## Abstract

A single-cycle (3 fs) 100 petawatt laser pulse is obtained theoretically by dramatically increasing the spectrum, accordingly reducing the pulse duration, of the optical parametric chirped pulse amplification (OPCPA) with a new designed wide-angle non-collinear OPCPA (WNOPCPA). While comparing with two other recent popular methods of the energy-further-increased single-beam femtosecond petawatt laser and the spatiotemporally coherent combination of multiple-beam femtosecond petawatt lasers, we believe that the proposed method is another choice for sub-exawatt lasers.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the past three decades, the techniques of chirped pulse amplification (CPA) and then optical parametric chirped pulse amplification (OPCPA) have increased the artificial peak power to terawatt (10^{12} W) and then petawatt (10^{15} W) [1,2]. For the next step to exawatt (10^{18} W), there are two possible choices, one is further increasing the pulse energy of recent femtosecond petawatt lasers, and the other is reducing the pulse duration. For the first case, due to the limitation of damage and nonlinearity, the pulse energy scaling is based on the beam size expanding in space. Right now, the beam size in a petawatt laser has already reached to the maximum of optical elements, and, accordingly, the coherent combination provides another solution [3]. However, the problem is that a perfect coherent combination of several femtosecond petawatt laser beams in both space and time would be greatly challenged in technology and economics. For the second case, the shortest pulse duration is limited by the spectral bandwidth of gain materials, especially in a high gain/energy amplifier. For example, in a Ti:sapphire CPA petawatt laser, the bandwidth of the output spectrum usually is around 50–80 nm [4,5], and from the Fourier-transform the shortest pulse duration is around 15–25 fs. In an OPCPA laser, the gain bandwidth of the type I BBO or LBO crystal could reach to around 200–250 nm [6], and accordingly the shortest pulse duration is around 5–10 fs. For the further reducing, the pulse duration cannot break the Fourier-transform limitation. Generally, the spectral bandwidth of a femtosecond mode-locking laser pulse could be dramatically broadened by propagating it through a strongly nonlinear device, such as bulk glass, photonic crystal fiber, and hollow-core fiber (HCF) [7,8]. However, such setups cannot support a high pulse energy (recently in the level of milli-joule), which can only be used as a frontend. Up to now, several novel techniques have already been presented to enhance the spectral bandwidth of optical parametric amplification (OPA) or OPCPA, however these methods still cannot meet the high energy requirement. For example, the method of achromatic phase-matching (APM) needs a specific pump (i.e., another broadband CPA laser with angular dispersion) [9], which limits the energy scaling. The method of cascaded OPA/OPCPA pumped by different wavelengths (second and third harmonics) also faces a similar problem [10]. To propose a possible solution, in this paper we developed a new designed wide-angle non-collinear OPCPA (WNOPCPA) to fill this gap, and according to our simulation a single-cycle (3 fs) 100 petawatt laser pulse might be achieved based on recently available optical elements.

## 2. New designed WNOPCPA

The WNOPCPA with a divergent monochromatic pump has already been demonstrated to enhance the gain bandwidth of OPCPA [11–13]. Figure 1(a) shows, in the crystal a collimated signal would interact with different pump rays propagating along different directions (i.e., different phase-matching angles). Different pump rays would contribute to different gain spectrum of the signal, pump rays at the upper side of the center lead to a gain red shift, and ones at the lower side lead to a gain blue shift. In this case, the gain spectrum of the signal is broadened. However, this method cannot be utilized in a large-aperture OPA/OPCPA. Figure 1(a) shows when the pump and signal beams are big, only the center signal ray would interact with every pump ray, and others only interact with parts of pump rays. And, consequently, the ultra-broadband gain spectrum only appears at the signal center. If further increasing the beam aperture, because of a small non-collinear angle (1°–3°) and a small convergent/divergent angle (<1°), even the center signal ray cannot get the bandwidth enhancement. Thereby, it can be found that the previous WNOPCPA can only apply to a very small aperture OPCPA, and the pulse energy would be seriously limited. Besides that, the interaction length between the signal ray and each pump ray is very short, which could also limit the signal gain.

To overcome the above disadvantage, here we develop a new designed WNOPCPA, which could apply to a large aperture OPCPA. Figure 1(b) shows, we introduce the traditional beam homogenizer [14], which consists of a lens-array pair and a Fourier lens (usually is used in the square/hexagon flat-top beam shaping), into the WNOPCPA, and then the shaped pump would have multiple-direction wave-vectors at each position in the nonlinear crystal. The geometrical layout is like this: the first lens-array divides the input beam into several beamlets [4 in Fig. 1(b)], and the second lens-array and the Fourier lens superimpose the image of each beamlet onto the Fourier plane. In this case, optical rays from different beamlets arrive at the Fourier plane along different directions. And this condition also applies to other planes before and after the Fourier plane [see Fig. 1(b)]. Therefore, the requirement of WNOPCPA is satisfied. Here, while comparing with the previous WNOPCPA, the only difference is that the direction angles of the pump rays are discrete instead of continuous, which will be explained in the following section.

When the focal length of the first lens-array is half of that of the second one and is positioned at the front Fourier plane of the second one, the divergent angle *α* in the crystal satisfies [14]

*M*and

*p*

_{LA}are the total number and the aperture of the micro-lenses in the lens-array,

*f*

_{FL}is the focal length of the Fourier lens, and

*n*is the refractive index. The angle gap

*Δα*between two adjacent discrete pump directions is around

*α*/(

*M*−1). In engineering, both

*α*and

*Δα*can be optimized conveniently.

Using the small signal gain approximation, we simulate the signal gain spatial-spectral distribution based on a 527 nm pump, a type I BBO crystal and a type I LBO crystal. Figure 1(c) illustrates the pump and signal angles in the type I BBO (upper) and the type I LBO (lower) crystals. In BBO, when the pump direction *θ*_{p} is 24°, at the signal direction of *θ*_{s} = 26.42°, Fig. 2(a) shows a broadband gain appears. When *θ*_{p} is reduced to 23.8° or increased to 24.2°, at the same signal direction of *θ*_{s} = 26.42°, the gain bandwidth is narrowed and has a red or a blue shift, respectively. At the other signal direction of *θ*_{s} = 21.58°, the red and blue shifts induced bandwidth enhancement is low. Similarly, Fig. 2(b) shows, in LBO, when the pump direction *ϕ*_{p} is 14.5°, the broadband gain appears at the signal direction of *ϕ*_{s} = 15.9°. When *ϕ*_{p} is reduced to 14.3° or increased to 14.7°, the gain bandwidth at the same signal direction of *ϕ*_{s} = 15.9° is narrowed and has a red and a blue shift, respectively. At the other signal direction of *ϕ*_{s} = 13.1°, the red and blue shifts induced bandwidth enhancement is slightly low. In this case, we choose the signal direction of *θ*_{s} = 26.42° in BBO and *ϕ*_{s} = 15.9° in LBO for the signal gain bandwidth enhancement. Figures 2(c) and 2(d) show the gain spectrum of BBO and LBO when the divergent angle *α* is 0.24° and the angle gap *Δα* is 0.04°, i.e., in BBO the pump direction *θ*_{p} varies from 24° to 23.76° every 0.04° and in LBO the pump direction *ϕ*_{p} varies from 14.5° to 14.26° every 0.04°. All of angles here are described inside the crystal. We can find that the divergent angle *α* and the angle gap *Δα* determines the bandwidth and the profile of the enhanced spectrum, respectively, and, by increasing *α* and reducing *Δα*, we can broaden the gain spectrum and smooth its profile. However, the problem is that if *α* is too big and *Δα* is too small, the energy conversion efficiency from pump to signal would be reduced. Because, the gain spectrum corresponding to each pump beamlet is narrow [see Figs. 2(c) and 2(d)], and for the case of OPCPA, the effective temporal overlap between each pump pulse and the chirped signal pulse is small. The result is that the conversion efficiency decreases with increasing the number of pump beamlets. And, another problem is the pump interference going to be discussed in the following section. Consequently, in this paper we choose a 0.24° divergent angle *α* and a 0.08° angle gap *Δα*, i.e., *θ*_{p} = 23.76°, 23.84°, 23.92° and 24° for type I BBO and *ϕ*_{p} = 14.26°, 14.34°, 14.42° and 14.5° for type I LBO.

## 3. Pump interference and solution

Another key problem is the interference among different pump beamlets [14]. Because the phase-matching is only in one plane [see Fig. 1(c)], we choose a cylindrical lens-array pair and a cylindrical Fourier lens to avoid 2-dimensional interference. Figure 3(a) shows the interference fringes when the divergent angle *α* is 0.24° and the angle gap *Δα* is 0.04° (i.e., 7 beamlets). If we increase the angle gap *Δα* from 0.04° to 0.08° (i.e., 4 beamlets), the fringe density is increased by 2 times [see Figs. 3(a) and 3(b)]. If remaining the angle gap *Δα* of 0.08° unchanged and reducing the divergent angle *α* from 0.24° to 0.16° (i.e., 3 beamlets), the fringe density remains unchanged but the bright fringe width is increased [see Figs. 3(b) and 3(c)]. Consequently, by increasing the angle gap *Δα* and reducing the number of beamlets contributing to the interference, both the bright fringe density and width could be increased, and accordingly the interference modulation would be reduced dramatically. Besides, just like a previously similar study of two-beam-pumped NOPCPA [15], the pump-signal non-collinear geometry and the pump spatial walk-off could smear the interference fringe modulation and improve the signal beam quality.

For the peak intensity of the interference fringes, while comparing with the intensity of each beamlet, the interference would increase the peak intensity by *M*^{2} times, where *M* is the number of beamlets. Figure 3 shows the peak intensity is increased by 49, 16 and 9 times in the case of 7, 4 and 3 beamlets interference. For a uniform pulse in time and a uniform beam in space, the peak intensity of interference fringes is given by

*E*

_{p}is the total pump energy,

*T*is the pump duration,

*S*is the pump area, and

*M*is the number of pump beamlets. If we delay a part of beamlets from the rest ones in time, Eq. (2) should be revised as where

*N*is the number of beamlets really contributing to the interference. Based on the parameters are going to be used in the following section, we make a simple calculation. The pump pulse duration is 1 ns, and a pump energy of 0.3 J and 1000 J is irradiated onto a 5×5 mm

^{2}BBO and a 130×130 mm

^{2}LBO, respectively. Table 1 shows the interference induced peak intensity or energy fluence for different numbers of beamlets, which increases linearly with increasing the beamlet number. However, for the case of 3 of 4 beamlets [i.e.,

*M*= 4,

*N*= 3, and 3(4) in Table 1] interfering with one another, Table 1 shows that, in BBO, the peak intensity of 2.7 GW/cm

^{2}is lower than 3.6 GW/cm

^{2}in the case of 3 “beamlet number”, and similarly, in LBO, the peak energy fluence of 13.3 J/cm

^{2}is lower than 17.8 J/cm

^{2}in the case of 3 “beamlet number”. The reported multiple-shot damage threshold of a BBO crystal for 8 ns 532 nm pulses is 32 GW/cm

^{2}[16], and that of a LBO crystal for 5.5 ns 532 nm pulses is 60 J/cm

^{2}[17]. In the long-pulse regime (τ > 20 ps), the damage threshold fits well by a τ

^{1/2}dependence, where τ is the pulse duration [18]. Consequently, we can estimate that the 1 ns 532 nm damage threshold of BBO and LBO would be around 11.3 GW/cm

^{2}and 25.6 J/cm

^{2}, respectively. And, the calculated results of 3(4) beamlets (i.e., 2.7 GW/cm

^{2}in BBO and 13.3 J/cm

^{2}in LBO) are lower than the damage thresholds. In this case, we choose the configuration of 3(4) beamlets, a 0.24° divergent angle

*α*, and a 0.08° angle gap

*Δα*(i.e., total pump directions are

*θ*

_{p}= 23.76°, 23.84°, 23.92° and 24° for type I BBO and

*ϕ*

_{p}= 14.26°, 14.34°, 14.42° and 14.5° for type I LBO). Only three beamlets (i.e.,

*θ*

_{p}= 23.76°, 23.84° and 23.92° for type I BBO and

*ϕ*

_{p}= 14.26°, 14.34° and 14.42° for type I LBO) interfere with one another [see Fig. 3(c)].

Figure 4(a) shows a time delay line is introduced to temporally separate the pulse of the pump beamlet 4 (i.e., *θ*_{p} = 24° in BBO and *ϕ*_{p} = 14.5° in LBO) from the pulses of the pump beamlets 1, 2 and 3 (i.e., *θ*_{p} = 23.76°, 23.84° and 23.92° in BBO and *ϕ*_{p} = 14.26°, 14.34° and 14.42° in LBO) to obtain the configuration of 3(4) beamlets, i.e., only pump beamlets 1, 2 and 3 interfere with one another. While comparing with Fig. 1(b), the chirped signal pulse should be further stretched due to the double of the overall pump duration. Figure 4(b) shows the pump beamlets 1, 2, 3 and 4 correspond to the signal gain spectrum ①, ②, ③ and ④, respectively. Because the signal pulse is deeply chirped, its spectral profile actually is its temporal profile. Then, in time, the pulses of the pump beamlets 1, 2 and 3 overlap with the parts of the chirped signal pulse corresponding to the spectrum ①, ② and ③, and that of the pump beamlet 4 overlaps with the part of the chirped signal pulse corresponding to the spectrum ④. Thereby, the pump interference induced signal beam modulation only influences the signal spectrum parts ①, ② and ③, which only contains a part of energy/power. Then, according to all of above solutions and phenomena, the adverse influence of the pump interference on the signal beam quality would be reduced a lot and become weak, which would be similar with the experimental results of two previous works [15,19].

## 4. Simulation method of WNOPCPA

Refer to the theoretical model of the two-beam-pumped NOPCPA [15], the interaction of one signal light, *M* pump lights and *M* idler lights satisfy the 2*M *+ 1 waves coupled-wave equations, and, in the frequency domain, which is given by

*k*

_{s}

*k*

_{p,m}and

*k*

_{i,m}are the wave vectors of signal, number

*m*pump and number

*m*idler (i.e., full dispersions of the material are considered),

*χ*

_{m}is the number

*m*second-order nonlinear coefficient which is direction-dependent, and

*n*is the refractive index which is wavelength and direction dependent,

*α*

_{m}is the number

*m*non-collinear angle between signal and number

*m*pump (i.e., non-collinear angle is considered),

*β*

_{m}is the number

*m*angle between number

*m*pump and number

*m*idler which is also wavelength dependent,

*ρ*is the pump walk-off angle (i.e., pump walk-off is considered) [20]. In theory, the simulation of 2

*M*+ 1 waves optical parametric coupling here is exactly the same with the previous 5 waves optical parametric coupling in the case of two-beam-pumped NOPCPA and discussed in Ref. [15].

## 5. 100 petawatt concept design

#### 5.1 Broadband amplification

Based on the above optimization and model, in this section we will propose a simple concept design of a single-cycle 100 petawatt laser. A milli-joule ultra-broadband seed is generated by a Ti:sapphire femtosecond frontend and HCF, as shown in Fig. 5(a), which right now is a very mature technique [7,8]. The generated seed is assumed to have a 4^{th} order super-Gaussian frequency profile with a spectral range from 600 to 1200 nm. After a pulse stretcher, it is chirped and stretched to 2 ns, and a spectral phase control (e.g., chirped mirrors, glass wedges, liquid-crystal modulators, deformable mirrors, acousto-optic modulators, etc.) is used to accurately adjust the final output pulse to get the minimum duration. Taking account of the efficiency of the whole setup, the pulse energy of the chirped ultra-broadband seed injected into the first amplifier (AMP1) is chosen as 0.1 mJ.

The AMP1 is based on a type I BBO crystal, and the four pump directions *θ*_{p} are 23.76°, 23.84°, 23.92° and 24°. The signal is along the direction of *θ*_{s} = 26.42°. The pump wavelength is 527 nm, the pump duration is 1 ns, and the total pump energy is 300 mJ which is homogenously distributed to four pump beamlets. Both pump and signal have a 5×5 mm^{2} beam aperture, and then the pump intensity along each direction is 0.3 GW/cm^{2}. The beamlet of *θ*_{p} = 24° is 1 ns delayed, and then the chirped signal duration should be 2 ns (i.e., 300 nm/ns chirped ratio). Figure 5(b) shows the evolutions of the signal spectral intensity [see (i)], the signal nonlinear spectral phase [see (ii)] [21,22], and pump, signal and idler energies [see (iii) and (iv)] along with the BBO length *z*. At the length of *z* = 15.2 mm, the maximum signal energy of 34 mJ is obtained. Figure 5(b)(i) shows that the spectrum here is not continuous, which is due to the discrete pump directions.

Here, we should emphasize that the signal nonlinear spectral phase (or called optical parametric phase [21,22]) illustrated in Fig. 5(b)(ii) and following figures is the spectral phase of the signal only induced by the nonlinear interaction, while the initial spectral phase (or called temporal dispersion) and the material dispersion in the crystals have already been removed. Generally, the profile of the spectral phase from the stretcher and transmission materials is smooth and easy to be controlled [23], and here we only discuss the signal nonlinear spectral phase to show the compression difficulty of the final pulse.

The 34 mJ signal pulse from AMP1 is then injected into the main amplification chain, which contains four-stage amplifiers (AMP2, AMP3, AMP4, and AMP5). The pump wavelength and duration are still 527 nm and 1 ns, respectively. Figure 6(a) shows the total pump energies in four-stage amplifiers are 10 J, 500 J, 1000 J, and 1000 J, respectively, and the effective crystal apertures are 20×20 mm^{2}, 130×130 mm^{2}, 130×130 mm^{2}, and 130×130 mm^{2}, respectively. Then, the pump intensities along each direction in four-stage amplifiers are 0.625 GW/cm^{2}, 0.740 GW/cm^{2}, 1.480 GW/cm^{2}, and 1.480 GW/cm^{2}, respectively. Considering large apertures and high pump intensities, we choose type I LBO crystals here, which right now has a maximum effective aperture of 130×130 mm^{2} [4]. The four pump directions *ϕ*_{p} are 14.26°, 14.34°, 14.42° and 14.5°, and the beamlet of *ϕ*_{p} = 14.5° is also 1 ns delayed for the configuration of 3(4) beamlets. The signal is along the direction of *ϕ*_{s} = 15.9°, and the beam sizes of signal and pump are carefully expanded to match that of the crystal in each amplifier. Figure 6(b) shows the variations of the signal spectral intensity [see (i)], the signal nonlinear spectral phase [see (ii)], and pump, signal and idler energies [see (iii) and (iv)] with the LBO length *z*. In AMP2, at the length of *z* = 17 mm, the maximum signal energy of 1.1 J is obtained, which is injected into AMP3. In AMP3, at the length of *z* = 16.5 mm, the maximum signal energy is 57 J, which is input into AMP4. In AMP4, the maximum signal energy of 190 J is obtained at the length of *z* = 5.2 mm and is further injected into the last amplifier of AMP5. In AMP5, the maximum signal energy goes to 370 J at the length of *z* = 3.2 mm. Table 2 gives the detailed parameters of each amplifier from AMP1 to AMP5.

Figures 7(a)–7(f) illustrate the intensity spectrum and the corresponding nonlinear spectral phase of the seed [see (a)] and the outputs of AMP1 to AMP5 [see (b)–(f)]. It can be found that, instead of a continuous spectrum, only four separated spectrum [i.e., 770–950 nm, 1020–1070 nm, 1090–1130 nm and 1155–1185 nm in (f)] are amplified due to the discrete pump directions. From AMP1 to AMP5, while comparing with the shorter wavelength part [i.e., 770–950 nm in (f)], the intensities and the widths of three longer wavelength parts [i.e., 1020–1070 nm, 1090–1130 nm and 1155–1185 nm in (f)] increase gradually [see (b)–(f)], we think it is due to gain saturation. The nonlinear spectral phase degrades from AMP1 to AMP5. The peak-to-valley value of the output in Fig. 7(f) is around 6π rad, which is small and within the control range of available elements [24]. And the requirement of the spectral resolution could also be satisfied by available elements [24], although the profile is not linear. Once the spectral phase is well controlled, Fig. 7(g) shows the Fourier-transform limited pulse would be 3 fs, i.e., a single optical cycle. If an efficiency of around 80% grating compressor is considered, a 300 J compressed pulse could be obtained, and accordingly, the peak power would reach to 100 petawatt. Figure 7(g) also gives the Fourier-transform limited pulse only corresponding to the short wavelength part of 770–950 nm, and the pulse duration is increased to 6 fs.

#### 5.2 Grating compressor

The grating compressor is a key element, especially for a broadband femtosecond petawatt-class laser. In this section, according to the above results, we propose an engineering design for the proposed single-cycle 100 petawatt laser. The chirped ratio introduced by the grating stretcher of 300 nm/ns (i.e., 600–1200 nm and 2 ns) remains unchanged, although the profile of the final spectrum is not continuous. Based on 800 groove/mm golden gratings and 0.2 J/cm^{2} energy fluence [25], Fig. 8(a) shows the evolutions of the beam size (square beam), the small grating size, the large grating size, the slant distance within each parallel grating pair and the grating-beam gap for various incident angles. When a 42° incident angle is chosen, the geometry of a four-grating compressor is satisfied, i.e., the grating-beam gap is larger than zero and no beam-grating cut appears. Figure 8(b) shows the engineering layout simulated by the software of ZEMAX [26], Table 3 lists the detailed parameters, and the required largest golden grating size is 400 mm × 800 mm which recently is available.

## 6. Discussions

For the discontinuous spectrum of the final output [see Fig. 7(f)], it is due to the discrete pump directions (4 in this paper) for the consideration of both the conversion efficiency and the pump interference. Actually, discontinuous spectrum have already been widely used in the generation of single-cycle lights [27], and the mainly adverse influence is the slight distortion of the pulse profile [see Fig. 7(g)]. While comparing with the difficulty of generating a single-cycle pulse, this factor right now can be neglected. For example, if we use the general method of OPCPA, the shortest pulse would be 6 fs [see Fig. 7(g)], and then the required pulse energy should be increased to 600 J. Because the femtosecond damage threshold of golden gratings exhibits no clear dependence on pulse width [25], the required grating size would be beyond the available largest one, and accordingly a high-energy pulse compression is greatly challenged.

For the signal beam quality, by using a temporally delayed pump [see Fig. 4(a)] and optimizing pump angles and the number of pump beamlets [see Fig. 3(c)], the width of interference dark fringes is reduced a lot. This spatial modulation is also smeared by two effects of the pump-signal non-collinear geometry and the pump spatial walk-off. Figure 4 shows only three parts of long wavelength spectrum are affected, and Figs. 7(b)–7(f) show that the power/energy of these three long wavelength parts is not the majority. Consequently, the output beam quality should be further improved. And, the detail of this will be continuously studied in our next-step work.

For the accurate optimization of the shortest pulse duration, the proposed method would be challenged due to the short pulse of only a single optical cycle (3 fs). Recently, such problems have already been solved in small-scale single-cycle lasers, and in this case we believe in technology it won’t seriously influence the feasibility of the proposed concept design. For the ultra-broadband spectral range, i.e., from 770 nm to 1185 nm here, we suppose two cascade spectral phase control elements with different spectral ranges could be considered [24]. Meanwhile, the spatiotemporal coupling distortion is another potential problem due to both ultra-short pulses and large beams [28,29], which should be further considered in engineering.

For the potential damage of the lens-array pair, the focal lens of the first lens-array should be chosen as half of that of the second one, and then the energy fluence at two lens-arrays are same. The chosen material is fused silica, which has a very high reported damage threshold of 420 J/cm^{2} for nanosecond pulses [30].

Anyway, a single-cycle 100 petawatt laser would be a very complex and systematic technology and facility. In this paper, we only propose a possible method, and in future it should be continuously improved and optimized.

## 7. Conclusion

In conclusion, by re-designing WNOPCPA to meet the requirement of large aperture pump and signal beams, we theoretically propose a new concept design of a single-cycle (3 fs) 100 petawatt laser. Different from previous methods, the spectral bandwidth is dramatically broadened, and consequently the pulse duration is reduced to only 3 fs. According to our simulation, based on recently available crystals and gratings, the amplified pulse energy could reach to 370 J, and, after pulse compression, a 100 petawatt peak power would become possible. We believe that, while comparing with a 6–10 fs, 600–1000 J, high-energy 100 petawatt laser or a 10-beam coherent combination of femtosecond 10 petawatt lasers, the proposed single-cycle 100 petawatt is feasible in engineering and cheap in economics.

## Funding

JST-Mirai Program, Japan (JPMJMI17A1).

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