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