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Abstract
Optical Kerr effects induced by the propagation of high peak-power laser beams through real atmospheres have been a topic of interest to the nonlinear optics community for several decades. This paper proposes a new analytical model for predicting the filamentation/light channel onset distance in real atmospheres based on modulation instability model considerations. The normalized intensity increases exponentially as the beam propagates through the medium. It is hypothesized that this growth can be modeled as a weighted ratio of the Gaussian beam diameter at range to the lateral coherence radius and can be used to set the power ratio for an absorbing, turbulent, nonlinear media to estimate the beam collapse distance. Comparison of onset distance predictions with those found from computer simulation and deduced from field experiments will be presented. In addition, this model will be used with an analytical approach to quantify the expected radius of light channels resulting from self-focusing both with and without the production of a plasma filament. Finally, this paper will describe a set of 1.5-micron, variable focal length USPL field experiments. Comparisons of theoretical radius calculations to measurements from field experiments will be presented.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
During the propagation of ultra-short pulse laser (USPL) beams in a nonlinear environment, their high peak power ${P_{peak}}$ (if greater than a certain critical threshold called the critical power ${P_{crit}}$) can create a dynamic interaction involving optical Kerr self-focusing, photo-ionization, diffraction, defocusing and a number of nonlinear mechanisms [1–4]. This interaction results in what is called filamentation.
The optical Kerr effect causes an increase in index of refraction that is proportional to the propagating laser intensity. This refractive index increase is responsible for the nonlinear optical effects of self-focusing, self-phase modulation and modulational instability. As the whole beam or a portion of the beam cross section self-focuses, the peak intensity increases near the center of the area undergoing self-focusing, which further increases the refractive index, giving rise to continued self-focusing. This process continues until it is arrested by other effects such as ionization, pulse broadening due to turbulence, energy loss to particulate scattering, or the generation of new frequencies. The production of a plasma channel or filament begins when the near infrared (NIR) laser beam’s focused laser intensity reaches 1013 to 1014 W/cm2. This produces electron densities of the order 1016 cm−3. The initiated plasma channel acts to lower the refractive index and causes the tightly focused light channel to defocus. The dynamic interplay between Kerr self-focusing and plasma defocusing can reoccur if there is optical coherence across a transverse area of the beam profile that contains a power of Pcrit or greater. A graphical representation of this process is depicted in Fig. 1. The filaments can extend to around tens of meters in length and contain essentially the critical power of the medium within core diameters in the high 10s to low 100s of microns at NIR wavelengths. [1,5]. Moreover, the number of filaments created appears to be proportional to the power ratio $PR = {{{P_{peak}}} / {{P_{crit}}}}$[1–3].
Kerr self-focusing also can result in the generation of light channels, which can occur concurrent with or exclusive of the production of plasma filaments. Light channels are self-guided beams that keep a relatively constant diameter and propagate without causing air ionization, i.e., a balance between nonlinear self-focusing and diffraction. The channels have millimeter-sized diameters and have been observed to propagate through air for hundreds of meters without a significant increase in diameter [3,6–8].
Filamentation has been suggested for many applications [1–13], but lossy, turbulent media engineering models successfully validated with computer simulation and/or experimentation are important for assessing USPL system performance in those applications under varying atmospheric conditions. To date, successful engineering modeling has been limited to Filamentation Onset Distance (FOD) estimation [14]. Filament and light channel radii have yet to be successfully modeled in lossy, turbulent media with validation.
The purpose of this paper is three-fold. First, we will describe a set of 1.53-micron, variable focal length USPL field experiments presenting filamentation results. Second, we propose a new ad hoc engineering models for predicting the filamentation/light channel onset distance in real atmospheres based on Modulation Instability (MI) that is dependent on atmospheric loss and key turbulence parameters. We hypothesize that this growth can be modeled as a weighted ratio of the Gaussian beam diameter at range to the lateral coherence radius that can be used in an absorbing, turbulent, nonlinear medium to estimate the filamentation/light channel onset distance. Analytical predictions of onset distances from this model are found to be consistent with computer simulations and experimental results. Third, this model will be used in a modification of the radius approach cited by Stotts, et al. [14] to create an ad hoc engineering model to predict light channel radii. Comparison of theoretical light channel radius calculations with field experiment measurements will be given and will be seen to be in reasonable agreement.
2. USPL experimental equipment layout and transmitted beam characteristics
A set of field experiments were conducted adjacent to the Kennedy Space Center (KSC) runway. Figure 2 shows the schematic of the experimental equipment layout. A USPL beam propagated from the laser transmitter through a Florida coastal atmosphere to a metal target board 1006 meters away. Because of the target /sensors setup, one could not easily move the target location so it was fixed for the duration of the experiment. The initial laser beam was sent through a variable focal length telescope set by a transmitter stage setting control, which also created a specific transmitter exit aperture diameter. The experiments were conducted at pre-selected power ratios$PR$and focal length values. The USPL used in these experiments was an Optical Parametric Chirped-Pulse Amplification (OPCPA) laser tuned to an output wavelength of 1.53 microns. Output energies per pulse ran from 30 millijoules (mJ) to 100 mJ and laser pulse widths ranged from 100s of femtoseconds to a few picoseconds.
Figure 3 depicts (a) the transmitter focal lengths and their associated output beam diameter and (b) the focused beam waist curves as a function of telescope stage settings. For the analysis in this paper, we assume that the telescope output beam has an intensity profile of the form
where${a_0} = a(0)$ is the e-1-intensity beam waist in Fig. 3(b) at its associated focal lengths.
Fig. 3. (a) Transmitter Focal Length and Output Beam Diameter as Function of Telescope Stage Setting and (b) Focused Beam Waist Curves as Function of Propagation Distance.
Visible and Short Wavelength InfraRed (SWIR) cameras were used to capture image sequences of the visible and SWIR radiance impinging a target board. The visible camera was used to detect any third harmonic generation (THG) light that occurred simultaneously [15]. THG was observed to be green, which differentiates it from white light continuum. We identified beam collapse by the coincidence of THG in visible images and areas of high intensity irradiance on the SWIR camera. This method ensured we were measuring non-diffraction-limited focus spots caused by the optical Kerr effect. Figure 4 shows an example of the two synchronized SWIR (left) and Visible (right) frames for PR = 20 and f = 1688.7 m. The amplitude of the THG emissions tracked with the magnitude of the light channel signal, i.e., the higher the peak of the light channel beam profile, the stronger the green emission became. To our knowledge, this is the first-time simultaneous SWIR and visible image data has been shown for USPL beam propagation through lossy turbulence measured at a 1-kilometer range.
Finally, a suite of ground truthing instrumentation was used by personnel from the Townes Institute, University of Central Florida (UCF), to characterize the field experimental environments simultaneous with the data collections. This instrumentation suite included a weather station, a visibility meter, and a BLS-900 long range scintillometer.
3. USPL data analysis
The optical phenomena seen in our experiments were either filaments with light channel or light channels alone. Nonlinear phenomena were assumed to occur where both the relative intensity is large over a small spot area and the strongest Third Harmonic Generation (THG) emissions was present in the same relative location in their respective cameras as seen in Fig. 4. The intensity threshold value where the strongest green emission occurs appeared to be at ${2 / 3}$ of the peak light channel intensity.
To make comparisons with the analytical work to come, we curve-fitted the light intensity to a Gaussian profile to find the ${e^{ - 1}} - $intensity beam radius. Because this asymmetry of experimental setup systematically distorted the images of light impinging the target board, independent curve-fits were done for the horizontal (H) and vertical (V) axes of light channel projections following Chin, et al. [16]. (Note: We did not correct the keystone effect in the horizontal direction caused by the SWIR and VIS cameras having opposite angular offset of 20 degrees relative to the optical axis of the experiment, there is a slight offset seen in the images). Figure 5 contain example cross-sections of SWIR beam profiles with their Gaussian curve-fits in the vertical and horizontal directions for PR=20.42. These values are smaller than the diffraction-limited beam.
Fig. 5. Data Frames and their Curves in the Vertical and Horizontal Directions for PR = 20 and $f$ = 1689 m.
4. Filament radius as a function of wavelength
Given the assumptions that, (1) the power within the filament equals critical power of the medium, (2) the refractive index is due equally to Kerr focusing and plasma defocusing, and (3) only multi-photon ionization is involved, the filament radius is given by
5. Modulation instability and beam collapse
Filaments and light channels have been hypothesized to be characterizable using modulation instability (MI) theory [11–13]. According to Bliss et al., linearized instability theory predicts that small intensity modulations superimposed on a uniform background will undergo exponential growth of the form $\exp \{{g{\kern 1pt} z} \}$, where$g$ is the growth rate [12]. To the author’s knowledge, no theories on the form of g have been proposed whose filamentation results in lossy, turblent media have been compared successfully with experimentations. This subsection proposes an ad hoc engineering model for predictig Filamentation Onset Distance (FOD) in that type of media.
The accepted equation for estimating the filamentation onset distance (FOD) in nonlinear medium is the Marburger equation [17–19]. Its fundamental form is given by
where ${z_{sf}}$ is the filamentation onset distance, ${z_r} = {{\pi W_0^2} / \lambda }$is the Rayleigh Range, and ${W_0}$is the initial ${e^{ - 2}}$-intensity beam radius. Stotts, et al. showed how the above can be used to estimate the FOD in a lossy, turbulent nonlinear medium [14]; specifically, the FOD for a lossy, turbulent nonlinear medium, is found by solving the following equationThe mutual coherence function is the fundamental means for characterizing the transverse spatial coherence radius of propagating laser beams [19]. One might hypothesize that the growth rate coefficient of the MI model is proportional to Gaussian beam diameter at range divided by the lateral spatial coherence radius ${{2W} / {{\rho _0}}}$, which is a modification of the radius ratio in the condition for self-focusing in turbulence proposed by Peñano et al. [13]. Specifically, we hypothesize that the ratio of peak power to critical power over a horizontal range just above the earth’s surface in a turbulent atmosphere is given by
whereTo calculate the FOD in a lossy, turbulent atmosphere, Eq. (6) is inserted into Eq. (3) to yield
Referring to Eq. (14), we have for large ${P{R^\ast }}$,
Figure 7(a) and Fig. 7(b) are comparisons of the variable and fixed focal length FODs versus analytical FOD estimates, respectively, using the above procedure, with the growth rate parameter ${m_0} = 0.093$ and system and computer simulation data given in [14]. These figures show good agreement between theory and simulation. (Similarly, it can be shown for super Gaussian beams that ${m_0} = 0.025$ (lens) and ${m_0} = 0.7$ (collimated beam) will yield good agreement with the computer simulation data of [14]).
Fig. 7. FOD Comparison of Theoretical Predictions with ${m_0} = 0.093$ and (a) the HELCAP Variable Focal Length Computer Simulation Results from [14], and(b) the HELCAP Fixed Focal Length Computer Simulation Results from [14].
6. Light channel radius estimation
Light channel radii and lengths are key characteristics of interest to many researchers. Durand, et al. experimental results showed their radii of the order of a couple of millimeters and a length of ∼300 m for PR∼13 at a 1 km range [6]. Others have reported similar size light channels. Durand, et al. also noted the number of filaments created per laser pulse drops to a few rather than being proportional to PR.
Radius equations are one means of determining when filamentation onset occurs, e.g., Marburger [18] and Petrishchev [21]. In the presence of atmospheric loss and turbulence, it has been noted that the filamentation onset distance has statistical fluctuations. It should be noted that for larger loss and turbulence, filaments/light channels do not always occur [2,13] In other words, at any distance in z, Kerr-induced optical phenomena can be in various stages of development, or non-existent.
Here we compare the beam collapse profiles from computer simulations and two reported filamentation radius equations. The two selected radius equations for the comparison were (1) the Marburger lossy radius equation [17],
The computer simulation data was produced using MZA’s Wave Train software [14]. This code quantifies the propagation of a Gaussian beam through a lossy, turbulent, nonlinear medium. These simulations assume$\lambda = 1\,\mu m$, ${W_0} = 10\,cm$and${F_0} = 1.5\,km$. The beam is assumed azimuthally symmetric, as are the Kerr-effect phase screens [14]. For computational efficiency, the simulations used the quasi-discrete Hankel transform [23]. The propagation grid had 2500 points in the radial coordinate with spacing 0.4 mm. The nonlinear beam had a power ratio $PR = 30$and a volume extinction coefficient $\alpha = 0.1\,k{m^{ - 1}}$. In the simulations, the nonlinear beam propagates in multiple steps, accumulating Kerr phase and reducing its irradiance based on the extinction coefficient at each step. The distance of each step is set so that the maximum value of the Kerr effect phase delay does not exceed 0.1 radians. Also, each step can be no further than 12% of the expected collapse range computed by the Marburger loss equations. After each step, beam collapse was tested based on a threshold ratio of the normalized nonlinear beam irradiance to the normalized linear beam irradiance. When the beam has collapsed, the simulation stops, which was at 968 m in this case. Near the source, the first few steps were about 11 m, and they were only 1.8 cm near the collapse at the end of the path.
Comparing the two equations and the computer simulation showed no agreement between any two data sets. In the spirit of developing ad hoc engineering models, the $a(0 )$term then was replaced in Eq. (18) by the linearly focused beam radius expression ${a_L}(z )$ from Gaussian beam theory [17]. This change caused Eq. (18) to better agree but this new equation again did not follow the collapse rate profile predicted from computer simulation. Raising the power in Eq. (18) from 4 to 16 yielded
A secondary effect we see from Eq. (19) in Fig. 8 is that the radius at the FOD does not collapse to zero, but to a finite value like light channels do. Given that fact, Eq. (19) is postulated to quantify the ${e^{ - 1}} - $intensity radius for light channels in lossy turbulent nonlinear media. This is because atmospheric loss and/or turbulence degrades the laser intensity to values that are insufficient for air ionization and only allow light channels to be generated.
For the remainder of the paper, we will continue to use ${m_0} = 0.093$ in our data comparisons with experimental results.
7. Comparison of model results with USPL field experiments
Table 1 gives the $C_n^2$ and visibility measurements taken during the PR = 20 and PR = 12 data runs, respectively. Since the KSC Test site is on the East Coast of Florida, we assumed that the atmospheric loss can be derived from MODTRAN transmittance program using the Navy Maritime Model and the UCF visibility measurements. The visibility parameter VIS used in MODTRAN is equal to 1.3 times the UCF visibility [24]. The resulting volume extinction coefficients for the measured visibilities also are given in this table. These values are consistent with those reported in other field experiments [25].
To confirm the validity of our proposed model, we compared its FOD predictions against probability density (PD) results derived from the Naval Research Laboratory (NRL) PyCAP computer software, which is the Python-based High Energy Laser Code for Atmospheric Propagation (HELCAP) program. Each simulation had 100 realizations to generate the simulation statistics. Figure 9(a) and Fig. 9(b) depict a comparison between the PyCAP FOD simulation results and the MI model FOD predictions for PR = 20 and 12, respectively, using the measured environmental parameters in Table 1. The red dashed line in these figures is where the target board was located. Table 2 is a summary of the PyCAP simulation results and the proposed model predictions for PR = 20 & 12. These figures and their associated tables show good agreement between the analytical model estimates and the median values from the simulation.
Fig. 9. Comparisons of MI model predictions to PyCAP simulation predictions for (a) PR = 20 and (b) PR = 12 where the numbers in the flags are analytical median values and red dash line indicates target board location. Distributions are color-coded according to the lens focal length.
Table 1. Refractive Index Structure Parameter and Visibility measurements take using KSC PR = 20 & 12 Run Data and the associated MODTRAN6 / Maritime Volume Extinction Coefficient
Table 2. Number of Light Channels per Camera Frame and MI Model Filament Onset Distance Predictions Versus Filament Onset Distance PyCAP Simulation Results for KSC PR = 20 & 12
Ideally, to compare the experimental data to the simulations and model, one would collect data at multiple distances along the line of propagation of the laser beam. This would require the ability to move the target board and suite of sensors during experimentation. This capability was not available for these experiments. However, we can compare the rate of filamentation observed for varying transmitter focal lengths to the predicted filamentation probability density for FODs predicted by our model and by simulations. Table 2 shows the number of light channels per individual image (frame) for all the PR = 20 and 12 images sequences. For FODs predicted by our model to occur at distances less than the transmitter to target board distance (before the target board), we observed that the number of light channels decreases as the focal length decreases. This is consistent with what one would expect from inspection of Fig. 9. As expected, longer focal lengths produced larger FODs, and the filamentation probability densities at the target board position decrease as model predicted FODs decrease or in other words as the distance between the expected filament onset position and the position of the target board increases. This consistency between observations, simulations and our model provide further confidence in the model.
Table 2 also shows each frame contained either one or two light channels, no more. Previous filamentation modeling work showed that the number should be around the PR value [3]. Durand, et al. confirmed this fact in their PR=16.7 experiment at a150 m range, but also showed that number reduces to around 4 filaments per frame for PR∼13.5 at 1 km range [6]. Clearly, our numbers are much lower than either PR, but are more in line with Durand, et al.'s results for a 1 km range. If the previously defined intensity threshold was reduced from 2000 to zero, the number of filaments per image frame was 5 or less, which is on the order of the number of filaments Durand, et al. reported.
Durand, et al. also reported light channels of 300 m lengths for PR ≈ 13 in their 1 km experiment [6]. (Shorter lengths were shown for short experimental distances. [6].) The light channel data taken at KSC were consistent with their work. For example, no light channels were observed for the 834 m focal length runs that had a FOD greater than 300 meters forward of the target board according to our simulations.
Tables 3 and 4 compare the Eq. (20) predictions using published ground-truth measurement to the curve-fitted light channel radii of experimental results in the vertical and horizontal directions, respectively, for PR = 20 & 12. These tables show reasonable agreement between the model estimates and field data. The results support the previously reported millimeter size for light channels and that the radius decreases as PR decreases. In Eq. (20), this behavior comes from the ${({1 - {z / {{F_0}}}} )^2}$ dominating Eq. (20) and decreasing as the value of z increases when PR decreases.
Table 3. MI Optical Channel Radius Predictions Versus Vertical Curve-fitted Radius Estimates from KSC PR = 20 & 12 Run Data
Table 4. MI Model Optical Channel Radius Versus Horizontal Curve-Fitted Radius Estimates from KSC PR = 20 & 12 Run Data
8. Summary
This paper described a set of 1.53-micron, variable focal length USPL field experiments presenting filamentation results. It proposed a new ad hoc engineering models for predicting the filamentation/light channel onset distance in real atmospheres based on Modulation Instability (MI) exponential gain term dependent on atmospheric loss and key turbulence parameters. It hypothesized that this growth can be modeled as a weighted ratio of the Gaussian beam diameter at range to the lateral coherence radius and can be used to set the power ratio for an absorbing, turbulent, nonlinear medium to estimate the filamentation/light channel onset distance. Comparison of analytical predictions of onset distances with those found from computer simulations were given and were consistent with our experimental results. This term also was used in a modification of the radius approach cited by Stotts, et al. [12] to create an ad hoc engineering model to predict light channel radii. Comparison of theoretical light channel radius calculations and with field experiment measurements were given and found of be in reasonable agreement. The implication of these results is the simplicity these equations provide for predicting the onset of filamentation and beam radii. Though not meant as a replacement for detailed simulation, the equations provide a tool, validated by physics-based simulations, for future experimentation and applications, for which time-consuming simulations are not feasible.
Funding
U.S. Naval Research Laboratory; Office of Naval Research; Defense Advanced Research Projects Agency.
Acknowledgement
The views, opinions, and/or findings contained in this article are those of the authors and should not be interpreted as representing the official views or policies, either expressed or implied, of the DARPA, the Department of the Navy, or of the Department of Defense.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors, e.g., Dr. G. DiComo, NRL, upon reasonable request.
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