1. 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)

2. 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

3. 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?

4. Spherical waves and plane waves

5. What is imaging? X X

6. Optical path and OPD Index of refraction Plane Wave variations
Distorted Wavefront
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

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

8. Optical invariant ( = Lagrange invariant)

9. Lagrange invariant has important consequences for AO on large telescopes
From Don Gavel

10. Fraunhofer diffraction equation (plane wave)
Diffraction region
Observation region
From F. Wildi “Optique Appliquée à l’usage des ingénieurs en microtechnique”

11. Fraunhofer diffraction, continued
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!

12. Looking at the far field (step 1)

13. Looking at the far field (step 2)

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

16. Diffraction pattern from MMT-AO

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

18. 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

19. 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

20. Examples of PSF’s and their Optical Transfer Functions
Seeing limited PSF
Seeing limited OTF
Intensity 
-1
l / D
l / r0
r0 / l
D / l
Diffraction limited PSF
Diffraction limited OTF
Intensity
-1 
l / D
l / r0
r0 / l
D / l

21. telescope primary mirror Pair of matched off-axis parabola mirrors
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
Pair of matched off-axis parabola mirrors
Deformable mirror
collimated
Science camera
Wavefront sensor (plus optics)
Beamsplitter

22. 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!

24. Piston
Tip-tilt

25. Astigmatism
(3rd order)
Defocus

26. Trefoil
Coma

27. “Ashtray”
Spherical
Astigmatism
(5th order)

28. Tip-tilt is single biggest contributor
Units: Radians of phase / (D / r0)5/6
Focus, astigmatism, coma also big
High-order terms go on and on….
Reference: Noll76