**4. A practical example**

2 2 0 0 1 1

*<sup>E</sup> Trdr Trdr r r*

*r r*

aa

 aa

é ù <sup>=</sup> ê ú - - ë û ò ò (15)

= +- ê ú - ë û ò ò (16)

*c*

= =

2 2 0 0 1 1

*r r*

= =

 a

*r r r r*

*<sup>E</sup> Trdr Trdr T*

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

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

1

æ ö <sup>=</sup> ç ÷ + è ø

*<sup>E</sup> <sup>F</sup> AE AE*

11 22

1 1

2 11 22

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 high-

*F A E A AE AE*

æ ö <sup>=</sup> ç ÷ + è ø

é ù

*r r r c*

*c*

1

n

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

*c*

s

1

n

isotropic thermoelastic material and the stress tensor obtained.

,1

*x*

,2

*x*

s

s

s

*3.2.2. Draw-induced stresses*

and

axial elastic stresses can be expresses as

viscosity glasses, respectively.

q

> To illustrate the application of the theory and also correlate it with experimental data, we will consider a common example of LPFG writing using NIR radiation: a standard single-mode fiber, SMF-28 [58], consisting of a core of 3.5 mol% Ge-doped SiO2, is irradiated by a CW 10.6 μm wavelength CO2 laser. For fused silica, *n* ≈ 1.45 (in the near-infrared), *χ*(*TF*) ≈ 0.0568 GPa-1 and *p* ≈ 0.22, which allows simplifying equation (19):

$$
\Delta\mathfrak{u} \approx -6.35 \times 10^{-8} \,\sigma \tag{20}
$$

Also, in this case, equations (17) and (18) represent, respectively, the residual axial elastic stresses at the core (low-viscosity glass) and the cladding (high-viscosity glass) [46].

The particular conditions used in both simulation and experimental works, as well as the obtained results, will be detailed next.

#### **4.1. Simulation**

(17)

(18)

The simulation of the writing process was made implementing a 3D FEM model using the COMSOL Multiphysics program. Whenever relevant, the material's dependence with temperature was considered and the proper geometry and FEM parameters defined.

Besides COMSOL, two other programs were used: Matcad and a simulation tool developed in MatLab by Baptista [60] based on the three layer model developed by Erdogran [14,60]. The latter was used to apply the refractive index data obtained in the FEM model to simulate the transmission spectrum of a LPFG. Matcad was used to solve equations (8) and (11) and compare the temperature data with that obtained through the FEM model.

#### *4.1.1. Fiber's characteristics*

When light interacts with matter, one of the main parameters is the absorption coefficient, aT. As it can be deduced from the formulae of section 3, it plays a major role in the process of heating the fiber. Besides its dependence with the wavelength of the light, it varies with temperature. This variation is important and for the 10.59 μm CO2 laser wavelength (λ1), within 298 K–2,073 K temperature range can be obtained by [61]:

$$a\_r \left( T \right) = \frac{4\pi}{\lambda\_\uparrow} \left[ 1.82 \times 10^{-2} - 10.1 \times 10^{-5} \left( T - 273.15 \right) \right] \tag{21}$$

For the thermal conductivity, heat capacity, density and emissivity, the temperature depend‐ ence was modelled using native COMSOL functions for a Corning fused silica glass (7940) [48]. The doping effect on most of the parameters was disregarded mainly because the Ge concen‐ tration in the fiber's core is very low [62]. However, for the Young's modulus and Poisson's ratio (Figure 5), the function behaviour was extrapolated [63]. Also, both the heat transfer coefficient and reflectivity were considered constant and equals to 418.68 W m−2 K−1 [45] and 0.15 [51], respectively.

**Figure 5.** Variation of (a) Young's module and (b) Poisson's ratio with temperature for both fused silica (from COM‐ SOL materials library) cladding and Ge-doped fused silica (extrapolated) core glasses.

### *4.1.2. Implementation*

latter was used to apply the refractive index data obtained in the FEM model to simulate the transmission spectrum of a LPFG. Matcad was used to solve equations (8) and (11) and

When light interacts with matter, one of the main parameters is the absorption coefficient, aT. As it can be deduced from the formulae of section 3, it plays a major role in the process of heating the fiber. Besides its dependence with the wavelength of the light, it varies with temperature. This variation is important and for the 10.59 μm CO2 laser wavelength (λ1), within

( ) ( ) 2 5


For the thermal conductivity, heat capacity, density and emissivity, the temperature depend‐ ence was modelled using native COMSOL functions for a Corning fused silica glass (7940) [48]. The doping effect on most of the parameters was disregarded mainly because the Ge concen‐ tration in the fiber's core is very low [62]. However, for the Young's modulus and Poisson's ratio (Figure 5), the function behaviour was extrapolated [63]. Also, both the heat transfer coefficient and reflectivity were considered constant and equals to 418.68 W m−2 K−1 [45] and

**Figure 5.** Variation of (a) Young's module and (b) Poisson's ratio with temperature for both fused silica (from COM‐

SOL materials library) cladding and Ge-doped fused silica (extrapolated) core glasses.

ë û (21)

<sup>4</sup> 1.82 10 10.1 10 273.15 *Ta T <sup>T</sup>*

compare the temperature data with that obtained through the FEM model.

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

298 K–2,073 K temperature range can be obtained by [61]:

1

l

p

*4.1.1. Fiber's characteristics*

0.15 [51], respectively.

The physical problem was mathematically solved using the FEM model implemented using the COMSOL Multiphysics 3.5 program to create the transient heat conduction and (mechan‐ ical) stress-strain models under the conditions of this study. In order to introduce some of the complexity of stress-related issues regarding the processing of the optical fibers, the residual axial elastic stresses were implemented considering Equations (9) and (10) and the total resulting stress was obtained adding the thermally-induced residual stresses obtained with the program.

As illustrated in Figure 4, the implemented geometry consists of a set of (concentric) cylinders with radius of curvatures accordingly with the characteristics of the core and cladding of the fiber previously described. To avoid the influence of the external bounda‐ ries on the irradiated and analysed zones, the overall length for the geometry was set as 13 mm. However, to reduce the computational load and loosen the mesh dimensions in the volumes not affected heat source, the cylinders were implemented as three separate sets; the central one, where the laser incidence will be simulated, has a 1.7mm length. The outer set of cylinders are asymmetric since the second laser shot will be simulated just in the positive x-direction.

Table 1 presents the 3D geometry data and the mesh statistics and Figure 6 shows the imple‐ mented mesh, with particular focus on the central irradiation zone. Both outer boundary surfaces are defined as thermally isolated, being one of them fixed. The ambient temperature was considered to be 295 K and equal to the external temperature, Text in Equation (3).


**Table 1.** Geometry data and mesh statistics.

**Figure 6.** Image of the mesh implemented at the central zone of the FEM geometry. The coloured region corresponds to the central geometry with finer mesh (Table 1).

#### **4.2. Experimental methodologies**

The implemented irradiation methodology combines a translation stage to move the fiber synchronized with a one-dimensional scan over a cylindrical lens [32]. Figure 7 shows a schematic of the setup and Figure 8 a picture of its implementation. The light source is a Synrad 48-2 CO2 laser emitting a 3.5 mm diameter CW laser beam with a wavelength of 10.6 μm. Two mirrors direct the beam towards the focusing lens, a ULO Optics ZnSe cylindrical lens (50 mm focal length) which produces a 0.15 mm x 1.75 mm (measured using the knife-edge method [64]) elliptical spot on the fiber with its longer axis perpendicular to the fiber's axis. One of the mirrors is a galvanometric mirror (Cambridge Technology 6860) which allows a scan over the lens (and, consequently, over the surface of the fiber).

A linear translation stage moves axially the fiber so the periodic refractive index change is accomplished. During the process, one of its ends is fixed and a weight (typically ~50 g ± 0.5 g) hangs on the other, thus creating a tension. Thus, this system acts like a XY writing system. The advantage of the translation stage is its long range with micrometric precision (10 cm with a repeatability of 1 μm). However, it has the disadvantage of having relative low speed when compared with a galvanometric system. Combining the two systems, we have the benefit of having a long X-axis range with a fast Y-axis range and, by moving the laser spot very fast over the fiber (Y-axis), lower interaction times can be achieved. Also, since the laser is not always over the fiber, it's possible to have the laser emitting continuously, which prevents the transients during the laser start-up and allows easier control of its power. Uncertainties regarding the irradiation data are: power, ±0.5 W; duration, ±1 ms.

To monitor the LPFG fabrication process, a broad band light source (Thorlabs S5FC1005S) and an OSA (Agilent 86140B) were used. The irradiated zones were analysed using an optical microscope (Zeiss AxioScope A1) with a maximum amplification of 1,000×. The LPFG period is limited by the laser spot size and by the translation stage minimum movement. In the considered setup, since the translation stage can have 1 μm steps, the laser spot size (150 μm) constitutes the major limitation.

**Figure 7.** (a) Top and (b) lateral schematic views of the optical setup. BS – beam splitter; CL-cylindrical lens; GM – galvanometric mirror; M – mirror; OF – optical fiber; PM – power meter; TS – translation stage.

**Figure 8.** Picture of the setup implemented. The inset shows a detail view of the laser output area. BS – beam splitter; CL – cylindrical lens; GM – galvanometric mirror; L – laser; M – mirror; OF – optical fiber; PM – power meter; TS – translation stage.

#### **4.3. Results and analysis**

**Figure 6.** Image of the mesh implemented at the central zone of the FEM geometry. The coloured region corresponds

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

The implemented irradiation methodology combines a translation stage to move the fiber synchronized with a one-dimensional scan over a cylindrical lens [32]. Figure 7 shows a schematic of the setup and Figure 8 a picture of its implementation. The light source is a Synrad 48-2 CO2 laser emitting a 3.5 mm diameter CW laser beam with a wavelength of 10.6 μm. Two mirrors direct the beam towards the focusing lens, a ULO Optics ZnSe cylindrical lens (50 mm focal length) which produces a 0.15 mm x 1.75 mm (measured using the knife-edge method [64]) elliptical spot on the fiber with its longer axis perpendicular to the fiber's axis. One of the mirrors is a galvanometric mirror (Cambridge Technology 6860) which allows a scan over the

A linear translation stage moves axially the fiber so the periodic refractive index change is accomplished. During the process, one of its ends is fixed and a weight (typically ~50 g ± 0.5 g) hangs on the other, thus creating a tension. Thus, this system acts like a XY writing system. The advantage of the translation stage is its long range with micrometric precision (10 cm with a repeatability of 1 μm). However, it has the disadvantage of having relative low speed when compared with a galvanometric system. Combining the two systems, we have the benefit of having a long X-axis range with a fast Y-axis range and, by moving the laser spot very fast over the fiber (Y-axis), lower interaction times can be achieved. Also, since the laser is not always over the fiber, it's possible to have the laser emitting continuously, which prevents the transients during the laser start-up and allows easier control of its power. Uncertainties

To monitor the LPFG fabrication process, a broad band light source (Thorlabs S5FC1005S) and an OSA (Agilent 86140B) were used. The irradiated zones were analysed using an optical microscope (Zeiss AxioScope A1) with a maximum amplification of 1,000×. The LPFG period is limited by the laser spot size and by the translation stage minimum movement. In the considered setup, since the translation stage can have 1 μm steps, the laser spot size (150 μm)

to the central geometry with finer mesh (Table 1).

lens (and, consequently, over the surface of the fiber).

regarding the irradiation data are: power, ±0.5 W; duration, ±1 ms.

**4.2. Experimental methodologies**

constitutes the major limitation.

An example of the temperature distribution resulting from considering 6 W laser power, irradiation duration of 0.6 s and 47 g weight (*F*=0.461 N) is shown in Figure 9(a). The result of simulating a second irradiation (equivalent to have a LPFG period), 1 s after the first, at a distance of 500 μm, is shown in Figure 9(b).

Figure 10 shows the equivalent plots of temperature with time for the irradiated front, core (middle) and the back surfaces (Figure 4), at x=y=0 m and x=500 μm, y=0 m. These plots show that the (spatial) proximity between shots raises the temperature even when they are not under direct irradiation. As the distance reduces (shorter LPFG periods), this secondary heating increases (Figure 11). This is particularly important when the second shot is applied because

**Figure 9.** Temperature distribution in the implemented 3D geometry for the laser irradiation of an optical fiber (*P*=6 W; *ton*=0.6 s; *F*=0.461 N; *Δt*=1 s; *Δx*=500 μm), at (a) *t*=0.6 s and (b) *t*=2.2 s. Colour bar values are in K.

of the possibility of annealing, which could (totally or partially) relieve the previously induced internal stresses.

Similarly, as we analyse the temperature distribution at the fiber's axial direction plotted in Figure 12(a) (for the same conditions of Figure 10), the superposition of thermally affected areas in both shots underlines one critical aspect when writing LPFGs: the influence of the grating's period on the pitch width, and consequent "softening" of the spatial refractive index gradient. As it can be seen in Figure 12(b), as the period decreases, the interception's xcoordinate values decreases and the temperature at that point increases.

The practical consequence of this behaviour it's not clear at this time and further research is necessary, but these phenomena can be responsible by the reduction of the success rate in producing LPFGs with shorter periods found in other studies [65].

The importance of using the FEM simulation instead of the analytical equation (11) or the approximated integral equation (8) is observed in Figure 13 for the temperature at the core (middle). In fact, disregarding radiative and convective losses, adding to the important variation on the parameters values with temperature, deviates the solution by a significant amount (about 200 K at its maximum values, between each solution).

As expected, as the laser power increases, the temperature also increases. From the data obtained using the FEM model and plotted in Figure 14, a 3rd order polynomial can be used to relate these two parameters. Considering

$$T\left(P\right) = a\_3 P^3 + a\_2 P^2 + a\_1 P + a\_0 \tag{22}$$

Writing of Long Period Fiber Gratings Using CO2 Laser Radiation http://dx.doi.org/10.5772/59153 303

**Figure 10.** Plots of the temperature evolution during laser irradiation and cooling at (a) x=0 m and (b) x=Δx. (*P*=6 W; *ton*=0.6 s; *F*=0.461 N; *Δt*=1 s; *Δx*=500 μm).

of the possibility of annealing, which could (totally or partially) relieve the previously induced

**Figure 9.** Temperature distribution in the implemented 3D geometry for the laser irradiation of an optical fiber (*P*=6 W;

Similarly, as we analyse the temperature distribution at the fiber's axial direction plotted in Figure 12(a) (for the same conditions of Figure 10), the superposition of thermally affected areas in both shots underlines one critical aspect when writing LPFGs: the influence of the grating's period on the pitch width, and consequent "softening" of the spatial refractive index gradient. As it can be seen in Figure 12(b), as the period decreases, the interception's x-

The practical consequence of this behaviour it's not clear at this time and further research is necessary, but these phenomena can be responsible by the reduction of the success rate in

The importance of using the FEM simulation instead of the analytical equation (11) or the approximated integral equation (8) is observed in Figure 13 for the temperature at the core (middle). In fact, disregarding radiative and convective losses, adding to the important variation on the parameters values with temperature, deviates the solution by a significant

As expected, as the laser power increases, the temperature also increases. From the data obtained using the FEM model and plotted in Figure 14, a 3rd order polynomial can be used to

( ) 3 2 *T P aP aP aP a* 3 2 10 = + ++ (22)

coordinate values decreases and the temperature at that point increases.

*ton*=0.6 s; *F*=0.461 N; *Δt*=1 s; *Δx*=500 μm), at (a) *t*=0.6 s and (b) *t*=2.2 s. Colour bar values are in K.

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

producing LPFGs with shorter periods found in other studies [65].

amount (about 200 K at its maximum values, between each solution).

relate these two parameters. Considering

internal stresses.

**Figure 11.** Maximum temperature at *x*=0 m when the laser is irradiating at *x*=*Δx* for different values of *Δx* (LPFG peri‐ od). (*P*=6 W; *ton*=0.6 s; *Δt*=1 s; *F*=0.461 N).

(with the laser power, *P*, in W, and the temperature, *T*, in K), Table 2 gives the equivalent parameters for the different zones of the fiber.

For the considered example, at *t*=0.6 s, axial residual thermal stresses values along *z*-axis were determined as having a maximum of about −0.8 MPa. Axial elastic stresses act in the opposite direction and were calculated as being σx,1=35 MPa and σx,2=0.153 MPa. This results in refractive index changes (difference between final and initial values) for the core and cladding of the order of −2 × 10−4 and 4 × 10–6, respectively. Figure 15 shows the calculated (maximum) changes for different laser power. Under the considered conditions, it is clearly observed that for the cladding, a well-defined step occurs between 4.5 W and 5 W. Regarding the core, although its refractive index shows minor variations, it is possible to observe the beginning of the contri‐ bution of thermal stresses around 5 W.

**Figure 12.** (a) Plots of the temperature distribution at the fiber's axial direction simulated for *t*=0.6 s and *t*=2.1 s, with *Δx=*500 μm (the label shows the position, in mm, and the temperature, in K, of the interception between plots), and (b) the variation on the interception point for different values of *Δx* (LPFG period). (*P*=6 W; *ton*=0.6 s; *Δt*=1 s; *F*=0.461 N).

**Figure 13.** Plots of the temperature evolution during laser irradiation and cooling obtained using equations (11), ana‐ lytical, and (8), integral, and the FEM model. (*P*=6 W; *ton*=0.6 s; *F*=0.461 N).

**Figure 14.** Maximum temperature calculated for different laser powers, at *x*=0 m. (*P*=6 W; *ton*=0.6 s; *F*=0.461 N).


**Table 2.** Coefficients for equation (22), depending on the analysis point being considered (Figure 4).

**Figure 12.** (a) Plots of the temperature distribution at the fiber's axial direction simulated for *t*=0.6 s and *t*=2.1 s, with *Δx=*500 μm (the label shows the position, in mm, and the temperature, in K, of the interception between plots), and (b) the variation on the interception point for different values of *Δx* (LPFG period). (*P*=6 W; *ton*=0.6 s; *Δt*=1 s; *F*=0.461 N).

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

**Figure 13.** Plots of the temperature evolution during laser irradiation and cooling obtained using equations (11), ana‐

lytical, and (8), integral, and the FEM model. (*P*=6 W; *ton*=0.6 s; *F*=0.461 N).

Another parameter to take in consideration is the draw force due to the weight applied on the fiber. Figure 16 shows the behaviour of the refractive index change with the applied force (and corresponding weight), considering *P*=6 W and *ton*=0.6 s. This parameter affects mainly the core and a linear relation (with *F* in N) can be obtained:

$$
\Delta \mathfrak{m}\_{\rm com} = \left( -4.9388 \, F + 0.0166 \right) \times 10^{-4} \tag{23}
$$

A microscope photograph of an example of an irradiated fiber under the experimental conditions considered in this work is shown in Figure 17. The imaged zone comprises one 500 μm period of a 25 mm grating irradiated with 6 W, for a duration of 0.6 s (each pulse) and subjected to a weight of 47 g. It is also visible a (small) micrometric deformation of the fiber in each irradiated region. Figure 18 shows the spectral transmission of the written LPFG, comparing the experimental data with the simulated spectrum, obtained using the refractive index changes calculated by the FEM model and using the simulation tool developed by Baptista [59].

In spite the relative spectral transmission data agreement, the experimental work demonstrat‐ ed that different operational parameters can influence the resulting LPFG. In particular, the way the weight is positioned and laser power fluctuations can easily change the final result.

**Figure 15.** Calculated (maximum) refractive index change at the core and cladding for different applied laser powers. (*ton*=0.6 s; *F*=0.461 N).

**Figure 16.** Calculated refractive index change (maximum change for core and cladding) for different applied draw ten‐ sions. (*P*=6 W; *ton*=0.6 s; *F*=0.461 N).

In general, the process involves monitoring the transmission spectrum and iterative action on the length of the LPFG so a well-defined resonance is obtained. In our setup, the feedback on the emitted laser power reduces the problem, but not completely.

Although theoretically we could simulate considering different laser powers and weights, experimentally it was observed that using lower laser powers (<5 W) no LPFGs were obtained. Also, using higher laser powers (>8 W) or higher weights (typically >60 g, F > 0.6 N) tapering occurs, a phenomena not included in our model.

The values obtained by the model are in agreement with those estimated by other authors for the refractive index modulations necessary for achieving a fiber optic grating. Temperatures

**Figure 17.** Picture showing two irradiated zones from a 25 mm LPFG with 500 μm period written on a SMF-28 optical fiber. (*P* ≈ 6 W; *ton*=0.6 s; *Δt*=1 s; *F* ≈ 0.5 N)

**Figure 18.** Experimentally obtained and simulated relative normalized spectral transmission under the same condi‐ tions considered in Figure 17.

calculated are similar to those obtained by other authors for arc-induced LPFG (e.g. in the range 1,100 K–1,400 K according to [56]) and the refractive index changes are within the overall range mentioned in other works [46,52,56]. Also, the behaviour of the refractive index change as the applied drawing force increases complies with recent experimental indications that the refractive index of the core decreases while the opposite occurs in the cladding, and that this change occurs primarily in the core [66,67].

In general, the process involves monitoring the transmission spectrum and iterative action on the length of the LPFG so a well-defined resonance is obtained. In our setup, the feedback on

**Figure 16.** Calculated refractive index change (maximum change for core and cladding) for different applied draw ten‐

**Figure 15.** Calculated (maximum) refractive index change at the core and cladding for different applied laser powers.

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

Although theoretically we could simulate considering different laser powers and weights, experimentally it was observed that using lower laser powers (<5 W) no LPFGs were obtained. Also, using higher laser powers (>8 W) or higher weights (typically >60 g, F > 0.6 N) tapering

The values obtained by the model are in agreement with those estimated by other authors for the refractive index modulations necessary for achieving a fiber optic grating. Temperatures

the emitted laser power reduces the problem, but not completely.

occurs, a phenomena not included in our model.

sions. (*P*=6 W; *ton*=0.6 s; *F*=0.461 N).

(*ton*=0.6 s; *F*=0.461 N).

Nevertheless, a complete model of the complex physical phenomena involved, in particular regarding the refractive index change dependence on stress, requires further research. It's expected that future work will focus on experimental measurements of temperature, stresses and refractive index changes induced by the MIR laser radiation. Also, although published works can contribute in assessing the validity of the results, the influence of specific charac‐ teristics of the fibers is a well-recognized issue. In particular, the effect of pre-existing stresses (typically from the fiber manufacture or preparation), differences in the materials, or other unaccounted phenomena can influence the performance of the FEM model when compared with real data. Similarly, the impact of the several approximations considered (e.g., transverse stresses are neglected), unaccounted phenomena like eventual changes on the glass polariza‐ bility and using standard material data must be analysed in detail, as well as the influence of the experimental data uncertainties on the model.
