Performance - Introduction to Image Stabilization - page 63

Performance 63

The curve A represents the “blurring” of the image due to diffraction of the tele-
scope only. Curve B displays image motion due to seeing, calculated using Eq. 6.1.
Curve C , the blurring function, has been obtained as an interpolation between the
two extreme cases; i.e., diffraction-limited or seeing-limited. It is noteworthy that
different interpolation expressions do not modify the position of the minimum sig-
nificantly. The minimum of the curve is roughly at a value of D / r 0 ~3. The total
spread angle, curve D ,isan ad hoc expression of curves B and C . Since the total
spread angle is not defined as a standard deviation, the equality D 2 = B 2 + C 2 is not
valid.
The ratio D / r 0 becomes a particularly important term in describing the effect of
image motion on an image. Image motion is produced by the global tilt in the atmo-
spheric turbulence, which is dominated by disturbances larger than the telescope
aperture. This allows the angular variation, σ , to be written as (Hardy 1998)

5
3

2

D
r

λ

2

σ = 0 182 . , (6.1)

α

D

0

where D is the aperture, r 0 is Fried's parameter, and λ is the wavelength.

6.3 Strehl Ratio
A diffraction-limited imaging system produces an image of an unresolved object
whose shape is defined by the Fourier transform of the entrance aperture. For circu-
lar apertures, the resulting image is an Airy function. Figure 6.3 shows two super-
imposed Airy functions, one a high-fidelity image and the other corresponding to a
slightly aberrated image. Notice that the minima of both functions are at the same
place, but the height of the point spread function (PSF) has changed. Clearly, using
the term diffraction-limited to refer to the location and visibility of the minima is
not a sufficient measure of performance in the presence of small aberrations.
A more sensitive measure of the performance of an optical system in the pres-
ence of small aberrations is to compare the height of the PSF to the ideal case. This
comparison, the Strehl ratio, is an image plane measure of the performance of an
optical system. The most common approach is to compare the ratio of the intensity
at the center of the PSF to that of an optimum or ideal system.
The effect of small wavefront aberrations on the final image is to move light out
of the focused point, reducing the peak height. This can be quantified, resulting in the
development of a relationship between the PSF height and the wavefront error. The
intensity of the light in the PSF can be determined using the Fresnel-Kirchoff diffrac-
tion integral, which was derived in Born and Wolf (1999). Hardy (1998) shows how
to quantize the reduction of the peak height, which is given as

2

2

Aa
R

* 2

I = π , (6.2)

2

λ