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Abstract
Techniques to generate a targeted temporal waveform with high accuracy are desirable to extend the application range for pulse shapers. In this study, a target energy adjustment mechanism is applied to the input-output iterative Fourier transform algorithm (IFTA). It is numerically demonstrated that, considering multi-pulse temporal waveforms, the developed algorithm provides a suitable spectral phase modulation pattern and improves the shape of the temporal waveform compared to that of the input-output IFTA.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Pulse shapers for short laser pulses [1,2] are extensively used in fields such as laser processing [3–6], nonlinear optical microscopy [7–11], ultrafast imaging [12,13], optical fiber communication [14,15], and THz pulse generation [16–18]. In these studies, several types of multi-pulse waveforms are widely used. In the field of laser processing, for example, airy pulse has been reported as a useful multi-pulse waveform to realize narrow and deep hole drilling for SiO2 [5]. A multi-pulse waveform with a particular temporal gap has also been reported to show that an ablation volume is much higher than that by a conventional pulse waveform [3]. Therefore, accurate and flexible control of each peak intensity in a multi-pulse temporal waveform is an essential technology to improve the performance of the applications.
As a method to control multi-pulse temporal waveforms, the Fourier synthesis method, which is also known as spectral pulse shaping, is widely utilized [1,2]. In this method, the temporal waveform of an incident laser pulse is optically Fourier-transformed, and it is converted into a spectral waveform. The spectral waveform is modulated with, for instance, a spatial light modulator. The modulated spectral waveform is then inversely Fourier-transformed, reconstructing temporal waveform. The shape of the reconstructed temporal waveform depends mainly on the spectral phase modulation pattern. In many cases, however, an appropriate phase modulation pattern for a targeted temporal waveform cannot be analytically derived. Therefore, phase retrieval algorithms [19–21] aimed at optimizing a spectral phase modulation pattern are used for realizing the targeted temporal waveform [22,23].
The various phase optimization algorithms are divided into the following families: the iterative Fourier transform algorithms (IFTAs), genetic algorithms, and simulated annealing algorithms. The IFTAs are commonly used because they converge to a solution faster than the other algorithms [22,23]. M. Hacker et al. [22] has given a detailed report on the comparison with an IFTA and genetic algorithm. Since the IFTAs still tend to fall into a local solution when a complicated temporal waveform is required [22], further development for the IFTAs is desired to find a better solution that is close to the global solution.
In this study, a target energy adjustment mechanism is introduced into an input-output IFTA (IOA) [19,22]to make the IOA more robust. The target energy adjustment mechanism adjusts an energy miss-match between a target temporal waveform and a calculated temporal waveform in every iteration in the IOA. It is numerically demonstrated that, for multi-pulse waveforms, the developed algorithm finds spectral phase modulation patterns to improve the shape of the temporal waveform compared to that of the IOA.
2. IFTAs to calculate a spectral phase modulation pattern
Figure 1 shows a basic flow diagram of IFTAs to calculate the spectral phase modulation pattern $\varPhi (\omega )$ for shaping pulse waveforms [22,23]. This algorithm Fourier transforms an optical field in each iteration (forward and backward) between a spectral domain and a temporal domain. During the iterations, this algorithm replaces the intensity pattern in each domain. In the temporal domain, the intensity pattern is replaced with a target intensity pattern. This is a user-designed targeted-temporal-intensity pattern. In the spectral domain, the spectral intensity pattern is replaced by a spectral intensity pattern of the input beam ${A_0}(\omega )$. In this way, the processes from (II) to (IX) are repeated, and the spectral phase modulation pattern $\varPhi (\omega )$ is calculated.
Fig. 1. Flow diagram for a basic IFTA to calculate the spectral phase modulation pattern $\varPhi (\omega )$. ${A_0}(\omega )$ and ${\varPhi _0}(\omega )$ are the initial spectral intensity patterns of the input beam and the initial spectral phase pattern, respectively. Both patterns are used as initial parameters to start the algorithm.
The replacement procedure shown in (V) determines the performance of the IFTA. In the most basic IFTA proposed by Gerchberg and Saxton (GS), the temporal intensity pattern is entirely replaced with a fixed target intensity pattern ${E_{TARGET,\; 0}}(t )$. ${E_{TARGET,\; 0}}(t )$ is the initially given desired pattern. Although the GS method is widely used to calculate the spectral phase modulation pattern, it can fall into a local solution. Finding a better solution using the GS method becomes particularly difficult when a complicated temporal intensity pattern is set into the algorithm as a ${E_{TARGET,0}}(t )$. As a more robust algorithm, IOA has been proposed, which uses an advanced replacement mechanism given by Eq. (42) in [20]. The IOA replaces the temporal intensity pattern with a modified temporal intensity pattern. The modified pattern, ${E_{TARGET,n}}(t )$, is calculated by an equation expressed as follows:
The performance of the IOA differs greatly, depending on the calculation conditions. Here, methodology on how to measure the IOA performance is first explained. Thereafter, two significant dependencies that affect IOA performance are also described.
When measuring the IOA performance, a root means square (RMS) error of ${a_n}(t )$ to ${E_{TARGET,\; 0}}(t )$ is often used [22]. The RMS error indicates how close the $\; {a_n}(t )$ is to ${E_{TARGET,0}}(t )$. For instance, zero RMS error indicates that ${a_n}(t )$ and ${E_{TARGET,0}}(t )$ are the same. The measurement of RMS error is useful in monitoring the IFTA performance during the iterations.
The IOA has two main dependencies. The first one is the dependency on the number of iterations. The value of the RMS error decreases as the number of iterations is increasing, and it converges to a specific value [22]. Therefore, termination condition of iteration is generally defined as the following: (1) when the fixed number of iterations has been completed; (2) when the RMS value becomes better than a threshold. In this study, we used condition (1) as the termination condition.
The second one is that the RMS value depends on the initial phase pattern [22,24]. An IFTA trial hardly gives a good solution that has a small RMS value. One approach to this problem is to select the best solution with the smallest RMS error from the ones calculated in many IOA trials with different initial phase patterns.
It is defined that the RMS value as a function of n and ${\varPhi _0}(\omega )$ is as follows:
In this study, the number of iterations is set as $n = 100$. This is enough iteration number that can make RMS error plateau [22,23]. In order to obtain a statistically valid solution, 50 trials, $\textrm{k} = \{{1,2 \ldots ,{\; }50} \}$, were performed. The different 50 phase patterns used in the trials are generated using a random function. The best solution with the smallest RMS error is selected from the calculated 50 solutions. In this paper, the 50 trials and best selection processes are set as an optimization protocol.
Figure 2 shows typical $RMS(n,{\varPhi _{0,k}}(\omega ))$ plots calculated in IOA. In this calculation, it is assumed that the initial spectral intensity ${A_0}(\omega )$ is a squared hyperbolic secant function with an FWHM of 5 nm. The center wavelength of the spectrum is 800 nm. ${E_{TARGET,0}}(t )$ is set as ten pulses with a gap of 1 ps and with the same peak values. The $\gamma $ is set as 0.4. An averaged plot of 50 trials is shown in Fig. 2. The error bar shows a standard deviation for the 50 trials at each iteration. The averaged plot decreases because of the increasing number of iterations, and it becomes plateau when the number of iterations is over 60. Although the error bar also becomes a plateau as an increasing number of iterations, the error bar at $\textrm{n} = 100$ is relatively large compared to the RMS value. This shows that the initial phase of dependency is significant in this calculation.
Fig. 2. Typical RMS plots calculated in IOA. A waveform of ten pulses with the same peak value and with a gap of 1 ps is set as the ${E_{TARGET,\; 0}}(t )$.
An example of the calculated spectral phase pattern, ${\varPhi _{IOA}}(\omega )$ is shown in Fig. 3(a). This is the best phase pattern in the 50 trials at $\textrm{n} = 100$ as mentioned above. The corresponding temporal waveform is shown in Fig. 3(b). It demonstrates that the IOA produces a spectral phase pattern that realizes a waveform with ten pulses. In the temporal waveform, however, significant variations in each peak value can be seen. It is also seen that there are higher-order intensities (HOI) at the outer side of the ten pulses as shown in Fig. 3(c). To the best of our knowledge, the HOI cannot be removed with phase-only modulation.
Fig. 3. Spectral waveform calculated by IOA to realize ten pulses with a gap of 1 ps and the corresponding temporal waveforms. The blue line shown in (c) is a magnified plot that corresponds to the dotted area.
3. Target energy adjustment mechanism
${E_{TARGET,0}}(t )$ is designed such that its total energy is equal to total energy of temporal intensity ${a_n}(t )$ in the IOA. Therefore, the energy relationship can be expressed by the following equation:
As mentioned above, the HOI is generated at the outer side of targeted peaks in ${a_n}(t )$. In this case, the total energy in the range of the target pulses is slightly lower than that of ${E_{TARGET,0}}(t )$, as shown in Eq. (4). The s and e indicate the range of the target pulses.
From the viewpoint of energy mismatch, ${E_{TARGET,0}}(t )$ is not an appropriate target pattern. The total energy of HOI, ${E_{HOI}}$, can be defined as Eq. (5).
For the compensation of the energy difference caused by ${E_{HOI}}$, we introduced a target energy adjustment mechanism (TEAM), which is defined in Eq. (6,) and introduced it into the replacement procedure (V) in the temporal domain as shown in Fig. 1. As described in Eq. (6), the TEAM is implemented by searching for a coefficient $\mathrm{\beta }$ to minimize the RMS error between $\beta \;{E_{TARGET,0}}(t )$ and ${a_n}(t )$. Based on the calculated $\mathrm{\beta }$, an appropriate target pattern can be regenerated.
We introduced the TEAM into the IOA. It is referred to as the IOA-TEAM. The replacement procedure of IOA in the temporal domain shown in Eq. (1) can be extended by introducing the $\mathrm{\beta }$ as expressed in Eq. (7):
4. Simulation for multi-pulse waveform generation tables
In this section, as an explanation of this IOA-TEAM, the following two are described respectively: (1) the operating functionalities of TEAM as shown in Eq. (6), (2) the effectiveness of the IOA-TEAM for the target pattern of the multi-pulse temporal waveform.
The TEAM compensates for the energy mismatch by finding an appropriate coefficient $\mathrm{\beta }$. The $\mathrm{\beta }$ increases when ${E_{HOI}}$ decreases according to Eq. (5). Figure 4 shows a typical example of plots of $\mathrm{\beta }$ and ${E_{HOI}}$, which are calculated in IOA-TEAM. It shows that $\mathrm{\beta }$ and ${E_{HOI}}$ have an inverse correlation. We confirm that a similar trend is found in other target cases.
Fig. 4. Plotted energy variations of ${E_{HOI}}$ during iteration calculation in IOA-TEAM. The coefficient $\mathrm{\beta }$ is also plotted in the dotted line. Calculation parameters used in this calculation are the same as those in Fig. 3. Both plots of ${E_{HOI}}$ and $\mathrm{\beta }$ are the average values of the 50 trials.
The differences between IOA and IOA-TEAM are described here while showing the typical calculation results of the spectral phase patterns. The calculated spectral phase pattern ${\varPhi _{IOA - TEAM}}(\omega )$ and the corresponding temporal waveform are shown in Figs. 5(a) and (b) respectively. The calculation parameters such as ${E_{TARGET,0}}(t )$, ${A_0}(\omega )$, $\gamma $, and a set of 50 initial phase patterns are the same as those in Fig. 3. The IOA-TEAM can produce a different spectral phase pattern ${\varPhi _{IOA - TEAM}}(\omega )$ from the pattern ${\varPhi _{IOA}}(\omega )$ produced by the IOA. An enlarged view is shown in Fig. 5(c) to clearly show the difference between the results obtained using IOA and IOA-TEAM.
Fig. 5. (a) Spectral phase pattern, ${\varPhi _{IOA - TEAM}}(\omega )$ which is calculated by IOA-TEAM. (b) Corresponding temporal waveform. An enlarged view of the phase patterns is shown in (c) to clarify the difference in the spectral phase patterns between ${\varPhi _{IOA - TEAM}}(\omega )$ and ${\varPhi _{IOA}}(\omega )$. Calculation parameters used to derive the ${\varPhi _{IOA - TEAM}}(\omega )$, and ${\varPhi _{IOA}}(\omega )$ are the same.
Concerning the shape of the temporal waveforms, the differences in the spectral phase patterns mainly appear in peak variance of the pulses in the temporal waveforms. We define a standard deviation for the pulse peak values, $\sigma $, as a measure of the shape quality of the multi-pulse waveform. $\sigma $ is calculated for the multi-pulse waveforms normalized to have an average peak power of 1. The standard deviations for the temporal waveforms shown in Fig. 3(b) and Fig. 5(b) are defined as ${\sigma _{IOA}}$ and ${\sigma _{IOA - TEAM}}$ respectively. Each value is shown in the corresponding figure. The difference reveals that, in the case of the ten pulses waveform, IOA-TEAM can find a better solution than IOA because IOA-TEAM has a lesser peak variation than IOA.
Reducing the peak variations with the use of IOA-TEAM becomes effective for various multi-pulse waveforms. The effectiveness is estimated in the calculation of the ratio ${\sigma _{IOA}}/{\; }{\sigma _{IOA - TEAM}}$ for each number of the multi-pulse waveform as shown in Fig. 6(a). We repeatedly calculated the optimization protocol (50 IOA trials and best selection) for 30 times to show the error bars of the improvement ratio. It shows that, in the range of three pulses to ten pulses, improvement between 1% to 15% is statistically confirmed. Figure 6(b) shows the ratio of improvement over 1.0. It shows that, in most cases, the improvement ratio exceeds 1.0. The ratio of improvement below 1.0 was just 10% in the case of eight pulses and 20% in the case of ten pulses. Therefore, it is shown that the developed IOA-TEAM is valid and useful for multi-pulse waveform control. In the case of two pulses, the improvement ratio actually calculated was 0.36${\pm} $0.38, which was far below 1.0.
Fig. 6. (a) Improvement ratio of ${\sigma _{IOA}}/{\; }{\sigma _{IOA - TEAM}}$ calculated as a function of a number of pulses. The ${\sigma _{IOA}}$ and ${\sigma _{IOA - TEAM}}$ are standard deviations for peak values for the temporal waveform calculated by IOA and IOA-TEAM, respectively. The gaps of the pulses are all set as 1 ps. The values (${\sigma _{IOA}}/{\; }{\sigma _{IOA - TEAM}}$) exceeding 1.0 shows that IOA-TEAM can find a more appropriate phase solution than IOA. (b) Ratio of improvement over 1.0. (c) Difference between ${E_{HOI,IOA - TEAM}}$ and ${E_{HOI,IOA}}$ calculated as a percentage of the total energy of the temporal waveform.
One more thing that has to be considered is the HOI which is an undesirable artifact in a pulse waveform. The impact of newly introduced TEAM on HOI is estimated by calculating ${E_{HOI,IOA - TEAM}}$ and ${E_{HOI,IOA}}$. ${E_{HOI,IOA - TEAM}}$ is the total energy of HOI calculated using IOA-TEAM, and ${E_{HOI,IOA}}$ is that of HOI calculated using IOA. The difference between ${E_{HOI,IOA - TEAM}}$ and ${E_{HOI,IOA}}$ is calculated as a percentage of the total energy of a temporal waveform as expressed in the equation below:
The $\delta {E_{HOI}}$ means the amount of change in the HOIs with or without the TEAM. The positive value of $\delta {E_{HOI}}$ indicates that the introduction of TEAM increases the HOI. The plots of $\delta {E_{HOI}}$ are shown in Fig. 6(c). The increase in the HOI is about 0.22% in average. This is the maximum value in the case of the six-pulse waveform, and in other cases, the value was smaller than 0.2%. Comparing the plots of Fig. 6(a) to those of Fig. 6(c), it can be understood that the pulse peak variance is improved by approximately up to 15% even though the $\delta {E_{HOI}}$ hardly changes. Therefore, introducing TEAM is considered as a cost-effective approach.
5. Conclusions
We have proposed a target energy adjustment mechanism (TEAM) and the introduction of TEAM into the input-output iterative Fourier transform algorithm (IOA). We evaluated the performance of a TEAM-introduced iterative Fourier transform algorithm (IOA-TEAM) using a numerical calculation. In a demonstration, multi-pulse waveforms with from two pulses to ten pulses have been calculated. As a result, it was confirmed that a standard deviation for peak intensity values of the multi-pulse waveforms was improved by up to approximately 15% with and without TEAM.
The developed TEAM can be applied to other phase retrieval algorithms, and can be widely used for pulse shaping studies to generate a more accurate temporal waveform for both multi-pulse waveform and other shapes of the waveforms. The TEAM-introduced IOA used in this study is one-dimensional. However, the method can be easily extended to higher dimensions, and used for hologram design, phase retrieval problems, etc.
Funding
Japan Science and Technology Agency; Adaptable and Seamless Technology Transfer Program through Target-Driven R and D.
Disclosures
The authors declare no conflicts of interest.
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