Lecture 3: Optical Transfer Function Flashcards
period
d_0 = 1/f_0
width D where D/d_0
> > 1
diffracted amplitude
fourier transform of A(x)
diffracted order at
x=0 +/- Z_0 λ/d
where z_0 is the object distance
objects can be decomposed into
its fourier components
these are not transmitted perfectly to the image
high spatial frequencies are filtered out (removed) at
the system’s aperture
leads to loss of resolution and contrast in the image
another perspective
Two small patches separated by s
Interference of two sources (c.f. Young’s slits)
point sources centred on +/- s/2
aperture function for point sources centred on +/- s/2
A(x)=δ(x-s/2)+δ(x+s/2)
spatial frequency in image
fi=s/zi lambda
zi=image distance
what if spacing corresponds to diffraction orders and let zi=z0=z
spatial frequency in image fi=f0
also di=d0
For a particular spatial frequency to be present in the image, the aperture must
be large enough to accommodate the spacing between the 0th and 1st diffracted orders, s=lambda z f0, of the corresponding spatial frequency in the object.
Spatial frequency only resolved if
diffraction orders corresponding
to that frequency pass through the aperture of system.
otherwise info about frequency is lsot
minimum requirement for spatial frequency to be resolved
at least two orders to be supported
by the system
Structure of object resolved if
first diffraction order
propagates through the optical system
Fidelity of the image increases with
the number of diffraction orders propagating through the optical system
Object can be considered to be made up of
many point sources
image of each is spread out
Image is convolution of
the geometrical image with
the Point Spread Intensity
Actual image is the convolution of
the ideal image predicted by
geometrical optics with the Point Spread Intensity
Start with Image Amplitude and Image Intensity as functions of spatial coordinates. Image amplitude:
Convolution of Point Spread Amplitude with Geometrical Amplitude,
Start with Image Amplitude and Image Intensity as functions of spatial coordinates. Image intensity
Convolution of Point Spread Intensity with Geometrical Intensity
Fourier transform of convolution is
the product of the Fourier transforms
fourier transform of product is
convolution of the fourier transforms
spatial angular frequency
kx=2pifx
in radians per metre
Image intensity is the convolution of
the geometric image intensity with the Point Spread
Intensity
get frequency content of image
take fourier transform
fourier transform of intensity
frequency spectrum of image = optical transfer function x frequency spectrum of geometrical image
filtering/degradation of system attributed to
H(kx,ky)
H(kx,ky) transmits
low spatial frequencies
H(kx,ky) attenuates
high spatial frequencies
H (optical transfer function) describes how
optical system filters or modifies spatial frequencies that would otherwise contribute to the image
magnitude of OTF |H| corresponds to
contrast
modulation transfer function (MTF)
Phase of OTF, arg(H) corresponds to
distortion
ie a shift in position
phase transfer function (PTF)
atmospheric turbulence
- Pockets of air with
different temperatures,
densities and refractive
indices - Corrugated wavefronts
- Dominant cause of
aberrations for large-scale
telescopes
atmospheric turbulence: airy
small aperture
short exposure
atmospheric turbulence: speckle
large aperture
short exposure
atmospheric turbulence: time average
large aperture
long exposure
atmospheric turbulence: telescope
well-defined
static
analytical
atmospheric turbulence: atmosphere
random
changing statistical
convolution
(f⊗g)(z)
cross-correlation
(f★g)(z)
auto-correlation
(f★f)(z)
power spectrum=
fourier transform of autocorrelation
fourier transform of power=
autocorrelation of amplitude spectrum
amplitude transfer function (ATF)
H(fx,fy)= F{h}
optical transfer function
H(fx,fy) = F{|h|^2} / normalisation
aurtocorrelation of ATF, what are p and q?
spatial frequency variables
H(fx,fy)≡
scaled version of aperture / pupil function
autocorrelation of pupil, H(f’x,f’y)=
area of overlap / total overlap
With Fourier Optics we can
understand
how an optical system filters frequencies present within the object
the optical transfer function expresses
how that frequency
content in filtered at the
aperture of the optical system
atmospheric turbulence requires a
statistical description
the optical transfer function can also be understood to be the
autocorrelation of the pupil function
