Astronomical Applications Department, U.S. Naval Observatory Beam Comp Example Page 12
17
96
C2
(
)
+ +
a b b2 a2
(
)
-
+
b2 a b a2
f4
1
16
C2
(
)
+
a2 b2
(
)
+
b4 a4
f6
11
512
C2
%4 %3
f8
+
-
+
11
1536
C2
(
)
+
a2 b2
(
)
+ +
a b b2 a2
(
)
-
+
b2 a b a2
(
)
-
+
b4 a2 b2 a4
f10
19
8192
C2
%2 %1
f12
-
+
d
2 f (
)
-
C
1 (
)
+
C
1
1
2
C
(
)
+
a2 b2
(
)
-
C
1
f
+
-
1
96
C
(
)
+ +
a b b2 a2
(
)
-
+
b2 a b a2
(
)
- +
16 17 C
f3
+
1
256
(
)
+
a2 b2
(
)
+
b4 a4
(
)
-
16 C 1 (
)
-
C
1
f5
1
2560
%4 %3 (
)
-
+ +
64 C 4 55 C2
f7
-
+
1
6144
(
)
+
a2 b2
(
)
+ +
a b b2 a2
(
)
-
+
b2 a b a2
(
)
-
+
b4 a2 b2 a4
(
)
+ -
44 C2 3 60 C
f9
-
1
57344
%2 %1 (
)
-
+
133 C2 198 C 8
f11
+
2
:=
%1
-
+
-
+
-
+
b6 b5 a a2 b4 b3 a3 a4 b2 b a5 a6
:=
%2
+
+ +
+
+
+
a6 b5 a b6 a4 b2 b a5 a2 b4 b3 a3
:=
%3
-
+
-
+
b
4
b
3
a a
2
b
2
b a
3
a
4
:=
%4
+
+
+ +
b4 a2 b2 b a3 a4 b3 a
>
cost(OPD_avg);
+
+
283 additions 1236 multiplications 39 divisions
Notice that the aperture-averaged OPD is second order in the perturbations. Also, note the
cross term. Hence, we should be able to find values of
and
that minimize the averaged
OPD. We can get a feel for the sensitivity of the averaged OPD by considering a numerical
example:
>
evalf( subs( a=5, b=10, f=100, d=20, C=10, OPD_avg ) );
+
+
.5864543026
2
234.9479888
23731.50613
2
With OPD and
expressed in microns, and
in arc seconds, this is
>
subs( Delta=10^(-4)*Delta, psi=evalf(arcsec)*psi, "*10^4 );
+
+
.00005864543026
2
.001139059994
.005577955379
2
Zernike Series Expansion of the Wavefront
