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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)
y11  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….