Stimulated rotational Raman scattering (SRRS) in the ultraviolet region () has been observed at the Nike laser over extended propagation paths in air during high power operation. Although this phenomenon is not significant for standard operating configurations at Nike, broadening of the laser spectrum and far-field focal profiles has been observed once the intensity-path length product exceeds a threshold of approximately . This paper presents experimental results and a new theoretical evaluation of these effects. The observations suggest that significantly broader spectra can be achieved with modest degradation of the final focal distribution. These results point to a possible path for enhanced laser-target coupling with the reduction of laser-plasma instabilities due to broad laser bandwidth produced by the SRRS.
For the realization of laser direct drive inertial confinement fusion (ICF), intense high-quality laser beams are generated, transported, and focused for the precise implosion of a spherical fuel pellet [1,2]. Stimulated rotational Raman scattering (SRRS) can occur when a laser beam excites rotational quantum states of air molecules along the propagation path [3,4]. The resulting emission of light into Stokes and anti-Stokes wings around the incident light degrades the beam quality and has been typically viewed as problematic. With sufficient growth, this process can spoil an implosion by degrading the scale and uniformity of the final focal profile, and reducing the total drive energy that impacts the target. SRRS can grow from ambient fluctuations or can be seeded by laser light either from within the spectrum of a broadband primary beam or from a copropagating secondary beam. The growth rate of SRRS depends on the composition of the transport media, the path length, and the intensity of the primary laser beam and secondary seed beam, if the latter is present. SRRS is widely regarded as a deleterious effect for ICF laser systems, and it is suppressed on very large lasers such as the National Ignition Facility by using inert gas (such as argon) filled tubes to transport the beams to the target chamber . Here we present SRRS obtained with the Nike krypton fluoride (KrF) laser which utilizes angularly multiplexed laser beams with long paths in air. While SRRS was found not to be a constraint on Nike normal operation due to the low intensities in the beams, significant SRRS could be generated by lengthening the optical paths in air. The results and simulations indicate that properly seeded SRRS may be a viable path to obtain sufficient laser bandwidth on target to suppress laser plasma instability while maintaining the required target illumination quality.
SRRS has been long studied within the high power laser community devoted to ICF development. Henesian et al.  conducted one of the earliest studies at the Nova laser with a 1.053 μm test beam and found approximately 1% conversion of the light into SRRS with many Stokes orders observed and with the general conclusion that beam quality greatly suffered once the process occurred, especially for shorter pulses ( or less). An observed intensity-path length threshold (IL) in these initial studies was . Similar studies were undertaken prior to the upgrade of the Omega laser at the Laboratory for Laser Energetics of the University of Rochester . This research confirmed that the conversion efficiency into SRRS modes for a solid-state laser frequency-tripled to 351 nm was 1% or lower for standard operation when the gain length product was kept below . The more restrictive criterion ensures that SRRS has not been considered a problem for this facility. In addition, smoothing by spectral dispersion (SSD) was shown to reduce SRRS , which grew primarily within the beam’s hot spots due to phase aberrations back in the laser. Recent numerical simulations indicate that SSD accomplishes this by partially smoothing those nonuniformities [9–11].
At roughly the same time as the Omega studies, SRRS was also investigated at the Aurora laser at the Los Alamos National Laboratory (LANL). Unlike the above solid-state laser systems, Aurora was an excimer laser with an active medium based on a krypton–fluorine (KrF) gas mixture pumped by an electron beam. This laser operated in the ultraviolet (248 nm), so the SRRS was expected to be more vigorous than the previous two cases. Changes in the output spectra and in the focusability were observed, but these results were only presented in summary form at a few conferences [12,13] and in short laboratory memoranda [12,14] near the end of the facility’s operation.
This paper reports new experimental and theoretical results on the Nike KrF laser that are oriented toward exploring SRRS as a means of improving laser-target coupling, rather than simply minimizing its occurrence. While based on technology similar to Aurora, Nike incorporates a beam smoothing technique: echelon-free induced spatial incoherence (ISI). Thus, while the spectral and temporal properties are similar to the Aurora work, Nike’s focal spot uniformity is higher. In addition, the initial motivation for investigating SRRS arose from target experiments to characterize laser plasma instabilities driven by a KrF laser . Laser plasma instabilities (LPIs) occur when the laser light focused onto target couples to waves in the plasma corona up to roughly the critical electron density, where the electron plasma frequency is approximately equal to the laser frequency. Mitigation of these instabilities are of vital importance to laser-driven ICF. For the LPI studies at Nike, the effect of SRRS on the input laser, particularly the total energy and the focal profile, were of interest for the evaluation of the experiments.
Observations of time-integrated and time-resolved spectra and time-integrated focal profiles are detailed in this paper. The SRRS experiments at Nike motivated a substantial effort to develop a new simulation code for SRRS. The paper includes a description of the first stages of the theoretical work and its comparison with the experimental results. In particular, the observations and simulations for the work at Nike when combined with the programmatic importance of LPIs has led to the re-examination of applying SRRS to the generation of ultrabroadband laser output. This latter work will build on other work initiated at the Nova laser  but appears particularly promising when added to the other advantages of KrF as a driver for ICF [17–19].
The next section of the paper discusses the Nike KrF laser. Section 3 describes the series of measurements made at NRL. Section 4 presents the data analysis methods. Section 5 has the results of the analysis. Section 6 presents an overview of the ongoing theoretical work. Section 7 contains the comparison between simulation and measurement. Section 8 summarizes the paper and the present path forward from the current results.
2. NIKE LASER FACILITY
The Nike laser at the Naval Research Laboratory (NRL) is the highest energy KrF laser in operation. The design and operation of the laser was detailed in a recent review paper , but some key aspects of the system should be briefly mentioned (See Fig. 1). Owing to the short energy storage time () for this lasing medium and limits on the gain for a given amplifier volume, the system architecture uses time and angular multiplexing of a long train of beams through the final amplifiers to create output energies of up to 4 kJ (2.5 kJ in routine operation). After the final amplifier, the relative beam-beam time delays are removed, and the beams are transported to the target chamber to create a high-energy pulse (0.4–12 ns long). Forty-four beams are directed into the chamber from a single array of mirrors and final focus lenses while a smaller set of twelve beams uses separate optical arrays to reach the target chamber. The latter “backlighter” array was designed to generate x-rays to radiograph the evolution of planar targets irradiated with the larger beam array. The longest path differential among the beams occurs between the first and last beams through the final amplifier and is approximately 74 m. The space required for this demultiplexing was halved with a single fold of the optical path prior to the output mirror arrays. The SRRS experiments used the large propagation region after the final amplifier to facilitate observable growth.
Another key function for Nike’s optical system is to enable beam smoothing via echelon-free ISI [20–23]. In brief, the optical system relays an image of a pinhole uniformly illuminated by a multimode KrF oscillator from the low-energy front end output to the target chamber. This feature has several important impacts on the current study. First, as part of the multiplexing mentioned above, there is a variation of beam size along the system. Each of the last two amplifiers double-pass their inputs to maximize their output, and each incorporates a focusing mirror to meet the required beam size . At the final amplifier, the beam cross-section is 60 cm square while along the final demultiplexing path, the beam is collimated to a 15 cm square cross-section. The variation in beam size will impact the estimation of the path-length and intensity product discussed in Section 5 and the modeling discussed in Section 6. The beam collimation was of sufficient quality that the beams can be propagated for distances two or more times farther than the original design, thus facilitating the measurements described in Sections 3–5.
Nike’s architecture places primary control of the pulse shape and far-field profile in the front end of the laser system. The time-multiplexing optics have been designed with standard pulse lengths of 4 ns and 5 ns for the main and backlighter beams, respectively. When the pulse length for either is reduced, the final two amplifiers store energy for a corresponding period between the fixed arrival times of the beam train. Additional laser energy is released at the leading edge (less than ) of the subsequent pulse. For 0.4 ns pulses, a power enhancement of has been demonstrated by this mechanism. Similarly, the standard main beam focal spot full width at half-maximum (FWHM) of Nike is 750 μm. The ability to reduce the focal spot diameter by a factor for a given pulse has been used to examine the effect of SRRS on the far-field profile without large changes to the optical hardware.
3. SRRS MEASUREMENTS AT NIKE
Observations in this paper encompass three separate series using different laser and diagnostic configurations. The first indications of SRRS were noted as a distortion of the laser spectrum during the highest intensity shots in one of the longer-path beams. Because the primary goals for that experimental campaign were related to high intensity laser-target interactions, laser operations were dictated by the target requirements and were not varied to examine SRRS growth. Two subsequent studies dedicated to SRRS were made that altered the propagation distances and intensity of selected beams. The first of these concentrated on the output spectrum, while the second expanded to examine changes in the far-field profile.
For all three studies, diagnostics were used to monitor beam energy, focal profile, laser spectrum, and pulse shape. The final output energy was measured with full-aperture absorbing calorimeters (Scientech model UV5SPL). These detectors were placed as close to the target chamber as possible, typically at the vacuum window of the target chamber. The focal profile, output spectrum, and pulse shapes were collected at selected demulitplexing mirrors used to remove the time delay from the beams. A small amount of light passed through the mirror (4–8%) to an optical train that transported, attenuated, and focused light onto sensors. For the focal profile and output spectrum, a single main beam (beam 38) was chosen for a reference. An equivalent on-target focal profile was recorded on a cooled time-integrated, ultraviolet-sensitive CCD camera (Photonics Star I ). Within the profile diagnostic, a small portion of light was extracted and directed to the entrance slit of a 1 m Czerny–Turner spectrometer (Minute Man model 305SMP) to record a time-integrated spectrum with the same model of CCD camera. In contrast to the focal profile and spectrum, the time history of the laser power was monitored for up to six main beams. Each pulse-shape station used two to four high-speed phototubes (Hamamatsu model R1328) with beam splitters, laser bandpass filters, and varied optical attenuation to send signals over a range of sensitivities to fast oscilloscopes (1–6 GHz bandwidth). The multiple channels were used to reconstruct a beam’s pulse shape. For a four-channel station, a high dynamic range of up to can be achieved, although the uncertainties at a chosen intensity level is limited by the given channel signal-noise of approximately 1–2%. The time resolution is chiefly limited by the high-speed cables between the detectors and digitizers to 100–200 ps.
Diagnostics for the SRRS measurements evolved from the initial study where measurements were made near the target chamber. Figure 2 shows a schematic of the beam paths for the three stages. For all three studies, a large aperture beam pickoff was located in the target area to capture light from the test beam and direct it onto diagnostics. The first measurements simply directed the beam through attenuators and a lens onto an optical fiber. The fiber transported the light to a ¼ m Czerny–Turner spectrometer coupled to a Hamamatsu streak camera. A sample time-resolved spectrum from a slightly shortened backlighter pulse is shown in Fig. 3(a). The SRRS signal mainly occurs toward longer wavelengths at the leading edge of this pulse where the increased power output and the long path length of the beam were sufficient for SRRS growth. It is interesting to note that in the long pulse case, the SRRS signal disappears after the initial burst of higher laser power output decays. This behavior suggested that a systematic study of SRRS would be possible if the path length of a beam is increased by factors of . Because the chosen beam had one of the longest paths in the Nike architecture, these initial results also indicate that SRRS is only a concern for Nike within a subset of beams when the laser is operated with pulses that create high power output.
After the initial observation, dedicated shots were made where a selected beam was directed to travel the length of the propagation bay a second time before passing to the same diagnostic station in the target area for a total distance of 178 m. This change doubled the beam’s total propagation distance and was accomplished by altering the alignment of large optics in the propagation bay (see Fig. 2). A movable optical stand was added in the propagation bay to place a 95% reflector in the beam path at selected locations. Because sufficiently high laser power is needed for SRRS, it was assumed that SRRS growth for the transmitted beam was not important for the distance between this partial reflector and the diagnostic station in the target area. The higher energy reflected beam was directed onto one of the standard large-area calorimeters to determine the beam energy. Shots were made with three different positions of the stand. The maximum path length for SRRS was taken to be the case where the reflector was entirely removed from the system so the SRRS growth could occur until the final diagnostic pickoff in the target area. The beam intensity was sufficiently low that the two-photon absorption coefficient  in the window between the propagation bay and target area and in the final diagnostic optics was not a concern.
For this first set of dedicated SRRS shots, the target area diagnostic station was altered. The fiber coupling used for the streak camera during the initial observations was replaced with a “fiber-free” system to attenuate and focus the light onto the entrance slit of the streak spectrometer. To examine the potential influence of the fiber, a secondary instrument was added to the diagnostic train. Behind a 4% reflector used to attenuate the beam prior to the spectrometer input, additional pickoff optics were added to attenuate and focus light onto the same type of fiber used in the first measurements. The fiber coupled the light to 1 m spectrometer (Minute Man model 305SMP) with a standard UV sensitive CCD camera (Photonics Star I). The two target area instruments yielded similar spectral broadening toward lower frequency compared to the unperturbed result from the spectrometer closer to the final amplifier, as shown in Fig. 3(b).
The third set of SRRS measurements used a slightly different beam arrangement from the first two studies. The goal was to determine the increase in the relative energy fraction in the wings of the focal profile as a function of SRRS growth. Focal profile diagnostics were included in the propagation bay and in the target area. The test beam was selected to take advantage of a standard beam 38 diagnostic for the main beam focal profile in the propagation bay. The total path length to the target area was extended by placing a full aperture 95% reflector at an unused beam station in the final turning array. Beam 38 was redirected from its usual final turning mirror onto the new reflector, which placed it onto transport optics to reach the diagnostic station in the target area. This longer path nearly doubled the standard path from 60–102 m (both distances are relative to the origin specified between the output of the 60 cm amplifier and the recollimation array). Additional optics were added to the target area diagnostics to focus a sample of the beam onto a CCD camera (Photonics Star I). The image from this new camera was then compared to the image from the (standard) reference profile diagnostic for shots with low and high intensities. Because the total beam path to the target area for this temporary laser configuration was similar to the long path length of beams where low levels of SRRS were observed for high intensity operation, this setup allowed a study relevant to the original target experiments. A limitation of this approach was the fixed size of the CCD sensor. If low levels of scattered light covered the entire area of the sensor, the signal could either be too low to measure or indistinguishable from noise. The continued spectroscopic observations were crucial to evaluate the level of SRRS growth on each shot.
Two ancillary measurements were taken to support the above studies. First, a critical parameter is the initial total energy in the test beam. As the configuration for the laser changed in the various studies and after a correction for a partial reflector, there was a need to reference the energy measured at a given location to the energy emitted at the final amplifier. Correction of the calorimeter value at any position required measurement of energy as a function of propagation distance. During the second SRRS series, such data were collected with suitable (nonmoving) reference detectors for shot–shot normalization. The average result from several trials yielded an exponential attenuation factor of . This value cannot be simply related to a single physical process, e.g., Rayleigh scattering in air, because many processes can contribute to the measured energy losses, such as imperfect beam collimation, diffraction losses, and low-level scatter at each optic in the system.
Finally, for the first and second studies, the test beam had a much longer path than the reference beam used to monitor the laser spectrum and far-field profile closer to the 60 cm amplifier output. There was a question on those shots to the degree of SRRS growth at intermediate distances—distances longer than the optical path from the final amplifier to the standard beam diagnostic station and significantly shorter than the distance to the target area diagnostic station. To check that the test beams had spectra similar to the reference beam, the time-integrated spectrometer was moved from the target area to the propagation bay with its fiber-optic collection system. Data along the beam path showed that the initial spectrum for the longer paths showed little or no SRRS growth at distances slightly beyond that of the reference station. From these observations, it is believed that the basic phenomena to be examined in Sections 4–6 are representative of the general behavior of the Nike beams and not indicative of peculiarities of a particular beam path chosen for a given measurement.
4. DATA ANALYSIS
A sample Nike output spectrum from the reference diagnostic in the propagation bay is shown in Fig. 4(a). The spectrum has a FWHM slightly below 0.2 nm () and shows a small asymmetry with somewhat higher emission at longer wavelengths. Sample spectra from the target area spectrometers are shown in Fig. 4(b) for a weak shot with small SRRS and two stronger shots. These two spectrometers have lower resolution than the reference spectrometer, but both are sufficient to monitor SRRS. Because the standard Nike output spectrum is not a simple symmetric function such as a Gaussian or a Lorentzian, the higher resolution spectra were fit with a skew-Gaussian distribution,4(a) for a spectrum taken from the standard reference spectrometer closer to the final amplifier. The skew Gaussian gives a much better fit to the wings of the distribution where the effects of SRRS are expected to appear. The more complicated distribution greatly decreases the residual for the longer wavelength side, both in absolute magnitude and shot–shot variation. The absolute magnitude of the error on the short wavelength side of the spectrum is not greatly improved, but the shot–shot variation is. With the skew-Gaussian model, the shot–shot variations in the residuals from both wings are similar.
A more accurate fit was important for two stages in the experiments. First, spectra close to the final amplifier never showed significant distortions due to SRRS growth. Although the path to this instrument was not varied in the shots, the output energies varied with the pulse shapes. The fit residuals associated with the skew-Gaussian model, however, stayed close to those characteristics of the lower energy shots where SRRS would not be expected. The lack of growth at the standard diagnostic position provides confidence that the observed spectral changes are associated with a phenomenon dependent on path length. A second important application was in the reduction of the first data where SRRS occurred during the beginning of a long laser pulse. The period for SRRS growth was combined with the longer period without SRRS in the time-integrated data. Although time-resolved spectra were not available for many of the early shots, the pulse shape diagnostics were available and were interpreted as a measure of the time history of the spectrally integrated beam energy. The early time SRRS-broadened spectra were estimated by normalizing each time-integrated spectrum to its total area, and then subtracting the above fixed model function as scaled by the estimated energy fraction during the lower intensity portion of the pulse. This method for correction appears to work well for the cases of SRRS near threshold and with increasing overcorrection for the cases with strong SRRS due to uncertainty in the energy fraction to be assigned to the lower intensity (below threshold) period of the pulse. The data obtained from these analyses support the onset of SRRS discussed in Section 5.
The energy measurements for the test beam had varied locations along the optical path over the three investigations. The recorded values consequently need to be referenced to a single point to facilitate comparisons. As noted in Section 2, the 60 cm square beam exiting the final amplifier converges onto a collimation optic to perform the time demultiplexing required to reach high energy on target. The impact of this geometry on SRRS will be discussed in Section 6, but, for the presentation of the measurements, the varying intensity has been accommodated by defining the starting point for SRRS growth at a position 13.13 m prior to the recollimation array. This distance reproduces the product of the path length and average beam intensity over the convergence region for a constant profile. The average beam energy along the path for each shot can be calculated from
The data analysis for the focal profile study required a slightly more involved process than the spectra. An examination of the sample data demonstrated that the spot size varied slightly from shot to shot, especially in the low-intensity portions of the distribution. The outer regions of the focal spot were affected by the optimization of the optical chain (system aberrations) and by turbulence along the long air paths. Although wavefront optimization by lens tilting  minimized the former problem prior to the SRRS shots, slightly noncircular spots and low-level scattered light remained a statistical problem that required sufficient data to allow averaging to yield a representative distribution.
An entirely statistical approach was taken to find an azimuthally averaged focal intensity as a function of radius for each shot. The first step was to find the baseline noise and peak signal within an image by looking at a histogram of the counts in each pixel. The number of pixels with a given count level peaked at very low values with a long tail that eventually ended at the maximum count level for a given recording. The distribution for the low count levels had a near Gaussian shape and was believed to be dominated by the noise in the CCD system. A Gaussian fit to the low-level counts provided an estimate of the baseline counts. The count value corresponding to the location of the peak of the Gaussian fit was subtracted from each pixel value, and then all pixels with counts below (where is the standard deviation from the fit) were set to zero to avoid contaminating the subsequent integration with counts unrelated to the laser light.
The next stage was to average the processed spot image around the peak value to estimate the intensity as a function of radius. Owing to the noise near the peak, the center of the profile was defined by finding the boundary for pixels with counts within 7.5% of the maximum value. A fit of a nearly circular ellipse to this contour provided values for the center point, relative scale of the axes, and the rotation of the two axes [Fig. 5(a)]. Count values were then summed for all pixels within the boundary for each contour. The process was stopped when the radius reached a value where the integration result no longer increased due to the zero values imposed above. The focal spot needed to be aligned close to the center of the sensor to restrict the summation to regions entirely within the sensor edges. A sample plot of the integrated values versus minor axis is shown in Fig. 5(b) where the vertical axis has been normalized to the total number of counts in the image. As this plot represents a polar integration of the profile, the derivative of this result versus radius yields an averaged representation of the counts as a function of radius. A sample output from such an analysis is shown in Fig. 5(b) for a standard flat-top Nike profile.
5. EXPERIMENTAL RESULTS
Typical SRRS growth for several IL values is shown in Fig. 4(b) for time-integrated spectra from 0.4 ns long pulses. SRRS conversion is seen to remain low until a threshold value of is reached resulting in a rapid change where 20–30% of the spectral energy has shifted toward the Stokes wing. Time-resolved spectra for short pulse, high intensity operation are shown in Fig. 6. The three streak images show time histories for increasing functions of intensity-path-length product. The first shows a short pulse where no SRRS appeared while the subsequent two cases show growth. For shots with significant SRRS, there appears to be a time delay of 0.1–0.2 ns prior to the appearance of light in the Stokes wing. The expansion of the spectrum is also seen to persist for a similar period after the laser pulse as well. This behavior is consistent with results reported for a longer wavelength laser .
The results for the profile modifications due to SRRS are summarized in an intensity-path-length plot and as a direct comparison of profiles. For the profile measurements, there appears to have been a greater shot-to-shot variation, so the result can be grouped into the two categories of above and below the threshold as determined by the spectral measurements. The fraction of excess energy in the wings compared to a Gaussian fit to the averaged profile appears to grow a modest amount from to once the threshold value of the intensity-path length product is crossed. If the measured profiles in the two cases are averaged, the increase in the wings of the distribution is much clearer, as seen in Fig. 7.
In the comparison of the profiles, a sample result for low intensity measurements taken at the focal plane in the target chamber is included. This measurement is slightly different from those made for the SRRS study in that it used a scintillator to convert the UV laser light to a visible wavelength required for the particular CCD camera used. This profile was included to demonstrate that the basic shape of the profiles from the SRRS studies appears consistent within the main portion of the profile. Although extra emission in the wings is observed for the below threshold case, this increase is believed to be the combined effect of stray light from the amplifier chain and scattering with the profile diagnostic in the target area used during the SRRS study.
The relatively low level of defocusing due to the growth of SRRS is particularly relevant for use at the Nike laser for target experiments. Figure 8 shows results from a tandem Wadsworth spectrometer that resolves on target emission spatially and spectrally on a CMOS detector . The data were taken from planar polystyrene targets where the overlapped intensity reached . In all of the cases shown, there appears to be no substantial amount of laser light away from the bright central region, even when the detector was allowed to have a saturated image. These data, combined with the below threshold results , demonstrated that for typical operation of the Nike laser, the estimated error in the intensity does not need to take into account the growth of SRRS. For high intensity operation required for some laser-plasma instability studies, some corrections should be made for the longer path beams.
6. THEORETICAL MODEL FOR SRRS
The theoretical description of SRRS begins with the full Maxwell wave equation for the total optical field driven by a nonlinear polarization whose THz frequency shifts result from the numerous rotational Raman transitions in and . Unlike the usual treatments of Raman scattering [27,28], this formulation envelopes only to the carrier frequency pair and located at the space–time spectral peaks of the incident ISI light, as follows:29] 30] (GVD), is the electronic (instantaneous) part of the nonlinear refractive index, is the intensity, 31,32]. This model includes population redistributions of the states in strongly driven molecules, but it is difficult to treat accurately because it must also include all the collisional relaxation rates, which are not accurately known.
At atmospheric pressure and the modest fluences found in our experiments (), the model simplifies greatly because the populations remain essentially unchanged from their initial thermally distributed values , which are independent of because each level is degenerate. We can also ignore the small AC Stark shifts , especially when dealing with broadband continuum light. The equations of the excitations in Eq. (5) then simplify to the independently driven harmonic oscillator equations34]: 6) includes not only the Raman Stokes and anti-Stokes frequency shifts, but also lower frequency components that account for the Raman-induced noninstantaneous part of the nonlinear refraction.
It is clear from Fig. 4 that the long tails on the input spectrum, with spectral intensities of at the Stokes frequencies around the output Stokes hump at , can easily self-seed the SRRS process. These features, plus the fact that the chaotic ISI light keeps the SRRS process in a perpetual low gain transient regime, eliminates the necessity of including a high divergence random source term in our SRRS model [7,10] and thus avoids the necessity of very large spatial arrays, such as .
The computational strategy first specifies the complete space–time behavior of the incident field and thus intensity at the entrance plane , and then solves Eqs. (4)–(6) to propagate to the exit plane . It begins by Fourier transforming Eq. (6) to the frequency domain with spectral intensity , solving for the corresponding total excitation,4) to calculate the field at the first interior grid plane . Equation (4) is integrated by a split-step algorithm, which effectively lumps the contributions on its right-hand side into thin windows spaced by a distance to give the intermediate solution
To model the input chaotic ISI field we begin with an array of independent Gaussian-distributed complex random numbers of equal RMS value, filter them in the frequency domain using the incident beam’s measured spectrum and a Gauss-fit to its far-field profile to obtain an intermediate distribution, and then transform to the domain and filter it using the measured incident pulseshape and apodized near-field profile. In these simulations, we typically choose and . The width of the far-field profile is specified by the FWHM angular width of the ISI field angle spectrum; in these simulations, times diffraction limit (XDL), which corresponds to about 50 μrad with our FWHM collimated Nike beams.
To benchmark the code against analytic theory and provide insight into the important role played by Stokes–anti-Stokes (SA) coupling, we also apply Eqs. (4)–(6) to the simple case where the incident light is comprised of only two narrowband apodized plane waves: a strong axial pump beam at (offset) frequency and a weak probe beam propagating along an off-axis angle at Stokes frequency resonant with one of the molecular transitions. Specifically, we study the dependence of the steady-state SRRS convective gain on phase mismatch due to the transverse angular divergence term and the GVD term in Eq. (4). In the limit where the SA amplitudes remain much smaller than the pump , the total steady-state amplitude and corresponding excitation are well approximated by12) and (14) into Eq. (4) and again retain only first order terms to obtain the equations for and . The pump amplitude satisfies 35–37]. The solution shown below confirms that the usual exponential Stokes growth and concomitant anti-Stokes suppression can occur only when the phase mismatch becomes large.
In the case where , the solutions for the absolute amplitude ratios and are
Figure 9 shows the code simulations for using the square apodized beams of the ISI model, but replacing the chaotic ISI fields by the intersecting narrowband plane waves described above, and compares the steady-state results to Eqs. (19). The incident Stokes seed beam is tuned to the frequency of the transition and can propagate at off-axis angles , thereby avoiding significant walkoff over the chosen propagation path. Its incident intensity is that of the axial pump beam, chosen to give a maximum steady-state power gain of only to avoid significant pump depletion. Figure 9(a) shows an example of the on-axis exit plane intensity,38,39]). Figure 9(b) shows the corresponding time-resolved absolute spectral amplitudes of the pump (), Stokes (), anti-Stokes (), and a very small second Stokes () contribution, using a 28 ps FWHM Gaussian filter centered as shown in Fig. 9(a). Figure 9(c) shows the excellent agreement between Eqs. (19) and the spectral simulations at different values, and clearly illustrates the SRS gain limitations imposed by SA coupling. (The slight deviation of the far left point is likely caused by small pump depletion.)
Unlike this simple case, the following simulations deal entirely with incident ISI beams that are temporally and spatially broadband. Under these conditions, the SRRS response never reaches steady-state but remains in a perpetual transient state with much lower convective gains. The resulting dependence of output on SA coupling will therefore be less dramatic than those shown in Fig. 9(c), but it will still show significant effects on the spectral shapes.
7. COMPARISON OF THEORY AND EXPERIMENT
Table 1 lists the parameters used in all the following ISI simulations. The attenuation coefficient was directly measured in the Nike propagation bay, while the GVD coefficient was calculated from Edlen’s dispersion formula for dry air . There appear to be no direct measurements of the electronic in air at 248 nm, thus requiring pulse-widths , so we estimated it by subtracting the total of all the contributions in Eq. (16b) from measurements at 10 ps . There are also no direct measurements of the polarizability anisotropies in either or at 248 nm, although they can be calculated from Eq. (7) and measured gain coefficients of any one transition in each molecule. To avoid the gain reduction issues shown in Fig. 9(c), these measurements will eventually be carried out using narrow-band 248 nm light with either linearly polarized plane pump and Stokes beams crossing at well-defined angles or with oppositely circularly polarized pump and seed beams. Here we chose and values based on Sellmeier equations used by Rokni , giving a reasonable fit to our measured spectra.
Finally, the XDL value was estimated by using a 2D Gaussian fit to our measured far-field profiles out of the 60 cm amplifier. These profiles result from a combination of the ISI divergence created at the Nike front-end with XDL random phase aberrations introduced primarily around the 60 cm amplifier. Our simulations make no attempt to model these aberrations more directly. For the cases of interest here, where the ISI beam divergence dominates over their point-spread width, our earlier simulations of nonSRRS beams show that the aberrations mainly cause persistent large scale length 20–40% nonuniformities in the near field . Test runs in which such nonuniformities were artificially introduced into the beam gave output spectra and far-field profiles almost identical to those presented here as long as the spatially averaged fluences were the same. As seen in Ref. , this would not be the case for simulations with coherent light [7,9,10], where the phase aberrations dominated and the near-field nonuniformities can easily exceed .
In all these simulations, we have decreased the calculation time and memory requirements by lumping the converging portion of the SRRS path between the 60 cm amplifier and recollimator mirrors into a shorter segment (13.1 m) with the same beam width and same average intensity (adjusted for linear attenuation) as the main portion.
The most beneficial effect of spectral broadening is its ability to reduce the optical coherence time , which scales as the inverse bandwidth. In the simulations shown here, we estimate using the expression 
The simulated azimuthally averaged time-integrated far-field profiles are compared to the measured profiles in Fig. 7(a). These profiles are proportional to the near-field spatial spectra, which are broadened by the SRRS process, just as the temporal spectra. At the output, there is good agreement with the measurements above the noise level, something not included in the simulations; however, the simulations show a larger amount of broadening, even at the half-power points. This is consistent with the collimated beam approximation, which allows disproportionately higher SRRS gains at larger off-axis angles [Fig. 9(c)] and corresponding far-field transverse displacements. Far-field broadening is a potential issue because it can affect beam focusability. However, these beam widths are around only 30 XDL, so it should be relatively less important for the XDL far-field profiles required for laser-fusion applications; moreover, a small amount of power-dependent broadening may be beneficial by counteracting the effects of beam power imbalance on target intensity nonuniformity.
Figure 10 shows the simulated ISI incident plane () pulse-shapes, which are averaged over the flat portion of the near-field aperture, and the near-field profiles at the incident and exit () planes, which are averaged around the pulse peak. Except for a linear attenuation, the average pulse-shape in Fig. 10(a) remains nearly constant along the entire propagation path. The near-field profiles shown in Fig. 10(b) also remain similar at the entrance and exit planes, but the exit plane profile shows the higher spatial frequencies acquired by the SRRS and a small amount of beam walkoff and diffraction around the edges. None of our measurements or simulations show any evidence of the intense multiple near-field filamentation that has been observed in spatially coherent 248 nm beams [45,46], despite that our average beam power was times higher than the critical power for self-focusing. Unlike the spatially coherent case, the important parameter for ISI light is the effective power within a single coherence zone, whose width is approximately the whole beam width/XDL; the effective power in our 30 XDL ISI beam is therefore , which is well below the critical power. This suppression of filamentation in ISI beams was predicted by earlier 3D simulations . All nonlinear refractive processes (Raman and electronic) are included in Eqs. (4)–(6) of our code; their most important effects are spectral and far-field spatial broadening.
Figure 11 compares measured and simulated time-integrated output spectra to the spectra measured at the input reference station [Fig. 4(a)]. The SRRS process approximately doubles the spectral width, giving an output coherence time close to the 0.26 ps value from the measured spectrum, as compared to the input coherence time . The overall agreement between the simulated and measured spectra is good, but the simulations generally tend to overestimate the anti-Stokes contributions and to underestimate the hump.
Figure 12 shows simulated time-resolved output spectra using 20 ps FWHM Gaussian filters to localize around several selected times near the pulse peak, along with estimated coherence times at each location. Because of the resonant response of the molecules and the chaotic temporal behavior of the ISI light, the transient buildup phase requires . The delay in exciting the higher Stokes orders is in qualitative agreement with the streak measurements in Fig. 6 and with earlier measurements on Nova . As expected, the spectral broadening and resulting coherence time reduction is significantly larger beyond the buildup phase than seen in the time-integrated spectra.
8. SUMMARY AND FUTURE WORK
Detailed experimental and theoretical studies of stimulated rotational Raman scattering in the deep UV have begun at the Nike laser. This work extends previous studies with KrF lasers and has a long-term objective of extending laser spectral bandwidth without significantly degrading the focusability of the beam. Greater laser bandwidth would help mitigate laser plasma instabilities that are problematic for applications such as inertial confinement fusion. The current paper presented detailed measurements of spectral broadening due to SRRS. A first-principles model developed at NRL has been incorporated into a numerical code that can accurately reproduce these observations with parameters drawn from the literature. The experimental onset of growth of the Stokes wings has been observed for intensity-path-length products exceeding . Measured changes in the focal distribution for beams experiencing significant SRRS growth were observed to have small changes in the outer portions of their focal distributions. These effects appear to not be significant for standard Nike operation.
Ongoing efforts will refine the parameters used in the simulations, assess steady-state SRRS behavior, and explore means to exploit SRRS for the ICF application. For the latter, it will be particularly important to find means of maximizing the laser bandwidth while minimizing any degradation in the beam quality. One of the first objectives is to go beyond the collimated beam approximation and take explicit account of the beam convergence in the path between the 60 cm amplifier and recollimator mirrors. Although the 13.1 m equivalent collimated segment is only a small part of the 102 m overall SRRS propagation path used in these simulations, it could become more important in subsequent experiments that down-collimate to smaller beams or propagate through a focal waist to reduce the SRRS growth regions. A second objective is to generalize to a full tensor formulation of the theory  to examine depolarization effects on our linearly polarized input beam and explore the possibility of exploiting the higher SRRS gain allowed by combining mixed circular polarizations between the seed and pump lasers.
U.S. Department of Energy (DOE); National Nuclear Security Administration (NNSA).
The authors would like to acknowledge informative SRRS discussions with J. Reintjes and N. Kurnit. They would also like to thank Dr. L. Y. Chan and the Nike technical staff, B. Jenkins, S. Terrell, L. Granger, and S. Krafsig, for their support during these experiments.
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