Engineering equations for the filamentation collapse distance in lossy, turbulent, nonlinear media

Larry B. Stotts; Joseph R. Peñano; Jason A. Tellez; Jason D. Schmidt; Vincent J Urick

doi:10.1364/OE.27.025126

Engineering equations for the filamentation collapse distance in lossy, turbulent, nonlinear media

Larry B. Stotts, Joseph R. Peñano, Jason A. Tellez, Jason D. Schmidt, and Vincent J Urick

Optics Express
Vol. 27,
Issue 18,
pp. 25126-25141
(2019)
•https://doi.org/10.1364/OE.27.025126

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Larry B. Stotts, Joseph R. Peñano, Jason A. Tellez, Jason D. Schmidt, and Vincent J Urick, "Engineering equations for the filamentation collapse distance in lossy, turbulent, nonlinear media," Opt. Express 27, 25126-25141 (2019)

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- Nonlinear Optics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: June 28, 2019
- Revised Manuscript: August 4, 2019
- Manuscript Accepted: August 5, 2019
- Published: August 21, 2019

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Abstract

The propagation of high peak-power laser beams in real atmospheres has been an active research area for a couple of decades. Atmospheric turbulence and loss will induce decreases in the filamentation self-focusing collapse distance as the refractive index structure parameter and volume extinction coefficient, respectively, increase. This paper provides a validated analytical method for predicting the filamentation onset distance in lossy, turbulent, nonlinear media. It is based on a modification of Petrishchev’s and Marburger theories. It postulates that the ratio of the peak power to critical power at range in turbulence is modified by a multiplicative, rather than additive, gain factor. Specifically, this new approach modifies the Petrishchev’s turbulence equation to create the required multiplicative factor. This is necessary to create the shortened filamentation onset distance that occurs when a laser beam propagates through the cited nonlinear medium. This equation then is used with the Marburger distance and the Karr et al loss equations to yield the filamentation onset distance estimate in lossy, turbulent, nonlinear environment. Theory validation is done against two independent sets of computer simulation results. One comes from the NRL’s HELCAP software and the other from MZA’s Wave Train modeling software package. This paper also shows that once the zero-turbulence onset distance is set based on link loss, the addition of turbulence creates essentially the same PDFs at similar median distances for each loss case. This result had not been previously reported. This is the first quantitative comparison between closed form equations and computer simulation results characterizing filament generation in a lossy, turbulent, nonlinear medium.

1. Introduction

The propagation of Ultra-Short-Pulse Laser (USPL) beams in the atmosphere continues to be an active research topic [1–8]. The high peak power of the laser (if above a certain critical threshold) still creates a dynamic interaction between beam diffraction and nonlinear optical effects to induce one or more filaments in this channel, but with some unique different properties induced by the atmospheric turbulence. For example, the filamentation onset distance shortens for moderate to strong turbulence [9,10]. In addition, once the filament is formed in this environment, it appears to be unaffected by the turbulence surrounding it in [3,11]. Ackermann et al showed this fact under extreme turbulent conditions (> 5 times the refractive index structure parameter that is observed in the Earth’s typical daytime atmosphere) [11]. This means that the initiation of filamentation in terms of position and pointing stability appear to be the only processes affected by turbulence. The presence of a converging, diverging and collimating lenses in the laser transmitter’s telescope also change the collapse distance and its Probability Density Functions (PDF) as well. Stotts et al recently provided a validated engineering equation for predicting the filamentation self-focusing distance in lossless, turbulent nonlinear media with and without a transmitter lens [12].

In real atmospheres, both atmospheric loss and turbulence are present. This would suggest that there will be dynamic interaction between loss, turbulence, diffraction and nonlinear effects in the creation of filaments. Research has been reported on laser propagation through a lossy, nonlinear medium [11,13,14]. Ackerman et. al. showed that filaments can survive their interaction with aerosols via both computer simulation and experimentation [11]. Karr et al provided an analytical equation for predicting the filamentation onset distance for lossy media, with and without the use of a transmitter lens [13]. Their equation is a modified version of the Marburger loss equation derived in [15]. To date, there has not been much reported in the literature on the characterization of laser propagation through lossy, turbulent, nonlinear media.

This paper provides a validated analytical method for predicting the filamentation onset distance in lossy, turbulent, nonlinear media. This approach is based on the alternations of the equations from the Petrishchev’s and Marburger theories. The key change is that the ratio of peak power to critical power at range in turbulence is increased by a multiplicative factor derived from a modification of the Petrishchev’s turbulence equation [16] as presented by Stotts et al [12]. This new power ratio then uses the Marburger distance and the Karr et. al. loss equations [13] to create an estimate of the filamentation onset distance. This paper also shows that once the zero-turbulence onset distance is set based on link loss, the addition of turbulence creates essentially the same PDFs at similar median distances for each loss case. This result had not been previously reported. Theory validation is done against two independent sets of computer simulation results. One comes from the Naval Research Laboratory (NRL) High Energy Laser Code for Atmospheric Propagation (HELCAP) software and the other from the MZA’s Wave Train modeling software package. It should be noted that our study is limited to those cases for which group velocity dispersion (GVD) is not important, i.e., longer pulses or pulses that are pre-chirped to compensate for GVD [17]. These results of this paper may provide a guide for further numerical simulations, further theoretical developments, application assessments and field experiments.

2. Initial insight from HELCAP simulations of laser propagation through lossy, turbulent, nonlinear media

In real atmospheres, laser beams are degraded by transmission loss because of molecular and aerosol scattering and absorption, and atmospheric turbulence. Figs. 1 shows the atmospheric extinction loss from the former for a 150 meter link located on the Kennedy Space Center (KSC) runway in mid-October 2017 [18]. This graph puts the volume extinction coefficient between $0.1\,k{m^{ - 1}}$ and $0.3\,k{m^{ - 1}}$ for wavelengths in the Near and Shortwave Infrared (NIR/SWIR) spectral region.

Fig. 1. Extinction Rate versus Hour of the Day Measured on the Kennedy Space Center runway in mid-October 2017 [18]. (Figure published with permission from Originator, Dr. Chris C. Davis, and the Optical Society of America.)

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Figure 2 depicts the atmospheric Refractive Index Structure Parameter $C_n^2$ measured at the KSC runway at the same time [18]. The turbulences is strong most of the time at this site, with a few intervals of weaker turbulence before sunrise and near sunset.

Fig. 2. Refractive Index Structure Parameter versus Hour of the Day Measured on the Kennedy Space Center runway in mid-October 2017 [18]. (Figure published with permission from Originator, Dr. Chris C. Davis, and the Optical Society of America.)

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NRL has reported computer simulation data and analysis for filamentation generation using the HELCAP capability for several years [2,17,19–22]. HELCAP is a fully time-dependent, 3D code for modeling the propagation of continuous and pulsed High Energy Laser (HEL) beams through various atmospheric environments. It includes the effects of aerosol and molecular scattering, aerosol heating and vaporization, thermal blooming due to both aerosol and molecular absorption, and atmospheric turbulence. Specifically, HELCAP solves the nonlinear Schrödinger-like equation

(1)$$\frac{{\partial A}}{{\partial z}} = \frac{{ic}}{{2{\omega _0}}}\nabla _ \bot ^2A + \left[ {i\frac{{{\omega_0}}}{c}({\delta {n_T} + \delta {n_{TB}}} )- \frac{1}{2}({{\alpha_{TB}} + \beta } )} \right]A + \sum\limits_j {{S_j}} $$

where $A = A({x,\;y,\;z,\;t} )$ is the complex electric field amplitude of the laser beam, ${\nabla _ \bot }$ is the gradient operator in the transverse direction, ${\omega _0} = {{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} \lambda }} \right.} \lambda }$, $\lambda $ is the laser wavelength, c is the speed of light in the medium, $\delta {n_T}$ and $\delta {n_{TB}}$ are the change in refractive index induced by turbulence and thermal blooming, respectively, and ${\alpha _{TB}}$ and $\beta $ are the combined absorption and scattering coefficients from both molecular and aerosol entities, respectively. The terms denoted by $\sum\limits_j {{S_j}} $ represent other physical processes that are included in the code but are not important for the parameter regime considered here. Atmospheric turbulence is modeled in the usual manner using phase screens for which the scale sizes of the index fluctuations are described by a Kolmogorov spectrum characterized by the parameter $C_n^2$. A more detailed description of the code can be found in [19–21]. Fig. 3 is an example result from a HELCAP simulation. It shows the laser fluence distribution near the collapse distance of two example beams with powers just above, and well-above the critical self-focusing power [22].

Fig. 3. Fluence contours generated by the HELCAP code for beams with (a) ${P \mathord{\left/ {\vphantom {P {{P_{crit}} = 1.5}}} \right.} {{P_{crit}} = 1.5}}$ and b) ${P \mathord{\left/ {\vphantom {P {{P_{crit}} = 20}}} \right.} {{P_{crit}} = 20}}$ near the onset of filamentation. Taken from Ref. [22].

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Filamentation onset distance is the range z at which the beam’s fluence becomes an order of magnitude larger than the limit of linear focusing in vacuum. In practice, the fluence increases very sharply around the nonlinear focus and, in the absence of a focusing arrest mechanism (e.g., plasma formation) the numerical accuracy of the code breaks down due to the small-scale structure generated within the beam. Given the above information, let’s now look at some HELCAP filamentation onset distance simulations for conditions similar extinction rates, but lower turbulence levels, reported for KSC above.

In our HELCAP simulation runs, we set ${P_{Peak}}/{P_{crit}}\, = \,10,$ $\lambda = 0.8\mu m$ and ${W_0} = \,15\,cm$, where ${P_{peak}}$ is the peak power of the laser at the transmitter, ${P_{crit}}$ is the critical power of the nonlinear medium, $\lambda $ is the laser wavelength, and ${W_0}$ is the transmitted ${e^{ - 2}} - $ intensity beam radius for a linearly propagating Gaussian beam. We also will used $C_n^2$ values of $1x{10^{ - 16}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}},$ $1x{10^{ - 15}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}$ and $1x{10^{ - 14}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}},$ and volume extinction coefficients equal $\alpha \, = \,0.0\,k{m^{ - 1}},$ $0.1\,k{m^{ - 1}}$ and $0.2\,k{m^{ - 1}}$.

Figures. 4, 5, and 6 show the collapse distance PDFs as a function of propagation distance created by HELCAP for the cited $C_n^2$ and extinction coefficient values. The transmitter lens focal lengths chosen for the three extinction losses are 5600 m, 4650 m and 3950 m, respectively, so that the collapse distance in zero turbulence is ∼4 km in all three cases.

Fig. 4. Filamentation Onset Distance Probability Density Function as a function of propagation distance and various values of the Refractive Index Structure Parameter for a lossless medium and a Zero Turbulence Collapse Distance of 4 km.

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Fig. 5. Filamentation Onset Distance Probability Density Function as a function of propagation distance and various values of the Refractive Index Structure Parameter and a volume Extinction Coefficient of 0.1 inverse-kilometers and a Zero Turbulence Collapse Distance of 4 km.

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Fig. 6. Filamentation Onset Distance Probability Density Function as a function of propagation distance and various values of the Refractive Index Structure Parameter and a volume Extinction Coefficient of 0.2 inverse-kilometers and Zero Turbulence Collapse Distance of 4 km.

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Looking at these graphs, once the zero-turbulence onset distance is set to 4 km, the addition of turbulence in each situation creates essentially the same PDFs at similar median distances. Stotts et al predicted the non-monotonical shift to shorter distances in a lossless, turbulent nonlinear atmosphere by recognizing that the peak power to critical power at range needs to be modified by a multiplicative, rather than an additive, gain factor derived from Petrishchev’s turbulence equations [12]. This gain was hypothesized to come from the constructive interference interacting with the Kerr effect. These facts suggests that one can get the filament onset predictions for a lossy, turbulent nonlinear atmosphere by using the equations from the Stotts et. al. paper [12] and the modified Marburger loss equation of Karr et. al. [13,14] with the proper choice of the power ratio ${P_{Peak}}/{P_{crit}}.$

3. Analytical equations for the filamentation self-focusing collapse distance in lossy, turbulent media

In Karr et al, [13,14] and Houard et al [23], the Marburger self-focusing distance equation was reconfirmed to be

(2)$${z_{sf}} = \frac{{0.367{z_r}}}{{\sqrt {{{\left( {\sqrt {{{{P_{peak}}} \mathord{\left/ {\vphantom {{{P_{peak}}} {{P_{crit}}}}} \right.} {{P_{crit}}}}} - 0.852} \right)}^2} - 0.0219} }}, $$

where

(3)$${z_r} = ka_0^2$$

is the Rayleigh Range. In Eq. (2), ${a_0}$ is the transmitted ${e^{ - 1}} - $ intensity beam radius and $k = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right.} \lambda }$ is the wavenumber of the laser. Eq. (2) provides a good estimate of the filamentation onset distance for a Gaussian input beam propagating through a non-linear medium.

When the transmitter has a lens at its output aperture, the filamentation onset distance then is given by the Talanov equation,

(4)$$z_{sf}^{\prime} = {{{z_{sf}}f} \mathord{\left/ {\vphantom {{{z_{sf}}f} {({{z_{sf}} + f} )}}} \right.} {({{z_{sf}} + f} )}}$$

with f being the focal length of the transmitter lens [25]. Durand et al [7], Karr et al, [13,14] and Houard et al [23] all reconfirmed the validity of Eq. (4) to predict the filamentation onset distance when a transmitting lens is present.

When the nonlinear medium has a loss, the Karr et al [13,14] found that the beam radius collapses into filamentation following the equation

(5)$${a^{16}}(z )= a_L^{16}(z )\left\{ {1 - \left( {\frac{2}{{{\alpha^2}z{{_{sf}^{\prime}}^2}}}} \right)[{1 - ({1 + \alpha z} )\exp \{{ - \alpha z} \}} ]} \right\}$$

where ${a_L}(z )$ is linearly focused beam radius from Gaussian beam theory. Eq. (4) was found to match MZA simulation results in [12] better than the beam radius equation derived by Zemlyanov and Geints [24] and that from the standard Marburger equation [15]. Eq. (5) assumes a lens in at the transmitter aperture; when there is no lens, the Marburger distance ${z_{sf}}$ replaces $z_{sf}^{\prime}$ in Eq. (5). In both cases, one solves for the new collapse distance by finding the value of z that sets the equations inside the curly brackets in Eq. (5) to zero. Karr et al, [13] compared Eq. (5) against computer simulation results and found the agreement excellent.

Let us now turn to the filamentation in turbulence. Stotts et al. postulated that the ratio of peak laser power to the medium’s critical power at range, $P_{peak}^\ast /{P_{crit}}$, is the product of the initial ratio of peak laser power at range to critical power times a multiplicative factor, or specifically,

(6)$$P_{peak}^\ast /{P_{crit}} = ({{P_{Peak}}/{P_{crit}}} )[{1 + {m_0}{k^2}a_0^2{{({{{{a_0}C} \mathord{\left/ {\vphantom {{{a_0}C} {{\ell_0}}}} \right.} {{\ell_0}}}} )}^{{2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}} ], $$

where

(7)$$C = 4.38\, \ell _0^{ - {1 \mathord{\left/ {\vphantom {1 3}} \right.} 3}}C_n^2(h )[{1 - {{\{{1 + 17.5{{({{{{a_0}} \mathord{\left/ {\vphantom {{{a_0}} {{\ell_0}}}} \right.} {{\ell_0}}}} )}^2}} \}}^{ - {1 \mathord{\left/ {\vphantom {1 6}} \right.} 6}}}} ], $$

with the parameter ${\ell _0}$ being the inner scale of the turbulence (e.g., ∼1 mm) [14]. Eq. (6) is a modification of the Petrishchev’s turbulence equation [16] as presented by Stotts et al [12]. In their paper, the Stotts et al substituted Eq. (5) into the Dawes and Marburger version of Eq. (2) to estimate the new filamentation onset distance $z_{sf}^\ast $[12], i.e.,

(8a)$$z_{sf}^\ast \, = \,\frac{{0.367{z_r}}}{{\sqrt {{{\left( {\sqrt {{{P_{peak}^\ast } \mathord{\left/ {\vphantom {{P_{peak}^\ast } {{P_{crit}}}}} \right.} {{P_{crit}}}}} - 0.852} \right)}^2} - 0.0219} }},$$

(8b)$$\approx \frac{{0.367{z_r}}}{{\left( {\sqrt {{{P_{peak}^\ast } \mathord{\left/ {\vphantom {{P_{peak}^\ast } {{P_{crit}}}}} \right.} {{P_{crit}}}}} - 0.852} \right)}}$$

where Eq. (8b) being the Dawes and Marburger version [26]. If a lens is involved, then the new onset distance equals

(9)$$z_{sf}^{\ast \ast } = \frac{{z_{sf}^\ast f}}{{z_{sf}^\ast + f}}, $$

with $z_{sf}^\ast $ is given by Eq. (9). The symbol ${m_0}$ is the multiplicative parameter that scales the turbulence term in Eq. (6) under different transmitter configurations. Comparisons between the above theory and computer simulation results generated by NRL’s HELCAP simulation were excellent for specific values of ${m_0}$ chosen for the cases of no lens, lenses and with a collimated beam [12].

Given the success of the above equations and the insight from the previous section, it is proposed that filamentation onset distance can be obtained from the use of Eqs. (8) and (9), with the appropriate form of Eq. (6).

4. Comparisons between theory and computer simulation results for the filamentation self-focusing collapse distances in lossy, turbulent media

Figure 7 is a scatter plot comparing predictions made from Eq. (9), using Eq. (6), ${\ell _0}\, = \,1\,mm$ and ${m_0} = 0.006$, with the computer simulation results in Figs. 4, 5, and 6. (The solid black line in this figure and others to come is the data trendline.) Reasonable agreement between the analytical equation predictions and HELCAP-derived data is shown in this figure. It shows that the collapse distance essentially is set primarily by the lens focal length and turbulence level once the zero-turbulence collapse distance is set by the extinction loss. This allows us to use the equation set as proposed in the last section. Let make more comparisons.

Fig. 7. Comparison of Median Collapse Distances from Eqs. (6), (8) [LHS] and (9), and HELCAP Computer Simulation Results.

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Figures 8 and 9 depict the same graphs as Figs. 4 and 5, respectively, except the lens focal lengths are set to the values that create zero turbulence collapse ranges of 3.608 km, and 3.807 km for $\alpha \, = \,0.0\,k{m^{ - 1}}$ and $0.1\,k{m^{ - 1}}$, respectively. These results again come from HELCAP computer simulations. The focal lengths chosen for the three extinction losses are 4840 m, 4400 m and 3950 m, respectively, to yield the previously cited collapse distances in zero turbulence.

Fig. 8. Filamentation Onset Distance Probability Density Function as a function of propagation distance and various values of the Refractive Index Structure Parameter and a volume Extinction Coefficient of 0.0 inverse-kilometers and a Zero Turbulence Collapse Distance of 3.6 km.

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Fig. 9. Filamentation Onset Distance Probability Density Function as a function of propagation distance and various values of the Refractive Index Structure Parameter and a volume Extinction Coefficient of 0.1 inverse-kilometers and a Zero Turbulence Collapse Distance of 3.8 km.

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Figure 10 is a scatter plot comparing predictions made from Eqs. (6), (8) [LHS] and (9), with ${\ell _0}\, = \,1\,mm$ and ${m_0} = 0.006$, and the computer simulation results in Figs. 6, 8 and 9. Reasonable agreement again is shown.

Fig. 10. Comparison of Median Collapse Distances from Eqs. (6), (8) [LHS] and (9), and HELCAP Computer Simulation Results

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In the HELCAP computer simulations, the laser beam is assumed to have a Gaussian beam profile [14]. Some newer laser transmitter systems being developed have Super Gaussian spatial-profiled laser beams. Specifically, they will have a Gaussian-like spatial profile of the form

(10)$$I(r )= {I_{peak}}\,{e^{ - 2\,{{({{{{r^2}} \mathord{\left/ {\vphantom {{{r^2}} {W_0^2}}} \right.} {W_0^2}}} )}^N}}}$$

with $N$ being an integer. Fig. 11 shows a comparison of a $30\,cm$- diameter hard aperture (“flat top”) profile with a Super Gaussian beam profile with $N = 6$ and ${W_0}\, \approx \,14\,cm.$ In this paper, independent computer simulations using the Wave Train software are provided assuming a ${D_{ap}}\, = \,28.4\,cm$ flat top rather than a Gaussian radial profile. In these simulations, the hard aperture is translated into a Super Gaussian profile assuming $N = 6$ and

(11)$${W_0}\, = \,{{{D_{ap}}} \mathord{\left/ {\vphantom {{{D_{ap}}} {{2^{{3 \mathord{\left/ {\vphantom {3 2}} \right.} 2}}}}}} \right.} {{2^{{3 \mathord{\left/ {\vphantom {3 2}} \right.} 2}}}}}$$

[27]. Let’s review the Wave Train approach to beam propagation characterization.

Fig. 11. Comparison of Hard Aperture (red) versus Super Gaussian Beam (blue) Profiles.

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MZA developed Wave Train to model beam propagation characteristics like HELCAP does but uses a modular component approach to computer simulation. Fig. 12 shows a typical component layout of the key system and atmospheric entities in Wave Train to model propagation in turbulent, non-linear atmospheres. Comparisons between HELCAP and Wave Train computer simulations have shown similar results. Filamentation onset distance determination is the same as done with HELCAP results. Let now look at Wave Train computer for the above “super Gaussian” beam profile.

Fig. 12. Wave Train component layout for modeling propagation in turbulent, non-linear atmospheres.

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Figure 13 depicts the PDFs for various ratios of ${P_{Peak}}/{P_{crit}}$ and their associated lens focal lengths, designated by $F,$ with the refractive index structure parameter $C_n^2$ set to $1x{10^{ - 17}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}.$ Fig. 14 illustrates the PDFs for the same ratios of ${P_{Peak}}/{P_{crit}}$ and focal lengths, but where the refractive index structure parameter $C_n^2$ now is set to $1x{10^{ - 16}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}$. The laser wavelength is $\lambda = 1\mu m$ and $\alpha = 0.0\,k{m^{ - 1}}$ in both cases. These plots represent 1000 Wave-Optics realizations in Wave Train. Table 1 provides the average filamentation self-focusing collapse distances for these PDFs.

Fig. 13. Graph for the Filamentation Onset Distance Probability Density Function as a function of propagation distance for various ratios of Peak Power to Critical Power for $C_n^2 = 1x{10^{ - 17}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}.$

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Fig. 14. Graph for the Filamentation Onset Distance Probability Density Function as a function of propagation distance for various ratios of Peak Power to Critical Power for $C_n^2 = 1x{10^{ - 16}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}.$

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Table 1. Average filamentation self-focusing collapse distances from the plots in Figs. 13 and 14.

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Figure 15 is a scatter plot comparing the collapse distances results using Eqs. (6), (8) and (9), and the average collapse distances given in Table 1. The agreement is good, except for the ∼10% deviation for ${P_{Peak}}/{P_{crit}}\, = \,4.07$ at $C_n^2 = 1x{10^{ - 16}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}$. This deviation is slightly larger than the statistical error of the calculation, but it exhibits a reasonable error for an engineering equation.

Fig. 15. Comparison of Average Collapse Distances from Eqs. (6), (8) [LHS] and (9), and Wave Train Computer Simulation Results

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Figures 16 and 17 are plots for collapse distance PDFs as a function of propagation distance and two sets of refractive index structure parameters and ${P_{Peak}}/{P_{crit}}$, respectively, with $\alpha \, = \,0.1\,k{m^{ - 1}}.$ The values of ${P_{Peak}}/{P_{crit}}$ in the former figure are 5 and 10, while the values of ${P_{Peak}}/{P_{crit}}$ in the latter figure are 1.5 and 8. The values of $C_n^2$ are $1x{10^{ - 17}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}},$ $1x{10^{ - 16}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}},$ $1x{10^{ - 15}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}$ and $1x{10^{ - 14}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}$ in Fig. 16 and $1x{10^{ - 17}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}},$ $1x{10^{ - 15}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}},$ $1x{10^{ - 14}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}$ and $1x{10^{ - 13}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}$ in Fig. 16. In these plots, $\lambda = 1.53\mu m$ and ${D_{ap}} = \,30\,cm$. These results come from Wave Train computer simulations. The transmitter lens focal lengths chosen for the single extinction losses to cause the collapse distance in zero turbulence is 4 km and 1 km, respectively. Table 2 provides the average filamentation self-focusing collapse distances for these PDFs.

Fig. 16. Filamentation Onset Distance Probability Density Function as a function of propagation distance and various values of the Refractive Index Structure Parameter and a volume Extinction Coefficient of 0.1 inverse-kilometers and a Zero Turbulence Collapse Distance of 4 km and Peak Power to Critical Power Ratios of 5 and 10.

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Fig. 17. Filamentation Onset Distance Probability Density Function as a function of propagation distance and various values of the Refractive Index Structure Parameter and a volume Extinction Coefficient of 0.1 inverse-kilometers and a Zero Turbulence Collapse Distance of 1 km and Peak Power to Critical Power Ratios of 1.5 and 8.

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Table 2. Average filamentation self-focusing collapse distances from the plots in Figs. 16 and 17.

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Figure 18 is a scatter plot comparing predictions made from Eqs. (6) and (8), with ${\ell _0}\, = \,1\,mm$ and ${m_0} = 0.003$, and the computer simulation results in Figs. 16 and 17. Fig. 19 replots the comparison using only the Fig. 17 data only. The change in ${m_0}$ comes from the use of a “flat top” rather than the Gaussian beam profile used in the HELCAP computer simulations. Again, reasonable agreement is depicted between the analytical equation predictions and Wave Train-derived data.

Fig. 18. Comparison of Average Collapse Distances from Eqs. (6) and (8) [LHS], and Wave Train Computer Simulation Results using Figs. 16 and 17 Data.

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Fig. 19. Comparison of Average Collapse Distances from Eqs. (6) and (8) [LHS], and Wave Train Computer Simulation Results using Fig. 17 Data Only.

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5. Summary

This paper provided an analytical approach for predicting the filamentation self-focusing collapse distance created by a Long Pulse USPL beam propagating through lossy, turbulent, nonlinear media. One first calculates the effect of turbulence on the USPL Peak Laser Power using the equation

(12)$$P_{peak}^\ast /{P_{crit}} = ({{P_{Peak}}/{P_{crit}}} )[{1 + {m_0}{k^2}a_0^2{{({{{{a_0}C} \mathord{\left/ {\vphantom {{{a_0}C} {{\ell_0}}}} \right.} {{\ell_0}}}} )}^{{2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}} ], $$

where

(13)$$C = 4.38\, \ell _0^{ - {1 \mathord{\left/ {\vphantom {1 3}} \right.} 3}}C_n^2(h )[{1 - {{\{{1 + 17.5{{({{{{a_0}} \mathord{\left/ {\vphantom {{{a_0}} {{\ell_0}}}} \right.} {{\ell_0}}}} )}^2}} \}}^{ - {1 \mathord{\left/ {\vphantom {1 6}} \right.} 6}}}} ], $$

The result of Eq. (12) then is substituted into the following equation

(14)$$z_{sf}^\ast \, = \,\frac{{0.367{z_r}}}{{\sqrt {{{\left( {\sqrt {{{P_{peak}^\ast } \mathord{\left/ {\vphantom {{P_{peak}^\ast } {{P_{crit}}}}} \right.} {{P_{crit}}}}} - 0.852} \right)}^2} - 0.0219} }},$$

to yield an estimate of the nonlinear medium’s Marburger distance. To account for the effect of extinction loss on the Marburger distance, the result of Eq. (14) is inserted into the loss equation

(15)$${a^{16}}(z )= a_L^{16}(z )\left\{ {1 - \left( {\frac{2}{{{\alpha^2}z{{_{sf}^{\ast \ast }}^2}}}} \right)[{1 - ({1 + \alpha z} )\exp \{{ - \alpha z} \}} ]} \right\}$$

and then solved for value of z that makes the equation within the curly brackets zero. This value of z is the filamentation onset distance for the system, turbulence and extinction loss parameters involved. In Eq. (15), we have

(16)$$z_{sf}^{\ast \ast } = \left\{ {\begin{array}{{c}} {z_{sf}^\ast \quad \quad \quad \quad \quad ;without\,a\,transmitter\,lens}\\ {{{z_{sf}^\ast \,f} \mathord{\left/ {\vphantom {{z_{sf}^\ast \,f} {({z_{sf}^\ast + f} )\quad \quad ;with\,a\,transmitter\,lens}}} \right.} {({z_{sf}^\ast + f} )\quad \quad ;with\,a\,transmitter\,lens}}} \end{array}} \right..$$

All parameters in the above equations are defined in the paper. Approach validation was done against two independent sets of computer simulation results. One comes from the NRL HELCAP software and the other from the MZA’s Wave Train modeling software package. This approach leveraged the fact that once the zero-turbulence onset distance is set based on link loss, the addition of turbulence creates essentially the same PDFs at similar median distances for each loss case. This result had not been previously reported. These results may provide a guide for further numerical simulations, further theoretical developments, application assessments and field experiments.

Funding

Defense Advanced Research Projects Agency.

Acknowledgments

The authors would like to thank Dr. Antonio Oliver for his comments and suggestions regarding the contents of this paper. 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 the Department of Defense.

References

1. S. Tochitsky, E. Welch, M. Polyanskiy, I. Pogorelsky, P. Panagiotopoulos, M. Kolesik, E. M. Wright, S. W. Koch, J. V. Moloney, J. Pigeon, and C. Joshi, “Megafilament in air formed by self-guided terawatt long-wavelength infrared laser,” Nat. Photonics 13(1), 41–46 (2019). [CrossRef]

2. J. Peñano, J. P. Palastro, B. Hafizi, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017). [CrossRef]

3. D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016). [CrossRef]

4. C. Jeon, J. Lane, S. Rostami, L. Shah, M. Baudelet, and M. Richardson, “Laser Induced Filament Propagation Through Adverse Conditions,” in Propagation Through and Characterization of Atmospheric and Oceanic Phenomena, OSA Technical Digest (online) (Optical Society of America, 2016), paper Tu2A.3 (2016).

5. A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016). [CrossRef]

6. A. V. Mitrofanov, A. A. Voroninl, D. A. Sidorov-Biryukov, A. Pugžlys, E. A. Stepanov, G. Andriukaitis, T. Flöry, S. Ališauskas, A. B. Fedotov, A. Baltuška, and A. M. Zheltikov, “Mid-infrared laser filaments in the atmosphere,” J. Sci. Res. Rev. 5(1), 1–5 (2015). [CrossRef]

7. M. Durand, A. Houard, B. Prade, A. Mysyrowicz, A. Durécu, B. Moreau, D. Fleury, O. Vasseur, H. Borchert, K. Diener, R. Schmitt, F. Théberge, M. Chateauneuf, J. Daigle, and J. Dubois, “Kilometer range filamentation,” Opt. Express 21, 26836–26845 (2013). [CrossRef]

8. A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78(3), 033804 (2008). [CrossRef]

9. S. A. Shlenov, V. P. Kandidov, O. G. Kosareva, A. E. Bezborodov, and V. Yu. Fedorov, “Spatio-temporal control of femtosecond laser pulse filamentation in the atmosphere,” Proc. SPIE 6733, International Conference on Lasers, Applications, and Technologies 2007: Environmental Monitoring and Ecological Applications; Optical Sensors in Biological, Chemical, and Engineering Technologies; and Femtosecond Laser Pulse Filamentation, 67332M (2007).

10. S. A. Shlenov and A. I. Markov, “Control of filamentation of femtosecond laser pulses in a turbulent atmosphere,” Quantum Electron. 39(7), 658–662 (2009). [CrossRef]

11. R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006). [CrossRef]

12. L. B. Stotts, J. Peñano, and V. Urick, “Engineering equation for filamentation self-focusing collapse distance in atmospheric turbulence,” Opt. Express 27(11), 15159–15171 (2019). [CrossRef]

13. T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Engineering equations for characterizing non-linear laser intensity propagation in air with loss,” Opt. Express 26(4), 3974–3987 (2018) [CrossRef] ; Thomas Karr, Larry B. Stotts, Jason A. Tellez, Jason D. Schmidt, and Justin D. Mansell, “Engineering equations for characterizing nonlinear laser intensity propagation in air with loss: erratum,” Opt. Express, 26(7), 8417 (2018).

14. T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Propagation of infrared ultrashort pulses in the air,” Proc. SPIE 10684, 1068414 (2018). [CrossRef]

15. J. H. Marburger, “Self-Focusing Theory,” in R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds., Self-focusing: Past and Present / Fundamentals and Prospects (Topics in Applied Optics, Springer Science + Business Media, 1975) Chap. 2.

16. V. Petrishchev, “Application of the Method of Moments to Certain Problems in the Propagation of Partially Coherent Light Beams,” Radiophys. Quantum Electron. 14(9), 1112–1119 (1971). [CrossRef]

17. P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002). [CrossRef]

18. C. Wu, J. R. Rzasa, J. Ko, D. A. Paulson, J. Coffaro, J. Spychalsky, R. F. Crabbs, and C. C. Davis, “A multi-aperture laser transmissometer system for long-path aerosol extinction rate measurement,” Appl. Opt. 57(3), 551–559 (2018). [CrossRef]

19. P. Spangle, J. R. Pefiano, and B. Hafizi, “Optimum Wavelength and Power for Efficient Laser Propagation in Various Atmospheric Environments,” NRL/MR/6790-05-8907.

20. P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002). [CrossRef]

21. J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004). [CrossRef]

22. J. Peñano, B. Hafizi, A. Ting, and M. H. Helle, “Theoretical and numerical investigation of Filament onset distance in atmospheric turbulence,” J. Opt. Soc. Am. B 31(5), 963–971 (2014). [CrossRef]

23. A. Houard, V. Jukna, G. Point, Y.-B. André, S. Klingebiel, M. Schultze, K. Michel, T. Metzger, and A. Mysyrowicz, “Study of filamentation with a high-power high repetition rate ps laser at 1.03 µm,” Opt. Express 24(7), 7437–7448 (2016). [CrossRef]

24. A. A. Zemlyanov and Y. É. Geints, “Evolution of Effective Characteristics of Laser Beam of Femtosecond Duration upon Self-Action in a Gas Medium,” Opt. Spectrosc. 104(5), 772–783 (2008). [CrossRef]

25. V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

26. E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179(3), 862–868 (1969). [CrossRef]

27. L. C. Andrews, A Field Guide to Atmospheric Optics, 2^nd Edition, 17 (SPIE Press, 2019).

References

View by:
Article Order
|
Year
|
Author
|
Publication

S. Tochitsky, E. Welch, M. Polyanskiy, I. Pogorelsky, P. Panagiotopoulos, M. Kolesik, E. M. Wright, S. W. Koch, J. V. Moloney, J. Pigeon, and C. Joshi, “Megafilament in air formed by self-guided terawatt long-wavelength infrared laser,” Nat. Photonics 13(1), 41–46 (2019).
[Crossref]
J. Peñano, J. P. Palastro, B. Hafizi, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017).
[Crossref]
D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]
C. Jeon, J. Lane, S. Rostami, L. Shah, M. Baudelet, and M. Richardson, “Laser Induced Filament Propagation Through Adverse Conditions,” in Propagation Through and Characterization of Atmospheric and Oceanic Phenomena, OSA Technical Digest (online) (Optical Society of America, 2016), paper Tu2A.3 (2016).
A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]
A. V. Mitrofanov, A. A. Voroninl, D. A. Sidorov-Biryukov, A. Pugžlys, E. A. Stepanov, G. Andriukaitis, T. Flöry, S. Ališauskas, A. B. Fedotov, A. Baltuška, and A. M. Zheltikov, “Mid-infrared laser filaments in the atmosphere,” J. Sci. Res. Rev. 5(1), 1–5 (2015).
[Crossref]
M. Durand, A. Houard, B. Prade, A. Mysyrowicz, A. Durécu, B. Moreau, D. Fleury, O. Vasseur, H. Borchert, K. Diener, R. Schmitt, F. Théberge, M. Chateauneuf, J. Daigle, and J. Dubois, “Kilometer range filamentation,” Opt. Express 21, 26836–26845 (2013).
[Crossref]
A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78(3), 033804 (2008).
[Crossref]
S. A. Shlenov, V. P. Kandidov, O. G. Kosareva, A. E. Bezborodov, and V. Yu. Fedorov, “Spatio-temporal control of femtosecond laser pulse filamentation in the atmosphere,” Proc. SPIE 6733, International Conference on Lasers, Applications, and Technologies 2007: Environmental Monitoring and Ecological Applications; Optical Sensors in Biological, Chemical, and Engineering Technologies; and Femtosecond Laser Pulse Filamentation, 67332M (2007).
S. A. Shlenov and A. I. Markov, “Control of filamentation of femtosecond laser pulses in a turbulent atmosphere,” Quantum Electron. 39(7), 658–662 (2009).
[Crossref]
R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006).
[Crossref]
L. B. Stotts, J. Peñano, and V. Urick, “Engineering equation for filamentation self-focusing collapse distance in atmospheric turbulence,” Opt. Express 27(11), 15159–15171 (2019).
[Crossref]
T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Engineering equations for characterizing non-linear laser intensity propagation in air with loss,” Opt. Express 26(4), 3974–3987 (2018); Thomas Karr, Larry B. Stotts, Jason A. Tellez, Jason D. Schmidt, and Justin D. Mansell, “Engineering equations for characterizing nonlinear laser intensity propagation in air with loss: erratum,” Opt. Express, 26(7), 8417 (2018).
[Crossref]
T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Propagation of infrared ultrashort pulses in the air,” Proc. SPIE 10684, 1068414 (2018).
[Crossref]
J. H. Marburger, “Self-Focusing Theory,” in R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds., Self-focusing: Past and Present / Fundamentals and Prospects (Topics in Applied Optics, Springer Science + Business Media, 1975) Chap. 2.
V. Petrishchev, “Application of the Method of Moments to Certain Problems in the Propagation of Partially Coherent Light Beams,” Radiophys. Quantum Electron. 14(9), 1112–1119 (1971).
[Crossref]
P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002).
[Crossref]
C. Wu, J. R. Rzasa, J. Ko, D. A. Paulson, J. Coffaro, J. Spychalsky, R. F. Crabbs, and C. C. Davis, “A multi-aperture laser transmissometer system for long-path aerosol extinction rate measurement,” Appl. Opt. 57(3), 551–559 (2018).
[Crossref]
P. Spangle, J. R. Pefiano, and B. Hafizi, “Optimum Wavelength and Power for Efficient Laser Propagation in Various Atmospheric Environments,” NRL/MR/6790-05-8907.
P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002).
[Crossref]
J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004).
[Crossref]
J. Peñano, B. Hafizi, A. Ting, and M. H. Helle, “Theoretical and numerical investigation of Filament onset distance in atmospheric turbulence,” J. Opt. Soc. Am. B 31(5), 963–971 (2014).
[Crossref]
A. Houard, V. Jukna, G. Point, Y.-B. André, S. Klingebiel, M. Schultze, K. Michel, T. Metzger, and A. Mysyrowicz, “Study of filamentation with a high-power high repetition rate ps laser at 1.03 µm,” Opt. Express 24(7), 7437–7448 (2016).
[Crossref]
A. A. Zemlyanov and Y. É. Geints, “Evolution of Effective Characteristics of Laser Beam of Femtosecond Duration upon Self-Action in a Gas Medium,” Opt. Spectrosc. 104(5), 772–783 (2008).
[Crossref]
V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).
E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179(3), 862–868 (1969).
[Crossref]
L. C. Andrews, A Field Guide to Atmospheric Optics, 2nd Edition, 17 (SPIE Press, 2019).

2019 (2)

S. Tochitsky, E. Welch, M. Polyanskiy, I. Pogorelsky, P. Panagiotopoulos, M. Kolesik, E. M. Wright, S. W. Koch, J. V. Moloney, J. Pigeon, and C. Joshi, “Megafilament in air formed by self-guided terawatt long-wavelength infrared laser,” Nat. Photonics 13(1), 41–46 (2019).
[Crossref]

L. B. Stotts, J. Peñano, and V. Urick, “Engineering equation for filamentation self-focusing collapse distance in atmospheric turbulence,” Opt. Express 27(11), 15159–15171 (2019).
[Crossref]

2018 (3)

T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Engineering equations for characterizing non-linear laser intensity propagation in air with loss,” Opt. Express 26(4), 3974–3987 (2018); Thomas Karr, Larry B. Stotts, Jason A. Tellez, Jason D. Schmidt, and Justin D. Mansell, “Engineering equations for characterizing nonlinear laser intensity propagation in air with loss: erratum,” Opt. Express, 26(7), 8417 (2018).
[Crossref]

T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Propagation of infrared ultrashort pulses in the air,” Proc. SPIE 10684, 1068414 (2018).
[Crossref]

C. Wu, J. R. Rzasa, J. Ko, D. A. Paulson, J. Coffaro, J. Spychalsky, R. F. Crabbs, and C. C. Davis, “A multi-aperture laser transmissometer system for long-path aerosol extinction rate measurement,” Appl. Opt. 57(3), 551–559 (2018).
[Crossref]

2017 (1)

J. Peñano, J. P. Palastro, B. Hafizi, M. H. Helle, and G. P. DiComo, “Self-channeling of high-power laser pulses through strong atmospheric turbulence,” Phys. Rev. A 96(1), 013829 (2017).
[Crossref]

2016 (3)

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

A. Houard, V. Jukna, G. Point, Y.-B. André, S. Klingebiel, M. Schultze, K. Michel, T. Metzger, and A. Mysyrowicz, “Study of filamentation with a high-power high repetition rate ps laser at 1.03 µm,” Opt. Express 24(7), 7437–7448 (2016).
[Crossref]

2015 (1)

A. V. Mitrofanov, A. A. Voroninl, D. A. Sidorov-Biryukov, A. Pugžlys, E. A. Stepanov, G. Andriukaitis, T. Flöry, S. Ališauskas, A. B. Fedotov, A. Baltuška, and A. M. Zheltikov, “Mid-infrared laser filaments in the atmosphere,” J. Sci. Res. Rev. 5(1), 1–5 (2015).
[Crossref]

2014 (1)

J. Peñano, B. Hafizi, A. Ting, and M. H. Helle, “Theoretical and numerical investigation of Filament onset distance in atmospheric turbulence,” J. Opt. Soc. Am. B 31(5), 963–971 (2014).
[Crossref]

2013 (1)

M. Durand, A. Houard, B. Prade, A. Mysyrowicz, A. Durécu, B. Moreau, D. Fleury, O. Vasseur, H. Borchert, K. Diener, R. Schmitt, F. Théberge, M. Chateauneuf, J. Daigle, and J. Dubois, “Kilometer range filamentation,” Opt. Express 21, 26836–26845 (2013).
[Crossref]

2009 (1)

S. A. Shlenov and A. I. Markov, “Control of filamentation of femtosecond laser pulses in a turbulent atmosphere,” Quantum Electron. 39(7), 658–662 (2009).
[Crossref]

2008 (2)

A. A. Zemlyanov and Y. É. Geints, “Evolution of Effective Characteristics of Laser Beam of Femtosecond Duration upon Self-Action in a Gas Medium,” Opt. Spectrosc. 104(5), 772–783 (2008).
[Crossref]

A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78(3), 033804 (2008).
[Crossref]

2006 (1)

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006).
[Crossref]

2004 (1)

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004).
[Crossref]

2002 (2)

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002).
[Crossref]

1971 (1)

V. Petrishchev, “Application of the Method of Moments to Certain Problems in the Propagation of Partially Coherent Light Beams,” Radiophys. Quantum Electron. 14(9), 1112–1119 (1971).
[Crossref]

1970 (1)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

1969 (1)

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179(3), 862–868 (1969).
[Crossref]

Ackermann, R.

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006).
[Crossref]

Ališauskas, S.

André, Y.-B.

Andrews, L. C.

L. C. Andrews, A Field Guide to Atmospheric Optics, 2nd Edition, 17 (SPIE Press, 2019).

Andriukaitis, G.

Baltuška, A.

Baudelet, M.

C. Jeon, J. Lane, S. Rostami, L. Shah, M. Baudelet, and M. Richardson, “Laser Induced Filament Propagation Through Adverse Conditions,” in Propagation Through and Characterization of Atmospheric and Oceanic Phenomena, OSA Technical Digest (online) (Optical Society of America, 2016), paper Tu2A.3 (2016).

Bergé, L.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Berti, N.

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

Bezborodov, A. E.

S. A. Shlenov, V. P. Kandidov, O. G. Kosareva, A. E. Bezborodov, and V. Yu. Fedorov, “Spatio-temporal control of femtosecond laser pulse filamentation in the atmosphere,” Proc. SPIE 6733, International Conference on Lasers, Applications, and Technologies 2007: Environmental Monitoring and Ecological Applications; Optical Sensors in Biological, Chemical, and Engineering Technologies; and Femtosecond Laser Pulse Filamentation, 67332M (2007).

Borchert, H.

Bourdon, P.

Brunetti, M.

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

Chateauneuf, M.

Coffaro, J.

Couairon, A.

Crabbs, R. F.

Daigle, J.

Davis, C. C.

Dawes, E. L.

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179(3), 862–868 (1969).
[Crossref]

DiComo, G. P.

Diener, K.

Dubois, J.

Durand, M.

Durécu, A.

Eeltink, D.

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

Fedorov, V. Yu.

Fedotov, A. B.

Fleury, B.

Fleury, D.

Flöry, T.

Franco, M.

Gateau, J.

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

Geints, Y. É.

Gordon, D. F.

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004).
[Crossref]

Hafizi, B.

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004).
[Crossref]

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002).
[Crossref]

P. Spangle, J. R. Pefiano, and B. Hafizi, “Optimum Wavelength and Power for Efficient Laser Propagation in Various Atmospheric Environments,” NRL/MR/6790-05-8907.

Helle, M. H.

Hermelin, S.

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

Houard, A.

Jeon, C.

Joshi, C.

Jukna, V.

Kandidov, V. P.

Kapetanakos, C. A.

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004).
[Crossref]

Karr, T.

T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Propagation of infrared ultrashort pulses in the air,” Proc. SPIE 10684, 1068414 (2018).
[Crossref]

Kasparian, J.

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006).
[Crossref]

Klingebiel, S.

Ko, J.

Koch, S. W.

Kolesik, M.

Kosareva, O. G.

Kurz, H. G.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Lane, J.

Lombard, L.

Mansell, J. D.

T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Propagation of infrared ultrashort pulses in the air,” Proc. SPIE 10684, 1068414 (2018).
[Crossref]

Marburger, J. H.

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179(3), 862–868 (1969).
[Crossref]

J. H. Marburger, “Self-Focusing Theory,” in R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds., Self-focusing: Past and Present / Fundamentals and Prospects (Topics in Applied Optics, Springer Science + Business Media, 1975) Chap. 2.

Marchiando, N.

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

Markov, A. I.

S. A. Shlenov and A. I. Markov, “Control of filamentation of femtosecond laser pulses in a turbulent atmosphere,” Quantum Electron. 39(7), 658–662 (2009).
[Crossref]

Méjean, G.

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006).
[Crossref]

Metzger, T.

Michau, V.

Michel, K.

Mitrofanov, A. V.

Moloney, J. V.

Moreau, B.

Mysyrowicz, A.

Palastro, J. P.

Panagiotopoulos, P.

Paulson, D. A.

Pefiano, J. R.

P. Spangle, J. R. Pefiano, and B. Hafizi, “Optimum Wavelength and Power for Efficient Laser Propagation in Various Atmospheric Environments,” NRL/MR/6790-05-8907.

Peñano, J.

L. B. Stotts, J. Peñano, and V. Urick, “Engineering equation for filamentation self-focusing collapse distance in atmospheric turbulence,” Opt. Express 27(11), 15159–15171 (2019).
[Crossref]

Peñano, J. R.

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004).
[Crossref]

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002).
[Crossref]

Petrishchev, V.

V. Petrishchev, “Application of the Method of Moments to Certain Problems in the Propagation of Partially Coherent Light Beams,” Radiophys. Quantum Electron. 14(9), 1112–1119 (1971).
[Crossref]

Pigeon, J.

Pogorelsky, I.

Point, G.

Polyanskiy, M.

Polynkin, P.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Prade, B.

Pugžlys, A.

Richardson, M.

Robert, C.

Rostami, S.

Rzasa, J. R.

Salmon, E.

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006).
[Crossref]

Schmidt, J. D.

T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Propagation of infrared ultrashort pulses in the air,” Proc. SPIE 10684, 1068414 (2018).
[Crossref]

Schmitt, R.

Schmitt-Sody, A.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Schultze, M.

Shah, L.

Shlenov, S. A.

S. A. Shlenov and A. I. Markov, “Control of filamentation of femtosecond laser pulses in a turbulent atmosphere,” Quantum Electron. 39(7), 658–662 (2009).
[Crossref]

Sidorov-Biryukov, D. A.

Skupin, S.

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Spangle, P.

P. Spangle, J. R. Pefiano, and B. Hafizi, “Optimum Wavelength and Power for Efficient Laser Propagation in Various Atmospheric Environments,” NRL/MR/6790-05-8907.

Sprangle, P.

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004).
[Crossref]

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002).
[Crossref]

Spychalsky, J.

Stepanov, E. A.

Stotts, L. B.

L. B. Stotts, J. Peñano, and V. Urick, “Engineering equation for filamentation self-focusing collapse distance in atmospheric turbulence,” Opt. Express 27(11), 15159–15171 (2019).
[Crossref]

T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Propagation of infrared ultrashort pulses in the air,” Proc. SPIE 10684, 1068414 (2018).
[Crossref]

Talanov, V. I.

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

Tellez, J. A.

T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Propagation of infrared ultrashort pulses in the air,” Proc. SPIE 10684, 1068414 (2018).
[Crossref]

Théberge, F.

Ting, A.

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004).
[Crossref]

Tochitsky, S.

Urick, V.

L. B. Stotts, J. Peñano, and V. Urick, “Engineering equation for filamentation self-focusing collapse distance in atmospheric turbulence,” Opt. Express 27(11), 15159–15171 (2019).
[Crossref]

Vasseur, O.

Voroninl, A. A.

Welch, E.

Wolf, J. P.

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006).
[Crossref]

Wright, E. M.

Wu, C.

Yu, J.

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006).
[Crossref]

Zemlyanov, A. A.

Zheltikov, A. M.

Appl. Opt. (1)

J. Opt. Soc. Am. B (1)

J. Sci. Res. Rev. (1)

JETP Lett. (1)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

Nat. Photonics (1)

New J. Phys. (1)

A. Schmitt-Sody, H. G. Kurz, L. Bergé, S. Skupin, and P. Polynkin, “Picosecond laser filamentation in air,” New J. Phys. 18(9), 093005 (2016).
[Crossref]

Opt. Express (4)

L. B. Stotts, J. Peñano, and V. Urick, “Engineering equation for filamentation self-focusing collapse distance in atmospheric turbulence,” Opt. Express 27(11), 15159–15171 (2019).
[Crossref]

Opt. Lett. (1)

R. Ackermann, G. Méjean, J. Kasparian, J. Yu, E. Salmon, and J. P. Wolf, “Laser filaments generated and transmitted in highly turbulent air,” Opt. Lett. 31(1), 86–88 (2006).
[Crossref]

Opt. Spectrosc. (1)

Phys. Plasmas (1)

J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11(5), 2865–2874 (2004).
[Crossref]

Phys. Rev. (1)

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179(3), 862–868 (1969).
[Crossref]

Phys. Rev. A (3)

D. Eeltink, N. Berti, N. Marchiando, S. Hermelin, J. Gateau, M. Brunetti, J. P. Wolf, and J. Kasparian, “Triggering filamentation using turbulence,” Phys. Rev. A 94(3), 033806 (2016).
[Crossref]

Phys. Rev. E (2)

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002).
[Crossref]

Proc. SPIE (1)

T. Karr, L. B. Stotts, J. A. Tellez, J. D. Schmidt, and J. D. Mansell, “Propagation of infrared ultrashort pulses in the air,” Proc. SPIE 10684, 1068414 (2018).
[Crossref]

Quantum Electron. (1)

S. A. Shlenov and A. I. Markov, “Control of filamentation of femtosecond laser pulses in a turbulent atmosphere,” Quantum Electron. 39(7), 658–662 (2009).
[Crossref]

Radiophys. Quantum Electron. (1)

V. Petrishchev, “Application of the Method of Moments to Certain Problems in the Propagation of Partially Coherent Light Beams,” Radiophys. Quantum Electron. 14(9), 1112–1119 (1971).
[Crossref]

Other (5)

P. Spangle, J. R. Pefiano, and B. Hafizi, “Optimum Wavelength and Power for Efficient Laser Propagation in Various Atmospheric Environments,” NRL/MR/6790-05-8907.

L. C. Andrews, A Field Guide to Atmospheric Optics, 2nd Edition, 17 (SPIE Press, 2019).

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Figures (19)

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14.
Fig. 15.
Fig. 16.
Fig. 17.
Fig. 18.
Fig. 19.

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Fig. 3. Fluence contours generated by the HELCAP code for beams with (a)

${P \mathord{\left/ {\vphantom {P {{P_{crit}} = 1.5}}} \right.} {{P_{crit}} = 1.5}}$

and b)

${P \mathord{\left/ {\vphantom {P {{P_{crit}} = 20}}} \right.} {{P_{crit}} = 20}}$

near the onset of filamentation. Taken from Ref. [22].

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Fig. 7. Comparison of Median Collapse Distances from Eqs. (6), (8) [LHS] and (9), and HELCAP Computer Simulation Results.

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Fig. 10. Comparison of Median Collapse Distances from Eqs. (6), (8) [LHS] and (9), and HELCAP Computer Simulation Results

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Fig. 11. Comparison of Hard Aperture (red) versus Super Gaussian Beam (blue) Profiles.

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Fig. 12. Wave Train component layout for modeling propagation in turbulent, non-linear atmospheres.

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Fig. 13. Graph for the Filamentation Onset Distance Probability Density Function as a function of propagation distance for various ratios of Peak Power to Critical Power for

$C_n^2 = 1x{10^{ - 17}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}.$

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Fig. 14. Graph for the Filamentation Onset Distance Probability Density Function as a function of propagation distance for various ratios of Peak Power to Critical Power for

$C_n^2 = 1x{10^{ - 16}}\,{m^{ - {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}.$

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Fig. 15. Comparison of Average Collapse Distances from Eqs. (6), (8) [LHS] and (9), and Wave Train Computer Simulation Results

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Fig. 18. Comparison of Average Collapse Distances from Eqs. (6) and (8) [LHS], and Wave Train Computer Simulation Results using Figs. 16 and 17 Data.

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Fig. 19. Comparison of Average Collapse Distances from Eqs. (6) and (8) [LHS], and Wave Train Computer Simulation Results using Fig. 17 Data Only.

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Tables (2)

Table 1. Average filamentation self-focusing collapse distances from the plots in Figs. 13 and 14.

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Table 2. Average filamentation self-focusing collapse distances from the plots in Figs. 16 and 17.

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Equations (17)

Equations on this page are rendered with MathJax. Learn more.

\frac{\partial A}{\partial z} = \frac{i c}{2 ω_{0}} \nabla_{⊥}^{2} A + [i \frac{ω_{0}}{c} (δ n_{T} + δ n_{T B}) - \frac{1}{2} (α_{T B} + β)] A + \sum_{j} S_{j}

z_{s f} = \frac{0.367 z_{r}}{\sqrt{{(\sqrt{P_{p e a k} / P_{c r i t}} - 0.852)}^{2} - 0.0219}},

z_{r} = k a_{0}^{2}

z_{s f}^{'} = z_{s f} f / (z_{s f} + f)

a^{16} (z) = a_{L}^{16} (z) {1 - (\frac{2}{α^{2} z {_{s f}^{'}}^{2}}) [1 - (1 + α z) \exp {- α z}]}

P_{p e a k}^{*} / P_{c r i t} = (P_{P e a k} / P_{c r i t}) [1 + m_{0} k^{2} a_{0}^{2} {(a_{0} C / ℓ_{0})}^{2 / 3}],

C = 4.38 ℓ_{0}^{- 1 / 3} C_{n}^{2} (h) [1 - {1 + 17.5 {(a_{0} / ℓ_{0})}^{2}}^{- 1 / 6}],

z_{s f}^{*} = \frac{0.367 z_{r}}{\sqrt{{(\sqrt{P_{p e a k}^{*} / P_{c r i t}} - 0.852)}^{2} - 0.0219}},

\approx \frac{0.367 z_{r}}{(\sqrt{P_{p e a k}^{*} / P_{c r i t}} - 0.852)}

z_{s f}^{* *} = \frac{z_{s f}^{*} f}{z_{s f}^{*} + f},

I (r) = I_{p e a k} e^{- 2 {(r^{2} / W_{0}^{2})}^{N}}

W_{0} = D_{a p} / 2^{3 / 2}

P_{p e a k}^{*} / P_{c r i t} = (P_{P e a k} / P_{c r i t}) [1 + m_{0} k^{2} a_{0}^{2} {(a_{0} C / ℓ_{0})}^{2 / 3}],

C = 4.38 ℓ_{0}^{- 1 / 3} C_{n}^{2} (h) [1 - {1 + 17.5 {(a_{0} / ℓ_{0})}^{2}}^{- 1 / 6}],

z_{s f}^{*} = \frac{0.367 z_{r}}{\sqrt{{(\sqrt{P_{p e a k}^{*} / P_{c r i t}} - 0.852)}^{2} - 0.0219}},

a^{16} (z) = a_{L}^{16} (z) {1 - (\frac{2}{α^{2} z {_{s f}^{* *}}^{2}}) [1 - (1 + α z) \exp {- α z}]}

z_{s f}^{* *} = {\begin{matrix} z_{s f}^{*}; w i t h o u t a t r a n s m i t t e r l e n s \\ z_{s f}^{*} f / (z_{s f}^{*} + f); w i t h a t r a n s m i t t e r l e n s \end{matrix} .

CONDITIONS	$C_{n}^{2} = 1 x 10^{- 17} m^{- 2 / 3}$	$C_{n}^{2} = 1 x 10^{- 16} m^{- 2 / 3}$
$f = - 10 k m; 1 / f = - 0.1 k m^{- 1}$	$4.89 k m \pm 328 m$	$4.20 k m \pm 485 m$
$f = \infty k m; 1 / f = 0 k m^{- 1}$	$5.01 k m \pm 200 m$	$4.75 k m \pm 459 m$
$f = 10 k m; 1 / f = 0.1 k m^{- 1}$	$5.00 k m \pm 98 m$	$4.98 k m \pm 309 m$

$C_{n}^{2}$	$P_{p e a k} / P_{c r i t}$	$C o l l a p s e D i s t a n c e$
$1 x 10^{- 14} m^{2 / 3}$	$5$	$3510 m \pm 593.6 m$
	$10$	$2607 m \pm 405.4 m$
$1 x 10^{- 15} m^{2 / 3}$	$5$	$3968.6 m \pm 303.5 m$
	$10$	$3242.0 m \pm 205.0 m$
$1 x 10^{- 16} m^{2 / 3}$	$5$	$4005.8 m \pm 85.6 m$
	$10$	$3661.9 m \pm 118.2 m$
$1 x 10^{- 17} m^{2 / 3}$	$5$	$4001.9 m \pm 27.5 m$
	$10$	$3913.4 \pm 56.6 m$

CONDITIONS	$C_{n}^{2} = 1 x 10^{- 17} m^{- 2 / 3}$	$C_{n}^{2} = 1 x 10^{- 16} m^{- 2 / 3}$
$f = - 10 k m; 1 / f = - 0.1 k m^{- 1}$	$4.89 k m \pm 328 m$	$4.20 k m \pm 485 m$
$f = \infty k m; 1 / f = 0 k m^{- 1}$	$5.01 k m \pm 200 m$	$4.75 k m \pm 459 m$
$f = 10 k m; 1 / f = 0.1 k m^{- 1}$	$5.00 k m \pm 98 m$	$4.98 k m \pm 309 m$

$C_{n}^{2}$	$P_{p e a k} / P_{c r i t}$	$C o l l a p s e D i s t a n c e$
$1 x 10^{- 14} m^{2 / 3}$	$5$	$3510 m \pm 593.6 m$
	$10$	$2607 m \pm 405.4 m$
$1 x 10^{- 15} m^{2 / 3}$	$5$	$3968.6 m \pm 303.5 m$
	$10$	$3242.0 m \pm 205.0 m$
$1 x 10^{- 16} m^{2 / 3}$	$5$	$4005.8 m \pm 85.6 m$
	$10$	$3661.9 m \pm 118.2 m$
$1 x 10^{- 17} m^{2 / 3}$	$5$	$4001.9 m \pm 27.5 m$
	$10$	$3913.4 \pm 56.6 m$