Lecture 2: Diffraction and PSF Flashcards
fraunhofer diffraction
diffraction from an aperture
plane-parallel waves incident upon aperture
viewed in far field
fraunhofer condition
viewed in far field
L»D^2/lambda
spatial frequency
fx in cycles per metre
diffraction angle frequency
f’x in units of cycles per radian
mathematical model of single slit
A(x) = 1 when between -a and a, 0 elsewhere
the diffraction patter is given by
the fourier transform of the aperture
bessel function
Up=U0[2J1(kαa)/(kαa)]
zeroes at kαa=1.22pi, 2.23pi, 3.24pi
where does α=1.22 lambda/D relationship come from
first zero at kαa=1.22pi
2pi/lambda D/2 α=1.22pi
geometrical objects: image formed of object at
point of convergence of rays - image plane
geometrical optics: one-to-one mapping
every point on the image corresponds to a point on the object
geometrical optics: spacing in the image plane governed by
the magnification of the system
when imaging an object at infinity, the image plane is
simply the focal plane
image formation in general
optical systems has imperfect focusing
all rays not brought to single point
aberration eg spherical aberration
image of point is spread out
image formation fundamental limit
system free from aberration
optical system had an aperture
gives rise to diffraction
image of point is spread out
object can be considered to be
made up of many point sources
image of each point is spread out
image is convolution of
the geometrical image with the point spread function intensity
convolution
integral that gives the changing overlap between two functions as one is shifted over the other
can be interpreted as blurring or blending
example of convolution
convolution of an array of delta-functions with a sinc-squared function is an array of sinc-squared functions
actual image is the convolution of
the ideal image predicted by geometrical optics with the point spread function intensity
imaging only degraded by
diffraction ie effects of aberrations are negligible, said to be ‘diffraction limited’
I_{PSF} of ideal telescope
central airy disc with surrounding airy rings
can characterise resolution by FWHM of I_{PSF} but…
I_{PSF} has rings
shape of I_{PSF} can be irregular
strehl ratio
S=I_{psf}(0) / I_{airy}(0)
central intensity of PSF compared to that of ideal PSF intensity (airy disc)
diffraction limit of a circular aperture is
α=1.22lambda/D
point spread function tells us
how the light from a point source spreads out in an image
for an ideal telescope, the PSF is
simply the airy pattern
