Comparative Analysis of the Methods for Fiber Bragg Structures Spectrum Modeling

Abstract

The work is dedicated to a comparative analysis of the following methods for fiber Bragg grating (FBG) spectral response modeling. The Layer Sweep (LS) method, which is similar to the common layer peeling algorithm, is based on the reflectance and transmittance determination for the plane waves propagating through layered structures, which results in the solution of a system of linear equations for the transmittance and reflectance of each layer using the sweep method. Another considered method is based on the determination of transfer matrices (TM) for the FBG as a whole. Firstly, a homogeneous FBG was modeled using both methods, and the resulting reflectance spectra were compared to the one obtained via a specialized commercial software package. Secondly, modeling results of a π-phase-shifted FBG were presented and discussed. For both FBG models, the influence of the partition interval of the LS method on the simulated spectrum was studied. Based on the analysis of the simulation data, additional required modeling conditions for phase-shifted FBGs were established, which enhanced the modeling performance of the LS method.

1. Introduction

Mathematical modeling of the reflection and/or transmission spectra of passive and active elements of fiber-optic circuits is one of the main approaches used in the design of both fiber communication lines and fiber-optic sensor systems. Mathematical modeling allows not only to predict the results of a certain impact on a fiber-optic system, but also to optimize individual elements of such systems. One of the most commonly used elements of fiber-optic sensor systems, as well as data transmission networks, is a periodic or quasi-periodic structure formed in the core of an optical fiber by changing the refractive index. In particular, a wide class of passive fiber-optic structures, such as fiber Bragg gratings (FBGs), belongs to this type of elements. Over the past decade, in addition to classical FBGs, numerous fiber-optic structures have become widespread: FBGs with one or more phase shifts, apodized and chirped FBGs, superstructured FBGs, FBGs combined with open or closed Fabry–Perot resonators (FPRs), etc.

The basic components of the abovementioned complex or combined fiber-optic structures are FBGs and FPRs. Combining these basic elements in different configurations makes it possible to obtain various sensor designs with specific reflection spectra and different properties and characteristics (Figure 1). An internal (or external) FPR (Figure 1a) can be used as a high-precision point temperature sensor [1]. A conventional uniform FBG (Figure 1b) is also used as a sensing element for temperature, deformation, vibration, etc. measurements [2]. Two FBGs recorded sequentially (Figure 1c) or one over the other [3,4] (Figure 1d) have found applications as addressed fiber Bragg structures [5]. An FBG-based structure with a discrete phase shift (Figure 1e) is used as a high-precision Bragg-type sensor due to the presence of an ultra-narrow transparency window in its reflectance spectrum, the wavelength position of which can be measured with higher accuracy than the shift of the entire FBG spectrum [6]. The FBG structures with two or more phase shifts can also be used as addressed [5] (Figure 1f) or even multi-addressed [7] structures. The sensing elements based on the combination of FBG and FPR (Figure 1g) represent a promising solution for gas concentration measurement systems [8,9,10].

To enhance the performance of the designed elements of the fiber-optic data transmission or sensor systems, it is necessary to have reliable methods and tools that enable spectral characteristics modeling of fiber-optic structures with high accuracy.

The basic element of such tools, which is of particular interest, is the mathematical apparatus that makes it possible to reconstruct the FBG reflection spectrum. The current paper investigated and compared various methods of FBG spectral response modeling. The first method is based on determination of the reflectance and the transmittance of plane waves propagating through a layered structure similarly to the layer peeling algorithm [11], which results in the solution of a system of linear equations for the transmittance and reflectance of each layer using the sweep method (further referred to as the Layer Sweep (LS) method). The second technique is based on the formulation of transfer matrices for the entire FBG structure [12,13] (Transfer Matrix (TM) method). The FBG transfer matrix is often reduced to a matrix with Chebyshev polynomials of the second kind [14], which requires their calculation for each wavelength. Modeling techniques based on recurrence relations, including the ones using Chebyshev polynomials, were deliberately excluded from consideration, since they do not provide performance benefits in comparison with the LS method.

In this paper, firstly, a homogeneous non-apodized FBG was modeled using both methods, and the resulting reflectance spectra were compared to the one obtained via a specialized commercial software. Secondly, modeling results of a π-phase-shifted FBG are presented. For both FBG models, the influence of the partition interval of the LS method on the simulated spectrum was studied. It was established that the asymmetry of the refractive index profile partition near the phase shift relative to its center causes the asymmetry of the spectral response of the simulated grating, resulting in the increased simulation error. Therefore, in the case of phase-shifted FBG, in order to increase the simulation accuracy, it is not enough to decrease the partition interval of the LS method, but it is also required to fulfill certain conditions of the refractive index definition in the model, which are discussed in the manuscript.

2. Modeling Methods

The interference scheme for recording FBGs, as one of the most commonly used schemes for recording fiber Bragg gratings, implements a change in the refractive index along the length of the grating according to the harmonic law [15,16]:

$$n\left(z\right)=n_0+\frac{\mathsf{\Delta }n}{2}+\frac{\mathsf{\Delta }n}{2}\cdot \sin \left(\frac{2\pi }{\Lambda }z\right) ,$$

(1)

where Λ is the FBG period, n₀ is the refractive index of the optical fiber core, and Δn is the value of the induced refractive index, which generally can be represented as a function of the grating length, for example, as the Gaussian apodization function [15,17]:

$$\mathsf{\Delta }n\left(z\right)=\mathsf{\Delta }n\cdot \exp \left(-\frac{1}{2}{\left(\frac{z-z_{\mathrm{M}}}{\sigma }\right)}^2\right),$$

(2)

where z_M is the FBG midpoint and σ is the apodization parameter defining the ratio p of the induced refractive index at the FBG edges to its value at the midpoint:

$${\sigma }^2=-\frac{1}{8}\frac{H^2}{\ln p} ,$$

(3)

where H is the FBG length. The FBG is formed in the optical fiber; therefore, the refractive index of the core of the optical fiber is the minimum value of the refractive index in the FBG interval, and the maximum will be the sum of the refractive indices of the fiber and the induced refractive index.

Thus, the initial parameters for the FBG modeling are: the length of the grating H, the refractive index of the optical fiber core n₀, the induced refractive index Δn, the FBG period Λ, and the apodization parameter σ.

2.1. Layer Sweep Method

Any medium inside the optical fiber with a known variation of the refractive index can be represented as a set of thin homogeneous “films”, in each of which the refractive index can be considered constant [18]. Thus, the entire FBG is divided by a one-dimensional coordinate grid {z_i}, i = 1, N (Figure 2) into homogeneous layers, in which the refractive index is determined as an integral average over the layer thickness:

$$n_{i+1}=\frac{1}{z_{i+1}-z_i}{\displaystyle \underset{z_i}{\overset{z_{i+1}}{\int }}n(z)dz} .$$

(4)

It must be noted that the length of the partition interval Δz = z_i+1 − z_i may significantly affect the performance of the modeled structure, which is discussed in Section 3. Substituting (1) and (2) into (4), we obtain a function that describes the layer-by-layer dependence of the refractive index on the coordinate z:

$$n_{i+1}=\frac{1}{z_{i+1}-z_i}{\displaystyle \underset{z_i}{\overset{z_{i+1}}{\int }}\left[n_0+\frac{1}{2}\cdot \mathsf{\Delta }n\cdot \exp \left(-\frac{1}{2}{\left(\frac{z-z_{\mathrm{M}}}{\sigma }\right)}^2\right)\cdot \left(1+\sin \left(\frac{2\mathsf{\pi }}{\Lambda }z\right)\right)\right]dz} .$$

(5)

Equation (5) is not an integral-elementary function, but integration in (5) is not required in this case, since the refractive index of the medium must be constant inside each layer, and expression (5) does not take this requirement into account, since the exponential multiplier that normalizes the amplitude of the refractive index variation implies that its amplitude within the layer changes along the coordinate z. The constancy of the induced refractive index within each layer implies the constancy of the exponent in (5) over the layer thickness. Therefore, the exponential multiplier in (5) must be replaced by the average value of the refractive index, which, on the interval from [z_i, z_i+1] can be taken as its value in the middle of the interval:

$$\exp \left(-\frac{1}{2}{\left(\frac{z-{\left(z_{\mathrm{M}}\right)}_k}{{\sigma }_k}\right)}^2\right) \simeq \exp \left(-\frac{1}{2}{\left(\frac{\frac{z_{i+1}+z_i}{2}-{\left(z_{\mathrm{M}}\right)}_k}{{\sigma }_k}\right)}^2\right) +\mathrm{o}(\mathsf{\Delta }{z}_i),$$

(6)

where Δz_i = z_i+1 − z_i, and o(x) is the error function, which denotes that the specified function decreases faster than its argument. Substituting (6) into (5), we obtain:

$$n_{i+1}=\frac{1}{z_{i+1}-z_i}{\displaystyle \underset{z_i}{\overset{z_{i+1}}{\int }}\left[n_0+\frac{1}{2}\mathsf{\Delta }n\left(\exp \left(-\frac{1}{2}{\left(\frac{z_{i+1}+z_i-2z_{\mathrm{M}}}{2\sigma }\right)}^2\right)+\mathrm{o}(\mathsf{\Delta }{z}_i)\right)\left(1+\sin \left(\frac{2\mathsf{\pi }}{\Lambda }z\right)\right)\right]dz} .$$

(7)

After that, the multiplier of the amplitude becomes independent of the integration variable and the integral can be taken analytically; thus, the dependence of the refractive index over all the layers of an arbitrary layered structure can be expressed as follows:

$$n_{i+1}=n_0+\frac{\mathsf{\Delta }n}{2}\exp \left(-\frac{1}{2}{\left(\frac{z_{i+1}+z_i-2z_{\mathrm{M}}}{2\sigma }\right)}^2\right) \cdot \left(1-\frac{\Lambda }{\mathsf{\pi }\left(z_{i+1}-z_i\right)}\sin \left(\frac{\mathsf{\pi }\left(z_{i+1}+z_i\right)}{\Lambda }\right)\sin \left(\frac{\mathsf{\pi }\left(z_{i+1}-z_i\right)}{\Lambda }\right)\right)+\mathrm{o}(\mathsf{\Delta }{z}_i) .$$

(8)

In order to model the reflection spectrum of such a structure, it is necessary to initialize the parameters of the layers based on relations (8), with further application of the classical algorithms of backward and forward sweeps for the layered structure calculation. The electric and magnetic components of the electromagnetic field at each point of each layer are the superposition of two waves (Figure 3) moving in different directions—the direct and the reflected ones:

$$\begin{array}{l}{E}_i(z)=t_i\cdot \exp \left(-j\left(k_{i}z+\mathsf{\omega }\mathsf{\tau }\right)\right)+r_i\cdot \exp \left(j\left(k_{i}z-\mathsf{\omega }\mathsf{\tau }\right)\right) ,\\ H_i(z)=\frac{t_i}{w_i}\cdot \exp \left(-j\left(k_{i}z+\mathsf{\omega }\mathsf{\tau }\right)\right)-\frac{r_i}{w_i}\cdot \exp \left(j\left(k_{i}z-\mathsf{\omega }\mathsf{\tau }\right)\right), i=\overline{0,N+1} ,\end{array}$$

(9)

where τ is the time, t_i and r_i are the transmittance and reflectance of the i-th layer; k_i is the wavenumber and w_i is the impedance for the i-th layer. The first summand in (9) for both the electric and magnetic components describes the wave moving in the direction of propagation, and the second term describes the waves reflected from the interfaces of the media and moving against the direction of propagation.

The wave number and wave impedance depend on the radiation wavelength, permittivity, and permeability of the medium:

$$k_i=2\mathsf{\pi }f\sqrt{{\mathsf{\epsilon }}_{\ast }{\mathsf{\mu }}_{\ast }}=\frac{2\mathsf{\pi }c\sqrt{{\mathsf{\epsilon }}_{\ast }{\mathsf{\mu }}_{\ast }}}{\lambda }\sqrt{{\mathsf{\epsilon }}_i{\mathsf{\mu }}_i}=\frac{2\mathsf{\pi }}{\lambda }\sqrt{{\mathsf{\epsilon }}_i{\mathsf{\mu }}_i}, w_i=\sqrt{{\mathsf{\mu }}_i{\mathsf{\mu }}_{\ast }/{\mathsf{\epsilon }}_i{\mathsf{\epsilon }}_{\ast }} ,$$

(10)

where λ = c/f is the wavelength, ε_i is the permittivity and μ_i is permeability of the i-th layer, respectively, ε_∗ and μ_∗ are the permittivity and permeability of the vacuum, respectively, and c is the speed of light in vacuum.

The condition for the continuity of electromagnetic waves in space presupposes the equality of electric and magnetic fields at every point in space, including the interfaces between the layers:

$$\{\begin{array}{l}{t}_i\cdot \exp \left(-j{k}_i{z}_i\right)+r_i\cdot \exp \left(j{k}_i{z}_i\right)=t_{i+1}\cdot \exp \left(-j{k}_{i+1}{z}_i\right)+r_{i+1}\cdot \exp \left(j{k}_{i+1}{z}_i\right)\\ \frac{t_i\cdot \exp \left(-j{k}_i{z}_i\right)-r_i\cdot \exp \left(j{k}_i{z}_i\right)}{w_i}=\frac{t_{i+1}\cdot \exp \left(-j{k}_{i+1}{z}_i\right)-r_{i+1}\cdot \exp \left(j{k}_{i+1}{z}_i\right)}{w_{i+1}}\end{array}, i=\overline{0,N}.$$

(11)

where λ = c/f is the wavelength, ε_i is the permittivity and μ_i is permeability of the i-th layer.

It must be noted that the common multiplier exp(−jωτ), which describes the variations of the electric and magnetic waves over time, is reduced in the left and right parts of Equation (11). As a result, the relations (11) give a system of 2(N + 1) linear equations for determination of 2(N + 2) unknown transmission coefficients t_i and the reflection coefficients r_i in each of the (N + 2) layers.

An additional analysis of the system of Equation (11) makes it possible to exclude from them the permittivity ε_∗ and permeability μ_∗ of the vacuum, using the following relation:

$$c\cdot \sqrt{{\mathsf{\epsilon }}_{\ast }{\mathsf{\mu }}_{\ast }}\equiv 1 ,$$

(12)

where c is the speed of light. In the system of Equation (11), the left and right parts are reduced by the square root of the product of the absolute permittivity and permeability, and the wave number is reformulated in terms of the wavelength.

Permittivity ε and permeability μ are related to the refractive index as follows:

$$n=\sqrt{\mathsf{\epsilon }\mathsf{\mu }}.$$

(13)

In order to close the resulting system of equations, it is necessary to supplement the system (11) with two more relations. As we postulated above, all radiation coming from the source to the zeroth layer (“Layer 0”) passes through it without loss, and the reflection from the second boundary of the last layer (“Layer N + 1”) is absent due to the infinite remoteness of the boundary. These conditions make it possible to supplement the system of equations with two missing relations for the transmission coefficients of the zeroth layer and the reflection of the last layer, respectively:

$$t_0=1, r_{N+1}=0 .$$

(14)

Combining (11) and (14) we obtain a closed system of 2(N + 1) linear equations for 2(N + 1) unknown transmission and reflection coefficients of each of the layers, which can be written in matrix form:

$$\left[\begin{array}{cccccccccc}1& -1& -1& 0& 0& 0& 0& \dots & 0& 0\\ -w_1& -w_0& w_0& 0& 0& 0& 0& \dots & 0& 0\\ 0& a_1& b_1& c_1& d_1& 0& 0& \dots & 0& 0\\ 0& e_1& f_1& g_1& h_1& 0& 0& \dots & 0& 0\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0& 0& 0& a_i& b_i& c_i& d_i& \dots & 0& 0\\ 0& 0& 0& e_i& f_i& g_i& h_i& \dots & 0& 0\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0& 0& 0& 0& 0& 0& 0& a_N& b_N& c_N\\ 0& 0& 0& 0& 0& 0& 0& e_N& f_N& g_N\end{array}\right]\times \left[\begin{array}{c}{r}_0\\ t_1\\ r_1\\ t_2\\ r_2\\ \dots \\ t_i\\ r_i\\ \dots \\ t_{N+1}\end{array}\right]=\left[\begin{array}{c}-1\\ -w_1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right] ,$$

(15)

where the following denotations are introduced:

$$\begin{array}{llll}{a}_i=\exp \left(-j{\mathsf{\kappa }}_i{z}_i\right)\hfill & b_i=\exp \left(j{\mathsf{\kappa }}_i{z}_i\right)\hfill & c_i=-\exp \left(-j{\mathsf{\kappa }}_{i+1}{z}_i\right)\hfill & d_i=-\exp \left(j{\mathsf{\kappa }}_{i+1}{z}_i\right)\hfill \\ e_i=w_{i+1}\exp \left(-j{\mathsf{\kappa }}_i{z}_i\right)\hfill & f_i=-w_{i+1}\exp \left(j{\mathsf{\kappa }}_i{z}_i\right)\hfill & g_i=-w_i\exp \left(-j{\mathsf{\kappa }}_{i+1}{z}_i\right)\hfill & h_i=w_i\exp \left(j{\mathsf{\kappa }}_{i+1}{z}_i\right)\hfill \end{array} .$$

(16)

A system of linear equations formulated in the matrix form can be solved using the Gaussian method by reducing the matrix of coefficients to a triangular form. Given the specific shape of the matrix elements, each internal row contains only four elements; moreover, rows containing elements in the same positions are grouped by two. Thus, the coefficient matrix is a five-diagonal matrix containing, in addition to the main diagonal, two over-diagonals, and two under-diagonals. Besides that, the two over- and under-diagonals further from the central diagonal are sparse. For systems of linear equations with a matrix of coefficients of three-, four-, and five-diagonal form, the sweep method can be applied, which involves the successive transformation of the matrix rows, thus bringing it to a triangular form. Reduction the coefficient matrix to a diagonal form immediately allows one to determine the value of the unknown standing at the top of the triangle. The matrix of coefficients in (15) can be reduced with equal ease both to the upper triangular form and to the lower triangular form. In the case of reducing the matrix to the upper triangular form, the value of the transmittance of the last layer is automatically calculated. In the case of the lower triangular form, it becomes possible to calculate the reflection coefficient of the zeroth layer.

The reduction of the matrix of coefficients in (15) to the lower triangular form can be performed using the backward sweep formulas for the recurrence relation, setting the initial value of γ in the following form:

$${\gamma }_N=1+\frac{w_N-w_{N-1}}{w_N+w_{N+1}}\exp \left(-j2k_N\left(z_N-z_{N-1}\right)\right) ,$$

(17)

then, in the reverse cycle, the remaining values of γ_i are determined using the recursive relation:

$${\gamma }_i=1+\frac{\left(w_i-w_{i+1}\right){\gamma }_{i+1}+2w_{i+1}}{\left(w_i+w_{i+1}\right){\gamma }_{i+1}-2w_{i+1}}\exp \left(-j2k_i\left(z_i-z_{i-1}\right)\right), i=\overline{N-1,-1,1} ,$$

(18)

which ultimately allows calculating the transmittance and reflectance for the zeroth layer:

$$\{\begin{array}{l}{t}_0^{\mathrm{A}}=1\\ r_0^{\mathrm{A}}=\frac{\left(w_1+w_0\right){\gamma }_1-2w_1}{\left(w_1-w_0\right){\gamma }_1-2w_1}\end{array} .$$

(19)

2.2. Transfer Matrix Method

The reflectance spectrum of a homogeneous non-apodized FBG is modeled by the formulation of the resulting transmission matrix in the following form [12,13,19]:

$${T}\left(\lambda \right)=\left[\begin{array}{cc}\cosh \left(\gamma \cdot H\right)-j\frac{\mathsf{\Delta }\mathsf{\beta }}{\gamma }\sinh \left(\gamma \cdot H\right)& -j\frac{\mathsf{\kappa }}{\gamma }\sinh \left(\gamma \cdot H\right)\\ j\frac{\mathsf{\kappa }}{\gamma }\sinh \left(\gamma \cdot H\right)& \cosh \left(\gamma \cdot H\right)+j\frac{\mathsf{\Delta }\mathsf{\beta }}{\gamma }\sinh \left(\gamma \cdot H\right)\end{array}\right] ,$$

(20)

where H is the length of the FBG, and the parameters γ, κ, and Δβ are determined by the following formulas:

$$\begin{array}{c}\mathsf{\Delta }{n}_{\mathrm{eff}}=\frac{\mathsf{\Delta }n}{2}, n_{\mathrm{eff}}=n_0+\mathsf{\Delta }{n}_{\mathrm{eff}}, \mathsf{\kappa }=\mathsf{\pi }\frac{\mathsf{\Delta }{n}_{\mathrm{eff}}}{\lambda },\\ \mathsf{\Delta }\mathsf{\beta }=\frac{2\mathsf{\pi }{n}_{\mathrm{eff}}}{\lambda }-\frac{\mathsf{\pi }}{\Lambda }, \gamma =\sqrt{{\left|\mathsf{\kappa }\right|}^2-\mathsf{\Delta }{\mathsf{\beta }}^2} . \end{array}$$

(21)

The dependence of T(λ) on the wavelength λ in (20) is implicit and expressed in the terms of κ, Δβ, and γ, which are the functions of the wavelength.

The transfer matrix of the FBG with one discrete phase shift (Figure 1e) is formulated as the product of three transfer matrices, two of which (the first and the third ones) are the transfer matrices of homogeneous FBGs with the length of H₁ and H₂, respectively, and the second matrix describes the section of the optical fiber with the length h forming the following phase shift:

$$T_{\mathsf{\pi }-\mathrm{FBG}}\left(\lambda \right)=T\left(\lambda ,H_1\right)\times T_{\mathsf{\phi }}\left(\lambda ,h\right)\times T\left(\lambda ,H_2\right) ,$$

(22)

where T_φ is the transfer matrix of the π-phase shift:

$$T_{\mathsf{\phi }}\left(\lambda ,h\right)=\left[\begin{array}{cc}\exp \left(-j\frac{2\mathsf{\pi }{n}_{eff}}{\lambda }\cdot h\right)& 0\\ 0& \exp \left(j\frac{2\mathsf{\pi }{n}_{eff}}{\lambda }\cdot h\right)\end{array}\right] .$$

(23)

In (23), the length h corresponds to the phase shift of the grating: h = Λ/2.

The reflection coefficient at each wavelength is defined as the square of the ratio modulus of the resulting transfer matrix elements:

$$R\left(\lambda \right)={\left|\frac{T_{1,0}}{T_{0,0}}\right|}^2 .$$

(24)

3. Results and Discussion

3.1. Modeling Parameters of Homogeneous FBG

In order to verify the implementation of the layer sweep (LS) and transfer matrix (TM) methods, a homogeneous non-apodized FBG was modeled and the results were compared with the spectrum obtained using specialized software package Optiwave OptiGrating 4.2.2. The spectra simulated via the OptiGrating were taken as reference, since it is a commercial software widely used in industry.

To compare the simulation results, we chose a homogeneous non-apodized FBG 6 × 10⁻³ m long with a Bragg wavelength of 1550 nm, a refractive index n₀ = 1.4682, and an induced refractive index Δn = 2 × 10⁻⁴. The chosen Bragg wavelength corresponds to the commonly used C-band wavelength range [20]. The refractive index of fiber core n₀ is chosen according to the specifications of the common optical fiber SMF-28 at 1550 nm [21], while the induced refractive index lies within the typical range for the FBG [12,22,23]. The effective refractive index of the FBG model is defined as the average value of the refractive index over the fiber section containing the grating:

$$n_{\mathrm{eff}}=n_0+\frac{\mathsf{\Delta }n}{2} ,$$

(25)

which gives the average value of the effective refractive index of the FBG equal to n_eff ≈ 1.4683. The FBG period is determined by the Wulff–Bragg relation:

$$\Lambda =\frac{{\lambda }_{\mathrm{Br}}}{2\cdot n_{\mathrm{eff}}}=\frac{{\lambda }_{\mathrm{Br}}}{2\cdot \left(n_0+\mathsf{\Delta }n/2\right)} ,$$

(26)

which is equal to Λ = 5.278213 × 10⁻⁷ m. The defined FBG parameters are listed in

Table 1. Modeling parameters of the homogeneous non-apodized FBG.

Parameter	Value
Grating length H, m	6 × 10−3
Refractive index of the fiber n0, RIU	1.4682
Induced refractive index Δn, RIU	2 × 10−4
Grating period Λ, m	5.278213 × 10−7

.

3.2. Results of Homogeneous FBG Modeling

An important note should be made regarding the choice of the partitioning interval used in the Layer Sweep method, since the modeling results using the LS method may significantly vary depending on the chosen partition interval. In order to establish the optimal interval, the following approach can be taken. A decreasing sequence of partition intervals {Δz_n} is assigned so that

$$\mathsf{\Delta }{z}_n=\frac{1}{2^n}\mathsf{\Delta }{z}_0 .$$

(27)

We require that two spectra obtained with different partition intervals Δz_n and Δz_n+1 differ from each other by an amount not exceeding a small arbitrary pre-defined value ε. The spectrum obtained with the partition interval Δz_n is denoted as a function depending on the wavelength and the partition interval Φ(λ, Δz_n). Then, there is an integer number N such that for all n greater than N, and for any partition {λ_i} of an arbitrary interval [λ_Br − Δλ, λ_Br + Δλ], where i takes the values from 1 to M, the norm of the relative differences of the two spectra obtained for partitions z_n and z_n+1 differ by an amount not exceeding the given value ε.

$$\begin{array}{c}\exists N,M\in \mathbb{Z};\forall \mathsf{\epsilon },{\lambda }_{\mathrm{Br}},\mathsf{\Delta }\lambda \in ℝ:\\ \mathsf{\Delta }{z}_n=2^{-n}\mathsf{\Delta }z,\forall n\ge N;{\lambda }_i={\lambda }_{\mathrm{Br}}-\mathsf{\Delta }\lambda +i\frac{2\mathsf{\Delta }\lambda }{M},i=\overline{0,M}\Rightarrow \\ \Rightarrow {\displaystyle \sum _{i=0}^M\Vert \frac{\mathsf{\Phi }\left({\lambda }_i,\mathsf{\Delta }{z}_{n+1}\right)-\mathsf{\Phi }\left({\lambda }_i,\mathsf{\Delta }{z}_n\right)}{\mathsf{\Phi }\left({\lambda }_i,\mathsf{\Delta }{z}_n\right)}\Vert } \le \mathsf{\epsilon } .\end{array}$$

(28)

In other words, the partition interval is successively halved each time until the difference between the spectra obtained for two successive intervals becomes less than the pre-defined small value ε. However, as it will be shown further, the difference between the modeled and the reference spectra does not always monotonically decrease with the decrease of the partition interval, especially in non-homogeneous FBG structures, such as π-phase-shifted FBGs. In such cases, it is required to take into account additional criteria of the structure partitioning.

Figure 4 presents the results of the homogeneous FBG modeling with the parameters listed in the Section 3.1 and the partition interval Δz = 0.04 × Λ.

As it can be seen from Figure 4, the results of both methods were very close to the spectrum obtained using the proprietary software, although having slightly lower reflectance values in general. For further illustration, the deviations of spectra obtained using each method were plotted against the wavelength, which is presented in Figure 5. The root-mean-square deviations (RMSD) were calculated for both methods according to the equation

$$\mathrm{RMSD}=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(R({\lambda }_i)-R_{\mathrm{ref}}({\lambda }_i)\right)}^2}},$$

(29)

where N = 201 is the number of wavelength values for which the reflectance R is estimated using LS or TM methods, and R_ref is the reference values of FBG reflectance derived from the OptiGrating software. In this case, RMSD_LS = 0.0043 and RMSD_TM = 0.0024 for the LS and TM methods, respectively.

It should be noted that the deviation peaks of the LS and TM methods did not coincide besides the central wavelength, which is the evidence of differences in the mathematical modeling approaches. Nevertheless, the deviations of the modeled FBG spectra across the whole relevant wavelength range can be considered negligibly small for most applications, since the deviations do not exceed −40 dB, while the typical signal-to-noise ratio of the FBG interrogation device is between 30 to 40 dB [24]. Therefore, the results confirm the correctness of the modeling methods implementation.

In addition, the RMSD values of the LS method were obtained for various partition intervals Δz, and the corresponding dependence is presented in Figure 6.

The results shown in Figure 6 indicate that the RMSD_LS values in the case of homogeneous FBG modeling decrease monotonically with the decrease of partition interval Δz. The data points were approximated by the second order polynomial (blue line in Figure 6) with the following coefficients: 1.184, 1.311 × 10⁻³, 2.366 × 10⁻³ and the norm of residuals equal to 3.087 × 10⁻⁵. Thus, with the vanishing partition interval of the homogeneous FBG model, the RMSD_LS converges to the value slightly lower than the RMSD_TM.

3.3. Results of a π-Phase-Shifted FBG Modeling

As it was mentioned before, the fiber Bragg structures comprising a phase shift (Figure 1e) are of particular interest for sensing applications since their spectral response includes an ultra-narrow transparency window, the wavelength position of which can be measured with higher accuracy than the shift of the entire FBG spectrum. Such a fiber Bragg structure can be represented as two identical FBGs located sequentially with the distance between them equal to the phase π-shift of their periodic refractive index variation. The modeling parameters of the sub-gratings are listed in

Table 2. Modeling parameters of the π-phase-shifted FBG.

Parameter	Value
Sub-grating length H, m	5 × 10−3
Refractive index of the fiber n0, RIU	1.4682
Induced refractive index Δn, RIU	2 × 10−4
Sub-grating period Λ, m	5.278213 × 10−7

. The simulated spectra of the π-phase-shifted FBG are shown in Figure 7, while the wavelength dependences of the spectra deviations from the OptiGrating data are presented in Figure 8. The partition interval was equal to Δz = 0.04 × Λ. In this case, RMSD_LS = 3.89 × 10⁻³ and RMSD_TM = 3.68 × 10⁻³ for the LS and TM methods, respectively.

As can be seen from the Figure 7 and Figure 8, the results of both methods are also very close to the spectrum obtained using the specialized software. The RMSD values were obtained for various partition intervals as well, which is demonstrated in Figure 9.

Unlike the previously considered example, the dependence of the RMSD_LS values on the partition interval Δz in the case of π-phase-shifted FBG modeling is not monotonical and cannot be approximated with an acceptable precision. There are certain values of the partition interval, at which the RMSD_LS significantly increases.

In order to establish the cause of such phenomenon, the modeled FBG refractive index profiles near the phase shift with different partition intervals Δz and the resulting spectra were compared to each other, as it is illustrated in Figure 10, where the intervals Δz = 0.13 × Λ and Δz = 0.12 × Λ are considered as an example.

As it can be seen from Figure 10, the spectrum obtained using the LS method with the partition interval Δz = 0.12 × Λ (Figure 10a, red solid line) has an asymmetric shape, which results in the increased RMSD in comparison with the spectrum acquired with the interval Δz = 0.13 × Λ despite the lower value of the partition interval. In this case, the displacement of transparency window of the phase-shifted FBG is comparable to its FWHM, which is not acceptable, since such structures are often used for high-resolution sensing [6]. If we compare the refractive index profiles of the grating (Figure 10b,c), we can see that in the case of the interval Δz = 0.12 × Λ (Figure 10b), the partitioning of the refractive index profile near the phase shift is asymmetrical relative to the center of the shift, while the partitioning with the interval Δz = 0.13 × Λ gives symmetrical profile relative to the phase shift center. The same effect was observed at all of the partition intervals indicated in Figure 9, at which the RMSD_LS exceeded 0.02.

Therefore, based on the analysis of the refractive index profile of the grating model, one can conclude that the asymmetry of the refractive index profile partition near the phase shift relative to its center causes the asymmetry of the spectral response of the simulated grating, which results in the significant increase of the RMSD_LS values at certain partition intervals, as it was reported in Figure 9. In the next sub-section of the paper, the authors propose a solution to eliminate the discussed negative effect in phase-shifted FBG simulation.

3.4. Amendment to Layer Sweep Method for Phase-Shifted FBG Modeling

In order to eliminate the spectral response asymmetry of the phase-shifted FBG modeled using the Layer Sweep method, it is necessary to ensure the symmetry of the partitioned refractive index profile of the FBG relative to the center of the phase shift. For that, the authors introduce an additional condition to the choice of the partition interval, according to which each FBG period must contain only integer and odd number of partition points. The simulation results obtained using the proposed approach are presented in Figure 11, where the value of Δz = (1/7) × Λ is taken as the partition interval.

The spectrum obtained using the LS method shown in Figure 11a closely replicates the spectrum derived from the OptiGrating software despite rather coarse partition of the refractive index profile depicted in Figure 11b. In the considered case, the RMSD_LS = 0.0151, which is significantly lower even than the deviation demonstrated at Δz = 0.03 × Λ (see Figure 9). The RMSD values for various partition intervals corresponding to the introduced condition are shown in Figure 12.

The data points can be approximated by the fourth-order polynomial (blue line in Figure 12) with the following coefficients: −27.929, 9.476, −0.2491, 3.734 × 10⁻³, 3.667 × 10⁻³ and the norm of residuals equal to 2.123 × 10⁻⁵. Hence, the RMSD of the phase-shifted FBG spectrum modeled using the LS method with integer number of partition points in the FBG period decrease monotonically with the reduction of the partition interval. With the vanishing partition interval, the RMSD_LS of the phase-shifted FBG model converges to the value slightly lower than the RMSD_TM similarly to the homogeneous FBG modeling.

4. Conclusions

The results of the FBG modeling using both the Layer Sweep (LS) and Transfer Matrix (TM) methods are generally close to the ones obtained using the specialized software. However, in order to mitigate the deviation of the LS method spectrum from the reference one, it is necessary to decrease the partition interval of the refractive index profile.

It was also established that at certain values of the partition interval of the LS method, the deviations of the modeled phase-shifted FBG spectrum significantly increase due to its asymmetrical distortion. Based on the analysis of the refractive index profile of the grating model, it was concluded that the asymmetry of the refractive index profile partition near the phase shift relative to its center causes the asymmetry of the spectral response of the simulated grating.

A solution to eliminate the discussed negative effect in phase-shifted FBG simulation was proposed, which presupposes an additional condition to the choice of the partition interval, according to which each FBG period must contain only integer (odd) number of partition points.

It must be noted that the Layer Sweep method offers more flexibility in terms of the modeled fiber optic structure configuration, since it can simulate any refractive index profile of the fiber medium, such as two or more FBGs recorded one over the other (also known as the moiré recording) [4], which is not possible to model using the transfer matrix (TM) approach, while the latter offers benefits in terms of computational performance, especially when small partition interval is required to provide the desired accuracy of the LS method implementation.

Further research will be dedicated to the modeling of other types of fiber-optic structures, including the combined structures incorporating FBGs and Fabry–Perot interferometers, moiré-recorded FBGs, etc., as well as to the enhancement of the LS method computational performance. The latter can be achieved, for instance, by applying the variable partition interval.