Page 1 Adaptive Optics in the VLT and ELT era François Wildi Observatoire de Genève Credit for most slides : Claire Max (UC Santa Cruz) Optics for AO

Goals of these 3 lectures 1)To understand the main concepts behind adaptive optics systems 2)To understand how important AO is for a VLT and how indispensible for an ELT 3)To get an idea what is brewing in the AO field and what is store for the future

Content Lecture 1 Reminder of optical concepts (imaging, pupils. Diffraction) Intro to AO systems Lecture 2 Optical effect of turbulence AO systems building blocks Error budgets Lecture 3 Sky coverage, Laser guide stars Wide field AO, Multi-Conjugate Adaptive Optics Multi-Object Adaptive Optics

Simplest schematic of an AO system COLLIMATING LENS OR MIRROR FOCUSING LENS OR MIRROR BEAMSPLITTER PUPIL Optical elements are portrayed as transmitting, for simplicity: they may be lenses or mirrors WAVEFRONT SENSOR

Spherical waves and plane waves

What is imaging? XX An imaging system is a system that takes all rays coming from a point source and redirects them so that they cross each other in a single point called image point. An optical system that does this is said “stigmatic”

Plane Wave Distorted Wavefront Optical path and OPD Index of refraction variations The optical path length is The optical path difference OPD is the difference between the OPL and a reference OPL Wavefronts are iso-OPL surfaces

Spherical aberration Through-focus spot diagram for spherical aberration Rays from a spherically aberrated wavefront focus at different planes

Optical invariant ( = Lagrange invariant)

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 In the “far field” (Fraunhofer limit) the diffracted field U 2 can be computed from the incident field U 1 by a phase factor times the Fourier transform of U 1 U 1 (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)

What is the ‘ideal’ PSF? The image of a point source through a round aperture and no aberrations is an Airy pattern

Details of diffraction from circular aperture and flat wavefront 1) Amplitude 2) Intensity First zero at r = 1.22 / D FWHM / D

Imaging through a perfect telescope (circular pupil) With no turbulence, FWHM is diffraction limit of telescope, ~ / D Example: / D = 0.02 arc sec for = 1  m, D = 10 m With turbulence, image size gets much larger (typically arc sec) FWHM ~ /D in units of /D 1.22 /D Point Spread Function (PSF): intensity profile from point source

Diffraction pattern from LBT FLAO

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 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 ) convolved with

Examples of PSF’s and their Optical Transfer Functions Seeing limited PSF Diffraction limited PSF Intensity   Seeing limited OTF Diffraction limited OTF / r 0 / D r 0 / D / r 0 / D /  -1

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

Spherical “Ashtray” Astigmatism (5th order)

Units: Radians of phase / (D / r 0 ) 5/6 Reference: Noll76 Tip-tilt is single biggest contributor Focus, astigmatism, coma also big High-order terms go on and on….