Modification of a standard incoherent optical system by a cubic-pm phase mask produces intermediate images that are insensitive to misfocus. Conceptually simple filtering techniques applied to these intermediate images form a complete system that images with high resolution and large depth of field. The cubic-pm mask, in normalized coordinates, is given by
where the constant 
controls the phase deviation.
The OTF of the
incoherent system related to this function can be approximated as
See Appendix A for a derivation of this result. The approximation of the
OTF is independent of misfocus. This can be inferred from the ambiguity function
related to the
cubic-pm mask, with 
, shown in figure 3.
The cubic-pm ambiguity function has uniform non-zero values
distributed about the u-axis. Radial
lines through the origin of this ambiguity function have nearly the same values
as a function of angle, for a broad range of angles.
Hence, the cubic-pm mask should form an
extended depth of field incoherent optical system. Figure 4 shows
a comparison between the stationary phase approximation and the actual calculated OTF using
(2). The smooth curve in this figure is the approximation of
the magnitude of the OTF;
the other is the calculated magnitude of the OTF. For this figure, the constant
of (6) was also selected as 90, while the misfocus parameter
was set to 15.
The approximation holds for other
values of misfocus as well as for the phase of the OTF. See figure
5. This figure is a plot of the magnitude of three misfocused OTFs related to
the cubic-pm
mask, with 
. The misfocus values of the three OTFs are 
,
and 
. These OTFs are nearly constant with misfocus and have no zeros.
This is what makes it possible to use one focus independent digital filter
to restore the intermediate image.
Note the dramatic variation of the OTFs of the
standard optical system, with the same misfocus values, shown in figure
6. Also notice that the vertical scale in figure
6 is different from that of figure 5.
In order to illustrate the performance of the optical/digital cubic-pm system for extended depth of field imaging, we present two methods of comparison. These are the simulated measurement of the full width at half maximum amplitude (FWHM) of the PSF as a function of misfocus, and simulated imaging of a spoke target at different misfocus values. Comparison is made to the standard optical system in both cases.
Figure 7 illustrates the FWHM criterion applied to the standard optical system
and cubic-pm optical/digital
system. The width of the standard system, with no misfocus, has been
normalized to unity. The width of the PSF from the cubic-pm system, after
focus independent digital filtering,
is essentially constant out to the normalized misfocus value of
.
From (3) we can show that the normalized misfocus
of is nearly 29 times that of the Hopkins criterion for misfocus
[16] where
. As expected, the width of the PSF of the standard
system greatly increases with misfocus. The width of the unfiltered or
intermediate PSF of the cubic-pm system would be much wider than that of
the infocus PSF from the standard system.
Figure 8 illustrates simulated imaging of a spoke target with
the cubic-pm optical/digital system, along with a comparison of images from the standard
optical system. The cubic-pm optical/digital
system includes both the formation of the incoherent intermediate image as well as
focus independent digital filtering of this image. Without digital
filtering the intermediate images would be unrecognizable. The digital filter used for this example was
a simple
inverse filter that, when combined with the intermediate OTF of (7), resulted
in a triangular system OTF, in a least squares sense. The left column of this figure simulates imaging a spoke
target with a standard optical system
under varying misfocus. The right column shows a
simulation of the same imaging conditions using the cubic-pm optical/digital system.
the cubic-pm system.
The term ``mild
misfocus'' corresponds to 
, or about 5 times the Hopkins
criterion for misfocus. The term ``extreme misfocus'' corresponds to
or about 29 times the Hopkins limit. The image of
the spoke target from the standard system is severely degraded for even mild misfocus.
The images
from the cubic-pm system are essentially constant with misfocus while the
image quality
is
nearly the same as that from the standard system with no
misfocus.
Only a single digital filter is used for all values of misfocus with the
cubic-pm system.
No single filter can be applied to the misfocused images from
the standard system to correct for the effects of misfocus.
These simulations assumed a noise free optical/digital system. In practice, ``restoration'' of the intermediate image through digital filtering will alter the noise properties of the final image. As in other restorative schemes, a signal-to-noise ratio (SNR) or dynamic range premium is required at the image. Different filtering schemes require different SNR premiums. The simple inverse filtering used here requires the largest premium. Other more complex filtering schemes would require less. The least squares inverse filter used for the simulations of figures 7 and 8 has a transfer function which is given in figure 9. From (7) the phase of this filter is approximately cubic. The zero spatial frequency component of this filter is normalized to unity. With this filter, the maximum magnification of any spatial frequency component is approximately 20 dB. An exaggerated estimate of the required SNR premium for this simple filter is then approximately 20dB, or the required extra dynamic range would be approximately 3.5 bits.
An algorithm independent measure of the increase in performance of the cubic-pm optical/digital system over the standard system can be found from the Fisher information of misfocus. Fisher information is a measure used to describe the information content of a given signal pertaining to a certain parameter[18][17]. For an ideal focus-invariant system the Fisher information of misfocus would be zero. In other words, the ideal focus-invariant system would produce an image that contains no information pertaining to the focus state. Such an image would not be a function of misfocus. A system whose OTF has a large variation with misfocus cannot employ a single focus-independent digital filter to correct for misfocus. A focus-dependent digital filter can be used if the focus state is known a priori.
Assume that a general incoherent system is imaging a point object, or one with a flat spatial frequency spectrum. We can show that the Fisher information of misfocus from this assumed scenario is
where 
is the traditional notation for the Fisher information of the
misfocus parameter , and
is the OTF.
A ratio of the Fisher information related to the standard system over the
Fisher information related
to the cubic-pm system can be used as a measure of performance of the
cubic-pm system. When this ratio is greater than unity, or 0 dB, the
theoretical variation with misfocus of the standard system exceeds that of the
cubic-pm system.
Again,
the cubic-pm system was chosen with the constant 
from (6)
as equal to 90. This ratio of the Fisher information
is given in figure
10. For example, the variation of the OTF of the standard
system
at misfocus of 
will, on average,
be 20 dB larger than the variation of the OTF for the
cubic-pm optical/digital system. Increasing the constant
of the cubic-pm system increases this difference in
the variation of the OTF; decreasing 
decreases the difference.
The misfocus value where the Fisher information of misfocus is equal for
the standard and cubic-pm system is monotonically related to the parameter
. Other methods of characterizing the performance of the cubic-pm
optical/digital system are currently under investigation.
