2 Impact of the B-integral on the compression ability of chirped lasers

Ultra-high-peak-power lasers are typically generated through CPA/OPCPA technology. In this process, a low-energy broadband seed pulse is first stretched into a nanosecond pulse by a stretcher. Subsequently, the nanosecond pulse is amplified through several pieces of gain media or parametric crystals, and then compressed to its minimum pulse width using a compressor. During this process, the nanosecond laser pulse with a certain chirp passes through multiple transmission elements. Each time it passes through a transmission element, a nonlinear phase shift accumulates due to the nonlinear refractive index of the transmission medium, which is given by the B-integral $B=\sum {\omega}_0{n}_2{I}_0L/c$ , where ω ₀ is the laser frequency, c is the speed of light, I ₀ is the laser intensity, n ₂ is the nonlinear refractive index of the medium and L is the length of the medium. Since this nonlinear phase shift is related to the intensity, the local frequency at each position of the laser pulse will change and new frequency elements will be generated, which is known as self-phase modulation (SPM). SPM disrupts the linear chirp of the laser pulse, generating higher-order dispersion, making it impossible for the compressor to compress the laser to its minimum pulse width. Therefore, in short-pulse laser systems, considering the impact of the spectral phase, it is necessary to control the B-integral accumulated during the amplification process of the laser pulse.

Assuming that the laser pulse has a certain chirp after passing through the stretcher and a B-integral is introduced during the amplification process, then before entering the compressor, the temporal waveform of the laser can be written as follows:

(1) $$\begin{align}E(t)={E}_0\exp \left(-\left(1+ iC\right)\frac{t^2}{2{T}_{\mathrm{p}}^2}+ iB\exp \left(-\frac{t^2}{T_{\mathrm{p}}^2}\right)\right).\end{align}$$

Here, E ₀ represents the peak laser intensity, C is the coefficient of the chirp rate, ${T}_{\mathrm{p}}={T}_{\mathrm{h}}/\sqrt{4\ln 2}$ is the 1/e pulse width of the chirped laser intensity and T _h is the full width at half maximum of the laser intensity. The relationship between T _h and C is as follows: $\Delta \omega {T}_{\mathrm{h}}/2\pi =0.441\sqrt{1+{C}^2}$ , where $\Delta \omega =2\pi c\Delta \lambda /{\lambda}_0^2$ is the spectral width (full width at half maximum) of the laser pulse, ${\lambda}_0=2\pi c/{\omega}_0$ is the central wavelength of the laser and ∆λ is the bandwidth (full width at half maximum) of the laser. In experiments, a more commonly used quantity is the chirp amount D, which is defined as the ratio of the laser’s full width at half maximum to its bandwidth: $D={T}_{\mathrm{h}}/\Delta \lambda =0.441\sqrt{1+{C}^2}{\lambda}_0^2/c{\left(\Delta \lambda \right)}^2$ .

Taking the derivative of the phase term, the instantaneous frequency at each position of the laser pulse is obtained as follows:

(2) $$\begin{align}\displaystyle \begin{array}{c}\omega =\dfrac{t}{T_{\mathrm{p}}^2}\left(C+2B\exp \left(-\dfrac{t^2}{T_{\mathrm{p}}^2}\right)\right).\end{array}\end{align}$$

Performing a Taylor expansion of the B-integral term in the laser waveform:

(3) $$\begin{align}\displaystyle \begin{array}{c}E(t)={E}_0{e}^{iB}\exp \left(-\dfrac{t^2}{2{T}_{\mathrm{p}}^2}-i\left(C+2B\right)\dfrac{t^2}{2{T}_{\mathrm{p}}^2}+ iB\dfrac{t^4}{T_{\mathrm{p}}^4}+\cdots \right).\end{array}\end{align}$$

Its Fourier transform is approximately given by the following:

(4) $$\begin{align}\displaystyle \begin{array}{c}\overset{\sim }{E}\left(\omega \right)\approx {\overset{\sim }{E}}_{0\omega}\exp \left(-\dfrac{T_{\mathrm{p}}^2}{1+i\left(C+2B\right)}\dfrac{\omega^2}{2}+ iB\dfrac{6{T}_{\mathrm{p}}^4}{{\left(1+i\left(C+2B\right)\right)}^4}\dfrac{\omega^4}{24}+\cdots \right).\end{array}\end{align}$$

When the B-integral is neglected, adjusting the compression grating to compensate for the second-order dispersion $\mathrm{GDD}=-\frac{C{T}_{\mathrm{p}}^2}{1+{C}^2}$ can minimize the pulse width of the output pulse. When the intensity increases, if no additional compensation is made by moving the compression grating under a certain B-integral, the pulse width of the output pulse will be broadened at high laser energy. Considering a chirped laser pulse with a central wavelength of 1000 nm, we calculated the B-integral corresponding to the decrease of the compressed laser peak intensity to 90% under different laser bandwidths and chirp amounts. We found that within a wide range of laser bandwidths and chirp amounts (∆λ = 5–50 nm, D = 10–500 ps/nm), when the B-integral reaches approximately 0.96, the laser intensity drops to 90%. It can be seen that if the grating is set at low energy and no adjustment was made at high energy (for example, in single-shot lasers), the B-integral should be designed to be less than 1.

However, in experiments, we can often achieve a more ideal compression effect by fine-tuning the spacing of the compression grating to compensate for the additional second-order dispersion induced by the B-integral. We assume that the additional second-order dispersion required to compress the high-energy laser to the limit pulse width is $\mathrm{GDD}^{\prime}=\alpha B\frac{T_{\mathrm{p}}^2}{1+{C}^2}$ , where α is an adjustable parameter. The B-integral corresponding to the decrease of the compressed laser peak intensity to 90% and the corresponding α values under different laser bandwidths and chirp amounts are shown in Figure 1. It can be seen that when the laser bandwidth and chirp amount are small, the tolerable B-integral is high. When the laser bandwidth or chirp amount increases (∆λ > 4 nm, D > 20 ps/nm), the tolerable B-integral gradually drops to a fixed value of approximately 1.7, and the corresponding α value gradually stabilizes at around 0.444. It can be seen that if the second-order dispersion can be fine adjusted, it is only necessary to control the B-integral to be less than 1.7.

Figure 1 For Gaussian pulses, (a) the maximum tolerable B-integral and (b) the corresponding α value when the laser pulse has different bandwidths and different chirp amounts.

For picosecond lasers, it is reasonable to assume that their spectrum follows a Gaussian function. However, for femtosecond lasers with broader spectral widths, the laser spectra are often shaped to approximate a rectangular distribution through spectral shaping techniques. In this case, after passing through a stretcher, the temporal waveform of the laser pulse is also similar to rectangular distribution. Under such circumstances, the introduction of the B-integral does not generate new frequencies in the central part of the pulse because the strength of the SPM effect is related to the gradient of the laser intensity. Therefore, the influence of the B-integral induced during the amplification process on the compressed femtosecond lasers should be weakened. To verify this, we consider a chirped laser pulse with an n _SG-order super-Gaussian distribution in the frequency domain: $\overset{\sim }{E}\left(\omega \right)={\overset{\sim }{E}}_0\exp \left(-\frac{\ln (2)}{2}{\left(2\frac{\omega -{\omega}_0}{\Delta \omega}\right)}^{2{n}_{\mathrm{SG}}}+i\frac{\mathrm{GDD}}{2}{\left(\omega -{\omega}_0\right)}^2\right)$ . Firstly, the value of GDD is estimated using the analytical formula in Gaussian light, but this value is not applicable to super-Gaussian light. Therefore, the value of GDD needs to be finely adjusted so that the inverse Fourier transform of $\overset{\sim }{E}\left(\omega \right)$ has a specified full width at half maximum ${T}_{\mathrm{h}}=D\Delta \lambda$ in the time domain. Then, a certain B-integral is added to the laser pulse in the time domain, and the second-order dispersion GDD’ required to compress the laser to the shortest width and the corresponding laser intensity are recalculated. For a super-Gaussian laser pulse, when the laser intensity decreases to 90%, the corresponding B-integral is taken as the maximum tolerable B-integral. When the laser pulse has different bandwidths ∆λ and chirp amounts D, the maximum tolerable B-integrals for n _SG = 2, 3, 4 and 5 are as shown in Figures 2(a)–2(d), respectively. It can be seen that for second-order super-Gaussian laser pulse, when the chirp amount and the bandwidth of the laser pulse are large enough, because of the central region of the laser’s temporal waveform becoming flatter, the tolerable B-integral will be much higher than that of a Gaussian pulse and will finally stabilize at around 5.72. However, for third-, fourth- and fifth-order super-Gaussian pulses, the tolerable B-integral decreases to around 4.11, 3.16 and 2.83, respectively. This is because for a high-order super-Gaussian pulse, the edges of the laser are steeper and the high-order dispersions generated at the laser edges are unable to be compensated by the compressor. Therefore, a large amount of energy at the laser edges is wasted, which leads to the decrease of the maximum B-integral.

Figure 2 For super-Gaussian pulses with (a) n _SG = 2, (b) n _SG = 3, (c) n _SG = 4 and (d) n _SG = 5, the maximum tolerable B-integral when the laser pulse has different bandwidths and different chirp amounts.

In femtosecond laser systems, an AOPDF (acousto-optic programmable dispersive filter) is commonly used to compensate dispersion up to the fifth order. In this configuration, the fourth-order dispersion induced by the B-integral can also be compensated within the compensation range. In the simulations, we found that within the parameter range of interest, the fourth-order dispersion generated by the B-integral typically far exceeds the compensation capability of the AOPDF (100,000 fs⁴). Therefore, we do not consider the scenario where the fourth-order dispersion induced by the B-integral is fully compensated by the AOPDF in this study.

3 Small-scale self-focusing effect caused by the B-integral

In addition to causing high-order dispersion in the time domain, the introduction of the B-integral also leads to modulation instability of the laser pulse in the spatial domain, that is, small-scale self-focusing. Small-scale self-focusing often imposes stricter requirements on the B-integral, which will be discussed through the Bespalov-Talanov (B-T) theory^[ Reference Bespalov and Talanov ^{20 ]}. The surface quality of optical elements can be described by the power spectral density (PSD), which is defined as the square of the amplitude of the Fourier elements of the surface undulation of the element:

(5) $$\begin{align} \begin{array}{c}{S}_{\mathrm{ph}}\left({k}_x\right)=\dfrac{{\left(\int z(x)\exp \left(-i{k}_xx\right)\mathrm{d}x\right)}^2}{\Delta {k}_x}.\end{array}\end{align}$$

For amplitude noise, z(x) is the transmitted/reflected laser amplitude at each measurement point; for phase noise, z(x) is the measured thickness at each measurement point. Here, we only consider phase noise. For phase noise, the PSD of an optical element surface generally satisfies the following power-law relationship^[ Reference Ginzburg, Kochetkov, Mironov, Potemkin, Silin and Khazanov ^{21 ]}:

(6) $$\begin{align}\displaystyle \begin{array}{c}{S}_{\mathrm{ph}}\left(\theta \right)=\dfrac{S_0}{k_x^b}.\end{array}\end{align}$$

Referring to the element parameters provided by the NIF device^[ Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig ^{22 ]}, we use the following parameters as the initial PSD distribution of surface distortion of reflective optical elements: ${S}_0=10\ {\mathrm{nm}}^2\cdot \mathrm{mm};\ b=1.55,\ {k}_x/2\pi <1\ {\mathrm{mm}}^{-1}; b=2.55,\ {k}_x/2\pi >1\ {\mathrm{mm}}^{-1}$ . With these parameters, the peak to valley (PV) value of the surface distortion is 60 nm.

When an optical element introduces phase noise into a laser pulse, due to slightly different transmission directions of noise at different spatial frequencies, the phase noise and amplitude noise of the laser will be mutually transformed during transmission in vacuum. However, before the noise leaves the main laser (i.e., self-filtering^[ Reference Mironov, Lozhkarev, Luchinin, Shaykin and Khazanov ^{23 –} Reference Ginzburg, Kochetkov, Mironov, Potemkin, Silin and Khazanov ^{25 ]}), the power spectrum of the noise remains unchanged. After incident on a transmission element with the Kerr effect, due to the stronger laser intensity positions having a larger refractive index, $n={n}_0+{n}_2I$ , these positions tend to converge to the surrounding energy, so the laser pulse will change from a uniform distribution to a distribution with some modulation, and the initial noise of the laser pulse will be amplified, that is, the SSSF. The growth multiple of the SSSF is related to the B-integral as follows:

(7) $$\begin{align}G&=\left({\cosh}^2\left(\eta B\right)+\frac{4{\Theta}^2}{\eta^2}{\sinh}^2\left(\eta B\right)\right){\sin}^2{\varphi}_{\mathrm{in}}\\[4pt]&\quad+\left({\cosh}^2\left(\eta B\right)+\frac{\eta^2}{4{\Theta}^2}{\sinh}^2\left(\eta B\right)\right){\cos}^2{\varphi}_{\mathrm{in}}\nonumber\\[4pt]&\quad+\left(\frac{2\Theta}{\eta }+\frac{\eta }{2\Theta}\right)\cosh \left(\eta B\right)\sinh \left(\eta B\right)\sin 2{\varphi}_{\mathrm{in}},\nonumber\end{align}$$

(8) $$\begin{align}\displaystyle \begin{array}{c}\Theta =\dfrac{n_0{\theta}^2}{4{n}_2{\left|{E}_0\right|}^2},\end{array}\end{align}$$

(9) $$\begin{align}\displaystyle \begin{array}{c}\eta =2\sqrt{\Theta -{\Theta}^2},\end{array}\end{align}$$

where $\theta ={k}_x/\left({n}_0{k}_0\right)$ is the tilt angle of the noise within the near-field and ${\varphi}_{\mathrm{in}}$ is the initial phase of the noise.

It can be seen that the growth rate of the SSSF is related to the initial phase ${\varphi}_{\mathrm{in}}$ of the noise. Since the phase of the noise is usually random when it reaches the Kerr medium, the initial phase can be averaged to obtain the average growth multiple of the SSSF:

(10) $$\begin{align}\displaystyle \begin{array}{c}{G}_{\mathrm{av}}=1+\dfrac{2}{\eta^2}{\sinh}^2\left(\eta B\right).\end{array}\end{align}$$

Considering the self-filtering effect during laser transmission, that is, when the laser is transmitted in the pipe, the noise elements gradually spread out of the laser aperture and are blocked by the edge of the pipe or the downstream optical elements, resulting in a reduction of noise energy. When considering the self-filtering effect, the amplification factor of the noise should be corrected. Let the laser pulse be emitted from the source plane and then propagate freely to the target plane at a distance of L. The visible angle is defined as ${\theta}_{\mathrm{v}}={w}_0/L$ , where w ₀ is the waist of the laser pulse. Then, the ratio of noise between the source plane and the target plane, that is, the noise transmission rate, is given by the following^[ Reference Ginzburg, Kochetkov, Mironov, Potemkin, Silin and Khazanov ^{25 ]}:

(11) $$\begin{align}\displaystyle \begin{array}{c}T\left(\theta \right)=\dfrac{2}{\pi }{\int}_0^1\operatorname{Re}\left(\operatorname{arccos}\left(\frac{{\left(\theta /{\theta}_{\mathrm{v}}\right)}^2+{x}^2-1}{2 x\theta /{\theta}_{\mathrm{v}}}\right)\right)x\mathrm{d}x.\end{array}\end{align}$$

Therefore, the PSD of spatial noise when it reaches the target plane after starting from the noise source plane and then being amplified in the Kerr medium can be expressed as follows:

(12) $$\begin{align}\displaystyle \begin{array}{c}{S}_{\mathrm{out}}\left(\theta, B\right)=T\left(\theta \right){G}_{\mathrm{av}}\left(\theta, B\right){S}_{\mathrm{ph}}\left(\theta \right).\end{array}\end{align}$$

We take the Nd:glass laser as an example. For phosphate glass, typical parameters are n ₀ = 1.5, n ₂ = 3.35 × 10^–20 m²/W, and saturation energy E _s = 3.1 J/cm². After calculating the PSD of the input and output pulse, we convert the PSD back into random phase noise. Then we multiply this phase noise by a super-Gaussian distributed optical field amplitude and let it freely propagate for a certain distance: ${L}_{\mathrm{free}}=\frac{\pi c}{2{\omega}_0{n}_2I}\approx 1\ \mathrm{m}$ ; the phase noise is transformed into energy noise with a certain amplitude and phase. A simulation result with a laser intensity of 1 GW/cm², a laser diameter of 100 mm (fifth-order super-Gaussian distribution), a self-filtering path length of 1 m and a B-integral of 2 is shown in Figure 3.

Figure 3 Simulation result of the PSD curves (a) and the intensity distribution (b) of the input pulse (blue solid line) and the output pulse (red solid line) with a laser intensity of 1 GW/cm², a laser diameter of 100 mm (fifth-order super-Gaussian distribution), a B-integral of 2 and a self-filtering path length of 1 m.

We take the PV value of the near-field modulation reaching 1.8 as the criterion to calculate the maximum allowable B-integral under different noise energies S ₀. For each case, we conducted 100 simulations and took the average PV value. The calculation results are shown in Figure 4, where the blue, red and yellow lines correspond to the laser intensities of 1, 5 and 10 GW/cm², respectively, and the solid and dashed lines correspond to the self-filtering lengths of 1, 10 and 20 m, respectively. It can be seen that for a single standard optical element ( ${S}_0=10\ {\mathrm{nm}}^2\cdot \mathrm{mm}$ ), the maximum tolerable B-integral is approximately 2.5, and it slightly increases as the laser intensity increases. Through the self-filtering effect, the maximum B-integral can be increased by 0.1–0.3, and the higher the laser intensity, the more the B-integral increases by the self-filtering effect. This is because when the laser intensity is high, the amplified noise is mainly the high-frequency components, which are more easily filtered out by self-filtering. When the noise energy reduces to one-quarter of the original, the maximum tolerable B-integral increases to about 3.5. When the noise energy increases to four times the original (i.e., continuously reflected by 16 standard optical elements), the maximum tolerable B-integral decreases to 1.7. It should be noted that the optical path length L required for self-filtering is related to the laser waist w ₀. If the laser waist is doubled, to maintain the same viewing angle, the optical path length L also needs to be doubled to achieve the same self-filtering effect.

Figure 4 For rectangular-time-profile narrowband Nd:glass lasers, the B-integral required for the near-field modulation to reach 1.8 under different initial noise energies S ₀.

In the above calculation, only the spatial distribution of the laser pulse was considered. The temporal distribution was not taken into account. That is, it was assumed that the laser had a rectangular distribution in the time domain, and the chirp of the laser pulse can be neglected. With these assumptions, the amplification distribution of the spatial noise at different times is the same. Therefore, the above conclusion is only applicable to narrowband nanosecond pulses that have a rectangular distribution in the time domain. However, for picosecond lasers, although their bandwidth is relatively narrow, the laser pulse often has a Gaussian or Lorentzian distribution in the time domain. In this case, the amplification distribution of the spatial noise at different times is completely different. For femtosecond lasers, the laser pulse has different local frequencies at different times. In this case the amplification distribution of the spatial noise at different times is also different too. We first consider Nd:glass picosecond lasers; let the temporal waveform of the laser pulse be $E(t)\propto \exp \left(-\frac{\ln (2)}{2}{\left(2\frac{\omega (t)-{\omega}_0}{\Delta \omega}\right)}^{2{n}_{\mathrm{SG}}}\right)$ , $\omega (t)=\frac{C}{T_{\mathrm{p}}^2}t$ , $\Delta \lambda =5\;\mathrm{nm}$ , ${n}_{\mathrm{SG}}=1$ . For each moment within the pulse width, we assume that the phase noise has the same initial distribution, and calculate the final distribution of the noise at each moment. Then we transfer the phase modulation into intensity modulation by letting the laser pulse freely propagate L _free. Finally, we accumulate the laser intensities at each moment and calculate the near-field fluence modulation. We calculated the B-integral required for the PV value of the fluence modulation to reach 1.8 under different noise energies, and calculated 100 times for each case and took the average value. The calculation results are shown in Figure 5, where the blue, red and yellow lines correspond to the laser intensities of 1, 5 and 10 GW/cm², respectively, and the solid and dashed lines correspond to the self-filtering lengths of 1, 10 and 20 m, respectively. It can be seen that for a single standard optical element, the maximum tolerable B-integral is 2.9. When the noise energy reduces to one-quarter of the original, the maximum tolerable B-integral increases to around 3.8; when the noise energy increases to four times the original, the maximum tolerable B-integral decreases to around 1.9. In all three cases, the tolerable B-integral remains basically unchanged as the laser intensity increases. The self-filtering effect can slightly increase the maximum B-integral by 0.1–0.3. Although self-filtering can filter out high-spatial-frequency noise, for the characteristic optical intensity we used in the simulation, a freely propagating length of tens of meters is not sufficient to effectively filter out the SSSF noise.

Figure 5 For picosecond Nd:glass lasers, the B-integral required for the near-field modulation to reach 1.8 under different noise energies S ₀.

Next, we consider the case of broadband femtosecond lasers. Taking the Ti:sapphire laser as an example, its specific parameters are n ₀ = 1.76, n ₂ = 4.77×10^–20 m²/W, and saturation energy E _s = 0.9 J/cm². For broadband lasers, although their spectrum is wide and they often have a rectangular distribution, due to the different instantaneous frequencies at each moment, the SSSF at different moments also has slightly different amplification distributions. Assuming that the parameters of the femtosecond laser are $\Delta \lambda =40\;\mathrm{nm}$ , ${n}_{\mathrm{SG}}=4$ , and that the laser pulse has the same initial surface distortion at each moment, after the same processing as before, due to the slight difference in local wavelength at each moment, the amplification distributions of noise at different moments will be slightly different, and thus a slightly different final surface distortion will be obtained at each moment. Finally, the surface distortion is converted into wavefront phase distortion, $\varphi \left(x,y,t\right)=z\left(x,y,{k}_0(t)\right)\cdot {k}_0(t)$ , and the laser pulse is allowed to freely propagate a distance L _free. After that, the near-field fluence modulation is calculated (averaged over 100 calculations). We take the fluence modulation reaching 1.8 as the criterion and calculate the B-integral with different intensities. The calculation results are shown in Figure 6. It can be seen that for a single standard optical element, the maximum tolerable B-integral in this case is 2.5, and it slightly increases when the laser intensity increases. When the noise energy reduces to one-quarter of the original, the maximum tolerable B-integral increases to around 3.3, and this value significantly increases when the laser intensity increases. When the noise energy increases to four times the original, the maximum tolerable B-integral decreases to around 1.5, and it remains basically unchanged when the laser intensity increases. Through the self-filtering effect, the maximum B-integral can be increased by 0.1–0.3. Overall, the calculation results of the Ti:sapphire laser in Figure 6 are not much different from those of the rectangular narrowband neodymium glass laser shown in Figure 4. Compared with narrowband lasers, broadband lasers can reduce the fluence modulation caused by the SSSF through the spectral smoothing effect, and thus the tolerable B-integral can be increased. However, because the wavelength of the Ti:sapphire laser is shorter than the wavelength of the Nd:glass laser, the same surface distortion of optical elements means a larger wavefront phase distortion for the Ti:sapphire laser, which offsets the benefits brought by the spectral smoothing effect.

Figure 6 For femtosecond Ti:sapphire lasers, the B-integral required for the near-field modulation to reach 1.8 under different noise energies S ₀.

In the development of Beamlet (a nanosecond Nd:glass laser system as the prototype of NIF architecture), researchers found that the maximum value of B-integral should not be larger than 1.8 to avoid severe near-field modulation caused by the SSSF^[ Reference Wegner and Burkhart ^{15 ]}, which was consistent with our numerical results. In the design of picosecond and femtosecond laser systems, this criterion was adopted despite some researches^[ Reference Zhang, Wen, Fu, Zhang, Deng and Fan ^{17 ,} Reference Khazanov ^{18 ]} suggesting that this criterion can be relaxed. We find that this is true for picosecond Nd:glass laser systems; for femtosecond Ti: sapphire laser systems we need to consider the number of optical components in the system, which was usually less than in the Nd:glass laser systems.

4 Influence of the B-integral on laser focusability

Other than causing the SSSF in the near-field, the B-integral also induces wavefront phase distortion, which does not increase the intensity modulation in the near-field, but will degenerate the focal spot in the far-field. The transmission or reflection spatial filter can suppress the growth of the laser intensity modulation in the near-field and protect the optical elements from laser damage. The wavefront phase distortion can also be compensated by a deformable mirror. However, this compensation is not perfect. The laser intensity profile varies at different moments, causing different wavefront phase distortions at different moments, but the deformable mirror can only compensate for the wavefront phase distortion at a certain moment. Besides, the unit size of the deformable mirror is limited, which means it can only compensate for the phase distortion in the low-spatial-frequency region. In this section, we use the (time-integrated) Strehl ratio of the far-field as the standard to calculate the influence of the B-integral on the laser focusing capability under different cases, and accordingly provide the B-integral criterion for broadband lasers. The specific calculation method is as follows: assume the central wavelength of the laser is 1 μm. For the amplitude noise of a laser electric field, its PSD can be approximately described by Equation (6). To simulate the filtering effect of the spatial filter, we ignore the intermediate and high-frequency parts of the PSD function and generate a spatial intensity distribution using the low-frequency part with a wave number less than 2π × 0.03 mm^–1, which cannot be filtered out by the pinhole of spatial filters. We increase the laser diameter to 300 mm (fifth-order super-Gaussian distribution) to better simulate the low-frequency distortion, with a step size of dx = dy = 2 mm. A spatial intensity distribution generated by the above method is shown in Figure 7.

Figure 7 Spatial intensity distribution generated by the simulation program.

After generating the spatial distribution, in the temporal domain, we set the laser profile as a chirped laser pulse with an n _SG-order super-Gaussian distribution. After passing through the Kerr medium, the laser accumulates a certain value of the B-integral. The laser pulse is then compressed back to the transform limit, and a phase compensation is given to the laser pulse through a deformable mirror. Theoretically, the phase to be compensated by the deformable mirror should be the phase of the compressed laser at the moment of the highest intensity, which can achieve the best compensation effect. However, in actual experiments, this phase cannot be measured. Therefore, we choose the phase distribution at the central wavelength of the laser pulse before compression as the phase to be compensated by the deformable mirror^[ Reference Ginzburg, Kochetkov, Potemkin and Khazanov ^{24 ]}. The unit sizes of the deformable mirror are 10, 20 and 30 mm, respectively. For a broadband laser pulse under the paraxial approximation, the far-field fluence distribution F _f(x _f,y _f) at the focal plane after passing through an ideal lens can be calculated through the Fresnel–Kirchhoff integral^[ Reference Martyanov, Mironov, Starodubtsev, Soloviev, Kochetkov, Ginzburg, Shaikin and Khazanov ^{26 ]}:

(13) $$\begin{align}\displaystyle \begin{array}{c}{S}_0\left(\omega, {x}_0,{y}_0\right)=\dfrac{1}{\sqrt{2\pi }}\int E\left(t,{x}_0,{y}_0\right){e}^{- i\omega t}\mathrm{d}t,\end{array}\end{align}$$

(14) $$\begin{align}\displaystyle \begin{array}{@{}l}{S}_{\mathrm{f}}\left(\omega, {x}_{\mathrm{f}},{y}_{\mathrm{f}}\right)\\\quad\!\!=\dfrac{- ik\left(\omega \right)}{2\pi f}{e}^{- ik\left(\omega \right)f- ik\left(\omega \right)\frac{x_{\mathrm{f}}^2+{y}_{\mathrm{f}}^2}{2f}}\iint {S}_0\left(\omega, {x}_0,{y}_0\right){e}^{ik\left(\omega \right)\frac{x_0{x}_{\mathrm{f}}+{y}_0{y}_{\mathrm{f}}}{f}}\mathrm{d}{x}_0\mathrm{d}{y}_0,\end{array}\end{align}$$

(15) $$\begin{align}\displaystyle \begin{array}{c}{F}_{\mathrm{f}}\left({x}_{\mathrm{f}},{y}_{\mathrm{f}}\right)=\int {\left|{S}_{\mathrm{f}}\left(\omega, {x}_{\mathrm{f}},{y}_{\mathrm{f}}\right)\right|}^2\mathrm{d}\omega. \end{array}\end{align}$$

The values of F _f(0,0) in the far-field focal spot with a certain value of the B-integral and without B-integral are calculated respectively, and the ratio of the two is the (time-integrated) Strehl ratio. The calculation results of the Strehl ratio under different B-integrals and different deformable mirror unit sizes are shown in Figure 8. We take the Strehl ratio decreasing to 0.8 as the maximum tolerable B-integral. It can be seen that when the laser bandwidth is 5, 10, 20 and 40 nm, the maximum tolerable B-integral without a deformable mirror is 2.2, 1.9, 1.8 and 1.7, respectively. After adding a deformable mirror with a unit size of 30 mm, the maximum tolerable B-integral increases to 3.2, 3.7, 3.9 and 4, respectively. When the unit size of the deformable mirror is 20 mm, the maximum tolerable B-integral further increases to 3.6, 4.7, 5.2 and 5.4, respectively. When the unit size of the deformable mirror is 10 mm, the maximum tolerable B-integral is 4.4, more than 6, more than 6 and more than 6, respectively. If the deformable mirror is not installed, the Strehl ratio decreases more severely when the B-integral increases for wider laser bandwidths and larger n _SG. This is because the leading and trailing edges of broadband lasers with larger n _SG have higher intensities, and under the same B-integral, the leading and trailing edges accumulate larger nonlinear phase shifts, thus having more severe wavefront phase distortions. However, after adding a deformable mirror, the Strehl ratio of the laser pulse shows a significant improvement. Moreover, the wider the laser bandwidth and the larger the n _SG, the better the correction effect of the deformable mirror. This is because before compression, the temporal profile of a broadband laser is closer to a rectangular pulse with larger n _SG, which enhances the correction effect of the deformable mirror. For the considered laser bandwidth range, using a deformable mirror with a unit size of 30 mm can achieve a satisfactory improvement in the Strehl ratio.

Figure 8 The variation of the Strehl ratio in the far-field with the B-integral under different laser bandwidths, temporal super-Gaussian orders and deformable mirror element sizes (no deformable mirror, 10 mm, 20 mm and 30 mm).

The criterion that the total B-integral of the system must be less than 3.5 to ensure wavefront distortion does not degrade laser far-field focusing capability originates from LLNL’s simulation^[ Reference Williams and Auerbach ^{16 ]} of transmission and focal spot of fundamental (1ω) and third harmonic (3ω) waves under 1 ns square pulse. After the LLNL established these criteria, subsequent designs of similar systems adopted them as a standard. However, the LLNL’s simulations had several limitations. Firstly, the LLNL did not use an authentic low-frequency wavefront model. Instead, an artificially constructed Gaussian random phase model was employed. Secondly, the simulation assumed a simplified deformable mirror capable of perfectly correcting all low-frequency wavefront aberrations, which is unrealistic in practical systems. Thirdly, the simulations were restricted to nanosecond square pulses and do not generalize to picosecond or femtosecond pulse scenarios. In our simulations, we utilized a more realistic low-frequency phase model and a deformable mirror model. Our simulation results are applicable not only to nanosecond pulses but also to picosecond and femtosecond pulses, providing design criteria for the upper limit of the total B-integral in nanosecond, picosecond and femtosecond laser systems.