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Zernike Polynomials
The Zernike polynomials were derived in the early 20
th
century by Dutch physicist Frits Zernike
(famous for the invention of the phaseconstrast microscope for which he won the 1953 Nobel
Physics Prize).
The classic treatment of Zernike polynomials for the description of atmospheric
turbulence is Noll, R.J. 1976, J. Opt. Soc. Am., 66, 207.
This is the form most often seen in the
astronomical literature, but beware that there are many other forms in use, and it can get quite
confusing.
The Zernike polynomials are a set of orthogonal polynomials defined on a unit disk.
They can
be expressed in either Cartesian (x,y) or Polar (r,
θ
) coordinates, but the polar form is most
common:
,
,
0
1
( )
2 cos(
),
m
0
1
( )
2 sin(
),
m
0
1
( ), m=0
m
jeven
n
m
jodd
n
jn
Zn
R
r
m
R
r
m
R
r
θ
=+
≠
≠
where
()
/
2
2
0
(1
)(
)
!
![(
) / 2
]! [(
) / 2
]!
s
nm
m
ns
n
s
Rr
r
snm
s nm
s
−
−
=
−−
=
+−
∑
The values of
n
and
m
are always integers, and satisfy
mn
≤
and
even
−=
.
The index
j
is the mode ordering number, and is a function of n and m.
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 Spring '08
 MARTINI
 Astronomical

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