Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical Computational Comparison, Laser Cladding versus Electron Beam Cladding

*Mihai Oane, Ion N. Mihăilescu and Carmen-Georgeta Ristoscu*

## **Abstract**

Laser cladding processing can be found in many industrial applications. A lot of different materials processing were studied in the last years. To improve the process, one may evaluate the phenomena behaviour from a theoretical and computational point of view. In our model, we consider that the phase transition to the melted pool is treated using an absorption coefficient which can underline liquid formation. In the present study, we propose a semi-analytical model. It supposes that melted pool is in first approximation a "sphere", and in consequence, the heat equation is solved in spherical coordinates. Using the Laplace transform, we will solve the heat equation without the assumption that "time" parameter should be interpolated linearly. 3D thermal graphics of the Cu substrate are presented. Our model could be applied also for electron cladding of metals. We make as well a comparison of the cladding method using laser or electron beams. We study the process for different input powers and various beam velocities. The results proved to be in good agreement with data from literature.

**Keywords:** laser cladding, electron beam cladding, heat equation, computer simulations

## **1. Introduction**

The laser processing of materials is a continuous subject of study from a practical and theoretical point of view [1–3]. Laser cladding is a very important application in laser processing [4]. Laser cladding started in the 1980s and was widely implemented in industry. Meanwhile, the application of the laser cladding has exploded especially in 3D additive manufacturing at a relatively low production cost. From theoretical point of view, the mentioned application was studied in Refs. [5, 6]. In the present study, we want to generalize the existent theory to laser beams with different transverse multimode intensities. We will use the heat diffusion equation for the melted pool [5, 6]. We note the depth of the melted pool with *H*. It is reasonable to consider that the depth of the melted pool varies between 0.2 and 2.5 mm, for a CO2 incident laser beam of 1 KW. The speed of laser is considered to

vary from 0 to 100 mm/s. The main purpose of engineering technology science is to achieve the best quality of the product with the maximum use of facilities and resources. In these terms, laser cladding is a very delicate process. In consequence, all kinds of modelling are welcome. In general, we find in literature a lot of experiments, but for a laboratory which wants to start to build up a laser cladding set-up for the first time, theoretical and computer modelling are essential for the experimental success.

The powder attenuation is defined as the following ratio:

$$X\_p = \frac{P\_L - P'\_L}{P\_L} \tag{1}$$

We obtain:

Such conditions lead to:

*DOI: http://dx.doi.org/10.5772/intechopen.88710*

*T r*ð Þ¼ *; θ; p; t T r*ð Þ¼ *; θ; m; t*

*∂ ∂μ* <sup>1</sup> � *<sup>μ</sup>*<sup>2</sup> � � *<sup>∂</sup><sup>T</sup>*

*<sup>∂</sup><sup>μ</sup>* � *<sup>m</sup>*<sup>2</sup>

<sup>1</sup> � *<sup>μ</sup>*<sup>2</sup> � � *<sup>∂</sup><sup>T</sup>*

� �

*∂μ*

<sup>þ</sup> *<sup>λ</sup>* � *<sup>m</sup>*<sup>2</sup>

� �

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical…*

ð<sup>2</sup>*<sup>π</sup>* 0

Now, in order to eliminate the variable *θ*, we assume that:

<sup>1</sup> � *<sup>μ</sup>*<sup>2</sup> � � *<sup>∂</sup>F*<sup>3</sup>

where *Pnm* are the associated Legendre polynomials. To eliminate the variable *r*, we have to consider that:

> 2 *r ∂T*e

*∂*2 e*v ∂r*<sup>2</sup> þ 1 *r ∂*e*v ∂r*

*∂*2 *F*3 *∂r*<sup>2</sup> þ 1 *r ∂F*<sup>3</sup> *∂r*

j jj *<sup>F</sup>*<sup>3</sup> *<sup>r</sup>*¼<sup>0</sup> <sup>&</sup>lt; <sup>∞</sup> *si k <sup>∂</sup>F*<sup>3</sup>

� �

*∂μ*

*<sup>∂</sup><sup>r</sup>* � *n n*ð Þ <sup>þ</sup> <sup>1</sup>

*<sup>r</sup>*<sup>2</sup> *<sup>T</sup>*<sup>e</sup> <sup>¼</sup> <sup>1</sup>

� *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 2 � �<sup>2</sup>

and the heat equation is the following:

<sup>þ</sup> *<sup>λ</sup>*<sup>2</sup> � *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

*∂r* � � � � *r*¼*a*

*<sup>r</sup>*<sup>2</sup> <sup>e</sup>*<sup>v</sup>* <sup>¼</sup> <sup>1</sup>

" #

*γ ∂*e*v ∂t*

2 � �<sup>2</sup> *r*

*<sup>F</sup>*3*<sup>T</sup>* <sup>¼</sup> *<sup>∂</sup> ∂μ*

*∂ ∂μ*

*∂*2 *T*e *∂r*<sup>2</sup> þ

2*m* � 1

*<sup>T</sup>*<sup>e</sup> <sup>¼</sup> <sup>e</sup>*<sup>v</sup> r*<sup>1</sup>*=*<sup>2</sup>

where *<sup>γ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>*

The theory says that:

We have:

**139**

(

<sup>1</sup> � *<sup>μ</sup>*<sup>2</sup> *<sup>T</sup>*

2 *r ∂T ∂r* þ 1 *r*2

and

*∂*2 *T ∂r*<sup>2</sup> þ

where

We have:

where

*F*1ð Þ¼ *φ cos mφ* for *p* ¼ *2m* (7)

*F*2ð Þ¼ *φ sin mφ* for *p* ¼ *2m* � 1 (8)

¼ 1 *γ ∂T*

*T r*ð Þ *; <sup>θ</sup>; <sup>φ</sup>; <sup>t</sup> <sup>F</sup>*1*<sup>p</sup>*ð Þ *<sup>φ</sup> <sup>d</sup>φ; p* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>*

� *<sup>m</sup>*<sup>2</sup>

1 � *μ*<sup>2</sup>

*λ* ¼ *n n*ð Þ þ 1 ð Þ *n* ¼ 0*;* 1*;* 2*;* 3…*:* and *F*<sup>3</sup> ¼ *Pnm*ðcosðθÞÞ (13)

*k ∂T*e *<sup>∂</sup><sup>t</sup>* � *Q r*ð Þ *; <sup>θ</sup>; <sup>p</sup>; <sup>t</sup>*

�

2*m* � 1

<sup>1</sup> � *<sup>μ</sup>*<sup>2</sup> *<sup>T</sup>* (11)

� �*F*<sup>3</sup> <sup>¼</sup> <sup>0</sup> (12)

*<sup>∂</sup><sup>t</sup>* � *Q r* <sup>f</sup>ð Þ *; <sup>γ</sup>; <sup>p</sup>; <sup>t</sup>* (14)

� *Q r* e ð Þ *; γ; p; t* (15)

*F*<sup>3</sup> ¼ 0 (16)

þ *hF*<sup>3</sup> ¼ 0 (17)

*<sup>k</sup>* (9)

(10)

where *PL* is the laser power and *P*<sup>0</sup> *<sup>L</sup>* is the transmitted laser power, which is in interaction with the work piece surface. We will focus to determine the temperature in the melted pool. For this we will choose a more realistic model (regarding the "time" parameter), writing the heat carried into the melted pool by the expression:

$$Q\_P = I(X, Y, Z) \cdot \left(a\_P X\_P + a\_P X\_P (1 - a\_P)(1 - a\_W)(h(t) - h(t - t\_0)) \right) = Q\_P \left(r, \theta, \varphi\right) \tag{2}$$

where *t0* is the exposure time, *α<sup>P</sup>* is powder absorption coefficient and *α<sup>W</sup>* is the workpiece absorption coefficient.

## **2. The analytical model**

The novelty of the proposed model is that we consider the melted pool like a sphere with diameter *H*. Using the Laplace transform, we will solve the heat equation avoiding making the assumption that "time" parameter should be interpolated linearly. The heat equation in spherical coordinates is:

$$\frac{\partial^2 T}{\partial r^2} + \frac{2}{r} \frac{\partial T}{\partial r} + \frac{1}{r^2} \left[ \frac{\partial}{\partial \mu} \left( 1 - \mu^2 \right) \frac{\partial T}{\partial \mu} \right] + \frac{1}{r^2} \cdot \frac{1}{(1 - \mu^2)} \frac{\partial^2 T}{\partial \rho^2} = \frac{1}{\chi} \frac{\partial T}{\partial t} - \frac{Q\_{\mathbb{P}}(r, \theta, \rho, t)}{k} \tag{3}$$

In Eq. (3),*T* is temperature variation; *r*, *θ* and *φ* are the spherical coordinates; *γ* is thermal diffusivity; and *k* represents the thermal conductivity. For simplicity, we will note for the rest of the present study *Q* ¼ *QP*.

We have the following relationships:

$$T|\_{t=0} = \mathbf{0} \text{ and } \mu = \cos \theta \tag{4}$$

The boundary conditions are:

$$k\left.\frac{\partial T}{\partial r}\right|\_{r=a} + h \cdot T = \mathbf{0} \tag{5}$$

where *a* = *H*/2 is the irradiated sphere radius and *h* is the thermal transfer coefficient.

We have the following relationships that are necessary to eliminate the variable *φ*:

$$\frac{\partial^2 F\_1}{\partial \rho^2} + m^2 F\_1 = 0 \text{ and } F\_1|\_{\rho=0} = F\_1|\_{\rho=2\pi} \tag{6}$$

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical… DOI: http://dx.doi.org/10.5772/intechopen.88710*

We obtain:

$$F\_1(\rho) = \cos m\rho \text{ for } p = 2m \tag{7}$$

and

vary from 0 to 100 mm/s. The main purpose of engineering technology science is to achieve the best quality of the product with the maximum use of facilities and resources. In these terms, laser cladding is a very delicate process. In consequence, all kinds of modelling are welcome. In general, we find in literature a lot of experiments, but for a laboratory which wants to start to build up a laser cladding set-up for the first time, theoretical and computer modelling are essential for the experi-

*Xp* <sup>¼</sup> *PL* � *<sup>P</sup>*<sup>0</sup>

interaction with the work piece surface. We will focus to determine the temperature in the melted pool. For this we will choose a more realistic model (regarding the "time" parameter), writing the heat carried into the melted pool by the expression:

*Q <sup>P</sup>* ¼ *I X*ð Þ� *; Y; Z αPXP* þ *αPXP*ð Þ 1 � *α<sup>P</sup>* ð Þ 1 � *α<sup>W</sup>* ð*h t*ðÞ� *h t*ð Þ � *t*<sup>0</sup> Þ ¼ *QP* ð ð Þ *r; θ; φ*

where *t0* is the exposure time, *α<sup>P</sup>* is powder absorption coefficient and *α<sup>W</sup>* is the

The novelty of the proposed model is that we consider the melted pool like a sphere with diameter *H*. Using the Laplace transform, we will solve the heat equation avoiding making the assumption that "time" parameter should be interpolated

In Eq. (3),*T* is temperature variation; *r*, *θ* and *φ* are the spherical coordinates; *γ* is thermal diffusivity; and *k* represents the thermal conductivity. For simplicity, we

*∂*2 *T <sup>∂</sup>φ*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *γ ∂T*

*<sup>T</sup>*j*<sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> 0 and *<sup>μ</sup>* <sup>¼</sup> cos *<sup>θ</sup>* (4)

*<sup>φ</sup>*¼<sup>0</sup> ¼ *F*1j

þ *h* � *T* ¼ 0 (5)

*<sup>φ</sup>*¼2*<sup>π</sup>* (6)

*<sup>∂</sup><sup>t</sup>* � *QP*ð Þ *<sup>r</sup>; <sup>θ</sup>; <sup>φ</sup>; <sup>t</sup>*

*<sup>k</sup>* (3)

*L PL*

*<sup>L</sup>* is the transmitted laser power, which is in

(1)

(2)

The powder attenuation is defined as the following ratio:

where *PL* is the laser power and *P*<sup>0</sup>

*Nonlinear Optics - From Solitons to Similaritons*

workpiece absorption coefficient.

linearly. The heat equation in spherical coordinates is:

will note for the rest of the present study *Q* ¼ *QP*. We have the following relationships:

> *∂*2 *F*1 *<sup>∂</sup>φ*<sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup>

<sup>1</sup> � *<sup>μ</sup>*<sup>2</sup> *<sup>∂</sup><sup>T</sup>*

*∂μ*

*k ∂T ∂r r*¼*a*

where *a* = *H*/2 is the irradiated sphere radius and *h* is the thermal transfer

We have the following relationships that are necessary to eliminate the

*F*<sup>1</sup> ¼ 0 and *F*1j

þ 1 *<sup>r</sup>*<sup>2</sup> � <sup>1</sup> 1 � *μ*<sup>2</sup> ð Þ

*∂ ∂μ*

The boundary conditions are:

**2. The analytical model**

*∂*2 *T ∂r*<sup>2</sup> þ 2 *r ∂T ∂r* þ 1 *r*2

coefficient.

variable *φ*:

**138**

mental success.

$$F\_2(\rho) = \sin m\rho \text{ for } p = 2m - 1 \tag{8}$$

Such conditions lead to:

$$\frac{\partial^2 \overline{T}}{\partial r^2} + \frac{2}{r} \frac{\partial \overline{T}}{\partial r} + \frac{1}{r^2} \left\{ \frac{\partial}{\partial \mu} \left( 1 - \mu^2 \right) \frac{\partial \overline{T}}{\partial \mu} - \frac{m^2}{1 - \mu^2} \overline{T} \right\} = \frac{1}{\chi} \frac{\partial \overline{T}}{\partial t} - \frac{\overline{Q(r, \theta, p, t)}}{k} \tag{9}$$

where

$$\overline{T}(r,\theta,p,t) = \overline{T}(r,\theta,m,t) = \int\_0^{2\pi} T\left(r,\theta,\varphi,t\right) F\_{1p}(\rho) \,d\rho; p = \begin{cases} 2m\\ 2m-1 \end{cases} \tag{10}$$

Now, in order to eliminate the variable *θ*, we assume that:

$$F\_3 \overline{T} = \frac{\partial}{\partial \mu} \left[ \left( \mathbf{1} - \mu^2 \right) \frac{\partial \overline{T}}{\partial \mu} \right] - \frac{m^2}{\mathbf{1} - \mu^2} \overline{T} \tag{11}$$

We have:

$$\frac{\partial}{\partial \mu} \left[ (\mathbf{1} - \mu^2) \frac{\partial F\_3}{\partial \mu} \right] + \left( \lambda - \frac{m^2}{\mathbf{1} - \mu^2} \right) F\_3 = \mathbf{0} \tag{12}$$

where

$$\lambda = n(n+1)(n=0,1,2,3...) \text{ and } F\_3 = P\_{nm}(\cos(\theta)) \tag{13}$$

where *Pnm* are the associated Legendre polynomials. To eliminate the variable *r*, we have to consider that:

$$\frac{\partial^2 \bar{T}}{\partial r^2} + \frac{2}{r} \frac{\partial \bar{T}}{\partial r} - \frac{n(n+1)}{r^2} \tilde{T} = \frac{1}{k} \frac{\partial \bar{T}}{\partial t} - \widetilde{Q}\ (r, \chi, p, t) \tag{14}$$

where *<sup>γ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>* 2*m* � 1 ( and the heat equation is the following:

$$\tilde{T} = \frac{\tilde{v}}{r^{1/2}} \frac{\partial^2 \tilde{v}}{\partial r^2} + \frac{1}{r} \frac{\partial \tilde{v}}{\partial r} - \frac{\left(n + \frac{1}{2}\right)^2}{r^2} \tilde{v} = \frac{1}{\gamma} \frac{\partial \tilde{v}}{\partial t} - \tilde{Q}(r, \gamma, p, t) \tag{15}$$

We have:

$$\frac{\partial^2 F\_3}{\partial r^2} + \frac{1}{r} \frac{\partial F\_3}{\partial r} + \left[ \lambda^2 - \frac{\left(n + \frac{1}{2}\right)^2}{r} \right] F\_3 = 0 \tag{16}$$

The theory says that:

$$\left| F\_3 \right| \vert\_{r=0} < \infty \text{ si } k \left. \frac{\partial F\_3}{\partial r} \right|\_{r=a} + hF\_3 = 0 \tag{17}$$

The obtained result is:

$$F\_3 = f\_{n+\frac{1}{2}}(\lambda a) \tag{18}$$

situations, for example, different transverse modes (for laser beam), various veloc-

For electron beam processing [7], one may consult the Katz and Penfolds

In **Figure 1**, the thermal field for Gaussian laser beam is presented, when *V* = 0 mm/s, *P* = 1 KW, *H* = 2 mm and the substrate is Cu. In **Figure 2** the thermal field for Gaussian laser beam is given when *V* = 10 mm/s, *P* = 1 KW and *H* = 2 mm. In **Figure 3** the thermal field for TEM03 laser beam is presented, when *V* = 0 mm/s, *P* = 2 KW and *H* = 3 mm. **Figure 4** shows the thermal field for TEM03 laser beam, when *V* = 10 mm/s, *P* = 4 KW and *H* = 4 mm. **Figure 5**

represents the thermal field for TEM03 laser beam, when *V* = 100 mm/s, *P* = 10 KW

If one compares **Figures 1** and **2**, the differences in the spatial distribution

comparison of **Figures 3**–**5** shows that for TEM03 we do not have significant changes in thermal profile but a proportional increase of the incident power

In **Figures 6** and **7**, we have as scanning source an electron beam of power *P* = 1 KW. If in **Figure 6** *V* = 0 mm/s, while in **Figure 7** the speed is *V* = 10 mm/s.

As observed from **Table 2**, Cu behaves very similarly with Au, Ag and Al from a thermal point of view [10]. Accordingly, we may consider that **Figures 1**–**7** are

*The thermal field for a Gaussian laser beam when V = 0 mm/s, P = 1 KW and H = 2 mm. The substrate is*

of thermal field for the two cases can be seen. On the other hand, the

Our simulations show a decrease of thermal field with the increase of

also meaningful if we use substrates from Au, Ag or Al.

absorption law [8] and also Tabata-Ito-Okabe absorption law [9].

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical…*

ities, incident powers and values of H.

*DOI: http://dx.doi.org/10.5772/intechopen.88710*

and *H* = 3 mm.

scanning velocity.

with *H*.

**Figure 1.**

**141**

*supposed to be from Cu.*

and

$$k\left(J\_{n-\frac{1}{2}}(\lambda\_m a) - J\_{n+\frac{1}{2}}(\lambda\_m a)\right) + hJ\_{n+\frac{1}{2}}(\lambda\_m a) = 0\tag{19}$$

To eliminate the temporal variable, we use the direct and reverse Laplace transform. Thus we obtain [1]:

$$\begin{split} T(r,\theta,\varphi,t) &= \frac{1}{r!} \sum\_{m=0}^{\infty} \sum\_{n\_x=1}^{\infty} \frac{1}{C\_{mn} \cdot C\_{nx}} \cdot \frac{1}{\lambda\_{\text{max}}^2} \left[ 1 - e^{-\lambda\_{\text{min}}^2 t} - \left( 1 - e^{-\lambda\_{\text{min}}^2 (t-t\_0)} \right) \cdot H(t-t\_0) \right] \\ &\cdot \cdot I\_{n+\frac{1}{2}}(\lambda\_{\text{m}} r) \left[ P\_{n\,\,m}(\cos \theta) \cos(m\phi) \cdot \left( \left[ \int\_0^a \int\_0^{\theta\_{\text{max}}} \right] \cdot \frac{E(E\_0,r,\cos\theta)}{C} \cdot r^{\frac{1}{2}} \cdot I\_{n+\frac{1}{2}}(\lambda\_{\text{m}} r) \right. \\ &\cdot P\_{n\,\,m}(\cos \theta) \cdot \cos(m\phi) dr d\theta d\rho \right] \\ &\cdot P\_{n\,\,m}(\cos \theta) \cdot \cos(m\phi) dr d\theta d\rho \Big] + P\_{n\,\,m}(\cos \theta) \cdot \sin(m\phi) \\ &\cdot \left[ \left( \int \int \int \int \cdot \frac{E(E\_0,r,\cos\theta)}{C} \cdot r^{\frac{1}{2}} \cdot I\_{n+\frac{1}{2}}(\lambda\_{\text{m}} r) \cdot P\_{n\,\,m}(\cos \theta) \cdot \sin(m\phi) dr d\theta d\rho \right) \right] \end{split} (20)$$

In the above relationship:

$$\begin{array}{c} \mathbf{C} \\ m \quad n = \int\_{-1}^{+1} \left[ P\_{m \quad n} \left( \right. \right. \right]^2 d\mu = \frac{2i}{2} \frac{(n+m)!}{n+1 \quad (n-m)!} \end{array} \tag{21}$$

and

0

0

0

$$\delta = \begin{cases} 2\,\text{for } m = 0\\ 1\,\text{for } m \neq 0 \end{cases} \tag{22}$$

but also

$$\mathcal{C}\_{ns} = \frac{1}{2} \text{ a } \left[ \mathcal{I}'\_{n + \frac{1}{2}}(\lambda\_{ns} a) \right] \tag{23}$$

The laser beam as compared to electron beam may be considered to be a sum of decoupled transverse modes, and one can write using a superposition of different transverse modes:

$$I = \sum\_{i,m,n} p\_i I\_{mn} \tag{24}$$

where *pi* are real numbers chosen in such a way to obtain the wanted laser intensity (from spatial distribution and intensity values' point of view).

### **3. Simulations and comments**

Let us consider the cladding processing on a Cu substrate. The input parameters corresponding to **Figures 1**–**7** are collected in **Table 1**. We have chosen various

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical… DOI: http://dx.doi.org/10.5772/intechopen.88710*

situations, for example, different transverse modes (for laser beam), various velocities, incident powers and values of H.

For electron beam processing [7], one may consult the Katz and Penfolds absorption law [8] and also Tabata-Ito-Okabe absorption law [9].

In **Figure 1**, the thermal field for Gaussian laser beam is presented, when *V* = 0 mm/s, *P* = 1 KW, *H* = 2 mm and the substrate is Cu. In **Figure 2** the thermal field for Gaussian laser beam is given when *V* = 10 mm/s, *P* = 1 KW and *H* = 2 mm. In **Figure 3** the thermal field for TEM03 laser beam is presented, when *V* = 0 mm/s, *P* = 2 KW and *H* = 3 mm. **Figure 4** shows the thermal field for TEM03 laser beam, when *V* = 10 mm/s, *P* = 4 KW and *H* = 4 mm. **Figure 5** represents the thermal field for TEM03 laser beam, when *V* = 100 mm/s, *P* = 10 KW and *H* = 3 mm.

If one compares **Figures 1** and **2**, the differences in the spatial distribution of thermal field for the two cases can be seen. On the other hand, the comparison of **Figures 3**–**5** shows that for TEM03 we do not have significant changes in thermal profile but a proportional increase of the incident power with *H*.

In **Figures 6** and **7**, we have as scanning source an electron beam of power *P* = 1 KW. If in **Figure 6** *V* = 0 mm/s, while in **Figure 7** the speed is *V* = 10 mm/s. Our simulations show a decrease of thermal field with the increase of scanning velocity.

As observed from **Table 2**, Cu behaves very similarly with Au, Ag and Al from a thermal point of view [10]. Accordingly, we may consider that **Figures 1**–**7** are also meaningful if we use substrates from Au, Ag or Al.

#### **Figure 1.**

The obtained result is:

form. Thus we obtain [1]:

2 4

1 *r* 1 2 X∞ *m*¼0

� *Pn m*ð Þ� *cosθ* cosð Þ *mφ drdθdφ*

In the above relationship:

*T r*ð Þ¼ *; θ; φ; t*

� *Jn*þ<sup>1</sup> 2

� *a*ð

0 @

0

and

but also

transverse modes:

**140**

**3. Simulations and comments**

2 ð *π* *θmax* ð

0 �

0

*k Jn*�<sup>1</sup> 2

*Nonlinear Optics - From Solitons to Similaritons*

X∞ *n, <sup>s</sup>*¼<sup>1</sup>

ð Þ *λnsr Pn m*ð Þ *cosθ* cosð Þ� *mφ*

*E E*ð Þ <sup>0</sup>*;r;* cos *θ <sup>C</sup>* � *<sup>r</sup>*

ð Þ� *<sup>λ</sup>nsa Jn*þ<sup>1</sup>

1 *Cmn* � *Cns*

� �

� 1 *λ*2 *ns*

*a*ð

0 @

!

3 <sup>2</sup> � *Jn*þ<sup>1</sup> 2

*C m n*¼ Ð þ1 �1

2 ð *π*

0

0

and

*F*<sup>3</sup> ¼ *Jn*þ<sup>1</sup> 2

> 2 ð Þ *λnsa*

To eliminate the temporal variable, we use the direct and reverse Laplace trans-

1 � *e* �*λ*<sup>2</sup>

*θmax* ð

0 �

þ *Pn m*ð Þ� *cosθ* sin ð Þ *mφ*

½ � *Pm n*ð Þ *<sup>μ</sup>* <sup>2</sup> *<sup>d</sup>μ*<sup>¼</sup> <sup>2</sup>*<sup>δ</sup>* <sup>2</sup> *<sup>n</sup>*þ<sup>1</sup>

*<sup>δ</sup>* <sup>¼</sup> <sup>2</sup> *for m* <sup>¼</sup> <sup>0</sup> 1 *for m* 6¼ 0

> <sup>2</sup> *a J*<sup>0</sup> *<sup>n</sup>*þ<sup>1</sup> 2 ð Þ *λnsa* h i

*<sup>I</sup>* <sup>¼</sup> <sup>X</sup> *i, m, <sup>n</sup> pi*

intensity (from spatial distribution and intensity values' point of view).

where *pi* are real numbers chosen in such a way to obtain the wanted laser

Let us consider the cladding processing on a Cu substrate. The input parameters

corresponding to **Figures 1**–**7** are collected in **Table 1**. We have chosen various

The laser beam as compared to electron beam may be considered to be a sum of decoupled transverse modes, and one can write using a superposition of different

(

*Cns* <sup>¼</sup> <sup>1</sup>

þ *hJn*þ<sup>1</sup> 2

*nst* � <sup>1</sup> � *<sup>e</sup>*

*E E*ð Þ <sup>0</sup>*;r;* cos *θ <sup>C</sup>* � *<sup>r</sup>*

ð Þ� *λnsr Pn m*ð Þ� *cosθ* sin ð Þ *mφ drdθdφ*

ð Þ *n*þ*m* ! ð Þ *n*�*m* !

�*λ*<sup>2</sup> *ns:*ð Þ *t*�*t*<sup>0</sup> � �

> 3 <sup>2</sup> � *Jn*þ<sup>1</sup> 2 ð Þ *λnsr*

h i

ð Þ *λa* (18)

ð Þ¼ *λnsa* 0 (19)

� *H t*ð Þ � *t*<sup>0</sup>

1 A #

*Imn* (24)

(20)

(21)

(22)

(23)

*The thermal field for a Gaussian laser beam when V = 0 mm/s, P = 1 KW and H = 2 mm. The substrate is supposed to be from Cu.*

#### **Figure 2.**

*The thermal field for a Gaussian laser beam when V = 10 mm/s, P = 1 KW and H = 2 mm. The substrate is supposed to be from Cu.*

**Figure 4.**

**Figure 5.**

**143**

*supposed to be from Cu.*

*supposed to be from Cu.*

*The thermal field for a TEM03 laser beam when V = 10 mm/s, P = 4 KW and H = 4 mm. The substrate is*

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical…*

*DOI: http://dx.doi.org/10.5772/intechopen.88710*

*The thermal field for a TEM03 laser beam, when V = 100 mm/s, P = 10 KW and H = 3 mm. The substrate is*

#### **Figure 3.**

*The thermal field for a TEM03 laser beam when V = 0 mm/s, P = 2 KW and H = 3 mm. The substrate is supposed to be from Cu.*

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical… DOI: http://dx.doi.org/10.5772/intechopen.88710*

#### **Figure 4.**

**Figure 2.**

**Figure 3.**

**142**

*supposed to be from Cu.*

*supposed to be from Cu.*

*Nonlinear Optics - From Solitons to Similaritons*

*The thermal field for a Gaussian laser beam when V = 10 mm/s, P = 1 KW and H = 2 mm. The substrate is*

*The thermal field for a TEM03 laser beam when V = 0 mm/s, P = 2 KW and H = 3 mm. The substrate is*

*The thermal field for a TEM03 laser beam when V = 10 mm/s, P = 4 KW and H = 4 mm. The substrate is supposed to be from Cu.*

#### **Figure 5.**

*The thermal field for a TEM03 laser beam, when V = 100 mm/s, P = 10 KW and H = 3 mm. The substrate is supposed to be from Cu.*

**Figure 6.**

*The thermal field for a Gaussian electron beam when V = 0 mm/s, P = 1 KW and H = 2 mm. The substrate is supposed to be from Cu.*

**4. Conclusions**

**Table 1.**

**Table 2.**

**Figure no. Beam**

**type**

*DOI: http://dx.doi.org/10.5772/intechopen.88710*

*The input parameters for the Figures 1–7.*

**Element Thermal diffusivity γ [cm2**

*The thermal parameters for: Cu, Au, Ag and Al.*

**Mode Velocity [mm/s]**

**Incident power [kW]**

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical…*

**Figure 1** Laser TEM00 0 1 2 1.14 3.95 **Figure 2** Laser TEM00 10 1 2 1.14 3.95 **Figure 3** Laser TEM03 0 2 3 1.14 3.95 **Figure 4** Laser TEM03 10 4 4 1.14 3.95 **Figure 5** Laser TEM03 100 10 3 1.14] 3.95 **Figure 6** Electron TEM00 0 1 2 1.14 3.95 **Figure 7** Electron TEM00 10 1 2 1.14 3.95

Cu 1.14 3.95 Au 1.22 3.15 Ag 1.72 4.28 Al 1.03 2.4

**Melted pool depth H [mm]**

**Thermal diffusivity γ [cm2 /s]**

**/s] Thermal conductivity k [W/cmK]**

**Thermal conductivity k [W/cmK]**

**Acknowledgements**

**Conflict of interest**

The authors declare no conflict of interest.

2016.

**145**

Our major conclusions are as follows: apart from Gaussian case, the increase in velocity of the other transversal modes does not affect too much the thermal profile; and second the large difference between the electron cladding and laser cladding is that in electron cladding an increase of beam velocity affects in an important amount the values of the thermal fields. The higher the velocity of electron beam, the lower the thermal fields at the surface sample. Our major conclusions are in good agreement with experimental data from literature; see, for example, references [11, 12]. On the other hand, it is known that there are some limitations in laser cladding, for example, high initial capital cost, high maintenance cost and presence of heat affected zone. For electron cladding, one can conclude that the cost is reduced as there are no involved mechanical cutting force, work holding and fixturing.

The authors acknowledge the support of MCI-OI under the contract POC G 135/

**Figure 7.** *The thermal field for a Gaussian electron beam when V = 10 mm/s, P = 1 KW and H = 2 mm. The substrate is supposed to be from Cu.*

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical… DOI: http://dx.doi.org/10.5772/intechopen.88710*


#### **Table 1.**

*The input parameters for the Figures 1–7.*


**Table 2.**

**Figure 6.**

**Figure 7.**

**144**

*supposed to be from Cu.*

*supposed to be from Cu.*

*Nonlinear Optics - From Solitons to Similaritons*

*The thermal field for a Gaussian electron beam when V = 0 mm/s, P = 1 KW and H = 2 mm. The substrate is*

*The thermal field for a Gaussian electron beam when V = 10 mm/s, P = 1 KW and H = 2 mm. The substrate is*

*The thermal parameters for: Cu, Au, Ag and Al.*

## **4. Conclusions**

Our major conclusions are as follows: apart from Gaussian case, the increase in velocity of the other transversal modes does not affect too much the thermal profile; and second the large difference between the electron cladding and laser cladding is that in electron cladding an increase of beam velocity affects in an important amount the values of the thermal fields. The higher the velocity of electron beam, the lower the thermal fields at the surface sample. Our major conclusions are in good agreement with experimental data from literature; see, for example, references [11, 12]. On the other hand, it is known that there are some limitations in laser cladding, for example, high initial capital cost, high maintenance cost and presence of heat affected zone. For electron cladding, one can conclude that the cost is reduced as there are no involved mechanical cutting force, work holding and fixturing.

## **Acknowledgements**

The authors acknowledge the support of MCI-OI under the contract POC G 135/ 2016.

### **Conflict of interest**

The authors declare no conflict of interest.

*Nonlinear Optics - From Solitons to Similaritons*

**References**

0, monograph

monograph

[1] Oane M, Ticos D, Ticoş CM. Charged Particle Beams Processing Versus Laser Processing. Germany: Scholars' Press; 2015. pp. 60-61, ISBN: 978-3-639-66753-

*DOI: http://dx.doi.org/10.5772/intechopen.88710*

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical…*

irradiation with a few MeV electron beam: Experiment versus theoretical simulations. Nuclear Instruments and Methods in Physics Research B. 2014;

**318**:232-236. DOI: 10.1016/j.

[9] Oane M, Toader D, Iacob N,

Ticos CM. Thermal phenomena induced in a small tungsten sample during irradiation with a few MeV electron beam: Experiment versus simulations. Nuclear Instruments and Methods in Physics Research B. 2014;**337**:17-20. DOI: 10.1016/j.nimb.2014.07.012

[10] Bauerle D. Laser Processing and Chemistry. 2nd ed. Berlin: Springer; 1996; ISBN 10: 354060541X/ISBN 13:

[11] Wirth F, Eisenbarth D, Wegener K. Absorptivity measurements and heat source modeling to simulate laser cladding. Physics Procedia. 2016;**83**:

1424-1434. DOI: 10.1016/j.

[12] Meriaudeau F, Truchetet F, Grevey D, Vannes AB. Laser cladding process and image processing. Journal of Lasers in Engineering. 1997;**6**:161-187

phpro.2016.08.148

nimb.2013.09.017

9783540605416

[2] Oane M, Peled A, Medianu RV. Notes on Laser Processing. Germany: Lambert Academic Publishing; 2013. pp. 7-9, ISBN: 978-3-659-487-48739-2,

[3] Oane M, Medianu RV, Bucă A. Chapter 16: A parallel between laser irradiation and electrons irradiation of solids. In: Radiation Effects in Materials. IntechOpen; 2016. pp. 413-430, ISBN: 978-953-51-2417-7, monograph

[4] Steen WM, Mazumder J. Laser Material Processing. London, Dordrecht, Heidelberg, New York: Springer; 2010, ISBN: 978-1-84996-

[5] Cline HE, Anthony TR. Heat treating and melting material with a scanning laser or electron beam. Journal of Applied Physics. 1977;**48**(9):3895-3900.

061-8, monograph

DOI: 10.1063/1.324261

[6] Picasso M, Marsden CF,

DOI: 10.1007/BF02665211

[7] Mul D, Krivezhenko D, Zimoglyadova T, Popelyukh A, Lazurenko D, Shevtsova L. Surface hardening of steel by electron-beam cladding of Ti+C and Ti+B4C powder compositions at air atmosphere. Applied Mechanics and Materials. 2015;**788**: 241-245. DOI: 10.4028/www.scientific.

net/AMM.788.241

**147**

[8] Oane M, Toader D, Iacob N,

Ticos CM. Thermal phenomena induced in a small graphite sample during

Wagniere JD, Frenk A, Rappaz M. A simple but realistic model for laser cladding. Metallurgical and Materials Transactions B. 1994;**25**(B):281-291.

## **Author details**

Mihai Oane<sup>1</sup> , Ion N. Mihăilescu<sup>2</sup> and Carmen-Georgeta Ristoscu<sup>2</sup> \*

1 Electrons Accelerators Laboratory, National Institute for Lasers, Plasma and Radiation Physics, Măgurele-Ilfov, Romania

2 Lasers Department, National Institute for Lasers, Plasma and Radiation Physics, Măgurele-Ilfov, Romania

\*Address all correspondence to: carmen.ristoscu@inflpr.ro

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Thermal Fields in Laser Cladding Processing: A "Fire Ball" Model. A Theoretical… DOI: http://dx.doi.org/10.5772/intechopen.88710*

## **References**

[1] Oane M, Ticos D, Ticoş CM. Charged Particle Beams Processing Versus Laser Processing. Germany: Scholars' Press; 2015. pp. 60-61, ISBN: 978-3-639-66753- 0, monograph

[2] Oane M, Peled A, Medianu RV. Notes on Laser Processing. Germany: Lambert Academic Publishing; 2013. pp. 7-9, ISBN: 978-3-659-487-48739-2, monograph

[3] Oane M, Medianu RV, Bucă A. Chapter 16: A parallel between laser irradiation and electrons irradiation of solids. In: Radiation Effects in Materials. IntechOpen; 2016. pp. 413-430, ISBN: 978-953-51-2417-7, monograph

[4] Steen WM, Mazumder J. Laser Material Processing. London, Dordrecht, Heidelberg, New York: Springer; 2010, ISBN: 978-1-84996- 061-8, monograph

[5] Cline HE, Anthony TR. Heat treating and melting material with a scanning laser or electron beam. Journal of Applied Physics. 1977;**48**(9):3895-3900. DOI: 10.1063/1.324261

[6] Picasso M, Marsden CF, Wagniere JD, Frenk A, Rappaz M. A simple but realistic model for laser cladding. Metallurgical and Materials Transactions B. 1994;**25**(B):281-291. DOI: 10.1007/BF02665211

[7] Mul D, Krivezhenko D, Zimoglyadova T, Popelyukh A, Lazurenko D, Shevtsova L. Surface hardening of steel by electron-beam cladding of Ti+C and Ti+B4C powder compositions at air atmosphere. Applied Mechanics and Materials. 2015;**788**: 241-245. DOI: 10.4028/www.scientific. net/AMM.788.241

[8] Oane M, Toader D, Iacob N, Ticos CM. Thermal phenomena induced in a small graphite sample during

irradiation with a few MeV electron beam: Experiment versus theoretical simulations. Nuclear Instruments and Methods in Physics Research B. 2014; **318**:232-236. DOI: 10.1016/j. nimb.2013.09.017

[9] Oane M, Toader D, Iacob N, Ticos CM. Thermal phenomena induced in a small tungsten sample during irradiation with a few MeV electron beam: Experiment versus simulations. Nuclear Instruments and Methods in Physics Research B. 2014;**337**:17-20. DOI: 10.1016/j.nimb.2014.07.012

[10] Bauerle D. Laser Processing and Chemistry. 2nd ed. Berlin: Springer; 1996; ISBN 10: 354060541X/ISBN 13: 9783540605416

[11] Wirth F, Eisenbarth D, Wegener K. Absorptivity measurements and heat source modeling to simulate laser cladding. Physics Procedia. 2016;**83**: 1424-1434. DOI: 10.1016/j. phpro.2016.08.148

[12] Meriaudeau F, Truchetet F, Grevey D, Vannes AB. Laser cladding process and image processing. Journal of Lasers in Engineering. 1997;**6**:161-187

**Author details**

Măgurele-Ilfov, Romania

Radiation Physics, Măgurele-Ilfov, Romania

*Nonlinear Optics - From Solitons to Similaritons*

provided the original work is properly cited.

\*Address all correspondence to: carmen.ristoscu@inflpr.ro

, Ion N. Mihăilescu<sup>2</sup> and Carmen-Georgeta Ristoscu<sup>2</sup>

1 Electrons Accelerators Laboratory, National Institute for Lasers, Plasma and

2 Lasers Department, National Institute for Lasers, Plasma and Radiation Physics,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*

Mihai Oane<sup>1</sup>

**146**

**Chapter 7**

**Abstract**

Optical Sensor for Nonlinear and

In this chapter, the main foundations for the conception, design, and the project

Classical electrodynamics is the basis for the analysis and formulation of electromagnetic waves. From the equations of Maxwell, it is possible to obtain the equation of the movement of the electric and magnetic fields whose solution describes the propagation of the electromagnetic wave. This formal treatment was originally developed by Maxwell [1], who verified that the electromagnetic wave propagated with the speed of light provided that the optics could be described from the electromagnetism. The medium through which the electromagnetic wave propagates responds in various ways to the electromagnetic field. This response depends on how the atoms and molecules are arranged spatially composing the constituent medium and how the interaction or scattering of the electromagnetic wave through the medium will occur. In other words, the way the medium responds to the electromagnetic excitation is contained in the middle polarization due to the propagation of the electromagnetic wave. It is in this context that some recent analyses have discovered some solutions from the nonlinear response of the medium to the propagation of the electromagnetic wave which may lead to an approach of some quantum effects from a

The propagation of optical pulses through waveguides such as optical fibers can give rise to nonlinear optical effects and quantum effects. The appropriate modeling of these effects can be used for the development of sensors to the optical fiber whose resolution can be regulated properly. In addition, the method allows the selection of propagation modes by selecting the desired modes by knowing the band

The development of sensors to the optical fiber is based on the propagation of optical pulses through waveguides like optical fibers and photonic crystals. The propagation of the pulses through waveguides can generate nonlinear and quantum

of optical sensors that explore the effects of nonlinear and quantum optics are presented. These sensors have a variety of applications from the design of waveguides with self-selection of propagation modes to signal processing and quantum computing. The chapter seeks to present formal aspects of applied modern optics

**Keywords:** optical sensor, nonlinear optics, quantum optics, optical fiber,

nonlinear treatment of electromagnetism in the material medium.

gap of the waveguide or the photonic crystal.

Quantum Optical Effects

*Antônio Carlos Amaro de Faria*

in a detailed, sequential, and concise manner.

optical signal processing

**1. Introduction**

**149**

**Chapter 7**
