**2. Theory**

This section presents the optical properties of FBG and LPG. Consequently, the basics of simulation models are provided. Coupled-mode theory and the transfer matrix methods are the two techniques used for the simulation of FBG and LPG [13, 14, 17, 20, 22–28]. Thus, the physical mechanism of the grating electric field interaction is given and aims to provide the reader with insight into the operation principles of FBG and LPG. The gratings are inscribed into the core of a step index optic fiber; consequently, the step index optic fiber case is analyzed.

Optical fiber mainly consists of a core, cladding and a protective layer called the primary plastic buffer coating. The optical fiber acts as a waveguide for optical frequencies and is normally cylindrical in shape. The core is a dielectric cylinder surrounded by the cladding to form a larger dielectric cylinder [13, 14]. The optical fiber has a uniform refractive index up to the core-cladding conjunction, where it undergoes a sharp change in refractive index. The refractive index of the core and the cladding is given as *nco* and *ncl*, respectively, the relation *nco* > *ncl* being valid [13, 36, 37]. This is the necessary condition for total internal reflection to occur. The structure and material of the fiber confines the electromagnetic waves to a direction parallel to the axis and also affects the transmission properties of the optic fiber [13, 20]. The light transmitted through the fiber is confined due to total internal reflection within the material. There are practically no electromagnetic fields outside the cladding because of exponential decay within this region [13]. The difference between *nco* and *ncl* is very small, of 10−4 order. This is the necessary condition for the weakly guiding approximation to be applicable. This approximation results into a mathematical apparatus, which allows descrip‐ tion of fiber grating processes as a byproduct. **Figure 1** shows the schematics of a typical optical fiber layout. For a single-mode step index fiber commonly used for grating to be inscribed into the core, the core radius is in the range 1.5–5 μm, the cladding radius being typically of 62.5 μm.

**Figure 1.** The schematics of a typical optical fiber layout.

optic fibers: the fiber Bragg grating (FBG) ones and the long-period grating (LPG) kind [5–7]. In literature, FBGs are considered as short-period grating (300–700 nm), while LPGs as long ones (10–1000 μm). For both FBGs and LPGs, the amplitude of the core refractive index is extreme‐ ly small, in the range 0.0001–0.0005 or even smaller [5–12]. It is important to mention that only step index optic fibers for which the weakly guiding approximation relying on a very small difference between the values of the core and cladding refractive index, *nco*–*ncl*, is applicable are analyzed [5–7, 11–15]. Related to this, it has to be underlined the fact that the amplitude of the spatial periodic variation of the refractive index inscribed in the core is smaller than *nco*–*ncl* [11– 15]. The basic functions as sensors and/or wavelength filter of both FBG and LPG are accom‐ plished by controlled, observed and measured variations of optical fiber refractive indexes of the core (*nco*) andcladding (*ncl*)to whichthe refractive index ofthe ambient(*namb*)is added, where the optical fiber is mounted. Consequently, the spectral characteristics that can be observed in fiberreflection(FBG)andtransmission(LPG)gratingswillbedescribed[13–19].Foranimproved designof experimental setupsdedicatedtothe above-mentionedapplications,itisobvious that, for both FBG and LPG, the principles for understanding and tools for designing fiber gratings are emphasized [11–20]. The emphasized understanding principles and designing tools are applicable for the wide variety of optical properties that are possible in fiber gratings [19–28]. There are given examples related to the large number of fiber grating subtypes of both FBG and LPG, considering uniform, apodized, chirped, discrete phase-shifted and superstructure gratings; symmetric and tilted gratings; and cladding-mode and radiation-mode coupling

Both FBGs and LPGs are manufactured in single-mode silicate optic fiber by modifying in a periodic manner its core refractive index using UV-irradiation delivered by Ar or other UV laser [20]. Most commonly, the LPG is created by altering the core in a periodic manner, but another class of manufacturing methods physically deforms the fiber to create the required optical modulation [34–40]. These include the following: irradiation from a carbon dioxide laser, radiation with femtosecond pulses and writing by electric discharge, ion implantation, periodic ablation and/or annealing, corrugation of the cladding, micro-structuring of tapered

This section presents the optical properties of FBG and LPG. Consequently, the basics of simulation models are provided. Coupled-mode theory and the transfer matrix methods are the two techniques used for the simulation of FBG and LPG [13, 14, 17, 20, 22–28]. Thus, the physical mechanism of the grating electric field interaction is given and aims to provide the reader with insight into the operation principles of FBG and LPG. The gratings are inscribed into the core of a step index optic fiber; consequently, the step index optic fiber case is analyzed.

Optical fiber mainly consists of a core, cladding and a protective layer called the primary plastic buffer coating. The optical fiber acts as a waveguide for optical frequencies and is normally cylindrical in shape. The core is a dielectric cylinder surrounded by the cladding to form a

gratings [20–33].

92 Modeling and Simulation in Engineering Sciences

**2. Theory**

fibers and dopant diffusion into the fiber core.

It is assumed that a perturbation to the effective refractive index of the guided mode(s) can be defined as follows [13]:

$$\delta n\_{c\mathcal{Y}}\left(z\right) = \delta n\_{c\mathcal{Y}max}\left(z\right)\left\{1 + \upsilon \cdot \cos\left[\frac{2\pi}{\Lambda}z + \varphi\left(z\right)\right]\right\}\tag{1}$$

where *δneffmean* is the amplitude of the perturbation, evaluated as the index change spatially averaged on a grating period, *v* is the fringe visibility of the index change, Λ is the nominal period and *φ(z)* is the grating chirp, which represents a variation of nominal period. In the case of a step index fiber without a grating, the core power confinement factor, Γ, is defined. For uniformity across the core-induced grating, with an induced index change *δnco* created in core, for the propagation mode, the following relation is defined [13, 21–28, 36]:

$$
\delta \mathfrak{N}\_{\rm eff} \cong \Gamma \cdot \delta \mathfrak{n}\_{\rm co} \tag{2}
$$

Since FBG or LPG is manufactured starting from single-mode light is propagating along the core as LP01 modes for which an effective index parameter b is introduced. It is useful to introduce the normalized frequency, *V*, a parameter synthetically characterizing the geomet‐ rical and optical materials fiber properties [13, 36, 37]. *V* is defined as follows:

$$V = \left(\frac{2\pi}{\lambda}\right) \cdot a\_{co} \cdot \sqrt{n\_{co}^2 - n\_{cl}^2} \tag{3}$$

where *aco* is the core radius. The effective index parameter is a solution to the dispersion rela‐ tion [20]:

$$V\sqrt{1-b}\cdot\frac{J\_{l-1}\left(V\sqrt{1-b}\right)}{J\_l\left(V\sqrt{1-b}\right)} = -V\sqrt{b}\cdot\frac{K\_{l-1}\left(V\sqrt{b}\right)}{K\_l\left(V\sqrt{b}\right)}\tag{4}$$

where *l* is the Azimuthal order of the mode LP01. In Eq. (4), *Jl* are the Bessel functions of the first kind and *Kl* are the modified Bessel functions of the second kind. The effective index *neff* is related to through the relation [13, 20–28]:

$$b = \frac{\mathfrak{n}\_{eff}^2 - \mathfrak{n}\_{cl}^2}{\mathfrak{n}\_{co}^2 - \mathfrak{n}\_{cl}^2} \tag{5}$$

Once *b* and *V* are known, Γ can be determined from

$$\Gamma = \frac{b^2}{V^2} \left[ 1 - \frac{J\_l^2 \left( V \sqrt{1-b} \right)}{J\_{l+1} \left( V \sqrt{1-b} \right) J\_{l-1} \left( V \sqrt{1-b} \right)} \right] \tag{6}$$

A fiber grating, FBG or LPG, is the periodic variation of refractive index within the core of a step index single-mode optical fiber. In **Figures 2** and **3**, the schematics of the two considered types of an optical fiber with a grating written in the core of the fiber are shown. The core inscribed refractive index changes can be described as cylinders. The refractive index changes of a fiber grating usually have a near sinusoidal variation. Firstly, the simple case of a uniform grating fiber grating is considered.

**Figure 2.** Propagation in a FBG.

of a step index fiber without a grating, the core power confinement factor, Γ, is defined. For uniformity across the core-induced grating, with an induced index change *δnco* created in core,

*eff co*

 d

Since FBG or LPG is manufactured starting from single-mode light is propagating along the core as LP01 modes for which an effective index parameter b is introduced. It is useful to introduce the normalized frequency, *V*, a parameter synthetically characterizing the geomet‐

where *aco* is the core radius. The effective index parameter is a solution to the dispersion rela‐

( ) 1 1 <sup>1</sup>

*J V b K Vb*

*JV b KVb*

first kind and *Kl* are the modified Bessel functions of the second kind. The effective index *neff*

2 2 2 2 *eff cl co cl*

( ) ( ) ( )

1 1

1

*n n*

*n n*

1 1


A fiber grating, FBG or LPG, is the periodic variation of refractive index within the core of a step index single-mode optical fiber. In **Figures 2** and **3**, the schematics of the two considered types of an optical fiber with a grating written in the core of the fiber are shown. The core inscribed refractive index changes can be described as cylinders. The refractive index changes

*V J V bJ V b* + é ù - ê ú G= -

*l l l*

*JV b b*

*l l*


*l l*

*n n* @G× (2)

è ø (3)

( )



are the Bessel functions of the

(6)

for the propagation mode, the following relation is defined [13, 21–28, 36]:

d

rical and optical materials fiber properties [13, 36, 37]. *V* is defined as follows:

l

( )


( )

*b*

<sup>2</sup> <sup>2</sup>

1

*V b V b*

1

Once *b* and *V* are known, Γ can be determined from

2

1

is related to through the relation [13, 20–28]:

94 Modeling and Simulation in Engineering Sciences

where *l* is the Azimuthal order of the mode LP01. In Eq. (4), *Jl*

tion [20]:

<sup>2</sup> 2 2 *V a nn co co cl* p

æ ö = ×× - ç ÷

**Figure 3.** Propagation in a LPG.

A fiber grating produces coupling between two fiber modes [13, 14, 20, 21]. The quantitative analysis of this phenomenon is achieved using coupled-mode theory. It is helpful to consider a qualitative analysis of the basic interactions of interest. A fiber grating is simply an optical diffraction grating, at each refractive index change junction, refraction and reflection occur, and thus its effect upon a light wave incident on it can be described by the familiar grating equation. The diffraction of the light incident on the diffraction grating at an angle *θ*1 can be described by the equation from Snell's law [13, 14]:

$$n \cdot \sin \theta\_2 = n \cdot \sin \theta\_1 + m \cdot \frac{\lambda}{\Lambda} \tag{7}$$

where *m* determines the diffraction order and *θ*2 is the angle of the diffracted wave λ and ^ are the wavelength of the incident light and the period of the diffraction grating, respectively. Eq. (6) can predict only the directions *θ*<sup>2</sup> into which constructive interference occurs, but it is not useful for determining the wavelength at which a fiber grating most efficiently couples light between two modes.

A fiber grating's main function is based on coupling between the modes propagating through the fiber, modes which can travel in opposite directions or in the same direction—this being the basic criterion for classification of fiber gratings into two main types: (a) modes traveling in the opposite directions, denoted as the short-period gratings, the fiber Bragg gratings (FBG), working as the reflection gratings; (b) modes traveling in the same direction, denoted as Long-Period Gratings (LPG), working as transmission gratings.

In **Figure 2**, the mode reflection by a Bragg grating is schematically presented. The incident mode has a bounce angle of *θ*2 and becomes the same mode traveling in the opposite direction with a bounce angle of *θ*2 = - *θ*1. It is worth to underline that the entire process is taking place only inside the core. For the incident and diffracted rays, the propagation constants are calculated as follows [13, 14]:

$$\begin{aligned} \beta\_1 &= \frac{2\pi}{\lambda} n\_{\epsilon g'1}, \ n\_{\epsilon g'1} = n\_{\epsilon o} \sin \theta\_1 \Rightarrow \beta\_1 = \left(\frac{2\pi}{\lambda}\right) n\_{\epsilon o} \sin \theta\_1\\ \beta\_2 &= \frac{2\pi}{\lambda} n\_{\epsilon g'2}, \ n\_{\epsilon g'2} = n\_{\epsilon o} \sin \theta\_2 \Rightarrow \beta\_2 = \left(\frac{2\pi}{\lambda}\right) n\_{\epsilon o} \sin \theta\_2 \end{aligned} \tag{8}$$

Eq. (7) can be rewritten in terms of the propagation constant of the incident beam and the reflected/diffracted light as follows [13, 14, 20–28]:

$$
\beta\_2 = \beta\_1 + m \frac{2\pi}{\Lambda} \tag{9}
$$

where the subscripts 1 and 2 describe the incident and reflected/diffracted propagation constant. For first-order diffraction, which usually dominates in a fiber grating, *m* = −1. Eq. (9) is modified to [13, 14, 20]

$$
\beta\_2 = \beta\_1 - \frac{\lambda}{\Lambda} \tag{10}
$$

For the bound core modes, the following relation is fulfilled:

$$
\mathfrak{m}\_{\rm cl} < \mathfrak{m}\_{\rm eff} < \mathfrak{m}\_{\rm co} \tag{11}
$$

In order to be rigorous, for the cladding modes, a relation similar to Eq. (11) is obtained by considering the value of optic fiber ambient medium, usually air, refractive index:

where *m* determines the diffraction order and *θ*2 is the angle of the diffracted wave λ and ^ are the wavelength of the incident light and the period of the diffraction grating, respectively. Eq. (6) can predict only the directions *θ*<sup>2</sup> into which constructive interference occurs, but it is not useful for determining the wavelength at which a fiber grating most efficiently couples light

A fiber grating's main function is based on coupling between the modes propagating through the fiber, modes which can travel in opposite directions or in the same direction—this being the basic criterion for classification of fiber gratings into two main types: (a) modes traveling in the opposite directions, denoted as the short-period gratings, the fiber Bragg gratings (FBG), working as the reflection gratings; (b) modes traveling in the same direction, denoted as Long-

In **Figure 2**, the mode reflection by a Bragg grating is schematically presented. The incident mode has a bounce angle of *θ*2 and becomes the same mode traveling in the opposite direction with a bounce angle of *θ*2 = - *θ*1. It is worth to underline that the entire process is taking place only inside the core. For the incident and diffracted rays, the propagation constants are

1 11 11 1

*nn n n*

= = Þ= ç ÷

= = Þ= ç ÷

2 1

2 1

b b

For the bound core modes, the following relation is fulfilled:

b b= +

2 2 , sin sin

*eff eff co co*

qb

qb

Eq. (7) can be rewritten in terms of the propagation constant of the incident beam and the

2 *m* p

where the subscripts 1 and 2 describe the incident and reflected/diffracted propagation constant. For first-order diffraction, which usually dominates in a fiber grating, *m* = −1. Eq. (9)

> l

 p

æ ö

 l

è ø æ ö

 p

 l

è ø

 q

> q

<sup>L</sup> (9)

= - <sup>L</sup> (10)

*cl eff co nn n* < < (11)

(8)

2 22 22 2

*nn n n*

2 2 , sin sin

*eff eff co co*

Period Gratings (LPG), working as transmission gratings.

p

l

p

l

reflected/diffracted light as follows [13, 14, 20–28]:

b

b

between two modes.

96 Modeling and Simulation in Engineering Sciences

calculated as follows [13, 14]:

is modified to [13, 14, 20]

$$1 < n\_{e\emptyset} < n\_{cl} \tag{12}$$

Fiber modes that propagate in the negative (−*z*) direction are described by negative *β* values. Using Eq. (9) and observing that *β*<sup>2</sup> < 0, the resonant wavelength is obtained for reflection of a mode of index *neff*1 into reflection of a mode of index *neff*2 as defined by the relation:

$$\mathcal{A} = \left(\boldsymbol{n}\_{\boldsymbol{c}\|\boldsymbol{1}} + \boldsymbol{n}\_{\boldsymbol{c}\|\boldsymbol{2}}\right) \cdot \boldsymbol{\Lambda} \tag{13}$$

Normally, the two counter propagating fiber modes have propagation constants with the same absolute value and the following relation is defined:

$$
\mathfrak{n}\_{\text{eff}} = \mathfrak{n}\_{\text{eff}1} = \mathfrak{n}\_{\text{eff}2} \tag{14}
$$

From (14), the familiar result for Bragg reflection peak wavelength is obtained:

$$
\mathcal{A}\_{\rm B} = \mathcal{Z} \cdot \boldsymbol{n}\_{\rm eff} \cdot \boldsymbol{\Lambda} \tag{15}
$$

In **Figure 3**, the diffraction is schematically presented by a transmission of a fiber core mode with a bounce angle *θ*1 on the grating into a cladding co-propagating fiber mode with an angle *θ*<sup>2</sup> [13, 14, 20–28]. Since, in the case illustrated in **Figure 1**, both incident core and transmitted cladding fiber modes propagate in positive +*z* direction, it follows that *β* ‼ 0. As a consequence, Eq. (9) predicts the resonant wavelength of an absorption peak for a trans‐ mission grating as follows:

$$\mathcal{A} = \left(\boldsymbol{\mathfrak{n}}\_{\mathfrak{gl}^1} - \boldsymbol{\mathfrak{n}}\_{\mathfrak{gl}^2}\right) \cdot \boldsymbol{\Lambda} \tag{16}$$

The Bragg condition required for a (fundamental) mode to couple to another mode (backward propagating or forward propagating) results from two requirements [20, 21]:


$$
\vec{K} + \vec{k}\_l = \vec{k}\_f \tag{17}
$$

$$K = \left| \vec{K} \right| = \frac{2\pi}{\Lambda} \tag{18}$$

Coupled-mode theory is used for quantitative information about the diffraction efficiency and spectral characteristics of fiber gratings by assuming the approximation of a weakly guiding fiber [13, 14, 36, 37]. Implicitly, it is assumed that the propagating fiber modes have slowly varying along the *z* direction amplitudes. Also it is assumed that a fiber mode has a transverse component of the electric field defined as a superposition of the modes labeled "*j*" traveling in the +*z* and −*z* directions such that [13, 20, 36]

$$\vec{E}\_{\text{i}}(\mathbf{x}, y, z, t) = \sum\_{\text{j}} \left[ A\_{\text{j}}(\mathbf{z}) \exp(i\beta\_{\text{j}}z) + B\_{\text{j}}(\mathbf{z}) \exp(-i\beta\_{\text{j}}z) \right] \cdot \vec{e}\_{\text{j}}(\mathbf{x}, y) \exp(-i\alpha t) \tag{19}$$

where *Aj* (*z*) and *Bj* (*z*) are slowly varying amplitudes of the *j* th mode, respectively. Eq. (19) describes the transverse mode electric fields of the bound core or radiation *LPil* modes, as given in [8], or of the cladding modes. Into an ideal uniform optical fiber, the modes are orthogonal and hence do not exchange energy; the presence of a dielectric perturbation such as a grating causes the modes to be coupled such that the amplitudes *Aj* (*z*) and *Bj* (*z*) of the *j*th mode evolution along the *z* axis are defined according to

$$\frac{dA\_{\rangle}}{dz} = i \sum\_{k} A\_{k} \left( K\_{k\rangle}^{\prime} + K\_{k\rangle}^{z} \right) \exp\left[ i \left( \mathcal{J}\_{k} - \mathcal{J}\_{\rangle} \right) z \right] + i \sum\_{k} B\_{k} \left( K\_{k\rangle}^{\prime} - K\_{k\rangle}^{z} \right) \exp\left[ -i \left( \mathcal{J}\_{k} + \mathcal{J}\_{\rangle} \right) z \right] \tag{20}$$

$$\frac{d\mathcal{B}\_{\rangle}}{dz} = -i\sum\_{k} A\_{k} \left( K\_{\mathbb{k}\rangle}^{\prime} - K\_{\mathbb{k}\rangle}^{z} \right) \exp\left[ i \left( \mathcal{J}\_{k} + \mathcal{J}\_{\uparrow} \right) z \right] - i \sum\_{k} B\_{k} \left( K\_{\mathbb{k}\rangle}^{\prime} + K\_{\mathbb{k}\rangle}^{z} \right) \exp\left[ -i \left( \mathcal{J}\_{k} - \mathcal{J}\_{\uparrow} \right) z \right] \tag{21}$$

In (20) and (21), two coupling coefficients are introduced: transverse and longitudinal. The transverse coupling coefficient between modes *k* and *j* is given by

$$K\_{\mathbb{W}}^{\prime}\left(\mathbf{z}\right) = \frac{\alpha}{4} \iint\_{\boldsymbol{\alpha}} d\boldsymbol{x} d\boldsymbol{y} \Delta\boldsymbol{\varepsilon}\left(\mathbf{x}, \mathbf{y}, \mathbf{z}\right) \cdot \vec{e}\_{\boldsymbol{\alpha}}\left(\mathbf{x}, \mathbf{y}\right) \cdot \vec{e}\_{\boldsymbol{\mu}}^{\ast}\left(\mathbf{x}, \mathbf{y}\right) \tag{22}$$

where Δ ε is the perturbation to the electric permittivity, which has a very small value in the weakly guiding approximation. When *δn* < < *n*, the Δ ε perturbation can be approximated as *Δε* ≅2*nδn*. The longitudinal coupling coefficient *Kkj <sup>z</sup>*(*z*) is analogous to *Kkj <sup>t</sup>* (*z*), but for slow longitudinally varying fiber modes approximation, the condition *Kkj <sup>z</sup>*(*z*)< <sup>&</sup>lt; *Kkj <sup>t</sup>* (*z*) is fulfilled and thus this coefficient is usually neglected.

In most fiber gratings, the induced index change *δn* (*x, y, z*)is approximately considered as uniform across the core and nonexistent outside the core. Thus, it becomes possible to define index by an expression similar to Eq. (1), but with *δneffmean*(*z*) replaced by *δnco*(*z*). As a conse‐ quence, it becomes convenient to define two new coefficients [13, 14, 20]

$$
\sigma\_{kj}\left(\mathbf{z}\right) = \frac{\alpha \mathfrak{m}\_{\alpha}}{2} \cdot \delta n\_{\text{comam}}\left(\mathbf{z}\right) \iint\_{\text{core}} d\mathbf{x} dy \cdot \vec{e}\_{\text{el}}\left(\mathbf{x}, y\right) \cdot \vec{e}\_{\text{\tiny $\mu$ }}^{\*}\left(\mathbf{x}, y\right) \tag{23}
$$

$$
\kappa\_{\boldsymbol{k}\boldsymbol{\up}}\left(\boldsymbol{z}\right) = \frac{\boldsymbol{\upupsilon}}{2} \sigma\_{\boldsymbol{k}\boldsymbol{\up}}\left(\boldsymbol{z}\right) \tag{24}
$$

where σ is a "DC" (period-averaged) coupling coefficient and κ is an "AC" coupling coefficient, then the general coupling coefficient can be written as follows:

$$K\_{kj}'(z) = \sigma\_{kj}\left(z\right) + 2\kappa\_{kj}\left(z\right) \cdot \cos\left[\frac{2\pi}{\Lambda}z + \varphi\left(z\right)\right] \tag{25}$$

Eqs. (20)–(23) are the coupled-mode equations forming a set used to describe fiber grating spectra below.

#### **2.1. FBG reflection spectra**

*Kk k i f* + = r r r

<sup>2</sup> *K K*

in the +*z* and −*z* directions such that [13, 20, 36]

98 Modeling and Simulation in Engineering Sciences

*j*

where *Aj*

*dA*

*dz*

*dB*

*dz*

(*z*) and *Bj*

p= = <sup>L</sup>

Coupled-mode theory is used for quantitative information about the diffraction efficiency and spectral characteristics of fiber gratings by assuming the approximation of a weakly guiding fiber [13, 14, 36, 37]. Implicitly, it is assumed that the propagating fiber modes have slowly varying along the *z* direction amplitudes. Also it is assumed that a fiber mode has a transverse component of the electric field defined as a superposition of the modes labeled "*j*" traveling

*<sup>t</sup>* ( ,,, ) *<sup>j</sup>*( )exp( *j j* ) ( )exp( *j jt* ) ( , exp ) ( )

( )exp ( ) ( )exp ( ) *<sup>j</sup> t z t z k kj kj k j k kj kj k j*

( )exp ( ) ( )exp ( ) *<sup>j</sup> t z t z k kj kj k j k kj kj k j*

*i AK K i z i BK K i z*

In (20) and (21), two coupling coefficients are introduced: transverse and longitudinal. The

( ) ( ) ( ) ( ) \* ,, , , <sup>4</sup> *<sup>t</sup> K z dxdy x y z e x y e x y kj kt jt*

where Δ ε is the perturbation to the electric permittivity, which has a very small value in the weakly guiding approximation. When *δn* < < *n*, the Δ ε perturbation can be approximated as

e

=- - + - + - - éù é ù

*i AK K i z i BK K i z*

å å ëû ë û (20)

å å ëû ë û (21)

= D ×× òò r r (22)

*<sup>z</sup>*(*z*) is analogous to *Kkj*

= + - + - -+ éù é ù

 + -× - åë û <sup>r</sup> <sup>r</sup> (19)

describes the transverse mode electric fields of the bound core or radiation *LPil* modes, as given in [8], or of the cladding modes. Into an ideal uniform optical fiber, the modes are orthogonal and hence do not exchange energy; the presence of a dielectric perturbation such as a grating

*E xyzt A z i z B z i z e xy i t* = é ù bb

(*z*) are slowly varying amplitudes of the *j*

causes the modes to be coupled such that the amplitudes *Aj*

*k k*

*k k*

transverse coupling coefficient between modes *k* and *j* is given by

w

*Δε* ≅2*nδn*. The longitudinal coupling coefficient *Kkj*

¥

b b

b b

evolution along the *z* axis are defined according to

<sup>r</sup> (18)

 w

(*z*) and *Bj*

b b

b b

th mode, respectively. Eq. (19)

(*z*) of the *j*th mode

*<sup>t</sup>* (*z*), but for slow

(17)

In the FBG case, the dominant interaction in the fiber grating is the reflection of a mode *A*(*z*) into an identical counter-propagating mode; at the Bragg resonance wavelength, Eqs. (20) and (21) are simplified by retaining only terms that involve the particular modes [13], neglecting terms on the right-hand sides of Eqs. (20) and (21) that contain rapidly oscillating *z* dependence, since these have low contributions to the variations of the mode amplitude. The resulting equations can be written as follows:

$$\frac{d\mathcal{R}(z)}{dz} = i\hat{\sigma}\mathcal{R}(z) + i\kappa\mathcal{S}(z) \tag{26}$$

$$\frac{d\mathcal{S}(z)}{dz} = -i\hat{\sigma}\mathcal{S}(z) - i\kappa^\*\mathcal{R}(z) \tag{27}$$

In (26) and (27), as a starting hypothesis, it is assumed that *R*(*z*) and *S*(*z*) are defined as follows:

$$R(z) \equiv A\left(z\right) \cdot \exp\left(i\delta z - \frac{\rho}{2}\right) \tag{28}$$

$$\mathcal{S}\left(z\right) \equiv \mathcal{B}\left(z\right) \cdot \exp\left(-i\delta z + \frac{\varphi}{2}\right) \tag{29}$$

In these equations, the "AC" coupling coefficient from (23) and the general "DC" self-coupling coefficient appear. The "AC" coupling coefficient is defined as follows:

$$
\hat{\sigma} \equiv \delta + \sigma - \frac{1}{2} \frac{d\rho}{dz} \tag{30}
$$

The detuning δ, considered independent of *z* for all gratings, is defined as follows:

$$\mathcal{S} \equiv \beta - \frac{\pi}{\Lambda} = \beta - \beta\_{\mathcal{D}} = 2\pi n\_{\text{eff}} \left( \frac{1}{\lambda} - \frac{1}{\lambda\_{\mathcal{D}}} \right) \tag{31}$$

where *λ<sup>D</sup>* is the "design wavelength" for Bragg scattering by an infinitesimally weak grating with a period *δneff* →. The "design wavelength" *λD* is defined as follows:

$$\mathcal{A}\_{\rm D} = \mathcal{D} n\_{\rm eff} \Lambda \tag{32}$$

When *δ* = 0, *λD* fulfill the Bragg condition, i.e. the following relation is accomplished

$$
\mathcal{A}\_{\mathsf{D}} = \mathcal{A}\_{\mathsf{B}} \tag{33}
$$

The "DC" coupling coefficient σ is defined in Eq. (23). Absorption loss in the grating is described by a complex coefficient *σ* ^. The power loss coefficient *α* is the proportional to imaginary part of the complex coefficient *σ* ^, being defined as follows:

$$a = 2\operatorname{Im}(\sigma) \tag{34}$$

Light not reflected by the grating experiences a transmission loss *TL* expressed in dB/cm as follows:

( ) ( ) exp <sup>2</sup> *Rz Az iz*

( ) ( ) exp <sup>2</sup> *Sz Bz iz*

<sup>1</sup> <sup>ˆ</sup> <sup>2</sup>

The detuning δ, considered independent of *z* for all gratings, is defined as follows:

sds

coefficient appear. The "AC" coupling coefficient is defined as follows:

p

 bb

with a period *δneff* →. The "design wavelength" *λD* is defined as follows:

db

100 Modeling and Simulation in Engineering Sciences

described by a complex coefficient *σ*

imaginary part of the complex coefficient *σ*

º × -+ ç ÷

In these equations, the "AC" coupling coefficient from (23) and the general "DC" self-coupling

*d dz* j

1 1 <sup>2</sup> *D eff*

l l

*n*

 p

where *λ<sup>D</sup>* is the "design wavelength" for Bragg scattering by an infinitesimally weak grating

æ ö º- =- = - ç ÷ <sup>L</sup> è ø

2 *D eff* l

When *δ* = 0, *λD* fulfill the Bragg condition, i.e. the following relation is accomplished

*D B* l l

a

The "DC" coupling coefficient σ is defined in Eq. (23). Absorption loss in the grating is

 s *D*

j dæ ö º× - ç ÷

> j dæ ö

è ø (28)

è ø (29)

º+- (30)

= L *n* (32)

= (33)

^. The power loss coefficient *α* is the proportional to

= 2Im( ) (34)

^, being defined as follows:

(31)

$$TL = 10 \cdot \log\_{10} \langle e \rangle a \tag{35}$$

The derivative describes possible chirp of the grating period, where φ(*z*) is defined using different variation laws. For a single-mode Bragg reflection grating, the following simple relations are useful:

$$
\sigma = \frac{2\pi}{\mathcal{N}} \delta n\_{cflmax} \tag{36}
$$

$$
\kappa = \kappa^\* \tag{37}
$$

$$
\hat{\sigma} = \frac{\pi}{\mathcal{N}} v \delta n\_{\text{effmax}} \tag{38}
$$

If the grating is uniform along *z*, then *δneffmean* is a constant, meaning no chirping of the grating, consequently *<sup>d</sup><sup>φ</sup> dz* =0, and thus *κ*, *σ* and *σ* ^ are constants. Thus, Eqs. (26) and (27) form a system of coupled first-order ordinary differential equations with constant coefficients and appropri‐ ate boundary conditions for which closed-form solutions can be found. As the boundary conditions, for a grating of length *L*, it is assumed that a forward-going wave incident from *z* → -∞, the grating reflectivity, is unitary, *R*(*z*=-*L*/2)=1, and that no backward-going wave exists for *z* larger or equal to *L*/2, *S*(*z*=*L*/2)=0. The amplitude and power reflection coefficients ρ and *r*, respectively, can then be shown to be defined as

$$\rho = \frac{-\kappa \sinh\left(L\sqrt{\kappa^2 - \hat{\sigma}^2}\right)}{\hat{\sigma}\sinh\left(L\sqrt{\kappa^2 - \hat{\sigma}^2}\right) + i\sqrt{\kappa^2 - \hat{\sigma}^2}\cosh\left(L\sqrt{\kappa^2 - \hat{\sigma}^2}\right)}\tag{39}$$

$$r = \frac{\sinh^2\left(L\sqrt{\kappa^2 - \hat{\sigma}^2}\right)}{\cosh^2\left(L\sqrt{\kappa^2 - \hat{\sigma}^2}\right) - \frac{\hat{\sigma}^2}{\kappa^2}}\tag{40}$$

From (40), it is found that the maximum reflectivity for a Bragg grating, *r*max, is defined as

$$r\_{\text{max}} = \tanh^2 \left( \kappa L \right) \tag{41}$$

This value occurs when *σ* ^ =0, or at the wavelength *λ*max, which is defined as

$$\mathcal{A}\_{\text{max}} = \left( 1 + \frac{\delta \mathfrak{m}\_{effmax}}{n\_{\text{eff}}} \right) \mathcal{A}\_{\text{D}} \tag{42}$$

#### **2.2. LPG transmission spectra**

In the LPG case, the coupled-mode equations are rearranged in the sense that near the peak resonance wavelength at which mode "1" of amplitude A1(z) is strongly coupled to a copropagating mode "2" with amplitude A2(z), Eqs. (20) and (21) may be simplified by keeping only terms that involve the amplitudes of these two modes and then making use of the synchronous approximation of modes. The resulting equations can be written as follows:

$$\frac{d\mathcal{R}(z)}{dz} = i\hat{\sigma}\mathcal{R}(z) + i\kappa\mathcal{S}(z) \tag{43}$$

$$\frac{dS(z)}{dz} = -i\hat{\sigma}S(z) + i\kappa^\*R(z)\tag{44}$$

where the new amplitudes R(z) and S(z) are defined as

$$R\left(z\right) \equiv A\_1 \exp\left[-i\frac{\left(\sigma\_{11} + \sigma\_{22}\right)z}{2}\right] \exp\left(i\delta z - \frac{\rho}{2}\right) \tag{45}$$

$$S(z) \equiv A\_z \exp\left[-i\frac{(\sigma\_{11} + \sigma\_{22})z}{2}\right] \exp\left(-i\delta z + \frac{\sigma}{2}\right) \tag{46}$$

and where 11 and 22 are "DC" coupling coefficients [13, 14]. From Eqs. (36), (37) and (38), the "AC" cross-coupling coefficient, κ, and, *σ* ^, a general "DC" self-coupling coefficient are defined as

$$
\boldsymbol{\kappa} = \boldsymbol{\kappa}\_{21} = \boldsymbol{\kappa}\_{12}^\* \tag{47}
$$

and

( ) <sup>2</sup>

^ =0, or at the wavelength *λ*max, which is defined as

*D*

 l

(41)

(43)

\* =- + (44)

j d

j d

^, a general "DC" self-coupling coefficient are defined

\* = = (47)

(42)

(45)

(46)

k

*eff*

In the LPG case, the coupled-mode equations are rearranged in the sense that near the peak resonance wavelength at which mode "1" of amplitude A1(z) is strongly coupled to a copropagating mode "2" with amplitude A2(z), Eqs. (20) and (21) may be simplified by keeping only terms that involve the amplitudes of these two modes and then making use of the synchronous approximation of modes. The resulting equations can be written as follows:

> ( ) <sup>ˆ</sup> ( ) ( ) *dR z i Rz iSz*

= + s

( ) <sup>ˆ</sup> ( ) ( ) *dS z i Sz i Rz*

( ) ( 11 22 ) <sup>1</sup> exp exp 2 2

( ) ( 11 22 ) <sup>2</sup> exp exp 2 2

é ù <sup>+</sup> æ ö º -ê ú ç ÷ - +

and where 11 and 22 are "DC" coupling coefficients [13, 14]. From Eqs. (36), (37) and (38), the

21 12

 k

*Sz A i i z* s s

kk

é ù <sup>+</sup> æ ö º -ê ú ç ÷ -

*Rz A i i z* s s

s

 k

 k

*z*

ê ú è ø ë û

*z*

ê ú è ø ë û

*dz*

*dz*

where the new amplitudes R(z) and S(z) are defined as

"AC" cross-coupling coefficient, κ, and, *σ*

as

*n n* d

æ ö = + ç ÷ è ø

max *r L* = tanh

max <sup>1</sup> *effmean*

l

This value occurs when *σ*

102 Modeling and Simulation in Engineering Sciences

**2.2. LPG transmission spectra**

$$
\hat{\sigma} = \delta + \frac{\sigma\_{11} - \sigma\_{22}}{2} - \frac{1}{2} \frac{d\varphi}{dz} \tag{48}
$$

Here the detuning, , which is assumed to be constant along z, is defined as

$$\delta \equiv \frac{1}{2} (\beta\_1 - \beta\_2) - \frac{\pi}{\Lambda} \tag{49}$$

or

$$\mathcal{S} \equiv \pi \Delta n\_{\mathcal{eff}} \left( \frac{1}{\mathcal{A}} - \frac{1}{\mathcal{A}\_{\mathcal{D}}} \right) \tag{50}$$

In Eqs.(49) and (50), λD is the design wavelength for an infinitesimally weak grating; as for Bragg gratings, λD is defined as follows:

$$
\mathcal{A}\_{\rm D} \equiv \Delta n\_{\rm eff} \,\Lambda \tag{51}
$$

As for the Bragg grating case, δ = 0 corresponds to the grating resonance condition predicted by the qualitative picture of grating diffraction, schematically presented in **Figures 2** and **3**.

In the usual case of a uniform grating, *σ* ^ and κ are constants. In the LPG case, unlike for a Bragg grating reflection of a single mode, here the coupling coefficient generally cannot be simply defined as in (38). For coupling between two different modes in the LPG case of transmission gratings, the overlap integrals (23) and (24) must be evaluated numerically. Like the analogous Bragg grating, Eqs. (43) and (44) are coupled first-order ordinary differential equations with constant coefficients. Thus, closed-form solutions can be found when appropriate initial conditions are specified for a grating of length L. The transmission can be found by assuming only one mode is incident from z → - ∞, and assuming that *R*(0) = 1 and *S*(0) = 0. The power bar and cross-transmission, *t*=and *t*×, respectively, can be defined as follows:

$$t\_- = \frac{\left|\mathcal{R}\left(z\right)\right|^2}{\left|\mathcal{R}\left(0\right)\right|^2} = \cos\left(z\sqrt{\kappa^2 + \hat{\sigma}^2}\right) + \frac{\hat{\sigma}^2}{\hat{\sigma}^2 + \kappa^2}\sin\left(z\sqrt{\kappa^2 + \hat{\sigma}^2}\right) \tag{52}$$

$$t\_{\times} = \frac{\left|\mathcal{S}(z)\right|^2}{\left|\mathcal{R}(0)\right|^2} = \frac{\kappa^2}{\hat{\sigma}^2 + \kappa^2} \sin\left(z\sqrt{\kappa^2 + \hat{\sigma}^2}\right) \tag{53}$$

The maximum cross-transmission (which occurs when *σ* ^ =0) is defined as

$$t\_{\times, \text{max}} = \sin^2 \left( \kappa L \right) \tag{54}$$

and it occurs at the wavelength

$$\lambda\_{\text{max}} = \frac{1}{1 - \left(\sigma\_{11} - \sigma\_{22}\right) \frac{\Lambda}{2\pi}} \lambda\_{\text{p}} \tag{55}$$

For coupling between a core mode "1" and a cladding mode "2" with an induced index change in the core only, σ11 = σ from Eqs. (36), σ22 ≪ σ11and for low cladding confinement factor, λmax can be approximated as

$$\mathcal{A}\_{\text{max}} \equiv \left( 1 + \frac{\delta \mathfrak{m}\_{effmax}}{\Delta \mathfrak{m}\_{eff}} \right) \mathcal{A}\_{\text{D}} \tag{56}$$

In Eq. (56), it is assumed that δ*neffmean*, the induced change in the core-mode effective index, is much smaller than Δ*neff* which is the common case. Analyzing Eqs. (56) and (42), a major difference is observed between the FBG and LPG cases, difference which consists in the fact that the wavelength of maximum coupling in a long-period cladding-mode coupler grating shifts toward longer wavelengths as the grating is being written many times more rapidly, meaning a longer grating period, than the shift occurring in the Bragg grating case.

#### **3. Simulation results**

Because of their various and important sensing and communication applications, the FBGs and LPGs are intensively studied in the last 20 years. Since the first reported results concerning their characteristics, fabrication and engineering their applications, in time, it became more and more clear that FBG and LPG simulation models are urgently needed for a proper design of their applications, especially the sensing ones. The design of FBG's and LPG's sensing applications involves a large number of input parameters or parameters having large variation domains. In time, more or less accurate FBG and LPG simulation models were reported in literature [13, 14, 17, 20-28, 36-40]. These FBG and LPG simulation models are crucial for design of their applications.

( )

*t z*

s k´ == +

k

,max *t L* sin

( ) max

s s

max <sup>1</sup> *effmean*

meaning a longer grating period, than the shift occurring in the Bragg grating case.

1

l

l

11 22

For coupling between a core mode "1" and a cladding mode "2" with an induced index change in the core only, σ11 = σ from Eqs. (36), σ22 ≪ σ11and for low cladding confinement factor, λmax

*eff*

In Eq. (56), it is assumed that δ*neffmean*, the induced change in the core-mode effective index, is much smaller than Δ*neff* which is the common case. Analyzing Eqs. (56) and (42), a major difference is observed between the FBG and LPG cases, difference which consists in the fact that the wavelength of maximum coupling in a long-period cladding-mode coupler grating shifts toward longer wavelengths as the grating is being written many times more rapidly,

Because of their various and important sensing and communication applications, the FBGs and LPGs are intensively studied in the last 20 years. Since the first reported results concerning their characteristics, fabrication and engineering their applications, in time, it became more and more clear that FBG and LPG simulation models are urgently needed for a proper design of their applications, especially the sensing ones. The design of FBG's and LPG's sensing applications involves a large number of input parameters or parameters having large variation

*n n* d

æ ö @ + ç ÷ <sup>D</sup> è ø

2

p

*D*

 l

*D*

 l

1

*S z*

*R*

The maximum cross-transmission (which occurs when *σ*

and it occurs at the wavelength

104 Modeling and Simulation in Engineering Sciences

can be approximated as

**3. Simulation results**

( ) ( ) <sup>2</sup> <sup>2</sup>

( ) <sup>2</sup>

k

2 22 sin <sup>ˆ</sup> <sup>ˆ</sup> <sup>0</sup>

2 2

<sup>+</sup> (53)

^ =0) is defined as

´ = (54)

<sup>=</sup> <sup>L</sup> - - (55)

(56)

k s

> However, in spite of the complicated mathematical apparatus defined in Section 2 used for describing the FBG or LPG mode of operation, there are several ideas which a researcher, using or designing FBG and LPG application, has to keep in mind:


Nevertheless, there are several steps to be accomplished in development of an accurate FBG or LPG simulation model, based on which a practical simulation algorithm can be defined. The first steps are identical for FBG and LPG simulation cycles. The first two identical steps of FBG and LPG simulations consist of:


In this stage, the FBG and LPG simulation cycles separate into different ways of evolution. In the FBG case:

**‐** STEP 3 FBG—evaluation of fiber short-period grating reflectivity spectrum in the domain including the Bragg wavelength by solving the system of differential equations defined from coupled-mode theory applied for core counter-propagating modes, i.e. using Eqs. (39)–(42). The obtained reflectivity spectrum will depend on the grating length, period and if it is uniform or has a variable period according to a predefined law on z along the grating (sine, sinc, positive tanh or Blackman) but keeping a constant amplitude of *δneff*, this being the chirping technique, or it is apodized, meaning that the period is constant and the amplitude of *δneff* is defined by a variation law on *z* as the argument (also sine, sinc, positive tanh or Blackman functions are applicable). Once the Bragg grating reflec‐ tivity spectrum is obtained, it is possible to correlate its spectral shift with mechanical or thermal load applied on the FBG device, meaning to use it as a sensor. FBG chirping or apodization is used for smoothing the reflectivity spectrum wings.

In this moment, two ideas have to be underlined:


The LPG case:

Nevertheless, there are several steps to be accomplished in development of an accurate FBG or LPG simulation model, based on which a practical simulation algorithm can be defined. The first steps are identical for FBG and LPG simulation cycles. The first two identical steps of

**‐** STEP 1—the usual gathering of input data, meaning core and cladding diameters, core and cladding refractive index values, to which the environment refractive index has to be added in LPG case and is only informal for FBG model. Any data concerning the geometry of the fiber grating has to be considered in this stage. For example, if fiber grating is

**‐** STEP 2—evaluation of fiber core effective value of the refractive index. This task is achieved by graphically or numerically solving the dispersion Eq. (4) for b and using Eq. (5). For more strictness, the confinement factor can be calculated using (6). The results of Step 2 consist of variation curves with wavelength of core effective refractive index,

In this stage, the FBG and LPG simulation cycles separate into different ways of evolution. In

**‐** STEP 3 FBG—evaluation of fiber short-period grating reflectivity spectrum in the domain including the Bragg wavelength by solving the system of differential equations defined from coupled-mode theory applied for core counter-propagating modes, i.e. using Eqs. (39)–(42). The obtained reflectivity spectrum will depend on the grating length, period and if it is uniform or has a variable period according to a predefined law on z along the grating (sine, sinc, positive tanh or Blackman) but keeping a constant amplitude of *δneff*, this being the chirping technique, or it is apodized, meaning that the period is constant and the amplitude of *δneff* is defined by a variation law on *z* as the argument (also sine, sinc, positive tanh or Blackman functions are applicable). Once the Bragg grating reflec‐ tivity spectrum is obtained, it is possible to correlate its spectral shift with mechanical or thermal load applied on the FBG device, meaning to use it as a sensor. FBG chirping or

**‐** The matrix transfer theory, which is based on dividing the fiber grating into a number of segments of short length (in comparison to the propagating light wavelength), the light propagating through these segments, the output signal (power) of one being the input for the next one and so on, the input of the first segment being the initial conditions for coupled-mode differential equations. Finally, in the matrix transfer theory, an iteration equation is defined, and solving it leads to the reflectivity spectrum. For FBG, coupled-

**‐** Tilted FBG consisting of a Bragg grating with pitches having an angle with z fiber axis represents an attempt to couple the core specific processes to the environment via

cladding. The analysis of tilted FBG is beyond the purposes of this Chapter.

supposed to be bent, or elongated, or longitudinally compressed or not.

normalized frequency *V* and, eventually, of confinement factorΓ.

apodization is used for smoothing the reflectivity spectrum wings.

mode and matrix transfer theories conduct to similar results.

In this moment, two ideas have to be underlined:

FBG and LPG simulations consist of:

106 Modeling and Simulation in Engineering Sciences

the FBG case:


**Figure 4.** The variations of core and clad refractive indices versus propagating radiation wavelength.

In the following, several examples of FBG and LPG optical characteristic simulations devel‐ oped in the above-described steps are presented. In the FBG case, the presented examples are obtained for uniform, chirped or apodized, the grating reflectivity being the main target. For LPG, its transmission characteristics are to be simulated. In the presented examples, simulation was performed considering the geometry and refractive index core and cladding values characteristic for Fibercore SM750 optical fiber (core radius 2.8 μm, *nco* = 1.4575, cladding radius 62.5 μm, *ncl* = 1.4545). Fibercore SM750 optical fiber is commonly used as host for FBG or LPG. For both FBG and LPG sensors design, the first step consists in simulation of core and cladding effective refraction indices of propagation modes. **Figure 4** displays the results obtained in simulation of core and cladding refractive indices variations versus the wavelength of light propagating through the fiber. There are analyzed possible values of refractive index for which the radiation modes can propagate through the optic fiber.

Results obtained in the analyzed FBG cases are presented in **Figures 5**-**10**. The primary task accomplished by simulation consists in defining the variation of the FBG reflectivity with wavelength around the Bragg resonance wavelength. In simulation of chirped and/or apo‐ dized Bragg grating, for its period variation law, sine, sinc, positive tanh and Blackman profiles were considered.

**Figure 5.** Variation of FBG reflectivity versus wavelength for a uniform Bragg grating displaying different grating strength *kL*. Length of the grating *L* = 1 mm, grating visibility *v* = 1, number of grating pitches *N* = 10,000, grating am‐ plitude ∆ *neff* = 1e−4, design wavelength *λD* = 1550 nm.

**Figure 6.** Effect of change in refractive indices on reflection spectra of uniform Bragg gratings. Length of the grating *L* = 1 mm, grating visibility *v* = 1, number of grating pitches *N* = 10,000, grating amplitudes 2206 = 20e−4, 15e−4 and 10e−4, design wavelength *λD* = 1550 nm.

For both FBG and LPG sensors design, the first step consists in simulation of core and cladding effective refraction indices of propagation modes. **Figure 4** displays the results obtained in simulation of core and cladding refractive indices variations versus the wavelength of light propagating through the fiber. There are analyzed possible values of refractive index for which

Results obtained in the analyzed FBG cases are presented in **Figures 5**-**10**. The primary task accomplished by simulation consists in defining the variation of the FBG reflectivity with wavelength around the Bragg resonance wavelength. In simulation of chirped and/or apo‐ dized Bragg grating, for its period variation law, sine, sinc, positive tanh and Blackman profiles

**Figure 5.** Variation of FBG reflectivity versus wavelength for a uniform Bragg grating displaying different grating strength *kL*. Length of the grating *L* = 1 mm, grating visibility *v* = 1, number of grating pitches *N* = 10,000, grating am‐

**Figure 6.** Effect of change in refractive indices on reflection spectra of uniform Bragg gratings. Length of the grating *L* = 1 mm, grating visibility *v* = 1, number of grating pitches *N* = 10,000, grating amplitudes 2206 = 20e−4, 15e−4 and 10e−4,

the radiation modes can propagate through the optic fiber.

plitude ∆ *neff* = 1e−4, design wavelength *λD* = 1550 nm.

design wavelength *λD* = 1550 nm.

were considered.

108 Modeling and Simulation in Engineering Sciences

**Figure 7.** Reflection of a Gaussian profile chirped Bragg grating. Length of the grating *L* = 50 mm, grating visibility *v* = 1, number of grating pitches *N* = 10,000, grating amplitude = 20e−4, design wavelength *λD* = 1550 nm.

**Figure 8.** Apodization of a chirped grating using different profiles. Length of the grating *L* = 50 mm, grating visibility *v* = 1, number of grating pitches *N* = 10,000, grating amplitude ∆ *neff* = 20e−4, design wavelength *λD* = 1550 nm.

**Figure 9.** Variations of the resonance wavelength versus LPG period, calculated for the first 10 possible clad propaga‐ tion modes. LPG length *L* = 75 mm, grating amplitude ∆ *neff* = 25e−4.

**Figure 10.** LPG transmission spectra simulated for an optic fiber in normal state. LPG period = 400 μm, LPG length *L* = 75 mm, grating amplitude ∆ *neff* = 25e−4.

**Figure 11.** LPG transmission spectra simulated for a bent optic fiber in normal state. LPG period = 400 μm, LPG length *L* = 75 mm, grating amplitude ∆ *neff* = 25e−4.

Results obtained in the analyzed LPG cases are presented in **Figures 9**–**11**. In **Figure 9** are presented simulation results obtained regarding the variations of the resonance wavelengths of the absorption peaks in the LPG transmission spectra, peaks defined using (16)-the "second key task to be accomplished" in the design of LPG fiber sensors. In **Figures 10** and **11** are presented the LPG transmission spectra simulated for an unperturbed fiber and for a bent fiber.

#### **4. Conclusions**

The chapter refers to a broad research domain concerning the optic fiber and fiber grating physics. One of the two main purposes of the chapter consists in presenting the theoretical tools and simulation procedures used for analysis of optical properties of short-period FBG and fiber LPG. The second purpose of the chapter consists in providing the basics of simulation models. Examples of simulation results obtained using coupled-mode theory, verified using the transfer matrix theory in the FBG case, are presented. The presented simulation results are in fairly good agreement with experimental and simulation results presented in literature.
