Method for determining a phase correction to apply to an optical element using focal plane intensity measurements

Abstract

A method of determining a phase correction to apply to a corrective element, comprising the steps of: acquiring low-resolution images; determining pixel intensity values in the low-resolution images; adding pixels one at a time in order of decreasing intensity; back-propagating each image to a pupil plane to obtain an estimate; and measuring a coupling efficiency to phase points to determine an optimum phase.

Description

METHOD FOR DETERMINING A PHASE CORRECTION TO APPLY TO ANOPTICAL ELEMENT USING FOCAL PLANE INTENSITY MEASUREMENTSFIELD

[0001] Aspects of the disclosure relate to optical communications systems and methods.BACKGROUND

[0002] Free-space optical communication is of interest for ground-to-satellite optical links provides significantly greater bandwidth than is possible with current radio-frequency technology [1], For such links, however, atmospheric turbulence induces wavefront distortions in amplitude and phase which degrades the coupling of the signal. Adaptive optics (AO) systems may be used to compensate for such wavefront distortions. In strong turbulence regimes, existing AO solutions may fail to guarantee coupling.

[0003] Prior art wavefront correction methods employs a Shack-Hartmann wavefront sensor (SHWFS), in which the incoming wavefront is decomposed into a grid of -100-1000 spots. The spots are recorded with a camera at high speed (~100Hz-1000Hz), and the relative position of the spots can be used to infer the wavefront error of the incoming beam. In another method, sensorless adaptive optics (AO) combine wavefront sensing and correction into a series of iterative updates. The incoming wavefront is incident on a deformable mirror. The state of the deformable mirror (DM) is changed, and the resulting change in the brightness of the focal spot is recorded. This process is repeated iteratively until a maximum brightness focal spot is achieved. Typically, about 100 iterations are required for convergence. Modal wavefront sensing is yet another approach in which a hologram is used to split the beam into -100 copies, each experiencing a small amount of extra aberration. The relative brightness of each spot imaged onto a camera can be used to infer the error of the whole wavefront. Alternatively, instead of a hologram, the extra aberration can be applied sequentially and read out with a single element detector (e.g. a photodiode).

[0004] The drawback of SHWFS and modal wavefront sensing is that these methods use cameras, and while these methods works well in the visible wavelength range, they are typically restricted to speeds under 1kHz, which is not fast enough to keep up with the dynamics of turbulent atmosphere. Furthermore, these methods require cameras operating at telecom wavelengths (~1550nm), which are substantially expensive.

[0005] For sensorless AO, the number of iterations (typically -50-100) poses a severe restriction on the correction speed. For a typical DM with a settling time of 1ms, this results ina total correction time of ~50-100ms. Even with a fast mirror settling time of ~100ps, the total correction time of 5-10ms is unacceptably stow for turbulent phase distortions, which typically require sensing rates greater than 1kHz. Intensity based AO is another approach in which a corrective phase map is constructed from modes related to speckle intensity measurements in the focal plane[2, 4], The mode amplitudes are obtained from the intensity values, while the phases are optimized iteratively to maximize the focal spot power, similar to sensorless AO. The intensity data input allows faster convergence with fewer iterations compared to sensorless AO. However, intensity based AO described herein requires a fairly high-resolution focal plane image and an additional image processing step to locate the principal speckles in the image, and these two factors severely limit the rate at which the correction can be updated.

[0006] A single-pixel imaging approach to acquire the focal plane images may be advantageous in providing greater optical integration and allowing operation at wavelengths where array -based image sensors are costly or have tow sensitivity [3], However, to allow operation at kHz rates, the image acquisition time must be reduced. Previous reports of the intensity -based AO algorithm [2, 3] implemented a local maximum search of relatively high- resolution focal plane images to find principal speckles.

[0007] For modal wavefront sensing, one may use a single photodetector, alleviating the camera issue above. However, modal wavefront sensing suffers from the so-called in turbulence, and the output of modal wavefront sensing approaches becomes highly inaccurate. This is because the current state of the art wavefront reconstruction algorithms in modal wavefront sensing are linear approximations to a fundamentally nonlinear relationship between phase error and detected intensity.SUMMARY

[0008] In one example, a method of determining a phase correction for application to a corrective element, the method comprising the steps of: a) measuring an intensity of each pixel of a focal plane image at tow resolution comprising a plurality of pixels a plurality of pixels measure a focal plane image; b) sorting the plurality of pixels from highest to lowest intensity, wherein each pixel m of the plurality of pixels is associated with a pixel positionand an amplitude Amgiven by the square root of the pixel’s measured intensity; c) for a first pixel, m=0, having the highest intensity, obtaining a first iteration of an estimate of a complex field in a pupil plane Eod) for each m>0, iteratively performing iterative steps of: i. calculating a trial complex field value Etrial for each trial phase value cpz in {c o,ii. calculating a current phase value of Etriai*, where * denotes complex conjugate; iii. applying a phase map on the corrective element; iv. measuring a coupled power Pi for the current phase trial value; v. determining an optimum phase value (popt for maximizing the coupled power by fitting the measured points Pi (<pz); vi. updating the estimate of the complex field as Em; vii. applying the phase value of Em* on the corrective element; viii. terminating the iterative steps after a predefined number of pixels or when the coupled power reaches a target value, otherwise repeating steps i to viii; and e) repeating steps a) to d) to adapt to changing atmospheric phase perturbation.

[0009] In another example, a method for calculating an adaptive optics correction, comprising the steps of: with an image capture apparatus, acquiring a plurality of images comprising a plurality of pixels; measuring pixel intensity values in each of the plurality of images; sorting each of the plurality of pixels in order of highest pixel intensity value to lowest pixel intensity value; for a first pixel having the highest pixel intensity value, obtaining a first iteration of an estimate of a complex field in a pupil plane; adding pixels one at a time in order of decreasing intensity and iteratively back- propagating each of the plurality of images to a pupil plane to obtain an updated estimate; and measuring a coupling efficiency to phase points to determine an optimum phase.

[0010] In another example, a system for determining a phase correction to apply to a corrective element, the system comprising: an image capture apparatus for acquiring a plurality of images comprising a plurality of pixels; a processing unit for executing instructions stored in a computer readable medium to at least perform the steps of:measuring pixel intensity values in each of the plurality of images; sorting each of the plurality of images in order of highest pixel intensity value to lowest pixel intensity value; for a first pixel having the highest pixel intensity value, obtaining a first iteration of an estimate of a complex field in a pupil plane; adding pixels one at a time in order of decreasing intensity and iteratively back- propagating each of the plurality of images to a pupil plane to obtain an updated estimate; and measuring a coupling efficiency to phase points to determine an optimum phase.

[0011] In another example, a method for calculating an adaptive optics correction, comprising the steps of: acquiring low-resolution images and processing the acquired low resolution images to determine the pixel intensity values. The image resolution and extent are optimized using the reciprocal Fourier optics relationships to minimize the number of data points required. Unlike in prior art methods, the methods described herein use low resolution focal plane data, without the need for peak finding, and uses back-propagation techniques to achieve a given coupling efficiency with fewer DM adjustments, thereby increasing the adaptive optics corrective bandwidth. Using low resolution images results rather than high resolution images results in significantly higher image acquisition rates. Furthermore, even faster imaging, higher optical power per pixel, and improved efficiency, may be realized by using measuring select pixels in a region of interest within the image. Accordingly, this method may be implemented on a low-resolution sensor or integrated with a tracking camera sensor; or may be used with a single-pixel camera, or even no camera sensor (e.g. scanning the tip-tilt mirror built into the AO to get the low-resolution focal plane data).

[0012] In another example, there is provided a method for obtaining an adaptive optics correction via a Fourier transform modified intensity-based AO algorithm without the need to perform image processing of a high-resolution focal plane image. Instead, the image resolution and extent of the focal plane data are intentionally chosen according to the telescope aperture size and maximum expected turbulence strength, resulting in a low-resolution image. The pixel intensity values are used directly, without performing speckle finding image processing, as in the prior art methods. The AO correction is obtained via a Fourier transform of the focal plane image, with the pixel phases optimized iteratively to maximize the focal spot power.

[0013] Additionally, simulations have showed that the adaptive optics correction obtained via a Fourier transform converges more rapidly, with fewer iterations, compared to the correction constructed from speckle intensity values in prior art intensity based AO methods.BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Several exemplary embodiments of the present disclosure will now be described, by way of example only, with reference to the appended drawings in which:

[0015] Fig.1 shows a schematic of a turbulence compensation system;

[0016] Fig. 2a shows a phase screen with D I rO = 10 over a circular aperture;

[0017] Fig. 2b shows intensity values in a central 8x8 pixel region of interest (ROI) of a focal plane;

[0018] Fig. 2c shows a phase screen estimate;

[0019] Fig. 3 shows a coupling efficiency as a function of the number n of pixels added to a trial focal plane image to obtain the phase screen estimate;

[0020] Fig. 4 shows a flow chart with example steps for determining a phase correction to apply to corrective element;

[0021] Fig. 5a shows a 16x16 focal plane image;

[0022] Figs. 5b, d, f show 16x16 focal plane images; and Figs. 5c, e, g show the corresponding phase screen estimates after carrying out the example steps of Fig. 4;

[0023] Fig. 6a shows the coupling efficiency; and

[0024] Fig. 6b shows the optimum phase is estimated by fitting a sine curve to the phase points.DETAILED DESCRIPTION

[0025] The following detailed description refers to the accompanying drawings. Wherever possible, the same reference numbers are used in the drawings and the following description to refer to the same or similar elements. While embodiments of the disclosure may be described, modifications, adaptations, and other implementations are possible. For example, substitutions, additions, or modifications may be made to the elements illustrated in the drawings, and the methods described herein may be modified by substituting, reordering, or adding stages to the disclosed methods. Accordingly, the following detailed description does not limit the disclosure. Instead, the proper scope of the disclosure is defined by the appended claims.

[0026] Moreover, it should be appreciated that the particular implementations shown and described herein are illustrative of the invention and are not intended to otherwise limit the scope of the present invention in any way. Indeed, for the sake of brevity, certain subcomponents of the individual operating components, conventional data networking, application development and other functional aspects of the systems may not be described in detail herein. Furthermore, the connecting lines shown in the various figures contained herein are intended to represent exemplary functional relationships and / or physical couplings between the various elements. It should be noted that many alternative or additional functional relationships or physical connections may be present in a practical system.

[0027] In one example, Fig.l shows a schematic of an adaptive optics correction system 10 in which a laser beam 10 from a laser 12 passes through aturbulent atmosphere that distorts the amplitude and the phase of the laser beam 10. The distorted laser beam 10 then travels to an adaptive optics (AO) corrective element 14 located in the pupil plane and to a beam splitter 16 which transmits a first portion 17 of the distorted beam 10 to focal plane of an image capture apparatus 18, and reflects a second portion 19 of the distorted beam 10 to a single mode fiber 20. The system 10 also comprises various optical elements 22, such as lenses, to shape the laser beam 10, 17 or 19. The image capture apparatus 18 produces a first data input associated with pixel intensity values of captured focal plane images is received by a computing device 30 with instructions 32 stored in a computer readable medium 33 and executable by a processor 34, and a second data input, which is a real-time acquisition of the power coupled into the single mode fiber 18, is also received by the computing device 30. Using the two data inputs, the processor 34 executes the instructions 32 to generate a corrective phase map via back- propagation techniques and optimization methods. The corrective phase map is then applied to the adaptive corrective element 14 to compensate for the phase distortions caused by the turbulence.

[0028] In one example of this disclosure, instead of processing a high-resolution focal plane image to locate speckle maxima, as described in [4], the pixel intensity values are used directly via a Fourier transform of the focal plane image, and the pixel phases are optimized iteratively to maximize the focal spot power. Furthermore, a low-resolution image may be employed, resulting in faster processing and imaging, as described above.

[0029] In one example, atmospheric turbulence was simulated using Kolmogorov phase screens generated using the method of [5] with five low spatial frequency subharmonic layers.The strength of the phase screen is set by D I rO, where D is the aperture and rO is the atmospheric coherence diameter. Fig. 2a shows a phase screen with D I rO = 10 over a circular aperture. The phase screen is numerically propagated to the focal plane using fast fourier transforms (FFT). The focal length f is chosen so as to make the pixel size somewhat larger (47%) than the diffraction limited resolution limit k 'f / D.

[0030] Fig. 2b shows intensity values in the central 8x8 pixel region of interest (ROI) of the focal plane. By masking the central 8x8 pixel ROI and setting all other pixels to zero, then back-propagating the complex-valued masked focal plane to the pupil plane, a phase screen estimate is obtained, as shown in Fig. 2c. This provides a limiting case for the phase screen estimation from the retained number of pixels where the phases are known. The ROI masking of the focal plane results in a low-pass fdtered estimate of the phase screen in which higher spatial frequencies are absent, suitable for application on a typical AO corrective element 14, such as a deformable mirror. The slightly larger pixel size than the diffraction limit results in distortion of the phase along the four edges of Fig. 2c, but due to the circular aperture this only affects a small fraction of the aperture area. To evaluate this estimate for use in an AO correction, the estimate is subtracted from the original phase screen and the coupling efficiency is calculated based on a back-propagated Gaussian mode [3], For the known-phase case this coupling efficiency is i] = 0.518, compared to i] = 0.006 for the uncorrected phase screen.

[0031] Generally, for an intensity-based image of the speckle pattern, the pixel amplitudes are obtained as the square root of the intensities, but the phases are unknown. In one example of this disclosure, instead of processing the image to locate speckle maxima, the focal plane pixel phase values are optimized directly by simulating an optimization method similar to the algorithm of [2], The pixels are sorted in descending order of amplitude, and the first pixel is assigned an arbitrary phase and added to a trial focal plane image with all other pixels set to zero. Next, starting from the second highest intensity pixel, the remaining pixels are added one at a time with trial phase values (n / 2, n, 3JC / 2). For each phase value, the complexvalued trial image is back-propagated numerically to the pupil plane to obtain a phase map estimate. This estimate is then subtracted from the original phase screen, simulating its application as an AO correction, and the coupling efficiency is calculated to simulate a measurement of the received power coupled into an optical fiber 18, as shown in Fig. 6a, for example. The optimum phase is estimated by fitting a sine curve to at least three phase points, as shown in Fig. 6b, for example. Although the coupling efficiency behavior as a function ofphase, while periodic over 2n, is not truly sinusoidal, this fitting method was found to provide an adequate estimate for constructing the AO correction.

[0032] Fig. 3 shows the coupling efficiency as a function of the number n of pixels added to the trial focal plane image to obtain the phase screen estimate, ending when all 64 pixels in the image have been added. For comparison, the coupling efficiency is also shown as a solid line for the phase screen estimate using the pixel phases known from the original phase screen propagation, but keeping only the n highest amplitude pixels. Throughout the optimization procedure, the coupling efficiency obtained using the phase optimization is at most 5% lower than that obtained using the exact known pixel phases. The convergence to a high coupling efficiency value requires fewer phase optimization points compared to the results reported in [3] where similar coupling efficiencies for a starting phase screen with D I ro = 10 were obtained after processing 100 speckles of the prior art. This is likely due to the increased accuracy obtained using the FFT -based back-propagation, compared to the sum of plane waves employed in [2,3], The low-pixel count ROI allows the present method to be employed with a faster imaging rate on either array-based image sensors where restricting the ROI allows an increase in the frame rate, or using a single pixel camera based on a digital micromirror device (DMD) [3], Currently available DMDs achieve frame rates above 100 kHz in restricted ROI mode, which would allow an image acquisition rate above 1 kHz with 64 patterns required for the 8x8 image used in this disclosure.

[0033] Looking at Fig. 4, there is shown a flow chart 100 with example steps for determining a phase correction for application to a corrective element 14. In step 101, a focal plane image comprising of a plurality of pixels is acquired, and in one example, the intensity values of a selected number of pixels in a region of interest are measured (step 102). The focal plane image at low resolution. In step 104, the selected number of pixels in the region of interest are sorted from highest to lowest intensity, and each pixel m of the plurality of pixels is associated with a pixel positionand an amplitude Amgiven by the square root of the pixel’s measured intensity. In step 106, for a first pixel, m=0, having the highest intensity, a first iteration of an estimate of a complex field in a pupil plane Eo, is obtained, with Eo = Ao eJkm r, where km= (2n xmI Ef, 2n ymI Ef) and X = wavelength; f = focal length; E^, Eare functions of position r.

[0034] Next, for each m>0, the following steps are performed iteratively: in step 108, a trial complex field value Etriai for each trial phase value q)z in {q)o, c i, 2, ... } is calculated, wherein the trial complex field value Etriai = Em.i + AmeJ km reJ <pZ.

[0035] In step 110, a current phase value of Etrial* is calculated, where * denotes complex conjugate, i.e. the sign of the phase is reversed to apply correction.

[0036] In step 112, a phase map is applied on the corrective element 14 of AO system 10.

[0037] In step 114, a coupled power Pi for the current phase trial value is measured.

[0038] In step 116, an optimum phase value q)opt for maximizing the coupled power is calculated by fitting the measured points E / (c / ). The coupled power is periodic over 2TI radians in the trial phase. The coupled power may be fitted via any suitable fitting method. In one example, a parabolic fit was employed and in another example a sinusoidal fit was employed.

[0039] In step 118, the estimate of the complex field Emis updated, and Em= Em-i + Amej km rej <popt

[0040] In step 120, the phase value of Em* is applied on the corrective element 14 of the AO system 10, where * denotes complex conjugate, i.e. the sign of the phase is reversed to apply the correction.

[0041] In step 122, determining whether each of the predefined number of pixels has been added or whether a target value of the coupled power has been reached; if yes, terminating the iterative steps (step 124) otherwise the process returns to step 108.

[0042] The entire process (steps 101-124) may be restarted to adapt to changing atmospheric phase perturbation.

[0043] In Fig. 5a there is shown 16x16 focal plane image for use in the method as described above. Generally, an attempt at peak finding in this 16x16 focal plane image using prior art methods would fail, as such the pixel intensity values are used instead. Figs. 5b, d, f show 16x16 focal plane images and Figs. 5c, e, g show the corresponding phase screen estimates after carrying out the example steps of Fig. 4, such as trying a few phase values, numerically back propagating each image to the pupil frame to obtain an estimate for correction, fitting the measured coupling, and calculating the optimum phases. For example, Fig. 6a shows the coupling efficiency p, and the optimum phase is estimated by fitting a sine curve to the phase points, as shown in Fig. 6b.

[0044] It should be appreciated that the computing environment and components configured to facilitate the systems and methods described herein is merely an example and that alternative or additional components are envisioned. The computing system 30 comprises at least one processor such as processor 34 or graphics processing units (GPU) 36, at least one memory device such as memory 33, input / output (I / O) module 42 and communicationsinterface 44, which are in communication with each other via centralized circuit system 46. Although computing system 30 is depicted to include only one processor 34, computing system 30 may include a number of processors therein. In an embodiment, memory 33 is capable of storing machine executable instructions 32, data models and process models. A database may be coupled to computing system 30 and stores data, such as pre-processed data, image data, model output data and audit data. Further, the processor 34 is capable of executing the instructions 32 in memory 33 to implement aspects of processes described herein. For example, processor 34 may be embodied as an executor of software instructions 32, wherein the software instructions 32 may specifically configure processor 34 to perform algorithms and / or operations described herein when the software instructions 32 are executed. Alternatively, processor 34 may be execute hard-coded functionality. The computing environment may be software (e.g., code segments compiled into machine code), hardware, embedded firmware, or a combination of software and hardware, according to various embodiments.

[0045] In one implementation, processor 34 may be embodied as a multi-core processor, a single core processor, or a combination of one or more multi-core processors and one or more single core processors. For example, processor 34 may be embodied as one or more of various processing devices, such as a coprocessor, a microprocessor, a controller, a digital signal processor (DSP), a processing circuitry with or without an accompanying DSP, or various other processing devices including integrated circuits such as, for example, an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a microcontroller unit (MCU), a hardware accelerator, a special-purpose computer chip, Application-Specific Standard Products (ASSPs), System-on-a-chip systems (SOCs), Complex Programmable Logic Devices (CPLDs), Programmable Logic Controllers (PLC), Graphics Processing Units (GPUs), and the like. For example, some or all of the device functionality or method sequences may be performed by one or more hardware logic components.

[0046] Memory 33 may be embodied as one or more volatile memory devices, one or more non-volatile memory devices, and / or a combination of one or more volatile memory devices and non-volatile memory devices. For example, memory 33 may be embodied as magnetic storage devices (such as hard disk drives, floppy disks, magnetic tapes, etc.), optical magnetic storage devices (e.g., magneto-optical disks), CD-ROM (compact disc read only memory), CD-R (compact disc recordable), CD-R / W (compact disc rewritable), DVD (Digital Versatile Disc), BD (BLU-RAY™ Disc), and semiconductor memories (such as mask ROM,PROM (programmable ROM), EPROM (erasable PROM), flash ROM, RAM (random access memory), etc.).

[0047] I / O module 42 facilitates provisioning of an output to a user of computing system 30 and / or for receiving an input from the user of computing system 30, and send / receive communications to / from the various sensors, components, and actuators of computing environment 40. I / O module 42 may be in communication with processor 34 and memory 33. Examples of the I / O module 42 include, but are not limited to, an input interface and / or an output interface. Some examples of the input interface may include, but are not limited to, a keyboard, a mouse, a joystick, a keypad, a touch screen, soft keys, a microphone, and the like. Some examples of the output interface may include, but are not limited to, a microphone, a speaker, a ringer, a light emitting diode display, a thin-film transistor (TFT) display, a liquid crystal display, an active-matrix organic light-emitting diode (AMOLED) display, and the like. In an example embodiment, processor 34 may include I / O circuitry for controlling at least some functions of one or more elements of I / O module 42, such as, for example, a speaker, a microphone, a display, and / or the like. Processor 34 and / or the I / O circuitry may control one or more functions of the one or more elements of I / O module 42 through computer program instructions 32, for example, software and / or firmware, stored on a memory, for example, the memory 33, and / or the like, accessible to the processor 34.

[0048] In an embodiment, various components of computing system 30, such as processor 34, memory 33, I / O module 42 and communications interface 44 may communicate with each other via or through a centralized circuit system 46. Centralized circuit system 46 provides or enables communication between the components of computing system 30. In certain embodiments, centralized circuit system 46 may be a central printed circuit board (PCB) such as a motherboard, a main board, a system board, or a logic board. Centralized circuit system 46 may also, or alternatively, include other printed circuit assemblies (PCAs) or communication channel media.

[0049] Communications interface 44 enables computing system 30 to communicate with other entities over various types of wired, wireless or combinations of wired and wireless networks, such as for example, the Internet. In at least one example embodiment, communications interface 44 includes a transceiver circuitry for enabling transmission and reception of data signals over the various types of communication networks. In some embodiments, communications interface 44 may include appropriate data compression and encoding mechanisms for securely transmitting and receiving data over the communicationnetworks. Communications interface 44 facilitates communication between computing system 30 and I / O peripherals.

[0050] Centralized circuit system 46 may be various devices for providing or enabling communication between the components (34-44) of computing system 30. In certain embodiments, centralized circuit system 46 may be a central printed circuit board (PCB) such as a motherboard, a main board, a system board, or a logic board. Centralized circuit system 46 may also, or alternatively, include other printed circuit assemblies (PCAs), communication channel media or bus.

[0051] A plurality of user computing devices and data sources may be coupled to computing system 30 via a communication network, such as the Internet.

[0052] It is noted that various example embodiments as described herein may be implemented in a wide variety of devices, network configurations and applications.

[0053] Those of skill in the art will appreciate that other embodiments of the disclosure may be practiced in network computing environments with many types of computer system configurations, including personal computers (PCs), industrial PCs, desktop PCs), hand-held devices, multi-processor systems, microprocessor-based or programmable consumer electronics, network PCs, server computers, minicomputers, mainframe computers, and the like. Embodiments may also be practiced in distributed computing environments where tasks are performed by local and remote processing devices that are linked (either by hardwired links, wireless links, or by a combination thereof) through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.

[0054] In another implementation, the computing environment follows a cloud computing model, by providing an on-demand network access to a shared pool of configurable computing resources (e.g., servers, storage, applications, and / or services) that can be rapidly provisioned and released with minimal or nor resource management effort, including interaction with a service provider, by a user (operator of a thin client).

[0055] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations of particular inventions. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in thecontext of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0056] Accordingly, the above description of example implementations does not define or constrain this disclosure. Other changes, substitutions, and alterations are also possible without departing from the spirit and scope of this disclosure.

[0057] REFERENCES[1] Calvo, R. M., et al., “Optical technologies for very high throughput satellite communications.” Free-Space Laser Communications XXXI. Vol. 10910. SPIE, 2019.[2] C. E. Carrizo, R. M. Calvo, and A. Belmonte, “Intensity-based adaptive optics with sequential optimizationfor laser communications.” Optics express 26.13 (2018): 16044-16053.[3] M. Pashazanoosi, M. Taylor, O. Pitts, C. Flueraru, A. Orth, S. Hranilovic, “Maximizing atmospheric-disturbed fiber coupling efficiency with speckle-based phase retrieval and a single-pixel camera.” Applied Optics 62.23 (2023): G43-G52.[4] R. Lane, A. Glindemann, and J. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media, vol. 2, no. 3, p. 209, 1992.[5] Calvo, R. M., et al., “Method for determining altering parameters for altering the optical features of an optical element for compensation of distortions in a beam for optical freespace Communication”, EP3493430A1.[6] M. A. A. Neil, M. J. Booth, and T. Wilson, "New modal wave-front sensor: a theoretical analysis," J. Opt. Soc. Am. A, JOSAA 17, 1098- 1107 (2000).[7] M. a. A. Neil, M. J. Booth, and T. Wilson, "Closed-loop aberration correction by use of a modal Zemike wave-front sensor," Opt. Lett., OL 25, 1083-1085 (2000).[8] A. Zepp, S. Gladysz, K. Stein, and W. Osten, "Simulation-based design optimization of the holographic wavefront sensor in closed-loop adaptive optics," Light: AM 3, 384-399 (2022).[9] A. Zepp, S. Gladysz, K. Stein, and W. Osten, "Optimization of the holographic wavefront sensor for open-loop adaptive optics under realistic turbulence. Part I: simulations," Appl. Opt., AO 60, F88-F98 (2021).

[0010] B. R. M. Norris, J. Wei, C. H. Beters, A. Wong, and S. G. Leon-Saval, "An allphotonic. focal-plane wavefront sensor," Nat Commun 11, 5335 (2020).

[0011] C. E. Carrizo, R. M. Calvo, and A. Belmonte, "Intensity-based adaptive optics with sequential optimization for laser communications." Optics express 26.13 (2018): 16044- 16053.

Claims

CLAIMS:

1. A method of determining a phase correction for application to a corrective element, the method comprising the steps of: f) measuring an intensity of each pixel of a focal plane image at low resolution comprising a plurality of pixels a plurality of pixels measure a focal plane image; g) sorting the plurality of pixels from highest to lowest intensity, wherein each pixel m of the plurality of pixels is associated with a pixel positionand an amplitude ftm: h) for a first pixel, m=0, having the highest intensity, obtaining a first iteration of an estimate of a complex field in a pupil plane Eo; i) for each m>0, iteratively performing iterative steps of: i. calculating a trial complex field value Etrial for each trial phase value cp / in {cpo,ii. calculating a current phase value of Etnai*, where * denotes complex conjugate; iii. applying a phase map on the corrective element; iv. measuring a coupled power Pi for the current phase trial value; v. determining an optimum phase value (popt for maximizing the coupled power by fitting the measured points Pi (cp / ); vi. updating the estimate of the complex field as Em; vii. applying the phase value of Em* on the corrective element; viii. terminating the iterative steps after a predefined number of pixels or when the coupled power reaches a target value, otherwise repeating steps i to viii; and j) repeating steps a) to d) to adapt to changing atmospheric phase perturbation.

2. The method of claim 1, wherein the amplitude Amis the square root of the pixel’s measured intensity.

3. The method of claim 2, wherein the complex field in the pupil plane, Eo = Ao eJ km r, where km= (2n xmlkf, 2nymI X ), where Z = wavelength; f= focal length. Em, Etrial are functions of position r.

4. The method of claim 3, wherein the trial complex field value Etrial = Em-i + AmeJ km re j<?i5. The method of claim 4, wherein the estimate of the complex field is updated as Em= Em-i + Am eJ km->-ej <fopt6. The method of claim 5, wherein the image resolution and extent of the focal plane data are chosen according to a telescope aperture size and a maximum expected turbulence strength.

7. The method of claim 5, wherein the correction is obtained via a Fourier transform of the focal plane image, with the pixel phases optimized iteratively to maximize the focal spot power.

8. A method for calculating an adaptive optics correction, comprising the steps of: with an image capture apparatus, acquiring a plurality of images comprising a plurality of pixels; measuring pixel intensity values in each of the plurality of images; sorting each of the plurality of images in order of highest pixel intensity value to lowest pixel intensity value; for a first pixel having the highest pixel intensity value, obtaining a first iteration of an estimate of a complex field in a pupil plane; adding pixels one at a time in order of decreasing intensity and iteratively back- propagating each of the plurality of images to a pupil plane to obtain an updated estimate; and measuring a coupling efficiency to phase points to determine an optimum phase.

9. The method of claim 8, wherein each of the plurality of images is a focal plane image at low resolution comprising a plurality of pixels a plurality of pixels measure a focal plane image.

10. The method of claim 9, wherein each pixel m of the plurality of pixels is associated with a pixel position (xm,ym) and an amplitude Amgiven by the square root of the pixel’s measured intensity.

11. The method of claim 10, further comprising the steps of, for a second pixel having the second highest pixel intensity value and each of the remaining sorted pixels:i. calculating a trial complex field value for each trial phase value; ii. calculating a current phase value of the trial complex field value; iii. applying a phase map on the corrective element; iv. measuring a coupled power for the current phase trial value; v. determining an optimum phase value for maximizing the coupled power by fitting the measured points; vi. updating the estimate of the complex field; and vii. applying the phase value of the estimate of the complex field on the corrective element.

12. The method of claim 11 , wherein the complex field in the pupil plane, Eo = Ao eJ km r, where km= (2n X„, l'kf, 2nymI X ), where Z = wavelength; f= focal length. Em, Etriai are functions of position r.

13. The method of claim 12, wherein the image resolution and extent of the focal plane data are chosen according to a telescope aperture size and a maximum expected turbulence strength.

14. The method of claim 13, wherein the correction is obtained via a Fourier transform of the focal plane image, with the pixel phases optimized iteratively to maximize the focal spot power.

15. A system for determining a phase correction to apply to a corrective element, the system comprising: an image capture apparatus for acquiring a plurality of images comprising a plurality of pixels; a processing unit for executing instructions stored in a computer readable medium to at least perform the steps of: measuring pixel intensity values in each of the plurality of images; sorting each of the plurality of images in order of highest pixel intensity value to lowest pixel intensity value; for a first pixel having the highest pixel intensity value, obtaining a first iteration of an estimate of a complex field in a pupil plane;adding pixels one at a time in order of decreasing intensity and iteratively back- propagating each of the plurality of images to a pupil plane to obtain an updated estimate; and measuring a coupling efficiency to phase points to determine an optimum phase.

16. The system of claim 15, wherein the image resolution and extent of the focal plane data are chosen according to a telescope aperture size and a maximum expected turbulence strength.

17. The system of claim 16, wherein each of the plurality of images is of low resolution.

18. The system of claim 15, wherein the correction is obtained via a Fourier transform of the focal plane image, with the pixel phases optimized iteratively to maximize the focal spot power.

19. The system of claim 18, further comprising a step of measuring select pixels in a region of interest within the image.

20. The system of claim 19, further comprising a step of generating a correction phase map to apply to the corrective element.

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