spectra of three FBGs written at different speeds.

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Fig. 6
Fig. 6 Reflection spectra of three FBGs written at the same speed (20 m/min).

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Equations (37)

Equations on this page are rendered with MathJax. Learn more.

n tot = ∑ c=1 N cn n c ϕ c
δ n tot = ∑ c=1 N cn ϕ c δ n c
n 2 −1 n 2 +2 M ρ v = R LL
ρ v ∂n ∂ ρ v = ( n 2 −1)( n 2 +2) 6n
ρ v (x,t)= ρ v 0 +Δ ρ v (x,t)
δn(x,t)= ∂n ∂ ρ v ⋅δ ρ v (x,t)
∂ ρ v ∂t =− ∂ ∂x ( ρ v u ,t )
δ ρ v (x,t)≈− ρ v 0 u ,xt δt
δ n c (x,t)=−δt⋅ u ,xt ( n c 2 −1)( n c 2 +2) 6 n c
δ n tot =−δt⋅ u ,xt ∑ c=1 N c ϕ c ( n c 2 −1)( n c 2 +2) 6 n c
n(x,t)= n 0 +δ n tot (x,t)
Δn(x)=Ω(x)Π(x)⋅Δ n 0 [1+cos( 2π Λ(x) x+ φ 0 )],           x 0 <x< x 0 + L B
Ω(x,t)=1−δt⋅ u ,xt ϕ CC ( n CC 2 −1)( n CC 2 +2) 6 n 0 n CC
Ω(x,t)=1+ ∑ m=1 N m G m sin(2π f um t− 2π λ um x)
G m =δt⋅ ϕ CC ( n CC 2 −1)( n CC 2 +2) 6 n 0 n CC ⋅ B m
Δn(x)=Δ n 0 [1+ ∑ m=1 N m G m sin(2π f um t− 2π λ um x)]   [1+ηcos( 2π Λ x+ φ 0 )]
≈Δ n 0 { 1+ηcos( 2π Λ x+ φ 0 )+ ∑ m=1 N m 1 2 η G m [ cos( 2π Λ m − x+2π f um t+ φ 0 ' )+cos( 2π Λ m + x−2π f um t+ φ 0 '' ) ] }
G(x)= t(x) δt ={ x/ l f  ,                       0≤x≤ l f     1  ,                                      l f ≤x≤ l 0 1−(x−  l 0 )/ l f  ,             l 0 ≤x≤ l 0 + l f
E k = ρ 2 ∫ 0 L { [ w ,t +c w ,x ] 2 + [ u ,t +c(1+ u ,x )] 2 } dx
E p = ∫ 0 L { T 0 ( u ,x + 1 2 w ,x 2 )+ 1 2 EA ( u ,x + 1 2 w ,x 2 ) 2 } dx
− ∂ ∂t ∂L ∂ w ,t − ∂ ∂x ∂L ∂ w ,x =0
− ∂ ∂t ∂L ∂ u ,t − ∂ ∂x ∂L ∂ u ,x =0
ρ u ,tt +2ρc u ,xt +ρ c 2 u ,xx =EA( u ,xx + w ,x w ,xx )
        ρ w ,tt +2ρc w ,xt +(ρ c 2 − T 0 ) w ,xx =EA{ 3 2 w ,x 2 w ,xx + w ,xx u ,x + w ,x u ,xx }
ρ u ,tt +2ρc u ,xt +(ρ c 2 −EA) u ,xx =0
u(x,0)= u ,t (x,0)=0,        0<x<L
u ,x (0,t)−hu(0,t)=0,         0<t<∞
u ,t (l,t)=g(t),         0<t<∞
R( θ p )= R 0 +δR
v( θ p )=2π f s R( θ p )=2π f s R 0 +2π f s R 0 ⋅δR
g(t)=2π f s δR⋅ ∑ i=1 ∞ δ(t−i/ f s ) ,         0<t<∞
ξ=x/L,     τ=t/ ρ L 2 T 0 ,    U=u/L V=c/ T 0 ρ ,     v 1 = EA T 0
U ,ττ +2V U ,ξτ −( v 1 2 − V 2 ) U ,ξξ =0
U(ξ,τ)= f 1 (ξ− v 1 τ)+ f 2 (ξ+ v 1 τ)
u(x,t)= ∑ m=1 N m A um sin(2π f um t− 2π λ um x)
u ,t (x,t)= ∑ m=1 N m 2π f um A um cos(2π f um t− 2π λ um x)
u ,tx (x,t)= ∑ m=1 N m 4 π 2 f um A um λ um sin(2π f um t− 2π λ um x)

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