Principles and performance comparison of phase
masks for the alleviation of optical aberration
Gonzalo Muyo and Andrew R. Harvey
School of Engineering and Physical Sciences, Heriot-Watt University,
Edinburgh, EH14 4AS, Scotland, United Kingdom
Email: a.r.harvey@hw.ac.uk
Abstract. Since Dowski et al. [Appl. Optics 34, 1995] designed a special phase mask
that, in combination with digital signal processing, produced increased depth of field in
an imaging system, there has been a growing interest in the development of imaging
systems based on this hybrid technique. The fundamental element of this improved
imaging system is the refractive phase mask, which codes or distorts the transmitted
wavefront so that the point spread function (PSF) remains practically unaffected by
defocus. In contrast to traditional imaging systems, the detected image appears
blurred, due to the particular intensity distribution of the PSF. In order to accurately
obtain a diffraction-limited image it is necessary to restore the intermediate image
using digital post processing. Extensive research has been carried out in designing
phase masks with optimum defocus-invariant properties. In this paper, we present a
unique physical insight into the functioning principles of various phase masks (cubic,
logarithmic, quartic, etc) and provide a comparison of their performance. There are
two main groups of phase masks: asymmetric and rotationally symmetric forms.
Although asymmetric phase masks offer a superior defocus tolerance, severe
artefacts in the restored image can be present. Whilst rotationally symmetric phase
masks allow simpler manufacturing process to be used, the optical resolution is
reduced and the performance differs between positive and negative values of defocus.
1. Introduction
In the last decade, there has been a considerable development of hybrid optical/digital
techniques to provide additional degrees of freedom or system trade space in the design and
manufacture of imaging systems ([1]-[4]).
The fundamental concept of the hybrid imaging techniques involves a pupil-plane phase
mask designed to encode or distort the incoherent wavefront so that the point spread
function is practically unaffected by variations in the magnitude of aberrations in the imaging
optics and across a region near the image focal plane. The encoded PSF is notably large in
comparison to the PSF of a traditional diffraction-limited optical system and therefore the
image formed at the detector appears to be blurred. Post-detection digital signal processing
is used to decode or remove the blur of the recorded image and obtain a high-quality image.
The improved performance of hybrid imaging systems is generally limited by noise
amplification in the digitally processed image.
The loss in signal-to-noise ratio is
substantially dependent on the shape and surface sag of the phase masks.
In this paper we present a performance comparison of various phase masks against
defocus aberration and provide a physical insight into the functioning principles of pupil plane
encoding. By decomposing the optical transfer function (OTF) into the complex amplitudes
of the component interferograms of a pupil-plane-encoding system we can obtain analytical
solutions of the performance parameters and understand the physical principles which
enable increased tolerance to defocus. Although the OTF decomposition method has been
implemented analytically in a 1D imaging system with a cubic phase mask [5], it can be
extended in the same way into a 2D system provided that pupil function is rectangularly
separable. Unfortunately, for 2D optical systems with non-separable pupil functions the OTF
decomposition is restricted mainly to numerical evaluation, nevertheless in some cases it still
possible to obtain valuable insight and basic performance expressions [7].
2. Phase mask comparison
2.1. Antisymmetric phase mask
Phase mask that provide increased defocus invariance and which have been derived
assuming rectangularly separable condition have been found to be antisymmetric ([1], [8],
[9]). As we have demonstrated in Ref. [6], antisymmetric profiles favor a greater tolerance to
defocus. The derivation of this type of mask involves the use the stationary phase
approximation (applied to the equations of the ambiguity function or the point spread
function) and results in linearly separable cubic or logarithmic phase functions. Despite
having different derivation methods, the logarithmic phase mask is practically identical in
profile and performance to its predecessor the cubic phase mask. The cubic phase mask is
described as f  x    x 3 , where  sets the maximum wavefront deviation and yields the
amount of defocus invariance. In addition, Prasad et al. [2] developed a new technique
based on Fisher information metrics that generalizes the phase equation of the cubic mask to
include higher polynomial orders and non-rectangularly separable functions (petal like shape)
such as f  x, y    x 3  y 3   x 2 y  xy 2 , where  and  are real variables that control




the optical path difference.
As an example of the performance of these phase masks, restored images of an out-offocus object (W 20=3) are shown in Fig.1. For comparison, the detected image of a
conventional optical system is shown in Fig.1(a) together with the restored images with a
cubic phase mask, see Fig.1(b), and petal phase mask, Fig.1(c). Both phase masks
introduce the same peak-to-valley optical path difference (OPD=10).
(a)
(b)
(c)
Figure 1. Restored images of an out-of-focus Lena (a) for a pupil plane encoding imaging
system with cubic (b) and petal (c) phase masks.
Clearly the use of phase mask and digital post-detection processing has enabled sharp
images to be recovered for this value of defocus. The image quality is clearly improved,
although it can be observed that artifacts are introduced during restoration. The restored
image with the cubic phase mask contains ringing or replica of most of the edges, due to the
phase mismatch between the convolution and restoration kernels. The presence of ringing is
practically negligible for the petal phase mask. In contrast to the cubic case, the image
becomes distorted and some features are washed out. The intensity and extend of the
artifacts increase with defocus.
The origin of the artifacts lays in the disparity between the phase of the deconvolution
kernel and the phase of the point spread function. This can be analyzed by decomposing the
OTF of each phase mask as it varies with defocus. The components of the decomposed OTF
at each spatial frequency are plotted in the complex plane forming a geometric shape that
yields a representation of the interferometric processes that result in defocus aberration
invariance. The real and imaginary components of the OTF phasors are mapped into the
abscissa and coordinate axes respectively. For the case of the cubic phase mask combined
with defocus, the OTF for spatial frequency =0.25 (in normalized units) is mapped onto a
spiral as illustrated in Fig. 2. The full length along the spiral (the line integral) represents the
maximum value of the OTF for this spatial frequency, as is obtained for diffraction-limited
imaging. The phase mask causes the OTF contributions to curl into this spiral and the
resultant magnitude and phase of the OTF is described by the vector between the inside
ends of the spiral. Increasing defocus causes the OTF to rotate and for the length of the
spiral to unwrap from around one focus and to wrap onto the alternate extreme. The distance
between the foci remains invariant and it can be appreciated that for modest degrees of
defocus, the MTF is practically constant. The same representation can be done for the petal
phase mask, see Fig. 2. For the same wavefront distortion as for the cubic, the spiral of the
petal phase mask is much more tightly wound; which results in lower modulation transfer
functions (MTF).
MTF
w 20 0.
0.25
MTF
w 20 0.
1
0.25
1
(a)W 20=0
(b) W 20=1.5
(c) W 20=3
(d)W 20=0
(e) W 20=1.5
(f) W 20=3
Figure 2. Pupil plane encoding MTFs and OTFs depicted as spirals for cubic (top) and petal
(bottom) phase masks for various amounts of defocus (normalized spatial frequency =0.25,
and =5).
Noise amplification is introduced during the inverse filtering process of the encoded
image. The MTF of an optical system with a phase mask is highly suppressed in comparison
with that of a diffraction-limited system. It can be appreciated from the MTF of the petal
phase mask that the noise amplification will be much greater than for the cubic. The noise
amplification was calculated assuming uncorrelated Gaussian white noise and inverse
filtering. The dependence of the noise amplification on the angular direction in the Fourier
space led to calculate various statistics: The noise amplification was evaluated for the
horizontal direction, for the diagonal and finally for the entire Fourier space.
For phase masks that offer the same defocus invariance the noise amplification is more
favorable for the cubic phase mask. For example, on average the overall noise gain for the
petal mask is 10dB more than for the cubic assuming they both have the same depth of field
performance.
(a) cubic
(b) peta
Figure 3. Noise amplification in dB as a function of the optical path difference introduced by
the cubic (a) and petal (b) phase masks when a inverse filter is applied.
Clear aperture
2.2. Rotationally symmetric phase mask
Radially symmetric quartic and logarithmic aspheric phase masks ([3], [5]), also enable
defocus aberration mitigation. The performance attained by these kinds of phase mask
cannot equal that of the antisymmetric, but are suitable under modest amounts of aberrations
and some times can be used without digital signal processing. Rotationally symmetric phase
mask can be designed for various degrees of defocus; however, the amount of defocus
tolerance defines the practical cutoff frequency. Thus, high tolerance to defocus will result in
a very low cutoff frequency and therefore very large PSFs. With asymmetric phase mask, in
principle, the cutoff frequency remains unaffected. In addition, the OTFs of rotationally
symmetric phase mask contain frequency bands of contrast reversal at low spatial
frequencies and for moderate values of defocus. In the regions where there is no phase
reversals it is possible to obtain high-quality images that do not require post-detection signal
processing, as shown in Fig.4. The top row shows the USAF target image by an optical
system that suffers from various amounts defocus aberration (W 20=0.5, 1, 2); the bottom
row are the unprocessed images after introducing a quartic phase mask with a peak-to-valley
optical path difference of about 8. It can be observed that the image quality remains
practically constant throughout the defocus range and most of the features remain visible.
(b) W 20=1
(b) W 20=2
(c) W 20=0.5
(d) W 20=1
(b) W 20=2
Quartic
(a) W 20=0.5
Figure 4. Simulated images of the USAF target of an optical system suffering from
defocus (top) and with a quartic phase mask inserted in the aperture stop (bottom).
The loss in contrast can be overcome by digital signal processing.
The decomposition of the OTF for the quartic phase mask shows clearly the functioning
principles and limitations of rotationally symmetric masks, see Fig. 5. Similar results are
obtained for the logarithmic aspheric mask derived by Chi et al. [3] when is designed to
image objects placed at a certain distance from the optical system to infinity. The quartic
phase mask designed in this example gives a practical defocus invariance for values ranging
0<W 20<3 and spatial frequencies below <0.5. As shown in Fig. 5, for a given spatial
frequency (=0.25, in normalized units) the infinitesimal components or phasors of the OTF
are mapped into spirals which no longer counteract the effect of defocus by unwrapping one
end and wrapping the other, as it is the case for antisymmetric masks, but by folding the
entire length of the curve. This effect results in a diminished range of defocus invariance in
comparison to the cubic phase mask and produces OTFs which are not symmetrical under
positive and negative values of defocus. An important advantage of the quartic mask is that
the imaginary component of the OTF is always zero; this is clearly observed in the spiral
representation where the OTF is contained in the real axis (horizontal axis) for any value of
defocus. Let us recall that for the antisymmetric phase masks the phase varies with defocus
and artifacts are introduced in the inverse filtered image unless the deconvolution kernel is
adjusted to the right phase.
OTF
w20
OTF
2.5
w20
2.5
0.25
1
(a) W 20=-2.5
0.25
(d) W 20=1
1
(b) W 20=-1
(c) W 20=0
(e) W 20=2.5
(f) W 20=3.5
Figure 5. Quartic OTFs depicted as spirals for various degrees of defocus ranging from
-2.5<W20<3.5 (normalized spatial frequency =0.25).
3. Conclusions
There are some important differences and similarities between the radially symmetric and the
antisymmetric phase masks. In both cases the MTF is suppressed and can be restored to
that of a diffraction-limited imaging system by inverse filtering. However, where as the mean
amplitude of the OTF is approximately invariant with varying defocus for both types of phase
masks, for the antisymmetric phase masks the phase also varies and this means that
artefacts are introduced in the restored image. In addition, antisymmetric phase mask
produce antisymmetric PSF which blur the detected image so that digital post processing is
required to obtain a high-quality image. The most important limitations in the use of
antisymmetric phase masks is the presence of artifacts noise amplification in the restored
image, however they offer a greater defocus invariance for positive and negative values of
defocus.
We have presented an understanding of the formation of the OTF which involves a
geometrical representation of its interferogram components. The representation can be used
as a tool to effectively synthesize phase masks that increase the tolerance to aberration
since the method only involves the equation of a line.
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