Calculation of an Optical setup for a LIBS system

In this work we will represent the necessary calculation for the optical setup on a LIBS spectroscopic system, as well as the parameters for the beam characterization. On the realized simulation, the beam was deal with the wave optics theory and its propagation with the paraxial approximation. It was described as a beam of Gaussian intensity profile and its propagation was acquired through the ABCD’s law. Our results have showed the necessary lens to hold a LIBS sign for different kinds of materials, depending on their  bond cleavage . Also, we have showed a table with typical values of focus lens, used in LIBS setups, and its respective irradiance.

In this work we present the calculation achieved for an optical LIBS system setup supposing the Gaussian profile of the laser beam. To do so, we applied the Maxwell's Equations resolution in cylindrical coordinates and the Gaussian beam propagation method in optical systems. Thereby, we have found the best optical setup which will allow us the suitable power density (Gw/cm²) to reach the breakdown of different sample matrixes.

2.-Gaussian beam and its propagation
From Maxwell's four equations, that describe wave light behavior, we can derive the wave equation for electric field ⃗ , given by eq. (1): where = 2 is the wave vector, is the magnetic permeability of the medium, is the electric permittivity and is the wave frequency. For simply, we'll treat the case in which the medium is homogeneous and non-magnetic, i. e., the vector ⃗ is a constant.
Writing the laplacian operator in cylindrical coordinates and separating it into transverse portion to beam propagation ( , ) and parallel to beam propagation (z), we achieved the wave equation resolution e we found the electric field expression [16,17].
As the wave intensity is proportional to the square of electric field magnitude, thus the light intensity ( , ), resultant from ⃗ wave equation, is given by eq.
For = ( ), the beam intensity decreases 1/ 2 from its maximum value and this distance is called beam ray. Then the parameter 0 is called waist and its relation with ( ) is given by eq. (3). Note that, in the origin of beam propagation = 0, the beam reaches its minimum value 0 .
The parameter 0 = 2 2 = 0 2 is called Rayleigh's length and represents the beam focusing region, where 0 < ( ) < √2 0 . For small 0 , there is a short focalization and for great 0 , there is a long focalization. Figure 1 illustrates the Gaussian profile of a laser beam. To find the beam ray ( ), the waist 0 and the curvature ray of the wavefront ( ) as the beam propagates and interacts with the optical elements (mirrors and lens), we need to apply the ABCD's law, widely used in geometric optics and that, with the correct construction of parameters which describe the beam, can be utilized in the Gaussian wave [17]. To do so, it is defined the parameter ( ), given by eq. (4), which is a complex number. Note that its real part gives information about the beam wavefront ( ) and its imaginary part depends on the beam ray where is the refraction index of the propagation medium and is the wavelength in the vacuum.
In accord with the matrix optics, each optical element can be represented by a unitary matrix 2 2 and its elements are called A, B, C and D in the form ( ).
When the beam strikes an optical element, the new parameter 2 ( ) is given by eq.
(5) and thus it is possible to find a new beam ray ( ) and the new wavefront ( ).
In this way, in an optical system composed of various elements, we only need to apply this routine in each element or to calculate the total matrix of the optical system, given by the multiplication of individual matrixes, to characterize the Gaussian beam, i. e., to find its ray ( ) and the ray of the wavefront ( ).

3.-Calculation for the optical LIBS system setup
To determine the best optical setup of a LIBS system, we have solved the following problem: assuming the beam arrives collimated at the focusing lens of the LIBS system, which is the beam waist 0 after the lens. Besides, we need to answer if the joined 0 is enough to presents a power density (irradiance) able to cause the material ablation and to create the LIBS signal. Figure 2 illustrates the calculated situation. As the beam is collimated when strikes the lens, its wavefront ray 1 → ∞.
After the lens, we apply the ABCD's law once more, eq. (5), for a translation with distance , whose matrix is ℳ = ( 1 0 1 ) and we obtain that the parameter 3 satisfies the relation given by eq. (6).
As Figure 2 shows, at 3 plan the wavefront ray is 3 → ∞, therefore the real part of eq. (6) is null. This information takes us to find the translation distance as a function of the lens focusing distance and the Rayleigh's length 01 , eq. (7).
Note that, if 01 ≫ , we have = , i. e., we are at geometric optics regime.
Matching the imaginary part of eq. (6) with imaginary part of eq. (4), we have: Substituting the values of and in the equation above, we achieved the relation between waist 0 and its initial ray 1 , given by eq. (9). To represent a LIBS system, we use a pulsed Nd:YAG laser, with pulse width of 10 nanoseconds, in fundamental harmonics (λ = 1064 nm) and energy about 50 mJ per pulse. Figure 3 shows the variation of the lens focusing distance value and the calculation of light irradiance at 0 , i. e., where the analyzed sample was placed.
The irradiance calculation was achieved dividing the beam power (  For sample matrixes utilized in LIBS spectroscopy, bond cleavage following material ablation happened in the range of 5 to 10 GW/cm². In this case, we can overcome this value with the initial use of a lens with focusing distance of to 20 cm. If in the experiment it was necessary a higher or lesser irradiance in the sample, we would change it easily according to the calculation realized and the Figure 3. In Table 1, we show the values of 0 and irradiance for some values usually used in LIBS assemblies.

4.-Conclusion
Utilizing the Gaussian beam formalism to treat the laser used in Laser Induced Breakdown Spectroscopy (LIBS) and the ABCD's law to model its propagation in the optical system, it was possible to find the values of focusing distance more suitable for LIBS setup. The simulation was reliable enough to change the optical assembly in the laboratory without compromise its validity. With a lens of = 7,5 cm we achieved an irradiance in the sample of approximately 246 GW/cm², which is enough to cause material ablation and consequently to obtain a LIBS signal.
The realized simulation in this work was necessary to know the beam parameters before and after it strikes the sample. Thus, the optical assembly of the LIBS system has all of its parameters known, that permits more agility and efficiency in the measurements and sample changes.