Optimization of the pulse width and injection time in a double-pass laser amplifier
1 Introduction
Among the various laser parameters, laser pulse width is a crucial factor to be selected carefully, depending on the application[ 1 , 2 ]. In particular, nanosecond and millisecond pulses have contributed significantly to applications in daily lives and practical laser work. For example, nanosecond lasers can be used for ultraviolet marking, thin film patterning and PC board cutting, as their pulse widths are in the moderate regime, offering a balanced throughput and high-quality material processing[ 2 ]. Microsecond laser pulses have been widely applied to various medical treatments[ 3 – 6 ]. Nd:YAG laser pulses with a width of microseconds and 1064 nm wavelength are useful for treating keloid, hypertrophic or surgical scars. Moreover, millisecond laser pulses have been used extensively in laser machining, e.g., for cutting, welding and drilling of machinery materials throughout the fields of automobile, shipbuilding and aerospace industries [ 7 – 9 ]. Recently, the high-energy laser system with pulse shaping techniques from nanosecond to millisecond timescales has been actively studied for these applications[ 10 ]. Therefore, as a tool for simulating laser amplifiers, the Frantz–Nodvik (F–N) equation needs to be extended to include diverse pulses without the limitation of their lengths and achieve more accurate results. In addition, as the user demands for laser applications grow, designing a laser amplifier that can maximize the output energy while minimizing the related costs is a vital issue. For achieving the latter, we employed a double-pass structure as a laser amplifier. This choice was made based on the following. First, in a double-pass amplifier – since the input pulse passes through a single gain medium twice – one gain medium has the same effect as using two. Therefore, it can aid the maximization of the input pulse energy. Second, because of requiring only half the amount of parts compared with amplifiers producing similar output energies, it can achieve a higher level of compactness and cost-effectiveness. Regarding the former, if the input pulse width reaches levels comparable to the fluorescence lifetime of the laser-active ions in the gain medium, spontaneous emission and pump energy variation must be considered during the amplification. However, there has been no research carried out considering spontaneous emission and pump energy variation simultaneously with the pulse overlap in a simulation of the pulse amplification. The F–N equation has been used to calculate the amplified pulse energies for the arbitrary input pulses passing through the laser gain medium[ 11 ]. Since the spontaneous emission was neglected in the derivation of the equation, it can be applied only to calculations with this condition satisfied. Neglecting spontaneous emission during the amplification means that the population of the excited state can return to the ground state only through stimulated emission. For example, in the case of inputting an input pulse with a duration of 1 s into a Nd:YVO4 amplifier[ 12 ], the fluorescence lifetime of the laser-active ions (90 s) is significantly longer than the duration of the amplification. Therefore, the pulse amplification ends before the excited ions spontaneously return to their ground state. In this case, the derivation conditions of the equation are satisfied. Besides, due to the short amplification time, no additionally incoming pump energy needs to be considered during the amplification. Strictly speaking, the existing F–N equation can only be applied if the input pulse width is significantly shorter than the fluorescence lifetime of the laser-active ions. In this paper, we have extended the F–N equation to consider dynamically spontaneous emission and pump energy ignored during conventional simulations of the pulse amplification. The suggested method was applied to a flash-lamp-pumped Nd:YAG double-pass amplifier structure, while simultaneously considering the pulse overlap effect as well. We changed the input pulse width and controlled its injection time. We have analyzed how these parameters can maximize the output pulse energy through our simulation results. Finally, we have verified that the optimal input pulse width and injection time can be realized within the given conditions to obtain the maximum output pulse energy.
Fig. 1. (a) Scheme of a double-pass laser amplifier: , , and are the equivalent fluences of the input and output energy in the forward propagating direction in gain media 1 and 2, respectively. Similarly, , , and are the equivalent fluences of the input and output energy in the backward propagating direction in gain media 1 and 2, respectively. The temporal lengths and calculated from the geometry of the amplifier determine the time delay between each input. (b) The front part of the amplified input pulse is reflected on the mirror and is overlapped with the rear part of the input pulse in gain media 1 and 2, according to and .
2 Modified Frantz–Nodvik equation
Equation (
Fig. 2. (a) Numerical F–N equation can be used to calculate the amplification during the whole duration of the input pulse according to the temporal sequence. , , and are the input and output fluences, effective pump and spontaneous emission, respectively. (b) Temporal profile of the flash-lamp pulse whose effective pump energy is J. (c) Variation of the stored energy in the gain medium pumped by a flash lamp: the spontaneous emission shows an exponential attenuation described by the fluorescence lifetime ( ) of the laser-active ions. The solid green line was obtained by considering spontaneous emission, while the green dashed line without.
As shown in Figure
Meanwhile, in the case of pulsed Nd:YAG lasers with a pulse width of tens of microseconds, or even milliseconds, the input pulse is not sufficiently short compared with the fluorescence lifetime of the trivalent neodymium ions (
s) used to dope the YAG crystal. The gray curve in Figure
In Equation (
The spontaneous emission can be expressed as a partial time derivative of the number density of the excited state,
[
14
]. It means that the excited ions exhibit an exponential attenuation described by their fluorescence lifetime
. Therefore, the spontaneously emitted fluence can be described numerically by the last term of Equation (
3 Simulation method and parameters
The proposed method was applied to a double-pass amplifier with the aim of maximizing its output pulse energy. In Figure
Fig. 3. Simulation parameters: the input pulse width, given by its FWHM : 2 ns to 280 s (red) and the input pulse injection time, : s to s (dashed).
Fig. 4. The extraction efficiency as a function of the input pulse width, where s (blue), s (orange), s (red), s (green) and s (purple) (a) without considering spontaneous emission, while (b) with considering. (c), (d) Trend of the stored energy in Nd:YAG rod1 and Nd:YAG rod2 during the amplification, where and ns–280 s. (e) Optimal pulse width and injection time of the input pulse resulting in the maximum extraction efficiency. (f) Normalized input and output pulse shapes at the optimal input conditions in (e).
4 Simulation results and discussion
Figure
In Sections I and II, the input pulse width is remarkably short compared with the fluorescence lifetime of Nd:YAG. Since the amplification time is concise, the stored energy variation of the gain medium is mainly influenced by the stimulated emission and the extraction efficiency depends on the pulse overlap effect in the double-pass amplifier rather than spontaneous emission and pump energy variation. In Section I, the shorter the pulse width, the higher the extraction efficiency because the pulse overlap in the double-pass amplifier is reduced, leading to an increase in the output pulse energy. On the other hand, the extraction efficiency tends to stay almost constant with a varying input pulse width in Section II. As the input pulse width is longer, larger energy is lost due to the pulse overlap, but the pump energy that could not be used in short pulse amplification can be utilized due to extended amplification time. Therefore, this has the effect of increasing the output pulse energy. That is why the extraction efficiency gradually decreases and remains constant as the input pulse width increases throughout Sections I and II. Consequently, when the input pulse width is within a sufficiently short range compared with the fluorescence lifetime of Nd:YAG, a pulse as short as possible is advantageous for high extraction efficiency with reducing overlap loss and rapid depletion of stored energy in gain media as shown in Figures
Differently from Sections I and II, in Section III, the input pulse width becomes comparable to the fluorescence lifetime of Nd:YAG. Hence, the pulse overlap effect, spontaneous emission, and pump energy variation are complexly involved in the extraction efficiency. On the left-hand side of Section III in Figure
5 Conclusions
We have studied the design of a laser amplifier that can maximize its output pulse energy. For this purpose, we have chosen a double-pass amplifier structure that can utilize one laser gain medium twice, leading to economic benefits of its manufacturing and a compact design. Second, a pulse amplification simulation method, considering spontaneous emission and pump energy variation numerically, has been proposed by extending the F–N equation. Finally, we set the simulation parameters to the super-Gaussian input pulse width and the injection time, while considering the pulse overlap effect during the amplification. We observed the extraction efficiency and the stored energy variation in the gain medium to obtain the highest achievable output pulse energy. With the considered parameters, for pulse widths below ns, i.e., for pulse widths significantly shorter than the fluorescence lifetime of the laser-active ions, the result fulfilled our intuitive expectations. Shorter pulses injected when the stored energy of the gain medium becomes maximum can produce higher possible output pulse energy through abrupt depletion of the upper-state ions while reducing the energy wasted by spontaneous emission and temporal overlap. Furthermore, as an intermediate region, input pulse widths between 100 ns and s do not exhibit a noteworthy influence on the output pulse energy. Finally, beyond s pulse widths, we obtained the optimal input pulse width and injection time resulting in the maximum output pulse energy. Sometimes the original F–N equation is limited for the input pulse width range in specific amplifier conditions by the assumptions made for its derivation, e.g., neglecting spontaneous emission. The proposed method allows for a more accurate amplification simulation without the limitation of pulse widths by considering dynamically spontaneous emission and pump energy. Therefore, even if the structure and conditions of the amplifier are different from the demonstrated Nd:YAG double-pass structure, the method is applicable to other amplifiers and enables us to get another meaningful result. Consequently, this paper is expected to be useful for designing pulsed laser amplifiers with the aim of generating more energy under the same conditions.
[1] 1.R. D.Schaeffer ,Ind. Laser Solut. Manuf.32 , 1 ( 2017 ).
[2] 2.R.Patel ,J.Bovatsek , and H.Chui ,Ind. Laser Solut. Manuf.32 , 3 ( 2017 ).
[3] 3.A.Rossi ,R.Lu ,M. K.Frey ,T.Kubota ,L. A.Smith , and M.Perez ,J. Drugs Dermatol.12 , 11 ( 2013 ).
[9] 9.H.Wang ,H.Lin ,C.Wang ,L.Zheng , and X.Hu ,J. Eur. Ceram. Soc.37 , 4 ( 2017 ).
[11] 11.L. M.Frantz and J. S.Nodvik ,J. Appl. Phys.34 , 8 ( 1963 ). 10.1063/1.1702744
[13] 13.J.Jeong ,S.Cho , and T. J.Yu ,Opt. Express25 , 4 ( 2017 ).
[14] 14.W.Koechner ,Solid-State Laser Engineering ( Springer , New York , 2006 ).
[15] 15.J.Sung ,S.Lee ,T.Yu ,T.Jeong , and J.Lee ,Opt. Lett.35 , 18 ( 2010 ).
[16] 16.T.Yu ,S.Lee ,J.Sung ,J.Yoon ,T.Jeong , and J.Lee ,Opt. Express20 , 10 ( 2012 ).
[19] 19.S.Hwang ,T.Kim ,J.Lee , and T.Yu ,Opt. Express25 , 9 ( 2017 ).
Article Outline
Daewoong Park, Jihoon Jeong, Tae Jun Yu. Optimization of the pulse width and injection time in a double-pass laser amplifier[J]. High Power Laser Science and Engineering, 2018, 6(4): 04000e60.