Effective Fresnel-number concept for evaluating the relative focal shift in focused beams

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1 Martíne-Corral et al. Vol. 15, No. / February 1998/ J. Opt. Soc. Am. A 449 Eective Fresnel-number concept or evaluating the relative ocal shit in ocused beams Manuel Martíne-Corral, Carlos J. Zapata-Rodrígue, Pedro Andrés, and Enrique Silvestre Departamento de Optica, Universidad de Valencia, Burjassot, Spain Received April 5, 1997; revised manuscript received August 7, 1997; accepted August 11, 1997 We report on an analytical ormulation, based on the concept o eective Fresnel number, to evaluate in a simple way the relative ocal shit o rotationally nonsymmetric scalar ields that have geometrical ocus and moderate Fresnel number. To illustrate our approach, certain previously known results and also some new ocusing setups are analytically examined Optical Society o America [S (98)0090-8] OCIS codes: , INTRODUCTION The evaluation o the electromagnetic ield diracted by a coherently illuminated circular aperture has constituted the aim o several research eorts over the last ew decades. Classical studies, 1, which are based on the Debye integral representation o ocused ields, reveal that when a monochromatic, uniorm, converging spherical wave is diracted by a circular aperture, the irradiance distribution in the ocal region is symmetric about the geometrical ocus, and the point o maximum irradiance is located at the ocal point. However, since the description in the ocal region by this representation is valid only when the Fresnel number o the ocusing geometry is much higher than unity, 3 these studies could not explain the shit toward the aperture o the irradiance maximum, which, under certain circumstances, was ound to appear by some researchers during the 1970 s. 4 6 It was in the early 1980 s when, on the basis o the Fresnel diraction integral, it was analytically established, 7 9 and experimentally veriied, 10 that the magnitude o the shit suered by the axial irradiance peak, i.e., the magnitude o the so-called ocal-shit eect, is governed by the Fresnel number o the ocusing geometry. More recently, it has been recognied that the ocal-shit eect is also present in systems with an obscured pupil, 11,1 in axially superresolving setups, 13 and, in general, in any type o diracting screen. 14,15 It has also been shown that this eect appears not only along the optical axis but also on any line directed toward the geometrical ocus o the incident spherical wave ront. 16 Speciically, in Re. 16 it is shown that the irradiance distribution along a line is determined by the projection o the pupil unction onto the line. So both the irradiance proile and the magnitude o the ocal shit depend on the selected line. On the other hand, it is also well established that the ocal-shit eect appears not only or uniormly illuminated diracting screens but also when some nonuniorm ocused beams illuminate a circular aperture. In this sense it was ound that when a monochromatic Gaussian beam is ocused by a thin lens, the point o maximum irradiance is not located at the geometrical ocus but is displaced toward the lens More recent studies have shown that this ocal-shit eect is governed by both the so-called Gaussian Fresnel number, 0 which is associated with the width o the incident beam, and the truncation parameter, 1, which evaluates the ratio between the radius o the lens and the beam width. The goal o this paper is to report on an analytical ormulation to evaluate the relative ocal shit or the general class o rotationally nonsymmetric scalar ields that have ocus in the sense o geometrical optics. The ormulation, which is based on an extension o the concept o eective Fresnel number applied to any ocused beam, permits us to evaluate in a quite simple way the ocal shit that appears in both converging and diverging beams and or truncated and nontruncated ocusing geometries. For describing our approach, in Section we ormulate the basic theory or evaluating the axial irradiance distribution corresponding to a ocused beam. In Section 3 we deine the eective Fresnel number over the exit plane associated with any ocused beam, and we obtain a simple analytical ormula, which explicitly depends on this parameter, or evaluating the relative ocal shit. In Section 4 we analye the role o the eective Fresnel number o a ocused beam, which is shown to be related to the eective width o the beam. Finally, in Section 5 we illustrate our approach by examining the ocal shit in some highly exempliying ocusing geometries.. AXIAL IRRADIANCE DISTRIBUTION IN FOCUSED BEAMS Let us start by considering a ocused beam, i.e., a monochromatic scalar wave ield that has ocus in the sense o geometrical optics. Thereore its amplitude distribution U(r, ) in a given plane transversal to the propagation direction, reerred to as the reerence plane or the remainder o the present paper, can be expressed as the product o a spherical wave o ocal length and a real nonnegative two-dimensional unction t(r, ). In mathematical terms, Ur, expik exp i k r tr,, (1) /98/ $ Optical Society o America

2 450 J. Opt. Soc. Am. A/ Vol. 15, No. / February 1998 Martíne-Corral et al. where k / represents the wave number o the light. Note that this rather general ormalism corresponds to the beams that provide an irradiance peak in the ocus o a spherical wave and describes, among others, two quite typical ocusing geometries in optics: (1) the case o a purely absorbing diracting screen, with or without radial symmetry, illuminated by a monochromatic spherical wave and () the situation corresponding to a truncated or nontruncated spherical Gaussian beam. Two examples o these situations are illustrated in Fig. 1. To evaluate the axial amplitude distribution o the ocused beam, we particularie the Fresnel Kirchho diraction ormula or the axial points, 7,9 i.e., h expik i 0 0 tr, exp i rr drd, () where denotes the axial coordinate as measured rom the paraxial ocal point F. At this point it is convenient to point out that the axial distances involved in Eq. () are directed. Their direction is determined by the point o the arrow (see Fig. 1). Note that both and are positive distances in Fig. 1. This act allows us to deal with diverging ocusing beams ( 0) and to describe the virtual diraction region ( 0). It is convenient to perorm two mathematical manipulations o Eq. (). First, we carry out the angular integration to obtain h expik t i 0 r 0 exp i rr dr, (3) where t 0 r 1 tr, d (4) 0 is a radially symmetric unction that stands or the aimuthal average o t(r, ). O course, any pair o ocused beams that yield the same aimuthally averaged unction t 0 (r) generate an identical axial irradiance distribution. 3,4 The second manipulation consists in the geometrical mapping r, t 0 r q 0, (5) which explicitly converts the integral in Eq. (3) into a onedimensional (1-D) Fourier transorm, namely, h q 0 exp i d, (6) where some irrelevant premultiplying constant phase actors have been omitted. Note that the lower limit in the integral has been extended to, since the unction q 0 () is identically ero or 0. Next, we recognie that the scale actor o the 1-D Fourier transorm is related to the axial variable through the relation u, (7) which is closely related to the axial coordinate deined by Lommel. Thus the axial amplitude distribution o the ocused beam can be expressed as h Qu 1 u q 0 expiud. (8) Finally, since our aim in this work is to calculate the position o the axial point o maximum irradiance, we express the axial behavior o the ocused beam in terms o the normalied axial irradiance distribution, which is obtained by simply dividing the squared modulus o Eq. (8) by the irradiance at the paraxial ocal point u 0. Mathematically, Fig. 1. Two typical examples o scalar ields that have geometrical ocus: (a) radially nonsymmetric diracting screen illuminated by a monochromatic spherical wave with ocal length, (b) truncated spherical Gaussian beam. I N u 1 u q 0 expiud q 0 d. (9)

3 Martíne-Corral et al. Vol. 15, No. / February 1998/ J. Opt. Soc. Am. A 451 From Eq. (9) it ollows that the axial irradiance distribution o the ocused beam is governed by the product o two terms that can be understood, rom the point o view o the Huygens Fresnel principle, 9,13 in the ollowing way. The irst term, which involves the 1-D Fourier transorm o q 0 (), describes at any axial point the intererence by the dierent Huygens spherical wavelets proceeding rom all points o the reerence plane. Since t(r, ), and consequently q 0 (), are real and positive unctions, the Huygens wavelets arrive in phase at the geometrical ocus, and then the maximum o this term is achieved at the origin. However, as the secondary wavelets propagate, their amplitude suers an attenuation that is proportional to the inverse covered distance. This attenuation is described in Eq. (9) by the term (1 u). It is then apparent that the competition between these two terms produces a displacement o the axial irradiance peak toward the reerence plane, resulting in the wellknown ocal-shit eect, whose magnitude we evaluate in Section RELATIVE FOCAL-SHIFT FORMULA To obtain the relative position o the greatest value o the axial irradiance distribution, one should obtain the roots o the equation d I N /du 0 (10) and select rom among them the absolute maximum. However, in many cases o interest an exact analytical result cannot be obtained by this method. To avoid this drawback, we propose to expand the normalied irradiance distribution into a Taylor series. Taking into account that the region o interest is located in the vicinities o the geometrical ocus (u 0), we may neglect the third- and higher-order terms and restrict the series to a quadratic approximation, i.e., I N u I N 0 I N 0u I N 0 u. (11) Now, by straightorward (albeit cumbersome) calculation, we obtain by making use o the moment theorem 5 the ollowing approximated parabolic expression or the normalied axial irradiance distribution: where I N u 1 4u 4 u, (1) m m 0 m 1/ 1 m 0 (13) stands or the standard deviation o the mapped unction q 0 () and m n q 0 n d, (14) denotes the nth moment o q 0 (). By virtue o Eq. (1), we ind that the position and the height o the maximum o the quadratic axial irradiance distribution are given, respectively, by u max N, (15) e N e I N u max N e 1, (16) where we have introduced a new parameter, named here the eective Fresnel number o the ocused beam, deined as N e /. (17) This parameter characteries any ocused beam at the reerence plane and is proportional to the standard deviation o q 0 (), which can be interpreted as a measure o the eective width o the beam. Now, by combining Eqs. (7) and (15), we ind that the value o the relative ocal shit is given by 1 N. (18) e This relevant ormula, which is a generaliation o the results obtained or both the circular aperture 7,9 and the Gaussian beam, 0,1 indicates that any scalar ield belonging to the general class o ocused beams suers a relative ocal shit that, within the second-order approximation, is proportional to the inverse square o the eective Fresnel number o the beam. So it is apparent that, independently o both the proile and the scale o the unction t(r, ) and the sign o the ocal length, any pair o ocused beams having the same value or N e, and consequently the same eective width, exhibit the same relative ocal shit. The minus sign in Eq. (18) indicates that the distances and have opposite sign. Thereore, either in a converging ( 0) or in a diverging ( 0) ocusing geometry, the displacement o the axial irradiance peak is always toward the reerence plane. Note that in the diverging case the virtual ocal shit can be viewed by ocusing a microscope with low magniying power behind the reerence plane, assuming that the optical instrument has a numerical aperture large enough not to introduce signiicant diraction eects on its own account. It is important to remark that, to the best o our knowledge, the virtual ocal shit has been reerred to only by Nye 6 when he studied the light diracted by small unstopped lenses. For illustrating the variation o the ocal-shit eect with the value o the eective Fresnel number o the ocused beam, in Fig. we have plotted the relative ocal shit against N e. From this igure we iner that the modulus o the relative ocal shit greatly increases as N e decreases, being, or example, o the order o 0.1% when N e 10 and.5% when N e. Concerning the accuracy o Eqs. (16) and (18), it can be easily shown (see Appendix A) that the relative error in the evaluation o the relative ocal shit, resulting rom the parabolic approximation, is given by N e 1, (19) whereas the error in the estimation o the height o the irradiance peak is

4 45 J. Opt. Soc. Am. A/ Vol. 15, No. / February 1998 Martíne-Corral et al. Fig.. Relative ocal shit as a unction o the eective Fresnel number o the ocused beam. Fig. 3. Relative error associated with the ocal shit determined by Eq. (18) versus the eective Fresnel number o the ocused beam. N e I max N e 1. (0) 3 We conclude rom Eqs. (19) and (0) that both errors depend solely on the eective Fresnel number o the ocused beam. In particular, the variation o the ratio ( )/ with N e is as indicated in Fig. 3. The examination o the curve in Fig. 3 reveals that the error is low or moderate values o N e (or example, when N e.0, the error is less than 5%; and it is even less than 1% i N e 4.5). On the contrary, or N 1, ( )/ increases dramatically as N decreases. 4. EFFECTIVE FRESNEL-NUMBER ANALYSIS In Section 3 we established that the position and the height o the axial irradiance peak or a general ocused beam are determined by a single parameter, the eective Fresnel number o the ocusing geometry. Hence it seems to be convenient to investigate more the role o the eective Fresnel-number parameter. To this end we now recall the interpretation o the axial irradiance distribution in terms o the Huygens Fresnel superposition principle, as was done in Section. Note that, arising rom this principle, when we deal with a large pupil unction t(r, ) that involves a huge number o Fresnel ones in the process o Huygens wavelet intererence, the 1-D Fourier transorm in Eq. (9) provides a very sharp unction centered at the geometrical ocus u 0. So its value is negligible unless u is very small. In this way the actor (1 u) is approximately unity in the region where the 1-D Fourier transorm is nonero, and consequently it can be ignored. In this case the axial irradiance distribution is ixed only by the intererence term and is symmetric around the geometrical ocus, since the Fourier transorm o a real unction is Hermitian. However, when t(r, ), and consequently q 0 (), are narrow, which implies that a small number o Fresnel ones are involved in the intererence process, the Fourier transorm gives rise to a unction that decreases smoothly in the neighborhood o the paraxial ocal point. Now the attenuation term (1 u) is not unity in the whole region in which the Fourier transorm is clearly dierent rom ero. The presence o this term, whose value increases with negative values o the product u, shits the irradiance peak toward the reerence plane, resulting in the ocalshit eect. It is then clear that the axial irradiance distribution o the ocused beam, and thus the relative ocal shit, are determined by both the number o Fresnel ones involved in the intererence process and their relative contribution. In other words, the axial pattern is governed by the eective width o the beam at the reerence plane. In this context we conclude that the eective Fresnel number, which is proportional to the standard deviation o the unction q 0 (), evaluates in a certain way the eective number o Fresnel ones in the ocusing geometry involved in the above process, and thereore it also takes into account the importance o the relative contribution o each individual one in the inal result. 5. APPLICATION TO SEVERAL FOCUSING GEOMETRIES We start by considering the case o an annular aperture o inner and outer radii a (0 1) and a, respectively, illuminated by a monochromatic spherical wave o ocal length. According to Eq. (17), the eective Fresnel number or this ocusing geometry is N e 1 a 1. (1) Note that the above eective Fresnel number is indeed, apart rom a proportionality actor 1/1, equal to the classical Fresnel number or annular ocusing setups, 11,15 which is given by N 1 a, () which implies that, because o the close connection between these concepts, our ormalism will allow us to reproduce the classical results, as we show next. By substituting the value o N e into Eqs. (16) and (18), we ind that the relative position and the height o the axial irradiance peak are

5 Martíne-Corral et al. Vol. 15, No. / February 1998/ J. Opt. Soc. Am. A N, (3a) N I N N 1, (3b) which are expressed in terms o the classical Fresnel number. We emphasie that, i we simply set 0, the above equations are valid, in particular, or describing the ocal shit or the quite typical case o the circular aperture. However, although equivalent equations have been derived elsewhere 7,9 or converging illumination, it was not recognied that they are also valid or describing the virtual ocal-shit eect when the circular aperture is illuminated by a diverging monochromatic spherical wave. As a second example we consider the case o a spherical Gaussian beam that illuminates a circular aperture o radius a, being the radius o curvature o the beam at the aperture plane. To carry out this study, it is convenient to introduce the truncation parameter o the ocusing geometry, deined by a/, (4) where stands or the width o the Gaussian beam at the reerence plane. The expression or N e in this case is N e 1 1/ exp. (5) 1 exp Now, by substituting Eq. (5) into Eq. (18), we ind that the relative ocal shit corresponding to a truncated spherical Gaussian beam is given, in terms o the truncation parameter, by Up to now we have illustrated our approach by reexamining, with our approach, the general case o the annular aperture or the truncated Gaussian beam, which have indeed been studied in the literature. Now we go one step urther and investigate the axial behavior or two diracting screens, reported in the optical literature or certain applications, when they are illuminated with a monochromatic spherical wave ront o ocal length. The irst screen under study is that o mapped transmittance q s 0 4/a 0.5 i 0 a, (9) 0 otherwise which produces a superresolving axial irradiance pattern. 8 The mapped transmittance or the second selected diracting screen is q a 0 1 q 0 s i 0 a, (30) 0 otherwise which produces an axial apodiation eect. 9 The proiles o both screens in the coordinate and in the radial coordinate are shown in Fig. 4. It is straightorward to obtain analytically the standard deviation, and conse- 1 exp/1 exp. (6) As in the previous example, next we analye some particular case. First, or strong truncation, the width o the incident beam is much higher than the radius o the circular aperture. In this case the value o the truncation parameter is almost ero. By perorming the limit when tends to ero in Eqs. (5) and (6), we achieve the eective Fresnel number and the relative ocal shit corresponding to the uniormly illuminated circular aperture. In the other extreme case, corresponding to an unapertured Gaussian beam (weak truncation), the coeicient tends to ininity. Here the limit value or the eective Fresnel number is N e /. (7) This expression is just the same as that o the classical Fresnel number deined or ocused Gaussian beams. 0 By substitution o this value into Eq. (18), the relative ocal shit or unapertured Gaussian beams is 4, (8) which reproduces in a quite good approximation the previously known result. 7 Fig. 4. Amplitude transmittance o the axially superresolving diracting screen (solid curves) and the axially apodiing parabolic ilter (dashed curves): (a) -space representation, (b) radial coordinate representation.

6 454 J. Opt. Soc. Am. A/ Vol. 15, No. / February 1998 Martíne-Corral et al. quently the corresponding eective Fresnel number, or these quadratic unctions. We have s N e 3 a 0, (31a) N a e 1 0 a. (31b) When we take into account Eq. (18), the relative position o the axial irradiance peak is given, respectively, by s max, (3a) 0 3 a a 0. (3b) a Note that i all geometric parameters in both optical setups are equal, it is apparent that 3. There- a s ore the axially apodiing diracting screen is three times more sensitive to the ocal-shit eect than that which produces axial superresolution. This is what one would expect, since, as we mentioned in Section 4, the wider the central lobe o the axial-distribution irradiance, the greater the amount o the ocal shit. In this work we recognie that this general statement can be quantiied by means o the eective Fresnel-number parameter. 6. CONCLUSIONS We have presented a quite simple analytical ormulation or evaluating, in a second-order approximation, the relative ocal shit or any ocused beam with or without radial symmetry. The proposed ormula depends solely on a beam parameter named the eective Fresnel number o the ocused beam. This important result means that any pair o ocused beams with the same eective Fresnel number show the same relative ocal shit, independently o their geometric parameters and their transverse proile. Moreover, our ormalism also permits us to quantiy the virtual ocal-shit eect or any diverging ocused beam. Finally, we have illustrated our approach by discussing the ocal-shit eect shown by dierent ocusing geometries. APPENDIX A The error involved in the approximation carried out in Eq. (11), i.e., the modulus o the dierence between the exact axial irradiance and the value provided by the quadratic approximation, is given by the absolute value o the so-called Lagrange remainder, namely, RI N ;0,u I Nu 0 u 3, (A1) 6 where u 0 is an axial-coordinate value between 0 and u. I we assume that the normalied irradiance distribution is a slowly varying unction and that the axial coordinate u under investigation is close to the paraxial ocus, the approximation I N u 0 I N (A) may be applied. I we particularie under this assumption the value o the Lagrange remainder or u u max, we obtain that the error in the estimation o the height o the irradiance peak, which is derived by inserting Eq. (15) and relation (A) into Eq. (A1), is N e I max I N u max N e 1. (A3) 3 For obtaining the error in the evaluation o the relative ocal shit, irst we express this parameter in terms o I N (u max ) by combining Eqs. (16) and (18). In this way Eq. (18) can be written as 1 1. I N u max (A4) Now, by merely applying standard error propagation techniques, we obtain 1 N e N e 1. (A5) Finally, the relative error in the determination o the relative ocal shit is achieved by simply dividing Eq. (A5) by the absolute value o Eq. (18). Thus ACKNOWLEDGMENTS N e 1. (A6) Part o this work was presented at the 80th Optical Society o America Annual Meeting, held in Rochester, N.Y., in October This work was supported by the Dirección General de Investigación Cientíica y Enseñana Superior (grant PB C0-01), Ministerio de Educación y Ciencia, Spain. C. J. Zapata-Rodrígue grateully acknowledges inancial support rom this institution. REFERENCES AND NOTES 1. E. Collet and E. Wol, Symmetry properties o ocused ields, Opt. Lett. 5, (1980).. M. Born and E. Wol, Principles o Optics, 6th ed. (Pergamon, Oxord, 1980), Sec E. Wol and Y. Li, Conditions or the validity o Debye integral representation o ocused ields, Opt. Commun. 39, (1981). 4. D. A. Holmes, J. E. Korka, and P. V. Avionis, Parametric study o apertured ocused Gaussian beams, Appl. Opt. 11, (197). 5. A. Arimoto, Intensity distribution o aberration-ree diraction patterns due to a circular aperture in large F-number optical systems, Opt. Acta 3, (1976). 6. M. A. Gusinow, M. E. Riley, and M. A. Palmer, Focusing in a large F-number optical system, Opt. Quantum Electron. 9, (1977). 7. Y. Li and E. Wol, Focal shits in diracted converging spherical waves, Opt. Commun. 39, (1981). 8. J. J. Stamnes and B. Spjelkavik, Focusing at small angular apertures in the Debye and Kirchho approximations, Opt. Commun. 40, (1981). 9. M. P. Givens, Focal shits in diracted converging spherical waves, Opt. Commun. 41, (198). 10. Y. Li and H. Plater, An experimental investigation o di-

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