Overcoming gas ionization limitations with divided-pulse nonlinear compression

G. W. Jenkins; C. Feng; J. Bromage

doi:10.1364/OE.402772

1. Introduction

Maturing Yb laser technology has many useful properties for high peak and average power applications, but the relatively narrow bandwidth of Yb-doped gain materials only supports pulses of the order of picoseconds to hundreds of femtoseconds depending on the doped material [1,2]. Nonlinear compression in gas-filled hollow-core fiber (HCF) has been very successful for the spectral broadening and subsequent temporal compression of high-average-power Yb systems to ∼10-fs pulses for nonlinear applications such as high-harmonic generation [3] and difference-frequency generation [4]. However, pulse energy in an HCF is limited and does not take full advantage of Yb’s excellent energy scaling. In this paper, we examine the limits on HCF energy scaling and analyze a method to scale to higher energy: divided-pulse nonlinear compression.

In HCF nonlinear compression, a glass cladding guides light within a gas-filled hollow core. The gas provides the nonlinearity to broaden the pulse’s spectrum through self-phase modulation (SPM), and after the fiber, the pulse can be compressed to a shorter duration with devices such as chirped mirrors and prisms [5]. Scaling to higher pulse energies and larger broadening factors is limited by self-focusing and ionization as summarized by Vozzi et al. [6]. As the pulse propagates through the gas, it will accumulate a nonlinear phase,

(1)$${\varphi _{\textrm{NL}}} = \frac{{{n_2}{\omega _0}}}{{c{A_{\textrm{eff}}}}}{P_0}{L_{\textrm{eff}}},$$

where n₂ is the nonlinear index of the gas, ω₀ is the central angular frequency of the pulse, c is the speed of light, A_eff is the effective mode area of the fundamental mode of the fiber, P₀ is the peak power of the pulse, and L_eff is the effective length of the fiber. The effective length is the fiber length (L) reduced by fiber losses (α) according to ${L_{\textrm{eff}}} = {{({1 - {e^{ - \alpha L}}} )} / \alpha }.$ The nonlinear phase will broaden the pulse’s spectrum by approximately the factor F

(2)$$F = \frac{{\Delta {\omega _{\textrm{SPM}}}}}{{\Delta {\omega _0}}} = \sqrt {1 + \frac{4}{{3\sqrt 3 }}\varphi _{\textrm{NL}}^2} ,$$

where Δω₀ and Δω_SPM represent the pulse’s bandwidth before and after spectral broadening in the HCF, respectively. These expressions indicate that to obtain extremely high broadening factors, we should maximize the nonlinearity n₂, the peak power P₀, the fiber length L_eff, and minimize the fiber core size A_eff.

As we optimize parameters to obtain the maximum spectral broadening, we reach thresholds that degrade the output pulse. One such limit is self-focusing, which sets a limit on the nonlinearity and peak power. For large n₂ and P₀, the beam profile changes, resulting in significant energy coupling into higher-order modes of the fiber. Energy in higher-order modes will reduce the quality of the output pulse through spatiotemporal couplings, modal dispersion, and increased fiber losses. The steady-state peak power in the second-order mode was given by Tempea and Brabec [7,8]. Tempea and Brabec chose a second-mode power threshold of ${{{P_2}} / {{P_1}}}\textrm{ < }9\%,$ which limits the nonlinearity to

(3)$${n_2}{P_0}\textrm{ < }0.15\lambda _0^2,$$

and

(4)$${\varphi _{\textrm{NL}}}\textrm{ < }0.3\pi {\lambda _0}\frac{{{L_{\textrm{eff}}}}}{{{A_{\textrm{eff}}}}},$$

where λ₀ is the central wavelength of the pulse. A useful property of gas-filled HCF’s is that the nonlinearity can be tuned by setting the gas pressure (p), so this threshold can be met regardless of the pulse’s peak power. A gas’s nonlinear index is nearly proportional to its pressure with constant of proportionality κ₂. To keep the nonlinearity below Tempea and Brabec’s 9% threshold, the gas pressure must be kept below p_9%,

(5)$${n_2} = {\kappa _2}p,$$

(6)$$p\textrm{ < }{p_{9\%}} = 0.15\frac{{\lambda _0^2}}{{{\kappa _2}{P_0}}}.$$

Once the gas pressure is set to control self-focusing, Eq. (4) indicates that the spectral broadening can still be increased by optimizing the fiber area A_eff, as depicted in Fig. 1. However, gas ionization inside the HCF sets a limit on the peak intensity of the pulse, and often requires fiber operation to occur at diameters larger than the optimum. Therefore, given an input pulse duration and energy, the spectral broadening achievable is entirely determined by the thresholds of self-focusing and ionization. The only ways to increase spectral broadening further are to increase the length of the fiber (increase L_eff) or to add a second broadening stage.

Fig. 1. Spectral broadening factors computed from Eq. (2) with gas pressure set at the self-focusing threshold from Eq. (6). There is clearly an optimum fiber diameter for each fiber length that maximizes spectral broadening. For fibers smaller than this optimum, fiber losses increase significantly and L_eff becomes small. For fibers larger than this optimum, the large fiber area A_eff reduces the spectral broadening. For high-energy-intensity applications, gas ionization forces fiber operation to occur at diameters larger than the optimum.

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Multiple ideas have been explored to overcome the self-focusing and ionization limits. For one, fibers with very large diameters can be used to avoid or reduce gas ionization [9,10]. The reduced spectral broadening from the large fiber diameter must be compensated by increasing the fiber length, so these fibers tend to be very long. For another, plasma effects can be reduced by using circular polarization [11]. The ionization rate from a circularly polarized pulse is less than the ionization rate from a linearly polarized pulse with the same energy/power/intensity. Therefore, a smaller-radius fiber can be used and more broadening achieved. Another method is to apply a pressure gradient along the length of the fiber [12]. Low pressure at the entrance of the fiber, where pulse intensity is greatest, reduces self-focusing and ionization. High pressure at the exit of the fiber compensates energy losses throughout the fiber and keeps SPM high over the full length of fiber.

Fig. 2. Illustration of divided-pulse nonlinear compression in a hollow-core fiber with (a) one pulse, (b) two pulses, and (c) four pulses. Birefringent plates with extraordinary axis “e” and ordinary axis “o” can be used to separate the pulses temporally, and identical birefringent plates and a polarizer recombine the pulses. Red arrows indicate the pulse’s polarization, which is chosen so the polarization in the fiber is always vertical and horizontal. The distorted pulse shape after the fiber indicates an arbitrary reshaping by nonlinear processes in the fiber.

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We investigate a more-scalable method to overcome ionization limits—divided-pulse nonlinear compression (DPNLC). In DPNLC, a high-energy pulse is divided into multiple low-energy pulses that are broadened in the fiber, as depicted in Fig. 2. After spectral broadening in the fiber, the low-energy pulses are recombined back into one high-energy pulse. DPNLC has been demonstrated in glass fibers to keep peak power below self-focusing thresholds [13] and demonstrated in HCF with good efficiency [14–16]. In gas-filled HCF we can use the gas pressure to address the self-focusing threshold, which frees up the divided pulses to overcome ionization thresholds. After the pulses are divided, the pressure of the gas can be increased so that the nonlinearity (n₂P₀) is the same for both the single-pulse and divided pulses. At the same time, the intensity of the divided pulses will be lower than the intensity of the single pulse, leading to less ionization. With less ionization, smaller fibers and higher pulse energies can be used, increasing HCF spectral broadening capabilities. Additionally, DPNLC is more scalable than other methods because the pulse division can be scaled to 2^N pulses, where N is number of birefringent plates. In this paper, we demonstrate that scaling by simulating Kerr and plasma effects in HCF to show how the onset of ionization degrades the output pulse. Our simulations show that DPNLC can shift the onset of ionization to smaller fiber diameters and improve the compression.

2. Model

To simulate the pulse division and recombination, we assume an input pulse that is Gaussian in time and divided into one, two, or four pulses by zero, one, or two birefringent plates, respectively (as shown in Fig. 2). The divided pulses have equal amplitude and are separated by a group delay of 3.5× the pulse duration. After the fiber, the pulses are recombined by applying the same group delay to the leading pulses and summing the fields.

If the pulses are still in phase after the fiber, the output polarization is linear and the pulses are considered recombined perfectly. However, if phase artifacts appear in some pulses but not others, the pulses will interfere and not be recombined perfectly. The imperfect recombination is seen as an elliptical polarization that varies across the pulse [17] and parasitic side pulses that appear when recombining four or more pulses [18]. Only the energy that is in the main pulse and can pass through the polarizer is considered recombined, following the definition for recombination from Guichard [18]. In this paper, the gas ionization will be the only effect that reduces recombination efficiency. Other effects that can reduce the recombination efficiency are unequal nonlinear phase accumulated by unequally divided pulses, cross-phase modulation (XPM) between pulses, and differential group delay dispersion (GDD) in the birefringent plates [18]; but all of those effects are small in the simulations presented here. We simulate equally divided pulses, separate the pulses by 3.5× the pulse duration to eliminate XPM effects, and do not see much effect from differential group delay dispersion due to the small bandwidth of our picosecond pulses. Indeed, simulations that include the XPM and differential GDD effects but omit plasma effects predict >99% recombination efficiency for all simulations presented here when calcite birefringent plates with a thickness of 6-mm (and 12-mm for four-pulse division) are used to divide the pulses. Experimentally, recombination efficiencies of 95% have been observed [14,15] so the low recombination losses predicted here are reasonable.

Next, the beam profile at the input of the fiber is taken to be Gaussian with optimum 1/e² radius for coupling into first-order mode (w_opt = 0.643a, where a is the fiber core’s radius) and the beam is projected onto 20 modes that were simulated along length of fiber. The modes, propagation constants, and loss coefficients of an HCF were given by Marcatili and Schmeltzer [19]

(7)$${F_m}({r,\theta } )= {J_0}\left( {\frac{{{u_m}r}}{a}} \right),$$

(8)$${\beta ^{(m )}}(\omega )= \sqrt {k{{(\omega )}^2} - {{\left( {\frac{{{u_m}}}{a}} \right)}^2}} \approx k(\omega )\left\{ {1 - \frac{1}{2}{{\left[ {\frac{{{u_m}}}{{k(\omega )a}}} \right]}^2}} \right\},$$

(9)$${\alpha ^{(m )}}(\omega )= \frac{{u_m^2}}{{k{{(\omega )}^2}{a^3}}}\frac{{{\nu ^2} + 1}}{{\sqrt {{\nu ^2} - 1} }},$$

where F_m gives the beam profile of the mth mode in the radial (r) direction (rotational symmetry is assumed), J₀ is the zeroth-order Bessel function, u_m is the mth zero of J₀(r), β^(m) is the propagation constant of the mth mode, k(ω) is the free-space propagation constant evaluated with the gas’s refractive index [k(ω) = n(ω)ω/c], α^(m) is the loss coefficient of the mth mode, and ν is the ratio of refractive indexes between the glass cladding and gas core $[{{{\nu = {n_{\textrm{clad}}}} / {n(\omega )}}} ].$ The carrier wave ${e^{i\,[{\beta_0^{(0)}z - {\omega_0}t} ]}}$ of the fundamental mode is removed from the field and the full field can be built from its modes as

(10)$${A_{\textrm{pol}}}({r,z,t} )= \sum\limits_m {{A_{\textrm{pol}}}^{(m )}} ({z,t} ){F_m}(r ){e^{i\Delta {\beta _0}^{{{(m )}_z}}}} + \textrm{c}\textrm{.c}\textrm{.},$$

where A_pol = A_H,A_V gives the field in horizontal and vertical polarization respectively, z is the coordinate along the length of the fiber, and $\Delta \beta _0^{(m )} = \beta {({{\omega_0}} )^{(m )}} - \beta {({{\omega_0}} )^{(1 )}}$ is the wave-vector mismatch between the modes. As depicted in Fig. 2, one-pulse simulations simulate one pulse with horizontal polarization, two-pulse simulations simulate one pulse in horizontal and one pulse in vertical polarization, and four-pulse simulations have two pulses in each the horizontal and vertical polarizations.

To simulate fiber propagation, we modified the model from Horak and Poletti [20] and propagated each mode according to

(11a)$$\frac{{\partial A_H^{(m )}}}{{\partial z}} = \hat{D}A_\textrm{H}^{(m )} + \frac{{i{\omega _0}}}{c}\frac{{{e^{ - i\Delta \beta _0^{(m )}z}}}}{{{K_{mm}}}}\int\!\!\!\int {rdrd\theta F_m^ \ast } [{{{\hat{N}}_\textrm{H}}{A_\textrm{H}}} ],$$

and

(11b)$$ \frac{{\partial A_V^{(m )}}}{{\partial z}} = \hat{D}A_V^{(m )} + \frac{{i{\omega _0}}}{c}\frac{{{e^{ - i\Delta {\beta ^{(m )}}z}}}}{{{K_{mm}}}}\int\!\!\!\int {rdrd\theta _m^\ast } [{{{\hat{N}}_V}{A_V}} ],$$

where ${\left|{{K_{mm}} = \int\!\!\!\int {rdrd\theta |{{F_m}} |} } \right|^2}$ is the self-overlap integral of mode m. The operator $\hat{D}$ represents dispersion and linear losses for each mode and can be written in the frequency domain as

(12)$$\hat{D} = i[{{\beta^{(m )}}(\omega )- \beta_0^{(1 )} - \beta_1^{(1 )}\omega } ]- \frac{{{\alpha _0} + {\alpha ^{(m )}}}}{2},$$

where $\beta _1^{(1 )}(\omega )= { {{{\partial \beta {{(\omega )}^{(1 )}}} / {\partial \omega }}} |_{{\omega _0}}}$ is the group delay of the fundamental mode and shifts the simulation to the reference frame of the fundamental mode, α₀ represents gas absorption, and the dispersion properties are evaluated with index data from Börzsönyi [21]. The operator $\hat{N}$ represents the Kerr and plasma effects and can be written as

(13)$${\hat{N}_{\textrm{pol}}} = \left( {1 + \frac{i}{{{\omega_0}}}\frac{\partial }{{\partial t}}} \right)\frac{{3{\chi ^{(3 )}}}}{{8{n_0}}}{|{{A_{\textrm{pol}}}} |^2} - \left( {1 - \frac{i}{{{\omega_0}}}\frac{\partial }{{\partial t}}} \right)\frac{{\omega _\textrm{p}^2(A )}}{{2{n_0}\omega _0^2}} - \frac{{{\mu _0}c{I_\textrm{p}}}}{{{n_0}}}\frac{{W(A )[{{\rho_0} - \rho (A )} ]}}{{{{|A |}^2}}},$$

where n₀ is the gas index at the central frequency, χ⁽³⁾ is the nonlinear polarization of the gas, μ₀ is the free-space permeability, I_p is the ionization energy of the gas, ρ₀ is the gas atomic density, and A is amplitude of full field $A = \sqrt {{{|{{A_V}} |}^2} + {{|{{A_\textrm{H}}} |}^2}} $ used to calculate the ionization rate W(A). The first term in $\hat{N}$ includes the Kerr effect with self-steepening, the second term calculates the change in refractive index of gas due to plasma formation, and the third term calculates energy lost to photoionization. XPM between the two polarizations is not included because XPM was shown to be negligible as long as the pulses were separated by more than 3× the pulse width [18]. The ionization rate is calculated according to Keldysh theory as written in [22]. The plasma density is calculated at each step according to

(14)$$\rho ({r,t} )= {\rho _0}\left[ {1 - {e^{ - \int_{ - \infty }^t {W(A )\textrm{d}\tau } }}} \right].$$

and then used to calculate plasma frequency according to

(15)$$\omega _\textrm{p}^2({r,t} )= \frac{{{e^2}\rho }}{{{m_\textrm{e}}{\varepsilon _0}}},$$

where e is the electron charge, m_e is the electron mass, and ${\varepsilon _0}$ is the free-space permittivity. The maximum fractional ionization in a single step of these simulations was 3% so higher-order plasma effects should be negligible.

Equation (11) was integrated with a split-step Fourier algorithm with embedded fourth-order Runge–Kutta as given in Balac [23] and the code was tested against theoretical and experimental results to ensure it was modeling the effects correctly [4,7,16,24].

3. Results

We simulated HCF spectral broadening with parameters for the laser system and HCF currently installed in our lab. The laser emits 10-mJ pulses with a 1-ps FWHM duration and 1030-nm central wavelength. The fiber is 1.8-m long and filled with xenon gas (xenon parameters I_p = 12.1 eV and κ₂ = 5 × 10⁻²³ m²/W bar [25]). Xenon was chosen because its low ionization threshold made plasma effects apparent even with our relatively low-intensity pulses. The xenon pressure was set to half of p_9% from Eq. (6) so that in the absence of ionization, coupling to higher modes was negligible. This pressure is 0.17, 0.34, and 0.68 bar for one, two, and four pulses, respectively. Then by varying the fiber core diameter, we varied the strength of the plasma effects. We found that the plasma effects became significant enough that noticeable intensity was found in the first ten modes so simulations were carried out with 20 modes.

3.1 Plasma effects on pulse energy

Simulations with one pulse show significant energy loss from plasma effects, as shown in Fig. 3. Due to intrinsic fiber losses, some energy loss is unavoidable, and the ideal pulse would only lose energy to the intrinsic loss of the fundamental mode. The ideal energy is plotted as a black dotted line for reference. The output energy tracks the ideal energy line well for large fiber diameters, but with the onset of ionization around 580 μm, there is a sharp loss of energy.

Fig. 3. Simulated output energy after HCF and divided pulse recombination stages for a range of fiber diameters. Onset of ionization is clearly visible around 580 μm, 400 μm, and 300 μm for one, two, and four pulses, respectively, and results in large energy losses. By dividing to two or four pulses, smaller diameter fibers can be used without energy loss.

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To demonstrate improvements from DPNLC, we simulated spectral broadening with two and four divided pulses. The 10-mJ pulse is split into two 5-mJ pulses for the two-pulse case, and split into four 2.5-mJ pulses for four-pulse case. As shown in Fig. 3, DPNLC shifts the onset of ionization to smaller fibers and improves the output energy. Using two pulses shifts the onset of ionization to a diameter of around 400 μm and using four pulses shifts the onset to around 300 μm. Therefore, with DPNLC, smaller fiber diameters can be used which leads to larger broadening factors following Eq. (4).

Energy is lost through three channels: intrinsic linear fiber losses given by Eq. (9), energy lost to ionize the gas atoms [third term in Eq. (13)], and recombination losses when the divided pulses cannot be recombined efficiently. The energy lost through each of these channels is plotted in Fig. 4 to compare their relative impacts. It is apparent that energy lost to ionize the gas is minimal. Energy is lost primarily to increases in linear losses and reduced recombination efficiency after the onset of ionization. The linear fiber losses increase after the onset of ionization because the generated plasma defocuses the pulse as commonly observed in free space [22]. The HCF fails to confine the defocused pulses because it lacks total internal reflection, so a significant amount of energy will be defocused out of the fiber. Energy is lost to reduced recombination efficiency because the trailing pulses pass through a gas–plasma mixture ionized by the first pulse and acquire phase artifacts from the index difference. We did not try to quantify or compensate this phase artifact because loss of recombination efficiency began around the same fiber diameter that linear losses degraded the output energy, so the fiber should never be operated there.

Fig. 4. Energy losses for (a) two and (b) four pulses. For diameters smaller than the onset of ionization, plasma effects cause both linear and recombination losses to spike sharply. The energy lost directly to ionization is relatively small throughout. For one pulse (not plotted), recombination losses are by definition zero and almost all the energy losses can be attributed to linear fiber losses.

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3.2 Plasma effects on pulse duration

Simulations with one pulse show that plasma effects can significantly broaden the pulse’s spectrum but make pulse compression unstable. In Fig. 5(a) we plot the on-axis, transform-limited pulse duration (measured as FWHM of intensity) after spectral broadening in the HCF. The plasma effects cause huge spectral broadening and quickly reduce the transform-limited pulse duration to about 15 fs. We assume that we can compensate the spectral phase up to second order with chirped mirrors and have plotted the compressed pulse duration on the same figure. The figure shows that the Kerr effect can be compressed well but plasma effects add much more instability to the pulse compression. The plasma effects create a large blue-shifted tail in the spectrum and add highly complicated phase ripples, as shown in Fig. 5(c). The phase ripples are not compensated well with a fit up to the second order in phase; therefore, for our applications, the plasma effects must be avoided. If the output bandwidth is the only figure of merit, it may be advantageous to drive the fiber past the onset of ionization and take advantage of the large spectral broadening that plasma effects provide.

Fig. 5. (a) Compressed pulse duration for one pulse broadened by HCF. For large fiber diameters, the pulse can be compressed near its transform limit and the pulse duration decreases as fiber diameter decreases. After the onset of ionization, the spectrum quickly broadens to a transform limit of 15 fs but the spectral phase compensation is much worse. (b) Spectral phase for the output of a 600-μm fiber shows that the Kerr effect can readily be compensated by second-order spectral phase where the spectral intensity is high. (c) Spectral phase for the output of a 400-μm fiber shows that plasma effects create large blue-shifted components of the spectrum that are not fit well by the second-order spectral phase.

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DPNLC improves the pulse compression by eliminating the plasma effects. We computed the same transform-limited and compressed pulse durations with two and four divided pulses after the fiber and recombination stages and display the results in Fig. 6. DPNLC makes the Kerr effect dominate down to smaller fiber diameters, and subsequent recompression allows us to reach shorter pulse durations without encountering the highly unstable spectral phases from plasma effects. With one pulse, we could only compress down to about 150 fs where the Kerr effect dominates. Figure 6 shows that can be improved to 110 fs and 70 fs with two and four pulses, respectively.

Fig. 6. On-axis transform-limited and compressed pulse durations after spectral broadening in the fiber for (a) two pulses and (b) four pulses. Note that compression for the four-pulse case is so poor after the onset of ionization that the compressed durations for fiber diameters 300 μm and 310 μm are >250 fs and do not appear on the plot.

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3.3 Peak power enhancement via DPNLC

The final figure of merit that shows the advantages of DPNLC is the peak power of the compressed pulse. Figure 7 plots the peak power of the pulse after compression with up to the second-order spectral phase. Because smaller-diameter fibers provide more spectral broadening, the general trend is that the smaller fibers give a higher peak power. However, after the onset of ionization, significant energy is lost, and the peak power falls. DPNLC with four pulses improves the optimum peak power achievable to about 90 GW, while only 60 GW were achievable with one pulse.

Fig. 7. Peak power of compressed pulse after fiber. Energy loss from plasma effects is so significant that peak power is maximized by using divided pulses and removing plasma effects even though the plasma creates a large increase in bandwidth.

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4. Conclusion

Our simulations show how DPNLC can be used to overcome plasma effects in an HCF. Plasma effects lead to significant energy loss, mostly from pulse defocusing, but DPNLC keeps the pulse intensity below ionization thresholds. At the same time, DPNLC allows use of smaller fibers so larger broadening factors can be obtained and higher peak powers can be achieved. Importantly, our simulations show that the divided pulses can be efficiently recombined for all fiber diameters greater than the onset of ionization.

We also note that these results are not limited to HCF. Spectral broadening techniques in gas-filled multipass cells are limited by similar self-focusing and ionization thresholds [26]. Implementing DPNLC as outlined in the background of this section would be useful to overcome those ionization thresholds as well.

Funding

National Nuclear Security Administration (DE-NA0003856).

Acknowledgements

We thank John Palastro for his patient discussions about the plasma modeling, and the Center for Integrated Research Computing (CIRC) at the University of Rochester for providing computational resources and technical support.

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

Disclosures

The authors declare no conflicts of interest.

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Overcoming gas ionization limitations with divided-pulse nonlinear compression

Abstract

1. Introduction

2. Model

3. Results

3.1 Plasma effects on pulse energy

3.2 Plasma effects on pulse duration

3.3 Peak power enhancement via DPNLC

4. Conclusion

Funding

Acknowledgements

Disclosures

References

Cited By

Figures (7)

Equations (16)

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