TOP: Introduction

FORWARD: Multi-Conjugate AO

4. Laser Guide Stars

4.1. Sky coverage of AO with NGS

Most of the actual astronomical AO systems use natural guide stars (NGSs) to measure the wave-fronts. This imposes a severe restriction on the choice of targets. Alternatively, if some set of targets for observation is selected randomly on the sky, the probability to find suitable guide stars (called sky coverage) may be low.

Guide stars must be selected within the isoplanatic patch $\theta_0$ of the target. For a given distance $\theta$ between the GS and target, the residual wave-front error due to anisoplanatism is estimated as

\langle \epsilon_{\rm iso}^2 \rangle =
\left( \frac{\theta}{\theta_0} \right) ^2 \propto {\lambda}^{-2}
\end{displaymath} (1)

On the other hand, the photon noise error is inversely proportional to the photon flux, which is related to the stellar magnitude m:

\langle \epsilon_{\rm phot}^2 \rangle \propto \lambda^{-3.6} 10^{-0.4m}
\end{displaymath} (2)

Fig. 3.10 From Roddier (1999) In the above formulas, $\lambda$ is the imaging wavelength. It is clear that the sky coverage is a strong function of $\lambda$: at longer wavelengths, the isoplanatic patch is larger, while the flux needed to measure wave-fronts is lower. In the Figure 3.10 and Table from the Roddier (1999) book the typical magnitudes of guide stars and their distances from the object as restricted by anisoplanatism are plotted for various photometric bands (caution: the optimistic turbulence altitude of 1 km was assumed for this plot!). The labeled curves show the probability to find a suitable guide star at medium Galactic latitude (dashed line - at Galactic pole). Both the required photon flux and the isoplanatic patch size depend on the turbulence profile. It means that difficult objects can be observed with AO only under favorable seeing. Instead of liberating astronomers from the dependence on seeing, AO makes this dependence even more critical!

Question: The sky coverage of some AO system at some wavelength was 10%. How will it improve if the sensitivity of the WFS is increased 2.5 times? Or at the wavelength 2 times longer?

Question: What would be the sky coverage of AO with NGS if all turbulence were concentrated in a thin layer near the ground?

LGS principle The probability to find a guide star is estimated by combining AO parameters with the model of star density in the sky which decreases away from the Milky way, i.e. with increasing Galactic latitude (cf. Bahcall and Soneira, ApJ V. 246, P. 122, 1981). According to this model, near Galactic pole there are about 400 stars per square degree with R-magnitudes between 14.5 and 15.5, or about 600 stars per square degree brighter than R=15. Going 1 magnitude fainter (brighter) increases (decreases) the number of stars per magnitude interval twice. Stellar density at low galactic latitudes is at least a factor of 2 higher than at Galactic pole. See the graph on this figure for more details (full, long and short dash lines refer to the Galactic plane, middle latitude and Galactic pole).

For the K band (wavelength 2.2 microns) under favorable seeing the sky coverage may be higher than 0.5 in the Galactic plane. But at optical wavelengths the sky coverage of AO with NGSs is hopelessly low.

The idea to use artificial laser guide stars (LGSs), also called laser beacons, for Adaptive Optics appeared in the end of 70-s, although its first open publication is dated 1985. The existing two types of LGS use either the Rayleigh scattering from air molecules or the fluorecence of sodium atoms in the mesosphere, and are called Rayleigh and sodium LGSs, respectively.

As we shall see, LGSs do not solve the problem of sky coverage completely. Faint astronomical objects need long exposure times, hence tip and tilt aberrations must still be sensed using natural guide stars. Additional degradation of the AO performance (so-called cone effect) comes from the fact that LGS is at finite altitude, whereas scientific target is at infinity.

4.2. Cone effect

Cone effect

The laser spot is formed at some finite altitude H above the telescope: H=10-20 km for Rayleigh LGS or 90 km for sodium LGS. A turbulent layer at altitude h will be sampled differently by the laser and stellar beams. There are three distinct effects:

It turns out that the last factor is the most important one. While the laser wave-front is compensated by the AO, the stellar wave-front has a residual error due to the cone effect:

\langle \epsilon_{\rm cone}^2 \rangle =
\left( \frac{D}{d_0} \right) ^{5/3}.
\end{displaymath} (3)

Here D is the telescope diameter, $d_0$ is a new parameter characterizing the cone effect. To the first approximation, it is not independent but rather related to the isoplanatic patch size $\theta_0$:

d_0 \approx 2.91 \theta_0 H.
\end{displaymath} (4)

Taking $\theta_0$=2.5 arcsec (at 0.5 microns), we obtain $d_0$=3.2 m for a sodium LGS and 0.7 m for a 20-km Rayleigh LGS (more rigorous formulas lead to $d_0$ of 4 m and 1 m, respectively, for the same conditions).

Question: Scale the above $d_0$ to the wavelength of 2.2 microns. What is the maximum telescope size that can be used at this wavelength with a Rayleigh LGS?

Clearly, the cone effect is a serious limitation precluding the use of LGSs at large telescopes and at short wavelengths, precisely where they are most needed! Several complicated schemes were proposed (but not implemented) that aim to reduce the cone effect by using several laser beams and segmenting the aperture of the telescope, effectively reducing D. We deliberately do not discuss them, because the best solution for the cone effect is to use multiple laser beams for reconstructing the 3-dimensional turbulence perturbations (this is called tomography, cf. the next chapter). If this goal is achieved, the upper layers can be re-scaled, and cone effect will be removed or at least reduced.

Sodium LGSs have a clear advantage with respect to the cone effect, they are chosen for most of astronomical LGS AO systems.

4.3. Tilt problem and solutions

LGS does not sense tip-tilt

Military LGS AO systems were used for applications requiring short exposure time, hence the overall image motion (tip and tilt) were of no importance. Laser beam is deflected by the atmosphere twice, on its way upwards and downwards, whereas stellar beam experiences only one deflection. It means that tip and tilt can not be sensed with the LGS. If LGS is projected from the main telescope, the upward and downward tilts compensate completely and the LGS image is stable in the telescope focal plane.

The standard solution for long-exposure imaging with LGS consists in using an additional NGS for tip-tilt compensation. We may consider it like a S-H WFS with only one sub-aperture equal to the telescope aperture. The number of photons is increased (larger surface, longer time constant), hence fainter stars can be used, promising increased sky coverage. The isoplanatic patch for tilt is larger than $\theta_0$, also favoring the sky coverage. These considerations are now developed in a quantitative way.

Differential tilt It is not a simple matter to compute the relative angular motion $\Delta \alpha$ between the NGS and the target as a function of the angular distance $\theta$ between them. The result depends on the turbulence profile. Supposing that the target and NGS are separated along the x direction, the first-approximation formulas are:

\langle \Delta \alpha_x^2 \rangle \approx
0.0472 \left( \fr...
\left( \frac{D}{r_0}\right)^{-1/3}.
\end{displaymath} (5)

For the tilt in y-direction the formula is the same, but the coefficient is smaller, 0.0157. It means that tilt anisoplanatism will mostly act in the direction of the NGS, elongating the PSF. In these formulas the diffraction image size $\lambda/D$ is isolated to show that, although the differential tilt indeed decreases at large telescopes, the diffraction limit decreases, too. To achieve a given Strehl ratio, the differential tilt must be restricted to a given (small) fraction of the Airy disk, and in this case the dependence on telescope diameter is only to the -1/6 power.

Now we remember that the Zernike coefficients for tip and tilt are related to the image motion like $a_2  = \Delta \alpha_x  \frac{\pi D}{2 \lambda},
\;\;\; a_3 = \Delta \alpha_y  \frac{\pi D}{2 \lambda},$ and that the sum of the two squared Zernike coefficients gives the residual phase variance caused by the tip-tilt anisoplanatism:

\langle \epsilon_{\rm TA}^2 \rangle \approx
0.1 \left( \frac{\theta}{\theta_0}\right)^2
\left( \frac{D}{r_0}\right)^{-1/3}.
\end{displaymath} (6)

Taking, for example, 8 m telescope and $r_0$=0.15 m, the Strehl ratio will be reduced by the tilt anisoplanatism by a factor of 0.37 (1 square radian error) for a separation of $\theta = 6 \theta_0$, or only 15 arcseconds (at 0.5 microns). The gain in the maximum separation between the object and the NGS brought by the laser is only 6 times for this example.

Question: In the above example, compute the maximum separation between NGS and target that corresponds to the Strehl reduction of 0.9 due to tilt anisoplanatism at the wavelengths of 0.5 and 2.2 microns.

The phase error caused by the photon noise in the measurements of the tilt is computed in the same way as in the S-H WFS (see Chapter 3.2). It is proportional to the $D^2/n$, where n is the number of photons, which is itself proportional to $D^2$. The net result is that despite the increased number of photons collected by a larger telescopes, the magnitude limit should not increase (in fact it does, but only slightly, because the tilt time constant depends on D). AO with LGS at large telescopes does not have big advantages in sky coverage as compared to smaller telescopes.

Question: Estimate the gain in limiting magnitude of tilt NGS between 10 m and 100 m telescopes (for the same Strehl ratio).

The errors associated with the tilt measurement are not limited to the two components discussed above but include also contributions from finite temporal bandwidth, from image structure of the guide star and from the correlation of tilt with higher-order Zernike terms.

Sky coverage with LGS

Of course, despite all the problems, the sky coverage with LGS is significantly increased with respect to NGS. Again, it depends on the turbulence profile. Some estimates and comparisons can be found, e.g. in SPIE, V. 3353, P. 364, 1998. For example, the Strehl ratio of 0.4 in K band can be achieved at the galactic pole with a probability of 0.1% using NGS and with a probability of 2% using LGS. These numbers improve to 4% and 80% at average Galactic latitude.

The ``tilt problem'' is considered as a major obstacle for an LGS-assisted AO. Many ingenious solutions were proposed to overcome it, some are listed below.

4.4. Laser guide stars: Rayleigh

The beam of a pulsed laser is focused at altitudes H between 10 and 20 km above ground, the return signal is obtained as a light back-scattered by the air density fluctuations. This Rayleigh scattering is more efficient at short wavelength (the cross-section is proportional to $\lambda^{-4}$), which explains the blue color of the clear sky and the increased extinction of blue stellar light. The return flux would be proportional to $H^{-2}$ at a constant air density, in fact it decreases with altitude more steeply because the air gets rarefied.

Most of the light is scattered at low altitudes. Beam focusing is not sufficient to define the desired LGS altitude. To do this, the return signal is gated in time (range gating). The length of the exposure typically corresponds to the L=1-2 km range in altitude. Gating is achieved by a fast electro-optical shutter in front of the WFS.

Question: Compute the exposure time needed for 1.5 km gating and the maximum pulse repetition rate for H=20 km.

Elongation of LGS image

The LGS with range gating is a "pencil" of light of the length L at altitude H. When it is viewed off-axis from some distance b, e.g. by a periferal sub-aperture of a S-H WFS, the image will be elongated by the angle $Lb/H^2$ (see the Figure). This elongation is important and eventually restricts both the range L and the distance b between the launching and main telescope.

Question: Rayleigh LGS at H=20 km is projected from a 8 m telescope with the range L=2 km. Compute the angular elongation of the LGS image seen from the perifery of the pupil.

The Rayleigh LGS at the Starfire Optical Range (SOR) is based on the 200 W copper vapor laser emitting the green light. More return is obtained at shorter (ultraviolet) wavelengths. For example, a 50 W laser emitting at 0.35 micron gives the return flux around 11000 photons per square meter per millisecond. This corresponds to a visual stellar magnitude 10. Laser light is monochromatic, while stellar flux in a given photometric band depends on the bandwidth. It means that the correspondence between LGS return flux and magnitudes is not unique, but depends on the adopted photometric system.

The beam quality (measured by wave-front distortion or by beam divergence) of the powerful pulsed lasers is bad, significantly worse than diffraction-limited. On the other hand, the photon noise in a S-H WFS strongly depends on image size $\beta$ (increasing $\beta$ twice requires 4 times more photons to compensate). In order to reduce $\beta$ to acceptable level, the current Rayleigh LGS systems project the laser beam through the main telescope aperture. The disadvantage of this solution (apart a technical problem of separating the powerful laser beam from the detector of the faint return signal) is the fluorecence of the elements of the telescope, which increases the background level in the scientific beam. No Rayleigh LGS AO system has yet been used for observations of really faint objects.

Question: Suppose that the divergence of the laser beam is 5 times more than diffraction limit, and the wavelength is 0.35 micron. What is the minimum diameter of the projection telescope needed to achieve a 1 arcsecond LGS size?

The major problem of Rayleigh LGS is the cone effect: a single LGS without tomography is not suitable even in the infrared at the 8 m class telescopes. On the positive side, under-corrected high-level turbulence leads to a (bad) correction of a larger field of view. If only ground layer needs to be corrected (to achieve improved seeing over the whole telescope field), the Rayleigh LGS is the best solution: the signal from multiple LGSs (or from single spinning beam) may be averaged.

UV lasers for Rayleigh LGS are relatively cheap and reliable. The UV beam is invisible and not harmful to most optical devices. These are the considerations that push some researchers to pursue the development of Rayleigh LGS. R. Angel proposed to increase the return flux and to remove the elongation problem by a fast re-focusing in order to track the upward-propagating pulse (this experiment will be performed soon).

4.5. Laser guide stars: Sodium

Sodium profile variations

The sodium layer at an altitude of about 90 km and with a thickness of about 10 km surrounds the Earth. It is most likely formed by micro-meteorite ablation. Parameters of the layer (total number of atoms, mean altitude, profile) change seasonally, but also on time scales of days, hours and even minutes. Strong sporadic layers appear sometimes and then dissipate in few hours. The sodium profile variations during 5 hours are plotted in the Figure. On the average, there are some $10^{13}$ sodium atoms per square meter.

Question: Compute the linear size of a 1 arcsecond sodium LGS.

Question: What is the angular size of the out-of-focus stellar images in 10 m and 100 m telescopes focused on the sodium layer?

The sodium atoms can be excited by a laser beam tuned to the D2 line (wavelength 0.5890 microns) and radiate at the same wavelength. The natural width of the sodium D2 line is determined by the thermal motion of the atoms in the mesosphere and by the hyper-fine structure of the D2 line itself (2 unequal peaks), it is about 3 GHz.

Only less than one half of the atoms can be excited to the upper state, because with increasing laser power the simulated emission becomes increasingly important. The flux density needed to excite 1/4 of atoms is called the saturation level. If the flux density of laser light in the mesosphere is much less than the saturation level, the return flux from the LGS is proportional to the laser power, otherwise it becomes saturated.

The efficiency of pumping (return flux per Watt of laser power) and the saturation flux are dependent on the spectrum of the laser. For monochromatic laser the saturation intensity is 6.3 mW per square cm. It increases to 1.9 W if the whole 3 GHz bandpass is excited. Thus, for a 10 W monochromatic laser focused into 1 arcsec spot the saturation is already important.

Both pulsed and continuous-wave (CW) lasers are used to create sodium LGSs. The spectral width of the pulsed-laser line is, roughly, inverse of the pulse duration. By varying the pulse parameters (or by modulating the flux of a CW laser) the width of the laser line can be matched to the width of the atmospheric sodium line, which alleviates the saturation problem and increases the return flux. The complexity of the physical processes governing the interaction of sodium atoms with laser radiation is substantial, it is not easy to compute the return flux for a particular laser.

For reference, a CW laser of 1 W power produces a star of 11-th magnitude (500 photons per square meter per millisecond). At first sight it might appear more than enough for AO correction in the infrared. In reality more power is typically asked in order to compensate for the finite size of LGS, for less-than-average sodium concentration, for worse-than-average seeing, etc. Pulsed lasers are less efficient and require more average power to achieve the same return flux (during the short pulse, the flux density at the sodium layer easily reaches saturation).

Sodium LGS and Rayleigh cone

Part of the laser beam is returned by Rayleigh scattering: an image consists of the sodium LGS sitting atop of the ``Rayleigh cone'' (see the Figure obtained with the ALFA LGS AO). Range gating can be used with pulsed lasers to cut off the cone, but it is impossible with CW lasers. Fortunately, if the laser beam is launched from the top of telescope tube, the secondary mirror blocks most of the Rayleigh cone. Remaining upper part of the Rayleigh cone is spatially separated from the LGS, hence can be isolated in the image plane of S-H WFS.

Beam projector behind the secondary mirror

Beam quality of sodium lasers is good, so they can be projected from small apertures. In order to obtain the smallest LGS size, the diameter of the beam projector (or launch telescope) must be around 3-4 $r_0$. The optimum location of the beam projector is behind the secondary (less spot elongation, better rejection of Rayleigh cone). Laser beam must be transported from the laser itself (which can be located in the dome or at the Nasmith focus) to the beam projector by means of mirrors or optical fiber.

Question: Estimate the maximum elongation of sodium LGS in a 8 m telescope.

Current sodium lasers are  10 times more expensive than Rayleigh lasers of equal power and are not robust. The problem is of course that here we need a laser for the sodium D2 wavelength. The most frequent type is dye laser, where the lasing medium is a solution of an organic dye, optically pumped by a radiation of a shorter wavelength from another, more powerful laser. These lasers are known to be capricious, hence need qualified technical assistance. Solid-state pulsed lasers have been used as well. Gemini team is co-financing a technological program for the development of robust solid-state sodium lasers.

Changing parameters of the mesospheric sodium layer influence the operation of the sodium LGS, e.g. the return flux. The altitude variations translate into focus variations, which are compensated by AO. It means that the amount of differential defocusing between science and LGS beams will change, hence it must be controlled. The NGS channel used for tit-tilt sensing must also include focus control (which need be only fast enough to follow the sodium height variations). Additionally, appearance of sporadic layers influences the shape of elongated image spots in the S-H WFS. One of the proposed solutions to deal with spot elongation problem is to project two beams from the sides of the telescope, at position angle of 90 degrees. The LGS is in the form of cross, each arm of the cross has small size in one direction and thus gives accurate slope measurements in this direction.

Some information on the astronomical AO systems using LGSs is summarized in the Table below.

LGS AO systems
Telescope LGS type and power Year
  In operation:  
SOR, 1.5 m Rayleigh (CuV), 200 W, 10 kHz 1989
Lick, 3 m Sodium, 20 W, 10 kHz 1996
ALFA, 3.5 m Sodium, 4 W, CW 1998, de-commissioned
Keck II, 10 m Sodium, 20 W, 20 kHz 2001
Gemini-N, 8 m Sodium, 10 W 2003
Gemini-S, 8 m 5xSodium, 50 W 2004?
VLT-ESO, 8 m Sodium 2003?

4.6. Operational aspects of LGS

Adaptive optic systems are complex. LGSs add another level of complexity to the design and operation of AO. On one hand, LGSs improve the sky coverage and permit the observations otherwise impossible. But they involve additional sources of error (cone effect, tilt anisoplanatism) which result in a reduced performance compared to NGS AO. This additional degradation strongly depends on the level of high-altitude turbulence, hence monitoring of turbulence profile is important for efficient operation of LGS-assisted AO. Other specific operational constraints are listed below.

Estimation of the overall observing efficiency of LGS-assisted AO taking into account all these factors lead to a duty cycle of less than 0.5. Satellite restrictions are particularly troublesome and cause segmentation of available observing time into small pieces. This does not make astronomers very happy. There are as yet no important scientific results obtained with LGS AO to date (2001).

Summary. The sky coverage of an AO system using NGSs can be estimated, given the seeing conditions, the imaging wavelength and the WFS sensitivity. Sky coverage is improved significantly with LGS, but it remains low at short wavelengths, because of the ``tilt problem'' and the need to find a tip-tilt NGS. Additionally, the performance of LGS-based AO systems is degraded by the cone effect, making LGS not useful at large telescopes and/or at short wavelengths without additional complications like tomography. The two current LGS concepts, Rayleigh and sodium, were described, and some operational problems specific to LGS AO were enumerated.

TOP: Introduction

FORWARD: Multi-Conjugate AO