Generalized Jinc functions and their application to focusing and diffraction of circular apertures

Transcription

1 Qing Cao Vol. 20, No. 4/April 2003/J. Opt. Soc. Am. A 66 Generalized Jinc fnctions and their application to focsing and diffraction of circlar apertres Qing Cao Optische Nachrichtentechnik, FernUniversität Hagen, Universitätsstrasse 27/PRG, Hagen, Germany Received October 7, 2002; revised manscript received December 2, 2002; accepted December 2, 2002 A family of generalized Jinc fnctions is defined and analyzed. The zero-order one is jst the traditional Jinc fnction. In terms of these fnctions, series-form expressions are presented for the Fresnel diffraction of a circlar apertre illminated by converging spherical waves or plane waves. The leading term is nothing bt the Airy formla for the Franhofer diffraction of circlar apertres, and those high-order terms are directly related to those high-order Jinc fnctions. The trncation error of the retained terms is also analytically investigated. We show that, for the illmination of a converging spherical wave, the first 9 terms are sfficient for describing the three-dimensional field distribtion in the whole focal region Optical Society of America OCIS codes: , , , , INTRODUCTION The diffraction problem of circlar apertres illminated by converging spherical waves or plane waves is an important topic,2 in optical science, becase this kind of wave phenomenon is freqently encontered in varios optical systems. It is well-known that the Franhofer diffraction of circlar apertres has a Jinc-fnction distribtion 3,4 (i.e., the Airy pattern distribtion) with the radial coordinate. However, this kind of simple field distribtion appears only at the geometrical focal plane for the case of converging spherical-wave illmination. And for the case of plane-wave illmination, it is also valid only for those transverse planes that are far from the apertre plane. In all the other regions, especially in the focal region of a converging spherical wave, the diffracted field has a rather complicated strctre. To investigate this kind of complicated diffraction problem, people have developed varios nmerical and analytical methods. At present, it is possible to se nmerical methods to implement an accrate simlation, bt one cannot obtain as mch physical insight as with analytical methods. Among the analytical methods, the classical Debye theory employs the Lommel fnctions to describe the three-dimensional field distribtion in the focal region for high-fresnel-nmber focsing systems. However, it is not applicable to low-fresnel-nmber focsing systems. By se of an appropriate variable transform, Li and Wolf 5 and Li 6 extended the Lommel-fnction description to general paraxial focsing systems with arbitrary Fresnel nmbers. The case of plane-wave illmination was also investigated in Ref. 6 as a separate treatment. Recently, Wang et al. 7 attempted to derive a simple closed-form expression for the Fresnel diffraction of circlar apertres illminated by spherical waves or plane waves. Unfortnately, their reslt was demonstrated to be incorrect. 8,9 As an alternative, Overfelt and White 9 developed the exponential polynomial description for the same problem. Obviosly, their treatment can be applied to focsing and diffraction of circlar apertres illminated by converging spherical waves or plane waves. Compared with the Lommel-fnction description, this new description eliminated the need to split the comptational problem into two different regions. In another context, Wang et al. 0 developed a novel analytical tool named moment expansion to investigate the depth of focs. Their analysis is mainly based on the Forier transform pairs of the partial derivatives in real space and the corresponding moments in the spatialfreqency domain. In principle, their expression [see Eq. (7) of Ref. 0] can be sed to analytically describe an arbitrary paraxial focsing system, provided that the farfield distribtion at the focal plane is known and has all the high-order partial derivatives. As a special case, the diffraction problem in the focal region of circlar apertres illminated by converging spherical waves can also be treated by this method. Compared with the Lommelfnction and exponential polynomial descriptions, the moment expansion method has a more explicit connection with the far field. This property can be easily fond from Eq. (7) of Ref. 0, where the leading term is jst the far field. However, their work is not appropriate for a planewave illmination, becase, in this case, there is no geometrical focal plane within a finite region at all. This drawback comes from the inconsistency between the asymptotic far-field behavior, which is proportional to z for large z in this case, and the moment expansion method, which actally ses the variable z f (becase the coordinate origin sed in Ref. 0 is located at the focal point), where z is the longitdinal distance from the apertre plane and f is the crvatre radis of the converging spherical waves. For the illmination of a plane wave, the variable z f is not appropriate becase f in this case. Considering the continos change of the physical behaviors with the change of the crvatre /2003/ $ Optical Society of America

2 662 J. Opt. Soc. Am. A/ Vol. 20, No. 4/ April 2003 Qing Cao f of the incident spherical waves, one can dedce that it is also inconvenient to se the moment expansion method 0 to describe the focsing systems with long focal length (i.e., f is small). In this paper, we shall modify the moment expansion method for focsing and diffraction of circlar apertres illminated by converging spherical waves or plane waves. This modified treatment employs the sitable variables z and z f (in the concrete employment, we shall se the corresponding Fresnel nmber) to replace z f. We shall also define and analyze a family of generalized Jinc fnctions. In terms of them, or treatment can lead to an elegant series-form expression, which is valid for arbitrary spherical-wave (inclding planewave) illmination. The paper is organized as follows. In Section 2, we define the family of generalized Jinc fnctions and otline their main properties. In Section 3, the series-form expression for the diffraction problem of circlar apertres is presented in terms of this family of fnctions, and we analyze the trncation error of the retained terms and check the validity. And, in Section 4, we conclde this paper and discss some related problems. As we shall show below, compared with other analytical methods, or treatment has the following for advantages: () It has a more explicit connection with the far field and therefore provides an explicit insight showing how the defocsed field distribtions gradally deviate from the far-field distribtion. (2) The case of planewave illmination is atomatically inclded as a special case. Therefore a separate treatment 6 for the case of plane-wave illmination is no longer needed. (3) Or treatment allows a simple analytical estimate for the trncation error of the retained terms. (4) The generalized Jinc fnctions sed in or treatment are one-variable fnctions, which are simpler than the two-variable Lommel fnction,5,6 and the two-variable exponential polynomials. 9 Becase of these advantages, this approach is more sitable for describing the focsing and diffraction problem of circlar apertres. 2. GENERALIZED JINC FUNCTIONS It is well-known that the Franhofer diffraction of circlar apertres has the simple Jinc-fnction distribtion 3,4 (i.e., the Airy pattern distribtion). The Jinc fnction is given by Jinc J, () where J (v) is the first-order Bessel fnction of the first kind and is the variable. From the properties of Bessel fnctions, one knows that the Jinc fnction can also be written as the integral form Jinc 2 vj 0 vdv, (2) 0 where J 0 () is the zero-order Bessel fnction of the first kind and v is the integral variable. In fact, in the derivation of the Jinc-fnction distribtion of the Franhofer diffraction of circlar apertres, the form of Eq. (2) is encontered before that of Eq. (). We now define a family of generalized Jinc fnctions as the following forms: Jinc n v 2n J 0 vdv, (3) 2n2 0 where the order n is a nonnegative integer, namely, n 0,, 2,.... From Eq. (3), one can see that the zeroorder one is jst the traditional Jinc fnction. By se of the properties of Bessel fnctions and the mathematical indctive method, one can prove the following closed-form expression (see Appendix A): n n! J m Jinc n m0 2 m n m! m, (4) where J m () is the (m )th-order Bessel fnction of the first kind and 0!. In particlar, the first several fnctions are given by [besides Jinc 0 (), given by Eq. ()] Jinc J Jinc 2 J Jinc 3 J 2 J 2 2, (5) 4 J J J 3 3, (6) 24 J J 4 4. (7) Let s now investigate some important properties of this family of fnctions: () The nth-order Jinc fnction has n sbterms, and each sbterm has the factor J m ()/ m, where the order m of the Bessel fnction in the nmerator is jst the power of in the denominator. (2) The first sbterm of each Jinc fnction is always J ()/. This property is more explicitly shown in Eqs. (5) (7). (3) From the asymptotic behavior J n 2 cos n 2 4 for large, one can dedce that, for each high-order Jinc fnction, all the other sbterms decrease faster than the first sbterm when becomes large. As a conseqence, we obtain the important property that all the Jinc fnctions approach Jinc 0 () for large, i.e., lim Jinc n Jinc 0 J. (8) (4) Ptting the relation lim 0 J 0 (v) into Eq. (3) and integrating, one can easily prove that Jinc n 0 (9) 2n when 0. (5) Throgh a great nmber of observations, it seems that the principal maximm of each Jinc fnction always appears at the point 0. These maximm vales, jst as shown in Eq. (9), decrease with the increase of the order n. We also observe that, jst as

3 Qing Cao Vol. 20, No. 4/April 2003/J. Opt. Soc. Am. A 663 Fig.. Fnctional crves of Jinc 0 (), Jinc (), and Jinc 2 (). where U(R, z) is the diffracted field at the z z plane, U 0 (r) is the incident field at the apertre plane, is the wavelength in free space, k 2/ is the wave nmber in free space, and j is the imaginary nit. Note that in Eq. () we have ignored the factor 2j exp( jkz)exp jkr 2 /(2z). Sbstitting the complex amplitde distribtion U 0 (r) expjkr 2 /(2 f ) of the converging spherical wave into Eq. () and employing the normalized coordinates r/a and R/a, one can obtain U, z N exp jn 2 2 J 0 2N d, 0 (2) where N a 2 /(z), N 2 N N, and N a 2 /(f ), which is the Fresnel nmber of the apertre illminated by a converging spherical wave with a crvatre radis f. We refer to N as the Fresnel nmber of the apertre itself, becase it corresponds to the case of plane-wave illmination. As we show below, the difference N 2 between N and N plays an important role for describing the defocsed field distribtions. For clarity, we write N 2 as the form Fig. 2. Asymptotic behavior of Jinc n () for large n. For comparison, Jinc 0 () and Jinc 30 () have been amplified 22 and 62 times, respectively. partly shown in Fig., all the Jinc fnctions have similar crves and these crves gradally change with the change of the order n. In fact, it is this similarity that stimlates s to call them the generalized Jinc fnctions. (6) We prove that, when the order n approaches, the normalized Jinc n () fnctions approach J 0 (); concretely (see Appendix B), lim Jinc n n 2n J 0. (0) This trend is clearly shown in Fig. 2, where the normalized Jinc 30 () is more similar to J 0 () than the normalized Jinc 0 (). 3. FOCUSING AND DIFFRACTION OF CIRCULAR APERTURES Consider a circlar apertre of radis a, as shown in Fig. 3, illminated by a converging sperical wave with a crvatre radis f. We denote by r, R, and z the radial coordinate at the apertre plane, the radial coordinate at the observation plane, and the distance between these two transverse planes, respectively. It is known that, when the paraxial condition is satisfied, one can se the Fresnel approximation to describe the diffracted field (both near field and far field) of a circlar apertre. 2 In terms of the above-mentioned coordinate parameters, one can express the Fresnel diffraction formla for rotationally symmetric fields trncated by a circlar apertre as N 2 a2 z f. (3) From Eq. (3), one can see that N 2 0 when z f and that its absolte vale increases when the observation plane gradally deviates from the focal plane. As a conseqence, it can be expected that the field distribtion U(, z) gradally deviates from the far-field Airy pattern distribtion with the increase of N 2. Similarly to the approach taken in the moment expansion method, 0 we now expand the factor exp( jn 2 2 ) in Eq. (3) as exp jn 2 2 n0 n! N 2 n 2n. (4) Unlike the treatment of Ref. 0, or expansion is not abot the variable z f bt abot N 2, which is proportional to the variable z f. Sbstitting this expansion into Eq. (2), one can expand the field distribtion U(, z) as the series form of N 2 : j n U, z n0 U n, z, (5) U n, z jn n! N 2 n N Jinc n 2N. (6) UR, z au 0 rexp jk r2 J z 0 2z 0 krr rdr, z () Fig. 3. Schematic view of the system configration.

4 664 J. Opt. Soc. Am. A/ Vol. 20, No. 4/ April 2003 Qing Cao As shown in Eq. (6), the leading term is jst the far-field Franhofer diffraction of a circlar apertre, and those high-order terms are directly related to those high-order Jinc fnctions. These properties provide an explicit physical pictre showing how the defocsed field distribtion at the observation plane gradally deviates from the far-field distribtion with the deviation of N 2 from 0. It is interesting that all the even-order terms are pre real and all the odd-order terms are pre imaginary. It is more interesting that the leading term is independent of N 2 in the expressive form. In other words, this term is independent of the crvatre radis f (or N ) in the expressive form. Perhaps one gesses that this is a mistake, becase it is well-known that the far-field distribtion of a circlar lens appears at the geometrical focal plane and this far-field distribtion has a scale factor f (or N ). In fact, this is not a mistake. The reason is that all the high-order terms disappear at the focal plane (i.e., N 2 0), and the leading term atomatically has the scale factor f (or N ) at the focal plane becase N N in this case. From Eq. (6), one can find that, as an important advantage of or analytical treatment, the case of plane-wave illmination has been atomatically inclded as the special case of N 2 N. Obviosly, this advantage partly reslts from the appropriate choice of the variables z f and z (the corresponding Fresnel nmbers are N 2 and N, respectively). In addition, from Eq. (6), one can see that, jst as is known, the diffracted field has an asymmetric distribtion abot the focal plane. This asymmetric distribtion leads to the well-known focal shift phenomenon. 3 6 It is worth mentioning that the first two terms of Eq. (6) for the special case of plane-wave illmination (i.e., N 0 and N 2 N ) have been sed for the focsing analysis 7 of a pinhole photon sieve, 8 which is a new class of diffractive optical element for focsing and imaging of soft x rays. In Ref. 7, the first two terms were called the far-field term and the qasi-far-field correction term. In fact, it is the sccessfl employment of the first two terms for the special case of plane-wave illmination in Ref. 7 that stimlates s to derive Eq. (6), which consists of infinite terms. It is well-known that the on-axis field distribtion U(0, z) can be presented in a closed form. Ptting the relation J 0 (0) into Eq. (2), one can derive that U0, z N exp jn 2. (7) j2n 2 By se of the expansion exp( jn 2 ) n0 j n (N 2 ) n /n!, one can re-express U(0, z) as U(0, z) n0 j n n N N n 2 /2(n )!. As a check of or analytical treatment, one can directly derive the same seriesform expression for U(0, z) by inserting the relations Jinc n (0) /2(n ) into Eq. (6). From Eq. (7), one can easily obtain the on-axis intensity distribtion I(0, z): I0, z U0, z 2 N 2 N 2 N 2 2 sin 2 N 2 2, (8) where we have written N as the form N 2 N. As a limit case, the on-axis principal maximm appears at N 2 0 for very large N, becase in this case the factor sin 2 (N 2 /2)/N 2 2 changes mch faster than the factor (N 2 N) 2. It is well-known that this reslt can be correctly predicted by the classical Debye theory. As another limit case, the on-axis principal maximm appears at N 2 when N 0 (i.e., plane-wave illmination), becase I(0, z) sin 2 (N 2 /2) in this case. In all other cases, the on-axis principal maximm appears in the interval 0 N 2. In addition, from Eq. (8), one can dedce that the on-axis intensity decreases from the principal maximm to zero when N 2 2 provided that N 2. For this reason, we reasonably refer to the region corresponding to 2 N 2 2 as the focal region. When N 2, this definition has direct meaning. However, when N 2, this definition shold be modified as N N 2 2, becase the on-axis intensity I(0, z) already decreases to zero when N 2 N (note that this corresponds to z ). In the practical employment of Eqs. (5) and (6), one needs to trncate the series. Therefore it is desirable to analytically provide the trncation error of the retained terms. It is appropriate to define the normalized sqare trncated error S M of the retained terms as S M 0 U, z V M, z 2 d 0 U, z 2 d, (9) where M is the nmber of retained terms and V M (, z) is the sm of the first M terms, i.e., V M (, z) M n0 U n (, z). When the series goes into the fast convergent region, those higher terms can be ignored compared with the term U M (, z), which is the first of those discarded terms. Based on this approximation, one can obtain U, z V M, z U M, z (20) when the series has gone into the fast convergent region. We now re-express Eq. (2) as the Forier Bessel transform form 3,9 U, z 2N W J 0 2N d, (2) 0 where W( ) exp( jn 2 2 )/2 for and W( ) 0 elsewhere. This means that U(, z) and W( ) are a pair of Forier Bessel transforms. Similarly, one can find that U M (, z) and W M ( ) are another pair of Forier Bessel transforms, where W M ( ) j M (N 2 ) M 2M /(2M!) for and W M ( ) 0 elsewhere. By se of the Parseval theorem, 3,9 one can derive the following reslt: S M 0 0 W M 2 d W 2 d N 2 2M 2M M! 2. (22)

5 Qing Cao Vol. 20, No. 4/April 2003/J. Opt. Soc. Am. A 665 Fig. 4. Relation between the nmber M 0 of the terms needed and N 2. Eqation (22) explicitly shows how the trncation error S M decreases with the increase of the nmber of retained terms for a given N 2 vale (note that S M is an even fnction of N 2 ). It is interesting that S M is independent of N. This is de to the fact that we expand only the factor exp( jn 2 2 ) related to N 2 in Eq. (2) and keep the factor J 0 (2N ) related to N in Eq. (2) nchanged. As is pointed ot above, this treatment can atomatically inclde the case of plane-wave illmination. We have observed that the approximate field distribtion V M (, z) of the first M terms and the exact field distribtion calclated by direct nmerical integration in Eq. (2) are completely indistingishable when S M 0 5. If one chooses S M 0 5 as the critical vale, then, from Eq. (22), one can calclate the nmber M 0 of the terms that shold be retained, where M 0 corresponds to the case S M0 0 5 for the first time. Exactly speaking, S M0 0 5 bt S M The relation of M 0 to N 2 is shown in Fig. 4 for the range N 2 2. From Fig. 4, one can see that fewer than 20 terms are necessary when N 2 2. This actally covers the whole focal region, where the on-axis intensity decreases from the principal maximm to zero. Therefore the first 9 terms are sfficient for describing the three-dimensional field distribtion in the whole focal region for the illmination of a converging spherical wave. To nderstand this statement better, we draw the transverse field distribtion U(, z) at the plane corresponding to N 2 2 (i.e., the bondary of the focal region) for N 5. In the concrete comptation, N has been expressed as N 2 N. Therefore, for a given N vale, U(, z) depends only on and N 2 in the expressive form. From Fig. 5, one can see that the field distribtion determined by the first 9 terms are completely indistingishable from that calclated from the exact nmerical integration in Eq. (2). In particlar, the zero field vale at the point 0 is accrately described by the first 9 terms of or analytical expression. Fig. 5. Transverse field distribtion at the plane corresponding to N 2 2: (a) real part, (b) imaginary part. The Fresnel nmberischosenschthatn 5. The solid lines are the exact reslts, and the stars are the analytical reslts of the first 9 terms. 4. CONCLUSIONS AND DISCUSSION We have defined a family of generalized Jinc fnctions and analyzed their main properties. In terms of these fnctions and the appropriate variables N 2 and N, an elegant series-form expression has been presented for focsing and diffraction of a circlar apertre illminated by converging spherical waves or plane waves. The trncation error of the retained terms has also been analytically investigated. In particlar, we have shown that the first 9 terms are sfficient for describing the threedimensional field distribtion in the whole focal region (we refer to the region corresponding to 2 N 2 2) for the illmination of a converging spherical wave. Compared with the Lommel-fnction,5,6 and exponential polynomial 9 descriptions, or method has the following advantages: () Or treatment has a more explicit connection with the far field. As shown in Eq. (6), the zero-order term is jst the far-field Franhofer diffraction of a circlar apertre, and those high-order terms are directly related to those high-order Jinc fnctions. These properties provide an explicit physical insight showing how the defocsed field distribtions gradally deviate from the far-field distribtion. This advantage partly reslts from sing a treatment similar to that in the moment expansion method. 0 (2) The case of plane-wave illmination is atomatically inclded as the special case of N 0. As a conseqence, one does not need to change the coordinate variable for the case of plane-wave illmination any longer. This advantage partly reslts from the sitable employment of the variables z f and z instead of z f. (3) As we showed above, or treatment can lead to an analytical expression for the trncation error of the retained terms. This property provides a great convenience in practical employment. (4) The generalized Jinc fnctions sed in or treatment are onevariable fnctions, bt the Lommel fnction and the exponential polynomials are two-variable fnctions. In addition, it is worth mentioning that, like the exponential polynomial description, 9 or analytical treatment does not need to split the comptation problem into two different regions.

6 666 J. Opt. Soc. Am. A/ Vol. 20, No. 4/ April 2003 Qing Cao The drawback of or treatment is that the series converges slowly when the absolte vale of N 2 becomes large. For this case, the asymptotic soltion presented by Sothwell 20 is sggested. This asymptotic soltion and or treatment are complementary with each other. The former is sitable for large N 2 vales, and the latter is sitable for small N 2 vales. It shold be mentioned that, very recently, Janssen 2 and Braat et al. 22 developed the extended Nijboer Zernike approach for the comptation of optical pointspread fnctions. We note that some similar mathematical problems on those integrals that are related to Bessel fnctions appeared in their papers, too. By comparing Eq. (3) here with Eq. (3) of Ref. 2, one can find that the generalized Jinc fnctions investigated here are related to the fnctions T n0 with f 0 and even integers n in Ref. 2. However, the explicit closed-form expressions are given in different forms. In other words, Eq. (4) here is different from the expression T n0 given by Eq. (4) of Ref. 2 for the case of f 0 and even integers n. The former is mch more compact than the latter, becase, for this case, we se relatively simpler mathematics. Of corse, the se of more complicated mathematics in Ref. 2 is mainly de to the more general sbject. As we showed in Section 2, the expression given by Eq. (4) here is particlarly sitable for investigating the important properties of the generalized Jinc fnctions, sch as the asymptotic behaviors for large variable and the similar crves among this family of fnctions. One may also note that Eq. (6) here is similar to Eq. (B4) of Ref. 22. They become more similar if one sbstittes the relation exp( jf ) n0 ( jf ) n /n! into Eq. (B4) of Ref. 22 and reorganizes those terms according to the power of f, where f is a parameter sed in Refs. 2 and 22 (note that this parameter is different from the focal length f sed in the present paper). In fact, after this procedre, one can find the generalized Jinc fnctions in the expression. This consistency frther checks the reslts of Eq. (6) in this paper and Eq. (B4) in Ref. 22 against each other. Obviosly, this check is helpfl for both eqations becase they are both derived from complicated mathematics (in particlar, the latter is derived from a more complicated mathematical backgrond). However, it shold be emphasized that the contents of the present paper are completely different from those of Refs. 2 and 22 except for the two similar mathematical problems mentioned above. v 0 2n J 0 vdv 2n J v 2n0 2n J vdv. (A2) Frther employing the relation J (v)dv dj 0 (v) and integration by parts in Eq. (A2), one can obtain Eq. (A). We se the mathematical indctive method to prove Eq. (4):. In the first stage, one can easily prove that Jinc 0 () J ()/ by se of the relation 0 vj 0 (v)dv J (). Obviosly, as the special case of n 0, Jinc 0 () satisfies Eq. (4). 2. In the second stage, we assme that Eq. (4) holds for the (n )th-order Jinc fnction. That is, we assme that n n! Jinc n m0 2 m n m! J m m, (A3) where n In the third stage, we prove that Eq. (4) also holds for the nth-order Jinc fnction if it holds for the (n )th-order Jinc fnction. Sbstitting Eq. (A) into Eq. (3), one can obtain Jinc n J 2n 2 J 0 L, (A4) L 4n2 2 Jinc n. (A5) Ptting the assmption of Eq. (A3) into Eq. (A5), one can obtain n 2n2 m n! J m L m0 n m! m3. (A6) Sbstitting the relation 2nJ n ()/ J n () J n () into the last sbterm (i.e., the sbterm corresponding to m n ) of Eq. (A6), one can obtain L 2 n n! J n n 2 n n! J n n n2 2n2 m n! J m m0 n m! m3. (A7) APPENDIX A For readability, we first derive the eqality 0 v 2n J 0 vdv 2n J 2n 2n J 0 4n 20 v 2n J 0 vdv, (A) After combining the second term (2) n n!j n ()/ n and the last sbterm (corresponding now to m n 2) of the smmed expression on the right-hand side of Eq. (A7) and sing the relation 2(n )J n ()/ J n () J n2 (), one can frther obtain L n mn m0 n n! 2 m n! J m n m! m 2 J n2 n n! n n3 2n2 m n! n m! J m m3, (A8) thogh it can be fond elsewhere. By se of the relation vj 0 (v)dv dvj (v) and integration by parts, one can obtain which incldes two smmed expressions and one isolated term. We repeat this process again and again. At each step, we first combine the isolated term, which can be

7 Qing Cao Vol. 20, No. 4/April 2003/J. Opt. Soc. Am. A 667 written in the form (2) i n!j i ()/n (i )! i2, and the last sbterm of the latter smmed expression, which can be written in the form 2n(2) i n!j i ()/(n i)! i2, where i is an integer that starts from i n 2 for the first step and ends at i for the last step. By se of the relation 2iJ i ()/ J i () J i (), one can prove that the sm of this combination is 2 i n! J i n i! i 2 i n! J i n i! i. We then pt the first part into the former smmed expression and let the second part alone (i.e., a new isolated term). After each step, the former smmed expression increases by one sbterm, and the latter smmed expression decreases by one sbterm. After n 2 steps [starting from Eq. (A8)], one can finally obtain n 2 m n! J m L m n m! m 2n 2 J 0. (A9) Note that the second smmed expression has disappeared completely. It is worth mentioning that Eqs. (A7) and (A8) already reach Eq. (A9), respectively, when n and n 2. Sbstitting Eq. (A9) into Eq. (A4), one can immediately obtain Eq. (4). This finishes the demonstration process based on the mathematical indctive method. APPENDIX B Changing the integral variable from v to v v/, one can rewrite Eq. (3) as Jinc n v 0 2n J 0 v dv. Eqation (B) can be frther expressed as (B) Jinc n gv J 0 v dv, (B2) 2n where g(v ) 2(n )v 2n for 0 v and g(v ) 0 elsewhere. Obviosly, g(v ) (v ) when n, where ( ) expresses the Dirac fnction. Sbstitting this relation into Eq. (B2), one can obtain lim Jinc n v J 0 v dv, n 2n (B3) which can directly lead to Eq. (0). ACKNOWLEDGMENT The athor is indebted to Jürgen Jahns for many fritfl discssions. The athor may be reached by at REFERENCES AND NOTES. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 975), Chap J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 986). 3. J. W. Goodman, Introdction to Forier Optics, 2nd ed. (McGraw-Hill, New York, 996), Chap A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 986), Sec Y. Li and E. Wolf, Three-dimensional intensity distribtion near the focs in system of different Fresnel nmbers, J. Opt. Soc. Am. A, (984). 6. Y. Li, Three-dimensional intensity distribtion in low- Fresnel-nmber focsing systems, J. Opt. Soc. Am. A 4, (987). 7. P. Wang, Y. X, W. Wang, and Z. Wang, Analytical expression for Fresnel diffraction, J. Opt. Soc. Am. A 5, (998). 8. J. C. Hertley, Analytical expression for Fresnel diffraction: comment, J. Opt. Soc. Am. A 5, (998). 9. P. L. Overfelt and D. J. White, Analytical expression for Fresnel diffraction: comment, J. Opt. Soc. Am. A 6, (999). 0. Y.-T. Wang, Y. C. Pati, and T. Kailath, Depth of focs and the moment expansion, Opt. Lett. 20, (995).. M. Abramowitz and I. A. Stegn, eds., Handbook of Mathematical Fnctions with Formlas, Graphs, and Mathematical Tables (Wiley, New York, 972). 2. W. H. Sothwell, Validity of the Fresnel approximation in the near field, J. Opt. Soc. Am. 7, 7 4 (98). 3. J. H. Erkkila and M. E. Rogers, Diffracted fields in the focal volme of a converging wave, J. Opt. Soc. Am. 7, (98). 4. Y. Li and E. Wolf, Focal shifts in diffracted converging spherical waves, Opt. Commn. 39, 2 25 (98). 5. J. J. Stamnes and B. Spjelkavik, Focsing at small anglar apertres in the Debye and Kirchhoff approximations, Opt. Commn. 40, 8 85 (98). 6. M. P. Givens, Focal shifts in diffracted converging spherical waves, Opt. Commn. 4, (982). 7. Q. Cao and J. Jahns, Focsing analysis of the pinhole photon sieve: individal far-field model, J. Opt. Soc. Am. A 9, (2002). 8. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelng, S. Harm, and R. Seemann, Sharper images by focsing soft x-rays with photon sieve, Natre (London) 44, (200). 9. E. Lalor, Conditions for the validity of the anglar spectrm of plane waves, J. Opt. Soc. Am. 58, (968). The Parseval theorem is the special case f(x, y) g(x, y) of Theorem IV of this reference. 20. W. H. Sothwell, Asymptotic soltion of the Hygens Fresnel integral in circlar coordinates, Opt. Lett. 3, (978). 2. A. J. E. M. Janssen, Extended Nijboer Zernike approach for the comptation of optical point-spread fnctions, J. Opt. Soc. Am. A 9, (2002). 22. J. Braat, P. Dirksen, and A. J. E. M. Janssen, Assessment of an extended Nijboer Zernike approach for the comptation of optical point-spread fnctions, J. Opt. Soc. Am. A 9, (2002).