# Design of Third-Order Dispersion Compensation for the SG PW Laser System Using a Birefringent Crystal

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{3}needs to compensate for the multi-PW design. The compensation scheme is designed using a quartz crystal of d = 6.5 mm, θ = 90$\xb0$, ϕ = 17$\xb0$, and T = 21 °C, corresponding to the thickness, inclination angle, in-plane rotation angle, and temperature, respectively. Furthermore, we show a principle-proof experiment offline and measure the GVD and TOD by the Wizzler, which is based on theoretical simulations. These results can be applied to independently and continuously control the TOD of short-pulse laser systems.

## 1. Introduction

^{20}W/cm

^{2}with a diameter of 20.8 μm in 50% cycle, which were achieved in 2016. The SG PW laser is generally used to heat and generate energetic protons in various physical experiments, such as fast ignition, indirect drive physics, laboratory astrophysics, and high energy density physics. Furthermore, to meet the demand for further physical applications, the design of a multi-PW (5–10 PW) with a pulse duration of 150 fs at 1053 nm has recently been approved, which is similar to the aforementioned design [19,20,21] and can be updated from the existing SG PW laser system. The amplified pulse, stretched by the Offner stretcher [22], is recompressed with a four-grating compressor, which can cancel out the GVD introduced by the transmission and amplifier materials in the system simultaneously. However, the residual TOD and FOD can influence the intensity contrast and shape fidelity of the compressed pulse when the pulse duration decreases and hence require compensation. The methods of compensating for the TOD include object-image-grating self-tiling [6] and fiber [8,9,10,12,16], prism pair [13], and acousto-optic programmable dispersion filter (AOPDF) [14,15], which are limited by the damage threshold, apertures, and low transmission, and are mainly set in a position of relatively low energy before amplification, leading to the lower energy of the input pulses. On the other hand, the hybrid gratings scheme [7] and Gires–Tournois interferometer (GTI) mirrors [11] can be used to compensate for the TOD with a large aperture but it is difficult to process and is not convenient to adjust. Therefore, finding a method to support high energy with a large aperture and convenient operation is required, especially for the TOD compensation of a multi-PW design based on the existing SG PW laser system. Birefringent crystals have been widely used to achieve gain-narrow compensation in CPA amplifiers [23,24]; however, to the best of our knowledge, few applications and considerations of birefringent crystals have been reported in detail for the introduced spectral phase and dispersions, especially in the experiment. Therefore, we investigated the design of TOD compensation for the SG PW laser system using a birefringent crystal. The remainder of this paper is organized as follows.

^{6}fs

^{3}TOD compensation for the SG PW laser system is designed using a quartz birefringent crystal with d = 6.5 mm, θ = 90°, ϕ = 17°, and T = 21 °C. Finally, the dispersions of GVD and TOD introduced by the birefringent crystals are investigated offline using a Wizzler for a proof-principle experiment, as described in Section 3.3.

## 2. Numerical Method

_{0}are the laser and center frequencies, respectively. ${\phi}_{0}$ is the absolute phase, φ

_{1}is the group delay (GD), and φ

_{2}and φ

_{3}are GVD and TOD, respectively. The dispersion of each order introduced by the crystal can be controlled by the thickness d, inclination angle θ, in-plane rotation angle ϕ, and temperature T.

## 3. Results and Discussion

#### 3.1. Residual TOD and FOD and Influence on the Multi-PW Design

^{3}and $-3.6\times {10}^{8}$ fs

^{4}for a pulse duration of 180 fs, and $1.2\times {10}^{2}$ fs

^{3}and $-8\times {10}^{6}$ fs

^{4}for a pulse duration of 80 fs. These are mainly caused by the pulse with a longer duration according to a narrower bandwidth, which demands a larger GVD to achieve the same stretched pulse duration of 3.3 ns, and needs to pass through a longer optical path in the stretcher and compressor, thereby introducing more aberrations for the TOD and FOD. The analysis dispersion results are consistent with implementing a stretcher and compressor for the SG PW laser with a seed pulse duration of 180 fs. They can also provide a design basis for upgrading to multi-PW.

^{2}, $1.54\times {10}^{5}$ fs

^{3}, and $-1.69\times {10}^{5}$ fs

^{4}. An adjustment compressor is used to compensate for the above GVD; however, this process can introduce extra TOD and FOD simultaneously. The change in the distance for adjusting the compressor gratings, GVD, TOD, and FOD for pulse durations of 180 fs (dotted line in blue) and 80 fs (solid in black line) with adjustable pulse durations are shown in Figure 4a–d. To achieve an FTL compressed pulse, the adjusted compressor needs to afford a GVD of −$8.8\times {10}^{4}$ fs

^{2}and introduce an additional but the same TOD of $1.3\times {10}^{6}$ fs

^{3}and FOD of $-3\times {10}^{7}$ fs

^{4}simultaneously. After compensating for the GVD of the transmission materials, the final TOD and FOD for the existing SG PW laser system are $1.3\times {10}^{6}$ fs

^{3}and $-3.9\times {10}^{8}$ fs

^{4}for a pulse duration of 180 fs and $1.3\times {10}^{6}$ fs

^{3}and $-3.8\times {10}^{7}$ fs

^{4}for a pulse duration of 80 fs. The values agree with the implementation of pulse duration controlled by the adjustment compressor in the SG PW laser and can provide a basis for the multi-PW design.

^{3}and FOD of $-3.9\times {10}^{8}$ fs

^{4}, according to the gain narrowing effect and compressed FTL pulse durations of 388 fs, 233 fs, and 166 fs, respectively. A comparison of the contrast with the compensation of TOD and the residual FOD is shown in Figure 5b; the B integral is neglected in the simulation. Similarly, Figure 5c shows the intensity contrast for bandwidths of 7 nm, 11 nm, and 15 nm, according to the compressed FTL pulse durations of 388 fs, 105 fs, and 77 fs, respectively, and includes the TOD of $1.3\times {10}^{6}$ fs

^{3}and FOD of $-3.8\times {10}^{7}$ fs

^{4}; only the residual FOD is shown in Figure 5d. From the comparison results of Figure 5, we can see that the residual TOD and FOD had no significant effect on the contrast for the existing SG PW laser with a bandwidth of 3 nm after gain narrowing [18], but high-order dispersion should be considered if the pulse bandwidth is broadened and compressed to the FTL. The simulated result for the bandwidth of 3 nm can afford the measured contrast of 10

^{−8}, by which there is no additional TOD dispersion adjustment in the SG PW laser. However, when the pulse bandwidth is broadened to over 7 nm, and the duration is shorter than 150 fs in the multi-PW design, the TOD can significantly distort the pulse shape and the pulse is no longer symmetric, which can cause the peak power of the pulse to decrease. Therefore, the TOD must be compensated, but the FOD is negligible. Furthermore, when the pulse duration is less than 100 fs, the FOD should be considered to further improve the intensity contrast, which is consistent with the results obtained in [32].

#### 3.2. Design of TOD Compensation Using the Birefringent Plate

^{3}for the multi-PW with a 150-fs level design. As detailed in Figure 6a, we first analyze the variation of the maximum amplitude of TOD and free-spectral range (FSR) with the length of the birefringent plate using Equations (1)–(7), where the parameters are θ = 90°, ϕ = 17°, and T = 21 °C. Similar to the GTI, the maximum amplitude of the TOD increased, but the FSR decreased with increasing plate length. Figure 6b shows the GVD and TOD change with ϕ, in which the length of the plate is 6.5 mm, and the other parameters are the same as in Figure 6a. In addition, the TOD can be adjusted from positive to negative with the scope of $\pm 2\times {10}^{6}$ fs

^{3}, which can compensate for the residual TOD of $1.3\times {10}^{6}$ fs

^{3}for the multi-PW design when ϕ is set to 17°. Furthermore, the GVD is small and can be neglected. Additionally, the transmission, GVD, and TOD curves changing with wavelength are also shown in Figure 6c,d, respectively, when the birefringent plate is set to the compensation TOD of $1.3\times {10}^{6}$ fs

^{3}. As shown above, the dispersion scheme can support the bandwidth of approximately 7 nm and the TOD of $1.3\times {10}^{6}$ fs

^{3}, which can be used to compensate for the demanded TOD for the multi-PW design. In addition, similar to the results of crystal cascade increasing the bandwidth [33], we observe from Figure 6 that the birefringent crystal can also be cascade designed to afford larger bandwidth with the same TOD simultaneously in the dispersion compensation. From the results, the residual TOD for multi-PW design can be compensated by choosing the thickness and controlling the inclination angle, the in-plane angle, and the temperature of the birefringent plate independently.

#### 3.3. Experiment

^{2}and the TOD of $5\times {10}^{6}$ fs

^{3}from the laser, resulting in more deviation, especially when measuring the peaks and valleys for TOD. Figure 8c shows the spectrum and phase with a GVD of $-6\times {10}^{4}$ fs

^{2}and TOD of $5\times {10}^{6}$ fs

^{3}from the initial laser without inserting quartz, and Figure 8d shows the GVD of $-8\times {10}^{4}$ fs

^{2}and TOD of $-1.3\times {10}^{7}$ fs

^{3}when using the birefringent quartz crystal, according to the valleys in Figure 8b. In addition, the spectrum in Figure 8d is modulated mainly by the longer crystal, which can limit the FSR.

## 4. Conclusions

^{3}was designed using a quartz crystal. Furthermore, a principle–proof experiment was investigated offline to measure the GVD and TOD by a Wizzler, which is based on theoretical simulations. These results provide a feasible method for controlling and compensating the TOD with a temperature-regulated birefringent crystal, which to the best of our knowledge, is achieved for the first time. In addition, they are important for supporting updates of the multi-PW (5–10 PW) at the 150-fs level.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Moses, E.I.; Lindl, J.D.; Spaeth, M.L.; Patterson, R.W.; Sawicki, R.H.; Atherton, L.J.; Baisden, P.A.; Lagin, L.J.; Larson, D.W.; MacGowan, B.J.; et al. Overview: Development of the National Ignition Facility and the Transition to a User Facility for the Ignition Campaign and High Energy Density Scientific Research. Fusion Sci. Technol.
**2017**, 69, 1–24. [Google Scholar] [CrossRef] - Galletti, M.; Oliveira, P.; Galimberti, M.; Ahmad, M.; Archipovaite, G.; Booth, N.; Dilworth, E.; Frackiewicz, A.; Winstone, T.; Musgrave, I.; et al. Ultra-broadband all-OPCPA petawatt facility fully based on LBO. High Power Laser Sci. Eng.
**2020**, 8, e31. [Google Scholar] [CrossRef] - Zhang, J.; Wang, W.M.; Yang, X.H.; Wu, D.; Ma, Y.Y.; Jiao, J.L.; Zhang, Z.; Wu, F.Y.; Yuan, X.H.; Li, Y.T.; et al. Double-cone ignition scheme for inertial confinement fusion. Philos. Trans. R. Soc. London. Ser. A Math. Phys. Eng. Sci.
**2020**, 378, 20200015. [Google Scholar] [CrossRef] [PubMed] - Mariscal, D.; Ma, T.; Wilks, S.C.; Kemp, A.J.; Williams, G.J.; Michel, P.; Chen, H.; Patel, P.K.; Remington, B.A.; Bowers, M.; et al. First demonstration of ARC-accelerated proton beams at the National Ignition Facility. Phys. Plasmas
**2019**, 26, 43110. [Google Scholar] [CrossRef] - Hornung, J.; Zobus, Y.; Boller, P.; Brabetz, C.; Eisenbarth, U.; Kuhl, T.; Major, Z.; Ohland, J.B.; Zepf, M.; Zielbauer, B.; et al. Enhancement of the laser-driven proton source at PHELIX. High Power Laser Sci. Eng.
**2020**, 8, e24. [Google Scholar] [CrossRef] - Li, Z.; Rao, D.; Leng, Y.; Chen, L.; Dai, Y. Third-order dispersion compensation for petawatt-level lasers employing object-image-grating self-tiling. Quantum Electron.
**2015**, 45, 891. [Google Scholar] - Zaouter, Y.; Papadopoulos, D.N.; Hanna, M.; Druon, F.; Cormier, E.; Georges, P. Third-order spectral phase compensation in parabolic pulse compression. Opt. Express
**2007**, 15, 9372. [Google Scholar] [CrossRef] - Marhic, M.E.; Kagi, N.; Chiang, T.K.; Kazovsky, L.G. Cancellation of third-order nonlinear effects in amplified fiber links by dispersion compensation, phase conjugation, and alternating dispersion. Opt. Lett.
**1995**, 20, 863. [Google Scholar] [CrossRef] [PubMed] - Liu, S.; Liu, X. Mutual compensation of the higher-order nonlinearity and the third-order dispersion. Phys. Lett. A
**1997**, 225, 67. [Google Scholar] [CrossRef] - Zhang, R.; Yang, X.; Cai, Y.; Hu, W.; Zhang, Z. Residual third-order dispersion compensation in femtosecond pulses transmission using a phase modulator. In Proceedings of the Optical Transmission, Switching, and Subsystems VI, Hangzhou, China, 26–30 October 2008; Volume 7136. [Google Scholar]
- Niu, H.L.; Shen, W.D.; Li, C.S.; Zhang, Y.G.; Xie, C.; Yu, P.; Yuan, W.J.; Liu, B.W.; Hu, M.L.; Wang, Q.Y.; et al. Dispersive mirrors for high third-order dispersion compensation in femtosecond amplification fiber laser system. Appl. Phys. B
**2012**, 108, 609. [Google Scholar] [CrossRef] - Kane, S.; Squier, J. Grating Compensation of Third-Order Material Dispersion in the Normal Dispersion Regime: Sub- 100-fs Chirped-Pulse Amplification Using a Fiber Stretcher and Grating-Pair Compressor. IEEE JQE
**1995**, 31, 2052. [Google Scholar] [CrossRef] - Yang, Q.; Xie, X.; Kang, J.; Zhu, H.; Guo, A.; Gao, Q. Independent and continuous third-order dispersion compensation using a pair of prisms. High Power Laser Sci. Eng.
**2014**, 2, e38. [Google Scholar] [CrossRef] [Green Version] - Verluise, F.; Laude, V.; Cheng, Z.; Spielmann, C.; Tournois, P. Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: Pulse compression and shaping. Opt. Lett.
**2000**, 25, 575. [Google Scholar] [CrossRef] [PubMed] - Lureau, F.; Matras, G.; Chalus, O.; Derycke, C.; Morbieu, T.; Radier, C.; Casagrande, O.; Laux, S.; Ricaud, S.; Rey, G.; et al. High-energy hybrid femtosecond laser system demonstrating 2 × 10 PW capability. High Power Laser Sci. Eng.
**2020**, 8, 8. [Google Scholar] [CrossRef] - Tsuda, H.; Okamoto, K.; Ishii, T.; Naganuma, K.; Inoue, Y.; Takenouchi, H.; Kurokawa, T. Second- and third-order dispersion compensator using a high-resolution arrayed-waveguide grating. IEEE Photonics Technol. Lett.
**1999**, 11, 569. [Google Scholar] [CrossRef] - Zhu, J.; Zhu, J.; Li, X.; Zhu, B.; Ma, W.; Lu, X.; Fan, W.; Liu, Z.; Zhou, S.; Xu, G.; et al. Status and development of high-power laser facilities at the NLHPLP. High Power Laser Sci.
**2018**, 6, 6. [Google Scholar] [CrossRef] [Green Version] - Xu, G.; Wang, T.; Li, Z.; Dai, Y.; Lin, Z.; Gu, Y.; Zhu, J. 1 kJ Petawatt Laser System for SG-II-U Program. Rev. Laser Eng.
**2008**, 36, 1172. [Google Scholar] [CrossRef] [Green Version] - Gaul, E.; Cheriaux, G.; Antipenkov, R.; Batysta, F.; Borger, T.; Friedman, G.; Greene, J.T.; Hammond, D.; Heisler, J.; Hidinger, D.; et al. Hybrid OPCPA/Glass 10 PW laser at 1 shot a minute. In Proceedings of the Conference on Lasers and Electro-Optics, San Jose, CA, USA, 13–18 May 2018. [Google Scholar]
- Gaul, E.W.; Martinez, M.; Blakeney, J.; Jochmann, A.; Ringuette, M.; Hammond, D.; Borger, T.; Escamilla, R.; Douglas, S.; Henderson, W.; et al. Demonstration of a 1.1 petawatt laser based on a hybrid optical parametric chirped pulse amplification/mixed Nd:glass amplifier. Appl. Opt.
**2010**, 49, 1676. [Google Scholar] [CrossRef] [PubMed] - Cheriaux, G.; Antipenkov, R.; Batysta, F.; Borger, T.; Friedman, G.; Greene, J.T.; Hammond, D.; Heisler, J.; Jochmann, A.; Kepler, M.; et al. Progress on ELI-Beamlines 10 PW Laser System. Rev. Laser Eng.
**2018**, 46, 125. [Google Scholar] [CrossRef] - Zhang, Z.; Song, Y.; Sun, D.; Chai, L.; Sun, H.; Wang, C. Compact and material-dispersion-compatible Offner stretcher for chirped pulse amplifications. Opt. Commun.
**2002**, 206, 7. [Google Scholar] [CrossRef] - Heebner, J.E.; Acree, R.J.; Alessi, D.A.; Barnes, A.I.; Bowers, M.W.; Browning, D.F.; Budge, T.S.; Burns, S.; Chang, L.S.; Christensen, K.S.; et al. Injection laser system for Advanced Radiographic Capability using chirped pulse amplification on the National Ignition Facility. Appl. Opt.
**2019**, 58, 8501. [Google Scholar] [CrossRef] [PubMed] - Zhang, T.; Li, D.; Wang, T.; Cui, Y.; Zhang, T.; Wang, L.; Zhang, J.; Xu, G. Spectral shaping of picosecond petawatt laser system based on lithium niobate birefringent crystal. Acta Phys. Sin.
**2021**, 70, 084202. [Google Scholar] [CrossRef] - Preuss, D.R.; Gole, J.L. Three-stage birefringent filter tuning smoothly over the visible region: Theoretical treatment and experimental design. Appl. Opt.
**1980**, 19, 702. [Google Scholar] [CrossRef] [PubMed] - Zhu, X. Explicit Jones transformation matrix for a tilted birefringent plate with its optic axis parallel to the plate surface. Appl. Opt.
**1994**, 33, 3502. [Google Scholar] [CrossRef] - Burneau, A.; Humbert, B. Temperature effect on a tilted birefringent filter in a tunable laser: A limitation for Raman spectroscopy. J. Appl. Phys.
**1989**, 66, 5702. [Google Scholar] [CrossRef] - Batysta, F.; Antipenkov, R.; Borger, T.; Kissinger, A.; Green, J.T.; Kananavicius, R.; Cheriaux, G.; Hidinger, D.; Kolenda, J.; Gaul, E.; et al. Spectral pulse shaping of a 5 Hz, multi-joule, broadband optical parametric chirped pulse amplification frontend for a 10 PW laser system. Opt. Lett.
**2018**, 43, 3866. [Google Scholar] [CrossRef] - Zhang, Z.; Sun, H. Calculation and evaluation of dispersions in a femtosecond pulse amplification system. Acta Phys. Sin.
**2001**, 50, 1080. [Google Scholar] [CrossRef] - Tian, J.; Sun, J.; Wei, Z.; Wang, Z.; Ling, W.; Huang, X.; Liu, L.; Wei, X.; Zhang, J. Theoretical and experimental studies on large-ratio stretching of femtosecond pulse with Offner triplet stretcher. Acta Phys. Sin.
**2005**, 54, 1200. [Google Scholar] [CrossRef] - Treacy, E. Optical pulse compression with diffraction gratings. IEEE JQE
**1969**, 5, 454. [Google Scholar] [CrossRef] - Liu, X.; Wang, C.; Wang, X.; Lu, X.; Bai, P.; Liu, Y.; Li, Y.; Liu, K.; Yu, L.; Leng, Y.; et al. Dispersion Management in 10-PW Laser Front End. Optics
**2020**, 1, 191–201. [Google Scholar] [CrossRef] - Brown, M. Increased spectral bandwidths in nonlinear conversion processes by use of multicrystal designs. Opt. Lett.
**1998**, 23, 1591. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Schematic of the dispersion compensation based on the birefringent plate. P1, P2: Polarizers; HWP: Half-wave plate.

**Figure 3.**Absolute value of the GVD, TOD, and FOD for pulse durations of 180 fs and 80 fs, which are stretched to 3.3 ns and compressed to FTL, together with the differences of dispersion between stretcher and compressor. (

**a**) The GVD and difference for the pulse duration of 180 fs; (

**b**) the TOD and difference for the pulse duration of 180 fs; (

**c**) the FOD and difference for the pulse duration of 180 fs; (

**d**–

**f**) are similar as above for the pulse duration of 80 fs.

**Figure 4.**Distance of adjustment compressor gratings, the GVD, TOD, and FOD change with pulse durations by the adjustable compressor for the pulse durations of 180 fs (solid black lines) and 80 fs (dotted blue lines), respectively. (

**a**) Distance; (

**b**) GVD; (

**c**) TOD; (

**d**) FOD.

**Figure 5.**(

**a**) Intensity contrast for bandwidths of 3 nm, 5 nm, and 7 nm, which includes the TOD of $1.3\times {10}^{6}$ fs

^{3}and FOD of $-3.9\times {10}^{8}$ fs

^{4}; (

**b**) the contrast only with the FOD, with other parameters the same as in (

**a**); (

**c**) the contrast for bandwidths of 7 nm, 11 nm, and 15 nm including the TOD of $1.3\times {10}^{6}$ fs

^{3}and FOD of $-3.8\times {10}^{7}$ fs

^{4}; (

**d**) the contrast only considering the FOD, with other parameters the same as in (

**c**).

**Figure 6.**(

**a**) Maximum amplitude of TOD and FSR changing with the length of the birefringent plate; (

**b**) The GVD and TOD change with ϕ, in which the length of the plate is 6.5 mm; (

**c**) The transmission curve for TOD compensation; (

**d**) The GVD and TOD curves changing with wavelength for the TOD compensation parameters.

**Figure 7.**Schematic diagram of principle–proof experiment for dispersion control, which uses a quartz birefringent crystal and measured by Wizzler; HWP is the half waveplate; M1 and M2 are reflector mirrors; P1 and P2 are polarizers.

**Figure 8.**(

**a**) Experimental measurement results of GVD represented by the dotted black line, compared with simulated results shown by the solid red line; (

**b**) the measured and simulated TOD; (

**c**) the initial spectral and phase measurement with the residual GVD of $-6\times {10}^{4}$ fs

^{2}and TOD of $5\times {10}^{6}$ fs

^{3}from the laser without inserting quartz; (

**d**) the spectral and phase with the GVD of $-8\times {10}^{4}$ fs

^{2}and the TOD of $-1.3\times {10}^{7}$ fs

^{3}, according to the valleys in Figure (

**b**).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, D.; Wang, T.; Yin, X.; Li, J.; Yu, H.; Wang, L.; Lu, X.; Xu, G.
Design of Third-Order Dispersion Compensation for the SG PW Laser System Using a Birefringent Crystal. *Appl. Sci.* **2022**, *12*, 4078.
https://doi.org/10.3390/app12084078

**AMA Style**

Li D, Wang T, Yin X, Li J, Yu H, Wang L, Lu X, Xu G.
Design of Third-Order Dispersion Compensation for the SG PW Laser System Using a Birefringent Crystal. *Applied Sciences*. 2022; 12(8):4078.
https://doi.org/10.3390/app12084078

**Chicago/Turabian Style**

Li, Dawei, Tao Wang, Xiaolei Yin, Jiamei Li, Hui Yu, Li Wang, Xingqiang Lu, and Guang Xu.
2022. "Design of Third-Order Dispersion Compensation for the SG PW Laser System Using a Birefringent Crystal" *Applied Sciences* 12, no. 8: 4078.
https://doi.org/10.3390/app12084078