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Abstract
We simulate a four-channel coherent beam combination (CBC) system designed for femtosecond petawatt lasers. Typical behavior characteristics of the CBC system are revealed. Key parameters relevant to engineering are identified. A simple automatic alignment method is also demonstrated with the virtual CBC system. This work may benefit relevant projects around the world.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Before Maiman [1] demonstrated the first operational laser in 1960, people already realized that the laser when focused could reach very high intensity approaching or even beyond that at the surface of sun [2]. Today, the reported highest focused intensity of laser has already surpassed the sun for many orders of magnitude [3]. The high intensity of lasers can create extreme physical conditions in laboratory, facilitating the fundamental and application research in many fields, such as ICF [4–6], particle acceleration [7,8], lab astrophysics [9], QED [10,11], etc. Despite these great achievements since 1960, the pursuit of higher intensity never stops. A global effort towards making bigger and more powerful lasers is continuing.
The fact that modern lasers can be staggeringly intense is mainly due to a technique called chirped pulse amplification (CPA) [12], and a later variant called optical parametric chirped pulse amplification (OPCPA) [13]. In this technique, a weak laser pulse is first temporally stretched by gratings to nanosecond level so that during amplification stage the laser does not damage the optical elements; then the stretched pulse is amplified in gain medium, followed by temporal compression with several optical gratings to pico- or femto-second level. Current technologies can deliver peak power above petawatt (PW, 1015W) level [14–22]. Further improvement is mainly limited by the size and damage threshold of optical gratings in the compressor. While scientists are trying to improve the manufacturing of gratings [23,24], several methods have been proposed to circumvent this problem for higher power. One is using tiled gratings [25–32]. The other is through coherent beam combination (CBC) [33–40].
The experimental study of coherent beam combination has lasted for a long time [41]. However, it is mainly used for achieving high average power with fiber lasers. Work involving combining femtosecond sources is rare [42–47], particularly combining large beams (approaching meter size) with very high power.
The Shanghai Extreme Light (SEL) facility is planning to be built as a 100 PW laser with 1500J pulse energy and 15fs pulse duration. In a preliminary design, the architecture will be based on OPCPA. After the last amplification stage, the nanosecond 500mm-by-500mm beam will be 2500J. Direct compression will damage the grating. In order to achieve this goal, CBC is one option. First, the amplified beam is split into 4 beams with similar energy. Then, the four beams are compressed separately. Last, they are tiled into 2-by-2 beam matrix, covering about 1m-by-1m area, before final focusing in a vacuum chamber for performing physics experiment. The aim for the CBC system is to achieve the spatiotemporal overlap of the 4 beams at the focus with minimal optical aberrations.
One challenge in building such CBC system originates from the size of the beam. The 500mm-by-500mm cross section needs meter-sized optics. The combined 4-beam requires even larger ones. As the optics becomes larger, the surface quality is hard to control. On the other hand, beams with larger size are more sensitive to small deformation of the optics. The beam will suffer from large aberration during interaction with the optics. Deformable mirrors (DM) are usually used to correct such aberrations. However, complete elimination is difficult. The next challenging part is to achieve and maintain the interferometric precision of the CBC system which consists of many meter-sized optics with heavy weight. Vibrations, air flows, thermal drift, input beam fluctuations all affect the coherent combination process. Besides these challenges, the cost for building such system is enormous, which limits the trial-and-error practice.
Considering the above difficulties, it would be beneficial to carry out extensive numerical modeling on the CBC process to identify the various factors that influence the performance. There have been many researches on this topic in recent years [48–52], which generally determine the requirements on the allowable mismatch of the time delay, pointing stability, dispersion, wavefront aberration, etc. among the beams to be combined. These researches bring many insights into the physical process. In this work, we take a different approach by building a realistic model of a CBC system and simulating its behavior. By doing so, we establish a new framework to study femtosecond petawatt (fs-PW) CBC system and relevant problems; reveal some behavior characteristics of the CBC system yet to be built; identify key parameters relevant to engineering; pave the way to design monitor and control methods with the help of the virtual CBC system.
The rest of the paper is organized as follows. Section 2 describes the model, including the key parameters of the laser, the CBC system configurations and methods of numerical simulations. The influence of input beam imperfections on focusing efficiency (defined by ratio of actual peak intensity to ideal peak intensity, also called Strehl ratio) is also studied. Section 3 studies the various imperfections of the CBC system separately, including delay, beam pointing, wavefront curvature and grating compressor mismatches among the four paths. In section 4, a CBC system with many kinds of imperfection is first generated. The system performance is analyzed, and its optimization is discussed. The paper ends with a conclusion in section 5.
2. Baseline model
The laser pulse before CBC system is about 2500J/4ns. The transverse section is 500mm-by-500mm super-Gaussian profile of order 20. The spectrum convers from 820nm to 1060nm, corresponding to a Fourier transform-limited duration of about 14fs as shown in Fig. 1.
Fig. 1. Input pulse parameters. (a, b) Beam transverse intensity profile; (c) Spectrum; (d) Fourier-transform-limited temporal intensity profile.
The input pulse is split by three 50:50 beam splitters into four beams each with one quarter input energy. The four beams are then directed to different 4-grating compressors for temporal compression. The line density of the gratings is 1480 lines/mm. The input angle is 62 degree. The slant distance between the first and second grating for 910nm is 1.3m. After compression, the four beams are reflected on several mirrors (one of them will be a DM mirror for each beam) before joining together on a larger mirror. Then, the tiled beam matrix is directed downwards to a lower floor and to a focusing off-axis parabola mirror (OAP). The reflected focal length is 3m; the off-axis angle is 20 degree.
Figure 2 shows a configuration of the CBC system. This configuration adopts a two-layer architecture. For each layer, two pulse compressor modules are placed side by side. To balance the dispersion induced by transmission through beam splitters, glass plates with same thickness are inserted at proper places. The gap between the up-layer and down-layer optics is set at 10cm to take into account the mechanical support. It leads to large gaps between the four beams, and results in lower peak intensity at the focus. To alleviate this problem, we narrow the corresponding gaps between mirrors M#5. This may require special design of the optics and mechanical support in practice. It should be noted that configuration in Fig. 2 only includes basic components for simplicity of analysis (common components like image-relay module is not included here; but adding them is straightforward). The cross section of the beam envelops after combination is shown in Fig. 3(a) with proper adjustment of the beam gaps. The envelop sides of the beams do not align perfectly with the coordinate axes as the input beams do, which is caused by height change of M#5.
Fig. 2. (a) Configuration of the coherent combination system; (b) beam splitting module; (c) pulse compressor module. ‘BS’ indicates beam splitter; ‘M’ reflective mirror; ‘GP’ glass plate for chirp compensation; ‘G’ optical grating; ‘P’ path; ‘FP’ focusing parabola; ‘#’ indicates the path number from 1 to 4.
Fig. 3. (a) Cross section of beam envelops after combination; (b) ray matrix distribution on a spherical plane before the focal point; (c) spectral phase induced by the beam combination system.
While each of the four compressors can operate independently, the path lengths need to be synchronized. Through changing the locations of mirrors M#3 along X direction, we can adjust path lengths until they are balanced. Meantime, the locations of other mirrors are fixed, and the beam direction after M#5 points to the minus direction of Y axis. To do so, the path lengths traversed by the beams need to be calculated.
To model and simulate the CBC system, vectorial ray tracing method [53,54] is used from the input plane to the target. If the target is the focal point, the optical path length can be calculated; if the target is a spherical plane before the focal point and after the OAP, the field distribution at the focus can be calculated with further diffraction integral. The basic steps are 1) place the various optics at reasonable locations with proper orientations; 2) each optics ‘generates’ a series of optical surfaces; 3) optical surfaces in sequences form the paths for the rays; 4) the rays are traced along the paths, during which the optical path length and location on grating for each ray is recorded for later analysis; 5) optimize the configuration so that the four optical path lengths are equal and the pulses are compressed. In step 4, the input rays are first transformed from the lab reference to the reference attached to the optics; then the location, direction, polarization state of the output rays are calculated according to the characteristics of the optical surfaces; lastly the output rays are transformed back to lab reference, and the optical path lengths are calculated and recorded. Figure 3(b) shows the ray matrix distribution on a spherical surface 0.1m away from the focal point. With the rays on the spherical plane, the vector field components near the focal point can be calculated with the Stratton-Chu integral [54]:
The ray density for integral is chosen such that the final result of the field distribution converges. If the spatiotemporal information of the pulse is needed, rays with wavelengths covering the whole spectrum are needed. 52 wavelengths are used to resolve the spectrum.
Three points need to be pointed out regarding the implementation of the simulation. First, the curved optical surfaces, whether a parabola mirror or deformed flat mirror, are described by the surface function and its gradient function. The precision of the solution for the intersection between a ray and a surface is sub-nanometer as required by the interferometric precision. Second, to trace the polarization state, the response of the surface to the ‘s’ and ‘p’ polarization of input wave needs to be specified. However, due to the unknown surface characteristics these data are not available. We implement this aspect in the following way to capture the essence of polarization transformation. The Fresnel coefficients are calculated for the interface between ambient environment and bulk glass or metal that makes up the mirrors. Then, the phase is kept, and the amplitude is changed according to a prescribed reflectivity or transmission efficiency. The situation for gratings is more complex, particularly when the grating is in so-called ‘conical mountings’ [55] relative to the rays. Since the rays may not deviate much from the expected way, the response of the grating surface to the polarization state is treated in the same method. Third, in the simulation the system is in vacuum which means the dispersion of gas is not considered.
With the above method, the spectral phase induced by the combination system is calculated and shown in Fig. 3(c). Due to the dispersion balance by glass plates, the spectral phases of the four paths in the ideal situation are the same. We assume the input pulse before entering the combination system has exactly the opposite of the phase in Fig. 3(c), so that all the four beam are compressed to Fourier-transform-limited duration. If there are no glass plates for dispersion balance in Fig. 2, then only one or two beam can be perfectly compressed in principle. The large mismatch in high-order spectral phase among the beams reduce the combination efficiency and peak intensity, even if the pulses are compressed to the shortest.
The path lengths can be synchronized with the help of ray tracing. Path 1 is fixed. Mirrors M#3 of path 2-3 are adjusted along X axis. Two factors are monitored during the adjustment, i.e., the group delay (in femtosecond) at 910nm relative to path 1 and the piston phase relative to path 1. For the ideal baseline model, the precision of the adjustment is as high as 0.1nm, so that both factors are down to 10−4 level.
We assume there is no surface deformation on the optics. Figures 4(a)–4(c) shows the vectorial fluence distributions at the focal spot produced by the ideal system, which are normalized by the peak value. The dominant component with polarization along Y axis concentrates around the optical axis. The black-dashed circle of 1um radius indicates potential interaction region of the PW laser with other beam. The polarization components along X, Z axes are outside the interaction region. Although several orders of magnitude weaker compared with the dominant one, these components may produce non-negligible effects in some experiments. Figure 4(d) shows the isosurfaces (50% of peak intensity) of spatiotemporal intensity of the combined pulse at from −10um to 10um along X axis relative to the focus at 0um. The pulse maintained a well-localized profile over this extended area. As a comparison, Fig. 4(e) shows the fluence distribution at the focus obtained with traditional scalar diffraction theory, which matches quite well with Fig. 4(c). As the fluence is one important information obtained in experiment, Figs. 4(f)–4(h) show the distributions of one single beam and 2-beam combination. The side lobes in Fig. 4(h) is more intense than that in Fig. 4(g) because the two beams are diagonal ones and have larger gaps. These patterns may serve as a reference in practice during alignment of the system. If there is 1500J pulse energy on the dominant polarization component within the observation window of the simulation (20um${\times}$20um${\times}$600fs), the ideal peak intensity would be 1.0692${\times}$1024W/cm2. In the following discussions, we only consider the dominant polarization component for simplicity and clarity.
Fig. 4. (a-c) Fluence distributions of 4-beam combination at the focus with different polarizations; (d) isosurfaces (50% of peak intensity) of the spatiotemporal intensity of the 4-beam-combined pulse at from −10um to 10um along propagation axis relative to the focal point at 0um; (e) fluence distribution of 4-beam at the focus obtain with scalar Fourier transform method; (f-h) fluence distributions of the dominant component at the focus with different number of beams. The values of the fluence are normalized to the peak number of the ideal 4-beam combination at the focus.
2.1 Influence of input pointing fluctuations
One concern about the large-scale CBC system is how the pointing fluctuation of input beam affects the performance due to the different path each sub-beam propagates along. Commercial terawatt laser system delivers beams with pointing fluctuations within 5urad. For a single beam, this may lead to transverse shift of focal spot less than 15um (5urad*3m) without changing field distribution much. Most targets are bigger than 15um, so the pointing fluctuation may not be a problem. For a CBC system even in its ideal form (the baseline model), performance is quite different because of interference among the 4 beams.
Figure 5 shows the spatiotemporal field distributions and fluence at 4 different point angles of input beam. Three isosurfaces are shown in each figure, corresponding to 20%, 50% and 80% of the peak intensity Imax of the ideal situation as shown in Fig. 5(a). When the input beam deviates just 0.5urad, a significant change in field distribution emerges. A side lobe with peak intensity between 20% and 50% of Imax appears at about 5um away from the central part. The central part also shifts a little, and the peak intensity (still higher than 80% of Imax) drops due to loss of energy to the side lobe. As the beam pointing deviates more, we can see a growth in the side lobe and weakening of the central part. However, their locations barely change. When the deviation is 0.9urad as shown in Fig. 5(c), the field distributions of the two parts are the same, and the peak intensity is between 50% and 80% of Imax. Note that the peak of the central part is still within the 1um circle indicated in fluence map, which might be an advantage because for single beam situation 0.9urad deviation would already shift the peak intensity outside the circle. As the beam deviates to 1.8urad, we see a complete cycle of field transformation in Fig. 5(d). The peak intensity is close to the ideal situation. Further pointing deviation results in repeating the transformation cycle until the mismatch among 4 beams is too large to keep the cycle. Pointing fluctuation along other directions leads to similar behavior.
Fig. 5. Spatiotemporal field distributions at the focal point under different input pointing deviations. (a) no deviation; (b) 0.5urad; (c) 0.9urad; (d) 1.8urad. Three isosurfaces in each figure corresponds to 20%, 50% and 80% of the peak intensity in (a).
The restriction on input beam pointing deviation may depend on specific experimental requirements according to the above results. If the beam is required to hit the region of black circle in Fig. 5 with more than 80% focusing efficiency, the maximum deviation is about 0.5urad. At 0.9urad pointing deviation, the field distribution may be restored to ideal situation through delay adjustment of each paths.
2.2 Influence of input wavefront curvature
Another common imperfection of the input beam that affects the combination efficiency is the wavefront curvature. In practice, the curvature may be partially corrected with deformable mirrors. However, it is difficult to eliminate all the curvatures, particularly for meter-level large beams. Here we study the influence of input wavefront curvature on focusing efficiency. As the beam cross section is square, we use Zernike polynomials on square aperture to represent curvature (see Table 14 in [56]). To properly induce curvature in the input beam, the perfect input pulse is reflected with small incident angle upon a reflective mirror before entering the CBC system. The surface of the mirror is deformed using synthesis of the square Zernike polynomials.
Figure 6(a) shows an example of mirror surface curvature based on the 4th order polynomial. The root-mean-square (RMS) value of the curvature is 0.125$\lambda$ ($\lambda$ is the central wavelength 910nm). The peak-to-valley (PV) value is about 0.6$\lambda$. Due to reflection with small incident angle, the PV value of the beam wavefront induced by the surface curvature is about 1.2$\lambda$, twice of the mirror surface as shown in Fig. 6(b). The influence of each polynomial separately from the 4th to 11th order (indicated by S4 to S11) on peak intensity at nominal focus with increasing RMS values are shown in Fig. 6(c). The peak intensities drop almost linearly with RMS values. At small RMS values, all kinds of curvature give similar performance. As RMS increases, the performance begins to diverge and form groups according to the symmetry and gradient characteristics of the polynomials. To get a focusing efficiency better than 80%, the wavefront of the input beam should have RMS value smaller than 0.08$\lambda$ (twice of 0.04$\lambda$ in Fig. 6(c)).
Fig. 6. (a) Mirror surface curvature based on the 4th order polynomial; (b) beam wavefront at 910 nm induced by the mirror surface curvature; (c) focusing efficiency of different curvatures with RMS values. The curvatures are represented with orthonormal polynomials on square aperture from 4th order to 11th order and indicated by S4 to S11.
In conclusion, we build a baseline model of four-channel CBC system for femtosecond petawatt lasers. The influence of input beam imperfections on focusing efficiency is also studied. This serves as the starting point for further investigating the CBC system with various imperfections.
3. Imperfect configuration
The perfect synchronization among the four beam is achieved with path length adjustment accuracy of sub-nanometers. Few commercial translation stages can meet the requirement. There are 49 large optics in the configuration in Fig. 2. Any of them can affect the performance of the system, such as the pointing direction, surface curvature. Unlike traditional researches on CBC system which start from imperfect beams, our research start from imperfect optics which is more realistic.
3.1 Delay among beams
Time delays induced by various mirrors can be lumped at a single mirror in the path. Here the delay is induced by translating mirror M#3 along X direction in each path while keeping the ray direction from M#5 fixed. The combination of changes of the four paths can be arbitrary. Figure 7(a) shows 3 patterns (the first and fourth beams are diagonal) of peak intensity change with mirror translations d. In the simplest case (blue-dotted line in Fig. 7(a)), only mirror M23 is moved, and the peak intensity first drops with d and then increases with d from 10nm to 340nm. The change of Imax with d is periodic with cycle of about 460nm; the periodicity lasts about 4 to 5 cycles before the Imax approaches constant. The cycle is about half of the central wavelength. Due to small incident angle, it corresponds to one wavelength change in optical path. The other two patterns in Fig. 7(a) have the same periodicity although with different curve shapes. The spatiotemporal distribution and fluence profile of the worst situation in Fig. 7(a) is shown in Fig. 7(b), illustrating the potential complexity even with simple delay mismatch pattern. According to Fig. 7(a), to maintain the focusing efficiency over 80%, the differences among the locations of mirrors M#3 along X axis should be controlled within 70nm. This corresponds to pulse group delay of about 0.5fs.
Fig. 7. (a) Change of focusing efficiency with mirror translations d mismatch among four paths; [0, d, 0, 0] indicates that mirror M23 moves d along X direction while other paths are fixed; the 1st (2nd) and 4th (3rd) beams are diagonal; (b) spatiotemporal distribution of the worst situation in (a); (c-f) histograms of 500 samples in which M#3 locations in the four paths are randomly chosen on the condition that the differences among the paths are within 50 nm, 100 nm, 150 nm, and 240 nm, respectively.
Figures 7(c)–7(f) show histograms of 500 samples in which M#3 locations in the four paths are randomly chosen on the condition that the differences among the paths are within 50nm, 100nm, 150nm, and 240nm, respectively. To achieve focusing efficiency over 80% in a robust way, the location mismatch should be less than 100nm as shown in Fig. 7(d). This suggests that mechanical translation control should have precision and stability better than 100nm, although specific values also depend on incident angle of the beam.
3.2 Pointing deviation
To get a clear picture of how pointing deviation affect focusing efficiency, only path 1 is changed with the other three paths fixed. Figure 8(a) shows the location change of peak fluence of path 1 with the pointing deviation (+/− 2.5urad revolving Z axis) of a reflective mirror. In the central part with small deviation, the location changes linearly with mirror angle. When the deviation is large, the peak begins to shift out of the observation window, and then the side lobe becomes the intense part. In the central part, the peak shifts about 6um with 1urad deviation. This is consistent with the fact that the beam deviates twice of the mirror due to reflection, and the focal length is 3m. Figure 8(b) shows the peak intensity of the combined four paths, in which only path 1 is changed. In particular, the pointing deviation is induced separately by three different mirrors. Mirror M11 is before the grating compressor; M13 and M15 are after the gratings and at different heights. The results of the three mirrors are almost the same (accurate to 0.1% level), indicating that in the deviation range of +/− 2.5urad the location of the mirror does not make a difference, and the pointing deviation can be lumped at a single mirror. The peak intensity drops quickly until about 1.5urad deviation, and then increases and approaches constant value which corresponds to 3-beam coherent combination. Pointing deviation revolving Y axis leads to similar results.
Fig. 8. (a) Peak location of a single path versus pointing deviation of a reflective mirror; (b) change of peak intensity with pointing deviation of a reflective mirror in path 1 while the other three paths are fixed. Blue-dotted line indicates results induced by mirror M11 alone; red-circle line by M13; black-star line by M15. (c, d) Histograms of 500 samples in which pointing directions of all four paths are randomly chosen revolving Z or Y axis within +/− 0.3urad and +/− 0.5urad, respectively.
According to Fig. 8(b), pointing deviation among paths should be within 0.5urad to achieve a focusing efficiency higher than 80%. This conclusion is further justified in Figs. 8(c) and 8(d) which show histograms of 500 samples in which pointing directions of all four paths are randomly chosen revolving Z or Y axis within +/− 0.3urad and +/− 0.5urad, respectively. It should be noted that the average pointing direction should also be limited to 0.5urad relative to the center; otherwise patterns similar to section 2.1 emerges. This suggests that mechanical pointing control should have precision and stability better than 0.5urad.
3.3 Mirror surface curvature
In this section, we only consider mirrors with 45 degree incident angle as most mirrors are used in this way. Through comparison (not shown here), it is found that in the parameter ranges of the simulations surface curvatures on mirrors before and after gratings lead to similar results. In the following, all curvatures are induced on mirrors M#1. Figure 9(a) shows the changing of focusing efficiency with increase of curvature RMS values for four kinds of deformations specified by single orthonormal polynomials on rectangular aperture from the 4th order to 7th order (indicated by S4 to S7). The curvatures on the four mirrors M#1 are the same. Due to 45 degree incident angle, the PV value of surface curvature is about the same as that of beam wavefront in units of nanometer. Therefore, focusing efficiency at 0.1$\lambda$ is close to that at 0.05$\lambda$ in Fig. 6(c). According to Fig. 9(a), RMS value should be less than 0.08$\lambda$ to achieve efficiency higher than 80%. Due to randomness, the efficiency will be even lower. In Figs. 9(b)–9(d), surface curvature is generated using combination of 8 polynomials with random coefficients while keeping RMS fixed. 500 samples are calculated for each RMS value of 0.02$\lambda$, 0.04$\lambda$ and 0.06$\lambda$, respectively. It can be seen that RMS should be less than 0.06$\lambda$, which is also the upper limit for the beam wavefront. This is consistent with the conclusion in section 2.2 for which randomness is not considered. It should be noted that the above results on curvature represent the overall effect. If two mirrors have curvature RMS value r, the overall RMS would be $\sqrt 2 r$ in average. In this section, very high order deformations are not considered for analysis simplicity, but adding them is straightforward.
Fig. 9. (a) Focusing efficiency versus increase of curvature RMS values for four kinds of deformations specified by single Zernike polynomials on rectangular aperture from 4th order to 7th order, indicated by S4 to S7; the curvatures on the four mirrors M#1 are the same. (b-d) Histograms of 500 samples in which surface curvature is generated using combination of 8 polynomials with random coefficients while keeping RMS fixed at 0.02$\lambda$, 0.04$\lambda$ and 0.06$\lambda$, respectively.
3.4 Compressor mismatch
There are many degrees of freedom for the grating compressor. Each grating can tilt, tip, rotate revolving the normal, translate along normal, and be curved on the surface. To sort out these factors, we first study the situation where only one grating in path 1 is changed while the other paths are ideal. Figure 10(a) shows the focusing efficiency dependence of the first grating translation along its normal. This translation not only change pulse group delay, but also the dispersion. Therefore, no periodic change as Fig. 7(a) appears. The angular changes of the first grating on efficiency is shown in Fig. 10(b). While the tip and rotation have similar effects, the tilt has much larger influence on efficiency. Angular changes of the second grating have similar effects. The effects of surface curvatures of the first and second grating are shown in Figs. 10(c) and 10(d). The surface curvatures are represented by single orthonormal polynomials on rectangular aperture from the 4th order to 11th order (indicated by S4 to S11). Due to the difference of size and the way light is incident upon the grating, the effects of the curvatures are different between the two gratings. In general, efficiencies by the second grating are higher than the first one because the second grating is larger and thus flatter than the first one with same RMS value. However, the efficiencies of the second grating have larger variations among different deformations. Compared with reflective mirrors, the influence on efficiency by gratings have similar dependence on the various deviations. This similarity is further justified in Fig. 11 which shows the results of the influence of random deviations in four paths on focusing efficiency. Note that the irregularity of the distribution in Fig. 11(b) originates from the discrete sampling of angles at four values for each type of deviation, and the distribution is obtained through random combinations. The point is that focusing efficiency begins to drop below 80% in this deviation range.
Fig. 10. (a) Focusing efficiency dependence of the first grating translation along its normal; (b) different angular changes of the first grating on efficiency; (c, d) effects of different surface deformations of the first (c) and second (d) grating on efficiency. The surface deformations are represented by single orthonormal polynomials on rectangular aperture from the 4th order to 11th order (indicated by S4 to S11).
Fig. 11. Histograms of 500 samples in which (a) the first grating in four paths randomly shift in [−50 nm, 50 nm]; (b) the first grating in four paths randomly tilt, tip or rotate in [−0.3urad, 0.3urad]; (c, d) surface curvature of the first (c) and second (d) gratings in four paths are generated using combination of 8 polynomials with random coefficients while keeping RMS fixed at 0.06$\lambda$.
In this section, the influence of different imperfections on focusing efficiency are investigated. Although the imperfections are attributes of the optics, the results should be interpreted as the minimal allowable aberrations of the beams to achieve a focusing efficiency like better than 80%. In this sense, our results agree well with previous work [48–51].
4. A case with many imperfections
There are 49 optics in the configuration in Fig. 2. Every one of them can have the above imperfections, and some of them have two surfaces. In this section, we simulate a situation in which all optics but the last parabola induce some random unknown imperfections; and then simulate the process of automatic alignment with a simple (yet not optimal) method in order to improve the focusing efficiency. The parameters of deformations are based on results in previous sections, which means the configuration is already nearly well aligned.
For the beam splitters and glass plates, we assume the front and back surfaces have the same surface curvature. The RMS of surface deformation is 0.06$\lambda$ ($\lambda$=910nm), and the deformation is synthesized with the first 8 orthonormal polynomials having random unknown uniformly-distributed coefficients. The random angular deviation for each optics is uniformly distributed in [−0.3urad, 0.3urad]; for ordinary optics there are two angles; for gratings there are three angles. Each optics is also translated randomly along its normal in [−50nm, 50nm]. The surface of mirror M#3 in each path is reserved for DM functions, i.e., we optimized the surface shape of M#3 to improve focusing efficiency. The angular direction of M#5 is reserved for pointing alignment for each path. The translation of M#5 along its normal is reserved for synchronization between the paths. To accelerate the simulation process, we only trace rays at 910nm, and the feedback in the automatic alignment is the fluence profile of 910nm beam at the focus. In experiment, the feedback will be the fluence profile of the pulse with whole spectrum. The pointing deviations of the four paths are first corrected by adjusting the orientation of M#5 successively until the peak of fluence coincides with the center of observation area for each path. Then, the positions of M#5 along their normals are adjusted simultaneously to maximize the peak of the fluence of the combined four beams with Nelder-Mead (NM) algorithm [57]. Last, the surface curvatures (in terms of coefficients of the first eight square Zernike polynomials) of M#3 in each path are optimized separately to maximize their fluence peaks with NM algorithm. This completes one cycle of optimization. After that the system performance is evaluated by the spatiotemporal field distribution at the focus. It should be noted that traditional wavefront correction system uses wavefront sensors to determine the deformation and then drives the DM accordingly. In this work, to facilitate the simulation, we use the above method. On the other hand, if it is difficult to deploy the wavefront sensor in experiment the above method could be an alternative solution.
Figure 12 shows the fields before (1st row) and after (2nd and 3rd row) optimization. The first 4 columns show the fluence distributions of the four paths separately; the last column shows the coherently-combined spatiotemporal field distribution. Before optimization, the focusing efficiency is about 14%, and the location is far from the center of observation window. After pointing and synchronization optimization (2nd row in Fig. 12), the focusing efficiency increases to 37% and the peak intensity is at the center. As expected, the peak value of fluence for each path does not change. When the wavefront in each path is further optimized, the fluence peak sees a large increase (3rd row in Fig. 12), and the focusing efficiency increases to 72.6%. Figure 13 shows the evolution of various parameters during wavefront optimization process. The fluence peak of each path increases quickly in the first 50 iterations and then begins to saturate except path 1. The saturated values for the paths are different from each other. Figure 13(b) shows the RMS of the DM in each path during optimization. Three of them converge to similar values. The normalized coefficients of the first eight square Zernike polynomials are shown in Figs. 13(c)–13(f). From these values, we can estimate the overall deformations of each path induced by their various optics. In practice, these data can be helpful in choosing proper DMs.
Fig. 12. Field distributions before (1st row) and after (2nd and 3rd row) optimization. The first 4 columns show the fluence distributions of the four paths separately (units in micrometer); the last column shows the coherently-combined spatiotemporal field distribution. The isosurfaces in the last column corresponds to 10% and 50% of the ideal peak intensity.
Fig. 13. Evolutions of relevant parameters of each path during optimization process: (a) peak fluence normalized by the ideal value; (b) RMS of surface deformations; (c-d) normalized coefficients of the first eight square Zernike polynomials used to synthesize the deformations for each path.
Further improving the focusing efficiency can be achieved by increasing the iteration times at the cost of time consumption. On the other hand, as shown in Fig. 12 the fluence peak of the beam deviates from the center after wavefront correction. Therefore, a second cycle of optimization process can be implemented. However, since the wavefront optimization process already saturates, another round along the same track will be inefficient. After only the pointing and synchronization are optimized for the second time, the focusing efficiency is increased to 80.7%. To efficiently improve the wavefront, novel methods need to be investigated, such as a clever choice of initial simplex for NM algorithm.
In conclusion, we simulate a realistic CBC configuration and its automatic alignment, which demonstrates the capability of our model in simulating complex configurations and their behavior. The optimization method works well and can be implemented in experiment. In large complex system placed in vacuum chamber which also demands interferometric precision, automatic control is very important. Work in this section can provide valuable insights. Besides, once the surface deformation data of the mirrors to be used in experiment are available, the system performance can be evaluated using the simulation model. Other important issues such as pulse contrast [58] are not covered, but will be discussed in future works.
5. Conclusion
A four-channel coherent beam combination (CBC) configuration is designed for femtosecond petawatt (fs-PW) lasers and is extensively modeled through numerical simulations. The behavior characteristics of the CBC system are revealed, and key parameters relevant to engineering are identified. Furthermore, with the virtual CBC system a simple automatic alignment method is demonstrated to optimize a realistic configuration. Future work may include building more realistic (for example using the surface deformation data of the mirrors to be used in experiment) or innovative configurations and investigating the monitor and control methods. This work may benefit relevant projects around the world.
Funding
National Natural Science Foundation of China (61521093, 61635012,11604351); Program of Shanghai Academic/Technology Research Leader (18XD1404200).Major Project Science and Technology Commission of Shanghai Municipality (2017SHZDZX02);Strategic Priority Research Program of The Chinese Academy of Sciences (XDB1603).
Acknowledgments
The authors acknowledge Prof. Jun Liu, Prof. Xiaoyan Liang, Dr. Chun Peng, Dr. Lianghong Yu, Dr. Peng Wang, and Dr. Xiong Shen for helpful discussions.
Disclosures
The authors declare no conflicts of interest.
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