propagation of ultrashort, intense laser pulses through the atmosphere · 2019. 5. 14. · andrews...

Propagation of Ultrashort, Intense Laser Pulses Through the Atmosphere

Joe PeñanoPlasma Physics Division, Naval Research Laboratory

Presentation to the Mid-Atlantic Senior Physics Group19 December 2018

Distribution A: Approved for public release

Collaborators

2

Michael Helle, Gregory DiComo, AliaksandrMamonau1, Scott Melis2, Richard Fischer, John Palastro3, Dmitri Kaganovich, Benjamin Rock, Bahman Hafizi, Luke Johnson, Antonio Ting1,4

1: RSI, Inc.2: Georgetown University3: Rochester LLE4: University of Maryland

Andreas Schmitt-Sody, Jennifer Elle, and Adrian Lucero

Samantha Gregory, and Technical Staff

laser parameters

peak power

intensity

pulse duration

average power

GW to PW

1012 to 1023 W/cm2

10-14 to 10-12 seconds

1 W- 1 kW

comparison of electric field strengths

Elaser[V/cm] = 27.5 I1/2 [W/cm2] • laser electric field: Elaser ~ 108 to 1013 V/cm

• atomic electric field (over the barrier ionization): Eatom ~ q/4pe0a2 ~ 5x109 V/cm

• Schwinger field (potential difference of a rest mass across a Compton wavelength):

What is an ultrashort laser pulse?

Esch =me2c3 / q! ≈1016 V/cm

3

• positron and gamma ray sources

• laser wakefield acceleration of electrons

• radiation pressure acceleration of ions

• generation of ultrabroadband radiation

a laser pulse propagating through plasma ponderomotivelydrives an electrostatic wave that can accelerate electronsto MeV-GeV energies

W. Lu et al., PRSTAB, 10 (2007).

a laser pulse incident on an overdense target acts as alight sail/piston, ponderomotively accelerating ions

S. V. Bulanov et al., Phys. Plasmas, 17 (2010).

a laser pulse propagating through a nonlinear mediumundergoes octave-spanning spectral broadening andgeneration of far out-of-band radiation

a laser pulse incident on a counterpropagating multi-GeV electron beam yields gamma rays that pair-produce

extreme light Infrastructure website

S. Varma, thesis defense (2011).

Fundamentally different interactions than long pulses

4

Motivation for this talk

55/13/19

• Much of our understanding of optical propagation through atmospheric turbulence is based on the medium’s linear response to the optical field.

• Relatively few studies of turbulence and nonlinear filamentation and nonlinear self-focusing*. The majority of these studies investigated cases where the peak laser power is larger than the self-focusing power of air (filamentation regime).

- Kandidov et al.: modeling of nonlinear self-focusing (NLSF) in turbulence using phase screens. [Quantum Electron. 29, 911 (1999)]

- Chin et al.: characterized filament wander in turbulence [Appl. Phys. B 74, 67–76 (2002)]- Salame et al.: laboratory experiments over regions of extended turbulence [Appl. Phys.

Lett. 91, 171106 (2007)]

- Houard et al.: characterized the competition between self-focusing and modulational instability, investigate filament stability in turbulence [Phys. Rev. A 78, 033804 (2008)]

- Penano et al.: calculation of filamentation onset/suppression in turbulence [JOSA B 31, 963 (2014)]

• In this study, we present a way to propagate long ranges through strong atmospheric turbulence that avoids the “filamentation” regime.

6

• turbulence: random fluctuations of air currents in the form of eddies driven by large scale thermal gradients

• Eddy scale distribution is described by the modified von Karman spectrum:

2 20

22 2 11/6

0

( ) .033( )

k

n nek C

k L

-

-F = +

!

log 1

0(F

n/C n

2 ) [m

11/3

]

log10(k) [m-1]

10L-

10-!

0L : outer scale (1 – 100 m)

0! : inner scale (~1-10 mm)2nC : refractive index structure constant (meteorological strength)

Characterization of Atmospheric Turbulence

6Andrews and Phillips, Laser Beam Propagation through Random Media

• Rytov variance parameterizes the optical effect: �2R = 1.23C2nk7/6L11/6

�2R 1(optically weak) (optically strong);

• Transverse coherence length of a plane wave: ⇢0 = (1.46k2C2nL)�3/5AAACCnicbVC7TsMwFHXKq5RXgJHFUCGVgZKU8liQKrowMBSJPqQ+Isd1WquOE9kOUhV1ZuFXWBhAiJUvYONvcNsM0HKke3V0zr2y73FDRqWyrG8jtbC4tLySXs2srW9sbpnbOzUZRAKTKg5YIBoukoRRTqqKKkYaoSDIdxmpu4Py2K8/ECFpwO/VMCRtH/U49ShGSkuOud8S/cCx4BXM2fni+aBTgGWH63571ImPT0/ORo6ZtfLWBHCe2AnJggQVx/xqdQMc+YQrzJCUTdsKVTtGQlHMyCjTiiQJER6gHmlqypFPZDuenDKCh1rpQi8QuriCE/X3Rox8KYe+qyd9pPpy1huL/3nNSHmX7ZjyMFKE4+lDXsSgCuA4F9ilgmDFhpogLKj+K8R9JBBWOr2MDsGePXme1Ap528rbd8Vs6TqJIw32wAHIARtcgBK4ARVQBRg8gmfwCt6MJ+PFeDc+pqMpI9nZBX9gfP4AlMqXAQ==AAACCnicbVC7TsMwFHXKq5RXgJHFUCGVgZKU8liQKrowMBSJPqQ+Isd1WquOE9kOUhV1ZuFXWBhAiJUvYONvcNsM0HKke3V0zr2y73FDRqWyrG8jtbC4tLySXs2srW9sbpnbOzUZRAKTKg5YIBoukoRRTqqKKkYaoSDIdxmpu4Py2K8/ECFpwO/VMCRtH/U49ShGSkuOud8S/cCx4BXM2fni+aBTgGWH63571ImPT0/ORo6ZtfLWBHCe2AnJggQVx/xqdQMc+YQrzJCUTdsKVTtGQlHMyCjTiiQJER6gHmlqypFPZDuenDKCh1rpQi8QuriCE/X3Rox8KYe+qyd9pPpy1huL/3nNSHmX7ZjyMFKE4+lDXsSgCuA4F9ilgmDFhpogLKj+K8R9JBBWOr2MDsGePXme1Ap528rbd8Vs6TqJIw32wAHIARtcgBK4ARVQBRg8gmfwCt6MJ+PFeDc+pqMpI9nZBX9gfP4AlMqXAQ==AAACCnicbVC7TsMwFHXKq5RXgJHFUCGVgZKU8liQKrowMBSJPqQ+Isd1WquOE9kOUhV1ZuFXWBhAiJUvYONvcNsM0HKke3V0zr2y73FDRqWyrG8jtbC4tLySXs2srW9sbpnbOzUZRAKTKg5YIBoukoRRTqqKKkYaoSDIdxmpu4Py2K8/ECFpwO/VMCRtH/U49ShGSkuOud8S/cCx4BXM2fni+aBTgGWH63571ImPT0/ORo6ZtfLWBHCe2AnJggQVx/xqdQMc+YQrzJCUTdsKVTtGQlHMyCjTiiQJER6gHmlqypFPZDuenDKCh1rpQi8QuriCE/X3Rox8KYe+qyd9pPpy1huL/3nNSHmX7ZjyMFKE4+lDXsSgCuA4F9ilgmDFhpogLKj+K8R9JBBWOr2MDsGePXme1Ap528rbd8Vs6TqJIw32wAHIARtcgBK4ARVQBRg8gmfwCt6MJ+PFeDc+pqMpI9nZBX9gfP4AlMqXAQ==AAACCnicbVC7TsMwFHXKq5RXgJHFUCGVgZKU8liQKrowMBSJPqQ+Isd1WquOE9kOUhV1ZuFXWBhAiJUvYONvcNsM0HKke3V0zr2y73FDRqWyrG8jtbC4tLySXs2srW9sbpnbOzUZRAKTKg5YIBoukoRRTqqKKkYaoSDIdxmpu4Py2K8/ECFpwO/VMCRtH/U49ShGSkuOud8S/cCx4BXM2fni+aBTgGWH63571ImPT0/ORo6ZtfLWBHCe2AnJggQVx/xqdQMc+YQrzJCUTdsKVTtGQlHMyCjTiiQJER6gHmlqypFPZDuenDKCh1rpQi8QuriCE/X3Rox8KYe+qyd9pPpy1huL/3nNSHmX7ZjyMFKE4+lDXsSgCuA4F9ilgmDFhpogLKj+K8R9JBBWOr2MDsGePXme1Ap528rbd8Vs6TqJIw32wAHIARtcgBK4ARVQBRg8gmfwCt6MJ+PFeDc+pqMpI9nZBX9gfP4AlMqXAQ==

7

Turbulence and its effect on laser propagation

�2R = 3.25 km path

Simulated laser intensity (linear propagation) Turbulence refractive index

• Median turbulence over near-surface, horizontal path*:

Diffraction-limited spot

*C.I. Moore, et al., Proc. of SPIE Vol. 5892(SPIE, Bellingham, WA, 2005):

C2n = 3⇥ 10�15m�2/3AAACEXicbVC7SgNBFJ31GeNr1dJmMAhpjLuJoo0QTGMZwTwgL2Ynk2TI7Owyc1cMy/6Cjb9iY6GIrZ2df+PkUWjigYFzz7mXO/d4oeAaHOfbWlpeWV1bT22kN7e2d3btvf2qDiJFWYUGIlB1j2gmuGQV4CBYPVSM+J5gNW9YGvu1e6Y0D+QdjELW8klf8h6nBIzUsbOljmzn8RUuNIH7TGPXaccn7nmCm8AeIPYTU+ZPC0nHzjg5ZwK8SNwZyaAZyh37q9kNaOQzCVQQrRuuE0IrJgo4FSxJNyPNQkKHpM8ahkpitrfiyUUJPjZKF/cCZZ4EPFF/T8TE13rke6bTJzDQ895Y/M9rRNC7bMVchhEwSaeLepHAEOBxPLjLFaMgRoYQqrj5K6YDoggFE2LahODOn7xIqvmc6+Tc27NM8XoWRwodoiOURS66QEV0g8qogih6RM/oFb1ZT9aL9W59TFuXrNnMAfoD6/MH5zGbMA==AAACEXicbVC7SgNBFJ31GeNr1dJmMAhpjLuJoo0QTGMZwTwgL2Ynk2TI7Owyc1cMy/6Cjb9iY6GIrZ2df+PkUWjigYFzz7mXO/d4oeAaHOfbWlpeWV1bT22kN7e2d3btvf2qDiJFWYUGIlB1j2gmuGQV4CBYPVSM+J5gNW9YGvu1e6Y0D+QdjELW8klf8h6nBIzUsbOljmzn8RUuNIH7TGPXaccn7nmCm8AeIPYTU+ZPC0nHzjg5ZwK8SNwZyaAZyh37q9kNaOQzCVQQrRuuE0IrJgo4FSxJNyPNQkKHpM8ahkpitrfiyUUJPjZKF/cCZZ4EPFF/T8TE13rke6bTJzDQ895Y/M9rRNC7bMVchhEwSaeLepHAEOBxPLjLFaMgRoYQqrj5K6YDoggFE2LahODOn7xIqvmc6+Tc27NM8XoWRwodoiOURS66QEV0g8qogih6RM/oFb1ZT9aL9W59TFuXrNnMAfoD6/MH5zGbMA==AAACEXicbVC7SgNBFJ31GeNr1dJmMAhpjLuJoo0QTGMZwTwgL2Ynk2TI7Owyc1cMy/6Cjb9iY6GIrZ2df+PkUWjigYFzz7mXO/d4oeAaHOfbWlpeWV1bT22kN7e2d3btvf2qDiJFWYUGIlB1j2gmuGQV4CBYPVSM+J5gNW9YGvu1e6Y0D+QdjELW8klf8h6nBIzUsbOljmzn8RUuNIH7TGPXaccn7nmCm8AeIPYTU+ZPC0nHzjg5ZwK8SNwZyaAZyh37q9kNaOQzCVQQrRuuE0IrJgo4FSxJNyPNQkKHpM8ahkpitrfiyUUJPjZKF/cCZZ4EPFF/T8TE13rke6bTJzDQ895Y/M9rRNC7bMVchhEwSaeLepHAEOBxPLjLFaMgRoYQqrj5K6YDoggFE2LahODOn7xIqvmc6+Tc27NM8XoWRwodoiOURS66QEV0g8qogih6RM/oFb1ZT9aL9W59TFuXrNnMAfoD6/MH5zGbMA==AAACEXicbVC7SgNBFJ31GeNr1dJmMAhpjLuJoo0QTGMZwTwgL2Ynk2TI7Owyc1cMy/6Cjb9iY6GIrZ2df+PkUWjigYFzz7mXO/d4oeAaHOfbWlpeWV1bT22kN7e2d3btvf2qDiJFWYUGIlB1j2gmuGQV4CBYPVSM+J5gNW9YGvu1e6Y0D+QdjELW8klf8h6nBIzUsbOljmzn8RUuNIH7TGPXaccn7nmCm8AeIPYTU+ZPC0nHzjg5ZwK8SNwZyaAZyh37q9kNaOQzCVQQrRuuE0IrJgo4FSxJNyPNQkKHpM8ahkpitrfiyUUJPjZKF/cCZZ4EPFF/T8TE13rke6bTJzDQ895Y/M9rRNC7bMVchhEwSaeLepHAEOBxPLjLFaMgRoYQqrj5K6YDoggFE2LahODOn7xIqvmc6+Tc27NM8XoWRwodoiOURS66QEV0g8qogih6RM/oFb1ZT9aL9W59TFuXrNnMAfoD6/MH5zGbMA==

• Strong turbulence regime can be realized within several kilometers:

(~1 um wavelength)

8

Self-channeling through turbulence

D ≤ ℓ0P ≈ PNL Beam spot

8

• Self-channeling concept

- Use small spot size (smaller than the coherence length or inner scale) to avoid spreading by turbulence

J. Peñano, J.P. Palastro, B. Hazi, M.H. Helle, and G.P. DiComo, Phys. Rev. A, 96(1):013829, 2017.

- Propagate near the critical power to cancel diffractive spreading

Channeling can persist for many Rayleigh lengths

spreading is due to small scale turbulence wander is due to large scale turbulence

keddy R > 1keddyR

9

z = 0

z = 15 ZR

propagation direction

Simulations Indicate Effectiveness at Long Range (z >> ZR)

100+ independent realizations of turbulence for a given Cn2

9

• Simulations solve NLSE in (x,y,z,t) including GVD, extinction, and turbulence

2ik

✓@

@z+ ↵

◆� k�2

@2

@⌧2+r2?

�A = �2k20 (�nT + n2I)A

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

• Characterize channeling by how much energy remains within the initial beam radius, W0.

-10 -5 0 5 10x [cm]

-10

-5

0

5

10

y [cm]

z = 1. km

-10 -5 0 5 10x [cm]

-10

-5

0

5

10

y [cm]

z = 5. km

-10 -5 0 5 10x [cm]

-10

-5

0

5

10

y [cm]

z = 7. km

z = 15 ZR

10% 80% 86%J. Peñano, J.P. Palastro, B. Hazi, 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.

10

Channeling is dependent on relationship between spot size (W0), inner scale (l0), and coherence length (r0)

10

(1)

(2)

(3)

no turbulence

P<<

PNL

(1)

(3)

0

1 normalizedintensity

(i) W0 < (`0, ⇢0)AAAB/nicbVDLSgMxFM34rPU1Kq7cBItQQUpGBF24KLpxWcE+oDMMmTTThmaSIckIZSj4K25cKOLW73Dn35i2s9DWAxcO59zLvfdEKWfaIPTtLC2vrK6tlzbKm1vbO7vu3n5Ly0wR2iSSS9WJsKacCdo0zHDaSRXFScRpOxreTvz2I1WaSfFgRikNEtwXLGYEGyuF7mE7RPAaVn3KeYjOfDWQIToN3QqqoSngIvEKUgEFGqH75fckyRIqDOFY666HUhPkWBlGOB2X/UzTFJMh7tOupQInVAf59PwxPLFKD8ZS2RIGTtXfEzlOtB4lke1MsBnoeW8i/ud1MxNfBTkTaWaoILNFccahkXCSBewxRYnhI0swUczeCskAK0yMTaxsQ/DmX14krfOah2re/UWlflPEUQJH4BhUgQcuQR3cgQZoAgJy8AxewZvz5Lw4787HrHXJKWYOwB84nz85+JO4AAAB/nicbVDLSgMxFM34rPU1Kq7cBItQQUpGBF24KLpxWcE+oDMMmTTThmaSIckIZSj4K25cKOLW73Dn35i2s9DWAxcO59zLvfdEKWfaIPTtLC2vrK6tlzbKm1vbO7vu3n5Ly0wR2iSSS9WJsKacCdo0zHDaSRXFScRpOxreTvz2I1WaSfFgRikNEtwXLGYEGyuF7mE7RPAaVn3KeYjOfDWQIToN3QqqoSngIvEKUgEFGqH75fckyRIqDOFY666HUhPkWBlGOB2X/UzTFJMh7tOupQInVAf59PwxPLFKD8ZS2RIGTtXfEzlOtB4lke1MsBnoeW8i/ud1MxNfBTkTaWaoILNFccahkXCSBewxRYnhI0swUczeCskAK0yMTaxsQ/DmX14krfOah2re/UWlflPEUQJH4BhUgQcuQR3cgQZoAgJy8AxewZvz5Lw4787HrHXJKWYOwB84nz85+JO4AAAB/nicbVDLSgMxFM34rPU1Kq7cBItQQUpGBF24KLpxWcE+oDMMmTTThmaSIckIZSj4K25cKOLW73Dn35i2s9DWAxcO59zLvfdEKWfaIPTtLC2vrK6tlzbKm1vbO7vu3n5Ly0wR2iSSS9WJsKacCdo0zHDaSRXFScRpOxreTvz2I1WaSfFgRikNEtwXLGYEGyuF7mE7RPAaVn3KeYjOfDWQIToN3QqqoSngIvEKUgEFGqH75fckyRIqDOFY666HUhPkWBlGOB2X/UzTFJMh7tOupQInVAf59PwxPLFKD8ZS2RIGTtXfEzlOtB4lke1MsBnoeW8i/ud1MxNfBTkTaWaoILNFccahkXCSBewxRYnhI0swUczeCskAK0yMTaxsQ/DmX14krfOah2re/UWlflPEUQJH4BhUgQcuQR3cgQZoAgJy8AxewZvz5Lw4787HrHXJKWYOwB84nz85+JO4AAAB/nicbVDLSgMxFM34rPU1Kq7cBItQQUpGBF24KLpxWcE+oDMMmTTThmaSIckIZSj4K25cKOLW73Dn35i2s9DWAxcO59zLvfdEKWfaIPTtLC2vrK6tlzbKm1vbO7vu3n5Ly0wR2iSSS9WJsKacCdo0zHDaSRXFScRpOxreTvz2I1WaSfFgRikNEtwXLGYEGyuF7mE7RPAaVn3KeYjOfDWQIToN3QqqoSngIvEKUgEFGqH75fckyRIqDOFY666HUhPkWBlGOB2X/UzTFJMh7tOupQInVAf59PwxPLFKD8ZS2RIGTtXfEzlOtB4lke1MsBnoeW8i/ud1MxNfBTkTaWaoILNFccahkXCSBewxRYnhI0swUczeCskAK0yMTaxsQ/DmX14krfOah2re/UWlflPEUQJH4BhUgQcuQR3cgQZoAgJy8AxewZvz5Lw4787HrHXJKWYOwB84nz85+JO4

long-range channeling (> 10 ZR0)rms spot increases, central hot-spot remains intactnonlinear self-focusing is not effectiveW0 >> (`0, ⇢0)

AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkjJiKCrUnTjsoJ9QGcYMmmmDc0kQ5IRSu3CX3HjQhG3/oY7/8a0nYVWD1w4nHMv994TpZxpg9CXU1haXlldK66XNja3tnfc3b2WlpkitEkkl6oTYU05E7RpmOG0kyqKk4jTdjS8nvrte6o0k+LOjFIaJLgvWMwINlYK3YN2iGCtBis+5TxEp74ayBCdhG4ZVdEM8C/xclIGORqh++n3JMkSKgzhWOuuh1ITjLEyjHA6KfmZpikmQ9ynXUsFTqgOxrP7J/DYKj0YS2VLGDhTf06McaL1KIlsZ4LNQC96U/E/r5uZ+DIYM5FmhgoyXxRnHBoJp2HAHlOUGD6yBBPF7K2QDLDCxNjISjYEb/Hlv6R1VvVQ1bs9L9ev8jiK4BAcgQrwwAWogxvQAE1AwAN4Ai/g1Xl0np03533eWnDymX3wC87HN8VnlAI=AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkjJiKCrUnTjsoJ9QGcYMmmmDc0kQ5IRSu3CX3HjQhG3/oY7/8a0nYVWD1w4nHMv994TpZxpg9CXU1haXlldK66XNja3tnfc3b2WlpkitEkkl6oTYU05E7RpmOG0kyqKk4jTdjS8nvrte6o0k+LOjFIaJLgvWMwINlYK3YN2iGCtBis+5TxEp74ayBCdhG4ZVdEM8C/xclIGORqh++n3JMkSKgzhWOuuh1ITjLEyjHA6KfmZpikmQ9ynXUsFTqgOxrP7J/DYKj0YS2VLGDhTf06McaL1KIlsZ4LNQC96U/E/r5uZ+DIYM5FmhgoyXxRnHBoJp2HAHlOUGD6yBBPF7K2QDLDCxNjISjYEb/Hlv6R1VvVQ1bs9L9ev8jiK4BAcgQrwwAWogxvQAE1AwAN4Ai/g1Xl0np03533eWnDymX3wC87HN8VnlAI=AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkjJiKCrUnTjsoJ9QGcYMmmmDc0kQ5IRSu3CX3HjQhG3/oY7/8a0nYVWD1w4nHMv994TpZxpg9CXU1haXlldK66XNja3tnfc3b2WlpkitEkkl6oTYU05E7RpmOG0kyqKk4jTdjS8nvrte6o0k+LOjFIaJLgvWMwINlYK3YN2iGCtBis+5TxEp74ayBCdhG4ZVdEM8C/xclIGORqh++n3JMkSKgzhWOuuh1ITjLEyjHA6KfmZpikmQ9ynXUsFTqgOxrP7J/DYKj0YS2VLGDhTf06McaL1KIlsZ4LNQC96U/E/r5uZ+DIYM5FmhgoyXxRnHBoJp2HAHlOUGD6yBBPF7K2QDLDCxNjISjYEb/Hlv6R1VvVQ1bs9L9ev8jiK4BAcgQrwwAWogxvQAE1AwAN4Ai/g1Xl0np03533eWnDymX3wC87HN8VnlAI=AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkjJiKCrUnTjsoJ9QGcYMmmmDc0kQ5IRSu3CX3HjQhG3/oY7/8a0nYVWD1w4nHMv994TpZxpg9CXU1haXlldK66XNja3tnfc3b2WlpkitEkkl6oTYU05E7RpmOG0kyqKk4jTdjS8nvrte6o0k+LOjFIaJLgvWMwINlYK3YN2iGCtBis+5TxEp74ayBCdhG4ZVdEM8C/xclIGORqh++n3JMkSKgzhWOuuh1ITjLEyjHA6KfmZpikmQ9ynXUsFTqgOxrP7J/DYKj0YS2VLGDhTf06McaL1KIlsZ4LNQC96U/E/r5uZ+DIYM5FmhgoyXxRnHBoJp2HAHlOUGD6yBBPF7K2QDLDCxNjISjYEb/Hlv6R1VvVQ1bs9L9ev8jiK4BAcgQrwwAWogxvQAE1AwAN4Ai/g1Xl0np03533eWnDymX3wC87HN8VnlAI=

`0 < W0 < ⇢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

(ii)

(iii)

W0 /

0̀ =1

AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8xVkR9CIEvXiMYB6QrMvspJMMmX0wM6uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1090VJFJoQ+m3s7S8srq2Xtgobm5t7+yW9vYbOk4VxzqPZaxaAdMoRYR1I4zEVqKQhYHEZjC8mfjNR1RaxNG9GSXohawfiZ7gzFjpoenT0w5K6VNyRVy/VKYVOgVZJG5OypCj5pe+Ot2YpyFGhkumddulifEypozgEsfFTqoxYXzI+ti2NGIhai+bXj0mx1bpkl6sbEWGTNXfExkLtR6Fge0MmRnoeW8i/ue1U9O79DIRJanBiM8W9VJJTEwmEZCuUMiNHFnCuBL2VsIHTDFubFBFG4I7//IiaZxVXFpx787L1es8jgIcwhGcgAsXUIVbqEEdOCh4hld4c56cF+fd+Zi1Ljn5zAH8gfP5Ax1nkPE=AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8xVkR9CIEvXiMYB6QrMvspJMMmX0wM6uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1090VJFJoQ+m3s7S8srq2Xtgobm5t7+yW9vYbOk4VxzqPZaxaAdMoRYR1I4zEVqKQhYHEZjC8mfjNR1RaxNG9GSXohawfiZ7gzFjpoenT0w5K6VNyRVy/VKYVOgVZJG5OypCj5pe+Ot2YpyFGhkumddulifEypozgEsfFTqoxYXzI+ti2NGIhai+bXj0mx1bpkl6sbEWGTNXfExkLtR6Fge0MmRnoeW8i/ue1U9O79DIRJanBiM8W9VJJTEwmEZCuUMiNHFnCuBL2VsIHTDFubFBFG4I7//IiaZxVXFpx787L1es8jgIcwhGcgAsXUIVbqEEdOCh4hld4c56cF+fd+Zi1Ljn5zAH8gfP5Ax1nkPE=AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8xVkR9CIEvXiMYB6QrMvspJMMmX0wM6uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1090VJFJoQ+m3s7S8srq2Xtgobm5t7+yW9vYbOk4VxzqPZaxaAdMoRYR1I4zEVqKQhYHEZjC8mfjNR1RaxNG9GSXohawfiZ7gzFjpoenT0w5K6VNyRVy/VKYVOgVZJG5OypCj5pe+Ot2YpyFGhkumddulifEypozgEsfFTqoxYXzI+ti2NGIhai+bXj0mx1bpkl6sbEWGTNXfExkLtR6Fge0MmRnoeW8i/ue1U9O79DIRJanBiM8W9VJJTEwmEZCuUMiNHFnCuBL2VsIHTDFubFBFG4I7//IiaZxVXFpx787L1es8jgIcwhGcgAsXUIVbqEEdOCh4hld4c56cF+fd+Zi1Ljn5zAH8gfP5Ax1nkPE=AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8xVkR9CIEvXiMYB6QrMvspJMMmX0wM6uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1090VJFJoQ+m3s7S8srq2Xtgobm5t7+yW9vYbOk4VxzqPZaxaAdMoRYR1I4zEVqKQhYHEZjC8mfjNR1RaxNG9GSXohawfiZ7gzFjpoenT0w5K6VNyRVy/VKYVOgVZJG5OypCj5pe+Ot2YpyFGhkumddulifEypozgEsfFTqoxYXzI+ti2NGIhai+bXj0mx1bpkl6sbEWGTNXfExkLtR6Fge0MmRnoeW8i/ue1U9O79DIRJanBiM8W9VJJTEwmEZCuUMiNHFnCuBL2VsIHTDFubFBFG4I7//IiaZxVXFpx787L1es8jgIcwhGcgAsXUIVbqEEdOCh4hld4c56cF+fd+Zi1Ljn5zAH8gfP5Ax1nkPE=

W0 /`

0 =10

AAAB+HicbVBNS8NAEJ34WetHox69LBbBU92IoBeh6MVjBfsBbQib7aZdutmE3Y1QQ3+JFw+KePWnePPfuG1z0NYHA4/3ZpiZF6aCa4Pxt7Oyura+sVnaKm/v7O5V3P2Dlk4yRVmTJiJRnZBoJrhkTcONYJ1UMRKHgrXD0e3Ubz8ypXkiH8w4ZX5MBpJHnBJjpcCttAN81mNCBBhdIw8HbhXX8AxomXgFqUKBRuB+9foJzWImDRVE666HU+PnRBlOBZuUe5lmKaEjMmBdSyWJmfbz2eETdGKVPooSZUsaNFN/T+Qk1noch7YzJmaoF72p+J/XzUx05edcpplhks4XRZlAJkHTFFCfK0aNGFtCqOL2VkSHRBFqbFZlG4K3+PIyaZ3XPFzz7i+q9ZsijhIcwTGcggeXUIc7aEATKGTwDK/w5jw5L8678zFvXXGKmUP4A+fzBwbUkVw=AAAB+HicbVBNS8NAEJ34WetHox69LBbBU92IoBeh6MVjBfsBbQib7aZdutmE3Y1QQ3+JFw+KePWnePPfuG1z0NYHA4/3ZpiZF6aCa4Pxt7Oyura+sVnaKm/v7O5V3P2Dlk4yRVmTJiJRnZBoJrhkTcONYJ1UMRKHgrXD0e3Ubz8ypXkiH8w4ZX5MBpJHnBJjpcCttAN81mNCBBhdIw8HbhXX8AxomXgFqUKBRuB+9foJzWImDRVE666HU+PnRBlOBZuUe5lmKaEjMmBdSyWJmfbz2eETdGKVPooSZUsaNFN/T+Qk1noch7YzJmaoF72p+J/XzUx05edcpplhks4XRZlAJkHTFFCfK0aNGFtCqOL2VkSHRBFqbFZlG4K3+PIyaZ3XPFzz7i+q9ZsijhIcwTGcggeXUIc7aEATKGTwDK/w5jw5L8678zFvXXGKmUP4A+fzBwbUkVw=AAAB+HicbVBNS8NAEJ34WetHox69LBbBU92IoBeh6MVjBfsBbQib7aZdutmE3Y1QQ3+JFw+KePWnePPfuG1z0NYHA4/3ZpiZF6aCa4Pxt7Oyura+sVnaKm/v7O5V3P2Dlk4yRVmTJiJRnZBoJrhkTcONYJ1UMRKHgrXD0e3Ubz8ypXkiH8w4ZX5MBpJHnBJjpcCttAN81mNCBBhdIw8HbhXX8AxomXgFqUKBRuB+9foJzWImDRVE666HU+PnRBlOBZuUe5lmKaEjMmBdSyWJmfbz2eETdGKVPooSZUsaNFN/T+Qk1noch7YzJmaoF72p+J/XzUx05edcpplhks4XRZlAJkHTFFCfK0aNGFtCqOL2VkSHRBFqbFZlG4K3+PIyaZ3XPFzz7i+q9ZsijhIcwTGcggeXUIc7aEATKGTwDK/w5jw5L8678zFvXXGKmUP4A+fzBwbUkVw=AAAB+HicbVBNS8NAEJ34WetHox69LBbBU92IoBeh6MVjBfsBbQib7aZdutmE3Y1QQ3+JFw+KePWnePPfuG1z0NYHA4/3ZpiZF6aCa4Pxt7Oyura+sVnaKm/v7O5V3P2Dlk4yRVmTJiJRnZBoJrhkTcONYJ1UMRKHgrXD0e3Ubz8ypXkiH8w4ZX5MBpJHnBJjpcCttAN81mNCBBhdIw8HbhXX8AxomXgFqUKBRuB+9foJzWImDRVE666HU+PnRBlOBZuUe5lmKaEjMmBdSyWJmfbz2eETdGKVPooSZUsaNFN/T+Qk1noch7YzJmaoF72p+J/XzUx05edcpplhks4XRZlAJkHTFFCfK0aNGFtCqOL2VkSHRBFqbFZlG4K3+PIyaZ3XPFzz7i+q9ZsijhIcwTGcggeXUIc7aEATKGTwDK/w5jw5L8678zFvXXGKmUP4A+fzBwbUkVw=

W0/`0 = 0.1AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU92IoBeh6MVjBfsBbQib7aRdutmE3U2hhP4TLx4U8eo/8ea/cdvmoK0PBh7vzTAzL0wF14aQb6e0tr6xuVXeruzs7u0fuIdHLZ1kikGTJSJRnZBqEFxC03AjoJMqoHEooB2O7md+ewxK80Q+mUkKfkwHkkecUWOlwHXbAbnogRABwbeY1LzArZIamQOvEq8gVVSgEbhfvX7CshikYYJq3fVIavycKsOZgGmll2lIKRvRAXQtlTQG7efzy6f4zCp9HCXKljR4rv6eyGms9SQObWdMzVAvezPxP6+bmejGz7lMMwOSLRZFmcAmwbMYcJ8rYEZMLKFMcXsrZkOqKDM2rIoNwVt+eZW0LmueTezxqlq/K+IooxN0is6Rh65RHT2gBmoihsboGb2iNyd3Xpx352PRWnKKmWP0B87nD3XCkZQ=AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU92IoBeh6MVjBfsBbQib7aRdutmE3U2hhP4TLx4U8eo/8ea/cdvmoK0PBh7vzTAzL0wF14aQb6e0tr6xuVXeruzs7u0fuIdHLZ1kikGTJSJRnZBqEFxC03AjoJMqoHEooB2O7md+ewxK80Q+mUkKfkwHkkecUWOlwHXbAbnogRABwbeY1LzArZIamQOvEq8gVVSgEbhfvX7CshikYYJq3fVIavycKsOZgGmll2lIKRvRAXQtlTQG7efzy6f4zCp9HCXKljR4rv6eyGms9SQObWdMzVAvezPxP6+bmejGz7lMMwOSLRZFmcAmwbMYcJ8rYEZMLKFMcXsrZkOqKDM2rIoNwVt+eZW0LmueTezxqlq/K+IooxN0is6Rh65RHT2gBmoihsboGb2iNyd3Xpx352PRWnKKmWP0B87nD3XCkZQ=AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU92IoBeh6MVjBfsBbQib7aRdutmE3U2hhP4TLx4U8eo/8ea/cdvmoK0PBh7vzTAzL0wF14aQb6e0tr6xuVXeruzs7u0fuIdHLZ1kikGTJSJRnZBqEFxC03AjoJMqoHEooB2O7md+ewxK80Q+mUkKfkwHkkecUWOlwHXbAbnogRABwbeY1LzArZIamQOvEq8gVVSgEbhfvX7CshikYYJq3fVIavycKsOZgGmll2lIKRvRAXQtlTQG7efzy6f4zCp9HCXKljR4rv6eyGms9SQObWdMzVAvezPxP6+bmejGz7lMMwOSLRZFmcAmwbMYcJ8rYEZMLKFMcXsrZkOqKDM2rIoNwVt+eZW0LmueTezxqlq/K+IooxN0is6Rh65RHT2gBmoihsboGb2iNyd3Xpx352PRWnKKmWP0B87nD3XCkZQ=AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU92IoBeh6MVjBfsBbQib7aRdutmE3U2hhP4TLx4U8eo/8ea/cdvmoK0PBh7vzTAzL0wF14aQb6e0tr6xuVXeruzs7u0fuIdHLZ1kikGTJSJRnZBqEFxC03AjoJMqoHEooB2O7md+ewxK80Q+mUkKfkwHkkecUWOlwHXbAbnogRABwbeY1LzArZIamQOvEq8gVVSgEbhfvX7CshikYYJq3fVIavycKsOZgGmll2lIKRvRAXQtlTQG7efzy6f4zCp9HCXKljR4rv6eyGms9SQObWdMzVAvezPxP6+bmejGz7lMMwOSLRZFmcAmwbMYcJ8rYEZMLKFMcXsrZkOqKDM2rIoNwVt+eZW0LmueTezxqlq/K+IooxN0is6Rh65RHT2gBmoihsboGb2iNyd3Xpx352PRWnKKmWP0B87nD3XCkZQ=


W0/`0 = 10AAAB+HicbVBNS8NAEJ34WetHox69LBbBU92IoBeh6MVjBfsBbQib7aZdutmE3Y1QQ3+JFw+KePWnePPfuG1z0NYHA4/3ZpiZF6aCa4Pxt7Oyura+sVnaKm/v7O5V3P2Dlk4yRVmTJiJRnZBoJrhkTcONYJ1UMRKHgrXD0e3Ubz8ypXkiH8w4ZX5MBpJHnBJjpcCttAN81mNCBBhdIw8HbhXX8AxomXgFqUKBRuB+9foJzWImDRVE666HU+PnRBlOBZuUe5lmKaEjMmBdSyWJmfbz2eETdGKVPooSZUsaNFN/T+Qk1noch7YzJmaoF72p+J/XzUx05edcpplhks4XRZlAJkHTFFCfK0aNGFtCqOL2VkSHRBFqbFZlG4K3+PIyaZ3XPFzz7i+q9ZsijhIcwTGcggeXUIc7aEATKGTwDK/w5jw5L8678zFvXXGKmUP4A+fzBwbUkVw=AAAB+HicbVBNS8NAEJ34WetHox69LBbBU92IoBeh6MVjBfsBbQib7aZdutmE3Y1QQ3+JFw+KePWnePPfuG1z0NYHA4/3ZpiZF6aCa4Pxt7Oyura+sVnaKm/v7O5V3P2Dlk4yRVmTJiJRnZBoJrhkTcONYJ1UMRKHgrXD0e3Ubz8ypXkiH8w4ZX5MBpJHnBJjpcCttAN81mNCBBhdIw8HbhXX8AxomXgFqUKBRuB+9foJzWImDRVE666HU+PnRBlOBZuUe5lmKaEjMmBdSyWJmfbz2eETdGKVPooSZUsaNFN/T+Qk1noch7YzJmaoF72p+J/XzUx05edcpplhks4XRZlAJkHTFFCfK0aNGFtCqOL2VkSHRBFqbFZlG4K3+PIyaZ3XPFzz7i+q9ZsijhIcwTGcggeXUIc7aEATKGTwDK/w5jw5L8678zFvXXGKmUP4A+fzBwbUkVw=AAAB+HicbVBNS8NAEJ34WetHox69LBbBU92IoBeh6MVjBfsBbQib7aZdutmE3Y1QQ3+JFw+KePWnePPfuG1z0NYHA4/3ZpiZF6aCa4Pxt7Oyura+sVnaKm/v7O5V3P2Dlk4yRVmTJiJRnZBoJrhkTcONYJ1UMRKHgrXD0e3Ubz8ypXkiH8w4ZX5MBpJHnBJjpcCttAN81mNCBBhdIw8HbhXX8AxomXgFqUKBRuB+9foJzWImDRVE666HU+PnRBlOBZuUe5lmKaEjMmBdSyWJmfbz2eETdGKVPooSZUsaNFN/T+Qk1noch7YzJmaoF72p+J/XzUx05edcpplhks4XRZlAJkHTFFCfK0aNGFtCqOL2VkSHRBFqbFZlG4K3+PIyaZ3XPFzz7i+q9ZsijhIcwTGcggeXUIc7aEATKGTwDK/w5jw5L8678zFvXXGKmUP4A+fzBwbUkVw=AAAB+HicbVBNS8NAEJ34WetHox69LBbBU92IoBeh6MVjBfsBbQib7aZdutmE3Y1QQ3+JFw+KePWnePPfuG1z0NYHA4/3ZpiZF6aCa4Pxt7Oyura+sVnaKm/v7O5V3P2Dlk4yRVmTJiJRnZBoJrhkTcONYJ1UMRKHgrXD0e3Ubz8ypXkiH8w4ZX5MBpJHnBJjpcCttAN81mNCBBhdIw8HbhXX8AxomXgFqUKBRuB+9foJzWImDRVE666HU+PnRBlOBZuUe5lmKaEjMmBdSyWJmfbz2eETdGKVPooSZUsaNFN/T+Qk1noch7YzJmaoF72p+J/XzUx05edcpplhks4XRZlAJkHTFFCfK0aNGFtCqOL2VkSHRBFqbFZlG4K3+PIyaZ3XPFzz7i+q9ZsijhIcwTGcggeXUIc7aEATKGTwDK/w5jw5L8678zFvXXGKmUP4A+fzBwbUkVw=

�2R = 3AAAB9XicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgl6EohePVewHtNuSTbNtaJJdkqxSlv4PLx4U8ep/8ea/MW33oK0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOgoUYTWScQj1QqwppxJWjfMcNqKFcUi4LQZjG6mfvORKs0i+WDGMfUFHkgWMoKNlbodzQYC9+67FXSFznrFklt2Z0DLxMtICTLUesWvTj8iiaDSEI61bntubPwUK8MIp5NCJ9E0xmSEB7RtqcSCaj+dXT1BJ1bpozBStqRBM/X3RIqF1mMR2E6BzVAvelPxP6+dmPDST5mME0MlmS8KE45MhKYRoD5TlBg+tgQTxeytiAyxwsTYoAo2BG/x5WXSqJQ9t+zdnZeq11kceTiCYzgFDy6gCrdQgzoQUPAMr/DmPDkvzrvzMW/NOdnMIfyB8/kD0QyRZA==AAAB9XicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgl6EohePVewHtNuSTbNtaJJdkqxSlv4PLx4U8ep/8ea/MW33oK0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOgoUYTWScQj1QqwppxJWjfMcNqKFcUi4LQZjG6mfvORKs0i+WDGMfUFHkgWMoKNlbodzQYC9+67FXSFznrFklt2Z0DLxMtICTLUesWvTj8iiaDSEI61bntubPwUK8MIp5NCJ9E0xmSEB7RtqcSCaj+dXT1BJ1bpozBStqRBM/X3RIqF1mMR2E6BzVAvelPxP6+dmPDST5mME0MlmS8KE45MhKYRoD5TlBg+tgQTxeytiAyxwsTYoAo2BG/x5WXSqJQ9t+zdnZeq11kceTiCYzgFDy6gCrdQgzoQUPAMr/DmPDkvzrvzMW/NOdnMIfyB8/kD0QyRZA==AAAB9XicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgl6EohePVewHtNuSTbNtaJJdkqxSlv4PLx4U8ep/8ea/MW33oK0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOgoUYTWScQj1QqwppxJWjfMcNqKFcUi4LQZjG6mfvORKs0i+WDGMfUFHkgWMoKNlbodzQYC9+67FXSFznrFklt2Z0DLxMtICTLUesWvTj8iiaDSEI61bntubPwUK8MIp5NCJ9E0xmSEB7RtqcSCaj+dXT1BJ1bpozBStqRBM/X3RIqF1mMR2E6BzVAvelPxP6+dmPDST5mME0MlmS8KE45MhKYRoD5TlBg+tgQTxeytiAyxwsTYoAo2BG/x5WXSqJQ9t+zdnZeq11kceTiCYzgFDy6gCrdQgzoQUPAMr/DmPDkvzrvzMW/NOdnMIfyB8/kD0QyRZA==AAAB9XicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgl6EohePVewHtNuSTbNtaJJdkqxSlv4PLx4U8ep/8ea/MW33oK0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOgoUYTWScQj1QqwppxJWjfMcNqKFcUi4LQZjG6mfvORKs0i+WDGMfUFHkgWMoKNlbodzQYC9+67FXSFznrFklt2Z0DLxMtICTLUesWvTj8iiaDSEI61bntubPwUK8MIp5NCJ9E0xmSEB7RtqcSCaj+dXT1BJ1bpozBStqRBM/X3RIqF1mMR2E6BzVAvelPxP6+dmPDST5mME0MlmS8KE45MhKYRoD5TlBg+tgQTxeytiAyxwsTYoAo2BG/x5WXSqJQ9t+zdnZeq11kceTiCYzgFDy6gCrdQgzoQUPAMr/DmPDkvzrvzMW/NOdnMIfyB8/kD0QyRZA==


J. Peñano, J.P. Palastro, B. Hazi, 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.

2D (x,y) beams

11

Channeling is degraded by field incoherence

(a)

(b)

⇢0

⇢0

FWHM

FWHM

W

W

11




Solid curves: theoretical plane wave coherence radius for W0/`0 = 0.1

AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU92IoBeh6MVjBfsBbQib7aRdutmE3U2hhP4TLx4U8eo/8ea/cdvmoK0PBh7vzTAzL0wF14aQb6e0tr6xuVXeruzs7u0fuIdHLZ1kikGTJSJRnZBqEFxC03AjoJMqoHEooB2O7md+ewxK80Q+mUkKfkwHkkecUWOlwHXbAbnogRABwbeY1LzArZIamQOvEq8gVVSgEbhfvX7CshikYYJq3fVIavycKsOZgGmll2lIKRvRAXQtlTQG7efzy6f4zCp9HCXKljR4rv6eyGms9SQObWdMzVAvezPxP6+bmejGz7lMMwOSLRZFmcAmwbMYcJ8rYEZMLKFMcXsrZkOqKDM2rIoNwVt+eZW0LmueTezxqlq/K+IooxN0is6Rh65RHT2gBmoihsboGb2iNyd3Xpx352PRWnKKmWP0B87nD3XCkZQ=AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU92IoBeh6MVjBfsBbQib7aRdutmE3U2hhP4TLx4U8eo/8ea/cdvmoK0PBh7vzTAzL0wF14aQb6e0tr6xuVXeruzs7u0fuIdHLZ1kikGTJSJRnZBqEFxC03AjoJMqoHEooB2O7md+ewxK80Q+mUkKfkwHkkecUWOlwHXbAbnogRABwbeY1LzArZIamQOvEq8gVVSgEbhfvX7CshikYYJq3fVIavycKsOZgGmll2lIKRvRAXQtlTQG7efzy6f4zCp9HCXKljR4rv6eyGms9SQObWdMzVAvezPxP6+bmejGz7lMMwOSLRZFmcAmwbMYcJ8rYEZMLKFMcXsrZkOqKDM2rIoNwVt+eZW0LmueTezxqlq/K+IooxN0is6Rh65RHT2gBmoihsboGb2iNyd3Xpx352PRWnKKmWP0B87nD3XCkZQ=AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU92IoBeh6MVjBfsBbQib7aRdutmE3U2hhP4TLx4U8eo/8ea/cdvmoK0PBh7vzTAzL0wF14aQb6e0tr6xuVXeruzs7u0fuIdHLZ1kikGTJSJRnZBqEFxC03AjoJMqoHEooB2O7md+ewxK80Q+mUkKfkwHkkecUWOlwHXbAbnogRABwbeY1LzArZIamQOvEq8gVVSgEbhfvX7CshikYYJq3fVIavycKsOZgGmll2lIKRvRAXQtlTQG7efzy6f4zCp9HCXKljR4rv6eyGms9SQObWdMzVAvezPxP6+bmejGz7lMMwOSLRZFmcAmwbMYcJ8rYEZMLKFMcXsrZkOqKDM2rIoNwVt+eZW0LmueTezxqlq/K+IooxN0is6Rh65RHT2gBmoihsboGb2iNyd3Xpx352PRWnKKmWP0B87nD3XCkZQ=AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU92IoBeh6MVjBfsBbQib7aRdutmE3U2hhP4TLx4U8eo/8ea/cdvmoK0PBh7vzTAzL0wF14aQb6e0tr6xuVXeruzs7u0fuIdHLZ1kikGTJSJRnZBqEFxC03AjoJMqoHEooB2O7md+ewxK80Q+mUkKfkwHkkecUWOlwHXbAbnogRABwbeY1LzArZIamQOvEq8gVVSgEbhfvX7CshikYYJq3fVIavycKsOZgGmll2lIKRvRAXQtlTQG7efzy6f4zCp9HCXKljR4rv6eyGms9SQObWdMzVAvezPxP6+bmejGz7lMMwOSLRZFmcAmwbMYcJ8rYEZMLKFMcXsrZkOqKDM2rIoNwVt+eZW0LmueTezxqlq/K+IooxN0is6Rh65RHT2gBmoihsboGb2iNyd3Xpx352PRWnKKmWP0B87nD3XCkZQ=


• Channeling of the central hot spot is lost when the coherence length of the field becomes smaller than the beam rms radius. (beam and “reservoir” become incoherent)

• Coherence length of the channeling Gaussian beam is characteristic of that of a plane wave.


12

Longitudinal Compression of Laser Pulses in Air

2/1221)(

÷÷÷

ø

ö

ççç

è

æ÷÷ø

öççè

æ+÷÷

ø

öççè

æ+=

TToo Z

zZz

TzT b

• Pulse duration after distance z:

22 2/ boT TZ =

time

• Arranging spectral content of the pulse so frequency decreases from front to back (negative chirp) results in longitudinal compression (for wavelengths ~ 1µm where b2(w)>0)

vg(ω)

front back

β2 (ω)> 0

β0 =T02β2

∂T∂z z=0

Compression of a chirped pulse

12

• Air is a dispersive medium at optical frequencies, i.e., frequencies have different group velocities

• Ultrashort pulses have sufficient bandwidth such that atmospheric dispersion can compress or stretch the pulse significantly over hundreds of meters of propagation

13

0 2 4 6 8 10 12 140

1

2

3

4

5

6

Distance [z/Zr0]

diffr

actio

n-lim

ited

radi

us

Nor

mal

ized

uni

ts

RMS r

adius

P/Pcrit

FWHM radius

IntensityRMS pulse duration

Laser intensity animation

Radi

al c

oord

inat

e [a

.u.]

t – z/vg [psec]~50% energy lost

In the presence of extinction, the pulse must be chirped to preserve peak power and maintain channeling

T (z) = T0h(1 + �0z/ZT )

2 + (z/ZT )2i1/2

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

• Pulse duration vs. z:

• Optimal chirp: �0 ⇡ ↵T0E0/(�2PNL)AAACGHicbVDLSgMxFM34rPU16tJNsAh1084UQZdFEVyIVOgLOmW4k6ZtaGYmJBmxDP0MN/6KGxeKuO3OvzF9LLT1wOUezrmX5J5AcKa043xbK6tr6xubma3s9s7u3r59cFhXcSIJrZGYx7IZgKKcRbSmmea0KSSFMOC0EQyuJ37jkUrF4qiqh4K2Q+hFrMsIaCP5dtELqAbfwR4IIeMn07noA64a6cZ3ivmZX8IVP72/G535ds4pOFPgZeLOSQ7NUfHtsdeJSRLSSBMOSrVcR+h2ClIzwuko6yWKCiAD6NGWoRGEVLXT6WEjfGqUDu7G0lSk8VT9vZFCqNQwDMxkCLqvFr2J+J/XSnT3sp2ySCSaRmT2UDfhWMd4khLuMEmJ5kNDgEhm/opJHyQQbbLMmhDcxZOXSb1UcJ2C+3CeK1/N48igY3SC8shFF6iMblEF1RBBz+gVvaMP68V6sz6tr9noijXfOUJ/YI1/AEXRngE=AAACGHicbVDLSgMxFM34rPU16tJNsAh1084UQZdFEVyIVOgLOmW4k6ZtaGYmJBmxDP0MN/6KGxeKuO3OvzF9LLT1wOUezrmX5J5AcKa043xbK6tr6xubma3s9s7u3r59cFhXcSIJrZGYx7IZgKKcRbSmmea0KSSFMOC0EQyuJ37jkUrF4qiqh4K2Q+hFrMsIaCP5dtELqAbfwR4IIeMn07noA64a6cZ3ivmZX8IVP72/G535ds4pOFPgZeLOSQ7NUfHtsdeJSRLSSBMOSrVcR+h2ClIzwuko6yWKCiAD6NGWoRGEVLXT6WEjfGqUDu7G0lSk8VT9vZFCqNQwDMxkCLqvFr2J+J/XSnT3sp2ySCSaRmT2UDfhWMd4khLuMEmJ5kNDgEhm/opJHyQQbbLMmhDcxZOXSb1UcJ2C+3CeK1/N48igY3SC8shFF6iMblEF1RBBz+gVvaMP68V6sz6tr9noijXfOUJ/YI1/AEXRngE=AAACGHicbVDLSgMxFM34rPU16tJNsAh1084UQZdFEVyIVOgLOmW4k6ZtaGYmJBmxDP0MN/6KGxeKuO3OvzF9LLT1wOUezrmX5J5AcKa043xbK6tr6xubma3s9s7u3r59cFhXcSIJrZGYx7IZgKKcRbSmmea0KSSFMOC0EQyuJ37jkUrF4qiqh4K2Q+hFrMsIaCP5dtELqAbfwR4IIeMn07noA64a6cZ3ivmZX8IVP72/G535ds4pOFPgZeLOSQ7NUfHtsdeJSRLSSBMOSrVcR+h2ClIzwuko6yWKCiAD6NGWoRGEVLXT6WEjfGqUDu7G0lSk8VT9vZFCqNQwDMxkCLqvFr2J+J/XSnT3sp2ySCSaRmT2UDfhWMd4khLuMEmJ5kNDgEhm/opJHyQQbbLMmhDcxZOXSb1UcJ2C+3CeK1/N48igY3SC8shFF6iMblEF1RBBz+gVvaMP68V6sz6tr9noijXfOUJ/YI1/AEXRngE=AAACGHicbVDLSgMxFM34rPU16tJNsAh1084UQZdFEVyIVOgLOmW4k6ZtaGYmJBmxDP0MN/6KGxeKuO3OvzF9LLT1wOUezrmX5J5AcKa043xbK6tr6xubma3s9s7u3r59cFhXcSIJrZGYx7IZgKKcRbSmmea0KSSFMOC0EQyuJ37jkUrF4qiqh4K2Q+hFrMsIaCP5dtELqAbfwR4IIeMn07noA64a6cZ3ivmZX8IVP72/G535ds4pOFPgZeLOSQ7NUfHtsdeJSRLSSBMOSrVcR+h2ClIzwuko6yWKCiAD6NGWoRGEVLXT6WEjfGqUDu7G0lSk8VT9vZFCqNQwDMxkCLqvFr2J+J/XSnT3sp2ySCSaRmT2UDfhWMd4khLuMEmJ5kNDgEhm/opJHyQQbbLMmhDcxZOXSb1UcJ2C+3CeK1/N48igY3SC8shFF6iMblEF1RBBz+gVvaMP68V6sz6tr9noijXfOUJ/YI1/AEXRngE=

T 0(z) = E0(z)/P�1NLAAACAnicbVDLSsNAFJ3UV62vqCtxM1jEurAmIuhGKIrgQqRCX9DGMJlO2qGTBzMToYbgxl9x40IRt36FO//GSZuFth643MM59zJzjxMyKqRhfGu5mdm5+YX8YmFpeWV1TV/faIgg4pjUccAC3nKQIIz6pC6pZKQVcoI8h5GmM7hI/eY94YIGfk0OQ2J5qOdTl2IklWTrW7W90sM+PIOXaT+s2vHNdXIXH5iJrReNsjECnCZmRoogQ9XWvzrdAEce8SVmSIi2aYTSihGXFDOSFDqRICHCA9QjbUV95BFhxaMTErirlC50A67Kl3Ck/t6IkSfE0HPUpIdkX0x6qfif146ke2rF1A8jSXw8fsiNGJQBTPOAXcoJlmyoCMKcqr9C3EccYalSK6gQzMmTp0njqGwaZfP2uFg5z+LIg22wA0rABCegAq5AFdQBBo/gGbyCN+1Je9HetY/xaE7LdjbBH2ifPzcAlMM=AAACAnicbVDLSsNAFJ3UV62vqCtxM1jEurAmIuhGKIrgQqRCX9DGMJlO2qGTBzMToYbgxl9x40IRt36FO//GSZuFth643MM59zJzjxMyKqRhfGu5mdm5+YX8YmFpeWV1TV/faIgg4pjUccAC3nKQIIz6pC6pZKQVcoI8h5GmM7hI/eY94YIGfk0OQ2J5qOdTl2IklWTrW7W90sM+PIOXaT+s2vHNdXIXH5iJrReNsjECnCZmRoogQ9XWvzrdAEce8SVmSIi2aYTSihGXFDOSFDqRICHCA9QjbUV95BFhxaMTErirlC50A67Kl3Ck/t6IkSfE0HPUpIdkX0x6qfif146ke2rF1A8jSXw8fsiNGJQBTPOAXcoJlmyoCMKcqr9C3EccYalSK6gQzMmTp0njqGwaZfP2uFg5z+LIg22wA0rABCegAq5AFdQBBo/gGbyCN+1Je9HetY/xaE7LdjbBH2ifPzcAlMM=AAACAnicbVDLSsNAFJ3UV62vqCtxM1jEurAmIuhGKIrgQqRCX9DGMJlO2qGTBzMToYbgxl9x40IRt36FO//GSZuFth643MM59zJzjxMyKqRhfGu5mdm5+YX8YmFpeWV1TV/faIgg4pjUccAC3nKQIIz6pC6pZKQVcoI8h5GmM7hI/eY94YIGfk0OQ2J5qOdTl2IklWTrW7W90sM+PIOXaT+s2vHNdXIXH5iJrReNsjECnCZmRoogQ9XWvzrdAEce8SVmSIi2aYTSihGXFDOSFDqRICHCA9QjbUV95BFhxaMTErirlC50A67Kl3Ck/t6IkSfE0HPUpIdkX0x6qfif146ke2rF1A8jSXw8fsiNGJQBTPOAXcoJlmyoCMKcqr9C3EccYalSK6gQzMmTp0njqGwaZfP2uFg5z+LIg22wA0rABCegAq5AFdQBBo/gGbyCN+1Je9HetY/xaE7LdjbBH2ifPzcAlMM=AAACAnicbVDLSsNAFJ3UV62vqCtxM1jEurAmIuhGKIrgQqRCX9DGMJlO2qGTBzMToYbgxl9x40IRt36FO//GSZuFth643MM59zJzjxMyKqRhfGu5mdm5+YX8YmFpeWV1TV/faIgg4pjUccAC3nKQIIz6pC6pZKQVcoI8h5GmM7hI/eY94YIGfk0OQ2J5qOdTl2IklWTrW7W90sM+PIOXaT+s2vHNdXIXH5iJrReNsjECnCZmRoogQ9XWvzrdAEce8SVmSIi2aYTSihGXFDOSFDqRICHCA9QjbUV95BFhxaMTErirlC50A67Kl3Ck/t6IkSfE0HPUpIdkX0x6qfif146ke2rF1A8jSXw8fsiNGJQBTPOAXcoJlmyoCMKcqr9C3EccYalSK6gQzMmTp0njqGwaZfP2uFg5z+LIg22wA0rABCegAq5AFdQBBo/gGbyCN+1Je9HetY/xaE7LdjbBH2ifPzcAlMM=

(compression rate equal to energy loss rate)


ZT = T20 /(2|�2|)

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

;

Validation Experiments at Increasing Distances

Lab environment at NRLkTFL laser (5 mJ, 35 fsec, 1 kHz, 800 nm)Nonlinear channeling in turbulence

AFRL FacilityPheenix laser (40 TW, 35 fsec, 10 Hz, 800nm)Nonlinear channeling in deep turbulence (Rytov > 5)

Carderock FacilityAstrella laser (7 mJ, 35 fsec, 1 kHz, 800 nm)Nonlinear propagation through turbulencePulse compressionControl of nonlinear focal range

2015-16

2016

2017

15

Laboratory Turbulence Generator for high-power laser propagation experiments*

turnbucklesTOP

VIEW

• 5 orders of magnitude range of turbulence• Kolmogorov spectrum over the inertial sub-range• Ideal for high-power experiments (no phase plates to damage)

Powe

r Spe

ctru

m o

f Acc

elera

tion

(m3 /

s4)

*G. DiComo et al. "Implementation of a long range, distributed-volume, continuously variable turbulence generator," Applied Optics 55, 5192 (2016).S. Pond, R. W. Stewart, and R. W. Burling, Journal of Atmospheric Sciences, vol. 20, pp. 319-324 (1963).

16

Experimental Facility for Validation at ~850m

• Continuous indoor path with low ambient turbulence

• Distributed turbulence with tunable Rytovfrom 10-3 to >10

• Two weeks to build

1 km

16

• Laser operation during graveyard shift over 2 months

• Laser intensity profile and pulse length fully characterized

• Integrated Vitara seed laser, revolution pump laser, STAR regenerative amplifier and sealed, compact stretcher/compressor

• HASS* verified (ruggedized for portability)• All major sub-systems thermally-stabilized for reliable

long-term performance • Water-only cooled Ti:Sapphire rod assembly for improved

beam quality and thermal management (M2 < 1.25, stability

USPL(Saturated)

HeNeDiagnostic

Weak Turbulence (!R ~ 0.05) Strong Turbulence (!R ~ 5)

Vide

o Ca

mer

a1

kHz,

12bi

t Cam

era

Self-Channeling Observed in Strong Turbulence

5 cm 5 cm

15 cm 15 cm

High-power beam remains small (

Experimental measurement of temporal profile

19

• Built SHG FROG to measure complex field (Can measure low intensities)

• Pulse was characterized at laser and received end of range

• Nonlinear effects such as self-phase modulation and self-steepening can be observed for the high power pulse

initial compressed (exp)

compressed (sim)

initial compressed (exp)

compressed (sim)

20

Simulated Beam Profile(contrast enhanced)

1x zoom

Scintillation Index

Hot Spot Beam Size

Experiments validate nonlinear self-focusing models in strong turbulence

RMS spot size

FWHM spot size

Initial rms spot

Initial rms spot

Experimental Beam Profile(contrast enhanced)

1x zoom

Simulated Beam Profile(contrast enhanced)

1x zoom

5 cm

simulation

expe

rimen

t

21

Beam wander is consistent with linear physics

22

Self-focusing of non-Gaussian beams

• For a given rms spot size, Gaussian beams have the longest diffraction length compared with any other beam shape

• Experimentally generated laser beams are never purely Gaussian, nor perfectly coherent

• What are the nonlinear self-focusing properties of non-Gaussian, partially coherent beams? Do they afford any propagation advantages?- Bessel beams- Partially coherent beams

23

The Critical Power of an Arbitrary Beam Shape

• Define the critical power as the condition M2 = 0AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9CIUvXgRKtgPaGPZbDft0s0m7E6EEvojvHhQxKu/x5v/xm2bg7Y+GHi8N8PMvCCRwqDrfjsrq2vrG5uFreL2zu7efungsGniVDPeYLGMdTughkuheAMFSt5ONKdRIHkrGN1M/dYT10bE6gHHCfcjOlAiFIyilVp3j1VyRdxeqexW3BnIMvFyUoYc9V7pq9uPWRpxhUxSYzqem6CfUY2CST4pdlPDE8pGdMA7lioaceNns3Mn5NQqfRLG2pZCMlN/T2Q0MmYcBbYzojg0i95U/M/rpBhe+plQSYpcsfmiMJUEYzL9nfSF5gzl2BLKtLC3EjakmjK0CRVtCN7iy8ukWa14bsW7Py/XrvM4CnAMJ3AGHlxADW6hDg1gMIJneIU3J3FenHfnY9664uQzR/AHzucPaIqOSg==AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9CIUvXgRKtgPaGPZbDft0s0m7E6EEvojvHhQxKu/x5v/xm2bg7Y+GHi8N8PMvCCRwqDrfjsrq2vrG5uFreL2zu7efungsGniVDPeYLGMdTughkuheAMFSt5ONKdRIHkrGN1M/dYT10bE6gHHCfcjOlAiFIyilVp3j1VyRdxeqexW3BnIMvFyUoYc9V7pq9uPWRpxhUxSYzqem6CfUY2CST4pdlPDE8pGdMA7lioaceNns3Mn5NQqfRLG2pZCMlN/T2Q0MmYcBbYzojg0i95U/M/rpBhe+plQSYpcsfmiMJUEYzL9nfSF5gzl2BLKtLC3EjakmjK0CRVtCN7iy8ukWa14bsW7Py/XrvM4CnAMJ3AGHlxADW6hDg1gMIJneIU3J3FenHfnY9664uQzR/AHzucPaIqOSg==AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9CIUvXgRKtgPaGPZbDft0s0m7E6EEvojvHhQxKu/x5v/xm2bg7Y+GHi8N8PMvCCRwqDrfjsrq2vrG5uFreL2zu7efungsGniVDPeYLGMdTughkuheAMFSt5ONKdRIHkrGN1M/dYT10bE6gHHCfcjOlAiFIyilVp3j1VyRdxeqexW3BnIMvFyUoYc9V7pq9uPWRpxhUxSYzqem6CfUY2CST4pdlPDE8pGdMA7lioaceNns3Mn5NQqfRLG2pZCMlN/T2Q0MmYcBbYzojg0i95U/M/rpBhe+plQSYpcsfmiMJUEYzL9nfSF5gzl2BLKtLC3EjakmjK0CRVtCN7iy8ukWa14bsW7Py/XrvM4CnAMJ3AGHlxADW6hDg1gMIJneIU3J3FenHfnY9664uQzR/AHzucPaIqOSg==AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9CIUvXgRKtgPaGPZbDft0s0m7E6EEvojvHhQxKu/x5v/xm2bg7Y+GHi8N8PMvCCRwqDrfjsrq2vrG5uFreL2zu7efungsGniVDPeYLGMdTughkuheAMFSt5ONKdRIHkrGN1M/dYT10bE6gHHCfcjOlAiFIyilVp3j1VyRdxeqexW3BnIMvFyUoYc9V7pq9uPWRpxhUxSYzqem6CfUY2CST4pdlPDE8pGdMA7lioaceNns3Mn5NQqfRLG2pZCMlN/T2Q0MmYcBbYzojg0i95U/M/rpBhe+plQSYpcsfmiMJUEYzL9nfSF5gzl2BLKtLC3EjakmjK0CRVtCN7iy8ukWa14bsW7Py/XrvM4CnAMJ3AGHlxADW6hDg1gMIJneIU3J3FenHfnY9664uQzR/AHzucPaIqOSg==

• The ensemble averaged beam spot size, w(z), along the propagation axis z is

w2(z) =2

P

Zr2hI(r, z)id2r � 2hR2c(z)i

(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)

• The spot size satisfies the following mathematical identity @2w

@z2=

�20n20⇡

2

M4(z)

w3(null)(null)(null)(null)

where the ensemble beam quality (spreading angle normalized to the the Gaussian spreading angle) is:

• In the absence of turbulence, it can be shown that M4 is a conserved quantity. If M4can be evaluated at z=0 then we will know how the ensemble of beams will propagate.

M4(0) =n20⇡

2

�0

(�14

✓@w2

@z

��0

◆2+

w(0)2

2

@2w2

@z2

��0

)

(null)(null)(null)(null)

24

@2w2

@z2=

2

P

Zr2

⌧@2I

@z2

�d2r =

2

P

cn08⇡

Zr2

*��@A

@z

��2

+A?@2A

@z2+ c.c.

+d2r


• Only the 2nd term is important in evaluating M4 (the first term represents a “lens” contribution which does not affect M4):

M4(0) =n20⇡

2

�0

(�14

✓@w2

@z

��0

◆2+

w(0)2

2

@2w2

@z2

��0

)


The Critical Power of an Arbitrary Beam Shape (cont.)

• The physics of the laser propagation is introduced through the nonlinear Schodingerequation (NLSE)

@A

@z=

i

2k0

✓r2?A+ 2k20

n20n2c

8⇡|A|2 A

◆


diffraction self-focusing

• Using a Gaussian beam as the initial condition in this formalism recovers the usual expression for the self-focusing power:

PG = �2/(2⇡n0n2)

AAACB3icbVDLSsNAFJ3UV62vqEtBBotQNzUJgm6EogtdVrAPaGKYTCbt0MkkzEyEErpz46+4caGIW3/BnX/jtM1CqwcuHM65h5l7gpRRqSzryygtLC4tr5RXK2vrG5tb5vZOWyaZwKSFE5aIboAkYZSTlqKKkW4qCIoDRjrB8HLid+6JkDTht2qUEi9GfU4jipHSkm/uN/0reA5dpiMhunOOa46bUsh9S49z5JtVq25NAf8SuyBVUKDpm59umOAsJlxhhqTs2VaqvBwJRTEj44qbSZIiPER90tOUo5hIL5/eMYaHWglhlAg9XMGp+jORo1jKURzozRipgZz3JuJ/Xi9T0ZmXU55minA8eyjKGFQJnJQCQyoIVmykCcKC6r9CPEACYaWrq+gS7PmT/5K2U7etun1zUm1cFHWUwR44ADVgg1PQANegCVoAgwfwBF7Aq/FoPBtvxvtstWQUmV3wC8bHN9eOlro=AAACB3icbVDLSsNAFJ3UV62vqEtBBotQNzUJgm6EogtdVrAPaGKYTCbt0MkkzEyEErpz46+4caGIW3/BnX/jtM1CqwcuHM65h5l7gpRRqSzryygtLC4tr5RXK2vrG5tb5vZOWyaZwKSFE5aIboAkYZSTlqKKkW4qCIoDRjrB8HLid+6JkDTht2qUEi9GfU4jipHSkm/uN/0reA5dpiMhunOOa46bUsh9S49z5JtVq25NAf8SuyBVUKDpm59umOAsJlxhhqTs2VaqvBwJRTEj44qbSZIiPER90tOUo5hIL5/eMYaHWglhlAg9XMGp+jORo1jKURzozRipgZz3JuJ/Xi9T0ZmXU55minA8eyjKGFQJnJQCQyoIVmykCcKC6r9CPEACYaWrq+gS7PmT/5K2U7etun1zUm1cFHWUwR44ADVgg1PQANegCVoAgwfwBF7Aq/FoPBtvxvtstWQUmV3wC8bHN9eOlro=AAACB3icbVDLSsNAFJ3UV62vqEtBBotQNzUJgm6EogtdVrAPaGKYTCbt0MkkzEyEErpz46+4caGIW3/BnX/jtM1CqwcuHM65h5l7gpRRqSzryygtLC4tr5RXK2vrG5tb5vZOWyaZwKSFE5aIboAkYZSTlqKKkW4qCIoDRjrB8HLid+6JkDTht2qUEi9GfU4jipHSkm/uN/0reA5dpiMhunOOa46bUsh9S49z5JtVq25NAf8SuyBVUKDpm59umOAsJlxhhqTs2VaqvBwJRTEj44qbSZIiPER90tOUo5hIL5/eMYaHWglhlAg9XMGp+jORo1jKURzozRipgZz3JuJ/Xi9T0ZmXU55minA8eyjKGFQJnJQCQyoIVmykCcKC6r9CPEACYaWrq+gS7PmT/5K2U7etun1zUm1cFHWUwR44ADVgg1PQANegCVoAgwfwBF7Aq/FoPBtvxvtstWQUmV3wC8bHN9eOlro=AAACB3icbVDLSsNAFJ3UV62vqEtBBotQNzUJgm6EogtdVrAPaGKYTCbt0MkkzEyEErpz46+4caGIW3/BnX/jtM1CqwcuHM65h5l7gpRRqSzryygtLC4tr5RXK2vrG5tb5vZOWyaZwKSFE5aIboAkYZSTlqKKkW4qCIoDRjrB8HLid+6JkDTht2qUEi9GfU4jipHSkm/uN/0reA5dpiMhunOOa46bUsh9S49z5JtVq25NAf8SuyBVUKDpm59umOAsJlxhhqTs2VaqvBwJRTEj44qbSZIiPER90tOUo5hIL5/eMYaHWglhlAg9XMGp+jORo1jKURzozRipgZz3JuJ/Xi9T0ZmXU55minA8eyjKGFQJnJQCQyoIVmykCcKC6r9CPEACYaWrq+gS7PmT/5K2U7etun1zUm1cFHWUwR44ADVgg1PQANegCVoAgwfwBF7Aq/FoPBtvxvtstWQUmV3wC8bHN9eOlro=

25

Propagation of an initially Gaussian beam near the critical power

z/ZR0

z/ZR0

W/W

0I/

I 0

P/PG = 0.8

0.95

1

1.1

P/PG =

0.8

0.95

1

1.1

(a)

(b)

FIG. 1: (a) Normalized rms spot size (W/W0) and (b) peak intensity (I/I0) versus normalized

propagation distance (z/ZR0) for initially collimated Gaussian beams with P/PG = 0.8, 0.95, 1,

and 1.1. Curves are results of simulations that solve Eq. (5).

far beyond the center of the beam and must be properly resolved to correctly calculate W .

The oscillatory behavior seen at the edge of the plots is a numerical artifact due to the

periodic boundary conditions of the simulation and a small amount of laser energy reaching

the edges of the simulation domain.

6

z/ZR0

z/ZR0

W/W

0I/

I 0

P/PG = 0.8

0.95

1

1.1

P/PG =

0.8

0.95

1

1.1

(a)

(b)

FIG. 1: (a) Normalized rms spot size (W/W0) and (b) peak intensity (I/I0) versus normalized

propagation distance (z/ZR0) for initially collimated Gaussian beams with P/PG = 0.8, 0.95, 1,

and 1.1. Curves are results of simulations that solve Eq. (5).

far beyond the center of the beam and must be properly resolved to correctly calculate W .

The oscillatory behavior seen at the edge of the plots is a numerical artifact due to the

periodic boundary conditions of the simulation and a small amount of laser energy reaching

the edges of the simulation domain.

6

RMS spot size Peak intensity

26

Partially-Coherent Beams

• The Gaussian-Schell model is a way to generate a partially-coherent field

ha?(~k)a(~k0)i = ⇢2c

⇡e�k

2⇢2c�(~k0 � ~k)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)(null)

A(~r) =

✓8⇡I0cn0

◆✓Za(~k)ei

~krd3k

◆e�r

2/R20


where a(k), the angular correlation function of the field, and the intensity are

I(~r) =cn08⇡

hA?(~r)A(~r)i(null)(null)(null)(null)

• By applying the formalism described in the previous slides, we obtain for the normalized spreading angle

M4(0) = n20

✓1 +

R202⇢2c

� PPcrit

◆


focusingdefocusing

• The defocusing term due to incoherence is identical in form for propagation in turbulence with the coherence length replaced by the Fried parameter (the atmospheric coherence length)

Pcrit =�20

2⇡n0n2(null)(null)(null)(null)

27

Bessel beams with OAM in finite apertures

Power of a Bessel beam: Pmn = ⇡R20I0J2m�1(jmn)

• For apertured Gaussian with intensity IG to have equivalent power:

IGIB

=2R20W 20

J2m�1(jmn)

1� exp(�2R20/W 20 )

Figures show intensity profiles of Bessel beams (solid curves) and apertured Gaussian beams (dashed curves) with equivalent RMS spot size and power.

I/I G

r/R0

m = 0n = 1

I/I G

r/R0

m = 0n = 5

m = 5n = 5

I/I G

r/R0

m = 5n = 1

I/I G

r/R0

• For both beams to have equivalent RMS radius: W20

2� R

20 exp(�2R20/W 20 )

1� exp(�2R20/W 20 )=

⌦r2↵mn

Field envelope: A =✓8⇡I0n0c

◆ 12

Jm(rjmn/R0) exp(im✓)(1�H(r �R0))jmn is the nth root of Jm;aperture radius: R0

Comparison with apertured Gaussian beam:

28

PB/P

G

Radial mode number, n

m = 03510

Self-focusing power (PB) of a Bessel beam transmitted through an aperture of radius R0

PG = �2/(2⇡n0n2) is the self-focusing power of a Gaussian beam

PB =PG2

j2mnJ2m�1(jmn)

R jmn0 dx

hx4 (Jm�1(x)� Jm+1(x))

2 + m2

x J2m(x)

i

R jmn0 dxxJ

4m(x)

Self-focusing power of Bessel beams with OAM in finite apertures

29

It is seen that self-focusing occurs mainly within the central lobe, with the intensity profiles

of the outer rings remaining relatively undisturbed. Figure 8(b) shows that there is excellent

agreement between the self-focusing distance of the central lobe of various n modes obtained

from simulations and from Eq. (26).

n

simulationstheory

(a)

(b)

100

10-1

10-2

10-3

10-4

FIG. 8: (a) Normalized intensity vs. transverse coordinate at z = 0 (dashed curve), and at z = LB1,

i.e., the self-focusing length, for the m = 0, n = 7 mode with power P = PB. (b) LB1, normalized

to the Rayleigh length of the aperture, vs. n, for m = 0, and P = PB, obtained from simulations

(points), and Eq. (26) (dashed curve).

V. CONCLUSIONS

Acknowledgements

The authors acknowledge useful discussions with. This work is supported by the NRL

Base Program.

13

Bessel beams are very unstable to filamentation

• For Bessel beams at the “critical power”, filamentation is governed by smaller-scale filamentation.

• For example, the m=0 mode has a filamentation distance given by:

PB

/PG

n

m = 03510

FIG. 6: Self-focusing power versus radial mode number n for various for various angular momentum

states m.

standpoint, for a laser with a given maximum power, P , Fig. 6 indicates the optical mode

that the laser should produce to maximize its propagation e�ciency, i.e., having P ⇡ PB.

Figure 7 shows the results of simulations for the propagation of various apertured Bessel

beams in the linear regime (P

PG, the self-focusing distance of the central lobe is

LB1 ⇡ 0.257(⇡r21/�)/p

PB1/PG � 1, (26)

where r1 = j01R0/j0n. Apart from the numerical constant, Eq. (26) is the self-focusing

length of a collimated Gaussian beam with spot size r1 and power PB1, derived under the

assumption that the intensity profile evolves self-similarly.

11

which can be much shorter than the Rayleigh length of the whole beam.

Self-focusing of an apertured, high-order Bessel beam at the “critical power”

PB ⇡ 40PGAAAB+3icbVDLSgMxFM3UV62vsS7dBIvgqsyIoMtSF7ocwT6gHYZMmmlDM0lIMtIy9FfcuFDErT/izr8xbWehrQcuHM65l3vviSWj2njet1Pa2Nza3invVvb2Dw6P3ONqW4tMYdLCggnVjZEmjHLSMtQw0pWKoDRmpBOPb+d+54koTQV/NFNJwhQNOU0oRsZKkVsNoibsIymVmMArDwbRXeTWvLq3AFwnfkFqoEAQuV/9gcBZSrjBDGnd8z1pwhwpQzEjs0o/00QiPEZD0rOUo5ToMF/cPoPnVhnARChb3MCF+nsiR6nW0zS2nSkyI73qzcX/vF5mkpswp1xmhnC8XJRkDBoB50HAAVUEGza1BGFF7a0Qj5BC2Ni4KjYEf/XlddK+rPte3X+4qjWaRRxlcArOwAXwwTVogHsQgBbAYAKewSt4c2bOi/PufCxbS04xcwL+wPn8AdczkwA=AAAB+3icbVDLSgMxFM3UV62vsS7dBIvgqsyIoMtSF7ocwT6gHYZMmmlDM0lIMtIy9FfcuFDErT/izr8xbWehrQcuHM65l3vviSWj2njet1Pa2Nza3invVvb2Dw6P3ONqW4tMYdLCggnVjZEmjHLSMtQw0pWKoDRmpBOPb+d+54koTQV/NFNJwhQNOU0oRsZKkVsNoibsIymVmMArDwbRXeTWvLq3AFwnfkFqoEAQuV/9gcBZSrjBDGnd8z1pwhwpQzEjs0o/00QiPEZD0rOUo5ToMF/cPoPnVhnARChb3MCF+nsiR6nW0zS2nSkyI73qzcX/vF5mkpswp1xmhnC8XJRkDBoB50HAAVUEGza1BGFF7a0Qj5BC2Ni4KjYEf/XlddK+rPte3X+4qjWaRRxlcArOwAXwwTVogHsQgBbAYAKewSt4c2bOi/PufCxbS04xcwL+wPn8AdczkwA=AAAB+3icbVDLSgMxFM3UV62vsS7dBIvgqsyIoMtSF7ocwT6gHYZMmmlDM0lIMtIy9FfcuFDErT/izr8xbWehrQcuHM65l3vviSWj2njet1Pa2Nza3invVvb2Dw6P3ONqW4tMYdLCggnVjZEmjHLSMtQw0pWKoDRmpBOPb+d+54koTQV/NFNJwhQNOU0oRsZKkVsNoibsIymVmMArDwbRXeTWvLq3AFwnfkFqoEAQuV/9gcBZSrjBDGnd8z1pwhwpQzEjs0o/00QiPEZD0rOUo5ToMF/cPoPnVhnARChb3MCF+nsiR6nW0zS2nSkyI73qzcX/vF5mkpswp1xmhnC8XJRkDBoB50HAAVUEGza1BGFF7a0Qj5BC2Ni4KjYEf/XlddK+rPte3X+4qjWaRRxlcArOwAXwwTVogHsQgBbAYAKewSt4c2bOi/PufCxbS04xcwL+wPn8AdczkwA=AAAB+3icbVDLSgMxFM3UV62vsS7dBIvgqsyIoMtSF7ocwT6gHYZMmmlDM0lIMtIy9FfcuFDErT/izr8xbWehrQcuHM65l3vviSWj2njet1Pa2Nza3invVvb2Dw6P3ONqW4tMYdLCggnVjZEmjHLSMtQw0pWKoDRmpBOPb+d+54koTQV/NFNJwhQNOU0oRsZKkVsNoibsIymVmMArDwbRXeTWvLq3AFwnfkFqoEAQuV/9gcBZSrjBDGnd8z1pwhwpQzEjs0o/00QiPEZD0rOUo5ToMF/cPoPnVhnARChb3MCF+nsiR6nW0zS2nSkyI73qzcX/vF5mkpswp1xmhnC8XJRkDBoB50HAAVUEGza1BGFF7a0Qj5BC2Ni4KjYEf/XlddK+rPte3X+4qjWaRRxlcArOwAXwwTVogHsQgBbAYAKewSt4c2bOi/PufCxbS04xcwL+wPn8AdczkwA=

30

Using a gas lens to create a Bessel beam

• A gas or plasma can be used to create a radial refractive index variation that can act as a lens with very high damage threshold for high-intensity laser pulses.

• NRL recently created a vortex flow gas lens that acts as a negative axicon, and can be used in conjunction with a focusing lens to create Bessel beams.

Lens schematic

3D printed lens

Formation of annular beam (focuses to a Bessel beam)

31

NRL’s newly-renovated ultrashort pulse laser facility

• 350 sq. ft. ISO 4 cleanroom• 40TW 5Hz high energy laser• 0.5TW 1kHz repetition rate laser • Tunable OPA

– Currently: 1.1-2.6um – 200nm-15um July 2018

• Laser-driven electron and ion accelerators

• Ultrafast laser-matter interactions and propagation• Novel radiation sources

A unique facility that enables high-precision, pump-probe experiments involving any combination of photons at almost arbitrary wavelengths (THz thru X-ray) and high energy particles (electrons and ions)

32

Summary Remarks

• Nonlinear self-channeling is achieved when the laser pulse power is close to the self-focusing power and the transverse dimensions are

smaller than the turbulence coherence scale.

• Experiments demonstrate– Self-channeling over many Rayleigh lengths achieved– Self-channeling in deep turbulence– Self-channeling maintained over distances where temporal spreading

(GVD) becomes important

– quantitative validation of simulations

• Future work to include propagation of partially coherent, intense beams in atmospheric turbulence

32

Program Support and Funding

33

Funding for this work has been provided by

• Joint Technology Office

• Office of Naval Research

• Internal USNRL 6.1 funding

Refereed Publications

G. DiComo et al. "Implementation of a long range, distributed-volume, continuouslyvariable turbulence generator," Applied Optics 55, 5192 (2016).

J. Palastro et al. "Reciprocity breaking during nonlinear propagation of adapted beams through random media," Optics Express 24, 18817 (2016).

J. Peñano et al. “Self-Channeling of High-Power Laser Pulses Through Strong Atmospheric Turbulence,” Phys. Rev. A 96, 013829 (2017).

W. Nelson, J.P. Palastro et al. ”Using an incoherent target-return to adaptively focus through atmospheric turbuelnce," Optics Letters 41, 1301 (2016).

B. Hafizi, J. Peñano, J.P. Palastro, R.P. Fischer, and G. DiComo, “Laser beam self-focusing in turbulent dissipative media,” Optics Letters 42, 298 (2017).

J. Peñano et al. “Theoretical and numerical investigation of filament onset distance in atmospheric turbulence,” JOSA B 31, 963 (2014).

34

propagation of ultrashort, intense laser pulses through the atmosphere · 2019. 5. 14. · andrews...

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