In relation to far-field diffraction phenomena, thephysical interpretation of the Fraunhofer diffractionphenomena is a common practice in theoretical re-search, experimental optical setups, and educationalissues to use optical components or screen aperturesof rectangular or circular shapes because their math-ematical treatment is quite simple. Classical andmodern optics textbooks1,2 explain Fraunhofer dif-fraction phenomena based on the rectangular or cir-cular aperture shapes to avoid unnecessarymathematical details that may obscure its physicalinterpretation. These basic shapes are also used inother optical phenomena such as the description ofthe optical transfer function of an optical system thatuses coherent or incoherent light sources.3 Far-fielddiffraction patterns of arbitrary aperture shapes in-cluding apertures with specific geometrics are not
easy to solve in mathematical closed form, and nu-merical algorithms such as the Fast Fourier Trans-form (FFT) are convenient and handy tools.
The use of aperture shapes related to circular ge-ometry has its roots in astronomical instrumentationsuch as the early heliometers used to measure thediameters of the larger planets. A systematic treat-ment4 of the far-field diffraction patterns produced bycircular sector apertures or annular versions of them,as well as a brief description of historically relateddiffraction studies, has provided analytical solutionsin terms of a series of Bessel functions. In Ref. 4experimental photographs are shown for circular sec-tors with angular sizes of 60°, 90°, 120°, 180°, 240°,270°, and 330° and for annular sectors with angularsizes of 60° and 90° and an inner-to-outer radius ra-tio equal to 0.395; also, numerical computation ofdiffraction patterns from their analytical solutionsare given for circular sector apertures with angularsizes of 60° and 180° (semicircular aperture) and foran annular 60° sector with the same radii ratio asbefore. In addition, the same study points out the factthat two symmetrical circular sectors produce inter-ference fringes in a fashion similar to that of a Youngwavefront division interferometer with extended op-posing openings. An alternative simplified treatment
G. Urcid ([email protected]) and A. Padilla are with the OpticsDepartment, Instituto Nacional de Astrofísica, Optica y Elec-troníca, Tonantzitla, Puebla, 72000, Mexico.
based on an incomplete cylindrical function of firstorder in Poisson form together with trigonometricand exponential functions gives a symbolically com-pact closed solution for the diffraction pattern’scomplex amplitude produced by a circular sector,5although no specific cases are discussed. A theoreticaland experimental study has also been made of theimaging properties of multiple circular sector aper-tures with different angular size5 for transparent andopaque sectors.6 An aperture with a semicircularshape7 makes use of an incomplete Struve function offirst order and presents a few limiting cases and as-ymptotic approximations for specific angular fre-quency values and geometrical aperture sizes. Later,an extensive study of the far-field diffraction pro-duced by sector stars used as test targets in opticalfrequency analysis was based on the cosine Fourierseries expansion of the transmission function andgave a detailed description for stars that comprised 1(semicircle), 2, 3, 4, 5, 6, and 72 sectors.8 Recently itwas suggested that the diffraction pattern of a semi-circular opening can be determined as a byproductobtained from an integral expression of the Zernikeradial polynomials used to compute the Fraunhoferdiffraction pattern of a Hilbert mask.9
Our purpose in this paper is to present a coherenttreatment for finding analytical solutions given as aseries of Bessel functions for the intensity light dis-tributions of the far-field diffraction produced bysector-type apertures related to the circular geometryincluding their numerical approximate computationas well as experimental confirmation. The class ofcircular sector apertures that we work with includesthe single annular sector, the double symmetricalannular sector, and the annular sector wheel formedby interleaved transparent sectors of equal angularsize equivalent to a star target. Our study covers allthe specific cases mentioned in the publications citedabove.4–9 It may also be considered an explicit exten-sion of the treatment initiated by Mahan et al.4 forsectors with constant amplitude and phase as well asan alternative mathematical approach to finding theFraunhofer diffraction pattern for a wheel of sectorsas presented in Ref. 8. The results presented here willserve as the basis for further research on the far-fielddiffraction effects of circular-type apertures, which,by themselves, might be considered for further inves-tigation in relation to the performance of optical sys-tems used as unconventional pupil functions.
Our exposition is organized as follows: in Subsec-tion 2.A a description of the class of circular-typeapertures is given to provide a unified treatment forthe analytical expressions developed in Section 3. InSubsection 2.B some mathematical background isprovided that supports the algebraic manipulationsto be developed in the following sections of the manu-script. Different forms of expression for the analyticalsolutions are given in Section 3: The first subsectionsettles the framework for the far-field diffractionsolutions based on the polar representation of thetwo-dimensional (2D) Fourier transform, and twoadditional subsections provide general formulas that
are basically a series whose general term is the prod-uct of an angular and a radial function coefficient.Section 4 develops in detail the symbolic expressionsthat use a series of Bessel functions of the first kind ofincreasing integral orders for the amplitude of diffrac-tion patterns produced by the aperture functions givenin Section 2. In Section 5 we make some pertinentremarks that should be helpful for the numerical ap-proximate evaluation of the analytic solutions estab-lished in Section 4, and several examples illustrate thecomputed diffraction patterns. The FFT algorithm isused in that section to check the validity of the math-ematical expressions presented in Section 4 as well toexplain their approximate computation. In Section 6we discuss some interesting properties derived fromthe mathematical formulas in previous sections anddescribe briefly a simple optical–digital system setupfor experimentally observing the Fraunhofer diffrac-tion patterns produced by this class of circular-relatedapertures. Finally, in Section 7 we close the paper withour conclusions on the research presented here anddiscussion of some future research possibilities relatedto far-field diffraction mathematical calculations.
2. Definitions and Mathematical Preliminaries
A. Circular Sectors and Related Apertures
Polar coordinates are used because the apertureshapes considered here are subsets of a circle; thusthe transmission or aperture function that definesany one of these circular-type openings is given by
f�r, ����1 r � �R1, R2�, �� �t�1� ��2t�1, �2t�
0 otherwise ,
(1)
where r and � are spatial polar coordinates in the planeof the aperture, R1 and R2 denote, respectively, theinner and outer radii of two concentric circles centeredat the origin, and ��1 � · · · � �2�� is a finite set ofangle values that determine the number of sectors inthe aperture; notice that f�r, �� is real and even withrespect to �. The initial angle �1 ����, 0� is followed counterclockwise by the angles�t � �0, 2�� for t � 2, . . . , 2�, where �2� � 2� isthe final angle. Table 1 summarizes the types of aper-ture related to the circular geometry that is used in theremaining sections. The minus goes with the first sub-script, and the plus is for the second subscript.
If R1 � 0, complete sectors are obtained; otherwise,R1 0, annular sectors result and W � R2 � R1 0 gives the corresponding width, as shown inFig. 1. If � � 2, then � � �2 � �1 � �4 � �3, and forR1 � 0 we obtain a double symmetrical sector ofangular size equal to �; otherwise we have a doublesymmetrical annular sector of width W (see Fig. 2). Fora �-sector wheel the angular size is taken to be thesame for all sectors; thus � � �t�1 � �t for t� 1, . . . , 2�, where �2��1 � 2� � �1. Note that eachaperture has the bisecting line of its first sector lyingon the � � 0 direction and the value of � must be in
the admissible range of angular sizes for each type.Therefore the first sector is chosen to have x-axissymmetry. In the trivial case of a two-sector wheel,� � ��2. Figure 3 displays an example of a wheelformed from four interleaved complete or annular sec-tors, each with � � ��4 � 45°. The circular apertureresults as a limiting case of each aperture type; i.e., ifangular size � � 2�, �, 0, when � � 1, 2, or � → �.
The apertures listed in Table 1 share a common setof geometrical properties. Based on the value of �, itis not difficult to see that the aperture function given
in Eq. (1) has �-fold symmetry in the sense that ro-tation through an angle of 2��� rad about the pole�r, �� � �0, 0� leaves the aperture in the same posi-tion; after � rotations the first sector returns to itsinitial place. Hence
f�r, ��� f r, � 2�� . (2)
Fig. 1. Single complete sector of radius R and single annularsector of radii R1 � R2: Angular size, � � (0, 2�], �1,2 � ���2, initialand final angles. The mirror line is the x axis.
Fig. 2. Double symmetrical complete sector of radius R and dou-ble symmetrical annular sector of radii R1 � R2: Angular size, � �(0, �]; �1 � ��2, initial angle; �4 � � �/2 final angle. Themirror lines are the x and y axes.
A mirror line M� is a line across which the aperturehas reflection symmetry, and there are � of them. Fora single-sector �� � 1� type of aperture the x axisspecified in the polar plane by the directions ��0, �1�� �0, �� is the only mirror line, M0. For a symmetri-cal double sector �� � 2�, the x and y axes with re-spective directions given by ��0, �2� � �0, �� and��1, �3� � ���2, 3��2� are the mirror lines denotedM0 and M1. For a �-sector wheel, each mirror line M�
is specified by the two directions ���, �� � ��, where�� � ���� for � � 0, . . . , � � 1; we identify M� withthe direction ��. Thus it is not difficult to verify, usingEq. (1), that
f�r, ��� f�r, 2M�� ��. (3)
A border line B� is a line that divides on opposing
sides, relative to the pole, a transparent from anopaque sector. For a single sector the border semili-nes are B0,1 � ��2. For a symmetrical double sectorthe border lines are given by B0, 1 � ���2, � ���2�, and for a �-sector wheel there are � borderlines described by the relation B� � �� � ��2�� ��2��2� � 1� for � � 0, . . . , � � 1.
B. Mathematical Background
The normalized Jinc function,2 defined by Jinc�x�� 2J1�x��x for x 0, where J1 is a Bessel function ofthe first kind and first order, has the property thatJinc�x� � 1.0 when x → 0. The function given byJ1�x��x for x 0 is also known as the Jinc function byother authors.10–12 In this case, Jinc�x� � 0.5 whenx → 0; we obtained this limit value at the origin byapplying L’Hopital’s rule and Bessel recurrence iden-tity Jn�1�x� � Jn�1�x� � 2J�n�x� and by knowing thatJ0�0� � 1 and J2�0� � 0. When J1�x��x is used, thesquare of the function 2 Jinc�x� describes the Airyintensity pattern produced by diffraction at a circularaperture. To simplify several mathematical resultsgiven in the following sections, we shall use the nota-tion Jinc�x� to represent specifically the quotient givenby J1�2�x��2�x. As has already been found,4,5,7,8 thefar-field intensity patterns produced by diffraction atapertures with circular geometry can be described byBessel functions of the first kind of integral order, byincomplete cylindrical functions combined with ele-mentary functions; or by Fourier series expansionswith Bessel functions as coefficients.
The material needed to develop our treatment isrelated to Bessel functions. A theoretically useful wayof defining the Bessel functions of the first kind ofintegral order is to use the generating function intro-duced by Schlomilch,13 (Chap. 2, p. 14) given by
exp�x�t� t�1��2�� �n���
�
tnJn�x�. (4)
Changing the variable t � i exp�i�� and substitutingthe result into Eq. (4) results in
exp�ix cos ��� �n���
�
in exp�in�� Jn�x�. (5)
Using the change of index m � �n, the fact that i�1
� �i, and the identity J�n�x� � ��1�nJn�x�, we canwrite the last sum in Eq. (5) as
�n���
�
inexp�in��Jn�x��J0�x�� �n�1
�
in[exp�in��� exp��in��]Jn(x), (6)
from which the conjugate Jacobi13 identity is ob-tained, i.e.,
exp��ix cos ���J0(x)� 2�n�1
�
(�i)nJn(x)cos n�
��n�0
�
�n(�i)nJn(x)cos n�, (7)
Fig. 3. Complete four-sector wheel of radius R and annular four-sector wheel of radii R1 � R2: Angular size, � � ��4 � 45°; �1 ���8, initial angle; �8 � 13��8, final angle. The mirror lines arethe x and y axes and the two diagonal dashed–dotted lines.
where Neumann’s factor13 �n is used to abbreviatefurther mathematical expressions; recall that �n �1 for n � 0 and �n � 2 for n 0. In Eq. (7) theimaginary powers ��i�n for n � 1, 2, . . . , are re-duced to the set of values � 1, i� when n is an evenor an odd positive integer, respectively. Hence theterms of the series can be collected to give the realand imaginary parts denoted, respectively, � and �.Hence
� �exp��ix cos ��� � �n�0
�
��1�n� 2n J2n�x�cos 2n�,
(8)
� �exp��ix cos ��� � �n�1
�
��1�n� 2n�1 J2n�1�x��cos �2n� 1��. (9)
If the argument of a function is to be repeated severaltimes in the discourse below, we substitute for it acentered dot �·�, and if a lengthy mathematical ex-pression is also repeated we shall use a bullet (●) inits place to save space. It will be understood thatthese symbolic markers correspond to the argument’sexpression displayed for the first time in a previousformula.
3. Far-Field Diffraction Solutions
A. Fourier Transform in Polar Coordinates
In classic scalar diffraction theory, the Fraunhoferapproximation, or far-field condition, imposed onKirchhoff’s integral reduces the integral to a 2D scaledFourier transform applied to the aperture function.The scaling constant lumps together a couple of phasefactors as well as the amplitude of the incident wave,its wavelength, and the observation distance. Inas-much as phase factors do not contribute to the inten-sity pattern and the remaining real parameters onlyscale the transform, we omit this constant in furtherexpressions. At the plane of the aperture, r and � arepolar coordinates used to describe the spatial extent ofan opening with circular geometry; similarly, at theplane of observation, � and � represent frequency po-lar coordinates for the corresponding diffraction pat-tern. The 2D Fourier transform2,10,14 of aperturefunction f�r, �� is defined by
��f�r, ����F��, ��
��0
��0
2�
f�r, ��exp��i2�r�
� cos������rdr d�. (10)
For the aperture types listed in Table 1, the allowableangular size defined by � and radii R1 and R2 are thethree parameters that control each opening spatialextent. The associated aperture functions f�r, �� arenot circularly symmetric because they are definedpiecewise with respect to �. Their Fourier transforms
must be computed by use of the additivity propertyfor nested integrals. After plugging Eq. (1) intoEq. (10) we have
F��, ����R1
R2��t�1
� ��2t�1
�2t
exp��i2�r�cos��
���� d� rdr. (11)
There are several mathematical approaches that canbe applied to manipulate Eq. (11), and it would bedifficult to decide which one is better than the others.We base our discourse on the application of the Jacobiidentity given by Eq. (7) to the complex exponential inthe inner integral of Eq. (11). In comparison withother mathematical techniques, it seems to be intui-tive and straightforward, and, for the apertures con-sidered here, it has the advantage of providinggeneral solutions by decomposing the Fourier trans-form into separable factors in each frequency polarcoordinate. Therefore we develop the technique fur-ther behind this approach. Using the auxiliary vari-able � � 2�� and integrating term by term withrespect to �, we find that
F��, ����R1
R2�t�1
� ��2t�1
�2t
��n�0
�
�n��i�nJn��r�cos n��
����d�rdr,
��n�0
�
�n��i�n�R1
R2
rJn��r�dr
� ��t�1
� ��2t�1
�2t
cos n�����d��. (12)
Equation (12) makes explicit the fact that the Fouriertransform is an infinite series whose general nthterm can be expressed as the product of a radialfunction coefficient denoted �n��� and an angularfunction coefficient represented by �n���. For eachn � 0, these coefficients are given by
�n�����R1
R2
rJn�2��r�dr, (13)
�n���� �t�1
� ��2t�1
�2t
cos n�����d�. (14)
Therefore, using the coefficients in Eqs. (13) and (14),we can write the 2D-Fourier transform of Eq. (12)more compactly as follows:
from which the real and imaginary parts are given by
� �F��, ��� � �n�0
�
��1�n�2n�2n����2n���, (16)
� �F��, ��� � �n�1
�
��1�n�2n�1�2n�1���� �2n�1���. (17)
Equations (15)–(17) are the most general formulas inthe present context from which specific cases ofcircular-related apertures will be treated. Thus,squaring the real and imaginary parts and addingthem gives the far-field intensity distribution pro-duced by diffraction. The procedure based on the ap-plication of the Jacobi identity to arrive at Eq. (15)has been used or suggested partially in somewhatdifferent but equivalent forms by other research-ers.2,4,15
Two alternative expressions for the radial functioncoefficient and explicit computation of the angularfunction coefficient follow next. As �2t � �2t�1 � �for any value of t, it is clear from Eq. (14) that angularfunction coefficient �0 equals ��; for n 0, afterintegration, we have used the fact that sin�x� is anodd function:
�n����1
�2���2 �2�R1�
2�R2�
rJn�r�dr
�1
2��n�rect�r� r0�����, (18)
�n����1n �
t�1
�
�sin n����2t�1�� sin n(���2t)�,
n 0. (19)
The first expression, for �n���, is a simplified versionof the finite integral of Eq. (13), and the second sym-bolic form is a scaled version of the Hankel transformof order n of a rectangle function centered at r0� �R1 � R2��2.2 It is important to remark that,whereas the angular integral is readily solved, theradial integral is more involved and less known incommon practice, except for n � 0, which is the firstterm of F��, ��; i.e., �0��� can be transformed into adifference of Jinc� � functions by means of the Hankeltransform of order 0, better known as the Fourier–Bessel transform.10,16 Thus
�n(�i)n�n(�)�n(�)|n�o����R22 Jinc�R2��
�R12 Jinc�R1���. (20)
Observe that Eq. (20) gives the amplitude of the dif-fraction pattern produced by a ring of width W scaledby the factor ��. As is shown in Section 4 below, the
radial function coefficient �n��� for n 0 can be ex-pressed in closed form. However, for the exact solu-tions presented in Subsection 3.B, the integral formof the radial function coefficient given in Eq. (18) willbe used to make explicit the presence of Bessel func-tions of higher order as well as the nature of thecomputation required for determining the diffractionpattern. The radial integrals are the same for eachaperture function considered here hence the changesseen in the diffraction patterns are produced mainlyby the angular coefficients.
B. Single and Double Sectors
For a single annular sector of angular size �, theangular function coefficient given in Eq. (14) can besimplified as follows. Here � � 1 and, from the fourthcolumn in Table 1,�1 ��0.5� and�2 � 0.5�. There-fore
�n����1n �sin n����1�� sin n����2��,
sin n����1�� sin n��� 0.5��,
sin n����2�� sin n��� 0.5��.
Applying the trigonometric identity sin�� � �� �sin�� � �� � 2 cos � sin �, with � � n� and �� 0.5n�, gives
�n����2n sin
n�2 cos n�. (21)
Splitting the first term and substituting the expres-sions for the radial and angular integrals yield, forthe amplitude of the far-field diffraction pattern pro-duced by a single annular sector (ss) for � � �0, 2��,
Fss��, �; �����R22 Jinc�R2���R1
2 Jinc�R1���
�1
����2 �n�1
� ��i�n
n sinn�2 cos n�
� �2�R1�
2�R2�
rJn�r�dr, (22)
where we have made explicit the dependence on an-gular size � of the sector. From Eq. (22) the completering is obtained for � � 2� because sin�n��2�� sin n� � 0 and the second term vanishes; there-fore, as expected,
Fss��, �; 2��� 2� �R22 Jinc�R2���R1
2 Jinc�R1���.(23)
In a similar way, for doubly symmetrical annularsectors the angular function coefficient given inEq. (14) can be reduced to a simpler expression. Here� � 2, and from the fourth column of Table 1, the
angle values are �1,2 � �0.5� and �3,4 � � � 0.5�.Therefore
�n����1n �
t�1
2
�sin n����2t�1�� sin n����2t��,
sin n����1, 2�� sin n�� 0.5��,sin n����3, 4�� sin n���� 0.5��.
Applying the same trigonometric identity as beforewith � � n� or � � n�� � �� and � � 0.5n�, andrecalling that cos n� � ��1�n, we obtain
�n����2n sin
n�2 �cos n�� cos n������ ,
�2n sin
n�2 cos n� �1� ��1�n�. (24)
Splitting the first term and substituting the expres-sions for the radial and angular integrals yield theamplitude of the far-field diffraction pattern pro-duced by a double symmetrical annular sector (ds) for� � �0, ��:
Fds��, �; ��� 2��R22 Jinc�R2���R1
2 Jinc�R1���
�1
����2 �n�1
� ��i�n�1� ��1�n�n sin
n�2
� cos n� �2�R1�
2�R2�
rJn�r�dr. (25)
Again, we have made explicit the dependence on an-gular size � of both sectors. In this case the completering is obtained for � � �; then sin�n��2�� sin n���2�, which is zero for n even. Factor �1 ���1�n� is zero for n odd. Therefore the productsin n���2��1 � ��1�n� in the second term of theFourier transform always vanishes and, because �� 2, the result is the same as in Eq. (23), i.e.,Fds��, �; �� � Fss��, �; 2��.
C. Sector Wheels
For a wheel aperture we consider the specific case inwhich a uniform division into 2� sectors is achievedover a complete turn. However, only half this numberwill be interleayed openings, each with the same an-gular size equal to � � ���, where � � 2. Again, thebase angle is taken �1 � �0.5� and, as listed in thethird row of Table 1, the remaining points of subdi-vision of the circle are described by �t � �t � 3�2��for t � 2, . . . , 2�. In addition, the initial and finalangles that describe the annular sectors are given by��2t�1, �2t�, where �2t � �2t�1 � � for t � 1, . . ., �.
The angular function coefficient given in Eq. (14) issimplified, considering that
�n����1n �
t�1
�
�sin n����2t�1�� sin n����2t��,
sin n����2t�1, 2t�� sin n���� 2��t� 1� 0.5���.
On subtraction of the sin�·� functions and taking �� n�� � 2��t � 1�� and � � n���2� in the previoustrigonometric identity, the angular coefficient as afunction of � is given by
�n����2n sinn�
2�Sn����, (26)
where
Sn����� �
t�1
�
cos n��� 2�� �t� 1��
��� cos n� ⇔ n � 0 �mod ��0 ⇔ n � 0 �mod �� (27)
has been introduced to economize notation in furtherexpressions related to this type of aperture. We de-termine the right-hand side of Eq. (27) by taking thereal part of the sum on the left-hand side when it isexpressed as a sum of complex exponentials and eval-uating it for integer multiples of �; if � is not a divisorof n, then the sum of the corresponding geometricprogression of exponentials is zero. Substitution ofthe radial and angular integrals in to Eq. (15) andseparating the first term of the Fourier transformyield the amplitude of the far-field diffraction patternproduced by a wheel with � equal angular size annu-lar sectors (ws):
Fws��, �; �����R22 Jinc�R2���R1
2 Jinc�R1���
�1
����2 �n�1
� ��i�n
n sin n�2�Sn
����
� �2�R1�
2�R2�
rJn�r�dr. (28)
Note that the limit of this expression when � → �reduces to the first term of Fws��, �� becausen��2� → 0, although Sn
���� � � cos n� → �. There-fore, increasing the number of annular sectors willeventually approach the pattern obtained with a ringaperture. However, even for � �� 0, the wheel is nota full ring or circular aperture, and its amplitudeapproaches half the value of the limit cases for asingle or a double annular sector, i.e., Fws��, �; ��� 0.5Fds ��, �; �� � 0.5Fss��, �; 2��. An alternativeexpression for Eq. (28) that involves the sinc�·� func-tion is
To complete the closed-form solutions of the Fouriertransforms Fss�·�, FdS�·�, and Fws�·� given, respec-tively, in Eqs. (22), (25), and (28) or (29), it is neces-sary to evaluate the radial function coefficient. In thissection we describe the way of finding the radial in-tegral in terms of a series of Bessel functions.
The common approach to finding the radial func-tion coefficient is based in the following integral re-lationship, which expresses a whole family of Besselintegrals as an infinite series of Bessel functions ofthe first kind and increasing orders. It is valid for��� � �� � 1, where � and are unrestricted orderand power values17:
�0
r
t�J�(t)dt� z���½(���� 1)���½(���� 1)� �
k�0
�
(�� 2k
� 1)��½(���� 1)� k���½(���� 3)� k�
J��2k�1(r).
(30)
In the present case � � 1 and � � n 0 is an integerindex; thus the condition ��� � �� � n � 1 �1 isclearly satisfied, and substitution of these values intoEq. (30) gives
�0
r
tJn�t�dt� r��1� �n�2��
��n�2� �k�0
�
�n� 2k
� 1����n�2�� k�
���n�2�� k� 2�Jn�2k�1�r�.
(31)
We apply the gamma factorization property ��a� 1� � a��a� once to outside factor ��1 � �n�2�� andtwice to inner factor ���n�2� � k � 2�, and, afteralgebraic simplification, Eq. (31) is reduced to
�0
r
tJn�t�dt� 2rn �k�0
� �n� 2k� 1��n� 2k� 2��n� 2k�
� Jn�2k�1�r�, (32)
� 2rn �k�0
� �n, k
�n, k2 � 1
J��(n, k)�r�, (33)
where �n, k � n � 2k � 1 simplifies forthcoming ex-pressions and the functional notation ��n, k� is usedinstead of �n, k to denote the variable-order index of
the Bessel function of the first kind in the given se-ries; note that n � 2k � �n, k and n � 2k � 2 � �n, k
� 1. The fractional coefficient in front of Bessel func-tion J��n, k� in Eq. (33) never vanishes because, for n 0 and k � 0, �n, k
2 � 1 0. From Eq. (33) it turns outthat the Bessel definite integral of the radial functioncoefficient in Eq. (18) is evaluated as follows:
�2�R1�
2�R2�
tJn�t�dt� 4�n� �k�0
� �n, k
�n, k2 � 1 �R2J�(n, k)
� �2�R2���R1J�(n, k)�2�R1���.(34)
Equation (34) can be plugged into Eqs. (22), (25), and(28) of Section 3 to yield another set of complete an-alytical solutions for the far-field amplitude distribu-tions produced by diffraction at the given aperture.As an example, we write in full the solution for eachaperture type for complete sectors, i.e., when R1� 0 and R2 � R.
A. Single Sector
For � � �0, 2��,
Fss��, �; ����R2 Jinc�R���4R�� �
n�1
�
��i�n
� sinn�2 cos n��
k�0
� �n, k
�n, k2 � 1
� J��n, k��2�R��. (35)
B. Double Symmetrical Sector
For � � �0, ��,
FdS(�, �; �)� 2�R2 Jinc(R�)�4R�� �
n�1
�
(�i)n[1� (�1)n]
� sinn�2 cos n��
k�0
� �n, k
�n, k2 � 1
J�(n, k)(2�R�).
(36)
C. Sector Wheel
For � � 2,
Fws��, �; ����R2 Jinc�R���4R�� �
n�1
�
��i�n
� sin n�2� Sn
���� �k�0
� �n, k
�n, k2 � 1
� J��n, k��2�R��. (37)
5. Approximate Diffraction Patterns
A. Computational Aspects
In Sections 3 and 4 we described symbolic expres-sions that give the complex amplitudes for the far-field diffraction produced by the circular-relatedapertures considered here. The analytic solution
based on the radial function coefficient given in Eq.(34) as a series of Bessel functions of the first kind,and increasing integral orders, when plugged intoEq. (22), (25), or (28), has the advantage that its realand imaginary parts can be further simplified by useof Cauchy’s rule for series multiplication. Anotherreason that supports our specific choice of mathemat-ical technique to solve this kind of diffraction problemis that fast numerical computation of Bessel func-tions of integral orders for small and large real argu-ments has received considerable attention in theliterature as well as support in most mathematicalsoftware packages.18–20
Simplification of the real and imaginary parts ofamplitude Eqs. (35)–(37) is based on Eqs. (16) and(17) and a specialized rule for series multiplication.For example, the series multiplication in the real andimaginary parts of the far-field diffraction amplitudeproduced by a single complete sector is emphasizedbelow:
��Fss�� �·�� �·� �n�1
�
��i�n sin n�
� cos 2n �·��k�0
� �2n, kJ�(n, k)�·��2n, k
2 � 1, (38)
��Fss�� �·�� �·� �n�1
�
��1�n sin�2n� 1��
2 cos�2n
� 1� �·��k�0
� �2n�1, kJ�(2n�1, k)�·��2n�1, k
2 � 1. (39)
Expressions with the same structure are easily de-duced for the other circular-type apertures. In each ofthese formulas the index of the general term of thesecond series is a compound index in the sense that itdepends on the index of the general term of the firstseries. Hence we use the following rule for seriesmultiplication:
�n�1
�
an�k�0
�
bn, k� �n�1
�
cn� �n�1
�
�k�1
n
akbk, n�k. (40)
For the real part of Fss, the general term of the firstseries is an � ��1�n sin n� cos 2n�·�; therefore ak
� ��1�k sin k� cos 2k�·�. The general term for thesecond series is bn, k � �2n, kJ��2n, k��·����2n, k
2 � 1�, which,after substitution and algebraic simplification, givesa result that is independent of k, i.e.,
bk, n� k��2k, n� k
�2k, n� k2 � 1
J��2k, n� k��·�
�2n� 1
4n�n� 1�J2n�1�·�. (41)
For the imaginary part of FSS, ak is the same asfor the real part; however, bn, k � �2n�1, kJ��2n�1, k��·����2n�1, k
2 � 1� and, again, the simplified expression
does not depend on k, as shown below:
bk, n� k��2k�1, n� k
�2k�1, n� k2 � 1
J�(2k�1, n� k)�·��2n
4n2� 1J2n�·�.
(42)
The computed value of bn, k, which turns to be inde-pendent of k, can be denoted bn*, and thereforeEq. (40) can be written as
�n�1
�
an�k�0
�
bn, k� �n�1
�
cn� �n�1
�
bn* �k�1
n
ak, (43)
making clear that the product of the two series thatappear in both the real and the imaginary parts ofEqs. (35)–(37) are finally reduced to a single series ofBessel functions whose coefficients are finite sumsof angular coefficients. With the help of Eq. (43) aswell as Eqs. (41) and (42), we write in full the solu-tions for the case of annular sectors:
The inner finite sums in Eqs. (48) and (49) can befurther simplified based on the behavior of the prod-ucts between the sin�·� and S�·� functions. In relationto Eq. (48); observe that if � is a divisor of 2k, then2k � q� for some q � Z�; hence Sq�
� � � cos q��·�. Forq even, i.e., q � 2p with p � Z�, sin(·)S(·) vanishesbecause sin�k���� � sin�p�� � 0. However, if q is odd,say, q � 2p � 1, then sin�k���� � sin�p� � ��2� ���1�p�1; thus sin�·� S�·� has a value given by��1�p�1S�2p�1��
� ���. In other words, the sum has non-zero terms if and only if 2k � �2p � 1��, and thiscondition implies that � must necessarily be an evennumber. From the previous condition, k � ��2p� 1��2 is always a positive integer; in addition, for agiven n in the outer series, the end value of p isobtained from 2n � ��2p � 1�. Therefore, for � aneven number, we obtain
�k�1
n
��1�k sink�� S2k
� ����� �p�1
Q
�-1)1�2�(2p�1)�p�1
� cos�2p� 1���, (50)
where Q � <�2n � ���2�=; if � is an odd number, thissum is zero. In a similar way, let � in Eq. (49) be adivisor of 2k � 1, then 2k � 1 � q� for some q � Z�.The argument k��� � ��2� of the sin�·� functionreduces to q��2. Again, if q � 2p, then sin�p��� 0°; and if q � 2p � 1, then sin�p� � ��2� ���1�p�1. The condition for nonzero terms is now 2k� 1 � �2p � 1��, from which � must be odd, conse-quently, k � ���2p � 1� � 1��2 is a positive integer.For a given n in the outer series, the final index valuefor p results from 2n � 1 � �2p � 1��. Hence for � anodd number we get
�k�1
n
��1�k sink��
��
2�S2k�1� ���
�� �p�1
Q*
�-1)½��(2p�1)�1��p�1 cos�2p� 1���, (51)
where Q* � <�2n � 1 � ���2�=. If � is an evennumber it cannot be the divisor of any odd number;thus S2k�1�
� ��� � 0, and the sum is zero.
B. Examples of Single and Double Sectors
A numerical approximate computation of the Fraun-hofer diffraction patterns produced by single circular
Fig. 4. Single complete sectors for angular sizes �; � 60°, 120°,180°, 240°, 300° and their respective far-field diffraction intensitypatterns.
or annular sectors can be made by use of eitherEq. (35) or Eqs. (44) and (45) with appropriate valuesfor the scaling constants, R1 and R2. The specific val-
ues used in all the examples shown below are 2� asan amplitude scale factor, R1 � 0.214 and R2� 0.535 for annular sectors, and R1 � 0 for completesectors. Because we use an angular step size of ��� 15° � ��12, 24 single complete or annular sectorpatterns and 12 double symmetrical complete or an-nular sector patterns were calculated. The Cartesianspatial frequency variables u and v were taken overthe domain ��2�, 2�� � ��2�, 2�� with a step sizeof ��32 to generate a grid of 129 � 129 points. Therelations given by �2 � u2 � v2 and � � tan�1�v�u�such that � � �0, 2�� were used to map the u and vfrequencies to the polar plane. Owing to the conju-gate symmetry of the Fourier transform, it is neces-sary to perform calculations only over the domain�0, 2�� � �0, 2��, and, to approximate the series bya finite sum; n � 8 terms were sufficient to produceacceptable results.
Figure 4 displays 5 of 24 single complete sector open-ings, together with their corresponding far-field dif-fraction patterns. The intensity patterns Iss��, �; ��are determined from Fss��, �; ��F*ss��, �; �� or from�2�Fss��, �;��� � �2�Fss��, �; ��� and were computedto include only a few small diffraction orders becauseour major concern here is to emphasize the fine struc-ture of the pattern near the central spot. Image pat-terns are displayed with the bisecting line of theirfirst sector lying on the x axis or the � � 0 direction,in agreement with the geometries depicted in Figs.1–3. In addition, for purposes of visualization, eachpattern has been enhanced with a logarithmic con-trast law given by log10�� � Iss��, �; ���, where � is asuitable constant taken from the interval �10�3, 1�.Similarly, Fig. 5 shows a specific annular version forthe same five openings where the radii ratio is
Fig. 5. Single annular sectors for angular sizes � � 60°, 120°,180°, 240°, 300° and their respective far-field diffraction intensitypatterns.
Fig. 6. Fraunhofer diffraction patterns for the semicircle and thesemiannulus and their contour map representations normalizedintensity from inner to outer contours, 1, 0.8, 0.6, 0.4, and 0.2.
R1�R2 � 2�5. As proved above, the expected limitcases for diffraction will correspond to that pro-duced from a circle or a ring. Another useful graphic
representation is provided by the level contour mapof the diffraction pattern; Fig. 6 illustrates the con-tour maps associated with the semicircular and
Fig. 7. Double symmetrical complete sectors for angular sizes �� 30°, 60°, 90°, 120°, 150°, and their respective far-field diffractionintensity patterns.
Fig. 8. Double symmetrical annular sectors for angular sizes � �30°, 60°, 90°, 120°, 150° and their respective far-field diffractionintensity patterns.
semiannular sectors. With as few as four or fivecontour levels the architecture in each diffractionpattern is clearly defined in the vicinity of the cen-tral spot. The inner-to-outer contours have normal-ized intensity values from 1.0 to 0.2 in 0.2 steps andare easily distinguishable by their different graylevels; intensity values less than 0.2 are shown asblack, making clear the complicated net of the zero-level contours embedded within the low-level inten-sity regions.
For double symmetrical complete or annular sec-tors, Eq. (36) or Eqs. (46) and (47) can be used tocalculate their Fourier transforms. Figure 7 shows5 of 12 double symmetrical complete sector open-ings, together with their corresponding far-field dif-fraction patterns. Similarly, Fig. 8 shows theirrespective annular versions where the radii ratio isagain R1�R2 � 2�5. It is interesting to observe, fromboth figures, the transition from open zero contours toclosed zero contours, the open zero contours occur forsector angular sizes � in the range �0, �� and may beregarded as fringe-type contours as a result of ampli-tude division interference produced by the two sym-metrical opposite openings that form the wholeaperture, for the angular step size selected, however,closed zero contours begin to appear when �� 7��12, and the effects of interference fringes arelost owing to the increasing extension in eachopposite opening until the gaps are completely filledto form the circular or ring aperture. In addition,Fig. 9 displays the contour map representationfor double complete and annular sectors with� � ��2.
C. Examples of Sector Wheels
We have divided the examples in this subsection intothree categories, as follows: Wheels with � even are
Fig. 9. Fraunhofer diffraction patterns for double symmetricalcomplete and annular sectors with � � ��2, and their contour maprepresentations, normalized intensity from inner to outer con-tours, 1, 0.8, 0.6, 0.4, and 0.2.
Fig. 10. Complete sector wheels for an even number of sectorswith � � 2, 4, 6, 8, 10 and their respective far-field diffractionintensity patterns.
shown in Figs. 10–12, wheels with � odd are dis-played in Figs. 13–15, and wheels with larger num-
bers of sectors are illustrated in Fig. 16. Forcomparison purposes, the apertures of each categorywere generated for both complete and annular sectorswith the same angular step size used at the beginningof Subsection 5.B as well as the same radii ratioR1�R2. It turns out, from the analysis made in Subsec-tion 5.A, that the wheels with even numbers of sectorshave real amplitude because their imaginary parts arezero and therefore have null phase; however, thewheels with odd numbers of sectors possess complexamplitude and therefore have varying phase. Noticethat the complex amplitude’s real part is formed onlyby the first term with the Jinc�·� functions because theangular coefficients in the Bessel series are all zero.This fact explains why the even-sector wheels lookless blurry and have a better contrast than the odd-sector wheels, as can be readily observed from thefigures provided. We checked all our examples nu-merically, using the FFT algorithm by synthesizingthe different types of aperture as white images em-bedded in a black screen of size 1024 � 1024 pixels;the values in pixels for radii R1 and R2 were adjustedto match their selected values for the approximatecomputation from the analytical solutions.
6. Pattern Properties and Optical Setup
A. Symmetries and Special Values
1. Conjugate, Fold, and Reflection SymmetryBecause f�r, �� is a real function, it can be shown fromthe polar form of the Fourier transform given in Eq.(10) that the amplitude field is conjugate symmetricwith respect to �. Using Eq. (2) together with Eqs. (50)
Fig. 11. Annular sector wheels for an even number of sectors with� � 2, 4, 6, 8, 10 and their respective far-field diffraction intensitypatterns.
Fig. 12. Fraunhofer diffraction patterns for the complete andannular four-sector wheel and their contour map representations;normalized intensity from inner to outer contours, 1, 0.8, 0.6, 0.4,and 0.2.
and (51) yields, for � even, cos�2p � 1���� � 2����� cos�2p � 1���; thus F has �-fold symmetry, as canbe seen from Figs. 7 and 8 for double symmetrical
sectors and from Figs. 10 and 11 for wheels with aneven number of sectors; however, for � odd, cos�2p� 1���� � ���� � �cos�2p � 1���, where the minus
Fig. 13. Complete sector wheels for an odd number of sectors with� � 3, 5, 7, 9, 11 and their respective far-field diffraction intensitypatterns.
Fig. 14. Annular sector wheels for an odd number of sectors with� � 3, 5, 7, 9, 11 and their respective far-field diffraction intensitypatterns.
has no effect when we square the imaginary part;hence F has 2�-fold symmetry, as illustrated in Figs. 4and 5 for single sectors and in Figs. 13 and 14 forwheels with odd numbers of sectors. In addition, fromthe reflection symmetry property of the aperture func-tion as stated in Eq. (3) and application of the rotationproperty10 of the Fourier transform, F has the samemirror lines as f and, consequently, F is reflection sym-metric across each one of these lines. The symmetriesdescribed above can be summarized as
F*(�, ���)�F(�, �),
F�, � 2�� �F��, ��, � even; (52)
F�, � �
��F��, ��, � odd; (53)
F��, 2M�����F��, ��, �� 0, . . . , �� 1.(54)
2. Pole IntensitiesFrom Eqs. (22), (25), and (28) we obtain the maxi-mum intensity at the central spot in the diffractionpattern by letting � � 0, � may take any value. Ineach equation the second term is zero and, consider-ing the limit value of the first term when � → 0, weobtain the corresponding amplitude value for eachaperture; hence
Fig. 15. Fraunhofer diffraction patterns for the complete andannular three-wheel sector and their contour map representations;normalized intensity from inner to outer contours, 1, 0.8, 0.6, 0.4,and 0.2.
Fig. 16. Complete and annular sector wheels for � � 16, 24, 36and their respective far-field diffraction intensity patterns.
from which the intensity ratios between a complete orannular sector-type aperture and a circular or ringaperture are given by
Iss�0, �; ��Iss�0, �; 2��
� �2�2
,
Ids�0, �; ��Ids�0, �; ��
� ��2
,
Iws�0, �; ��4Iws�0, �; ��
�14. (56)
For single- and double-sector openings the intensityratio depends on angular size � selected, and forsector wheels it is always constant because for anyvalue of � the area of the given aperture is alwayshalf of the area of the circle or ring. Furthermore, if� � �, then Iss�·; ���Iss�·; 2�� � 1�4 and, if �� ��2, then Ids�·; ��2��Ids�·; �� � 1�4, which showsthat the semicircular sector as well as the doublesymmetrical sector with a 90° angular size coincideswith the trivial cases of one-sector and two-sectorwheels.
3. Evaluation across Mirror LinesWe now consider the case for which � 0 and � takesvalues that correspond to the directions of mirror lineM� and its nearest orthogonal direction, defined byM�
� � M� � ��2 for � � 0, . . . , � � 1. Computationis based on Eqs. (44)–(49), which give the real andimaginary parts of the amplitude field. To avoid un-necessary repetitions of complete formulas we focusour attention on the inner finite sums that depend on�. For a single-type aperture, the only mirror line isM0 � 0, and its orthogonal direction is M0
� � ��2.Thus, if � � 0, then cos 2k� � cos �2k � 1�� � 1 and,if � � ��2, then cos2k� � ��1�k and cos�2k � 1��� 0. Therefore
��Fss��, 0; ������·��1�� �
n�1
�
�·� �k�1
n
��1�k sin k�,
��Fss��, 0; ����8�� �
n�1
�
�·� �k�1
n
��1�k
� sin�2k� 1��
2 , (57)
��Fss�,�
2 ; � ���·� �1�� �
n�1
�
�·� �k�1
n
sin k�,
��Fss�,�
2 ; � � 0. (58)
For a double-type aperture, mirror lines M0 � 0 andM1 � ��2 are mutually orthogonal because M0
�
� ��2 and M0� � � (equivalent to 0). These values
were evaluated above; hence
��Fds��, 0; ���� 2��·��2�� �
n�1
�
�·��k�1
n
��1�ksin k�,
��Fds�,�
2 ; � � 2��·��2�� �
n�1
�
�·��k�1
n
sin k�,
(59)
and ��Fds��, 0; ��� � ��Fds��, ��2; ��� � 0. The fi-nite trigonometric sums displayed in Eqs. (57)–(59)are simplified to21
�k�1
n
��1�k sin k�� sinn� 1
2 �����sinn2 ��
���sec�
2 , (60)
�k�1
n
��1�k sin �2k� 1��
2 �12��1�n sin n� sec
�
2 ,
(61)
�k�1
n
sin k�� sinn� 1
2 � sinn2 � csc
�
2 (62)
and require special care for evaluation when eithercos���2� � 0 or sin���2� � 0. As an example of thissituation, we consider the explicit determination ofthe intensity profile of the mirror line in a semian-nular sector opening. In this case the angular size is� � � and the mirror line is � � 0. Thus, we use thesingle annular solution given by Eq. (57) togetherwith Eq. (61). In Eq. (61), L’Hopital’s rule is applied tothe right hand side to give
lim�→�
��1�n sin n�2 cos���2�
��n. (63)
Therefore the desired intensity profile along the mir-ror line is given by
metrical complete and annular sectors with �� ��2, Fig. 17 shows in each part the normalizedintensity profiles along their first mirror line M0� 0 (solid curves) and along its nearest orthogonaldirection M0
� � ��2 (dashed curves). To emphasizelow-level intensities as well as the distribution ofrelative maxima and minima the vertical scale is log-arithmic in base 10; the horizontal scale is the closedinterval ��2�, 2��. Figure 17 illustrates the factthat, for single- and double-sector-type apertures, theM0 and M0
� intensity profiles are always different. Inthe semicircular aperture, the pattern intensity pro-file along the direction parallel to its diameter is,except for a scaling constant, the same as the Airyprofile corresponding to the diffraction pattern of acircular opening with the same radius.
In a sector wheel, the mirror lines are given byM� � ���� for � � 0, . . . , � � 1. Here we use Eqs.(50) and (51), recalling that, for � even,��Fws��, �; ��� � 0 and, for � odd, the second term ofthe real part vanishes; thus ��Fws��, �; ��� � ��·�.In any case, if � � ���� then cos�2p � 1���� cos ��2p � 1��, which equals �1 for � even andequals �1 if � is odd. Therefore we get
� even: ��Fws�,�
��; � ���·�
�
�� �n�1
�
�·�
� �p�1
Q
��1�½��2p�1��p�1 ��Fws�,�
��; � � 0; (65)
� odd � �Fws�,�
��; � ���·�,
� �Fws�,�
��; � �
�
�� �n�1
�
�·�
� �p�1
Q*
��1�½���2p�1��1��p�1 . (66)
From Eq. (65), clearly shows that, for a wheel withan even number of sectors, the intensity distributionalong all even mirror lines is the same but is differentfrom the common profile shared by all odd-mirrorlines, as can be appreciated from Figs. 10 and 11. Inthis case, notice that each even mirror line bisectstwo transparent opposite sectors, whereas each oddmirror line goes through two opaque opposite sectors.On the contrary, squaring the imaginary part ofEqs. (66) always gives the same result for any mirrorline; hence all mirror lines associated with a wheelwith an odd number of sectors will possess the sameintensity profile, a fact that can be seen from Figs. 13and 14. Recall that this type of sector wheel has 2�-fold symmetry and that each mirror line bisects twoopposite sectors, one transparent and the otheropaque. Figure 18 shows in each part the patternintensity profiles for complete and annular versionsof wheels with three and four sectors. The solidcurves refer to their first mirror line M0 � 0, and thedashed curves are the corresponding orthogonal di-rection M0
� ���2. Observe that the two profiles differ
Fig. 17. Pattern intensity profiles along mirror line. M0 � 0 (solid curves) and its orthogonal direction M0�� ��2 (dashed curves). Top row,
semicircle and semiannulus; bottom row, double symmetrical complete and annular sectors with � � 90°. The Vertical scale is logarithmic.
for � � 3 because M0� is parallel to a border line in the
wheel aperture; however, they are the same for �� 4.
B. Optical–Digital System Setup
The pattern field images obtained from numericalapproximation by use of the analytical formulas werealso verified experimentally by means of an optical–digital implementation that uses a coherent pro-cessor with a 2f spherical wavefront single-lensarchitecture, as shown in Fig. 19. This configurationallows us to scale the Fraunhofer diffraction patternif the aperture screen is displaced properly betweenthe lens and the output plane. The light source is aHe–Ne laser with � 543 nm whose output beam isspatially filtered; a 50 mm diameter convex lens withfocal length f � 500 mm was employed as the trans-forming lens, so the central replica of the multiplespectra falls exactly on the digital camera sensor area(CCD), leaving out the high-order replicas of lowerintensity produced by the grid structure of the liquid-crystal display (LCTV). The CCD camera has physi-cal dimensions of 8.8 � 6.6 mm, and the LCTV is7.68 mm� 5.76 mm. Each diffractive mask is clippedto a size of 480 � 480 pixels and displayed in theLCTV by means of a video electronic interface con-nected to a video multiplexer attached to the micro-computer. The paraxial conditions are satisfiedbecause the ratio of the lens area to the LCTV area isapproximately 45:1. The resultant diffraction pat-
terns captured by the CCD can be observed in anauxiliary black-and-white (aux. BW) monitor one byone with proper timing or in batch mode at a maxi-mum video rate of 18 frames�s. Otherwise, they canbe saved in the microcomputer’s local storage for fur-ther analysis.
7. Conclusions
The research reported in this paper gives completeanalytical expressions for circular sectors and severalrelated apertures, such as single, double symmetri-cal, and wheels of sectors, in their complete and an-nular versions. This paper extends previous researchon classical scalar diffraction theory applied to thestudy of far-field diffraction phenomena produced bycircular-type openings. The exposition given here
Fig. 18. Pattern intensity profiles along mirror line M0 � 0 (solid curves) and its orthogonal direction M0� � �/2 (dashed curves). Top row,
complete and annular three wheel; bottom row, complete and annular four wheel. The vertical scale is logarithmic.
Fig. 19. Optical–digital Fourier transformer. Lens L has a focallength of 500 mm, and the LCTV can be displaced to the right of Lto scale the transform.
provides a general theoretical framework based onJacobi’s identity involving Bessel functions of thefirst kind with real arguments. These results not onlycan be used as examples in the symbolic calculation ofFraunhofer diffraction patterns but also can form thebasis for studies of optical phenomena constrained tothis kind of geometry. Circular-type openings can beused as an alternative path for analyzing and under-standing light-diffraction phenomena besides thecommon practice that considers mainly circular orrectangular shapes. The examples and illustrationsprovided demonstrate the validity of the computedapproximate solutions obtained from their analyticalcounterparts. These results were verified with a FFTnumerical algorithm as well as by experimental ob-servation with an optoelectronic Fourier processor.
Finally, we remark that the theoretical study pre-sented here is useful for several reasons. Closed-formmathematical solutions of diffraction pattern distri-butions produced by apertures with specific geome-tries provide a better understanding of and insightinto pattern characteristics such as energy distribu-tion at low and high frequencies, shape of zero con-tours, and axis of symmetry with the same irradianceprofile. Experimental optical setups and fast numer-ical algorithms do not offer these advantages in aclear and systematic way. Circular-type apertureshave been applied in optics for different purposes; forexample, a holographic volume display proposes theuse of a double symmetrical sector as an encodingpupil of different perspectives in a single three-dimensional object22; binary masks with a finite num-ber of sectors, defined by a specific shape function;were used to synthesize gray-level apodizers that in-crease the axial irradiance depth without distortingthe in-focus impulse response too much.23 A more-recent application in confocal scanning systems con-siders phase-only three-level pupils with a real innerdisk and complex-conjugate outer semiannuli whosecomputation requires a Fraunhofer diffraction pat-tern of a semicircular aperture.24 Another importantfuture application of the treatment given here is re-lated to studies of optical imaging in the presence ofaberrations for which an excellent description hasbeen already given for circular and annular pupils.25
We appreciate the useful comments made by theanonymous reviewers. The authors are grateful to theSistema Nacional de Investigadores of the Consejo,Nacional de Cienga y Tecnología (National Council ofScience and Technology), Mexico City, for its partialfinancial support of this research.
References1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge
U. Press, 1999).
2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed.(McGraw-Hill, 1996).
3. C. S. Williams and O. A. Becklund, Introduction to the OpticalTransfer Function (SPIE Press, 2002).
4. A. I. Mahan, C. V. Bitterli, and S. M. Cannon, “Far-field dif-fraction patterns of single and multiple apertures bounded byarcs and radii of concentric circles,” J. Opt. Soc. Am. 54, 721–732 (1964).
5. M. M. Agrest and M. S. Maksimov, Theory of Incomplete Cy-lindrical Functions and Their Applications (Springer-Verlag,1971).
6. R. A. Lessard and S. C. Sorn, “Imaging properties of sector-shaped apertures,” Appl. Opt. 11, 811–817 (1972).
7. A. Barna, “Fraunhofer diffraction by semicircular-apertures,”J. Opt. Soc. Am. 67, 122–123 (1977).
8. J. Komrska, “Fraunhofer diffraction from sector stars,” J. Mod.Opt. 30, 887–925 (1983).
9. C. J. R. Sheppard, S. Campbell, and M. D. Hirschhorn,“Zernike expansion of separable functions of Cartesian coordi-nates,” Appl. Opt. 43, 3963–3966 (2004).
10. R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall,1994).
11. Q. Cao, “Generalized Jinc functions and their application tofocusing and diffraction of circular apertures,” J. Opt. Soc. Am.A 20, 661–667 (2003).
12. E. W. Weisstein, “Jinc function,” MathWorld, a Wolfram Webresource (2005), http:��mathworld.wolfram.com�JincFunc-tion.html.
13. G. N. Watson, Treatise on the Theory of Bessel Functions, 2nded. (Cambridge U. Press, 1966).
14. G. E. Andrews, R. Askey, and R. Roy, Special Functions,Vol. 71 of Encyclopedia of Mathematics and Its Applications(Cambridge U. Press, 1999).
15. A. W. Lohmann, Optical Information Processing (UniversityCalifornia Press, 1978), Vol. 1.
16. A. Papoulis, Systems and Transforms with Applications inOptics (McGraw-Hill, 1968).
17. Y. L. Luke, “Integrals of Bessel functions,” in Handbook ofMathematical Functions with Formulas, Graphs, and Mathe-matical Tables, M. Abramovitz and I. A. Stegun, eds. (Dover,1970), pp. 479–494.
18. D. W. Lozier, “Software needs in special functions,” J. Comput.Appl. Math. 66, 345–358 (1996).
19. S. Zhang and J. Jin, Computation of Special Functions (Wiley,1996).
20. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery, Numerical Recipes in C, The Art of ScientificComputing, 2nd ed. (Cambridge U. Press, 2002).
21. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series,and Products, 5th ed., A. Jeffrey, ed. (Academic, 1994).
22. M. Trivi, H. J. Rabal, N. Bolognini, E. E. Sicre, and M. J.Garavaglia, “Three-dimensional display through speckle ste-reograms,” Appl. Opt. 25, 3776–3780 (1986).
23. A. W. Lohmann, J. Ojeda-C., and A. Serrano-H., “Synthesis ofanalog apodizers with binary angular sectors,” Appl. Opt. 34,317–322 (1995).
24. M. Kowalcyzk, C. J. Zapata-R., and M. Martinez-C., “Asym-metric apodization in confocal scanning systems,” Appl. Opt.37, 8206–8214 (1998).
25. V. N. Mahajan, Optical Imaging and Aberrations: Part II,Wave Diffraction Optics, Vol. PM103 of SPIE Press Mono-graphs (SPIE Press, 2001).
