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Probability density function estimation for filament creation in lossy, turbulent, nonlinear media

Larry B. Stotts, Antonio Oliver, and Joseph R. Peñano

Open Access Open Access

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. Previous work has focused on estimating the Filamentation Onset Distance (FOD) in real atmospheres but not its statistical variance. This paper describes two ad hoc engineering models for predicting the FOD Probability Density Function (PDF) for lossy, turbulent, nonlinear media. Specifically, these models characterize the FOD variation with turbulence. One model uses a log-normal PDF with mean and variance proportional to the Rytov Variance. The other uses a gamma PDF employing the same mean and variance equations. These two PDFs will be compared to previous computer simulation results. Both show reasonable agreement between PDF predictions and computer simulation results for long-range filamentation. In fact, both give similar results, and there is no preference given to the data comparisons presented.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Controlled, long range propagation and filamentation of high peak power laser pulses in real atmospheres has potential application to light detection and ranging [1], remote laser-induced breakdown spectroscopy [2], and triggering and guiding of high voltage discharges [3]. However, filamentation onset is not easily predicted in such a lossy, turbulent, nonlinear medium because the turbulence of real atmospheres causes a statistical distribution of onset distances for lasers pulses launched with identical beam parameters [4–13]. In recent work, ad hoc models for filamentation onset distance (FOD) have been developed and validated by computer simulation, and in some cases by experimental results [4,6], but these efforts did not attempt to predictably estimate the statistical FOD variance. Larkin et al. examined turbulent, short-range propagation of high peak power laser beams in a laboratory setting. In their experiments of beams propagated with diameters about the inner scale of turbulence, the mean value of histograms of collapse distances appeared largely unaffected by turbulence strength, though the variance of the collapse distance increased with increasing turbulence strength [5]. Stotts et al., found that once the zero-turbulence FOD is set based on link loss, the addition of turbulence creates essentially the same PDFs at the median distances for each loss case [4,6]. They also showed that turbulence and loss influence light channel radius [4,6].

This paper describes two ad hoc engineering models for predicting the PDF for lossy, turbulent, nonlinear media. Their intent is to characterize the FOD variation with turbulence. One model uses a log-normal PDF with mean and variance proportional to the Rytov variance. The other uses a gamma PDF employing the same mean and variance equations. PDFs from this model will be compared to previous computer simulation results [3].

2. Filamentation process

During the propagation of ultra-short pulse laser 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, photoionization, diffraction, defocusing and a number of nonlinear mechanisms [4–24]. The ratio of ${{{P_{peak}}} / {{P_{crit}}}}$ is called the Power Ratio (PR). 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. A graphical representation of this process for a lossless, non-turbulent, nonlinear medium is depicted in Fig. 1. Nicholson provided a nice summary of the process, and it will be synopsized below [18].

 figure: Fig. 1.

Fig. 1. High level filamentation process layout.

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After ionization in the plasma channel begins, the transverse beam profile is distinctly separated into two regions: the filament region and the light channel region. 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. The filament radius is much smaller than that of a light channel and is wavelength dependent. This plasma channel contains charge densities of the order 1016 cm−3.

The light channel region, which acts as an energy reservoir, can extend to several millimeters in diameter and carries a much lower overall intensity, about 1010 to 1011 W/cm2 in intensity However, it often contains most of the total pulse energy. The light channels have millimeter-sized diameters and have been observed to propagate through air for hundreds of meters without a significant increase in diameter [15,17]. Its radius also is wavelength dependent. In a nonlinear medium, the plasma channel will experience a reduction in the refractive index and the enclosed light will tend to move out from the beam center. At the same time, the light in the light channel will continue to self-focus, moving towards the beam center. It is the balance of these two processes that allow sustained filament propagation [15].

As discussed in the literature, these filamentation processes will be stable when propagating through adverse conditions that may be encountered in long-range air propagation [20–24]. In most situations where turbulence is present, the FOD occurs at distances less than the lossy Marburger distance, with a statistical spread about its median.

In this paper, we specifically examine the statistical nature of FOD and how it may be represented using different models. FOD is complicated by atmospheric turbulence, which is characterized by random variations in the refractive index. These variations can give rise to phase perturbations that result in a spatially and temporally distorted optical field. Distortions across the wave front of a beam can initiate localized areas of higher intensity that evolve into self-focusing centers that eventually become filaments. In the absence of turbulence, an ideal laser beam with very good beam quality will produce a single filament/optical channel located on the optical axis at a single FOD. In the presence of turbulence, the beam can break up into non-synchronized, multiple filaments, numbering up to approximately PR [12] at random (x, y)-locations across the wavefront, and at multiple FODs. In addition, the randomness of the turbulence causes FOD to be statistical in nature, by which we mean that multiple beams launched with identical initial parameterization will exhibit a distribution of onset distances after propagating through the turbulent atmosphere. It is this distribution of onset distances that we characterize utilizing PDFs in the analyses to follow. Simulation results have recently shown quantitively that stronger atmospheric turbulence gives rise to a wider distribution of onset distances [6–8]. Further, the turbulence can become strong enough to prevent filamentation completely.

3. Naval Research Laboratory's High Energy Laser Code for Atmospheric Propagation (HELCAP)

The Naval Research Laboratory (NRL) has reported computer simulation data and analysis for filamentation generation using their High Energy Laser Code for Atmospheric Propagation (HELCAP) capability for several years [6–9]. HELCAP is a fully time-dependent, three-dimensional 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

$$\frac{{\partial A}}{{\partial z}} = \frac{i}{{2{k_0}}}\nabla _ \bot ^2A + \left[ {i\frac{{{\omega_0}}}{c}({\delta {n_T} + \delta {n_{TB}} + \delta {n_{Kerr}} + \delta {n_{Plasma}}} )- \frac{1}{2}\alpha } \right]A + \sum\limits_j {{S_j}}$$
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, ${\omega _0} = {{2\pi c} / \lambda }$, $\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 $ is the volume extinction coefficient. The change in refractive index due to the Kerr nonlinearity and plasma formation is given by ${\delta _{Kerr}}\, = \,{n_2}\,I$ and ${\delta _{Plasma}}\, = \, - {{\omega _p^2} / {({2\omega_0^2} )}},$ respectively, with ${n_2}$e in is the nonlinear Kerr refractive index, $I\, = \,{{c\,{{|A |}^2}} / {({8\pi } )}}$ is the laser beam intensity, $\omega _p^2\, = \,{{4\pi \,{n_e}} / {{m_e}}}$ is the plasma frequency-squared and ${m_e}$ is the electron mass. The parameter ${n_e}$ is the plasma density and is calculated using a rate equation involving models for photoionization, and electron reattachment and recombination. The other term denoted by $\sum\limits_j {{S_j}} $ represent other physical processes that are included in the code. In writing Eq. (1) it is assumed that the propagation medium is air and the linear refractive index equals unity. 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 refractive index structure parameter $C_n^2$. Figure 2 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 [6].

 figure: Fig. 2.

Fig. 2. Fluence contours generated by the HELCAP code for beams with (a) $PR \approx 1.6$ and (b) $PR \approx 21.1$ near the onset of filamentation. Taken from Ref. [6].

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The FOD derived using HELCAP is the range at which peak intensity of the beam increases by a factor of 500. 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.

In our HELCAP simulation runs, we set $PR\, = \,10, \; \;\lambda = 0.8\mu m$ and ${W_0} = \,15\,cm$, where ${P_{crit}} = 0.948\;{P_{NL}}$, ${P_{NL}}$ is the non-linear power of the medium 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 $1{\times}{10^{ - 17}}\,{m^{ - {2 / 3}}}, 1{\times}{10^{ - 16}}\,{m^{ - {2 / 3}}}, \; \;1{\times}{10^{ - 15}}\,{m^{ - {2 / 3}}}$ and $1{\times}{10^{ - 14}}\,{m^{ - {2 / 3}}},$ and volume extinction coefficient $\alpha $ values equal $\alpha \, = \,0.0\,k{m^{ - 1}}, \; \;0.1\,k{m^{ - 1}}$ and $0.2\,k{m^{ - 1}}$. (In the case of multi-filamentation in the propagation direction, we only considered the onset of the first occurring filament in establishing the FOD.) 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 [6].

In the next section, two PDF models for the FOD will be proposed and validated.

4. FOD PDF model

For this study, it is assumed that the FOD PDF can be characterized by the lognormal distribution model, which is given by:

$${p_{\ln }}(\mu )= \frac{1}{{\mu \sqrt {2\pi \sigma _{\ln }^2} }}\exp \left\{ { - \frac{{{{({{{\log }_e}[\mu ]- {\mu_{\ln }}} )}^2}}}{{2\sigma_{\ln }^2}}} \right\}$$
where $\mu > 0,$ with $\mu $ and ${\mu _{\ln }}$ being dimensionless parameters defined as
$$\mu = {\log _e}[{{{FOD} / {\,1\,m}}} ]$$
and
$${\mu _{\ln }} = {\log _e}[{{{\overline {FOD} } / {\,1\,m}}} ],$$
respectively. The log normal variance is given by
$${\sigma _{\ln }} \approx 0.1778\sqrt {\sigma _R^2}$$
where
$$\sigma _R^2 = 1.23\,C_n^2\,{k^{{7 / 6}}}{\overline {FOD} ^{{{11} / 6}}}.$$
Eq. (6) is the Rytov Variance evaluated at the median value of the FOD PDF, $\overline {FOD} ,$ where $C_n^2$ is the refractive index structure parameter of the atmosphere, $k = {{2\pi } / \lambda }$ is the laser wavenumber and $\lambda $ is the laser wavelength.

As noted in [6], the PDF variance only directly depends on turbulence and not on atmospheric loss. The proportionality of the PDF variance to the Rytov Variance should not be surprising. Many of the key effects from turbulence are related to this parameter, e.g., long- and short-term beam radius, the various scintillation indices, aperture averaging.

An alternative PDF model to the log- normal distribution used in Eq. (2) is the Gamma distribution. The specific form of the FOD gamma PDF is hypothesized to be

$$p(FOD) = \frac{{\beta _0^\gamma {{({{FOD} / {1m}})}^{\gamma - 1}}}}{{\Gamma (\gamma )}}{e^{ - {\beta _0}({{FOD} / {1m}})}}$$
where dimensionless parameters $\gamma $ and $\beta $ are defined as
$$\gamma = {1 / {\sigma _{\ln }^2}}$$
and
$${\beta _0} = {{[{{{\overline {FOD} } / {\,1\,m}}} ]} / {\sigma _{\ln }^2}},$$
respectively. In Eq. (7), $\Gamma (x )$ is the Gamma Function.

To calculate the $\overline {FOD} ,$ we start with a modified version of the Marburger self-focusing distance

$$z_{sf}^\ast \, = \,\frac{{0.367{z_r}}}{{\sqrt {{{\left( {\sqrt {P{R^\ast }} - 0.852} \right)}^2} - 0.0219} }},$$
where
$$P{R^\ast } = PR\,\exp \{{{m_0}\,({{{2W} / {{\rho_0}}}} )} \},$$
$$\frac{{{\rho _0}}}{{2W}} = 0.35\sqrt {\frac{\Lambda }{q}} {\left[ {\frac{8}{{3({b + 0.62{\Lambda ^{{{11} / 6}}}} )}}} \right]^{{3 / 5}}},\,{\ell _0} \ll {\rho _0} \ll {L_0},$$
$${\rho _0} = \left\{ \begin{array}{lr} {\sqrt {\,\frac{3}{{1 + \Theta + {\Theta ^2} + {\Lambda ^2}}}\,} \,{{[{1.87\,C_n^2\,{k^2}\,z\,\ell_0^{ - {1 / 3}}} ]}^{ - {1 / 2}}}},& {\rho_0} \ll {\ell_0}\\ {{{\left[ {\frac{8}{{3({b + 0.62{\Lambda ^{{{11} / 6}}}} )}}} \right]}^{{3 / 5}}}\,{{[{1.46\,C_n^2\,{k^2}\,z} ]}^{ - {3 / 5}}}},& {\ell_0} \ll {\rho_0} \ll L \end{array} \right.,$$
$$W = \sqrt {\,W_0^2[{\Lambda _0^2 + \Theta _0^2} ]} ,$$
$$q = 1.56\,{({C_n^2\,{k^{{7 / 6}}}{z^{{{11} / 6}}}} )^{{6 / 5}}},$$
$$b = \left\{ {\begin{array}{cc} {{{({1 - {\Theta ^{{8 / 3}}}} )} / {[{1 - \Theta } ]}}} &{\Theta \ge 0} \\ {{{({1 + {{|\Theta |}^{{8 / 3}}}} )} / {[{1 - \Theta } ]}}} &{\Theta < 0} \end{array}}\right.,$$
$$\Theta = {{{\Theta _0}} / {({\Lambda _0^2 + \Theta _0^2} )}}, $$
and
$$\Lambda = {{{\Lambda _0}} / {({\Lambda _0^2 + \Theta _0^2} )}}$$
[25,26]. In the above equations, ${m_0} = 0.093$ is the exponential growth for Gaussian beams; z is the propagation distance; ${z_r} = {{\pi W_0^2} / \lambda }$ is the Rayleigh Range; W is the ${e^{ - 2}} - $ intensity beam radius at z for a linearly propagating Gaussian beam; ${\rho _0}$ is the spatial coherence radius for the propagated Gaussian beam; ${\ell _0}$ and ${L_0}$ are the inner and outer scales of the turbulence, respectively; $\Lambda $ and $\Theta $ are the propagated Fresnel Ratio and refraction parameter, respectively; ${\Lambda _0} = {{\lambda z} / {\pi W_0^2 = }}{z / {k{a^2}}}$ is the Fresnel Ratio; ${\Theta _0} = ({1 - {z / f}} )$ is the refraction parameter; and f is the transmitter lens focal length [1]. Equation (12) is applicable for weak to strong turbulent atmospheric conditions [25,26].

We substitute Eq. (10) into

$$z_{sf}^{{\ast}{\ast} } = \left\{ {\begin{array}{c} {z_{sf}^\ast \quad \quad \quad \quad \quad ;without\,a\,transmitter\,lens}\\ {{{z_{sf}^\ast \,f} / {({z_{sf}^\ast{+} f} )\quad \quad ;with\,a\,transmitter\,lens}}} \end{array}} \right..$$
Equation (19) is the Talanov equation that covers the self-focusing situations where the lens is defocusing $({f\, < \,0} )$ or focusing $({f\, > 0} )$ or collimating the transmitted laser beam [6,7]. This equation then is used in the following equation
$$\left\{ {1 - \left[ {\frac{2}{{{\alpha^2}z{{_{sf}^{{\ast}{\ast} }}^2}}}} \right][{1 - ({1 + \alpha z} )\exp \{{ - \alpha z} \}} ]} \right\} = 0.$$
The value of z that solves eq. (20) is $\overline {FOD}.$

As noted in earlier data comparisons, there is a slight difference (5-10% on the average) between the PDF median value and $\overline {FOD} .$ Since we are only interested in how well the model PDF profile fits the PDF profile from the simulations, we will perform a log-normal curve fit of the simulation data and use its estimate of $\overline {FOD} $ in our two proposed models. The log-normal curve fitting was performed using the PDF curve fitting routine contained in MATLAB.

Figures 3, 4, 5 and 6 show HELCAP data-generated PDFs and PDF predictions derived from Eq. (2), Eq. (7) and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 17}}\,{m^{ - {2 / 3}}}$, $C_n^2 = {10^{ - 16}}\,{m^{ - {2 / 3}}}$, $C_n^2 = {10^{ - 15}}\,{m^{ - {2 / 3}}}$ and $C_n^2 = {10^{ - 14}}\,{m^{ - {2 / 3}}}$, respectively, and $\alpha = 0.2\,k{m^{ - 1}}$.

 figure: Fig. 3.

Fig. 3. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 17}}\; {m^{ - 2/3}}$ and $\alpha = 0.2\; k{m^{ - 1}}$.

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 figure: Fig. 4.

Fig. 4. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 16}}\; {m^{ - 2/3}}$ and $\alpha = 0.2\; k{m^{ - 1}}$.

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 figure: Fig. 5.

Fig. 5. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 15}}\; {m^{ - 2/3}}$ and $\alpha = 0.2\; k{m^{ - 1}}$.

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 figure: Fig. 6.

Fig. 6. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 14}}\; {m^{ - 2/3}}$ and $\alpha = 0.2\; k{m^{ - 1}}$.

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Figures 7, 8, 9 and 10 exhibit the comparisons of HELCAP data-generated PDFs and PDF predictions derived from Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 17}}\,{m^{ - {2 / 3}}}$, $C_n^2 = {10^{ - 16}}\,{m^{ - {2 / 3}}}$, $C_n^2 = {10^{ - 15}}\,{m^{ - {2 / 3}}}$ and $C_n^2 = {10^{ - 14}}\,{m^{ - {2 / 3}}}$, respectively, and $\alpha = 0.1\,k{m^{ - 1}}$.

 figure: Fig. 7.

Fig. 7. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = 10^{ - 17}\; m^{ - 2/3}$ and $\alpha = 0.1\; km^{-1}$.

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 figure: Fig. 8.

Fig. 8. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 16}}\; {m^{ - 2/3}}$ and $\alpha = 0.1\; k{m^{ - 1}}$.

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 figure: Fig. 9.

Fig. 9. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 15}}\; {m^{ - 2/3}}$ and $\alpha = 0.1\; k{m^{ - 1}}$.

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 figure: Fig. 10.

Fig. 10. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 14}}\; {m^{ - 2/3}}$ and $\alpha = 0.1\; k{m^{ - 1}}$.

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Figures 11, 12, 13 and 14 show comparisons of HELCAP data-generated PDFs and PDF predictions derived from Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 17}}\,{m^{ - {2 / 3}}}$, $C_n^2 = {10^{ - 16}}\,{m^{ - {2 / 3}}}$, $C_n^2 = {10^{ - 15}}\,{m^{ - {2 / 3}}}$ and $C_n^2 = {10^{ - 14}}\,{m^{ - {2 / 3}}},$ respectively, and $\alpha = 0.0\,k{m^{ - 1}}$.

 figure: Fig. 11.

Fig. 11. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 17}}\; {m^{ - 2/3}}$ and $\alpha = 0.0\; k{m^{ - 1}}$.

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 figure: Fig. 12.

Fig. 12. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 16}}\; {m^{ - 2/3}}$ and $\alpha = 0.0\; k{m^{ - 1}}$.

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 figure: Fig. 13.

Fig. 13. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 15}}\; {m^{ - 2/3}}$ and $\alpha = 0.0\; k{m^{ - 1}}$.

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 figure: Fig. 14.

Fig. 14. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 14}}\; {m^{ - 2/3}}$ and $\alpha = 0.0\; k{m^{ - 1}}$.

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From examination of these figures, both PDF models agree with each other for weak turbulence and are close for the higher turbulence levels. Nakagimi showed that the gamma distribution is a good representation of the log-normal distribution in weak-to-moderate turbulence cases, which these comparisons seem to confirm [27].

In comparing these models with the curve fits and the data histogram PDFs, we find that the two models produce PDF profiles in reasonable agreement with the simulated data. For $\alpha = 0.2\,k{m^{ - 1}},$ all the profiles seem to track well. For $\alpha = 0.1\,k{m^{ - 1}},$ the $C_n^2 = {10^{ - 17}}\;{m^{ - {2 / 3}}}$ model profiles are more peaked than the curve fits. The widening of the data curve fit comes from the absence of data in the one bin near the median. The $C_n^2 = {10^{ - 16}}\;{m^{ - {2 / 3}}}$ and $C_n^2 = {10^{ - 15}}\;{m^{ - {2 / 3}}}$ model profiles are a reasonable approximation of the curve fits. On the other hand, the data curve fit is more peaked than the $C_n^2 = {10^{ - 14}}\;{m^{ - {2 / 3}}}$ model. The narrowing of the data curve fit probably comes from the absence of data in the second bin near the start of the PDF. For $\alpha = 0.0\,k{m^{ - 1}},$ the $C_n^2 = {10^{ - 17}}\;{m^{ - {2 / 3}}}$ model profiles are more peaked than the curve fits. The widening of the data curve fit again probably comes from the absence of data in the one bin near the median. For the remaining $\alpha = 0.0\,k{m^{ - 1}}$ PDF comparison, the profiles seem to track well. Larger simulation data sets would be expected to fill in those histogram bins that lack data and provide closer profile comparisons. Given that, the authors surmise the agreement is reasonable to use these models for engineering purposes. In other words, both models will be useful tools for many experimental design or trade studies.

5. Summary

Previous research in the filamentation process has focused on estimating the FOD in real atmospheres but not its statistical variance. This paper described two ad hoc engineering models for predicting the PDF for lossy, turbulent, nonlinear media. The intent is to characterize the dependence of FOD variation on turbulence. One model uses a log-normal PDF with mean and variance proportional to the Rytov Variance. The other uses a gamma PDF employing the same mean and variance definitions. Model PDF profiles were compared to data histograms and associated curve fits of previously reported HELCAP computer simulation results and showed reasonable agreement. These models are adequate for providing estimates of the FOD PDF for application trade studies and experiment design for the atmospheric loss and turbulence levels considered here.

Funding

U.S. Naval Research Laboratory; Office of Naval Research.

Acknowledgement

The authors would like to thank Dr. Larry C. Andrews, the University of Central Florida, for his extremely helpful technical discussions on atmospheric turbulence. 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 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 upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (14)
Fig. 1.
Fig. 1. High level filamentation process layout.

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Fig. 2.
Fig. 2. Fluence contours generated by the HELCAP code for beams with (a) $PR \approx 1.6$ and (b) $PR \approx 21.1$ near the onset of filamentation. Taken from Ref. [6].

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Fig. 3.
Fig. 3. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 17}}\; {m^{ - 2/3}}$ and $\alpha = 0.2\; k{m^{ - 1}}$.

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Fig. 4.
Fig. 4. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 16}}\; {m^{ - 2/3}}$ and $\alpha = 0.2\; k{m^{ - 1}}$.

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Fig. 5.
Fig. 5. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 15}}\; {m^{ - 2/3}}$ and $\alpha = 0.2\; k{m^{ - 1}}$.

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Fig. 6.
Fig. 6. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 14}}\; {m^{ - 2/3}}$ and $\alpha = 0.2\; k{m^{ - 1}}$.

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Fig. 7.
Fig. 7. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = 10^{ - 17}\; m^{ - 2/3}$ and $\alpha = 0.1\; km^{-1}$.

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Fig. 8.
Fig. 8. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 16}}\; {m^{ - 2/3}}$ and $\alpha = 0.1\; k{m^{ - 1}}$.

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Fig. 9.
Fig. 9. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 15}}\; {m^{ - 2/3}}$ and $\alpha = 0.1\; k{m^{ - 1}}$.

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Fig. 10.
Fig. 10. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 14}}\; {m^{ - 2/3}}$ and $\alpha = 0.1\; k{m^{ - 1}}$.

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Fig. 11.
Fig. 11. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 17}}\; {m^{ - 2/3}}$ and $\alpha = 0.0\; k{m^{ - 1}}$.

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Fig. 12.
Fig. 12. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 16}}\; {m^{ - 2/3}}$ and $\alpha = 0.0\; k{m^{ - 1}}$.

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Fig. 13.
Fig. 13. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 15}}\; {m^{ - 2/3}}$ and $\alpha = 0.0\; k{m^{ - 1}}$.

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Fig. 14.
Fig. 14. Comparisons among the HELCAP data-generated PDF, and predictions using Eq. (2), Eq. (7), and a log-normal curve fit of HELCAP data for $C_n^2 = {10^{ - 14}}\; {m^{ - 2/3}}$ and $\alpha = 0.0\; k{m^{ - 1}}$.

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Equations (20)

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

∂ A ∂ z = i 2 k 0 ∇ ⊥ 2 A + [ i ω 0 c ( δ n T + δ n T B + δ n K e r r + δ n P l a s m a ) − 1 2 α ] A + ∑ j S j
p ln ( μ ) = 1 μ 2 π σ ln 2 exp ⁡ { − ( log e [ μ ] − μ ln ) 2 2 σ ln 2 }
μ = log e [ F O D / 1 m ]
μ ln = log e [ F O D ¯ / 1 m ] ,
σ ln ≈ 0.1778 σ R 2
σ R 2 = 1.23 C n 2 k 7 / 6 F O D ¯ 11 / 6 .
p ( F O D ) = β 0 γ ( F O D / 1 m ) γ − 1 Γ ( γ ) e − β 0 ( F O D / 1 m )
γ = 1 / σ ln 2
β 0 = [ F O D ¯ / 1 m ] / σ ln 2 ,
z s f ∗ = 0.367 z r ( P R ∗ − 0.852 ) 2 − 0.0219 ,
P R ∗ = P R exp ⁡ { m 0 ( 2 W / ρ 0 ) } ,
ρ 0 2 W = 0.35 Λ q [ 8 3 ( b + 0.62 Λ 11 / 6 ) ] 3 / 5 , ℓ 0 ≪ ρ 0 ≪ L 0 ,
ρ 0 = { 3 1 + Θ + Θ 2 + Λ 2 [ 1.87 C n 2 k 2 z ℓ 0 − 1 / 3 ] − 1 / 2 , ρ 0 ≪ ℓ 0 [ 8 3 ( b + 0.62 Λ 11 / 6 ) ] 3 / 5 [ 1.46 C n 2 k 2 z ] − 3 / 5 , ℓ 0 ≪ ρ 0 ≪ L ,
W = W 0 2 [ Λ 0 2 + Θ 0 2 ] ,
q = 1.56 ( C n 2 k 7 / 6 z 11 / 6 ) 6 / 5 ,
b = { ( 1 − Θ 8 / 3 ) / [ 1 − Θ ] Θ ≥ 0 ( 1 + | Θ | 8 / 3 ) / [ 1 − Θ ] Θ < 0 ,
Θ = Θ 0 / ( Λ 0 2 + Θ 0 2 ) ,
Λ = Λ 0 / ( Λ 0 2 + Θ 0 2 )
z s f ∗ ∗ = { 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 .
{ 1 − [ 2 α 2 z s f ∗ ∗ 2 ] [ 1 − ( 1 + α z ) exp ⁡ { − α z } ] } = 0.
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