Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-fresnel-number optical system

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1 Zapata-Rodríguez et al. Vol. 7, No. 7/July 2000/J. Opt. Soc. Am. A 85 Gaussian imaging transormation or the paraxial Debye ormulation o the ocal region in a low-fresnel-number optical system Carlos J. Zapata-Rodríguez, Pedro Andrés, Manuel Martínez-Corral, and Laura Muñoz-Escrivá Department o Optics, University o Valencia, E-4600 Burjassot, Valencia, Spain Received June 25, 999; revised manuscript received March 20, 2000; accepted March 22, 2000 The Debye ormulation o ocused ields has been systematically used to evaluate, or example, the pointspread unction o an optical imaging system. According to this approximation, the ocal wave ield exhibits some symmetries about the geometrical ocus. However, certain discrepancies arise when the Fresnel number, as viewed rom ocus, is close to unity. In that case, we should use the Kirchho ormulation to evaluate accurately the three-dimensional amplitude distribution o the ield in the ocal region. We make some important remarks regarding both diraction theories. In the end we demonstrate that, in the paraxial regime, given a deocused transverse pattern in the Debye approximation, it is possible to ind a similar pattern but magniied and situated at another plane within the Kirchho theory. Moreover, we may evaluate this correspondence as the action o a virtual thin lens located at the ocal plane and whose ocus is situated at the axial point o the aperture plane. As a result, we give a geometrical interpretation o the ocal-shit eect and present a brie comment on the problem o the best-ocus location Optical Society o America [S (00) ] OCIS codes: , 0.220, INTRODUCTION The knowledge o the three-dimensional light distribution in the vicinities o the ocus is o particular importance, or example, in estimating the transverse resolution power and the tolerance in the setting o the receiving plane in an optical imaging system. 2 The properties o the out-o-ocus monochromatic images o a point source given by a diraction-limited optical system with a circular exit pupil were treated by Debye, 3 who established that the ield is a superposition o plane waves whose propagation vectors all inside the geometrical cone ormed by drawing straight lines rom the ocal point through the edge o the aperture. 4 Also, he derived certain general eatures o the diracted ield both near and ar away rom the ocus. For example, he ound that the amplitude, and hence also the intensity, possesses inversion symmetry about the ocus, where the point o maximum intensity in the ocal region is located. 5 This result was later extended to the more general class o monochromatic scalar wave ields that have a ocus in the sense o geometrical optics. 6 In a number o publications that appeared in recent years it was demonstrated that the classic Debye theory regarding the amplitude distribution in the ocal region does not predict correct results under all circumstances. Using the Kirchho approximation, which assumes that the ield inside the aperture is set equal to the ield that would exist there in the absence o the aperture and vanishes outside the aperture, Arimoto, 7 and later Stamnes and Spjelkavik 8 and Li and Wol, 9 ound that the intensity distribution about the geometrical ocal plane no longer exhibits the well-known symmetry properties. Moreover, the point o maximum intensity o the diracted wave may not be at the geometrical ocus o the incident wave but may be located closer to the aperture. Experimental evidence o this phenomenon has been published elsewhere. 0, This situation may be better understood i we bear in mind that the Debye approximation results when, in addition to the Kirchho approximation, we make the assumption that the aperture is ininitely distant rom the ocal region. 4 Wol and Li 2 derived a simple condition under which the Debye integral representation may be expected to give a good approximation o the structure o a ocused ield. For low-angular-aperture systems this condition may be replaced by the requirement that the number o Fresnel zones in the aperture, when viewed rom the geometrical ocus, be large compared with unity. Simultaneously, they showed 9 that the relative ocal shit, that is, the ratio o the shit o the point o maximum intensity to the distance between the geometrical ocus and the plane o the aperture, depends only on the Fresnel number. Recently, the concept o the eective Fresnel number that may be applied to any rotationally nonsymmetric scalar ield that has a paraxial ocus was ormulated. 3 In the case o a circular clear aperture, the expression or the ield in the ocal region based on the Fresnel Kirchho approximation may be expressed, as with the paraxial Debye approximation, in terms o the Lommel unctions, but with arguments that are scaled by a certain actor. 8,4 In the present paper we demonstrate that, rom the basis that both ormulations are mathematically identical (even with aberrated or apodized ocused beams), given a deocused diraction pattern in the Debye approximation, we ind a similar transverse pattern /2000/ $ Optical Society o America

2 86 J. Opt. Soc. Am. A/ Vol. 7, No. 7/ July 2000 Zapata-Rodríguez et al. in the Kirchho approximation but magniied and located at another position. In Section 4 we interpret this threedimensional mapping as the action o a negative thin lens with a ocal length given by the radius o curvature o the incident wave ield and situated at the ocal plane. Finally, in Section 5 we treat the ocal-shit eect rom a geometrical point o view, and we discuss the concept o the plane o best ocus in image-orming optical systems. 2. DESCRIPTION OF THE FIELD IN THE FOCAL VOLUME Let us start by considering a scalar monochromatic spherical wave, emerging rom an opaque screen o radius a. Let F be the geometrical ocus o the wave (see Fig. ), assumed to be located on the normal o the aperture, through its center O, at a distance rom it. According to the Huygens Fresnel principle, 3 the wave ield at any point P that is not too close to the plane o the aperture is, as predicted by the Fresnel Kirchho diraction theory, given by UP expik i W expiks AS s ds, () where k 2/ is the wave number, s is the distance between the point o observation P and a typical point Q on the spherical wave ront passing through the center o the aperture, and the integration extends over the wave ront. In Eq. () we have neglected the well-known obliquity actor that takes values close to unity when the wave ield is evaluated in the vicinities o the geometrical ocus. The unction A(S) stands or the amplitude distribution o the exiting wave ield that makes possible the study o general types o ocused ields, e.g., diracted spherical waves in the presence o aberrations 5 and those arising rom the ocusing o Gaussian laser beams. 6 First we will determine the diracted ield at the ocal plane o the spherical beam. For that purpose, it is usual to use the paraxial approximation, which gives suitable results, assuming that a, a/ 2. (2) The approximation is based on the binomial expansion o the distance s in the exponent o Eq. (), giving s x 0 y 0 x y 0. (3) 2 However, or the s appearing in the denominator, the error introduced by dropping all terms but is generally acceptably small. The resultant expression or the ield at the ocal plane thereore becomes U 0 x 0, y 0 i 2 exp i k 2 x 0 2 y 0 2 A, exp i k x 0 y 0 dd. As is well known, on the ocal plane we observe the Fraunhoer diraction pattern o the ield emerging rom the plane o the aperture. Also, a quadratic phase actor appears, which is not accessible by direct observation o the intensity distribution o the ocal ield. It would seem more interesting to obtain the amplitude distribution o the wave ield in adjacent planes o the ocal volume. In the Fresnel regime, the amplitude distribution may be obtained by means o the two-dimensional convolution o the wave ield at the ocal plane and the unit impulse response associated with the ree-space propagation; 7 that is, UP U 0 x 0, y 0 iz exp i k 2z x x 0 2 y y 0 2 dx 0 dy 0, where z is the distance between the point o observation P and the ocal plane. Equation (5) is itsel the Fresnel Kirchho diraction integral that can be applied over arbitrarily short distances z i the optical beam is truly paraxial. We should mention that an inessential linear phase actor exp(ikz), which cannot be observed in the intensity distribution o the wave ield, has been removed. When we substitute Eq. (4) into Eq. (5), we can express the wave ield at point P o the ocal region in terms o the amplitude distribution at the illuminated opaque screen. For that purpose, we use the expression 8 (4) (5) expiax 2 ixdx 2 2a exp i 4a 2. (6) Fig.. Schematic diagram o the ocusing setup. Ater a somewhat long but straightorward calculation, we inally obtain the Fresnel Kirchho diraction equation or the amplitude distribution in the ocal region:

3 It has been shown 5 that the Debye integral given in Eq. (9) presents some important symmetry properties. For instance, when A(, ) is a real unction, the amplitude, and hence the intensity, possesses inversion symmetry about the ocus. Also, the phase has inversion antisymmetry, apart rom an additive term o hal a period. 6 Ad- Zapata-Rodríguez et al. Vol. 7, No. 7/July 2000/J. Opt. Soc. Am. A 87 UP i z exp k ditionally, when the unction A(, ) is positive, the point i 2 z x2 y 2 o maximum intensity in the ocal region is located at the ocal point. Now we will ind the restrictions that we must impose to ensure that the Debye approximation will give a rea- A, exp i k 2 z z 2 2 k exp i z x y dd. (7) Finally, the above expression reduces to that given in Eq. (4) when the axial coordinate z is replaced by its value at the ocal plane, that is, z DEBYE APPROXIMATION It is usual to evaluate the diracted ield o a truncated spherical wave within the Debye approximation, according to which the ield in the ocal region is a superposition o plane waves whose propagation vectors all inside the geometrical cone ormed by drawing straight lines rom the ocal point through the edge o the aperture. Under the paraxial regime, the ield at the ocal plane given in Eq. (4) generates the diraction ormula predicted by the Debye theory when the quadratic phase actor outer to the integral is removed, resulting in U D 0 x 0, y 0 A, i 2 exp i k x 0 y 0 dd; (8) that is, the Debye approximation agrees with the Kirchho ormulation when the quadratic phase actor in Eq. (4) may be neglected. When the amplitude distributions at planes adjacent to the ocal plane in the ocal region are evaluated, we may use Eq. () and later consider the Debye assumptions concerning the ocusing properties o three-dimensional waves given above. However, the resultant amplitude distribution may be obtained instead by ree-space propagating the wave ield at the ocal plane to the observation plane situated at a distance z, with the aid o the twodimensional convolution given in Eq. (5). Thus we substitute the approximated expression in Eq. (8) into Eq. (5), giving U D P A, exp i k i 2 2 z 2 2 sonably good prediction o the diracted ield. This requirement is satisied when the quadratic phase actor in Eq. (4) may be neglected. For that purpose, we note that a diraction-limited optical system, such as that represented schematically in Fig., concentrates most o the light energy at the ocal plane in an area about the ocus given by x 2 0 y ( /a) 2. For example, in the case o a circular clear pupil, the energy encircled within this area represents 90.64% o the emerging radiation energy passing through the aperture. This means that the most representative points contributing to the diracted ield at the ocal plane are those satisying the above inequality. When the phase corresponding to the quadratic term in Eq. (4) is evaluated, the maximum value is given by exp(i/n), where N a 2 / (0) stands or the Fresnel number o the ocusing geometry, that is, the number o hal-waves covered by the diracting aperture as viewed rom the geometrical ocus. When the Fresnel number is much higher than unity, it is clear that the quadratic phase actor does not introduce a noticeable variation in the phase o the ocal plane, which implies that this term may be ignored. Hence it is concluded that the Debye integral representation o spherical waves should be applied only when the Fresnel number o the ocusing geometry is large compared with unity. Another interesting point is the act that the Debye approximation results when, in addition to the Kirchho approximation, we make the assumption that the aperture is ininitely distant rom the ocal region. A telecentric optical system ulills this severe restriction, which implies that Eqs. (8) and (9) hold or this case. 9 However, we will observe the previously mentioned symmetries about the geometrical ocus when the wave ield is ocused onto a region whose axial magnitude is much lower than the ocal length. In relation to this point, we can estimate the ocal depth o an imaging optical setup in terms o the wavelength and the numerical aperture o the system, N.A. sin, as the quantity 3,20 z 4 sin 2 /2 a 2, () exp i k x y dd. (9) where we have introduced the paraxial approximation. Then we should impose the inequality z to guarantee that the Debye approximation is valid. Taking into account that we can express the Fresnel number in terms o the ocal depth and the ocal length, i.e., N /z, (2)

4 88 J. Opt. Soc. Am. A/ Vol. 7, No. 7/ July 2000 Zapata-Rodríguez et al. we inally obtain that, as deduced previously with alternative reasoning, the Fresnel number should be constrained to values much higher than unity. 4. THREE-DIMENSIONAL MAPPING FOR THE DEBYE REPRESENTATION OF THE FOCAL FIELD We have shown that to ind the ocal wave ield o a truncated spherical beam it is possible to employ either the Kirchho ormulation or the Debye approximation. The latter gives suitable results when the wave ield is ocused mostly on a region about the geometrical ocus whose lateral and axial dimensions are much smaller than the aperture diameter and the ocal length 4 : meanwhile, noticeable inaccuracies exist when the ocal region increases in size. Most investigators have stressed the act that both theories show such departures. 8,2 However, we will demonstrate that the two theories provide the same collection o irradiance transverse patterns that constitute the ocal region but with a dierent scale and position. By comparing Eqs. (7) and (9), we observe that both diraction ormulas give an amplitude transverse pattern by perorming a two-dimensional Fourier transorm o the product o the unction A(, ) and a quadratic phase actor. This product is usually called the deocused pupil unction or the optical system. 2 Thus, when evaluating a given intensity transverse pattern located at a distance z rom the ocal plane, provided by the Kirchho ormula in Eq. (9), we may obtain the same transverse distribution within the Debye theory, in a unique plane placed at a distance z D z z (3) but applying a lateral magniication given by Mz z. (4) Such coordinate transormations are successully utilized in space-variant imaging systems that are to be modeled as an aocal telecentric space-invariant system. 22 The replicated transverse irradiance distributions are obtained when the two deocused pupil unctions in Eqs. (7) and (9) coincide, while the lateral magniication arises rom the ratio o the two scales in both Fourier transorm kernels. Consequently, the ocal volume given by the Debye theory is then deormed in such a way that it conserves the same transverse structure, but both the lateral magniication and the axial distribution o the wave ield are altered (see Fig. 2). To interpret adequately the transormation that the ield undergoes in the ocal region within the Debye approximation, we ocus our attention on the amplitude transverse distribution at the ocal plane o the ocusing setup given by both theories. According to Eqs. (4) and (8), the two expressions dier in the use o a quadratic phase term that multiplies the spectrum o the pupil unction A(, ). In agreement with the Kirchho boundary conditions, a convergent thin lens with a ocal length given by located at the back ocal plane o the optical Fig. 2. Three-dimensional mapping o the ocal volume provided by (a) the paraxial Debye ormulation, thus giving (b) the Fresnel Kirchho representation o the ocal wave ield. system would compensate the phase modulation o the quadratic actor and produce a collection o amplitude transverse patterns as given by the Debye approximation. Conversely, to obtain the amplitude distribution in the ocal plane given by the Fresnel Kirchho theory, we may employ the paraxial Debye approximation and the action o a divergent thin lens o ocal length, that is, with the ocal point located on the axial point o the diracting opaque screen. Also, when the transverse pattern o adjacent planes in the ocal region is evaluated, it may be obtained analytically in terms o the Debye ormula and later virtually introduce the action o a divergent thin lens. The Gaussian imaging transormation undergone by the diraction amplitude distribution given by the Debye ormula may be mathematically represented as ollows: k Ux, y, z exp i 2 z x2 y 2 M UD x M, y M, z M. (5) We should mention that in the previous threedimensional mapping there exist a actor /M and a quadratic phase actor accompanying the amplitude distribution given by the Debye approximation. The irst term is associated with the energy conservation law, and the second one is somewhat irrelevant, since it is not observable when intensity is being detected. The point-spread unction o a low-angular-aperture diraction-limited optical imaging system is usually described in terms o the paraxial Debye integral 23 given in Eq. (9). It may be demonstrated that, in general, the light is concentrated in planes neighboring the ocal plane; hence the deormation experienced in the ocal volume is unnoticeable. It may be demonstrated that in this region the lateral magniication is closer to unity, M(z), and the axial translation o the irradiance transverse patterns is negligible, z D z. However, we should remark that this kind o transormation in the diracted wave ield around the ocal plane always occurs. In Fig. 3 we show the symmetries that the ocal waves exhibit when the Fresnel number is much higher than

5 Zapata-Rodríguez et al. Vol. 7, No. 7/July 2000/J. Opt. Soc. Am. A 89 unity and hence when the Debye approximation holds. However, when N decreases, the intensity in the ocal region begins a process o deormation as deduced rom Eq. (5), which becomes more noticeable as the aperture plane comes closer to the ocal region. As a result, we observe that the point o maximum intensity o the diracted wave may not be at the geometrical ocus o the This is the so- incident wave but closer to the aperture. called ocal-shit eect. 5. GEOMETRICAL INTERPRETATION OF THE FOCAL-SHIFT EFFECT Much attention has been addressed to the problem concerning the evaluation o the relative ocal shit, that is, the ratio o such a shit o the point o maximum axial intensity to the distance between the geometrical ocus and the plane o the aperture. 3,24 This interest is well justiied because the determination o the plane at which an imaging system comes to the best ocus is o great importance. On the other hand, similar considerations should be taken into account in discussing how to optimally ocus a laser beam to illuminate a distant target. In this context, Li and Wol 9 recognized that the magnitude o the relative ocal shit depends solely on the Fresnel number o the ocusing geometry. Moreover, they presented a ormula or the rapid evaluation o the relative ocal shit: 2 /2N 2. (6) This ormula gives the ocal shit accurately to within % when N 2. However, or lower values o the Fresnel number, Eq. (6) gives a rough estimation o the relative ocal shit. Now we consider the mapping o Eq. (5) that should be applied on the paraxial Debye ormulation when the Fresnel number o the ocusing setup is comparable to unity. Thus it is possible to obtain the location o the plane provided by the Debye ormulation whose conjugate plane corresponds to the transverse pattern in the Kirchho approximation that includes the point o maximum intensity along the optic axis. By substituting Eq. (6) into Eq. (3), where the axial coordinate z is given by the ocal displacement, we may determine that this plane is then situated at a distance rom the ocal plane o the spherical beam given by z D 2 2 z N, (7) Fig. 3. Diagram o isophotes corresponding to the impulse response o an optical imaging system with a circular clear pupil o radius a mm when the wavelength is given by 500 nm and the Fresnel number is (a) high (N 500), (b) moderate (N 0), and (c) low (N 3). The continuous white line passes through the axial point o the pupil plane, which gives a rough idea o the relative distance between the ocal plane and the aperture plane. where in addition we have made use o Eq. (2). From Eq. (7) we iner that the plane belonging to the Debye representation that images to the plane in the Kirchho ormulation o the ield that contains the point o maximum intensity on the optical axis is located inside the ocal depth. This is true in the range o validity o Eq. (6), that is, or moderate and high values o the Fresnel number. This act ensures that the plane o best ocus, that is, the plane in which the ocused beam radius reaches its minimum transverse extension, suers a deocus aberration that may be neglected. At this point we note that there exists a certain controversy about the appropriate deinition o beam radius and hence o plane o best ocus. We point out that Mahajan 6,25 used a criterion or the spot size based on the encircled energy. In particular, the maximum o encircled energy occurs

6 90 J. Opt. Soc. Am. A/ Vol. 7, No. 7/ July 2000 Zapata-Rodríguez et al. where maximum central irradiance is reached or small spot sizes, which agrees with the reasoning presented here. When the Fresnel number decreases to values lower than unity, the distance z D o the plane containing the beam waist is located outside the ocal region given in the Debye regime. Following Parker Givens, 26 we may conclude that when we deal with an image-orming optical system, we are not able to ensure that the ocus o the optical system is eectively shited to the point o maximum intensity along the axis. However, it should be noted that imaging systems generally have very large Fresnel numbers, and hence the maximum axial irradiance is observed at the geometrical ocus. In the case in which the optical system is used to ocus light on a target we should consider a dierent criterion, such as the encircled energy. In particular, we point out that maximum encircled energy is obtained when the beam is ocused on the target, even though a higher axial irradiance is obtained at a point closer to the diracting aperture. As highlighted by Li and Wol, 4 a somewhat paradoxical aspect o this situation is that as the Fresnel number N decreases, with a and kept ixed, the distance increases; i.e., the geometrical ocus F moves arther away rom the aperture. However, the point o maximum intensity moves in the opposite direction, i.e., closer to the aperture. 6. DISCUSSION AND CONCLUSIONS We have proved that both the Debye and the Kirchho ormulations used or the evaluation o ocal waves give the same collection o transverse intensity patterns, except or location and magniication. Deviations rom both theories in the ocal region arise when the Fresnel number o the ocusing geometry is close to unity. However, we claim that the great majority o research perormed in the Debye regime are still valuable, even when N is comparable to unity. A simple example represents the great variety o strategies or increasing the power resolution o an imaging system, 27 or the threedimensional intensity distribution o the point-spread unction that characterizes the diraction behavior o an apodized setup may be applied when the Fresnel number, when viewed rom ocus, is close to unity. However, the similarity o the Debye and the Kirchho representations o the wave ield allows us to point out that there may exist a great discrepancy between the concept o plane o best ocus and the plane containing the point o maximum intensity along the axis or Fresnel numbers lower than unity. From our point o view, i the optical system is being used or imaging purposes, we should stress that the two planes do not coincide. To determine the plane o best ocus, it is desirable to select a plane belonging to the ocal region in the Debye ormulation that, when the three-dimensional mapping o Eq. (5) is perormed, has the narrower transverse spot light. However, it is possible to demonstrate that, as we will detail in a uture paper, the adjacent planes in the ocal volume oer practically the same imaging abilities but are characterized by a higher magniication, as given by Eq. (4). ACKNOWLEDGMENTS This work was supported by the University o Valencia (grant UV99-34). L. Muñoz-Escrivá grateully acknowledges the inancial support rom this institution (Cinc Segles Program). REFERENCES. P. Jacquinot and B. Roizen-Dossier, Apodization, in Progress in Optics, E. Wol, ed. (North-Holland, Amsterdam, 964), Vol. 3, pp M. Martínez-Corral, P. Andrés, and J. Ojeda-Castañeda, On-axis diractional behavior o two-dimensional pupils, Appl. Opt. 33, (994). 3. M. Born and E. Wol, Principles o Optics, 4th ed. 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7 Zapata-Rodríguez et al. Vol. 7, No. 7/July 2000/J. Opt. Soc. Am. A For an extensive discussion on spatial invariance and thecorrect use o the point-spread unction in optical systems o inite Fresnel number, see C. J. R. Sheppard, Imaging in optical systems o inite Fresnel number, J. Opt. Soc. Am. A 3, (986). 24. Y. Li, A high-accuracy ormula or ast evaluation o the eect o ocal shit, J. Mod. Opt. 38, (99). 25. V. N. Mahajan, Uniorm versus Gaussian beams: a comparison o the eects o diraction, obscuration, and aberrations, J. Opt. Soc. Am. A 3, (986). 26. M. Parker Givens, Focal shits in diracted converging spherical waves, Opt. Commun. 4, (982). 27. C. J. R. Sheppard and Z. S. Hegedus, Axial behavior o pupil-plane ilters, J. Opt. Soc. Am. A 5, (988).