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 msomayaj@engr.smu.edu. M. P. Christensen’s e-mail address is mpc@engr.smu.edu.

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

��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

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

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

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

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

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

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

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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