Optical images of astronomical objects such as stars produced by ground telescopes are blurred, moving, or scintillating due to Earth’s atmosphere [1]. This is due to the optical refractive index variations caused by atmospheric turbulent mixing. The atmospheric disturbance limits the achievable angular resolution of ground telescopes regardless of the aperture size, and this is commonly known as seeing or astronomical seeing at a particular site. For even excellent sites under the best seeing conditions, large-aperture telescopes are not able to resolve objects any better than those with an aperture of ~20 cm, even though they efficiently collect light.

An adaptive optics (AO) system is typically an auxiliary instrument to a ground telescope and it has shown great promise for improving astronomical seeing beyond the limits imposed by atmospheric turbulence [2]. AO systems compensate the wavefront distortion introduced by the atmosphere by introducing controllable counter wavefront distortion that both spatially and temporally follows that of the medium. A large ground telescope with an adaptive optics system typically consists of a telescope, relay optics, a tip/tilt mirror, a deformable mirror, a scientific camera, a wavefront sensor, a laser guide star, and a data processing or control system (Fig. 1). A laser guide star (LGS), also known as an artificial guide star, is required as a point reference source of light for measuring and correcting wavefront distortions when any bright stars, known as natural guide stars (NGSs), are not available within the isoplanatic angle, i.e. the angle at which the AO corrections are valid. A solution to create a laser guide star is the sodium-beacon approach, which is of interest in this paper [3-5]. This approach focuses laser light of the sodium D2 line (589 nm) to excite a layer of sodium atoms that are present in the mesosphere at an altitude of ~90 km, which then appear as a star. The laser guide star can serve as a wavefront reference in the same way as a natural guide star except that natural reference stars, which could be much fainter than NGSs, are still required for image position (tip/tilt) information.

We are currently investigating the feasibility of a 1.6 m telescope with a laser-guide star adaptive optics (AO) system. The telescope, if successfully commissioned, would be the first dedicated adaptive optics observatory in South Korea. The 1.6 m telescope is an f/13.6 Cassegrain telescope with a focal length of 21.7 m. The AO system consists of a tip/tilt secondary mirror, a deformable mirror, two scientific cameras (CCD and IR detector), a Shack-Hartman wavefront sensor, a laser guide star, and a data processing or control system (Fig. 1).

This paper first presents the system design of the AO system based on the seeing conditions measured at the Bohyun Observatory, South Korea, which is one of the telescope site candidates. We then investigate the imaging performance of the telescope in terms of the Strehl ratio predicted at four wave bands (V/I/J/K/L centered at 0.55, 0.79, 1.26, 2.22, and 3.4 μm) for NGS and LGS cases. The prediction considers a wide range of parameters and error sources, including the strength and profile of the atmospheric turbulence, the fitting error caused by the finite spatial resolutions of the wavefront sensor and deformable mirror, wavefront sensor noise propagating through the wavefront reconstruction algorithm, servo lag resulting from the finite bandwidth of the control loop, and the anisoplanatism for a given constellation of natural and/or laser guide stars [6-10].

The telescope is an f/13.6 Nasmyth-Cassegrain telescope on an alt-azimuth mount, which is a simple two-axis mount for supporting and rotating an instrument about two perpendicular axes - one vertical and the other horizontal. As in the Cassegrain telescope, the light falls on a concave primary mirror and then is reflected toward a convex secondary mirror. A small 45° tilted tertiary flat mirror, placed on the altitude axis, focuses the light through a hole in the middle of the altitude bearing to one of the sides of the telescope called the Nasmyth focus. Table 1 lists the major specifications and Fig. 2 shows the optical layout and a 3D model of the telescope.

The telescope is equipped with a laser launch telescope with a sodium laser, an AO system set, and two scientific cameras (a CCD and an IR detector). The laser launch telescope with the laser head is mounted on the center frame, the laser electric control box is mounted on one of the Nasmyth platforms, and the optical bench including the AO system is set on the other Nasmyth port. The secondary mirror is mounted on a tip/tilt platform so it works as a tip/tilt mirror and also as an IR chopping mirror for subtracting IR backgrounds from IR observations [11], as shown in Fig. 3.

The theoretical resolution limit of a telescope is given by Rayleigh criterion [12]:

where θ is the angular resolution, λ the wavelength of incoming light, and D the telescope aperture. A telescope’s effective angular resolution is further limited by atmospheric disturbance. The limited resolution or quality of the optical imaging at a particular site is referred to as seeing or astronomical seeing, which is commonly described in two ways: (1) the full width at half maximum of a star’s intensity distribution at the focus of a telescope in angle (θ_seeing), (2) the Fried parameter (r_o) representing the spatial size of a typical lump of uniform air within the turbulent atmosphere. The limited angular resolution or seeing θ_seeing is related to the Fried parameter (r_o) as follows [2]:

The Fried parameter (r_o) is then related to the spatial and temporal variation of the refractive index of the air that the beam comes through. The variation of the refractive index is typically measured by the refractive index structure parameter , where the average is often determined as a function of local differences in the temperature, moisture, and wind velocity at discrete points. The Kolmogorov model of turbulence states that the Fried parameter (r_o) is a function of the index structure parameter and the wavelength, as given below [13].

Here, is the refractive index structure parameter at altitude h, the observed wavelength is λ, and the observed angle is ς. Roddier similarly defined the coherence time or critical time constant (τ_o) from the Fried parameter and the average velocity of the turbulence (V_o), which corresponds to the duration over which the standard deviation of the phase fluctuations at a given point is of the order of 1 rad [14, 15].

The Greenwood frequency (f_G) is then defined as the frequency or bandwidth required for optimal correction with an adaptive optics system [16]. The Greenwood frequency is a measure of how fast the AO system must respond. Typically, the closed-loop bandwidth of the AO system is practically taken to be about 1/10~1/12.5 of the wavefront sampling frequency. Analogous to the Greenwood frequency, a fundamental tracking frequency (f_T) is also defined as the frequency or bandwidth required for optimal tip/tilt correction [17].

The time-averaged atmospheric disturbance is independent of the viewing direction because the turbulence and its structure function are statistically the same everywhere in the field. But the instantaneous atmospheric phase aberrations do depend on the viewing direction. Hence there is an angular limitation called the isoplanatic angle θ₀ within which effective AO corrections can still be made. The isoplanatic angle is defined as the angle for which the RMS (root-mean-squared) wavefront phase error has increased by 1 radian. Analogous to the isoplanatic angle, the isoplanatic tilt angle (θ_TA) for tip/tilt sensing can be similarly defined as follows [18];

The above equation is approximated by the following equation. H is a characteristic average turbulence altitude. The averaging is done by weighting the profile with h^5/3; as a result a relatively high H ~ 5 km is obtained for typical conditions.

One of the site candidates for the 1.6 m telescope is the Bohyun observatory located at 36.1648°N and 128.977°E with altitude 1124 m. Astronomical seeing was monitored at the site for a year starting in June of 2014 by a seeing monitor called SLODAR (SLOpe Detection And Ranging) [10, 19-21]. The SLODAR measured the vertical profile of with the total seeing (r_o) at 500 nm. Since the SLODAR was remotely operated without on-site human presence, the SLODAR was cautiously operated only under specific weather conditions: no rain/snow and humidity under 70%. In addition, the measurement was further hindered by cloud coverage since the SLODAR requires a binary star with a specific angular separation at the observation direction. As results, the seeing data were recorded over 23 nights in total for the one year campaign. Due to the small number of observation nights, it is premature to draw conclusions on the seeing conditions but it is still valid to present the baselines. Figure 4 shows the temporal variation of the total seeing (r_o) and refractive index structure parameter at altitude h, measured at 500 nm over one night (20 Nov. 2014).

A weather station, located beside the SLODAR, continuously recorded environmental conditions including temperature, humidity, and ground wind speed over the year. The average ground speed was 2.58 m/sec with a standard deviation (sigma) of 0.89 m/sec and the instant maximum speed was 20.8 m/sec. However, the measurements were adequate for predicting the average velocity of the turbulence (V_o), which requires a wind profile up to ~15 km beyond a strong wind shear layer occurring near the tropopause at 8~12 km height [22]. García-Lorenzo et al. reported experimental prediction of the average velocity of the turbulence (V_o) from the mean ground velocity (V_ground) [23, 24] as follows, where the average velocity of the turbulence (V_o) at the site is estimated with two sigma values to be around 6.76~9.72 m/sec with a nominal value of 8.29 m/sec, which is in good accordance with measurements performed at other astronomical telescope sites.

Table 2 lists the statistical predictions of seeing conditions at some optical bands (V/R/I/J/H/K/L/M centered at 0.55, 0.64, 0.79, 1.22, 1.65, 2.20, 3.55, and 4.77 μm), which are derived from the SLODAR measurements at 0.5 μm over the 23 nights. The total seeing (r_o,λ1) at one wavelength (λ₁) can be calculated from the total seeing (r_o,λ2) measured at another wavelength (λ₂) as follows:

The adaptive optics system is a laser-guide star (LGS) adaptive optics system for the 1.6 m telescope. The AO system consists of a tip/tilt secondary mirror, a deformable mirror, two scientific cameras (CCD and IR detector), a Shack-Hartman wavefront sensor, a sodium laser, and a data processing or control system. The adaptive optics system set is installed on an optical bench located on one of the Nasmyth ports, as shown in Fig. 5.

The incoming beam from the 1.6 m telescope is first beam-split into the guide camera and the main AO beam path. The guide camera is a wide field camera of 0.3° × 0.2° that serves as a finder-scope for aiming purposes. The main AO path beam is then re-collimated into a parallel beam of 24.5 mm diameter with recollimating optics, i.e. a parabolic mirror. The parabolic mirror also conjugates the exit pupil to the deformable mirror. The phase-corrected beam by the deformable mirror (DM) is then forwarded into a spectral beam splitter that reflects the whole beam but the V band centered at 589 nm, i.e. the sodium laser wavelength with 78.5 nm bandwidth for wavefront sensing (WFS). The visible and IR parts of the reflected beam are then reimaged on a low-noise electron-multiplying charge-coupled device (EMCCD) and a scientific instrument, respectively. Two scientific instruments are considered at this time, an IR imaging detector and an IR high resolution spectrograph.

Figure 6 shows three representative operation concepts (or modes) of the 1.6 m telescope with the adaptive optics system. First, the telescope can observe a bright target while wavefront sensing with it. Second, the telescope can observe a less bright target while wavefront sensing from a separate natural bright star called a natural guide star (NGS). Third, the telescope observes a faint target with wavefront sensing with a laser guide star. In this case, we need another bright natural star for tip/tilt sensing since the angular anisoplanatism blocks the tip/tilt sensing with the laser guide star. We name these three scenarios ‘None’, ‘NGS only’, and ‘LGS + NGS’, respectively.

The system design of adaptive optics systems has been reported with the first order prediction of the system performance [1-9, 23-27]. Based on the first order prediction, we can find optimal values for the adjustable system-design parameters such as number of actuators, WFS sample rate, etc. Based on the seeing conditions in Table 2, the major optical components were optimally selected among commercially available components. Figure 7 shows pictures of the selected major AO components with their product numbers. Table 3 summarizes some key parameters of the components. It is worth mentioning here that beam reduction takes place during the conjugation from the telescope entrance pupil to the deformable mirror surface, as in Fig. 5. The Fried parameter of the seeing (r₀) is conjugated to the mirror with the beam reduction ratio (BR) as follows:

where r_0,DM is the conjugated Fried parameter on the DM surface.

The first order performance estimation can be accomplished by several methods. The most frequently employed method is a statistical tolerance analysis method called the root squared sum (RSS) method. Here, is the standard deviation of the entire system, denotes the standard deviation of the i^th part, and n is the number of parts. In optics, and are the most commonly described terms of the root-mean-squared wavefront error (RMS WFE). Other important optical performance criteria, such as the Strehl ratio, can be predicted from the total RMS wavefront error as follows,

In adaptive optics systems, the main error sources can be mostly divided into three groups: residual errors in telescope instrumental factors in the adaptive optics system and external factors. The instrumental factors receive contributions from the components within an adaptive optics system. The instrumental errors include the wavefront fitting error of a deformable mirror , temporal errors due to pure time delays and the limited bandwidth of the feedback loop , and measurement error of the wavefront sensor. The wavefront sensing error then consists of the aliasing error and the noise error The aliasing error is often called reconstruction error, which reflects the fact that the measurement device is only sensitive to low spatial frequency and the noise error is introduced by centroiding errors in the wavefront sensor, which is related to measurement of the signal-to-noise ratio. The external factors include the structure and dynamics of the atmosphere and the characteristics of the star or beacon used as the wavefront sensor. The external errors include the focal anisoplanatism due to finite distance of a laser guide star, angular anisoplanatism due to the angular separation between a guide star and an observation star, and chromatism error due to atmospheric dispersion. Each term in the following equation can be estimated from the AO system and atmospheric parameters, as tabulated in Table 4.

As tabulated in Table 5, we analyzed nine cases, three of each adaptive optics observation mode, as in Fig. 6. Each of the three cases applies stellar magnitudes of 5, 10, and 15 as the brightness of the observation target. In the case of the NGS only mode, the natural guide star (NGS) was assumed as 5 stellar magnitude and 1 arcsec angular separation. In the case of the LGS + NGS mode, all the guide stars were assumed as 5 stellar magnitude but with 1 and 4 arcsec angular separations, respectively.

Some errors mentioned in the above section depend on the number of photons arriving at the wavefront sensor. The total number (N) of photons arriving at the telescope in the visible region of the spectrum from a star of stellar magnitude m is given approximately by

where A is the area of the telescope in cm². QE is the quantum efficiency of the WFS and TR is the total transmission of the optics. IT is the integration time of each detector and Δλ is the bandwidth in nm of the WFS. In this simulation, we assumed that the total optics transmittance is 50% and the WFS bandwidth (Δλ) is 80 nm, which are common values in adaptive optics systems. In addition, the chromatism error is due to the index fluctuations with respect to the wavelength. Since the correction given by the DM is proportional to the refractive index at the sensing wavelength, the residual wavefront for another wavelength is not null and is directly proportional to the input signal itself, with an attenuation coefficient depending on the index values of the two wavelengths. However, this proportionality is fully predictable and therefore we can reduce its effects in the following calculations and the effect is hence not considered.

Table 6 lists all the error terms and their expected WFE and Strehl ratio based on the RSS method. In this analysis, we commonly applied the worst (2σ) seeing conditions, as tabulated in Table 2. From the results, we can expect a Strehl ratio >0.3 in most seeing conditions with guide stars.

We presented a schematic layout of a 1.6 m telescope with a laser-guide star adaptive optics (AO) system. The AO system was designed based on the astronomical seeing conditions measured over a year at the Bohyun observatory, South Korea. Following an extensive investigation into the errors sources of the adaptive optics system with a sodium laser guide star, we concluded that we can achieve a Strehl ratio >0.3 over most of the seeing conditions with a NGS or the LGS guide star.

In this study, the laser guide star was assumed to be of 5 stellar magnitude based on previous experimental results from other astronomical observatories located at latitude similar to the Bohyun observatory. However, the intensity of the laser guide star depends on the density of sodium atoms at the mesosphere, which strongly varies locally and temporally. Further study is under preparation to predict the sodium density above the observatory and to predict the AO performance accordingly.