Transcript Document
Adaptive Optics in the VLT and ELT era Optics for AO François Wildi Observatoire de Genève Credit for most slides : Claire Max (UC Santa Cruz) Page 1 Simplest schematic of an AO system BEAMSPLITTER PUPIL WAVEFRONT SENSOR COLLIMATING LENS OR MIRROR FOCUSING LENS OR MIRROR Optical elements are portrayed as transmitting, for simplicity: they may be lenses or mirrors What optics concepts are needed for AO? • Design of AO system itself: – What determines the size and position of the deformable mirror? Of the wavefront sensor? – What does it mean to say that “the deformable mirror is conjugate to the telescope pupil”? – How do you fit an AO system onto a modest-sized optical bench, if it’s supposed to correct an 8-10m primary mirror? • What are optical aberrations? How are aberrations induced by atmosphere related to those seen in lab? Spherical waves and plane waves What is imaging? X X xi si Mx xo so so Ma si Optical path and OPD Plane Wave Index of refraction variations • The optical path length is Distorted Wavefront n( z)dz Z • The optical path difference OPD is the difference between the OPL and a reference OPL • Wavefronts are iso-OPL surfaces Spherical aberration Rays from a spherically aberrated wavefront focus at different planes Through-focus spot diagram for spherical aberration Optical invariant ( = Lagrange invariant) y11 y2 2 Lagrange invariant has important consequences for AO on large telescopes From Don Gavel Fraunhofer diffraction equation (plane wave) Diffraction region Observation region From F. Wildi “Optique Appliquée à l’usage des ingénieurs en microtechnique” Fraunhofer diffraction, continued 1 j (t kR) U 2 ( x , y ) e U1 ( x, y) exp j kx x ky y ds R aperture can be complex • In the “far field” (Fraunhofer limit) the diffracted field U2 can be computed from the incident field U1 by a phase factor times the Fourier transform of U1 • U1 (x1, y1) is a complex function that contains everything: Pupil shape and wavefront shape (and even wavefront amplitude) • A simple lens can make this far field a lot closer! Looking at the far field (step 1) Looking at the far field (step 2) Details of diffraction from circular aperture and flat wavefront 1) Amplitude First zero at r = 1.22 / D 2) Intensity FWHM /D Diffraction pattern from MMT-AO What is the ‘ideal’ PSF? • The image of a point source through a round aperture and no aberrations is an Airy pattern The Airy pattern as an impulse response • The Airy pattern is the impulse response of the optical system • A Fourier transform of the response will give the transfer function of the optical system • In optics this transfer function is called the Optical Transfer Function (OTF) • It is used to evaluate the response of the system in terms of spatial frequencies Define optical transfer function (OTF) • Imaging through any optical system: in intensity units Image = Object Point Spread Function convolved with I ( r ) = O PSF dx O( x - r ) PSF ( x ) • Take Fourier Transform: F ( I ) = F (O ) F ( PSF ) • Optical Transfer Function is the Fourier Transform of PSF: OTF = F ( PSF ) Examples of PSF’s and their Optical Transfer Functions Seeing limited OTF Intensity Seeing limited PSF /D / r0 D/ Diffraction limited OTF Intensity Diffraction limited PSF r0 / -1 /D / r0 r0 / D/ -1 Concept Question: what elementary optical calculations would you have to do, to lay out this AO system? (Assume you know telescope parameters, DM size) telescope primary mirror Deformable mirror Pair of matched offaxis parabola mirrors Wavefront sensor (plus optics) Science camera Beamsplitter Zernike Polynomials • Convenient basis set for expressing wavefront aberrations over a circular pupil • Zernike polynomials are orthogonal to each other • A few different ways to normalize – always check definitions! Piston Tip-tilt Astigmatism (3rd order) Defocus Trefoil Coma “Ashtray” Spherical Astigmatism (5th order) Units: Radians of phase / (D / r0)5/6 Tip-tilt is single biggest contributor Focus, astigmatism, coma also big Reference: Noll76 High-order terms go on and on….