Engineering equations for characterizing non-linear laser intensity propagation in air with loss

Thomas Karr; Larry B. Stotts; Jason A. Tellez; Jason D. Schmidt; Justin D. Mansell

doi:10.1364/OE.26.003974

1. Introduction

Research in filamentation creation in real atmospheres continues with several recent papers publishing experimental results [1–5]. During the propagation of an ultra-short pulse laser beam in the atmosphere, the beam’s inherent high peak power (if above a certain critical threshold) can create dynamic interaction between beam diffraction and non-linear optical effects like the Kerr Effect to create a linear string of high optical intensity in propagation direction, which can be periodic in nature. This phenomenon is known as filamentation. The general mathematical characterization of this effect comes from the solution of the Non-Linear Schrödinger Equation. There is no closed form solution to this equation for real atmospheres. Subset solutions areusually found by its numerical evaluation using techniques like the Split Step Fresnel Method (SSFM) [6] and Finite Difference Method [7]. Even with advanced computation programs like MATLAB and Mathematica, this can be very time consuming if many variations need to be evaluated.

The key parameter of interest is the distance that filamentation occurs, better known as the self-focusing distance. Arguably, J. H. Marburger is the researcher most associated with the mathematical characterization of this phenomenon, e.g., his seminal papers on self-focusing. This characterization includes the effects of transmitter lens and absorption on the self-focusing distance of the laser beam. Unfortunately, his characterizations have limited, or incomplete, verification with either computer simulation or experimental data. This paper provides a validation of many of his characterizations, or the modification of, using computer simulation. Specifically, we will provide a validated set of engineering equations for characterizing the self-focusing distance from a laser beam propagating through non-turbulent air with loss, and with three source configurations: (1) no lens, (2) converging lens or (3) diverging lens.

2. Comparison of the engineering equation with simulation results - lossless atmosphere

Dawes and Marburger (D&M) [8, 9] cited the following equation for characterizing laser propagation through non-linear media:

\frac{1}{ρ^{*}} \frac{\partial}{\partial ρ^{*}} (ρ^{*} \frac{\partial E_{0}^{*}}{\partial ρ^{*}}) + i \frac{\partial}{\partial z^{*}} E_{0}^{*} + k^{2} a^{2} \frac{ε_{2}}{2 ε_{0}} | E_{0}^{*} | E_{0}^{*}^{2} = 0

where

ρ^{*} = ρ / a

z^{*} = z / 2 k a^{2}

and

E_{0}^{*} = k a \sqrt{\frac{ε_{2}}{2 ε_{0}}} E_{0} .

Equation (1) is the Non-Linear Schrödinger Equation attributed to Kelley [10]. In this equation, we assume

k = 2 π / λ

with

λ

being the laser wavelength, a Gaussian beam of the form

E = E_{0} e x p {- r^{2} / (2 a^{2})}

at

z = 0

where

a

is the

e^{- 1}

beam-intensity radius, and the parameters

ε_{0}

and

ε_{2}

are the free space permittivity and nonlinear dielectric constant in electrostatic units (esu) [11] described by Chiao, Garmire and Townes [12].

D&M numerically evaluated Eq. (1) at $ρ = 0$ . Figure 1 reproduces their set of plots using SSFM. (All computer simulation in this paper uses a form of the SSFM method described in chapters 6-8 in one of the authors’ book [8], modified with the Kerr effect and automated identification of beam collapse.) This figure depicts the on-axis intensity ( $ρ = 0$ ) versus the normalized longitudinal distance $z^{*} \sqrt{P_{p e a k} / P_{2}}$ for several powers in the input beam relative to their “dynamical critical power” for self-focusing. D&M showed that for all practical purposes, $P_{2} \approx P_{c r i t}$ , where $P_{c r i t}$ is the critical power of the non-linear atmosphere given by

P_{c r i t} \approx 1.4884 λ_{0}^{2} (\frac{c n_{0}}{128 n_{2}})

in esu units (erg per second), where

λ_{0} = 2 π \sqrt{ε_{0}} k^{- 1}

and

n_{2} = ε_{2} / 2 .

In these last equations, D&M added

\sqrt{ε_{0}} = n_{0} \approx 1

to the numerator and a factor 2 to the denominator, respectively, to the critical power cited in Goldberg, Talanov, and Erm [13]. Today, we use meters-kilograms-seconds (mks) units rather than esu [11]. In the mks units (Watts), the critical power equals

P_{c r i t} \approx 3.77 \frac{λ^{2}}{8 π n_{0} n_{2}},

[9, 14]. Here,

n_{2}

is the nonlinear index of refraction [15] or coefficient of intensity-dependent refractive index [16]. Typically, we find in the literature that

n_{0}

is around

1.0003

and the nonlinear refractive index

n_{2} (m^{2} / W)

is nominally between

3 x 10^{- 23} m^{2} / W

and

5 x 10^{- 23} m^{2} / W

for air [16, 17].

Fig. 1 Normalized intensity versus normalized propagation distance as a function of $P_{p e a k} / P_{c r i t}$

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The self-focusing distance was shown by Marburger to be given by

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

where

z_{r} = k a^{2}

is the Rayleigh Range [3, 6]. For

P_{p e a k} / P_{c r i t} \geq 1.5

, D&M found the following analytical expression,

\frac{I (z)}{I (0)} = {[1 - {(\frac{z}{z_{f}})}^{2}]}^{- γ},

was a good approximation for the data found in their Fig. 1 (like ours). The exponent

γ

was observed to be a parameter that depends on

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

. For

P_{p e a k} / P_{c r i t} > 100

, they further stated that

γ \approx 0.5

, which was originally proposed by Kelley [10]. Unfortunately, no other values for

γ

were provided and its evaluation was left to the reader using a coarse graph in the paper. Using our results for normalized intensity equation in Fig. 1, we derived an analytical equation for the exponent parameter as function of the peak laser power to critical power ratio; specifically, we used the curve-fitting routine in MATLAB and our data to yield

γ = {\sum_{n = 0}^{8} a_{n} (\log_{10} [P_{p e a k} / P_{c r i t}])}^{n}

where

\begin{array}{l} {a_{n}; n = 0, ..., 8} = {0.23805, - 2.4559, 10.027, 3.0896, - 20.010, 18.467, \\ - 3.3566, - 4.608, - 0.30433} \end{array}

Figure 2 depicts Eq. (12) as a function of $P_{p e a k} / P_{c r i t}$ showing both original points used for the curve fit and other arbitrary values of $P_{p e a k} / P_{c r i t}$ . Figure 3 shows a comparison of Eq. (11) using Eq. (12), and the SSFM self-focusing results like those in Fig. 1, and with additional curves, as function of the normalized propagation distance $z^{*}$ . This figure shows good agreement for $P_{p e a k} / P_{c r i t} > 3.33$ and moderate agreement for lower ratio values.

Fig. 2 Exponent parameter $γ$ as a function of $P_{p e a k} / P_{c r i t}$

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Fig. 3 Normalized Intensity versus normalized propagation distance $z^{*}$ as a function of $P_{p e a k} / P_{c r i t}$

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3. Engineering equation and simulation results - lossless atmosphere and a lens

In Boyd et. al., Marburger discuss how the self-focusing distance is affected by a transmitter lens [8]. It essentially followed the work of Talanov [18], but not entirely. The latter provided the correct solution, which was based on the Lens Law equation [19]. Specifically, Talanov stated that the new self-focusing distance using a lens is characterized by

\frac{1}{z_{f}^{*}} = \frac{1}{f} + \frac{1}{z_{f}}

where

z_{f}^{*}

is the modified (new) self-focusing distance [18].

Again, following Marburger’s Eq. (8).44) [8], Eq. (11) implies that the beam radius should be proportional to

\sqrt{(z / z_{r})^{2} + [1 - {(z / z_{f}^{*})}^{2}]} .

Figure 4 depicts the self-focusing distance derived from Eq. (15) and SSFM computer simulation results as a function of telescope power under the same assumptions of Fig. 4. The equation and simulation results agree well. Durand, et.al. published a recent paper demonstrating self-focusing distances out to 1 km using differing focal lengths and beam diameters and claimed that Eq. (14) was the equation that they found valid for the modified self-focusing distance in their field trials [4]. Thus, even in a real atmosphere, this equation appears to provide reasonable prediction of the self-focusing distance when the ranges are short (within the near-field of

7 k m

Rayleigh Range) for the level of atmospheric loss and turbulence they experience, which are probably low in both cases. For longer paths, atmospheric loss and turbulence will have an effect; namely causing a distribution of shorter beam collapse ranges. Papers by Peñano, and his colleagues demonstrate this effect for a turbulent atmosphere [20, 21]. For example, for

P_{p e a k} / P_{c r i t} \approx 1.5

and a 14.2 cm-radius/collimated beam, the average self-focusing distance will be between

100 %

and

80 %

of the Marburger distance for

C_{n}^{2}

-ranges from

1 x 10^{- 17} m^{(- 2 / 3)}

to

1 x 10^{- 15} m^{(- 2 / 3)}

, respectively [16]. However, by increasing

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

to

10

, the average self-focusing distance range reduces further, from 80% of the Marburger distance for

C_{n}^{2} = 1 x 10^{- 17} m^{(- 2 / 3)}

; down to ~60% to

C_{n}^{2} = 1 x 10^{- 15} m^{(- 2 / 3)}

; and down to ~33% to

C_{n}^{2} = 1 x 10^{- 14} m^{(- 2 / 3)}

[20]. Higher values of

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

reduce the average self-focusing distance even further. A full treatment of the beam collapse range starting from the differential equation is outside the scope of the paper. Nonetheless, this and the analytical equations in this paper should prove useful engineering tools for low

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

, short ranges and low turbulence values. For beams diameters larger than 14.2 cm radius and wavelengths in the Vis-to-LWIR band that can be propagated long distances through the atmosphere, an optical path < 20 km is all in the near field. In addition, Adaptive Optics (AO) is known to work well for correcting near-field weak-to-moderate atmospheric optical turbulence. As a relevant example, recent numerical simulations of a collimated Gaussian beam with

a = 3 c m

and AO phase compensation under turbulence ranging from

C_{n}^{2} = 1 x 10^{- 13}

to

1 x 10^{- 16}

, the resulting mean collapse ranges were within 5% of the predicted non-turbulent collapse ranges. In other words, these results suggest that AO has the potential to mitigate much of the optical degradation induced by turbulence when using moderate to large beam sizes, visible-IR wavelengths, and moderate path lengths with weak-to-moderate turbulence. Thus, our analytic tools should approximate the distance where self-focusing should occur in numerical simulations, and thereby provide a guide to more detailed numerical studies and open-air experiments.

Fig. 4 Comparison of modified self-focusing distance derived from Eq. (17) and computer simulation results versus telescope power for various values

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Let’s now make a point about the number of filaments generated at the self-focusing distance. Durand, et.al. also validated that the number of filaments decrease with range, asymptotically decreasing to the value on the order of $P_{p e a k} / P_{c r i t}$ at the longer ranges [4]. They cited this latter correlation agreed with theoretical predictions of Couairon and A. Mysyrowicz [22]. This fact should be true for our analytical expressions as well.

As final note to this section, we should mention the effect of lens aberrations on filament generation. Equation (1) can be employed to account for an initial beam field that contains azimuthally symmetric aberrations, like defocus and spherical aberration. However, for the more general problem where azimuthal symmetry cannot be assumed, Eq. (1) is written as

\frac{1}{ρ^{* 2}} \frac{\partial^{2} E_{0}^{*}}{\partial ϕ^{2}} + \frac{1}{ρ^{*}} \frac{\partial}{\partial ρ^{*}} (ρ^{*} \frac{\partial E_{0}^{*}}{\partial ρ^{*}}) + i \frac{\partial}{\partial z^{*}} E_{0}^{*} + k^{2} a^{2} \frac{ε_{2}}{2 ε_{0}} | E_{0}^{*} | E_{0}^{*}^{2} = 0

where

ϕ

is the azimuthal angle. With the added first term, this form of the differential equation can be used to account for both symmetrical and asymmetrical aberrations at the laser source. Marburger does not have any detailed models and analysis of this effect. However, the literature has reported intentional use of aberrations like astigmatism to extend the self-focusing range [22–25]. A detailed look at this aspect is beyond the intent of this paper (looking at the validity of Marburger’s various models), but the authors plan to investigate this effect in the future.

4. Engineering equation and simulation results - lossy atmosphere and a lens

In many of the recent published experiments, lenses are used to extend the self-focusing range [4, 5] to help overcome the atmospheric loss that is expected to degrade the self- focusing range. In Boyd et. al. [8], Marburger stated an analytical expression for the beam radius for laser beam propagation through a lossy, non-linear atmosphere, and then suggested how it could be modified if a transmitter lens is employed. We now will see how well these equations agree with computer simulation.

When the atmosphere is lossy, Eq. (1) has the term $i α^{*} E^{*} / 2$ added to the left-hand side of the equation, with $α^{*} = 2 k a^{2} α$ and $α$ being the volume extinction coefficient [9]. Figure 5 illustrates SSFM-based normalized intensities versus the normalized propagation distance $z^{*}$ for various value of $α^{*}$ for $P_{p e a k} / P_{c r i t} = 333$ . This graph resembles Fig. 5 in the D&M paper [9].

Fig. 5 Normalized intensity versus normalized propagation distance $z^{*}$ for various value of normalized extinction coefficient $α^{*}$ for $P_{p e a k} / P_{c r i t} = 333$

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In Boyd et. al. [8], Marburger provided a generalized approach for the normalized on-axis intensity based on the numerical analysis he did in 1969 with Dawes [9]. Specifically, he stated that when absorption is present, the propagated beam size $a$ must obey the following equation:

\frac{d a^{4}}{d z} = - \frac{4 P}{k^{2} P_{1}} z \exp {- α z},

[8]. He proposed the following solution [8]

a^{4} = a_{0}^{4} {1 - (\frac{2}{α^{2} z_{f}^{2}}) [1 - (1 + α z) \exp {- α z}]},

where

a_{0}

is the original laser beam size, based the work of Kaiser, Laubereau, Maier and Giordmaine [26]. Marburger claimed [6] that Eq. (18) agreed well with both the experimental results of Kaiser et. al. [26] and the computer results he generated with Dawes for large values of

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

[9].

Let

ζ_{0} = f / (z_{f} + f),

which yields

z_{f}^{*} = z_{f} f / (z_{f} + f) = z_{f} ζ_{0} .

When both a lens and absorption are present in the medium, Marburger states that the Eq. (17) needs to be modified using a lens transformation, e.g.,

\frac{d a^{4}}{d ζ} = - \frac{4 P}{k^{2} P_{1}} ζ \exp {- α ζ} .

The lens transformation he suggests does not yield the Talanov self-focusing distance when the extinction coefficient goes to zero. This is not surprising with our previous findings using this transformation. However, if we let the lens transformation be

ζ = z ζ_{0}

, then Eq. (21) becomes

\frac{d a^{4}}{d z} = - \frac{4 P}{k^{2} P_{1} ζ_{0}^{2}} z \exp {- α z / ζ_{0}},

accounting for the transformation’s Jacobian. The possible solution becomes

a^{4} = a_{0}^{4} {1 - (\frac{2}{α^{2} z_{f}^{2}}) [1 - (1 + α z / ζ_{0}) \exp {- α z / ζ_{0}}]},

Besides satisfying Eq. (22), Eq. (23) leads to the Eq. (15) as the extinction coefficient goes to zero for very large

z_{r}

. It appears we have a consistent theory. Let us now look at the accuracy of this equation under nominal atmospheric loss in typical atmospheric transmittance windows. We will use typical volume extinctions coefficients in real atmospheres as our metric.

Visibility (or visual range) corresponds to the horizontal range at which radiation at 0.55 μm is attenuated to 0.02 times its transmitted level. For a given wavelength $λ$ , the general volume extinction coefficient is related to visibility $z_{v}$ through Koschmeider equation:

α_{v} \approx \frac{3.912}{z_{v}} {(\frac{λ_{c}}{0.55 μ m})}^{q} \to q = {\begin{matrix} 1.6, z_{v} > 50 k m \\ 1.3, 6 k m \leq z_{v} \leq 50 k m \\ 0.585 z_{v}^{1 / 3}, z_{v} < 6 k m \end{matrix}

where

α_{v} (m^{- 1})

and

z_{v} (m)

are the visibility extinction (Mie scattering) coefficient and visibility range, respectively, and

λ_{c}

is the optical wavelength of interest [27, 28]. Figure 6 illustrates this volume extinction coefficients as function of visibility for several visibility ranges. In this figure, the visibilities

3 k m, 8 k m, 15 k m, 23.5 k m

and

60 k m

represent hazy, light haze, clear, standard clear and exceptionally clear atmospheric conditions, respectively. From this figure, the infrared volume extinction coefficient

α

is less than

0.1 k m^{- 1}

under the various clear sky conditions for

λ > 1 μ m

; this also is true under light haze conditions for

λ > 2 μ m .

Let us now look at the accuracy of this equation under nominal atmospheric loss in typical atmospheric transmittance windows. We will use typical volume extinctions coefficients in real atmospheres as our metric.

Fig. 6 Sea-Level volume extinction coefficients $α$ versus wavelength as a function of atmospheric visibility

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Figures 7-9 show a comparison of Eq. (23) and MZA Wave Train-derived (commercial SSFM software) self-focusing distances as a function of telescope power $(\equiv 1 / f o c a l l e n g t h)$ at various ratios of Peak Power-to-Critical Powers for $α = 0.1 k m^{- 1},$ $0.3 k m^{- 1}$ and $0.5 k m^{- 1}$ , respectively. In these figures, we have $a = 3 c m$ and $λ = 1 μ m .$

Fig. 7 Comparison of Eq. (23) and computer simulation results versus telescope power for various values of $P_{p e a k} / P_{c r i t}$ for $α = 0.1 k m^{- 1}$

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Fig. 8 Comparison of Eq. (23) and computer simulation results versus telescope power for various values of $P_{p e a k} / P_{c r i t}$ for $α = 0.3 k m^{- 1}$

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Fig. 9 Comparison of Eq. (23) and computer simulation results versus telescope power for various values of $P_{p e a k} / P_{c r i t}$ for $α = 0.5 k m^{- 1}$

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In Fig. 7, all theoretical and simulation results agree reasonably well for clear sky conditions when $α = 0.1 k m^{- 1}$ , with slightly less agreement for the lower $P_{p e a k} / P_{c r i t}$ values. In Fig. 8, for $α = 0.3 k m^{- 1}$ , the agreement between theory and simulation degrades when the telescope power is greater than 1 and $P_{p e a k} / P_{c r i t} < 10$ . Otherwise, the agreement between the two are reasonably good, except for $P_{p e a k} / P_{c r i t} = 2.5$ curve. In Fig. 9, for $α = 0.5 k m^{- 1},$ the agreement between theory and simulation again degrades when the telescope power is greater than 1 and $P_{p e a k} / P_{c r i t} < 10$ .

The conclusion from the above is that the curves agree with Marburger’s comment that the self-focusing equation given by Eq. (25) agrees with simulation when large values of $P_{p e a k} / P_{c r i t}$ are used, but we now add further caveat that the agreement degrades when absorption and a non-divergent lens are present. Hence, it is not a very useful equation for hazy atmospheres and focusing lens.

Recalling our discussion on using a transmitting lens in a lossless atmosphere, let us now substitute Talanov self-focusing distance, Eq. (14), in place of the Marburger Distance in Eq. (18) so that we have

a^{4} = a_{0}^{4} {1 - (\frac{2}{α^{2} z_{f}^{'}^{2}}) [1 - (1 + α z) \exp {- α z}]},

and compared this equation with our simulation results. This equation predicts the beam collapse range accurately with initially focused, collimated, and defocused beams in lossy air. However, it does not properly account for the beam’s focusing or defocusing width as it propagates. Other models, such as that by Zemlyanov and Geints, have been published that capture this effect, although they are accurate only for collimated and focused beams with very low loss [29, 30].

Figures 10-13 show a comparison of this new equation with the SSFM-derived self-focusing distances as a function of telescope power at various $P_{p e a k} / P_{c r i t}$ for $α = 0.1 k m^{- 1}, 0.3 k m^{- 1}, 0.5 k m^{- 1}$ and $1 k m^{- 1}$ , respectively. In these figures, we againhave $a = 3 c m$ , $λ = 1 μ m$ . Figures 10-12 show much better agreement between theory and the computer simulation results, for all, $P_{p e a k} / P_{c r i t}$ shown, than we saw in Figs. 7-9, respectively. In addition, Fig. 13 shows this excellent agreement between theory and simulation results extend to an even higher attenuation rate; namely, a volume extinctions coefficient $α = 1.0 k m^{- 1}$ that represents a haze extinction coefficient. We see that from Fig.6 that Eq. (25) is valid for hazy atmospheric conditions over range of $P_{p e a k} / P_{c r i t}$ in the $1 - 4 μ m$ region, a significant improvement over the Marburger model.

Fig. 10 Comparison of Eq. (25) using the Talanov self-focusing distance and computer simulation results versus telescope power for various values of $P_{p e a k} / P_{c r i t}$ for $α = 0.1 k m^{- 1}$

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Fig. 11 Comparison of Eq. (25) using the Talanov self-focusing distance and computer simulation results versus telescope power for various values of $P_{p e a k} / P_{c r i t}$ for $α = 0.3 k m^{- 1}$

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Fig. 12 Comparison of Eq. (25) using the Talanov self-focusing distance and computer simulation results versus telescope power for various values of $P_{p e a k} / P_{c r i t}$ for $α = 0.5 k m^{- 1}$

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Fig. 13 Comparison of Eq. (25) using the Talanov self-focusing distance and computer simulation results versus telescope power for various values of $P_{p e a k} / P_{c r i t}$ for $α = 1.0 k m^{- 1}$

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5. Summary

This paper validates a set of engineering equations for characterizing the self-focusing distance from a laser beam propagating through both lossless and lossy, non-linear atmospheres under three source configurations: (1) no lens, (2) converging lens or (3) diverging lens. Some validated equations follow Marburger and others do not, requiring modification of the original theory. These equations may be useful in experiment design and analysis, as well as in any trade-study.

6. Disclaimer

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 or of the Department of Defense.

Funding

Defense Advanced Research Projects Agency.

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Engineering equations for characterizing non-linear laser intensity propagation in air with loss

Abstract

Corrections

1. Introduction

2. Comparison of the engineering equation with simulation results - lossless atmosphere

3. Engineering equation and simulation results - lossless atmosphere and a lens

4. Engineering equation and simulation results - lossy atmosphere and a lens

5. Summary

6. Disclaimer

Funding

References and links

Cited By

Figures (13)

Equations (25)

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