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The role of the Higher-Order Kerr Effects (HOKE) in intensity clamping is experimentally investigated. We fail to observe any evidence of HOKE-based intensity clamping in a tight geometrical focusing experiment. We introduce a polarization-based technique that can distinguish between spectral components from the leading and trailing edges of the pulse. The results of this time-resolved measurement support the ionization theory of intensity clamping.
© 2013 Optical Society of America
It has been known for a long time that it is difficult to create high laser intensities in dense gases [1–8]. While the intensities that can be created in a vacuum depend only on the aberrations and f/# of the focusing optic and the optical quality of the incoming beam, and can exceed 1022 W/cm2 [9], the maximum obtainable intensity in an atmospheric-density gas is “clamped” by a variety of nonlinear effects [10–13]. Positive group-velocity dispersion can cause a pulse to split in two when focused, and may be the primary intensity-clamping mechanism in solid dielectrics [14–16]. Nonlinear absorption may also play a large role in solids and liquids [17, 18]. However, in gases, the dominant clamping mechanism has long been believed to be plasma defocusing [10]. Numerical simulations have repeatedly shown that plasma defocusing limits laser intensities to the 1014 W/cm2 level, even when the same beam focused in vacuum would produce an intensity orders of magnitude higher [1, 3, 4, 19].
Dense gases can add to or even replace geometrical focusing by the nonlinear mechanism of “self-focusing”, provided the pulse exceeds a critical instantaneous power [20], Pcr. Self-focusing is due to the lowest order Kerr effect, which produces an intensity-dependent refractive index of n(I) = n0 + n2I. Because n2 is positive for most materials in the near infrared, the refractive index will be higher in the higher intensity regions of a spatial mode, which will tend to focus the beam. If the pulse carries sufficient power, the focusing term will exceed diffraction, and the beam will collapse. Values for Pcr in air have been cited as 1.7 [21] to 10 GW [22], with some dependence on the pulse spatial mode [10]. A pulse subjected to the lowest-order Kerr effect (n2I), and no additional nonlinearities, satisfies a cubic nonlinear Schrödinger equation. This equation predicts focusing to an infinite intensity in a finite propagation distance [23, 24]. Experimental measurements have pegged the maximum intensity in the self-focusing geometry to the mid-1013 W/cm2 level [11, 12], indicating additional nonlinearities must arrest the beam collapse. The nonlinearity primarily responsible for intensity clamping in the self-focused case has again traditionally been taken to be plasma defocusing [25]. A dynamic balance between self-focusing and plasma defocusing has been used to explain the formation of filaments, long strings of plasma formed when pulses with a power exceeding Pcr are either weakly focused (large f/# lens) or fully self-focused (no lens) into air [10, 26].
However, several recent works have challenged the role of plasma defocusing in arresting the beam collapse [25, 27–31]. Loriot et al. proposed that higher order Kerr terms (n4I2, n6I3, etc.), which have the opposite sign from the lowest order Kerr term (n2I), defocus the pulse and clamp the intensity before it rises to the point where ionization becomes significant [25, 30, 32]. In this model, the Kerr effect is responsible for both the focusing and the intensity clamping observed in filaments. While the Higher-Order Kerr Effect (HOKE) model was developed to explain the intensity clamping in filaments; if correct, it should also be relevant for intensity clamping in geometrical focusing [32]. The key question is whether the HOKE terms begin to defocus the pulse at an intensity above or below that for which ionization becomes significant, and this question is independent of how the high intensity is generated.
To make this last point clear, we have plotted the predicted change in refractive index from ambient (Δn) as a function of intensity for a few different models. As argued in Rankin et al. [5], the point where Δn changes from positive to negative is a good approximation to the clamping intensity, regardless of the focusing mechanism. This is because multiphoton ionization increases very rapidly with intensity following its onset, such that a minor increase in intensity significantly strengthens the defocusing lens. The two ionization models in Fig. 1 assume a 70 fs (FWHM in intensity) Gaussian pulse, and the labeled intensity is the peak intensity. However, the clamping intensity was found to be very insensitive to the pulse shape; a triangular pulse with a rising linear ramp and vertical decay gave the same threshold. This pulse shape was modeled because it approximates the envelope developed in optical shock formation [20, 33]. The only way to increase the clamping intensity was to use a pulse with a very steep rising edge. Such a pulse cannot survive long-distance propagation in a Kerr medium, because the high-intensity portions propagate more slowly than the low intensity portions and reduce the slope of the leading edge [20].
Fig. 1 Plots of refractive index of air as a function of intensity, according to the models of Mlejnek [34], Skupin [35], and Loriot [27]. Δn = 0 is the intensity where the pulse stops focusing and starts defocusing. For the Mlejnek and Skupin models, a Gaussian pulse of width 70 fs was used, and the intensities are for the pulse peak.
The nonlinear mechanism that clamps the intensity will also modify the pulse spectrum. If plasma is responsible, the leading half of the pulse will see a time-decreasing refractive index and hence be blueshifted [36–38]. Ions do not recombine on a femtosecond timescale, however, so the pulse trailing half will be defocused by the plasma and will not be intense enough to add to its density. The trailing half should therefore retain the original laser spectrum, as it does not see a time-varying index. Conversely, the Kerr effect for ultrashort pulses is based primarily on an instantaneous or nearly instantaneous nonlinearity [39]. The lowest order Kerr term will result in the pulse leading half being stretched and redshifted, while the trailing half will be compressed and blueshifted by the same amount, for a temporally symmetric pulse. Any defocusing Kerr terms will have the opposite effect: the leading half will be blueshifted, and the trailing half redshifted. The key difference between HOKE intensity clamping and plasma intensity clamping is the persistence of the nonlinearity: HOKE responds instantly (or nearly so) to the intensity, while ionization builds up continuously on the pulse leading edge but then remains nearly constant as the pulse wanes.
To test the role of HOKE in intensity clamping, we bring 1 mJ, 70 fs, 800 nm pulses from a chirped-pulse amplifier to a tight focus in air with a 60 mm focal length achromatic lens. The spectrum of the output pulse is examined with a Czerny-Turner spectrometer, and compared with that of the input pulse. Care is taken to avoid introducing any additional nonlinearities or color cast in the detection system. A few percent of the output pulse is reflected off a neutral density filter at near-normal incidence, and the reflection is scattered off a white card. The spectrometer acceptance angle exceeds the scattering spot size on the card, so any spatial inhomogeneity in the beam is integrated. The card is vibrated to reduce coherent artifacts. We also scan the chirp of the pulse by lengthening the compressor stage [40], to ensure that our results are not chirp-dependent (second order dispersion can lead to an asymmetric pulse if third-order dispersion is also present [41]). The results are presented in Fig. 2; the input pulse spectrum is presented below the white line. The figure shows that both the input pulse and the focused pulse contains negligible power for wavelengths longer than 820 nm (the color scale is logarithmic), for pulses of any chirp. The intensity clamping is not associated with any significant amount of redshift, as would occur if HOKE played a major role. There is a very significant blueshift (wavelengths shorter than 770 nm), however, characteristic of ionization. The “striations” in the blueshifted portion of the spectrum occur because the blueshift occurs at varying rates as the pulse moves through the focus; if the amount the pulse was blueshifted was linear in time, they would disappear [6].
Fig. 2 Spectrum of the pulse after passing through a tight geometrical focus in air. The input spectrum is shown below the white line. The second-order dispersion (chirp, β2) is adjusted by changing the compressor length.
Figure 2 shows that the pulse experiences only blueshift at 1 mJ regardless of chirp. Figure 3(a) shows the spectral centroid as a function of pulse energy. At very low pulse energies, it retains its input value (798 nm), but it shifts monotonically toward the blue for increasing pulse energy. Pulse blueshifting occurs in this geometry independent of the chirp and energy values; we have not observed any redshifting. The increasing magnitude of the blueshift may be connected the elongation of the ionized region (Fig. 3(b)). We have tracked the self-luminous plasma ball with a camera as the pulse energy is increased. It is found that the left (lens side) edge moves toward the focusing lens, while the right (far side) remains nearly fixed, leading to an asymmetric extension of the plasma in the direction of the incident beam [42].
Fig. 3 The left panel shows the spectral centroid as a function of pulse energy, while the right panel shows the position of the left edge (lens side) of the luminous plasma ball.
Figure 2 shows that the pulse blueshifting is much stronger than any redshift, ruling out instantaneous nonlinearities (such as any order of Kerr effect) as the cause. However, optical shock is also known to asymmetrically broaden the pulse spectrum, as creating an envelope that changes quickly within a single optical cycle requires wavelengths shorter than the optical cycle, but not longer [33]. For media such as air with a focusing lowest order Kerr index, optical shock will occur on the pulse trailing half, as the waning tail overruns the peak. Conversely, plasma defocusing will occur primarily on the pulse leading half, as the free electrons created by the pulse leading edge will prevent the trailing half from focusing to a high intensity. We therefore need a method for distinguishing the rising and falling halves of a pulse. We accomplish this with a longer, but polarization-varying, “twisted pulse”, as shown in Fig. 4. By using thick quartz waveplates (0.4495 mm, or five waves at 800 nm), the different group delays along the fast and slow axes cause the pulse to lengthen and develop a time-varying polarization state. This is one form of “polarization gating” [43]; alternately, it can be thought of as a pump-probe measurement with a partly overlapping pump and probe. The leading and trailing halves of the twisted pulse (or equivalently, the pump and the probe) can then be separated with an ordinary polarizer and sent to a spectrometer, to see if blueshifting occurs on the leading or trailing edge.
Fig. 4 Setup for the twisted pulse experiment. Light entering the experiment is linearly polarized at 45° (a). A series of 1-5 quartz waveplates, each with a phase thickness of five waves, is inserted into the beam with the fast axes vertical. Each plate introduces a group delay between the s and p components of Δt = 14 fs. After the waveplates, the pulse is “twisted” with an s-polarized leading edge and a p-polarized trailing edge (b). The pulse is focused by a lens to produce the plasma, then recollimated by an identical lens. The pulse now has a strongly blueshifted leading edge due to plasma blueshifting, but the trailing edge shows little spectral change (c). A small portion of the pulse energy is reflected off the front surface of a near-normal neutral density filter; the very small angle of incidence insures the polarization is not affected (d). A polarizer then selects either the s (leading) or p (trailing) component; here we are selecting s (e). The remaining radiation scatters off a vibrating screen, and is picked up by a spectrometer.
The results of the experiment are presented in terms of the branching ratio, R(λ) = Is(λ)/(Ip(λ) + Is(λ)), where Is(λ) is the spectral intensity of s (leading edge) component, while Ip(λ) is the spectral intensity of the p (trailing edge) component. A branching ratio well above 0.5 indicates that those wavelengths occur in the leading component, while a ratio less than 0.5 indicates association with the trailing component. The results, for varying numbers of waveplates, are presented in Fig. 5. The steady increase in branching ratio for the “far blueshifted” components as more waveplates are added is a clear sign that these wavelengths are generated on the leading edge. Conversely, the wavelengths that were originally present in the beam (780 – 820 nm) are reduced on the leading edge, as the spectral power in this region has been blueshifted to shorter wavelengths. Wavelengths in the region from 780 to 790 nm were the most strongly depleted. The reason for this is not known; it is possible the pulse used in this set of experiments had a small negative chirp, such that the bluer portions of the laser spectrum arrived first.
Fig. 5 Spectral intensity fraction R(λ) = Is(λ)/(Ip(λ) + Is(λ)) as a function of number of waveplates in the input beam: more waveplates correspond to a longer delay between the s and p portions of the pulse.
A few technical issues could complicate the interpretation of these results. The twisted pulse is longer than the pulse coming into the waveplates, by up to 70 fs. However, Fig. 2 shows that the only effect of increasing pulse length (setting a non-zero β2) is to decrease the magnitude of the blueshift; the overall spectral shape remains similar. While the group delay (Δt = 14 fs per plate) can be stated with some confidence, uncertainty as to the precise carrier wavelength leads to some uncertainty as to the ellipticity of the center section of the pulse, especially with higher numbers of inserted waveplates. This does not affect the main result of the experiment, however. When the polarizer is set for s, it has the highest sensitivity to the leading edge of the pulse – the part which has no overlap with any portion of the p “probe”. The temporal region where the s and p portions do overlap will appear in the spectrum in both polarizer positions, regardless of whether this region is best described as a “linear pulse at 45 degrees” or an “elliptical pulse”. These overlap portions will largely disappear when the ratio R(λ) = Is(λ)/(Ip(λ) + Is(λ)) is taken, irrespective of the ellipticity. Finally, molecules can exhibit dynamic alignment with the laser field, potentially leading to birefringence so that a twisted pulse exhibits different dynamics than independent s and p pulses. To test this, the power passing through the polarizer was verified to be the same for s and p in all cases, to an accuracy of a few percent. This indicates that any pulse-induced birefringence (which would shift power from one component to the other) is very weak and cannot materially affect our result.
A key assumption in our interpretation is that ionization occurs predominantly in the first half of the laser pulse. Additional important assumptions are that self-phase modulation and optical shock play little role in the tight focusing geometry. To validate these assumptions, we have conducted a numerical simulation of the Nonlinear Schrodinger Equation (NLSE) governing pulse propagation in the presence of a Kerr nonlinearity and ionization [10, 21, 44]. For the ionization model, we have used the Perelemov, Popov, and Terent’ev (PPT) model [45], which shows good agreement with the experimental data for molecular oxygen and nitrogen over a broad range of intensities (1013 – 1015 W/cm2) [46]. For the lowest order Kerr effect, we used an n2 of 3.2 x 10−19 cm2/W [10]. The 70 fs (FWHM in intensity) pulse was divided into 81 time slices, and each one was propagated through the geometrical focus individually, interacting with the ions created by all earlier time slices. To keep the simulation tractable, the possibility of mixing between the time slices was ignored. This method of simulation allows the pulse to be temporally reshaped through focusing / defocusing of individual time slices [47], but excludes the possibility of GVD-induced mixing between neighboring time slices (“time focusing”). This omission is well justified in the tight-focusing case, because the length of the interaction region (60 mm from lens to focus) is much shorter than the scale length for GVD in neutral air (100 meters for a 70 fs pulse, k2 = 0.2 fs2/mm [21]) or even fully-ionized air plasma (200 mm for a 70 fs pulse, k2 = −22 fs2/mm). The scale length for self-steepening at 1014 W/cm2 in air is ~0.5 meters [20], while the length of the region at or above this intensity is only a few hundred microns, allowing this term to be neglected as well.
The results of this simulation for a 1 mJ pulse focused with a perfect f = 60 mm lens to a focal radius (e−2 in intensity) of 3.3 μm are shown in Fig. 6 and Fig. 7. Figure 6 shows that, as claimed, nearly all of the oxygen and nitrogen ions are created in the first half of the pulse. This result was found to be quite insensitive to both pulse length and energy, indicating that the somewhat longer pulses used in the twisted pulse measurement should not change this core result. By following the slice-by-slice intensity evolution, we have determined the origin of the effect to be the defocusing of later time-slices by the ions created at the beginning of the pulse. From the pulse peak onward, the only new ions created are in a small crescent around the edge of the existing plasma ball, and only 4% of the total ions are created by the second half of the pulse. Figure 7 directly verifies the intensity clamping; the maximum intensity achievable with the full PPT ionization model saturates at 4.8 x 1014 W/cm2, in contrast to the 8.7 x 1016 W/cm2 that this same pulse would deliver in a vacuum. This value is very similar to the simulation results of [4] in argon, and the experimental results of [2] in various gases. We have also modeled the intensity clamping with self-focusing turned off (n2 = 0), and obtained nearly the same result (Fig. 7). The key to understanding the insignificance of self-focusing in the tight geometrically-focused case is to realize that only time slices with a power above Pcr (3.3 GW for n2 = 3.2 x 10−19 cm2/W) can self-focus [48]. Due to the strong geometrical focusing, very early time slices with instantaneous powers well below Pcr can create ions in the focus. Subsequent time slices extend the ionization region asymmetrically back toward the focusing lens [49] (Fig. 3(b)). When the time slices with power above Pcr finally arrive, they are refracted by the plasma ball before any significant self-focusing can occur. The comparative insignificance of n2 in our tight-focusing geometry distinguishes our experiments from the plethora of results obtained with much looser focusing. In loose or non-focused experiments, early time slices lack sufficient intensity to produce ions, so that self focusing can significantly increase the intensity over the vacuum case. Additionally, the very long interaction lengths in these experiments allow n2-induced self-phase modulation to redshift the pulse leading edge, producing the Stokes-shifted radiation observed in loose-focusing [50] but not tight-focusing experiments [37].
Fig. 7 Maximum intensities as a function of axial position for four different pulse propagation models. “Vacuum” sets the Kerr index and ionization to zero. “PPT and Kerr” uses the full PPT model as well as Kerr self-focusing; “PPT w/o Kerr” sets n2 = 0. “MPI” uses lowest order perturbation theory as well as self-focusing.
A simulation similar to ours was recently used to challenge the existence of intensity clamping in a geometrically-focused experiment [51]. However, the simulation of Kiran et al. [51] used a “pure Multiphoton Ionization (MPI)” model, which is the low-intensity approximation to the PPT model. Additionally, they considered only oxygen molecules, with an 8-photon ionization cross-section of 3 x 10−96 s−1 cm16/W8. The results of their simulation indicated that the air becomes fully ionized at intensities in the mid-1013 W/cm2, allowing the remainder of the pulse to continue focusing without interference from plasma defocusing, and producing intensities exceeding 1015 W/cm2 [51]. However, the MPI approximation is only valid at low intensities, and overestimates the ionization rate by many orders of magnitude at high intensities [10]. We have repeated their simulation with an 8-photon oxygen cross section of 2.81 x 10−96 s−1 cm16/W8 and an 11-photon nitrogen cross section of 6.31 x 10−140 s−1 cm20/W11 [10]. With a pure MPI model, we found, as they did, a saturation of plasma defocusing; but comparing this result with the full PPT model (Fig. 7) shows that this is a consequence of drastically overestimating the ionization rate at higher intensities. Kiran et al. also presented experimental evidence for the saturation of plasma defocusing in the form of a blueshifted “edge” which monotonically increases with pulse energy; they argued that if the intensity really saturated, the blueshift ought to saturate as well. We have also observed a monotonically increasing blueshift with pulse energy (Fig. 3(a)), but interpret it differently. The magnitude of the blueshift depends on both the ionization rate (dn/dt) and the length of space over which dn/dt < 0 [37]. As seen in Fig. 3(b), the plasma length also continuously elongates as the pulse energy increases, providing a mechanism which can continue to increase the overall pulse blueshift even after the saturation intensity has been reached.
In summary, we have demonstrated with simple experiments that HOKE plays little role in intensity clamping, at least for tight-focusing geometries. Because the active defocusing mechanism depends primarily on the intensity, rather than the method for producing that intensity, we expect our results to also be relevant to the cases of weak geometrical focusing and pure self-focusing. Furthermore, we have demonstrated a simple method for distinguishing dynamics on the leading and trailing halves of the pulse. We expect this method to be even more useful in weak focusing geometries when additional nonlinearities can play more role; for example, it may be able to track the formation of an optical shock through the appearance of blueshifting on the pulse trailing edge.
Acknowledgments
We would like to acknowledge the supports from US Air Force Office of Scientific Research.
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