2022 Volume 63 Issue 4 Pages 436-441
In this paper, we present results of numerical analysis of phase-shifted fiber Bragg gratings aimed at slowing down the group velocity of light propagating through these structures. Using coupled-mode theory and transfer matrix method we model the impact of several parameters such as length of the grating, refractive index modulation depth and phase shift in the periodicity of Brag gratings to calculate transmission spectral properties and to maximize the final group delay. By introducing irregularity into a periodic structure of refractive index, Fabry-Perot like cavity formed in the grating results in a favorable narrow dip in resonance spectra characteristics. Correspondingly, we observe a significant spike in group delay at the same wavelength. Simulations results obtained by numerical approach show a strong need for parameter optimization in process of tailoring gratings behavior. Considerable attention must be put on width of spectral region suitable for large group delay realization and suitable interrogation schemes need to be implemented when adopting studied structures for sensors applications.
Fig. 5 Comparison of group delay for 5 mm uniform and π phase shifted gratings. With selected index modulation we observe a rapid peak centered around 13 × 10−4, for a phase shift grating. Uniform grating shows stable and predictable increase.
Over the last few decades fiber Bragg gratings (FBGs) have been extensively studied in order to harness advantages of their use in real life systems as well as improve the basic understanding of underlying physics related to light propagation in these structures. The light-matter interaction in a periodic modulation of refractive index of the fiber core first observed in 1978 enabled the field of sensing with optical sensors to develop rapidly as well as offering new architectures of many fiber optics network elements used today. High sensitivity, small size, electromagnetic interference immunity, multiplexed sensor technologies utilizing multiple gratings embedded on a single optical fiber, reliability and stability in harsh environment, low cost and mature fabrication methods are just some of the often-emphasized advantages of FBG sensors used mostly in industrial strain and temperature measuring, structural health monitoring or biomedicine.1) A FBG ultrasonic system developed for acoustic emission (AE) structure condition monitoring uses optical fiber to overcome drawbacks of conventional AE monitoring systems, along with interferometric methods and other different approaches.2) FBGs in their most simple character being a wavelength-specific filters can reflect a desired wavelength which is tuned by geometrical dimensions of grating, properties of used fiber and parameters of index modulation. Accurate combination of these variables will result in a behavior tailored to the custom application, possibly a sensor, a delay line replacement or an optical network element such as wavelength converter.3,4) Furthermore, different long-term sensing methods suitable for structural health monitoring are being researched and developed continuously. Specific structural parameters such as fatigue behavior could be evaluated by different smart solutions improving on some of FBG based sensors limitations.5)
Structural slow light produced in uniform and phase shifted gratings offers a passive and relatively simple means of creating a resonant wavelength where light propagating through this region experiences multiple reflections and as a result effective increase in group delay is observable. Sensors based on such structures benefiting from slow light use present increased sensitivity at the cost of very narrow wavelength range over which these may be utilized. Coupled with the requirement for operation in transmission, integration of slow light sensors and suitable interrogation schemes becomes a limited solution for fabricated using conventional fiber. Wavelength selectivity of this enhancement accounts for careful incorporation of structures enhanced by effect of slow light.6)
Apart from a large market share dominated with optical sensors based on FBGs many other important applications such as single-frequency lasers, coherent telecommunications or LIDAR technology attract a lot of interest in exploiting the FBGs advantages. In a single-frequency lasing domain the two most common solutions are distributed feedback fiber (DFB) lasers and distributed Bragg reflector (DBR) fiber lasers. Both methods are based on a short-length linear cavity to achieve robust single-frequency operation. Recently published results presented a direct inscription of a strong π-phase shifted Bragg grating structure into the Er/Yb co-doped phosphate glass fiber using 5 cm long phase-shift phase mask and KrF excimer laser. To increase its photosensitivity before writing process the fiber was hydrogen loaded and inscribed FBG was then thermally annealed for stability improvement. π-phase shifted phase mask offers better positioning of phase gap and is also more suitable in case of additional grating apodization.7,8) Several other techniques have been proposed, such as use of homogeneous low voltage electron beam irradiation (HLEBI) which demonstrated the enhancement in elasticity of 250 µm diameter optical fiber constructed with both core and cladding silica glasses covered with acryl-urethane sheath. In addition, notable effects have been observed on both tensile strength and fracture strain of irradiated optical fiber. Both parameters affected by selected treatment could interfere strongly with conditions used for grating inscription process.9)
Many different phase-shifted FBG inscription techniques have been proposed over the years. Good repeatability and flexibility can be obtained by implementing phase-shift phase mask method, various post processing techniques or repeated exposure, leading to formation of period overlapping region in the grating. Femtosecond laser technology ushered a point-by-point technique which in combination with high precision translation stage proves to be highly versatile approach.10)
In this paper we focused on structural and optical properties of FBGs, comparing slow light in uniform FBGs and PSFBGs. Optimization of physical and spectral characteristics considering their fabrication limitations is emphasized. Theory of phase shifted gratings is presented and employed for following group delay and reflection spectra characteristics simulations done in MATLAB® environment.
The uniform and phase-shifted fiber Bragg gratings have been modelled using an algorithm written and implemented for Matlab programming platform.
Using Maxwell’s equations to describe the behavior of fiber gratings we obtain a set of coupled mode equations governing the coupling between forward and backward propagating modes. Analytical solution exists only for a lossless uniform grating case. Other gratings require a piece-wise approach dividing the gratings into several subsections. In our next steps we will consider a FBG to be an optical device realized as a short section of fiber with periodic modulation of core refractive index. Spatial period will establish whether the structure supports coupling between core modes or coupling between one core mode and one or more cladding modes. Due to the discontinuity at the layer interfaces a portion of the forward propagating light is reflected while the remaining part continues through. Bragg gratings consider spatial periods on the order of sub-micron scale which enables the coupling between forward and backward propagating core modes.
Modelling of uniform FBG effective refractive index profile inscribed into the core described with respect to the direction of z along the fiber longitudinal axis and an average refractive index before the inscription n0 is performed by the following expression:
| \begin{equation} n(z) = n_{0} + \varDelta n_{\textit{DC}} + \varDelta n_{\textit{AC}}\cos \left(\frac{2\pi z}{\varLambda} \right), \end{equation} | (1) |
(a) Refractive index modulation in a uniform fiber Bragg grating and (b) refractive index modulation with inserted π phase shift.
Transmission of an FBG remains unaltered outside of its reflection bandgap and whole structure acts as a filter suitable for wavelength selection of choice. Properties of chosen fiber as well as fabrication parameters such as FBG length, index modulation amplitude, and the spatial period will influence the bandgap and its position. Mode interactions have been described by coupled mode theory using following expressions.
The forward (incident):
| \begin{equation} F(z) = F_{0}(\mathrm{z})\exp\left(i\delta z - \frac{\phi}{2} \right) \end{equation} | (2) |
| \begin{equation} B(z) = B_{0}(\mathrm{z})\exp\left(- i\delta z + \frac{\phi}{2} \right) \end{equation} | (3) |
Propagating modes accounting for the dominant interaction in FBG, where F0 and B0 are slowly varying-amplitudes are being coupled together through the coupled-mode equations:
| \begin{equation} \frac{dF(z)}{dz} = i\hat{\sigma}F(z) + i\kappa B(z), \end{equation} | (4) |
| \begin{equation} \frac{dB(z)}{dz} = - i\kappa F(z) - i\hat{\sigma}B(z), \end{equation} | (5) |
| \begin{equation} \kappa = \frac{\pi}{\lambda}\varDelta n_{\textit{AC}}, \end{equation} | (6) |
| \begin{equation} \hat{\sigma} = \delta + \sigma - \frac{1}{2}\frac{d\phi}{dt}, \end{equation} | (7) |
| \begin{equation} \sigma = \frac{2\pi \varDelta n_{\textit{DC}}}{\lambda}, \end{equation} | (8) |
| \begin{equation} \delta = \beta - \frac{\pi}{\varLambda}, \end{equation} | (9) |
Calculation of interaction between propagating modes has been represented by a 2 × 2 transfer matrix Ti:
| \begin{equation} \begin{bmatrix} F_{i}\\ B_{i} \end{bmatrix} = T_{i} \begin{bmatrix} F_{i - 1}\\ B_{i - 1} \end{bmatrix} = \begin{bmatrix} T_{11} & T_{12}\\ T_{21} & T_{22} \end{bmatrix} \begin{bmatrix} F_{i - 1}\\ B_{i - 1} \end{bmatrix} , \end{equation} | (10) |
| \begin{equation} T_{11} = \cosh (\gamma_{B}\Delta z) - \mathrm{i}\frac{\hat{\sigma}}{\gamma_{B}}\sinh (\gamma_{B}\Delta z), \end{equation} | (11) |
| \begin{equation} T_{12} = - \mathrm{i}\frac{\kappa}{\gamma_{B}}\sinh (\gamma_{B}\Delta z), \end{equation} | (12) |
| \begin{equation} T_{21} = T_{12}*, \end{equation} | (13) |
| \begin{equation} T_{22} = T_{11}*. \end{equation} | (14) |
For non-uniform gratings, coupled-mode equations are calculated using numerical approach applied for a selected fiber region and appropriate boundary conditions are specified. We consider the modelled grating (depicted in Fig. 2) of length L and assume a forward going wave passing through the grating structure while there is no backward going wave for z ≥ L, such that F(L) = 1 and B(L) = 0.
The schematics of a fiber Bragg grating.
The whole section is divided along its length L into N segments represented by uniform FBG sub-gratings of the length L ≫ Λ. For each segment, the 2 × 2 transfer matrix coupling the forward and backward electric fields at its output and input at specific wavelength is calculated. FBG is a multiple-wave interferometer of the length L introducing dispersion in the reflected/transmitted light signals.
The phase shift matrix is given as:
| \begin{equation} F_{ps} = \begin{bmatrix} e^{- i\frac{\phi}{2}} & 0\\ 0 & e^{i\frac{\phi}{2}} \end{bmatrix} \end{equation} | (15) |
| \begin{equation} \begin{bmatrix} F_{L}\\ B_{L} \end{bmatrix} = [F_{N}][F_{N - 1}]\ldots [F_{N}]\ldots [F_{2}][F_{1}] \begin{bmatrix} F_{0}\\ B_{0} \end{bmatrix} \end{equation} | (16) |
Calculating the group delay related to the slope of the phase spectrum a steep change in the phase of the transmission/reflection spectrum will result in an increase, where θρ is the phase of the field reflection r and the group delay τg is then the derivative of θρ with respect to the frequency ω:
| \begin{equation} \tau_{g} = \frac{d\theta_{p}}{d\omega} = \frac{d}{d\omega}\arg (r) = - \frac{\lambda^{2}}{2\pi c}\frac{d\theta_{p}}{d\lambda} = \frac{n_{g}L}{c} \end{equation} | (17) |
Broadening of the optical pulse in optical fiber due to the time dispersion limits the usable bandwidth and causes inter symbol interference and subsequent information loss. Hence the use of single mode fibers where only one propagated mode is supported for long distance connections is favorable. However single mode fibers do suffer from group velocity dispersion. Group velocity is defined with use of group index ng as follows:
| \begin{equation} v_{g} = \frac{c}{n_{g}}, \end{equation} | (18) |
| \begin{equation} D = \frac{- 2\pi c}{\lambda^{2}}\beta_{2}, \end{equation} | (19) |
| \begin{equation} \beta_{2} = \frac{d^{2}\beta}{d\omega^{2}}. \end{equation} | (20) |
Remember that optical fiber is a dispersive medium where different spectral components of pulse travel at different velocity due to the variation of the refractive index of the fiber core and cladding with frequency. It is possible to design dispersion compensation fibers by altering the waveguide parameters.16) It is possible to obtain low dispersion slow light in optimized structures formed in various materials where it is used to enhance the properties of different nonlinear devices.17)
By introducing a T parameter in our modelling approach, we can effectively change the position of phase shifting region. Using the simple ratio depicted in Fig. 3, every grating segment and its length is properly defined. As described above, several techniques for phase changing region inscription have been developed.
The schematics of a phase shifted FBG. Position of phase shifting region is modelled by introduction of T parameter used in all calculations in this paper.
Presence of the slow light resonance in the middle of phase shifted fiber Bragg gratings connected to phase irregularity and manipulation of its position is presented in Fig. 4. Large group delay and high transmissivity are the two most considered features when designing a PSFBG based sensor or time delay element, however number of factors important in achieving such properties require careful evaluation process of different parameter combinations.
Reflectivity spectra of a phase shifted FBG for different phase shift values. Index modulation ΔnAC = Δn is fixed.
To effectively model the effect of changing phase shift and study the formed cavity numerically, it is often easier to focus on a certain parameter first while keeping others fixed. Regarding results published in Ref. 18) stating the strongly favorable 7-fold increase in calculated group delay, number of complementary features need to be considered. It is possible to change the position of center of reflectivity notch on a wavelength spectrum keeping its physical position during fabrication and index modulation ΔnAC = Δn unchanged but by adjusting the amount of phase shift introduced into the structure. Phase shift of π which translates into half spatial period of grating skip has been numerically modeled and compared to several other values in Fig. 4. By numerically examining the decrease in the amount of phase shift we observe the shift the reflectivity notch to shorter wavelengths on the sub nanometer scale, accompanied with lower values of maximum reflectivity compared to basic π phase shift and broadening of its bandgap. Observed changes are reciprocal for the increase of phase shift. The interval of usable values is limited from both sides, as zero and 2π will result in a spectrum identical to uniform grating.
Shown in Fig. 5 is the comparison between uniform and phase shifted fiber in terms of the group delay maximum within a selected index modulation interval. Light traveling multiple times back and forth inside the cavity formed between two uniform grating structures experiences multiple passes through this region experiencing enhanced phase change and decrease in group delay. Notice the sharp growth (and subsequent decrease) of phase shifted grating group delay values (blue line) compared to almost stable growth of uniform grating group delay. PSFBG results shows a clear benefit of phase shifting region in otherwise uniform structure.
Comparison of group delay for 5 mm uniform and π phase shifted gratings. With selected index modulation we observe a rapid peak centered around 13 × 10−4, for a phase shift grating. Uniform grating shows stable and predictable increase.
To investigate the group delay for gratings with different index modulation, we performed further simulations focusing on the transmissivity of preferred area. Large increase in group delay comes with two major concerns, one being usable wavelength span and transmissivity threshold suitable for chosen interrogation setup. Attention must be put on width of spectral region suitable for large group delay realization. In many cases the FWHM of central wavelength peak is on the order of units of picometers or smaller. Such narrow resonances set a demanding condition for implementing these gratings into real world systems.
Compared to interferometric interrogation techniques used with conventional FBG sensors, the complex nature of phase shifted grating reflectivity spectrum and the narrow, pronounced transmission peak located at the middle need a different concept of detection. Accurate measurement of wavelength shift of transmission peak, sensitivity characteristics and resolution of sensor are all influenced by the spectral notch created by grating irregularity. Transmission spectrum of a simulated PSFBG shown in Fig. 6 shows a decrease in the amount of transmitted light with increasing index modulation. A direct relation to a sensitivity of a slow-light sensor have been observed by previous publications and is often referred to as figure of merit or sometimes effective group delay.19)
Transmissivity and group delay for π-phase shifted grating. Note the sudden decrease in transmissivity corresponding with the maximum value of group delay at smaller index modulation rendering the right-hand part of selected interval unsuitable for interrogation.
Various techniques aimed at fabrication and sensitization of a fiber superseding the common telecommunication fiber exist and offer a possibility of writing strong gratings. By using a standard 3% Ge dopant concentration it is possible to obtain index modulation of approximately 3 × 10−5. This parameter can be influenced by varying the writing exposure time, using a fiber with different dopants ratios or by applying a combination of post write methods. Considering the germanium-doped silica fiber, numerous studies of its structure show the increase of photosensitivity being a direct outcome of increasing the concentration of GeO defects. One way of achieving this is hydrogen loading where fiber is infused with hydrogen in a sealed vessel. In combination with pulsed laser sources there is an initial growth of index change noticeable for high-germania-doped fibers (around 8%) which differs from low germania content fibers showcasing a complex nature of appropriate fiber selection.20) Untreated fibers behave differently under long exposure times and together with initial grating erasure caused by long exposure results in definition of different grating types.
For our next part we were interested in the width of the group delay resonance outside of our previously studied interval. We focused on a single value of index modulation of 4.27 × 10−4 and investigated shorter gratings with length of 1 to 10 mm with 1 mm increase. In observation of these results in Fig. 7 it is apparent that chosen index modulation combined with increasing length translates to full width of half maximum declining characteristic of group delay resonance, alongside the increase in actual group delay (not pictured here). Preferred slow light operation at the very center of the reflection dip benefits from increasing modulation depth as we proved in a previous section, where, considering the fabrication techniques, prolonged exposure to laser irradiation leads to larger changes in the refractive index. The potential sensing components based on a central wavelength spectral shift in FBG structures are simple to construct but present a suitable interrogation setup resolution challenge. One way of solving this situation present a use of spectral analyzer with very fine step implementing stimulated Brillouin scattering to scan selected bandgap, taking advantage of an extremely sharp filtering effect and other related precautions to examine slow light resonances with their actual width.
PSFBG notch FWHM two consecutive values ratios.
Clear connection between the FWHM of bandpass in reflection spectrum and usable width of group delay peak exists. Our aim was to identify conditions for increase of obtainable and operational delay with respect to grating parameters. One benefit of introducing a narrow reflection dip/transmission peak is sensitivity increasement of the sensor based on this structure. In our model, uniform FBG supporting slow light areas in form of well described formation of pronounced narrow side-lobes resonances on both sides of the main peak, was directly compared with same structure with inserted phase shift. Simulated values do show an advantage of phase shifting region, however compared directly to previously studied uniform gratings results, we see no straightforward method for phase-shifted gratings exists.
In conclusion we have numerically examined selected parameters of phase shifted FBGs intended for slow light realization and directly compared these with their uniform counterparts. Here we have provided a closer analysis of selected mechanisms behind the slow light maximum formation and its properties at the central wavelength because of inserting phase shifting region into uniform grating. Main objective was to investigate high resonances areas by adjusting certain parameters of FBGs namely the grating length and modulation depth. Results show that with increasing index modulation, there is an apparent increase of the time delay and a noticeable advantage of phase shifted FBG over uniform gratings with the maximum calculated around 13 × 10−4 for selected 5 mm length. However, no definite answer to achieving a simple case with only beneficial results is to be found as subsequent increase results in subpar group delay compared to uniform structure. Adopted numerical methods chosen for investigation of this behavior were implemented in MATLAB® showing good agreement with multiple previously published data, while also providing new study of parameters influence and overall tuning. Lack of simple solution providing group delay enhancement without detrimental effects on other parameters of grating proves that these structures accommodate complex light-matter interactions causing unpredicted behavior at shorter lengths.
This work was partly supported by the Slovak Research and Development Agency under the project APVV-17-0631.
