Engineering equation for filamentation self-focusing collapse distance in atmospheric turbulence

Larry B. Stotts; Joseph Peñano; Vincent J. Urick

doi:10.1364/OE.27.015159

1. Introduction

The propagation of high peak-power, femtosecond laser beams in nonlinear media continues to be an active research topic [1–7]. During the propagation of such ultra-short pulse laser (USPL) beams in this environment, their high peak power (if above a certain critical threshold) can create dynamic interaction between beam diffraction and nonlinear optical effects like the Kerr Effect to create a linear string of high optical intensity in the propagation direction, which can be periodic in nature. This phenomenon is known as filamentation. A remarkable feature of these filaments is that along most of the propagation distance more than 10% of the pulse energy is localized in the near-axis area with the diameter about 100 microns for near infrared (NIR) wavelengths [8,9].

With real atmospheres that involve light absorption and scattering, filamentation still occurs. Ackerman et al [10] pointed out that filaments can survive their interaction with aerosols via both computer simulations [11] and experimentation [12]. Karr et al showed that the original and modified Marburger equations could be used to quantify the filamentation self-focusing distance in a lossy medium with and without a lens in the transmitter system, respectively [13,14]. Two insights from this work is that filamentation distance increases with Rayleigh Range and/or by decreasing the peak laser power to critical power ratio. Thus, the use of a reasonable sized telescope and a less ambitious laser system could help bring some applications to fruition sooner.

On the other hand, a turbulent atmosphere affects the creation of filaments differently to that found with a lossy atmosphere. An interesting aspect to this process is that the filament, once formed in turbulence, is unaffected by the turbulent region it exists in [8,10]. 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) [10]. This means that the initiation of filamentation in terms of position and pointing stability are the only processes affected by turbulence. In the former case, the collapse distance shortens, and its Probability Density Function (PDF) spreads, as the turbulence increases [7,13,14]. The presence of a lens in the laser transmitter’s telescope also affects the collapse distance and its PDF as well.

Houard et. al. noted that two distinct situations in filamentation generation has been observed [6]. In their 2004 paper, Penano et al. reported that air turbulence can create an increase in the filamentation onset distance with chirped laser pulses when the refractive index structure parameter, $C_{n}^{2}$ , is increased [15]. However, Kandidov et al. previously showed using numerical simulations with $C_{n}^{2} = 1 x 10^{- 11} m^{- 2 / 3}$ that turbulence should lead to a shorter collapse distances [16]. Shlenov et. al. clarified the situation in their 2007 paper when they noted that the filamentation onset distance increases when the turbulence is weak, but quickly shifts to distances closer to the transmitter as $C_{n}^{2}$ increases [17]. Houard et. al. explains that the presence of turbulence creates filaments from self-focusing and the collapse of the whole beam exhibit different features from those resulting from Modulational Instability (MI) of the beam inhomogeneity [6]. They state that an understanding of the competition between beam self-focusing and MI in turbulent air is required to achieve control of longitudinal features of filaments. In their 2009 paper, Shlenov et. al. discussed various methods for controlling the onset of filamentation of a high-power laser pulse in extended vertical atmospheric paths [18]. They showed that an increase in $C_{n}^{2}$ leads to an earlier formation of 'hot' spots when the initial pulse power exceeds the critical self-focusing power by an order of magnitude and more. They also found via numerical experiments that the use of broad focused beams is preferable for achieving the minimal standard deviation of a distance for the filament onset. Thus, the understanding of filamentation distance shortening with increasing turbulence is the problem many researchers are interested in. However, because of all the complexity involved in computer simulations of laser propagation in stressing turbulence, it would be useful to have an analytical expression for the filament self-focusing distance, like that reported earlier for the lossy medium case [13].

This paper provides an engineering equation for the filamentation self-focusing collapse distance from an USPL beam propagating through turbulent air, with validating comparisons to computer simulations results [7]. These results may provide a guide for further numerical simulations and field experiments.

2. The effect on filamentation onset distance by turbulence

In an atmospheric channel, turbulence is associated with the random velocity fluctuations of the “viscous fluid” comprising that channel. Unfortunately, these fluctuations are not laminarin nature, but rather are random sub-flows called turbulence eddies [19–21], which are generally unstable. In 1941, Kolmogorov, and simultaneously Obukhov, analyzed an isotopic velocity field to derive a set of inertial-subrange predictions for the forms of the velocity vector field and velocity structure functions that bear his name [21]. Both researchers independently found from their analyses that these eddies can be contained in a large range of discrete eddies, or turbules. Specifically, a continuum of cells for energy transfer from a macro scale, known as the outer scale of turbulence $L_{0}$ , down to micro scale, known as the inner scale of turbulence $l_{0}$ , was hypothesized [21]. This is illustrated in Fig. 1 showing the energy process. Near the surface (< 100 m), the outer scale $L_{0}$ is on the order of the height above ground level (AGL)of the observation point, and the inner scale $l_{0}$ typically is on the order 1-10 mm [21]. This range of turbules affects the local temperature and pressure structure, causing changes in the medium’s refractive index. The twinkling of the stars at night is a manifestation of the dynamic nature of the refractive index changes induced by these cells.

Fig. 1 Kolmogorov cascade theory of turbulence.

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The index of refraction for air depends mostly on the local pressure $P (r)$ and temperature $T (r)$ , and for the optical wavelength $λ \approx 0.5 μ m$ , is written as

n (r) = n_{0} + n_{1} (r) \approx 1 + (0.79 x 10^{- 6} m i l l i b a r s / K e l v i n) [P (r) / T (r)]

where

n (r)

is the refractive index at the vector location inside the argument,

n_{0} (r)

is the mean refractive index of air,

n_{1} (r)

is the statistical deviation of the refractive index

n (r)

induced by the pressure and temperature fluctuations, and

r

is a vector location in space [21]. For a standard atmosphere where

T = 15^{\circ} C = 288.25 K

and

P = 101.325 k P a = 1013.25 m i l l i b a r s

, the refractive index of air is

1.00027767

.

Because turbulence is a random process, it must be analyzed statistically. The turbulent fluctuations in the channel are best described by the structure function

D_{n} (r_{1}, r_{2}) = 〈 {| n (r_{1}) - n (r_{2}) |}^{2} 〉,

where

D_{n} (r_{1}, r_{2})

is the refractive index structure function and

〈 \dots 〉

represents ensemble averaging.

For homogeneous and isotropic turbulence, the above refractive index structure function only depends on the modulus of the vector separation between the two vector locations. This means that Eq. (2) can be rewritten as

D_{n} (r_{1}, r_{2}) = D_{n} (| r_{1} - r_{2} |) = D_{n} (ρ),

where

ρ = | r_{1} - r_{2} |

. From the Kolmogorov-Obukhov analyses, the refractive index structure function can be written as a two-thirds power law

D_{n} (ρ) = C_{n}^{2} (h) ρ^{2 / 3} for l_{0} < ρ < L_{0},

where

C_{n}^{2} (h)

is called the refractive index structure parameter [21]. This parameter is a function of atmospheric temperature and pressure, wavelength and height above ground. It is largest near the ground and falls off dramatically with increasing height above ground. It also follows a diurnal cycle with peaks at mid-day, near constant values at night and lowest values near sunrise and sunset. Weak turbulence typically has a refractive index structure parameter around

10^{- 17} m^{- 2 / 3}

and strong turbulence has values around

10^{- 12} m^{- 2 / 3}

or greater [21]. Fortunately, all the key parameters used in turbulence analysis, e.g., beam wander, are computed using this parameter and it is used to characterize filament generation in turbulence [1–14]. Let’s now turn to the characterization of the self-focusing collapse distance in turbulence [longitudinal wander of the collapse distance], the focus of this paper.

The current view of longitudinal beam wander, substantiated by computer simulation, is that the self-focusing collapse distance is statistically reduced by the turbulence [6,7,13]. It is a result of Modulation Instability (MI) [6,7]. Filamentation also exhibits a very different focusing profile than that achieved with a lens [6]. Figure 2 is an illustration of the difference between the two. Specifically, we compared the short-term beam radius $W_{S T}$ induced by turbulence for a focused beam with the radius created by a self-focusing nonlinear beam and lens for the same system and environmental circumstances. In this figure, we used the self-focusing theory developed in this paper and the Andrews and Phillips $W_{S T}$ equation, which is given by

W_{S T} = {\begin{matrix} W \sqrt{1 + 1.33 σ_{R}^{2} Λ^{5 / 6} [1 - 0.66 {(Λ_{0}^{2} / 1 + Λ_{0}^{2})}^{1 / 6}]} \\ W \sqrt{1 + 0.35 σ_{R}^{2} Λ^{5 / 6}} \end{matrix} \begin{matrix} C o l l i m a t e d B e a m \\ F o c u s e d B e a m \end{matrix}

where

W = \sqrt{W_{0}^{2} [{(λ L / π W_{0}^{2})}^{2} + {(1 - L / f)}^{2}]} = \sqrt{W_{0}^{2} [Λ_{0}^{2} + Θ_{0}^{2}]},

Λ = Λ_{0} / (Λ_{0}^{2} + Θ_{0}^{2}),

and

σ_{R}^{2} = 1.23 C_{n}^{2} k^{7 / 6} L^{11 / 6}

where

Λ_{0} = λ z / π W_{0}^{2} = z / k a^{2}

is the Fresnel Ratio,

λ

is transmitting laser wavelength,

W_{0}

is the

e^{- 2}

-intensity beam radius,

a

is the

e^{- 1}

-intensity beam radius,

Θ_{0} = (1 - L / f)

is the diffraction parameter,

f

is the focal length of the transmitter lens,

k = 2 π / λ

is the laser wavenumber and

L

is the link distance [17, Chap 6]. Figure 2 was created using

P_{p e a k} / P_{c r i t} = 10

,

W_{0} = 14.2 c m

,

λ = 1.53 μ m

and

f = 5 k m

, where

P_{p e a k}

is the peak power of the laser and

P_{c r i t} = 3.79 λ^{2} / 8 π n_{0} n_{2}

is the medium’s critical power [24,25]. Here,

n_{0}

is the average refractive index of the base medium and

n_{2}

is the nonlinear index of refraction [22] or coefficient of intensity-dependent refractive index [23].

Fig. 2 Qualitative comparison between USPL filamentation and lens focusing in turbulence.

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As clearly seen in Fig. 2, filamentation imposes a dramatic drop in beam radius compared to a lens, causing a long filament of very large, nearly uniform, intensities in a small lateral confine (e.g., on the order of a 100 microns diameter for NIR wavelengths). The size of the filament appears not to be dependent on $C_{n}^{2}$ ; only the self-focusing collapse distance is [8,10]. That is why all the filaments in Fig. 2 are threads, overlapping each other on the optical axis. When there is a lens at the transmitter, filamentation follows the same rules as without a lens. This is counter to what we see in convention lens focusing of a laser beam. In the conventional focusing case, lenses create a more gradual reduction in beam size, for which the beam size approaches a minimum but increases after this [focus] point, e.g., has a “depth of focus”. In other words, it exhibits a gradient in beam intensity before and after the lens focal length distance. The unperturbed beam size is set by the lens aperture size and becomes larger with increasing $C_{n}^{2}$ , unlike filamentation.

Let now turn back to turbulence for the moment to better understand Eq. (5). In that equation, $σ_{R}^{2}$ represents the Rytov variance. In general, theoretical investigations into wave propagation are classified in terms of either: (1) weak irradiance fluctuations or (2) strong irradiance fluctuations, both of which are characterized by the Rytov variance or its standard deviation. The specific criteria are:

\begin{matrix} W e a k i r r a d i a n c e f l u c t u a t i o n s : σ_{R}^{2} < 1 a n d σ_{R}^{2} Λ^{5 / 6} < 1 \\ S t r o n g i r r a d i a n c e f l u c t u a t i o n s : σ_{R}^{2} > 1 a n d σ_{R}^{2} Λ^{5 / 6} > 1 \end{matrix} .

Obviously, the use of higher wavelength lasers will reduce the Rytov variance, thereby reducing the degrading effects on laser beams by turbulence. On the other hand, Eq. (8) suggests that the Rytov variance increases without limit as

C_{n}^{2}

and/or the path length increases. This is not true and there is a clear limit. This can be seen in Fig. 3. This figure depicts observed scintillation versus predicted scintillation from Rytov theory [26]. It is clear from this figure that the scintillation peaks when the Rytov Standard Deviation (SD) reaches a value of around 0.3. When the Rytov SD is around 0.3 or less, little scintillation is to be expected, i.e., weak irradiance fluctuations. Above 0.3, larger scintillation (fading) is likely. However, when it exceeds 1.0, we begin to enter the saturation regime and the high scintillation goes down slightly for increasing Rytov SD. Physically, when turbulence is mild [

σ_{χ} < 0.3

], the laser beam will experience wave front tilt/tip from larger turbules, which are micro-radian deflections inducing minor scintillation. Under higher turbulence condition [

0.3 < σ_{χ} < 1

], the laser beam begins to propagate through numerous small turbules which has the possibility of one, or more, of them deflecting their portion of the beam into different same lateral locations at the link distance

L

. The scintillation increases, but still is in the linear regime, i.e., strong irradiance fluctuations. When we have very strong to severe turbulence, the turbulence break up the beam into many small beams, which are mixed over the link distance plane, causing the slight asymptotically decreasing scintillation shown for

σ_{χ} > 1

. This regime also is known as Deep Turbulence. Let now turn back to filamentation.

Fig. 3 Observed scintillation versus predicted scintillation [19].

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Peñano, et. al. explained MI’s influence on filamentation and related their discussion to a set of computer simulation results in [7]. We provide a modified form of the MI discussion here. The turbulent beam interaction for self-focusing is postulated to be governed by

Q^{2} (z) = 4 (P_{p e a k} / P_{c r i t}) {(ρ_{0} / 2 W)}^{2}

where

ρ_{0}

is the spatial coherence radius [17, Chap 6]. The ratio of the spatial coherence radius and the diffractive beam diameter is given by,

\frac{ρ_{0}}{2 W} = 0.35 \sqrt{\frac{Λ}{q}} {[\frac{8}{3 (a_{T H} + 0.62 Λ^{11 / 6})}]}^{3 / 5} l_{0} ≪ ρ_{0} ≪ L_{0},

with

a_{T H} = {\begin{cases} (1 - Θ^{8 / 3}) / [1 - Θ] \\ (1 - {| Θ |}^{8 / 3}) / [1 - Θ] \end{cases} \begin{matrix} Θ \geq 0 \\ Θ < 0 \end{matrix},

Θ = Θ_{0} / (Λ_{0}^{2} + Θ_{0}^{2}),

and

q = 1.22 {(σ_{R}^{2})}^{6 / 5}

Figure 4 shows the ratio of the spatial coherence radius and the diffractive beam diameter as a function of the Fresnel ratio

Λ_{0}

for converging, collimated and divergent beams. In this figure, we have assumed

q = 0.2

. This figure shows that the largest ratio of coherence radius to beam diameters occurs in the region

1 \leq Λ_{0} \leq 3

. For a converging beam, the interval of largest ratios is slightly truncated while for a diverging beam that interval is extended slightly.

Fig. 4 Plot of the ratio of the Spatial Coherence Radius and the Diffractive Beam Diameter versus the Fresnel Ratio $Λ_{0}$ for converging, collimated and divergent beams assuming $q = 0.2$ .

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Returning to our discussion on $Q^{2} (z)$ , Eq. (10) implies that

Q^{2} (z) \propto 1 / q \approx 1 / (1.5 C_{n}^{2} k^{7 / 6} L^{11 / 6})

For

Q^{2} ≫ 1

, the turbulence is weak and a Gaussian-profiled beam collapses on-axis producing a single filament. As

Q^{2}

decreases to unity, the turbulence becomes stronger for fixed values of wavenumbers and link distances. If

P_{p e a k} > P_{c r i t}

, one can image the speckle sizes decreasing relative to its normal turbulence diameter, the Fried Parameter

r_{0} (= 2.1 ρ_{0})

,because of increased optical effects induced by the Kerr effect. The effective power ratio increases because the critical power decreases with the effective refractive index change created by the reduced speckle size, and a higher power ratio occurs causing a shortened collapse distance and a much-increased PDF spreading. For

Q^{2} < 1

, the turbulence can be so strong that the filamentation will occur in the propagation regime nearer the transmitter than its unperturbed value for

P_{p e a k} \approx P_{c r i t}

, if filamentation occurs at all. Much larger values of

P_{p e a k} / P_{c r i t}

will continue to have shortened self-focusing distances and different PDF spreading as the turbulence increases. (Fortunately, real atmospheres have an upper limit to the possible refractive index structure parameter; Figure 5 illustrates typical

C_{n}^{2}

measurements close to the upper limits to that possible in the real atmospheres [20]).

Fig. 5 Ground level C_n² values at Hollister Airport during June 7-9, 2011 measured by a BLS 900 Scintec scintillometer [20].

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3. Analytical equation development for the filamentation onset distance affected by turbulence

As noted above, one of the key parameters of interest in characterizing filamentation in nonlinear media is the self-focusing collapse distances. Arguably, J. H. Marburger is the researcher most associated with the mathematical characterization of this event, e.g., his seminal papers on self-focusing [24,25]. His characterization includes the effects of transmitter lens and absorption on the self-focusing distance of the laser beam. Unfortunately, his characterizations had limited, or incomplete, verification by either computer simulation or experimental data. Karr et. al. provided a validation of many of his characterizations, or their modification of, using computer simulation [13,14]. Specifically, these authors provided a validated set of engineering equations for characterizing the self-focusing distance from a laser beam propagating through non-turbulent air with loss and initial convergence or divergence [13]. Their work did not cover the cases when turbulence is present. Others have investigated it.

Houard et al investigated Petrishchev’s analytical work on quantifying filamentation onset in atmospheric turbulence [6]. They stated that Petrishchev determined that the distance for filamentation could be obtained by finding the zeros of the function

1 + (1 - P_{p e a k} / P_{c r i t}) (z / z_{r})^{2} + (k^{3} W_{0}^{4} C / 8) {(z / z_{r})}^{3} = 0

where

z_{r} = π W_{0}^{2} / λ

is the Rayleigh Range,

P_{c r i t}

again is the critical power for filamentation,

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

and

a = W_{0} / \sqrt{2}

again is the

e^{- 1}

-intensity beam radius.

Houard et al compared the distance required for MI to amplify narrow perturbation by a factor of $10^{4}$ ; the classical Marburger self-focusing distance [13,14] and the solutions to Eq. (16) for $C_{n}^{2} = 10^{- 13} m^{- 2 / 3}$ as a function of $P / P_{c r i t}$ . Here, $l_{0} = 1 m m$ . This plot is shown in Fig. 6. It is apparent from this figure that the Petrishchev equation always predicts a larger self-focusing collapse distance than the Marburger Equation for large values of $P_{p e a k} / P_{c r i t}$ . This agrees with the example given in Petrishchev’s paper [23]. This trend is counter to the simulation results of Peñano, et. al. presented in [7]. Their results show that the filamentation self-focusing distance decreases with increasing turbulence, except for light turbulence and specific lens configurations. Equation (16) can be modified to yield different results.

Fig. 6 Plot of the Houard et al’s MI Distance Estimate, the classical Marburger self-focusing distance and Eq. (6) solutions as a function of Peak Power to Critical Power Ratio.

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Rewriting Petrishchev’s equation, we see that

(z / z_{r})^{2} + (1 - [P_{p e a k} / P_{c r i t}] {(z / z_{r})}^{2}) + (k^{3} W_{0}^{4} C / 8) {(z / z_{r})}^{3} =

or

(z / z_{r})^{2} + (1 - {(z / {z_{r} / \sqrt{[P_{p e a k} / P_{c r i t}]}})}^{2}) + (k^{3} W_{0}^{4} C / 8) {(z / z_{r})}^{3} = 0.

In the absence of turbulence, Eq. (19) should revert to the standard Marburger result. This suggests that Eq. (19) maybe should be rewritten as

(z / z_{r})^{2} + (1 - {(z / z_{s f})}^{2}) + (k^{3} W_{0}^{4} C / 8) {(z / z_{r})}^{3} \approx 0

with

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

which is the definition of the Dawes & Marburger self-focusing distance [25]. Figure 7 compares solutions to the collimated beam form of Eq. (20) with the median collapse distances derived from the PDFs in [7] using the NRL High Energy Laser Code for Atmospheric Propagation (HELCAP). The HELCAP modeling capability has shown good agreement between its simulation results and data from a previously published laboratory-scale experiment [7]. This also is a good set of data for an equation comparison as the Rytov SD for their modelled cases range from Weak Turbulence to Deep Turbulence, as seen in Fig. 8. This modified equation still predicts larger self-focusing collapse distances, except for the lowest

P / P_{c r i t}

, but the estimates are much closer to the simulation results than in Fig. 6.

Fig. 7 Comparison of median self-focusing distance with modified Petrishchev Equation, Eq. (22).

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Fig. 8 Rytov Standard Deviation versus Link Range for the Peñano, et. al. Computer Simulations [7].

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At the end of his paper, Petrishchev estimates the power ratio for a collimated beam propagating through a Tartarskii-spectrum turbulent atmosphere [27, p 67], which is given by

P_{p e a k} / P_{c r i t} = 1 + 0.75 k^{2} a^{2} {(a C / l_{0})}^{2 / 3} .

In this form, the power ratio linearly increases with increasing turbulence level. This additive model is like that proposed by Peñano, et. al. [7]. This is good as this function should produce a smaller collapse distances as the turbulence increases, but it does not (See Fig. 6). Equation (10) suggests that the increase in

P_{p e a k} / P_{c r i t}

is created by a multiplicative factor scaled by an integrated turbulence factor. Given that fact and our desire to get the zero-turbulence level to trend to the Marburger result, we set the right side of Eq. (22) as a multiplier of the initial peak-power-to-critical-power ratio. That is, we write

P_{p e a k}^{*} / P_{c r i t} = (P_{P e a k} / P_{c r i t}) [1 + 0.75 k^{2} a^{2} {(a C / l_{0})}^{2 / 3}] .

where

P_{p e a k}^{*} / P_{c r i t}

is the new power ratio induced by the turbulence. To now calculate the filamentation onset distance, we would insert the result from Eq. (23) into the Marburger self-focusing distance equation given in Eq. (23) to solve for

z_{s f}

. Unfortunately, this new approach does not provide a much better comparison plot than found in Fig. 7. Since Petrishchev derived the 0.75 factor for an additive rather than multiplicative increase in the peak to critical power ratio, the constant may need to change to better characterize what is happening in the turbulent nonlinear medium. This change follows what Andrews et. al. showed to better quantify the effect of adaptive optics on the Strehl Ratio. Namely, they modelled the improvement by putting a non-unity multiplicative factor in front of the

{(2^{3 / 2} W_{0} / r_{0})}^{2}

in the Andrews/Phillips Strehl Ratio equation [20]. This change fill experimental field results. Likewise, one might change the 0.75 factor in the second term in the square bracketed portion of Eq. (23) to better account for the Kerr effect on the turbulence. Rewriting Eq. (23) with a factor

m_{0}

replacing the 0.75 in the second term on the left, we have

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^{2} {(a C / l_{0})}^{2 / 3}] .

Figure 9 is a comparison between the median self-focusing distances derived from [7] with the modified Petrishchev power ratio equation, Eq. (24), with

m_{0}

= 0.02 and the assumption of a collimated beam. This approach provides distance predictions more in line with the computer simulations results. Figure 10 depicts a comparison between the median self-focusing distances from [7] with the modified Petrishchev power ratio equation, Eq. (24), with

m_{0}

= 0.008 for the various lens configurations. The change in

m_{0}

between these two situations comes from the assumption that the Kerr effect is influence differently when a lens is used versus the case when it is not present. In any event, the chosen

m_{0}

values appear to create the correction to the power ratio

P_{p e a k} / P_{c r i t}

necessary to predict the computer simulation results. Equation (24), with the appropriate value of

m_{0}

when a lens is present or not, appears to be an analytical means for estimating the filamentation self-focusing collapse distance when turbulence in present in the optical channel.

Fig. 9 Comparison of Peñano, et. al. median self-focusing distances with modified Petrishchev Equation, Eq. (24) with $m_{0}$ = 0.02.

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Fig. 10 Comparison of Peñano, et. al. median self-focusing distance with modified Petrishchev Equation, Eq. (24) with $m_{0}$ = 0.008.

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4. Summary

This paper provided the first validated analytical equation for predicting the nonlinear self-focusing collapse distance based on a modification of Petrishchev’s and Marburger theories. It shows that the estimate of the peak power to critical power at range in turbulence is modified to be the product of the transmitted peak power to critical power ratio times a multiplicative factor derived from Petrishchev’s turbulence equations. This insight comes from the fact that the turbulent channel is multiplicative rather than additive channel. This estimate is used in the Marburger distance equation to yield a predicted self-focusing collapse distance. This approach was compared to previous NRL’s HELCAP computer simulation results and showed good agreement. The HELCAP simulations capability has shown good agreement between its results and a previously published laboratory-scale experiment. These results may provide a guide for further numerical simulations, more formal theoretical developments and field experiments.

Funding

U. S. Defense Advanced Research Projects Agency (DARPA).

Acknowledgements

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 Defense Advanced Research Projects Agency, the Department of the Navy or of the Department of Defense.

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Engineering equation for filamentation self-focusing collapse distance in atmospheric turbulence

Abstract

1. Introduction

2. The effect on filamentation onset distance by turbulence

3. Analytical equation development for the filamentation onset distance affected by turbulence

4. Summary

Funding

Acknowledgements

References

Cited By

Figures (10)

Equations (24)

Optics Express