Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6, 4089-4100 2024 Publisher: Learning Gate DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate © 2024 by the authors; licensee Learning Gate * Correspondence: zainabalihussein4@gmail.com A theoretical study of the output coupler reflectivity effect on the characteristics of Er+3 laser pulse at passively q-switching Zainab Ali Hussein1*, Abdul- Kareem Mahdi Salih2 1,2Department of Physics, College of Science, University of Thi-Qar, Thi-Qar, Iraq; zainabalihussein4@gmail.com (Z.A.H.) karimmahdisalih@yahoo.co.uk (A.K.M.S.). Abstract: The objective of this study deal the effect of the output coupler mirror reflectivity of laser system on behavior and characteristics (duration and energy) of erbium passive Q-switching laser pulse was studied theoretically. In the methods of the study, virtual passive Q-switching laser system consist of major elements such as erbium doped silica as an active medium, Cr+4: YAG crystal used as a saturable absorber, and the mirrors of resonator. Mathematical rate equations model has been used to describe the physical relationship between this element has been solved numerically by Rung - Kutta – Fehlberg method. The results showed regarding the pulse behavior, the increasing value of output coupler mirror reflectivity lead to decreasing of rising and falling time of pulse, so the emission of pulse occur in an advanced time. While regarding the pulse characteristics, the pulse was distinguishable by short duration and high energy, resulting high power of pulse when the output coupler mirror reflectivity increasing. Regarding of results discussion, the study attributes the results to a decrease of the initial and final population inversion density of active medium that results from the increasing value of output coupler mirror reflectivity. Therefore, it is possible to conclude in order to improve the characteristics of erbium passive Q-switching laser pulse; it is appropriate increase the value of output coupler mirror reflectivity within acceptable limits. Keywords: Cr+4:YAG, Er +3 doped silica, High power pulses, Laser, Passive Q-switched. 1. Introduction Q-switching is excellent technique used in applications which demand high laser intensities. It is mainly applied with solid state lasers in order generation pulses characterized by short duration (10-9 - 1012) second with high power [1,2]. Various applications use this technique extensively, examples include fine material processing, spectroscopy, biomedical treatment, precise distance measurement, and others. [3-5]. There are two types of Q-switching technique, the first type is the active Q-switching, in this type the losses are modulated with an active control element, typically either mechanical, acousto-optic, and electro-optic element [6,7]. The second type is the passive Q-switching technique[8,9]. In this type, an non-linear optical element (saturable absorber material(SA)) added inside the optical cavity[10- 12], this element has the characteristic of self-action (passively) without any external action[13,14], it is shows at the initial time high absorption activity of laser photons which are oscillating inside the system cavity, it is an excellent crystal for passive Q-switching in the wavelength range from 800 nm to 1200 nm, because of its good ratio between its ground and excited levels cross-sections, the absorption activity decreases with the time by semi-exponential relationship until reach to the optical bleaching state and allows oscillating photons to transmittance through it [15,16]. In the virtual optical system of this study, the erbium (Er+3) doped silica material has been used as a active medium, Figure 1 illustrates an intra-4f shell transition from its first excited state ( 213 4I ) to the ground state ( 215 4I ) of the Er+3 ion in its trivalent state [17]. Er+3 application in optoelectronics has 4090 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate caused it to attract a lot of attention,]. While the chromium (Cr+4) doped crystal of YAG has been used as SA material, it is having a great deal of attention as passive Q-switches. The Cr+4 YAG crystals have a wide absorption cross section and a low saturable absorbtion, this results in greater photochemical and thermal stability, and a higher damage threshold [18.19]. Figure 2 shows the energy level of Cr+4 [20]. In the present study, we studied the reflectivity of the output coupler mirror effect on the duration, energy, and the power of Er+3 passive Q-switching pulse. (4) Figure 1. Energy level diagram of Er+3 [17]. 4091 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate Figure 2. Energy level diagram of Cr+3 [20]. 2. Theory Coupled rate equations model as the following equations [21] has been used in this study: )]) 1 (ln()(2)(2)(2[ )()( 2 lossessaesgssagsamam r L R tNltNltNl t dt td +−−−=    (1) am amp tN tNtcR dt tdN   )( )()( )( −−= (2) rgssags sa esgs tntl tn dt tdn   /)()(2 )()( −= (3) rgssags sa eses tntl tn dt tdn   /)()(2 )()( +−= (4) Where: (cm-3) represents the photons number density, clamr /2= (s) refer to the transit time for one round –trip, , rl (cm) is the optical cavity length, am (cm2) is the emission cross section of active medium, c (ms-1) is the speed of light, gs (cm2) is the absorption cross section of SA ground-state, lam (cm) is the AM length, lsa (cm) is the length of SA, ngs (cm-3) represents the ions population of SA ground state, N (cm-3 ) is the active medium population inversion density, nes (cm-3) represents the ions population of SA exited state, R2=( )1RR 1/2 is the geometric mean of the cavity, 1R is the total reflectivity of first mirror, R is the output coupler mirror, lossl is the dissipative optical losses for round –trip. es (cm2) is the absorption cross section of SA excited-state, is the population reduction factor equal 2, 1 for 3 levels and 4 level of active medium system respectively, pR is the rate of optical 4092 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate pumping, sa (s) is the lifetime of the excited level of SA, am (s) refer to the fluorescence lifetime of the upper laser level. SA has a significantly shorter microseconds lifetime compared to the fluorescence life of the upper laser level [22]. The build-up time for Q-switched laser pulses is typically short, The spontaneous decay in AM and SA can be neglected. The pumping rate is very long compare with the build-up time of Q-switched laser pulses, then can be neglected [23]. then Eq.(2), Eq.(3), and Eq.(4) can be reformulation as the below respectively: )()( )( tNtc dt tdN am−= (5) rgssags gs tntl dt tdn  /)()(2 )( −= (6) rgssags es tntl dt tdn  /)()(2 )( = (7) At the initial time, the photons density number inside the optical cavity is minimum. Also most of SA molecules are in the ground state ( gsn ), then can be consider sogs nn  , 0esn , where esgsso nnn += is the total number of SA molecules. Also, at the initial time the SA absorption activity is very high. From Eq.(1) can be consider ( 0dtd ) while cannot consider 0)( =t . Then; 0)) 1 (ln(22 =+−− losssosagsoamam L R nlNl  (8) The spatial variation of pulse energy per unit aria (E) at any point in the length of the SA (within the longitudinal direction of the SA) when the pulse passes through the SA can be expressed by [24]: (9) seEso gs gs es so n hv E hvn dz dE     − − −−−= )]exp(1)[1( The transmission of SA is known as small-signal transmission or initial transmission (To) at low energy. This situation can be taken into account and consider )/1()exp( hvE hv E gs gs   −− , and substituted into Eq. (9) to get: Ennn dz dE essoessogsso ][  −−= Given that gs  greater than es . the term which include es can be neglected, then: = sal gsso E E dznE 0 max min ]ln  The optimization of Emax occur when the SA became bleaching to allowed maximum transmission of photons, then can be estimates hvE maxmax  . It is possible to estimate that Emin optimization occurs when the SA has high absorption activity or small signal transmission of photons hvTE o maxmin  : sagsso oo ln ThvT hv E E    === ) 1 ln(lnln max max min max )exp( asgssoo lnT −= (10) 4093 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate sagsso o ln T 2) 1 ln( 2 = (11) Substituted Eq.(11) into Eq. (8), can be get the initial value of population inversion density as the follow: am loss o o l L RT N 2 ) 1 (ln) 1 ln( 2 ++ = (12) At maximum of  , in Eq.(1) can be consider ( 0 dt d ), soes nn  , or gsn can be neglected. Threshold population inversion density can be estimated in term of To and  , where (    = gs es ) as the following expression: amam loss o th l L RT N   2 ) 1 ln() 1 ln( 2 ++ = (14) 3. Results and Discussion By computer program which prepared in this study, the set of rate equations (1, 5-7) was solved numerically by Rung- Kutta -Fehlberg method. The Table 1 shows the input data which used in computations. Table 1. Computations input data. Parameter Reference Parameter Reference cml am 25= [25] 2191025.2 cmes −= [26] cmlr 300= 2191075.8 cmgs −= 22010575.0 cmam −= ssa 6100.4 − [23] sam 310545.5 −= %952 =R [25] 1= Figure 3 represents the time variation of population inversion density (PID)as a function of R for three cases of laser passive Q-switching pulses generation.The figure shows that the initial population inversion density (IPID) and final population inversion density (FPID) values decreases as the R value increases. for order verify this behavior for both of IPID and FPID, several cases of several R values have been studied as in Figure 4 and Figure 5 , they are shown the decrease in each of IPID and FPID values when R increasing and enhanced the Figure 3. The study explains that, the increasing of R value lead to increasing the number of laser photons reflected from the output mirror to the AM, resulting in two important results. The first, the state of accumulation of ions at the excited laser level was prevented by the strong photons' interaction with these ions and forced to move to the lower level and increased stimulated emission, which caused a decrease in the IPID. The second result is a high number of ions discharge from the excited laser level in less time, which caused high decrease in the FPID. 4094 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate The low value of FPID which is shown in Figure 5 has resulted in the release of a large number of laser photons despite the low IPID value lead to obtained laser pulses characterized by high photons densities as the R value increased as shown in Figure 6. Also observed in Figure 6 is the emission of the pulse occurring in an advanced time as the R value increases, It is possible to note that the times at which the pulses in Figure 6 reach their peaks are 550 ns, 516 ns, and 495 ns approximately at R = 0.76, 0.84,and 0.94 respectively. To verify this behaviour of pulses which is shown in Figure 6, several cases of several R values have been studied as in Figure 7, it is shown the increase in the maximum value of pulse photons when R increasing and enhanced the Figure 6. The study explains this du to the decrease in the FPID values Figure 8 shows an increase in the pulse energy with increasing value of R. The study explains this because of the increase in the maximum value of the pulse photons, as shown in Figure 6 and 7. Figure 3. Profile of population inversion density as a function of time. Figure 4. Initial population inversion density as a function of R. 4095 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate Figure 5. Final population inversion density as a function of R. Figure 6. Photons density of pulses as a function of R. 4096 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate Figure 7. Maximum photons density as a function of R. Figure 8. Pulses energy as a function of R. Figure 9 Represents a decrease in the value of the pulse rising time. The study explains this due to rapid pulse construction by the increase in the photons density with increasing value of R. Figure 10 represents the decrease in the pulse falling time with increasing of R, this means that the population inversion density its final value at an earlier time when the value of R increases ,and this is reinforced by figure 3. where it is observed The FPID reaches its minimum value at time 828 ns, 796 ns, and 758 ns approximately when R =0.76, 0.84,and 0.96 approximately. 4097 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate Figure 9. The rising time of many pulses as a function of R. Figure 10. The fulling time of many pulses as a function of R. Figure 8. The rising time as a function of R Figure 11 represents decrease in the value of the pulse duration with an increase in value of R. This is due to the decrease in the rising and fulling time values of the pulse as shown in figure (9) and (10), respectively. Figure 11 represents an increase in the pulse power, which is due to the decrease in the duration time as shown in figures (11) and the increasing in energy of pulse as shown in Figure (8). 4098 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate Figure 11. Duration time of many pulses as a function of R. Figure 12. power of many Pulses as a function of R. Figure 8. The rising time as a function of R 4. Conclusions The increasing of the output coupler reflectivity of laser system mirror lead to improve the characteristics of passive Q-switching laser pulses. High photons density have been emitted related to decreasing of final population inversion density with R increasing; caused stimulated emission process has a greater intensity resulting in a high-energy pulse, and the time needed to emit it decreases. 4099 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 4089-4100, 2024 DOI: 10.55214/25768484.v8i6.2899 © 2024 by the authors; licensee Learning Gate Improving pulse characteristics was also appearing through short duration resulting in pulse being characterized by high power making it more important in practical applications. Acknowledgements: We are deeply appreciative of the provision of essential resources and facilities by University of Thi- Qar, College of Science, Department of Physics, which facilitated the smooth execution of our research activities. Copyright: © 2024 by the authors. 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