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Adaptive Optics

Acronym: AO

Definition: a class of techniques where wavefront distortions are actively compensated

More general term: optics

German: adaptive Optik

Categories: general opticsgeneral optics, methodsmethods

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Cite the article using its DOI: https://doi.org/10.61835/gps

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Summary: This in-depth article on adaptive optics explains

  • the fundamental principles of adaptive optics, being based on wavefront sensing and wavefront correction devices such as deformable mirrors,
  • the function and challenges of control systems used in open-loop and closed loop correction strategies,
  • various technical details such as the quantification of optical aberrations, the basic requirements for correction (which depend on the application), and aspects of the placements of sensor and corrector, and
  • the application of adaptive optics in astronomical observatories, vision science and ophthalmology, microscopy, free-space optical communications and directed energy weapons.

The article also has a bibliography containing, among others, some useful review papers.

Adaptive optics is a technological solution to a fundamental problem in optical science: the distortion (aberration) of wavefronts. These are typically caused by random inhomogeneities (spatial variations) in optical systems or in the atmosphere; in the latter case, the distortions can also be time-dependent due to air motion (turbulence). Adaptive optics attempts to solve this problem by correcting these distortions, often in real time. Such techniques are applied in advanced imaging methods to provide clearer, sharper images, obtaining the native resolution of the instrument, as is limited e.g. by diffraction). Also, there are some other application areas, e.g. where optimum beam focusing is required.

The design and implementation of adaptive optics involves a complex interplay of physics, engineering, and computational algorithms. This multidisciplinary technology relies on advances in fields such as optics, control systems, signal processing, and more. It is a dynamic and vibrant field that continues to evolve and find new applications.

The Principle of Adaptive Optics

Adaptive optics operates on the principle of measuring wavefront distortions and compensating for these distortions to deliver a corrected optical wavefront. While the optical setups using that principle can differ in many respects, a system typically contains the following key components:

  • a wavefront sensor,
  • a wavefront correction device, and
  • a control system, forming the drive signals of the correction device from the wavefront sensor readings.

These components may be combined into an open-loop or closed-loop control system. Here, an open loop means that the sensor detects the uncorrected wavefronts, i.e., cannot “see” how well the correction works. A closed loop means that the sensor detects the corrected wavefronts, thus having the chance to eventually achieve near-perfect compensation despite adverse aspects such as, for example, a possible nonlinearity of the phase corrector system. However, one then needs to take care that dynamic instabilities are avoided. Often, one then applies an integrating control system, exhibiting a response function which decreases substantially with increasing frequency.

Note also that there are also sensorless imaging systems, i.e., not requiring a wavefront sensor. Here, the required wavefront correction is obtained by varying the effects of the correction device (with some set of settings) on the obtained images themselves – using some preexisting knowledge of the image, e.g. that it should contain well localized features.

The mentioned components are explained in the following.

1. Wavefront Sensors

Shack-Hartmann wavefront sensor
Figure 1: Optical setup of a Shack–Hartmann wavefront sensor.

Wavefront sensors are devices that measure the shape of an optical wavefront. The most commonly used type of wavefront sensor in adaptive optics systems is the Shack–Hartmann wavefront sensor. This sensor consists of an array of microlenses that focus the incoming light onto a detector plane (a focal plane array). Each microlens creates a focal spot on the detector, and the local wavefront direction can be inferred from the position of such a spot, i.e., from the deviation of its position from the nominal position.

Other types of wavefront sensors, in particular pyramid wavefront sensors, are also used in certain applications. Pyramid sensors offer advantages in terms of sensitivity and dynamic range, making them suitable for demanding applications with strong wavefront distortion.

A critical detail is the number of sensor elements, which limits the number of degrees of freedom. Obviously, this number must be at least equal to the number of aberration modes to be corrected (see below), and is often actually chosen several times higher.

2. Wavefront Corrector

A central component of an adaptive optics system is the wavefront correction device, which physically alters the optical wavefront to counteract the measured distortions. It is essentially a phase modulator with some spatial resolution – a spatial light modulator.

The most common type of wavefront corrector is the deformable mirror (Figure 2). Typically, such a mirror has a substantial number of actuators that can deform the surfaces in various ways to correct a wide range of wavefront distortions. Deformable mirrors used in adaptive optics systems come in several forms, including continuous-faceplate deformable mirrors, segmented mirrors, and micro-electromechanical system (MEMS) mirrors. Each of these types has its specific advantages and limitations, and the choice of a deformable mirror depends on the specific requirements of the application.

segmented mirrors vs. continuous surface deformable mirrors
Figure 2: Different kinds of deformable mirrors: (a) segmented mirror, (b) continuous surface deformable mirror.

The phase excursions for the reflected light are directly related to the spatial displacements according to <$\delta \varphi = 2 \: k \: \delta z = (4\pi / \lambda) \: \delta z$> (where the factor 2 is due to the double passage of light). The included wavelength dependence is ideal if the aberrations are caused by refractive index variations without any wavelength dependence – which is at least a reasonable approximation in the case of atmospheric turbulence, for example. A hypothetical spatial light modulator that produces wavelength-independent phase changes would be substantially less suitable for various applications.

Another variant are deformable phase plates, which can be operated in transmission rather than in reflection. They contain a deformable membrane above a liquid, which allows for modulation of the optical path length in the liquid:

setup of a deformable phase plate
Figure 3: Setup of a deformable phase plate, where the optical path length in a liquid can be modulated using a transparent electrostatic actuator array. Source: Phaseform.

There are also liquid crystal spatial light modulators (LCSLM), which are based on the liquid crystal modulator technology commonly used in displays. Note, however, that in this case, the wavelength dependence of the achieved phase changes may be problematic, and such modulators typically operate with polarized light. On the other hand, they offer high spatial resolution, and some of them operate in transmission rather than reflection, which can be advantageous. However, many modulators are based on liquid crystal on silicon (LCOS) technology, which is also reflective.

Several details of the correction device must be considered for use in an AO system. In particular:

  • Typically, each actuator of a deformable mirror primarily controls the height elevation of the mirror surface in its immediate vicinity, but it also has some influence on the surface shape further away. Similar phenomena may occur to some extent with other types of corrective devices.
  • The achievable optical phase changes must be large enough. (Some applications require particularly large phase shifts.)
  • For high-precision correction, it may be necessary to carefully characterize the detailed response of each input channel. This is because different actuators of a deformable mirror may have varying effects due to random variations in imperfections caused by mechanical coupling. It may even be necessary to periodically perform a calibration procedure that effectively measures the influence functions of all correction channels.
  • For high-speed correction, it is necessary to consider not only the limited bandwidth of the actuator but also its full frequency-dependent amplitude and phase response.
  • Of course, the effects of a corrective device strongly depend on its precise placement within an optical system. See below for further explanation.

3. Control Systems

The control system in an adaptive optics setup is responsible for generating control signals for the correction device based on the wavefront measurements obtained by the wavefront sensor. Ideally, it should provide accurate compensation while fully utilizing the speed offered by the correction device. This task is not trivial for several reasons:

  • One must map a given number of sensor measurements to a given number of actuators (e.g., in a deformable mirror), taking into account the known effect of each actuator. This is often far from a simple one-to-one mapping, since the actuators may not only have slightly different responses, but also affect overlapping regions of the wavefronts. A large control matrix, which may be sparse depending on the situation, is required.
  • To fully utilize the bandwidth of the correction device, it is important to achieve the optimal temporal shape of the correction signals, considering the complete frequency-dependent amplitude and phase response. (Note that the bandwidth limitations of the wavefront sensor are usually much less relevant.)
  • In cases where the actuators occasionally reach their travel limits, it may be necessary to optimize the behavior of the control system for such situations. Also, other possible types of nonlinearities should be appropriately addressed.

For these reasons, the design of a control system often involves the use of sophisticated algorithms and high-speed signal processors to achieve rapid and precise correction of wavefront distortion. Modern digital technology provides powerful components for such systems, which can also be fine-tuned during operation. This allows for achieving maximum control speed and accuracy in a closed-loop system while avoiding any dynamic instability.

There are several common control algorithms used in adaptive optics systems, including integrator-based controllers, adaptive filters, and model-based controllers. The selection of a control algorithm depends on the specific requirements of the application, including the level of wavefront distortion, the desired speed of correction, and the limitations of the correction device.

Some Technical Details

Characterization of Optical Aberrations

A simple measure of the strength of aberrations is the root mean square (r.m.s.) deviation of the optical path length. This usually needs to be compensated to a level well below <$\lambda / 10$>.

In some cases, the peak-to-valley aberrations are given, which are more directly related to the required dynamic range of a wavefront corrector, although they are in some ways less representative than the r.m.s. deviation.

Such single quantities carry very little information; they do not provide any details about the spatial shape or the spatial variation of aberrations. Therefore, more sophisticated methods have been developed to indicate the extent of aberration present in a system that requires compensation. The most common form is a decomposition into aberration modes based on Zernike polynomials. Here, the wavefront distortion is characterized by a function <$W(\rho, \varphi)$> which specifies the deviation of the optical path length as a function of the radial coordinate <$\rho$> (normalized to the pupil radius) and the azimuthal angle <$\varphi$>. This is then viewed as a linear superposition of Zernike polynomials <$Z_n^m(\rho, \varphi)$> with amplitude coefficients <$a_n^m$>:

$$W(\rho, \varphi) = \sum_{n = 0}^{\infty} \sum_{m = -n}^{+n} a_n^m \: Z_n^m(\rho, \varphi)$$

Different terms in that equation correspond to various types (“modes”) of aberrations such as piston, tilt, defocus, astigmatism or coma, as explained in the article on optical aberrations.

In adaptive optics, one can design a system to directly address Zernike modes (modal control), compensating for them up to a certain maximum order. Alternatively, one can use the Zernike model of aberrations for analysis purposes, but not directly in the manner mentioned above. For instance, it can be employed to work with spatial zones of the modulator (zonal control). Of course, this decision has a profound impact on signal processing.

High-end applications, such as astronomical telescopes sometimes require the compensation of hundreds or even thousands of Zernike terms. This, of course, necessitates wavefront sensors and correctors with a correspondingly large number of channels. Nevertheless, the r.m.s. deviations can be moderate, for example, limited to a few micrometers. In retinal imaging, the corrections are an order of magnitude stronger but require only a moderate number of Zernike terms (e.g., a few dozen terms).

Correction Requirements

The requirements for wavefront correction can vary greatly from one application to another. The following aspects must be considered:

  • The required wavefront accuracy. This is usually on the order of <$\lambda / 10$>, although smaller values may still give slightly better performance (with diminishing returns).
  • The number of aberration modes (e.g. Zernike modes) is related to the required number of sensor elements and actuators for the wavefront corrector. With a wide range of modes, it is possible to compensate for more complex types of aberrations. For example, a large telescope usually requires a substantial number of modes.
  • The required control bandwidth is another important parameter. While in some cases (such as microscopy), the aberrations may have little time dependence, in ophthalmology, a control bandwidth of at least a few Hertz is required, while telescopes benefit from a larger bandwidth of, for example, 50 Hz. Realized with an integrating control system, this may require an update rate on the order of 1 kHz.

Placement of Wavefront Sensor and Correction Device

The following considerations are necessary mainly because optical distortions do not simply remain unchanged along the optical path in the system, but rather evolve in a complicated manner. In particular, phase distortions are eventually transformed into intensity variations in the image plane, which is exactly the problem that AO aims to address.

Typically, an adaptive optical system contains a single wavefront sensor and a single wavefront corrector. This configuration is called single-conjugate adaptive optics (SCAO). The correct placement of these components in the optical system is critical to achieving good performance. This can be understood by considering Fourier optics principles. In general, one will try to position both the wavefront sensor and the wavefront corrector in planes that are conjugate to those where the aberrations are largely generated. This positioning offers the best chance of properly compensating for the aberrations. For a telescope, for example, one often (but not always) uses pupil-conjugate planes. This is because atmospheric propagation mainly causes phase changes in that plane, which in turn lead to intensity changes in the image plane and distort the image. Trying to compensate for this with a phase modulator in the image plane would not work. However, sometimes a deformable secondary mirror is used, which, unfortunately, is not in the pupil conjugate plane.

In principle, even better performance can be achieved with multiconjugate adaptive optics (MCAO) (see e.g. Ref. [8]), which essentially means using wavefront sensors (and correctors) in at least two different planes that are not mutually conjugate. This way, one can better compensate for aberrations originating from different places – for example, from atmospheric turbulence in a wider range of directions, as well as imperfect components of an imaging system. Of course, this approach greatly increases system complexity and cost, and is therefore rarely employed.

Applications of Adaptive Optics

The potential of adaptive optics is being utilized in a wide range of fields, each presenting its own distinct challenges and requirements. Some of these are discussed below.

Astronomy

Well before the year 2000, large ground-based astronomical telescopes reached a performance that was no longer diffraction-limited, but severely limited by the turbulent nature of the Earth's atmosphere. In other words, it was much worse than what would be theoretically possible for a large telescope in the absence of atmospheric distortions. This is true even for telescopes placed on high mountains in regions with more favorable atmospheric conditions. The main problem is variations in refractive index caused by temperature inhomogeneities, which affect the density of the air. These effects are most pronounced for telescopes that operate at short wavelengths of light. The problem is compounded by turbulence, which causes rapid variations in optical distortions.

One way to overcome this obstacle was to develop large space-based telescopes, such as the Hubble Telescope and the James Webb Space Telescope. For several reasons, however, it remained highly desirable to solve this problem by significantly reducing the effects of atmospheric distortion in ground-based telescopes. For example, ground-based telescopes can have much larger mirrors than space-based telescopes, and they can be optimized incrementally due to their greater accessibility during operation.

Adaptive optics has become the key solution to the aforementioned problem [1], and is now used in many observatories.

One of the challenges in achieving high-quality image correction is accurately measuring time-dependent wavefront distortions. A fundamental principle involves utilizing distant stars as virtual point sources, which deliver fairly flat wavefronts that are then distorted by our atmosphere. For accurate wavefront correction, a guide star must be (a) sufficiently close to the direction of observation (so that the corresponding isoplanatic patch contains the objects of interest) and (b) sufficiently bright for accurate wavefront measurements. Since a star may not always be available for observation, an alternative option is to use an artificial guide star, which is created by sending an appropriate laser beam into the sky (→ laser guide stars). One possibility is to create a sodium beacon, where the laser beam's wavelength is precisely tuned to an absorption resonance of sodium atoms at 589.2 nm. These atoms are found at about 90 km altitude in the atmosphere and emit fluorescent light after absorbing the laser light. There are also Rayleigh beacons, which use Rayleigh scattering of light from a pulsed laser, typically at a shorter wavelength, such as in the green spectral range. Laser guide stars have greatly improved the sky coverage, although they are not ideal, primarily due to the cone effect.

Several types of wavefront correctors have been optimized for use in astronomical telescopes [6]. Although large telescopes are often equipped with active optics based on a multi-segment primary telescope mirror, the associated positioning control is usually far too slow to be used for adaptive optics. Therefore, an additional fast wavefront corrector with a large number of actuators (hundreds or even thousands) and high speed is required.

Traditional AO systems with a single laser guide star can only correct for distortions within a limited area of the sky. Multi-conjugate adaptive optics (MCAO) systems, however, have been developed which use multiple guide stars and deformable mirrors to correct distortions over a wider area. As a result, they enable the study of larger astronomical regions.

Some prominent examples of large observatories using adaptive optics technology:

  • The Keck Observatory on Mauna Kea, an active volcano in Hawaii, with its twin 10-meter telescopes has been among the first to be equipped with adaptive optics systems since 1999. These have been regularly upgraded to improve their performance [3].
  • The 5.1-meter Hale Telescope at the Palomar Observatory in California was also one of the first telescopes to be equipped with an adaptive optics system and has been regularly upgraded.
  • The 8.2-meter Subaru Telescope, located at Mauna Kea, Hawaii, is equipped with an AO system based on a wavefront sensor with 36 photon-counting avalanche photodiode modules, as well as a bimorph wavefront correcting deformable mirror with 36 driving electrodes. The system has been in service since 2002.
  • The Large Binocular Telescope (LBT), located in Arizona, USA, features an innovative AO system that includes an adaptive secondary mirror with 672 actuators and a high-order pyramid wavefront sensor. This system has achieved performances never seen before on large ground-based optical telescopes, with images showing a contrast as high as 10-4.
  • The Gemini South Telescope in Cerro Pachón, Chile has a multi-conjugate adaptive optics system (GeMS) and was the first sodium-based multi-Laser Guide Star (LGS) adaptive optics system. It uses five laser guide stars and two deformable mirrors. The system has been in regular operation since 2011, producing near-infrared images close to the diffraction limit [8, 9].
  • The Very Large Telescope (VLT) in Chile, which is operated by the European Southern Observatory (ESO), has been equipped with adaptive optics systems for many of its instruments. These systems have enabled the VLT to capture some of the most detailed images of the night sky.
  • The Giant Magellan Telescope (GMT) utilizes a set of seven 8.4-m mirrors and is currently being constructed at the Las Campanas Observatory in Chile. The objective is to achieve an image resolution that exceeds that of the Hubble Space Telescope. Adaptive optics will be based on a deformable secondary mirror containing over 7000 voice coil actuators and will utilize various techniques, including six laser guide stars, depending on the observed objects.

Given the importance and high cost of astronomical installations, substantial investments were made to implement this technology with high precision. Using highly advanced AO technology, all of these observatories now provide significantly improved images of galaxies, planets, and other celestial bodies compared to previous telescopes. This has allowed for a much closer approach to the final limit of diffraction-limited resolution for large telescopes. This has facilitated the detailed study of a wide range of astronomical phenomena. For instance, AO has enabled astronomers to examine the surfaces of planets in our solar system, observe the formation of stars in distant galaxies, and study the supermassive black hole at the center of our galaxy, the Milky Way.

Vision Science and Ophthalmology

In ophthalmology, it is often necessary to capture precise high-resolution images of the retina, which is the light-sensitive organ in the eye. For example, this technology can help improve the diagnosis and treatment of various retinal diseases, including retinal detachment, glaucoma, diabetic retinopathy, age-related macular degeneration, and retinitis pigmentosa. Furthermore, high image resolution, which enables imaging at the cellular level, is crucial for fundamental vision research. For instance, it has played a significant role in accurately identifying various types of photoreceptors.

Similar to astronomy, one faces the problem of (also somewhat time-dependent) image aberrations, but in this case, the aberrations are primarily caused by the cornea and the lens of the eye. Of course, such aberrations also limit the eye's vision, but here we are focusing on the challenge of diagnosis. Adaptive optics is used to achieve significant improvements in the quality of retinal images by compensating for aberrations. AO retinal imaging can reveal pathological changes in the retina at an early stage, enabling early intervention and enhancing patient outcomes. Besides, the measurements quickly and reliably determine the necessary details for prescription glasses.

Quite different methods of adaptive optics in-vivo retinal imaging have been developed:

  • Various methods are based on a wavefront sensor and a correction device, which are integrated into some kind of imaging system:
    • Flood illumination refers to capturing images using a flash of incoherent light and imaging the retina with a camera. This is done immediately after compensating for wavefront errors using an additional infrared laser source. A non-ideal aspect is that aberrations measured at one (infrared) wavelength do not allow perfect compensation at other (visible) wavelengths used for imaging.
    • In scanning laser ophthalmoscopy (SLO), one uses a laser source both for aberration measurement and imaging with a raster scan. The principle of confocal scanning microscopes is employed to achieve significant longitudinal resolution and minimize the effects of light scattered outside the plane of interest.
    • Optical coherence tomography (OCT) is another option [10], using a spatially coherent light but temporally incoherent source, such as a lower-power wavelength-swept laser or a superluminescent source. While the longitudinal resolution of OCT is inherently excellent, the transverse resolution can also be significantly improved using AO. In any case, a low-power beam (often in the near infrared) is sent through the eye's pupil, and light returning from the retina is analyzed. Note that the retinal details are not apparent within the pupil plane, where the wavefront measurement and correction are performed. Therefore, it is possible to use the same beam for wavefront analysis and imaging. The pupil is usually dilated to reduce image blurring by diffraction. That would normally increase the detrimental effects of eye aberrations, but those can be largely compensated for with AO.
  • Sensorless AO takes a different approach, not requiring a wavefront sensor. Instead, it iteratively adjusts the correction device in order to optimize specific sharpness metrics of the obtained images. Sensorless AO can be simpler and less expensive than traditional AO. However, the iterative process is significantly slower than wavefront sensing. Also, the image quality metrics may not always guide the system towards optimal correction; it depends significantly on the object's specific details.
  • Computational AO is a newer method that utilizes computational techniques to correct wavefront aberrations. Instead of relying on hardware components like wavefront sensors and deformable mirrors, computational AO uses algorithms to computationally correct the aberrations present in retinal images. One approach involves capturing a series of images with different defocus settings, a method called “computational optical sectioning”, and then combining them to create a high-resolution image. Requiring less optical hardware, computational AO can be a cost-effective method that simplifies the imaging setup, but it can be computationally intensive and may not achieve the same resolution as traditional AO.

In addition to retinal imaging, AO has also been applied in the field of refractive surgery, such as Laser-Assisted In Situ Keratomileusis (LASIK), and other vision correction methods. These procedures aim to reshape the cornea in order to correct refractive errors such as myopia (nearsightedness), hyperopia (farsightedness), and astigmatism. AO systems can be used to measure the patient's wavefront aberration with high precision before surgery, which helps guide the reshaping of the cornea. After surgery, AO can be used to evaluate the effectiveness of the procedure and identify any remaining aberrations.

Furthermore, AO has been used in research to investigate the finer aspects of ocular optics, offering valuable insights into the individual contributions of various eye structures to overall visual acuity. For example, AO has been used to investigate the effect of the eye's tear film on visual quality, to study the impact of age and disease on the optical properties of the cornea and lens, and to identify photoreceptors and their defects in patients with limited color vision.

Microscopy

Microscopy is an area where a significant portion of image imperfections can be attributed to aberrations, either in the optical system or in the samples themselves. The latter becomes relevant for microscopy techniques where images are taken at some (often variable) depth inside a sample. This is the case in confocal scanning microscopes and in two-photon excitation fluorescence microscopes. There are several technical variants of adaptive optics designed for specific use cases, including both traditional methods of microscopy and advanced techniques such as confocal scanning microscopy and fluorescence microscopy.

Due to the challenge of accurately measuring the necessary wavefront corrections, sensorless techniques are frequently employed. However, the effectiveness of these methods largely depends on the objects being imaged. A helpful aspect is that the aberrations in microscopy samples are usually not time-dependent, which allows sensorless methods to work.

Free-space Optical Communications and Directed Energy

Adaptive optics has important applications in systems where laser beams need to be sent through the atmosphere, where atmospheric distortions have detrimental effects. Some examples:

  • Free-space optical communications over long distances can benefit from adaptive optics. In the simplest case, only the direction of a laser beam is stabilized, but more advanced beam optimization techniques can also be used. However, cost constraints often make it difficult to implement advanced techniques.
  • Particular challenges are encountered in the defense sector, for example, in the use of adaptive optics in directed-energy weapons systems to deliver high-energy laser beams with minimal distortions over long distances. The ability to focus a high-energy beam onto a small target area is crucial for the effectiveness of these systems, and adaptive optics may play a key role in achieving this. While it may be impractical to accurately measure the wavefront distortions caused over a long distance to a distant target, it is possible to compensate for distortions that originate within the laser system being used. One example of such distortions is time-dependent thermal lensing in the power amplifier. Another challenge is related to the high beam intensities. Several optimizations are required to achieve the necessary power handling. For example, one may place a deformable mirror in an expanded section of the laser beam before it passes through the final optical amplifier.

Challenges and Future Directions

Despite the significant advancements and widespread adoption of adaptive optics over the years, there are still various challenges that need to be addressed. Some examples:

  • Magnitude of compensated distortions: One of the main challenges is correcting significant phase aberrations, especially in applications like microscopy, where the sample itself introduces substantial distortion. Improving the dynamic range of wavefront correctors while preserving other important characteristics is an area of ongoing research.
  • Integration: Many optical systems were not originally designed with adaptive optics in mind, and retrofitting these systems with adaptive optics can be challenging and costly. The more compact wavefront sensors and wavefront correctors are, the better the chances of integrating them into various devices.
  • Usability: The operation of adaptive optics systems requires specialized training and expertise, which can hinder widespread adoption. Efforts to develop user-friendly adaptive optics systems with automated operation are crucial for expanding the reach of this technology.
  • Cost: The cost of adaptive optics equipment, especially deformable mirrors and high-speed wavefront sensors, can be prohibitive. Developing lower-cost techniques could significantly broaden the potential areas of application.

Continued research and development in this field should lead to the creation of more effective and economically viable adaptive optics systems. Further advances in wavefront sensors, wavefront correctors, optical setups, and control strategies can be expected.

More to Learn

Encyclopedia articles:

Bibliography

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(Suggest additional literature!)

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