Introduction
Chapter 6 Fiber Bragg Grating Based Humidity Sensor Employing
6.2 Theoretical Analysis
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wavelength shift scheme for the first time to the best of author’s knowledge, rather than the traditional strain induced Bragg wavelength shift scheme reported in the literature. Sensor development is focused to achieve a throughout linear response over a wide dynamic RH range and an optimum sensitivity. To design the sensor, FBG structure is suitably modified by reducing fiber diameter in the grating region to a lowest possible working limit (~ a few microns). Response characteristic of the sensor against ambient environmental parameter, resulting from RH variations, is theoretically investigated. Afterwards, sensor is experimentally investigated by exposing it to the repeat cycles of increasing and decreasing RH variations over a sufficiently long time period. A linear response over a dynamic range as wide as ~3 – 94% RH with a good sensitivity of ~ 0.08 pm/%RH during increasing as well as decreasing humidity is observed. Proposed sensor is more than ten times faster than the commercial sensor. Maximum fiber output variation is observed to be of the order of only 10-
4 during the repeatability and reversibility tests. Importantly, experimental response characteristic of the sensor is observed to be in accordance to the theoretical analysis. In addition, a relatively better resolution (~ 0.6%RH), accuracy (~ ±2.5%RH), average discrepancy (~ ±0.2 pm) and response time (~ 0.2s) is observed during humidification and dessication cycles.
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Effective-refractive-index of the fundamental mode in a standard uniform SMF is also independent of the RI of the medium surrounding the fiber cladding. However, if the refractive-index is varied in the medium surrounding the fiber cladding at a constant temperature and without straining the fiber, as is the case of varying RH from lowest to the highest possible values in the present case, and if the fiber diameter in the cladding region is sufficiently reduced, neff, and hence, B will get modulated accordingly. In this way, the resulting Bragg wavelength modulation would be a direct consequence of effective-index- variation as dictated by the RH variation. This is the principle exploited for developing RH sensor in the present work. In order to theoretically study the response characteristics of FBG that exits in a fiber of reduced cladding diameter and is exposed to the refractive-index variations corresponding to 0% RH (dry air with refractive index 1) to an RH value close to 99%, doubly-clad fiber model reported in [137] is employed. This model is best suited for the present case to find a distinct relationship between the modal propagation constant and nout. Once is known for a given nout, B can be easily calculated through B = 2neff = 2(/k0), where k0 is free space wave number. Simulated fiber structure under doubly-clad fiber model for thinned FBG is depicted in the inset of Fig. 6.1. Here a and b are the core and inner- cladding radii respectively, whereas nco, ncl, and nout are the refractive indices of core, inner- cladding and the surrounding medium (outer cladding) respectively. For the azimuthal order m, the radial dependence 𝜓(𝑟) of the axial field components are expressed as [137]
𝜓 = 𝐴′0𝐽𝑚(𝑢𝑎𝑟) For 𝑟 ≤ 𝑎 (6.1) 𝜓 = 𝐴′1𝐼𝑚(𝑣′ 𝑟𝑏) + 𝐴′2𝐾𝑚(𝑣′ 𝑟𝑏) For 𝑎 ≤ 𝑟 ≤ 𝑏 if 𝛽 > 𝑘0𝑛2 (6.2) 𝜓 = 𝐴′3𝐾𝑚(𝑣𝑟𝑏) For 𝑟 ≥ 𝑏 (6.3) where Jm are ordinary Bessel functions, Km and Im are modified Bessel functions; β is the propagation constant, which is greater than k0ncl for guided modes; u = a(k02nco2 2)1/2, v =
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b(2 k02nout2)1/2, and v = b(2 k02ncl2)1/2. A’0 is a normalization coefficient which is determined by applying the boundary conditions. Under the weekly-guiding approximation, continuity of the transverse field components across the two interfaces (between core and inner cladding, and between inner and outer cladding) leads to a set of four equations:
𝐴0′𝐽𝑚(𝑢) − 𝐴1′𝐼𝑚(𝑣′𝑐) − 𝐴2′𝐾𝑚(𝑣′𝑐) = 0 (6.4) 𝑢𝐴0′𝐽′𝑚(𝑢) − 𝑣′𝑐𝐴1′𝐼′𝑚(𝑣′𝑐) − 𝑣′𝑐𝐴2′𝐾′𝑚(𝑣′𝑐) = 0 (6.5) 𝐴1′𝐼𝑚(𝑣′) + 𝐴2′𝐾𝑚(𝑣′) − 𝐴′3𝐾𝑚(𝑣) = 0 (6.6) 𝑣′𝐴1′𝐼′𝑚(𝑣′) + 𝑣′𝐴2′𝐾′𝑚(𝑣′) − 𝑣𝐴3′𝐾′𝑚(𝑣) = 0 (6.7) where Z’= dZ/dr (Z is Bessel function: 𝐽𝑚, 𝐼𝑚 or 𝐾𝑚) and c = a/b. Equating the determinant of 44 coefficient matrix of these equations to zero (to ensure a nontrivial solution) leads to the dispersion equation for the guided mode (β > k0ncl) of the doubly-clad fiber as following [𝐽̂𝑚(𝑢)−𝐾̂𝑚(𝑣
′𝑐)][𝐾̂𝑚(𝑣)+𝐼̂𝑚(𝑣′)]
[𝐽̂𝑚(𝑢)+𝐼̂𝑚(𝑣′𝑐)][𝐾̂𝑚(𝑣)−𝐾̂𝑚(𝑣′)]
=
[𝐼̂[𝐼̂𝑚+1(𝑣′𝑐)𝐾̂𝑚+1(𝑣′)]𝑚+1(𝑣′)𝐾̂𝑚+1(𝑣′𝑐)] (6.8) where
𝑍̂
𝑚(𝑥) =
𝑥𝑍𝑍𝑚(𝑥)𝑚+1(𝑥) and x is u/vc/v. Replacing the value of 𝑍̂𝑚(𝑥) and substituting m
= 0 for the fundamental mode (LP01) in equation (6.8), dispersion equation simplifies to [𝑣
′𝑐𝐾1(𝑣′𝑐)𝐽0(𝑢)−𝑢𝐽1(𝑢)𝐾0(𝑣′𝑐)]
[𝑣′𝑐𝐼1(𝑣′𝑐)𝐽0(𝑢)+𝑢𝐽1(𝑢)𝐼0(𝑣′𝑐)]
=
[𝑣[𝑣′′𝐾𝐼1(𝑣′)𝐾0(𝑣)−𝑣𝐾1(𝑣)𝐾0(𝑣′)]1(𝑣′)𝐾0(𝑣)+𝑣𝐾1(𝑣)𝐼0(𝑣′)]
(6.9) This transcendental (dispersion) equation ((Eq. (6.9)) is solved numerically in MATLAB software where the fiber parameters of SMF-28 (a = 4.15μm, nco = 1.460, nclad = 1.4564) are used owing to fact that the FBG used in the experiment is written in this standard fiber. While solving Eq. (6.9), nout is varied from 1 (corresponding to 0%RH) to 1.456 (clad refractive- index) at a fixed cladding diameter. Solutions obtained from the above analysis are used to simulate the response of FBG towards the surrounding RI variations. Figure 6.1 shows the simulated response of FBG (B vs nout) for the optical fiber having cladding radii (b) equal to 62.5 (corresponding to the unetched fiber), 12, 10, 8, 6.75 and 6μm in the FBG region.
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Figure 6.1: Bragg wavelength of a FBG existing in a reduced cladding diameter fiber versus outer medium refractive-index for different cladding radii. Upper inset shows the refractive index variation along a cross section of a double cladding and weekly guiding fiber. Lower inset shows effective-refractive-index of the thinned FBG versus outer medium refractive-index for cladding radius of 6m.
As can be observed from Fig. 6.1, FBG in a normal (unetched) fiber is insensitive to the variations in the surrounding RI (and hence to the surrounding RH). Decreasing the cladding diameter in the FBG region increases its sensitivity towards the surrounding refractive- indices. In the present calculation, highest overall sensitivity towards the surrounding RI is witnessed for the fiber having a cladding radius of 6m in the FBG region. Further, this sensitivity is observed to be very low for the surrounding RI variations in between 1 to 1.07.
Sensitivity increases very slowly and linearly afterwards till surrounding RI approaches to 1.4; and afterwards, sharply and very significantly as surrounding RI approaches to nclad. Basically, the sensor response is dictated by neff seen by the fundamental mode against the
1535.0 1535.2 1535.4 1535.6 1535.8 1536.0 1536.2 1536.4
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70
Bragg Wavelengthshift (nm)
Outer Medium Refractive Index (nout) 62.5µm
12µm 10µm 8µm 6.75µm 6µm
nco
ncl
nout
2a 2b
1.4561 1.4566 1.4571 1.4576
1.00 1.20 1.40
neff
nout 6 µm
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surrounding RI variations. Hence, neff versus nout is also plotted, as an example, for b = 6μm in the inset of Fig. 6.1. If the fundamental mode is confined within the fiber with its evanescent tail dying-off sufficiently well below the clad-outer medium interface, the effective-index (neff) seen by the mode will have no influence of the surrounding medium refractive-indices. This is exactly the case of unetched fiber in weekly-guiding approximation. However, when the fiber is sufficiently etched in its cladding region down towards the core, average RI in the cladding region gets significantly affected. Lower the average RI in the cladding region (as is the case with the lowest value of nout = 1), the tighter is the confinement of the fundamental mode to the fiber core; which in turn, leads to a week influence of the surrounding RI on the effective-index (neff) seen by the mode. As nout
increases, average RI also increases. This leads to a lesser confinement of the fundamental mode to the fiber core and a greater interaction with the outer medium in the thinned cladding region, and that way, a greater influence of nout on neff seen by the mode. Any influence of nout
on neff gets accordingly imprinted on the reflected Bragg wavelength, B. The observed theoretical response of the FBG that exists in an etched fiber region is used to analyze the experimental findings in the subsequent section.