Fiber Bragg Grating Sensors

As we shall see in more detail later, the interest in using FBGs as sensors arises, because both n and Λ are aected if the ber is strained or its temperature changed,* and hence, these parameters can be monitored using a measurement of the precise wavelength reected by the grating, but a more rigorous analysis is needed to extract further information. A common approach is to use coupled wave theory to obtain an analytic solution (Kogelnik 1976). Here we shall simply quote the major results. e reectivity of a grating of length l with an index modulation of constant amplitude is given by R l k l l = + κ ψ ψ ψ ψ sinh sinh cosh , ∆ (17.2) where Δk = k-π/λ is called the detuning wave vector and provides a measure of how much the Bragg condition is violated, ψ = (κ2−Δκ2)1/2, with κ being the coupling constant given by κ pi η λ = ∆n , (17.3) where Δn is the amplitude of the index modulation. e quantity η is a measure of the grating eciency and represents how much of the guided power is actually in the core overlapping the grating. In terms of the ber parameters, η is given by Russell et al. (1993) η λ pi ≈ −     −( )1 2 2 2a n n , (17.4) where λ0 is the free space wavelength a the ber core radius n1 the core index n2 the cladding index If we turn our attention to Equation 17.2, if the incident light is Bragg matched so that Δk = 0, the reectivity simplies to R = tanh ( l), 2 κ (17.5) so the reectivity increases asymptotically toward unity as either the length or the strength (Δn) of the grating is increased. It turns out that strong gratings may have a high reectivity over a wide range of wavelengths, as shown in Figure 17.3. For sensing applications, weaker gratings are usually used (reectivity 95% or less) to avoid the broadening of the reection peak displayed by stronger gratings, which could reduce the accuracy of the Bragg wavelength determination.