Image Processing Reference
In-Depth Information
64
Chapter 6
Figure 6.3
Two PSFs showing the location of the minima, with one whose peak is
lower. The ratio of the peak heights is the Strehl ratio.
where
I
*
is the intensity,
A
is the peak height,
a
is the radius of the aperture,
R
is the
radius of the Gaussian reference sphere, and
is the wavelength.
As a result, the Strehl ratio (
S
) can be written as
λ
I
I
1
2
2
p
∫∫
ik
φ
S
==
edd
ρρθ
,
(6.3)
*
π
where
I
p
is the intensity of the aberrated peak,
k
is the wave number,
ϕ
is the wave-
front phase, and
are polar coordinates.
This can be approximated as
ρ
and
θ
(
)
2
S
=−
1
σ
,
(6.4)
p
where
p
is the standard deviation of the wavefront phase.
This form of the Strehl ratio is only useful for very small aberrations on the or-
der of a 10
th
wave rms. Marechal (1947) derived an alternative for the Strehl ratio
for small aberrations, which has the form of the first two terms of an exponential
expansion. Thus, Eq. 6.5 shows the extended Marechal approximation of the Strehl
ratio, which is commonly used and valid for phase errors out to about 2 rad rms:
σ
