Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2021; 5(2): 101-113
Published online April 25, 2021 https://doi.org/10.3807/COPP.2021.5.2.101
Copyright © Optical Society of Korea.
Kyohoon Ahn , Sung-Hun Lee, In-Kyu Park, Hwan-Seok Yang
Corresponding author: kyohoon.ahn89@gmail.com, ORCID 0000-0002-1094-852X
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Adaptive optics (AO) systems are becoming more complex to improve their optical performance and enlarge their field of view, so it is a hard and time consuming process to set up and optimize the components of AO systems with actual implementation. However, simulations allow AO scientists and engineers to experiment with different optical layouts and components without needing to obtain and prepare them physically. In this paper, we introduce a new AO simulation named the Korea Adaptive Optics Simulation (KAOS), independently developed by LIG Nex1. We verified the performance of KAOS by comparing with other AO simulation tools. In the comparison simulation, we confirmed the results from KAOS and other AO simulation tools were very similar. Also, we proposed a laser tomography AO system with five Rayleigh laser guide stars (LGSs) optimized by using KAOS to overcome the disadvantages of the AO system with a single sodium LGS for the satellite imaging system. We verified the performance of the proposed AO system using KAOS, and the simulation result showed averaged Strehl ratio of 0.37.
Keywords: Adaptive optics, Atmospheric turbulence, Laser guide star, Numerical simulation, Wavefront sensing and correction
OCIS codes: (010.1330) Atmospheric turbulence; (010.7350) Wave-front sensing; (220.1080) Active or adaptive optics
Adaptive Optics (AO) systems have reached the state-of-the-art level, which successfully can deliver diffraction-limited imaging performance [1]. Early AO systems can only reach the telescope’s diffraction limit over a narrow field with a single Wavefront Sensor (WFS) and a single Deformable Mirror (DM) as referred to Single-Conjugated AO (SCAO). To overcome the limitation of the early AO systems, current AO systems have been developed in two main directions: (a) increasing the imaging field of view and sky coverage, and (b) improving wavefront correction for bright targets. The first approach is to create an artificial star in the field of interest by using a bright (5 W–100 W) laser beam, so that the wavefront can be measured along a direction where no bright star is present [2]. After that, the AO systems has been addressed by tomographic AO configurations including Ground-Layer AO (GLAO), Multi-Conjugated AO (MCAO), Multi-Object AO (MOAO), and Laser Tomography AO (LTAO). These tomographic AO systems have multiple DMs, WFSs and Laser Guide Star (LGS)s to increase the AO corrected field of view and wavefront correction. GLAO systems deliver significant images sharpening over degree-scale fields by correcting only wavefront aberrations induced at low altitude [3]. MCAO systems deliver diffraction-limited imaging over a continuous arcminute-sized field of view using multiple DMs, each optically conjugated to a separate altitude [4]. With each DM correcting for atmospheric aberrations induced at the corresponding altitude, the overall wavefront correction is valid over a larger angle than is possible with a single DM. Also, MCAO systems require the multiple WFSs along multiple directions to measure the wavefront error induced from multiple layers. MOAO systems deliver the tomographic information in a different manner and instead of providing one continuously corrected field of view, corrects for a finite number of individual lines of sight within a large field of regard. Multiple WFSs of the MOAO systems calculate the optimum correction for each science direction, and a different DM is used to perform correction for each line of sight to each science target [5]. LTAO systems deliver diffraction-limited imaging over a narrow field of view and mitigate LGS focus anisoplanatism also known as the cone effect. LTAO systems are similar to MOAO, where multiple LGSs are used as guide stars, though usually only one direction is corrected for [6]. The use of multiple LGSs compensates for the cone effect as the overlap of observed turbulence provides the entire cylinder observed by the science target. This improves performance of LGS AO systems and will be vital for extremely large telescopes and AO systems with Rayleigh LGS that have large cone effect error.
As mentioned above, recent AO systems are becoming more optically complex to improve their optical performance. These AO systems require expensive components such as multiple DMs, WFSs, and a Real-Time Controller (RTC), which are non-trivial to setup and optimize. Also, it is a hard and time-consuming process to setup and optimize the components of AO systems with actual implementation. However, simulations allow AO scientists and engineers to experiment with different optical layouts and components without needing to obtain and prepare them physically. So, simulations play a number of important roles in the development, creation and operation of AO systems. Novel and optimized concepts can be explored quickly and performance gains estimated by allowing AO scientist to decide whether such a concept will provide suitable gains. Also, the parameters for existing AO systems can be optimized in a systematic way as it is possible to replicate a configuration in simulation and alter only one variable at a time.
In this paper, we introduce a new AO simulation named Korea Adaptive Optics Simulation (KAOS), independently developed by LIG Nex1, and we verified the performance of KAOS by comparing with other AO simulation tools. Also, we propose a LTAO system with five Rayleigh LGSs optimized by using KAOS to overcome the disadvantage of the AO system with a single sodium LGS. We already tested the AO system with a single sodium LGS for a satellite imaging system, which shows a good optical performance [7]. However, sodium LGS has a disadvantage for use in South Korea. The disadvantage is that we, unfortunately, have only a few weeks to use the AO system because the summer in South Korea has a heavy rainy season and high humidity, even though a greater sodium density is observed during the winter than summer. Also the sodium LGS has a common disadvantage that is the variation of the sodium density with season and time, it causes that the variation of the LGS magnitude. Therefore, we proposed the LTAO system with five Rayleigh LGSs that can be operated for all four seasons without the variation of the LGS magnitude, and optimized by using KAOS. The design and configuration of KAOS are presented in Section II. Also, we compared with other AO simulation tools to verify KAOS, and these results are presented in Section III. Lastly, the simulation result of the proposed LTAO system with five Rayleigh LGSs for the satellite imaging system is discussed in section IV.
KAOS is written entirely in the Matlab^{®} programming language and software suite designed by MathWorks for science computing, available for Linux, Mac OS X and Windows [8]. Because of the compatibility of the Matlab^{®} (MathWorks, MA, USA), KAOS works well regardless of the user’s operating system or environment, and does not require the installation of additional modules. We also have designed it to be easy, be extremely modular, and flexible to use to enable development of new AO concepts. So, the modules can be used in a “stand-alone” fashion, and a monolithic end-to-end simulation too. The modules of KAOS consist of an atmospheric turbulence, LGSs & Line of Sight (LOS), WFSs, DMs, and reconstruction & correction. The flow of the end-to-end simulation of KAOS is shown in Fig. 1, and each module is detailed in the following sections. Figure 2 shows the Graphical User Interface (GUI) of the end-to-end simulation of KAOS, (a) is the phase screen observed by WFS, (b) is the detector image of tip-tilt WFS, (c) is the detector image of five SH-WFSs, (d) is the satellite image before AO correction, (e) is the residual wavefront error, (f) is the graph of Strehl ratio, and (g) is the satellite image after AO correction.
KAOS generates one or more phase screens to simulate the atmospheric turbulence above a telescope. We used two methods to generate the phase screens [9]. Firstly, by generating very large phase screens using a Fast Fourier Transfer (FFT) method, that are then moved across the telescope with a given direction and wind speed. However, we assumed that the phase screens are independent of time variation following “Frozen Flow” [10]. This method is preferred for long time simulations, because the simulated phase screens are generated by using the FFT method which results in a screen that is periodic and continuous across its opposite edges. As multiple phase screens at different heights and with different wind speeds and directions are used, this potential problem from periodic edges of the FFT method can be mitigated.
The other method is to generate smaller, uncorrelated phase screens for each AO loop. These are generated with additional sub-harmonics to better approximate the low-order spatial frequency in the atmospheric turbulence, which are often not present in the FFT method [9]. This method is called by a subharmonic method and can be very useful when applied to applications that require the random uncorrelated screens. Figure 3 shows the phase screens generated by the FFT method and the subharmonic method. We noticed that the phase screen generated by the FFT method clearly showed the periodic and continuous phase values across its opposite edges, and the phase screen generated by the subharmonic method clearly showed the low-order spatial frequency (tip/tilt and focus) in the atmospheric turbulence.
One of the main function of LGSs & LOS modules is to simulate the propagation of light through the atmospheric turbulence in a specific direction determined by the LGS, and to simulate the cone effect induced by the LGS as show in Fig. 4. Once the altitude and position of the LGS is set, the phase and complex amplitude of light through the phase screens in a specific direction are simulated by using LOS function. This can be achieved using either a geometry ray tracing method or using an angular spectrum method. The geometrical ray tracing method means that phase from each phase screen layer is simply summed to get the final wavefront distortion. On the other hand, the angular spectrum method means physical light propagation, and it is implemented in KAOS using Matlab^{®} code derived from Schmidt [9]. Figure 5 shows the original phase (a), and LOS phase (b) simulated by LGSs & LOS module, and these two phase screens are slightly different because of the LOS and cone effect of the LGS. From this result, we confirmed that the propagation of light through atmospheric turbulence and the cone effect can be effectively simulated by using LGSs & LOS module.
Also, KAOS has the function that calculates the number of return photons from the LGS to simulate the photon noise and readout noise for the WFS module. It can be calculated separately according to the type of the LGS (‘Sodium’ and ‘Rayleigh’) [11]. Using this function, a user can optimize the power, repetition rate, and altitude of the LGS.
In WFSs module, two types of the WFS are simulated, Shack-Hartmann WFS (SH-WFS) and Pyramid WFS (PyWFS), and the principles of the wavefront measurements of each are shown in Figs. 6(a) and 6(b), respectively. Micro-lens array of the SH-WFS divide the incident wavefront, and each of the beams in the sub-apertures focus onto the detector. SH-WFS measures the slope of the incident wavefront using the displacement of the spots on the detector [12]. PyWFS uses a pyramid prism in the image plane to create four sub-beams on the detector. Using these four sub-beams, PyWFS measures the slope of the incident wavefront [13]. Also, the noise of the detector can be simulated according to the number of return photons as shown in Fig. 7, and multiple WFSs for the tomographic AO system can be easily simulated by the modular nature of the simulation design. In tomographic AO simulation, each of the WFSs measures the phase through the phase screens in a specific direction simulated by LGSs & LOS module. Additionally, the tip-tilt WFS of the AO system can be simulated as the SH-WFS that has only 2 × 2 sub-apertures in this module.
DMs module generates the interaction matrices, which is consistent with the influence functions of the actuators. The influence functions are a grid of the actuators where each influence function is the shape of the DM where only one actuator is activated and modeled by 2-dimensional Gaussian functions [14]. This simulates a thin mirror surface when pushed or pulled by an actuator. Also, a tip-tilt mirror is modeled with only the tip-tilt modes as the influence functions. Figure 8 shows three examples of the simulated Gaussian influence functions by the DMs module. In Fig. 8, the center one is the influence function of the center actuator for the DM, which has 9 × 9 actuator array and diameter of 100 mm, and the left and right one is the influence functions of the next right actuator, respectively. Finally, the interaction matrices can be generated by using pseudo-inverse of the influence functions.
In KAOS, reconstruction & correction module is used to reconstruct the wavefront from the slope measured by WFSs, and create a command matrix using the interaction matrix computed by a DMs module. This module has two types of the reconstruction method, one is Zonal method using the Southwell algorithm [15–17], the other is a modal method using a Zernike basis [18]. After that, using the reconstructed wavefront, the command matrix can be calculated by a simple Matrix-Vector-Multiply using the interaction matrix. Finally, the correction process is simulated by using the command matrix and influence functions of the DMs. Also, this module calculates the residual wavefront errors, Strehl ratios, Point Spread Function (PSF)s before and after AO correction [19]. The PSF is used to simulate the satellite image through the AO system using the convolution theory. Figure 9 shows the residual wavefront errors, PSFs, simulated satellite images according to their Strehl ratios. Additionally, a Learn & Apply algorithm [20] is applied for tomographic AO simulation.
When a new AO simulation is introduced, it is important to ensure that the results from the new AO simulation are accurate. In generally, the performance of the AO simulation is verified against current and accepted AO simulation tools. There are a number of AO simulation tools, written in a number of different programming languages and providing a variety of functions [21–24]. The most widely used AO simulations of them are Yorick Adaptive Optics (YAO) [25], Python Adaptive Optics Simulation (PyAOS, also called by SOAPY) [26]. These AO simulations show great performance, flexibility, and ease of use. Here, KAOS is compared with YAO and SOAPY because KAOS benchmarked them.
The comparison simulations consist of two scenarios. The first scenario is AO performance with increasing SH-WFS sub-apertures as shown in Fig. 10. This scenario confirms that the simulation is correctly reproducing the fitting error expected when a DM with a finite spatial resolution is used to correct the wavefront errors with high spatial orders present. The parameters for the first scenario are listed in Table 1. As shown in Fig. 10, Strehl ratios of the YAO, SOAPY, KAOS are very close, it also indicates that the generation and scaling of atmospheric turbulence in KAOS is correct.
TABLE 1 The parameters used in the first AO simulation scenario
Parameter | Value |
---|---|
Simulation phase elements | 128 |
Number of turbulence layers | 4 |
Integrated seeing strength, r0 (cm) | 18.6 |
Altitude of turbulence layers (km) | 0, 5, 10, 15 |
Fractional layer strength | 0.5, 0.3, 0.1, 0.1 |
Layer wind speeds (m/s) | 10, 10, 15, 20 |
Frame rate (Hz) | 400 |
Whole phase screen diameter (m) | 256 |
Telescope primary diameter (m) | 8 |
Number of sub-apertures | 8 × 8–18 × 18 |
Pixels per sub-aperture | 10 |
Number of DM actuators | 9 × 9–19 ×19 |
Guide star position | On-axis |
Science target wavelength (nm) | 1,650 |
The second scenario is AO performance with increasing focus anisoplanatism as shown in Fig. 11. This scenario confirms that the simulation is correctly reproducing the cone effect error according to the altitude of the LGS. Three altitudes of LGS are simulated, with 15 km and 25 km being typical values for Rayleigh LGS, and 90 km as the mean altitude of sodium LGS. For each guide star configuration, a single turbulence layer is simulated, and its height increases. The parameters for the first scenario are listed in Table 2. As shown in Fig. 10, the layer is at low altitudes, the cone effect error is small, Strehl ratios are high for all guide star altitudes. As the layer increases, all simulations show that Strehl ratios decrease, with the reduction dependent upon the altitude of the guide star. There is a discrepancy in the simulation results. This discrepancy is currently under the investigation, however, all simulations obviously show the reduction of Strehl ratio because of the cone effect error.
TABLE 2 The parameters used in the second AO simulation scenario
Parameter | Value |
---|---|
Simulation phase elements | 128 |
Number of turbulence layers | 1 |
Integrated seeing strength, r0 (cm) | 18.6 |
Altitude of turbulence layers (km) | 0–20 |
Fractional layer strength | 1 |
Layer wind speeds (m/s) | 10 |
Frame rate (Hz) | 400 |
Whole phase screen diameter (m) | 256 |
Telescope primary diameter (m) | 8 |
Number of sub-apertures | 8 × 8 |
Pixels per sub-aperture | 10 |
Number of DM actuators | 9 × 9 |
Guide star position | On-axis |
Science target wavelength (nm) | 1,650 |
As mentioned in Section I, LGS is used to improve sky coverage of AO systems. In particular, the AO systems for the spy satellite imaging system also require the bright LGS because the brightness of the spy satellite is insufficient to measure the wavefront error. The most common ways of creating an artificial star in the sky are sodium and Rayleigh methods. The sodium method creates an artificial star on a thin layer of sodium atoms that are present between 80 and 100 km above the Earth’s surface [27]. The strongest of these lines for the sodium atom is the D2 line, centered at 589 nm, so the laser of sodium LGS has a wavelength of 589 nm. The Rayleigh LGS is created by propagating a beam into the atmosphere and observing the light backscattered from molecules in the atmosphere [28]. Therefore, the laser of the Rayleigh LGS is pulsed and synchronized with a shutter in front of the WFS so that only light at a specified altitude is observed. As the atmospheric air pressure decreases with altitude, the scattered return also decreases. This limits the altitude of Rayleigh LGS to around 20-25 km [29]. Rayleigh LGS is normally green laser with Nd:YAG [30] or Yb:YAG [31], and Ultra-Violet (UV) laser also can be used [29].
More commonly, sodium LGS is used for AO systems because the cone effect of the LGS is much smaller than of Rayleigh LGS. The wavefront variance
where
From these equations, the sodium LGS AO system has smaller wavefront error due to the cone effect than the Rayleigh LGS AO system because of the altitude of the LGS.
However, sodium LGS has a disadvantage for use in South Korea. The disadvantage is that we, unfortunately, have only a few weeks when we can use the AO system because the summer in South Korea has a heavy rainy season and high humidity, even though a greater sodium density is observed during the winter than summer [32]. Also, the sodium LGS has two common disadvantages: the variation of the sodium density with season and time causes variation of the LGS magnitude, and the other disadvantage of the sodium LGS is the thick depth of the sodium layer, which is of the order of 10 km. When using an SH-WFS, this thick depth exhibits itself as an elongation of the WFS and reduces the WFS performance.
To overcome the disadvantages of the sodium LGS of the AO systems for the satellite imaging system, we propose the LTAO system with multiple Rayleigh LGSs in this paper. Because Rayleigh LGSs use the backscattering from molecules in the atmosphere, the AO system can be operated for all four seasons without the variation of the LGS magnitude. The spot elongation also can be mitigated by the shutter in front of the WFS. The biggest issue of the Rayleigh LGS is the cone effect, however, it can be mitigated by using multiple LGSs. In principle, the cone effect error for the multiple LGSs case can then be approximated by [11]
where
where
To simulate the LTAO system with multiple Rayleigh LGSs, we used the same optical components and simulation parameters presented in [7], and just increased the number of the SH-WFSs and the LGSs, and changed the type of the LGS from sodium to Rayleigh. Also, we set the magnitude of all LGSs as 6 stellar magnitude, being a typical value for the LGS. Firstly, we simulated with the parameters of current AO system with a single sodium LGS to compare the simulation result from KAOS with actual measurement data. The parameters used in the AO simulations also are detailed in Table 3. Figure 12 shows the comparison of Strehl ratio between the simulation result and actual measurement data. In this figure, the green lines are from the simulation result before (dashed line) and after (solid line) AO correction, and the blue lines are averaged value of Strehl ratio from the simulation (dashed line) and actual measurement (solid line). We confirmed that the simulation result from KAOS is very close to the averaged value of actual measurement data. The difference between averaged Strehl ratios is only 1.12%. From this result, we notice that the simulation result from KAOS is also well matched with actual measurement result.
TABLE 3 The parameters used in section IV for AO simulation
Parameter | Value |
---|---|
Simulation phase elements | 256 |
Number of turbulence layers | 4 |
Integrated seeing strength, r0 (cm) | 7 |
Altitude of turbulence layers (km) | 0, 5, 10, 15 |
Fractional layer strength | 0.5, 0.3, 0.1, 0.1 |
Layer wind speeds (m/s) | 10, 10, 15, 20 |
Frame rate (Hz) | 2,000 |
Whole phase screen diameter (m) | 256 |
Telescope primary diameter (m) | 1.6 |
Number of sub-apertures | 18 × 18 |
Pixels per sub-aperture | 10 |
Number of DM actuators | 19 × 19 |
Number of SH-WFSs and LGSs | 5 |
Science target wavelength (nm) | 880 |
After that, we simulated with the parameters of the AO system with a single Rayleigh LGS to compare with the AO system with Sodium LGS. We remained the magnitude of the LGS and changed the altitude of the LGS to 20 km being a typical value for Rayleigh LGS. As confirmed in the Section III, we expected that Strehl ratio of the AO system with a single Rayleigh LGS was lower than the AO system with a single sodium LGS because of the cone effect. Figure 13 shows Strehl ratio of the AO system with a single Rayleigh LGS, the green lines are Strehl ratio before (dashed line) and after (solid line) AO correction, and the blue dash-dotted line is averaged value of Strehl ratio after AO correction. Most of Strehl ratio after AO correction is still less than 0.2, not much improvement than before the AO correction. We also compared the satellite images before and after the AO correction to verify the optical performance of the AO system. Figure 14 shows the comparison of simulated satellite images between uncorrected (left) and corrected (right) by the AO system with a single Rayleigh LGS. We confirmed that the satellite image after AO correction is not much different than before the AO correction, and concluded that the AO system with a single Rayleigh LGS is insufficient to use for the satellite imaging system because of a large cone effect.
Lastly, we simulated the LTAO system with five Rayleigh LGSs to confirm the AO performance of this AO system. There are two reasons why we designed the AO system with five Rayleigh LGSs. The first one is that at least three LGSs are required to mitigate the cone effect, and the second one is that five LGSs are optimized number of the LGS to use the Learn & Apply algorithm for LTAO [20]. Figure 15 shows Strehl ratio of the AO system with five Rayleigh LGSs, the green lines are Strehl ratio before (dashed line) and after (solid line) AO correction, and the blue dash-dotted line is averaged value of Strehl ratio after AO correction. In contrast to the case of the AO system with a single Rayleigh LGS, most of Strehl ratio after the AO correction is increased to above 0.3, which is similar to the case of the AO system with a single Sodium LGS. Also, as shown in Fig. 16, we confirmed that the satellite image after the AO correction was much more improved sharpness, contrast, and resolution compare to the satellite image before the AO correction. From these results, we verified that the cone effect of Rayleigh LGS is effectively mitigated by using multiple LGSs, and proposed AO system is appropriate to use for the satellite imaging system.
Consequently, we proposed a LTAO system with five Rayleigh LGSs and optimized this AO system by using KAOS to overcome the disadvantages of the AO system with a single sodium LGS for the satellite imaging system. We expected that this proposed system not only offers more available date to use the AO system under atmospheric conditions of South Korea, but also shows the similar optical performance with the sodium LGS AO system. We also presented a new AO simulation written by the Matlab^{®} program language, which is named KAOS. The key features of KAOS are simple, flexible, extensible, and easy to use AO simulation. Also, the modules of KAOS are introduced in each of the sections. Because these modules are designed to be extremely modular, these can be used in a “stand-alone” fashion and a monolithic end-to-end simulation as well. To verify the performance of KAOS, we compared it with current and accepted AO simulations, and confirmed that the simulation results are very close. We also confirmed that the simulation result from KAOS is very close to averaged values of actual measurement data from AO system with sodium LGS of a 1.6 m telescope. Because KAOS is still under development, we will continuously update the features of the modules to provide users with more options. In the near future, the atmosphere module will be added a new generation method of atmospheric turbulence, and to the reconstruction & correction module will be added another reconstruction method for tomographic AO system such as GLAO, MOAO, and MCAO. Also, we will improve the computational performance of KAOS using multi-core processors and Graphics Processing Unit (GPU) to achieve our ultimate goal to make KAOS as real-time AO simulation in the future works.
Curr. Opt. Photon. 2021; 5(2): 101-113
Published online April 25, 2021 https://doi.org/10.3807/COPP.2021.5.2.101
Copyright © Optical Society of Korea.
Kyohoon Ahn , Sung-Hun Lee, In-Kyu Park, Hwan-Seok Yang
Laser R&D Team, LIG Nex1, Yongin 16911, Korea
Correspondence to:kyohoon.ahn89@gmail.com, ORCID 0000-0002-1094-852X
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Adaptive optics (AO) systems are becoming more complex to improve their optical performance and enlarge their field of view, so it is a hard and time consuming process to set up and optimize the components of AO systems with actual implementation. However, simulations allow AO scientists and engineers to experiment with different optical layouts and components without needing to obtain and prepare them physically. In this paper, we introduce a new AO simulation named the Korea Adaptive Optics Simulation (KAOS), independently developed by LIG Nex1. We verified the performance of KAOS by comparing with other AO simulation tools. In the comparison simulation, we confirmed the results from KAOS and other AO simulation tools were very similar. Also, we proposed a laser tomography AO system with five Rayleigh laser guide stars (LGSs) optimized by using KAOS to overcome the disadvantages of the AO system with a single sodium LGS for the satellite imaging system. We verified the performance of the proposed AO system using KAOS, and the simulation result showed averaged Strehl ratio of 0.37.
Keywords: Adaptive optics, Atmospheric turbulence, Laser guide star, Numerical simulation, Wavefront sensing and correction
Adaptive Optics (AO) systems have reached the state-of-the-art level, which successfully can deliver diffraction-limited imaging performance [1]. Early AO systems can only reach the telescope’s diffraction limit over a narrow field with a single Wavefront Sensor (WFS) and a single Deformable Mirror (DM) as referred to Single-Conjugated AO (SCAO). To overcome the limitation of the early AO systems, current AO systems have been developed in two main directions: (a) increasing the imaging field of view and sky coverage, and (b) improving wavefront correction for bright targets. The first approach is to create an artificial star in the field of interest by using a bright (5 W–100 W) laser beam, so that the wavefront can be measured along a direction where no bright star is present [2]. After that, the AO systems has been addressed by tomographic AO configurations including Ground-Layer AO (GLAO), Multi-Conjugated AO (MCAO), Multi-Object AO (MOAO), and Laser Tomography AO (LTAO). These tomographic AO systems have multiple DMs, WFSs and Laser Guide Star (LGS)s to increase the AO corrected field of view and wavefront correction. GLAO systems deliver significant images sharpening over degree-scale fields by correcting only wavefront aberrations induced at low altitude [3]. MCAO systems deliver diffraction-limited imaging over a continuous arcminute-sized field of view using multiple DMs, each optically conjugated to a separate altitude [4]. With each DM correcting for atmospheric aberrations induced at the corresponding altitude, the overall wavefront correction is valid over a larger angle than is possible with a single DM. Also, MCAO systems require the multiple WFSs along multiple directions to measure the wavefront error induced from multiple layers. MOAO systems deliver the tomographic information in a different manner and instead of providing one continuously corrected field of view, corrects for a finite number of individual lines of sight within a large field of regard. Multiple WFSs of the MOAO systems calculate the optimum correction for each science direction, and a different DM is used to perform correction for each line of sight to each science target [5]. LTAO systems deliver diffraction-limited imaging over a narrow field of view and mitigate LGS focus anisoplanatism also known as the cone effect. LTAO systems are similar to MOAO, where multiple LGSs are used as guide stars, though usually only one direction is corrected for [6]. The use of multiple LGSs compensates for the cone effect as the overlap of observed turbulence provides the entire cylinder observed by the science target. This improves performance of LGS AO systems and will be vital for extremely large telescopes and AO systems with Rayleigh LGS that have large cone effect error.
As mentioned above, recent AO systems are becoming more optically complex to improve their optical performance. These AO systems require expensive components such as multiple DMs, WFSs, and a Real-Time Controller (RTC), which are non-trivial to setup and optimize. Also, it is a hard and time-consuming process to setup and optimize the components of AO systems with actual implementation. However, simulations allow AO scientists and engineers to experiment with different optical layouts and components without needing to obtain and prepare them physically. So, simulations play a number of important roles in the development, creation and operation of AO systems. Novel and optimized concepts can be explored quickly and performance gains estimated by allowing AO scientist to decide whether such a concept will provide suitable gains. Also, the parameters for existing AO systems can be optimized in a systematic way as it is possible to replicate a configuration in simulation and alter only one variable at a time.
In this paper, we introduce a new AO simulation named Korea Adaptive Optics Simulation (KAOS), independently developed by LIG Nex1, and we verified the performance of KAOS by comparing with other AO simulation tools. Also, we propose a LTAO system with five Rayleigh LGSs optimized by using KAOS to overcome the disadvantage of the AO system with a single sodium LGS. We already tested the AO system with a single sodium LGS for a satellite imaging system, which shows a good optical performance [7]. However, sodium LGS has a disadvantage for use in South Korea. The disadvantage is that we, unfortunately, have only a few weeks to use the AO system because the summer in South Korea has a heavy rainy season and high humidity, even though a greater sodium density is observed during the winter than summer. Also the sodium LGS has a common disadvantage that is the variation of the sodium density with season and time, it causes that the variation of the LGS magnitude. Therefore, we proposed the LTAO system with five Rayleigh LGSs that can be operated for all four seasons without the variation of the LGS magnitude, and optimized by using KAOS. The design and configuration of KAOS are presented in Section II. Also, we compared with other AO simulation tools to verify KAOS, and these results are presented in Section III. Lastly, the simulation result of the proposed LTAO system with five Rayleigh LGSs for the satellite imaging system is discussed in section IV.
KAOS is written entirely in the Matlab^{®} programming language and software suite designed by MathWorks for science computing, available for Linux, Mac OS X and Windows [8]. Because of the compatibility of the Matlab^{®} (MathWorks, MA, USA), KAOS works well regardless of the user’s operating system or environment, and does not require the installation of additional modules. We also have designed it to be easy, be extremely modular, and flexible to use to enable development of new AO concepts. So, the modules can be used in a “stand-alone” fashion, and a monolithic end-to-end simulation too. The modules of KAOS consist of an atmospheric turbulence, LGSs & Line of Sight (LOS), WFSs, DMs, and reconstruction & correction. The flow of the end-to-end simulation of KAOS is shown in Fig. 1, and each module is detailed in the following sections. Figure 2 shows the Graphical User Interface (GUI) of the end-to-end simulation of KAOS, (a) is the phase screen observed by WFS, (b) is the detector image of tip-tilt WFS, (c) is the detector image of five SH-WFSs, (d) is the satellite image before AO correction, (e) is the residual wavefront error, (f) is the graph of Strehl ratio, and (g) is the satellite image after AO correction.
KAOS generates one or more phase screens to simulate the atmospheric turbulence above a telescope. We used two methods to generate the phase screens [9]. Firstly, by generating very large phase screens using a Fast Fourier Transfer (FFT) method, that are then moved across the telescope with a given direction and wind speed. However, we assumed that the phase screens are independent of time variation following “Frozen Flow” [10]. This method is preferred for long time simulations, because the simulated phase screens are generated by using the FFT method which results in a screen that is periodic and continuous across its opposite edges. As multiple phase screens at different heights and with different wind speeds and directions are used, this potential problem from periodic edges of the FFT method can be mitigated.
The other method is to generate smaller, uncorrelated phase screens for each AO loop. These are generated with additional sub-harmonics to better approximate the low-order spatial frequency in the atmospheric turbulence, which are often not present in the FFT method [9]. This method is called by a subharmonic method and can be very useful when applied to applications that require the random uncorrelated screens. Figure 3 shows the phase screens generated by the FFT method and the subharmonic method. We noticed that the phase screen generated by the FFT method clearly showed the periodic and continuous phase values across its opposite edges, and the phase screen generated by the subharmonic method clearly showed the low-order spatial frequency (tip/tilt and focus) in the atmospheric turbulence.
One of the main function of LGSs & LOS modules is to simulate the propagation of light through the atmospheric turbulence in a specific direction determined by the LGS, and to simulate the cone effect induced by the LGS as show in Fig. 4. Once the altitude and position of the LGS is set, the phase and complex amplitude of light through the phase screens in a specific direction are simulated by using LOS function. This can be achieved using either a geometry ray tracing method or using an angular spectrum method. The geometrical ray tracing method means that phase from each phase screen layer is simply summed to get the final wavefront distortion. On the other hand, the angular spectrum method means physical light propagation, and it is implemented in KAOS using Matlab^{®} code derived from Schmidt [9]. Figure 5 shows the original phase (a), and LOS phase (b) simulated by LGSs & LOS module, and these two phase screens are slightly different because of the LOS and cone effect of the LGS. From this result, we confirmed that the propagation of light through atmospheric turbulence and the cone effect can be effectively simulated by using LGSs & LOS module.
Also, KAOS has the function that calculates the number of return photons from the LGS to simulate the photon noise and readout noise for the WFS module. It can be calculated separately according to the type of the LGS (‘Sodium’ and ‘Rayleigh’) [11]. Using this function, a user can optimize the power, repetition rate, and altitude of the LGS.
In WFSs module, two types of the WFS are simulated, Shack-Hartmann WFS (SH-WFS) and Pyramid WFS (PyWFS), and the principles of the wavefront measurements of each are shown in Figs. 6(a) and 6(b), respectively. Micro-lens array of the SH-WFS divide the incident wavefront, and each of the beams in the sub-apertures focus onto the detector. SH-WFS measures the slope of the incident wavefront using the displacement of the spots on the detector [12]. PyWFS uses a pyramid prism in the image plane to create four sub-beams on the detector. Using these four sub-beams, PyWFS measures the slope of the incident wavefront [13]. Also, the noise of the detector can be simulated according to the number of return photons as shown in Fig. 7, and multiple WFSs for the tomographic AO system can be easily simulated by the modular nature of the simulation design. In tomographic AO simulation, each of the WFSs measures the phase through the phase screens in a specific direction simulated by LGSs & LOS module. Additionally, the tip-tilt WFS of the AO system can be simulated as the SH-WFS that has only 2 × 2 sub-apertures in this module.
DMs module generates the interaction matrices, which is consistent with the influence functions of the actuators. The influence functions are a grid of the actuators where each influence function is the shape of the DM where only one actuator is activated and modeled by 2-dimensional Gaussian functions [14]. This simulates a thin mirror surface when pushed or pulled by an actuator. Also, a tip-tilt mirror is modeled with only the tip-tilt modes as the influence functions. Figure 8 shows three examples of the simulated Gaussian influence functions by the DMs module. In Fig. 8, the center one is the influence function of the center actuator for the DM, which has 9 × 9 actuator array and diameter of 100 mm, and the left and right one is the influence functions of the next right actuator, respectively. Finally, the interaction matrices can be generated by using pseudo-inverse of the influence functions.
In KAOS, reconstruction & correction module is used to reconstruct the wavefront from the slope measured by WFSs, and create a command matrix using the interaction matrix computed by a DMs module. This module has two types of the reconstruction method, one is Zonal method using the Southwell algorithm [15–17], the other is a modal method using a Zernike basis [18]. After that, using the reconstructed wavefront, the command matrix can be calculated by a simple Matrix-Vector-Multiply using the interaction matrix. Finally, the correction process is simulated by using the command matrix and influence functions of the DMs. Also, this module calculates the residual wavefront errors, Strehl ratios, Point Spread Function (PSF)s before and after AO correction [19]. The PSF is used to simulate the satellite image through the AO system using the convolution theory. Figure 9 shows the residual wavefront errors, PSFs, simulated satellite images according to their Strehl ratios. Additionally, a Learn & Apply algorithm [20] is applied for tomographic AO simulation.
When a new AO simulation is introduced, it is important to ensure that the results from the new AO simulation are accurate. In generally, the performance of the AO simulation is verified against current and accepted AO simulation tools. There are a number of AO simulation tools, written in a number of different programming languages and providing a variety of functions [21–24]. The most widely used AO simulations of them are Yorick Adaptive Optics (YAO) [25], Python Adaptive Optics Simulation (PyAOS, also called by SOAPY) [26]. These AO simulations show great performance, flexibility, and ease of use. Here, KAOS is compared with YAO and SOAPY because KAOS benchmarked them.
The comparison simulations consist of two scenarios. The first scenario is AO performance with increasing SH-WFS sub-apertures as shown in Fig. 10. This scenario confirms that the simulation is correctly reproducing the fitting error expected when a DM with a finite spatial resolution is used to correct the wavefront errors with high spatial orders present. The parameters for the first scenario are listed in Table 1. As shown in Fig. 10, Strehl ratios of the YAO, SOAPY, KAOS are very close, it also indicates that the generation and scaling of atmospheric turbulence in KAOS is correct.
TABLE 1. The parameters used in the first AO simulation scenario.
Parameter | Value |
---|---|
Simulation phase elements | 128 |
Number of turbulence layers | 4 |
Integrated seeing strength, r0 (cm) | 18.6 |
Altitude of turbulence layers (km) | 0, 5, 10, 15 |
Fractional layer strength | 0.5, 0.3, 0.1, 0.1 |
Layer wind speeds (m/s) | 10, 10, 15, 20 |
Frame rate (Hz) | 400 |
Whole phase screen diameter (m) | 256 |
Telescope primary diameter (m) | 8 |
Number of sub-apertures | 8 × 8–18 × 18 |
Pixels per sub-aperture | 10 |
Number of DM actuators | 9 × 9–19 ×19 |
Guide star position | On-axis |
Science target wavelength (nm) | 1,650 |
The second scenario is AO performance with increasing focus anisoplanatism as shown in Fig. 11. This scenario confirms that the simulation is correctly reproducing the cone effect error according to the altitude of the LGS. Three altitudes of LGS are simulated, with 15 km and 25 km being typical values for Rayleigh LGS, and 90 km as the mean altitude of sodium LGS. For each guide star configuration, a single turbulence layer is simulated, and its height increases. The parameters for the first scenario are listed in Table 2. As shown in Fig. 10, the layer is at low altitudes, the cone effect error is small, Strehl ratios are high for all guide star altitudes. As the layer increases, all simulations show that Strehl ratios decrease, with the reduction dependent upon the altitude of the guide star. There is a discrepancy in the simulation results. This discrepancy is currently under the investigation, however, all simulations obviously show the reduction of Strehl ratio because of the cone effect error.
TABLE 2. The parameters used in the second AO simulation scenario.
Parameter | Value |
---|---|
Simulation phase elements | 128 |
Number of turbulence layers | 1 |
Integrated seeing strength, r0 (cm) | 18.6 |
Altitude of turbulence layers (km) | 0–20 |
Fractional layer strength | 1 |
Layer wind speeds (m/s) | 10 |
Frame rate (Hz) | 400 |
Whole phase screen diameter (m) | 256 |
Telescope primary diameter (m) | 8 |
Number of sub-apertures | 8 × 8 |
Pixels per sub-aperture | 10 |
Number of DM actuators | 9 × 9 |
Guide star position | On-axis |
Science target wavelength (nm) | 1,650 |
As mentioned in Section I, LGS is used to improve sky coverage of AO systems. In particular, the AO systems for the spy satellite imaging system also require the bright LGS because the brightness of the spy satellite is insufficient to measure the wavefront error. The most common ways of creating an artificial star in the sky are sodium and Rayleigh methods. The sodium method creates an artificial star on a thin layer of sodium atoms that are present between 80 and 100 km above the Earth’s surface [27]. The strongest of these lines for the sodium atom is the D2 line, centered at 589 nm, so the laser of sodium LGS has a wavelength of 589 nm. The Rayleigh LGS is created by propagating a beam into the atmosphere and observing the light backscattered from molecules in the atmosphere [28]. Therefore, the laser of the Rayleigh LGS is pulsed and synchronized with a shutter in front of the WFS so that only light at a specified altitude is observed. As the atmospheric air pressure decreases with altitude, the scattered return also decreases. This limits the altitude of Rayleigh LGS to around 20-25 km [29]. Rayleigh LGS is normally green laser with Nd:YAG [30] or Yb:YAG [31], and Ultra-Violet (UV) laser also can be used [29].
More commonly, sodium LGS is used for AO systems because the cone effect of the LGS is much smaller than of Rayleigh LGS. The wavefront variance
where
From these equations, the sodium LGS AO system has smaller wavefront error due to the cone effect than the Rayleigh LGS AO system because of the altitude of the LGS.
However, sodium LGS has a disadvantage for use in South Korea. The disadvantage is that we, unfortunately, have only a few weeks when we can use the AO system because the summer in South Korea has a heavy rainy season and high humidity, even though a greater sodium density is observed during the winter than summer [32]. Also, the sodium LGS has two common disadvantages: the variation of the sodium density with season and time causes variation of the LGS magnitude, and the other disadvantage of the sodium LGS is the thick depth of the sodium layer, which is of the order of 10 km. When using an SH-WFS, this thick depth exhibits itself as an elongation of the WFS and reduces the WFS performance.
To overcome the disadvantages of the sodium LGS of the AO systems for the satellite imaging system, we propose the LTAO system with multiple Rayleigh LGSs in this paper. Because Rayleigh LGSs use the backscattering from molecules in the atmosphere, the AO system can be operated for all four seasons without the variation of the LGS magnitude. The spot elongation also can be mitigated by the shutter in front of the WFS. The biggest issue of the Rayleigh LGS is the cone effect, however, it can be mitigated by using multiple LGSs. In principle, the cone effect error for the multiple LGSs case can then be approximated by [11]
where
where
To simulate the LTAO system with multiple Rayleigh LGSs, we used the same optical components and simulation parameters presented in [7], and just increased the number of the SH-WFSs and the LGSs, and changed the type of the LGS from sodium to Rayleigh. Also, we set the magnitude of all LGSs as 6 stellar magnitude, being a typical value for the LGS. Firstly, we simulated with the parameters of current AO system with a single sodium LGS to compare the simulation result from KAOS with actual measurement data. The parameters used in the AO simulations also are detailed in Table 3. Figure 12 shows the comparison of Strehl ratio between the simulation result and actual measurement data. In this figure, the green lines are from the simulation result before (dashed line) and after (solid line) AO correction, and the blue lines are averaged value of Strehl ratio from the simulation (dashed line) and actual measurement (solid line). We confirmed that the simulation result from KAOS is very close to the averaged value of actual measurement data. The difference between averaged Strehl ratios is only 1.12%. From this result, we notice that the simulation result from KAOS is also well matched with actual measurement result.
TABLE 3. The parameters used in section IV for AO simulation.
Parameter | Value |
---|---|
Simulation phase elements | 256 |
Number of turbulence layers | 4 |
Integrated seeing strength, r0 (cm) | 7 |
Altitude of turbulence layers (km) | 0, 5, 10, 15 |
Fractional layer strength | 0.5, 0.3, 0.1, 0.1 |
Layer wind speeds (m/s) | 10, 10, 15, 20 |
Frame rate (Hz) | 2,000 |
Whole phase screen diameter (m) | 256 |
Telescope primary diameter (m) | 1.6 |
Number of sub-apertures | 18 × 18 |
Pixels per sub-aperture | 10 |
Number of DM actuators | 19 × 19 |
Number of SH-WFSs and LGSs | 5 |
Science target wavelength (nm) | 880 |
After that, we simulated with the parameters of the AO system with a single Rayleigh LGS to compare with the AO system with Sodium LGS. We remained the magnitude of the LGS and changed the altitude of the LGS to 20 km being a typical value for Rayleigh LGS. As confirmed in the Section III, we expected that Strehl ratio of the AO system with a single Rayleigh LGS was lower than the AO system with a single sodium LGS because of the cone effect. Figure 13 shows Strehl ratio of the AO system with a single Rayleigh LGS, the green lines are Strehl ratio before (dashed line) and after (solid line) AO correction, and the blue dash-dotted line is averaged value of Strehl ratio after AO correction. Most of Strehl ratio after AO correction is still less than 0.2, not much improvement than before the AO correction. We also compared the satellite images before and after the AO correction to verify the optical performance of the AO system. Figure 14 shows the comparison of simulated satellite images between uncorrected (left) and corrected (right) by the AO system with a single Rayleigh LGS. We confirmed that the satellite image after AO correction is not much different than before the AO correction, and concluded that the AO system with a single Rayleigh LGS is insufficient to use for the satellite imaging system because of a large cone effect.
Lastly, we simulated the LTAO system with five Rayleigh LGSs to confirm the AO performance of this AO system. There are two reasons why we designed the AO system with five Rayleigh LGSs. The first one is that at least three LGSs are required to mitigate the cone effect, and the second one is that five LGSs are optimized number of the LGS to use the Learn & Apply algorithm for LTAO [20]. Figure 15 shows Strehl ratio of the AO system with five Rayleigh LGSs, the green lines are Strehl ratio before (dashed line) and after (solid line) AO correction, and the blue dash-dotted line is averaged value of Strehl ratio after AO correction. In contrast to the case of the AO system with a single Rayleigh LGS, most of Strehl ratio after the AO correction is increased to above 0.3, which is similar to the case of the AO system with a single Sodium LGS. Also, as shown in Fig. 16, we confirmed that the satellite image after the AO correction was much more improved sharpness, contrast, and resolution compare to the satellite image before the AO correction. From these results, we verified that the cone effect of Rayleigh LGS is effectively mitigated by using multiple LGSs, and proposed AO system is appropriate to use for the satellite imaging system.
Consequently, we proposed a LTAO system with five Rayleigh LGSs and optimized this AO system by using KAOS to overcome the disadvantages of the AO system with a single sodium LGS for the satellite imaging system. We expected that this proposed system not only offers more available date to use the AO system under atmospheric conditions of South Korea, but also shows the similar optical performance with the sodium LGS AO system. We also presented a new AO simulation written by the Matlab^{®} program language, which is named KAOS. The key features of KAOS are simple, flexible, extensible, and easy to use AO simulation. Also, the modules of KAOS are introduced in each of the sections. Because these modules are designed to be extremely modular, these can be used in a “stand-alone” fashion and a monolithic end-to-end simulation as well. To verify the performance of KAOS, we compared it with current and accepted AO simulations, and confirmed that the simulation results are very close. We also confirmed that the simulation result from KAOS is very close to averaged values of actual measurement data from AO system with sodium LGS of a 1.6 m telescope. Because KAOS is still under development, we will continuously update the features of the modules to provide users with more options. In the near future, the atmosphere module will be added a new generation method of atmospheric turbulence, and to the reconstruction & correction module will be added another reconstruction method for tomographic AO system such as GLAO, MOAO, and MCAO. Also, we will improve the computational performance of KAOS using multi-core processors and Graphics Processing Unit (GPU) to achieve our ultimate goal to make KAOS as real-time AO simulation in the future works.
TABLE 1 The parameters used in the first AO simulation scenario
Parameter | Value |
---|---|
Simulation phase elements | 128 |
Number of turbulence layers | 4 |
Integrated seeing strength, r0 (cm) | 18.6 |
Altitude of turbulence layers (km) | 0, 5, 10, 15 |
Fractional layer strength | 0.5, 0.3, 0.1, 0.1 |
Layer wind speeds (m/s) | 10, 10, 15, 20 |
Frame rate (Hz) | 400 |
Whole phase screen diameter (m) | 256 |
Telescope primary diameter (m) | 8 |
Number of sub-apertures | 8 × 8–18 × 18 |
Pixels per sub-aperture | 10 |
Number of DM actuators | 9 × 9–19 ×19 |
Guide star position | On-axis |
Science target wavelength (nm) | 1,650 |
TABLE 2 The parameters used in the second AO simulation scenario
Parameter | Value |
---|---|
Simulation phase elements | 128 |
Number of turbulence layers | 1 |
Integrated seeing strength, r0 (cm) | 18.6 |
Altitude of turbulence layers (km) | 0–20 |
Fractional layer strength | 1 |
Layer wind speeds (m/s) | 10 |
Frame rate (Hz) | 400 |
Whole phase screen diameter (m) | 256 |
Telescope primary diameter (m) | 8 |
Number of sub-apertures | 8 × 8 |
Pixels per sub-aperture | 10 |
Number of DM actuators | 9 × 9 |
Guide star position | On-axis |
Science target wavelength (nm) | 1,650 |
TABLE 3 The parameters used in section IV for AO simulation
Parameter | Value |
---|---|
Simulation phase elements | 256 |
Number of turbulence layers | 4 |
Integrated seeing strength, r0 (cm) | 7 |
Altitude of turbulence layers (km) | 0, 5, 10, 15 |
Fractional layer strength | 0.5, 0.3, 0.1, 0.1 |
Layer wind speeds (m/s) | 10, 10, 15, 20 |
Frame rate (Hz) | 2,000 |
Whole phase screen diameter (m) | 256 |
Telescope primary diameter (m) | 1.6 |
Number of sub-apertures | 18 × 18 |
Pixels per sub-aperture | 10 |
Number of DM actuators | 19 × 19 |
Number of SH-WFSs and LGSs | 5 |
Science target wavelength (nm) | 880 |