High-order laser guide star performance for the Euro50 telescope is analyzed. Focus anisoplanatism residual errors using multiple LGSs are obtained by numerical simulations. Results calculated for annual mean conditions at the La Palma site show that, considering only focus anisoplanatism, it will be possible to obtain a Strehl ratio of 0.72~0.82 at 2.2 µm using 13 LGSs. A Strehl ratio of 0.60~0.70 can be obtained at 1.25 µm using 21 LGSs. The dependence of wavefront measurement error on pulsed laser power is also investigated. The variance of the measurement error is lower than 0.2 rad2 when the laser power is larger than 14 W.
©2005 Optical Society of America
It is well known that adaptive optics (AO) is a powerful tool for overcoming the effect of atmospheric turbulence on astronomical images. To perform AO correction for the atmosphere, a bright guide star is needed, but, as sky coverage studies demonstrate, candidates for bright guide stars are rare. As a consequence laser guide stars (LGSs) have been investigated as substitutes for natural guide stars (NGSs) [1,2,3]
Use of an LGS allows photon-noise errors to be made small, even for very high system servo bandwidths, if the laser beacon can be made sufficiently bright and concentrated in an area equal to the seeing disc. In reality, the images of laser spots are larger than the short exposure seeing disc for star images by a factor of at least √2 . Thus, more photons are needed to maintain the same signal-to-noise ratio for an LGS as compared to a NGS. In addition, rays from the LGS and from the NGS do not traverse the same turbulence, leading to wrong correction of the NGS, so-called focus anisoplanatism (FA). FA becomes more critical for shorter wavelengths, larger telescope apertures and lower altitude LGSs. FA has been thoroughly studied, and the best solution for eliminating it seems to be using multiple LGSs.
This paper firstly investigates the FA residual variance resulting from use of multiple LGSs based on resonant scattering in the sodium layer. Sodium LGSs are chosen because they provide a high altitude reference source, thus reducing the FA error in itself. Possible methods for reducing FA are tomographic mapping of the atmosphere, wavefront stitching or wavefront merging as proposed for the Euro50 . Next we analyze the wavefront measurement variance induced by finite light level in the wavefront sensor (WFS). These two factors are discussed together because they are closely related for astronomical telescopes.
2. Focus anisoplanatism residual error with multiple laser guide stars
FA is a consequence of the finite LGS altitude. The spherical wavefront that comes from an artificial star does not pass through the same portion of the atmosphere as the plane wavefront of the natural star located at an infinite distance from the ground. The mean square wavefront error is given by 
where d 0 is a parameter which depends only on wavelength λi and the turbulence profile at the telescope site. d 0 represents the largest diameter over which near-diffraction-limited operation can be obtained. Typical values are d 0~4(20)m at λi=0.5(2)µm. D is the telescope aperture. For a 50-m telescope, use of a single LGS will lead to a large FA. To reduce FA, it is necessary to use an array of LGSs to probe the atmosphere properly. The major motivation for using more than one beacon can easily be understood from Eq. (1). If the aperture is broken up into several smaller contiguous sections of width s, each with its own laser beacon for wavefront sensing, then the FA error will reduce to =(s/d 0)5/3. If each sub-section has its own LGS for wavefront sensing, then wavefront merging is an option. The LGS wavefront merging solution proposed for the Euro50 telescope is illustrated in Fig. 1 [5,7]. A two-dimensional Hanning window is used as weighting function. The width of the Hanning windows is two times the distance between the down-projected LGSs. For less than four LGSs contributing to the merged wavefront, the weights are renormalized. By weighting the LGS wavefront this way, the base of each LGS cone is artificially reduced.
The residual wavefront errors using both 13 LGSs and 21 LGSs were calculated by numerical simulations. The down-projected patterns of 13 LGSs and 21 LGSs are shown in Fig. 2. The distance between the nearest guide stars is 12.5m and 11.18m, respectively.
The atmospheric model used for the calculation is given in Table 1 . This is a seven-layer turbulence model for the Observatorio del Roque de los Muchachos(ORM) site. Table 1 lists values of Fried’s parameter r 0 at λ=0.5 µm. The atmospheric turbulence was assumed to obey von Karman statistics with the power spectrum given below
where κ0=1/L 0 and L 0 is the outer scale of the atmospheric turbulence. All seven layers were assigned a common outer scale.
Neglecting scintillation, propagation of the spherical wavefronts from the LGSs was simulated by magnifying each layer by a factor H/(H-hlay ) using the Fourier scaling theorem, before summing all the phase screens to obtain the phase perturbation ϕLGS(r) on the telescope pupil. H is the altitude of the sodium layer and hlay is the altitude of the atmospheric layer. Likewise, propagation of a plane wavefront from a NGS was simulated by direct summation of all the phase screens to obtain the phase perturbation ϕP(r) in the telescope pupil. The residual wavefront variance that is due to the FA is then given by
where R is the telescope’s radius, W(r/R) is a pupil function equal to 1 if r<Rand to zero otherwise. (r) is the wavefront obtained by merging wavefronts from LGSs using the Hanning window.
Fifty statistically independent sets of wavefronts were calculated at 0.589 µm. By computing the average value of with piston removed, the residual variance was obtained for 13 LGSs and 21 LGSs, respectively. The results are summarized in Table 2 and Table 3 for an infinite outer scale of turbulence and for 20m outer scale of turbulence, respectively. As a measure of image quality, we also use the Strehl ratio (SR ). The relation of SR to the variance is given by the Maréchal approximation SR =exp(-σ2). This equation is valid for phase errors up to about 1.5 rad rms corresponding to SR >0.1. The Strehl ratios SR for the two LGS numbers are given in Table 2 and Table 3. The variances were scaled inversely proportional to λ2 reflecting the need for more LGS at lower wavelengths.
3. Analysis of wavefront measurement error
3.1 Hartmann sensor measurement error
The random measurement errors in a Hartmann sensor depend on the size of the spots, which are images of the reference source produced by the subaperture lenslets. The spot size is determined by four factors: the subaperture dimension d, the angular diameter of the source θ, Fried’s parameter r 0, and the sensor wavelength λ. The resulting expression for the standard deviation of the one-axis measurement error in Hartmann sensors for r 0<d, in radians rms of phase difference per subaperture, is 
where SNR is the signal-to-noise ratio of the detected signal. For an unintensified CCD detector, the signal-to-noise ratio is 
where nc is number of photons from the reference source counted in the sensor integration time, e is read noise per pixel in electrons rms, and m is the number of pixels per subimage on the CCD detector.
The phase error, which is scaled from the beacon wavelength λb to the observing wavelength λi is given below:
Expression (6) shows that the measurement error is inversely proportional to the signal-to-noise ratio. The photon count and signal-to-noise ratio improve as the integration time increases resulting in a smaller measurement error. On the other hand, a longer integration time increases the temporal error. So for each set of operating conditions, there is an optimum integration time topt for which the sum of the time-dependent errors is minimal. For the Euro50 telescope, the planned integration time is 1.6 ti =ms. As mentioned the measurement error is also related to the size θ of the laser spots. This will be discussed in the next section.
3.2 Laser spot elongation
The atmospheric sodium layer is roughly 10~15 km thick. This causes the LGSs to be elongated, for subapertures, which are not positioned along the principal directions of the LGS launcher located behind the secondary . The apparent size of a Hartmann spot formed by a subaperture at the edge of the telescope is given by the expression:
where ΔH is the thickness of the sodium layer (10~15 km), H is the altitude of the sodium layer (~92 km) and D is the diameter of the considered telescope. With D=50m we get θspot=6~9 arcsec, which is smaller than the isoplanatic angle of 15 arcsec for K-band operation. Several methods have been proposed to eliminate spot elongation. Beckers et al  have suggested a rapid refocusing system aimed at removing the perspective elongation of LGSs for the Euro50 telescope. The rapid refocusing system can be changed in 16 equal steps. For a maximum elongation of approximately 9 arcsec for the Euro50 telescope there will be 16 steps of 9/16=0.56 arcsec each approximately equal to 0.6 arcsec size of the on-axis sodium beacon image. Hence we assumed the size of the laser spot in the sodium layer to be 0.6 arcsec in the following calculations.
3.3 Backscattered flux versus laser power
To utilize the rapid refocusing system aimed at removing the perspective elongation of the LGSs, a sum-frequency pulsed Neodymium YAG was proposed for LGS generation. Each pulse (macropulse) consists of several micropulses. The micropulses have duration τp and repetition rate Rp . The corresponding quantities for the macropulses are τM and RM . The number of backscattered photons received per unit area per macropulse is given by 
where Cs is the sodium column density in atoms/cm2, T 0 is the atmospheric transmission from ground to the mesosphere, z is the altitude of the sodium layer and NB (x, y) is the number of backscattered photons per steradian per atom per macropulse, which can be approximated as NB (x, y)=(τ MRp )p 2(x, y)/4π where τ MRp is the number of micropulses in a macropulse. p 2(x, y) is the excitation probability of the radiating transition after a single micropulse. It can be approximately expressed as [10,11]
Here A=6.25×107sec-1is the radiative decay rate, δνD ≈1 GHz is the Doppler width, Ip (x, y) is the beam intensity of a temporally Gaussian micropulse and the relation of Ip to the average laser power P is given by with being the FWHM spatial width of the beam. Therefore, the number Rt (photons/cm2/s) of backscattered photons (using linear polarized laser radiation) is given by
Calculations were based on the laser format selected for the Euro50 telescope: τp=0.6ns, Rp =100MHz, τM=3µs, RM =10KHz, w=25cm and Cs =3.6×109atom/cm2.
Combining expression (6) and (10), we can estimate the variance of the wavefront sensor measurement error as a function of laser power at λi=2.2µm. For the Euro50 telescope, the wavefront sensor subaperture size is d=0.781 m and the pixel number of each subimage is 32×32. The integration time of the wavefront sensor is 1.6ms. The atmospheric transmission (one way) is 0.7 and the total transmission of the optical system (including telescope, wavefront sensor, etc) is 0.4. The detector quantum efficiency is chosen as 0.8. A rms readout noise of 3 electrons per pixel is assumed. In addition, the wavefront variance is also calculated for a subimage format of4×4pixels. Results are given in Fig. 3. Inspection of Fig. 3 shows that the average laser power should be larger than 14 W to keep the measurement variance below 0.2 rad2 for the 32×32 pixels format. In this case, the Strehl ratio SR is about 0.81 when only the measurement error is considered. However, the measurement variance is under 0.2 rad2 when the average laser power is larger than 3 W for the 4×4 pixels format. The 32x32 pixel format was chosen to accommodate the elongated spots, and the above results show that when the elongation is compressed, it will be an advantage to use a smaller pixel format for localizing the spots. It is seen that to reduce the laser beacon power requirements, it is essential to use a detector with minimum readout noise.
We have studied the effect of focus anisoplanatism on the image quality for the Euro50 telescope. Our aim has been to investigate how high order system performance can be achieved for an AO system in the K-band using multiple LGSs. Using 13 LGSs, the Strehl ratio is reduced to 0.72~0.82 on-axis at 2.2 µm under annual mean conditions. For 21 LGSs, the corresponding value is 0.60~0.70 at 1.25 µm under annual mean conditions.
We have also studied the relation of wavefront measurement error to pulsed laser power based on the sodium abundance at the site of the Euro50 telescope. The variance of the measurement error is lower than 0.2 rad2 when the average laser power is larger than 14 W for a 32×32 pixel format wavefront sensor with 3 e/pixel read out noise.
When FA and wavefront measurement errors are combined, Strehl ratios of 0.59 with 13 LGSs and 0.70 with 21 LGSs at 2.2µm can be obtained under annual mean conditions using a pulsed laser with an average power of 14 W and a pulse format given by a macro pulse duration of 3 µsec with a repetition rate of 10 kHz and a micro pulse duration of 0.6 nsec with a repetition rate of 100 MHz.
The calculations were based on geometrical propagation through properly scaled and shifted phase screens and hence scintillation was not taken into account. Preliminary calculations of the discrepancy between geometrical and Fresnel propagation however indicates the wavefront variance contribution from scintillation to be around 0.02 rad2 for r0=15 cm and λ=0.589 µm, and hence it has been neglected. Another effect not taken into account is atmospheric dispersion, which will lead to erroneous correction unless properly accounted for.
Although this paper only deals with FA related to a single axial science star, it is of course the intension to use the many LGSs to provide wide field correction in a multi conjugate adaptive optics mode.
Obviously, in addition to the above error sources, a multitude of other effects must be considered to estimate the global performance of the telescope and adaptive optics.
This work was done while Chengyu Fan was at the Lund Observatory, Sweden. Thanks to Dan Moraru for useful introduction to use Beowulf Cluster.
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