Paraxial light distribution in the focal region of a lens: a comparison of several analytical solutions and a numerical result

Transcription

1 Jornal of Modern Optics, 214 Vol. 61, No. S1, S57 S67, Paraxial light distribtion in the focal region of a lens: a comparison of several analytical soltions and a nmerical reslt Yang W and Damien P. Kelly Institt für Mikro-nd Nanotechnologien, Technische Universität Ilmena, Ilmena, Germany (Received 23 April 214; accepted 3 Jly 214) The distribtion of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for nderstanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical soltions in the neighborhood of the focs, when a plane wave is incident on a focsing lens whose finite extent is limited by a circlar apertre. For example, in Born and Wolf s treatment of this problem, two different, bt mathematically eqivalent analytical soltions, are presented that describe the 3D field distribtion sing infinite sms of U n and V n type Lommel fnctions. An alternative soltion expresses the distribtion in terms of Zernike polynomials, and was presented by Nijboer in More recently, Cao derived an alternative analytical soltion by expanding the Fresnel kernel sing a Taylor series expansion. In practical calclations, however, only a finite nmber of terms from these infinite series expansions is actally sed to calclate the distribtion in the focal region. In this manscript, we compare and contrast each of these different soltions to a nmerically calclated reslt, paying particlar attention to how qickly each soltion converges for a range of different spatial locations behind the focsing lens. We also examine the time taken to calclate each of the analytical soltions. The nmerical soltion is calclated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of ρ and then nmerically integrated. This prodces an infinite set of replicas in the diffraction plane, that are located in circlar rings centered at the optical axis and each with radii given by 2πm/ ρ, where m is the replica order. These circlar replicas are shown to be fndamentally different from the replicas that arise in a Cartesian coordinate system. Keywords: propagating methods; diffraction; analytical soltion 1. Introdction The complex amplitde distribtion in the focal region of a converging lens remains an important problem in optics today [1]. Althogh the problem is well known and has been analyzed, sing many different techniqes [2 13], the most sitable approach depends on the natre of the problem being investigated. When an optical system is being analyzed, where the lens has a low nmerical apertre, a scalar analysis is often sfficient to accrately model the behavior of the focsed field. In other problems, however, for example; the design of DVD optics [14], a fll vectorial treatment is reqired to nderstand and design the necessary optics [15 17]. In the past and increasingly in more recent times (de to the widespread availability of relatively inexpensive compting power), nmerical calclation techniqes have been sed to examine the finer details of the form of the electromagnetic distribtion by solving complex vectorial diffraction integrals or sing grid or mesh techniqes sch as Finite Element Methods or the Finite Difference Time Domain method [18]. Traditionally, however, efforts were made to find analytical soltions to scalar diffraction problems so that insight into the characteristics of the problem cold be broght ot. For example, in Nijboer s approach, it was shown how an analytical soltion cold be sed to balance higher order aberrations against lower order aberrations so that the overall imaging performance of the lens approached the ideal diffraction limited case. Other approaches based on orthogonal expansions are otlined here [19,2]. Sch an insight wold not natrally arise from a nmerical investigation. Another observation is that with increased comptational power, nmerical approaches often face comptational limits, with apparently straight-forward diffraction calclations. Hence there are several good reasons for investigating diffraction problems sing analytical techniqes: (i) more insight into the diffraction process is provided, (ii) a correct analytical soltion serves as an excellent way of testing the predictions of nmerical calclations in specific cases, and (iii) sometimes it may be desirable to se a combination of analytical Corresponding athor. yang.w@t-ilmena.de 214 The Athor(s). Pblished by Taylor & Francis. This is an Open Access article. Non-commercial re-se, distribtion, and reprodction in any medim, provided the original work is properly attribted, cited, and is not altered, transformed, or bilt pon in any way, is permitted. The moral rights of the named athor(s) have been asserted.

2 S58 Y. W and D.P. Kelly and nmerical techniqes to best model a given diffraction problem. In scalar theory the Kirchhoff Fresnel diffraction integral is often sed to model the propagation of coherent monochromatic light in free space. In many cases, however, the less restrictive paraxial approximation still provides sfficiently accrate reslts. In this instance, the Fresnel transform can be sed to model the diffraction and the propagation of light. In this manscript, we assme that the paraxial approximation is valid and se the Fresnel transform to model the diffraction and propagation of light in free space. When examining the diffraction of monochromatic light from a circlar apertre, Lommel [1] introdced two fnctions to solve the integral in In 1947, Nijboer [21,22] gave another soltion of the integral sing Zernike polynomials. With this soltion, we can calclate the 3D field distribtion behind a perfect converging lens. This soltion can also be extended to inclde the effects of aberration. In 22, Cao [9] developed another series expansion, and sed it to solve the diffraction integral. These analytical soltions are all mathematical eqivalent. However, none of them are a closed form soltion and hence reqire an infinite sm of series terms. To calclate the reslt from one of these soltions, in practice, reqires trncating the infinite series when a desired accracy has been achieved. As we shall see each soltion, has different convergence properties, which vary depending on the spatial location in the otpt plane. In this manscript, we find some of these soltions are stable and converge qickly in some regions, however, do not perform so well in other locations. We analyze the reasons behind this and provide gidelines on how to choose the appropriate soltion for a given spatial location. In order to compare the performance of each of these analytical soltions, it was necessary to measre their accracy against a common benchmark. We therefore also examine the nmerical integration of the diffraction integral in polar coordinates. The radial variable is discretized in S niform steps, ρ, over the range: ρ 1. By increasing S, we improve the accracy of the nmerical soltion. Here, we find that this discretization process prodces an infinite set of replicas in the otpt diffraction domain that can overlap with each other if S is not large enogh. When that happens, the replicas add coherently leading to erroneos nmerical reslts. We derive an expression describing the effect of this sampling process and find that the replicas are located in concentric circles abot the origin with radii of λz/ ρ, where λ is the wavelength of the light, z the propagation distance and m is the replica order (the replica radii are given here in physical coordinates, however, later in the paper, we find it more convenient to se a normalized coordinate system). These replicas have fndamentally different properties from the replicas that arise in a Cartesian coordinate system, see for example Ref. [1]. This theoretical Figre 1. Schematic view of the diffractive geometric. reslt may have significant implications for the iterative design of optical elements in polar coordinate systems and nderstanding the loss of information that occrs when sing nconventional sensors that measre at discrete radial locations. We se Figre 1 to illstrate the optical system that we wish to analyze. A plane wave is incident on a perfect converging spherical lens. In the geometrical approximation, the focsed light wold converge to an ideal point sorce at the focs. In paraxial wave optics, however, diffraction introdced by the finite extent of the focsing lens apertre, cases the complex amplitde distribtion in the focal plane to spread ot over the plane. If one traces the intensity distribtion in the focal plane, moving radially ot from the focs, one observes a bright central lobe which changes to series of bright and dark rings as one moves ot along the plane. We wish to examine the distribtion over the entire focal volme, which reqires a more complex analytical soltion. We begin or analysis by defining the Fresnel transform in same manner as Ref. [23] (see Chapter 4), A(x, y, z) = eikz iλz S [ ( A x, y ) 2z e ik (x x) 2 +(y y) 2] dx dy, where k = 2π/λ is the wavenmber and i is the imaginary nit. We now wish to describe the operation of a perfect converging thin lens. Again from Ref. [23] in Chapter 5 we see that, A (x, y ) = e i 2 k f (x 2 +y 2 ), (2) where f is the focal length of the lens. De to the symmetrical property of the complex field, we se a cylindrical coordinate system instead of the Cartesian coordinate system, with following relations: x = r cos φ, (3) y = r sin φ, (4) x = R cos θ, (5) y = R sin θ. (6) Sbstitting Eqations (2) (6) into Eqation (1), then sing a property of the Bessel fnction (see Ref. [23], Chapter 2, pp 28, 2 3), (1)

3 Jornal of Modern Optics S59 J (α) = 1 2π e iα cos(θ φ) dθ (7) 2π we arrive at the following reslt, A(r, z) = 2π iλz e jk ( ) z+ r2 a 2z e iπ R2 λ ( ) ( 1 z 1 f ) 2π Rr J RdR, λz (8) where J is the Bessel fnction of the first kind. We introdce the normalize parameter ρ, ρ = R/a 1. (9) We also se two normalized optical coordinates,vto make the integral concise, ( = 2πa2 1 λ f 1 ), (1) z v = 2πar λz. (11) Sbstitting Eqations (9) (11) into Eqation (8), we get, A(,v)= e jkc2πa2 λf T (,v). (12) iλf where c = f (v 2 λ 2 + 2π 2 a 2 )/π(2πa 2 λf ) and T (,v) is, T (,v)= e iρ2 2 J (vρ)ρdρ. (13) Using Eqation (12), we can calclate the complex amplitde of the points behind the lens. In the next section, however, we first concentrate on finding a soltion to the integral T (,v), sing for different analytical methods. As shown in Figre 1, the diffraction field can be divided into two areas: if > v, it is in the illmination area; And if < v, it is in the geometrical shadow area. Note that, if = v, the point is at the bondary between the illmination area and the shadow area, and that the focal point is located at the coordinates =, v =. It seems reasonable to expect that the varios analytical soltions have different properties in each of these regions. We have organized the manscript in the following manner: In Section 2, we derive for different bt mathematically eqivalent analytical soltions for T (,v). Then in Section 3, the relative performance of each of these soltions is compared, i.e. in relation to their nmerical accracy, speed of convergence for a range of different spatial locations behind the focsing lens and the speed of comptation. In Section 4, we examine the implementations of the nmerical soltion in detail. Becase the integral is represented discretely, it prodces infinite a set of replicas near the original, the replicas are discssed and analyzed. We finish with a brief conclsion. 2. Analytical soltion for T(,v) In this section, we derive for different analytical soltions for the diffraction integral, Eqation (13). Figre 2. A contor plot of T nm 2 in the focal region. (The color version of this figre is inclded in the online version of the jornal.) I. The first Lommel soltion T (,v)is separated into real and imaginary parts [1], where T (,v)= C(,v)+ is(,v), (14) C(,v)= S(,v)= ( 1 J (vρ) cos J (vρ) sin Using integration by parts, A(x)B (x)dx = A(x)B(x) ) 2 ρ2 ρdρ, (15) ) ρdρ. (16) ( 1 2 ρ2 A (x)b(x)dx. (17) Where A(x) = cos( 1 2 ρ2 ) and B (x) = J (vρ), and sing the property of Bessel fnction (Ref. [24], p 18), d [ ] x n+1 J n+1 (x) = x n+1 J n (x). (18) dx we get that, C(,v)= 1 ( ) 1 [J v 1 (v) cos 2 ( ) ] 1 + ρ 2 J 1 (vp) sin 2 ρ2 dρ. (19) Using the Eqation (18) and integration by parts again, we get finally, ( ) ( ) cos 12 sin 12 C(,v)= U 1 (,v)+ U 2 (,v), (2)

4 S6 Y. W and D.P. Kelly where U n (,v) is the first Lommel fnction, as discssed in Refs. [24,25]: ( U n (,v)= ( 1) s ) n+2s Jn+2s (v). (21) v s= Similar we get that, ( ) ( ) sin 12 cos 12 S(,v)= U 1 (,v)+ U 2 (,v), (22) Sbstitting Eqations (2) (22) into Eqation (14), we get the first of or analytical soltion of Eqation (13), T 1 (,v)= U 1(,v)+ U 2 (,v)i e i 2, (23) where the sbscript 1 indicate that this is the first soltion of the integral. Note that the term J n+2s (v) in the first Lommel fnction converge to zero. In the illmination area, < v,theterm ( ) n+2s v converge to zero, too. It, therefore, is appropriate to se this soltion in the illmination area [26,27]. II. The second Lommel soltion We now provide a soltion to the Eqation (13) sing the second Lommel fnction. Integration by parts is still sed. Bt with A(x) = J (vρ) and B (x) = cos( 1 2 ρ2 ) [1,28], or sing alternatively the eqations from Ref. [24], Chapter 16, pp , V 1 (,v) V (,v)i e i 2 = 1 J (vρ)e iρ2 2 ρdρ, (24) J (vρ)e iρ2 2 ρdρ = i e iv2 2. (25) V n (,v)is the second Lommel fnction, ( V n (,v)= ( 1) s v ) n+2s Jn+2s (v). (26) s= Using Eqations (25) (24), the soltion is, T 2 (,v)= i e iv2 2 V 1 (,v) V (,v)i e i 2. (27) Using the second Lommel fnction, the term ( ) v n+2s appears instead of ( ) n+2s v in Eqation (21), this indicates the soltion converges in the shadow area. So we propose to se this soltion in the geometric shadow area [26,27]. III. Nijboer s soltion with Zernike polynomials We now trn ot attention to employing the third soltion. In Eqation (13), we have the term e iρ2 2 = e i 4 e i 4 (2ρ2 1). (28) and from Ref. [25] and Ref. [29], we note that, π e ic(2ρ2 1) = i n (2n + 1)J 2c n+ 1 (c)p n (2ρ 2 1), 2 n= (29) where P n is the Legendre polynomial, which is related to the Zernike polynomial (Ref. [22]), as follows P n (2ρ 2 1) = R2n (ρ). (3) R2n (ρ) is the Zernike polynomials, n m 2 Rn m (ρ) = ( 1) k (n k)! k! ( m+n k= 2 k )! ( n m 2 k )! ρn 2k, (31) Sbstitting Eqations (28) (3) into Eqation (13), and introdcing the parameter c = /4, T (,v)= e i 2π 4 i n (2n + 1)J n+ 1 ( ) 2 4 n= R 2n (ρ)j (vρ)ρdρ. (32) Another relationship, which is important in the diffraction theory of aberrations [22], is that R2n (ρ)j (vρ)ρdρ = ( 1) n J 2n+1(v). (33) v Sbstitting Eqation (33) into Eqation (32) gives third soltion for Eqation (13), T 3 (,v) = e i 4 n= 2π ( i)n (2n + 1)J n+ 1 2 ( 4 ) J2n+1 (v). v (34) IV. Cao s soltion Recently, Cao defined a family of generalized Jinc fnctions w n (v) as follows [9], w n (v) = 1 v v 2n+2 t 2n+1 J (t)dt. (35) The zero order(n = ) of this fnction is the traditional Jinc fnction. We rearrange the integral as follows, with ρ = t/v, w n (v) = ρ 2n+1 J (vρ)dρ. (36) This can also be rewritten as the form of a polynomial [9], w n (v) = n ( 2) m n! (n m)! m= J m+1 (v) v m+1. (37) Using the Taylor series expansion for e 1 2 ip2, it can be shown that ) n ( 1 2 i T (,v)= ρ 2n+1 J (vρ)dρ. (38) n! n=

5 Jornal of Modern Optics S61 Figre 3. Intensity along the focal plane ( = ) when five series terms are sed in the calclations. (The color version of this figre is inclded in the online version of the jornal.) Figre 4. Intensity distribtion along the optical axis (v = ) retaining five series terms. (The color version of this figre is inclded in the online version of the jornal.) Sbstitting Eqations (36) (37) into Eqation (38) we get a forth soltion of the form, ) n ( 1 2 i n T 4 (,v)= ( 2) m n! J m+1 (v) n! (n m)! v m+1. n= m= (39) Applying these for soltions, we can calclate the intensity distribtion, to within a constant mltiplicative factor, at any plane behind the lens, I T (,v)= T(,v) 2. (4) 3. A comparison of the soltions It mst be emphasized that, all of the for analytical soltions, derived in the previos section, are mathematically eqivalent. They all involve smming over an infinite nmber of Bessel fnctions to solve the integral exactly. In practice, however, only a finite nmber of terms can be sed in the calclation and so the rate of convergence of each soltion is of very real practical importance. It is shown that the rate at which each method converges has strong spatial dependence and we establish a relationship between the spatial location and the convergence rate for each soltion. In order to compare the relative performance of each of the different analytical soltions, we have fond it sefl to compare the reslts to those fond by directly nmerically integrating the diffraction integral Eqation (13). We refer to this nmerical soltion as T N, and present one reslt for sch a calclation in Figre 2. In Section 4, we detail how this calclation is implemented nmerically. We first, however, compare the analytical soltions for given nmbers of series terms along different cross-sections throgh the focs. We then examine how many series terms are reqired for each soltion to meet the same convergence criteria at a set of specific points, P 1, P 2 and P 3,see Figre 2. Varios cross-sections and spatial locations are Figre 5. Intensity at the bondary ( = v ) of the illmination area. Again only five series terms are sed in the calclations. (The color version of this figre is inclded in the online version of the jornal.) identified in Figre 2. In Section 3.3, we examine the differences between each analytical soltion and the nmerically calclated reslts and prodce a set of error maps so that the significants of these errors can be visalized. In Section 3.4, we examine the time taken to calclate the reslts sing each of the analytical soltions A comparison of the analytical soltions along several cross-sections throgh the focal region We begin or comparison of the analytical soltions by examining their predictions along three different crosssections, each of which passes throgh the focal point. Three different cross-sections are shown in Figre 2 and defined as: A. along the focal plane ( = ), B. along the optical axis (v = ) and C. along the bondary of the illmination area ( = v ). A. Along focal plane ( = ) For each soltion only the first five terms of the analytical series is sed for the calclation and we examine T (,v) 2 over the range 3 <v<3. We note that when =,

6 S62 Y. W and D.P. Kelly Figre 6. Intensity at the bondary ( = v ) of the illmination area with ten terms. (The color version of this figre is inclded in the online version of the jornal.) Figre 7. Converge process at P 1 as a fnction of series terms. (The color version of this figre is inclded in the online version of the jornal.) one mst perform a limiting operation in order to arrive at the correct reslt for the first Lommel fnction, second Lommel fnction and Zernike polynomial soltions. This is becase of the presence of the 1/ term. In practice, we se =.1 close to =. Figre 3 shows the log of the intensity distribtion in the focal plane. The soltions, T 1 2, T 3 2 and T 4 2 converge qickly and closely agree with each other in Figre 3. However the second Lommel fnction, T 2 2, does not converge. As when, and v, theterm(v/) n+2s in Eqation (26) reqires that more terms be inclded so as to achieve convergence. B. Along the optical axis (v = ) As in the previos calclation along the focal plane, we again se the first five series terms in each of the analytical soltions. The calclation range is 3 < < 3. Again de to the presence of the 1/v term, we only calclate the reslt for v =.1 instead of v =. The comparison is shown in Figre 4. In this region, T 2 2 and T 3 2 converge qickly (brown-yellow distribtion). Bt for the other two, five terms are not sfficient to ensre that the analytical soltion converges. The T 4 soltion converges over the range 6 < v < 6 when 5 terms are sed while the T 1 2 soltion performs better converging over the range 9 < v < 9 (see the green and ble plots in Figre 4 respectively). Therefore, we conclde that it is better to se T 2 2 or T 3 2 near the optical axis, i.e. when v =. C. Along the bondary of the illmination area ( v = ) In this region, we have v =, and the calclation range is 3 <v= < 3. The reslts from the for analytical soltions are presented in Figre 5. The reslt predicted by T 4 2 converges (green plot) for spatial locations near the focal region, however, does poorly for most other spatial locations along the cross-section. In fact each soltion converges at different rates along this cross-section. To examine this featre in more detail, we calclate the intensity distribtions again, however, now sing ten terms, see Figre 6. It is clear that the T 3 2 distribtion does not Figre 8. Converge process at P 2 as a fnction of series terms. (The color version of this figre is inclded in the online version of the jornal.) change, however, the additional terms improve the convergence of the other three soltions. So we see that the T 3 2 soltion converges most qickly along this bondary region while the T 4 2 soltion performs the worst in this region Convergence rate at three specific spatial locations: P 1,P 2 and P 3 In the previos section, we examined the performance of the analytical soltions along three different cross-sections of the focal distribtion all of which pass throgh the focs. In this section, we wish to examine in more detail how each soltion converges as a fnction of the nmber of series terms retaining. We choose three specific spatial locations as indicated in Figre 2. P 1. Near the focal plane =5, v =25 From Figre 3 we see that at the focal plane the performance of the T 1, T 3 and T 4 soltions are good, while the T 2 soltion does not converge. A qestion arises as to whether we can make T 2 converge by simply inclding more series terms? In the Figre 7, we calclate the intensity at point P 1 ( = 5, v = 25) near the focal plane, and we see that the T 1, T 3 and T 4 soltions converge rapidly. Increasing the

7 Jornal of Modern Optics S63 Figre 9. Converge process at P 3 as a fnction of series terms. (The color version of this figre is inclded in the online version of the jornal.) Figre 11. Error map of soltion T 2,log( T 2 T N ) is plotted. (The color version of this figre is inclded in the online version of the jornal.) Figre 1. Error map of soltion T 1,log( T 1 T N ) is plotted. (The color version of this figre is inclded in the online version of the jornal.) nmber of terms sed with the T 2 soltion does not seem to help and the soltion does not converge. This lack of convergence is nmerical in natre and is related to the nmber of significant figres that can be sed to represent a rational nmber in a compter. The floating-point machine reals follow the IEEE standard Doble Format sing 53 bits of machine storage (inclding one hidden bit) with a machine epsilon of 2 52 (which is approximately ) [3], and hence the inaccracy is of the order In most cases, this difference does not case nmerical instability, however here, if v, the second Lommel soltion prodces vales that are larger than 1 16.IntheT 2 series soltion at P 1, sccessive series terms are very large and opposite in sign. Adding large nmbers of opposite sign means that the reslting small rond-off errors can qickly lead to nmerical stability problems. While there are means of improving the nmerical accracy, they are not prsed here and we say that second Lommel fnction does not provide a nmerically stable soltion in this region. Figre 12. Error map of soltion T 3,log( T 3 T N ) is plotted. (The color version of this figre is inclded in the online version of the jornal.) P 2. Near the optical axis =25, v =5 Now we trn to look at the point near the optical axis = 25, v =5. From Figre 8 we see that both the T 2 and T 3 soltions converge qickly needing only a few series terms to reach a final stable vale. The T 1 and T 4 soltions converge more slowly. If we compare the manner in which the T 2 soltion in Figre 7, and the T 1 and T 4 soltions in Figre 8, converge, we can see a similar trend. Althogh the T 1 and T 4 soltions converge for all the cases presented here, we expect from this observation that similarly there are regions where these soltions will fail de to similar nmerical stability reasons. We have not fond a simple means of defining these nstable regions. Hence we adopt the rle that if the soltion does not stabilize as the series are increased, one shold choose an alternative soltion, see Section 3.3.

8 S64 Y. W and D.P. Kelly Figre 13. Error map of soltion T 4,log( T 4 T N ) is plotted. (The color version of this figre is inclded in the online version of the jornal.) Figre 15. Error of nmerical soltion with increasing nmber of samples s. (The color version of this figre is inclded in the online version of the jornal.) Figre 14. Comparison of the comptation time for the for analytical methods. (The color version of this figre is inclded in the online version of the jornal.) Figre 16. The diffractive field with replicas at the focal plane sing 3 samplings. (The color version of this figre is inclded in the online version of the jornal.) P 3. At the bondary of geometric shadow =25, v =25 This point is located away from the optical axis and the focal plane, and we can see that more terms are reqired to achieve the same level of accracy (see Figre 9). The T 1 and T 2 soltions need abot 14 terms to reach the correct answer. The T 3 soltion converges the fastest, needing only ten terms. The T 4 soltion needs more than 3 terms to converge. We note the natre of the convergence is similar in form to T 2 in Figre 7. As we move ot frther along crve C, and away from the focs, this T 4 soltion becomes increasingly nstable and will eventally reach a point where it does not converge de to the digital limit discssed before Error maps In this section, we determine regions where the soltions converge well and where the maximm errors are to be fond. We compare the for analytical soltions, each cal- clated sing only ten terms, with a nmerical soltion T N. As we show later, the error in this nmerical soltion is of the order 1 6, and so we can provide a definite error vale for the analytical soltions. In Figres 1 13 the reslting errors can be seen and regions of convergence identified. We make the following observations: (1) The T 1 (,v) soltion converges well within the illminated region behind the lens. (2) The opposite appears to hold for the T 2 (,v) soltion, where the largest errors are within the illminated region. It does however converge well in the geometrical shadow. (3) The Nijboer soltion provides the most stable soltion over the ranges we have examined. The error increases as one move away from the focs of the lens.

9 Jornal of Modern Optics S65 (4) The Cao soltion seems to be have a performance that is a combination of the Lommel soltions. It is not as robst as the Nijboer and the errors appear to be symmetrical abot the = plane. From these reslts, we conclde that while all the soltions are mathematically eqivalent they reqire a finite nmber of series terms to be sed in practical calclations. When only a finite nmber of terms are sed, the Nijboer soltion has the most robst convergence properties Comparison of the calclation time In addition to the accracy, the total time taken to calclate a nmerical reslt from the analytical soltions is an important parameter. Here we examine how the comptation time varies for each of the for analytical soltions. We calclate 1 1 points distribted in the focal region 2 < < 2, 2 <v<2. The compter we sed is an Intel(R) Core(TM) i7-26k, and the compting platform is Matlab. The comptation time for the for soltions are plotted as a fnction of the nmber of series terms, see Figre 14. The T 4 soltion takes the most time of all, then follow the Nijboer s soltion and two Lommel soltions, the time spent of the three soltions are similar. The Nijboer s soltion performs very good, not only the fast comptation time bt also the relative stable convergence properties, and noted that the time spent can be significantly improved sing lookp tables and other optimizing steps, see [15]. 4. Nmerical method to solve the integral In this section, we examine in detail how the integral in Eqation (13) can be calclated nmerically. We represent the variable ρ with S samples over the range <ρ<1, in steps of ρ = 1/S, giving s the radial spatial vector [ρ 1,ρ 2,ρ 3,...ρ S ], which is defined as ρ n = n/s. This reslts in the following discrete expression, S TN S (,v)= ρ e i 2 ρ2 n J (vρ n )ρ n, (41) n=1 and it is sed to generate the plot in Figre 2. We expect that, the more samples S we take, the more accrate or nmerical soltion becomes, however it takes obviosly more time. We take S = 2, as the reference, and see how the error E(s) = TN s T 2, N changes with increasing nmber of samples in Figre 15. We can see that as S increases the error drops dramatically. In order to ensre the high accracy of the nmerical soltion, we take S = 1 samples, which keeps the accracy to a level with 1 6 from Figre 15. This soltion is sed to generate Figre 2. Also this soltion (which is correct to a level of 1 6 ) is sed to prodce the error plots in Section 3.3. Using nmerical method, we do not need to worry abot the converge property at different locations, and the time spent (abot 4 s sing 1 samples) is in the same level of the corresponding analytical soltion T 1, T 2 and T 3 (abot 2 s sing ten terms), which is a little different from the sitation when solving high apertre vectorial diffraction integrals sing series expansion [31]. In that case, direct integration is more qick than the analytical soltions. However the nmerical approach prodces an infinite set of replicas in the diffraction plane. The replicas prodced by this nmerical integration are shown to be fndamentally different from the replicas that arise in a Cartesian coordinate system (x, y), which occrs in practice in digital holography and we refer the reader to the following pblications for more detail [32,33]. The errors introdced by the replicas are shown to be located along an infinite set of concentric circles, each centered at v =, and with a radis 2πm/ ρ, m = 1, 2, 3,... We begin or analysis of these radial replicas by rewriting Eqation (41) as follows, T N (,v)= ρ e i 2 ρ2 J (vρ)δ T (ρ)ρdρ, (42) where δ T (ρ) is the Dirac delta train fnction, δ T (ρ) = δ(ρ m ρ), (43) and δ(ρ) is a Dirac delta fnction [34], which we se to define the location of each sample. In Eqation (42)wese the well-known Poisson formla for Dirac delta train [35], we get, T N (,v)= δ(ρ m ρ) = 1 ρ ei2π m ρ ρ, (44) e iρ2 2 J (vρ)e i2π m ρ ρ ρdρ. (45) By changing integral limits and introdcing P(ρ), we rewrite Eqation (45) as a Hankel transform of zero order, T N (,v)= e iρ2 2 J (vρ)e i2π ρ m ρ P(ρ)ρdρ = {[ ] } H e iρ2 2 P(ρ) e i2π ρ m ρ (v), where P(ρ) is defined as, { 1 <ρ<1; P(ρ) = otherwise. We note that Eqation (13) can also be written as, (46) T (,v)= H {e iρ2 2 P(ρ)}(v). (47) From [36], Chapter 17, we get the convoltion property of Hankel transform, H {2π f 1 (x) f 2 (x)}(y) = F 1 (y) F 2 (y), (48)

10 S66 Y. W and D.P. Kelly and the Hankel transform of exponential fnction, H {e βr β }(v) =, (49) (v 2 + β 2 ) 3 2 where Re{β} >. Then we get, H {e i2π ρ m ρ }(v) = lim H {e a+i2π ρ ρ }(v) a + = lim a + m m ρ a i2π, (5) [v 2 + (a i2π ρ m )2 ] 3 2 Sbstitting Eqation (47), (48) and (5) into Eqation (46), T N (,v)= = 1 2π H {e iρ2 2 P(ρ)e i2π m ρ ρ }(v) i2π ρ m T (,v). [v 2 (2π ρ m )2 ] 3 2 (51) Examining Eqation (51) we see that, there is an infinite nmber of poles located at v = 2πm/ ρ, m =, ±1, ±2,..., and the radial distance between adjacent poles is 2π/ ρ. Taking S = 3 samples, setting v = (i.e. we are at the focal plane), we plot the reslt of Eqation (41) in Figre 16. In Figre 16, we can clearly see a set of concentric circles (centred at v =, radis 2πm/ ρ), which is agreement with the location of the poles in Eqation (51). Provided that these replicas are well separated from each other, the nmerical soltion provides a highly accrate reslt. When v =, the replicas introdced by the sampling operation remain at these locations, v = 2πm/ ρ. However, the extent of T (,v)increases de to defocs. If we consider the point of view of power, the power is defined as 2π T (,v) 2 vdv. We know that at the inpt plane, the power is P in = πa 2. Becase power is conserved, the power in each diffraction plane mst remain constant. However, the effect of sampling acts to prodce an infinite set of replicas in the diffraction plane. Using the variable P to refer to the power of the zero order replica, we find that the power in the annlar region, (2k + 1) π/ ρ ρ (2k + 3) π/ ρ, is2p, where k =, 1, 2,... This is becase when m = in Eqation (51), negative and positive replicas overlap spatially dobling the power in these annlar regions, see the negative and positive vales for the index m in Eqation (51). In the plot in Figre 16, N = 3, samples were sed giving P =.97P in. Becase the power associated with the replicas may not necessary be well separated, it leads to nmerically erroneos reslts. If the inpt field is sampled at a sfficiently high rate the power within the first replica window, i.e. when ρ π/ ρ, see Figre 16, P will be closed to P in, and we can be assred that the calclation has been correctly implemented. 5. Conclsions In this manscript, we set orselves the task of examining the complex amplitde distribtion in the region behind a perfect converging spherical lens. The analysis assmes that the paraxial approximation is valid and therefore sed the Fresnel transform to describe the process of diffraction. By first simplifying the reslting analytical soltion the doble integral (integration with respect to the polar angle) was redced to a single integral over the R. The soltion of this latter integral is more involved and for different bt mathematically eqivalent analytical expressions were derived in terms of an infinite series expression. We then proceeded to examine the properties of these different soltions for three different cross-sections throgh the focs. The convergence properties as a fnction of the nmber of terms reqired in each analytical soltion was examined for three different spatial locations. The soltion sing the first Lommel fnction, was fond to perform well within geometrical shadow, and converges well when v. In geometrical illmination region more terms were needed so that the soltion converged to the correct answer. The second Lommel soltion T 2 performs in a contrary manner, working well within the geometrical illmination volme and particlarly well near the optical axis. However, this soltion failed along focs plane perpendiclar to the optical axis, becase of the limited machine precision in comptational software program. The comptation time for these two soltions is most efficient of all. For Cao s soltion, we fond it is similar to the soltion sing first Lommel fnction, performing well in geometrical shadow region. However in geometrical illmination it does not work as well as the first Lommel soltion, reqiring more terms to converge. Nijboer s soltion with Zernike polynomials has the most robst soltion of all the approaches: it generally reqires a lower nmber of series terms to converge at nearly all of the locations examined here. By sing this soltion, it is not necessary to worry if the spatial location is in the illmination region or in the shadow region, and by the comptation time it performs also very good, see Section 3.4. A nmerical soltion for the diffraction integral, was examined and sed to prodce error maps presented in Section 3. This nmerical approach take a little more time than soltions T 1, T 2, and T 3, and it prodces the replicas in the diffraction domain which redce the accracy of the calclation. These replicas are cased by the sampling process, and located in along concentric circles (radii = 2πk/ ρ), with an origin at the optical axis. The more samples we take, the farther each replica located, and the more comptation time is reqired. It is possible to implement alternative nmerical soltions that are based on the fast Forier transform. This topic has been addressed by several other athors, see for example [13] and is not prsed here. Note that, we have sed the Fresnel transform, nder the Fresnel approximation the soltion is only valid if the

11 Jornal of Modern Optics S67 nmerical apertre NA is small, for high NA focsing systems NA >.5 vector diffraction theory need to be sed [37]. Fnding Yang W is spported by Photograd (Thüringer Landesgradiertenschle für Photovoltaik). Damien P. Kelly is now Jnior-Stiftngsprofessor of Optics Design and is spported by fnding from the Carl-Zeiss-Stiftng [FKZ: /121/1]. References [1] Born, M.; Wolf, E. Principles of Optics, 6th ed.; Pergamon Press, Oxford, 198; Chapters 8 9. [2] Sommargren, G.E.; Weaver, H.J. App. Opt. 199, 29, [3] Li, Y.; Wolf, E. Opt. Comm. 1981, 39, [4] Wolf, E.; Li,Y.Opt. Comm. 1981, 39, Focsing of vortex beams: Lommel treatment. [5] Sheppard, C.J.R.; Hrynevych, M. J. Opt. Soc. Am. A 1992, 9, [6] Sheppard, C.J.R. Opt. Lett. 2, 25, [7] Sheppard, C.J.R. Opt. Lett. 213, 38, [8] Sheppard, C.J.R. J. Opt. Soc. Am. A 214, 31, [9] Cao, Q. J. Opt. Soc. Am. A 22, 2, [1] Kelly, D.P. J. Opt. Soc. Am. A 214, 31, [11] Mielenz, K.D. J. Res. Nat. Inst. Stand. Technol. 1998, 13, [12] Janssen, A.J.E.M.; Braat, J.J.M.; Dirksen, P. J. Mod. Opt. 24, 51, [13] Stammes, J.J. Wave in Focal Region: Progation, Diffraction and Focsing of light, Sond and Water Waves, 1st ed.; Taylor & Francis, New York, NY, [14] van Haver, S. The Extended Nijboer-Zernike Diffraction Theory and its Applications; Technical University Delft, Delft, 21. [15] van Haver, S.; Janssen,A.J.E.M. J. Er. Opt. Soc. Rap. Pblic 213, 8, [16] Wolf, E. Proc. R. Soc. London A 1959, 253, [17] Richards, B.; Wolf, E. Proc. R. Soc. London A 1959, 253, [18] Iserles, A. A First Corse in the Nmerical Analysis of Differential Eqations; Cambridge University Press, Cambridge, 28. [19] Siegman, A.E. Lasers; University Science Books, Sasalito, CA, 1986; Chapter 16. [2] Frieden, B.R. Prog. Opt. 1971, 9, [21] Nijboer, B.R.A. Physica X 1943, 8, [22] Nijboer, B.R.A. Physica XIII 1947, 1, [23] Goodman, J. Introdction to Forier optics, 3th ed.; Mc Graw Hill, New York, NY, 1996; Chapters 2, 4 5. [24] Watson, G.N.A Treatise on the Theory of Bessel Fnctions; Cambridge University Press, Cambridge, 1992; Chapters 2, 16. [25] Gray, A.; Mathews, G.B.; MacRobert, T.M. A treatise on Bessel Fnctions, 2nd ed.; Macmillan: London, 1922; Chapter 14. [26] Kelly, D.P.; Hennelly, B.M.; Sheridan, J.T.; Rhodes, W.T. Opt. Comm. 26, 263, [27] Kelly, D.P.; Hennelly, B.M.; Sheridan, J.T.; Rhodes, W.T. Opt. Comm. 26, 263, [28] Lommel, E. Abh. Bayer. Akad. 1885, 15, [29] Abramowitz, M.; Stegn, I.A. Handbook of Mathematical Fnctions: With Formlas, Graphs, and Mathematical Tables; Dover, New York, NY, 1964; Chapter 1. [3] Noyes, J.L. Nmerical Comptation with Mathematica: Basic Analysis and Visalization; Wittenberg University Press, 29; Chapter 1. [31] Török, P.; Hewlett, S.J.; Varga, P. J. Mod. Opt. 1997, 44, [32] Kelly, D.P.; Healy, J.J.; Hennelly, B.M.; Sheridan, J.T. J. Er. Opt. Soc. Rap. Pblic 211, 6, [33] Kelly, D.P.; Hennelly, B.M.; Pandey, N. Opt. Eng. 29, 48, [34] Lohmann, A.W. Optical Information Processing; Universitätsverlag Ilmena, Ilmena, 26; Chapter 6. [35] Stern, A. Signal Process. 26, 86, [36] Polarikas, A.D., Ed. The Handbook of Formlas and Tables for Signal Processing; CRC Press LLC, Boca Raton, FL, 1999; Chapter 17. [37] Sherif, S.S.; Foreman, M.R.; Török, P. Opt. Express. 28, 16,