Adaptive optical pre-compensation is seen as crucial for free-space laser communication in order to overcome the influence of atmospheric turbulence, particularly with respect to Earth-to-GEO feederlinks. This paper presents an experimental investigation into adaptive optical pre-compensation under large point-ahead-angles. We detail the design and realization of a free-space laser communication experiment over a 1.0 km horizontal path using a divergent beacon beam and a focussed signal beam, propagating in opposite directions. We describe the design and development of our experimental setup and measurement campaign using real turbulence. The median isoplanatic angle was calculated to be 0.16 mrad, while an increase in the received optical power through pre-compensation could be demonstrated for point-ahead-angles in the range of 0.13 mrad to 0.27 mrad.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
In the race towards very high throughput satellite (VHTS) systems, optical communication is a promising candidate to replace existing RF technologies, offering higher data rates, higher security and increased capacity . In this scenario, optical feederlinks would act as the backbone of the communication network, delivering huge amounts of data bidirectionally between Earth-based optical ground stations and geostationary satellites. Advances are being made in the development of this technology, however, the obstacle of losses due to the Earth’s atmosphere remains to be fully addressed. Site diversity will address cloud coverage by using a network of multiple optical ground stations [2,3], whereas adaptive optics (AO) is necessary to compensate for link efficiency losses due to atmospheric turbulence. As turbulence is strongest close to the ground, the beam travelling in the upward direction (uplink) is distorted at the very beginning of its propagation path leading to significant fading. This is combined with beam wander and smaller receiver apertures in space, making AO compensation of the uplink very challenging as compared to that of the beam travelling from the satellite downwards (downlink)  where large receiving apertures and traditional correction methods established in astronomy have already resulted in successful demonstration of AO post-compensation .
Adaptive optical pre-compensation describes the method of distorting an outgoing wavefront before propagation through a turbulent medium in order to optimize its wavefront at the target. It is seen as essential for the correction of beam wander of optical feederlinks . In an Earth-to-GEO feederlink scenario, it is performed by measuring the wavefront of a beacon beam which has propagated through the atmosphere, such as the downlink. The measured distorted wavefront is then applied to the uplink beam. After propagation through the same atmospheric turbulence, the uplink beam emerges with an optimal target wavefront due to the principle of reciprocity . In this way, simultaneous downlink post-compensation and uplink pre-compensation can be implemented, simplifying system architectures and maximizing bidirectional link efficiencies. It is imperative that the downlink and uplink propagate along the same atmospheric path. Limitations to this condition arise from the temporal evolution of the atmosphere and the relative movement of the satellite to the ground, as described by the point-ahead-angle (PAA).
Investigations into the viability of pre-compensation have accelerated in recent years with the push towards Earth-to-GEO optical communication. Simulation has been extensive [8–10], however, experimental demonstrations have been largely confined to laboratory tests using artificial turbulence [11, 12] and experiments using real turbulence have not investigated the PAA . Our previous work successfully demonstrated a 7.1 dB increase in the received intensity over a 0.5 km path under a small PAA of 28 μrad . To the best of our knowledge, experimental work on pre-compensation under larger PAAs has not been reported to date. In this paper, we investigate pre-compensation under PAAs up to 0.32 mrad in a horizontal propagation geometry.
In the development plan of AO technology for feederlinks, on-ground experimentation and validation is an essential precursor to integration in an Earth-to-GEO satellite experiment. To this end, we designed and conducted a proof-of-principle experiment to determine the feasibility of pre-compensation under PAAs larger than the isoplanatic angle. For this purpose we developed a “ground terminal” representing an optical ground station and a “satellite terminal” representing a satellite based communication partner. In section 2 we describe our experiment concept and design, in section 3 we introduce the design of the ground and satellite terminals and describe our experimental execution and measurement campaign, in sections 4 and 5 we detail our experimental results and discussions and in section 6 we present our conclusions.
2. Experiment concept
Given the baseline scenario of an Earth-to-GEO link, it is our aim to try to reproduce such turbulence conditions along a horizontal path. While it is not possible to influence the distribution of turbulence itself, we have adapted the beam geometry so as to approximate the development of the turbulence effect.
The influence of optical turbulence on a signal is dependent on the path’s length, the turbulence along it and the size of the beam. In the horizontal direction, we assume constant turbulence, described by the refractive index structure parameter, , whereas in the vertical direction, turbulence strength quickly reduces with elevation, becoming negligible within the first 20 km to 50 km of the 36 000 km GEO path . The Fried parameter, r0, is dependent on this profile, however, the resulting wavefront error is dependent on the ratio of the beam diameter, D, to the Fried parameter, D/r0. The rate of change of the influence of atmospheric turbulence can therefore be tailored by changing the turbulence strength i.e. the value of over the path, or by modifying the ratio of D/r0. For that reason, we proposed in a previous experiment , that by focussing the uplink beam on a target, we could achieve a beam geometry where the turbulence at the beginning of the path near the ground terminal would have the greatest influence on the wavefront and that the effective turbulence strength with respect to the induced wavefront aberration would decrease over the propagation distance.
The PAA for a GEO satellite is approximately 18 μrad. The accepted angular limit over which AO implementation is beneficial is described by the isoplanatic angle, θ0. Under good seeing conditions, the PAA is similar in size to θ0, but it decreases rapidly with stronger turbulence. While optical ground station locations are largely chosen for their good seeing, strong turbulence cannot always be avoided and it is therefore important to investigate AO pre-compensation under large PAAs with respect to θ0. Using the example of the ESA optical ground station on Mount Teide, the isoplanatic angle at 1064 nm is (15.7 ± 6.4) μrad [11,15], meaning that the isoplanatic angle is typically smaller than the PAA. This highlights the need for experiment beyond the PAA. The isokinetic angle, θTA, dependent on the telescope diameter, gives a larger bound under which tip-tilt pre-compensation can be performed . The relationship of θTA and θ0 to has been calculated and is depicted in Fig. 1 for the measurement campaign conditions. For our ground terminal telescope diameter, the wavelength of 1064 nm and anticipated turbulence conditions, θTA is roughly twice as large as θ0. Having already successfully demonstrated pre-compensation under a small PAA, we designed this experiment to investigate pre-compensation under large PAAs in the region of θ0 to θTA. We chose to separate the receiving and transmission apertures of the satellite terminal to enable measurement over 1.0 km of PAAs up to approximately 0.6 mrad, a displacement in the focal plane from a minimum of 123 mm up to a maximum of 596 mm.
3. Experimental execution
3.1. Terminal design
Figure 2 shows a sketch of the experimental setup of the ground and satellite terminals. In the following paragraph, the relevant elements can be identified in Fig. 2 through the designations in parentheses. The ground terminal comprises of a Newton telescope, the AO-box and corresponding interface optics. It is the more complex of the two terminals as it contains all of the AO elements. The AO-box is a stand-alone device which can be coupled with a given optical telescope using the correct interface optics. The entrance aperture (EP) of the AO-box is conjugated to the deformable mirror (DM), tip-tilt mirror (TTM), Shack-Hartmann wavefront sensor (WFS) and optical telescope aperture. There are three optical signals within the box, all at 1064 nm: the downlink coming from the satellite terminal, the uplink coming from within the box and an internal calibration source. The calibration source allows for the checking of the AO-box alignment independent of external optics and provides a means by which the interface optics of the ground terminal can be aligned. To meet a possible Earth-to-GEO scenario, the uplink and downlink are inversely circular polarized so that they are suitable for polarization separation while still being independent from the relative angle between satellite terminal and ground terminal. In operation, the downlink is aligned to the calibration source, using the interface optics. Motorization of the mirrors (M1 and M2) and the inclusion of two alignment cameras (PSF Cam 1 and Beam Cam), one to observe the point spread function and one the beam profile, have allowed for a degree of remote operation and are a step towards automation. Details on the unimorph deformable mirror, tip-tilt mirror, wavefront sensor and optical system are reported in [11,14]. For the implementation of a PAA, we used a point-ahead mirror (PAM) with an actuation range that corresponds to ±0.7 mrad at the ground terminal exit pupil.
The telescope used was the 30 cm ASA N300 12″ Astrograph from Astro Systeme Austria. The AO-box was mounted on this telescope which was in turn mounted on the ASA Direct Drive DDM85 Standard Mount with motorized right ascension and declination axes and Autoslew software. Thus, motorized alignment and the finest pointing of the entire ground terminal towards the satellite terminal is enabled, with great stability and repeatability at optimum position. See Fig. 3.
On the right side of Fig. 2, a sketch of the satellite terminal components can be seen. The satellite terminal contains two separate optical systems for transmission and receiving on a single breadboard. In the transmission path, the downlink beam is collimated with a (1/e2) diameter of 4.4 mm that broadens due to diffraction and atmospheric turbulence so that it is larger than the telescope aperture at the ground terminal. The satellite terminal transmission path also includes a camera (PSF Cam 2) for alignment and measurement of the uplink beam without a PAA. A VIS-camera (VIS Cam) aligned to the downlink optical axis allows for coarse pointing of the satellite terminal towards the ground terminal in the visible regime, before switching on the laser.
The uplink receiving optics of the satellite consist of two folding mirrors which direct the uplink towards a lens which concentrates all received light onto a photo diode (power-meter) for analysis (Thorlabs PM100 power meter and S122C sensor head). The first folding mirror (M3) is mounted on a linear stage which has a displacement range of between 123 mm and 596 mm from the downlink transmission axis. It also acts as an aperture stop thanks to its small 25.4 mm size. This can be seen in Fig. 4. There are no polarization optics in the receiving paths of the satellite terminal. The effective angular size of our measurement sampling is 18/25 μrad (X/Y) so that any pointing inaccuracies rapidly decrease the measured power.
Coarse pointing of the two terminals was performed in the VIS range, where the satellite terminal and ground terminal were pointed towards one another using the integrated VIS camera and a finderscope, respectively. The downlink laser was then switched on and an infrared viewer was used to ensure that it was centred on the 30 cm telescope aperture. Fine tuning of the telescope pointing was performed using the Autoslew software so that the downlink signal could be detected in the focal plane of the telescope. The motorized interface mirror optics (M1 and M3) were then used in combination with the beam camera (Beam Cam), PSF camera (PSF Cam 1) and finally the wavefront sensor (WFS) of the AO-box to align the downlink beam to the internal calibration source. It was checked that they were co-linear, and any defocus of the downlink was corrected using the collimation lens of the interface optics. The uplink signal was then switched on and the point-ahead-mirror (PAM) was used to correct pointing towards the satellite terminal. This was evaluated with an infrared viewer and the received power at the satellite terminal.
3.2. AO implementation
The control loop of the AO system has been developed and detailed in previous publications . In overview, the control loop uses wavefront measurements from a Shack-Hartmann Sensor to control two adaptive optical elements – a 40 actuator deformable mirror and a fast piezo tip-tilt mirror. The AO-box’s internal calibration source was used to generate a reference wavefront. After recording this reference wavefront, both the deformable mirror as well as the tip-tilt mirror were characterized by recording the respective actuator influence functions. The downlink correction efficiency decreases with successive deformable mirror modes and it was determined that correction of the first 24 modes alone (plus tip and tilt) was most efficient due to these modes’ large singular values . In order to calculate the voltages in the kHz-regime, an FPGA-based real-time setup was used for the control loop implementation.
The AO system operated in two modi.
- Real-time wavefront correction was performed in the “live AO” modus in which the control actively measured the wavefront and operated the adaptive mirrors at a frequency of 1.0 kHz. It is in this modus that AO pre- and post-compensation could be performed based on the live atmospheric turbulence measurements.
- The “static bias” modus was generated from the most recent live AO control sequence. It calculates and applies the average applied voltage to each actuator based on the last 1000 signals of the live AO measurement, equivalent to the last 1 s of operational time. This provides a static surface form to the deformable and tip tilt mirrors. It enabled a type of “active” as opposed to adaptive correction, as it provided an average wavefront and thereby some correction of residual system aberrations and lower order modes such as tip, tilt and defocus. This enabled more accurate pointing and focusing, and comparison with the live AO case.
3.3. Measurement campaign
The measurement campaign took place on a laser test range in Germany. The two link terminals were positioned close to the ground (1–2 m) in open-fronted garage-type housing to protect the operators and apparatus from the elements. The landscape between the terminals was lightly undulating and predominantly green. Measurements were taken in the evening in the absence of daylight while afternoons were used for terminal alignment and pre-testing. Conditions were mainly calm, with occasional snow or mist which typically dissipated after nightfall. All measurements presented here were recorded on the same day, where temperatures were in the range of 2.4 °C to 2.6 °C with high humidity and transverse winds of approximately (0.8 ± 0.2) m s−1.
The measurement routine was as follows: after alignment and AO-loop initiation without PAA, the minimal displacement of the receiving optics was implemented at 123 mm, translating into a PAA of 0.12 mrad. Measurements were then taken in 50 mm intervals up to a maximum value of 323 mm. For each PAA value measurement, the satellite terminal receiving mirror was first re-positioned. Using live AO, the point-ahead mirror in the ground terminal was driven in incremental steps until the spot could be observed on the receiving mirror using an infrared viewer. The mirror optics in the satellite terminal uplink path were then aligned in order to maximize the power received on the power meter. The live AO was then switched off and the static bias as described in sec. 3.2 was applied. The spot position on the satellite terminal was then verified to ensure that there was no pointing offset between the two AO control states. Measurements were recorded in the sequence “live AO” followed by the corresponding generated “static bias”. For each measurement, whether with live AO or static bias, the wavefront at the ground terminal was recorded at a rate of 1.0 kHz and a power-meter reading was also recorded at the satellite terminal at a rate of (320 ± 9) Hz for (3.1 ± 0.1) ms. The emitted power was approximately 5.2 mW in the uplink direction and 4.8 mW in the downlink direction.
In order to evaluate AO efficacy independent of the uplink measurements, ground terminal downlink wavefront measurements were analyzed by performing a Zernike deconstruction  and fitting the variance of the coefficients to turbulence theory  thereby approximating the effective value of D/r0. Analysis of all downlink wavefront measurements in the ground terminal recorded without the static bias provided the uncorrected value of D/r0 to be 4.66 ± 0.92. With AO correction, D/r0 was reduced to 0.47 ± 0.13 over the course of all measurements, an improvement by a factor of 10, indicating that AO post-compensation was highly effective in the correction of the downlink wavefront. Figure 5 shows the D/r0 values of the final measurements. The fluctuations in the static bias values indicate the changing turbulence conditions. The level of correction is however consistent, with D/r0 in the range of 0.4 to 0.5 for the AO measurements.
The goal of this experiment is to provide a proof-of-principle of pre-compensation through demonstration of a brighter and more stable uplink signal at the satellite terminal. Pre-compensation efficacy was therefore evaluated using the measured optical power at the satellite terminal. Figure 6 depicts the measured optical power for PAAs in the range of 0.12 mrad to 0.32 mrad with live AO compensation in blue and static bias in orange. First, consider the measurements with the static bias, which do not apply a dynamic correction and should therefore be independent of PAA. Nonetheless, the received optical power fluctuated in its overall intensity. In particular, for 0.17 mrad and 0.22 mrad the optical power was higher than for the other measured values. This could be explained by a change of turbulence conditions, a conjecture which is further supported by the analysis of the downlink discussed above, where the average values D/r0 under static bias are also slightly lower. The values of the received power still falls within the error bars of the other measurements.
With the live AO control, the increase in the received power as compared to the measurements with the static bias is immediately evident. The measured power is greater for all values of PAA up to 0.32 mrad for which no significant difference can be seen. One can observe a trend of decreasing power with increasing PAA for all measurements apart from the smallest value of the PAA. While there is an improvement using live AO for 0.12 mrad as compared to the static bias, the absolute value is smaller than for the larger PAA of 0.17 mrad. Turbulence conditions can rapidly change and it is possible that turbulence was stronger for this measurement. The corresponding D/r0 measurement for the bias case (Fig. 5) indicate the worst turbulence conditions of all measurements. While the absolute values of the fluctuations in the power readings are larger with AO, they have reduced proportionally. The standard deviations of the recordings without AO are in the range of 30 % to 90 %, with AO this reduces to 14 % to 24 % for the measurements up to 0.22 mrad, increasing to a maximum of 50 % for 0.32 mrad where pre-compensation is no longer effective. Consequently, we can conclude that the received signal at the satellite terminal was not only of higher intensity but also more stable through AO pre-compensation.
In Fig. 7, we present the above data as ratios of the power measurements with live AO PAO to those with static bias PBias. The error bars are obtained through error propagation methods and only include the fluctuations in the received power. Systematic errors such as those which arise through the changing turbulence conditions are not included. A maximum four-fold increase in the received power for 0.17 mrad is evident, as well as the decline in the efficiency with increasing PAA.
In Fig. 8, the cumulative distribution functions (CDF) of the power measurements for each value of the PAA are depicted. The specified threshold, depicted with the dotted line, indicates when the static bias curve has reached a value of 1.0. With increasing PAA, the absolute power measured with live AO was larger than the maximum power received under static bias for 61%, 100%, 95%, 43% and 0% of recordings. These results are an unambiguous demonstration of the increased efficiency provided by AO pre-compensation, even under a PAA.
It is important to note that the comparison of live AO and static bias is not directly equivalent to that of AO and no AO. The static bias provides a type of active compensation based on the average turbulence correction over the last 1000 AO measurements. It was our experience that in addition to correcting residual system aberrations, this provided a good correction of the lower modes, particularly of tip, tilt and defocus, which made the general measurement routine easier as it enabled reliable pointing even without live AO. Consistent measurement at the satellite terminal without live AO was not possible without implementation of the static bias. We therefore conclude that the static bias already demonstrates a better efficiency as compared to a completely static system devoid of AO and that this could be a useful AO application, particularly under strong turbulence conditions where measurement is otherwise not possible. Moreover, it could be further improved upon, given that in our system there may be residual errors in the uplink path which are not seen and therefore not corrected by the ground terminal AO. We are therefore of the opinion that the absolute values in increased efficiency of this demonstration of AO pre-compensation are conservative as the static bias already demonstrates active pre-compensation.
The recorded values of fell over the range of 1.3 × 10−16 m−2/3 to 8.3 × 10−16 m−2/3. The median value of corresponded to a value of the isoplanatic angle of 0.16 mrad so that it can be assumed that measurements were chiefly recorded outside of the isoplanatic range. Nonetheless, a significant improvement was demonstrated. As has been forecast elsewhere , the drop-off in optical quality may not be as severe as theory predicts since anisoplanatism is not as severe for lower spatial frequencies and lower modes are more easily corrected through AO. These results are a first demonstration of AO pre-compensation under large PAAs using real turbulence. The results are auspicious and future steps will require further testing under varying turbulence conditions but also with varying degrees of correction. The drop-off in the compensation efficiency is scientifically significant but is also of great interest to the laser communication and feederlink community and more definitive comparison with the theoretical isoplanatic and isokinetic models is required. Our next steps include the implementation of fiber coupling so that we are ready for experiment with an optical payload.
This paper details the design, execution and results of an experimental investigation into the viability of pre-compensation under a PAA over a horizontal test range. Two communication terminals were designed and manufactured to represent an optical ground terminal and optical satellite terminal, separated by a 1.0 km distance. The ground station consisted of an optical telescope and an AO-box, which was capable of simultaneous post-compensation of an incoming downlink beam and pre-compensation of an outgoing uplink beam. A point-ahead-mirror allowed for angular separation of each beam to recreate a PAA. The satellite terminal had a separate transmission aperture for the downlink and receiving aperture for the uplink. The receiving aperture could be moved along a linear stage for measurement under varying PAA. A divergent downlink beam and converging uplink beam was used. The efficiency of the pre-compensation was evaluated using power measurements of the received uplink. The experiment was conducted at 1064 nm with separation of the uplink and downlink signals through polarization.
In the experiment, measurements were recorded over PAAs in the range of 0.12 mrad to 0.37 mrad in steps of 0.05 mrad. A comparison was made between live AO and a static bias which applied a fixed form to the adaptive optical elements based on the average corrected wavefront of the last 1000 measurements of the live AO. The static bias was used as it provided an active compensation of residual aberrations and was useful for correction of lower modes such as tip, tilt and defocus, providing reliable pointing and allowing for measurements with and without live AO. The static bias therefore already provided an improvement upon a case without any AO pre-compensation. In the comparison of live AO and static bias, we were clearly able to demonstrate an increase in the received power using AO and a more stable signal. Moreover, a decline in the efficiency of the pre-compensation with increasing PAA was shown as is consistent with turbulence theory. A maximum ratio of 4.2 (6.2 dB) was recorded using pre-compensation under an angle of 0.17 mrad which is already in the region of the approximated median isoplanatic angle of 0.16 mrad.
Based on these results, we have confirmed the viability of pre-compensation under a PAA and anticipate that this will have significant results in the design of Earth-to-GEO feederlinks and generally in free space laser communication applications.
Bundesministerium für Wirtschaft und Energie and Deutsches Zentrum für Luft- und Raumfahrt (50 YH 1717)
This project has been managed by the German Aerospace Center (DLR). We would like to extend huge gratitude to Astrosysteme Austria (ASA) GmbH for providing us with the Astrograph N300 12″ Newton telescope and the ASA Direct Drive DDM85 Mount for our measurement campaign.
1. A. K. Majumdar and J. C. Ricklin, eds., Free-Space Laser Communications, vol. 2 (Springer-Verlag, 2008), 1st ed. [CrossRef]
2. A. Gharanjik, K. P. Liolis, B. Shankar, and B. E. Ottersten, “Spatial multiplexing in optical feeder links for high throughput satellites,” IEEE Global Conference on Signal and Information Processing (IEEE, 2014), pp. 1112–1116.
3. B. Roy, S. Poulenard, S. Dimitrov, R. Barrios, D. Giggenbach, A. L. Kernec, and M. Sotom, “Optical feeder links for high throughput satellites,” in 2015 IEEE International Conference on Space Optical Systems and Applications (ICSOS), (IEEE, 2015), pp. 1–6.
5. M. Chen, C. Liu, D. Rui, and H. Xian, “Performance verification of adaptive optics for satellite-to-ground coherent optical communications at large zenith angle,” Opt. Express 26, 4230–4242 (2018). [CrossRef] [PubMed]
6. P.-D. Arapoglou and N. Girault, “Optical feeder link architectures for very hts: Issues and possibilities,” presented at the International Conference on Space Optics (ICSO), Chania, Greece, 9–12 October 2018.
7. J. H. Shapiro, “Reciprocity of the turbulent atmosphere*,” J. Opt. Soc. Am. 61, 492–495 (1971). [CrossRef]
8. J. D. Barchers and D. L. Fried, “Optimal control of laser beams for propagation through a turbulent medium,” J. Opt. Soc. Am. A 19, 1779–1793 (2002). [CrossRef]
9. N. H. Schwartz, N. Védrenne, V. Michau, M.-T. Velluet, and F. Chazallet, “Mitigation of atmospheric effects by adaptive optics for free-space optical communications,” Proc. SPIE 7200, 72000J (2009). [CrossRef]
10. C. Robert, J.-M. Conan, and P. Wolf, “Impact of turbulence on high-precision ground-satellite frequency transfer with two-way coherent optical links,” Phys. Rev. A 93, 033860 (2016). [CrossRef]
11. N. Leonhard, R. Berlich, S. Minardi, A. Barth, S. Mauch, J. Mocci, M. Goy, M. Appelfelder, E. Beckert, and C. Reinlein, “Real-time adaptive optics testbed to investigate point-ahead-angle in pre-compensation of earth-to-geo optical communication,” Opt. Express 24, 13157–13172 (2016). [CrossRef] [PubMed]
12. R. Biérent, M.-T. Velluet, N. Védrenne, and V. Michau, “Experimental demonstration of the full-wave iterative compensation in free space optical communications,” Opt. Lett. 38, 2367–2369 (2013). [CrossRef] [PubMed]
13. T. Weyrauch and M. A. Vorontsov, “Atmospheric compensation with a speckle beacon in strong scintillation conditions: directed energy and laser communication applications,” Appl. Opt. 44, 6388–6401 (2005). [CrossRef] [PubMed]
14. A. Brady, R. Berlich, N. Leonhard, T. Kopf, P. Böttner, R. Eberhardt, and C. Reinlein, “Experimental validation of phase-only pre-compensation over 494 m free-space propagation,” Opt. Lett. 42, 2679–2682 (2017). [CrossRef] [PubMed]
15. B. García-Lorenzo and J. J. Fuensalida, “Statistical structure of the atmospheric optical turbulence at teide observatory from recalibrated generalized scidar data,” Mon. Notices Royal Astron. Soc. 410, 934–945 (2011). [CrossRef]
16. R. J. Sasiela, A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals, vol. 807 (Lincoln Laboratory, Massachusetts Institute of Technology, 1988). [CrossRef]
17. R. K. Tyson, Principles of Adaptive Optics (CRC Press, 2011).
18. V. N. Mahajan and G. ming Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007). [CrossRef]
19. R. J. Noll, “Zernike polynomials and atmospheric turbulence*,” J. Opt. Soc. Am. 66, 207–211 (1976). [CrossRef]
20. A. Quirrenbach, The Effects of Atmospheric Turbulence on Astronomical Observations, (The Center for Adaptive Optics, University of California, 2003).