Allen R. Crockett
Math E-118 Research Project
January 4, 2005
ABSTRACT
We will look at the 2-dimensional Fourier Transform, how it relates to the resolution of
stars as seen through an optical telescope, and how astronomers are currently proposing
different aperture shapes and masks to help identify possible Earth-like planets up to
20pc away.
DIFFRACTION
Diffraction is a general wave phenomenon, formed at some point in space at some instant
by the sum of the amplitudes of multiple waves that may have different phases. For
waves of nearly (within 5%) the same wavelength, constructive interference at some
point results from waves whose phases differ by an integer multiple of their wavelength.
Similarly, total destructive interference results from the sum
of two or more waves whose phases differ by integer multiples of half wavelengths.
Between these two outcomes, a continuous range of amplitude sums is possible.
The diffraction pattern formed by a distant star on the image plane of a telescope--often
referred to as the objects point spread function (PSF)--is a consequence of the wave-like
nature of light. To simplify our discussion of the PSF formed by a star, we will treat the
wave fronts of emitted light as if they are parallel plane waves traveling in phase with
some uniform intensity (that is, the square of the time average of the amplitudes is
constant) and some uniform wavelength (550 nm).
It is helpful to mention that plane waves of light obey Huygens’s Principle, which models
a plane wave as the superposition of a continuum of spherical waves, each spreading out
isotropically. It is this model of a plane wave that allows us to more easily understand
diffraction: because the spherical, constituent waves emitted by each point in the aperture
travel different distances to points in the image, diffraction results (see figure 1).
AIRY DISK
The PSF of a star in the image plane of a telescope with a conventional, circular aperture
takes the form of an Airy Disk--a pattern of concentric rings of alternating brightness,
with the intensity of each bright lobe decreasing within an envelope that nearly resembles
the sinc-squared function (see figure 2). In general, the width, shape, and intensity
envelope of a star’s PSF places a practical limit on how closely the PSFs of two
neighboring stars can form on the image plane (see figure 3). That is, the wider and more
intense the PSF of a star, the greater the angular separation needed in the image plane for
an observer to distinguish the two patterns. So the narrower the PSF, the better the
resolution obtained by the telescope.
THE FOURIER INTERPRETATION OF THE PSF
To understand how the resolution of a telescope can be improved, it helps to understand
that the PSF of a star is directly related to the shape of the telescope's aperture.
Let E0 represent the uniform amplitude of the electric field across a telescopes aperture
at some instant. Then the magnitude of the electric field at a point P(X,Y) in the image
plane is given by
Ep 
E0 i (t kr )
e
dxdy
r
aperture

In the preceding equation omega and k are the angular frequency and angular spatial
frequency, respectively, of the electromagnetic wave (traveling transversely to the
incoming wave fronts); P(x,y) is a point on the open aperture, and r is the distance from
P(x,y) to P(X,Y).
In words, the value of the electric field at P(X,Y) is the sum of all the amplitudes of the
waves emitted from all points on P(x,y). Waves traveling from different points on the
aperture plane travel different distances to get to the same point on the image plane, and
each distance contributes to the overall interference there.
Now let R be the distance from the center of the aperture to P(X,Y). We can reasonably
approximate r:
r  R 2 (1  2 xXR2yY )  R (1  2 xXR2yY )  R(1 
Assuming xX + yY << R 2 , we rewrite E p :
Ep 
E0
R
e i (t kR )
e
i ( x kX
 y kY
)
R
R
dydx
aperture
If we consider an aperture function
A(x,y) = {1 for x 2  y 2  a 2 , 0 otherwise},
xX  yY
R2
)
then the form of E p above is the Fourier Transform of A(x,y). Note that A describes a
cylinder of radius a and unit height. This is a profound result. The action of light passing
through the aperture of some shape naturally transforms the continuous spatial
frequencies of the shape's geometry to the angular frequencies seen in the resulting image.
Calculating the above integral in cylindrical coordinates (and introducing some Bessel
functions), we find that the minimum angle in the image plane required for optical
resolution of two points seen through a circular aperture of diameter D is given by:
 min  1.22 D
The equation above is known as the Rayleigh Criteria for resolution. This result agrees
with our intuitive notion of the action of the Fourier Transform: if f(x) has a high spatial
frequency (it's 'skinny'), then its Fourier Transform has a low angular frequency, and
vice-versa.
What is the intensity I(x,y) along the x or y axis of the image plane (recall that intensity is
the square of the time average of the electric field)? This is not an easy question to
answer, since I(x,y) involves complicated Bessel functions. However, if we realize that a
single-dimension aperture--a thin slit--of width d has its resolution given by
 min 

d
and if we recall that the Fourier Transform of a square pulse is given by the sinc function,
then perhaps to a liberal approximation we can say that a profile of the Airy Disk PSF
roughly looks like sinc. This fact will become significant in the next section, where we
consider the PSF of the square aperture.
THE NASA-JPL TERRESTRIAL PLANET FINDER MISSION
The Terrestrial Planet Finder (TPF) is an ongoing joint NASA and JPL project to identify
extra-solar, Earth-like planets. Originally the mission was proposed to use an infra red
interferometer composed of 6 to 8 free-flying elements in orbit. Out of concern that this
proposal is expensive and risky, an orbiting optical telescope of roughly the same size as
the Hubble Space Telescope was proposed as an alternative. Currently both designs are
planned for launch, with the optical telescope scheduled for 2014 and the interferometer
to follow a few years later.
One of the most acute problems facing the optical telescope is that of resolution. The
mission would like to investigate extrasolar systems in the 10-20 pc distance range (1pc
~= 3.25 light years). To find a possible Earth-like planet, it would have to share orbital
characteristics with Earth with respect to orbital raduis and planetary
mass. This implies that the angular separation between the planet and its star would be
roughly 0.01 arcseconds (the diameter of Neptune as seen from Earth is 2 arcseconds). To
make matters worse, since planets to not emit their own light, the relative brightness
between the star and planet would be roughly 1 : 10 10 . A circular aperture would require
an (unpractical) width of roughly 10 meters to meet these imaging needs.
To address this issue of resolution, in 2001 Costas Papaliolios and Peter Nisenson
proposed the use of a telescope with a square aperture. Here the aperture function is given
by A(x,y) = A(x)A(y), and so the Fourier Transform of A(x,y) turns out to be
proportional to the product of the transforms of A(x) and A(y). Thus along the x and y
axes, the PSF will decrease in intensity from the central peak as the square of the sinc
function (similar to the case of the circular aperture along all axes), and the diagonal axes
where |x| = |y| decrease as sinc^4. So for planets along the diagonal in the image, a planet
with relative brightness of 10 9 could be detected (see figure 4), but when noise effects
are accounted for, the planet is barely detectible.
Other, more exotic aperture shapes have been proposed, such as the
single prolate spheroidal wave function shaped-pupil aperture (figure 5), elliptical
apertures with multiple openings, (figure 6), concentric ring apertures (figure 7), and
concentric ring apertures with supporting spiders (figure 8).
In each of the apertures pictured, we notice that the symmetry in the aperture is retained
in the PSF.
REFERENCES
Hect, Eugene. Optics:Third Edition. Addison-Wesley, 1998.
Kasdin, N. J., Spergel, D. N., & Littman, M. G. Extrasolar Planet Finding via Optimal
Apodized and Shaped Pupil Coronographs. 2002, Applied Optics.
Nisenson, P., & Papaliolios, C. Detection of Earth-like Planets Using Apodized
Telescopes. 2001, Astrophysical Journal.
Terrestrial Planet Finder Mission Offical Web Site. http://tpf.jpl.nasa.gov
Vanderbei, R. J., Spergel, D. N., Kasdin, N. J. Spiderweb Masks for High-Contrast
Imaging. 2003, Astrophysical Journal.
Vanderbei, R. J. The Terrestrial Planet Finder Mission: Detecting and Characterizing
Earth-like planets Orbiting Nearby Stars. 2003.