Modeling and Implementing Nonlinear Equations in Solid-State Lasers for Studying their Performance

In this paper, the effect of radius variation of beam light on output efficacy of SFD Yttrium aluminium borate laser doped with Neodymium ion, which is simultaneously a non-linear and active laser crystal, is investigated in a double-pass cavity. This is done with a concave lens that concentrates (Reduction of optical radius within nonlinear material) as much optical laser as possible, resulting in increasing the laser efficiency, second harmonic and the population inversion difference. In this study, we first developed five discrete differential equations describing the interactions of 807 nm pump beam, 1060nm laser beam and 530nm second harmonic beam. Output efficiencies of laser and second harmonic beams at pumping power of Pp =20W and beam radius of 5μm have been presented. Meanwhile, in this paper, the first experiment for creating second harmonic in solid state lasers was fully described with a figure and its procedure was investigated and then the equations (second harmonic and laser and population inversion) were studied. Radius variation of beam light aims at increasing laser output efficacy and improving second harmonic and population inversion. The analytic methods which have been solved the discrete differential equations via Matlab.

Frequency-doubler materials are very useful.These materials can double the frequency of the laser beam (i.e. they halve the wavelength).By this method, the output laser with 1.06μ wavelength can be converted to a radiation having 0.53μ wavelengths [7].
The laser consists of three components including active material, a pumping design and an intensifier [8].

2-1-The first experiment to generate the second harmonic beam in lasers
In Figure 1, the frequency region of accessible lasers can be developed using nonlinear optical materials, such as harmonic generators and optical parameter oscillators.
The ruby laser beam at  = 0.694  wavelength is projected onto a quartz crystal and a ruby laser beam at  1 = 0.347 wavelength is produced.Two beams (λ and λ1) are separated by a prism which are appeared on a photographic plane.

3-Discrete Equations of Second Harmonic and Laser Beams and Inverse Population
Forward difference for the discretization of the equations corresponding progressive fields has been used as bellow: Where, ψ s + and ψ  + are functions of the second harmonic wave and progressive lasers, respectively.∝  and Ln are the absorption coefficient of the second harmonic beam and interaction length, respectively.k is a variable and Δz is another variable depends on z (path length) and k. also  = √−1.
In the above equation, ΔN is the inverse population and   is a variable depends on laser length.
A backward difference for discretization of the equations corresponding with retrograde fields has been used as bellow: Where: δ is cross section of laser emission of Nd 3+ ions and ∝  is absorption coefficient of the laser.
Tp = 1-Rp= 0.74 and Pp are passed power using the left mirror for the pump beam, and the power of the pump beam, respectively.η q = 0.26 is quantum efficacy.Wp and Wl are the angular frequency of the pump and laser beams, respectively.ϑ p , ϑ l and ϑ s are the frequencies of the pump, the laser, the second harmonic beams.The reflection through the mirrors leads to Rsb=0.03,Rlb =Rlf = Rsf =0.998.l and a are the crystal length and radius.I= 5mm and h is the Hamiltonian constant [13].
L2: is interaction length of No. 2.

4-Output Efficacy of Forward Second Harmonic Beam along Z-Direction for Beam Radius of rp = 5μm
Figure 2 shows output of forward second harmonic beam in distance along crystal length for 20 w pump power.Output mirror (left side) has small reflection power and is a good crossing.
Output efficacy of forward second harmonic beam along Z=5 mm for pumping power of Pp=20W and beam radius of rp=5mm is 0.634 %.Reflection is RSF = 0.988.

5-Output Efficacy of Backward Second Harmonic Beam along Z-Direction for Beam Radius of rp = 5 μm
Figure 3 shows that backward second harmonic beam loses some of its energy in backward (toward left side) via optical absorption in crystal (equation 7 manifests it).Backward second harmonic absorption assists in creating population reversion in laser.Increasing population backward leads to increasing infrared light 1060 nm which means increasing output of backward second harmonic beam.
Output efficacy of backward second harmonic beam along Z=5 mm for pumping power of Pp=20W and beam radius of rp=5mm is 0.6327 %.Reflection is RSb = 0.03.

6-Output Efficacy of Forward Laser Beam along Z-Direction for Beam Radius of rp = 5 μm
Figure 4 shows output of forward laser beam.According to this figure, 20% of pumping power is transferred to laser that is expected for solid-state lasers.
Output efficacy of forward laser beam along Z=5 mm for pumping power of Pp=20W and beam radius of rp=5mm is 34.1%.Reflection is RLF = 0.988.

7-Output efficacy of backward laser beam along z-direction for beam radius of rp = 5 μm
Figure 5 shows output of backward laser beam as z for 20w power.This figure also shows than high value of pumping power is transferred to laser beam.
Output efficacy of backward laser beam along Z=5 mm for pumping power of Pp=20W and beam radius of rp=5mm is 34.03 %.Reflection is RLB = 0.998.

8-The Population Difference in Z-Direction for Beam Radius of rp = 5μm
Figure 6 shows that population reverse depends on pumping power and has direct relationship with second harmonic beam efficacy.
The population difference along Z =5mm for pumping power of PP=20W and beam radius of rp = 5mm is 1.262 ×10 24 .

9-Experimental Method
Experimental efficacy of transferring 530 nm beam to 1060 nm in NYAB crystal by Jaque et.al is as follow [15,16]:

10-Conclusion
NYAB solid-state laser is an active non-linear crystal which has been studied in recent years due to its ability in producing green light laser.High effective non-linear coefficient and high damage threshold provides the possibility of being polluted with high value of ND ion, possibility of making pumped laser with diode in green area via self doubler (SFD) NYAB crystal has been the most efficient self doubler laser.
According to computations in MATLAB, thermal effects prevents the increasing opf pumping power.
In this study, five discrete differential equations describing the conversion of 807 nm pump beam to 1060 nm laser beam and then 530 nm second harmonic beam have been used in Yttrium aluminum borate crystal doped with neodymium ion (the most efficient self-doubler crystal).Calculations were done in MATLAB software.Finally, the efficacies of the second harmonic and the laser beams as well as the inverse population difference increased for beam radius of rp=5μm.This method is an interesting idea to achieve high efficacies in studied laser.

Figure 1 .
Figure 1.Generation of the second harmonic beam by nonlinear materials [1].

Figure 2 .
Figure 2. Efficacy of forward second harmonic beam in z-direction (path length).

Figure 3 .
Figure 3. Efficacy of backward second harmonic beam along z-direction (path length).

Figure 4 .
Figure 4. Output efficacy of forward laser beam along z direction (path length).

Figure 5 .
Figure 5. Output efficacy of backward laser beam in z-direction (path length).