Transcript Optical Transfer Function is the Fourier Transform of PSF

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
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 and 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
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
Spherical waves and plane waves
What is imaging?
X
X
• An imaging system is a system that takes all rays coming
so
x
s
i
i
from a point source
so that they cross
Mthem
M x and
 redirects

a  
si
xo
scalled
o
each other in a single point
image
point. An optical
system that does this is said “stigmatic”
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)
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
First zero at
r = 1.22  / D
2) Intensity
FWHM
/D
Imaging through a perfect telescope
(circular pupil)
With no turbulence,
FWHM is diffraction limit
of telescope,  ~  / D
FWHM ~/D
Example:
1.22 /D
in units of /D
Point Spread Function (PSF):
intensity profile from point source
 / D = 0.02 arc sec for
 = 1 mm, D = 10 m
With turbulence, image
size gets much larger
(typically 0.5 - 2 arc sec)
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
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
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….