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Adaptive opticsastronomy

The main basic expressions related to Gaussian beams were mathematically obtained in the previous paragraph. We will now describe their physical signification.

The size of the beam at the origin, w0, is minimal : the beam will diverge from this point (see figure 11). This minimal dimension is called “beam waist” (the waist is the radius of the spot. The diameter is of course given by 2 w0).

The divergence of a Gaussian beam is inversely proportional to the size of its waist. In the framework of Gaussian optics, “collimating a beam” is the same thing as “having a big waist”.

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For a tightly focused laser beam (w0 = 10 µ) and a 1 µm wavelength, we find ZR = 314 µm and a divergence (half-angle) of 1,8 degrees.

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Science and Engineering Adaptive Optics Interferometry Staff Research Bibliography Contact 65-1120 Mamalahoa HighwayKamuela, Hawaii 96743USA(808) 885-7887

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If we consider a “big” waist (1 mm), we find ZR = 3,14 m and a divergence (half-angle) of 0,018 degrees. We then obtain a so-called “collimated beam”.

The Gaussian characteristics of the beam are essentially important in the vicinity of the beam waist. Indeed, when z increases, the complex radius of curvature becomes close to R and the wave could be considered spherical.

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Next Generation Wavefront Controller References: SPIE Marseille 2008 paper: E. Johansson, et al., KAON 598 Performance on Keck II, June 2007 M. van Dam et al., KAON 489 Note:The Keck I & II AO systems were upgraded with a new wavefront sensor and a new wavefront controller. The other documents below refer to the system in its previous state, although a lot of the information is still useful and valid. Natural Guide Star AO References: Keck NGS AO first light paper: P.L. Wizinowich et al. 2000, PASP, Vol. 112, 315. Keck NGS AO performance: M.A. van Dam et al. 2004, Appl. Opt., Vol. 43, 29 Laser Guide Star AO References: The W. M. Keck Observatory LGS-AO System: Overview. [PDF 2.16Mb] Wizinowich, P., et al. 2006, PASP, 118, 297-309 The W. M. Keck Observatory LGS-AO System: Performance Characterization. [PDF 217Kb] M.A. van Dam et al. 2006, PASP, 118, 310-318 Observing with NIRC2 Observing with OSIRIS

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When z increases, the beam expands in the transverse direction while its amplitude on the z-axis decrease (energy conservation). The profile shape remains Gaussian.

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Let's take again the origin at the waist position w0, corresponding to a plane wave (infinite radius of curvature). We have defined the Rayleigh length : .

The Rayleigh length is the distance (from the waist) where the beam area is twice the beam area at the waist (the radius is times bigger). This parameter is useful to define a “collimated” beam : over this length, the beam size is nearly constant (between and ) - see figure 11.

w(z) is the dimension of the laser spot (the “radius” if the spot is circular) in the plane perpendicular to the propagation, at a distance z from the origin. Precisely, it is the radius (at 1/e for the amplitude, or 1/e² for the intensity) of the transverse Gaussian profile at the z abscissa.