-
Frequency analysis of the wavefront-coding
odd-symmetricquadratic phase mask
Manjunath Somayaji and Marc P. Christensen
A mathematical analysis of the frequency response of the
wavefront-coding odd-symmetric quadraticphase mask is presented. An
exact solution for the optical transfer function of a
wavefront-coding imagerusing this type of mask is derived from
first principles, whose result applies over all misfocus values.
Themisfocus-dependent spatial filtering property of this imager is
described. The available spatial frequencybandwidth for a given
misfocus condition is quantified. A special imaging condition that
yields anincreased dynamic range is identified. © 2007 Optical
Society of America
OCIS codes: 110.4850, 070.6110.
1. Introduction
In recent years, researchers have developed a com-putational
imaging technique called wavefront cod-ing to create imaging
systems that are capable ofachieving extended depths of field.1
Wavefront codingenables these systems to operate over an
extendeddepth of field by modifying the light field at the
ap-erture of these optical systems so as to permit imag-ing even in
the presence of considerable misfocus. Inthe original version of
this technique as it was intro-duced in Ref. 1, a cubic phase
modulation (cubic-pm)mask is placed at the exit pupil of a
conventionalimager, thereby transforming the spatial
frequencyresponse of the imager. The optical transfer function(OTF)
of such a system varies little in magnitudewith misfocus and
contains no nulls within its pass-band. The optically formed image
captured by thedetector therefore exhibits a uniform blur that
islargely independent of misfocus. This intermediateimage can then
be digitally processed with a simplelinear restoration filter to
yield a final image withgreatly improved depth of focus. Figure 1
describesone such wavefront-coding system.
Once wavefront coding with the cubic phase mask
demonstrated that enhanced imaging performancecould be achieved
by deliberately blurring an imagein a calculated way, researchers
began to look formethods to optimize the nature of the blur to
maxi-mize the quality of the postprocessed image. Variousimage
quality metrics were subsequently proposed tomeasure imager
performance and tailor the pupilphase profile to obtain even better
results than thecubic phase mask.2–6 Wavefront coding was also
ex-tended to circular and annular pupils, and numerouscircular
phase profiles were studied.5–10 Even thoughcircular apertures are
a staple of a majority of imag-ers, rectangularly separable systems
retain the ad-vantage of faster image reconstruction due to
thereduced computational overhead associated with theseparate
signal processing along each of the imageplane dimensions.
Moreover, the ambiguity functions(AFs) of rectangularly separable
systems may betreated in just two dimensions, a feature available
tophase profiles with circular apertures only if theyexhibit radial
symmetry.11
The appeal of wavefront coding as a new para-digm for optical
system design lies in the elegance ofits simplicity and its ability
to address multipleissues simultaneously. It has been
demonstratedthat wavefront-coding techniques reduce complex-ity in
optical design12 and are capable of correctingor minimizing the
effects of a host of aberrationssuch as Petzval curvature,
astigmatism, chromaticaberration, spherical aberration, and
temperature-related misfocus.13–17 In this work, the behavior of
awavefront coding imager incorporating a phasemask with an
odd-symmetric quadratic phase pro-file is examined by conducting a
mathematical anal-ysis of its spatial frequency response.
Development
The authors are with the Department of Electrical
Engineering,Southern Methodist University, 6251 Airline Road,
Dallas, Texas75275-0338, USA. M. Somayaji’s e-mail address is
[email protected]. M. P. Christensen’s e-mail address is
[email protected].
Received 30 May 2006; revised 6 September 2006; accepted
20September 2006; posted 20 September 2006 (Doc. ID 71356);
pub-lished 21 December 2006.
0003-6935/07/020216-11$15.00/0© 2007 Optical Society of
America
216 APPLIED OPTICS � Vol. 46, No. 2 � 10 January 2007
-
of an exact analytical representation of the OTF ofa cubic-pm
system allowed a comprehensive spatialfrequency analysis of such an
imager.18 Knowledgeof the exact OTF of a cubic-pm imager helped
quan-tify its available spatial frequency bandwidth andenabled the
system design and trade-off analysis ofthis system. Here, a similar
mathematical analysison the frequency response of a wavefront
codingimager with an odd-symmetric quadratic phasemodulation
element is performed. An exact repre-sentation of the OTF of such a
system is presented,and the available spatial frequency bandwidth
andspecial imaging conditions that yield an enhanceddynamic range
of this imager are described. Theimproved noise handling ability
due to this enhanced
dynamic range could make this odd-symmetric qua-dratic phase
modulation element an attractive alter-native to the cubic phase
mask in applications wherenoise issues dominate over other system
performancerequirements.
2. Odd-Symmetric Quadratic Phase Mask
Research on rectangularly separable systems hasshown that a
phase plate that extends the depth offield must have an
odd-symmetric phase profile19;that is, the phase function must
satisfy the condition
��x, y� � ����x, � y�. (1)
The above condition implies that a phase plate thatyields an
extended depth of focus must itself lack anyfocusing power. The
results have been used to derivephase plates with a logarithmic
contour that havesimilar properties as those of the cubic phase
mask.19A family of phase masks that have also been proposedas good
candidates for enhancing the depth of fieldare described along one
dimension by the phase func-tion15,20,21:
��x� � � sgn�x�|x|�, �1 � x � 1, � � 0, � � 2.(2)
In the above equation, x is the normalized pupil
planecoordinate, � is a positive design constant that con-trols the
phase deviation and hence the strength ofthe phase mask, and � is a
positive real power. Thesignum function in Eq. (2) contributes to
the oddsymmetry of the phase profile and is given by
sgn�x� ��1, x � 00, x � 0�1, x 0
. (3)
The properties of imaging systems with fractionalvalues of �
have been studied,20 and numerical inves-tigations into the
behavior of the modulation transferfunction (MTF) of a mask with �
� 2 have beenconducted.21
In this work, the OTF of a member of the family ofphase masks
described by Eq. (2) is mathematicallyevaluated. Specifically, an
analytical expression forthe OTF for a phase mask with � � 2 is
derived andthe result is exploited to evaluate the available
spa-tial frequency bandwidth of this imaging system for agiven
value of misfocus. This phase mask is hereintermed the
odd-symmetric quadratic (OSQ) phasemask. This work also identifies
a special imagingcondition that yields an increased dynamic range
ofthe imager.
A. Phase Mask Partitioning
The phase profile of Eq. (2) for � � 2 may be expressedin
conjunction with the definition of the signum func-tion given by
Eq. (3) as
Fig. 1. Wavefront coding system incorporating a cubic phasemask.
(a) A wavefront coding imager utilizes a cubic-pm elementwhose (b)
2D phase profile achieves an extended depth of field. (c)The
corresponding OTF varies little in magnitude with misfocus.The plot
in (c) shows the magnitude of the OTF for three misfocusvalues � �
0, � �, � 5��. The smooth curve in (c) representsthe approximate
OTF as described in Ref. 1.
10 January 2007 � Vol. 46, No. 2 � APPLIED OPTICS 217
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��x� ����x2, �1 � x 0�x2, 0 � x � 1 . (4)The generalized pupil
function of the wavefront codingimager with the above phase profile
is then given by
P�x� ��1
�2exp�j� � ��x2� � P��x�, �1 � x 0
1
�2exp�j� � ��x2� � P��x�, 0 � x � 1
0, otherwise
.
(5)
Here, � is the misfocus parameter and is defined as1
��L2
4 1f � 1do � 1dc. (6)In the above equation, L is the width of
the aperture,f is the focal length, do is the object distance from
thefirst principal plane of the lens, dc is the distance ofthe
image capture plane from the second principalplane of the lens, and
� is the wavelength of light.
B. Optical Transfer Function of the Odd-SymmetricQuadratic Phase
Mask System
Following the techniques developed in the analyticalevaluation
of the OTF of a cubic-pm imager,18 theOTF of a rectangularly
separable OSQ phase masksystem along one dimension may similarly be
calcu-lated from first principles by analytically evaluatingthe
autocorrelation of the generalized pupil func-tion.22 Evaluating
the OTF of the imager whose pupilfunction is given in Eq. (5) from
first principles re-quires careful consideration of the area of
overlap inthe autocorrelation integral due to the partitioning
ofthe pupil function. For a general phase mask P�x�that is
partitioned into two sections P��x� and P��x�about x � 0, the
autocorrelation process may be splitinto four distinct regions and
the OTF written as
Here, the spatial frequency u is expressed in normal-ized form
as u � fX�2fo such that �1 � u � 1 withinthe diffraction limit. fX
is the unnormalized spatialfrequency and 2fo � L�di is the
diffraction-limitedcutoff frequency of the imager, with di being
the dis-tance of the diffraction-limited imaging plane fromthe
second principal plane of the lens. Incorporatingthe actual values
of P��x� and P��x� from Eq. (5) intoEq. (7), the OTF of the OSQ
phase mask system maybe expressed as
H�u, � �
�
�u�1
u�1
P��x � u�P�*�x � u�dx, �1 � u � �12
��u�1
u
P��x � u�P�*�x � u�dx ��u
�u
P��x � u�P�*�x � u�dx ���u
1�u
P��x � u�P�*�x � u�dx, �12 � u � 0
�u�1
�u
P��x � u�P�*�x � u�dx ���u
u
P��x � u�P�*�x � u�dx ��u
1�u
P��x � u�P�*�x � u�dx, 0 � u �12
�u�1
1�u
P��x � u�P�*�x � u�dx,12 � u � 1
.
(7)
H�u, � �
12 �
�u�1
u�1
exp�j�4ux � 2��x2 � u2��dx, �1 � u � �12
12 �
�u�1
u
exp�j4u� � ��x�dx �12 �
u
�u
exp�j�4ux � 2��x2 � u2��dx �12 �
�u
1�u
exp�j4u� � ��x�dx, �12 � u � 0
12 �
u�1
�u
exp�j4u� � ��x�dx �12 �
�u
u
exp�j�4ux � 2��x2 � u2��dx �12 �
u
1�u
exp�j4u� � ��x�dx, 0 � u �12
12 �
u�1
1�u
exp�j�4ux � 2��x2 � u2��dx,12 � u � 1
.
(8)
218 APPLIED OPTICS � Vol. 46, No. 2 � 10 January 2007
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Equation (8) may be further simplified by exploitingthe symmetry
of the kernel of the integrals aboutu � 0. The OTF may therefore be
restated as
For ease of notation, the OTFs within the two distinctregions in
Eq. (9) are referred to as the centerHC�u, � and tails HT�u, � such
that
HC�u, � �12 �
|u|�1
�|u|
exp�j4u� � ��x�dx
�12 �
|u|
1�|u|
exp�j4u� � ��x�dx
�12 �
�|u|
|u|
exp�j�4|u|x � 2��x2 � u2�
� sgn�u��dx, 0 � |u| �12, (10)
HT�u, � �12 �
|u|�1
1�|u|
exp�j�4|u|x � 2��x2 � u2�
� sgn�u��dx,12 � |u| � 1. (11)
The tails of the OTF are evaluated first. The right-hand side of
the above equation may be rewrittenafter completing the square in
the exponent as
HT�u, � �12 exp�j2�u21 � 2�2sgn�u����
|u|�1
1�|u|
exp�j �24�� x � �|u|2
� sgn�u��dx, 12 � |u| � 1. (12)
Applying a change of variables on the integral, theabove
equation may be expressed with the new lim-its as
HT�u, � �14��1�2exp�j2�u21 � 2�2sgn�u����
aT�u�
bT�u�
exp�j �2 �T2sgn�u��d�T,12 � |u| � 1, (13)
where the integration limits aT�u� and bT�u� are givenby the
relation
aT�u� � 4�� 1�2�|u|1 � �� 1�,
bT�u� � 4�� 1�2�1 � |u|1 � ��. (14)Next, Euler’s identity is
applied on the kernel to yield
HT�u, � �14��1�2exp�j2�u21 � 2�2sgn�u�����
aT�u�
bT�u�
cos�2 �T2d�T � j sgn�u���
aT�u�
bT�u�
sin�2 �T2d�T�, 12 � |u| � 1.(15)
The integrals are identified as Fresnel cosines andFresnel
sines. The OTF at the tails may then beexpressed as
H�u, � �
12 �
|u|�1
�|u|
exp�j4u� � ��x�dx �12 �
|u|
1�|u|
exp�j4u� � ��x�dx
�12 �
�|u|
|u|
exp�j�4|u|x � 2��x2 � u2�sgn�u��dx, 0 � |u| �12
12 �
|u|�1
1�|u|
exp�j�4|u|x � 2��x2 � u2�sgn�u��dx,12 � |u| � 1
. (9)
10 January 2007 � Vol. 46, No. 2 � APPLIED OPTICS 219
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HT�u, � � �8�1�2exp�j2�u21 � 2�2sgn�u���
1
�2��C�bT�u�� � C�aT�u��� � j sgn�u�
� �S�bT�u�� � S�aT�u���,12 � |u| � 1.
(16)
The operators C�·� and S�·� represent the Fresnelcosine integral
and Fresnel sine integral, respec-tively. To evaluate the central
portion of the OTF, theterm HC�u, � is further split into its three
constitu-ent sections, each section representing one of the
in-tegration expressions. Therefore, HC�u, � � I1�u, �� I2�u, � �
I3�u, �, where
These three terms are evaluated separately and thencombined to
form the OTF at the central portion ofthe spatial frequency range.
Performing the integra-tion operation on I1�u, � results in
I1�u, � �12 �
|u|�1
�|u|
exp�j4u� � ��x�dx
�12
exp�j4u� � ��x�j4u� � �� �|u|�1
�|u|
. (18)
Let �C1 � 1�2 � |u| so that �|u| � �C1 � 1�2 and|u| � 1 � ��C1 �
1�2; Eq. (18) then becomes
I1�u, � �12j
14u� � ���
exp�j4u� � ����C1 � 1�2��
� exp�j4u� � �����C1 � 1�2��, (19)
which may be rewritten as
I1�u, � �exp��j2u� � ���
4u� � ��
��exp�j4u� � ���C1� � exp��j4u� � ���C1�2j �.(20)
Euler’s identity is once again invoked on the termwithin the
curly parentheses to obtain
I1�u, � � exp��j2u� � ����sin�4u� � ���C1�4u� � �� �.(21)
By first multiplying the numerator and denominatorof Eq. (21) by
�C1, then multiplying and dividing thedenominator as well as the
argument of the sine func-tion by �, and finally reinstating the
value of �C1, theterm I1�u, � is obtained as
I1�u, � � 12 � |u|sinc�4u� � � ��12 � |u|�� exp��j2u� � ���.
(22)
Next, the term I2�u, � is calculated. This term issimilar to the
integral encountered in the analysis ofHT�u, � except for the
limits of the integral. There-fore it is possible to write
I2�u, � �12 exp�j2�u21 � 2�2sgn�u����
�|u|
|u|
exp�j �24�� x � �|u|2sgn�u��dx.(23)
Application of a change of variable in the integral ofthe above
equation allows this expression to be re-stated as
I2�u, � �14��1�2exp�j2�u21 � 2�2sgn�u����
aC�u�
bC�u�
exp�j �2 �C2sgn�u��d�C, (24)where the integration limits aC�u�
and bC�u� are givenby the relation
I1�u, � �12 �
|u|�1
�|u|
exp�j4u� � ��x�dx, 0 � |u| �12
I2�u, � �12 �
�|u|
|u|
exp�j�4|u|x � 2��x2 � u2�sgn�u��dx, 0 � |u| �12
I3�u, � �12 �
|u|
1�|u|
exp�j4u� � ��x�dx, 0 � |u| �12
. (17)
220 APPLIED OPTICS � Vol. 46, No. 2 � 10 January 2007
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aC�u� � 4�� 1�2|u|� � 1,bC�u� � 4�� 1�2|u|� � 1. (25)
As in the case of HT�u, �, Euler’s identity is appliedon the
kernel to yield
I2�u, � �14��1�2exp�j2�u21 � 2�2sgn�u�����
aC�u�
bC�u�
cos�2 �C2d�C � j sgn�u���
aC�u�
bC�u�
sin�2 �C2d�C�. (26)The integrals in the above equation are then
ex-pressed as Fresnel cosines and Fresnel sines. Theresulting
expression for I2�u, � is then
I2�u, � � �8�1�2exp�j2�u21 � 2�2sgn�u���
1
�2��C�bC�u�� � C�aC�u��� � j sgn�u�
� �S�bC�u�� � S�aC�u���. (27)
The term I3�u, � is alike in nature to I1�u, � and maybe
evaluated similarly. Performing the integration onI3�u, � results
in
I3�u, � �12 �
�u�
1��u�
exp�j4u� � ��x�dx
�12
exp�j4u� � ��x�j4u� � �� ��u�
1��u�. (28)
Let �C3 � 1�2 � |u| so that 1 � |u| � �C3 � 1�2 and|u| � ��C3 �
1�2. Equation (28) then becomes
I3�u, � �12j
14u� � ���
exp�j4u� � ����C3 � 1�2��
� exp�j4u� � �����C3 � 1�2��, (29)
which may be expressed as
I3�u, � �exp�j2u� � ���
4u� � ��
��exp�j4u� � ���C3� � exp��j4u� � ���C3�2j �.(30)
Utilizing Euler’s identity on the term within the
curlyparentheses yields
I3�u, � � exp�j2u� � ����sin�4u� � ���C3�4u� � �� �.(31)
The above equation may be expressed in terms ofa sinc function
similar to that seen in Eq. (22).Straightforward algebraic
manipulation produces
I3�u, � � 12 � |u|sinc�4u� � � ��12 � |u|�� exp�j2u� � ���.
(32)
Combining the results for I1�u, �, I2�u, �, andI3�u, � gives the
expression for HC�u, � as
HC�u, � � 12 � |u|sinc�4u� � � ��12 � |u|�� exp��j2u� � ��� �
�8�1�2
� exp�j2�u21 � 2�2sgn�u���
1
�2��C�bC�u�� � C�aC�u��� � j sgn�u�
� �S�bC�u�� � S�aC�u��� � 12 � |u|
� sinc�4u� � � ��12 � |u|�� exp�j2u� � ���, 0 � |u| �
12. (33)
Equations (16), (22), (27), and (32) together form theOTF of the
OSQ phase mask. The magnitude of thisOTF at u � 0 is obtained by
evaluating HC�u, � at thezero-frequency location. Direct inspection
of Eqs. (22)and (32) at u � 0 reveals that I1�0, � � I3�0, �� 1�2.
Similarly, inspecting Eq. (25) shows thataC�0� � bC�0� � 0, which
when incorporated into Eq.(27) indicates that I2�0, � � 0. From
these results, itis seen that
HC�0, � � 1. (34)
Therefore the normalized OTF of the odd-symmetricquadratic phase
mask along one dimension isgiven by
10 January 2007 � Vol. 46, No. 2 � APPLIED OPTICS 221
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where aT�u� and bT�u� are as shown in Eq. (14), andaC�u� and
bC�u� are as described in Eq. (25). Since theOTF is described by
different equations at differentregions along the spatial frequency
axis, it is neces-sary to verify continuity at the boundaries of
theseregions, namely, at |u| � 1�2. Equations (22) and(32) indicate
that I1�u, � � I3�u, � � 0 at |u| �1�2. It therefore suffices to
show that I2�u, � �HT�u, � at this spatial frequency value. The
onlydifference in these two equations lies in the argu-ments of the
Fresnel integrals and hence the condi-tions aC�u� � aT�u� and bC�u�
� bT�u� at |u| � 1�2 aresufficient to prove equality of I2�u, � and
HT�u, �.From Eqs. (14) and (25), it is seen that, at |u|� 1�2,
aC�12� aT�12� 4�� 1�2 2� � 12,bC�12� bT�12� 4�� 1�2 2� � 12.
(36)
The OTF is hence continuous at |u| � 1�2. A quicksanity check on
Eq. (35) may also be performed toverify three key properties of an
OTF, namely,22
1. H�0, 0� � 1;2. H��fx, �fy� � H*�fx, fy�;3. |H�fx, fy�| �
|H�0, 0�|.
The first property is validated by Eq. (34). Visualinspection of
the plot of the OTF in Fig. 2 supports thethird property of the
OTF. The Hermitian symmetryrequired by the second property is
easily verified byinspecting HC�u, � and HT�u, � in Eq. (35).
Thetriangle and the sinc functions in I1�u, � and I3�u, �are real
and have even symmetry about u � 0. Her-mitian symmetry is then
imparted by the termsexp��j2u� � ��� and exp�j2u� � ��� in I1�u, �
andI3�u, �, respectively. In I2�u, � the arguments aC�u�and bC�u�
are also real and even symmetric aboutu � 0. Hermitian symmetry is
then supplied by thesignum function in the complex exponential
term.Since I1�u, �, I2�u, �, and I3�u, � are all Hermitian
symmetric, their sum HC�u, � also exhibits the sameproperty. In
the tails, HT�u, � can similarly be shownto exhibit Hermitian
symmetry since aT�u� and bT�u�are real and even symmetric about u �
0, and thesignum function in the complex exponential
termcontributes to the required property.
Figure 2 depicts the intensity point-spread func-tion (PSF) and
MTF of the OSQ phase mask imagerfor three different misfocus
values. The MTF plots ofthe OSQ phase mask imager shown in Fig. 2
indi-cates that the height of the MTF is fairly constantwithin the
passband for different values of defocusexcept when || � �. The
passband of this imagingsystem when || � is defined as the length
alongthe spatial frequency axis bounded by the zero-crossing points
of the function aT�u� or bT�u�, depend-
H�u, � �
12 � |u|sinc�4u� � � ��12 � |u|�exp��j2u� � ��� �
�8�1�2exp�j2�u21 � 2�2sgn�u��
�1
�2��C�bC�u�� � C�aC�u��� � j sgn�u��S�bC�u�� � S�aC�u���
�12 � |u|sinc�4u� � � ��12 � |u|�exp�j2u� � ���, 0 � |u| � 12
�8�1�2exp�j2�u21 � 2�2sgn�u���
1
�2��C�bT�u�� � C�aT�u��� � j sgn�u��S�bT�u�� � S�aT�u���,
12 � |u| � 1
0, otherwise
,
(35)
Fig. 2. Intensity PSF and MTF of an OSQ phase mask system.(a),
(b), (c) Intensity PSFs for misfocus values of � 0, �15�, and�30�,
respectively. (d) Normalized 1D MTF as a function of thenormalized
spatial frequency u. The value of � is set at 30�.
222 APPLIED OPTICS � Vol. 46, No. 2 � 10 January 2007
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ing on whether the misfocus is positive or
negative,respectively. In the special case where the magnitudeof
misfocus equals the strength of the OSQ phasemask, the MTF is
significantly raised compared withits counterparts at other defocus
values, albeit with alower spatial frequency bandwidth. Figure 2
also sug-gests that increasing the magnitude of misfocus de-creases
the available spatial frequency bandwidth.However, to confirm that
the bandwidth variation ismonotonic with respect to the magnitude
of misfocusover all misfocus values, it is essential to study theAF
plots for this phase mask system.
Figure 3 depicts the magnitude of the AF for theOSQ phase mask
system. The region where imagingis possible takes on a
double-diamond shape in theAF magnitude plot, with the raised MTFs
markingthe boundaries of the operating region. Equation (35)shows
that the MTF within the operating regionother than when || � � and
at u � 0 has a station-ary value of ���8��1�2; that is, it is
independent ofmisfocus. The MTF and AF of the OSQ phase
maskdescribed herein support the previously researchedbehavior of
this system.21 The plots of magnitude ofthe AF seen in Fig. 3
demonstrate that raising themagnitude of misfocus reduces the
available spatialfrequency bandwidth. An expression for this
band-width as a function of the misfocus parameter is eval-uated
next.
Fig. 3. Plots of magnitude of the AF of OSQ phase mask
systems.An absence of nulls inside the passband of the MTF marks
theoperating region of this imager. The MTFs at the boundaries of
thisregion display a higher dynamic range than those for other
misfo-cus values within the area where imaging is possible.
Fig. 4. Effect of misfocus on the bandwidth of OSQ phase mask
systems. The left column represents a system with � � 70� and
nomisfocus � � 0�. The right column is for the same system �� �
70��, but with � �30�. (c), (d) Location where the MTF drop
occurscorresponds to the zero-crossing points of bT�u� as seen in
(a) and (b).
10 January 2007 � Vol. 46, No. 2 � APPLIED OPTICS 223
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3. Available Bandwidth of the Odd-SymmetricQuadratic Phase Mask
System
It is seen from Fig. 3 that misfocus present in thesystem has a
low-pass filtering effect on its OTF, withincreasing magnitudes of
misfocus serving to dimin-ish the available spatial frequency
bandwidth. Sincethe filtering effect is low pass in nature, an
analysisof the tails of the OTF is first performed to
obtaininformation about the roll-off point of the OTF on thespatial
frequency axis.
Examining the expression for the tails of the OTFin Eq. (35)
reveals that all the frequency-dependentterms in the OTF expression
that contribute to itsmagnitude manifest themselves in the Fresnel
inte-grals. Figures 4(a) and 4(b) demonstrate the effect ofdefocus
on the arguments aT�u� and bT�u� of theseFresnel integrals in the
tails of the OTF. When aT�u�and bT�u� are large and on opposite
sides of the hor-izontal axis, the real and imaginary parts of
theFresnel integral terms each vary about unity. Forapplications in
which the misfocus is negative, Figs.4(c) and 4(d) show that the
available bandwidth onthe MTF plots extend up to the zero-crossing
points ofthe function bT�u� seen in Figs. 4(a) and 4(b). On
theother hand, when the misfocus is a positive quantity,the
bandwidth is determined by the zero-crossingpoints of the function
aT�u�. The available spatialfrequency bandwidth of the OSQ phase
mask imageris then given by
uc ��
� � ||, || � �, � � 0. (37)
As seen in Eq. (37), the expression for the availablespatial
frequency bandwidth is valid as long as themagnitude of misfocus is
less than or equal to thestrength of the phase mask.
Figure 5 illustrates the relationship between avail-able spatial
frequency bandwidth and the zero-crossing points of the function
bT�|u|�. Threedifferent values of defocus are shown. The top
plotshowing the function bT�|u|� illustrates the move-ment of the
zero-crossing points toward u � 0 as themagnitude of � is
increased. The other two plots showthe corresponding frequency
cutoffs from the MTFand AF perspectives.
4. Imaging under the Special Case of |�| � �
The plots depicting the arguments of the Fresnel in-tegrals in
Fig. 5 also show that when || � �, bC�u�is zero. The terms, aT�u�
and bT�u� are on the sameside of the horizontal axis with bT�u� � 0
occurringat |u| � 1�2. Therefore in this special case andbeyond �||
� ��, the quantity HT�u, � does notcontribute to the spatial
frequency bandwidth of theimager. Equation (37) indicates a
normalized band-width of 1�2 when || � �; that is, the spatial
fre-quency bandwidth of the imager is one half
itsdiffraction-limited bandwidth under this condition. Ittherefore
falls upon the central portion of the OTF toprovide any available
bandwidth in this scenario.
The raising of the MTF when the magnitude ofmisfocus equals the
strength of the OSQ phase maskmay be better understood by an
examination of theanalytical expression for HC�u, �. The value of
mis-focus is taken to be a negative quantity and thereforethe
special case of � � � is considered, where� �� 1. The analysis for
positive misfocus values en-tails only minor changes as outlined
below. Undersuch a condition, it is seen from Eq. (33) that
thearguments of the sinc and complex exponential termsin the
constituent expression representing I3�u, � re-
Fig. 5. Available bandwidth of the OSQ phase mask imager andthe
relationship between the zero-crossing points of the
functionbT�|u|� and the spatial frequency bandwidth of this imager
as seenfrom the AF magnitude plot. The expression for available
spatialfrequency bandwidth given by Eq. (37) is valid as long as
the radiallines fall within the double-diamond pattern seen in the
AF plot.The MTF at the edges of the AF magnitude plot are raised
andhave a normalized bandwidth of uc � 1�2.
224 APPLIED OPTICS � Vol. 46, No. 2 � 10 January 2007
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duce to zero since � � � 0. The magnitudes of theseterms
therefore reduce to unity, leaving only the tri-angle function of
(1�2 � |u|), which is a real-valuedquantity that takes on a
magnitude of 1�2 at thezero-frequency location and zero at |u| �
1�2. Mean-while, I1�u, � is a complex-valued term (except atu � 0,
where it is real) whose magnitude is rapidlydecaying as a function
of |u|, due to the relativelylarge frequency of its sinc term.
I1�u, � thus contrib-utes little to the magnitude of the MTF except
at thezero frequency where its value equals 1�2.
When the magnitude of misfocus equals �, Eq. (25)indicates that
bC�|u|� � 0 along the spatial frequencyaxis for 0 � |u| � 1�2, as
seen in Fig. 5. The real andimaginary parts of the contribution of
the Fresnelintegrals in I2�u, � therefore vary about 1�2,
result-ing in a stationary value of 1�2���8��1�2 for this term.It
must be noted that this magnitude is one half thestationary height
of the MTF seen inside the pass-band when || �. For large values of
�, this mag-nitude of I2�u, � is much smaller than 1�2. Theprimary
contribution to the elevated dynamic rangeof the OTF thus comes
from I3�u, � and the shape ofthe MTF is nearly that of a triangle.
In applicationswhere the misfocus is a nonnegative quantity,
theroles of I1�u, � and I3�u, � are reversed, as are thoseof
aC�|u|� and bC�|u|� in I2�u, �.
Figure 6 illustrates imaging under the special casefor || � �.
This particular example is shown for
� ��. The magnitudes of the three constituentterms of HC�u, �
when the defocus magnitude equalsthe strength of the OSQ phase mask
are shown. Itmust be noted that the sum of the plotted
quantitiesdoes not represent the magnitude of HC�u, �, as|I1�u, �|
� |I2�u, �| � |I3�u, �|is generally greaterthan |I1�u, � � I2�u, �
� I3�u, �| except when thephases of these three constituent terms
are zero. Fig-ure 6 also shows the raised MTF under this
specialcondition, and the corresponding radial line for theworking
value of the misfocus on the AF magnitudeplot. It may be noted that
this radial line runs throughthe middle of the dark band at the
boundaries of theoperating region of the imager. In applications
wherethe misfocus is positive, the special case of � � wouldbe
represented by a radial line through the middle ofthe other dark
band on the AF magnitude plot with aslope of identical magnitude
but opposite sign.
5. Conclusions
An analytical approach to evaluating the frequencyresponse of
the OSQ phase mask provides a mathe-matical formulation of its OTF
and paves the way forthe design of wavefront-coding imagers that
could beuseful in various applications. Obtaining a mathe-matical
expression for the OTF of an OSQ phasemask imager leads to the
determination of the avail-able spatial frequency bandwidth of this
system as afunction of its working value of misfocus. Such
ananalysis also identified the special imaging conditionthat
yielded an enhanced dynamic range. In futurework, imaging
configurations will be designed thatexploit the enhanced dynamic
range of this specialcondition and obtain improved noise
characteristicsin applications such as form factor enhancement
andaberration correction.
The authors gratefully acknowledge the support ofthe Defense
Advanced Research Projects Agency(DARPA) through grant
N00014-05-1-0841 with theOffice of Naval Research.
Fig. 6. Imaging with an OSQ phase mask system when || � �.The
case of � �� is depicted. (a) Magnitudes of the
individualcomponents of HC(u, �). The triangular shape of I3�u, �
causes anelevation in the MTF, resulting in an increased dynamic
rangewhen � ��. The horizontal dotted line indicates the
stationaryvalue of the MTF for imaging conditions of || � and has
aheight of ���8��1�2. The stationary magnitude of I2�u, � when|| �
� is one half the height of this dotted line. (b) MTF.(c)
Corresponding radial line for � �� on the AF magnitude plot.The
value of � was taken as 30�.
10 January 2007 � Vol. 46, No. 2 � APPLIED OPTICS 225
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