Sensors play an important role in shaping and monitoring human health. Exploration of methods to use Fiber Bragg Grating (FBG) with enhanced sensitivity has attracted great interest in the field of medical research. In this paper, a novel apodization function is proposed and performance evaluation and optimization of the same have been made. A comparison was conducted between various existing apodization functions and the proposed one based on optical characteristics and sensor parameters. The results evince the implementation of the proposed apodization function for vital sign measurement. The optical characteristics considered for evaluation are Peak Resonance Reflectivity level, Side Lobes Reflectivity level and Full Width Half Maximum (FWHM). The proposed novel apodization novel function has better FWHM, which is narrower than the FWHM of uniform FBG. Sensor characteristics like a quality parameter, detection accuracy and sensitivity also show improvement. The proposed novel apodization function is demonstrated to have a better shift in wavelength in terms of temperature and pulse measurement than the existing functions. The sensitivity of the proposed apodized function is enhanced with a Poly-dimethylsiloxane coating of varying thickness, which is 6 times and 5.14 times greater than uniform Fiber Bragg grating and FBG with the proposed novel apodization function, respectively, enhancing its utilization in the field of medicine.

Fiber Bragg Gratings (FBGs) is a fiber with a core refractive index induced with periodic variation in the direction of the fiber axis for a length called grating length. A specific wavelength is reflected back due to this periodic grating and it is known as Bragg wavelength (λ_{B}). The periodicity (Λ), modulation of the index (

Reference | Existing apodization | Proposed apodization | Application |
---|---|---|---|

[ |
Uniform, Blackman, Nutall, Gaussian, Hamming, Sinc | Dispersion compensation | |

[ |
Uniform, Gaussian, Blackman, Sine, Tanh Types I & II, Cauchy, Barthan Welch, Cones | Sensing application | |

[ |
Sine, Hamming, Tanh, Bartlett, Blackman, Sinc, Gaussian, Cauchy, Raised sine | - | Sensing application |

[ |
Uniform, Raised sine, Sinc, Gaussian, Nutall | - | Temperature measurement in DWDM systems. |

[ |
Gaussian, Tanh4z | Structural health Monitoring | |

[ |
Sinc, Gaussian, Raised cosine | - | - |

Proposed | Uniform, Gaussian, Barthan, Sine, Welch, Cones, Hamming, Blackman, Nutall, Bessel | Medical sensing applications |

The organization of the present work is as follows:

The refractive index variation of the Fiber Bragg Grating with periodic variation along its axis is given as
_{core} is the refractive index of the core region, z is the direction of propagation, Λ is the grating periodicity, Δn is the refractive index modulation. Bragg wavelength (λ_{B}), is given as
_{eff} is the core effective refractive index. Using couple mode theory, the reflectivity of uniform grating FBG is given as

where ^{2} = ^{2}. _{core}/λ. At centre wavelength, maximum reflectivity happens as Δβ = 0.

The strain-based shift in Bragg wavelength is given as
_{e} is the effective photoelastic coefficient, n_{eff} is the effective refractive index and v_{f} is the Poisson ratio. p_{11} and p_{12} are Pockel’s piezo coefficients for strain optic tensor. For silica, p_{11} = 0.121, p_{12 }= 0.27 and v_{f} = 0.17.

The wavelength shift due to the change in temperature is given as
_{f} is thermal expansion coefficient and α_{n} is thermo optic coefficient and its corresponding values are 5.5 × 10^{–7} ^{–6}

The propagation of electromagnetic waves inside the fiber core can be represented using coupled mode theory (CMT). The electric field distribution in forward and backward propagating waves can be defined by equations as shown below, respectively.

For 0 ≤ z ≤ L. By applying boundary conditions B(0) = Bo and A(L) = AL and solving for closed form solutions,

Scattering matrix is used to express the reflected and transmitted wave as

Substituting values of

To obtain the general solution of coupled mode equation, an effective method known as the Transfer matrix method (T-matrix) has been used, in which a single grating is subdivided into a series of separate gratings (N) of uniform type and described aa s square 2 * 2 matrix. This provides greater flexibility and the precision is based on the number of sections, N. The T-matrix is described as

The reflectivity of FBG is expressed as

The apodization functions considered for comparison of performance evaluation are listed in this section. Apodization functions are chosen from the literature [

Uniform

Gaussian

Barthan

Sine

Welch

Cones

Hamming

Blackman

Nutall

Bessel [

J_{0}-Bessel function of the first kind of zero order and L is the grating length.

In this work, we propose a new apodization function for better sensor characteristics and reflectivity level. The above mentioned characteristics can be attained by using a compact transform which results in narrow bandwidth with high resolution. The strategy to make a transform compact is to increase the power of the exponent or multiply with the increasing functions. The former one is implemented to optimize the apodization function, maximizing its usage as a sensor. The proposed apodization function is given in

The schematic representation of the proposed novel apodized FBG based sensor system is shown in

The coating on uniform FBG enhances its sensitivity. The schematic diagram of UFBG with coating is shown in

Polymer | Refractive index | Thermo-optic coefficient (/°C) | Thermal expansion coefficient (/°C) | Poisson’s ratio | Reference |
---|---|---|---|---|---|

PDMS | 1.399 | −4.5 × 10^{–4} |
3 × 10^{–4} |
0.4950 | [ |

PC | 1.585 | −0.9 × 10^{–4} |
1.7 × 10^{–4} |
0.3182 | [ |

PMMA | 1.48 | −1.3 × 10^{–4} |
2.2 × 10^{–4} |
0.34 | [ |

Using Opti-Grating software, simulations to investigate the optical properties of various existing apodization functions with the proposed apodization function are evaluated by considering a single mode fiber (silica) with 1.46 as refractive index of core and 1.45 as refractive index of cladding. The center wavelength of 1550 nm is considered. The length of the grating has been varied from 1 mm to 15 mm and index depth is varied from 1 × 10^{–4} to 2.5 × 10^{–4} for performance evaluation.

An increase in the reflectivity level of the main lobe, for varying grating length with Δn = 0.0001 and index modulation with grating length 10 mm, is shown in

In cones apodization function, left and right-side lobe levels increase by 22.84% and 22.43%, respectively, for increasing index change and by 63.02% and 71.04%, respectively, for increasing grating length. Likewise, for Gaussian function, left and right-side lobe level increases by 29.67% and 22.04%, respectively, for increasing index change and by 59.19% and 66.55% respectively for increasing grating length. The proposed apodization function shows 22.39% and 22.92% increase in the left and right-side lobe reflectivity levels, respectively, for increasing index modulation and 68.15% and 74.56% increase in left and right-side lobe reflectivity levels respectively for increasing grating length.

Apodization types | Reflectivity |
MSL |
SLSR |
FWHM |
Slope |
---|---|---|---|---|---|

Uniform | −0.5648 | −14.0133 | −17.7356 | 0.12 | −125.8200 |

Gaussian | −2.9001 | −35.4861 | −32.5860 | 0.06 | −67.4465 |

Barthan | −0.5952 | −14.3129 | −13.7177 | 0.12 | −115.8685 |

Sine | −1.9503 | −29.6796 | −27.7293 | 0.12 | 373.9926 |

Welch | −1.7587 | −28.7008 | −26.9421 | 0.12 | 225.6871 |

Cones | −2.7796 | −34.9280 | −32.1484 | 0.12 | 140.8413 |

Hamming | −2.7266 | −35.2340 | −32.5074 | 0.06 | 20.3894 |

Blackman | −4.1295 | −34.7823 | −30.6528 | 0.06 | −43.2555 |

Nutall | −4.0572 | −16.8327 | −12.7755 | 0.12 | 86.1890 |

Bessel | −2.3881 | −34.9419 | −32.5538 | 0.12 | 186.9966 |

Type of apodization | FWHM (nm) | Sensitivity |
Detection accuracy | Quality parameter |
Wavelength shift(Δλ_{B})/ με |
Wavelength shift (Δλ_{B})/°C |
---|---|---|---|---|---|---|

Uniform | 0.12 | 1.6783 | 13057 | 13.9858 | 1.2pm | 12pm |

Gaussian | 0.06 | 4.8888 | 26114 | 81.4667 | 1.2pm | 12pm |

Barthan | 0.12 | 1.7450 | 13057 | 14.5418 | 1.2pm | 12pm |

Sine | 0.12 | 3.9112 | 13057 | 32.5933 | 1.2pm | 12pm |

Welch | 0.12 | 3.6736 | 13057 | 30.6133 | 1.2pm | 12pm |

Cones | 0.12 | 4.7757 | 13057 | 39.7975 | 1.2pm | 12pm |

Hamming | 0.06 | 4.7309 | 26114 | 78.8483 | 1.2pm | 12pm |

Blackman | 0.06 | 5.7904 | 26114 | 96.5067 | 1.2pm | 12pm |

Nutall | 0.12 | 5.7316 | 13057 | 47.7633 | 1.2pm | 12pm |

Bessel | 0.12 | 4.4017 | 13057 | 36.6808 | 1.2pm | 12pm |

Parameter | Uniform FBG | Proposed apodization | Comments (%) |
---|---|---|---|

FWHM (nm) | 0.12 | 0.07 | ↓41.67 |

MSL (dB) | −14.0133 | −34.6852 | ↑148 |

SLSR (dB) | −17.7356 | −30.3563 | ↑71.16 |

Sensitivity (AU/RIU) | 1.6783 | 5.9083 | ↑252 |

Quality parameter |
13.9858 | 84.4043 | ↑503 |

Detection accuracy | 13057 | 22383 | ↑71.42 |

Induced periodic refractive index modulation along the fiber core is given as
_{0} is the averaged index change.

Optimization of grating length (L) and index modulation (Δn) is essential to determine the maximum reflectivity of peak resonance. To achieve 100% reflectivity, a proper selection of ‘L’ and ‘Δn’ should be made. In this study, simulation was performed for various values of index modulation and the grating length and observations are tabulated as shown in

Grating length (mm) | Reflectivity of main lobe (dB) | |||
---|---|---|---|---|

Δn = 0.0001 | Δn = 0.00015 | Δn = 0.0002 | Δn = 0.00025 | |

5 | −127.3710 | −123.9490 | −4.2684 | −2.9585 |

10 | −4.3289 | −2.0968 | −1.0252 | −0.5018 |

15 | −2.1575 | −0.7402 | −0.2534 | −0.0866 |

20 | −1.1044 | −0.2648 | −0.0631 | −0.0150 |

25 | −0.5734 | −0.0950 | −0.0158 | −0.0026 |

30 | −0.3004 | −0.0345 | −0.0039 | 0 |

35 | −0.1584 | −0.0125 | 0 | 0 |

40 | −0.0839 | −0.0045 | 0 | 0 |

45 | −0.0447 | −0.0017 | 0 | 0 |

50 | −0.0239 | 0 | 0 | 0 |

Sensitivity enhancement with PDMS coating for varying thickness is shown in

Apodization functions are implemented to enhance the performance of sensors by suppressing side lobes and making FHWM narrower. The proposed novel apodization is manifested to have suitable performance parameters to implement it as a sensor in the field of medicine. Narrow FWHM, highest sensitivity (3.52 times more compared to UFBG) and detection accuracy (71.42% more than UFBG), make it a preferable solution for sensing employment. Enhancement in sensitivity as 1.4 pm/με and 14 pm/°C is shown by the proposed novel apodization for strain and temperature measurement respectively. With optimization, maximum reflectivity of 100% has been achieved for 15 mm of grating length and index modulation of 0.00025 for the proposed apodization function. Enhancement in temperature sensitivity of 72 pm/°C was achieved by using PDMS coating, which is 5.14 times greater than the proposed apodization without coating, enabling its implementation for monitoring vital signals in human beings.

_{2}layers