Addressed fiber Bragg structure

The concept of addressed fiber Bragg structures was introduced in 2018 by Airat Sakhabutdinov[1] and developed in collaboration with his scientific adviser, Oleg Morozov. The idea emerged from the earlier works of Morozov and his colleagues,[2][3] where the double-frequency optical radiation from an electro-optic modulator was used for the definition of the FBG central wavelength based on the amplitude and phase analysis of the beating signal at the frequency equal to the spacing between the two components of the probing radiation. This eliminates the need for scanning the FBG spectral response while providing high accuracy of measurements and reducing the system cost.[1][2] AFBS has been developed as a further step towards simplification of FBG interrogation systems by transferring the shaping of double-frequency probing radiation from the source modulator to the sensor itself.[1]

Thus far, two types of AFBS with different mechanisms of forming double-frequency radiation have been presented: 2π-FBG and 2λ-FBG.

Figure 2: An interrogation scheme for two addressed fiber Bragg structures (2π-FBG-type): 1 - a wideband optical source; 2.1 and 2.2 - addressed fiber Bragg structures; 3 - an optical filter with a pre-defined linear inclined frequency response; 4, 7 - photodetectors; 5, 8 - analog-to digital converters; 6, 9 - fiber-optic splitters; 10 - fiber-optic coupler; a - e - optical spectra.

Fig. 2 presents the block diagram of the interrogation system for two AFBSs (2π-FBG-type) with different address frequencies Ω₁ and Ω₂. A broadband light source 1 generates continuous light radiation (diagram a), which corresponds to the measurement bandwidth. The light is transmitted through the fiber-optic coupler 9, then enters the two AFBSs 2.1 and 2.2. Both AFBSs transmit two-frequency radiations that are summed into a combined radiation (diagram b) using another coupler 10. At the output of the coupler, a four-frequency radiation (diagram c) is formed, which is sent through a fiber-optic splitter 6. The splitter divides the optical signal into two channels – the measuring channel and the reference channel. In the measuring channel, an optical filter 3 with a pre-defined linear inclined frequency response is installed, which modifies the amplitudes of the four-frequency radiation into the asymmetrical radiation (diagram d). After that, the signal is sent to the photodetector 4 and is received by the measuring analog-to-digital converter (ADC) 5. The signal from the ADC is used to define the measurement information from the AFBS. In the reference channel, the signal (diagram e) is sent to the reference photodetector 7 for the optical power output control, and then it is received by the reference ADC 8. Thus, the normalization of output signal intensity is achieved, and all subsequent calculations are performed using the relations of the intensities in the measuring and reference channels.[5][6]

Assume that the response from each spectral components of AFBSs is represented by a single harmonic, then the total optical response from the two AFBSs can be expressed as:[1][4]

${\displaystyle F(t)=\left[{\sum _{i=1}^{N}A_{i}\sin(\omega _{i}t)+B_{i}\sin((\omega _{i}+\Omega _{i})t)}\right]^{2},}$

where A_i, B_i are the amplitudes of the frequency components of the i-th AFBS; ω_i, is the frequency of the left spectral components of the i-th AFBS; Ω_i is the address frequency of the i-th AFBS.

The luminous power received by the photodetector can be described by the following expression:

${\displaystyle P(t)=\sum _{i=1}^{N}\sum _{k=1}^{N}\left({A_{i}A_{k}\cos((\omega _{i}-\omega _{k})t)+A_{i}B_{k}\cos((\omega _{i}-\omega _{k}-\Omega _{k})t)+ \atop B_{i}A_{k}\cos((\omega _{i}-\omega _{k}+\Omega _{i})t)+B_{i}B_{k}\cos((\omega _{i}-\omega _{k}+\Omega _{i}-\Omega _{k})t)}\right).}$

By narrowband filtering of the signal P(t) at the address frequencies, a system of equations can be obtained, using which the central frequencies of the AFBSs can be defined:

${\displaystyle \sum _{i=1}^{N}\sum _{k=1}^{N}\left({A_{i}A_{k}\exp \left(-{\frac {(\Omega _{j}-|\omega _{i}-\omega _{k}|)^{2}}{2\sigma ^{2}}}\right)+A_{i}B_{k}\exp \left(-{\frac {(\Omega _{j}-|\omega _{i}-\omega _{k}-\Omega _{k}|)^{2}}{2\sigma ^{2}}}\right)+ \atop B_{i}A_{k}\exp \left(-{\frac {(\Omega _{j}-|\omega _{i}-\omega _{k}+\Omega _{i}|)^{2}}{2\sigma ^{2}}}\right)+B_{i}B_{k}\exp \left(-{\frac {(\Omega _{j}-|\omega _{i}-\omega _{k}+\Omega _{i}-\Omega _{k}|)^{2}}{2\sigma ^{2}}}\right)}\right)=D_{j},j={\overline {1,N}},}$

where D_j is the amplitude of the signal at the address frequencies Ω_j, and the exponential multipliers describe the bandpass filters at the address frequencies.