Temporal properties of the Zernike expansion coefficients of turbulence-induced phase aberrations for aperture and source motion

Whiteley, Matthew R.; Roggemann, Michael C.; Welsh, Byron M.

doi:10.1364/josaa.15.000993

Cited by 27 publications

(40 citation statements)

References 12 publications

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“…The formulas of Whiteley et al (1998) express these covariances through the integrals of the triple products of Bessel functions. This calculation was carefully optimized using some term re-arrangement, pre-computing, etc., gaining 10-100 times in computing speed compared to the straightforward integration (Appendix C).…”

Section: Covariance Codementioning

confidence: 99%

“…However, they considered only covariances of the polynomials of the same number. More general formulae for the covariances between different modes were later derived by Whiteley et al (1998). Basically, for each layer the covariances are expressed through a one-dimensional integral I involving triple products of Bessel functions J n (x) combined with powers: The dimensionless integration variable x is actually a product of the spatial frequency f and aperture radius, x = fD/2.…”

Section: Appendix C: Computation Of Zernike Mode Covariancesmentioning

confidence: 99%

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Optimized modal tomography in adaptive optics

Tokovinin

Louarn

Viard

et al. 2001

A&A

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Abstract. The performance of modal Multi-Conjugate Adaptive Optics systems correcting a finite number of Zernike modes is studied using a second-order statistical analysis. Both natural and laser guide stars (GS) are considered. An optimized command matrix is computed from the covariances of atmospheric signals and noise, to minimize the residual phase variance averaged over the field of view. An efficient way to calculate atmospheric covariances of Zernike modes and their projections is found. The modal covariance code is shown to reproduce the known results on anisoplanatism and the cone effect with single GS. It is then used to study the error of wave-front estimation from several off-axis GSs (tomography). With increasing radius of the GS constellation Θ, the tomographic error increases quadratically at small Θ, then linearly at larger Θ when incomplete overlap of GS beams in the upper atmospheric layers provides the major contribution to this error, especially on low-order modes. It is demonstrated that the quality of turbulence correction with two deformable mirrors is practically independent of the conjugation altitude of the second mirror, as long as the command matrix is optimized for each configuration.

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Section: Covariance Codementioning

confidence: 99%

Section: Appendix C: Computation Of Zernike Mode Covariancesmentioning

confidence: 99%

Optimized modal tomography in adaptive optics

Tokovinin

Louarn

Viard

et al. 2001

A&A

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“…So in order to best mimic the later applications one of the guide stars is used to provide TT-information the other one to provide high order information. The modal basis used are the Zernike polynomials as there exists an analytical formulation (see [13], [14]) for the projection between different directions. The later application can use any sort of mode set as the necessary matrices can be calculated numerically and then are stored (see 2) i.e.…”

Section: Simulationsmentioning

confidence: 99%

“…The other possibility to derive the off-axis PSF is to first project the WFS signal into the desired off-axis direction (see e.g. [13]) via correlations between the wavefront on-axis a e-mail: peterd@mpia.de Post processing and off-axis and then use the standard on-axis algorithm. These approaches should be similar as the ATF is build just using these correlations (see [10]).…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of the point spread function for the ground layer adaptive optics system ARGOS

Peter

Gässler

2010

1st AO4ELT Conference - Adaptive Optics for Extremely Large Telescopes

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Abstract. Present and future adaptive optics (AO) systems aim for the correction of the atmospheric turbulence over a large field of view combined with large sky coverage. Thus they use multiple laser beacons on sky. Still there will always be variations in the shape of the point spread function (PSF) with field angle. To overcome this drawback, a good knowledge of the PSF over the field is required. In the past, several algorithms of PSF retrieval have been proposed and tested. This includes on-axis PSF retrieval from wavefront sensor (WFS) data as well as a correction over the field using a reference PSF. The schemes have been set up for both an AO system using a single natural guide star as well as a system using a laser beacon. For these algorithms to work, a detailed knowledge of the atmospheric structure, i.e. Cn 2 profile, is crucial. In the context of the ARGOS system, a GLAO system proposed for the LBT with three Rayleigh laser beacons per eye of the telescope, we build a reconstruction scheme which uses the multiple guide stars to directly measure the atmospheric profile from the WFS data itself. Here we show how the conventional PSF-retrieval algorithms must be changed to make up for the difference between the multiple beacon GLAO used in ARGOS and the typical single source on-axis schemes. We present first results of PSF-reconstruction using the data from two WFS and investigate in the accuracy of this procedure.

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“…Due to the finite temporal response of the AO system, a time delay exists between the beacon measurements at time t -'7-and the tilt correction applied to the outgoing laser beam at time t. The measured beacon tilt coefficients, designated ab2 (t -T) and ab3 (t -r), are determined by the projection of the beacon phase b(Rp, t -i-) onto the appropriate Zernike polynomials: ab(t -r) = f dW(pb(Rfl,t -r)Z(p, for i = 2,3. (5) Due to wind motion of the turbulence within the delay period r, abj(t-r) will be decorrelated from a02(t) [10].…”

Section: Theorymentioning

confidence: 99%

<title>Finite outer scale considerations for beam-steering optimization</title>

Whiteley

1999

SPIE Proceedings

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Finite turbulence outer scale attenuates turbulence power at low spatial frequencies, providing a finite variance for turbulence-induced phase. The temporal properties of low spatial-frequency aberrations such as tilt are affected directly by finite outer scale turbulence models. When optimal estimation techniques are used in beam-steering applications, the estimator form and its performance are related directly to the correlative properties of tilt and higher-order modes as determined by the value of Lo/D. In this work, we consider the effect of finite turbulence outer scale on the performance of optimal tilt compensation incorporating time-delayed tilt plus higher-order modes. The results indicate that while smaller values of Lo/D provide for higher Strehi ratios after tilt compensation, the Strehi ratio gain obtained through optimal compensation is reduced with decreasing values of Lo/D. For the relative aperture diameter D/ro = 5 and using timedelayed tilt plus higher-order modes through Zernike 15, the Strehl ratio gain with time delay vr/D = 0.5 is approximately 15 for L0/D = 100. Under the same conditions, the Strehl ratio gain for L0/D = 2 is approximately 3.

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Temporal properties of the Zernike expansion coefficients of turbulence-induced phase aberrations for aperture and source motion

Cited by 27 publications

References 12 publications

Optimized modal tomography in adaptive optics

Optimized modal tomography in adaptive optics

Reconstruction of the point spread function for the ground layer adaptive optics system ARGOS

<title>Finite outer scale considerations for beam-steering optimization</title>

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