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Abstract
Pulse energy transferred from the peak to wings is inevitable in the process of ultrashort optical pulse amplification due to the accumulation of high nonlinearity, which degrades the temporal pulse quality. The energy-related saturable absorbers are applied to remove the wings of high-energy pulses without causing a severe loss of the peak in this research. A cascaded-saturable-absorbers system is proposed, and the effects of the cascaded order, low-intensity loss coefficient, saturable fluence, relaxation time and input fluence distribution on the pulse wings are numerically investigated, respectively. The obtained results can provide a general guidance of design for high-energy ultrashort pulse cleaning.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-energy ultrashort pulse lasers have been indispensable and effective tools in the field of micromachining, especially for the hard and brittle materials e.g., sapphire, glass and ceramics [1–8]. The characteristics of the ultrashort interaction time between laser pulses and materials prevents from heat accumulated, which is critical to the process. However, pulse energy transferred from the valuable peak to the unnecessary wings in the process of ultrashort pulse amplification is inevitable, resulting in a deterioration of the processing quality. Self-phase modulation (SPM) accumulated in the chirped-pulse amplifiers, as well as higher-order dispersion (HOD) are two primary causes for pulse distortion, which broaden the pulses after compression and degrade the effective energy [9–11]. The third-order dispersion and nonlinear phase shifts could be compensated with each other to some extent, presented by some previous experiments [12–14]; besides, the SPM-induced phase shift in the amplification stages is equivalent to the effect of a dispersive device with corresponding dispersion orders, theoretically demonstrated in the Ref. [15]. However, the wings induced by SPM and HOD required to be dealt with separately in practice—HOD could be compensated by Grisms and chirped multilayer coatings, while SPM could be solved to some degree by controlling the phase of input pulses [16–21]. Therefore, the way to directly and effectively reduce the pulse wings regardless of the causes is desiderate.
The saturable absorbers (SA) that generally have a high absorption on the low-intensity part, while could be bleached by the high-intensity part of pulses, are likely to be used for pedestals removal of ultrashort pulses. Such as Yogo et al. demonstrated an efficient insertable pulse cleaning module that uses a saturable absorber (SA) pair with a compensating multi-pass amplifier to suppresses the amplified spontaneous emission (ASE) level of the pulse pedestal [22]. As nonlinear components, SAs are widely used in mode-locked lasers to generate ultrashort pulses [23–26]. They could be generally classified as fast absorbers and slow absorbers according to the relaxation time in comparison with the duration of input pulses [27,28]. For fast absorbers whose response to input pulses is instantaneous, the absorption acts like intensity-dependent. As a result, fast absorbers would reduce the intensity both in the leading and trailing edge of the pulses, leading to such great energy loss in the peak that it is unacceptable for this application. However, for slow absorbers whose relaxation time are much longer than the input pulse width, the absorption is related to the integral of time-dependent intensity i.e., the accumulated energy before the given moment. As a consequence, slow absorbers would not have a great influence on the peak and trailing edge of pulses if the absorption on the front edge has been deeply saturable. Therefore, such energy-related SAs could be applied to the situations where the leading edge of pulses are unnecessary pedestals.
In this study, numerical simulations are carried out to investigate the absorption characteristics of SAs on the wings of high-energy ultrashort pulses. A scheme of cascaded-SAs system is proposed and studied in order to promote the limited impact of a single SA on pedestals. The effects of low-intensity absorption coefficient and saturable fluence on the wings is also investigated. Moreover, the effect of relaxation time on the trailing edge of pulses is studied and optimized. Furthermore, the influence of input fluence distribution on pedestals is investigated. The proposed method to eliminate the wings of high-energy pulses is simple but effective, therefore cleaner ultrashort pulses with higher energy could be pursued by the use of this technique.
2. Theoretical modeling
The SA is considered as a two-level system whose differential equation can be written as [29]
In comparison with SAs used in other fields especially in mode-locked lasers where the beam needs to be focused onto a tiny spot of tens of micrometer, this application favors an extremely large beam size to avoid damage depending on the input pulse energy, e.g., ${r_0} \approx$2 mm is sufficient for an input pulse energy of 50 µJ. The effect of two-photon absorption (TPA) generally needs to be considered in mode-locked lasers due to the small beam size. In this application, the TPA-induced absorption at the peak can be expressed as
where $\beta$ and d are the TPA coefficient and absorber layer thickness, correspondingly. E and ${t_p}$ represent the pulse energy and pulse width, respectively. For typical parameters $\beta =$ 2.5 × 10−10 m/W, $d =$3 µm, ${r_0} \approx$2 mm, ${t_p} =$300 fs and $E =$50 µJ, ${A_{TPA}}$ is calculated to be 4 × 10−3 at the peak, therefore the TPA-induced absorption is negligible in this application.Nonlinear phase shift (mainly induced by SPM) in the amplification stages is equivalent to the effect of a dispersive device with corresponding dispersion orders $\beta _n^{SPM}$[15], which can be written as
The complex amplitude of the output pulse reflected or transmitted (depending on the types of SAs, in this study the transmission-type SAs are all assumed) from a SA can be written as
For cascaded SAs, i.e., a bunch of SAs being passed through in turn, the output amplitude becomes
The effect of saturable absorption on input pulse are evaluated in two ways: the percentage loss in the peak and wings on the one hand, and peak-sidelobe ratio (PSLR) which is defined as the intensity ratio of the peak over the highest subpeak on the other hand. Here are the definitions
where the energy of the peak is ${E_p} = {\int_{{t_1}}^{{t_2}} {|{U(t )} |} ^2}{\rm d}t$ with ${t_1}$ and ${t_2}$ the lower and upper limit of the integral for the peak, so the energy left in the wings is ${E_w} = {\int_{ - \infty }^\infty {|{U(t )} |} ^2}{\rm d}t - {E_p}$. The subscript i and o stand for the input and output pulse, respectively.3. Simulation results and discussions
3.1 Effect of single SA and cascaded SAs
The effect of a single SA on the pulse wings is firstly investigated. A TOD of 0.02 ps3 is applied to a chirp-free Gaussian pulse with full width at half maximum of 300 fs. As a result, 6.7% of the pulse energy is transferred to the front-edge wings and leads to a PSLR of 8.3. The energy of the input pulse is fixed at 50 µJ in this study, thus the parameters are close to the output of some practical high-energy fiber CPA systems. The beam with a flat-top intensity distribution and a radius of 2 mm, while the SA with an identical size to the beam is assumed. As a consequence, the input fluence is 398 µJ/cm2. In order to minimize the loss of energy and intensity in the peak as far as possible, the relaxation time of the SA should be much longer than the input pulse width, where we set it to 30 ps. The saturable fluence of the SA is set to 100 µJ/cm2. Such parameters of SA used in the simulation are also commercially available. The effect of a single SA on input pulse with the aforementioned parameters and the low-intensity absorption coefficient as high as 0.9 is demonstrated in Fig. 1(a), where the front-edge pedestals of the output pulse (red solid) show no obvious reduction in comparison with the input pulse (black bold). The energy loss of the peak is extremely low ($\Delta {E_p} = $0.6%) but the PSLR is only increased to 11 in the case. A limited result ($\Delta {E_w} = $40%) of reducing the front-edge wings is obtained by the action of a single SA. Therefore, more SAs are expected to be implemented to make a greater difference to the pedestals. Thanks to the characteristic of the energy-related absorption, the effect of cascaded slow absorbers on the front-edge wings is additive and would not lead to huge losses in the peak when optimizing the parameters.
The cumulative effects of cascaded SAs on the front-edge wings is illustrated in Fig. 1(b). The parameters of SAs used in the simulation are as follows: ${A_0} = $ 0.7, ${F_{sat}} = $100 µJ/cm2, $R = {r_0} = $2 mm, $\tau = $30 ps. Meanwhile, the parameters of input pulse are consistent with the case of a single SA. All cascaded SAs are assumed identical for simplicity, while better performances could be obtained by optimizing the parameters of each SA. The simulation result indicates that both $\Delta {E_w}$ and PSLR ascend with the increase of the cascaded orders, however, the energy loss of the peak boosts at the same time. For N = 6 (magenta), 99% of the energy in the wings is absorbed and the PSLR is promoted by a factor of 33, but it causes an energy loss of 7.6% in the peak, which is intolerant since the magnitude is even greater than the one of the whole pedestals. For N = 4 (red), the energy loss of the peak is 3.5%, which could be acceptable since it is about 50% of the energy in the front-edge wings, and majority of the energy in pedestals (86%) is absorbed. For N = 1, 2 (blue and green), however, the reduction of the wings is apparently insufficient. The simulation result indicates that a better impact of the cascaded SAs on the removal of front-edge wings could be realized if N is optimized for the certain parameters of input pulse and SAs.
The scheme of a four-stage cascaded-SAs system is illustrated in Fig. 2, where the high-energy ultrashort pulses generated from the amplification-compression stages are injected into four transmission-type SAs. Each SA is passed through once, thus constituting a four-stage cascaded-SAs system. The sequences of SAs the pulses passing through are SA1→SA2→SA3→SA4. The proposed cascaded-SAs system could be expanded to higher orders by increasing the number of transmission-type SAs in Fig. 2.
3.2 Effect of low-intensity loss coefficient and saturable fluence
Indicated by Eq. (2), the time-dependent absorption $A(t )$ is proportional to low-intensity absorption coefficient ${A_0}$. In the action of cascaded number N, $A(t )$ increases with the Nth power of ${A_0}$, as a result, the absorption on the front-edge pedestals mounts dramatically. The effect of a four-stage cascaded SAs system on the front-edge wings with the change of ${A_0}$ is illustrated in Fig. 3(a), where the parameters of input pulse are the same with the ones presented in 3. 1. The parameters of four identical SAs are as follows: ${F_{sat}} = $100 µJ/cm2, $\tau = $30 ps, $R = $2 mm. A minor increase of PSLR induced by the small ${A_0}$(magenta and blue solid) and an excessive energy loss of the peak caused by the large ${A_0}$(red solid curve) are both unsatisfied. Therefore, a modest ${A_0}$ that introduces a great reduction of the wings and a minor loss of the peak is favored. The green solid curve stands for the preferable case, where the energy loss of the peak is less than 50% of the pulse energy in the wings of input pulse, i.e., $\Delta {E_p} \approx $ 3.5%; and the energy reduction of the front-edge pedestals is over 80%. Furthermore, the ascending absorption in the trailing edge with the increase of ${A_0}$ indicates a larger ${A_0}$ is well-suited for removing the trailing edge if it is unnecessary, illustrated by the absorption curves (dash curves).
The impact of ${F_{sat}}$ on $A(t )$ is not intuitive from the Eq. (2). However, it can be directly deduced from the approximate slow absorber model ${{\partial A} / {\partial t}} ={-} {{A(t )\cdot I(t )} / {{F_{sat}}}}$ with the solution $A(t )= {A_0} \cdot \exp \left[ { - \int_{ - \infty }^t {{{I({t^{\prime}} )} / {{F_{sat}}}}} {\rm d}t^{\prime}} \right]$ that, $A(t )$ grows with the increase of ${F_{sat}}$, where the slow-absorber model can be obtained by neglecting the second term of Eq. (1) for $\tau \gg {t_p}$. The effect of saturable fluence on front-edge wings is illustrated in Fig. 3(b), where a four-stage cascaded-SAs system is used. The parameters of input pulse are the same with the ones presented in 3. 1, and the parameters of four identical SAs are as follows: ${A_0} = $0.6, $\tau = $30 ps, $R = $2 mm. The simulation result accords well with the approximate slow-absorber model—the larger the ${F_{sat}}$, the higher the $A(t )$, as well as the larger amount the reduction of wings and peak. As a consequence, improper value of ${F_{sat}}$ would also lead to insufficient reduction of the front-edge wings (magenta solid) or excessive loss of the peak (green and red solid).
The above simulation results show that $A(t )$ boosts with the increase of ${A_0}$ or ${F_{sat}}$, however, in consideration of the difficulty in producing SAs with so large ${A_0}$ and ${F_{sat}}$ simultaneously, the effect of different combinations of ${A_0}$ and ${F_{sat}}$ on the wings that may also lead to excellent result is investigated. The cases of equal $\Delta {E_w}$(80%) induced by different combinations of ${A_0}$ and ${F_{sat}}$ are illustrated in Fig. 4(a). Indicated by the simulation result, having the same impact on the front-edge wings, SAs with larger ${A_0}$ and smaller ${F_{sat}}$ at the same time is preferable since that would lead to less energy loss of the peak. Moreover, the cases of identical $\Delta {E_p}$(3.5%) achieved by different combinations of ${A_0}$ and ${F_{sat}}$ are illustrated in Fig. 4(b). Paying the same price for losing a certain amount of energy in the peak, SAs with the combination of larger ${A_0}$ and smaller ${F_{sat}}$ are also preferred because it would have a greater influence on the front-edge wings and lead to a higher PSLR.
Fig. 4. Effect of the combination of low-intensity loss coefficient and saturable fluence on the wings to achieve the goal of (a) $\Delta {E_w} = $80% and (b) $\Delta {E_p} = $3.5%.
3.3 Effect of relaxation time
SAs could be generally classified as slow absorbers ($\tau \gg {t_p}$) and fast absorbers ($\tau \ll {t_p}$) according to the relaxation time in comparison with the duration of input pulses. Indicated from the solution of slow absorber presented in 3. 2, the time-dependent absorption $A(t )$ is inversely proportional to the integral of time-dependent intensity $\int_{ - \infty }^t {I({t^{\prime}} )} {\rm d}t^{\prime}$, i.e., pulse energy before t, thus $A(t )$ of the slow absorbers acts like energy-dependent. In comparison, for $\tau \ll {t_p}$ the dependency ${{\partial A} / {\partial t}}$ in Eq. (1) can be neglected, so the approximate model of fast absorbers could be written as ${{[{{A_0} - A(t )} ]} / \tau } = {{A(t )\cdot I(t )} / {{F_{sat}}}}$with the solution $A(t )= {{{A_0}} / {[{1 + {{I(t )\cdot \tau } / {{F_{sat}}}}} ]}}$. As a result, fast absorbers are intensity-dependent such that the leading and trailing edge of the peak would suffer from great losses even deeply saturable. Therefore, fast absorbers are not suitable for this application.
The effect of relaxation time on front-edge wings is illustrated in Fig. 5(a), where the parameters of input pulse are the same with the ones presented in 3. 1. A four-stage cascaded-SAs system is used in the simulation and the parameters of four identical SAs are as follows: ${A_0} = $0.6, ${F_{sat}} = $100 µJ/cm2, $R = $2 mm. For $\tau \ll {t_p}$(magenta), the absorption curve shows a nearly symmetric absorption characteristic in the leading and trailing edge of pulse, leading to a huge loss of the peak. For $\tau \gg {t_p}$(yellow and red), the absorption of both wings and peak exhibit minor difference with the increase of $\tau $. For $\tau $ and ${t_p}$ approximately in the same order of magnitude (blue and green curves), the absorption of the trailing edge becomes increasingly obvious with the increase of $\tau $. Therefore, a shorter $\tau $ is expected to play some positive role in the presence of unnecessary trailing-edge wings. A tradeoff between increasing the absorption of trailing-edge wings and reducing the loss of the peak is demonstrated in Fig. 5(b). The input pulse with symmetric wings in the front and trailing edges is generated from a linear-chirped Gaussian pulse (de-chirped duration of 300 fs) that firstly accumulates a nonlinear phase shift of π rad and is subsequently compressed by second-order dispersion with a suitable value. As a result, 10% of the energy is transferred to the wings of the leading and trailing edge equally. The other parameters of input pulse and cascaded-SAs system are the same as the above mentioned. The simulation result indicates a suitable shorter $\tau $ could make some difference in reducing the trailing-edge wings, however, a much too short $\tau $ would still lead to significant loss of the peak.
Fig. 5. Effect of relaxation time on (a) front-edge wings and (b) wings in both leading edge and trailing edge
4. Effect of input fluence distribution
In the aforementioned simulations the input pulse with a flat-top profile is assumed, therefore the input fluence distribution along the beam axis is spatially averaged. However, in some cases the fluence profile deviating from the flat-top distribution, a great difference would occur. For SAs with finite size, the input fluence distribution influences the absorption characteristics in two ways: the one hand the input energy out of the area of SA is filtered out, on the other hand the integral of time-dependent intensity $\int_{ - \infty }^t {I({t^{\prime}} )} {\rm d}t^{\prime}$ is reduced if it is not spatially averaged and needs to be reconsidered. Therefore, the output amplitude with consideration of a non-averaged input fluence could be rewritten as
where $h(R )= {{\int_0^R {F(r )\cdot 2{2\pi }r2{\rm d}r} } / {\int_0^\infty {F(r )\cdot 2{2\pi }r2{\rm d}r} }}$ is the ratio of input energy on the absorber. A circular SA with the radius R ($R \ge {r_0}$ in this application) concentric with the beam is assumed. The time-dependent absorption of Eq. (2) is rewritten asFig. 6. Effect of input fluence distribution in the form of super-Gaussian function on front-edge wings.
5. Conclusions
In conclusion, we demonstrate the way to remove unnecessary wings of high-energy ultrashort pulses. In order to have a great effect on the reduction of high-energy pulse wings, we select the SAs parameter with a high low-intensity absorption coefficient and large saturable fluence are better choice. Further, a great improvement of the performance on the reduction of pulse wings could be realized by increasing the order of the proposed cascaded-SAs system. Generally, a four-stage cascaded-SAs system constituted by four SAs is sufficient to have a significant impact on the pulse wings. The preferable relaxation time depends on the goal to be realized—for the case in which large energy loss of the peak is unacceptable the relaxation time should be as long as possible, while for the case where the primary goal is to reduce the trailing-edge wings but place less emphasis on the loss of the peak, the relaxation time should be in the same order of magnitude as the pulse width. Moreover, the input fluence with a flat-top distribution is the optimal choice because no pulse energy is filtered out by the SAs for the case of identical-size SA to the input beam. The input fluence distribution with a high-order super-Gaussian profile is also well suited for this application.
Funding
Higher Education Institutions from Shenzhen Science, Technology Innovation Commission (20200925162216001); Special Funds for the Major Fields of Colleges and Universities by the Department of Education of Guangdong Province (2021ZDZX1023); Basic and Applied Basic Research Foundation of Guangdong Province (2021B1515120013); State Key Laboratory of Information Photonics and Optical Communications (IPOC2020A002); Natural Science Foundation of Guangdong Province (No. 2022A1515011434); The Open Projects Foundation of State Key Laboratory of Optical Fiber and Cable Manufacture Technology (No. SKLD2105); Key-Area Research and Development Program of Guangdong Province (2020B090922006); National Natural Science Foundation of China (62005081); Key-Area Research and Development Program of Foshan City (2120001009232).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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