Effects of laser beam propagation and saturation on the spatial shape of sodium laser guide stars

Fabien Marc; Hugues Guillet de Chatellus; Jean-Paul Pique

doi:10.1364/OE.17.004920

Optics Express
Vol. 17,
Issue 7,
pp. 4920-4931
(2009)
•https://doi.org/10.1364/OE.17.004920

Effects of laser beam propagation and saturation on the spatial shape of sodium laser guide stars

Fabien Marc, Hugues Guillet de Chatellus, and Jean-Paul Pique

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Fabien Marc, Hugues Guillet de Chatellus, and Jean-Paul Pique, "Effects of laser beam propagation and saturation on the spatial shape of sodium laser guide stars," Opt. Express 17, 4920-4931 (2009)

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Abstract

The possibility to produce diffraction-limited images by large telescopes through Adaptive Optics is closely linked to the precision of measurement of the position of the guide star on the wavefront sensor. In the case of laser guide stars, many parameters can lead to a strong distortion on the shape of the LGS spot. Here we study the influence of both the saturation of the sodium layer excited by different types of lasers, the spatial quality of the laser mode at the ground and the influence of the atmospheric turbulence on the upward propagation of the laser beam. Both shape and intensity of the LGS spot are found to depend strongly on these three effects with important consequences on the precision on the wavefront analysis.

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Fig. 1. Methodology of the present study. The calculation is based on two steps: i) propagation of the laser beam to the mesosphere, ii) calculation of the fluorescence spot (i.e. LGS) from the laser intensity spot. The size of each map is 300cm × 300cm. Data are normalized according to the color scale (right).

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Fig. 2. Laser spectra considered in this study. SM, PM and ML stand respectively for single mode, phase modulated and modeless lasers. The parameters of the phase modulation function are given in [10]. The black solid line is the profile of the sodium D₂ line at the temperature of the mesosphere.

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Fig. 3. Number of photons emitted per steradian, per sodium atom and per laser pulse as a function of the laser power density for different laser types (50 ns pulse duration, linear polarization). The numerical values are given by the density matrix code Beacon [10].

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Fig. 4. Evolution of the spot broadening with the laser power for different laser formats (SM: single-mode, PM: phase modulation and ML: modeless laser).

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Fig. 5. Variation of the figure of merit of the LGS with the laser power for different laser format (SM: single-mode, PM: phase modulation and ML: modeless laser).

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Fig. 6. Variation of the HWHM of the LGS spot with the laser pulse energy for a laser with M² =1. Solid lines: single mode laser, dashed lines: phase-modulated laser, dotted lines: modeless laser. Green data: r₀ = 5 cm, red data: r₀ = 10 cm and black data: r₀ infinite. Error bars correspond to the standard deviation of the data defined in Table 1.

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Fig. 7. Variation of the HWHM of the LGS spot with the laser pulse energy for a laser with M² =1.3. Solid lines: single mode laser, dashed lines: phase-modulated laser, dotted lines: modeless laser. Green data: r₀ = 5 cm, red data: r₀ = 10 cm and black data: r₀ infinite. Error bars correspond to the standard deviation of the data defined in Table 1.

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Fig. 8. Variation of the intensity of the LGS spot with the laser pulse energy for a laser with M² =1. Solid lines: single mode laser, dashed lines: phase-modulated laser, dotted lines: modeless laser. Green data: r₀ = 5 cm, red data: r₀ = 10 cm and black data: r₀ infinite.

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Fig. 9. Variation of the intensity of the LGS spot with the laser pulse energy for a laser for M² =1 (black) and M² = 1.3 (blue). r0 is infinite. Solid lines: single mode laser, dashed lines: phase-modulated laser, dotted lines: modeless laser.

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Tables (3)

Table 1. Summary of the different simulations performed for an accurate statistical treatment for each laser type (SP, PM and ML) and each pulse energy (0.05, 0.1, 0.2, 0.5, 1 and 2 mJ). The pulse width is fixed to 50 ns.

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Table 2. Surface broadening for 2mJ/50ns laser pulses for different laser formats and seeing conditions.

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Table 3. Return flux at the ground per m² of telescope produced per 2 mJ laser pulses for different laser formats and seeing conditions. The atmospheric transmission is supposed to be 100%.

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Equations (9)

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M^{2} = \frac{π r_{1 / e^{2}} θ}{λ}

σ {(I)}^{2} = \frac{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} I (x, y) [{(x - x_{0})}^{2} + {(y - y_{0})}^{2}] dxdy}{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} I (x, y) dxdy}

A (x, y) = A_{0} e^{\frac{x^{2} + y^{2}}{r_{1 / e^{2}}^{2}}}

I (x, y) = A (x, y) A^{*} (x, y) = I_{0} e^{- 2 \frac{x^{2} + y^{2}}{{r_{1 / e^{2}}}^{2}}} = I_{0} e^{- \log (2) \frac{x^{2} + y^{2}}{{r_{1 / 2}}^{2}}}

r_{1 / e^{2}} = r_{1 / 2} \sqrt{\frac{2}{\log (2)}}

S = π {r_{1 / 2}}^{2}

σ (I) = \frac{r_{1 / e^{2}}}{2} = \frac{r_{1 / 2}}{\sqrt{2 \log (2)}}

W^{2} (ρ) = \frac{0.023 e^{- ρ^{2}} L_{i}^{2}}{r_{0}^{5 / 3} {(ρ^{2} + \frac{1}{L_{0}^{2}})}^{11 / 6}}

Φ = \frac{d_{Na} s}{D^{2}} \sum_{i = 1}^{512} \sum_{j = 1}^{512} n (i, j)

	M² = 1	M² = 1.3
r₀ = ∞	1 atm. × 1 phase map	1 atm. × 10 phase maps
r₀ = 10 cm	50 atm. × 1 phase maps	50 atm. × 10 phase maps
r₀ = 5 cm	50 atm. × 1 phase maps	50 atm. × 10 phase maps

	M² = 1	M² = 1.3
r₀ = ∞	1 atm. × 1 phase map	1 atm. × 10 phase maps
r₀ = 10 cm	50 atm. × 1 phase maps	50 atm. × 10 phase maps
r₀ = 5 cm	50 atm. × 1 phase maps	50 atm. × 10 phase maps

		r₀ infinite	r₀ = 10 cm	r₀ = 5 cm
M² = 1	SM	52	190	300
M² = 1	ML	170	390	620
M² = 1.3	SM	62	200	310
M² = 1.3	ML	180	400	620

		r₀ infinite	r₀ = 10 cm	r₀ = 5 cm
M² = 1	SM	52	190	300
M² = 1	ML	170	390	620
M² = 1.3	SM	62	200	310
M² = 1.3	ML	180	400	620

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