Laser propagation in plasma is a fundamental and important issue in laser plasma interactions, which is related to a number of applications such as the fast ignition scheme for inertial confinement fusion ^[1,2], the laser wakefield acceleration of electrons ^[3–8], and lightning channeling in air ^[9,10]. Usually, these applications require that intense laser pulses can stably propagate over a large distance in plasma. On this issue, many theoretical and experimental studies have been performed in the last 30 years ^[11–24]. Self-focusing of an ultrashort intense laser pulse in a tenuous plasma was investigated theoretically in Refs. [11–15], and the well-known critical laser power required for self-focusing was found ^[13,14], where is the plasma electron density, is the critical density, and is the laser frequency. Since then, there have been a lot of studies on this topic when the laser power is around , e.g., laser channeling in underdense plasmas ^[16–18], laser guiding in plasma channels ^[19,20], and propagation of multi-laser beams in plasmas ^[21–24].

Meanwhile, ultrashort ultraintense laser technology has been developing quickly. A few Petawatt (PW) laser systems are available nowadays ^[25]. The extreme light infrastructure (ELI) will be able to provide hundreds of PW laser beams with intensity as high as . In this case, the laser power will be much higher than . It is interesting to investigate how such laser pulses can stably be self-guided. Actually, there have been a lot of laser wakefield acceleration (LWFA) experiments conducted with laser power about 10  for 1 GeV-scale electron beam generation ^[26–29].

In this paper, we focus on the propagation of extremely high power lasers. It is shown that there is an upper-limit power for self-guided propagation of laser pulses in underdense plasma. This is caused by the transverse ponderomotive force of the laser pulses, expelling local plasma electrons and creating an electron-free channel in a certain area. This effect can lead to defocused propagation of the laser pulses similar to in the vacuum, which may already occur at tens of . Here, we call such a phenomenon ponderomotive defocusing, which suggests that ponderomotive forces may not help laser self-focusing in a high laser power regime.

The outline of the paper is as follows. First, the ponderomotive defocusing is demonstrated through a set of two-dimensional (2D) particle-in-cell (PIC) simulations in Section 2. Then both the upper-limit critical laser power and the lower-limit critical plasma density for self-focusing are derived theoretically in Section 3. The results are checked by 2D PIC simulations in Section 4. Finally, the paper is summarized in Section 5. It should be noted that ponderomotive defocusing has been demonstrated in our paper ^[30] by three-dimensional PIC simulations and the upper-limit critical laser power has been derived. In the current paper, we will present a more detailed investigation by 2D PIC simulations and give a more detailed derivation of the upper-limit power.

We first demonstrate the ponderomotive defocusing by the results of a set of 2D PIC simulations, shown in Figure 1(a) (a similar 3D PIC demonstration can be seen in ^[30]). In the simulations, 1 wavelength laser pulses propagate along the direction. They are linearly polarized along the direction and their vector potential takes the form

 ((1))

Figure 1(a) shows the spacial distributions of the laser electric fields at propagation distances of 0.25 and 2 , respectively, where is the Rayleigh length. It is shown in the second row that the laser pulse with the power of 10 (8.8 TW) propagates with self-focusing for several in the plasma. However, when the laser power is increased to (219 TW) [ is an upper-limit power defined by Equation (13)], self-focusing does not appear, and the evolution of the laser pulse is very close to that in the vacuum, as observed in the first and third rows. We call this phenomenon ponderomotive defocusing in a plasma. This has resulted from the complete expulsion of all local electrons by the transverse ponderomotive force of the extremely intense laser pulse, as shown in the first picture in Figure 1(b). Therefore, the laser propagates as in the vacuum.

One expects that there is a laser power threshold above which the laser pulse starts to experience ponderomotive defocusing in a plasma. This threshold can be given according to balance of the transverse ponderomotive force with the electrostatic (ES) force. The ES force is formed by charge separation resulting from expulsion of local plasma electrons by the transverse ponderomotive force. The ES force counterworks the transverse ponderomotive one, which prevents ponderomotive defocusing. One can assume that the ES force is equal to the transverse ponderomotive one at some radius and that the plasma electrons are completely expelled within the column with radius . If is smaller than the laser spot radius , one can consider that the ES force is able to succeed in preventing the occurrence of ponderomotive defocusing. Then, one can find the laser power threshold for ponderomotive defocusing through the conditions of balance of the ES force with the ponderomotive one at .

In the following, we derive this power threshold. For this purpose, one needs to derive the ponderomotive force in a highly relativistic case. Note that the ponderomotive force have been derived in weak and moderate relativistic cases with the electron longitudinal velocity not so close to ^{[12,14,20,35,36]}. Here, we try to derive the ponderomotive force expressed approximately by the laser parameters in a highly relativistic case, where the longitudinal electron momentum may be much larger than the transverse one. Set that the laser pulse propagates along the direction and has linear polarization along the direction, with the vector potential

 ((2))

 ((3))

 ((4))

 ((5))

 ((6))

 ((7))

 ((8))

According to Equations (6) and (8), one can construct a fast varying Hamiltonian . Consider that in a tenuous plasma , and then is the function of for a given electron, since the time of interaction of the electron with the laser pulse is at the order of the laser duration usually, within which the laser waveform does not vary much. Then one can obtain a conversed quantity , which gives . It can be easily obtained that ^[31,32] , , and . To give the transverse ponderomotive force , one needs to get the slowly varying momentum , which is very difficult. Here, we take the 0-order approximation assuming , insert it into the expression of , and obtain . Taking the laser vector potential as Equation (2), one can obtain the transverse ponderomotive force at the laser pulse peak ():

 ((9))

 ((10))

Through , one can derive . Then one can obtain the upper-limit critical power for self-focusing or the power threshold for ponderomotive defocusing:

 ((11))

 ((12))

In the 2D slab geometry, is reduced by a factor ^[33,34], is enhanced by a factor 2, and , where the laser vector potential has been taken as Equation (1). Then, one can rewrite Equations (11) and (12) in the 2D slab geometry as

 ((13))

 ((14))

 ((15))

It should be pointed out that our model holds when the longitudinal electron momentum is important, which is justified, in particular, for an ultrashort ultraintense laser pulse. While the longitudinal electron momentum is neglected, one can assume and derive the laser amplitude ^[35,36], when the ES force and the ponderomotive force are balanced at . In this case, one can derive  TW and in 3D geometry, as well as in the 2D slab geometry  TW and . It is obtained that , and usually is smaller than in the underdense plasma case. Taking the laser and plasma parameters from this Letter, one can calculate . We will take the critical power and density as and because they show better agreement with the simulation results presented below.

We fix the laser spot radius at and vary the plasma density as well as the laser power. The evolution of the laser intensity with the propagation distance is plotted Figure 2. The plot with the initial plasma density illustrates clearly that self-focusing does not occur at any laser power. For a larger , the evolution curve of the laser intensity is closer to the one in the vacuum. Notice that the curve with nearly coincides with the one in the vacuum. When the plasma density is increased to (with ), occurrence of self-focusing is observed at , as shown in Figure 2(b). As the power is enhanced to 2, 4 and 8 , the corresponding curves at the beginning phase are close to the one in the vacuum. After a distance of defocusing, self-focusing appears because the self-focusing condition is satisfied with the reduced laser intensity and the increased laser spot radius. This distance of defocusing grows with the increase of the initial laser power. Similar results can also be seen in the plot with the plasma density of , although stronger self-focusing is observed at . In particular, when is up to 5000 , the laser evolves like in the vacuum in the whole distance of 5 . Here, in the simulations we judge if a laser pulse self-focuses or not according to the evolution curve at the beginning phase.

Then we take the laser spot radius as 4 and 16 , respectively, and the results are displayed in Figures 3 and 4. It is found that self-focusing begins to occur at the plasma densities of 5 and 6 , respectively, for cases with laser spot radiuses of 4 and 16 ; ponderomotive defocusing starts to be observed obviously at laser power of and 4 , respectively, for the cases with 4 and 16 (this value is about 2 for the case with 8 ). One can also see from Figures 2 and 3, and 4 that, for a smaller laser spot radius, the curve with the same initial laser power, which takes the unit as the respective or , approaches the one in the vacuum within a longer distance. These indicate that Equations (13) and (14) can better predict the threshold of ponderomotive defocusing for a smaller laser spot radius . This is because, for a smaller , the transverse ponderomotive force expels the electrons outside of faster and more easily, and thus the assumption is better that the electrons are completely expelled within the column with radius . This can be observed from the second and third pictures in Figure 1(b).

Besides, one can see from Figure 4(c) that the laser pulses attenuate at large propagation distances due to the light absorption by the plasmas. One notices that is as large as 804 for the laser pulse with . The light absorption effect will also becomes important when the plasma density is high, which can be observed in Figure 6.

Next, we fix the laser intensity and vary as well as the plasma density to check Equations (13) and (14) in another way. As mentioned above, is required for self-focusing. When , this gives that according to Equations (13) and (14). This prediction is confirmed by Figure 5. For , self-focusing begins to occur at ; for , it begins at ; for , it begins at . For a more intense pulse, a higher density threshold is required, e.g., for . The validity of is confirmed by our PIC simulations, which reveal that, when is taken as a few of , the pulses with , 16 and start to self-focus at the beginning stage, and then they attenuate fast due to light absorption in relatively high density plasmas.

In summary, we have shown that there is an upper limit of the laser power for self-focusing in plasma, which is a function of the initial spot size of the laser pulse and the plasma electron density. Self-focusing occurs only when the laser power is above . Otherwise, the laser pulse experiences ponderomotive defocusing due to expulsion of local plasma electrons by the transverse ponderomotive force. It is also found that there is a lower limit of the plasma density for self-focusing, below which self-focusing does not occur for any laser power. These are verified by 2D PIC simulations. The present study provides guidance for future experimental designs when the self-guided propagation of laser pulses over a long distance is required, such as in laser wakefield acceleration with laser power at the 100 TW level or above.