**2. CO2 laser induced LPFG**

The use of CO2 lasers to write LPFGs has emerged as an important alternative. Compared with other LPFG fabrication methods, this irradiation technique provides many advantages, including high thermal stability, more flexibility, lower insertion loss, lower cost. This section addresses the application of the CO2 laser irradiation in creation of LPFG and the physical principles involved in this process.

The first results of the application of the 10.6 μm wavelength radiation emitted by CO2 lasers for the fabrication of LPFG in a conventional fiber were published in 1998 [26-28]. Since then, different experimental methodologies have been described. The most common is the point-topoint technique using a static asymmetrical irradiation with a CO2 laser emitting in a specific mode (continuous wave, CW, or pulsed) and a lens focusing the beam on the fiber. Akiyama [28] used continuous wave emission, while Davis [26] applied laser pulses with powers of about 0.5W.

In Figure 3 it is presented a schematic representation of a typical LPFG fabrication system based on the point-to-point technique employing a CO2 laser. The optical fiber with its buffer stripped is placed on a motorized translation stage. In order to keep the fiber straight during the writing process, a small weight is applied at the end of the fiber and the laser beam is focused on the fiber. In this technique the periodicity of the LPFG writing is accomplished by moving the fiber along its axial direction via a computer controlled translation stage, which also controls the CO2 laser beam emission. A broadband source and an optical spectrum analyzer (OSA) are employed to monitor the evolution of the spectrum during the laser irradiation. This method has the advantage of requiring a simpler setup, although the irradi‐ ation occurs on just one of the sides of the fiber. Also, the translation stage movement can generate vibrations that may be transmitted to the fiber, affecting the quality and repeatability of the LPFG. This problem can be solved using a system where the beam delivery system is moving instead of the fiber [29] or if the fiber is maintained static and a two-dimensional galvanometric mirrors system scans the beam over its surface [29-30]. Some authors reported hybrid methods, combining point-by-point and scanning [31,32]. In this chapter we will consider the method presented by Alves *et al.* [32] that combines a translation stage to move the fiber synchronized with a one-dimensional scan over a cylindrical lens.

In section 2 it will be described the fabrication process for applying the MIR radiation, starting with a review on the use CO2 lasers in the creation of LPFGs. Experimental work is presented as well as the physical principles that may be responsible to induce the periodic refractive

Section 3 will address the subject of simulating the thermal mechanical processes involved in the process. Analytical and numerical models will be analysed and compared. In particular, a 3D finite element method (FEM) model will be presented, including the temperature depend‐

In section 4, it is presented a practical example of writing LPFGs on a Ge-doped fiber using a CO2 laser. A comparison between calculated data and experimental data is made, and future

The use of CO2 lasers to write LPFGs has emerged as an important alternative. Compared with other LPFG fabrication methods, this irradiation technique provides many advantages, including high thermal stability, more flexibility, lower insertion loss, lower cost. This section addresses the application of the CO2 laser irradiation in creation of LPFG and the physical

The first results of the application of the 10.6 μm wavelength radiation emitted by CO2 lasers for the fabrication of LPFG in a conventional fiber were published in 1998 [26-28]. Since then, different experimental methodologies have been described. The most common is the point-topoint technique using a static asymmetrical irradiation with a CO2 laser emitting in a specific mode (continuous wave, CW, or pulsed) and a lens focusing the beam on the fiber. Akiyama [28] used continuous wave emission, while Davis [26] applied laser pulses with powers of

In Figure 3 it is presented a schematic representation of a typical LPFG fabrication system based on the point-to-point technique employing a CO2 laser. The optical fiber with its buffer stripped is placed on a motorized translation stage. In order to keep the fiber straight during the writing process, a small weight is applied at the end of the fiber and the laser beam is focused on the fiber. In this technique the periodicity of the LPFG writing is accomplished by moving the fiber along its axial direction via a computer controlled translation stage, which also controls the CO2 laser beam emission. A broadband source and an optical spectrum analyzer (OSA) are employed to monitor the evolution of the spectrum during the laser irradiation. This method has the advantage of requiring a simpler setup, although the irradi‐ ation occurs on just one of the sides of the fiber. Also, the translation stage movement can generate vibrations that may be transmitted to the fiber, affecting the quality and repeatability of the LPFG. This problem can be solved using a system where the beam delivery system is moving instead of the fiber [29] or if the fiber is maintained static and a two-dimensional galvanometric mirrors system scans the beam over its surface [29-30]. Some authors reported

work towards a full understanding of the physical processes is foreseen.

290 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

index modulation in the fiber's core.

ence of the fiber's main parameters.

**2. CO2 laser induced LPFG**

principles involved in this process.

about 0.5W.

**Figure 3.** Schematic diagram of a typical LPFG fabrication system based on the point-to-point technique using a CO2 laser.

Since the silica glass has strong absorption around the wavelength of the CO2 laser radiation, the beam intensity is gradually attenuated along the incident direction, resulting in asymmetric RI modulation. Such distribution could cause coupling of the core mode to both the symmetric cladding modes and the asymmetric cladding modes [33]. As a result, high fiber grating birefringence and high polarization-dependent loss can become inevitable [30].

In order to solve the birefringence problem, different methods have been proposed. Single– side and symmetric exposures to the laser radiation were compared by Oh *et al.* [34], demon‐ strating that the polarization-dependent loss of the single-side exposure (1.85 dB at 1534 nm) could be significantly reduced to 0.21 dB by applying the second method. Nevertheless, due to its simplicity, the single-size exposure is the most common applied methodology. Grubsky [35] proposed a simple fabrication method for obtaining high-quality LPFGs with a CO2 laser using a reflector to make the fiber's exposure axially uniform. Zhu *et al.* [36] used a high frequency CO2 laser system to write LPFGs with uniform RI modulation by introducing twist strains to the fiber and then exposing the two sides of the fiber to the laser radiation. Experi‐ mental results showed that twisted LPFGs (T-LPFGs) exhibited clear spectra, low insertion losses, and low polarization-dependent losses when compared with those created by exposure to a single-side CO2 laser beam [37].

Gratings inscription was also achieved through the use physical deformations (or geometrical deformation). Wang *et al.* [38] proposed a new method for writing an asymmetric LPFG by means of carving periodic grooves on one side of a fiber with a focused CO2 laser beam. The periodic grooves do not cause a large insertion loss because these grooves are totally confined within the cladding and have no influence on the light transmission in the fiber's core. The grooves enhance the efficiency of grating fabrication and introduce unique optical properties (extremely high strain sensitivity). In 2006, Su *et al.* [39] demonstrated the possibility of producing long-period grating in multimode fibers by deforming the geometry of the fiber periodically with a focused CO2 laser beam and applied them to strain measurements.

Besides conventional single-mode fibers (SMFs), CO2 laser have been used to write LPFGs in other types of fiber, such as boron doped SMF [40,41], and photonic crystal fibers (PCFs) [42,44].

There have been many studies focused on the understanding of the physical mechanisms involved in the CO2 laser writing process for different kinds of fibers. Most of the existing works consider that the main mechanisms responsible for creating a refractive-index change in the CO2 laser irradiation-induced LPFGs are residual stress relaxation processes [26,29,35,40-44]. In these processes, heat created by the absorption of laser energy in the material play an important role and, as will be explained next, modelling the writing process requires considering both thermal and mechanical processes.

#### **3. Thermo-mechanical modelling**

As mentioned before, the main effect behind LPFG fabrication using CO2 lasers is heating, where the refraction index change is achieved by irradiating a fiber submitted to a tensile stress. The high absorption of the glass material to the MIR radiation emitted by these lasers leads to an excess of energy due to the excitation of the lattice which is transformed into heat, increasing the material's temperature from its surface to its bulk by heat conduction. This effect depends on the irradiation time and on the thermal diffusivity of the material, it is localized and periodically induced in the fiber's length, being responsible for the creation of the gratings.

Considering the temperature, *T*, changing with time, *t*, (transient regime) due to the action of a heat source *Q*(*r,t*), the resulting energy balance leads to the heat conduction equation:

$$\left[\frac{\partial\rho}{\partial t} + \rho\nabla\cdot\vec{u}\right]\left[\mathbf{C}\_p\,dT + \rho\mathbf{C}\_p\left(\frac{\partial T}{\partial t} + \vec{u}\cdot\nabla T\right) - \nabla\cdot\mathbf{K}\nabla T = q\left(T\right) + Q\left(r, t\right) \tag{2}$$

where *r* represents the geometric coordinates (depending on the geometry) and being *ρ* the density, *ū* the velocity vector, *Cp* the specific heat and *K* the thermal conductivity. The convec‐ tive and radiative heat flux is represented by [45]:

$$\sigma(T) = h\left(T\_{\text{ext}} - T\right) + \sigma\_{\text{B}}\varepsilon \left(T\_{\text{amb}}{}^{4} - T^{4}\right) \tag{3}$$

being *Text* the external temperature, *Tamb* the ambient temperature, *h* the heat transfer coefficient, *σB* the Stefan-Boltzmann constant and *ε* the surface emissivity.

If enough energy is applied, differences in the thermal expansion coefficients and viscosity of core and cladding lead to residual thermal stresses and draw-induced residual stresses, and refractive index change results from frozen-in viscoelasticity [46]. The analysis of these effects is complex and highly dependent on the physical characteristics of the different materials composing the optical fiber. For simplifying the subject, we will consider silica-based single mode fibers since they are at the base of most LPFGs manufactured using CO2 lasers. Also, from the different irradiation methodologies explained in the previous section, we will consider the coordinate referential illustrated in Figure 4. The main interfaces between regions of interest in the fiber, illustrated in Figure 4(b), are represented by a point in the upper surface (relative to laser incidence), two points in the cladding/core interface and one point in the bottom interface. In order to simplify the plots regarding calculated data in section 4, and since early work demonstrated low variation in temperature between the cladding/core interfaces [47-49], we use the core's middle point instead.

**Figure 4.** (a) Schematic of coordinates used in this work and (b) optical fiber cross-section indicating the considered referential, the interfaces between the different regions, and points of interest: A – irradiated surface, B – core/cladding interface (upper), C – core/cladding interface (lower), D – bottom surface and E – middle point. The origin of the refer‐ ence system is in the middle of the laser line.

#### **3.1. Laser heating modelling**

grooves enhance the efficiency of grating fabrication and introduce unique optical properties (extremely high strain sensitivity). In 2006, Su *et al.* [39] demonstrated the possibility of producing long-period grating in multimode fibers by deforming the geometry of the fiber periodically with a focused CO2 laser beam and applied them to strain measurements.

Besides conventional single-mode fibers (SMFs), CO2 laser have been used to write LPFGs in other types of fiber, such as boron doped SMF [40,41], and photonic crystal fibers (PCFs) [42,44]. There have been many studies focused on the understanding of the physical mechanisms involved in the CO2 laser writing process for different kinds of fibers. Most of the existing works consider that the main mechanisms responsible for creating a refractive-index change in the CO2 laser irradiation-induced LPFGs are residual stress relaxation processes [26,29,35,40-44]. In these processes, heat created by the absorption of laser energy in the material play an important role and, as will be explained next, modelling the writing process

As mentioned before, the main effect behind LPFG fabrication using CO2 lasers is heating, where the refraction index change is achieved by irradiating a fiber submitted to a tensile stress. The high absorption of the glass material to the MIR radiation emitted by these lasers leads to an excess of energy due to the excitation of the lattice which is transformed into heat, increasing the material's temperature from its surface to its bulk by heat conduction. This effect depends on the irradiation time and on the thermal diffusivity of the material, it is localized and periodically induced in the fiber's length, being responsible for the creation of the gratings. Considering the temperature, *T*, changing with time, *t*, (transient regime) due to the action of a heat source *Q*(*r,t*), the resulting energy balance leads to the heat conduction equation:

*p p* ( ) ( , ) *<sup>T</sup> u C dT C u T K T q T Q r t t t*

( ) ( ) ( ) 4 4 *ext B amb qT hT T T T* = -+ s e

where *r* represents the geometric coordinates (depending on the geometry) and being *ρ* the density, *ū* the velocity vector, *Cp* the specific heat and *K* the thermal conductivity. The convec‐

being *Text* the external temperature, *Tamb* the ambient temperature, *h* the heat transfer coefficient,

If enough energy is applied, differences in the thermal expansion coefficients and viscosity of core and cladding lead to residual thermal stresses and draw-induced residual stresses, and

ê ú + Ñ× + + ×Ñ -Ñ× Ñ = + ç ÷ ëû èø ¶ ¶ <sup>ò</sup> r r (2)

(3)

 r

éù æö ¶ ¶

*σB* the Stefan-Boltzmann constant and *ε* the surface emissivity.

tive and radiative heat flux is represented by [45]:

requires considering both thermal and mechanical processes.

292 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

**3. Thermo-mechanical modelling**

r r

> When considering an homogeneous isotropic material, the condition of mass conservation, and introducing the thermal diffusivity *k*, given by *K*/(*ρCp*), Equation (2) can be simplified to:

$$\left(\frac{\partial T}{\partial t} + \vec{u} \cdot \nabla T\right) - k \nabla \cdot \nabla T - q\left(T\right) = \frac{\mathcal{Q}\{r, t\}}{\rho \, \mathbb{C}\_p} \tag{4}$$

For a laser beam incident on a surface and propagating in the *z*-direction:

$$Q(r,t) = a\_r \left(1 - R\right) I(r) \varphi(t) \tag{5}$$

where *aT* is the attenuation coefficient of the material, *R* its reflectance and *I*(*r*) the irradiance. For continuous wave emission with a duration *ton*:

$$\varphi(t) = \begin{cases} ^{0, \text{if } t \le 0 \lor t > t\_{\text{on}}} \\ ^{1, \quad \text{if } 0 < t \le t\_{\text{on}}} \end{cases} \tag{6}$$

Accordingly with the irradiation geometry schematized in Figure 4, and considering that the laser beam has an elliptical Gaussian distribution at the surface being irradiated, then [50]:

$$I(\boldsymbol{r}) = I(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}) = \frac{2a\_r P}{\pi \, r\_x \, r\_y} \exp\left[ -2\left(\frac{\boldsymbol{x}^2}{r\_x^2} + \frac{\boldsymbol{y}^2}{r\_y^2}\right) - a\_r \left|\boldsymbol{z}\right|\right] \tag{7}$$

where *rx* and *ry* are the values of the ellipse's semi-minor and semi-major axis, respectively.

If one neglects radiative and convective losses and considers temperature dependent material properties, then it is possible to obtain analytical expressions to the temperature. Typically [47,50,51], the heat equation is solved numerically using the Green's function method and the temperature can be obtained [49]:

$$\begin{split} T\left(\mathbf{x}, y, z, t\right) &= \frac{\left(1 - R\right)P}{4\pi K r\_{\mathbf{x}} r\_{y}} \int\_{0}^{\sqrt{a\_{l}}} \Psi\left(\mathbf{x}, y, s\right) \cdot \left[\exp\left(a\_{r} \left|z\right|\right) \text{erfc}\left(\frac{a\_{r}s}{2} + \frac{\left|z\right|}{s}\right) + \\ &+ \exp\left(-a\_{l} \left|z\right|\right) \cdot \sqrt{a^{2} + b^{2}} \cdot \text{erfc}\left(\frac{a\_{l}s}{2} - \frac{\left|z\right|}{s}\right) \right] ds \end{split} \tag{8}$$

with

$$\Psi\left(\mathbf{x}, \mathbf{y}, \mathbf{s}\right) = \frac{a\_\Gamma \mathbf{s}}{\mathbf{s}^2} \cdot \exp\left[\frac{\mathbf{x}^2}{r\_x^2 + \mathbf{s}^2} - \frac{\mathbf{y}^2}{r\_y^2 + \mathbf{s}^2} + \frac{\left(a\_\Gamma \mathbf{s}\right)^2}{4}\right] \tag{9}$$

From (8) an on-axis approximation can be used [51] and the temperature rise can be approxi‐ mated through simple analytical expressions:

$$
\Delta T \left( 0, 0, \infty \right) = \frac{\left( 1 - R \right) P}{2 \sqrt{\pi \tau\_x r\_y K}} \tag{10}
$$

for the steady state condition (t >> rx 2 ry 2 / 4k) and

where *aT* is the attenuation coefficient of the material, *R* its reflectance and *I*(*r*) the irradiance.

0, if t 0 t > t 1, if 0 < t t

£ Ú £

Accordingly with the irradiation geometry schematized in Figure 4, and considering that the laser beam has an elliptical Gaussian distribution at the surface being irradiated, then [50]:

*x y x y*

*r r r r*

where *rx* and *ry* are the values of the ellipse's semi-minor and semi-major axis, respectively.

If one neglects radiative and convective losses and considers temperature dependent material properties, then it is possible to obtain analytical expressions to the temperature. Typically [47,50,51], the heat equation is solved numerically using the Green's function method and the

> *k t <sup>T</sup> T*

*R P a s z*


*a z a b erfc ds*

*T T x y*

+ +

*a s x y a s*

*T*

*a s z*

æ öù

*s*

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

é ù

ë è ø

( )

<sup>+</sup> ë û (9)


= = - +- ê ú ç ÷

*on on*

2 2

é ù æ ö

ç ÷ ë û è ø

*T*

(6)

(7)

(8)

( )

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

*a P <sup>x</sup> <sup>y</sup> Ir I xyz a z*

<sup>2</sup> , , exp 2 *<sup>T</sup>*

p

( ) ( ) ( ) ( )

2 2

+ - × +× - ç ÷ú ç ÷ú è øû

<sup>2</sup> 22 22 , , exp <sup>4</sup> <sup>1</sup>

( ) (<sup>1</sup> ) 0,0,

Y = × -+ ê ú

*s rsrs*

From (8) an on-axis approximation can be used [51] and the temperature rise can be approxi‐

2 *x y*

p*rrK*

*R P*

<sup>1</sup> ,,, , , exp 4 2

*T xyzt x y s a z erfc Kr r <sup>s</sup>*

exp <sup>2</sup>

4 0

( )

ò

*T*

( )

*r r*

*x y*

*T*

*x y*

p

( )

mated through simple analytical expressions:

*xys*

ìï = í ïî

j*t*

294 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

For continuous wave emission with a duration *ton*:

temperature can be obtained [49]:

with

$$
\Delta T \left( 0, 0, t \right) = \frac{\left( 1 - R \right) P}{\pi^{3/2} K \sqrt{r\_x} \ r\_y} \tan^{-1} \left( \sqrt{\frac{4kt}{r\_x}} \right) \tag{11}
$$

under transient conditions. Although Yang *et al.* [51] proved that these equations can be used to study the thermal transport in CO2 laser irradiated fused silica, when modelling the effects in writing LPFGs, the full 3D temperature distribution is required and a numerical approach is mandatory [48]. Using a dedicated Finite Element Method (FEM) program, although computationally demanding, allows not only considering all the physical phenomena, but also including the temperature dependence of the materials and even the simulation of consecutive periods. If two laser shots are considered, one centred in (0,0,0) and at *t*=0 s, and the other at *x*=*Δx* and *t*=*t2*=*ton+Δt*. In the latter case (*t*=*t2*), equations (6) and (7) are altered accordingly to

$$\log\left(t^{\prime}\right) = \begin{cases} ^{0, \text{if } t \le 0 \vee t^{\prime} > t\_{\text{on}}} & \text{with } t^{\prime} = t - \left(t\_{\text{on}} + \Delta t\right) \\ ^{1, \text{}} & \text{if } 0 < t^{\prime} < t - \left(t\_{\text{on}} + \Delta t\right) \end{cases} \tag{12}$$

$$I(r) = I(\mathbf{x}, y, z) = \frac{2a\_r P}{\pi r\_x r\_y} \exp\left[ -2\left(\frac{\left(\mathbf{x} - \Delta\mathbf{x}\right)^2}{r\_x^2} + \frac{y^2}{r\_y^2}\right) - a\_r \left|z\right| \right] \tag{13}$$

#### **3.2. Residual elastic stresses modelling**

Residual elastic stresses are considered those that are frozen into the fiber [52] and have an important impact on the production of LPFG since they affect the refractive index of an optical fiber. When using the considered MIR radiation, two categories of residual elastic stresses must be considered: thermal and draw-induced stresses.

#### *3.2.1. Residual thermal stresses*

Residual thermal stresses appear as the optical fiber is cooled down from high temperatures and regions with different thermal expansion coefficients contract differently in time. Dopants introduced increase the differences in viscosity and thermal expansion coefficients.

A solution for the resulting residual thermal stresses of an initially unstressed axisymmetric cylinder heated at a given temperature, *T*, can be obtained using the radial coordinate *r* [52,53]

$$
\sigma\_x = \frac{E}{1-\nu} \left[ \frac{2\nu}{r\_c^2} \int\_{r=0}^{r\_c} aTrdr - aT \right] \tag{14}
$$

$$
\sigma\_r = \frac{E}{1-\nu} \left[ \frac{1}{r\_c^2} \int\_{r=0}^{r\_c} \alpha T r dr - \frac{1}{r^2} \int\_{r=0}^{r} \alpha T r dr \right] \tag{15}
$$

$$
\sigma\_{\theta} = \frac{E}{1-\nu} \left| \frac{1}{r\_c^2} \int\_{r=0}^{r\_c} \alpha T r dr + \frac{1}{r^2} \int\_{r=0}^{r} \alpha T r dr - \alpha T \right| \tag{16}
$$

being *rc* the radius (cladding or core), *E* the Young's modulus and υ the Poisson's ratio. This solution is valid when the elastic properties can be considered constant, which doesn't applies for optical fibers. Thus, and similarly to the approximated solution for heating, given by equations (8) and (9), not taking in consideration the temperature dependence of the different parameters can lead to non-accurate results. Again, the use of numerical methods, with particular focus on FEM, are appropriated and give the opportunity to combine both heating and thermal stresses models. In this case, by solving equation (4), the thermally-induced residual stresses, σT, can be obtained considering the constitutive equations for a linear isotropic thermoelastic material and the stress tensor obtained.

#### *3.2.2. Draw-induced stresses*

Residual stress effects on the refractive indices of fibers were reported for the first time by Hibido *et al.* in 1987 [54,55] regarding undoped silica-core single mode fibers. Considering a fiber with core and cladding, having different viscosities due to different dopants concentra‐ tions, during the draw process the higher viscosity glass will solidify first and support the draw tension. The low-viscosity glass solidifies conforming with the elastically stretched highviscosity glass. Then, when the draw force is released at room temperature, the high-viscosity glass cannot contract due to the already solidified low-viscosity glass. The resulting residual axial elastic stresses can be expresses as

$$
\sigma\_{x,1} = F\left(\frac{E\_1}{A\_1 E\_1 + A\_2 E\_2}\right) \tag{17}
$$

and

$$
\sigma\_{x,2} = \frac{F}{A\_2} \left( \frac{A\_1 E\_1}{A\_1 E\_1 + A\_2 E\_2} \right) \tag{18}
$$

being *F* the draw tension, *A* the cross-section area and *E* the Young's module for the considered regions. The indexes 1 and 2 in these equations represent the regions of low-viscosity and highviscosity glasses, respectively.

Similarly to these stresses, frozen-in inelastic strains were also found if a fiber is rapidly cooled to room temperature while under tension (as in the considered case) [46,52]. Using the equivalent elastooptic relations [52], the isotropic perturbation on the refractive, *Δn*, index is

$$
\Delta \mathfrak{m} = -\frac{\mathfrak{m}^3}{6} \mathcal{X} \left( T\_{\mathbb{F}} \right) p \,\sigma \tag{19}
$$

In the later expression, *n* is the nominal refractive index, *χ*(*TF*) is the relaxation compressibility at the fictive temperature, *TF*, *p* is the orientation average elastooptic coefficient, and σ the overall residual stresses (in MPa) in the fiber's axial direction. Accordingly with Yablon [52], stresses in the other directions can be neglected.

Besides stress-related refractive index change, localized heating can induce micro-deformation of the fiber and changes in the glass structure. The latter is likely to occur in the core for which the fictive temperature (the glass structure doesn't change below the fictive temperature) is lower [56,57]. As reported for a Ge-doped core (e.g. the fictive temperature ranges from 1150 K to 1500 K.
